Materials Sciences and Applicatio n, 2011, 2, 729-738
doi:10.4236/msa.2011.27101 Published Online July 2011 (
Copyright © 2011 SciRes. MSA
Structural and Electronic Properties Calculations
of AlxIn1–xP Alloy
Mohammed Ameri1*, Ali Bentouaf1, Mohammed Doui-Aici2, Rabah Khenata3,4, Fatima Boufadi1,
Amina Touia1
1Département de Physique, Faculté des Sciences, Université Djillali Liabès, Sidi-Bel-Abbés, Algérie; 2Laboratory Applied Materials
(AML), Research Center (Ex: CFTE), Route de Mascara, University of Sidi-Bel-Abbes, Algeria ; 3Laboratoire Quantum Physics and
Mathematical Modeling of Matter (M LPQ3), University de Mascara, Mascara, Algeria; 4Department of Physics and Astronomy,
Faculty of Science, King Saud University, Riyadh, Saudi Arabia.
Received December 23rd, 2010; revised March 21st, 2011; accepted May 3rd, 2011.
The equilibrium structure and the electronic properties of III-V zinc-blende AlP, InP semiconductors and their alloy
have been studied in detail from first-p rinciples calculations. A full-potential linea r muffin-tin-orbita l (FP-LMTO) me-
thod has been used in conjunction with both the local-density approximation (LDA) and the generalized-gradient ap-
proximation (GGA) to investigate the effect of increasing the concentration of aluminum on the structural properties
such as the lattice constants and the bulk moduli. Besides, we report the concentration dependence of the electronic
band structure, the direct-indirect band gap crosso vers and bowing. Using the approach of Zunger and co-workers the
microscopic origins of the gap bowing were also explained. A reasonable agreement is found in comparing our results
with other theoretica l calculations.
Keywords: AlP, InP, Semiconductors, FP-LMTO, Bowing, Alloys
1. Introduction
Understanding the electronic properties of semiconductor
alloys plays a vital role in developing new technologies.
The advantage of alloying is that the alloy properties,
such as band gap, can be tuned by varying the alloy
composition to meet the specific requirements of modern
device applications [1-3]. With the advent of small-
structure systems, such as quantum wells and superlat-
tices, the effects of alloy compositions, size, device ge-
ometry, doping and controlled lattice strain can be com-
bined to achieve maximum tenability [4].
AlxIn1-xP alloy provides wide bandgap energy in the
non-nitride III-V semiconductors and has been wide ap-
plied in electronic and photonic devices. The parent (bi-
nary) compounds such as aluminum phosphide (AlP) and
indium phosphide InP, are non-centrosymmetric cubic
semiconductors with zinc-blende structures based on the
space group F43m [5,6]. Recently, these compounds
have attracted a great deal of attention, [5-41] expecting
fabrication of important electronic devices. Indeed, InP is
a very promising material for solar cells and high-per-
formance computing and communications [7-9]. Simi-
larly, AlP, with the largest direct gap of the III-V com-
pound semiconductors, is undoubtedly the most “exotic”.
Usually, this material is alloyed with other binary mate-
rials for applications in electronic devices such as light-
emitting diodes (e.g. aluminium gallium indium phosphi-
de) [10].
Motivated by the technological importance of these
materials, III-phosphides have been the subject of vari-
ous theoretical investigations, from empirical [42] to first
principles based on the density functional theory (DFT)
[43,44]. Most of these studies have been undertaken us-
ing the pseudo-potential [45] or the full-potential lin-
earized-augmented plane wave (FP-LAPW) method
which considered to be one of the most accurate methods
for calculating the structural and the electronic properties
of solids, within the local density approximation (LDA)
[46] or the generalized gradient approximation (GGA)
[47]. Nevertheless, to the best of our knowledge, the (FP-
LMTO) method has not yet been used to study the struc-
tural and the electronic properties of AlxIn1–xP alloy.
Below, we report the results obtained in the study of
the variation of different structural and electronic pa-
rameters such as lattice constant, bulk modulus, band gap
and effective masses with the alloy fraction using the
Structural and Electronic Properties Calculations of Al In P Alloy
730 x1–x
(FP-LMTO) method. In our calculations, we have
adopted the “special quasirandom structures” (SQS) ap-
proach [48,49] which is based on the observation that
(for any given composition) atomic disorder mainly af-
fects the electronic properties of an alloy through the
short-range atomic structure. In fact, Zunger and co-
workers have introduced “SQS” approach by the princi-
ple of close reproduction of the perfectly random net-
work for the first few shells around a given site.
