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Magnetic field dependence of the critical current in YBa_2Cu_3O_{7-\delta}/Au/Nb ramp-zigzag Josephson junctions

We study the critical current I_c dependence on applied magnetic field H for multifacet YBa_2Cu_3O_{7-\delta}-Au-Nb ramp-type zigzag Josephson junctions. For many experiments one would like to apply a homogeneous field in the junction plane. However, even tiny misalignments can cause drastic deviations from homogeneity. We show this explicitly by measuring and analyzing I_c vs. H for an 8 facet junction, forming an array of 4\times(0-\pi)-segments. The ramp angle is \theta_r=8^\circ. The facet width is 10\,\mum. H is applied under different angles \theta relative to the substrate plane and different angles \phi relative to the in-plane orientation of the zigzags. We find that a homogeneous flux distribution is only achieved for an angle \theta_h\approx 1^\circ - 2^\circ and that even a small misalignment \sim 0.1^\circ relative to \theta_h can cause a substantial inhomogeneity of the flux density inside the junction, drastically altering its I_c vs. H interference pattern. We also show, that there is a dead angle \theta^*_d relative to \theta_h of similar magnitude, where the average flux density completely vanishes.


A. Introduction
Large (in a geometrical sense) Josephson junctions (JJs) are studied intensively since decades, e.g. in the context of Josephson fluxon physics. Interesting and important physics is related to junctions where some regions obey the usual Josephson relation, while other regions incorporate an additional phase jump of π, which can be viewed as a negative critical supercurrent density j c < 0. These 0 -π junctions can be fabricated in different ways, e.g. by connecting a d-wave superconductor such as YBa 2 Cu 3 O 7−δ (YBCO) [1][2][3][4] or Nd 2−x Ce x CuO 4−δ (NCCO) 4,5 to a conventional superconductor like Pb or Nb, or by using biepitaxial grain boundaries in YBCO 6-10 , or superconductor-ferromagnet-superconductor (SFS) or superconductor-insulator-ferromagnet-superconductor (SIFS) junctions [11][12][13] . In the YBCO-Nb or NCCO-Nb structures the Josephson junction is often of the ramp type and the barrier forms a zigzag line parallel to the crystallographic a and b axes 2-5 , see Fig. 1. This type of junction has been important for determining the symmetry of the order parameter of the cuprate superconductors 1,14 .
In addition, several properties make zigzag junctions very interesting for ongoing studies. In particular, halfinteger vortices (semifluxons) can form spontaneously at the corners of the zigzag line and, thus, quasi-onedimensional vortex crystals can be realized 3,15 . Also, under special conditions the zigzag junctions can be used to create ϕ 0 16,17 or ϕ junctions, with an arbitrary value of the ground state phase ϕ [18][19][20][21] . Such junctions, if long compared to the Josephson penetration depth λ J , can carry mobile fractional vortices (splintered vortices) having many unusual properties 19,[22][23][24][25] . Apart from these research areas, aiming mostly at long junctions, ramp junctions are also interesting for superconducting electronics, e.g. in the context of self-biased RSFQ circuits 26 or in the context of superconducting quantum interference filters 27,28 . In all cases a good understanding of the implications of the zigzag ramp geometry is required.
Studies of Josephson junctions in many cases require the application of a magnetic field H which, in theoretical studies, is usually considered to be oriented "parallel" to the junction plane, leading to a homogeneous flux density µ 0 H in the absence of self fields generated by the Josephson currents. Most Josephson junctions have a simple geometry where the barrier layer and also the superconducting layers are oriented parallel to the substrate plane. Then, "parallel" simply means parallel to the substrate plane (θ = 0 • , using the coordinates defined in Fig. 1 (a)). For ramp junctions the junction plane (barrier layer) is tilted relative to the substrate plane by the ramp angle θ r (θ r = 8 • for the junction we study here), while the superconducting layers are partially parallel to the substrate plane and partially bent along the ramp, cf. Fig. 1 (b). "Parallel" is thus not defined well. As it will be discussed in detail in Sec. C, there is an angle θ h , with 0 < θ h < θ r , where the applied field creates an almost homogeneous and uncompressed flux density in the junction. A magnetic field applied under this angle should be referred to as parallel.
Experimental studies on the zigzag junctions have usually been performed with the magnetic field applied perpendicular to the substrate plane [2][3][4][5] . One reason for this was the finding that, when aligning the field roughly par- allel to the barrier layer, apart from a field scaling factor due to flux focusing, no essential difference to the perpendicular field orientation was observed. Theoretical interference patterns I c vs. H, calculated under the assumption that the applied field causes a homogeneous flux density B in the junction plane, and the experimental patterns agreed only qualitatively: the critical current I c was maximum when the flux per (0 -π) -segment roughly equalled one flux quantum [2][3][4][5] . However, in almost any other respect experimental and theoretical I c vs. H curves were not even similar.
