**Electronic band structure of the ordered Zn _{0.5}Cd_{0.5}Se
alloy calculated by the semi-empirical tight-binding
method considering second-nearest neighbor**

**Estructura electrónica de bandas de la aleación
ordenada de Zn _{0.5}Cd_{0.5}Se calculada por el método
semi-empírico de enlace fuerte teniendo en cuenta
interacción a segundos vecinos**

Pontificia Universidad Javeriana, Cra. 7 No. 40-62, Bogotá, Colombia

jsalcedo@fisica.ugto.mx; salcedo.juan@javeriana.edu.co

Recibido: 23-07-2007: Aceptado: 14-10-2008:

**Abstract**

Usually, semiconductor ternary alloys are studied via a pseudo-binary approach in which the semiconductor
is described like a crystalline array were the cation/anion sub-lattice consist of a random distribution of the
cationic/anionic atoms. However, in the case of reported III-V and II-VI artificial structures, in which an
ordering of either the cations or the anions of the respective fcc sub-lattice is involved, a pseudo-binary
approach can no longer be employed, an atomistic point of view, which takes into account the local
structure, must be used to study the electronic and optical properties of these artificial semiconductor
alloys. In particular, the ordered Zn_{0.5}Cd_{0.5}Se alloy has to be described as a crystal with the simple-tetragonal
Bravais lattice with a composition equal to the zincblende random ternary alloy. The change of symmetry
properties of the tetragonal alloy, in relation to the cubic alloy, results mainly in two effects: i) reduction of
the banned gap, and ii) crystal field cleavage of the valence band maximum. In this work, the electronic
band structure of the ordered Zn_{0.5}Cd_{0.5}Se alloy is calculated using a second nearest neighbor semi-empirical
tight binding method. Also, it is compared with the electronic band structure obtained by FP-LAPW (fullpotential
linearized augmented-plane wave) method.

**Key words**: band gap narrowing; electronic band structure; ordered alloys; Semi-empirical thigh binding
method; ZnCdSe alloy.

**Resumen**

Aunque la descripción de las aleaciones ternarias semiconductoras se hace tradicionalmente asumiendo la
aproximación de compuesto pseudo-binario. Para el caso de aleaciones artificiales de compuestos II-VI y
III-V, en las cuales se ha reportado un ordenamiento inducido por el crecimiento, una aproximación de este
tipo no es aplicable, de modo que, con el fin de hacer una descripción adecuada de las propiedades ópticas
y electrónicas de dichas aleaciones artificiales, se debe asumir una descripción atomística que tenga en
cuenta la estructura local. En particular, para la aleación ordenada de Zn_{0.5}Cd_{0.5}Se, el cambio de simetría
implica que se debe usar una estructura tetragonal simple, dando lugar, principalmente, a dos efectos: i)
disminución de la brecha prohibida del material y ii) un desdoblamiento en el máximo de la banda de
valencia. En este trabajo se calcula la estructura de bandas de la aleación ordenada de Zn_{0.5}Cd_{0.5}Se usando
la aproximación semi-empírica de enlace fuerte teniendo en cuenta interacción a segundos vecinos y se
compara con la estructura de bandas obtenida por el método FP-LAPW (full-potential linearized augmentedplane
wave). Se obtiene una buena concordancia de las principales características entre las estructuras de
bandas calculadas por el método *semi-empírico* y el método *ab initio*.

**Palabras clave**: aleaciones ordenadas; aleaciones de ZnCdSe; Disminución de la brecha de energía; Estructura
electrónica de bandas; Modelo de enlace fuerte.

**INTRODUCTION**

From the traditional point of view, when two zincblende
binary compounds AC and BC are mixed homogeneously
obtaining a random ternary alloy A_{1-x}B_{x}C, the ternary II-VI
and III-V semiconductors alloys are treated as pseudobinary
compounds (Bernard and Zunger, 1987), in which,
traveling along the [001] direction, a sequence of cationanion
planes can be found. The A and B cations, in a pseudobinary
alloy, are randomly distributed in each cation plane.
In particular, for the II-VI pseudo-binary zincblende
Zn_{0.5}Cd_{0.5}Se alloy, the Se layers are alternating with
Zn_{0.5}Cd_{0.5} layers with a random distribution in average of
the same amount of Cd and Zn atoms. However, ordering
of isovalent A_{0.5}B_{0.5}C semiconductors alloys has been
widely observed and studied (Kuan *et al*., 1985; Su *et al*.,
1994; Lu *et al*., 1987; Wei and Zunger, 1991). That is how,
at least four ordered structures related to the zincblende
structure have been found to date (see Table 1): CuPt
(Gomyo *et al*., 1987), CuAu (Mowgray *et al*., 1992),
femetinite (Wang, 1989), and chalcopyrite (Jen *et al*., 1986).

