Spin structure of spin-orbit split surface states in a magnetic material revealed by spin-integrated photoemission

The emergence of ferromagnetism in Rashba systems, where the evolving exchange interaction enters into competition with spin-orbit coupling, leads to a nontrivial spin-polarized electronic landscape with an intricate momentum-dependent spin structure, which is challenging to unveil. Here, we show a way to disentangle the contributions from the effective spin-orbit and exchange fields and thus to gain knowledge of the spin structure in ferromagnetic Rashba materials, which is required for spintronic applications. Our approach is based exclusively on spin-integrated photoemission measurements combined with a two-band modeling. As an example, we consider the mixed-valent material ${\mathrm{EuIr}}_2{\mathrm{Si}}_2$ which, while being nonmagnetic in the bulk, reveals strong ferromagnetism at the iridium-silicide surface where both spin-orbit and exchange magnetic interactions coexist. The combined effect of these interactions causes a complex band dispersion of the surface state which can be observed in photoemission experiments. Our method allows us to comprehensively unravel the surface-state spin structure driven by spin-orbit coupling at the ferromagnetic surface. This approach opens up opportunities to characterize the spin structure of ferromagnetic Rashba materials, especially where dedicated spin-resolved measurements remain challenging.

Figure 1

ARPES view on the Fermi surface with the surface state highlighted in blue (a) and its electron band dispersions $E\left(\mathbf{k}\right)$ (b) around the $\overline{M}$ point for paramagnetic and ferromagnetic Si-terminated surfaces of ${\mathrm{EuIr}}_2{\mathrm{Si}}_2$. Dashed lines indicate the border of the surface Brillouin zone. The red arrow indicates the direction of magnetization which stems from ordered Eu $4f$ moments.

Figure 2

(a), (b) Measured Fermi surfaces and band dispersions $E^{\pm }\left(\mathbf{k}\right)$ in the FM phase. (c) $\mathbf{B}·\mathbf{J}$ product, showing a triple winding of the effective SO field. (d) Normalized $S_x$ component of spin for the ${\alpha }^{+}$ state in the PM phase. (e), (f) Experimentally obtained spin structure in the PM phase (e) at the Fermi level and (f) at the binding energy of 0.16 eV. Here and further, the $k$ values are measured relative to the $\overline{M}$ point.

Figure 3

(a) The ARPES data for the FM phase of ${\mathrm{EuIr}}_2{\mathrm{Si}}_2$ measured along the $y$ axis in Fig. 1 through the $\overline{M}$ point ($\overline{X}\overline{M}\overline{X}$ direction). (b) The spin component $S_x$ in the PM phase, qualitatively determined from the ARPES data as $S_x^{\pm }\propto \pm B_x$ using Eq. (9), compared with the spin component $S_x$, derived from the DFT calculations. Solid lines indicate Fermi contours.

Figure 4

(a), (b) Measured Fermi surfaces and dispersions for the ${\alpha }^{\pm }$ states in the PM phase. (c) The absolute value of the effective SO field obtained from the measured dispersion using Eq. (10). (d), (e) Spin components in the FM phase extracted from the ARPES data by means of Eq. (13). (f) The strength of exchange coupling derived from experiment using Eq. (11). Blue color indicates regions of undefined $J$. (g), (h) Spin expectation values for the ${\alpha }^{+}$ band derived from the DFT calculation. Dashed line in (g) shows the measured Fermi contour in the PM phase, similar to the solid lines in (b)–(f). Solid lines in (h) show the calculated constant energy contours at $-0.2$ eV in the FM phase. (i) Spin texture obtained from ARPES with the use of Eq. (13).