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We consider a structure consisting of two parallel arrays of long Josephson junctions sharing a common electrode that allows inductive coupling between the arrays. Cavity modes excitations exhibiting 2D synchronization as well as solitonic excitations are inferred from the mathematical model of the system. The stable coherent solitonic solutions of the coupled sine-Gordon equations describing the system is found to consist of a fluxon-antifluxon string travelling in the structure.

We present measurements of the magnetic field dependent complex resistivity at 48 GHz in a nearly-optimally doped thin YBa 2 Cu 3 O 7-δ film. Below Tc the application of the magnetic field leads to a detectable magnetoresistivity. The real part is clearly measurable from our lowest temperature up to Tc, while the imaginary part can be distinguished from zero only for T>87 K . The magnetoresistivity arises from the combined effects of fluxon motion and field-induced superfluid depletion. We discuss whether our data can be described or not by models for the magnetic response in an s-wave or d-wave superconductor.

The numerical problems associated to the calculation of the phase shift due to a flux line lattice (FLL) in a thin superconducting specimen are presented and discussed. In order to be able to adequately treat the periodic case, the Fourier transform of the phase shift due to a flux tube is needed. This result is obtained by means of a new approach, where the phase shift is calculated from the beginning in the reciprocal space, treating separately each Fourier coefficient of the magnetic vector potential. The equivalence with the old approach is demonstrated and the utility of the new formulation for calculating the phase shift of closely packed FLL shown.

A single magnetic fluxon moving at a high velocity in a Josephson multilayer (e.g., high-temperature superconductor such as BSCCO) can emit electromagnetic waves (Cherenkov radiation), which leads to formation of novel stable dynamic states consisting of several bunched fluxons. We find such bunched states in numerical simulation in the simplest cases of two and three coupled junctions. At a given driving current, several different bunched states are stable and move at velocities that are higher than corresponding single-fluxon velocity. These and some of the more complex higher-order bunched states and transitions between them are investigated in detail., 6 pages + 6 Figures, to be published in Phys. Rev. B on July 1, 2000

The shape of Josephson vortex (fluxon) and fluxon distribution in stacked Josephson junctions (SJJs) at low parallel magnetic fields were studied. Fluxon solution for arbitrary number of SJJs is derived. For comparison with HTSC, large scale numerical simulations with N =600 long SJJs are presented. Due to electromagnetic coupling of SJJs, multiple quasi-equilibrium fluxon configurations (modes) exist in SJJs for given external conditions. In particular, a single row mode was demonstrated to be more favorable than a regular fluxon lattice at low fields. The existence of such modes results in strong fluctuations of perpendicular transport properties of SJJs. Analytic results are compared with numerical simulations and experimental data for low- T c and high- T c SJJs.

Flux dynamics of layered high Tc superconductors are considered with special emphasis on the small oscillation modes. In particular we find the dispersion relation for the plasma modes and discuss the spectra to be observed in microwave experiments.

We ropose a Structure where homopolar fluxons are expected to have an attractive interaction energy on a physically intuitive basis and such an interaction is recovered in the proposed mathematical model of the system. The inferred coherent motion of fluxons is then found experimentally in arrays of long Josephson junctions. The dynamics are also discussed with the help of numerical simulations.

In this contribution, we report (i) on the observation of commensurability effects between regular arrays of artificial defects and the vortex lattice in high-temperature superconductor (HTS) films and (ii) on the reduction of low-frequency 1/ f noise in high- T c superconducting quantum interference devices (SQUIDs) caused by these regular arrays of artificial defects. Arrays of submicrometer holes (antidots) with periodicity down to 0.5 μm were patterned into YBCO thin films without deterioration of superconducting properties. Commensurability effects between the antidot and vortex lattice were observed via resistive and inductive measurements, which clearly prove the presence of an attractive interaction between vortices and antidots and the existence of multiquanta vortices. Consequently, the attractive interaction between antidots and vortices was used to reduce the vortex motion in superconducting devices, thus leading to a strong reduction of 1/ f noise. For a proof of the principle, YBCO films with antidot lattices (5 μm spacing) were mounted in flip-chip configuration onto bicrystal rf-SQUIDs. Commensurability between vortex lattice in film, fluxons in the bicrystal boundary and the antidot array was observed in the form of a strong reduction and increase of the 1/ f noise at discrete magnetic inductions, which are predicted by simple geometric considerations. The transfer of this very promising technology for 1/ f noise reduction to standard HTS SQUIDs used in applications (e.g., NDE, MGC) is presently under way.

