Strongly anisotropic spin and orbital Rashba effect at a tellurium – noble metal interface

We study the interplay of lattice, spin, and orbital degrees of freedom in a two-dimensional model system: a flat square lattice of $\mathrm{Te}$ atoms on a $\mathrm{Au}\left(100\right)$ surface. The atomic structure of the Te monolayer is determined by scanning tunneling microscopy and quantitative low-energy electron diffraction. Using spin- and angle-resolved photoelectron spectroscopy and density functional theory, we observe a Te-Au interface state with highly anisotropic Rashba-type spin-orbit splitting at the $\overline{\mathrm{X}}$ point of the Brillouin zone. Based on a profound symmetry and tight-binding analysis, we show how in-plane square lattice symmetry and broken inversion symmetry at the Te-Au interface together enforce a remarkably anisotropic orbital Rashba effect which strongly modulates the spin splitting.

Figure 1

Structural data for the $(2\times 2)$ phase of $\mathrm{Te}$ on $\mathrm{Au}\left(100\right)$. (a) LEED pattern at $40\mathrm{eV}$ electron energy. Golden arrows mark the $(1\times 1)$ unit cell of the unreconstructed $\mathrm{Au}\left(100\right)$ surface and the green square the $(2\times 2)$ Te superstructure cell. (b) STM image ($U=-0.27\mathrm{V}$, $I=0.13\mathrm{nA}$) showing the perfectly ordered $(2\times 2)$ lattice. Te atoms are assigned to the bright protrusions. (c) Selected best-fit LEED-I(V) spectra of the $(2\times 2)$-Te structure with single-beam $R$ factors close to the overall $R$-factor value of $R_{\mathrm{P}}=0.085$. The fit has a redundancy of $\rho =29.8$. The complete data set comprises a total energy range of $8720\mathrm{eV}$ and is presented in the SM [44]. (d) Ball model of the LEED best-fit structure in side view along the path indicated by the arrow in (e). (e) Top view. Parameters (atomic distances in $\mathrm{pm}$) given in (d) and (e) correspond to the LEED best-fit (red) and the relaxed DFT structure (green) for comparison.

Figure 2

Band structure of $\mathrm{Te}/\mathrm{Au}\left(100\right)$ along (a), (b) a $\overline{\mathrm{\Gamma }}\overline{\mathrm{X}}\overline{\mathrm{M}}$ path [see the inset in (a)] and (e) isoenergy contours at given binding energies. Bands denoted as ${\alpha }_{\pm }$, $\beta$, and $B$ correspond to adsorbate-induced interface states and substrate bulk bands, respectively. ARPES data in (a) were taken with circularly polarized light ($\mathrm{left}+\mathrm{right}$) at $h\nu =38\mathrm{eV}$. The dot size and red-blue color code in the DFT slab calculation (b) reflect the calculated spin polarization perpendicular to ${\mathbf{k}}_{\parallel }$. We plotted $\langle S_{\perp }\rangle$ as $\langle S_x\rangle$ along $k_x$ ($\overline{\mathrm{X}}\overline{\mathrm{\Gamma }}$) and $-\langle S_y\rangle$ along $k_y$ ($\overline{\mathrm{X}}\overline{\mathrm{M}}$), where blue (red) stands for positive (negative) values, respectively. (c) Spin-resolved energy distribution curves ($h\nu =38\mathrm{eV}$, $p$ polarization) taken at momenta indicated as dashed lines in (a). (d) DFT-calculated partial charge $|{\mathrm{\Psi }}_{k=\overline{\mathrm{X}}}{\left(\mathbf{r}\right)|}^2$ of ${\alpha }_{\pm }$ at the $\overline{\mathrm{X}}$ point.

Figure 3

Anisotropy of the Rashba effect around the $C_{2v}$-symmetric $\overline{\mathrm{X}}$ point. (a) Spin splitting extracted from the experimental data [see Fig. 2] along $\overline{\mathrm{X}}\overline{\mathrm{M}}$ and $\overline{\mathrm{X}}\overline{\mathrm{\Gamma }}$ lines as a function of $k_{\parallel }$. (b) Same as in (a) but obtained from DFT [see Fig. 2]. (c) Rashba parameter ${\alpha }_{\mathrm{R}}$ as a function of ${\varphi }_k$ (see text), where $\overline{\mathrm{X}}$ is the origin and $0^{\circ }$ and ${90}^{\circ }$ being along $\overline{\mathrm{\Gamma }}$ and $\overline{\mathrm{M}}$, respectively. (d) Same as in (c) but obtained from the utilized tight-binding model as a function of different ${\gamma }_2^{\pm 1}$ values (red ${\gamma }_2^{\pm 1}=0$, black ${\gamma }_2^{\pm 1}=0.2$, and cyan ${\gamma }_2^{\pm 1}=0.3$). Markers in (c) and (d) correspond to data points extracted from the experimental data and TB model, respectively. Solid lines are fits to those (see text).

Figure 4

(a) Momentum-space textures in a $d$-orbital square lattice. Parity of a respective texture can be identified along specific high-symmetry lines. (b)–(e) Band structure plots obtained from the tight-binding model as a function of varying ISB parameters ${\gamma }_l^{m_l}$. The top panels in (b)–(e) are when SOC is not being taken into account (color scale adjusted compared to the lower panel for better visualization; see SM Fig. S8 [44] for a direct comparison). The bottom panels show the outcome when SOC is considered.