Abstract
Topological insulators are a newly discovered class of materials in which helical conducting modes exist on the surface of a bulk insulator^{1,2,3,4,5,6}. Recently, theoretical works have shown that breaking gauge symmetry^{7} or timereversal symmetry^{8} in these materials produces exotic states that, if realized, represent substantial steps towards realizing new magnetoelectric effects^{9,10} and tools useful for quantum computing^{11}. Here we demonstrate the latter symmetry breaking in the form of ferromagnetism arising from the interaction between magnetic impurities and the Dirac fermions^{12,13}. Using devices based on cleaved single crystals of Mndoped Bi_{2}Te_{3−y}Se_{y}, the application of both soliddielectric and ionicliquid gating allows us to measure the transport response of the surface states within the bulk bandgap in the presence of magnetic ions. By tracking the anomalous Hall effect we find that the surface modes support robust ferromagnetism as well as magnetoconductance that is consistent with enhanced onedimensional edgestate transport on the magnetic domain wall. Observation of this evidence for quantum transport phenomena demonstrates the accessibility of these exotics states in devices and may serve to focus the wide range of proposed methods for experimentally realizing the quantum anomalous Hall effect^{8,10} and states required for quantum computing^{14,15}.
Main
The discovery of topological insulators has attracted wide interest owing to the exotic spin and charge properties of their edge modes. Among threedimensional topological insulators, particular interest has been focused on a family of spin–orbitcoupled semiconductors based on Bi: Bi_{1−x}Sb_{x}, Bi_{2}Te_{3} and Bi_{2}Se_{3} (refs 6, 16, 17). An active area of research since their discovery has been the breaking of timereversal symmetry (TRS) in these systems^{12,13}. The application of a magnetic field B is predicted to give rise to a halfinteger quantum Hall effect on each surface akin to graphene but with spin and valley degeneracy removed^{18}. Alternatively, the introduction of ferromagnetism is predicted to produce a quantum anomalous Hall effect^{8} (QAHE) and a topological magnetoelectric effect^{9}, both of which are hallmarks of a topologically nontrivial state.
Original theoretical proposals involving broken TRS through ferromagnetism focused on depositing magnetic insulators on the surface of topological insulators to proximitize the surface states^{14,15}. More recently, it has been shown that magnetic ions substituted for Bi in the parent compound can realize TRS breaking even in the absence of bulk ferromagnetic ordering^{19}. In analogy to bulk dilute magnetic semiconductors, such as (Ga,In)_{1−x}Mn_{x} As, where holelike carriers mediate a ferromagnetic interaction among localized spins^{20}, one further scenario is ferromagnetic ordering mediated by the surface Dirac electrons^{12,13}. As shown in Fig. 1a, we expect the exchange coupling J between the z components of the Dirac electron spin σ and the localized spin S to compete energetically with the Rashbalike spin texture, locking σ perpendicular to the momentum k, namely the Hamiltonian H has the form
where 2πℏ is the Planck constant, v_{F} is the Fermi velocity, is the surface unit normal, σ_{z} is the z component of the electron spin and n_{S} is the areal density of localized spins with average z component (ref. 10). Accordingly, starting from k = 0 (the Dirac point), the ground state will be TRS broken with finite σ_{z} generating an energy gap Δ_{E} (representing a finite mass in the Dirac spectrum and spontaneous lowering of the electronic energy), as shown schematically in Fig. 1b. Here, we report such a ferromagnetic response through the interaction between local spins and surface Dirac electrons when the chemical potential μ lies in the bulk electronic bandgap. The observed anomalous Hall effect (AHE) and concurrent magnetoconductance stand contrary to the behaviour of conventional dilute magnetic semiconductors and are consistent with the surfacestatemediated ferromagnetism approaching the QAHE ground state.
The starting point of our study is growth of single crystals of Mn_{x}Bi_{2−x}Te_{3−y}Se_{y} using the Bridgman method. As shown for x = 0.04 and y = 0.12 (carrier density n_{3D} = 6×10^{19} e^{+} cm^{−3}) in Fig. 1c, these crystals are bulk ptype ferromagnets with the easy axis along c and a transition temperature T_{C}≈13 K. Also shown are the associated rise in the B = 0 extrapolated Hall conductivity Δσ_{x y}(T) (the remnant value in the case of hysteretic σ_{x y}(B)) that tracks the magnetization M(T) and an anomaly in σ_{x x}(T) (σ_{3D} in Fig. 1c) near T_{C}. Generally, σ_{x y} can be decomposed as σ_{x y} = σ_{x y}^{N}+σ_{x y}^{A}, where σ_{x y}^{N} is due to the normal Lorentz force Hall effect and σ_{x y}^{A} is due to the AHE (ref. 21). The AHE term follows the direct proportionality σ_{x y}^{A} = S_{H}(T)M, demonstrated in the inset of Fig. 1d with an appropriate choice of the scaling constant S_{H}. We will use Δσ_{x y} to track σ_{x y}^{A} and M also in the microcrystalbased devices where magnetization cannot be directly measured.
