Josephson Current-phase Relation Research Articles

From nonreciprocal to charge-4e supercurrent in Ge-based Josephson devices with tunable harmonic content

Hybrid superconductor(S)-semiconductor(Sm) devices bring a range of functionalities into superconducting circuits. In particular, hybrid parity-protected qubits and Josephson diodes were recently proposed and experimentally demonstrated. Such devices leverage the nonsinusoidal character of the Josephson current-phase relation (CPR) in highly transparent S-Sm-S junctions. Here, we report an experimental study of superconducting quantum-interference devices (SQUIDs) embedding Josephson field-effect transistors fabricated from a SiGe/Ge/SiGe heterostructure grown on a 200-mm silicon wafer. The single-junction CPR shows up to three harmonics with gate-tunable amplitude. In the presence of microwave irradiation, the ratio of the first two dominant harmonics, corresponding to single and double Cooper-pair transport processes, is consistently reflected in relative weight of integer and half-integer Shapiro steps. A combination of magnetic-flux and gate-voltage control enables tuning the SQUID functionality from a nonreciprocal Josephson-diode regime with 27% asymmetry to a π-periodic Josephson regime suitable for the implementation of parity-protected superconducting qubits. These results illustrate the potential of Ge-based hybrid devices as versatile and scalable building blocks of superconducting quantum circuits. Published by the American Physical Society 2024

Light-modulated graphene-based φ 0 Josephson junction and −φ 0 to φ 0 transition

We investigate the chiral edge states-induced Josephson current–phase relation in a graphene-based Josephson junction modulated by the off-resonant circularly polarized light and the staggered sublattice potential. By solving the Bogoliubov–de Gennes equation, a φ 0 Josephson junction is induced in the coaction of the off-resonant circularly polarized light and the staggered sublattice potential, which arises from the fact that the center of-mass wave vector of Cooper pair becomes finite and the opposite center of-mass wave vector to compensate is lacking in the nonsuperconducting region. Interestingly, when the direction of polarization of light is changed, −φ 0 to φ 0 transition generates, which generalizes the concept of traditional 0–π transition. Our findings provide a purely optical way to manipulate a phase-controllable Josephson device and guidelines for future experiments to confirm the presence of graphene-based φ 0 Josephson junction.

Anomalous Josephson current through a driven double quantum dot

Josephson junctions based on quantum dots offer a convenient tunability by means of local gates. Here we analyze a Josephson junction based on a serial double quantum dot in which the two dots are individually gated by phase-shifted microwave tones of equal frequency. We calculate the time-averaged current across the junction and determine how the phase shift between the drives modifies the current-phase relation of the junction. Breaking particle-hole symmetry on the dots is found to give rise to a finite average anomalous Josephson current with phase bias between the superconductors fixed to zero. This microwave gated weak link thus realizes a tunable ``Floquet ${\ensuremath{\varphi}}_{0}$ junction'' with maximum critical current achieved for driving frequencies slightly off resonance with the subgap excitation energy. We provide numerical results supported by an analytical analysis for infinite superconducting gap and weak interdot coupling. We identify an interaction-driven $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition of anomalous Josephson current as a function of driving phase difference. Finally, we show that this junction can be tuned so as to provide for complete rectification of the time-averaged Josephson current-phase relation.

Interference phenomena in Josephson junctions with ferromagnetic bilayers: Spin-triplet correlations and resonances

