Cooper Pair Tunneling Research Articles

Correlating Josephson supercurrents and Shiba states in quantum spins unconventionally coupled to superconductors

Local spins coupled to superconductors give rise to several emerging phenomena directly linked to the competition between Cooper pair formation and magnetic exchange. These effects are generally scrutinized using a spectroscopic approach which relies on detecting the in-gap bound modes arising from Cooper pair breaking, the so-called Yu-Shiba-Rusinov (YSR) states. However, the impact of local magnetic impurities on the superconducting order parameter remains largely unexplored. Here, we use scanning Josephson spectroscopy to directly visualize the effect of magnetic perturbations on Cooper pair tunneling between superconducting electrodes at the atomic scale. By increasing the magnetic impurity orbital occupation by adding one electron at a time, we reveal the existence of a direct correlation between Josephson supercurrent suppression and YSR states. Moreover, in the metallic regime, we detect zero bias anomalies which break the existing framework based on competing Kondo and Cooper pair singlet formation mechanisms. Based on first-principle calculations, these results are rationalized in terms of unconventional spin-excitations induced by the finite magnetic anisotropy energy. Our findings have far reaching implications for phenomena that rely on the interplay between quantum spins and superconductivity.

A density-functional theory study of the Al/AlOx/Al tunnel junction

The aluminum oxide tunnel junction is a key component of the majority of superconducting quantum devices. For high-quality, reproducible, and scalably manufacturable qubits, the ability to fabricate Josephson junctions (JJs) with a targeted critical current and high uniformity is essential. We use first-principles modeling to assess fundamental aspects of the atomic structure of both amorphous and crystalline aluminum oxide tunnel junctions and relate the structure to predicted performance metrics. We use modified ab initio molecular dynamics to develop realistic models of the tunnel junction, from which interface roughness and local thickness fluctuations are analyzed in an unbiased manner by training a neural network to identify the boundary between metal and oxide. We show that the effective thickness of the insulating part of the junction can be different from the apparent physical thickness. We calculate the rate of Cooper pair tunneling for the atomically resolved electrostatic potential using direct numerical solution in 3D, which shows a channeling effect that impacts the junction critical current. The predicted critical current is a useful JJ design parameter that can be accessed from the ab initio calculations without fitting parameters. To assess the limits of uniformity and fabrication choices (e.g., oxidation vs epitaxy), we compare the amorphous junctions to crystalline models, which show order of magnitude more efficient tunneling compared to the amorphous case, underlining the connection between atomistic structure and Cooper pair tunneling efficiency. Further, this work provides a foundation for ab initio materials design and evaluation to help accelerate future development of improved tunnel junctions.

Transport properties of proximitized double quantum dots

We study the sub-gap spectrum and the transport properties of a double quantum dot coupled to metallic and superconducting leads. The coupling of both quantum dots to the superconducting lead induces a non-local pairing in both quantum dots by the Andreev reflection processes. Additionally, we obtain two channels of Cooper pair tunneling into a superconducting lead. In such a system, the direct tunneling process (by one of two dots) or the crossed tunneling process (by both quantum dots at the same time) is possible. We consider the dependence of the Andreev transmittance on an inter-dot tunneling amplitude and the coupling between a quantum dot and the superconducting lead. We also consider the occurrence of interferometric Fano-type line shapes in the linear Andreev conductance spectra.

Critical current for an insulating regime of an underdamped current-biased topological Josephson junction

We study analytically an underdamped current-biased topological Josephson junction. First, we consider a simplified model at zero temperature, where the parity of the non-local fermionic state formed by Majorana bound states (MBSs) localized on the junction is fixed, and show that a transition from insulating to conducting state in this case is governed by single-quasiparticle tunneling rather than by Cooper pair tunneling in contrast to a non-topological Josephson junction. This results in a significantly lower critical current for the transition from insulating to conducting state. We propose that, if the length of the system is finite, the transition from insulating to conducting state occurs at exponentially higher bias current due to hybridization of the states with different parities as a result of the overlap of MBSs localized on the junction and at the edges of the topological nanowire forming the junction. Finally, we discuss how the appearance of MBSs can be established experimentally by measuring the critical current for an insulating regime at different values of the applied magnetic field.

