Table Of ContentAll-electron study of InAs and GaAs wurtzite: structural and electronic properties
Zeila Zanolli and Ulf von Barth
Department of Phisics, Lund University, S¨olvegatan 14 A, 223 62 Lund, Sweden
(Dated: February 6, 2008)
The structuraland electronic properties of thewurtzite phaseof theInAsand GaAscompounds
are, for the first time, studied within the framework of Density Functional Theory (DFT). We
7
used the full-potential linearized augmented plane wave (LAPW) method and the local density
0
approximation (LDA) for exchange and correlation and compared the results to the corresponding
0
pseudopotential calculations.
2
FromthestructuraloptimizationofthewurtzitepolymorphofInAswefoundthatthec/aratiois
n somewhatgreaterthantheidealoneandthattheinternalparameteru/chasavalueslightlysmaller
a
than the ideal one. In the all-electron approach the wurtzite polymorph has a smaller equilibrium
J
volumeperInAspairandahigherbindingenergywhencomparedtothezinc-blendephasewhereas
5 thesituationisreversedinthepseudotreatment. Theenergydifferencesare,however,smallerthan
the accuracy of standard density-functional codes (∼ 30 meV) and a theoretical prediction of the
]
relative stability of the two phases cannot bemade.
i
c In order to investigate the possibility of using an LDA calculation as a starting point for many-
s
body calculations of excitations properties, we here also present the band-structures of these ma-
-
l terials. The bands are calculated with and without relativistic effects. In InAs we find that the
r
t energy gaps of both polymorphs are positive when obtained from a non-relativistic calculation and
m negative otherwise.
. For both semiconductors, we determine the spin-orbit splittings for the zinc-blende and the
at wurtzite phases as well as thecrystal-field splittings for thenew wurtzite polymorphs.
m
PACSnumbers: 71.15.Ap,71.15.Mb,71.20.-b,71.55.Eq
-
d
n
I. INTRODUCTION tal study exists and only little theoreticalwork has been
o
c done.
[ Nanowires(NW:s)areattractinganincreasingamount In this article we investigate the structural and elec-
2 of attention in the scientific community because they of- tronic properties of these compounds by using all-
v fer the possibility of investigating fundamental physical electron calculations based on Density-Functional The-
6 properties at the nano scale and because of the numer- ory (DFT) [9]. Indeed, this approach does not require a
6 ous possible applications in electronic and photonic de- knowledgeofmaterialparametersandenablesustocom-
0
vices. In particular, InAs-based nanowires have already pute numerous properties ranging from the equilibrium
0
been used to improve electronics in 1D as, for instance, lattice constant to the cohesive energy of the system.
1
6 in the case of resonant tunneling diodes [1] and single- DFT in the Local Density Approximation (LDA)
0 electron transistors [2]. These NW:s, typically grown [10, 11] provides an accurate description of the ground
/ on an (111)B InAs substrate via Chemical Beam Epi- state electronic structure of solids and such calculations
t
a taxy (CBE), have a wurtzite crystal structure in con- are adequate tools for structural studies. The compari-
m
trastto the stable phaseatnormalpressureandtemper- son of the calculated band structure to experimental ex-
- aturewhichisthezinc-blendestructure(zb,3C)withthe citation energies is, strictly speaking, not allowed but,
d 2
space group F43m (T ). Despite the successes achieved nevertheless,providesimportantinformationontheelec-
n d
so far with InAs-based NW:s, it has recently been real- tronic properties of the material. It is, for instance, well
o
c ized that their wurtzite structure is an important factor known that the experimental band gaps are usually un-
: in the interpretationofexperimentaldata obtainedfrom derestimated by DFT. The excited-state properties of a
v
measurements of photo-luminescence (PL) [3, 4], photo- many-electron system, such as the band structure, are
i
X current (PC) [5] or other electronic properties [6]. better described within Many-Body Perturbation The-
r The situation is similar in the case of GaAs NW:s. ory (MBPT) as, for instance, by the GW approximation
a
When the (111)B GaAs surface is chosen as a substrate, [12, 13] to the quasiparticle self-energy Σ. Still, a DFT
the resulting NW:s consist of alternating wurtzite and calculation is a necessary step toward the GW calcula-
zinc-blendsegments[7]. Hence,bothsegmentscontribute tion, since the LDA eigenenergiesandeigenfunctions are
to the optical and electronic properties of such a NW. usually taken to be the zero:th order approximation for
Theoretical studies based on total-energy calculations the perturbative expansion. In particular, the LDA pro-
[8]predictthatthewurtzitepolymorph(wz,2H),having videsanaccurateHartreepotentialwhichalwaysmustbe
space group P63mc (C46v), is a metastable high-pressure treated to infinite order in an extended system. Hence,
modification of both the InAs and GaAs compounds. thepresentbandstructurecalculationsallowustoinves-
Hence, we are faced with new materials,InAs and GaAs tigatehowwellthe LDAcanworkasastartingpointfor
in the wurtzite phase, for which no previous experimen- subsequent many-body calculations.
