The introduction of magnetism has proved to be an incisive method to experimentally access the exotic nature of topological insulators (TIs). Early theoretical work predicted ordering of the magnetic spins at the surface of TIs would open a gap at the Dirac point in the protected surface electronic spectrum15,16. This gap was observed by angle-resolved photoemission spectroscopy (ARPES) experiments in both bulk17 and surface18 magnetically doped TIs. Such bulk19 and surface20 magnetic orderings were also observed to interact with the electronic transport of the surface modes. The nature of the ground state within the electronic gap was predicted to be characterized by a zero-field quantized Hall response (the QAH effect) in electronic transport6 accessible under the condition of sufficient electronic localization7. This line of inquiry culminated in the experimental observation of the QAH state in ferromagnetic TIs at low temperature8. With this discovery, significant interest has been focused on understanding the nature of the QAH state and its relation to conventional quantum Hall (QH) states.

Here we establish an understanding of these symmetry-broken TI surface modes within the general context of the stable ground states of two-dimensional (2D) systems. In the presence of disorder and absence of magnetic fields, it is known that as the temperature T goes to zero that 2D systems do not support electrical conduction21. With the introduction of a magnetic field, there is an additional stable state characterized by vanishing longitudinal conductance and quantized Hall conductance ne2/h (n is a non-zero integer, e is the electronic charge and h is Planck’s constant): the QH liquid22. 2D systems subject to time-reversal symmetry (TRS) breaking due to ferromagnetism rather than external magnetic fields have also been shown theoretically to support stable insulating and QH liquid states23. We are thus led to consider the surface states of magnetically doped TIs as 2D transport systems in the presence of symmetry-breaking ferromagnetism. As we show below, in the presence of magnetic order these states can be successfully understood as stabilized QH liquids and are fully characterized by the universal delocalization response known in QH systems.

We have grown thin films of the topological insulator Crx(Bi1 − ySby)2 − xTe3 on semi-insulating InP(111) substrates using molecular beam epitaxy (Methods). The films presented here have a thickness t of approximately 8 nm, as estimated from the calibrated growth rate and a fixed Bi/Sb ratio set by y = 0.8. For this y value, it has been shown by ARPES that the Dirac point of the surface states is isolated within the bulk electronic band gap24. To demonstrate the effect of the magnetic dopant Cr, we compare results for the growth of pristine (x = 0) and doped (x = 0.22) films. Figure 1a, b shows X-ray diffraction patterns for the pristine and doped films, respectively; in each case all peaks can be identified with (0 0 0 n) diffraction of (Bi1 − ySby)2Te3 or the InP (n n n) peaks. Figure 1c shows a detailed view of the (0 0 0 15) peak, exhibiting Laue fringes consistent with our estimated t. A compression of the c-axis lattice parameter is apparent in the Cr-doped film, reducing from 3.047 nm to 3.026 nm, suggestive of Cr replacing Bi. The left and right insets of Fig. 1c show the topography of the pristine and doped films, respectively, taken by atomic force microscopy. The pristine film shows an almost atomically flat surface—that is, it is comprised almost entirely of two quintuple layer (QL) levels separated by the QL thickness (1 nm)—whereas the surface of the doped film shows approximately three QL levels. From this comparison we conclude that the introduction of Cr increases disorder in the film, but the film nevertheless retains the expected crystalline structure and quality.

Figure 1: Development of ferromagnetic topological insulator film devices.
figure 1

a,b, X-ray diffraction patterns for Crx(Bi1 − ySby)2 − xTe3 withy = 0.8 for the pristine (x = 0) (a) and optimally magnetically doped (x = 0.22) (b) cases. c, Detailed view of the (0 0 0 15) peak. The left and right insets show the atomic force microscope topography of 2 μm × 2 μm areas of the pristine and doped films, respectively. d, Resistance R as a function of temperature T for the films. The inset shows a lithographically defined device used in this study along with a schematic of the electrostatic gate structure. e,f, Longitudinal resistance Rxx and transverse resistance Ryx, respectively, of the Cr-doped film as a function of magnetic field B, showing hysteresis associated with the onset of ferromagnetic ordering near T = 40 K. g, Gate voltage VT − V0 (relative to the zero-gap voltage) dependence of pristine and Cr-doped films at T = 2 K and B = 5 T, showing ambipolar behaviour.

