STUDY OF ENERGY BAND DIAGRAMS OF

GROUP III-NITRIDE HETEROSTRUCTURES

by

SENDILL GNANAESWARAN, B.E.

A THESIS

IN

ELECTRICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

ELECTRICAL ENGINEERING

Approved

Chairperson of the Committee

Accepted

Dean of the Graduate School

December, 2003

ACKNOWLEDGEMENTS

Firstly, I thank my advisors Dr. Sergey Nikishin and Dr. Tim Dallas for all

the help, guidance and support they have provided, not only for this work, but,

throughout my term as a graduate student at Texas Tech University.

I would like to thank Dr. Gregory Snider, Associate Professor, Department

of Electrical Engineering, University of Notre Dame, for helping me understand

the simulation program better and Sten Heikman, a PhD Student from the

Electrical and Computer Engineering, University of California, Santa Barbara, for

helping me with the simulations.

I acknowledge the encouragement that I have received from all my

friends. They have been really supportive during difficult times. Finally, I thank

my parents for providing me with a good education. The support, encouragement

and love, I have received from my family has been a real motivation for me and

has always inspired me to do my best.

11

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

LIST OF TABLES v

LIST OF FIGURES vi

CHAPTER

1. INTRODUCTION 1

2. PHYSICAL PROPERTIES OF GROUP III-NITRIDES 5

2.1 Crystal and band stmcture 5

2.2 Bandgaps of group Ill-Nitride materials 7

2.3 Heterojunctions and their relation to band structure 8

2.4 Polarization fields 9

2.5 AlGaN/GaN heteroj unction 13

2.6 AIGaN/GaN Quantum wells 18

2.7 Quantum Microcavity 23

3. SOLVING SCHRODINGER AND POISSON EQUATIONS 24

4. RESULTS AND DISCUSSIONS 36

4.1 Effect of AlGaN barrier thickness on the energy band diagram and the sheet carrier density 37

4.2 Effect of AI mole fraction in AIGaN/GaN structures 41

4.3 Sheet carrier density dependence on the AlGaN barrier thickness for various values of the alloy composition 44

4.4 Critical thickness dependence on alloy composition 45

in

4.5 Formation of 2DHG 46

4.6 Effect of GaN cap layer on the sheet density 48

4.7 Thick GaN-capped GaN/AlGaN/GaN stmctures 50

4.8 hiN/GaN Structures 52

5. CONCLUSIONS 54

BIBLIOGRAPHY 56

IV

LIST OF TABLES

2.1 Lattice constants. Piezoelectric Constants and Spontaneous polarization of Wurtzite group Ill-nitrides 13

3.1 Parameters used to model group Ill-nitrides 35

LIST OF FIGURES

2.1 Wurtzite crystal structure

9 9 The schematic band structure of the Wurtzite GaN along Kz direction and in the kx-ky plane near F point

2.3 Bandgap parameters of hexagonal (a-phase) GaN, AIN, InN and their alloys versus lattice constant ao 7

2.4 Energy band diagram for an ideal abrupt heterojunction of type I between two different semiconductor materials labeled A and B 8

2.5 Spontaneous polarizations in AlJn^Ga^_^_^,N alloys according

to a Vegard-like rule 11

2.6 Crystal stmcture of Wurtzite Ga-face and N-face GaN 14

2.7 Polarization induced sheet charge density and directions of SP and PZ polarization in Ga-face and N-face strained and relaxed AIGaN/GaN heterostmcture 15

2.8 Schematic diagram of a nominally undoped Al o.isGao.gsN/GaN HFET stmcture 16

2.9 Mobility at two sheet concentration as a function of temperature 18

2.10 Time-integrated photoluminescence spectra of a series of GaN/AlGaN quantum wells 20

2.11 Schematic picture of the energies and wavefunctions of electrons and holes in a strained quantum well with a piezoelectric field 21

2.12 Comparison of the measured energy positions and decay times of the low-energy lines in GaN/AlGaN SQW's with a calculation based on piezoelectric fields 22

3.1 An example of an input file to the 1-D Poisson solver program 32

4.1 Single AIGaN/GaN heterostmcture 37

VI

4.2 Simulated energy band diagram of AlGaN(x=0.30)/GaN heterostructure with AlGaN layer thickness = 2.5 nm 38

4.3 Simulated values for A/gjGao7 A thickness Vs 2DEG density 39

4.4 Energy band diagram illustrating the surface donor model 40

4.5 Simulated values for A/ojGao^N thickness Vs 2DEG density 41

4.6 AIGaN/GaN heterostructure with fixed barrier thickness and varying mole fraction of Al 42

4.7 Simulated band diagram of A/gojGao 95 A' /GaA^ heterostmcture 42

4.8 Sheet carrier density in the AIGaN/GaN stmcture as a function of AlGaN barrier composition x 43

4.9 Sheet carrier density dependence on the AlGaN barrier thickness for various values of the alloy composition 44

4.10 Critical thicknesses as a function of Alloy composition in

AlGaN 46

4.11 GaN/AIGaN/GaN heterostmcture used for simulations 46

4.12 Simulated band diagram of GaN/AIGaN/GaN heterostmcture for low values of GaN cap layer thickness 47

4.13 Simulated band diagram of GaN/AIGaN/GaN heterostmcture

after 2DHG is formed 48

4.14 Dependence of the sheet density on GaN cap layer thickness 49

4.15 Thick -capped GaN/AlGaN/GaN stmcture used in this simulation 50

4.16 Sheet density dependence on the AlGaN barrier thickness. 51

4.17 Simulated band diagrams of Thick -capped GaN/AlGaN/GaN stmcture 51

4.18 Single InGaN/GaN heterostmcture 52

4.19 Simulated values for/no jG^o 7 A thickness Vs2DEG density 53

Vll

CHAPTER 1

INTRODUCTION

Extensive research is being done in the field of Semiconductor technology

and the development has been rapid in the past few years. Primary

semiconductors fill the last decade were Silicon (Si) and Group III-V materials

such as Gallium Arsenide (GaAs) and Aluminum Arsenide (AlAs). Although

these materials had extensive application, they were limited in their usage

associated with their narrow bandgaps (1.1 eV for Si and 1.4 eV for GaAs). Due

to the fact that electrons can easily travel from the valence band to the conduction

band in a material with a narrow bandgap, it made these materials unappealing for

high temperature and high power applications.

In the 1970s, considerable interests were shown on group Ill-nitrides. But

during that time, developing low-ohmic p-type group Ill-nitrides failed. But as

technology improved, in the late 1980's such low-ohmic p-type group Ill-nitrides

were developed initiating back the interest in the field [1].

The group Ill-nitrides Gallium Nitride (GaN), Aluminium Nitride (AIN)

and Indium Nitride (InN) with related alloys form an interesting class of wide

bandgap materials. These materials found special usage in Optronics as well as in

electronics due to the fact the entire spectral region from UV to red can be

covered with III-N optical devices. These materials form a continuous alloy

system made up of Indium Gallium Nitride(InGaN), Indium Aluminium Nitride

(InAIN), and Aluminium Gallium Nitride (AlGaN) whose direct opfical bandgaps

for the hexagonal Wurtzite phase range from 0.7 eV [2] for a-InN and 3.4 eV for

a-GaN to 6.2 eV for ot-AlN. The bandgap of InN was found out recenfiy to be 0.7

eV. Previously, it was thought to be 1.9 eV. This was not possible with III-V

material system based on GaAs, AlAs, GaP, InAs and related alloys.

Also due to the wide bandgap, group Ill-Nitride transistors are superior to

the corresponding ones made from Si and other III-V materials in terms of other

factors, such as:

• Group Ill-Nitride transistors have higher mechanical and thermal

stability.

