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1 Direct observation of Josehson vortex cores S1. Descrition of the Ginzburg-Landau simulations The framework for the theoretical simulations in this aer is the henomenological Ginzburg-Landau (GL) theory. To incororate the variation of electron mean free ath l e across the islands in the GL equations, we used the exressions for GL coefficients α and β in the dirty limit, i.e..3. ħ ħ (1.1) (1.) where l ξ =l e /ξ is the ratio of the electron mean free ath and the BCS coherence length. In this case, the two GL equations can be written in the following dimensionless form: A ψ. ψ (1.3) A=- ψ-ψ A (1.) where all lengths are scaled to ξ, enetration deth λ is defined as /, the vector otential A is exressed in /, and the order arameter is in units of /. In the case that samle has non-uniform thickness d(x,y), but is sufficiently thin to assume the constant order arameter in the z-direction, Eq. (1.1.3) has to be exanded on the left side by the term A, as derived and used in Ref. 9. The simulations are then imlemented on the exact geometry of the islands (on a Cartesian grid, see Ref. 3) using STM maing from the exeriment, with grid sacing of 1 nm. Note that equations ( ) need to be solved self-consistently and both contain higher order derivatives, hence this is a demanding comutation. With grid also being large (simulation region was tyically 1. 1.μm ), arallelization of the comutations is necessary, and for that we have used CUDA and GPU comuting. The mean-free-ath was taken to be twice the thickness, i.e., tyical of ultrathin films. Then, for each value of the alied magnetic field, the calculations are initialized starting either from the fully suerconducting state in all islands (zero-field cooled) or the normal state (field-cooled), where all found solutions were subsequently back-tracked in increasing and decreasing magnetic field. In such a way, full set of ossible vortex states is obtained and comared to exerimental images. NATURE PHYSICS 1

2 S. Descrition of the calculations of the suerconducting correlations mas The microscoic Usadel theory is the aroriate microscoic aroach to describe suerconducting correlations induced by roximity effect in a diffusive metallic medium. However, the required calculations are technically very difficult, and at this stage of develoment, only a few regular geometries, as that resented in Fig., may be solved numerically. The calculations of realistic 3D systems within the framework of Usadel formalism are still beyond the state of the art. This is why we used instead the Ginzburg-Landau equations in order to reroduce the comlex vortex arrangements observed exerimentally. As the Ginzburg-Landau theory cannot describe the suerconducting correlations induced by roximity effect in the normal region where the order arameter is null we elaborated a heuristic extension to Ginzburg-Landau as exlained below. The general idea of our calculations is the following: the Ginzburg-Landau theory enables us reroducing the vortex configurations in the suerconducting islands and thus access to the hase ortrait of the order arameter inside the islands in the resence of an alied magnetic field. These hase ortraits have been used as inut for the simulations of the suerconducting correlations in-between the islands in the roximity regions. Gauge Invariance: We took a secial attention to the fact that any correct model should be gauge invariant. Indeed, the hase or vector-otential ortraits have no hysical meaning when taken searately, each ortrait deends on the gauge choice. However, observable hysical henomena are gauge invariant. For instance, the currents circulating in a suerconductor deends on both hase and vector-otential, as given by the famous equation ħ, yet they must be indeendent on the choice of the gauge. Namely, one defines the vector otential such that but the choice for the vector otential is not unique, since one can always take, where is any differentiable function of, and still get the same field,. The articular choice of gauge will have some consequences on the wave-function. For examle, for a single charge q in an electromagnetic field the Schrödinger equation is given by. As the choice of a articular gauge should not modify the hysics, the Schrödinger equation has to be gauge invariant. Hence if for a vector otential one gets the wave function, then for another gauge, the wave function will become, with. With this understanding, we ħ return to the exression for the current density in a suerconductor, ħ. If we relace and, we obtain, i.e. the current is indeed gauge invariant as it ħ must be. In the case of a suerconductor, the order arameter follows similar rules as for the Schrödinger equation. In Ginzburg-Landau or Usadel formalisms there is a similar gauge invariance as for Schrödinger equation excet that q=-e due to the charge -e of the Cooer airs. Thus, in a suerconductor the current density deends on the hase NATURE PHYSICS