Effects of Rashba-spin–orbit coupling on superconducting boron-doped nanocrystalline diamond films: evidence of interfacial triplet superconductivity

Among the many remarkable properties of diamond, the ability to superconduct when heavily doped with boron has attracted much interest in the carbon community. When considering the nanocrystalline boron doped system, the reduced dimensionality and confinement effects have led to several intriguing observations most notably, signatures of a mixed superconducting phase. Here we present ultra-high-resolution transmission electron microscopy imaging of the grain boundary and demonstrate how the complex microstructure leads to enhanced carrier correlations. We observe hallmark features of spin–orbit coupling (SOC) manifested as the weak anti-localization effect. The enhanced SOC is believed to result from a combination of inversion symmetry breaking at the grain boundary interfaces along with antisymmetric confinement potential between grains, inducing a Rashba-type SOC. From a pronounced zero bias peak in the differential conductance, we demonstrate signatures of a triplet component believed to result from spin mixing caused by tunneling of singlet Cooper pairs through such Rashba-SOC grain boundary junctions.

Current correlations of Cooper-pair tunneling into a quantum Hall system

We study Cooper pair transport through a quantum point contact between a superconductor and a quantum Hall edge state at integer and fractional filling factors. We calculate the tunnelling current and its finite-frequency noise to the leading order in the tunneling amplitude for dc and ac bias voltage in the limit of low temperatures. At zero temperature and in case of tunnelling into a single edge channel both the conductance and differential shot noise vanish as a result of Pauli exclusion principle. In contrast, in the presence of two edge channels, this Pauli blockade is softened and a non-zero conductance and shot noise are revealed.

Electric field control of radiative heat transfer in a superconducting circuit

Heat is detrimental for the operation of quantum systems, yet it fundamentally behaves according to quantum mechanics, being phase coherent and universally quantum-limited regardless of its carriers. Due to their robustness, superconducting circuits integrating dissipative elements are ideal candidates to emulate many-body phenomena in quantum heat transport, hitherto scarcely explored experimentally. However, their ability to tackle the underlying full physical richness is severely hindered by the exclusive use of a magnetic flux as a control parameter and requires complementary approaches. Here, we introduce a dual, magnetic field-free circuit where charge quantization in a superconducting island enables thorough electric field control. We thus tune the thermal conductance, close to its quantum limit, of a single photonic channel between two mesoscopic reservoirs. We observe heat flow oscillations originating from the competition between Cooper-pair tunnelling and Coulomb repulsion in the island, well captured by a simple model. Our results highlight the consequences of charge-phase conjugation on heat transport, with promising applications in thermal management of quantum devices and design of microbolometers.

Resonant Andreev reflections probed by photon-assisted tunnelling at the atomic scale

Tunneling across superconducting junctions proceeds by a rich variety of processes, which transfer single electrons, Cooper pairs, or even larger numbers of electrons by multiple Andreev reflections. Photon-assisted tunneling combined with the venerable Tien-Gordon model has long been a powerful tool to identify tunneling processes between superconductors. Here, we probe superconducting tunnel junctions including an impurity-induced Yu-Shiba-Rusinov (YSR) state by exposing a scanning tunneling microscope with a superconducting tip to microwave radiation. We find that a simple Tien-Gordon description describes tunneling of single electrons and Cooper pairs into the bare substrate, but breaks down for tunneling via YSR states by resonant Andreev reflections. We develop an improved theoretical description which is in excellent agreement with the data. Our results establish photon-assisted tunneling as a powerful tool to analyze tunneling processes at the atomic scale which should be particularly informative for unconventional and topological superconductors.