For junction geometries where the barrier layer is oriented either parallel or perpendicular to the substrate plane it was shown previously that for a field orientation perpendicular to the substrate plane (θ = 90 • ) the flux density in the junction barrier becomes inhomogeneous in the case of a homogeneous applied field [29][30][31][32][33] . This leads, e.g., to a striking difference of I c vs. H for SIFS multifacet 0 -π junctions (SIFS-MJJs), when measured in fields applied parallel and perpendicular to the substrate plane 32,34 . Moreover, at least for junctions with a barrier layer oriented parallel to the substrate plane (for which θ h = 0) and H applied under an arbitrary angle θ there is a "dead angle" θ d where the magnetic flux caused by the parallel (θ = 0 • ) and perpendicular (θ = 90 • ) components of H cancel, leading to a critical current which almost does not modulate with H 30,32,35 . The dead an-gle θ d can be very close to zero, making proper junction alignment parallel to H very difficult, if not impossible.
Obviously, ramp zigzag junctions are considerably more complex than conventional overlap junctions or SIFS-MJJs and need separate consideration. A systematic study under oblique fields seems necessary, having in mind that this type of junction is very useful for many future investigations. In the present study we have chosen a YBa 2 Cu 3 O 7 -Au-Nb junction with 8 facets (4×(0 -π -)segments), each facet being 10 µm long. We have investigated its I c (H) dependence as a function of θ and also the in-plane angle φ, where φ = 0 • corresponds to H applied along the facets oriented in y direction (cf. Fig. 1). Below we show that the field component perpendicular to θ h leads to a periodically modulated flux density profile. The average value of the flux density caused by this component is enormously compressed, by a factor ≈ 100 compared to B = µ 0 H. As a consequence, the dead angle To achieve a more or less homogeneous flux density B, the field H must be aligned better than some 0.1 • relative to θ h and, to achieve 1 Φ 0 or more per (0 -π) -segment, µ 0 H values of more than 3 mT are required. These conditions require quite dedicated experimental setups. Alternatively, realistic theoretical analyses should be based on the case of θ ≈ 90 • .

B. Samples and Measurement Techniques
The YBCO layer was grown by pulsed laser deposition (PLD) on a [001]-oriented SrTiO 3 (STO) single crystal substrate. The substrate temperature T s during the deposition of the 65 nm thick YBCO thin films was 770 • C and the oxygen pressure P O2 was 20 Pa. A 60 nm thick STO insulation layer on top of YBCO was also deposited by PLD at T s = 760 • C and P 02 = 9 Pa. The targets were ablated using a KrF excimer laser at a repetition rate of 2 Hz. After thin film deposition, the chamber was vented with oxygen up to 0.5 bar and the sample was cooled down to room temperature, with an annealing step at 450 • C for 30 min. The zero-resistance transition temperatures T c of the YBCO films were between 88 and 90 K. The YBCO/STO bilayer was patterned using optical lithography and Ar-ion milling under an angle of 30 • between surface normal and ion beam. To ensure a constant ramp angle for all junction orientations the sample was rotated about the axis normal to the surface during milling. With these parameters a ramp angle of θ r = 8 • was obtained. After removal of the photoresist the sample surface was cleaned in a soft Ar rf-plasma, in situ. Before the next deposition steps, an interlayer of 4 unit cells of YBCO was grown by PLD with the same deposition and annealing parameters as before to provide recrystallization 36 . According to Ref. [36], the thin YBCO interlayer is expected to become superconducting only on the YBCO ramp area, but not on the STO substrate and insulation layer. With electron beam evaporation a 9 nm Au barrier was deposited, followed by a sputtered Nb layer of thickness d Nb = 100 nm. The plasma-cleaning and the last three depositions steps were done without breaking the vacuum. The Nb and Au layer were patterned by a final photo lithography and Ar-ion milling step. The resulting JJ is sketched in Fig. 1. The samples were measured at T = 4.2 K in a magnetically and electrically shielded cryostat. The mounted sample typically had a misalignment of the applied magnetic field relative to the substrate plane θ off below 1 • . An external field µ 0 H of up to 3.5 mT could be applied and continuously rotated with respect to θ via two perpendicular coils operated in linear combination. The inplane angle φ was varied by mounting the sample with proper orientation relative to the in-plane field axis, resulting in a misalignment error φ off ∼ 2 − 3 • .