• They can be operated in high temperature region.

Other important features of group Ill-Nitride devices are,

• They have large piezoelectric constants.

• There is a possibility of passivation by forming thin layers of

Ga203 or AI2O3 with bandgaps of approximately 4.2 eV and 9 eV.

• AIN is an important material with a variety of applications such as

passive barrier layers, high-frequency acoustic wave devices, high-

temperature windows, and dielectric optical enhancement layers in

magneto-optic multilayer stmctures.

• The presence of spontaneous and piezoelectric polarization at the

interface of two group Ill-Nitride materials.

• The high electron drift velocities of GaN are used to fabricate

high-power transistors based on AIGaN/GaN heterostmctures.

Recent research on Ill-V nitrides has paved the way for the realization of

high-quality crystals on GaN, AlGaN and GaInN, and of p-type conduction in

GaN and AlGaN. In Mg-doped p-type GaN, Mg acceptors are deacfivated by

atomic hydrogen which produced by NH3 gas used as the N source during GaN

growth. High-brightness blue LEDs have been fabricated on the basis of these

results, and luminous intensities over 1 cd has been achieved. These LEDs are

now commercially available. These have applications in areas such as full color

display, lighting, indicator lights, and traffic signals. Continuously operating

purplish-blue laser diodes (LDs) are also commercialized. Such short-wavelength

coherent sources are essential for high-density optical storage systems because the

diffraction limited optical storage density increases to a first extent quadratically

as the probe laser wavelength is reduced.

The basic stmcture of such transistors consists of a very thin layer of a

Group Ill-Nitride, usually in nanometers, sandwiched between two barrier layers

of another Group Ill-Nitride. The bandgap difference between these layers will

play a major role in the working of the transistor. This stmcture is called a

quantum well. Varying certain stmcture parameters such as the composition of the

AlGaN or AlInN barrier layers, the well width and bandgap discontinuity, we can

modify and control factors such as carrier confinement and transition energy of

quantum wells which invariably controls the output of the devices [3].

The purpose of this thesis is to study the effect of piezoelectric and

spontaneous polarization on the energy band diagram of GaN/AlGaN

heterostmctures and the formation of 2-dimensional electron gas (2DEG) and 2-

dimensional hole gas (2DHG). It will be shown that these polarization effects play

a major role in the GaN/AlGaN heterostmctures. Also the dependence of sheet

carrier density on factors such as the barrier width of the AlGaN layer and the

composition of the AlGaN will be studied.

The stmcture of the study is as follows. In the second chapter properties of

group Ill-nitrides relevant to this study such as the formation of quantum wells

will be outiined. In Chapter 3, the solution of Schrodinger-Poisson equations

using a nonuniform mesh and the working of G. L. Snider's computer program

"1-D Poisson solver: a Band Diagram Calculator'" [4] will be discussed. In

Chapter 4, Energy band diagrams of GaN/AlGaN heterostmctures will be studied

using the above mentioned computer program followed by the study of the sheet

carrier density's dependence on the factors such as the alloy composition of

AlGaN and barrier thickness of AlGaN. The effect of GaN cap layer on the

AlGaN is also studied in this chapter.

CHAPTER 2

PHYSICAL PROPERTIES OF GROUP III-NITRIDES

The study of energy band diagram of group Ill-nitrides require knowledge

of certain physical properties of the materials used and the heterostructures

formed by these materials. In this chapter the physical properties and the

formation of heterostmctures are discussed briefly.

2.1 Crystal and band stmcture

Although zinc-blende structure of Group Ill-nitrides has advantages over

the Wurtzite stmctures in electrical properties, Wurtzite stmctures have high

crystal quality and the Wurtzite stmctures have shown better results in

optoelectronics application and that being the primary application of Group III-

nitride devices, we take for granted that the groups ni-nitrides in this study has

the Wurtzite crystal stmcture. Figure 2.1 shows the Wurtzite crystal stmcture. The

lattices constants for GaN, AIN and InN are given in Table 3.1. These constants

vary strongly with their chemical compositions giving rise to large lattices

mismatch in the heterojunction.

The schematic band stmcture of the Wurtzite GaN along Kz direction and

in the kx-ky plane near F point is given in Figure 2.2. The band stmctures of GaN

and other group Ill-nitrides have direct band gaps at the centre of the Brillouin

zone ( r point). A Brillouin zone is a property of a crystal. This geometrical shape

can be considered to contain the valence electrons of the crystal. Its planes define.

Figure 2.1 Wurtzite crystal structures [5]

m)jk

Figure 2.2 The schematic band structure of the Wurtzite GaN along K direction and in the kx-ky plane near F point [5]

in momentum space (k-space), the location of the band gap. Wave vectors lying

within the zone are in the same energy band: there is a jump in energy before the

state given by the shortest vector in the next Brillouin zone. The band stmcture

over a small k range around band extrema is concentrated on because the electric

and optical properties are generally governed by this local E (k) relationship. The

valence band is split into three sub-bands and the relative energies of the energy

band maxima are determined by a combination of spin-orbit splitting and axial

crystal field strength. In Figure 2.2 these three bands are labeled as HH (Heavy),

LH (Light) and CH (Crystal-field split-off).

2.2 Bandgaps of group Ill-Nitride materials

As discussed earlier, the bandgaps of group Ill-Nitrides ranges from as

low as 0.7 eV for InN to 6.2 eV for AIN. Figure 2.3 shows the bandgaps of

various alloys of group III-Nifrides. By varying the alloy composition, we can get

7-1

B-

5 •

1 4 -a

" 3 -R

1 2 -n

1 •

.AIN

.41GaN\ \ .

\ \ . -AUnN

GiiN ^^-v...^^^ \ .

GaI^r""~^..../\^^

3.1 3.2 3.3 3.4

Lattice Constant a, (A)

~ " ~ ^ Mf

3.5 3.6

Figure 2.3 Bandgap parameters of hexagonal (a-phase) GaN, AIN, InN and their alloys versus lattice constant ao

a material with the desired bandgap. For different applications, different bandgap

materials are required and this can be obtained by using group Ill-Nitride with

different alloy compositions.

2.3 Heterojunctions and their relation to band structure

A heterojunction is basically a p-n junction in a semiconductor between

materials of different composition. Normal junctions are between p and n type

versions of the same material. But in this case we refer to a junction formed

between two group Ill-nitrides usually a GaN/AlN interface or a GaN/AlGaN

interface. Since they are two different materials, the band stmcture is

discontinuous from one material to the other and the band alignment across the

interface is typically of type I, i.e. the band gap of the lower bandgap material is

positioned energetically within the bandgap of the wider bandgap semiconductor.

Figure 2.4 shows an ideal type I heterojunction. But due to the presence

^ E a E„ J_J TT

Figure 2.4 Energy band diagram for an ideal abrupt heterojunction of type I between two different semiconductor materials labeled A and B

of strain fields in multilayer structures, and also the presence of strong

polarization fields makes the band diagram complicated. Moreover the band

offset can be defined properly only if we know the precise strain field. More work

is required in this field to determine the exact band offset.

2.4 Polarization fields

The usual growth direction for hexagonal III-N materials is along the

polar [0001] axis, for which the crystal lacks inversion symmetry. This will result

in the formation of polarization fields. There are two kinds of polarization fields.

They are:

• Spontaneous polarization (SP),

• Piezoelectric polarization (PZ).

