Topological Hall Effect and Berry Phase in Magnetic Nanostructures

We discuss the anomalous Hall effect in a two-dimensional electron gas subject to a spatially varying magnetization. This topological Hall effect does not require any spin-orbit coupling and arises solely from Berry phase acquired by an electron moving in a smoothly varying magnetization. We propose an experiment with a structure containing 2D electrons or holes of diluted magnetic semiconductor subject to the stray field of a lattice of magnetic nanocylinders. The striking behavior predicted for such a system (of which all relevant parameters are well known) allows one to observe unambiguously the topological Hall effect and to distinguish it from other mechanisms.

Figure 1

The proposed structure consisting of a triangular lattice of magnetic nanocylinders on top of 2D diluted magnetic semiconductor.

Figure 2

Distribution of the $z$-component of dipolar field $B/4\pi M_s$ inside the semiconductor film for the triangular lattice of magnetic nanocylinders, for a zero external field. The black solid circles correspond to the lines with $B_z=0$. Dashed lines correspond to the lines with $B_z=0$ under a uniform external magnetic field $B_{\mathrm{e}\mathrm{x}\mathrm{t}}/4\pi M_s=+0.058$.

Figure 3

Topological field $B_t(\mathbf{r})$ (in units of ${\varphi }_0$ per unit cell area) for the triangular lattice of magnetic nanocylinders. Black lines correspond to $B_t=0$.

Figure 4

Dependence of the Hall conductivity on external magnetic field (schematically). The slope corresponds to the contribution of the normal Hall effect; ${\sigma }_{xy}^0$ is the Hall conductivity corresponding to a topological flux per unit cell equal to ${\varphi }_0$.