Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.cplire.ru/html/lab234/pubs/2009-03.pdf Дата изменения: Mon Jan 11 18:09:09 2010 Дата индексирования: Tue Oct 2 08:36:27 2012 Кодировка: IBM-866 Поисковые слова: п п п п п п п п п п п п п п п п п

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, Unitр INFM, Universitр 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
й2009 The American Physical Society

144521-1


MONACO et al.

PHYSICAL REVIEW B 79, 144521 2009

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 н 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

144521-2


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

140ђ 160ђ 176ђ 107ђ

JJ G

Geometry annular

ri m 5

T 490

T 310

A

A

/

A

M

0.63

145ђ

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 = 90ђ 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

144521-3


MONACO et al.

PHYSICAL REVIEW B 79, 144521 2009

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 1ђ 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 3ђ 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 = 10ђ .

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 AнIII 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 = 10ђ . 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 50ђ see Fig. 4 b and later on quickly increases until it reaches an absolute maximum when 140ђ see Fig. 4 d . In Fig. 4 c we also report the transverse = 90ђ 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,

144521-4


STATIC PROPERTIES OF SMALL JOSEPHSON TUNNEL...

PHYSICAL REVIEW B 79, 144521 2009

(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 = 50ђ, c = 90ђ transverse field , and d = 140ђ. 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 н5 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 н5 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 8ђ off the

in-plane direction M 172ђ 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 = 90ђ , the skewness gradually disappears and, keeping increasing toward 180ђ, 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 = 15ђ . 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

144521-5


MONACO et al.

PHYSICAL REVIEW B 79, 144521 2009

(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 п 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 90ђ belong to the same order

144521-6


STATIC PROPERTIES OF SMALL JOSEPHSON TUNNEL...

PHYSICAL REVIEW B 79, 144521 2009

(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 = 15ђ. 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 145ђ, 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 п = x / L and п = y / W with the junction x y center coinciding with the axis origin. Our task is to find out the Josephson phase distribution п , п over the barrier area xy -1 п 1 and -1 п 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 п , п . 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 п ,п = H п ,п + H п ,п , 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 junction plane Hy =0 . According to Eq. 7 , due to the linearity of Eq. 1 , also the phase distribution can be written as the sum of two terms п , п and п ,п , xy xy п ,п = xy п ,п + xy п ,п , xy 8

^ ^ = H z and =H z. provided that Once п , п is known, it will be possible to calculate the xy junction critical current Ic as14 Ic = I
0

sin
1 -1

2

+ cos

2

,

9

in which the brackets junction area 4 f п , п = xy

denote spatial averages over the ^1 ^ xy dx -1dy f п , п .

144521-7


MONACO et al.

PHYSICAL REVIEW B 79, 144521 2009

FIG. 9. Color online Comparison of the magnetic patterns recorded for the annular junction quoted in Table I п r J in a transverse open squares referred to top horizontal scale and in-plane closed circles referred to the bottom horizontal scale applied magnetic field. The logarithmic vertical scale helps the data comparison in the lower current range. The solid line corresponds to a Besseltype fit according to Eq. 23 .

FIG. 10. Color online Magnetic field range A vs in degrees for the annular junction quoted in Table I п r . The exJ perimental data are presented by closed circles, while the open stars are the result of calculations described in Secs. IV. The solid line arises from the approximate analytical expression 27 . The inset sketches the Lyngby-type annular junction and its orientation with respect to the chosen Cartesian coordinates.

