Light Extraction Efficiency in Iii- Nitride Light-Emitting Diodes and Piezoelectric Properties in Zno Nanomaterials

LIGHT EXTRACTION EFFICIENCY IN III- NITRIDE LIGHT-EMITTING DIODES AND PIEZOELECTRIC PROPERTIES IN ZNO NANOMATERIALS

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.

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

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

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

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

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

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-

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

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

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

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

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

Light Extraction Efficiency in III-nitride Light-Emitting Diodes and Piezoelectric Properties in ZnO Nanomaterials

Abstract

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.

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

13 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

14 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

6 Bevacqua first demonstrated ternary GaAs1-xPx p-n junctions emitting red light . 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

15 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

16 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

23 (θc) can go through the top surface . The cone-shaped solid angle is called escape cone.

With some simple derivation, it can be shown that the light extraction efficiency is only

2 about 1/4n . 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.

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

æ 1 ö æw ö2 Ñ´ç Ñ´ H(r)÷ = ç ÷ H(r) (1) èe(r) ø è c ø

Hence, the problem of solving Maxwell’s equations becomes an eigenvalue problem.

Equation (1) can also be written as

2    H(r)    H(r) (2)  c 

18 together with H(r)  0, [(r)E(r)]  0 , we can solve the electric and magnetic field everywhere in the medium, E(r) can be written as

i E(r)    H(r) (3)  0 (r)

The above equations are based on the following assumptions: first, the field strength should be small so that the displacement of field D(r) has a linear relation with the electric field E(r) ; second, the material should be macroscopic and isotropic so that and

are related by  0 multiplied by a scalar dielectric function (r,); 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

ik×r H(r) = H0 ×e (4)

2 2 2 where H 0 is a constant vector and H 0 k  0 . The eigenvalues are ( / c )  k /  , which gives the dispersion relation,   c k /  .

Considering the electromagnetic waves propagating in the periodic dielectric medium with discrete translational symmetry in just one direction, that is (r)  (r  layˆ) , l is an integer. As is described by Bloch’s theorem, the eigenfunctions should have the form of

ik y H (r)  eikx x  e y u(y, z) (8) kx ,k y u(y, z)  u(y  la, z) is a periodic function. For a three dimensional periodic system, the more general solution is

ikr H k (r)  e uk (r) (9) where k is the Bloch wave vector which lies in the first Brillouin Zone and

uk (r) = uk (r+ R) 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 f or angular frequency w = 2p f and wave vector k . 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.   c k /  . There is a certain frequency value corresponding to a certain wave number. The spectrum is a continuous range of frequency.

The behavior of light in a periodic structure will have some unique properties. The

ikr electromagnetic modes are Bloch modes like H k (r)  e uk (r) instead of simple

ik×r harmonics like H(r) = H0 ×e 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.

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

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

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:

æ 1 ö æw ö2 Ñ ´ç Ñ´ H(r)÷ = ç ÷ H(r) èe (r) ø è c ø

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.

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

The solution to this equation is a linear combination of in-plane harmonic waves40:

i(k||+G)×r E = åEGe G

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

22 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

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

24 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.

(a)

Figure 1.2.1-1 (a) Wurtzite crystal structure of ZnO61. (b) The formation of electric dipole under external strain62.

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.

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.

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

28 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

29 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:

H E   (2.1a) t E  H  E   (2.1b) t

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, E x , E y , Ez and H x , H y , H z . This leads to a set of six scalar equations as follows in equations 2.2a-f 77:

H x 1  Ey Ez      (2.2a) t   z y 

H y 1  E E    z  x  (2.2b) t   x z 

H z 1  Ex Ey      (2.2c) t   y x 

E 1  H H  x  z y     Ex  (2.2d) t   y z 

E y 1  H x H z     E y  (2.2e) t   z x 

E 1  H H  z  y x     Ez  (2.2f) t   x y  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

(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: (i, j,k)  (ix, jy,kz) (2.3) and any function of space and time can be represented as: F n (i, j,k)  F(i , j ,k ,nt) (2.4) where   x  y  z 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: F n (i, j,k) F n (i 1/ 2, j,k)  F n (i 1/ 2, j,k)   O 2  (2.5a) x  F n (i, j,k) F n1/ 2 (i, j,k)  F n1/ 2 (i, j,k)   Ot 2  (2.5b) t t

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.

Figure 2.1.2-1 A unit cell of Yee’s lattice with specified position of the field components77.