2
Although the LDA cannot describe the excited-state of the plane waves by KMAX, the number of basis func-
properties, it can nevertheless provide a good descrip- tions is determined by the product RMTKMAX, where
tion of the properties of the occupied states [14]. This RMT is the smallest muffin-tin radius in the unit cell.
means that we can extract useful informations from the The angular-momentum cut-off of the expansion of the
valence part of the band structure as, for instance, the wave functions in spherical harmonics within the atomic
spin-orbitand-forthe wurtzitecompounds -the crystal spheres is called l .
MAX
field splittings. Starting from the original APW method, several im-
Moreover, the all-electron band structures, calculated provedversions of the method were developed as, for in-
forthefirsttimeforthewurtzitephase,offerbenchmarks stance, the Linearized APW method (LAPW) [25], and,
for the corresponding pseudopotential calculations. with the addition of local orbitals to the basis func-
Related studies reported in the literature are per- tions, the LAPW+LO [26], and the APW+lo [27] meth-
formed within DFT, use pseudopotentials [15, 16], and ods. Within the WIEN2k code it is possible to use a
include the In 4d and the Ga 3d electrons among the mixed LAPW+LO and APW+lo basis [28] to get the
core states. These approximations lead to an underes- advantageofboth schemes. For the expansioninside the
timation of the equilibrium lattice parameters and to a atomic spheres we have chosen the APW+lo basis for
less accuratedescriptionofthe band-structure. Here, we l = 0,1,2, the LAPW+LO for higher l:s and we have
focus on a study of the ground state properties of InAs used lMAX =10 as the highest l value.
and GaAs in the wurtzite phase at the DFT/LDA level To ensure that no charge leaks outside the atomic
with a full potential description of the atomic species. sphereswehavechosen−9.0RyforInAsand−6.0Ryfor
For the calculations of quasiparticle band structures of GaAsas the energywhichseparatesthe coreandthe va-
thesecompoundswereferthe readertoasubsequentpa- lencestates. IntheInAscase,thischoiceguaranteesthat
per [17]. the In 4s, As 3d and all higher-energy states are treated
The article is organized as follows. After the descrip- as band states, i.e., 36 valence electrons per InAs pair.
tion of the theoretical background underlying the calcu- Instead, in GaAs, the valence band states start from the
lations, we present the results concerning the structure Ga 3d and the As 3d, leading to 28 valence electrons per
optimization of both the zinc-blende and the wurtzite GaAs pair.
phases of InAs. Then, we discuss the band structure For both the InAs and the GaAs polymorphs we have
obtained in the non-relativistic and in the scalar rela- performed convergence studies to determine the opti-
tivistic approximations, the latter with and without the mum value of RMTKMAX and the number of k-points
inclusionofthespin-orbit(S-O)interaction. IntheGaAs inthewholeBrillouinzone(BZ).Wehavefoundthatfor
case, instead, we use the experimental lattice constants RMTKMAX = 11 the total energy is converged to within
taken from TEM measurements on GaAs NW:s [18] to less than 10 meV for both phases. The number of k-
allow for a direct comparison with experiment. Indeed, points which ensures convergence is 1000 (10×10×10)
since the deformationpotential for the GaAs gap is very in the zb and 405 (9 × 9× 5) in the wz cases, respec-
large [19], small changes in the lattice constant have a tively. The chosennumber of k-points corresponds to 73
strong influence on the band gap. Finally, we present k-points in the irreducible wedge of the Brillouin zone
the band structures for the GaAs compound calculated (IBZ) for the zb structure and to 95 k-points for the wz
inthescalarrelativisticapproximationwithandwithout structure. These k-meshes are used in a modified tetra-
the spin-orbit interaction. hedronintegrationscheme[29]todeterminewhichstates
areoccupied. TheFourierexpansionofthe electronden-
sityintheinterstitialregion(i.e. valencechargedensity)
II. METHODS OF INVESTIGATION is cut-off at GMAX = 14, leading to 913 and 6524 plane
waves for the zb and the wz structures, respectively.