We study the Josephson effect in planar $\mathrm{S}{\mathrm{F}}_{1}{\mathrm{F}}_{2}\mathrm{S}$ junctions that consist of conventional $s$-wave superconductors (S) connected by two metallic monodomain ferromagnets (${\mathrm{F}}_{1}$ and ${\mathrm{F}}_{2}$) with an arbitrary transparency of interfaces. We solve the scattering problem in the clean limit based on the Bogoliubov--de Gennes equation for both spin-singlet and odd in frequency spin-triplet pairing correlations. We calculate numerically the Josephson current-phase relation $I(\ensuremath{\phi})$. While the first harmonic of $I(\ensuremath{\phi})$ is completely generated by spin-singlet and short-range spin-triplet superconducting correlations, for noncollinear magnetizations of ferromagnetic layers the second harmonic has an additional long-range spin-triplet component. Therefore, for a strong ferromagnetic influence, the long-range spin-triplet contribution to the second harmonic dominates. We find an exception due to the geometric resonance for equal ferromagnetic layers when the first harmonic is strongly enhanced. Both first and second harmonic amplitudes oscillate with ferromagnetic layer thicknesses due to $0\text{\ensuremath{-}}\ensuremath{\pi}$ transitions. We study the influence of interface transparencies and find additional resonances for finite transparency of the interface between ferromagnetic layers.

Microwave response of NbSe2 van der Waals Josephson junctions

van der Waals Josephson junctions utilize the interface between two superconducting ${\mathrm{NbSe}}_{2}$ crystal flakes as a tunnel barrier. We fabricate such junctions by a dry-transfer procedure and characterize them by electric transport experiments at various temperatures and under microwave irradiation at frequencies up to 224 GHz. We observe integer Shapiro steps in the current-voltage characteristics indicating a sinusoidal Josephson current-phase relation and the possibility to use this kind of junction even in the terahertz frequency range.

Quantum phase transitions in superconductor–quantum-dot–superconductor Josephson structures with attractive intradot interaction

We theoretically study the superconducting proximity effect in a quantum dot coupled to two superconducting leads when the intradot interaction between electrons is made attractive. Because of the superconducting proximity effect, the electronic states for the embedded quantum dot are either spin-polarized states with an odd occupation number or BCS-like states with an even occupation number. We show that in the presence of an external magnetic field, the system can exhibit quantum phase transitions of fermion parity associated with the occupation number. In this work, we adopt a self-consistent theoretical method to extend our considerations beyond the so-called superconducting atomic limit in which the superconducting gap for the leads is assumed to be the largest energy scale. The method enables us to numerically investigate the electronic structure of the dot as results of the attractive interaction. For energy phase diagrams in the regime away from the atomic limit, we find a reentrant behavior where a BCS-like phase of the dot exists in an intermediate range of the hybridization strength between the quantum dot and the leads. We also consider Josephson current phase relations and identify a number of examples showing $0-\pi$ phase transitions that may offer important switching effects.

A topological Josephson junction platform for creating, manipulating, and braiding Majorana bound states

As part of the intense effort toward identifying platforms in which Majorana bound states can be realized and manipulated to perform qubit operations, we propose a topological Josephson junction architecture that achieves these capabilities and which can be experimentally implemented. The platform uses conventional superconducting electrodes deposited on a topological insulator film to form networks of proximity-coupled lateral Josephson junctions. Magnetic fields threading the network of junction barriers create Josephson vortices that host Majorana bound states localized in the junction where the local phase difference is an odd multiple of π, i.e. attached to the cores of the Josephson vortices. This enables us to manipulate the Majorana states by moving the Josephson vortices, achieving functionality exclusive to these systems in contrast to others, such as those composed of topological superconductor nanowires. We describe protocols for: (1) braiding localized Majorana states by exchange, (2) controlling the separation and hence the coupling of adjacent localized Majorana states to effect non-Abelian rotations via hybridization of the Majorana modes, and (3) reading out changes in the non-local parity correlations induced by such operations. These schemes utilize current pulses and local magnetic field pulses to control the location of vortices, and measurements of the Josephson current–phase relation to reveal the presence of the Majorana bound states. Finally, we present brief discussions of readout schemes and viable experimental settings for realizing the platform.

