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Dirac-fermion-mediated ferromagnetism in a topological insulator

Abstract

Topological insulators are a newly discovered class of materials in which helical conducting modes exist on the surface of a bulk insulator1,2,3,4,5,6. Recently, theoretical works have shown that breaking gauge symmetry7 or time-reversal symmetry8 in these materials produces exotic states that, if realized, represent substantial steps towards realizing new magnetoelectric effects9,10 and tools useful for quantum computing11. Here we demonstrate the latter symmetry breaking in the form of ferromagnetism arising from the interaction between magnetic impurities and the Dirac fermions12,13. Using devices based on cleaved single crystals of Mn-doped Bi2Te3−ySey, the application of both solid-dielectric and ionic-liquid 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 one-dimensional edge-state 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 effect8,10 and states required for quantum computing14,15.

Main

The discovery of topological insulators has attracted wide interest owing to the exotic spin and charge properties of their edge modes. Among three-dimensional topological insulators, particular interest has been focused on a family of spin–orbit-coupled semiconductors based on Bi: Bi1−xSbx, Bi2Te3 and Bi2Se3 (refs 6, 16, 17). An active area of research since their discovery has been the breaking of time-reversal symmetry (TRS) in these systems12,13. The application of a magnetic field B is predicted to give rise to a half-integer quantum Hall effect on each surface akin to graphene but with spin and valley degeneracy removed18. Alternatively, the introduction of ferromagnetism is predicted to produce a quantum anomalous Hall effect8 (QAHE) and a topological magnetoelectric effect9, both of which are hallmarks of a topologically non-trivial state.

Original theoretical proposals involving broken TRS through ferromagnetism focused on depositing magnetic insulators on the surface of topological insulators to proximitize the surface states14,15. 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 ordering19. In analogy to bulk dilute magnetic semiconductors, such as (Ga,In)1−xMnx As, where hole-like carriers mediate a ferromagnetic interaction among localized spins20, one further scenario is ferromagnetic ordering mediated by the surface Dirac electrons12,13. 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 Rashba-like spin texture, locking σ perpendicular to the momentum k, namely the Hamiltonian H has the form

where 2π is the Planck constant, vF is the Fermi velocity, is the surface unit normal, σz is the z component of the electron spin and nS 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 σz generating an energy gap ΔE (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 surface-state-mediated ferromagnetism approaching the QAHE ground state.

Figure 1: Ferromagnetism in a magnetically doped topological insulator.
figure 1

a, Schematic of a Dirac electron with spin σ and momentum k on the surface of a topological insulator with an average z component of local spins . In the TRS broken state, the competition between the exchange interaction and the Rashba-like surface texture yields a finite component of σ in the z direction. b, Breaking of TRS opens a gap in the surface state spectrum starting at the Dirac point located at the time reversal invariant momentum Γ. c, Bulk crystals of MnxBi2−xTe3−ySey show ferromagnetic ordering below TC = 13 K, as seen in M(T). Both σx x and Δσx y (the spontaneous component of σx y) show anomalies at TC. The inset shows scaling of M and the anomalous part of the Hall conductivity σx yA. d, Device structure for measurement of microcrystals. A back gate is formed by the SiO2/Si substrate and a top gate with the deposition of ionic liquid. e, Atomic force micrograph (false colour) of Ti/Au contacts made by electron beam lithography for device B. The crystal thickness is 7 nm. f, Measurements of Δσx y for device A indicate a ferromagnetic transition. Application of the back-gate voltage V B is shown to tune the magnetic response with varying of the transition temperature TC.

