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Spontaneous Flux and Magnetic Interference Patterns in 0-pi Josephson Junctions

The spontaneous flux generation and magnetic field modulation of the critical current in a 0-pi Josephson junction are calculated for different ratios of the junction length to the Josephson penetration depth, and different ratios of the 0-junction length to the pi-junction length. These calculations apply to a Pb-YBCO c-axis oriented junction with one YBCO twin boundary, as well as other experimental systems. Measurements of such a junction can provide information on the nature of the c-axis Josephson coupling and the symmetry of the order parameter in YBCO. We find spontaneous flux even for very short symmetric 0-pi junctions, but asymmetric junctions have qualitatively different behavior.

Although many measurements now support the idea that the gap in the high-temperature cuprate superconductors has dominant d x 2 −y 2 character [1], an orthorhombic superconductor such as Y Ba 2 Cu 3 O 7−δ (YBCO) cannot be described as a purely tetragonal d-wave. In the orthorhombic symmetry group the d x 2 −y2 − and s−wave basis function belong to the same irreducible representation, so that one expects that the gap can be described as d x 2 −y 2 −wave with some s−wave admixture, d + αs. The deviation from purely d-wave behavior is supported by the observations [2,3] of Josephson pair tunneling from Pb into c-axis oriented YBCO, although the variation of the Josephson tunneling strength with twin density, the relative areas of the twins, and the possible role of Josephson screening currents, remain open questions [4]. It has also been proposed that these results can be explained by the development of a complex order parameter at twin boundaries [5], or mixing of states with s and d x 2 −y 2 symmetry at the Pb-YBCO interface [6]. Part of the problem of interpreting these experiments is the complex geometry presented by highly twinned materials. Here we consider the simpler case of a c-axis YBCO-Pb tunnel junction in which the YBCO has only one twin boundary in the junction. Experimental efforts on this system are currently underway [7].
In the idealized geometry we describe below ( Fig. 1), the Pb/YBCO junction which contains a single twin boundary can be described as a "0 − π" junction, where the pair transfer integral has a relative pair phase of 0 in the left part of the junction (length L 0 ) and π in the right part of the junction (length L π ). Here we are interested in understanding the behavior of such "0 − π" Josephson junctions. While we describe our geometry in terms of c-axis tunneling in YBCO-Pb junctions, our results apply equally well to the 0 − π a-b plane YBCO-Pb planar junctions [8] and grain boundary junctions [9,10] used for phase sensitive tests of the symmetry of the high-T c superconductors. In particular, while traditionally one considers the magnetic interference pattern, we will also be interested in spontaneous flux, flux generated by the self-screening Josephson currents in the absence of a drive current and an external field H e , which can be directly probed by, for example, a scanning SQUID microscope [9]. As noted in Ref. [11], the two measurements are complementary, since while the magnetic field interference patterns are only strikingly different between a 0 − π junction and a conventional 0−0 junction in the short junction limit L < ∼ λ J , appreciable spontaneous flux only appears in the 0 − π junction in the long junction limit L > ∼ λ J .
The maximum Josephson current I c which a junction can carry versus an external magnetic field H e applied parallel to the plane of the junction is called the "magnetic interference pattern". The interference pattern for a short (L ≪ λ J ) 0 − π junction has been discussed by Wollman et al. [8]. Xu et al. [11] have estimated the crossover to the limiting case of a long (L ≫ λ J ) 0 − π junction. With the approximations made by Xu et al., the magnetically modulated critical current of a long 0 − π junction is basically identical to the conventional 0 − 0 junction due to entrance of a half-flux quantum vortex, and the solution which contains spontaneously generated flux ceases to be a global minimum of the free energy for short junctions. Our exact numerical solutions, however, show that there is still a "dip" in the center of the diffraction patterns even for junctions as long as 10λ J , and that the symmetric junction should contain spontaneously generated magnetic flux even for L < λ J .
