Tailoring superconducting states in superconductor-ferromagnet hybrids

Figure 1 shows for all SF samples a schematic sketch of the magnetic domains in FePd along the out-of-plane direction (c-direction) of the L1₀-phase and the formation of closure domains, together with the measured domain pattern (obtained by MFM) and the magnetic hysteresis loops M(μ₀H_ext) at 300 K. The top (blue) layer in figure 1(a) denotes the superconducting Nb layer and the bottom is the FePd layer where the magnetic domains and their orientations are depicted with different colours and arrows, respectively. Depending on the strength of PMA, a formation of closure domains is assumed, which have also been reported earlier [27, 28] on MBE grown FePd thin films. The quality factor Q denotes the strength of magnetocrystalline anisotropy in the ferromagnetic layer: for Q < 1, the sample has an in-plane easy magnetization axis and weak PMA, whereas for Q > 1 the easy magnetization axis is out-of-plane with high PMA [24, 29]. Q is calculated by the ratio of the magnetocrystalline anisotropy constant K_u and the shape anisotropy constant K_sh, which can be determined by measuring the magnetic hysteresis in-plane and out-of-plane to the sample surface. For details we refer to [24].

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Sample SF_high shows a maze domain structure and strong PMA with Q = 2.0 ± 0.1, see figures 1(b) and (c). Sample SF_low with Q = 0.95 ± 0.02 has an in-plane easy axis with low PMA, evolving in parallel organized stripe domains after in-plane oscillating demagnetization. Also SF_mid shows a parallel formation of magnetic domains but with high PMA and Q = 1.30 ± 0.02, resulting from the combination of two FePd layers with different degree of PMA. All samples comprise a maze domain structure after saturation in an in-plane or out-of-plane magnetic field (for details on the domain configuration see reference [24]).

Figure 2(a) shows a high-angle-annular-dark-field (HAADF) STEM image of SF_mid with its sharp interface between FePd and Nb, marked by the white dotted line. The red and yellow dots denote Fe and Pd atoms organized in monolayers in the L1₀-ordered phase, respectively. The HAADF image of the reference sample SIF_high (see figure 2(b)) reveals that FePd and Nb layers are well separated by MgO, preventing proximity effects at the interface between the S and F layers.

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Due to the high Curie temperature of FePd (∼723 K [30]), the hysteresis loop of SF_high at room temperature exhibits the behaviour of an F thin film with a maze domain structure and with the easy axis aligned along the out-of-plane direction (see figure 3): coming from saturation, the magnetization drops fast while cylindrical domains in opposite direction to H_ext ('reversed domains') nucleate. As H_ext is reduced, the cylindrical domains evolve into a maze structure which results in a linear magnetization dependency [27, 31]. The inset in figure 3 shows the small but finite difference in the linear part of the hysteresis loops for its branches with increasing and decreasing H_ext.

The magnetic domain structure and saturation magnetization do not change significantly in the temperature (T) range from 6 to 10 K (near the critical temperature T_c = 6.958 ± 0.001 K for SF_high). Down to a temperature of 6 K, the saturation field is clearly visible and not altered by the onset of superconductivity. Below T_c, an additional signal is observed due to the magnetic field repulsion of the superconducting Nb layer. Here, the hysteresis loop exhibits an overlap of the ferromagnetic signal from the FePd layer and the superconducting response from Nb. Similar results were obtained for superconducting MgB₂ and ferromagnetic Co composites studied by Altin et al [32], who have interpreted their hysteresis loops as a sum of respective single superconducting and ferromagnetic signals.

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From the magnetization measurements reported in figure 3 we conclude that superconductivity and ferromagnetism coexist below T_c and that the domain formation and its magnetic field dependence remain unchanged. Hysteresis loops measured for SF_low and SF_mid exhibit a similar behaviour to that observed for SF_high (see supplementary figure S3).

The resistivity dependence on an out-of-plane H_ext, ρ(μ₀H_ext), at given T for SF_high, SF_low and SF_mid is shown in figures 4(a)–(c), respectively. To confirm that the magnetoresistance features in ρ(μ₀H_ext) originate from stray fields, we plot in figure 4(d) the ρ(μ₀H_ext) curves for the high-PMA reference sample SIF_high with an insulating layer between S and F. For this reference sample, any magnetoresistance feature in ρ(μ₀H_ext) must originate due to stray field effects other than to a superconducting proximity effect, which is suppressed by the presence of an I layer at the S/F interface. The measurement loops start at the negative saturation field −H_sat of the samples. Subsequently, H_ext is ramped to +H_sat (red lines) and then back to −H_sat (black lines).

