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. 2017 Apr 5;7:45664. doi: 10.1038/srep45664

Emergent Momentum-Space Skyrmion Texture on the Surface of Topological Insulators

Narayan Mohanta 1,a, Arno P Kampf 1, Thilo Kopp 2
PMCID: PMC5381213  PMID: 28378779

Abstract

The quantum anomalous Hall effect has been theoretically predicted and experimentally verified in magnetic topological insulators. In addition, the surface states of these materials exhibit a hedgehoglike “spin” texture in momentum space. Here, we apply the previously formulated low-energy model for Bi2Se3, a parent compound for magnetic topological insulators, to a slab geometry in which an exchange field acts only within one of the surface layers. In this sample set up, the hedgehog transforms into a skyrmion texture beyond a critical exchange field. This critical field marks a transition between two topologically distinct phases. The topological phase transition takes place without energy gap closing at the Fermi level and leaves the transverse Hall conductance unchanged and quantized to e2/2h. The momentum-space skyrmion texture persists in a finite field range. It may find its realization in hybrid heterostructures with an interface between a three-dimensional topological insulator and a ferromagnetic insulator.

Breaking of time-reversal symmetry (TRS) in three-dimensional (3D) topological insulators (TIs)1,2,3,4 has led to fascinating new topological phenomena. Among them are the quantum anomalous Hall effect (QAHE)5,6,7,8,9,10, the inverse spin-galvanic effect11, axion electrodynamics12,13, and the half-quantum Hall effect on the surface with conductance σxy = e2/2h14. In TIs, strong spin-orbit coupling locks the electron’s spin to its momentum and forces the surface states to form a helical spin texture in momentum space15,16. Advances in angle-resolved photoemission spectroscopy (ARPES) have facilitated to observe these textures in spin-resolved spectra17,18,19,20,21,22. The two routes to break the TRS and to gap the surface state of a 3D TI are either the doping with transition-metal ions as magnetic impurities23,24 or the magnetic proximity effect of a magnetic insulator (MI) adlayer or substrate25,26. In magnetically doped TIs, Dirac semi-metallic surface states acquire a gap and reveal a hedgehog-like spin texture23; their Hall conductance is quantized in units of e2/h5,6,27.

The isostructural tetradymite compounds Bi2Se3, Bi2Te3 and Sb2Te3 belong to the class of strong TIs with an odd number of massless Dirac cones at selected surfaces15,28. Bi2Se3 has a band gap of 0.3 eV and only one massless Dirac cone in the surface-band dispersion, if the crystal is cleaved along the (111) direction29,30. Ab initio GW calculations have challenged the results of earlier band structure calculations and concluded that the band gap is direct31. Experimentally, ARPES19,31,32,33,34 or scanning tunneling microscopy35 leave this issue still unsettled. Typically, Se vacancies at the surface shift the Fermi level towards the conduction band36, but further doping by Ca counteracts this shift and can move the Fermi level back to the Dirac point37. The real-space structure of Bi2Se3 consists of stacked layers. In this stacking, Bi and Se alternate to form five-layer blocks which are coupled via van der Waals interactions; it is therefore well suited for preparing thin films or heterostructures. A structure of five such layers, typically referred to as the ‘quintuple’ layer, repeats along the (111) direction38,39.

Here, we focus on a slab geometry for a 3D TI, in which the exchange field acts on only one of the surface layers. This choice naturally applies to a geometry, in which a TI slab is attached to a ferromagnetic insulator. We adopt a previously-developed strategy to describe a slab of Bi2Se3, stacked with Nz quintuple layers along the z-direction40. The formalism, as outlined in the Method section, straightforwardly allows to examine the layer-resolved electronic dispersion of the slab with respect to the transverse momenta. The low-energy bands of Bi2Se3 result from four bonding and anti-bonding Pz orbitals with total angular momenta Jz = ±1/2. Below, we will refer to the Jz eigenvalues in short as “spin”. If one of its surfaces is exposed to a magnetic field or exchange coupled to a ferromagnetic insulator, such a slab has the same hedgehog spin-texture in momentum space as in magnetically doped TIs. However, at a critical field strength, the hedgehog texture transforms into a skyrmion texture. This topological transition is signalled by a discrete change in the skyrmion counting number. It originates from a field-induced degeneracy point of a surface and a bulk band which, thereafter, interchange their spatial characters. Remarkably, the spin-texture transition leaves the Hall conductance σxy = e2/2h unchanged. The skyrmion “spin” texture remains stable over a finite range of exchange fields similar to the real-space skyrmion lattices in chiral magnets in an external magnetic field41.

