First-principles study of the phonon replicas in the photoluminescence spectrum of $4H$-SiC

The free-exciton photoluminescence (PL) spectrum of $4H$-SiC exhibits a detailed fine structure due to the different phonons involved in the indirect gap transition. Here a first-principles calculation of these phonons, their symmetry labeling and their contribution to the photoluminescence spectrum is presented. The calculation uses phonons and electron-phonon coupling matrix elements computed via density functional perturbation theory and energy bands and optical phonon matrix elements calculated in density functional theory. The results are in excellent agreement with the experiment for the phonon energies and the polarization dependence of the spectrum. The relative intensities are also in fair agreement if we allow for some phonons within a few meV to be interchanged. There is however a remarkable discrepancy that the experimental spectrum shows a distinct behavior for phonons with energy below $\sim 55$ meV and above that energy, which is absent in the theory. The experimental PL lines corresponding to phonon energies below 55 meV are about a factor 5–10 smaller in intensity. This is not found in our calculations. The calculations show a similar peak distribution as the experiment in this range but with intensities comparable to those above 55 meV. This indicates that another mechanism outside the scope of the electron-phonon mediated transitions is operative for photon energies of the PL lines closer to the indirect exciton gap than this 55 meV cutoff, which reduces the overall intensity of these lines. We propose that this may result from competition between the phonon-assisted PL and trapping of the electron in the available unoccupied hexagonal site N-donor shallow level at 53-meV binding energy. As part of this study, we also present the phonon dispersions and density of states in $4H$-SiC and the electronic band structure including quasiparticle corrections.

Figure 1

The crystal structure of $4H$-SiC. The smaller brown spheres and larger blue spheres denote C atoms and Si atoms, respectively. Note that there are eight atoms per cell, four Si-C pairs in ABCA stacking. The atoms included in the figure occurring in neighboring cells are shown to emphasize the tetrahedral coordination.

Figure 2

Projection of the Brillouin zone on the $c$ plane with reciprocal lattice vectors ${\mathbf{b}}_1$, ${\mathbf{b}}_2$, in relation to the real-space primitive vectors: ${\mathbf{a}}_1$, ${\mathbf{a}}_2$. Of the three equivalent $M$, $M^{\prime }$, $M^{″}$ points, the $M$ point along the ${\mathbf{b}}_1$ direction, which coincides with the Cartesian $x$ axis is the one used to give the optical matrix elements in Appendix. The mirror planes and $x,y$ axes are shown in dashed red lines.

Figure 3

The LDA band structure of $4H$-SiC.

Figure 4

The $\mathrm{QS}GW$ band structure of $4H$-SiC.

Figure 5

The phonon dispersion along different k-directions and phonon density of states of $4H$-SiC.

Figure 6

The simulated photoluminescence spectra compared with experimental data (black lines). The experimental lines are offset vertically by an arbitrary amount. For $E\parallel c$, they correspond to Fig.–7(c) of Ref. [5], and for $E\perp c$, they correspond to Fig.–7(d). The dark green lines correspond to Fig.–7(a) ($E\parallel c$) and Fig.–7(b) ($E\perp c$) which were for samples with higher N doping emphasizing the $P$ lines. The violet lines (using LDA bands) and cyan dashed lines (using $GW$ bands) are our calculated spectra based on Tables 4 and 5 with a Gaussian broadening of 0.5 meV. The blue lines correspond to the LDA results but shifted up by 9 meV and represent the corresponding $P$ lines. For $E\parallel c$, the red and orange lines correspond to the experimental spectra $\times 10$ for the low phonon energy region. FE =Free exciton, N-BE is Nitrogen bound exciton