Ferromagnetic planar Josephson junction with transparent interfaces: a φ junction proposal
We calculate the current phase relation of a planar Josephson junction with a ferromagnetic weak link located on top of a thin normal metal film. Following experimental observations we assume transparent superconductor-ferromagnet interfaces. This provides the best interlayer coupling and a low suppression of the superconducting correlations penetrating from the superconducting electrodes into the ferromagnetic layer. We show that this Josephson junction is a promising candidate for an experimental junction realisation.
pacs:85.25.Cp, 74.78.Fk, 74.45.+c, 74.50.+r
A junction [1, 2] is a Josephson junction with a doubly degenerate ground state, in which the Josephson phase takes the values or () . This junction being closed into a ring is able to self-generate a fractional flux /(2), where is the magnetic flux quantum.
In this sense the junction is a generalisation of the junction  which has a Josephson phase or in its ground state. It has been experimentally demonstrated that the junction improves the performance and simplifies the design of classical and quantum circuits [5, 6, 7]. Since the junction offers the possibility to choose a special value of the phase in the ground state it may further optimize these circuits.
The initial junction proposal  investigated grain-boundary junctions, which were analysed experimentally in . From then on junctions were studied more and more intensively and many other systems appeared as possible candidates for the realisation of junctions, e.g. [2, 3, 9, 10, 11, 12, 13, 14]. Only recently, an experimental evidence of a junction made of and parts [2, 11, 12] was reported . One half of the junction had the Josephson phase in its ground state and the other half the phase . This was realised  by connecting two superconductor-insulator-ferromagnet-superconductor (SIFS) junctions in parallel. The advantage of this concept is that it is based on the technology already developed for the fabrication of - junctions [16, 17].
On the other hand this junction concept is difficult to realise experimentally because, e.g., a step in the thickness of the F layer must be realised with very high accuracy [11, 12, 15]. A completely other method, the “ramp-type overlap” (RTO) junction, was proposed by Bakurskiy et al. . It only requires one small SFS junction located on a thin normal (N) metal layer, see figure 1. This basic setup provides a miniaturized junction. Moreover, this type of junction has already been realised experimentally for the analysis of the double proximity effect .
A simple model  to show that the RTO junction can be used as a junction requires its current-phase relation (CPR). By writing it in terms of a sine series
where is the Josephson phase, the amplitudes have to obey the conditions 
The RTO junction, schematically shown in figure 1, can fulfil these conditions because the current flows between the S electrodes through the F metal and the N layer. In this way the properties of an SFS and SNS junction are combined. The SFS junction can have a negative [20, 21] amplitude in (1), while the SNS junction has a positive [20, 22] amplitude in (1). By adding both the total amplitude can be minimized and a dominant negative amplitude from the SNS part is obtained to fulfil conditions (2). Since supercurrents in SFS junctions are rather small, the SNS contribution has to be reduced. This is done by using only a thin normal metal film.
In the present paper we investigate an RTO junction which has, differently from the one proposed in , transparent SF interfaces in order to amplify the SFS contribution to the total current. This assumption has already successfully been used to describe various experiments [19, 23, 24]. As a result, we obtain slightly smaller system sizes for the junction realisation than , where weakly transparent interfaces were assumed. Moreover, our approach provides a better penetration of the superconducting correlations into the F layer which may increase the Josephson current. In the framework of transparent SF interfaces we cannot use linearised equations for the SFS part, as it was done in . Therefore, we use non-linearised equations in the SFS and SNS part for our analytical approach.
We derive the CPR in the “dirty” limit. For this purpose, we combine the solution of the Usadel equations in the N film  with the solution of the Usadel equations in the SFS layer . The resulting current phase relation consists of three parts: (i) a contribution from the SFS layer, (ii) a contribution from the N film and (iii) a composite SNFS term.
The paper is organized as follows. In section 2 we introduce the model of the considered Josephson junction in terms of Usadel equations. The analytical expression of the CPR of our system is based on this model and presented in section 3. In section 4 we use this expression together with realistic system parameters to discuss its applicability as junction. Finally, an appendix provides a detailed derivation of the composite SNFS current.
The considered Josephson junction is sketched in figure 1. It consists of an SFS junction located on a normal metal film. The F layer has a thickness and a length while the N layer has a thickness and is considered as infinitely long. We have chosen the and axis in directions parallel and perpendicular to the plane of the N film, respectively.
For the calculation of the current flowing from one superconducting electrode to the other we determine the Green’s functions describing our system. We consider the “dirty” limit [19, 23, 24], in which the elastic scattering length is much smaller than the characteristic decay length, we can use the Usadel equations  to model our system. We write them in the form 
in the N and F layer, respectively. Here, and are the Usadel Green’s functions in the parametrization . The frequencies contain the Matsubara frequencies at temperature , where , and the exchange field of the ferromagnetic material which is assumed to be zero in the N layer. The decay lengths
of the superconducting correlations are defined via the critical temperature of the superconductor (we use ) and the diffusion coefficients and in the normal and ferromagnetic metal, respectively.
We assume that superconductivity in the S electrodes is not suppressed by the neighbouring N and F layers. This assumption is valid in our case of transparent SF interfaces with the conditions for the suppression parameters
Here, and are the resistances and areas of the SN and SF interfaces. The values of describe the resistivity of the N, F, and S metals.
