Josephson junction with magnetic-field tunable current-phase relation
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.
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 .
The static sine-Gordon equation that describes the behavior of the Josephson phase in a 0- JJ is
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
We write the critical current density as
is the average critical current density, is the total length of the junction, and . The function is defined as
that results in
Then we divide Eq. (1) by and normalize the coordinate to the Josephson length calculated using , i.e.,
Thus, we obtain a normalized sine-Gordon equation for the phase difference
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
We look for a solution of Eq. (11) in the form
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
The constant terms (zero order of in Eq. (14)) are
The terms of first order of in Eq. (14) are
Numerical calculations show that the two terms have an extremely weak effect on solutions of Eq. (16). We neglect these terms and obtain for
We treat solutions of Eq. (17) by using the matching continuity (at ) and boundary (at , ) conditions
The applied field is normalized by , i.e.,
where is the effective magnetic thickness of the JJ. We integrate Eq. (17) once and obtain
The second integration results in
The integration constant can be obtained using the condition
where the coefficients and are given by
In the case of equal lengths of 0 amd parts () we find
The energy corresponding to the current-phase relation (29) is given by
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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