Low-velocity dynamics of a dc-driven discrete sine-Gordon chain is studied in the damped case. It is shown that resonant steps of the average velocity emerge due to the resonant phase locking between the kink motion and its radiated linear modes in its wake. A mean-field description is given, which can predict precisely the numerical results. The mechanism and formula proposed in this paper can be directly applied to the discussion of fluxon dynamics of one-dimensional Josephson junction arrays and Josephson junction ladders.

Through a measurement of flux profiles, the inductive method of Campbell has not only been used for determining the bulk critical current density of large sized samples but has also been extended to study the coupling strength of the domains in melt textured NdBa 2 Cu 3 O 7− δ superconductors. An analysis of the profiles measured for the slow (1°C/h) and fast cooled (40°C/h) samples through peritectic temperature, T p clearly showed that the domain boundaries in the fast cooled samples are decoupled in the presence of external dc fields. The observed features of flux profiles are correlated to the microstructural features of the samples. It is proposed that this technique can be used routinely to assess the quality of large size melt textured samples for high current applications like magnetic bearings or quasi permanent magnets.

We experimentally study and compare the perpendicular (c-axis) transport properties of layered high-${T}_{c}$ superconductor (HTSC) ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+x}$ and low-${T}_{c}$ (LTSC) Nb/Cu multilayers. Both for HTSC and LTSC samples similar anomalous features were observed. (i) The c-axis critical current ${I}_{c}$ is multiple valued as a function of temperature T and in-plane magnetic field H. (ii) ${I}_{c}$ exhibits extremely large fluctuations and the probability distribution ${P(I}_{c})$ has multiple maxima. (iii) The ${I}_{c}(H)$ patterns are aperiodic. (iv) In the dynamic state, the flux-flow branches in current-voltage characteristics consist of multiple closely spaced sub-branches and exhibit large fluctuations due to switching between the sub-branches. Experimental data are in qualitative agreement with numerical simulations for the stack of long strongly coupled Josephson junctions. All this is taken as an evidence for the existence of multiple quasiequilibrium Josephson fluxon modes in our samples. From the study of LTSC multilayers we have found one-to-one correspondence between the anomalous behavior and appearance of stacked Josephson junctions in the two-dimensional state.

We present a systematic study of the Cherenkov radiation of Josephson plasma waves by fast moving fluxon in a stack of coupled long Josephson junctions for different fluxon modes. It is found that at some values of parameters current-voltage characteristic may exhibit a region of the back-bending on the fluxon step. In the opposite limit the emission of the Cherenkov radiation takes place. In the annular junctions of moderate length the interaction of the emitted waves with fluxon results in the novel resonances which emerge on the top of the fluxon step. We present more exact formulas which describe the position of such resonances taking into account difference between junction and non-linear corrections. The possibility of direct detection of the Cherenkov radiation in junctions of linear geometry is discussed.

The interaction of a Josephson fluxon with an externalmagnetic field-induced potential in a long Josephson junctionis investigated experimentally. The thermal activation of thefluxon from a potential well is observed and experimentsprobing its predicted quantum properties are discussed. Amethod for engineering a magnetic double-well potential for afluxon is proposed and the use of the coupled fluxon statesfor quantum computation is suggested.

