light extraction efficiency in iii- nitride light-emitting diodes and piezoelectric properties
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LIGHT EXTRACTION EFFICIENCY IN III- NITRIDE LIGHT-EMITTING DIODES AND PIEZOELECTRIC PROPERTIES IN ZNO NANOMATERIALS by JUNCHAO ZHOU Submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical Engineering and Computer Science CASE WESTERN RESERVE UNIVERSITY August, 2016
Transcript of light extraction efficiency in iii- nitride light-emitting diodes and piezoelectric properties
LIGHT EXTRACTION EFFICIENCY IN III-
NITRIDE LIGHT-EMITTING DIODES AND
PIEZOELECTRIC PROPERTIES IN ZNO
NANOMATERIALS
by
JUNCHAO ZHOU
Submitted in partial fulfillment of the requirements for the degree of
Master of Science
Department of
Electrical Engineering and Computer Science
CASE WESTERN RESERVE UNIVERSITY
August, 2016
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the thesis of
Junchao Zhou
candidate for the Master of Science degree*.
Committee Chair
Dr. Hongping Zhao
Committee Member
Dr. Christian A. Zorman
Committee Member
Dr. Philip Feng
Committee Member
Dr. Roger H. French
Date of Defense
May 31st. 2016
*We also certify that written approval has been obtained for any proprietary material
contained therein.
Acknowledgements
I would like to express my sincere gratitude to my advisor Dr. Hongping Zhao, for her
dedicated help on my Master’s study and related research, for her patience, motivation, and
immense knowledge. Her guidance helped me throughout my research and writing of this
thesis.
Then, I would like to thank Dr. Zorman for his constructive advice for my thesis work
as well as his help on my thesis project experiment. His insights helped me overcome the
difficulties in my experiment.
I would also like to thank Dr. French, Dr. Feng and Dr. Zorman for being on my thesis
committee member and providing insightful comments. Also, I would like to thank Dr.
Ming-chun Huang for the discussion on the applications of ZnO in piezoelectric devices.
In addition, I would like to thank my group members for their advice on my research
projects. I would like to thank Lu Han for her guidance on using the experimental
equipment and Subrina Rafique for providing the materials for experiment.
Finally, I would like to thank my family for their unlimited support and encouragement.
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Table of Content
List of Tables……………………………………………………………………….……..4
List of Figures…………………………………………………………………………5
Abstract……………………………………………………………………………..12
Chapter 1: Introduction……………………………………………………………..13
1.1. InGaN Quantum Wells Light-Emitting Diodes: Problems and Solutions.13
1.1.1.Light Emitting Diodes for Solid State Lighting………………………….14
1.1.2.Problem of Light Extraction in Planar Light-Emitting Diodes (LEDs)…….16
1.1.3.Approaches to Enhance Light Extraction Efficiency for III-nitride LEDs…18
1.1.4.Electromagnetic Guided Modes in Periodic Dielectric Medium………..18
1.1.5.Band Structure and Guided Modes of Photonic Crystals………………..20
1.1.6.Bloch Modes and Light Extraction……………………………………..21
1.2. ZnO Piezoelectric Devices……………………………………………………..24
1.2.1.Piezoelectric Property of ZnO…………………………………………...25
1.2.2.ZnO Piezoelectric devices………………………………………………….26
1.3. Thesis Organization……………………………………………………………27
Chapter 2: Finite-Difference Time-Domain (FDTD) Method for Calculating
Light Extraction Efficiency of Light-Emitting Diodes………………….28
2.1. FDTD Method………………………………………… ……………...……….28
2.1.1. Introduction…………………………………………………………….28
2.1.2. Three-Dimensional FDTD Method and Yee’s Mesh……………………29
2.2. Computational Model…………………………………………….……………34
2.2.1. Light Extraction Efficiency Calculation Method………………………..34
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2.2.2. Photonic Crystal Band Structure Simulation Methodology…………….37
Chapter 3: Analysis of Light Extraction Efficiency for Thin-Film-Flip-Chip
(TFFC) InGaN Quantum Wells (QWs) Blue Light-Emitting Diodes with
Different Structural Design………………………………………………40
3.1. Introduction of InGaN Quantum Wells Blue LEDs…………………………..40
3.1.1. Structure of InGaN QWs Blue LEDs…………………………………….40
3.1.2. Thin-Film-Flip-Chip Technology……………………………………...42
3.1.3. Emission Polarization of InGaN QWs Blue LEDs…………………….....45
3.1.4. Fabrication of Photonic Crystals on TFFC LEDs…………………….46
3.2. Band Structure of 2D Photonic Crystals……………………………….……….49
3.2.1. 2D Simulation of Hexagonal PC…………………………………………49
3.2.2. Physical Meaning of Photonic Band Gap………………………………..53
3.2.3. Transmittivity and Reflectivity of 2D PC Slab………………………….56
3.3. Effect of P-GaN Layer Thickness on Light Extraction Efficiency for Conventional
TFFC InGaN LEDs…………………………………………………………….60
3.3.1. Emission Enhancement by Constructive Interference: Micro-Cavity
Effect……………………………………………………………………..60
3.3.2. FDTD Analysis of the Effect of P-GaN Layer Thickness…………….63
3.4. Effect of Photonic Crystals on Light Extraction Efficiency of Blue LEDs……65
3.4.1. The Simulation Model……………………………………………...……65
3.4.2. Effect of Photonic Crystal Depth d………………………………………70
3.4.3. Effect of the Filling Factor f………………………………………….…..74
3.4.4. Effect of the Lattice Constant a………………………………………….77
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3.4.5. Effect of Dipole Source Position…………………………………………80
3.5. Effect of Cone-Shaped Periodic Nanostructure on Light Extraction Efficiency of
Blue LEDs…………………………………………..……………………….82
3.5.1. Effect of Sharp-tip Cones………………………………………………..83
3.5.2. Effect of Truncated Cones…………………………………………….86
3.5.3. Effect of Dipole Source Position………………………………..………88
3.6. Summary of Light Extraction Efficiency Enhancement for InGaN Blue TFFC
LEDs…………………………………………………………………………89
Chapter 4: Piezoelectric Properties in ZnO Nanomaterials……………………...….90
4.1. Simulation of ZnO Nanostructured Materials……………………………….90
4.2. Transfer of ZnO Nanostructures Grown by Chemical Vapor Deposition ……94
4.3. Wet Etching of PDMS………………………………………………………99
4.4. Dry etching of PDMS……………………………………………………….104
4.5. Summary of ZnO Piezoelectric Force Sensor…………………………….…106
Chapter 5: Conclusions and Future Work……………………………………..….107
5.1. Conclusions ………………………………………….……………………….107
5.2. Future Work……………………………………..…………………………108
Appendix………………………………………………...……………………………109
References…………………………………………………………………...………111
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List of Tables
Table 3.1 Comparison between FDTD simulation and experimental result………………69
Table 3.4.5-1 Weighted average of light extraction efficiency………………………….81
Table 3.5.3-1 Weighted average of LEE for LEDs with truncated cones……………….88
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List of Figures
Figure 1.1.2-1 The illustration of the total internal reflection. amb=ambient,
SC=semiconductor. k0 is the wave number in air………………………………………....17
Figure 1.1.5-1 Dispersion relation for TE mode of a squared lattice photonic crystal of air
holes computed by FDTD and Effective index method…………………………………..21
Figure 1.1.6-1 Left: Two-dimensional photonic crystal using a square lattice. Vector r is
an arbitrary vector. Right: The Brillouin zone of the square lattice, centered at the origin
(Γ). k is an arbitrary in-plane wave vector. The irreducible zone is the light blue triangular
wedge. The special points at the center, corner, and face are conventionally known as Γ,
Μ, and Χ………………………………………………………………………………….22
Figure 1.1.6-2 Ewald construction for a Bloch mode. The wave vector k|| of the main
harmonic is coupled to other harmonics k|| + G by the RL (gray dots). Here, one of the
RL points is in the air circle (inner circle) and radiates to air, while two are in the
substrate circle (outer circle)………………………………………………………………………………23
Figure 1.2.1-1 (a) Wurtzite crystal structure of ZnO61. (b) The formation of electric dipole
under external strain……………………………………………………………………...25
Figure 2.1.2-1 A unit cell of Yee’s lattice with specified position of the field
components………………………………………………………………………………32
Figure 2.2.1-1 Calculating light extraction efficiency using FDTD method: (a) simulation
region setting; (b) Determining the total power emitted from a dipole source using a power
box……………………………………………………………………………….……….36
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Figure 2.2.2-1 Simulation region settings for calculating photonic crystal band structure
using Lumerical’s FDTD solutions. The orange square is the simulation region of one unit
cell. Yellow cross is the field monitor…………………………………………………….38
Figure 2.2.2-2 Recorded signals of a guided mode in a two-dimensional hexagonal
photonic crystal. (a) The recorded time signal; (b) Fourier transform of (a)……………..39
Figure 3.1.1-1 A typical structure of InGaN-based LED grown on sapphire substrate…..42
Figure 3.1.2-1 A typical structure of thin-film flip-chip GaN-based LED on a metal
mirror…………………………………………………………………………………….44
Figure 3.1.2-2 A brief schematic diagram of the fabrication process for the LLO-LEDs. (a)
laser processing, (b) separation, (c) etching of undoped GaN, (d) TFFC LED…………..44
Figure 3.1.3-1 The edge-emitting spectrum of blue InGaN/GaN MQWs LED at 455 nm..46
Figure 3.1.4-1 Illustration of processing flow for the formation of photonic crystals on
LEDs. (a) e-beam resist by spin coating or deposition; (b) patterning by direct write e-beam
lithography; (c) Dry etching of GaN surface; (d) e-beam resist lift-off………………….48
Figure 3.1.4-2 Nanoimprint process for the formation of photonic crystals on LEDs.
(a)form a NIP polymer resist layer by spin coating; (b) the pattern of the stamp is
transferred to polymer resist by imprinting; (c) dry etching of GaN surface; (d) NIP
polymer resist lift-off…………………………………………………………………….49
Figure 3.2.1-1 TE (left) and TM (right) mode light in photonic crystals. Here the two-
dimensional photonic crystals are considered as infinite in the vertical direction……….50
Figure 3.2.1-2 Band structure for photonic crystals of pillars. The left side is for TE mode
and the right side is for TM mode. Photonic crystals of different R/a ratio were analyzed.
The horizontal coordinate is Bloch wave vector in the first Brillouin zone along M-Gamma-
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K-M direction with 15 data points in each direction. The vertical coordinate represents the
Bloch mode frequency normalized by c/a……………………………………………….51
Figure 3.2.1-3 Band structure for photonic crystals of air holes. The left side is for TE mode
and the right side is for TM mode. Photonic crystals of different R/a ratio were analyzed.
The horizontal coordinate is Bloch wave vector in the first Brillouin zone along M-Gamma-
K-M direction with 15 data points in each direction. The vertical coordinate represents the
Bloch mode frequency normalized by c/a……………………………………………….52
Figure 3.2.2-1 Schematic illustration of a PC periodic in one dimension……………….53
Figure 3.2.2-2 (a) Band structure of GaN/InGaN multilayer slab. (b) Band structure of
GaN/air multilayer slab. The right side depicts the energy distribution of the Bloch mode
for band 1 and 2 at the zone edge…………………………………………………………55
Figure 3.2.2-3 Bloch mode profile of hexagonal photonic crystal of pillars for R=0.2a. The
dashed circles represent pillars and the other region is air. (a) the top of band 1 at K point,
f=0.393 c/a. (b) the bottom of band 2 at K point, f=0.6 c/a. TM mode…………………..56
Figure 3.2.3-1 Theoretical and simulation results for transmittivity and reflectivity for
conventional GaN-based blue LEDs. (a) TE mode; (b) TM mode………………………58
Figure 3.2.3-2 Effect of photonic band gap on transmission coefficient. (a) The band
structure of 2D PC slab of pillars at R=0.2a; (b) Possible diffraction options of incident
light; (c) The transmission coefficient of the PC slab with a=207nm, r=0.2a, TM mode for
different wavelength…………………………………………………………………...…59
Figure 3.3.1-1 Illustration for the interference of original top emitting light and the light
reflected by the mirror……………………………………………………………………61
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Figure 3.3.2-1 The effect of p-GaN layer thickness on the light extraction efficiency for
TFFC GaN LEDs. (a) a schematic for the structure of the TFFC GaN LED, n-GaN
thickness is 3 μm. (b) dependence of light extraction efficiency on p-GaN layer thickness
for TFFC GaN LED with flat surface at λ=460nm. Solid triangular dots and dashed line
represent the FDTD simulation results and the fitting curve, respectively………………64
Figure 3.4.1-1 The simulation model for thin-film flip-chip GaN-based LED. r is the radius
of the pillar or air holes; a is the lattice constant; d is the depth of the photonic crystal…67
Figure 3.4.1-2 Top view of the simulation region. The orange square region is the
simulation region from top-down view. The structural parameters of photonic crystals are
arbitrarily chosen. ………………………………………………………………….……67
Figure 3.4.1-3 Light extraction efficiency as a function of simulation time. The structural
parameters of the photonic crystal are a=600nm, r=199nm, d=200nm, f=0.4. Absorption
loss is neglected……………………………………………………….………………….69
Figure 3.4.1-4 The Archimedean A13 lattice. a is the distance between the center of a hole
to the center of its neighbor. a’ the base vector of the larger hexagonal unit cell…………69
Figure 3.4.2-1 The dependence of light extraction efficiency on PC depth d for TE
polarized TFFC InGaN QWs LEDs with optimized p-GaN thickness (330nm for
λpeak=460nm). (a) PC of pillars; (b) PC of air holes. Both are hexagonal lattice……….…72
Figure 3.4.2-2 (a) Transmittivity and (b) reflectivity of hexagonal PC of pillars for different
thickness of the PC layer for TE polarized TFFC InGaN QWs LEDs with optimized p-
GaN thickness (330nm for λpeak=460nm). Light is in Γ-K direction…………………...…73
9
Figure 3.4.3-1 Light extraction efficiency of TFFC PC GaN LED for TE polarized emission
as a function of filling factor f with optimized p-GaN thickness (330nm for λpeak=460nm).
