Bulk dissipation in the quantum anomalous Hall effect

In three-dimensional topological insulators, strong spin–orbit interactions give rise to 2D Dirac-like surface states with spin-momentum locking, surrounding a fully gapped 3D bulk. An exchange gap opens in this Dirac spectrum when time-reversal symmetry is broken through introduction of ferromagnetism by magnetic doping.^1–3 The result is a Chern insulating state where a single chiral edge mode connects the gapped 2D Dirac bands. By tuning the Fermi level (E_F) into the exchange gap, dissipationless 1D conduction at zero external magnetic field is observed. This is known as the quantum anomalous Hall (QAH) effect. Similar to the quantum Hall (QH) effect, transport measurements of the QAH effect are predicted to yield zero longitudinal resistivity ρ_xx = 0 and a quantized Hall resistivity ρ_yx = h/e², where h is Planck’s constant and e is the electron charge.

The QAH system is a vehicle for tailored control of the spin and momentum of electrons. As parts of new heterostructures, QAH materials could have applications in dissipationless spintronics^4,5 and quantum computing.^6,7 Furthermore, QAH systems could replace the quantum Hall (QH) systems used today as resistance standards. Needing no large magnetic field, QAH-based resistance metrology would be more portable and economical. QAH metrological standards would further allow for simultaneous measurement of a resistance standard alongside a Josephson voltage standard in a single cryostat, something that is not currently possible due to the magnetic fields required for measuring QH devices.⁸ 

Three-dimensional thin-film magnetic topological insulators (3D MTIs) have indeed demonstrated the QAH effect.^9–17 Cr- and V-doped (BiSb)₂Te₃ (Cr-BST and V-BST) samples have shown Hall resistances equal to h/e² to part-per-million precision.^18,19 However, these samples are not perfect. In all instances, this quantization was accompanied by a finite longitudinal resistivity ρ_xx ≠ 0, even at the lowest temperatures and excitation currents, with rapid increases at higher temperatures and currents. This dissipation hinders technological applications of QAH materials as it demands operation at temperatures and current densities far lower than can be used with typical QH systems.

Despite the high ferromagnetic Curie temperature (∼20 K for Cr- and V-doped BST samples^{3,10–16,20,21}), quantization of the Hall resistivity ρ_yx to even 5% accuracy is typically limited to temperatures below 200 mK, with the exception of films with elaborate five-layer modulation of Cr concentration; these have exhibited ρ_yx = 0.97 h/e² at 2 K.¹⁴ In all these cases, the amount of dissipation measured at these temperatures is far greater than one would expect given the large exchange gap, which has been shown to be on the order of tens of meV.^1,21,22 These discrepancies may relate to the presence of residual conduction channels that add dissipation to the chiral edge conduction.¹⁵ Understanding the types of dissipative conduction channels present could enable targeted material improvements to increase the temperature at which QAH can be observed.

One proposed source of dissipation in thin-film MTIs is backscattering in quasihelical edge modes, which are predicted to exist on the sidewalls of thick QAH insulator films.^12,23,24 Having no perpendicular magnetization, the sidewalls do not have an exchange gap. Thin sidewalls are gapped due to transverse quantum well confinement, but thick sidewalls could support counterpropagating modes. In Cr-BST, these quasihelical edge modes are predicted to exist for all samples with a thickness greater than three quintuple layers (QLs).²³ Yet, the QAH phase in Cr-BST requires a minimum thickness of four QLs due to the competition between the magnetic exchange and the hybridization between top and bottom surface states.²⁵ As quasihelical edge modes are a significant source of dissipation, Cr-BST-based QAH insulators therefore could not host truly dissipationless transport at any thickness, regardless of material improvements. On the other hand, dissipative conduction arising from disorder-based bulk processes, such as variable range hopping^10,26 or thermal activation of bulk and surface carriers,⁹ could plausibly be addressed by reducing disorder. Transport measurements to date of magnetically substituted BST films have been interpreted to support either bulk-dominated¹⁸ or edge-dominated²⁴ dissipation, so there is a lack of consensus.

