We propose an alternative explanation to the oscillations of the flux-flow
resistance found in several previously published experiments with BiSrCaCuO
stacks. It has been argued by the previous authors that the period of the
oscillations corresponding to the field needed to add one vortex per two
intrinsic Josephson junctions is associated with a moving triangular lattice of
vortices (out-of-phase mode), while the period corresponding to one vortex per
one junction is due to the square lattice (in-phase mode). In contrast, we show
that both type of oscillations may occur in a single-layer Josephson junction
and thus the above interpretation is inconsistent

A. V. Ustinov

I. Kakeya

I. O. Kulik

N. F. Pedersen

T. Hatano

English

arXiv

Crossref

Origin of flux-flow resistance oscillations in
                    
                      
                        
                          Bi

We propose an alternative explanation to the oscillations of the flux-flow resistance found in several previously published experiments with Bi2Sr2CaCu2O8+y stacks. It has been argued by the previous authors that the period of the oscillations corresponding to the field needed to add one vortex per two intrinsic Josephson junctions is associated with a moving triangular lattice of vortices (out-of-phase mode), while the period corresponding to one vortex per one junction is due to the square lattice (in-phase mode). In contrast, we show that both type of oscillations may occur in a single-layer Josephson junction and thus the above interpretation is inconsistent

Ustinov, A.V.