(a) PC of pillars; (b) PC of air holes. Both are hexagonal lattice………………….……..75
Figure 3.4.3-2 Transmittivity and reflectivity of hexagonal PC of pillars for different filling
factor of PC for TE polarized TFFC InGaN QWs LEDs with optimized p-GaN thickness
(330nm for λpeak=460nm). (a) light is in Γ-K direction; (b) light is in Γ-M direction……..76
Figure 3.4.4-1 Light extraction efficiency of TFFC PC GaN LED for TE polarized emission
as a function of lattice constant a with optimized p-GaN thickness (330nm for
λpeak=460nm). (a) PC of pillars; (b) PC of air holes. Both are hexagonal lattice………….79
Figure 3.4.5-1 Illustration of dipole source position changing along Γ-K direction. a is the
lattice constant…………………………………………………………………..……….80
Figure 3.4.5-2 Source position dependence analysis of light extraction efficiency for TE-
polarized TFFC PC GaN LED. The position of the dipole source is changed along Γ-K
direction. The optimized p-GaN thickness is 330nm for λpeak=460nm. The depth of PC
layer is 200nm. (a) PC of pillars; (b) PC of air holes. Both are hexagonal lattice……….81
Figure 3.5.1-1 The effect of cones on light extraction efficiency for InGaN-MQW TFFC
LEDs. (a) The simulation model. (b) The light extraction efficiency as a function of bottom
radius r/R……………………………………………………………………………….84
Figure 3.5.1-2 The effect of cones on light extraction efficiency for InGaN-MQW TFFC
LEDs as a function of etching depth d. The lattice constant a is 600nm, d/rbot = 0.74…..85
Figure 3.5.2-1 The effect of truncated cones on light extraction efficiency for InGaN-
MQW TFFC LEDs as a function of rtop/rbot for different filling factor f……………….87
10
Figure 3.5.3-1 Light extraction efficiency as a function of dipole source position for
InGaN-MQW TFFC LED at fixed a=800nm and d=200nm. Four curves are rbot=230nm,
297nm, 350nm, 400nm. The top radius rtop is chosen according to the peak values from
Figure 3.5.2-1…………………………………………………………….………………88
Figure 4.1-1 Simulation of ZnO nanowire and hexagonal nanowall piezoelectric effect. The
radius and length of the nanowire are 50 nm and 1200 nm respectively. The side length and
the wall thickness of the hexagonal nanowall are 500 nm and 100 nm while the height is
1000nm. The potential of bottom surface is set to ground potential. The top surface is
applied with an external load. (a)(c) Fx=0 nN, Fy = 0 nN, Fz = -80 nN; (b)(d) Fx=0 nN,
Fy= 80 nN, Fz=0 nN…………………………………………………….………………..92
Figure.4.1-2 Output voltage of a single ZnO nanowire with different height under various
external load conditions. (a)(b)(c) show the results of ZnO nanowires with radius of 25nm,
50nm, and 200nm. (d) shows the schematic of an array of ZnO nanowires as a force
sensor…………………………………………………………………………………….94
Figure 4.2-1 The flow chart of the transfer process of ZnO to a conductive substrate. ...95
Figure 4.2-2 Transfer of ZnO nanostructure onto PDMS stamp. (a) The cured PDMS was
peeled off from ZnO sample. (b) (c) (d) are microscope images of original sample, sample-
after-peel-off-PDMS and the surface of PDMS stamp, respectively……………………..97
Figure 4.2-3 The area of ZnO nanowires that were not transferred to PDMS stamp. (a) The
sample after peeling off PDMS sheet. (b) the surface of PDMS stamp. ………………..98
Figure 4.3-1 (a) The thickness of PDMS stamp before doing wet etching. (b) The PDMS
stamp bonded to a glass substrate………………………………………………………..99
11
Figure 4.3-2 Etching rate of PDMS using TBAF/NMP solution. a) the PDMS sample with
a beam on top before etching; b) etched for 30min in TBAF in NMP with a ratio of 1:3; c)
etched for 1 hour in TBAF in NMP with a ratio of 1:3; d) etched for another 30 min after
c) in TBAF in NMP with a ratio of 1:6…………………………………………………..102
Figure 4.3-3 The width of the PDMS beam as a function of etch time. Stage I &II: TBAF:
NMP=1:3; Stage III: TBAF:NMP=1:6…………………………………………..……..103
Figure 4.3-4 Top morphology of PDMS with ZnO nanostructure underneath. (a) The top
surface texture before wet etching. (b)(c)(d) top morphology of PDMS etched for
1h30min…………………………………………………………………………..…….103
Figure 4.4-1 (a) PDMS spin-coated on a silicon wafer; (b) The thickness of the PDMS thin
film as a function of RIE etching time. The film thickness was measured using a thin film
measurement system from FILMETRICS…………………………………………..….104
Figure 4.4-2 The surface of PDMS stamp before and after RIE etching. (a) The PDMS
surface morphology after wet etching. (b) the wet-etched PDMS surface morphology after
dry etching………………………………………………………………………..……..105
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Light Extraction Efficiency in III-nitride Light-Emitting Diodes
and Piezoelectric Properties in ZnO Nanomaterials
Abstract
By
JUNCHAO ZHOU
III-nitride semiconductors based light-emitting diodes (LEDs) for solid state lighting
represent the most promising next generation lighting technology. Total internal reflection
significantly limits the light extraction efficiency in III-nitride LEDs due to the large
refractive index of GaN. In this thesis, the light extraction efficiency for III-nitride LEDs
was calculated and analyzed using 3D finite difference time domain (FDTD) method.
Particularly, detailed studies focused on understanding the mechanisms of enhancing light
extraction efficiency in LEDs by using photonic crystals. We found that the effect of
photonic crystals does not rely on the photonic band gap effect but is more related to
scattering and diffraction. Besides, we also studied the effects of cones and truncated cones
on the enhancement of light extraction efficiency in III-nitride LEDs, which can be used
as alternatives to PCs. In addition, ZnO represents another type of wide-bandgap material.
The wurtzite crystal structure of ZnO possesses strong piezoelectric properties. In the
second part of this thesis, we investigated a low cost and scalable approach using PDMS
to fabricate a ZnO piezoelectric device based on ZnO nanomaterials grown via chemical
vapor deposition (CVD) method.
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Chapter 1: Introduction
Gallium nitride (GaN) and Zinc oxide (ZnO) are two important wide-bandgap
semiconductor materials. GaN has received great attention since the first high-power GaN-
based blue light-emitting diode (LED) was demonstrated in 1991. The success of using
GaN-based LEDs as a blue light source led to the fast development of solid-state lighting.
ZnO is one of the metal oxide materials and exhibits several favorable properties, including
good transparency, high mobility and wide bandgap. These properties are used in various
applications such as transparent electrodes in liquid crystal displays, thin-film transistors
and LEDs. ZnO also exhibits strong piezoelectric property. Piezoelectric energy harvesters
based on ZnO nanowires was first demonstrated in 2006. Combining with other flexible
materials, ZnO piezoelectric nanogenerators can be used to power electronics for wearable
devices.
1.1. InGaN Quantum Wells Light Emitting Diodes: Problems and
Solutions
Lighting is essential for our daily lives after Edison first popularized incandescent light
bulbs in the late 1800s. Incandescent light source is a heat source in nature. The filament
of an incandescent light bulb operates at 3500K and glows white-hot, radiating white-light.
This will result in energy loss inevitably since large amount of energy is emitted as heat.
Only 5% of the input electrical energy is converted to visible light for an incandescent light
bulb. The typical efficacy and lifetime of an incandescent light bulb is only 15 lm/W and
1000 h respectively. The second generation of light source is fluorescent light tube. The
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inner surface of the fluorescent light tube is coated with phosphor, which can emit visible
light once exposed to ultra-violet (UV) light. The efficacy of a fluorescent light tube is 60-
100 lm/W and the efficiency is typically 25%. Incandescent and fluorescent light source
are two main light sources used around us. It is estimated that about 20% to 30% of the
U.S. electrical use is for lighting, but only about 25% of such electrical energy is converted
into light1. Thus, it is important to seek more efficient lighting technology to save energy.
Light-emitting diode (LED) is considered as the most promising next generation solid
state lighting. It exceeds other traditional lighting technologies in efficacy and lifetime.
With the technology development of white-light LED, the efficacy of color-mixed LED
packages is expected to go beyond 250 lm/W by the year of 20252. A report conducted by
McKinsey&Company showed a bright future for LED market 3 . White-light LED is
believed to replace the traditional light sources in our homes and work places. This is
beneficial in energy-saving and reducing carbon dioxide release. However, achieving high
efficiency white-light LED still has a long way to go. The total electric-to-light conversion
efficiency of current white-light LEDs is still far below the theoretical limit. Low crystal
quality and high refractive index of GaN are two main reasons for the low efficiency.
1.1.1. Light-Emitting Diodes For Solid State Lighting
Russell Ohl first discovered p-n junctions accidentally while he was testing several
silicon crystal samples at Bell Telephone Laboratories in 19404,5. Almost during the same
period, the quantum theory of solids was developed, providing foundations for
understanding the physical process occurring during the light emission in semiconductor
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p-n junction diode. Thereafter, researches on p-n junction electroluminescence led to the
real development of today’s high-brightness LEDs.
By investigating the electronic band structure of semiconductor materials such as Si,
Ge and SiC, it is found that they cannot be made to efficient LEDs since their bandgap is
indirect. Electron-hole recombination process in indirect bandgap materials is very
inefficient. Later in 1950s, binary compound semiconductor materials consisting of group-
III and group-V elements were investigated for light generation. However, these materials
such as GaAs and InAs have band-gap energies in the infrared region, which are not
suitable for lighting application. GaP, which is able to emit visible orange light, is an
indirect band-gap material, so just like Ge and Si, light emission is not efficient. Thus, new
materials were required to get the desired visible light emission.
With the development of new technologies for materials growth such as liquid phase
epitaxy (LPE) and metal-organic chemical vapor deposition (MOCVD), novel materials
were created with the ability to emit light in the visible range. In 1962, Holonyak and
Bevacqua first demonstrated ternary GaAs1-xPx p-n junctions emitting red light6. At that
time, the efficacy of such red-light LED was only 0.1 lm/W7. With the maturity of material
epitaxy growth technologies, researchers were able to create other ternary and quaternary
alloys such as AlGaAs, GaAlAsP and AlInGaP. The efficacy of LEDs based on these
materials first exceeded 100 lm/W in late 19904. However, the bandgap energy of these
materials is in the range of infrared to yellow, which is not enough for producing white
light.
There are generally two ways to generate white light. One way is by stimulating a
phosphor with ultra-violet light (UV). Another is by combining light of three different
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colors - red, green and blue. Hence, semiconductors capable of producing light from blue
to UV are needed. Early attempts to produce blue light were focused on SiC8,9. However,
such devices were inefficient (0.03% power efficiency), since SiC has an indirect
bandgap10. GaN is an alternative to SiC. GaN is a direct-bandgap material with a 3.45 eV
bandgap, which corresponds to near-ultraviolet emission (364 nm). The bandgap can be
tuned by adding Al to form AlGaN quantum wells so that blue light can be emitted.
Although GaN was identified as a suitable material for blue LEDs in the late 1960s by H.P.
Maruska, J.E. Berkeyheiser and coworkers, no success was achieved to fabricate high
efficient GaN blue LED at that time11,12. This is because high quality p-type GaN is hard
to obtain. In the late 1990s, Akasaki and coworkers at Nagoya University successfully
synthesized high-quality GaN using the MOCVD technique and demonstrated light
emission from GaN p-n junction LED using Mg-doped GaN as the p-type material13,14,15.
Shortly afterwards, in 1991, Shuji Nakamura at Nichia Corporation fabricated high-power
p-n junction blue LEDs using GaN films grown with GaN buffer layers for the first
time16,17,18. And they showed external quantum efficiency as high as 0.18%, which was
almost ten times higher than that of SiC blue LEDs17. Nowadays the external quantum
efficiency of blue GaN LEDs can achieve as high as 80% or higher19.
1.1.2. Problems of light extraction in planar light-emitting diodes
The external quantum efficiency (EQE) of an LED can be divided into three main parts:
(1) injection efficiency, (2) internal quantum efficiency (IQE) and (3) light extraction
efficiency (LEE). The injection efficiency and IQE largely depend on the quality of the
crystal. Injection efficiency and IQE together measure how efficient photons can be
17
generated from the input electrical power. Light extraction efficiency tells how efficiently
the light generated in the LED can be emitted out of the semiconductor material. While the
internal quantum efficiency of GaN based LEDs can go beyond 80% in experiments and
is predicted to be capable of reaching 90% 20,21,22, the total external quantum efficiency is
still low due to limited light extraction efficiency. The main cause of low light extraction
efficiency is total internal reflection (TIR). As shown in Figure 1.1.2-1, according to Snell’s
law θc=asin(namb/nSC), only light with incident angle (θSC) smaller than the critical angle
(θc) can go through the top surface23. The cone-shaped solid angle is called escape cone.
With some simple derivation, it can be shown that the light extraction efficiency is only
about 1/4n2. As for GaN (nGaN=2.5), there are only 4% of the generated photons that can
go out of the semiconductor material to the air.
Figure 1.1.2-1 The illustration of the total internal reflection.
amb=ambient, SC=semiconductor. k0 is the wave number in
air23.
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1.1.3. Approaches to Enhance Light Extraction Efficiency for III-nitride
LEDs
Several methods were proposed to increase the LED light extraction efficiency. The
first type of mechanism is to reduce total internal reflection such as integrating a
hemisphere on the chip24, concave hemisphere patterned ITO layers25, nano-texturing the
thin film surface26,27 and surface roughening28,29. Another way is to use light coupling and
diffraction in periodic structure such as periodic micropit30, micro-lens arrays31,32, and
photonic crystal19,33,34,35. Among all the proposed approaches, the photonic crystal (PC) is
the most efficient for enhancing the light extraction efficiency. All of the above schemes
can be combined with micro-cavity effect36, chip shaping37,38, and high-reflective bottom
mirror39 to further increase the light extraction efficiency.
1.1.4. Electromagnetic Guided Modes in Periodic Dielectric Medium
Applying Maxwell’s equations into dielectric medium, one can derive the master
equation
(1)
Hence, the problem of solving Maxwell’s equations becomes an eigenvalue problem.
Equation (1) can also be written as
(2)
Ñ´1
e(r)Ñ´H(r)
æ
èç
ö
ø÷=
w
c
æ
èç
ö
ø÷
2
H(r)
)()(
2
rHrH
c
19
together with , , we can solve the electric and magnetic field
everywhere in the medium, can be written as
(3)
The above equations are based on the following assumptions: first, the field strength
should be small so that the displacement of field has a linear relation with the electric
field ; second, the material should be macroscopic and isotropic so that and
are related by multiplied by a scalar dielectric function ; third, we ignore
any frequency dependence of the dielectric constant; fourth, we mainly focus on
transparent materials whose dielectric constant can be considered as purely real and
positive. Gallium nitride is an isotropic material and is transparent in the wavelength range
around 460nm, which meets with the conditions above.
To solve the above eigenvalue problem to obtain the electromagnetic modes allowed
for the dielectric medium, we need to take the symmetric property of the dielectric medium
into consideration. If the system has continuous translational symmetry, e.g. free space, we
know that the modes have the form of plane waves
(4)
where is a constant vector and . The eigenvalues are ,
which gives the dispersion relation, .
Considering the electromagnetic waves propagating in the periodic dielectric medium
with discrete translational symmetry in just one direction, that is , is an
integer. As is described by Bloch’s theorem, the eigenfunctions should have the form of
0)( rH 0)]()([ rEr
)(rE
)()(
)(0
rHr
rE
i
)(rD
)(rE )(rE
)(rD 0 ),( r
H(r) =H0 ×eik×r
0H 00 kH /)/( 222 kc
/kc
)ˆ()( yla rr l
20
(8)
is a periodic function. For a three dimensional periodic system, the
more general solution is
(9)
where is the Bloch wave vector which lies in the first Brillouin Zone and
for all lattice vector R.
1.1.5. Band Structure and Guided Modes of Photonic Crystals
The behavior of a wave is mainly determined by its frequency or angular frequency
and wave vector . Solving the eigenvalue problem described in section 1.1.4
gives the relation between and , which will eventually form a whole spectrum of
modes allowed to propagate in the medium. For the light propagating in an infinite isotropic
medium, its frequency has a linear relation with wave number e.g. . There is
a certain frequency value corresponding to a certain wave number. The spectrum is a
continuous range of frequency.
