digitalassets.lib.berkeley.edu€¦ · 1 Abstract Understanding Electrothermal Physics in Complex...

Understanding Electrothermal Physics in Complex Oxide Thin Films

by

Shishir Pandya

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Engineering-Materials Science and Engineering

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Lane W. Martin, ChairProfessor Chris Dames

Professor Ramesh Ramamoorthy

Spring 2018


Copyright 2018

by

Shishir Pandya

1

Abstract


by

Shishir Pandya

Doctor of Philosophy in Engineering-Materials Science and Engineering

University of California, Berkeley

Professor Lane W. Martin, Chair

Advances in materials displaying strong electrothermal phenomena can enable next-generation devices that besides powering themselves utilizing ambient waste heat can alsocool down electronics that, in times to come, will dissipate nearly all the input energy intoheat. Pyroelectrics is one such class of materials that exhibits a complex interplay of elec-trical polarization and heat that has the potential for waste-heat energy conversion andsolid-state cooling. While pyroelectricity and its thermodynamic converse, the electrocaloriceffect have been known for decades, there remains a significant gap in the fundamentalunderstanding of how to control and exploit this electrothermal phenomena, including sig-nificant differences between theoretical and experimental observations, under utilization ofadvances in the design and synthesis of complex-oxide ferroelectric thin films, and inade-quate characterization techniques (particularly for thin films). The research presented here,therefore, aims to bridge this gap by employing a comprehensive approach to the design,synthesis, and characterization of complex oxide thin films and explore routes to enhanceelectrothermal responses. First, new insights into the nature of growth of these complexferroelectric materials will the developed. This will include understanding the role of epi-taxial strain in exerting control over the film morphology and the ferroic domain structure.Next, a direct technique for measuring both pyroelectric and electrocaloric effect in epitax-ial ferroelectric thin films will be demonstrated. Combined, this will facilitate a thoroughinvestigation of how domain structures effect the electrothermal and electromechanical sus-ceptibility in thin films of the canonical ferroelectric PbZr1−xTixO3. Next, the potential ofthin-film ferroelectrics in high-power waste-heat harvesting will be explored in the relaxorferroelectric 0.68Pb(Mg1/3Nb2/3)O3-0.32PbTiO3 using solid-state thermodynamic cycles thatmimic traditional gas-phase thermodynamic cycles. Using field- and temperature-dependentpyroelectric measurements, the role of polarization rotation and field-induced polarizationin mediating the large pyroelectric effect will be explained. Finally, moving beyond the pro-totypical pervoskite ferroelectrics like PbZr1−xTixO3 and (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3,electrothermal response in new lead-free and CMOS-compatible ferroelectric HfO2 will beexplored.

i

To Ankita, my Love, Papa, Mummy & Amit

ii

Contents

Contents ii

List of Figures v

List of Tables xv

List of Abbreviations xvi

List of Symbols xvii

1 Introduction to Electrothermal Phenomena in Ferroic Materials 11.1 Fundamentals of Electrothermal Interactions-Symmetry and Phenomena . . 2

1.1.1 Pyroelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Electrocaloric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Measuring Electrothermal Susceptibilities-Background and Challenges . . . . 61.2.1 Measuring Pyroelectric Susceptibilities . . . . . . . . . . . . . . . . . 61.2.2 Measuring Electrocaloric Susceptibilities . . . . . . . . . . . . . . . . 7

1.3 Enhancing Electrothermal Susceptibilities . . . . . . . . . . . . . . . . . . . 81.4 Energy Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.1 Pyroelectric Energy Conversion . . . . . . . . . . . . . . . . . . . . . 101.4.2 Electrocaloric Solid-State Cooling . . . . . . . . . . . . . . . . . . . . 12

1.5 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Synthesis and Structural Characterization of Complex Oxide Thin Films 162.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Growth of Epitaxial PbZr0.2Ti0.8O3 Thin Films . . . . . . . . . . . . . . . . 182.3 Structural Characterization of PbZr0.2Ti0.8O3 Thin Films . . . . . . . . . . . 19

2.3.1 Atomic Force Microscopy Study of PbZr0.2Ti0.8O3 Thin Films . . . . 192.3.2 X-Ray Diffraction Studies of PbZr0.2Ti0.8O3 Thin Films . . . . . . . . 20

2.4 Strain Relaxation and its Implications on Growth Mode . . . . . . . . . . . . 212.4.1 Quantification of Strain Relaxation . . . . . . . . . . . . . . . . . . . 212.4.2 Predicting Surface Strains using Linear Elastic Theory . . . . . . . . 232.4.3 Evidence of Misfit Dislocation at Film-Substrate Interface . . . . . . 26

iii

2.4.4 Island Coarsening with Increasing Thickness . . . . . . . . . . . . . . 272.4.5 Mechanism behind the Topography Evolution under Surface Strain . 29

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Electrothermal Characterization of Complex Oxide Thin Films 313.1 Background and Challenges in Measuring Electrothermal Susceptibilities . . 323.2 Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Theory of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.1 Pyroelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.2 Electrocaloric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.1 Pyroelectric Measurements . . . . . . . . . . . . . . . . . . . . . . . . 433.4.2 Electrocaloric Measurements . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Effect of Ferroelastic Domains on Electrothermal and ElectromechanicalSusceptibilities 504.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Evolution of Ferroelastic Domain Structures with Epitaxial Strain . . . . . . 51

4.2.1 Synthesis and Structural Characterization . . . . . . . . . . . . . . . 524.2.2 Mechanism Behind the Coexistence of Mixed-Phase Domain Structures 56

4.3 Probing the Effect of Ferroelastic Domain Structures on Electrothermal Sus-ceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.1 Pyroelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.2 Electrocaloric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4 Probing Electromechanical Response in Mixed-Phase Domain Structures . . 744.4.1 Understanding Ferroelastic Switching in Mixed-Phase Domain Struc-

tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4.2 Deterministic Control of Ferroelastic Switching . . . . . . . . . . . . 77

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Pyroelectric Energy Conversion using Complex Oxide Thin Films 805.1 Introduction and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Identifying materials with large pyroelectric response . . . . . . . . . . . . . 825.3 Growth and Characterization of PMN-32PT Thin Films . . . . . . . . . . . 825.4 Pyroelectric Response of PMN-32PT Thin Films . . . . . . . . . . . . . . . . 835.5 Pyroelectric Energy Conversion using PMN-32PT Thin Films . . . . . . . . 885.6 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Electrothermal Effects in Novel Ferroelectric Materials - HfO2 946.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

iv

6.2 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.3 Understanding the Pyroelectric Effect in Si-doped HfO2 Thin Films . . . . . 97

6.3.1 Measurement of Pyroelectric Susceptibility . . . . . . . . . . . . . . . 976.3.2 Role of Defect Dipoles on Ferroelectric and Pyroelectric Susceptibility 100

6.4 Understanding the Electrocaloric Effect in Si-doped HfO2 Thin Films . . . . 1026.4.1 Measurement of Electrocaloric Susceptibility . . . . . . . . . . . . . 1026.4.2 Role of Defect Dipoles on Electrocaloric Susceptibility . . . . . . . . . 104

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Summary and Suggestions for Future Work 1067.1 Summary of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.1.1 Summary of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.1.2 Summary of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.1.3 Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.1.4 Summary of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.1.5 Summary of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.1.6 Summary of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Bibliography 115

A Pulsed Laser Deposition of Complex Oxide Thin Films 134

B Description of Heat Transport Models 137

v

List of Figures

1.1 Schematic indicating the range of thermo-electro-mechanical coupling effects inferroelectric materials. (Adapted from J. F. Nye Physical Properties of Crystals,1964) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 (a) Free energy as a function of polarization (for a second-order phase transition)in the paraelectric phase (T > Tc) shown as a dashed-line and in the ferroelectricphase (T < Tc) shown as a bold line. (b) Spontaneous polarization as a function oftemperature. (Adapted from K. Rabe et al. Physics of Ferroelectrics: A ModernPerspective, (2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 (a) A schematic of the pyroelectric Ericsson cycle showing the various isothermaland isoelectric processes. The total energy density per cycle is given by the areaof the closed loop,

∮EdP . (b) A schematic illustration of electrocaloric cooling

showing the various isentropic and isoelectric processes. . . . . . . . . . . . . . . 11

2.1 Atomic force microscopy (AFM) images and extracted line profiles (black dashedline in AFM image) for heterostructures with PbZr0.2Ti0.8O3 thickness of (a),(b)25 nm, (c),(d) 40 nm, (e),(f) 55 nm, and (g),(h) 150 nm, respectively. For (b),(d), (f), and (h), the vertical scale is in units of nanometers and the horizontalscale in units of microns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 (a) X-ray diffraction pattern of PbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001) or PZT/SRO/STO (001) heterostructures for films with (bottom to top) 25 nm, 55 nmand 150 nm PZT film thicknesses. (b) Phi scan about the SrTiO3 110- (blue) andPZT 110- (red) diffraction conditions for a 55 nm PbZr0.2Ti0.8O3/SrRuO3/SrTiO3

(001) heterostructure revealing full in-plane epitaxy. . . . . . . . . . . . . . . . . 212.3 Symmetric reciprocal space maps about the pseudocubic 002-diffraction condition

for PbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001) heterostructures with PbZr0.2Ti0.8O3

film thickness of (a) 25 nm, (b) 55 nm, and (c) 150 nm, respectively. Thinnerfilms (25 nm and 55 nm) do not show the presence of the PbZr0.2Ti0.8O3 200-diffraction condition suggesting the absence of any in-plane domain formationunlike the thicker films. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Asymmetric reciprocal space mapping studies about the 103-diffraction conditionsof heterostructures with PbZr0.2Ti0.8O3 thicknesses of (a) 25 nm, (b) 55 nm, and(c) 150 nm. Diamond symbols (black, open) represent the bulk lattice parameter. 23

vi

2.5 In-plane strain relaxation (%, red circles) and surface ripple density (µm−1, bluesquares) as a function of film thickness. . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 (a) Calculated variation of compressive in-plane strain at the film surface as afunction of film thickness. The data, from top to bottom, represent the case forpseudomorphically-strained (blue), 40 nm thick (purple), 100 nm thick (magenta),and fully-relaxed (red) heterostructures and show periodic strain modulations onthe surface of 40 nm and 100 nm thick films due to the buried misfit dislocationarray (T-symbol). (b) Schematic illustration of the variation in the adatom bind-ing energy on the surface where red and blue areas are regions with stronger andweaker binding energy, respectively. (c) Schematic illustration of the in-planelattice parameters on the surface of a film under a partially relaxed biaxial strain- red and blue areas are under less and more compressive strain, respectively.(d) The measured topography of a 55 nm thick PbZr0.2Ti0.8O3 heterostructureshowing aligned wedding-cake structures. . . . . . . . . . . . . . . . . . . . . . . 25

2.7 (a) Cross-section, dark-field TEM image of the same heterostructure taken underthe g = (200) two beam condition along the [012] zone axis showing the presenceof dislocations (white arrows point to dislocations located at the PbZr0.2Ti0.8O3/SrRuO3 interface) and corresponding strain fields. (b) Zoom-in image of thestrain-induced contrast from the dislocation strain fields. (c) The representativeelectron diffraction pattern taken along [012] zone axis. . . . . . . . . . . . . . . 26

2.8 Atomic force microscopy images and illustration of island identification procedurefor heterostructures with thicknesses of (a) 55 nm, (b) 65 nm, (c) 90 nm, and (d)150 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.9 Schematic illustration of the thickness evolution of the surface structure. (a) Inthin, unrelaxed films, step-flow growth persists; the arrows indicate the possibilityof attachment of adatoms either directly to the step edges or after being reflectedby the descending edge due to the ES barrier (inset shows the adatom potentialenergy landscape). (b) As the thickness increases, there is an onset of a kineticinstability due to dislocation formation and nucleation of islands in the capturezones (inset shows the adatom potential energy landscape is lowered, red color,due to the dislocation strain field). (c) With further increased film thickness,formation of an ordered array of dislocations induces ordered, multiterrace islandformation. (d) Continued film growth, leads to formation of multi-terrace islandsand island coarsening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 (a) Schematic of the process to fabricate the electrothermal device for measuringpyroelectric and electrocaloric effects in thin films. (b) Section-view through theview A-A depicting various layers in the heterostructure. The film under testhere is PbZr0.2Ti0.8O3 (red), while the two SrRuO3 layers (blue) connect to thetop and bottom metal contacts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

vii

3.2 (a) DC Current-voltage characteristics, (b) ferroelectric hysteresis loops, and (c)dielectric permittivity and loss tangent as a function of frequency for a 150 nmthick PbZr0.2Ti0.8O3 device obtained at 300 K. . . . . . . . . . . . . . . . . . . . 35

3.3 (a) Schematic of the setup for pyroelectric and 3-omega measurement. (b)Schematic of the relationship between the applied heating current [IH(t)] at an-gular frequency 1ωH , dissipated power [QH(t)], and the resulting temperatureoscillation [TH(t)] of the metal heater line at 2ωH . (c) Schematic of the tempera-ture field during a pyroelectric measurement and a simplified thermal circuit.(d)Measured temperature oscillation θFE and thermal phase lag φ between the heat-ing current (and, therefore, the dissipated power) and the temperature change inthe ferroelectric as a function of frequency. . . . . . . . . . . . . . . . . . . . . . 36

3.4 (a) Schematic of the setup for electrocaloric measurement using the top metal lineas a resistance thermometer (sensor). (b) Schematic of the relationship betweenthe applied voltage [VE(t)] across the ferroelectric capacitor at angular frequencyωE, the power generated [QE(t)], and the resulting temperature change [TE(t)]due to the ECE at the same frequency ωE. (c) Schematic of the temperature fieldduring an electrocaloric measurement. (d) Computed temperature oscillationθSensor and thermal phase lag between the temperature oscillation of the on thetop sensing metal line and the ECE power as a function of frequency of voltagecycling across the ferroelectric for an assumed Σ = 100 µC m−2 K−1 and anapplied electric field amplitude of 50 kV cm−1. . . . . . . . . . . . . . . . . . . . 39

3.5 Pyroelectric measurements of PbZr0.2Ti0.8O3 thin film. (a) Measured total current(iTotal) in the frequency-domain and time-domain (inset) for an applied heatingcurrent (IH,1ωH ) equal to 16 mA at a frequency of 1 kHz. (b) Phase relationship ofthe measured current (at 2ωH) and the temperature oscillation (also at 2ωH) withrespect to the phase of the applied heating current depicting that the measuredcurrent leads the temperature oscillation by ∼90◦. (c) Measured pyroelectriccurrent and pyroelectric coefficient as a function of heating frequency. At lowfrequencies (<4 Hz), π →

(∂P∂T

)E,σ

, while at high frequencies (>500 Hz) π →(∂P∂T

)E,ε

due to secondary contribution to pyroelectricity (reduction by ∼30%).

(d) Pyroelectric hysteresis loops as a function of background DC electric field, at1 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

viii

3.6 Electrocaloric measurements of PbZr0.2Ti0.8O3 thin film. (a) Measured differentialvoltage across the top metal sensor in the frequency-domain for an applied electricfield of magnitude 50 kV cm−1 at a frequency ωE/2π equal to 98,147 Hz and asensing voltage VS equal 7.07 V at a frequency ωS/2π equal to 2,317 Hz. Insetshows the voltage response at frequency ωS−ωE and ωS +ωE which is a measureof the change in the resistance and thus θSensor of the top metal. (b) Averagemeasured temperature oscillation in the ferroelectric (θFE) due to the ECE andthe resulting calculated value of Σ. (c) Electrocaloric loops as a function ofbackground DC electric field using an AC electric field amplitude of 50 kV cm−1.Up-poled state (red) results in the positive ECE while the down-poled state (blue)results in the negative ECE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.7 Sensitivity of the calculated electrocaloric temperature change at the top metalsensor, θSensor as a function of selected parameters in the thermal model as func-tion of frequency of electric field cycling for a 150 nm PbZr0.2Ti0.8O3 heterostructure. 48

4.1 Analytical phenomenological modeling-based temperaturestrain phase diagramrevealing the stability regimes for c, c/a, and a1/a2 domain structures. Strainpositions for the substrates used are shown as dashed lines. . . . . . . . . . . . . 52

4.2 Wide-angle θ-2θ scans for 40 nm thick PbTiO3 films grown on SrTiO3 (001),DyScO3 (110), GdScO3 (110), and SmScO3 (110) substrates. . . . . . . . . . . . 53

4.3 2D RSM studies about the (a) SrTiO3 001-diffraction condition (suggesting anearly monodomain c structure), (b) the DyScO3 110-diffraction condition (con-sistent with a c/a domain structure), and (c) the SmScO3 110-diffraction condi-tion (showing a1/a2 domain structures). . . . . . . . . . . . . . . . . . . . . . . 54

4.4 3D, synchrotron-based RSM about the 110-diffraction peak of the GdScO3 sub-strate (capturing 001-diffraction peaks of the PbTiO3) indicating the extractedlattice parameters and crystallographic tilts. . . . . . . . . . . . . . . . . . . . . 55

4.5 Topography (a-d) and out-of-plane piezoresponse amplitude (in-plane piezore-sponse amplitude as inset) (e-h) of PbTiO3 films grown on SrTiO3 (a,e), DyScO3

(b,f), GdScO3 (c,g), and SmScO3 (d,h) substrates. . . . . . . . . . . . . . . . . . 564.6 (a) 3D representation of the topography of PbTiO3 hierarchical domain architec-

tures grown on GdScO3 (110). Inset shows line trace (red-dashed line) indicatingthe topography and crystallographic tilts. (b) Vertical PFM (amplitude) responsesuperimposed on 3D representation of the topography of the hierarchical PbTiO3

domain structure. Inset shows a schematic representation of the domain structurewithin a c/a region as indicated by the shaded box. (c) Lateral PFM (amplitude)response superimposed on 3D representation of the topography of the hierarchicalPbTiO3 domain structure. Inset shows a schematic representation of the domainstructure within an a1/a2 region as indicated by the shaded box. . . . . . . . . 57

4.7 Schematics of the lowest energy domain structures computed from the phase-fieldstudies for misfit strain values ranging from -2.5% to 2.5%. . . . . . . . . . . . . 58

ix

4.8 (a) Landau, (b) elastic, (c) electric, and (d) gradient energy contributions to thetotal energy of the lowest energy structures computed from phase-field studies. . 59

4.9 (a) Strain-energy landscape (square points, black, solid curve) and the sum ofLandau and elastic energy density within the constituent domain structure (cir-cles, dashed line) including: c (blue circles), c/a (red circles), and a1/a2 (greencircles). (b) Evolution of domain fractions (%) with misfit strain. A hierarchi-cal domain architecture of coexisting c/a and a1/a2 is observed within a strainrange of 0.2-0.8%.(c) Strain accommodation in the different domain structuressuch that within the region of negative curvature (violet regions, e.g. at 0.6%strain), the material partitions itself into two domains structures: c/a domainsat a strain of 0.1% (right inset) and a1/a2 domains at a strain of 1.35% strain (leftinset). (d) The resulting temperature-strain phase diagram showing the regionof hierarchical domain structure between c/a and a1/a2 regions. . . . . . . . . . 60

4.10 Analytical phenomenological GLD-based temperature-strain phase diagram re-vealing the stability regimes for c, c/a, and a1/a2-domain structures. Regionwithin the dashed-line schematically represents the region where a mixed-phaseor a coexisting c/a, and a1/a2-domain structures is expected under equilibrium.The dots show the pseudocubic strain positions for the different substrates. . . . 62

4.11 Wide-angle θ-2θ scans for 120 nm thick PbZr0.2Ti0.8O3 films grown on TbScO3

(110), GdScO3 (110), SmScO3 (110), NdScO3 (110) and PrScO3 (110) substrates. 634.12 Topography (a-e) and out-of-plane piezoresponse amplitude (in-plane piezore-

sponse amplitude as inset) (f-j) of PbZr0.2Ti0.8O3 films grown on TbScO3 (a,f),GdScO3 (b,g), SmScO3 (c,h), NdScO3 (d,i), and PrScO3 (e,j) substrates. . . . . 64

4.13 AFM topography of the PbZr0.2Ti0.8O3 film grown on TbScO3 (a,b), GdScO3

(c,d), and PrScO3 (e,f) at 300 K and 450 K, respectively. . . . . . . . . . . . . . 654.14 (a) Polarization-electric field hysteresis loop and, (b) dielectric permittivity (εr)

for oxide metal (40 nm SrRuO3)-ferroelectric (120 nm PbZr0.2Ti0.8O3)-oxide metal(20 nm Ba0.5Sr0.5RuO3) tri-layer heterostructures on TbScO3, GdScO3, SmScO3,NdScO3, and PrScO3 substrates with εm = −1.26%,−1.02%,−0.57%,−0.02%and +0.26%, respectively (brown, orange, yellow, light green and dark greendata, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.15 (a) Measured temperature oscillation, θFE and thermal phase lag, φ between theheating current and the temperature change in the ferroelectric as a functionof the frequency. (b) Measured pyrocurrent and (c) pyroelectric coefficient as afunction of frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

x

4.16 (a) Pyroelectric coefficient, π as a function DC electric field measured at roomtemperature and 1 kHz, for oxide metal (40 nm SrRuO3)-ferroelectric (120 nmPbZr0.2Ti0.8O3)-oxide metal (20 nm Ba0.5Sr0.5RuO3) tri-layer heterostructures onTbScO3, GdScO3, SmScO3, NdScO3, and PrScO3 substrates with εm = -1.26%,-1.02%, -0.57%, -0.02% and +0.26%, respectively (brown, orange, yellow, lightgreen and dark green data, respectively). (b) Experimentally measured π (reddata) and estimated πint (black dashed-line) for different strain states. (c) Cal-culated πext for all strain states corresponding to c, c/a, and mixed phase het-erostructures. (d) Temperature-dependent change of normalized π ( πT

πT=303K) for

heterostructures with εm = −1.26%,−1.02%,−0.57%,−0.02% and +0.26%, re-spectively (brown, orange, yellow, light green and dark green data, respectively). 67

4.17 Synchrotron-based RSM (showing the k − l plane about h = 0) about the 220-diffraction peak of the NdScO3 substrate showing the 002- and 200-peak ofPbZr0.2Ti0.8O3 corresponding to the tilted c and untilted a domain variant at(a) 300 K, (b) 373 K, and (c) 473 K. (d-f) Representative h− k slices shown fordifferent l corresponding to c-domains (d), substrate (e), and a-domains (f). (g)Integrated intensity on the h − k plane for every l at 300 K (blue data), 373 K(yellow data), and 473 K (red data). The inset shows the volume fraction of c-and a-domains as a function of temperature extracted by fitting the peaks in (g)and calculating the area under the corresponding peaks. . . . . . . . . . . . . . 69

4.18 (a) Measured electrocaloric coefficient, Σ showing conventional (inverse) ECE inthe down- (up-) poled state under positive (negative) saturation DC electric fieldfor heterostructures with εm = −1.26%,−1.02%,−0.57%,−0.02% and +0.26%,respectively (brown, orange, yellow, light green and dark green data, respec-tively). (b) Measured Σ (red data) and estimated Σint (black dashed-line) fordifferent strain states. (c) Calculated Σext for all strain states corresponding toc, c/a, and mixed phase heterostructures. (d) Schematic showing the change inthe polarization (from c domains and interconversion of a → c domains) withincreasing temperature under an applied electric field. (e) Schematic showing thechange in the polarization (from c domains and interconversion of c/a → a1/a2

domains) with increasing temperature under an applied electric field. . . . . . . 73

xi

4.19 Band excitation piezoresponse spectroscopy (BEPS) studies on 80 nm PbTiO3/BaSr0.5Ru0.5O3/GdScO3 heterostructures with coexisting hierarchical c/a anda1/a2 domain architectures. (a) Schematic illustration of a c/a region, and (b)the corresponding out-of-plane piezoresponse hysteresis loop obtained from sucha region. (c) Contact resonance frequency loops with various polarization statesthrough the switching process indicated. (d,e) Topographic images of the regionprior to (d) and following (e) switching revealing an unchanged domain struc-ture. Inset shows a binary image of the 75 nm region directly under the PFMtip. (fg) Schematic illustration of an a1/a2 region (f), and corresponding out-of-plane piezoresponse hysteresis loop (g) and contact resonance frequency loop (h)taken in such a region revealing a three-state ferroelastic switching process (c+,a, and c−) with high (low) piezoresponse and low (high) contact resonance statescharacteristic of c (a) domains. Throughout the switching process the polariza-tion states are indicated. (i,j) Topographic images of the region prior to (i) andfollowing (j) switching revealing the nucleation of a c domain within an a1/a2

domain structure. Inset shows a binary image of the 75 nm region directly underthe PFM tip. Scale bars in all images are 500 nm. . . . . . . . . . . . . . . . . . 75

4.20 Scanning probe microscopy-based photothermal-excitation contact resonancemeasurements showing (a) topography and (b) contact resonance map forPbTiO3/ Ba0.5Sr0.5RuO3/SmScO3 (110) heterostructures. . . . . . . . . . . . . . 76

4.21 Superimposed topography and vertical PFM (amplitude) response for a 40 nmthick PbTiO3 film grown with εs = 0.7% predominantly a1/a2 structure. a)Shows the initial a1/a2 domain structure and an atomically terraced surface (asindicated by the line trace in height). b) Reveals a deterministically written pat-tern of c/a domains within the a1/a2 region wherein large surface displacementsare observed (line trace in height shows ≈ 0.5 nm changes). c) Shows that suchstructures can be erased by application of an in-plane electric field, restoring thesurface to the original smooth state (line trace in height). The inset height linetraces reveal the evolution of the surface topography and indicate the potentialfor reversible electromechanical strains of 1.25%. d) Electrically written ferroe-lastic pattern (Cal logo) on a 10 µm2 as-grown region. The Cal logo, which is aregistered trademark and intellectual property of the Regents of the Universityof California, is reproduced with permission. . . . . . . . . . . . . . . . . . . . . 78

5.1 (a) Wide angle θ − 2θ scans for a 150 nm 0.68Pb(Mg1/3Nb2/3)O3-0.32PbTiO3

(PMN-PT)/25 nm Ba0.5Sr0.5RuO3 (BSRO)/NdScO3 (110) heterostructure show-ing the film is (001)-oriented, single-phase and of high quality. Inset shows Kiessigfringes about the primary diffraction peaks, again suggesting pristine interfacesand high quality. (b) Off-axis RSM about the 332-diffraction conditions revealingthat both PMN-PT and BSRO have the same in-plane lattice parameter as thesubstrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

xii

5.2 (a) Polarization-electric field hysteresis loop measured at room temperature and1 kHz for the PMN-32PT heterostructure. (b) Dielectric permittivity (εr) asa function of temperature and frequency (1-10 kHz) showing a systematic shiftof the maxima (Tmax) to lower temperatures with decreasing measurement fre-quency. (c) Inverse dielectric permittivity (at 1 kHz) as a function of temperaturerevealing the deviation from Curie-Weiss behavior (black straight-line) to extractthe Burns temperature (Tb = 259◦C, shown as black dashed-line). . . . . . . . . 84

5.3 (a) Sinusoidally varying current at frequency 1ω is used to heat the heater linevia the outer thermal probe pads. Voltage drop at 3ω across the inner thermalprobe pads is measured to extract the amplitude of temperature oscillation in theheater line. (b) Measured temperature oscillation θFE and thermal phase lag φbetween the heating current ih(t) (IRMS = 20mA) at an angular frequency 1ωh(and therefore the dissipated power) and the temperature change in the relaxorferroelectric as a function of frequency using the 3ω measurement. . . . . . . . . 85

5.4 (a) Measured pyroelectric current (rms) for a sinusoidal oscillation of temper-ature with amplitude θFE = 10K at 2 kHz as a function of dc electric field.(b) Extracted pyroelectric coefficient π = iP (AdT

dt)−1 plotted as a function of dc

electric field. (c) Variation of εr (open red circles) and loss tangent (filled redsquares) with applied dc electric field. Blue data in b and c show results for aPbZr0.2Ti0.8O3 thin film measured in the same configuration. (d) FoMPEC as afunction of applied dc electric field. The measured FoMPEC for a PbZr0.2Ti0.8O3

thin film (blue) and literature values for LiNbO3 (black) are provided for com-parison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.5 (a) Pyroelectric measurements conducted under zero-field with an oscillating actemperature with amplitude θFE = 2.5K as a function of background tempera-ture. (b) Pyroelectric measurements under applied dc electric fields with dashedlines showing the MC to T transition temperature for various dc fields. . . . . . 87

5.6 Pyroelectric measurements under field-heating (FH1→2, red) and -cooling (FC2→3,blue). The difference between the FH and FC (orange) provides a quantitativemeasurement of the contribution of polarization rotation to the net pyroelectricsusceptibility. Repeated pyroelectric measurements under field-heating (FH3→4,shaded red circles) retraces the path of previous field-cooled state confirming thatthe structure remains quenched in the T phase and does not relax back to the Mphase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xiii

5.7 (a) Schematic of the pyroelectric Ericsson cycle showing the various isothermaland isoelectric processes. The total energy density per cycle is given by the areaof the closed loop,

∮E.dP (shaded region). (b) Example of the implementation

of the Ericsson cycles wherein the temperature of the heterostructure is sinu-soidally varied by ∆T = 70K (blue curve) at 40Hz by applying a sinusoidallyvarying current (red curve) at 20Hz that locally heats the heterostructure viaJoule heating. (c) A periodic electric field E(t), phase-synced (delayed by φ)with the heating current is applied across the relaxor ferroelectric to completethe cycle. (d) In response to the periodically-varying temperature and electricfield, the resulting current (blue curve) is measured and the change in polariza-tion is calculated by integrating the current (orange curve). (e) Experimentallyobtained changes in polarization as a function of electric field for ∆T = 10−90Kwhile keeping Tlow = 25◦C with ∆E = 267kV cm−1 at f = 40Hz. . . . . . . . . . 89

5.8 Change in polarization as a function of electric field (pyroelectric Ericsson cycle)for (a) ∆T = 27 − 67K while maintaining Tlow = 25◦C with ∆E = 267kV cm−1

at f = 400Hz, and (b) ∆T = 31 − 56K while maintaining Tlow = 25◦C with∆E = 267kV cm−1 at f = 1000Hz. . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.9 (a) Energy density, (b) power density, and (c) scaled efficiency extracted fromvarious pyroelectric Ericsson cycles as a function of cycle frequency [40 Hz (blue),400 Hz (orange), and 1 kHz (red)] and ∆T while keeping Tlow = 25◦C with∆E = 267kV cm−1. (d) Comparison of the current work with experimentally-achieved energy and power densities from a number of prior studies of pyroelectricenergy conversion using thermodynamic cycles. The superscripts B, C, and Fcorrespond to bulk-ceramic, single-crystal, and thin-film samples, respectively. . 92

6.1 Schematic illustration showing the fabrication of the electrothermal devicethrough the various processing steps (a-e). (f) Cross-section view of the finaldevice showing the various layers in the heterostructure. . . . . . . . . . . . . . 96

6.2 (a) Measured temperature oscillation (θHeater) and the thermal phase lag (φ)between the heating current and the temperature change in the heater line asa function of frequency. (b) Scaling of the 1st harmonic and (c) 3rd harmonicvoltage signal with varying the heater current as a function of heating frequency. 98

6.3 (a) P-E measurements showing a pinched or constricted hysteresis loop. (b)Pyroelectric current (ip) and (c) pyrocoefficient (π) measured as a function of thebackground DC electric field, at 1 kHz. The black arrows in the figures show thestarting point of the measurement. . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.4 (a) P-E measurements conducted after cycling the device for 2 (red), 2k (yellow)and, 200k (blue) cycles. (b) Measured pyroelectric coefficient (π) as a function ofthe number of wake-up cycles. The red dashed-line shows an empirical logarithmicfit to the measured data. (c) Pyroelectric-retention measurements conducted onsamples after for 2 (red), 2k (yellow), and 200k (blue) wake-up cycles. . . . . . . 100

xiv

6.5 (a) Measured temperature change of the top metal line (θSensor) and the calculatedvalue of the electrocaloric coefficient (Σ) on samples after for 2 (red), 2k (yellow),and 200k (blue) wake-up cycles. (b) Comparison of Σ with π as a function ofnumber of wake-up cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

A.1 Schematic illustration of the Pulsed Laser Deposition process . . . . . . . . . . . 135

B.1 (a) Schematic showing the difference in the effective heating cross-section (2beffL)and the area collecting pyroelectric current (2belectrodeL). (b) Example FEA 2Dcontour plot of the temperature amplitude, θ(x, z), for the specific geometrystudied and at a heating frequency f = 1000 Hz . (c) Typical line trace θ(x)through the ferroelectrics mid-z plane. . . . . . . . . . . . . . . . . . . . . . . . 138

B.2 Correction factorbeffbheater

as a function of frequency for the specific geometry stud-ied. Changing the thermophysical properties of the materials will cause the lineto shift up/down and may affect the steepness of the decay, but the general trend(decreasing as a function of frequency) will remain. . . . . . . . . . . . . . . . . 141

xv

List of Tables

2.1 Measured island density as a function of increasing film thickness. . . . . . . . . 28

4.1 Thermophysical properties (at 300 K) of the various thin films and substratesused in the thermal transport calculations. a Ref. [35]; b Ref. [205]; c Ref. [206];d Measured using differential 3ω method on a single-crystal sapphire substrate; e

Ref. [207]; f Measured using 3ω method . . . . . . . . . . . . . . . . . . . . . . 72

6.1 Thermophysical properties (at 300 K) of the various thin films and substratesused in the thermal transport calculations. a Ref. [284]; b Ref. [285]; c Ref. [286];d Ref. [36]; e Ref. [207]; f Ref. [287] . . . . . . . . . . . . . . . . . . . . . . . . . 102

xvi

List of Abbreviations

PEE Pyroelectric Effect

ECE Electrocaloric Effect

EC Electrocaloric

PE Pyroelectric

TSC Thermally Stimulated Currents

PZT Lead Zirconate Titanate

PMN Lead Magnesium Niobate

PMN-PT Lead Magnesium Niobate-Lead Titanate

P(VDF-TrFE) Poly[(vinylidenefluoride-co-trifluoroethylene]

FE Ferroelectric

PEC Pyroelectric Energy Conversion

ES Ehrlich-Schwoebel

STM Scanning Tunneling Microscopy

UV Ultra Violet

SRO Strontium Ruthenate

BSRO Barium Strontium Ruthenate

STO Strontium Titanate

AFM Atomic Force Microscopy

PFM Piezoresponse Force Microscopy

RSM Reciprocal Space Map

LGD Landau-Ginzburg-Devonshire

BE-SS Band-Excitation Switching Spectroscopy

FoM Figure of Merit

FoMPEC Figure of Merit for Pyroelectric Energy Conversion

MPB Morphotropic Phase Boundary

PNR Polar Nano Regions

rms Root Mean Square

T -phase Tetragonal Phase

R-phase Rhombohedral Phase

M -phase Monoclinic Phase

Mc-phase Monoclinic-c Phase

FH Field Heating

FC Field Cooling

xvii

List of Symbols

P PolarizationPS Spontaneous polarizationPr Remanent polarizationPi0 Spontaneous polarization along the i-directionπi Pyroelectric coefficient along the i-directionEk Electric field along the k directionεr Relative permittivityε0 Permittivity of free spaceεzk Component of the dielectric permitivity tensordzkl Component of the piezoelectric tensorσkl Component of the stress tensorukl Component of the strain tensorµzijkl Component of the flexoelectric tensorαij Component of the thermal expansion tensorC Volumentric heat capacityTc Curie temperatureT Temperaureip Pyroelectric currentA Area of capacitora In-plane lattice constantc Out-of-plane lattice constantλ WavelengthQx In-plane dimension in the reciprocal spaceQy Out-of-plane dimension in the reciprocal space1ω Principal angular frequency2ω 2nd Harmonic angular frequency3ω 3rd Harmonic angular frequencyΣ Electrocaloric coefficient∆S Isothermal Entropy change∆T Adiabatic temperature changeRFE Resistance of ferroelectric layerIH,1ωH Heating currentωH Frequency of heating currentR0 Room temperature resistance of heating lineθdc DC temperature riseθH Amplitude of AC temperature oscillation on the heater lineφ Thermal phase lag between the applied heat current and temperature oscillation

xviii

V3ω Amplitude of 3rd harmonic voltageλS Thermal penetration depthDS Thermal diffuvisity of the substratedFE Thickness of the ferroelectric layerZT Thermal impedance of the top thermal circuitZB Thermal impedance of the bottom thermal circuitθFE Amplitude of AC temperature oscillation of the ferroelectric layerqH Thermal flux into the heterostructureQE Dissipated electrocaloric powerVE Amplitude of voltage applied for ECE measurementωE Frequency of dissipated electrocaloric powerQJ Dissipated power due to Joule heatingθSensor AC temperature oscillation of the top sensor lineZ Generic transfer functionRe(Z) Real part of the thermal transfer functionIm(Z) Imaginary part of the thermal transfer functionRSensor Resistance of the sensor lineVS Amplitude of sensing voltageωS Frequency of sensing voltageVA Voltage measured across the inner thermal pads of the thermal circuitVB Voltage across the potentiometerRP Resistance of the potentiometerRB Resistance of the ballast resistorG Volumetric power generation created by entropy changedqtot Differential thermal flux from a volume with diffential thickness dzitot Total measured current across the ferroelectric during periodic heatingiup Total measured current across the ferroelectric in the up-poled stateidown Total measured current across the ferroelectric in the down-poled stateiparasitic Parasitic current due to cross-coupling between the thermal and the electrical circuitSTx Sensitivity of T to xk Thermal conductivityfLandau Landau energyfElastic Elastic energyfElectric Electrical energyfGradient Gradient energyεm Misfit strainapc Pseudocubic lattice parameterπint Intrinsic contribution to pyroelectric susceptibilityπext Extrinsic contribution to pyroelectric susceptibilityΣint Intrinsic contribution to electrocaloric susceptibility

xix

Σint Extrinsic contribution to electrocaloric susceptibilityφc Fraction of c domainsφa Fraction of a domainsSc Dipolar entropy change in c domainsSa Dipolar entropy change in a domainsSβ→δ Entropy change due to phase change from β to δ〈P a→c

z 〉 Average change in polarization due to a to c switching⟨P

c/a→a1/a2z

⟩Average change in polarization due to a to c switching

Tmax Temperature corresponding to the dielectric maxima in relaxorsTb Burns temperature in relaxorsTlow Lower temperature during PECThigh Higher temperature during PECElow Lower electric field during PECEhigh Lower electric field during PEC∀ Volume of the ferroelectric filmW Workdone

xx

Acknowledgments

The journey to a PhD is long and without the motivation, guidance and support offriends and colleagues, this journey would not have been rewarding and enjoyable. Thissection is too short to acknowledge all the people who have contributed to my success. Mysincere gratitude to everyone who has helped me over the last few years of my research.

