Consistent set of band parameters for the group-III nitrides AlN, GaN, and InN

0.46 MB·English

by  Patrick Rinke

(accepted at Phys. Rev.B)
Consistent set of band parameters for the group-III nitrides AlN, GaN, and InN
Patrick Rinke,1,∗ M. Winkelnkemper,1,2 A. Qteish,3 D. Bimberg,2 J. Neugebauer,4 and M. Scheffler1
1Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6,D-14195 Berlin, Germany
2Institut fu¨r Festk¨orperphysik, Technische Universit¨at Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany
3Department of Physics, Yarmouk University, 21163-Irbid, Jordan
4Max-Planck-Institut fur Eisenforschung, Department of Computational Materials Design, D-40237 Du¨sseldorf, Germany
(Dated: February 2, 2008)
8 We have derived consistent sets of band parameters (band gaps, crystal field-splittings, band
0 gap deformation potentials, effective masses, Luttinger and E parameters) for AlN, GaN, and
P
0 InN in the zinc-blende and wurtzite phases employing many-body perturbation theory in the
2 G W approximation. TheG W methodhasbeencombinedwithdensity-functionaltheory(DFT)
0 0 0 0
calculations in the exact-exchange optimized effective potential approach (OEPx) to overcome the
n
limitationsoflocal-densityorgradient-correctedDFTfunctionals(LDAandGGA).Thebandstruc-
a
J turesinthevicinityoftheΓ-pointhavebeenusedtodirectlyparameterizea4x4k·p Hamiltonian
tocapturenon-parabolicitiesintheconductionbandsandthemorecomplexvalence-bandstructure
2
of the wurtzite phases. We demonstrate that the band parameters derived in this fashion are in

] very good agreement with the available experimental data and provide reliable predictions for all
i parameters which havenot been determined experimentally so far.
c
s
- PACSnumbers: 71.15.Mb,71.20.Nr,78.20.Bh
l Keywords: GW,exact-exchange, DFT,group-III-nitrides,bandparameters,kp-theory
r
t
m
t. I. INTRODUCTION Hamiltonian is parameterized to reproduce the G0W0
a band structure in the vicinity of the Γ-point. Since the
m parameters of the k p method are closely related or,
The group III-nitrides AlN, GaN, and InN and their ·
- alloys have become an important and versatile class of in some cases, even identical to basic band parameters,
d many key band parameters can be directly obtained us-
semiconductor materials, in particular for use in opto-
n
ing this approach.
o electronic devices andhigh-powermicrowavetransistors.
c Currentapplicationsinsolidstatelighting[lightemitting Thekp-modelHamiltonianistypicallyparameterized
[ diodes (LEDs) and laser diodes (LDs)] range from the ·
forbulk structuresandisthen applicableto heterostruc-
1 visible spectrum1,2,3,4 to the deep ultra-violet (UV)5,6, tures with finite size (e.g., micro- and nanostructures)
v while future applicationsas,e.g.,chemicalsensors7,8,9 or within the envelope-function scheme.24 Ideally, the pa-
1 in quantum cryptography10 are being explored.
rameters are determined entirely from consistent experi-
2
For future progress in these research fields reliable mental input. For the group-III-nitrides, however, many
4
material parameters beyond the fundamental band gap, of the key band parameters have not been conclusively
0
. like effective electron masses and valence-band (Lut- determined until now, despite the extensive research ef-
1 tinger or Luttinger-like) parameters, are needed to aid fort in this field.25,26 In a comprehensive review Vur-
0
interpretationof experimentalobservationsand to simu- gaftman and Meyer summarized the field of III-V semi-
8
late(hetero-)structures,like,e.g.,optoelectronicdevices. conductors in 2001 and recommended up-to-date band
0
: Material parameters can be derived from first-principles parameters for all common compounds and their alloys
v electronic-structuremethodsforbulkphases,butthesize including the nitrides.27 Only two years later they real-
i
X and complexity of structures required for device simula- ized that it is striking how many of the nitride proper-
r tions currently exceeds the capabilities of first-principles ties have already been superseded, not only quantitatively
a electronic-structuretoolsbyfar. Tobridgethisgapfirst- but qualitatively.26 They proceeded to remedy that obso-
principle calculations can be used to parameterize sim- lescence, by providing a completely revised and updated
plifiedmethods,likethekp method,11,12,13,14 theempir- description of the band parameters for nitride-containing
ical tight-binding (ETB) m· ethod,15,16,17,18 or the empir- semiconductors in2003.26Whilethisupdateincludesev-
ical pseudo-potential method (EPM)19, which are appli- idencesupportingarevisionofthebandgapofInNfrom
cable to large-scale heterostructures at reasonable com- its former value of 1.9eV to a significantly lower value
putational expense. around 0.7eV,28,29,30,31,32 they had to concede that in
In this Article we use many-body perturbation theory many cases experimental information on certain param-
in the G W approximation,20—currently the method of eters was simply not available.26 This was mostly due
0 0
choice for the description of quasiparticle band struc- to growth-related difficulties in producing high quality
tures in solids21,22,23—in combination with the k p samples for unambiguous characterization. In the mean-
approach,11,12,13,14 to derive a consistent set of mater·ial timethequalityof,e.g.,wurtziteInNsampleshasgreatly
parameters for the group-III nitride system. The k p- improved25 andeventhegrowthofthezinc-blendephase
·
2
hasadvanced.33 Nevertheless,manyofthebasicmaterial 1.2 wz-InN
propertiesofthegroup-IIInitridesarestillundetermined
0.8
or, at least, controversial.
On the theoretical side, certain limitations of density- eV] 0.4 LDA OEPx(cLDA) OOEEPPxx((ccLLDDAA))
functional theory (DFT) in the local-density or gen- gy [ 0.0 + G0W0
eralized gradient approximation (LDA and GGA, ner
E-0.4
respectively)—currently the most wide-spread ab-initio
electronic-structure method for poly-atomic systems— -0.8
have hindered an unambiguous completion of the miss-
-1.2
ing data. To overcome these deficiencies we use M Γ AM Γ AM Γ A
8 4 8 4 8 4
G W calculations based on DFT calculations in the
0 0
exact-exchange optimized effective potential approach
FIG. 1: LDA Kohn-Sham calculations incorrectly predict
(OEPx) to determine the basic band parameters. We wurtzite InN to be a metal with the wrong band ordering
have previously shown that the OEPx+G0W0approach attheΓpoint. InOEPx(cLDA)thebandgapopensandInN
providesanaccuratedescriptionofthequasiparticleband correctly becomes a semiconductor, thus providing a more
structure for GaN, InN and II-VI compounds23,34,35,36. suitablestartingpointforsubsequentquasiparticleenergycal-
The quasiparticle band structure in the vicinity of the culations in theG0W0 approximation.
