Electronic structure and optical properties of zinc-blende GaN
Introduction
GaN and III–V nitrides are semiconductor materials recently used to develop microelectronic devices and optoelectronic devices. Along with SiC, diamond and other semiconductor materials, they have become known as third-generation semiconductor materials after Ge, Si, GaAs and InP. GaN has some fascinating properties that do not exist in silicon-based semiconductor materials, such as a wide band gap, high thermal conductivity, high breakdown voltage, high melting point and chemical stability. These make it useful for short wavelength optoelectronic devices, high temperature devices and high frequency high power devices. In recent years, most research focused on its physical properties [1], [2], [3], [4].
The optical and electrical properties of optoelectronic materials are mainly characterized by the dielectric function, refractive index, optical conductivity and absorption coefficient. These optical constants are determined by the energy band structure near the Fermi level, the carriers’ concentration and the mobility rate. Consequently, it is necessary to study the electronic structure of photoelectric materials.
GaN has wurtzite and zinc-blende structures. Although many theoretical and experimental studies of the electronic structure of zinc-blende GaN [5], [6], [7], [8], [9], [10] have been carried out, work on the optical properties have analyzed only the absorption coefficient and dielectric function. In this work, we used computer simulation to investigate the properties of band structure, density of states (DOS), dielectric function, refractive index, absorption spectra, reflective spectra, optical conductivity and the energy loss function of GaN using the plane-wave pseudopotential method based on the density functional theory (DFT).
Zinc-blende GaN belongs to the F-43M (216) space group; the lattice constant can be described with a = b = c = 0.45 nm [11] and α = β = γ = 90°. In the 1/4 place of the body diagonal are N atoms. In eight corners and six face-center are Ga atoms. Each cell contains four Ga atoms and four N atoms.
All calculations were performed with the quantum mechanics program Cambrige Serial Total Energy Package (CASTEP) [12], based on DFT. All calculations were performed in a 2 × 2 × 2 supercell. The Broyden–Flecher–Goldfarb–Shanno (BFGS) algorithm was used to relax the structure of the crystal model. The valence electronic wave function is expanded in a plane wave basis vector. The final sets of energies were computed with an energy cutoff of 400 eV. The convergence was set to energy change below 2 × 10−6 eV/atom, force less than 0.005 eV/nm, the convergence tolerance of a single atomic energy below 1 × 10−5 eV/atom, stress less than 0.05 GPa, and change in displacement less than 0.0001 nm. All calculations were performed with a plane-wave pseudopotential method based on DFT combined with the generalized gradient approximation (GGA) [13], [14]. The integral in the Brillouin zone was sampled with the Monkhorst–Pack [15] scheme and special k points of high symmetry. The number of k points is 7 × 7 × 4. All calculations were carried in reciprocal space with Ga:3D4s4p and N:2s2p as the valence electrons. The scissors operator correction was used for the optical properties calculation, referring to the experimental data to improve the calculation accuracy.
In the linear response range, the macro-optical response functions of solids often are described by the complex dielectric function or by the complex refractive index: ,
In discussing the interaction between light and solid, one usually uses adiabatic approximation and single-electron approximation. Since the transition frequencies both in-band and between bands are much larger than the phonon frequency in the calculation of electronic structure and the method used is single-electron approximation, the phonon participation in the indirect transition process can be ignored, with only the electronic excitation considered. According to the definitions of direct transition probabilities and Kramers–Kronig dispersion relations, one can deduce the imaginary and the real parts of the dielectric function, absorption coefficient, reflectivity and complex optical conductivity [16], [17], [18].
Consider the theoretical formula:where n is the refractive index, k is the extinction coefficient, ɛ0 is the vacuum dielectric constant, λ0 is the wavelength of light in vacuum, C and V are the conduction band and valence band respectively, BZ is the first Brillouin zone, K is the electron wave vector, a is the unit direction vector of the vector potential A, MV,C is the transition matrix element, ω is the angular frequency, and EC(K) and EV(K) are the intrinsic energy level of the conduction band and valence band respectively.
Eqs. (3), (4), (5), (6), (7) provide a theoretical basis to analyze the band structures and optical properties of crystal, which reflect the mechanism of spectrum production when electrons transit energy levels. In a sense, these optical constants capture the physical characteristics of a material and link with the micro model of physical processes and the microscopic electronic structure of solids.
Section snippets
Results and discussion
To get the stability structure of the model, using BFGS methods in a geometric optimization, we calculated the lattice parameters of GaN, as shown in Table 1.
The unit cell parameters are in agreement with the experimental values and other calculated results as shown in Table 1. This shows that the method used is reliable.
Conclusions
The band structure, DOS and optical properties of zinc-blende GaN have been calculated based on a first-principles plane-wave pseudopotential method. The results show that zinc-blende GaN is a typical direct band gap semiconductor. Its valence bands are composed mainly of Ga3d, N2s and N2p electronic states. The conduction bands are mainly composed of Ga4s and Ga4p electronic states. The electrical transport properties of GaN and the types of carriers are mainly composed of Ga4s and N2p
Acknowledgments
This work is supported by the National Natural Science Foundation of China (60871012), Shandong Province Natural Science Foundation (ZR2010FL018) and Shandong Province Higher Educational Science and Technology Program (No. J10LG74).
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