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Topological phases of Bi(111) bilayer in an external exchange field

Using first principles methods, we investigate topological phase transitions as a function of exchange field in a Bi(111) bilayer. Evaluation of the spin Chern number for different magnitudes of the exchange field reveals that when the time reversal symmetry is broken by a small exchange field, the system enters the time-reversal broken topological insulator phase, introduced by Yang {\it et al.} in Phys. Rev. Lett. 107, 066602 (2011). After a metallic phase in the intermediate region, the quantum anomalous Hall phase with non-zero Chern number emerges at a sufficiently large exchange field. We analyze the phase diagram from the viewpoint of the evolution of the electronic structure, edge states and transport properties, and demonstrate that different topological phases can be distinguished by the spin-polarization of the edge states as well as spin or charge transverse conductivity.

Using first principles methods, we investigate topological phase transitions as a function of exchange field in a Bi(111) bilayer. Evaluation of the spin Chern number for different magnitudes of the exchange field reveals that when the time reversal symmetry is broken by a small exchange field, the system enters the time-reversal broken topological insulator phase, introduced by Yang et al. in Phys. Rev. Lett. 107, 066602 (2011). After a metallic phase in the intermediate region, the quantum anomalous Hall phase with non-zero Chern number emerges at a sufficiently large exchange field. We analyze the phase diagram from the viewpoint of the evolution of the electronic structure, edge states and transport properties, and demonstrate that different topological phases can be distinguished by the spin-polarization of the edge states as well as spin or charge transverse conductivity.

I. INTRODUCTION
Topological insulators (TIs) have drawn quite intensive attention recently owing to the gapless surface/edge states, topologically protected against perturbations which do not close the bulk gap. 1,2 Similarly to graphene, 3 exciting physical properties are expected if the TI surface/edge states are made gapped, resulting in e.g. half-quantized surface Hall effect, 4 topological magneto-electric effects, 5,6 and Majorana zero states. 7 It is suggested that such effects can be triggered via the proximity effect of TIs to magnetically ordered materials or superconductors, in which case new terms which impose or break certain symmetries are introduced into the effective Hamiltonian.
One particularly interesting situation arises if the timereversal symmetry (T ) in a TI is broken via a controlled perturbation. The definition of the Z 2 topological invariant, used to classify two-dimensional (2D) insulators into ordinary insulators and TIs, is hinged on the presence of the time-reversal symmetry in the system. 8 It still remains an open question of how to generalize the Z 2 topological invariant when T is broken. In such a case, it is suggested 9 that it is plausible to use the so-called spin Chern number: 10 where C + and C − are the Chern numbers for the spin "up" and "down" manifolds of the occupied states. Since the spin-orbit coupling (SOC) in general induces spin mixing, the spin Chern number is well defined only when the system exhibits a gap in the spectrum of σ z in addition to being a band insulator. In this case C s proves to be equivalent to the Z 2 number when time-reversal symmetry is present. 11 For models of TIs 10 and graphene, 9 the spin Chern number picture proved to be valid, while a more detailed analysis of this concept in real materials investigated from first principles is still lacking. Owing to T symmetry, the surface/edge states of TIs are spin degenerate, making it hard to manipulate and exploit their transport properties in experiments based on magnetic detection. While the quantum Hall effect usually requires a very large magnetic field to emerge, in this sense, the quantum anomalous Hall (QAH) effect in ferromagnets would be much more suited for future spintronics. The QAH effect, which has attracted a lot of attention recently but so far has not been verified experimentally, occurs in 2D Chern insulators 16 with broken T -symmetry, exhibiting overall spin-polarized edge states carrying quantized electric charge. Up to now, it is believed that TIs can be turned into Chern insulators via magnetic doping 12,13 but it still remains an experimentally unsolved problem despite recent advances in this direction. 14 In this light, it is necessary to explore how tunable 2D TIs are under T -broken perturbations, in particular, exchange interactions.