Our next step is to probe the surface transport properties of these crystals. Reducing the crystal thickness t to between 5 and 10 nm greatly suppresses the bulk conductance channel and allows for manipulation of μ by electrostatic gates. As shown in Fig. 1d, crystals are cleaved onto heavily doped Si substrates with a 300 nm SiO_{2} overlayer acting as a back gate with voltage V _{B}. Electrical contacts are made by electron beam lithography (shown in the atomic force microscope image in Fig. 1e) and finally an ionic liquid contacted in a sidegate configuration is deposited to act as a top gate with voltage V _{T} (ref. 22). The act of cleavage and fabrication induces defects that tend to e^{−}dope devices^{23}. With no applied gate voltage, the twodimensional carrier density derived from σ_{x y}^{N} for device A in Fig. 1f is n_{2D} = 2.4×10^{13} e^{−} cm^{−2}. The critical value of n_{2D} to reach the bottom of the conduction band is estimated to be using the Fermi surface area observed in Bi_{2}Te_{3} at the onset of bulk conduction^{17}, so that μ is in the vicinity of the conduction band edge in Fig. 1b. Despite the change from the bulk crystals, in particular from ptype to ntype conduction, we observe an abrupt increase in Δσ_{x y}(T), indicating a ferromagnetic transition qualitatively similar to the bulk crystals. In this class of Mndoped compounds, ntype bulk carriers have not been observed to mediate ferromagnetism^{19,24}, which makes the robust ferromagnetism here unexpected. Fig. 1f also shows that T_{C} can be tuned by V _{B}, which we use below to further investigate this unconventional n_{2D} dependence of the ferromagnetism.
We report the results of five devices A–E with different V _{B} and V _{T} covering the range 3×10^{12} e^{+} cm^{−2}<n_{2D}<6×10^{13} e^{−} cm^{−2}, where n_{2D}(V _{B})≡n_{2D}(0 V)+η V _{B} with the calibrated gate efficiency η = 8.9×10^{10} e^{−} cm^{−2} V^{−1} (Supplementary Section SB and Fig. S3). V _{T} is fixed for each device throughout the experiment. Beginning with device A, in Fig. 2a (T = 2 K) we see an enhancement of the spontaneous σ_{x y} with depletion of e^{−} carriers. Starting at large positive V _{B} = +100 V (n_{2D} = 3.3×10^{13} e^{−} cm^{−2}), the hysteresis becomes progressively more pronounced as V _{B} is lowered. In Fig. 2b we show the result for device B after application of V _{T} = −3 V, resulting in a lower n_{2D} = 6×10^{12} e^{−} cm^{−2}. The growth of σ_{x y}^{A} continues with depletion of carriers, seeming to saturate at the lowest V _{B} where σ_{x y}^{N} changes sign. For the case of larger n_{2D} where μ lies above the bulk conduction band minimum, we do not observe any sign of σ_{x y}^{A} up to 5.5×10^{13} e^{−} cm^{−2} (device E). This again confirms that the bulk ntype carriers cannot mediate ferromagnetism.
Next, the typical T dependence of σ_{x y}(B) is shown for device C in Fig. 2c. Here, on decreasing T below 12 K there is an abrupt onset attributed to σ_{x y}^{A}. We can estimate T_{C} by examining either the remnant Δσ_{x y}(T) as in Fig. 1c or using an Arrott plot (Supplementary Fig. S4). In Fig. 2d, T_{C} for each device is plotted as a function of n_{2D}. The phase boundary separating the ferromagnetic and paramagnetic state is dome shaped, closing at a critical density n_{c} = (3.8±0.5)×10^{13} e^{−} cm^{−2}, indicated in the inset of Fig. 2d. This finding and its implications, which we next explicate, are the main result of this paper.
