Plasmon manipulation by exchange magnetic field in two-dimensional spin-orbit coupled electronic systems: A higher-order relativistic k.p study

Abstract

A higher-order relativistic k.p model is developed to describe plasmon excitations in two-dimensional (2D) electronic systems with spin-orbit coupling (SOC) and magnetic-exchange interactions. Derived entirely from ab initio band structure, the model allows for a non-Rashba spin-momentum locking and enables a direct coupling of the exchange field to the real spin of electrons. Using the BiTeI trilayer (hexagonal C3v symmetry) and the Si-terminated surface state of TbRh2Si2 (cubic C4v symmetry) as prototypes, we show that the exchange field induces strong, symmetry-dependent modifications of the band structure and plasmon dispersion. In BiTeI, it breaks the sixfold symmetry and leads to anisotropic, nonreciprocal plasmon modes, while in TbRh2Si2 it suppresses the characteristic triple spin winding and alters the plasmon damping. The results reveal that the interplay between SOC and exchange magnetism enables magnetic control of collective charge excitations in 2D spin-orbit systems beyond the Rashba paradigm.

Figures (18)

• Figure 1: Band structure of the BiTeI trilayer (a) and the Si-terminated centrosymmetric 31-layer slab (b) simulating the Si-terminated (001) surface of the paramagnetic TbRh$_2$Si$_2$. The high-symmetry k-lines are shown in graphs (e) for BiTeI and (f) for TbRh$_2$Si$_2$ by green arrows. Highlighted in cyan are the lowest conduction-band state of the trilayer, graph (a), and the Rh-related higher-energy surface state, graph (b). These states serve as prototypes for the model 2D systems (see text). The atomic structure of the trilayer (c) and of the upper half of the 31-layer slab (d) are shown along with the corresponding 2D Brillouin zones (BZ) (e) and (f).
• Figure 2: Upper row: Green arrows show the directions in the 2D BZ along which the eigenenergies in the lower row are presented. Red arrows show the direction of the exchange field $\bm{\mathcal{J}}$. Lower row: Band structure of BTI by the two-band $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian with the parameters listed in Table \ref{['tab:table1']} for the PM and FM phases with the exchange field along $\mathbf{\hat{z}}$ (a), $\mathbf{\hat{x}}$ (b), and $\mathbf{\hat{y}}$ (c). The respective values of the exchange parameter $\mathcal{J}$ are shown in the graphs. The green and orange lines correspond to the outer and inner branches of the split 2D states, respectively. The red numbers label the the positions $\mathcal{E}_{a}$, $\mathcal{E}_{\mathrm{R}}$, and $\mathcal{E}_{b}$ of the Fermi level $\mathcal{E}_{\mathrm{F}}$ for which the calculations of the dielectric function will be presented in the following.
• Figure 3: Spin-resolved Fermi contours (upper panels) and the non-orthogonality $\delta^{\lambda}_{\mathbf{k}}$ defined in Eq. (\ref{['delta_nonorth']}) as a function of $\varphi_{\mathbf{k}}$ (lower panels) for PM and FM BTI. In the FM phase, a $z$-directed exchange field with $\mathcal{J}_z = 250$ meV is implied. In the upper panels, the colored areas highlight the deviation of $\mathbf{S}^{\shortparallel \lambda}_{\mathbf{k}}$ from the classical Rashba in-plane spin at a given $\mathbf{k}$-point in the contour: the border of the areas is given by $|\mathbf{k}|+R\sin\delta^{\lambda}_{\mathbf{k}}$ with $R$ being the scaling factor. The green areas and lines correspond to the outer branch of the split 2D state, while the orange ones to the inner branch. The contours are calculated at the Fermi energy $\mathcal{E}_{\mathrm{F}}$ indicated by 1 in Fig. \ref{['fig2']}.
• Figure 4: Spin-resolved Fermi contours for the energies marked by red numbers in Fig. \ref{['fig2']} for three FM phases of BTI. For the $z$-directed field, in the cases 1 and 2 the exchange-interaction parameter is $\mathcal{J}=250$ meV, while in the case 3 it is $\mathcal{J}=500$ meV. For the $x$- and $y$-directed fields, $\mathcal{J}=35$ meV in all cases.
• Figure 5: Same as in Fig. \ref{['fig2']} but for TRS. The exchange field is along $\mathbf{\hat{z}}$ (a), along the diagonal $(\mathbf{\hat{x}}+\mathbf{\hat{y}})/\sqrt{2}$ (b), and along $\mathbf{\hat{x}}$ (c).