Pedersen, Niels Falsig

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Origin of flux-flow resistance oscillations in Bi2Sr2CaCu2O8+y: Possibility of Fiskesteps in a single junctionUstinov, A.V.; Pedersen, Niels FalsigPublished in:Physical Review B (Condensed Matter and Materials Physics)Link to article, DOI:10.1103/PhysRevB.72.052502Publication date:2005Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Ustinov, A. V., & Pedersen, N. F. (2005). Origin of flux-flow resistance oscillations in Bi2Sr2CaCu2O8+y:Possibility of Fiske steps in a single junction. Physical Review B (Condensed Matter and Materials Physics),72(5), 052502. DOI: 10.1103/PhysRevB.72.052502Origin of flux-flow resistance oscillations in Bi2Sr2CaCu2O8+y: Possibility of Fiske stepsin a single junctionA. V. UstinovPhysikalisches Institut III, Universität Erlangen-Nürnberg, D-91058, Erlangen, GermanyN. F. PedersenOersted-DTU, Section of Electric Power Engineering, The Technical University of Denmark, DK-2800 Lyngby, DenmarkReceived 13 April 2005; published 8 August 2005We propose an alternative explanation to the oscillations of the flux-flow resistance found in several previ-ously published experiments with Bi2Sr2CaCu2O8+y stacks. It has been argued by the previous authors that theperiod of the oscillations corresponding to the field needed to add one vortex per two intrinsic Josephsonjunctions is associated with a moving triangular lattice of vortices out-of-phase mode, while the periodcorresponding to one vortex per one junction is due to the square lattice in-phase mode. In contrast, we showthat both type of oscillations may occur in a single-layer Josephson junction and thus the above interpretationis inconsistent.DOI: 10.1103/PhysRevB.72.052502 PACS numbers: 74.50.r, 74.72.h, 85.25.CpRecently, a lot of attention has been focused on the pos-sible use of single crystals of Bi2Sr2CaCu2O8+y as generatorsof electromagnetic radiation in the THz range.1 A crucialrequirement for this purpose is realizing coherent in-phaseoscillations of intrinsic Josephson junctions which belong todifferent atomic layers of such crystals. Promising experi-ments in that direction were reported by Hirata et al.2 andshortly thereafter by Kakeya et al.3 and Hatano et al.4 Theseexperiments all showed oscillations in the flux-flow voltageand flux-flow resistance when a large magnetic field of or-der several Tesla was applied parallel to the ab plane in thepresence of a small bias current in the c-direction of order afew percent of the critical current. The observed flux-flowvoltage oscillations typically showed two different oscilla-tion periods. At the lowest magnetic fields the period wasHT=0 / 2sL corresponding to one extra flux quantum pertwo layers in the stack. Here 0 is the magnetic flux quan-tum, s and L are the thickness and the length of the junction,respectively. At higher magnetic fields there was a transitionto a period HS=0 / sL, i.e., corresponding to an extraflux quantum in every layer.These exciting experiments were interpreted bothanalytically5 and numerically6,7 by several authors. One ofthe important questions to be answered was whether the fluxlattice correspond to a triangular lattice antiphase orderingwith possible cancellation of the sum voltage at the junctionend or a square lattice in-phase ordering leading to a largesum voltage at the junction end. Figure 1 shows schemati-cally a Bi2Sr2CaCu2O8+y stack with flux ordering in a trian-gular lattice and square lattice. Obviously, the latter case ishighly preferable for applications to THz generation of elec-tromagnetic waves; however simple intuition would suggesttriangular ordering since fluxons of same polarity naturallyrepel each other. An intuitive interpretation of the experimen-tally observed oscillations in the flux-flow voltage wouldsuggest that an oscillation period HT corresponds to trian-gular ordering while a period HS corresponds to a squarelattice. Numerical simulations1,6,7 have shown that both tri-angular and square lattices are possible, but their relation toregions of HT oscillations and regions of HS oscillationsis not simple and details are still a matter of debate.In this paper, we present fairly standard numerical simu-lations corresponding to the simplest case of a single-layerJosephson junction. Even for this case, where there is notriangular nor square lattice ordering as our stack consists ofonly one junction, we find that both the HT and HS pe-riods appear much the same way as in experiments and nu-merical simulations for Bi2Sr2CaCu2O8+y stacks with manylayers and flux lattice ordering. After presenting the numeri-cal simulations we provide a qualitative explanation in termsof the well-known Fiske modes.8The system under investigation is described by coupledsine-Gordon equations of the form9SJ = 2x2; S =1 SS 1 S 0    0 S 1 SS 1 1withFIG. 1. Schematic drawing of the triangular lattice a corre-sponding to the out-of-phase fluxon mode and the square lattice bassociated with the in-phase fluxon mode.PHYSICAL REVIEW B 72, 052502 20051098-0121/2005/725/0525024/$23.00 ©2005 The American Physical Society052502-1Ji 2it2+ it+ sin i −  . 2Here = 1/R / 2eI0C1/2 is the dissipation parameter R,I0, and C are the normal resistance, the critical current andthe capacitance per unit length, respectively,  is the currentnormalized to the critical current I0 of the individual junc-tions. The normalized coupling term among the junctionsin the stack reads S=−	L / d sinht /	L, where d=d+2	L cotht /	L, t is the thickness of one superconduct-ing layer, and 	L is the London penetration depth.9 Time t isnormalized to the inverse of the Josephson plasma frequencyp= 2eI0 / CJ1/2 and spatial coordinate x is normalizedwith respect to the Josephson length 	J=  / 2e0I0d+2	L1/2. The magnetic field gives rise to boundary condi-tions of the form9 ixx=0,=H	JI0 h . 