The electronic structure calculations presented here
are performed within DFT using the Full-Potential Lin-
earizedAugmentedPlaneWave(FP-LAPW)method[20] III. STRUCTURE OPTIMIZATION
as implemented in the WIEN2k software package [21].
For the exchange-correlation energy in the Local Spin The parameters obtained in the convergence study
Density Approximation (LSDA) [9, 10, 11] we have cho- wereusedtocarryoutthestructureoptimizationofboth
sen the electron gas data of Ceperley and Alder [22] as InAs polymorphs in the way described below.
parametrized by Perdew and Wang [23]. Duetothecubicsymmetry,theequilibriumlatticecon-
The APW method [24] is a procedure to solve the stant a of the zinc-blende compound is determined by
zb
Kohn-Sham equation based on the division of the unit the minimization of the total energy of the system with
cell into two spatial regions: non-overlapping muffin-tin respect to the volume of the unit cell. The energy vs
spheres centered at the atomic sites of radius RMT and volume curve is then fitted to the Murnaghan equation
a remaining interstitial region. The basis set used con- of state [30] giving the equilibrium parameters for the
sist of atomic partial waves and plane waves in the two volume V0 (and hence one lattice constant), the binding
regions, respectively. Denoting the cut-off wave number energy E0 per pair of atoms, the bulk modulus B, and
3
wurtzite
zinc-blende
Ry) Ry)
Total Energy ( Energy/InAs pair (
otal
T
Deviation from ideal c/a (%)
Volume/InAs pair ( a.u.3 )
FIG. 1: Deviation from the ideal c/a ratio for InAs in the
wurtzite phase.
FIG. 2: Total energy versus volume normalized to one InAs
pair for the zinc-blende (diamonds) and the wurtzite (cir-
cles) polymorphs. The continuous lines are the Murnaghan
TABLE I: All-electron structural and energetic parameters fit curves.
for the two InAs polymorphs. The equilibrium volume and
thebinding energy refer toone InAspair.
anion pair of the wurtzite structure is smaller than that
V0(a.u.3) B(GPa) B′ E0 (Ry) of the zinc-blende by 2.3703 a.u.3, corresponding to a
wz 369.8984 59.6014 4.6170 -16272.35981
volume reduction of 0.637 %.
zb 372.2687 61.2977 4.1660 -16272.35959
Theresultobtainedforthebindingenergyissomewhat
surprising. Contrary to the observed stability of the zb
phase, we find that the binding energy of an InAs pair
is larger in the wz than in the zb case by 3 meV. This is
′
its pressure coefficient B =(dB/dp)p=0 . clearly seen in the energy vs volume curves, normalized
On the other hand, the wurtzite phase is character-
to one InAs pair,shownin Figure 2 for both phases. We
ized by two lattice parameters, namely a and c, and
wz have checked this result by also calculating the total en-
by the internal parameter u. Consequently, the struc-
ergyinthegeneralizedgradientapproximation(GGA)by
ture optimization of the wurtzite polymorph is carried
Perdew-Burke-Ernzerhof(PBE)[32]. Inthe GGA calcu-
out in four steps ordered according to the importance
lationweusedthesamelatticeconstantandconvergence
of the parameter under investigation. At first, the equi- parameterasinthecaseoftheLDA.Thewztotalenergy
librium volume is determined by assuming the ideal c/a was found to be 28 meV lower than that of the zb. We
ratio. This gives a first estimate of the equilibrium a
wz note that the difference in the equilibrium total energies
which is then used in the determination of the devia-
calculated within the LDA is actually smaller than the
tion of c/a from the ideal value. Then, the volume
wz typical accuracy of calculations carried out within stan-
optimization is repeated using the new c and awz finally dard DFT codes, i.e. ∼30 meV. We thus conclude that
producing the equilibrium values. As the last step, the
the relative stability of the two phases cannot be deter-
internal parameter u is determined via force minimiza-
mined within the present accuracy of the calculations.