This allows us to use the rigid boundary conditions 
where is the absolute value of the order parameter in the superconductor.
and at the interfaces of the superconductor they are
Additionally we use
at the NF interfaces, where
is defined analogous to (6).
Finally we calculate the total current
by integrating the standard expressions  for the current densities of the N and F part over the junction cross section along the axis. This leads us to
The width of the junction along the axis is supposed to be small compared to the Josephson penetration depth. We have chosen the position for the integration over the junction cross section since the component of the current densities vanishes there because of the symmetry of the considered junction geometry.
In order to calculate the current from (13) we cannot simply add the current through the N layer calculated by Bakurskiy et al.  to the SFS current calculated by Buzdin et al.  because we have to take into account a composite SNFS current which appears due to a penetration of superconductivity from the N layer into the F layer. Therefore, we split the current into a contribution due to a direct penetration of superconductivity into the F layer and the additional part . This leads us to
In the following three sections we derive the expressions of these three currents using the scaling
3.1 Current in the N layer
In this layer we adopt the current
with the definitions
from . Its derivation is based on the assumption and an infinitely long N layer. It is calculated with the help of the solution (40) of the non-linear Usadel equations which depends only on the coordinate because the thickness is assumed to be small.
3.2 Current in the F layer
is a result of . It also has been calculated with the help of a solution of the non-linear Usadel equations because is assumed. Additionally the condition is required.
3.3 Composite NF current
into a function , which corresponds to currents only flowing in the F layer, and a function , which corresponds to currents flowing through the N layer into the F layer. The second function is obtained by linearising the Usadel equations (3) in the F layer. Then we connect it to the N layer solution via the boundary conditions.
The superposition (26) of the solution of the non-linear Usadel equation with the solution of the linearised Usadel equation is valid because we distinguish in the F part between two cases: (i) at near the boundaries to the S regions the Green’s function is very dominant due to a transparent boundary between the S and the F part, that is ; (ii) at , that is away from the boundaries the contribution of decays exponentially. Therefore, the contribution from the N part is dominant .
In this section we estimate the geometrical parameters , and , see figure 1, for which the considered Josephson junction obeys the junction conditions (2). We use the analysing scheme of  and finally compare our results with the ones obtained in .
We split the sine series amplitudes
of the total current (16), scaled according to (17), into parts originating from the current of the N layer (18), the F layer (23) and the composite NF current (27). There is no amplitude because we have a pure sinusoidal CPR (23) in the F layer.
In our calculations we chose the temperature . We make this choice because far away from the critical temperature the CPR has larger deviations from the form  which results in a larger second harmonic . As S electrode material we chose Nb with because it is commonly used in superconducting circuits.
Our first step is to find suitable parameters . For this purpose we analyse the amplitudes (28) and (29) as a function of for different values of for the same parameters as in : , , , , and . Figure 2 shows three typical examples: (a) , (b) and (c) . The first (a) and last (c) examples correspond to limiting cases where it is difficult to realize a junction because the intervals of where conditions (2) hold are not large. These intervals of are highlighted by bold lines. In between the two limiting values for this line becomes longer. Figure 2 (b) shows an optimum situation because there is a wide range of which yields a junction configuration.
Here, we use the constant with , and
From (30) we find the minimum value and maximum values of .
For summarising our suggestion of the geometrical configuration of a junction we use the value for Cu as N layer, a strongly diluted ferromagnet such as FePd or the CuNi alloy with and as F metal. Our set of parameters then become , and , which we compare to the values , and of .
Since we use the same N layer configuration, the value for is the same. But the suggested regime for differs. A change in this direction was expected because we only need a thin F layer since the transparency of our interfaces already amplifies our SFS current contribution. The possible range for the length of the F part is smaller in our case but the whole junction configuration is still experimentally feasible.
We have shown that the considered Josephson junction with a ferromagnetic weak link located on a thin normal metal film is a good candidate for a junction realisation. By choosing transparent SF interfaces we obtained slightly different system sizes for the junction existence compared to a junction with weakly transparent interfaces.
The current was split into a contribution through the N layer, the F layer and a composite term which described the current flowing through the N and F parts of the junction simultaneously. We performed our calculations in the “dirty” limit, that is, the currents are obtained from solutions of the non-linear Usadel equations.
Since our case of a large interface transparency corresponds better to the experimental situation [23, 24, 19] than weakly transparent interfaces  it is important to note that a smaller thickness and length of the F layer have to be chosen than predicted in . We are looking forward to experiments realising this junction and its application in classical and quantum devices.
Appendix A NF current derivation
In this appendix we derive the current (27) which flows through the N and F part of the junction, sketched in figure 1, simultaneously. We first linearise the Usadel equations (3) and then combine the solution with the Green’s functions from  and .
For the linearisation of (3) we assume the superconducting correlation coming from the N part into the F part as rather small. Then, the Green’s function can also be assumed to be small. Using we obtain the linearised Usadel equation 
with the definitions
Its solution in the form of a series
and a Fourier coefficient , already obeys the boundary condition (8) at the upper border ().
This equation is already fulfilled by the solution
In order to obtain the Fourier coefficient which fixes the solution from (34) we use the boundary condition (11) where we neglect the term assuming . Now, we replace the Green’s function by and insert
Our last step is to calculate the current . Therefore, we insert the Green’s function (26), which contains the Green’s functions from (34) and (37), into the definition (15) of the F layer current. Due to the condition it reduces to a sum , where the NF current is defined by
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