Spin filtering via resonant reflection of relativistic surface states

A microscopic approach is developed to the scattering of surface states from a nonmagnetic 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.

Figure 1

(a) CECs of Rashba-split surface states. ${\mathbf{k}}_{\mathrm{r}}^{\pm },{\mathbf{k}}_{\mathrm{t}}^{\pm }$, and ${\mathbf{k}}_{\mathrm{i}}^{\pm }$ are the Bloch vectors of reflected, transmitted, and incident waves. (b) A finite-thickness slab with a linear defect in the topmost layer. (c) Supercell geometry: topmost layer with a repeated row of impurity atoms. The two boxes indicate two asymptotic regions. (d) Upper panel: CEC at $E-E_{\overline{\mathrm{\Gamma }}}=1.5\mathrm{eV}$ for a 12-fold supercell. Lower panel: The dots are the dispersion $E\left(K_y\right)$ of the solutions ${\mathrm{\Phi }}_{{\mathbf{K}}^n}$ for $K_x=0$. The shaded area shows the $k_y$-projected states of the ideal surface. The spin-orbit split states due to the defect are highlighted in blue (true bound state) and red (resonance when inside the gray area). In the 1.5-eV CEC, the arrows indicate the bound state ($B$) and the resonance ($R$). (e) Bloch vectors $k_x$ extracted from the eigenvalues $exp\left(i\mathbf{\tau }\mathbf{k}\right)$ of the host lattice translation operator for the three supercells for the well ($\epsilon =1.04$, upper quadrant) and the barrier ($\epsilon =0.96$, lower quadrant). (f) Density profiles of four scattering states at $E-E_{\overline{\mathrm{\Gamma }}}=0.7\mathrm{eV}$ for $\gamma ={44}^{\circ },{51}^{\circ },{61}^{\circ }$, and ${65}^{\circ }$ for the well $\epsilon =1.07$ in a 12-fold supercell. The dashed lines are their asymptotic representations in the zeroth and first cells continued up to the defect.

Figure 2

Transmission probability as a function of the angle of incidence for the (a) $\epsilon =0.96$, (b) 1.04, and (c) 1.07 for $E-E_{\overline{\mathrm{\Gamma }}}=1.5$, 0.7, and 0.4 eV. The shades of red (blue) show $T^{+}\left(T^{-}\right)$ for eightfold, tenfold, and 12-fold supercells. The solid lines are the continuous model fit of $T^{+}$ (red) and $T^{-}$ (blue) obtained with $U=0.27,-0.33$, and $-0.55\mathrm{eV}$ for $\epsilon =0.96$, 1.04, and 1.07, respectively. The parameters $\alpha$ and $m^{*}$ for the presented energies are listed in Table 1.

Figure 3

(a) Electronic structure of the Rashba system with linear defect. The solid lines show the bound states split off from the Rashba continuum. The width of the blurred red line shows the $k_y$ width of the resonance. The Rashba continuum of the ideal system $E^{\pm }(\mathbf{k})$ [cf. the gray area in Fig. 1] is shown by the sign of the $k_y$-projected $x$-spin spectral density $S_x^{\mathrm{tot}}$. Light red indicates $S_x^{\mathrm{tot}}>0$, and light blue indicates $S_x^{\mathrm{tot}}<0$. (b) Band structure and side-view geometry of a BiTeI trilayer with a nanostripe. The shaded area covers the $k_y$-projected states of the clean trilayer. The localization of the trilayer states on the atoms beneath the stripe are shown by red (${\sigma }_x^{\uparrow }$) and blue (${\sigma }_x^{\downarrow }$) fat bands. Light red (light blue) fat bands correspond to ${\sigma }_x^{\uparrow }\left({\sigma }_x^{\downarrow }\right)$ dangling bond states localized on the stripe.