H=250 Oe. Supplementary Figure 1 Magnetic domains: Room temperature 4 x 4 µm 2 MFM phase

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1 1 Supplementary Information Supplementary Figures (b) () 1.6± 1 µm 1 µm 1 µm H290 Oe (d) (e) (f) H410 Oe -1.5± 1 µm 1 µm H250 Oe 1 µm H590 Oe Supplementary Figure 1 Magneti domains: Room temperature 4 x 4 µm 2 MFM phase images, aquired in lift mode, of the, (b), () 200 bilayers of Co(2nm)/Pd(2nm) (Sample ) and (d), (e), (f) 50 bilayers of Co(2 nm)/pd(2nm) (Sample ) in external magneti fields applied perpendiular to the surfae. Oe, (b) H290 Oe, () H410 Oe, (d) Oe, (e) H250 Oe, (f) H590 Oe. Sale bars are 1 µm. MFM images have been aquired after deposition of the l 2 O 3 layer.

2 2 1.6 Pb film 1.5 (b), Γ (mev) 0.8 Γ CS (K) 0.5 Pb film Pb film H400 Oe 1.6 K (Co-Pd)/Pb Supplementary Figure 2 Comparison between referene Pb film and (Co-Pd)/Pb systems: emperature dependene of the gap (red dots) and of the quasipartile lifetime Γ (blue dots) for the referene 30 nm Pb film, derived by the CS DOS fitting with the Dynes formula, as desribed in the main text. he error bars represent the standard deviation obtained from the fit. he ontinuous blak line is the CS gap equation orresponding to 6.8 K. (b) Comparison between the tunnelling spetrum on the Pb referene film at 1.6 K in zero applied magneti field, a tunnelling spetrum aquired on the same film at the same temperature in a magneti field H400 Oe applied perpendiular to the film surfae and a tunnelling spetrum aquired on a (Co-Pd)/Pb (Sample ) system in a zero applied external field. In all ases spetra have been aquired in loations far from vorties and with the same tunnelling onditions: V-10 mv, I100 p. he urves have been normalized to the value of ondutane at V-10 mv.

3 3 2 Sample 2 (b) Sample 1 1 b z /4M b z /4M insulator surfae superondutor surfae x(nm) insulator surfae superondutor surfae x(nm) Supplementary Figure 3 Stray Field Profiles in (Co-Pd)/Pb systems: Calulated magneti field distribution for the Ferromagnet-Insulator-Superondutor systems for Sample (ferromagnet thikness D800 nm, magneti domain width w200 nm) and for (b) Sample (ferromagnet thikness D200 nm, magneti domain width w300 nm). In both graphs the red line represents the stray field at the top of the insulator s surfae (at a distane z10 nm from the ferromagnet s surfae) and the blue line is the magneti profile at the surfae of the superondutor (at a distane z40 nm from the surfae of the ferromagnet). his alulation is based on the formula given in Supplementary note 1.

4 K 1.5 (d) 1.5 K 0.12 C 100 nm C 1.6 (b) 5.0 K 1.1 (e) 5.0 K nm C () 5.5 K 1.1 (f) 5.5 K nm C Supplementary Figure 4 Emergene of superondutivity in the (Co-Pd)/Pb Sample :, (b), () LDOS maps at the Fermi energy aquired at Oe and at 1.5 K, (b) 5.0 K, () 5.5 K. ll maps have been normalized to the maps aquired simultaneously at V-10 mv. he san areas in, (b) and () are 450 x 450 nm 2 and the sale bars are 100 nm. he white dashed line in indiates the approximate positions of the domain walls as inferred from the vortex onfigurations at different fields (Fig. 2 in the main text).he graphs in (d), (e) and (f) show the tunnelling spetra aquired at different loation aross the domain wall (Positions,, C) of the orresponding LDOS map taken at the same temperature. ll spetra have been aquired with the same tunnelling onditions V- 10 mv and I100 p and normalized at V-10 mv.