We next compare the electronic transport properties of the films. Figure 1d shows the resistance R as a function of temperature T for both films. Decreasing from T = 300 K, the pristine film exhibits a non-metallic R(T) to T 80 K, followed by metallic behaviour to 10 K, and finally an upturn to 2 K. As has been previously discussed, these regimes arise when the chemical potential μ resides in the bulk band gap of a TI such that, at high temperature, excitation of carriers into the bulk bands dominates transport, giving way to an intermediate temperature regime dominated by metallic surface conduction, and finally quantum corrections to conduction at the lowest temperatures25. On doping with Cr, the most prominent difference observed is an enhancement in R(T), peaking at a critical temperature TC = 45 K. As shown in Fig. 1e, f, this TC corresponds also to the onset of hysteresis in the longitudinal Rxx(B) and transverse Ryx(B) magnetotransport for the Cr-doped films, suggesting that TC can be associated with the magnetic ordering temperature of the films. The behaviour in Fig. 1e, f is that of a typical ferromagnetic metal, where Rxx has a peak at the coercive field HC (here, HC = 0.16 T at T = 2 K) and Ryx is dominated by the anomalous Hall effect26. Above TC we can estimate the electron mobility μe of the films; at T = 80 K we have μe 270 cm2 V−1 s−1 (in comparable conditions the pristine films exhibit μe approaching 700 cm2 V−1 s−1). We note that the parameters t and x are optimized for this experiment by examining R(T) and Ryx(B) so as to retain metallic behaviour with the largest anomalous Hall response (Supplementary Methods).

To have in situ control of μ, we patterned the films into Hall bars with deposited Ti/Au top gates and Al2O3 gate dielectrics (Methods). An image of a patterned device is shown as the left inset of Fig. 1d, along with a schematic representation of the vertical structure in the right inset. For the films here, we are able to reach charge neutrality (μ balanced between electron-like and hole-like) at a film-dependent top gate voltage VT, which we hereafter refer to as V0. The results for both the pristine and doped films at T = 2 K and B = 5 T are shown in Fig. 1g as a function of VT relative to V0. For the pristine film, we see ambipolar behaviour, indicated by the sign change in Ryx(VT) at VT = V0 and the simultaneous peak in Rxx(VT); (ref. 25). For the doped film, we also observe a peak in Rxx(VT), although it markedly less symmetric, along with a simultaneous variation in Ryx(VT). Here, we define V0 at the peak in Ryx(VT), although no sign change occurs because of the large contribution of the anomalous Hall response. The observation of a maximum in the anomalous Hall response at charge neutrality is consistent with previously reported behaviour19.

Cooling to lower temperatures, we observe behaviour characteristic of the QAH effect. In Fig. 2, measurements of a doped film at VT = V0 and T = 50 mK are shown up to a magnetic field of B = 14 T. Ryx exhibits a value of 0.98 ± 0.003h/e2, whereas the longitudinal resistance falls with increasing B, reaching values <0.03h/e2 at B = 14 T. The inset of Fig. 2 shows the response at low magnetic field, highlighting a remnant anomalous Hall resistance of 0.98 h/e2. These results are similar to those previously reported8, although here a larger B is required to suppress Rxx (TC and HC are also different, see above). It seems that whereas a substantial B is needed to support dissipation-less longitudinal transport, the Hall response is robust even in vanishing B, with the slight reduction from h/e2 probably due to remnant conduction channels20.

Figure 2: Quantum anomalous Hall effect.
figure 2

Longitudinal resistance Rxx and transverse resistance Ryx of Cr-doped film as a function of magnetic field B at T = 50 mK and with an optimally tuned gate voltage V0. Ryx exhibits a value of 0.98 ± 0.003h/e2 whereas the longitudinal resistance falls with increasing B, reaching values <0.03h/e2 at B = 14 T. The inset shows the response at low magnetic field.