Bismuth is the heaviest atom with a stable isotope and strong spin-orbit coupling makes it an important ingredient of the newly discovered TIs such as Bi 2 Se 3 18 and Bi 1−x Sb x . 8 Ultrathin Bi(111) films, which can be produced experimentally on different substrates, 19,20 are also predicted to be 2D TIs. 21,22,24 In our study we take Bi(111) bilayer as a representative of the latter class of TIs. In this work, we investigate the topological phase transitions in a 2D TI -the Bi(111) bilayer 21 − in an external exchange field using first principles methods. We found that at small exchange fields the bilayer is in the time-reversal broken topological insulator phase, which can be characterized by non-trivial values of spin Chern numbers, transverse spin Hall conductivity, and spin polarization of the edge states. Increasing the magnitude of the exchange field further drives the system first into a metal and then into a QAH phase. Detailed analysis of the electronic structures reveals that spin-mixing is important for closing the bulk gap and Chern number exchange between valence and conduction bands. At last, we analyze the spin-polarization of the metallic edge states in different phases, show that its sign can be controlled by the strength of the exchange field, and prove that its magnitude is large enough to be observed in scanning tunneling microscopy experiments.

II. METHOD
In a Bi(111) bilayer two layers of Bi atoms form a honeycomb lattice when projected onto the plane of the film. The relaxed bulk in-plane lattice constant and the distance between the two layers were 4.52Å and 1.67Å, respectively (see also Fig. 4). Our theoretical investigations are based on density functional theory. 25 We apply the local density approximation 26 to the exchange-correlation potential and use the full-potential linearized augmented plane wave method (FLAPW) as implemented in the FLEUR code. 27 The self-consistent calculations with SOC were carried out with the cut-off parameter k max of 3.8 bohr −1 and 50 k-points in the full two-dimensional Brillouin zone (BZ). The muffin-tin radius of 2.5 bohr was used. The band structure of the system with SOC is presented in Fig. 2(f). The Wannier functions technique was used on top of self-consistent DFT calculations to derive an accurate tight-binding Hamiltonian of the system. [28][29][30] Moreover, we have constructed the matrix elements ψ mk | σ α | ψ nk of the Pauli matrices σ α (α = x, y, z), where ψ nk are the occupied Bloch wave functions. These matrices were evaluated from the DFT calculations and transformed into the real space representation in terms of maximally localized Wannier functions (MLWFs). 30 This allowed us to take into account an exchange field applied perpendicularly to the surface of the bilayer by adding a σ z · B term on top of the original Hamiltonian, and to provide an ability to accurately evaluate spin polarization P α (see caption of Fig. 4 for details). In general, applying an exchange field lifts the spin degeneracy, where spin-up (spin-down) states get higher (lower) in energy spectra, and the difference of the occupation of spin-up and spin down leads to the spin-polarization P α which can be positive (negative) along the direction of exchange fields. In Fig. 2 in which the summation goes over all occupied states and Ω n (k) is the Berry curvature of the n-th band, given by with u nk as the lattice-periodic part of ψ nk with energy eigenvalues ǫ nk ,v ν (ν = x, y) is the velocity operator. To evaluate the spin Hall conductivity, we have replacedv x with the spin-velocity operatorŝ z ν = {v ν , σ z } in Eq. (3). To calculate the spin Chern number, we followed the procedure of Ref. [10]. First, the σ z matrix for the occupied states ψ mk | σ z | ψ nk (m, n are the band indices of occupied states at each k) is constructed and diagonalized. Note that due to the spin-flip part of the spin-orbit coupling 38 , the eigenvalues of σ z matrix are not necessarily ±1. Then, using the "spin-up" or "spin-down" eigenvectors |φ ± of the σ z matrix, the occupied states are projected into the "spin-up" and "spin-down" manifolds ψ ± = φ ± (k) | ψ(k) , where |ψ(k) denotes the eigenvector of the occupied states. At last, the spin Chern number is evaluated by integration of the Berry curvature for each manifold where the derivatives of the wave functions ψ ± with respect to k are obtained using the finite difference methods. 33 For the spin Chern number, it is only well defined if the spectrum of σ z is gapped. 10 In our calculations, we observed that in the range of exchange field considered (0 eV ≤ B ≤ 1 eV) the spectrum of σ z remains finite which justifies the usage of spin Chern numbers in this work.

