transmission electron microscopy and thermal …

TRANSMISSION ELECTRON MICROSCOPY AND THERMAL RESIDUAL

STRESS ANALYSIS OF AlN CRYSTAL

by

RAC. G. LEE., B.S.

A Thesis

In

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

Approved

Dr. J. Chaudhuri

Dr. A. Idesman

Dr. Y. Ma

John Borrelli Dean of the Graduate School

May, 2007

Texas Tech University, Rac. G. Lee, May 2007

ii

ACKNOWLEDGEMENTS

I would like to acknowledge the support and guidance of my committee chair, Dr. J.

Chaudhuri, who donated countless hours to investigate the high energy gab

semiconductor used in this work. Thanks for the help during the long afternoons in the

office, week ends, and most importantly thanks for your open-mind.

Appreciation is also given to the remaining members of my advisory committee,

Dr. A. Idesman, Dr. Y. Ma, for their time and effort.

Thanks to my wife, Sun Jin Jang, who is everything to me, and I dedicate this

work to my three months old daughter, A Hyun Lee (Katie Lee). Most of all, I would like

to give special thanks to my parents for always supporting my decisions.


iii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS.......................................................................................ⅱ

ABSTRACT ............................................................................................................Ⅴ

LIST OF TABLES.....................................................................................................Ⅵ

LIST OF FIGURES ...................................................................................................Ⅶ

CHAPTER

Ⅰ INTRODUCTION .............................................................................1

1.1 Preamble ......................................................................................1

1.2 Objective ......................................................................................1

Ⅱ MODELING OF RESIDUAL STRESS FOR AlN CRYSTAL

GROWN ON TUNGSTEN SUBSTRATE........................................2

2.1 Introduction..................................................................................2

2.2 Problem formulation and sample geometry.................................3

2.3 Effect of the meshes.....................................................................9

2.4 Effect of the size of AlN film ......................................................10

2.5 Effect of the thickness of the AlN film........................................18

2.6 Effect of the thickness of the substrate ........................................20

2.7 Effect of the interaction of the islands .........................................21

2.8 Effect of the different orientation of the grain .............................24

2.9 Conclusions..................................................................................26

Ⅲ MODELING OF RESIDUAL STRESS FOR AlN CRYSTAL

GROWN ON POSSIBLE CRUCIBLES ...........................................28

3.1 Introduction..................................................................................28


iv

3.2 Sample geometry and properties..................................................29

3.3 Sapphire (Al2O3) substrate...........................................................31

3.4 TaC and NbC substrate ................................................................32

3.5 α-SiC (6H) and W substrate.........................................................34

3.6 Conclusions..................................................................................39

Ⅳ THE ANALYSIS OF BULK AlN CRYSTAL GROWN ON

TUNGSTEN SUBSTRATE..............................................................40

4.1 Introduction..................................................................................40

4.2 Sublimation method .....................................................................40

4.3 Surface morphology by SEM.......................................................44

4.4 EDAX result using SEM..............................................................46

4.5 Dislocation Study by Etching ......................................................49

4.6 HRTEM images of AlN crystal ...................................................51

4.7 CBED technique and experimental data ......................................59

4.8 Conclusions..................................................................................62

BIBLIOGRAPHY......................................................................................................63

APPENDIX ............................................................................................................65

A. Thermal stress data of AlN using W substrate..........................................65

B. Thermal stress data of AlN using possible crucibles ................................81

C. The diffraction pattern of AlN crystal.......................................................86


v

ABSTRACT

Presented in this thesis is an investigation into the residual thermal stress

distribution in an AlN single crystal (film), grown using W (substrate) as a crucible

material, and appropriate crucibles, sapphire, tantalum carbide, niobium carbide and

silicon carbide, are also investigated. An optimal choice of crystal growth conditions

results in the formation of coalesced boundaries known as island structures. A finite

element model has been used that accounts for different arrangements of these island

structures. The model is based on the dimensionless coordinates that significantly reduces

the number of variables to be computed. Experimentally grown aluminum nitride crystal

with rough and smooth surface was examined by Transmission Electron Microscopy

(TEM) and Scanning Electron Microscopy (SEM). The special technique, Convergent

Beam Electron Diffraction (CBED), was used to determine the polarity of the rough and

smooth surface, and High Resolution Transmission Electron Microscopy (HRTEM)

image was used to investigate the dislocation in aluminum nitride crystal.


vi

LIST OF TABLES

2.1 Elastic Constants......................................................................................5

2.2 Thermal Expansion Coefficients..............................................................5

2.3 The Minimum and Maximum Values of the Stress for 2-D case ............14

2.4 The Minimum and Maximum Values of the Stress for 3-D case ............18

3.1 Elastic Constants for AlN, Al2O3, SiC, TaC, NbC and W ......................30

3.2 Thermal Expansion Coefficients for AlN, Al2O3, SiC, TaC, NbC, W ...31

3.3 The Values of the Thermal Stress for Various Substrate ........................38


LIST OF FIGURES

2.1 Sample Geometry.....................................................................................7

2.2 Effect of the Meshes ................................................................................10

2.3 The Distribution of Stress Component for 2-D Case...............................13

2.4 The Variation of Stress Component for 2-D and 3-D Cases ...................14

2.5 (a) The Distribution of Stress Component σ11 and σ22 for 3-D Case.......15

2.5 (b) The Distribution of Stress Component σ33 and σ12 for 3-D Case.......16

2.5 (c) The Distribution of Stress Component σ13 and σ23 for 3-D Case.......17

2.6 The Distribution of Stress Component for H

L

dd 1 =32 and 2.......................19

2.7 Variation of Stress Component for H

L

dd 1 = 32, 16, 8, 4 and 2 ...................20

2.8 Variation of Stress Component for H

S

dd

= 10, 5 and 2 .............................21

2.9 The Distribution of the Stress and Comparison of Different H

L

dd 2 ............23

2.10 (a) The Variation of the Stress in Different Orientations of the Grain ..25

2.10 (b) Three Dimensional Distribution of (a) ............................................26

3.1 Sample Geometry.....................................................................................30

3.2 The Illustration of the Stress Component for Al2O3 Substrate................32

3.3 (a) The Illustration of the Stress Component for TaC Substrate .............33

3.3 (b) The Illustration of the Stress Component for Al2O3 Substrate ..........34

3.4 (a) The Illustration of the Stress Component for SiC Substrate ..............35

3.4 (b) The Illustration of the Stress Component for W Substrate ................36

vii


3.5 The Comparison of the Stress of W, NbC, TaC, Al2O3 and SiC............37

3.6 The Comparison of the σ33 Stress of W, NbC, TaC, Al2O3 and SiC.....38

4.1 Sketch of the AlN Crystal Growth Process..............................................41

4.2 Actual AlN Image and Structure..............................................................43

4.3 SEM Images of Rough and Smooth Surface ...........................................45

4.4 The Description of Bragg Angle..............................................................46

4.5 EDAX Result of the Rough and Smooth Surface....................................48

4.6 Optical Microscope Image of the AlN, Before and After Etching ..........49

4.7 Optical Microscope and SEM Image of Etched AlN...............................50

4.8 Actual Image of TEM and Electron Diffraction ......................................51

4.9 The Optical Microscope Image of TEM Samples ...................................52

4.10 Kikuchi Map for Hexagonal Structure...................................................53

4.11 HRTEM Image of Rough Surface and Slip Phenomenon .....................55

4.12 HRTEM Image of Rough Surface and Diffraction Pattern ...................56

4.13 HRTEM Image of Smooth Surface and Diffraction Pattern..................57

4.14 CBED Pattern Formation.......................................................................58

4.15 CBED Images of AlN Crystal................................................................60

4.16 Comparison of Experimental CBED with Simulation...........................61

A.1 The Distribution of the Stress Component for H

L

dd 1 =4 for 2D Case .......66


L

dd 1 =6 for 2D Case .......67


L

dd 1 =8 for 2D Case .......68

viii


A.4 The Distribution of the Stress for H

L

dd 1 =4 for 3D Case, fixed ..........69 Hd


L



L



L

dd 1 =32 for 3D Case, fixed ........72 1Ld


L



L

dd 1 =8 for 3D Case, fixed ..........74 1Ld


L



L



L

dd 2 =0 for 3D Case.....77


L

dd 2 =0.2 for 3D Case..78


L

dd 2 =1 for 3D Case.....79

A.15 The Distribution of the Stress Component for Mis-oriented Grain ......80

B.1 The Distribution of the Stress Component for AlN Grown on Al2O3......82

B.2 The Distribution of the Stress Component for AlN Grown on TaC .......83

B.3 The Distribution of the Stress Component for AlN Grown on NbC.......84

ix


x

B.4 The Distribution of the Stress Component for AlN Grown on SiC ........85

C.1 The Diffraction Pattern of AlN using [0001] Zone Axis ........................87

C.2 The Diffraction Pattern of AlN using [11-20] Zone Axis.......................88

C.3 The Diffraction Pattern of AlN using [1-100] Zone Axis.......................89


1

CHAPTER 1

INTRODUCTION

1.1 Preamble

Presented in this thesis is an investigation into the thermal residual stress and

transmission electron microscope (TEM) analysis of an AlN crystal. The system for

calculation of thermal residual stress was modeled using a finite element method,

ABAQUS, and AlN crystal grown on W (tungsten) substrate was investigated using a

transmission electron microscope (TEM) and Gatan Digital Micrograph image analysis

program.

1.2 Objective

The objective of the work presented in this thesis is to find out the best growth

condition of AlN and prevent micro crack to make a high quality single AlN crystal. The

various crucibles such as tungsten, sapphire, tantalum carbide, niobium carbide and

silicon carbide were considered for the substrate of AlN crystal for the calculation of

thermal residual stress.

Additionally, it is another objective of this work to analyze experimentally grown

AlN crystal using Transmission Electron Microscopy (TEM) and Scanning Electron

Microscopy (SEM). For TEM analysis, diffraction pattern, Convergent Beam Electron

Diffraction (CBED), Energy Dispersive X-ray Analysis (EDAX) and High Resolution

Transmission Electron Microscope (HRTEM) image were used.