Below we discuss results from our most homogeneous sample. Fig. 2 shows current voltage characteristics, measured at three values of applied field. The current voltage characteristics were hysteretic for critical currents > 2 µA, with a junction resistance of R ≈ 6.6 Ω. To measure I c the bias current I was ramped up at fixed magnetic field until the junction switched to its resistive state. A voltage criterion V cr = 2 µV was used to determine I c , leading to a maximum over-estimate of I c by V cr /R ∼ 0.3 µA in the nonhysteretic regime. In the hysteretic regime I c is underestimated by some 0.1 µA due to premature thermal activation.  The ramp areas of all facets form a zigzag line parallel to the a,b axes of the YBCO film.
To investigate the homogeneity of the critical current densities of individual facets j i c , we have imaged the current distribution of the entire zigzag junction at zero magnetic flux using low temperature scanning electron microscopy (LTSEM). Details of the method can be found in Ref. 4. In brief, the pulsed focused electron beam, which is scanned across the sample, causes local heating δT < 1 K on a lateral length scale of ≈ 1-3 µm, which determines the spatial resolution of this imaging technique. The measured integral quantity is the voltage V across the junction, which is biased slightly above I c . δT results in a local reduction of |j c (T )|. The corresponding change of the overall I c of the zigzag junction causes a slight change of δV (x, y), which depends on the beam position (x, y) on the sample surface. For a bias current slightly above I c and B = 0, δV (x, y) ∝ −j c (x, y). This imaging technique requires non-hysteretic current voltage characteristics, which for our junction is not the case for T = 4.2 K, cf. Fig. 2. Therefore, the δV (x, y) image of the YBCO-Nb zigzag junction in Fig. 3 (b) has been taken at T = 8.0 K. The junction has been biased at I = 2.3 µA. A field µ 0 H = −0.34 µT was applied to compensate the residual field in the LTSEM setup. The δV image clearly shows the alternating sign of supercurrent flow across neighboring facets. On the scale of the spatial resolution of our imaging technique neither defects nor j c asymmetries between facets are visible. Furthermore, the critical current densities of the facets seems to be quite homogeneous.

Single-Facet Ramp Junction
Before we address I c (H) of zigzag JJs let us first theoretically consider a JJ with a single facet oriented parallel to the x axis, (see Fig. 4). The JJ area extends from x = 0 to a length L along the x direction. The bottom YBCO electrode grows in thickness along y, reaching its maximum thickness d max YBCO = 65 nm at y = d max YBCO /tan(θ r ) = 462 nm, which is the projection of the junction width W = d max YBCO /sin(θ r ) = 467 nm to the y axis. Let us first estimate how the magnetic flux density distributes inside the junction when H is applied perpendicular to the substrate plane (θ = 90 • ). The situation is sketched qualitatively in Fig. 4. We first note that d max YBCO is well below the YBCO in-plane magnetic penetration depth λ YBCO ∼ 150 nm. Near the edge of the YBCO film located at y = 0 magnetic field lines can cross the YBCO film on the scale of the Pearl length 37-39 Λ = 2λ 2 YBCO /d YBCO > ∼ 700 nm. This implies that over the whole junction area field lines can penetrate the YBCO film freely for any angle θ.
By contrast, the thickness of the Nb film is comparable to the respective penetration depth (λ Nb ∼ 80 nm) and approximately constant over the ramp area. Some field lines will cross the Nb film at its left edge (cf. Fig. 4 (a)) on a length scale of λ Nb . However, due to the idle region this edge is far away from the junction area. Thus, no field line will cross the Nb film inside the junction area and the resulting flux density is essentially oriented parallel to the ramp (θ = θ r ). Further, the screening currents preventing magnetic field lines from crossing the Nb layer lead to a strongly enhanced flux density in the junction barrier, cf. Fig. 4 (a). Along x this effect is strongest at the center of the junction (x = L/2). It disappears at the edge of the Nb film, cf. Fig. 4 (b, c).