To examine transport behaviour in the vicinity of the QAH state, we show the detailed evolution of Rxx and Ryx at B = 14 T at various temperatures as a function of VT in Fig. 3a, b, respectively. A nearly dissipation-less Rxx is observed over a narrow VT range around V0 3 V at T = 50 mK, which quickly weakens with increasing T in a metallic fashion. There is a notable asymmetry for Rxx on the electron and hole sides, but both sides become non-metallic at sufficiently large |VT|. For Ryx we observe a plateau over a slightly larger VT domain, which recedes with both increasing T or for VT significantly away from V0. Mapping these curves into conductivities produces a systematic view of this evolution. In Fig. 3c–e we show parametric plots of (σxy(VT), σxx(VT)), with VT being the parameter, for various B for T = 700 mK, 200 mK, and 50 mK, respectively. At the lowest T (Fig. 3e), the evolution from low VT begins from large σxx towards an apparent critical point at (e2/h, 0), although with a minor deviation consistent with Ryx being slightly below h/e2 and the finite remnant Rxx. Interestingly, as VT passes through V0 (in the vicinity of the critical point) and further increases, curves at all B collapse onto a single line nearly described by a semicircle of radius e2/2h centred at (e2/2h, 0), shown as a dashed line in Fig. 3c–e. This recalls the semicircular law derived to describe the transition between adjacent QH states or to the insulator27,28. The breakdown of this behaviour seems to be connected with the electron–hole asymmetry discussed above; we suggest that the asymmetry of the electronic structure with bulk valence band states being in closer proximity to the Dirac point may be the origin of this. As more readily expected from theory, the deviation from the semicircular behaviour increases with increasing T (Fig. 3c, d). This is a preliminary connection of the present QAH state to known QH behaviour.

Figure 3: Transport response near the quantum anomalous Hall regime.
figure 3

a, VT dependence of the longitudinal resistance Rxx at B = 14 T. b, VT dependence of the transverse resistance Ryx at B = 14 T. ce, Parametric plots of (σxy(VT), σxx(VT)) for various values of B at T = 700 mK, 200 mK, and 50 mK, respectively. The direction of flow is clockwise for negative to positive gate voltage, as depicted by the arrow in e. A semicircle of radius e2/2h centred at (e2/2h, 0) is shown as a dashed line in each panel.

We can further probe the connection of the present system to QH states by examining the detailed temperature dependence of the conductivity tensor. In Fig. 4a we show Rxx(T) for several VT for B = 0 (after application of a large B to saturate the magnetization M). A crossover from non-metallic to metallic behaviour across h/e2 is observed as VT crosses +4 V. In Fig. 4b we show Ryx(T) measured under the same conditions, where we observe a change from increasing to decreasing Ryx(T) with decreasing T as VT is increased across +4 V. This apparently complex crossover between two phases (one insulating and one metallic) is resolved by viewing these results from the perspective of the renormalization group (RG) properties of QH states.

Figure 4: Temperature driven renormalization group flow.
figure 4

a, Temperature dependence of the longitudinal resistance Rxx at B = 0 (after magnetic training) for various values of top gate voltage VT. b, Temperature dependence of the transverse resistance Ryx at B = 0 (after magnetic training) for various values of top gate voltage VT. c, Calculation of renormalization group flow for a 2D electron system with insulating and quantum Hall stable points. d, Flow in (σxy(VT), σxx(VT)) observed at B = 0 for various VT. On decreasing T, the system evolves towards lower σxx. The thin lines are renormalization group flow lines calculated as discussed in the text.

As noted above, in the absence of a TRS-breaking magnetic field, 2D systems flow towards the insulating ground state as T approaches zero (or equivalently as the system size L diverges), a notion based on single parameter scaling analysis21. In the presence of broken TRS, however, the scaling function (the so-called β function) involves two parameters. Theoretically, the behaviour of the conductivity tensor based on these β functions under the condition of diverging L under RG flow is suggested to be characterized by flow lines dictated by stable and unstable critical points in the parameter space of σxx and σxy (ref. 9). The law of corresponding states for QH systems, the rules that govern the symmetry relations under which QH states are identical and thus dictate the phase diagram of QH systems13, plays a key role in allowing us to quantitatively draw a comparison between experiment and this theoretical description.

The symmetries embodied by the law of corresponding states have been shown to impose a symmetry corresponding to the Γ0(2) modular subgroup on the associated conductivity tensor14. By writing the complex conductivity σ = xx + σxy, one can describe the behaviour of σ in the upper half of the complex plane under RG flow. Here, we employ work motivated by the proposed duality of QH systems and supersymmetric Yang–Mills theory rooted in their common Γ0(2) symmetry12. In analogy to RG flow studied in the latter29, it is postulated that the relevant β functions are complex analytical functions of σ, from which the following function f(x) can be derived

