Spin Filtering via Resonant Reflection of Relativistic Surface States

A microscopic approach is developed to scattering of surface states from a non-magnetic linear defect at a surface with strong spin-orbit interaction. Spin-selective reflection resonances in scattering of Rashba-split surface states by an atomic stripe are theoretically discovered in a proof-of-principle calculation for a model crystal potential. Spin-filtering properties of such linear defects are analyzed within an envelope-function formalism for a perturbed surface based on the Rashba Hamiltonian. The continuous Rashba model is found to be in full accord with the microscopic theory, which reveals the essential physics behind the scattering resonance. The spin-dependent reflection suggests a novel mechanism to manipulate spins on the nanoscale.

Scattering of spin-orbit coupled electrons by extended defects arises in many spintronics-related phenomena, such as spin transport, accumulation, and filtering, which underlie the manipulation of spin currents in spin-based devices [1,2]. Furthermore, a detailed understanding of reflection and transmission of relativistic electrons is important for the unambiguous interpretation of scanning tunneling spectroscopy of spin-orbit split surface states [3,4]. Similar problem arises in ballistic transport through interfaces where powerful ab initio methods exist for scattering of bulk electrons from surfaces, such as multiple scattering [5], embedded Green-function [6], or Bloch-waves formalism [7]. These methods are, however, not directly applicable to scattering of surface states because of a complicated structure of the incident and reflected waves in the asymptotic (unperturbed) region. Therefore, scattering of surface states has been considered either within a tight-binding scheme [8] or within a k · p theory combined with continuity conditions for the envelope function [9][10][11][12] (see also the application to spin dependent transport in nanowires [13,14]). However, in the k · p method the smoothness of the envelope spinor function generally conflicts with current conservation [15], which leads to a qualitatively incorrect separation of the probability current into the spin-orbit and classical-momentum contributions [16]. The tightbinding formalism, on the other hand, is not well suited for free-electron-like motion along the surface. This calls for a more universal approach to scattering of twodimensional (2D) states, which could be formulated in an ab initio framework.
We present a method to microscopically calculate the scattering of spin-orbit split 2D states from a linear (1Dperiodic) defect. "Microscopic" means that the system is defined by the crystal potential V (r), and the wave functions satisfy the Schrödinger equation in real space. Therefore, the method can be straightforwardly transferred to ab initio calculations. Here, we report a proofof-principle calculation of the transmission of Rashbasplit states through atomically thin defects. We study spin-filtering properties of the defects and discover spindependent reflection resonances for certain scatterers.
Previous studies of the effect of spin-orbit coupling on the scattering of 2D states included multibeam spinpolarized reflection from a lateral barrier [17][18][19], spin accumulation at the edges of semi-infinite systems [9,10,20,21], spin selective refraction at an interface of two 2D media [22], spin dependent transmission of electrons incident from a non-relativistic medium through a barrier with spin-orbit coupling [11,23], and a semi-classical reflection from a smooth barrier [24]. The above studies relied on an envelope-function description of the surface states using effective Hamiltonians. By contrast, here, the perturbed surface is treated fully microscopically: the scattering problem is reduced to a supercell band structure problem, which naturally involves both the propagating and all the required evanescent 2D waves and yields a detailed description of scattering, beyond the envelope-function picture. Still, the resonant properties of the scatterer can be related to the parameters of a k·p Rashba model for the perturbed surface. This demonstrates the generality of the phenomenon and suggests a way to its experimental realization.
Typical constant energy contours (CEC) of Rashbasplit states comprise two circles centered atΓ with spin oriented along k × n for the inner circle (of radius R + ) and along −k × n for the outer circle (R − ), where n is the surface normal and k is the 2D Bloch vector. We denote the unperturbed states by | k χ ξ , where χ = ± indicates chirality and ξ = r/t is the propagation direc- x (a.u.) 0th cell 1st tion along x, see Fig 1(a). Consider a defect created by substituting a row of atoms (along y axis) by a different atom, Fig. 1(b). For a surface state | k ± i incident from the left half-plane the scattering solution | Ψ far from the defect contains two transmitted | k ± t and two reflected | k ± r waves, Fig. 1(a). The crystal momentum along y is conserved, so k + ty = k − ty = k + ry = k − ry = k χ iy . For R − > k − iy > R + there is only one transmitted and one reflected wave. In the unperturbed region, | Ψ contains also evanescent waves, and depending on how fast they decay away from the defect the scattering state | Ψ can be obtained from band structure solutions Φ K n for a smaller or larger supercell, Fig. 1(c). The method works as follows: At a given k χ iy , the surface states perturbed by the periodic defect give rise to four (or two) supercell eigenfunctions Φ K n with the supercell crystal momentum K n y = k χ iy , n = 1, 2, 3, 4 (n = 1, 2 if k − iy > R + ), see Fig. 1(d). Far from the scatterers [in the asymptotic region denoted 0th cell in Fig. 1(c)] the functions Φ K n (r) can obviously be decomposed into a sum of the four unperturbed surface states | k χ ξ . The latter are obtained as eigenfunctions of the translation operator of the ideal surfaceT Θ(r) = Θ(r + τ ) = exp(iτ k)Θ(r) in terms of the supercell solutions: Θ = n c n Φ K n . The functions Θ are defined everywhere in the crystal, and in the asymptotic region they coincide with the unperturbed surface states: Θ χ ξ (r) = r | k χ ξ . The full scattering solution is then a linear combination Ψ = χξ a χ ξ Θ χ ξ defined by the condition that Ψ contain only one right-traveling wave in the 0th supercell and no left-traveling waves in the 1st supercell (the next asymptotic region), Fig. 1(c). Note that Ψ is valid everywhere, including the vicinity of the defect.
