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Physical Review B

Theoretical and numerical study on multistacked Josephson junctions with a parametric modulation of their critical currents

Xia Zhao, Shigeki Sakai

Published Oct 3, 2005• 3 citations
Publisher

Abstract

Flux-flow cavity-resonance properties in twofold Josephson junctions with different critical currents are investigated. While the critical current ${J}_{21}$ of one of junctions is fixed, the critical current ${J}_{10}$ of the other junction is parametrically increased up to extremely large values. Numerical simulations for inductively coupled twofold Josephson junctions are performed under several magnetic fields, and we observe resonance voltage steps at various ratios $M\ensuremath{\equiv}{J}_{10}∕{J}_{21}$. Obtained theoretical eigenequations include the parameter ${J}_{10}$ that is different from the equations by the existent conventional theory for the homogeneous system. Almost all data points on a resonance voltage vs $M$ domain obtained by the numerical simulations exist on the theoretical curves of eigenfrequency in voltage unit vs $M$. At $M\ensuremath{\rightarrow}\ensuremath{\infty}$ the twofold Josephson junctions can be regarded as a single junction. The lower critical-current $({J}_{21})$ junction works as a single junction with a new effective barrier thickness, and cavity resonance occurs in this single junction. At $M\ensuremath{\rightarrow}1$ the theoretical eigenfrequencies in the voltage unit asymptotically approach the resonance voltages of the antiphase and in-phase modes of the conventional system. For the range with intermediate $M$ values, there are no asymptotic approximate solutions, but the agreement of the results of the present theory with the numerical simulation results is excellent. The theory of the twofold Josephson junctions is able to extend to that of multifold junction with general index number $N$. At $N=9$, numerical simulations are performed and the agreement of its results with the theoretical results is good. At $M\ensuremath{\rightarrow}\ensuremath{\infty}$, the $N$-fold Josephson junction can also be regarded as a single junction with a new effective barrier thickness, ${d}_{N}^{\ensuremath{''}}$.