^ Being H = Hxx for naked small JTJs , 2 that, with our normalization, becomes п ,п = h п cos , xy y

is given by Eq. 10

sively make use of Eq. 12 , inserting--for each junction geometrical configuration--the proper expression from Ref. 9.
A. Overlap-type junctions

in which h =2 0HaWde / 0, since Ha cos = Hx. If one wants to consider the effect of the idle region, still h Ha, but the proportionality constant needs proper correction. H and their corresponding have been found in Ref. 9 for naked small JTJs having the most common electrode conHz figurations. Generally speaking, it was found = Ha sin , so that we are allowed to write п ,п = h xy п ,п sin , xy 11

п , п containing the spatial part of xy with h Ha and п , п . It is important to stress that the proportionality conxy stant between h and Ha is not known a priori, being a nontrivial and still unknown function of several geometrical junction features such as the widths and the thicknesses of the two electrodes, their separation, and their configuration.9 Therefore h remains the only free parameter when comparing the experimental data to their theoretical counterparts. Equation 8 can be now rewritten in terms of Eqs. 10 and 11 as п ,п = h п cos xy y +h п ,п sin . xy

From the numerical analysis of the magnetic scalar potential induced in the barrier plane of a planar JTJ by a transverse field, we were able to derive approximate and simple expressions for the Josephson phase distribution in the bar2 / x2 rier area that satisfy the Laplace equation 2 2 + / y =0 . These heuristic expressions were found to be markedly dependent on the junction aspect ratio = L / W. For an overlap-type junction in a unitary transverse magnetic field, we found п ,п xy sin п y cosh п x , sinh 13

Inserting Eq. 13 into Eq. 12 , we observe that h, is an odd function of п , henceforth sin h, = 0; furthermore, y considering that h, is an even function of п , the calculation x Eq. 9 of the magnetic diffraction pattern reduces to
1 1

Ic h,

=I

0 0

п dx
0

п dy cos

h,

.

14

Being both h and h proportional to the intensity of the applied field Ha, their ratio = h / h depends uniquely on geometrical parameters including the junction magnetic thickness de . Therefore, the last equation can be conveniently cast in its final form,
h,

п ,п = h п cos xy y

+

п ,п sin xy

,

12

in which, h has simply been renamed h and the indices h and have been added to explicitly indicate that the Josephson phase distribution depends on the field strength and orientation. In the remaining part of this section, we will exten-

The above integral has been numerically evaluated as a function of the reduced field h and for several values of , setting I0 = 3.9 mA, = 1, and = 0.85 in order to make a comparison with the experimental data for the square overlap junction of Figs. 4--the calculated Ic vs h are reported as dotted lines and a proper horizontal scaling was chosen to match the in-plane field range R = 1.29 mT. By construction, for = 0, Eq. 14 returns the Fraunhofer 0 the calculated Ic h , at a quantitative level, pattern. For is only consistent with the experimental data, the discrepancies being more evident for large-field values. However our calculations can grasp most of the pattern small field fea-

144521-8


STATIC PROPERTIES OF SMALL JOSEPHSON TUNNEL...

PHYSICAL REVIEW B 79, 144521 2009

tures: in particular, they can nicely reproduce the dependence 1 see the open of the pattern width R on , as far as stars in Figs. 5 a н5 d . In fact, for = 4, to use expression 13 only allows us to reproduce the correct value of M .
B. Cross-type junctions

=- п r

d Hr п, r

+

0

,

18

in which 0 is an integration constant. By resorting again to the superposition principle, we can readily write the analogous of Eqs. 7 and 8 for annular junctions as H and
r

For a cross-type small naked JTJ with aspect ratio was heuristically found,9 sinh п x sinh п / y п ,п = sin п xy y + sin п x . cosh cosh 1/

, it

=H

r

+H

r

19

15

=

+

,

20

This approximate expression, beside satisfying the Laplace equation, has the proper symmetry properties required by the п , п = 1/ п ,п . We will consider first the relxy yx problem evant case of a square cross junction, i.e., = 1. When this is the case, by retaining the first terms in the Taylor expansion of the trigonometric and hyperbolic functions, the above expression 15 reduces to п ,п = пп x y xy 16

provided / =- пHr and r / =- пHr . r It is well known23 that when an external field is applied in the plane of an electrically short п r J annular junction, it ^ fully penetrates the barrier H = Hxx, whose radial component Hr = Hx cos , through Eq. 18 , leads to = h sin , 21

