First-principles calculations of the effective mass parameters of Al xGa1-xN and Zn xCd1-xTe alloys

First-Principles Calculations of the Effective Mass Parameters of Al_xGa_1-xN and Zn_xCd_1-xTe Alloys

R. de Paiva, R. A. Nogueira, C. de Oliveira,

Departamento de Física, ICEx, UFMG,

C.P.: 702, CEP: 13.081-970, Belo Horizonte, MG, Brazil

H. W. Leite Alves, J. L. A. Alves,

Departamento de Ciências Naturais - FUNREI,

C.P.: 110, CEP: 36300-000, 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: 05389-970, São Paulo, SP, Brazil

Received on 23 April, 2001

First-principles calculations of electronic band structures of the ordered cubic alloys Al_xGa_1-xN 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 light-emitting 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_xGa_1-xN, 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 II-VI semiconductor compounds ZnTe and CdTe and their alloys Zn_xCd_1-xTe 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 zinc-blende (cubic) materials and their alloys on the basis of first-principles band calculations and then link the electronic band calculations with the effective-mass theory. Therefore, we focus on electronic structures around the valence-band maximum (VBM) and the conduction-band minimum (CBM) and obtain the electron effective masses, hole effective masses, and, equivalently, the Luttinger-like parameters [5]. As a result, we present significant parameters of these materials, which can be used for the design of opto-electronic devices.

II Theoretical Framework

AlN and GaN usually have wurtzite structure and zinc-blende structure as well; ZnTe and CdTe have the zinc-blende structure. In the present study we consider the zinc-blende phase for all compounds and perform electronic structure calculations by means of the ab initio full-potential linearized augmented plane-wave (FLAPW) method [6,7]. We use the Ceperley-Alder [8] of local-density approximation (LDA) for the exchange-correlation term. We do not consider relativistic effects.

The Ga(3d)-, Zn(3d)-, Cd(4d)-and Te(4d)-electrons are treated as part of the valence-band states, since they are relatively high in energy even though they constitute a well-localized narrow band. Inside the muffin-tin spheres, the angular momentum expansion is truncated at = 9 for the wave functions. We use the parameter R_mtK_max = 9 which yields to a set of about 10⁴ LAPW basis functions. The charge density was self-consistently determined using 63 k-points 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 zinc-blende structure and compositions x = 0.0; 0.25; 0.50; 0.75; 1.00, we adopt a very approximate model based on an eight-atoms 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 zinc-blende 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 Vegard-type law [9]. The muffin-tin sphere radii for all atoms were taken with the same value 0.21 a, since the use of the full-potential 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 = ²½²/2m^*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 g₁ = 1/2(1/ + 1/); g₂ = 1/4(1/ - 1/) and g₃ = 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 first-principles 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 spin-resonance experiment [10], 0.21m_o by a first-principle 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 heavy-hole effective mass along the growth (z) direction of a multiple-quantum-well 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 first-principles calculations, the energy dispersion of the lowest conduction band has some anisotropy for directions. On the other hand the hole masses have non-negligible -directional dependence which is important when designing devices like laser diodes (LD's). So the Luttinger parameters g_i, in the zinc-blende structure, may be very useful for technological applications. The effective masses of Ga_1-xAl_xN alloys exhibit a stronger dependence on the compositon than the Cd_1-xZn_xTe alloys. This behavior may reflect the dehibridization of the d-electrons of Ga atom with the top of valence band as the composition x increases. In the case of the Cd_1-xZn_xTe 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_1-xZn_xTe 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_1-xAl_xN 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_1-xZn_xTe: 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 first-principles FLAPW band calculations within the LDA for the zinc-blende Ga_1-xAl_xN and Cd_1-xZn_xTe 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 CENAPAD-MG-CO for computational support.

• [1] S. Nakamura, and G. Fasol, The Blue Laser Diode, Springer: Berlin(1997).
• [2] Y. Wu, B. P. Keller, S. Keller, D. Kapolnek, P. Kosodoy, S. P. Denbaars and J. K. Mishra. Appl. Phys. Lett. 69, 1438(1996).
• [3] R. Dornhaus and G. Nimitz, Narrow-Gap Semiconductors, Eds. G. Höhler and E. A. Nickisch (Springer, Berlin) pp 119(1983).
• [4] J. P. Faurie, J. Reno and M. Boukerche, J. Crystal Growth 72, 111(1985).
• [5] R. Enderlein, G. M. Sipahi, L. M. R. Scolfaro and J. R. Leite. Phys. Stat. Sol(b) 206, 623(1998).
• [6] E. Wimmer, H. Krakauer, M. Weinert and A. J. Freeman, Phys. Rev. B24, 864(1981).
• [7] X. Zhun, S.B.Zhang, S.G.Louie, and M.L.Cohen, Phys. Rev. Lett. 63, 2112(1989).
• [8] P. Blaha, K. Schwarz, P. Sorantin, S. B. Trickey, Comp. Phys. Commun. 59, 399(1990).
• [9] A. B. Chen and A. Sher,''Semiconductor Alloys: Physics and Materials Engineering'' Plenum Press (1996).
• [10] M. Fanciulli, T. Lei, and T. D. Moustakas, Phys. Rev. BB48, 15144(1993).
• [11] K. Miwa and A. Fukumoto, Phys. Rev. B48, 7897(1993).
• [12] W. J. Fan, M. F. Li, T. C. Chong, and J. B. Xia, J. Appl. Phys. 79, 188(1996).
• [13] F. Long, P. Harrison, W. E. Hagston, J. Appl. Phys. 79, 6939(1996).

Publication Dates

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

History

• Received
  23 Apr 2001