One-dimensional quantum waveguide theory of Rashba electrons

Abstract

The ballistic spin transport in one-dimensional waveguides with the Rashba effect is studied. Due to the Rashba effect, there are two electron states with different wave vectors for the same energy. The wave functions of two Rashba electron states are derived, and it is found that their phase depend on the direction of the circuit and the spin directions of two states are perpendicular to the circuit, with the +π/2 and −π/2 angles, respectively. The boundary conditions of the wave functions and their derivatives at the intersection of circuits are given, which can be used to investigate the waveguide transport properties of Rashba spin electron in circuits of any shape and structure. The eigenstates of the closed circular and square loops are studied by using the transfer matrix method. The transfer matrix M(E) of a circular arc is obtained by dividing the circular arc into N segments and multiplying the transfer matrix of each straight segment. The energies of eigenstates in the closed loop are obtained by solving the equation det[M(E)−I]=0. For the circular ring, the eigenenergies obtained with this method are in agreement with those obtained by solving the Schrödinger equation. For the square loop, the analytic formula of the eigenenergies is obtained first. The transport properties of the AB ring and AB square loop and double square loop are studied using the boundary conditions and the transfer matrix method. In the case of no magnetic field, the zero points of the reflection coefficients are just the energies of eigenstates in closed loops. In the case of magnetic field, the transmission and reflection coefficients all oscillate with the magnetic field; the oscillating period is Φm=hc/e, independent of the shape of the loop, and Φm is the magnetic flux through the loop. For the double loop the oscillating period is Φm=hc/2e, in agreement with the experimental result. At last, we compared our method with Koga’s experiment.