Part-per-million quantization and current-induced breakdown of the quantum anomalous Hall effect

E. J. Fox, I. T. Rosen, Yanfei Yang, George R. Jones, Randolph E. Elmquist, Xufeng Kou, Lei Pan, Kang L. Wang, and D. Goldhaber-Gordon

Phys. Rev. B 98, 075145 – Published 27 August 2018

Abstract

In the quantum anomalous Hall effect, quantized Hall resistance and vanishing longitudinal resistivity are predicted to result from the presence of dissipationless, chiral edge states and an insulating two-dimensional bulk, without requiring an external magnetic field. Here, we explore the potential of this effect in magnetic topological insulator thin films for metrological applications. Using a cryogenic current comparator system, we measure quantization of the Hall resistance to within one part per million and, at lower current bias, longitudinal resistivity under 10 $m Ω$ at zero magnetic field. Increasing the current density past a critical value leads to a breakdown of the quantized, low-dissipation state, which we attribute to electron heating in bulk current flow. We further investigate the prebreakdown regime by measuring transport dependence on temperature, current, and geometry, and find evidence for bulk dissipation, including thermal activation and possible variable-range hopping.

Received 20 September 2017
Revised 12 May 2018

DOI:https://doi.org/10.1103/PhysRevB.98.075145

Physics Subject Headings (PhySH)

Metrology Quantum anomalous Hall effect Topological phases of matter

Topological insulators

Condensed Matter, Materials & Applied PhysicsInterdisciplinary PhysicsGeneral Physics

Authors & Affiliations

E. J. Fox^1,2, I. T. Rosen^2,3, Yanfei Yang⁴, George R. Jones⁴, Randolph E. Elmquist⁴, Xufeng Kou^5,6, Lei Pan⁵, Kang L. Wang⁵, and D. Goldhaber-Gordon^1,2,*

¹Department of Physics, Stanford University, Stanford, California 94305, USA
²Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA
³Department of Applied Physics, Stanford University, Stanford, California 94305, USA
⁴National Institute of Standards and Technology (NIST), Gaithersburg, Maryland 20899-8171, USA
⁵Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA
⁶School of Information Science and Technology, ShanghaiTech University 201210, China

^*To whom correspondence should be addressed: goldhaber-gordon@stanford.edu

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 98, Iss. 7 — 15 August 2018

