Planar Josephson Tunnel Junctions in a Transverse Magnetic Field
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Planar Josephson Tunnel Junctions in a Transverse Magnetic Field ∗ R. Monaco, M. Aaroe, V. P. Koshelets, and J. Mygind Istituto di Cibernetica del C.N.R., 80078, Pozzuoli,Italy and Unita’ INFM-Dipartimento di Fisica, Universita’ di Salerno, 84081 Baronissi, Italy. † Department of Physics, B309, Technical University of Denmark, DK-2800 Lyngby, Denmark. ‡ Institute of Radio Engineering and Electronics, Russian Academy of Science,Mokhovaya 11, Bldg 7, 125009, Moscow, Russia. § (Dated: November 1, 2018)Traditionally, since the discovery of the Josephson effect in 1962, the magnetic diffraction patternof planar Josephson tunnel junctions has been recorded with the field applied in the plane of thejunction. Here we discuss the static junction properties in a transverse magnetic field where de-magnetization effects imposed by the tunnel barrier and electrodes geometry are important. Mea-surements of the junction critical current versus magnetic field in planar Nb-based high-qualityjunctions with different geometry, size and critical current density show that it is advantageous touse a transverse magnetic field rather than an in-plane field. The conditions under which this occursare discussed. PACS numbers:
INTRODUCTION
It is well known that a magnetic field H modulates thecritical current I c of a Josephson Tunnel Junction (JTJ)[1]. Indeed the occurrence of such diffraction phenomena I c ( H ) is one of the most striking behaviors of JTJs[2]. Inthe case of a planar JTJ, it was Josephson himself [3] whopointed out that the gradient of the Josephson phase φ ,which is the difference between the complex wavefunctionphases in the electrodes, can be expressed as ∇ φ = 2 πd e µ Φ H × n , (1)where n is a unit vector normal to the insulating barrierseparating the two superconducting electrodes, µ is thevacuum permeability and Φ = h/ e is the magnetic fluxquantum. If the two superconducting films have thick-nesses d , and London penetration depths λ L , and t j is the barrier thickness, then the effective magnetic pen-etration d e is given by[4]: d e = t j + λ L tanh d λ L + λ L tanh d λ L , which, in the case of thick superconducting films ( d i >>λ Li ), reduces to d e ≈ λ L + λ L .Since Rowell[5] in 1963 made the first experimental veri-fication of Eq.1, a large number of theoretical and ex-perimental papers have been devoted to the study ofmagnetic diffraction patterns, in various Josephson junc-tions. Nowadays the I c ( H ) curves for planar JTJs hav-ing the most common geometrical and electrical param-eters is fully understood (see, for example, Cpt.4 and 5of Ref.[1]). It is important to point out that nearly allwork was done with the external magnetic field appliedin the barrier plane. In fact, since Eq.1 states that φ is insensitive to transverse fields, this is the most obviouschoice of the magnetic field orientation.For the reasons above, a magnetic field parallel to thebarrier of planar JTJs is applied in the practical applica-tions of the Josephson magnetic diffraction phenomenasuch as, for example, the suppression of the d.c. Joseph-son effect in SIS mixers for photon detection[6] and inspecially shaped JTJs for particle detection[7], the mag-netic biasing of a flux flow oscillator[8], the tuning ofresonant fluxon oscillators[9]. TRANSVERSE MAGNETIC FIELD
In 1975 Rosenstein and Chen[10] first reported on theeffect of a transverse magnetic field on the critical currentof a JTJ with overlap geometry. Among other things,they showed that the value of the junction critical field H c at which the magnetic diffraction pattern first goesto zero, changes with the inclination of the field with re-spect to the barrier plane, the minimum being obtainedwhen the field is transverse. This was the first experi-mental observation that transverse fields could be moreefficient that parallel ones in modulating the Josephsoncurrent. Soon after, Hebard and Fulton[11] correctly in-terpreted the findings of Ref.[10] in terms of stationaryscreening currents which develop when a superconductoris subjected to an external magnetic field. To better un-derstand the mechanism through which also a transversefield is able to modulate the critical current of a planarJTJ, let us consider first a single isolated superconduct-ing film immersed in a uniform static magnetic field H ⊥ perpendicular to its surface. This system has received acontinuous interest over the years and here we only recallthe main features. For a deep treatment of this topic weremand to Ref.