3 but also on the vector otential. At zero magnetic field it is always ossible to take as a ossible gauge; in that case one can consider only the hase of the order arameter for the calculation of roximity effect or whatever. However, in our exeriments the Josehson vortices are measured in a finite magnetic field, and it is thus imossible to find a gauge in which the vector otential is zero everywhere. For simle geometries it is ossible to choose a vector otential arallel to the edges of the samle and to get a simle hase ortrait. For examle, in a cylindrical suerconductor in Meissner state, if one uses the symmetric gauge B, then the hase of the order arameter is constant, and one gets a simle roortionality! However, for a collection of islands with irregular shaes, the hase ortrait becomes more comlicated since it is not ossible to find a gauge for which the vector otential is arallel to all the edges. We show below significantly different hase ortraits calculated for two different gauges, without any consequence for the Cooer air density or for the currents, which remain identical. The Ginzburg-Landau calculations shown in the article are erformed in the symmetric gauge, i.e. for a vector otential B, with the origin taken at the centre of the ma (indicated by a star in the sulied below Fig. S.1a). With this articular gauge we get a articular hase ortrait, resented as colour ma in Fig. S.1a. Another calculation, with a shifted origin of the symmetric gauge, is shown in Fig..1b. As exected, the hase ortrait looks comletely different. However, the air density ma (shown in Fig. S.a below) remains identical to the one (see Fig. 3a of the main manuscrit) calculated using the centred gauge hase ortrait. This is an exected result, because the hysical quantities are gauge invariant. In addition, we resent in Fig. S.b the current ma calculated using the shifted gauge hase ortrait, Fig. S.1b. One can see that the currents are always arallel to the edges, even though the hase ortrait is unclear in this resect. a b Figure S.1: Two different hase ortraits at mt, calculated by Ginzburg- Landau simulations using the symmetric gauge centered on the star: (a) at the middle of the ma, (b) intentionally shifted out of the centre. NATURE PHYSICS 3

4 a b Figure S.: (a) The air density calculated for the shifted gauge shown in Fig. S.1b. This air density ortrait is identical to that shown in Fig.3a of the main manuscrit, which was obtained for the centered gauge, Fig. S.1a. This demonstrates the indeendence of the calculated suerconducting state on the choice of the gauge. In (b) we show the currents which are also the identical for both gauges, while the corresonding hase ortraits shown in Figs. S.1(a,b) are clearly different. The currents are roerly flowing arallel to the edges of each suerconducting island, as exected. Gauge invariant correlation mas: The rincile of calculation of the gauge invariant correlation mas is the following: For any given location in the roximity region between islands, we consider exonentially decaying artial contributions originating from all the oints located on the edges of the islands. These henomenological evanescent waves reresent decaying suerconducting correlations in the roximity region; they vanish outside the islands on a length scale of the roximity coherence length. Each artial correlation contributes with its amlitude and hase to the total correlation function, i.e. all the artial correlations interfere in according to their relative amlitude and gauge-invariant hase difference:,, (.1) where, is a gauge invariant hase, and the local hase on the erimeter of the island i, is the quantum of flux for Cooer airs. The local amlitude of the correlations is given by the summation of evanescent NATURE PHYSICS

5 waves on circuit around all the islands indexed by i. In our calculations we took nm, according to the conclusion of our revious study 17. The correlation mas in the regions outside the islands dislay color-coded (S.3b, S.b). Inside the islands the mas are comleted with the calculated Ginzburg- Landau local density of Cooer airs (see S1). In the Fig. S.3 we show the Ginzburg- Landau hase ortrait calculated for a field of mt (left anel) and the corresonding correlation ma (right anel). It should be noted that the Ginzburg-Landau hases for each island i are defined u to an arbitrary island hase for each island. Such free rotation of the hase of an angle for each island is allowed, when the islands are comletely indeendent from each other - as in our Ginzburg-Landau simulation (the island hase ortraits were simly calculated all together, using the same gauge for every island). The true hase ortraits realized in each island are obtained by shifting by,. The free rotation of island hases is allowed for indeendent islands, but once the islands are couled by Josehson links the hases should be fixed. By rotating the island hases, or more recisely, by rotating the island hase difference (since for each indeendent air of islands forming a junction we have only one free arameter), one tunes the exact osition of Josehson vortex cores. Note that this rotation of the island hases does not influence the Abrikosov vortex configurations inside each island but it does affect the osition of Josehson vortices in the roximity regions. As a result, neglecting the island hase issue leads to a oor matching between the correlation mas and the exerimental data (as it is the case between Fig. S.3b and Fig. 1c of the main manuscrit). Formally, there is one free arameter er junction. In ractice however, the choice of the islands hase difference is very limited because of strong hysical and geometrical constraints. In order to fix the islands hase difference for each junction we alied two constraints: First, in the absence of external loos, the total Josehson current across a given junction should be zero. Among the several solutions that give zero net current one chooses the only one which has minimum kinetic energy. The automatic rocedure used for the hase difference adjustment is described below in more details, but first let us give some hysical intuition on the two conditions required: minimum kinetic energy and not net current. Let us illustrate the condition of minimum kinetic energy with junctions J3 and J because they have well defined locations at which the inter-island distance is the shortest, corresonding to a significantly higher Josehson energy as comared to other locations of the same junction. When Josehson vortices aear, they naturally avoid occuying these locations. Indeed, the vortex core centres corresond to a local hase difference. If we use in local sense the well-known exression for the free energy,, the free energy would be locally maximized by utting a vortex core in a constriction. On the other hand, utting a Josehson link there,, is very favorable because it minimizes the free energy. It means that close to constrictions between two islands where the local Josehson energy is high, the system will favour the hase shift close to. NATURE PHYSICS