Single channel Josephson effect in a high transmission atomic contact

The Josephson effect in scanning tunneling microscopy (STM) is an excellent tool to probe the properties of a superconductor on a local scale. We use atomic manipulation in a low temperature STM to create mesoscopic single channel contacts and study the Josephson effect at arbitrary transmissions. We observe significant deviations from the Ambegaokar-Baratoff formula relating the critical current to the order parameter starting from transmissions of τ > 0.1. Using the full current-phase relation, we model the Josephson effect in the dynamical Coulomb blockade regime, where the charging energy of the junction capacitance cannot be neglected, and find excellent agreement with the experimental data. Projecting the current-phase relation onto the charge transfer operator shows that at high transmission, non-linear behaviour arises and multiple Cooper pair tunneling may occur. Our model includes these deviations, which become non-negligible in Josephson-STM, for example, when scanning across single adatoms.

Electrical conductivity of YBa2Cu3O7-δ single crystals under conditions of anionic ordering in Cu(1)O1-δ layers

The influence of thermocycling annealing processes on the oxygen ordering degree (order parameter) in YBa2Cu3O7-δ single crystals has been studied. It has been shown that an increase in the critical onset temperature of the transition to the superconducting state during thermocycling annealing procedures is consistent with the decrease of the σс/σаb parameter. This fact indicates a redistribution of the electronic density between the structurally inhomogeneous Cu(2)O2 and Cu(1)O1-δ planes, due to the formation of oxygen long-range order in the O(4)–Cu(1)–O(4) linear groups along the (b) crystal structure axis of the unit cell, and elimination of oxygen defects in the square nets of the Cu(2)O2 planes. The existence of the critical value of the conductivity anisotropy σс/σаb, below which its behavior does not correlate with the change of Тс, has been confirmed. In this case an increase in Тс and orthorhombic distortion of the crystal structure during isothermal annealing are caused by the amplification of the “interlayer” interaction between the Cu(2)О2 and Cu(1)О1-δ planes. As a result, the contribution of the Cu(1)О1-δ chain layers to the electron state density at the Fermi level increases. These layers can acquire superconducting properties due to tunneling of Cooper pairs from the Cu(2)О2 planes resulting in the formation of the induced superconductivity in these planes.

Electronic and Cooper pair tunneling signatures in the electronic spectra of superconductors coupled to quantum dot nano-junction

We present here calculations on electronic spectra of a system consisting of a quantum dot connected to superconducting leads on its either side. In these calculations the generated spectra has contributions from both, the single particle tunneling, and from the tunneling Cooper pair of electrons. In this, the Anderson model is used for the theoretical construct, and hence, the single particle tunneling, the Josephson tunneling, the energy states of the quantum dot and the super conducting order parameter for BCS superconductors are principally considered. The Green's function equation of motion technique is used to obtain the single particle spectral function. Our results from the numerical computations in the present work indicate, under certain conditions, the occurrence of breaking of Cooper pairs at the junctions between the superconducting leads and the quantum dot. These are seen to be, understandably, depending on the interaction terms included in our model. Our present work suggests a way to understand the interplay of single particle and the ‘Cooper pairs tunneling’ in the resultant electronic spectra in the light of extended Anderson model applicable to the system. It thus presents a theoretical understanding on the transmission of supercurrents through nanoscopic junctions, such as that considered here.

Photon-assisted tunnelling of zero modes in a Majorana wire

Hybrid nanowires with proximity-induced superconductivity in the topological regime host Majorana zero modes (MZMs) at their ends, and networks of such structures can produce topologically protected qubits. In a double-island geometry where each segment hosts a pair of MZMs, inter-pair coupling mixes the charge parity of the islands and opens an energy gap between the even and odd charge states at the inter-island charge degeneracy. Here, we report on the spectroscopic measurement of such an energy gap in an InAs/Al double-island device by tracking the position of the microwave-induced quasiparticle (qp) transitions using a radio-frequency (rf) charge sensor. In zero magnetic field, photon assisted tunneling (PAT) of Cooper pairs gives rise to resonant lines in the 2e-2e periodic charge stability diagram. In the presence of a magnetic field aligned along the nanowire, resonance lines are observed parallel to the inter-island charge degeneracy of the 1e-1e periodic charge stability diagram, where the 1e periodicity results from a zero-energy sub-gap state that emerges in magnetic field. Resonant lines in the charge stability diagram indicate coherent photon assisted tunneling of single-electron states, changing the parity of the two islands. The dependence of resonant frequency on detuning indicates a sizable (GHz-scale) hybridization of zero modes across the junction separating islands.

