Ab initio calculations of excitons in GaN

The optical absorption and excitonic properties of wurtzite GaN are investigated by means of an ab initio approach taking into account electron-hole correlations. This is done by solving the Bethe-Salpeter equation, using the results of density functional theory calculations as a starting point. Our main focus is the calculation of the binding energies of bound excitons, as well as their dependence on structural parameters and details of the electronic structure.

Figure 1

Imaginary part of the dielectric function ${\epsilon }_2\left(\omega \right)$ calculated within the BSE formalism (full line) and the DFT with scissor shift one (dotted line) for two polarizations of the incoming light (perpendicular and parallel in the upper and lower panels, respectively). The polarization is given with respect to the $c$ direction of the hexagonal unit cell. A broadening of $0.2\mathrm{eV}$ was used to generate the BSE curves. It should be noted that due to the relatively small number of $\mathbf{k}$ points and the broadening the details around the gap edge are not reliable.

Figure 2

Schematic energy-level diagram of band splitting under action of the crystal-field and spin-orbit interactions in wurtzite crystals (from Ref. 41). To the left, splitting induced only by the crystal field; to the right, splitting induced only by the spin-orbit interaction. The combined case is shown in the middle. The gaps $E_g\left(A\right)$, $E_g\left(B\right)$, and $E_g\left(C\right)$ which enter in the calculations of the binding energies of $A$, $B$, and $C$ excitons, respectively, are indicated in the middle. Note, however, that our basic BSE calculations did not include spin-orbit coupling (see discussion in text). In GaN our fully relativistic DFT calculations show that the final splitting of ${\Gamma }_5\to {\Gamma }_7$, ${\Gamma }_9$ is $\approx 5\mathrm{meV}$.

Figure 3

(Color online) The calculated exciton spectra of GaN relative to the fundamental optical band gap $E_g$ for light polarization perpendicular (top panel) and parallel (lower panel) to the crystallographic $c$ direction. Both degeneracy and oscillator strength are shown. The dashed lines represent $A$ and $B$ excitonic levels (degenerate), solid lines represent $C$ excitons, bars show the oscillator strength for corresponding levels. For $\mathbf{E}\Vert \mathbf{c}$ the binding energy is not equal to $E_g-E^{\lambda }$ (see text, and Fig. 2).

Figure 4

(Color online) Absolute values of the weights $A_{vc\mathbf{k}}^{\lambda }$, defining the excitonic wave function plotted in the reciprocal space for the three highest valence and the lowest conduction band. The $\mathbf{k}$ points lie in the basal plane of the hexagonal BZ and are shown as small black dots (the $\Gamma$ point being being at the center of each plot). The sampled $k$ points span a fraction of $1∕12\times 1∕12$ of the BZ in this plane with a mesh of $11\times 11$ points. The absolute values of the expansion coefficients $\mid A_{vc\mathbf{k}}^{\lambda }\mid$ are represented by the colors and sizes of the filled circles (the color map and diameters are, in fact, scaled as the square root of these absolute values in order to improve the visibility of small weights) and range from 0 (no circle) to about 0.22 (largest and darkest circles). The scale is kept constant for all figures.

Figure 5

(Color online) Excitonic wave function weights for the sum of the $A$ and $B$ excitons in the ground and the first excited states. The same representation and scale as for Fig. 4 is used (except that the maximum value of the weights is about 0.25 here).

Figure 6

(Color online) Binding energies for the ground state of bound excitons as a function of the relative change of the unit cell volume with respect to the experimental one (for constant $c∕a$ and $u$). These are shown for excitons $A$ and $B$ with perpendicularly polarized light (circles) and exciton $C$ with parallelly polarized light (diamonds). The inset shows the volume dependence of the optical gap $E_g$. $\Delta V=0$ corresponds to the experimental equilibrium volume.

Figure 7

(Color online) Dependence of the bound exciton binding energies on (a) the band gap, and (b) the conduction band mass, plotted for excitons $A$, $B$ $(\mathbf{E}\perp \mathbf{c})$, and $C$ $(\mathbf{E}\Vert \mathbf{c})$. The band gap was varied by simply rigidly shifting the conduction band and the conduction band mass by rescaling the bandwidth. The calculations were done for the experimental structural parameters.