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Materials Science in Semiconductor Processing

Materials Science in Semiconductor Processing 16 (2013) 1619–1628

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Density functional and tight binding theories of electronicproperties of II–VI heterostructures

H. Hakan Gürel a,1, Hilmi Ünlü a,b,n

a İstanbul Technical University, Informatics Institute, Computational Science and Engineering Department, Maslak, 34469 Istanbul, Turkeyb İstanbul Technical University, Faculty of Science and Letters, Department of Physics, Maslak, 34469 Istanbul, Turkey

a r t i c l e i n f o

Available online 5 July 2013

Keywords:DFT-MBJLDAWien2kNN sp3d5 and 2NN sp3sn TB models Cd(Zn)S(Se)/Cd(S, Se, Te) heterostructures

01/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.mssp.2013.05.021

esponding author at: İstanbul Technical Uand Letters, Department of Physics, MasTel.: +90 212 285 3201; fax: +90 212 285 6ail address: hunlu@itu.edu.tr (H. Ünlü).esent address: Kocaeli University, TechnolInformation Systems Engineering, 41380 İz

a b s t r a c t

We present comparative calculations of the electronic structure of Cd and Zn based groupII–VI compounds and their heterostructures based on the density functional and tightbinding theories. The first principles density functional theory (DFT) uses the modifiedBecke–Johnson exchange potential with LDA correlation potential (MBJLDA) and the semi-empirical tight binding theory uses the first nearest neighbor (NN) sp3d5 and secondnearest neighbor (2NN) sp3sn basis with spin–orbit coupling of II cation (Cd, Zn) and VIanion (S, Se, Te) atoms for calculating the electronic structure of Cd and Zn based II–VIcompounds and their heterostructures. The results of DFT with MBJLDA functional and NNsp3d5 and 2NN sp3sn TB models are found to be in excellent agreement with measuredband gaps of CdX and ZnX (X¼S, Se, Te) based group II–VI compounds and their CdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe heterostructures. We conclude that the NN sp3d5 TBmodel gives much more physical insight than the (2NN) sp3sn TB model, making use of thefictitious s* state unnecessary.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Cadmium and zinc based group II–VI compounds andtheir ternary/binary heterostructures have received consider-able interest among the material scientists and deviceengineers over the years. These semiconducting compoundsare known to have a considerable potential for makingoptical and photovoltaic devices (e.g., light emitting diodes,laser diodes, infrared detectors, photovoltaic devices, andquantum dots [1,2]). Therefore, accurate and reliable physicalmodeling of their structural properties (e.g., lattice constantsand bulk modulus) and electronic properties (e.g., band gap,effective mass, etc.) is essential for precise controlling of theirgrowth conditions and better predictions of their potential inutilizing them for the fabrication of high performance

All rights reserved.

niversity, Faculty oflak, 34469 Istanbul,386.

ogy Faculty, Depart-mit, Turkey.

electronic and optical devices [1]. The theoretical calculationsof structural and electronic properties of semiconductors areoften carried out by: (i) ab-initio methods, such as thedensity functional theory (DFT), and (ii) empirical methods,such as the nonlocal empirical pseudo-potential method(EPM), the k.p. method or by semi-empirical tight bindingmethods with various orbital bases (e.g., sp3, sp3sn, sp3snd5

and sp3d5).It is a well known fact that the conventional first principle

density functional theory (DFT) allows one to calculate theelectronic structure from first principles, in which there is noneed for empirical fitting parameters. Although the DFTcalculations yield satisfactory results for the structural prop-erties of semiconductors [2], they fail to produce satisfactoryresults for the electronic properties of the semiconductorswhen compared with the experiment [3,4] and the nonlocalempirical pseudo-potential theory [5]. This is attributed tothe self-interaction errors and discontinuity of the exchangecorrelation potential in calculating band gaps with conven-tional DFT. Although the GW approximation or hybridfunctional overcomes the band gap problem, they still resultin band gap error on the order of 10–20% compared to the

H. H. Gürel, H. Ünlü / Materials Science in Semiconductor Processing 16 (2013) 1619–16281620

experiment [3]. Recently, Tran and Blaha [9] proposed a newpotential called the modified Becke–Johnson density func-tional (MBJLDA) that combines the Becke–Johnson exchangepotential and the local density approximation (LDA) correla-tion potential in the DFT for band structure calculations.

The semi-empirical tight binding model, with variousorbital bases (e.g., sp3, sp3sn,sp3snd5, and sp3d5), is knownto realistically describe the electronic properties of semi-conductors. However, there is still a question about the useof the fictitious s* state in the TB model (such as sp3sn andsp3snd5) in the electronic structure calculations. Since theinclusion of sn excited state is done by modeling theaverage of p–d interactions, the sp3sn tight binding para-meterization does not permit the inclusion of excited d-orbitals to the sp3 basis and therefore, the actual contribu-tion of the excited d-states to the band structure calcula-tions is not reliably and accurately reflected.

