Introduction

When a two-dimensional electron gas (2DEG) is subjected to a strong perpendicular magnetic field (B⊥), the energy spectrum evolves into discrete Landau levels (LLs). As a result, the electron motion is localized by the cyclotron orbits inside the 2DEG system, while the one-dimensional chiral states are formed at the edge, thus giving rise to a quantized Hall conductance1,2,3. Ever since the discovery of the quantum Hall effect (QHE)4, enormous efforts have been made to elucidate the universal behaviour of this quantum transport phenomenon5,6,7. Within the framework of 2D localization theorem, the renormalization group (RG) flow of the system can be well-described in the conductance plot, where stable points appear at (σxx, σxy)=(0, ne2/h) (σxx is the longitudinal conductance, σxy is the transverse conductance, e is the electron charge, h is the Planck constant and the integer n is in accordance with the nth LL). In the meanwhile, the quantum phase transition between neighbouring LLs follows a continuous semicircle centred at (σxx, σxy)=(0, (n−1/2)e2/h) with the scaling behaviour dictated by the localization length8,9,10,11. In the extreme quantum limit when B⊥ is so large that the ground LL exceeds the Fermi level (EF), the 2DEG system will be driven into the so-called quantum Hall insulator state where the longitudinal resistance diverges as the temperature approaches absolute zero, yet the Hall resistance saturates towards the quantized value corresponding to the ground LL state7,12,13,14.

Following the footprint of QHE, it was suggested that by introducing appropriate magnetic exchange field and large intrinsic spin–orbit coupling, similar dissipationless chiral edge conduction could also occur in non-zero first Chern number (C1) ferromagnetic (FM) insulators without the assistance of external magnetic field15,16,17,18,19,20,21. Given the same broken symmetry and non-trivial band topology, the corresponding quantum anomalous Hall effect (QAHE) may be regarded as a special derivative of QHE17,22,23. Accordingly, soon after the first experimental demonstrations of QAHE in magnetic topological insulators (TIs)24,25,26,27,28, a microscopic network model of quantum percolation was applied to investigate the universal scaling of the QAHE plateau transition around the coercive field29. Generally, it was proposed that in an ideal 2D magnetic TI system, the competition between the FM exchange field and the quantum confinement-induced hybridization would change the band topology, therefore leading to the phase transition from the C1=1 QAHE state to the C1=0 insulating state. Unfortunately, because of the major challenge of growing high-quality magnetic TIs (that is, well-defined single-crystalline structure with atomically smooth surfaces and extremely low-defect density) within the 2D hybridization limit24,25,26, experimental observations of the zero Hall plateau state in a quantum anomalous Hall insulator and the plateau transition from σxy=± e2/h to σxy=0 in the QAHE regime have not been reported yet. Meanwhile, although there was an earlier attempt to interpret QAHE following the RG flow argument26, the universality of this distinct QAHE phase remains to be ascertained.

In this article, we use Cr-doped (BiSb)2Te3 films to study QAHE and its related phase diagram. Due to the highly insulating bulk state achieved by the non-equilibrium molecular beam epitaxy (MBE) growth, we observe the giant longitudinal resistance peak and zero Hall conductance plateau at the coercive field in the six quintuple-layer (QL) (Cr0.12Bi0.26Sb0.62)2Te3 film. Such zero Hall plateau manifests the presence of the unique QAHE insulating state, which may be understood within the magnetic multi-domain network context. Moreover, by studying the angle-dependent quantum transport behaviours, the 2D massive Dirac fermion-featured QAHE phase diagram is mapped out to show that the QAHE state with the first Chern number C1=1 is transitioned into the C1=0 insulating state, and the conductance tensor (σxx, σxy) follows a universal semicircle relation, regardless of the applied magnetic field strength. The presence of the zero Hall plateau and the realization of the metal-to-insulator switching in the 2D QAHE regime expand the insights of the critical quantum transport phenomena, and may also pave the way for low-power dissipation spintronics applications.

