Theory

The setup of the underlying experiment is shown in Figure 1a. It consists (bottom-up) of an EuS substrate, a superconducting (Al) film, and a normal metal film that is separated from the superconductor by an oxide layer. The normal layer acts as the tunnel probe to measure the differential conductance of the superconductor and is assumed not to influence the system properties. Since the size of the detector electrode is not small (unlike the tip of a scanning tunneling microscope) and the FI affects the whole superconductor, we assume that the magnetization can be modeled by one magnetization direction that results from averaging over the internal magnetic structure. In the language of the circuit theory [38] this means that we can reproduce the whole system with a single node as depicted in Figure 1b. The superconductor is represented by the node that has a “source” (of coherence) term (marked with Δ) and “leakage” (of coherence) term that is characterized by the Thouless energy of the superconductor (εTh) and its normal-state conductance G. Additional pseudo terminals model the spin mixing angles δφ induced by the FI (top), the external field H (right).

Figure 1: (a) The experimental setup of the FI–S bilayer. The differential conductance is measured with help of the top normal contact (N), which is separated by an insulating barrier (I). (b) Circuit diagram to represent the FI–S bilayer in the quantum circuit theory. The superconductor is represented by the node, the Δ-source term, the εTh-leakage term, and its normal-state conductance G. The ferromagnetic insulator adds the δφn-term with magnetization directions κn for each channel, and in accordance with the experiment an external exchange field is added, here represented by the H-pseudo terminal.

To describe the FI–S bilayer as illustrated in Figure 1b within the circuit theory [38], we use the formalism for the boundary conditions for spin-dependent connectors developed in [30], which agrees with the results of [31]. This boundary condition (BC) for the Usadel equation was derived, and it was shown how this BC can be applied to a ferromagnetic insulator–superconductor bilayer system in the limit that the thickness d of the superconducting film is small compared to the coherence length. We define the spin-dependent Green’s function of the superconductor as Here, σ = ± denotes the spin index, and are Pauli matrices in Nambu space. We obtain the following equation that determines the Green’s function of the superconductor:

(1)

Here, εσ = ε + σμBH is the energy including the applied Zeeman field, εTh = ℏD/d2 is the Thouless energy, Gq = e2/h is the conductance quantum, and G = σNA/d is the conductance of the film (in the direction perpendicular to the interface of cross section A). D and σN are the diffusion constant and the normal-state conductivity of the film, respectively. Note, that due to the normalization condition for quasiclassical Green’s functions one has Due to the small coercivity of EuS the assumption of only one magnetization direction as in [30] is reasonable. This is why the Green’s functions decouple in spin space.

In the following, we will only use a single spin mixing angle δφn = δφ for simplicity to illustrate the results. However, the theory is not restricted to this case. Thus, we replace the sum over the channel index n in the matrix current conservation with the number of channels, that is, However, in a phenomenological way we assume that, effectively, only a certain fraction r ∈ [0,1] of scattering channels contributes to the spin mixing effect. Alternatively, we may say that spin mixing only occurs with a certain probability r. The corresponding equation then reads

(2)

Hence, the strength of the magnetic proximity effect can be expressed by the dimensionless parameter ε’ = rN(Gq/G)(εTh/kBTc), where kB is the Boltzmann constant, and Tc is the critical temperature of the bulk superconductor. Using the conductivity the density of states at the Fermi energy of the free electron gas, and for the number of channels per area, we can simplify this to where vF is the Fermi velocity. With the definition ξ0 = ℏvF/πΔ(T = 0) of the superconducting coherence length and the approximation Δ(T = 0) ≈ 1.76 kBTc, one finds ε’ ≈ 0.69rξ0/d. The parameter ε’ becomes smaller for increasing film thickness and decreasing fraction of spin-active channels.

With the above definitions, the BCS self-consistency relation is given by:

(3)

We defined the cutoff energy ΩBCS related to the upper limit of the phonon spectrum. In the following, we use ΩBCS = 100kBTc, and the coupling constant λ, which can be eliminated for the bulk superconductor in favor of the critical temperature Tc. After solving the fully self-consistent problem in order to obtain Δ, the differential tunnel conductance (measured as shown in Figure 1a) is found from the standard definition with the Fermi distribution f and the normal-state conductance GN of the tunnel probe. The density of states is given by Note, that the actual total density of states per volume is given as

We now discuss the self-consistency relation for different values of the parameter ε’. Figure 2 shows the phase diagram for values ε’ = [100–0.1], which for a d = 10 nm aluminium layer roughly translates into fractions r = [1–0.001]. The plotted curves are the phase boundaries between superconducting and normal state, with the superconducting phase at low temperature and small δφ. In general, the critical value δφc increases with decreasing ε’. For small ε’, spin mixing can no longer completely suppress superconductivity for T = 0 (note that the boundary conditions are periodic in δφ and hence the maximum spin mixing is reached at δφ = π). For small ε’ or high temperature, the phase transition is of second order. For larger ε’ and low temperature, the phase transition becomes of first order. In this case, the self-consistency relation becomes multi-valued, and a coexistence region appears. The solid and dashed lines represent the lower and upper boundary of the coexistence region, respectively. The coexistence region becomes larger for larger ε’ and correspondingly smaller δφc. In this regime, the effect of spin mixing is similar to a Zeeman splitting [39,40].

