Landauer-Büttiker study of the anomalous Hall effect

We report on Landauer-Büttiker studies of anomalous Hall transport in a two-dimensional electron gas with Rashba spin-orbit coupling and a magnetization provided by localized magnetic moments. Our system is described by a discretized tight-binding model in a four-terminal geometry. We consider both the case of magnetically disordered systems as well as ballistic transport in disorder-free systems with spatially homogeneous magnetization. In the latter case we investigate both out-of-plane and in-plane magnetizations. We numerically establish a close connection between singularities in the density of states and peaks in the Hall conductance close to the lower band edge. Consistent with previous theoretical studies based on diagrammatic perturbation expansions, these peaks occur at Fermi energies where only the lower dispersion branch is occupied. Moreover, for large magnetization the Hall conductance is, along with the density of states, suppressed. This numerical finding can be understood from analytical properties of the underlying model in the limit of an infinite system.

Figure 1

Energy dispersion ${\epsilon }_{\pm }\left(\vec{k}\right)$ for $\Delta =0.2{\epsilon }_R$ (solid lines) and $\Delta =1.5{\epsilon }_R$ (interrupted lines). Left panel: perpendicular magnetization $\vec{\Delta }=\left(0,0,\Delta \right)$. Right panel: in-plane magnetization $\vec{\Delta }=\left({\Delta }_x,{\Delta }_y,0\right)$ with the wave vector $\vec{k}$ being orthogonal to $\vec{\Delta }$, $\vec{\Delta }\cdot \vec{k}=0$. The energies are given in units of the Rashba energy ${\epsilon }_R$ while the wave vector is measured in units of the inverse “Rashba wave length” $k_R=m^{\ast }\alpha /{\hslash }^2$.

Figure 2

(Color online) Four-terminal Hall bridge. The central region is described by the discretized Hamiltonian (12) incorporating Rashba spin-orbit coupling and magnetic impurities.

Figure 3

(Color online) Hall conductance and DOS for a Zeeman coupling $\Delta =0.001t$ and a Rashba parameter $\lambda =0.01t$ as a function for Fermi energy ${\epsilon }_f∊\left[-4t-\Delta ,4t+\Delta \right]$ and a linear system size of $N=30$. The inset shows the behavior of the Hall conductance and the DOS close to the lower band edge. Both quantities are characterized by a simultaneous peak.

Figure 4

(Color online) DOS (red dashed line) in units $1/t{a}^2$ and Hall conductance (blue solid line) in units of $e^2/h$ near the lower band edge for the same system size and Rashba coupling as in Fig. 3 but various Zeeman couplings $\Delta$. In all cases an obvious correspondence between the extrema DOS and Hall conductance occurs.

Figure 5

(Color online) Hall Conductance near the bottom of the band for a Rashba parameter of $\lambda =0.01t$ and various Zeeman couplings $\Delta$. The linear system size varies from $N=30$ (blue, right), $N=35$ (red), $N=40$ (green), $N=45$ (yellow), and $N=50$ (black, left).

Figure 6

(Color online) Dependence of the position ${\epsilon }_N^{\ast }$ of the Hall conductance peak on the system size for same data sets as in Fig. 5. The Rashba parameter is $\lambda =0.01t$.

Figure 7

(Color online) Energy position ${\epsilon }^{\ast }$ of the Hall conductance peak at the lower band edge extrapolated to an infinite system versus the magnetic coupling $\Delta$: the dependence is well fitted by a linear function.

Figure 8

(Color online) Hall conductance for a Rashba parameter of $\lambda =0.01t$ and various directions of magnetization with the nonzero components of $\vec{\Delta }$ stated in the legend. For magnetization along the $x$ direction no Hall current occurs.

Figure 9

(Color online) DOS (red dashed line) in units $1/t{a}^2$ and Hall conductance (blue solid line) in units of $e^2/h$ near the lower band edge for an in-plane magnetization polarized along the $y$ direction, for a system of linear size $N=30$ and for three different magnitudes of the Zeeman splitting. This picture is analogous to Fig. 4 for the case of a perpendicular magnetization. Even in the case of an in-plane magnetization we observe the correspondence between the extrema DOS and Hall conductance.

Figure 10

(Color online) Hall conductance near the bottom of the band for a Rashba parameter of $\lambda =0.01t$ in the case of an in-plane magnetization along the $y$ direction and for various Zeeman couplings ${\Delta }_y$. The linear system size varies from $N=30$ (blue, right), $N=35$ (red), $N=40$ (green), $N=45$ (yellow), and $N=50$ (black, left). This picture can be directly compared with Fig. 5 for the case of a perpendicular magnetization.

Figure 11

(Color online) Hall conductance and DOS for a system of linear dimension $N=40$ and for a perpendicular magnetization of amplitude ${\Delta }_z=0.01t$ (blue dashed line) and an in-plane magnetization ${\Delta }_y=0.01t$ (green solid line).

Figure 12

(Color online) Hall conductance close to the lower band edge for different fractions of magnetically occupied sites for a Rashba parameter of $\lambda =0.01t$ and an average magnetization of $\Delta =0.001t$. The linear size of the system is $N=40$.

Figure 13

(Color online) Hall conductance maxima for systems of linear dimension $N=30$ (blue, right), $N=40$ (green), $N=50$ (black, left) and for several fractions of magnetically occupied sites.