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PHYSICAL REVIEW B 79, 144521 2009

Static properties of small Josephson tunnel junctions in an oblique magnetic field
R. Monaco*
Istituto di Cibernetica del CNR, 80078, Pozzuoli, Italy and Dipartimento di Fisica, Unit INFM, Universit di Salerno, 84081 Baronissi, Italy

M. Aaroe and J. Mygind
DTU Physics, B309, Technical University of Denmark, DK-2800 Lyngby, Denmark

V. P. Koshelets
Kotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Science, Mokhovaya 11, B7, 125009, Moscow, Russia Received 15 January 2009; revised manuscript received 11 March 2009; published 22 April 2009 We have carried out a detailed experimental investigation of the static properties of planar Josephson tunnel junctions in presence of a uniform external magnetic field applied in an arbitrary orientation with respect to the barrier plane. We considered annular junctions, as well as rectangular junctions having both overlap and cross-type geometries with different barrier aspect ratios. It is shown how most of the experimental findings in an oblique field can be reproduced invoking the superposition principle to combine the classical behavior of electrically small junctions in an in-plane field together with the small junction behavior in a transverse field that we recently published R. Monaco et al., J. Appl. Phys. 104, 023906 2008 . We show that the presence of a transverse field may have important consequences, which could be either voluntarily exploited in applications or present an unwanted perturbation. DOI: 10.1103/PhysRevB.79.144521 PACS number s : 74.50. r

I. INTRODUCTION

One of the earliest experiments involving Josephson junctions and magnetic fields has been the measurement of the magnetic diffraction pattern,1 i.e., the dependence of the junction critical current Ic on the amplitude of an externally applied magnetic field Ha. Traditionally, since the discovery of the Josephson effect in 1962, the magnetic diffraction pattern Ic Ha of planar Josephson tunnel junctions JTJs has been recorded with the magnetic induction field applied in the junction plane to avoid the huge computational complications of taking demagnetization effects into account, when a transverse magnetic component is present. A number of important results have been derived from experiments under these assumptions--a prominent example being the determination of the London penetration depth2 L from which one derives the Josephson penetration depth3 J which sets the JTJ electric length scale. Nowadays, every textbook on the Josephson effect deserves at least one chapter to the magnetic diffraction phenomena. The simplest case is that sketched in Fig. 1 of a rectangular JTJ placed in a uniform and constant external magnetic field parallel to one of the barrier edges. Let us choose the coordinate system such that the tunnel barrier lies in the z = 0 plane and let 2L and 2W be the junction dimensions along the x and y directions, respectively. Finally, let us assume that the JTJ is electrically small, meaning that its dimensions are both smaller than the Joabsence of selfsephson penetration depth 2L ,2W J fields and that its Josephson current density JJ is constant over the barrier area. If the externally applied field Ha is along the x direction ^ Ha =Hxx then the magnetic field H inside the junction is constant and equal to the external value, i.e., H Hx ,0,0 . By integrating the Josephson equation3 relating the Josephson phase to the magnetic induction field in the barrier H,
1098-0121/2009/79 14 /144521 12

x, y =

2d

e 0

0

H x, y

^ z,

1

in which de is the junction magnetic thickness, 0 is the vacuum permeability, and 0 = h / 2e is the magnetic-flux quantum, we readily obtain the spatial dependence of the Josephson phase, = Hxy , 2

with =2 de 0 / 0. Equation 2 leads to the well-known Fraunhofer-type magnetic diffraction pattern,3 Ic Hx = I
0

sin Hx/H H x/ H c

c

,

3

where I0 =4JJWL is the zero-field junction critical current and Hc = 0 / 2 0deL is the so-called (first) critical field, i.e., the smallest field value for which the Josephson current vanishes. Barone and Patern?4 generalized Eq. 3 to the case of an arbitrary orientation of the external magnetic field in the ^ ^ junction plane Ha =Hxx +Hyy. In such a case, still H = Ha and the resulting magnetic diffraction pattern will be I c H x, H y = I
0

sin Hx/H Hx/Hcx

cx

sin Hy/H Hy/Hcy

cy

,

4

with Hcx = 0 / 2 0deL and Hcy = 0 / 2 0deW. Unfortunately, the last equation cannot be easily generalized to the case of ^ ^ ^ an arbitrary applied field orientation Ha =Hxx +Hyy +Hzz, simply because when Hz 0 then H Ha. The effect of a transverse magnetic field has been first considered in 1975 by Hebard and Fulton5 in order to provide a correct interpretation to some experimental data published in the same year.6 ^ They observed that a transverse applied field Ha =Hzz induces Meissner surface demagnetizing currents js feeding the
2009 The American Physical Society

