Spin-polarized voltage probes for helical edge state: a model study

TL;DR

This paper models spin-polarized voltage probes for helical edge states in quantum spin Hall systems, analyzing their effectiveness, stability, and potential for measuring spin-resolved voltages, including a feasibility study with disorder effects.

Contribution

It introduces two models of spin-polarized voltage probes for helical edge states and analyzes their stability and practical implementation in quantum Hall systems.

Findings

Optimal probe-edge coupling enhances spin-resolved voltage measurement.

Probe stability is maintained under Gaussian fluctuations in coupling and polarization.

Edge of quantum anomalous Hall state can serve as an effective spin-polarized voltage probe.

Abstract

Theoretical models of a spin-polarized voltage probe (SPVP) tunnel-coupled to the helical edge states (HES) of a quantum spin Hall system (QSHS) are studied. Our first model of the SPVP comprises $N_{P}$ spin-polarized modes (subprobes), each of which is locally tunnel-coupled to the HES, while the SPVP, as a whole, is subjected to a self-consistency condition ensuring zero average current on the probe. We carry out a numerical analysis which shows that the optimal situation for reading off spin-resolved voltage from the HES depends on the interplay of the probe-edge tunnel-coupling and the number of modes in the probe ($N_P$). We further investigate the stability of our findings by introducing Gaussian fluctuations in {\it{(i)}} the tunnel-coupling between the subprobes and the HES about a chosen average value and {\it{(ii)}} spin-polarization of the subprobes about a chosen direction of the net polarization of SPVP. We also perform a numerical analysis corresponding to the situation where four such SPVPs are implemented in a self-consistent fashion across a ferromagnetic barrier on the HES and demonstrate that this model facilitates the measurements of spin-resolved four-probe voltage drops across the ferromagnetic barrier. As a second model, we employ the edge state of a quantum anomalous Hall state (QAHS) as the SPVP which is tunnel-coupled over an extended region with the HES. A two-dimensional lattice simulation for the quantum transport of the proposed device setup comprising a junction of QSHS and QAHS is considered and a feasibility study of using the edge of the QAHS as an efficient spin-polarized voltage probe is carried out in presence of an optimal strength of the disorder.