Zero-Field Dissipationless Chiral Edge Transport and the Nature of Dissipation in the Quantum Anomalous Hall State

The quantum anomalous Hall (QAH) effect is predicted to possess, at a zero magnetic field, chiral edge channels that conduct a spin polarized current without dissipation. While edge channels have been observed in previous experimental studies of the QAH effect, their dissipationless nature at a zero magnetic field has not been convincingly demonstrated. By a comprehensive experimental study of the gate and temperature dependences of local and nonlocal magnetoresistance, we unambiguously establish the dissipationless edge transport. By studying the onset of dissipation, we also identify the origin of dissipative channels and clarify the surprising observation that the critical temperature of the QAH effect is 2 orders of magnitude smaller than the Curie temperature of ferromagnetism.

Figure 1

Local two-terminal and three-terminal measurements in the QAH regime. (a) Magnetic field (${\mu }_{0}H$) dependence of two-terminal local resistances ${\rho }_{12,12}$, ${\rho }_{13,13}$, and ${\rho }_{14,14}$. (b) The peak value near the coercive field ($H_c$) of two-terminal resistances ${\rho }_{12,12}$, ${\rho }_{13,13}$, and ${\rho }_{14,14}$ as a function of the spacing length $l(\mathrm{mm})$ between the voltage electrodes. The inset photograph shows the Hall bridge device. (b) Schematic layout of the device applicable for panels (d) and (e) and also for ${\rho }_{14,14}$ in (a). The current flows from 1 to 4. The red and blue lines indicate the chiral edge current for magnetization into ($M<0$) and out of the plane ($M>0$), respectively. (d),(e) ${\mu }_{0}H$ dependence of three-terminal resistances ${\rho }_{14,13}$, ${\rho }_{14,12}$ (d) and ${\rho }_{14,15}$, ${\rho }_{14,16}$ (e).

Figure 2

Local and nonlocal three-terminal measurements in the QAH regime. (a) Chiral edge conduction channels when the current flows from 3 to 1. (b) ${\mu }_{0}H$ dependence of local and nonlocal three-terminal resistances ${\rho }_{31,32}$, ${\rho }_{31,34}$, ${\rho }_{31,35}$, and ${\rho }_{31,36}$ measured at $T=25\mathrm{mK}$ with $V_g=V_g^0$.

Figure 3

Temperature dependence of chiral edge transport in the QAH regime. (a) ${\mu }_{0}H$ dependence of the nonlocal resistance ${\rho }_{26,35}$ measured at $V_g=V_g^0$ from 25 mK to 20 K. (b) Chiral edge conduction channels when the current flows from 2 to 6, and the nonlocal voltage measured between 3 and 5. (c) Temperature dependence of the zero-field longitudinal sheet resistance ${\rho }_{14,23}$ (0) (blue curve) and Hall resistance ${\rho }_{14,35}$ (0) (red curve). (d) Temperature dependence of zero-field nonlocal signal ${\rho }_{26,35}$ (0) for $M<0$ (red curve) and $M>0$ (blue curve), respectively. (e) Temperature dependence of nonlocal resistance ${\rho }_{26,35}$ peaks in the PPT as seen in (a) going between two magnetization orientations.

Figure 4

Gate bias dependence of chiral edge transport in the QAH regime. (a)–(j) ${\mu }_{0}H$ dependence of longitudinal resistance ${\rho }_{14,23}$ and Hall resistance ${\rho }_{14,35}$ (a)–(e), as well as the nonlocal signal ${\rho }_{26,35}$ (f)–(j) at various $V_g\mathrm{s}$. (k) $V_g$ dependence of the zero-field longitudinal sheet resistance ${\rho }_{14,23}$ (0) (blue curve) and Hall resistance ${\rho }_{14,35}$ (0) (red curve). (l) $V_g$ dependence of zero-field nonlocal signal ${\rho }_{26,35}$ (0) for $M<0$ (red curve) and $M>0$ (blue curve), respectively. (m) $V_g$ dependence of the ratio between nonlocal resistance ${\rho }_{26,35}$ (0) for $M>0$ and longitudinal sheet resistance ${\rho }_{14,23}$ (0). Note that the nonlocal resistance ${\rho }_{26,35}$ (0) for $M>0$ indicates the dissipation channels in the voltage probe side. (n) The schematic diagrams identify three channels in the sample: dissipationless edge channels (red arrows), dissipative edge channels (blue arrows), and dissipative bulk channels (green arrows). (o) The schematic band dispersions of the sample. The horizontal dashed-dotted line indicates the Fermi level position for $V_g>V_g^0$ and $V_g<V_g^0$, respectively. The excitation gap is defined as the energy difference between the bottom of the nonchiral edge mode and the maximum of the bulk valence band as indicated.