tion. We obtained u/c = 0.3748, which is very close to
As a further check, we have also performed the struc-
the ideal value (0.375).
tureoptimizationbyusingUltraSoft(US)pseudopoten-
The deviation from the ideal c/a ratio is obtained
wz tials and plane waves [33], as implemented in the VASP
by calculating the total energy of the wz polymorph for
code [34]. The latter calculations are done by treating
different values of the ratio and then fitting the so ob-
the In 4d states as either valence (d-val) or core (d-core)
tained curve to a 4th order polynomial (Fig. 1). In this
states. In the pseudopotential calculations we have used
way, we found c/a = 1.64202, which is slightly larger
aMonkhorst-Pack[35]gridcenteredonthe Γ point with
(by 0.5528 %) than the ideal value p8/3. The resulting a 11×11×11 and 8×8×8 mesh in reciprocalspace for
equilibrium value for c is 7.00118 ˚A. The fact that the the zb and the wz phase, respectively. The convergence
equilibrium value of c/a is largerthan the ideal one sug- study gives the following kinetic energy cutoffs: 262 eV
geststhatthewurtzitephaseisnotthestablepolymorph (d-val, for both polymorphs) and 222 eV and 202 eV (d-
of InAs, in agreement with the rule stated in Ref. [31]. core)forthe zb andwz cases,respectively. The structure
The coefficients of the Murnaghan fit of the energy optimizationwasperformedbyminimizationofthetotal
vs volume curves are summarized in Table I for both energy. In the wurtzite case we have assumed the ideal
polymorphs. By comparing the results obtained for two values for c/a and u/c. We have thus found that the
phases we find that the equilibrium volume per cation- zb phase, having a binding energy lower than the wz by
4
searchforthatvalueoftheinternalparameteruforwhich
TABLE II: Lattice constants for InAs in the zinc-blend and
the force on the nuclei vanishes. We searched for this
wurtzitephases. USdenotesUltraSoft pseudopotentials cal-
point starting from the wz structure with the equilib-
culationswiththeIn4delectronsincludedamongthevalence
riumc anda lattice constants. Thesearchwascarried
(d-val)orcore(d-core)states. Thec/aandu/cvalueswereas- wz
sumedtobetheidealones(i.e. 1.633and0.375,respectively) out by means of a minimization routine from the PORT
in pseudopotential calculations. The percentage deviation of library[40]choosing0.5mRy/Bohrasatoleranceonthe
c/a from the ideal value is 0.5528% in the full potential cal- force. We obtained u/c = 0.37481, i.e. 0.05% smaller
culation. than the ideal value.
azb (˚A) awz (˚A) c (˚A) c/a u/c
exp. 6.0542 4.2839 6.9954 1.633 0.3750
all-electr. 6.0428 4.2638 7.0018 1.642 0.3748 IV. BAND STRUCTURES
USd-val 6.0329 4.2663 6.9669 ideal ideal
US d-core 5.8023 4.1060 6.7051 ideal ideal
All the LDA band structures presented in this arti-
cle are calculated by solving the Kohn-Sham equation
[10] at the experimental lattice parameters, along lines
ink-spacewhichincludehighsymmetrypoints. Wehave
∼ 18 meV, is correctly predicted to be the stable phase
performed such calculations for both the wz and the zb
of InAs. We also found that the equilibrium volume per
polymorphs of InAs and GaAs.
InAspairofthewzphaseissmallerthanthezbby0.08%.
Themotivationsforcalculatingthefull-potentialLDA
Clearly, the differing results is connected to the differ-
bandstructuresareasfollows. Theall-electronbands(i)
ent method of calculation, i.e. all-electron versus pseu-
allowustoinvestigatehowwelltheLDAcanfunctionasa
dopotential calculations. These tests thus provide an
zero:th order approximationfor the perturbation expan-
indication of the accuracy obtainable from calculations
sionsofMBPT;(ii)allowustoassesstheeffectsofdiffer-
based on Ultra-Soft paseudopotentials. In our opinion,
entrelativisticapproximations;(iii)provideabenchmark
both sets of results must be considered correct in the
forpseudopotentialcalculationsbasedonLDA; (iv)pro-
sense that the total energy difference between them is of
vide a good approximationto the electronic structure of
the same magnitude as the overall accuracy in the total
the occupied states (i.e. valence band states), thus al-
energies from standard DFT codes.
lowing us to estimate the spin-orbit and the crystal field
The calculated equilibrium lattice constants (in both
splittings for the new wurtzite structures.
the full and pseudo potential method) are displayed in
Table II together with the experimental values. The The WIEN2k code allows for the inclusion of rela-
experimental values for the wurtzite lattice constants tivistic effects at different levels of approximation: non-
are taken from TEM measurements [4]. For the zinc- relativistic, scalar relativistic without spin-orbit interac-
blende structure the experimental parameter measured tionandapproximatefully-relativisticwith theinclusion
at room temperature [36] has been extrapolated to zero of the spin-orbit interaction. The core states are treated
degrees Kelvin. We thus find that the all-electron calcu- fully relativistically [41], unless otherwise specified. In
lations and the pseudopotential calculations with the In thecaseofthevalencestatestherelativisticeffectsarein-
4d treated as valence electrons give very similar results cludedviaascalarrelativisticapproximation[42]where-
andtheyslightlyunderestimate(∼0.01−0.02˚A)theex- afterthespin-orbitinteractioncanbeaddedbyusingthe
perimentalvaluesforthelatticeparameters,asisusually scalarrelativisticeigenfunctionsasabasis[43,44]. Inor-
the case within the LDA. The results of the pseudopo- der to overcomethe problemassociatedwith the averag-
tential calculation with the In 4d electrons frozen into ing procedure inherent in the scalar relativistic approx-
the core are, instead, rather far from the experimental imation additional local orbitals of definite j character
values. This shows the importance of treating the d- have been added to the basis set [45].
electrons as valence states in order to obtain a realistic Whenthespin-orbitcouplingistakenintoaccount,the
descriptionofthe InAscompound. Inthewzcasethere- corresponding Hamiltonian is diagonalized using scalar
sults reported in the literature [15] are calculated using relativistic eigenfunctions in a range of 20 Ry centered
HGHpseudopotentials[37]constructedbytreatingthed around the top of the valence band. This is necessary
electrons as core states. The results are: awz =4.192 ˚A, because the S-O splitting in InAs has a strength compa-
u = 0.3755, c = 0.6844 ˚A. Also results for the zb case rable to the energy gapand can, thus, not be considered
arereportedinthesamearticleusingthesameapproach, as a “small” correction to the scalar relativistic bands.
leadingtoazb =5.921˚A.Othercalculationsbasedonthe In the case of GaAs the S-O spitting is smaller than the
all-electronapproachhavegiventheresultsazb =6.063˚A energygap(atlowtemperature,forthezbphase: 0.33eV
[38] and azb =6.084 ˚A [39] in the case of InAs in the zb and1.519eV,respectively[36]), butstillthe effectisim-
phase. portantandsoweusethesameapproximationasforthe
The last step in the structural optimization is the InAs.
5
FIG.3: FullpotentialbandstructuresofInAsinthewurtzitephaseindifferentrelativisticapproximations: (a)non-relativistic,
(b) scalar-relativistic, and (c) scalar-relativistic with theinclusion of spin-orbit coupling.
A. InAs band structures investigation has a small energy gap. This is indeed the
case in InAs, for which the experimental energy gap of
the zb polymorph at zero Kelvin is 0.415 eV [36].
Atfirst,thebandstructuresofInAsinboththezband
We should also consider the fact that the binding en-
the wz phases were calculated within the non-relativistic
ergy of the In 4d states is under-estimated by the LDA
approximation. We then found that InAs is correctly
[47]. As a consequence the d states are too close to the
predicted to be a semiconductor with energy gaps of
VBM, resulting in a too large repulsion between the p
0.395 eV and 0.444 eV in the zb and the wz cases, re-
andthe dstates. Thishasthe effectofshifting the VBM
spectively. We are grateful to Prof. Almbladh for cor-
toward higher energies, hence contributing to the reduc-
roborating these results using his own LAPW code [46].
tion of the energy gap.