III. CHERN NUMBERS AND THE PHASE DIAGRAM
Here, we calculate the Chern numbers and analyze the topological phases of Bi(111) bilayer as a function of the externally added exchange field B, presenting the phase diagram, calculated band gap and AHC in Fig. 1. When B = 0, the system is a 2D TI with Z 2 number of 1 (nontrivial). The Chern numbers for the spin-up (minority) and spin-down (majority) manifolds are C ± = ∓1, respectively, leading to a spin Chern number (cf. Eq. (1)) in agreement with previous calculations. 22 In this case, the spin Chern number is equivalent to the Z 2 number. Introducing and increasing the exchange field breaks the T -symmetry and causes an exchange splitting between the spin-up and spin-down valence and conduction bands, bringing thus the minority valence bands towards majority conduction bands, 32 Fig. 2(b). Further increase of the exchange field leads to a closure of the gap at B = 0.31 eV, Fig. 2(c). As evident from Fig. 2(a)-(c), until the bulk gap is closed, the original inverted band structure at the Γ-point remains topologically nontrivial. This is confirmed by calculating the spin Chern number C ± = ∓1 for 0 ≤ B ≤ 0.31 eV, identifying the topological phase of Bi(111) bilayer in this range of the exchange field as the T -broken, or, TRB TI phase (c.f. Fig. 1), analogous to that considered in graphene. 9 To verify that the T -broken phase could display non-trivial topological transport properties, we calculate the transverse spin Hall conductivity (SHC) as a function of the spin-orbit strength in the system, λ, and present the results in Fig. 3. Consider first the bilayer at B = 0. At small SOC strength the system is in a trivial insulator phase, which can be characterized with a zero SHC. Upon increasing λ, the bilayer first goes through a metallic phase, and at around λ = 0.7 of the SOC strength of Bi atoms a transition to the TI phase with SOC-driven band inversion occurs, which is accompanied by a nonzero SHC of the magnitude of around −0.7 e 4π . 22 The deviation from the quantized value of − e 4π is due to the nonvanishing spin-non-conserving part of the spin-orbit interaction: artificially switching it off in our calculations 38 we indeed acquire a quantized value of − e 4π for the SHC (dashed line in Fig. 3 Hall heterostructures. [34][35][36] . By looking at the SHC for the Bi bilayer at B = 0.15 eV, we observe an almost identical behavior to that without the field, with only modifications in the boundaries of the metallic region, SHC in the metallic regime, and a slight difference in the value of the SHC at large λ. Again, by neglecting the spin-non-conserving part of SOC, we restore a quantized value of − e 4π for the SHC in the TRB TI phase, which clearly manifests a topologically non-trivial phase, as far as the topological transport is concerned. Note that for both phases, TI and TRB TI, the transverse charge conductivity in exactly zero when λ lies outside of the metallic region. The ability to characterize the Tbroken TI phase with observable in an experiment nonzero spin Hall conductivity could be particularly important for distinguishing this phase from a trivial insulator, since, as we shall see in the next section, the topological protection of the edge states in the system for B = 0 is lifted. Returning now to the phase diagram in Fig. 1, we observe that the size of the band gap decreases linearly as the exchange field is increased, and for B in between 0.31 and 0.42 eV the bulk gap is closed with Bi(111) bilayer in the metallic phase, see Fig. 1. However, increasing the exchange field further reopens the gap, which results in the occurrence of a topologically non-trivial phase. For B ≥ 0.42 eV the spin Chern number of the occupied majority states reverses sign, leading to a zero spin Chern number C s , while the Chern number C acquires a value of −2. That is, in this range of the exchange field the system is in the QAH, or, Chern insulator phase. It is worth to recall here that in case of a metal in which the spinnon-conserving part of the spin-orbit interaction is hy-pothetically switched off, 38 the intrinsic anomalous Hall conductivity is given by the sum of the Hall conductivities for spin-up and spin-down bands, while the spin Hall conductivity is given by their difference (spin is a good quantum number in such a case). 