Most transport studies of QAH materials have employed Hall bar geometries, in which components of the resistivity tensor ρ are extracted straightforwardly. Components of the conductivity tensor σ are then computed using the relations $σxx=ρxx/ρyx2+ρxx2$ and $σxy=ρyx/ρyx2+ρxx2$⁠. In the Hall bar geometry, both bulk and edge conduction contribute to the measured resistivity. In the Corbino geometry, no edges connect the source and drain contacts, eliminating edge contributions from conduction measured with transport. While insensitive to the Hall voltage, transport in Corbino devices is an intensive measurement of the bulk conductivity σ_xx, which is invaluable when studying dissipation in systems exhibiting topological edge transport.^27–30 

Here, we present Corbino geometry measurements of dissipation in the QAH state in a thin-film sample of Cr-BST. By comparing transport in Corbino and Hall geometries, we conclude that dissipation in the QAH state is dominated by bulk processes at all values of temperature and electric field.

Three Corbino devices were fabricated from a 6-QL (Cr_0.12Bi_0.26Sb_0.62)₂Te₃ film grown by molecular beam epitaxy. A chip from the same wafer was used in 2018 to fabricate Hall bars, which demonstrated part-per-million quantization of the Hall resistance.¹⁸ In both cases, nearly identical process flows were used (full fabrication details can be found in Sec. I of the supplementary material).

A drawback of the Corbino geometry is non-uniform distribution of radial electric fields across the width of the annulus. Here, the Corbino width is defined as W_Corbino = r_o − r_i, the difference between the outer contact radius r_o and the inner contact radius r_i. The electric field becomes less uniform as the disk becomes wide compared to the radius of the inner contact (i.e., outside the limit W_Corbino ≪ r_i). To gain insight into how this non-uniformity affects transport, three devices, C₁, C₂, and C₃, were chosen to have a fixed width of 100 µm with varying inner and outer radii [C₁ (r_i = 100 μm, r_o = 200 µm) and C_2,3 (r_i = 200 μm, r_o = 300 µm)]. Using the AC + DC voltage-biased measurement setup shown in Fig. 1, the longitudinal conductivity,

was measured simultaneously with the differential conductivity. Note that we define $σxx=σxxVDC$ as a function of bias, which is not necessarily equivalent to the local conductivity, specifically when the I–V characteristic is non-linear.

We first verified the presence of the QAH state. In a Hall bar geometry, this is done by measuring a quantized Hall resistance, ρ_yx = h/e², and a vanishing longitudinal resistance. The Corbino geometry, however, does not allow for simple Hall measurements.³¹ Measurement of a minimum in the radial current, signifying a fully insulating bulk, was instead used to verify QAH. The radial current decreased in two stages: first, when the sample was magnetized, opening an exchange gap with spatially uniform sign; and second, when the Fermi level was tuned near the center of the exchange gap using the gate.

The sample was cooled to a base temperature of 30 mK in a dilution refrigerator and the source–drain current, I_AC, was measured under a 10 µV rms AC bias and zero DC bias. At gate voltage V_g = 0, I_AC ≈ 4 nA initially after cooling the sample from room temperature. Next, the sample was magnetized by applying a perpendicular field μ₀H = 0.5 T and then reducing the field to zero [Fig. 2(a)]. After magnetization with V_g = 0, I_AC ≈ 20 pA. Upon tuning V_g [Fig. 2(b)], we observed further reduction in I_AC and a wide range of gate voltages (−14 V ≤ V_g ≤ −4.5 V), over which I_AC < 1 pA, indicating that the QAH state was well-formed.²⁷ Further investigation demonstrated that heating of the sample and gate voltage modulation³² led to partial demagnetization (see Sec. IV of the supplementary material). For this reason, all subsequent measurements were taken under a 0.5 T external magnetic field applied perpendicularly to the plane of the sample, to maintain the magnetization.

The Hall bar devices were fabricated in August 2017 while the Corbino devices were fabricated in June 2018. Aging studies have shown that long term exposure of thin-film TIs to atmospheric conditions can destroy the insulating bulk character.³³ To limit the effects of aging, our samples were coated with PMMA after growth and stored in an N₂ dry box prior to device fabrication. After fabrication, additional protection was provided by the 40 nm alumina top-gate dielectric.