3Index i=1, . . . ,N stands here for the junction number in thestack and  is the normalized length of the system.Figure 2 shows the result of a simulation of the current-voltage characteristics for only one junction in the stack, i.e.,N=1. It shows the flux-flow branch of the junction with nor-malized length =40 and damping constant =0.1 placed inmagnetic field h=4. The characteristics is calculated by ris-ing the bias current  from zero to 0.85 and then decreasingit back to zero. The voltage V is given in normalized unitschosen such that the voltage spacing between neighboringFiske stepsV =0c¯2L4is equal to unity here c¯ is the Swihart velocity and L=	J isthe physical length of the junction. The current-voltagecharacteristics displays fine structure due to the Fiske steps,which gather around the flux-flow voltage, also known asEck peak.10 The hysteresis in this voltage region is due to thecoexistence of several Fiske resonances at a given current .Some parts of these Fiske resonances are located inside thehysteresis and are not displayed in the plot.Using the above junction parameters, we calculated thedependence of the flux-flow resistance on magnetic field.Figure 3 presents the differential flux-flow resistance dV /dversus magnetic field h at the fixed bias current =0.3. Itclearly shows flux-flow resistance oscillations. In order torelate the period of oscillations with the number of vorticesin the junction we show a grid in magnetic field with a pe-riod h=2 /, which approximately corresponds to addingone vortex in the junction. This can be seen from the simplefact that at the critical field h=2 the normalized spacing be-tween vortices penetrated into the junction is equal to , andtheir number rises proportionally to h. In Fig. 3 we see thatmost oscillations have a characteristic period in h corre-sponding to adding a half flux quantum into the junction,with a tendency of doubling the period at higher fields.As another illustration, in Fig. 4 we show both static re-sistance V / and differential resistance dV /d versus mag-netic field h for the same junction but at a lower bias current=0.2. The oscillations of dV /d thin curve have two char-acteristic periods, which are found at different ranges ofmagnetic field h. The oscillation period at low fields corre-sponds to about or less than half flux quantum, while athigher fields we find very clear oscillations which accountfor one flux quantum into the junction. The crossover fromone regime to another occurs at magnetic field h	3. Belowthis field the differential resistance at =0.2 is lower as it isdetermined by the Eck peak see also Fig. 2 composed ofindividual Fiske steps. At h3 the Eck peak shifts to highercurrents and the differential resistance levels at the resistiveslope determined by the loss parameter  of the junction.The static junction resistance upper curve in Fig. 4 alsochanges in this range but its oscillations are much less pro-nounced and can only be clearly seen on the magnified scale.We suggest the following explanation for the two oscilla-FIG. 2. Current-voltage characteristics of a long junction N=1 with parameters =40, =0.1, and h=4. Arrows indicateswitching between branches for rising and decreasing bias current.FIG. 3. Color online Differential flux-flow resistance dV /dversus magnetic field h for the single junction N=1 with =40and =0.1 at constant bias current =0.3.BRIEF REPORTS PHYSICAL REVIEW B 72, 052502 2005052502-2tion periods of the flux-flow resistance in a single-barrierJosephson junction. At high enough magnetic field the resis-tance at a low bias current follows oscillations of the criticalcurrent of the junction and thus have a characteristic periodof one flux quantum. At the same time, the resistance mea-sured at a higher current or lower field but the same biascurrent follows the oscillations due to the Fiske steps, whichenvelope is associated with the Eck peak sometimes calledas flux-flow or velocity matching step. The matter is that—for a single junction—the Fiske steps induce a variation ofthe resistance with characteristic period corresponding tohalf flux quantum.Fiske modes in a Josephson junction8 are linear cavitytype excitations with resonance angular frequencies given byn = n2, n = 1,2,3, . . . 5in normalized units. The corresponding to wave vectors arekn=n /. In experiments these Fiske modes are visible ascurrent singularities in the current-voltage curve with a volt-age spacing given by Eq. 4. The amplitude of the Fiskesteps oscillate with the magnetic field in a typical Besselfunction like pattern such that the even numbered steps havemaxima together with maxima in the critical current, whilemaxima in the odd numbered steps correspond to minima inthe critical current. References 11 and 12 give an approxi-mate analytical form for the current-voltage curve for asingle Josephson junction with Fiske steps. The current-voltage characteristics is approximately written as12,h =  + −+  hh2 − kn2sinh − kn2 2·22 − kn22 + 22. 