Electrical control of magnetization in superconductor/ferromagnet/superconductor junctions on a three-dimensional topological insulator

Strong dependence of the Josephson energy on the magnetization orientation in Josephson junctions with ferromagnetic interlayers and spin-orbit coupling opens a way to control magnetization by Josephson current or Josephson phase. Here we investigate the perspectives of magnetization control in superconductor/ferromagnet/superconductor (S/F/S) Josephson junctions on the surface of a 3D topological insulator hosting Dirac quasiparticles. Due to the spin-momentum locking of these Dirac quasiparticles a strong dependence of the Josephson current-phase relation on the magnetization orientation is realized. It is demonstrated that this can lead to splitting of the ferromagnet's easy axis in the voltage driven regime. We show that such a splitting can lead to stabilization of an unconventional fourfold degenerate ferromagnetic state.

Thermoelectric Effect in Two-Band Single-Junction Superconducting Rings

Based on Ginzburg-Landau theory, we study the thermoelectric effect of two-band superconducting ring of inductance L with a microbridge structure. The phase difference of two order parameters in the ring satisfies the sine-Gordon equation, and from its soliton solution and the Josephson current-phase relation in the two-band microbridge, we obtain the analytical expression of the generated magnetic flux as a function of applied temperature gradient. Our results show that in the regime LIc > Φ0/2π with Ic the critical supercurrent in the microbridge and Φ0 = h/2e, this single-junction system will exhibit the irreversible behavior and the induced fractional flux transition can be easily observed in the typical two-band superconductor MgB2.

Chiral current-phase relation of topological Josephson junctions: A signature of the 4π -periodic Josephson effect

The $4\ensuremath{\pi}$-periodic Josephson effect is an indicator of Majorana zero modes and a ground-state degeneracy which are central to topological quantum computation. However, the observability of a $4\ensuremath{\pi}$-periodic Josephson current-phase relation (CPR) is hindered by the necessity to fix the fermionic parity. As an alternative to a $4\ensuremath{\pi}$-periodic CPR, this paper proposes a chiral CPR for the $4\ensuremath{\pi}$-periodic Josephson effect. This is a CPR of the form $J(\ensuremath{\phi})\ensuremath{\propto}C\phantom{\rule{0.16em}{0ex}}|sin(\ensuremath{\phi}/2)|$, describing a unidirectional supercurrent with the chirality $C=\ifmmode\pm\else\textpm\fi{}1$. Its nonanalytic dependence on the Josephson phase difference $\ensuremath{\phi}$ translates into the $4\ensuremath{\pi}$-periodic CPR $J(\ensuremath{\phi})\ensuremath{\propto}sin(\ensuremath{\phi}/2)$. The proposal requires a spin-polarized topological Josephson junction which is modeled here as a short link between spin-split superconducting channels at the edge of a two-dimensional topological insulator. In this case, $C$ coincides with the Chern number of the occupied spin band of the topological insulator. This paper details three scenarios of achieving a chiral CPR: By only Zeeman-like splitting, by Zeeman splitting combined with bias currents, and by an external out-of-plane magnetic field.

Fractional magnetic flux transition in two-band single-junction superconducting rings

Based on Ginzburg–Landau (GL) theory, we study the electromagnetic properties of two-band superconducting ring with a microbridge structure. The phase difference of two-order parameters in two-band superconductors satisfies the sine-Gordon equation, and from its soliton solution, the linear relation between the phase difference at both ends of the junction and total magnetic flux in the ring can be obtained. Then with the Josephson current-phase relation in the two-band superconducting microbridge, we establish the dependence of the circulating current and total magnetic flux on the applied external magnetic field. Our analysis shows that the soliton solution and the fractional flux quantization in two-band superconductors can be verified by measuring the properties of this single-junction structure.

Josephson effect in junctions of conventional and topological superconductors.

