The role of spin-orbit coupling in topologically protected interface states in Dirac materials

We highlight the fact that two-dimensional materials with Dirac-like low energy band structures and spin-orbit coupling will produce linearly dispersing topologically protected Jackiw-Rebbi modes at interfaces where the Dirac mass changes sign. These modes may support persistent spin or valley currents parallel to the interface, and the exact arrangement of such topologically protected currents depends crucially on the details of the spin-orbit coupling in the material. As examples, we discuss buckled two-dimensional hexagonal lattices such as silicene or germanene, and transition metal dichalcogenides such as MoS$_2$.

We demonstrate the general principle that inversion of the band mass in Dirac-like condensed matter systems will drive the formation of topologically protected interface modes which may carry spin or valley current. As examples, we discuss buckled 2D hexagonal lattices such as silicene or germanene, and transition metal dichalcogenides such as MoS2. The common features among these systems in the bulk are a Dirac-like low energy effective Hamiltonian, a mass gap, and spin-orbit coupling. If spatial variation of the mass gap can be controlled to the extent that the sign of the mass is reversed, the bulk states on each side of the transition will carry different topological invariants which implies that linearly dispersing interfaces states must be present near the region where the mass is zero. The details of the spin-orbit coupling will then dictate the nature of the resulting topologically protected edge currents.
A major issue in contemporary condensed matter physics is the attempt to produce and control currents which are polarized in one or more of several spin-like degrees of freedom, and which can be used for switching and possibly quantum computing applications. Initially, the real electron spin was proposed for this, leading to the moniker "spintronics" [1] and many advances in commercially relevant technologies. Recently, the isolation of (2D) hexagonal crystals with their two inequivalent valleys and two inequivalent lattice sites have lead to the suggestion of using these degrees of freedom in an analogous way in so called "valleytronics" [2] and "pseudospintronics" [3] applications. In general, one way to create polarization in one of these spin-like degrees of freedom is to break a symmetry of a system such that the degeneracy of the two spin-like flavors are lifted and transport is only permitted for one flavor, for example using the giant magneto-resistance effect [4]. Another way is to utilize differences in how each flavor behaves under an external perturbation [5]. A third way is to utilize the topological properties of materials in order to engineer spatially localized states which can be manipulated via the topology of the underlying material. One such example of this is the 2D surface state which forms at the interface between bulk Bi 2 Se 3 crystals and the vacuum [6,7], where the change in the Z 2 invariant from the crystal to the vacuum implies that there must be a local closing of the band gap and associated interface mode. These interface modes are protected against disorder because the change in the topological nature of the material on either side of the interface is only permitted if metallic surface states are present.
In this Letter, we propose a localized and controllable version of the phenomenon of current polarization enforced by topological change in Dirac Materials [8], but in our system a single piece of crystalline material is used and the topological change may be modified dynamically by external gating. The system we consider is a hexagonal 2D lattice where the electrons near the Fermi surface are best described as massive Dirac fermions. We shall demonstrate that in the generic case where there is a band gap at the Fermi energy and a finite quasiparticle mass, arranging a system such that there is an interface where the Dirac mass is positive on one side and negative on the other will ensure that topologically protected modes are present at the interface. Further, we shall show that these interface modes may carry intrinsic valley or spin current which is dispersionless and topologically protected. These currents can, in principle, be utilized for electronic control and switching applications, and may lead to significant efficiency improvements over conventional charge switching technology. We shall then give two examples of 2D hexagonal crystals where this effect may be manifested.