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For SF_high and its corresponding SIF_high with high PMA, three resistivity minima are clearly visible at T across the superconducting transition: one sharp minimum near zero field and two broad minima at H_ext = ±100 mT. In section 4 we will show that these minima correspond to the formation of DWS and RDS states, respectively. As T is progressively increased across the superconducting transition, the broad resistivity minima vanish and only the superconducting state near zero applied field survives. This behaviour suggests that the state of the system around zero-field has a higher T_c compared to the superconducting states corresponding to the broad minima at ±100 mT. Finite resistivity values in the DWS and RDS states are due to percolation effects and the resistivity shown in figure 4 originates from an overlap of superconducting and non-superconducting domains. As T is increased above the superconducting transition, the superconducting state is destroyed. It must be noted that, at T at the bottom of the superconducting transition, the resistivity drops to zero in the whole range between the superconducting upper critical fields ±B_c2, as the sum of the applied field and the stray fields (B_d) satisfies the condition |μ₀H_ext + B_d| < B_c2.

In SF_mid also one sharp resistivity minimum is observed at 7 K, vanishing at lower T, as shown in figure 4(c). At 6.3 K (with T_c = 6.223 ± 0.001 K of SF_mid), two broad resistivity minima appear at the same H_ext = ±100 mT as for SF_high and SIF_high.

In contrast, the minimum near zero applied field at the highest temperature shown in figure 4(b) for SF_low is less sharp and no well resolved resistivity minima are observed at ±100 mT even at lower temperatures.

The dependence of T_c upon a constant external field μ₀H_ext is shown in figure 5 and is extracted from resistivity measurements in dependence of temperature ρ(T), measured independently from the magnetotransport measurements shown in figure 4. We define T_c here as the maximum of the first derivative of ρ(T). In section 4, the magnetic field dependence of T_c reported in figure 4 is compared with a theoretical model by Aladyshkin et al [15], proving that such dependence can be ascribed to DWS and RDS generated by stray fields.

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The data in figure 5 suggest that all samples with high magnetocrystalline anisotropy and Q > 1 show a parabolic dependence of μ₀H_ext near T_c, with a sharp transition into the linear T dependence expected for a bulk superconductor in an external field, at H_ext ≈ 600 mT. The linear behaviour of T_c versus H_ext for a bare S film is also confirmed by the data which we collect on a reference sample of Nb grown on MgO(001) substrate with similar thickness as that used for the samples SF_high, SF_low and SF_mid (see dashed line in figure 5(a)). This effect is reduced in SF_low with a smooth transition to bulk superconductivity. The parabolic temperature dependence near T_c becomes better visible in figure 5(b): after a first increase in T_c, ${\mu }_{0}^{2}{H}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{2}$ follows a linear temperature dependence. Unfortunately, for SF_mid only measurements very near T_c were performed, but they show the same trend as for SF_high.

From μ₀H_ext in figure 5(a) the highest critical field at T = 0 K for an out-of-plane applied field, B_c,⊥(0), can be extracted from the linear part measured above H_ext ≈ 600 mT. From this B_c,⊥(0) value, the Ginzburg-Landau coherence length at T = 0 K parallel to the sample surface, ξ_GL,||(0), is calculated using B_c2,⊥(0) = $\frac{{{\Phi}}_{0}}{2\pi {\xi }_{\mathrm{G}\mathrm{L},{\Vert}}^{2}\left(0\right)}$ for anisotropic coherence lengths [33, 34]. Table 1 shows the Q-values, Nb-thicknesses d_Nb, FePd domain width D_FePd (obtained by MFM), T_c values, ${B}_{\text{c,}\perp }^{{\ast}}\left(0\right)$, and ξ_GL,||(0) of all samples. We use the notation ${B}_{\text{c}}^{{\ast}}$ other than B_c2 since we observe critical field values much higher than those reported in the literature (e.g. 1 T for a 40 nm-thick Nb bare film [33]). The high ${B}_{\text{c}}^{{\ast}}$ values can be explained as due to DWS, as discussed in section 4.

