Josephson junction with magnetic-field tunable current-phase relation

# Josephson junction with magnetic-field tunable current-phase relation

A. Lipman The Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel    R. G. Mints The Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel    R. Kleiner    D. Koelle    E. Goldobin Physikalisches Institut and Center for Collective Quantum Phenomena in LISA, Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany
July 14, 2019
###### Abstract

We consider a 0- Josephson junction consisting of asymmetric and regions of different lengths and having different critical current densities and . If both segments are rather short, the whole junction can be described by an effective current-phase relation for the spatially averaged phase , which includes the usual term , a negative second harmonic term as well as the unusual term tunable by magnetic field . Thus one obtains an electronically tunable current-phase relation. At this corresponds to the Josephson junction.

Josephson junction
###### pacs:
74.50.+r, 85.25.Cp

## I Introduction

Recently we proposedGoldobin et al. (2011) to implement a Josephson junction (JJ) Buzdin and Koshelev (2003) with magnetic-field tunable current-phase relation (CPR) based on an 0- JJ with the 0 and segments of different length . This proposal was made keeping in mind YBaCuO-Nb ramp zigzag JJ technologySmilde et al. (2002); Hilgenkamp et al. (2003) (or a similar oneAriando et al. (2005) with NdCeCuO-Nb) established recently in our group alsoScharinger et al. (2012). However, in experiment we were more successfulSickinger et al. (2012) in employing superconductor-insulator-ferromagnet-superconductor (SIFS) 0- JJsWeides et al. (2006, 2010); Kemmler et al. (2010), where the lengths of 0 and segments are equal, but critical current densities and in the 0 and parts are different.

Therefore, in this paper we present a more general theory, which describes an effective JJ made of asymmetric and regions of different lengths and having different critical current densities and .

## Ii Model

The static sine-Gordon equation that describes the behavior of the Josephson phase in a 0- JJ is

 Φ02πμ0dJϕ′′−jc(x)sinϕ=−j. (1)

Here is the magnetic flux quantum, is the specific inductance (per square) of the superconducting electrodes forming the JJ and is the bias current density. The prime denotes the partial derivatives with respect to coordinate . We assume that the critical current density has the form of a step-function

 jc=jc,0>0, 0≤x≤L0, (2) jc=jc,π<0, −Lπ≤x<0. (3)

We write the critical current density as

 jc(x)=⟨jc(x)⟩[1+g(x)], (4)

where

 ⟨jc⟩=1L∫L0−Lπjc(x)dx=1L(jc,0L0+jc,πLπ) (5)

is the average critical current density, is the total length of the junction, and . The function is defined as

 g(x)=jc(x)⟨jc(x)⟩−1 (6)

that results in

 g(x)={g0,0

where

 g0=(jc,0−jc,π)Lπjc,0L0+jc,πLπ;gπ=−(jc,0−jc,π)L0jc,0L0+jc,πLπ. (8)

Then we divide Eq. (1) by and normalize the coordinate to the Josephson length calculated using , i.e.,

 (9)

Thus, we obtain a normalized sine-Gordon equation for the phase difference

 (10)

where is the normalized bias current density. It is worth mentioning that can be positive as well as negative. Below, for the same of simplicity, we assume . Thus, Eq. (10) becomes

 ϕ′′−[1+g(x)]sinϕ=−γ. (11)

In the case the substitution converts Eq. (10) to the same Eq. (11).

We look for a solution of Eq. (11) in the form

 ϕ(x)=ψ+ξ(x)sinψ, (12)

where

 ψ=⟨ϕ(x)⟩ (13)

is a constant average phase, while describes the deviation of the phase from the average value, i.e., . Further we assume that the deviation is small, i.e., . Then we plug the relation (12) into Eq. (11), expand it in series in , and keep the terms of zero and first order. We get

 ξ′′sinψ−[1+g(x)][1+ξ(x)cosψ]sinψ=−γ. (14)

The constant terms (zero order of in Eq. (14)) are

 γ=sinψ+⟨ξ(x)g(x)⟩cosψsinψ. (15)

The terms of first order of in Eq. (14) are

 ξ′′−g(x)={ξ+ξ(x)g(x)−⟨ξ(x)g(x)⟩}cosψ. (16)

Numerical calculations show that the two terms have an extremely weak effect on solutions of Eq. (16). We neglect these terms and obtain for

 ξ′′−g(x)=0. (17)

We treat solutions of Eq. (17) by using the matching continuity (at ) and boundary (at , ) conditions

 ξπ(0)=ξ0(0),ξ′π(0)=ξ′0(0), (18)
 ξ′π(−lπ)sinψ=h,ξ′0(l0)sinψ=h. (19)

The applied field is normalized by , i.e.,

 h=2HHc1,Hc1=Φ0πΛλJ, (20)

where is the effective magnetic thickness of the JJ. We integrate Eq. (17) once and obtain

 ξ′0(x)=g0(x−l0)+hsinψ,0
 ξ′π(x)=gπ(x+lπ)+hsinψ,−lπ

The second integration results in

 ξ0(x)=g0(x22−l0x)+hxsinψ+C, (23) for 0
 ξπ(x)=gπ(x22+lπx)+hxsinψ+C, (24) for −lπ

The integration constant can be obtained using the condition

 C=l0−lπ2(g0l0+gπlπ3−hsinψ). (25)

We use Eqs. (7), (23), and (24) and obtain the average in the form

 ⟨ξ(x)g(x)⟩=Γ0+Γhhsinψ, (26)

where the coefficients and are given by

 Γ0 = −l20l2π3(jc,0−jc,π)2(jc,0l0+jc,πlπ)2, (27) Γh = l0lπ2jc,0−jc,πjc,0l0+jc,πlπ. (28)

Using Eqs. (15) and (26) we find the current-phase relation in the form

 j=⟨jc⟩(sinψ+Γ0sinψcosψ+hΓhcosψ). (29)

It is worth noting that there is a simple relation between the coefficients and . Indeed, it follows from Eqs. (27) and (28) that

 Γ0=−43Γ2h. (30)

In the case of equal lengths of 0 amd parts () we find

 Γ0=−l212(jc,0−jc,πjc,0+jc,π)2,Γh=l4jc,0−jc,πjc,0+jc,π. (31)

The energy corresponding to the current-phase relation (29) is given by

 U(ψ)=⟨jc⟩(1−cosψ+hΓhsinψ+Γ02sin2ψ). (32)

## Iii Conclusions

We have extended our previous results Goldobin et al. (2011) to the case of arbitrary critical current densities more relevant for experimentSickinger et al. (2012). The dependence (29) of the CPR on the phase and applied field is the same as in our previous studyGoldobin et al. (2011). The difference is in the formulas (28) for and .

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