I would like to first thank my research advisor Prof. Lane Martin who has shapedthe researcher in me. His constant motivation to strive for excellence, achieve the best,and perform at the peak has brought out the best of me. With just the right amount ofguidance and feedback, that never swayed too heavily on either being too pushy or beingtotally hands-off, he provided me the absolute perfect spot to learn, hone my skills and thinkcreatively like an independent researcher. Working with Lane, I also learned how to lookat the bigger picture, scout for impactful ideas and quite-importantly how to communicatescience.

I am equally thankful to Prof. Chris Dames, working with whom I developed themuch-needed aptitude to be scientifically-rigorous and attentive to details. I am indebted tohis guidance and willingness to collaborate on the major themes of this dissertation.

I would also like to thank Prof. Ramesh Ramamoorthy for the stimulating discussions,brainstorming and, although quite short but valuable collaboration. More than just science,I picked up how his passion and drive to achieve the best can pull resources in line to achievethe best results.

I would like to thank my qualifying exam committee: Prof. Mark Asta, Prof. OscarDubon, Prof. Chris Dames and Prof. Ramesh Ramamoorthy for their crucial feedback andinsights on my work.

My learning would not have been complete without the peer interaction in the group.I would like to thank my colleagues and friends in the Martin group. Thank you Anoop,Josh, Ruijuan, Liv, Sungki, Eric, Brent, Vengadesh and Zuhuang for getting me started inthe group. Thank you Ran for always being the closest cousin in research. The endlessdiscussions on all kinds of topics over the long lunches have been memorable. Thank youSahar, Arvind, Jieun, Anirban, Gabe, Eduardo, Josh, Lei and Abel. I learned a good loadof science from each of your research.

Thank you Bikram for helping me get started with the core of my research. I amdeeply indebted to you.

Thanks to the members of the Ramesh group: Ajay, James, Claudy, Yeonbae, Sujit,Nikita, Arnoud, and Bhagwati for even more discussions across broader topics of interest.

xxi

My sincere thanks to the Dames group: Josh, Christian, Jason, Vivek, and Geoff for thevaluable insights on thermal transport and Suraj and Charles from Sayeef’s group.

Thanks to the wonderful teachers and professors from my years at Kanpur. Thankyou Prof. Anish Upadhyaya and his group for providing me the first research opportunityas an undergraduate at IIT Kanpur.

Thanks to my friends from UIUC and Berkeley. Thanks Mayank, Prakalp and Pranavfor being great housemates and for our daily dinner hangouts at UIUC. Thanks Neelay,Chimi, Bikash and Oleksa for making Berkeley feel like home.

Deepest thanks to you, Ankita for always being there to motivate me. This PhDwould not have been completed without your unwaivering love and support. Thanks forlistening to my rants from bad days at work and for talking sense into me when I neededit the most. My heartfelt thanks for bearing with the long stretch of distance and time. Ifsomeone has worked equally hard for this work, it’s you. I can’t wait to get married andstart our lives together.

Finally, I owe this work to my parents. Thank you for teaching me the value ofeducation, hard work, and persistence and for providing everything I needed to get to whereI stand today. Thanks for always letting me know what is more important in life. I hope Ihave made you proud.

1

Chapter 1

Introduction to ElectrothermalPhenomena in Ferroic Materials

In this Chapter, I provide a brief introduction to electrothermal phenomena in com-plex oxide ferroelectrics. Starting with a discussion of the symmetry considerations requiredfor these electrothermal interactions, the pyroelectric and electrocaloric effect in materialswill be formally introduced. This will be followed by a discussion of how such effects aremeasured in materials with an emphasis on what makes the measurement of such effectschallenging in the case of thin films. The nature of both these electrothermal susceptibili-ties in ferroelectrics, specifically, in the vicinity of phase transitions, will then be discussed.This will include a brief discussion of the possible routes by which these electrothermalsusceptibilities have been or can potentially be enhanced. Finally, two major applications,pyroelectric energy conversion and electrocaloric cooling will be introduced that have notonly motivated this dissertation but many more. Following the technical background, a briefchapter-by-chapter summary of the work to be covered will be provided.

2

1.1 Fundamentals of Electrothermal

Interactions-Symmetry and Phenomena

1.1.1 Pyroelectric Effect

The inherent coupling between the polarization and temperature in certain ferroicmaterials makes them an important class of electrothermal systems. Susceptible to changesin temperature, these materials can exhibit the so-called pyroelectric effect (PEE) where achange in the temperature engenders a change in the electrical dipole moment. This canbe understood with reference to the potential-energy landscape of the ions constituting thedipole moment. Any excitation caused by an increase in the lattice temperature, forces theions into a higher energy level in the potential well, resulting in the change in the meanequilibrium position in the lattice (due to the anharmonic nature of the potential well)and hence a change in the electrical dipole moment.[1] This electrical dipole moment underzero applied electric field (referred to as the spontaneous polarization (PS) must conformto the point-group symmetry of the crystal. It therefore follows immediately that the PEEcannot exist in a centrosymmetric crystals possessing an inversion symmetry (11 out of the 32possible point-group symmetries).[2] Of the remaining 21 non-centrosymmetric point groups,only 20 (with the exception of group 432), exhibit the piezoelectric effect, i.e., generationof electric charge due to an applied mechanical (direct piezoelectric effect) or generation ofmechanical strain upon application of an electric field (converse piezoelectric effect). Sincepolarization is a vector property (first-rank tensor), only 10 of the 20 point groups exhibitingthe piezoelectric effect constitute the polar class that exhibit the PEE.[2]

Quantitatively, the PEE is parametrized as a vector π, or the pyroelectric coefficient,which is defined as the rate of change of polarization with temperature. In most materials,this change is due to a change in PS or the permanent dipole moment of the system and thePEE herein is referred to as the primary PEE. Since the PEE is manifested as a change inthe surface charge density due to a change in temperature, the PEE can also be generatedin a variety of materials in other ways that include:[3] 1) Temperature-dependent change indielectric permittivity : A dielectric material under an electric field can show a pyroelectricresponse if the dielectric permittivity strongly changes with temperature. 2) Piezoelectriceffect : When piezoelectric materials are heated, the propensity to expand results in thermalstress (strain) in a mechanically constrained (free) system which, via the piezoelectric effect,results in the change in the surface charge density or electric displacement. This effect isreferred to as the secondary PEE (T → ε→ P , Figure 1.1).[4] 3) Flexoelectric effect : Evencentrosymmetric materials when subjected to inhomogeneous strains (for example, a straingradient) can polarize via the flexoelectric effect[5, 6] and result in temperature-dependentchanges in the surface charge density. Non-homogeneous strains can also be generated undera spatially non-uniform temperature and in this case, the developed pyroelectricity is referred

3

Figure 1.1: Schematic indicating the range of thermo-electro-mechanical coupling effects inferroelectric materials. (Adapted from J. F. Nye Physical Properties of Crystals, 1964)

as the tertiary PEE.[7] As such, the net pyroelectric coefficient along the z-direction, cantherefore, be expressed as

πz =∂Pz∂T

=∂Pz0∂T

+ ε0Ek∂εzk∂T

+∂ (dzklσkl)

∂T+∂(µzijk

∂ujk∂i

)∂T

(1.1)

where Pz0 is the spontaneous polarization along the z-direction, εzk is the componentof the dielectric tensor, Ek is the applied electric field along the k-direction, dzkl and σkl arethe components of the piezoelectric and stress tensors, µzijk and ujk are the components ofthe flexoelectric tensor and the strain tensor, respectively.

A subset of pyroelectrics, that can be switched via an externally applied electricfield, constitutes another group of crystals called ferroelectrics. The polar ferroelectric phaseis related to its prototypical higher-symmetry, non-polar phase (referred as the paraelectricphase) by means of a reversible structural displacement that occurs as the materials is cooledthough the transition temperature known as the Curie temperature (Tc) (Figure 1.2a). Below(Tc), the spontaneous polarization tends to increase with decreasing temperature (Figure1.2b), resulting in the PEE and ECE that forms the basis of a wide variety of applicationsincluding waste-heat energy harvesting,[4–6] infrared imaging,[1] and electron emission.[8, 9]

As the ferroelectric material cools through Tc, the emergence of PS results in a largeelectrostatic energy due to uncompensated bound charges if the polarization is uniformly

4

Polarization

>

<

Temperature

Po

lari

zati

on

Curie Temp.

En

erg

y

a b

Figure 1.2: (a) Free energy as a function of polarization (for a second-order phase transition)in the paraelectric phase (T > Tc) shown as a dashed-line and in the ferroelectric phase(T < Tc) shown as a bold line. (b) Spontaneous polarization as a function of temperature.(Adapted from K. Rabe et al. Physics of Ferroelectrics: A Modern Perspective, (2007).

polarized. The resulting depolarizing field, therefore, tends to destabilize the polarization.In ferroelectrics, this excess electrostatic energy is accommodated by the formation of fer-roelectric domains with polarization pointing in antiparallel directions. These domains areseparated by 180◦ domain walls across which the direction of polarization changes fromone domain to the other. In addition to the electrostatic energy, the excess elastic energyparticularly in clamped thin-films and mechanically-constrained systems, can result in theformation of ferroelastic domains. These domain structures can have a profound effect onthe pyroelectric susceptibilities. The primary PEE in ferroelectrics consisting of ferroelasticdomain structures can therefore be divided into two contributions: (1) intrinsic contribu-tions that arise from the temperature-dependent change in PS within a ferroelectric domain,and (2) extrinsic contributions that arise because of the temperature-dependent movementof ferroelastic domain walls resulting in a change in the fraction of the ferroelastic domains.This can be expressed mathematically as

∂ 〈Pi0〉∂T

= φi∂ 〈P 〉∂T

+ 〈P 〉 dφidT

(1.2)

where 〈Pi0〉 is the net polarization in the i-direction, 〈P 〉 is the spontaneous polar-ization within the domain polarized in the i-direction, and φi is the fraction of ferroelasticdomains spontaneously polarized in the i-direction.

5

1.1.2 Electrocaloric Effect

Electrothermal systems wherein the order parameter is strongly coupled to temper-ature, can, in turn, have their entropy tied to various thermodynamic fields (e.g., stress,magnetic- or electric-field). For instance, if one takes an expanded spring and releases thetension,[10] or moves a piece of nickel away from a magnet,[11] or removes an electric fieldfrom a Rochelle salt crystal,[12] all these materials will spontaneously cool down. These ex-amples depict the mutual convertibility of work from an external perturbation. While mostof the caloric effects (elastocaloric and magnetocaloric) have predominantly been studiedin intermetallics and alloys, the electrocaloric effect has typically been studied in ceramicoxides. The electrocaloric effect (ECE) is defined as the isothermal change in entropy or theadiabatic change in temperature upon the application or withdrawal of an externally-appliedelectric field in a material that crystallize in one of the ten polar point groups. Advances inmaterials displaying these effects could provide alternatives to thermoelectrics[13] for low-power, integrated solid-state cooling.[14–16]

By thermodynamic analysis, the net change in the entropy of a crystal due to changein the elastic stress, electric field, and temperature can be derived by the following tensorequation[17]

dS =

(∂S

∂Ek

)σ,T

dEk + αEkij dσij +Cσ,Ek

TdT (1.3)

The net electrocaloric coefficient can thus be expressed as

Σ =dS

dEk=

(∂S

∂Ek

)σ,T

+ αEkijdσijdEk

(1.4)

where αEkij is the thermal expansion tensor at a constant electric field. The first termon the right-hand side of the equation is the primary contribution while the second termis the secondary contribution to ECE due to a field-induced piezoelectric strain. It shouldbe noted that in the above expression the volumetric heat capacity, Cσ,Ek is assumed to beconstant under an electric field and stress which is not necessarily the case especially nearphase transitions where contributions to the entropy change due to a phase change shouldalso be taken into consideration.

Akin to the extrinsic contributions to the primary PEE due to temperature-dependentmovement of ferroelastic domain walls, the switching of ferroelastic domains under an elec-tric field can result in extrinsic contributions to ECE. The primary ECE can therefore beexpressed as

6

∂S

∂E= φi

∂Si

∂E+∂Sβ→δ

∂E(1.5)

where ∂Si

∂Eis the dipolar entropy change in the domains polarized along the i-direction

and with volume fractions φi and ∂Sβ→δ

∂Erepresents the entropy change due to a phase change

from a generic phase β to δ under the same electric field constituting the extrinsic contribu-tion to ECE.

While the ECE has been known for decades, it has remained one of the least studiedand understood ferroic phenomena. Part of the reason includes the significantly small mag-nitude of temperature change associated with this effect that it was (until about a decadeago) deemed useless for any practical application. The dearth of work on electrocaloricstherefore resulted in fundamental discrepancies in our understanding of these effects. Forexample, three of the most famous, classic texts on ferroelectrics (Fatuzzo & Merz,[18] Mit-sui et al.,[19] Jona & Shirane[20]) disagree with each other as to what temperature regimethey believe the ECE is permitted to occur (above or below Tc, or both). Another exampleis whether the PEE,

(∂P∂T

)E

a correct measure of the ECE,(∂S∂E

)T

in thin films, where sucha discrepancy has led to significant differences between the theoretically predicted and theexperimentally observed values of the electrocaloric temperature change. At the same time,inadequate measurement techniques for thin films have further crippled accurate characteri-zation of the ECE. The result is a lack of fundamental understanding about the mechanismsunderlying these electrical-thermal interactions and, subsequently, how to control these in adeterministic fashion to enhance overall material response.

1.2 Measuring Electrothermal

Susceptibilities-Background and Challenges

1.2.1 Measuring Pyroelectric Susceptibilities

The pyroelectric susceptibility of a ferroelectric is typically characterized by measur-ing the pyroelectric current generated in response to a known temperature change. Thepyroelectric current (ip) depends on the rate of change of temperature

(dTdt

)as ip = πAdT

dt,

where π is the pyroelectric coefficient and A is the area of the ferroelectric capacitor. Asfuture nanoscale electronics, energy-conversion devices, and on-chip solid-state refrigeratorsincreasingly require thin films, the challenges associated with measuring the electrothermalproperties in thin films has become a pressing concern in both advancing the fundamentalphysics and to enable new functionalities in materials. The area A probed for measuring

7

the electrothermal response in thin films is limited by the presence of defects like pinholes,etc. causing the measured ip to be significantly smaller and prone to electrical noise for thesame

(dTdt

)in comparison to the bulk counterparts. Further, the intrinsic defects in thin

films results in thermally stimulating currents (TSC) due to a temperature-induced releaseof charge from trap states within the ferroelectric.[21, 22] In turn, the techniques used forcharacterizing the PEE in the bulk like the Byer-Roundy method[23] are rendered less usefuland as a matter of fact quite imprecise for thin films. Even the more sophisticated phase-sensitive periodic heating method[24, 25] can be difficult to implement given the limitationsin the heating frequency and the poor accuracy in estimating

(dTdt

)in thin films. Ways

to circumvent these limitations were proposed by using either modulated laser-based[26] ormicrofabricated resistive heater-based[27] approaches. However, the challenges associatedwith modeling the heat transport and therefore in the estimation of the temperature changehas kept this method largely reserved within a few research groups. As a result, even aftermore than 200 years of quantitative characterization of the PEE, research/review papersexclusively on the measurement protocols are still found useful and worthy of publication.[3,28]

1.2.2 Measuring Electrocaloric Susceptibilities

Since the report of the large ECE in antiferroelectric PbZr0.95Ti0.05O3 ceramic thinfilms[29] and organic P(VDF-TrFE) thin films,[30] there has been a marked increase in theresearch to probe ECE in ferroelectric materials with a quest to find more efficient coolingtechnologies on par with the magnetocaloric effect.[31] Despite these successful and excitingresults there is discrepancy regarding the accuracy and methodologies of the approaches usedto quantify the electrocaloric responses in these materials. Typically, to quantify ECE, thepolarization is measured as a function of electric field and temperature to extract

(∂P∂T

)E

and the electrocaloric response is inferred through the thermodynamic Maxwell relations[(∂S∂E

)T

=(∂P∂T

)E

]. The isothermal entropy change and the adiabatic electrocaloric temper-

ature change is estimated (indirectly) using the integral

∆Siso =

∫ E2

E1

(∂P

∂T

)E

dE (1.6)

∆Tad = −∫ E2

E1

(T

C(T,E)

)E

(∂P

∂T

)E

dE (1.7)

Such indirect measurements, however, require a careful understanding of the bound-ary conditions of the materials since the equivalence of ECE and PEE necessarily holds trueonly in mechanically-free/unclamped systems or systems under a constant stress boundarycondition. This is of concern in case of thin films that are clamped to the substrate and

8

are free to expand only in the direction normal to the surface place. Alternatively, directmeasurements of electrocaloric responses have been attempted, but the inherent difficultiesdue to the small size and the extremely small thermal mass of thin-film heterostructures re-sults in the electrocaloric temperature change that is usually well below the resolution of anyoff-the-shelf thermocouple or a calorimeter. Additionally, since ECE is a high-field response,one must carefully consider the effects such as Joule heating (from current flow through thematerial) which can alter the temperature change in direct measurements. Comparisons ofdirect and indirect methods have shown that in some cases the two techniques yield similarresults,[32–34] but can also vary dramatically in others,[35–37] especially in relaxor ferro-electrics where the compositional disorder keeps the system in non-equilibrium.[38] Anothercategory of polar materials where the use of indirect technique can be potentially misleadingincludes antiferroelectrics. Here, the measurement of an average polarization as a measure ofthe order parameter is questionable since the appropriate order parameter is the staggeredpolarization and not the average.[39] Such indirect measurement schemes have been a causefor concern in the field and the development of measurement capabilities that allow reli-able and direct measurements of ECE in thin-films is essential to undisputedly understandthe magnitude and mechanisms of this electrothermal response. Over the last few years, re-searchers have been adapting new measurement strategies for the direct characterization ECEin thin films. This includes the use of scanning thermal microscopy and infrared imaging,[40]differential scanning calorimetry,[38] and laser-based thermo-reflectance measurements.[26]These techniques once fully developed and vetted provide promise to probe the role of strain,crystal and domain structure on the electrocaloric susceptibility in ferroelectric materials.

1.3 Enhancing Electrothermal Susceptibilities

The second-order derivatives of the free energy typically diverge in the vicinity ofphase transition,[41] and therefore perching the material at the brink of its phase transi-tion has served as a conventional route to enhance electrothermal response in materials.[42]Consequently, much of the work in the literature has focused on studying the nature of elec-trothermal susceptibility in the proximity of a ferroelectric-to-paraelectric phase transitionin turn establishing ferroelectrics as prime candidates exhibiting large PEE and ECE. Forexample, some of the earliest works reported large electrocaloric temperature changes nearthe Tc of potassium dihydrogen phosphate (KDP, Tc ∼ 123 K)[43], triglycine sulphate (TGS,Tc ∼ 323 K),[44] and Pb0.99Nb0.02(Zr0.75Sn0.20Ti0.05)0.98O3 (PNZST, Tc ∼ 434 K).[45] More re-cent investigations have also revealed that the maximum electrocaloric change indeed occursat the Tc of ferroelectric materials.[46] However, under the application of an electric field, Tccan shift to higher temperatures. If one looks at the electrocaloric responsivity (∆TEC/E),different results were found. In bulk ferroelectrics, it was found that the temperature depen-dence of ∆TEC/E exhibited a maxima at Tc regardless of the magnitude of electric field.[47]

9

On the contrary, in some thin films no maxima in ∆TEC/E was found despite applying themaximum possible electric field.[48] Similarly, in the case of relaxor ferroelectrics, the max-imum value of ∆TEC/E was found near the tricritical point.[49] Similarly, PEE was foundto be maximized near the ferroelectric-to-paraelectric transition of Pb(Mg1/3Nb2/3)-basedceramics.[50] Similar to the case of ECE, the PEE under an applied bias is expected is showa maxima above the Tc. In general, significant research has therefore been invested in ex-ploring routes to controllably tune Tc (via chemistry,[51–53] hydrostatic pressure[54, 55] orepitaxial strain[56–59] in case of thin films) to a desired range of temperature.

In addition to the ferroelectric-to-paraelectric transition, the ferroelectric-to- ferro-electric transition also results in the divergence of the electrothermal susceptibilities as, forexample, has been shown in the pyroelectric[60] and electrocaloric[61] response of BaTiO3

ceramic and single crystal, respectively. About these ferroelectric-to-ferroelectric phase tran-sitions, ferroelectrics undergo a change in the structural symmetry and cause a rotation ofspontaneous polarization. This phenomena has been observed in numerous relaxor ferro-electrics[62–64] and systems exhibiting a morphotropic phase boundary.[65, 66] A change inthe surface charge density due to rotation of polarization can therefore result in anomaliesin the electrothermal susceptibility. The rotation of polarization about the ferroelectric-to-ferroelectric phase transition under high-field also results in interesting electrocaloric phe-nomena. ECE which conventionally results in an increase in the lattice entropy under anapplied electric field can result in an inverse effect across these phase boundaries wherethe application of electric field results in a reduction of lattice entropy. For example, therhombohedral-to-orthorhombic and orthorhombic-to-tetragonal phase transitions in single-crystal BaTiO3 result in an inverse ECE as very recently shown experimentally[61] as wellas using first-principle-[67] and molecular-dynamics-based simulations.[68]

The competition between the polar phases also extends to systems that exhibit co-existing ferroelectric and antiferroelectric phases. Recently, it was shown that field-inducedantiferroelectric-to-ferroelectric phase transitions can result in substantial caloric effects.[69–71] Direct experiments on PbZrO3 ceramics[72] and phenomenological Landau-Kittel based-model for Ba-doped PbZrO3[73] have reported that the application of electric field belowthe critical field for a field-driven ferroelectric transition, to an antiferroelectric disrupts thedipolar ordering in the two oppositely-polarized sublattices. This results in an increase in thedipolar entropy and therefore in an inverse ECE. With increasing magnitude of electric field(above the critical field), the conventional ECE due to the ferroelectric ordering countersthe inverse ECE from the antiferroelectric ordering. Therefore, the net ECE goes to zeroif the entropy changes in the two phases completely cancel out. With much higher appliedelectric field, the antiferroelectric undergoes a field-driven transition into a ferroelectric suchthat the ECE becomes conventional. Phase transitions in antiferroelectrics, therefore offersunique and interesting ways to tune the polarity of the electrocaloric response. In turn, somestudies have proposed refrigeration cycles that can make use of the alternating polarity ofthe ECE to improve the refrigeration capacity.[74, 75]

10

So far, I have discussed how the competition between polar phases with differentsymmetries under a temperature or electric-field perturbation can result in a change in thepolar structure/order at a unit-cell level. At a more macroscopic level (∼ tens to hundredsof nanometers), temperature and electric field can also alter the mesoscopic domain struc-ture in ferroelectrics. These domain structures can have a profound effect on ferroelectricsusceptibilities. Simply, the presence of domain walls in BaTiO3[76] and PbZr0.2Ti0.8O3[77]was reported to result in a stationary or frozen contribution to the dielectric permittivity. Inaddition, the field-dependent irreversible motion of domain walls (constituting the extrinsiccontribution) can further enhance the dielectric permittivity.[78] Similarly, the piezoelec-tric effect was found to be strongly dependent on the irreversible domain wall motions inbulk ceramics[79] of barium titanate and lead zirconate titanate and thin films of lead zir-conate titanate.[80] The role of such domain structures in controlling the electrothermalsusceptibilities (extrinsic contributions), however, has remained comparatively understudiedin part due to experimental challenges associated with the measurement of these physicalphenomena which have, instead, driven almost all the work to date to be done via computa-tions. Phenomenological Ginzburg-Landau-Devonshire-based thermodynamic formulationshave calculated the extrinsic contribution to PEE[81, 82] and ECE[83] due to temperature-and field-dependent motion of domain walls, respectively. It was found that these extrinsiccontributions, that were often ignored in the previous calculations, have an appreciable im-pact on PEE and ECE and both the effects are maximized at the phase boundaries (betweenc and c/a, and between c/a and a1/a2). Later, phase-field studies[84, 85] predicted that theECE is maximum under a tensile strain in the c/a phase as a consequence of the competi-tion between the enhancement of ECE and the reduction of c-domain fraction. A systematicexperimental study across the entire strain-induced phase space is therefore necessary tounderstand the true magnitude and nature of the electrothermal response in polydomainferroelectrics where domain interconversion under both changes in temperature and electricfield can result in sizable electrothermal responses.

1.4 Energy Conversion

1.4.1 Pyroelectric Energy Conversion

The coupling between polarization and heat in pyroelectric materials allows for theconversion of temporal fluctuations in temperature into useful electrical energy. In contrastto the steady-state energy conversions using the Seebeck effect in thermoelectrics,[86] py-roelectrics require a temporal change in the temperature to get a steady flow (alternating)of electric current. This idea was first implemented using a direct conversion cycle wherea periodic heating and cooling of a pyroelectric material drove an alternating current in aload resistor.[87] Later, more popular approaches involved the use of thermodynamic cycles

11

Electric field

Po

lariza

tio

n

1

2

3

4

Isoelectric

heating

Isothermal

polarization

Isothermal

depolarization

Isoelectric

cooling

Win

Wout

Wout

Win

Qin

Qout

Qin

Qout

Entropy T

em

pe

ratu

re

3

2

1

4

Isoelectric

heat rejection

Isoelectric

heat absorption

Isentropic

depolarization

Isentropic

polarization

Qin

Qout

Pyroelectric Energy Conversion Electrocaloric Coolinga b

Win

Wout W

out

Win

Figure 1.3: (a) A schematic of the pyroelectric Ericsson cycle showing the various isothermaland isoelectric processes. The total energy density per cycle is given by the area of the closedloop,

∮EdP . (b) A schematic illustration of electrocaloric cooling showing the various

isentropic and isoelectric processes.

that mimicked gas-phase cycles for energy conversion including, for example, Stirling-likecycles with two isothermal and two isodisplacement processes,[88, 89] Carnot cycles withtwo isothermal and two adiabatic processes.[90] It was however, the adaptation of an Eric-sson cycle (or the Olsen cycle) which includes two isothermal and two isoelectric processes(Figure 1.3a), that revealed the potential of extracting the maximum potential work usingpyroelectrics out of all the cycles studied to date.[91] This cycle begins with the applicationof an electric field at a constant temperature that increases the polarization of the ferroelec-tric (1 → 2). This is followed by an increase in the temperature at constant electric field(2 → 3). Next, the ferroelectric is depolarized isothermally by reducing the electric field(3 → 4). Finally, the ferroelectric is cooled at a constant electric field (4 → 1). The net

workdone includes the workdone due to the pyroelectric effect (E23

∫ T3T2πdT + E41

∫ T1T4πdT )

and dielectric effect (∫ E2

E1εEdE +

∫ E4

E3εEdE). The net heat input, likewise, includes the en-

ergy to change the polarization (T12

∫ E2

E1ΣdE+T34

∫ E4

E3ΣdE) and to increase the temperature

of the ferroelectric (∫ T3T2CdT +

∫ T1T4CdT ). It should be noted that the physical properties

π, Σ, ε and C are functions of electric-field and temperature and the integrals for work andheat should be taken into account. The efficiency of pyroelectric energy conversion is simplythe ratio of the net workdone to the heat input.

Earlier work on pyroelectric energy conversion (PEC) using the Eric-

12

sson cycle has mostly focused on bulk pyroelectric materials. Key exam-ples include bulk-ceramic (Pb1−xLax(Zr0.65Ti0.35)1−x/4O3), [92] and single-crystal(Pb0.99Nb0.02(Zr0.68Sn0.25Ti0.07)0.98O3, [87] Pb(Zn1/3Nb2/3)0.955Ti0.045O3, [93, 94] andPb(Mg1/3Nb2/3)0.68Ti0.32O3).[95] Limited work on thin films include (P(VDF-TrFE), [96–98]and BaTiO3[99]). To cycle the temperature, a wide variety of techniques were employed.Bringing pyroelectric materials in physical contact with hot and cold metal blocks andimmersing the pyroelectric material alternately in thermal baths maintained at high andlow temperatures have remained the most common approaches. Other techniques includedpumping hot/cold fluids for heat exchange or using liquid-based thermal interfaces[98]that permits a change in the thermal conductance between the pyroelectric material andheat source/sink. Combined, these materials/approaches have produced maximum energydensity, power density, and scaled efficiency values of ∼1 J cm−3,[92] ∼30 W cm−3,[99]and 5.4%,[100] respectively to date. The ultimate performance of these prior studies waslimited by aspects related either to the materials themselves (i.e., they had intrinsicallylow pyroelectric coefficients) and/or to the functional form of the material (i.e., inabilityto apply large electric fields and slow thermal cycling due to a large thermal mass). Inturn, to propel the field forward, novel routes (field-induced domain structure and phaseevolution, field-induced defect reorientation, etc.) to enhance the PEE are needed. Further,since PEE is inherently a surface phenomenon, PEC in thin film version of materials needsto be explored that can permit operation at both higher electric field and allow for muchfaster thermal transport.

1.4.2 Electrocaloric Solid-State Cooling

Vapor-compression cooling technology has stood the test of time and has been in usefor virtually all the air-conditioning and refrigeration needs. However, the growing demandfor clean cooling solutions and, in particular, for the ones that are suited for modern wearableelectronics and mobile computing devices, has motivated research to look for newer ways tocool things. In turn, in the last few years, significant progress has been made in exploringroutes to cool things using the elastocaloric,[101] magnetocaloric[31] and electrocaloric ef-fects.[14] Similar to the case of pyroelectric energy conversion, electrocaloric cooling can beachieved by a sequence of thermodynamic cycles (Figure 1.3b). Briefly, the electrocaloriccooling cycle implements adiabatic depolarization (akin to adiabatic demagnetization)[102]and consists of the following steps. First, the dielectric material is polarized adiabatically(1 → 2). This causes the lattice entropy and therefore the temperature to increase. Next,the heat is rejected to a sink while keeping the electric field constant (2→ 3). Next, the fieldis removed adiabatically (3→ 4). This increases the dipolar entropy and in turn reduces thelattice entropy and the temperature. Finally, heat is absorbed from the cooling load (source)to complete the cycle (4→ 1). This process, in fact is the exact opposite (anticlockwise in-stead of clockwise) implementation of the pyroelectric Ericsson cycle and converts electrical

13

work into thermal energy.

The earliest implementation of electrocaloric cooling used the strong temperature-dependence of the dielectric permittivity (under both zero- and large electric field ) of bulkSrTiO3 to extract heat at 4 K and reject it to a reservoir at 15 K.[103] About a decade later,the order-disorder relaxor PbSc0.5Ta0.5O3 (with ∆T = 1.3 - 1.5 K at 25 kV cm−1) was used asthe working body of an electrocaloric cooler.[104] To efficiently use an electrocaloric material,it is required that the heat transfer be fast enough so that heat can diffuse through the entirethickness of the material and the material undergoes a complete heat absorption/rejectionstep.[105] However, the slower heat transport in these materials (due to low thermal con-ductivities) combined with the lower breakdown field of the bulk dielectrics limited the useof ECE materials in their bulk-functional form and propelled the research community toexplore the potential of thin-film ECE materials. Additionally, significant efforts were madeto circumvent the challenges associated with the slower/poor heat transport in the elec-trocaloric system by using thin-film heat switches,[106, 107] switchable liquid-based thermalinterfaces,[108] co-axially rotating electrocaloric rings,[109] to more recent designs involvingelectrostatic actuation.[14]

1.5 Organization of the Dissertation

In the remainder of this dissertation, scientific and engineering work focused on suchelectrothermal material and effects are organized into six chapters, two appendices and asection for references. These chapters cover the following.

In Chapter 2, the synthesis of thin-film versions of the canonical ferroelectricPbZr1−xTixO3 will be discussed. It will be shown, how growth via pulsed-laser deposi-tion can give rise to high-quality heteroepitaxial systems and how this process in particularcan be used to control the growth kinetics and give rise to interesting topographical featuresmostly observed in metals. A wide variety of structural characterization techniques com-monly used to study ferroelectric thin films will be introduced. These techniques, rangingfrom X-ray diffraction-based macroscopic measurement of crystal structure, orientation andepitaxial relationship to more localized probe of the atomic-level topography, piezoresponseand electron diffraction that will be used throughout the body of this dissertation will beintroduced.

In Chapter 3, the challenges in characterizing the electrothermal susceptibility in thinfilms will be discussed. In turn, a novel electrothermal characterization technique is devel-oped and presented for measuring both PEE and ECE in epitaxial ferroelectric thin films. Anelectrothermal test platform is demonstrated where localized high-frequency periodic heat-ing and highly-sensitive thin-film resistance thermometry allows for the direct measurement

14

of pyrocurrents < 10 pA and electrocaloric temperature changes as small as 2 mK usingthe “2-omega” and an adapted “3-omega” technique, respectively. The development of sucha reliable measurement technique will facilitate a deeper understanding of the mechanismsunderlying and magnitude of these responses and inspire confidence in the design of devices;opening the door to a new community of researchers to study such electro-thermal effects ina robust manner.

In Chapter 4, the role of epitaxial strain and the resulting domain structures isstudied on the electrothermal and electromechanical response in ferroelectric thin films ofPbZr1−xTixO3. Beginning with phenomenological models, the stability of various ferroelasticdomain structures under a range of epitaxial strain conditions is discussed. Further, usingphase-field studies a new regime in the temperature-strain phase space is identified wheretwo well-known domain structures can coexist and compete thereby providing facile con-trol of the structure under external perturbations in temperature and electric field. Usingthe techniques described in Chapter 3, the electrothermal susceptibilities of heterostructureswith varying domain structures are investigated and the role of domain structures in theenhancement of both PEE and ECE via the extrinsic contribution is measured. Finally,ferroelectric switching behavior is investigated using scanning-probe-based measurementsand the resulting multi-step switching process concomitant with large electromechanical re-sponse is discussed in the light of a facile interconversion of the different ferroelastic variants.Ultimately, a deterministic and non-volatile writing/erasure of large-area patterns is demon-strated as a pathway to improved electromechanical functionalities.

In Chapter 5, pyroelectric susceptibilities due to a temperature-and electric field-dependent change in the crystal symmetry of a canonical relaxor-ferroelectric system (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 is investigated. The role of electric field in enhancing thezero-field pyroelectric response is discussed within the framework of field-induced rotation ofpolarization and field-induced enhancement of the average polarization. The resulting largevalues of the π in conjunction with the dramatically quenched value of εr under the appliedelectric field yields significant figure of merit for pyroelectric energy conversion. Ultimately,solid-state, thin-film devices are demonstrated that convert heat into electrical energy usingpyroelectric Ericsson cycle, and optimized to yield the highest energy density, power densityand efficiency (fraction of Carnot) values reported to date that put pyroelectrics on par withthermoelectrics.

In Chapter 6, the electrothermal response in new, lead-free and silicon-compatiblebinary oxide ferroelectric HfO2 thin films are investigated. Like many other ferroelectrics,there exists a significant dearth of literature on the pyroelectric and electrocaloric responseof HfO2-based ferroelectrics. Here, I measure the pyroelectric and electrocaloric response(for the first time) in ultra-thin versions of these materials and correlate the response to theinherent defect structure in the films.

15

In Chapter 7, I will summarize the major findings of the current work and providesuggestions for future work to build on the findings presented here.

Appendix A covers the details of pulsed-laser deposition and gives an overview ofthin-film-growth mechanisms with special attention to the epitaxial growth of complex oxidethin films.

Appendix B provides details of the thermal transport models developed and employedto characterize pyroelectric and electrocaloric effects in thin-film heterostructures.

16

Chapter 2

Synthesis and StructuralCharacterization of Complex OxideThin Films

In this Chapter, I report the synthesis and structural characterization of complexoxide thin films of the type to be used in this dissertation. Using the prototypical ferroelec-tric PbZr0.2Ti0.8O3 as an example, I introduce pulsed-laser deposition as a non-equilibriumgrowth technique to synthesize epitaxially-strained version of functional oxides thin films.In particular, this Chapter will introduce the growth process while highlight the obser-vation of strain-induced growth instabilities and nanoscale-surface patterning as the epi-taxial strain perturbs the growth dynamics. Using scanning-probe force microscopy, agrowth-mode transition from 2D-step flow to self-organized, nanoscale 3D-island formationin PbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001) heterostructures are demonstrated as the kineticsof the growth process respond to the evolution of strain. Using, X-ray diffraction-basedreciprocal space mapping (RSM) studies and transmission electron microscopy (TEM), Ishow how the increase in the relaxation of epitaxial strain results in the formation of misfitdislocations at the buried film-substrate interface. In turn, a periodic, modulated strain fieldis generated that alters the adatom binding energy resulting in inhomogeneous adsorptionkinetics causing preferential growth at certain sites. Ultimately, this strain-induced kineticinstability drives a transition from 2D growth to long-range, periodically-ordered arrays ofso-called “wedding-cake” 3D nanostructures which self-assemble along the in-plane [100] and[010].

17

2.1 Introduction

Next-generation thin-film devices will require the production of ultrathin films andsuperlattices with atomically-sharp interfaces and, potentially, self-patterned 3D nanostruc-tures to enable a range of applications.[110–113] In turn, the ability to control and manipulatethe nature of growth processes in epitaxial films, for instance between 2D-growth modes forplanar structures to self-organized 3D nanostructures, is critical. In order to achieve suchcontrol, an understanding of the physics of the complex atomistic processes that occur atthe growth-front is essential. Such control can be obtained by understanding the kineticsof growth instabilities in non-equilibrium growth processes. For example, in the growth ofPt/Pt (111)[114] and Ag/Ag (111),[115] multi-terrace islands can arise as a consequence oflimited interlayer mass transport kinetics due to an Ehrlich-Schwoebel (ES) energy barrierwhich restricts the diffusion of adatoms down a step edge and biases growth towards multi-terrace islands.[116–118] Quantitative estimates of this barrier have been extracted fromscanning tunneling microscopy (STM) studies[114, 119] and kinetic Monte-Carlo[120] andphase-field simulations.[121] The ES barrier can also cause a morphological Bales-Zangwillinstability,[122] which is diffusional in origin and promotes waviness of the vicinal terraceedge[123] and can destabilize step-flow growth producing step meandering and multi-terraceisland formation.[124]

At the same time, research has suggested that the growth kinetics of heteroepitaxialthin-film systems can be strongly dependent on epitaxial strain. In both metal[125, 126]and semiconductor[127] systems, the diffusion of adatoms can be strongly influenced by thesurface strain. For instance, the activation energies for Ag self-diffusion can vary greatlydepending on the strain state with considerably lower values being observed for diffusion ona strained, pseudomorphic Ag monolayer on Pt (111) (60 meV) as compared to unstrainedAg (111) surfaces (97 meV).[115] Additionally, relaxation of epitaxial strain via misfit dislo-cation formation can also affect the surface diffusion and nucleation. Again, in Ag/Pt (111)heterostructures, after 2 monolayers of Ag are deposited, the large lattice mismatch (4.3%)is accommodated by the formation of stress relief patterns where the misfit dislocations onthe surface repel the adsorbed adatoms and partition them into 3D nanostructures.[128] InSi1−xGex films on Si (001), strain relief manifests in a series of morphological changes. Atsub-monolayer coverages surface reconstruction begins to form.[129, 130] Later, 3D islandswith well-defined facets that are coherent with the substrate lattice begin to form. Thedriving force for such facet formation is the effective strain relief along the crystallographicdirection that compensates the increase in surface energy.[131] Strain relief can also manifestby cross-hatching of the surface or formation of undulations or “ripples.” Such structureshave been explained by equilibrium calculations as the result of balancing the strain energyfrom the buried misfit dislocations and the surface energy[132, 133] and by incorporatingstrain relaxation due to dislocation glide in the film interior and lateral surface transportthat eliminates surface steps.[134, 135]

18

On the other hand, the evolution of surface morphology in the growth of complex-oxide thin films has been studied far less. The limited work includes experimental study ofa growth mode transition from layer-by-layer to step-flow growth in SrRuO3/SrTiO3 (001)heterostructures[136] and theoretical study of the evolution of growth mode under competingelastic and kinetic forces.[137] Recently, researchers have reported evidence of surface cross-hatch morphologies in SrTiO3/NdGaO3 (110) heterostructures[138] which were attributedto the anisotropic strain relaxation with 60◦ misfit dislocation formation that caused lateralsurface step flow. On the other hand, in La0.5Ca0.5MnO3/SrTiO3 (001) heterostructures,[139]it was suggested that the interaction of the depositing adatoms with the dislocation stressfield was responsible for the cross-hatching.

Building off of this work in metals, semiconductor, and oxide thin films, this Chap-ter will demonstrate the presence of misfit dislocations at a buried interface producing aperiodic, modulated strain field that alters the adatom binding energy landscape and leadsto a kinetic instability that drives a transition from 2D- to ordered, 3D-island growth inPbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001). The result is long-range, periodic, ordered arrays of“wedding cake” 3D nanostructures along the in-plane [100] and [010]. The spatially inho-mogeneous surface strain state drives deterministic changes in the surface morphology andpotentially provides a useful experimental pathway to understand and control the dynamicsof epitaxial thin-film growth processes.

2.2 Growth of Epitaxial PbZr0.2Ti0.8O3 Thin Films

Thin films of PbZr0.2Ti0.8O3, a prototypical tetragonal ferroelectric (a = 3.95 A andc = 4.13 A at 300 K)[140] with thickness ranging from 25-150 nm were grown on 25 nmSrRuO3/SrTiO3 (001) (cubic, a = 3.905 A at 300 K) substrates using pulsed-laser deposition(details of the pulsed-laser deposition process and the setup are in Appendix A). Briefly, inpulsed-laser deposition, a UV (λ = 248 nm) excimer laser with a pulse width of 10-50 nsand an energy of 50-100 mJ ablates the target (ceramic or single-crystal) with the desiredcomposition resulting in a power of ∼ 107 W. This intense power of the laser plume producesa plasma plume that is directed towards a heated substrate upon which the film is deposited.For oxides, the deposition is done in an oxidizing atmosphere (typically with dynamic partialpressures of oxygen in the 1-200 mTorr range) at temperatures typically ranging from 500-900◦C.

SrRuO3 (an orthorhombic oxide metal with a pseudocubic lattice constant of 3.93A)[141] was used as a lattice-matched bottom electrode for electrical characterization of theferroelectric heterostructure. SrRuO3 was grown at 645◦C in an oxygen pressure of 100mTorr with a laser fluence of 1.30 J cm−2 and repetition frequency of 10 Hz. Subsequently,PbZr0.2Ti0.8O3 was grown at 635◦C in an oxygen pressure of 200 mTorr with a laser fluence

19

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

(b) (d) (f) (h)

Figure 2.1: Atomic force microscopy (AFM) images and extracted line profiles (blackdashed line in AFM image) for heterostructures with PbZr0.2Ti0.8O3 thickness of (a),(b) 25nm, (c),(d) 40 nm, (e),(f) 55 nm, and (g),(h) 150 nm, respectively. For (b), (d), (f), and (h),the vertical scale is in units of nanometers and the horizontal scale in units of microns.

of 1.35 J cm−2 and repetition frequency of 3 Hz. Following growth, the heterostructure wascooled to room temperature at 5◦C min−1 in a static ∼700 Torr pressure of oxygen.

2.3 Structural Characterization of PbZr0.2Ti0.8O3 Thin

Films

2.3.1 Atomic Force Microscopy Study of PbZr0.2Ti0.8O3 ThinFilms

Topographic study of the films was carried out using atomic force microscope (AFM)(MFP-3D, Asylum Research). AFM studies of surface topography reveal that 25 nm thickheterostructures (Figure 2.1a) exhibit atomically-smooth, terraced surfaces indicative ofgrowth in a predominantly step-flow growth mode with some formation of 2D islands onthe flat terraces which eventually coalesce with the vicinal step edges. A line-trace acrossthis surface confirms that the individual steps are a single unit cell in height (Figure 2.1b).As the thickness of the film increases to 40 nm (Figure 2.1c), a change in the surface struc-ture occurs whereby localized multi-terrace islands, which are ordered along the [100] and[010], are observed to form (the multi-layer island formations can be seen in the line profile,Figure 2.1d). By a film thickness of 55 nm (Figure 2.1e), a highly-ordered array of multi-

20

terrace islands is observed (confirmed by the line trace, Figure 2.1f). The ordered nature ofthe multi-terrace islands gives the appearance of a cross-hatched or ripple-like pattern thatruns along the [100] and [010]. Detailed analysis of the linear density of these features wascompleted by measuring several areas of the same size over several areas on several differentheterostructures of each thickness. The linear density of the cross-hatched or ripple-like fea-tures increases from an average of 0.78 µm−1 for 40 nm thick films to an average of 4.53 µm−1

for 55 nm thick films. When the heterostructure thickness reaches 150 nm (Figure 2.1g),the multi-terrace islands are reminiscent of so-called “wedding-cake” structures (due to theiruncanny resemblance)[142] with a large number of terraces (Figure 2.1h). The tilted struc-ture of ferroelastic domains in the PbZr0.2Ti0.8O3[143] can also be seen superimposing on themulti-terrace island structure (Figure 2.1g). These multi-terrace islands in the PbZr0.2Ti0.8O3

heterostructures bear a striking resemblance to the features observed in homoepitaxial metalfilms.[114, 115] Again, in metal films, such features were explained solely by the presence ofan ES barrier which limits interlayer mobility. Despite some similarities, there are a few keydifferences between the observations in the PbZr0.2Ti0.8O3 heterostructures as compared tothat in the metal films. First, we note that multi-terrace island formation does not occurin the PbZr0.2Ti0.8O3 heterostructures until a film thickness of 30-40 nm as opposed to theonset in sub-monolayer deposition for metal films.[114] Second, in contrast to metal sys-tems where there is no long-range order to the island formation, the multi-terrace islands inPbZr0.2Ti0.8O3 are found to be aligned along the [100] and [010] and to occur in bands withan average periodicity that scales inversely with the film thickness.

2.3.2 X-Ray Diffraction Studies of PbZr0.2Ti0.8O3 Thin Films

To understand the mechanism underlying both the delayed onset and ordering of thesemulti-terraced island features, we probed the structure of the films using high-resolution X-ray diffraction (X’Pert MRD Pro equipped with a PIXcel detector, Panalytical). θ − 2θdiffraction patterns reveal that all films are single-phase (Figure 2.2a) and epitaxial (Figure2.2b). As the film thickness increases from 25 nm to 150 nm, the 002-diffraction peak forthe PbZr0.2Ti0.8O3 shifts to higher Bragg angles indicating a decrease in the out-of-planelattice parameter possibly due to strain relaxation. Further insight is gained from the RSMstudies about the 002- and 103-diffraction conditions. Symmetric RSM studies about the002-diffraction condition confirm that the morphological features observed herein are not theresult of ferroelastic domain formation due to the absence of diffraction peaks correspondingto the a domains (i.e., the 200-diffraction peak) (Figure 2.3a-b) that only shows up at muchlarger thickness (150 nm, Figure 2.3c). Asymmetric RSM studies about the 103-diffractioncondition revealed that the strain in the films steadily relaxes with increasing film thickness(Figure 2.4a-c). It can be seen in the thinnest substrate are all found to possess the same Qx

values for their diffraction peaks which is indicative of coherently-strained films with in-planelattice parameters that match that of the substrate. As the film thickness is increased (i.e.,

21

Figure 2.2: (a) X-ray diffraction pattern of PbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001) or PZT/SRO/STO (001) heterostructures for films with (bottom to top) 25 nm, 55 nm and 150 nmPZT film thicknesses. (b) Phi scan about the SrTiO3 110- (blue) and PZT 110- (red) diffrac-tion conditions for a 55 nm PbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001) heterostructure revealingfull in-plane epitaxy.

55 nm and 150 nm, Figure 2.4b and 2.4c, respectively), however, the diffraction peak for thePbZr0.2Ti0.8O3 is found to exhibit intensity that is smeared to smaller Qx values revealingthat the in-plane lattice parameter of the film is larger than that of the substrate and thebottom electrode and that the film has partially relaxed towards its bulk in-plane latticeconstant. It should also be noted that while SrRuO3 bottom electrode of identical thicknesswas grown under identical growth conditions, there is smearing of the Qx values of SrRuO3

with increasing thickness of PbZr0.2Ti0.8O3 film indicating lattice distortion in the SrRuO3

layer possibly as a result of strain relaxation of the PbZr0.2Ti0.8O3 film.

2.4 Strain Relaxation and its Implications on Growth

Mode

2.4.1 Quantification of Strain Relaxation

To quantify the magnitude of strain relaxation, we calculated an average diffractionpeak position (θavg, ωavg) weighted by the intensity. The effective in-plane lattice parametera‖ and in-plane strain relaxation can thus be calculated as

22

Figure 2.3: Symmetric reciprocal space maps about the pseudocubic 002-diffraction con-dition for PbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001) heterostructures with PbZr0.2Ti0.8O3 filmthickness of (a) 25 nm, (b) 55 nm, and (c) 150 nm, respectively. Thinner films (25 nm and55 nm) do not show the presence of the PbZr0.2Ti0.8O3 200-diffraction condition suggestingthe absence of any in-plane domain formation unlike the thicker films.

afilm‖ =λ

2sin(θavg)sin(θavg − ωavg)√h2 + k2 (2.1)

StrainRelaxation =afilm‖ − asubstrate‖

afilm,bulk‖ − asubstrate‖

(2.2)

In turn, it can be seen that the PbZr0.2Ti0.8O3 film monotonically relaxes with in-creasing film thickness (red data, Figure 2.5). The surface ripple density (blue data, Figure2.5), however, first increases - trending similarly to the strain relaxation in Regime I - beforeabruptly peaking and then rapidly decreases with a subsequent increase in the film thickness(thus trending counter to the strain relaxation in Regime II).

23

0.19 0.21

0.54

0.55

0.56

0.57

0.58

0.59

0.60

0.19 0.21Qx

0.19 0.21

0.54

0.55

0.56

0.57

0.58

0.59

0.60

Qy

25 nm 55 nm 150 nm

SrTiO3 103

SrRuO3 103

(a) (b) (c)SrTiO3 103

PbZr0.2Ti0.8O3

103

SrRuO3 103

SrTiO3 103

SrRuO3 103

PbZr0.2Ti0.8O3

103

PbZr0.2Ti0.8O3

103

Figure 2.4: Asymmetric reciprocal space mapping studies about the 103-diffraction condi-tions of heterostructures with PbZr0.2Ti0.8O3 thicknesses of (a) 25 nm, (b) 55 nm, and (c)150 nm. Diamond symbols (black, open) represent the bulk lattice parameter.

2.4.2 Predicting Surface Strains using Linear Elastic Theory

Strain relaxation in lattice mismatched heterostructures is usually accommodated bythe formation of misfit dislocations during growth.[144, 145] We propose that the growthinstability and formation of spatially-ordered islands is the result of the stress fields from a2D array of edge dislocations at a buried interface and the subsequent preferential nucleationof adatoms driven by inhomogeneous surface strains.[146, 147] Let us consider an array ofuniformly-spaced, semi-infinite edge dislocations running parallel to and at a distance lfrom the free surface of a semi-infinite body. For this system, the Burgers vector b = a <100 >.[148–150] The in-plane stress at an arbitrary point can be calculated by summing thecontributions from all the dislocations in the array. The normal in-plane stress σxx along thefree surface is given by[151]

σxx(x

h, z = 0) = 8k0

bl

h2

{∞∑

n=−∞

(xh− n)2[

(xh− n)2 + ( l

h)2]2}

(2.3)

24

Figure 2.5: In-plane strain relaxation (%, red circles) and surface ripple density (µm−1,blue squares) as a function of film thickness.

where z = 0 implies the stress at the film surface and x is an arbitrary point in theplanar space, k0 = µ/2π(1 − ν) where µ and ν are the shear modulus and Poisson ratio,respectively, h is the dislocation periodicity, b is the magnitude of the Burgers vector, and lis the distance between the free surface and the dislocation array. The total stress at the filmsurface is thus calculated by superimposing the compressive stress imposed by the epitaxialstrain and that due to the presence of a dislocation array at the interface as calculated above(Equation 2.3). The resultant in-plane surface strain can thus be plotted (Figure 2.6a).

It can be seen that the pseudomorphically-strained heterostructures exhibit no straininhomogeneity. As the film thickness increases and the heterostructures begins to relax,however, two effects are observed: 1) the mean compressive strain reduces due to an av-erage strain relaxation and 2) a surface strain modulation due to the periodic strain fieldsfrom the buried dislocation network is formed. It can also be seen that in thinner films,the amplitude of strain modulation is larger in comparison to the thicker films. Such mod-ulations in the surface strain state can affect the binding energy of the adatoms. Withinlinear elastic theory, the binding energy varies linearly with external strain εext and can bewritten as Ead = E0

ad + Aσεext,where E0ad is the binding energy on an unstrained surface,

A is the surface area, and σ is the surface stress tensor induced by the adatom.[152] Thisdifference in the binding energy due to external strain can be understood as the additionalelastic energy equal to the work done by the adatom-induced stress against the appliedstrain. In the present case, relaxation of a film by the formation of buried dislocations at

25

Figure 2.6: (a) Calculated variation of compressive in-plane strain at the film surfaceas a function of film thickness. The data, from top to bottom, represent the case forpseudomorphically-strained (blue), 40 nm thick (purple), 100 nm thick (magenta), and fully-relaxed (red) heterostructures and show periodic strain modulations on the surface of 40 nmand 100 nm thick films due to the buried misfit dislocation array (T-symbol). (b) Schematicillustration of the variation in the adatom binding energy on the surface where red andblue areas are regions with stronger and weaker binding energy, respectively. (c) Schematicillustration of the in-plane lattice parameters on the surface of a film under a partiallyrelaxed biaxial strain - red and blue areas are under less and more compressive strain, re-spectively. (d) The measured topography of a 55 nm thick PbZr0.2Ti0.8O3 heterostructureshowing aligned wedding-cake structures.

the film-substrate interface causes strain modulation in two orthogonal directions resultingin an inhomogeneous deformation of the surface creating local regions of maximum and min-imum biaxial compressive strain (depicted in blue and red respectively, Figure 2.6b). Thisinhomogeneity in the surface strain locally alters the binding energy of the adatoms andcreates an inhomogeneous potential energy landscape (Figure 2.6c). Quantification of thismodulation of the local binding energy is possible, but difficult. For instance, first-principlescalculations have been used to examine the strain dependence of adatom binding energiesin both homo- and heteroepitaxial growth of single-component metal and semiconductorsystems.[125, 152] Such quantification for the current multi-component system under studyin this work, however, has not been completed due to the lack of first-principles calculationsof the surface stress tensor for a multi-component system with complex adsorbing species.One can, however, argue qualitatively that the adatom binding energy will alter in a peri-odic manner following the distribution of the dislocation network and, therefore, the surfacestrain modulation. It should be noted, that depending upon the nature of the surface stresstensor, the adatom binding energy could either increase or decrease with the strain and thusany qualitative representations should be viewed with this limitation in mind. In response

26

Figure 2.7: (a) Cross-section, dark-field TEM image of the same heterostructure takenunder the g = (200) two beam condition along the [012] zone axis showing the presenceof dislocations (white arrows point to dislocations located at the PbZr0.2Ti0.8O3/ SrRuO3

interface) and corresponding strain fields. (b) Zoom-in image of the strain-induced contrastfrom the dislocation strain fields. (c) The representative electron diffraction pattern takenalong [012] zone axis.

to such a spatially-varying surface strain (Figure 2.6b) and adatom binding energy (Figure2.6c), the film surface can evolve into a spatially-patterned, ordered array of multi-terraceislands (Figure 2.6d) wherein 3D “wedding cake” structures are found to form in regions ofthe least compressive strain which gives rises to the highest binding energy (preference forstrong adatom binding).

2.4.3 Evidence of Misfit Dislocation at Film-Substrate Interface

To confirm the presence of misfit dislocations in the heterostructure, transmissionelectron microscopy studies were completed on a heterostructure with a 55 nm thickPbZr0.2Ti0.8O3 film using JEOL 3010 microscope at the National Center for Electron Mi-croscopy at Lawrence Berkeley National Laboratory. Cross-sectional, dark field imagingtaken under the g = (200) two beam condition along the [012] zone axis reveals the presenceof edge-type misfit dislocations at the PbZr0.2Ti0.8O3/SrRuO3 interface (Figure 2.7a). Amagnified region of this image reveals contrast characteristic of the distortion of the latticearound the dislocation cores (Figure 2.7b); in other words, we can directly image the strainfields produced by the dislocations. It can be seen that such a distortion extends close to thefilm surface and manifests as periodic strain oscillations as shown previously in the modelabove. By measuring several regions of the cross-section sample, the average strain modula-tion was found to be periodic with a period of ∼ 200 nm corresponding to the surface rippledensity of ∼ 5 µm−1.

27

Figure 2.8: Atomic force microscopy images and illustration of island identification proce-dure for heterostructures with thicknesses of (a) 55 nm, (b) 65 nm, (c) 90 nm, and (d) 150nm.

2.4.4 Island Coarsening with Increasing Thickness

Having established a connection between the multi-terrace islands that form at thesurface of the growing film and the misfit dislocations and strain modulations in the het-erostructures, it is also important to understand how the growth process evolves as the strainis relaxed and the strain modulations subside (Regime II, Figure 2.5). To do this, we focuson films in a thickness regime where dislocations have already formed and the multi-terraceislands are present to understand how further film growth drives a change in film morphology.In particular, we have extracted the top terrace island density from AFM studies of filmsvarying from 55 nm to 150 nm in thickness to gain insight into the nature of island coarsen-ing (Figure 2.8a-d). It can be seen that the density of the top terrace islands decreases withincreasing film thickness (Figure 2.8e) indicating that as the strain is relaxed, the local vari-ations in the binding energy are reduce, thus diminishing the preferential growth in specificstrain-induced capture zones. As the binding energy profile is flattened across the surface,diffusion of mass from the islands to the valleys is enabled, thus resulting in coalescenceof the islands. It should be noted, however, that the despite a driving for force island-to-valley diffusion, the presence of the ES barrier restricts the extent to which surface diffusioncan occur (as it restricts downward funneling) and thus prevents complete annihilation ofisland-like features and terraces on the surface. The net effect is, in turn, that the number oftop-terrace islands increases (again, predominantly due to the ES barrier) while the islandsthemselves begin coalesce.

28

Film thickness (nm) Island density (µm−2)55 20.0±1.665 8.1±0.890 4.0±0.7150 2.7±0.1

Table 2.1: Measured island density as a function of increasing film thickness.

Figure 2.9: Schematic illustration of the thickness evolution of the surface structure. (a) Inthin, unrelaxed films, step-flow growth persists; the arrows indicate the possibility of attach-ment of adatoms either directly to the step edges or after being reflected by the descendingedge due to the ES barrier (inset shows the adatom potential energy landscape). (b) As thethickness increases, there is an onset of a kinetic instability due to dislocation formation andnucleation of islands in the capture zones (inset shows the adatom potential energy land-scape is lowered, red color, due to the dislocation strain field). (c) With further increasedfilm thickness, formation of an ordered array of dislocations induces ordered, multiterraceisland formation. (d) Continued film growth, leads to formation of multi-terrace islands andisland coarsening.

29

2.4.5 Mechanism behind the Topography Evolution underSurface Strain

The observed surface morphology evolution can be explained as follows (Figure 2.9).In thin films (i.e., 25 nm thick heterostructures) which have not relaxed, the surface in-planestrain resulting from the lattice misfit is spatially homogeneous enabling persistent stepflow growth (Figure 2.9a). As relaxation begins, the resulting strain modulations patternthe surface with an inhomogeneous adatom binding energy profile. As a result, there arelocal regions wherein there is stronger binding of the adatoms to the surface which, inturn, reduces the local desorption rate, gives rise to preferential growth in these specificcapture zones,[153] and disrupts the coherence of the step-flow growth (Figure 2.9b). Thisis consistent with kinetic Monte Carlo simulations[154] which suggest that periodic strainfields can alter the binding energy of the adatoms and experimental work on the growthof self-assembled Ge quantum dots which demonstrated that the quantum dots preferablygrow first over the intersection of two perpendicular buried dislocations, second at a singledislocation line, and finally away from dislocations.[155] With subsequent deposition (i.e.,40 nm thick heterostructures), the film can potentially grow in three different ways withadatoms 1) attaching to the existing vicinal step edges, 2) attaching at the edges around thefreshly nucleated 2D islands, or 3) nucleating on top of the 2D island and growing as stable“wedding-cake” structures. AFM (Figure 2.6d) confirms that the second and the third modesare preferred as a regular pattern of almost circular, multi-terrace islands grow in 2D periodicarrays. Once the surface is patterned with inhomogeneous strain, the vicinal steps do notadvance as parallel edges and can grow outwards as finger-like protrusions thereby markingthe transition from stable to unstable step-flow growth and finally ordered multi-terraceisland formation (Figure 2.9c). The fact that the islands are multi-terraced additionallysuggests the existence of an ES barrier that inhibits down-step diffusion; adatoms thatland on top of these islands can only desorb or nucleate as stable clusters that grow as anew terrace. With further growth, strain modulations on the surface are dampened withincreasing thickness (Figure 2.6a). Hence, the difference in the binding energy between thecompressive strain maxima and minima decreases. In turn, the adatoms are less stronglyconfined to the capture zones, reducing nucleation and accentuating the coalescence of theexisting islands (Figure 2.8).

2.5 Conclusion

The results presented here demonstrate that by gaining control of the nature of epi-taxial strain relaxation, we are provided a route to produce self-patterned film surfaces astemplates for fabrication of novel self-assembled nanostructures. Secondly, the preferenceof certain systems to produce defects along specific crystallographic directions provides a

30

second level of control to produce ordered and periodic nanoscale structures. In summary,we showed that the strain fields from buried misfit dislocation arrays can strongly affect thedynamics of nucleation and growth in epitaxial thin films. A transition from 2D to 3D-islandgrowth in PbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001) is observed with increasing film thickness.This transition is driven by the evolution of a periodic misfit-dislocation-induced strain field,in two orthogonal directions, that creates local binding energy variations which, in turn,give rise to local capture zones for preferential adatom nucleation. As the film thicknessincreases, a reduction in the surface strain modulation gives rise to a more homogeneousbinding energy profile which, in turn, diminishes the influence of the local capture zones andresults in a coalescence of the island features.

31

Chapter 3

Electrothermal Characterization ofComplex Oxide Thin Films

In this Chapter, I report a direct technique developed for measuring both pyroelectricand electrocaloric effects in epitaxial thin films. Using the same 150-nm-thick PbZr0.2Ti0.8O3

thin films synthesized in Chapter 2 as a model test material, here, an electrothermal testplatform is demonstrated wherein localized high-frequency (approximately 1 kHz) periodicheating and highly sensitive thin-film resistance thermometry allows for the direct measure-ment of pyrocurrents (< 10 pA) and electrocaloric temperature changes (< 2 mK) using the“2-omega” and an adapted “3-omega” technique, respectively. Frequency-domain, phase-sensitive detection permits the extraction of the pyrocurrent from the total current, which isoften convoluted by thermally-stimulated currents. The wide-frequency-range measurementsemployed in this Chapter further show the effect of secondary contributions to pyroelectric-ity due to the mechanical constraints of the substrate. Similarly, measurement of the elec-trocaloric effect on the same device in the frequency domain (approximately 100 kHz) allowsfor the decoupling of Joule heating from the electrocaloric effect. Using one-dimensional,analytical heat-transport models, the transient temperature profile of the heterostructure ischaracterized to extract pyroelectric and electrocaloric coefficients.

32

3.1 Background and Challenges in Measuring

Electrothermal Susceptibilities

The pyroelectric coefficient π is typically characterized by measuring the pyroelectriccurrent (iP = Aπ dT

dt, where A is the area of the capacitor and dT

dtthe temperature ramp rate)

generated in response to a known temperature transient. Most techniques used to measurepyroelectric properties were developed to probe bulk ceramics or single crystals, includinglaser-induced heating[156] and continuous ramp-rate heating studies.[157] These techniquesare adequate to estimate π for large samples (or large area capacitors), but lack the precision,as a consequence of poor temperature accuracy, non-uniform heating, and contributionsfrom thermally-stimulated currents[21, 22], to measure small-volume samples (and small-areacapacitors).[3] For all materials, but particularly for thin-film samples, measurement methodsthat can separate out deleterious or spurious signals are key to accurately measuring the truepyroelectric nature. In turn, the work on thin films has turned to phase-sensitive pyroelectricmeasurement techniques[24, 25], including those based on microfabricated resistive heater-based measurements[27] and modulated laser-based approaches[26] to more accurately assessthe pyroelectric response of these small-volume samples.

The ECE, on the other hand, is parametrized by the isothermal entropy change(∆Siso = −

∫ E2

E1ΣdE) where Σ =

(∂S∂E

)T

is the electrocaloric coefficient. In the case ofbulk materials, the traditional approach to measuring this quantity is to measure either thechange in temperature using thermometry[34] or the heat flux using calorimetry.[38, 158]Similar to the PEE, the last few decades have seen growing interest in investigating theECE in thin films and, in turn, a need for more advanced measurement approaches hasdeveloped. Driven by advances in thin-film epitaxy, researchers can now synthesize ferro-electric thin films and superlattices with atomic-level control over composition and epitaxialstrain.[58, 159] It is thought that such thin-film versions of ferroelectrics can undergo signif-icantly larger (order-of-magnitude) temperature changes as they can be driven considerablyharder than their bulk counterparts owing to the fact that considerably higher fields can beapplied to these thin versions of materials.[160] In turn, reports of large ECE in thin filmsof PbZr0.95Ti0.05O3,[29] (1-x)PbMgyNb1−yO3-(x)PbTiO3,[29, 161] and SrBi2Ta2O9[162] haverejuvenated interest in the study of ECE.

Despite this growing interest, these electrothermal responses (i.e., PEE and ECE)in thin films remain considerably under-studied compared to dielectric, piezoelectric, andferroelectric effects. This discrepancy in study (and, in turn, understanding) is primarilyrelated to the fact that direct (and accurate) measurements of the temperature changesare difficult. The small thermal mass of thin-film-based devices and the presence of thesubstrate leads to non-adiabatic conditions where the film loses heat to the substrate with atime constant smaller than the characteristic time constant for the application (or removal)

33

of the electric field.[163] These inherent difficulties in measurement have, therefore, resultedin the majority of work to date relying on so-called ”indirect” methods to measure ECEIn such indirect approaches, the polarization is measured as a function of electric field atdifferent temperatures to extract

(∂P∂T

)E

and the ECE is inferred via application of the

Maxwell relation(∂P∂T

)E,σorε

=(∂S∂E

)T,σorε

. This allows one to estimate the isothermal dipolar

entropy change (∆Siso) and, in turn, the adiabatic temperature change (∆Tad) as[164, 165]

∆Siso = −∫ E2

E1

(∂P

∂T

)E

dE (3.1)

∆Tad =

∫ E2

E1

(T

C(T,E)

)(∂P

∂T

)E

dE (3.2)

where C(T,E) is the volumetric heat capacity. Many recent reports of large ECE,including ∆T = 12 K for PbZr0.95Ti0.05O3 (∆E = 480; kV cm−1, initial temperature of 220◦

C),[29] ∆T = 12 K for the ferroelectric copolymer P(VDF-TrFE),[30] and colossal ∆T = 45.3K at the transition temperature of films of the relaxor ferroelectric Pb0.8Ba0.2ZnO3[69] haveutilized this indirect approach. Correct application of this approach, however, requires thatthe appropriate mechanical boundary conditions of the material are considered; particularlyin the case of thin films where the mechanical clamping to the rigid substrate breaks thethermodynamic equivalence of the Σ and π.[35] Ultimately, the development of reliablemeasurement techniques for thin films is needed to truly understand the mechanism andmagnitude of these responses and to inspire more confidence for the design of electrocaloricdevices.

3.2 Device Design

Measurement of both the PEE and the ECE on the same thin-film heterostructurerequires the ability to heat the ferroelectric thin film and measure iP flowing across theferroelectric in a capacitor circuit and to apply electric fields across the ferroelectric andmeasure the temperature change in the ferroelectric layer. This calls for an independentferroelectric circuit, comprised of a rectangular capacitor-like geometry with symmetric topand bottom electrodes, and a thermal circuit, comprised of a narrow thin-film metal stripacting as a heater wire and four-point probes for thermal sensing.

Here, as a demonstration of the potential of this approach, I focus on measuring theelectrothermal response of a 150-nm-thick thin film of PbZr0.2Ti0.8O3 epitaxially grown on20-nm-thick SrRuO3/SrTiO3 (001) substrate via pulsed-laser deposition using the processdescribed in Chapter 2. Following growth, the electrothermal characterization devices are

34

Device isolation

Deposit electrical

insulator

Deposit metal

contacts

Deposit top electrode

(a) (b)

A

A

SrRuO3 (90 nm)

SrRuO3 (20 nm)

PbZr0.2Ti0.8O3 (150 nm)

SrTiO3

Pt (100 nm)

SiNx (200 nm)

Z

XY

Figure 3.1: (a) Schematic of the process to fabricate the electrothermal device for measuringpyroelectric and electrocaloric effects in thin films. (b) Section-view through the view A-Adepicting various layers in the heterostructure. The film under test here is PbZr0.2Ti0.8O3

(red), while the two SrRuO3 layers (blue) connect to the top and bottom metal contacts.

produced via a multistep, microfabrication process. Briefly, the ferroelectric heterostruc-ture is lithographically patterned and ion-milled to define the bottom electrode and theferroelectric “active” layer (Figure 3.1a). This step removes the bottom electrode (SrRuO3)everywhere except under the active layer and the bottom electrode probe pad. This greatlyreduces the risk of electrical shorting of the top and the bottom electrode during wire bond-ing. In this work, 90-nm-thick SrRuO3 is then selectively deposited as a symmetric topelectrode using a MgO hard-mask process[166] to establish a rectangular ferroelectric ca-pacitor geometry (300 µm × 20 µm). It should be noted that the SrRuO3 deposited overthe ferroelectric or bottom-electrode mesa does not contact the SrRuO3 deposited on theion-milled region of the substrate to be used later for establishing an electrical contact pad.This is done purposefully to ensure that the SrRuO3 does not get deposited on the sidewallsof the ferroelectric layer and electrically short to the bottom electrode. Next, to electricallyisolate this ferroelectric circuit from the thermal circuit, a 200-nm-thick blanket layer of SiNx

is deposited using plasma-enhanced chemical vapor deposition (SiH4 + NH3 based). SiNx ischosen as it exhibits a low dielectric permittivity (εr ≈ 7) and a thermal conductivity of 2W m−1 K−1 measured via the differential 3-omega method.[167] The deposition temperatureof the nitride is limited to 350◦C to limit decomposition (i.e., Pb, Ru, or O loss) of thefilms. This nitride layer is subsequently patterned and selectively etched using reactive-ionetching in SF6 plasma, to obtain an electrically insulating film on top of the ferroelectriccapacitor. Etching the SiNx from everywhere around the ferroelectric capacitor reduces thelateral dissipation of heat, which makes the top sensor line more sensitive to measuring smalltemperature changes. Finally, a 100-nm-thick platinum thin-film resistance heater and ther-mometer is lithographically patterned in the shape of a thin strip with four probe pads (twoouter and two inner pads; see the geometry in Figure 3.1a) to define the thermal circuit(Figure 3.1b). Platinum contact pads for the top and the bottom electrode are also definedin this step. While the platinum pad for the bottom electrode contacts the bottom SrRuO3

35

100 1,000 10,0000

100

200

300

400

Die

lectr

ic P

erm

ittivity (ε

r)

Frequency (Hz)

0.00

0.05

0.10

0.15

0.20

Lo

ss T

an

ge

nt

(c)

-3 -2 -1 0 1 2 310

-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Cu

rre

nt (A

/cm

2)

Voltage (V)

(a)

-200 -100 0 100 200-100

-75

-50

-25

0

25

50

75

100

Po

lari

za

tio

n (µ

C/c

m2)

Electric Field (kV/cm)

1 kHz10 Hz1 Hz

33 mHz

(b)

Figure 3.2: (a) DC Current-voltage characteristics, (b) ferroelectric hysteresis loops, and(c) dielectric permittivity and loss tangent as a function of frequency for a 150 nm thickPbZr0.2Ti0.8O3 device obtained at 300 K.

as the device gets wire bonded, the platinum pad for the top electrode contact runs overthe SiNx to contact the top SrRuO3 without shorting with the bottom electrode (see thehighlighted region in Figure 3.1b).

