The peculiarities of the phase dynamics of stacks of coupled Josephson junctions with topologically trivial and nontrivial barriers have been investigated numerically and their comparative analysis is carried out. The effect of coupling and dissipation parameters on the parametric resonance in the breakpoint region is shown. It is found that the dependence of the breakpoint voltage on the dissipation parameter demonstrates a minimum due to the changing of frequency of longitudinal plasma wave. We have shown that in case of the stack with nontrivial barriers the observed minimum shifts along the $β$ to the $\sqrt{2}$ in comparison with trivial barriers stack. We assume that the found features may be used for the experimental determination of Majorana fermions in the stack of JJs with the nontrivial barriers.
We study the breakpoint region on the outermost branch of current-voltage characteristics of the stacks with different number of intrinsic Josephson junctions. We show that at periodic boundary conditions the breakpoint region is absent for stacks with even number of junctions. For stacks with odd number of junctions and for stacks with nonperiodic boundary conditions the breakpoint current is increased with number of junctions and saturated at the value corresponding to the periodic boundary conditions. The region of saturation and the saturated value depend on the coupling between junctions.…
Eutectic Sr$_2$RuO$_4$-Ru samples with $μ$m-sized Ru-metal inclusions support inhomogeneous superconductivity above the bulk transition of Sr$_2$RuO$_4$ in the so-called 3-Kelvin phase. In Pb/Ru/Sr$_2$RuO$_4$ Josephson junctions as realized by Maeno et al., a Pb film is indirectly coupled to the superconductor Sr$_2$RuO$_4$ mediated by the proximity-induced superconducting Ru-inclusions, yielding an extended Josephson contact through the interface between Ru and Sr$_2$RuO$_4$. Motivated by this experimental setup, we formulate a sine-Gordon model for the Josephson phase of the interface, assum…
We study Josephson junctions made of semiconducting nanowires with Rashba spin-orbit coupling, where superconducting correlations are induced by the proximity effect. In the presence of a suitably directed magnetic field, the system displays the anomalous Josephson effect: a nonzero supercurrent in the absence of a phase bias between two superconductors. We show that this anomalous current can be increased significantly by tuning the nanowire into the helical regime. In particular, in a short junction, a large anomalous current is a signature for topologically nontrivial superconductivity in t…
A planar Josephson junction with a normal metal attached on its top surface will form a hollow nanowire structure due to its three dimensional nature. In such hollow nanowire structure, the magnetic flux induced by a small magnetic field (about 0.01T) will tune the system into topologically non-trivial phase and therefore two Majorana zero-modes will form at the ends of the nanowire. Through tuning the chemical potential of the normal metal, the topologically non-trivial phase can be obtained for almost all energy within the band. Furthermore, the system can be conveniently tuned between the t…
Josephson vortices are imaged magneto-optically due to their decoration with pancake vortices in $Bi_{2}Sr_{2}CaCu_{2}O_{8+d}$ single crystals. Peculiarities of interaction between the pancake and Josephson vortices (JV) depending on the values of crossing fields and temperature are studied based on the observations of these decoration patterns. Evidences of the period-doubling in the decoration patterns compared to the JV stack period, migration of JV lines between neighboring stacks, and transitions between different JV configurations are reported. Imaging of the pancake/Josephson vortex dec…
We present a unified description of a junction between $s$-wave (BCS) superconductors and a junction between $p$-wave superconductors in a topologically nontrivial phase, which relies on a scattering state expansion. We compute Josephson current and thermal noise in the two kinds of junction and exhibit some characteristic features for a junction of two topological superconductors hosting Majorana zero-energy modes.
