Giant anisotropic magnetoresistance in a quantum anomalous Hall insulator

When a three-dimensional ferromagnetic topological insulator thin film is magnetized out-of-plane, conduction ideally occurs through dissipationless, one-dimensional (1D) chiral states that are characterized by a quantized, zero-field Hall conductance. The recent realization of this phenomenon, the quantum anomalous Hall effect, provides a conceptually new platform for studies of 1D transport, distinct from the traditionally studied quantum Hall effects that arise from Landau level formation. An important question arises in this context: how do these 1D edge states evolve as the magnetization is changed from out-of-plane to in-plane? We examine this question by studying the field-tilt-driven crossover from predominantly edge-state transport to diffusive transport in Crx(Bi,Sb)2−xTe3 thin films. This crossover manifests itself in a giant, electrically tunable anisotropic magnetoresistance that we explain by employing a Landauer–Büttiker formalism. Our methodology provides a powerful means of quantifying dissipative effects in temperature and chemical potential regimes far from perfect quantization.


Supplementary Note 1 -Device characterization: Device A
Device A is patterned from a 10 QL film of Cr 0.1 (Bi 0.5 ,Sb 0.5 ) 1.9 Te 3 , grown on STO, and capped with a thin film of Al that naturally oxidizes upon exposure to atmosphere. We note that for Cr x (Bi,Sb) 2 x Te 3 films of this thickness, the top and bottom surfaces are not hybridized. [1].

Large-field Hall measurements
The large non-linearity in the Hall resistivity ⇢ xy beyond the coercive field, particularly when gated near charge neutrality, makes it di cult to extract accurate numbers for carrier concentration and mobility. This can be seen in the large field Hall data for the main device of the paper, device A, shown in Supplementary Figure 1. These are measurements performed in a Quantum Design PPMS system with a 9T superconducting magnet. We attempt to extract some approximate numbers for the Hall resistivity at large fields in Supplementary Figure 1 (beyond 1T). For a ferromagnet, the transverse resistivity is given where R H is the Hall resistance, H is the external applied field, M is the magnetization, and f (⇢ xx ) is some function of the longitudinal resistivity. In this system, for the field range above 1T, the magnetization has reached saturation, and the percentage change in ⇢ xx is very small. Therefore, one can assume the field dependence of the second term in the above expression to be very small, and approximate it to be a constant. Now, by a simplistic linear fit to the large field hall we can obtain rough estimates for single carrier density and mobility. When the film is not gated (V G -V G 0 = 100V, left panel), we extract a 2D electron density n = 7.7⇥10 12 cm 2 , and a mobility µ = 161.44 cm 2 /Vs. The observation of edge state transport despite such low mobilities is one of the remarkable properties of the QAHE.
Evidence of edge state transport: Gate and temperature dependence of ⇢ xx The drop in ⇢ xx , as ⇢ xy continues to rise, both in their temperature dependence (Supplementary Figure 2) and gate voltage dependence (Supplementary Figure 3) , are indicative of 1D edge transport dominating over di↵usive transport. This metallic temperature dependence of ⇢ xx is unique to QAH physics. Other magnetically doped (and undoped) topological insulator thin films systems outside the QAH regime have typically shown an insulating behavior in this temperature range, often associated with e-e interactions [2]. Supplementary   Figure 2 demonstrates that as dissipative channels are populated by pushing the chemical potential towards the bulk conduction band, a typical insulating behavior is recovered. Additionally, gate sweeps (Supplementary Figure 3) show that the drop in ⇢ xx , observed as we tune into the magnetic gap, is weakened at higher temperatures. This is possibly due to the increased thermal excitation of dissipative channels.

Gate dependence of AHE
The gate dependence of the AHE and the corresponding longitudinal MR is shown in Supplementary Figure 4. While previous reports have demonstrated the quantum anomalous Hall e↵ect at dilution fridge temperatures, we already observe a zero-field Hall resistivity ⇢ xy tantalizingly close to quantization (⇠ 0.95h/e 2 ), even at 280mK. The squareness of the AHE is clearly indicative of the out-of-plane magnetic anisotropy: an important prerequisite for accessing the QAHE regime. While initial reports of the QAHE achieved a vanishing ⇢ xx upon application of an external field [3,4], our device instead shows a rise in resistance at charge neutrality as also reported in thicker Cr x (Bi,Sb) 2 x Te 3 films [5]. The typical negative MR is however recovered when the chemical potential is tuned towards the Supplementary Figure 4). The coercive field is seen to be constant over this voltage range.

