Electronic structure and optical properties of Zn X(X=O,S,Se,Te):A density functional study S.Zh.Karazhanov,1,2P.Ravindran,1A.Kjekshus,1H.Fjellvåg,1and B.G.Svensson3 1Centre for Material Science and Nanotechnology,Department of Chemistry,University of Oslo,P.O.Box1033Blindern,
N-0315Oslo,Norway
2Physical-Technical Institute,2B Mavlyanov Street,Tashkent700084,Uzbekistan
3Department of Physics,University of Oslo,P.O.Box1048Blindern,N-0316Oslo,Norway
͑Received5July2006;revid manuscript received15November2006;published6April2007͒
Electronic band structure and optical properties of zinc monochalcogenides with zinc-blende-and wurtzite-
type structures were studied using the ab initio density functional method within the local-density approxima-
玉溪怎么辨别真假
tion͑LDA͒,generalized-gradient approximation,and LDA+U approaches.Calculations of the optical spectra
白鸽绿豆汤
have been performed for the energy range0–20eV,with and without including spin-orbit coupling.Reflec-
tivity,absorption and extinction coefficients,and refractive index have been computed from the imaginary part
of the dielectric function using the Kramers-Kronig transformations.A rigid shift of the calculated optical
spectra is found to provide a goodfirst approximation to reproduce experimental obrvations for almost all the
zinc monochalcogenide phas considered.By inspection of the calculated and experimentally determined
band-gap values for the zinc monochalcogenide ries,the band gap of ZnO with zinc-blende structure has
been estimated.
DOI:10.1103/PhysRevB.75.155104PACS number͑s͒:71.15.Ϫm,71.22.ϩi
I.INTRODUCTION
The zinc monochalcogenides͑Zn X;X=O,S,Se,and Te͒
are the prototype II-VI miconductors.The compounds
are reported to crystallize in the zinc-blende-͑z͒and wurtzite ͑w͒-type structures.The Zn X-z phas are optically isotropic, while the Zn X-w phas are anisotropic with c as the polar
axis.Zn X phas are a primary candidate for optical device
technology such as visual displays,high-density optical
memories,transparent conductors,solid-state lar devices,
photodetectors,solar cells,etc.So,knowledge about optical
properties of the materials is especially important in the
design and analysis of Zn X-bad optoelectronic devices.
Optical parameters for some of the Zn X phas have
widely been studied experimentally in the past.Detailed in-
formation on this subject is available for ZnO-w,1–9ZnS-w,9
ZnS-z,9–11ZnSe-z,9,10and ZnTe-z,9,10,12,13and e the sys-
tematized survey in Ref.14.However,there are no experi-
mental data on optical properties of ZnSe-w,ZnTe-w,and
ZnO-z.Furthermore,there is a lack of consistency between
some of the experimental values for the optical spectra.This
is demonstrated in Fig.1,which displays reflectivity spectra
for ZnO-w measured at T=300K by three different groups.
Dielectric-respon functions were calculated using the
Kramers-Kronig relation.As is en in Fig.1,intensity of the
imaginary part of the dielectric function͑⑀2͒and reflectivity ͑R͒corresponding to the fundamental absorption edge of ZnO-w are higher8than tho at the energy range10–15eV, while in Ref.14it is vice versa.The optical spectra in Fig.1 measured using the linearly polarized incident light for elec-tricfield͑E͒parallel͑ʈ͒and perpendicular͑Ќ͒to the c axes are somehow clo to tho of Ref.7using nonpolarized incident light.
Using the experimental reflectivity data,a full t of op-
tical spectra for ZnO has been calculated15for the wide en-
ergy range0–26eV.Density functional theory16͑DFT͒in the local-density approximation17͑LDA͒has also been ud to calculate optical spectra for ZnO-w͑Ref.18͒and ZnS-w ͑Ref.18͒by linear combination of atomic orbitals and for ZnS-z19and ZnSe-z19by lf-consistent linear combination of Gaussian orbitals.The optical spectra of ZnO͑including excitons͒has been investigated20by solving the Bethe-Salpeter equation.Band-structure studies have been per-formed by linearized-augmented plane-wave method plus lo-cal orbitals͑LAPW+LO͒within the generalized gradient and LDA with the multiorbital mean-field Hubbard potential ͑LDA+U͒approximations.The latter approximation is found to correct not only the energy location of the Zn3d electrons and associated band parameters͑e also
Refs.21 and22͒but also to improve the optical respon.Despite the shortcoming of DFT in relation to underestimation of band gaps,the locations of the major peaks in the calculated en-ergy dependence of the optical spectra are found to be in good agreement with experimental data.