In particular, in the CuAu ordered structure, a sequence of
A-C-B-C-A-C-... planes along the [001] azimuth is found.
Sometimes, this structure is described as an (AC)_{1}(BC)_{1}
superlattice. However, it is not a true superlattice. It is a
crystal with the simple tetragonal Bravais lattice and the
same A_{0.5}B_{0.5}C composition of a zincblende random ternary
alloy. Due to changes in symmetry, local ordering, and, in
particular, to the change from zincblende unit cell -with
space group *T*^{2}_{d} - to simple tetragonal primitive cell -with
space group *D*^{5}_{2d} - predicted and observed changes in material
properties such as band gap reduction, valence band
splitting, polarization dependence of optical transitions,
vibrational spectrum, and others may be expected (Salcedo-Reyes and Hernández-Calderón, 2005). In the case of a
pseudo-binary alloy most of the optical, structural and
electronic properties are correctly described by the virtual
crystal approximation (VCA). However, it is evident that
in the case of an ordered alloy, with x=0.5, the VCA approach
can no longer be employed to explain the physical
properties and a more suitable crystalline structure must
be considered.

On the other hand, the Semiempirical Tight-Binding (STB)
is one example of the so- called *simplified quantum
mechanical methods*, in which a compromise between the
computational efficiency and the physical correctness of
the approximation is used. The usefulness of these
*approximated* methods comes from the balance between
theoretical rigor and pragmatism, speed, and accuracy. That
is, despite the generality and transferability of the method
is limited, the heavy computational effort of first-principles
calculations is avoided by replacing difficult integrals, *i.e*.
the so called two centers (Coulombic) integrals, by
empirical parameters to fit experimental results. In general
terms, in the STB method the solution to the timeindependent
single electron Schrödinger equation is
assumed as a linear combination of atomic orbitals centered
at each lattice point. The atomic orbitals are assumed to be
very small at distances exceeding the lattice constant (this
is what is meant by tight-binding), and, therefore,
practically all matrix elements are approached by analytical
functions of the inter-atomic separation and of the atomic
environment. In section 2, the STB method, taking into
account all first and second nearest-neighbor interaction,
is applied in order to obtain the tight binding parameters
(TBP) of the binary compounds ZnSe and CdSe. In section
3, the tetragonal STB Hamiltonian is deduced and, in
addition to those second nearest neighbor TBP, the
electronic band structure of the ordered ZnCdSe alloy is
obtained, and then, in section 4 it is compared with the
band structure obtained by FP-LAPW.

**TIGHT BINDING METHOD FOR ZINCBLENDE ZnSe
AND CdSe BINARY COMPOUNDS**

In the STB method the basis of the system is assumed as a lineal combination of quasi-atomic functions centered in each lattice point. That is, the tight binding basis is written as

where the numbers *n* runs over the atomic orbitals, the *N*
wavevectors *k* lie in the first Brillouin zone (FBZ), the site
index, *b*, is either *a* for anions or *c* for cations, and the
anion/cation positions are *R** _{i}*. The Schrödinger equation
for the Bloch function , written in terms of the tight
binding basis, is

where λ is the band index. In this work one *s* and three *p*
orbitals (*p _{x},p_{y}*, and

In order to calculate the TBP, symmetry properties of the
zincblende structure are used. From the irreducible representation
(IR) of the wave vector group, *g _{0}*(

where the plus sign corresponds to the energy of the conduction
band, in each ** k**-value, and the minus sign to the
valence band. For the

A group of twelve equations and seventeen unknown quantities,
then, have to be solved. The parameters *E ^{c}_{sp}* and

**TIGHT BINDING METHOD FOR THE ORDERED
Zn _{0.5}Cd_{0.5}Se**

In order to calculate the electronic band structure of the
tetragonal Zn_{0.5}Cd_{0.5}Se alloy, their crystal structure has to
be taken into account. In this case each cation (Cd, for
example) has four anions (Se) first-nearest neighbors, to, and eight cations of the same specie (Cd) and four cations
of the other specie (Zn) as second-nearest neighbors, to, as is shown in Figure 2.

