The band structure of 2 × 2 × 2 GaN super-cell model can be calculated, which is shown in Figure 3. As can be seen, both the bottom of the conduction band and the top of the valence band are located on the G point of the Brillouin zone, which indicates that GaN crystal belongs to direct band-gap that is consistent with the experiment value, and its band-gap width is 1.654 eV that is different from the experimental value 3.40 eV. Because the KohnSham equation based on GGA functional ignores the impact of the surrounding electrons while used to compute the band gap, the band width is undervalued, but the result does not affect the analysis of the band-gap variation with different doping concentration in the same structure and calculation conditions. Moreover, the scissors operator should be added in order to make up the errors of the calculated band-gap widths.

The calculated electron density of states of GaN is given in Figure 4, which can be seen that the valence band of GaN is divided into three parts, including the lower valence band of −17 eV ~ −11 eV, the upper valence band of −7 eV ~ 0 eV, and the conduction band of 1.6 eV ~ 7 eV. The partial state density curves of the atoms are shown in Figure 5, wherein s, p, d, and f denote electronic sub-shell respectively. It can be seen that the lower valence band of GaN is aroused by the 3d state of Ga atoms and 2s state of N atoms, while the conduction band part is mainly composed of the 4s state and 4p state of Ga atoms and the amount provided by N is relatively less and can be ignored. The peak of electronic density of states appearing in −13 eV is provided by the 3d state of Ga atoms while the peak of electronic density of states in −15 eV is supplied by the 2s state of N atoms.

Considering the relationship between the absorption

coefficient and the complex refractive index [6], the interaction between phonon and electron of the crystal can be expressed by the time-dependent perturbation including in the wave function of the electronic ground state, and the mutual interference of the phonon and the electric field can cause the transition between the electronic occupied state and unoccupied state, which can be related to the connection between the density of states of the conduction band and the valence band and result in the spectra generated by the excited states. Because the absorption coefficient α of the semiconductor is related to the imaginary part of its dielectric constant and the reflectivity coefficient R is related to the real part of its dielectric constant and and of GaN must be first calculated out, which can be given by the following formula [10,11]:

In the formula, K is the wave vector of electron, C and V are the valence band and conduction band, BZ is the first Brillouin zone, Ψ is the wave function, û is electric vector, and are the intrinsic energy levels on conduction band and valence band.

By calculating 2 × 2 × 2 intrinsic GaN super-cell model in Figure 1, the complex dielectric function and the absorption spectrum of GaN crystal are shown in Figures 6 and 7. From the imaginary part-dashed line in Figure 6, it can be seen that 5 peaks appear at the range of 3 eV ~ 23 eV that are located at 4.5 eV, 6.8 eV, 7.7 eV, 9.5 eV, 13 eV, separately. The main peak at 7.7 eV that broadens in 5 eV ~ 12 eV is caused by the transitions between the remote bands. The peak at 4.5 eV corresponds to the direct transitions from the bottom of valence band to the top of conduction band, which is brought mainly about by the transitions of 3p state orbit electrons of Ga atoms. Three dielectric peaks in 7.7 eV, 9.5 eV, 13 eV may be caused by the transitions from lower valence band to the upper valence band, which may be from 3d state of Ga atoms and 2s state of N atoms in lower valence band or the 3p state of Ga atoms and 2p state of N atoms in upper valence band. Secondly, the peak in 20 eV is very weak, which is mainly caused by transitions from the lower valence band to the conduction band, since the atoms involving the transition mode are very few. Comparing Figure 6 with Figure 7, the peak positions in the absorption spectra are very close

to ones of the complex dielectric function, but the difference of the peak strength are relative large. The strongest peak of the absorption spectrum situates at 9.5 eV approximately, which is from the transitions of 3d state electron of Ga and 2p state electron of N in Figure 5.

By using the equations above, the transmittance (α is the absorption coefficient, is the normalized reflection coefficient R) can be deduced and calculated, as shown in Figure 8, from which it can be seen that the change of transmittance is irregular before 350 nm and reaches gradually smooth state in the range of visible light after 350 nm, and the transmisivity can achieve 92% ~ 94%.

A series of band structures of Al_xGa_1−xN crystal models mixed with different Al component of 25%, 50%, 75%, and 100% are calculated respectively, and the band-gap widths are listed in Table 1. Because the conduction bands are underestimated by the error of the calculation

Table 1. The band-gap under the different concentrations of Al compositions.

software, the band gap values of the theoretical calculation, namely “Origin values” are very low, and the Test values are the ones modified by adding the scissor difference.

The curve of density of states of Al_0.5Ga_0.5N crystal is computed as Figure 9, and the band-gap width is about 2.396 eV which is counted from the conduction band of 3 eV ~ 7 eV to the upper valence band of −6 eV ~ 0 eV while the part of the lower valence band is −15 eV ~ −11 eV. The conduction band is caused by the 4s state and 4p state electrons of Ga atoms and 2p state electron of N atoms, and the upper valence band is aroused by the 4s state electrons of Ga atoms and 2p state electrons of N atoms while the lower valence band is mainly dominated by 3d state electron of Ga atoms. It can be seen that the energy band is mainly distributed in the range of −15 eV ~ 7 eV, and the peak at −13 eV is very strong which has nothing to do with the surrounding valence bands. By comparison in detail, the energy band structure of Al_xGa_1−xN above is similar to one of the intrinsic GaN, and the band positions are almost unanimous and the band widths just become bigger, which is formed by the movement of conduction band to the high energy region and the a little shift of Fermi level to the top of valence band, so that the conductivity weakens. But in the structure, two new peaks of electronic density of states appear in the positions of −14 eV and −11.8 eV which should be brought about by Al doping.

The imaginary part of dielectric function of Al_0.5Ga_0.5N is calculated and shown in Figure 10. It is found that 4

peaks are degenerated by 5 peaks of GaN while the peak positions of Al_0.5Ga_0.5N move slightly to high energy region, and the main peak is located at 8 eV whose peak value did not change basically, which is resulted from that the optical band gap become wide after the Al doped and shows the electronic transitions of Al_0.5Ga_0.5N is more different than GaN. As can be seen from the above curves, the dielectric peak values slightly increase and the peak position keeps moving to the high energy region with the amount increasing of Al doping.

Figure 11 gives out the absorption spectrum of Al_xGa_1−xN. As can be seen, the more the number of Al atoms which are doped in GaN is, the higher the energy appearing the light absorption peak is. While the positions of the main peaks slightly move to the right along with the increase of Al doping, the positions of the peaks at 4.5 eV, 13 eV and 20.1 eV are not almost changed, and the intensities of peaks at 4.5 eV and 20.1 eV weaken and even disappear, but the peak value at 13 eV fluctuates. It indicates that the performance before and