In conventional dilute magnetic semiconductors, the mean field theory ordering temperature scales as , where N_{s} is the number density of local spins and is the density of states at the Fermi energy^{20,25}. It is therefore understood that an increase in n would lead to an increase in T_{C} (refs 26, 27); this is contrary to the observation here. As expressed by equation (1), the surface Dirac electrons in the bulk bandgap mediate a ferromagnetic coupling among the local moments (Mn spins) that favours polarization of the carrier spin (σ_{z}) and the averaged moments () in competition with the Rashba energy. In this way, T_{C}(n_{2D}) as shown in Fig. 2d may be seen as primarily a reflection of the onset of TRS breaking rather than the conventional carrier number’s role in ordering.
We can compare this result with the mean field value calculated for ferromagnetic ordering of magnetic impurities on the surface of a topological insulator given in ref. 12:
where a_{0} is the lattice constant and Λ is a cutoff energy associated with the termination of the Dirac surface band. Equation (2) indicates a maximum T_{C} at the Dirac point that smoothly decreases with increasing , in accord with our observation identifying T_{C} = 0 at μ(n_{c}) = Λ. Whereas equation (2) predicts a steadily increasing T_{C} as μ approaches the Dirac point, we observe a saturation at the lowest n_{2D}. This can be understood as the natural consequence of finite charge inhomogeneity in our devices. Such inhomogeneity effectively supports a residual carrier density n^{*} as has been discussed in the context of the minimum conductivity at the Dirac point^{28}. Here, from analysis of n_{2D}(V _{B}), we estimate n^{*}≈0.5×10^{13} e^{−} cm^{−2} (see Supplementary Section SB), in good agreement with the onset of saturation in Fig. 2d. Evaluating equation (2) (see Supplementary Section SC), we find T_{C}≈13 K in the vicinity of the Dirac point, within 20% of our observed value. We note that an enhanced Van Vleck susceptibility in the bulk bands has been proposed as a mechanism for generating magnetism on a comparable energy scale^{8}; the present results do not rule out the existence of such a phenomenon, but the agreement between equation (2) and the observed T_{C}(n_{2D}) strongly suggests that the surface electron scenario is relevant here.
We now turn to the B dependence of the longitudinal conductivity σ_{x x}. As shown for device A in Fig. 3a, starting at V _{B} = +100 V (large n_{2D}) we observe the conventional butterfly pattern associated with increased scattering at domains walls^{29}, but as we move to lower V _{B} (smaller n_{2D}) we observe a sign reversal, that is, an enhanced conductivity in the vicinity of the magnetization reversal. In terms of T dependence, this trend is clearly illustrated for the low/highdensity regime in Fig. 3d and Fig. 3e, respectively, where for T<T_{C} this enhancement/decrement tracks the M reversal. We interpret this anomalous behaviour at low n_{2D} in terms of an enhancement of a domainwall conductance as μ approaches Δ_{E}. Magnetic topological insulators are anticipated to have the unique property that their domain walls trap chiral conducting modes. Viewed equivalently as either the trapped edge mode of an unfolded integer quantum Hall state^{18} or a bound mode analogous to solitons in doped polyacetylene^{30}, a mode with conductance σ_{DW} = σ_{x y}^{A} is predicted to exist along an isolated boundary of M reversal. Consistent with this, the conductivity enhancement also appears in the virgin behaviour as shown in Fig. 3b. Defining Δσ_{x x}^{v} as the difference between the virgin and trained state at B = 0, we see a systematic enhancement of Δσ_{x x}^{v}(n_{2D}) for ingap states (Fig. 3f). In Fig. 3c we depict a hypothetical domain structure of the virgin state (or that during reversal of M) to illustrate the appearance of these chiral conducting modes along domain walls. Future experiments in devices where the domain structure can be controlled will serve to test this interpretation (see Supplementary Section SD).