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Ref. 9. This paper is constructed as follows. In Sec. II we will present the samples used for the measurements and describe the experimental setup. Section III will report on the experimental results obtained for those samples whose barrier has a rectangular shape overlap-type junctions in Sec. III A and cross-type junctions in Sec. III B . Section III C will be devoted to the annular JTJs. Then, in Sec. IV we will discuss how to generalize the theoretical analysis of the effect of a transverse magnetic field to the case of an oblique field. Finally, the discussion and the interpretation of the measurements will be given in Sec. V, while the conclusions will be presented in Sec. VI.
FIG. 1. Sketch of a rectangular planar Josephson tunnel junction. The tunnel barrier lies in the z = 0 plan. An oblique field Ha is applied in the x-z plane and forms an angle with respect to the x direction. II. SAMPLES

interior of the junction and so generating a magnetic field H ^ in the barrier plane such that js = z H. The problem of finding the js distribution in a single superconducting field subjected to a transverse magnetic field has been analytically solved for different film geometries.7,8 However, a planar JTJ is made by two overlapping superconducting films separated by a thin dielectric layer and the Meissner current distributions on the interior surfaces top surface of the bottom film and bottom surface of the top film require numerical approaches even for the more tractable electrode configurations. Recently,9 the magnetic field distribution H in the barrier of small planar JTJs has been numerically obtained in the case when an external magnetic field is applied perpendicular to the barrier plane Ha 0,0, Hz . The simulations allowed for heuristic analytical approximations for the Jofrom which the dependence sephson static phase profile of the maximum Josephson current Ic Hz on the applied field amplitude was calculated for the most common electrode geometrical configurations overlap, cross, and annular junctions . Unfortunately, the theoretical findings could not be tested against experimental results due to the insufficiency of data available in the literature. One of the aims of this paper is to fill this vacancy. We have measured the transverse magnetic diffraction patterns of several planar JTJs with the most common geometrical configurations and compared the results with their expected counterparts. More generally, we have recorded the Ic Ha when the applied field is oblique, that is, has nonzero inplane and transverse components. To avoid complications and without loss of generality, we have chosen the in-plane component to be along one of the electrode axis--more specifically--along the x direction, ^ ^ Ha =Hxx +Hzz , so that, as shown in Fig. 1, the applied field forms an angle with respect to the x-y plane, that is, Hx = Ha cos , Hz = Ha sin , 6
2 with Ha = H2 + Hz . We will demonstrate that the experimenx tal oblique magnetic diffraction patterns can be nicely reproduced by properly extending the theoretical framework of

5

High-quality Nb / Al-Alox / Nb JTJs were fabricated on 0.35-mm-thick silicon substrates using the trilayer technique in which the junction is realized in the window opened in a SiO2 insulator layer--details of the fabrication process can be found in Ref. 10. The so-called passive or idle region, i.e., the distance of the barrier borders to the electrode borders, was on the order of 1 2 m for all the junctions. The thickness of the SiO2 insulator layer was 400 nm. The demagnetization currents strongly depend on the electrode thicknesses relative to the London penetration depth. For our samples the nominal thicknesses of the base, top, and wiring Nb layers were 200, 100, and 500 nm, respectively. Considering that the London penetration depth for Nb film is L 90 nm,2 we see that our samples satisfy the thick-film approximation. For all samples, the high quality has been inferred by a measure of the I-V characteristic at T = 4.2 K. In fact, the subgap current Isg at 2 mV was small compared to the current rise Ig in the quasiparticle current at the gap voltage Vg, typically Ig 20Isg; the gap voltage was as large as Vg = 2.8 mV. The geometrical and electrical at 4.2 K parameters of the seven samples quoted in this paper are listed in Table I. For the rectangular junctions #A-F, beside their dimensions 2L and 2W along the x and y directions, respectively, we also report the junction aspect ratio = L / W. As shown in Ref. 9, this geometrical parameter turns out to be crucial for the magnetic field line distribution in the barrier of a JTJ subjected to a transverse magnetic field. All the samples belonged to the same fabrication batch except sample #C . Let us observe that for the overlap-type junctions #A and #B, the zero-field critical current I0 was as large as the theoretical value 0.7 Ig predicted for strong-coupling Nb-Nb JTJs, indicating the absence of self-field effects. The critical current density has been calculated as11 Jc = 0.7 Ig / A in which Ig is the measured quasiparticle current step at the gap voltage and A is the junction nominal area A =2L 2W for rectangular junctions and A = r2 o - r2 for the annular junction . The Josephson critical current i density was Jc = 3.9 kA / cm2 for all samples, except for sample #C having Jc =80 A / cm2. The values of the barrier magnetic thickness de =2 L 180 nm has been used to calculate the Josephson penetration depth J = 0 / 2 0deJc. In the thin-film limit, J can be better determined by using the expression for de found by Weihnacht.12 Accordingly, all samples had J 6 m, except sample #C which had J