DOI
10.1103/physrevb.72.134501

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Flux-flow cavity-resonance properties in twofold Josephson junctions with different critical currents are investigated. While the critical current J21{J}_{21}J21​ of one of junctions is fixed, the critical current J10{J}_{10}J10​ of the other junction is parametrically increased up to extremely large values. Numerical simulations for inductively coupled twofold Josephson junctions are performed under several magnetic fields, and we observe resonance voltage steps at various ratios M\ensuremath≡J10∕J21M\ensuremath{\equiv}{J}_{10}∕{J}_{21}M\ensuremath≡J10​∕J21​. Obtained theoretical eigenequations include the parameter J10{J}_{10}J10​ that is different from the equations by the existent conventional theory for the homogeneous system. Almost all data points on a resonance voltage vs MMM domain obtained by the numerical simulations exist on the theoretical curves of eigenfrequency in voltage unit vs MMM. At M\ensuremath→\ensuremath∞M\ensuremath{\rightarrow}\ensuremath{\infty}M\ensuremath→\ensuremath∞ the twofold Josephson junctions can be regarded as a single junction. The lower critical-current (J21)({J}_{21})(J21​) junction works as a single junction with a new effective barrier thickness, and cavity resonance occurs in this single junction. At M\ensuremath→1M\ensuremath{\rightarrow}1M\ensuremath→1 the theoretical eigenfrequencies in the voltage unit asymptotically approach the resonance voltages of the antiphase and in-phase modes of the conventional system. For the range with intermediate MMM values, there are no asymptotic approximate solutions, but the agreement of the results of the present theory with the numerical simulation results is excellent. The theory of the twofold Josephson junctions is able to extend to that of multifold junction with general index number NNN. At N=9N=9N=9, numerical simulations are performed and the agreement of its results with the theoretical results is good. At M\ensuremath→\ensuremath∞M\ensuremath{\rightarrow}\ensuremath{\infty}M\ensuremath→\ensuremath∞, the NNN-fold Josephson junction can also be regarded as a single junction with a new effective barrier thickness, dN\ensuremath′′{d}_{N}^{\ensuremath{''}}dN\ensuremath′′​.
Flux-flow cavity resonances in intrinsic Josephson junctions (IJJs) with long c-axis periodicity by artificial critical-current (Jc) modulation are studied numerically and theoretically. For an n-1–n-1–n-1–n IJJ system with n-layer high-Jc and one-layer low-Jc alternately stacked, numerical simulation confirms fluxon penetration only in the low-Jc junctions under proper low magnetic fields. The simulation also shows pronounced cavity-resonance steps in the I–V curves of the low-Jc junctions, meaning that fluxon dynamics can be generated under much lower magnetic fields, compared to the fields for usual IJJs with homogeneous Jc. A theoretical method for describing the flux-flow cavity-resonance properties is presented. The general disperse k–ω relationship shows that, at low-k regions, the critical-current-modulated junction system can be regarded as simple homogeneous stacked junctions with a new effective thickness and a new inductive coupling strength. For general-k cases, the cavity-resonant voltage steps on the I–V curves at various magnetic fields can be well indexed by integers, which means excellent agreement between the theoretical analysis and the numerical simulations.
We describe a variety of models to characterize the behavior of several types of underdamped Josephson systems.We also present numerical results based on these models.We first discuss a stack of small junctions in a single mode resonant cavity.Such a stack is believed to be appropriate to model some high temperature cuprate superconductors such as Bi 2 Sr 2 CaCu 2 O 8+x .Our results show self-induced-resonant-steps in the current voltage (IV) curve.These steps represent a coherent phase locking between the junctions in the stack and the resonant cavity.Next we generalize these results to treat a single long junction coupled to a single mode resonant cavity.We find a number of features not seen in stacks of small junctions.Our numerical results show that if the electromagnetic fields of the cavity mode are spatially periodic, we again see steps in the IV curve and furthermore, for sufficiently large coupling, the cavity frequency is shifted by coupling to the long junction.We also discuss how a stack of long junctions could be used to model a layered high temperature superconductor.We then turn to some properties of single junctions in the "quantum" regime where the phase/number non-commutation is important.This quantum regime is crucial to all proposals to use such junctions as solid state quantum bits (qubits).We show that such qubits, when subjected to external a.c.perturbations, exhibit not only . . . . . . . . . .
Resonances in the current-voltage characteristics of interferometers with two Nb-NbOx-Pb window junctions have been experimentally investigated. The data are compared with the theoretical curves computed both for the voltage and for the magnetic field dependence of the resonance amplitudes. Very good agreement is found in all the range of values of the damping parameter Γ=I0 R/Vr, where Vr is the resonant voltage and I0 and R are the maximum Josephson current and the parallel resistance of each junction. New features that cannot be explained in terms of the existing theory have been observed in devices with larger values of the parameter β=2πLI0/Φ0, where L is the loop inductance, and Φ0 is the flux quantum.
A system of two vertically stacked Nb/Al-AlOx/Nb long Josephson junctions is studied experimentally. The magnetic coupling between the junctions is provided by their common electrode of the thickness d∼λL. The current-voltage characteristics of the stack displays resonant steps which are understood to be the phase-locked flux-flow resonances of the two junctions. These modes can be tuned by the magnetic field in a broad voltage range which corresponds to the Josephson frequencies between 150 and 600 GHz. Our observations suggest a possibility of application of the stacked long Josephson junctions as local oscillators for millimeter and submillimeter wave circuits.
We investigated numerically IV-characteristics and power of emission from stacks with various quantities of long Josephson junctions (up to 6 junctions) which interacted inductively with each other. Parameters of junctions were chosen close to those for MoRe-Si(W)-MoRe heterostructures. We set Gaussian spread of about 0.01% of critical currents along junctions. Electrical properties of a stack consisted of three junctions was investigated in details. Zero-fi eld steps at voltages corresponding to frequencies of various modes of electromagnetic waves in the stack were found in the IV-characteristic. We showed that positions of zero-fi eld steps in IV-curves were in good agreement with predictions of the theory. The highest maximum of power of emission corresponded to the so-called in-phase mode at which all voltages over junctions in the stack oscillate in-phase. Considering stacks with many junctions, we showed that power of emission at the voltage of the resonance which corresponds to the in-phase mode is proportional to the square of quantity of long junctions in the stack that is the characteristic of coherent emission.
During last decade, considerable efforts were made to achieve coherent emission from stacks of many Josephson junctions. It is known that strong emission from a junction in the presence of external magnetic field appears at the so-called Fiske steps in the IV-characteristic at voltages which correspond to frequencies of geometrical resonances. However, it is possible to obtain resonant steps in long junctions without external magnetic field. The periodical movement of fluxons is excited due to some disorder in the distribution of critical currents along junctions. The so-called zero-field steps are formed in the IV-curve due to the interaction of fluxons with oscillations of voltage at Josephson frequencies. We investigated numerically IV-characteristics and the dependence of the average square of ac voltage at the end of the stack of two long Josephson junctions on the average voltage. Junctions interacted inductively with each other. We introduced not only the Gaussian distribution of critical currents along junctions but also the Gaussian distribution of coefficients of the interaction between junctions (mutual inductances). Zero-field steps in the IV-characteristic were found at voltages which corresponded to frequencies of in-phase collective modes in the stack as well as to frequencies of uncoupled junctions. Zero-field steps appeared in the hysteretic region of the IV-curve. There appeared also jumps of voltage from the resistive branch to the zero-field step. We showed that there existed distributions of mutual inductances along junctions which provided jumps to voltages at which the average square of ac voltage at the end of the stack (which is proportional to power of emission) was larger than that for the stack with the uniform distribution of mutual inductances.