Anomalous Josephson Hall effect charge and transverse spin currents in superconductor/ferromagnetic-insulator/superconductor junctions

Interfacial spin-orbit coupling in Josephson junctions offers an intriguing way to combine anomalous Hall and Josephson physics in a single device. We study theoretically how the superposition of both effects impacts superconductor/ferromagnetic insulator/superconductor junctions' transport properties. Transverse momentum-dependent skew tunneling of Cooper pairs through the spin-active ferromagnetic insulator interface creates sizable transverse Hall supercurrents, to which we refer as anomalous Josephson Hall effect currents. We generalize the Furusaki-Tsukada formula, which got initially established to quantify usual (tunneling) Josephson current flows, to evaluate the transverse current components and demonstrate that their amplitudes are widely adjustable by means of the spin-orbit coupling strengths or the superconducting phase difference across the junction. As a clear spectroscopic fingerprint of Josephson junctions, well-localized subgap bound states form around the interface. By analyzing the spectral properties of these states, we unravel an unambiguous correlation between spin-orbit coupling-induced asymmetries in their energies and the transverse current response, founding the currents' microscopic origin. Moreover, skew tunneling simultaneously acts like a transverse spin filter for spin-triplet Cooper pairs and complements the discussed charge current phenomena by their spin current counterparts. The junctions' universal spin-charge current cross ratios provide valuable possibilities to experimentally detect and characterize interfacial spin-orbit coupling.

Quantum phases of a one-dimensional Majorana-Bose-Hubbard model

Majorana zero modes (MZM-s) occurring at the edges of a 1D, p-wave, spinless superconductor, in absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically-protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the fate of the topological phase in the presence of phase-fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on mesoscopic superconducting islands at zero temperature. Cooper-pair and MZM-assisted single-electron tunneling, together with finite charging energy of the mesoscopic islands, give rise to a rich phase-diagram of this model. We show that the system can be in either a Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper-pairs or a second gapless phase. In contrast to the LL of Cooper-pairs, this second phase is characterized by nonlocal string correlation functions which decay algebraically due to gapless charge-$e$ excitations. The three phases are separated from each other by phase-transitions of either Kosterlitz-Thouless or Ising type. Using a Jordan-Wigner transformation, we map the system to a generalized Bose-Hubbard model with two types of hopping and use DMRG to analyze the different phases and the phase-transitions.

Superconductivity-driven negative interfacial magnetization in YBa2Cu3O7−δ/SrTiO3/La0.67Sr0.33MnO3 heterostructures

Using spin-polarized neutron reflectivity experiments, we demonstrate an unusual proximity behavior when a superconductor (SC) and a ferromagnet (FM) are coupled through an insulator (I) in YBa2Cu3O7−δ (SC)/SrTiO3 (I)/La0.67Sr0.33MnO3 (FM) heterostructures. We have observed an unexpected magnetic reversal confined to the interface region of the FM below the superconducting transition temperature. The magnetization of the interfacial FM layer at the I/FM interface was found to be aligned opposite to the magnetization of the rest of the FM layer. This result indicates that the Cooper pairs tunnel across the insulator, interact with the local magnetization in the interfacial region (extending ∼30 Å) of the FM, and then modify the magnetization at the interface. This unexpected magnetic behavior cannot be explained on the basis of the existing theoretical models. However, the length scale associated here clearly suggests the long-range proximity effect as a result of tunneling of Cooper pairs. The magnetic exchange field-effect across SC/I/FM interfaces driven by tunneling may serve as the basis for application in superconducting spintronic devices.