It should be noted that the error in calculation of the band gap by DFT within LDA and generalized-gradient approxi-mation͑GGA͒is more vere in miconductors with strong Coulomb correlation effects than in other solids.21–25This is due to the mean-field character of the Kohn-Sham equations and the poor description of the strong Coulomb correlation and exchange interaction between electrons in narrow d bands͑viz.,the potential U͒.Not only the band gap͑E g͒but also the crystal-field͑CF͒and spin-orbit͑SO͒splitting ener-gies͑⌬CF and⌬SO͒,the order of states at the top of the valence band͑VB͒,the location of the Zn3d band and its width,and the band dispersion are found21,22,26,27to be incor-rect for ZnO-w by the ab initio full potential͑FP͒and atomic-sphere-approximation͑ASA͒linear muffin-tin orbital ͑LMTO͒methods within the pure LDA͑Refs.26and27͒and by the projector-augmented wave͑PAW͒method within LDA and GGA.21,22Thefindings were ascribed21,22to strong Coulomb correlation effects.DFT calculations within LDA plus lf-interaction correction͑LDA+SIC͒and LDA +U are found21,22,26to rectify the errors related to⌬CF and
PHYSICAL REVIEW B75,155104͑2007͒
⌬SO ,order of states at the top VB,and width and location of the Zn 3d band,as well as effective mass.In other mi-conductors,in which the Coulomb correlation is not suffi-ciently strong,the ⌬CF and ⌬SO values derived from DFT calculations within LDA are found to be quite accurate.This was demonstrated for diamondlike group IV ,z -type group III-V ,II-VI,and I-VII miconductors,28w -type AlN,GaN,and InN,29using the LAPW and V ASP -PAW,the w -type CdS and CdSe,27z -type ZnSe,CdTe,and HgTe,30using the ab initio LMTO-ASA,and z -and w -type ZnSe and ZnTe ͑Refs.21and 22͒as well as z -type CdTe,31using the V ASP -PAW and FP LMTO methods.Although the SO splitting at the top of VB is known to play an important role in electronic structure and chemical bonding of
牛字拼音
miconductors,
21,22,26,28–30,32,33there is no systematic study of the role of the SO coupling in optical properties of the materials.
Several attempts have been undertaken to resolve the DFT eigenvalue problem.One such approach is the utilization of the GW approximation ͑“G”stands for one-particle Green’s function as derived from many-body perturbation theory and “W”for Coulomb screened interactions ͒.Although GW re-moves
most of the problems of LDA with regard to excited-state properties,it fails to describe the miconductors with strong Coulomb correlation effects.For example,two studies of the band gap of ZnO calculated using the GW correction underestimated E g by 1.2eV ͑Ref.34͒and overestimated it by 0.84eV.35Calculations for Zn,Cd,and Hg monochalco-genides by the GW approach showed 36that the band-gap underestimation is in the range 0.3–0.6eV.Combination of exact-exchange ͑EXX ͒DFT calculations and the optimized-effective GW potential approach is found 37to improve the agreement with the experimental band gaps and Zn 3d en-ergy levels.Band gaps calculated within the EXX treatment are found to be in good agreement with experiment for the
s -p miconductors.38,39Excellent agreement with experi-mental data was obtained 39also for locations of energy levels of the d bands of a number of miconductors and insulators such as Ge,GaAs,CdS,Si,ZnS,C,BN,Ne,Ar,Kr,and Xe.Another means to correct the DFT eigenvalue error is to u the screened-exchange LDA.40Compared to LDA and GW,this approximation is found to be computationally much less demanding,permitting lf-consistent determination of the ground-state properties and giving more correct band gaps and optical properties.Other considered approaches for ab initio computations of optical properties involve electron-hole interaction,41partial inclusion of dynamical vertex cor-rections that neglect excitons,42and empirical energy-dependent lf-energy c
orrection according to the Kohn-Sham local-density theory of excitation.19However,the simplest method is to apply the scissor operator,43which displaces the LDA eigenvalues for the unoccupied states by a rigid energy shift.Using the latter method,excellent agree-ment with experiments has been demonstrated for lead monochalcogenides 44and ferroelectric NaNO 2.45However,the question as to whether the rigid energy shift is generally applicable to miconductors with strong Coulomb correla-tion effects is open.
