PHYSICAL REVIEW B 15 FEBRUARY 1999-IIVOLUME 59, NUMBER 8
Density-functional calculations for III-V nitrides using the local-density approximationand the generalized gradient approximation
C. Stampfl* and C. G. Van de WalleXerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, California 94304
~Received 7 October 1998!
We have performed density-functional calculations for III-V nitrides using the pseudopotential plane-wavemethod where thed states of the Ga and In atoms are included as valence states. Results obtained using boththe local-density approximation~LDA ! and the generalized gradient approximation~GGA! for the exchange-correlation functional are compared. Bulk properties, including lattice constants, bulk moduli and derivatives,cohesive energies, and band structures are reported for AlN, GaN, and InN in zinc-blende and wurtzitestructures. We also report calculations for some of the bulk phases of the constituent elements. The perfor-mance of our pseudopotentials and various convergence tests are discussed. We find that the GGA yieldsimproved physical properties for bulk Al, N2 , and bulk AlN compared to the LDA. For GaN and InN,essentially no improvement is found: the LDA exhibits overbinding, but the GGA shows a tendency forunderbinding. The degree of underbinding and the overestimate of the lattice constant as obtained within theGGA increases on going from GaN to InN. Band structures are found to be very similar within the LDA andGGA. For the III-V nitrides, the GGA therefore does not offer any significant advantages; in particular, noimprovement is found with respect to the band-gap problem.@S0163-1829~99!06107-X#
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I. INTRODUCTION
The group-III nitrides~AlN, GaN, InN, and their alloys!have attracted much attention in recent years due to tgreat potential for technological applications~see e.g., Refs1–5, and references therein!. In the wurtzite~ground-state!structure, AlN, GaN, and InN have direct energy band gof 6.2, 3.4, and 1.9 eV, respectively,3 ranging from the ultra-violet ~UV! to the visible regions of the spectrum. This implies that the AlxGa12xInN alloy system can be used to fabricate optical devices operating at wavelengths ranging frred into the UV. In addition, AlN and GaN have a higmelting point, a high thermal conductivity, and a large bumodulus.6 These properties, as well as the wide band gaare closely related to their strong~ionic and covalent! bond-ing. These materials can therefore be used for shwavelength light-emitting diodes~LED’s! laser diodes, andoptical detectors, as well as for high-temperature, hipower, and high-frequency devices. Bright and highly ecient blue7 and green8 LEDs are already commercially avaiable, and diode lasers have been reported, emitting inblue-violet range initially under pulsed conditions9 and sub-sequently under continuous operation.10
In order to help understand and control the materialsdevice properties, theoretical studies can be most valuablgrowing number of first-principles calculations have beperformed for these materials over the past few years. Mof these calculations are based on density-functional theemploying the local-density approximation~LDA !, either inan all-electron formalism or using the pseudopotential plawave approach. A number of studies have also been caout usingab initio Hartree-Fock methods; however, themethods are much more computationally demanding thanLDA, and they significantly overestimate the band gap. Iwell known that the LDA leads to an underestimate of t
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band gaps in semiconductors,11,12 as well as to overbindingAn additional problem for GaN and InN is that the LDpredicts that the Ga 3d and In 4d states overlap with the N2s band forming two sets of bands.6 Recent experimentshave shown, however, that the 3d bands of GaN lie severaeV below the N 2s band.13–17 The same problems may bexpected for InN. This has been explained as being duneglect in the LDA of a combination of self-interaction anfinal-state screening effects.13
Use of the generalized gradient approximation~GGA! indensity-functional-theory calculations is currently receiviincreasing attention as a possible improvement overLDA. The GGA has generally been found to improve tdescription of total energies, ionization energies, electronfinities of atoms, atomization energies of molecules,18–20andproperties of solids.21–24 Improvements have also been rported for adsorption energies of adparticles on surfaces25,26
and for reaction energies.27,28 Furthermore, the GGA hasbeen shown to be crucial in obtaining activation energconsistent with experiment for H2 dissociation.29,30The rela-tive stability of structural phases also appears to be bedescribed for magnetic31 and nonmagnetic systems.32,33 Re-cent studies by Dufek and co-workers34,35 for transition-metal oxides reported a significant improvement in the bastructure when using the GGA. In an earlier publicatiohowever, Leung, Chan, and Harmon31 reported no significantchange in the band structure between LDA and GGA resfor the same materials. Thus the effect of the GGA onband structure is still unclear.
Given the large ionicity and wide band gap of thenitrides, it is important to investigate the effects that tGGA may have on the electronic structure, in particulwhether it would lead to an improvement in the band gSince the GGA affects binding energies in other systeone may also expect a difference in defect formation en
gies depending on the LDA or GGA treatment; the issuedefect formation is of prime interest in the nitrides.36,37 As afirst step, we have performed a comprehensive study ofbulk materials in the present work. To our knowledge thhas only been one published calculation for the groupnitrides employing the GGA~Ref. 38!: in that work onlyselected lattice constants were reported.