37,38 Evidently, such a correspondence between the Chern number and the spin Chern number, i.e., their representation as the sum and the difference of the Chern numbers for spin up and spin down bands, holds true for Bi(111) bilayer with gapped spectrum of σ z , based on the argument that a unitary transformation (eigenvectors of σ z in our case) of the occupied states will keep the Chern number invariant. 39 Examining the electronic structure of the system, presented in Fig. 2, reveals that it is the spin-mixing accompanied by the exchange of the Chern number between valence and conduction bands that leads to the phase transition from the T -conserved TI phase to the QAH phase through the T -broken TI and metallic phases. The occurrence of the metallic phase has not been predicted previously e.g. for graphene 9 or Bi 2 Se 3 . 15 It emerges due to an overlap in energy of bands with opposite spin character in the vicinity of the Γ-point (cf. Fig. 2(c)), which is characteristic of a material with a non-monotonous dispersion of the bands on both sides of the Γ-point. Such a metallic phase cannot be achieved e.g. for pure Dirac bands, which have a local maximum (minimum) at the high symmetry (in this case) Γ-point, as is evident from Fig. 2(c).
Our transport calculations according to Eq. (3) show that a finite AHC develops in the metallic and QAH phases, see Fig. 2. Microscopically, the non-vanishing AHC can be traced back to the development of the singular Berry curvature in k space as the B-field is increased, see Fig. 2(g)-(j). For B below 0.31 eV the Berry curvature has a non-trivial distribution in the BZ with the regions of positive and negative values, comparatively small in magnitude ( Fig. 2(g)), while the BZ integral of the Berry curvature amounts to zero.
At the onset of the metallic phase there is a noticeable hybridization between the conduction and valence bands of opposite spin, which gives rise to a very large negative contribution to the Berry curvature around the points in the BZ at the Fermi level (E F ) where the orbital character and the spin character of the valence and conduction bands is changing, or, in terminology of Ref. [38], along a "hot loop" in the BZ, Fig. 2(c)-(e) and (h)-(j). In such a situation, the Berry curvature and the AHC, proportional to the integrated value of the Berry curvature over the BZ, are dominated by the spin-flip transitions mediated by the non-spin-conserving part of the spin-orbit interaction. 38 At the Γ-point, the sign of the Berry curvature reverses for B > 0.3 eV, accompanying the switch in parity of valence and conduction bands.
In the metallic phase, the AHC as a function of B is monotonously increasing (Fig. 1), while the radius of the hot loop is increasing owing to the shift of the hybridization point between the valence and conductance bands away from Γ (Fig. 2). When the gap is reopened for B > 0.42 eV, the AHC reaches a value of −2 e 2 h , with the Chern number C = −2, see Fig. 2. In the QAH phase, we observe from our calculations that out of six valence bands, the topmost two bands dominate the contribution to the AHC for exchange fields up to 0.60 eV. When the exchange field is increased further, deeper spin-up bands will be pushed closer to the Fermi energy and take the spin-down character derived from the conduction bands (cf. Fig. 1(e)), also providing contribution to the Berry curvature.

