Josephson Tunnel Junctions in a Magnetic Field Gradient
JJosephson Tunnel Junctions in a Magnetic Field Gradient a) R. Monaco, J. Mygind, and V. P. Koshelets Istituto di Cibernetica del CNR, Comprensorio Olivetti, I-80078, Pozzuoli,Italy and Dipartimento di Fisica, Universit `a di Salerno, 84081 Baronissi,Italy b)2) DTU Physics, B309, Technical University of Denmark, DK-2800 Lyngby,Denmark Kotel’nikov Institute of Radio Engineering and Electronics,Russian Academy of Science, Mokhovaya 11, Bldg 7, 125009, Moscow,Russia. (Dated: 13 November 2018)
We measured the magnetic field dependence of the critical current of high quality
N b -based planar Josephson tunnel junctions in the presence of a controllable non-uniformfield distribution. We found skewed and slowly changing magnetic diffraction patternsquite dissimilar from the Fraunhofer-like ones typical of a homogeneous field. Ourfindings can be well interpreted in terms of recent theoretical predictions [R. Monaco,
J. Appl. Phys. , 033906 (2010)] for a uniform magnetic field gradient leading toFresnel-like magnetic diffraction patterns. We also show that Fiske resonances canbe suppressed by an asymmetric magnetic field profile.PACS numbers: 03.70.+k, 05.70.Fh, 03.65.Yz a) Submitted to Appl. Phys. Letts. b) Electronic mail: [email protected] a r X i v : . [ c ond - m a t . s up r- c on ] J a n t is commonly believed that a Faunnhofer-like magnetic diffraction pattern (MDP) char-acterized by its periodic and well-pronounced minima is the hallmark of the d.c. Josephsoneffect in ideal tunnel junctions, weak links, Dayem bridges etc. However, this is onlytrue if the applied magnetic field is uniform over the junction area. Experimentally this israrely satisfied: in fact, the presence of the tip of a magnetic force microscope, of trappedAbrikosov vortices, of magnetic dots or strips, the self-field of nearby wires and/or electrodes,the mounting out of a solenoid axis etc., make the magnetic field distribution in the Joseph-son device environment non-uniform. Recently the consequences of a uniform magneticfield gradient on the static properties of both electrically short and long planar Joseph-son tunnel junctions (JTJs) have been studied analytically and numerically, respectively.In both cases marked differences from the ideal text-book case of perfectly homogeneousmagnetic field were predicted: short junctions exhibit zeros-free Fresnel-like MDPs charac-terized by small-amplitude non-periodic damped oscillations slowly approaching a non-zerolarge-field asymptotic value and by symmetry with respect to current or field inversion.Upon increasing the junction normalized length l the MDPs show a progressive break ofthese symmetries and only the symmetry with respect to the current and field inversion isretained. In this Letter, we report measurements of the magnetic field dependence of thecritical current and of the amplitude of resonant cavity modes (Fiske steps) of high-qualitywindow-type N b/Al − Al ox /N b JTJs in presence of a static non-uniform magnetic field. Webelieve that our finding can be very useful to correctly interpret those non-Fraunhofer-likeMDPs which may erroneously be attributed to low sample quality; in addition we providehints for a deeper understanding of both the static and dynamics properties of Josephsondevices subjected to non-uniform magnetic fields.Our design is sketched in Fig.1(a) together with the coordinate system used in thiswork; two independent 5 µm wide and 0 . µm thick N b control lines run adjacent to thelong dimension of an asymmetric (2 × µm window-type JTJ, 2 µm away from theborder of the junction base electrode. In the vicinity of the junction center they bentat right angle and run parallel to each other separated by a 2 µm gap. At a distance d = 600 µm from the first bending the lines bend back and run again in opposite directions.In case the left and right control line currents, I LCL and I RCL respectively, have the sameamplitudes and the directions indicated by the left clockwise and right counterclockwisearrows in Fig.1(a), then an asymmetric field profile is realized; since the magnetic fields2 a) (b)