Results

QAHE in the Cr-doped (BiSb)2Te3 sample

High-quality Cr-doped (BiSb)2Te3 films were firstly grown on semi-insulating GaAs (111)B substrates via MBE30. The growth condition and chemical composition of the film were carefully optimized such that extremely low-bulk carrier density was achieved and the Fermi level was already within the surface gap without additional electric field tuning (See Supplementary Note 1). To generate the hybridization surface state gap m0 (refs 19, 31, 32), the film thickness used in this work was precisely chosen as 6 QL, as highlighted in Supplementary Fig. 1. After careful in situ surface passivation33, we carried out standard four-point magneto-transport measurements on the six QL (Cr0.12Bi0.26Sb0.62)2Te3 Hall bar device with dimensions of 2 × 1 mm, as illustrated in Fig. 1a. Fig. 1b shows the quantization of the anomalous Hall resistance (Ryx=h/e2∼25.81 kΩ) up to T=0.26 K. The sign of the quantized Ryx with respect to the magnetization direction is consistent with the chiral conduction property, as elaborated in ref. 25. Likewise, the magnetic field-dependent longitudinal resistance (Rxx) results are plotted in Fig. 1c (The complete temperature-dependent data of Ryx and Rxx are discussed in Supplementary Note 2). The striking feature here is that at the base temperature T=0.02 K, Rxx increases dramatically from almost a vanishing value (Rxx_min∼20 Ω) at zero field to a giant peak (Rxx_PEAK∼380 kΩ) at the coercive field (HC=± 0.12 T); both Rxx_PEAK and the corresponding magneto-resistance (MR) ratio (>106%) are much larger than those reported previously24,25,26,27,28. It is noted that the steep divergence of Rxx at zero LL was also observed in graphene under high magnetic field34,35,36. However, the underlying mechanisms are quite different. For the graphene case, it was believed that the increased exchange interaction at high B⊥ would split the spin degeneracy of graphene around the Dirac point, and lift the original fourfold degenerate Dirac cone by a Zeeman gap34,35,36,37,38. In contrast, both the giant Rxx_PEAK at small HC and its temperature-dependent behavior shown in our 2D magnetic TI film (Fig. 1d and Supplementary Fig. 2) can be understood from the magnetic multi-domain network model at the QAHE phase transition regime, as we will explain in detail below.

Figure 1: Quantum anomalous Hall effect in the six QL (Cr0.12Bi0.26Sb0.62)2Te3 film.
figure 1

(a) Schematic representation of the mm-sized Hall bar structure and four-point Hall measurements based on the MBE-grown magnetic TI thin film. (b) Quantum anomalous Hall results at T=0.26 K. The Hall resistances are quantized to be±h/e2 where the signs are determined by the chirality of the edge conduction. (c) Temperature-dependent magneto-resistance results (from 0.02 to 13 K). At the coercive field, the peak of Rxx quickly diverges at lower temperatures. (d)Temperature-dependent Rxx_PEAK and Ryx extracted from (b,c). The anomalous Hall resistance Ryx becomes quantized up to 0.3 K, and the giant Rxx_PEAK resolved at 0.02 K is ∼400 kΩ, the largest value obtained among all reported QAHE systems.

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Capturing the e2/h-to-zero QAHE plateau transition

Following ref. 29, we re-plot the QAHE data on the basis of conductance in Fig. 2 by using the reciprocal resistance-to-conductance (ρ–σ) tensor conversion given by

Figure 2: Quantum phase transition of quantum anomalous Hall effect.
figure 2

(a) Magnetic field-dependent σxy at different temperatures. Zero Hall plateaus at ±HC are developed between the two QAHE states up to 0.33 K. (b) Magnetic field-dependent σxx at different temperatures. Even when the film already deviates from the perfect QAHE state at T=0.33 K, both the zero σxy plateaus and double-peaked σxx still persist. (c) Schematic representation of the multi-domain network formed during the magnetization reversal process. The upward yellow arrows and downward green arrows denote the up (ΔM>|m0|) and down (ΔM<-|m0|) magnetic domains, respectively. (d) Temperature-dependent transition slope S=()max extracted from a. When T<0.1 K, S∝T-κ follows the power low scaling behaviour with κ=0.22.

