Subharmonic Shapiro Steps Research Articles

Shapiro steps and chaos in the Frenkel-Kontorova model with substrate lateral vibration.

Numerical simulations are used to examine the dynamics of the dc-driven Frenkel-Kontorova model with an oscillation substrate subjected to lateral periodic excitations in overdamped and underdamped cases, respectively. The results reveal that the system exhibits frequency locking and chaotic behaviors due to the fact that the lateral vibration of the substrate potential introduces an additional frequency and degree of freedom into the system. In the overdamped case, we show that the appearance of subharmonic Shapiro steps can be attributed to the deformation of the substrate potential or inertia. The characteristics of the steps are significantly affected by the amplitude and frequency of the lateral vibration. When the vibration frequency is relatively high, the change of the width of the first harmonic Shapiro step with increasing amplitude exhibits a Bessel-type oscillation, but the oscillation deviates from the Bessel curve at lower frequencies. In the amplitude dependence, although the oscillatory behavior of the critical depinning force at the high frequency is anomalous, local maxima (minima) of the first step width correspond to local minima (maxima) of the critical depinning force, and the largest Lyapunov exponent obtained in the pinned state represents a mirror relationship of the critical depinning force. In contrast to the overdamped system, the underdamped one exhibits both subharmonic Shapiro steps and chaotic behaviors. We show the increased inertia of the latter system plays an important role in suppressing the emergence of subharmonic steps, which is opposite to the result of the former. When the dc force changes, chaos appears not only between adjacent subharmonic Shapiro steps but also on some specific steps where chaos should be avoided. The variation regularity of the first step width and the critical depinning force is thus annihilated in vibration amplitude and frequency dependence. However, superlubricity can be achieved by careful adjustment of vibration amplitude and frequency. The findings can serve as a theoretical guideline for technological applications such as device building and voltage standards.

Subharmonic Shapiro steps in the a.c. Josephson effect for a three-dimensional Weyl semimetal WTe2

We experimentally study electron transport between two superconducting indium leads, coupled to a single WTe2 crystal, which is a three-dimensional Weyl semimetal. We demonstrate Josephson current in 5 μm long In-WTe2-In junctions, as confirmed by the observation of integer (1, 2, 3) and fractional (1/3, 1/2, 2/3) Shapiro steps under microwave irradiation. The demonstration of the fractional a.c. Josephson effect indicates the multivalued character of the current-phase relationship, which we connect with the Weyl topological surface states contribution to the Josephson current. In contrast to topological insulators and Dirac semimetals, we do not observe periodicity in the a.c. Josephson effect for WTe2 at different frequencies and power, which might reflect the chiral character of the Fermi arc surface states in a Weyl semimetal.

Features of Long YBCO Josephson Junctions Fabricated by Preliminary Topology Mask

We present the results of an experimental and theoretical study of YBCO long Josephson junctions on bicrystal YSZ substrates under the influence of an external magnetic field and under the combined action of a magnetic field and microwave radiation. Samples of YBCO films have been fabricated using the original technology of a preliminary topology mask, leading to critical current densities of order ${\text{0.6}}\,{{\text{MA}\cdot \text{cm}}}^{-2}$ at a temperature of 6 K. Detailed measurements of the samples were made in the temperature range of 6–90 K, and the wide variety of subharmonic Shapiro steps has been observed and investigated depending on the magnetic field.

The Influence of the McCumber Parameter on Parametric Amplification of High-Frequency Radiation by Josephson Junctions

The influence of the internal capacity of a Josephson junction on the parametric amplification of external electromagnetic radiation was studied in terms of a resistively and capacitively shunted junction model. The influence of the McCumber parameter on parametric amplification was clarified. It is shown that the additional regions of amplification can occur near subharmonic Shapiro steps in the case of Josephson junctions with internal capacity.