Prior to any electrothermal measurements, we investigate the room-temperaturecurrent-voltage, ferroelectric, and dielectric response of the thin-film heterostructures.Current-voltage measurements reveal that the devices exhibit a symmetric and highly-resistive response (RFE > 108 Ohms) and that the electrode contacts are rectifying,[168]altogether confirming the high-quality, low-leakage characteristics of the heterostructures(Figure 3.2a). Ferroelectric hysteresis loops are measured between 0.033 and 1000 Hz (Figure3.2b), and the devices exhibit a robust ferroelectric response with low-loss, square ferroelec-tric hysteresis loops, large remnant polarization (Pr ≈ 77 µC cm−2), and loop closure evenat the lowest measurement frequencies; again confirming the excellent quality of the ferro-electric films and devices. Low-field dielectric permittivity measurements are also conductedon the same devices (Figure 3.2c). A near-frequency-independent value of the dielectricpermittivity (∼ 248 ± 2.7) is measured across the frequency range of 102 - 105 Hz at roomtemperature, consistent with expectations for these heterostructures.[81]

3.3 Theory of Measurement


The pyroelectric measurement (Figure 3.3a) relies on the application of a sinusoidally-varying current [IH,1ωH = I0H cos(ωHt)] at frequency ωH across the outer thermal probe

36

A

~

VV

3ωH

IH,1ω

H

iP,2ω

H

Top

Bottom

1ωH

2ωH+ DC

2ωH+ DC

θH

1 10 100 10000

2

4

6

8

θF

E (

K)

Electrical frequency, ωΗ/2π (Hz)

-90

-45

0

45

90

Th

erm

al p

ha

se

la

g, φ (°)

(c)

Electrode

Ferroelectric

Electrode

Insulator

Heater

Substrate

(a)

(d)

(b)

Q

θ

Z

θ

Z

θ∞=0

B

FE

T

H

H

IH(t)

QH(t)

TH(t)

φ

Figure 3.3: (a) Schematic of the setup for pyroelectric and 3-omega measurement. (b)Schematic of the relationship between the applied heating current [IH(t)] at angular fre-quency 1ωH , dissipated power [QH(t)], and the resulting temperature oscillation [TH(t)] ofthe metal heater line at 2ωH . (c) Schematic of the temperature field during a pyroelectricmeasurement and a simplified thermal circuit.(d) Measured temperature oscillation θFE andthermal phase lag φ between the heating current (and, therefore, the dissipated power) andthe temperature change in the ferroelectric as a function of frequency.

pads. The driving current at frequency ωH dissipates power equal to I2H,1ωH

R0 =12I2

0H [1 + cos(2ωHt)], where R0 is the resistance of the heater line (∼ 30 Ohms). Because theresponse in the thermal domain is linear, periodic Joule heating causes a temperature rise ofthe line at dc (θdc) and ac with amplitude θH at a frequency 2ωH (Figure 3.3b), and a phaselag (φ) that is determined by geometry and material properties (i.e., thermal conductivityand specific heat capacity).

θ = θdc + θHcos(2ωHt+ φ) (3.3)

This oscillating temperature causes the resistance of the line to vary asR = R0(1+αθ),where α is the temperature coefficient of resistance. Finally, when mixed with the inputcurrent, the resulting voltage drop across the line is, according to Ohm’s Law,

37

V = R0I0Hcos(ωHt)[1 + αθdc] +R0I0H

2αθHcos(ωHt+ φ) +

R0I0H

2αθHcos(3ωHt+ φ) (3.4)

Using a lock-in amplifier (Stanford Research, SR830), we measure the third harmonicof the voltage signal and thus determine the average heater temperature amplitude as[169]

θH = 2V3ω

R0I0Hα= 2

dT

dR

V3ω

I0H

(3.5)

where V3ω is the amplitude of the third harmonic voltage.

For a simple heater-on-substrate geometry, the heat generated in the top metal heaterline diffuses into the substrate a characteristic distance (often referred to as penetration depthor thermal wavelength) defined as[170]

λS =

√DS

2ωH(3.6)

where DS is the thermal diffusivity of the substrate. The heating frequency waschosen so that the heat completely propagates through the film stack while still ensuringthat the substrate remains semi-infinite such that the thermal waves completely dampenwithin the substrate (Figure 3.3c). Thus, the natural bounds on the thermal wavelengthare λFE > 3dFE and λS < 1

3dS, where the subscripts FE and S denote the ferroelectric

film and substrate, respectively, and d is the thickness of the relevant component. Key tothe sensitivity and accuracy of this approach is the fact that the thermal impedance of thetop thermal circuit ZT (comprising the heater line, the electrical insulator and the top oxidemetal electrode) is much smaller than that of the bottom circuit ZB (comprising the bottomoxide metal electrode and the substrate), thereby minimizing the error between θH and θFE

The measured θH and the thermal flux (qH) into the heterostructure (i.e., the acpower dissipated per unit area of the heater) are used with a one-dimensional thermal trans-port model[171] to calculate θFE (see Appendix B for details). This scheme is applied tocalculate θFE at 5 equally-spaced nodes within the ferroelectric layer. We found that thedifference in the temperature amplitude between the top (just below the ferroelectric andtop electrode interface) and bottom nodes (just above the ferroelectric and bottom electrodeinterface) corresponds to 1.6% of the average θFE. Hence, any tertiary PEE[172] induced bytemperature gradients in the film can be neglected and we can use the lumped parameterapproach where the ferroelectric layer is considered to be isothermal at any instant. A sim-ilar approach[27] calculates θFE using qH and works from the bottom of the substrate up to

38

the ferroelectric layer. We, however chose to work from the top down so as to maximize thenumber of measured quantities in the model to compensate for any unideal deviation fromtheory.

The mean temperature oscillation of the ferroelectric, averaged across the 5 nodes,θFE, as a function of frequency is thus extracted and plotted (Figure 3.3d). It can be seenthat θFE decreases and the phase between the dissipated power (as a result of heating)and the temperature (φ) increasingly lags as the frequency of ac heating current on the topmetal line increases (Figure 3.3d). The phase lag behaves as expected, approaching 0◦ and-17.5◦ in the low- and high-frequency limits, respectively, reflecting the gradual transitionfrom cylindrical to one-dimensional heat transport. Determination of thermal phase lag isparticularly important for extracting iP which can be convoluted with thermally-stimulatedcurrents that are in-phase with the temperature change.

With the known thermal state of the heterostructure in response to periodic Jouleheating, we can proceed to extract iP and π. The current is measured through the bottomelectrode using the current input (106 V/A gain) of a lock-in amplifier (Stanford Research,SR830) (Figure 3.3a). The top electrode is held at ground potential which shields anycapacitive coupling between the heating and the ferroelectric measurement circuits. Since iPis proportional to the rate of change of temperature, it leads the temperature change by 90◦,and is extracted by taking the component of the measured total current which is out-of-phase(leads by 90◦) with the temperature change.[24, 25] Thereafter, π can be calculated usingthe formula, iP = Aπ dT

dt.


Measuring the ECE relies on the ability to sense small temperature changes when anelectric field is applied and removed across the ferroelectric. A sinusoidally-varying electric

field, E =(

V0dFE

)sin(ωEt), when applied to the bottom electrode of the ferroelectric capacitor

as depicted (Figure 3.4a), results in an adiabatic temperature change via the ECE

∆Tad =

∫ E2

E1

(T

C(T,E)

)E

ΣdE (3.7)

The temperature change via the ECE, unlike the case of Joule heating which happensat DC and 2ω, happens at ω. This can be understood by calculating the ECE power (QE)

QE = C(T,E)AdFEdT

dt(3.8)

39

V

VE(ω

E)

Top

Bottom

ωE

θFE

(c)

Electrode

Ferroelectric

Electrode

Insulator

Sensor

Substrate

(a)

(d)

(b)

Q

θ

Z

θ

Z

θ∞=0

B

FE

T

Sensor

VE(t)

QE(t)

TE(t)

FEQ

rr

eeeee

QQc Qc QQQcFE

QcFE

QQQcFE

QQQFEFEFE

eeee

103

104

105

106

107

108

0.001

0.01

0.1

θsensor (K)

Frequency, ωE/2π (Hz)

-360

-270

-180

-90

0

Phase, θSensor-Q

E(t) (°)

~

VE(ω

E±ω

S)

~

V

VS(ω

S)

VB(ω

E±ω

S)

RP

RB

RB

ωE

Figure 3.4: (a) Schematic of the setup for electrocaloric measurement using the top metalline as a resistance thermometer (sensor). (b) Schematic of the relationship between theapplied voltage [VE(t)] across the ferroelectric capacitor at angular frequency ωE, the powergenerated [QE(t)], and the resulting temperature change [TE(t)] due to the ECE at the samefrequency ωE. (c) Schematic of the temperature field during an electrocaloric measurement.(d) Computed temperature oscillation θSensor and thermal phase lag between the temperatureoscillation of the on the top sensing metal line and the ECE power as a function of frequencyof voltage cycling across the ferroelectric for an assumed Σ = 100 µC m−2 K−1 and an appliedelectric field amplitude of 50 kV cm−1.

40

where A is the area of the heat source (ferroelectric layer). Using equation (3.7), equation(3.8) can be simplified to

QE = TΣAωEV0E cos(ωEt) (3.9)

Hence, the electrocaloric power and the resultant temperature change occur at thesame frequency, ωE (Figure 3.4b). Power dissipation due to Joule heating on the other handis given by

QJ =V 2

0E

2RFE

(1− cos(2ωEt)) (3.10)

where RFE is the resistance of the ferroelectric. Thus, Joule heating still happens at DC and2ωE thereby making ECE measurements in the frequency-domain at ωE effective in filteringout contributions from Joule heating.

The temperature oscillation of the top sensor line (θSensor) can be related to theECE power from the underlying ferroelectric film (QE) via a generic thermal transfer functionin the frequency domain, Z[173]

θSensor(t) = Q0E [Re(Z) cos(ωEt) + Im(Z) sin(ωEt)] (3.11)

where Q0E = TΣAωEV0E from equation (3.9). This temperature change perturbs the elec-trical resistance of the top metal heater/sensor line as

RSensor = R0,Sensor [1 + α∆TDC + αθSensor] (3.12)

where R0,Sensor is the resistance of the sensor line at 300 K (∼ 10 Ohms) and α is thetemperature coefficient of resistance. To sense the temperature in the top metal line, asensing signal of the form VS sin(ωSt) is applied across the outer thermal pads (Figure 3.4a).The voltage source is converted to a current source by inclusion of a ballast resistor (RB,which is 102 times larger than RSensor) in series. Other approaches include the use of acommercially-available AC current source (for example, Keithley 6221A) or a custom-builtV-to-I circuit using operational amplifiers.[174] Voltage (VA) is measured across the innerpads of the thermometer and has a form

41

VA =VS sin(ωSt)

RB +RSensor

RSensor (3.13)

Since RBRSensor

≥ 102, using equation (3.11) and (3.12), equation (3.13) can be expressed as

VA =VS sin(ωSt)

RB

RSensor[1 + α∆TDC +

αQ0E(Re(Z) cos(ωEt) + Im(Z) sin(ωEt))] (3.14)

The response frequencies ωS ± ωE (due to the product of two sinusoidal functions)contain the information regarding the temperature oscillation of the sensor. The voltagesignal corresponding to ωS is particularly large and can be 1000-times larger than the onecorresponding to ωS ± ωE resulting in a significantly larger background. Therefore, a com-mon mode cancellation circuit using a Wheatstone bridge has been implemented in ourmeasurement scheme (Figure 3.4a). The sensing signal is simultaneously applied across apotentiometer which is adjusted such that the resistance of the potentiometer RP ≈ R0Sensor

and, therefore, the voltage drop (VB) across RP nullifies the ωS component of the differentialvoltage (VA−VB) that is fed to the lock-in amplifier. Note that the ωS±ωE signal only comesfrom the temperature-dependent response due to the ECE (RSensor) and is absent in the po-tentiometer (RP ). Therefore, only the ωS background is attenuated and the measurementof the much smaller ωS ± ωE signal becomes possible.

The measured differential voltage (VωS±ωE) is related to the temperature oscillationof the top sensor via[167, 169]

θSensor = 21

αVωSVωS±ωE (3.15)

This temperature change of the top sensor line is used to calculate Σ as well as the averagetemperature oscillation in the ferroelectric, θFE. This requires knowledge of the volumetricpower generation (G) created by the entropy change and can be related to Σ as[35]

G = TdS

dE

dE

dt= TΣ

(V0E

dFE

)ωE cos(ωEt) (3.16)

The resulting heat flux can be determined from the measured temperature changeon the top sensor line and the thermal properties of all the layers in the heterostructure. Dueto the ECE, heat flux is generated homogeneously in the ferroelectric and diffuses in both

42

directions (i.e., into the substrate and into the top sensor line). To model this heat transport,we solve a 1D problem for the heat flux generated in each differential thickness (dz) at aheight z in the ferroelectric.[35] We defined G to be the power generated per unit volumeand note the differential flux dqtot generated from each volume is dqtot = Gdz. The totaltemperature oscillation sensed by the metal sensor is the sum of the contributions from theentire thickness of the ferroelectric (Figure 3.4c). This is a continuous domain, so summationis replaced by integration and the amplitude of the temperature oscillations sensed by themetal line is defined by

θSensor = G

∫ dFE

0

ξdz (3.17)

where ξ is an algebraic expression calculated using the elements of the thermal transportmatrices and the appropriate boundary conditions (see Appendix B for details). Usingequation (3.16), equation (3.17) simplifies to

θSensor = TΣ

(V0E

dFE

)ωE cos(ωEt)

∫ dFE

0

ξdz (3.18)

Equation (3.18) allows the calculation of Σ from the measured value of θSensor. Further,the average temperature oscillations of the ferroelectric, θFE, can be calculated using asimilar approach to find the temperature amplitudes at the top and bottom surface of theferroelectric then taking their average.

In our measurements, high-frequency electric-field oscillations are required to gen-erate enough flux to produce measurable temperature change. The effect of drivingfrequency on the resulting temperature change on the top sensor line, for an assumedΣ = 100 µCm−2K−1 and an applied electric field oscillation of 50 kV cm−1, is calculatedfrom the above thermal models and shown for reference (Figure 3.4d). It can be seen thatat lower frequencies (< 3 × 106 Hz), the temperature change on the top metal sensor islimited due to low thermal flux [refer to equation (3.18)]. At high frequencies (> 3 × 106),the temperature change diminishes due to the limited thermal-penetration depth, placing anupper limit on the frequency range of the experiment. It is, Therefore, desirable to performECE measurements at frequencies in the range close to 2× 105 − 1× 106 Hz.

43

3.4 Results and Discussion

3.4.1 Pyroelectric Measurements

Following the theory of pyroelectric measurement in section IIIA (Figure 3.3), thepower spectrum of the measured total current (iTotal) is provided (Figure 3.5a) for a heatingcurrent (IH,1ωH ) equal to 16 mA at 1 kHz. It can be seen that a dominant current peak occursat frequency 2ωH as a result of the temperature oscillations at the same frequency in theferroelectric due to the driving current at 1ωH in the heater line (inset, Figure 3.5a). It shouldbe noted that the presence of the electrically-insulating SiNx layer and electrical shieldingvia the top electrode reduce, but do not completely suppress, the capacitive coupling. Thismanifests as the observed response at ωH (Figure 3.5a); however, our ability to measure iP(measured at 2ωH frequency) is not affected by the non-zero capacitive coupling. The basisof this argument is twofold. First, measurements with zero heating, but with capacitive biasacross the heterostructure, yield 2ω current which is ∼103 smaller than the case when heatingoccurs in conjunction with the capacitive bias. Second, we measured the total current withthe polarization of the ferroelectric in the up-poled state (iup) and the down-poled state(idown). Since the switching of the polarization only reverses the direction of iP (manifestedas a phase reversal by 180◦) and does not change the capacitively-coupled parasitic current,iup and idown can be expressed as

iup = iP + iParasitic , (3.19)

idown = −iP + iParasitic (3.20)

By taking the sum and difference of equations (3.19) and (3.20), we are able tosubtract out any contribution due to parasitic-coupling which constitutes less than 4% ofthe total measured current. These two checks confirmed that the current due to parasiticcapacitance is negligible. This, however, does not prove that the current is completelypyroelectric in origin. Our measurements show that the measured current leads the actualtemperature change in the ferroelectric (Figure 3.3d) by∼ 90◦ (Figure 3.5b). Deviations fromthe expected phase can be attributed to thermally-stimulated currents which flow in-phase(zero lag) with the temperature change. The real iP is extracted by taking the componentof the total current which is out-of-phase with the temperature change and is plotted as afunction of the frequency of heating current (red circles, Figure 3.5c). Apparently π has afrequency dependence (black squares, Figure 3.5c). At low frequencies (< 4 Hz), π is seento begin to converge to a value of ∼ 280 µCm−2K−1, while at higher frequencies (> 500Hz) it asymptotes to a lower value of ∼ 195 µCm−2K−1. This reduction (∼ 30%) can be

44

1,000 10,000

-90

-60

-30

0

30

60

2ωΗ

i To

tal2 (

dB

m)

Electrical Frequency, ω/2π (Hz)

0.0 0.5 1.0 1.5 2.0-30

-20

-10

0

10

20

30

i Tota

l (n

A)

I H,1

ωH

(m

A)

Time (ms)

-9

-6

-3

0

3

6

9

1 10 100 1,000-45

0

45

90

135

∼90°

Temperature oscillation (θFE

)

Ph

ase

(d

eg

ree

s)

Electrical frequency, ωΗ/2π (Hz)

Measured current (iTotal

)

-75 -50 -25 0 25 50 75-300

-200

-100

0

π (

µC

/m2K

)

DC Electric field (kV/cm)

-2

-1

0

1

2

i P,2

ω (n

A) P

P

(d)

1 10 100 1,000

0.01

0.1

1

10

100

1,000

Pyro

cu

rre

nt i P

,2ω (

nA

)

Electrical frequency, 1ωΗ/2π (Hz)

0

-50

-100

-150

-200

-250

-300

Pyro

Co

effic

ien

t π

(µ

C/m

2K

)

(c)

εfilm

εsub

(b)(a)

ε

Figure 3.5: Pyroelectric measurements of PbZr0.2Ti0.8O3 thin film. (a) Measured total cur-rent (iTotal) in the frequency-domain and time-domain (inset) for an applied heating current(IH,1ωH ) equal to 16 mA at a frequency of 1 kHz. (b) Phase relationship of the measuredcurrent (at 2ωH) and the temperature oscillation (also at 2ωH) with respect to the phase ofthe applied heating current depicting that the measured current leads the temperature oscil-lation by ∼90◦. (c) Measured pyroelectric current and pyroelectric coefficient as a function ofheating frequency. At low frequencies (<4 Hz), π →

(∂P∂T

)E,σ

, while at high frequencies (>500

Hz) π →(∂P∂T

)E,ε

due to secondary contribution to pyroelectricity (reduction by ∼30%). (d)

Pyroelectric hysteresis loops as a function of background DC electric field, at 1 kHz.

explained by the secondary contribution[4] to the total pyroelectricity due to the thermalexpansion mismatch between film and substrate and the resulting thermal-stress-induced,piezoelectric-driven polarization change. At low frequency, the thermal penetration depth(λS) is large compared to the lateral dimensions of the device, therefore, the ferroelectricfilm expands and contracts in-plane with the substrate. As a result, the total π under aconstant stress boundary condition can be expressed as

π =

(∂P

∂T

)E,σ

=

(∂P

∂T

)E,ε

+

(∂P

∂σ

)E,T

(∂σ

∂ε

)E,T

(∂ε

∂T

)E,σ

(3.21)

At higher frequencies, the lateral dimensions of the film do not change as the

45

film is clamped to the substrate which cannot expand laterally due to the reduced ther-mal penetration depth. Therefore, the contribution from the lateral thermal expansion(∂P∂σ

)E,T

(∂σ∂ε

)E,T

(∂ε∂T

)E,σ

is suppressed and the magnitude of π decreases.

Changes in temperature not only change the permanent dipole moment (intrinsicportion of primary pyroelectricity), but also alter the dielectric permittivity and hence thesurface charge density under an applied electric field. Particularly near a dielectric anomaly(for example, near a phase transition), the dielectric constant changes sharply with tem-perature and can result in either a significant suppression or enhancement of iP dependingupon the poled-state of the ferroelectric. Full characterization of π therefore requires one tomeasure iP as a function of background electric field. In our measurements, this is achievedby applying a constant bias voltage to the top electrode while the bottom electrode remainsconnected to the input of the lock-in amplifier. The measured pyroelectric hysteresis loopsare provided (Figure 3.5d). When the ferroelectric is up-poled, increasing the temperatureresults in a “positive” iP (red data, Figure 3.5d). With increasing positive bias (greater thanthe coercive field) on the top electrode, the polarization switches and causes iP to reverseits direction (blue data, Figure 3.5d). When the direction of electric field bias is reversed(i.e., negative bias greater than the coercive field is applied), the polarization switches backand iP reverses direction again, giving a characteristic hysteresis-like pyroelectric loop. Itshould be noted that the slope of iP vs. electric field is close to zero away from the switchingnear the coercive fields indicating that the contribution from the temperature-dependentdielectric permittivity is negligible as the dielectric permittivity does not change stronglywith temperature far from the transition temperature (TC > 450◦C).

3.4.2 Electrocaloric Measurements

Measurements were performed on the same device on which the pyroelectric mea-surements were conducted. The frequency (ωE) of the applied electric field of magnitude(V0EdFE

)= 50 kV cm−1 was chosen to be 98,147 Hz while that of the sensing voltage (ωS) was

chosen to be 2,317 Hz with a magnitude VS = 7.07 V . The choice of frequency was deter-mined so as to avoid any resonance between the signal to be measured and power supplynoise (harmonics of 60 Hz) or any higher-order harmonics of the sensing signal frequency. Afull frequency-domain voltage response is plotted (Figure 3.6a). As discussed above, the ECEresponse in the frequency-domain occurs at ωS − ωE and ωS + ωE corresponding to 95,830and 100,464 Hz. This can be clearly seen against the low background (insets, Figure 3.6a).To further validate our technique, we performed the same measurement with varying mag-nitudes of applied electric field. We note that the magnitude of the maximum electric fieldnever exceeded the coercive field of switching for the ferroelectric so that the ECE response(change in temperature) remains linear with applied electric field [refer to equation (3.18)].The average temperature change in the ferroelectric layer is found to scale linearly with the

46

-75 -50 -25 0 25 50 75-20

0

20

40

60

80

Σ (

µC

/m2K

)

DC Electric field (kV/cm)

-0.008

-0.004

0.000

0.004

0.008

θF

E (K

) P

P

96,000 98,000 100,00010

-7

10-6

10-5

10-4

10-3

10-2

10-1

A-B

(V

)

Frequency ω /2π (Hz)

2x10-6

4x10-6

2x10-6

4x10-6

ωE−ω

S

(a)

ωE+ω

S

20 30 40 50 600.000

0.002

0.004

0.006

0.008

0.010

θ FE (

K)

Electric field magnitude (kV/cm)

0

20

40

60

80

100

120

EC

E C

oe

ffic

ien

t Σ

(µ

C/m

2K

)(b) (c)

Figure 3.6: Electrocaloric measurements of PbZr0.2Ti0.8O3 thin film. (a) Measured differen-tial voltage across the top metal sensor in the frequency-domain for an applied electric fieldof magnitude 50 kV cm−1 at a frequency ωE/2π equal to 98,147 Hz and a sensing voltage VSequal 7.07 V at a frequency ωS/2π equal to 2,317 Hz. Inset shows the voltage response atfrequency ωS − ωE and ωS + ωE which is a measure of the change in the resistance and thusθSensor of the top metal. (b) Average measured temperature oscillation in the ferroelectric(θFE) due to the ECE and the resulting calculated value of Σ. (c) Electrocaloric loops as afunction of background DC electric field using an AC electric field amplitude of 50 kV cm−1.Up-poled state (red) results in the positive ECE while the down-poled state (blue) resultsin the negative ECE.

applied electric field (red squares, Figure 6b). While frequency-selective measurements atωS ± ωE inherently remove any Joule heating that happens at ωS ± 2ωE, this measurementadditionally proves that the ECE response is free from Joule heating which would havecaused θFE to scale quadratically with the driving electric field. Using the thermal transportmodel explained above, Σ has been extracted from the measured temperature change of thetop metal sensor line (black circles, Figure 3.6b). A near-field-independent value of Σ iscalculated; further proving the linearity of the response in the electric-field regime withoutany polarization switching. We do observe slight deviations at very low magnitude of electricfield which correspond to low signal to noise ratio.

We further measured the ECE response as a function of background DC electricfield (Figure 3.6c), and, similar to the PEE, a hysteresis-like ECE loop is measured. Posi-tive ECE is the increase/decrease in the temperature of the material when an electric fieldis applied/removed. This is often the case in dielectrics or ferroelectrics above the Curietemperature. Tetragonal ferroelectrics such as PbZr0.2Ti0.8O3 below the Curie temperature,however, can exist in either of two stable polarization states. It is the relative directionof the polarization vector and the electric field vector that determines the nature (whetherpositive or negative) of the ECE. When the ferroelectric is up-poled, applying an electricfield parallel to the direction of polarization such that the application of field further re-inforces the electrical dipoles results in a positive ECE (red data, Figure 6c). Applicationof negative electric field greater than the coercive field results in polarization switching. In

47

this down-polarized state, the positive electric field excitation (which is now anti-parallel tothe direction of polarization) tends to misalign the dipoles thereby increasing the dipolarentropy and hence decreasing the temperature. In this regime, the ECE is negative (bluedata, Figure 3.6c). It is worth mentioning that Joule heating, which under zero backgroundelectric field occurs only at DC and 2ωE, has a component at ωE under a finite backgroundelectric field

QJ =

(V 2

0E

2RFE

+V 2DC

RFE

)+

2V0EVDCRFE

sin(ωEt)

− V 20E

2RFE

cos(2ωE) (3.22)

Should Joule heating be significant, the measured voltage response at ωS ± ωE will havecontributions from Joule heating and the shape of the ECE loop will no longer be square-like and the otherwise constant temperature change and Σ (zero slope) exhibit a finite slope(proportional to (2V0E

RFE) with increasing temperature change with applied background electric

field. Our data (in Figure 3.6c) does not show any measurable slope under backgroundelectric field suggesting a highly insulating device as suggested by the measured electricalproperties (Figure 3.2).

3.5 Sensitivity Analysis

The thermal model used for the PEE measurements is not sensitive to any materialparameter due to the thin layers and low frequencies. Instead, the predominant error in thecalculation of the average temperature change in the ferroelectric comes from the calculationof the heater temperature from the 3-omega equations.

To evaluate the accuracy of the ECE measurement due to propagation of errorsfrom the various thermophysical properties, we quantify the sensitivity of the measuredtemperature response of the top metal line sensor (during ECE measurement) in the thermalmodel as[175]

STx =∂ lnθ

∂ lnx(3.23)

where x is one of the parameters (e.g., thermal conductivity, specific heat capacity, thicknessof a particular layer) in the thermal model. With this definition of sensitivity, a value of

48

104

105

106

107

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2 kPZT

kSRO

CPZT

dSiNx

kSub,

CSub

CSRO d

SRO

CSiNx

kSiNx

Se

nsitiv

ity, S

T X

Electric field frequency ωE/2π (Hz)

Figure 3.7: Sensitivity of the calculated electrocaloric temperature change at the top metalsensor, θSensor as a function of selected parameters in the thermal model as function offrequency of electric field cycling for a 150 nm PbZr0.2Ti0.8O3 heterostructure.

S = −0.1, for example, means that a 10% increase in parameter x will result in a 1% decreasein θ. Equation (3.23) is evaluated for the following layer properties: thermal conductivityk, volumetric heat capacity C, and layer thickness d as function of frequency (Figure 3.7).At low frequencies (3 < ×105 Hz), ECE measurements are most sensitive to k and C of thesubstrate while at high frequency (3 > ×105 Hz), C and d of SiNx and PbZr0.2Ti0.8O3 arethe most important parameter.

3.6 Conclusion

In conclusion, I demonstrated a reliable technique for direct measurement of thePEE and ECE in thin films using a microfabricated electrothermal test platform. Periodicheating via the top metal line resulted in localized heating of the ferroelectric and, in turn, ina flow of an external current which was separated into pyroelectric and thermally stimulatedcurrents using frequency-domain phase-sensitive detection. Further, wide-frequency rangemeasurement depicted the role of the secondary contribution to PEE (∼ 30%) due to sub-strate clamping. The same device was used to measure ECE where high frequency electricfield oscillation across the ferroelectric resulted in the EC temperature change which wassensed by the same top metal line now functioning as a resistance thermometer. Frequency-

49

domain measurements allowed decoupling of Joule heating from the ECE. Using 1-D heattransport models, the thermal state of the heterostructure was characterized during themeasurement of the PEE and the ECE to extract π and Σ.

50

Chapter 4

Effect of Ferroelastic Domains onElectrothermal and ElectromechanicalSusceptibilities

In this Chapter, I will investigate the role of epitaxial strain and, in turn, the effectof ferroelastic-domain structures on the electrothermal (both PEE and ECE) and electrome-chanical susceptibility in thin films of the canonical tetragonal ferroelectric PbZr1−xTixO3

system. Using a combination of scanning-probe-based microscopy, X-ray diffraction, andphase-field modeling, I will first discuss how epitaxial strain (both compressive and ten-sile) governs the evolution of ferroelastic domain structure. In particular, I will report on anovel strain-induced domain-structure instability and competition where an external pertur-bation (temperature and electric field) can provide a route to enhanced electrothermal andelectromechanical susceptibility.

Using the high-frequency heating and temperature-sensing capability of an inte-grated thin-film device architecture (as discussed in Chapter 3), the domain-structure con-tributions (the so-called “extrinsic” contributions) to the electrothermal susceptibilities thatarise exclusively from the temperature- and field-dependent changes in the ferroelastic do-main population will be isolated. In particular, I will focus on the hierarchical mixed-phasedomain structure where the facile interconversion among different ferroelastic variants canresult in significant enhancement of the electrothermal susceptibility. Further, using band-excitation piezoresponse spectroscopy with acoustic detection, I will show how the inherentcompetition between the ferroelastic variants results in a two-step, three-state switchingprocess resulting in large electromechanical strains and large variations in elastic modulus.

51

4.1 Introduction

The complex interplay between the various thermodynamic forces [electric-field (E),temperature (T ), and stress (σ)] and the displacements [polarization (P ), entropy (S), andstrain (ε)] in ferroic materials enables electrothermal and electromechanical susceptibilitiesuseful for a range of applications.[14, 176, 177] Key to such applications is the ability tomanipulate and control the T−, E−, or σ−dependence of P , S, or ε in ferroic materials.Typically, large electrothermal and electromechanical susceptibilities have been achieved byperching the material in the vicinity of a phase transition[42] driven either by chemistry[178,179] or temperature[30, 180] where most ferroic susceptibilities diverge. Large response canalso arise at chemically-induced phase boundaries (i.e., morphotropic phase boundaries)where interconversion of nearly degenerate phases is possible. At such boundaries, applica-tion of an appropriate stimulus (E, T , or σ) allows one to control the constituent phase frac-tions thereby providing an extrinsic mechanism that can enhance the ferroic susceptibilities.For instance, a temperature-induced monoclinic-to-tetragonal phase transition, resulting ina rotation of polarization, has recently been shown to enhance the pyroelectric susceptibilityin (1-x)PbMg1/3Nb2/3O3-xPbTiO3 (x = 0.27-0.35).[178] Similarly, enhanced electromechan-ical response has been found in PbZr1−xTixO3 (x = 0.48), (1-x)PbMg1/3Nb2/3O3-xPbTiO3

(x = 0.295),[63] rare-earth doped BiFeO3,[181–183] and other systems.[184]

More recently, advances in the growth of ferroic thin films[58, 159] has providedepitaxial strain as another route by which to control ferroic susceptibilities by altering ferro-electric order[56, 57, 185] and manipulating ferroelastic-domain structures.[65, 186, 187] Therole of such domain structures in controlling the electrothermal susceptibilities, however, hasremained comparatively understudied in part due to experimental challenges (as noted inChapter 3) which have, instead, driven almost all the work to date to be done via mod-eling including phenomenological Ginzburg-Landau-Devonshire-based thermodynamic for-mulations[81–83] and phase-field studies.[84, 85] Similarly, the electromechanical responsein systems exhibiting strain-induced structural/domain-variant competition has remainedless pervasive with most work focusing on the electromechanical response in strain-inducedcoexisting tetragonal and rhombohedral phases in BiFeO3.[65, 188] Thus, it is desirable toexplore alternate strain-based pathways with a potential for large field-driven response.

4.2 Evolution of Ferroelastic Domain Structures with

Epitaxial Strain

Analytical phenomenological and phase-field models have been used to predict thestrain and field evolution of domain structures in numerous ferroelectrics.[189–193] In tetrag-

52

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

300

600

900

1200

)K(

erut

are

pm

eT

Strain % (εm)

000

c

c/a

SrT

iO3

5 0

DyS

cO

3

Paraelectric

0 0

Gd

ScO

3

5

Sm

ScO

3

a1/a2

Figure 4.1: Analytical phenomenological modeling-based temperaturestrain phase diagramrevealing the stability regimes for c, c/a, and a1/a2 domain structures. Strain positions forthe substrates used are shown as dashed lines.

onal PbZr1−xTixO3 system specifically, a systematic evolution from monodomain and out-of-plane polarized c domains at large compressive strains, to polydomain c/a domain structures(consisting of alternating c domains and in-plane polarized a domains) at small compressiveor tensile strains, and finally to completely in-plane polarized a1/a2 domains at large tensilestrains has been predicted and observed (Figure 4.1).[81, 194, 195] Beyond these structures,phase-field models have suggested a complex domain structure coexistence near the c/a -a1/a2 boundary.[192] Such domain-structure coexistence is reminiscent of phase competition,and thus, could provide routes to enhanced electrothermal and electromechanical suscepti-bilities.

4.2.1 Synthesis and Structural Characterization

Pulsed-laser deposition was employed to grow 40 nm thick PbTiO3 films on SrTiO3

(001), DyScO3 (110), GdScO3 (110), and SmScO3 (110) substrates corresponding to latticemisfit strains of -1.3%, -0.2%, 0.3% and 0.7%, respectively (note that all values are relative tothe pseudocubic lattice parameter of PbTiO3; dashed lines, Figure 4.1). Following growth,detailed X-ray diffraction patterns were obtained for all the heterostructures. θ-2θ scans

53

Lo

g In

ten

sity (

Arb

. U

nits)

20 25 30 35 40 45 50

2θ (°)

SrTiO3001

SrTiO3002

DyScO3110

DyScO3220

GdScO3110

GdScO3220

SmScO3110

SmScO3220

PbTiO3002

PbTiO3 001

PbTiO3200

PbTiO3100/010

PbTiO3100/010

Figure 4.2: Wide-angle θ-2θ scans for 40 nm thick PbTiO3 films grown on SrTiO3 (001),DyScO3 (110), GdScO3 (110), and SmScO3 (110) substrates.

