Electronic Band Structures of the Highly DesirableIII–V Semiconductors: TB-mBJ DFT Studies

GUL REHMAN,1,2 M. SHAFIQ,1,2 SAIFULLAH,1,2 RASHID AHMAD,3

S. JALALI-ASADABADI,4 M. MAQBOOL,5 IMAD KHAN,1,2

H. RAHNAMAYE-ALIABAD,6 and IFTIKHAR AHMAD1,2,7

1.—Center for Computational Materials Science, University of Malakand, Chakdara, Pakistan.2.—Department of Physics, University of Malakand, Chakdara, Pakistan. 3.—Department ofChemistry, University of Malakand, Chakdara, Pakistan. 4.—Department of Physics, Faculty ofScience, University of Isfahan (UI), Hezar Gerib Avenue, Isfahan 81746-73441, Iran. 5.—Departmentof Physics and Astronomy, Ball State University, Muncie, IN 47306, USA. 6.—Department of Physics,Hakim Sabzevari University, Sabzevar, Iran. 7.—e-mail: ahma5532@gmail.com

The correct band gaps of semiconductors are highly desirable for their effec-tive use in optoelectronic and other photonic devices. However, the experi-mental and theoretical results of the exact band gaps are quite challengingand sometimes tricky. In this article, we explore the electronic band structuresof the highly desirable optical materials, III–V semiconductors. The mainreason of the ineffectiveness of the theoretical band gaps of these compoundsis their mixed bonding character, where large proportions of electrons resideoutside atomic spheres in the intestinal regions, which are challenging forproper theoretical treatment. In this article, the band gaps of the compoundsare revisited and successfully reproduced by properly treating the density ofelectrons using the recently developed non-regular Tran and Blaha’s modifiedBecke–Johnson (nTB-mBJ) approach. This study additionally suggests thatthis theoretical scheme could also be useful for the band gap engineering of theIII–V semiconductors. Furthermore, the optical properties of these compoundsare also calculated and compared with the experimental results.

Key words: III–V semiconductors, optical materials, electronic bandstructure, non-regular TB-mBJ

INTRODUCTION

Semiconductor-based technologies are playingleading roles in the development of human civiliza-tion. The III–V compound semiconductor family isone of the most commonly used groups of semicon-ductors due to their unique nature of a wide range ofdirect band gaps and high temperature stability.These unique properties make III–V semiconductorsextremely useful in electronic, optoelectronic andphotonic devices like transistors, photodetectors fromUV to far-IR, phosphor materials in optical displays,satellite receivers, digital versatile disks (DVDs) andcell phones. Most III–V semiconductors exist in thezinc-blende structure, but the III-nitrides are found

in thewurtzite phase; however, theirmetastable zinc-blende structures have also been reported.1

Althoughdensity function theory (DFT)-basedmeth-ods are appropriate techniques to investigate theelectronic band structures of semiconductors and insu-lators; however, each DFT approximation has somelimitations. Therefore, knowledge of the various DFTapproximations is essential for the reproduction of theexperimental band gaps of semiconductors and insula-tors. The band gap of a semiconductor is a keyparameter and itsminor change can significantly affectthe application of the compound in optoelectronic orthermoelectric devices.2 Hence, the understanding ofthe nature of the band gaps of the existing materialsand their exact values whether by experiments orcalculations are not only crucial for their technologicalapplications but also for their band gap tailoring.(Received August 13, 2015; accepted March 24, 2016;

published online May 5, 2016)