6 The hase differences that we automatically adjusted within our crude roximityextended GL aroach satisfy both conditions on the minimum kinetic energy and no net inter-island current. For instance, let us look at the islands in the lower right corner of Fig. S.3 and Fig. S.. We see that once the hases are adjusted (Fig. S.) a clear Josehson bridge is stabilized where the inter-island distance is the shortest, while without any adjustment (Fig. S.3) there is a clear current imbalance. The latter situation is not hysically correct since it would lead to charge accumulation in islands. The total current could have also been nullified by utting a vortex core in the constrictions, but in that case the resence of a node in the Josehson link will naturally give a much higher energy as exlained above, and thus is not favourable. We have adoted this simle rule when adjusting the hase differences automatically. (a) Suerconducting correlations (b) High Low Figure S.3: (a) ortrait for the islands at mt as calculated by Ginzburg-Landau equations. (b) Corresonding correlation ma obtained directly, without the island hase adjustments. (a) Suerconducting correlations (b) High Low Figure S.: (a) ortrait for the islands at mt calculated with Ginzburg- Landau equations with additional global hase shifts added to each island. (b) The corresonding correlation ma fits well the exerimentally observed roximity Josehson vortices. NATURE PHYSICS

7 As Ginzburg-Landau theory is valid only inside suerconducting regions, it does not catch the roximity henomena regions where the GL order arameter is zero and thus, there is no ossibility to evaluate the contribution of roximity regions to the total free energy within standard GL aroach. However, since the condensation energy is zero in the roximity region, and the magnetic energy negligible in such a small system (see Sec.3), the main contribution of the roximity region in the energy balance is the kinetic energy of Josehson currents. The strength of our heuristic correlation function aroach [Eq..1] is recisely here: It allows us to comute Josehson suercurrents in the N-arts and their kinetic energy with no adjustable arameters. We define a current in the N regions as:, (.) where. We also define a kinetic energy as:., (.3) Then we calculate the net current circulating in between each junction as function of the inter-island hase difference. The figure S. shows the lines through which the currents have been calculated. The table S. shows the current hase and the kinetic energy hase relations for each junction at mt (in a window). For each junction, there are two hases with no net current, one corresonding to a minimum and the other to a maximum of kinetic energy. By convention the zero of the island hase difference has been fixed in lots to the hase that fulfilled the two criteria: not net current and minimum kinetic energy. By this way it was ossible to adjust automatically the hase differences in each island at mt, 1mT and 18mT. The current and kinetic energy as function of the hase difference for the three junctions of figure are shown in tables S.7 and S Figure S.: Correlation ma for the islands at mt with the lines used in order to calculate the total current flowing in each Josehson junction. The number associated with each junction is in corresondence with the tables below. NATURE PHYSICS 7

8 Junction Current Kinetic Energy Current a.u. Current a.u. Current a.u. Current a.u. Current a.u. Current a.u. Current a.u Table S.: Current and kinetic energy as function of the hase difference for each Josehson junction at mt. Kinetic Energy a.u. Kinetic Energy a.u. Kinetic Energy a.u. Kinetic Energy a.u. Kinetic Energy a.u. KineticEnergy a.u NATURE PHYSICS Kinetic Energy a.u

9 Junction Current Kinetic Energy 1 3 Current a.u. Current a.u. Current a.u Kinetic Energy a.u. Kinetic Energy a.u. Kinetic Energy a.u Table S.7: Current and kinetic energy as function of the hase difference at 1 mt for Josehson junctions 1 to 3. NATURE PHYSICS 9

10 Junction Current Kinetic Energy Current a.u. Current a.u. Current a.u Kinetic Energy a.u. Kinetic Energy a.u. Kinetic Energy a.u Table S.8: Current and kinetic energy as function of the hase difference at 18 mt for Josehson junctions 1 to 3. 1 NATURE PHYSICS

