Computational search for direct band gap silicon crystals

Due to its abundance, silicon is the preferred solar-cell material despite the fact that current silicon materials have indirect band gaps. Although the band gap properties of silicon have been studied intensively, until now, no direct band gap silicon-based material has been found or suggested. We report here the discovery of direct band gap silicon crystals. By using conformational space annealing, we optimize various crystal structures containing multiple (10 to 20) silicon atoms per unit cell so that their electronic structures become direct band gap. Through first-principles calculations, we identify many direct and quasidirect band gap crystal structures, which exhibit excellent photovoltaic efficiency.

Figure 1

Bond length, bond angle, and dihedral angle distributions are shown for cubic diamond, clathrate-I, D12, D63, D76, and D135.

Figure 2

Bond length, bond angle, and dihedral angle distributions are shown for D262, Q135, I926, ${\mathrm{NaSi}}_6$ clathrate without Na (Ref. [22]), Si-III (BC8), and Si-XII (R8).

Figure 3

Atomic structures of (a) D135, (b) Q135, and (c) I926 are shown. No coordination defects are found.

Figure 4

(a) Total energy (eV/atom) vs atomic volume curves plotted for cubic diamond, $\beta$-Sn, bct5, D135, Q135, and I926. (b) The characteristic band gap size dependence on the external pressure is shown for D135, Q135, I926, and diamond Si. D135 undergoes a transition from direct band gap to indirect band gap at pressures above 106 kbar. Also, at pressures below $-51$ kbar, D135 becomes indirect. Black symbols represent the indirect band gap of diamond Si, which decreases more rapidly than D135 as pressure increases.

Figure 5

The phonon dispersion and phonon density of states are shown for (a) D135, (b) Q135, and (c) I926. The phonon density of states of amorphous Si ($a-\mathrm{Si}$) is taken from Ref. [32].

Figure 6

The thermal stability of D135 and Q135 is examined by performing first-principles molecular dynamics simulations for 200 ps at temperatures 500 and 900 K for D135 and Q135, respectively. Potential energy fluctuations are obtained for a supercell containing 96 atoms (eight unit cells).

Figure 7

The PBE electronic band structures of (a) D135, (b) Q135, and (c) I926 are shown near the Fermi level. The Bravais lattices of D135, Q135, and I926 are base-centered monoclinic.

Figure 8

The imaginary part of the dielectric function ${\varepsilon }_2\left(\omega \right)$ is shown as a function of photon energy. Data for D135 (red), Q135 (blue), and I926 (yellow) are shown in color and the solar spectrum (Ref. [37]) in gray. Diamond Si, amorphous Si, and polycrystalline Si are respectively denoted as $c\text{−}\mathrm{Si}$, $a\text{−}\mathrm{Si}$, and $p\text{−}\mathrm{Si}$. The absorption of I926 is shown to be as good as $a\text{−}\mathrm{Si}$ probably due to rather small difference between the direct and indirect band gaps ($E_g^d-E_g^i=0.24$ eV).

Figure 9

(a) The spectroscopic limited maximum efficiency (SLME) [38] is calculated as the function of film thickness $L$ for D135, D63, Q130, Q135, Q78, Q1102, Q465, and I926. For D135, SLME reaches 29.71% at the film thickness of $L$ = 0.5 $\mu m$. (b) The photovoltaic efficiency of D135 is shown along with other designed silicon crystals (Table 2) and top five systems (${\mathrm{CuInTe}}_2$, ${\mathrm{CuInSe}}_2$, ${\mathrm{AgInTe}}_2$, ${\mathrm{AgInSe}}_2$, and ${\mathrm{CuGaSe}}_2$) proposed by Yu and Zunger [38]. For comparison, the theoretical upper limit (Shockley-Queisser efficiency limit) is also shown.

Figure 10

The variation of the quasiparticle G${}_0$W${}_0$ band gap is plotted with respect to the energy cutoff for D135.

Figure 11

For D135, the variation of the imaginary part of the dielectric function is plotted with respect to (a) the $\vec{k}$-point grid and (b) the number of bands considered in the BSE calculations. In (b), $n_1$ and $n_2$ in parentheses ($n_1,n_2$) denote the numbers of occupied and unoccupied bands around the Fermi level, respectively.