Analytic and numeric computation of edge states and conductivity of a Kane-Mele nanoribbon

Highlights: • Analytical calculations of edge states and robustness of QSH phase. • Electronic band structure of ZGNR in presence of spin-orbit interactions. • LDOS and charge transport study. • Competition of Rashba and intrinsic spin-orbit coupling in QSH phase. We compute analytic expressions for the edge states in a zigzag Kane-Mele nanoribbon (KMNR) by solving the eigenvalue equations in presence of intrinsic and Rashba spin-orbit couplings. Owing to the P-T symmetry of the Hamiltonian the edge states are protected by topological invariance and hence are found to be robust. This is not the case where either of the spin-orbit couplings in the Kane-Mele Hamiltonian is switched off. We have done a systematic study for each of the above cases, for example, a pristine graphene, graphene with an intrinsic spin-orbit coupling, graphene with a Rashba spin-orbit coupling, a Kane-Mele nanoribbon and supported our results on the robustness of the edge states by analytic computation of the electronic probability amplitudes, the local density of states (LDOS), band structures and the conductance spectra.