Abstract
Firstprinciples calculations of electronic band structures of the ordered cubic alloys Al xGa1xN and Cd xZn1xTe are carried out. The band structures are used to provide e ective masses and Luttinger parameters which are useful in the parametrization of theories based on e ective hamiltonians.
FirstPrinciples Calculations of the Effective Mass Parameters of Al_{x}Ga_{1x}N and Zn_{x}Cd_{1x}Te Alloys
R. de Paiva, R. A. Nogueira, C. de Oliveira,
Departamento de Física, ICEx, UFMG,
C.P.: 702, CEP: 13.081970, Belo Horizonte, MG, Brazil
H. W. Leite Alves, J. L. A. Alves,
Departamento de Ciências Naturais  FUNREI,
C.P.: 110, CEP: 36300000, São João del Rei, MG, Brazil
L. M. R. Scolfaro, and J. R. Leite
LNMS, Departamento de Física dos Materiais e Mecânica  USP
C.P.: 66.318, CEP: 05389970, São Paulo, SP, Brazil
Received on 23 April, 2001
Firstprinciples calculations of electronic band structures of the ordered cubic alloys Al_{x}Ga_{1x}N and Cd _{x} Zn_{1}_{x} Te are carried out. The band structures are used to provide e ective masses and Luttinger parameters which are useful in the parametrization of theories based on e ective hamiltonians.
I Introduction
The wide bandgaps of GaN and AlN make them ideal materials for the construction of blue/green lightemitting devices (LED^{'}s and lasers) and of transistors intended to operate at high power and temperatures [1,2]. The alloying between GaN and AlN, producing Al_{x}Ga_{1x}N, allows the ''engineering of the band gap'' from 6.28 eV for AlN to 3.44 eV for GaN by controlling the composition x.
On the other hand the IIVI semiconductor compounds ZnTe and CdTe and their alloys Zn_{x}Cd_{1x}Te have important applications such as infrared detectors, solar cells and other devices [3,4].
Despite the recent intensive experimental and theoretical study of these materials some of the fundamental parameters which are essential to theoretical models used to account for the behavior of the alloys are either unknown or the subject of some debate. In particular, although the effective mass approximation is used extensively throughout in the literature, the actual electron and hole effective masses for the alloys in the whole range of x, are unknown. The transport and optical phenomena usually are governed by the band structures in the immediate vicinity of the Brillouin zone center. Thus the effective mass approximation turn to be an appropriate method to make the analysis of the electronic properties of materials amenable. The purpose of the present study is to obtain the electronic structure of those zincblende (cubic) materials and their alloys on the basis of firstprinciples band calculations and then link the electronic band calculations with the effectivemass theory. Therefore, we focus on electronic structures around the valenceband maximum (VBM) and the conductionband minimum (CBM) and obtain the electron effective masses, hole effective masses, and, equivalently, the Luttingerlike parameters [5]. As a result, we present significant parameters of these materials, which can be used for the design of optoelectronic devices.
II Theoretical Framework
AlN and GaN usually have wurtzite structure and zincblende structure as well; ZnTe and CdTe have the zincblende structure. In the present study we consider the zincblende phase for all compounds and perform electronic structure calculations by means of the ab initio fullpotential linearized augmented planewave (FLAPW) method [6,7]. We use the CeperleyAlder [8] of localdensity approximation (LDA) for the exchangecorrelation term. We do not consider relativistic effects.
The Ga(3d), Zn(3d), Cd(4d)and Te(4d)electrons are treated as part of the valenceband states, since they are relatively high in energy even though they constitute a welllocalized narrow band. Inside the muffintin spheres, the angular momentum expansion is truncated at = 9 for the wave functions. We use the parameter R_{mt}K_{max} = 9 which yields to a set of about 10^{4} LAPW basis functions. The charge density was selfconsistently determined using 63 kpoints in the irreducible wedge of the first Brillouin zone. The total energy criterion for convergence was 10^{7} Ry.
In order to simulate the ordered alloys with zincblende structure and compositions x = 0.0; 0.25; 0.50; 0.75; 1.00, we adopt a very approximate model based on an eightatoms supercell with cubic symmetry. Large supercells that would be necessary to allow a more continuous variation of the composition x would be more computationally demanding. The alloy consists of a cation sublattice where the metals are interchanged (Ga and Al; Zn and Cd) and of an anion sublattice (N;Te), both originated from the zincblende structure. The cubic supercells were built by placing the anions at the positions (0,0,0), (0,1/2,1/2), (1/2,0,1/2) and (1/2,1/2,0); Zn(Al) atoms replacing successively the Cd(Ga) atoms initially placed at the positions (1/4,1/4,1/4), (1/4,3/4,3/4), (3/4,1/4,3/4) and (3/4,3/4,1/4) simulate the compositions x = 0.25; 0.50; 0.75; 1.00, respectively. The lattice parameters, a, were optimized for each composition x by means of total energy and force calculations. The results were found to obey a Vegardtype law [9]. The muffintin sphere radii for all atoms were taken with the same value 0.21 a, since the use of the fullpotential ensures that the calculation is independent of the choice of the sphere radii.