FIG. 1. (a) Sketch (not in scale) of a (2 × µm window-type Josephson tunnel junctions inthe asymmetric magnetic field profile H ( x ) generated by two properly biased independent controllines. The junction area is white, the base electrode black and the top electrode gray. In our design a = 2 x = 5 µm and d = 600 µm . (b) Computed magnetic field profiles along the junction length inthe RCL, symmetric and asymmetric modes (dashed, solid and dotted line, respectively), for thelayout depicted in (a). at the junction extremities have the same amplitude, but opposite directions, H L = − H R ,we call asymmetric mode. In the opposite case, we end up with a symmetric profile fieldsimilar to the classical uniform field generated by a single control line running aside thewhole junction length (or by a long solenoid); we will refer to it as the symmetric mode.By playing separately with I LCL and I RCL , one can achieve any desired magnetic boundarycondition. The largest achievable field values are set by the control line critical currents whichdepend on the temperature and the geometrical parameters. The control line technique hasbeen used to produce local magnetic fields for digital applications of Josephson circuitssince 1969 . It is useful to remind that the current I CL in each control line produces amagnetic field perpendicular to the junction plane which induces Meissner screening currentsin the superconducting junction electrodes; these circulating currents, in turn, produce amagnetic field in the junction plane proportional to I CL . The details of how a transversefield modulates the critical current of planar JTJs with different geometrical configurationscan be found in Ref.6. Considering that the distance d between the parallel arms of eachcontrol line is orders of magnitude larger than the separation, a , between the junction3 IG. 2. Normalized MDP obtained in the uniform field generated by a long coil (open circles andlower horizontal scale) compared with the computed i c ( h a ) for a one-dimensional overlap junctionwith normalized length l = 4 in a uniform normalized field h a (dashed line and upper horizontalscale). The inset compares on a logarithmic scale the MDPs obtained in the uniform field of asolenoid (open circles and lower horizontal scale) and by the control lines in the symmetric mode(dotted line and upper horizontal scale). T = 4 . K and I c (0) = 4 . mA . and the control lines, the magnetic field distribution H ( x, y = a ) along the junction longdimension generated by each control line can be evaluated, to a very good approximation,by applying the Laplace equation to a semi-infinite wire carrying a current ± I CL , originatingat x = ± x and running along the x axis. For the right control line (RCL) operated alonewe get H RCL ( x, a ) = H ( ∞ , a )[0 . . − ( x − x ) /a ] with H ( ∞ , a ) = I CL / (4 πa ), asshown by the dashed line in Fig.1(b) in which the solid and dotted lines show, respectively,the field profile for the asymmetric and symmetric modes. We observe that the largest fieldchanges occur in a region in the middle of the junction whose size is determined by thedistance 2 x between the left and right control lines; in our case this separation is muchsmaller than the junction physical length. Quite similar field distributions were computedwith 3D magneto-static Comsol Multiphysics simulations taking into account the correctionto the free-space solution due to the presence of close superconducting electrodes.In the following, we will present the data of one representative sample out of few testedones all having Josephson current density of 2 . kA/cm at T = 4 . K and (2 × µm bar-rier area. We underline that the overlap geometry is the only one for which is strictly properto speak of zero-field behavior; for other geometries, current-bias self-induced fields are not4egligible. Overlap-type JTJs with asymmetric electrode configuration have often been con-sidered in the literature but, to our knowledge, only from a theoretical point of view; thetwo overlapping electrodes make the bias current distribution very uniform over the junctionlength and at the same time create a very large (asymmetric) idle region which dresses thejunction and strongly influences its behavior with respect to that of bare junctions (in-creased Josephson penetration depth and Swihart velocity). Preliminarily, we characterizethe sample by measuring its MDP in the (conventional) uniform field of a long solenoidwhose axis lies in the barrier plane and is perpendicular to the junction long dimension, i.e.,along the y -direction of Fig.1(a). Such a test MDP is shown, in normalized units by thesolid circles of Fig.2; i c = I c /I c (0) with I c (0) = 4 . mA at T = 4 . K . As the dotted lineindicates, the experimental data are best fitted by the numerically computed in-plane MDPof an overlap junction when its normalized length l = L/λ j is set to 4, indicating that thesample Josephson penetration depth is λ j ≈ µm . (For a naked junction with no idle regionthe calculated λ j would be approximately equal to 7 µm , with the N b