(Figure 4.2) show that the films are phase-pure and the orientation of the film systematicallyevolves with the substrate-induced epitaxial strain. Films grown under compressive strain asis the case on SrTiO3 (001) and DyScO3 (110) substrates (orange and blue data, respectively,Figure 4.2) reveal a single film peak corresponding to 001-oriented (c domain) PbTiO3. Notethat there is no peak corresponding to the a domains on these substrates, as they are expectedto be highly tilted with respect to the substrate surface. For films grown on GdScO3 (110)substrates, with small tensile strain (∼0.3%), peaks corresponding to both 001- and 100/010-oriented PbTiO3 (i.e., both c and a domains, respectively) are observed (green data, Figure4.2). Lastly, PbTiO3 films grown on SmScO3 (110) substrates (corresponding to a tensilestrain of ∼0.7%) are found to be almost entirely 100-oriented PbTiO3 (red data, Figure 4.2).

More insights into the structure of these heterostructures can be gained from sym-metric X-ray reciprocal space mapping (RSM) studies (Figure 4.3) about the 001-diffractioncondition for films grown on SrTiO3 substrates, and about the 110-diffraction condition forfilms grown on the various orthorhombic scandate substrates. PbTiO3/SrTiO3 heterostruc-tures reveal an intense peak (labeled PbTiO3 001, Figure 4.3a) corresponding to a nearlymonodomain c domain structure, with a very small a-domain fraction (identified as thefaint streak highlighted in the white box). For heterostructures grown on DyScO3 (110)

54

0.231

0.238

0.252

0.259

0.266

0.245Ql(Å

-1)

SrTiO3

001

DyScO3

110 SmScO3

110

PbTiO3001

(c-oriented)

c

tilted a PbTiO3100

(a-oriented)

0 0.007-0.007

Qh (Å-1)

0 0.007-0.007

Qh (Å-1)

0 0.007-0.007

Qh (Å-1)

tilted a

Figure 4.3: 2D RSM studies about the (a) SrTiO3 001-diffraction condition (suggesting anearly monodomain c structure), (b) the DyScO3 110-diffraction condition (consistent with ac/a domain structure), and (c) the SmScO3 110-diffraction condition (showing a1/a2 domainstructures).

substrates, a significantly higher peak intensity corresponding to the tilted a domains is ob-served, which, in combination with the untilted c domain peak, is characteristic of a densec/a domain structure (Figure 4.3b). For heterostructures grown on SmScO3 (110) substrates,a single peak (labeled PbTiO3 100) corresponding to the a-domains is observed, indicativeof an almost entirely a1/a2 domain structure (Figure 4.3c). However, for heterostructuresgrown on GdScO3 (110) substrates, 3D-RSMs reveal the hierarchical architecture comprisedof a nanoscale mixture of a highly-tilted c/a domain structures and untilted a1/a2 domainstructures (Figure 4.4). An RSM about the 110-diffraction condition of the GdScO3 substratereveals film peaks corresponding to 00l -oriented c domains (out-of-plane lattice parameterof 4.104 A, crystallographic tilt of 1.17◦), and two versions of in-plane polarized a domains:(1) an untilted variant (corresponding to the classic a1/a2 structure, with out-of-plane lat-tice parameter of 3.926 A) and (2) a tilted variant (corresponding to the a domains in thec/a structure with an out-of-plane lattice parameter of 3.911 Aand a crystallographic tilt of0.65◦).

The structure of all the heterostructures was also probed using scanning-probe-based studies of topography (Figure 4.5a-d) and piezoresponse force microscopy (PFM, ver-tical amplitude, inset Figure 4.5e-h; lateral amplitude, insets Figure 4.5e-h). In all casesthe films were found to be single-phase, but the surface topography and the correspondingdomain structures were dramatically different. Consistent with the expectations of analyti-cal phenomenological models, the topography and domain structure of the PbTiO3/SrTiO3

55

Figure 4.4: 3D, synchrotron-based RSM about the 110-diffraction peak of the GdScO3

substrate (capturing 001-diffraction peaks of the PbTiO3) indicating the extracted latticeparameters and crystallographic tilts.

(Figure 4.5a and 4.5e) and PbTiO3/DyScO3 (Figure 4.5b and 4.5f) heterostructures revealpredominantly c and c/a domain structures, respectively, wherein the a domain fraction in-creases with decreasing compressive strain. Similar studies of PbTiO3/GdScO3 heterostruc-tures (Figure 4.5c and 4.5g), however, reveal the emergence of a complex hierarchical do-main architecture consisting of a mixture of c/a and a1/a2 domain structures that is notpredicted in analytical phenomenological models. Further increasing the tensile strain, thePbTiO3/SmScO3 heterostructures (Figure 4.5d and 4.5h) show an a1/a2 domain structurewhich has a small fraction of c/a domains indicating its nearness to the phase boundarybetween the hierarchical regime and the regime of a1/a2 domain structure (inset, Figure4.5h).

Detailed analysis of the topography (Figure 4.6a) and PFM (Figure 4.6b and 4.6c)studies of the PbTiO3/GdScO3 heterostructures reveal the nature of the hierarchical-domainarchitecture. Two distinct regions are observed. First, the regions with elevated, saw-tooth-like topography and domain walls along the <100>pc (schematic inset, Figure 4.6b) arefound to be a type of distorted c/a domain structure. A line trace across this band (inset,Figure 4.6a) reveals that the c and a domains are tilted by ∼1.17◦ and ∼0.75◦ relative tothe substrate surface, respectively. The second region, corresponding to the recessed, flattopography, exhibits reduced out-of-plane and enhanced in-plane piezoresponse with 90◦ do-main walls along the <110>pc (Figure 4.6c); typical of a1/a2 domain structures (schematic

56

Figure 4.5: Topography (a-d) and out-of-plane piezoresponse amplitude (in-plane piezore-sponse amplitude as inset) (e-h) of PbTiO3 films grown on SrTiO3 (a,e), DyScO3 (b,f),GdScO3 (c,g), and SmScO3 (d,h) substrates.

inset, Figure 4.6c). Further insights into the various components of the hierarchical-domainarchitectures can be obtained by comparing features in the nanoscale topography and piezore-sponse images with 3D, synchrotron-based RSM studies. These tilt angles match well withthe scanning-probe-based measurements for the different regions (inset, Figure 4.6a). Addi-tionally, the large difference in the out-of-plane lattice parameters of the a domains withinthe c/a and the a1/a2 domain structures is indicative of a complex strain accommodationin these hierarchical-domain architectures.

4.2.2 Mechanism Behind the Coexistence of Mixed-PhaseDomain Structures

To understand the energetics and the nature of strain distribution in thehierarchical-domain architectures, and to illuminate the mechanism behind the formationof this complex structure, phase-field models were employed. Briefly, in the phase-field mod-eling of the single-layer PbTiO3 thin films, the evolution of the polarization was obtained bysolving the time-dependent Landau-Ginzburg-Devonshire (LGD) equations:

57

Figure 4.6: (a) 3D representation of the topography of PbTiO3 hierarchical domain ar-chitectures grown on GdScO3 (110). Inset shows line trace (red-dashed line) indicating thetopography and crystallographic tilts. (b) Vertical PFM (amplitude) response superimposedon 3D representation of the topography of the hierarchical PbTiO3 domain structure. Insetshows a schematic representation of the domain structure within a c/a region as indicated bythe shaded box. (c) Lateral PFM (amplitude) response superimposed on 3D representationof the topography of the hierarchical PbTiO3 domain structure. Inset shows a schematicrepresentation of the domain structure within an a1/a2 region as indicated by the shadedbox.

∂Pi(x, t)

∂t= −L ∂F

∂Pi(x, t), i = 1, 2, 3 (4.1)

where Pi is the polarization vector, x and t are position the time, and L is thekinetic coefficient related to the domain-wall mobility. The contributions to the total freeenergy F include the Landau bulk energy, elastic energy, electric (electrostatic) energy, andgradient energy, i.e.,

F =

∫(fLandau + fElastic + fElectric + fGradient)dV (4.2)

58

Figure 4.7: Schematics of the lowest energy domain structures computed from the phase-field studies for misfit strain values ranging from -2.5% to 2.5%.

Detailed expressions for each of the energy-density contributions, and the materialconstants for PbTiO3 used in the simulations are collected from the literature and theseinclude the Landau potentials, elastic constants, electrostrictive coefficients, backgrounddielectric constants, and gradient energy coefficients.[191, 192, 196, 197]

3D phase-field simulations of PbTiO3 were done using a realistic 3D geometrysampled on a fine grid mesh of (128∆x1)(128∆x2)(32∆x3) with ∆x1 = ∆x1 = ∆x1 = 1 nm.The thickness of the substrate, film, and air are (10∆x3), (20∆x3), and (2∆x3), respectively.Periodic boundary conditions are applied in the in-plane direction (x1 and x2) while the thinfilm is simulated by properly choosing the mechanical and electrical boundary conditionalong the out-of-plane direction (x3). Random noise is used as the initial setup to simulatethe thermal fluctuation during the annealing process.

The models were focused on the strain range from -2.5 to 2.5% to obtain thelowest-energy structure (Figure 4.7). It can be seen that the domain structure evolves frommonodomain c (≤ -1%) to c/a (≈ -0.5 to 0%) with decreasing compressive strain. Movingto tensile strain, at small values the nucleation of a1/a2 domain structures can be noted;corresponding to the emergence of the hierarchical-domain architectures (≈ 0.2 to 0.8%).

59

0.00

0.07

0.14

-0.75

-0.68

-0.61

Landau

Elastic

-2 -1 0 1 2

-0.07

0.00

0.07

-0.07

0.00

0.07

Electric

Gradient

Strain (%)

Indiv

idual contr

ibutio

n to T

ota

l E

nerg

y (

arb

. units)

c

c/a

Mix

a1/a2

Figure 4.8: (a) Landau, (b) elastic, (c) electric, and (d) gradient energy contributions tothe total energy of the lowest energy structures computed from phase-field studies.

Further increasing the tensile strain causes an increase in the fraction of a1/a2 domainstructures within the hierarchical-domain architecture until eventually transforming into aclassical a1/a2 domain structure (≥ 1%).

The nature of the energy landscape can be gained by probing how the variousenergies of the system (i.e., Landau, elastic, gradient, and electrostatic; Figure 4.8) evolvewith misfit strain. It can be seen that the Landau (Figure 4.8a) and the elastic (Figure4.8b) energies are the two components primarily responsible for the evolution of the doubleminima observed in the total energy landscape. The electrical (Figure 4.8c) and gradient(Figure 4.8d) energies are, on the other hand, relatively flat in the mixed-phase region, butotherwise monotonically increase with strain. The dominant energies are therefore the elasticand Landau components and thus the focus is on the sum of these two energies within eachconstituent domain structure.

This combined elastic and Landau energy (multi-color line, Figure 4.9a) combine

60

a

b

c

d

Figure 4.9: (a) Strain-energy landscape (square points, black, solid curve) and the sum ofLandau and elastic energy density within the constituent domain structure (circles, dashedline) including: c (blue circles), c/a (red circles), and a1/a2 (green circles). (b) Evolutionof domain fractions (%) with misfit strain. A hierarchical domain architecture of coexistingc/a and a1/a2 is observed within a strain range of 0.2-0.8%.(c) Strain accommodation in thedifferent domain structures such that within the region of negative curvature (violet regions,e.g. at 0.6% strain), the material partitions itself into two domains structures: c/a domainsat a strain of 0.1% (right inset) and a1/a2 domains at a strain of 1.35% strain (left inset). (d)The resulting temperature-strain phase diagram showing the region of hierarchical domainstructure between c/a and a1/a2 regions.

61

to drive the formation of the two local minima corresponding to monodomain c (-2.5 to -1%,blue circles, Figure 4.9a) and the a1/a2 (1.0 to 2.5%, green circles, Figure 4.9a) domain struc-tures. With decreasing compressive strain (-0.5 to 0.2%), a domains form in the c matrix,reducing the combined elastic and Landau energies (red circles, Figure 4.9a) and producinga classic c/a domain structure. With increasing tensile strain, the combined elastic andLandau energies for both the c/a and a1/a2 domain structures are nearly equivalent; hencethey coexist (violet region, Figure 4.9a). In turn, the overall lowest strain-energy landscape(black line, Figure 4.9a) is characterized by the presence of two local minima separated bya region of negative curvature (akin to a strain-induced spinodal instability).[198, 199] Itcan be seen that the hierarchical domain architectures are observed within this region ofnegative curvature and this region corresponds to a strain-energy landscape which is nearlydegenerate. This coexistence, in turn, is found to be formed by a decomposition of the ma-terial into two domain structures which effectively partitions the average biaxial strain byforming the appropriate mixture of c/a domains (which stay clamped to an in-plane strainof 0.1%) and a1/a2 domains (which exhibit strains between 1-2% with increasing tensilestrain) such that the overall strain balances to the value set by the substrate (≈ 0.2 to 0.8%,Figure 4.9c). For instance, the hierarchical-domain architecture observed at a misfit strain of0.6% results from a partitioning of the material into c/a and a1/a2 domain structures withan in-plane strain of ∼0.1% and ∼1.35%, respectively (inset, Figure 4.9c). In this regime,any change in epitaxial strain or, by extension, any external perturbation (e.g., electric field,temperature, stress) can be accommodated by changing the relative c/aa and a1/a2 domainfractions while keeping the total energy essentially unchanged. Said another way, by con-straining the material with the appropriate biaxial strain conditions, it is possible to producestructural competition between different domain structures (analogous to phase competitionin MPB systems) in single-phase PbTiO3. This approximate regime of hierarchical-domainarchitectures can thus be represented on a revised temperature-strain phase diagram (Figure4.9d).

4.3 Probing the Effect of Ferroelastic Domain

Structures on Electrothermal Susceptibilities


Depending upon the strain regime in the revised temperature-strain phase diagram,the effect of temperature on the domain structures can be broadly classified into four cate-gories, (1) within the pure c phase, the monodomain structure present at room temperatureis preserved upon heating (shaded in orange, Figure 4.10); (2) within the c/a phase, in-creasing the temperature drives an increase in the fraction of c-oriented domains (shaded in

62

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

300

600

900

1200

)K(

e rut

are

pm

eT

Strain % (εm)

000

c-phase

a1/a2-type

c/a-phase

TbS

cO

3

1 01

GdS

cO

3

0 5

Sm

ScO

3

Paraelectric

0N

dS

cO

3

0 5

PrS

cO

3

Mixed-phasea a

Figure 4.10: Analytical phenomenological GLD-based temperature-strain phase diagramrevealing the stability regimes for c, c/a, and a1/a2-domain structures. Region within thedashed-line schematically represents the region where a mixed-phase or a coexisting c/a,and a1/a2-domain structures is expected under equilibrium. The dots show the pseudocubicstrain positions for the different substrates.

yellow, Figure 4.10); (3) within the mixed phase, where there is a coexistence of c/a- anda1/a2-domain structures, increasing the temperature drives an increase in the fraction ofa1/a2 at the expense of the c/a domain structure (shaded in light green, Figure 4.10); and(4) within the a1/a2 phase, the completely in-plane polarized domain structure is preservedwith increasing temperature (shaded in dark green, Figure 4.10). In the current work, we ex-amine the electrothermal susceptibilities along the out-of-plane ([001]- or z-) direction, thus,we focus on the first three domain structure variants (i.e., the c, c/a, and mixed phases).As a result, we will drop the subscript [001] or z from πz, Σz, and Ez from the rest of thearticle for simplicity.

To achieve a finer control over the c, c/a, and mixed-phase domain structures,120 nm PbZr0.2Ti0.8O3 / 20 nm Ba0.5Sr0.5RuO3 / TbScO3 (110), GdScO3 (110), SmScO3

(110), NdScO3 (110), and PrScO3 (110) heterostructures were grown using pulsed-laserdeposition. These substrates provide a lattice mismatch (εxx, εyy) with PbZr0.2Ti0.8O3

of (-1.27%, -1.26%), (-1.08%, -0.97%), (-0.69%, -0.45%), (-0.18%, 0.13%), and (+0.08%,+0.43) along [010] and [100], respectively (note that the misfit strain is calculated us-ing the room-temperature pseudocubic lattice parameter of PbZr0.2Ti0.8O3[145] and the

63

20 25 30 35 40 45 50

2θ

)s ti

nu .

brA(

ytis

net

nI

TbScO3

220

TbScO3

110

GdScO3

220

GdScO3

110

SmScO3

220

SmScO3

110

NdScO3

220NdScO3

110

PrScO3

220

PrScO3

110

PZT

002

PZT

002

PZT

002

PZT

002

PZT

002

PZT

200

PZT

200

PZT

001

PZT

001

PZT

001

PZT

001

PZT

001

PZT

100

PZT

100

εm=+0.26%

εm=-0.02%

εm=-0.57%

εm=-1.02%

εm=-1.26%

Figure 4.11: Wide-angle θ-2θ scans for 120 nm thick PbZr0.2Ti0.8O3 films grown on TbScO3

(110), GdScO3 (110), SmScO3 (110), NdScO3 (110) and PrScO3 (110) substrates.

anisotropic in-plane lattice parameters of the substrates[200]). Henceforth, we will use theaverage biaxial misfit strain (εm = 0.5(εxx + εyy)) to refer to a particular strain state.Ba0.5Sr0.5RuO3 (apc=3.97 A, calculated as a mean of SrRuO3 and BaRuO3)[201] was chosenas a lattice-matched bottom electrode. Room-temperature X-ray diffraction line scans (Fig-ure 4.11) reveal that the heterostructures are epitaxial and single-phase. It can also be seenthat the PbZr0.2Ti0.8O3 is predominantly (001)-or c-oriented when grown on TbScO3 (110)(εm = −1.26%), GdScO3 (110) (εm = −1.02%), and SmScO3 (110) (εm = −0.57%) due to thecompressive strain. With increasing tensile strain for films on NdScO3 (110) (εm = −0.02%;note that εyy is positive) and PrScO3 (110) (εm = +0.26%), there is a systematic evolutionof the films towards more (100)- or a-oriented structures. These trends are also evidentfrom the room-temperature domain structure imaged via atomic force microscopy (Figure4.12a-e) and piezoresponse force microscopy (Figure 4.12f-j).

To probe the effect of temperature on the c, c/a, and mixed-phase domain struc-tures, atomic-force microscopy-based topography measurements were conducted on represen-

64

(a)

(f)

(b)

(g)

(c)

(h)

(d)

(i)

(e)

(j)

Figure 4.12: Topography (a-e) and out-of-plane piezoresponse amplitude (in-plane piezore-sponse amplitude as inset) (f-j) of PbZr0.2Ti0.8O3 films grown on TbScO3 (a,f), GdScO3

(b,g), SmScO3 (c,h), NdScO3 (d,i), and PrScO3 (e,j) substrates.

tative structures on TbScO3 (110) (c), GdScO3 (110) (c/a), and PrScO3 (110) (mixed-phase)at 450 K (Figure 4.13). For heterostructures grown on TbScO3 (110), increasing the temper-ature from 300 K to 450 K does not alter the c-domain structure (Figure 4.13a and 4.13b).For the heterostructures grown on GdScO3 (110), increasing the temperature from 300 K to450 K causes a gradual reduction in the fraction of a domains (Figure 4.13c and 4.13d). Forthe heterostructures grown on PrScO3 (110), increasing the temperature from 300 K to 450K causes the a1/a2 domain structure to grow at the expense of the c/a domain structure,effectively reducing the net fraction of c domains (Figure 4.13e and 4.13f); this is oppositeto the trend observed in films under compressive strain within the c/a-phase regime (Figure4.13b).

With this understanding of how the domain structures evolve in response to tem-perature in the different strain regimes, devices with in situ heating/temperature-sensingability for measurement of both PEE and ECE in thin-film heterostructures were fabricatedusing the established process in Chapter 3.[36] Prior to any electrothermal measurements,polarization-electric field hysteresis loops were measured (Figure 4.14a). As expected, as thefraction of out-of-plane switchable c domains is reduced with increasing tensile strain (from-1.26% to +0.26%,), the remanent polarization (Pr) systematically diminishes from 72 µCcm−2) to 8 µC cm−2). The corresponding increase in the a-domain fraction also manifestsas an increase in the measured out-of-plane dielectric permittivity (εr) (Figure 4.14b) as εris higher for a domains than for the c domains.[202]

To study the effect of ferroelastic domains on the pyroelectric susceptibility, let us

65

(e)(a) (c)T

=4

50

KT

=3

00 K

(d)(b) (f)

cc-c-phase cc/c/cccccc a/a-aa--phase Mixeded-ddddd-dddddd phase

c c ↑ a a ↓ cc/c/cccccc a ↓ aa1a /a1//aaa222 ↑

Figure 4.13: AFM topography of the PbZr0.2Ti0.8O3 film grown on TbScO3 (a,b), GdScO3

(c,d), and PrScO3 (e,f) at 300 K and 450 K, respectively.

-150 -100 -50 0 50 100 150 200-90

-60

-30

0

30

60

90

( n

oita

ziral

oP

µC

cm

-2)

Electric field (kV cm-1)

= −1.26%(a) (b)

= −1.02%= −0.57%= −0.02%= +0.26%

102 103 104 1050

100

200

300

400

500

600

700

800

( yti

vittimr

ep

cirtc

elei

Dε r

)

Frequency (Hz)

0.0

0.2

0.4

0.6

0.8

1.0

Lo

ss ta

ng

en

t

= −1.26%

= −1.02%

= −0.57%

= +0.26%

= −0.02%

Figure 4.14: (a) Polarization-electric field hysteresis loop and, (b) dielectric permittivity(εr) for oxide metal (40 nm SrRuO3)-ferroelectric (120 nm PbZr0.2Ti0.8O3)-oxide metal (20nm Ba0.5Sr0.5RuO3) tri-layer heterostructures on TbScO3, GdScO3, SmScO3, NdScO3, andPrScO3 substrates with εm = −1.26%,−1.02%,−0.57%,−0.02% and +0.26%, respectively(brown, orange, yellow, light green and dark green data, respectively).

66

10 100 10000.1

1

10

i ,tn

erru

cor

yP

p (

nA

)

Heating current frequency, ωH/2π (Hz)

10 100 10000

1

2

3

4

5

6

θF

E (

K)


-180

-135

-90

-45

0

Th

erm

al p

ha

se

la

g, ф (°)

(a) (b)

= −1.26%

= −1.02%

= −0.57%

= −0.02%

= +0.26%

= −1.26%

= −1.02%

= −0.57%

= −0.02%

= +0.26%

10 100 10000

-50

-100

-150

-200

-250

-300

-350

.ffe

oC

ory

Pπ (μ

C m

-2 K

-1)


(c)= −1.26%

= −1.02%

= −0.57%

= −0.02%

= +0.26%

Secondary contributionsà0

Figure 4.15: (a) Measured temperature oscillation, θFE and thermal phase lag, φ betweenthe heating current and the temperature change in the ferroelectric as a function of the fre-quency. (b) Measured pyrocurrent and (c) pyroelectric coefficient as a function of frequency.

first recall the various contributions to which include: (1) intrinsic contributions that arisefrom the temperature-dependent change in the spontaneous polarization within a ferroelec-tric domain, (2) extrinsic contributions that arise because of the temperature-dependentmovement of domain walls resulting in a change in the fraction of the ferroelastic domains,and (3) secondary contributions that arise because of the thermal-stress-induced polarizationchange.[3, 81] As such, the total pyroelectric response can be written as:

π =∂ 〈Pz〉∂T

=∂ 〈Pz0〉∂T

+∑i

d∗ijkcjklmαlm = φcd 〈P 〉dT

+ 〈P 〉 dφcdT

+∑i

d∗ijkcjklmαlm (4.3)

where 〈Pz〉 and 〈Pz0〉 are the average net polarization and spontaneous polarizationin the out-of-plane or z-direction, respectively, and φc is the equilibrium c-domain fraction.d∗ijk, cjklm, and αlm are the components of the direct piezoelectric, elastic constant, andthermal expansion tensor, respectively. The first and the second term on the right-handside constitute the intrinsic (πint) and the extrinsic (πext) pyroelectric response (collectivelyconstituting the primary contribution) while the third term constitutes the secondary con-tribution. Since, we are interested the role of ferroelastic domains (πext), we will work tosystematically eliminate (or understand) the contributions from the intrinsic and secondarycontributions.

Pyroelectric measurements were conducted by periodically oscillating the temper-ature of the heterostructure by applying a sinusoidal heating current of 10 mA (rms) atvarious frequencies (f = ω

2π) across the microfabricated heater line. This heating current

drives temperature oscillations (dTdt

) in an area (A) of the heterostructure at frequencies (2f)with an amplitude that is measured using the 3ω method[36] (Figure 4.15a). The resultingpyroelectric current (ip) (Figure 4.15b) was measured using phase-sensitive detection[24, 25]

67

1

-200 -150 -100 -50 0 50 100 150 200

0

-50

-100

-150

-200

-250

-300

Pyro

Co

eff. π (μ

C m

-2K

-1)

DC Electric field (kV cm-1)

= +0.26%

= −0.02%= −0.57%

= −1.02%

= −1.26%

300 325 350 375 400

1.0

1.1

1.2

1.3

1.4

.ffe

oC

ory

P d

ezil

amr

oN

πT/π

T=

30

3 K

Temperature (K)

= −1.26%

= −1.02%

= −0.57%

= −0.02%

= +0.26%

(a) (d)

-1.5 -1.0 -0.5 0.0 0.5

-50

0

50

πext (μ

C m

-2 K

-1)

Strain (%)

0

-100

-200

-300

π (μ

C m

-2 K

-1)

c c/a Mixed-phase

(b)+

(c)

Continuous c/aàa1/a2

conversion

Towards Tc

Decay as à0

+

−

2

Figure 4.16: (a) Pyroelectric coefficient, π as a function DC electric field measuredat room temperature and 1 kHz, for oxide metal (40 nm SrRuO3)-ferroelectric (120 nmPbZr0.2Ti0.8O3)-oxide metal (20 nm Ba0.5Sr0.5RuO3) tri-layer heterostructures on TbScO3,GdScO3, SmScO3, NdScO3, and PrScO3 substrates with εm = -1.26%, -1.02%, -0.57%, -0.02% and +0.26%, respectively (brown, orange, yellow, light green and dark green data,respectively). (b) Experimentally measured π (red data) and estimated πint (black dashed-line) for different strain states. (c) Calculated πext for all strain states corresponding to c,c/a, and mixed phase heterostructures. (d) Temperature-dependent change of normalizedπ ( πT

πT=303K) for heterostructures with εm = −1.26%,−1.02%,−0.57%,−0.02% and +0.26%,

respectively (brown, orange, yellow, light green and dark green data, respectively).

and π was calculated as π = ip(AdT

dt

)−1(Figure 4.15c). It can be seen that with increas-

ing heating frequency, the measured value of π decreases for all the heterostructures. Atlower heating frequencies, the thermal penetration depth into the heterostructure is largesuch that it allows the ferroelectric film to undergo lateral thermal expansion along withthe substrate resulting in a finite secondary contribution that can be calculated by knowingthe relative thermal mismatch between the film and the substrate.[81] With increasing heat-ing frequency, however, thermal penetration is reduced and the ferroelectric film becomesprogressively clamped to the substrate which is no longer sensing the temperature change,thus resulting in negligible thermal expansion and effectively zero secondary contributionto π.[36] In our measurements, these secondary contributions can be effectively “turned-offabove 1 kHz heating frequencies. Thus, in the remainder of the discussion we will reportmeasurements of π at 1 kHz that exclude these contributions.

The primary contribution, under an electric field, also includes the contributionfrom temperature-induced changes in the dielectric permittivity

(ε0E

∂εz∂T

), To probe this

contribution, π was measured as a function of background DC electric field (Figure 4.16a).Here, we see a butterfly-shaped pyroelectric loop as the direction of the pyroelectric currentswitches when the ferroelectric switches. The contribution from temperature-induced changein the dielectric permittivity is manifested as a reduction in π from the zero DC electric-field

state to one under a finite electric field. This is because(∂〈Pz0〉∂T

)is negative while

(ε0E

∂εz∂T

)

68

is positive as εz increases with increasing temperature below Tc. Since, the heterostructuresunder tensile strain exhibit a higher dielectric constant (Figure 4.2b), the contribution fromthe temperature-induced change in the dielectric permittivity is higher (notice the higherslope between the representative points 1 and 2 for all the heterostructures).

Next, to separate πext from the measured primary contributions, we will first es-timate πint. This is done using the following assumption. For a pure c-phase heterostruc-ture with remanent polarization (Pr,−1.26%), the pyroelectric coefficient (π−1.26%) is purelyintrinsic due to the complete absence of any ferroelastic domains and therefore there isno possible change in the domain structure with temperature around room temperature.Thus, πint for any heterostructure with remanent polarization (Pr,εm) can be estimated as

πint,εm = πint,−1.26%Pr,εm

Pr,εm=−1.26%(black dashed line, Figure 4.16b). It should be noted that this

assumption can be applied here because the Curie temperature for all heterostructures ismuch higher than 750 K, and at room temperature, πint should scale with Pr. By subtract-ing πint from the measured primary contribution (red data, Figure 4.16b), πext be calculated(blue data, Figure 4.16c). For the c-phase heterostructures, πext is equal to zero as thefraction of c domains does not change with temperature and the measured is simply πintwhich mathematically is a negative quantity since 〈P 〉 decreases with T . For the c/a-phaseheterostructures, φc increases with temperature. Therefore, πext is positive and thus counterto πint. On the other hand, for the mixed-phase heterostructures, φc decreases with tempera-ture. Therefore, πext adds to πint thereby reinforcing the pyroelectric response. For example,in the heterostructure with misfit strain εm = +0.1%, πext from the temperature-dependentchange in the domain structure accounts for approximately -80 µC m−2 K−1 of the total π≈ -185 µC m−2 K−1; thus contributing as much as 43% of the total pyroelectric response.The mixed-phase heterostructures which exhibit a large negative πext, therefore, can providea new strain-induced route to enhance the PEE at a domain length-scale potentially usefulfor small-pitch, high-density, nano-patterned pyroelectric detector arrays.[203]

Further understanding of πext, especially for the mixed-phase heterostructures, canbe developed by probing the temperature-dependent change in π (Figure 4.16d). For an easeof understanding, we show how the normalized value of π (πT/π30◦C) changes with temper-ature. For the c-phase heterostructures with zero πext, the measured π simply increases asthe temperature approaches the ferroelectric-to-paraelectric phase transition (which happensabove 750 K and is not shown here due to experimental constraints) where the susceptibilitydiverges (brown data, Figure 4.16d). For the c/a-phase heterostructures, π also increases;however, under a competing response of πint and πext. As the temperature increases, thefraction of c domains in the c/a-phase heterostructures first increases (positive πext) beforetransitioning into a fully c-phase structure (Figure 4.13b) and then the response mimicsthe response of a c-phase heterostructure (orange and yellow data, Figure 4.16d). The re-sponse for the mixed-phase heterostructures, on the other hand, is fundamentally differentfrom the other two. Here, the phase-competition between c/a- and a1/a2-domain struc-tures causes significant domain reconfiguration resulting in a net increase in the fraction of

69

3.00 3.05 3.10 3.15

rev

o ytisn

etnI

det

arg

etnI

h-k

pla

ne (

Arb

. U

nits

)

4π/c (Å-1)

300 K

373 K

473 K

c-domains

a-domains

NdScO3

300 400 5000.0

0.2

0.4

0.6

0.8

1.0

noitc

arf ni

am

oD

Temperature (K)

c-domains

a-domains

(e) (f)

h k

l

3.0

3.1

3.2

4π/c

-0.1 0.0 0.1

3.0

3.1

3.2

4π/c

4π/a

473 K

3.0

3.1

3.2

4π/c NSO

220 Untilted a1/a2

Tilted a in c/a

Tilted c

300K(a)

(c)

(d)

(g)373 K(b)

Figure 4.17: Synchrotron-based RSM (showing the k− l plane about h = 0) about the 220-diffraction peak of the NdScO3 substrate showing the 002- and 200-peak of PbZr0.2Ti0.8O3

corresponding to the tilted c and untilted a domain variant at (a) 300 K, (b) 373 K, and (c)473 K. (d-f) Representative h−k slices shown for different l corresponding to c-domains (d),substrate (e), and a-domains (f). (g) Integrated intensity on the h − k plane for every l at300 K (blue data), 373 K (yellow data), and 473 K (red data). The inset shows the volumefraction of c- and a-domains as a function of temperature extracted by fitting the peaks in(g) and calculating the area under the corresponding peaks.

a domains with increasing temperature. This negative πext adds to πint and causes a muchsteeper enhancement of π with increasing temperature (light-green and dark-green data,Figure 4.16d). Since the c/a constituent, and therefore the fraction of c domains in themixed-phase heterostructures continuously reduces with temperature, a transition into thea1/a2 phase causes the pyroelectric response to decay (brown data, Figure 4.16d) and finallygo to zero without exhibiting any divergence in π.

To rigorously quantify πext within the mixed-phase heterostructures, where theycontribute to the net pyroelectric susceptibility, we measured the temperature-dependent

70

change in the c-domain population (which is responsible for π) using synchrotron-basedX-ray diffraction. For brevity we highlight such evolution in the heterostructure grown onNdScO3 (110) with εm=-0.02% wherein there is a measurable quantity of c domains in thetemperature range of interest. Three-dimensional reciprocal space mapping studies weremeasured about the on-axis 220-diffraction condition of the NdScO3 substrate at 300 K,373 K, and 473 K. A two-dimensional slice (k − l plane at h = 0) at 300 K reveals filmpeaks corresponding to 00l -oriented c domains and two versions of the in-plane polarizeda domains: (1) an untilted variant (corresponding to the a1/a2-domain structure) and (2)a tilted variant (corresponding to a domains in the c/a-domain structure) (Figure 4.17a).Increasing the temperature to 373 K (Figure 4.17b) and 473 K (Figure 4.17c) results in aprogressive increase in the intensity of the untilted a1/a2-domain structure revealing a netincrease in the fraction of the a domains. To accurately quantify the change in the fractionof c and a domains, we integrated the intensity on the h − k plane across all measuredvalues of l (Figure 4.17d-f) to get an integrated intensity (Figure 4.17g) that accounts forthe crystallographic tilts and allows for a more accurate quantification (in comparison to θ-2θX-ray line scans or scanning-probe-based image analysis) of the volume fraction of the c anda domains. By fitting the integrated intensity curve with mixed Lorentzian and Gaussianpeaks and calculating the area under the peaks, the fraction of c and a domains was extracted(inset, Figure 4.17g). To calculate the magnitude of πext (πext = 〈P 〉 dφc

dT), 〈P 〉=72 µC cm−2

corresponding to the monodomain c phase, was taken. This was under the assumption thatin the mixed-phase heterostructures, the average biaxial misfit (tensile) strain partitions intoa less tensile component accommodated predominantly by the c/a phase and a more tensilecomponent accommodated by the a1/a2 phase.[186] Using the measured value of dφc

dT(inset,

Figure 4.17g) between 300 K and 373 K, an average πext equal to approximately -128 µCm−2 K−1 can be estimated; suggesting that strain-/temperature-dependent changes in thedomain population can significantly contribute to π and can be used as a promising routeto enhance pyroelectric susceptibility in thin-film systems.


The strain- and temperature-dependent changes in the domain configurations haveimplications for the ECE as well. Similar to the case of PEE, we will break down thecontributions to Σ into (1) intrinsic contributions that arise from the electric-field-dependentisothermal change in the entropy or an adiabatic change in temperature within a ferroelectricdomain, (2) extrinsic contributions that arise because of electric-field-dependent ferroelasticdomain conversion, and (3) secondary contributions that arise because of the piezoelectricity-induced elastocaloric effect (or piezocaloric effect). As such, the total electrocaloric responsecan be written as:

71

Σ =∂S

∂E=∂S0

∂E+ ΣiCdijkγjk = φc

dSc

dE+ φa

dSa

dE+dSβ→δ

dE+ ΣiCdijkγjk (4.4)

where dSc

dEand dSa

dEis the dipolar entropy change in the c and a domains with volume

fractions φc and φa, respectively, under an applied electric field (E). Together, these two

terms constitute the intrinsic response (Σint). The third term, dSβ→δ

dE, exclusively represents

the entropy change due to a phase change from a generic phase β to δ under the sameelectric field constituting the extrinsic contribution to ECE (Σext). C, dijk, and γjk are thevolumetric heat capacity, and components of the converse piezoelectric and Grneisen tensor,respectively. Their product (last term, right-hand side) constitutes the secondary responseresulting from the piezocaloric effect.

