Bulk versus surface contributions to the Rashba spin splitting of Shockley surface states

摘要

Shockley surface states of a bulk crystal with both time reversal and space inversion symmetries exhibit the Rashba spin splitting due to the broken space inversion symmetry at the surface. Since the evanescent states in the bulk region are doubly degenerate with respect to spin degrees of freedom even in the presence of spin-orbit interaction (SOI), one might think that the entire spin splitting occurs via the SOI in the surface region where the potential energy deviates from the bulk one. In the present work, we elucidate why this is not the case. Namely, in the presence of SOI, the complex energy bands are modified such that a pair of evanescent states having the same energy and the same complex wave number become two distinct solutions of the Schrodinger equation that do not satisfy the same boundary condition. Since the tail of the surface-state wave function is expressed as a superposition of these evanescent waves, the bulk region also contributes to the Rashba spin splitting. We illustrate this effect by a first-principles calculation for Au(111) and by a simplified sp-band model Hamiltonian on a square lattice.