Abstract
To date, spin generation in three-dimensional topological insulators is primarily modelled as a single-surface phenomenon, attributed to the momentum-spin locking on each individual surface. In this article, we propose a mechanism of spin generation where the role of the insulating yet topologically non-trivial bulk becomes explicit: an external electric field creates a transverse pure spin current through the bulk of a three-dimensional topological insulator, which transports spins between the top and bottom surfaces. Under sufficiently high surface disorder, the spin relaxation time can be extended via the Dyakonov–Perel mechanism. Consequently, both the spin generation efficiency and surface conductivity are largely enhanced. Numerical simulation confirms that this spin generation mechanism originates from the unique topological connection of the top and bottom surfaces and is absent in other two-dimensional systems such as graphene, even though they possess a similar Dirac cone-type dispersion.
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Introduction
Topological insulators (TIs) have attracted world-wide attention because of their intriguing fundamental physics and exciting application opportunities in spintronics1. Three-dimensional (3D) TIs2,3 are of particular technological importance as the unique spin generation can be realized in single crystals rather than in complex heterogeneous structures4. TIs are considered as efficient spin generators5, yet the spin generation is generally regarded as a pure surface phenomenon. Namely, the electronic momentum and spin are locked at the TI surface, and a net charge current leads to a net spin polarization at the surface, whose magnitude is directly proportional to the charge current6. In this view, all physics occur independently at the top and bottom surfaces of a TI and the role of the bulk is passive, which simply separates the top and bottom surfaces. The surface conductivity is understood through density of states and scattering rate, just like in other two-dimensional (2D) systems such as graphene and 2D electron gas. The conductivity behaviour governs the spin generation on the surface of a 3D TI, and spin accumulation is merely a side product of conductivity. Although this interpretation of spin generation in TIs is most mathematically straightforward, it is far from satisfactory in the sense that the most amazing feature of a TI—surface-bulk correspondence does not explicitly enter this physical picture.
On the other hand, there is an alternative viewpoint of spin generation. The external electric field induces a transverse pure spin current through the bulk, which acts as a bridge for transporting spins between top and bottom surfaces. Opposite spins are thus accumulated on the two surfaces, which lead to charge current in the same direction of the electric field because of the opposite chirality of the momentum-spin textures on the top and bottom surfaces (Fig. 1a). An empirical formula for the bulk spin current can be written down as
(a) An electric field induces a transverse pure spin current in the bulk. Consequently, opposite spins accumulate on the top and bottom surfaces, leading to a charge current according to the chiral momentum-spin texture. The small cylindrical arrows denote spins. The hollow vertical arrows indicate spin current. The long horizontal blue arrows indicate charge current. (b) The anomalous behaviour of transport coefficients proposed in this article. At a sufficiently high disorder level, conductivity σxx, electro-spin susceptibility κyx and spin relaxation time τs should all have positive dependence on the disorder, in contrast with the well-known negative dependence in the low disorder limit.
where js is the spin current density, E is the electric field and σs is the spin Hall conductivity tensor7. A system that is electrically insulating but can carry a pure spin current is termed a spin Hall insulator8. The bulk of a 3D TI has been demonstrated to be a spin Hall insulator because of its Z2 topological order9.
Analogous to Hall effect, the transverse spin Hall current leads to surface spin accumulation in a slab geometry. Yet unlike electric charge, spin is usually a nonconserved quantity in a spin Hall insulator. The ultimate spin accumulation induced on the surface closely depends on the spin relaxation mechanism. In the low disorder limit 
with μ being the Fermi level, 
being the momentum relaxation time, it has been demonstrated that the spin relaxation time 
on the surface of a 3D TI is identical to the momentum relaxation time 
because of the momentum-spin locking, and the traditional Dyakonov–Perel spin relaxation is absent13. (d) The DC conductivity of a single surface (σxx(0)) plotted against the Fermi level position (μ). The impurity concentration c varies from 0.002 to 0.2. (e) Conductivity (σxx(0)) plotted against the impurity concentration (c). The Fermi level position was fixed at μ=0.13 eV.
where αi(i=x, y, z), β are the Dirac matrices, a and az are the lattice constants in the xy and z directions, Δ is the mass term and A, Az, B, Bz are nearest-neighbour hop** amplitudes. In the slab configuration, inverse Fourier transform is performed in z-direction to comply with the finite thickness.