In this work,electronic structure and optical properties of the Zn X -w and -z phas have been studied in the energy range from 0to 20eV bad on first-principles band-structure calculations derived from DFT within the LDA,GGA,and LDA+U .
IIPUTATIONAL DETAILS
Experimentally determined lattice parameters have been ud in the prent ab initio calculations ͑Table I ͒.The ideal positional parameter u for Zn X -w is calculated on the as-sumption of equal nearest-neighbor bond lengths:27
u =
13ͩa c
ͪ
2
+14
.͑1͒
The values of u for the ideal ca agree well with the experi-mental values u *͑e Table I ͒.Self-consistent calculations were performed using a 10ϫ10ϫ10mesh according to the Monkhorst-Pack scheme for the Zn X -z phas and the ⌫-centered grid for the Zn X -w phas.
A.Calculations by
V ASP
package
Optical spectra have been studied bad on the band-structure data obtained from the V ASP -PAW package,55which solves the Kohn-Sham eigenvalues in the framework of the DFT ͑Ref.16͒within LDA,
17GGA,56and the simplified ro-tationally invariant LDA+U .23,24The exchange and correla-tion energies per electron have been described by the Perdew-Zunger parametrization 57of the quantum Monte Carlo results of Ceperley and Alder.58The interaction be-tween electrons and atomic cores is described by means of non-norm-conrving pudopotentials implemented in the V ASP package.55The pudopotentials are generated in accor-dance with the PAW ͑Refs.59and 60͒method.The u of the PAW pudopotentials address the problem of
inad-
FIG.1.Reflectivity spectra R ͑͒for ZnO-w determined experi-mentally at 300K in Refs.9and 14͑solid circles ͒,Ref.8͑open circles ͒,and Ref.7͑solid lines ͒,along with the imaginary part of the dielectric-respon function ͓⑀2͔͑͒calculated using the Kramers-Kronig relation.The results of Ref.7͑open circles ͒are ud for both E ʈc and E Ќc ,becau no polarized incident light was ud in the experiments.
KARAZHANOV et al.PHYSICAL REVIEW B 75,155104͑2007͒
equate description of the wave functions in the core region ͑common to other pudopotential approaches61͒,and its ap-plication allows us to construct orthonormalized all-electron-like wave functions for Zn3d and4s and s and p valence electrons of the X atoms under consideration.LDA and GGA pudopotentials have been ud,and the completelyfilled micore Zn3d shell has been considered as valence states.
It is well known that DFT calculations within LDA and GGA locate the Zn3d band inappropriately clo to the top-most VB,hybridizing the O p band,falsifying the band dis-persion,and reducing the band gap.Nowadays,the problem is known to be solved by using the LDA+SIC and LDA +U.21,22,26,62–64For the DFT calculations within LDA+U, explicit values of the parameters U and J
are required as input.In previous papers,21,22we have estimated the values of the U and J parameters within the constrained DFT theory65and in a miempirical way by performing the cal-culations for different values of U and forcing it to match the experimentally established66location of the Zn3d bands. Bad on the results,21,22the values of the parameters U and J listed in Table I are chon to study the optical spectra.
B.Calculations by MINDLAB package
For investigation of the role of the SO coupling in elec-tronic structure and optical properties of Zn X,DFT calcula-tions have been performed using the MINDLAB package,67 which us the full potential linear muffin-tin orbital͑FP LMTO͒method.For the core charge density,the frozen-core approximation is ud.The calculations are bad on LDA with the exchange-correlation potential parametrized accord-ing to Gunnarsson-Lundquist68and V osko-Wilk-Nussair.69The ba geometry in this computational method consists of a muffin-tin part and an interstitial part.The basis t is comprid of linear muffin-tin orbitals.Inside the muffin-tin spheres,the basis functions,charge density,and potential are expanded in symmetry-adapted spherical harmonic functions together with a radial function and a Fourier ries in the interstitial.