By comparing conductivity as a function of temperature and gate voltage on the two sets of devices, we confirmed that aging had not led to significant sample degradation. Figure 2(c) shows σ_xx as a function of 1/T for devices C₁ and C₃ as well as the corresponding data for a 100-μm-wide Hall bar measured in 2018. For temperatures 100 mK < T < 1.2 K, and with V_g tuned within the regime minimizing I_AC (−14 V ≤ V_g ≤ −4.5 V), σ_xx follows thermally activated behavior $σxx∝e−T0/T$⁠. Here, conductance was Ohmic under the 10 μV rms AC excitation.

The fitted thermal activation temperature scale T₀ reaches a maximum of 727 mK at $Vgopt=$ −8.8 V for device C₁ and 729 mK at $Vgopt=$ −8.5 V for device C₃. While these values are far below what has been seen in scanning tunneling microscopy measurements of a cleaved Cr-BST bulk crystal,²¹ where the average exchange gap size is 30 meV, they are remarkably similar to what was measured in the 2018 Hall bar device both in peak value and peak location [Fig. 2(d)]. The similar optimum gate voltages indicate that, despite fabrication at different times, aging did not significantly shift the native Fermi level between devices; the similarity of the Arrhenius activation scales indicates that the magnetically induced exchange gap did not shrink.

We next considered the behavior of the devices at $Vg=Vgopt$ as a function of source–drain voltage bias, V_DC. At base temperature, no Ohmic regime could be observed; we observed highly nonlinear I–V characteristics at the lowest currents we could measure (∼0.3 pA). An abrupt increase in dissipation followed at higher bias as the sample underwent breakdown of the QAH state. This behavior was present in all three Corbino devices [Fig. 3(a)].

Figure 3(b) shows σ_xx(V_DC) of device C₃ at various temperatures. At low temperatures (T < 50 mK), a clear transition at breakdown was present. For T > 50 mK, Ohmic conduction was observed at low bias, and the transition at breakdown was increasingly smeared until a nearly continuous evolution emerged above ∼100 mK. This behavior was consistent with the temperature dependence of breakdown observed in the Hall bar samples (see Fig. 3a in Ref. 18).

There is no universal method for quantifying the voltage onset of breakdown; even in QH systems, where there is often a discontinuity in the current–voltage characteristic, various criteria have been used to define where breakdown begins.^{28,29,34–36} Here, we define the critical breakdown voltage V_c as the source–drain voltage at which σ_xx = 5 × 10⁻⁴ e²/h, which roughly corresponds to the dissipation measured in each device when the differential conductivity reaches its peak value at breakdown (Fig. S3 of the supplementary material) so that the current rises rapidly with the increasing voltage [Fig. 3(a)].

Despite their geometrical differences, we find similar values of V_c in all three Corbino devices with $VcC1=2.9$ mV, $VcC2=2.9$ mV, and $VcC3=3.0$ mV. For a material with uniform conductivity in the Corbino geometry, the radial electric field varies as

and reaches a maximum value of $Emax=VDCri⁡lnro/ri$ at r_i, the edge of the inner contact. For a given value of V_DC, E_max is 17% larger in C₁ than in C₂ and C₃. If breakdown occurred when the maximum electric field in the QAH insulator reached a critical value, we should therefore observe $VcC2,3$ larger than $VcC1$ by 17%. This was not the case: $VcC2,3$ were roughly equal to $VcC1$⁠, and the I–V characteristics of the three devices were similar throughout the breakdown transition without normalizing for device geometry (we emphasize that all devices had equal W_Corbino).

The similarity in unnormalized I–V characteristics between the devices suggests that even if differences in radial field distributions are playing some role in the fine details of σ_xx at breakdown, it appears to be the average electric field,

across the annulus that is most relevant for breakdown, at least for the two Corbino geometries used here. The extent to which this finding holds in general for QAH Corbino disks will be the topic of future work; the limited literature on breakdown in QH Corbino geometries does not show this.^27,28,37

We want to compare transport in the Hall and Corbino geometries as a function of a single bias parameter. In the QAH regime, the Hall angle θ_H ≈ 90° and the electric field is orthogonal to the direction of dissipationless current flow.²⁹ In a Hall bar, this electric field corresponds to the transverse field E_y, and in a Corbino disk, this electric field corresponds to the radial (source–drain) field E_r. Because we cannot directly probe the local value of electric field, we must appropriately scale extensive measurements to correct for differences in geometry and measurement configuration (current vs voltage bias). As was done in Ref. 18, we assume for this purpose a uniform distribution of current in the bulk for the Hall bar devices.