The spontaneous polarization exists in polar semiconductors with a

Wurtzite or lower symmetry crystal stmcture and is related to the deviation of the

crystal lattice parameters from the ideal values for the stmcture, thereby creating

molecular dipoles in the material building a polarization field just like that formed

in ferroelectrics [6]. This field has a fixed direction along the [0001] c-axis in the

Wurtzite lattice. Therefore the field resulting from spontaneous polarization will

point along the growth direction, and this

• Maximizes spontaneous polarization effects in these systems,

• Renders the problem effectively one-dimensional.

It is represented by,

P =P C sp ^ sp '^

where P^ is the scalar value of spontaneous polarization. Several theoretical

calculations have been performed to get this value but experimentally this value is

not yet found accurately. For a ternary alloy such asAlJnpa^_^_ ,N, Vegard like

mle is followed. The spontaneous polarization of a ternary alloy is given by,

n, (X, y)=X.P,;'"+y.?:;"+(1 - X - yyp^;".

Figure 2.5 shows that for a given lattice constant, a wide interval of

spontaneous polarizations (hence spontaneous fields), is accessible varying the

alloy composition. In particular, consider a GaN/AlJn^,Ga^_^_^,N MQW, where

the composition is chosen so that the alloy is lattice matched to GaN, which we

assume to be also the substrate (or buffer) material (dashed-dotted line in the

figure). Then, piezoelectric polarization vanishes, but spontaneous polarization

remains. It can be seen that SP effects are likely to be very important for the AI-

containing stmctures when compared to Ga-In-N stmctures.

10

-0.D20

-0.0-10 c .2

I ^ -0,060 o Q. W 3 o c -0,080 « R to

-0.1 ^ .

GaN.,

/

AIN

• V f :

1 _ ^ _ InW

AUn,Ga, , . ^N^ /^

^ LATTICE MATCHED TO GaN

l.C© 3,15 325 3.35 3.45 3.55 3.6S

LATTICE CONSTANT (A)

Figure 2.5 Spontaneous polarizations in Al^ln Ga^_ N alloys according to a

Vegard-like rule [7]

The other type of polarization field, the piezoelectric polarization occurs

due to the presence of strain in the system. When two layers are joined together to

form a heterojunction, the difference in the lattice constant between the two

materials will lead to a strain. This strain also occurs due to the difference in the

thermal expansion coefficients in the layers during cool down after growth. This

leads to elastic strain in the layers. The piezoelectric polarization field can be

represented as:

Pr,z=e®£ pz (2.1)

where ^ij is the piezoelectric tensor and ^kl is the strain tensor. The component

along the c-axis is given by,

11

Ppz -C = (^33^3 + ^31 (^1 + ^2 ))-C (2.2)

where f, and £•-, are the in-plane strain given by,

£^=£^=(a-aj/a, (2.3)

and 3 the strain along the c-axis being,

^ 3 = ( C - C j / C ^ (2.4)

Here a^ and c are unstrained lattice parameters.

In a planar biaxially strained Wurtzite system, f, and e^ are related by,

c ^3 - 2^1 ^ (2.5)

where Cj3 and ^33 are elastic tensor components. Hence equation (2.2) can be

written as:

c Pp,=2£,{e,,-e^,-^) (2.6)

C33

The calculated values of lattice constants and piezoelectric constants are

given in Table 2.1.

12

Table 2.1 Lattice constants, Piezoelectric Constants and Spontaneous polarization of Wurtzite group lll-nitrides

Crystal Structure

Piezoelectric Constant ei4

Piezoelectric Constant ei5

Piezoelectric Constant 631

Piezoelectric Constant 633

Spontaneous polarization

Lattice Constant, a

Lattice Constant, c

Units

C/m^

C/m

C/m

C/m^

C/m

Angstroms

Angstroms

GaN

Wurtzite

0.375 [8]

-0.4S[11]

-0.60[11]

1.46/7/7

-O.OSlfllJ

3.189 [10]

5.1S5 [10]

AIN

Wurtzite

0.92 [9]

-0.58 [9]

-0.48 [9]

1.55 [9]

-0.029[11]

3.11 [10]

4.98 [10]

InN

Wurtzite

0.375 [8]

-0.57[11]

Q.91[ll]

-0.032[11]

3.54 [10]

5.10 [10]

2.5 AIGaN/GaN heteroiunction

The presence of SP and PZ polarization, influence the potential across the

heterojunction and this plays an important part in Heterojunction Field Effect

Transistors (HFET). The value of the polarization field depends on the polarity at

the interface of AlGaN and GaN. GaN has unique "up" and "down" directions:

One side—the "up" side—of a GaN crystal will be all Ga atoms, while the

opposite face—the "down" side—will be all N atoms. Both GaN's growth

mechanism and its electrical properties are different for the two surfaces and the

13

polarity of SP also depends on this. Schematic diagram of Ga-faced Wurtzite GaN

and N-faced Wurtzite GaN are given in Figure 2.6.

Ga-tace N-face

a t!) Substrate

Figure 2.6 Schematic diagram of the crystal structure of Wurtzite Ga-face and N-face GaN [12]

When AlGaN is grown over a GaN buffer layer which is relaxed, this

layer will experience a biaxial strain field because of the difference in the thermal

expansion coefficient in the buffer layer. As in the figure, the Pjp polarization

vector will point towards the substrate for the Ga-face and hence the SP and PZ

fields will add up in the same direction. Ga-face means Ga on the top position of

the {0001} bilayer, corresponding to the [0001] polarity. The interface charge is

given by.

rr—\P — (P _ P \ — ( p^"^^'^ 4. pAlOaN s_^ pOaN , pGaN s ^ — ^^^ ~ \^AlGaN ^GaN^~^^sp pz > ^^ sp ^ ' pz '

(2.7)

14

is formed. This value is positive and the value is in the order ono"aif\ This

makes it easier for the interface charge to attract electrons from the AlGaN layer

or a metal contact. This forms a two-dimensional electron gas (2DEG) near this

interface. Figure 2.7 shows the polarity of sheet charge density due to the

polarization fields.

a)

-t-CF

b)

+CT

c)

-<T

6a-face T §

la 1'^SP (T)

{J'sp

Substrate

1 ^SP ^PF n

i'" Substrate

+""1" ^ ( ' 'sp

Substrate

AiQaN relaxed

GaN relaxed

AlGaN tensile strain

GaN ri^axed

GaN compressive

strain

AlGaN relaxed

N-face §

1 e f..

Substrate

1 A e jfsP

Substrate

f ^ y"' f APsp

Substrate

-a

-a

Figure 2.7 Polarization induced sheet charge density and directions of SP and PZ polarization in Ga-face and N-face strained and relaxed AIGaN/GaN heterostructure [12]

However, in case of an N-face, the SP field and the PZ field are reversed

and the fixed interface charge will be negative. But in this case the holes are

15

attracted to the interface and hence a two-dimensional hole gas (2DHG) is

formed. But an addition of another thin GaN layer on top of the AlGaN will form

a 2DEG in its interface.

The expected polarization charge for a device similar to a HFET is given in

Figure 2.8. High sheet carrier concentration was observed on this structure as a

consequence of the piezoelectric polarization present in this structure. Figure

2.8(c) shows these polarization fields.

(a) •••'Schqtiky

_ ^ (b)

SOOAi-AlaijGaoar.N

(-GaN

Figure 2.8 (a) Schematic diagram of a nominally undoped Al 0.15 Ga o.gs N/GaN HFET structure. (b) Conduction-band energy diagram (solid line) Calculated for this structure; the dotted line represents the Fermi level, and e 0h is the Schottky barrier height, (c) Schematic diagram of

piezoelectrically induced and free-carrier charge distribution. [13]

To find the energy band diagram of heterostructures, the Schrodinger

equation and the Poisson equation have to be solved simultaneously and the

resulting electron charge density is given by:

n(x) = CT(JC)

^^^^ \eO,(x) + E,(x)-AEc(x)] de

(2.8)

16

where x is the Al composition in the AlGaN layer, a is the interface charge, O^is

the Schottky barrier height, E,, is the Fermi energy at the interface between the

GaN layer and the AlGaN layer with respect to the GaN conduction band edge

and Aii^ is the conduction band offset at the heterojunction. By varying the

composition value, the width of the barrier and other parameters the sheet charge

density can be increased well above lO'^cm". In the next chapter, a detailed study

of solving the Schrodinger equation and the Poisson equation simultaneously is

done.