2

CHAPTER 2

MODELING OF RESIDUAL STRESS FOR AlN CRYSTAL GROWN ON TUNGSTEN SUBSTRATE

2.1 Introduction

Residual thermal stress distribution in AlN single crystal, grown on tungsten as a

crucible material, was investigated using a numerical study. It has been demonstrated that

a 3-D formulation instead of a 2-D formulation predicts more accurate values of stress.

Dimensionless coordinates were used to essentially simplify the stress analysis and

reduce the number of calculations. In addition, a thermo elasticity approach simplifies the

study of stresses for a non-stationary temperature field.

The analysis on the interaction of the neighboring island in order to simulate

coalescence of island growth indicates stress concentration at the boundaries of the

islands which could produce threading dislocations and hence polycrystalline growth.

The analysis of the effect of the mis-orientation of the neighboring grains on the residual

thermal stress in the film has shown that a large stress can develop at the grain boundary

and can lead to grain boundary cracking.


2.2 Problem formulation and sample geometry

In this section, the modeling of residual stresses in a sample of heterostructures

(consisting of a film and a substrate) during its cooling from growth temperature (>2000

oC) to room temperature (20 oC) is described. We assume that there are no stresses at the

growth temperature. The residual stresses are induced during cooling to room

temperature by the difference in the thermal expansion coefficients of the film and the

substrate and can be obtained as a result of the solution of the corresponding thermo

elastic problem. A system of equations includes the kinematical equations, the

generalized Hooke’s law and the equilibrium equations. Since tungsten and AlN have

body-centered cubic and hexagonal wurtzite crystal structures, respectively, we will

consider these two crystal lattices. Both cubic and hexagonal lattices can be described as

transversely isotropic materials (material properties of hexagonal lattices are isotropic in

the x1x2 - plane) (J. F. Nye et al. 1957). The generalized Hooke’s law for transversely

isotropic thermo elastic materials in the local material Cartesian system can be written in

the matrix form as follows.

,

000

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 0 0 0 0

33

22

11

23

13

12

33

22

11

66

55

44

332313

232212

131211

23

13

12

33

22

11

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

θ

θ

θ

ε

ε

ε

εεεεεε

σσσσσσ

cc

cccccccccc

(1)

are components of the stress, the total strain and the thermal strain,

are the elastic coefficients. For hexagonal crystal lattices, there are only five independent

stiffness coefficients, ; e.g. . All other coefficients can be expressed

θεεσ ijijij ,, ijc

ijc 5513123311 ,,,, ccccc

3


in terms of these five coefficients as follows: , , , 556613231122 cccccc ===

. Also, . For cubic crystals the generalized Hooke’s law can

be represented by Equation (1) with three independent coefficients ; e.g., .

All other coefficients can be expressed in terms of these ones as follows:

2/)( 121144 ccc −= θθ εε 1122 =

ijc 441211 ,, ccc

446655121323113322 , , ccccccccc ====== . Also, . θθθ εεε 332211 ==

The following expressions are valid for the thermal strain

)( )()( 0θθθαθε θ −= iiii , i=1,2,3 (2)

where )(θαii are the thermal expansion coefficients and depends on the current

temperature θ (i. e. there are no summation over the repeated index i). θ and 0θ are the

growth and room temperatures, respectively. In the present study, thermo elastic

problems are considered; i.e., the stresses depend on the initial and final states and are

independent of intermediate states during cooling and the thermal expansion coefficients

at the initial (growth) and final (room) temperatures are only needed. The residual

stresses are induced due to the difference between the thermal expansion coefficients of a

film (i.e. AlN) and a substrate (i. e. W). Therefore it can be shown that the problem can

be solved if zero thermal strain is prescribed for one material; e.g., for the substrate, and

the thermal strain equal to the difference in the thermal strains in the film and the

substrate is prescribed for the second material; e.g., for the film:

)(])(- )([)()( 0θθθαθαεεε θθθ −=−=Δ sii

fiisiifiiii (3)

Indices s and f correspond to the film and the substrate, respectively. This result follows

from the principle of superposition; i.e., the deduction of a homogeneous thermal strain

4


5

from the total strains does not change the stress distribution since a homogeneous thermal

strain does not induce stresses. All material parameters are given in Tables 2.1and 2.2.

Table 2.1. Elastic Constants (GPa) Crystal C11 C12 C13 C33 C44 C66

AlN (wurtzite) 410.5 148.5 98.9 388.5 131 124.6 W (bcc) 501 198 198 501 151.4 151.4

Source : (Robert R. Reeber et al. 2001) for AlN and (H. B. Huntington) for W

Table 2.2. Thermal expansion coefficients at 1700 K (α, 10-6 K-1) Crystal α 11 α 22 α 33

AlN 6.6 6.6 5.8 W 5.7 5.7 5.7

Source : (H. B. Huntington. P 70)

Geometry of the AlN film grown on tungsten substrate is modeled as a single or

several thin islands placed on a large substrate. A representative sample size was

determined by simulating the stresses with different thickness ratios between the film and

the substrate and also for different planar dimensions of both the film and the substrate.

Based on the convergence of the overall stress-strain response, it was determined that the

thickness of the substrate should be 10 times the thickness of the film. In addition, planar

dimensions of the substrate in directions perpendicular to the thickness, are much larger

then the corresponding dimensions of the islands. A 2-D formulation, with the plane-

stress state and similar to the approach by W. M. Ashmawi (W. M. Ashmawi et al. 2004)

that corresponds to a thin sample, as well as a general 3-D formulation are considered.

For simplicity, rectangular islands (i. e. rectangles and parallelepipeds in the 2-D and 3-D


6

cases, respectively) are studied (more complicated shapes of islands can be analyzed

without any difficulty as well).

Due to symmetry, one half of a sample in the 2-D case and one quarter of a

sample in the 3-D case are considered for problems with one island (Figure 2.1). For the

case of several islands, a symmetric sample with one plane of symmetry for the 2-D case

(Figure 2.1 (b)) and two planes of symmetry for the 3-D case (Figure 2.1 (c)), in addition

to above configuration, are considered. This geometrical symmetry along with the

boundary conditions reduces the dimension of the system for finite element applications

and does not affect the analysis of residual stresses. For the 3-D case, the Cartesian axes

notations are designated as x1 and x2 are the in-plane directions and x3 is the thickness

direction. For the 2-D case, x1 along the in-plane direction and x2 along the thickness

direction are. Normal displacements and tangential forces are zero on planes and lines of

symmetry; all other surfaces are free of stresses.


χ3

χ1

χ2

dHdL1

dW

dS

• •A B

7

(a)

SLd

Hd

Sd

1Ld

χ1

χ3

(Island Ⅰ)

Substrate

(b) (c)

Figure 2.1. Sample geometry: the sample consists of a substrate and one or three islands. (a) Sketch of the entire sample with one island. Due to the symmetry, (b) one half of the sample geometry for the 2-D case and (c) one quarter of the sample for the 3-D case are used for all calculations and reporting results. An additional island is shown with the dotted lines in Figure 1 (c) which is the representative of 3 islands for the whole sample.

The introduction of the dimensionless coordinates e.g.H

X

dd 1 ,

H

X

dd 2 and

H

X

dd 3

essentially simplifies the analysis, where is the height or thickness of the island. It

can be shown that for the linear thermo elastic problems considered, strains and stresses

in the sample depend on the ratio

Hd

H

L

dd 1 in the 2-D case and on the ratios

H

W

dd

and H

L

dd 1 in the


3-D case, and are independent of the absolute value of the island height, ( and

are the dimensions of the island along the coordinate axes x

Hd 1Ld Wd

1 and x2, respectively. The

dimensions of the substrate along the x1 and x2 axes are and , respectively. In the

2-D and 3-D cases stresses

SLd SWd

ijσ can be expressed in terms of dimensionless quantities as:

),,,,,,,,( 21321 θεσσ iif

mnsmn

HHH

S

H

L

H

L

H

Lijij cc

dx

dx

dd

dd

dd

dd

Δ= (4)

and

),,,,,,,,,( 321321 θεσσ iif

mnsmn

HHHH

S

H

L

H

L

H

Lijij cc

dx

dx

dx

dd

dd

dd

dd

Δ= (5)

respectively, where and are the elastic coefficients for the film and the substrate,

and the strain, , is given by Equation (3). For simplicity, and are taken as

equal to each other for the all calculations. The axes of material symmetry used in

Equation (1) are directed in the common Cartesian system as follows: the 11-axis

coincides with the x

fmnc s

mnc

θε iiΔ Wd 1Ld

1 -axis ( i. e. a-axis [1-210] for hexagonal and a-axis [100] for cubic),

the 22-axis coincides with the x2 –axis (i. e. direction [01-10] for hexagonal and b-axis

[ 010] for cubic), and the 33-axis coincides with the x3-axis( i. e. c-axis [0001] for

hexagonal and c-axis [001] for cubic). For the 2-D case 22-axis coincides with the x2-axis

(i. e. c-axis [0001] for hexagonal and c-axis [001] for cubic).

8

Finite element calculations were performed using the commercial finite element

code ABAQUS (ABAQUS. 2004) with quadrilateral quadratic elements. Due to non-

homogeneous stress distributions, non-uniform meshes with refinement near the interface

between the film and substrate are generated for the improved accuracy. Approximately

3200 quadrilateral elements for AlN and 48000 quadrilateral elements for W were used.


Due to discontinuities in thermo mechanical properties across the interface

between the film and the substrate, stresses across the interface are also discontinuous

and have to be analyzed in the film and in the substrate separately along the interface.

To determine the optimum growth condition for the AlN crystal, several

problems as follows are studied; the effect of the growth of the island or film in the in-

plane direction, the effect of different thicknesses of the island and the tungsten substrate,

misorientaion of grains in AlN, and the effect of the interaction between islands. The

results of these calculations are presented in the following sections.

