Superfluid spin transport in ferro- and antiferromagnets

This paper focuses on spin superfluid transport, observation of which was recently reported in antiferromagnet ${\mathrm{Cr}}_2{\mathrm{O}}_3$ [Yuan et al., Sci. Adv. 4, eaat1098 (2018).]. This paper analyzes the role of dissipation in transformation of spin current injected with incoherent magnons to a superfluid spin current near the interface where spin is injected. The Gilbert damping parameter in the Landau–Lifshitz–Gilbert theory does not describe dissipation properly, and the dissipation parameters are calculated from the Boltzmann equation for magnons scattered by defects. The two-fluid theory is developed similar to the two-fluid theory for superfluids. This theory shows that the influence of temperature variation in bulk on the superfluid spin transport (bulk Seebeck effect) is weak at low temperatures. The scenario that the results of Yuan et al. are connected with the Seebeck effect at the interface between the spin detector and the sample is also discussed. The Landau criterion for an antiferromagnet put in a magnetic field is derived from the spectrum of collective spin modes. The Landau instability starts in the gapped mode earlier than in the Goldstone gapless mode, in contrast to easy-plane ferromagnets where the Goldstone mode becomes unstable. The structure of the magnetic vortex in the geometry of the experiment is determined. The vortex core has the skyrmion structure with finite magnetization component normal to the magnetic field. This magnetization creates stray magnetic fields around the exit point of the vortex line from the sample, which can be used for experimental detection of vortices.

Figure 1

Long-distance spin transport. (a) Spin injection to a spin-nonsuperfluid medium. (b) Spin injection to a spin-superfluid medium. (c) Geometry of the experiment by Yuan et al. [20]. Spin is injected from the left Pt wire and flows along the ${\mathrm{Cr}}_2{\mathrm{O}}_3$ film to the right Pt wire, which serves as a detector. The arrowed dashed line shows a spin-current streamline. In contrast to (a) and (b), the spin current is directed along the same axis $z$ as a magnetization parallel to the external magnetic field $\mathit{H}$.

Figure 2

Angle variables $\theta$ and ${\theta }_0$ for the case when the both magnetizations are in the plane $xz$ (${\phi }_0=\phi =0$).

Figure 3

Precession of magnetization $\mathit{m}$ around the direction of the magnetic field $\mathit{H}$ along the path around the vortex axis. (a) The geometry of the experiment [20] with the magnetic field (the axis $z$) in the plane of the ${\mathrm{Cr}}_2{\mathrm{O}}_3$ film. The vortex axis is normal to the film (the axis $y$). (b) Precession of the magnetization $\mathit{m}$ is shown in the plane $xz$ (the plane of the film). The path around the vortex axis (dashed lines) is inside the vortex core where the total magnetization is not parallel to $\mathit{H}$ ($\theta \ne 0$).