It is very important to note that the Hamiltonian matrix has to be invariant under the chosen origin. That is, it has the same form when the origin coincides with one cation or when it coincides with one anion.

In this work, the ZnSe and CdSe first and second nearest
neighbor TBP's are used to *simulate* the parameters of the
ordered alloy. The *16x16* ordered STB-Hamiltonian matrix,
in the *sp ^{3}* basis, is

The matrixes on the diagonal are

The functions *g _{1}, g_{2}, g_{3}*, and

and

The off-diagonal matrixes are

where

and

Also

and

with

and

In all cases, the ZnCd parameters are the average between
the corresponding ZnSe-CdSe parameters, α, β, γ, α', β', g
γ' are like in the zincblende case, and *a* the lattice constant
calculated from the Vegard's law.

**RESULTS**

Since there is no experimental information on optical transitions
of the ordered Zn_{0.5}Cd_{0.5}Se alloy, the band structure
obtained by STB is compared to the electronic band structure
obtained by the FP-LAPW method (that do not include
spin-orbit interaction), as it is implemented in the
Wien97 code (Blaha *et al*., 1997), in the Fig 3, along the
[100] direction of the FBZ of the simple tetragonal lattice,
and in the Fig. 4, along the [110] direction. It is remarkable
the good qualitative agreement between both methods.

**SUMMARY**

An improved STB calculation of the band structure of the
Zn_{0.5}Cd_{0.5}Se ordered alloy, employing a *sp ^{3}* basis that takes
into account second nearest neighbor interaction, necessary
to describe the alloy ordering, was made. A complete
description of the electronic band structure of the tetragonal
alloy employing both, the STB and FP-LAPW methods,
was done obtaining numerical values for crystal field
splitting of the valence band maximum and for the band
gap reduction in relation to the band gap of the disordered
alloy. A good qualitative agreement is observed between
the obtained STB band structure (do not include spin-orbit
interaction) and the band structure calculated by FP-LAPW.

**ACKNOWLEDGEMENTS**

The author gratefully acknowledges I. Hernández-Calderón at Centro de Investigación y de Estudios Avanzados del IPN, Centro de Investigación y de Estudios Avanzados del IPN, Cinvestav (México) and H. Méndez at Pontificia Universidad Javeriana (Colombia) for helpful discussions and D. Olguín at Centro de Investigación y de Estudios Avanzados del IPN, Centro de Investigación y de Estudios Avanzados del IPN, Cinvestav (México) for his advice in the FP-LAPW calculations.

**REFERENCES**

BERNARD, J.E. and ZUNGER, A. Electronic structure of ZnS,
ZnSe, ZnTe, and their pseudobinary alloys. *Physical
Review B*, 1987, 36, 3199-3228.

BLAHA, P.; SCHWARTZ, K. and LUITZ, J. Wien97, Vienna University
of Technology, 1997. (Improved and updated
Unix version of the original copyrighted WIEN code,
which was originally published in Blaha P., Schwarz
K., Sorantin P. and Trickey S. B., Full potential, linearized
augmented plane wave programs for crystalline
systems, *Computer Physics Communications*,
1990, 59, 399-415.

CONTINENZA, A.; MASSIDDA, A. and FREEMAN A.J., Structural
and electronic properties of bulk ZnSe, *Physical Review
B*, 1994, 38, 12996-13001.

FLESZAR, A.; HANKE, W.; WEIGAND, W.; KUMPF, C.; HESKE, C.;
UMBACH, E.; SCHALLENBERG, T. and MOLENKAMP, L.W.,
Valence-band electronic structure of ZnSe(001) thin
films: Theory and experiment, *Physical Review B*,
2004, 125308-125318.

GOMYO, A.; KOBAYASHI, K.; KAWATA, S.; HINO, I.; SUZUKI, T. and
YUASA, T., Studies of Ga_{x}In_{1-x}P layers grown by
metalorganic vapor phase epitaxy; Effects of V/III ratio
and growth temperature, *Journal of Crystal Growth*,
1986, 77, 367-373.

GOMYO, A.; SUZUKI, T.; KOBAYASHI, K.; KAWATA, S.; HINO, I. and
YUASA, T. Evidence for the existence of an ordered
state in Ga_{0.5}In_{0.5}P grown by metalorganic vapor phase
epitaxy and its relation to band-gap energy. *Applied
Physics Letters*, 1987, 50, 673-675.

