Abstract
Topological insulators are bulk electronic insulators with symmetryprotected gapless modes on their surfaces^{1,2,3,4,5}. Breaking timereversal symmetry in these systems via ferromagnetism induces a gapped state characterized by a zero magnetic field quantized Hall response—the quantum anomalous Hall (QAH) state^{6,7,8}. A key question that has remained experimentally unexplored is the relationship of this new type of quantum Hall (QH) state with the previously known orbitally driven QH states. Here, we show experimentally that a ferromagnetic topological insulator exhibiting the QAH state is well described by the global phase diagram of the QH effect. By mapping the conductivity tensor in the vicinity of the QAH phase, we find evidence for quantum criticality and delocalization behaviour^{9,10} that can quantitatively be described by the renormalization group properties of the QH ground state^{11,12}. This result demonstrates that this QAH state can be understood within the context of the law of corresponding states that governs the QH state^{13,14}.
Main
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 spectrum^{15,16}. This gap was observed by angleresolved photoemission spectroscopy (ARPES) experiments in both bulk^{17} and surface^{18} magnetically doped TIs. Such bulk^{19} and surface^{20} 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 zerofield quantized Hall response (the QAH effect) in electronic transport^{6} accessible under the condition of sufficient electronic localization^{7}. This line of inquiry culminated in the experimental observation of the QAH state in ferromagnetic TIs at low temperature^{8}. 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 symmetrybroken TI surface modes within the general context of the stable ground states of twodimensional (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 conduction^{21}. With the introduction of a magnetic field, there is an additional stable state characterized by vanishing longitudinal conductance and quantized Hall conductance ne^{2}/h (n is a nonzero integer, e is the electronic charge and h is Planck’s constant): the QH liquid^{22}. 2D systems subject to timereversal symmetry (TRS) breaking due to ferromagnetism rather than external magnetic fields have also been shown theoretically to support stable insulating and QH liquid states^{23}. We are thus led to consider the surface states of magnetically doped TIs as 2D transport systems in the presence of symmetrybreaking 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 Cr_{x}(Bi_{1 − y}Sb_{y})_{2 − x}Te_{3} on semiinsulating 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 gap^{24}. 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 Xray diffraction patterns for the pristine and doped films, respectively; in each case all peaks can be identified with (0 0 0 n) diffraction of (Bi_{1 − y}Sb_{y})_{2}Te_{3} 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 caxis lattice parameter is apparent in the Crdoped 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.
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 nonmetallic 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 temperatures^{25}. On doping with Cr, the most prominent difference observed is an enhancement in R(T), peaking at a critical temperature T_{C} = 45 K. As shown in Fig. 1e, f, this T_{C} corresponds also to the onset of hysteresis in the longitudinal R_{xx}(B) and transverse R_{yx}(B) magnetotransport for the Crdoped films, suggesting that T_{C} 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 R_{xx} has a peak at the coercive field H_{C} (here, H_{C} = 0.16 T at T = 2 K) and R_{yx} is dominated by the anomalous Hall effect^{26}. Above T_{C} we can estimate the electron mobility μ_{e} of the films; at T = 80 K we have μ_{e} ≍ 270 cm^{2} V^{−1} s^{−1} (in comparable conditions the pristine films exhibit μ_{e} approaching 700 cm^{2} V^{−1} s^{−1}). We note that the parameters t and x are optimized for this experiment by examining R(T) and R_{yx}(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 Al_{2}O_{3} 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 electronlike and holelike) at a filmdependent top gate voltage V_{T}, which we hereafter refer to as V_{0}. 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 V_{T} relative to V_{0}. For the pristine film, we see ambipolar behaviour, indicated by the sign change in R_{yx}(V_{T}) at V_{T} = V_{0} and the simultaneous peak in R_{xx}(V_{T}); (ref. 25). For the doped film, we also observe a peak in R_{xx}(V_{T}), although it markedly less symmetric, along with a simultaneous variation in R_{yx}(V_{T}). Here, we define V_{0} at the peak in R_{yx}(V_{T}), 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 behaviour^{19}.
Cooling to lower temperatures, we observe behaviour characteristic of the QAH effect. In Fig. 2, measurements of a doped film at V_{T} = V_{0} and T = 50 mK are shown up to a magnetic field of B = 14 T. R_{yx} exhibits a value of 0.98 ± 0.003h/e^{2}, whereas the longitudinal resistance falls with increasing B, reaching values <0.03h/e^{2} 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/e^{2}. These results are similar to those previously reported^{8}, although here a larger B is required to suppress R_{xx} (T_{C} and H_{C} are also different, see above). It seems that whereas a substantial B is needed to support dissipationless longitudinal transport, the Hall response is robust even in vanishing B, with the slight reduction from h/e^{2} probably due to remnant conduction channels^{20}.
