Multiterminal Junctions Research Articles

Majorana-mediated thermoelectric transport in multiterminal junctions

Majorana-mediated thermoelectric transport in multiterminal junctions

Theory of universal diode effect in three-terminal Josephson junctions

We theoretically study the superconducting diode effect in a three-terminal Josephson junction. The diode effect in superconducting systems is typically related to the presence of a difference in the critical currents for currents flowing in the opposite direction. We show that in multi-terminal systems this effect occurs naturally without the need of the presence of any spin interactions and is a result of the presence of a relative shift between the Andreev bound states carrying the supercurrent. On an example of a three-terminal junction, we demonstrate that the non-reciprocal current in one of the superconducting contacts can be induced by proper phase biasing of the other contacts, provided that there are at least two Andreev bound states in the system and the symmetry of the system is broken. This result is confirmed in numerical models describing the junctions in both the short- and long-regime. By optimizing the geometry of the junction, we show that the efficiency of the realized superconducting diode exceeds 35%. We relate our predictions to recent experiments on multi-terminal junctions, in which non-reciprocal supercurrents were observed.

Testing superconducting pairing symmetry in multiterminal junctions

An approach to distinguish p-wave from s-wave superconducting pairing symmetry and thus to select potential platforms for Majorana fermions is proposed in terms of electronic transport differences in a four terminal junction consisting of superconducting (S) and normal (N) terminals in the diffusive regime. The Keldysh Green’s function equations are derived in the θ-parametrisation, incorporating terms previously neglected in the literature. A stable procedure to solve these equations is presented. The supercurrent and differential conductance between two superconducting electrodes were calculated in the Keldysh–Usadel approximation. The N-terminals can be used to manipulate the energy distribution functions of electrons in the junction in order to control the overall charge transport. Our results provide a new experimental test to detect potential p-wave superconductivity. In fact, we show that the differential conductance of junctions containing p-wave superconductors is distinctly different from the differential conductance in junctions with s-wave superconductors, whereas the supercurrent through the junction is qualitatively similar. This is of importance for the search for Majorana fermions since it may help to design experiments to detect signatures of p-wave symmetry, which may lead to potential platforms for Majorana fermions.

Evidence for 4e charge of Cooper quartets in a biased multi-terminal graphene-based Josephson junction

In a Josephson junction (JJ) at zero bias, Cooper pairs are transported between two superconducting contacts via the Andreev bound states (ABSs) formed in the Josephson channel. Extending JJs to multiple superconducting contacts, the ABSs in the Josephson channel can coherently hybridize Cooper pairs among different superconducting electrodes. Biasing three-terminal JJs with antisymmetric voltages, for example, results in a direct current (DC) of Cooper quartet (CQ), which involves a four-fermion entanglement. Here, we report half a flux periodicity in the interference of CQ formed in graphene based multi-terminal (MT) JJs with a magnetic flux loop. We observe that the quartet differential conductance associated with supercurrent exhibits magneto-oscillations associated with a charge of 4e, thereby presenting evidence for interference between different CQ processes. The CQ critical current shows non-monotonic bias dependent behavior, which can be modeled by transitions between Floquet-ABSs. Our experimental observation for voltage-tunable non-equilibrium CQ-ABS in flux-loop-JJs significantly extends our understanding of MT-JJs, enabling future design of topologically unique ABS spectrum.