Γ-pointisthenusedtoparameterizea4 4kp Hamilto-
× ·
niantodetermineband-dispersionparameters,likeeffec-
tivemasses,Luttingerparameters,etc. Thisallowsus to eigenvaluedifferenceinLDAandGGA.ForInNthiseven
take the non-parabolicity of the conduction band, which resultsin anoverlapbetweenthe conductionandthe va-
isparticularlypronouncedinInN29,37,andthemorecom- lence bandsandthus aneffectively metallic state,as dis-
plex valence band structure of the wurtzite phases into played in Fig. 1. It goes without mentioning that a k p
·
account properly. parameterization derived from this LDA band structure
This paper is organized as follows: In Section II we would not appropriately reflect the properties of bulk
briefly introduce the G W approach and its application InN.
0 0
to the group-III nitrides, followed by a discussion of cer- Many-bodyperturbationtheoryintheGW approach20
tain key parameters of the quasiparticle band structure, presents a quasiparticle theory that overcomes the defi-
such as the fundamental band gaps (and their depen- ciencies of LDA and GGA and provides a suitable de-
denceonthe unit-cellvolume)andthecrystal-fieldsplit- scription of the quasiparticle band structure of weakly
ting energies (Section III). In Section IV we present correlated solids, like AlN, GaN and InN.21,22,23 Most
our recommendations for the band dispersion parame- commonly,the Green’sfunctionG andthescreenedpo-
0
ters (k p parameters) of the wurtzite and zinc-blende tentialW requiredin theGW approach(henceforthde-
· 0
phases of AlN, GaN and InN. A detailed discussion of noted G W ) are calculated from a set of DFT Kohn-
0 0
the parametersetsisgiveninSectionIVBtogetherwith Sham single particle energies and wave functions. The
a comparisonto experimental values and parameter sets DFT ground state calculation is typically carried out in
obtained by other theoretical approaches. Our conclu- the LDA or GGA and the quasiparticle corrections to
sions are given in Sec. V. the Kohn-Sham eigenvalues are calculated in first order
perturbation theory (LDA/GGA + G W ) without re-
0 0
sorting to self-consistency in G and W.38
II. QUASIPARTICLE ENERGY While the LDA+G W approach is now almost rou-
0 0
CALCULATIONS tinely applied to bulk materials,21,22,23 G W calcu-
0 0
lations for GaN and InN have been hampered by
A. GW based on exact-exchange DFT the deficiencies of the LDA. For zinc-blende GaN the
LDA+G W band gap of 2.88 eV39,40 is still too low
0 0
The root of the deficiencies in LDA and GGA for de- compared to the experimental 3.3 eV,41,42,43 while for
scribing spectroscopic properties like the quasiparticle InNtheLDApredictsametallicgroundstatewithincor-
bandstructurecanbefoundinacombinationofdifferent rect band ordering. A single G0W0iteration proves not
factors. LDA and GGA are approximate (jellium-based) to be sufficient to restore a proper semiconducting state
exchange-correlation functionals, which suffer from in- andonlyopensthebandgapto0.02-0.05eV,44,45 which
completecancellationofartificialself-interactionandlack is stillfarfromthe experimentalvalue of 0.7eV.28,29,30
∼
the discontinuity of the exchange-correlation potential Here we apply the G W approach to DFT calcu-
0 0
with respect to the number of electrons. As a con- lations in the exact-exchange optimized effective po-
sequence the Kohn-Sham (KS) single-particle eigenval- tential approach (OEPx or OEPx(cLDA) if LDA cor-
ues cannot be rigorouslyinterpreted as the quasiparticle relation is included). In contrast to LDA and GGA
band structure as measured by direct and inverse pho- the OEPx approach is fully self-interaction free and
toemission. This becomes most apparent for the band correctly predicts InN to be semiconducting with the
gap,whichisseverelyunderestimatedbytheKohn-Sham right band ordering in the wurtzite phase34,46 as Fig. 1
3
a0 [˚A] c0 [˚A] c0/a0 u lattic constants consistent with the G0W0calculations
zb-AlN 4.370 the crystal structure would have to be optimized within
zb-GaN 4.500 the G0W0formalism, too. However, G0W0total en-
zb-InN 4.980 ergy calculations for realistic systems have, to our kon-
wz-AlN 3.110 4.980 1.6013 0.382 wledge, not been performed, yet, and the quality of the
wz-GaN 3.190 5.189 1.6266 0.377 G W totalenergy for bulk semiconductorshas not been
0 0
wz-InN 3.540 5.706 1.6120 0.380 assessed so far.59 The alternative ab initio choices, LDA
and OEPx(cLDA), give different lattice constants60 and
wouldthus introduceuncontrolablevariationsinthe cal-
TABLE I: Experimental lattice parameters (a , c and u)
0 0
culated band parameters that would aggrevate a direct
adopted in this work (see text), for zinc blende (zb) and
wurtzite (wz) AlN,GaN, and InN.50 comparison.
The thermodynamically stable phase of InN at the
usual growth conditions is the wurtzite phase. Reports
demonstrates. For II-VI compounds, GaN and ScN we
ofasuccessfulgrowthofthezinc-blendephasehavebeen
have previously illustrated the mechanism behind this
band gap opening in the OEPx approach,23,34,35,36,47 scarce. Recently, high-quality films of zb-InN grown on
indium oxide have been obtained by Lozano et al.33 We
which brings the OEPx Kohn-Sham band gap much
adopt their lattice constant of 4.98˚A,33 which is in good
closer to the experimental one. Combining OEPx with
G W 48 in this fashion yields band gaps for II-VI com- agreement with previous reports of 4.98˚A,61 4.986˚A,62
0 0 and 5.04˚A63 for wz-InN grown on different substrates.
pounds, Ge, GaN and ScN in very good agreement with
experiment.23,34,35,36,47 ForwurtziteInNthebandgapof Forzb-AlNreportsofsuccessfulgrowthareevenscarcer.
