International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 5, May 2013
ISSN 2319 - 4847
Electronic structure of InxGa1-xAs nanocrystals
alloy using Ab-initio Density functional theory
calculation coupled with LUC method
Mohammed T. Hussien1, Akram H. Taha2 and Thekra Kasim3
1&3
2
Department of Physics, College of Science, University of Baghdad
Department of Physics, Faculty of Science & Health, Koya University,
ABSTRACT
Ab-initio Density function theory (DFT) method coupled with Large Unit Cell (LUC) method has been used to simulate
electronic structure of InxGa1-xAs nanocrystals alloy for three Indium concentrations x=0.25, 0.50, and 0.75. Gaussian 03
package has been used throughout this study to calculate some of the physical properties such as energy gap , lattice constant ,
valence and conduction band as well as density of state . Results show that the lattice constant increases with the increasing in
the Indium concentration in the alloy, while the energy gap deceases due to the size effect. The total energy, cohesive energy,
electron affinity and ionization as well as ionicity for these concentrations have been reported.
Keywords: InXGa1-XAs nanocrystals, Ab-initio (DFT) , (LUC)
1. INTRODUCTION
A nanomaterial is a term that includes all nanosized materials. Nanomaterials semiconducting materials have been a
subject of intense study for last several years due to their size dependent physical and chemical properties [1-4].InGaAs
belongs to the InGaAsP quaternary system that consists of indium arsenide (InAs), gallium arsenide (GaAs), and
gallium phosphate (GaP). These binary materials and their alloys are all III-V compound semiconductors. A
comprehensive of band parameters for the technologically important III-V zinc blende compound semiconductor is
given in ref. [5].
In general III-V semiconductors provide the materials basis for a number of well-established commercial technologies.
Energy gap and whether the bandgap is “direct” or ”Indirect” is the main factor that gives semiconductors the optical,
electrical, and magnetic properties. The energy bandgap of the quaternary system (InGaAsP) range from 0.33eV
(InAs) to 2.25eV (GaP), with InP (1.29eV) and GaAs (1.43 eV) falling between. A semiconductor will only detect light
with photon energy larger than the band gap, in other word, with the wavelength shorter than the cutoff wave length
associated with the band gap. This wavelength cutoff works out to 3.75 µm for InAs and 0.55 µm for GaP with InP at
0.96 µm and at 0.87 µm for GaAs. The progress made in physics and technology of semiconductors depends mainly on
two families of materials, the group IV elements and the III-V compounds. For all of the ternary alloys, such as
InGaAs, the dependence of the energy gap on alloy composition is assumed to fit the quadratic form [6]
E g ( A1 X B X C )  (1  x) Eg ( AC )  [ E g ( BC )  b]  bx 2
The value of (b) is a measure of the deviation from the linear interpolation curve and is often referred to be as the
bowing parameter.
InxGa1-xAs alloy has been of great interest for short wave infrared (SWIR) detector applications [7]. The SWIR InGaAs
cameras are a good complement to thermal imaging cameras, SWIR cameras can be used to identify and recognize
objects in cooler environment than those cameras which detect only warm bodies (cars, people, etc...) [8]. Indium
gallium arsenide (InGaAs) is used in high-power and high-frequency electronics [9] because of its superior electron
velocity with respect to the more common semiconductors silicon and gallium arsenide. InAs single and multiple
quantum dot (QD) layers have been explored for their potential use in the implementation of GaAs-based optical
devices such as Lasers, semiconductor optical amplifiers (SOAs) and saturable absorber mirrors operating at
telecommunication wavelengths (1300-1550nm) which is the same InGaAs band gap, also makes it
the detector material of choice in optical fiber communication at those mentioned wavelengths [10-13].
2. THEORY
One of the most important goals of physics is to describe the physical properties of interacting many-particle systems.
This can be done by deriving the properties of many-particle systems from the quantum mechanical laws this requires
the solution of the Schrödinger equation in 3N spatial variables and N spin variables. Solutions of the Schrödinger
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 5, May 2013
ISSN 2319 - 4847
equation yield the electron state and their characteristic for a given arrangement of the atoms [14]. The non-relativistic
Hamiltonian for a system consisting n electrons and N nuclei is given by (all equations are expressed in atomic units)

1 N 2 1
H   I 
2 I
2
n
ZI N ZI Z J
1


i   
rIJ
i
I rIi
I J
i  j rij
n
n
N
2
i
(1)
Where the lower case indexes the electron and the upper case indexes the nuclei. r is a distance between the objects
specified by the subscripts.
After applying the Born-Oppenheimer approximation, the Hamiltonian is obtained and neglecting relativity is

