GROUP III-NITRIDE HETEROSTRUCTURES
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
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
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
2.7 Quantum Microcavity 23
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
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
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:
stability.
• 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.
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 •
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:
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.,
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
£^=£^=(a-aj/a, (2.3)
^ 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:
C33
The calculated values of lattice constants and piezoelectric constants are
given in Table 2.1.
Crystal Structure
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.
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)
(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'
•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
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,
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'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.
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^
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.
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.
- 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(
...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)
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)
L~'BL-'Ly/ = L-'LLAy/ = AL'LLY (3.13)
or
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
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-
(3.18)
where
30
and
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
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
4*10'
4*10'^
250 10 1 1 1
35
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
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.
u.uut+uj - (
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.
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.
18 n
16 14
12 - 10 -
8 - 6 -
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
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
S 2- 1—1
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
</» c 8 •
* j
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
Thickness>Critical Thickness
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
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
t 6 •
IS ^ 2 •
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
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.
VI
2 -
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
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.
O -3
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
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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