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:

n1/ 2 H x (i, j 1/ 2,k 1/ 2)  t H n1/ 2 (i, j 1/ 2,k 1/ 2)  E n (i, j 1/ 2,k 1)  E n (i, j 1/ 2(2.6a,k) x (i, j 1/ 2,k 1/ 2) y y n n )  Ez (i, j,k 1/ 2)  Ez (i, j 1,k 1/ 2)

n1/ 2 H y (i 1/ 2, j,k 1/ 2)  t H n1/ 2 (i 1/ 2, j,k 1/ 2)  E n (i 1, j,k 1/ 2)  E n (i, j,k 1(2.6b/ 2) y (i 1/ 2, j,k 1/ 2) z z n n )  Ex (i 1/ 2, j,k)  Ex (i 1/ 2, j,k 1)

n1/ 2 H z (i 1/ 2, j 1/ 2,k) 

n1/ 2 t n n H z (i 1/ 2, j 1/ 2,k)  Ex (i 1/ 2, j 1,k)  Ex (i 1/ 2, j(2.6c,k) (i 1/ 2, j 1/ 2,k) n n )  E y (i, j 1/ 2,k)  E y (i 1, j 1/ 2,k)

n1   (i 1/ 2, j,k)t  n Ex (i 1/ 2, j,k)  1  Ex i 1/ 2, j,k   (i 1/ 2, j,k)  (2.6d t  H n1/ 2 (i 1/ 2, j 1/ 2,k)  H n1/ 2 (i 1/ 2, j 1/ 2,k)  (i 1/ 2, j,k) z z ) n1/ 2 n1/ 2  H y (i 1/ 2, j,k 1/ 2)  H y (i 1/ 2, j,k 1/ 2)

  (i, j 1/ 2,k)t  E n1 (i, j 1/ 2,k)  1  E n i, j 1/ 2,k y   (i, j 1/ 2,k)  y   (2.6e t n1/ 2 n1/ 2  H x (i, j 1/ 2,k 1/ 2)  H x (i, j 1/ 2,k 1/ 2) )  (i, j 1/ 2,k) n1/ 2 n1/ 2  H z (i 1/ 2, j 1/ 2,k)  H z (i 1/ 2, j 1/ 2,k)

  (i, j,k 1/ 2)t  E n1 (i, j,k 1/ 2)  1  E n i, j,k 1/ 2 z   (i, j,k 1/ 2)  z   (2.6f t n1/ 2 n1/ 2  H y (i 1/ 2, j,k 1/ 2)  H y (i 1/ 2, j,k 1/ 2) )  (i, j,k 1/ 2) n1/ 2 n1/ 2  H x (i, j 1/ 2,k 1/ 2)  H x (i, j 1/ 2,k 1/ 2) 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,

n1/ 2 H x (0,1/ 2,1/ 2) is obtained from the value of itself in one-time step before together with

n n n n E y (0,1/ 2,1) , E y (0,1/ 2,0) , Ez (0,0,1/ 2) and Ez (0,1,1/ 2) in half-time step before.

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:

(2.7) S  E  H 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:

S  EH (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

34 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:

E B (2.9) S  max max 20 Since E (2.10)  c B the average intensity can also be written in terms of just the electric field maximum as: E 2 (2.11) S  max 2c0 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 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

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: 1  (2.12) Power( f )   real(P( f )) ds 2 surface   1 where P( f ) is the complex Poynting vector P( f )  E  H * whose real part gives the 2 time-averaged rate of energy flow78. The frequency dependent Poynting vector  could be calculated from electric field component E( f ) 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.

  2 (2.12) P( f )  n f  0 E( f ) 0

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 (2.13) Ek (x)  uk (x)e where uk (x) is a periodic function of x, with period a. This means uk (x  a)  uk (x) . 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.

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 2.2.2-2 Recorded signals of a guided mode in a two- dimensional hexagonal photonic crystal. (a) The recorded time signal; (b) Fourier transform of (a).

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

40 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.

Figure 3.1.1-1 A typical structure of InGaN-based LED grown on sapphire substrate.

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

42 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.

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.

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.

Figure 3.1.3-1 The edge-emitting spectrum of blue InGaN/GaN MQWs LED at 455 nm92.

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

46 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.

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

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.

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 3.1.4-2 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.

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)

49 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.

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

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

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

ck (3.1) (k)  n 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.

Figure 3.2.2-1 Schematic illustration of a PC periodic in one dimension.

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

53 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.

(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.

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.

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 (3.2) (n1 cosi  n2 cost ) RTE  2 n1 cosi  n2 cost  (3.3) 4n1n2 cosi cost TTE  2 n1 cosi  n2 cost 

TM mode: 2 (3.4) (n2 cosi  n1 cost ) RTM  2 n2 cosi  n1 cost  (3.5) 4n1n2 cosi cost TTM  2 n1 cost  n2 cosi 

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

57 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.

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. 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.