1
H el  
2
n
n
N
  i2  
i
i
I
n
ZI
1

rIi i j rij
(2)
Now the expectation value of Hamiltonian operator due to the Hartree-Fock representation is given by
N
N
1
E HF   HF Hˆ  HF   H i   ( J ij  K ij )
2 i , j 1
i 1
Where H i defines the contribution due to the kinetic energy and the electron-nucleus attraction i.e
  1

 
H i   i* ( x )   2 Veff ( x )  i ( x ) dx
 2

the coulomb integrals J ij is given by
1 * 



 
J ij    i ( x1 )  i* ( x1 )
 j ( x 2 ) j ( x 2 ) dx1 dx 2
r12
 1


 
* 
And the exchange integrals K ij is K ij    i ( x1 )  j ( x1 )
 i ( x 2 ) *j ( x 2 ) dx1 dx 2
r12
All these integrals are real, and J ij  K ij  0 .
The unrestricted Hartree-Fock is based upon the use of a single determinant total wavefunction in which orbitals of the
same n, L, and mL values but different ms. values are regarded as being independent [15]. This method is the most
common molecular orbitals method for open shell molecules where the number of electrons of each spin is not equal. It
is an extremely convenient method because it is a very natural extension of the conventional Hartree-Fock method, and
the many-electron wave function is written in the form of a single determinant so that calculations are relatively easy.
However, because the unrestricted Hartree-Fock method allows the α and β spins to have different wave functions the
total wave function obtained is not an eigenfunction of S2 and hence the validity of the method must be questioned. It
has been suggested that [16-18] this objection may be overcome by determining the one-electron functions by the
unrestricted Hartree-Fock method and then eliminating by means of the appropriate projection operators those parts of
the function corresponding to the unwanted values of S2.
Density functional theory predicts a great variety of molecular properties: molecular structure, atomization energies,
vibrational frequencies, electrical and magnetic properties as well as ionization energies….etc. The central quantity in
DFT is the electron density ρ(r), it is defined as the integral over the spin coordinates of all electrons and over all but
one of the spatial variables (x=r,s)

 
 2


 (r )  N  ......  ( x1 , x 2 , ....., x N ) ds1 dx 2 ......dx N
(3)
Kohn and Sham introduced the following separation of the functional F[ρ]
F [  ]  Ts [  ]  J [  ]  E xc [  ]
(4)
Where Ts do not equal to the true kinetic energy of the system, and Exc is the exchange-correlation energy and can be
defined through equation (4) as
E xc [  ]  T [  ]  Ts [  ] Eee [  ]  J [  ]
(5)
. Then the energy of the interacting system will be
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E[  ]  Ts [  ]  J [  ]  E xc [  ]  E Ne [  ]

1  (r1 )  ( r2 )  
 