11 3. Theoretical descrition of the roximity effect and the Josehson vortices In this section we discuss in detail the theoretical framework used to characterize the roximity effect in the wetting layer in the absence of magnetic field and to comute the roximity Josehson vortices shown in the manuscrit. 3.1 Tunnelling sectra in the normal state of the wetting layer Before discussing the suerconducting roximity effect in the wetting layer (WL), it is necessary to rovide a theoretical descrition of the tunnelling sectra in the normal state of this disordered D metal. As exlained in the manuscrit (see also Ref. 17), the local sectra in the WL far away from any suerconducting island exhibit a zero-bias anomaly that resembles the diffusive anomalies found in tunnelling junctions between disordered conductors 3. Those anomalies also aear in ultrasmall tunnel junctions due to the interaction of the tunnelling electrons with the electromagnetic environment in which the junctions are embedded, an effect referred to as dynamical Coulomb blockade (DCB) 31. Indeed, it has been shown that these two henomena are closely related and their theoretical descritions are very similar 3-3. Based on this analogy, and following Ref. 17, we have made used of the DCB theory in ultrasmall junctions 33 to describe the tunnelling sectra in the normal art of the WL. Within this theory, the tunnelling current is given by I (V) e WLti (V) tiwl (V), (3.1) where the inelastic tunnelling rates can be exressed as WLti (V) 1 de e R T den WL (E) f (E)[1 f (E E ev )]P( E ) (3.) and tiwl (V) WLti (V). Here, R T is the tunnelling resistance, f(e) is the Fermi function, n WL (E) is the normalized local density of states (LDOS) of the WL, and P(E) is the robability for an electron to emit the energy E into the electromagnetic environment. It is worth stressing that in Eq. (3.), we have neglected the energy deendence of the ti LDOS. The P(E) function is given 33 P(E) 1 dt exj(t) iet / h, (3.3) where J(t) (t) () () is the equilibrium correlation function of the hase which can be exressed in terms of the total imedance of the circuit Z( )as d Re Z( ) J(t) coth R K k B T [cos(t) 1] i sin(t), (3.) where R K h / e is the resistance quantum. We model our STM tunnel junction by an RC circuit where the total imedance is given by Z( ) 1/[iC WL 1/R WL ]. Here, C WL and R WL are the effective caacitance and resistance of the WL, resectively (see Ref. 17). To describe the tunnelling sectra in the normal art of the WL, we have used this model with C WL and R WL as adjustable arameters, while the LDOS of the WL was assumed to be the non-interacting one, i.e. n WL (E) 1.In Fig. S3.1 we show the best fit to the exerimental results within this theory, which was obtained for C WL = 1.3 af and R WL = 1.73 k. Let us remark that this fit was obtained using an effective temerature of. K, which is slightly higher than the nominal base temerature in the exeriments. As one can see, the DCB theory NATURE PHYSICS 11

12 describes qualitatively the ZBA found exerimentally. Figure S3.1: The symbols corresond to a normalized characteristic conductance sectrum measured in the WL away from the Pb islands. The red solid line corresonds to the best fit obtained with the DCB theory described in text with C WL = 1.3 af, R WL = 1.73 kand an effective temerature of. K. 3. Proximity effect in the WL in the absence of magnetic field: Usadel equations Following Ref. 17, our aroach to describe the roximity effect in the WL is based on a combination of the DCB theory described in the revious subsection with the socalled Usadel equations 8. The key idea is to use the roximity LDOS comuted with the hel of the Usadel aroach and introduce it into Eqs. (3.) and (3.1) to comute the local tunnelling sectra in the WL. In what follows, we describe in detail how the Usadel equations were used and solved in ractice, focusing on the case where no magnetic field is alied. In this discussion, we shall closely follow Ref. 3. The Usadel equations summarize the quasi-classical theory of suerconductivity in the diffusive limit 8, where the mean free ath is much smaller than the suerconducting coherence length. In the case of the WL, this length is given by D /, where D is the diffusion constant of the WL and is the energy ga in the suerconducting leads. Within the quasi-classical theory, all equilibrium roerties are described in terms of a momentum averaged retarded Green s function Ĝ( R, E),which deends on osition R and energy E. This roagator is indeed a matrix in electron-hole sace g f Ĝ f g. (3.) Our goal here is to describe the roximity effect in the WL in the regions between two nearby Pb islands, which effectively form suerconductor-normal metalsuerconductor (SNS) junctions. In order to describe these junctions, we consider in ractice a one-dimensional SNS junction, where S is a BCS suerconducting reservoir with a constant ga Δ, corresonding to a Pb island, and N is a diffusive normal wire of 1 NATURE PHYSICS