Multiterminal ballistic Josephson junctions coupled to normal leads

Multiterminal Josephson junctions have aroused considerable theoretical interest recently and numerous works aim at putting the predictions of correlations among Coopers (i.e., the so-called quartets) and simulation of topological matter to the test of experiments. This paper is motivated by recent experimental investigation from the Harvard group reporting $h/4e$-periodic quartet signal in a four-terminal configuration containing a loop pierced by magnetic flux, together with inversion controlled by the bias voltage, i.e., the quartet signal can be larger at half-flux quantum than in zero magnetic field. Here, we theoretically focus on devices consisting of finite-size quantum dots connected to four superconducting leads and to a normal lead. In addition to presenting numerical calculations of the quartet signal within a simplified modeling, we reduce the device to a non-Hermitian Hamiltonian in the infinite-gap limit. Then, relaxation has the surprising effect of producing sharp peaks and log-normal variations in the voltage sensitivity of the quartet signal in spite of the expected moderate fluctuations in the two-terminal DC Josephson current. The phenomenon is reminiscent of a resonantly driven harmonic oscillator having amplitude inverse proportional to the damping rate, and of the thermal noise of a superconducting weak link, inverse proportional to the Dynes parameter. Perspectives involve quantum bath engineering multiterminal Josephson junctions in the circuits of cavity quantum electrodynamics.

Multi-Terminal Nanotube Junctions: Modeling and Structure-Property Relationship

Carbon nanotube based multi-terminal junction configurations are of great interest because of the potential aerospace and electronic applications. Multi-terminal carbon nanotube junction has more than one carbon nanotube meeting at a point to create a 2D or 3D structure. Accurate atomistic models of such junctions are essential for characterizing their thermal, mechanical and electronic properties via computational studies. In this work, computational methodologies that uses innovative Computer-Aided Design (CAD) based optimization strategies and remeshing techniques are presented for generating such topologically reliable and accurate models of complex multi-terminal junctions (called 3-, 4-, and 6-junctions). This is followed by the prediction of structure-property relationship via study of thermal conductivity and mechanical strength using molecular dynamics simulations. We observed high degradation in the thermal and mechanical properties of the junctions compared to pristine structures which is attributed to high concentration of non-hexagonal defects in the junction. Junctions with fewer defects have better thermal transport capabilities and higher mechanical strengths, suggesting that controlling the number of defects can significantly improve inherent features of the nanostructures.

Spatial spin flipping and spin switching phenomena on a Y-shaped graphene nanoribbon ferromagnetic junction with Rashba spin orbit coupling and strain

We theoretically study a controllable spatial spin converter and spin valve by using a Y-shaped even-chain zigzag graphene nanoribbon (ZGR) junction with Rashba spin orbit coupling (SOC) and strain. By calculating the spin-dependent conductance spectra of different tunneling terminals through the multi-terminal Landauer formula, we find that the spin up electron can flip its spin orientation, and convert to spin down one on the left electrode, the same spin converting phenomena can be inhibited on the right terminal. The above spin flipping and its valve phenomena are derived from the interactions of the Rashba SOC, strain and magnetization layout. When the strain’s orientation is along (perpendicular) to the zigzag chain’s direction, the valley valve effect of the even-chain ZGR is remained, the Rashba SOC takes little effect on the spatial spin switching, one can get 100% polarized spin up (down) electrons at different terminals of the Y-shaped ZGR due to the valley valve effect. When the strain is changed to other direction, the valley valve effect is partly destroyed, Rashba SOC can enhance the spin flipping conductance and break the ON/OFF transport states of spin valve. Our investigations might be useful in designing a multi-parameter controllable spin valve and spin detector based on a multi-terminal graphene nanoribbon junction.

Conductance quantization in topological Josephson trijunctions

Here, the authors predict that topological Josephson trijunctions generically display a nonzero quantized transconductance. This result originates from a combination of two different aspects of topology: topological superconductivity in the leads, thus hosting Majorana end modes, and a synthetic topological band structure in the multiterminal Josephson junction. Many groups are currently studying multiterminal junctions using various superconducting hybrid platforms. An experimental check of this prediction would constitute an important milestone on the roadmap toward demonstrating Majorana braiding in trijunctions.

Ballistic Graphene Cooper Pair Splitter.

We report an experimental study of a Cooper pair splitter based on ballistic graphene multiterminal junctions. In a two transverse junction geometry, namely the superconductor-graphene-superconductor and the normal metal-graphene-normal metal, we observe clear signatures of Cooper pair splitting in the local as well as nonlocal electronic transport measurements. Our experimental data can be very well described by our beam splitter model. These results open up possibilities to design new entangled state detection experiments using ballistic Cooper pair splitters.