Hence, within the LDA, the energy gap of the wz poly-
morphis49meVlargerthatthatofthezb. Inthezbcase, Anothersourceoferrorcanbefoundinthescalarrela-
tivistictreatmentofInAs. Indeed,inthissemiconductor,
thethreep-likestatesconstitutingthevalencebandmax-
imum (VBM) are degenerate at the Γ point. In the wz therelativisticeffectsareofthesameorderofmagnitude
asthe energygap, ascanbe seenby consideringthe fact
case two of these states show a crystal field splitting of
91 meV. The resulting band structure for the wz poly- thatthespin-orbitsplittingatzeroKelvinis0.38eV[36].
morph is displayed in Fig. 3a. We have, therefore, improved our relativistic descrip-
tionofthesystembyincludingthe spin-orbitinteraction
Thescalarrelativisticbandstructuresofthetwopoly-
as a correction to the scalar relativistic band structure.
morphs were then calculatedwithout including the spin-
This approach led to spin-orbit splittings in the zb and
orbit interaction. Within this approach the InAs energy
the wz cases of 0.355 eV and 0.336 eV, respectively. In
gap is not present anymore. Instead, we found a “nega-
addition, we obtained a crystal field splitting of 48 meV
tivegap”or“wrongbandordering”forbothpolymorphs.
in the wz polymorph. Unfortunately, the inclusion of
Indeed, in the zb case, at the Γ point, the three p-like
spin-orbitinteractiondidnot solvethe problemwith the
degenerate states at the top of the valence band have
“wrong band ordering”. We obtained the “negative en-
a higher energy than the s-like conduction band mini-
m−0u.m46(7CeBVM. T),hreeswuzltpinhgasine eaxnheibnietrgtyhegasapmoef Γpr1ocb−leΓm15avn=d ewrzgyphgaaspess”, r−es0p.4e5ct3iveeVly.anTdh−e 0b.a5n0d6setVrufcotrurtehewzitbhatnhdetihne-
the calculatedenergygapis Γ1c−Γ1v =−0.359eV.The cfolurstihone wofztphoelysmpionr-pohrb.iTt hineterreasuctltiosnfoirstshheowInnAisncFaisge.a3rce
crystal field splitting can be identified as a splitting of
collected in Tab. III.
84 meV between the the Γ6v and Γ1v valence band lev-
els. ThescalarrelativisticbandstructureforInAsinthe
wz phase is shown in Fig. 3b.
B. GaAs band structures
The main reason for this un-physical result has to be
ascribed to the notorious limitations of the LDA in re-
producing band structures. Since the calculated energy The band structure calculations for GaAs were per-
gapsaretypically underestimated,this mayresultinthe formed by using the values of lattice constants mea-
“negative gap” problem when the semiconductor under sured via TEM [18] on single NW:s having a diameter
6
TABLE III: The experimental and calculated energy band
gaps, spin-orbit splittings and crystal field splittings (only
for wz) for the two polymorphs of the InAs. The tabulated
crystalfieldsplittingiscalculatedwiththeinclusionoftheS-
Ointeraction. AlltheenergiesareineVandtheexperimental
values are from Ref. [36].
Egap Egap Egap S-O Crystal
non-rel scal. rel scal rel splitt. field
w.out S-O with S-O splitt.
zb 0.395 -0.467 -0.453 0.336 none
zb (exp) 0.542 0.415 0.38 none
wz 0.444 -0.359 -0.506 0.355 48
FIG. 4: Scalar-relativistic band structure of GaAs in the
of 110 nm. We used the lattice constants a =5.653 ˚A,
zb wurtzite phase.
a =3.994˚A,c=6.528˚Aandu/c=3/8,i.e. the ideal
wz
u/c value. As mentioned in the introduction, since the
deformation potential of GaAs is very strong the calcu-
lated band gap depends in a critical way on the lattice
constant. Hence, in order to allow a better comparison
withexperiment,wechoosetoworkwiththe experimen-
tal lattice parameters.