The paper is divided in three parts. In Section 2, we
briefly describe the computational techniques used in this
study. The most relevant results obtained for the ground-
state properties as well as the bandgap, optical bowing
and effective masse are presented and discussed in Sec-
tion 3. Finally, in Section 4 we summarize the main con-
clusions of our work.
2. Computational Details
The calculations reported here were carried out using the
ab-initio full-potential linear muffin-tin orbital (FP-
LMTO) method [50,51] as implemented in the Lmtart
code [52]. The exchange and correlation potential was
calculated using the local density approximation (LDA)
[46] and the generalized approximation (GGA) [47]. This
is an improved method compared to previous (LMTO)
methods. The FP-LMTO method treats muffin-tin
spheres and interstitial regions on the same footing,
leading to improvements in the precision of the eingen-
values. At the same time, the FP-LMTO method, in
which the space is divided into an interstitial regions (IR)
and non overlapping muffin-tin spheres (MTS) sur-
rounding the atomic sites, uses a more complete basis
than its predecessors. In the IR regions, the basis func-
tions are represented by Fourier series. Inside the MTS
spheres, the basis functions are represented in terms of
numerical solutions of the radial Schrödinger equation
for the spherical part of the potential multiplied by
spherical harmonics. The charge density and the potential
are represented inside the MTS by spherical harmonics
up to lmax = 6. The integrals over the Brillouin zone are
performed up to 35 special k-points for binary com-
pounds and 27 special k-points for the alloys in the irre-
ducible Brillouin zone (IBZ), using the Blöchl’s modi-
fied tetrahedron method [53]. The self-consistent calcu-
lations are considered to be converged when the total
energy of the system is stable within 10–5 Ry. In order to
avoid the overlap of atomic spheres the MTS radius for
each atomic position is taken to be different for each case.
Both the plane waves cut-off are varied to ensure the
total energy convergence. The values of the sphere radii
(MTS), number of plane waves (NPLW), used in our
calculation are summarized in Table 1.
3. Results and Discussions
3.1. Structural Parameters
To investigate the structural properties of AlP and InP
compounds and their alloys in the cubic structure, we have
started our FP-LMTO calculation with the zinc-blende
structure and let the calculation forces to move the atoms
to their equilibrium positions. We have chosen the basic
cubic cell as the unit cell. In the unit cell there are four C
anions , three A and one B, two A and two B, and one A
and three B cations, respectively, for x = 0.25, 0.50 and
0.75. For the considered structures, we perform the
structural optimization by calculating the total energies
for different volumes around the equilibrium cell volume
V0 of the binary AlP, InP compound and their alloy. The
Table 1. The plane wave number PW, energy cut-off (in Ry) and the muffin-tin radius (MTS) (in a.u.) used in calculation for
binary AlP and InP and their alloy in zinc ble nde (ZB) str uc ture.
PW Ecut total (Ry) MTS (a.u)
0 5064 12050 92.120 156.381 In 2.453 2.510
P 2.357 2.412
0.25 33400 65266 131.543 197.867 Al 2.395 2.411
In 2.395 2.411
P 2.348 2.939
0.50 33400 65266 136.726 205.941 Al 2.349 2.393
In 2.349 2.393
P 2.303 2.346
0.75 33400 65266 142.279 215.299 Al 2.280 2.340
In 2.280 2.340
P 2.280 2.294
1 5064 12050 105.387 184.707 Al 2.248 2.264
P 2.248 2.264
Copyright © 2011 SciRes. MSA
Structural and Electronic Properties Calculations of Al In P Alloy 731
calculated total energies are fitted to the Murnaghan’s
equation of state [54] to determine the ground state prop-
erties such as the equilibrium lattice constant a, and the
bulk modulus B. The calculated equilibrium parameters (a
and B) are given in Table 2 which also contains results of
previous calculations as well as the experimental data.