Only a few of the published calculations have goneyond the LDA: for wurtzite and zinc-blende AlN,39 and forwurtzite39 and zinc-blende GaN~Refs. 39–41! using aGWapproach, and for wurtzite and zinc-blende GaN ussimple quasiparticle schemes.42 The calculations employingthe GW approximation to the quasiparticle self-enershowed its effects not only on the band gaps but also onposition of the N 2s band and the bandwidth. Quasiparticcalculations essentially overcome the underestimate ofband gap as obtained using the LDA, and yield band strtures in much better agreement with experiment; theyhowever, time consuming and do not, as yet, produce sconsistent total-energy values. TheGW calculations for GaNalso did not include thed states as valence states, but treathem as part of the pseudopotential core. For completewe mention two other recently introduced approachesaim to obtain an improved electronic structure of wide-bagap semiconductors:~i! the use of self-interaction- anrelaxation-corrected pseudopotentials,43 and~ii ! a scheme in-volving generalization of the LDA known as the ‘‘screenexchange’’ method.38,44
In the present study we perform density-functional-thecalculations for AlN, GaN, and InN, using the pseudopotetial plane-wave method and treating the Ga and Ind states asvalence, where we employed both the LDA and GGA for texchange-correlation functional. We report lattice constabulk moduli and derivatives, cohesive energies, and bstructures for AlN, GaN, and InN in the zinc-blende awurtzite structures. We also present results for some ofbulk phases of the constituent elements.
Before undertaking extensive calculations for a new stem, it is mandatory to perform various tests to assessquality of the calculations and to establish acceptable bsets. Comprehensive information about the performanceaccuracy of our pseudopotentials is provided here, includan investigation of ghost states,45 logarithmic derivatives,and transferability.46 We compare our results with experment where possible, and with other first-principles calcutions, where we have made an effort to collect as manypossible of theab initio results.
The paper is organized as follows. In Sec. II we givebrief description of the calculational method, and in Secs.IV, V, and VI we report results for nitrogen~and the N2dimer!, AlN ~and bulk Al!, GaN, and InN, respectively. Section VII discusses the stability of the zinc-blende and wurite structures, and Sec. VIII contains the conclusions.
II. CALCULATIONAL METHOD
We use density-functional-theory and the local densapproximation47 as well as the generalized gradient appromation of Perdew et al.18 ~PWII! for the exchange-correlation functional. The wave functions are expandedplane-wave basis set, and we use an optimized tight-bind
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initialization scheme to improve the convergence of tstrong N 2p, Ga 3d, and In 4d potentials, which are in-cluded as valence states. Details of the method and progcan be found in Ref. 48.
We useab initio fully separable soft pseudopotentials crated by the scheme of Troullier and Martins49 in which weinclude the GGA in the creation of the respectipseudopotentials50,51 as well as in the self-consistent totaenergy calculations. This approach is to be distinguishfrom the use of LDA pseudopotentials in an otherwise seconsistent GGA total energy calculation, i.e., whereexchange-correlation energy is treated in the GGA butpseudopotentials are not~inconsistent treatment of the GGAsee Ref. 51!, or from apost-LDA treatment where the electronic total energy is first minimized within the LDA anthen corrected perturbatively for the GGA exchangcorrelation energy. In the present work the GGA is thtreated in a fully consistent way. Relativistic effects are takinto account for the Ga and In atoms using weighted spaveraged pseudopotentials. Specific details concerningergy cutoff andk-point sampling for the investigated systems are described in the corresponding sections alongthe results.
III. NITROGEN
Essential tests for the pseudopotential plane-wave meinvolve the pseudopotential itself, e.g., logarithmic derivtives, ghost states, and transferability, as well as the physproperties of the systems of interest. We tested a numbedifferent nitrogen pseudopotentials, in particular, we varthe reference electronic configuration and the cutoff radiir c ,and considered the inclusion or absence of the 3d scatteringchannel. The LDA and GGA pseudopotentials that wecided to use were based on best agreement with experimresults for the bond length, binding energy, and vibratiofrequencies of the N2 dimer, while still requiring a manageable basis set for the total-energy calculations. These potials were generated in the non-spin-polarized ground-svalence electronic configuration, 2s22p3, with cutoff radiir c
s5r cp51.37a0 . In the total-energy calculations we take th
2p channel as local. We found that including thed channel,generated in the electronic configuration 2s22p33d0 or2s12p1.753d0.25, resulted in bond lengths that were somwhat too short, and binding energies and frequencieswere too large with respect to experiment, with the latelectronic configuration yielding the largest deviations.52 Inthis work we discuss mainly the GGA pseudopotentials,we also performed analogous tests for all the LDA pseupotentials; the quality of the results was similar in both cas
For computational efficiency it is convenient to transforthe semilocal form of the pseudopotential operator intofully separable nonlocal form as introduced by Kleinman aBylander.53 Transferable pseudopotentials should clospreserve the all-electron atomic scattering properties as gby the logarithmic derivatives at some radius outside the cregion over the range of valence energies relevant to checal bonding. In the left panel of Fig. 1 we show the logaritmic derivatives of the all-electron radial wave function~solidcurve! and the pseudo-wave-functions~semilocal, dashedline; separable, dot-dashed line! demonstrating the close
uceors
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PRB 59 5523DENSITY-FUNCTIONAL CALCULATIONS FOR III-V . . .
agreement of the pseudopotential and all-electron resover the relevant energy range, and the apparent absen‘‘ghost states.’’ When using the separable form, it is imptant to ensure that problems associated with ghost stateavoided. These states cannot always be easily identifiedinspection of the logarithmic derivatives so we usedscheme of Gonze, Stumpf, and Scheffler45 as implemented inthe programFHIPP.50 For all pseudopotentials discussedthe present paper, we verified that no ghost states wpresent. Figures 2~a! and 2~b! show, respectively, the ionicpseudopotential, and the pseudoelectron and all-electrondial wave functions. The N 2p potential is quite deep, resulting in the need for a large plane-wave cutoff, as we will sbelow. We also note that the ionic pseudopotential exhismall short-ranged oscillations near the origin; we makeattempt to remove these, with the understanding that thoscillations are largely filtered by means of the plane-wabasis energy cutoff.50,51
Pseudopotentials are constructed so that they will repduce the all-electron calculation in the reference configu
FIG. 1. Logarithmic derivatives@d ln R(r )/dr , whereR(r ) isthe radial wave function# vs energyE of the all-electron radial wavefunction ~solid curve! and the ~GGA! pseudo-wave-functions~semilocal, dashed line; separable, dot-dashed line! for the nitrogenatom ~left panel! and the aluminum atom~right panel!.