胪怎么读C.Calculation of optical properties
From the DFT calculations,the imaginary part of the di-electric function⑀2͑͒has been derived by summing transi-tions from occupied to unoccupied states for energies much larger than tho of the phonons:
⑀2ij͑͒=Ve
2
2បm22
͵d3k͚
nnЈ
͗kn͉p i͉knЈ͘
ϫ͗knЈ͉p j͉kn͘f kn͑1−f knЈ͒␦͑⑀knЈ−⑀kn−ប͒.
͑2͒Here,͑p x,p y,p z͒=p is the momentum operator,f kn the Fermi distribution,and͉kn͘the crystal wave function correspond-ing to the energy⑀kn with momentum k.Since the Zn X-w phas are optically a
nisotropic,components of the dielectric function corresponding to the electricfield parallel͑Eʈc͒and perpendicular͑EЌc͒to the crystallographic c axis have been considered.The Zn X-z phas are isotropic;con-quently,only one component of the dielectric function has to be analyzed.
The real part of the dielectric function⑀1͑͒is calculated using the Kramer-Kronig transformation.The knowledge of
TABLE I.Theoretically and experimentally͑in brackets͒determined unit-cell dimensions a and c,vol-umes V,ideal u͓calculated by Eq.͑1͔͒,and experimental u*,as well as values of the parameters U and J from Refs.21and22,were ud in the prent calculations.For w-type structure,a=b.For the z-type structure, a=b=c and all atoms are infixed positions.
Pha
a
͑Å͒
c
͑Å͒
V
͑Å3͒u*u
U
͑eV͒
J
͑eV͒
ZnO-w a 3.244͑3.250͒ 5.027͑5.207͒45.82͑47.62͒0.3830.38091
ZnS-w b,c 3.854͑3.811͒ 6.305͑6.234͒81.11͑78.41͒0.3750.37561
ZnSe-w a,d 4.043͑3.996͒ 6.703͑6.626͒94.88͑91.63͒0.3750.37181
ZnTe-w e,f 4.366͑4.320͒7.176͑7.100͒118.47͑114.75͒0.3750.37371
商业银行信用风险ZnO-z g 4.633͑4.620͒99.45͑98.61͒81
ZnS-z h,i 5.451͑5.409͒161.99͑158.25͒91
ZnSe-z a 5.743͑5.662͒189.45͑181.51͒81
ZnTe-z i,j 6.187͑6.101͒236.79͑227.09͒81
Reference46.
b Reference18.
c Reference47.
d Reference48.
e Reference49.
f Reference50.
g Reference51.
h Reference52.
i Reference53.
j Reference54.
ELECTRONIC STRUCTURE AND OPTICAL PROPERTIES…PHYSICAL REVIEW B75,155104͑2007͒
both the real and imaginary parts of the dielectric tensor allows one to calculate other important optical spectra.In this paper,we prent and analyze the reflectivity R ͑͒,the absorption coefficient ␣͑͒,the refractive index n ͑͒,and the extinction coefficient k ͑͒:
R ͑͒=
鱼丸怎么做好吃ͯ
ͱ⑀͑͒−1ͱ⑀͑͒+1
ͯ
2
,͑3͒
␣͑͒=ͱ
2ͱ⑀12͑͒+⑀22
͑͒−2⑀1͑͒,
͑4͒
n ͑͒=
ͱͱ⑀12
͑͒+⑀22͑͒+⑀1͑͒
2
,͑5͒
k ͑͒=
ͱ
ͱ⑀12͑͒+⑀22
͑͒−⑀1͑͒
2
.͑6͒
Here,⑀͑͒=⑀1͑͒+i ⑀2͑͒is the complex dielectric function.
The calculated optical spectra yield unbroadened functions and,conquently,have more structure than the experimental ones.44,45,70,71To facilitate a comparison with the experimen-tal findings,the calculated imaginary part of the dielectric function has been broadened.The exact form of the broad-ening function is unknown.However,analysis of the avail-able experimentally measured optical spectra of Zn X shows that the broadening usually increas with increasing excita-tion energy.Also,the instrumental resolution smears out many fine features.The features have been modeled using the lifetime broadening technique by convoluting the imagi-nary part of the dielectric function with a Lorentzian with a full width at half maximum of 0.002͑ប͒2eV,increasing quadratically with the photon energy.The experimental reso-lution was simulated by broadening the final spectra with a Gaussian,where the full width at half maximum is equal to 0.08eV.