Subharmonic Shapiro steps of sliding colloidal monolayers in optical lattices

We investigate theoretically the possibility to observe dynamical mode locking, in the form of Shapiro steps, when a time-periodic potential or force modulation is applied to a two-dimensional (2D) lattice of colloidal particles that are dragged by an external force over an optically generated periodic potential. Here we present realistic molecular dynamics simulations of a 2D experimental setup, where the colloid sliding is realized through the motion of soliton lines between locally commensurate patches or domains, and where the Shapiro steps are predicted and analyzed. Interestingly, the jump between one step and the next is seen to correspond to a fixed number of colloids jumping from one patch to the next, across the soliton line boundary, during each ac cycle. In addition to ordinary ‘integer’ steps, coinciding here with the synchronous rigid advancement of the whole colloid monolayer, our main prediction is the existence of additional smaller ‘subharmonic’ steps due to localized solitonic regions of incommensurate layers executing synchronized slips, while the majority of the colloids remains pinned to a potential minimum. The current availability and wide parameter tunability of colloid monolayers makes these predictions potentially easy to access in an experimentally rich 2D geometrical configuration.

Josephson junction detectors for Majorana modes and Dirac fermions

We demonstrate that the current-voltage (I-V) characteristics of resistively and capacitively shunted Josephson junctions (RCSJs) hosting localized subgap Majorana states provides a phase sensitive method for their detection. The I-V characteristics of such RCSJs, in contrast to their resistively shunted counterparts, exhibit subharmonic odd Shapiro steps; such steps occur even in the absence of any $2\pi$ periodic terms in the current-phase relation of these junctions. These steps, owing to their subharmonic nature, exhibit qualitatively different properties compared to harmonic odd steps of conventional junctions. In addition, the RCSJs hosting Majorana bound states also display an additional sequence of steps in the devil staircase structure seen in their I-V characteristics; such sequence of steps make their I-V characteristics qualitatively distinct from that of their conventional counterparts. A similar study for RCSJs with graphene superconducting junctions hosting Dirac-like quasiparticles reveals that the Shapiro step width in their I-V curves bears a signature of the transmission resonance phenomenon of their underlying Dirac quasiparticles; consequently, these step widths exhibit a $\pi$ periodic oscillatory behavior with variation of the junction barrier potential. We discuss experiments which can test our theory.

Farey sequence in the appearance of subharmonic Shapiro steps.

The largest Lyapunov exponent has been examined in the dynamical-mode locking phenomena of the ac+dc driven dissipative Frenkel-Kontorova model with deformable substrate potential. Due to deformation, large fractional and higher order subharmonic steps appear in the response function of the system. Computation of the largest Lyapunov exponent as a way to verify their presence led to the observation of the Farey sequence. In the standard regime, the appearance of half-integer and other subharmonic steps between the large harmonic steps, and their relative sizes follow the Farey construction. In the nonstandard regime, however, the half-integer steps are larger than harmonic ones, and Farey construction is only present in the appearance of higher order subharmonic steps. The examination of Lyapunov exponents has also shown that regardless of the substrate potential or deformation, there was no chaos in the system.

Structured chaos in a devil's staircase of the Josephson junction

The phase dynamics of Josephson junctions under external electromagnetic radiation is studied through numerical simulations. Current-voltage characteristics, Lyapunov exponents and Poincare sections are analyzed in detail. It is found that the subharmonic Shapiro steps at certain parameters are separated by structured chaotic windows. By performing a linear regression on the linear part of the data, a fractal dimension of D = 0.868 is obtained, with an uncertainty of +-0.012. The chaotic regions exhibit scaling similarity and it is shown that the devil's staircase of the system can form a backbone that unifies and explains the highly correlated and structured chaotic behavior. These features suggest a system possessing multiple complete devil's staircases. The onset of chaos for subharmonic steps occurs through the Feigenbaum period doubling scenario. Universality in the sequence of periodic windows is also demonstrated. Finally the influence of the radiation and Josephson junction parameters on the structured chaos is investigated and it is concluded that the structured chaos is a stable formation over a wide range of parameter values.