Due to the presence of the polarization field a high sheet electron

concentration is achieved. The other key factor for achieving a good channel

conductance of HFET devices is high electron mobility. Electron mobility is a

measure of electron scattering. It is expressed as the ratio of electron drift velocity

and electric field or the ratio of carrier concentration and conductivity of the

semiconductor. Its unit is cm'^INs. The mobility of a semiconductor depends on

various scattering mechanism and is also limited by optical phonon scattering and

other defects in the structure. In a GaN/AlGaN heterojunction, the various

scattering mechanism limiting the mobility can be expressed in terms of the

scattering time of these mechanisms. Mathematically, it can be expressed as.

1 1 1 1 1 1 — = + + + +

^for ^ph "^imp ^alloy ^int ^disl

17

where r^,,is the scattering of phonons including deformation potential.

piezoelectric scattering and polar optical phonon scattering mechanisms, r,„, is

the coulomb scattering with impurities, -r„„„ , is the scattering with alloy potential

fluctuations at the interface, r,„, denotes the influence of the roughness of the

interface, r , ; is the dislocation scattering. Figure 2.9 shows the mobility Vs

temperature graph of a GaN/AlGaN heterostructure for two different sheet carrier

concentrations.

10'

I o

o

10^

fls-

• ' -• f

a ., •

•

/ =8x10 ^

^ ^ ^ »

/?s=2.6xi0^^ain' / a

K

-2** a '

X ^ a j - 1 . 8

•j-1.6

10 ' 10^ Temperature (K)

Figure 2.9 Mobility at two sheet concentration as a function of temperature [14]

2.6 AIGaN/GaN Quantum wells

A quantum well is a semiconductor heterostructure fabricated to

implement quantum effects in electronic and photonic applications. It is typically

an ultra-thin layer of narrower bandgap semiconductor sandwiched between two

layers of larger bandgap semiconductor. Here holes are free to move in the

direction perpendicular to the crystal growth direction but not in the direction of

crystal growth, and hence are confined. Quantum wells are important in

semiconductor lasers because they allow some degree of freedom in the design of

the emitted wavelength through adjustment of the energy levels within the well by

careful consideration of the well width. Solving the Schrodinger equation for a

finite potential will produce values of the energy levels within the well.

An ideal Multiple quantum well (MQW) consists of alternating layers of

GaN and AlGaN of certain widths. Let us assume that Fermi level at the top of the

structures lines up with the position in the GaN buffer layer, then the electric field

in the MQW's caused by spontaneous and piezoelectric polarization fields is

given by,

r / pAlGaN , pAlGaN \ _ / pGaN , pGaN N-I

T-GaN _ rpGaN , jpGaN _ i ^^^sp pz ^ ^ sp pz -1

^tot - ^ s p '^^pz - ^ AlGaN , a 4.^ c^ "•AIGaN/GaN "^^GaN^AlGaN

(2.9)

and,

r / pGaN , pGaN N _ / pAlGaN , pAlGaN \- |

r'AlGaN _ rpAlGaN , p'AlGaN _ i ^^ sp pz > ^ sp pz 'i

^tot - ^ s p ^^pz -^GaN , , "-GaN/AlGaN ^^AIGaN/GaN

(2.10)

19

where d^^^^md d^,^^_^ are the well width of the GaN and AlGaN layers,

respectively, £-g ^ andf , ^^ and are the dielectric constants of the GaN and

AlGaN layers respectively. The two polarization fields add up in the same

direction, hence the electric fields are large even for small values of AIN

composition in the AlGaN layer. These polarization fields influence the optical

properties of the heterostructures. Figure 2.10 shows the PL data of a

GaN/ Al^^fia^^^N heterostructures with various thickness of the GaN layer as

mentioned.

GaN/AloisGaossN DH's T = 5K

3.3 3.4 energy (eV)

Figure 2.10 Time-integrated photoluminescence spectra of a series of GaN/AlGaN quantum wells. The dashed line indicates the position of the GaN bandgap [15]

As can be seen from the figure, there is a drastic downshift of the PL peak

energy with the QW width due to the polarizations fields present. For the 10 nm

QW sample, the exciton peak is at around 3.3 eV which is well below the

20

bandgap of the GaN material. This is called quantum confined stark effect

(QCSE). As shown is Figure 2.11, the potential in the QW also reduces the

overlap between the electron and the hole wavefunction. This in turn will reduce

the oscillator strength for the excitonic recombination process.

AlGaN

Figure 2.11 Schematic picture of the energies and wavefunctions of electrons and holes in a strained quantum well with a piezoelectric fleld

A comparison of the calculated energies and oscillator strengths with the

experimental data is shown in Figure 2.12 for GaN/AlGaN QW's. For this figure,

the energetic positions were taken from spectra at the longest possible delay time

after excitation, i.e., with screening of the field being as small as possible. The

calculated curves include only a single adjustable parameter, which is the

magnitude of the piezoelectric field. In the present case, a field of 350 kV/cm

consistently explains both the red-shift and the dramatic increase of the decay

time with increasing well width.

21

10~

10^

10^

I 10= > .

O 10 CD

10

3.70

0, 3.60

I 3.50

^ 3.40 cd

Q- 3.30

3.20

GaN/Alo^^GaoggN SQW's

T = 5K

0.0 2.0 4.0 6.0 8.0 10.0 well width (nm)

Figure 2.12 Comparison of the measured energy positions (dots) and decay times (squares) of the low-energy lines in GaN/AlGaN SQW's with a calculation based on piezoelectric fields. The

triangles give the values for the respective higher-energy emission lines [16]

Moreover, Figure 2.12 also explains the origin of the higher energy line

in the thick layers: It is due to spatially direct transitions in the strained GaN well

layer. The dashed-dotted line was calculated on the basis of the known strain of

about 0.4 % of the GaN layer and coincides almost perfectly with the measured

position of the corresponding luminescence peaks.

22

2.7 Quantum microcavity (OMO

When a QW structure is formed inside an optical cavity, quantum

microcavties (QMC) are formed. These QMC's are one-dimensional planar

structures grown by layer-by-layer epitaxial techniques. The cavity plays the role

of a "defect" in periodic stack of layers providing strong localization of light

along growth direction. In this structure, the exciton and the photon can be

controlled simultaneously and coupled exciton-photon particles are called

exciton-polaritons. In strong coupling regime, this interaction leads to the creation

of a new type of quasi particles in solids, so-called "microcavity polaritons." Their

optical properties are of great fundamental interest because these quasiparticles

possess properties of light (photons) and matter (excitons) at the same time. Such

QMC's are used in advanced light emitters.

In an AIGaN/GaN QMC, there is a huge rabi splitting between the

polariton branches of the QW excitons due to the high oscillator strength in GaN.

Since the exciton binding energy is large, a very strong cavity eigenmode

splittings are expected and this is of prime importance for optical devices based

on nitride QMC's.

23

CHAPTER 3

SQLVING SCHRODINGER AND POISSON EQUATIONS

The study of energy band structures of heterostructures needs a detailed

knowledge of optical and transport properties of the heterostructures. These

properties can be found by solving self-consistentiy Poisson's and Schrodinger's

equations for the electron wave functions [17].