To examine transport behaviour in the vicinity of the QAH state, we show the detailed evolution of R_{xx} and R_{yx} at B = 14 T at various temperatures as a function of V_{T} in Fig. 3a, b, respectively. A nearly dissipationless R_{xx} is observed over a narrow V_{T} range around V_{0} ≍ 3 V at T = 50 mK, which quickly weakens with increasing T in a metallic fashion. There is a notable asymmetry for R_{xx} on the electron and hole sides, but both sides become nonmetallic at sufficiently large V_{T}. For R_{yx} we observe a plateau over a slightly larger V_{T} domain, which recedes with both increasing T or for V_{T} significantly away from V_{0}. Mapping these curves into conductivities produces a systematic view of this evolution. In Fig. 3c–e we show parametric plots of (σ_{xy}(V_{T}), σ_{xx}(V_{T})), with V_{T} 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 V_{T} begins from large σ_{xx} towards an apparent critical point at (e^{2}/h, 0), although with a minor deviation consistent with R_{yx} being slightly below h/e^{2} and the finite remnant R_{xx}. Interestingly, as V_{T} passes through V_{0} (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 e^{2}/2h centred at (e^{2}/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 insulator^{27,28}. 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.
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 R_{xx}(T) for several V_{T} for B = 0 (after application of a large B to saturate the magnetization M). A crossover from nonmetallic to metallic behaviour across h/e^{2} is observed as V_{T} crosses +4 V. In Fig. 4b we show R_{yx}(T) measured under the same conditions, where we observe a change from increasing to decreasing R_{yx}(T) with decreasing T as V_{T} 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.
As noted above, in the absence of a TRSbreaking 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 analysis^{21}. In the presence of broken TRS, however, the scaling function (the socalled 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 σ_{xx} and σ_{xy} (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 systems^{13}, 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 Γ_{0}(2) modular subgroup on the associated conductivity tensor^{14}. By writing the complex conductivity σ = iσ_{xx} + σ_{xy}, 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 Γ_{0}(2) symmetry^{12}. In analogy to RG flow studied in the latter^{29}, 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 ϑ_{i}(x) are Jacobi theta functions of the ith kind, such that for x = e^{iπσ} 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 (e^{2}/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 methods^{9,11} 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 Γ_{0}(2) symmetry serves as the basis for an alternate derivation^{30}.
Turning to a comparison with our experimental results, we plot (σ_{xy}(V_{T}), σ_{xx}(V_{T})) for B = 0 at various V_{T} in Fig. 4d. As T is decreased (corresponding to increasing L), (σ_{xy}(V_{T}), σ_{xx}(V_{T})) seems to flow to one of two stable fixed points at (0,0) and (e^{2}/h, 0), with the pattern suggesting an unstable fixed point in the vicinity of e^{2}/h(0.5, 0.55). This unstable fixed point coincides closely with that expected for the delocalization transition of extended states producing universal singularities^{31}. 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 systems^{10}; this is clear evidence that the QAH state obeys the same symmetry rules. This flow is controllable with V_{T} 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 percolation^{10}, 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 origin^{11,26}. 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 proximity^{13}. 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 previously^{10}, 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 system^{32}. 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.
Methods
Thin films were grown by molecular beam epitaxy on epiready semiinsulating 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 lowtemperature measurements, films are capped ex situ after growth with Al_{2}O_{3} 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 ebeam evaporation of Ti/Au. Care is taken to use low T < 150 °C processing to avoid degradation of the films. R_{xx} and R_{yx} reported here are the longitudinal (sheet) resistance and transverse (Hall) resistance, respectively. Measurements on devices are performed at low frequency (1–3 Hz) with lockin amplifiers and voltage and current preamplifiers 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.
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Acknowledgements
We are grateful to N. Nagaosa, B.J. Yang and A. F. Young for fruitful discussions, and M. Nakano, T. Hatano, S. Shimizu, M. Kubota and S. Ono for technical support. This research is supported by the Japan Society for the Promotion of Science (JSPS) through the ‘Funding Program for World Leading Innovative R&D on Science and Technology (FIRST Program),’ initiated by the Council for Science and Technology Policy (CSTP), and by JPSP GrantinAid for Scientific Research, Nos. 24224009, 24226002 and 25871133. This work was carried out by joint research of the Cryogenic Research Center, the University of Tokyo.
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J.G.C. and R.Y. grew and characterized the films. J.G.C., Y.K. and J.F. performed the lowtemperature measurements. J.G.C. analysed the data and wrote the paper with contributions from all authors. A.T., K.S.T., M.K. and Y.T. contributed to discussion of the results and guided the project. Y.T. conceived and coordinated the project.
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Checkelsky, J., Yoshimi, R., Tsukazaki, A. et al. Trajectory of the anomalous Hall effect towards the quantized state in a ferromagnetic topological insulator. Nature Phys 10, 731–736 (2014). https://doi.org/10.1038/nphys3053
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DOI: https://doi.org/10.1038/nphys3053
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