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

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.

Introduction

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 electrodynamics^12,13, and the half-quantum Hall effect on the surface with conductance σ_xy = 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 space^15,16. Advances in angle-resolved photoemission spectroscopy (ARPES) have facilitated to observe these textures in spin-resolved spectra^{17,18,19,20,21,22}. 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 impurities^23,24 or the magnetic proximity effect of a magnetic insulator (MI) adlayer or substrate^25,26. 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/h^5,6,27.

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 surfaces^15,28. 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) direction^29,30. Ab initio GW calculations have challenged the results of earlier band structure calculations and concluded that the band gap is direct³¹. Experimentally, ARPES^{19,31,32,33,34} 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) direction^38,39.

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 N_z 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 P_z orbitals with total angular momenta J_z = ±1/2. Below, we will refer to the J_z 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 σ_xy = 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 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 h_z 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 S_z(k) projected into the surface layer (enumerated as l_z = 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 l_z = 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 h_z = 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 S_z in the near vicinity of the surface Brillouin zone center, the point (|k| = 0). Increasing h_z 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.

“Spin” texture S(k) (top row) and the z-component of the “spin” expectation value S_z(k) (in units of ħ/2) (bottom row) in the vicinity of the Dirac point at the surface Brillouin zone center on the exchange-coupled surface of the Bi₂Se₃ slab. (a) The texture is hedgehog-like at the field value h_z = 0.1 eV. (c) At h_z = 0.3 eV, the texture is skyrmion-like. (b) The texture at h_z = 0.273 eV is close to the topological transition. (d) At h_z = 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)). S_z at the 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.

(a,b) Band dispersions of the Bi₂Se₃ slab near the point. The unsplit and exchange-split surface bands are marked in red and green, respectively, for field strengths (a) h_z = 0.2 eV (hedgehog phase), and (b) h_z = h_zc1 = 0.273 eV at the transition. The arrow indicates the special radius for the avoided level crossing of the two-surface bands. (c) Expanded view of the spectrum in (b) near R_k = R_{k s}. (d) The energy gaps at the 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 for the green and orange bands in (a,b) as a function of the layer index l_z for (e) h_z = 0.2 eV (hedgehog phase) and (f) h_z = 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 in the exchange-split occupied surface band (the green band in Fig. 2(c)) at the top surface (l_z = 1) of the slab, where the integral is extended to the hexagonal surface Brillouin zone. is the normalized “spin” expectation value which ensures the quantization of the skyrmion number. N as a function of the exchange field strength h_z is shown in Fig. 3(a). Indeed, N = 1/2 for exchange fields below the critical value h_zc1 = 0.273 eV, identifying more precisely that the hedgehog phase has the “spin” texture of a half-skyrmion (or meron). At h_zc1, the skyrmion number switches to −1, indicating the (anti)-skyrmion character of the texture for , and N = 0 beyond h_zc2. 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.

(a) The skyrmion number N as a function of the exchange-field strength h_z. The dashed vertical lines at the critical fields h_zc1 and h_zc2 bound the field range in which the skyrmion “spin” texture appears. Inset: Hall conductance versus h_z. throughout the finite-field range. (b) The z-component of the “spin” expectation value S_z (in units of ħ/2) and (c) the polar angle versus the distance from the point for different values of h_z. The symbols and colors used in (c) refer to the same parameters as in (b). (d) The variation of the characteristic radii R_H (blue circles) and R_S (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 σ_xy for the full slab via the Kubo formula⁴⁵

where m and n are the band indices, are the velocity operators and n_f 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 field^{32,40,46,47,48,49}. σ_xy takes a finite value even for h_z = 0 due to the tiny energy gap at the point. Therefore, to isolate the effect of the TRS breaking exchange field, we evaluate and plot in the inset of Fig. 3(a). The dependence of 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, σ_xy takes the quantized half-integer value e²/2h¹⁴. σ_xy changes its sign when the magnetic-field direction is reversed⁸. Remarkably, σ_xy 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 h_z is particularly evident within a circular region around the point. Characteristic momentum-space radii R_H and R_S can be determined at which the polar angle of the “spins” has changed by 90° or 180° for the hedgehog and the skyrmion pattern, respectively, upon moving radially outward from the point. Figure 3(b) shows the variation of S_z in the occupied part of the exchange-split surface band with respect to and thereby identifies the special radius inside which the characteristic hedgehog and skyrmion textures form. At R_{k s}, |S(k)| sharply drops to nearly zero. S(k) in the unsplit surface band has a complementary pattern beyond R_{k s} (see Supplementary Information, Fig. S1). As discussed above (see also Fig. 3(b)), changes sign at the critical field h_zc1.

As illustrated in Fig. 3(c), the polar angle , calculated in the occupied part of the exchange-split surface band, continuously varies from at R_k = 0 to at R_k = R_H for the hedgehog texture, and from at R_k = 0 to at R_k = R_S for the skyrmion texture. The plateaus, appearing at and for the skyrmion-“spin” texture, establish a distinctive difference to the typical spatial structure of skyrmions in chiral magnets⁴⁴. With increasing h_z, the characteristic radius R_H for the hedgehog texture increases slowly within the field range 0 < h_z < h_zc1, while the radius R_S for the skyrmion texture increases rapidly within the field range as shown in Fig. 3(d). R_H and R_S even exceed further out than the special radius R_{k s}. Beyond h_zc2, 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 (h_z = 0.2 eV) and at the critical field h_zc1 = 0.273 eV, respectively, along the direction in the hexagonal surface Brillouin zone. Upon increasing h_z, the top occupied bulk band (orange) rises up in energy and touches the exchange-split surface band (green) at the point for h_z = h_zc1, as depicted in Fig. 2(b). The former turns back towards the lower-energy bulk bands upon further increasing h_z.

The exchange-split and unsplit surface bands have an avoided level crossing at , as visible in Fig. 2(c). This observation clarifies the role of the special radius R_{k s} 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 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 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 h_z = h_zc1, a bulk and a surface states become degenerate at the point. Figure 2(e,f) show the squared amplitude of the wave functions at the , calculated for the occupied exchange-split surface band and the top occupied bulk band, as a function of the layer index l_z for h_z = 0.2 eV (hedgehog phase) and h_z = 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 meV^51,52. 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 splittings^25,53. 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 ω_c in semiconductors are of the order ω_c ~ 10¹¹ × H[Tesla] Hz. Specifically, for Bi₂Se₃, an inverse scattering rate τ_s ~ 5.1 × 10⁻¹⁴ s was inferred from de-Haas-van Alphen experiments⁵⁴. So ω_cτ_s < 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]

where , index α labels the four bonding and antibonding states of P_z orbitals in the following order: , , , ; these orbitals form the low-energy bands of Bi₂Se₃. The superscripts denote the parity²⁸, l_z is the layer index, and the arrows represent the total angular momentum eigenvalues J_z = ±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

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

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 l_z = 1. The total Hamiltonian matrix for the slab, of dimension 4N_z × 4N_z, therefore, has the tridiagonal structure

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

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