Phonon Engineering:Phonon Engineering: an introduction – part 1an introduction part 1

C M Sotomayor Torres, P.-O. Chapuis, F Alzina, D Dudek, J Cuffe and L Schneider

Catalan Institute of Nanotechnology (CIN2-ICN-CSIC)

NISP Summer School on “Energy Harvesting at the micro- and nano-scale”

1

NISP Summer School on Energy Harvesting at the micro- and nano-scaleAvigliano Umbro TR, Italy, 1st-8th August 2010

Collaborators• N Kehagias, V Reboud - ICNg ,

• J Ahopelto, M Prunnila – VTTp ,

• E H El Boudouti – U Oujda & IEMN, U Lillej ,

• B Djafari-Rouhani - IEMN, U Lillej ,

•A Zwick, J Groenen, F Poinsotte, A Mlayah – U Paul , , , ySabatier (LPST now part of CNRS-CEMES)

2

SupportSupport

www.nanopack.org

3www.tailphox.org

The Phononic and Photonic N t t G (P2N)Nanostructures Group (P2N)

Vi tFrancesc VincentReboud

NikolaosKehagias

TimothyKehoe

FrancescAlsina Oliver

ChapuisDamian Dudek

And

Clivia M Sotomayor

TorresJohn Yamila Lars Naomi Emigdio

4

TorresCuffeGarcia Schneider Baruch

gChavez

OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities

Ph tifi• Phonon rectifiers• Inelastic light scattering

- Confined acoustic phonons• Thermal transport (part 2-O Chapuis)p (p p )

– the 3-omega methodConclusions

5

Conclusions

Phonons in solidsKey excitation in energy and momentum relaxation in, e.g, semiconductor quantum wells wires and dots (1 THz ~ 4 meV)

(meV)

semiconductor quantum wells, wires and dots (1 THz 4 meV)

Typical dispersion relations

TOLO

65-30of optical phonons and

TALA

TO

acoustic phonons TA

up to~30

p

/2a k in bulk material

6

Focus on acoustic phononsFocus on acoustic phononsReasons to study THz acoustic phonon engineering:

- Imaging with nm resolution acoustic wavelength ~ 5nm at 1 THz5nm at 1 THz

- Thermal dissipation- Thermoelectricityy- Optoelectronic devices modulation at high speed- Phonon caustics

C t l f d h- Control of decoherence- Detrimental role in nanoelectronic devices (mobility)- NanometrologyNanometrology- Design of density of phonon states- A candidate for a non-charge state variable

7

Phonon anharmonic decayPhonon anharmonic decay

Optical phonons Acoustic phonons(f V)(10’s of meV) (few meV)

optac ~ 5 ps in Si

e-opt ph~ 100s fsOptical phonon

Acoustic hOptical phonon

emissionhigh-field Joule heating

phonons carry heat away from hot spots

8

hot spots

“Phonon bottleneck” in deep etched quantum dots

- Intrinsic model of luminescence yield in decoupled 1- and 0-D semiconductors.

wires and dots- wires and dots decoupled from surrounding (unlikesurrounding (unlike self-organised ones)

9 H Benisty, C M Sotomayor Torres & C Weisbuch, Phys Rev B 44 (19) 10945 (1991)

Effect of phonon confinement on ZT of quantum wells

A Balandin and K L Wang 1998

See also, M.S. Dresselhaus et al, Adv Mat 19, 1043

10

(2007).

Electronic, Photonic and Phononic crystals

Common key parameters: periodicity contrast filling factor geometry

11 Adapted from Sigalas et al Z. Kristallogr, vol. 220, pp. 765–809, 2005

Common key parameters: periodicity, contrast, filling factor, geometry

OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities

Ph tifi• Phonon rectifiers• Inelastic light scattering

- Confined acoustic phonons• Thermal transport p

– the 3-omega methodConclusions

12

Conclusions

Phonons in Low Dimensional Systems– Interface phonons – Zone-folded acoustic phononso e o ded cous c p o o s– Geometrically confined surface modes– Localised and or confined phononsLocalised and or confined phonons– Surface acoustic waves– Acoustic cavity modesAcoustic cavity modes

– …

Treat as standing waves (eg, confined) or propagating waves (eg., zone-folding in superlattices)

“Phonon Raman scattering in semiconductors, quantum wells and superlattices: Basic results and applications “ Tobias Ruf, Springer (Berlin, New York) 1998.