Majorana-Weyl crossings in topological multiterminal junctions

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 nanowires and with quantum-spin Hall insulators.

Topological supercurrents interaction and fluctuations in the multiterminal Josephson effect

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 $2\pi$ and $4\pi$ periodic levels leads to the appearance of finite energy Majorana-Weyl nodes. This topological regime in the junction can be characterized by a quantized nonlocal conductance that measures the Chern number of the corresponding bands. In addition, we calculate current-phase relations, variance, and cross-correlations of topological supercurrents in multiterminal contacts and discuss the universality of these transport characteristics. At the technical level these results are obtained by integrating over the group of a circular ensemble that describes the scattering matrix of the junction. We briefly discuss our results in the context of observed fluctuations of the gate dependence of the critical current in topological planar Josephson junctions and comment on the possibility of parity measurements from the switching current distributions in multiterminal Majorana junctions.

Supercurrent Flow in Multiterminal Graphene Josephson Junctions.

We investigate the electronic properties of ballistic planar Josephson junctions with multiple superconducting terminals. Our devices consist of monolayer graphene encapsulated in boron nitride with molybdenum-rhenium contacts. Resistance measurements yield multiple resonant features, which are attributed to supercurrent flow among adjacent and nonadjacent Josephson junctions. In particular, we find that superconducting and dissipative currents coexist within the same region of graphene. We show that the presence of dissipative currents primarily results in electron heating and estimate the associated temperature rise. We find that the electrons in encapsulated graphene are efficiently cooled through the electron-phonon coupling.

Quantum energy exchange and refrigeration: a full-counting statistics approach

We formulate a full-counting statistics description to study energy exchange in multi-terminal junctions. Our approach applies to quantum systems that are coupled either additively or non-additively (cooperatively) to multiple reservoirs. We derive a Markovian Redfield-type equation for the counting-field dependent reduced density operator. Under the secular approximation, we confirm that the cumulant generating function satisfies the heat exchange fluctuation theorem. Our treatment thus respects the second law of thermodynamics. We exemplify our formalism on a multi-terminal two-level quantum system, and apply it to realize the smallest quantum absorption refrigerator, operating through engineered reservoirs, and achievable only through a cooperative bath interaction model.

Quantum transport in coupled Majorana box systems

We present a theoretical analysis of low-energy quantum transport in coupled Majorana box devices. A single Majorana box represents a Coulomb-blockaded mesoscopic superconductor proximitizing two or more long topological nanowires. The box thus harbors at least four Majorana zero modes (MZMs). Setups with several Majorana boxes, where MZMs on different boxes are tunnel-coupled via short nanowire segments, are key ingredients to recent Majorana qubit and code network proposals. We construct and study the low-energy theory for multi-terminal junctions with normal leads connected to the coupled box device by lead-MZM tunnel contacts. Transport experiments in such setups can test the nonlocality of Majorana-based systems and the integrity of the underlying Majorana qubits. For a single box, we recover the previously described topological Kondo effect which can be captured by a purely bosonic theory. For several coupled boxes, however, non-conserved local fermion parities require the inclusion of additional local sets of Pauli operators. We present a renormalization group analysis and develop a nonperturbative strong-coupling approach to quantum transport in such systems. Our findings are illustrated for several examples, including a loop qubit device and different two-box setups.

Josephson effect in multiterminal topological junctions

We study the Josephson effect in a trijunction formed by two topological superconductor (TS) wires and a conventional $s$-wave superconductor. Using a boundary Green's function formalism, analytical results for the current-phase relation are obtained in various limiting cases by modeling the TS wires via the low-energy limit of a Kitaev chain. We show that Josephson transport critically depends on the spin canting angle $\theta$ between the boundary spin polarizations of the TS wires, which in turn suggests that the spin structure of Majorana states can be accessed through supercurrent measurements. We also extend the boundary Green's function approach to a more microscopic spinful wire model and thereby compute the dependence of $\theta$ on experimentally accessible parameters such as the Zeeman field and/or the chemical potential. Furthermore, we show that the equilibrium current-phase relation between both TS wires exhibits a robust $4\pi$-periodicity since the conventional superconducting lead effectively locks the fermion parity of the trijunction.

