Electronic and spin properties of Rashba billiards

Abstract

Ballistic electrons confined to a billiard and subject to spin-orbit coupling of the Rashba type are investigated, using both approximate semiclassical and exact quantum-mechanical methods. We focus on the low-energy part of the spectrum that has negative eigenvalues. When the spin precession length is smaller than the radius of the billiard, the low-lying energy eigenvalues turn out to be well described semiclassically. Corresponding eigenspinors are found to have a finite spin polarization in the direction perpendicular to the billiard plane. Spin-dependent phenomena in semiconductor nanostructures have become the focus of strong interest recently [1,2]. In nonmagnetic systems, intriguing effects can arise from the presence of spin-orbit coupling. Structural inversion asymmetry in semiconductor heterostructures has been shown [3] to give rise to a spin splitting of the same type as was discussed in an early paper by Rashba [4]. Its tunability by external gate voltages [5,6] has motivated the theoretical design of a spin-controlled field-effect transistor [7]. Novel spin properties arise from the interplay between Rashba spin splitting and further confinement of two-dimensional electrons in quantum wires [8,9,10,11] or dots [12,13,14,15,16,17]. Spinorbit coupling has also been shown to affect the statistics of energy levels and eigenfunctions as well as current distributions [18,19].In this work, we study a Rashba billiard, i.e., noninteracting ballistic electrons moving in finite 2D regions whose dynamics is affected by the Rashba spin-orbit coupling. The Rashba spin-orbit coupling strength can be conveniently measured in terms of a wave-number scale 2k so , which corresponds to the Fermi-wave-vector difference for the two spin-split subbands. Typical values for the spin-orbit-induced spin precession length L so = π/k so are of the order of a few hundred nanometers [1]. The relevant parameter characterizing a Rashba billiard of size L is k so L. Taking L = 10µm for a typical size of quantum dots, the relevant parameter in Rashba billiards can be as large as 70. Furthermore, the tunability of the Rashba spin-orbit coupling strength is a convenient tool to induced changes of the billiard's energy spectrum without applying external magnetic fields.Below we present interesting features of the energy spectrum for Rashba billiards, focusing especially on its negative energy eigenvalues. We will show that the density of states (DOS) is singular at the bottom of the spectrum. This singular behavior occurs independently of the billiard's shape and is most striking if the Rashba parameter is large. We have found that for a circular shape, the DOS has additional singularities at negative energies. We obtain analytic results for their positions. Their corresponding eigenspinors have a finite spin projection in the direction perpendicular to the billard plane, which is the direct result of imposing hard-wall boundary conditions.Our central quantity of interest is the Green's function for Rashba billiards. Having obtained i...

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