We tested this choice by calculating the relativistic
band structure (without taking into account the spin-
orbit interaction) of GaAs in the zb phase with both the
TEM lattice constant and the equilibrium lattice con-
stant a = 5.6 ˚A taken from Ref. [48]. The result-
zb,eq
ing band gaps are 280 meV and 493 meV, respectively.
Hence,adecreaseinthelattice constantof0.94%results
in an increase of the band gap of 43.2%.
The scalar relativistic band structures of both the zb
and the wz phases of GaAs were calculated using the
same procedure as in the case of InAs. The main dif-
ference with respect to InAs is that GaAs is correctly
FIG. 5: Scalar-relativistic band structure of GaAs in the
predicted to be a semiconductor, even though the LDA
wurtzite phase with theinclusion of spin-orbit coupling.
gapisverysmallcomparedtoexperiment. Thisbehavior
canbeexplainedwithargumentsanalogoustothoseused
in the discussion of the InAs band structures. The band
Without the S-O interaction we find that the band
structures calculated for the wz polymorph without and
structure of the wz phase is characterizedby a band gap
withtheinclusionofthespin-orbitinteractionareshown
of327meVandacrystalfieldsplittingof131meV.When
in Figs. 4 and 5, respectively.
theS-Ointeractionistakenintoaccount,thebandgapis
decreased to 167 meV for the zb and to 211 meV for the
wz phase. The calculatedspin-orbit splitting is 339 meV
TABLE IV: The experimental and calculated energy band and 340 meV in the zb and the wz cases, respectively,
gaps, spin-orbit splittings and crystal field splittings (only
and the crystalfield splitting in the wz band structure is
for wz) for the two polymorphs of the GaAs. The tabulated
79 meV. All these results are collected in Tab. IV.
crystalfieldsplittingiscalculatedwiththeinclusionoftheS-
Ointeraction. AlltheenergiesareineVandtheexperimental
values are from Ref. [36].
V. SUMMARY AND CONCLUSIONS
Egap Egap S-O Cryst. field
w.out S-O with S-O splitt. splitting We have reported full-potential all-electron studies of
zb 0.280 0.167 0.339 none InAs andGaAs in both the wurtzite andthe zinc-blende
zb (exp) 1.629 1.519 0.33 none phases. The structuraloptimization of both polymorphs
wz 0.327 0.211 0.340 79
ofInAsiscarriedoutandtheresultisingoodagreement
with the TEM measured lattice parameters. We have
7
foundthat,withinthe presentaccuracyofstandardden- Instead GaAs is correctly predicted to be a semiconduc-
sity functional calculations, it is not possible to decide tor in the scalar relativistic approximation, even tough
which of the phases of InAs is the stable one. Our to- the calculated band gaps are considerably smaller than
tal energydifference between the phases is ∼3 meV and the experimental one.
much smaller than the numerical accuracy of the code Spin-orbit splittings and crystalfield splittings for the
(∼30 meV). new polymorph of InAs and GaAs have been calculated
In the case of GaAs, the band structures were calcu- for the first time.
lated at the experimental lattice constants measured via
TEM on NW:s having partially wurtzite structure.
The LDA band structures have been computed for
both polymorphs of InAs and GaAs using different rel-
Acknowledgments
ativistic approximations. At first, results from the non-
relativisticapproachhavebeenpresentedforInAs. Then
we performed calculations using an approximate rela- WethanktheNanoquantanetworkofExcellence(con-
tivistic model in which the core states are treated fully tract number NMP4-CT-2004-500198) for support. We
relativisticallywhilethevalencestatesaredescribedbya alsothank thePhoton-MediatedPhenomena(PMP)Re-
scalarrelativisticmodel. Spin-orbitinteractionwasthen search Training Network (contract number HPRN-CT-
added as a correction. When the bands are calculated 2002-00298) for supporting this research. Finally, we
relativistically we find that InAs is wrongly predicted to thank Prof. C.-O. Almbladh for assistance and helpful
be a zero-gap semiconductor while InAs is a finite gap comments during the course of this work and Dr. Jakob
semiconductorwithinthenon-relativisticapproximation. B. Wagner for TEM imaging.
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