# Comment on “Band gap bowing and electron localization of GaXIn1-XN” [J. Appl. Phys. 100, 093717 (2006)]

**ABSTRACT** Some previous density functional theory (DFT) calculations of the band gap of wurtzite and cubic InN, before the work of Lee and Wang [J. Appl. Phys. 100, 093717 (2006)], are in agreement with the screened-exchange findings of these authors and with experiment. These previous findings point to an intrinsic capability of DFT, in the local density approximation, to correctly describe the band gap of semiconductors. These comments also discuss some recent results [Phys. Rev. B 76, 037101 (2007)] on an extensive hybridization of the In 4d and N 2s bands that is lost when the d electrons are included in the core. Our discussions in these comments indicate that when the two inherently coupled equations of DFT are both solved self-consistently, the resulting bands, including low-lying conduction ones, appear to have much more physics content than previously believed.

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Comment on “Band gap bowing and electron localization of GaXIn1−XN”

†J. Appl. Phys. 100, 093717 „2006…‡

D. Bagayoko,a?L. Franklin, G. L. Zhao, and H. Jin

Department of Physics, Southern University and A & M College, Baton Rouge, Louisiana 70813, USA

?Received 10 August 2007; accepted 2 March 2008; published online 2 May 2008?

Some previous density functional theory ?DFT? calculations of the band gap of wurtzite and cubic

InN, before the work of Lee and Wang ?J. Appl. Phys. 100, 093717 ?2006??, are in agreement with

the screened-exchange findings of these authors and with experiment. These previous findings point

to an intrinsic capability of DFT, in the local density approximation, to correctly describe the band

gap of semiconductors. These comments also discuss some recent results ?Phys. Rev. B 76, 037101

?2007?? on an extensive hybridization of the In 4d and N 2s bands that is lost when the d electrons

are included in the core. Our discussions in these comments indicate that when the two inherently

coupled equations of DFT are both solved self-consistently, the resulting bands, including low-lying

conduction ones, appear to have much more physics content than previously believed.

© 2008 American Institute of Physics. ?DOI: 10.1063/1.2908179?

In their article on the “Band gap bowing and electron

localization of GaXIn1−XN,” Lee and Wang reported1results

of their local density approximation ?LDA? and screened-

exchange local density functional approximation ?sX-LDA?

calculations for GaXIn1−XN and for wurtzite and cubic GaN

and InN. Their computations employed a plane-wave basis

and norm-conserving pseudopotentials. In their LDA calcu-

lations for GaN and InN, the d electrons were included in the

core. For their LDA calculations, the effects of these d elec-

trons on the valence bands were included within the nonlin-

ear core corrections of Louie et al.2The inclusion of these d

electrons in the valence was found to reduce the band gaps

by approximately 0.3 and 0.2 for cubic GaN and InN, respec-

tively. In the sX-LDA calculation, including the d electrons

in the valence was reported to lead to a significant error in

the pseudopotentials.

Lee and Wang reported sX-LDA results1that agree well

with measured values for the band gaps of wurtzite and cubic

?i.e., zinc blende? GaN and InN, as shown in Table I below.

Their LDA calculations led to negative band gaps for w-InN

and c-InN, while the band gap for GaN was more than 1 eV

lower than the corresponding sX-LDA value. According to

Lee and Wang, previous LDA band gaps did not agree with

experiment; this view is different from ours in light of LDA

band gaps reported by Bagayoko and co-workers for w-GaN

?Ref. 3? and for w-InN ?Ref. 4? and c-InN.5The calculations

of Bagayoko and co-workers employed local density func-

tional potentials and utilized Gaussian orbitals in a linear

combination of atomic orbital ?LCAO? formalism. A key dif-

ference between their work and that of Lee and Wang con-

sists of their utilization of the Bagayoko, Zhao, and Williams

?BZW? method while implementing the LCAO formalism.