Noise cross-correlation and Cooper pair splitting efficiency in multi-teminal superconductor junctions

We analyze the non-local shot noise in a multi-terminal junction formed by two Normal metal leads connected to one superconductor. Using the cross Fano factor and the shot noise, we calculate the efficiency of the Cooper pair splitting. The method is applied to d-wave and iron based superconductors. We determine that the contributions to the noise cross-correlation are due to crossed Andreev reflections (CAR), elastic cotunneling, quasiparticles transmission and local Andreev reflections. In the tunneling limit, the CAR contribute positively to the noise cross-correlation whereas the other processes contribute negatively. Depending on the pair potential symmetry, the CAR are the dominant processes, giving as a result a high efficiency for Cooper pair split. We propose the use of the Fano factor to test the efficiency of a Cooper pair splitter device.

Fluctuating-bias controlled electron transport in molecular junctions

We consider the problem of transport through a multiterminal molecular junction in the presence of a stochastic bias, which can also be used to describe transport through fluctuating molecular energy levels. To describe these effects, we first make a simple extension of our previous work [Phys. Rev. B 91, 125433 (2015)] to show that the problem of tunneling through noisy energy levels can be mapped onto the problem of a noisy driving bias, which appears in the Kadanoff-Baym equations for this system in an analogous manner to the driving term in the Langevin equation for a classical circuit. This formalism uses the nonequilibrium Green's function method to obtain analytically closed formulas for transport quantities within the wide-band limit approximation for an arbitrary time-dependent bias and it is automatically partition free. We obtain exact closed formulas for both the colored and white noise-averaged current at all times. In the long-time limit, these formulas possess a Landauer-B\uttiker--type structure which enables the extraction of an effective transmission coefficient for the transport. Expanding the Fermi function into a series of simple poles, we find an exact formal relation between the parameters which characterize the bias fluctuations and the poles of the Fermi function. This enables us to describe the effect of the temperature and the strength of the fluctuations on the averaged current which we interpret as a quantum analog to the classical fluctuation-dissipation theorem. We use these results to convincingly refute some recent results on the multistability of the current through a fluctuating level, simultaneously verifying that our formalism satisfies some well-known theorems on the asymptotic current. Finally, we present numerical results for the current through a molecular chain which demonstrate a transition from nonlinear to linear $I\text{\ensuremath{-}}V$ characteristics as the strength of fluctuations is increased, as well as a stochastic resonance effect in the conductance of this system.

Reprint of : Block-determinant formalism for an action of a multi-terminal scatterer

The scattering theory of electron transport allows for a compact and powerful description in terms of gˇ2=1 Green functions, the so-called circuit theory of quantum transport. A scatterer in the theory is characterized by an action, most generally a Keldysh one, that can be further used as a building bock of theories describing statistics of electron transport, superconducting correlations, time-dependent and interaction effects. The action is usually used in the form suitable for a two-terminal scatterer.Here we provide a comprehensive derivation of a more general form of the action that is especially suitable and convenient for general multi-terminal scatterers. The action is expressed as a determinant of a block of the scattering matrix obtained by projection on the positive eigenvalues of the Green functions characterizing the reservoirs. We start with traditional Green function formalism introducing gˇ2=1 matrices and give a first example of multi-terminal counting statistics. Further we consider one-dimensional channels and discuss chiral anomaly arising in this context. Generalizing on many channels and superconducting situation, we arrive at the block-determinant relation. We give the necessary elaborative examples reproducing basic results of counting statistics and super-currents in multi-terminal junctions.