“Ph i S i d t N t t ” Ed J P L b t J P l d C M

13

“Phonons in Semiconductor Nanostructures”, Eds J-P Leburton, J Pascual and C M Sotomayor Torres, Kluwer Publishing, The Netherlands, 1993

Folded acoustic phonons in superlattices

F iFor a review on vibrations in superlattices, see B. Jusserand and M. Cardona, in Light Scattering in Solids, edited by M Cardonaby M. Cardona and G. Güntherodt(Springer-Verlag,

14 From Lecture notes of B Jusserand, Son et Lumiere 2006

( p g gHeidelberg, 1989), p. 49.

Surface Phonons in semiconductor cylinders 1Surface Phonons in semiconductor cylinders 1(nh)2/()2=( m nh)/( m nh),

El t t ti ti d l f R i d

with nh in terms of modified Bessel functions and derivatives.

Electrostatic continuum model of Ruppin and Engelman (1970) with geometry determined by boundary conditions, neglecting retardation effects.

GaAs pillar 100 nm diameter 700 nm high

15 M Watt et al Semicond Sci Technol 5, 285-290 (1990)

700 nm high.

Surface Phonons in semiconductor cylinders 2GaAs cylinders of 80 nm diameter and 250 nm high.

Contribution of surface phonons to the RamanContribution of surface phonons to the Raman signal appear between the TO and LO phonons as expected.

Pillars coated with SiN: surface phonon frequencies

16

surface phonon frequencies decrease.

M Watt et al Semicond Sci Technol 5, 285-290 (1990)

OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities

Ph tifi• Phonon rectifiers• Inelastic light scattering

- Confined acoustic phonons• Thermal transport p

– the 3-omega methodConclusions

17

Conclusions

Phononin crystal types

Typical phonon frequency range in solids:

Phononin crystal types

Typical phonon frequency range in solids:(0 – 45 THz)

– Sonic crystals (≤ kHz)y ( )– Ultrasonic crystals (kHz – MHz)

Hypersonic crystals (GHz THz)– Hypersonic crystals (GHz - THz)– Mixed optical and phononic crystals (several

TH )THz)

18

Phononic crystals– Acoustic and elastic analogues of photonic crystals

y

– ‘stop bands’ in phonon spectrum (phonon mirrors); – ‘negative refraction’ of phonons (phonon caustics)g p (p )– Good theory available: Multiple scattering theory

for elastic and acoustic waves See for example:for elastic and acoustic waves. See, for example:Kafesaki & Economou PRB 60, 11993 (1999), Li t l PRB 62 2446 (2000)Liu et al PRB 62, 2446 (2000)Psaroba et al PRB 62, 278 (2000).

A d f d t b 2006 2008And for a database 2006-2008:http://www.phys.uoa.gr/phononics/PhononicDatabase.html

19

2D infinite phononic crystal: air holes in siliconmatrix (B Djafari-Rouhani Y Pennec IEMN U Lille)matrix (B Djafari Rouhani, Y Pennec, IEMN, U Lille)

Sq are H l H b

y 1.0triangular, f=0.6

y 1.0square, f=0.3

y 1.0honeycomb, f=0.3

Square Hexagonal Honeycomb

ed fr

eque

ncy

0.4

0.6

0.8

ed fr

eque

ncy

0.4

0.6

0.8

ed fr

eque

ncy

0.4

0.6

0.8

wavenumber

redu

ce0.0

0.2

X J Xwavenumber

redu

ce

0.0

0.2

M X Mwavenumber

redu

ce

0.0

0.2

X J Xwavenumber

ncy

0 8

1.0triangular, f=0.85

wavenumber

ncy

0 8

1.0square, f=0.6

wavenumber

ency

0.8

1.0honeycomb, f=0.6

uced

freq

ue

0 2

0.4

0.6

0.8

uced

freq

ue

0 2

0.4

0.6

0.8

duce

d fre

que

0 2

0.4

0.6

20 wavenumber

red

0.0

0.2

X J Xwavenumber

red

0.0

0.2

M X Mwavenumber

red

0.0

0.2

X J X

Imaging with phononic crystals - I

21 S Yang, J Page, et al, PRL 93, 024301 (2004)

Imaging with phononic crystals - II

22 S Yang, J Page, et al, PRL 93, 024301 (2004)

OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities

Ph tifi• Phonon rectifiers• Inelastic light scattering

- Confined acoustic phonons• Thermal transport p

– the 3-omega methodConclusions

23

Conclusions

Length scales in SiLength scales in SiPhonon MPFi b lk Si 41 @ RT D b d lin bulk Si = 41 nm @ RT Debye model