),()(, zyeeyikxik
kkyx
yxurH
),(),( zlayzy uu
)()( rurH k
rk
k
ie
k
uk(r) = uk(r+R)
f
w = 2p f k
k
/kc
21
The behavior of light in a periodic structure will have some unique properties. The
electromagnetic modes are Bloch modes like instead of simple
harmonics like in a continuous isotropic medium. Because of the
periodicity, the system cannot tell from k to k + G (G is the reciprocal lattice vector). So
the spectrum can be restricted to the first Brillouin zone. For each value of k, there will be
an infinite set of modes with discretely spaced frequencies, which can be labeled as a band
index n, as shown in Figure 1.1.5-140.
1.1.6. Bloch Modes and Light Extraction
A crystal is formed by a large amount of atoms arranged in a highly ordered periodic
structure. Photonic crystals are analogous to atomic crystals. However, the periodic unit
consists of macroscopic media with different dielectric constants instead of atoms. Figure
)()( rurH k
rk
k
ie
H(r) =H0 ×eik×r
Figure 1.1.5-1 Dispersion relation for TE mode of a squared lattice photonic crystal of air
holes computed by FDTD and Effective index method.40
22
1.1.6-1 shows an example of two-dimensional photonic crystal. The yellow circles can be
either rods or holes with different dielectric constants from the red region.
By solving the Maxwell equations, we can get the master equation40:
where ε(r) is the permitivity function of the material; Η(r) is the magnetic field; ω is the
angular frequency of the wave and c is the speed of light in vacuum.
The solution to this equation is a linear combination of in-plane harmonic waves40:
which is interpreted as Bloch’s theory. E is the electric field, G is reciprocal lattice (RL)
vector and k|| is the in-plane wave vector. There is a fundamental mode k|| which will carry
most of the total energy of Bloch mode. Figure 1.4 shows how the photonic crystal
Ñ´1
e r( )Ñ´H(r)
æ
èçç
ö
ø÷÷=
w
c
æ
èç
ö
ø÷
2
H(r)
E = EGei(k||+G)×r
G
å
Figure 1.1.6-1 Left: Two-dimensional photonic crystal using a square lattice. Vector r is an
arbitrary vector. Right: The Brillouin zone of the square lattice, centered at the origin (Γ). k is an
arbitrary in-plane wave vector. The irreducible zone is the light blue triangular wedge. The special
points at the center, corner, and face are conventionally known as Γ,Μ,and Χ.40
23
reciprocal lattice can help redirect the light to the escape cone. The inner circle is the
projection of the extraction cone and the larger circle is the substrate cone. If the wave
vector is inside the extraction cone, then such light can be emitted outwards to air. If the
wave vector is in the substrate cone, then such light can enter the substrate region. If there
is a reciprocal lattice point in the extraction cone, the in-plane wave vector k|| can be
redirected to the extraction cone by adding or subtracting a reciprocal lattice vector G. If
the substrate is replaced with a mirror, then light can be reflected back and be diffracted by
the PC again, increasing the light interaction with the PC.
A good photonic crystal structure should be able to extract light coming from all
different directions and have strong coupling efficiency41. Thus, it is important to optimize
the PC lattice structure and the parameters such as hole diameter, depth and lattice constant.
Figure 1.1.6-2 Ewald construction for a Bloch mode. The wave vector k|| of the main
harmonic is coupled to other harmonics k|| + G by the RL (gray dots). Here, one of the
RL points is in the air circle (inner circle) and radiates to air, while two are in the
substrate circle (outer circle).41
24
1.2. ZnO Piezoelectric Devices
Advancements of technology in the field of wearable devices and wireless sensor
networks allow gathering information about a human body’s health condition more easily
for enhancing the life quality of human kinds42-48. These devices currently rely on the use
of electrochemical batteries for supplying electrical power. However, the lifespan of
batteries couldn’t support the sustainable operation of these devices. Using batteries also
limits the weight and miniaturization of wearable electronic devices. Thus, it is significant
to seek an alternative energy source which can be made small and has a long lifespan.
Mechanical energy would be a promising energy source for the continuous operation of
micro bio-sensors, nanorobotics and wearable personal electronics since small mechanical
vibrations commonly exist in the operation condition of these electronics49,50. Efficient,
clean, varied shaped, convenient for maintenance, piezoelectric energy harvesters have
attracted lots of attention. The most commonly used piezoelectric material is ceramic PZT
which has a high energy conversion efficiency. However, being ceramic in nature, bulk or
thin-film PZT would crack easily under twisting, wrapping, pressing or distortion. Also
PZT cantilever devices only provide high efficiency when the external vibration frequency
matches its resonant frequency which is not popular in natural low-frequency agitations
found in our living environment51. To make PZT flexible, PZT nanofibers have been
fabricated52. Another common piezoelectric material is PVDF polymer. Harvesting energy
from respiration 53 , walking 54 and shoulder straps of the backpack 55 can be realized.
However, compared with ceramic piezoelectric material, the power output of PVDF was
much lower. Since piezoelectric ZnO nanowire was first demonstrated to be an energy
harvesting nanogenerator in 2006 56 , many studies on ZnO nanogenerator have been
25
conducted. Several advantages of ZnO such as combining both semiconducting and
piezoelectric properties, relatively bio-safe and biocompatible, and multitudinous
configurations of nanostructures56, made ZnO more useful for powering nanodevices57-60.
1.2.1. Piezoelectric Property of ZnO
The origin of the piezoelectric effect of ZnO is attributed to its wurtzite crystal structure.
Figure 1.2.1-161,62 shows the minimum non-repeatable cell of ZnO wurtzite structure,
where tetrahedrally coordinated O2- and Zn2+ are stacked layer by layer. This minimum cell
will form a hexagonal unit cell after performing a 3-fold symmetry operation with
a=0.3296 and c=0.52065 nm). Without stress or strain, the charge-center of positive Zn2+
cations and the charge-center of 4 negative O2- are overlapped with each other, showing no
electric field. Under an external strain along c-axis, the structure will deform so that the
charge-centers of cations and anions separate, resulting in an electric dipole as shown in
Figure 1.2.1-1 (b). Because of the ordered structure of the ZnO crystal, all of the dipoles
are aligned in the same direction. In a macroscopic view, all the dipoles added together will
produce a static electric field inside the material, giving rise to an electromotive force. This
force will cause free charges to collectively move to one side of the ZnO material,
producing an inverse electric field. With a low doping of free charge carriers, the strained
ZnO structure will preserve a considerable electric potential.
26
1.2.2. ZnO Nanostructure Piezoelectric Devices
A variety of ZnO nanogenerators have been investigated including lateral ZnO
nanowires 63 , vertical nanowire arrays 64 - 67 , nanosheets or nanowalls 68 , 69 and hybrid
piezoelectric structure70. Since ZnO nanomaterial could generate electrical signal under
external force, self-powered force/pressure sensor could be made for robotics and other
medical purposes. Quantization of socket/stump interface force such as pressure and shear
force is critical for the design of the prosthetic socket to reduce discomfort, pain, skin
irritation, pressure ulceration and associated infection. Previously, force sensing resistors
are used to interpret the pressure71,72. However, this type of sensor requires batteries as an
external power supply. Here, we propose to use ZnO as a force sensor with a particular
design to interpret the force such as pressure and shear force at the interface between two
objects. Such sensors can be used in prosthetic socket to measure the force at socket/stump
interface without external power needed.
Figure 1.2.1-1 (a) Wurtzite crystal structure of ZnO61. (b) The formation of
electric dipole under external strain62.
(a)
27
1.3. Thesis Organization
This thesis is composed of 4 chapters. Chapter 1 introduces III-nitride LEDs, the
challenges and solutions and ZnO piezoelectric properties and devices. Chapter 2 gives an
introduction to the FDTD method and the simulation methodology for light extraction
efficiency of LEDs and band structure of photonic crystals. Chapter 3 focuses on the
analysis of light extraction efficiency for InGaN-MQW LEDs as well as the explanation of
effect of photonic crystals and other periodic structures. Chapter 4 discusses the
piezoelectric property of ZnO nanostructure as well as the proposed fabrication process for
ZnO piezoelectric devices using CVD-grown ZnO nanowires and nanowalls.
28
Chapter 2: Finite-Difference Time-Domain (FDTD) Method
for Calculating Light Extraction Efficiency of Light-Emitting
Diodes
2.1. Finite-Difference Time-Domain Method
2.1.1. Introduction
Numerical modeling of electromagnetic waves could provide insightful information for
the design of antenna, radar, satellite, photonic devices, medical imaging and other
applications. Maxwell’s equations are the basis for describing the behavior of
electromagnetic waves. However, the actual analytic solution of the Maxwell equations is
complicated and is not realistic to provide a whole field profile for a large space with mixed
boundary conditions. With the development of computer technology, fast computational
speed and more memory space broaden the applications of computational electromagnetics.
For realistic problems, computational electromagnetics typically makes reasonable and
efficient approximations to Maxwell’s equations. All the computational methods for
electromagnetics can be divided into two categories – time domain and frequency domain.
Based on these two concepts, three simulation algorithms are most widely used to provide
near perfect results. The three methods are finite element method (FEM), method of
moments (MoM) and FDTD, respectively. FEM and MoM are applied to frequency domain
while FDTD works in time domain. Each one has its own advantages and disadvantages.
MoM normally only discretizes the surface of antenna and scatterer and as a result, it is
more popular in solving antenna radiation and scattering problems73. FEM uses tetrahedral
meshes in three dimensions and thus is more powerful in dealing with complex
29
geometries74. On the other hand, FDTD only uses rectangular grids for meshing which
makes it difficult to deal with structures with oblique or curved boundaries compared with
FEM74. However, the FDTD method solves the Maxwell’s equations with a series of time-
stepping functions explicitly, that is, the current field values for a certain grid point are
obtained based on the previous values of itself and its neighbor grid points. This makes
FDTD method more efficient than FEM for simulating time-evolution problems. For FEM,
it is impossible to explicitly derive formulas for updating the fields in time-domain in the
general case74.
To simulate the light extraction efficiency of LEDs requires collecting the extracted
energy of light continuously. This involves calculating electromagnetic fields in a time-
evolution manner. The structure of most LED chips is a rectangular shape which doesn’t
require complex meshing technique. The time-stepping formulas of FDTD makes it
efficient for simulating the light extraction efficiency of LEDs and the weakness of using
rectangular mesh could be avoided when dealing with LEDs of simple rectangular structure.
In this work, commercial electromagnetic solver ‘FDTD solutions.2015b.527’ from
‘Lumerical Solutions, Inc’ was used to calculate the light extraction efficiency for GaN-
based blue LEDs75.
2.1.2. Three-dimensional FDTD method and Yee’s Cell
Applying Maxwell’s equations to get the x, y, z components of electromagnetic fields
with finite differences in space was first introduced by Kane S. Yee in 196676. This method
was then developed and refined in all its theoretical and computational aspects by many
other researchers. The major attraction of this method attributes to the power and simplicity
30
it provides. Using FDTD, the propagation of electromagnetic waves and its interaction with
materials can be directly shown visually.
As explained in appendix A, in an isotropic medium, Maxwell’s curl equations in
differential forms can be written as:
t
HE (2.1a)
t
EEH (2.1b)
In equation 2.1, vectors E and H represent the magnitude and direction of the electric and
magnetic field, which can be further separated into three spatial components in Cartesian
coordinate system, xE , yE , zE and xH , yH , zH . This leads to a set of six scalar equations
as follows in equations 2.2a-f 77:
y
E
z
E
t
H zyx
1 (2.2a)
z
E
x
E
t
Hxzy
1 (2.2b)
x
E
y
E
t
H yxz
1 (2.2c)
x
yzx Ez
H
y
H
t
E
1 (2.2d)
y
zxyE
x
H
z
H
t
E
1 (2.2e)
z
xyz Ey
H
x
H
t
E
1 (2.2f)
Based on the above set of differential form of Maxwell’s equations, electromagnetic fields
extending in space and time could be solved in three spatial directions. The value of electric
31
(or magnetic) field components could be obtained from magnetic (or electric) field
components in other directions along with the previous value of itself.
Applying finite difference methods to Maxwell’s equations requires performing
segmentation to space and time. The space, typically the simulation region, is divided into
tiny grids, which should be small enough than certain fractions of wavelength according to
the sampling theorem. If the dimension of the system is too large compared with the
wavelength, it would be unrealistic for doing the simulation since the computer resources,
such as memory space and computing speed, are limited.
In the Cartesian (rectangular) coordinate system, a grid point can be defined using
Yee’s notation as:
),,(),,( zkyjxikji (2.3)
and any function of space and time can be represented as:
),,,(),,( tnkjiFkjiF n (2.4)
where zyx is the space increment, t is the time increment, while i , j , k
and n are integers. Thus, assuming second-order approximation for space and time, the
partial derivatives of a function could be expressed as77:
2),,2/1(),,2/1(),,(
O
kjiFkjiF
x
kjiF nnn
(2.5a)
22/12/1 ),,(),,(),,(
tOt
kjiFkjiF
t
kjiF nnn
(2.5b)
32
In order to apply equation (2.5a) to all the space derivatives in equation (2.2), specific
positions of the components of E and H are chosen by Yee in a unit cell of the lattice as
illustrated in Figure 2.1.2-1.
With spatially defined field components together with equation (2.5a) and (2.5b), explicit
finite difference approximation of equation 2.2 can be expressed as77:
)2/1,1,()2/1,,(
),2/1,()1,2/1,()2/1,2/1,(
)2/1,2/1,(
)2/1,2/1,(
2/1
2/1
kjiEkjiE
kjiEkjiEkji
tkjiH
kjiH
n
z
n
z
n
y
n
y
n
x
n
x
(2.6a
)
Figure 2.1.2-1 A unit cell of Yee’s lattice with specified position of the field
components77.
33
)1,,2/1(),,2/1(
)2/1,,()2/1,,1()2/1,,2/1(
)2/1,,2/1(
)2/1,,2/1(
2/1
2/1
kjiEkjiE
kjiEkjiEkji
tkjiH
kjiH
n
x
n
x
n
z
n
z
n
y
n
y
(2.6b
)
),2/1,1(),2/1,(
),,2/1(),1,2/1(),2/1,2/1(
),2/1,2/1(
),2/1,2/1(
2/1
2/1
kjiEkjiE
kjiEkjiEkji
tkjiH
kjiH
n
y
n
y
n
x
n
x
n
z
n
z
(2.6c
)
)2/1,,2/1()2/1,,2/1(
),2/1,2/1(),2/1,2/1(),,2/1(
,,2/1),,2/1(
),,2/1(1),,2/1(
2/12/1
2/12/1
1
kjiHkjiH
kjiHkjiHkji
t
kjiEkji
tkjikjiE
n
y
n
y
n
z
n
z
n
x
n
x
(2.6d
)
),2/1,2/1(),2/1,2/1(
)2/1,2/1,()2/1,2/1,(),2/1,(
,2/1,),2/1,(
),2/1,(1),2/1,(
2/12/1
2/12/1
1
kjiHkjiH
kjiHkjiHkji
t
kjiEkji
tkjikjiE
n
z
n
z
n
x
n
x
n
y
n
y
(2.6e
)
)2/1,2/1,()2/1,2/1,(
)2/1,,2/1()2/1,,2/1()2/1,,(
2/1,,)2/1,,(
)2/1,,(1)2/1,,(
2/12/1
2/12/1
1
kjiHkjiH
kjiHkjiHkji
t
kjiEkji
tkjikjiE
n
x
n
x
n
y
n
y
n
z
n
z
(2.6f
)
As illustrated in Figure 2.1 and equations (2.6a-f), the components of E and H are
interlaced with each other and are evaluated at alternate half-time steps. For example,
)2/1,2/1,0(2/1n
xH is obtained from the value of itself in one-time step before together with
)1,2/1,0(n
yE , )0,2/1,0(n
yE , )2/1,0,0(n
zE and )2/1,1,0(n
zE in half-time step before.