We use a typical 3D TI Bi2Se3 as our prototype and adopt parameters as obtained in ref. 2 to best fit the band structure of Bi2Se3. The resultant band structure of the surface is shown in Fig. 2b, which clearly has a Dirac cone near the Γ point. Owing to the z-inversion symmetry of the slab, all bands are doubly degenerate.
To account for surface disorder, atoms in the top and bottom layers of the slab are subject to a typical kind of impurity—vacancies. Each site at the surface has a probability of c to be occupied by a vacancy where the on-site energy is brought to infinity so as to forbid electrons from this site. As c may not be small, the first Born approximation does not apply. Here we adopted the coherent potential approximation (CPA) method for binary alloys14,15 in computing the Green’s function G(k, ω) and self-energy Σ(ω). A typical spectral function −(1/π)ImG(k, ω) obtained by CPA at impurity concentration c=0.001 is plotted in Fig. 2c. The evolution of the spectral function with increasing impurity concentration is consistent with results obtained in ref. 13. Subsequently, transport coefficients were calculated via the standard linear response theory. More details of the numerical simulation can be found in the Methods section.
Figure 2c,d shows the electrical conductivity calculated from the CPA Green’s function via the Kubo–Greenwood formalism. With an impurity concentration c ranging from 5 × 10−4 to 0.5, the Fermi level dependence of conductivity gets weaker and the magnitude of conductivity reaches a minimum at around c=0.006. Further increasing the impurity concentration leads to an increase of conductivity at a given Fermi level position. Such anomalous increase of conductivity with impurity concentration is difficult to understand based on a single surface model16,17, which suggests the essential role of the bulk of a 3D TI in surface conduction. In the following, we reveal that the anomalous increase of conductivity is a signature of a different type of spin dynamics and manifests a spin generation mechanism in 3D TIs.
Spin generation via bulk spin current
To start discussions on spin dynamics, we notice that spin is not a predefined quantity in Hamiltonian (equation 2). Although common 3D TIs such as Bi2Se3 are known to have chiral spin texture on the surface states, the spin polarization is not 100% (ref. 18). Nevertheless, one can always talk about a pseudo-spin, which is defined to exactly match the energy eigenstates and has all essential features of the real spin19. Here we take the definition
It can be verified that such definition satisfies all symmetry requirements of the real spin and shows a chiral spin texture near the Γ point, as shown in Fig. 3a. Note that a unique spin polarization can be specified for all points in the k-space except for those time-reversal invariant momenta (TRIM), where the Kramers theorem asserts the degeneracy of the two opposite spin polarizations.
(a) The chiral spin texture over the entire BZ for the conduction band of the top surface. The spin orientations on TRIM points (red dots) are degenerate due to the Kramers theorem. (b) The DC spin Hall conductivity 
obtained from numerical simulation plotted against the Fermi level position μ at different impurity concentrations c. It is seen that the bulk spin current is independent of both Fermi level position and impurity concentration. (c) Energy dispersion for the top surface near the Γ point along Γ−X direction. The spin polarization of the coloured dots is in the + y direction. (d) The evolution of electronic wave functions along the coloured dots in c. As the magnitude of wave vector becomes larger, the wave function of an electron gradually evolves from being localized near the surface to extensive in the bulk. (e) A schematic plot for the spin generation mechanism in the extended BZ view. The drift motion along x direction in k space gives rise to spin transfer in the z direction of the real space, which results in a pure spin current through the bulk. The red/green arrows pointing into/out of the page indicate spin polarization. The purple arrows indicate the direction of drift motion of electrons under an electric field in the +x direction. The dashed horizontal line indicates the Fermi level position. The schematic drawing under the black dots denote the electronic wave functions. The dashed boxes denote the true surface state regions. It is essential that the Fermi level lies within the box regions for the spin transfer mechanism to apply.
