Cavity-mediated superconductor–ferromagnetic-insulator coupling

A recent proof of concept showed that cavity photons can mediate superconducting (SC) signatures to a ferromagnetic insulator (FI) over a macroscopic distance [A. T. G. Janssønn et al., Phys. Rev. B 102, 180506(R) (2020)]. In contrast with conventional proximity systems, this facilitates long-distance FI–SC coupling, local subjection to different drives and temperatures, and studies of their mutual interactions without proximal disruption of their orders. Here we derive a microscopic theory for these interactions, with an emphasis on the leading effect on the FI, namely, an induced anisotropy field. In an arbitrary practical example, we find an anisotropy field of $14–16\mu \mathrm{T}$, which is expected to yield an experimentally appreciable tilt of the FI spins for low-coercivity FIs such as Bi-YIG. We discuss the implications and potential applications of such a system in the context of superconducting spintronics.

Figure 1

Illustration of the setup. A thin ferromagnetic insulator and thin superconductor are placed spaced apart inside a rectangular, electromagnetic cavity. The FI is subjected to an aligning external magnetic field ${\mathbf{B}}_{\mathrm{ext}}$. The cavity is short along the $z$ direction, and long along the perpendicular $xy$ directions, causing cavity modes to separate into a bandlike structure. The FI and the SC are, respectively, placed in regions of maximum magnetic ($z=L_z$) and electric ($z=L_z/2$) cavity field of the ${\ell }_z=1$ modes, as defined in Sec. 2b1 and illustrated above by the colored field cross section on the right wall.

Figure 2

Illustration of the 123 coordinate system. $\mathbf{Q}$ is the photon momentum vector, and $\mathbf{q}$ is its component in the $xy$ plane. $\theta$ (single line) is the polar and $\phi$ (double line) the azimuthal angle associated with $\mathbf{Q}$ in relation to the $xyz$ basis. The 123 axes result from a rotation of the $xyz$ axes by an angle $\theta$ about the $y$ axis, followed by a rotation by an angle $\phi$ about the original $z$ axis. In the illustration, the 1 axis points somewhat outwards and downwards, the 2 axis points somewhat inwards and is confined to the original $xy$ plane, and the 3 axis aligns with $\mathbf{Q}$.

Figure 3

Feynman diagrams [54] of the bare cavity coupling to the FI and SC, and the resulting terms in the FI and SC effective actions after integrating out the cavity photons, where $G^{\mathrm{cav}}$ is the photon propagator.

Figure 4

Illustration of the setup used in the example given in Sec. 3a. A small, square FI and SC are placed spaced apart in the $y$ and $z$ directions inside a comparatively large cavity. Only a small portion of the cavity length in $y$ is utilized as the contributions by the various mediating cavity modes add constructively only over short distances. The FI and SC are nevertheless separated by hundreds of $\mu \mathrm{m}$, 2–5 orders larger than typical effectual lengths in proximity systems.

Figure 5

The magnitude and direction (arrows) of the effective anisotropy field [Eq. (72)] at $T=1\mathrm{K}$ as a function of the supercurrent momentum $\mathbf{P}$, for the simple case of a small FI ($l_x^{\mathrm{FI}}=l_y^{\mathrm{FI}}=10\mu \mathrm{m}$) relative to the cavity ($L_x=L_y=10\mathrm{cm}, L_z=0.1\mathrm{mm}$). The SC dimensions are $l_x^{\mathrm{SC}}=l_y^{\mathrm{SC}}=50\mu \mathrm{m}$, with a depth of $d_{\mathrm{SC}}=10\mathrm{nm}$. The FI and SC center points are separated by (a) $L_y^{\mathrm{sep}}=140\mu \mathrm{m}$ and (b) nothing (placed directly over each other). Observe the change in both the strength and direction of the anisotropy field. The plots were produced using Python with the numpy and matplotlib libraries.

Figure 6

Illustration of the setup with the SC placed at the magnetic antinode. The SC is subjected to an aligning external in-plane magnetic field ${\mathbf{B}}_{\mathrm{ext}}^{\mathrm{SC}}$. This setup is otherwise identical to the one illustrated in Fig. 1.

Figure 7

Absolute value (contour plot) and direction (arrows) of the averaged anisotropy field as a function of applied field strength and direction. The anisotropy field points opposite the applied field over the SC, following a $cos\varphi$ and $sin\varphi$ dependence for the $x$ and $y$ component, respectively. The inset shows the absolute value of the in-plane projection as a function of the field strength. The temperature is set to $T=0.5T_{c0}$. The cavity dimensions are $L_x=L_y=L=10\mathrm{cm}$ and $L_z=1\mathrm{mm}$, and the FI and SC have sides of length $0.001L$ in the $x$ and $y$ directions, and are placed at the center of the cavity. The thickness of the SC is $d_{\mathrm{SC}}=10\mathrm{n}\mathrm{m}$.