Tunneling spectroscopy of Andreev energy levels in a quantum dot coupled to a superconductor

A two-site Kitaev chain in a two-dimensional electron gas

Artificial Kitaev chains can be used to engineer Majorana bound states (MBSs) in superconductor-semiconductor hybrids1-4. In this work, we realize a two-site Kitaev chain in a two-dimensional electron gas by coupling two quantum dots through a region proximitized by a superconductor. We demonstrate systematic control over inter-dot couplings through in-plane rotations of the magnetic field and via electrostatic gating of the proximitized region. This allows us to tune the system to sweet spots in parameter space, where robust correlated zero-bias conductance peaks are observed in tunnelling spectroscopy. To study the extent of hybridization between localized MBSs, we probe the evolution of the energy spectrum with magnetic field and estimate the Majorana polarization, an important metric for Majorana-based qubits5,6. The implementation of a Kitaev chain on a scalable and flexible two-dimensional platform provides a realistic path towards more advanced experiments that require manipulation and readout of multiple MBSs.

Correlation between two distant quasiparticles in separate superconducting islands mediated by a single spin

Controlled coupling between distant particles is a key requirement for the implementation of quantum information technologies. A promising platform are hybrid systems of semiconducting quantum dots coupled to superconducting islands, where the tunability of the dots is combined with the macroscopic coherence of the islands to produce states with non-local correlations, e.g. in Cooper pair splitters. Electrons in hybrid quantum dots are typically not amenable to long-distance spin alignment as they tend to be screened into a localized singlet state by bound superconducting quasiparticles. However, two quasiparticles coming from different superconductors can overscreen the quantum dot into a doublet state, leading to ferromagnetic correlations between the superconducting islands. We present experimental evidence of a stabilized overscreened state, implying correlated quasiparticles over a micrometer distance. We propose alternating chains of quantum dots and superconducting islands as a novel platform for controllable large-scale spin coupling.

Phase-dependent Andreev molecules and superconducting gap closing in coherently-coupled Josephson junctions

The Josephson junction (JJ) is an essential element of superconducting (SC) devices for both fundamental and applied physics. The short-range coherent coupling of two adjacent JJs forms Andreev molecule states (AMSs), which provide a new ingredient to engineer exotic SC phenomena such as topological SC states and Andreev qubits. Here we provide tunneling spectroscopy measurements on a device consisting of two electrically controllable planar JJs sharing a single SC electrode. We discover that Andreev spectra in the coupled JJ are highly modulated from those in the single JJs and possess phase-dependent AMS features reproduced in our numerical calculation. Notably, the SC gap closing due to the AMS formation is experimentally observed. Our results help in understanding SC transport derived from the AMS and promoting the use of AMS physics to engineer topological SC states and quantum information devices.

Spin-filtered measurements of Andreev bound states in semiconductor-superconductor nanowire devices

Semiconductor nanowires coupled to superconductors can host Andreev bound states with distinct spin and parity, including a spin-zero state with an even number of electrons and a spin-1/2 state with odd-parity. Considering the difference in spin of the even and odd states, spin-filtered measurements can reveal the underlying ground state. To directly measure the spin of single-electron excitations, we probe an Andreev bound state using a spin-polarized quantum dot that acts as a bipolar spin filter, in combination with a non-polarized tunnel junction in a three-terminal circuit. We observe a spin-polarized excitation spectrum of the Andreev bound state, which can be fully spin-polarized, despite strong spin-orbit interaction in the InSb nanowires. Decoupling the hybrid from the normal lead causes a current blockade, by trapping the Andreev bound state in an excited state. Spin-polarized spectroscopy of hybrid nanowire devices, as demonstrated here, is proposed as an experimental tool to support the observation of topological superconductivity.

Superior thermoelectric properties through triangular triple quantum dots (TTQD) attached to one metallic and one superconducting lead

We theoretically investigate the thermoelectric transport properties of triangular triple quantum dots (TTQD) with the central quantum dot coupled to one metallic and one superconducting lead. The system shows significantly superior thermoelectric performance over parallel coupled triple quantum dots and those coupled to two conventional metallic leads. The thermoelectric coefficients strongly depend on the ratio of superconducting gap to interdot coupling, as well as asymmetry and interference effects. The thermopower exhibits single-platform and double-platform structures for different ratios of superconducting gap to interdot coupling. The thermopower and figure of merit achieve quite remarkable values near the superconducting gap edges where the single-particle tunnelling occurs. For symmetric coupling, the maximal figure of merit might reach the order of 102 when the superconducting gap is about half that of the interdot coupling. Moreover, the figure of merit can be further greatly enhanced by appropriately matching the electrode coupling asymmetry and interdot coupling asymmetry.