In Section 2.1, we will briefly describe the density func-tional theory (DFT) that uses the modified Becke-Johnsonexchange potential with local density approximation (LDA)correlation potential (called MBJLDA) [6] for the electronicstructure calculations. In Section 2.2, we briefly describe thesemi-empirical nearest neighbor (NN) sp3d5 and secondnearest neighbor (2NN) sp3sn tight binding parameterizations,with spin–orbit coupling of II cation (Cd, Zn) and VI anion (S,Se, Te) atoms, based on the parametrization presented in Refs.[15–18]. In Section 3, we present the results of our presentcalculations for the electronic properties of CdX and ZnX(X¼S, Se, Te) compounds and their heterostructures (CdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe).

2. Band structure calculation methods

2.1. Density functional theory with the modifiedBecke–Johnson potential

The conventional density functional theory (DFT) utilizesthe variation method to calculate the ground state propertiesof many-body system and allows one to determine the

Fig. 1. The crystal structures of C

structural and electronic properties of semiconductors fromfirst principles. The conventional DFT calculations give satis-factory ground state properties [6] (such as total energiesand also lattice constant and bulk modulus) but giveunsatisfactory electronic properties (such as band gaps andeffective masses) [3]. The DFT calculations yields band gapsthat are very small as compared with the predictions of thenonlocal pseudo-potential theory [5] and the experiment[3].Different approaches (such as GW approximation or hybridfunctional) have been implemented in the DFT to improve itsband gap predictions. Even GW and hybrid functional resultin band gap error on the order of 10–20% according toexperimental data [3]. The exchange-correlation potentialcontribution to the DFT band gap is shifted by using a scissoroperator, in accordance to the suggestion of Fiorentini andBalderschi [7]. The difference between the LDA calculatedand measured band gaps (ΔE) scales with the opticaldielectric constant (ΔE≅9:1=ε∞) [7,8].

Recently, Tran and Blaha [6] proposed a new potentialcalled modified Becke–Johnson density functional (MBJLDA)that combines the Becke–Johnson exchange potential andthe local density approximation (LDA) correlation potentialin the DFT for band structure calculations. The MBJLDAfunctional is a exchange-correlation (XC) potential that isobtained as functional derivative of the XC-energy functionalExc with respect to the electron density ρs(V

MBJx;s ¼ δEx=δρs),

taken from LDA. In our recent study [4], we compared theresults of the MBJLDA functional based DFT band gapcalculations with GW (RPA-type screening of the electron–-electron interaction) and G′W′ (GW-RPA with G and W re-calculated by updating the eigenvalues) results. For CdS,CdSe and CdTe, we compared MBJLDA band energies withour previous uncorrected LSDA results [8] and other LDAresults as parameterized by Perdew and Zunger [13]. Ourobservation was that the MBJLDA functional based DFTresults were much more satisfactory than the LSDA resultsfor conduction bands and band gaps of CdS, CdSe and CdTe.The DFT calculations with MBJLDA functional yield band gapsare in good agreement with the experiment and the pseudo-potential theory [3,10–12].

dSe and ZnS compounds.

H. H. Gürel, H. Ünlü / Materials Science in Semiconductor Processing 16 (2013) 1619–1628 1621

In this work we used a full potential linear augmentedplane wave method to calculate the electronic properties ofzinc blende CdX and ZnX (X¼S, Se, Te) which uses the socalled modified Becke–Johnson density functional(MBJLDA), proposed by Tran and Blaha [9]. We used theMBJLDA implementation available in the Wien2K [14]package for the DFT calculations of the electronic structureof CdX and ZnX (X¼S, Se, Te) group II–VI compounds andtheir heterostructures (CdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe). Fig. 1 shows the crystal structure of CdSe and ZnScompounds for visualizing the problem. CdX and ZnX (X¼S,Se, Te) have a zinc-blende crystal structure (space group F-43m No: 216) in which the Cd/Zn atoms are located at (0, 0,0) and S (or Se) at (0.25, 0.25, 0.25). A converged results areobtained using 286 k points in the first Brillouin zone withRMTKmax¼8.50, where RMT represents the smallest muffin-thin radius and Kmax is the maximum size of reciprocallattice vectors. LSDA, GGA and MBJLDA implemented in theWIEN2k package code are used for exchange and correla-tion potentials in our present calculations. SCF iterations arerepeated until total energy converges to a point less than10−4 Ryd. In Section 3, we will show that DFT with MBJLDAperforms well in reproducing the electronic properties ofCdX and ZnX (X¼S, Se, Te) II–VI compounds and CdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe heterostructures.