Petrovetal. firstachievedtogrowAlNinthezinc-blende
0.7eV and the non-parabolicity of the conduction band
(CB)34 (shown in Fig. 1) strongly supports the recent phase and reported a lattice constant of a0=4.38 ˚A.64
experimental findings.28,29,30,37,49 In addition we have ThiswaslaterrefinedbyThompsonetal. toa=4.37˚A,65
whichisthe valueweadoptinthis work. Forzb-GaNwe
shown that the source for the startling, wide interval of
follow the work of Lei et al.66,67 and chose a =4.50 ˚A.
experimentally observed band gaps can be consistently 0
explained by the Burstein-Moss effect (apparent band
Although wurtzite is the phase predominantly grown
gapincreasewithincreasingelectronconcentrationinthe
for InN, reported values for the structural parameters
conduction band) by extending our calculations to finite
still scatter appreciably.68 In order to determine the
carrier concentrations.34
effect of the lattice constants on the band gap (E )
g
and the crystal-field splitting (∆ ) we have explored
CR
the range between the maximum and minimum val-
B. Computational Parameters
ues of a and c /a reported in Ref. 68 by perform-
0 0 0
ingOEPx(cLDA)+G W calculationsatthevalueslisted
0 0
The LDA and OEPx calculations in the present work
in Tab. II. Since u remains undetermined in Ref. 68
were performed with the plane-wave, pseudopotential
we have optimized it in the LDA. Neither u, E , nor
code S/PHI/nX,51 while for the G W calculations we g
0 0 ∆ depends sensitively on the lattice constants in this
have employed the G W space-time method52 in the CR
0 0 regimeandwehavethereforeadoptedthemeanvaluesof
gwst implementation.53,54,55 Local LDA correlation is a =3.54 ˚A, c =5.706 ˚A(c /a =1.612) and u=0.380 (the
0 0 0 0
added in all OEPx calculations. Here we follow the
LDA-optimized value) for the remainder of this article.
parametrization of Perdew and Zunger56 for the corre-
lation energy density of the homogeneous electron gas Forwz-AlNandwz-GaNthelatticeconstantsaremore
based on the data of Ceperley and Alder.57 This com- established.26,69 For wz-AlNwe adoptSchulz and Thier-
bination will in the following be denoted OEPx(cLDA). mann’s values of a=3.110˚A, c=4.980˚A, and u=0.382,70
Consistent pseudopotentials were used throughout, i.e. which are close to those reported by Yim et al.71 Schulz
exact-exchangepseudopotentials58 for the OEPx(cLDA) and Thiermann also provide a value for the internal pa-
and LDA ones for the LDA calculations. The cation rameter u, which is identical to the one we obtain by
d-electrons were included explicitly.23,46 For additional relaxing u in the LDA at the experimental a and c
0 0
technical details and convergenceparameters we refer to parameters. The same is true for wz-GaN. Schulz and
previous work.23,34 Thiermann’s values of a=3.190 ˚Aand c=5.189 ˚A70 are
close to those first reported by Maruska and Tietjen,72
but in addition offer a value of u=0.377, which corre-
C. Lattice Parameters sponds to our LDA-relaxed value at the same lattice pa-
rameters. Note, that the lattice parameters of wz-InN
All calculations are carried out at the experimental andwz-GaN have been refinedcomparedto our recently
lattice constants reported in Tab. I and not at the ab publishedcalculations.34Theinfluenceoftheadjustment
initio ones to avoid artificial strain effects in the derived onthe differentbandparameterswillbe discussedwhere
parameter sets. For an ab initio determination of the necessary.
4
OEPx(cLDA)+G W OEPx(cLDA)
0 0
a (˚A) c /a u ∆ (eV) E (eV) ∆ (eV) E (eV)
0 0 0 CR g CR g
3.535 1.612 0.380 0.067 0.71 0.079 1.01
3.540 1.612 0.380 0.066 0.69 0.079 1.00
3.545 1.612 0.380 0.065 0.68 0.079 0.98
3.540 1.611 0.380 0.065 0.70 0.078 1.00
3.540 1.612 0.380 0.066 0.69 0.079 1.00
3.540 1.613 0.380 0.068 0.69 0.081 0.99
TABLEII:Bandgap(E ),andcrystalfieldsplitting(∆ )forwurtziteInNintherangeoftheexperimentallyreportedvalues
g CR
of the structural parameters a and c /a (Ref. 68) and udetermined in LDA for a =3.54 ˚Aand c /a =1.612.
0 0 0 0 0 0
III. BAND GAPS, CRYSTAL-FIELD B. Crystal-Field Splitting
SPLITTINGS, AND BAND-GAP DEFORMATION
POTENTIALS
Experimental values for the crystal-field splitting,
We will now discuss the quasiparticle band structure ∆CR, of wz-GaN scatter between 0.009 and 0.038eV
of AlN, GaN and InN in their zinc-blende and wurtzite (Tab. III). The OEPx(cLDA)+G0W0value of 0.033eV
phases in terms of certain key band parameters such supports a crystal-field splitting within this range.
as the band gap (E ), the crystal field splitting (∆ )
g CR
in the wurtzite phase and the band-gap volume defor- Theoretical87 andexperimental73 investigationsofwz-
mation potentials α . At the end of this section we AlNagreeuponthe factthatthe crystal-fieldsplittingof
V
will draw a comparisonbetween LDA and OEPx(cLDA) AlN is negative. Our calculations also yield a negative
based G0W0calculations for AlN. valueof∆CR =−0.295eV.Thisresultsupportsacrystal-
field splitting in AlN below 0.2eV, as reported by
Chenetal.,73 ratherthanasma−llnegativevaluebetween
0.01 and 0.02, as implied by the results of Freitas et
−al.88 For w−z-InN a crystal-field splitting between 0.019
and 0.024eV has been reported recently.83 This value is
A. Band Gaps
significantlysmallerthantheOEPx(cLDA)+G W value
0 0
of 0.07eV.