Using the same device structure as for measuring the PEE, ECE measurementswere conducted by applying an electric-field perturbation to measure the electrocaloric-temperature change. Briefly, a sinusoidally varying electric-field with an amplitude of 83kV cm−1 at 98.147 kHz was applied across the ferroelectric heterostructure. In responseto the AC electric-field perturbation, the resulting sinusoidal temperature oscillation in theferroelectric was sensed using the top metal line now functioning as a thin-film resistancethermometer. An AC sensing current with a magnitude of 2 mA at a frequency of 2.317kHz was sourced through the top line and the electrocaloric response (manifested as anAC voltage) was measured at 98.147-2.317=95.830 kHz. Using the temperature coefficientof resistance, the temperature change in the top metal line was calculated and this wassubsequently used to calculate Σ after solving the 1-D heat transport model using the ther-mophysical properties of the heterostructure (Table 1).[36] Since, the ferroelectric thin filmis clamped on the substrate, the secondary contribution due to the piezocaloric effect can beneglected[204] and the measured value Σ should only account for the primary contributions(Σ = Σint + Σext).

ECE measurements were conducted by sweeping the background DC bias to boththe positive and negative saturation bias to measure the response in both the up- and thedown-poled state. It can be seen that Σ, for all the heterostructures, exhibits a ferroelectrichysteresis-like response as the DC electric is swept (Figure 4.18a). Conventional ECE (i.e., arise in temperature with increasing electric field) is observed in all the heterostructures whenthe device is poled-down (with polarization equal to +Pr) and the application of a positiveelectric field reduces the dipolar entropy and results in an increase in the temperature. Whenthe sample is poled-up (with polarization equal to−Pr), applying a positive field increases thedipolar entropy and results in the inverse ECE (i.e., a decrease in temperature with increasingelectric field). Further, it can be seen that the ECE for the c-phase heterostructures ismaximum and with increasing tensile strain the magnitude of Σ diminishes. To understandthe role of strain and the resulting domain structures on Σ, we have estimated the intrinsic

72

Film/Substrate Thermal conductivity Specific heat capacityk (Wm−1K−1) C(T ) (Jm−3K−1)

PbZr0.2Ti0.8O3 1.5 a 2.7× 106(25◦C)b

SrRuO3 3.9 3.0× 106 c

Ba0.5Sr0.5RuO3 Assumed as 3.9 Assumed as 3.0× 106

SiNx 2.0 d 1.16× 106 e

TbScO3 2.82 f Assumed as 2.7× 106

GdScO3 3.34 f 2.7× 106 a

SmScO3 3.49 f Assumed as 2.7× 106

NdScO3 3.86 f Assumed as 2.7× 106

PrScO3 3.61 f Assumed as 2.7× 106

Table 4.1: Thermophysical properties (at 300 K) of the various thin films and substratesused in the thermal transport calculations. a Ref. [35]; b Ref. [205]; c Ref. [206]; d Measuredusing differential 3ω method on a single-crystal sapphire substrate; e Ref. [207]; f Measuredusing 3ω method

contribution to the ECE as Σint,εm = Σint,−1.26%Pr,εm

Pr,εm=−1.26%(black dashed line, Figure 4.18b)

akin to πint. This holds true because the application of electric field to a pure c-phaseheterostructure does not result in any domain conversion and, therefore, Σext = 0 (Figure4.18c). For the other heterostructures (c/a and mixed phase), Σext can be calculated bysimply subtracting Σint from the measured total Σ (Figure 4.18c).

It can be seen that Σext is positive for the c/a-phase heterostructures and is negativefor the mixed-phase heterostructures (Figure 4.18c). This can be understood as follows. Σext

explicitly accounts for the change in the entropy due to domain conversion under an appliedelectric field. The application of an electric field can stabilize a phase with either a lower ora higher entropy (note that the phase stable at a higher temperature has a higher entropythan the one stable at a lower temperature). Σext will therefore be negative for the formerand positive for the latter cases. For the c/a-phase heterostructures, the a domains aresusceptible to transform to c domains under an applied electric field. Since the electric fieldstabilizes the higher-entropy c phase, Σext, captured as dSa→c

dE, is positive (inverse). On the

other hand, for the mixed-phase heterostructures, Σext is negative (conventional) as the fieldstabilizes the lower-entropy phase (mixed phase over the a1/a2 phase). The electrocaloricresponse can alternatively be understood by looking at how, under an applied electric field,the component of polarization along the direction of field (Pz) changes with increasing tem-perature. To understand Σext in particular, let us focus only on the volume of a or c domainsthat change with temperature under an applied out-of-plane electric field. In the c/a-phaseheterostructure, an increase in temperature causes a a→c transformation resulting in anincreasing in the out-of-plane polarization 〈P a→c

z 〉 and therefore a positive slope of(dPdT

)E

73

(a)

-100 -75 -50 -25 0 25 50 75 100

-200

-100

0

100

200

. ffe

oC

EC

EΣ

(μ

C m

-2K

-1)


-15

-10

-5

0

5

10

15

Te

mp

era

ture

ch

an

ge

(m

K)

-1.5 -1.0 -0.5 0.0 0.5

-50

0

50

Σext (μ

C m

-2 K

-1)

Strain (%)

0

-100

-200

Σ (μ

C m

-2 K

-1)

-1

c

-0.5

c/a

0.0 0.5

Mixed-phase

(b)

Σ +

Σ

(c)

= 0!

≠ 0

= 0!

1 2

≠ 0

(d)

+Σ

−Σ

(e)

= −1.26%

= −1.02%

= −0.57%

= −0.02%

= +0.26%

Figure 4.18: (a) Measured electrocaloric coefficient, Σ showing conventional (inverse) ECEin the down- (up-) poled state under positive (negative) saturation DC electric field forheterostructures with εm = −1.26%,−1.02%,−0.57%,−0.02% and +0.26%, respectively(brown, orange, yellow, light green and dark green data, respectively). (b) Measured Σ(red data) and estimated Σint (black dashed-line) for different strain states. (c) CalculatedΣext for all strain states corresponding to c, c/a, and mixed phase heterostructures. (d)Schematic showing the change in the polarization (from c domains and interconversion ofa → c domains) with increasing temperature under an applied electric field. (e) Schematicshowing the change in the polarization (from c domains and interconversion of c/a → a1/a2

domains) with increasing temperature under an applied electric field.

(Figure 4.18d). Qualitatively, as per the Maxwell relation,(dSdE

)T,σ

=(dPdT

)E,σ

, this corre-

sponds to a positive Σext. In the mixed-phase heterostructure, an increase in temperaturecauses a c/a→a1/a2 transformation resulting in a decrease in the out-of-plane polarization⟨P

c/a→ a1/a2z

⟩and therefore a negative slope of

(dPdT

)E

(Figure 4.18e) and hence negative Σext.

Thus, Σext competes with the conventional Σint in the c/a-phase heterostructures while itcomplements the conventional Σint in the mixed-phase heterostructures.

74

4.4 Probing Electromechanical Response in

Mixed-Phase Domain Structures

4.4.1 Understanding Ferroelastic Switching in Mixed-PhaseDomain Structures

The potential for electric-field driven response was explored via PFM-based BE-SS[208, 209] which allows for the visualization of the switching process (Figure 4.19). To con-duct this experiment, we applied a bipolar triangular switching waveform using a scanning-probe tip in a square-grid (3 µm spacing between points) and measured the piezoresponse,phase, cantilever loss, and cantilever contact resonance (indicative of elastic modulus) ateach voltage step. Prior to, and following switching, PFM and topography scans were ob-tained and image registry techniques were employed to observe any structural changes. Fromthis analysis, two characteristic switching processes were observed; each of which revealed ahigh degree of correlation to the initial domain structure. First, switching in a purely c/aregion (Figure 4.19a) reveals piezoresponse loops with a sharp, single positive and negativecoercive bias, consistent with ferroelectric switching for an out-of-plane polarized, tetragonalferroelectric (Figure 4.19b). The voltage-contact resonance frequency loops (Figure 4.19c),again indicative of the elastic modulus of the film, reveal a butterfly loop shape with soft-ening of the lattice during the switching, and hardening during saturation, consistent withprior reports.[210–212] Specifically, as positive bias is applied, there is a rapid change inthe piezoresponse as the film switches, resulting in elastic softening up to a bias of ∼4 V.Further increasing the voltage gives rise to saturation of the piezoresponse and, correspond-ingly, elastic hardening, indicating that the structure has fully reoriented under the appliedfield. As the bias is decreased towards 0 V, the resonance frequency remains essentiallyconstant until, under negative bias, an identical switching process in the opposite directionis observed. Looking at the initial and final topographic images (Figure 4.19d and 4.19e,respectively) we note no change in the topography and domain structures.

Focusing now on to the second type of switching, occurring when applying a biasin an a1/a2 domain region (Figure 4.19f), an atypical switching process is observed. Thepiezoresponse hysteresis loops exhibit intermediate stable states with a reduced piezoresponse(labeled 2 and 4, Figure 4.19g) relative to the saturation states (labeled 5 and 3, Figure4.19g). Corresponding analysis of the voltage-contact resonance loops (Figure 4.19h), revealsan unconventional response wherein an initial elastic hardening to a stiffer intermediatestate is observed (we note that this type of response persists through multiple switchingcycles). In particular, upon increasing the bias, there is a rapid and abrupt increase in thecontact resonance, indicative of a transition to a mechanically stiffer state (1→2, Figure4.19h), which corresponds to the intermediate, low piezoresponse state (2, Figure 4.19g).

75

Ch

an

ge

in

Co

nta

ct

Re

so

na

nc

e (

Hz)

Pie

zo

res

po

ns

e (

a.u

.) b

Final

Initial

Final

Initial

d

e

i

j

Voltage (V)

Pie

zo

res

po

ns

e (

a.u

.)

2

3

4

5

10 5-5-10 10

gf

a

Voltage (V)8

HardeningS

oftening

13

4

5

a

a

Hardening

cc

Softening

2

0 2-2-4-6-8

0

375

-375

-750

4 6

c

h

-375

0

375

750

1500

Figure 4.19: Band excitation piezoresponse spectroscopy (BEPS) studies on 80 nm PbTiO3/BaSr0.5Ru0.5O3/GdScO3 heterostructures with coexisting hierarchical c/a and a1/a2 domainarchitectures. (a) Schematic illustration of a c/a region, and (b) the corresponding out-of-plane piezoresponse hysteresis loop obtained from such a region. (c) Contact resonancefrequency loops with various polarization states through the switching process indicated.(d,e) Topographic images of the region prior to (d) and following (e) switching revealing anunchanged domain structure. Inset shows a binary image of the 75 nm region directly underthe PFM tip. (fg) Schematic illustration of an a1/a2 region (f), and corresponding out-of-plane piezoresponse hysteresis loop (g) and contact resonance frequency loop (h) taken insuch a region revealing a three-state ferroelastic switching process (c+, a, and c−) with high(low) piezoresponse and low (high) contact resonance states characteristic of c (a) domains.Throughout the switching process the polarization states are indicated. (i,j) Topographicimages of the region prior to (i) and following (j) switching revealing the nucleation of a cdomain within an a1/a2 domain structure. Inset shows a binary image of the 75 nm regiondirectly under the PFM tip. Scale bars in all images are 500 nm.

76

356

358

357

kHz

Figure 4.20: Scanning probe microscopy-based photothermal-excitation contact reso-nance measurements showing (a) topography and (b) contact resonance map for PbTiO3/Ba0.5Sr0.5RuO3/SmScO3 (110) heterostructures.

Considering the anisotropy in stiffness (C11 ≈ 105 GPa, C33 ≈ 230 GPa) and piezoelectriccoefficients (d33 ≈ 83.7 pm V−1, d11 ≈ 0 pm V−1) in tetragonal PbTiO3,[202] the transitionobserved corresponds to a 90◦ ferroelastic switching from up-polarized c to in-plane polarizeda. To validate this observation, contact resonance measurements (without bias) in purelyc and a domain regions were completed and reveal an increase in the contact resonancefrequency of ∼2000 Hz from c to a (Figure 4.20). Thus, this increase in elastic modulusrepresents the direct detection of 90◦ ferroelastic switching from an up-polarized c+ to in-plane polarized a. Continuing to increase the positive bias results in an abrupt reduction ofthe contact resonance indicative of a transition to a mechanically softer state (2→3, Figure4.19h) which corresponds with the saturation of the piezoresponse (3, Figure 4.19g). Thisindicates a second 90◦ ferroelastic switch from the in-plane polarized a to a down-polarizedc- domain. As the field is subsequently reduced, the region under the tip remains down-polarized c−, until reaching the negative coercive field, wherein it undergoes a transitionto in-plane polarized a as indicated by a rapid increase in the contact resonance (3→4,Figure 4.19h) and the intermediate state with low piezoresponse (4, Figure 4.19g). Uponfurther increasing the negative bias, the material transitions from the in-plane polarizeda to an up-polarized c+ as indicated by a rapid decrease in the contact resonance (4→5,Figure 19h) and saturation of the piezoresponse (5, Figure 4.19g). Comparison of the initial(Figure 4.19i) and final (Figure 4.19j) topographic images reveals the nucleation of a stablec domain within the a1/a2 matrix. These studies illuminate the role of domain-structurecompetition in reducing clamping effects, which results in an acoustically detected two-step,three-state ferroelectric switching process, and facilitates the interconversion of ferroelasticdomain structure variants.

77

4.4.2 Deterministic Control of Ferroelastic Switching

Such facile electric-field-driven ferroelastic switching provides a promising routeto controllably reconfigure domain architectures important for a range of applications. Todemonstrate the potential of this system, we focus on a hierarchical domain architecturesconsisting almost entirely of a1/a2 domain structures (as found in films at εs = 0.7%). As-grown, the films reveal a flat topography (height, Figure 4.21a) and a diminished out-of-planepiezoresponse (color scale, Figure 4.21a) corresponding to a1/a2 domain structures. A biaspattern (7 V) was applied using the PFM tip to ferroelastically write an array of out-of-planepolarized c domain lines which protrude from the surface (height, Figure 4.21b) and exhibitenhanced out-of-plane piezoresponse (color scale, Figure 4.21b). To erase these c domains,in-plane electric fields (∼100 kV cm−1) were applied using interdigitated electrodes restoringthe as-grown topography (height, Figure 4.21c) and out-of-plane piezoresponse (color scale,Figure 4.21c). Line traces across the switched regions (height, Figures 4.21a-c) reveal thatthe electric-field induced ferroelastic interconversion from a to c is accompanied by ∼0.5 nmchanges in the surface topography; corresponding to a large and recoverable out-of-planeelectromechanical strains of ∼]1.25%. The versatility of this effect is demonstrated by elec-trically writing an arbitrary pattern over a large area (resembling the Cal R© logo, Figure4.21d). Such deterministic, reversible, and non-volatile control of ferroelastic switching in aclamped thin film is promising to control the dielectric, optical, acoustic, etc. properties ofmaterials at the nanoscale which has implications for a range of applications including na-noelectromechanical systems,[176] reconfigurable metamaterials,[213, 214] piezotronics,[177]smart/adaptable surfaces and modules,[215] and much more. For example, in addition to thelarge changes in topography and piezoresponse, the inherent anisotropy in PbTiO3 meansthat these same ferroelastic interconversions give rise to a reversible 23% change in the localYoung’s modulus of the material. Such effects point to a rich potential for exotic nanoscaledesign of materials and mesostructures.

4.5 Conclusion

In summary, the effect of epitaxial strain on the evolution of ferroelastic domainstructures in a canonical tetragonal ferroelectric PbZr1−xTixO3 was investigated. We showedhow epitaxial strain can induce a domain-structure competition in PbZr1−xTixO3 giving riseto a rich spectrum of domain-structures and the resulting physical responses. We illustratedthe role of ferroelastic domains on the electrothermal susceptibilities and experimentallyshowed how their response fundamentally varies with the imposed epitaxial strain boundarycondition. The so-called extrinsic effect was found to suppress the electrothermal susceptibil-ities in the compressive strain regime. However, in the tensile strain regime, the temperature-and field-induced interconversion of the domain structures in the mixed-phase domain ar-

78

Δh=0.5 nm

ε=1.25%

a

d

b

c

500 nm

1 μm

Figure 4.21: Superimposed topography and vertical PFM (amplitude) response for a 40 nmthick PbTiO3 film grown with εs = 0.7% predominantly a1/a2 structure. a) Shows the initiala1/a2 domain structure and an atomically terraced surface (as indicated by the line tracein height). b) Reveals a deterministically written pattern of c/a domains within the a1/a2

region wherein large surface displacements are observed (line trace in height shows ≈ 0.5 nmchanges). c) Shows that such structures can be erased by application of an in-plane electricfield, restoring the surface to the original smooth state (line trace in height). The insetheight line traces reveal the evolution of the surface topography and indicate the potentialfor reversible electromechanical strains of 1.25%. d) Electrically written ferroelastic pattern(Cal logo) on a 10 µm2 as-grown region. The Cal logo, which is a registered trademarkand intellectual property of the Regents of the University of California, is reproduced withpermission.

79

chitecture enhanced the same susceptibilities.

Further, we demonstrated that the field-induced ferroelastic interconversions cangive rise to large electromechanical responses. Such effects are made possible by the nearly en-ergetically degenerate, hierarchical domain architectures of coexisting c/a and a1/a2 domainstructures which drives an acoustically detectable two-step, three-state ferroelastic switchingprocess. This allows for deterministic writing and erasure of ferroelastic domain-structurevariants with large out-of-plane electromechanical strains (∼1.25%) and tunability of thelocal piezoresponse and elastic modulus (>23%). Such structures, in turn, are useful for de-terministic control of nanoscale modulations in thermal, electrical, mechanical, and opticalsusceptibilities and motivate the exploration of other systems wherein similar approachescan be used to induce hierarchical ferroelastic domain architectures capable of exhibitingnew functionalities and improved performance.

80

Chapter 5

Pyroelectric Energy Conversion usingComplex Oxide Thin Films

In this Chapter, I explore pyroelectric energy conversion from low-grade thermalsources that exploits strong field- and temperature-induced polarization susceptibilities inthe relaxor ferroelectric 0.68Pb(Mg1/3Nb2/3)O3-0.32PbTiO3. Electric-field-driven enhance-ment of pyroelectric response (as large as −550µCm−2K−1) and suppression of dielectricresponse (by 72%) yield large figures of merit for pyroelectric energy conversion. Field- andtemperature-dependent pyroelectric measurements highlight the role of polarization rotationand field-induced polarization in mediating these large effects. Solid-state, thin-film devicesthat convert low-grade heat into electrical energy are demonstrated using pyroelectric Eric-sson cycles, and optimized to yield maximum energy density, power density, and efficiencyof 1.06 J cm−3, 526 W cm−3, and 19% of Carnot, respectively; the highest values reportedto date and equivalent to the performance of a thermoelectric with an effective ZT ≈ 1.16for a temperature change of 10 K.

81

5.1 Introduction and Challenges

In the United States nearly 68% of the primary energy produced in 2016 was wastedas heat.[216] For example, it is projected that a single next-generation exascale supercom-puter will consume 5-10% of the total power output from the average coal-fired power plantand turn virtually all that energy into heat.[217] Considerable efforts are, therefore, under-way to harvest some of this waste heat. The majority of research in this regard has focusedon thermoelectrics (materials that convert a temperature gradient into electrical energy viathe Seebeck effect).[86] For low-grade waste heat (< 100◦C) thermoelectrics are inherentlylimited by their low ZT, a dimensionless figure of merit (FoM) that requires high electricaland low thermal conductivity; a combination that is difficult to achieve.[218] For example,the widely-studied Bi2Te3-based thermoelectrics have ZT ≈ 1 which corresponds to 17%scaled efficiency (the ratio of the absolute to the Carnot efficiency) for low-grade heat scav-enging.[219] Despite some commercial success with thermoelectrics, researchers continue toexplore alternative technologies for waste-heat energy conversion; including approaches thattake advantage of phenomena such as thermo-osmotic and -galvanic effects.[220, 221] ThisChapter demonstrates the potential of an approach based on pyroelectric materials.

The pyroelectric effect describes the temperature-dependent evolution of sponta-neous polarization P (resulting in changes in the surface charge density of the material) andis parameterized by the pyroelectric coefficient, π = ∂P

∂T. In contrast to the steady-state oper-

ation of thermoelectrics, pyroelectric energy conversion requires thermodynamic cycles thatmake use of a temporally-varying thermal profile to extract electrical work from the pyroelec-tric. Such cycles mimic gas-phase cycles and a number of different thermal-electrical cyclesfor pyroelectric energy conversion have been proposed including, for example, Stirling-likecycles with two isothermal and two isodisplacement processes, [88, 89] Carnot cycles withtwo isothermal and two adiabatic processes,[90] and an adaptation of the Ericsson cycle,an Olsen cycle,[91] which includes two isothermal and two isoelectric processes.[91, 92, 99,100, 222, 223] Early work in the exploration of these cycles revealed that these adaptedEricsson cycles can extract the maximum potential work from a pyroelectric out of all of thethermodynamic cycles studied to date.[87] In turn, much of the work in this field since thistime has employed these adapted Ericsson cycles for pyroelectric energy conversion[222, 223]and these approaches have produced maximum energy density, power density, and scaledefficiency values of ∼1 J cm−3,[92] ∼ 30 W −3,[99] and 5.4%,[100] respectively. Based onthese thermal-electrical cycles, researchers have also established that a high-performancepyroelectric energy conversion material should have a high π and low relative permittivity(εr), a trade-off parameterized by the FoM for pyroelectric energy conversion (FoMPEC)which is π2

ε0εr(ε0 is the permittivity of free space).[32] In turn, optimization of pyroelec-

tric performance requires independently enhancing π and suppressing εr, which is difficultin conventional materials where dielectric and pyroelectric susceptibilities are generally en-hanced by the same generic features (i.e., proximity to phase transitions driven by chemistry,

82

temperature, strain, etc.).[82, 224]

5.2 Identifying materials with large pyroelectric

response

Phase transitions with enhanced ferroic susceptibilities are a hallmark of relaxorferroelectrics like (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 where changing the ferroelectric PbTiO3

content results in a change in the lattice symmetry from rhombohedral, R (x ≤ 0.31), totetragonal, T (x ≥ 0.35), via a bridging low-symmetry monoclinic, M (0.31 < x < 0.35),phase constituting the morphotropic phase boundary (MPB).[225, 226] In proximity to theMPB, different ferroelectric phases are nearly energetically degenerate, resulting in facilerotation of polarization under applied stimuli (i.e., temperature,[226] stress,[227] electricfields[228]). While electric-field-induced polarization rotation has been shown to result ingiant electromechanical response,[63, 64] such field-coupling to pyroelectricity is poorly stud-ied. The limited work in this regard has shown the possibility for enhancement of π underapplied bias,[50, 229] however, the mechanism for such effects remains unclear with argu-ments ranging from the importance of a diffuse first-order-like phase transition between theferroelectric and relaxor phases,[230] to contributions from the secondary pyroelectric effectvia field-induced piezoelectricity.[231, 232] Understanding such electric field/temperaturesusceptibility of polarization in relaxor ferroelectrics is crucial to establishing these materi-als as promising candidates for pyroelectric energy conversion.

5.3 Growth and Characterization of PMN-32PT Thin

Films

150 nm PMN-0.32PT / 20 nm Ba0.5Sr0.5RuO3 heterostructure was synthesized onsingle-crystalline NdScO3 (110) via pulsed-laser deposition. PMN-0.32PT growth was carriedout a deposition temperature of 600◦C in a dynamic oxygen pressure of 200 mTorr usinga laser fluence of 1.8 J cm−2, and a laser repetition rate of 2 Hz. The Ba0.5Sr0.5RuO3

bottom electrode layer was grown a temperature of 750◦C in a dynamic oxygen pressure of20 mTorr by ablating a Ba0.5Sr0.5RuO3 target (Praxair) at a laser fluence of 1.85 J cm−2 anda laser repetition rate of 3 Hz. Following growth, the heterostructures were cooled to roomtemperature in a static oxygen pressure of 760 Torr at 10◦C min−1.

Following the growth of the PMN-32PT/BSRO heterostructure, X-ray diffractionθ−2θ scans were conducted that reveal that the heterostructures are single-phase and are epi-

83

20 25 30 35 40 45 50

)stin

u .br

A( ytisn

etnI

2

NdScO3110

PMN-PT001

BaSrRuO3001

0.24 0.25 0.26

0.244

0.248

0.252

a-1

en

alp-f

o-tu

o (

Å-1

)

a-1in-plane (Å

-1)

PMN-PT103

NdScO3

332

BSRO103

44 45 46

002

002 220

BaSrRuO3

NdScO3

PMN-PTa b

θ

Figure 5.1: (a) Wide angle θ − 2θ scans for a 150 nm 0.68Pb(Mg1/3Nb2/3)O3-0.32PbTiO3

(PMN-PT)/25 nm Ba0.5Sr0.5RuO3 (BSRO)/NdScO3 (110) heterostructure showing the filmis (001)-oriented, single-phase and of high quality. Inset shows Kiessig fringes about theprimary diffraction peaks, again suggesting pristine interfaces and high quality. (b) Off-axisRSM about the 332-diffraction conditions revealing that both PMN-PT and BSRO have thesame in-plane lattice parameter as the substrate.

taxial with (001)-orientation (Figure 5.1a). Further, off-axis RSM about the 332-diffractioncondition of NdScO3 shows that the heterostructure is coherently strained (compressivestrain of -0.5%) (Figure 5.1b).

Electrothermal devices were fabricated using the approach described in Chapter3.[36] Polarization-electric field hysteresis loops, measured from such devices, reveal high po-larizability and behavior characteristic of relaxor ferroelectrics[233] (Figure 5.2a). Additionalfeatures of relaxor behavior are evident in temperature-dependent εr which shows frequencydispersion below its maximum (Tmax = 150◦C) (Figure 5.2b) and deviation from Curie-Weissbehavior below the Burns temperature (Tb = 259◦C, above which the short-range-ordereddomains or polar nano-regions (PNRs) dissolve and PMN-0.32PT is a non-polar paraelec-tric)[234] (Figure 5.2c).

5.4 Pyroelectric Response of PMN-32PT Thin Films

Pyroelectric measurements were conducted by periodically oscillating the temper-ature of the heterostructure by applying a sinusoidal heating current of 19 mA (rms) at a

84

50 100 150 200 250 300800

1000

1200

1400

1600

1800

( yti

vi tt im r

ep

cir tc

ele i

Dr)

Temperature ( C)

10 kHz

1 kHz

-300 -150 0 150 300

-30

-20

-10

0

10

20

30

( n

oita

ziral

oP

C c

m-2

)

Electric Field (kV cm-1)

1 kHz

a b

0 50 100 150 200 250 3006

7

8

9

10

11

01(

ytivitti

mre

p e

sre

vnI

-4r)

Temperature ( C)

1 kHz

Td=259°C

c

µ

ε ε

° °

-1

Figure 5.2: (a) Polarization-electric field hysteresis loop measured at room temperatureand 1 kHz for the PMN-32PT heterostructure. (b) Dielectric permittivity (εr) as a functionof temperature and frequency (1-10 kHz) showing a systematic shift of the maxima (Tmax)to lower temperatures with decreasing measurement frequency. (c) Inverse dielectric per-mittivity (at 1 kHz) as a function of temperature revealing the deviation from Curie-Weissbehavior (black straight-line) to extract the Burns temperature (Tb = 259◦C, shown as blackdashed-line).

frequency of 1 kHz on the microfabricated heater line. This heating current perturbs thetemperature of the heterostructure by an amplitude θFE = 10K at a frequency of 2 kHz andis measured using the 3ω-method (Figure 5.3).[36] The resulting pyroelectric current (iP )was measured as a function of dc electric field (red data, Figure 5.4a) and π is extracted fromthe relationship iP (AdT

dt)−1 (red data, Figure 5.4b). Under zero bias, π ≈ −100µCm−2K−1;

however, under application of a dc electric field, iP (Figure 5.4a) and π (Figure 5.4b) arefound to be dramatically enhanced such that | π |> 550µCm−2K−1; far surpassing the field-induced enhancement of π in a classic ferroelectric (e.g., (001)-oriented PbZr0.2Ti0.8O3 thinfilm are provided for comparison; blue curve, Figure 5.4b). We further measured the tun-ability of εr with dc electric field and found that εr is dramatically suppressed (by ∼72%)with field (red open circles, Figure 5.4c) due to the suppression of the extrinsic polarizationcontribution from the presence of PNRs and consistent with prior observations in relaxorferroelectrics.[235] In addition, the loss tangent due to charge leakage within the capacitoris also suppressed with increasing dc electric field (red filled squares, Figure 5.4c). From apractical standpoint, strongly suppressed εr (and loss tangent) in conjunction with signifi-cantly enhanced π, results in an enhancement of the FoMPEC (red data, Figure 5.4d) by ∼5times in comparison to commonly used materials [e.g., LiNbO320 (black data, Figure 5.4d)and PbZr0.2Ti0.8O332 (blue data, Figure 5.4d)]. The question, then, is: what is the role ofthe dc electric field in enhancing π to such an extent?

To probe the effect of electric field on this strongly field- and temperature-coupledrelaxor ferroelectric, a series of field- and temperature-dependent pyroelectric measurementswere conducted. For all studies, the heterostructures were first heated to above the Tb to

85

1 10 100 10000

10

20

30

FE (

K)

Electrical frequency, (Hz)

-90

-45

0

45

90

Th

erm

al p

ha

se

la

g,

()

φ°

θ

ωH/2π

~

IHeat

(1ω)

V

V(3ω)

ba

Figure 5.3: (a) Sinusoidally varying current at frequency 1ω is used to heat the heaterline via the outer thermal probe pads. Voltage drop at 3ω across the inner thermal probepads is measured to extract the amplitude of temperature oscillation in the heater line. (b)Measured temperature oscillation θFE and thermal phase lag φ between the heating currentih(t) (IRMS = 20mA) at an angular frequency 1ωh (and therefore the dissipated power) andthe temperature change in the relaxor ferroelectric as a function of frequency using the 3ωmeasurement.

300◦C before being cooled to room temperature in zero applied field. Once at room temper-ature, pyroelectric measurements were conducted as a function of dc electric field (0, 13.3,26.7, and 40 kV cm−1) with increasing sample temperature using an oscillating ac temper-ature with amplitude θFE = 2.5K. The magnitude of oscillation was deliberately chosento be small to capture any temperature-dependent changes in the pyroelectric susceptibilityas the background temperature was varied. Under zero field, two distinct anomalies in πare observed (Figure 5.5a). We attribute the first anomaly at 82◦C to a M (specificallyMC wherein the polarization is confined to the 010) to T phase transition.[236] The secondanomaly at 216◦C likely corresponds to a T to cubic phase transition. Single-crystal PMN-0.32PT shows such a transition at 150◦C;[236] however, in the compressively strained (-0.5%)films used here, the transition temperature could easily be shifted higher. Upon probing thesame pyroelectric response for heterostructures under applied dc electric fields of 13.3, 26.7,and 40 kV cm−1, we observe field-induced enhancement of π at all temperatures (Figure5.5b). The first anomaly in π, corresponding to the MC to T phase transition, progressivelyshifts to lower temperatures (as extracted from the derivative of the temperature-dependentdata) with increasing field indicating a field-induced rotation of polarization[228, 237] whichcould contribute to the enhancement of π. At the same time, since the same electric fieldcan also enhance the average magnitude of the polarization of the material either by aligningthe polarization within the domains or re-alignment/growth of the domains,[238, 239] thequestion arises as to whether the enhancement of π comes primarily from field-induced po-

86

Pyro

cu

rre

nt (n

A)

Ba0.5

Sr0.5

RuO3

PMN-0.32PT

SrRuO3

SiNx

Pt

NdScO3

t

nei

ciffe

oc

ory

P (

C m

-2K

-1)

μ0

-100

-200

-300

-400

-500

-600

PbZr0.2Ti0.8O3

PMN-0.32PT

b

-300 -200 -100 0 100 200 3000

-400 -200 0 200 400

-100

-50

0

50

100

-300 -200 -100 0 100 200 300


Fo

MP

EC

(

0rε

ε)

π2

-1-1

0

20

40

60

80

100

PbZr0.2Ti0.8O3

PMN-0.32PTd

LiNbO3

0

250

500

750

1000

( yti

vittimr

ep

cirtc

elei

Dr)

0

2

4

6

8

Lo

ss T

an

ge

nt (1

0-2)

-300 -200 -100 0 100 200 300


c

PbZr0.2Ti0.8O3

PMN-0.32PT

ε

a

Figure 5.4: (a) Measured pyroelectric current (rms) for a sinusoidal oscillation of temper-ature with amplitude θFE = 10K at 2 kHz as a function of dc electric field. (b) Extractedpyroelectric coefficient π = iP (AdT

dt)−1 plotted as a function of dc electric field. (c) Variation

of εr (open red circles) and loss tangent (filled red squares) with applied dc electric field.Blue data in b and c show results for a PbZr0.2Ti0.8O3 thin film measured in the same con-figuration. (d) FoMPEC as a function of applied dc electric field. The measured FoMPEC fora PbZr0.2Ti0.8O3 thin film (blue) and literature values for LiNbO3 (black) are provided forcomparison.

87

0 50 100 150 200 250 300

-100

-150

-200

-250

-300

-350 [001]

[100 ]

[010 ]

[001]

[100 ]

[010 ]

[001]

[100 ]

[010 ]

tn

eiciff

eo

c or

yP

(C

m-2

K-1

)μ

π

Temperature ( C)°

E = 0 kV cm-1

a

0 50 100 150 200 250

[001 ]

[100 ]

[010 ]

[001 ]

[100 ]

[010 ]

[001 ]

[100 ]

[010 ]

E = 13.3 kV cm-1

E = 26.7 kV cm-1

E = 40 kV cm-1

-150

-300

-450

-600

-750

Temperature ( C)°

b

Figure 5.5: (a) Pyroelectric measurements conducted under zero-field with an oscillatingac temperature with amplitude θFE = 2.5K as a function of background temperature. (b)Pyroelectric measurements under applied dc electric fields with dashed lines showing the MC

to T transition temperature for various dc fields.

larization rotation, field-induced enhancement of the average polarization, or a combinationthereof?

To separate these effects, π was measured while heating under a dc electric field of40 kV cm−1 (field heating, FH) through the MC to T phase transition to 125◦C (red data,Figure 5.6) and upon cooling to room temperature under application of the same field (fieldcooling, FC) (blue data, Figure 5.6). In this situation, the transition back to the MC phaseis quenched by the applied electric field and the sample remains in the T phase (in effectcontributions to π from polarization rotation would be “turned-off”). To confirm that it isindeed “locked” in the T phase, π was measured again while heating (FH) after the initialFH and FC cycle and, in this case, the MC to T phase transition is not observed confirmingthat the sample remained in the T phase. By comparing the difference between the FH (red)and FC (blue) curves, the contribution to π due to polarization rotation (orange data, Figure5.6) can be quantified at any temperature. For example, polarization rotation contributes amaximum of ∼16% of the total π near the phase transition. Repeating the same experimentat a higher dc electric field (67 kV cm−1) yielded the same fractional contribution frompolarization rotation. In turn, this implies that majority of the large response under biasarises from field-induced enhancement of the average polarization. Ultimately, the combinedtemperature- and field-induced changes in polarization magnitude and direction yield largepyroelectric responses that make this material promising for pyroelectric energy conversion.

88

tn

e iciff

eo

c or

yP

(C

m-2

K-1

)μ

π

25 50 75 100 125

0

-50

-300

-350

-400

-450

-500

Field-induction

Rotation

FH-FC

Temperature ( C)°

E = 40 kV cm-1

c FH1à2

FC2à3

1

2

3

4

FH3à4

Figure 5.6: Pyroelectric measurements under field-heating (FH1→2, red) and -cooling(FC2→3, blue). The difference between the FH and FC (orange) provides a quantitative mea-surement of the contribution of polarization rotation to the net pyroelectric susceptibility.Repeated pyroelectric measurements under field-heating (FH3→4, shaded red circles) retracesthe path of previous field-cooled state confirming that the structure remains quenched in theT phase and does not relax back to the M phase.