To illustrate the general point of topological modes confined by mass inversion, the Hamiltonian for the bulk of a generic hexagonal lattice can be written for a single valley and spin as where ∆ is the Dirac mass,k x = −i∂ x andk y = −i∂ y are the wave vector operators, v is the Fermi velocity, and τ x,y,z are Pauli matrices in the sublattice space. We begin by demonstrating that changing the sign of the band mass ∆ → −∆ in the bulk will alter the topological properties of this toy system. The topological quantum number associated with a Hamiltonian such as Eq. (1) is the Chern number C(∆) [9]. It may be expressed as an integral of the Berry curvature over the Brillouin zone and for a system described by , the Chern number can be calculated from [10,11] In this equation, we have split the integration into two contributions. The first, C (1) , comes from wave vectors near to the gapless point, and the second, C (2) , from the rest of the Brillouin zone where we assume there are no further gap closings. A cut-off Λ distinguishes these two regions,d = d/ |d|, and ǫ is the three-dimensional Levi-Civita symbol. We examine the change in the Chern number δC when the Dirac mass is inverted by calculating δC = C(∆) − C(−∆) so that a finite value of δC indicates a change in the system's topological properties. We know that the region of k-space near the gapless point is the only place where the conduction and valence bands come near to each other, and that the gapped part of a Hamiltonian cannot introduce a change in the Chern number. Hence, δC may be computed solely from the low-momentum part of Eq. (2). Hence, and, in the limit ∆ → 0 and Λ ≫ ∆/(hv), we obtain Therefore, since there is a change in the Chern number when the sign of ∆ is reversed, the boundary between regions of a system with ∆ > 0 and ∆ < 0 hosts a topologically protected interface state. We now demonstrate the localization of a linear "interface mode" in the region where the band mass changes sign. We allow the mass term to become inhomogenous, such that it is constant in the y direction, but has linear slope in the x direction with ∆(0) = 0 then ∆(r) = ∆x/R so that 1/R characterises the gradient of the change in the mass. The spectrum of this system is found by rotating the Hamiltonian with the unitary operator U = e iπτx/4 . Sincek y commutes with the Hamiltonian, we replace it with the eigenvalue k y and hence H ′ = −ihv∂ x τ x +hvk y τ z − ∆xτ y /R. The off-diagonal elements can now be written in the form of ladder operators a = α/2(x+∂ x /α) and a † = α/2(x−∂ x /α) with α = ∆/(hvR) which act on the harmonic oscillator functions Φ n as a † Φ n = √ n + 1Φ n+1 , aΦ n = √ nΦ n−1 , and aΦ 0 = 0. We can construct by inspection an eigenvector of the form Ψ = e ikyy Φ 0 0 which has dispersion ε =hvk y . This mode behaves as zero gap semiconductor despite the mass gap in the bulk system, and is topologically protected by the change in the Chern number. The precise details of the manifestation of the topologically protected interface states will depend on the microscopic characteristics of the material in which they are realized, so we give two examples of physical systems which fulfil the requirements outlined above. The first, silicene, has already been discussed in the context of interface modes [12,13], although valley currents were not mentioned in that context. Theoretical proposals for transport through normal-ferromagnetic-normal junctions [14] and gated regions [15] which produce spin or valley currents have been reported, but this is a fundamentally different mechanism from the interface state transport we discuss. For silicene, the τ z term in the Hamiltonian represents the asymmetry between the on-site potential of the two sublattices in the hexagonal crystal. Because silicene exhibits a buckled structure [as illustrated in Fig. 1(a)], applying a transverse electric field adjusts the static potential on the two sublattices in different magnitude, hence modifying the mass. Silicene is known to have a rich phase diagram in the non-interacting regime [13] in which varying the transverse electric field causes the 2D bulk to undergo a phase transition from a quantum anomalous Hall phase to a band insulator phase due to the change in the Dirac mass. In common with all materials that have a 2D hexagonal lattice, silicene exhibits the valley pseudospin in addition to the real electron spin and lattice pseudospin. The low-energy effective Hamiltonian for electrons of spin s = ±1 in valley ξ = ±1 of silicene is [16] H Sil ξs =hv(k x τ x − ξk y τ y ) + ξsλτ z + where v is the Fermi velocity associated with the Dirac spectrum, the Pauli matrices τ x,y,z are in the sublattice space, λ parameterizes the strength of the spin-orbit coupling, lE z is the contribution to the band gap induced by external gating, andk i = −ih∂ xi is the operator for the electron wave vector. The spin-orbit coupling term enters with the τ z Pauli matrix because the warping of the lattice which generates the coupling is opposite on the two sublattices [16]. The inhomogeneous electric field is assumed to define an interface oriented along the y axis at x = 0, and is represented as E z = xE/R. The effective mass of the Dirac fermions is negative when lE z (x) < −ξsλ, and positive otherwise. Therefore, for bands with ξs = 1, there are interface modes confined near x = −λR/(El) and a further two modes with ξs = −1 near x = λR/(El). This is a dif-ference from our toy example discussed above, where all bands were localized near x = 0, and is due to the combination of the intrinsic spin-orbit coupling in silicene and the electric field in creating the total Dirac mass. To find the dispersion of