With the above definition of spins, the bulk spin current density 
denoting the transport of y-spins in the z-direction can be defined as
where 
and 
are spin projection operators in the +y and −y directions, vz is the velocity operator in the z direction and Ω3 is the volume of the slab serving as a normalization factor. With the above expression, the spin Hall conductivity 
defined by
can be calculated via the standard linear response theory. Owing to the even symmetry of spin current js under time reversal 
, the resultant expression for the spin Hall conductivity is different from that for the electrical conductivity (Kubo–Greenwood formula), but contains a term which involves all states below the Fermi level. This term has been thoroughly reviewed in ref. 20 for the calculation of electrical conductivity in a 
-symmetry broken system and also discussed in a recently published article21 for the calculation of spin Hall conductivity. The emergence of this term in our system indicates the non-dissipative nature of the spin current, which has already been demonstrated possible for a wide class of traditional semiconductors22,23. We leave the details of derivation to the Methods section and plot the calculated spin Hall conductivity 
in Fig. 3b. It is clear that the magnitude of the spin Hall conductivity is independent of both the Fermi level position (must be within the bulk bandgap) and the surface impurity concentration.
Although the existence of a bulk spin current in 3D TI has been predicted analytically through topological argument9, a visualization of the spin transfer mechanism is not yet available so far. Neither has its relevance to the transport behaviour of the gapless surface states been studied ever. In the following, we present an intuitive picture of the spin transfer in a 3D TI slab and uncover its close relationship with the surface-bulk correspondence of a 3D TI.
We notice that despite the chiral spin texture over the entire Brillouin zone (BZ), only states with small magnitude of momentum are truly localized on the surface. Figure 3c,d shows the evolution of electronic wave functions as the wave vector k approaches the BZ boundary from the Γ point. It is seen that, beyond a certain point, electronic wave functions become extended through the entire bulk and the surface band has essentially merged into bulk bands. States beyond this merging point should be classified as bulk states although they lie on the same branch of energy sub-band as true surface states.
Imagine applying a weak electric field to this system in the +x direction and examine the Γ−X line in the extended BZ view. Because of the inversion symmetry in the z direction, all bands are doubly degenerate. We notice, however, in order for the spin texture to be continuous, every top surface branch must be connected to the adjacent bottom surface branch and vice versa, as shown in Fig. 3e. This alternating structure exists across all TRIM points in our system, and is distinctively different from a normal band, which smoothly connects to itself at the BZ boundary. Consider an electron on the bottom surface with its spin polarized in +y direction. Under the driving of the electric field, this electronic state drifts to −x direction in k space and merges into the bulk valance band. Upon further drifting, this electron finally enters the top surface with its spin in +y direction unchanged. Simultaneously, an electron with spin polarized in the −y direction will drift from the top surface to bottom surface. The drift motion across the X point is similar to the Klein tunnelling of Dirac Fermions in the sense that, in order for a certain spin to be continuous, the electron must tunnel to another band rather than return to its original band. Overall, each of these processes corresponds to a unit spin-pair exchange between the bottom surface and the top surface. Thus, a longitudinal electric field induces a transverse pure spin current through the bulk, which plays the role of a spin injector for the two surfaces, as described in Fig. 1a. During this process, it is essential that the Fermi level lies within the gap, because there exists another pair of merging points near the conduction band edge. If the Fermi level is above these points as well, there would be an opposite process that leads to the cancelation of net spin current. This is of course consistent because the system in this case is not a TI any more.