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Consistent with Fig. 1b,c, the QAHE in the conductance plot is also manifested by the quantized (σxy=±e2/h, σxx→0) at zero magnetic field, as shown in Fig. 2a,b. Remarkably, two intermediate plateaus with the Hall conductance σxy∼0 are clearly developed at HC=±0.12 T when T=0.02 K. In the meantime, the longitudinal conductance σxx exhibits the double-peaks behaviour, and its minimum value at HC reflects the insulating behaviour during the magnetization reversal process. Moreover, the temperature dependence of the zero Hall plateaus is also investigated. Fig. 2a,b present the measured σxy and σxx data at several different temperatures (that is, 0.02– 0.33 K). It is seen that with increasing temperature, the zero Hall plateaus near HC gradually narrow (Fig. 2a), while the σxx double-peaks widen at high temperatures, and the minimum of σxx at HC also gradually becomes larger, indicating the increase of thermally activated bulk carrier conduction at higher temperatures (Fig. 2b). When the sample further warms up, both these intermediate features cannot be resolved any more at T=1.9 K.

To further validate our observations, we investigate more Cr-doped (BiSb)2Te3 thin films with different thickness (6 and 10 QL, see Supplementary Note 3) and Cr doping concentrations (10–15%, see Supplementary Note 4), and the relevant transport data are shown in Supplementary Figs 3 and 4, respectively. It is seen that while the quantized QAHE states are observed in all samples, the zero Hall plateau state only persists in the six QL films and disappears when the thickness increases to 10 QL. Together with Fig. 2, these thickness-dependent features strongly suggest that the signatures of physics behind the quantized QAHE state (σxy=±e2/h, σxx→0) and the zero Hall plateaus (σxy∼0, σxx→0) are different, and they may be understood as the following based on the mean field theory29. In principle, for a 2D magnetic TI system, there are two mechanisms to open the surface state gap: one is the hybridization gap m0 between the top and bottom surface states because of quantum confinement, and the other is the exchange field gap ΔM introduced by the FM ordering along z-direction19,29,39. In the quantized QAHE state with |ΔM|>|m0|, all the magnetic domains in the magnetic TI film are well-aligned along the same direction, and the corresponding first Chern number of the system is C1=ΔM/|ΔM|=±1. Consequently, there is only one single chiral edge state propagating along the sample edge, so that (σxy=±e2/h, σxx→0). On the other hand, the zero Hall plateau occurs around the coercive field during the magnetization reversal process, where many upward and downward domains coexist in a random manner40,41, as schematically shown in Fig. 2c. Under the mean field approximation, now |ΔM|<|m0| and the first Chern number of the system is C1= 0. Accordingly, σxy develops the zero plateaus in the -|m0|<ΔM<|m0| regime, and σxx approaches zero.

Recently, some of the authors proposed a microscopic network model to describe the critical behaviour of the QAHE plateau transition in the magnetic TI system29. It was suggested that zero plateaus in σxy could occur and σxx would show two peaks around HC. The basic physical picture is that, at the coercive field, the chiral edge states that are located at the magnetic domain boundaries may tunnel into each other when the spatial decay length of edge states is larger than the distance between them. Under such circumstances, the QAHE plateau transition at HC can be mapped into the network model of the integer QHE plateau transition in the lowest LL. Therefore, the phase transition from the quantized QAHE state to the zero Hall plateau should show critical properties, where the universal temperature and size scaling behaviour of σxy and σxx were proposed29.

Following the above suggestion29, temperature-dependent σxy slope S=()max is displayed in Fig. 2d to study the scaling behaviour of the QAHE plateau transition. Although S is found to monotonically decrease versus temperature (which is consistent with the theoretical proposal), we should point out that there are some quantitative differences between the theoretical proposal and our experimental results. First, unlike the simulation results29, both zero σxy plateaus and double-split σxx persist even when the system has already deviated from the perfect quantization case at 0.33 K (that is, σxy=0.992 e2/h and σxx picks up a relative large background signal of 0.127 e2/h). Second, the temperature dependence of S seems not to follow the predicted simple S∝T−κ relation in the entire temperature range (0.02–0.33 K); yet when T<0.1 K, the power law scaling is fitted quite well with κ=0.22, which is just one half of κ=0.42 measured in the QHE transition10. The possible reasons for the above differences may be because of the fact that the coercivity shifts when 0.1 K<T<0.33 K compared to T<0.1 K, and extra thermally activated bulk carriers would also contribute to the transport (that is, Rxx increases dramatically when T>0.1 K, as highlighted in Supplementary Fig. 5, and detailed temperature-dependent resistance behaviours are discussed in Supplementary Note 5)27, therefore complicating the slope of σxy in high-temperature regime (Also, the complicated magnetic domain dynamics at the coercive field may also affect the temperature scaling41, which is not included in the original network model for QHE plateau transition). More importantly, it is noted that κ is not a universal exponent; instead, it strongly depends on the microscopic details of the randomness in magnetic domains29. Specifically, κ is defined to be p/2ν, where p is the exponent of temperature dependence of the inelastic scattering length with Lin∝T−p/2, and ν is the real universal critical exponent29. For QHE systems, ν is found to be a constant of 2.4, p=2 for high-mobility sample and p =1 for ‘dirty sample’42. In our case, by assuming the same ν =2.4, we obtain p=1 which seems to be consistent with our sample property (that is, low carrier mobility25). Nevertheless, the exact values of both ν and p in the QAHE systems remain to be further investigated.