TheOEPx(cLDA)+G W bandgapsforthethreema-
0 0
terials and two phases are reported in Tab. III together The crystal-field splitting is known to be sensitive to
with the LDA and OEPx(cLDA) values for compari- lattice deformations, such as changes in the c0/a0 ratio
son. For GaN and InN the OEPx(cLDA)+G W band or the internallattice parameteru.89,90,91 Therefore,the
0 0
gaps have been reported previously in Ref. 34. There discrepancy between experiment and theory might stem
we have also argued that the wide interval of exper- from the uncertainties of the lattice parameters of wz-
imentally observed band gaps for InN can be con- InN (cf Sec. IIC). However, varying the c0/a0 ratio or
sistently explained by the Burstein-Moss effect. The the unit cell volume within the experimental range dis-
OEPx(cLDA)+G0W0value of 0.69eV for wz-InN85 sup- cussed in Section IIC yields values for ∆CR which are
ports recent observationsof a band gapat the lower end always larger than 0.06eV (Tab. II), leaving only the
of the experimentally reported range. For zinc-blende internal lattice parameter u as possible source of error.
InN, which has been explored far less experimentally, This parameter is – at least for GaN – known to have a
our calculated band gap of 0.53 eV also agrees very large influence on the crystal-field splitting.91 Although
wellwiththe recentlymeasured(andBurstein-Mosscor- the LDA-optimized u values are in very good agreement
rected) 0.6 eV.79 withexperimentalvaluesforGaNandAlN,experimental
confirmation of the u parameter of InN is still pending.
For GaN the band gaps of both phases
We therefore calculated the crystal-field splitting of wz-
are well established experimentally and our
InN for different values of u (and a and c fixed at the
OEPx(cLDA)+G W calculated values of 3.24eV86 0 0
0 0 values listed in Tab. I) between 0.377 and 0.383. Gen-
and 3.07eV agree to within 0.3eV.
erally, ∆ decreases with increasing u, but even for u
CR
For AlN experimental results for the band gap of as large as 0.383, the crystal-field splitting is still larger
the wurtzite phase scatter appreciable, whereas for zinc than 0.05eV The discrepancy between the experimen-
blende only one value has – to the best of our knowl- talreportandthe OEPx(cLDA)+G W calculations can
0 0
edge – been reported so far. Contrary to GaN, the hence not be attributed to the uncertainties in the lat-
OEPx(cLDA)+G W gaps for AlN are larger than the tice parametersandhasto remainunsettled forthe time
0 0
experimentally reported values. being.
5
param. OEPx(cLDA)+G W Exp. LDA OEPx(cLDA)
0 0
(this work) (this work, for comparison)
wurtzite
wz-AlN E 6.47 6.0-6.3b 4.29 5.73
g
∆ −0.295 -0.230c −0.225 −0.334
CR
wz-GaN E 3.24 3.5e 1.78 3.15
g
∆ 0.034 0.009-0.038e 0.049 0.002
CR
wz-InN E 0.69 0.65-0.8g 1.00
g
∆ 0.066 0.019-0.024h 0.079
CR
zinc blende
zb-AlN EΓ−Γ 6.53 4.29 5.77
g
EΓ−X 5.63 5.34a 3.28 5.09
g
zb-GaN E 3.07 3.3d 1.64 2.88
g
zb-InN E 0.53 0.6f 0.81
g
aReference65
bReferences7173,74,75,76,77,and78
cReference73.
dReferences41,42,and43
eReference26andreferencestherein.
fReference79
gReferences29,80,81,30,82,28,and83
hReference83
TABLEIII:Bandgaps(E )andcrystal-fieldsplittings(∆ )84 forthewurtziteandzinc-blendephasesofAlN,GaN,andInN.
g CR
All values are given in eV.
OEPx(cLDA)+G W LDA OEPx(cLDA) (LDA and OEPx(cLDA)) calculations, i.e., the band
0 0
(thiswork) (this work, for compar.) gaps vary stronger with volume deformations. Hydro-
wurtzite static band gap deformation potentials obtained from
AlN αwVz −9.8 −8.8 −8.9 LDA+Ucalculationshaverecentlybeenreportedfor the
GaN αwVz −7.6 −6.8 −6.5 wurtzite phases of GaN and InN92. Unlike in OEPx,
InN αwVz −4.2 −3.0 where an improved description of the p-d hybridization
zinc blende is achieved by the full removal of the self-interaction
AlN αzVbΓ−Γ −10.0 −9.1 −9.1 for all valence states, the LDA+U approach reduces
αzVbΓ−X −1.8 −0.6 −0.9 the p-d repulsion by adding an on-site Coulomb corre-
GaN αzΓb −7.3 −6.4 −6.1 lation U only to the semicore d-electrons. With refer-
InN αzVb −3.8 −2.6 encetotheOEPx(cLDA)+G W deformationpotentials,
0 0
the LDA+U improves upon LDA for GaN (αLDA+U=-
V
TABLEIV:Volumebandgapdeformationpotentials(αV)for 7.7eV), but worsensfor InN (αLVDA+U=-3.1eV,αLVDA=-
the wurtzite and zinc-blende phases of AlN, GaN, and InN. 4.2 eV in Ref. 92).
The volume deformation potentials can be transformed into
Experimentally the band gap deformation potential
pressure deformation potentials using thebulk moduli of the
is usually measured as a function of the applied pres-
respective materials (see text). All values are given in eV.
sure, which aggravates a direct comparison to our cal-
culated volume deformation potentials. However, since
B = dP/dlnV, where B is the bulk modulus and P
−
C. Band Gap Deformation Potentials the pressure, the pressure deformation potential α can
P
be expressed in terms of α according to α = α /B.
