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We compute analytic expressions for the edge states in a zigzag Kane-Melenanoribbon (KMNR) by solving the eigenvalue equations in presence of intrinsicand Rashba spin-orbit couplings. Owing to the P-T symmetry of the Hamiltonianthe edge states are protected by topological invariance and hence are found tobe robust. We have done a systematic study for each of the above cases, forexample, a pristine graphene, graphene with an intrinsic spin-orbit coupling,graphene with a Rashba spin-orbit coupling, a Kane-Mele nanoribbon andsupported our results on the robustness of the edge states by analyticcomputation of the electronic probability amplitudes, the local density ofstates (LDOS), band structures and the conductance spectra.

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