One-dimensional Josephson Junction Research Articles

Boundary effects on Josephson current through one-dimensional Josephson junction arrays

Josephson junctions hybridized with an array of superconducting islands are studied. Assuming large on-site Coulomb interactions, a particle number and a Josephson critical current are calculated as functions of voltages of two leads based on a hard-core boson model, which is equivalent to a spin system with boundary fields. After the Wigner-Jordan transformation, an additional factor depending on the parity of the particle number appears at the boundary. This factor neglected in the previous issues crucially determines properties of the junction. The boundary effects due to this factor are discussed by studying resonant tunneling peaks in critical currents and phase dependence of Josephson currents.

Quantum phase transitions in one-dimensional long Josephson junction stacks in parallel magnetic fields

We report the effect of quantum and thermal fluctuations on stability of mutual phase locking in one-dimensional long Josephson junction devices, involving layered superconductors. Accounting for both the induction coupling and the charging effect, we determined the zero-temperature $(T=0)$ phase diagram, using renormalization-group analysis, and found that the in-phase mode is stable, but some out-of-phase modes are unstable against quantum fluctuations. At finite T, all stable phase-locking modes (at $T=0)$ are unstable, but stability is still maintained within a finite length, which decreases inversely with T. In ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+y},$ this length for the in-phase mode is roughly 1000 \ensuremath{\mu}m at 1 K.

Temperature-dependent resistance of a finite one-dimensional Josephson junction array

We study theoretically the temperature and array-length dependences of the resistance of a finite one-dimensional array of Josephson junctions. We use both analytic approximations and numerical simulations, and conclude that within the self-charging model, all finite arrays are resistive in the low-temperature limit. A heuristic analysis shows qualitative agreement with the resistance obtained from Monte Carlo simulations, establishing a connection between resistance and the occurrence of vortices in the corresponding 1+1D XY model. We compare our results with recent experiments and conclude that while the self-charging model reproduces some of the experimental observations, it underestimates the superconducting tendencies in the experimental structures.

Quantum Complementarity for the Superconducting Condensate and the Resulting Electrodynamic Duality

Experiments with small capacitance Josephson junction arrays are described that demonstrate the quantum behavior of the phase of the superconducting condensate. This quantum behavior is based on the complementarity of phase and number for a condensate state of many bosons. A key factor to observation of this quantum behavior is the coupling of the superconducting condensate state to dissipation in the electrodynamic environment of the Josephson junction. It is shown how one-dimensional Josephson junction arrays can be used to design an environment with very weak coupling to dissipation, allowing for measurement of condensate with well defined number. The well defined number is manifest in the Coulomb blockade of Cooper pair tunneling.

Scaling Analysis of Magnetic-Field-Tuned Phase Transitions in One-Dimensional Josephson Junction Arrays

We have studied experimentally the magnetic field-induced superconductor-insulator quantum phase transition in one-dimensional arrays of small Josephson junctions. The zero bias resistance was found to display a drastic change upon application of a small magnetic field; this result was analyzed in context of the superfluid-insulator transition in one dimension. A scaling analysis suggests a power law dependence of the correlation length instead of an exponential one. The dynamical exponents $z$ were determined to be close to 1, and the correlation length critical exponents were also found to be about 0.3 and 0.6 in the two groups of measured samples.

Quantum phase transition in one-dimensional Josephson junction arrays

We describe experiments on one-dimensional arrays of small capacitance Josephson junctions which show how the Coulomb blockade of Cooper pair tunneling is influenced by changing the Josephson coupling energy in situ with an externally applied magnetic flux. We show how the zero bias resistance of the array is affected by the length of the array, and we use the length scaling of this resistance to infer that a quantum phase transition occurs as the Josephson coupling energy is changed. The data are qualitatively analyzed in terms of a theoretical model of the quantum phase transition which uses a mapping to the two-dimensional XY model.

Satellite flux-flow steps in a long Josephson junction

Multifluxon dynamics in the presence of an external rf field is investigated both analytically and numerically for a long one-dimensional Josephson junction. A simple model reported recently for the flux-flow mode is extended to the case when the external rf field is applied at the junction boundaries. It is shown that the interaction between the flux-flow mode and plasma waves induced by the external rf radiation results in additional (satellite) steps in the $I\ensuremath{-}V$ characteristic. According to a simple analytical formula, the satellite steps appear around the main flux-flow step and are separated by the voltage corresponding to the second harmonic of the external frequency.

Localized modes in finite and phase-inhomogeneous Josephson tunnel junctions

We have investigated the spectrum of localized modes in both finite and phase inhomogeneous long one-dimensional Josephson junctions in the presence of an external magnetic field. For finite junctions these excitations correspond to evanescent waves localized near the edges. Both the frequency and the characteristic decay lengths depend on the magnetic field. In particular, a different behavior is expected for the Meissner state and the mixed state in which wholesale penetration of Josephson vortices occurs. The related problem of the spectrum of modes localized near phase inhomogeneities occurring well inside a long junction has been also tackled. This problem is relevant to the case in which Abrikosov vortices penetrate the electrodes and become pinned near the junction plane.

