We investigate the spin-orbit (SO) interaction in two-dimensional electron gases in quantum wells with two subbands. From the 8 8 Kane model, we derive a new intersubband-induced SO term which resembles the functional form of the Rashba SO but is nonzero even in symmetric structures. This follows from the distinct parity of the confined states (even or odd) which obliterates the need for asymmetric potentials. We self-consistently calculate the new SO coupling strength for realistic wells and find it comparable to the usual Rashba constant. Our new SO term gives rise to a nonzero ballistic spin-Hall conductivity, which changes sign as a function of the Fermi energy (" F ) and can induce an unusual Zitterbewegung with cycloidal trajectories without magnetic fields. The rapidly developing field of spintronics has generated a great deal of interest in spin-orbit (SO) coupling in semiconductor nanostructures [1]. For an n-doped zincblende semiconductor quantum well with only the lowest subband occupied, i.e., in a strictly 2D situation, there are two main contributions to the interaction of the spin and orbital degrees of freedom of electrons. One contribution is the Dresselhaus term, which results from the lack of inversion symmetry of the underlying zinc-blende lattice [2] and is to lowest order linear in the crystal momentum [3]. This linearity is shared by the other contribution known as the Rashba term [4], which is due to structural inversion asymmetry and can be tuned by an electric gate across the well [5]. These two contributions can lead to an interesting interplay in spintronic systems [6].In this Letter we consider yet another type of electronic SO coupling which, as we show, occurs in III-V (or II-VI) zinc-blende semiconductor quantum wells with more than one subband. We derive a new intersubband-induced SO interaction which resembles that of the ordinary Rashba model; however, in contrast to the latter, ours is nonzero even in symmetric structures (Fig. 1). We self-consistently determine the strength of this new SO coupling for realistic single and double wells and find it comparable to the Rashba constant [Figs. 2(a) and 2(b)]. We have investigated the spin-Hall effect and the dynamics of spinpolarized electrons due to this new SO term. We find (i) a nonzero ballistic spin-Hall conductivity which changes sign as a function of " F and (ii) an unusual Zitterbewegung [7] with cycloidal trajectories without magnetic fields (Fig. 3). As derived below, for a symmetric well with two subbands our 4 4 electron Hamiltonian is
Recently, we have found an additional spin-orbit (SO) interaction in quantum wells with two subbands [Phys. Rev. Lett. 99, 076603 (2007)]. This new SO term is non-zero even in symmetric geometries, as it arises from the intersubband coupling between confined states of distinct parities, and its strength is comparable to that of the ordinary Rashba. Starting from the $8 \times 8$ Kane model, here we present a detailed derivation of this new SO Hamiltonian and the corresponding SO coupling. In addition, within the self-consistent Hartree approximation, we calculate the strength of this new SO coupling for realistic symmetric modulation-doped wells with two subbands. We consider gated structures with either a constant areal electron density or a constant chemical potential. In the parameter range studied, both models give similar results. By considering the effects of an external applied bias, which breaks the structural inversion symmetry of the wells, we also calculate the strength of the resulting induced Rashba couplings within each subband. Interestingly, we find that for double wells the Rashba couplings for the first and second subbands interchange signs abruptly across the zero bias, while the intersubband SO coupling exhibits a resonant behavior near this symmetric configuration. For completeness we also determine the strength of the Dresselhaus couplings and find them essentially constant as function of the applied bias.Comment: 16 pages, 12 figure
Recently, we have introduced a novel inter-subband-induced spin-orbit (s-o) coupling [Phys. Rev. Lett. 99, 076603 (2007); cond-mat/0607218] arising in symmetric wells with at least two subbands. This new s-o coupling gives rise to an usual zitterbewegung -i.e. the semiconductor analog to the relativistic trembling motion of electrons -with cycloidal motion without magnetic fields. Here we complement these findings by explicitly deriving expressions for the corresponding zitterbewegung in spin space.
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