5 K (b) 5.0 K () 4.88 K (d) 4.3 K nm 100 nm 100 nm (e) 5.2 K (f) 5.0 K (g) 4.88 K nm Supplementary Figure 5 Emergene of superondutivity in the (Co-Pd)/Pb Sample :, (b), (), (d) LDOS maps at the Fermi energy aquired at Oe and at 5.2 K, (b) 5.0 K, () 4.88 K and (d) 4.3 K. ll maps have been normalized to the maps aquired simultaneously at V-10 mv. he olour sale (representing the normalized zero bias ondutane) is the same for all images. he san areas are 300 x 300 nm 2 for, (b), () and 600 x 600 nm 2 for (d). he maps in, (b) and () have been aquired at the loation indiated with a dashed square in (d). he graphs in (e), (f) and (g) show the tunnelling spetra aquired at different loation aross the domain wall (along the dashed line from to every 15 nm) of the orresponding LDOS map at the same temperature. he olours used for the tunnelling spetra are the same used to indiate the loations where the spetra were aquired (oloured dots on the orresponding LDOS map). ll spetra have been aquired with the same tunnelling onditions V-10 mv and I100 p and they have been normalized to the value of ondutane at V-5 mv.

6 6 Supplementary Note 1 Magneti Field Profile in Ferromagnet-Insulator-Superondutor Systems he distribution of stray fields for a stripe domain struture in a ferromagneti film an be alulated analytially. If the film is parallel to the xy plane the z omponent of the stray field in the diretion perpendiular to the film surfae is 1 : ( t id) πt π bz ( x, z) Im 4M 0 ln tan ln tan (1) 2w 2w where t x + iz, D is the ferromagnet thikness and w is the stripe domain width. In a Ferromagnet-Insulator-Superondutor system if the London penetration depth is larger than the superondutor thikness the stray field is almost not affeted by the presene of the superondutor and Supplementary Equation (1) an be used to alulate the stray field at the top of the superondutor s surfae. In Supplementary Fig. 3 the stray field profiles for Sample and Sample are reported at different distanes z from the ferromagnet s surfae. Supplementary Note 2 Spatial variation of the loal density of states in (Co-Pd)/Pb systems In Supplementary Fig. 4 and Supplementary Fig. 5 we desribe the loal variations of tunnelling spetra in the (Co-Pd)/Pb Sample and respetively. In Supplementary Fig. 4, (b), and () we report the LDOS at the Fermi energy aquired at different temperatures aross the superonduting temperature. In Supplementary Fig. 4(d), (e) and (f) we report typial ondutane spetra aquired aross the domain wall. In Sample the spetra aross the domain wall are very similar with very small hanges of the ZC values. In partiular very lose to the ritial temperature at 5.5 K (i.e. / 0.92) the

7 7 sample is already in the superonduting state with tunnelling spetra that show oherene peaks and a value of ZC very onsistent in the field of view ranging from 0.9 to 0.94 (Supplementary Fig. 4() and Supplementary Fig. 4(f)). In Supplementary Fig. 4() the LDOS map at the Fermi energy shows a very uniform state. We do not observe on this sample the variation in ZC that are observed instead in Sample at similar values of redued temperature. In this ase, indeed, as explained in the main text the narrow dimension of the magneti domains favours the overlap of the superonduting nulei aross the magneti domains and a more homogeneous superonduting state emerges. In Supplementary Fig. 5 we report the LDOS maps at the Fermi energy (Supplementary Fig. 5, (b), () and (d)) and loal tunnelling spetra aquired at different temperatures on Sample (Supplementary Fig. 5(e), (f) and (g)). he loal spetra aquired aross regions above two stripes of opposite polarities show that at 5.2K (i.e. / 0.94), there is a strong variation of loal spetra with ZC values ranging from 0.82 to 1 (Supplementary Fig. 5 and (e)) as opposed to Sample. y reduing further the temperature at 5.0 K (i.e. / 0.90), the ZC values are still inhomogeneous with values ranging from 0.7 to 1 (Supplementary Fig. 5(b) and (f)). t 4.88 K (i.e. / 0.88), the ZC values range from 0.65 to 1 (Supplementary Fig. 5() and (g)). Finally at 4.3 K, we an learly distinguish single vorties along eah magneti domain. herefore, in Sample while dereasing the temperature below the superonduting order parameter nuleates first at the regions above the domain wall.

8 8 Supplementary Referenes 1. Sonin, E.. Comment on Ferromagneti film on a superonduting substrate. Phys. Rev. 66, (2002).