Proximity effect for superconductor/ferromagnetic nanostructure containing spin vortex

We theoretically study the effect of skyrmions and chiral bobbers [1] on a superconductor (S)/ferromagnetic (F) heterostructure critical temperature. In recent years many very small scale (1--100 nm) magnetic skyrmions and skyrmion lattices have been experimentally discovered in specially created materials [2,3]. In contrast to the previously discovered micron-scale spin vortices, such structures can have a significant effect on the critical temperature of the superconducting transition.

We have studied the S/F systems in the dirty limit, since this approximation is consistent with most experimental systems. We started with rotation of the matrix Usadel function $\hat{F}\left( \mathbf{r},\omega \right)$ [4] in spin space. The transformed Usadel equation has the form

$$\frac{D_{f}}{2}{\hat{\mathcal{D}}}^{2}\tilde{F}\left( \mathbf{r},\omega \right) - \left| \omega \right|\tilde{F}\ \left( \mathbf{r},\omega \right) - \frac{i I}{2}\text{sgn}\omega\left\{ \tilde{F}\left( \mathbf{r},\omega \right),\hat{\sigma}_3 \right\} = 0,$$

where $\hat{\mathcal{D}}\hat{f} = \nabla\hat{f} + [ \hat{\mathbf{A}},\hat{f} ]$, $\hat{\mathbf{A}} = \hat{U}\nabla{\hat{U}}^{- 1}$, $D_{f}$ is the diffusion constant, $\omega$ is the Matsubara frequency, $I$ is the effective exchange field. The rotation matrix $\hat{U}$ is chosen in such a way that in the transformed equation the term responsible for the interaction with localized spins becomes diagonal. After transformation the gradient transforms into the extended derivative $\hat{\mathcal{D}}$ in the boundary conditions. Self-consistency equation is invariant under such transformations. This procedure simplifies the boundary value problem in many practical cases.

For complex spin textures, such as spin vortices and domain walls, the Usadel equation does not reduce to an equation with constant coefficients after the unitary rotation. However, the transformed Usadel function phase changes much weaker along the SF boundary. This allows us to use an approximate approach to solve boundary value problem by neglecting some terms in the Usadel equation for the ferromagnetic layer. This approach allowed us to obtain a quantitative estimation of the effect on the superconducting critical temperature for almost any spin texture.

Using our approach, we calculated the critical temperature for S/F systems containing conical magnetization, Neel, Bloch, chiral skyrmions, chiral bobbers. As expected, the impact on critical temperature near the magnetic inhomogeneity is determined by its scale compared to the superconducting coherence length. According to our calculations, the described effect is very sensitive to the thickness of the superconducting layer and the border transparency. By special choice of layer thicknesses, it is possible to achieve that superconductivity occurs only in the spin vortex localization region. In this case the critical temperature is about 10--20\% of the bulk superconductor critical temperature.

The significant effect of nanoscale spin vortices on the critical temperature, combined with topological stability and low current density required for their movement [3], makes it possible effectively to use such systems as superconducting spin valves.

[1] A.B. Borisov, Physics-Uspekhi, \textbf{63}, 269 (2020).

[2] N. Romming \emph{et al}, Science, \textbf{341}, 636 (2013).

[3] J. Baumard \emph{et al}, Physical Review B, \textbf{99},
014511(2019).

[4] Y.V. Fominov, A. F. Volkov, K. B. Efetov, Physical Review B,
\textbf{75},104509 (2007).