34
Based on this model, Maxwell’s equations are solved discretely in time and space.
When the initial condition of the simulation regions is known and a particular boundary
condition is defined, the electric and magnetic fields can be calculated continuously in time
everywhere in the simulation region.
2.2. Computational Method
2.2.1. Light Extraction Efficiency Calculation Method
From classical electromagnetics, the energy flow of an electromagnetic wave per unit
area is given by:
HES (2.7)
where S is termed the Poynting vector, and indicates both the direction of the energy flow
and its magnitude per unit area. Since the electric field E and magnetic field H
of the
electromagnetic waves are always perpendicular to each other, the magnitude of the energy
flow is given by:
EHS (2.8)
This expression gives you the intensity of the energy flow at any point in the
electromagnetic wave at one specific instant of time. However, this is not particularly
useful. Since the frequency of the light wave is high, approximately 1014 cycles per second,
this means that the intensity of the wave cycles through maximum and minimum values
every 10-14 s. The recorded electric and magnetic field could be at the maximum or
minimum point at that specific instant of time, which could not be used to represent the
actual energy flow over t . A much more useful expression would be the average intensity
of the wave per unit area per unit time. Since the electromagnetic wave could be
35
represented as sine or cosine functions, the average value will have a factor of ½ to the
maximum magnitude of Poynting vector. Therefore, the average intensity of an
electromagnetic wave is:
0
maxmax
2
BES
(2.9)
Since
cB
E
(2.10)
the average intensity can also be written in terms of just the electric field maximum as:
0
2
max
2 c
ES
(2.11)
The settings of the simulation region is shown in Figure 2.2.1-1 (a). Due to the
limitations of computer memory, the simulation region could not be set too large. In this
work, the length of the simulation region is about 10 m in x and y direction and 5 m in
z direction. The mesh step is 15 nm. The required memory space is about 1.5G. However,
an actual LED chip is typically several millimeters wide and a couple hundred microns
thick. Therefore, in order to simulate the light extraction of an LED, perfect mirror
boundary is used for the four sides to mimic the infinite length of x and y dimension
compared with the vertical dimension. Perfect matching layer (PML) boundary is used to
absorb the light to imitate an open space.
The light extraction efficiency could be defined as the ratio of total extracted light
power to the total emitted light power of the active region. To collect the extracted light
from the LED, an electromagnetic (EM) field monitor was placed above the surface of the
LED as shown in Figure 2.2.1-1 (a). The field monitor will automatically collect the
36
Poynting vectors for each unit cell of the mesh. The extracted power from LED top surface
can be obtained by integrating the Poynting vectors over the monitor plane.
A dipole source is placed at the center of the active region to generate light to the
simulation region. A box of power monitors is placed surrounding the dipole to record the
generated light power by integrating the Poynting vectors on the six field monitors, as
shown in Figure 2.2.1-1 (b).
The collected power from a field monitor is:
surface
dsfPrealfPower ))((2
1)(
(2.12)
where )( fP
is the complex Poynting vector *2
1)( HEfP
whose real part gives the
time-averaged rate of energy flow78. The frequency dependent Poynting vector )( fP
could be calculated from electric field component )( fE
based on the plane wave
approximation, as follows:
Figure 2.2.1-1 Calculating light extraction efficiency using FDTD method: (a)
simulation region setting; (b) Determining the total power emitted from a dipole
source using a power box.
37
2
0
0 )()( fEfnfP
(2.12)
2.2.2. Photonic Crystal Band Structure Simulation Methodology
Calculation of the photonic band structure is a critical step for the design of photonic
crystals. The formation of photonic band gaps depends on the structural parameters of
photonic crystals. By varying the design of photonic crystals, we can control the photonic
band gap to fall into different range of wavelengths. In that wavelength region, the
propagation of electromagnetic waves is prohibited. This phenomenon can be used in many
applications, such as fabrication of integrated optical circuits for telecommunication
systems. For light-emitting diodes with photonic crystals on the top surface, if the
wavelength of light is in the band gap, the light couldn’t propagate in the photonic crystal
layer when it arrives at the interface of photonic crystal. Hence, such light could only
propagate through the photonic crystal to the air or be reflected back. If the transmission is
increased beyond the critical angle, light extraction could also be increased.
The electromagnetic waves guided in a periodic structure is determined by Bloch’s
theorem. For simplicity, consider a structure that is periodic in x direction, with period a,
the Bloch modes can be written as:
ikx
kk exuxE )()( (2.13)
where )(xuk is a periodic function of x, with period a. This means )()( xuaxu kk . To
calculate the band structure of two-dimensional photonic crystals using FDTD, the
simulation region is set as shown in Figure 2.2.2-1.
38
Since the Bloch modes are periodic, using only one unit cell is enough to get the band
structure of photonic crystals. For hexagonal lattices, it is necessary to include multiple
unit cells to create a structure that is periodic in the X, Y, Z direction since FDTD
simulation region is always rectangular. Random positioned dipole sources are used to
inject energy into the simulation region. A series of random positioned field monitors are
used to record the electromagnetic field over time. Bloch boundary conditions with a
unique value of k is used to restrict the angle of propagation of electromagnetic waves. The
guided mode, whose frequency is supported by the structure in the propagation direction,
can propagate for a longer time while light at other frequencies will decay quickly. Each
field monitor can record the time signals of light waves at a fixed position. Adding all the
recorded signals together can include all the information of a guided mode. A simple
Fourier Transform was used to get the frequency of the guided mode. The time signal and
the transformed frequency spectrum is shown is Figure 2.2.2-2.
Figure 2.2.2-1 Simulation region settings for calculating photonic crystal band
structure using Lumerical’s FDTD solutions. The orange square is the
simulation region of one unit cell. Yellow cross is the field monitor.
39
Figure 2.2.2-2 Recorded signals of a guided mode in a two-
dimensional hexagonal photonic crystal. (a) The recorded time
signal; (b) Fourier transform of (a).
40
Chapter 3: Analysis of Light Extraction Efficiency for Thin-
Film-Flip-Chip (TFFC) InGaN Quantum Wells (QWs) Blue
Light-Emitting Diodes with Different Structural Design
In this chapter, the analysis of light extraction efficiency for thin-film flip-chip (TFFC)
InGaN quantum wells (QWs) light-emitting diodes (LEDs) was conducted using finite-
difference-time-domain method. A brief study on the band structure of photonic crystals
(PCs) demonstrated the guided waves in PCs as well as the forbidden range of wavelength
due to the bandgap of PCs. The photonic bandgap effect on light transmission and
reflection was also studied. Two explanations of the mechanism of enhancing light
extraction efficiency by photonic crystals were proposed. The light extraction efficiency of
TFFC InGaN-QWs LEDs with photonic crystals were calculated and compared to that of
the conventional TFFC InGaN-QWs LEDs with flat surface. Structural parameters of
photonic crystals and p-GaN layer thickness were studied systematically for increasing the
light extraction efficiency. A comparison between cylindrical and cone-shaped
nanostructure was performed. The position of light dipole source was considered to
estimate light extraction efficiency under actual emission condition.
3.1. Introduction of InGaN Quantum Wells Blue LEDs
3.1.1. Structure of InGaN Quantum Wells Blue LEDs
GaN has a direct bandgap of 3.39 eV, corresponding to an optical wavelength of 366
nm, which is in the UV range79. To get visible blue light requires tuning the bandgap of
GaN to a smaller value. The ternary semiconductor alloy, InxGa1-xN, has a band gap
41
ranging from 0.7 eV to 3.40 eV, depending on the indium mole fraction80,81,82. A fraction
of x=1 corresponds to pure InN, which has a band gap of 0.7 eV. Blue light-emitting diodes
based on InGaN were demonstrated to emit high-brightness light for the first time in 1994
by Shuji Nakamura of Nichia Corporation. Because of the higher efficacy of InGaN-based
LEDs than SiC-based LEDs and the ability to tune the band gap, InGaN quantum wells are
employed as the main light emitters for solid state lighting in the near ultraviolet, blue and
green spectral region.
A typical configuration of III-nitride InGaN LED device is shown in Figure 3.1.1-1.
The LED chip is usually grown by MOCVD technology. Due to the lack of native bulk
substrates, current InGaN (a ~ 3.2-3.3 Å) alloys are lattice-matched to c-plane GaN (a ~
3.2 Å) grown on c-sapphire (a ~ 2.75 Å) substrate for cost consideration. Before growing
thick n-type GaN layer, a thin GaN buffer layer is first grown on sapphire substrate because
many defects such as dislocations will occur at the interface between substrate surface and
nitride layers. Then, InGaN multiple quantum wells with GaN barriers are grown on n-
GaN layer and capped with thinner p-type doped GaN layer. After growth, an etching step
is required to expose n-GaN to deposit n-electrode. Finally, p-electrode is deposited on p-
GaN layer. Nickle, titanium and gold are typical metal materials used for Ohmic contacts.
42
3.1.2. Thin-Film-Flip-Chip Technology
Sapphire is the most commonly used substrate for epitaxial growth of GaN film.
However, due to the lattice and thermal-expansion coefficient mismatch between sapphire
and GaN, high-density structural dislocations could occur in the grown material, limiting
the quality of the GaN film83. In addition, poor thermal and electrical conductivity of
sapphire substrates also limit the device performance, resulting in increased operating
voltages84. For LEDs fabricated on sapphire substrates, all electrodes must be made on the
top surface. This process requires etching the top p-type GaN down to the n-type GaN,
increasing the complexity for fabrication and packaging schemes. This configuration also
causes an inevitable current crowding effect near the edge of the contact, leading to
efficiency droop in LEDs 85 , 86 . Surface patterning on the p-GaN surface also faces
constraints since the thickness of p-GaN is only a couple of hundred nanometers. Therefore
Figure 3.1.1-1 A typical structure of InGaN-based LED grown on
sapphire substrate.
43
the thin thickness of p-GaN limits the etching depth reducing the positive effect of surface
patterns.
Thin-film flip-chip technology, combined with wafer bonding technique, can be used
to transfer GaN to other metal substrates and eliminate the constraint of sapphire substrate.
Also, the thicker n-GaN layer is more suitable for fabricating surface patterns. A typical
thin-film flip-chip LED is shown in Figure 3.2. Current commercial blue LEDs, utilizing
thin-film-flip-chip package design, possess high light extraction efficiency in comparison
with that of conventional LED packages87,88. A backside mirror of metal such as copper or
silver can be integrated to the p-GaN side to reflect the downward-emitting light back to
the top surface, increasing the total emitted light to the outside space.
Laser lift-off (LLO) process is the most efficient way to separate the GaN film from
sapphire substrate. A brief process flow for fabrication of thin-film flip-chip GaN LEDs is
shown in Figure 3.3. The detailed optical process for lifting off GaN films was discussed89.
It can be briefly explained as follows. GaN film can be detached from a sapphire substrate
by illuminating the interface with a pulsed ultraviolet laser which induces localized thermal
decomposition of the GaN. The localized temperature at the interface could be as high as
800 °C, resulting in the effusion of nitrogen gas. The generated nitrogen gas expands and
separates the two interface, realizing GaN film lift-off. Before making the n-contact on top
surface, a further reactive ion etching (RIE) is used to remove the undoped GaN to expose
n-GaN layer.
44
Figure 3.1.2-1 A typical structure of thin-film flip-chip GaN-
based LED on a metallic mirror.
Figure 3.1.2-2 A brief schematic diagram of the fabrication process for the LLO-
LEDs. (a) laser processing, (b) separation, (c) etching of undoped GaN, (d) TFFC
LED.
45
3.1.3. Emission Polarization of InGaN QWs Blue LEDs
InGaN quantum wells have been the most common active layers for LEDs emitting
light from blue to near-ultraviolet. Because of the large lattice mismatch between InGaN
and GaN, compressive strain is induced in InGaN quantum wells grown by MOCVD. This
compressive strain leads to large spontaneous and piezoelectric polarization in the quantum
well, preventing high performance InGaN-GaN quantum well90. Several studies on the
spontaneous luminescence modes of InGaN/GaN MQWs have confirmed TE mode
dominating in the emission91,92,93. A polarization ratio of TE/TM intensities was reported
to be 1.9 for 460 nm blue LEDs92. The experimental result of electroluminescence
intensities for TE and TM polarizations in the spontaneous emission spectra is shown in
Figure 3.1.3-192.
Theoretical studies using the k·p method also suggested TE mode dominating in the
emission spectrum94. The band edge emission of InGaN QW comes from the transition of
electrons from the bottom of conduction band to the top of valence band. Three valence
bands from top to bottom are heavy hole (HH), light hole (LH), and crystal-field split-off
hole (CH). The electron transition from conduction band to HH/LH corresponds to TE
polarized light, while the electron transition from conduction band to CH corresponds to
TM polarized light. Since the energy difference between the bottom of conduction band
and CH is larger, the recombination rate of electrons with CH will be lower. Thus the TE
mode dominates in the spontaneous emission spectrum of InGaN QWs.
46
3.1.4. Fabrication Techniques of Photonic Crystal LEDs
As mentioned in section 1.1, total internal reflection is the primary issue that limits the
total efficiency for GaN-based LEDs. Due to the large refractive index mismatch between
GaN (n~2.4 at 450nm) and air (n=1), a large portion (nearly 90%) of the emitted photons
from the quantum wells are trapped inside the chip with a critical angle of about 24.5°
leading to very low light extraction efficiency. Thus, extracting the trapped photons out is
considered to be crucial for the improvement of the light extraction efficiency. The
methods to improve the light extraction efficiency can be classified as chip-shaping, metal
reflection layer on the bottom, flip-chip packaging and surface structuring. Among all the
surface structures, two-dimensional photonic crystals of periodic air holes and pillars are
demonstrated to be the most efficient way to improve the light extraction efficiency with a
combination of other methods such as thin-film flip-chip packaging and bottom metal
reflection layer95. Photonic crystals consist of fine cylindrical holes or pillars. However,
the actual fabricated air holes or pillars do not have exact cylindrical shape. This will
Figure 3.1.3-1 The edge-emitting spectrum of blue
InGaN/GaN MQWs LED at 455 nm92.
47
destroy the periodicity of photonic crystals. Thus, it is interesting to study the effect of non-
cylindrical periodic structures such as cones and truncated cones. There are mainly two
ways to fabricate photonic crystals on GaN film, electron beam lithography and
nanoimprint lithography.
