Cubic Rashba Effect in the Surface Spin Structure of Rare-Earth Ternary Materials

Spin-orbit interaction and structure inversion asymmetry in combination with magnetic ordering is a promising route to novel materials with highly mobile spin-polarized carriers at the surface. Spin-resolved measurements of the photoemission current from the Si-terminated surface of the antiferromagnet ${\mathrm{TbRh}}_2{\mathrm{Si}}_2$ and their analysis within an ab initio one-step theory unveil an unusual triple winding of the electron spin along the fourfold-symmetric constant energy contours of the surface states. A two-band $\mathbf{k}·\mathbf{p}$ model is presented that yields the triple winding as a cubic Rashba effect. The curious in-plane spin-momentum locking is remarkably robust and remains intact across a paramagnetic-antiferromagnetic transition in spite of spin-orbit interaction on Rh atoms being considerably weaker than the out-of-plane exchange field due to the Tb $4f$ moments.

Figure 1

(a) Fermi contours from ARPES for PM and AFM phases. Red and blue arrows indicate color schemes used for in-plane spin components derived from DFT calculation in graphs (b) and (c). Calculated and measured band structure for Si-terminated TRS for PM (b) and AFM (c) phases. The inset in (b) shows the spin-orbit splitting of the $\alpha$ state: black arrows indicate the 35 meV splitting. Black arrows in (c) indicate the 140 meV splitting of the $\alpha$ state in AFM phase. Violet-brown palette shows calculated surface-projected bulk states. For the surface states the in-plane spin polarization is shown in red and blue, while yellow and green highlight the positive and negative spin component $S_z$, respectively. Spin-resolved ab initio CECs for the state $\alpha$ in the PM (d) and AFM (e) phase calculated at the binding energy of 0.23 eV. The in-plane spin orientation, ${\mathbf{S}}_{\parallel }$, is indicated by pink (purple) arrows for the inner (outer) contour. In AFM phase, the color of the spin vector $\mathbf{S}={\mathbf{S}}_{\parallel }+S_z\widehat{\mathbf{z}}$ shows the sign of $S_z$ as in Fig. 1.

Figure 2

SR-ARPES measurements of state $\alpha$ for PM (a) and AFM (c) phases of TRS. Measured $k$ points are denoted 1 and 2 for PM (b) and 1–6 for AFM (e)–(f). Red and blue symbols show spin-up $J^{\uparrow }$ and down $J^{\downarrow }$ EDCs for $J_X$ (a) and for $J_Y$ (c). Yellow and green symbols in (c) are $J_Z^{\uparrow }$ and $J_Z^{\downarrow }$. Brown symbols are the net spin $J^{\uparrow }\text{−}{J}^{\downarrow }$. Calculated EDCs for AFM (c) are shown with lines of the same color code. EDCs are energy broadened and $k$ averaged along the analyzer slit to mimic the experimental energy resolution and acceptance angle. Lines in graph (a) are a guide for the eye. Energy momentum maps in graph (d) show calculated net spin photocurrent without broadening and averaging. Theoretical data in (c) and (d) are averaged over the opposite magnetization directions. The $XYZ$ axes of the analyzer relative to SBZ are shown in graphs (b), (e), and (f) by brown arrows (see Supplemental Material [27] for details of geometry). In (b), (e), and (f), arrows (pink for inner and purple for outer CEC) are a guide to relate EDCs in (a) and (c) to spin polarization in ground state.

Figure 3

(a) Effective spin-orbit magnetic fields ${\mathcal{B}}_{\mathrm{R}}^{(1)}$ (${\mathrm{R}}^{(1)}$ effect, blue arrows), ${\mathcal{B}}_{\mathrm{R}}^{(3)}$ (“heavy-hole” ${\mathrm{R}}^{(3)}$ effect, green arrows), and ${\mathcal{B}}^{(3)}$ (${\mathrm{R}}^{(3)}$ effect for the state $\alpha$, orange arrows) as a function of polar angle ${\phi }_{\mathbf{k}}$. The legends on the left show the rotation direction of the vectors ${\mathcal{B}}_{\mathrm{R}}^{(1)}$, ${\mathcal{B}}_{\mathrm{R}}^{(3)}$, and ${\mathcal{B}}^{(3)}$ with the anticlockwise rotation of $\mathbf{k}$ (gray dashed arc arrow). (b) Band structure of the states $\alpha$ and $\beta$ by the Hamiltonian (1) presented by red ($S_y>0$) and blue ($S_y<0$) bands. Graphs (c) and (d) show spin-resolved CECs by the $\mathbf{k}·\mathbf{p}$ model at the binding energy of 0.2 and 0.7 eV, respectively [indicated by dashed gray lines in graph (b)], with the ${\mathbf{S}}_{\parallel }$ orientation shown by pink (purple) arrows for the inner contour and by purple (pink) for outer contour of the state $\alpha$ ($\beta$).