Spin Edge Helices in a Perpendicular Magnetic Field

We present an exact solution to the problem of the spin edge states in the presence of equal Bychkov-Rashba and Dresselhaus spin-orbit fields in a two-dimensional electron system, restricted by a hard-wall confining potential and exposed to a perpendicular magnetic field. We find that the spectrum of the spin edge states depends critically on the orientation of the sample edges with respect to the crystallographic axes. Such a strikingly different spectral behavior generates new modes of the persistent spin helix–spin edge helices with novel properties, which can be tuned by the applied electric and magnetic fields.

Figure 1

Two physically different configurations of the 2DES, restricted with a hard-wall confining potential at $x=0$ (filled areas). The spin edge states propagate in the skipping orbits along $[1\overline{1}0]$ and [110] axes, respectively, parallel and perpendicular to the direction of electron spins. The Fermi contours in the absence of the magnetic field are shown in the momentum plane with arrows indicating the directions of spin.

Figure 2

The energy spectrum of spin edge states in the presence of Bychkov-Rashba and Dresselhaus SOI of equal strengths. The dashed and solid curves correspond to the (a) and (b) configurations in Fig. 1.

Figure 3

Spin edge helices in a perpendicular magnetic field. Panels (a),(c) [(b),(d)] correspond to the (a) configuration [the (b) configuration]. Panels (a),(b) [(c),(d)] correspond to the limit of strong [weak] magnetic fields. The bottom (top) of each figure represents the helical structures in the edge channels [in the quasibulk Landau states].