To account for these effects we may represent the resulting flux density profile inside the junction barrier, oriented along θ = θ r , as where the field compression factor f 90,f ≫ 1. The normalization constant N 90,f is chosen such that the spatial average of N 90,f (a 90,f + P 90,f (x)) equals 1. The index '90' indicates perpendicular direction of the applied field (θ = 90 • ) and the subscript 'f' indicates the single facet scenario. The spatially dependent field profile P 90,f (x) > 0, to be specified later, is maximum in the center of the junction and is assumed to reach zero at its edges. Without idle regions B 90,f would drop to almost zero at the junction edges, i.e. a 90,f ≈ 0. The effect of the idle regions is to make B 90,f nonzero here. This is parametrized by a 90,f . Using the spatial average P 90,f of P 90,f (x), B 90,f can be regrouped as where P 90,f (x) = N 90,f (P 90,f (x) − P 90,f ) has vanishing average. The case of "parallel" fields requires some discussion. For theoretical considerations "parallel" should refer to an angle (relative to the substrate plane) where the flux density in the junction is homogeneous. We denote this direction as θ h . In the absence of idle regions a natural choice would be the ramp plane, i.e., θ h = θ r . Screening currents in the idle regions, however, will deform the flux density similar to the case of perpendicular fields, leading to an ambiguity of what "parallel" actually means.
Let us consider a field H applied at θ = 0 • , i.e. parallel to the substrate plane. Screening currents in the vicinity of the ramp cause B to be (almost) parallel to the Nb film, i.e. tilted by θ r within the area of the YBCO ramp. These screening currents also cause a slight field compression towards the center of the ramp. For the case of θ = 0 • we thus expect a flux density profile for the component along θ r of the form where the subscript '0' stands for θ = 0 • . f 0,f > ∼ 1 represents field compression. N 0,f normalizes the field profile to 1. P 0,f (x) has an absolute value which is maximum in the center of the facet and zero at its edges. The constant a 0,f has been introduced to account for idle region effects.
By decomposing H into components perpendicular and parallel to the ramp one realizes that P 0,f ≤ 0 at least in the absence of idle regions. The screening currents in the idle regions reduce |P 0,f (x)|.
Similar as B 90,f , B 0,f (x) can be regrouped as where If H is oriented within the (y, z) plane at an arbitrary angle θ relative to the y axis the total flux density in the junction is with B 0,f (x) and B 90,f (x) as defined in Eqs. (4) and (2). The factors cos(θ) and sin(θ) arise from a decomposition of H into components parallel and perpendicular to the substrate plane.
If the spatial dependences of P 0,f (x) and P 90,f (x) are similar, the ratio p = | P 0,f / P 90,f | is about constant. Then, there is an angle θ h = −arctan(f 0,f p/f 90,f ) where the flux density B θ,f penetrating the junction is homogeneous along x and given by The angle θ h might be referred to as "field applied parallel to the ramp plane" 35 . In the absence of idle regions we expect θ h ≈ θ r . In their presence θ h is reduced. We did not perform an explicit calculation, but a guess is to consider a field line, which starts at the YBCO ramp edge at y = 0 [point • 1 in Fig. 4 (a)] and touches the edge of the overlapping Nb film [point • 2 in Fig. 4 (a)], which, for φ = 45 • (the relevant angle for the multifacet system), is at the projected lengthỹ ≈ 4 µm and z = 125 nm. The corresponding angle is θ h = 1.8 • .
A similar argument will hold for a multifacet system. In the data shown below we determined the misalignment angle relative to θ h ("parallel" alignment, homogeneous field) as θ * off = 1.68 • at φ = 45 • and as θ * off = 0.85 • at φ = 0 • . We typically mount our samples with an offset angle θ off < 1 • relative to the substrate plane. Thus, we can put a limit θ h < ∼ 2 • which is fully compatible with the above estimate of θ h but rules out θ h = θ r .
Finally, to account for the fact that θ h cannot be precisely determined experimentally we introduce an angle θ * = θ − θ h . By definition, the applied field is "parallel" for θ * = 0 and "perpendicular" for θ * ≈ θ = 90 • .
For the single facet case the flux density resulting from a field applied at an angle θ * is: with . f ⊥,f denotes the field compression. As in Eq. (2), P ⊥,f (x) ≈ P 90,f (x) is a spatially varying function with zero average, having its maximum in the center of the junction plane at x = L/2. For the parallel field component (θ * = 0 • ) we have explicitly used that the flux density is homogeneous and essentially no flux compression occurs.