13 length L describing the WL. In this model, we assume that the interfaces with the central wire are erfectly transarent and we neglect the inverse roximity effect in the S reservoirs. Ignoring inelastic interactions and hase-breaking mechanisms, the roagator Ĝ( R, E) satisfies the stationary Usadel equation, which in a normal wire N reads 8 D ĜĜ Eˆ 3,Ĝ, (3.) where ˆ 3 is the Pauli matrix in electron-hole sace. Equation (3.) must be sulemented by the normalization condition Ĝ. In order to solve numerically the Usadel equation, it is convenient to use the so-called Riccati arameterization 3, which accounts automatically for the normalization condition. In this case, the retarded Green s functions are arameterized in terms of two coherent functions ( R, E) and ( R, E) as follows i 1 Ĝ 1 1. (3.7) Using their definition in Eq. (3.7) and the Usadel equation (3.), one can obtain the following transort equations for these functions in the normal wire region f x i x E i, (3.8) E T x f i x E i E T. (3.9) Here, x is the dimensionless coordinate that describes the osition along the N wire and ranges from (left S lead) to 1 (right S lead) and E T D is the Thouless energy of the L wire. The exressions for f, g, f, and g are obtained by comaring Eq. (3.) with Eq. (3.7). Notice that Eqs. (3.8) and (3.9) coule the functions with and without tilde. However, for the system under study one can show that the symmetry ( R, E) ( R, E) holds and therefore, only Eq. (3.8) needs to be solved. Now, we have to rovide the boundary conditions for Eq. (3.8). For erfect transarency, such conditions at the ends of the N wire result from the continuity of the Green s functions at the interfaces:, (3.1) (x, E) (x 1, E) S (E) / E R i (E R ) where E R E i. In summary, the main task is to solve Eq. (3.8) with the boundary conditions of Eq. (3.1). This is a tyical two-oint boundary value roblem that we solved numerically using the so-called relaxation method as described in Ref. 37. Once the numerical solution for the coherent function is obtained, one can construct the retarded Green s function from Eq. (3.7) and comute the normalized local density of states (LDOS) in the WL as n WL 1 Img(x, E). (3.11) NATURE PHYSICS 13

14 This local DOS was finally introduced into Eq. (3.) in the revious subsection to comute the local tunnelling sectra in the WL. In Fig. S3. we show examles of the LDOS (anel a) and the corresonding tunnelling sectra (anel b) in the middle of the normal wire of SNS junctions of different lengths. As one can see in anel a, the main feature of the LDOS is the aearance of a mini-ga that decreases as the junction length increases. This mini-ga is constant throughout the whole normal wire and for long junctions ( L ) it scales with the Thouless energy as (see Ref. 3 and references therein). In Fig. S3.b we dislay the corresonding tunnelling sectra comuted combining the LDOS obtained from the Usadel equations with the DCB theory, as described above. In articular, the values of the arameters defining the electromagnetic environment were taken from the fit of Fig. S3.1. As one can see, the main effects of the inelastic interaction with the environment are the rounding of the coherent eaks close to the mini-ga edges and the introduction of states inside the mini-ga, which is more ronounced as the junction length increases. Figure S3.: a, Local density of states in the middle of a SNS junction as a function of energy comuted with Usadel aroach described in the text. The different curves corresond to different values of the junction length in units of the diffusive coherence length. The energy ga of the suerconducting leads was set to = 1. mev. b, The tunnelling sectra comuted with the results of anel a using the DCB described in subsection 3.1. The arameters defining the electromagnetic environment were taken from the fit of Fig. S3.1: C WL = 1.3 af, R WL = 1.73 k and T =. K. The tunnelling sectra described above rovide the basis to understand our exerimental results for the roximity sectra in the absence of magnetic field. To illustrate this fact, we show in Fig. S3.3a the exerimental results obtained in the middle of the Josehson junction J with a length of nm (see Figs. 1a & d in the manuscrit) along with the theoretical sectrum for a junction length L. As one can see, our theoretical model rovides a nice qualitative descrition of the exerimental sectrum. Similar qualitative agreements for the other junctions discussed in the manuscrit can be found by simly changing the junction length in the calculations. This is illustrated in Fig. 3.3b, where we show the exerimental result for the junction J1 (location C) which has an aroximate junction length of 7 nm, together with theoretical curve for L. This analysis suggests that coherence length of the WL is 1 nm, which 1 NATURE PHYSICS

15 imlies an aroximate value of cm /s for the diffusion constant, which confirms that we are dealing with a rather disordered metal. Figure S3.3: a, The symbols corresond to the exerimental tunnelling sectrum of junction J (see manuscrit). The red solid line is the theoretical sectrum obtained with the combination of the DCB theory and the Usadel equations for a junction of length equal to. The other arameters of the model used to comute this curve are: = 1. mev, C WL = 1.3 af, R WL = 1.73 k and T =. K. b, The same as in anel a, but for the junction J1 (location C). The theoretical curve corresonds to a length. 3.3 Magnetic-field-induced roximity Josehson vortices: D Usadel equations The roximity Josehson vortices induced by an external magnetic field shown in Fig. a and the corresonding LDOS of Fig. b were calculated following Refs. 7 & 38, as we now roceed to briefly exlain. In the absence of electronic correlations (no DCB), the descrition of the Josehson vortices in diffusive SNS junctions can be done within the framework of the Usadel equations. The difference with resect to the revious subsection resides in the fact that in the resence of a erendicular magnetic field, one cannot longer ignore the variation of the suerconducting correlations along the transverse direction (arallel to the interfaces), i.e. the roblem becomes intrinsically two-dimensional. To roceed, we consider a SNS junction, where N is a diffusive thin film (the wetting layer) of length L and width W couled to two identical suerconducting reservoirs with an energy ga. The film lies in the xy-lane, where x [, L] and y [W /,W / ] and it is subjected to a uniform external magnetic field H Hẑ erendicular to the film. Since the thickness of the WL is much smaller than the London enetration deth, we can assume that the field enetrates comletely in the normal region. Moreover, as in the revious subsection, we assume that the SN interfaces are erfectly transarent NATURE PHYSICS 1