2.2. Semi-empirical NN sp3d5 and 2NN sp3sn tight bindingmodels

In this sub-section, we briefly describe the semi-empirical first nearest neighbor (NN) sp3d5 and secondnearest neighbor (2NN) sp3sn tight binding (TB) models weused in the calculation of electronic structure of CdX andZnX (X¼S, Se, Te) compounds and their heterostructures.We begin with the general solution of the SchrödingerEquation written in matrix form [15]:

∑β½HαβðkÞ−SαβðkÞE�uβ ¼ 0 ð1Þ

where E is the energy eigenvalue of the Hamiltonianmatrix Hαβ , given as

Hαβ ¼ ⟨χαðkÞjHjχβðkÞ⟩¼ εαβ þ ∑i≠0

Iαβð0; iÞeikri þ Hso ð2Þ

where the diagonal elements εαβ are the atomic orbitalenergies of on-site energy for β orbital (s, p, d) at theatomic site α (cation (Cd, Zn) and anion (S, Se, Te) atoms).They represent the intra-atomic integrals that coupleatomic orbitals located in the same unit cell. The offdiagonal elements Iαβð0; iÞ represent the nearest neighbor(NN) interaction integrals that couples atomic orbitalslocated in different unit cells. Sαβ ¼ ⟨χαðkÞjχβðkÞ⟩ is theoverlap integral between the atomic-like orbitals and istaken to simplify the NN sp3d5 and 2NN sp3sn TB para-meterizations. jχðkÞ⟩ is the basis function formed by thelinear combination of s, p and d orbitals of cation andanion atoms with wave function coefficient uβ .

In the 2NN sp3sn TB model, the Hamiltonian matrix Hαβ

has 13 independent matrix elements; six diagonal elements

(on-site atomic energies: Esa; Esc; Epa; Epc; Esna; Esnc) andseven off-diagonal elements (interaction integrals:Ess; Exx; Esapc ; Escpa ; Exy; Esnp; Epsn ). The inclusion of spin–orbit coupling to the sp3sn basis set adds two more TBparameters. The spin–orbit interaction is given by two para-meters; λa ¼ ⟨xa↑jHsojza↓⟩ for anion atom and λc ¼⟨xc↑jHsojzc↓⟩ for cation atom. The role of λa is to reproducethe bulk zone center splitting between the split-off band andthe light-hole and heavy-hole bands. The inclusion of 2NNinteractions (εsx and εxy) in the sp3sn basis set adds two moreinteraction TB parameters and that the size of the Hαβ

becomes a 20�20 one for the cubic structure that isdiagonalized for each k vector.

In the first nearest neighbor (NN) sp3d5 TB model, theHamiltonian matrix Hαβ has 19 independent matrix ele-ments: nine on-site and 10 off-site elements. The ninestateatomic-like (s; x; y; z; xy; yz; zx; x2−y2; 3z2−r2) basisset describes each cation and anion atoms of semiconduc-tors. The inclusion of spin–orbit coupling to the sp3d5 basisset, in which the spin–orbit interaction is given by twoparameters; λa ¼ ⟨xa↑jHsojza↓⟩ for anion and λc ¼ ⟨xc↑jHsojzc↓⟩for cation atom, adds two extra TB parameters. Therefore, thesp3d5 Hamiltonian matrix has total of 21 independent matrixelements.

The Hamiltonian matrix elements in both the NN sp3d5

and 2NN sp3sn TB models are conventionally determinedby fitting the calculated energy band dispersion curves tothose of the nonlocal pseudo-potential theory for bulksemiconductors [6] and experimental data [3]. One startswith estimating the values of on-site and off-site matrixelements (εαβ and Iαβð0; iÞ) in the Hamiltonian matrix Hαβ

and then carry out a least-squared error minimizationfitting procedure at a number of high symmetry points inthe valence and conduction band dispersion curves to fitband gap energies at Γ, X and L high symmetry points ofthe first Brillouin zone, obtained from the empirical non-local pseudo-potential method. We have previouslydescribed the finding of the optimized tight bindingparameters in the 2NN sp3sn and NN sp3d5 to calculatethe electronic band structure of semiconductors [15–18]and will not repeat the same procedure here. We shallsimply use these optimized tight binding parameters weobtained earlier [15–18].

2.3. Strain effects in Cd(Zn)S(Se)/Cd(Zn) (S, Se, Te) II–VIheterostructures

As further application of the MBJLDA functional based DFTand the semi-empirical tight binding theory with NN sp3d5

and 2NN sp3sn basis, we briefly described in Section 2.2, theeffect of alloy composition and interface strain on electronicproperties of Cd and Zn based group II–VI ternary/binaryheterostructures is also carried out by using the so calledModified Virtual Crystal Approximation (MVCA) [16,19,20]. Inthe MVC approximation one first formulates the bond lengthof ternary. The effect of composition disorder on the electronicproperties of ternary semiconductors is implemented in theMBJLDA functional based DFT and in semi-empirical NN sp3d5

and 2NN sp3sn TB models in terms of the host bond lengthand the distorted bond length by the substitutional impuritywithout any adjustable parameter [16,19,20].

Table 1Fundamental band gaps of CdS, CdSe, CdTe, ZnS, ZnSe and CdTe bulksemiconductors calculated by using program package WIEN2k [14] withmodified Becke–Johnson local density approximation (MBJLDA), the

semi-empirical 2NN sp3sn and NN sp3d5 tight binding models, comparedwith available experimental data [23].