Results

“Spin” texture

In the absence of a magnetic or an exchange field, two degenerate Dirac cones appear in the spectrum near the center of the surface Brillouin zone, the Inline graphic point; the corresponding states are spatially confined to the top or the bottom surface. With the Bi2Se3 specific parameter set adopted from ref. 40, the Dirac point is not precisely located at the Fermi energy, but this has no influence on the results presented below. Once the TRS is broken by a finite field of strength hz in one of the two surfaces of the slab, the two-fold degeneracy is lifted in all the bands and the surface state, which experiences the exchange field, acquires a gap. In Fig. 1, we plot the momentum-space “spin” texture S(k) and the z-component of the “spin” expectation value Sz(k) projected into the surface layer (enumerated as lz = 1), which is subject to the exchange field, in the vicinity of the Dirac point in the 2D surface Brillouin zone. S(k) is evaluated as the sum of the lz = 1 contributions from the two surface-centered bands, top and bottom (marked in red and green in Fig. 2(a,b)). The resultant of the two bands is taken here, because the surface bands hybridize away from the Brillouin zone center (see below and the Supplementary Information). The “spin” texture in the selected surface layer changes qualitatively upon increasing the exchange field. The texture in Fig. 1(a) for hz = 0.1 eV is “hedgehog”-like. A similar pattern was detected in the spin-resolved ARPES experiments on Mn doped Bi2Se323. For larger field strength, the momentum-space “spin” structure transforms into a skyrmion-like texture as shown in Fig. 1(c). Most noticeable is the sign change of Sz in the near vicinity of the surface Brillouin zone center, the Inline graphic point (|k| = 0). Increasing hz further leads to yet another qualitative change of the “spin” texture. At first sight, the texture in Fig. 1(d) appears to have changed only quantitatively in comparison with Fig. 1(c). But as the analysis below will reveal, the topological character of these textures is indeed qualitatively different.

Figure 1. Hedgehog and skyrmion “spin” textures.

Figure 1

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“Spin” texture S(k) (top row) and the z-component of the “spin” expectation value Sz(k) (in units of ħ/2) (bottom row) in the vicinity of the Dirac point at the surface Brillouin zone center Inline graphic on the exchange-coupled surface of the Bi2Se3 slab. (a) The texture is hedgehog-like at the field value hz = 0.1 eV. (c) At hz = 0.3 eV, the texture is skyrmion-like. (b) The texture at hz = 0.273 eV is close to the topological transition. (d) At hz = 0.4 eV, the texture has lost its skyrmion structure. The values for the critical exchange fields are determined from the evaluation of the skyrmion counting number (see text and Fig. 3(a)). Sz at the Inline graphic point changes discontinuously from a positive to a negative value upon crossing the critical exchange field at which the transition from a hedgehog to a skyrmion texture takes place.

Figure 2. Electronic structure across the topological transition.

Figure 2

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(a,b) Band dispersions of the Bi2Se3 slab near the Inline graphic point. The unsplit and exchange-split surface bands are marked in red and green, respectively, for field strengths (a) hz = 0.2 eV (hedgehog phase), and (b) hz = hzc1 = 0.273 eV at the transition. The arrow indicates the special radius Inline graphic for the avoided level crossing of the two-surface bands. (c) Expanded view of the spectrum in (b) near Rk = Rks. (d) The energy gaps Inline graphic at the Inline graphic point of the exchange-split surface band (green) and between the occupied part of this band and the top occupied bulk band (brown). The squared amplitude of the wave function Inline graphic for the green and orange bands in (a,b) as a function of the layer index lz for (e) hz = 0.2 eV (hedgehog phase) and (f) hz = 0.3 eV (skyrmion phase).