Let us consider a 7-layer slab with the geometry of a Au(111) surface with an overlayer. The atoms are represented by a 3D regular muffin-tin potential, which is expanded in a truncated 3D Fourier series and included into the microscopic Hamiltonianp 2 + V (r) + βσ · [ ∇V (r) ×p ], with β scaled such that the Rashba splitting of the surface states be close to that in Au(111). The supercell band structure is calculated on a rectangular k-mesh with ∆K x = ∆K y = 0.0056 a.u. −1 , and the functions Φ K n for a given energy and K y are obtained by triangular interpolation. A typical constant energy contour is shown in the upper panel of Fig. 1(d). The artificial periodicity of the defect gives rise to spectral gaps, so for certain K y there are no solutions Φ. However, for a given K y one can always choose a supercell for which the solution exists.
We will consider two types of defects: barrier and well. For a barrier, the potential at the impurity site U D is shallower than the potential U S at the host atom, and for a well it is deeper. Computationally, the muffin-tin potentials U D are linearly scaled: U D (r) = ǫU S (r). The Bloch vectors of the unperturbed surface states extracted from the eigenvalues exp(iτ k) are shown in Fig. 1(e). The good agreement between the three supercells both for a well and for a barrier demonstrates that the evanescent waves are negligible in the asymptotic region already for the 8-fold supercell. Figure 1(f) shows the density pro-  Table I. files of the outer-circle surface states at E − EΓ = 0.7 eV scattered by a well ǫ = 1.07 for four angles of incidence γ. Although the unperturbed surface states are derived from 0th cell, the asymptotic representation is seen to be valid over a much wider region (see, especially, γ = 44 • ). The transmission probability T ± as a function of γ is shown in Fig. 2 for a barrier, ǫ = 0.96, and for two wells, ǫ = 1.04 and 1.07. Here T + and T − stand for the incident wave in the inner and in the outer circle, respectively. The colored symbols are the microscopic calculations, with shades of red used for T + and blue for T − . The voids in the curves correspond to the gaps in the supercell band structure, and in approaching the gap the transmission sometimes shows a spurious growth [see vertical arrows in Figs. 2(a) and 2(b)]. This happens when two of the Bloch vectors K n y are close to the edge of the Brillouin zone, and the numerical method finds the two solutions Φ K n linearly dependent. Such artifacts are recognized by an accuracy criterion, and they are easily sorted out because they occur at different angles for different supercells. Most important is the strikingly different behavior of the transmission probability T − for the two types of defects: for a barrier, T − steadily decreases, Fig. 2(a), whereas for a well it shows a sharp minimum followed by a maximum, see Figs. 2(b) and 2(c). By contrast, T + steadily decreases in both cases. To understand this behavior, let us consider the contribution of evanescent waves to the scattering states Ψ. Their weight can be inferred from the deviation of the density profile |Ψ| 2 in Fig. 1(f) from the left and right asymptotics (dashed lines) continued up to the scatterer. For small angles the weight of the evanescent waves is negligible, and it starts growing when k − iy exceeds R + because the evanescent waves replace the missing propagating solutions of the inner circle. This point manifests itself by a cusp maximum in T − , e.g., at 0.4 eV around γ = 45 • in Fig. 2(a). In approaching the minimum [see γ = 61 • in Fig. 1(f)] the density around the defect steeply grows and then rapidly decreases with increasing γ. This happens because the defect causes a sharp perturbation of the potential V (r) (comparable to the lattice period), which is known to give rise to a bound state localized at the defect and energetically split off from the band continuum [25]. For the Rashba states that are bounded only from below a barrier does not produce any bound states. By contrast, a well-like perturbation produces two structures [ Fig. 1(d)]: bound state B and its spin-orbit counterpart resonance R [highlighted red in Fig. 1(d)]. The hybridization of the incident wave of the outer circle with the resonance -the inner branch of the Rashba split 1D impurity state -gives rise to the asymmetric T − (γ) feature.