Electron beam lithography (EBL) is a highly-developed technique used to fabricate
extremely fine patterns in the modern electronics industry for integrated circuits. It has
been used to fabricate two-dimensional photonic crystals on GaN-based LEDs in
laboratories for many years. A brief process for the fabrication of photonic crystals on
LEDs is shown in Figure 3.1.4-1. Before using e-beam lithography, an e-beam resist layer
is first formed on the surface of the target material. A periodic pattern of holes and pillars
on the e-beam resist can be defined by exposing to electron beam using positive or negative
e-beam resist. In the case of positive resists, electron beam can change the chemical
structure of the resist so that it becomes more soluble in the developer solution. The defined
soluble area is then washed away by the developer solution, leaving the desired pattern of
open windows with bare underlying material. Dry etching such as inductively coupled
plasma (ICP) etching and reactive ion etching (RIE) can be used to etch the surface of the
GaN to fabricate photonic crystals. After dry etching, the remaining resist is removed and
the material is cleaned.
48
Nanoimprint technology (NIP) is more promising to be used for commercial mass
production of sub-micron size structure over large areas than electron beam
lithography96,97,98. The brief process for fabricating two-dimensional photonic crystal GaN
LEDs using nanoimprint technology is shown in Figure 3.1.4-2. A silicon hard stamp with
desired micro-patterns is first fabricated using laser interference lithography (LIL) and
RIE97,98. Prior to imprinting, a soft polymer imprint resist is prepared on the surface of GaN
for transferring the pattern to GaN. During the imprinting process, a temperature of about
120 °C is kept for the solidification of the polymer. The imprinted polymer resist serves as
either an etch mask for the subsequent GaN layer etching or a lift-off mask for Cr
deposition, which is also an etch mask for GaN layer etching.
Figure 3.1.4-1 Illustration of processing flow for the formation of photonic crystals on
LEDs. (a) e-beam resist by spin coating or deposition; (b) patterning by direct write e-beam
lithography; (c) Dry etching of GaN surface; (d) e-beam resist lift-off.
49
3.2. Band Structure of 2D Photonic Crystal (PC) (Pillars and air holes)
3.2.1. 2D Simulation of Hexagonal PC
The simulation methodology for PCs band structure has already been discussed in
section 2.2.2. The electromagnetic guided modes supported in the two-dimensional PCs
are in the form of Bloch modes. Due to the unique periodic structure of PCs, the
propagation of light could be restricted in a few directions for a specific range of frequency
in the spectrum. Such a restricted region is photonic band gap. A full photonic band gap
can only be realized in 3D PCs for light propagating in all directions. 2D PCs only have
incomplete band gaps, which only restrict one mode of the light, transverse electric (TE)
Figure 3.1.4-2 Nanoimprint process for the formation of photonic crystals on LEDs.
(a)form a NIP polymer resist layer by spin coating; (b) the pattern of the stamp is
transferred to polymer resist by imprinting; (c) dry etching of GaN surface; (d) NIP
polymer resist lift-off.
50
or transverse magnetic (TM) mode. However, 2D PCs are much easier for fabrication and
practical implementation for many applications.
In this work, 2D PCs are used on the top surface of GaN LED to increase the light
extraction efficiency. Two types of PCs are analyzed, pillars and air holes. Band structure
of the PCs of two types are separated into TE and TM mode. TE light is the light with
electric field polarized parallel to the top surface. Thus no electric field is in the vertical
direction. Conversely, TM light is the light with magnetic field polarized parallel to the top
surface. Figure 3.7 shows the two conditions. The band structure for pillar-type and airhole-
type PCs are shown in Figure 3.8. The band structure is calculated for different R/a ratios.
Pillar-type PCs have a complete band gap (region between blue dashed line) for TM
polarization. Airhole-type PCs have a complete band gap (region between blue dashed line)
for TE polarization. So the pillar-type structure favors TM light and the airhole-type
structure favors TE light. For PC of pillars with R/a = 0.2, a band gap from 0.393 to 0.527
(c/a) is found. While for PC of air holes with R/a = 0.4, a band gap from 0.356-0.375 (c/a)
is found.
Figure 3.2.1-1 TE (left) and TM (right) mode light in photonic
crystals. Here the two-dimensional photonic crystals are considered
as infinite in the vertical direction.
51
Figure 3.2.1-2 Band structure for photonic crystals of pillars. The left side is for
TE mode and the right side is for TM mode. Photonic crystals of different R/a ratio
were analyzed. The horizontal coordinate is Bloch wave vector in the first
Brillouin zone along M-Gamma-K-M direction with 15 data points in each
direction. The vertical coordinate represents the Bloch mode frequency normalized
52
Figure 3.2.1-3 Band structure for photonic crystals of air holes. The left side is for TE
mode and the right side is for TM mode. Photonic crystals of different R/a ratio were
analyzed. The horizontal coordinate is Bloch wave vector in the first Brillouin zone along
M-Gamma-K-M direction with 15 data points in each direction. The vertical coordinate
represents the Bloch mode frequency normalized by c/a.
53
3.2.2. Physical Meaning of Photonic Band Gap
In order to understand the physical meaning of photonic band gap, we can investigate
a one-dimensional photonic grating for simplicity, as shown in Figure 3.2.2-1. Materials
with refractive index n1 and n2 are arranged interlacing with each other with a lattice
constant a. For a bulk homogeneous medium (that is n1 = n2), we already know that the
speed of light is reduced by the refractive index. All the modes satisfy the dispersion
relation, given by
n
ckk )(
(3.1)
We can set an arbitrary lattice constant a for such bulk homogeneous medium. Then the
above dispersion relation could be folded back into the first Brillouin zone. There is no
band gap for a bulk optical medium and all frequencies are supported in such medium.
In order to generate a photonic band gap, a periodic contrast in dielectric constant is
needed. The contrast in the dielectric constant of two materials will affect the size of the
band gap. In Figure 3.2.2-2, the band structure [ )(k ] of two different multilayer films are
Figure 3.2.2-1 Schematic illustration of a PC periodic in one dimension.
54
investigated, GaN/InGaN multilayers and GaN/air multilayers. The refractive index
contrast of GaN (2.48) and InGaN (2.28) is small, which gives a band structure similar to
bulk material. For GaN/air multilayer slab, an obvious band bending near the 1st Brillouin
zone edge is observed. The propagation speed of light near the zone edge is decreasing and
becomes zero at the edge. The Bloch mode with a wave vector at the edge of 1st Brillouin
zone will in the form of a standing wave. Such a mode cannot transfer energy. Also, a wide
band gap between the bottom and upper band was found. No higher order mode is found
in the band gap. All frequencies of light in the band gap is forbidden in such multilayer
medium. The right side of Figure.3.2.2-2 depicts the energy distribution of the Bloch mode
of the top of band 1 and the bottom of band 2 with a wave vector k at the 1st Brillouin zone
edge. For band 1 whose frequency is lower than the band gap, more energy is concentrated
in the high-n region. In contrast, for band 2, more energy is concentrated in the low-n region.
So the origin of the band gap comes from the difference in the energy distribution of
Bloch modes. The periodic characteristic of Bloch modes leads to the periodic energy
distribution. If for a certain frequency, there is no existing periodic energy distribution that
can support the frequency of light, then such light cannot propagate in the periodic medium.
55
(a)
(b)
Figure 3.2.2-2 (a) Band structure of GaN/InGaN multilayer slab. (b) Band structure
of GaN/air multilayer slab. The right side depicts the energy distribution of the Bloch
mode for band 1 and 2 at the zone edge.
56
Figure 3.2.2-3 depicts the Bloch mode profile of a 2D hexagonal photonic crystal of
pillars, R=0.2a, at K point. Figure 3.12 (a) and (b) are the Bloch mode profiles for the top
of bottom band and bottom of upper band at K point. It confirms the energy distribution
obtained from 1D simulation. For 2D photonic crystal, at lower frequencies, most of the
mode energy is concentrated in the higher index region while for higher frequencies, most
of the mode energy is in the air region.
3.2.3. Transmission and Reflection Coefficients of 2D PC Slab
In the previous sections, band structure of 2D photonic crystal is simulated. By
carefully selecting the value for the lattice constant a, we could manipulate the frequency
range of the band gap and let the emitted-light frequency of interest fall into the band gap.
Figure 3.2.2-3 Bloch mode profile of hexagonal photonic crystal of pillars for R=0.2a.
The dashed circles represent pillars and the other region is air. (a) the top of band 1 at
K point, f=0.393 c/a. (b) the bottom of band 2 at K point, f=0.6 c/a. TM mode.
57
It is anticipated that the light extraction efficiency could be enhanced if the light frequency
is in the photonic band gap99. However, the effect of the photonic band gap in increasing
the light extraction efficiency for LEDs was not clearly proved.
For a dielectric medium with a flat surface, the transmission and reflection coefficients
are provided by the Fresnel equations. In terms of transmitted and reflected energy, the
relation of energy fraction with the incident angle is given for TE and TM mode.
TE mode:
2
21
2
21
coscos
)coscos(
ti
ti
TEnn
nnR
(3.2)
2
21
21
coscos
coscos4
ti
ti
TEnn
nnT
(3.3)
TM mode:
2
12
2
12
coscos
)coscos(
ti
ti
TMnn
nnR
(3.4)
2
21
21
coscos
coscos4
it
ti
TMnn
nnT
(3.5)
Light propagates from medium 1 to medium 2. In our simulation, n1 is chosen to be 2.48
and n2 is 1. If n1>n2, then θi should be smaller than the critical angle θc. For conventional
GaN-based blue LEDs emitting at 460 nm, the transmittivity and reflectivity is shown in
Figure 3.12. The dashed lines are calculated results from equations (3.2)-(3.5) and the solid
dots represent the FDTD simulation results. The theoretical curve shows good agreement
with the simulated curve, indicating the correctness of our model. For normal incidence,
around 80% of emitted light can be extracted to the air. When increasing the incident angle,
the transmittivity of TE-mode light decreases until it reaches 0, however, the transmittivity
of TM mode first increases and then decreases to 0. When the incident angle reaches the
58
critical angle θc, light is totally reflected back and guided in the GaN slab. As mentioned
in section 3.1.3, TE-like emission dominates in InGaN-MQWs blue LEDs. In this case,
light can only be extracted out efficiently for a small incident angle. Even for light within
the extraction cone, there is always a portion of light reflected back, reducing the light
extraction efficiency. A bottom mirror can be used to redirect the reflected light back to
(a)
(b
)
Figure 3.2.3-1 Theoretical and simulation results for
transmittivity and reflectivity for conventional GaN-based
blue LEDs. (a) TE mode; (b) TM mode.
59
the top surface. With several reflections, all light within the extraction cone can be
extracted out if one is ignoring the absorption loss, which increases with the optical path
length and multiple reflections.
From Figure 3.2.3-1, it is clear that transmittivity becomes zero when the incident angle
is beyond the critical angle. So in order to increase the light extraction efficiency, it is
important to increase the transmittivity for incident angles larger than the critical angle.
Figure 3.2.3-2 Effect of photonic band gap on transmission coefficient. (a) The
band structure of 2D PC slab of pillars at R=0.2a; (b) Possible diffraction
options of incident light; (c) The transmission coefficient of the PC slab with
a=207nm, r=0.2a, TM mode for different wavelength.
60
To see the effect of photonic band gap on light extraction, a 2D PC with a=207 nm,
r=0.2a was constructed on top of the LED chip computationally. Figure 3.2.3-2 (a) plots
the band structure of PC of pillars with r=0.2a. Since PC of pillars favors TM mode for
obtaining large band gap, TM mode was used in the simulation. Such photonic crystal
shows a photonic band gap in the range of [0.4, 0.5]c/a. Figure 3.2.3-2 (b) shows three
possible ways for light diffracted by the PC layer. If the photon energy is in the photonic
band gap, then such light can be guided in the photonic crystal layer, which forbids the way
(2) in Figure 3.2.3-2 (b). So light can only be extracted out or reflected back by the PC
layer. Figure 3.2.3-2 (c) plots the transmission as a function of incident angle for different
light wavelength. It shows that the light is still totally reflected back by the top surface even
when the light is in the band gap of PC layer. So there is no relation of PC band gap with
light extraction. The effect of PC on light extraction is more related to the diffraction and
scattering effect.
3.3. Effect of p-GaN Layer Thickness on Light Extraction Efficiency for
Conventional TFFC InGaN LEDs
3.3.1. Emission Enhancement by Constructive Interference: Micro-
Cavity Effect
In the history of solving the issue of extracting light as efficiently as possible from a
high-index material (n>2), an important idea is to use a high reflectivity mirror to redirect
spontaneous emission toward the top emission surface100. The reflected light can produce
constructive interference with the original upward emitting light, enhancing the brightness
61
and efficiency. The intensity of the interference light has been calculated theoretically for
a monochromatic emitting dipole source101.
The position of a dipole source relative to the high reflectivity mirror is shown in Figure
3.3.1-1. Assume that the dipole source has a vacuum wavelength λ, angular frequency ω,
and is put in a medium of index n. The associated wavevector of the dipole source is k =
nω/c. For simplicity, the source lifetime, polarization and orientational effects are
neglected and the medium is assumed to extend infinitely to the top. With this assumption,
the common form of the formula for the interference of light over the light intensity could
be expressed as follows101:
)(2cos21)(1)()( 22
0
2)(22
0
2 rrEreEE i (3.1)
Figure 3.3.1-1 Illustration for the interference of original top
emitting light and the light reflected by the mirror.
62
where )(0 E is the far-field electric field without the bottom mirror, r is the reflection
coefficient of the metal-type mirror and )(2 is the phase shift due to the optical path
difference between the reflected light and the original light. The optical path difference is
related to the source-to-mirror distance t, the incident angle θ, and associated wave vector
k = 2πn/λ. Without considering the phase shift caused by the reflection at the mirror
interface, the expression for the phase shift solely due to the optical path difference can be
written as follows:
)cos(22 kt (3.2)
For light propagating from high-index medium to low-index medium, only the light with
incident angle within the critical angle can be extracted out. The critical angle c of an
interface between a high-index medium (n>2.3) and air is small. Thus, to an acceptable
approximation, )cos( could assume to be 1 when < . For constructive interference,
the optical path difference has to satisfy the relation mkt 22 with an integer m for +r
and a half-integer m for –r. For a perfect lossless mirror ( 1r ) and under the constructive
interference condition, the final output intensity would have a 4 times enhancement
)(4)( 0 EE . This enhancement corresponds to a source-mirror distance t having the
following relation with the wavelength of light:
n
mt
2
(3.3)
c
63
3.3.2. FDTD Analysis of the Effect of P-GaN Layer Thickness
For the thin-film flip-chip GaN-based LEDs with a bottom reflective mirror, it is
necessary to select a proper thickness for the p-GaN thin layer to achieve the constructive
interference effect. Once the p-GaN layer reaches a proper thickness, the output intensity
is predicted to be enhanced by a factor of 4. To simulate the effect of p-GaN layer thickness
on the light extraction efficiency, the simulation model is shown in Figure 3.3.2-1(a). Since
TE mode dominates in the emission spectrum of InGaN-MQWs blue LEDs, a TE polarized
dipole source with wavelength of λpeak=460nm is placed at the center of the active region.