The finite difference method (FDM) is a simple and efficient method for

solving ordinary differential equations (ODEs) in problem regions with simple

boundaries. FDM can be used to solve for the Schrodinger equation. The method

requires the construction of a mesh defining local coordinate surfaces. For each

node of this mesh, the unknown function values are found, replacing the

differential equations by difference equations. These values gives the vector

solution for \|/ and a matrix formulation if the Schrodinger equation:

Av|/ = A,v|/ (3.1)

where A is the matrix operator and A, the energy eigenvalues. Usually a uniform

mesh size is selected but this means that this method wasn't effective. We need a

small mesh when the wavefunction is changing rapidly and a large mesh during a

slow change in the wavefunction for the ideal speed. Moreover, careful

calculations are also required at the junction of two different mesh sizes and

destroying the symmetry of the matrix A, making it more difficult to calculate.

Dr. Snider et al. came up with a method to making it easier to solve the

24

Schrodinger equations with different mesh sizes while keeping the symmetry of

the matrix, hi the mid 1990s, Snider used this mathematical model to make a

simulation program to calculate the energy band diagrams of group III-V

heterostructures. For my study of the group Ill-nitrides I modeled group Ill-nitride

materials GaN, AIN and AlGaN in this program to simulate the energy band

structures and others properties of these heterostructures. Since his mathematical

model was the basis of his program, all the mathematical equations used by him

are seen in this chapter. In the second part of the chapter, the working of his

program and the modeling of the group Ill-nitrides are discussed.

The one-dimensional, one-electron Schrodinger equation is given as

^ ^ ' ^ ^ \l/{x) + V{x)y/{x) = Ey/{x) (3.2) / 1 J \

2 dx m*{x) dx

where ^is planck's constant divided by In, m* is the effective mass, y/ is the

wave function, E is the energy and V is the potential energy.

The 1-D Poisson equation is given by

Ar,,(.)AW).-^[^-.w-"W] ,3.3, dx\ ax J £p

where is e^ the dielectric constant, ^ is the electrostatic potential, TV is the

ionized donor concentration, and n is the electron density distribution.

25

The potential energy V is related to the electrostatic potentuil ^ as

y{x) = -q^{x) + E^{x) (3.4)

where Af is the pseudopotential energy due to the band offset at the

heterointerface. The wave function y/{x) is related to the electron density n(x) by

m

n{x) = Yy/]{x)y/^{x)n, 0 5) k=\

where m is the number of bound states and Uk is the electron occupation for each

state.

The electron concentration for each state is given by

- ml. [ ^ .^F

where E^ is the eigenenergy.

These are the basic equations required to solve for finding the solution to

the Schrodinger and the Poisson equations. We start with a trial potential value for

V(x) and then calculate the wave functions and the corresponding eigenenergies

Ek and then the electron density distribution n(x) is calculated by using Eqs. (3.5)

and (3.6). Then this calculated value of n(x) and a given donor concentration

ND(X) is used in Eq. (3.3) to calculate. Then a new potential energy V(x) is

26

obtained from Eq. (3.4). More iteration will yield self-consistent solutions for

V(x) and n(x).

The Schrodinger and the Poisson equations are solved numerically by

using a three-point finite difference scheme. The Schrodinger equation becomes

t?^ '^^¥,.x-¥d 2(^y,-^,_,)

"C,.^,(^+v,) "C.v.c/'.+v,) ^^¥i (3.7)

and the corresponding matrix form is

(3.7')

where

Ay

-.i(

{--

I

2 1

0...Otherwise

...j = i + \

...j = i-\ (3.8)

Here i represent the grid point on the mesh. As discussed earlier, the use of

nonuniform mesh size here will destroy the symmetry if the matrix A and

complicate the computations. But if we use

L] = {h,+h,_,)ll (3.9)

27

Eq. (3.8) becomes.

2

, m h

1 1

1}

, '" h.. , V ,1/2 ' - ' /

—r...y = / - l I}.

-Av..-A,-,+^-7 = '' 0...Otherwise

(3.8')

Then we set

5,=^A, (3.10)

with corresponding matrix notation

B=MA (3.10')

where M is the diagonal matrix whose elements are l}. .Since A is a triadiagonal

matrix, B will also be one. Also since B is symmetric (from Eq. (3.8') and (3.9)),

we can also say.

By/ = MAy/ = XMy/ (3.11)

Lets say L is a diagonal matrix with elements Li, then M obtained from the FDM

is a diagonal matrix of the form

M=LL (3.12)

28

Using Eq. (3.10') its showed that,

L~'BL-'Ly/ = L-'LLAy/ = AL'LLY (3.13)

or

H<^ = X^ (3.13')

where

H = L-'BL-' (3.14)

and

y/ = L-'<^. (3.15)

H is a symmetric and triadiagonal matrix. Instead of solving Eq. (3.1), Eq. (3.13')

is solved to get the eigenvalue >! corresponding to the eigenfunction O . Then Eq.

(3.15) is used to calculate the wavefunction y/ from the eigenfunction O .

If there are two wavefunctions in the potential well, then they are related

to each other by,

^'^^'=So.:':o <'•'*>

29

where ^^ and y/^ are the two wavefunctions and are orthonormal to each other.

The nonlinear Poisson equation is solved numerically using the Newton's

method. Using Eq. (3.3) we get,

d_(dl dx\ ' dx J

.1 d ( A^rh \ d5(j) + lN,{x)-n{x)\=- „,, • ^ . . . . . p

e, \ dx\ dx ) £^ k=\ dE^

dn +—Y.y^l¥k -^{y^k \q^¥k)

(3.17)

A differential equation describing the incremental potential change S(l>, at

every step, satisfies the above equation provided certain assumptions are made.

They are:

• The variation of wavefunction with respect to 5(1) is very small.

• The donors are completely ionized.

The above differential equation is very hard to solve, hence a similar

method used to solve the Schrodinger method is used here too. Hence Eq. (3.17)

becomes.

qm-

V '^o k=\ (1-+c,_M-.

(3.18)

where

30

^'=E<^';^y+^(^D,-",)/£•„ (3.19) J=\

and

2f - ^ • ' ^ •...j = i + l

h,(h,+h,_,)

h{h^+h^_^)

-CiM-C.^_,...j = i

[ Q...Otherwise

We can say that:

C'J^ = - ^ (3.21)

where C is a triadiagonal, nonsymmetric and nXn matrix,<5(;Z> is the «xl vector

containing the corrected potential at each point which must be added to the former

potential profiles, and ^ is a nxl vector with the Poisson error at each point also

taken into account. Then Grout's reduction method is used on Eq. (3.21) to solve

for the Poisson equation.

The computer program "1-D Poisson solver: a Band Diagram

Calculator"/^/ is based on the above mathematical calculations for solving the

Schrodinger and the Poisson equations. This program can calculate the band

diagrams for multiple bias voltages and can be also used to measure the C-V

characteristics. Dr. G. L. Snider wrote this program with group III-V materials

31

such as GaAs, AlAs in mind but it works for other materials as well. Some of the

features of this program are,

• It also calculates hole and electron concentrations.

• Mobile charge concentrations are calculated using Boltzmann statistics.

• Semiconductors are represented by their name.

• Calculate the parameters of a ternary based on the given x value.

• The semiconductors are arranged according to their family.

• Structure can be simulated only in thermal equilibrium.

• Calculates the wavefunctions.

Apart from these, three possible boundary conditions can be defined on

the surface. These are Schottky barrier, ohmic contact and energy band slope=0.

