Long-junction Limit Research Articles

Magnetic field oscillations of the critical current in long ballistic graphene Josephson junctions

We study the Josephson current in long ballistic superconductor-monolayer graphene-superconductor junctions. As a first step, we have developed an efficient computational approach to calculate the Josephson current in tight-binding systems. This approach can be particularly useful in the long-junction limit, which has hitherto attracted less theoretical interest but has recently become experimentally relevant. We use this computational approach to study the dependence of the critical current on the junction geometry, doping level, and an applied perpendicular magnetic field $B$. In zero magnetic field we find a good qualitative agreement with the recent experiment of M. Ben Shalom et al. [Nat. Phys. 12, 318 (2016)] for the length dependence of the critical current. For highly doped samples our numerical calculations show a broad agreement with the results of the quasiclassical formalism. In this case the critical current exhibits Fraunhofer-like oscillations as a function of $B$. However, for lower doping levels, where the cyclotron orbit becomes comparable to the characteristic geometrical length scales of the system, deviations from the results of the quasiclassical formalism appear. We argue that due to the exceptional tunability and long mean free path of graphene systems a new regime can be explored where geometrical and dynamical effects are equally important to understand the magnetic field dependence of the critical current.

Parity dependent Josephson current through a helical Luttinger liquid

We consider a superconductor–two dimensional topological insulator–superconductor junction and study how the 2π- and 4π-periodic Josephson currents are affected by the electron–electron interaction. In the long-junction limit the supercurrent can be evaluated by modeling the system as a helical Luttinger liquid coupled to superconducting reservoirs. After having introduced bosonization in the presence of the parity constraint we turn to consider the limit of perfect and poor interfaces. For transparent interfaces, where perfect Andreev reflections occur at the boundaries, the Josephson current is marginally affected by the interaction. However, if strong magnetic scatterers are present in the weak link, the situation changes dramatically. Here Coulomb interaction plays a crucial role both in low and high temperature regimes. Furthermore, a phase-shift of Josephson current can be induced by changing the direction of the magnetization of the impurity.

Fermion-Parity Anomaly of the Critical Supercurrent in the Quantum Spin-Hall Effect

The helical edge state of a quantum spin-Hall insulator can carry a supercurrent in equilibrium between two superconducting electrodes (separation L, coherence length ξ). We calculate the maximum (critical) current I(c) that can flow without dissipation along a single edge, going beyond the short-junction restriction L << ξ of earlier work, and find a dependence on the fermion parity of the ground state when L becomes larger than ξ. Fermion-parity conservation doubles the critical current in the low-temperature, long-junction limit, while for a short junction I(c) is the same with or without parity constraints. This provides a phase-insensitive, dc signature of the 4 π-periodic Josephson effect.

Impurity-assisted Andreev reflection at a spin-active half metal-superconductor interface

The Andreev reflection amplitude at a clean interface between a half-metallic ferromagnet (H) and a superconductor (S) for which the half metal's magnetization has a gradient perpendicular to the interface is proportional to the excitation energy $\varepsilon$ and vanishes at $\varepsilon=0$ [B\'{e}ri {\em et al.}, Phys.\ Rev.\ B {\bf 79}, 024517 (2009)]. Here we show that the presence of impurities at or in the immediate vicinity of the HS interface leads to a finite Andreev reflection amplitude at $\varepsilon=0$. This impurity-assisted Andreev reflection dominates the low-bias conductance of a HS junction and the Josephson current of an SHS junction in the long-junction limit.

Proximity dc squids in the long-junction limit

We report the design and measurement of superconducting/normal/superconducting (SNS) proximity dc squids in the long-junction limit, i.e., superconducting loops interrupted by two normal metal wires, which are roughly a micrometer long. Thanks to the clean interface between the metals, a large supercurrent flows through the device at low temperature. The dc squidlike geometry leads to an almost complete periodic modulation of the critical current through the device by a magnetic flux, with a flux periodicity of a flux quantum $h/2e$ through the SNS loop. In addition, we examine the entire field dependence, notably the low and high field dependences of the maximum switching current. In contrast to the well-known Fraunhofer-type oscillations typical of wide junctions, we find a monotonous Gaussian extinction of the critical current at high field. As shown by Cuevas and co-workers [Phys. Rev. B 76, 064514 (2007); Phys. Rev. Lett. 99, 217002 (2007)], this monotonous dependence is typical of long and narrow diffusive junctions. We also find in some cases a puzzling reentrance at low field. In contrast, the temperature dependence of the critical current is well described by the proximity effect theory, as found by Dubos et al. [Phys. Rev. B 63, 064502 (2001); Phys. Rev. Lett. 87, 206801 (2001)] on SNS wires in the long-junction limit. The switching current distributions and hysteretic $IV$ curves also suggest interesting dynamics of long SNS junctions with an important role played by the diffusion time across the junction.