FIG. 2. Ionic GGA pseudopotential~a! and all-electron andpseudopotential~dashed line! wave function~b! for the nitrogenatom.~c! and ~d! Same as~a! and ~b! but for the aluminum atom.
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tion. However, the pseudopotentials should also yield acrate results in a wide range of atomic environments, i.e., tshould be transferable. In order to achieve this, it is necsary that the pseudopotential reproduces the all-electronsults~total energy and eigenvalues! to within the accuracy ofthe underlying frozen-core approximation, for different vlence electron densities of the atom~e.g., excited atomic configurations! and over a desired energy range. We thereftest the transferability of the pseudopotential by monitorthe pseudo-atom ‘‘hardness’’ in a variety of electronic cofigurations. To do this we compare the change in eneeigenvalues and excitation~neutral charge! and ionization~positive charge! energies as a function of electron occuption as obtained using pseudopotential and all-electronculations. In Fig. 3 we plot thedifferenceof these quantitiesbetween the pseudopotential and all-electron results. Inleft panel, emptying of the N 2p state is considered, and ithe right panel excitation~or electron transfer! of electronsfrom the 2s into the 2p level. It can be seen that the eigevalues and excitation energies of the pseudopotential dincreasingly from those of the all-electron potential for largdeviations from the reference electronic configurati(2s22p3). Given that we are considering rather large ioniztion and excitation energies~a maximum of 3.23 H and 0.85H, respectively! the magnitude of the deviation is quitsmall, indicating good transferability for normal physical a
FIG. 3. Deviations in the excitation energies (Epp-Eae, where‘‘pp’’ stands for pseudopotential and ‘‘ae’’ for all electron! andenergy eigenvalues (D2s,D2p) of the nitrogen pseudoatom~GGA!as a function of occupation compared to all-electron results wrespect to the ground-state configuration. The left panel showssults as a function of occupation of the 2p state~with a constant 2soccupation of 2 electrons! and the right panel shows results asfunction of electron transfer from the 2s to the 2p state~plottedwith respect to 2s occupation!.
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5524 PRB 59C. STAMPFL AND C. G. VAN de WALLE
plications. These results can be compared, for examplethose in Refs. 46, 51, 54, and 55.
The pseudopotential should also yield accurate physproperties of the N2 dimer. In Figs. 4~a!–4~d! we show, re-spectively, the calculated bond length, binding energy, vibtional frequency, and total energy as a function of the enecutoff Ecut. The equilibrium bond-length, vibrational frequency, and total energy are obtained using a third-orpolynomial fit to the total energy versus N2 bond-lengthcurve. Corrections to the theoretical values of the bindenergy for zero-point energies are not included; theseexpected to be on the order of a few tenths of an eV. Zepoint energies are also not included in the cohesive enerreported in subsequent sections.
Although the absolute value of the total energy is nconverged at 50 Ry, the other properties seem reasonwell converged at this cutoff.Differencesof total energiesare known to converge notably faster than the absolute egies. Even a 40-Ry cutoff yields reasonable results, butenergy cutoffs lower than 40 Ry, the results exhibit a cllack of convergence.
Values of the calculated physical properties are listedTable I ~obtained using a 15-bohr cubic supercell andenergy cutoff of 70 Ry with theG-point for the k-spacesampling!. The binding energy~defined here as a positivvalue! is obtained as the energy difference of twice the toenergy of a~spherical! N pseudoatom and the total energythe N2 dimer. The spin-polarization energy of the atomground state of the free N atom is taken into account;energy was calculated to be 2.893 eV using the LDA a3.151 eV using the GGA~Ref. 62!; that of the free N2 dimeris negligible. The present results agree well with previoLDA and GGA calculations. Compared to the LDA resulthe GGA yields very similar, but slightly longer bonlengths, slightly lower frequencies, and significantly smabinding energies that are closer to experiment. Similar tre
FIG. 4. Convergence of the~a! bond length,~b! binding energy,~c! frequency, and~d! total energy for N2 as a function of cutoffenergyEcut . Solid and dashed lines represent LDA and GGAsults, respectively.
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have been reported for other small molecules~see, for ex-ample, Ref. 19!.
IV. ALUMINUM NITRIDE
In this section we first discuss the performance of ourpseudopotential and calculations for bulk Al, and then dscribe results for AlN in the zinc-blende and wurtzite strutures.