proposed by Miller et al.14 in 1985. The transverse magnetic pattern of a square cross junction can be computed through Eq. 9 , by inserting Eq. 15 into Eq. 8 and setting = / 2. The resulting Ic Hz is shown by the open squares in Fig. 6. Very similar results are obtained, if Eq. 15 is replaced by Eq. 16 . For large fields, the calculations are well fitted by an inverse proportionality law Ic 1 / Hz, while the best fit to the experimental data results in a quadratically -2 decreasing dependence Ic Hz . Again we come to the conclusion that the empirical expression taken from Ref. 9 is only valid in the small field range, in this case, as far as were Ic Hz 0.2I0. Also for cross junctions, the Ic Ha , calculated and the dependencies of R were extracted and shown by the open stars in Figs. 8. In evaluating the integral 9 , the parameter in Eq. 12 was chosen to be equal to its experimental counterpart R / R. As expected, the calculated are symmetric with respect to = 90ђ. R
C. Annular junctions

r is an odd periodic function, with h = Hxп. As far as the calculation of the maximum critical current reduces to the following integration: Ic = I
0 0

d cos

.

22

Inserting as in Eq. 21 , we obtain the in-plane magnetic modulation pattern Ic h = I0 J0 h , 23

In this section, we will examine the static behavior of small annular JTJs in the presence of an oblique magnetic field. Denoting the inner and outer ring radii, respectively, as ri and ro, we assume that the annular junction is unidimensional, i.e., the ring mean radius п = ri + ro / 2 is much larger r than the ring width r = ro - ri. Using polar coordinates r and such that x = r cos and y = r sin , the Josephson magnetic Eq. 1 can be split into = H, =- Hr , 17

in which J0 is the zero-order Bessel function of first kind . In deriving Eq. 23 , it was assumed that the Josephson current density is uniform over the barrier area and that no magnetic flux is trapped in between the junction electrodes +2 = . The static properties of an annular junction in a transverse field have been numerically investigated in Ref. 9 for a Lyngby-type annular JTJ obtained by two films having the same widths;24 it was found that in a first approximation, Hr sinusoidally depends on resulting in a Bessel-type transverse magnetic pattern. In other words, small differences are expected in comparing the shapes of the in-plane and transverse magnetic diffraction patterns of an annular junction, as shown by the logarithmic graph in Fig. 9. However, a small amplitude third harmonic has to , be added in order to correctly reproduce Hr H
r

Hz cos +3 cos 3 ,

24

with the coefficient much smaller than unity. The last expression readily provides a more realistic dependence = h sin + sin 3 , 25

r

r

where Hr and H are the radial and tangential components of the magnetic field in the ring plane, respectively. With the annulus unidimensional, we can neglect the radial depenr dence of the Josephson phase,23 i.e., r , = п , ; henceforth,

with h Hz. Being Hx = Ha cos and Hz = Hasin , by inserting Eqs. 21 and 25 into Eq. 20 , the Josephson phase in the presence of an arbitrary oblique magnetic field applied 2 with amplitude Ha = H2 + Hz and orientation x = arctan Hx / Hz becomes

144521-9


MONACO et al.

PHYSICAL REVIEW B 79, 144521 2009

h,

= h sin

cos

+

sin +

sin 3 sin

, 26

in which, again, h has simply been renamed h and = h / h . In order to reproduce the experimental findings reported in Sec. III C, we have computed the Ic h , , inserting Eq. 26 in Eq. 22 , being still h, - =- h, . The has been taken from the experimental A / A value of ratio A = 0.63 from Table I, while the value of was determined from the best fit of the experimental transverse magnetic pattern = 0.02. The results of such calculations for several values are displayed in Fig. 10 with open stars; the agreement with the experimental data is excellent. Indeed, the third harmonic correction is mainly needed to reproduce the peak in A ; in fact, with set to zero, the field amplitude dependence in Eq. 26 would be undetermined for = -arctan 1 / , resulting in an unphysical independence of Ic on Ha. However, far enough from this critical angle, we can set = 0 in Eq. 26 and the integral 22 trivially results in a Bessel-type behavior with critical field Hc or pattern width A given by
A

=

A

cos

+

sin

.