5.5 Pyroelectric Energy Conversion using

PMN-32PT Thin Films

Having understood the mechanism behind the large electric-field enhancement ofπ, we proceed to demonstrate the potential of directly converting heat into electrical energyvia pyroelectric Ericsson (or Olsen) cycles. The ideal cycle begins with an isothermal (Tlow)change of electric field (Elow → Ehigh; 1 → 2, Figure 5.7a) that polarizes the system. Thisis followed by an isoelectric (Ehigh) absorption of heat (Tlow → Thigh; 2 → 3, Figure 5.7a).Next, the electric field is isothermally (Thigh) reduced (Ehigh → Elow; 4 → 4, Figure 5.7a).Finally, the heat is expelled isoelectrically (Elow) (Thigh → Tlow; 4→ 1, Figure 5.7a). In ourdevices, this cycle is implemented by applying carefully chosen time-dependent, spatially-uniform temperatures and electric fields to the relaxor ferroelectric. Akin to the pyroelectricmeasurements above, applying a sinusoidally-varying current at 1ω that locally heats theheterostructure via Joule heating (red curve, Figure 5.7b) results in the temperature of theheterostructure varying sinusoidally at 2ω with an amplitude that depends on the magnitude

89

0 100 200 300 400

-3

0

3

6

9

( n

oita

ziral

oP

C c

m-2

)


= 115°CTlow

= 25°C

40 Hz

1

2

3

4

e0

4

8

12

16

20

24

( n

oita

ziral

oP

C c

m-2

)

1 2 =

3 4 =

1

2

3

4

4 1 = 2 3 =

a

μμ

∆

Thigh

ThighT

TlowT

ElowE E

highE

Tlow

Thigh

20

40

60

80

100

-10

-5

0

5

10

0 10 20 30 40 50-300

-150

0

150

300

-60

-30

0

30

60

0

100

200

300

400

500

1

2 3

4

( P

C c

m-2

)μ

∆

He

atin

g

cu

rre

nt (m

A)

E-f

ield

(kV

cm

-1)

Cu

rre

nt (n

A)

Te

mp

(ºC

)

Time (ms)

b

c

d

∆T~5%

Ф

Figure 5.7: (a) Schematic of the pyroelectric Ericsson cycle showing the various isothermaland isoelectric processes. The total energy density per cycle is given by the area of theclosed loop,

∮E.dP (shaded region). (b) Example of the implementation of the Ericsson

cycles wherein the temperature of the heterostructure is sinusoidally varied by ∆T = 70K(blue curve) at 40Hz by applying a sinusoidally varying current (red curve) at 20Hz thatlocally heats the heterostructure via Joule heating. (c) A periodic electric field E(t), phase-synced (delayed by φ) with the heating current is applied across the relaxor ferroelectric tocomplete the cycle. (d) In response to the periodically-varying temperature and electric field,the resulting current (blue curve) is measured and the change in polarization is calculated byintegrating the current (orange curve). (e) Experimentally obtained changes in polarizationas a function of electric field for ∆T = 10 − 90K while keeping Tlow = 25◦C with ∆E =267kV cm−1 at f = 40Hz.

90

of the heating current (e.g., ∆T = 70K; blue curve, Figure 5.7b). A synchronized periodicelectric field E(t) is applied across the heterostructure in a nearly square-wave fashion (Figure5.7c). In our devices, isothermal polarization and depolarization is achieved by switchingthe electric field exactly where the temperature achieves its extrema thereby approximatingsteps 1→ 2 and 3→ 4 of the ideal Ericsson cycle (Figure 5.7a). In other words, the electricfield changes from high to low or vice-versa while the temperature goes through a maximumor a minimum with less than 5% change in the temperature. This was achieved by delayingthe application of electric field by a phase φ with respect to the heating current to account forthe thermal phase lag using the measured phase lag from the 3ω method (Figure 5.3b).Thissmall, but finite, 5% temperature-change window was chosen so as not to change the electricfield using a step function that results in a very large capacitive current saturating our currentto voltage amplifier. In response to the periodically-varying temperature and electric field,the resulting current shows two features, respectively: 1) a background pyroelectric currentand 2) dielectric current spikes (blue curve, Figure 5.7d). The measured total current canthen be numerically integrated to calculate the change in polarization, ∆P (t) (orange curve,Figure 5.7d). However, it should be noted that the electrode area for collecting the dielectriccurrent (Aelectrode) is larger than the area getting heated or the effective area for collectingthe pyroelectric current (Aheater), therefore the net change in polarization is calculated byseparately integrating the current using the respective areas as

∆P (t) =1

Aheater

∫ t

0

ip(t)dt+1

Aelectrode

∫ t

0

id(t)dt (5.1)

To separate the current contribution from the change in capacitance (dielectriccurrent id(t)) versus the pyroelectric current (ip(t)), the heating is turned off (ih(t) = 0)to measure just the dielectric current only. Since the dielectric current does not changedramatically as a function of temperature under large applied field (due to quenched dielectricpermittivity), this dielectric current is subtracted from the total measured current to get thepyroelectric current. Using equation 5.1, ∆P (t) is calculated and further plotting ∆P (t) vsE(t) gives the pyroelectric Ericsson cycle with the cyclic loop (Figure 5.7a).

Using this approach, plots of ∆P (t) vs. E(t) yield a familiar representation ofthe pyroelectric Ericsson cycle such as, for example, how the response changes for variousmagnitudes of ∆T = 10− 90K (maintaining Tlow = 25◦C, ∆E = 267kV cm−1, and cycle fre-quency f = 40Hz; Figure 5.7e). We further conducted pyroelectric energy conversion cyclesat higher frequencies also; 400 Hz (Figure 5.8a) and 1000 Hz (Figure 5.8b). Examination ofdata at all cycling frequencies reveals that upon increasing the magnitude of ∆T , the area ofthe cycle in P−E space increases as more heat is converted into electrical work (

∮E.dP ).The

thin-film geometry of our devices also enables application of both larger electric fields andfaster temperature oscillations as compared to bulk ceramics or single crystals, allowing us toexplore the pyroelectric energy conversion potential across a range of experimental conditions

91

0 100 200 300 400

-3

0

3

6

9

( n

oita

ziral

oP

C c

m-2

)


0 100 200 300 400

-3

0

3

6

9

( n

oita

ziral

oP

C c

m-2

)


400 Hz

1

2

3

4

a

1000 Hz

1

2

3

4

b

μ∆

μ∆

= 92°C

= 25°CTlow

Thigh

Tlow

Thigh

Tlow

Thigh

= 81°C

= 25°CTlow

Thigh

Figure 5.8: Change in polarization as a function of electric field (pyroelectric Ericssoncycle) for (a) ∆T = 27 − 67K while maintaining Tlow = 25◦C with ∆E = 267kV cm−1 atf = 400Hz, and (b) ∆T = 31−56K while maintaining Tlow = 25◦C with ∆E = 267kV cm−1

at f = 1000Hz.

(∆T = 10− 110K, ∆E = 33− 400kV cm−1, and cycle frequency f = 10− 2000Hz). Follow-ing an initial optimization within these parameter ranges, the PMN-0.32PT thin films areshown to provide record-breaking pyroelectric energy conversion, including: 1) The largestenergy density of 1.06 J cm−3 (at ∆T = 90K, ∆E = 267kV cm−1, and f = 40Hz; Figure5.9a) which is made possible because of the large field induced value of π and the abilityto apply large electric fields maximizing the electrical work (

∮E.dP ). 2) The largest power

density of 526 W cm−3 (at ∆T = 56K, ∆E = 267kV cm−1, and f = 1000Hz; Figure 5.9b)which is realized because the power density scales directly with cycling frequency, which canbe increased because the thermal time constant of the thin-film geometry is small. 3) The

largest scaled efficiency of 19% [Thigh∆T

∮E.dP∫ Thigh

TlowC(T )dT

, where (Thigh∆T

)−1 is the Carnot efficiency,∮E.dP is the net work done, and

∫ ThighTlow

C(T )dT is the calculated heat input to the active

material with a temperature-dependent volumetric heat capacity C(T )][240] (at ∆T = 10K,∆E = 267kV cm−1, and f = 40Hz; Figure 5.9c); a significant improvement over the highestreported value of ∼ 5.4% for the same ∆T 15. It should be noted that for the current work,C(T ) is assumed to change only with temperature[241, 242] and is assumed to be constantwith electric field[243] over the range studied herein.

For comparison, the energy outputs from the current work are compared to a num-ber of experimentally-investigated energy and power densities extracted from previous studiesof pyroelectric energy conversion (Figure 5.9d). These works employed a variety of thermal

92

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0yti

sn

ed

ygr

en

E(J

cm

-3)

T (K)

0 20 40 60 80 1000

200

400

600

mc

W(

r d

en

sity

ew

oP

-3)

T (K)

0 20 40 60 80 1000

5

10

15

20

(%

)

T (K)

a b

c

40 Hz400 Hz

1000 Hz

40 Hz400 Hz

1000 Hz

40 Hz400 Hz

1000 Hz

∆∆

∆

Sca

led

effic

ien

cy

This work

PMN-0.32PTF

BaTiO3F

PLZTB

PZN-PTC

P(VDF-TrFE)F

PZSTC

PZN-PTC

PMN-PTC

mc W(

r densityewoP -3)

ytis

ne

dy

gre

nE

(J c

m-3

)

0.01 0.1 1 10 100 10000.0

0.2

0.4

0.6

0.8

1.0 d

Figure 5.9: (a) Energy density, (b) power density, and (c) scaled efficiency extracted fromvarious pyroelectric Ericsson cycles as a function of cycle frequency [40 Hz (blue), 400 Hz(orange), and 1 kHz (red)] and ∆T while keeping Tlow = 25◦C with ∆E = 267kV cm−1. (d)Comparison of the current work with experimentally-achieved energy and power densitiesfrom a number of prior studies of pyroelectric energy conversion using thermodynamic cy-cles. The superscripts B, C, and F correspond to bulk-ceramic, single-crystal, and thin-filmsamples, respectively.

cycling methods, from pumping hot/cold fluids to immersing the pyroelectric material alter-nately in hot and cold thermal baths,to study bulk-ceramic (Pb1xLax(Zr0.65Ti0.35)1x/4O314),single-crystal (Pb0.99Nb0.02(Zr0.68Sn0.25T 0.07)0.98O317, Pb(Zn1/3Nb2/3)0.955Ti0.045O3,[93, 94]Pb(Mg1/3Nb2/3)0.68T 0.32O3),[95] and thin-film (P(VDF-TrFE),[96–98] BaTiO3)[99] samples.As compared to the current work, the ultimate performance of these prior studies was lim-ited by aspects related either to the materials themselves (i.e., they had intrinsically lowpyroelectric coefficients) and/or to the functional form of the material (i.e., inability to ap-ply large electric fields and slow thermal cycling). In the current work, we have created afield-tuned, intrinsically-large pyroelectric coefficient in conjunction with a thin-film devicegeometry. This thin-film geometry allows a number of important advantages over work inbulk materials, including: 1) significantly larger sweeps of electric field (achieved with muchlower voltage (V ) in comparison to bulk materials) thus maximizing field-induced polariza-

93

tion changes and work, 2) high-frequency, high-temperature-amplitude cycling to increasepower and work, respectively; and 3) significantly less heat input (Qin) is required to increase

the temperature of the lattice (since it scales with volume (∀) as Qin = ∀∫ ThighTlow

C(T )dT )while still driving the same changes in the surface charge density with the applied volt-age (V), which is equivalent to the electrical work done (which scales with area (A) asW = ∀

∮E.dP = A

∮V.dP since electric field is simply due to the applied voltage). Thus,

said another way, the advantage of doing pyroelectric energy conversion with thin films isthat the effects scale with just the area of the device, not the volume. Ultimately, this placespyroelectric thin films (here corresponding to an effective ZT = 1.16) as a very promisingcandidate for low-quality waste heat energy harvesting, on par with thermoelectrics.

5.6 Conclusions and outlook

Large | π |> 550µCm−2K−1 under an application of dc electric field is obtainedin PMN-0.32PT thin films owing to a strong field-induced enhancement of the average po-larization and polarization rotation. The same electric field also suppresses εr by ¿70%which, in turn, enhances the FoMPEC by ∼ 5 times in comparison to zero-field values andstandard pyroelectric materials. Ultimately, implementation of solid-state pyroelectric Er-icsson cycles yields a maximum energy density, power density, and scaled efficiency of 1.06J cm−3, 526 W cm−3, and 19%, respectively; the highest values reported to date for pyro-electrics. The implications of this work are multifold: First, these results suggest that therecould be novel routes to enhance pyroelectric response in materials. Much as field-inducedpolarization enhancement and rotation were used herein to augment the response, otherfield-induced routes to large effects should also be considered (e.g., field-induced domain andphase structure evolution, field-induced defect reorientation and motion, etc.). This could,in turn, greatly expanding the number of candidate materials for pyroelectric energy conver-sion. Second, this work reveals the importance of expanding the study of pyroelectric energyconversion in thin-film versions of materials since this process scales inherently with materialarea, not volume. At the same time, thin films permit operation at both high electric fieldsand thermal oscillation frequencies which can further enhance energy and power outputs.Finally, this work suggests that pyroelectric energy conversion can potentially compete withthermoelectrics, in particular, for energy harvesting from low-grade waste heat.

94

Chapter 6

Electrothermal Effects in NovelFerroelectric Materials - HfO2

In this Chapter, I investigate the pyroelectric and electrocaloric response in ultra-thin films of Si-doped HfO2 that have recently been shown to exhibit ferroelectricity. Zero-field pyroelectric response confirms that the material does exhibit broken-inversion symmetry.Further, field-dependent pyroelectric measurements reveal electrical-switching of polarizationsupporting the existence of ferroelectricity in these thin films of Si-doped HfO2. In addition,the pyroelectric response is found to exhibit a wake-up behavior like the wake-up phenomenonobserved in the ferroelectric polarization with electric-field cycling. From the measurement ofthe polarization-electric field hysteresis loops, this wake-up phenomenon is attributed to thepresence of defect dipoles which explains the measured pyroelectric response. Finally, directelectrocaloric measurements are on these Si-doped HfO2 thin films reveal an electrocaloriccoefficient ∼4 times larger in magnitude as compared to the pyroelectric coefficient. Thisenhancement is explained using the plausible role played by defect dipoles that contributeto additional configurational or dipolar entropy.

95

6.1 Introduction

While research continues to mount evidence to prove the existence of intrinsic fer-roelectricity in the simple binary oxide HfO2, pyroelectricity, the requisite property thatconfirms a polar phase in which ferroelectricity can occur has remained largely unexplored.Probing the temperature-dependent changes in the spontaneous polarization (the pyroelec-tric effect or PEE) would therefore, not only prove if the point-group symmetry of the crystalpermits ferroelectricity,[244] but also gauge the extent of the susceptibility of the polarizationto any change in the temperature. Prior reports[245] using indirect measurement techniqueshas claimed the existence of an exorbitantly large PEE in Si-doped HfO2 attributed to atemperature-driven phase transition, while more recent work using a phase-sensitive detec-tion technique[24, 25] on Zr-doped HfO2[246] and Si-doped HfO2[247, 248] has shown thatPEE in these simple fluorite structures is rather modest. Furthermore, since any pyroelec-tric material also exhibits the electrocaloric effect (ECE), there is growing interest to explorehow much an electric field can perturb the dipolar entropy and therefore the temperaturein this CMOS-compatible system. Recent work on Zr-doped HfO2 thin films, using indirectmeasurement techniques (which are often misleading as they are readily confounded by spu-rious contributions in thin films),[163] have also suggested the existence of a large ECE.[249]Despite this observations, however, a plausible mechanism for the large values has remainedelusive calling into question the veracity of such reports. A detailed study, leveraging theextensive infrastructure developed in this thesis, is therefore essential to develop a deeperphysical insight into the electrothermal response and to explore the potential for pyroelectricenergy conversion[91, 250] and electrocaloric solid-state cooling[14] in this new lead-free andCMOS-compatible ferroelectric.

HfO2 has various polymorphs in the bulk depending on the pressure and temper-ature. Under atmospheric pressure, temperature-induced phase transitions from monoclinic(P21/c) to tetragonal (P42/nmc) and finally to a cubic (Fm3m) phase occur at 1973 and 2773K, respectively.[251] All these phases possess a center of inversion symmetry and are thereforenon-polar. A polar orthorhombic phase (Pca21), energetically very close to the non-polarmonoclinic phase at room temperature,[252] can however be stabilized under different me-chanical[253–255] and chemical environments. First stabilized with Mg-doping,[256] differentdopants including Al,[257] Gd,[258] La,[259] Si,[260] Sr,[261] and Y[262] have been shown tostabilize ferroelectricity in HfO2-based systems. In the case of Si-doped HfO2, the ferroelec-tric polarization, however, requires a so-called “wake-up” process[263] wherein an initiallyconstricted ferroelectric hysteresis behavior relaxes and the remnant polarization enhancesafter electric-field cycling.[264] This is analogous to the phenomena observed in BaTiO3 singlecrystals[265, 266] and PbZr1−xTixO3 ceramics,[267, 268] where point defects (vacancies) canoccupy energetically favorable sites in the bulk lattice,[265, 266] domain walls,[269] or grainboundaries[270] and form defect dipoles or complexes that create a restoring force favoringa certain direction of spontaneous polarization. Recent work on thin films of Si-doped HfO2

96

Selective deposition of Pt

Blanket deposition of SiNx &

selective SF6 reactive ion etch

Selective deposition of Pt

SF6 reactive ion etch

(a) (f)

Si

TiN (bottom electrode)

TiN (top electrode)

SiNx

Pt

Pt

Si:HfO2

(b)

(c)

(d)(e)

Figure 6.1: Schematic illustration showing the fabrication of the electrothermal devicethrough the various processing steps (a-e). (f) Cross-section view of the final device showingthe various layers in the heterostructure. .

has reported the presence of defect dipoles due to non-homogeneously-aggregated oxygen va-cancies that can cause the as-grown ferroelectric domains to preferentially polarize in order

to screen the internal bias field.[271] Since, the pyroelectric susceptibility[πi =

(∂〈Pi〉∂T

)E

]strongly depends on the net polarization [〈Pi〉], any change in 〈Pi〉 due to the presenceof defects can significantly alter the pyroelectric response. Furthermore, since the defectscan alter the permissible configurational states for the polarization,[272, 273] the additionaldipolar entropy change can constitute a novel mechanism for electrocaloric susceptibility[Σi =

(∂S∂Ei

)T

].

In this work, we investigate the PEE and ECE in 10-nm-thick films of Si-dopedHfO2 (Si:HfO2). High-frequency localized heating was employed to measure the pyroelectriccoefficient (π) directly using phase-sensitive detection (Chapter 3).[36] By systematicallywaking-up the polarization using bipolar electric field cycling, we investigate the evolutionof π. By measuring the wake-up-cycle-dependent ferroelectric-electric field hysteresis loops,the presence of defect dipoles in thin films of Si:HfO2 is established. In turn, the roleof defect dipoles in effecting π is illustrated. Finally, direct measurements are conducted tomeasure the electrocaloric susceptibility (Σ) for the first time in these ultrathin Si:HfO2 films.Comparison between π and Σ reveals a significant deviation from the Maxwell relations (Σis larger than ∼4 times larger than expected for the measured value of π), suggesting thatdefect dipoles/complexes can potentially contribute to the high-field electrocaloric responsethrough field-driven rearrangement.

97

6.2 Device Fabrication

Electrothermal devices were fabricated using the as-received rapid-thermal-annealed 10 nm TiN/10 nm 5.88 mole% Si-doped HfO2/ 10 nm TiN heterostructure ondoped Si wafers (see footnote; Figure 6.1). Briefly, 30 nm platinum was first sputtered onthe as received metal-ferroelectric-metal tri-layer heterostructure (Figure 6.1a) after litho-graphically patterning a 30020 µm2 rectangular top electrode (Figure 6.1b) to enhance theelectrical conductivity of an otherwise poorly conducting TiN top electrode. The top TiNwas etched away (using SF6 plasma-based reactive ion etching) using the above platinum asa hard mask from everywhere except the active device (30020 µm2 rectangular area) to definethe ferroelectric capacitor geometry with the bottom TiN and silicon wafer serving as thebottom electrode and the platinum on the top TiN serving as the top electrode (Figure 6.1c).A 200-nm-thick blanket layer of SiNx was deposited using plasma-enhanced chemical vapordeposition (SiH4 + NH3 based) and was selectively etched (using SF6 plasma-based reac-tive ion etching) leaving an electrically-insulating layer on the ferroelectric capacitor (Figure6.1d). Finally, a 50-nm-thick layer of platinum was sputtered to define the thermal charac-terization circuit (akin to a 4-point probe pattern) and the contact pad to access the topelectrode (Figure 6.1e). Note that the thermal characterization circuit does not electricallyshort with the capacitor circuit underneath (Figure 6.1f).

6.3 Understanding the Pyroelectric Effect in

Si-doped HfO2 Thin Films

6.3.1 Measurement of Pyroelectric Susceptibility

To probe the pyroelectric response in the Si-doped HfO2 heterostructures, a 10mA (rms) sinusoidal heating current was sourced into the microfabricated platinum heatingline. 3ω-measurements were conducted to estimate the resulting temperature change in theferroelectric (Si-doped HfO2) (Figure 6.2a). Since the substrate (doped-silicon wafer) is elec-trically conducting, it is necessary to ensure that there is no electrical cross-talk between thethermal circuit and the top/bottom electrode of the ferroelectric capacitor. Therefore, ad-ditional experiments were conducted to check if the response at the 3rd harmonic frequencyscales with the cube of current[173] and that the calculation of the amplitude of the oscil-lating temperature change is accurate. It can be seen that the response at the fundamentalfrequency (1ω) scales linearly (Figure 6.2b) and the one at the 3rd harmonic (3ω) scales asper the cubic law (Figure 6.2c) above 1 kHz. Henceforth, the pyroelectric measurementsreported in this Chapter are conducted using a 10 mA (rms) heating current at a frequency

98

1000 10000

-0.001

0.000

0.001

3ω

Vrm

s


1000 10000-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1ω

Vrm

s


10 100 10000.0

0.2

0.4

0.6

0.8

1.0

θr

eta

eH

(K

)


-90

-45

0

45

90

Th

erm

al p

ha

se

la

g, ф (o)

10 mA

12.5 mA

15 mA

17.5 mA

In-phase

Out-of-phase

In-phase

Out-of-phase

10 mA

12.5 mA

15 mA

17.5 mA

(a) (b) (c)

Figure 6.2: (a) Measured temperature oscillation (θHeater) and the thermal phase lag (φ)between the heating current and the temperature change in the heater line as a function offrequency. (b) Scaling of the 1st harmonic and (c) 3rd harmonic voltage signal with varyingthe heater current as a function of heating frequency.

of 1 kHz (1ω). This heating current results in a temperature oscillation of 0.61 K (rms) at 2kHz (2ω) as calculated using the measured value of the temperature coefficient of resistance.

Subsequently, π is calculated using π = ip(AdT

dt

)−1where ip is the measured pyroelectric

current and A is the area of the sinusoidally-heated cross-section of the device.

Polarization-electric field hysteresis loops were first measured on the as-grown de-vice (having subjected the heterostructure to zero electric-field cycles or a DC bias) using aPrecision Multiferroic Tester (Radiant Technologies) that employs a virtual ground method.The ferroelectric hysteresis loops reveal a distinct “pinched” characteristic (Figure 6.3a) ashas been observed in the prior reports on Si-doped HfO2 thin films.[260, 263, 274, 275] Thispinched or constricted hysteresis loop has often been attributed to antiferroelectric behaviorin Si-doped HfO2 thin films[274, 276–278] in contradiction to the proposed explanations at-tributing the double-hysteresis behavior to the defect-induced internal bias fields.[279] Thepresence of either an antiferroelectric behavior or defect-induced pinning/back-switching offerroelectric domains should manifest in the pyroelectric response as well. Antiferroelectricordering should result in a zero value of π under zero bias while back-switching of ferroelec-tric domains due to defect-pinning should result in a reduction of π below the coercive field.To investigate this phenomenon, pyroelectric measurements were conducted while sweepingthe background DC electric field. It can be seen that the magnitude of both ip (Figure 6.3b)and π (Figure 6.3c) under zero DC electric field (shown as a shaded red circle and markedas the starting point of the field sweep using a black arrow) is non-zero confirming that thepinched hysteresis loop is not due to antiferroelectricity in the material. As the magnitudeof the DC electric field in increased, the magnitude of both ip (Figure 6.3b) and π (Figure6.3c) increases in comparison to the zero-field response. This field-induced enhancement inπ is possibly due to the increase in the net polarization due to 180◦ switching of ferroelectricdomains. It should, however, be noted that the zero-field response of π after saturation undera high-bias does not trace back the original path (compare ip and π under zero electric field

99

-4000 -2000 0 2000 4000

-30

-20

-10

0

10

20

30

( n

oita

ziral

oP

μC

cm

-2)

Electric Field (kV cm-1)

1 kHz

(a)

-3000 -2000 -1000 0 1000 2000 3000-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

)A

n( tn

erru

cor

yP

DC Electric Field (kV cm-1)

-2000 -1000 0 1000 20000

-5

-10

-15

-20

-25

-30

-35

,tn

eicif f

eo

co r

yP

π (μ

C m

-2 K

-1)

DC Electric Field (kV cm-1)

(b) (c)

Figure 6.3: (a) P-E measurements showing a pinched or constricted hysteresis loop. (b)Pyroelectric current (ip) and (c) pyrocoefficient (π) measured as a function of the backgroundDC electric field, at 1 kHz. The black arrows in the figures show the starting point of themeasurement.

with the starting values shown as a shaded red circle). This suggests that the switched fer-roelectric domains do not completely switch back after the electric-field is removed, in-turn,suggesting that the polarization in the ferroelectric has progressively woken-up as the biasis applied.

The field-dependent pyroelectric measurements additionally enable the quantifica-tion of the contribution from the temperature-dependent change in the relative dielectricpermittivity (εr). Under a background DC electric field (E) applied along the direction ofpolarization (z direction), π is enhanced by additional contributions which can be mathe-matically expressed as

π =∂P

∂T=∂Pr∂T

+ ε0Eεr∂T

(6.1)

where P is the net polarization under an electric field and Pr is the remanent po-larization. Depending on the sign of ( εr

∂T), the contribution from the temperature-dependent

change in εr can either enhance or diminish the π. This contribution is particularly im-portant for pyroelectric energy conversion where the thermodynamic cycles usually involvetemperature changes under electric field. From the current work, it can be seen, that underan applied electric field the magnitude of ip (Figure 6.3b) and π (Figure 6.3c) first increasesto a maximum and then starts to diminish. The initial increase is due to an increase in thedegree of alignment of the ferroelectric domains. Once the domains are aligned, increasein the electric field results in the reduction of π due to the competing contribution fromthe (ε0E

εr∂T

) term which is mathematically positive (permittivity increase with temperature)while the (∂Pr

∂T) term which is mathematically negative.

100

-4000 -2000 0 2000 4000

-30

-20

-10

0

10

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30

( n

oita

ziral

oP

μC

cm

-2)

Field (kV cm-1)

200k cycles2k cycles

2 cycles

1 10 100 1000 10000 1000000

-5

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,tn

e ic iff

eo

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π (μ

C m

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Number of wake-up cycles

(a) (b)

100 1000 100000

-5

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, tn

eici ff

eo

co r

yP

π (μ

C m

-2K

-1)

Time (sec)

2 cycles

2k cycles

200k cycles

(c)

Figure 6.4: (a) P-E measurements conducted after cycling the device for 2 (red), 2k (yellow)and, 200k (blue) cycles. (b) Measured pyroelectric coefficient (π) as a function of the numberof wake-up cycles. The red dashed-line shows an empirical logarithmic fit to the measureddata. (c) Pyroelectric-retention measurements conducted on samples after for 2 (red), 2k(yellow), and 200k (blue) wake-up cycles.

As shown earlier (Figures 6.3b and 6.3c), the application of a E enhances π due to anincrease in P . Since the polarization can be woken-up with electric-field cycling, pyroelectricmeasurements were conducted as a function of the number of the wake-up cycles. To achievea systematic increase in Pr the devices were field-cycled using a sinusoidal electric field withan amplitude of 4 MV cm−1 at a frequency of 1 kHz. The ferroelectric hysteresis loopsfor the as-grown state and after 2,000 and 200,000 cycles are shown (red, yellow, and bluecurves, respectively, Figure 6.4a). Immediately after the wake-up, π was measured (Figure6.4b). It can be seen that both Pr and π increase with the number of electric-field cycles;Pr increases from ∼5 µC cm−2 (red data, Figure 6.4a) to ∼10 µC cm−2 (blue data, Figure6.4a) and π increases from ∼-13 µC m−2 K−1 to ∼-27 µC m−2 K−1 (dashed line shows anempirical logarithmic dependence,[263] Figure 6.4b). Both the pinched hysteresis loop andan imprinted hysteresis loop (as seen in Figure 6.4a) are signatures of the presence of defectdipoles in oxide ferroelectrics.[265, 280] Therefore, a possible explanation to the observedferroelectric and pyroelectric susceptibility with field-cycling in the current work can berelated to the presence of defect dipoles.

6.3.2 Role of Defect Dipoles on Ferroelectric and PyroelectricSusceptibility

In general, a defect dipole is a reference to any defect complex formed out ofoppositely-charged point defects in a lattice that couple such that their association reducesthe free energy of the system.[281, 282] In ferroelectrics, such defects have been reported tobe comprised of some acceptor defect (e.g., an oxygen vacancy pair)[264] or a donor defect

101

(e.g., a cation vacancy pair).[280] Recent work on Si-doped HfO2 has revealed the presence ofcharged oxygen vacancies[283] and the possibility of the formation of a defect dipole/complexdue to an interfacial chemical reaction between the HfO2 and TaN (or TiN).[271] The ob-served ferroelectric switching response can, therefore, be interpreted by considering a scenariowhere the defect dipoles (with a net polarization PD) contribute to the overall polarization(Pr = Pi + PD) where Pi is the intrinsic polarization of Si-doped HfO2. Since the dipolarinteraction between the defect dipoles and the intrinsic polarization in the lattice energeti-cally favors a parallel alignment of PD and Pi,[265, 280, 281] a much smaller measured valueof Pr in the as-grown state (red data, Figure 6.4a) indicates that the defect dipoles have anear-equal probability of pointing in the up- and down-direction. Consequently, a certainfraction of Pi within the range of interaction with the defect dipole remains coupled or prefersto remain parallel-aligned with PD.

During a single ferroelectric switching process, the defects do not switch due to theirslower migration kinetics. In turn, they provide a restoring force to the adjacent intrinsicpolarization that had switched under the applied electric field, to back-switch once the fieldis removed causing a pinched hysteresis loop.[265] The near-equal distribution of the up-and down-pointing defect dipoles causes the net built-in potential across the ferroelectric tobe zero and therefore a negligible horizontal offset or imprint (red data, Figure 6.4a). Afterrepeatedly electrically cycling the ferroelectric, however, the as-grown defect orientationcan be altered. The imprint in the measured ferroelectric hysteresis loops after 200,000cycles (blue data, Figure 6.4a) is indicative of a net non-zero built-in potential across theferroelectric suggesting that the probability of the up- and the down-pointing defect dipoleis no longer near-equal and that the defect dipoles are now largely pointing in a singlepreferred direction. A positive horizontal offset along the electric-field axis indicates thatunder zero applied electric-field, an up-pointing or −Pr-state is favored. Back-switching nowhappens after poling the ferroelectric in the +Pr state only. Consequently, the pinching inthe ferroelectric hysteresis loop is reduced. The pyroelectric susceptibility, likewise, showsan increase as Pr increases (Figure 6.4b).

To check the stability of the pyroelectric response after waking-up the polarization,pyroelectric-retention measurements were conducted. Here, after poling the ferroelectricwith a bipolar triangular pulse after a varying number of wake-up cycles, π was measured asa function of time. It can be seen that π decreases with time for all cases of wake-up cycling(2, 2,000, and 200,000 cycles shown in red, yellow, and blue color, respectively, Figure 6.4c).This suggests that the up-pointing alignment of the defect dipoles slowly relaxes back to thestate where some of the defect dipoles have flipped to be down-pointing.

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Film/Substrate Thermal conductivity Specific heat capacityk (Wm−1K−1) C(T ) (Jm−3K−1)

Si:HfO2 1.0 a 2.76× 106 b

TiN 2.7 c 3.47× 106 c

SiNx 2.0 d 1.16× 106 e

Si 120 f Assumed as 1.6× 106 f

Table 6.1: Thermophysical properties (at 300 K) of the various thin films and substratesused in the thermal transport calculations. a Ref. [284]; b Ref. [285]; c Ref. [286]; d Ref.[36]; e Ref. [207]; f Ref. [287]

6.4 Understanding the Electrocaloric Effect in

Si-doped HfO2 Thin Films

6.4.1 Measurement of Electrocaloric Susceptibility

With an understanding of PEE, we proceed to investigate ECE in Si-doped HfO2

thin films. Using the same device structure as for measuring the PEE, ECE measurementswere conducted by applying an AC electric-field to measure the electrocaloric-temperaturechange. Briefly, a sinusoidally varying electric-field was applied across the ferroelectric het-erostructure at 98.147 kHz. In response to the AC electric-field perturbation, the resultingsinusoidal temperature oscillation in the ferroelectric was measured using an AC sensing cur-rent with a magnitude of 2 mA at a frequency of 2.317 kHz applied across the top line. Theelectrocaloric response (manifested as an AC voltage) was measured at 98.147-2.317=95.830kHz. Using the temperature coefficient of resistance, the AC temperature change in the topmetal line (θSensor) was calculated and subsequently used to calculate Σ after solving the1-D heat transport model using the thermophysical properties of the heterostructure (Table1).

Beginning with the as-grown heterostructures (exhibiting ferroelectric hysteresisloop response shown as red data, Figure 6.4a), θSensor was measured as a function of increas-ing AC electric-field perturbation. The maximum bipolar electric field applied to perturbthe entropy of the system was limited to 1.25 MV cm−1. This was done for two reasons; 1)to ensure that the ferroelectric does not switch and, 2) to ensure that the polarization in theferroelectric is not woken-up by applying a bipolar electric field. It can be seen that θSensorscales linearly with the applied electric-field (red squares, Figure 6.5a). Using the thermaltransport model (see Appendix B), Σ was calculated (shaded red circles, Figure 6.5b). Next,the sample was cycled with electric field 2,000 times to wake up the polarization (see fer-

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0.7 0.8 0.9 1.0 1.1 1.2 1.3

1.5

2.0

2.5

3.0

θS

en

so

r)

Km(

Electric field magnitude (MV cm-1)

0

-20

-40

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-140

EC

E C

oe

ffic

ien

t Σ

(μ

C m

-2K

-1)

1 10 100 1000 10000 1000000

-10

-20

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-130

-140

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eiciff

eo

cor

yP

π (μ

C m

-2K

-1)

Number of wake-up cycles

0

-20

-40

-60

-80

-100

-120

-140

EC

E c

oe

ffic

ien

t, Σ

(μ

C m

-2K

-1)

(a) (b)

200k cycles2k cycles

2 cycles

Figure 6.5: (a) Measured temperature change of the top metal line (θSensor) and the calcu-lated value of the electrocaloric coefficient (Σ) on samples after for 2 (red), 2k (yellow), and200k (blue) wake-up cycles. (b) Comparison of Σ with π as a function of number of wake-upcycles.

roelectric hysteresis loop shown in yellow, Figure 6.4a). Similar to the as-grown sample,θSensor scaled linearly (yellow squares, Figure 6.5a) with the applied AC electric field whileΣ remained field-independent (shaded yellow circles, Figure 6.5a). Finally, the same samplewas subjected to 200,000 electric field wake-up cycles (see ferroelectric hysteresis loop shownin blue, Figure 6.4a) and θSensor (blue squares, Figure 6.5a) and Σ were measured (shadedblue circles, Figure 6.5a). It can be seen that Σ does not change with repeated electric-fieldcycling.

