The effect of normal and insulating layers on 0-π transitions in Josephson junctions with a ferromagnetic barrier

Using the Usadel approach, we provide a formalism that allows us to calculate the critical current density of 21 different types of Josephson junctions (JJs) with a ferromagnetic (F) barrier and additional insulating (I) or/and normal (N) layers inserted between the F layer and superconducting (S) electrodes. In particular, we obtain that in SFS JJs, even a thin additional N layer between the S layer and F layer may noticeably change the thickness d F ?> of the F layer at which the 0-π transitions occur. For certain values of d F , ?> a 0-π transition can even be achieved by changing only the N layer thickness. We use our model to fit experimental data of SIFS and SINFS tunnel junctions.

We consider an FJJ consisting of two thick superconducting (S) electrodes with a ferromagnetic (F) film between them; see figure 1(a). This canonical arrangement was considered in many theoretical works [2,3]. The key property of this structure is the possibility of having negative critical current density J c in some ranges of F layer thickness d . F The transition from positive to negative J c corresponds to the transition from the 0 to the π ground state of the JJ. For applications, one tries to choose such a thickness d , F for which the (absolute) value of J c in the π domain is as high as possible. This is usually the case inside the first π domain along the F axis. For the simple SFS structure, shown in figure 1(a), the boundaries of π domains and the whole J d c F ( )dependence is known [24]. However, experimental π JJs often include extra insulating (I) layers [25,26] and/or normal (N) layers [26,27] between SF or FS layers. The purpose of the additional I layer(s) is to enlarge the characteristic voltage, especially in the π state. It was shown [28,29] that the presence of extra insulating layers shifts the first 0-π transition to smaller values of d . F There are also several reasons to consider N layer(s), as follows. First, a so-called 'dead' layer exists in many sputtered ferromagnetic films. The dead layer is a surface layer of the ferromagnet, which behaves as a non-magnetic metal. It usually appears due to the surface roughness or the mutual dissolution of atoms at the interface between N and F layers. Such a dead layer is inherent, for example in NF interfaces involving Cu and its alloys with d 3 metals, which are very popular as spacers. Usually it is naively assumed that the dead layer makes the effective F layer thinner and adds an extra N layer (non-magnetic F). Many experimental data match the theory only if one assumes such a dead layer of finite thickness [24,26,[30][31][32][33][34]. However, sometimes such a naive fit gives questionable results because it does not take into account the correct boundary conditions at all interfaces.
Second, an N layer between F and S is often technologically necessary to produce high-quality JJs [24,26,27,[30][31][32][33][35][36][37][38], for example by preventing diffusion between F and S films [39]. The presence of an N layer in FJJs was not taken into account in any theoretical work [44,[40][41][42][43]45] (see also [2,3] for review) in spite of numerous experiments. We show that this is reasonable only if the F and N metals behave fully identically, except for their magnetic properties. Otherwise, the presence of the thin N layer changes the boundary conditions, which affects the dependence of the Josephson current J c on d .
F Recent experiments [46], which use a new continuous in situ technology allowing the deletion of this layer, actually exhibit a change of the 0-π transition points in the J d The overall effect of these extra I and N layers is not studied in detail. Therefore, we present a formalism in the following, which allows us to calculate the critical current density of FJJs with additional I and/or N layers inserted between SF and/or FS layers. The heterostructures under question can be constructed by selecting one of the items of figure 1(b) and inserting it by following one of the arrows into figure 1(a). At the other arrow position we insert either the same or another item from figure 1(b). In this way we obtain 21 possible configurations of layers in FJJs.
The article is organized as follows. In section 2 we describe our model based on the Usadel equations supplemented with Kupriyanov-Lukichev boundary conditions. Different types of interlayer boundaries are analysed. Section 3 presents the obtained dependencies of the critical current density on the F layer thickness as well as the analysis of the 0-π transitions in the framework of a linear approximation. We use our formalism in section 4 to fit experimental data of SINFS and SIFS junctions. Section 5 concludes this work. Details of the calculation can be found in the appendix.