2.3 Effect of the meshes

For the considered problems, the stress reaches a maximum value at the corner of

the interface between the island and the substrate. In other words, the maximum stress

around the corner obtained with the finite element method (or any numerical method) is

mesh-dependent. For example, the distribution of the normal stress in the film along the

x1 direction at the interface for a 3-D problem with one island is given in Figure 2.2 for

the uniform coarse meshes (with 20 nodes along the interface), uniform fine meshes (with

80 nodes along the interface), and non-uniform fine meshes with refinement in the

vicinity of the corner (with 100 nodes along the interface). For the uniform coarse meshes,

the maximum normal stress at the corner is 1.789 GPa. For the uniform fine meshes the

stress is 2.458 GPa and for the non-uniform fine meshes, the stress becomes 2.844 GPa.

It can be seen that stresses for the coarse and fine meshes coincides except a small region

near the corner (i. e. 0.1 from the corner). Therefore, in order to obtain mesh

independent results, we will consider stresses up to the distance 0.1 from the corner

Hd

Hd

9


for all analysis. Another possibility to avoid mesh-dependent results is to use non-linear

models (e.g., elastoplastic models).

0 2 4 6 80.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

S-S

11 (G

Pa)

X1/dH

Figure 2.2. The distribution of in-plane stress component, 11σ in the film along the x1 axis at the interface for different finite element meshes. Curves A, B and C correspond to the non-uniform fine mesh with mesh refinement near the corner (100 nodes along the interface), the uniform fine mesh (80 nodes along the interface), and uniform coarse mesh ( 20 nodes along the interface), respectively.

2.4 Effect of the size of the AlN film

In this section, the effect of the planar dimensions of the AlN film on residual

thermal stress is analyzed. A thick and large substrate is considered e. g. the thickness

and planar dimensions of the substrate are 10 and 24 times the thickness of the island

(H

S

dd

= 10, H

SL

dd

= 24 for the 2-D case and H

S

dd

= 10 and H

SL

dd

= H

SW

dd

= 24 for the 3-D case,

where dS, dSL and dSW are the thickness and planar dimensions of the substrate, 10


respectively, and dH is the thickness of the island (Figure 2.1)). The dimensions of the

island were varied as follows: H

L

dd 1 = 4, 6 and 8 for both 2-D and 3-D cases where dL1 is

the length of the film along the x1 direction (see Figure 2.1 (b) for the 2-D case, and

Figure 2.1(c) for the 3-D case). For example, for the island thickness, = 0.1 mm, these

ratios correspond to the following planar dimensions of the island = = 0.4, 0.6

and 0.8 mm, respectively, and the substrate dimensions are = 1mm and = =2.4

mm.

Hd

1Ld Wd

Sd SLd SWd

For the 2-D formulation (plane stress case), Figures 2.3 (a), (b) and (c) show the

distribution of the normal and shear components of the stresses in the film and substrate

for H

L

dd 1 as 8 . Figure 2.4 (a) shows the comparison of variation of σ11 in the film along the

x1 axis at the interface and Table 2.3 summarizes the stress values at the center and corner

in the film and the substrate along the interface for all three cases. For the 3-D

formulation, Figures 2.5 (a), (b) and (c) show the distribution of stresses for H

L

dd 1 as 8,

Figure 2. 4 (b) shows the comparison of variation of σ11 in the film along the x1 axis at

the interface and Table 2.4 summarizes the stress values at the center and corner in the

film and the substrate along the interface for all three cases. The stress components are

almost constant along the interface except at the corner where there is a stress

concentration and the stress increases enormously. The in-plane stresses (i. e. σ11 and σ22)

in AlN are tensile and higher than the yield strength of AlN (i. e. 0.3 GPa (I. Yonenaga et

11


12

al.)) while the normal component along the thickness direction (i. e. σ33) is almost zero

and shear component (i. e. σ13) is compressive.

In-plane stress in AlN increases as the islands become larger and may lead to the

formation of microscopic cracks especially at the corner. The stresses in the tungsten

substrate are compressive except at the corner, where the normal stress becomes tensile

and the shear stress is zero. The reversal of stress across the interface (compressive in the

substrate and tensile in the film) may cause delamination or separation of the film.

The values of all stress components in the 3-D case are much higher as compared

to those in the 2-D case. This indicates that stresses in 2-D calculations are

underestimated and should be used for very rough estimations. Therefore, all results that

follow are analyzed using the 3-D formulation.


(a)

(b)

(c)

Figure 2.3. The distribution of stress components (a) σ11, (b) σ22, and (c) σ12 for H

L

dd 1 = 8

for the 2-D case.

13


-1 0 1 2 3 4 5 6 7 8 90.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

CBAS

-S11

(GP

a)

X1/dH

A: dL1/dH = 4B: dL1/dH = 6C: dL1/dH = 8

-1 0 1 2 3 4 5 6 7 8 90.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

CB

A

S-S

11 (G

Pa)

X1/dH

A: dL1/dH = 4B: dL1/dH = 6C: dL1/dH = 8

(a) (b) Figure 2.4. The variation of stress component, σ11, in the film along x1 direction at the

interface (i. e. from the center to the edge) for H

L

dd 1 = 4, 6 and 8 for (a) the 2-D case and

(b) 3-D case.

Table 2.3. The minimum and maximum values of the stress components σ11, σ22, σ12, at the interface in AlN and W for the 2D case

Stress 1Ld / = 4 Hd 1Ld / = 6 Hd 1Ld / = 8 Hd

σ11 (GPa)(center,corner) (0.492 ~ 0.805) (0.530 ~ 0.812) (0.549 ~ 0.813)

σ22 (GPa)(center,corner) (-0.020 ~ 0.505) (-0.006 ~ 0.514) (-0.001 ~ 0.515)

AlN

(film) σ12 (GPa)(center,corner) (0 ~ -0.563) (0 ~ -0.569) (0 ~ -0.570)



W σ12 (GPa)(center,corner) (0 ~ -0.480) (0 ~ -0.480) (0 ~ -0.487)

14


(a)

(b)

Figure 2.5. (a). The distribution of stress components (a) σ11 and (b) σ22 , for H

L

dd 1 = 8 for

the 3-D case.

15


(c)

(d)

Figure 2.5. (b). The distribution of stress components (c) σ33 and (d) σ12 for H

L

dd 1 = 8 for

the 3-D case.

16


(e)

(f)

Figure 2.5. (c) The distribution of stress components (e) σ13 and (f) σ23 for H

L

dd 1 = 8 for the

3-D case.

17


Table 2.4. The minimum and maximum values of the normal stress components, σ11 , σ22, and σ33, and shear stress component, σ13, at the interface between AlN and W for the 3D case

Stress 1Ld / = 4 Hd 1Ld / = 6 Hd 1Ld / = 8 Hd

σ11(GPa) (center, corner) (0.738 ~ 1.339) (0.819 ~ 1.438) (0.852 ~ 1.480)

σ22(GPa) (center, corner) (0.739 ~ 1.053) (0.819 ~ 1.137) (0.853 ~ 1.154)

σ33(GPa) (center, corner) (-0.063 ~ 0.199) (-0.019 ~ 0.298) (-0.006 ~ 0.355)

AlN ( film)

σ13(GPa) (center, corner) (0 ~ - 0.611) (0 ~ -0.629) (0 ~ -0.631)




W

σ13(GPa) (center, corner) (0 ~ -0.442) (0 ~ -0.461) (0 ~ -0.466)

2.5 Effect of the thickness of the AlN film

In this section the effect of the thickness of the AlN film on residual thermal

stress is analyzed. The thickness of the film, dH, was varied as 0.25, 0.5, 1.0, 2.0 and 4

mm (i. e. H

L

dd 1 = 32, 16, 8, 4 and 2 with = 8mm). Figures 2.6 (a) and (b) show the

distribution of the in-plane stress component, σ

1Ld

11, for the film thickness 0.25 and 4 mm,

respectively, and Figure 2.7 shows the comparison of the variation σ11 in the film along

the x1 direction across the interface. The stress decreases as the film thickness increases

and the stress is reduced by a factor of two when there is a sixteen fold increase in the

thickness.

18


(a)

(b)

Figure 2.6. The distribution of the stress component, σ11, for (a) H

L

dd 1 = 32 and (b)

H

L

dd 1 = 2

for 3-D case.

19


-1 0 1 2 3 4 5 6 7 8

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

E

D

C

BA

S-S

11 (G

Pa)

X1/dH

A: dL1/dH=32B: dL1/dH=16C: dL1/dH=8D: dL1/dH=4E: dL1/dH=2

Figure 2.7. The variation of the stress component, σ11, in the film along the x1 direction at the interface for different thicknesses of the film. Curves A, B, C, D and E correspond to

the following ratios H

L

dd 1 = 32, 16, 8, 4 and 2, respectively.

2.6 Effect of the thickness of the substrate

The thickness of the substrate can be one of the parameters that could be varied

for the investigation of the optimization of the crystal growth. Therefore, to understand

the effect of the thickness of the substrate on thermal residual stress in the film,

calculations of thermal residual stress with various thicknesses of the substrate were

performed. Residual stresses with variable thickness of the substrate (i. e. H

S

dd

=10, 5,

and 2 with the thickness of the film, dH, as 0.1 mm, and the thickness of the substrate, ds,

as 1.0, 0.5, and 0.2 mm) were calculated. Figure 2.8 shows the variation of the σ11 stress

component in the film at the interface along the x1 axis. The decrease in the substrate

thickness from 1.0 to 0.2 mm leads to the decrease in the in- plane stress by 35 %. Thus

formation of microscopic cracks and delamination in the film might be avoided by using

a thinner substrate. 20


0 2 4 6 80.450.500.550.600.650.700.750.800.850.900.951.001.051.101.151.201.251.30

C

B

A

S-S

11 (G

Pa)

X1/dH

A: ds/dH=10B: ds/dH=5C: ds/dH=2

Figure 2.8. The variation of the normal stress in the x1 direction (σ11) along the interface in the film for different thicknesses of the substrate. Curves A, B, and C correspond to

the following ratios, H

S

dd

=10, 5, and 2, respectively.