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

Figure 3.3.1-1 Illustration for the interference of original top emitting light and the light reflected by the mirror. 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:

2 2 2i () 2 2 2 (3.1) E ()  E0 ()1 re  E0 ()1 r  2r cos2()

61 where E0 () 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:

2  2ktcos() (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 2kt  2m with an integer m for +r and a half-integer m for –r. For a perfect lossless mirror ( r 1) and under the constructive interference condition, the final output intensity would have a 4 times enhancement

E()  4E0 () . This enhancement corresponds to a source-mirror distance t having the following relation with the wavelength of light:

m (3.3) t  2n

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

 460 which gives t   (nm)  92.7(nm) . 2n 22.48

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

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.

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

65 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.

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.

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 3.4.1-2 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.

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.

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 80.9% 68%-78% Efficiency

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

71 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.

72 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,

73 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:

퐴푟푒푎 표푓 푝𝑖푙푙푎푟푠 표푟 푎𝑖푟 ℎ표푙푒푠 (3.4) 푓 = 푊ℎ표푙푒 푠푢푟푓푎푐푒 푎푟푒푎 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:

푟 = 0.5251 × √푓 × 푎 (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.

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.

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

75 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

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. 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

76 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

77 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.

(a)

(b)

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.

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.

80 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

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 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.

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.

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.

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

84 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.

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.

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.

3.5.3. Considering Dipole Position for LEDs with Truncated Cones

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.

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.

Table 3.5.3-1 Weighted average of LEE for LEDs with truncated cones Top & bottom rbot=230nm, rbot=297nm, rbot=350nm, rbot=400nm, radius rtop=115nm rtop=106nm rtop=175nm rtop=57nm Average 60.7% 63.7% 64.6% 58.5% (weight)

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.

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.

⃗(푏) The mechanical equilibrium equation under an external body force 푓푒 is ⃗(푏) ∇ ∙ 휎 = 푓푒 where 휎 is the stress tensor, which is related to strain 휀, electric field 퐸⃗⃗, and electric displacement 퐷⃗⃗⃗ by constitutive equations: 휎 = 푐 휖 − 푒 퐸 { 푝 푝푞 푞 푘푝 푘 퐷푖 = 푒푖푞휖푞 + 휅푖푘퐸푘

90 where 푐푝푞 is the linear elastic constant, 푒푘푝 is the linear piezoelectric coefficient, and 휅푖푘 is the dielectric constant. According to the 퐶6푣 symmetry of a ZnO crystal, the above three

constant can be expressed as:

In 푐푝푞 matrix, c11 = 209.714 GPa, c12 = 121.14 GPa, c13 = 105.359 GPa, c33 = 211.194 GPa,

2 2 c44 = 42.3729 GPa. In 푒푘푝 matrix e31 = -0.567005 C/m , e33 = 1.32044 C/m , e15 = -

2 0.480508 C/m . In 휅푖푘 matrix, 휅11= 8.91 휅33 = 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

91 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.

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.

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.

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 4.2-1. 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.

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 Cͦ 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.

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.

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.

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.

99 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

100 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.

101

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.

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

4.4. Dry etching of PDMS Since PDMS contains Si-O-Si bonds (siloxane bonds), the RIE etch chemistry is similar

113 to that required by etching silicon or silicon dioxide . 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

104 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

105

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).

106

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.

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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.

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Appendix A. Electromagnetism in dielectric medium The electromagnetic fields in the macroscopic medium are described by the Maxwell equations. In SI units, they are

¶B Ñ×B = 0 Ñ ´ E+ = 0 ¶t (1) ¶D Ñ×D = r Ñ ´ H - = J ¶t 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

D =e0E+ P (2) The relation between B and H is given by

B H = - M (3) m0 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

P =e0 cE (4) Thus

D =e0 (1+ c)E =eE (5) where e =e0er is the permittivity and er = (1+ c) the relative permittivity of the material.

109

Similarly, we have

B = m0mH (6) For most materials, if there are no magnetic materials around, then the relative magnetic permeability m is often close to unity and we can further simplify the relation to be

B = m0H.

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

¶H(r,t) Ñ×H(r,t) = 0 Ñ´ E(r,t)+ m = 0 0 ¶t (7) ¶E(r,t) Ñ×[e(r)E(r,t)]= 0 Ñ´ H(r,t)-e e = 0 0 r ¶t The solution to equations (7) is the combination of basic harmonic modes and can be expressed mathematically as

-iwt E(r,t) = E(r)e (8) H(r,t) = H(r)e-iwt 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

Ñ´E(r)-iwm H(r) = 0 0 (9)

Ñ´H(r)+iwe0e(r)E(r) = 0 We can decouple these two equations by replacing E(r) in the first equation and get

æ 1 ö æw ö2 Ñ´ç Ñ´ H(r)÷ = ç ÷ H(r) (10) èe(r) ø è c ø Simplest version:

110

According to the Maxwell’s equations, the governing equation Master Equation for the behavior of magnetic field can be written as

æ 1 ö æw ö2 Ñ´ç Ñ´ H(r)÷ = ç ÷ H(r) èe(r) ø è c ø

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