The boundary value problem
The basic Josephson junction configuration we consider is sketched in figure 1(a). It consists of two thick S electrodes enclosing an F layer of the thickness d F along the x axis. Our model allows to consider an additional I or N layer at the SF interfaces as well as I layers at the SN or NF interfaces, as illustrated by figure 1(b).
We calculate the critical current density J c of these configurations by determining their Green's functions in the 'dirty' limit. In this limit, the elastic electron scattering length is much smaller than the characteristic decay length of the superconducting wave function. We determine the Green's functions with the help of the Usadel equations [47], which we use similar to [3] in the form in the N and F layer, where F j and G j are the Usadel Green's functions, while F F .
)at the temperature T, and T c is the critical temperature of the superconductor. By using the definition T 1 m c h t p º ( ) we take, similar to [28], the spin-flip scattering time m t into account. This approach requires a ferromagnet with strong uniaxial anisotropy, for example, Cu alloys with transition metals, which are used in many experiments. Equation (1) should be satisfied for any integer number n. The scaled exchange energy h H T c p º ( ) of the ferromagnetic material, where the energy H describes the exchange integral of the conducting electrons, is assumed to be zero in the N layer.
In our model we use the coherence lengths of the superconducting correlations, which are defined with the help of the diffusion coefficients D N and D F in the normal and ferromagnetic metal, respectively. We use the scaling defined by k 1.
B  º º The decay length H x of superconducting correlations in the ferromagnet is usually in the order of nm. This is ) to consider the supercurrent as a result of interference of anomalous Green's functions induced from the superconducting banks. It was shown [28] that this ansatz is valid even for small distances d , F H x that is, in the region of the first 0-π transition. It is convenient to consider this problem in theta parametrization [48] F G e sin , cos , 3 where j j is independent of the coordinate x. It corresponds to the phase 2 j j f º  of the order parameter of the S banks for the right and left superconducting electrode, respectively, while j q satisfies the sine-Gordon-type differential equation Since we assume that the superconductivity in the S electrodes is not suppressed by the neighbouring N and F layers, we obtain analogous to Vasenko et al [28] at the interfaces of the superconductor, where Δ is the absolute value of the order parameter in the superconductor. The validity of this assumption depends on the values of the suppression parameters r r and S r describe the resistivity of the N, F and S metals, respectively.
The Kupriyanov-Lukichev boundary condition [49,50] at the superconducting interface, shown in figure 1(c), is at the NF boundary. Here we defined x F,N F NF q q º ( )and x .
N,F N NF q q º ( ) Additionally, we use the differentiability condition The suppression parameters are defined analogous to (6), but not restricted to only small or large values. In order to finally extract the critical current density J c from the current phase relation J J sin flowing through our device, with the help of the Green's function F F in the F layer. Here we chose the position x = 0; see figure 1(a), in order to simplify the calculation.

Critical current density
In this section we rewrite expression (12) to be able to directly calculate the critical current densities of all SFS Josephson junctions of the type sketched in figure 1(a), which may include each of the layers, shown in figure 1(b) at the SF interfaces. In order to solve the Usadel equations (1) in the F layer we use the ansatz [28,51] F x x x e sin e sin , 13  Then the solution x F q -( ) will turn out to be most dominant in the left side of the F part and to decay exponentially in the right side of the junction. Therefore, it has practically no overlap with the solution x F q + ( ) which is dominant in the right side of the F layer.
We obtain both solutions x F q -( ) and x F q + ( ) by integrating the differential equation (4) for j F = twice. The first integration results in Here g  are the integration constants. In the F layer we can assume small superconducting correlations 1 F q  to linearise the denominator of the left-hand side of (15), which leads us to the equation The rewritten integration constants c  are given by the boundary conditions at the right and left ferromagnetic interfaces as The constants c  will be determined in the next section.