Josephson effects in three-band superconductors with broken time-reversal symmetry

In superconductors with three or more bands, time-reversal symmetry (TRS) may be broken in the presence of repulsive interband couplings, resulting in a pair of degenerate states characterized by opposite chiralities. We consider a Josephson junction between a three-band superconductor with broken TRS and a single-band superconductor. Phenomena such as asymmetric critical currents, subharmonic Shapiro steps, and symmetric Fraunhhofer patterns are revealed theoretically. Existing experimental results are discussed in terms of the present work.

Size effect of the subharmonic Shapiro steps on the interference phenomena in the Frenkel-Kontorova model with realistic substrate potentials

Size effect of the subharmonic Shapiro steps on the behavior of harmonic ones is studied in the ac driven overdamped Frenkel-Kontorova model with different types of physically important realistic substrate potentials. The properties of harmonic Shapiro steps are directly correlated with the size of halfinteger steps; and according to that in their amplitude dependence, three distinctive types of behavior have been classified. Our results on other realistic systems show that these classifications are not universal, and that not only the size of halfinteger steps but also the type of the substrate potential determines the properties and correlation between the Shapiro steps. While for some potentials such as variable and double-barrier potential, system evolves through different types of amplitude dependence as the size of halfinteger steps changes, in the case of double-well potential, the system maintains the standard behavior (the behavior for small halfinteger steps) even in the presence of large halfinteger steps.

Publisher’s Note: Frequency dependence of the subharmonic Shapiro steps [Phys. Rev. E83, 056604 (2011)

Received 5 February 2013DOI:https://doi.org/10.1103/PhysRevE.87.029903©2013 American Physical Society

Effects of noise on interference phenomena in the presence of subharmonic Shapiro steps

Existence and properties of the Shapiro steps in the presence of noise have been studied in the ac driven overdamped Frenkel-Kontorova model with deformable substrate potential. The results have shown that the influence of noise on interference phenomena is determined by the presence and the size of half-integer steps. Since half-integer steps are more affected by noise, and the properties of harmonic steps are directly correlated with the size of half-integer steps, noise will have completely different effects on the harmonic steps if half-integer steps are present. As temperature increases, in the amplitude dependence of the step width, deviation from the well-known Bessel-like oscillations has been observed, and three different types of behavior have been classified. In the influence of noise on the frequency dependence of the steps, appearance of oscillations and strong influence of half-integer steps on their form have been observed.

Frequency dependence of the subharmonic Shapiro steps

Frequency dependence of the subharmonic Shapiro steps has been studied in the ac driven overdamped Frenkel-Kontorova model with deformable substrate potential. As potential gets deformed, in addition to the harmonic steps, subharmonic steps appear in the number and size that increase as the frequency of the external force increases. It was found that size of both harmonic and subharmonic steps strongly depend on the frequency where in the high-amplitude limit oscillatory dependence appears. When expressed as a function of period these oscillations of the step size with frequency have the same form as the oscillations of the step size with amplitude. Deformation of the potential has strong influence on these oscillations, and as in the case of amplitude dependence, with the increase of deformation, the same three distinctive types of behavior have been classified.

Subharmonic shapiro steps and assisted tunneling in superconducting point contacts.

We analyze the current in a superconducting point contact of arbitrary transmission in the presence of a microwave radiation. The interplay between the ac Josephson current and the microwave signal gives rise to Shapiro steps at voltages V = (m/n)h(omega)(r)/2e, where n,m are integer numbers and omega(r) is the radiation frequency. The subharmonic steps (n not equal 1) are a consequence of multiple Andreev reflections (MAR) and provide a signature of the peculiar ac Josephson effect at high transmission. Moreover, the dc current exhibits a rich subgap structure due to photon-assisted MARs.

