Multiband $k \cdot p$ theory for hexagonal germanium

TL;DR

This paper develops a comprehensive multiband p model for hexagonal germanium, accurately describing its band structure, effective masses, and spin-related properties, aiding future optoelectronic and spintronic applications.

Contribution

It introduces a self-consistent 10-band p model including spin-orbit coupling for hexagonal germanium, with new inverse effective mass parameters and insights into band hybridization.

Findings

Accurate band curvature and effective mass parameters obtained.

Hybridization of conduction bands explained by spin-orbit coupling.

Predicted electron and hole g-factors under magnetic field.

Abstract

The direct bandgap found in hexagonal germanium and some of its alloys with silicon allows for an optically active material within the group-IV semiconductor family with various potential technological applications. However, there remain some unanswered questions regarding several aspects of the band structiure, including the strength of the electric dipole transitions at the center of the Brillouin zone. Using the $\mathbf{k\cdot p}$ method near the $\Gamma$ point, including 10 bands, and taking spin-orbit coupling into account, we obtain a self-consistent model that produces the correct band curvatures, with previously unknown inverse effective mass parameters, to describe 2H-Ge via fitting to {\it ab initio} data and to calculate effective masses for electrons and holes. To understand the weak dipole coupling between the lowest conduction band and the top valance band, we start from a spinless 12-band model and show that when adding spin-orbit coupling, the lowest conduction band hybridizes with a higher-lying conduction band, which cannot be explained by the spinful 10-band model. With the help of L\"owdin's partitioning, we derive the effective low-energy Hamiltonian for the conduction bands for the possible spin dynamics and nanostructure studies and in a similar manner, we give the best fit parameters for the valance-band-only model that can be used in the transport studies. Finally, using the self-consistent 10-band model, we include the effects of a magnetic field and predict the electron and hole g-factor of the conduction and valance bands.