Magnetic Domain Wall Floating on a Spin Superfluid

We theoretically investigate the transfer of angular momentum between a spin superfluid and a domain wall in an exchange coupled easy-axis and easy-plane magnetic insulator system. A domain wall in the easy-axis magnet absorbs spin angular momentum via disrupting the flow of a superfluid spin current in the easy-plane magnet. Focusing on an open geometry, where the spin current is injected electrically via a nonequilibrium spin accumulation, we derive analytical expressions for the resultant superfluid-mediated motion of the domain wall. The analytical results are supported by micromagnetic simulations. The proposed phenomenon extends the regime of magnon-driven domain-wall motion to the case where the magnons are condensed and exhibit superfluidity. Furthermore, by controlling the pinning of the domain wall, we propose a realization of a reconfigurable spin transistor. The long-distance dissipationless character of spin superfluids can thus be exploited for manipulating soliton-based memory and logic devices.

Figure 1

A bilayer of an easy-$z$-axis magnet exchange coupled (with the coupling strength $g$) to an easy-$x\text{−}y$-plane magnet. A $z$-polarized spin current is injected from an incoherent spin source and propagates as a superfluid spin current through the easy-plane magnet. This spin current is $\propto \nabla \phi$, where $\phi$ is the azimuthal angle of the spin order parameter within the $x\text{−}y$ plane. This spin current is interrupted and absorbed by a domain wall in the easy-axis magnet, where it is converted into its sliding motion at speed $v$.

Figure 2

The model of a domain wall of width $\lambda$ coupled to a spin superfluid. The domain wall divides the bilayer into three regions: (I) up domain, (III) down domain, and (II) the domain wall. A spin current, $J_s^{\mathrm{sh}}$, is injected on the left by converting a charge current, $j$, into a spin accumulation via the spin Hall effect. Upon reaching the domain-wall region, a portion of this spin current, $J_{\mathrm{\Phi }}$, is absorbed from the easy-plane magnet by the domain wall. The resultant dynamics of the domain wall is characterized by the generalized coordinates $X$ and $\mathrm{\Phi }$, parametrizing its position and the associated azimuthal angle. The dynamics of the spin superfluid pumps a spin current, $J_s^p$, back to the contact. (Bottom panel) The corresponding superfluid spin current flowing in the easy-plane magnet, as obtained by plotting $-At{\partial }_x\varphi$.

Figure 3

(a) For a given exchange coupling $\tilde{g}\equiv g/Kt$, two regimes for domain-wall motion are obtained. A steady-state regime with linearly increasing velocity ($\tilde{v}$) and oscillatory motion above a critical value of the injected spin current ${\tilde{J}}_s$. The broken line plots the analytical result from Eq. (7). (Inset) The critical ${\tilde{J}}_s$ increases linearly with $\tilde{g}$. The broken line shows the analytical result from Eq. (9). (b) The spin current detected at the right end of the bilayer ($J_s^{\mathrm{out}}$) exhibits nonlinear behavior in the presence of a pinned domain wall. When the injected spin current, $J_s^{\mathrm{in}}$, is below (above) a critical breakdown current, $J_s^{\mathrm{out}}=0$ ($J_s^{\mathrm{out}}\ne 0$). Solid and broken curves plot the nonlinear characteristics for $\lambda =10$ and 5 nm, respectively. The nonlinearity can be used to construct a transistor, as indicated by the vertical dashed-dotted line. Fixing $J_s^{\mathrm{in}}$ and changing $\lambda$ by an external gate switches the device from an off ($J_s^{\mathrm{out}}=0$) to an on ($J_s^{\mathrm{out}}\ne 0$) state. These off and on states are depicted schematically in the insets.