Effective Hamiltonians for Ge/Si core/shell nanowires from higher order perturbation theory

TL;DR

This work develops higher-order Schrieffer–Wolff perturbation theory to derive interpretable, transferable effective hole Hamiltonians for Ge/Si core/shell nanowires. Building on the Luttinger–Kohn framework with Bir–Pikus strain and cylindrical confinement, and implemented via Pymablock, the authors generate low-energy models up to fifth order in perturbations, capturing growth-direction–dependent orbital inversions, g^*-tensor renormalizations, and linear/cubic spin–orbit interactions. They show that small excitation gaps can drive orbital inversions and lead to divergent effective masses, yielding quasi-flat bands tunable by strain, electric, and magnetic fields, with potential applications in correlated states and Majorana-engineered devices. The results offer a practical, symmetry-aware toolkit for designing Ge-based nanowire devices with controlled spin-orbit coupling and mass dispersion, relevant for quantum information platforms and topological superconductivity.

Abstract

We theoretically explore the electronic structure of holes in cylindrical Germanium/Silicon core/shell nanowires using a perturbation theory approach. The approach yields a set of interpretable and transferable effective low-energy models for the lowest few sub-bands up to fifth order for experimentally relevant growth directions. In particular, we are able to resolve higher order cross terms e.g., the dependency of the effective mass on the magnetic field. Our study reveals orbital inversions of the lowest sub-bands for low-symmetry growth directions, leading to significant changes of the lower order effective coefficients. We demonstrate a reduction of the direct Rashba spin-orbit interaction due to competing symmetry effects for low-symmetry growth directions. Finally, we find that the effective mass of the confined holes can diverge yielding quasi flat bands interesting for correlated states. We show how one can tune the effective mass of a single spin band allowing one to tune the effective mass selectively to its divergent points.

Figures (24)

• Figure 1: Illustration of the crystallographic directions and their bases considered in this work. By aligning the bases with high symmetry directions of the crystal, the effective $g^*$-tensor (c.f. Sec. \ref{['sec:g_fac']}) and SOI tensor (c.f. Sec. \ref{['sec:rashba']}) become purely diagonal and off-diagonal respectively (cf. App. \ref{['app_symmetry']}). Only in $[112]$ the basis cannot fully be aligned with symmetry directions leading to cross terms between orthogonal directions.
• Figure 2: Schematic of the Ge/Si core/shell nanowire. The relevant parameters for strain (c.f. eq. \ref{['LK:BP']}) are the core- and shell-radius, $R_c$ and $R_s$ respectively. They tune an aggregated parameter we call $\delta_\epsilon$.
• Figure 3: Exact diagonalization of the full Luttinger-Kohn Hamiltonian in the presence of a magnetic field $\vec{B}||x$ for the $[110]$ and $[112]$ growth directions. Reducing the symmetry from cylindrical to cubic causes orbital inversions in the $[110]$ direction and interlaced orbital states in the $[112]$ direction. In the former case, a perturbative treatment based on the lowest two eigenstates of the isotropic model is not applicable. In the latter case, the interlacing of orbitals prevents the definition of a well-defined effective g-factor.
• Figure 4: Excitation gap between the lowest two orbitals. In the low strain regime, the orbital inversion due to the cubic symmetry terms leads to a reduction of the excitation gap to up to $50\%$ for crystal orientations other than $[001]$.
• Figure 5: Exact diagonalization (black) and perturbation theory results (blue) for the parallel, $g^*_{zz}$, and perpendicular, $g^*_{xx}$, components of the $[001]$$g^*$-tensor to different orders in perturbation theory. The strain exerted by the Bir-Pikus Hamiltonian, eq.\ref{['LK:BP']}, changes the energy gap between the ground and excited orbitals. This yields a dependency of the $g^*$-tensor on the renormalized strain parameter, $\tilde{\delta}_\epsilon$.