Here we are primarily interested in determining the behavior of a 0 − π junction for intermediate values of L/λ J and different L 0 /L π asymmetries. We believe that the dependence on L/λ J of both the field-modulated critical current and the spontaneous flux generation of a Pb-YBCO junction with one twin boundary can provide important information on the c-axis coupling.
In the following, we discuss a numerical method for calculating the spontaneous flux generation and magnetic interference patterns for arbitrary junction lengths L/λ J and junction asymmetries L 0 /L π . We then compare results for the critical current I c versus external magnetic field H e for the case of a traditional 0 −0 junction with the symmetric 0 −π case as a function of the reduced junction length L/λ J . Following this, we focus on the Josephson screening currents by examining the flux generated by these self-currents in the absence of an external current and magnetic field. As previously noted, this self generated flux can be probed using a scanning SQUID microscope [9], providing important information on the nature of the c-axis Josephson coupling. We then turn to the interesting case in which the areas of the two twin regions differ and again look at the self-flux versus L/λ J . There is a qualitative difference between the results for the spontaneous flux generation which occurs in asymmetric junctions compared to symmetric junctions. We show briefly how these results can be extended to a junction with multiple twin boundaries. We conclude by discussing what this type of experiment can tell us about the Pb-YBCO c-axis Josephson tunneling.

II. THE CALCULATION
The geometry and gap structure which we envision is illustrated in Fig. 1. In this idealized geometry [12], the upper YBCO strip contains a twin boundary which separates the YBCO-Pb junction into two regions. As schematically illustrated in Fig. 1, we have taken the phase of the larger lobe of the d + αs gap in region 0 < x < L 0 to be positive.
Then, if the phase is locked across the grain boundary as one expects [13], the larger lobe in region L 0 < x < L will be negative. Thus the pair transfer integral between the YBCO and the Pb leads to a relative Josephson pair phase of 0 for 0 < x < L 0 and π for L 0 < x < L.
Following the notation of Owen and Scalapino [12], we describe a junction ( Fig. 1) with width w small compared to λ J in the y-direction, of length L in the x-direction, carrying a total current I tot in the x-direction, in an external magnetic field H e oriented in the ydirection. We include the effects of the 0 and π junctions by introducing an extra phase angle θ(x) which takes the value 0 or π. The superconducting phase difference across the junction is then just the solution of the Sine-Gordon equation: Applying Ampere's Law with a contour of integration around the perimeter of the junction in the xy plane leads to the boundary condition Contours of integration in the yz plane circling the leads at x = 0 and x = L lead to The Josephson penetration depth is given by where d is the sum of the Pb and YBCO λ ab penetration depths plus the thickness of the insulator layer between the two superconductors, and j c is the Josephson critical current density. The current per unit length through the junction is given by: The gradient of the phase is in turn related to the field H in the junction by: If we redefine the parameters H(x) and I tot as and Defining a dimensionless coupling parameter the differential equation turns into a difference equation on a grid of size ∆x: The boundary conditions are then described as difference equations, where n j is the total number of junctions: These coupled difference equations are solved using a relaxation method to find the solution φ(x). The free energy of this solution is given by: Written as a difference equation, the free energy for the vortex solution becomes: while the free energy of the no-vortex (φ = 0 everywhere) solution is: To obtain the critical current, the current I tot through the junction is increased in steps, iterating the solution until either the convergence criterion is satisfied, with ǫ normally chosen to be 10 −6 , or until the solution diverges with | φ(n j /2) |> we have plotted I c /I 1 versus Φ/Φ 0 , where I 1 = j c wL, Φ = H e dL, and Φ 0 = hc/2e is the superconducting flux quantum. In the short junction limit L ≪ λ J , the critical current can be written as: where θ(x) can be 0 or π, and the maximum value of the expression is obtained by