We have not yet made use of the fact that field lines can penetrate the YBCO film freely in the junction area. Thus, in y direction the flux through the junction is not conserved but varies along y. Alternatively, assuming a constant flux density along y, this can be rephrased in terms of an effective junction thickness t eff that varies along the y direction. Generally, t eff can be found via 40 Over the ramp area, the YBCO film thickness d YBCO grows along y from 0 to d max YBCO . Thus, t eff increases from ∼ 53 nm to ∼ 85 nm over the ramp. For further calculations we use a linearized Ansatz with −0.5 < y/W < 0.5. The average effective thickness t eff0 is given by Eq. (7), using d YBCO ≈ d max YBCO /2. Ignoring self-field effects caused by the Josephson current, the Josephson phase γ is calculated via The maximum supercurrent is obtained from where integration has to be performed over the junction area. The dependence of t eff on y causes dephasing, which becomes severe when the flux difference (along y) over the junction width becomes on the order of Φ 0 /2. In the above estimate t eff differs by ∆t eff /t eff0 ≈ 0.5 from its average value t eff0 (∼69 nm), and thus we expect the effect to become noticeable when the total flux through the junction becomes larger than about Φ 0 . However, the data discussed below for the 8 facet junction indicate ∆t eff /t eff0 ≈ 0.02. Also, using t eff0 as a free parameter, our data indicate a value for t eff0 which is close to 85 nm. I.e. only the "thick" part of the ramp seems to be relevant. Eq. (7) assumed isotropic superconductors and, thus, the YBCO anisotropy could lead to modifications. Whether or not this solves the discrepancy is unclear to us. Let us summarize the central results of this section: We expect that in the junction plane the magnetic flux density along x follows the shape given by Eq. (6). B is homogeneous and equal to µ 0 H if the field H is applied under an angle θ h (∼ 1 • −2 • for our geometry) relative to the substrate plane, which we denote as "parallel field" (θ * = θ − θ h = 0 • ). Perpendicular components of H lead to a compressed flux density which varies along x, having a maximum in the center of the junction at x = L/2. Due to the fact that flux lines can freely penetrate the YBCO film we expect that the total flux through the junction is not conserved but depends on y. This effect seems to be present in the zigzag junctions discussed below. However, its magnitude is at least an order of magnitude smaller than expected.

Multifacet Ramp Junction
To get an idea of the field focused flux density profile inside the junction for H applied at θ = 90 • , we have simulated the flux density distribution around two overlapping planar superconducting layers located in the (x, y) plane being separated by a distance d L along z, using 3D-MSLI 41 , see hatched areas in Fig. 5 (a). The ramp was not included. The layer shape corresponded to the experimental situation and included N = 8 facets of length L = 10 µm. We used d L = 500 nm. Smaller values of d L led to convergence problems. The layer thicknesses were d YBCO = 65 nm and d Nb = 100 nm, respectively. Fig. 5 (a) shows the geometry projected onto the (x, y) plane and the field lines calculated in the plane parallel to the films and situated in the middle of the d L = 500 nm gap between the films. The field was calculated along the edge of the YBCO film. B is strongest at the inner edges of the YBCO layer and weakest at the outer edges. A flux density profile B 90 (ξ) − we use the index '90' to indicate that H has been applied perpendicular to the layers − of the calculated in-plane magnetic field B projected onto the zigzag line is shown in Fig. 5 (b). ξ is a curvilinear coordinate along the zigzag edge of YBCO. The projection is in units of µ 0 H and the coordinate ξ runs along the facets, −0.5 ≤ ξ/N L ≤ 0.5. B 90 (ξ)/µ 0 H varies roughly sinusoidally, with one period per (0 -π) -segment (by contrast, one might have expected one period per facet, cf. Fig. 4 (c)). B 90 (ξ)/µ 0 H reaches a maximum value of about 4.5, which is in fact much less than the actual field compression (∼ 100) found in experiment. It also turned out that the B component perpendicular to the layers is of the order of the in-plane component -a feature which we expect to disappear in a more realistic scenario. Further, simulating I c (H) with the profile of Fig. 5 (b) gave strong differences to the experimental interference pattern for flux values larger than Φ 0 per (0 -π) -segment.
To simulate our zigzag junctions, as for the single facet case we use angles θ * = θ − θ h and φ to describe the angles between the applied field and the junction. In this case "perpendicular" refers to θ * = 90 • and differs from θ = 90 • by θ h . To account for this, below we use the index '⊥' to denote perpendicular direction. For the flux density generated by the perpendicular component of the applied field we follow the two-facet periodic shape obtained by the 3D-MSLI simulations but, in order to be more flexible, generalize it in the following way. Each (0 -π) -segment is described by a field profile a ⊥,f + P ⊥,f (x), with 0 < x < 2L. We used The resulting periodic pattern along ξ, i.e. the coordinate running along the zigzag line, is multiplied by an envelope function extending smoothly across the whole junction. We parametrized P ⊥,e via The overall shape of B ⊥ (ξ) is given by N ⊥ normalizes the spatial average of B ⊥ /µ 0 Hf ⊥ to 1, f ⊥ denotes field compression.