In order to relate the reflection resonance to phenomenologically relevant spin-orbit characteristics of the material let us consider a Rashba system with the Hamil-tonianĤ R = k 2 /2m * + α(k y σ x − k x σ y ). The relation k 2 = k 2 x + k 2 y determines whether a given branch is propagating or evanescent for a given k y and E [11]. The defect is represented by a potential barrier (well) V (x) = U FIG. 3. (a) Electronic structure of the Rashba system with linear defect. Solid lines show the bound states split off from the Rashba continuum. The width of the blurred red line shows the ky-width of the resonance. The Rashba continuum of the ideal system E ± (k) [cf. grey area in Fig. 1(d)] is shown by the sign of the ky-projected x-spin spectral density S tot x . Light red indicates S tot x > 0, and light blue S tot x < 0. (b) Band structure and side view geometry of a BiTeI trilayer with a nanostripe. Shaded area covers the ky-projected states of the clean trilayer. The localization of the trilayer states on the atoms beneath the stripe are shown by red (σ ↑ x ) and blue (σ ↓ x ) fat bands. Light red (light blue) fat bands correspond to σ ↑ x (σ ↓ x ) dangling bond states localized on the stripe.
for −l < x < l, with V (x) = 0 elsewhere. The width of the defect equals the width of the unit cell: 2l = τ x . The scattering solution is found by the condition of the continuity of the spinor wave function and flux across the defect [15]. Thus, the envelope-function formalism solves the problem without resorting to an artificial supercell. A four-wave representation in the perturbed region χξ d χ ξ |k χ ξ is matched to the wave function in the left and right half-planes at the boundaries x = ±l indicated in Fig 1(f).
Herek are the wave vectors of the eigenfunctions of the HamiltonianĤ R +U (with the same α and m * as for the unperturbed surface). The scattering problem then reduces to an 8 × 8 matrix equation M a = f for the vector a = (r ± , d ± r , d ± t , t ± ) T . HereM is the matching matrix, and r ± and t ± are the coefficients of the two reflected and two transmitted waves in the two unperturbed half-planes, see the legends in Fig. 1(f). The right-hand side f represents the incident wave | k χ i , and it has four non-zero components: the value and the flux (for both spins) at x = −l.
The transmitted current T χ = |t + χ | 2 + |t − χ | 2 is shown in Fig. 2 by solid lines. With U adjusted to fit the microscopic calculations the Rashba model perfectly reproduces the shape of the curves and the dependence of the position and the width of the resonance on the energy and on the scatterer. Surprisingly, the envelopefunction method originally designed for slowly varying potentials shows excellent performance for the atomic stripe. To establish the analogy with the microscopic picture, let us consider the eigenspectrum of the perturbed system. It is obtained by dropping the incident wave and finding zeros of real and imaginary part of the determinant of the matrixM . In Fig. 3(a), the ideal surface is presented by the energy-momentum distribution of the sign (↑ or ↓) of the k y -projected σ x -spin spectral density S tot The incident wave comes from the σ ↓ x continuum (blue area), which overlaps with the spectral resonance having σ ↑ x spin. Just at the resonance, T − (γ) sharply drops to zero and then steeply rises to unity, exactly as in the microscopic model, see Figs. 2(b) and 2(c). Thus, the scattering by the 1D defect is transparently expressed through the relation between the CECs of the host and the defect region. Here, an important ingredient is the spin non-conservation, so the effect does not occur, e.g., for Zeeman splitting.
For the above monoatomic stripes, practically all the transmitted current is carried by the same wave as is incident, so the spin orientation of the incident electron is preserved on transmission. The picture becomes very different for thicker stripes, in which the evanescent waves (complexk x ) do not participate in the transmission through the defect. The continuous Rashba model predicts that for barriers thicker than 1 nm the spin-flip transmission, i.e., | k ∓ t for | k ± i , becomes quite important, see Fig. 4. Another vivid feature of the nanosized barrier are the Fabry-Pérot oscillations of the transmission.
Finally, as a possible platform for the experimental realization of the discovered resonant reflection, we suggest the layered semiconductors of the BiTeX (X =I, Br, Cl) family, where giant Rashba-split surface states reside in an absolute gap [26][27][28][29][30]. Already a trilayer Te-Bi-X -the easily exfoliated structure element of these semiconductors -provides the desired 2D spin-orbit split valence and conduction states [31,32]. For holes and electrons of a stand-alone trilayer, a perturbation can be introduced by putting on it a nanostripe as shown in the inset of Fig. 3(b) [33]. This linear defect gives rise to 1D spin-orbit split bound states that split off from the valence band, as follows from our ab initio calcula-tion [34], see vertical arrows in Fig. 3(b). These states form the 1D Rashba channel that guides the holes. As seen in Fig. 3(b), the inner branch (red arrow) becomes the above-mentioned spectral resonance when it enters the projected continuum.
To summarize, we have developed a microscopic approach to scattering of relativistic surface states by a linear defect and found strong spin selectivity of electron transmission for well-like perturbations. Thereby, the transmitted spin current can be enhanced, which suggests a potential technique for non-magnetic spin filtering and spin injection.