InAs/MoRe Hybrid Semiconductor/Superconductor Nanowire Devices

Implementing superconductors capable of proximity-inducing a large energy gap in semiconductors in the presence of strong magnetic fields is a major goal toward applications of semiconductor/superconductor hybrid materials in future quantum information technologies. Here, we study the performance of devices consisting of InAs nanowires in electrical contact with molybdenum-rhenium (MoRe) superconducting alloys. The MoRe thin films exhibit transition temperatures of ∼10 K and critical fields exceeding 6 T. Normal/superconductor devices enabled tunnel spectroscopy of the corresponding induced superconductivity, which was maintained up to ∼10 K, and MoRe-based Josephson devices exhibited supercurrents and multiple Andreev reflections. We determine an induced superconducting gap lower than expected from the transition temperature and observe gap softening at finite magnetic field. These may be common features for hybrids based on large-gap, type II superconductors. The results encourage further development of MoRe-based hybrids.

Proposal for Detection of the 0^{'} and π^{'} Phases in Quantum-Dot Josephson Junctions

The competition between the Kondo correlation and superconductivity in quantum-dot Josephson junctions (QDJJs) has been known to drive a quantum phase transition between 0 and π junctions. Theoretical studies so far have predicted that under strong Coulomb correlations the 0-π transition should go through intermediate states, 0^{'} and π^{'} phases. By combining a nonperturbative numerical method and the resistively shunted junction model, we investigated the magnetic-field-driven phase transition of the QDJJs in the Kondo regime and found that the low-field magnetotransport exhibits a unique feature which can be used to distinguish the intermediate phases. In particular, the magnetic-field driven π^{'}-π transition is found to lead to the enhancement of the supercurrent which is strongly related to the Kondo effect.

Photon-Assisted Tunneling of High-Order Multiple Andreev Reflections in Epitaxial Nanowire Josephson Junctions

Semiconductor/superconductor hybrids exhibit a range of phenomena that can be exploited for the study of novel physics and the development of new technologies. Understanding the origin of the energy spectrum of such hybrids is therefore a crucial goal. Here, we study Josephson junctions defined by shadow epitaxy on InAsSb/Al nanowires. The devices exhibit gate-tunable supercurrents at low temperatures and multiple Andreev reflections (MARs) at finite voltage bias. Under microwave irradiation, photon-assisted tunneling (PAT) of MARs produces characteristic oscillating sidebands at quantized energies, which depend on MAR order, n, in agreement with a recently suggested modification of the classical Tien-Gordon equation. The scaling of the quantized energy spacings with microwave frequency provides independent confirmation of the effective charge, ne, transferred by the nth-order tunneling process. The measurements suggest PAT as a powerful method for assigning the origin of low-energy spectral features in hybrid Josephson devices.

Signatures of Andreev Blockade in a Double Quantum Dot Coupled to a Superconductor

We investigate an electron transport blockade regime in which a spin triplet localized in the path of current is forbidden from entering a spin-singlet superconductor. To stabilize the triplet, a double quantum dot is created electrostatically near a superconducting Al lead in an InAs nanowire. The quantum dot closest to the normal lead exhibits Coulomb diamonds, and the dot closest to the superconducting lead exhibits Andreev bound states and an induced gap. The experimental observations compare favorably to a theoretical model of Andreev blockade, named so because the triplet double dot configuration suppresses Andreev reflections. Observed leakage currents can be accounted for by finite temperature. We observe the predicted quadruple level degeneracy points of high current and a periodic conductance pattern controlled by the occupation of the normal dot. Even-odd transport asymmetry is lifted with increased temperature and magnetic field. This blockade phenomenon can be used to study spin structure of superconductors. It may also find utility in quantum computing devices that use Andreev or Majorana states.

Kondo Cloud in a Superconductor

Magnetic impurities embedded in a metal are screened by the Kondo effect, signaled by the formation of an extended correlation cloud, the so-called Kondo or screening cloud. In a superconductor, the Kondo state turns into subgap Yu-Shiba-Rusinov states, and a quantum phase transition occurs between screened and unscreened phases once the superconducting energy gap Δ exceeds sufficiently the Kondo temperature, T_{K}. Here we show that, although the Kondo state does not form in the unscreened phase, the Kondo cloud does exist in both quantum phases. However, while screening is complete in the screened phase, it is only partial in the unscreened phase. Compensation, a quantity introduced to characterize the integrity of the cloud, is universal, and shown to be related to the magnetic impurities' g factor, monitored experimentally by bias spectroscopy.

Nontopological zero-bias peaks in full-shell nanowires induced by flux-tunable Andreev states

A semiconducting nanowire fully wrapped by a superconducting shell has been proposed as a platform for obtaining Majorana modes at small magnetic fields. In this study, we demonstrate that the appearance of subgap states in such structures is actually governed by the junction region in tunneling spectroscopy measurements and not the full-shell nanowire itself. Short tunneling regions never show subgap states, whereas longer junctions always do. This can be understood in terms of quantum dots forming in the junction and hosting Andreev levels in the Yu-Shiba-Rusinov regime. The intricate magnetic field dependence of the Andreev levels, through both the Zeeman and Little-Parks effects, may result in robust zero-bias peaks-features that could be easily misinterpreted as originating from Majorana zero modes but are unrelated to topological superconductivity.