SF interface without or including an N layer
In the following we determine a constant TI c to replace c + or cin (18) in the case of no N layer at an SF interface, as shown for example in figure 1(c). The index TI stands for transparent or insulating.
We insert the integrated sine-Gordon equation (14) at the position x SF into the boundary condition (7) and obtain the relation (17) In the case 0, h  which means neglecting the effect of spin-flip scattering, this equation is a quartic equation in TI c and therefore exactly solvable. To find the solutions in this case we use the function solve of the MATLAB software. Afterwards we make use of (19) to select one of the four solutions. In the case 0 h ¹ we solve (20) numerically by using the function fsolve of the MATLAB software together with the solution of the limit 0 h  as the starting value.
In this way we find TI c for the determination of the critical current density (18) in the case of no N layer at the SF boundary. The case of a small parameter BSF g corresponds to a transparent SF interface, while a large one corresponds to an insulating interface [28,52].
Next, we determine a constant N c for the case of a thin N layer d N N x  between the superconductor and ferromagnet, as shown in figure 1(d).
By inserting the integrated sine-Gordon equation (14) for x x NF = into the boundary condition (9), we obtain the equation When we rewrite this equation using the definition sin 2 looks similar to (20). The main difference is that it reduces in the case 0 h  not to an equation of fourth order in . N c This is because we take an effect similar to the inverse proximity effect at the NF boundary into account; that is, the reduction of the superconducting correlations in the N layer due to the proximity of the F layer. Therefore, the value N, as we show in the appendix. However, we also show in the appendix that (22) reduces in the limit 0 h  together with 0 NF g  , which means assuming the conductivity of the N layer to be much larger than that of the ferromagnet, to an equation of fourth order in . N c Therefore, we make three steps in order to solve (22). First we determine its solution in the case , 0 NF h g  similar to the fourth-order case of (20). We then use this result as a starting value to solve (22) for only the limit 0 h  with the help of the function fsolve of the MATLAB software. This in turn leads to another starting value which we use to solve (22) with fsolve, but without any limiting case.
The solution N c of (22) can finally be used as c + or cfor the determination of the critical current density (18) in the case of an N layer at the SF interfaces. Small parameters BSN g and BNF g correspond to transparent SN and NF interfaces, while large ones correspond to insulating interfaces [28,52].

Discussion
In this section we first select FJJ configurations, where the N layer has the largest influence. We then analyse their critical current densities with the help of the formalism we derived in the previous section. Finally, we discuss the results with the help of solutions of the linearised differential equation.
We do not analyse configurations where a thin N layer (d N N x  ) is located between S and I layers, which gives only a negligible reduction of J c compared to the case without an N layer. This is because the superconducting condensate simply penetrates into the whole N layer. The same effect occurs when the thin N spacer separates the S and F layers and both (SN and NF) interfaces are transparent.
However, when the SN boundary has a very weak transparency or gets even insulating, that is, when the N layer is located between an I and an F layer, then the N layer(s) play(s) a more notable role depending on the relation of resistances NF g (11), as we will see in the following. Examples for the critical current density J d For a physical explanation of this behaviour, one can imagine that a decrease of the amplitude of the superconducting pair wave-function in the F layer is connected to a decrease of the function . F q In particular, the positions along the F layer where F q becomes zero correspond to sign reversals of the critical current density and are therefore directly linked to the thicknesses d F where a 0-π transition occurs. This picture already helps us to understand why an insulating layer at the SF interface shifts the 0-π transitions towards smaller values of d F [28,29]. This is because the I layer induces a decreasing shift to F q at the SF interface, as can be seen from (7) for 1.
BSF g  Since F q decreases monotonically from the interfaces into the F layer, this shift results in a shift of its zeros towards the interface. This in turn leads to a shift of the 0-π transitions to smaller d , F as can be seen by comparing, for example, the black lines in figures 2(a) and (b). By inserting an N layer at the IF interface, we can mitigate this effect. In fact, the function θ is still decreased by the I layer, but the decrease of its derivative q¢ may be smaller than in the case of a superconducting pair wavefunction that directly penetrates the F layer. This in turn leads to a shift of the 0-π transition back to larger d .  Table 1. Parameters for the calculation of the critical current densities (18) shown in figure 2. The parameters B g are responsible for the presence of an I layer, while the equation for the calculation of c  determines whether we consider an N layer or not. We keep the product BSN NF g g constant because its outcome (23) does not change during our analysis.
at the F interface. For d 0, N = (24) resembles (7). Therefore, we obtain, by using the values defined in table 1, the correct limiting results. Note that F q ¢ is negative in this case because the amplitude of the superconducting pair wave-function decreases when entering the F layer.
An increase of d N increases F q ¢ and therefore shifts the 0-π transitions towards larger d , F as shown by figures 2(b)-(d). Furthermore, from (24) it can be understood why a smaller value of NF g induces a larger increase of . F q ¢ This again shifts the 0-π transitions towards larger d , F as shown by figures 2(f)-(h). The same effect occurs in figure 2(e), but it has a different interpretation because the 0-π transitions are already shifted to large d F without an N layer, due to the absence of the I layer (black line). A small value of NF g does not change this situation significantly. However, if NF g increases and therefore F q ¢ decreases, the 0-π transitions get shifted to smaller d .