We present a theoretical analysis of the equilibrium Josephson current-phase relation in hybrid devices made of conventional s-wave spin-singlet superconductors (S) and topological superconductor (TS) wires featuring Majorana end states. Using Green’s function techniques, the topological superconductor is alternatively described by the low-energy continuum limit of a Kitaev chain or by a more microscopic spinful nanowire model. We show that for the simplest S–TS tunnel junction, only the s-wave pairing correlations in a spinful TS nanowire model can generate a Josephson effect. The critical current is much smaller in the topological regime and exhibits a kink-like dependence on the Zeeman field along the wire. When a correlated quantum dot (QD) in the magnetic regime is present in the junction region, however, the Josephson current becomes finite also in the deep topological phase as shown for the cotunneling regime and by a mean-field analysis. Remarkably, we find that the S–QD–TS setup can support φ0-junction behavior, where a finite supercurrent flows at vanishing phase difference. Finally, we also address a multi-terminal S–TS–S geometry, where the TS wire acts as tunable parity switch on the Andreev bound states in a superconducting atomic contact.

6π Josephson Effect in Majorana Box Devices.

We study Majorana devices featuring a competition between superconductivity and multichannel Kondo physics. Our proposal extends previous work on single-channel Kondo systems to a topologically nontrivial setting of a non-Fermi liquid type, where topological superconductor wires (with gap Δ) represent leads tunnel coupled to a Coulomb-blockaded Majorana box. On the box, a spin degree of freedom with Kondo temperature T_{K} is nonlocally defined in terms of Majorana states. For Δ≫T_{K}, the destruction of Kondo screening by superconductivity implies a 4π-periodic Josephson current-phase relation. Using a strong-coupling analysis in the opposite regime Δ≪T_{K}, we find a 6π-periodic Josephson relation for three leads, with critical current I_{c}≈eΔ^{2}/ℏT_{K}, corresponding to the transfer of fractionalized charges e^{*}=2e/3.

Signatures of topological phase transitions in Josephson current-phase discontinuities

Topological superconductors differ from topologically trivial ones for the presence of topologically protected zero-energy modes. To date, experimental evidence of topological superconductivity in nanostructures has been mainly obtained by measuring the zero-bias conductance peak via tunneling spectroscopy. Here, we propose an alternative and complementary experimental recipe to detect topological phase transitions in these systems. We show in fact that, for a finite-sized system with broken time-reversal symmetry, discontinuities in the Josephson current-phase relation correspond to the presence of zero-energy modes and to a change in the fermion parity of the groundstate. Such discontinuities can be experimentally revealed by a characteristic temperature dependence of the current, and can be related to a finite anomalous current at zero phase in systems with broken phase-inversion symmetry.

Electric control of the Josephson current-phase relation in a topological circuit

We study the current-phase relation of a topological ring-shape Josephson junction, where the ring structure is defined by one-dimensional topological interface states constructed in a two-dimensional honeycomb-lattice system. We show that control of the potential difference between the two ring arms can lead to a ${\ensuremath{\varphi}}_{0}$ Josephson junction. The physics origin is the superconducting electron- and holelike quasiparticles possessing a valley-dependent chirality and moving separately in the two ring arms. Our findings provide a purely electric way to consecutively manipulate the Josephson current-phase relation.

Modeling and simulations of quantum phase slips in ultrathin superconducting wires

We study quantum phase slips (QPS) in ultrathin superconducting wires. Starting from an effective one-dimensional microscopic model, which includes electromagnetic fluctuations, we map the problem to a (1+1)-dimensional gas of interacting instantons. We introduce a method to calculate the tunneling amplitude of quantum phase slips directly from Monte Carlo simulations. This allows us to go beyond the dilute instanton gas approximation and study the problem without any limitations of the density of QPS. We find that the tunneling amplitude shows a characteristic scaling behavior near the superconductor-insulator transition. We also calculate the voltage-charge relation of the insulating state, which is the dual of the Josephson current-phase relation in ordinary superconducting weak links. This evolves from a sinusoidal form in the regime of dilute QPS to more exotic shapes for higher QPS densities, where interactions are important.