The dependence of the light extraction efficiency on the p-GaN layer thickness is shown
in Figure.3.3.2-1 (b). The solid dots represent the results obtained from FDTD simulation,
and the dashed line is the fitting curve from equation (3.1), (3.2) and (3.3). The simulation
results has a good agreement with the theory from section 3.3.1 indicating the effectiveness
of the constructive interference on the light extraction efficiency enhancement.
In Figure 3.3.2-1 (b), at t=0, the light extraction efficiency reaches its minimum value,
indicating a destructive interference of the upward emitting light with the reflected light.
Since the optical path difference is 0 for t=0, it can be inferred that the reflection coefficient
of the reflective mirror is -1. A periodicity of around 92.5 nm for the p-GaN layer thickness
was obtained from the fitting curve. This periodicity confirmed the theoretical prediction
which gives )(7.92)(48.22
460
2nmnm
nt
.
From Figure 3.-, the large difference between the minimum and the maximum of the light
extraction efficiency emphasized the importance of optimizing the p-GaN layer thickness
for TFFC GaN LEDs. A typical thickness for the p-GaN layer is around 200 ~ 400 nm.
From the simulation results, the maximum for the light extraction efficiency occurs at
64
230nm and 330nm in the range of 200~400nm. For the subsequent simulations for TFFC
GaN LEDs, 330 nm is selected for the p-GaN layer thickness.
Figure 3.3.2-1 The effect of p-GaN layer thickness on the light extraction
efficiency for TFFC GaN LEDs. (a) a schematic for the structure of the
TFFC GaN LED, n-GaN thickness is 3 μm. (b) dependence of light
extraction efficiency on p-GaN layer thickness for TFFC GaN LED with
flat surface at λ=460nm. Solid triangular dots and dashed line represent the
FDTD simulation results and the fitting curve, respectively.
65
3.4. Effect of Photonic Crystals on Light Extraction Efficiency of Blue
LEDs
It is challenging to design a photonic crystal to reach the highest light extraction
efficiency for LEDs, since many parameters need to be optimized. For a certain type of
lattice (e.g. hexagonal lattice), depth d, filling factor f and lattice constant a will affect the
total light extraction efficiency enhancement. Filling factor f is defined as the ratio of the
area of pillars or air holes to the whole surface area. These three parameters are considered
to be independent on each other. So each parameter can be optimized separately and the
final design is a combination of three optimized parameters. When using photolithography
to define the two-dimensional periodic pattern, positive or negative photoresist can be used
to leave the exposure area open or closed. Then the subsequent dry etching of GaN can
produce a periodic pattern of pillars or air holes. So in the simulation, two types of the
nanostructure, pillars and air holes, need to be simulated.
3.4.1. The Simulation Model
Figure 3.4.1-1 shows the schematic diagram of our simulation model. Note that the
model represents a flip-chip structure. The simulation region is a rectangular region
containing a dipole source, field monitors and the LED structure. A single dipole source is
placed at the center of the quantum-well active region to mimic the light generation from
the quantum well. A small transmission box is placed around the dipole source to collect
the total emitted power by the dipole source. A plane field monitor is placed above the top
surface of the LED to collect the power emitted to the air. The distance of the monitor to
the top surface of the LED is at least λ/n to avoid collecting the power of evanescent mode
light. For the bottom boundary, we used metal boundary condition which functions as a
66
perfect mirror and employs the micro-cavity effect to enhance the emitted power. The p-
type GaN layer thickness is chosen to be 330nm to achieve constructive interference
between the upward-emitting light and the mirror-reflected light. The thickness of n-type
GaN is chosen to be 3 μm. Due to the limited memory space and computational speed of
the computer, the simulated structure of the LED has to be small. An actual LED chip can
be as large as 350 μm×350 μm which is unrealistic to simulate102. For our simulation
region, the size we chose is around 10 μm ×10 μm, which is much smaller than a real LED
chip. The mesh step is 15 nm. Since the chip size is much larger than the thickness of GaN
slab, the structure can be treated as infinite in x and y direction. Thus perfect mirror
boundary condition is used for the four sides to extend the small simulation region.
Although side emission could cause some loss to the emitted energy, this loss is pretty
small since the thickness of the chip is much smaller than the size so that it can be neglected
for our simulation. In addition, we neglected the absorption loss of the material just for
simplicity.
67
Figure 3.4.1-2 Top view of the simulation region. The orange square region is
the simulation region from top-down view. The structural parameters of
photonic crystals are arbitrarily chosen.
Figure 3.4.1-1 The simulation model for thin-film flip-chip GaN-
based LED. r is the radius of the pillar or air holes; a is the lattice
constant; d is the depth of the photonic crystal.
68
Figure 3.4.1-2 shows the top view of the simulation region. Since we used perfect
mirror boundary condition for the four sides, we can imagine that the structure in the region
is copied on the other side of the mirror and then the whole structure will just be like infinite
in the x, y direction. In order to maintain the periodicity of the structure after mirror-
symmetry operation, the side boundary should cut the structure at its symmetry axis, e.g.
the half-point of a pillar or air hole. Thus when changing the lattice constant a, the region
size need to be changed since the same region size could not satisfy the periodic
characteristic after mirror-symmetry operation for different lattice constant a. Although the
region size has to be changed, it is kept near 10 μm × 10 μm.
Figure 3.4.1-3 shows the simulated light extraction efficiency as a function of
simulation time. It is shown that the light extraction efficiency will reach its saturation level
at a certain value of the simulation time. This saturation is expected since the input dipole
source only emits a certain amount of energy at the beginning of the simulation. The
extracted energy to the air is collected by the EM field monitor and then absorbed by the
top PML absorbing boundary. When the simulation time is long enough, the remaining
energy in the simulation region will decrease and no further light will be extracted to the
air. So the energy collected by the monitor will decrease and become zero after a period of
simulation time. Since the light extraction efficiency is defined as dividing the extracted
energy by the total energy emitted from the dipole source, the light extraction efficiency at
the saturation level can be considered as the final light extraction efficiency for the device.
Note that the increase in light extraction efficiency is small after t=3000fs, we used this
value for all our simulation time for performing the simulation efficiently.
69
Figure 3.4.1-3 Light extraction efficiency as a function of simulation
time. The structural parameters of the photonic crystal are a=600nm,
r=199nm, d=200nm, f=0.4. Absorption loss is neglected.
Figure 3.4.1-4 The Archimedean A13 lattice. a is the distance
between the center of a hole to the center of its neighbor. a’ the
base vector of the larger hexagonal unit cell103.
70
To verify our model, we simulated the structure which used Archimedean A13 lattice
with a pitch of 455nm in the reference 19. In reference 19, thin-film flip-chip technology
was used for the fabrication of LED chip. The p-type GaN is bonded to a bottom metallic
mirror. The total thickness of GaN is about 700 nm. On the top surface, there is a photonic
crystal of Archimedean A13 lattice with a pitch of 455 nm, a filling factor f≈0.3 and a depth
of 250 nm. The Archimedean A13 lattice is shown in Figure 3.4.1-4 snatched from
reference103. A comparison of our simulation result and their experimental result is shown
in table 3.1. Our simulation gave the light extraction efficiency of 80.9% which is a little
bit higher than the maximum light extraction efficiency of 78% from the experiment. The
error is in the acceptable range. This indicates that our model is reasonable for simulating
the light extraction efficiency for LEDs.
Table 3.1 Comparison between FDTD simulation and experimental result
Archimedean A13 lattice FDTD simulation result Experimental result [19]
Light Extraction
Efficiency
80.9% 68%-78%
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3.4.2. Effect of Photonic Crystal Depth d
For the thin-film flip-chip GaN LEDs, the relatively thick n-type GaN layer is suitable
for deep etching without worrying about doing damage to the quantum well region. It is
expected that deeper etching could lead to stronger coupling of the emitted light with the
photonic crystal. However, deeper etching requires longer time and it will lower the quality
of the photonic crystal. Thus it is necessary to study the effect of etching depth d to achieve
higher light extraction efficiency while keeping the depth d relatively shallow.
Figure 3.4.2-1 shows the light extraction efficiency as a function of photonic crystal
depth d. The filling factor f is kept at 0.5 for both PC of pillars and PC of air holes. From
Figure 3.4.2-1, it is clear that light extraction efficiency first increases and then reach a
saturation level. When the thickness of PC layer reaches about 200nm, light extraction
efficiency almost stops increasing. This phenomenon holds for different lattice constant a.
We can make the comparison between the depth d and wavelength (λ/n ≈ 186nm) to
understand this phenomenon. For d << λ/n, the scattering or PC is in Rayleigh regime. In
this regime, the intensity of the scattered light increases dramatically when increasing the
size of the scatters. Typically, d needs to be smaller than a tenth of the wavelength which
is about 20 nm. Beyond the Rayleigh regime, the strength of the diffraction will increase
until the size of the scatters is close to the wavelength in the dielectric medium. The
dependence of diffraction losses on the depth d was also explained in the reference104 by
calculating the imaginary component of the wave vector. No clear saturation phenomenon
was illustrated in this reference. Since we only focus on a single mode (460nm) in the GaN
slab, our simulation results could be different from their calculation.
The transmittivity and reflectivity of photonic crystal layer can be used to clearly see
the effect of increasing the depth. Figure 3.4.2-2 shows the transmittivity and reflectivity
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for light (Γ-K direction) propagating from GaN to air where the top surface is covered with
a layer of hexagonal PC of pillars. In order to simulate the extremely thin layer of PC
(15nm), much smaller size of mesh (2 nm) is used for the PC layer while other area is still
15nm for the mesh to make the simulation more efficient. From Figure 3.4.2-2, the curve
Figure 3.4.2-1 The dependence of light extraction efficiency on PC depth
d for TE polarized TFFC InGaN QWs LEDs with optimized p-GaN
thickness (330nm for λpeak=460nm). (a) PC of pillars; (b) PC of air holes.
Both are hexagonal lattice.
73
of extremely thin layer of PC (15nm and 30nm) is similar to that of conventional LED with
flat surface. This is expected since the extremely thin layer of PC acts just as a small
perturbation to the flat surface. When increasing the depth d of PC layer, the effect of PC
on light extraction is stronger. Although the transmittivity for light within the extraction
cone is decreased, the transmittivity is enhanced for light of incident angle greater than the
critical angle breaking the total internal reflection. Since the critical angle is so small,
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increasing the transmission for larger incident angle will have more impact on light
extraction than the decreasing of transmission for light in the extraction cone. With a
bottom mirror, more light can be extracted out to air after several reflections within the
slab.
3.4.3. Effect of Filling Factor f
The filling factor f refers to the ratio of the area of pillars or air holes to the whole
surface area:
𝑓 =𝐴𝑟𝑒𝑎 𝑜𝑓 𝑝𝑖𝑙𝑙𝑎𝑟𝑠 𝑜𝑟 𝑎𝑖𝑟 ℎ𝑜𝑙𝑒𝑠
𝑊ℎ𝑜𝑙𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎
(3.4)
It is equal to the area ratio of pillars or air holes in one unit cell. The relation between filling
factor f and radius r of pillars or air holes for a hexagonal lattice with lattice constant a
could be obtained using the following function:
𝑟 = 0.5251 × √𝑓 × 𝑎 (3.5)
The effect of filling factor (f) on light extraction efficiency was studied for the TFFC InGaN
LED structure as shown in Figure 3.4.1-1. Figure 3.4.3-1 plots the light extraction
efficiency as a function of the filling factor (f) at λ=460nm for both PC of pillars and air
holes. The optimized p-GaN layer thickness and the etching depth of photonic crystals are
330nm and 200nm, respectively. The filling factor f is changed from 0.1~1. Note that a
filling factor of 1 equals to the conventional LED of flat surface. The study shows that
when increasing the filling factor f, light extraction efficiency first increases and then
decreases. Light extraction efficiency reaches its maximum when filling factor is in the
range of 0.3~0.7. Note that similar trend was found for different lattice constant a,
indicating that the effect of filling factor is independent on other parameters.
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Figure 3.4.2-2 shows the transmittivity and reflectivity of the PC interface for light
propagating (Γ-K and Γ-M direction) from GaN slab through PC of pillars to air for
different filling factor f. The lattice constant a and depth d are fixed at 800nm and 200nm.
In the region of large incident angles (θ>θc), the transmittivity is low for filling factor f
=0.1 and 0.9 while the transmittivity curve is similar for f=0.3, 0.5, 0.7. So, the light
Figure 3.4.3-1 Light extraction efficiency of TFFC PC GaN LED for TE polarized
emission as a function of filling factor f with optimized p-GaN thickness (330nm for
λpeak=460nm). (a) PC of pillars; (b) PC of air holes. Both are hexagonal lattice.
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extraction efficiency is low for small and large filling factor (f=0.1 and 0.9) while for f
=0.3~0.7, the difference in light extraction efficiency is less than 10% as shown from
Figure 3.4.2-1.
For a practical PC GaN LED, a filling factor too small or too large is not suitable
because the dielectric contrast is small. A small filling factor corresponds to a PC of pillars
or air holes of small radius. It is just like a small perturbation to a flat surface, resulting in
low light extraction efficiency due to total internal reflection. A large filling factor is
similar to the small filling factor, where the vacancy region is switched with the padding
region. As indicated from Figure 3.4.2-1 and 3.4.2-2, there is a large range of filling factor
Figure 3.4.3-2 Transmittivity and reflectivity of hexagonal PC of pillars for different
filling factor of PC for TE polarized TFFC InGaN QWs LEDs with optimized p-GaN
thickness (330nm for λpeak=460nm). (a) light is in Γ-K direction; (b) light is in Γ-M
direction.
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to achieve high light extraction efficiency. This is beneficial for practical fabrication since
it is not necessary to control precisely the size of pillars or air holes.
3.4.4. Effect of Lattice Constant a
The effect of a photonic crystal on the light extraction efficiency for TFFC GaN-based
LEDs was also studied by tuning the lattice constant a. As explained in section 1.1.6, if the
wave vector of the incident light is outside the extraction cone, such light can be coupled
back to the extraction cone by adding a certain reciprocal lattice vector G on the original
wavevector. From the Ewald construction view, a reciprocal lattice point is required to be
in the extraction circle so that there exists a reciprocal lattice vector G that can couple the
original wave vector into the extraction cone. Since the reciprocal lattice vector G is
inversely proportional to the lattice constant a, increasing lattice constant a could make
reciprocal points denser so that more reciprocal lattice points can be found inside the
extraction cone, leading to a higher possibility for coupling the wave vector to the cone.
However, if the lattice constant a is too large, the pillars or air holes might act just as a
local flat surface, decreasing the diffraction effect of the photonic crystal.
Figure 3.4.2-3 plots the light extraction efficiency as a function of the lattice constant
a for light of λ=460nm for both PC of pillars and air holes with an optimized p-GaN
thickness of 330nm. The depth of PC layer is 200 nm for all simulations. The filling factor
f = 0.1, 0.3, 0.4 and 0.5 were studied. Almost the same trend was found for different filling
factor f. Increasing the lattice constant a from 200nm to 600 nm leads to the increase in
light extraction efficiency. A maximum light extraction efficiency was found at a =
1300nm and a = 600nm for PC of pillars and PC of air holes, respectively. However, when
further increasing the lattice constant a beyond 1400nm, no obvious increase or decrease
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in light extraction efficiency was found. The light extraction efficiency reaches a saturation
level. This can be explained by the Ewald construction view. For small lattice constant a,
the reciprocal lattice is so large that no reciprocal lattice point is in the extraction cone,
resulting in low diffraction effect of PC. When increasing the lattice constant a, the
reciprocal lattice becomes smaller and the reciprocal lattice point starts to fall into the
extraction cone, leading to an increase in light extraction efficiency. Although more
reciprocal lattice points could be inside the extraction cone by further increasing lattice
constant a, the pillars or air holes are also becoming larger reducing their diffraction effect.