Journal of ELECTRONIC MATERIALS, Vol. 45, No. 7, 2016

DOI: 10.1007/s11664-016-4492-7! 2016 The Minerals, Metals & Materials Society

3314

In this article, we review the electronic bandstructures of the III–V semiconductors reported bydifferent density functional theory approaches likelocal density approximation (LDA), generalized gra-dient approximation (GGA), screen exchange LDA(sX-LDA), hybrid functional of Heyd–Scuseria–Ernz-erhof (HSE) and Green’s function (GW). Comparisonof the calculated band gaps with the experimentalresults show that every theoretical approximation issuccessful for some compounds up to a certain rangeand accuracy but none of these methods is effective inreproducing the precise experimental bandgaps of theentire family members of III–V semiconductors. Toresolve the issue of the theoretical band gaps of thesecompounds, we have revisited the problem with therecently proposedDFT-based Tran andBlaha3 orbitalindependent exchange semilocal potential which isobtained by modifying the Becke and Johnson (BJ)4

exchange potential called TB-mBJ. This exchangepotential overcomes some of the inherent limitationsof the DFT5,6 in the calculations of the electronic bandstructure of solids. The advantages and limitations ofthe TB-mBJ exchange potential are tested by calcu-lating the electric field gradients, magnetic moments,and band gaps for a variety of compounds.7 Theapproach was further improved by the reparameter-ization of the correction c-factor in the TB-mBJscheme called the non-regular TB-mBJ (nTB-mBJ)method,8 which can achieve band gaps for semicon-ductors as accurate as experimental band gaps. In thepresent studies, these techniques are used to repro-duce the correct electronicbandstructuresof the III–Vsemiconductors. Furthermore, the optical propertiesof these semiconductors are also calculated and com-pared with the experimental results.

COMPUTATIONAL DETAILS

In this work, all the calculations are performed self-consistently within the DFT5,6 by employing the aug-mented plane waves plus local orbitals (APW + lo)scheme with the exchange–correlation potential ofPBE-GGA9 and the exchange potential of TB-mBJ3,8

as embedded in the WIEN2k code.10 The TB-mBJexchange potential is defined as:

vTB!mBJx;r ðrÞ ¼ 1:023

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

X

Z

X

rqrðr0Þj jqrðr0Þ

dr0

s

! 0:012

!

%2ffiffiffiffiffiffiffiffiffiffiffiffiffiffipqrðrÞ3

pexrðrÞ=3

xrðrÞ

(

% 1þ 1

2xrðrÞ

" #e!xrðrÞ ! 1

$ %

þ 3

p

ffiffiffiffiffiffi5

12

r PNri¼1 rw'

i;r (rwi;rPNr

i¼1 wi;r

&& &&2

!12

9=

;

! 2

p

ffiffiffiffiffiffi5

12

r PNri¼1 rw'

i;r (rwi;rPNr

i¼1 wi;r

&& &&2

!12

ð1Þ

¼ c% vBJx;rðrÞ þ1

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10trðrÞ3qrðrÞ

s !

! 1

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10trðrÞ3qrðrÞ

s

ð2Þ

where X is the unit cell volume, qr(r) is the electron

charge density qrðrÞ ¼PNr

i¼1 wi;r

&& &&2' (

, xr is related to

the topology of the electron charge density and can beobtained from qr(r), rqr and r2qr are the gradientand Laplacian of the electron charge density, andtrðrÞ ¼ 1

2

PNri¼1 rw'

i;r (rwi;r is the electron’s kinetic-

energy density. The vTB!mBJx;r rð Þ exchange potential3

is represented versus the vBJx;r rð Þ exchange potential4

as shown in Eq. 1, where c is the expression given inthe first parenthesis of Eq. 1. Therefore, if c = 1.0,then, vTB!mBJ

x;r rð Þ ¼ vBJx;r rð Þ, consistent with Ref. 3.This clarifies that LDA or GGA calculations must beperformed to generate initial exchange-correlationenergy and the exchange energy functional Ex isfurther used to obtain TB-mBJ exchange potential,vTB!mBJx;r 6¼ dEx=dqr.