16 and we neglect the suression of the air otential in the leads near the interfaces. Additionally, we assume that the field does not affect the suerconductivity in the electrodes. In this case, the retarded Green s function Ĝ( R, E), which contains the information of the LDOS, satisfies the following equation in the N region D Ĝ Ĝ Eˆ 3,Ĝ ied A ˆ 3, ĜĜ. (3.1) Here, A is the vector otential and ˆ1 (ie/ ) Aˆ 3. In the revious equation we have already used the Coulomb gauge A. Again, Equation (3.1) must be sulemented by the normalization condition Ĝ. Now, we have to secify the boundary conditions. For the SN interfaces, since we assume erfect transarency, these conditions simly reduce to the continuity of Green s functions across the interfaces. On the other hand, for the metal-vacuum lower and uer borders in the normal wire, the aroriate conditions are ie yˆ1 A ˆ y 3 Ĝ ˆ, yw/ (3.13) which simly exress the fact that the current density in the y-direction vanishes at the edges of the samle. As exlained in the revious section, to solve the Usadel equations (3.1) it is convenient to use the Riccati arameterization, see Eq. (3.7). In this case, if we choose the gauge A Hyˆx and introduce the dimensionless coordinates x [,1] and y [1/,1/], the coherent function ( R, E) satisfies the following equation x L f W y i x L E W ( y ) i E T (3.1) yg x i y g, where HLWis the magnetic flux enclosed in the junction and is the flux quantum. There is another equation (couled to this one) for that can be obtained from Eq. (3.1) by exchanging by (and vice versa) and by changing the sign of the vector otential. Finally, we need to secify the boundary conditions for these coherent functions. Along the two SN interfaces the coherent functions adot their bulk BCS values, see Eq. (3.1). The corresonding conditions for the border of the normal film adot the following simle form: y y1/ y y1/. (3.1) Eq. (3.1) and the corresonding one for form a set of couled second-order nonlinear differential equations. As exlained in Refs. 7 & 38, this system of equations can be solved numerically using the so-called relaxation method for boundary value roblems 37, adated to the case of artial differential equations. With the numerical solution for the coherent functions, one can easily comute the LDOS in the WL using Eq. (3.11). In Fig. a of the manuscrit, we show an examle of the LDOS at the Fermi energy throughout a junction of length L and width W 1 for a magnetic flux enclosed in the junction equal to. This examle illustrates how the magnetic field modulates the LDOS, revealing the resence of Josehson vortices. In Fig. b, we show the corresonding LDOS as a function of energy at different locations in the middle of the wire. More examles of these LDOS mas for different values of the 1 NATURE PHYSICS

17 junction length and width can be seen in Refs. 7 & 38. Finally, in order to simulate the exerimental tunnelling sectra, we simly combine the LDOS obtained from the Usadel equations with the DCB theory, as exlained in the revious subsection [see Eq. (3.)]. An examle of the comuted tunnelling sectra revealing the resence of the roximity vortices is shown in Fig. c of the manuscrit. Again, the values of the arameters defining the electromagnetic environment used to comute the curves of Fig. c were those of the fit of Fig. S From fluxoid quantization to fully current-generated Josehson vortices Unlike in bulk suerconductors, in mesoscoic systems of a lateral size Λ (Λ, where is London enetration deth, is the thickness) the magnetic flux is not quantized, but the fluxoid is. To understand the oint, let us start with the second GL equation and aly it to a single vortex in a quantum condensate:, (3.1) where, is the electric charge, vector-otential of a field that rotates the condensate (which would be the magnetic field for suerconductors), and is the suerfluid velocity. In order to calculate the flux, let us consider a loo of a surface around the vortex core centre characterized by hase singularity. By taking of eq.(3.1), integrating over and using Stocks theorem we get:, (3.17) where is the ath element of the loo. The second term on the right side is the magnetic flux crossing the loo,. In suerconductors,, the hase makes a turn around a single Pearl-Abrikosov vortex,, and the flux becomes:, (3.18) where is indeed the flux quantum. Eq. (3.18) immediately shows that the flux around a vortex (Abrikosov, Pearl or Josehson one) is only quantized if the circulation of the currents through the loo is zero. Thus, in order to get the flux quantum, one needs to have. That is why in bulk suerconductors, loos of a size Λ from the vortex centre are considered to warrant. The size of our mesoscoic system is Λ and thus, the flux is not quantized there. Nevertheless, it is still ossible to choose secific loos over which. For the exerimental vortex mas at 1 mt and 18 mt such loos are resented in the figure S3.. Here light blue arrows reresent Abrikosov-Pearl vortex currents, and yellow arrows Meissner currents near island edges. The aths (dashed lines) are traced in a secific way in order to: (i)-either to have or (ii) to have by choosing the ath direction erendicular to the velocity vector. This automatically ensures in any location of the loo, ; and, NATURE PHYSICS 17