In general, for ferroelectrics with 180◦ domain structures, the domains where thespontaneous polarization is aligned with the electric field exhibit conventional ECE (applica-tion of field reduces the dipolar entropy or increases the lattice temperature). The domainswhere the polarization is anti-aligned with the applied electric field exhibit an inverse ECE(application of field increases the dipolar entropy or decreases the lattice temperature).[36]In the current scenario, the as-grown sample exhibits a reduced value of Pr suggesting thepresence of near-equal magnitude of up-pointing and down-pointing polarization in the sam-ple. Therefore, the ECE for such a case should exhibit a much lower value than measured.As such, ECE should have increased with the number of wake-up cycles, but in the currentcase ECE remains independent of the number of wake-up cycles. This is in sharp contrastto π which steadily increases as Pr increases (Figure 6.5b). Additionally, the magnitude ofΣ is ∼4 times larger than what is expected for the measured value of π. From the Maxwellsequivalence of π and Σ under a constant-stress boundary condition, and a higher magnitudeof π than Σ for mechanically clamped thin films,[35] the results (Figure 6.5) suggest thatthe ECE very likely involves additionally contributions to the change in the entropy due to

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the presence of defect dipoles.

6.4.2 Role of Defect Dipoles on Electrocaloric Susceptibility

A plausible explanation to the enhanced ECE in the presence of defect dipoles isexplored here. First, in the case of an equal distribution of up- and down-pointing defectdipoles (as in the case of the as-grown heterostructures with pinched ferroelectric hysteresisloops, red data in Figure 6.4a), the intrinsic polarization accesses more configurations thana homogeneously-poled ferroelectric, thus having a higher entropy. Under the applicationof an electric field, the intrinsic polarization is poled along the direction of the appliedelectric-field thus reducing the dipolar entropy. As the field is removed/reversed (withouta net switching of the ferroelectric), the back-switching of some fraction of the intrinsicpolarization makes the system configurationally disordered again resulting in a higher ECEin comparison to a case where the polarization would not have back-switched. Thus, inpresence of defect dipoles with a propensity to point equally in both the directions, theentropy under a zero-field state is higher than the case if there was no additional disorderimposed by the defect dipoles. Next, when the defect dipoles are largely aligned to pointin the same direction after the wake-up cycles (in the case of the heterostructure with anunpinched, but imprinted ferroelectric hysteresis loop, blue data in Figure 6.4a), applicationof electric field again aligns the intrinsic polarization along the direction of applied electricfield. As the electric field is removed/reversed, however, a significant fraction of the intrinsicpolarization back-switches again making the dipolar state more disordered than a scenariowhere there exists no back-switching. Thus, in both the cases, the presence of defect dipolesprovides more configurations in the zero-field state causing additional entropy changes.

6.5 Conclusion

To summarize, the pyroelectric and electrocaloric response was investigated in 10-nm-thick Si-doped HfO2 films. Pyroelectric measurements support the existence of fer-roelectricity which was evident from the switching of polarity of the pyroelectric current.Field-dependent pyroelectric measurements additionally provided the magnitude of contri-bution from the temperature-dependent change in the dielectric permittivity. By measuringthe ferroelectric susceptibility as a function of wake-up cycles, the presence of defect dipoleswas suggested. In turn, the wake-up phenomenon exhibited in the pyroelectric response wasexplained. Finally, direct electrocaloric measurements were conducted on HfO2-based thinfilms for the first time. The measured values of the electrocaloric coefficient were found tobe ∼4 times larger in magnitude in comparison to its thermodynamic-converse pyroelectric

105

coefficient. The enhancement in the ECE was explained using the plausible role played bydefect dipoles that contribute to additional configuration or dipolar entropy.

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Chapter 7

Summary and Suggestions for FutureWork

In this Chapter, I will summarize the key results from all the Chapters in thisdissertation. Building off of the work in the previous Chapters, I will discuss the possibleways in which the current work can be broadly extended into a few important studies.

7.1 Summary of Findings

7.1.1 Summary of Chapter 1

Chapter 1 served as an introduction to the major themes of this dissertation. Be-ginning with a systematic outline of crystal symmetry, a class of polar materials called py-roelectrics was formally introduced. The physical phenomena exhibited by these materialsconstitute the core of this dissertation. The thermodynamic converse of PEE, ECE, was thenintroduced. The various secondary contributions due to cross-coupled ferroic susceptibilitieswere then discussed. This was followed by an in-depth discussion of how electrothermaleffects have been/are measured in materials with an emphasis on what makes the character-ization of such effects extremely challenging in the case of thin films. The nature of both theelectrothermal susceptibilities was then discussed in case of ferroelectrics where these effectsare typically stronger in comparison to dielectric materials exhibiting pyroelectricity. Thediscussion then led itself into the possible routes and pathways by which electrothermal sus-ceptibilities can be enhanced. Starting with the nature of electrothermal susceptibilities inthe vicinity of ferroelectric-to-paraelectric phase transitions, various extrinsic contributionsthat can enhance PEE and ECE were introduced. This included a discussion of the struc-tural phase transitions in the ferroelectric phase where either rotation of polarization within

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a domain or reconfiguration of domain structures under an application of either temperatureor electric field can yield significant enhancement in the electrothermal susceptibilities. Fi-nally, the technological relevance of these materials was discussed with major focus on twobroad, but related applications, pyroelectric energy conversion and electrocaloric cooling.


Chapter 2 introduced the synthesis and structural characterization of complex-oxide thin films of the type predominantly used in this dissertation. Using the prototypicalferroelectric PbZr0.2Ti0.8O3 as an example, pulsed-laser deposition was introduced as a non-equilibrium growth technique to synthesize epitaxially-strained version of functional oxidethin films. In particular, this Chapter introduced the growth process while highlighting theobservation of strain-induced growth instabilities and nanoscale-surface patterning. Struc-tural characterization techniques like scanning-probe force microscopy, X-ray diffraction,and TEM, key to the characterization of the structure of ferroelectric thin films, were intro-duced. Using scanning-probe force microscopy, a growth-mode transition from 2D-step flowto self-organized, nanoscale 3D-island formation in PbZr0.2Ti0.8O3/SrRuO3/SrTiO3 (001)heterostructures was observed. With an understanding of the strain relaxation measuredusing X-ray RSMs it was found that with increasing film thickness more dislocations werenucleated causing more ripples on the surfaces. Using TEM, the formation of misfit dislo-cations at the buried film-substrate interface was confirmed. Using linear elastic theory, itwas shown that the array of misfit dislocations modulated the strain field that altered theadatom binding energy resulting in inhomogeneous adsorption kinetics causing preferentialgrowth at certain sites.


Chapter 3 introduced a direct technique developed for measuring PEE and ECE inepitaxial thin films. Here, an electrothermal-test platform was demonstrated wherein local-ized high-frequency periodic heating and highly sensitive thin-film resistance thermometryallow for the direct measurement of pyrocurrents (< 10 pA) and electrocaloric temperaturechanges (< 2 mK) using the “2-omega” and an adapted “3-omega” techniques, respectively.Frequency-domain, phase-sensitive detection permitted the extraction of the pyrocurrentfrom the total current, which is often convoluted by thermally-stimulated currents. Thewide-frequency-range measurements employed in this Chapter also illustrated the effect ofsecondary contributions to pyroelectricity due to the mechanical constraints imposed by thesubstrate. Similarly, measurement of the ECE on the same device in the frequency domainallowed for the decoupling of Joule heating from the ECE. A one-dimensional, analytical

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heat-transport model was developed to characterize the transient temperature profile of theheterostructure to extract π and Σ.


Chapter 4 investigated the role of epitaxial strain and, in turn, the effect offerroelastic-domain structures on the electrothermal (both PEE and ECE) and electrome-chanical susceptibility in thin films of the canonical tetragonal ferroelectric PbZr1−xTixO3

system. Using a combination of scanning-probe-based microscopy and X-ray diffraction,a systematic evolution of ferroelastic domain structure with epitaxial strain was observed.Phase-field models predicted the same structures as observed experimentally and additionallyprovided a detailed insight into the nature of competing energies (Landau, elastic, electric,and gradient) that lead to the formation of a region of negative curvature in the free-energyand strain landscape. It was found that in this region of negative curvature, the ferroelectricdomain structures exhibit a strain-induced spinodal instability that results in spontaneouspartitioning of epitaxial strain into competing c/a and a1/a2 domain structures. Thesecompeting domain structures were found to exhibit a large extrinsic contribution to theelectrothermal susceptibility where a continuous domain interconversion with respect to ei-ther temperature and electric field provided previously unexplored ways to enhance PEE andECE in ferroelectric thin films. Further, the energetically-degenerate states were leveraged toachieve large field-driven electromechanical response. Using band-excitation piezoresponseforce microscopy, a facile interconversion of different ferroelastic variants was acousticallydetected. The ferroelastic switching process was found to exhibit a two-step, three-stateswitching (out-of-plane polarized c+ in-plane polarized a out-of-plane polarized c− state)which was concomitant with large non-volatile electromechanical strains ( 1.25%). It was fur-ther demonstrated that deterministic, non-volatile writing/erasure of large area patterns ofthis electromechanical response is possible thus showing a new pathway to improved functionand properties.


Chapter 5 explored pyroelectric energy conversion from low-grade thermal sourcesthat exploited strong field- and temperature-induced polarization susceptibilities in the re-laxor ferroelectric 0.68Pb(Mg1/3Nb2/3)O3-0.32PbTiO3 thin films. Electric-field-driven en-hancement of pyroelectric response (as large as -550 µC m−2 K−1) and suppression of di-electric response (by 72%) were found to yield large figures of merit for pyroelectric energyconversion. Field- and temperature-dependent pyroelectric measurements revealed a quan-titative measure of the maximum contribution from polarization rotation. In turn, themeasurements revealed that the enhancement from the zero-bias pyroelectric response under

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a background DC bias arose primarily from the field-induced enhancement of the averagepolarization either by aligning the polarization within the domains or re-alignment/growthof the domains. Finally, solid-state, thin-film devices that convert low-grade heat into elec-trical energy were demonstrated using pyroelectric Ericsson cycles to yield an optimizedmaximum energy density, power density, and efficiency of 1.06 J cm−3, 526 W cm−3, and19% of Carnot, respectively; the highest values reported to date and equivalent to the per-formance of a thermoelectric with an effective ZT ≈ 1.16 for a temperature change of 10K.


Chapter 6 investigated PEE and ECE in ultra-thin films of Si-doped HfO2 thathave recently been shown to exhibit ferroelectricity. Zero-field pyroelectric response con-firmed that the material does exhibit broken-inversion symmetry. Further, field-dependentpyroelectric measurements revealed the electrical-switching of polarization supporting theexistence of ferroelectricity in these thin films of Si-doped HfO2. The pyroelectric responsewas found to exhibit a wake-up behavior like the wake-up phenomenon observed in the fer-roelectric polarization with electric-field cycling. Measurement of the polarization-electricfield hysteresis loops revealed signatures of the presence of defect dipoles that were found tobe the origin of the observed wake-up phenomenon in both the ferroelectric and pyroelectricresponse. Finally, direct electrocaloric measurements were performed on these doped HfO2

thin films systems for the first time. Measurement on Si-doped HfO2 revealed Σ values ∼4times larger in magnitude as compared to π. This enhancement is explained using the plau-sible role played by defect dipoles that contribute to additional configurational or dipolarentropy.

7.2 Suggestions for Future Work

Although the work presented in this dissertation spawned several interesting scien-tific ideas, there are a few that particularly demand investigation to advance the broad fieldof electrothermal physics in oxides. A few key ideas conceived during this dissertation arediscussed below.

1. Coupled Caloric Effects in Ferroic Thin Films

In multiferroic materials and heterostructure, caloric effects can exploit multiplesources of entropy and be driven by either single or multiple stimuli.[288] Over the last few

110

years, there has been a flurry of research on multiferroic/magnetoelectric materials. Re-searchers have been driven by the promise of coupling between the magnetic and electricalorder parameters and the potential to manipulate one with the other.[289] One promisingcandidate is the type 2-2 composite multiferroic where the two ferroic orders are coupled viastrain. An applied electric field creates a piezostrain in the ferroelectric, which produces acorresponding strain in the ferromagnet and a subsequent piezomagnetic/magnetostriction-induced change in the magnetization.[290] Coupling like this is particularly useful for mag-netocaloric materials which otherwise would require large magnetic fields (1-10 T) changingat the fast time scales (which is difficult to achieve with high magnetic fields).

Such coupled caloric effects can be achieved in FeRh/PMN-PT heterostructure.FeRh exhibits a first-order metamagnetic phase transition between the antiferromagnetic-ferromagnetic order resulting in a giant negative magnetocaloric effect[291] (∆ S = 20 JK−1kg−1 and ∆T = 13 K, in a 2 T magnetic field) near room temperature. FeRh is ahighly competitive magnetocaloric material as the transition is associated with unit-cellvolume expansion by 1% thereby tapping additional entropy change from the elastocaloriceffect. Prior work on magnetoelectric coupling between FeRh on BaTiO3[292, 293] and PMN-PT[294] substrates has revealed the possibility of full magnetic phase switch. However, theuse of single-crystal substrates in these works is not feasible due to cracking and an ultimatefailure of the driving piezoelectric on repeated cycling. Therefore, thin films with highlateral piezoelectric coefficient are most desirable. With this idea, thin films of FeRh can besynthesized on thin films of PMN-0.32PT (composition corresponding to MPB and knownto exhibit large piezoelectric effect). Electrothermal devices can be fabricated using theapproach developed in Chapter 3 of this dissertation. The only exception to the designwould be to replace the metallic top electrode with FeRh which will function not only as aconducting top electrode but also like a magnetocaloric layer actuated with the underlyingrelaxor ferroelectric PMN-PT.

2. Hybrid E- and H-Field Solid-State Cooling Cycles for Fully-ReversiblePhase Transitions

As noted above, FeRh exhibits a first-order metamagnetic phase transition andtherefore the magnetic-phase transition is associated with a hysteretic response. Such hys-teretic response is not desirable in energy-conversion applications where the area within thehysteresis loop corresponds to irrecoverable thermal losses. While the irreversibility and hys-teresis losses in other materials exhibiting a first-order magnetic transition has been reducedby chemical doping, such effects have remained elusive in case of FeRh. Recent work onFeRh on BaTiO3 single-crystal substrates has revealed that the application of electric (E)field in conjunction with a magnetic (H) field can reduce the magnetic hysteresis and allowfor full reversibility of the phase transition.[295] With the realization of the heterostructuredesirable for measuring the coupled magneto-electrocaloric effect (as described above), the

111

effect of applied H-field in conjunction with the E-field on the reversibility of the magneticphase transition can be studied. In turn, by optimizing the phase-difference between theapplied periodic E- and H-fields, the net workdone can be maximized.

3. Probing Electrothermal Susceptibility in Free-Standing Ferroelectric ThinFilms

1. Understanding the Contribution from Secondary PEE and ECE

Electrothermal susceptibilities in ferroic materials are strongly dependent on the stressand strain imposed on the system. The response in bulk and thin films is, therefore,different with respect to the mechanical boundary conditions. Often, the net magnitudeof the pyroelectric and electrocaloric response is reduced in the case of thin filmsdue to the secondary contributions from the piezoelectric effect. Therefore, a free-standing film in the form of a cantilever or a membrane can, in part, relieve thesubstrate-induced strain on the film and could therefore enhance the electrothermalsusceptibilities. Such a system can be designed in the following two ways: 1) Completelift-off of the ferroelectric heterostructure using a sacrificial layer and a subsequenttransfer onto a PMMA stamp[296] that reduces the effects of clamping, or 2) growthof the ferroelectric heterostructure onto a substrate that consists of a hole or a trench(fabricated via a through-etch) to reduce the clamping effect.

2. Improvement in Pyroelectric Energy Conversion

Besides these mechanical effects, free-standing thin films also changes the heat trans-port through the heterostructure. Removal of the substrate below the ferroelectric thinfilm imposes an adiabatic boundary at the free-standing surface. In turn, no heat isexchanged between the film and the substrate in the cross-plane configuration. Thissignificantly enhances the process of pyroelectric energy conversion where the absorbedheat is almost completely confined to the free-standing ferroelectric thin film and there-fore the magnitude of temperature rise in the thin film can be much more than in thecase with a substrate attached for the same heat input. Again, by adopting the fab-rication approach proposed above, the heat transport can be manipulated to confinethe temperature oscillation to just the functional ferroelectric layer and to extract themaximum workdone.

3. Design Improvement in the ECE Measurement Technique

The free-standing thin film form also permits a more sensitive characterization of ECE.As discussed in Chapter 3 of this dissertation, the heat transport during the charac-terization of ECE is bidirectional, i.e. heat is transported to the top metal sensorline and also down to the substrate. With the design of a free-standing device as pro-posed above, the adiabatic boundary at the free-standing thin-film surface will force

112

the heat transport to the unidirectional and therefore, again for the same magnitudeof the ECE heat flux the temperature rise of the top sensor line will be higher than thecurrent scenario. This will dramatically improve the sensitivity of the characterizationtechnique.

4. Probing ECE in Relaxor Ferroelectric Thin Films

In Chapter 5, it was shown how rotation of polarization can contribute to the netpyroelectric susceptibility in relaxor ferroelectric 0.68Pb(Mg1/3Nb2/3)O3-0.32PbTiO3 thinfilms. As shown in Chapter 5, such a polarization rotation can be achieved via an applica-tion of an electric field. The entropy change due to such a phase transition can result inextrinsic contributions to ECE. Using the same thin-film heterostructure as used in Chap-ter 5, temperature-dependent ECE measurements can elucidate how much can polarizationrotation under an applied electric field contributes to the electrocaloric susceptibility. Bysystematically investigating how ECE changes with the crystallographic orientation of thefilm ((001)-, (101)- or (111)-oriented) with respect to the application of electric field, theeffect of polarization rotation can be controlled, in turn, their contributions to the net ECEcan either be enhanced or completely turned off. It should be noted that direct measurementsof ECE in relaxor ferroelectrics is especially important because the compositional disorderkeeps the system in non-equilibrium and the use of Maxwell equation equating π and Σ isnot valid.

5. Understanding Electrothermal Susceptibilities in Compositionally-GradedFerroelectric Thin Films

Electrothermal effects have almost always been studied in the vicinity of phase tran-sitions[42] as the susceptibilities (second-order derivatives of the free energy) are maximizedabout the transition temperature. This has led the calorics community to explore ways totune the transition temperature near the desired temperature of application via composi-tion[297] or epitaxial strain[57] (in case of thin films). Little has been done, however, tofundamentally alter the polar order or the nature of the phase transition to produce largersusceptibilities across a wide range of temperature. This is important as any thermodynamiccycle of electrocaloric refrigeration requires a large entropy change over a broad temperaturerange (refrigeration capacity,[298] C =

∫ ThotTcold

∆SdT ). Relaxors constitute one such promis-

ing class of candidate materials with a broad phase transition.[299] In classic ferroelectrics,however, similar effects are not common, but can potentially be engineered through the pro-duction of compositionally-graded versions. In a simple scenario, the compositionally-gradedferroelectric can be thought of multiple ferroelectric layers stacked in series, each with its ownTC resulting in a material with broad transition temperature and, in turn, large susceptibil-ities throughout the entire temperature range. Such a scenario is, however, too simplistic.

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Gradients in polarization lead to non-vanishing electric field, the so-called depolarizationfield, across the entire thickness of the film that works to even out the polarization acrossthe film thickness.[300] However, in real materials the presence of additional charge screen-ing mechanisms (for example, space charges) can reduce the depolarization field causing thepolarization in any differential film thickness to be decoupled from the rest, resulting in TCbeing highly smeared across the TCs of the end member composition. The average polariza-tion across the entire ferroelectric in such a case is predicted (via GLD models) to changealmost linearly with temperature. This should result in enhanced and near-constant valueof electrothermal susceptibilities across a wide temperature range. However, PEE being asurface effect is sensitive to the top or bottom most composition of the graded ferroelectric.Experimental investigation can therefore develop a clear understanding of how the “loosely”coupled layers in a compositionally-graded ferroelectric effect its electrothermal susceptibil-ity and if such gradients in composition or strain can provide novel routes to enhance thePEE and ECE in thin-film ferroelectrics.

6. Probing Extrinsic PEE and ECE in Mixed Vortex-Ferroelectric Superlattices

The coexisting toroidal phase and the ferroelectric phase in symmetricPbTiO3/SrTiO3 superlattices[301] is subject to phase competition under a temperatureand/or electric-field stimuli. Temperature-induced phase transitions between the vortexand the a1/a2 phase allows for significant changes in the polarization that can result in largeextrinsic contributions to pyroelectric susceptibilities. It should, however, be noted that incase of superlattice structures, the pyroelectric measurement should be performed in the in-plane configuration to collect the pyroelectric current from all the polar layers. A device likethat developed in Chapter 3 can be used to measure the in-place pyroelectric current with afew modifications. These include a change in the ferroelectric electrode configurations whichshould now be embedded into the ferroelectric superlattice. The thermal transport model(both analytical and finite-element-based) should be appropriately modified. Additionally,the reversible electric-field induced switching between the vortex and the a1/a2 phase canallow for large entropic changes providing a novel route to electrocaloric susceptibility.

7. Role of Point Defects (Defect-Dipoles or Complexes) on ECE

In Chapter 6 of this dissertation, the possible role of defect-dipoles on the elec-trocaloric susceptibility was illustrated. Briefly, it was hypothesized that defect-dipoles gen-erate internal electric and elastic fields that have a strong impact on the accessible configura-tional or dipolar entropy. Recent work using molecular dynamics simulation has shown thatthe ECE in BaTiO3 at high temperature (above the Curie temperature) can be increased inthe presence of defect dipoles oriented parallel to the external electric field while a negative

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ECE can be induced by defect if the direction of the external electric field is anti-parallelto the orientation of defect dipoles.[272] Such a phenomena can be probed directly in thinfilms of BaTiO3 where a high density of defect dipoles can be controllably generated usinga high-fluence (high-energy) pulsed laser deposition.[302] Comparison of the ECE responsebetween the samples with and without defect dipoles will unequivocally illustrate the role ofdefect dipoles on ECE.

115

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Appendix A

Pulsed Laser Deposition of ComplexOxide Thin Films

In the last decade, complex oxide thin-films have established themselves as a class ofmaterials exhibiting a plethora of exotic functional properties. At the heart of the functionalproperties of such materials lies the ability to controllably synthesis them to realize thecorrect structure that enables the desired functional property. Among the various growthtechniques for the synthesis of functional complex-oxide thin films, pulsed-laser deposition(PLD) has rapidly grown to become a technique of choice since its use in the growth of hightemperature superconductors at Bellcore in 1987.[303] The concept of PLD, however, beganwhen the development of high-power lasers that triggered the idea of ablating materialsusing short wave-pulses with a low wavelength. With further developments in the vacuumtechnology and in-situ monitoring techniques like RHEED (Reflection High Energy ElectronDiffraction) and scanning probe microscopy during the growth process, PLD has establisheditself as a workhorse in thin-film epitaxy. As it stands today, it enables control of growth at aunit-cell level including non-equilibrium growth of superlattices and nanostructures, permitscontrol of stoichiometry, crystallinity, and epitaxial strain.[304]

The working principle of PLD is relatively simple. A laser pulse when incident ona ceramic material (referred to as the target) creates a plasma plume containing the atomic,molecular and ionic species of the target composition. Under a low pressure created insidea vacuum chamber, this plume is directed towards a heated substrate such that depositionoccurs on the substrate (Figure A.1). The process of plasma formation, propagation towardsthe substrate, plume chemistry, and the growth dynamics are quite complex and an optimalgrowth of the desired material composition, phase and crystallinity therefore requires controlover a wide variety of experimental parameters.[304] These include the temperature of theheated substrate, chemistry of the gas environment (ozone, oxygen, atomic oxygen, etc.),gas pressure, and energy density and flux from the laser pulse. Ultimately, the techniqueoffers a wide range of thermodynamic and kinetic variables to access the growth conditions

135

Viewport

Aperture

Focusing

Lens

Pulsed Excimer

Laser

O2 inlet

Target

Rotator

Pulsed Excimer

Laser

λ = 248 nm

Figure A.1: Schematic illustration of the Pulsed Laser Deposition process

of a wide variety of complex-oxide materials. The effect of various growth parameters willbe discussed in the rest of this Appendix.

The process of PLD begins with the interaction of an electromagnetic laser-pulsewith the material that depends on the optical, topological and thermodynamic propertiesof the interacting material (target). The electromagnetic energy of the incident laser pulsefirst converts into electronic excitation. The absorbed photons excite the electrons that arethermalized on a timescale of picoseconds followed by the melting of the target surface andconduction of heat into the bulk It should be noted that the incident energy from the laseris localized into a very small volume with an area less than 0.1 cm2 and a penetration depthof a few nanometers to a few tens of nanometers depending upon the wavelength of theincident laser and the refractive index of the target. Ultimately, vaporization occurs and themultiphoton absorption causes ionization of the gaseous phase forming plasma. This entireprocedure from absorption to ionization occurs within 10-100 picoseconds. This plasmaconsists of a mixture of atoms, molecules, electrons, ions, and even solid particulates. Witha typical pulse duration of 10-100 ns with a laser wavelength, λ = 248 nm (for KrF excimerlaser), the incident energy is above the ablation threshold of the constituents of the target.This makes the evaporation congruent, and PLD a highly efficient thin-film growth processwith near-stoichiometric transfer of material from the target to the substrate. However, ifany element of the target is more volatile than the rest, or if the plume is not thermalized,the less-volatile element will be enriched in the film resulting in a non-stoichiometry. Inaddition, if the sticking coefficient of an element is significantly different from the other oran element has a tendency to resputter, the stoichiometry is not preserved. Therefore, the

136

target material is enriched in the more volatile element (for example the use of 10% excessPb in the PbZr1−xTixO3, and ()1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 targets materials used forthis dissertation work).

After ablation, the otherwise dense plasma, rapidly expands under vacuum to createa highly directional plume of material directed towards the heated substrate due to coulombrepulsion and recoil from the target surface. This plume has two components. The firstcomponent has considerable thermal energy and tends to be off-stoichiometric. The shapeof this plume exhibits a cosθ distribution. The other component is more stoichiometric, andhas a lobe shape that has a cosn θ distribution (n = 5 to 10).[304] This plume has a largekinetic energy (∼ 100 eV). This large kinetic energy along with the high temperature of thesubstrate provides the adatoms enough mobility to diffuse on the surface. Growth, from thispoint onwards, can proceed in three different modes: 1) Volmer-Weber or island growth,2) Frank-Van der Merwe or layer-by-layer growth, and 3) Stranski-Krastanov growth.[305]Volmer-Weber island growth occurs when many stable clusters nucleate on the surface ofthe film and the subsequently deposited material has a strong preference to bond to thepreexisting nuclei compared to the substrate surface, making well defined islands. This ismost common when depositing dissimilar materials or onto a substrate with a high surfaceenergy. Frank-Van der Merwe layer-by-layer growth has the opposite premise. In layer-by-layer growth the deposited material prefers to bond to the substrate and/or a preexistingstep edge than to bond on top of a previously formed layer, tending to form 2D planar layers.The final growth mechanism Stranski-Krastanov growth represents a combination of thesetwo mechanism where initially the adsorbed adatoms prefer to grow layer-by-layer. After afew layers however, the preference for layered growth no longer exist, giving way to threedimensional island growth.

137

Appendix B

Description of Heat Transport Models

B.1 Pyroelectric Effect

B.1.1 One-Dimensional Heat Transport Model

The full solution to the heat diffusion equation for a heater-on-substrate geometryis solved in cylindrical coordinates. With the addition of thin films on top of the substrate,it is reasonable to approximate the heat transport through the films as one dimensional solong as the heater half-width, b (5 µm), is greater than the total thickness of the film stack(< 0.5 µm). An exact quantification of the error1[170] in making this assumption caps theerror in our model at less than 5%.

The steady, periodic solution to the one-dimensional heat diffusion equation is well-described by Carslaw and Jaeger,[171] so many of the details will be skipped here. However,the transport matrix method for practical application of the solution will be described indetail. Starting from the analytical solution to the heat diffusion equation, heat transportacross a single domain can be written via vector multiplication as

[θdqd

]=

[cosh(γd) − 1

kγsinh(γd)

kγsinh(γd) cosh(γd)

] [θ0

q0

](B.1)

where γ =√

jωD

, j = (−1),k is the material thermal conductivity, and d is the

layer thickness. Recursive application of this transport matrix scheme can thus be used todescribe one-dimensional heat flow across several domains. That is,

138

2

2

a b c

Figure B.1: (a) Schematic showing the difference in the effective heating cross-section(2beffL) and the area collecting pyroelectric current (2belectrodeL). (b) Example FEA 2Dcontour plot of the temperature amplitude, θ(x, z), for the specific geometry studied and ata heating frequency f = 1000 Hz . (c) Typical line trace θ(x) through the ferroelectrics mid-zplane.

[θNqN

]= MNMN−1...M1

[θ1

q1

](B.2)

where MN is the transport matrix for the specified layer, containing the appropriatelayer properties. Given any two of the temperatures or fluxes at either end of a domain, theother two can be found using this method.

In the PEE measurement scheme depicted in Figure 3.3c, we measured the tem-perature amplitude (complex value) at the top surface of the stack, θH , using the 3-omegamethod, and the heat flux at the top surface, qH , is just the AC power dissipated per unitarea. These values are used to walk down the stack and thus determine the temperatureamplitude at several positions within the stack. The key insight is to note that any singledomain may be divided into an arbitrary number of sub-domains, allowing for the calculationof desired values at any height within the stack. Note, also, that interfaces between layershave been modeled as thin layers (∼1 nm) with low thermal conductivity (1 W m−1 K−1)and small heat capacity (105 J m−3 K−1).

139

B.1.2 Accounting for Lateral Spreading of Heat within the 1-DHeat Transport Approximation in PyroelectricMeasurements

To be rigorous in the extraction of π, we also account for lateral heat spreading inthe films since the effective area of the ferroelectric heated via the oscillating temperatureon the top heater line with area 2bheaterL is not necessarily the same as the electrode area2belectrodeL used to measure the pyroelectric current (Figure B.1a). Therefore, an effectivecurrent-collecting area is defined as 2beffL, where beff is the effective half-width of the heaterline. The most robust way to eliminate heat spreading concerns would be to fabricate a thinfilm mesa so the heater, top electrode, and all layers in between have the same half-width.This would force the heat conduction to be truly one dimensional and bheater = belectrode.However, microfabricating a mesa of this sort can be quite challenging and was deemedimpractical in the present work, so instead a numerical finite element analysis (FEA) packagewas employed to evaluate the appropriate geometric correction to π (Figure B.1b).

The temperature amplitude of the ferroelectric film does not vary by more than afew percent through the films thickness (the z-direction, Figure B.1a), but the temperatureprofile does vary significantly along the x-direction. For a one-dimensional, non-uniformprofile such as this the total measured pyrocurrent is rigorously defined as

ip = πAdT

dt= πL2ω

(2

∫ belectrode

0

θ(x)dx

)(B.3)

where θ(x) is the temperature amplitude of the ferroelectric response at the z planehalfway through its thickness, the time derivative of the temperature amplitude (corrected forthe phase lag) has been taken explicitly giving a factor of 2ω, and the extra factor of 2 arisesbecause the integral only accounts for half of the electrode width. The measured temperatureamplitude used to determine π is 〈θFE〉. Note that 〈θFE〉 is determined from 〈θFE〉 and istherefore an estimate of the average temperature amplitude of the ferroelectric only out tobheater whereas ip is collected via the electrode with a larger half width (belectrode > bheater)necessitating a correction.

The goal of using FEA is to determine an effective half-width, beff , that properlycombines the disparate belectrode and bheater in such a way as to return the rigorous definitionof the pyrocurrent, ip. Mathematically, this requirement corresponds to

πL2ω2beff 〈θFE〉 = πL2ω

(2

∫ belectrode

0

θ(x)dx

)(B.4)

140

beff =

∫ belectrode0

θ(x)dx

〈θFE〉(B.5)

Because 〈θFE〉 is

〈θFE〉 =1

b

∫ bheater

0

θ(x)dx (B.6)

we have found the result sought

beff = bheater

∫ belectrode0

θ(x)dx∫ bheater0

θ(x)dx(B.7)

The above equation shows that beff is simply the heater half-width, bheater, mul-tiplied by the ratio of two integrals. Appropriately employing FEA can give θ(x) at anyheight in the stack and can therefore be used to solve for the numerical factor (the ratio ofthe two integrals) in beff .

Using a commercial FEA package, COMSOL Multiphysics R©, the 2D time-periodicheat diffusion problem can be solved and the temperature amplitude θ(x, z) can be used toevaluate the integrals. The COMSOL R© heat transfer module does not have a clean interfacefor working with time-periodic problems and the Coefficient Form PDE module proves to besignificantly easier to work with and simulations run noticeably faster. Given the variableof interest, θ, the generic coefficient form PDE equation in COMSOL R© is

ea∂2θ

∂t2+ da

∂θ

∂t+∇.(−c∇θ − αθ + γ) + β.∇θ + αθ = f (B.8)

Careful inspection of the equation above reveals that for ea = da = α = γ =β = f = 0, the PDE simply reduces to the time-periodic heat diffusion equation witha = ρc and c = k, where c is the volumetric heat capacity [J m−3 K−1] and k is thethermal conductivity [W m−1 K−1]. It is easiest to employ this FEA model in the half-space x ≥ 0, as symmetry about x = 0 is expected. This equation can then be used tosolve for the complex temperature amplitude, θ(x, z), with appropriate boundary conditions(i.e., uniform periodic heat flux from the heater and adiabatic boundaries elsewhere). The

141

Figure B.2: Correction factorbeffbheater

as a function of frequency for the specific geometrystudied. Changing the thermophysical properties of the materials will cause the line to shiftup/down and may affect the steepness of the decay, but the general trend (decreasing as afunction of frequency) will remain.

FEA solution was verified against the analytical solution for the average heater temperatureamplitude, θheater.[169] The magnitude of the complex integrals in Equation B.7 are foundand their ratio determined for the given system with properties specified in Table 1 and formany frequencies, as can be seen in Figure B.2.

B.2 Electrocaloric Effect

The 1-D heat transport model used for the PEE measurements can be extended toaccount for a heat source of finite thickness in the middle of a material stack in case of ECEmeasurements. A 1-D model is valid here because the heater (ferroelectric) half width (b=5µm) is much greater than the total thickness of the film stack (< 0.5 µm) and thus thereis minimal lateral spreading within the substrate. Beginning with the transport matricesdefined by equation (B.1) for each layer in the stack, we can build up a formalism for theinternal heat generation.

Considering the stack in Figure 3.4c, differential heat generated at an infinitesimal

142

layer dz at height z, diffuses up toward the surface of the stack and also into the substrate.This allows the stack to be divided into two sub domains: the upper (1) and lower (2) halfof the stack, relative to height z within the ferroelectric. From these considerations we havethree equations,

dqtot = dq1 + dq2 (B.9)

[θtdqt

]=

[A1 B1

C1 D1

] [θzdqz1

](B.10)

[θbdqb

]=

[A2 B2

C2 D2

] [θzdqz2

](B.11)

where the subscript t represents the properties of the bottom surface of the sensor,b the properties of the bottom surface of the substrate, z the properties of the ferroelectriclayer at the prescribed height of interest, 1) the total transport matrix (i.e., the recursivemultiplication of the M matrices for the relevant domains)) for the upper domain, 2) theanalogous transport matrix for the lower domain, and qtot is the total flux generated by theinfinitesimal layer.

The fluxes at the top surface (adiabatic) and the bottom surface (semi-infinite) arezero. This allows us to write:

C1θz +D1dqz1 = 0 (B.12)

C2θz +D2dqz2 = 0 (B.13)

Applying the conservation of flux, equation (B.9), to these equations leads to

θz =

(−D1D2

C1D2 + C2D1

)(B.14)

The volumetric heat generated within the ferroelectric when exposed to an oscillat-ing electric field, G, is defined by equation (B.11). Using equation (B.12), the temperatureresponse at any plane in the ferroelectric can be calculated by

143

θz = G

∫ dFE

0

(−D1D2

C1D2 + C2D1

)dz (B.15)

Equation (B.15) can be applied to the top and bottom surfaces of the ferro-electric layer to determine the average temperature oscillation in the ferroelectric, θFE =12

(θz=0 + θz=dFE).

We can further extend this matrix formalism to determine an explicit expressionfor the temperature at the top surface of the stack (i.e. the temperature measured by thesensor). From equation (B.10) and (B.12),

θt = A1θz +B1dqz1 = A1θz +B1

(−C1

D1

)θz (B.16)

Finally, combining (B.15) and (B.16),

θt = G

∫ dFE

0

(B1C1 − A1D1

C1D2 + C2D1

D2

)dz (B.17)

We see in equation (B.17) the complete expression used to relate the surface tem-perature measured by the heater line (θsensor) to Σ.

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