For arbitrary values of θ * and φ the flux density B through facets oriented parallel to x is The factors sin θ * and cos θ * cos φ arise from a projection of the applied field H. For facets oriented parallel to y one obtains with a homogeneous flux density B = µ 0 H along the zigzag line and B ⊥ (ξ) as defined in Eq. (14). Using this field profile (the actual parameters used are discussed in Sec. D, see also inset of Fig. 6 (a)) we solved Eqs. (9) and (10), with x replaced by ξ. The critical current density j c has been assumed to be constant in amplitude (homogeneous junction). It changes sign between adjacent facets. Our model contains the 7 parameters α f , α e , a ⊥,e , ∆t eff , t eff0 , f ⊥ and θ h . Further, in experiment there is the offset angle θ off relative to the substrate plane. As discussed in more detail in Sec. D, the shape of I c (H) for "perpendicular" fields is not very sensitive to small errors in θ or θ * when θ h is small and thus, from this curve one obtains α f , α e , a ⊥,e and ∆t eff /t eff0 with good accuracy. From the field modulation period in perpendicular field one further obtains the product f ⊥ t eff0 . A priori, t eff0 is not known precisely. From the angle dependent I c (H) measurements one can identify the dead angle θ * d relative to θ h with high accuracy. However, θ * = (θ − θ h ) = 0 • is much harder to identify if, as in our case, I c (H) is measurable only over a small number of modulation periods. θ * d depends on f ⊥ . Thus, f ⊥ needs to be determined by comparing a set of I c (H) curves near parallel orientation, finally also allowing to determine t eff0 and the offset angle θ * off relative to θ h . θ h itself, however, is hard to determine and has an uncertainty which is on the order of the offset angle θ off .

D. Results
Fig. 6 (a) compares measured and calculated interference patterns for θ * = 90 • . The horizontal scales of the calculated patterns are in units of H 0 /f ⊥ , where H 0 ≡ f ⊥ N Lt eff0 /Φ 0 , with the field compression factor for perpendicular field components f ⊥ , the average effective junction thickness t eff0 , the facet length L and the facet number N . The solid (black) line is for a relative variation of the effective junction thickness ∆t eff /t eff0 = 0.02, the dotted (blue) line is for ∆t eff = 0. The agreement between measured and calculated (for ∆t eff /t eff0 = 0.02) interference patterns is reasonable, although differences occur for normalized fields Hf ⊥ /H 0 > ∼ 5. For ∆t eff /t eff0 > 0.02, at high fields the I c maxima are suppressed in comparison with the ∆t eff /t eff0 = 0.02 case, while the I c minima are significantly lifted from zero, both in contrast to the measurements. Thus the margin on ∆t eff /t eff0 is relatively narrow. For the (0 -π)segments we used in Eq. (11) α f = 0.7 and a ⊥,f = 2.0.
The power α f was suggested by the 3D-MSLI simulations, cf. Sec. C. The calculated interference patterns, however did not depend strongly on this parameter. By  (13) and (14). For the black curve ∆t eff /t eff0 = 0.02, for the dotted (blue) line ∆t eff = 0. H0/f ⊥ = N Lt eff0 /Φ0. Critical current in the calculated plots is normalized to Ic0 = |jc|AJ , where jc is the critical current density and AJ is the junction area. Inset in (a) shows flux density profile B ⊥ normalized to µ0Hf ⊥ . contrast, the quite large value of a ⊥,f was necessary to achieve a reasonable agreement with experimental data. As a result, the periodic modulations of B ⊥ (ξ) are much shallower than suggested by the 3D-MSLI simulations. For the envelope function (Eqs. (12) and (13)) we used α e = 0.5 and a ⊥,e = 0.3. Normalization resulted in N ⊥ = 0.34. The corresponding field profile is shown in the inset of Fig. 6 (a). Finally, by matching the abscissas of the theoretical and measured interference patterns we find t eff0 f ⊥ = 8.5 µm. A thickness t eff0 = 69 nm calculated from geometry (film thicknesses) corresponds to f ⊥ = 120. In fact, for the I c (H) data at different values of θ * and φ this value turned out to be somewhat too large. Best agreement was obtained for f ⊥ ∼ 100, corresponding to t eff0 ∼ 85 nm.