27

The last approximate expression plotted as a the solid line in Fig. 10 exactly matches the experimental points, everywhere except near M = -arctan 1 / A 145ђ, where it goes to infinity. From measurements not reported in this paper, we found that Eq. 27 can be usefully applied to reproduce also the behavior of long п 10 J unidimensional Lyngby-type r annular junctions in the presence of an arbitrary oblique field.
V. DISCUSSION

A similar approach can be adopted to describe the behavior of overlap-type junctions having the aspect ratio smaller than unity. In fact, being -1 п 1 and -1 п 1, then x y sin п п , and, under the assumption yy 1, cosh п 1. x Therefore, Eq. 12 simplifies to a linear п dependence y п ,п = h cos xy + sin п, y

100 R. Equation 28 is plotted as a solid line in Figs. 5 a н5 c and it is seen that it nicely reproduces the experimental data not only when 1 see Fig. 5 b but also for =1 see Fig. 5 a . It also closely fits the results of a long 2W 12 J unidimensional JTJ see Fig. 5 c for which the calculations developed in Sec. IV A for electrically small junctions do not apply. Equations 27 and 28 indicate that for small unidimensional annular junctions and small overlap-type junctions with small aspect ratio, only two measurements are needed to forecast their static behavior in an arbitrary oblique field, that is, the in-plane and the transverse magnetic diffraction patterns. As far as cross-type samples JJ # E and JJ # F are concerned, we observe that the experimental data shown, respectively, in Figs. 5 a and 5 b are affected by a slight asymmetry with respect to = 90ђ, in contrast with the system symmetry properties even in the case of electrically long junctions . We explain this symmetry break in terms of tiny fabrication misalignments. In our fabrication line, once the base electrode has been etched away, each next layer is positioned with an accuracy better than 1 m in each direction . This accuracy, although smaller, is comparable with the idle-region dimension, resulting in an unavoidable-- albeit small--imperfection in positioning the junction area exactly in the centers of the films. Since the screening currents mainly flow along the electrode borders, the misalignment effect increases with the junction side dimension. This explains why the observed asymmetry is larger for junction #F. For the Lyngby-type annular junction, our theory reproduces the experimental data in a more than satisfactory fashion. However, the situation might be not so good for annular junctions made by electrodes of unequal widths as, for example, those used in Ref. 18, which require also the introharmonic in Eq. 24 . However, duction of the second when is as in Eq. 25 , exploiting the trigonometric equivalence sin 3x = sin x 1 + 2 cos 2x , the transverse magnetic pattern can be analytically shown to be given by the following even expression: Ic h = I0 J0 h +2 J2 h , in which J2 is the second-order Bessel function and was assumed to be much smaller than unity. Similarly, Ic h can be analytically worked out in the more general case of Eq. is far from M , more precisely, when 26 , as far as 1+1 / A tan . The weak point of our theoretical approach based on the superposition principle is the indetermination of the parameter h introduced in Eq. 11 . The knowledge of the magnetic field actually introduced into the barrier for a given transverse field Hz requires a careful experimental investigation with samples having a given junction geometry and different geometrical parameter of the connecting electrodes. However, while the superconducting film widths can be easily varied, it is difficult to realize samples with much different film thicknesses. Consequentially, when the shape of the Ic Hz is known a priori, the effective normalized field h remains to be determined from the direct measure of the transverse magnetic diffraction pattern.

resulting in a Fraunhofer-type magnetic diffraction pattern with Hc or, with our notation R, given by
R

=

R

cos

+

sin

=1/
R

cos
R

+

sin
R

.

28

It's seen that the critical width c given by
c

diverges at a critical angle

= - arctan

1

= - arctan

R R

.