A. Al
For the Al pseudopotential we again use the non-sppolarized electronic ground-state configuration to createLDA and GGA pseudopotentials, i.e., 3s23p13d0. The cut-off radii were taken to ber c
s51.80a0 , r cp52.10a0 , and r c
d
52.00a0 . In the total-energy calculations the 3d channel istaken as local. The right panel of Fig. 1 shows the logarmic derivatives of the all-electron radial wave function~solidcurve! and the pseudo-wave-functions~semilocal, dashedline; separable, dot-dashed line! for the Al atom, again show-ing a close tracking to the all-electron results in the relevenergy range. At higher energies~above 0.5 H! notable de-viations occur for thed channel, but this energy range is weabove that of interest in the present work. In Figs. 2~c! and2~d! we show the ionic pseudopotential, and the pseudoetron and all-electron radial wave functions. The much sofpotential of the Al atom is apparent, as reflected by the snificantly faster convergence of the physical propertiesbulk Al as a function of energy cutoff~see Fig. 5! as com-pared to that of N2 ~Fig. 4!. Results of the transferability testare collected in Table II; emptying of the valence electrowas considered here~positive ionization of the atom!. The
TABLE I. Calculated bond lengthb, frequencyn, and bindingenergyEb for the N2 dimer. The particular functional used is enclosed in brackets; for the values taken from Ref. 19, the functials are separated into exchange and correlation. The exchangecorresponds to Slater~Ref. 56! ~S! or Becke ~Ref. 57! (B). Forcorrelation, either the LSD~local electron spin density! theory ofVosko, Wilk, and Nusair~Ref. 58! ~VWN! or the gradient-correctedfunctional of Lee, Yang, and Parr~LYP! ~Ref. 59! was used. PWII~Ref. 18! is the GGA employed in the present work and PWI is tearlier GGA of Perdew and Wang~Ref. 60!. Present values arecalculated with an energy cutoff of 70 Ry in a 15-bohr cubic supcell using one special point (G). Experimental values are includefor comparison.
PRB 59 5525DENSITY-FUNCTIONAL CALCULATIONS FOR III-V . . .
values in Table II are sufficiently small to indicate satisfatory transferability~see Refs. 46, 51, 54, and 55!.
In Fig. 5 the convergence of the physical propertiesbulk Al is tested with respect to the energy cutoff. We otained the equilibrium geometry by computing the total eergy per atom in bulk, varying the lattice constant withabout 65% of the equilibrium value and using the Munaghan equation of state.63 From these data we also derivethe bulk modulus and its derivative. To calculate the cosive energy~defined here as a positive value!, we take intoaccount the spin-polarization energy of the free Al atowhich is calculated to be 0.136 eV for the LDA and 0.1eV for the GGA.62 At aboutEcut512 Ry the system may bregarded as being satisfactorily converged, i.e., the difences in the values of the lattice constant, cohesive eneand bulk modulus obtained at 12 and 40 Ry are20.005 Å,0.015 eV, and 0.014 Mbar, respectively, for the LDA, a20.008 Å, 0.019 eV, and 0.022 Mbar, respectively, fGGA. Our results indicate that the rates of convergencethe various physical properties are very similar for the LDand GGA.
TABLE II. Eigenvalue differences (DE3s ,DE3p) andionization/excitation energy differences (DEion/exc) ~in eV! for thealuminum atom between the pseudopotential~GGA! and all-electron calculations for various electronic configurations withspect to the ground-state configuration.
FIG. 5. Convergence of the~a! lattice constant,~b! cohesiveenergy,~c! bulk modulus, and~d! total energy for bulk Al as afunction of cutoff energyEcut . Solid and dashed lines represeLDA and GGA results, respectively.
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The calculated values obtained usingEcut540 Ry with182 k points in the irreducible part of the Brillouin zone acollected in Table III. It can be seen that good agreemenobtained with other LDA and GGA calculations, all of whicwere calculated using the pseudopotential plane-wmethod. Our GGA results show a 2.07% larger lattice cstant, an 11.26% smaller bulk modulus, and a 0.619smaller cohesive energy than our LDA results, and arebetter agreement with experiment.
B. AlN
The ground-state structure of AlN is wurtzite, but AlN haalso been reported to stabilize in the zinc-blende~cubic!structure~see Ref. 6 and references therein!. The zinc-blendeand wurtzite structures are schematically depicted in F6~a! and 6~b!. For the zinc-blende structure, determinationthe theoretical equilibrium geometry is straightforward sinthere is just one lattice constanta with two atoms per unit
cell, one at (0,0,0) and the other at (14 , 1
4 , 14 )a, with unit
vectors a5(0,12 , 1
2 )a, b5( 12 ,0,12 )a, and c5( 1
2 , 12 ,0)a. For
wurtzite there are four atoms per hexagonal unit cell. W
the unit vectorsa5( 12 ,A3/2,0)a, b5( 1
2 ,2A3/2,0)a, andc5(0,0,c/a)a, the positions of the atoms, in units ofa, b,
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TABLE III. Lattice constanta, bulk modulusB and derivativeB8, and cohesive energyEc , of bulk Al. Present values were obtained using an energy cutoff of 40 Ry and 182k points. Experi-mental results are included for comparison.
FIG. 6. Schematic illustration of~a! the zinc-blende structureand~b! the wurtzite structure. Larger and smaller spheres reprecations and anions, respectively.
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5526 PRB 59C. STAMPFL AND C. G. VAN de WALLE
andc are (0,0,0) and (23 , 13 , 1
2 ) for atoms of the first type, and
(0,0,u) and (23 , 1
3 ,u1 12 ) for atoms of the second type, whe
u is the dimensionless internal parameter. For the id
wurtzite structure,c/a5A83 andu5 3
8 .To determine the equilibrium geometry of the wurtz
phase, we optimize the independent parametersV ~volume of
FIG. 7. Convergence of the~a! lattice constant,~b! cohesiveenergy,~c! bulk modulus, and~d! total energy for AlN in the zinc-blende structure as a function of cutoff energyEcut . Solid anddashed lines represent LDA and GGA results, respectively.
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the unit cell!, c/a, and u as follows: In the first step, weassume the ideal wurtzite structure and determine the elibrium volume by varying the lattice constanta. Then, keep-ing the equilibrium volume fixed andu5 3
8 , the c/a ratio isvaried ~generally in the range of 1.593 to 1.663 in steps0.01! to find the optimum value. At the newc/a ratio weonce again vary the lattice constanta, to determine the newequilibrium volumeV8. Then, having foundc/a andV8, wevary the internal parameteru ~generally from 0.365 to 0.390in steps of 0.005! to minimize the total energy.