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STATIC PROPERTIES OF SMALL JOSEPHSON TUNNEL... TABLE I. Relevant electrical at T = 4.2 K and geometrical parameters junctions quoted in this paper. The Josephson critical current density was Jc except for sample #C having Jc =80 A / cm2 J 42 m . The experimental as follows: overlap-type junctions in Sec. III A, cross-type junctions in Sec. 2L JJ A B C D E F Geometry overlap overlap overlap overlap Cross Cross 10 5 4 20 10 20 2W m2 10 20 500 5 10 5 I0 mA 3.9 3.9 1.0 3.6 3.8 3.0 I0 mA 4.3

PHYSICAL REVIEW B 79, 144521 2009 of the rectangular and annular Nb / Alox / Nb Josephson tunnel = 3.9 kA / cm2 corresponding to J 6 m for all samples, results obtained for these samples will be presented in Sec. III III B, and annular junctions in Sec. III C.
R R

L/W 1 0.25 0.008 4 1 4 ro m 8

Ig mA 5.6 5.6 2.3 5.6 5.8 5.4 Ig mA 6.4

T 1290 550 12 2940 1080 420
A

T 1100 190 0.88 5400 810 2240
A

R

R

/

R

M

0.85 0.35 0.073 1.84 0.75 5.3

140 160 176 107

JJ G

Geometry annular

ri m 5

T 490

T 310

A

A

/

A

M

0.63

145

42 m. In other words, as far as the electrical length concerns, all samples can be classified as intermediate length junctions 2L ,2W J , except sample #C, that is, a long 2L J unidimensional 2W J overlap-type JTJ. We now come to the definition of the parameters R, R , and their ratio R whose experimental values are reported in Table I for the rectangular junctions A, A , and A for the annular junction . As already mentioned in Sec. I, it is well known that the magnetic diffraction pattern of an electrically small rectangular JTJ in the presence of an in-plane field perpendicular to one of the barrier edge follows the Fraunhofer pattern in Eq. 3 characterized by a periodic amplitude modulation. As depicted in Fig. 2, the value Hc of the applied field where first the critical current vanishes is called the first critical field. For those samples whose Ic Ha is still amplitude modulated but follows a different pattern for example, annular, circular, and rhombic junctions , the critical field Hc can still be defined as that value of external field Ha where first the critical current nulls Ic Hc = 0. Further, for

FIG. 2. Color online Definition of the parameter R as the width of the magnetic field range in which Ic Ha 2 / I0. By definition, for a Fraunhofer-type magnetic diffraction pattern, R coincides with the junction first critical field Hc.

those samples whose Ic Ha shows modulation lobes but never vanishes for example, the small junction with nonuniform tunneling current and long JTJs , the critical field can still be obtained extrapolating to zero the first modulation lobe. However, in those cases in which the critical current Ic is a monotonically decreasing function of the applied field Ha, the concept of critical field Hc looses its meaning and a new feature has to be introduced to characterize the behavior of Ic Ha for small fields. A theoretical example is offered by a Gaussian-shaped junction subjected to an in-plane magnetic field that is characterized by a Gaussian magnetic diffraction pattern.13 A practical example is given by a square cross junction in a transverse field, whose Ic Ha decreases with Ha with no measurable modulation.14 The experimental magnetic diffraction patterns that will be reported in Sec. III span all kinds of behaviors from Fraunhofer-type to 1 / Ha type with 0 . Therefore, as the new and universal figure of merit to characterize the response of the critical current to the externally applied field amplitude, we have chosen the width of the magnetic field range R in which Ic Ha 2 / I0 0.64I0 see Fig. 2 . Considering that, when Eq. 3 holds, Ic Hc / 2 = 2 / I0, the value of the prefactor stems from the requirement that the new merit figure R numerically equals the critical field Hc whenever the measured magnetic pattern follows a Fraunhofer dependence, i.e., R = Hc. In all other cases, generally speaking R Hc. In our notation, R and R are the merit figures of, respectively, an in-plane =0 and transverse = 90 magnetic diffraction patterns. With a similar reasoning, we define the parameter A for annular junctions as the width of the magnetic field I0, with 0.67. The slightly range R in which Ic Ha different prefactor stems from the fact that the in-plane diffraction pattern of a small annular junction with no trapped fluxon follows a Bessel-type dependence:15 Ic Ha = I0 J0 1Ha / Hc in which J0 is the zero-order Bessel function and 1 2.405 is its first zero. Now Hc = 0 / 0deC, where C is the ring mean circumference. The measurement of requires an external field smaller than the one required for Hc; henceforth, this new parameter

