Tunnel junction of helical edge states: Determining and controlling spin-preserving and spin-flipping processes through transconductance

TL;DR

This paper demonstrates how to determine and control spin-preserving and spin-flipping tunneling processes in helical edge states of 2D quantum spin Hall systems using transconductance measurements and gate voltages.

Contribution

It introduces a method to independently measure and control two tunneling channels in helical edge states through factorization of the scattering matrix and transconductance analysis.

Findings

Transmission coefficients can be extracted via transconductance measurements.

Gate voltages enable independent control of spin-preserving and spin-flipping processes.

Method is robust against disorder and finite junction length effects.

Abstract

When a constriction is realized in a 2D quantum spin Hall system, electron tunneling between helical edge states occurs via two types of channels allowed by time-reversal symmetry, namely spin-preserving ({p}) and spin-flipping ({f}) tunneling processes. Determining and controlling the effects of these two channels is crucial to the application of helical edge states in spintronics. We show that, despite the Hamiltonian terms describing these two processes do not commute, the scattering matrix entries of the related 4-terminal setup always factorize into products of p-terms and f-terms contributions. Such factorization provides an operative way to determine the transmission coefficient $T_p$ and $T_f$ related to each of the two processes, via transconductance measurements. Furthermore, these transmission coefficients are also found to be controlled independently by a suitable combination of two gate voltages applied across the junction. This result holds for an arbitrary profile of the tunneling amplitudes, including disorder in the tunnel region, enabling us to discuss the effect of the finite length of the tunnel junction, and the space modulation of both magnitude and phase of the tunneling amplitudes.