Coexistence of superconductivity and spin-splitting fields in superconductor/ferromagnetic insulator bilayers of arbitrary thickness

Ferromagnetic insulators (FI) can induce a strong exchange field in an adjacent superconductor (S) via the magnetic proximity effect. This manifests as spin splitting of the BCS density of states of the superconductor, an important ingredient for numerous superconducting spintronics applications and the realization of Majorana fermions. A crucial parameter that determines the magnitude of the induced spin splitting in FI/S bilayers is the thickness of the S layer $d$: In very thin samples, the superconductivity is suppressed by the strong magnetism. By contrast, in very thick samples, the spin splitting is absent at distances away from the interface. In this work, we calculate the density of states and critical exchange field of FI/S bilayers of arbitrary thickness. From here, we determine the range of parameters of interest for applications, where the exchange field and superconductivity coexist. We show that for $d>3.0{\xi }_s$, the paramagnetic phase transition is always of the second order, in contrast to the first-order transition in thinner samples at low temperatures. Here ${\xi }_s$ is the superconducting coherence length. Finally, we compare our theory with the tunneling spectroscopy measurements in several $\mathrm{EuS}/\mathrm{Al}/{\mathrm{AlO}}_x/\mathrm{Al}$ samples. If the Al film in contact with the EuS is thinner than a certain critical value, we do not observe superconductivity, whereas, in thicker samples, we find evidence of a first-order phase transition induced by an external field. The complete transition is preceded by a regime in which normal and superconducting regions coexist. We attribute this mixed phase to inhomogeneities of the Al film thickness and the presence of superparamagnetic grains at the EuS/Al interface with different switching fields. The steplike evolution of the tunnel-barrier magnetoresistance supports this assumption. Our results demonstrate on the one hand, the important role of the S layer thickness, which is particularly relevant for the fabrication of high-quality samples suitable for applications. On the other hand, the agreement between theory and experiment demonstrates the accuracy of our theory, which, originally developed for homogeneous situations, is generalized to highly inhomogeneous systems.

Figure 1

(a) Experimental setup and schematic view of the FI/S bilayer. The S layer in contact with the FI has a thickness $d$. See Sec. 5 for additional details on the experiment. (b) DoS of a homogeneous spin-split superconductor.

Figure 2

DoS for (a) a superconductor of intermediate thickness ($d=0.5{\xi }_0$) and (b) thick superconductor ($d=3{\xi }_0$) at different distances from the FI/S interface: $x=0$ (green), $x=d/2$ (red) and $x=d$ (blue). Note that the energy is normalized by the order parameter. The value of the exchange field is $ha/{\xi }_0=0.3\mathrm{\Delta }$, and we assume no magnetic impurities (${\tau }_{\mathrm{sf}}^{-1}=0$).

Figure 3

(a) DoS at $x=d$ for different values of the spin-flip relaxation rate. The samples have a thickness $d={\xi }_s$, and the value of the exchange field is $ha/{\xi }_s=0.5T_{c0}$. (b) Phase boundary for the second-order phase transition. The solid part of the lines represent the critical exchange field for the second-order transition, while the dashed lines correspond to the temperature range where the first-order phase transition occurs.

Figure 4

(a) Critical exchange field for different thicknesses of the superconductor and (b) temperature dependence of the critical thickness for different exchange fields in the magnetic impurity-free limit (${\tau }_{\mathrm{sf}}^{-1}=0$). The inset in (a) corresponds to the critical exchange field in the thin limit (homogeneous case). The dashed part corresponds to the temperature region for which the transition is of first order.

Figure 5

(a) The differential conductance of the S3 sample measured at 30 mK as a function of the external magnetic field and the voltage drop across the junction. The arrow indicates the direction of the magnetic field sweep. At $B=10\mathrm{mT}$ most of the bottom Al film undergoes a transition into the superconducting state. (b) $dI/dV$ curves at three different values of $B$ ($B=4, B=12$ and $B=20\mathrm{mT}$), indicated by dashed lines in (a). The solid lines correspond to the experimental data, whereas the dashed lines to the theoretical fitting. The values of the fitting parameters used in the theoretical model at $B=0$ are $\overline{h}=146\mu \mathrm{eV}, {\mathrm{\Delta }}_2=219\mu \mathrm{eV}$, and ${\tau }_{\mathrm{sf}}^{-1}=38.5\mu \mathrm{eV}$. (c) Effective exchange field, $\overline{h}$, and self-consistent order parameter, ${\mathrm{\Delta }}_2$ of the bottom Al layer divided by $\sqrt{2}$. The horizontal dashed lines represent the Chandrasekhar-Clogston critical field ${\mathrm{\Delta }}_{2,0}/\sqrt{2}$ in the absence of magnetic impurities. (d) Magnetoresistance of the bottom Al film adjacent to the EuS layer measured at $T=30$ mK.

Figure 6

(a) Magnetization loop measured for a continuous EuS film at 5 K, with the field applied in the in-plane direction. (b) Contour plot showing the resistance of Al wire adjacent to the EuS layer as a function of the external field and the temperature. The arrow indicates the sweep direction of the magnetic field. These measurements and the measurements shown in the next two panels were performed on the sample S3. (c) The hysteresis of the resistance of the Al wire adjacent to the EuS layer at $T=30$ mK. (d) The $B$-field dependence of the resistance of the Al wire adjacent to the EuS layer at different temperatures.