The expression for the density of the bias current, which compensates for the radiation loss of a fluxon moving in an S-I-S (superconductor-insulator-superconductor) tunnel junction with tunnel resonances-quantum resonant percolation trajectories (quantum jumpers), randomly formed in the disordered I layer [1], has been derived. The substantial physical difference between the described model and the model with microjumpers in the I layer [2] is that radiation friction acts on fluxon only in the narrow energy range of tunnel resonances.

Detailed measurements of both the Nernst coefficient, N e , and the current vs. voltage characteristics were carried out in a YBCO film by varying systematically the flux density, B , and the temperature, T . The Nernst coefficient was observed over the temperature region of a fairly wide superconducting vs. normal resistive transition under the applied flux density, B . The transport entropy of a fluxoid, S φ , estimated directly from the present measurements showed a small tail decreasing exponentially with increasing temperature, even above the mean-field critical temperature, T c ( B ), whereas the existing theory predicted S φ to become zero above T c ( B ). Then the expression for S φ above T c ( B ) was derived. The expression for the resistivity around the temperatures of the resistive transition was also derived, including the resistivity due to the flux flow. With the aid of these expressions, the observed Nernst coefficient was explained quantitatively by the present theory.

We present a review of the dynamical states of the long Josephson junction when dc and ac currents are considered as bias terms. When only the dc terms are present, depending on the value of the external magnetic field, the excitations consists of fluxon oscillations, cavity modes and travelling waves interactions with the Josephson effect. These three different modes generate well identified singularities in the current-voltage characteristics. We also report on the effect that ac current terms applied to the transmission line can have on these current singularities. In this case a theoretical analysis, based on the nature of dc excitations, has provided predictions for the ranges of locking of the external signal to junction’s oscillations. These predictions are in good agreement with experimental and numerical data that have been reported in the literature.

We derive an approximate analytic solution for a single fluxon in a double stacked Josephson junctions (SJJ's) for arbitrary junction parameters and coupling strengths. It is shown that the fluxon in a double SJJ's can be characterized by two components, with different Swihart velocities and Josephson penetration depths. Using the perturbation theory we find the second order correction to the solution and analyze its accuracy. Comparison with direct numerical simulations shows a quantitative agreement between exact and approximate analytic solutions. It is shown that due to the presence of two components, the fluxon in SJJ's may have an unusual shape with an inverted magnetic field in the second junction when the velocity of the fluxon is approaching the lower Swihart velocity.

Magnetic flux trapped on the surface of superconducting rotors of the Gravity Probe B (GP-B) experiment produces some signal in the SQUID readout. For the needs of GP-B error analysis and simulation of data reduction, this signal is calculated and analyzed in the paper. We first solve a magnetostatic problem for a point source (fluxon) on the surface of a sphere, finding the closed form elementary expression for the corresponding Green's function. Second, we calculate the flux through the pick-up loop as a function of the fluxon position. Next, the time dependence of a fluxon position, caused by rotor motion according to a symmetric top model, and thus the time signature of the flux are determined, and the spectrum of the trapped flux signal is analyzed. Finally, a multi-purpose program of trapped flux signal generation based on the above results is described, various examples of the signal obtained by means of this program are given, and their features are discussed., Comment: 14 pages, including 7 figures. Submitted to: "Journal of Applied Physics"

The change in the topography of the magnetic field associated to a superconducting fluxon due to the finite specimen thickness and its influence on the simulation of Lorentz and holographic interference images are presented and compared to the previous results obtained by using bulk models. Substantial differences between the two cases are demonstrated. These are due to the broadening of the field at the specimen surfaces, whereas the bending of the field lines within the specimen can be overlooked. It is also shown that the range of validity of one-dimensional modelling is too restricted to be practically useful when thickness effects are considered. Finally, the influence of the perturbed reference wave in the holographic interference images is investigated, showing that this effect, in addition to that due to thickness, should be taken into account for a correct interpretation of the experimental data.