V P V
−
For the hydrostatic band gap deformation potentials Experimentallyreportedvaluesforthebulkmodulusof
the band gaps have been calculated at different vol- wz-GaNscatterbetween1880and2450kbar.93,94,95,96,97
umes (V) between 2% around the equilibrium vol- Using these values, our volume deformation potential
±
ume V . In the explored volume range the band gaps of α =-7.6 eV would translates into a pressure defor-
0 V
vary linearly with ln(V/V ). The linear coefficient is mation potential in the range of 3.1 - 4.0 meV/kbar,
0
then taken as the hydrostatic volume deformation po- which is comparable to the experimentally determined
tential α . The calculated band gap deformation po- range of 3.7 and 4.7 meV/kbar.97,98,99,100,101,102 This
V
tentials are listed in Tab. IV for LDA, OEPx(cLDA), large uncertainty has been partially ascribed to the low
and OEPx(cLDA)+G W . We observe that for all com- quality of earlier samples and substrate-induced strain
0 0
pounds and phases the quasiparticle deformation po- effects.101 The fact that the pressure dependence of the
tential is larger in magnitude than that of the DFT band gap is sublinear (unlike the volume dependence)
6
further questions the accuracy of linear or quadratic fits 0.3 LDA
for the extraction of the deformation potentials in the eV) 0.2 LDA+G0W0
experiments.101 (M OEPx(cLDA)
For wz-InN experimentally reportedvalues are sparse. εCB0.1 OEPx(cLDA)+G0W0
ε-
Franssen et al. determined a hydrostatic pressure de- 0.0
formation potential of 2.2 meV/kbar103, while Li et al. 0.0
found 3.0 meV/kbar.104 This range agrees with our the-
-0.1
oretical one of 2.8 - 3.3 meV/kbar, using for the con-
versionof volume to pressure deformation potentials the V) -0.2
bulk modulus range of 1260 - 1480 kbar95,96 quoted in (eM-0.3
the literature. VB
ε
For wz-AlN we are only aware of one experi- ε- -0.4
mental study reporting a pressure deformation po-
-0.5
tential of 4.9 meV/kbar.76 With experimental bulk wz-AlN
moduli between 1850 and 2079kbar105,106,107 the -0.6K Γ A
OEPx(cLDA)+G0W0pressure deformation potential of 8 4
wz-AlN would fall between 4.7 and 5.3 meV/kbar strad-
dling the experimentally reported value. FIG. 2: Comparison between LDA (solid lines),
Toourknowledge,noexperimentalinformationonthe LDA+G0W0 (open circles), OEPx(cLDA) (dashed lines)
and OEPx(cLDA)+G W (solid circles) for wz-AlN. In both
deformation potential of zb-AlN and zb-InN are avail- 0 0
G W calculations bands are essentially shifted rigidly
able. For zb-GaN our computed volume deformation 0 0
compared to the LDA, whereas OEPx(cLDA) yields bands
potential of α =-7.3 eV translates to a pressure defor-
V with different dispersion.
mation potential range of 3.0 - 3.9 meV/kbar using the
same bulk modulus range as for wz-GaN. This range is
slightly below the experimentally reported range of 4.0
atesfromthe otherthreeapproaches,whichisconsistent
- 4.6 meV/kbar.102,108,109 Employing a semi-empirical
with the observation made for wz-InN in Fig. 1. We at-
approach to overcome the band gap underestimation of
tribute this behaviour to the approximate treatment of
the LDA (LDA-plus-correction (LDA+C), see also dis-
correlation in the OEPx(cLDA) approach and the fact
cussion in Section IVB1), Wei and Zunger found vol-
that the band structure in OEPx(cLDA) is a Kohn-
ume deformation potentials of 10.2eV [zb-AlN (Γ
− − Sham and not a quasiparticle band structure. While
Γ)], 1.1eV [zb-AlN (Γ X)], 7.4eV (zb-GaN) and
− − − theLDAbenefitsfromafortuitouserrorcancellationbe-
3.7eV (zb-InN)110 in good agreement with our full
− tweentheexchangeandthecorrelationpart111,thisisno
OEPx(cLDA)+G W calculations (see Tab. IV).
0 0 longer the case once exchange is treated exactly in the
OEPx(cLDA) scheme. Using a quasiparticle approach
with a more sophisticated description of correlation,like
D. Comparison between LDA+G0W0 and the GW method, then notably changes the dispersion
OEPx(cLDA)+G0W0 of the OEPx(cLDA) bands. As we will demonstrate
in the next Section this will lead to markedly different
For the materials presented in this article a meaning- band parameters not only for the conduction but also
ful comparison between LDA and OEPx(cLDA) based for the valence bands (cf Tab. VII). Against common
G W calculations can only be constructed for AlN for believe OEPx(cLDA) calculations without subsequent
0 0
reasonsgiveninSectionIIA. Figure2 displaysthe band G W calculationsmaythereforeprovideadistortedpic-
0 0
structure of wz-AlN in the four approaches discussed in ture andwe wouldadviseagainstderivingbandparame-
this Article. The “band gap problem” has been elimi- ters from OEPx or OEPx(cLDA) band structures alone.
nated from this comparison by aligning the conduction Third,unlikeinthe LDA+G W casethe G W correc-
0 0 0 0
bands at the minimum of the lowest conduction band tionstothe OEPx(cLDA)startingpointbecomek-point
(ǫ ) and the valence bands at the maximum of the dependent, a fact already observed for GaN and II-VI
CBM
hightest valence band (ǫ ). For this large gap ma- compounds.23 Most remarkably and in contrast to what
VBM
terial three main conclusions can be drawn from Fig. 2. weobserveforGaNandInN(seeSec. IVB2)thecorrec-
First, LDA and both G W calculations yield very sim- tionsaresuchthatthe banddispersionnowagreesagain
0 0
ilar band dispersions. Or in other words the G W cor- with that obtained from the LDA and the LDA+G W
0 0 0 0
rections to the LDA in the LDA+G W approach are approach. Note also that both the band gap and
0 0
not k-point dependent shifting bands almost rigidly. A the crystal field splitting still differ slightly between
rigid shift between conduction and valence bands is fre- LDA+G W and OEPx(cLDA)+G W for AlN (E :
0 0 0 0 g
quentlyreferredtoas“scissoroperator”. Fig.2,however, LDA+G W : 5.95 eV, OEPx(cLDA)+G W : 6.47 eV,
0 0 0 0
illustrates that this shift is not identical for all bands, ∆ : LDA+G W : -0.252 eV, OEPx(cLDA)+G W : -
CR 0 0 0 0
which cannot be attributed to a single scissor operator. 0.295eV). Unlike for GaNand InN, experimentaluncer-
Second, the dispersion obtained in OEPx(cLDA) devi- taintiesdo,atpresent,notpermitarigorousassessment,
7
whichofthetwoG W calculationsprovidesabetter de-
0 0
TABLE V: Recommended band parameters for the
scription for AlN (see also Sections IIIA and IIIB).
wurtzite phases of GaN, InN, and AlN derived from the
OEPx(cLDA)+G W band structures.