3

According to Eq. 1, the c-factor is linearly depen-dent on the square root of the average of jrqr =j qr. Inthe regular TB-mBJ calculations, the correction c-factor is allowed to converge self-consistently, asintroduced in the TB-mBJ scheme. Usually, the c-factor in the TB-mBJ is small and therefore itprovides smaller band gaps than the experimentalvalues for semiconductors. As the band gap of amaterial depends on its electron density, where thisdensity can be properly manipulated by the c-factor,therefore the electron density is optimized by opti-mizing the c-factor for every compound, and thisapproach of TB-mBJ calculations performed withthe optimized c-factor (copt) is called non-regularTB-mBJ (nTB-mBJ) calculations.

The input parameters, i.e., the Monkhorst–Packgrid for integrations in the irreducible Brillouinzone of the unit cell, muffin-tin radii, copt, separa-tion energy, cutoff values of the expansions of thewave functions (RMTKMax) and charge densities(GMax) used in the present calculations are providedin Table I. The convergence is ensured for less than1 mRy/Bohr on the exerted forces.

RESULTS AND DISCUSSIONS

The III–V semiconductors family covers a broadrange of band gaps from far infrared to ultraviolet,where most of these compounds have a direct bandgap nature. Furthermore, their band gaps can beengineered for specific applications. The optoelec-tronic applications are generally concerned withthe direct band gap nature of a semiconductoralong with its width, while narrow band gapsemiconductors are attractive materials for ther-moelectric applications. Hence, the band gap of acompound plays a key role in its applications inoptoelectronic, photonic and thermoelectricdevices. Therefore, it is essential to know the exact

Electronic Band Structures of the Highly Desirable III–V Semiconductors: TB-mBJ DFTStudies

3315

value of the band gap of an optical material for itsefficient use in high-tech electronic, photonic andoptoelectronic devices. The III–V semiconductorsfamily, especially III-nitrides, is leading the opto-electronic industry. The experimental band gaps ofthese semiconductors are summarized in Table II.As most of these semiconductors have a direct bandgap nature and direct band gap materials areoptically active as compared to indirect band gapmaterials which are optically inactive, so the III–Vfamily especially nitrides are leading the optoelec-tronic industry.

The calculated band gaps for these semiconduc-tors with different DFT approaches are summarizedand compared with the experimental results inTable II. The table clearly indicates that the LDAand GGA results except for few compounds are quiteunderestimated from the experimental values; forsome compounds the error is more than 50%.However, the calculated band gaps with the hybridfunctional HSE, GW and sX-LDA approaches arebetter than LDA and GGA but the comparison withthe experimental results reveals that for somecompounds by each method the error is larger than

Table I. Computational parameters

Compoundk-pointsmeshes

RMTKMAX

(Bohr 3 Ry1/2)Separationenergy (Ry)

GMAX

(Ry1/2)

Muffin-tinradii (Bohr)

c-factorIII V

BN 10 9 10 9 10 7 6 12 1.36 1.50 1.3159BP 10 9 10 9 10 7 6 12 1.75 1.83 1.1926BAs 10 9 10 9 10 7 6 12 1.65 2.12 1.1751BSb 10 9 10 9 10 7 6 12 1.76 2.38 1.1620AlN 12 9 12 9 6 7 6 12 1.82 1.73 1.3221AlP 10 9 10 9 10 7 6 12 2.26 2.05 1.1703AlAs 10 9 10 9 10 7 6 12 2.12 2.34 1.1835AlSb 10 9 10 9 10 7 6 12 2.24 2.5 1.1567GaN 12 9 12 9 6 7 6 12 1.91 1.64 1.3473GaP 10 9 10 9 10 7 6 12 2.44 2.00 1.2169GaAs 10 9 10 9 10 7 6 12 2.23 2.23 1.2277GaSb 10 9 10 9 10 7 6 12 2.34 2.46 1.2023InN 12 9 12 9 6 7 6 12 2.16 1.77 1.3137InP 10 9 10 9 10 7 6 12 2.50 2.03 1.2141InAs 10 9 10 9 10 7 6 12 2.45 2.33 1.2229InSb 10 9 10 9 10 7 6 12 2.5 2.5 1.2013

Table II. Theoretical and experimental fundamental band gaps (in eV) for III–V semiconductors (ZB and WZ)obtained by different methods

Compounds Type LDA sX-LDA18 GGA GW HSE TB-mBJ* nTB-mBJ* Expt.