2.7 Effect of the interaction of the islands (i. e. the island structure of the film).

For the analysis of the influence of the island structure of the film on the stress

distribution, the film consisting of three islands has been considered (Figure 2.1 (c)). The

lengths of the islands I and II were selected as H

L

dd 1 =4 and

H

L

dd 3 =8, respectively.

Preliminary study has found that the stresses in the islands are affected by the distance

between the islands, H

L

dd 2 , and the dimensions of islands

H

L

dd 1 and

H

L

dd 3 , and are

independent of the number of islands. The distances between the islands were taken

asH

L

dd 2 = 0 (infinitely small separation), 0.2, 1 and ∞, where dH is 1.0 mm. Figure 2.9 (a)

21


shows distribution of in-plane stress component, σ11, for H

L

dd 2 ≈ 0 and 2.9 (b) shows the

comparison of variation of σ11 in the film along the x1 axis at the interface for all four

cases. At dL2 =1 and ∞, the effect of the interaction between islands is very small, and the

stress in the islands is close to that for a single island. For = 0.2 and 0, the effect of

interaction increases the stress by 80% and 100% at the corner, respectively. This stress

build up could be the origin of the formation of threading dislocations and grain

boundaries leading to polycrystalline films.

2Ld

22


(a)

0 1 2 3 40.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

D

C

B

A

S-S

11

X1/dH

A: dL2/dH = 0B: dL2/dH = 0.2C: dL2/dH = 1D: dL2/dH = infinity

(b)

Figure 2.9. (a) The distribution of the normal stress component, σ11, for H

L

dd 2 ≈ 0.0 mm,

(b) The variation of σ11 in the film along the interface (x1 direction) for different values

of H

L

dd 2 .

23


2.8 Effect of the different orientation of the grain.

The presence of misoriented grains in the AlN will affect the stress distribution

and the probability of cracking due to the change in material properties across the grain

boundaries. For the analysis of stresses in the film with misoriented grains, we have

studied the film consisting of three equal size grains with different directions of material

axes of symmetry for each grain. Figure 2.10 (a) shows one half of the A grain and one B

grain i.e. the numerical scheme includes A and B grains that correspond to one A grain

and two B grains for the sample (Figure 2.1 (c)). Grain A and substrate have the same

orientation of material axes as that described in section 2. For the grain B, the axes of

material symmetry used in Equation (1) are directed in the common Cartesian system as

follows: the 11-axis coincides with the x1 –axis or a-axis [1-210], the 22-axis coincides

with the x3 –axis or c-axis [0001], and the 33-axis coincides with the x2-axis or direction

[10-10], respectively, for the hexagonal structure in AlN. The [0001] direction of

hexagonal structure is along the thickness direction in grain A and in-plane in grain B.

The ratio H

L

dd 1 was used as 18 where dL1 is the length along the interface and dH is the

thickness of the island (AlN film). Due to symmetry, = ½ . In Figure 2.10 (a),

curves A and B are corresponding to misoriented grtain and one grain with same

orientation respectively. Along the x

1Ld 2Ld

1 axis at the interface and three dimensional

distribution of σ11 is shown in the Figure 2.10 (b). When the c-direction of the AlN grain

is oriented along the in-plane direction (grain B), the in-plane stress is less as compared

to the stress when the c-direction is oriented along the growth or thickness direction

(grain A). There is a large increase of stress at the grain boundary (two fold or 100%),

24


25

[001] direction of substrate (cubic)

33-axis

[0001] direction of B grain[0001] direction of

A grain

which can lead to grain boundary cracking. Therefore, growth of polycrystalline film

with misoriented grains should be avoided.

(a)

Figure 2.10. (a). The variation of the in-plane stress component, σ11, in grains A and B along the interface (x1 direction)

A grain B grain

Tungsten substrate

11Ld

12Ld1Ld

22-axis

-2 0 2 4 6 8 10 12 14 16 18 20

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

B

A

S-S

11 (G

Pa)

X1/dH

11-

axis


(b)

Figure 2.10. (b). Three dimensional stress distribution of σ11.

2.9. Conclusions

The numerical study of the residual thermal stress distribution in AlN single

crystal, grown using tungsten as a crucible material, has been investigated. The necessity

of using a general 3-D formulation instead of a 2-D formulation to obtain more

conservative values of stress has been demonstrated. The introduction of the

dimensionless coordinates essentially simplifies the stress analysis and reduces the

number of calculations. The use of thermo elasticity for the stress analysis also simplifies

the study of stresses for a non-stationery temperature field.

The in-plane residual stress in AlN film grown along the [0001] direction is large

compared to the yield strength of AlN, while the stress along the thickness or growth

26


27

direction is zero. The stress is higher at the corner of the film which could cause

formation of micro cracks. The stress is tensile in the film and compressive in the

substrate. Separation or delamination of the film from the substrate could occur due to

reversal of the stress across the interface. The stress decreases as the thickness of the film

increases or the thickness of the substrate decreases. Thus a thinner substrate could be use

to avoid formation of microscopic cracks in the film.

The analysis on interaction of neighboring island in order to simulate coalescence

of island growth has shown that a large stress develops when the islands are too close to

each other. This could lead to formation of dislocations and hence grain boundaries in the

AlN film. The analysis of the effect of misorientation of the neighboring grains on the

residual thermal stress in the film has shown that a large stress can develop at the grain

boundary and can lead to grain boundary cracking.

In the future, we plan to apply more complicated non-linear models (e.g., thermo

elastoplasticity) for the more accurate prediction of the stress distribution.


28

CHAPTER 3

MODELING OF RESIDUAL STRESS FOR AlN CRYSTAL GROWN ON POSSIBLE CRUCIBLES

3.1 Introduction

In this chapter, the residual thermal stress distribution in AlN single crystal,

grown using different crucible materials, are investigated. An optimal choice of crystal

growth conditions results in the formation of coalesced boundaries known as island

structures. A finite element model has been used that accounts for different arrangements

of island structure. To achieve good quality aluminum nitride crystals by the sublimation-

recondensation technique, high growth temperatures, above 2373 K, are required. At this

high temperature, there will be such problems as durability of the furnace (B. Liu et al.)

and generation of the thermal stress. Sapphire, silicon carbide, tungsten, tantalum carbide

and niobium carbide were investigated as substrate materials to produce high purity

aluminum nitride single crystal. Comparison of residual stress in AlN grown using

different crucible materials will assist in better crystal growth conditions and hence

minimization of stress.


29

3.2 Sample geometry and properties

Geometry of samples of materials grown on different substrates is modeled as a

single island placed on a large substrate. Because the epitaxial islands can be grown on

all crystal orientations, comparing of the thermal residual stress with different growth

direction has been done by R.G. Lee (R. G. Lee. et al.). For this calculation, the epitaxial

growth direction of AlN island was assumed as [001] for cubic structure substrates and

[0001] for hexagonal structure substrates. The geometry of the sample and the values of

reference are shown in figure 3.1. Problem formulations are the same as Chapter 2, but in

this chapter, instead of using dimension less coordinates, sample height is fixed to 1mm

and length is 8mm for aluminum nitride Island and for substrate, sample height and

length were fixed to 10mm and 24mm respectively. Sample properties are listed in table

3.1 and 3.2. Three-dimensional model was considered with transversely isotropic

material properties for hexagonal structure and cubic structure. A quarter of a sample will

be taken for calculations due to symmetry, see figure 2.1. To achieve good quality bulk

AlN single crystals, the growth of AlN was modeled using temperature range from 2473

K (growth temperature) to 293 K (room temperature).


Figure 3.1. Sample geometry: the sample consists of a substrate and one island.

Table 3.1. Elastic Constants (GPa) Lattice Crystal Crystal

Structure Parameter(Å) C11 C12 C13 C33 C66

AlN wurtzite a=3.111 c=4.978

410.5 148.5 98.9 388.5 124.6

Al2O3 hexagonal a=4.7589 c=12.991 465 124 117 563 233

α-SiC(6H) hexagonal a=3.0806

* 2/)( 121144 ccc −=

c=15.1173 479.3 98.1 55.8 521.6 148.3

TaC* cubic a=4.4540(23°C) 621 155.3 166.8 NbC* cubic a=4.4691(20°C) 566.4 116.9 153.1

W cubic(bcc) a=3.1645(25°C) 501 198 151.4

* TaC and NbC are belongs to the group of RX-type crystals. RX-type crystals have the structure of NaCl, sodium chloride(Ralph W. G. Wyckoffpp 85~93). Atomic positions in the unit cube : R: (4a) 000 or ½ ½ 0; ½ 0 ½; 0 ½ ½ X: (4b) ½ ½ ½; or ½ 0 0 ; 0 ½ 0 ; 0 0 ½ Source : Elastic constants for W and Al2O3 (H. B. Huntington), AlN and SiC (Robert R. Reeber et al. 2001), NbC (H. M. Ledbetter) and Tac (L. Lopez de la Torre)

30


31

Table 3.2. Thermal expansion coefficients at 2000 K (α, 10-6 K-1) AlN Al2O3 α-SiC(6H) TaC NbC W

α 11 7.2 10.2 4.88 8.4 9.3 6.1 α 22 7.2 10.2 4.88 8.4 9.3 6.1 α 33 6.2 11.4 4.94 8.4 9.3 6.1

Source : thermal expansion coefficients for AlN and α-SiC(6H)( Robert R. Reeber. et al.), Al2O3 ,NbC, TaC and W(Y. S. Touloukian et al. 1977)

3.3 Sapphire (Al2O3) substrate

Figure 3.2. illustrates the stress component, σ11, using Al2O3 substrate. Sapphire

substrate has much higher thermal expansion coefficient and lattice parameter of the C-

direction [0001] is much bigger than aluminum nitride, hence it produces much higher

defect densities and thermal stress. The thermal stress of aluminum nitride grown on

sapphire shows under horizontal compressed stress σ11 (figure 3.5) and vertical tensile

stress σ33 (figure 3.6). Table 3.3 shows the stress component at the center ( B point from

figure 3.2 (a)) and corner (C point from figure 3.2 (a)), and the stresses are maximum at

the corner because of the stress concentration. Aluminum nitride grown on sapphire

substrate has the highest thermal stress than other substrate compared. However, the

substrate is under tensile stress near the interface between aluminum nitride and sapphire

substrate. This opposite stress component at the interface are arranged from -2.496 GPa

(σ11) (at the center of aluminum nitride) to 0.763 GPa (σ11) ( at the center of sapphire

substrate), and similar result are showing for all the substrate. At the end of the corner

point (c) of the substrate, the tensile stress 0.264 GPa (σ11) changes to the compressed

stress -0.479 GPa (σ11) at the vicinity of the corner because of the thermal expansion of

the aluminum nitride, thus gradually increasing to the zero stress which shows far from


the corner. The majority of the film area is under small shear stress σ12 and σ23. Shear

stress σ13 is plotted in figure 3.4 (c).