Characteristics of acoustic phonon transport and thermal conductance in multi-terminal graphene junctions

By using non-equilibrium Greens function method, we investigate the transmission rate of acoustic phonon and thermal conductance through a parallel multi-terminal graphene junctions, the relationship between the thermal-transport property in each terminal and the number of quantum terminals, the relationship between the thermal-transport property in each terminal and the relative position of quantum terminals in quantum structure, and also study the thermaltransport property in each terminal and the rough degree of edge structure. The results show that when the graphene chains (dimer lines) across the ribbon width are fixed, the increase of the number of the parallel multi-terminal graphene junctions can reduce the transmission rate of the phonons and the thermal conductance of each output terminal as well. This is because the increase of the number of the graphene junctions can lead to the decrease of the transverse dimension of the each output terminal, which enlarges the strength of the phonon scattering and results in the reduction of the phonon transmission. Owing to long distance scattering, the transmission rate of the phonons of the furthest distant output terminal is the smallest, and also the thermal conductance of the furthest output terminal is the smallest. On the contrary, the strength of the phonon scattering is the weakest for the closest output terminal. So the transmission rate of the phonons is the biggest, which induces the thermal conductance to be the biggest. The thermal conductance of the middle-output terminal depends sensitively on the structural parameters of each terminal. This is because mainly the relative position between the middle-output terminal and the phonon-input terminal is related closely to the structural parameters of each terminal, which can influence the strength of the phonon scattering and the transmission rate of the phonons. However, the thermal conductances in the top and bottom output terminals are just sensitively dependent on the structural parameters of the respective output terminal. This is because the relative position between the top (or bottom) output terminal and the phonon-input terminal is only related to the structural parameters of the respective output terminal. The rough edge structure can reduce obviously the transmission rate of the phonons, and the thermal conductance of the closest output terminal as well. The rough edge structure can modulate slightly the transmission rate of the phonons and the thermal conductance of the other output terminal. The total thermal conductance is related closely to the number of total graphene chains, the number of the multi-terminal graphene junctions, and the rough degree of edge structure. These results shed new light on the understanding of the thermal transport behaviors of multi-terminal junction quantum devices based on graphene-based nanomaterials in practical application.

Chirality and spin transformation of triplet Cooper pairs upon interaction with singlet condensate

We show that the fully polarized triplet $s$-wave component is characterized not only by the spin direction, but also by chirality. Interaction of a polarized triplet component and a singlet one results in creation of triplet Cooper pairs with opposite spin direction or of different chiralities. Such spin transformation leads to interesting phenomena in multiterminal magnetic Josephson junctions. We calculate the dc Josephson current ${I}_{\text{J}}$ in a multiterminal Josephson contact of the ${S}_{\text{m}}/n/{S}_{\text{m}}^{\ensuremath{'}}$ type with ``magnetic'' superconductors ${S}_{\text{m}}$ that generate fully polarized triplet components. The superconductors ${S}_{\text{m}}$ are attached to magnetic insulators (filters) which let pass electrons with a fixed spin direction only. The filter axes are assumed to be oriented antiparallel to each other. The Josephson current is zero in two-terminal Josephson junction, i.e., in $S/n/{S}_{\text{m}}$ or in ${S}_{\text{m}}/n/{S}_{\text{m}}^{\ensuremath{'}}$ contact. But in the three-terminal Josephson junction, with another $S$ superconductor attached to the normal wire, the finite current ${I}_{\text{J}}$ appears flowing from the $S$ superconductor to ${S}_{\text{m}}$ superconductors. The currents through the right (left) superconductors ${S}_{\text{m}}$ are opposite in sign, ${I}_{\text{R}}\ensuremath{\equiv}{I}_{\text{J}}={I}_{\text{c}}sin({\ensuremath{\chi}}_{\text{R}}+{\ensuremath{\chi}}_{\text{L}}\ensuremath{-}2\ensuremath{\chi})=\ensuremath{-}{I}_{\text{L}}$, where ${\ensuremath{\chi}}_{\text{L/R}}$ and $\ensuremath{\chi}$ are the phases of superconductors ${S}_{\text{m}},{S}_{\text{m}}^{\ensuremath{'}}$, and $S$, respectively. We discuss possibilities of experimental observation of the effect.