We mention here that a periodic modulation of B ⊥ (ξ) with one period per facet rather than one period per (0 -π) -segment also gave satisfactory agreement between calculated and measured interference patterns. Thus, from I c (H) we cannot unambiguously distinguish these scenarios. Fig. 6 (b) shows measured and calculated interference patterns near parallel orientation for φ = 45 • . For the calculation (θ * = 0 • ) we have assumed that the magnetic flux density along the zigzag line of the multifacet junction B (x) is homogeneous and not compressed, i.e. B (x) = µ 0 H. The field scale is given in units of H 0 √ 2. The factor √ 2 has been included to account for the fact that only a field µ 0 H/ √ 2 is applied per facet. The main I c maximum in this normalization roughly appears at N/2. The dotted (blue) interference pattern is for ∆t eff = 0 to emphasize the "ideal" case. ∆t eff /t eff0 = 0.02 (black line) yields only minor deviations in the field range shown. The experimental curve is at an angle θ * = −0.085 • . No curve has been measured at θ * = 0.0 • an angle, which is defined only after data analysis. Note that, in principle, the experimental and theoretical field scales are linked for a fixed value of f ⊥ . However, in order to compare the shape of I c (H), in Fig. 6 (b) we have additionally compressed the experimental field scale by a factor of 1.2 to account for the nonzero value of θ * . Further note that the experimental curve only covers a limited field range. This is the limit set by our experimental system. Comparing the curves at θ * = 90 • and θ * = 0 • one notices that for θ * = 0 • the amplitudes of the secondary I c maxima situated between the two main I c maxima follow a U-shaped dependence while for θ * = 90 • their amplitudes are about constant. Also, for θ * = 0 • and absolute values of fields higher than the main I c maxima, the I c maxima are strongly reduced compared to θ * = 90 • . These are the main differences in I c (H) at θ * = 0 • and θ * = 90 • . Thus, the overall difference of the interference pattern at θ * = 90 • compared to the ideal case is much less striking than in previous publications [2][3][4][5] , presumably pointing to a much higher inhomogeneity of the critical current density in previous generations of ramp-zigzag 0 -π JJs. Fig. 7 compares measured and calculated interference patterns for φ = 45 • and −1.128 • < θ * < 0.68 • , where the angles quoted here and also below refer to the calculated patterns. In the experimental curves the offset angle θ * off = 1.684 • relative to θ * = 0 is not subtracted. The agreement is fair for all angles shown. The only additional parameter required to fit the whole series of curves was θ * off . It was determined by comparing the calculated interference pattern at the dead angle, cf. Fig. 7 (l), to the measured one, cf. Fig. 7 (k). For arbitrary φ, the dead angle (relative to θ * = 0) is given by with θ * d ≈ −0.4 • for φ = 45 • . Near θ * d , variations in θ * on the order of 0.01 • already cause significant changes in I c vs. H so that, once f ⊥ is fixed, θ * off can be determined very precisely. Note that the experimental I c vs. H pattern of Fig. 7 (k) slightly modulates around 2.5 µA, while the calculated pattern in Fig. 7 (l) is almost at zero current. This is presumably caused by residual fields in θ * = θ−θ h = 0 corresponds to "parallel" field orientation, as defined in the text. Offset angle θ * off = 1.684 • relative to θ * = 0 is not subtracted in the experimental graphs. H0/f ⊥ = N Lt eff0 /Φ0. Critical current in the calculated plots is normalized to Ic0 = |jc|AJ , where jc is the critical current density and AJ is the junction area. Model parameters are αe = 0.5, a ⊥,e = 0.3, α f = 0.7, a ⊥,f = 2.0, f ⊥ = 100 and ∆t eff /t eff0 = 0.02, cf. Eqs. (11), (12), (13) and (14).
the cryostat, causing also the shift in the I c minimum relative to H = 0 in Fig. 7 (m). Also, the θ * dependence of the interference patterns strongly depends on f ⊥ . This parameter cannot be altered by more than some 5 % from 100 without substantial degradation of the fit qualitiy.
