A variant transfer matrix method suitable for transport through multi-probe systems

TL;DR

This paper introduces a new transfer matrix method for analyzing transport in multi-probe systems and applies it to study the quantum spin Hall effect in graphene, revealing phase boundaries and robustness conditions.

Contribution

A novel variant transfer matrix method is developed for multi-probe transport systems, enabling detailed numerical analysis of QSHE in graphene with spin-orbit couplings.

Findings

Integer QSHE occurs with intrinsic SO interaction and is affected by Rashba SO and disorder.

Energy gap for QSHE depends on the strength of spin-orbit couplings, peaking at Vso > 0.2t.

QSHE robustness varies within the energy gap depending on the type of spin-orbit interaction.

Abstract

We have developed a variant transfer matrix method that is suitable for transport through multi-probe systems. Using this method, we have numerically studied the quantum spin Hall effect (QSHE) on 2D graphene with both intrinsic (Vso) and Rashba (Vr) spin-orbit (SO) couplings. The integer QSHE arises in the presence of intrinsic SO interaction and is gradually destroyed by the Rashba SO interaction and disorder fluctuation. We have numerically determined the phase boundaries separating integer QSHE and spin Hall liquid. We have found that when Vso> 0.2t with t the hopping constant the energy gap needed for the integer QSHE is the largest satisfying |E|<t. For smaller Vso the energy gap decreases linearly. In the presence of Rashba SO interaction or disorders, the energy gap diminishes. With Rashba SO interaction the integer QSHE is robust at the largest energy within the energy gap while at the smallest energy within the energy gap the integer QSHE is insensitive to the disorder.