III.RESULTS AND DISCUSSION
A.Band structure
The optical spectra are related to band dispersion and probabilities of interband optical transitions.So,it is of in-terest to analyze the electronic structure in detail.Band dis-persions for Zn X -w and Zn X -z calculated by DFT within LDA and LDA+U are prented in Fig.2.The general fea-tures of the band dispersions are in agreement with previous studies ͑,Refs.26,62,and 72͒.It is en from Fig.2that the conduction-band ͑CB ͒minima for Zn X -w and Zn X -z are much more dispersive than the VB maximum,which shows that the holes are much heavier than the CB electrons in agreement with experimental data 73,74for the effective mass and calculated with FP LMTO and ͑Ref.26͒linear combination of atomic orbitals,18as well as with our findings.21,21Conquently,mobility of electrons is higher than that of holes.Furthermore,the features indicate that p electrons of X ͑that form the topmost VB states ͒are tightly bound to their atoms and make the VB holes less mobile.
Hence,the contribution of the holes to the conductivity is expected to be smaller than that of CB electrons even though the concentration of the latter is smaller than that of the former.The feature
s emphasize the predominant ionic na-ture of the chemical bonding.Another interesting feature of the band structures is that the VB maximum becomes more dispersive with increasing atomic number of X from O to Te.As noted in our previous contributions,21,22the band gaps of Zn X calculated by DFT within LDA,GGA,and LDA +U are underestimated and the question as to whether it is possible to shift the CB states rigidly was kept open.As found from the optical spectra discusd on the following ctions,rigid shifts of the CB states up to the experimen-tally determined locations can provide a good first approxi-mation for the stipulation of the band gap.So,for the band dispersions in Fig.2,we have made u of this simple way for correcting the band gaps calculated by DFT.The only problem in this respect was the lack of an experimental band-gap value for ZnO-z .To solve this problem,the experi-mental and calculated ͑by DFT within LDA ͒band gaps ͑E g ͒of the Zn X ries were plotted as a function of the atomic number of X .As en from Fig.3,E g for the Zn X -w phas are very clo to the corresponding values for the Zn X -z phas and the shape of the experimental
and calculated functional dependencies is in conformity.On this basis,the
FIG.2.Band dispersion for ZnO-w ,ZnS-w ,ZnSe-w ,ZnTe-w ,ZnO-z ,ZnS-z ,ZnSe-z ,and ZnTe-z calculated according to LDA ͑solid lines ͒and LDA+U ͑dotted lines ͒.The Fermi level is t to zero energy.
KARAZHANOV et al.PHYSICAL REVIEW B 75,155104͑2007͒
band gap of ZnO-z is estimated by extrapolating the findings for Zn X -z from ZnS-z to ZnO-z .This procedure gave E g Ϸ3.3eV for ZnO-z .
It is well known that not only band gaps are underesti-mated within LDA and GGA,but also band dispersions come out incorrectly,whereas location of energy levels of the Zn 3d electrons are overestimated ͑,Refs.20–22and 63͒.As also en from Fig.2,calculations within the LDA+U approach somewhat correct the location of the en-ergy levels of the Zn 3d electrons.The elucidation of the eigenvalue problem and the order of states at the topmost VB from LDA,GGA,and LDA+U calculations are discusd in Refs.20–22and 26and will not be repeated here.
Examination of Fig.2shows that the VB compris three regions of bands:first a lower region consist
s of s bands of Zn and X ,a higher-lying region of well localized Zn 3d bands,and on top of this a broader band dispersion originat-ing from X -p states hybridized with Zn 3d states.The latter subband is more pronounced in ZnO than in the other Zn X phas considered.The hybridization is most vere accord-ing to the LDA and GGA calculations,whereas the LDA +U calculations somehow suppress this and improve the band-gap underestimation.A more detailed discussion of the aspects is found in Refs.21and 22.
The SO splitting at the topmost VB is known to play an important role for the electronic structure and chemical bonding of solids.28,29,32In miconductors with z -type struc-ture,the SO splitting energy is determined as the difference between energies of the topmost VB states with symmetry ⌫8v and ⌫7v .28,29,32In the w -type compounds,the topmost VB is split not only by SO interaction but also by CF,giving ri to three states at the Brillouin-zone center.To calculate the
SO splitting energy for w -type phas,the quasicubic model of Hopfield 75is commonly ud.