We discuss the Josephson current between two noncentrosymmetric superconductors. The coexistence of superconducting order parameters between spin-singlet $Δ_{\text{S}}$ and helical p-wave spin-triplet $Δ_{\text{T}}$ enriches a variety of low-temperature behavior of Josephson current depending on their relative amplitudes. We will show that characteristic behaviors of the Josephson current for $Δ_{\text{S}} > Δ_{\text{T}}$ are clearly different from those for $Δ_{\text{S}} < Δ_{\text{T}}$. The topologically protected zero-energy surface bound states are responsible for the clear difference. We …
We study the peculiarities in current-phase relations (CPR) of the SIsFS junction in the region of $0$ to $π$ transition. These CPR consist of two independent branches corresponding to $0-$ and $π-$ states of the contact. We have found that depending on the transparency of the SIs tunnel barrier the decrease of the s-layer thickness leads to transformation of the CPR shape going in the two possible ways: either one of the branches exists only in discrete intervals of the phase difference $φ$ or both branches are sinusoidal but differ in the magnitude of their critical currents. We demonstrate …
We study theoretically the Josephson effect in junctions based on unconventional superconductors with diffusive barriers, using the quasiclassical Green's function formalism. Generalized boundary conditions at junction interfaces applicable to unconventional superconductors are derived by calculating a matrix current within the circuit transport theory. Applying these boundary conditions, we have calculated the Josephson current in structures with various pairing symmetries. A number of predictions are made: (a) nonmonotonic temperature dependence in d-wave superconductor/diffusive normal meta…
It is widely believed that superconducting junctions involving topological insulators and hosting Majorana-like bound states may exhibit unusual "fractional" ($4π$-periodic) ac Josephson effect. Accordingly, "fractional" Shapiro steps on the current-voltage characteristics of such junctions are expected to occur under external microwave radiation. Here, we microscopically evaluate Shapiro steps in topologically trivial highly transparent superconducting weak links. The key features recovered within our analysis -- including, e.g., the so-called "missing" Shapiro steps -- turn out to be similar…
In the tunnel limit, the current-phase relation of Josephson junctions can be expanded as $I(ϕ)=I_1\sinϕ+I_2\sin 2ϕ$. Standard BCS theory predicts that $I_1R_N\simΔ/e$ and $I_2/I_1\sim D$, where $R_N$ is the resistance of the junction in the normal state, $Δ$ is the superconducting gap, and $D\ll 1$ is the junction transparency. In the cuprates, the experimental value of $I_1R_N$ ($I_2/I_1$) is much smaller (larger) than the BCS prediction. We argue that both peculiarities of the cuprates can be explained by postulating quantum fluctuations of the pairing symmetry.
The phase-dependent bound states (Andreev levels) of a Josephson junction can cross at the Fermi level, if the superconducting ground state switches between even and odd fermion parity. The level crossing is topologically protected, in the absence of time-reversal and spin-rotation symmetry, irrespective of whether the superconductor itself is topologically trivial or not. We develop a statistical theory of these topological transitions in an N-mode quantum-dot Josephson junction, by associating the Andreev level crossings with the real eigenvalues of a random non-Hermitian matrix. The number …
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 prese…
We argue that the frustrated Josephson junction arrays may support a topologically ordered superconducting ground state, characterized by a non-trivial ground state degeneracy on the torus. This superconducting quantum fluid provides an explicit example of a system in which superconductivity arises from a topological mechanism rather than from the usual Landau-Ginzburg mechanism.
The rise of topological phases of matter is strongly connected to their potential to host Majorana bound states, a powerful ingredient in the search for a robust, topologically protected, quantum information processing. In order to produce such states, a method of choice is to induce superconductivity in topological insulators. The engineering of the interplay between superconductivity and the electronic properties of a topological insulator is a challenging task and it is consequently very important to understand the physics of simple superconducting devices such as Josephson junctions, in wh…
The Josephson effect describes the generic appearance of a supercurrent in a weak link between two superconductors. Its exact physical nature however deeply influences the properties of the supercurrent. Detailed studies of Josephson junctions can reveal microscopic properties of the superconducting pairing (spin-triplet correlations, $d$-wave symmetry) or of the electronic transport (quantum dot, ballistic channels). In recent years, considerable efforts have focused on the coupling of superconductors to topological insulators, in which transport is mediated by topologically protected Dirac s…
We compute the current voltage characteristic of a chain of identical Josephson circuits characterized by a large ratio of Josephson to charging energy that are envisioned as the implementation of topologically protected qubits. We show that in the limit of small coupling to the environment it exhibits a non-monotonous behavior with a maximum voltage followed by a parametrically large region where $V\propto 1/I$. We argue that its experimental measurement provides a direct probe of the amplitude of the quantum transitions in constituting Josephson circuits and thus allows their full characteri…
We study the Josephson effect in small one-dimensional (1D) Josephson junction arrays. For weak Josephson tunneling, topologically different regions in the charge-stability diagram generate distinct current-phase relationships (I$Φ$). We present results for a three-junction system in the vicinity of charge degeneracy lines and triple points. We explain the generalization to larger arrays, show that discontinuities of the I$Φ$ at phase $π$ persist and that, at maximum degeneracy, the problem can be mapped to a tight-binding model providing analytical results for arbitrary system size.