Table 1. Magnetic and superconducting properties of SF_high, SF_low and SF_mid and comparison Nb/MgO (this work) and Nb/Al₂O₃ (obtained from [33]) reference layers. Q denotes the strength of PMA as explained in the text, d_Nb is the Nb layer thickness, D_FePd the FePd domain width, T_c the critical temperature in zero applied field, and ${B}_{\text{c,}\perp }^{{\ast}}\left(0\right)$ the upper critical field in out-of-plane direction and ξ_GL,||(0) the parallel Ginzburg–Landau coherence length at T = 0 K.

Sample	Q	dNb(nm)	DFePd(nm)	Tc(K)	${B}_{\text{c,}\perp }^{{\ast}}\left(0\right)$(T)	ξGL,||(0)(nm)
SFhigh	2.0 ± 0.1	39 ± 2	120 ± 3	6.958 ± 0.001	3.60 ± 0.03	9.6 ± 0.1
SFlow	0.95 ± 0.02	32 ± 2	76 ± 3	4.605 ± 0.001	1.2 ± 0.1	17 ± 2
SFmid	1.30 ± 0.02	37 ± 2	107 ± 3	6.223 ± 0.001	4.0 ± 0.2	9.1 ± 0.7
Nb/MgO 1		41 ± 1		8.645 ± 0.001	2.51 ± 0.01	11.45 ± 0.07
Nb/MgO 2		28 ± 1		8.457 ± 0.001	1.0 ± 0.1	18 ± 3
Nb/Al2O3 [33]		40				10.4

In L1₀-ordered FePd films, the out-of-plane axis denotes the easy axis of magnetocrystalline anisotropy. Only SF_low exhibits an easy axis in the sample surface plane as confirmed by magnetization measurements (see figure 1). Figure 6 shows the resistivity as a function of an in-plane applied magnetic field ρ(μ₀H_ext) with H_ext applied along the 〈100〉 crystal axis at given T for all SF samples as well as for the reference sample SIF_low. The magnetic domain configuration comprises a maze domain structure after saturation, unless the sample is demagnetized. Hence, in these measurements we do not expect an in-plane magnetic anisotropy, which was confirmed by magnetization measurements performed as a function of the in-plane direction of H_ext (see supplementary figure S6).

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Samples SF_high and SF_mid with high out-of-plane magnetocrystalline anisotropy (Q > 1) show only one resistivity minimum near zero field. In contrast, SF_low shows two resistivity minima and one local resistivity maximum at the coercive field H_coerc at low temperatures. The SIF_low reference sample shows the behaviour of a conventional type-II superconducting layer with a broad resistivity minimum inside the range $\vert {\mu }_{0}{H}_{\text{ext}}+{B}_{\mathrm{d}}\vert {< }{B}_{\text{c}}^{{\ast}}$ and a sharp increase to the normal-state resistivity above ${B}_{\text{c}}^{{\ast}}$. Therefore, by comparing the data for SF_low and SIF_low, we conclude that the magnetoresistance features observed for SF_low cannot be due to stray fields, but must be instead related to a genuine S/F proximity effect (unlike for the out-of-plane data reported in figure 4).

Figure 7 shows the dependence of T_c and magnetization of SF_low as function of H_ext. T_c was obtained from ρ(T) measurements in a constant applied field after saturating the sample in a negative field of μ₀H_ext = −1.5 T. The two maxima in T_c and the minimum in T_c at H_coerc are directly related to the minima and the maximum in ρ(μ₀H_ext) in figure 6, respectively.

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Preliminary information on the formation of DWS and RDS can be obtained from resistivity measurements as a function of an external out-of-plane magnetic field ρ(μ₀H_ext), see figure 4. As mentioned in the introduction, superconductivity nucleates where the overlap of μ₀H_ext with the stray fields B_d of FePd leads to a minimum in the total magnetic field strength [10]. In zero applied field, the stray fields are smallest on top of domain walls. Hence, in zero applied field and near T_c, superconductivity will nucleate close to the domain walls [16]. An external magnetic field applied perpendicular to the sample surface induces RDS with a minimum in resistivity where the external applied magnetic field cancels out the stray fields of the magnetic domains in reverse direction [17]. This mechanism is schematically illustrated in figure 8.

