First-principle calculations of optical properties of wurtzite AlN and GaN

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Abstract

The imaginary part of the dielectric function of wurtzite AlN and GaN has been calculated in the long wavelength limit, using two different first-principle electronic structure methods. The first method is a full-potential linearized augmented plane wave method and the second is a full-potential linear muffin-tin orbital method. From the Kramers–Kronig dispersion relations the real part of the dielectric function has been obtained, taking into account a quasi-particle band-gap correction according to Bechstedt and Del Sole. Absorption due to optical phonons is treated as a delta function in the imaginary part of the dielectric function. Both the longitudinal as well as the transverse components of the dielectric function are presented, showing that the anisotropy is small in these materials. Although we use different correlation potentials in the two methods, the results are similar. We compare our calculated dielectric functions with spectroscopic ellipsometry and reflectance spectra measurements.

Introduction

The III-nitrides are wide-band-gap semiconductors with low compressibility, good thermal stability and with chemical and radiation inertness [1]. This makes the nitrides suitable for high-temperature optoelectronic devices, which can operate in harsh environments. Although, the world-wide interest for the nitrides has been increasing in the last decade, measurements of the optical properties have been almost absent until the last two, three years. Guo et al. [2] have measured the reflectance spectra on AlN, showing main optical transition peaks at 7.9, 9.0, 13.0, and 14.8 eV. Kawashima et al. [3] have measured the dielectric function in wurtzite GaN by the means of the spectroscopic ellipsometry, giving peaks at 6.20, 7.81, and 8.83 eV at a temperature of 120 K. Similar work has been presented for both wurtzite GaN and AlN by Wethkamp and Benedict with co-workers [4], [5]. The above measurements were done for E⊥c-axis only.

Calculations of the optical properties in both GaN and AlN have been reported. Xu et al. [6] have presented an orthogonalized linear-combination-of-atomic orbitals (OLCAO) calculation for different wurtzite structures. Christensen et al. [7] have presented the imaginary part of the dielectric function, using a linearized muffin-tin orbital (LMTO) calculation within the atomic sphere approximation (ASA). However, the spherically symmetric potential inside the spheres cannot fully describe the potential of wurtzite structures. The choice of the spherical radii affects the amplitude of the dielectric functions [7], and inclusion of empty spheres affects the total dielectric function [8]. Furthermore, ideal values of the lattice parameter c/a and u were used, which produce inaccurate crystal field for wurtzite materials [9], [10]. It is not clear how much these approximations influence the dielectric function, but by using ideal lattice parameters in AlN results in a 0.21 eV smaller band gap compared to similar calculation with optimized parameters [7]. In GaN the effect is not so pronounced, however. Solanki et al. [8] have also presented a LMTO-ASA calculation of the imaginary part of the dielectric function of AlN, using experimental values for the lattice parameters. Their wave functions were expanded to include s and p states only. Recently, Benedict et al. [11] have presented a first-principle calculation of GaN including the electron–hole interaction, showing that the two-particle exciton states contribute to the dielectric function.

In the present work, we have performed first-principle electronic structure calculations of wurtzite GaN and AlN, using two different computational methods, namely the full-potential linearized augmented plane wave (FPLAPW) method [12] (see also Ref. [13] for details in the calculations) and the full-potential linear muffin-tin orbital (FPLMTO) method [14]. For the FPLAPW calculation, we have chosen lattice constants for the relaxed crystal [15], and the exchange-correlation potential of Perdew et al. [16], which is derived within the generalized gradient approximation (GGA). For the FPLMTO calculation, we have chosen experimental values of the lattice constants [17] and the potential of Hedin and Lundquist [18], derived within the local density approximation (LDA). Both computational methods are described with a scalar-relativistic Hamiltonian [19].

The imaginary part of the dielectric function, ε2(ω), in the long wavelength limit have been obtained directly from the electronic structure, using the joint density of states and the optical matrix overlap. The real part of the dielectric function, ε1(ω), was calculated from the Kramers–Kronig dispersion relations. The longitudinal (‖) and the transverse (⊥) dielectric functions were calculated separately, where the longitudinal direction is along the c-axis and the transverse direction is the plane perpendicular to the c-axis. Furthermore, we have also investigated the effects of inclusions of electron–optical phonon interactions on the low-frequency dielectric function.

The band gap is underestimated both in the LDA as well as in the GGA. The FPLAPW (FPLMTO) calculation gives Eg=1.92 (1.99) eV in GaN and 4.20 (4.24) eV in AlN, whereas the experimental values are 3.50 and 6.28 eV, respectively [1]. Therefore, one cannot guarantee that the calculation of the dielectric function gives exact dispersion. However, it has been shown by Del Sole and Girlanda [20] that the LDA combined with the scissors-operator approximation describes the optical spectrum rather accurately. We have therefore made an estimate of the correction Δg to the band gap by using a quasi-particle method proposed by Bechstedt and Del Sole [21]. The calculation is presented in Ref. [15], and with the correction the band gap comes very close to the experimental values. The corrected FPLAPW band gaps are Eg+Δg=3.55 eV in GaN and 6.05 eV in AlN, and the corrected FPLMTO band gaps are Eg+Δg=3.45 eV in GaN and 6.15 eV in AlN.

Section snippets

Density of states

The density of states (DOS) were calculated with the FPLAPW method and by means of the modified tetrahedron method [22]. The resulting partial DOS of the s- , p- , and d-like states for the Ga, Al, and N atoms are presented in Fig. 1. In this figure, the band-gap correction is included. From the figure one can observe that the valence-band maximum is predominated by the p states of the N atoms. The conduction-band maximum is almost equally predominated by s and p states, and the d states are of

Dielectric functions

The complex dielectric function ε(ω)=ε1(ω)+iε2(ω) is presented in Fig. 2, Fig. 3 [solid lines for FPLAPW and dashed lines in (c) and (d) for FPLMTO]. The imaginary part was obtained directly from the electronic structure calculations, and the onset to absorption has been shifted by Δg in order to take into account the band-gap correction. Thereafter, the real part ε1(ω) was determined from the Kramers–Kronig dispersion relations (Fig. 3). This procedure has been shown to be the most proper way

Conclusion

We have calculated the dielectric function ε(ω)=ε1(ω)+iε2(ω) of wurtzite GaN and AlN, using a full-potential linearized augmented plane wave method and a full-potential linearized muffin-tin orbital method. Although we use different exchange-correlation potentials in the two methods, the resulting dielectric functions are similar. We have calculated the longitudinal and the transverse part of the dielectric function separately, showing that the anisotropy in the two nitrides is small. By

Acknowledgements

This work was financially supported by the Swedish Research Council for Engineering Sciences (TFR), the Swedish Natural Science Research Council (NFR), and the Brazilian National Research Council (CNPq).

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