[12] and references therein. We assumethat the film thickness d is larger than its London pene-tration depth λ L and that the field everywhere is muchsmaller than the critical field which would force the filminto the intermediate or normal state, i.e., that the film isin the flux-free Meissner regime. At the top and bottomfilm surfaces, the flux lines are excluded from the interiorof the film where H = . In fact, due to the screeningcurrents J s (= ∇ xH ), they bend as they approach thefilm surface, flow along the film surfaces, concentrate atthe film edges, and bent backward. Due to continuity, H n , the component of H normal to the surface, may betaken to be zero, while its tangential component H t de-cays exponentially inside the film on the scale of λ L .The knowledge of the distribution of the magnetic fieldlines around the film requires a self-consistent solution ofa magnetostatic problem combining the London equation( H + λ ∇ H = ) in the superconducting film and thefourth Maxwell equation ( ∇× H = ) in the empty spacearound the film with boundary conditions appropriate tothe film surface geometry. This problem can be solvedanalytically only for simple axially-symmetric cases suchas, for example, that of an ellipsoid of revolution with theaxis of revolution parallel to the applied field H ⊥ [13].If the film width w is much less than its length, butmuch greater than its thickness d , then we can approx-imate the film as a elliptical oblate cylinder of infinitelength whose cross-section has axes w and d ; with theapplied magnetic field H ⊥ directed along the minor axis,then H t , the component of H tangent to the surface,only depends on the angle β with respect to the mi-nor axis: H t /H ⊥ = 1 / √ cos β + ( d/w ) sin β , whosemaximum value w/d >> β = ± π/ w is comparable toits length, the film can be approximated by a disk whosediameter w is much greater than its thickness d , with theexternal field applied parallel to its axis. In this case, forsymmetry reasons, on the disk top and bottom surfacesthe magnetic field lines are radial and H t only dependson the distance r from the disk center; it is null at thecenter of the disk and increases as we move outward[15].The surface or sheet current density j s , defined as thescreening current density J s integrated over the speci-men thickness, equals in magnitude H t and is everywhereorthogonal to H . Numerical simulations carried out fora N b disk having λ L = 90 nm show that the shape of H t ( r ) only depends on the disk aspect ratio w/d , as faras d >> λ L . We found that at the disk border H t is sev-eral times larger than the applied field H ⊥ , as shown inFig.1 for three values of the disk aspect ratio w/d = 10,100 and 1000, with d = 1 µm . Both film approximationslead to conclude that a thin superconducting film of anygeometry in a transverse field produce a magnetic field:i) whose orientation on the film end surfaces is paral-lel to surfaces themselves; ii) whose direction near theborders of the film end surfaces is perpendicular to theborders themselves; iii) whose strength is proportionalto the transverse applied field intensity and exceeds its w=10(cid:181)mw=100(cid:181)m H t / H FIG. 1: Computed normalized magnetic induction field H t /H ⊥ at the border of a Nb superconducting disk in a per-pendicular field H ⊥ . The disk thickness is d = 1 µm anddiameters w = 10, 100 and 1000 µm . 2 r/w is the normalizeddisk radius. value near the borders of a film with a large aspect ratio w/d .Now, we can consider the situation in which two super-conducting films partially overlap to form a planar JTJ.If the tunnel barrier is very transparent, then the screen-ing currents cross the barrier and the two films act asone single fused film. In the opposite case the screeningcurrents in the two films are independent on each otherand no cross talk is allowed. In the intermediate situ-ations, a fraction of the screening currents in one filmcrosses the barrier and circulates in the other film andviceversa. It is clear that, for a given transverse field anda given junction and electrodes geometry, the transitionfrom the fused to the independent films regime can becontrolled via the barrier transparency, i.e., in our caseby the Josephson current density J c . The determinationof the magnetic field distribution in a system made bytwo superconducting film forming a planar JTJ is a verycomplex task, since it also involves the Josephson equa-tions. The analysis becomes even more difficult when theJTJ is biased (the distribution of the bias current peaksnear the film edges) and the junction dimensions exceedthe Josephson penetration depth λ J = p ~ / eµ d e J c .However, following Ref.[14], we can argue that in theindependent films regime, the intensity of the in-planemagnetic field felt by a planar JTJ placed at the bor-ders of two superconducting films can exceed by severaltimes the value of the applied transverse field. In fact,the screening currents flowing within a depth λ L,b in thetop surface of the bottom film and those flowing withina depth λ L,t in the bottom surface of the top film haveopposite directions. Consequently the associated mag-netic fields add in the barrier plane and a more efficientmodulation of the JTJ critical current is achieved.
