Rashba spin splitting of Shockley surface states on semi-infinite crystals

We study the Rashba spin splitting of the $L$-gap surface state on the (111) surfaces of Au, Ag, and Cu, and that of the topologically protected surface state on Sb(111) by a first-principles calculation based on the embedded Green's function technique. By taking advantage of semi-infinite surface geometry, we address several issues that are difficult to handle with the finite slab calculation. For Au(111), we investigate the nonlinear dependence on wave vector $\mathbf{k}$ of the spin-splitting energy, and also at what wave number each dispersion curve of the two spin-split bands is merged into a surface-projected bulk band. Then, we discuss why Cu(111) exhibits a larger Rashba effect than Ag(111) despite the fact that the atomic orbitals of Ag have larger spin-orbit matrix elements than those of Cu. We reveal that, while the wave function of the $L$-gap surface state has strong $p_z$ character (the $z$ axis is surface normal), the Rashba spin splitting occurs through the spin-orbit matrix element between the $d_{z^2}$ and $d_{uz}$ orbital components hybridized with $p_z$, where $d_{uz}$ denotes a linear combination of $d_{xz}$ and $d_{yz}$ with its orbital lobes pointing to the $\mathbf{k}$ direction (defined as the $u$ axis). Cu(111) exhibits a larger spin splitting than Ag(111), since the amount of both the $d_{z^2}$ and $d_{uz}$ components mixed in its surface-state wave function is much larger than that of Ag(111). In contrast to the (111) surfaces of noble metals, the spin splitting of the topologically protected surface state on Sb(111) arises from the spin-orbit matrix element between the $p_z$ and $p_u$ orbital components. We also discuss under what condition the energy dispersion curve of a surface state intersects the boundary line between a projected bulk band and a projected bulk band gap at a large angle, instead of gradually approaching the boundary line with increasing $\left|\mathbf{k}\right|$. Finally, we comment on a recent theoretical work of Krasovskii [Phys. Rev. B 90, 115434 (2014)] on the microscopic origin of the Rashba spin splitting.

Figure 1

(a) Intensity plot of $\mathbf{k}$-resolved DOS calculated in a first-layer MT sphere with radius $R=2.48$ a.u for Au(111) along $\overline{K}\text{−}\overline{\Gamma }\text{−}\overline{M}$ directions. Here and henceforth, bright and dark colors represent large and small values of $\rho (\mathbf{k},\epsilon )$, respectively, with the color scale being linear with ${log}_{10}\rho (\mathbf{k},\epsilon )$. (b) The same as panel (a) in a smaller region surrounding the projected bulk band gap including $E_F$ and $\overline{\Gamma }$. The unit of the wave number is a.u. (${\mathrm{bohr}}^{-1}$). (c) The $y$ component of the $\mathbf{k}$-resolved spin DOS calculated in a first-layer MT sphere for five $\mathbf{k}$ vectors along $\overline{\Gamma }\text{−}\overline{K}$. The small number near each line indicates $k_x$. The broadening parameter $\gamma$ is chosen as $1\times {10}^{-3}$, $3\times {10}^{-4}$, and $5\times {10}^{-4}$ a.u. for panels (a), (b), and (c), respectively.

Figure 2

(a) $\rho (\mathbf{k},\epsilon )$ calculated in a first-layer MT sphere for Au(111) at large $k_x$ values along $\overline{\Gamma }\text{−}\overline{K}$, where $\gamma =1\times {10}^{-5}$ a.u. (b) Energy dispersion of the $L$-gap surface states, ${\epsilon }_{\pm }\left(k_x\right)$. (c) Spin-splitting energy $\Delta \epsilon \equiv {\epsilon }_{+}\left(k_x\right)-{\epsilon }_{-}\left(k_x\right)$ for a semi-infinite Au(111) surface along the $\overline{\Gamma }\text{−}\overline{K}$ direction (solid line). For comparison, $\Delta \epsilon$ calculated by using a 35-layer slab is shown by dashed line.

Figure 3

Complex band structure of Au(111) at $\mathbf{k}=0$ ($\overline{\Gamma }$). Red lines are Bloch states with ${\kappa }_z=0$, while green lines indicate evanescent waves with ${\kappa }_z>0$. The $L$-point gap is located between a band maximum with energy ${\epsilon }_v$ and a band minimum with energy ${\epsilon }_c$. The complex band connecting the two points with the smallest ${\kappa }_z$ determines the decay length of the $L$-gap surface band.