Topological insulators (TIs) hold great promise for realizing zero-energy Majorana states in solid-state systems. Recently, several groups reported experimental data suggesting that signatures of Majorana modes in topological insulator Josephson junctions (TIJJs) have -- indeed -- been observed. To verify this claim, one needs to study the topological properties of low-energy Andreev-bound states (ABS) in TIs of which the Majorana modes are a special case. It has been shown theoretically that topologically non-trivial low-energy ABS are also present in TIJJs with doped topological insulators u…
The $π$-ring qubit array is described using quasiclassical approaches that are shown to be accurate and give clarity to the complex energy landscape of connected vortex qubits. Using the techniques, large arrays of Josephson junction systems can be designed, including phase shift devices. Herein, connected arrays of loops containing $π$ junctions are described. These techniques are useful for design of quantum computers based on superconducting technologies, hybrid quantum technologies and quantum networks.
We investigate phase-coherent thermal transport in Josephson junctions made from unconventional superconductors or superfluids. The thermal conductance is evaluated for one- and two-dimensional junctions within the Bogoliubov-de Gennes formalism. We analyze three different scenarios of junctions between two triplet superconductors with (i) helical pairing, (ii) unitary chiral and (iii) nonunitary chiral pairing. We find that the phase-dependent thermal conductance allows us to distinguish the different pairings and provides insight into the formation of topologically nontrivial Andreev bound s…
We study the Josephson effect for a topologically nontrivial superconducting (TS) wire with Majorana fermion end states and tunnel-coupled to s-wave BCS superconducting (S) electrodes. In an S-TS junction, no supercurrent flow is possible under fairly general conditions. For the S-TS-S junction, bulk TS quasiparticles must be accessible to have a Josephson effect. In the noninteracting case, we derive the exact current-phase relation (CPR) and find π-periodic behavior with negative critical current for weak tunnel couplings. Charging effects then cause the anomalous CPR $I(φ)=I_c \cosφ$, where…
The anomalous proximity effect in dirty superconducting junctions is one of most striking phenomena highlighting the profound nature of Majorana bound states and odd-frequency Cooper pairs in topological superconductors. Motivated by the recent experimental realization of planar topological Josephson junctions, we describe the anomalous proximity effect in a superconductor/semiconductor hybrid, where an additional dirty normal-metal segment is extended from a topological Josephson junction. The topological phase transition in the topological Josephson junction is accompanied by a drastic chang…
We investigate 1D quantum systems that support Majorana bound states at interfaces between topologically distinct regions. In particular, we show that there exists a duality between particle-hole and spin degrees of freedom in certain spin-orbit-coupled 1D platforms such as topological insulator edges. This duality results in a spin analogue of previously explored `fractional Josephson effects'---that is, the spin current flowing across a magnetic junction exhibits 4π-periodicity in the relative magnetic field angle across the junction. Furthermore, the interplay between the particle-hole and …
Josephson junctions with three or more superconducting leads have been predicted to exhibit topological effects in the presence of few conducting modes within the interstitial normal material. Such behavior, of relevance for topologically-protected quantum bits, would lead to specific transport features measured between terminals, with topological phase transitions occurring as a function of phase and voltage bias. Although conventional, two-terminal Josephson junctions have been studied extensively, multi-terminal devices have received relatively little attention to date. Motivated in part by…
We study the key features of the Josephson transport through a curved semiconducting nanowire. Based on numerical simulations and analytical estimates within the framework of the Bogoliubov-de Gennes equations we find the ground-state phase difference $φ_0$ between the superconducting leads tuned by the spin splitting field $h$ driving the system from the topologically trivial to the nontrivial superconducting state. The phase $φ_0$ vanishes for rather small $h$, grows in a certain field range around the topological transition, and then saturates at large $h$ in the Kitaev regime. Both the sub…
Wires made of topological insulators (TI) are a promising platform for searching for Majorana bound states. These states can be probed by analyzing the fractional ac Josephson effect in Josephson junctions with the TI wire as a weak link. An axial magnetic field can be used to tune the system from trivial to topologically nontrivial. Here we investigate the oscillations of the supercurrent in such wire Josephson junctions as a function of the axial magnetic field strength and different contact transparencies. Although the current flows on average parallel to the magnetic field we observe $h/2e…