Resonant Steps in the Discrete sine-Gordon Chain**The project supported in part by the Research Grant Council RGC, the Hong Kong Baptist University Faculty Research Grant FRG, and National Natural Science Foundation of China

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.

Superconducting and Insulating Behavior in One-Dimensional Josephson Junction Arrays

Experiments on one-dimensional small capacitance JosephsonJunction arrays are described. The arrays have a junctioncapacitance that is much larger than the stray capacitance ofthe electrodes, which we argue is important for observation ofCoulomb blockade. The Josephson energy can be tuned in situand an evolution from Josephson-like to Coulomb blockadebehavior is observed. This evolution can be described as asuperconducting to insulating, quantum phase transition. Inthe Coulomb blockade state, hysteretic current-voltagecharacteristics are described by a dynamic model which is dualto the resistively shunted junction model of classicalJosephson junctions.

Analytical description of the flux-flow mode in a long Josephson junction

The flux-flow mode in a long one-dimensional Josephson junction is studied analytically. An approximate steady-state solution of the perturbed sine-Gordon equation is derived in the form of a dense fluxon chain accompanied by small-amplitude plasma waves. Next, some time-averaged quantities are calculated, making it possible to evaluate the constant bias current and the constant voltage between the junction electrodes. The analytical results are compared with numerical simulations both for the field distribution within the junction and for the $I\ensuremath{-}V$ characteristics.

Flux-flow mode in the sine-Gordon system

An inverse transformation of the theta function is derived, making it possible to investigate a multiperiodic solution of the sine-Gordon equation in the limit of a dense sequence of overlapping solitons. A special case of a unidirectional soliton train interacting with small-amplitude quasi-linear oscillations is discussed as a simple model of the flux-flow state in a long one-dimensional Josephson junction. Approximate analytical solutions for the dispersion parameters are compared with numerical results.

Discreteness-induced resonances and ac voltage amplitudes in long one-dimensional Josephson junction arrays

New resonance steps are found in the experimental current-voltage characteristics of long, discrete, one-dimensional Josephson junction arrays with open boundaries and in an external magnetic field. The junctions are underdamped, connected in parallel, and dc biased. Numerical simulations based on the discrete sine-Gordon model are carried out, and show that the solutions on the steps are periodic trains of fluxons, phase locked by a finite amplitude radiation. Power spectra of the voltages consist of a small number of harmonic peaks, which may be exploited for possible oscillator applications. The steps form a family that can be numbered by the harmonic content of the radiation, the first member corresponding to the Eck step. Discreteness of the arrays is shown to be essential for appearance of the higher order steps. We use a multimode extension of the harmonic balance analysis, and estimate the resonance frequencies, the ac voltage amplitudes, and the theoretical limit on the output power on the first two steps.

Mapping of one-dimensional Josephson junction arrays onto cellular neural networks and their dynamics

It is shown that one-dimensional Josephson junction arrays can be mapped onto sworks under some restrictions. Analytic expressions to the stability of equilibria of these arrays are derived and applied to the behavior of vortex solutions. Relations between these static solutions and a particular solution of a partial differential equation are shown. By means of the theory of Hamiltonian systems, a qualitative theory for Josephson junction arrays is derived and applied in numerical simulations.

Bloch vortices in one-dimensional Josephson junction arrays

The quantum dynamics of vortices in very long and narrow arrays of Josephson junctions have been studied experimentally. The motion of the vortices through the one-dimensional array is homologous to the motion of a boson through a one-dimensional periodic potential. The external force on the vortices is proportional to the bias current. When the bias current exceeds a critical value, a negative differential resistance is observed. In the presence of external RF radiation the IV-characteristic exhibits steps at currents which scale with the RF frequency. The position of the steps also depends on the vortex density, which is probably caused by a collective effect of the vortices. These preliminary results strongly indicate the existence of a vortex bandstructure.

Electromagnetic waves in a Josephson junction in a thin film.

We consider a one-dimensional Josephson junction in a superconducting film with a thickness that is much less than the London penetration depth. We treat an electromagnetic wave propagating along this tunnel contact. We show that the electrodynamics of a Josephson junction in a thin film is nonlocal if the wavelength is less than the Pearl penetration depth. We find the integrodifferential equation determining the phase difference between the two superconductors forming the tunnel contact. We use this equation to calculate the dispersion relation for an electromagnetic wave propagating along the Josephson junction. We find that the frequency of this wave is proportional to the square root of the wave vector if the wavelength is less than the Pearl penetration depth.