EXPERIMENTS
In order to prove the advantages to use a transversefield rather than an in-plane one, we have measured thetransverse magnetic diffraction patterns I c ( H ⊥ ) of pla-nar high quality JTJs having different geometries, sizesand critical current densities. The samples were placedon the axis of a superconducting coil surrounded by aPb shield and a cryoperm can in order to attenuate theearth magnetic field. The magnetic field produced bythe coil was calculated through COMSOL MultiPhysicsnumerical simulations. Overlap type junctions
Fig.2 compares the diffraction patterns measured ina transverse field of two overlap-type junctions havingthe same geometry and dimensions, but different criticalcurrent density J c . The junctions have been fabricatedwith the trilayer technique in which the junction is real-ized in the window opened in an SiO insulating layer.The thicknesses of the base, top and wiring layer are 200,100 and 400 nm , respectively. Details of the fabricationprocess can be found in Ref.[16]. The junction length is L = 500 µm , while the width is equal to 4 µm . The baseand top electrode widths are 540 and 506 µm , respec-tively. The so called ’idle region’, i.e. the overlapping ofthe wiring layer onto the base electrode is about 3 µm allaround the barrier. In Fig.2 the junction critical currentshave been normalized to their maximum values in orderto make the comparison easier. The closed circles refer toa N b/Al ox /N b − N b tunnel junction having a critical cur-rent density J c = 60 A/cm , while the open circles referto a N b/AlN/N bN − N b sample with J c = 400 A/cm .As expected, considering that the field lines associated tothe screening current are perpendicular to the electrodesedges, the shape of these I c vs. H ⊥ curves looks veryalike to that expected for long one-dimensional overlap-type junctions when a uniform external field H k is ap-plied in the barrier plane in the direction perpendicularto the junction length. In fact, according to the analysisof Refs.[17, 18], for small field values (Meissner regime)the critical current I c decreases linearly with the appliedfield I c ( H k ) = I c (0)(1 − | H k | /H k c ), where H k c is the crit-ical field at which I c would vanish if flux quanta did notstart to enter the junction barrier. The skewness seenin the experimental I c − H ⊥ curves (being larger for thesample having the larger J c ) is due to the self-field pro-duced by the bias current I flowing in a close-by super-conducting strip in the chip circuitry. The skewness doesnot prevent us from measuring the junction critical crit-ical fields H ⊥ c determined by linearly extrapolating thebranches starting at maximum critical current to I = 0(dotted lines in Fig.1); in fact, when I = 0, the bias cur- -20 -15 -10 -5 0 5 10 15 20-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 I C / I C m a x H (A/m)
FIG. 2: Magnetic diffraction patterns measured in a trans-verse field H ⊥ of two overlap-type junctions L × W = 500 × µm . The closed circles refer to a Nb/Al ox /Nb JTJ hav-ing J c = 60 A/cm , d e = 180 nm and λ J ∼ µm . Theopen circles refer to a Nb/AlN/NbN with J c = 400 A/cm , d e = 240 nm and λ J ∼ µm . The dotted lines show theprocedure used to determine the junction critical field H ⊥ c . rent self-field effects vanish. The critical field H k c for along one-dimensional junction in presence of a in-planeexternal field applied perpendicular to the long junctiondimension L can be expressed as the sum of two terms: H k c = H Fc + H F Pc = Φ µ d e L + 2 λ J J c = Φ µ d e ( 1 L + 1 πλ J ) = Φ µ d e L + s J c πµ d e . (2)The first term H Fc in Eq.2 dominates in low J c sam-ples for which L << πλ j and corresponds to the criti-cal field of a point-like junction which exhibits the wellknown Fraunhofer diffraction pattern. The second term H F Pc ∝ p J c /d e becomes dominant in the high J c regimewhen L >> πλ j . It was first introduced by Ferrell andPrange[19] in order to describe the self-field limiting ef-fects in long inline-type junctions.Table.1 reports the values of J c and d e used to predictthe critical field H k c from to Eq.2. The last two columns Sample J c d e L/λ J H ⊥ c H k c = H Fc + H F Pc