where ϑi(x) are Jacobi theta functions of the ith kind, such that for x = eiπσ equation (1) has a constant complex phase ϕ along RG flows—namely, arg f = ϕ (ref. 12). We can then construct the RG flow diagram for the QH and insulating states by plotting the contours of arg f in σ. Several contours in the phase space covering the insulator and the ν = 1 filling QH state are shown in Fig. 4c, with arrows indicating the direction of flow with increasing length scale. In the vicinity of (0,0) and (e2/h, 0) there are two stable fixed points (denoted by circles with a centre dot) corresponding to insulating and QH ground states, respectively. The unstable point (circle with cross) indicates the transition regime or delocalization regime where extended states exist. These features reproduce the flows predicted by other methods9,11 and characterize the phase transitions of the 2D electron system with broken TRS. We note that setting ϕ = 0 recovers the condition for the semicircular law discussed above, which is not surprising as Γ0(2) symmetry serves as the basis for an alternate derivation30.

Turning to a comparison with our experimental results, we plot (σxy(VT), σxx(VT)) for B = 0 at various VT in Fig. 4d. As T is decreased (corresponding to increasing L), (σxy(VT), σxx(VT)) seems to flow to one of two stable fixed points at (0,0) and (e2/h, 0), with the pattern suggesting an unstable fixed point in the vicinity of e2/h(0.5, 0.55). This unstable fixed point coincides closely with that expected for the delocalization transition of extended states producing universal singularities31. This behaviour thus qualitatively reproduces the characteristics of RG flow towards the insulating and QH ground states in Fig. 4c. Quantitatively, we find that equation (1) describes flow lines that capture our experimental result with reasonably high accuracy and precision, shown as the thin black curves in Fig. 4d. Previous experimental work has captured aspects of this flow in heterojunction QH systems10; this is clear evidence that the QAH state obeys the same symmetry rules. This flow is controllable with VT even withB = 0, suggesting that spontaneous M in the QAH can drive identical behaviour to B in QH systems (similar behaviour is observed in a field, see Supplementary Methods).

Comparing to previous systems in which RG flow has been less successful in the regime of classical percolation10, one conclusion here is that quantum localization plays a key role in determining the observed behaviour. The agreement between the behaviours of the QAH effect and QH effect in terms of RG flow may not be entirely surprising given that both the anomalous Hall effect and QH effect are described by field theories that are topological in origin11,26. This observation motivates a number of directions for future study. One extension of this analogy is to the larger global phase diagram of QH states for the QAH effect, which would predict that the ν = 1 QAH state would be connected continuously to other integer and fractional QAH states and that other exotic insulating states may be in close proximity13. If there are indeed neighbouring states this suggests that the corresponding phase transitions would belong to the same universality class in the Anderson localization regime and would be of great interest for further study. As has been observed previously10, we would also predict this scaling would break down at elevated temperatures. More generally, experiments designed to detect the edge states and their properties associated with the QAH ground state would test our present understanding of this system32. We expect this framework for understanding the QAH effect will guide the discovery of new quantum states and phase transitions in the study of TIs and TRS breaking.


Thin films were grown by molecular beam epitaxy on epi-ready semi-insulating InP(111) at a substrate temperature of 260 °C after annealing at 380 °C. The beam flux of Te relative to Bi, Sb, and Cr is kept at a ratio of of 20:1 to suppress Te vacancies. The elemental compositions of Bi, Sb and Cr are estimated from the relative beam flux pressures checked before and after deposition. The growth rate of the films was approximately 0.2 QL min−1. Transport measurements for T > 2 K are performed on unpatterned films with an excitation current of 1 μA via electrical contacts made with Au wires and Ag paint in a commercial variable temperature cryostat. To correct for contact misalignment, the data in Fig. 1e, f are field symmetrized (unsymmetrized data is shown in Supplementary Fig. 7). For low-temperature measurements, films are capped ex situ after growth with Al2O3 deposited by atomic layer deposition (performed at T < 120 °C to a thickness of 25 nm) before subsequent processing by standard photolithography techniques and Ar ion milling (the etching is done with a brief exposure of less than 60 s). Electrical contacts and the top gate are made with e-beam evaporation of Ti/Au. Care is taken to use low T < 150 °C processing to avoid degradation of the films. Rxx and Ryx reported here are the longitudinal (sheet) resistance and transverse (Hall) resistance, respectively. Measurements on devices are performed at low frequency (1–3 Hz) with lock-in amplifiers and voltage and current pre-amplifiers with an excitation current limited to 1 nA. We use a commercial dilution refrigerator equipped with a superconducting magnet. For device measurements we do not perform field symmetrization.