6This approximation neglects high-order nonlinearities in thejunction and is originally expected11 to describe well the caseof short junction, i.e., 1. However, the comparison withlong junction data made by Cirillo et al.12 shows that Eq. 6can approximately account for the shape and for the maxi-mum current modulation of the Fiske singularities in long1 junctions when the field penetration overcomesMeissner shielding, i.e., at h2.The first term in Eq. 6 represents the Ohmic part of thecurrent-voltage characteristics, while the second term givesan infinite series of equidistant resonances. The height of theresonances is modulated by a slowly varying amplitude fac-tor and a fast Fraunhofer amplitude factor.12 The Fraunhoferfactor emphasizes the resonance closest to =h and dropsoff fast: If h is an even multiple of , the odd numberedFiske steps are enhanced and if h is an odd multiple of ,the even numbered Fiske steps survive.Equation 6 gives the current-voltage curve containingFiske steps with the magnetic field as a parameter. If weinstead assume a fixed bias current  and vary the magneticfield, Eq. 6 expresses the voltage V oscillations in animplicit form. In order to compare this analytical form withour simulations, we solved Eq. 6 for  at a given  numeri-cally. The obtained the dependence of the flux-flow resis-tance dV /d at constant bias current =0.2 versus magneticfield h is presented in Fig. 5. The qualitative agreement be-tween Figs. 3 and 4 obtained by full numerical simulation ofthe perturbed sine-Gordon equation and Fig. 5 emergingfrom the analytical formula 6 is strikingly good. Figure 5clearly displays two characteristic periods of oscillations,namely half flux quantum oscillations at low fields and oneflux quantum oscillations which become very explicit at highfields. The intermediate field range shows a complicatedbeating between two periods.The low field oscillations period from our simulationsshown in Figs. 3 and 4 is only approximately equal to thehalf flux quantum. We suppose that it is due to nonlinearityof the plasma wave dispersion curve in long Josephson junc-tion at low magnetic fields. We found that shorter junctionsFIG. 4. Color online Flux-flow static resistance V / thickcurve and differential resistance dV /d thin curve versus mag-netic field h for the same junction as in Fig. 3 at a lower bias current=0.2.FIG. 5. Color online Differential flux-flow resistance dV /d atconstant bias current =0.2 versus magnetic field h calculated di-rectly from Eq. 6 with parameters =40 and =0.1. The voltageis multiplied by a factor  / to make its scale identical to thevoltage normalization used in the previous figures.BRIEF REPORTS PHYSICAL REVIEW B 72, 052502 2005052502-3display more regular low-field oscillations with period whichis very close to the half flux quantum, h1/2= /, whichcorresponds to HT=0 / 2sL in physical units. The fieldperiod is inversely proportional to the junction length .It is worth pointing out that the current range in which theoscillations are observed entirely depends of the chosen dis-sipation constant . In experiments typical  is smaller thanthe value 0.1 used in our simulations, so one actually needsto apply smaller currents relative to the critical current atzero field to reach the same voltages.Thus, odd and even numbered Fiske resonances in the lastterm in Eq. 6 produce the “magic” half-flux-quantum oscil-lations corresponding to the magnetic field period HT=0 / 2sL even in a single Josephson junction. Although weinvestigated here Fiske steps with N=1, we note that Fiskesteps are also present in stacks with N1.13 Thus we arelead to suggest that also for N1 the flux-flow voltage os-cillations have their origin in the Fiske mode excitations.We conclude that for Bi2Sr2CaCu2O8+y stacks the flux-flow voltage oscillations with two different periods in a mag-netic field have their origin in the Fiske mode excitations.Thus the flux lattice ordering in either triangular or squarelattice is not directly related to the two periods of the oscil-lations. We speculate that Fiske modes also existing in stacksindirectly play a role for the flux lattice formation.We would like to acknowledge discussions with T. Ha-tano, S. Kim, A. E. Koshelev, S. Madsen, M. R. Samuelsen,H. B. Wang, and T. Yamashita.1 M. Tachiki, M. Iizuka, K. Minami, S. Tejima, and H. Nakamura,Phys. Rev. B 71, 134515 2005.2 S. Ooi, T. Mochiku, and K. Hirata, Phys. Rev. Lett. 89, 2470022002.3 I. Kakeya, M. Iwase, T. Yamamoto, and K. Kadowaki, cond-mat/0503498 unpublished; I. Kakeya, T. Wada, M. Machida, andK. Kadowaki, Physica C 378–381, 437 2002.4 T. Hatano et al., IEEE Trans. Appl. Supercond. 15, 948 2005.5 A. E. Koshelev, Phys. Rev. B 66, 224514 2002.6 M. Machida, Phys. Rev. Lett. 90, 037001 2003.7 N. F. Pedersen and S. Madsen, IEEE Trans. Appl. Supercond. 15,948 2005.8 M. D. Fiske, Rev. Mod. Phys. 36, 221 1964.9 S. Sakai, P. Bodin, and N. F. Pedersen, J. Appl. Phys. 73, 24111993.10 R. E. Eck, D. J. Scalapino, and B. N. Taylor, Phys. Rev. Lett. 13,15 1964.11 I. O. Kulik, Pis’ma Zh. Eksp. Teor. Fiz. 2, 134 1965 Sov. Phys.JETP 2, 84 1965; I. O. Kulik, Zh. Tekh. Fiz. 37, 157 1967Sov. Tech. Phys. Lett. 12, 111 1967.12 M. Cirillo, N. Grønbech-Jensen, M. R. Samuelsen, M. Salerno,and G. V. Rinati, Phys. Rev. B 58, 12377 1998.13 S. Kim, S. Urayama, H. B. Wang, T. Hatano, M. Nagao, S.Kawakami, Y. Takano, and T. Yamashita, presented at the con-ference “Plasma-2004,” Tsukuba, Japan.BRIEF REPORTS PHYSICAL REVIEW B 72, 052502 2005052502-4

Origin of flux-flow resistance oscillations in Bi2Sr2CaCu2O8+y: Possibility of Fiske steps in a single junction

The origin of flux-flow resistance oscillations in BiSrCaCuO: Fiske
  steps in a single junction?

The origin of flux-flow resistance oscillations in BiSrCaCuO: Fiske steps in a single junction?

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