To check convergence of the calculations as a functionenergy cutoffEcut we calculated the bulk properties of AlNin the zinc-blende structure as a function ofEcut. The cohe-sive energyEc is obtained as the difference between the toenergy of the bulk material,Etot
bulk ~per cation-anion pair!, andthat of the free atoms,Etot
atom. We choose to define this energy as positive, i.e.,Ec52Etot
bulk1( iEtotatom,i . The results are
shown in Fig. 7. A convergence behavior similar to thatthe N2 dimer can be observed in that below 40 Ry the phycal quantities are poorly converged. These results reflectfact that the N pseudopotential is dictating the rate of cvergence for AlN.
In Tables IV and V, our calculated bulk properties of Alin the zinc-blende and wurtzite structures are presentedcompared with experiment and with other publishedab initiocalculations. These results were obtained using an encutoff of 80 Ry with ten and 24k points in the irreduciblepart of the Brillouin zone, for the zinc-blende and wurtzistructures, respectively. Calculations for the zinc-blenstructure with 60k points in the irreducible part of the Brillouin zone showed almost identical results, as was the cfor GaN and InN.
plane-
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TABLE IV. Lattice constanta, bulk modulusB and derivativeB8, cohesive energyEc , and band gapEgG
of zinc-blende AlN, calculated at the theoretical lattice constant. Methods include pseudopotentialwave ~PPPW!, pseudopotential Gaussian basis~PP-GB!, all-electron~AE!, and Hartree-Fock~HF!. Presentvalues were obtained using an energy cutoff of 80 Ry and 10k points. Experimental results are included fcomparison.
PRB 59 5527DENSITY-FUNCTIONAL CALCULATIONS FOR III-V . . .
TABLE V. Lattice constantsa and c, c/a, internal parameteru, bulk modulusB and derivativeB8,cohesive energyEc , and band gapEg
G of wurtzite AlN, calculated at the theoretical lattice constants. Methinclude pseudopotential plane-wave~PPPW!, pseudopotential Gaussian basis~PP-GB!, all-electron~AE!, andHartree-Fock~HF!. Present values were obtained using an energy cutoff of 80 Ry and 24k points. Experi-mental values are included for comparison.
Method LDA calculation a ~Å! c ~Å! c/a u B ~Mbar! B8 Ec ~eV! EgG ~eV!
aThis result was obtained by optimizingc/a and u, but the equilibrium volume was taken to be thatexperiment.
bAll-electron results.cPseudopotential results.
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We sectioned the entries in Tables IV and V accordingthe calculational method for ease of comparison: pseudotential plane-wave, pseudopotential Gaussian basis,electron, and Hartree-Fock methods. Table IV~zinc-blendestructure! shows that the lattice constants and bulk modagree fairly well for all calculation methods. The largest dviation in lattice constant was reported in Ref. 69, whereobtained value was somewhat larger than the others. Wethat for zinc-blende AlN the band gap is indirect; the entrin Table IV correspond to the direct band gap atG.
From Table V~wurtzite structure! we can see that the HFmethods yield slightly larger lattice constants than the LDresults; this is a well-known effect. Table V shows no snificant difference in the results of the physical propertiesthe all-electron and pseudopotential methods for AlN. Wnote that the cohesive energies obtained by Sattaet al.70 aresignificantly larger than those of the present work~by 4.748and 4.746 eV for the zinc-blende and wurtzite structurrespectively!. This is surprising since both Sattaet al.’s andour approach takes spin-polarization of the free atomsaccount. For GaN~Tables VII and VIII! and InN ~Tables Xand XI! the agreement is much closer~with results differingby less than 0.36 eV!.
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We also calculated the heat of formationDH f of AlN inthe zinc-blende structure atEcut580 Ry to be 23.4 eV~LDA ! and23.0 eV~GGA!. The heat of formation is calculated as DH f5Etot
bulk AlN2Etotbulk Al21/2Etot
N2 ~i.e., DH f isnegative for a stable structure!. The experimental value is23.3 eV.6 The absolute value of the heat of formation of twurtzite structure will be larger by the zinc-blende/wurtzenergy difference, which we calculate to be'44 meV ~seeTable XII!.
Our lattice constants as obtained using the GGA are ab1.95% and 1.83% larger than the LDA values, for the zinblende and wurtzite structures, respectively. For the ziblende structure the LDA result is 1.3% smaller than expement, and the GGA result 0.55% larger. The values ofbulk moduli are also lower when calculated within the GGabout 8% smaller than the LDA results for both the zinblende and wurtzite structures. The cohesive energies astained by the GGA are 1.881 eV~zinc blende! and 1.883 eV~wurtzite! smaller than the LDA results, largely correctinthe overbinding of the LDA. The GGA values are therefoin significantly better agreement with experiment, as wascase for bulk Al and the N2 dimer.
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th
5528 PRB 59C. STAMPFL AND C. G. VAN de WALLE
In Fig. 8 the band structure of AlN in the zinc-blendstructure is displayed for calculations using the LDA~solidcurve! and GGA ~dashed curve!. The band structures arcalculated at the appropriate theoretical equilibrium lattconstants for the LDA and GGA, respectively. We see tthe band structures are very similar, except that the bandat G for the GGA result is about 0.61 eV ('13%! smallerthan the LDA result. The conduction bands in the GGA cculation are shifted down slightly in energy, but the shiftnot constant and depends on thek point and energy. Slighdifferences are also seen in the valence bands: in thisthe GGA bands lie higher in energy than those of the LDleading to slightly reduced bandwidths. The differencestween the LDA and GGA observed in Fig. 8 are primardue to the larger lattice constant obtained using the Gcompared to the LDA, i.e., to deformation-potential effecIf, instead, the experimental lattice constant is used, theculated band gap for the zinc-blende structure is the samwithin 0.02 eV for the LDA and GGA.