Spin relaxation on the surface
Unlike charge, spin is not a conserved quantity in our system. The spins injected onto the surface suffer from immediate relaxation. The scenario is slightly different from both the Hall effect and the 2D spin Hall effect. In 3D, the spin relaxation is actually necessary for the system to reach a steady state, as detailed in Supplementary Note 1. The ultimate spin density accumulated on the surface is determined by the spin relaxation time 
. In the following, we provide an intuitive physical picture of the spin relaxation dynamics under the eigenbasis defined by H0. More details of this picture can be found in Supplementary Note 2. This is essentially an interaction picture that splits the Hamiltonian into a free part H0 and an interaction part U. Because of the momentum-spin locking, each electron senses an effective magnetic field Beff according to its wave vector k. When an electron with its spin aligned with Beff suffers from a momentum change ħΔk because of the scattering of an impurity potential, its spin may no longer align with the new Beff. If scatterings are rare, that is, the time it takes for momentum to change by a unit amount is long, the adiabatic perturbation theory predicts that the new spin must evolve to the new energy eigenstate, that is, rotate to the direction of the new Beff. If scatterings are frequent, however, the spin does not have time to follow Beff and will precess about the instantaneous Beff, as shown in Fig. 4a. Frequent scatterings constantly change the precession axis and the spin ends up doing a random walk on a unit sphere, as shown in Fig. 4b,c. The more frequent momentum scattering is, the less effective the random walk is, and the spin will preserve its original direction for a longer time. Therefore, the spin relaxation time 
inversely depends on the momentum relaxation time τ, just like in the traditional Dyakonov–Perel mechanism24. One point to note is that as disorder is only present on the surface, states outside the dashed box of Fig. 3e are unaffected by scattering and the previously discussed spin transfer mechanism remains valid even under strong surface disorder.
(a) When scatterings are frequent, an electron suffers from rapid momentum change where the electronic spin does not have time to follow the instantaneous energy eigenstate given by adiabatic perturbation theory. Instead, the electronic spin precesses about the instantaneous energy eigenstate, which serves as an effective magnetic field Beff. (b) Frequent scatterings constantly change the precession axis of spin. During the interval of two consecutive scatterings the spin can only precess for a small angle. (c) The spin ends up doing a random walk on a unit sphere. The more frequent the scatterings are, the less efficient this random walk is, and consequently spin can preserve its original direction for a longer time. (d) The numerical simulation result for spin relaxation time 
and momentum relaxation time 
at Fermi level μ=0.13 eV. As expected, when disorder is high, 
and 
have an inverse dependence as in the traditional Dyakonov–Perel spin relaxation mechanism. (e) The simulated DC electro-spin susceptibility κyx(0) plotted against the Fermi level μ at different impurity concentrations. The inset is κyx(0) at a fixed Fermi level μ=0.13 eV versus impurity concentration c. It is clear that under high disorder, the accumulated spin density increases with impurity concentration.
If the above physical picture is correct, the ultimate spin density accumulated on the surface should increase with the increase of disorder. We calculated the spin relaxation time 
and electro-spin susceptibility κyx via standard linear response theory. The results are shown in Fig. 4d,e, which perfectly agree with the expectation. Combined with the fact that velocity operator is proportional to spin on the surface, it is not difficult to understand the anomalous increase of conductivity as well.
. Different from the 3D case, in the edge channel of a 2D TI, the spin (understood as pseudo-spin when necessary) is a conserved quantity, which does not relax. Consequently, the electro-spin susceptibility is infinite, which means no external field is needed to support the edge spin accumulation and charge current. The quantized and finite channel conductivity e2/h is actually a contact effect, while the channel itself is dissipationless
takes the form
. (ii) The components of S satisfy the anti-commutation rules, 
. (iii) S is a pseudo-vector. It transforms like a vector under in-plane (xy) rotation but does not flip sign under space inversion, βSiβ=Si. (4) To comply with the chiral surface spin texture, we require S be polarized along y-direction when ky=0. Thus, [Sy, H(kx, 0)]=0.
contains a factor of 1/Ω and thus the above expression is actually independent of the box size Ω.
, the above expression is independent of the box size Ω.