London penetrationset equal to 90 nm .) In the inset of Fig.2 we compare the same data with that obtained withthe control lines operated in the symmetric mode (dotted line and upper horizontal scale).The vertical log-scale is chosen to emphasize the suppression of few pattern lobes. This islikely due to the non uniformity of the field profile in the symmetric mode - see the well inthe dotted line of Fig.1(b). Incidentally we observe that, despite the boundary conditionsare the same, H L = H R , the MDPs are quite different. We remark that all the MDPs inFig.2 and its inset are symmetric with respect to inversion of either the junction bias currentor the applied magnetic field.The normalized MDP of the same sample in presence of the asymmetric field profile producedby the control line operated in the asymmetric mode is reported in Fig.3 (closed squares andlower horizontal scale) versus the common control line current amplitudes, I CL = I RCL = − I LCL . We observe a pronounced skewness despite the fact that the maximum I c valueoccurs for I CL = 0; the negative pattern (not shown here) reveals perfect symmetry withrespect to the simultaneous inversion of both the junction and control lines currents. In thesame plot we superimpose the numerically computed i c ( h a ) dependence (open circles andupper horizontal scale) for an uniformly biased overlap junction with l = 4 embedded ina linear zero-mean magnetic profile h ( x ) = 2 h a x/l , so that h ( ± l/
2) = ± h a , as reported inRef.3. We observe agreement only at a qualitative level. We believe that the discrepancy5 a) (b) FIG. 3. (a) Normalized MDP obtained with the control lines in the asymmetric mode (closed circlesand lower horizontal scale) and the numerically computed MDP i c ( h a ) for an overlap junction withnormalized length l = 4 in a zero-mean magnetic field profile h ( x ) = 2 h a x/l (dashed line and upperhorizontal scale). (b) Normalized MDP obtained by feeding just one control line (closed circlesand lower horizontal scale) and the numerically computed MDP i c ( h a ) for an overlap junction with l = 4 in a field profile: h ( x ) = h a ( x + l/ /l (dashed line and upper horizontal scale). can be ascribed to the fact that in our experiments the field gradient is unevenly distributedover the junction length, being rather ’squeezed’ near its center. However, in the well-known modeling of a one-dimensional JTJ the externally applied magnetic field only entersas the boundary conditions of a static sine-Gordon equation regardless of the particularfield profile ; our results indicate that this approach does not capture all important physicaldetails and should be used only for very long junctions when l >> i c ( h a ) dependence (open circles and upperhorizontal scale) for an overlap junction with l = 4 embedded in a uniform magnetic gradient: h ( x ) = h a ( x + l/ /l , so that h ( − l/
2) = 0 and h ( l/
2) = h a . As expected for the reasonsabove, we get just qualitative agreement.A inhomogeneous magnetic field drastically modifies also the dynamics of a planar JTJ.Since the large idle region prevents the zero-field resonant fluxon motion , we focused our6 a) (b) FIG. 4. (Color online) Magnetic field dependence of the amplitudes of Fiske steps (a) in a uniformfield and (b) in the inhomogeneous field generated by the control lines operated in the asymmetricmode. T ≈ K and I c (0) = 2 . mA . attention on the magnetic resonant modes which manifest in the junction current-voltagecharacteristic as current singularities called Fiske steps . Figs.4(a) and (b) display the mag-netic field dependence of the Fiske steps amplitudes for our sample in presence, respectively,of a uniform in-plane field and of the non-uniform field profile obtained in the asymmetricmode. These measurements were taken a T ≈ K [ I c (0) = 2 . mA ] since at lower tempera-ture it was difficult to latch on the low order Fiske steps. We observe that in presence of anasymmetric profile only the even resonances survive; the full suppression of the odd steps isconsistent with recent analytical calculations aimed to extend the Kulik theory for smalljunctions to a magnetic field distribution of type h ( x ) = h a x/l . We believe that the keyingredient of the odd step suppression is the asymmetry of the field profile. Surprisingly, all the resonances disappear in presence of the field profile proper of just one control line;indeed, Raissi et al. proposed that the reversing magnetic field sets extra conditions at thecenter of long JTJs which makes the creation of any standing electromagnetic waves impos-sible. Definitely, further investigation is needed to find a proper theoretical interpretationof these facts.To summarize, we have experimentally shown how the static and dynamic properties ofJosephson tunnel junctions change when one abandons the standard assumption of a homo-geneous magnetic field. New measurements with samples having both shorter and longernormalized length have already been planned. In the near future, the tuning of a shuttling7uxon by means of a magnetic field gradient will also deserve our attention. REFERENCES B. D. Josephson,
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