Figure 3.2. The illustration of the stress component, σ11, using Al2O3 substrate

3.4 TaC and NbC substrate

Figure 3.3 (a) and (b) illustrate the stress component, σ11 using tantalum carbide

and niobium carbide substrate. The advantage of using TaC and NbC is well presented

by B. Liu (B. Liu et al.). Compare to sapphire substrate, thermal expansion coefficients

are small, and lattice parameter is much closer to the aluminum nitride, but thermal

32


expansion coefficient are higher than aluminum nitride. Therefore aluminum nitride

grown on tantalum carbide (TaC) or niobium carbide (NbC) behave the same way as

grown on sapphire substrate. However, the range of stress is smaller than using sapphire

substrate. For the aluminum nitride grown on tantalum nitride substrate, the minimum

compressed stress area at the bottom corner is not concentrated. According to the figures

3.5 ,3.6 and Table 2.3, it is clear that the aluminum nitride grown on tantalum carbide

(TaC) and niobium carbide (NbC) produces much small thermal stress and will prevent

micro crack caused by thermal residual stress.

Figure 3.3 (a). The illustration of the stress component, σ11, using tantalum carbide substrate

33


Figure 3.3 (b). The illustration of the stress component, σ11, using niobium carbide substrate

3.5 α-SiC(6H) and W substrate

Figure 3.4 (a) and (b) illustrate the stress component, σ11 using silicon carbide and

tungsten substrate. Thermal expansion coefficient of the silicon carbide and tungsten are

smaller than aluminum nitride. Specially, thermal expansion coefficient of silicon carbide

shows much smaller than tungsten substrate, and lattice parameter of C-direction [0001]

is even bigger than sapphire. Using silicon carbide as a substrate increased coefficient of

thermal expansion mismatch thus higher thermal stress. The thermal stress of aluminum

nitride grown on silicon carbide or tungsten shows under horizontal tensile stress σ11

(figure 3.5) and vertical compressed stress σ33 (figure 3.6). The substrates, silicon

carbide and tungsten, are under compressive stress between aluminum nitride and

34


substrates. Comparing thermal residual stress of tungsten with silicon carbide, it is clear

that using tungsten as substrate produces smaller thermal stress for aluminum nitride

single crystal. In addition to the thermal residual stress, tungsten crucibles have been

chosen to avoid carbon contamination and resistant to attack by Al vapor at the high

temperature (G. A. Slack et al. 1976, 1977).

Figure 3.4 (a). The illustration of the stress component, σ11, using silicon carbide substrate

35


Figure 3.4 (b) . The illustration of the stress component, σ11, using tungsten substrate

36


0 2 4 6 8

-6

-4

-2

0

2

4

6

8

S-S1

1 (G

Pa)

Length, mm

W substrate NbC substrate TaC substrate SiC substrate Al2O3 substrate

(a)

0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5

6

S-S2

2 (G

Pa)

Length, mm


(b)

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

5

6

S-S

13 (G

Pa)

Length, mm


(c)

Figure 3.5. The variation of the stress component, (a) σ11, (b) σ22 and (d) σ13 in the film along x1 direction at the interface between island and substrate. Curves are corresponding to the following substrates, W, NbC, TaC, SiC and Al2O3, respectively.

37


Table 3.3. The minimum and maximum values of the normal stress components, σ11 , σ22, and σ33, and shear stress component, σ13, in the film at the interface between AlN and substrates.

Substrates Stress (GPa) Al2O3 α-SiC(6H) TaC NbC W σ11(GPa) (center~corner)

-2.49 ~ -5.91 1.91 ~ 4.61 -0.604 ~ -1.374 -1.03 ~ -2.41 0.53 ~ 1.25

σ22(GPa) (center~corner)

-2.49 ~ -4.44 1.91 ~ 3.41 -0.555 ~ -1.059 -1.03 ~ -1.83 0.53 ~ 0.94


0.01 ~ -4.20 -0.01 ~ 3.09 0.004 ~ -1.029 0.00 ~ -1.72 -0.004 ~ 0.87

AlN (film)


0 ~ 3.79 0 ~ -2.89 0 ~ 0.89 0 ~ 1.55 0 ~ -0.802

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

S-S

33 (G

Pa)

Thickness (From the top surface),mm

W NbC TaC SiC Al2O3

Figure 3.6. The variation of the normal stress component σ33 in the film along x3 direction (thickness direction) at the center of island. Curves are corresponding to the following substrates, W, NbC, TaC, SiC and Al2O3, respectively.

38


39

3.6 Conclusions

Suitability of several refractory materials such as tungsten (W), tantalum carbide

(TaC) and niobium carbide (NbC) for aluminum nitride (AlN) crystal growth was

discussed by B. Liu and it has been concluded that W is a durable and less expensive

crucible material. Thermal residual stress of AlN grown on sapphire, silicon carbide,

tungsten, tantalum carbide and niobium carbide were investigated to produce high purity

aluminum nitride single crystal. Thermal residual stress of AlN grown on silicon carbide

and tungsten shows compressive stress (σ 11) at the end of the island and, because of

thermal expansion coefficient, tungsten shows the least amount of thermal stress. When

sapphire, TaC and NbC were used as substrate, the thermal stress (σ 11) of AlN shows

tensile stress at the end of the island. Even though sapphire has hexagonal structure

which is same as AlN structure, it has the highest amount of thermal stress, thus it can be

the worst substrate. TaC and NbC will be the good substrate to grow AlN crystal. At the

interface, compressive and tensile stresses are conjunct, and this different stress state can

cause of the generation of dislocation and micro crack.


40

CHAPTER 4

THE ANALYSIS OF BULK AlN CRYSTAL GROWN ON TUNGSTEN SUBSTRATE

4.1 Introduction

The unique properties of the group III-nitrides, such as wide direct band gap, high

thermal conductivity, and high thermal stability have made GaN and AlN the most

serious candidates for the high-power and high-frequency electronic and deep ultraviolet

(UV) optoelectronic devices (B. Monemar. et al. 1999) (S. Strite et al. 1992) . AlN is one

of the most promising substrates for group III-nitride based devices due to lower lattice

constant and thermal expansion coefficient mismatch and hence reduced defect densities

and residual thermal stresses (J. H. Edgar et al. 2002). Currently, several groups are

producing bulk AlN crystals by the sublimation-recondensation technique (J. H. Edgar et

al. 2002) (J. Carlos Rojo et al. (2002) (R. Sclesser et al. 2002)

.

4.2 Sublimation method

Sublimation (one kind of physical vapor deposition, PVD) is the most successful

method to grow AlN bulk crystals so far (G. A. Slack et al. 1977). Vapor deposition can

be separated by two methods. One is Physical Vapor Deposition (PVD) and another is

Chemical Vapor Deposition (CVD). When the vapor state of the source material changes

to solid state at the substrate, different solidification procedure occurs by PVD and CVD

respectively. In detail, for PVD, sintered or melted composite materials are used as solid

target materials and heated by laser beam or thermal method to deposit to the substrate.


High Vacuum state is required to prevent vapor state target material to react with other

vapor molecules. The element attached to the substrate has the same composition with the

element arrived to the target by vapor state. Figure 4.1 shows the sketch of the AlN

crystal growth process. AlN powder source sublimes and re-condenses on a colder seed

crystal (R. Dalman, 2005).

Figure 4.1. Sketch of the AlN crystal growth process: Source; (R. Dalman, 2005)

41


42

The source material for AlN crystal growth in this study was sintered AlN. The

original AlN powders had an average agglomerated particle size of 1.8 mm, and

contained 0.9 wt% O and 0.06wt% C as the major impurities, as reported by the vendor.

The sintered AlN source was easy to handle and the distance between the source material

and the growth region did not change much in the initial growth period. Sintered AlN

source was prepared by heating AlN powders at 1960 C for up to 4 h. The sublimation

growth was conducted in a resistively heated tungsten furnace. The furnace was heated

by two tungsten wire mesh heating elements with a maximum operation temperature of

2400 C at one atmospheric pressure as claimed by the vendor. The heating elements

provided an axial temperature gradient of 3–5 C/mm between the source material and the

crystal growth region, which was the driving force for the sublimation growth. The

growth chamber consisted of a tungsten crucible contained in a concentric tungsten retort,

to prevent the escape of vaporized species during the growth. The growth temperature

was measured by an optical pyrometer focused on the top lid of the outside tungsten

retort, and was controlled by the furnace output power (Z. Gu. et al. 2006). Figure 4.2 (a)

shows the aluminum nitride boul grown by sublimation method and (b) illustrate the

crystal structure of aluminum nitride. Aluminum and nitride can be occupied by arbitrary.


(a)

(b)

Figure 4.2 aluminum nitride poly crystal grown on tungsten by sublimation method (a) and computed atomic structure (b).

