Extensible Gauge-Invariant FDM With Spin-Orbit Coupling for Quantum Devices

We present a novel derivation and implementation of the finite-difference method (FDM) that is gauge invariant and incorporates spin-orbit coupling for the study of quantum systems. This version of FDM is meant to assist in the design and simulation of quantum devices that utilize multiple internal degrees of freedom (e.g., spin) by providing a way to directly use the effective Hamiltonian, which is often used to mathematically describe such systems. Our derivation is validated via comparison with perturbation theory and a quasi-tight-binding calculation and is shown to reproduce the expected results to a high degree of accuracy. This implementation of the FDM is expected to be very useful due to both its generality as well as its relative ease of implementation.