Figure 4

(a) Imaginary part of complex wave number, ${\kappa }_z$, of the complex band connecting the lower and upper edges of the projected bulk band gap of Au(111) for several $\mathbf{k}$ vectors oriented in the $\overline{\Gamma }\text{−}\overline{K}$ direction. The real part of complex wave number is a constant $k_z=\pi /d_z$. The small number near each curve indicates the value of $k_x$. Open circles on each line indicate the energy positions of the ${\epsilon }_{\pm }$ bands. (b) Decay constant of the wave function of the $L$-gap surface states, ${\epsilon }_{\pm }$, for Au(111) as a function of $k_x$ along $\overline{\Gamma }\text{−}\overline{K}$.

Figure 5

Intensity plot of $\mathbf{k}$-resolved DOS calculated in a first-layer MT sphere for (a) Ag(111) and (b) Cu(111) surfaces along $\overline{\Gamma }\text{−}\overline{K}$ (left half) and $\overline{\Gamma }\text{−}\overline{M}$ (right half) directions. The MT sphere radii are 2.48 and 2.33 a.u. for Ag and Cu, respectively. The unit of wave number is a.u. (${\mathrm{bohr}}^{-1}$). For Cu(111), the first interlayer spacing is contracted by $1.27\%$ of the bulk value.

Figure 6

Rashba spin-splitting energy of the $L$-gap surface state on Ag(111) and Cu(111) along the $\overline{\Gamma }\text{−}\overline{K}$ direction. Two vertical bars near each line indicate positions of two Fermi wave numbers, $k_{F+}$ and $k_{F-}$.

Figure 7

$c(\mathbf{k},lm)$, $(l,m)$-decomposed charge of the $L$-gap surface state in the absence of SO coupling calculated within the MT sphere of a first-layer surface atom along the $\overline{\Gamma }\text{−}\overline{K}$ direction for (a) Au(111), (b) Ag(111), and (c) Cu(111). The vertical axis shows the ratio of $c(\mathbf{k},lm)$ to $c(\mathbf{k}=0,p_z)$ with $c(\mathbf{k}=0,p_z)$ being 0.111, 0.108, and 0.154 a.u. for Au, Ag, and Cu(111), respectively.

Figure 8

(a) Intensity plot of $\mathbf{k}$-resolved DOS calculated in a first-layer MT sphere for Sb(111) surface along $\overline{K}\text{−}\overline{\Gamma }\text{−}\overline{M}$ directions. The MT sphere radius $R$ is 2.64 a.u. Broadening parameter $\gamma$ is $1\times {10}^{-4}$ a.u. (b) The same as panel (a) with higher energy resolution ($\gamma =2\times {10}^{-5}$ a.u.) along $\overline{\Gamma }\text{−}\overline{M}$. (c) Intensity plot of $\mathbf{k}$-resolved DOS calculated in a first-layer MT sphere for Bi(111) surface along $\overline{\Gamma }\text{−}\overline{M}$. The unit of wave number is a.u. (${\mathrm{bohr}}^{-1}$).

Figure 9

(a) $c(\mathbf{k},lm)$, $(l,m)$-decomposed charge of the gap state in the absence of SO coupling calculated within the MT sphere of a first-layer atom along the $\overline{\Gamma }\text{−}\overline{M}$ direction for Sb(111). The vertical axis shows $c(\mathbf{k},lm)/c(\mathbf{k}=0,p_z)$ with $c(\mathbf{k}=0,p_z)$ being 0.257 a.u. (b) Spin-splitting energy of the two surface bands by the full EGF calculation (solid line) and by Eq. (19) (dashed line) along $\overline{\Gamma }\text{−}\overline{M}$.

Figure 10

(a) $\rho (\epsilon ,k_x)$ calculated in a first-layer MT sphere for Sb(111) at three $k_x$ values along $\overline{\Gamma }\text{−}\overline{K}$. Broadening parameter $\gamma =1\times {10}^{-5}$ a.u. (b) Real part of complex wave number, $k_z$, vs energy of complex bands for Sb(111) at $\mathbf{k}=(0.055,0)$ a.u. Complex bands with negative $k_z$ are not shown as complex band structure is symmetric with respect to $k_z=0$ along $\overline{\Gamma }\text{−}\overline{K}$. Two open circles on $c_1$ band indicate complex wave numbers associated with localized surface states shown in (a). (c) Imaginary part of complex wave number, ${\kappa }_z$, vs energy of complex bands for Sb(111) at $\mathbf{k}=(0.055,0)$ a.u. (d) Same as (b) at $\mathbf{k}=(0.065,0)$ a.u. (e) Same as (c) at $\mathbf{k}=(0.065,0)$ a.u.