Finally, we would like to point out that the insulating state in the 2D QAHE case is different from the quantum Hall insulator in conventional 2DEG systems in terms of resistivity. In particular, as T→0, the zero Hall plateau QAHE insulator approaches (ρxx→∞, ρyx→0), while the QHE insulator exhibits (ρxx→∞, ρyx→νh/e2) where ν is the lowest LL filing factor7,14. This is due to the differences of the band structures: for magnetic TI, the linear Dirac-cone-like surface states enable the Fermi level to be located at the Dirac point; yet for ordinary 2DEG with parabolic energy dispersion relation, no zero LL is allowed and the minimum value of ρyx thus has to saturate at a non-zero ground quantized value of νh/e2. Even for QHE in 2D Dirac fermion system (that is, graphene) with zeroth LL43, the observed zero-energy state (ρxx→∞, ρyx→0) at high magnetic field34,35,36 is microscopically different from the zero Hall plateau QAHE insulator discussed above. Although the measured ‘zero Hall plateaus’ in Supplementary Fig. 6 exhibit a flat linear slope with the values of σxy below 1 μS (since experimentally Rxx_PEAK is always finite), the e2/h-to-zero QAHE plateau transition is still manifested by the dramatic change of the slope of σxy (see Supplementary Note 6). Moreover, extra transport data in other six QL samples all show the zero Hall plateaus around HC, again suggesting the quantum phase transition between the C1=1 QAHE state and the C1=0 insulating state in the 2D hybridization regime.

Mapping the semicircle QAHE phase diagram

In light of the importance of QAHE phase transition, we further performed the angle-dependent measurements. It has been shown the Cr-doped magnetic TI develops robust out-of-plane FM order in the QAHE regime19,30,44. If the applied magnetic field is not perfectly perpendicular to the film, the induced in-plane magnetic component tilts the Cr magnetization, and thereby helps to modulate the strength of ΔM as well. Figure 3a,b show the magneto-transport results of the six QL (Cr0.12Bi0.26Sb0.62)2Te3 film as the tilted angle θ of the Hall bar device with respect to the magnetic field is varied from 90° (out-of-plane) to 180° (in-plane). We observe that with small magnetic field sweeping (−0.5 T<B<0.5 T) at T=0.02 K, the quantization of Ryx (Fig. 3a) and the zero Hall plateau (Supplementary Fig. 7) are relatively robust as long as θ<150°, as addressed in Supplementary Note 7. On the other hand, as both B and θ increase, the system starts to deviate from the QAHE state. In the extreme case when the film is rotated almost parallel with the magnetic field (θ=180°±5°), the measured in-plane MR reflects a rather insulating feature: the corresponding dashed purple curve in Fig. 3b strongly suggests the system is rapidly approaching toward an angle-induced insulating state with Rxx>400 kΩ. Note that due to the giant perpendicular anisotropy in the Cr-doped TI systems, out-of-plane magnetic domains might still be formed in this case when θ∼180° (refs 45, 46, 47). A more detailed experiment was carried out subsequently in which Rxx and Ryx were recorded when the six QL magnetic TI film was continuously rotated from 90° to 180°, under different fixed applied magnetic fields. From Fig. 3c,d, it is clear that the system undergoes the smooth quantum phase transition with respect to θ. Most importantly, we find that both Rxx and Ryx curves for B≥1 T tend to change between points [Rxx(90°)=0, Rxx(180°)∼15 h/e2] and [Rxy(90°)=h/e2, Ryx(180°)=0], while different magnetic fields only modulate the transition process in between: Ryx rolls off the h/e2quantization line more quickly and the divergence of Rxx occurs at smaller θ when B increases.