Fluxon motion in one-dimensional Josephson junction arrays

Current-voltage ( IV-) characteristics of linear and annular discrete arrays of underdamped small Josephson junctions with different values of single loop inductances are calculated numerically and compared with those obtained for continuum long Josephson junctions. The resonances between the moving fluxon and the linear waves caused by the array discreteness induce steps on the IV -curve which do not exist in the continuum case. The voltage position of the steps is found to be in good agreement with the kinematic approach based on the Frenkel-Kontorova model. Our results show that in Josephson fluxon devices the discreteness may lead to strong super-radiant emission of electromagnetic waves by moving fluxons.

Circularly symmetric fluxon generation in annular Josephson junctions.

We consider the generation of circularly symmetric (CS) fluxons in annular Josephson junctions by a current pulse and by alternative methods. The theory of fluxon generation by a current pulse in one-dimensional (1D) Josephson junctions developed by Kivshar and Malomed (KM theory) is sketched and extended to the case of CS-fluxon generations. It turns out from numerical simulations that CS fluxons can also be generated by a driving current pulse, like the counterparts in 1D long junctions. However, significant differences from the 1D case exist and are discussed in depth. Attention is drawn to such factors as the soliton return effect, size effect, fluxon stability against the driving current, and the applied magnetic field, which all play crucial roles in the generation process. An alternative method of CS-fluxon generation is decreasing adiabatically the driving current from a rotational state until it becomes unstable and switches to a CS-fluxon state. The features of both approaches are discussed. The plausibility of future physical experiments is addressed.

Quantum fluctuations in finite size Josephson junctions

An effective Josephson coupling energy for a one-dimensional Josephson junction is renormalized due to quantum fluctuations of the phase difference. In a long junction at T → 0 a Kosterlitz-Thouless phase transition takes place. The state with a logarithmically divergent phase-phase correlation function shows a nontrivial combination of phase disorder on a junction surface with phase order in the bulk. For finite size junctions the renormalized value of the Josephson coupling energy turns out to be strongly suppressed for small Josephson-to-charging energy ratio. The implications of this effect for Bloch oscillations are discussed.

Pinning phenomena and critical currents in disordered long Josephson junctions.

The critical current ${\mathit{I}}_{\mathit{c}}$ of a long one-dimensional (1D) Josephson junction in the presence of different types of structural disorder is investigated both analytically and numerically. It is shown that most properties of ${\mathit{I}}_{\mathit{c}}$ can be understood from the behavior of the elementary pinning force (PF) of a single defect, which we calculate exactly as a function of the external magnetic field ${\mathit{H}}_{\mathit{e}}$, pinning-center size, and strength. The following types of disorder are discussed: (i) For a given field, and pinning centers with equal strength, a unique arrangement of pins that maximizes ${\mathit{I}}_{\mathit{c}}$ is found. (ii) In the case of a periodic pinning-center lattice, we reproduce the commensurability peaks in the field dependence of the critical current, ${\mathit{I}}_{\mathit{c}}$(${\mathit{H}}_{\mathit{e}}$), previously reported by Oboznov and Ustinov [Phys. Lett. A 139, 481 (1989)]. In addition, we predict that a peak can be damped or disappear, if its position coincides with a field value at which the elementary PF vanishes. (iii) The most interesting effects appear in the presence of random disorder. Using the exact expression for the elementary PF, we extend the collective pinning analysis of Koshelev and Vinokur (Zh. Eksp. Teor. Fiz. 97, 976 (1990) [Sov. Phys. JETP 70, 547 (1990)]) to arbitrary fields and properties of the disorder, and compare the obtained predictions with the results of numerical simulations.The agreement between the two approaches is extremely good. In particular, we find that the appearance of a plateau in ${\mathit{I}}_{\mathit{c}}$(${\mathit{H}}_{\mathit{e}}$) for large fields depends strongly on the ratio r${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{0}$/\ensuremath{\lambda}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{j}}$ between the average pinning center size r${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{0}$ and the average Josephson penetration depth \ensuremath{\lambda}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{j}}$. If r${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{0}$/\ensuremath{\lambda}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{j}}$\ensuremath{\approxeq}1, there is no plateau at all, and in the case r${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{0}$/\ensuremath{\lambda}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathit{j}}$\ensuremath{\ll}1, a plateau is found up to fields for which the vortex spacing becomes of the order of r${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{0}$. Furthermore, we predict the possibility of a dimensional crossover from a 1D behavior at low fields to 0D behavior at large fields. Finally, we present a possible explanation of the experimentally observed plateau in the ${\mathit{j}}_{\mathit{c}}$(${\mathit{H}}_{\mathit{e}}$) dependence of granular high-${\mathit{T}}_{\mathit{c}}$ materials.