Compound Reference EΓg (eV) ELg(eV) EXg (eV) Relative errorin MBJLDA (%)for EΓg

CdS DFT-MBJLDA 2.66 4.28 4.71 4.312NN sp3snTB 2.476 3.983 4.341 –

NN sp3d5 TB 2.555 5.193 4.696 –

Experimenta 2.55 – – –

CdSe DFT-MBJLDA 1.89 3.40 4.04 3.852NN sp3snTB 1.887 3.097 3.784 –

NN sp3d5 TB 1.937 3.027 3.779 –

Experimenta 1.82 – – –

CdTe DFT-MBJLDA 1.56 2.36 3.03 –2.502NN sp3sn TB 1.59 2.46 3.57 –

NN sp3d5 TB 0.76 1.83 4.76 –

Experimenta 1.60 – – –

ZnS DFT-MBJLDA 3.66 4.48 4.42 –4.192NN sp3sn TB 3.680 4.810 5.103 –

NN sp3d5 TB 3.702 4.641 5.190 –

Experimenta 3.82 – – –

ZnSe DFT-MBJLDA 2.67 3.61 3.78 –5.322NN sp3sn TB 2.83 3.99 4.54 –

NN sp3d5 TB 1.91 2.14 5.04 –

Experimenta 2.82 – – –

ZnTe DFT-MBJLDA 2.22 2.45 2.71 –7.112NN sp3sn TB 2.39 2.80 3.43 –

NN sp3d5 TB 1.53 1.62 3.34 –

Experimenta 2.39 – – –

a Ref. [23].

H. H. Gürel, H. Ünlü / Materials Science in Semiconductor Processing 16 (2013) 1619–16281622

The bond length of A1−xBxC ternary constituent in anABC/AC heterostructure is written as sum of undistortedbond length (dVCA ¼ ð1−xÞd0AC þ xd0BC), determined accord-ing to the virtual crystal approximation (VCA) and thedistorted term drelax ¼ xð1−xÞδcðdBCðxÞ−dAC ðxÞÞ, due tocation–anion relaxation [16,19,20]:

dðxÞ ¼ dVCA þ drelax ¼ ð1−xÞd0AC þ xd0BCþxð1−xÞδcðdBCðxÞ−dACðxÞÞ ð3Þ

where d0AC and d0BC are the undistorted bond lengths of hostmaterials AC and BC. dACðxÞ and dBCðxÞ are the bond lengthsof AC and BC binaries in A1−xBxC ternary constituent.

dBCðxÞ ¼ ð1−xÞd0AC þ xd0BC þ ð1−xÞξBC:Aðd0BC−d0AC Þ ð4Þ

dACðxÞ ¼ ð1−xÞd0AC þ xd0BC þ xξAC:Bðd0AC−d0BCÞ ð5Þδc is the difference between dimensionless relaxation

parameters ξBC:A and ξAC:B.

δc ¼ ξAC:B−ξBC:A ¼1

1þ ðαAC=6αBCÞð1þ 10ðβAC=αAC ÞÞ−

11þ ðαBC=6εACÞð1þ 10ðβBC=αBC ÞÞ

ð6Þ

Eq. (3) can now be used to take into account thecomposition variations of the band structures of hetero-structures in the DFT with MBJLDA functional and thesemi-empirical theory, with NN sp3d5 and 2NN sp3sn basis.The diagonal elements in NN sp3d5 and 2NN sp3sn TBHamiltonian matrices for A1−xBxC ternary are expressed asa nonlinear function of alloy composition [16,19,20].

Eα=βðxÞ ¼ ð1−xÞEα=βðACÞ þ xEα=βðBCÞþxð1−xÞδc½Eα=βðACÞ−Eα=βðBCÞ� ð7Þ

where Eα=βðACÞ and Eα=βðBCÞ represent the fitted energiesof the s, p and d states of anion and cation atoms formingthe AC and BC compounds.

The interface strain that causes the modification of theenergy levels of heterostructure constituent can also beincorporated in Eq. (3). The elastic theory allows one todecompose the biaxial strain tensor into the sum of hydro-static and uniaxial strains. The hydrostatic strain correspondsto the relative volume change of the crystal; TrðεÞ ¼ 2ε⊥ þεjj ¼ΔV=V which leads to shifts of the conduction bandminimums relative to the average valence band maximum atΓ point. The uniaxial strain splits the heavy-hole, light-holeand split-off valence band edges relative to the averagevalence band edge [20]. The lattice constant of strainedepilayer along the interface is equal to that of the substrateand is expanded/or compressed in the direction parallel tothe interface. In the case of (001) heterepitaxy one has

εxx ¼ εyy ¼ εf jj ¼af jjaf0

−1� �

and

εzz ¼ εf⊥ ¼ af⊥af0

−1� �

¼ 2C12

C11

� �fεf jj ð8Þ

where εf⊥ and εf jj are the strain components perpendicularand parallel to interface. af⊥ is lattice constant of epilayerperpendicular to the interface.

af⊥ ¼ af0 1−2C12

C11

� �f

af ∥af0

−1� �" #

and af jj ¼ as

where af jj is the lattice constant of strained epilayer parallelto the interface equals to that of unstrained substrate(af jj ¼ as). af o is the unstrained lattice constant of epilayerand C11 and C12 are elastic stiffness constants of strainedepilayer.