43


44

4.3 Surface morphology by SEM When aluminum nitride was grown on tungsten substrate, it has been investigated

that the crystal has smooth and rough surfaces. In order to see it closely, scanning

electron microscope (SEM) image was taken (figure 4.3). The nature of the scattering can

result in different angular distributions. Scattering can be either forward scattering or

backward scattering. The forward scattered electrons are used for transmission electron

microscopy (TEM) and backward scattered electrons can be detected by SEM. When the

electron passes through the specimen, it may interact with the electron cloud, and make

small angular deviation. However, when it pass through the electron cloud and

approaches the nucleus, the electron (negative charge) may be attracted and scattered

through a large angle. Backward scattered electron consists of Secondary electrons from

within the specimen and Incoherent elastic backscattered electrons. As the specimen gets

thicker, fewer electrons are forward scattered and more are backscattered until primary

incoherent backscattering is detectable in bulk, nontransparent specimens. Because

backscattered electrons has high angle (> ~10°), the amount of energy loss is higher than

secondary electrons, which dedicates to the high resolution surface morphology. Figure

4.3 (a) was taken by secondary electrons and (b) was taken by backscattered electrons

which show rough surface morphology clearly.


(a)

(b)

Figure 4.3. Scanning Electron Microscopy (SEM) image of rough and smooth surface by (a) secondary Electrons and (b) back scattered electrons.

45


46

4.4 EDAX result using SEM Energy Dispersive X-ray Analysis (EDAX) technique has been used to investigate

the compositional analysis for polarity determination. Since the surface morphology

consists of rough and smooth surface, it was assumed that the surface polarity (N- polar

or Al- polar) makes this difference. When the incident beam hit the specimen, the

electron from the incident beam emit the electrons from the electron cloud and specific

amount of energy releases when electrons form next shell occupy emitted position of the

inner shell. The amount of energy released by the transferring electron depends on which

shell it is transferring from, as well as which shell it is transferring to. Thus, by

measuring the amounts of energy present in the X-rays being released by a specimen

during electron beam bombardment, the identity of the atom from which the X-ray was

emitted can be established. If the emitted x-ray satisfies Bragg’s angle (figure 4.4), then

the detector can catch the emitted x-ray with specific angle.

Figure 4.4. The description of Bragg angle. The path difference between reflected waves is AB + BC.

B

θ

θ θ

θ

A C

d

Incident plane wave

Scattered plane wave


47

n λ = AB + BC = 2d sin θ (6)

Equation (6) shows the path difference between reflected waves called Bragg’s Law in

figure 4.4. If we know the wave length (λ) for the incident electron and we can measure θ

experimentally, then we can calculate the interplanar spacing (d) in crystal. The X-rays

are generated in a region about 2 microns in depth (tear drop effect), and thus EDAX is

not a surface science technique. However, in this experiment, it has been used for the

guide line. Figure 4.5 shows the EDAX result by Hitachi S-4300 SEM. For EDAX

experiment, the region used for x-ray analysis was shown in figure 4.5 (a). From figure

4.5 (a), there are two different type of the surface such as smooth and rough surface.

Figure 4.5 (b) shows the EDAX result from smooth surface and 4.5 (c) are from the

rough surface. EDAX result shows that the smooth surface contains nitride and atomic

weight of aluminum is 26.62 % and 11.78% for nitride, respectively. Carbon and oxide

also detected and those elements are detected from the grid and epoxy.

However, no nitride or very small amount of nitride was detected in the rough surface.

Nitride peak is not showing in figure 4.5 (c), but it is probably over-wrapped by carbon

peak, because of small amount of nitride compared to the smooth surface.


(a)

(b)

(c)

Figure 4.5. Pointed aluminum nitride surface morphology (a) and EDAX result of smooth (b) and rough(c) surface.

48


4.5 Dislocation Study by Etching

In general, the shape and morphology of the etch pit depend on the type of defect,

the crystal orientation and polarity, the basic crystal structure and symmetry, and the

etchant and its composition. Usually, the planes exposed by etching have the slowest etch

rate. For hexagonal symmetry crystals, such as wurtzite structured GaN, AlN, and SiC

(6H–SiC or 4H–SiC), hexagonal pits are usually produced on the (0 0 0 1) planes. Such

pits can be delimited by six planes inclined to the c-axis (Z. Gu., 2006). For this study,

AlN crystal was etched in molten KOH at 405 °C for 2 minutes by Dr. Edgar’s group at

Kansas State University. Figure 4.6 (a) shows the surface of AlN crystal before etching

and (b) shows after etching

(a) (b)

Figure 4.6. Optical microscope image of the AlNsurface (a) before etching and (b) after etching in molten KOH at 405 °C for 2 minute.

Figure 4.7 shows (a) the optical microscope image with high magnification and

(b) scanning electron microscopy image in etch pit area. The etch pit density after etching

at 405 °C for 2 minute was approximately 10.4 * 1010 Cm-2, as determined by counting

49


the number of etch pits in a randomly chosen area. Etch pits were found to be of

hexagonal shape with a deviation of the apex from center. These dislocations are

associated with edge and screw dislocation. The final shape of the etch pit gives

information about the type of dislocation since the etch pit is centered around the

dislocation. If the dislocation line is normal to the etched surface, the etch pit will be a

perfect hexagonal with the apex at the center, which is the case of edge dislocation with

burger’s vector, 1/3<11-20> and 1/3<-1100>, and in case of screw dislocations, if the

screw dislocations have burger’s vector, [0001] or ½[0001], then the etch pit will be a

perfect hexagonal with the apex at the center also. On the other hand, If the dislocations

have the burger’s vector, 1/3<11-23> or 1/6<-2203>, then the dislocation line is oblique

to the etched surface and the apex of the pit will be off center. The study about the etch

pit formation for NaCl crystal was done by J. J. Gilman. (J. J. Gilman, et. al., 1958).

50

(a) (b) Figure 4.7 shows (a) the optical microscope image with high magnification and (b) scanning electron microscopy image in etch pit area.


4.6 HRTEM images of AlN crystal

The Transmission Electron Microscope (TEM) has been used in all areas of

biological, biomedical investigations and material science. For the crystallographer,

metallurgist or semiconductor research scientist, current high voltage/high resolution

TEMs, utilizing 200 keV to 1 MeV, have permitted the routine imaging of atoms,

allowing materials researchers to monitor and design materials with custom-tailored

properties. Figure 4.8 (a) shows the actual image of the equipment and (b) shows the

electron diffraction to project the image onto the screen.

(a) (b) Figure 4.8 (a) The actual image of the TEM and (b) shows the electron diffraction

to project the image onto the screen.

51


The defect and the quality of aluminum nitride crystal were examined by high

resolution transmission electron microscopy (HRTEM) and energy dispersive x-ray

analysis using ZEOL 2010F TEM which has field emission gun at the University of New

Mexico. Figure 4.9 shows the sample made by rough and smooth separately. Samples are

made by conventional method using disk cutter, dimple grinder and ion milling machine

(PIPS) to achieve very thin area. Both of the samples were tilted to the [11-20] zone axis.

(a) (b) Figure 4.9. The optical microscope image of the cross-sectional TEM sample for (a) smooth surface (b) rough surface. In order to take High Resolution Image, the beam source should be located in the

exact zone axis where we intend to study. Therefore, tilting sample is extremely

important technique in TEM. If the sample is thick enough to produce the Kikuchi line,

the sample will be tilted to the zone axis using Kikuchi line. The reason we form Kikuchi

pattern is that, if the specimen is thick enough, we will generate a large number of

scattered electrons which travel in all directions; i.e., they have been incoherently

scattered but not necessarily in-elastically scattered. They are sometimes referred to as

diffusely scattered electrons (David. B. Williams., Transmission Electron Microscopy,

52


vol 2, chapter 19). Figure 4.10 shows the Kikuchi map for Hexagonal Structure (Defect

analysis in electron microscopy, appendix 2)

(a)

(b)

Figure 4.10 shows the Kikuchi map for Hexagonal Structure for (a) two standard triangles and (b) centered on [0001] zone axis.

53


54

Figure 4.11 shows the TEM image taken from rough surface and indicate stacking

fault of the hexagonal system. Stacking sequence of perfect single crystal is

ABABAB…However, when the layer A slips to the 1/3<10-10> direction, C point of

figure 4.11 (b), from the perfect crystal, the next layer should be on a A point, thus

stacking sequence needs to be changed from ABABABAB...to ABABCACA...It is called

second order intrinsic stacking fault. First order intrinsic stacking fault occurs when the A

layer is eliminated and vacancies are produced in that area. After that, upper lattice (B)

needs to slip to the 1/3<10-10> direction to reduce the stacking energy. In this case,

stacking sequence needs to be changed from ABABAB…to ABABCBCB…We call the

stacking fault produced by vacancy agglomeration “intrinsic stacking fault”

(http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/backbone/r5_4_1.html). Another

possible stacking fault is extrinsic stacking fault produced by interstitial agglomeration.

In this case the stacking sequence needs to be changed from ABABAB….to

ABABCABAB…. Figure 4.12 and 4.13 show the TEM images taken rough and smooth

surface separately.


(a)

55

(b)

1/3<10-10>

A A

B B

C C

Figure 4.11. High resolution TEM image from the rough surface indicate (a) the stacking fault and (b) the slip phenomenon.


2 nm2 nm

(a)

(b) (c) Figure 4.12. (a) High Resolution TEM image from the rough surface. (b) Fast Fourier Transform (FFT) of the rough surface showing that the beam direction is [11-20]. (c) Sample profile shows the lattice distance of the c-direction [0001] is 0.493nm.

56


57

(a)

2 nm2 nm

(b) (c) Figure 4.13. (a) High Resolution TEM image from the smooth surface. (b) Fast Fourier Transform (FFT) of the smooth surface showing that the beam direction is [11-20]. (c) Sample profile shows the lattice distance of the c-direction [0001] is 0.499nm.


4.7 CBED technique and experimental data Convergent Beam Electron Diffraction (CBED) uses a convergent beam of

electrons to overcome the spatial-resolution limitations of Selected Area Diffraction

(SAD) and limit the area of the specimen which contributes to the diffraction pattern

instead of using a parallel beam which is usually used to SAD. Each spot then becomes a

disc within which variations in intensity can usually be seen. Such patterns contain a

wealth of information about the symmetry and thickness of the crystal. Figure 4.14

illustrates the Ray diagram showing CBED pattern formation. The big advantage of

CBED over all other diffraction technique is that most of the information is generated

from minuscule region beyond the reach of other diffraction method (David. B. Williams.,

Transmission Electron Microscopy, vol 2, chapter 20).