The LDA band structure compares well with that reportin Ref. 6. The band structure for AlN in the wurtzite pha~not shown! exhibits a qualitatively similar behavior: the drect band gap for the GGA result is found to be 0.49 e('10%! smaller than the LDA result.
V. GALLIUM NITRIDE
As for AlN, the ground-state structure of GaN is wurtzitStabilization of the zinc-blende structure has been repofor growth on~001! GaAs, cubic SiC, MgO, and~001!Si ~seeRef. 1 and references therein!.
The LDA and GGA Ga pseudopotentials were generain the ground-state valence electronic configurat3d104s24p1, with cutoff radii r c
s52.08,r cp52.30, and r c
d
52.08. To avoid ghost states it was necessary to take thschannel as local in the total-energy calculations. Thepanel of Fig. 9 shows that the logarithmic derivatives disp
FIG. 8. Band structure of zinc-blende AlN as obtained usingLDA ~solid curve! and the GGA~dashed curve!, at the theoreticallattice constants appropriate for the LDA and GGA.
etap
-
se,-
A.l-to
d
dn
4fty
good scattering properties, as indicated by the close agment of the all-electron and pseudopotential results inrelevant energy range. In Figs. 10~a! and 10~b! we show theionic pseudopotential and the pseudoelectron and all-elecradial wave functions. The depth of the Ga 3d potential in-dicates that a large energy cutoff is necessary to treat the3d states, as we will see below.
Results of the transferability or ‘‘hardness’’ tests are clected in Table VI. Similar to our tests for the Al atom, wconsider emptying of the valence states in accord withcationic nature of Ga in GaN. We also considered twocited electronic configurations. Good transferability is oserved; these values can be compared with those reporteRef. 54 in which the transferability of a Ga pseudopotenwas also considered. In that work, however, the Ga 3d statewas included in the core. The authors of Ref. 54 found tthe partial core correction scheme79 substantially improvedthe transferability, while without it the transferability was nvery satisfactory. A similar improvement when using t
e
FIG. 9. Logarithmic derivatives@d ln R(r )/dr , where R(r ) isthe radial wave function# vs energyE of the all-electron radial wavefunction ~solid curve! and the ~GGA! pseudo-wave-functions~semilocal, dashed line; separable, dot-dashed line! for the galliumatom ~left panel! and the indium atom~right panel!.
FIG. 10. Ionic GGA pseudopotential~a! and all-electron andpseudopotential~dashed line! wave function ~b! for the galliumatom.~c! and ~d! Same as~a! and ~b! but for the indium atom.
s,
nntueti
kuc4
c-ntoat-
DAt
th
PRB 59 5529DENSITY-FUNCTIONAL CALCULATIONS FOR III-V . . .
partial core correction has been reported for silicon.46 In thepresent work we explicitly treat thed states as valence stateresulting in good transferability.
For zinc-blende GaN we calculated the lattice constacohesive energy, bulk modulus, and total energy as a fution of energy cutoffEcut. Figure 11 shows that a cutoff of aleast 60 Ry is required to yield satisfactory results; the valof the bulk properties obtained using a 50-Ry cutoff are ssignificantly different from those at 60 Ry and higher.
In Tables VII and VIII the values of the various bulproperties are listed for the zinc-blende and wurtzite strtures, as obtained using an 80-Ry cutoff and ten and 2k
TABLE VI. Eigenvalue differences (DE3d ,DE4s ,DE4p) andionization/excitation energy differences (DEion/exc) for the galliumatom ~in eV! between pseudopotential~GGA! and all-electron cal-culations for various electronic configurations with respect toground-state configuration.
points in the irreducible part of the Brillouin zone, respetively. For calculating the cohesive energy we again take iaccount the spin-polarization energy of the constituentoms. For the Ga atom the values are 0.133 eV for the Land 0.182 eV for the GGA.62 We find that the lattice constan
e
FIG. 11. Convergence of the~a! lattice constant,~b! cohesiveenergy, ~c! bulk modulus, and~d! total energy, for GaN in thezinc-blende structure as a function of cutoff energyEcut . Solid anddashed lines represent LDA and GGA results, respectively.
tential
e
TABLE VII. Lattice constanta, bulk modulusB and derivativeB8, cohesive energyEc , and band gapEgG
of bulk zinc-blende GaN, calculated at the theoretical lattice constant. Methods include pseudopoplane-wave~PPPW!, pseudopotential Gaussian basis~PP-GB!, all-electron~AE!, and Hartree-Fock~HF!.Present values were obtained using an energy cutoff of 80 Ry and 10k points. Experimental values arincluded for comparison.
Method LDA calculation a ~Å! B ~Mbar! B8 Ec ~eV! EgG ~eV!
Expt. ~Refs. 6, 71, and 82! 4.50, 4.531 1.90 3.45,3.21
and
, 3.41
5530 PRB 59C. STAMPFL AND C. G. VAN de WALLE
TABLE VIII. Lattice constantsa, c, andc/a, internal parameteru, bulk modulusB and derivativeB8, cohesive energyEc , and band gapEg
G of bulk wurtzite GaN, calculated at the theoretical lattice constants. Methods include pseudopotential plane-wave~PPPW!, pseudopo-tential Gaussian basis~PP-GB!, all-electron~AE!, and Hartree-Fock~HF!. Present values were obtained using an energy cutoff of 80 Ry24 k points. Experimental values are included for comparison.