260 nm considering dispersion300 nm (Ju & Goodson APL 1999)300 nm (Ju & Goodson, APL 1999)

(cf Electron MFP = 7.6 nm)Dominant phonon wavelength

/ fd = vs / fd in Si d = 1.4 nm @ RT = 4000 nm @0.1

velocity ofsound 1.48 /kBT

To confine phonons in the strong regime at RT need structures with

24From A Balandin, UC Riverside

sound~ 1-10 nm lateral dimensions

Phonons in MEMS and NEMS

Predicted in-plane dispersion curves for dilatational modes in a 10 nm Si freedilatational modes in a 10 nm Si free

standing membrane Example of a MEMS / NEMS

90cm-1

A Erbe et al Appl Phys Lett 77

0

A. Erbe et al., Appl. Phys. Lett 77 (2000) 3102.

25 (Balandin & Wang PRB 581554 (1998))

Confined Acoustic Phonons: Phonon cavity approachConfined Acoustic Phonons: Phonon cavity approach

q in plane

q

q// in-plane (small)

SOISOIair

vsound

qz

+ SOI

BOX

V / Vtransverse longitudinal

qz

2V2/1V1

idensityPhonons in SOI/BOX Longitudinal acoustic mismatch = 0.69 (significant)

Vi=sound velocity

Transverse acoustic mismatch = 0.99 (almost negligible)

26

Cavity interference effects: phonons and electronsCavity interference effects: phonons and electrons

Interaction via deformation potential p

SurfaceDisplacement field

Dis

0

d

stance to surface

electrons planePlane Waves

Standing Waves

27

Waves WavesHuntzinger et al 2000, Cazayous et al 2002

Observation of confined modes: Raman scattering

EEi

Coupled optical and EsEip

phonon cavities

105

M. Trigo et al., PRL 89, 227402 (2002)

28

g , , ( )P. Lacharmoise et al., APL 84, 3274 (2004)N. D. Lanzillotti-Kimura, PRL 99, 217405 (2007)

From: A. Fainstein – presented at ICREA Workshop - May 2010 – Barcelona

Opto-mechanical structures

29 M Eichenfield et al. Optomechanical Crystals, Nature 462, 78-82 (2009

Optical forces control mechanical modes prospects for cooling, heating, …

OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities

Ph tifi• Phonon rectifiers• Inelastic light scattering

- Confined acoustic phonons• Thermal transport p

– the 3-omega methodConclusions

30

Conclusions

31

Considerations

Thermal rectification requires asymmetry and non-linearity.

Asymmetry as in pyramids in the plane or 2D arrays of offset trianglestriangles. Also coupled phonon cavities with mismatched modes.

Non-linearity: distorted distribution functions

Model by ray tracing or Monte Carlo techniques or displaced Bose-Einstein distribution or, better still, by Landauer-Buettiker theoryBuettiker theory.

32

Solid state rectifier

Maximum rectification = 7% for a mass ratio of 5

H=heater, s= sensorL=low massH hi hH= high mass

33 C W Chang et al, Science 314, 1121 (2006)

OutlineOutline• MotivationMotivation• Taxonomy of phonons• Phononic crystals• Phononic crystals• Phonon cavities

Ph tifi• Phonon rectifiers• Inelastic light scattering

- Confined acoustic phonons• Thermal transport p

– the 3-omega methodConclusions

34

Conclusions

Raman scattering in semiconductorsRaman scattering in semiconductorsRaman scattering is one type of inelastic light scattering.

cb

i id tScattered light

gES

Sincident

lightlScattered light

Bandgap luminescence

vbOthers: Rayleigh,

Bandgap luminescence,Unwanted in Raman scatt experiments

35

Others: Rayleigh, Brillouin scattering

Considerations

Momentum conservation: Typical laser line = 4880Åincident 4880Å

4.33(GaAs);2 nnk

Thus15106.5 cm

maximum momentum transfer is twice thismaximum momentum transfer is twice this,i.e., 1.1 x 106cm-1

The extent of the Brillouin Zone /awith a = lattice parameter = 5.65Å for GaAs

B.Z. boundary ~ 6 x107cm-1

Th i f B ZThus maximum wavevector transfer << B.Z. extent First order approximation: Raman scattering

probes excitations with q0

36

probes excitations with q0.