The lattice constants obtained within the LDA for the
parent binary system InP and AlP are respectively 0.17 %
and 0.03 % lower than the experimental value, while the
corresponding bulk modulus are 0.7% and 1.2% larger
than the experimental value, which is the usual level of
accuracy of the LDA. When comparing the results ob-
tained within GGA, the lattice constant are 2.6 % for InP
and 1.6 % for AlP larger than the experimental values and
the corresponding bulk modulus are 15.14% and 4.7%
smaller than the corresponding experimental values.
Hence it is safe to conclude that the LDA bulk modulus
and lattice constants is in fact in better agreement with the
experimental data than the GGA values. The calculated
bulks modulus using both approximation LDA and GGA
decreases in going from AlP to InP, suggesting the more
compressibility for InP compared to that for AlP.
Usually, in the treatment of alloys when the experi-
mental data are scare, it is assumed that the atoms are
located at the ideal lattice sites and the lattice constants
varies linearly with concentration x according to the
so-called Vegard’s law [55].
aABC xaxa
where AC and BC are the equilibrium lattice con-
stants of the binary compounds AC and BC respectively,
(AxB1–xC) is the alloy lattice constant. However, the
law was postulated on empirical evidence, several cases
of both positive and negative deviations from this law
have been documented [56,57]. Hence, it has been sug-
gested in the literature that the deviation from Vegard’s
law can be represented by a quadratic expression:
a a
 
1 1
aABC xaxaxxb
  (2)
where b, is the bowing parameter accounting for the de-
viation from linearity.
Figures 1 and 2, show the variation of the calculated
equilibrium lattice constants and the bulk modulus versus
concentration x for AlxIn1–xP alloy. Our calculated lattice
constants were found to vary almost linearly following
the Vegard’s law [55] with a marginal upward bowing
parameters equal to –0.07143 Å. In going from InP to
AlP, when the Al-content increases, the values of the
lattice parameters of AlxIn1–xP alloy decrease. This is due
to the fact that the size of the Al atom is smaller than that
of the In atom. Oppositely, one can see from Figure 2
that the value of the bulk modulus increases with the
increase of Al concentration. The deviation of the GGA
bulk modulus from the linear concentration dependence
with a downward bowing equal to +21.9259 GPa. The
bowing lattice parameters and the bulk modulus are
found to be equal to –0.168 Å and +13.9569 GPa by us-
ing LDA approximation. In view on Table 2, it is clear
Table 2. Computed lattice parameter a and bulk modulus B compared to experimental and other theoretical values of AlP
and InP and their alloy.
Lattice constant a(Ǻ) Bulk modulus B(GPa)
this work. exp. other calc. this work. exp. other calc.
0 5.8509 6.014 5.861b 5.942a,5.688c,5.869d71.5399 60.25 71b 68
,5.8686e ,5.8783g,5.729h,5.838l,73.60h ,71l ,62l
,5.968l ,5.729m,5.930n,74m ,76n ,76s
,5.6591o ,5.93s
0.25 5.7979 5.9092 73.2808 61.4585
0.5 5.687 5.7922 75.7209 65.5753
0.75 5.5749 5.6649 80.2789 72.4857
1 5.449 5.534 5.451e 5.471c ,5.462d ,5.44285g 87.067 81.89 86f 84.5
,5.467i ,5.41700h,5.508j,5.42k,81.52j ,86.5k ,89l
,5.520n ,5.4131o ,5.43p 90,46p ,90q ,88r
,5.40q ,5.48r
aRef. [11]; bRef. [12]; cRef. [13]; dRef. [14]; eRef. [17]; fRef. [18]; gRef. [19]; hRef. [20]; iRef. [23]; jRef. [24]; kRef. [25]; lRef. [26]; mRef. [27]; nRef. [29]; oRef.
[31]; pRef. [32]; qRef. [33]; rRef. [34]; sRef. [39].
Copyright © 2011 SciRes. MSA
Structural and Electronic Properties Calculations of Al In P Alloy
732 x1–x
Figure 1. Composition dependence of the calculated lattice
constants within GGA (solid circ le) and LDA (solid squares)
of AlxIn1–xP alloy compared with Vegard’s prediction (dot
Figure 2. Composition dependence of the calculated bulk
modulus within GGA (solid circle) and LDA (solid squares)
AlxIn1–xP alloy.
that the LDA yields higher values than the experiment
while GGA provides a good agreement.