The strain dependent off-site tight binding matrixelements in the semi-empirical NN sp3d5 and 2NN sp3sn

TB models obeys the universal scaling law [22], writtenhere for lattice constants as

Hsllm ¼Hllmða=a0Þ−ηllm ¼Hllmða0Þ 1þ 2

3ðεxx þ εyy þ εzzÞ

�

þ13ðε2xx þ ε2yy þ ε2zzÞ

�−ηllm=2ð9Þ

Here a and a0 are the strained and the unstrained values ofthe bulk lattice constant. The ηllm exponent (for thehopping interactions between α and β orbitals) needs tobe adjusted in order to fit the first pressure derivatives ofband gaps EgΓ ; EgL and EgX to their experimental and/ornonlocal pseudo-potential values, the fitting processdepends on the mapping of large number of orbitalcoupling parameters on the set of observables. Since there

H. H. Gürel, H. Ünlü / Materials Science in Semiconductor Processing 16 (2013) 1619–1628 1623

are not many analytical expressions available, ηllm is oftentaken to be constant (ηllm ¼ 2) for practical purposes.

The difficulty in reliable determination of the effects ofstrain on the conduction and valence band energy levelsby using the NN sp3d5 and 2NN sp3sn TB models, accordingto Eq. (9), and by the MBJLDA functional based DFT isovercome by using the so-called statistical thermodynamicmodel [21]. In this model the conduction electrons andvalence holes are treated as charged chemical particles.One expresses the conduction and valence band edges as afunction of pressure at any temperature

EclðT ; PÞ ¼ Ecl þ C0cPTð1−lnTÞ−

aclB

P−P2

2B−ð1þ B′ÞP3

6B2

" #ð10Þ

EvðT ; PÞ ¼ Ev þ C0vPTð1−lnTÞ−

avB

P−P2

2B−ð1þ B′ÞP3

6B2

" #ð11Þ

where P is the hydrostatic pressure. P ¼−2Bf Cf εf jj is forepilayer and P ¼−3Bsεsjj is for substrate withCf ¼ ðC11−C12Þ=C11 (001) for crystal growth. acl ¼−Bð∂Ecl=∂PÞ and av ¼ −Bð∂Ev=∂PÞ are the conduction andthe valence band deformation potentials at Γ, L and X highsymmetry points, and B is the bulk modulus with its

derivative B′¼ ∂B=∂P. ΔC0P ¼ C0

nP þ C0pP−C

00P is the heat

capacity of reaction for the formation of free electrons

Table 2Calculated energies (in eV) at symmetry points for CdS, CdSe and CdTe.

Reference Γ1v Γ15v Γ1c Γ15c L1v L1v

CdS Present (MBJLDA) −12.21 0.00 2.66 7.20 −11.85 −3.8Theory (sp3d5)a −12.41 0.00 2.56 6.00 −13.24 −3.4Theory (sp3d5)b −12.06 0.00 1.68 7.13 −12.08 −3.5Theory (sp3s*)a −12.23 0.00 2.48 6.39 −11.92 −4.7

CdSe Present (MBJLDA) −12.31 0.00 1.89 6.72 −11.99 −4.0Theory (sp3d5)a −12.02 0.00 1.94 4.72 −11.46 −5.1Theory (sp3d5)b −12.29 0.00 0.86 6.57 −12.44 −4.0Theory (sp3s*)a −11.51 0.00 1.89 5.71 −11.14 −5.2

CdTe Present (MBJLDA) −11.11 0.00 1.56 4.99 −10.81 −4.3Theory (sp3d5)b −10.66 0.00 0.76 4.96 −10.79 −4.0Theory (sp3s*)c −11.07 0.00 1.59 5.36 −10.03 −4.5

a Ref. [17].b Ref. [18].c Ref. [8].

Table 3Calculated energies (in eV) at symmetry points for ZnS, ZnSe and ZnTe.