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also turns out to be a very useful quantity whenever the junction critical field cannot be experimentally determined since it exceeds the irreversible field, i.e., when the Abrikosov vortices first enter into the superconducting films and become pinned in the junction.16 For our samples, the transverse irreversible field was about 5 mT 50 G . The ratios R = R / R and A = A / A provide a direct comparison between the Ic response to a transverse field rela1 means that the tive to the in-plane field; specifically, junction critical current modulates faster when the applied field is transverse. In a recent paper,17 we already provided an experimental proof that a transverse magnetic field can be much more capable than an in-plane one to modulate the critical current Ic of a planar JTJ with proper barrier and electrodes geometry requirements. This property was first obtained and exploited in the context of a detailed investigation of the phase symmetry breaking during fast normal-tosuperconducting phase transitions of long annular JTJs.18 Our setup consisted of a cryoprobe inserted vertically in a commercial LHe dewar. The cryoprobe was magnetically shielded by means of two concentric magnetic shields: the inner one made of Pb and the outer one of cryoperm. Inside the vacuum tight can of the cryoprobe, a nonmagnetic insert holds a chip mount with spring contacts to a Si chip with planar JTJs. With reference to the coordinate system in Fig. 1, the chip was positioned in the center of a long superconducting cylindrical solenoid whose axis was along the x direction within less than 1 of accuracy to provide an inplane magnetic field. In order to provide a transverse magnetic field, a superconducting cylindrical coil was placed 5 mm far from the chip with its axis oriented along the z direction within less than 3 of accuracy . Two independent low-noise dc current sources were used to feed the solenoid and the coil in order to expose our samples at magnetic fields having arbitrary magnitude and orientation in the x-z plane . The field-to-current ratio was 3.9 T / mA for the solenoid and 4.4 T / mA for the coil. These values have been numerically obtained from Comsol Multiphysics magnetostatic simulations in order to take into account the strong correction to the free-space solution due to the presence of the close fitting superconducting shield.19
III. MEASUREMENTS

FIG. 3. Color online Tridimensional plot showing for the square overlap junction =1 the recorded magnetic diffraction patterns Ic Ha for different values of the field orientation with = 10 .

pected, due to the absence of any measurable stray fields in our setup. For this reason, we will only present data for I+, c which we will simply call Ic. We stress that, in recording the Ic vs Ha curves, we took special care that the applied field never exceeded the reversible field, so it was not expected that the applied field penetrated the films. Furthermore, through measurements of the sample's I-V characteristic, it was verified that Ha was so small as not to affect the energy gap. Finally, the raw experimental data were postprocessed to take into account the difference in the solenoid and coil field-to-current factors.
A. Overlap-type junctions

In this section we present the experimental oblique magnetic diffraction patterns relative to planar JTJs having the seven different electrode configurations listed in Table I. Section III AIII C will concern samples having, respectively, overlap, inline, and annular geometry. The theoretical interpretation of our data sets will be given in Sec. IV. The angle that the external oblique field forms with barrier plane could be experimentally spanned in the interval - , ; is however, according to Eq. 6 , an angle rotation of equivalent to an inversion of the field direction, i.e., of the field amplitude Ha -Ha. For this reason, we will only present data for in the 0, interval with the amplitude Ha assuming both negative and positive values. Further, by denoting with I+ and I- the positive and negative critical c c currents, respectively, we always had I- Ha = I+ -Ha , as exc c