BN I 4.393 n/a 5.6030 7.1416 5.9023 5.8 6.22 6.253

BP I 1.1921 n/a 1.2520 n/a 2.1223 1.8 2.38 2.428

BAs I 1.2321 n/a 1.2320 n/a 1.8923 1.62 0.67 0.6713

BSb I 0.828 n/a 0.7520 n/a 1.3123 1.08 0.51 0.5132

AlN D 4.2814 5.52 5.5031 6.025 5.8124 5.63 6.16 6.221

AlP I 3.0614 2.36 1.5722 2.1515 2.5124 2.29 2.49 2.5113

AlAs I 1.7514 2.17 1.8629 2.7915 2.1623 2.12 2.25 2.313

AlSb I 1.2414 1.91 1.2127 2.2615 1.8523 1.78 1.65 1.6813

GaN D 1.7214 3.26 3.2026 3.1025 3.1424 3.2 3.48 3.513

GaP I 1.5014 2.43 1.6422 2.5615 2.4223 2.2 2.32 2.3513

GaAs D 0.303 1.19 0.3329 1.7517 1.1823 1.6 1.51 1.5213

GaSb D 0.3814 0.67 0.0327 0.7415 0.7023 1.0 0.82 0.8213

InN D 0.4814 n/a 0.9030 0.0115 0.7128 0.85 0.65 0.6921

InP D 0.3714 1.43 0.6822 1.4419 1.623 1.59 1.42 1.4228

InAs D 0.6414 0.45 0.028 0.3215 0.3623 0.66 0.42 0.4213

InSb D 0.7014 0.22 0.0027 0.3215 0.2823 0.48 0.24 0.2413

TB-mBJ* and nTB-mBJ* (Present)I Indirect, D Direct).

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10%; where the error is more than 10% is notacceptable in any experimental or theoretical stud-ies. The table clearly demonstrates that the calcu-lated band gaps through HSE functional areunderestimated for GaSb, InAs, BP and GaAs by14.6%, 14%, 11.6% and 22%, while overestimatedfor BAs, AlSb and InP by 50%, 10% and 13%,respectively. Similarly, the calculated band gaps bythe self-consistent GW method are underestimatedfor GaSb and InN by 9.7% and 98.5%, whereas theyare overestimated for InAs, InSb and BN by 23.8%,33% and 12.5%, respectively. The table also revealsthat the calculated band gaps for some of the III–Vsemiconductors by the screened exchange sX-LDAmethod are consistent with the experimental resultsbut are not available (n/a) for BN, BP, BAs, BSb andInAs compounds, while for AlSb the result is 13.6%overestimated and for AlN is about 10% underesti-mated. These results clearly indicate that the sX-LDA which is used to improve accuracy in the bandgap calculation, mixes the Thomas–Fermi screenedHartree–Fock exchange into the LDA11,12 is also notsuccessful in all the III–V compounds. Summarizingthe above discussion, we can say that each of theabove-mentioned theoretical approaches are suc-cessful for some compounds but not successful forother compounds. It needs to be mentioned herethat the GW method is computationally very expen-sive as compared to the other methods.

As mentioned in the computational details, thedistribution of the electronic charge density plays akey role in the band gap of a compound, whereas thedensity distribution depends upon the bondingnature. To display the bonding character in theIII–V semiconductors, the contour plots of theelectron charge density of aluminum nitride in the

(001) plane and boron nitride in the (111) plane areshown in Fig. 1. It is clear from the figure that theelectrons are not limited to their atomic spheres andtheir reasonable proportion resides in the intersti-tial region, hence the bonding nature of thesecompounds is predominantly covalent. It can beinferred from these two compounds that most of theIII–V semiconductors have dominant covalent char-acters. With this knowledge of the electron chargedensity distribution in the unit cell, now we are in abetter position to approach to the band gap problemof these semiconductors with a more appropriatetheoretical scheme.