It is well known that the SO splitting energy derived from ab initio calculations agrees well with experimental data only for some of the miconductors.This is demonstrated,for example,for all diamondlike group IV and z -type group III-V ,II-VI,and I-VII miconductors,28w -type AlN,GaN,and
InN,29Zn X -w and -z ͑X =S,Se,and Te ͒,21,22and CdTe.31However,the errors in estimated SO and CF splitting ener-gies by LDA calculations are significant for miconductors with strong Coulomb correlation effects,as ,for ZnO.21,22,26For such systems,DFT calculations within LDA+U ͑Refs.21,22,and 26͒are shown to provide quite accurate values for ⌬CF and ⌬SO .Overestimation of the p -d hybridization in various variants of the DFT can also lead to the wrong spin-orbit coupling of the valence bands.76,77
Systematic study of the SO coupling parameters was per-formed for zinc-blende II-VI miconductors ͑Ref.30͒using the TB and LMTO methods,as well as for all diamondlike and zinc-blende miconductors ͑Ref.28͒using the FLAPW method with and without the p 1/2local orbitals and the frozen-core PAW method implemented into V ASP .The cor-rections coming from the inclusion of the local p 1/2orbitals are found to be negligible for the compounds with light at-oms.Analysis of the results shows that the SO splitting energy coming from calculations using the V ASP -PAW shows good agreement with the experimental data.This result was also obtained 21recently for Zn X of wurtzite and zinc-blende structures.As demonstrated in Refs.21and 22the SO split-ting energy ͑⌬SO ͒increas when one moves from ZnO-z to ZnTe-z ,in agreement with earlier findings of Ref.28.
To study the role of the SO coupling in band dispersion,the prent ab initio calculations have been performed by V ASP and MINDLAB packages and spin-orbit splitting energy is found.The results are prented in Table II .Analysis of Table II shows that ͑⌬SO ͒calculated by MINDLAB is quite accurate.
As expected,band dispersions calculated with and with-out the SO coupling differ little when the SO splitting
energy
FIG.3.Band gaps for Zn X -w ͑circles ͒and Zn X -z ͑triangles ͒phas determined experimentally ͑filled symbols,from Refs.21and 22͒and calculated ͑open symbols ͒by DFT within LDA as a function of the atomic number of the X component of Zn X .
TABLE II.Calculated SO splitting energy ͑in meV ͒using the MINDLAB package along with the previous theoretical and experi-mental findings.ZnO-z ZnS-z ZnSe-z ZnTe-z –3166432914–3166432914−34a 66a 393a 889a −34b 66b 398b 916b −37c 64c 392c 898c −33d
64d 393d 897d 65e
420f
910f
LAPW,Ref.28.
b LAPW+p
1/2,Ref.28.
c V ASP -PAW,Ref.28.d
V ASP -PAW,Ref.21.e
断奶Experiment,Ref.78.f Experiment,Ref.79.
ELECTRONIC STRUCTURE AND OPTICAL PROPERTIES …PHYSICAL REVIEW B 75,155104͑2007͒
is small.However,the difference increas when one moves from ZnO to ZnTe.This feature is demonstrated in Table II and Fig.4for band dispersions of ZnO-z ,ZnO-w ,ZnTe-z ,and ZnTe-w calculated by V ASP with and without including the SO coupling.As is well known ͑,Refs.21,26,and 27͒,without the SO coupling,the top of the VB of Zn X -w is split into a doublet and a singlet state.In the band structure,the Fermi level is located at the topmost one ͑Fig.4͒,which is the zero energy.Upon inclusion of the SO cou-pling into calculations,the doublet and singlet states are split into three twofold degenerate states called A ,B ,and C states with energies E g ͑A ͒,E g ͑B ͒,and E g ͑C ͒,respectively,80ar-ranged in order of decreasing ,E g ͑A ͒ϾE g ͑B ͒ϾE g ͑C ͒.The center of gravity of the A ,B ,and C states,located at ͓E g ͑A ͒−E g ͑C ͔͒/3below the topmost A state,re-mains to be nearly the same as the topmost VB,correspond-ing to the ca without the SO c
oupling.26,27Conquently,to compare band structures calculated with and without the SO coupling,one should plot the band structure with the Fermi energy at the center of gravity of the A ,B ,and C states for the former and at the topmost VB for the latter.Hence,when the SO coupling is applied,the A and B states as well as the bottommost CB move upwards to ͓E g ͑A ͒−E g ͑C ͔͒/3in en-ergy,whereas the C state moves downwards to ͓E g ͑A ͒−E g ͑C ͔͒2/3compared to the center of gravity.Then,posi-tions of the lowest VB region calculated with and without the SO coupling remain nearly identical.
