Quantum geometry of common semiconductors

Abstract

The quantum geometric properties of typical diamond-type (C, Si, Ge) and zincblende-type (GaAs, InP, etc) semiconductors are investigated by means of the $sp^{3}s^{\ast}$ tight-binding model, which allows to calculate the quantum metric of the valence band states throughout the entire Brillouin zone. The global maximum of the metric is at the $Γ$ point, but other differential geometric properties like Ricci scalar, Ricci tensor, and Einstein tensor are found to vary significantly in the momentum space, indicating a highly distorted momentum space manifold. The momentum integration of the quantum metric further yields the gauge-invariant part of the spread of valence band Wannier function, whose value agrees well with that experimentally extracted from an optical sum rule of the dielectric function. Furthermore, the dependence of these geometric properties on the energy gap offers a way to quantify the quantum criticality of these common semiconductors.

Figures (3)

• Figure 1: Numerical results for the quantum geometric properties of the 5 indirect band gap semiconductors C, Si, Ge, SiC, and Gap obtained from the $sp^{3}s^{\ast}$ tight-binding model, plotted along the high symmetry line $L-\Gamma-X-(U,K)-\Gamma$. From top to bottom, we present the band structure $(\varepsilon_{n},\varepsilon_{m})$, quantum metric $g_{\mu\nu}$, Ricci tensor $R_{\mu\nu}$, Ricci scalar $R$, and Einstein tensor $G_{\mu\nu}$, with $\mu\nu=\left\{xx,xy,xz,yy,yz,zz\right\}$ indicated by different colors. The last row shows that imaginary part of dielectric function $\varepsilon_{2}(\omega)$, together with the experimental data available for some of these materials listed in Ref. Aspnes83 shown as dashed lines.
• Figure 2: Same as Fig. \ref{['fig:indirect_gap_SM_figure']}, but for the 5 direct band gap semiconductors GaAs, GaSb, InP, InAs, and InSb.
• Figure 3: (a) The quantum metric at the $\Gamma$ point $g_{xx}(\Gamma)$, (b) trace of fidelity number ${\rm Tr}{\cal G}_{\mu\nu}$, (c) the gauge-invariant part of the spread of valence band Wannier function $\Omega_{I}$, and (d) the dimensionless ratio $\Omega_{I}^{3/2}/V_{\rm cell}$ versus the gap at the $\Gamma$ point $\Delta_{\Gamma}$ for the ten semiconductors under investigation.