Method LDA calculation a ~Å! c ~Å! c/a u B ~Mbar! B8 Ec ~eV! EgG ~eV!
in the GGA is 1.59% and 1.63% larger than in the LDA fthe zinc-blende and wurtzite structures, respectively. Cospondingly, the bulk modulus is smaller by 18% for ziblende and 15% for wurtzite. For the zinc-blende structwe find that the LDA yields a slightly larger lattice constathan experiment~by 0.4%!, while that of the GGA is 2%
FIG. 12. Band structure of zinc-blende GaN as obtained usthe LDA ~solid curve! and the GGA~dashed curve!, at the theoret-ical lattice constants appropriate for the LDA and GGA.
e-
e
larger. In this case the LDA all-electron results yield latticonstants about 0.8% smaller than experiment. The coheenergies, similarly to what we found for AlN, are also sinificantly smaller using the GGA as compared to the LDby 1.926 and 1.923 eV for the zinc-blende and wurtzstructures, respectively. The GGA cohesive energies arslightly better agreement with experiment than the LDA vues, but indicate an underbinding as opposed tooverbinding of the LDA. It appears therefore that the GGdoes not bring about a significant improvement overLDA for GaN.
In Fig. 12 the zinc-blende band structure of GaN is dplayed as calculated using the LDA~solid curve! and the
TABLE IX. Eigenvalue differences (DE4d ,DE5s ,DE5p) andexcitation energy differences (DEion/exc) for the indium atom~ineV! between pseudopotential~GGA! and all-electron calculationsfor various electronic configurations with respect to the groustate configuration.
PRB 59 5531DENSITY-FUNCTIONAL CALCULATIONS FOR III-V . . .
GGA ~dashed curve!, at the respective theoretical lattice costants. As in the case of AlN, the band structures look rasimilar. The band gap is about 0.33 eV ('20%) smaller forthe GGA as compared to the LDA. Similar results are otained for the wurtzite structure~not shown!: the GGA yieldsa band gap approximately 0.31 eV ('18%) smaller than theLDA. This, as mentioned earlier, can be primarily attributto the larger GGA lattice constant.
FIG. 13. Convergence of the~a! lattice constant,~b! cohesiveenergy,~c! bulk modulus, and~d! total energy, for InN in the zinc-blende structure as a function of cutoff energyEcut . Solid anddashed lines represent LDA and GGA results, respectively.
er
-
VI. INDIUM NITRIDE
Indium nitride is perhaps the least studied of the groupnitrides. The equilibrium crystal structure is wurtzite but tzinc-blende structure also has been reported to form.6 Similarto the Ga pseudopotential construction, we created the Land GGA pseudopotentials for In assuming the ground-svalence electronic configuration 4d105s25p1 with cutoff ra-dii r c
s52.08a0 , r cp52.30a0 , and r c
d52.08a0 . For the total-energy calculations we again find it necessary to take thschannel as local to avoid ghost states. The right panel of9 contains the logarithmic derivatives of the GGA In potetial. They appear similar to those of Ga, and display goscattering properties. In Figs. 10~c! and 10~d! we show, re-spectively, the ionic pseudopotential, and the pseudoelecand all-electron radial wave functions. It can be noted tthe In 4d potential is shallower than that of Ga 3d anddeeper than that of N. Results of the transferability teststhe pseudopotential are given in Table IX. Again, good bhavior is seen.
The convergence of lattice constant, cohesive enebulk modulus, and total energy as a function of cutoff eneEcut for the zinc-blende structure is given in Fig. 13. We finthat the properties of InN converge slightly faster thanGaN, but an energy cutoff ofEcut550 Ry or more is re-quired. The values at 40 Ry cutoff are still notably differefrom those at 50 Ry and higher.
In Tables X and XI the structural parameters, bulk modand derivatives, cohesive energies, and band gaps are gfor the zinc-blende and wurtzite structures as calculatedthe present work and as taken from other publications.used an 80-Ry cutoff and ten and 24k points in the irreduc-ible part of the Brillouin zone for the zinc-blende and wurt
e-wave
or
TABLE X. Lattice constanta, bulk modulusB and derivativeB8, cohesive energyEc , and band gapEgG
of zinc-blende InN calculated at the theoretical lattice constant. Methods include pseudopotential plan~PPPW!, pseudopotential Gaussian basis~PP-GB!, all-electron~AE!, and Hartree-Fock~HF!. Present valueswere obtained using an energy cutoff of 80 Ry and 10k points. Experimental values are included fcomparison.
Method LDA calculation a ~Å! B ~Mbar! B8 Ec ~eV! EgG ~eV!
TABLE XI. Lattice constantsa, c, andc/a, internal parameteru, bulk modulusB and derivativeB8, cohesive energyEc , and band gapEg
G of wurtzite InN, calculated at the theoretical lattice constants. Methods include pseudopotential plane-wave~PPPW!, pseudopotentialGaussian basis~PP-GB!, all-electron~AE!, and Hartree-Fock~HF!. Present values were obtained using an energy cutoff of 80 Ry andkpoints. Experimental values are included for comparison.
