A variant transfer matrix method suitable for transport through multi-probesystems

Abstract

We have developed a variant transfer matrix method that is suitable for transportthrough multi-probe systems. Using this method, we have numerically studiedthe quantum spin Hall effect (QSHE) on 2D graphene with both intrinsic(Vso) andRashba (Vr) spin–orbit (SO) couplings. The integer QSHE arises in the presence of intrinsicSO interaction and is gradually destroyed by the Rashba SO interaction anddisorder fluctuation. We have numerically determined the phase boundariesseparating integer QSHE and spin Hall liquid. We have found that whenVso≥0.2t with t the hopping constant the energy gap needed for the integer QSHE is the largest satisfying|E|<t. Forsmaller Vso the energy gap decreases linearly. In the presence of Rashba SO interaction ordisorders, the energy gap diminishes. With Rashba SO interaction the integerQSHE is robust at the largest energy within the energy gap while at the smallestenergy within the energy gap the integer QSHE is insensitive to the disorder.