The results presented in Table II and Fig. 2demonstrate that the band gaps calculated withthe TB-mBJ scheme are however slightly betterthan those of the full potential LDA and GGAresults but are poorer than the HSE, GW and sX-LDA methods. The comparison of the results clearlyindicates that this method underestimates the bandgaps as compared to the experimental values, andthe reason for this underestimation of the band gapsof the III–V semiconductors, in the zinc blende andwurtzite structures, is the improper treatment of

Scale: n(r)

+0.0061

+0.0218

+0.0777

+0.2768

+0.9862

+3.5134

Scale: n(r)

+0.0430

+0.1063

+0.2628

+0.6502

+1.6083

+3.9784

(b)

(a)

BN (111)

AlN (001)

Fig. 1. The contour plot of the electron charge density of (a) alu-minum nitride (AlN) in the (001) plane and (b) boron nitride (BN) inthe (111) plane.

BN

AlN

GaN

AlPBP

GaPAlP

AlSb

GaAsInP

GaSbBSb

InNBAs

InSbInAs

(a)

(b)

Fig. 2. Comparison of the calculated band gaps by nTB-mBJ with (a)HSE, GW, TB-mBJ and experiments and (b) sX-LDA and experi-ments for the III–V semiconductors.

Electronic Band Structures of the Highly Desirable III–V Semiconductors: TB-mBJ DFTStudies

3317

the electrons’ charge density. The covalent behavioris responsible for a reasonable proportion of theelectron density in the interstitial region, whichcannot be properly treated by this method andconsequently one obtains underestimated band gapsfor these semiconductors.

The calculated band gaps by the nTB-mBJmethod are compared with the band gaps obtainedby LDA, GGA, sX-LDA, HSE, GW and TB-mBJ inTable II. It is obvious from these results that thenTB-mBJ reproduces better experimental bandgaps as compared to LDA, GGA, sX-LDA, HSE,GW and TB-mBJ schemes. The accuracy of the bandgaps by nTB-mBJ over HSE, GW, TB-mBJ and sX-LDA is clearly shown in Fig. 2. Table III shows thecomparison of the calculated band gaps for the III–Vsemiconductors using different flavors of the TB-mBJ, i.e., TB-mBJ, iTB-mBJ, TB-mBJAvg and nTB-mBJ, with the experimental results. These resultsclearly indicate the effectiveness of the nTB-mBJ ascompared to all the other TB-mBJ approaches.

The modified Becke–Johenson exchange potentialscheme similar to most of the other theoreticalapproaches has some advantages and disadvan-tages, discussed by David Koller and his coworkers.8

For many cases, the c-factor can be obtainedthrough a self-consistent TB-mBJ procedure whenthe set of Kohn–Sham equations are solved to reacha well-converged electron charge density. The TB-mBJ potential uses an average of rq ~rð Þj j=q ~rð Þ overthe unit cell,3 which does not make sense whenelectron charge density is low somewhere in theunit cell such as thin films, nanowires,37,38 com-pounds having vacancies,39 and clusters bondingthrough a weak van der Waals bond.40 In thesesystems, there are hollow spaces in their unit cellsand thereby rq ~rð Þj j=q ~rð Þ fails to give satisfactory c-factors due to the small electron charge densities inthese regions. In such cases, the TB-mBJ method

gives a small c-factor, which may not be suitable forthe corresponding systems.