18 eq. (3.18). In the figure, white dashed lines are zero-current lines - loos inside islands where Abrikosov vortex currents comensate Meissner ones, yellow dashed lines the aths between islands on which conditions are fulfilled. Figure S3.: Suercurrents and flux in the studied system at 1mT (left) and 18mT (right); (see exlanations in the text). One immediately notices that all drawn loos have the same area, and this area decreases when the field rises. Calculating the flux crossing these loos gives in each loo, and for each field. This formally roves the absence of any measurable diamagnetic moment roduced by Meissner currents or vortices which is the common roerty of ultrathin mesoscoic ( ) suerconductors. Thus, the only solid roerty of a vortex in any quantum condensate is the hase circulation around a vortex core,, where is the vorticity (sometimes called winding number)/ From eq. (3.17) one sees that hase circulation a vortex may be roduced either by currents or by field, or by combination of both:. (3.19) The currents of the mesoscoic device suggested in the main manuscrit ( ) do not roduce any significant magnetic field (as we demonstrate in sec. 3. below). If one sets the external magnetic field to zero, then, and the hase circulation over a secific ath deends only on currents. The geometry of the suggested device is secifically designed to ensure, and thus the required hase circulation is roduced by exclusively currents. 3.. Current-induced roximity Josehson vortices: D Usadel equations As exlained in the main manuscrit, our exerimental results clearly suggest the ossibility of generating roximity Josehson vortices by urely electrical means. This is indeed easy to show theoretically, as we now roceed to exlain. In articular, we shall exlain how the current-induced vortices shown in Fig. were actually comuted. 18 NATURE PHYSICS

19 Let us consider the D SNS described in the revious subsection. Now, let us assume that there is no external magnetic field, but instead there is suercurrent flowing along the SN interfaces. This suercurrent can be modelled by assuming that the order arameter in the leads has a sace-deendent hase that varies along the direction arallel to the interfaces (the y-direction in our case): (y) ex[i(y)]. In articular, we shall assume that the suerconducting hase has the form (y) / y / W, (3.) where is a constant hase difference, y is the transverse coordinate, W is the junction width, and / is in this case simly a arameter that determines the hase gradient. We write the hase difference in this way in order to be able to establish a comarison with the case in which there is a true magnetic field in the junction, see discussion below. Thus, the descrition of current-induced roximity vortices only requires solving the D Usadel equations [Eqs. (3.1) or (3.1)] in the absence of magnetic field, but with boundary conditions at the two SN interfaces that take into account the variation of the suerconducting order arameter according to Eq. (3.), which only requires relacing by (y) in the conditions of Eq. (3.1). Finally, we can consider two different situations. First, a symmetric situation where the suercurrents are flowing in oosite directions in both suerconducting leads. This can be modelled by assuming that the suerconducting hase in the left (L) and right (R) electrodes are of the form: R (y) (y)/ and L (y) (y)/. Second, we can consider an asymmetric situation where the suercurrent only flows along one of the electrodes. This can be described, e.g., by assuming that: R (y) (y) and L (y). In Fig. d & e of the manuscrit we show two examles of the LDOS at the Fermi energy for both arrangements (symmetric and asymmetric) for a junction with length L and width W. To conclude this section, let us say that in the wide-junction limit W >> L one can demonstrate the exact equivalence between the field- and the current-induced roximity Josehson vortices discussed in these last two subsections. The idea goes as follows. As it was shown in Refs. 7 & 38, the analysis of the field-induced vortices simlifies in the wide-junction limit. In this case, it is convenient to use the gauge A Hxŷ. Then, a dimensional analysis of the Usadel equations [Eq. (3.1)] shows that in this limit one can neglect the terms where the derivatives with resect to the y-coordinate aear. With the revious gauge, the magnetic field also disaears from the equations and its only effect is to change the suerconducting hase difference into the gauge-invariant combination of Eq. (3.), which justifies our choice for the arameterization of the hase gradient in Eq. (3.). Thus, in this limit the field- and current-induced vortices are identical, and both symmetric and asymmetric situations exhibit the same vortex structure. This is illustrated in Fig. S3. where we show the LDOS at the Fermi energy for a wide junction with L and W 1 in three different situations: (a) with an external magnetic field, (b) with currents flowing in both leads, and (c) with a current flowing only in one of the electrodes. In all cases and. NATURE PHYSICS 19

