Toggling Valley-Spin Locking and Nonlinear Optical Properties of Single-Element Multiferroic Monolayers via Light

Ferroicities in structural ferroic materials can effectively tune many different fascinating physical properties, leading to potential applications in the future. Usually, in order to break inversion symmetry in ferroelectrics, the material needs to be composed of multiple types of element. Here, we study a single element multiferroic material with time-reversal symmetry, monolayer $\mathrm{Bi}(110)$, which possesses multiple types of ferroic (ferroelectric and ferroelastic) order. By performing first-principles calculations combined with a group-theory analysis, we reveal that this multiferroic material exhibits various interesting properties, such as an out-of-plane persistent spin helix, valleytronics, and promising nonlinear optical properties, which depend on its orientation variant. We show further that under illumination with linearly polarized light, different orientation variants in the $\mathrm{Bi}(110)$ monolayer possess different free energies. Thus, one can use this noncontacting approach to control the thermodynamic stability of different types of order. The switching of the orientation variant is ultrafast and can occur in times on the order of picoseconds. During this process, no electron-hole pairs are generated. Therefore, one can eliminate waste heat during operation, ensuring good reversibility.

Figure 1

(a) Geometric structure of optimized $Pmn2\overline{1}$ $\mathrm{Bi}(110)$ monolayer (left panel, different viewing directions) and the phonon dispersion (right panel) along a high-symmetry q-path. (b) Geometric structure of $\mathrm{Bi}(110)$ monolayer in a phosphorenelike structure (Pmna, without corrugation and with no electrical polarization) and its phonon spectrum.

Figure 2

(a) Band dispersion of (left) pristine $\mathrm{Bi}(110)$ monolayer and (right) $\mathrm{Bi}(110)$ monolayer with a slight tensile strain. (b),(c) Spin textures (s_x, s_y, and s_z) of (b) conduction band and (c) valence band (around Γ) of pristine $\mathrm{Bi}(110)$ monolayer.

Figure 3

(a) Schematic plot of band dispersion and valleys coupled with left- and right-hand polarized light. The expectation value of s_z for each state is shown by colors (red to blue, spin-up to spin-down). (b) Coupling strengths |P₊| and |P₋|. (c) Variation of degree of circular polarization $\eta (\mathbf{k})$, which shows contrasting features in the two valleys. (d) z-component Berry curvature of the highest VB and lowest CB.

Figure 4

(a) Berry-curvature dipoles $D_x$ and $D_y$ as functions of the Fermi level E_F. The inset shows the k-resolved Berry curvature dipole D_x in k space when $E_F=-0.2\mathrm{eV}$. (b) z-component Berry curvature in k space. (c) Shift-current conductivity along the y direction when x-polarized (red curve) and y-polarized (blue curve) LPL is used for illumination. We use a scaling method (${\sigma }_{bb}^a(\omega )\times c/h$, where c and h are the thicknesses of the supercell and of the monolayer, respectively; see text) to eliminate the vacuum-space contribution. (d) k-resolved shift-current conductivities of selected peaks in (c).

Figure 5

(a) Schematic plot of low-frequency LPL illumination of a $\mathrm{Bi}(110)$ monolayer to induce ferroic-order switching. (b) Potential-energy profile without (blue curve) and with (red curve) optical illumination. TS, transition state. (c) Real part of frequency-dependent dielectric function and (d) optical absorption of FE1, FE2, and TS structures.

Figure 6

Spin texture of (a) lowest conduction band and (b) highest valence band around high-symmetry point Y $(0,\pi /b=0.69{\AA }^{-1})$.

Figure 7

Polarization-valley-spin locking in monolayer $\mathrm{Bi}(110)$. When the polarization is along the y direction (upper panel, ±P_y), the two unidirectional spin valleys lie along the k_y = 0 line (which is the horizontal axis of the band-structure plot, calculated by DFT). Polarization flipping leads to swapping of spin-up and spin-down. When the polarization direction is switched along the x direction (lower panel, ±P_x), the resulting band structure along k_x = 0 is plotted, and the two valleys are spin-up and spin-down, depending on the polarization direction.