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Near T_c, B_c2 of Nb is in the range of the stray field values of FePd, so that superconductivity can be suppressed or reinforced through small variations in H_ext. As a result, DWS and RDS can be observed as separate minima in the resistivity measurement as a function of field. In SF_high and SIF_high (see figures 4(a) and (d)), the higher T_c of the sharp minimum near zero field is attributed to a formation of DWS [37], similar to an increased T_c for surface superconductivity in a thin S sheath near its surface [15, 38]. As reported by Buzdin and Mel'nikov [39], in an S/F domain structured system with stray fields penetrating the S layer, superconductivity will be destroyed at high temperatures due to the pair-breaking effect of the stray fields. By lowering T across the S transition, superconductivity nucleates first at the boundaries of magnetic domains, where in-plane magnetic moments of the closure domains reduce the stray fields inside such areas. An increase in μ₀H_ext lowers the effect of DWS while at the same time RDS evolves due to the local reduction of stray fields generated by reversely magnetized out-of-plane domains [39]. Therefore, the three resistivity minima shown in figures 4(a) and (d) can be associated with DWS near H_ext = 0 and RDS at H_ext = ±100 mT. Upon a further reduction in T, superconductivity is stable above both the domain walls and the domains, resulting in one broad resistivity minimum within $\vert {\mu }_{0}{H}_{\text{ext}}+{B}_{\mathrm{d}}\vert {< }{B}_{\text{c}}^{{\ast}}$.

Sample SF_mid with medium magnetocrystalline anisotropy (Q = 1.3 ± 0.2) also clearly shows the DWS state at 7 K and the RDS states at ±100 mT at 6.3 K. At this temperature, the DWS at H_ext = 0 cannot be resolved due to the broad RDS transitions (see figure 4(c)).

In SF_low no well-resolved RDS states can be observed in figure 4(b) due to the low out-of-plane magnetic stray fields (Q = 0.95). The amount of the increase of T_c in the RDS states (and therefore the strength of the resistivity minima) depends strongly on the strength of the stray fields B_d [15]. The magnetic stray fields interact with both electrons in one Cooper pair via the Lorentz force. Due to the opposite momenta of the electrons in a spin-singlet Cooper pair, the Lorentz force leads to a circulation of the two electrons around the penetrating magnetic field (so called 'orbital effect') [40, 41]. This is as well the origin for the vortex state of a type-II superconductor, where the Cooper pairs circulate around each magnetic field vortex. A measure for the change in T_c due to orbital effects is given by [15].

$$\begin{equation}{\Delta}{T}_{\text{c}}^{\text{orb}}=2\pi {B}_{\mathrm{d}}{T}_{\text{c0}}{\xi }_{\text{GL}}{\left(0\right)}^{2}/{{\Phi}}_{0}.\end{equation} \tag{ 1 }$$

Here, ${\Delta}{T}_{\text{c}}^{\text{orb}}$ denotes the change in T_c due to orbital mechanisms, T_c0 the critical temperature in zero field, B_d the maximum stray field of the F layer, ξ_GL(0) the Ginzburg–Landau coherence length at T = 0 K and Φ₀ the magnetic flux quantum. The dependence of ${\Delta}{T}_{\text{c}}^{\text{orb}}$ on B_d can explain the difference in the evolution of RDS in the SF samples: strong out-of-plane anisotropy leads to large ${\Delta}{T}_{\text{c}}^{\text{orb}}$ values in the H_ext region corresponding to RDS (see figures 4(a) and (d)), whereas lower anisotropy results in smaller ${\Delta}{T}_{\text{c}}^{\text{orb}}$ values (see figures 4(b) and (c)).

Different resistivity values at the same field position in up and down oriented field ramping are caused by the hysteretic response of ferromagnetic FePd with a small but finite difference in magnetization at the same field position in field increasing and decreasing state. In the RDS state, superconductivity nucleates over domains in reverse direction to H_ext. Starting from +H_sat, the area of such reversed domains is smaller than the area of domains parallel to H_ext if H_ext > 0, and larger if H_ext < 0. Yang et al [10] observed the same resistivity hysteresis caused by a magnetic hysteresis of their ferromagnetic substrate BaFe₁₂O₁₉ using field-dependent MFM measurements.