JEN, H.R.; CHENG, M.J.; STRINGFELLOW, B. Ordered structures
in GaAs_{0.5}Sb_{0.5} alloys grown by organometallic vapor
phase epitaxy. Applied Physcis Letters, 1986, 48,
1603-1605.

KUAN, T.S.; KUECH, T.F.; WANG, W.I. and WILKIE L. Long-Range Order in Al_{x}Ga_{1-x}As. *Physical Review Letters*,
1985, 54, 201-204.

KURGANSKI, S.I.; FARBEROVICH, O.V. and DOMASHEVSKAYA, É.P.
Energy band structure of II-VI compounds. I. Calculation
by the modified OPW method and interpretation,
*Soviet Physics - Semiconductors*, 1980, 14,
775-780.

LEE, G.D.; LEE M.H. and IHM, J. Role of d electrons in the
zinc-blende semiconductors ZnS, ZnSe, and ZnTe,
*Physical Review B*, 1995, 52, 1459-1462.

LEY, L.; POLLAK, R.A.; MCFEELY, F.R.; KOWALCZYK, S.P. and
SHIRLEY, D.A. Total valence-band densities of states of
III-V and II-VI compounds from x-ray photoemission
spectroscopy, 1974, *Physical Review B*, 9,600-621.

LU, Q.; BUNKER, B.A.; LUO, H.; KROPF, A.J.; KEMMER, K.M. and
FURDYNA, J.K. X-ray study of atomic correlations in
Zn_{0.5}Cd_{0.5}Se_{0.5}Te_{0.5} epitaxial thin films. *Physical Review
B*, 1987, 55, 9910-9914.

MAGNUSSON, K.O.; NEUHOLD, G.; HORN, K. and EVANS, D.A.,
Electronic band structure of cubic CdSe determined
by angle-resolved photoemission: Cd 4d and valencelevel
states, *Physical Review B*, 1998, 57, 8945-8950.

MOWGRAY, D.J.; HOGG, R.A.; SKOINICK, M.S.; DELONG, M.C.;
KURTZ, J.M. and OLSON, J.M. Valence-band splitting in
ordered Ga_{0.5}In_{0.5}P measured by polarized photoluminescence
excitation spectroscopy. *Physical Review B*,
1992, 46, 7232-7235.

PANTELIDES, S.T. and HARRISON, W.A. Structure of the valence
bands of zinc-blende-type semiconductors, *Physical
Review B*, 1975, 11, 3006-3021.

PARMENTER, R.H. Symmetry Properties of the Energy Bands
of the Zinc Blende Structure, *Physical Review*, 1955,
100, 573-579.

SALCEDO-REYES, J.C. and HERNÁNDEZ-CALDERÓN, I. Symmetry
properties and electronic band structure of ordered
Zn0.5Cd0.5Se alloys. *Microelectronics Journal*, 2005,
36, 342-346.

SLATER, J.C. and KOSTER, G.F. Simplified LCAO Method for
the Periodic Potential Problem, *Physical Review*, 1954,
94, 1498-1524.

SU, L.C.; HO, L.H.; STRINGFELLOW, B. Effects of substrate
misorientation and growth rate on ordering in GaInP.
*Journal of Applied Physics*, 1994, 75, 5135-5141.

WANG, C.T. *Introduction to semiconductor technology:
GaAs and related compounds*. 1 edition, Wiley-Interscience, New York, 1990, 624 pages.

WANG, Y.R. and DUKE, C.B. Cleavage faces of wurtzite CdS
and CdSe: Surface relaxation and electronic structure,
*Physical Review B*, 1988, 37, 6417-6424.

WEI, S. and ZUNGER, A. Disorder effects on the density of
states of the II-VI semiconductor alloys Hg_{0.5}Cd_{0.5}Te,
Cd_{0.5}Zn_{0.5}Te, and Hg_{0.5}Zn_{0.5}Te. *Physical Review B*,
1991, 43, 1662-1677.

ZAKHAROV, O.; RUBIO, A.; BLASÉ, X.; COHEN, M.L. and LOUIE
S.G. Quasiparticle band structures of six II-VI compounds:
ZnS, ZnSe, ZnTe, CdS, CdSe, and CdTe, *Physical
Review B*, 1994, 50, 10780-10787.

ZAKHAROV, O.; RUBIO, A.; BLASÉ, X.; COHEN, M.L. and LOUIE,
S.G. Quasiparticle band structures of six II-VI compounds:
ZnS, ZnSe, ZnTe, CdS, CdSe, and CdTe, *Physical
Review B*, 1994, 50, 10780-10787.