The quantum anomalous Hall effect (QAHE)1,2,3,4,5,6 is a transport phenomenon where the Hall resistance is quantized to the von Klitzing constant due to the spontaneous magnetization of a ferromagnetic material even at zero magnetic field. Similar to the quantum Hall effect (QHE) under strong magnetic fields, the quantized Hall resistance of QAHE is supposed to be universal, independent of the details in the experimental realization7,8. However, the quantization accuracy of QAHE reported so far9,10,11 is much poorer than that of QHE. Here we demonstrate a precision of 10 parts per billion of Hall resistance quantization in QAHE. By directly comparing QAHE with QHE, we confirm that the quantization accuracy of QAHE satisfies the required level as a primary standard of electric resistance. We achieve this high accuracy of quantization by using a weak magnetic field supplied by a permanent disc magnet to align the magnetization domains. Our findings establish a milestone for developing a quantum resistance standard without strong magnetic fields.
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Source data are provided with this paper. The data that support the findings of this study are available from the corresponding authors on reasonable request.
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This research was supported by JST CREST (no. JPMJCR16F1 (M. Kawasaki)); MEXT/JSPS KAKENHI grant nos. JP15H05867 (M. Kawamura), JP15H05853 (A.T.), JP17H04846 (R.Y.), JP18H04229 (R.Y.), JP18H01155 (M. Kawamura), JP18H01156 (Y.O.) and JP18H05258 (N.-H.K.); and RIKEN-AIST Joint Research Fund (Y.O. and M. Kawamura).
The authors declare no competing interests.
Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work
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Extended Data Fig. 1 Device Structure.
Photo-mask pattern of the mesa and the Ohmic contacts for the 800 μm-wide Hall bar device.
Extended Data Fig. 2 Transport properties of the FMTI film.
a, Optical microscope image of a 400 μm-wide Hall bar. S, D, and G denote source, drain contacts and a top-gate electrode, respectively. Vy was measured between the electrodes U2 and L2, and Vx was measured between L2 and L3. b, Gate voltage dependence of ρyx and ρxx with a measurement current I = 1.0 μA. The data were taken at B = 0 T after magnetizing the sample at B = 1.0 T. The value of ρxx was multiplied by 10 for clarity. c, d, Field- and zero-field cooling traces of ρyx and ρxx. Magnetic field B = 0.2 T in the field cooling measurement.
Extended Data Fig. 3 Gate dependence of ρxx.
Gate voltage VG dependence of ρxx measured at I = 0.5 μA using the upper pair of the voltage contacts (U1-U5) of the 800 μm-wide Hall bar device. The values of ρxx are suppressed around − 2 V < VG < 0.5 V, indicating that the QAHE is well developed at VG = 0 V. Note that the measurement was performed below 0.5 V because a relatively large gate leakage was observed above 1.0 V.
Extended Data Fig. 4 One-to-one resistance bridge between the two QHEs.
a, b, Simplified circuit diagrams in the normal (a) and swapped (b) configurations. Here RX is RQAHE (RQHE1) and RS is RQHE1 + RQHE2(RQHE2) for the QAHE-QHE (QHE-QHE) comparison. c, Detailed circuit diagram of the resistance bridge for the comparison between QHE1 and QHE2.
Extended Data Fig. 5 Measurement sequence.
Single batch on-off sequence of the current I and the resulting measured voltage difference Vd. The current ramping rate was 0.16 μA/s. In the measurements of Fig. 4a, the single batch of the sequence took 45 s for I = 0.8 μA, and it took about 110 hours in total [45 s (single batch) × 2200 (number of points) × 2 (current polarity change) × 2 (circuit swapping)].
Extended Data Fig. 6 Uncertainty evaluation.
Relative uncertainty from possible source components in the normalized deviation ΔR/RK for the QAHE-QHE comparison with N = 2200 and I = 0.8 μA, and for the QHE-QHE comparison with N = 50 and I = 10 μA. In both cases, the dominant component of the uncertainty is dispersion from random effect.
Extended Data Fig. 7 Comparison of the relevant parameters.
Material: the materials and the layer structure of the FMTI film (the capping layer and the gate electrode are omitted for clarity). ΔR/RK: the normalized deviation in the values of the quantized anomalous Hall resistance. The current value I used for the precise measurement is also appended. (*Hall resistance measurements performed at I = 0.005 μA and 0.01 μA. **obtained by averaging the Hall resistance measurements for I < 0.09 μA.) W: Hall bar width. T0: activation energy (***measured in the 400 μm-wide Hall bar as described in Methods). ρxx: longitudinal resistivity. Method: experimental technique for the precision measurement of the Hall resistance.
Source Data Fig. 2
Numerical source data for Fig. 2.
Source Data Fig. 3
Numerical source data for Fig. 3c.
Source Data Fig. 4
Numerical source data for Fig. 4.
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Okazaki, Y., Oe, T., Kawamura, M. et al. Quantum anomalous Hall effect with a permanent magnet defines a quantum resistance standard. Nat. Phys. 18, 25–29 (2022). https://doi.org/10.1038/s41567-021-01424-8