The schottky barrier is used to calculate the effect of an applied bias, and the

energy band slope=0 condition can be used to simulate a certain region of interest.

The input to this program is a text file containing information about the

structure of the device. An example of an input file is given in Figure 3.1.

The first line following the "#" symbol represents a comment line. This

can be placed anywhere in the file. Starting from the second line, the structure

information is provided. The schottky in the surface represents that a schottky

barrier condition is used and vl is the applied bias whose value is specified in the

later stages of the input file. The topmost layer according to the input file is a

GaN layer of thickness (t) of 500 angstroms. All the values of thickness are in

32

# Example of an input file

surface schottky vl GaN t=500 Nd=lel7 AIN t=100 AlGaN t=200 x=0.3Nd=5el8 GaN t=500 Nd=lel7 substrate

fullyionized vl 0 temp=300K dy=0.5 maxiterations=400

Figure 3.1 An example of an input file to the ID Poisson solver program

angstroms if not specified otherwise. Here Nd represents the donor concentration

of the GaN layer. The GaN layer is followed by a AIN layer of 100 angstroms

thickness. Since neither the donor concentration nor the acceptor concentration

(Na) is specified, this layer is taken to be undoped. The next layer that follows is a

AlGaN layer of thickness 200 angstroms and donor concentration 5 X 10" ^ cm" .

Since AlGaN is a ternary, the composifion level x must be specified. Here x=0.3

corresponds to Al 0.3 Ga 0.7 N. The substrate is a GaN layer of 500 angstroms

thickness.

The fullyionized statement specifies that all the shallow dopants in the

structure are ionized. The value of the applied bias is set to 0 in the next line and

the temp represents the temperature and is set to 300 K in this example. The dy

and maxiterations respectively specifies the mesh spacing and the maximum

33

number of iterations to be done in case the answer does not converge. In this

example dy, the mesh size is uniform throughout the structure but non-uniform

mesh sizes can be used to by giving the mesh size next to each structure. There

other keywords that can be used in the program as well.

The original program as mentioned earlier was written with group III-V

materials in mind. Hence the group Ill-nitrides materials had to be modeled in the

materials file before running the simulation. Table 3.1 has the parameters used to

model GaN, AIN and InN. The parameters of the ternary AlGaN with a given x

value are calculated by the program by following a linear relationship between

GaN and AIN. Similarly the values of InGaN follow a linear relation between

GaN and InN.

Note that all the concentration levels are given as 0 cm'^.All the values

mentioned are default values. These values can be changed in the input file. After

the simulation of the input file, the output values are stored in files with an .out

extension. The result can be plotted using any plotting software.

34

Table 3.1 Parameters used to model group Ill-nitrides [18]

Parameters Energy Gap Conduction band offset Relative dielectric constant Electron effective mass Conduction band degeneracy Heavy hole effective mass Light hole effective mass Donor level Donor concentiation Acceptor level Default Acceptor concentration Deep donor level Deep Donor concentration Deep acceptor level Deep Acceptor concentration Barrier height Electron mobility Hole mobility Electron recombination time Hole recombination time Absorption coefficient

Units eV eV

eV cm'^ eV

cm' eV

cm'^ eV

cm'^ eV

cm^/Vs cm^/Vs

ns ns

GaN 3.4 0

8.9 0.20

1 1.3

0.19 5*10''

0 5*10"'

0 -5 0

160*10"' 0

1.7 1000 30 1 1

4*10'

AIN 6.2 1.96 8.5

0.48 1

3.53 3.53

5*10"' 0

5*10"' 0

160*10"' 0

160*10"' 0

3.1 135 14 1 1

4*10'^

InN 6.2 -0.9 9.5

0.11 1

1.63 0.27

15*10"' 0

230*10"' 0 5 0 -5 0 1

250 10 1 1 1

35

CHAPTER 4

RESULTS AND DISCUSSION

Transistors made up of AIGaN/GaN heterostructures have a great potential

in high-power high-frequency applications. As discussed eariier, the difference in

the spontaneous and piezoelectric polarization between AlGaN and GaN will

result in a fixed sheet of polarization charge at the interface and this charge tends

to attract high concentiation of electrons. Although the origin of electrons is

uncertain, one assumption is that electrons originate from donor-like states on the

AlGaN surface. Ibbetson et al. [19], argued that the number of ionized donors in

the AlGaN plus the number of ionized donor-like states on the surface and hence

the assumption that the electrons originated from the donor-like states on the

AlGaN surface. The above process is the formation of a two-dimensional electron

gas (2DEG) at the interface of AlGaN and GaN. Similarly a two-dimensional hole

gas (2DHG) can also be formed at the interface of the two materials with a

negative polarization charge [20].

These charges play an important role in the electric and optical properties

of nitride heterostructures. Hence a detailed study of the energy band diagram of

the heterostructure and an understanding of the polarization charges are required

to optimize the performance of these devices. In this chapter the simulated band

diagrams for different heterostructures are discussed. The variation of the sheet

carrier concentration with parameters like the well width and the composition of

the AlGaN layer are also studied.

36

— Effect of AlGaN barrier thickness on the energy band diagram and the sheet carrier density

Let's start with energy band diagram of a single AIGaN/GaN

heterostructure as given in Figure 4.1.

AlGaN ^0.30

GaN

Buffer layer

50 nm

Thickness of the AlGaN layer is vahed

Figure 4.1 A Single AIGaN/GaN heterostructure

In this heterostructure, the GaN and the AlGaN layers are undoped. The

AIN composition in AlGaN is 30% and the thickness of the AlGaN layer is varied

starting from a very small thickness to a large thickness while the thickness of the

buffer layer is made constant at 50 nm.

In the first example, the thickness of the AlGaN layer is taken to be 2.5

nm. The energy band diagram of that heterostructure is simulated and is given in

Figure 4.2. In this structure the surface state is at donor energy level, ED, which is

below that of the conduction band edge. We assume that this state is donor-like,

.i.e., it is neutral when occupied and positive when empty. Formation of 2DEG

depends on the occupancy of this state and thus on its energy relative to the Fermi

level, Ep. If the state is sufficiently deep then it lies below Ep, as in this case.

37

Hence there will be no formation of 2DEG and so for all practical purposes 2DEG

density should be zero. In Figure 4.2, AEc represents the conduction band offset.

Figure 4.2 Simulated energy band diagram of AlGaN(x=0.30)/GaN heterostructure with AlGaN layer thickness = 2.5 nm

As the AlGaN barrier width increases, Ep - ED decreases. After a certain

thickness known as the critical thickness, the donor energy reaches the Fermi

level. At this point, electrons from the occupied surface states will be able to

transfer to the empty conduction band states at the interface and this creates the

2DEG. The critical thickness is mathematically represented by,

tcR=iEj,-^Ec)£lq(Jp^ (4.1)

where f^„is the critical thickness and £ is the AlGaN relative dielectiic constant. *CJ?

The Fermi level remains at the donor energy till the surface state becomes empty.

With the increase in the barrier thickness, more electirons transfer from the surface

38

state. Hence with the increase in barrier thickness, the density of 2DEG

polarization charges also increases.

By following the same heterostructure as in Figure 4.1, and by increasing

the thickness of the AlGaN layer, the energy band diagrams of the device were

simulated and the 2DEG density was also found out for each structure. Figure 4.3

shows a graph between the thickness of the undoped AlGaN layer and the 2DEG

density at x=0.30. As we can see from the graph, the 2DEG density starts

increasing rapidly after a certain thickness of the AlGaN layer. This layer is the

critical thickness and marked as t^.^. This value is found out to be 3.5 nm for the

particular doping level.