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
QAH device and dependence of conductivity on gate voltage and temperature. (a) Schematic band structure diagram of a magnetic topological insulator in the QAH state, where $E$ is energy and $k$ the wave vector. A gapless chiral edge state is hosted in the exchange-induced gap in the Dirac spectrum of the topological surface states inside the 3D bulk gap. Quantization of the Hall resistance is expected when the Fermi level is in the surface-state gap. (b) Micrograph of top-gated Hall bar based on 6-nm-thick film of ( ${Cr}_{0.12} {Bi}_{0.26} {Sb}_{0.62})_{2} {Te}_{3}$ . A simplified schematic of the measurement scheme is overlaid. (c) Longitudinal and transverse conductivities of device 1, $σ_{x x}$ and $σ_{x y}$ , respectively, derived from lock-in amplifier measurements of the resistivity as a function of gate voltage $V_{g}$ for temperatures between 21 and 915 mK. At the lowest temperatures, the conductivities plateau over a wide range in $V_{g}$ at values consistent with $σ_{x y} = - e^{2} / h$ and $σ_{x x} \approx 0$ to within expected experimental error. (d) Fitted temperature scale $T_{0}$ for thermally activated conduction, $σ_{x x} \propto e^{- T_{0} / T}$ , as a function of $V_{g}$ in device 1, peaking at 780 mK. Inset, an Arrhenius plot of $σ_{x x}$ as a function of $1 / T$ at $V_{g} = - 7.4$ V with a fit (blue line) to thermal activation.
Reuse & Permissions
Figure 2
Cryogenic current comparator measurements. (a) Precise measurements of $ρ_{y x}$ in device 1 using a 100-nA current at 21 mK show a plateau of $δ ρ_{y x} = | ρ_{y x} | - h / e^{2} \approx 0$ for a range of $V_{g}$ , indicating the accuracy of quantization. Inset, a zoomed-in view of the Hall resistance deviations in the center of the plateau. In (b), the same data are plotted as $| δ ρ_{y x} |$ on a log scale. (c, d) $ρ_{x x}$ of device 1 measured as a function of $V_{g}$ for three different bias currents shows strong current and gate voltage dependence, displayed on linear (c) and log (d) scales. At 25 nA with $V_{g}$ near the center of the plateau, the resistivity is near vanishing and measurements approach the noise floor. (e) Simplified schematic of the cryogenic current comparator. (f) Measurements of $δ ρ_{y x}$ in a different Hall bar (device 2B) as a function of current $I_{x}$ , showing accurate quantization for $I_{x} \leq 90$ nA. In all plots, error bars show the standard uncertainty and are omitted when smaller than the marker.
Reuse & Permissions
Figure 3
Current-induced breakdown of the QAH effect. (a) Longitudinal voltage $V_{x x}$ in device 1 measured with the nanovoltmeter as a function of bias current $I_{x}$ with $V_{g} = - 7.5$ V, shown for lattice temperatures of 21 mK and 203 mK as measured by the mixing chamber plate thermometer. At base temperature and $I_{x} \leq 100$ nA, we observe an apparent power law $V_{x x} \propto I_{x}^{6.4}$ . After a sharp rise in $V_{x x}$ for $T = 21$ mK starting at around $I_{x} = 125$ nA, the curves for the two temperatures nearly overlap, consistent with runaway electron heating. (b) $ρ_{x x}$ of device 1 calculated from the data in (a), shown on a linear plot versus $I_{x}$ . (c) Hall resistance deviations $δ ρ_{y x}$ of device 1, measured on three separate upward sweeps of current (shown with different colors and symbols—data shown in blue and yellow were taken on the opposite pair of voltage contacts from that in red), plotted against separate measurements of $ρ_{x x}$ from (b). Almost all points below $ρ_{x x} = 1 k Ω$ , where $| δ ρ_{y x} | < 10^{- 4} e^{2} / h$ , fall within $| δ ρ_{y x} | < 0.03 ρ_{x x}$ , shown by the dashed lines. A detailed view of the data with low $ρ_{x x}$ is given in the Supplemental Material [29]. (Inset) $δ ρ_{y x}$ exhibits quadratic dependence on $ρ_{x x}$ at higher dissipation. (d) Longitudinal electric field $E_{x}$ vs current density $j_{x}$ for three Hall bars of varying size (devices 2A–2C), each at their optimum gate voltage, showing geometry-independent behavior at high current density and similar behavior even at low current density. The average transverse electric field $E_{y} = ρ_{y x} j_{x}$ shown on the top axis is calculated assuming $ρ_{y x} = h / e^{2}$ .
Reuse & Permissions
Figure 4
Temperature dependence of $σ_{x x}$ for $V_{g} = - 7.5$ V. (a) Arrhenius plot of the temperature dependence of $σ_{x x}$ in device 1 for five bias currents between 25 nA and 125 nA. The conductivity appears relatively temperature independent below 50 mK, while above 100 mK it may be thermally activated, although we cannot rule out variable-range hopping conduction based on the temperature dependence alone. The black solid line shows the fit to thermal activation for measurements with a lock-in amplifier at 5 nA ac bias current [Fig. 1, inset], and the dashed line extrapolates the fit to lower temperatures. (b) Detailed view of $σ_{x x}$ vs $1 / T$ for higher temperatures, with fits to activated conductivity (colored dashed lines) for measurements above 100 mK. (c) Fitted activation temperature scale $T_{0}$ as a function of current density $j_{x}$ or transverse electric field $E_{y} \approx (h / e^{2}) j_{x}$ . A fit (blue line) to $T_{0} (E_{y}) = T_{0} (0) - a e E_{y} / k_{B}$ for $j_{x} \leq 0.75$ nA/ $μ m$ yields $T_{0} (0) = 850 \pm 30$ mK and $a = 740 \pm 170$ nm.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics