Abstract
Geometry, both in momentum and in real space, plays an important role in the electronic dynamics of condensed matter systems. Among them, the Berry phase associated with nontrivial geometry can be an origin of the transverse motion of electrons, giving rise to various geometric effects such as the anomalous^{1}, spin^{2} and topological Hall effects^{3,4,5,6}. Here, we report two unconventional manifestations of Hall physics: a signreversal of the anomalous Hall effect, and the emergence of a topological Hall effect in magnetic/nonmagnetic topological insulator heterostructures, Cr_{x}(Bi_{1−y}Sb_{y})_{2−x}Te_{3}/(Bi_{1−y}Sb_{y})_{2}Te_{3}. The signreversal in the anomalous Hall effect is driven by a Rashba splitting at the bulk bands, which is caused by the broken spatial inversion symmetry. Instead, the topological Hall effect arises in a wide temperature range below the Curie temperature, in a region where the magneticfield dependence of the Hall resistance largely deviates from the magnetization. Its origin is assigned to the formation of a Néeltype skyrmion induced by the Dzyaloshinskii–Moriya interaction.
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Main
The geometry and topology in Hilbert space constitute a central issue in quantum physics, which has recently also shed a new light on the electronic states in solids. The wavefunctions are characterized by the Berry connection between two neighbouring points, in both momentum space and real space, which plays the role of the vector potential leading to the concept of emergent electromagnetic field (EEMF). Global topology of the manifold in Hilbert space is represented by topological integers. As integers cannot change continuously, it gives a certain stability to the system. For example, the Chern number, which is given by the integral of the emergent magnetic field over the first Brillouin zone, protects the surface or edge states supporting the dissipationless current flows (that is, bulk–edge correspondence). The onedimensional chiral edge mode in the quantum Hall effect and the Dirac surface state in threedimensional topological insulators (TI; refs 7,8) are the representative examples of this phenomenon. In addition, it has recently been proposed^{9,10} and observed^{11,12,13,14} that TI with doped magnetic ions—namely, Cr or V—produces the quantized version of the anomalous Hall effect (AHE) in the absence of an external magnetic field.
Topology in real space, on the other hand, is exemplified by skyrmion spin texture^{15,16,17} found in chirallattice magnets such as MnSi (ref. 15) or Fe_{1−x}Co_{x}Si (ref. 16). Here, the solid angle subtended by the spins forms an emergent magnetic field in real space, and its integral over the twodimensional space defines a topological integer called the skyrmion number. Namely, the skyrmion number counts the number of times the spin direction wraps around a unit sphere. This integer protects the skyrmion from annihilation and allows it to behave as a single particle.
Thus far, the EEMF in momentum and real spaces have mostly been studied separately. Recent advances in fabricating artificial structures of materials, however, enable the design of emergent phenomena considering both momentum and real spaces in a unified way. Here, we examined these geometric Hall effects in TI heterostructures composed of magnetic TI Cr_{x}(Bi_{1−y}Sb_{y})_{2−x}Te_{3} (CBST) and nonmagnetic TI (Bi_{1−y}Sb_{y})_{2}Te_{3} (BST; refs 11,12,13,18,19,20,21,22), which were grown on InP(111) substrates using molecularbeam epitaxy^{20,21,22}. By applying a fieldeffect transistor structure (FET, see Methods), the Fermi energy (E_{F}) of the TI heterostructures can be precisely controlled in a large energy range over the bulk bandgap. In particular, the quantum AHE realized in the magnetic TIs offers an interesting scenario where the two different physical mechanisms merge—namely, skyrmion formation around the doped carrier into the quantum Hall ferromagnet^{23} and skyrmion formation at the interface of magnets^{24,25,26,27}. First, we investigate the gate voltage (V_{G}) and temperature (T) dependence of the Hall effect for a 2nm CBST/5nm BST heterostructure as shown in Fig. 1a. With increasing V_{G} at 2 K (Fig. 1b), the hysteresis curves of the Hall conductivity σ_{xy} gradually change shape, demonstrating a maximum value at V_{G} = 0.2 V . In Fig. 1c, d, we plot the temperature dependence at selected voltages V_{G} = 0.2 V and −7.0 V. In contrast to the conventional temperature dependence of magnetic TI shown in Fig. 1c at V_{G} = 0.2 V, where E_{F} is close to the Dirac point, we find two notable features under hole accumulation at V_{G} = −7.0 V in Fig. 1d: first, signreversal of the anomalous Hall conductivity σ^{A}_{xy} (Hall conductivity under zero magnetic field) at 6 K and, second, nonmonotonous behaviour of σ_{xy} as a function of magnetic field. In this paper, we first study the former feature and discuss the latter afterwards, both from the viewpoint of the geometric Hall effect. The detailed data set of σ^{A}_{xy} in T–V_{G} parameter space (Fig. 1e) shows the signreversal roughly at negative V_{G} and in the 6–20 K region. Such a signreversal of σ^{A}_{xy} has not been observed in singlelayer CBST, which always gives a positive sign regardless of its E_{F} position and temperature^{11,12,13}, as exemplified in Fig. 2a. Hence, such a transition between positive and negative σ^{A}_{xy} is a distinct behaviour in the 2nm/5nm (Fig. 1d) and 3nm/5nm (Fig. 2b) heterostructures. Note that the longitudinal conductivity (approximately 10^{2} (Ω cm)^{−1}) is so low that each anomalous component cannot be of an extrinsic origin^{1}, such as skew scattering; therefore, an intrinsic origin should be explored in the electronic band structure characteristic of the TI heterostructures.
To address this issue, we theoretically study the lowenergy effective Hamiltonian of the TI heterostructures (see Methods). We consider two cases, a 10nm CBST single layer and a 5nm CBST/5nm BST heterostructure, as shown in Fig. 3a, d. The band structure obtained for CBST (Fig. 3b) shows gapped Dirac surface bands in addition to the bulk bands. In the case of CBST/BST (Fig. 3e), the Dirac band at the top surface of the CBST is gapped owing to the exchange coupling, whereas the Dirac band at the bottom surface of the BST remains gapless. Moreover, the top of the valence bands shows Rashba splitting because of the broken spatial inversion symmetry in the heterostructures (red lines in the lower inset of Fig. 3e), and these Rashba split bands are slightly gapped owing to the exchange coupling (black lines). In fact, such Rashba splitting in the bulk band of TI has also been observed in previous angleresolved photoemission spectroscopy (ARPES) studies^{28}. In Fig. 3c, f, we show the Hall conductivity obtained from the Berry curvature that encodes the nontrivial geometry in the momentum space. In both cases, σ_{xy} is maximized when E_{F} is within the gap of the surface bands. In addition, σ_{xy} in Fig. 3f shows a sharp negative peak at the top of the valence bands, denoted by a red triangle. This is attributed to the slightly gapped Rashba split bands on which the Berry curvature is concentrated. Moreover, the negative peak decreases for larger exchange coupling, which corresponds to the case of lower temperature (see Supplementary Information for the details). This means that a positive AHE component is dominant at lower temperatures. These theoretical arguments coincide well with the experimental results shown in Fig. 1e, g, in which the band energy scheme is based on the results of σ_{xy} at zero and high (14 T) fields, including ν = 1 and ν = 0 quantum Hall plateaux (Fig. 1f; ref. 20).
In addition to the signreversal, we observe anomalous magneticfielddependent behaviours in the Hall effect in 2nm CBST/5nm BST heterostructures, as shown in Figs 2c and 1d. To elucidate the hysteretic anomaly in the R_{yx} versus B curve we compared R_{yx}′ (= R_{yx} − R_{0}B) with the magnetization of the film (see Methods) in the top panel of Fig. 2d. Here, R_{0} is the ordinary Hall coefficient determined from the linear slope at high magnetic field (see Supplementary Information). In the conventional theory, R_{yx}′ is nothing other than the anomalous Hall term, which is proportional to the magnetization M. In reality, however, a large discrepancy between R_{yx}′ and M is discerned; we need to add the nontrivial Hall component (shown in the bottom panel of Fig. 2d) as well as the Mlinear anomalous Hall term to reproduce the observed R_{yx}′. As a plausible origin, we propose skyrmion formation from both theoretical and experimental aspects. Under skyrmion spin texture, the moving electrons experience the EEMF, giving rise to an additional Hall component termed the topological Hall effect (THE, note that ‘topological’ is defined in real space)^{3,4}, which is observed in the skyrmion phase of some chirallattice magnets^{5,6} as well as in frustrated magnets endowed with scalar spin chirality^{29,30}. The magneticfield dependence of THE shows a hysteresis behaviour; the magnetic field showing the maximal THE is observed to shift slightly to lower field as a whole by up to ∼0.02 T with decreasing sweep rate from 2 × 10^{−3} T s^{−1} to 3 × 10^{−5} T s^{−1} (see Supplementary Information for the details).
To confirm the possibility of skyrmion formation, we simulated the energetic stability of a skyrmion using a threedimensional tightbinding model (see Methods). We consider the following three cases: Bloch (Fig. 4a), Néel1 (Fig. 4b) and Néel2 (Fig. 4c) for a CBST single layer (Fig. 4d) and a CBST/BST heterostructure (Fig. 4h). In Fig. 4e–g, i–k, we show the formation energy of a skyrmion relative to that of the spincollinear ferromagnet as a function of the skyrmion radius R at various doping levels. For the CBST single layer in Fig. 4e–g, the energy is minimized always at R = 0, implying that the ferromagnetic state is the ground state regardless of E_{F}. For the electrondoped (Fig. 4i) or halffilling (Fig. 4j) cases of the CBST/BST heterostructure, the situation is similar. In the holedoped case in the heterostructure (Fig. 4k), by contrast, a negative energy region exists for the Néel2type skyrmion (R ≠ 0), denoted by a red triangle. The emergence of a stable condition for skyrmion formation can be explained as follows. The surface state of the TI exhibits inversion symmetry breaking and a fairly strong spin–orbit interaction, as exemplified by the spin–momentumlocking in the Dirac dispersion. Thus, the electrons at the surface state mediate the Dzyaloshinskii–Moriya (DM) interaction where the DM vector is pointing in the inplane direction^{24,25,26,27,31}. In the heterostructure, therefore, such an inplane DM vector originating from the top CBST surface favours the formation of a Néeltype skyrmion^{24,25,26,27}. However, in the singlelayer CBST, DM vectors exist both at the top and bottom surface, each pointing along opposite directions, so that skyrmion formation is disfavoured owing to the frustration. Hence, broken spatial inversion symmetry associated with the heterostructure is the essential requirement for skyrmion formation.
Theoretical verification of skyrmion formation at the holeaccumulated condition agrees well with the experimental situation at large negative V_{G} (−7.0 V), as shown in Fig. 1d. In 2nm/5nm heterostructures (Fig. 2c), the topological Hall components R^{T}_{yx} are identified as the green region, which is shown in Fig. 2e in the B–T parameter space. From this plot, we can see that skyrmions are formed in the course of magnetization reversal in a wide temperature range below the Curie temperature. The maximum THE, amounting to 140 Ω (lower panel of Fig. 2d), corresponds to the magnitude of an emergent magnetic flux density of 0.29 T, taking into consideration the normal Hall coefficient of 490 Ω T^{−1} (see Supplementary Information). However, in the 3nm CBST/5nm BST heterostructure, no THE is observed, although signreversal of AHE is discerned, as shown in Fig. 2b. (As for the reproducibility of such a critical CBSTthicknessdependent THE, see Supplementary Information.) This observation implies the importance of the DM interaction; as the CBST thickness increases, the relative strength of the exchange interaction to the DM interaction increases, such that skyrmion formation would become less favoured. From this viewpoint, further optimization of the magnetic/nonmagnetic TI heterostructure may open a way to control skyrmions on TI.
Methods
MBE thin film growth.
Thin films were grown by molecularbeam epitaxy (MBE) on insulating InP (111) substrates. The growth temperatures for BST and CBST are 220 °C and 200 °C, respectively. The Cr (x ∼ 0.2) content is estimated from the beam equivalent pressure (BEP) ratio through Cr/(Bi + Sb). The Te flux (BEP = 1.0 × 10^{−4} Pa) was oversupplied while keeping the Te/(Bi + Sb) ratio close to 20 to suppress Te vacancies. Under this condition, the growth rate is about 0.2 nm min^{−1}. Before the growth of the first layer, a monolayer Sb_{2}Te_{3} buffer layer was grown to give a better morphology and epitaxy. After epitaxial growth of the BST layer, annealing under exposure to the Te flux was performed in situ at 380 °C to get a smoother surface. The same procedure was employed for the following CBST layer. To suppress degradation of the film, an AlO_{x} capping layer with a thickness of approximately 3 nm was deposited at room temperature by an atomic layer deposition system immediately after removing the samples from the MBE vacuum chamber. Please note that Cr interdiffusion in the heterostructure is negligible, as confirmed by energydispersive Xray spectroscopy imaging (see Supplementary Information). A list of sample structures, such as film thickness, and Bare or FET (including applied gate voltages), is summarized in Supplementary Note 1 and Supplementary Table 1.
FET device fabrication.
The Hallbar device pattern was defined by a photolithographic technique and Ar ionmilling processes. After defining the Hallbar structure, AlO_{x} with a thickness of approximately 20 nm was deposited to serve as a gate capacitor. For Ohmiccontact electrodes and a top gate electrode, 5nm Ti/45nm Au were deposited by means of an electronbeam evaporator.
Transport and magnetization measurements.
Transport measurements for the bare films were conducted using the d.c. transport option of a physical property measurement system (PPMS, Quantum Design). FET devices were measured in the PPMS with a standard lockin technique at low frequency (∼7 Hz) and with a low excitation current (10 nA) to suppress heating effects. Lowtemperature (<2 K) measurements were performed using the ^{3}He option of the PPMS. Magnetization measurements for the bare film of 2nm CBST/5nm BST were conducted using a magnetic property measurement system (MPMS, Quantum Design). All the transport and magnetization data were antisymmetrized as a function of the magnetic field.
