Comparing electron-phonon coupling strength in diamond, silicon, and silicon carbide: First-principles study

Renormalization of the electronic band gap due to electron-phonon coupling in the tetrahedral semiconductors diamond, silicon, and cubic silicon carbide is studied from first principles. There is a marked difference between the coupling of the vibrational state to the valence band maximum and to the conduction band minimum. The strength of phonon coupling to the valence band maximum is similar between the three systems and is dominated by vibrations that change the bond length. The coupling strength to the conduction band minimum differs significantly in diamond, silicon carbide, and silicon. In diamond, the coupling is dominated by six small pockets of vibrational states in the phonon Brillouin zone that are ultimately responsible for the stronger electron-phonon coupling in this material. Our results represent a first step towards the development of an a priori understanding of electron-phonon coupling in semiconductors and insulators that should aid the design of materials with tailored electron-phonon coupling properties.

Figure 1

Electronic band structures along symmetry lines of the first electron BZ (blue, left side), and electronic densities of states (red, right side) of C, SiC, and Si. The dotted line shows the VBM. The wave vector axes have been scaled so that each plot has the same width.

Figure 2

Phonon dispersions along symmetry lines of the first phonon BZ (blue, left side), and phonon densities of states (red, right side) of C, SiC, and Si. The wave vectors axes have been scaled so that each plot has the same width.

Figure 3

ZP band gap corrections $\Delta \epsilon \left(\omega \right)$ as a function of the frequency $\omega$ of the vibrational modes for C, SiC, and Si. The vertical dashed lines separate regions of the phonon branches with different vibrational characters (see text for details). The numbers reported for each of these regions correspond to the integrated correction to the band gap for that region.

Figure 4

Charge density isosurfaces of the electronic state corresponding to the CBM (top three) and VBM (bottom three) for diamond (left), silicon carbide (center), and silicon (right). The structures are not to scale.

Figure 5

(a) Charge density of the electronic state corresponding to the VBM along a bond of C (solid black line), SiC (dashed red line), and Si (dashed-dotted green line). In each case we also give the integrated charge density along the bonds, averaged over the four bonds of the tetrahedral structure. For SiC, the C atom is at the right of the figure, where the charge density is larger. (b) The average bond density $\rho$ as a function of the inverse lattice constant $a^{-1}$.

Figure 6

ZP correction to the band gap of C and Si over 2-dimensional slices of the phonon BZ defined by ${\mathbf{c}}^{*}=\mathbf{0}$. For each $\mathbf{k}$ point we have summed over the six phonon branches present. The red solid circles indicate the region of strong electron-phonon coupling of the VBM, and the dashed blue circles indicate the region of strong coupling to the CBM. The overall ZP correction to the band gap is $-334$ meV for C and $-60$ meV for Si.

Figure 7

Schematic of the bond distortion in the C tetrahedra for the vibration at the reciprocal-space point (${\mathbf{a}}^{*}/3,-{\mathbf{b}}^{*}/3,\mathbf{0}$) in the phonon BZ. The charge density (blue) accumulates near the two stretched bonds due to its antibonding nature.

Figure 8

ZP band gap correction for C (black up triangles), SiC (red down triangles), and Si (green diamonds) as a function of lattice parameter $a$. The solid lines are a guide to the eye.