these interface modes, we apply the same series of manipulations as in our toy system. The dispersion of the interface modes in this case is ε Sil ξs = −ξhvk y . We can use a topological characteristic to describe the spectral asymmetry of the Hamiltonian which is enforced by the topological modes. We define the indices [17] where P s and P ξ are, respectively, the projection operators onto spin s and valley ξ. The notation Tr denotes a sum over all spins and valleys. This index counts the excess number of bands with positive energy. As shown in Fig. 2(a), applying this definition to the topological bands of Eq. (4) gives η K = 2sgn(k y ) and η K ′ = −2sgn(k y ) because the electrons in valley K and in valley K ′ have opposite sign of their energy for a given k y . This index indicates that there is a fundamental asymmetry in the valley distribution of the interface modes in silicene. Conversely, the spin characteristic is zero because of the degeneracy of the bands. We can also describe one-dimensional spin or valley currents along the interface in terms of the group velocity of electrons in each band. For a single band, where v ξs = dε Sil ξs /dk y , n ξs (k y ) is the occupation of the state with wave vector k y in the band with indices ξ and s, and k c is a cutoff wave vector of the order of the Brillouin zone size. Then, the total spin and total valley currents including the contributions from all topological bands are where we have adopted the convention that ↑-spin (K valley) electrons moving in the positive y direction give a positive contribution to the total spin (valley) current. In the case of silicene, the K valley electrons are all leftmoving (with v Ks = −v), while the K ′ electrons are all right moving (with v K ′ s = −v), clearly indicating that there is a valley current j ξ = −2v at half filling. The sign of this current can be reversed by inverting the gradient of the electric field. The higher energy bands are quadratic and degenerate in spin and valley so do not contribute to the current. If a finite chemical potential µ is introduced Silicene (a) so that the system is shifted away from half filling, a correction of −2µ/(hk c ) is added to the current. Sincē hvk c ≫ µ by definition, this effect is small indicating that the valley current is robust against realistic changes in the density. Silicene has been experimentally isolated [18], but the disadvantage of this material is that the spin-orbit coupling is relatively small (λ ≈ 4meV). However, if germanium is used to make the buckled hexagonal lattice then this increases by an order of magnitude. For the interface modes we are discussing, stronger spin-orbit coupling has the advantage of allowing a wider intrinsic gap between the bulk bands so that the interface modes are easier to detect.
The second system we present is transition metal dichalcogenides (TMDs) such as MoS 2 . These materials are constructed of a hexagonal bipartite lattice where a sheet of transition metal atoms on the A sublattice are surrounded by two sheets of chalcogenide atoms on the B sublattice, as shown in Fig. 1(b). The spin-orbit coupling comes primarily through the interaction with the heavier transition metal atom [19,20], and so the coupling is asymmetric in the lattice sites, in contrast to the case of silicene. Hence we expect that there may be differences in the manifestation of the topological modes. The low-energy effective Hamiltonian in the bulk is [21] H TMD ξs =hv(ξk x τ x + k y τ y ) + ∆ 2 τ z − ξsλ 2 (τ z − τ 0 ). (7) where the band mass ∆ ≈ 1.6eV is a parameter of the lattice. The band mass is assumed to vary in space, with the same one-dimensional form ∆(x) = ∆ 0 x/R as was taken in the silicene case [22]. The rotation of the Hamiltonian distributes the spin-orbit coupling term over τ y and τ 0 so that the parameter λ enters the dispersion of the interface modes as ε TMD ξs = ξhvk y +ξsλ/2. This indicates that the band spin asymmetry index is η s = 0 for all k y , but that the valley index η ξ is finite for |k y | > λ/(2hv) [see Fig. 2(b)]. This illustrates the fact that the arrangement of the interface modes is different from silicene because of the different nature of the spin-orbit coupling. A finite valley current exists with, j ξ = 2v + 2µ/(hk c ) which can also be reversed by switching the sign of the gradient of the inhomogenous band mass. In contrast with silicene, we also have an overall spin current with j s = −λ/(hk c ) which is set by the magnitude of the spin-orbit coupling. This current does not depend on the location of the chemical potential.
In conclusion, we have demonstrated that mass inversion is a theoretically viable technique for the creation and manipulation of topologically protected modes that may carry pseudo-spin polarized currents. In silicene, the interface modes carry valley current which can be controlled by local gating, and (if a mechanism for band inversion can be found) TMDs such as MoS 2 will exhibit both spin and valley currents at interfaces. The direction of these currents can be manipulated by reversing the sign of the gradient of the mass inhomogeneity. The topological protection we discuss is robust so long as there is no inter-valley scattering. This is because the derivation of Eq. (3) assumes that there is only one gapless point. However, for realistic systems with hexagonal lattice structure, there is a gapless point at each of the six Brillouin zone corners (i.e. one gapless point in each valley). We note that if the valleys are connected, the assumptions inherent in Eq. (3) are not satisfied. Thus, short-range scatterers such as lattice defects will be severely detrimental to the existence of interface modes.