Theoretical optical parameters for III-nitride semiconductors

Microelectronics Journal 34 (2003) 721–724 www.elsevier.com/locate/mejo

Theoretical optical parameters for III-nitride semiconductors E. Silva Pintoa,*, R. de Paivab, L.C. de Carvalhoa, H.W.L. Alvesa, J.L.A. Alvesa a

Departamento de Cieˆncias Naturais, Universidade Federal de Sa˜o Joa˜o del Rei, Caixa Postal 110, 36,301-160 Sa˜o Joa˜o del Rei, Minas Gerais; Brazil b Departamento de Fı´sica, Universidade Federal de Minas Gerais, Caixa Postal 702, 13081-970 Belo Horizonte, Minas Gerais, Brazil

Abstract The III-nitride compounds (GaN, AlN, BN, and InN) are semiconductor materials which are promising for application in optoelectronics. They find applications in light emitting diodes, laser diodes and luminescent alloys. In the present work we calculate by means of an ab initio method the optical response functions for these compounds in their cubic phase (zinc-blend). We obtain the absorption coefficients, aðEÞ; the dielectric constant, e ðEÞ; the reflectance, RðEÞ; and the index of refraction, nðEÞ from the calculated energy band structures of the semiconductors. The values are compared to the available values in the literature. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: III-Nitride compounds; Optoparameters; Electronic structure

1. Introduction The aim of this paper is to calculate the optical parameters of the III-nitrides, GaN, AlN, BN and InN in their cubic (zinc-blend) phase. These nitrides are semiconductor materials characterized by high ionicity, very short bond lengths, low compressibility, and big thermal conductivity. They find application in blue-light-emitting diodes, and lasers operating in the blue and ultraviolet regime, as well as in high temperature diodes and transistors. From an ab initio approach we obtain the energy band structures of these materials and from them the complex dielectric functions, e ðEÞ ¼ e 1 ðEÞ þ ie 2 ðEÞ; the absorption coefficients, aðEÞ; the reflectance, RðEÞ; and the complex refractive index, np ðEÞ ¼ nðEÞ þ ikðEÞ:

2. Theoretical framework The calculations were carried out using the full-potential linearized augmented plane wave method (FLAPW), based on the density functional theory as implemented in the WIEN 97 code [1]. We use the local density approximation (LDA) based on the Ceperley – Alder data for the free electron gas [2] the parameter Rkmax ¼ 9:0 in all cases and the muffin-tin radii of the atomic spheres were taken as ˚ (GaN), 1.0 A ˚ (InN), 0.91 A ˚ (AlN) and 0.75 A ˚ (BN). 0.93 A * Corresponding author.

The lattice parameters were optimized by energy ˚ (GaN), 4.36 A ˚ minimization having the values 4.47 A ˚ ˚ (AlN), 4.96 A (InN), and 3.59 A (BN). The total energy was converged within 1026 eV.

3. Results and discussion Figs. 1 – 4 show the curves aðEÞ; nðEÞ and kðEÞ; e 1 ðEÞ and e 2 ðEÞ and RðEÞ; respectively. Considering the calculations one should bear in mind that they employ LDA one-particle band structures (too small gaps), and local-field effects are not included. The FLAPW calculations give Eg ¼ 2:10 eV (GaN- GG), 3.25 eV ðAlN-GXÞ; 0.0 eV ðInN-GGÞ; 4.39 eV ðBN-GXÞ; whereas the experimental values are 3.50, 6.28, 1.90, 6.20 eV, respectively. Our calculations do not take into account the selfenergy corrections and excitonic effects. The imaginary part e 2 ðEÞ was calculated by considering summation over all conduction and valence bands, and over the first Brillouin zone as implemented in the WIEN97 code. The real part e 1 ðEÞ was then obtained from e 2 ðEÞ by the use of the Kramers – Kronig relations. The other parameters follow straightforwardly. However, the LDA combined with the so called scissors-operator approximation has been reported to describe the optical spectrum with accuracy [2,3], by taking into account the self-energy operator simply by shifting the conduction bands by

0026-2692/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0026-2692(03)00111-3

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E. Silva Pinto et al. / Microelectronics Journal 34 (2003) 721–724

Fig. 1. Optical absorption coefficients aðEÞ as a function of photon energy.

Fig. 2. Refractive index, n; and extinction coefficient k as a function of photon energy.

E. Silva Pinto et al. / Microelectronics Journal 34 (2003) 721–724

Fig. 3. e 1 ðEÞ and e 2 ðEÞ spectra as a function of photon energy. The dashed curves are from Ref. [4].

Fig. 4. Normal-incidence reflectivity, R; as a function of photon energy.

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E. Silva Pinto et al. / Microelectronics Journal 34 (2003) 721–724

the self-energy D: In Ref. [3] it is shown that a rigorous treatment of the non-locality of the self-energy operator requires a re-scaling of the matrix elements giving values for e 2 ðEÞ consistent which experiment. In Fig. 3, our theoretical results for e 1 and e 2 are compared to calculations by Gravilenko and Wu [4] using the FLAPW approach and including quasi particle (QP) corrections for GaN ðD ¼ 0:5 eVÞ and InN ðD ¼ 1:6 eVÞ: An overall agreement with the localization of the structures are achieved. The absolute values of e 1 and e 2 for GaN and InN are very much increased by inclusion of the QP correction. For AlN ðD ¼ 0:0 eVÞ and BN ðD ¼ 0:0 eVÞ our results are surprisingly too small than in Ref. [4]. As mentioned before our calculations do not take into account the self-energy corrections and excitonic effects. The self-energy correction tends to shift the spectrum towards higher energies while excitonic effects are expected to shift the spectrum in the opposite direction.

So, more detailed and refined theoretical calculations are desirable [5] for a better comparison with measurements.

Acknowledgements The authors are thankful to the Brazilian agencies FAPEMIG, CNPq, FAPESP for financial support, and to CENAPAD-MG/CO for computational support.

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