The parabolic temperature dependence in gure 5 can be interpreted as an indication for 2D superconducting behaviour [33], which arises when the thickness of the superconducting regions are small compared to the Ginzburg–Landau coherence length ξ_GL or the effective London penetration depth Λ of the thin Nb film [20, 42]. DWS can account for such behaviour: since the rotation axis of Cooper pairs is defined by the field direction, size constraints of the superconducting state in an out-of-plane applied field are given by lateral structures such as domain walls with a size smaller than the superconducting coherence length ξ_GL(T_c). At about 600 mT, the saturation magnetization of the magnetic domain structure in the FePd layer is reached (see figure 3, field increasing branch) and the domain-superconductivity of SF_high, SIF_high and SF_mid turns sharply into bulk superconductivity. The smooth transition of SF_low to bulk superconductivity in figure 5 originates from the non-collinear magnetic moments in this sample.

A confirmation of the existence of isolated DWS and RDS states can be derived from the dependence of T_c on μ₀H_ext as shown in figure 5(a). If the Ginzburg–Landau superconducting coherence length, a measure of the Cooper pair size, near T_c is smaller than half the domain size D_FePd, superconducting nuclei in the DWS and RDS states are well separated. Following [43], ξ_GL(T_c) can be calculated by (2):

$$\begin{equation}{\xi }_{\text{GL}}\left({T}_{\text{c}}\right)={\xi }_{\text{GL,ref}}\left(0\right)/\sqrt{1-\frac{{T}_{\text{c}}}{{T}_{\text{c,}\;\text{ref}}}}.\end{equation} \tag{ 2 }$$

Here, ξ_GL,ref(0) is the GL coherence length at T = 0 K and T_c,ref the critical temperature in zero field of the reference samples Nb/MgO.

For SF_high and SF_mid the condition 2ξ_GL,||(T_c) < D_FePd is fulfiled, and isolated DWS/RDS is possible. In SF_high 2ξ_GL,||(T_c) = 54.4 ± 0.5 nm is much larger than the estimated Bloch wall width of ${D}_{\text{DW}}\enspace =\enspace \pi \sqrt{A/{K}_{\text{u}}}\enspace =\enspace 8.1{\pm}0.6\enspace$nm [44] with A = 10⁻¹¹ J m⁻¹ for FePd thin films [45] and K_u = 1500 ± 200 kJ m⁻³ calculated from the hysteresis loops [24]. For a confinement of superconductivity on the domain walls this result supports a 2D superconducting behaviour as discussed above. From a comparison between ρ(μ₀H_ext) measurements of SF_high and its reference sample SIF_high (see figures 4(a) and (d)), we argue that the magnetoresistance features observed in these samples originate from stray fields and we support this claim by fitting a model derived from Aladyshkin et al [15]. Here, an S thin film is placed onto an F with PMA. The model is based on a phenomenological Ginzburg–Landau approach and takes into account the effect of an applied H_ext. The reported DWS in reference [15] is purely related to the stray fields generated by the F layer. The critical temperature for this case is given by:

$$\begin{equation}{T}_{\text{c}}\left(b\right)={\Delta}{T}_{\text{c}}^{\text{orb}}\left(\frac{1}{2}-{E}_{\mathrm{min}}\right){b}^{4}+{\Delta}{T}_{\text{c}}^{\text{orb}}\left(2{E}_{\mathrm{min}}-\frac{1}{2}\right){b}^{2}+{T}_{\text{c}}\left(0\right),\end{equation} \tag{ 3 }$$

with b = $\frac{{\mu }_{0}{H}_{\text{ext}}}{{B}_{\mathrm{d}}}$. E_min is an eigenvalue of the Ginzburg–Landau equation for the highest possible applied field with superconducting nucleation in special boundary conditions [38], in this case given by the domain-wall-superconductivity. The calculation is based on a model where the domain wall width D_DW is much smaller than the Ginzburg–Landau coherence length: D_DW ≪ ξ_GL [15]. Figure 9 shows μ₀H_ext(T_c) for SF_high close to T_c and the corresponding fit to equation (3). At μ₀H_ext = 0 mT in figure 9, T_c increases due to DWS over Bloch domain walls and closure domains with finite thickness, which are not assumed in the fit to equation (3).