Skyrmion number

In order to decisively identify the topological character of the “spin” textures in Fig. 1, we calculate the skyrmion number Inline graphic in the exchange-split occupied surface band (the green band in Fig. 2(c)) at the top surface (lz = 1) of the slab, where the integral is extended to the hexagonal surface Brillouin zone. Inline graphic is the normalized “spin” expectation value which ensures the quantization of the skyrmion number. N as a function of the exchange field strength hz is shown in Fig. 3(a). Indeed, N = 1/2 for exchange fields below the critical value hzc1 = 0.273 eV, identifying more precisely that the hedgehog phase has the “spin” texture of a half-skyrmion (or meron). At hzc1, the skyrmion number switches to −1, indicating the (anti)-skyrmion character of the texture for Inline graphic, and N = 0 beyond hzc2. The discontinuous changes of N decisively display the signals for topological phase transitions. N takes a finite value (1/2 or −1) in the exchange-split surface band (green band in Fig. 2(a–c)) only and is zero in the unsplit surface band (red band in Fig. 2(a–c)). N changes sign upon reversal of the magnetic-field direction. Two types of skyrmion lattices commonly appear in chiral magnets. They are either classified as Néel-type or Bloch-type skyrmion (see e.g. refs 42, 43, 44); both have the same skyrmion number, but they differ in their spin-winding pattern. A closer inspection of Fig. 1(a) reveals that the momentum-space texture emerging here is a Bloch-type skyrmion.

Figure 3. Analysis of the “spin” textures.

Figure 3

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(a) The skyrmion number N as a function of the exchange-field strength hz. The dashed vertical lines at the critical fields hzc1 and hzc2 bound the field range in which the skyrmion “spin” texture appears. Inset: Hall conductance Inline graphic versus hz. Inline graphic throughout the finite-field range. (b) The z-component of the “spin” expectation value Sz (in units of ħ/2) and (c) the polar angle Inline graphic versus the distance Inline graphic from the Inline graphic point for different values of hz. The symbols and colors used in (c) refer to the same parameters as in (b). (d) The variation of the characteristic radii RH (blue circles) and RS (red squares) of the hedgehog and the skyrmion “spin” textures, respectively.

Hall conductance

The obvious question arises whether the topological “spin” texture transitions are accompanied by a change in the Chern number and the associated Hall conductance. To address this question, we calculate σxy for the full slab via the Kubo formula45

graphic file with name srep45664-m6.jpg

where m and n are the band indices, Inline graphic are the velocity operators and nf denotes the Fermi-Dirac distribution function. The energy gap in thin slabs of 3D TIs is not truly closed at the Dirac point due to a finite size effect even in the absence of a TRS breaking magnetic field32,40,46,47,48,49. σxy takes a finite value even for hz = 0 due to the tiny energy gap at the Inline graphic point. Therefore, to isolate the effect of the TRS breaking exchange field, we evaluate and plot Inline graphic in the inset of Fig. 3(a). The dependence of Inline graphic on the number of layers is discussed in the Supplementary Information. As expected for our current set up, which is equivalent to an interface between a 3D TI slab and a ferromagnetic insulator, σxy takes the quantized half-integer value e2/2h14. σxy changes its sign when the magnetic-field direction is reversed8. Remarkably, σxy does not change at the critical exchange fields, at which the topological “spin” texture transitions take place. We thus encounter the unusual example for topological phase transitions without an energy gap-closing at the Fermi level and without a change in the Chern number. Examples for the former aspect have been presented in ref. 50.

