Scaling of the Quantum Anomalous Hall Effect as an Indicator of Axion Electrodynamics

We report on the scaling behavior of V-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ samples in the quantum anomalous Hall regime for samples of various thickness. While previous quantum anomalous Hall measurements showed the same scaling as expected from a two-dimensional integer quantum Hall state, we observe a dimensional crossover to three spatial dimensions as a function of layer thickness. In the limit of a sufficiently thick layer, we find scaling behavior matching the flow diagram of two parallel conducting topological surface states of a three-dimensional topological insulator each featuring a fractional shift of $\frac{1}{2}{e}^2/h$ in the flow diagram Hall conductivity, while we recover the expected integer quantum Hall behavior for thinner layers. This constitutes the observation of a distinct type of quantum anomalous Hall effect, resulting from $\frac{1}{2}{e}^2/h$ Hall conductance quantization of three-dimensional topological insulator surface states, in an experiment which does not require decomposition of the signal to separate the contribution of two surfaces. This provides a possible experimental link between quantum Hall physics and axion electrodynamics.

Figure 1

${\rho }_{xy}$ (a) and ${\rho }_{xx}$ (b) of several ${\mathrm{V}}_{0.1}({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x{)}_{1.9}{\mathrm{Te}}_3$ layers as a function of the external magnetic field. The varied parameters are the Sb content $x$, the layer thickness $d$, and substrate type. The data acquired on the five layers with $d\approx 9\mathrm{nm}$ are displayed as solid lines. Four of them are grown on Si(111) with $x$ values of 0.69 (cyan), 0.79 (green), 0.80 (navy), 0.86 (magenta). One is grown on InP(111) with $x=0.79$ (purple). The measurements of two thinner layers with $d=8\mathrm{nm}$ and $x=0.78$ (orange) and $d=6\mathrm{nm}$ and $x=0.79$ (dark yellow), grown on Si(111), are displayed as dashed lines.

Figure 2

Flow diagram mapping ${\sigma }_{xx}$ to ${\sigma }_{xy}$ of layers with $d\approx 9\mathrm{nm}$ from the external magnetic field measurement (see Fig. 1) is shown. The black dashed line represents the flow diagram of the IQHE, the green dashed line the predicted flow diagram of a single topological surface state, and the red dashed line two parallel conducting topological surface states. The color code of the measurement data is the same as in Fig. 1.

Figure 3

Flow diagram mapping ${\sigma }_{xx}$ to ${\sigma }_{xy}$ for four different fixed gate voltage values $V_G=-9\mathrm{V}$ (cyan), $V_G=3\mathrm{V}$ (green), $V_G=6.2\mathrm{V}$ (blue), and $V_G=9\mathrm{V}$ (magenta), for a representative 9-nm-thick ${\mathrm{V}}_{0.1}({\mathrm{Bi}}_{0.21}{\mathrm{Sb}}_{0.79}{)}_{1.9}{\mathrm{Te}}_3$ layer exhibiting perfectly quantized transport (green in Fig. 1, Fig. 2, and Fig. 5). The black dashed line represents the flow diagram of the IQHE and the red dashed line two parallel conducting topological surface states.

Figure 4

Flow diagram mapping ${\sigma }_{xx}$ to ${\sigma }_{xy}$ for four different fixed temperature values $T=155\mathrm{mK}$ (magenta), $T=530\mathrm{mK}$ (green), $T=1\mathrm{K}$ (cyan), and base temperature of the dilution refrigerator (blue) for a representative 9-nm-thick ${\mathrm{V}}_{0.1}({\mathrm{Bi}}_{0.21}{\mathrm{Sb}}_{0.79}{)}_{1.9}{\mathrm{Te}}_3$ layer exhibiting perfectly quantized transport (green in Fig. 1, Fig. 2, and Fig. 5). The black dashed line represents the flow diagram of the IQHE and the red dashed line two parallel conducting topological surface states.

Figure 5

Flow diagram mapping ${\sigma }_{xx}$ to ${\sigma }_{xy}$ for three different layers with $d=9\mathrm{nm}$ (green), $d=8\mathrm{nm}$ (orange), and $d=6\mathrm{nm}$ (dark yellow) and similar composition, as extracted from the external magnetic field measurement (see Fig. 1). The black dashed line represents the flow diagram of the IQHE and the red dashed line two parallel conducting topological surface states.