The starting point of our study is growth of single crystals of MnxBi2−xTe3−ySey using the Bridgman method. As shown for x = 0.04 and y = 0.12 (carrier density n3D = 6×1019 e+ cm−3) in Fig. 1c, these crystals are bulk p-type ferromagnets with the easy axis along c and a transition temperature TC≈13 K. Also shown are the associated rise in the B = 0 extrapolated Hall conductivity Δσx y(T) (the remnant value in the case of hysteretic σx y(B)) that tracks the magnetization M(T) and an anomaly in σx x(T) (σ3D in Fig. 1c) near TC. Generally, σx y can be decomposed as σx y = σx yN+σx yA, where σx yN is due to the normal Lorentz force Hall effect and σx yA is due to the AHE (ref. 21). The AHE term follows the direct proportionality σx yA = SH(T)M, demonstrated in the inset of Fig. 1d with an appropriate choice of the scaling constant SH. We will use Δσx y to track σx yA and M also in the microcrystal-based 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 SiO2 over-layer 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 side-gate 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 devices23. With no applied gate voltage, the two-dimensional carrier density derived from σx yN for device A in Fig. 1f is n2D = 2.4×1013 e cm−2. The critical value of n2D to reach the bottom of the conduction band is estimated to be using the Fermi surface area observed in Bi2Te3 at the onset of bulk conduction17, 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 p-type to n-type conduction, we observe an abrupt increase in Δσx y(T), indicating a ferromagnetic transition qualitatively similar to the bulk crystals. In this class of Mn-doped compounds, n-type bulk carriers have not been observed to mediate ferromagnetism19,24, which makes the robust ferromagnetism here unexpected. Fig. 1f also shows that TC can be tuned by V B, which we use below to further investigate this unconventional n2D dependence of the ferromagnetism.

We report the results of five devices A–E with different V B and V T covering the range 3×1012 e+ cm−2<n2D<6×1013 e cm−2, where n2D(V B)≡n2D(0 V)+η V B with the calibrated gate efficiency η = 8.9×1010 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 σx y with depletion of e carriers. Starting at large positive V B = +100 V (n2D = 3.3×1013 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 n2D = 6×1012 e cm−2. The growth of σx yA continues with depletion of carriers, seeming to saturate at the lowest V B where σx yN changes sign. For the case of larger n2D where μ lies above the bulk conduction band minimum, we do not observe any sign of σx yA up to 5.5×1013 e cm−2 (device E). This again confirms that the bulk n-type carriers cannot mediate ferromagnetism.

Figure 2: AHE and TC tuned by carrier density.
figure 2

a, Hall conductivity σx y at T = 2 K for device A at several back-gate voltages V B, offset vertically 0.04 e2/h. The observed anomalous Hall response grows on depletion of carriers. b, After application of V T = −3 V, device B shows a further enhanced spontaneous σx y whereas the ordinary Hall response σx yN changes from n-type to p-type at the most negative V B. Here, a vertical offset of 0.4 e2/h is used. c, Temperature dependence of σx y in device C. There is a sharp onset of σx yA below 12 K. See Fig. 1f for the case of device A. d, Estimates of TC for devices A–E. Contrary to conventional dilute magnetic semiconductors, the transition is suppressed on increasing the carrier density. The error bars reflect the discreteness of the measurement in T. The inset depicts the critical chemical potential position μc for the onset of ferromagnetic behaviour, as estimated from σx yN relative to the conduction band (CB), valence band (VB) and surface states (SS).

Next, the typical T dependence of σx y(B) is shown for device C in Fig. 2c. Here, on decreasing T below 12 K there is an abrupt onset attributed to σx yA. We can estimate TC by examining either the remnant Δσx y(T) as in Fig. 1c or using an Arrott plot (Supplementary Fig. S4). In Fig. 2d, TC for each device is plotted as a function of n2D. The phase boundary separating the ferromagnetic and paramagnetic state is dome shaped, closing at a critical density nc = (3.8±0.5)×1013 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 Ns is the number density of local spins and is the density of states at the Fermi energy20,25. It is therefore understood that an increase in n would lead to an increase in TC (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 (σz) and the averaged moments () in competition with the Rashba energy. In this way, TC(n2D) 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 a0 is the lattice constant and Λ is a cutoff energy associated with the termination of the Dirac surface band. Equation (2) indicates a maximum TC at the Dirac point that smoothly decreases with increasing , in accord with our observation identifying TC = 0 at μ(nc) = Λ. Whereas equation (2) predicts a steadily increasing TC as μ approaches the Dirac point, we observe a saturation at the lowest n2D. 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 point28. Here, from analysis of n2D(V B), we estimate n*≈0.5×1013 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 TC≈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 scale8; the present results do not rule out the existence of such a phenomenon, but the agreement between equation (2) and the observed TC(n2D) strongly suggests that the surface electron scenario is relevant here.

