Quantum geometry of common semiconductors

The quantum metric of the valence band states in typical diamond-type (C, Si, Ge) and zinc blende-type (GaAs, InP, etc.) semiconductors is investigated by means of the sp3s∗ tight-binding model. The global maximum of the metric is at the Γ point, but the resulting differential geometric properties like Ricci scalar, Ricci tensor, and Einstein tensor are found to vary significantly throughout the momentum space, indicating a highly distorted momentum space manifold. The momentum integration of the quantum metric yields the gauge-invariant part of the spread of the valence band Wannier function, whose value agrees well with that estimated experimentally from the dielectric function. The dependence of these geometric properties on the energy gap further offers a way to quantify the quantum criticality of these common semiconductors.