3.2. Electronic Properties
The important features of the band structure (direct Γ-Γ
and indirect Γ-X band gaps) are given in Table 3. It is
clearly seen that the band gaps are on the whole underes-
timated in comparison with experiments results. This
underestimation of the band gaps is mainly due to the
fact that both the simple form of LDA or GGA do note
take into account the quasiparticle self energy correctly
[58] which make them not sufficiently flexible to accu-
rately reproduce both exchange and correlation energy
and its charge derivative. We worth also mention that in
general, it is far to say that the experimental data are well
reproduced by the calculation. On raison for this differ-
ence is that in our calculations we have assumed the
crystal to be at T = 0 K and thus do not include contribu-
tions from lattice vibrations that are present at room
temperature measurements. The calculated band gaps
for AlP compound and in good agreement with the
available theoretical results. This agreement disappears
for the case of InP compound. Figure 3 shows the plots
of the concentration variation of the direct gap (
and indirect gap (Γ-X) the studied alloys within both
LDA and GGA. Increasing Al content leads to a shift of
the conduction band (CB) upwards the Fermi energy
(EF) resulting an increase of the direct energy band gap
(. The calculated direct band gap values are 0.56
(0.26), 1.0 (0.83), 1.58 (1.45), 1.75 (1.74) and 3.36
(3.08) eV within of LDA (GGA) approach for x = 0.0,
0.25, 0.5, 0.75 and 1.0, respectively. The band structure
calculations in the present work yield a direct gap (ΓΓ)
for InP, while for AlP compound an indirect gap (Γ–X)
has been determined. Hence, one can expect that the
band gap of AlxIn1–xP alloys should undergo a crossover
between the direct and the indirect band in going from x
= 0 to x = 1. As shown in Figure 3, this crossover oc-
curs at x = 0.79 for LDA and at 0.82 for GGA. For both
approximation the predicted crossover value is twice
larger that those determined by Onton and co-authors
5.4 0.0 0.2 0.4 0.6 0.8 1.0
The calculated band gap versus concentrations was
fitted by a polynomial equation.
Ex xExEbxx
 (3)
where EAC and EBC corresponds to the of the AlP and InP
gaps for the AlxIn1xP alloy. The results are shown in
Figure 3 and are summarized as follows:
0.0 0.2 0.4 0.6 0.8 1.0
E0.680.0562.41 (LDA)
E=1.56+4.124.05Al InP
 
 
E12.96 4.011.39(GGA)
E1.73 3.813.69
 
It is clear from the above equations that the direct (Γ
Γ) and indirect (Γ X) band gaps versus concentra-
tion have a nonlinear behavior. The direct gap (Γ Γ)
has a downward bowing with a value of 1.396, while the
indirect gap (Γ X) has an upward bowing of –3.907.
These parameters are lower than those obtained using the
LDA (2.38 and –4.059).
The physical origins of gap bowing were investigated
following the approach of Zunger and co-workers [60],
which decompose it into three contributions. The overall
bowing coefficient at a given average composition x
measures the change in the band gap according to the
formal reaction
ACBCx xeq
 a
where and
a are the equilibrium lattice constant
Copyright © 2011 SciRes. MSA
Structural and Electronic Properties Calculations of Al In P Alloy 733
Table 3. Direct band gap energy of AlxIn1-xP alloys at different Al concentrations (all values are in eV).
Energy gap (eV) (Г-Г) Energy gap (eV) (Г-X)
this work. exp. other calc. this work. exp. other calc.
0 0.5647 0.2674 1.39
c, 1.39
a,1.98e,1.23f, 1.6479 1.8804 0.43g ,1.5522n, 2.19r
1.35d, 1.54g,1.67h ,0.62j
1.424l, ,0.85j,1.50j,1.232k,
1.350m 1.3831n,1.34r
0.25 1.0099 0.8315 2.3836 2.7014
0.5 1.5819 1.4578 2.3997 2.5271
0.75 1.578 1.742 2.5961 2.7014
1 3.3666 3.0881 3.63b , 3.44a,2.54e ,3.26f , 1.4658 1.6386 2.50b, 2.17
2.45d 2.55g ,3.073i, 3.3457n, ,2.52s ,1.44
3.11o ,3.62p ,3.073q ,2.500l ,1.4194n,1.41o,1.49t
aRef. [14]; bRef. [15]; cRef. [16]; dRef. [17]; eRef. [19]; fRef. [20]; gRef. [21]; hRef. [22]; iRef. [24]; jRef. [26]; kRef. [27]; lRef. [28]; mRef. [30]; nRef. [31]; oRef.