Reference Γ1v Γ15v Γ1c Γ15c L1v L1v

ZnS Present (MBJLDA) −12.97 0.00 3.66 7.15 −12.14 −5.9Theory (sp3d5)a −14.18 0.00 3.70 7.93 −13.82 −3.2Theory (sp3d5)b −10.77 0.00 3.44 6.70 −11.64 −3.0Theory (sp3s*)a −14.22 0.00 3.68 7.94 −13.33 −4.2

ZnSe Present (MBJLDA) −13.02 0.00 2.67 6.50 −12.28 −4.8Theory (sp3d5)b −12.54 0.00 1.91 6.21 −12.55 −4.2Theory (sp3s*)c −13.78 0.00 2.83 7.41 −13.32 −4.8

ZnTe Present (MBJLDA) −11.78 0.00 2.22 4.75 −10.96 −5.0Theory (sp3d5)b −10.98 0.00 1.53 4.57 −10.82 −4.5Theory (sp3s*)c −13.00 0.00 2.39 4.60 −12.02 −5.7

a Ref. [17].b Ref. [18].c Ref. [8].

and holes obtained from fitting the predicted band gapEglðT ; PÞ ¼ EclðT ; PÞ−EvðT ; PÞ to its measured value. Here

C0cP ¼ C0

nP−C00P ¼ C0

pP þ ΔC0P and C0

vP ¼ C0pP are the standard

heat capacities of conduction electrons and valence holes

C0nP ¼ C0

pP ¼ ð5=2Þk, where k is the Boltzmann's constant.

3. Results and discussion

In this section we present the results of our presentband structure calculations based on the WIEN2k packagesimulations, with MBJLDA functional based DFT, and thesemi-empirical tight binding theory, with NN sp3d5 and2NN sp3sn basis, for CdX and ZnX (X¼S, Se, Te) compoundsand CdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe heterostrcuc-tures. We will give a brief discussion of our findingsdisplayed in Tables 1–3 and Figs. 2–6.

3.1. Band structures of CdX and ZnX II–VI compounds

As shown in Table 1, our band structure calculationsusing the WIEN2k package simulation, based on DFT withMBJLDA functional, are in good agreement with those ofthe semi-empirical NN sp3d5 and 2NN sp3sn TB models forCdX and ZnX (X¼S, Se, Te) II–VI compounds. As thecomparison in Table 1 demonstrates, the MBJLDA

L3v L1c L3c X1v X3v X5v X1c X3c

9 −0.66 4.28 7.74 −11.75 −3.48 −1.68 4.71 5.354 −0.26 5.19 6.41 −13.48 −3.84 −0.49 4.70 6.608 −1.10 2.54 8.23 −11.87 −4.21 −2.04 5.03 5.443 −0.81 3.98 5.66 −11.94 −4.24 −1.60 4.34 5.331 −0.67 3.40 7.10 −11.89 −3.73 −1.70 4.04 4.506 −1.24 3.03 6.13 −11.28 −5.26 −2.51 3.78 5.684 −1.12 2.25 7.68 −12.38 −4.42 −2.07 4.45 4.932 −1.21 3.10 5.05 −11.01 −5.18 −2.68 3.78 4.260 −0.72 2.36 7.75 −10.72 −4.18 −1.79 3.03 3.103 −1.00 1.83 6.52 −10.75 −4.41 −1.86 4.16 4.241 −0.79 2.46 5.01 −9.48 −5.05 −1.98 3.77 3.95

L3v L1c L3c X1v X3v X5v X1c X3c

7 −0.79 4.48 7.62 −11.87 −4.05 −2.01 4.42 5.097 −0.53 4.64 8.99 −13.69 −3.47 −0.89 5.19 7.382 −0.92 3.97 8.54 −11.76 −3.54 −1.63 5.18 7.610 −0.67 4.81 8.88 −13.35 −3.55 −1.43 5.10 6.045 −0.79 3.61 6.91 −12.02 −4.38 −2.03 3.78 4.255 −1.24 2.14 8.02 −12.30 −4.99 −2.31 5.04 5.447 −1.90 3.99 6.95 −13.19 −5.70 −2.69 4.54 5.175 −0.86 2.45 8.19 −10.67 −4.86 −2.11 2.71 2.828 −1.09 1.62 6.51 −10.68 −5.05 −2.08 3.34 5.144 −0.44 2.80 3.95 −11.56 −5.36 −2.40 3.43 3.57

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functional based DFT yields remarkable improvement overLDA and GGA for calculating the band gaps of II–VIcompound semiconductors. The MBJLDA is a XC-potential,not a XC-energy functional, thus Exc is taken from LDA.Since there is no exchange functional Ex such thatVMBJx;c ¼ δEx=δρs, this method cannot be used to calculate

total energy. In order to overcome this obstacle, it ispreferred to use GGA or LDA based functionals for struc-tural properties, and then to use MBJLDA potential for thecalculation of electronic properties. Because of this reasonone can follow two ways to calculate electronic properties.The first option is to use experimental lattice constants inband structure calculations, as we did in our presentmanuscript.The second option is to perform geometrical optimizationby using the LDA or GGA and then using the calculatedlattice constant to perform MBJLDA for band structurecalculations. In order to test effects of lattice constants toband gap calculations, we performed some extra calcula-tions with MBJLDA. We reported that for CdSe usingexperimental lattice constant a¼6.08 we got 1.89 eV for bandgap with the relative error 3.84% with respect to the experi-ment. If we use our previous LDA result for lattice constant [4](a¼6.017) we got 1.95 eV for band gap with the relative error7.1% with respect to the experiment. We think that even thisresult is in acceptable range according to LDA or GGA bandgaps. We can say that the band gap results are highlydepended on lattice constants. Therefore, for a reliable andaccurate comparison of calculated results withmeasured band

Fig. 2. Energy band structures of CdS and ZnS bulk semiconductors along Γ−L(MBJLDA) DFT model (dark-solid line) compared with NN sp3d5 TB (red-solidparameters given in Tables 1 and 2 of Gürel et al. [17]. (For interpretation of theversion of this article.)

gaps we preferred to use the experimental lattice constantsin our band structure calculations.