:^50E+17 -

3 00E+12 •

_ 2.50E'-12 * 5 *

« 2.00E+12 •

1 150E+12-

S 1.00E+12 UI a " 5.00E+n •

u.uut+uj -(

/ /

/

/

/

1 10 20 30 ks. AG

Thickness (Angstroms)

/ f

' 50

1 •

' 60

Figure 4.3 Simulated values for Al^^-fia^jN thickness Vs 2DEG density

39

Figure 4.4 shows the simulated band diagram of the structure when the thickness

of the A/ojGag 7// layer is greater than the critical thickness. Here the 2DEG

formation is shown.

EF <

\ 2DEG

. V ' ^ - ^ - -

Partially-filled Surface donors

\

Ec

Ev

Figure 4.4 Energy band diagram illustrating the surface donor model with undoped Algfia^^N

barrier thickness greater than the critical thickness for the formation of 2DEG

As the thickness of the AI^^GGQ^N layer is increased, the 2DEG density

also increases and saturates after a certain thickness. Figure 4.5 shows a graph

between the thickness and the 2DEG density for thickness varying from 2.5 nm to

200 nm.

40

*'-S 1

a CM

o ci •a

den

•8 J3 OS O P4 Q <s

18 n

16 14

12 -10 -

8 -6 -

4 -2 -n -u

-2

^ f ^ ^

/ ^ y '

/ j

j \ I 1 J 1 J

1 1 i- 1 1

) 50 100 150 200 250

Thickness(nm)

Figure 4.5 Simulated values for Al^fia^-jN thickness Vs 2DEG density

As can be seen from the graph above, as the Al^fia^^N barrier thickness

is increased, the sheet density also increases but it reaches the value of the

polarization induced charge after some time and this is the maximum value the

sheet concentration can obtzdn for that particular composition.

4.2 Effect of Al mole fraction in AIGaN/GaN structures

In the previous example we had seen the effect of thickness on the sheet

carrier concentration at a fixed value of x. The value of x in the Al/}a^_^N barrier

layer is also an important factor in formation of 2DEG density. Here a single

heterostincture of AIGaN/GaN with a fixed layer thickness for both GaN and

AlGaN is considered while the mole fraction of Al is the variable. The thickness

of the AlGaN layer is fixed at 5 nm which is greater than the critical thickness.

The heterostructure used is given in Figure 4.6.

41

AiGa^_^N 5nm

GaN

Buffer layer 50 nm

Figure 4.6 AIGaN/GaN heterostructure with fixed barrier thickness and varying mole fi-action of Al

For a very small value of x (say 0.05% AIN), the value of the sheet carrier

density in the above heterostixicture was found out to be8.51x10'^ cm"\ This

density value is smaller than the value obtained for an alloy composition used in

practical purposes of 0.30-0.40. The simulated band diagram of the

heterostructure when x=0.05 is given in Figure 4.7.

Figure 4.7 Simulated band diagram of Al^^fiUQi^^N I GaN heterostructure

42

As can be seen from the energy band diagram, the 2DEG formation is not

as pronounced for a low value of alloy composition as in the case shown in Figure

4.4.

As the value of alloy composition is increased the sheet density also

increases and almost has a linear relationship between the two variables. The

dependence of sheet carrier density on the alloy composition value x is given in

Figure 4.8. All simulations were performed at temperature equal to 300K.

18 n

«-- 16-

S 12 -

•^ 10 -

J 8 -1 6 -^ 4 -

S 2-1—1

^' 0 -[

1

) 0.2 0.4 0.6 0.8

Alloy composition, x

1.2

Figure 4.8 Sheet carrier density in the AIGaN/GaN structure as a function of AlGaN barrier composition x

The two variables in the Figure 4.8 share a linear relationship with each

other. The slope of this graph was calculated from the simulated values to be

5.7xlO'^cm"^ which was similar to the result obtained by Smorchkova et al. [21],

of 5.45xlO'Vm"^ for a structure grown by both MOCVD and MBE.

43

4.3 Sheet carrier density dependence on the AlGaN barrier thickness for various values of the alloy composition

Figure 4.9 shows the dependence of the carrier sheet density on the

AlGaN layer thickness for various values of alloy composition ranging from

x=0.1 tox=l (Pure AIN).

18 1 .—. ? ' 6 -O

N 14 -a> ^ 12 -U) 2 ^ 1 0 -

</» c 8 •

•o i j 6 E n 4 . w

* j

<1» n 4> ^ • £

«" 0 -

,.-—-^

k = 0.8

/ r

•' / • / ' ' " / -

/ / / / if

1

0

_,--,.y

50 100 150

AlGaN thickness(nm)

x = 1 x = 0.5

s — Cl "^ X — U . i

u n 1 A — U. 1

' 200 250

Figure 4.9 Sheet carrier density dependence on the AlGaN barrier thickness for various values of the alloy composition

From the graph, it is clear that the sheet carrier density increases with both

the thickness of the barrier and the alloy composition value, x. For all values of x,

the sheet density starts saturating at a thickness little over 50 nm. For small

values of x, sheet carrier density increases rapidly right after critical thickness is

obtained and saturates with a value near the value of the polarization induced

charge. For x = 0.1, the maximum sheet carrier density that can be obtained

is 10.7 X10'^cm"^. For x = 0.5, the maximum value of sheet density is around and

44

for 16.7X10'^cm"^ and for x = 0.8 it is 16.8x 10'^cm"^ So as x increases, the

dependence of the sheet density on the barrier thickness is not as rapid as in the

case of low values of x. As x approaches 1, that is when AlGaN is almost an AIN

layer; the dependence on the thickness is almost negligible after a certain

thickness. Also the sheet carrier density for 200 nm AlGaN barrier when x = 0.8

and when x = 1 are almost the same at 1.7x10'^cm"^. Hence this value is the

maximum sheet carrier density that is obtained from the simulations of the

AIGaN/GaN heterostructure.

4.4 Critical thickness dependence on alloy composition

As seen in section 4.1, In a AIGaN/GaN the thickness of the AlGaN

barrier layer at which a 2DEG is formed is known as the critical thickness. The

critical thickness depends on factors such as the alloy composition of AlGaN and

the doping concentrations of the layers. In this section the dependence of the

critical thickness on the alloy composition for an undoped structure is studies.

The dependence of the critical thickness on the alloy composition is

shown in Figure 4.10. The critical thickness, for a very low value of x is very high

but it decreases rapidly as the alloy composition is increased. After a certain point

it decreases slowly and has the minimum critical thickness of 0.5 nm for a

AlN/GaN structure.

45

Figure 4.10 Critical thicknesses as a function of Alloy composition in AlGaN

4.5 Formation of 2DHG

Let's introduce a GaN cap layer on top of the AlGaN layer as shown in

Figure 4.10. Here the undoped AlGaN has an alloy composition value x =0.30

and a thickness greater than the critical thickness.

GaN

AlGaN x=0.30

GaN

Buffer layer

50 nm

f Thickness of the Cap i layer is varied

Thickness>Critical Thickness

Figure 4.11 GaN/AlGaN/GaN heterostructure used for simulations

46

Simulations were done for different values of thickness of the GaN cap

layer. When the GaN layer thickness is very small, the energy band diagram

shown in Figure 4.11 is obtained.

EF

/ \ 2DEG / \ / V—

/ \ / \

V N

GaN Alo3Ga^>i;f'—

^0°°i 20 nm

"•' GaN 100 nm

Ec

Ev

Figure 4.12 Simulated band diagram of GaN/AlGaN/GaN heterostructure for low values of GaN cap layer thickness

As the GaN cap layer thickness is increased, the valence band at the

interface between this cap layer and the AlGaN layer is shifted upwards and at a

certain thickness it reaches the Fermi level. This thickness depends on the

thickness of the AlGaN layer and the doping characteristics of the two layers. For

this particular heterostructure, the thickness was found to be around 15 nm. At

this junction a 2DHG is formed as shown in Figure 4.12.