B.General features of optical spectra of Zn X
Since optical properties of solids are bad on the band structure,the nature of the basic peaks in the optical spectra
can be interpreted in terms of the interband transitions re-sponsible for the peaks.Such an interpretation is available for miconductors with z -and w -type structures.11,14,81In order to simplify the prentation of the findings of this work,the labels E 0,E 1,and E 2of Ref.11͑from the reflec-tivity spectra ͒were retained in Table III and Fig.4.The subscript 0is ascribed to transitions occurring at ⌫,the sub-script 1to transitions at points in the ͓111͔direction,and the subscript 2to transit
ions at points in the ͓100͔direction ͑re-ferring to the k space for the z -type structure ͒.Assignment of the E 0,E 1,and E 2peaks to optical transitions at high-symmetry points is prented in Table III and Fig.4.
The optical spectra ⑀1͑͒,⑀2͑͒,␣͑͒,R ͑͒,n ͑͒,and k ͑͒calculated by DFT within LDA,GGA,and LDA+U are displayed in Figs.5–8and compared with available experi-mental findings.14The spectral profiles are indeed very simi-lar to each other.Therefore,we shall only give a brief ac-count mainly focusing on the location of the interband optical transitions.The peak structures in Figs.5–8can be explained from the band structure discusd above.
All peaks obrved by experiments ͑,Refs.11and 14͒are reproduced by the theoretical calculations.Becau of the underestimation of the optical band gaps in the DFT calculations,the locations of all the peaks in the spectral profiles are consistently shifted toward lower energies as compared with the experimentally determined spectra.Rigid shift ͑by the scissor operator ͒of the optical spectra has been applied,which somewhat removed the discrepancy between the theoretical and experimental results.In general,the cal-culated optical spectra qualitatively agree with the experi-mental data.In our theoretical calculations,the intensity of the major peaks are underestimated,while the intensity of some of the shoulders is overestimated.This result is in good agreement with previou
s theoretical findings ͑,Ref.19͒.The discrepancies are probably originating from the ne-glect of the Coulomb interaction between free electrons and holes ͑excitons ͒,overestimation of the optical matrix ele-ments,and local-field and finite-lifetime effects.Further-more,for calculations of the imaginary part of the dielectric-respon function,only the optical transitions from occupied to unoccupied states with fixed k vector are considered.Moreover,the experimental resolution smears out many fine features,and,as demonstrated in Fig.1,there is inconsis-tency between the experimental data measured by the same method and at the same temperature.However,as noted in the Introduction,accounting for the excitons and Coulomb correlation effects in ab initio calculations 20by the LAPW +LO within LDA+U allowed correcting not only the芝士烤土豆
energy
FIG. 4.Band dispersion for ZnO-z ,ZnO-w ,ZnTe-z ,and ZnTe-w calculated by the V ASP -PAW method within LDA account-ing for SO coupling ͑solid lines ͒and without SO coupling ͑open circles ͒.Topmost VB of the band structure without SO coupling and center of gravity of that with SO coupling are t at zero energy.Symmetry labels for some of the high-symmetry points are shown for ͑c ͒ZnTe-z and ͑d ͒ZnTe-w to be ud for interpretation of the origin of some of the peaks in the optical spectra of Zn X -w and Zn X -z .
TABLE III.Relation of the basic E 0,E 1,and E 2peaks in the optical spectra of Zn X to high-symmetry points ͑e Refs.11and 14͒in the Brillouin zone at which the transitions em to occur.Peak z type w type,E ʈc w type,E Ќc E 0⌫8→⌫6⌫1→⌫1⌫6→⌫1E 1L 4,5→L 6A 5,6→A 1,3
M 4→M 1
E 2
X 7→X 6
KARAZHANOV et al.PHYSICAL REVIEW B 75,155104͑2007͒