Characteristic radii

The characteristic “spin” texture in the exchange-split surface band in the surface layer with finite hz is particularly evident within a circular region around the Inline graphic point. Characteristic momentum-space radii RH and RS can be determined at which the polar angle Inline graphic of the “spins” has changed by 90° or 180° for the hedgehog and the skyrmion pattern, respectively, upon moving radially outward from the Inline graphic point. Figure 3(b) shows the variation of Sz in the occupied part of the exchange-split surface band with respect to Inline graphic and thereby identifies the special radius Inline graphic inside which the characteristic hedgehog and skyrmion textures form. At Rks, |S(k)| sharply drops to nearly zero. S(k) in the unsplit surface band has a complementary pattern beyond Rks (see Supplementary Information, Fig. S1). As discussed above (see also Fig. 3(b)), Inline graphic changes sign at the critical field hzc1.

As illustrated in Fig. 3(c), the polar angle Inline graphic, calculated in the occupied part of the exchange-split surface band, continuously varies from Inline graphic at Rk = 0 to Inline graphic at Rk = RH for the hedgehog texture, and from Inline graphic at Rk = 0 to Inline graphic at Rk = RS for the skyrmion texture. The plateaus, appearing at Inline graphic and Inline graphic for the skyrmion-“spin” texture, establish a distinctive difference to the typical spatial structure of skyrmions in chiral magnets44. With increasing hz, the characteristic radius RH for the hedgehog texture increases slowly within the field range 0 < hz < hzc1, while the radius RS for the skyrmion texture increases rapidly within the field range Inline graphic as shown in Fig. 3(d). RH and RS even exceed further out than the special radius Rks. Beyond hzc2, Inline graphic stops at a finite angle and the “spins” no longer sweep to the opposite direction indicating the loss of the texture’s skyrmion character.

Electronic spectra across the transition

To get more insight into the origin of the topological phase transition, we analyze the changes in the electronic structure across the transition. In Figs 2(a,b), the band dispersions of the slab are plotted in the hedgehog phase (hz = 0.2 eV) and at the critical field hzc1 = 0.273 eV, respectively, along the Inline graphic direction in the hexagonal surface Brillouin zone. Upon increasing hz, the top occupied bulk band (orange) rises up in energy and touches the exchange-split surface band (green) at the Inline graphic point for hz = hzc1, as depicted in Fig. 2(b). The former turns back towards the lower-energy bulk bands upon further increasing hz.

The exchange-split and unsplit surface bands have an avoided level crossing at Inline graphic, as visible in Fig. 2(c). This observation clarifies the role of the special radius Rks within which the hedgehog and skyrmion textures form. The hybridization between the two (top and bottom) surface bands of the slab is possible, because their corresponding wave functions extend towards the interior of the slab at momenta away from the Inline graphic point and therefore allow for a finite overlap (see also the Supplementary Information).

Figure 2(d) shows the variation of the energy gap at the Inline graphic point of the exchange-split surface band and the gap between the occupied part of this band and the top occupied bulk band. When the exchange field reaches hz = hzc1, a bulk and a surface states become degenerate at the Inline graphic point. Figure 2(e,f) show the squared amplitude of the wave functions at the Inline graphic, calculated for the occupied exchange-split surface band and the top occupied bulk band, as a function of the layer index lz for hz = 0.2 eV (hedgehog phase) and hz = 0.3 eV (skyrmion phase). Evidently, these states interchange their spatial character across the transition.