Angle tuned metal-insulator transition
A systematic study of the transition from metallic to insulating behavior, in the temperature dependence of ⇢ xx , is presented in Supplementary Figure 5. In contrast to the regime of perfect Hall quantization, the temperature scale for observing such signatures of edge transport is much higher, and is set by the position of the chemical potential, the size of the magnetic gap, and/or the thermal activation energy of in-gap impurity bands. In Supplementary Figure 5, the chemical potential of the film is kept fixed at charge neutrality, while the size of the magnetic gap is tuned by changing the strength of the perpendicular component of magnetization. Clearly, metallic behavior sets in at lower temperatures as the magnetization is tilted in-plane, and eventually beyond some critical angle, ⇢ xx only shows insulating behavior. These results clearly demonstrate magnetization control of edgetransport.
Out-of-plane field sweeps: x z plane The field sweeps of Supplementary Figure 6 at di↵erent polar angles in the x z plane confirm that our chosen field of 1T for the AMR measurements is well beyond the field range of any hysteresis. Also, a comparison of the perpendicular-to-plane and in-plane MR curves reveals the strong out-of-plane magnetic anisotropy. The sharp ⇢ xx peak in the perpendicular-to-plane sweep may be compared to the low-field ⇢ xx of the in-plane sweep. Both are very similar, since they represent the identical magnetic state of net zero magnetization. Similarly, the AHE curves of Supplementary Figure 6b demonstrate that even for sweeps at large tilt angles (75 o ), the zero-field remenant state is strongly out-of-plane, leading to a near quantized Hall resistivity. Slight o↵sets in the in-plane field sweeps lead to significantly large zero-field Hall resistivity. Therefore, we ensure that the deviation in our in-plane sweep is less than 0.1 o .

In-plane AMR
As the chemical potential is tuned from the conduction band towards the magnetic gap, the in-plane AMR shows a weak, non-monotonic gate dependence, unlike the out-of-plane AMR (Supplementary Figure 7). However, the cos 2 angular dependence is unchanged over the entire gating range. An additional data set from device B, presented in Supplementary Figure 9 shows the evolution of the out-of-plane AMR as the chemical potential is tuned from the magnetic gap, into the bulk valence band. This is in contrast to Device A in which we solely accessed the electron transport regime. The functional dependence of the AMR is unchanged even as the chemical potential is tuned into the bulk valence band, and we extract self-consistent fitting parameters over the entire voltage range. This further conveys the e↵ectiveness of our methodology in quantifying edge contributions to transport in temperature and chemical potential regimes far from perfect Hall quantization.

Supplementary Note 3 -Device C
Device C is the farthest from quantization (max. ⇢ xy ⇠ 0.48 h/e 2 ) of the devices studied in this work, and is plagued by a very large zero-field ⇢ xx ⇠ 2.79 h/e 2 (at V

Supplementary Note 4 -Landauer-Buttiker formalism for quantum anomalous Hall with dissipative channels
We employ a simplistic four-terminal Landauer-Buttiker formulism that accounts for edge channels in the presence of dissipation paths. The model geometry is depicted in Supplementary Figure 11. The conductance G ij is given by where T ij is the transmission co-e cient. We label the four terminals as 1, 2, 3 and 4 where 2 and 3 are the voltage terminals. The Landauer-Buttiker formula may then be expressed There transmission co-e cient takes the form T ij = ⌘ i,j+1 + t ij , where ⌘ is the contribution from edge modes that transmit between leads j and j + 1, and t ij accounts for dissipative contributions. For simplicity, we make the following assumptions: t 23 = t 32 = T By multiplying the matrices we obtain The measured longitudinal resistance in units of h/e 2 is then defined as The denominator of the Supplementary equation 7 has two terms, the first of which is associated with the edge states. One can see that in the limit of dominant edge transport, we have ⌘ T 1 , T 2 =) ⌘/T 2 ! 1 and therefore, R ! 0 as one would expect for pure chiral edge transport. For the other extreme of no edge transport we have ⌘ = 0, and R = 1/T 1 . Therefore, the second term in the denominator of Supplementary equation 7 corresponds to solely the dissipative resistance, R d = 1/T 1 . Supplementary equation 7 is now re-written as With this correlation between the dissipative transmission co-e cient and the dissipative resistance emerging, and based on typical geometric considerations, we now make the assumption T 1 ⇡ T 2 . This is reasonable since the two-point resistances of adjacent contacts may be expected to be of the same order. This also serves to reduce the number of free fitting parameters. The longitudinal resistance now reduces to the expression described in the main text after including the h/e 2 .