Therefore, various theoreticians have modifiedthe regular TB-mBJ in different ways. Koller et al.8

made some modifications in regular TB-mBJ byreparameterizing the coefficients of the c parame-ters,3 to define it for specific group of solids, i.e.,small and medium size band gap semiconductors.They fixed the c-value for two types of compoundshaving band gaps under 7 eV, where they used theP-present parameter for larger band gaps and P-semiconductor parameters for small band semicon-ductors. The P-present and P-semiconductorresults are shown in Table III in the columns ofiTB-mBJ(I) and iTB-mBJ(II), respectively. For bet-ter descriptions of the band gaps and bindingenergies, Jiang35 proposed to combine the regularTB-mBJ with the Hubbard U correction for local-ized d/f states. As LDA usually underestimates andGGA overestimates the lattice constant, thereforeCamargo et al.36 took the average of the latticeconstants calculated by LDA and GGA for theregular TB-mBJ calculation (TB-mBJ(avg)) for theimprovement of the regular TB-mBJ, but there washardly any development in the results. Davidet al.41 used body perturbation theory with theTB-mBJ, although their results for a few oxidecompounds were in agreement with the experi-mental values but with high computational costscompared with nTB-mBJ.

Consequently, in many compounds, the non-reg-ular TB-mBJ (nTB-mBJ) scheme could be used toovercome the low electron charge density problemby performing calculations with the optimized c-factor. More detailed justifications of the manuallyadapted c-factor of the TB-mBJ for the defectivespinel c-Al2O3 are presented in Ref. 37, which weavoid repeating here. Quite recently we have stud-ied and reproduced the electronic structure and

Table III. Fundamental band gaps (in eV) for III–V semiconductors (ZB and WZ) obtained by differentTB-mBJ methods

Compounds TB-mBJ iTB-mBJ(I) iTB-mBJ(II) TB-mBJAvg TB-mBJ + U nTB-mBJ* Expt.

BN 5.853 5.898 6.048 5.8536 – 6.22 6.253

BP 1.8934 – – 1.8336 – 2.38 2.428

BAs 1.7535 – – 1.7236 – 0.67 0.6713

BSb 1.1933 – – – – 0.51 0.5132

AlN 5.553 5.588 5.708 5.5336 – 6.16 6.221

AlP 2.323 2.278 2.218 2.3336 – 2.49 2.5113

AlAs 2.058 2.018 1.988 2.1736 – 2.25 2.313

AlSb 1.8133 – – 1.8036 – 1.65 1.6813

GaN 2.818 2.878 3.008 3.1336 3.3235 3.48 3.513

GaP 2.09034 – – 2.2436 2.3035 2.32 2.3513

GaAs 1.643 1.528 1.568 1.5236 1.5235 1.51 1.5213

GaSb 0.748 0.708 0.708 0.9036 0.7135 0.82 0.8213

InN 0.8834 – – 0.8236 – 0.65 0.6921

InP 1.608 1.578 1.598 1.5236 – 1.42 1.4228

InAs 0.598 0.578 0.608 0.5536 – 0.42 0.4213

InSb 0.278 0.248 0.248 0.3136 – 0.24 0.2413

Rehman, Shafiq, Saifullah, Ahmad, Asadabadi, Maqbool, Khan,Rahnamaye-Aliabad, and Ahmad

3318

band gap of the fcc-C60 compound in the presence ofweak van der Waals interactions by using the nTB-mBJ and our obtained results are in agreement withthe experimental data.40 For the present work, wehave also used the nTB-mBJ due to the fact that inthe presented electron charge densities there arealso regions with low electron charge densities. Inthis way, our results clearly show that the bandgaps are improved and are in better agreement withthe experiments. It is important to note that thec-factor cannot be increased without limit, because

after a critical c-value the results may be worse.Thus, it may not always be possible to obtain theexperimental band gap even by the nTB-mBJscheme. Therefore, by increasing the c-factor man-ually we are just making the TB-mBJ scheme moresuitable to use the stronger repulsive potential tophysically cure the low electron charge densityproblem and not fitting the band gaps with theexperimental data, as doing this may be out of thescope and/or control of the TB-mBJ method within anon-physical electron charge density.

