Bulk versus surface contributions to the Rashba spin splitting of Shockley surface states

Shockley surface states of a bulk crystal with both time reversal and space inversion symmetries exhibit the Rashba spin splitting due to the broken space inversion symmetry at the surface. Since the evanescent states in the bulk region are doubly degenerate with respect to spin degrees of freedom even in the presence of spin-orbit interaction (SOI), one might think that the entire spin splitting occurs via the SOI in the surface region where the potential energy deviates from the bulk one. In the present work, we elucidate why this is not the case. Namely, in the presence of SOI, the complex energy bands are modified such that a pair of evanescent states having the same energy and the same complex wave number become two distinct solutions of the Schrödinger equation that do not satisfy the same boundary condition. Since the tail of the surface-state wave function is expressed as a superposition of these evanescent waves, the bulk region also contributes to the Rashba spin splitting. We illustrate this effect by a first-principles calculation for Au(111) and by a simplified $sp$-band model Hamiltonian on a square lattice.

Figure 1

(a) Spin splitting $\mathrm{\Delta }\varepsilon$ of the $L$-gap surface state on Au(111) as a function of $k_x$, wave number along the $\overline{\mathrm{\Gamma }}-\overline{\mathrm{M}}$ line. $L$ denotes the number of Au layers included in the embedded surface region. (b) Planar-averaged charge density of the $L$-gap surface state on Au(111) at $\overline{\mathrm{\Gamma }}$ as a function of normal coordinate $z$. $d_z$ denotes the layer spacing between two neighboring Au(111) layers, and $z/d_z=i$ ($i=1,2,...,7$) indicates the positions of Au atomic planes. (c) Complex band structure of Au(111) at $k_x=0.1{\mathrm{bohr}}^{-1}$ along the $\overline{\mathrm{\Gamma }}-\overline{\mathrm{M}}$ line. Left and right panels show, respectively, the real and imaginary part of the $z$ component of wave number $q=k_z+i{\kappa }_z$. Red lines indicate Bloch states with ${\kappa }_z=0$, while pairs of green lines numbered as $1,2,3,...$ in both panels represent evanescent states with ${\kappa }_z>0$. For brevity, only a few bands are numbered. The decay length of the $L$-gap surface state is determined by the ${\kappa }_z$ value of the band “1.” Energy is measured relative to the Fermi energy $E_F$.

Figure 2

Energy dispersion with $(k_x,k_z)$ of bulk Bloch states for the three-band model Hamiltonian on a square lattice. (a) $\lambda =0$ and (b) $\lambda =0.5$. The other parameters common to both panels are ${\varepsilon }_s^a=0, {\varepsilon }_p^a=3, t_{ss}^{\sigma }=-1.0, t_{pp}^{\sigma }=0.75, t_{pp}^{\pi }=-0.25$, and $t_{sp}^{\sigma }=1.0$.

Figure 3

Intensity plot of the $k_x$-resolved DOS of the outermost layer, ${\rho }_1(k_x,\varepsilon )$, for semi-infinite surfaces with (a) $\lambda =0$, (b) $\lambda =0.25$, and (c) $\lambda =0.5$. The other parameters are the same as those in Fig. 2. See the vertical color scale for the actual DOS values. Spin-up and spin-down surface bands are indicated by red and blue arrows, respectively. Shaded regions represent surface-projected bulk bands. Imaginary energy $\gamma =5\times {10}^{-4}$.

Figure 4

Intensity plot of the $k_x$-resolved DOS of the outermost layer, ${\rho }_1(k_x,\varepsilon )$, for a semi-infinite surface with Hamiltonian parameters, ${\varepsilon }_s^a=0, {\varepsilon }_z^a={\varepsilon }_x^a=4, t_{ss}^{\sigma }=-1.0, t_{pp}^{\sigma }=t_{pp}^{\pi }=0.75, t_{sp}^{\sigma }=0.75$, and $\lambda =2.0$. The color scaling of the DOS is the same as that in Fig. 3. Spin-up and spin-down surface bands are indicated by red and blue arrows, respectively. Shaded regions represent surface-projected bulk bands. Imaginary energy $\gamma =5\times {10}^{-4}$.