0 0
IV. BAND DISPERSION PARAMETERS param. AlN GaN InN
mk 0.322 0.186 0.065
e
We will now turn our attention to band parameters m⊥e 0.329 0.209 0.068
A −3.991 −5.947 −15.803
that describe the band dispersion in the vicinity of the 1
A −0.311 −0.528 −0.497
Γ-point: theeffectivemasses,theLuttinger(-like)param- 2
A 3.671 5.414 15.251
eters, and the EP parameters. These parameters are ob- A3 −1.147 −2.512 −7.151
4
tained by means of the k p-method. The k p-method A −1.329 −2.510 −7.060
· · 5
is a well-establishedapproachthat permits a description A −1.952 −3.202 −10.078
6
of semiconductor band structures in terms of parame- A (eV˚A) 0.026 0.046 0.175
7
ters that can be accessed experimentally. Throughout Ek (eV) 16.972 17.292 8.742
P
this paper, we use four-band k p-theory, which is typ- E⊥ (eV) 18.165 16.265 8.809
· P
ically used to describe direct-gap materials, mostly in
its spin-polarized form as eight-band k p-theory. The
·
k p Hamiltonian and all relevant formulas are given in
· tion. Mostinterpolationschemesthatarefrequentlyem-
Appendix A. The k p-method is a widely accepted
· ployedto calculate the quasiparticle correctionsfor arbi-
technique for, e.g., the interpretation of experimental
trary band structure points are to no avail in this case,
data37,112 or modeling of semiconductor nanostructures
because they do not add new information to the fitting
and (opto-)electronic devices.12,13,14,113,114,115,116 Its ac-
problem at hand. Existing schemes to directly compute
curacy, however, depends crucially on the quality of the
theself-energyforbandstructureq-pointsnotcontained
input band parameters, like effective electron masses,
in the k-grid (see, for instance, Ref. 21 for an overview)
Luttinger-parameters, etc., which have to be derived ei-
are usually not implemented.
ther experimentally or from band structure calculations.
In the GW space-time method52 these problems are
As alluded to in the introduction, many important band
easilycircumvented,becausethe self-energyis computed
parameters of the group-III nitrides GaN, InN, and AlN
in real-space [Σ (r,r′;ǫ)]. By means of Fourier interpo-
are still unknown. In particular, the band structure of R
lation,
InN is currently the subject of active research in both
experiment and theory. Σ (r,r′;ǫ)= Σ (r,r′;ǫ)e−iq·R , (1)
In this paper, we use the k p-method to derive band q R
· R
dispersion parameters from OEPx(cLDA)+G W band X
0 0
structures. This approach has certain advantages over theself-energyoperatorcanbecalculatedatarbitraryq-
a simple parabolic approximation around the Γ point. points.53 The matrix elements φ Σ (ǫ)φ are then
nq q nq
h | | i
First, the k p band structure is valid, not only directly obtained by integration over r and r′. In this fashion
·
at the Γ-point, but also in a certain k-range around it. therelevantBrillouinzoneregionsforthebandstructure
This allows to extend the fit to larger k’s and thereby fittingcanbecalculatedefficientlywithoutcompromising
increases the accuracy of the fitted parameters. Second, accuracy.
the k p-method is capable of describing non-parabolic The k p Hamiltonian and all parameter relations are
bands·, such as the CB of InN34,37 and can therefore also given in·Appendices A and B. To determine the k p
·
beappliedtoaccuratelydeterminevaluesfortheeffective Hamiltonian for a given band structure with band gap
electron masses and E parameters in InN. E and crystal-field splitting ∆ we fit the parame-
P g CR
ters mi, A, γ, and Ei . This is achieved by least-
e i i P
square-root fitting of the k p band structure to the
·
A. Computational details OEPx(cLDA)+G0W0bandstructureinthevicinityofΓ.
For the wurtzite phases the directions Σ, Λ, T, and ∆
Foranaccuratefitofthekp parameterstothequasi- have been included in the fit, represented by 22 equidis-
particle band structure a sm· all reciprocal lattice vec- tant k-points from Γ to M8 (L8, K8) and 22 equidistant
tor spacing is required. Since most GW implementa- points from Γ to A. For the zinc-blende phases the di-
4
tions evaluate the self-energy Σ (the perturbation op- rections Σ, ∆, and Λ have been included, each with 22
erator that links the Kohn-Sham with the quasiparticle k-points from Γ to K (X, L).
8 8 8
system)inreciprocalspace,the matrixelementswithre-
specttotheKohn-Shamwavefunctions φ Σ(ǫqp)φ
required for the quasiparticle correctionhsnaqr|e onnlyq a|vanqili- B. Band parameters of GaN, AlN, and InN
able on the k-points of the underlying k-grid. A fine
sampling ofthe Γ-pointregionwouldthereforebe equiv- The parameters obtained by fitting to the
alent to using formidably large k-grids in the computa- OEPx(cLDA)+G W band structures are listed in
0 0
8
(a) wz-AlN (b) wz-GaN (c) wz-InN
eV) 0.4 Ok.EpPx(cLDA)+G0W0
( k.p (VM ’03)
M
B 0.2
C
ε
-
ε 0.0
0.0
)
V
e
-0.2
(

M
B
V-0.4
ε
-
ε
-0.6
K Γ A K Γ A K Γ A
8 4 8 4 8 4
FIG. 3: Band structure wz-AlN (a), wz-GaN (b), and wz-InN (c) in the vicinity of Γ. The graphs show the
OEPx(cLDA)+G W band structure (black circles), the corresponding k·p band structure (black solid lines), and the k·p
0 0
band structureusing the parameters recommended byVurgaftman and Meyer26,117 (VM’03) (red dashed lines).