Γ M K Γ-7-6-5-4-3-2-10123456789

101112

)V e(

y grenE

GGA

Γ M K Γ-7-6-5-4-3-2-10123456789

101112

TB-mBJ

Γ M K Γ-7-6-5-4-3-2-10123456789

101112

nTB-mBJ

EF

W L Γ X W K-7-6-5-4-3-2-10123456789

101112

)Ve (

ygrenE

GGATB-mBJnTB-mBJ

BN

W L Γ X W-7-6-5-4-3-2-10123456789101112

AlP

Γ M K Γ-7-6-5-4-3-2-10123456789101112

GaN

EF

(a) (b)

(d)

(c)

Fig. 3. Band structures for GaN by (a) GGA, (b) regular TB-mBJ (c), non-regular TB-mBJ, where (d) represents the comparison of the minima ofconduction bands and maxima of valance bands for BN, AlP and GaN using GGA (red), regular TB-mBJ (blue), and non-regular TB-mBJ (green).

Electronic Band Structures of the Highly Desirable III–V Semiconductors: TB-mBJ DFTStudies

3319

Although nTB-mBJ provides accurate band gapsfor the III–V semiconductors as compared to theother methods, it is necessary to check whether ornot the internal structures of the bands obtained bythe nTB-mBJ potential are consistent with theother approaches? To answer this question, thenTB-mBJ band structure for GaN is compared withthe GGA and TB-mBJ in Fig. 3a–c, while the rest ofthe three plots show the behavior of a single bandcalculated by GGA, TB-mBJ and nTB-mBJ for theminimum of the unoccupied conduction band andthe maximum of the occupied valance bands for AlP,GaN and BN. The comparison of these band

structures confirm that the electronic orbitals arenot affected by the stronger repulsive TB-mBJpotential; however, the conduction bands are ingeneral shifted to higher energies without dis-turbing the structure of the bands after manipulat-ing the electronic charge density.

This study shows that some of the bandstructures for the III–V semiconductors comefrom the interstitial region, though this contri-bution is smaller than the band structures withinthe muffin-tin (MT) spheres. However, it confirmsthat the electrons are not confined in the MTspheres, which cannot be properly treated by the

(a) (b)

(c) (d)

Fig. 4. The calculated imaginary part e2(x) of the frequency-dependent dielectric function of (a) B-V, (b) Al-V, (c) Ga-V and (d) In-V semicon-ductors obtained by nTB-mBJ potential.

Table IV. Direct optical, calculated and experimental band gaps (in eV) for III–V semiconductors (ZB andWZ)

Compounds BN BP BAs BSb AlN AlP AlAs AlSb GaN GaP GaAs GaSb InN InP InAs InSb

Optical 10.90 4.60 2.6 2.2 6.2 4.4 3.3 2.2 3.48 3.07 1.5 0.8 0.65 1.45 0.45 0.25nTB-mBJ 10.85 4.55 2.8 2.45 6.16 4.4 3.3 2.2 3.48 3.07 1.51 0.82 0.65 1.42 0.42 0.24Exp – – – – 6.221 – – – 3.513 – 1.5213 0.8213 0.6921 1.4228 0.4213 0.2413

Rehman, Shafiq, Saifullah, Ahmad, Asadabadi, Maqbool, Khan,Rahnamaye-Aliabad, and Ahmad

3320

APW-based LDA and GGA methods. This is dueto the fact that some orbitals are not fullylocalized inside the atomic spheres. This explainsthe better performance of the orbital-independentsemilocal TB-mBJ scheme as compared to theother theoretical approaches for the III–Vsemiconductors.