Recently we predicted that the Andreev bound state spectrum of 4-terminal Josephson junctions may possess topologically protected zero-energy Weyl singularities, which manifest themselves in a quantized transconductance in units of $4e^2/h$ when two of the terminals are voltage biased [R.-P. Riwar et al., Nature Commun. 7, 11167 (2016)]. Here, using the Landauer-Büttiker scattering theory, we compute numerically the currents flowing through such a structure in order to assess the conditions for observing this effect. We show that the voltage below which the transconductance becomes quantized i…
Josephson junctions made of closely-spaced conventional superconductors on the surface of 3D topological insulators have been proposed to host Andreev bound states (ABSs) which can include Majorana fermions. Here, we present an extensive study of the supercurrent carried by low energy ABSs in Nb/Bi$_2$Se$_3$/Nb Josephson junctions in various SQUIDs as we modulate the carrier density in the Bi$_2$Se$_3$ barriers through electrostatic top gates. As previously reported, we find a precipitous drop in the Josephson current at a critical value of the voltage applied to the top gate. This drop has be…
We study the emergent band topology of subgap Andreev bound states in the three-terminal Josephson junctions. We scrutinize the symmetry constraints of the scattering matrix in the normal region connecting superconducting leads that enable the topological nodal points in the spectrum of Andreev states. When the scattering matrix possesses time-reversal symmetry, the gap closing occurs at special stationary points that are topologically trivial as they carry vanishing Berry fluxes. In contrast, for the time-reversal broken case we find topological monopoles of the Berry curvature and correspond…
Transition metal dichalcogenides (TMDs) offer a unique platform to study unconventional superconductivity, owing to the presence of strong spin-orbit coupling and a remarkable stability to an in-plane magnetic field. A recent study found that when an in-plane field applied to a superconducting monolayer TMD is increased beyond the Pauli critical limit, a quantum phase transition occurs into a topological nodal superconducting phase which hosts Majorana flat bands. We study the current-phase relation of this nodal superconductor in a Josephson junction geometry. We find that the nodal supercond…
Topological superconductors which support Majorana fermions are thought to be realized in one-dimensional semiconducting wires coupled to a superconductor. Such excitations are expected to exhibit non-Abelian statistics and can be used to realize quantum gates that are topologically protected from local sources of decoherence. Here we report the observation of the fractional a.c. Josephson effect in a hybrid semiconductor/superconductor InSb/Nb nanowire junction, a hallmark of topological matter. When the junction is irradiated with a radio-frequency f in the absence of an external magnetic fi…
We study the combined effects of spin-orbit interaction, magnetic field, and Coulomb charging on the Josephson current-phase relation, I(φ), for a multi-level quantum dot tunnel-contacted by two conventional s-wave superconductors with phase difference φ. 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(φ=0). For a two-level dot with spin-orbit coupling and arbitrarily weak Zeeman field B, we find…
Multi-terminal superconducting Josephson junctions based on the proximity effect offer the bright opportunity to tailor non trivial quantum states in nanoscale weak-links. These structures can realize exotic topologies in multidimensions as, for example, artificial topological superconductors able to support Majorana bound states, and pave the way to emerging quantum technologies and future quantum information schemes. Here, we report the first realization of a three-terminal Josephson interferometer based on a proximized nanosized weak-link. Our tunneling spectroscopy measurements reveal tran…
We study the Josephson effect in the multiterminal junction of topological superconductors. We use the symmetry-constrained scattering matrix approach to derive band dispersions of emergent sub-gap Andreev bound states in a multidimensional parameter space of superconducting phase differences. We find distinct topologically protected band crossings that serve as monopoles of finite Berry curvature. Particularly, in a four-terminal junction the admixture of $2π$ and $4π$ periodic levels leads to the appearance of finite energy Majorana-Weyl nodes. This topological regime in the junction can be …
We discuss transport properties of fully spin-polarized triplet superconductors, where only electrons of one spin component (along a certain axis) are paired. Due to the structure of the order parameter space, wherein phase and spin rotations are intertwined, a configuration where the superconducting phase winds by $4π$ in space is topologically equivalent to a configuration with no phase winding. This opens the possibility of supercurrent relaxation by a smooth deformation of the order parameter, where the order parameter remains non-zero at any point in space throughout the entire process. D…