Figure 4.14. The CBED pattern formation. A convergent beam at the specimen results in the formation of disks in the BFP of the object lens.

58


59

Figure 4.15 shows the CBED image taken from (a) rough surface and (b) smooth surface.

Both CBED images are acquired from [11-20] beam direction and shows different

diffraction intensity for each disk. In order to determine symmetry, computational

simulation has been performed. For the simulation, Web-emap from the Center for Micro

analysis of Materials (CMM) was used with 200 electron potential (Kv), [11-20] zone

axis of aluminum nitride, [001] x-axis and 0.3 disk radius. Because CBED diffractions

need thicker sample region then other techniques such as HRTEM, EDAX and EELS,

where the best information is obtained from the thinnest specimens, 300, 400 and 500Å

of thickness was investigated in computational simulation. Figure 4.16 shows the

comparison of experimental CBED images with simulated CBED images. From the

simulation, bright spot indicates [0002] disk, and intensities depend on the sample

thickness. The diffraction intensity of Al is much stronger than nitride because of atomic

position. [0002] disk of smooth surface shows brighter than rough surface, and it might

be assumed that smooth surface is Al-polar crystal. In the Smooth surface, the intensity of

[000-2] disk which represents the N-polarity shows much darker than the intensity of

[0002] disk. Since [000-2] disk of rough surface has Al polar, it has higher intensity than

the intensity of [000-2] from smooth surface. For the simulation formulation, the Bloch

Wave Method was used, which was developed by J. M. Zuo (J. M. Zuo., 1998).


90.00 µm

(a) (b)

90.00 µm

100.00 µm

(c) (d) Figure 4.15. CBED image of aluminum nitride crystal from (a), (b) rough and (c), (d) smooth surface.

60


Figure 4.16. The comparison of experimental CBED images from rough and smooth surface of aluminum nitride crystal with simulated CBED images to determine sample symmetry. 4.8 Conclusions Aluminum nitride (AlN) crystal grown on tungsten (W) substrate was investigated

by Transmission Electron Microscope (TEM) and Scanning Electron Microscope (SEM).

Bulk AlN has rough and smooth surface because of the polarity of the sample. Energy

Dispersive X-ray Analysis (EDAX) shows more nitride in smooth surface and

Convergent Beam Electron Diffraction (CBED) result shows [0002] disk of rough

surface has more intensity because of the diffraction from Al atoms. Therefore it can be

concluded that the smooth surface has aluminum polarity (Al-Polar), and rough surface

has nitride polarity (N- Polar). From the rough surface, second order intrinsic dislocation

was observed.

61


62

BIBLIOGRAPHY

ABAQUS/CAE Finite Element Analysis, Version 6.5, ABAQUS, Inc, 2004. W. M. Ashmawi, M. A. Zikry, K. Wang and R. R. Reeber, “Modeling of Residual Stress for Thermally Strained GaN/Al2O3 Heterostructures”, Journal of Crystal Growth, 2004, Vol. 266, Pages 415~422. J. H. Edgar, L. Liu, B. Liu, D. Jhuang, J. Chaudhuri, M. Kuball, S. Rajasingam, “Bulk AlN crystal growth”: self-seeding and seeding on 6H-SiC substrates, Journal of Crystal Growth, 2002, Vol. 246, Pages 187~193. Helmut Foll, University of Kiel, Partial Dislocations and Stacking Fault: Internet Source; http://www.tf.unikiel.de/matwis/amat/def_en/kap_5/backbone/r5_4_1.html J.J. Gilman, W. G. Johnston, G. W. Sears, “Dislocation Etch Pit Formation in Lithium Fluoride”, Journal of Applied Physics, 1958, Vol 29, Number 5. Z. Gu, J. H. Edgar, D. W Coffey, J. Chaudhuri, L. Nyakiti, R. G. Lee, J. Wen, “Defect-Selective Etching of Scandium Nitride Crystals”, Journal of Crystal Growth, 2006, Vol. 293, No. 2, Pages 242~246. Z. Gu, L. Du, J. H. Edgar, N. Nepal, I. Y. Lin, H. X. Jiang, R. Witt, “Sublimation Growth of Aluminum Nitride Crystals”, Journal of Crystal Growth, 2006, Vol.297, Pages 105~110. H. B. Huntington, The Elastic Constants of Crystals, Academic Press, Reprinted From Solid State Physics, 1958, Page 70. H. M. Ledbetter, S. Chevacharoenkui, R. F. Davis, “Monocrystal Elastic Constants of NbC”, J. Appl. Phys, 1986, Vol. 60, No. 5.1. R. G. Lee, A. Idesman, L, Nyakity, J. Chaudhuri, “Modeling of residual stress for aluminum nitride crystal growth by sublimation”-to be submitted B. Liu, J. H. Edgar, Z. Gu, D. Zhuang, B. Raghothamachar, M. Dudley, A. Saura, M. Kuball and H. M. Meyer, “The Durability of Various Crucible Materials for Aluminum nitride Crystal Growth by Sublimation”, Mat. Res. Soc. Internet Jour. MIJ-NSR , 2004, Vol. 9, Art. 6. M. Loretto, And R. Smallman, Defect analysis in electron microscopy, appendix 2, 1975, Chapman and Hall, London.

http://www.tf.unikiel.de/matwis/amat/def_en/kap_5/backbone/r5_4_1.html


63

B. Monemar, “III-V nitrides-important future electronic materials”, Journal of Materials Science, Materials in Electronics, 1999, Vol.10, No. 4, Pages 227. J. F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, 1951, Pages 131~149. Robert R. Reeber, Kai Wang, “High Temperature Elastic Constant Prediction of Some Group ІІІ-Nitrides”, MRS Internet J. Nitride Semicond, 2001, Res. 6,3. Robert R. Reeber, Kai Wang, “Lattice Parameters and Thermal Expansion of Important Semiconductors and Their Substrates”, Mat. Res. Soc. Symp, 2000, Vol.622, T6.35.1~T6.35.6 J. Carlos Rojo, L. J. Schowalter, R. Gaska, M. Shur, M. A. Khan, J. Yang, D. D. Koleske, “Growth and Characterization of Epitaxial Layers on Aluminum Nitride Substrates Prepared from Bulk, Single Crystals”, Journal of Crystal Growth, 2002, Vol. 240, Page 508. R. Sclesser, R. Dalmau, R. Yakimova, Z. Sitar, Mater, “Growth of AlN Bulk Crystals from the Vapor Phase”, Res. Soc. Symp. Proc. 693, I9.4.1, 2002. G. A. Slack, T. F. McNelly, “AlN Single Crystals”, Journal of Crystal Growth, 1977, Vol.42, Page 560. G. A. Slack, T. F. McNelly, “ Growth of high purity AlN crystals”,Journal of Crystal Growth, 1976, Vol. 34, Page 263. S. Strite and H. Morkoc, “GaN, AlN and InN: A Review”, J. Vac. Sci. Technol, 1992, B 10, 1237. L. Lopez de la Torre, B. Winkler, J. Schreuer, K. Knorr, M. Avalos-Borja, “Elastic Properties of Tantalum Carbide (TaC)”, Solid State Communications, 2005, Vol 134, Issue 4, Pages 245~250. Y. S. Touloukian, R. K. Kirby, R. E. Taylor and T. Y. R. Lee (eds.), Thermo Physical Properties of Matter, Plenum Press, New York, 1977, Vol. 13. David. B. Williams, C. Barry Carter, Transmission Electron Microscopy, Plenum Press, New York, 1996, Vol 2, chapter 19. David. B. Williams, C. Barry Carter, Transmission Electron Microscopy, Plenum Press, New York, 1996, Vol 2, chapter 20. Ralph W. G. Wyckoff, Crystal Structures, Second Edition, 1963, Vol. 1, Interscience Publishers, Pages 85~93.


64

I. Yonenaga, “Hardness of bulk single-crystal GaN and AlN”, Mat. Res. Soc. Internet Jour. MIJ-NSR, 2002, Vol. 7, Art. 6. J. M. Zuo, Annual Reports of the HVEM Laboratory Kyushu University, 1998, No 22, Pages 3-10.


65

APPENDIX A

THERMAL STRESS DATA OF AlN USING W SUBSTRATE

Appendix A contains the completed result in graphical form for all the normal stress and in-plane stress of aluminum nitride grown on tungsten substrate.


A.1 2-D result of aluminum nitride grown on tungsten substrate

-1 0 1 2 3 4 50.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

S-S

11 (G

Pa)

X1/dH

0 1 2 3 4-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 1 2 3 4-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

S-S

33 (G

Pa)

X1/dH

0 1 2 3 4-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

Figure A. 1 : The distribution of the stress component for H

L

dd 1 = 4 for 2-D case

66


0 1 2 3 4 5 60.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

S-S

11 (G

Pa)

X1/dH

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 1 2 3 4 5 6-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

S-S

33 (G

Pa)

X1/dH

0 1 2 3 4 5 6-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12


L


67


0 2 4 6 80.30

0.35

0.40

0.45

0.50

0.55

0.60

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (a) σ 12


L


68


A.2 3-D result of AlN for the effect of the size

0 1 2 3 4

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

S-S

11 (G

Pa)

X1/dH

0 1 2 3 4

0.4

0.6

0.8

1.0

1.2

1.4

1.6

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

S-S

33 (G

Pa)

X1/dH

0 1 2 3 4-0.0016

-0.0014

-0.0012

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

0.0000

0.0002

S-S

12 (G

Pa)

X Axis Title

(c) σ 33 (d) σ 12

0 1 2 3 4-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

S-S

13 (G

Pa)

X1/dH

0 1 2 3 4

-0.0044

-0.0042

-0.0040

-0.0038

-0.0036

-0.0034

-0.0032

-0.0030

-0.0028

-0.0026

-0.0024

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23

Figure A. 4 : The distribution of the stress for H

L

dd 1 = 4 for 3-D case, fixed. Hd

69


0 1 2 3 4 5 60.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

S-S

11 (G

Pa)