Anomalous Josephson current, incipient time-reversal symmetry breaking, and Majorana bound states in interacting multilevel dots

We study the combined effects of spin-orbit interaction, magnetic field, and Coulomb charging on the Josephson current-phase relation, $I(\ensuremath{\varphi})$, for a multilevel quantum dot tunnel contacted by two conventional $s$-wave superconductors with phase difference $\ensuremath{\varphi}$. A general model is formulated and analyzed in the cotunneling regime (weak tunnel coupling) and in the deep subgap limit, fully taking into account interaction effects. We determine the conditions for observing a finite anomalous supercurrent ${I}_{a}=I(\ensuremath{\varphi}=0)$. For a two-level dot with spin-orbit coupling and arbitrarily weak Zeeman field $B$, we find the onset behavior ${I}_{a}\ensuremath{\propto}\mathrm{sgn}(B)$ in the presence of interactions, suggesting the incipient spontaneous breakdown of time-reversal symmetry. We also provide conditions for realizing spatially separated (but topologically unprotected) Majorana bound states in a double dot variant of this system. Here, Majoranas are predicted to leave a clear signature in the $2\ensuremath{\pi}$-periodic current-phase relation.

Anomalous Josephson current in superconducting topological insulator

We investigate the effect of helical Majorana fermions at the surface of superconducting topological insulators (STI) on the Josephson current by referring to possible pairing states of Cu-doped Bi2Se3. The surface state in the present STI has a spin helicity because the directions of spin and momentum are locked to each other. The Josephson current-phase relation in an STI/s-wave superconductor junction shows robust sin(2{\phi}) owing to mirror symmetry, where \phi denotes the macroscopic phase difference between the two superconductors. In contrast, the maximum Josephson current in an STI/STI junction exhibits a nonmonotonic temperature dependence depending on the relative spin helicity of the two surface states. Detecting these features qualifies as distinct experimental evidence for the identification of the helical Majorana fermion in STIs.

Coherent quantum transport of charge density waves

Recent experiments show oscillations of dominant period $h/2e$ in conductance vs magnetic flux of charge density wave (CDW) rings above 77 K, revealing macroscopically observable quantum behavior. The time-correlated soliton tunneling model discussed here is based on coherent, Josephson-like tunneling of microscopic quantum solitons of charge $2e$. The model interprets the CDW threshold electric field as a Coulomb blockade threshold for soliton pair creation, often much smaller than the classical depinning field but with the same impurity dependence (e.g., $\ensuremath{\sim}{n}_{i}^{2}$ for weak pinning). This picture draws upon the theory of time-correlated single-electron tunneling to interpret CDW dynamics above threshold. Similar to Feynman's derivation of the Josephson current-phase relation for a superconducting tunnel junction, the picture treats the Schr\"odinger equation as an emergent classical equation to describe the time-evolution of Josephson-coupled order parameters related to soliton dislocation droplets. Vector or time-varying scalar potentials can affect the order parameter phases to enable magnetic quantum interference in CDW rings or lead to interesting behavior in response to oscillatory electric fields. The ability to vary both magnitudes and phases is an aspect important to future applications in quantum computing.

Signature of the long range triplet proximity effect in the density of states

We study the impact of the long range spin-triplet proximity effect on the density of states (DOS) in planar SF${}_{1}$F${}_{2}$S Josephson junctions that consist of conventional superconductors (S) connected by two metallic monodomain ferromagnets (F${}_{1}$ and F${}_{2}$) with transparent interfaces. We determine the electronic DOS in F layers for arbitrary orientation of the magnetizations using the solutions of Eilenberger equations in the clean limit and for a moderate concentration of impurities in ferromagnets. We find that a fully developed long range proximity effect occurs in Josephson junctions with a highly asymmetric ferromagnetic bilayer. For orthogonal magnetizations, the effect manifests itself as an enhancement in DOS and as a dominant second harmonic in the Josephson current-phase relation. Distinctive variation of DOS in ferromagnets with the angle between magnetizations is experimentally observable by tunneling spectroscopy. This can provide an unambiguous signature of the long range spin-triplet proximity effect.