29

In other words, when = c, the effect of the transverse component of the applied magnetic field exactly cancels that of the in-plane component, so that the junction is virtually insensitive to the magnetic field. In practice, a full compensation is never achieved, although R M can be as large as

144521-10


STATIC PROPERTIES OF SMALL JOSEPHSON TUNNEL... VI. CONCLUSION

PHYSICAL REVIEW B 79, 144521 2009

In this paper, we examined the static properties of small planar Josephson tunnel junction in presence of a uniform external field applied at an arbitrary angle with respect to the barrier plane. This topic has been considered from both the experimental and theoretical point of view. We have presented the recorded oblique magnetic diffraction patterns of junctions with the most common electrode configuration, namely, overlap, cross, and annular geometries. These data, beside being original by themselves, also served as a test for the theoretical analysis of small JTJs in a purely transverse field that we recently proposed.9 Further, by invoking the superposition principle, the findings of Ref. 9 in a transverse field were combined with the classical knowledge for a JTJ in a parallel field to provide a general theory for any arbitrary oblique field. We stress that the theory has been developed assuming that junction was electrically small, naked, geometrically perfect, and made with purely diamagnetic superconducting films L = 0 and Hc = . Although our samples satisfied these conditions only to a rough approximation, the agreement between the experimental and theoretical results is more than satisfactory especially for annular junctions. For rectangular samples, the theoretical approach is again very good only for small values of the applied field and of the junction aspect ratio. In the other cases, our theoretical predictions fail to provide a correct description. This was to be expected because the approximate analytical expressions heuristically found in Ref. 9 were already observed to have the largest relative error near the junction corners. As already reported elsewhere,14 the importance of the Josephson phase at the junction corners grows with the amplitude of the external field. The main message of this paper is that even a small transverse field which has been largely ignored in the past can strongly influence the magnetic interference patterns. We explore the implications of this result in supposing systematic errors in previous experiments and in proposing different possible applications. Since in most of the applications the external field needed to modulate the critical current of a planar JTJ is applied in the barrier plane by means of a long solenoid or Helmholtz coil pairs, we want to stress the importance of the alignment of the solenoid or coil axis with the junction plane. It was believed that any possible tiny between the coil axis and the juncangular misalignment tion plane would result in only second-order errors, being Hx = 1 - 0.5 2 Ha. However, as we have shown, this is not necessarily true, since the unwanted and often unknown small transverse field component Hz = Ha might have an effect comparable or even larger than that of the in-plane

component, if the junction critical angle c is close to 180ђ. For example, in the case of sample JJ # C of Table I, a mis-4ђ would result in a systematic error of alignment more than 1 order of magnitude for the measurement of the first critical field. The consequence of the coil misalignment might have been underestimated, if not ignored, in many previous experiments dealing with JTJs in an external magnetic field, including those in which the exact knowledge of the field in the barrier plane is of capital importance. As a corollary, it also follows that in shielding a cryoprobe the same care has to be taken to minimize both the in-plane and the transverse stray fields. Furthermore, in planar superconducting quantum interference device SQUID applications, the magnetic field to be measured is applied perpendicular to the SQUID loop and its effect on the junction critical currents has never been considered. However, it might not be negligible, especially when the field is large and the junction s is placed close to the borders of the superconducting electrodes where the induced screening currents are larger--this is the case of step-edge or ramp-type junctions.25,26 In this paper, we have shown that for a given junction geometry the response to an externally applied magnetic field drastically depends on the field orientation; a property that might be exploited to design angle resolving instruments. Furthermore, considering that this response is different for JTJs having different geometries, it makes possible to design multijunction chips in which, for a given applied magnetic field, the critical current of some junctions is almost completely suppressed while that of other junctions remains unaffected. Our findings also suggest that two or more independent magnetic fields with different amplitudes and orientations can be applied to multijunction chips in order to obtain the proper critical current suppression required for samples having different geometrical configurations. By using Eq. 4 , the theory developed in Sec. IV for a uniform oblique field applied in the y = 0 plane can be easily extended to the most general case in which all three field components are nonzero. In other words, the oblique magnetic diffraction pattern can be theoretically predicted for the most common junction configurations, as far as the junction dimensions are smaller than Josephson penetration length. The case of long JTJ in a transverse field still remains an open question since it requires the solution of a tridimensional magnetostatic problem in the presence of external nonlinear currents.
ACKNOWLEDGMENT

We thank Pavel Dmitriev for the fabrication of the samples.