ima I c = 0.5I c0 whenever the flux through the facets oriented along x equals a multiple of Φ 0 . Then the critical current of these facets cancel and I c is given by the sum of the critical currents of the facets oriented along y. Curves of Fig. 8 (a)-(f) show the appearance of the first and second main I c maximum, which grows in amplitude for θ * → 0 • . Although the maximum field provided by our setup was not sufficient to resolve the main maxima for θ * = 0 • the evolution of the first and second I c main peaks are clearly visible for θ * < 0.336 • , with a good agreement between experimental and theoretical patterns. By lowering θ * one again runs through a dead angle (θ * d = −0.253 • ) and obtains somewhat strangely looking interference patterns for −1 • < θ * < θ * d . Finally, we address the effect of self fields generated by the Josephson currents. Such effects become prominent when the junction is 2-3 times longer than the Josephson length λ J . To estimate λ J we first ignore idle regions and assume that the supercurrent flows homogeneously across the junction area A J = N LW ≈ 37 µm 2 . We then find j c0 = I c0 /A J ≈ 140 A/cm 2 . Using λ J0 = [Φ 0 /2πµ 0 j c d eff ] 0.5 , with the effective magnetic junction thickness 40 we find λ J0 ≈ 20 µm and the normalized junction length l = N L/λ J ≈ 4. The idle region effect 42,43 leads to an increased Josephson length, λ J,i = δ · λ J0 , with δ = (1 + (d eff /d eff,i )(W i /W )) 0.5 . W i and d eff,i , respectively, are the width and effective magnetic thickness of the idle region. With d eff,i = 535 nm and W i = 3 µm one obtains δ ≈ 2.6 and λ J,i ≈ 50 µm. Thus, l ≈ 1.6, justifying the short junction approach taken above. An ambiguity, however, arises from the problem to refer the measured I c to the proper junction area. Above, we have used the whole ramp area. Alternatively, assuming that the current is dominantly carried by the in-plane currents on the YBCO side one might refer to a junction area which is projected perpendicular to the substrate plane. Then, j c increases by a factor of 7 and λ J decreases by a factor of 3, bringing the junction closer to the long junction regime. To distinguish these scenarios we simulated I c (H) based on the sine-Gordon equations for the case of θ * = 90 • and various values of the normalized junction length l. For the flux density profile the same shape as in the inset of Fig. 6 (a) has been used. Fig. 9 shows simulations of I c (H) patterns for different l and the calculation using the short junction model. Deviations from the short junction model that are incompatible with our experimental data become prominent near the main I c maxima for l > 3 (the case of l = 4 is shown in the graph). By contrast, I c (H) for l = 1.6 is almost indistinguishable from the short junction model and in agreement with the above estimate, using the ramp area instead of its projection.

E. Conclusions
In summary, we have investigated the interference patterns I c vs. H for a 8 facet YBCO-Au-Nb zigzag-ramp Josephson junction, with a facet length of 10 µm. The angle θ between the substrate plane and the applied field H was systematically varied and we also discussed two in-plane angles φ between H and the facet orientation (φ = 0 • and φ = 45 • ). All interference patterns could be understood from the fact that a field component perpendicular to the substrate causes a strong and spatially varying contribution to the flux density profile inside the junction. The junction by itself -admittedly our best junction -seems to be very homogeneous, with an essentially constant critical current density inside each facet. Particularly, no j c asymmetry between facets oriented along the in-plane x and y directions were observable. We have expected a strong dephasing effect on I c vs. H due to the fact that the flux penetrating the junction is not preserved. The effect is present but at least an order of magnitude weaker than expected. Our investigations also showed that H is applied "parallel" to the junction (in terms of an homogeneous flux density profile) for an angle θ h < ∼ 2 • and not for a field orientation about parallel to the ramp angle θ r = 8 • . These findings may contribute new knowledge to the general physics of ramp junctions.
Further, due to strong compression of the perpendicular field component by about a factor of 100, only for angles |θ * | ≡ |θ − θ h | ≪ 0.1 • the interference patterns were "ideal" in the sense that the flux density in the junction is essentially homogeneous. The effective junction thickness is only about 85 nm, leading to enormous fields (∼ 35 mT) that are required to produce a flux quantum per (0 -π) -segment for the case of θ * = 0 • . We also demonstrated, that there is a dead angle θ * d very close to "parallel" field orientation, where tremendous changes in I c (H) occur. These properties make it extremely hard to study zigzag junctions in "parallel" field configuration. Due to idle regions the Josephson length of our junction was on the order of 50 µm, requiring junction lengths of several 100 µm to study long junction effects. For such junctions the field compression will be even bigger than in our case, and, thus, the field alignment must be much better than 0.1 • relative to θ * = 0 to achieve a homogeneous flux density. Thus it seems that, for a study of (semi)fluxon physics 3,6,7,15 or ϕ-junction effects [18][19][20][21] in zigzag junctions magnetic fields should be oriented perpendicular to the substrate plane and the corresponding flux density profiles should be taken into account rather than being avoided by parallel field alignment.
We finally note that dead angles, as described here also appear in other types of Josephson junctions or Josephson junction arrays 30,32,35 . Whenever there is a strong compression of the perpendicular field component this angle may be very close to the parallel field orientation, making measurements in "parallel" field unreliable, favoring the perpendicular field orientation.