Precise Quantization of the Anomalous Hall Effect near Zero Magnetic Field

We report a nearly ideal quantum anomalous Hall effect in a three-dimensional topological insulator thin film with ferromagnetic doping. Near zero applied magnetic field we measure exact quantization in the Hall resistance to within a part per 10 000 and a longitudinal resistivity under $1\mathrm{\Omega }$ per square, with chiral edge transport explicitly confirmed by nonlocal measurements. Deviations from this behavior are found to be caused by thermally activated carriers, as indicated by an Arrhenius law temperature dependence. Using the deviations as a thermometer, we demonstrate an unexpected magnetocaloric effect and use it to reach near-perfect quantization by cooling the sample below the dilution refrigerator base temperature in a process approximating adiabatic demagnetization refrigeration.

Figure 1

Device demonstrating quantum anomalous Hall effect. (a) Photograph of 10-nm-thick film of $({\mathrm{Cr}}_{0.12}{\mathrm{Bi}}_{0.26}{\mathrm{Sb}}_{0.62}{)}_2{\mathrm{Te}}_3$ on a GaAs substrate, scratched by hand into a Hall bar shape, with indium metal Ohmic contacts. Schematic measurement setup included. (b) Longitudinal resistivity ${\rho }_{xx}$ and transverse resistivity ${\rho }_{yx}$ of the device at base temperature as a function of the applied magnetic field ${\mu }_{0}H$ in each sweep direction, forming a ferromagnetic hysteresis loop. As the field approaches zero from either starting point, ${\rho }_{yx}$ reaches its quantized value $h/e^2$ and ${\rho }_{xx}$ approaches zero. (c) Nonlocal and two-terminal measurements verifying edge-dominated transport. The insets show the measurements performed and chirality at each magnetization.

Figure 2

Precise quantization near zero applied field. (a) Longitudinal and transverse conductivities in hysteresis loops over field ranges smaller than the coercive field so as to maintain the starting magnetization $M_z$. When approaching zero field from either starting point, ${\sigma }_{xy}=e^2/h$ to 0.01% precision while ${\sigma }_{xx}$ reaches as low as $0.0002e^2/h$. (b),(c) Resistivities measured during the hysteresis loops, plotted parametrically, both before (b) and after (c) performing a correction for geometry. The inset of (b) shows the linear deviation of the two magnetization branches from a parabolic line, resulting from uneven spacing of the leads. The inset of (c) shows a close-up of the corrected resistivity data, with ${\rho }_{yx}$ quantized to $h/e^2$ within $3\mathrm{\Omega }$ whenever ${\rho }_{xx}<200\mathrm{\Omega }$.

Figure 3

Temperature dependence implying possible thermal activation. Parametric plot of ${\sigma }_{xx}$ versus ${\sigma }_{yx}$ at each magnetization while increasing temperature, with calculated renormalization group flow lines shown in gray. Inset: an Arrhenius plot of ${\sigma }_{xx}$ as a function of inverse temperature, showing roughly exponential dependence. The implied energy scale of the fit is $17\mu \mathrm{eV}$, or 200 mK.

Figure 4

Evidence for demagnetization cooling. (a) Inferred temperature of the TI film, extrapolated using the temperature fit in Fig. 3, during the hysteresis loop in Fig. 2. After a long wait, the temperature approaches the fridge thermometer reading $\sim 40\mathrm{mK}$, whereas sweeping $|H|$ down drives the temperature below 30 mK and sweeping $|H|$ up drives it above 90 mK. (b) Effect of upward field sweep on ${\sigma }_{xx}$ in three field and polarity regions, consistent with the demagnetization hypothesis. When $H<0$ (decreasing $|H|$), ${\sigma }_{xx}$ decreases when the field changes and creeps back up during a 5-min wait. When $H>0$ (increasing $|H|$), the opposite occurs regardless of magnetization. (c) A rudimentary demagnetization cycle, involving a slow field sweep to magnetize and a fast field sweep to zero to perform demagnetization cooling. Plotted during this cycle are the inferred temperature, ${\rho }_{xx}$ (now dropping below $2\mathrm{\Omega }$), and ${\sigma }_{xy}$, which remains within 0.01% of $e^2/h$ during the last three stages.