Calculation of band structures and anomalous Hall conductivity.
Band structures and anomalous Hall conductivity were calculated using the lowenergy effective Hamiltonian of the TI heterostructures given by
Here, σ_{i} and τ_{i} are the Pauli matrices for the spin and orbital degrees of freedom. τ_{z} = + 1 for Bi/Sb porbitals and τ_{z} = −1 for Te porbitals. The x and y directions are parallel to the TI thin films and periodic, whereas the z direction is perpendicular to the TI thin films and not periodic. Therefore, k_{z} should be regarded as the operator −i∂_{z}. We adopted the following parameters in numerical calculations: v_{F} = 5.0 × 10^{5} m s^{−1}, m = −300 meV, D_{1} = 60 eV Å^{2}, D_{2} = 20 eV Å^{2} inside the TI thin films^{8}. The asymmetry between the conduction and valence bands is captured by the D_{2} and J_{3} term. We introduced the potential asymmetry between BST and CBST by setting U(z) = 0 meV for BST and U(z) = −40 meV for CBST, and the exchange couplings in magnetic TI by setting J_{0} = J_{3} = 0 meV for BST and J_{0} = 5 meV and J_{3} = 1 meV for CBST. Here, the parameters J_{0} and J_{3} are symmetric and asymmetric parts of the exchange couplings for the two orbitals^{31}. The exchange coupling for Te porbitals should be slightly larger than that for Bi/Sb porbitals because Te atoms locate nearer to doped Cr ions. (This asymmetry is captured by setting J_{3} = 0.2J_{0}.) The anomalous Hall conductivity is obtained by integrating the Berry curvature B_{k} = ∇ × A_{k} over the momentum space, where A_{k} is the Berry connection A_{k} = −i〈ψ_{k}  ∂_{k}  ψ_{k}〉 with Bloch wavefunctions ψ_{k}. As seen in Fig. 3e, the valence top bands show Rashba splitting due to the broken inversion symmetry in the TI heterostructure. This Rashba splitting results in the concentration of the Berry curvature at the valence top bands and a large contribution to the anomalous Hall conductivity there.
Calculation of skyrmion stability.
The stability of a skyrmion was calculated by the threedimensional tightbinding model for TI, corresponding to the effective Hamiltonian in the continuum approximation given in equation (1),
where σ_{i} and τ_{i} are the Pauli matrices for the spin and the orbital degrees of freedom. The system size is L_{x} = L_{y} = 31 sites, L_{z} = 8 layers, and the periodic boundary condition for the x and y directions is imposed. In this model, the D_{2} and J_{3} terms break the particle–hole symmetry^{31} as mentioned above, and they lead to asymmetry between the electrondoped and holedoped cases. We have set D_{1} = 1, D_{2} = 0.2, t = 1, m = −1, which realizes the strong TI phase. The exchange couplings in magnetic TI are set as J_{0} = J_{3} = 0 for BST and J_{0} = 0.1 and J_{3} = 0.02 for CBST. This choice of the parameters is determined to reduce the finitesize effect, and hence the numerical results are not regarded as quantitatively corresponding to the real system, but rather as qualitative information.
The skyrmion configuration n(r) is set as
for r < R, and n_{z} = 1 for the outer region r < R. The skyrmion radius R is set as the variational parameter and φ determines the magnetic helicity or the skyrmion type—that is, the Bloch type for φ = (π/2), the Néel1 type for φ = 0 and the Néel2 type for φ = π. We note that R = 0 corresponds to the ferromagnetic state. One can interpret the obtained results presented in the main text from the facts that valence bands are more strongly coupled to the magnetic ions as a result of the J_{3} term, and the DM interaction is induced from the spin–momentumlocking in the surface state of TI.
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Acknowledgements
We thank J. Matsuno and T. Yokouchi for fruitful discussions and experimental support. This research was 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) on ‘Quantum Science on Strong Correlation’ initiated by the Council for Science and Technology Policy and by JSPS GrantinAid for Scientific Research(S) No. 24224009 and No. 24226002 from MEXT, Japan.
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K.Y. conducted the thin film growth and device fabrication with help from R.Y. K.Y. carried out transport/magnetization measurements. T.M. and R.W. carried out numerical calculations of the Hall conductivity and the skyrmion stability, respectively. K.Y., T.M., R.W., A.T., M.K., N.N. and Y.T. wrote the manuscript with contributions from all the authors. M.K., N.N. and Y.T. conceived and guided the project.
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Yasuda, K., Wakatsuki, R., Morimoto, T. et al. Geometric Hall effects in topological insulator heterostructures. Nature Phys 12, 555–559 (2016). https://doi.org/10.1038/nphys3671
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DOI: https://doi.org/10.1038/nphys3671
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