Figure 3: Angle-dependent transport measurements in the quantum anomalous Hall regime.
figure 3

(a) Quantum anomalous Hall results when the six QL magnetic TI film is rotated to θ=90°, 120° and 150°, respectively. The external magnetic field is swept between −1 T and +1 T. The sample temperature is 0.02 K. (b) Magneto-resistance results of the sample under different tilted angle θ at T=0.02 K. The film enters the QAHE insulating state when it is rotated parallel to the magnetic field. Angle-dependent (c) Ryx and (d) Rxx under different fixed B at T=0.02 K. All curves tend to converge at two critical points at (Rxx, Ryx)=(0, h/e2) and (∼15 h/e2, 0).

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We can further visualize the angle-assisted QAHE phase transition in Fig. 4. Significantly, when displayed in the σxy−σxx plot, all θ-dependent curves for B≥1 T, which have been manually shifted vertically by (B−1) × 0.1e2/h for comparison in Fig. 4a, follow a single continuous semicircle which is centred at (σxx, σxy)=(0, e2/2 h) with the radius of e2/2 h. Meanwhile, another interesting finding in Fig. 4a–c is the importance of the in-plane magnetic field B//=Bcosθ on the universal QAHE phase diagram. If we divide the conductance semicircle into three angle regions (red dots for (90°, 120°), green dots for (90°, 150°) and blue dots for (150°, 180°)), and track the evolutions of each component versus the applied magnetic field, we see that it is the strength of B// rather than the B///B⊥ ratio that determines the QAHE phase transition. As highlighted by the dashed curves in Fig. 4a, it is evident that even though B///B⊥=cotθ is always smaller than 1 within (90°, 120), the in-plane B// under high magnetic fields (B>1 T) still manage to force the conduction deviate from the dissipationless (0, h/e2) point, and such spread-out trend becomes more pronounced with increased B (and thus B//). In contrast, as long as the total magnetic field B is smaller than the critical magnetic field (B0∼1 T), the semicircle phase transition cannot be completed even when θ=180° (B//,max=B), as manifested in Fig. 4b. Finally, by combining both the field-dependent results of Fig. 2 and the angle-dependent data of Fig. 4a (θ⊆[90°, 270°], B=1 T) together in the σxy–σxx plot, we produce a single semicircle curve in Fig. 4d, illustrating the similar QAHE phase transition feature between the C1=0 zero Hall plateau state and the two C1=±1 QAHE states. Given that the scaling rule of localization in the multi-domain configuration is temperature-dependent29, the deviation from the two QAHE-characterized (0, 0) and (0, e2/h) points is indeed found to become more obvious with increasing T, as shown in Fig. 4e and Supplementary Fig. 8 (Relevant explanations are given in Supplementary Note 8).

Figure 4: Global phase diagram of quantum anomalous Hall effect.
figure 4

(a) Angle-assisted QAHE phase transition in the (σxx, σxy) plot. The six QL Cr-doped TI film is continuously rotated from 90°(out-of-plane) to 180° (in-plane) with respect to the applied magnetic field direction. All curves overlap with each other following the single semicircle relation. Data are shifted vertically by (B−1) × 0.1e2/h for convenient comparison. (b,c) Magnetic field-dependent QAHE phase diagram in the three-dimensional plot. When B//<1 T, the semicircle transition cannot be completed. (d) Comparison between the field-dependent Rxx–Rxy results (with the sweeping direction of the applied magnetic field from +0.3 T to −0.3 T) and the angle-dependent data in a (θ⊆[90°, 270°] and B=1 T). Identical QAHE phase transition between the C1=0 insulating state and the two C1=± 1 QAHE states is manifested. (e) Temperature-dependent (σxx, σxy) plot. As T increases, the intrinsic localization scaling rule diverts the system from the original QAHE and insulating states. The dotted lines link the data points with the same angles (θ⊆[90°, 270°] with each step of 10°) between 0.02 and 1.9 K.