III Results and Discussion
We calculate the bottom of the conduction band and the heavy hole and light hole bands at 21 points with ½½ ranging from 0.05(2p/a) to 0.05(2p/a). The corresponding effective masses can be fitted using E = ^{2}½^{2}/2m^{*}½^{.} The heavy hole and the light hole effective masses, and , and the electron effective masses, , were then determined along the [111], [100] and [110] directions, and are shown in Table I. Using g_{1} = 1/2(1/ + 1/); g_{2} = 1/4(1/  1/) and g_{3} = 1/4(1/ + 1/2/ we obtain the Luttinger parameters [5] which are shown in Table II.
IV Discussion
The literature is scarce in experimental and firstprinciples calculated values of the effective masses for the materials studied in this work. There is a broad general agreement between the results presented here and those of other calculations. For instance for GaN is 0.17m_{o} in our calculation, 0.15m_{o} obtained by electron spinresonance experiment [10], 0.21m_{o} by a firstprinciple pseudopotential calculation [11] and 0.13m_{o} by an empirical pseudopotential calculation. On the other hand our effective masses along the [100] direction for CdTe, = 0.13m_{o}, = 0.14m_{o} and = 0.42m_{o} are to be compared to the values = 0.11m_{o}, = 0.18m_{o} and = 0.60m_{o}, obtained by empirical pseudopotential calculations [12], while, in particular, the heavyhole effective mass along the growth (z) direction of a multiplequantumwell has been assigned values ranging from 0.4m_{o} to 0.6m_{o} [13]. The effective masses we obtain for the alloys are almost all unknown in the literature.
According to our firstprinciples calculations, the energy dispersion of the lowest conduction band has some anisotropy for directions. On the other hand the hole masses have nonnegligible directional dependence which is important when designing devices like laser diodes (LD's). So the Luttinger parameters g_{i}, in the zincblende structure, may be very useful for technological applications. The effective masses of Ga_{1x}Al_{x}N alloys exhibit a stronger dependence on the compositon than the Cd_{1x}Zn_{x}Te alloys. This behavior may reflect the dehibridization of the delectrons of Ga atom with the top of valence band as the composition x increases. In the case of the Cd_{1x}Zn_{x}Te alloys the hibridization of the Cd(4d)electrons with the top of the valence band is continously substituted by the Zn(3d)electrons. The effective masses of the Cd_{1x}Zn_{x}Te alloys have an almost linear dependence on x, showing small bowing factors. The calculated effective masses along the [110] direction are slightly smaller ( ~ 4%) than the calculated masses along the [100] and [111] directions, which are coincident. The calculated effective masses of the light holes along the [100] direction are bigger ( ~ 29%) than the calculated masses along the [111] and [110] directions which are almost coincident. On the other hand the calculated effective masses of the heavy holes along the [111] direction are bigger ( ~ 134%) than the masses along the [100] and [110] directions, which are also almost coincident.
The effective masses of the Ga_{1x}Al_{x}N alloy, except for x = 0.5, also have an almost linear behavior with the composition x, and exhibit the same features as described for the Cd_{1x}Zn_{x}Te: is about 5% smaller than » . > » ; >> » . The lost of symmetry occuring for x = 0.25; 0.50 or 0.75 should be responsible for some of these fluctuations. On the other hand, as mentioned before, these results also depends on the small size of our supercells.
V Conclusions
We have performed firstprinciples FLAPW band calculations within the LDA for the zincblende Ga_{1x}Al_{x}N and Cd_{1x}Zn_{x}Te alloys (x = 0.0; 0.25; 0.50; 0.75; 1.00). The electron effective mass, hole effective masses, or equivalently, the Luttinger parameters, were derived from the calculated band structures near the CBM and VBM. These results provide the basis for the study of the electronic structure and physical properties of alloys of varying compositions. A more detailed analysis of the structural, electronic and thermodynamic properties of these alloys will appear soon elsewhere.
Acknowledgements
The authors are thankful to the Brazilian agencies CNPq, FAPEMIG and CAPES for finantial support, and to CENAPADMGCO for computational support.
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Publication Dates

Publication in this collection
26 Nov 2002 
Date of issue
June 2002
History

Received
23 Apr 2001