 Ts [  ]  
dr1dr2  E xc [  ]   V Ne  (r ) dr
2
r12
(6)
N

1
1
 2 1
 2  
  i  2  i  2    i (r1 ) r  i (r2 ) dr1 dr2 
2 i
12
N
N
E xc [  ]    
i
A
ZA
 2 
 i (r ) dr1
r1A
The generalized gradient approximation (GGA) coupled with large unit cell method is used to evaluate the electronic
structure of InxGa1-xAs.
3.RESULTS AND DISCUSSION
The stoichiometry of InxGa1-xAs using geometrical optimization is shown in figure (1,2) for 16 and 64 core atoms. It is
found that all our samples with all concentrations x(=0.25, 0.50, and 0.75) have the prralelpiped shape for 16 and 54
core atoms, while for 8 and 64 core atoms the shape was cubic (primitive cell).
The lattice constant optimization of 16 and 64 core atoms is illustrated in fig. (3, 4) for one concentration only
In0.5Ga0.5As noting that we found lattice optimization for them but we will just review the optimization for only
In0.5Ga0.5As. The results for the value of lattice constant for the different concentration is in good agreement with Ref.
[19].
The total energy versus number of core atoms for the concentrations x=0.25, 0.50, and 0.75 is shown in figures (5). The
linearity is the main characteristic for all concentrations. The energy decrease as the number of core atom increase. For
each concentration the total energy decreases with the increasing of the number of core atoms [20].
Density of states for InxGa1-xAs core atoms 16 and 64 and for the concentrations x=0.50 are shown in figures (6,7).
The density of state increases with the increasing of the number of core atoms which is due to the increasing of the
number of orbital hence the degenerates of energy state. The dense of line become more with increasing the number of
atoms.
The density of states as a function of number of core atoms is shown in figure (8) for the three concentrations of
In25Ga75As, In50Ga50As and In75Ga25As respectively. The increase in the density of states is obvious as the number of
core atom increase for each concentration. On the other hand as the concentration of indium in the alloy increases the
density of states increases also. The variation of bands width (conduction & valence) with the number of core atoms
and the concentration is shown in fig.(9,10).
Finally the dependency between the energy gap and the concentration is illustrated in fig. (11). The effect of changing
of the concentration of Indium in the InxGa1-xAs alloy on the energy gap is shown in fig. (3.52). Three concentration
has been taken x=0.25, 0.50, and 0.75 the energy gap is decreasing with increasing In concentration in the alloy due to
the occurrence of exciton in the band gap which give rise to a new energy level to be there in the “forbidden region”.
When the size of the nanoparticle approaches that of an exciton, size quantization occurs. Each band has a width that
reflects the interaction between atoms, with a bandgap between the conduction and the valence bands that reflects the
original separation of the bonding and antibonding states. When the size of the nanoparticle approaches that of an
exciton, size quantization occurs.
Conclusions:
Using Ab-initio density functional theory (DFT) coupled with large unit cell (LUC) approximation to estimate the
electronic properties of InXGa1-XAs alloy give the main conclusions summarized as follow:
1. The total energy and cohesive energy depends on the number of core atoms , it decreases with increasing the number
of core atom. Also the total energy decrease with increasing the indium concentration in InXGa1-XAs alloy.
2. The energy gap varies with the indium concentration in the InXGa1-XAs alloy as well as the number of core atoms.
3. The valence and conduction bands decreases with increasing the number of core atoms.
4. The results show a strong shape that 8 and 64 core atoms have a cubic structure (Bravais cells) while (16 and54) core
atoms have a parallelepiped structure (primitive cell).
5. The density of states become more intense when the number of core atoms increase also the degeneracy increase.
Figure (1): 16 In0.5Ga0.5As core atoms
Volume 2, Issue 5, May 2013
Figure (2): 64 In0.5Ga0.5As core atoms
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Figure (3): Total energy Vs lattice constant for 16 atom of In 50Ga50As core nanocrystal
Figure (4): Total energy Vs lattice constant for 64 atom of In 50Ga50As core nanocrystal
Figure (5): The total energy as a function of the number of core atoms for different concentrations
x= 0.25, 0.50, and 0.75
Figure (6): Density of states of 16 core atoms for In50Ga50As. The bold lines represent the conduction
band while the ordinary lines represent the valence band, Eg=1.353 eV
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Figure (7): Density of states of 64 core atoms for In50Ga50As. The bold lines represent the conduction
band while the ordinary lines represent the valence band, Eg=1.45 eV
Figure (8): The density of states vs the number of core atoms for In 25Ga75As,
In50Ga50As and In75Ga25As
Figure (9): The relationship between the band width (conduction and valence) with the number of core atoms of
In50Ga50As nanocrystals
Figure (10): variation of band width (conduction & valence) with the concentration x=0.0, 0.25, 0.50, 0.75, and 1.0 of
InXGa1-XAs
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Figure(11): The energy gap versus the number of core atoms for different concentrations of In, 25, 50, and 75%.
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Volume 2, Issue 5, May 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 5, May 2013
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AUTHOR
Dr.Mohammed T.Hussein completed his Ph.D. at the physics department in laser spectroscopy from
Complutense University – Madrid-Spain in 1995. His research interests lie in the field of organic
semiconductor and molecular spectroscopy. He is currently a member of the Nanotechnology &
Optoelectronics Research Group at the Physics department of Baghdad University.
Akram H.Tahacompleted his M.Sc. in solid state physics/ Yarmouk university (HKJ), His work Assistant
lecture at university of science and technology then lecturer at Sau's university (Republik of Ymen) and
lecturer at Faculty of science and health.-physics dep-koya university.
Thekra Kasim her M.Sc. Solid state Physics/ University of Baghdad , her work Assist. Lecturer at physics
department /college of science / University of Baghdad then lecturer at physics department./University of
Baghdad, now Ph.D. Postgraduate student at physics department, college of science /University of Baghdad
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