Superconducting current–phase relation in high- Tc symmetrical bicrystal junction

The current–phase relation (CPR) of a YBa 2Cu 3O x Josephson junction (JJ) on a bicrystal sapphire substrate was determined using the dependences of the amplitudes of the harmonic and subharmonic Shapiro steps and the critical current dependences of the DC-SQUIDs on external magnetic field. The deviations of CPR from sinusoidal one, caused by asymmetric dc biasing, result in an appearance of subharmonic Shapiro steps and in asymmetric shape of the SQUID modulation of the I c( H) dependence. Precise measurements suppose sinusoidal CPR for symmetrical bicrystal junctions, while for asymmetric current flow in JJ the contribution of second harmonic in CPR increases monotonously.

Coherent low-energy charge transport in a diffusive S-N-S junction.

We have studied the current-voltage characteristics of diffusive mesoscopic Nb-Cu-Nb Josephson junctions with highly transparent Nb-Cu interfaces. We consider the low-voltage and high-temperature regime eV < epsilon(c) < k(B)T, where epsilon(c) is the Thouless energy. The observed excess current as well as the observed subharmonic Shapiro steps under microwave irradiation suggest the occurrence of low-energy coherent multiple Andreev reflection.

Phase dependence of the superconducting current in YBCO Josephson junctions on a bicrystal substrate

Dependences of the amplitudes of the harmonic and subharmonic Shapiro steps on an external monochromatic signal were used to study the current-phase dependence of high-temperature superconducting junctions on a bicrystal substrate. It is shown that for a symmetric definition of the transport current across the junction with an edge transparency of the order of \(\bar D = 10^{ - 4}\) and a mirror-symmetric bicrystal interface, the current-phase dependence is close to sinusoidal which differs from the theoretical predictions and is most likely caused by twinning of the high-temperature superconducting films of the electrodes forming the junction. A departure from symmetry in the definition of the transport current across the junction causes the current-phase dependence to deviate from sinusoidal, which increases with increasing degree of asymmetry. This change in the current-phase dependence is accurately described by a model which takes into account the formation of coupled Andreev states in junctions of superconductors with a \(d_{x^2 - y^2 } \) type of superconducting wave function.

High frequency detector response from an array of HTSC bicrystal Josephson junctions

YBaCuO Josephson junction arrays, fabricated on MgO bicrystal substrates were studied in wide frequency range at 7-26 GHz, 45-120 GHz, and at submillimeter 470-510 GHz waves. Subharmonic fractional Shapiro steps have been observed with amplitudes increasing with frequency of applied microwaves. Mm and submm wave harmonic and subharmonic selective detector response signals were registered when dc magnetic field was applied.

Subharmonic Shapiro steps and noise in high-T c superconductor Josephson junctions

An experimental investigation is made of the subharmonic Shapiro steps observed on the I-V curves of high-T c superconductor Josephson junctions and on the bias-voltage dependences of the rf noise and detector response when the junctions are subjected to external submillimeter radiation. Structures of this type are ordinarily described by a nonsinusoidal current-phase relation, which is why subharmonic steps appear. Numerical modeling of the processes occurring in a Josephson junction by means of a simple current-phase relation, as in the case of an SNS junction, gives good agreement with experiment. The width of the characteristic Josephson generation line of the junction was estimated on the basis of the noise dependences and the selective detector response. The width can be explained by taking into account the shot noise of the tunneling component of the conductivity. A model of the conductivity of a high-T c superconductor Josephson junction, consisting of a tunnel junction with microshorts possessing metallic conductivity, is discussed.

Spectral aspect of the onset of subharmonic Shapiro steps in a chain of Josephson junctions connected in parallel

A chain of parallel-connected Josephson elements described by a resistive model is considered as a model for high-temperature Josephson junctions. The calculated spectra of the Josephson generation can account for the experimentally observed subharmonic Sapiro steps on the current-voltage characteristic and for their aperiodic dependence on the external magnetic field.