Thin SOI film sample cross-section

Native oxide 3 nm

Buried (thermal) oxide (SiO2) 400 nm

SOI 40 nm

28 nmBuried (thermal) oxide (SiO2) 400 nm

Base Si wafer CZ p-type <100> 525 micrometer

37

Sample fromSotomayor Torres et al phys stat sol c , 1, 2609 (2004)

Raman spectra of thin SOI films

1500

1750

2000

ity (u

a)

Silicon thickness 34nm

500

750

1000

1250R

aman

Inte

nsity

Silicon thickness 34nm

Black - experimentRed simulation

300 K, 413.1

0 10 20 30 40 50 60 700

250

500

Ra Red - simulation

413.1 nm, // polarisation 500

600

700

sity

(ua)

200

300

400

Ram

an In

tens

ity

Silicon thickness 30.5nm

0 10 20 30 40 50 60 70

Δω (cm -1

)

0

100R

38

Δω (cm )J Groenen et al PRB 77, 045420 (2008)

Confined LA phonons vs. thin film thicknessConfined LA phonons vs. thin film thickness

39Sotomayor Torres et al unpublished

Simulations Raman spectra SOI thin filmS u at o s a a spect a SO t

• Photoelastic model 2)(Photoelastic model for scattering by LA phonos

* )().(...)(

zzzpEEdzI SLqz

EL (ES) : laser (scattered) fieldp(z) : photoelastic constant

Φ1(z) oxide

p(z) : photoelastic constantΦ(z) : phonon displacementΦ2(z) silicon

Φ3(z) oxide

Silicon buffer

40

J Groenen et al PRB 77, 045420 (2008)

• Vibrational part

- phonons displacement and stress boundary conditions )()(

)()(

/2

2/1

1

/2/1

SiOSiO

SiOxSiOx

zCzC

zz

p

{stress boundary conditions )()( /2/1 SiOxSiOx zz

Czz

C

ii- Hypothesis

{Phonons stationary waves

Free surface

ziqziq eBeAz 11111 )(

0)(1C Free surface

Dispersion relation

0)( /1

1 Oxairzz

C

vqsound velocity

Vac(oxide) =5970 m s-1

• Electronic part

Infinite silicon buffer

0)(zP

aczqz vq . Vac(oxide) =5970 m.s-1

Vac(silicon) =8433 m.s-1

{• Electronic part1)(0)(

zPzP

Si

Ox{41

J Groenen et al PRB 77, 045420 (2008)

Si l i f Sili hi k i iSimulations of Silicon thickness variationSOI simulations

silicon thickness variation

1500

300

(ua)

1500

man

(ua)

100

200

Inte

nsité

Ram

an (

ua

1000

Inte

nsité

Ram

a

0 10 20 30 40 50

Δω (cm-1

)

0

500

In

-50 -40 -30 -20 -10 0 10 20 30 40 500

42

Δω (cm-1

)F Poinsotte et al unpublished

6000

SOI simulationoxide thickness variationSimulation of oxide thickness variation

5000

6000

oxide : 0nm

4000

5000

(ua)

2nm

4nm

3000

sité

Ram

an (u

a

6nm

8nm

2000Inte

nsité 10nm

12nm

14nm

1000

14nm

16nm

18nm

-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70

(cm-1

)

0

18nm

20nm

43

Δω (cm-1

)F Poinsotte et al unpublished

SOI membranes and configuration

Back-scattering500 m

g

Laser spot

Forward scatteringspot

44Sotomayor Torres et al phys stat sol c , 1, 2609 (2004)

Si membrane

30 nm SOI + 400 nm BOX

45

30 SO 00 O

Examples of raw datap

Wavenumber (cm-1)

46

Wavenumber (cm )

Sotomayor Torres et al, unpublished

Simulations of Raman spectra of SOI membranes

T t SOI l it f ti h i

Simulations of Raman spectra of SOI membranes

Treat SOI layer as a cavity for acoustic phonons, ie, confined since longitudinal vs in Si = 8433 m/s (cf. 332 m/s in air at 0 C)m/s in air at 0 C).