We experimentally study the c-axis current-voltage characteristics (IVCs) of small area Bi/sub 2/Sr/sub 2/CaCu/sub 2/O/sub 8+x/ mesas containing only a few intrinsic stacked Josephson junctions (SJJs). In a magnetic field, H, parallel to the ab-plane, a Josephson flux-flow branch (FFB) appears in the IVCs. The FFB consists of multiple closely spaced but distinct sub-branches and exhibit a well defined Fiske step structure. When normalised to the applied magnetic field, the flux-flow branches for different H collapse into two universal curves representing two different Josephson flux-flow regimes characterised by different propagation velocities. A transition between the two regimes occurs within a narrow field interval and may indicate phase transition of the Josephson fluxon lattice in intrinsic SJJs.

Long Josephson junctions of width down to less than 0.3 /spl mu/m are fabricated using electron beam lithography. The junctions are made in niobium-aluminum-oxide trilayer technology using cross-linked PMMA for insulation. We measured the fluxon penetration field, the magnetic field period of the critical current modulation, and the Fiske step voltages of the junctions. A strong dependence of these quantities on the junction width is observed. Assuming a general-type relation between the spatial derivative of the phase and the spatial variation of the magnetic field along the plane of the junction, we derive a scaling relation between the measured quantities depending on the junction width. The derived relation is consistent with the experimental data.

We report the successful fabrication of a single intrinsic Josephson junction on a (Bi,Pb)-Sr-Ca-Cu-O thin film and investigate the current-voltage characteristics. Mesa structures are fabricated on the smooth surface of a high-quality thin film. The current-voltage characteristic along the c-axis shows a large distinct hysteresis and the superconducting gap edge structure. The estimated gap is about 75 mV at 4.2 K and is comparable with results of scanning tunneling spectroscopy. The obtained current-voltage curve is explained quite well by assuming d-wave symmetry for the superconducting order parameter neglecting nonequilibrium effects. We have observed voltage steps in the current-voltage characteristic induced by microwave irradiation. The voltage step shifts to higher voltages with increasing irradiation power. It is proposed that this behaviour is caused by fluxon motion.

Josephson transmission lines (JTLs) are commonly used as signal interconnects in single-flux quantum (SFQ) circuits. They are also used, however, to transform arbitrary waveforms into SFQ pulses. Here we use a low-temperature electro-optic sampling system with subpicosecond resolution to observe the response of a JTL to a picosecond input pulse. Previously, this system has been used to directly observe SFQ pulses on niobium transmission lines. In our experiment, niobium metal semiconductor-metal (MSM) photodiodes were connected to 3, 4, 19, and 20-stage JTLs. The MSM photodiodes provided picosecond pulses to excite the JTL. The JTL bias and input pulse amplitude were varied to observe fluxon interaction and time delay of the JTL stages. Observed JTL behavior is dominated by dipolar oscillations radiated from the first stages of the JTL. By varying the values of line inductance and junction capacitance, thus changing the oscillation periods, these oscillations were traced to the coupling of the junction capacitance and line inductance. This is a new feature of the JTL that has thus far not been observed or incorporated into simulations. Our results could have significant implications on the limit of SFQ circuits.

We have measured the current-voltage characteristics of overdamped long YBa/sub 2/Cu/sub 3/O/sub 7/ biepitaxial Josephson junctions. The velocity-matching step was observed in the current-voltage characteristics with an applied magnetic field. The presence of velocity-matching steps suggests that the biepitaxial Josephson junction is formed around a dielectric-like barrier between superconducting grains. Anomalous subharmonic voltage steps were found under microwave irradiation. The relations between anomaly steps and fluxon motion are discussed.

The numerical method was developed to simulate the fluxoids' behavior in high T c superconductor Bi-2212. Using this method, the critical current density J c , in the function of the pinning centers concentration and applied magnetic field has been calculated. To verify the validity of the method, the numerical results of J c were compared with the experimental data. Then, J c for an artificial lattice distribution of pinning centers was calculated, and the effect of the distribution of pinning centers on J c was considered.