The optical properties are directly related to theelectronic band gap of a semiconductor, the real andimaginary parts of the frequency-dependent dielec-tric function are used to calculate the opticalproperties of materials. The calculated imaginaryparts of the dielectric function for the III–V semi-conductors in the energy range 0–25 eV are pre-sented in Fig. 4. The observed offset points in thespectra of the imaginary parts express optical gapsof the semiconductors. These optical gaps arerelated to the direct band gaps of the materialsand are compared with nTB-mBJ and experimentalresults in Table IV. The table confirms that theobserved optical gaps are consistent with the avail-able direct experimental and our theoretical nTB-mBJ band gaps. Figure 4 shows that the offsetpoints (optical gap) shift towards a lower energyrange as we go from N to Sb except the In-V becauseof the small band gap of InN than InP. In the casesof Ga-V and In-V, the absorption peaks decrease by

replacing P to Sb, a decrease in pecks are alsoobserved from B to In except for nitrides havingthe lowest fluctuated peaks. The lowest absorptionpeak of Al, Ga, and In nitrides can be related to awurtzite crystal structure. The peak patterns inB-V and Al-V compounds are different than Ga-Vand In-V with the highest absorption peaks of BAsand AlSb, respectively. The behavior of theabsorption ranges are opposite to that of thepeaks, with high absorption ranges for wurtzite-based nitrides from around 15–25 eV. The calcu-lated real e1(x) and imaginary e2(x) parts of thedielectric function for Ga- and In-based III–Vcompounds in the ZB phase are compared withthe experiments42 in Figs. 5 and 6, respectively.The comparison shows good agreements in termsof peak position, peak height and structure shiftstowards lower energy in e2(x) when P is replacedby As and As by Sb. We noted that the large valueof e1(x) yields a small band gap, which can beexplained on the basis of the Penn model.43 Ingeneral, our calculated e1(x) and e2(x) are in goodagreement with the experimental results andconsistent with other theoretical results,44,45

though their theoretical results are scissors cor-rected, while our results are attributed to our useof nTB-mBJ potential.

(d)

(b) (e)

(c)

(a)

(f)

Fig. 5. The calculated imaginary part e2(x) (a), (b), (c) and real part e1(x) (d), (e), (f) of GaP, GaAs, and GaSb, respectively, of the dielectricfunction (solid curve) obtained by nTB-mBJ potential along with experimental data (dashed curves), Ref. 42.

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CONCLUSIONS

In summary, we reviewed the available experi-mental and theoretical band gaps of the III–Vsemiconductors and explored the corrected theoret-ical band gaps of the compounds by properly treatingthe electron density in the nTB-mBJ approach. Thecomparison of the band gaps by LDA, GGA, sX-LDA,HSE, GW and TB-mBJ schemes with the experi-mental values conclude that for most of thesecompounds the band gaps are severely underesti-mated by LDA and GGA, whereas the sX-LDA, HSE,GW and TB-mBJ approaches although providingbetter band gaps are not successful in reproducingthe correct experimental band gaps. The calculatedelectron charge densities for these semiconductorsshow covalent bonding; therefore, a sufficient num-ber of electrons is present in the interstitial regionthat cannot be properly treated by the LDA and GGAapproaches. We used the orbital independent semilo-cal TB-mBJ scheme, where this method providedbetter band gaps than LDA and GGA but which werepoorer than those of the sX-LDA, HSE and GWmethods. The corrected regular mBJ flavors, e.g.,iTB-mBJ, TB-mBJ + U and G0W0@TB-mBJ, arethough successful to some extent but failed toreproduce the experimental band gaps and are alsocomputationally expansive. This work finallyresolves this issue of the theoretical band gaps ofthe III–V semiconductors and concludes that the

correct band gaps can be obtained through the propertreatment of the electron charge density in thecompounds using the nTB-mBJ scheme. The calcu-lated dielectric functions also confirm the validity ofnTB-mBJ for the calculation of the band gaps of III–V semiconductors.

ACKNOWLEDGEMENT

We acknowledge the financial support from theHigher Education Commission of Pakistan (HEC),Project No. 20-3959/NRPU/R&D/HEC2014/119.

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