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Discussions

Conductance semicircles similar to those discussed in above section have been extensively investigated to describe the global phase diagram of QHE5,6,7,8,13,14, yet we emphasize here that the microscopic physics between the QHE and QAHE phase transitions are different. Specifically speaking, the QHE phase diagram is closely related to the LL quantization. The corresponding quantum Hall plateau transitions happen when the Fermi level crosses mobility edges, which are due to disorder-induced localization-delocalization transitions6. Concurrently, the quantum Hall insulator is achieved when the applied B⊥ is large enough to drive the ground LL overlaps with EF (refs 14, 35). In contrast, in the QAHE state, the zero quantized Hall plateau may be the result of the multi-domain formation and a network of chiral edge states at domain walls during the magnetization reversal process29. As a result, the phase transition to the zero Hall plateau state in the QAHE regime can be obtained at a much smaller magnetic field, as addressed in Figs 2 and 4. Furthermore, since the first Chern number in the QAHE state is determined by the competition between magnetic exchange gap and hybridization gap29, it is thus suggested that in three-dimensional magnetic TIs where higher subbands may participate into the band topology transition48, new QAHE phases with tunable C1 are expected, and relevant phase transition can thus be further modulated by film thickness.

In conclusion, we study the QAHE phase transition for the 2D hybridized magnetic TI system. We show that such QAH metal-to-insulator switching can only be achieved in high-quality samples with truly bulk insulating state and 2D quantum confinement. The observations of the zero Hall plateaus and double-split longitudinal conductance are consistent with the proposed microscopic multi-domain network model where the vanishing of the magnetic exchange gap ΔM at the coercive field causes the topology change, yet the temperature-dependent and size scaling behaviours of the QAHE plateau transition needs further investigations to reveal the nature of this exotic state of matter. It is noted that a recent work by Y. Feng el al. also reported the observation of the zero Hall plateau (although the sample was not fully quantized) in a back-gated 5QL Cr-doped magnetic TI sample49. At the same time, from both the field-dependent and angle-dependent magneto-transport results, we map out the global QAHE phase diagram which can be described by a single semicircle curve continuously connecting the (0, 0) and (0, e2/h) in the (σxx, σxy) conductance plot. Additionally, we achieve the QAHE insulator regime by making either (B=± μ0˙HC, θ=90°) or (B>B0, θ=180°) at relatively small magnetic fields. The discovered universal phase transition rule is significant for the understanding of the QAHE system and our results open new avenues for the exploration of novel QAHE-related phenomena and applications.

Methods

MBE growth

High-quality single crystalline Cr-doped (BixSb1-x)2Te3 films were performed in an ultra-high vacuum Perkin-Elmer MBE system. Semi-insulating (ρ>106 Ω cm) GaAs (111)B substrates were cleaned by acetone with ultrasonic for 10 min before loaded into the growth chamber. Then the substrates were annealed to 580 °C to remove the native oxide, under Se rich environment. During the growth, the GaAs substrate was maintained around 200 °C (growth temperature), with the Bi, Sb, Te and Cr shutters opened at the same time. Epitaxial growth was monitored by an in situ RHEED technique, where the digital RHEED images were captured using a KSA400 system built by K-space Associates, Inc. After the film growth, a 2 nm Al was evaporated to passivate the surface at room temperature.

Characterizations

For magneto-transport measurements at 0.02 K<T<0.33 K, we used a He3/He4 dilution refrigerator system equipped with a 18 T superconducting magnet at the SCM1 cell in NHMFL, Tallahassee, USA. For transport measurements at 1.9 K<T<300 K, we used the Quantum Design physical property measurement system (p.p.m.). We are able to systematically alter several experimental variables such as temperature, magnetic field, working frequency and rotation angle. Multiple lock-in-amplifiers, Lake Shore AC resistance bridges, and Keithley source meters are also connected with the physical property measurement system, enabling comprehensive and high-sensitivity transport measurements for the Hall bar devices.

Additional information

How to cite this article: Kou, X. et al. Metal-to-insulator switching in quantum anomalous hall states. Nat. Commun. 6:8474 doi: 10.1038/ncomms9474 (2015).