F
These effects are related partially to d N that may be small (d N N x  ) but mainly to the conducting properties of the N layer represented by NF g (11). Note that we neglected the effect of spin-flip scattering in figure 2; that is, we chose 0. h = An increase of η shifts all shown 0-π transitions towards larger d , F including the ones of junctions without an N layer [28,52]. It is not necessary to consider this effect in order to understand the role of N layers in FJJs. However, the described effect is important for the fitting of experimental results in section 4.
The influence of N layers on FJJs can be seen most clearly when they are inserted at IF interfaces and d F is kept constant, not far from a 0-π transition, while d N changes. In this way, the 0-π transition can be controlled by d N , as shown in figure 3. Here we consider an SIFIS junction which is in the 0 state for d 0.5 .
F F x = By adding N layers at the IF interfaces and increasing their thicknesses simultaneously, we tune the FJJ into the π regime. Figure 3 considers the same FJJ configuration as figure 2(d), where d N is fixed and d F changes.
To understand the role of the boundary parameters in the 0-π transition patterns in more detail, it is useful to analyse it in a simple linear approximation. This approximation can be used if both S electrodes have nontransparent interfaces, or if T T. c  Then we may assume that G 1, general solution of the Usadel equations (1) in the non-superconducting layers has the form pmk x exp , where p and q are real. The critical current density is given by the expression (12). For FJJs without an N layer, the critical current density has already been calculated in [28,45,52,53].
does not differ much from (25). We only obtain an additional real factor k d cosh , 2 N N -( ) but the position of the 0-π transitions is still defined by the term marked as the real part. Therefore, the positions of the 0-π transitions will be the same as in the SFS case (see figure 2(a)) for one extra N layer. The small boundary parameter SN g is needed in order to neglect the proximity effect in the S electrodes. However

Double-barrier structures SIFIS versus SINFNIS
In order to discuss the interplay of the N and I layers we jump to the description of the configurations shown by figure 2(d) which transforms into the two previous cases (25) and (28) for The 0-π transitions are defined by the zeros of the real part, which has the same form as in the case of SFS JJs with transparent interfaces (25). That is, the N layers have mitigated the effect of the I layers, which can be seen by comparing figures 2(d) with (a).

SIFIS versus SINFIS structures
The effect of a single N layer on a double-barrier SIFIS junction, shown in figure 2(c) and (g), is discussed in the following. The critical current density of the SINFIS junction with the same boundary parameters as in the section before is given by In this case, the 0-π transitions are defined by the zeros of the function qd cos F ( ) and located at the positions where d mm 2 , 0, 1, 2 ...; F H x p p = + = that is, they are also shifted towards larger d F in comparison with the ones of the SIFIS junction; see figures 2(c) and (g).
In our previous article [42] we obtained in fact the same expressions (28) and (33). There we assumed that the interface transparencies of both S electrodes are small, one of them due to the presence of an insulating barrier. In this way we analysed SI 1 FI 2 S and SI 1 NFI 2 S structures with rather different transparencies of the I 1 and I 2 barriers. We found in the linear approximation that the critical current density for an SI 1 NFI 2 S FJJ is the same as the one for an SI 1 FNI 2 S structure.

SIFS versus SINFS structures
If the structure contains only one insulating barrier, as in figure 2(b) and (f), we may use the tunnel Hamiltonian method, which, for the critical current density, yields the expression To use the linear approximation we shall assume that T is close to T c , and in order to neglect the proximity effect in the right S electrode we use the rigid boundary conditions , 1.
BSF SF g g  We also assume the N layer to be thin, d . x g  because the last value is determined by the large resistance of the I barrier. The solution weakly depends on d N because the suppression of the superconducting correlation along the thin N layer is negligible in comparison with that of the I barrier. However, the ratio of the N and F resistance, which defines via NF g (the derivative jump (10) at the NF interface), still plays a role. Then the 0-π transition takes place at d , F for which the equation

