Real-space study of the optical absorption in alternative phases of silicon

We introduce a real-space approach to understand the relationship between optical absorption and crystal structure. We apply this approach to alternative phases of silicon, with a focus on the ${\mathrm{Si}}_{20}$ crystal phase as a case study. We find that about 83% of the changes in the calculated low-energy absorption in ${\mathrm{Si}}_{20}$ as compared to Si in the diamond structure can be attributed to reducing the differences between the on-site energies of the bonding and antibonding orbitals as well as increasing the hopping integrals for specific Si-Si bonds.

Figure 1

Optical absorption spectra of diamond-Si (black) and ${\mathrm{Si}}_{20}$ (red) calculated using the DFT-RPA (a) and GW-BSE (b) approaches.

Figure 2

Isosurfaces of the calculated bonding (a) and antibonding (b) Wannier functions in diamond-Si. Gray spheres represent the silicon atoms forming the bond. Isosurface in (a) is 1.4 and 1.0 in (b). Red and blue colors indicate parts of the Wannier function with opposite signs.

Figure 3

Conventional unit cell of diamond-Si containing eight silicon atoms. Its primitive unit cell contains only two silicon atoms.

Figure 4

Conventional unit cell of ${\mathrm{Si}}_{20}$ containing 20 silicon atoms. Its primitive unit cell is the same as its conventional unit cell. The four distinct Si-Si bonds are labeled using the letters a, b, c, and d. Bonds forming a triangle are labeled with the letter a.

Figure 5

On-site energy of bonding $(e_i$, lower value) and antibonding $({\overline{e}}_i$, higher value) Wannier function in diamond-Si (left, black) and ${\mathrm{Si}}_{20}$ (right, red). Numbers indicate ${\overline{e}}_i-e_i$ in eV. In the case of ${\mathrm{Si}}_{20}$ we show ${\overline{e}}_i-e_i$ for all four types of bonds. (The origin of the energy scale is arbitrary.)

Figure 6

Hopping integrals between bonding Wannier functions, as a function of hopping distance for diamond-Si (black) and ${\mathrm{Si}}_{20}$ (red).

Figure 7

Hopping integrals between antibonding Wannier functions, as a function of hopping distance for diamond-Si (black) and ${\mathrm{Si}}_{20}$ (red).

Figure 8

Optical absorption spectra of diamond-Si (black), ${\mathrm{Si}}_{20}$ (solid red), and the three hybrid model systems (dashed, dotted-and-dashed and dotted red). These absorption curves have all been scissors-shifted to higher energy by 0.6 eV, based on the results of our $GW$-BSE calculation.