Spin-Orbit Twisted Spin Waves: Group Velocity Control

We present a theoretical and experimental study of the interplay between spin-orbit coupling (SOC), Coulomb interaction, and motion of conduction electrons in a magnetized two-dimensional electron gas. Via a transformation of the many-body Hamiltonian we introduce the concept of spin-orbit twisted spin waves, whose energy dispersions and damping rates are obtained by a simple wave-vector shift of the spin waves without SOC. These theoretical predictions are validated by Raman scattering measurements. With optical gating of the density, we vary the strength of the SOC to alter the group velocity of the spin wave. The findings presented here differ from that of spin systems subject to the Dzyaloshinskii-Moriya interaction. Our results pave the way for novel applications in spin-wave routing devices and for the realization of lenses for spin waves.

Figure 1

(a) Top plane: Raman incoming (${\mathbf{\kappa }}_i$) and outgoing (${\mathbf{\kappa }}_s$) photon wave vectors. $\mathbf{q}$ is the in-plane momentum of the spin wave probed by the Raman process. The amplitude and direction of $\mathbf{q}$ are controlled by $\theta$ and $\phi$, respectively. The magnetic field, parallel to ${\mathbf{e}}_z$, is always perpendicular to $\mathbf{q}$. Bottom plane: the spin-wave oscillation in real space is associated with an out-of-phase oscillation of the two Fermi disks in momentum space with respect to their equilibrium positions (gray circles). Electron spins remain parallel or antiparallel to $\mathbf{B}$. (b) Illustration of the spin-wave twisting caused by SOC. Bottom plane: the momentum-space motion twists the spins with respect to their equilibrium positions (gray vectors). A $z$-spin current parallel to ${\mathbf{q}}_0$ appears. Top plane: the spins now evolve in a moving wavelike reference frame (highlighted by the blue shading). Consequently, the spin waves are twisted with a phase ${\mathbf{q}}_0·\mathbf{r}$ (see text). [(c)–(e)] Electronic Raman spectra obtained by varying the momentum $\mathbf{q}$ for [(c) and (d)] $\phi =\pi /4$, $B=\pm 2\mathrm{T}$, and (e) $\phi =3\pi /4$, $B=2\mathrm{T}$. The low-energy Raman line, sharply peaked, is a signature of the spin wave. The smoother structure at higher energy is due to single-particle excitations. The spectrum highlighted in red for each case shows the spin-wave maximum energy.

Figure 2

[(a) and (b)] Lifting of the spin-wave chiral degeneracy by a momentum shift of the dispersions due to SOC: Momentum dispersion of energy (a) and linewidth (b) of the spin wave for $\phi =\pi /4$ and $B=\pm 2\mathrm{T}$. Dispersions are shifted by $q_s$ from $q=0$ with a mirror symmetry when inverting the magnetic field. (c) (circle) represents the $q_s$ dependence with $\phi$, which has been extracted from the dispersions measured for $\phi \in [(-\pi /4),(3\pi /2)]$. The red curve is a fit of $q_s(\phi )$ to the $x$ component of $-{\mathbf{q}}_0$ given by Eq. (3). [(d) and (e)] Universal linear relation between the linewidth and the energy of the spin wave: $(\eta -{\eta }_0)/{\eta }_2$ is plotted as a function of $\frac{2m^{*}}{{\hslash }^2}(\hslash \omega -Z)/S_{\mathrm{sw}}$; symbols of the same color are for a given in-plane angle $\phi$, but for various values of $q$. (d) $B=+1\mathrm{T}$; open (solid) symbols correspond to spin waves with wave vector $\mathbf{q}$ directed towards $-{\mathbf{e}}_x$ ($+{\mathbf{e}}_x$). (e) $B=+2\mathrm{T}$; solid symbols correspond, here, to the two extremal angles $\phi =\pi /4$, $3\pi /4$; open symbols are for other angles.

Figure 3

Optical gating of the spin-wave group velocity. The group velocity vector changes for $\phi =3\pi /4$ and $B=2\mathrm{T}$ as a function of momentum and electron density (note that the group velocity is purely longitudinal at $\phi =3\pi /4$): a spin wave with momentum $q=-0.5\mu {\mathrm{m}}^{-1}$ experiences an inversion of its group velocity when the density is changed from 1.8 to $2.7\times 10^{11}{\mathrm{cm}}^{-2}$. The red dots, where $q=q_s$, indicate a standing spin wave.