Discussion

An experimental detection of the skyrmion texture will be challenging using spin-resolved ARPES techniques. The real obstacle, however, to induce the topological transition is the required large exchange splitting. For the Bi2Se3 specific parameter set which we have used in our calculations, the required exchange field is more than four times larger than the so far observed splitting of ~50 meV in Bi2Se3 samples which are homogeneously doped with magnetic impurities24. At the TI/MI heterointerface of Bi2Se3/MnSe(111), the exchange splitting is only 7 meV51,52. Yet, the extraordinarily large g-factor of ~50 observed for the Dirac electrons in the Bi2Se3 surface states may render it possible to achieve unusually large exchange splittings25,53. We have verified that the critical field can be reduced by applying an electric field along z-direction (up to ~15% by a bias voltage of 0.1 V between the two open surfaces). The phenomenon of the topological transition is expected to be generic to other strong TIs as well. Therefore, the selection of a TI with a band gap, narrower than Bi2Se3, is another possible route to realize the anticipated topological transition or the “spin”-skyrmion texture in momentum space itself. Explicit calculations confirm the expectation that temperature effects are negligibly small for the observed phase transition because of the material’s sizeable energy gap of 0.3 eV. Hence, the transitions should robustly occur at room temperature and even beyond. For these temperatures, orbital effects arising from the magnetization of the surface will not be relevant, justifying a posteriori the ansatz that the exchange field couples only to the electron’s spin. Furthermore, the typical cyclotron frequencies ωc in semiconductors are of the order ωc ~ 1011 × H[Tesla] Hz. Specifically, for Bi2Se3, an inverse scattering rate τs ~ 5.1 × 10−14 s was inferred from de-Haas-van Alphen experiments54. So ωcτs < 1 even for magnetic fields near 100 T, indicating that the effects of orbital magnetic-field are unlikely to influence the surface electrons in Bi2Se3.

The encountered topological phase transition provides a new example where the energy gap at the Fermi level does not close across the transition. Remarkably, while the skyrmion counting number changes, the Hall conductance remains constant. The hedgehog to skyrmion phase transition in the momentum-space “spin” texture is yet another striking phenomenon to occur in three dimensional topological insulators.

Method

The Hamiltonian for a slab of Bi2Se3 is given by [ref. 40, Supplementary Information]

graphic file with name srep45664-m33.jpg

where Inline graphic, index α labels the four bonding and antibonding states of Pz orbitals in the following order: Inline graphic, Inline graphic, Inline graphic, Inline graphic; these orbitals form the low-energy bands of Bi2Se3. The superscripts denote the parity28, lz is the layer index, and the arrows represent the total angular momentum eigenvalues Jz = ±1/2 which result from spin-orbit coupling22.

A single quintuple layer, in the presence of a perpendicular exchange (or Zeeman) field, is effectively described by the Hamiltonian15

graphic file with name srep45664-m39.jpg

with Inline graphic, Inline graphic, Inline graphic Inline graphic, Inline graphic, a is the lattice constant in a layer, hz is the strength of the exchange field, and Hz describes the exchange coupling via the hz entries on the matrix diagonal. H1 accounts for the coupling between two neighboring layers and is expressed as

graphic file with name srep45664-m45.jpg

The parameters in H0 and H1 are taken from ref. 40: A0 = 0.8 eV, B0 = 0.32 eV, C0 = −0.0083 eV, C1 = 0.024 eV, C2 = 1.77 eV, M0 = −0.28 eV, M1 = 0.216 eV, M2 = 2.6 eV and a = 4.14 Å.

The exchange field is subsequently chosen to act only on the top surface layer of the slab with layer index lz = 1. The total Hamiltonian matrix for the slab, of dimension 4Nz × 4Nz, therefore, has the tridiagonal structure

graphic file with name srep45664-m46.jpg

The band dispersion Ek of the slab is obtained by solving the eigenvalue equation Inline graphic, where Inline graphic and Ek are the eigenvectors and eigenvalues of H(k), respectively. The “spin” expectation values, at the surface layer with exchange coupling, are computed using

graphic file with name srep45664-m49.jpg

where  Inline graphic with lz = 1, n labels the eigenenergies corresponding to the two surface bands, Inline graphic (Inline graphic) are the Pauli matrices, and ħ is the Planck’s constant. The results presented above are obtained for a slab of 15 quintuple layers.

Additional Information

How to cite this article: Mohanta, N. et al. Emergent Momentum-Space Skyrmion Texture on the Surface of Topological Insulators. Sci. Rep. 7, 45664; doi: 10.1038/srep45664 (2017).

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Material

Supplementary Information
srep45664-s1.pdf (547.9KB, pdf)

Acknowledgments

The authors gratefully acknowledge discussions with Daniel Braak. This work was supported by the DFG through TRR 80.

Footnotes

The authors declare no competing financial interests.

Author Contributions N.M. performed the calculations. N.M., A.P.K., and T.K. discussed the results and wrote the manuscript.

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