IV. EDGE STATES
We turn now to the properties of the edge states in Bi(111) bilayer zig-zag ribbon with the width of 80 atoms in y direction and periodic (infinite) in x direction. 21 The tight-binding Hamiltonian of the ribbon was constructed in terms of the MLWFs by a mere termination of the 2D (infinite) Hamiltonian after 40 unit cells in the y direction. For Bi(111) ribbon, when T symmetry is not broken, there are twelve edge states in total, as evident from Fig. 2(k). Six of them are located on each side of the ribbon, while out of these six there are three "spin-up" leftmovers and three "spin-down" right-movers. The axis of spin polarization of the edge states is determined by the details of the electronic structure and spin-orbit interaction, and can be also energy-dependent. 22 Applying the exchange field to the ribbon breaks the T -symmetry, lifts the spin degeneracy and polarizes the edge states along the direction of the applied field, see Fig. 2(l)-(o). A close inspection of the edge bands in the case of the TRB TI phase, Fig. 2(l) reveals that the edge states do not connect the valence and conduction bands in the system, as in the case of the edge states of a conventional TI, c.f. Fig. 2(k). This means that the Hamiltonian of the system can be transformed such that the spectrum at the edges exhibits a gap at the Fermi energy. Moreover, owing to the breaking of the T -symmetry, back-scattering is in general allowed which results in the dissipative transport properties of the edge states. 9 Noticeably, in the QAHE phase, Fig. 2(n,o) the connection of the valence and conduction bands of the bilayer via the edge states is restored again, resulting in a dissipationless longitudinal transport of the ribbon, realized by the topologically protected edge states.
Of particular interest here is the spatial distribution of the spin-polarization of the edge states around the Fermi energy, P z , which plays a very important role in spin-polarized scanning-tunneling microscopy (SP-STM) experiments. 43 We calculate this spin polarization as a function of the B-field in real space across the ribbon, and present the results in Fig. 4. Let us consider first the case of a very small exchange field B = 0.05 eV and look at the spin-polarization at E F across the ribbon (Fig. 4). We observe that the spin polarization is: (i) rather localized at the edges of the ribbon, (ii) reaches very large values, (iii) displays oscillations in sign across Atom no. the ribbon. These properties can be understood by referring to the electronic structure of the edge states for this value of the field, which is quite similar to that depicted in Fig. 2(l), only with the smaller separation between the upper (predominantly spin-up) and lower (predominantly spin-down) subgroups of two edge states, so that there are twelve states at E F all together. The direction and spin character of the edge states at E F is sketched in Fig. 4. We observe that owing to the presence of the non-zero exchange field and breaking of T symmetry, the edge-states get spin-polarized, while on each side of the ribbon the number of right and left movers is equal for B < 0.42. This results in zero transverse charge conductance, in accordance to the phase diagram, Fig. 1. For larger wavevectors |k x · a| ≈ π, see Fig. 2(l), the upper group of two states has a strong spin-up character, and for B = 0.05 eV (not shown) it crosses the Fermi energy, giving rise to a strong spin-polarization in the direction of the B-field. Since the states with larger k x (away from the bulk bands) are more localized at the edges of the ribbon than the states with smaller k x (closer to the bulk bands), 21 positive P z in Fig. 4 for this field is localized at the ribbon edges. Correspondingly, the states at E F close to Γ are the source of negative spin polarization, which is smeared more across the ribbon owing to stronger hybridization with the bulk states, thus giving rise to the change in the sign of P z away from the ribbon edges. We speculate that such oscillatory spatial dependence of P z should be a common phenomenon in a situation where the electronic structure of the edge states of a 2D TI exposed to an exchange field is non-trivial. The spatial distribution of the P z in the unit cell for this case is illustrated in the inset of Fig. 4.
Increasing the exchange field further pushes the tail of long wave length spin-up edge states out of the bulk gap completely around B = 0.15 eV, Fig. 2(l). In this case, there are only 8 edge states left, whose direction of the spin-polarization at the Fermi energy is opposite to the direction of the B-field. Since the Bi(111) bilayer exhibits a T -broken TI phase for this value of B, there are two left and two right movers on each side of the ribbon, with zero transported charge, Fig. 4. In correspondence to the predominantly spin-down character of the edge states in k space, the P z is large and negative in the vicinity of the edge of the ribbon. Noticeably, the spin polarization decays much slower with the distance towards the center of the ribbon, when compared to the case considered previously, owing to the proximity of the edge states in k space to the bulk states. We remark here that the predicted change in sign of P z is an interesting phenomenon, which can be exploited experimentally e.g. via tuning the strength of the exchange field by deposition of different adatoms 15,42 (see also comments in section V).