*roberto@sa.infn.it
1

3 4

2

J. M. Rowell, Phys. Rev. Lett. 11, 200 1963 ; M. D. Fiske, Rev. Mod. Phys. 36, 221 1964 . R. F. Broom, J. Appl. Phys. 47, 5432 1976 .

5

B. D. Josephson, Rev. Mod. Phys. 36, 216 1964 . A. Barone and G. Patern?, Physics and Applications of the Josephson Effect John Wiley & Sons, New York, 1982 . A. F. Hebard and T. A. Fulton, Phys. Rev. Lett. 35, 1310 1975 .

144521-11


MONACO et al.
6 7 8 9 10

PHYSICAL REVIEW B 79, 144521 2009
15 16 17 18 19 20 21 22 23 24 25

11 12 13 14

I. Rosenstein and J. T. Chen, Phys. Rev. Lett. 35, 303 1975 . L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media Addison-Wesley, Reading, 1960 . E. H. Brandt, Phys. Rev. B 72, 024529 2005 , and references therein. R. Monaco, M. Aaroe, J. Mygind, and V. P. Koshelets, J. Appl. Phys. 104, 023906 2008 . V. P. Koshelets, S. V. Shitov, A. V. Shchukin, L. V. Filippenko, I. L. Lapitskaya, and J. Mygind, "Integrated Flux-Flow Oscillators for Sub-MM Wave Receivers," Proceedings of the International Conference on Nonlinear Superconducting Devices and HTc Materials, Capri, Italy, 1994 , edited by R. D. Parmentier and N. F. Pedersen World Scientific, Amsterdam, 1995 , pp. 383н 401; Valery P. Koshelets, Sergey V. Shitov, Alexey V. Shchukin, and Lyudmila V. Filippenko, Appl. Phys. Lett. 69, 699 1996 . R. Cristiano, L. Frunzio, R. Monaco, C. Nappi, and S. Pagano, Phys. Rev. B 49, 429 1994 . M. Weihnacht, Phys. Status Solidi 32, K169 1969 . R. L. Peterson, Cryogenics 31, 132 1991 . S. L. Miller, K. R. Biagi, J. R. Clem, and D. K. Finnemore, Phys. Rev. B 31, 2684 1985 .

26

N. Martucciello and R. Monaco, Phys. Rev. B 54, 9050 1996 . M. Breitwisch and D. K. Finnemore, Phys. Rev. B 62, 671 2000 , and references therein. R. Monaco, M. Aaroe, J. Mygind, and V. P. Koshelets, J. Appl. Phys. 102, 093911 2007 . R. Monaco, J. Mygind, M. Aaroe, R. J. Rivers, and V. P. Koshelets, Phys. Rev. Lett. 96, 180604 2006 . M. Aaroe, R. Monaco, V. P. Koshelets, and J. Mygind unpublished . R. Monaco, G. Costabile, and N. Martucciello, J. Appl. Phys. 77, 2073 1995 . A. Franz, A. Wallraff, and A. V. Ustinov, J. Appl. Phys. 89, 471 2001 . J. G. Caputo, N. Flytzanis, and M. Devoret, Phys. Rev. B 50, 6471 1994 . N. Martucciello and R. Monaco, Phys. Rev. B 53, 3471 1996 . A. Davidson, B. Dueholm, B. Kryger, and N. F. Pedersen, Phys. Rev. Lett. 55, 2059 1985 . M. Podt, J. Flokstra, and H. Rogalla, Supercond. Sci. Technol. 16, 1394 2003 . H. J. H. Smilde, Ariando, D. H. A. Blank, H. Hilgenkamp, and H. Rogalla, Phys. Rev. B 70, 024519 2004 .

144521-12