A/cm nm A/m A/mNb/Al ox /Nb
60 180 ∼
10 3 . Nb/AlN/NbN
410 240 ∼
25 13 36 + 131 = 167TABLE I: Relevant parameters for two overlap-type Joseph-son tunnel junctions whose transverse magnetic diffractionpatterns are shown in Fig.2. allow the comparison between H ⊥ c and H k c . We observethat for both samples H k c > H ⊥ c , and the ratio H k c /H ⊥ c changes from about 30 to about 13 when we move fromthe low to the high J c junction. The data in Table.1 un-ambiguously indicate that a transverse field can be moreeffective than a in-plane one and, remembering that thetwo samples have the same geometrical details, the effectof a transverse field weakens as the junction transparencyincreases. Annular junctions
Further, we have measured the static properties of sev-eral
N b/Al ox /N b − N b annular JTJs having the samecritical current density J c = 60 A/cm ( λ J = 50 µm ), thesame width ∆ R = 4 µm , but different radius R rangingfrom 80 to 500 µm . The fabrication details are the sameas those for the overlap-type JTJs discussed previously.The diffraction patterns in transverse magnetic field willbe reported elsewhere[20]; here we focus our interest onthe values of the critical fields.The analogous of Eq.2 for a one-dimensional annularjunction having radius R in an in-plane field is[21]: H k c = H Bc + H MMc = 2 .
404 Φ πµ d e R + RJ c . (3)Again we have a contribution H Bc , independent on theJosephson current density J c , typical of small and in-termediate radius annular JTJs immersed in a uniformin-plane magnetic field which results in a periodic radialfield H r ( θ ) ∝ cos θ felt by the junction[22, 23]. In suchcase, the I c vs. H k curve follows a Bessel, rather than aFraunhofer, behavior (2.404 is the argument correspond-ing to the first minimum of the zero-order Bessel func-tion). The second term H MMc = RJ c in Eq.3 was numer-ically found by Martucciello and Monaco[21]; consideringthat H MMc /H Bc = ( R/λ J ) / . R >> R m = √ . λ J [9, 23]. For given d e and J c ,Eq.3 has a minimum when R = R m and linearly increasewith R when R >> R m . The last two columns of theTable.2 report, respectively, the transverse critical field H ⊥ c measured for four annular junctions having differentradii and the expected parallel critical field H k c accordingto Eq.3 with J c = 60 A/cm and d e = 180 nm .For all samples we observe once again that H k c > H ⊥ c ,and that the ratio H k c /H ⊥ c changes from about 25 toabout 250 when we increase the ring diameter, i.e. thetop and bottom film widths, confirming that the effectof a transverse field strengthens as the electrode widthsincreases. In particular, for the three largest rings, hav-ing the so called Lyngby geometry[24], i.e. the base andtop electrode widths match the ring diameter, this ratiois proportional to R . [The smallest ring ( R/λ J ≈ . R R/λ J H ⊥ c H k c = H Bc + H MMc mm A/m A/m . ∼ . . . ∼ . . ∼ . . . ∼
10 1 . Nb/Al ox /Nb annularJosephson tunnel junctions used in the experiments. has the base electrode width equal to 540 µm that is con-siderably larger than the ring diameter 160 µm .] CONCLUDING REMARKS
We have measured the transverse magnetic diffractionpattern of planar JTJs having different geometries, sizesand barrier transparencies. Our measurements clearlyindicate that a magnetic field is more effective to mod-ulate the junction critical current I c when applied per-pendicularly (rather than parallel) to the junction planeprovided the JTJ is fabricated close to the borders ofsuperconducting films with a large aspect ratio. This isdue to screening (or Meissner) currents induced by thetransverse field that circulate mainly on the film surfaceborders which in turn behave as intrinsic control lines.We suggest that a transverse magnetic field can be use-fully exploited in those applications where the Josephsoncritical current and the Fiske resonances need to be sup-pressed. ∗ Journal of Applied Physics vol.102, 093911 (2007) † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected][1] A. Barone and G. Patern`o
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