When their size becomes larger than the wavelength, the center area of a pillar or air hole
could be considered as a local flat surface, where light can be totally reflected back. In such
condition, light is diffracted to a large extent by the side surface of pillars or air holes. So
the competitive mechanism between decreasing the reciprocal lattice vector G and
increasing the size of pillars and air holes lead to no increase and decrease in light
extraction efficiency. A balance was reached between them, resulting in a saturation of
light extraction efficiency for lattice constant larger than 1400nm.
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Figure 3.4.4-1 Light extraction efficiency of TFFC PC GaN LED for TE
polarized emission as a function of lattice constant a with optimized p-
GaN thickness (330nm for λpeak=460nm). (a) PC of pillars; (b) PC of air
holes. Both are hexagonal lattice.
(a)
(b)
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3.4.5. Effect of Dipole Source Position
All of the previous simulations were conducted for the LED structure with a single
dipole source fixed at the center of the InGaN MQW region. This could lead to errors in
light extraction efficiency compared with an actual device since an actual MQW region
emits photons over all the positions. Thus it is necessary to simulate light extraction
efficiency for dipoles located at different positions in the MQW region. Due to the periodic
symmetry of hexagonal lattice, the dipole source is only needed to be changed inside the
small triangle as shown in Figure 3.4.5-1. For simplicity, the dipole source position is only
changed along Γ-K direction.
Figure 3.4.5-2 plots the light extraction efficiency as a function of dipole source
position. The dipole source position is changed from the center point to R=a/2 along Γ-K
direction. Beyond R=a/2, it can be considered as another period which should be equivalent
to the region [0, R]. It can be clearly seen that there are significant differences in light
Figure 3.4.5-1 Illustration of dipole source position changing
along Γ-K direction. a is the lattice constant.
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extraction efficiency when changing the source position. To approximate the light
extraction efficiency for an actual device, a weighted average of light extraction efficiency
over different source position could be used. The light extraction efficiency for source
position at dx could be used to approximate the light extraction efficiency for all the source
Figure 3.4.5-2 Source position dependence analysis of light extraction
efficiency for TE-polarized TFFC PC GaN LED. The position of the
dipole source is changed along Γ-K direction. The optimized p-GaN
thickness is 330nm for λpeak=460nm. The depth of PC layer is 200nm. (a)
PC of pillars; (b) PC of air holes. Both are hexagonal lattice.
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positions on the circle of radius dx. The weighted average of the light extraction efficiency
is shown in the table 4.3.5-1.
Table 3.4.5-1 Weighted average of light extraction efficiency
Pillars (f=0.5, d=200nm) a = 600nm a = 1000nm a = 1300nm
Light extraction efficiency 65.1% 69.1% 69.8%
Air holes (f=0.4, d=200nm) a = 600nm a = 1200nm a = 1300nm
Light extraction efficiency 67.2% 66.5% 60.7%
From table 3.4.5-1 listing the weighted average of the light extraction efficiency, the
highest light extraction efficiency is around 70% for photonic crystal of both pillars and air
holes. The light extraction efficiency is indeed different after changing the dipole source
position as shown in Figure 3.4.5-2. However, after taking account the weighted average
for different dipole source position, the final result of light extraction efficiency doesn’t
change much compared with putting the dipole source at the center.
3.5. Effect of Cone-Shaped Periodic Nanostructure on Light Extraction
Efficiency of Blue LEDs
In the previous sections, two-dimensional PC requires cylindrical shaped structure.
However, for the real fabrication, defects could lead to cone-shaped or truncated cone
structures. Such structures will break the periodicity of two-dimensional photonic crystal.
In this section, hexagonal periodic arrays of cone-shaped structure was studied for
increasing the light extraction efficiency of TFFC InGaN MQW LED.
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3.5.1. Effect of Cones
Hexagonal array of cones was studied as an alternative to the photonic crystal structure.
Cones can be considered as defects in photonic crystals. However, the angled side facets
could reduce the final incident angle to become smaller than the critical angle, decreasing
the effect of total internal reflection.
A single dipole source is placed at the center of the MQW region under the center of
one cone. Other parameters of the GaN LED chip are the same to that used in photonic
crystal LEDs settings, which are 330nm for p-GaN thickness, 3μm for n-GaN thickness.
Figure 3.5.1-1 (a) plots the model of the LED chip with cones on the top surface. Figure
3.5.1-1 (b) plots the light extraction efficiency as a function of cone’s bottom radius r when
fixing the height of the cone at 200nm as a comparison to the above photonic crystal depth.
It clearly shows that light extraction efficiency increases when increasing the bottom radius
r of the cones. This is expected since small cones can be considered as a perturbation to
the flat surface. A maximum value of light extraction efficiency was reached when
rbot/R=0.9, (R=a/2). Note that rbot/R=1 represent the closed-packed pattern. This concludes
that a non-closed pattern with rbot/R=0.9 shows higher performance than the closed-packed
ones.
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Figure 3.5.1-2 shows the light extraction efficiency enhanced by cones as a function of
etching depth d. The lattice constant a equals 600 nm. Because the etching direction of
cones is usually along a certain crystal direction, the ratio of etching depth d and bottom
Figure 3.5.1-1 The effect of cones on light extraction efficiency for
InGaN-MQW TFFC LEDs. (a) The simulation model. (b) The light
extraction efficiency as a function of bottom radius r/R.
85
radius rbot is kept as a constant in Figure 3.5.1-2. This constant is chosen manually so that
at a=600nm, d=200nm, rbot=270nm, light extraction efficiency will reach a maximum as
shown in Figure 3.5.1-1 (b), where rbot/R =0.9. This ratio of d/rbot may not be a true etching
angle of GaN. A ratio of 1 : 1 was shown in reference 105.
Figure 3.5.1-2 clearly shows that increasing the etching depth will increase the light
extraction efficiency. Light extraction efficiency will reach a maximum point at a = 600
nm, d = 200 nm, rbot = 270 nm, before the pattern becomes close-packed. This point is the
same maximum point in Figure 3.5.1-1 for a=600nm.
Figure 3.5.1-2 The effect of cones on light extraction efficiency for
InGaN-MQW TFFC LEDs as a function of etching depth d. The lattice
constant a is 600nm, d/rbot = 0.74.
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3.5.2. Effect of Truncated Cones
Truncated cones are the more general alternatives to photonic crystals. Etching the GaN
material using either e-beam lithography or nanoimprint technique will usually produce
cylinders with top and bottom radius mismatch. Similar to cones, the angled side facets
could reduce the effect of total internal reflection. Figure 3.5.2-1 plots the light extraction
efficiency as a function of rtop/rbot for different filling factors. Here, rtop stands for the
top radius while rbot stands for the bottom radius. A ratio of 0 refers to the cone structure
while a ratio of 1 refers to cylinders. In a transition from cones to cylinders, there is a peak
for light extraction efficiency. For a = 800 nm and rbot = 230 nm, rbot = 297 nm, rbot =
350 nm, the peak happens when rtop = 115 nm, 106 nm, 175 nm and 57 nm, respectively.
For rbot = 400 nm, the light extraction efficiency decreases when increasing the rtop/rbot
ratio.
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Figure 3.5.2-1 The effect of truncated cones on light extraction efficiency
for InGaN-MQW TFFC LEDs as a function of rtop/rbot for different
filling factor f.
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3.5.3. Considering Dipole Position for LEDs with Truncated Cones
Figure 3.5.3-1 shows the light extraction efficiency as a function of source position for
LEDs with truncated cones. The top and bottom radius correspond to the peaks in Figure
3.5.2-1. Generally, the light extraction efficiency decreases when the dipole source moves
away from the center point. The weighted averages of light extraction efficiency
considering dipole source positions are listed in Table 3.5.3-1. Considering the effect of
dipole source position, the maximum light extraction efficiency was achieved when rbot =
350nm, rtop = 175nm.
Figure 3.5.3-1 Light extraction efficiency as a function of dipole source
position for InGaN-MQW TFFC LED at fixed a=800nm and d=200nm.
Four curves are rbot=230nm, 297nm, 350nm, 400nm. The top radius rtop
is chosen according to the peak values from Figure 3.5.2-1.
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Table 3.5.3-1 Weighted average of LEE for LEDs with truncated cones
Top & bottom
radius
rbot=230nm,
rtop=115nm
rbot=297nm,
rtop=106nm
rbot=350nm,
rtop=175nm
rbot=400nm,
rtop=57nm
Average
(weight)
60.7% 63.7% 64.6% 58.5%
3.6. Conclusions: Light Extraction Efficiency Enhancement for InGaN-
MQW TFFC LEDs
The light extraction efficiency for TFFC InGaN QWs LEDs was studied by using 3D
FDTD method. GaN is a wide bandgap semiconductor material with high refractive index
(n~2.5). For light emitted from LED to the air, strong total internal reflection will limit the
total efficiency for III-nitride LEDs. Lots of solutions have been proposed to increase light
extraction efficiency among which photonic crystals exhibit higher enhancement. We
found that the photonic band gap effect has no relation with enhancing light extraction
efficiency. After performing a structural parameter optimization, we found that the etching
depth of 200nm is enough for high light extraction efficiency and the filling factor around
0.5 will give a maximum LEE. Small lattice constant a is not efficient due to the large
reciprocal lattice limiting the light coupling effect, but large lattice constant a will not give
higher efficiency because the pillars or air holes are also large reducing the scattering effect.
Since photonic crystals require fine periodicity with no defects, other structure like cones
and truncated cones will be good alternatives to photonic crystals with less requirement on
defects. We found the truncated cones can give higher light extraction efficiency than cones
and pillars.
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Chapter 4: ZnO Piezoelectric Devices
4.1. Simulation of ZnO Nanostrucutre Piezoelectric Properties
A piezoelectric potential is generated when a ZnO nanostructure is compressed or
stretched. Compressing and stretching leads to opposite electric charges across the surface
of the ZnO material. To understand the working mechanism of ZnO piezoelectric devices,
it is important to obtain the piezoelectric potential distribution on the surface of ZnO
nanostructure. The simulation of ZnO piezoelectric effect is performed by using COMSOL
Multiphysics®_5.1, a simulation tool based on finite-element method (FEM)106.
Besides using FEM simulations, a numerical approach is also derived for calculating
the piezoelectric potential distribution on the surface of ZnO nanostructure107. There are
three governing sets of equations for a static piezoelectric material, which are: mechanical
equilibrium equation, constitutive equation, geometrical compatibility equation, and Gauss
equation of electric field.
The mechanical equilibrium equation under an external body force 𝑓𝑒(𝑏)
is
∇ ∙ 𝜎 = 𝑓𝑒(𝑏)
where 𝜎 is the stress tensor, which is related to strain 𝜀 , electric field �⃗⃗� , and electric
displacement �⃗⃗⃗� by constitutive equations:
{𝜎𝑝 = 𝑐𝑝𝑞𝜖𝑞 − 𝑒𝑘𝑝𝐸𝑘
𝐷𝑖 = 𝑒𝑖𝑞𝜖𝑞 + 𝜅𝑖𝑘𝐸𝑘
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where 𝑐𝑝𝑞 is the linear elastic constant, 𝑒𝑘𝑝 is the linear piezoelectric coefficient, and 𝜅𝑖𝑘
is the dielectric constant. According to the 𝐶6𝑣 symmetry of a ZnO crystal, the above three
constant can be expressed as:
In 𝑐𝑝𝑞 matrix, c11 = 209.714 GPa, c12 = 121.14 GPa, c13 = 105.359 GPa, c33 = 211.194 GPa,
c44 = 42.3729 GPa. In 𝑒𝑘𝑝 matrix e31 = -0.567005 C/m2 , e33 = 1.32044 C/m2 , e15 = -
0.480508 C/m2. In 𝜅𝑖𝑘 matrix, 𝜅11= 8.91 𝜅33 = 7.77.
Figure.4.1-1 shows the 3D model of ZnO nanowire and a single unit of 3D hexagonal
network used in COMSOL. We studied the changes of output voltage when changing the
dimensions of ZnO nanowires and the hexagonal wall structure. The applied force on the
top surface of Figure.4.1 (a)(c) is along the negative z direction while in (b)(d) the force is
in positive y direction. In a real case, the result will be the combination of these two cases.
It showed that for force normal to the top surface, the piezoelectric potential is distributed
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along the vertical axis. However, for lateral deflection, the piezoelectric potential is across
the cross section of the nanostructure. The reason is that, under bending, the two opposite
sides undergo two different deformations - one is tension and the other is compression.
Figure.4.1-1 Simulation of ZnO nanowire and hexagonal nanowall piezoelectric
effect. The radius and length of the nanowire are 50 nm and 1200 nm respectively. The
side length and the wall thickness of the hexagonal nanowall are 500 nm and 100 nm
while the height is 1000nm. The potential of bottom surface is set to ground potential.
The top surface is applied with an external load. (a)(c) Fx=0 nN, Fy = 0 nN, Fz = -80
nN; (b)(d) Fx=0 nN, Fy= 80 nN, Fz=0 nN.
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According to the piezoelectric properties of ZnO, we can know that the output voltage
has a linear relationship with the nanostructure’s height and external load. However, we
can’t distinguish between the pressure and the shear force from the output voltage alone.
In order to separate the pressure and the shear force, we can grow ZnO nanowires or 3D
hexagonal network structure of different height in an array of cells. Figure.4.1-2 shows the
output voltage versus the height of ZnO nanowires. From Figure.2 (a) (b) (c), we can know
that the output voltage increases linearly with the height of ZnO nanowires and the pressure
(Fz) will change the slope of each line. Based on this, we can interpret the pressure from
the slope of each line. When applying a lateral force (or shear force, Fy), the slope won’t
change but only the voltage values will be increased linearly with the lateral force. Also,
Figure.4.1-2 shows that when increasing the radius of ZnO nanowire, the slope of the lines
under the same force condition will decrease while we can get more obvious changes due
to the lateral force relative to the line without the lateral force just like Figure.4.1-2 (c).
The results of ZnO 3D hexagonal network structure are similar to ZnO nanowires.
However, it will be more robust than ZnO nanowires under large external force.
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4.2. Transfer of ZnO Nanostructures Grown by Chemical Vapor
Deposition
The ZnO nanowall/nanowires structure were grown on sapphire by Chemical Vapor
Deposition (CVD) by one of my group members108. Due to the non-electrically conducting
nature of sapphire, the ZnO nanomaterials have to be transferred to another conductive
substrate which will serve as the bottom electrode. The whole transfer process is shown in
Figure.4.1-2 Output voltage of a single ZnO nanowire with different height under
various external load conditions. (a)(b)(c) show the results of ZnO nanowires with
radius of 25nm, 50nm, and 200nm. (d) shows the schematic of an array of ZnO
nanowires as a force sensor.