[32]; pRef. [36]; qRef. [37]; rRef. [38]; sRef. [23]; tRef. [35]
lattice constant of the alloy with the average composition
Figure 3. Energy band gap of AlxIn1–xP alloy as a function
of Al composition using LDA and GGA approximations.
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Figure 4. Calculated optical bowing parameter as a function
of composition vx within LDA (xCenter Square) and GGA
(-Center circle).
0.5 0.0 0.2 0.4 0.6 0.8 1.0
The Equation (5) is decomposed into three steps:
aaACaBCa (6)
 
 a (7)
1.0 The first step measures the volume deformation (VD)
effect on the bowing. The corresponding contributions
bVD to the bowing parameter represents the relative re-
sponse of the band structure of the binary compounds AC
and BC to hydrostatic pressure, which here arises from
the change of their individual equilibrium lattice con-
stants to the alloy value a = a(x). The second contribution,
0.0 0.0 0.2 0.4 0.6 0.8 1.0
Copyright © 2011 SciRes. MSA
Structural and Electronic Properties Calculations of Al In P Alloy
734 x1–x
the charge exchange (CE) contribution bCE, reflects the
charge transfer effect which is due to the different (aver-
aged) bonding behavior at the lattice constant a. The fi-
nal step measures changes due to the structural relaxation
(SR) in passing from the unrelaxed to the relaxed alloy
by bSR. Consequently, the total bowing parameter is de-
fined as
b = bVD + bCE + bSR. (9)
The general representation of the composition-depen-
dent band gap of the alloys in terms of binary compounds
gaps of the, EAC(aAC) and EBC(aBC), and the total gap
bowing parameter b is defined as:
Ex xEaxEabxx  (10)
This allows a division of the total gap bowing b into
three contributions according the following expressions:
EaEa EaEa
EaEaE a
All terms presented in Equations (11)-(13) are com-
puted separately via self-consistent band structure calcu-
lations. The different contributions to the gap bowing
were calculated using the LDA and the GGA schemes
and the results are given in Table 4. The calculated band
gap bowing coefficient for random AlxIn1–xP alloy ranges
from 1.3614 eV (x = 0.25) to 5.8396 (x = 0.75). Our re-
sult for x = 0.5 is higher than the experimental one and is
in excellent agreement with those obtained by Ferhat and
co-authors [61] using the full potential linearized aug-
mented plane wave. One can note that for x = 0.25 and
0.50 the main contribution to the gap bowing is due to
the volume deformation (VD) effect. The importance of
bVD can be correlated with the mismatch of the lattice
constants of the corresponding binary compounds. Con-
sequently, the main contribution to the gap bowing is
raised from the volume deformation effect. In the case of
x = 0.75, the contribution of the charge transfer bCE has
been found greater than those of the volume deformation
bVD. This contribution is due to the different electronega-
tivities of the In and Al or P atoms. Indeed, bCE scales with
the electronegativity mismatch. The contribution of the
structural relaxation is negligible and the band gap bowing
is due essentially to the charge exchange effect. Finally, it
is clearly seen that our LDA values for bowing parameters
are larger than the corresponding values within GGA.
3.3. Calculated Effective Masses
The knowledge of the electron and hole effective mass
values is indispensable for the understanding of transport
phenomena, exciton effects and electro-hole in semicon-
ductors. Therefore, it would be of much interest to de-
termine the electron and hole effective mass values for
the alloys for various Al content. We have computed the
electron effective mass at the conduction band minima
(CBM) and hole effecRtive mass at the valence band
maxima (VBM) for the studied alloy. The electron and
hole effective masses values are obtained from the cur-
vature of the energy band near the Γ-point at the CBM
and VBM for all concentration. At the Г-point the s-like
conduction band effective mass can be obtained through
a simple parabolic fit using the definition of the effective
mass as the second derivative of the energy band with
respect to the wave vector, k, via:
m* /m0 =– (ћ2/m0)·1/(d2E/d k2) (14)
Table 4. Decomposition of optical bowing into volume deformation (VD), charge exchange (CE) and structural relaxation (SR)
contributions (all values in eV).
this work. exp other cal.