We now turn to the B dependence of the longitudinal conductivity σx x. As shown for device A in Fig. 3a, starting at V B = +100 V (large n2D) we observe the conventional butterfly pattern associated with increased scattering at domains walls29, but as we move to lower V B (smaller n2D) 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/high-density regime in Fig. 3d and Fig. 3e, respectively, where for T<TC this enhancement/decrement tracks the M reversal. We interpret this anomalous behaviour at low n2D in terms of an enhancement of a domain-wall conductance as μ approaches ΔE. 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 state18 or a bound mode analogous to solitons in doped polyacetylene30, a mode with conductance σDW = σx yA 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 Δσx xv as the difference between the virgin and trained state at B = 0, we see a systematic enhancement of Δσx xv(n2D) for in-gap 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).

Figure 3: Sign reversal in magnetoconductivity driven by gating for device A.
figure 3

a, Back-gate voltage V B dependence of σx x, showing the onset of enhanced conductivity at the reversal of M on depletion of carriers at T = 2 K. b, Virgin curve and trained behaviour for V B = −100 V (n2D = 1.5×1013 e−2 cm−2). c, Schematic depiction of the domain structure in a magnetic topological insulator; domain walls across the opposite M domains support a chiral mode (shown in green). d, T dependence of σx x(B) through the ferromagnetic transition gated with V B = −100 V. The enhanced conductivity can be seen to track the M reversal. e, σx x(B) at high carrier density shows hysteresis of the conventional form, that is, suppressed conductivity attributed to carrier scattering at magnetic domain walls. f, The difference between the virgin and trained σx x (≡σx xv) exhibits a crossover from positive to negative values on increased carrier density.

Finally, we return to the detailed behaviour of Δσx y. Whereas equation (1) and Fig. 1b depict a disorder-free system, it has been shown that considering disorder in a magnetically doped topological insulator for μ outside the clean limit ΔE, a finite σx yA arises approaching the quantized value inside ΔE (ref. 10). On decreasing n2D, a huge enhancement in Δσx y is observed but we fail to obtain the quantized value e2/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 σx yobs = ρy xS/((ρx xS)2+(ρy xS/α)2), where S denotes the contribution of the surface channel and α = (ρx xS/ρx xB+1) with B denoting the bulk channel. As depicted by the calculated contours of σx yobs 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 σx x and σx yA studied in ferromagnetic systems is violated here21. For example, holding ρy xS = h/e2 and ρx xS<h/e2, this simple construction shows that in fact σx yobs may overshoot and approach e2/h from above as the metallic channel is suppressed (ρx xB>h/e2). We expect this counterintuitive result will be ubiquitous in doped magnetic topological insulators and other non-ideal 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 e2/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 zero10. The realization of this TRS broken state accessible by transport measurements is a significant step forward in the realization of dissipationless devices using topologically non-trivial electronic states.

Methods

Single crystals of MnxBi2−xTe3−ySey 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 X-ray diffraction is performed on the resulting crystals to confirm growth of a single phase; x and y are determined by energy-dispersive X-ray 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 b-plane and magnetic field applied along the c axis. For microcrystal measurement, crystals are cleaved using Scotch tape onto cleaned SiO2/Si wafers. As described for a number of materials including Bi2Se3 (ref. 23), the colour of the cleaved crystal can be used to identify t for crystals with t30 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, N-diethyl- N-(2-methoxyethyl)- N-methylammonium bis-(trifluoromethylsulphonyl)-imide (DEME-TFSI) as a top gate. Transport measurements are performed using standard four-probe 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, B-J. 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 World-Leading 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 single-crystal 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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Correspondence to Joseph G. Checkelsky.

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Checkelsky, J., Ye, J., Onose, Y. et al. Dirac-fermion-mediated ferromagnetism in a topological insulator. Nature Phys 8, 729–733 (2012). https://doi.org/10.1038/nphys2388

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