We begin with an intermediate length square overlap-type JTJ, namely, sample #A in Table I 2L =2W 1.6 J . Figure 3 is a tridimensional plot of the magnetic diffraction patterns values with = 10 . Since, for recorded for different this sample, Ic -Ha = Ic Ha , we only show the data for Ha 0. It is evident that Ic Ha smoothly, but drastically, changes with the field orientation . To be clearer, in Fig. 4 we report the magnetic patterns for four selected values. For = 0 the applied field is in the barrier plane z =0 and, as seen in Fig. 4 a , Ic Ha closely follows a Fraunhofer-type behavior, as expected. The small discrepancy between the experimental data closed circles and the Fraunhofer fit solid line can be ascribed to the fact that junction dimensions are slightly larger than the Josephson penetration depth. Increasing , in the beginning the junction critical field Hc or equivalently the width of the pattern main lobe R defined earlier first slowly decreases until it reaches an absolute minimum when 50 see Fig. 4 b and later on quickly increases until it reaches an absolute maximum when 140 see Fig. 4 d . In Fig. 4 c we also report the transverse = 90 magnetic pattern to evidence how much it differs from a Fraunhofer dependence. It is worth stressing 2m with integer m , the magnetic difthat whenever fraction pattern looses the modulation periodicity Hcn = nHc1 featuring the Fraunhofer behavior; more specifically,

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(a)

(c)

(b)

(d)

FIG. 4. Color online Magnetic diffraction patterns of the square overlap junction =1 for different values: a =0 in-plane field , b = 50, c = 90 transverse field , and d = 140. The experimental data are presented by closed circles. The solid lines, when present, are the best Fraunhofer fit, while the dotted lines are the results of the calculations described in Secs. IV. For = 0, by construction, the calculations reproduce the Fraunhofer shape.

the distance between two adjacent minima increases as we move to larger fields, i.e., Hcn nHc1. Each plot in Fig. 4 explicitly reports the corresponding position of the measured R. dependence of R normalized to R for the The sample #A is summarized in Fig. 5 a solid circles . R is reported in Figs. 5 b 5 d for the samples #B, #C, and #D, overlap-type JTJs having aspect ratios, respectively, = 0.25, 0.08, and 4. The insets in Figs. 5 a 5 d sketch for each sample its electrode configuration and its orientation with respect to the Cartesian coordinates chosen in the Sec. I see Fig. 1 . We used a vertical log scale for those samples 1. Each plot in Fig. 5 is characterized by an abhaving solute maximum achieved when = M . The M values quoted in the last column of Table I were found to monotonically depend on the R ratios which, in turn, scale with the ratios. In Sec. V we will discuss a simple theoretical approach aimed to find the dependence of R and the relationship between M and R as well. We like to point out that the only measurements similar to those reported in Fig. 5 can be found in a pioneering paper dated 1975 by Rosenstein and Chen.6 They measured the first and second junction critical fields in an oblique mag0.5 and netic field for an overlap-type planar JTJ having formed by two 300-nm-thick Pb electrodes of unequal widths . They found that both Hc1 and Hc2 reach their maximum values when the field orientation is about 8 off the

in-plane direction M 172 in our notation . This value is consistent with our findings.
B. Cross-type junctions

Figure 6 shows the transverse magnetic pattern of the square cross junction sample #E in Table I on a log-log plot. The inset displays the same data on linear scales. We observe that the critical current monotonically decreases as the external field is increased. The experimental data indicate -2 that for large fields Ic Hz , in contrast with the simple inverse proportionality suggested by Miller et al.14 In Figs. 7 a and 7 b , we report the oblique magnetic diffraction patterns of the two cross-type junctions quoted in Table I, respectively, JJ # E and JJ # F. For these samples, the in-plane patterns are skewed due to the self-field effects. The skewness is more pronounced for the asymmetric sample, having 2L 3 J and I0 = Ic Ha =0 = 3.6 mA of course I0 does not depend on . As we move from an in-plane field say =0 to a transverse one say = 90 , the skewness gradually disappears and, keeping increasing toward 180, the skewness changes its polarity; in other words, Ic Ha , / 2+ = Ic -Ha , / 2- . For this reason, we only present the Ic Ha , plots for in the / 2, range with = 15 . The two samples show a quite different dependence of the normalized pattern width R / R, as shown in Fig. 8. While the former one is characterized by a weak

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(a)

(c)

(b)

(d)