(a) zb-AlN (b) zb-GaN (c) zb-InN
1.0
eV) 0.8 Ok.EpPx(cLDA)+G0W0
( 0.6 k.p (VM ’03)
M
B 0.4
C
ε 0.2
-
ε
0.0
0.0
)
V
e -0.2
(

M-0.4
B
V -0.6
ε
-
ε
-0.8
L Γ X L Γ X L Γ X
8 8 8 8 8 8
FIG. 4: Band structure zb-AlN (a), zb-GaN (b), and zb-InN (c) in the vicinity of Γ. The graphs show the
OEPx(cLDA)+G W band structure (black circles), the corresponding k·p band structure (black solid lines), and the k·p
0 0
band structureusing the parameters recommended byVurgaftman and Meyer26,117 (VM’03) (red dashed lines).
Tab. V (wz) and Tab. VI (zb). The resulting k p band available experimental data and selected theoretical
·
structures are plotted in Figs. 3 and 4 (black solid lines) values, representing the state-of-the-art parameters
together with the respective OEPx(cLDA)+G W data up until the year of compilation (2003). We will also
0 0
(black circles). The excellent agreement of the k p and compare our results to more recent experimentally and
·
OEPx(cLDA)+G W band structures illustrates that theoretically derived parameters. (see Tab. VII)
0 0
the band structures of the wurtzite and zinc-blende In the following we will show that the parameters de-
phases of all three materials are accurately described by rived from the OEPx(cLDA)+G W calculations match
0 0
the k p-method within the chosen k-ranges. Addition- allavailableexperimentaldatatogoodaccuracy. Acom-
·
ally, the k p band structures based on the parameters parison to parameters derived by other, theoretical or
recommend·ed by Vurgaftman and Meyer26,117 (VM semi-empirical, methods will be presented thereafter.
’03) are shown (red dashed lines). As alluded to in
Before we proceed, however, we would like to empha-
the introduction, their recommendations are based on
size two points regarding the relation between the VB
9
valence-bandparameters,A,cannotbeobtaineddirectly
i
TABLE VI: Recommended band parameters for the zinc
experimentally, but can be related to the effective hole
blende phases of GaN, InN, and AlN derived from the
masses (see appendix B), which, in turn, can be mea-
OEPx(cLDA)+G W band structures. The effective hole
0 0
sured.
masses for the HH and LH band have been calculated from
theLuttinger parameters. (see appendix B) The available experimental values for the wurtzite
phasesarelistedinTab.VII. Forthethermodynamically
param. AlN GaN InN
metastable zinc-blende phases of GaN, AlN, and InN
m (Γ) 0.316 0.193 0.054
e hardlyanyexperimentalreportsontheirbanddispersion
γ 1.450 2.506 6.817
1 parametersareavailablesofar. Therefore,werestrictthe
γ 0.349 0.636 2.810
2
discussion to the wurtzite phases, for which experimen-
γ 0.597 0.977 3.121
3
m[001] 1.330 0.810 0.835 tal data on, at least, the effective electron masses are
hh
available. For wz-InN also E has been determined, by
m[110] 2.634 1.384 1.368 P
hh fitting a simplified kp-Hamiltonian to the experimental
m[h1h11] 3.912 1.812 1.738 data.37,112 Forwz-Ga·N,valuesforE 125,126 andalsosev-
m[001] 0.466 0.265 0.080 P
lh eralreports on the effective hole masses are available.131
m[110] 0.397 0.233 0.078
lh Wurtzite GaN. The OEPx(cLDA)+G W effective
m[111] 0.378 0.224 0.077 0 0
lh electron masses in wz-GaN (mk = 0.19m , m⊥ =
EP (eV) 23.844 16.861 11.373 e 0 e
0.21m ) are in very good agreement with experimen-
0
tal values, which scatter around m = 0.20m .131 How-
e 0
ever, our calculations predict an anisotropy of the elec-
parameters A and the effective hole masses in wurtzite
i tron masses of about 10 %, which is larger than values
crystals: (i) Two different sets of equations, connecting found experimentally. (< 1% - 6%132,133,134) Our E
P
the effective hole masses to the A parameters, are used
i values of 17.3eV and 16.3eV support those obtained by
in the literature. Reference 118 lists both; one is labeled Rodina and Meyer125 ( 18.3eV and 17.3eV), rather
“Near the band edge (k 0)” and the other “Far away ≈ ≈
than a largervalue of E 19.8eV reported recently by
→ P
from the band edge (k is large)”. The latter is widely Shokhovets et al.126 ≈
used to calculate the effective hole masses.119,120,121,122
Adetailedanalysisoftheeffectiveholemasseshasbeen
However,theexperimentallyrelevanteffectivemassesare presentedbyRodinaetal.127 Note,thatonlytheA-band
those close to Γ. Thus, we use the “Near the band
masses in their work have been extracted directly from
edge”equations(seeAppendix B)throughoutthis work.
experimental data. All other effective hole masses have
Quoted values differ from the original publications in
been calculated from the A-band effective masses and
cases where the original work uses the “Far away from
the spin-orbit and crystal-field splitting energies within
thebandedge”equations. (ii)TheLuttinger-likeparam-
the quasi-cubic approximation. The effective A-band
eters, A, are independent of the spin-orbit and crystal-
i masses derived in the present article (mk =1.88m and
field interaction parameters ∆ and ∆ ; the effec- A 0
SO CR m⊥ = 0.33m ) agree very well with the experimental
tive hole masses, however, differ for different ∆ and A 0
∆CR parameters. Only the A-band (C-band inSOAlN) values derived by Rodina et al. (mkA = 1.76m0 and
hole masses can be calculated from the Luttinger-like m⊥A =0.35m0). Adopting their values for the spin-orbit
parameters alone. All other hole masses depend addi- and crystal-field splitting parameters (∆SO = 0.019eV,
tionally on the choice of the spin-orbit and crystal-field ∆CR = 0.010eV), we also find good agreement for the
splitting energies.118 Thus, effective B- and C-band (A- B- and C-band masses. (see Tab. VII)
and B-band in AlN) hole masses derived from different Wurtzite AlN. The available experimental data on
sets of Luttinger-like parameters are comparable, only if the band dispersion in wz-AlN is limited to the ef-
the same ∆ and ∆ values are assumed. fective electron mass, which has been determined
