Magnetically confined bound states in Rashba systems

A Rashba nanowire is subjected to a magnetic field that assumes opposite signs in two sections of the nanowire, and, thus, creates a magnetic domain wall. The direction of magnetic field is chosen to be perpendicular to the Rashba spin-orbit vector such that there is only a partial gap in the spectrum. Nevertheless, we prove analytically and numerically that such a domain wall hosts a bound state whose energy is at bottom of the spectrum below the energy of all bulk states. Thus, this magnetically confined bound state is well-isolated and can be accessed experimentally. We further show that the same type of magnetic confinement can be implemented in two-dimensional systems with strong spin-orbit interaction. A quantum channel along the magnetic domain wall emerges with a nondegenerate dispersive band that lies energetically below the bulk states. We show that this magnetic confinement is robust against disorder and various parameter variations.

Figure 1

(a) The NW aligned in $x$ direction with uniform Rashba SOI vector ${\mathit{\alpha }}_R$ pointing along the $y$ axis. A magnetic field $\mathbf{B}$ is applied perpendicular to ${\mathit{\alpha }}_R$, i.e. along the $z$ axis, resulting in a partial gap in the bulk spectrum. To generate a magnetic domain wall hosting a bound state (red), the directions of $\mathbf{B}$ are chosen to be opposite for $x>0$ (green region) and for $x<0$ (yellow region). (b) Bulk spectrum for uniform magnetic field for two different regimes of relative strength between Zeeman energy ${\mathrm{\Delta }}_Z$ and Rashba SOI energy $E_{\mathrm{so}}$. A gap at zero momentum $k=0$ given by ${\mathrm{\Delta }}_Z$ separates the two bands. If ${\mathrm{\Delta }}_Z/E_{\mathrm{so}}<2$, the lowest band has a local maximum at $k=0$ and a local minimum around $k\sim k_{\mathrm{so}}$. If ${\mathrm{\Delta }}_Z/E_{\mathrm{so}}>2$, only a single global minimum exists at $k=0$.

Figure 2

(a) The lowest energy states of a Rashba NW with a Zeeman field (${\mathrm{\Delta }}_Z/E_{\mathrm{so}}=1.5$) that changes sign at $x=0$, obtained by numerical diagonalization of a system with 1300 sites. The lowest energy state (black line), which corresponds to the nondegenerate bound state at $x=0$, is energetically well separated by $\mathrm{\Delta }E$ from the bulk spectrum. (b) The probability density ${\left|\psi \left(x\right)\right|}^2$ of the bound state, (c) the $x$ component, and (d) the $z$ component of its spin polarization. The analytical (red lines) and numerical results (blue dots) are in excellent agreement.

Figure 3

(a) Energy separation $\mathrm{\Delta }E/E_{\mathrm{so}}$ between the bound state and the lowest bulk state as a function of ${\mathrm{\Delta }}_Z/E_{\mathrm{so}}$ for different sizes of the domain wall $\delta$. The different colors are assigned to points obtained by numerical simulation. Black solid line corresponds to the analytical result obtained in the sharp domain wall limit. (Inset) The magnetic field profile close to the interface for different value of $\delta$. The larger $\delta$, the smaller is $\mathrm{\Delta }E$. Also, the smoother the profile, the smaller is ${\mathrm{\Delta }}_Z$ at which the bound state merges with the continuum. (b) Energy separation $\mathrm{\Delta }E/E_{\mathrm{so}}$ as a function of the magnitude of magnetic field ${\tilde{\mathrm{\Delta }}}_Z$ and the angle $\varphi$ that the magnetic field forms with the $z$ axis in the presence of an additional parallel component. This shows the robustness of $\mathrm{\Delta }E$ towards tilting of the magnetic field.

Figure 4

(a) 2D Rashba layer placed in the $xy$ plane and $\mathbf{B}$ is applied perpendicular to the plane with opposite directions for $x>0$ (green region) and for $x<0$ (yellow region) creating a one-dimensional domain wall that hosts a quantum channel running along the $y$ axis. (b) Energy spectrum as function of $k_y$ in the absence (presence) of Rashba SOI along the $y$ axis is shown on left (right) panel. The red curves describe the spectrum of the nondegenerate quantum channel localized along the domain wall. (Inset) Probability density of the quantum channel states found numerically for a finite-size system.