47

Figure 4.13 Simulated band diagram of GaN/AlGaN/GaN heterostructure after 2DHG is formed

This phenomenon is explained as follows; When a GaN cal layer is added

to the AIGaN/GaN heterostructure, a negative polarization charge is introduced at

the interface between the GaN cap layer and the AlGaN layer. This causes a

decrease in the 2DEG density and an increase in the electric field in the AlGaN

layer. As the thickness of the cap layer is increased, the valence band at the upper

interface shifts upwards and reaching the Fermi level after a certain thickness. At

this point, a 2DHG is formed at the upper interface and this stops any increase of

the electric field in the AlGaN layer.

4.6 Effect of GaN cap laver thickness on the sheet density

The introduction of a thick GaN leads to the formation of 2DHG at the

interface between the cap layer and the AlGaN as shown earlier. This leads to a

change in the 2DEG sheet density. The dependence of the 2DEG density on the

thickness of this GaN cap layer is shown in Figure 4.14.

48

12 n

C a • a o 8-1—1

t 6 •

IS ^ 2 •

0 •

i

V ^ 2DEG

.-'•' 2DHG

f - " ' • • « •••• 1 1 ( ' —

3 50 too 150 200

GaN Cap layer thickness (nm)

250

Figure 4.14 Dependence of the sheet density on GaN cap layer thickness. Solid line represents 2DEG sheet density and the dotted line represents 2DHG density.

Both the 2DEG and the 2DHG sheet densities for various GaN layer

thicknesses are shown in the figure. When there is no cap layer the 2DEG density

is as its maximum. The introduction of the cap layer decreases this density and as

the thickness of the cap layer increases the 2DEG density decreases rapidly until a

2DHG is formed. For this particular structure this thickness was found to be 15

nm. After this point the 2DEG decreases slowly and becomes constant after some

point. The dotted line represents the 2DHG density in upper interface in the

structure. It remains 0 till the valence band reaches the Fermi level and the 2DHG

is formed at 15 nm. After this the 2DHG starts increasing as the cap layer

thickness is increased and this density saturates as the thickness is further

increased.

49

4.7 Thick GaN-capped GaN/AlGaN/GaN structures

For increasing GaN cap layer thickness, the 2DEG and 2DHG densities

saturate after a certain distance. In this section, the GaN cap layer was chosen to

be thick enough so that both t he densities are saturated and the effect of varying

AlGaN barrier thickness is studied. The heterostructure used is shown in Figure

4.14.

GaN

200 nm

AlGaN x=0.30

GaN

Buffer layer

I Thic[<ness of the AlGaN layer is varied

Figure 4.15 Thick-capped GaN/AlGaN/GaN structure used in this simulation

The dependence of the sheet densities on the thickness of the AlGaN layer

is shown in Figure 4.15. As the thickness is increased the densities are also

increased, rapidly at first and then it saturates after a certain thickness. Since a

thick cap is used, the surface effects were removed from the two-dimensional

carrier gases.

50

16 1

14 •

S 12-u

M

o 10 • 1—1 1 '• J 6 -*

VI

2 -

n -0

/ • • •

'

200

2DEG

2DHG

400 600 800 1000

AlGaN layer tliickness (lun)

1200

Figure 4.16 Sheet density dependence on the AlGaN barrier thickness. Solid line represents 2DEG sheet density and the dotted line represents 2DHG density.

BF

1*-GaN

200 nm lOOnm

20 nm

Figure 4.17 Simulated band diagrams of Thick-capped GaN/AlGaN/GaN structure

For large GaN cap layer as shown in Figure 4.16, the bands of the GaN

layer on both sides are almost flat. Hence the magnitudes of the 2DEG and the

2DHG densities are almost same as shown in Figure 4.15.

51

4.8 InN/GaN Structures

Next, the InN/GaN structures were studied. Similar to section 4.1, an

InGaN layer of fixed alloy composition of x = 0.30 were used on top of a GaN

buffer layer and the effect of varying InGaN thickness on the sheet carrier density

were plotted. The structure used is given in Figure 4.18.

Thickness '. Vufied

Figure 4.18 A Single InGaN/GaN heterostructure

Initially, the thickness was kept at a very small value. At this point, similar

to AIGaN/GaN structure, the surface state was well below the Fermi level and as

the thickness was increased, the surface state reached the Fermi level at a critical

thickness and 2DEG density was formed. The critical thickness was found out to

be 5 nm. The dependence of the 2DEG density on InGaN layer thickness is

plotted in Figure 4.19.

52

14-

<r 12 -

a 10 •

1 '' "ttf 1£ J 4 •

O -3

^ -

/

/

/

(1 50 100 150 200

ImOiiN layer tMckneiss (nm)

250

Figure 4.19 Simulated values for In^fiaQ^N thickness Vs 2DEG density

The density is 0 for small values of thickness. But as the thickness is

increased, Ep - ED decreases. Eventually, the surface state reaches the Fermi level

and after this point, as the thickness is increased, the density starts increasing

rapidly at first and then it saturates at a value close to the polarization induced

charge.

The maximum sheet carrier concentration that can be obtained using an

InGaN/GaN with x=0.30 was found to be 12x10'^cm~^. For a sunilar AIGaN/GaN

structure we can get a sheet carrier concentiration of ISxlO'^cm"^. This shows that

a better sheet carrier concentiation can be obtained with AIGaN/GaN stinctiire

compared to its corresponding InGaN/GaN structure.

53

CHAPTER 5

CONCLUSIONS

In summary. Band structures of AIGaN/GaN and GaN/AlGaN/GaN

structures were simulated and studied. In the AIGaN/GaN structure a 2DEG was

formed when the thickness of the AlGaN barrier reaches the critical thickness and

the critical thickness of an undoped Al^^Ga^^N /GaN structure were found to be

3.5 nm. The dependence of the critical thickness on other alloy compositions was

also shown. A 2DHG was formed on the upper interface of the GaN/AlGaN/GaN

structure when a thick GaN cap layer was used.

The sheet carrier concentration of theses structures and its dependencies

on factors such as the alloy composition and the barrier thickness were also

plotted and studied. The 2DEG density increased rapidly at first and then saturates

near the polarization induced charge, as the AlGaN barrier thickness was

increased and this value for a Al^^Gaf^jN /GaN structure was found to be around

1.6xlO"cm"^and the corresponding sheet densities for other alloy compositions

were plotted. The dependencies of the 2DEG density on the alloy composition for

a 5 nm AlGaN barrier was plotted and it was found to have a linear relationship

with the alloy composition with slope 5.7xlO"cm"^ which was similar to

experimental values. The 2DEG density decreases as the thickness of the GaN cap

layer increases while the 2DHG increases with the increase in the thickness and

both saturate after a certain thickness. For a considerably large GaN cap layer

54

thickness, the 2DEG and the 2DHG densities were almost equal and found to be

increasing with the increase in the AlGaN barrier thickness. Also an AIGaN/GaN

structure is better than its corresponding InGaN/GaN structure in terms of sheet

carrier density.

The major irregularity during these simulations was the presence of very

small values of sheet density even when the thickness of the AlGaN layer

thickness was lesser than the critical thickness. A major drawback of using this

software was that the dependence of mobilities on sheet carrier density could not

be studied. Since these two are very important factors in determining the

conductance of a HFET, a software that could study this dependence is required in

the future.

55

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