Dimensional Crossover-Induced Topological Hall Effect in a Magnetic Topological Insulator

We report transport studies of Mn-doped ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ topological insulator (TI) films with an accurately controlled thickness grown by molecular beam epitaxy. We find that films thicker than five quintuple layers (QLs) exhibit the usual anomalous Hall effect for magnetic TIs. When the thickness is reduced to four QLs, however, characteristic features associated with the topological Hall effect (THE) emerge. More surprisingly, the THE vanishes again when the film thickness is further reduced to three QLs. Theoretical calculations demonstrate that the coupling between the top and bottom surface states at the dimensional crossover regime stabilizes the magnetic Skyrmion structure that is responsible for the THE.

Figure 1

The Hall effect at various temperatures in eight-QL $\mathrm{Mn}\text{−}{\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) Schematic side view of the magnetic TI sample used in the transport measurement. The red arrows indicate the magnetic moments of Mn, and the extension of surface states into the bulk is displayed by the color gradient. (b) The blue and red arrows represent the opposite spin chirality of the top and bottom surface states, respectively, when they are decoupled. On the right side is the schematic surface band structure at the ultrathin limit with hybridization. (c) Temperature dependence of the Hall effect in the eight-QL $\mathrm{Mn}\text{−}{\mathrm{Bi}}_2{\mathrm{Te}}_3$ sample at $V_g=0\mathrm{V}$. The arrows mark the direction of the magnetic field sweep.

Figure 2

Temperature dependence of the Hall effect and the magnetoresistivity in five-, four-, and three-QL $\mathrm{Mn}\text{−}{\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) In the five-QL sample, the Hall effect exhibits the usual AHE behavior commonly observed in FM materials. (b) In the four-QL sample, an extra contribution to the Hall effect (marked by the green patches) appears simultaneous with the AHE. These broad humps are the characteristic behavior of the THE due to magnetic Skyrmions. (c) In the three-QL sample, the Hall effect recovers the usual squared-shaped AHE loop, and the THE features disappear completely. (d)–(f) The longitudinal resistivity ${\rho }_{xx}$ as a function of magnetic field in the five-, four-, and three-QL samples. The red and blue arrows mark the direction of the magnetic field sweep.

Figure 3

Evolution of the anomalous Hall effect and topological Hall effect in four-QL $\mathrm{Mn}\text{−}{\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) Gate voltage dependence of the Hall effect. The THE exists in the whole $V_g$ range and shows a maximum value near $V_g=-25\mathrm{V}$, which is close to the charge neutral point when the slope of normal Hall effect changes sign. (b) Evolution of the anomalous Hall resistivity (blue curve) ${{\rho }^0}_{yx}$ and topological Hall resistivity (red curve) ${{\rho }^T}_{yx}$ with the temperature. (c) The phase diagram of the four-QL $\mathrm{Mn}\text{−}{\mathrm{Bi}}_2{\mathrm{Te}}_3$. The broad area between the $H_C$ and $H^T$ lines represents the regime when magnetic Skyrmions are stable. (d) Gate voltage dependence of the AHE ${{\rho }^0}_{yx}$ (blue curve) and THE ${{\rho }^T}_{yx}$ (red curve), both showing a peak near the charge neutral point and electron-hole asymmetry.

Figure 4

Numerical calculation of the DM interaction and simulation of the Skyrmion stability. (a) The DM interaction strength as a function of the hybridization gap calculated with an inversion symmetry breaking potential $2U=20\mathrm{meV}$. The sharp peak indicates that the DM interaction is significant only for a narrow range of the hybridization gap corresponding to a specific film thickness. (b) Calculated DM interaction for various chemical potentials with the hybridization gap fixed at $\mathrm{\Delta }=50\mathrm{meV}$. The behavior is consistent with the gate voltage dependence of the THE shown in Fig. 3. (c) The energy of three different types of Skyrmion configuration relative to the FM state as a function of the Skyrmion size. The Néel2-type Skyrmion has an energy minimum for the size of four sites, indicating that it is the most stable configuration in the four-QL $\mathrm{Mn}\text{−}{\mathrm{Bi}}_2{\mathrm{Te}}_3$ film.