It is shown that there are two vortex states for the configuration of an Abrikosov vortex in the presence of a long Josephson junction. One state, dubbed `conventional', carries one magnetic-flux quantum Φ 0 , as does the generating Abrikosov vortex. The second and new vortex state entails in addition to the generating vortex two Josephson vortices with opposite polarity to that of the generating-vortex and its magnetic flux is − Φ 0 . As a function of the vortex core and Josephson junction–plane distance, the accompanying pinning potentials are quasi-degenerate in the near proximity to the Josephson junction plane, where the conventional-vortex-state pinning potential is the lower. Conditions facilitating the experimental observation of the new vortex state and a specific signature are briefly discussed.

We present measurements of the microwave resistivity at 48 GHz in YBa 2 Cu 3 O 7-δ film, as a function of the temperature and the magnetic field. The magneto-dissipation from 60 K up to a few kelvins below Tc finds a quantitative description in terms of free flux flow. Approaching Tc, the data are not consistent with a dissipative mechanism based solely on flux dynamic. Fluxon viscosity is measured as a function of the temperature, and the in-plane coherence length is estimated. The analysis of the zero-field transition at 48 GHz in terms of anisotropic renormalized fluctuations yields identical estimate.

The Foucault method, one of the classical Lorentz microscopy techniques, is here investigated in order to demonstrate its applicability to in-focus observations of superconducting fluxons. As the deflections involved are of the order of 10−5−10−6 rad, low-angle techniques, high brightness, coherent illumination, and a low aberration magnetic stage are needed. The first experimental results are presented and discussed in relation with a theoretical model for the fluxon which allows the interpretation of the main features of the observed patterns.

The main ideas and approximations underlying the model developed to interpret the principal features of holographic and Fresnel images of superconducting vortices, or fluxons, are briefly reviewed and applied to the exploitation of new experimental setups, like the standard and coherent Foucault techniques. It is also shown how the model can be implemented to interpret effects due to misalignment and/or distortion of the fluxon cores. Finally, the effect of the finite specimen thickness is also taken into account by resorting to the solution of the Ginzburg-Landau equations found by Clem, and by showing that through it a better agreement can be obtained between theoretical and experimental results.

The possible equilibrium states of an isolated linear vortex in an ordered 3D Josephson medium are analysed. The energy of pinning due to the cellular structure of the medium is calculated. The pinning energy of a linear vortex and its magnetic and Josephson energies are found as functions of the critical current of Josephson contacts and spatial period of the medium.

We perform numerical simulations of a kink-shaped soliton, a fluxon, propagating in arrays of Josephson junctions ordered according to the period-doubling sequence, the Fibonacci sequence, the paper-folding sequence, the Rudin-Shapiro sequence, and the Thue-Morse sequence. The equation of motion is the discrete sine-Gordon equation with additional terms describing dissipation and an injected bias current. With the use of an effective potential we explain the behavior of the fluxon when it gets pinned in different arrays. The potential also gives a qualitative understanding of the deviation of the velocity of a propagating fluxon compared with an earlier obtained formula. It turns out that the self-similarity of the underlying sequences is important for the detailed dynamics, but not for the speed of a propagating fluxon. Finally, we show how this effective potential can be used to arrange an array to have some desired properties.