The evolution of the edge states for 0.15 ≤ B ≤ 0.80 eV can be seen in Fig. 2(l)-(o). As the exchange field is increased, the magnitude of P z gradually decreases, since the conduction and valence bands become very close in energy, and the k F vectors of the edge states approach the Γ point. This is particularly clear for the edge states in the QAH phase at B = 0.8 eV, for which P z and its decay rate into the ribbon are very small, Fig. 4. For the regime of B in between 0.42 and 0.7 eV, i.e., at the onset of the QAH phase, the number of edge states is the same as that in the T -broken TI phase at B = 0.15 eV, but the emergent quantized charge conductance of the edge states proportional to C on each side of the ribbon is evident, Fig. 4. When B > 0.7 eV four more edge states are pushed out of the bulk gap window, and only four edge states crossing E F are left overall. In this case, we have a situation of two right/left movers on each side of the ribbon, with a small positive spin polarization, Fig. 4. The microscopic mechanism for spin-polarized electron conduction in these edge states without possibility to back-scatter remains and open and interesting topic for future studies.

V. SUMMARY
In this work, based on first principles calculations, we have considered the emerging topological phases of Bi(111) bilayer in an external exchange field B. We identify four different phases as the exchange field is varied: TI phase for B = 0, T -broken TI phase for small fields, metallic phase for intermediate fields, and Chern insulator, or, QAH phase for large exchange fields. We consistently identify each of the insulating phases in terms of the Chern number C, spin Chern number C s , spin and charge transverse conductivity. We attribute the origin of phase transitions to the spin-orbit mediated spin-mixing between valence and conduction bands whose topology in k space is controlled by the applied field.
In particular, we focus on the electronic structure and development of the edge states in a zigzag ribbon of Bi(111) bilayer as a function of the B-field. We show that the spin polarization of the edge states along the direction of the exchange field at the Fermi level can be significant in magnitude, and its sign, as well as its spatial distribution can be controlled by changing the magnitude of the applied field. Since reversing the direction of the exchange field leads to reversed propagation direction and reversed spin polarization of the edge states, the exchange field provides a tool to tune the properties of one-dimensional spin-polarized transport arising at the edges of insulators with non-trivial topological properties. We also speculate that the spin polarization of the edge states, which characterizes different topological phases, can be observed in e.g. scanning tunneling microscopy experiments.
We remark that the experimental realization of the tunable exchange field applied to a thin Bi(111) bilayer can be sought in two directions. One way lies in finding a suitable insulating substrate for deposition of Bi(111) which exhibits essential magnetism at least at the interface with the bilayer. In this case the direction of the Bfield can be easily (depending on the magnetocrystalline anisotropy energy) reversed, while the magnitude of the exchange field can be tuned via e.g. the magnetoelectic effect if the substrate used for deposition has multiferroic properties. Second route lies in deposition of transition metal adatoms on the Bi(111) surface. Recently, it has been demonstrated theoretically that such adatom deposition can induce considerable exchange splitting with magnitude as large as several tenth of eV on the topological insulator underneath, 42 and for Bi 2 Se 3 , the exchange field is large enough to get the system into the QAH phase. 15 . The magnitude of the exchange field in this case can be controlled by an appropriate choice of the transition metal, or by deposition of the system on an insulating non-magnetic substrate, which can cause essential modifications in the distance between the TI and deposited adatoms. We also note, that, generally speaking, the magnitude of the exchange field necessary to achieve the QAH phase in the phase diagram of Fig. 1 depends directly on the initial value of the gap in the TI phase without the field.

VI. ACKNOWLEDGMENTS
We acknowledge very helpful discussions with Emil Prodan, Sheng Li, Klaus Koepernik and Marjana Ležaić. This work was supported by the HGF-YIG Programme VH-NG-513 and by the DFG through Research Unit 912 and grant HE3292/7-1. Computational resources were provided by the Jülich Supercomputing Centre.