Under this assumption, we calculate the average transverse electric field E_avg in the Hall geometry as E_avg = E_y = ρ_yxj_x ≈ h/e²j_x. Here, j_x is the average current density j_x = I_x/W_hb in a Hall bar of width W_hb biased with current I_x. Note that the approximations θ_H ≈ 90° and ρ_yx ≈ h/e² hold even after breakdown: for the largest value of current bias used in this comparison, ρ_yx remained quantized to within 1% of h/e² and θ_H ≥ 80°.¹⁸ In the Corbino geometry, in place of E_r, which varies as a function of radius, we use E_avg of Eq. (3). Specifically, E_avg ≈ E_r only in the limit of a narrow disk (W_Corbino ≪ r_i), but the similar I–V characteristics of C₁ and C_2,3 [Fig. 3(a)] justify using E_avg as a representative parameter. Fine details regarding the dependence of σ_xx on the local electric field will require a systematic study of Corbino disks over a broader range of dimensions and will be the topic of future work.

Figures 4(a) and 4(b) show transport data for Corbino devices C₁, C₂, and C₃ and Hall bar devices 2A, 2B, and 2C (measured in Ref. 18) as a function of E_avg. For the Corbino devices, σ_xx was calculated using Eq. (1). For the Hall bar devices, $σxx=ρxx/(ρxx2+ρyx2)$⁠; ρ_yx was measured in DC using a commercial cryogenic current comparator system, and ρ_xx was measured directly using the system’s nanovoltmeter as described in Ref. 18. The input impedance of the nanovoltmeter is ∼100 GΩ, so leakage currents were insignificant in these measurements.³⁸ 

As previously mentioned, due to partial demagnetization, all Corbino disk data presented thus far were taken under a 0.5 T perpendicular magnetic field. This was not the case for the Hall bar data presented in Ref. 18, which were taken with zero external magnetic field. As seen in Fig. 4(a), application of the magnetic field reduced σ_xx by roughly a factor of 3 in the prebreakdown regime. Modest further reduction was observed as V_g was tuned to $Vg=Vgopt$⁠. Whether demagnetization has a significant effect on transport in Hall bar devices will be the topic of future work.³² The analysis that follows will focus on the overall shape of σ_xx(E_avg), which appears unaffected by the presence of the external magnetic field (see Sec. IV of the supplementary material).

If dissipation in the prebreakdown regime were due solely to backscattering in quasihelical edge states, no current would flow between the source and drain in the Corbino geometry. Our measurements show that this is not the case. Not only does current flow through the Corbino devices within the prebreakdown regime, but also the dependence of σ_xx on E_avg in the three Corbino devices is nearly identical to that in the three Hall bars [Fig. 4(b)]. In all devices, we observe an exponential dependence of σ_xx on E_avg. Fitting these low bias data to the function $σxx=σ0⁡expaeEavg/kBT$⁠, where e is the electron charge and T is our 30 mK base temperature, yields a = (600 ± 60) nm for all six devices (fitting parameters are presented in Table I of the supplementary material). The parameter a reflects a length scale within the field-assisted thermal activation³⁹ and non-Ohmic variable range hopping⁴⁰ models, both of which are consistent with the observed exponential behavior and are discussed extensively in our previous work.¹⁸ 

Since the conductivity scales with the electric field in the same way in Hall and Corbino device geometries, we infer that its dominant contributing process is shared between both. In the Hall geometry, both edge and bulk processes could reasonably be linked to the measured dissipation. However, in the Corbino geometry, edge processes do not contribute at all. Therefore, we argue that dissipative conduction observed in thin-film Cr-BST at low temperatures is bulk dominated. If quasihelical modes exist, they are strongly localized; otherwise, their conductance is insignificant compared to that of other dissipative channels. Our findings are inconsistent with the conclusions of a previous study of a V-BST film.²⁴ 

The exchange gap in QAH materials is expected to be of the order of tens of meV, but the observed transport gap is closer to 1 K.^9,15–19 Our Corbino geometry measurements show that quasihelical edge modes cannot explain this disparity. The disparity may instead result from charge fluctuations spatially modifying the Fermi level,⁴¹ regions with a locally reduced exchange gap due to magnetic disorder, or incomplete localization of midgap defect states.