This method3–8avoids a basis set and variational effect that

is inherently associated with self-consistent calculations that

employ a basis set and a variational approach of the

Rayleigh–Ritz type. The LDA-BZW results that also agree

well with experiment are shown in Table I. In particular, the

LDA-BZW predictions of 5.017 Å and 0.65 eV for the equi-

librium lattice constant and the band gap of c-InN have re-

cently been corroborated by Schörmann et al.9who found

5.01?0.01 Å and 0.61 eV for these quantities, respectively.

As summarized elsewhere,7the density fuctional theory

?DFT?-BZW calculations are totally ab initio and self-

consistent. The key distinction of BZW calculations is the

methodical search, beginning with the minimum basis set, of

the optimal basis set that guarantees the convergence of the

size of the basis vis-à-vis the description of the ground state

while it avoids unnecessarily large basis sets that lead to

further lowering of some unoccupied energy levels or bands

even though the occupied ones do not change. Such a low-

ering is ascribed to a direct consequence of the Rayleigh

theorem as opposed to some physical interactions.3–7

In light of these LDA-BZW results, the view that LDA

leads to significant errors in the band gaps of semiconduc-

tors, including w-InN and c-InN, does not hold for all LDA

calculations. While this view applies to many previous DFT

and LDA calculations that do not employ the BZW method,

including those of Lee and Wang,1it does not intrinsically

hold for DFT and LDA that have led to numerous, calculated

band gaps3–8,16in agreement with the experiment—without

scissors approximation of other ad hoc remedies to the band

gap problem.

Bagayoko et al.16recently reported the results of ab ini-

tio, self-consistent, LDA-BZW calculations for w-InN and

c-InN, where the d electrons were in and out of the valence.

The latter case means that the d electrons are included in the

core. In these ab initio, self-consistent calculations, drastic

differences were observed between the resulting band struc-

tures, except for the band gaps that only changed by 0.13 eV.

The extensive hybridization between In 4d and N 2s is lost

when the 4d electrons are moved from the valence to the

core. Consequently, the band gaps are increased by 0.13 eV

for both w-InN and c-InN while the bottoms of the groups of

4d and 2s bands are shifted upward by 1 and 0.71 eV, re-

a?Electronic mail: bagayoko@aol.com.

JOURNAL OF APPLIED PHYSICS 103, 096101 ?2008?

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103, 096101-1

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Page 2

spectively. Concomitantly, the tops of both groups of bands

are lowered by 0.64 eV for w-InN and c-InN. With the d

bands in the valence, the LDA-BZW calculated separation

between the minimum of these bands and the Fermi level is

15.685 eV for w-InN, practically in agreement with the ex-

perimental value17of 16.0?0.1 eV. The difference of

?0.1 eV jumps to 1.315 eV when the 4d electrons are in-

cluded in the core. These significant changes raise questions

relative to the suitability of including the d bands in the core,

despite the use of the nonlinear core corrections of Louie et

al.2to account for the effects on the valence bands of d

electrons in the core. Furthmüller et al. reported LDA bang

gaps for c-InN and w-InN that are negative and smaller than

those reported by Lee and Wang, even though they treated18

the d electrons as valence electrons. These authors performed

several other calculations18out of the scope of these com-

ments. Their LDA calculations found positive band gaps for

both c-InN and w-InN when the d electrons are frozen in the

core. They interpreted the large differences between the gaps

obtained with and without the d electrons in the core in terms

of pd repulsion. The reported differences18are 0.76 and

0.79 eV for w-InN and c-InN, respectively. Our approach

agrees with the work of Wei and Zunger19who found much

earlier that the d electrons should be treated as valence elec-

trons, even though we do not see any indication of pd repul-

sion. The agreement between the recently calculated LDA-

BZW, optical properties8of w-InN and their corresponding

experimental values, up to energies of 5.5–6 eV, supports

the view that the d electrons should be treated as valence

electrons.

We begin our discussions by pointing out that the BZW

method3–8essentially solves a system of two coupled equa-

tions self-consistently as opposed to guessing a solution ?i.e.,

a basis set? for one and solving the other self-consistently.