Josephson current through a ferromagnetic bilayer: Beyond the quasiclassical approximation

Based on the Bogoliubov-de Gennes equations, we provide an exact numerical solution for the critical current of Josephson junctions with a composite ferromagnetic bilayer. We demonstrate that for the antiparallel orientation of the magnetic moments of the bilayer, the presence of a potential barrier at the bilayer interface results in large oscillations of the critical current as a function of ferromagnet thickness and/or exchange field. Because of this, and remarkably, in the range of small exchange field and thicknesses, the magnetism leads to the increase of the critical current. This effect is well pronounced at low temperature but disappears near $T_c$. If the potential barrier is replaced by a spin-active barrier at the bilayer interface the conventional 0-$\pi$ transition, similar to the case of an uniform ferromagnetic Josephson junction, is observed. Strikingly, for a parallel orientation of the magnetic moments of the bilayer, the presence of the spin-active barrier restores the anomalous behavior---potential barrier in the antiparallel case. These behaviors result from the resonant tunneling of Cooper pairs across the composite barrier---an effect related to the spin-dependent Fermi vector in the presence of the ferromagnets' exchange field.

Voltage-controlled coupling of flux states in superconducting rings

Superconducting circuits represent a versatile playground to explore novel quantum phenomena and a platform to realize quantum bits (qubits). Superconducting qubits hinge on the coherent tunneling of Cooper pairs across the insulating barrier of a Josephson junction. This qubit implementation is, however, plagued with the variability of the fabricated junction, along with the need for magnetic tunability of the qubit transition frequency. Alternatively, a dual to the Josephson effect—referred to as quantum phase slips—allows for a coherent coupling of flux states and has been used to build phase-slip qubits. In this paper, we propose coupling flux states of a superconducting ring using a bias voltage. By solving the time-dependent Ginzburg–Landau equations, we demonstrate that the bias voltage locally suppresses the density of Cooper pairs, thereby forming a weak link where phase slips occur. Moreover, we show that a voltage-biased superconducting ring serves as an electrically-controlled magnetic memory. The work presented in this paper is an important step towards realizing electrically-tunable superconducting devices, such as qubits and SQUIDs.

Topological ordering in the Majorana toric code

At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically-ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase-transitions separating them. While the field theory provides details on the properties of the system close to the phase-transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of the current work. We describe a lattice gauge theory of the Majorana toric code in terms of $\mathrm{U}(1)$ matter fields coupled to an emergent $\mathbb{Z}_2$ gauge field. Subsequently, we use a generalized Wilson-loop order-parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott insulator and the charge-$2e$ superconductor phases, while showing that the toric code phase disappears in the charge-$e$ superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.

Theory of double Cooper-pair tunneling and light emission mediated by a resonator

Photon emission by tunneling electrons can be encouraged by locating a resonator close to the tunnel junction and applying an appropriate voltage-bias. However, studies of normal metals show that the resonator also affects how the charges flow, facilitating processes in which correlated tunneling of two charges produces one photon. We develop a theory to analyze this kind of behavior in Josephson junctions by deriving an effective Hamiltonian describing processes where two Cooper-pairs generate a single photon. We determine the conditions under which the transport is dominated by incoherent tunneling of two Cooper-pairs, whilst also uncovering a regime of coherent double Cooper-pair tunneling. We show that the system can also display an unusual form of photon-blockade and hence could serve as a single-photon source.

Charge control of blockade of Cooper pair tunneling in highly disordered TiN nanowires in an inductive environment

A central problem in understanding the superconductor-insulator transition in disordered superconductors is that the properties of grains and intergrain medium cannot be independently studied. Here we demonstrate an approach to the study of strongly disordered superconducting films by relying on the stochastic nature of the disorder probed by electrostatic gating in a restricted geometry. Charge tuning and magnetotransport measurements in quasihomogeneous TiN nanowires embedded in a superinductor environment allow us to classify different devices and distinguish between spontaneously formed Coulomb islands (with typical blockade voltage in the mV range) and homogeneous wires showing behavior indicative of coherent quantum phase slips (with significantly smaller blockade voltage).