The correct nature of high-TC superconductivity is outlined: (i) the cuprates do not superconduct in their cuprate-planes, but in their BaO, SrO, or interstitial-oxygen regions; (ii) doped Ba2YRuO6 and Sr2YRuO6, ruthenates without cuprate-planes, superconduct in their BaO or SrO layers; (iii) the rutheno-cuprates GdSr2Cu2RuO8 and Gd2−zCezSr2Cu2RuO10 have cuprate-planes which do not superconduct, but superconduct in their SrO layers; (iv) LaFeAsO1−xFx superconducts via its oxygen ions, as do related compounds; and (v) the organic compound κ-[BEDT-TTF]2Cu[NCS]2 superconducts along the S chains of the molecule. In YBa2Cu3O7, the superconductivity is consistent with observations by positive muon spin rotation and with analyses indicating an absence of Cu d-band contribution to the superconductivity. Hence only, the BaO layer superconducts. The superconductivity is s-wave in character (not d-wave), once fluxon depinning has been properly accounted for. The superconducting BaO layers are p-type and adjacent to the n-type cuprate-planes. The hole-pairing is not phononic, but Coulombic. Many experiments can be explained by understanding the characteristics of the high-TC mechanism, which contradicts the theories with d-wave superconductivity in the cuprate-planes.

Transitions between fluxoid states in asymmetric mesoscopic superconducting rings are investigated experimentally. We detected the fluxoid state transitions by the voltage change across current-biased small tunnel junctions attached to the ring. We found that the temperature above which the hysteresis and voltage jump in the transition disappear is lower in less symmetric rings, which is qualitatively consistent with the theoretical prediction that the one-dimensional vortex state is stabilized in asymmetric rings.

Magnetic flux quanta in Josephson junctions, often called fluxons, in many cases behave as solitons. A review of recent experiments and modelling of fluxon dynamics in Josephson circuits is presented. Classic quasi-one-dimensional junctions, stacked junctions (Josephson superlattices), and discrete Josephson transmission lines (JTLs) are discussed. Applications of fluxon devices as high-frequency oscillators and digital circuits are also addressed.

The increase in voltage noise observed by increasing magnetic field in granular high-Tc superconductors is explained in the framework of a percolative model. Noise is reproduced with computer simulations of a network of junctions, many of which are randomly shorted. The network configuration is representative of interconnections ~weak links! of grains in polycrystalline bulk samples. With respect to previous percolative models, we have demonstrated that the calculated noise intensity is modified when interdependence between different configurations in the time evolution of the network is considered. The concordance with the experimental dependence of noise on resistance at different magnetic field intensities in YBa2Cu3O72d and HgBa2CuO41d supports the hypothesis that magnetic field affects the fluxon speed and thus the switching rate of weak links. @S0163-1829~98!05633-1#

Interpreting their experimental data in terms of an approximate solution of the Lawrence - Doniach - Clem (LDC) model in the continuum limit (describing an isolated interlayer Josephson vortex), Moler et al (1998 Science 279 1193) have estimated the penetration depth in the direction normal to the layers of the compound (Tl-2201) to have the value . They thus concluded that only of the superconducting condensation energy in Tl-2201 would be due to the interlayer-tunnelling mechanism. We have studied the LDC model and found that it has no physical solution. Therefore in Tl-2201 is in need of re-examination.

The dynamics of bound states of fluxons in two coupled long Josephson junctions under fluctuations of the current is investigated. Mean values of the oscillation amplitude and kinetic energy of the bound state are calculated. The lifetime of the bound state is estimated.

The bunching of moving magnetic flux quanta (fluxons) of the same vorticity has been observed in experiments with annular Josephson tunnel junctions by means of spatially resolving measurements using low-temperature scanning electron microscopy.

A magnetic memory effect is observed in the absorption of electromagnetic waves of 20–70 MHz in YBa 2 Cu 3 O 7 at 77–300 K.

NdBa 2 Cu 3 O 7− δ superconductor is processed with varying Nd 4 Ba 2 Cu 2 O 10 (Nd-422) content, thereby varying systematically its microstructure. The samples are then investigated by studying the flux penetration into the samples as a function of magnetic field using a flat-band lock-in technique. Unlike in sintered samples, the flux profiles in the melt-processed samples are found to be linear, whose slopes increase with increasing Nd-422 content. The slope of the profile gives a direct, model-independent measure of the critical current density in the materials, and the specimens measured can be large enough to include several millimeter-sized domains. The critical current densities thus obtained are compared with those obtained using SQUID magnetometer, wherein the samples have to be much smaller and mostly single domain. The extent of flux penetration, reversible penetration depth, and the slope changes in the flux profiles at high fields are correlated to the microstructural features of the melt-textured samples.

