Effect of superconductivity on non-uniform magnetization in dirty SF junctions

TL;DR

The paper tackles how superconducting proximity in a tunnel SF junction with a thin disordered ferromagnet can destabilize uniform magnetization and induce a nonuniform texture. It develops a quasiclassical Usadel framework, derives a Landau-type free-energy functional with coefficients $\alpha$, $\beta$, and $\beta'$, and maps a phase diagram showing a resonance at $h=\gamma$ where proximity effects dominate. A helical magnetic texture is shown to be the energetically favored nonuniform state near the transition, with explicit scaling of the wavevector $q$ across regimes and an exponentially small $q$ at resonance. The results provide a quantitative description of the interplay between superconducting proximity and ferromagnetism, offering guidance for observing proximity-driven magnetic modulations in thin ferromagnetic layers coupled to superconductors.

Abstract

We study proximity effect in a tunnel junction between a bulk superconductor and a thin disordered ferromagnetic layer on its surface. Cooper pairs penetrating from the superconductor into the ferromagnet tend to destabilize its uniform magnetic order. The competition of this effect and the intrinsic magnetic stiffness of the ferromagnet leads to a second order phase transition between uniform and non-uniform magnetic states. Using the quasiclassical Usadel equation, we derive the Landau functional for this transition and construct the complete phase diagram of the effect. We identify a special point of "resonance" at which the characteristic energy scale of the proximity effect equals the exchange field of the ferromagnet. At this point, the uniform magnetic state is unstable even in the limit of large stiffness. We further explore the parameter regime far beyond the transition and determine the properties of the resulting strongly non-uniform magnetic state.

Figures (6)

• Figure 1: Schematic depiction of the SF junction. A bulk superconductor (blue) is brought into contact with a thin ferromagnetic layer (green). Cooper pairs from the superconductor penetrate into the ferromagnet and can establish there a nonuniform magnetic order.
• Figure 2: Phase diagram for the transition into a non-uniform magnetic state at zero temperature for different values of $\Delta$. Curves correspond to $\zeta = \alpha$ with $\alpha$ from Eq. (\ref{['alpha']}). The horizontal axis shows $\arctan(\gamma/h)$ in order to emphasize the symmetry $h \leftrightarrow \gamma$. On the vertical axis, the magnetic stiffness $\zeta$ is measured in units $\nu D \Delta$. The upper curve corresponds to Eq. (\ref{['alphalargegamma']}). The lowest curve approaches the limit of Eq. (\ref{['alphasmallgamma']}). Both limiting curves are symmetric under $\gamma \leftrightarrow h$ while the intermediate curves are very slightly asymmetric.
• Figure 3: Phase diagram for the transition into a non-uniform magnetic state in the limit $\gamma \gg \Delta$ at different temperatures. Transition occurs at $\zeta = \alpha$ with the value of $\alpha$ according to Eq. (\ref{['alpha']}). At zero temperature, the curve corresponds to Eq. (\ref{['alphalargegamma']}) and diverges at the point of resonance $h = \gamma$. For non-zero temperatures we also take into account suppression of the order parameter $\Delta(T)$ according to the standard BCS theory deGennesBook99. The horizontal axis is the same as in Fig. \ref{['fig::alpha delta']}. On the vertical axis, the magnetic stiffness $\zeta$ is measured in units $\nu D \Delta_0$ where $\Delta_0$ is the value of $\Delta$ at $T = 0$. Maximal possible value of $T$ is the critical temperature $T_C \approx 0.57 \Delta_0$.
• Figure 4: Dependence of $\beta$ [Eq. (\ref{['beta']}), upper panel] and $\beta'$ [Eq. (\ref{['betaprime']}), lower panel] on $\gamma/h$ for different values of $\Delta$ at zero temperature. Horizontal axis is the same as in Figs. \ref{['fig::alpha delta']} and \ref{['fig::alpha temperature']}. Parameter $\beta'$ diverges at $h > \gamma$ according to Eq. (\ref{["beta' zero temperature divergences"]}) hence only half of the horizontal axis in the lower panel is shown.
• Figure 5: Illustration of the helical state in the ferromagnetic layer, view from above. Wave vector $q$ points to the right and the direction of magnetization rotates in the horizontal plane with the aperture angle $\psi = \pi/2$, cf. Eq. (\ref{['n']}).