The Josephson current flowing in a junction between two superconductors is a striking manifestation of macroscopic quantum coherence, with applications in metrology and quantum information. This equilibrium current is related with the formation of Andreev states localized in the junction, whose energy depends periodically on the superconducting phase difference. Topology emerged as a guide for predicting exotic properties of Andreev states. In particular, topological superconductors host Majorana modes at their ends. Then, in a junction with such leads, the hybridization of two Majorana modes …
We analyze the Andreev spectrum in a four-terminal Josephson junction between one-dimensional topological superconductors in class D. We find that a topologically protected crossing in the space of three superconducting phase differences can occur between the two lowest Andreev bound states. This crossing can be detected through the transconductance quantization, in units of $2e^2/h$, between two voltage-biased terminals. Our prediction provides yet another example of topology in multi-terminal Josephson junctions. We discuss possible realizations of such junctions with semiconducting crossed …
Theory of reentrant superconductivity in Corbino Josephson junctions
Cornell University（康奈尔大学）；Weizmann Institute of Science（魏茨曼科学研究所）；Harvard University（哈佛大学）；Sungkyunkwan University (SKKU)（成均馆大学）；Institute for Basic Science (IBS)（基础科学研究院）；John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Columbia University（哥伦比亚大学）
We report anomalous enhancement of the critical current at low temperatures in gate-tunable Josephson junctions made from topological insulator BiSbTeSe$_2$ nanoribbons with superconducting Nb electrodes. In contrast to conventional junctions, as a function of the decreasing temperature $T$, the increasing critical current $I_c$ exhibits a sharp upturn at a temperature $T_*$ around 20$\%$ of the junction critical temperatures for several different samples and various gate voltages. The $I_c$ vs. $T$ demonstrates a short junction behavior for $T>T_*$, but crosses over to a long junction behavio…
The Majorana zero-energy modes (MZMs) residing at the boundary of topological superconductors have attracted a great deal of interest recently, as they provide a platform to explore fundamental physics such as non-Abelian statistics, as well as fault-tolerant quantum computation. Period doubling of Shapiro steps in a Josephson junction under microwave irradiation has been regarded as strong evidence for the emergence of the MZMs at the junction edges. However, questions remain as to how the Shapiro steps respond to the presence of a 4π-periodic Josephson current. In this study, we investigated…
We investigate Josephson flux-flow in annular Josephson tunnel junctions (AJTJs) under the application of magnetic fields generating finite-voltage steps in their current-voltage characteristics. Experimental data are presented for confocal AJTJs which are the natural generalization of the well studied circular AJTJs for which flux flow effects have never been reported. Displaced linear slopes, Fiske step staircases and Eck steps were sequentially recorded at $4.2\,K$ with high-quality Nb/Al-AlOx/Nb confocal AJTJs when increasing the strength of a uniform magnetic field applied in the plane of…
Proximity-induced superconductivity in three dimensional (3D) topological insulators forms a new quantum phase of matter and accommodates exotic quasiparticles such as Majorana bound states. One of the biggest drawbacks of the commonly studied 3D topological insulators is the presence of conducting bulk that obscures both surface states and low energy bound states. Introducing superconductivity in topological Kondo insulators such as SmB$_6$, however, is promising due to their true insulating bulk at low temperatures. In this work, we develop an unconventional Josephson junction by coupling su…
Majorana Bound States are predicted to appear as boundary states of the Kitaev model. Here we show that a pi-Josephson Junction, inserted in a topologically non trivial model ring, sustains a Majorana Bound State, which is robust with respect to local and non local perturbations. The realistic structure could be based on a High Tc Superconductor tricrystal structure, similar to the one used to spot the d-wave order parameter. The presence of the Majorana Bound State changes the ground state of the topologically non trivial ring in a measurable way, with respect to that of a conventional one.
Dynamical Steering and Unambiguous Signature of Majorana Corner Modes in Altermagnetic Josephson Junctions
Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics, South China Normal University（华南师范大学物理学院）；Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, South China Normal University（华南师范大学前沿物理研究院）
The energies of Andreev bound states (ABS) forming in a $N$-terminal junction are affected by $N - 1$ independent macroscopic phase differences between superconducting leads and can be regarded as energy bands in $N - 1$ periodic solid owing to the $2π$ periodicity in all phases. We investigate the singularities and peculiarities of the resulting ABS spectrum combining phenomenological and analytical methods and illustrating with the numerical results. We pay special attention on spin-orbit (SO) effects. We consider Weyl singularities with a conical spectrum that are situated at zero energy in…