varying 0 < ψ < 2π for each value of H e . For the 0 − 0 junction (θ(x) = 0) this expression reduces to: while for the symmetric 0 − π junction (θ(x) = 0, x < L/2; θ(x) = π, x ≥ L/2) this becomes: These equations are plotted as the solid lines in Figure 2. Also included in Figure 2a is the result of Owen and Scalapino for the reduced length L/λ J = 10. For the 0 − 0 junction the effect of increasing L is to reduce the height of the central peak in the magnetic interference pattern, and to reduce the amplitude of the successive oscillations as H e is increased. For the 0 − π junction, increasing L tends to reduce the depth of the minimum at H e = 0, and as L/λ J → ∞ the curves for the 0 − 0 junction and the 0 − π junction become identical as discussed by Xu et al. [11]. However, even for L/λ J = 10, we find, as shown in Fig.   2(b), that the critical current initially increases with flux until Φ/Φ 0 ∼ = 0.5. This differs from the L/λ J = 10 result shown in Ref. [11]. We believe that this difference results from the approximation made by Xu et al. that the presence of a π vortex simply changes the phase in the π section of the junction by π, effectively changing it into a 0-junction. This approximation is valid on length scales large compared to λ J , but, as shown in Fig. 2b, leads to an experimentally detectable difference in the solutions even for L as long as 10λ J . For short and medium length junctions, Xu et al. took the analytical long-junction expression and made the approximation that the effect of shortening the junction would be simply to cut off this expression. However, this is not a valid solution of the modified sine-Gordon equation and the boundary conditions. It also leads, as we show below, to dramatic differences in both the free energy and the total flux for the π-vortex solutions for short-to medium-length junctions.  Fig. 3b. However, for the asymmetric case, the state with no flux has the lowest free energy for short junctions, and as shown in Fig. 3b, there is no spontaneously generated flux, up to a critical value of L π /λ J < ∼ 1. The amount of spontaneously generated flux approaches Φ 0 /2 as L π gets larger, and rate of increase of flux at the onset of flux generation increases for less symmetrical junctions.
The onset of spontaneous flux generation is also apparent in the magnetic interference patterns, as shown in Fig. 4. Here we hold the asymmetry factor L π /L fixed at 0.25, and vary the length of the junction L π /λ J . For small L π /λ J there is no spontaneous flux generated, and the magnetic interference has a minimum at zero field. However, as L π /λ J approaches unity, there is an abrupt shift to a magnetic interference pattern with a maximum at zero field. The solid line in Fig. 4 is the short-junction limit calculated from Eq. 3.1, which is a good indicator of the actual interference pattern only until flux generation becomes energetically favorable.
The behavior of junctions with multiple twin boundaries can be calculated using the same techniques, as long as the junction width w is small compared to λ J . We contribution is dominant, giving the type of order parameters illustrated in Fig. 1, one should observe the behavior we have discussed. In particular, if the junction areas for the two twin regions are equal, one has the conventional minimum in the critical current at zero magnetic field as shown in Fig. 2(b), if L/λ J is sufficiently small, and provided the current density is uniform on a scale set by λ J . For the asymmetric case in which the areas of the two twin regions differ, more striking behavior should appear. There should be a sudden shift of the interference pattern from one with a minimum at zero field, to one with a maximum (Fig. 4), as L π /λ J goes through unity. One way to vary L/λ J would be by changing λ J with temperature. Another would be to apply a magnetic field H x to vary Alternatively, or in addition, one could use a scanning SQUID microscope to measure the spontaneous flux generated by the self-screening Josephson currents. For the case of equal areas, this spontaneous flux rises continuously as L π /λ J increases. However, for nonequal areas one finds no spontaneous flux generation below a critical value of L π /λ J ∼ 1 and then a sharp increase in flux should occur. We believe that if these types of behavior are observed, they would show that the Pb-YBCO c-axis Josephson coupling is consistent with a description of the high T c superconductors as having dominant d x 2 −y 2 symmetry.