For E_avg ⪆ 30 µV/μm, σ_xx begins to rise even more rapidly than the exponential dependence described above and breakdown occurs. Based on the linear scaling of critical current I_c with the Hall bar width, it was concluded in 2018 by some of the present authors that breakdown of the QAH phase in this material likely occurred through bulk conduction.¹⁸ The Corbino disk measurements presented here reinforce this conclusion, removing possible ambiguity.

In both geometries, σ_xx sharply increases by nearly two orders of magnitude as the QAH state breaks down, and the transition to the highly dissipative state becomes increasingly smeared out as the samples are heated above the base temperature [Fig. 3(b), and Fig. 3(b) in Ref. 18]. Section II B introduced a phenomenological definition of breakdown based on I–V curves and noted that the value of E_avg at breakdown in the Corbino devices appears consistent across multiple devices. We find that this critical field E_c = 29–30 µV/μm. In the Hall bars, E_c = 31 µV/μm is strikingly similar. The qualitative and quantitative consistency of breakdown phenomenology between the two device geometries confirms that breakdown occurs through bulk conduction. This finding echoes studies of QH breakdown incorporating Corbino devices in which breakdown has been found to be an intrinsic property of the electron system.^27–29,42

Far beyond breakdown (E_avg > 35 µV/μm), the conductivity measured in the Corbino devices falls somewhat below that in the Hall bar devices [Fig. 4(a) shows the most direct comparison under zero field, while Fig. 4(b) includes a larger set of devices]. A possible contributor to this difference is that the Corbino measurements, being two-terminal, are sensitive to contact resistance, which reduces the conductivity inferred from measurements.⁴³ 

We have shown that dissipation in Cr-BST is dominated by bulk processes both before and after breakdown of the QAH state. Furthermore, both dissipation in the prebreakdown regime and the trigger for breakdown are largely a function of the average transverse electric field in the devices studied here.

By comparing the conductivity in Hall and Corbino geometries, we have ruled out nonchiral edge modes as a substantial source of dissipation in the Cr-BST thin-films we measured, though we cannot rule out their existence altogether. Our comparative measurements protect against important sources of artifacts in sensitive studies of dissipation. First, the Corbino geometry directly probes σ_xx in the material’s two-dimensional bulk. Since our Corbino and Hall measurements show the same dissipation, this must occur in the bulk, not on the edges. Second, Corbino measurements do not suffer from systematic offsets in σ_xx caused by leakage through voltage preamplifiers, which is known to plague current-biased Hall bar measurements.^38,44 Third, given available current-to-voltage preamplifiers (for Corbino measurements) and voltage preamplifiers (for Hall bars), the Joule power at the measurement noise floor is about five orders of magnitude lower for the Corbino configuration. The methodology we have established here can be used to determine the nature of dissipation in other QAH materials, including V-BST,^15,16 Cr-BST/BST heterostructures,¹⁴ and MnBi₂Te₄,^45,46 a crucial step in the development of improved QAH materials. This method can also be applied to glean a precise understanding of the relationship between bulk and edge transport in other systems, including three-dimensional topological insulator films (expected to have sidewall conduction in parallel with 2D bulk top and bottom surface conduction), quantum spin Hall materials, and higher-order topological insulators.

See the supplementary material for full details regarding device fabrication, instrumentation, choice of breakdown threshold voltage V_c, preamplifier leakage currents in the Hall bar geometry, fitting parameters, and partial demagnetization of the Cr-BST thin-film.

We thank George R. Jones and Randolph E. Elmquist for their help in acquiring the 2018 data referenced throughout this report, and Molly P. Andersen and Steven Tran for helpful discussions. This research was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) (Contract No. DE-AC02-76SF00515) (device fabrication, measurements, and analysis). P.Z., L.P., and K.L.W. acknowledge support from the Army Research Office under Grant Nos. W911NF-16-1-0472 and W911NF-15-1-0561:P00001, and from the National Science Foundation under ERC-TANMS for material synthesis and material characterization. Measurement infrastructure was funded, in part, by the Gordon and Betty Moore Foundation (Grant Nos. GBMF3429 and GBMF9460), part of the EPiQS Initiative. During the early stages of this work, L.K.R. was supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1656518. I.T.R. acknowledges support from the ARCS Foundation. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation (NSF) (Award No. ECCS-1542152).

The data that support the findings of this study are available from the corresponding author upon reasonable request.