0.25 bVD 0.6354 0.6873
CE –0.0756 –0.1583
SR 0.8016 0.2234
b 1.3614 0.7524
0.5 bVD –0.039 0.7376 0.740b
CE 0.986 –0.09087 0.265b
SR –0.1312 0.2333 –0.172b
b 0.815 0.880 0.38a, 0.568a 0.834b
0.75 bVD 0.7286 0.7941
CE 5.2214 2.6625
SR –0.1104 –0.0399
b 5.8396 3.4167
aRef. [40]; bRef. [61]
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Structural and Electronic Properties Calculations of Al In P Alloy735
Table 5. Electron (), light hole () and heavy hole () effective mass (in units of free electron mass m0) of the ternary
alloys under investigation compared with the available experimental and theoretical predictions.
m *
m *
this work. other calc. this work. other calc. this work. other calc.
0 0.004 0.032 0.095a, 0.058a 0.351 0.603 0.389a, 0.477a 0.098 0.103 0.093a, 0.057a
b, 0.060b 0.430
b, 0.400b 0.097
a, 0.078a
b, 0.081b 0.47
b, 0.895a 0.104
b, 0.118b
b,0.610b 0.052
a, 0.074a
b, 0.90b 0.051
b, 0.950b
0.25 0.0042 0.047 0.804 0.770 0.219 0.367
0.50 0.032 0.042 4.126 0.7648 4.214 0.2602
0.75 0.0427 0.0428 0.857 0.8448 0.269 0.2870
1 0.167 0.024 0.176a, 0.170a 0.462 6.194 0.489a, 0.509a 0.159 0.182 0.187a, 0.181a
aRef. [31]; bRef. [41]; cRef. [62]
where m* is the conduction electron effective mass and
m0 is the free electron mass. We can calculate the curva-
ture of the valence band maximum using the following
approach: if the spin-orbit interaction were neglected, the
top of the valence band would have a parabolic behavior;
this implies that the highest valence bands are parabolic
in the vicinity of the Г-point. In this work, all the studied
systems satisfy this parabolic condition of the valence
band maximum at the Г-point. Within this approach, and
by using the appropriate expression of Equation (14) (us-
ing a plus sign instead of the minus sign in the prefactor),
we have computed the effective masses of the heavy and
light holes at the Г-point. The calculated electron and hole
effective mass values for the parent binary compounds InP
and AlP and their alloy are given in Table 5. Results from
earlier theoretical works are also quoted for comparison.
Our results for the binary compounds are in fairly good
agreement with the available theoretical data. We would
like mentioning here that the divergence of some values
should be expected since the computation of the effective
mass is very sensitive to the form of the energy band.
The highest curvature of the electronic band yields the
smallest effective mass of the charge carriers and the
highest conductivity. From Table 5 data, we can outline
that holes are much heavier than electrons, for all con-
centrations in AlxIn1–xP alloy, so carrier transport in this
alloy should be dominated by electrons.
4. Conclusions
We have performed first-principles calculations using
(FP-LMTO) method within the LDA and GGA for the
zinc-blende AlxIn1-xP alloy (x = 0.0; 0.25; 0.50; 0.75;
1.00). We have found that lattice parameter follows Ve-
gard’s law, the bulk modulus varies significantly with the
composition x and the electronic band structure has a
nonlinear dependence on the composition. We have cha-
racterized the deviation from the linear behavior by cal-
culating the optical bowing parameter. The main contri-
bution to the total bowing parameter arises from struc-
tural (volume deformation) and chemical effects. The
computed effective masses of the systems studied are
found comparable to those reported in literature. Our
results provide an estimate of this important compound.
5. Acknowledgements
The author Rabah KHENATA extends his appreciation
to the Deanship of Scientific Research at King Saud Uni-
versity for funding the work through the research group
project No RGP-VPP-088.
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