We note that the DFT with MBJLDA functional under-estimates band gap only between −2.50% and −7% forCdTe, ZnS, ZnSe, and ZnTe and overestimates up to �4% forCdS and CdSe [4]. In most cases, except ZnS, MBJLDA isin a better agreement with the experiment than GW and G′W′. Comparison of band gap energies at L, Γ and Xhigh symmetry points calculated by using the MBJLDAfunctional based DFT and semi-empirical NN sp3d5 and2NN sp3sn TB models is also given in Table 2 for CdS, CdSeand CdTe and in Table 3 for ZnS, ZnSe and ZnTe. The overallagreement between the MBJLDA functional based DFT andsemi-empirical NN sp3d5 and 2NN sp3sn TB models isremarkable. There is no need to shift/or correct thecalculated band gaps in the DFT calculations in order toget agreement with measured band gap data.

The comparison of the band structure dispersion curvesobtained by using the WIEN2k package simulations, withMBJLDA functional embedded in DFT, and the semi-empiricaltight binding theory, with NN sp3d5 and 2NN sp3sn basis, isalso shown in Fig. 2(a and b) for bulk CdS and ZnS and inFig. 3(a, b and c) for bulk CdSe, CdTe and ZnSe compounds,respectively. As displayed in Figs. 2(a and b) and 3(a, b and c),respectively, the MBJLDA functional based DFT and semi-empirical NN sp3d5 and 2NN sp3sn TB models accuratelyreproduce the band gaps and band dispersion curves at highsymmetry points when they are compared with empiricalpseudo-potential calculations for bulk II–VI compounds and

and Γ−X directions, obtained by using the modified Becke–Johnson LDAline) and 2NN sp3sn TB (dashed-blue line) models for the tight bindingreferences to color in this figure legend, the reader is referred to the web

Fig. 3. Energy band structures of CdSe, CdTe and ZnSe bulk semiconductors along Γ−L and Γ−X directions, obtained by using the modified Becke–JohnsonLDA (MBJLDA) DFT model (dark-solid line) compared with NN sp3d5 TB (red-solid line) and 2NN sp3sn TB (dashed-blue line) models for the tight bindingparameters given in Tables 1 and 2 of Gürel et al. [17]. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

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their heterostructures. The three different approaches almostequally well reproduce the band gaps at Γ, X and L highsymmetry points and valence band and conduction banddispersion curves in the entire Brillouin zone of bulk CdX andZnX (X¼S, Se, Te) II–VI bulk compounds and CdZnS/CdS,CdSTe/CdTe and ZnSSe/ZnSe heterostructures.

When we compare the results of semi-empirical NNsp3d5 and 2NN sp3sn TB models against the pseudo-potential theory predictions and the experiment, as canbe seen from Tables 1 and 2 and Figs. 2 and 3, they bothgive reasonable description of the energy levels in thevicinity of the bottom of the conduction band and top ofthe valence band of group II–VI compounds. However, theformer TB parameterization (NN sp3d5) is better than latter(2NN sp3sn) in accurately reproducing band gaps andvalence bands and conduction band dispersion curves atL high symmetry point of these compounds. The inclusionof excited d-states in the sp3 TB Hamiltonian significantlychanges the electron and hole energy levels and is neces-sary for the realistic description of the optical properties ofII–VI compounds and their ternaries. Furthermore, sincethe inclusion of fictitious sn excited state is done bymodeling the average of p–d interactions, the 2NN sp3sn

TB parametrization does not permit the inclusion ofexcited d-orbitals to the sp3 basis set. Consequently, the

actual contribution of the excited d-states in the bandstructure calculations is not reliably and accuratelyreflected in the 2NN sp3sn TB model. We should pointout that both NN sp3d5 and 2NN sp3sn TB models are semi-empirical and their basis is based on a good description ofthe band structures that are produced exactly by theMBJLDA functional based DFT, as verified by the pseudo-potential theory [5] and the experiment [23].