The first of these equations is known as the Kohn–Sham

equation and the second one gives the charge density in

terms of the wave functions of the occupied states. The full

description of the method is available from several

sources.3,6,7We are aware of reports on self-interaction20and

on derivative discontinuities21–23of the exchange correlation

energy as explanations of the widespread failure of DFT cal-

culations to yield correct band gaps. We underscore the fact

that self-interaction correction calculations20have mostly

overestimated the band gaps of semiconductors to date and

that the value of the discontinuity of the exchange correlation

potential ?Vxc? remains unknown. Perdew et al.21derived a

discontinuity for Vxcfrom a thought experiment that em-

ployed two atoms. Their generalization of their results, from

the two-atom thought experiment, to be applicable to solids

is a suggestion and not an established fact. Further, Perdew

and Levy22asserted that the discontinuity of Vxcis nonzero

in real semiconductors, but they did not claim to have proven

it to be nonzero. Sham and Schlüter23clearly stated that their

work does not demonstrate that the discontinuity of Vxcis

nonzero. Hence, it has yet to be established ?i.e., proven? in

the literature that the discontinuity of Vxcis nonzero for

semiconductors. One fact established by the ab initio, doubly

self-consistent DFT-BZW ?particularly LDA-BZW? calcula-

tions is that the referenced discontinuity of Vxcis much

smaller than previously intimated, at least for the several

semiconductors we studied. The zero discontinuity in LDA

could not otherwise lead to our many results that agree with

experiment. While we do not have the space to elaborate

upon it here, it should be noted that the agreement between

our calculated effective masses3–7and experiment points to

the correct shape and curvature of DFT-BZW bands. Addi-

tionally, the agreement between our DFT-BZW calculated

dielectric functions6,8and experiment denotes accurate en-

ergy separations between the calculated valence and conduc-

tion bands. These two additional features of DFT-BZW re-

sults clearly indicate that DFT ?i.e., DFT-BZW? eigenvalues,

even for low-lying, unoccupied states, may have much more

physics content than previously inferred—mostly based on

limited solutions ?i.e., of one equation? of a system of two

intrinsically coupled equations.

TABLE I. Band gaps, in eV, of wurtzite and cubic GaN and InN. In light of the well-established Burstein–Moss

effect in InN, we note that the experimental results of Refs. 14 and 15 are for carrier densities of 1.2?1019and

5?1019cm−3, respectively.

SystemsLDAa

LDAl

sX-LDAa

LDA-BZW Experiment

c-GaN

w-GaN

c-InN

w-InN

1.97

2.11

−0.22

−0.014

3.04

3.20

0.76

0.89

3.3e

3.39f, 3.5g

0.6h

0.8,i0.883,j0.89k

3.2b

0.65c

0.88d

0.43 ?−0.36?

0.55 ?−0.21?

aReference 1.

bReference 3.

cReference 5.

dReference 4.

eReference 10.

fReference 11.

gReference 12.

hReference 9.

iReference 13.

jReference 14.

kReference 15.

lReference 19. ?positive values are for d electrons outside the valence and self-interaction correction made to

partly account for their effects on the valence electrons; negative values are for d electron in the valence?.

096101-2Bagayoko et al.J. Appl. Phys. 103, 096101 ?2008?

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Page 3

In summary, these comments addressed a view relative

to a perceived inability of LDA to lead to correct band gaps

of semiconductors. This inability, as per the DFT-BZW cal-

culations, should rather be ascribed to other limitations of

computational methods as opposed to an intrinsic limitation

of LDA or DFT. We noted the good agreement between the

sX-LDA results of Lee and Wang with the experiments.

Based on the results we obtained for w-InN and c-InN, we

questioned the suitability of including d electrons in the core

as this approach leads to the loss of significant hybridization

that is germane to the physics of the systems. DFT-BZW

calculations indicate that the resulting bands, including low-

lying, unoccupied ones, have much more physics content

than previously believed.

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