20 Figure S3.: LDOS at the Fermi energy throughout the normal wire of a wide SNS junction with length L and width W 1 for three different situations: a, an external magnetic field alied to the junction, b, a current flowing in both suerconducting leads, and c, a current flowing through only one electrode. In all cases and. Notice that the longitudinal coordinate (x) and the transversal one (y) are normalized with the junction length and width, resectively. 3.. Irrelevance of the magnetic field generated by suerconducting leads in current-induced Josehson vortex device The minimal Josehson-vortex-generating device suggested in Fig. f of the main manuscrit contains two current-carrying suerconducting leads. There are also Josehson currents circulating across the junction. These currents unavoidably generate a magnetic field. The field intensity even doubles if both electrodes carry the same currents in oosite directions. However, in suerconductors the current-induced hase ortrait deends on the current density j=i/(w h) (where I is the total current in the wire, w the width of the wire, h its thickness), while the generated magnetic field deends on the total current I. Therefore, by reducing the thickness of the leads while keeing constant the current density j, the magnetic field may be strongly suressed and made negligible. Indeed, by Biot-Savart law, where is the vacuum ermeability, is a vector whose magnitude is the length of the differential element of the wire and its direction is that of the current, is the full dislacement vector from the wire element to the oint at which the field is being comuted. In ractice, a device having the dimensions of the junction J1 (as shown in Fig. ) and made of a tyical conventional suerconductor with a coherence length of ξ =3- nm and critical current density of j C =1 1 A/m will generate, when both leads carry the maximum ossible suercurrent, several Josehson vortices. The exact number of vortices deends NATURE PHYSICS

21 on coherence length ξ; at critical current density the hase gradients at the edge of each electrode are, i.e. the inter-vortex distance inside the junction can be made as short as ξ. With the same conditions, the currents will generate in the centre of the device a magnetic field of the order of.1 mt, which corresonds to the magnetic flux Φ over whole junction area Φ Φ, where Φ is the flux quantum. Note that the Josehson currents flowing across the junction are much weaker than the critical currents of suerconducting leads, the magnetic flux they roduce is much weaker than Φ. It is evident that such a tiny flux will not contribute significantly to the Josehson vortex formation (as justified in the section 3.). Hence the magnetic field lays no role in the formation of Josehson vortices in our device, and they are fully controllable by electric currents. Since the magnetic field lays no role, Josehson vortex cores can be ut very close to each other, down to the scale of ξ, allowing for their very high density. This could be useful for electronically integrable nano-scale Josehson devices. NATURE PHYSICS 1

22 References 9. Berdiyorov, G. R., Milošević, M. V., Baelus, B. J., and Peeters, F. M. Suerconducting vortex state in a mesoscoic disk containing a blind hole. Phys. Rev. B 7, 8 (); Berdiyorov, G. R., Misko, V. R., Milošević, M. V., Escoffier, W., Grigorieva, I. V., and Peeters, F. M. Pillars as antiinning centers in suerconducting films. Phys. Rev. B 77, (8); Berdiyorov, G. R., Milošević, M. V., and Peeters, F. M. Comosite vortex ordering in suerconducting films with arrays of blind holes. New J. Phys. 11, 13 (9). 3. Milošević, M. V., and Geurts, R. The Ginzburg Landau theory in alication. Physica C 7, 791 (1). 31. Single Charge Tunneling, edited by H. Grabert and M. Devoret, NATO ASI, Ser. B, Vol. 9 (Plenum, New York, 199). 3. Nazarov, Yu. V. Anomalies of current voltage characteristics of tunnel-junctions. Zh. Eks. Teor. Fiz. 9, 97 (1989) [Sov. Phys. JETP 8, 1 (199)]. 33. Ingold, G. L., and Nazarov, Yu. V. in Single charge tunneling, edited by H. Grabert and M. Devoret, NATO ASI, Ser. B, Vol. 9, ag. 1 (Plenum, New York, 199). 3. Rollbühler, J., and Grabert, H. Coulomb blockade of tunnelling between disordered conductors. Phys. Rev. Lett. 87, 18 (1). 3. Hammer, J. C., Cuevas, J. C., Bergeret, F. S. & Belzig, W. Density of states and suercurrent in diffusive SNS junctions: role of nonideal interfaces and sin-fli scattering. Phys. Rev. B 7, 1 (7). 3. Eschrig, M. Distribution functions in nonequilibrium theory of suerconductivity and Andreev sectroscoy in unconventional suerconductors. Phys. Rev. B 1, 91 (). 37. Press, W. H. et al., Numerical Recies (Cambridge University Press, 199). 38. Bergeret, F. S. & Cuevas, J. C. The vortex state and Josehson critical current of a diffusive SNS junction. J. Low. Tem. Phys. 13, 3 (8). NATURE PHYSICS