Finally, we return to the detailed behaviour of Δσ_{x y}. Whereas equation (1) and Fig. 1b depict a disorderfree system, it has been shown that considering disorder in a magnetically doped topological insulator for μ outside the clean limit Δ_{E}, a finite σ_{x y}^{A} arises approaching the quantized value inside Δ_{E} (ref. 10). On decreasing n_{2D}, a huge enhancement in Δσ_{x y} is observed but we fail to obtain the quantized value e^{2}/h (see Fig. 2a–c). This behaviour can be understood in terms of the role of remnant metallic conduction in parallel with the anomalous Hall surface state. Modelled as two parallel conductance channels, the observed Hall conductivity can be written as σ_{x y}^{obs} = ρ_{y x}^{S}/((ρ_{x x}^{S})^{2}+(ρ_{y x}^{S}/α)^{2}), where S denotes the contribution of the surface channel and α = (ρ_{x x}^{S}/ρ_{x x}^{B}+1) with B denoting the bulk channel. As depicted by the calculated contours of σ_{x y}^{obs} in Supplementary Section SE, the measured Hall response is acutely sensitive to changes in α even for a fixed Hall contribution from the surface state. Moreover, casting the Hall response in this manner makes it apparent that the standard universal scaling of σ_{x x} and σ_{x y}^{A} studied in ferromagnetic systems is violated here^{21}. For example, holding ρ_{y x}^{S} = h/e^{2} and ρ_{x x}^{S}<h/e^{2}, this simple construction shows that in fact σ_{x y}^{obs} may overshoot and approach e^{2}/h from above as the metallic channel is suppressed (ρ_{x x}^{B}>h/e^{2}). We expect this counterintuitive result will be ubiquitous in doped magnetic topological insulators and other nonideal systems approaching the QAHE ground state.
With this perspective of the approach to the QAHE state, it is clear that devices prepared with further suppressed bulk states or perhaps more plausibly bulk states with lower mobility will be necessary to observe the spontaneous e^{2}/h value. Despite this difficulty, the combination of an enhanced AHE for the Dirac surface modes and indications of edge transport along magnetic domains suggests the presence of the QAHE, which in principle fully develops as T approaches zero^{10}. The realization of this TRS broken state accessible by transport measurements is a significant step forward in the realization of dissipationless devices using topologically nontrivial electronic states.
Methods
Single crystals of Mn_{x}Bi_{2−x}Te_{3−y}Se_{y} are grown using 99.999% pure powders of Bi, Te and Se and a >99.9% pure powder of MnTe. After vacuum sealing in a quartz tube, the powders are heated in a Bridgman furnace to 800 °C and held for two days before pulling through a temperature gradient at 10 °C cm^{−1} at 2 mm h^{−1} for one week followed by furnace cooling. Powder Xray diffraction is performed on the resulting crystals to confirm growth of a single phase; x and y are determined by energydispersive Xray spectroscopy. Mn acts as a local magnetic ion replacing Bi with concentration x, whereas low levels of Se doping y are used to compensate hole carriers arising from crystalline defects and the Mn substitution. Bulk crystals are characterized using a commercial superconducting quantum interference device magnetometer to determine magnetic properties. For transport of bulk crystals, Au wires are attached with Ag paint to apply current in the a bplane and magnetic field applied along the c axis. For microcrystal measurement, crystals are cleaved using Scotch tape onto cleaned SiO_{2}/Si wafers. As described for a number of materials including Bi_{2}Se_{3} (ref. 23), the colour of the cleaved crystal can be used to identify t for crystals with t≲30 nm. Conventional electron beam lithography, electron beam evaporation and liftoff techniques are used to make electric contact (Ti/Au: 3/77 nm) in a Hall bar geometry. To avoid exposure to moisture, the device is loaded into the cryostat immediately after wire bonding to the contact pads and sealing with the ionic liquid N, Ndiethyl N(2methoxyethyl) Nmethylammonium bis(trifluoromethylsulphonyl)imide (DEMETFSI) as a top gate. Transport measurements are performed using standard fourprobe a.c. methods at low frequency (<13 Hz) and typical excitation current 200 nA. The magnetic field is directed along the c axis.
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Acknowledgements
We are grateful to N. Nagaosa, M. Kawasaki, F. Zhang, K. Nomura, BJ. Yang and S. Bahramy for fruitful discussions. This research is supported by the Japan Society for the Promotion of Science through the Funding Program for WorldLeading Innovative R&D on Science and Technology (FIRST Program), initiated by the Council for Science and Technology Policy.
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J.G.C. and Y.O. performed the singlecrystal growth and experiments. J.G.C. and J.Y. made devices and performed device experiments aided by Y.O. J.G.C. analysed the data and wrote the manuscript with contributions from all authors. Y.I. 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., Ye, J., Onose, Y. et al. Diracfermionmediated ferromagnetism in a topological insulator. Nature Phys 8, 729–733 (2012). https://doi.org/10.1038/nphys2388
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DOI: https://doi.org/10.1038/nphys2388
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