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The maximum stray field B_d = 108 mT corresponds to the minimum of the field increasing branch of ρ(μ₀H_ext) in figure 4(a) and is given as fixed parameter in the fit. A rough estimation of the stray field strength of FePd on the Nb surface (using equation (1) in the supplementary information from a model given by [46] with z = 40 nm distance, see figure S4) yields a value of B_d = 230 mT, which is higher than the measured value of B_d = 108 mT. This can have several reasons, e.g. a reduction of field penetration due to the superconducting screening. Also, the long-range order of the FePd L1₀-phase is incomplete as demonstrated by the HAADF STEM measurements. In figure 2(a) on the left side of the FePd phase, the layered structure of the L1₀-phase with Fe and Pd monolayers is clearly visible, whereas on the bottom right it cannot be distinguished between Fe and Pd. It indicates a planar defect, which is inclined to the observation direction. In figure S5 in the supplementary, a stacking fault between Fe and Pd planes from the left to the right part of figure S5 indicates the existence of further lattice defects reducing the long-range order of the L1₀-phase in the FePd layers. Another possible reason for this discrepancy in B_d is that the model used for stray field computation assumes infinitely-thin domain walls, which does not strictly apply to our FePd films.

All fit parameters are shown in table 2. The fitted value of E_min = 0.32 ± 0.04 is slightly lower than for surface superconductivity with E_min = 0.59 [38]. E_min can be converted into the highest critical field ${B}_{\text{c}}^{{\ast}}$ by ${E}_{\mathrm{min}}=\enspace -mc\alpha /\left(e\hslash {B}_{\text{c}}^{{\ast}}\right)$, with α being the first expansion coefficient from the Ginzburg Landau theory. Following the well known relation B_c3 = B_c2/E_min = B_c2/0.59 for surface superconductivity [38], this results into a higher critical field than for conventional surface superconductivity. Yang et al [10] observed a value of E_min = 0.37 for the nucleation of DWS, using the same model. They explained the difference to the surface superconducting value of E_min = 0.59 with their high domain wall width of D_DW = 200 nm, which exceeds ξ_GL, whereas the model is based on D_DW ≪ ξ_GL [15]. It shows that DWS with both, large and small domain wall width, can be described by equation (3), and that E_min for DWS differs slightly from E_min for surface superconductivity. Assuming that in the resistivity measurements the obtained critical field ${B}_{\text{c}}^{{\ast}}$ is related to the surface critical field B_c3, this can explain the high values of ${B}_{\text{c}}^{{\ast}}$ in table 1.

We conclude that the observed effects in figure 4 for SF_high and SF_mid in an out-of-plane applied magnetic field are arising from stray field generated, isolated DWS and RDS states. The higher T_c of the S/I/F structure compared to the S/F structure of SF_high clearly indicates a proximity coupling [35, 36], but the magnetoresistance behaviour can be purely explained based on DWS and RDS states other than based on spin-triplet generation.

Table 2. Fit parameters extracted from T_c(H_ext) of sample SF_high for a model assuming DWS-like behaviour.

Parameter	Value
Tc(0)	6.954 ± 0.001 K
${\Delta}{T}_{\text{c}}^{\text{orb}}$	0.15 ± 0.04 K
Emin	0.32 ± 0.04

For an external magnetic field H_ext applied in the sample surface plane, no magnetoresistance features related to DWS can be observed. In figure 6, only SF_low exhibits one sharp resistivity maximum near the coercive field H_coerc, which cannot be observed for the samples SF_high, SF_mid, and the reference sample SIF_low in the same applied field orientation.

Zdravkov et al [22] have measured the resistivity versus field characteristics in an S/F1/F2 spin-valve structure and predicted a sharp maximum in ρ(μ₀H_ext) at H_coerc of their sample due to the generation of LRTC (as defined in the introduction), which exhibit the highest density at H_coerc. This resistivity maximum corresponds to a minimum in T_c and arises due to the lower T_c value of LRTC compared to spin-singlet Cooper pairs in such a system [36]. In general, a transition of Cooper pairs into the proximity coupled layer lowers the T_c value [47] for both, spin-singlet and spin-triplet Cooper pairs. However, due to their spin alignment LRTC exhibit larger penetration depths ξ_F into the F layer. In an S/F1/F2 spin-valve, Fominov et al [36] predicted an additional reduction in T_c compared to the S/F1 bilayer for a non-collinear alignment of F1 and F2 which generates LRTC with S_z = ±1 and large ξ_F. In literature, different designations for ξ_F are used. Originally, ξ_F was defined by de Gennes [47] as 'coherence length'of Cooper pairs inside the proximity coupled layer. Other authors refer to ξ_F as 'characteristic length of superconducting correlation decay' [3] or 'penetration depth into F' [48], as it is used here.