SO CR
to be in the range of 0.29 to 0.45m .123 The
0
OEPx(cLDA)+G W values of mk =0.32m and m⊥ =
0 0 e 0 e
1. Comparison to experimental values 0.33m fall within this range.
0
Wurtzite InN. Experimentally derived effective elec-
Experimentally, the band structure of a semiconduc- tron masses in wz-InN scatter over a wide range (see
tor is accessible only indirectly, via band parameters Tab. VII). The most reliable seem to be those reported
like E , ∆ , ∆ , and the effective masses. Angle by Wu et al.37 and Fu et al.,112 since they explicitly ac-
g SO CR
resolved direct and inverse photoemission experiments, count for the high carrier concentration of their sam-
which would, in principle, directly probe the quasipar- ples and the non-parabolicity of the CB in their anal-
ticle band structure, are not accurate enough, yet, to ysis. Their effective electron masses of 0.05m0112 and
determine the band structure with sufficient accuracy. 0.07m037 in conjunction with values for EP of 9.7eV
The dispersion of the conduction band around the Γ and 10eV, respectively, are in good agreement with
point depends only on the effective electron masses and thosederivedfromtheOEPx(cLDA)+G0W0calculations
E parameters,whichareaccessibleexperimentally. The (mk =0.065m ,m⊥ =0.068m andEk =8.7eV,E⊥ =
P e 0 e 0 P P
10
TABLE VII: Band parameters of wurtzite group-III nitrides: Comparison to parameters from the literature. Listed are
experimentalparameters,theparametersrecommendedbyVurgaftmanandMeyer26 (VM’03),parametersusingtheLDAand
OEPx(cLDA),parameterscalculatedbyCarrierandWei87 (CW’05)usingLDA+C,andvaluesdeterminedusingtheempirical
pseudopotential method (EPM) byseveral different groups.
param. OEPx(cLDA)+G0W0 exp. VM’03a LDA OEPx(cLDA) LDA+Cb EPM
(recommended) (thiswork,forcomparison)
mke 0.32 0.29-0.45c 0.32 0.33 0.38 0.32 0.23d,0.24e,0.27f
m⊥e 0.33 0.29-0.45c 0.30 0.32 0.39 0.33 0.24d,0.25e,0.18f
AlN EPk (eV) 16.97 - − 16.41 16.89 − -
EP⊥ (eV) 18.17 - − 17.45 17.51 − -
mkC 3.13 - 3.57 3.46 3.72 3.43 2.38d,1.87-1.95e,2.04f
m⊥C 0.69 - 0.59 0.68 0.83 0.68 0.49d,0.43-0.48e,0.36f
mke 0.19 0.20g 0.20 0.15 0.23 0.20 0.14d,0.14e,0.16f
m⊥e 0.21 0.20g 0.20 0.17 0.26 0.22 0.15d,0.15e,0.12f
EPk (eV) 17.29 17.8-18.7h,19.8i − 12.93 16.14 − -
EP⊥ (eV) 16.27 16.9-17.8h,19.8i − 11.30 14.07 − -
GaN mkA 1.88 1.76j 1.89 1.92 2.20 2.04 1.89k,2.37d,1.45-1.48e,1.27f
m⊥A 0.33 0.35j 0.26 0.26 0.38 0.39 0.26k,0.49d,0.26-0.27e,0.20f
mkB 0.37l,0.92m 0.42j − − − 0.85 -
m⊥B 0.49l,0.36m 0.51j − − − 0.43 -
mkC 0.26l,0.19m 0.30j − − − 0.19 -
m⊥C 0.65l,1.27m 0.68j − − − 1.05 -
mke 0.065 0.07n,0.05o,0.04p,0.085q 0.07 − 0.11 0.06 0.072r,0.10e,0.14f
m⊥e 0.068 0.07n,0.05o,0.05p,0.085q 0.07 − 0.12 0.07 0.068r,0.10e,0.10f
InN EPk (eV) 8.74 10n,9.7o − − 9.22 − -
EP⊥ (eV) 8.81 10n,9.7o − − 8.67 − -
mkA 1.81 - 1.56 − 2.12 2.09 2.56-2.63r,1.35-1.43e,1.56f
m⊥A 0.13 - 0.17 − 0.21 0.14 0.14-0.15r,0.18-0.20e,0.17f
aVurgaftmanandMeyer,Ref.26.
bCarrierandWei,Ref.87.
cSilveiraetal.,Ref.123.
dFritschetal.,Ref.120.
eDugdaleetal.,Ref.124.
fPughetal.,Ref.122.
gReference26andreferencestherein.
hRodinaandMeyer,Ref.125.
iShokhovets etal.,Ref.126.
jRodinaetal.,Ref.127.
kRenetal.,Ref.121.
lCalculated using ∆SO = 0.019eV and ∆CR = 0.010eV
(Ref.127).
mCalculatedusing∆SO=0.016eVand∆CR=0.025eV(Ref.87).
nWuetal.,Ref.37.
oFuetal.,Ref.112.
pHofmannetal.,Ref.128.
qInushimaetal.,Ref.129.
rFritschetal.,Ref.130.
8.8eV).Ourcalculationsalsopredictananisotropyofthe inTab.VIIforthewurtzitephasesofGaNandAlN.Since
electron masses of about 5%. A similar anisotropy has the LDA predicts InN to be metallic, no LDA band pa-
been reported by Hofmann et al.128 (see Tab.VII) rameters could be derived for InN.
The effective electron masses of GaN in LDA are
smaller than in OEPx(cLDA)+G W and the experi-
0 0
2. Other parameter sets ment. The effective electron masses of a given material
are, to a first approximation, proportional to the fun-
damental band gap.136 Thus the underestimation of the
Local density approximation (LDA).
effectiveelectronmassesinLDAistosomedegreeanat-
For means of comparison we have also derived band pa-
ural side effect of the underestimation of the fundamen-
rametersfromLDAandOEPx(cLDA)calculationsinthe
tal band gap. Additional factors (e.g. self-interaction)
same way as for the OEPx(cLDA)+G W data. LDA
0 0
contributing to the deviationof the LDA band structure
band structures are frequently employed for fitting pa-
fromthequasiparticleonewerealludedtoinsectionIIA.
rameter sets,119,135 but we will demonstrate here, that
the LDA is not suitable to consistently determine allpa- The A-band hole masses in LDA show an increased
rameters for the group-III-nitrides accurately. The pa- anisotropy; the deviation from the experimental values
rametersderivedfromtheLDAbandstructuresarelisted increases.