Localized states at the Rashba spin-orbit domain wall in magnetized graphene: Interplay of Rashba and magnetic domain walls

It is well known that electronic states in graphene with a uniform Rashba spin-orbit interaction and uniform magnetization, e.g., due to exchange coupling to a magnetic substrate, display an energy gap around the Dirac $K$ and $K^{\prime }$ points. When the magnetization of graphene is nonuniform and forms a magnetic domain wall, electronic states localized at the wall emerge in the energy gap. In this paper we show that similar localized electronic states appear in the gap when the graphene is uniformly magnetized, while a domain wall appears in the Rashba spin-orbit interaction (i.e., opposite signs of the Rashba parameter on both sides of the wall). These electronic states propagate along the wall and are localized exponentially at the Rashba domain wall. They form narrow and nearly parabolic (at small wave vectors) bands, with relatively large effective electron mass. However, contrary to the magnetic domain wall, these states do not close the energy gap. We also consider the situation when the magnetic domain wall coexists with the Rashba domain wall, and both walls are localized at the same position. Electronic states due to the interplay of both domain walls are determined analytically and it is shown that the electronic states localized at the walls close the gap when a magnetic domain wall (symmetric or asymmetric) exists, independently of the Rashba parameter behavior.

Figure 1

Schematic picture of a uniformly magnetized graphene with a Rashba spin-orbit domain wall. The layer is in the $x,y$ plane, with the axis $y$ along the domain wall. In turn, the axis $z$ is normal to the layer.

Figure 2

(a) Dispersion curves of the edge states, $\varepsilon \left(k_y\right)$ (red dashed lines), for ${\lambda }_0=5$ meV and $M=10$ meV, and (b) the dependence of $\varepsilon (k_y=0)$ on the parameter ${\lambda }_0$. The solid black lines correspond to bulk bands. The dots in (a) indicate the points for which the spin polarization has been calculated, as will be discussed later. The vertical dashed line in (b) indicates the magnitude of ${\lambda }_0/M$ used in the following when calculating the spin polarization.

Figure 3

(a) The effective electron mass in the edge states, $m^{*}/m_0$, corresponding to small values of $k_y$ and presented as a function of magnetization $M$ for indicated values of Rashba parameter. (b) Bulk bands (colored areas) and the corresponding edge states (lines of the same color as the corresponding bulk bands) for different values of ${\lambda }_0$, as indicated in the legend. For simplicity, only the bulk and edge states of positive energy are shown here.

Figure 4

Spin-density components (a) $s_x$ and (b) $s_z$ corresponding to the edge states for indicated $k_y$ and $\varepsilon$, and associated with the Dirac points $K$ and $K^{\prime }$. The other parameters: ${\lambda }_0/M=0.7$ and $M=10$ meV. The red and blue lines correspond to the red and blue points marked on the dispersion curves in Fig. 2. The curves for the $K$ and $K^{\prime }$ Dirac points are equivalent.

Figure 5

The (a), (c) $x$ and (b), (d) $z$ components of spin density associated with the edge states localized at the Rashba domain wall, presented as a function of $k_y$ and $x$. The top (bottom) panels correspond to the upper (lower) energy branch. The $y$ component is not presented here as it vanishes for all values of $k_y$ and $x$.

Figure 6

The (a) $x$ and (b) $z$ components of the integrated spin in the edge states as a function of $k_y$. The red (blue) curves corresponds to the high (low) energy branch of the edge states. Here the energy of Rashba spin-orbit coupling is equal ${\lambda }_0=M/2$.

Figure 7

The total spin per unit length accumulated at the domain wall as a function of magnetization $M$. Here the Fermi level is assumed in the gap, so the total spin includes contribution from the edge mode of lower energy.

Figure 8

Schematic picture of a graphene-based hybrid system with nonuniform Rashba spin-orbit coupling and nonuniform magnetization (coexisting Rashba spin-orbit and magnetic domain walls).

Figure 9

Edge states for different parameters $[{\lambda }_l,{\lambda }_r]$ of Rashba coupling and different magnetizations $[M_l,M_r]$ on the left and right sides. Top panels: (a)–(d) $[{\lambda }_l,{\lambda }_r]=[-{\lambda }_0,{\lambda }_0]$, (a) $[M_l,M_r]=[M_0,M_0]$, (b) $[M_0/2,M_0]$, (c) $[-M_0/2,M_0]$, $[-M_0,M_0]$ (d). Bottom panels: (e)–(h) $[M_l,M_r]=[M_0,-M_0]$, (e) $[{\lambda }_l,{\lambda }_r]=[{\lambda }_0,{\lambda }_0]$, (f) $[{\lambda }_0/2,{\lambda }_0]$, (g) $[-{\lambda }_0/2,{\lambda }_0]$, (h) $[-{\lambda }_0,{\lambda }_0]$. Here, ${\lambda }_0=10\mathrm{meV}$ and $M_0=10\mathrm{meV}$. The insets show schematically the corresponding magnitudes of $[M_l,M_r]$ and $[{\lambda }_l,{\lambda }_r]$, respectively).

Figure 10

Distribution of the probability density for electron states with linear spectrum and localized at the double domain wall (coexisting magnetic and Rashba antisymmetric domain walls). (a)–(c) present the probability density for one of the edge modes (corresponding to the solution $A,B=1$), while (d)–(f) present the probability density for the second edge mode (corresponding to $C,D=1$). (a), (d) correspond to Dirac point $K$ while (b), (e) to $K^{\prime }$. In turn (c), (f) show the difference of the probability density in the $K$ and $K^{\prime }$ Dirac points for both edge modes.

Figure 11

The $x$ and $z$ components of the spin associated with the edge states, plotted as a function of $k_y$. Difference between the spins in the states corresponding to $k_y$ and associated with the $K$ and $K^{\prime }$ is clearly visible.

Figure 12

Electrical conductance due to the edge states as a function of $M_0$, calculated for the DW length $L=30\mu \mathrm{m}$, $T=0.1\mathrm{K}$, relaxation time ${\tau }_{\varepsilon }=10$ ps, chemical potential $\mu =10\mu \mathrm{eV}$ and ${\lambda }_0=10\mathrm{meV}$. The inset shows the absolute value of the corresponding electron velocity.