Method LDA calculation a ~Å! c ~Å! c/a u B ~Mbar! B8 Ec ~eV! EgG ~eV!
ite structures, respectively. We included the spin-polarizaenergy of the N and In atoms in obtaining the cohesiveergy, where the values for the indium atom were calculato be 0.126 eV for the LDA and 0.168 eV for the GGA.62
The values of the~negative! band gaps atG, given in TablesX and XI, were obtained by evaluating the band gap afunction of lattice constant, and extrapolating to the obtainequilibrium lattice constant.
The heat of formation of InN is found to be quite smawithin the LDA, namely,20.103 eV~obtained using an energy cutoff of 80 Ry!. Within the GGA, the value at 80 Ry ifound to be 0.394 eV~i.e., unstable!. Reported experimentavalues range from20.22 to21.49 eV.6 Growth of InN re-quires low temperatures~around 650 °C) due to the therminstability of InN which is consistent with the calculatesmall values of the heat of formation.
We find that our lattice constants as obtained usingGGA are 2.10% and 1.95% larger than those obtained uthe LDA for the zinc-blende and wurtzite structures, resptively. With respect to experiment, the zinc-blende LDA aGGA lattice constants are too large by 0.5% and 2.6%,spectively. The bulk moduli as obtained using the GGAabout 16% smaller for zinc blende, and 17% smallerwurtzite. The cohesive energies, similarly to what we foufor AlN and GaN, are also notably smaller for the GGA~by1.821 eV for zinc blende and 1.822 eV for wurtzite! as com-pared to the LDA. We note that the LDA/GGAdifferencesincohesive energies are very similar for AlN, GaN, and InN
In comparison with experiment we see that, as for Gathe GGA values are somewhat too small, whereas the Lvalues are too large. For InN the degree to which the Gunderbinds is larger than for GaN. Thus we find the tendeof the LDA to overbind decreases on going from GaNInN, while the tendency of the GGA to underbind~and over-estimate the lattice constant! increases on going from GaN tInN. The reason for this is at present unclear. It could
n-d
ad
eg-
-erd
,AAy
e
related to the pseudopotential treatment, for example, seing thef channel as local and allowing a nonlocal descriptifor each of thes, p, andd channels may improve the resultor it could be related to relativistic effects which increawith atomic number. In these respects, consistent,all-electroncalculations for the cohesive energies would beformative.
In Fig. 14 the zinc-blende band structure is displayedthe LDA ~solid curve! and GGA~dashed curve! calculations,at the theoretical lattice constants. In both cases InN istallic; neither exchange-correlation functional yields a potive band gap.
FIG. 14. Band structure of zinc-blende InN as obtained usthe LDA ~solid curve! and the GGA~dashed curve!, at the theoret-ical lattice constants appropriate for the LDA and GGA.
rgrltetohea
aono
th
ex-
lN,, asl-oxi-onthe
I-Varetheom-
er-theeve-s asesisndNnd-
theicelat-llera
-Vect
e-esftin-s.
-Ior
PRB 59 5533DENSITY-FUNCTIONAL CALCULATIONS FOR III-V . . .
VII. ENERGY DIFFERENCE BETWEEN WURTZITEAND ZINC BLENDE
From the calculations described above we obtain enedifferences between the wurtzite and zinc-blende structuwhich are given in Table XII. They are compared to resuof other first-principles calculations. We find a trend of dcreasing energy difference on going from AlN to InNGaN; this trend is the same as that found in all the otstudies. It can be seen, however, that there is considerscatter in the magnitude~and in two cases, thesign! of theenergy differences. These values are obviously quite smand sensitive to the technical details and approximatiused in the various calculation methods. In spite of this, mcalculations~with the two noted exceptions! find that thewurtzite structure is the ground-state configuration and
TABLE XII. Energy difference per cation-anion pair~in meV!between the wurtzite and zinc-blende structures of the groupnitrides. A negative value indicates the wurtzite structure is mstable.
aElectron correlation energy contributions included.bElectron correlation energy contributions omitted.
a
,
yess-
rble
ll,s
st
e
zinc-blende structure is metastable, in accordance withperiment.
VIII. CONCLUSIONS
We have calculated various physical properties of AGaN, and InN, in the zinc-blende and wurtzite structureswell as of the N2 dimer and bulk Al, using both the locadensity approximation and the generalized gradient apprmation for the exchange-correlation functional. In additiwe have reported tests of our pseudopotentials and ofconvergence of the total-energy calculations. For the IInitrides we find that using the GGA the lattice constants1.6–2.1 % larger, the bulk moduli 8–18 % smaller, andcohesive energies approximately 14–20 % smaller, as cpared to the LDA results. For AlN, N2 , and bulk Al, thisresults in a significant improvement in the physical propties obtained using the GGA. For GaN and InN, althoughLDA/GGA deviations are very similar to those of AlN, thGGA does not appear to bring about any essential improment, when compared with experiment. The GGA exhibittendency to underbind for these materials, which increaon going from GaN to InN. The underlying reason for thisunclear. The wurtzite/zinc-blende energy difference is fouto be largest for AlN and smallest for GaN, with that of Inin between. In each case the wurtzite structure is the groustate configuration, in agreement with experiment.
The band structures are found to be very similar inLDA and GGA, when calculated at the experimental lattconstant. When calculated at the appropriate theoreticaltice constants, some differences are found, with a smaband gap in the case of the GGA; this is essentiallydeformation-potential effect. We conclude that for the IIInitrides the GGA does not offer any advantage with respto the band-gap problem.
ACKNOWLEDGMENTS
This work was supported in part by DARPA under Agrement No. MDA972-96-3-014. C.S. gratefully acknowledgsupport from the DFG~Deutsche Forschungsgemeinscha!,and thanks A. P. Seitsonen and M. Fuchs for the sppolarization energy values and for stimulating discussion
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