Displacement field of acoustic vibrations in a slab of thi k t i ti l t

)(n thickness t is proportional to:

)cos( zt

n order of the confined frequencies can be derived fromn order of the confined frequencies can be derived from LA dispersion branch, considering discrete wave vectors q = n /t

47

q n /t

Simulations of Raman spectra of SOI membranes- cont.

t)(zE),(

)(tzu

ptzP zAcoustic vibration periodici i f i l i i

Simulations of Raman spectra of SOI membranes cont.

t)(z,E),( izptzP z

S variation of strain polarisation

field in presence of em wavep photo-elastic coefficient of slab

tzu ),( *),(

tzuz

ps photo elastic coefficient of slab

P(z,t) OK for anti-Stokes part.Obtain Stoke part by changing by

ztzuz

),( ),(

zzObtain Stoke part by changing by

Thus, scattered field:

2

2

22

2

2

2

2

2 ),(1),(),( tzPtzEsntzEs

22

0222 tctcz

Where n = slab index of refraction.

48

Raman spectra of 31.5 nm thick SOI membrane

Forward scatteringscattering

BackBack scattering

Wavenumber cm-1

49

J Groenen et al PRB 77, 045420 (2008)

Data from five SOI membranes

50Sotomayor Torres et al phys stat sol c , 1, 2609 (2004)

15 nm Si QW, expect shear, dilatation & flexural modes

2nd dilatation2 dilatation mode

3rd dilatation mode

51

A Balandin 2000 in-plane V-group 0 at cusps

Raman & Brillouin scattering Inelastic Light Scattering (I vs. )

vs k [~ Phononic Dispersion Relation vs. k [~ Phononic Dispersion Relation

Frequency Range

0.2 – 500 GHz> 90 GHz0.0067 – 16.6 cm-1> 3 cm-1

5282.7 ev – 2.07 meV> 0.37 meV

Conclusions and trends• Phonon engineering possible with phononic crystals, cavities, coupled cavities., p

• Phonon sources are needed for progress in the field

• A wealth of device-relevant physics expected• A wealth of device-relevant physics expected

• Nanofabrication (in 3D) and nanometrology developments neededneeded.

• “Heterogeneous” coupled cavities need better description with e g quantum mechanics and elasticity theorywith, e.g., quantum mechanics and elasticity theory.

• Phonon coherence studies in confined structures are expected to become important soon (Impulsive Ramanexpected to become important soon (Impulsive Raman scattering, see work by R Merlin, U Michigan)

53

References (see, for example, in addition to refs in slides)

An early work:D F Sheahan and R A Johnson, Crystal and mechanical oscillators, IEEE Trans Circuits and Systems, Vol CAS-22, pp 69-89, February 1975

On 2D Phononic Crystals:M.S. Kushwaha et al, Phys. Rev. Lett. 71, 2022 (1993)J. Vasseur et al, J. Phys.: Cond. Matter, 7, 8759 (1994)J Vasseur et al J Phys : Cond Matter 9 7327 (1997)J. Vasseur et al J. Phys.: Cond Matter, 9, 7327 (1997)Y. Pennec et al, Optics Express (2010)

On Light scattering:Book series «Topics in Applied Physics” volumes on “Light Scattering in Solids” Ed MBook series «Topics in Applied Physics , volumes on Light Scattering in Solids , Ed M. Cardona, Springer.

On modeling of light scattering and acoustic structures:E H El Boudouti et al Surface Science Reports 64 (2009) 471594E.H. El Boudouti et al., Surface Science Reports 64 (2009) 471594.

On phonon-photon interaction:Lecture Notes of Summer Schools Son et Lumiere 2006, 2008 and 2010, http://prn1.univ-lemans fr/prn1/siteheberge/Son et lumiere/index php?page=1lemans.fr/prn1/siteheberge/Son_et_lumiere/index.php?page=1M Maldovan and EL. Thomas. Appl Phys Lett, 88(25):251907, 2006.

On thermal rectifiers: C Dames J Heat Transfer vol 131 061301-1 to 061301-7 (2009)

54

C Dames, J Heat Transfer, vol 131, 061301-1 to 061301-7 (2009)