3.2. Band structures of Cd(Zn)S(Se)/Cd(Zn) (S, Se, Te) II–VIheterostructures

We will now discuss our band structure calculations forCdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe heterostructures.Figs. 4a and 5a and b, respectively, display the comparisonof calculated principal band gaps of ternaries of CdZnS/CdS,CdSTe/CdTe and ZnSSe/ZnSe heterostructures as a function ofalloy composition and interface strain calculated by using theMBJLDA functional based DFT and semi-empirical NN sp3d5

and 2NN sp3sn TB models, integrated with the statisticalthermodynamic model via Eqs. 10,11. Since the measuredband gaps are near 0 K we ignored the logarithmic term,which accounts for the electron–phonon interactions fortemperature dependence. Figs. 4b and 6a and b, respectivelyindicate that the interface strain effects on band gaps of

Fig. 4. Predicted composition and interface strain effects on the band gap energies (Fig. 3a) and valence and conduction band offsets (Fig. 3b) ofCd1−xZnxS=CdS heterostructures, obtained by using the modified Becke–Johnson LDA (MBJLDA) DFT model (dark-solid line) compared with NN sp3d5 TB(red-solid line) and 2NN sp3sn TB (dashed-blue line) models for the tight binding parameters given in Tables 1 and 2 of Gürel et al. [17]. (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Predicted composition and interface strain effects on the band gap energies of (Fig. 5a) ZnSxSe1−x=ZnSe and (Fig. 5b) CdSxTe1−x=CdTeheterostructures, obtained by using the modified Becke–Johnson LDA (MBJLDA) DFT model (dark-solid line) compared with NN sp3d5 TB (red-solid line)and 2NN sp3sn TB (dashed-blue line) models for the tight binding parameters given in Tables 1 and 2 of Gürel et al. [17]. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 6. Predicted composition and interface strain effects on the (Fig. 6a) band offsets of ZnSxSe1−x=ZnSe and (Fig. 6b) CdSxTe1−x=CdTe heterostructures,obtained by using the modified Becke–Johnson LDA (MBJLDA) DFT model (dark-solid line) compared with NN sp3d5 TB (red-solid line) and 2NN sp3sn TB(dashed-blue line) models for the tight binding parameters given in Tables 1 and 2 of Gürel et al. [17]. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

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ternary constituents of CdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe heterostructures can be quite large when the interfacestrain increases for the large deformation potential and highalloy composition. The predicted band gaps, especially at Γpoint, are in excellent agreement with the experiment [23].

The key property to understand the impact of hetero-structures on the performance of electronic and opticaldevices is to reliably and accurately determine the com-position and strain variation of their electronic energyband structure across the heterointerfaces. Using the DFTwith MBJLDA functional and semi-empirical NN sp3d5 and2NN sp3sn TB models we can determine the band offsets atthe A1−xBxC=BC heterointerface from [17,21]

ΔEv ¼Evε∞

� �BC−

Evε∞

� �ABC

; ΔEci ¼ EciABC−EciBC ¼ΔEgi−ΔEv

ð12Þwhere Ev ¼ EvðΓ15Þ is the top of the valence band at Γsymmetry point, Eci ¼ EΓ6c ; EL6c and EX6c are the bottom ofconduction bands and Egi ¼ EgΓ ; EgL and EgX are the bandgaps at Γ, L and X symmetry points are. ΔEgi ¼ EgiðABCÞ−EgiðBCÞ with band gaps EgiðABCÞ and EgiðBCÞ. ε∞ðABCÞ andε∞ðBCÞ are the optical dielectric constants of A1−xBxC ternaryand BC binary compounds. Figs. 4b, 5b and 6b, respectively,show the composition and strain effects on conduction andvalence band offsets of CdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe heterostructures calculated by using the MBJLDA func-tional based DFT and semi-empirical NN sp3d5 and 2NN sp3sn

TB models. Figs. 4–6b, respectively, indicate that the interface

strain effects on the conduction band offsets can be quite largewhen the interface strain increases for the large deformationpotential and high alloy composition.

4. Conclusion

We presented a comparative prediction of electronicstructure of CdX and ZnX (S, Se, Te) II–VI compounds andtheir heterostructures (CdZnS/CdS, CdSTe/CdTe and ZnSSe/ZnSe) obtained by the MBJLDA functional based DFT andsemi-empirical NN sp3d5 and 2NN sp3sn TB models. Wedemonstrated that the use of MBJLDA density functional inthe DFT gives a good agreement with measured band gapsand band dispersion curves obtained by the nonlocalpseudo-potential theory at Γ, X and L high symmetrypoints. We can conclude that the band gap results arehighly depended on lattice constants. To perform MBJLDAcalculations, first we need to perform LDA or GGA geo-metry optimizations or use directly experimental latticeconstants. More accurate lattice constants give us moreaccurate results and good agreement with measured bandgaps. Furthermore, the NN sp3d5 TB parameterizationadequately reproduces the band gaps and dispersioncurves at Γ, X and L high symmetry points. This suggeststhat adding any fictitious sn state to the sp3 set isunnecessary in the TB band structure modeling. Weconclude that the MBJLDA functional based DFT and theNN sp3d5 TB model have considerable potential in the

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design and optimization of group II–VI compounds andtheir ternaries are used for the fabrication of electronicand optical devices.

Acknowledgment

The authors would like to greatly acknowledge thecomputer usage of High Performance Computing Labora-tory of Informatics Institute at Istanbul Technical Univer-sity and the partial financial support by the TurkishScientific and Technical Research Council (TÜBITAK), withGrant no: 105T463.

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