The fluxon structure in a layered superconductor under an applied parallel constant magnetic field is examined within the model of stacked Josephson junctions. The interlayer phase difference ${\ensuremath{\varphi}}_{n,n+1}$, which is found as a solution of the system of nonlinear equations, corresponds to buckled chains of fluxons, alternating with the field strength. The calculated ${\ensuremath{\varphi}}_{n,n+1}$ is then used to obtain the spectrum of vortex and plasma small oscillations that has a fine band structure showing gaps.

Two boundary-value problems in plane geometry, modeling the field and current distribution produced by moving sources of magnetic field (fluxons) in surrounding normal metals, are solved analytically. In the first case a fluxon moves with a constant velocity in a superconducting plane separated by a vacuum gap from a half space filled with a metal having a low electrical conductivity. In the second case the half space is replaced by a thin metal layer of high conductivity. The problems are solved by perturbation techniques, the Fourier integral transform, and the Parseval identity. The latter is used to calculate the power dissipated in the metal. An estimate of the power dissipation is needed in order to calculate certain torques of magnetic origin in the Gravity Probe B project.

The ac properties of layered superconductors in the vortex state for the parallel orientation of the external magnetic field B are examined. We find that due to an intrinsic Josephson effect the ac characteristics depend on the kind of the steady-state fluxon structure. Both the collective mode {Omega}{sub ps} and the ac penetration depth {lambda}{sub {parallel}}{sup ac}({omega},{bold B}) are affected by the magnetic field in an anomalous way, because of the reactive properties of the vortex state{close_quote}s environment. {copyright} {ital 1998} {ital The American Physical Society}

Vortex states of icosahedral superconducting wire network are investigated by the de Gennes–Alexander theory. We obtained the magnetic field dependence of the transition temperature and fluxon arrangements in the network. Fluxon arrangements of icosahedral network have some different symmetry related to directions of the applied fields. In a magnetic field parallel to the fivefold rotational symmetry axis, we found that the order parameter vanishes at the central nodal point along the field axis. In a magnetic field parallel to the threefold rotational symmetry axis, anti-fluxon can appear at the central triangular loop.

Current-voltage ( I − V ) characteristics and radiation emission experiments at 80–120 GHz with 7- and 9-layer stacked Nb/AlAlO x /Nb Josephson junctions are reported. The I − V curves measured in magnetic field applied parallel to the layers display the flux-flow behavior modulated by resonances which are consistent with two-dimensional modes predicted by Kleiner. Voltage dependence of the radiation power indicates collective fluxon motion.

The microwave-induced phase-locking phenomena in the intrinsic Josephson junctions in Bi 2 Sr 2 CaCu 2 O y single crystal mesas have been investigated experimentally. For the mesas with the dimensions much larger than the Josephson penetration depth, obvious microwave-induced steps composed of fundamental and subharmonic steps have been successfully observed in the current-voltage characteristics. The voltages of both steps increase proportionally to the microwave power, and the amplitudes of the subharmonic steps show periodical change with the microwave frequency. These dynamical properties can be interpreted to be caused by the microwave-induced phase-locking fluxon motion in the intrinsic Josephson junctions.

The energy spectrum of electrons on a square lattice under a uniform magnetic field B0 and a twodimensional magnetic modulation B1 is calculated. When the flux of B0 is a rational multiple p/q of the fluxon, we find that the Bloch band is broken intoq subbands for even q and into 2q subbands for odd q. Symmetry of the modified Hofstadter spectrum is discussed. @S0163-1829~97!04239-2#