Figure 2: Phase diagram of the spin mixing angle δφ/π as a function of the normalized temperature T/Tc for values of ε’ = [100,0.1].

Now, we discuss the dependence of the density of states on the spin mixing angle δφ for different values of the parameter ε’. For the sake of clarity, the Zeeman splitting from the external field is ignored at this point. The changes in the density of states of the superconductor are dominated by two effects. On one hand, the initial peaks at the T = 0 superconductor gap Δ0 are spin-split into two separated peaks, each positioned depending on δφ and Δ. On the other hand, Δ self-consistently also depends on the spin mixing angle.

In Figure 3, we plot the density of states for T ≪ Tc with self-consistent Δ. For very thin layers (ε’ = 100) the peaks (initially at Δ) symmetrically split into their spin components. This behavior is also is similar to the Zeeman splitting in an applied field, as already measured, for example, in [41]. However, with decreasing ε’, the superconductivity persists for larger spin mixing angles and the behavior changes qualitatively until a completely different situation is found at ε’ = 0.1. Here, the outer peak position is nearly independent of δφ while only the inner peak moves towards (and across) the Fermi level. Another effect of larger spin mixing angles is that the inner peak is broadened, and finally becomes a wide and flat band. Besides this, the self-consistency relation for thin films produces the typical step-like first-order phase transition at the critical δφ (here always plotted for the upper branch of Figure 2), while especially in the case ε’ = 1 a significant shift of the peak positions is visible for larger spin mixing angles. For sufficiently small ε’, spin mixing has little effect on Δ.

Figure 3: Density of states in a superconductor in proximity to a ferromagnetic insulator indicated by the spin mixing angle δφ, with (from (a) to (d)) ε’ = 100,10,1,0.1, while the superconducting order parameter is evaluated self-consistently (T ≪ Tc).

Comparison of experiment and theory

To illustrate our model, we use it to fit experimental data obtained on a sample made of a superconducting aluminium film on top of the ferromagnetic insulator europium sulfide. Figure 4 shows a false-color scanning electron microscopy image of the sample, together with the experimental scheme. The sample was fabricated in a two-step procedure: First, a EuS film of 44 nm thickness was created by e-beam evaporation of EuS onto a Si(111) substrate heated to 800 °C. In a second fabrication step, aluminium/aluminium oxide/copper tunnel junctions were fabricated on the EuS film using e-beam lithography and shadow evaporation. The nominal aluminium film thickness was d = 10 nm. The differential conductance g = dI/dV of the tunnel junctions was measured as a function of the bias voltage V using standard low-frequency lock-in techniques in a dilution refrigerator at base temperatures down to T = 50 mK with an in-plane magnetic field B applied along the direction of the copper wires, as indicated in Figure 4. Details of film fabrication, magnetic properties, and experimental procedures can be found in [42,43].

Figure 4: False-color scanning electron microscopy image of the sample and experimental scheme.

Examples of the conductance spectra measured for different applied fields in one of the junctions are shown in Figure 5a. At small fields, the spectra exhibit a well-defined gap with negligible subgap conductance, indicating a defect-free tunnel barrier. Spin splitting of the density of states is clearly visible. The observed splitting greatly exceeds the expected splitting due to the Zeeman energy εZ = μBB (which is about 35 μeV at B = 0.6 T). The solid lines in Figure 5a are fits with our model. We have included orbital depairing in the fits, with an orbital depairing parameter [44]

(4)

for a thin film in an in-plane field. From known sample parameters we estimate Bc,orb ≈ 2 T and ε’ ≈ 70, which leaves us with Δ and δφ as free parameters. The fits give a good account of the observed spin splitting. The spin mixing angle extracted from the fits is plotted in Figure 5b. It is found to depend on the applied magnetic field over the entire field range. In contrast, the EuS magnetization is saturated above a few milliteslas in our film [42]. A similar dependence of the spin splitting on the applied field is commonly observed in EuS/Al structures [45,46], and the microscopic origin is yet unclear. A possible explanation are misaligned spins at the interface, which are nearly free and therefore gradually aligned by the applied field. The misaligned spins might be the result of partial oxidation of the EuS surface during sample transfer between our two fabrication steps. Lacking a microscopic model, we have attempted to fit the field dependence of δφ with a Brillouin function. The fit is shown as a line in Figure 5b. It is in reasonable agreement with the data up to about 0.6 T, with an effective angular momentum J ≈ 0.74ℏ. While the Eu2+ ions in EuS have J = 7/2 [47], the stable oxide of Eu is Eu2O3 with Eu3+ ions and J = 0 [48]. Therefore, a reduced effective J at the interface appears reasonable. Above 0.6 T, the data deviate downwards from the fit, and these data points were excluded from the fit. The deviation can be explained by Fermi-liquid renormalization of the effective spin splitting near the critical field [46,49], which is not included in our model.

Figure 5: (a) Differential conductance g as a function of the bias V for different applied magnetic fields B (symbols), and fits with our model (lines). (b) Spin mixing angle δφ (left axis) and pair potential Δ (right axis) as functions of the applied field B extracted from the fits (symbols), and fit of the spin mixing angle with a Brillouin function (line).