Comparison with experiment
To check our theory, we use data from SINFS JJs [26], based on Nb Al | 2 O 3 |Cu Ni | 0.6 Cu 0.4 |Nb heterostructures. These samples include a 2 nm Cu interlayer between the I and F layers. Using the same technology, new series of samples were produced, but the process was changed in order to delete the Cu layer. That is, we can compare SIFS and SINFS FJJs with the same layer properties, including the concentration of the NiCu alloy. In figure 4 we show a fit of experimental data of critical current densities for different F layer thicknesses d F of both types of junctions. Dots correspond to SIFS junctions and triangles correspond to SINFS junctions.
We calculated the critical current densities with the help of (18). In the case of the SIFS configuration we made use of (20) to calculate the parameter cand in the case of the SINFS configuration we used (22). For our fit we used the coherence lengths N x = 10 nm, F x = 7.60 nm and H x = 1.72 nm. Our exchange energy H/k B = 880 K is situated between the value 850 K corresponding to the alloy Ni 0.53Cu 0.47 [27] and the value 930 K of clean Ni [36]. The product H m t = 1/1.7 is similar to the one used by Weides et al [26]. Further values taken from this publication are the temperature T = 4.2 K, junction area A = (100 μ m) 2 and resistivity F r = 54 μΩcm.
Additionally, we used the damped critical temperature T c = 7.2 K of Nb and the resistivity N r = 0.66 μΩcm.
Together with the fit parameters BSF g = 0.1, BSN g = 90000, BNF g = 0.01 and NF g = 0.016 of figure 4 we obtain the boundary resistances R BSF = 4.10 n R , BSN W = 584 μΩ and R BNF = 0.41 nΩ, which are realistic values. As we have shown in figure 2(f), a small suppression parameter 1 NF g < results in a shift of the 0-π transition to larger d F for the sample with N layer. This effect explains the shift of the 0-π transition observed in the experiments on SIFS and SINFS FJJs. The difference in the amplitude of the curves is attributed to the different thicknesses of the I barrier in these two sample series.
This conclusion is also supported by experimental observations on SIsFS junctions [54][55][56]. These observations indicate that the introduction of a thin s interlayer, which should make a transition to the normal state if its thickness is of the order of the coherence length, shifts the 0-π transitions towards larger d . F

Conclusion
Using the Usadel equations, we have calculated the critical current densities of ferromagnetic Josephson junctions (FJJs) of different types, containing I and N layers at the SF interfaces, and compared them to critical current densities of structures without N layers. Such layers were technologically required in many FJJ experiments, but were not taken into account in previous models.
It was shown earlier [28,52,53] that insulating barriers decrease the critical current density and shift the 0-π transitions to smaller values of the ferromagnet thickness d . ( )minima are shifted because of different boundary conditions. When the dirty limit does not apply, the oscillation period of J d c F ( )may depend on many other parameters and does not have to be constant, but can change with the F layer thickness [57]. A multi-domain structured ferromagnet may also change J d c F ( ); for instance, the oscillation period decreases when the domain width increases [58].
If the transport properties of the N layer between the I and F layer are the same as those of the ferromagnet, not only the period of the J d   (20), and in the SINFS case we used (22). Fitting parameters are BSN g = 90000, BNF g = 0.01, BSF g = 0.1 and NF g = 0.016.
properties differ from those of the F layer. The smaller the value of , NF g the larger is the change of the J c amplitude and the shift of the 0-π transitions; see figures 2(f)-(h).
The situation is completely different in the case of transparent SF interfaces, that is, without an I layer in between. In this case, the additional thin normal layer with conductivity much larger than that of the ferromagnet ( 1 NF g  ) does not play any role. In the same setup, an N layer with transport properties similar to those of the ferromagnet ( 1 NF g » ) provides a shift of the 0-π transitions to smaller d ; F see figure 2(e). This process is explained in more detail after (24).
In summary, even a thin additional N layer may change the boundary conditions at the IF boundary depending on the value of . NF g We conclude that it can effectively mitigate the effect of the insulating barrier on the decaying oscillations of the critical current density J d .  q Inserting the result into (A.5) gives us N,F q as a function of N c , which itself, when inserted into (22), allows us to finally determine N c by solving the transcendental equation (22) numerically.
In the following we consider the limit , 0 NF h g  to reduce (22) to an equation of fourth order in .