X1/dH

0 1 2 3 4 5 60.4

0.6

0.8

1.0

1.2

1.4

1.6

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 1 2 3 4 5 6-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

S-S

33 (G

Pa)

X1/dH

0 1 2 3 4 5 6-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 1 2 3 4 5 6-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

S-S

13 (G

Pa)

X1/dH

0 1 2 3 4 5 6

-0.0014

-0.0012

-0.0010

-0.0008

-0.0006

-0.0004

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23


L


70


0 2 4 6 80.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8

0.6

0.8

1.0

1.2

1.4

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8

0.0

0.5

1.0

1.5

2.0

2.5

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8-0.00020

-0.00015

-0.00010

-0.00005

0.00000

0.00005

0.00010

0.00015

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8

0.00018

0.00020

0.00022

0.00024

0.00026

0.00028

0.00030

0.00032

0.00034

0.00036

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23


L


71


A.3 3-D result of AlN for the effect of the thickness of AlN

0 2 4 6 80.5

0.6

0.7

0.8

0.9

1.0

1.1

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8

-0.00008

-0.00006

-0.00004

-0.00002

0.00000

0.00002

0.00004

0.00006

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8

0.00008

0.00009

0.00010

0.00011

0.00012

0.00013

0.00014

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23


L

dd 1 = 32 for 3-D case, fixed. 1Ld

72


0 2 4 6 8

0.55

0.60

0.65

0.70

0.75

0.80

S-S

11 (G

Pa)

X1/dH

0 2 4 6 80.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8-0.00008

-0.00006

-0.00004

-0.00002

0.00000

0.00002

0.00004

0.00006

0.00008

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8-1.0

-0.8

-0.6

-0.4

-0.2

0.0

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8

-0.000015

-0.000010

-0.000005

0.000000

0.000005

0.000010

0.000015

0.000020

0.000025

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23


L


73


0 2 4 6 8

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

S-S

11 (G

Pa)

X1/dH

0 2 4 6 80.5

0.6

0.7

0.8

0.9

1.0

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8-0.00010

-0.00008

-0.00006

-0.00004

-0.00002

0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8-0.00002

-0.00001

0.00000

0.00001

0.00002

0.00003

0.00004

S-S

23 (G

Pa)

X1/dH

(a) σ 13 (a) σ 23


L


74


0 2 4 6 80.4

0.5

0.6

0.7

0.8

0.9

1.0

S-S

11 (G

Pa)

X1/dH

0 2 4 6 80.4

0.5

0.6

0.7

0.8

0.9

1.0

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8

0.0

0.5

1.0

1.5

2.0

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8-0.00015

-0.00010

-0.00005

0.00000

0.00005

0.00010

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8

-0.00002

-0.00001

0.00000

0.00001

0.00002

0.00003

0.00004

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23


L


75


0 2 4 6 8

0.4

0.5

0.6

0.7

0.8

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8

0.4

0.5

0.6

0.7

0.8

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8

0.000

0.001

0.002

0.003

0.004

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8-1.0

-0.8

-0.6

-0.4

-0.2

0.0

X1/dH

S-S

13 (G

Pa)

0 2 4 6 8-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

X1/dH

S-S

23 (G

Pa)

(e) σ 13 (f) σ 23


L


76


A.4 3-D result of AlN for the effect of the interaction of islands

0 2 4 6 8 10 120.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8 10 120.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8 10 12-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8 10 12

-0.00004

-0.00002

0.00000

0.00002

0.00004

0.00006

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8 10 12

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8 10 12

-0.00002

0.00000

0.00002

0.00004

0.00006

0.00008

S-S

23 (

GPa

)

X1/dH

(e) σ 13 (f) σ 23


L


77


0 2 4 6 8 10 12 14

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8 10 12 140.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8 10 12 14

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8 10 12 14-0.00010

-0.00008

-0.00006

-0.00004

-0.00002

0.00000

0.00002

0.00004

0.00006

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8 10 12 14-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8 10 12 14-0.00010

-0.00008

-0.00006

-0.00004

-0.00002

0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

S-S

23 (G

Pa)

X1/dH

(d) σ 13 (e) σ 23


L

dd 2 = 0.2 for 3-D case

78


-2 0 2 4 6 8 10 12 140.4

0.6

0.8

1.0

1.2

1.4

S-S

11 (G

Pa)

X1/dH

-2 0 2 4 6 8 10 12 14

0.5

0.6

0.7

0.8

0.9

1.0

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

-2 0 2 4 6 8 10 12 14

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

S-S

33 (G

Pa)

X1/dH

-2 0 2 4 6 8 10 12 14-0.00003

-0.00002

-0.00001

0.00000

0.00001

0.00002

0.00003

0.00004

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

-2 0 2 4 6 8 10 12 14-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

S-S

13 (G

Pa)

X1/dH

-2 0 2 4 6 8 10 12 14-0.00010

-0.00008

-0.00006

-0.00004

-0.00002

0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23


L


79


A.5 3-D result of AlN for the effect of the mis-oriented grain

-2 0 2 4 6 8 10 12 14 16 18 200.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

S-S

11 (G

Pa)

X1/dH

-2 0 2 4 6 8 10 12 14 16 18 20

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

-2 0 2 4 6 8 10 12 14 16 18 20-0.4

-0.2

0.0

0.2

0.4

0.6

S-S

33 (G

Pa)

X1/dH

-2 0 2 4 6 8 10 12 14 16 18 20

-1.0

-0.5

0.0

0.5

1.0

1.5

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

-2 0 2 4 6 8 10 12 14 16 18 20-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

S-S

13 (G

Pa)

X1/dH

-2 0 2 4 6 8 10 12 14 16 18 20

-0.0020

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23 Figure A. 15 : The distribution of the stress component for mis-oriented grain.

80


81

APPENDIX B

THERMAL STRESS DATA OF AlN USING POSSIBLE CRUCIBLES

Appendix B contains the completed result in graphical form for all the normal stress and in-plane stress of aluminum nitride grown on sapphire, tantalum carbide, niobium carbide, silicon carbide and tungsten substrate.


B.1 The distribution of the thermal residual stress of AlN grown on sapphire substrate

0 2 4 6 8-7

-6

-5

-4

-3

-2

-1

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8

-8

-7

-6

-5

-4

-3

-2

-1

0

1

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8

-0.0002

-0.0001

0.0000

0.0001

0.0002

0.0003

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8

0

1

2

3

4

5

6

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8

-0.00055

-0.00050

-0.00045

-0.00040

-0.00035

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23 Figure B.1 : The distribution of the thermal residual stress of AlN grown on sapphire substrate

82


B.2 The distribution of the thermal residual stress of AlN grown on TaC substrate

0 2 4 6 8-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8-2.0

-1.5

-1.0

-0.5

0.0

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8

-0.00006

-0.00004

-0.00002

0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8-0.00023

-0.00022

-0.00021

-0.00020

-0.00019

-0.00018

-0.00017

-0.00016

-0.00015

-0.00014

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23 Figure B.2 : The distribution of the thermal residual stress of AlN grown on TaC substrate

83


B.3 The distribution of the thermal residual stress of AlN grown on NbC substrate

0 2 4 6 8

-2.6

-2.4

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8-0.00015

-0.00010

-0.00005

0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8

0.0

0.5

1.0

1.5

2.0

2.5

S-S

13 (G

Pa)

X1/dH

0 2 4 6 8

-0.00046

-0.00044

-0.00042

-0.00040

-0.00038

-0.00036

-0.00034

-0.00032

-0.00030

-0.00028

-0.00026

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23 Figure B.3 : The distribution of the thermal residual stress of AlN grown on NbC substrate

84


B.4 The distribution of the thermal residual stress of AlN grown on SiC substrate

0 2 4 6 81.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

S-S

11 (G

Pa)

X1/dH

0 2 4 6 8

1.5

2.0

2.5

3.0

3.5

4.0

S-S

22 (G

Pa)

X1/dH

(a) σ 11 (b) σ 22

0 2 4 6 8-1

0

1

2

3

4

5

6

S-S

33 (G

Pa)

X1/dH

0 2 4 6 8-0.00025

-0.00020

-0.00015

-0.00010

-0.00005

0.00000

0.00005

0.00010

0.00015

S-S

12 (G

Pa)

X1/dH

(c) σ 33 (d) σ 12

0 2 4 6 8

-4

-3

-2

-1

0

S-S

13 (G

Pa)

X1/dH

0 2 4 6 80.00050

0.00055

0.00060

0.00065

0.00070

0.00075

0.00080

0.00085

S-S

23 (G

Pa)

X1/dH

(e) σ 13 (f) σ 23 Figure B.4 : The distribution of the thermal residual stress of AlN grown on SiC substrate

85


86

APPENDIX C

THE DIFFRACTION PATTERN OF AlN CRYSTAL

Appendix C contains the computational simulations of the diffraction pattern of aluminum nitride in graphical form for the important zone axis.


C.1 The diffraction pattern of aluminum nitride using [0001] beam direction

Figure C.1 : The diffraction pattern of aluminum nitride using [0001] beam direction

87


C.2 The diffraction pattern of aluminum nitride using [11-20] beam direction

Figure C.2 : The diffraction pattern of aluminum nitride using [11-20] beam direction

88


C.3 The diffraction pattern of aluminum nitride using [1-100] beam direction

Figure C.3 : The diffraction pattern of aluminum nitride using [1-100] beam direction

89

PERMISSION TO COPY

In presenting this thesis in partial fulfillment of the requirements for a master’s

degree at Texas Tech University or Texas Tech University Health Sciences Center, I

agree that the Library and my major department shall make it freely available for research

purposes. Permission to copy this thesis for scholarly purposes may be granted by the

Director of the Library or my major professor. It is understood that any copying or

publication of this thesis for financial gain shall not be allowed without my further

written permission and that any user may be liable for copyright infringement.

Agree (Permission is granted.)

____Rac Gyu Lee_________________________________ _3/ 23/ 2007_ ____ Student Signature Date Disagree (Permission is not granted.) _______________________________________________ _________________ Student Signature Date

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