FIG. 5. Color online Magnetic field range R vs in degrees for overlap junctions with different aspect ratios : a =1, b = 0.25, c = 0.08, and d = 4. The experimental data are presented by closed circles. The open stars result from the calculations described in Secs. IV, while the solid lines, when present, are the result of a simple theory developed in Sec. V. The insets sketch for each sample its electrode configuration and its orientation with respect to the chosen Cartesian coordinates.

dependence of R with a minimum when the applied field is transverse, the latter one shows a substantial variation with a maximum when the applied field is close to be transverse, 105 . more specifically, when

C. Annular junctions

FIG. 6. Color online Log-log graph of the transverse magnetic pattern Ic Hz of the cross square junction. The closed squares are the experimental data, while the open circles are the result of computations based on Eq. 15 . The dashed and solid lines are the large-field best fit of the experimental and computed data, respec-2 -1 tively, Hz and Hz . The inset shows the experimental data on linear scales.

In two recent papers,9,17 among other things, we have reported on the transverse magnetic diffraction patterns of ring-shaped Nb-based annular JTJs with radii ten times or more larger than the Josephson penetration depth. In this section, we present the results relative to a sample having the mean radius r J, namely, JJ # G in Table I. Figure 9 compares in a combined plot the transverse and the in-plane recorded magnetic diffraction patterns: respectively, the open squares referred to top horizontal scale Hz and the closed circles referred to the bottom horizontal scale Hx . The vertical logarithmic scale was needed to enhance the plot differences that would be otherwise barely observable using a vertical linear scale a part of the quite different horizontal scales . On a first order of approximation, both patterns closely follow the zero-order Bessel function behavior; the solid line in Fig. 9 is the best data fit using Eq. 23 with the first critical field as a unique fitting parameter. Figure 9 indicates that for this particular sample, a transverse field modulates the junction critical current about 1.5 times faster than an in-plane field. In Ref. 17 we have shown that this gain increases with the ring diameter and can be even larger than 100. As shown in Fig. 10, the annular range width A drastically depends on the external field orientation . In fact, although its values for = 0 and 90 belong to the same order

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(a) (a)

(b)

(b)

FIG. 7. Color online Tridimensional plots for the recorded oblique magnetic diffraction patterns Ic Ha , of two cross-type junctions: a square junction with = 1 and b asymmetric junction with = 0.25. The angular separation is = 15. For these samples Ic Ha , / 2+ = Ic -Ha , / 2- .

FIG. 8. Color online Magnetic field range R vs in degrees for the two cross-type junctions of Fig. 7: a = 1 and b = 0.25. The experimental data are presented by closed circles, while the open stars are the result of the calculations described in Secs. IV. The insets sketch for each sample its electrode configuration and its orientation with respect to the chosen Cartesian coordinates.

of magnitude A = 0.63 A, we see that A is peaked at 145, with A M 100 A 160 A ; in other words, M when M , the sample is practically insensitive to the external magnetic field, and an external field amplitude as large as the irreversible field is required to reduce the critical current Ic to 67% of its zero-field value I0.
IV. THEORY

In order to provide a theoretical interpretation of the experimental data presented in Sec. III, let us introduce the spatial normalized units = x / L and = y / W with the junction x y center coinciding with the axis origin. Our task is to find out the Josephson phase distribution , over the barrier area xy -1 1 and -1 1 of a small planar JTJ in a weak x y oblique applied magnetic field Ha. As a preliminary step, before resorting to Eq. 1 , we need to determine the magnetic field distribution over the barrier area H , . As the xy Maxwell equations are linear in the magnetic field, one can resort to the principle of superposition to calculate the field H. Thus, the effect of the oblique field Ha can be conveniently split into the sum of the effects of two orthogonal components, that is, the in-plane component H2 + H2 and x y the transverse one Hz. In other words, H , = H , + H , , xy xy xy 7 in which H and H are the barrier field distributions induced by in-plane and transverse external fields, respectively.

As mentioned in Sec. I, it has been traditionally assumed that when the external field lays in the barrier plane Hz = 0 and H =0 , it uniformly threads the oxide layer, so that H = H = Ha Hx , Hy . Today we know that this is only true to the first approximation for naked JTJs, since field focusing effects should to be considered in planar JTJ structures especially in the case of window junctions20,21 that are surrounded by a passive thick oxide layer--the so-called idle region.22 Our samples were designed to have the smallest possible idle region, so that field focusing effects could be neglected; consequently our theory has been developed under the simplifying assumption that the samples are naked. Further, as shown in Fig. 1, we will only consider magnetic field directions confined to the plane specified by the angle between the applied field and the junc