The ρ(μ₀H_ext) measurements and μ₀H_ext(T_c) of SF_low in this work (see figures 6 and 7, respectively) show exactly the same sharp resistivity maximum and a minimum in T_c at H_coerc as observed by Zdravkov et al [22]. Sample SF_low consists of high in-plane magnetic moments and low PMA, still showing a lateral magnetic domain structure. This indicates the formation of large closure domains and non-collinear magnetic moments as shown in the schematic drawing of figure 1(a). The equilibrium lateral domain thickness of SF_low was determined by MFM at room temperature to be D_FePd(SF_low) = 76 ± 3 nm. This is larger than the Cooper pair coherence length near the critical temperature of SF_low with ξ_GL,||(T_c) = 27 ± 4 nm (obtained from equation (2)), so that the magnetic inhomogeneity existing in SF_low can affect the superconducting parameters. The non-collinear alignment favours the generation of LRTC as discussed above. In contrast to Zdravkov et al, only one F layer, however, is present in our samples. Still we argue that LRTC with S_z = ±1 are generated with highest density at H_coerc as a result of the non-collinear magnetic texture present in our F films. They penetrate into the F layer over long distances and lower the density of spin-singlet pairs in the Nb layer, resulting in a lower T_c value [23].

The assumption of a generation of spin aligned triplet Cooper pairs in SF_low is additionally supported by the results of the reference sample SIF_low, which helps rule out other possible explanations for the resistivity maximum at H_coerc. Following Zdravkov et al [22], the magnetic domain formation or the generation of Abrikosov vortices with vortex movements in an applied field could also cause local maxima in ρ(μ₀H_ext). Both effects would be stray field generated, as vortices in a sufficiently thin Nb layer can only form in the out-of-plane direction (in our S/F bilayers, vortices could be generated by stray fields in the unsaturated F layer). The stray fields should still penetrate through the 7.5 nm thick insulating MgO layer, whereas proximity effects are suppressed by the insulator. The ρ(μ₀H_ext) curves of SIF_low show only a conventional suppression of superconductivity due to an applied field, as in a bare S layer. As a result, a stray field origin of the maximum in ρ(μ₀H_ext) at H_coerc of SF_low can be ruled out.

The measurements in an in-plane applied field indicate as well a possible spin-triplet Cooper pair formation in samples of higher anisotropy: while no local maxima at H_coerc are observed, both SF_high and SF_mid show a kink in ρ(μ₀H_ext) at field values near to the local maxima in ρ(μ₀H_ext) of SF_low. This can be a possible sign of spin-triplet generation with lower density compared to SF_low, and possibly also arising from the magnetic inhomogeneity due to closure domains forming between Nb and FePd.

In summary, we conclude that LRTC are generated at the S/F interface of SF_low due to proximity effects between the S layer and the lateral inhomogeneous magnetization of the F layer, with a width of magnetic domains larger than the Cooper pair coherence length of SF_low near T_c. The highest density of LRTC in relation to spin-singlet Cooper pairs is reached at the coercive field, where the magnetic moments acquire a maximum non-collinearity. This results in a ΔT_c as high as 100 mK in SF_low between the resistivity minima and the local maximum at H_coerc (see figure 10). However, the resistive transition widths in figure 10(b) are of the same order of magnitude as ΔT_c. In comparison, the resistive transition widths as also the maximum ΔT_c of SF_high in an out-of-plane applied field of data shown in figure 9 are an order of magnitude smaller than in SF_low as can be seen in figure S7 in the supplementary information.

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Large differences in the critical temperature under application of small applied fields are crucial for the engineering of spin-valve devices operating at cryogenic temperature which can switch between a high-T_c and a low-T_c state. Up to now, the highest reported reversible ΔT_c for fully-metallic F/S/F' trilayers is about 400 mK in a Ho/Nb/Ho spin valve structure [7]. As ΔT_c can raise significantly in F/S/F' trilayer structures compared to respective S/F bilayers, our value of 100 mK in the Nb/FePd bilayer has a great potential as starting point for the fabrication of F/S/F' trilayers with even larger ΔT_c values.