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Figure 4.2-1. First, a thick PDMS (Sylgard 184, Dow Corning) layer was deposited onto
the ZnO nanostructure. The thick PDMS layer can be peeled off together with the
embedded ZnO nanostructure in the following step. Then, attach the PDMS layer to a glass
substrate with silver conductive adhesive epoxy deposited on the top surface. Before
bonding the peeled PDMS sheet to the glass substrate, the thick PDMS layer (3~5 mm
thickness) was thinned down to less than 1 mm thickness beforehand, using a razor blade.
This preprocessing can decrease the time of wet etching required to further thin down the
upper PDMS layer. The wet etching of PDMS was done using a solution of TBAF (75 wt%
in water) and NMP. TBAF etches PDMS while NMP dissolve the etched product109. The
volume ratio of TBAF (75 wt% in water) to NMP was 1:3. Finally, RIE etching of PDMS
was employed to expose the top of ZnO nanostructure for the subsequent top electrode
integration. The PDMS base and the curing agent was mixed with a ratio of 10:1.
According to a previous report, the wet etching of PDMS is much faster than dry etching110.
The typical etching rate of wet etching is about 1.5 μm/min (~100μm/h). However, the
fastest rate of dry etching is about 20 μm/h. So wet etching is about 5 times faster than dry
etching.
Figure 4.2-1 The flow chart of the transfer process of ZnO to a conductive
substrate.
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Figure 4.2-1 (1)-(3) shows the steps for transferring ZnO nanostructure to PDMS. The
PDMS base and curing agent of 10:1 weight ratio were mixed thoroughly and poured onto
the ZnO sample. Before curing the PDMS, the whole unit shown in Figure4.2-1 (2) was
degassed and left in room temperature for 24 hours to let the PDMS go into the space
between ZnO nanostructures. After 24 hours at room temperature, the PDMS was cured at
125 ͦC for 30 min. Figure 4.2-2 shows the microscope image of the ZnO nanostructure
grown on sapphire and the ZnO nanostructure transferred onto PDMS. 50x magnification
was used. It is clear that the ZnO nanowires were transferred to the PDMS stamp and the
PDMS filled the space between the ZnO nanostructures. The big metal partical was Zn
crystal spun onto the surface of sapphire substrate during CVD growth of ZnO.
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Figure 4.2-2 Transfer of ZnO nanostructure onto PDMS stamp. (a) The cured PDMS
was peeled off from ZnO sample. (b) (c) (d) are microscope images of original
sample, sample-after-peel-off-PDMS and the surface of PDMS stamp, respectively.
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However, not all ZnO nanowires can be transferred to the PDMS stamp. Figure 4.2-3
gives an example that no ZnO nanowires were transferred onto PDMS stamp. Figure 4.2-
3 (b) corresponds to the same area of (a) indicated by the stripe area where no material was
inside.
Figure 4.2-3 The area of ZnO nanowires that were not
transferred to PDMS stamp. (a) The sample after peeling off
PDMS sheet. (b) the surface of PDMS stamp.
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4.3. Wet Etching of PDMS
In order to fabricate ZnO piezoelectric device, the top and bottom end of ZnO
nanostructure should be bonded to metal electrode. Thus, the following etching of PDMS
was needed to expose the top of ZnO nanostructure. Before doing wet etching of PDMS,
the PDMS stamp was bonded to a glass substrate using silver electrical conductive adhesive
epoxy. Figure 4.3-1 shows the PDMS stamp bonded to the glass substrate. Super glue was
Figure 4.3-1 (a) The thickness of PDMS stamp before doing wet
etching. (b) The PDMS stamp bonded to a glass substrate.
100
used to seal the edges of PDMS stamp to prevent etching the edges and bottom side of
PDMS stamp, otherwise the PDMS will fall off the glass substrate during wet etching.
Wet etching of PDMS was done in a solution of TBAF (Tetrabutylammonium fluoride,
75 wt% in water, Sigma Aldrich) in NMP (N-Methyl-2-pyrrolidone, Sigma Aldrich) with
a volume ratio of 1:3. The fluoride ion can react with Si-O bond to etch the PDMS and
NMP can carry away the etched product109,111. To test the etching speed of the etchant
solution, a small PDMS piece with a beam on top was used. Figure 4.3-2 shows the results
of etching rate test for PDMS wet etching using the solution of TBAF mixed in NMP.
Before etching, the original width of the beam was 1.96mm. After 30 min in the solution
of TBAF in NMP (1:3 volume ratio), the width decreased to 1.76mm. After another 30 min,
the width dropped to 1.65 mm. The etching rate was about 100 μm/30min at the beginning
and then decreased to 55 μm/30min. The decrease of the etching rate may be caused by the
decrease of TBAF concentration in the etchant solution. It is also possible that the etched
products were not carried away by NMP instantly, thus inhibiting subsequent etching.
Finally the beam was immersed in TBAF/NMP solution with a v/v ratio of 1:6, the width
reduced to 1.43 mm after etching for 30 min. The etching rate of TBAF/NMP solution of
1:6 volume ratio was 110 μm/30min. This result was not as high as the previous reported
speed which was approximately 180 μm/30min112. The reason for the difference is possibly
that the total amount of the solution was not large enough so that the etching rate drops fast.
Figure 4.3-3 shows the width of the beam as a function of etching time. The slop of stage
II drops a little indicating drops in etching rate.
Figure 4.3-4 shows the top morphology of the PDMS with ZnO nanostructure
underneath after wet etching. The wet etching time was 1h30min. Figure 4.3-3 (a) shows
101
the top surface of PDMS before etching. Incision marks caused by razor blade cutting could
be clearly seen on the top surface. After wet etching for 1h 30min, in Figure 4.3-3 (b) the
incision marks disappeared and the bottom silver epoxy layer could be seen through the
PDMS layer. Figure 4.3-3 (c)(d) shows the top surface with 50x and 100x magnification.
The large Zn source powders can be seen through the thin PDMS layer. It can be inferred
that the thickness of PDMS was decreased significantly. The wet etching increased the
surface roughness of PDMS and porous structure occurred on top surface. While the large
Zn powders can be seen through the PDMS layer, it is difficult to determine if any ZnO
nanostructure was exposed after wet etching.
102
Figure 4.3-2 Etching rate of PDMS using TBAF/NMP solution. a) the PDMS sample
with a beam on top before etching; b) etched for 30min in TBAF in NMP with a ratio
of 1:3; c) etched for 1 hour in TBAF in NMP with a ratio of 1:3; d) etched for another
30 min after c) in TBAF in NMP with a ratio of 1:6.
103
Figure 4.3-4 Top morphology of PDMS with ZnO nanostructure underneath. (a) The
top surface texture before wet etching. (b)(c)(d) top morphology of PDMS etched for
1h30min.
Figure 4.3-3 The width of the PDMS beam as a function of etch
time. Stage I &II: TBAF:NMP=1:3; Stage III: TBAF:NMP=1:6.
104
4.4. Dry etching of PDMS
Since PDMS contains Si-O-Si bonds (siloxane bonds), the RIE etch chemistry is similar
to that required by etching silicon or silicon dioxide113. Typically CF4/O2 plasma was used
to etch Si and SiO2 because fluorine atoms react with Si atoms to form volatile compounds
(SiFx). O2 plasma was found to be able to increase the amount of reactive fluorine atoms
present in the plasma by combining with carbon. Besides, it is possible that O2 can more
Figure 4.4-1 (a) PDMS spin-coated on a silicon wafer; (b)
The thickness of the PDMS thin film as a function of RIE
etching time. The film thickness was measured using a thin
film measurement system from FILMETRICS.
105
easily remove methyl groups which contain carbon and hydrogen. Apart from etching, O2
plasma is usually used to treat the surface of PDMS for bonding to another PDMS sheet or
other metals.
In order to measure the etching speed of RIE etching of PDMS, a thin PDMS film was
spin-coated onto a silicon wafer as shown in Figure 4.4-1 (a). The spin speed and time was
2000 rpm and 5 min, respectively. The original PDMS film thickness was estimated to be
12.5 μm according to the report114. Using a thin film measurement system, the thickness
was measured to be 12.271 μm. The PDMS film thickness was measured every 3 min after
RIE etching. The etching recipe was CF4:O2 = 3:1 (37:13 sccm), 260 mTorr, 150 W.
Figure 4.4-2 The surface of PDMS stamp before and after RIE etching.
(a) The PDMS surface morphology after wet etching. (b) the wet-etched
PDMS surface morphology after dry etching
106
Figure 4.4-1 (b) plots the PDMS film thickness as a function of etching time. The
measured thickness was 12.271 μm, 11.564 μm, 10.844 μm, 10.020 μm corresponding to
0, 3, 6, 9 min. The etching rate was about 14 μm/h. Starting from 6 min, the pressure was
decreased to 205 mTorr, but no significant changes occurred to the etching rate. The
etching speed was close to the reported value113. However, it is smaller than the maximum
value 20 μm/h, probably due to the lower power used.
Figure 4.4-2 shows the surface of PDMS after RIE etching. Before dry etching, the
wet-etched PDMS surface showed a roughed surface. The wet etchant caused PDMS
surface to decompose into small pieces and left a porous structure. After RIE etching, the
roughed surface was smoothed and the size of the holes was decreased. The large crystals
were possibly blown away by the plasma gas, leaving black holes on the PDMS surface.
4.5. Conclusion: Piezoelectric Properties in ZnO Nanomaterials
The wurtzite crystal structure endow piezoelectric effect in ZnO materials. Under
external load, internal electric field was generated along the ZnO nanowires and nanowalls.
The output voltage shows a linear relationship with ZnO nanowires/nanowalls height as
well as the external force. PDMS was introduced to transfer ZnO on sapphire to conductive
substrate. However, the fabrication of ZnO piezoelectric devices require top and bottom
electrode integration. A low cost and scalable fabrication process using PDMS was
demonstrated in this work. Etching process was introduced to remove the top PDMS layer
to expose ZnO nanowires/nanowalls. The wet etching of PDMS using TBAF in NMP
solutions shows higher etching rate than RIE etching (CF4:O2 gas). However, we found it
difficult to control the PDMS thickness using etching. The ZnO layer is too thin (~1 μm)
compared with PDMS layer (~500 μm).
107
Chapter 5: Conclusions and Future Work
5.1 Conclusions
In this work, the light extraction efficiency of III-nitride LEDs was investigated.
Photonic crystals, cones and truncated cones were employed on top of n-GaN surface to
enhance the light extraction efficiency. 3D-FDTD method was used to analyze the light
extraction efficiency for TFFC InGaN MQW LEDs. We found that the light extraction
efficiency enhancement from photonic crystals is not related to photonic band gap effect
but is more related to scattering and diffraction effect and light coupling effect. Based on
the study of the structural parameters of PCs, we found that 200 nm thickness is sufficient
for PCs to be efficient, the filling factor around 0.5 can give maximum light extraction
efficiency. Lattice constant that is smaller than 400 nm limits the light coupling effect while
lattice constant that is larger than 1500 nm reduces the scattering effect. The calculated
light extraction efficiency was about 70% for PC LEDs and 60 % for LEDs with truncated
cones. Truncated cones can serve as alternatives to photonic crystals with the advantages
of lower cost for fabrication and high tolerance of defect effects. In contrast, PCs are
typically fabricated with expensive e-beam lithography.
Piezoelectric properties of ZnO nanomaterials were investigated. ZnO nanowires and
nanowalls were grown on sapphire substrates. In order to fabricate ZnO piezoelectric
devices, electrically conductive substrate is required. A transfer process for ZnO
nanomaterials to a conductive substrate using PDMS was proposed in this work. The
transfer of ZnO onto the PDMS stamp is successful. However, the thick PDMS layer needs
to be removed to expose the ZnO nanomaterials top end for the top electrode integration.
108
Wet and RIE etching of PDMS was compared. However, it is hard to determine whether
the top of ZnO nano materials is exposed since the ZnO nanostructure is too thin.
5.2 Future Work
Based on the studies of light extraction efficiency of III-nitride LEDs, photonic crystals
proved a significant enhancement. However, the effect of the defects of photonic crystals
on the light extraction efficiency has not been well studied yet. This issue is critical as it is
challenge to fabricate defect-free photonic crystal during the RIE process.
For ZnO piezoelectric devices, well control of the thickness of the etched PDMS layer
need to be studied to expose the top end of the ZnO nanostructure. Longer RIE etching of
PDMS could be employed. Instead of monitor the thickness of PDMS, we can also design
an etch stop process. We can first spin coat a thin film of PDMS filling the space between
the ZnO nanostructures while exposing the top of ZnO nanostructure. And then we can
deposit a thin layer of parylene which cannot be etched by the wet etchant of PDMS so that
the wet etching will stop when reaching the parylene layer. Using this technique, we can
control the etching process and expose the top of ZnO nanomaterial. If the etching process
is optimized to expose the top surface of the ZnO nanomaterials, ZnO piezoelectric devices
fabrication can be implemented.
109
Appendix
A. Electromagnetism in dielectric medium
The electromagnetic fields in the macroscopic medium are described by the Maxwell
equations. In SI units, they are
where E and H are electrical and magnetic fields, D and B are the displacement and
magnetic induction fields, ρ and J are the free charge and current densities.
For a given dielectric medium, the structure and the distribution of composites don’t
change with time, and there are no free charge or currents. Only light propagates through
this material. Thus, we can have ρ=0 and J=0.
The relation between D and E is given by
(2)
The relation between B and H is given by
(3)
where P and M is polarization density and magnetization vector field in the material,
respectively.
For simplicity, we assume the dielectric medium is linear, homogenous and isotropic
so that P depends linearly on the electric field E, which is given by
(4)
Thus
(5)
where is the permittivity and the relative permittivity of the material.
Ñ×B= 0 Ñ´E+¶B
¶t= 0
Ñ×D= r Ñ´H-¶D
¶t= J
D =e0E+P
H =B
m0
-M
P =e0cE
D=e0(1+ c)E=eE
e =e0er er = (1+ c)
(1)
110
Similarly, we have
(6)
For most materials, if there are no magnetic materials around, then the relative magnetic
permeability is often close to unity and we can further simplify the relation to be
.
Using equations (5) and (6) to replace D and B in (1) and considering E and H’s
dependence on position vector r and time t, we can rewrite the Maxwell’s equations to
become
The solution to equations (7) is the combination of basic harmonic modes and can be
expressed mathematically as
We can insert equations (8) into equations (7). The two equations of (7) on the left side just
interpret that there are no point sources. Other two equations give the relation between E(r)
and H(r) and they are
We can decouple these two equations by replacing E(r) in the first equation and get
(10)
Simplest version:
B= m0mH
m
B= m0H
Ñ×H(r, t) = 0 Ñ´E(r, t)+m0
¶H(r, t)
¶t= 0
Ñ×[e(r)E(r, t)]= 0 Ñ´H(r, t)-e0er¶E(r, t)
¶t= 0
E(r, t) =E(r)e-iwt
H(r, t) =H(r)e-iwt
Ñ´E(r)- iwm0H(r) = 0
Ñ´H(r)+ iwe0e(r)E(r) = 0
Ñ´1
e(r)Ñ´H(r)
æ
èç
ö
ø÷=
w
c
æ
èç
ö
ø÷
2
H(r)
(7)
(8)
(9)
111
According to the Maxwell’s equations, the governing equation Master Equation for the
behavior of magnetic field can be written as
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