SUB-BRANCH SPLITTING IN JOSEPHSON-VORTEX-FLOW CHARACTERISTICS OF A STACK OF Bi2Sr2CaCu2O8+x INTRINSIC JOSEPHSON JUNCTIONS

We observed subbranch splitting in the low-bias vortex-flow branch of a dense-Josephson-vortex state in current-voltage characteristics of Bi 2 Sr 2 CaCu 2 O 8+x intrinsic Josephson junctions in the long-junction limit. We suggest that each subbranch corresponds to a plasma mode in serially coupled Josephson junctions, manifested in the Josephson vortex-flow current-voltage characteristics resonating with the plasma modes.

Collective resonance modes of Josephson vortices in a sandwiched stack ofBi2Sr2CaCu2O8+xintrinsic Josephson junctions

We observed splitting of the low-bias vortex-flow branch in a dense-Josephson-vortex state into multiple sub-branches in current-voltage characteristics of intrinsic Josephson junctions (IJJs) of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+x}$ single crystals in the long-junction limit. Each sub-branch corresponds to a plasma mode in serially coupled Josephson junctions. Splitting into low-bias linear sub-branches with a spread in the slopes and the inter-sub-branch mode-switching character are in good quantitative agreement with the prediction of the weak but finite inter-junction capacitive-coupling model incorporated with the inductive coupling. This suggests the importance of the role of the capacitive coupling in accurately describing the vortex dynamics in serially stacked IJJs.

Fluxon resonance steps in stacked intrinsic Josephson junctions of Bi 2Sr 2CaCu 2O 8 + x single crystals

We observed jumps in the low-bias fluxon-flow branch of current–voltage characteristics of serially stacked intrinsic Josephson junctions (IJJs) of Bi 2Sr 2CaCu 2O 8 + x (Bi-2212) single crystals in the long-junction limit. A stack of IJJs (1.5 μm wide and 17 μm long) was fabricated to be sandwiched between two Au electrodes. Measurements were made in relatively high magnetic fields up to 5 T. The jumps can be explained in terms of switching between different Josephson fluxon modes in resonance with the collective plasma oscillation modes, presumably induced by both inductive and capacitive coupling.

Infrared divergence in d.c. Josephson current through charge density wave

We calculate the d.c. Josephson current through charge density wave (CDW) region connected to two superconductors (S 1, S 2) by tunnel junctions. At T=0 and without impurities, we discover infrared divergence in the d.c. Josephson current caused by the phase mode in the long-junction limit: d/ v F≫ℏ/ Δ S, ℏ/ Δ CDW where T is temperature. These limits are expressed by using the energy gap in S region Δ S, the length d, the Fermi velocity v F, and the energy gap Δ CDW in CDW region. The critical current is proportional to ( Δ CDWℏ v F/ d) 1/2∝ Δ CDW( ξ CDW/ d) 1/2. At βΔ S( T), βΔ CDW( T)≫1 and βℏ v F/ d≪1, the T-dependence of the correction caused by the phase mode is T-linear where β=1/ k B T and k B is Boltzmann constant. At finite temperatures and with impurities, the correction caused by the phase mode becomes proportional to ln T . At finite temperatures and without impurities, the critical current is similar to the Ambegaokar–Baratoff formula which is proportional to Δ S( T) in the short-junction limit: d/ v F≪ℏ/ Δ S, ℏ/ Δ CDW.

Mesoscopic Fluctuations of the DC Josephson Current in a Dirty SNS Junction

We present theoretical studies on mesoscopic fluctuations of the dc Josephson current in superconductor (S)-dirty normal conductor (N)-superconductor junctions. The analytical expression for the Josephson-current fluctuation δ J 2 is obtained by using the thermal Green's function method, which is shown to be valid when the separation between two superconductors is much lager than the coherence length of the dirty superconductor. The effects of temperature and inelastic scattering on δ J 2 are examined. The results show that the characteristic temperature (size) for the reduction of δ J 2 is of the order of the Thouless correlation energy E c (the phase choherence length). From the current correlation function, the energy correlation range is estimated to be at about E c . In order to prepare a specific reference for the system under our consideration, we numerically examine the critical-current fluctuations δ J 2 c in the long-junction limit in a model system. Numerical results show that the magnitude of ...

Numerical Study on Josephson-Current Fluctuations in a Dirty SNS Junction: The Effect of Incomplete Andreev Reflection

Mesoscopic fluctuations of the DC Josephson current in a dirty SNS junction (S: superconductor, N: normal conductor) are studied numerically in the presence of barrier potentials at the NS interfaces, which reduce the Andreev reflection probability. The results show that the critical-current fluctuations in the long-junction limit are suppressed as the Andreev reflection probability decreases while those in the short-junction limit are relatively insensitive to reductions in the Andreev reflection probability. Therefore, the effect of incomplete Andreev reflection is significant in the long-junction limit. We show that the quantitative difference between our previous theory and the experimental results of Takayanagi et al. is well explained by taking the effect of incomplete Andreev reflection into account.