Influence of superconductivity on the magnetic moment of quantum impurity embedded in BCS superconductor

We study the influence of superconductivity on the formation of the localized magnetic moment for a single-level quantum impurity embedded in an s-wave Bardeen-Cooper-Schrieffer (BCS) superconducting medium, modeled by single-impurity Anderson Hamiltonian. We have combined Bogoliubov transformation with Green's function method within self-consistent Hartree-Fock Mean Field approximation to analyze the conditions necessary in metal (in the superconducting) for the formation of the magnetic moment at the impurity site for the low-frequency limit $|\omega|<<\Delta_{sc}$ as well as for the finite superconducting gap $\Delta_{sc}$. We have compared these results with other theoretical results and with the single-level quantum impurity embedded in the normal metallic host. Further we analyze the electronic spectral density of the quantum impurity embedded in superconducting host with the finite superconducting gap to study the sub-gap states as a function of impurity parameters.

Magnetic field effect on trivial and topological bound states of superconducting quantum dot

We investigate the properties of a quantum dot embedded between the normal and superconducting leads which is additionally side-attached to the topological superconducting nanowire, hosting the Majorana modes. This setup enables formation of the trivial (finite-energy) bound states induced in the quantum dot through the superconducting proximity effect, coexisting/competing with the topological (zero-energy) mode transmitted from the topological superconductor. We analyze their interplay, focusing on a role played by the external magnetic field. To distinguish between these bound states we analyze the qualitative and quantitative features manifested in the subgap charge tunneling originating under nonequilibrium conditions from the Andreev (particle to hole) scattering processes.

Large spatial extension of the zero-energy Yu-Shiba-Rusinov state in a magnetic field

Various promising qubit concepts have been put forward recently based on engineered superconductor subgap states like Andreev bound states, Majorana zero modes or the Yu-Shiba-Rusinov (Shiba) states. The coupling of these subgap states via a superconductor strongly depends on their spatial extension and is an essential next step for future quantum technologies. Here we investigate the spatial extension of a Shiba state in a semiconductor quantum dot coupled to a superconductor. With detailed transport measurements and numerical renormalization group calculations we find a remarkable more than 50 nm extension of the zero energy Shiba state, much larger than the one observed in very recent scanning tunneling microscopy measurements. Moreover, we demonstrate that its spatial extension increases substantially in a magnetic field.

Local magnetic moments coupled to superconductors can form subgap Yu-Shiba-Rusinov states. Here the authors show that Shiba states made with an InAs nanowire quantum dot have large spatial extent, which is beneficial for making Shiba chains that are predicted to host Majorana zero modes.

Anomalous Fano Resonance in Double Quantum Dot System Coupled to Superconductor

We analyze the influence of a local pairing on the quantum interference in nanoscopic systems. As a model system we choose the double quantum dot coupled to one metallic and one superconducting electrode in the T-shape geometry. The analysis is particularly valuable for systems containing coupled objects with considerably different broadening of energy levels. In such systems, the scattering of itinerant electrons on a discrete (or narrow) energy level gives rise to the Fano-type interference. Systems with induced superconducting order, along well understood Fano resonances, exhibit also another features on the opposite side of the Fermi level. The lineshape of these resonances differs significantly from their reflection on the opposite side of the Fermi level, and their origin was not fully understood. Here, considering the spin-polarized tunneling model, we explain a microscopic mechanism of a formation of these resonances and discuss the nature of their uncommon lineshapes. We show that the anomalous Fano profiles originate solely from the pairing of nonscattered electrons with scattered ones. We investigate also the interplay of each type of resonances with the Kondo physics and discuss the resonant features in differential conductivity.

Zeeman Tunability of Andreev Bound States in van der Waals Tunnel Barriers

Quantum dots proximity coupled to superconductors are attractive research platforms due to the intricate interplay between the single-electron nature of the dot and the many body nature of the superconducting state. These have been studied mostly using nanowires and carbon nanotubes, which allow a combination of tunability and proximity. Here we report a new type of quantum dot which allows proximity to a broad range of superconducting systems. The dots are realized as embedded defects within semiconducting tunnel barriers in van der Waals layers. By placing such layers on top of thin NbSe_{2}, we can probe the Andreev bound state spectra of such dots up to high in-plane magnetic fields without observing the effects of a diminishing superconducting gap. As tunnel junctions defined on NbSe_{2} have a hard gap, we can map the subgap spectra without a background related to the rest of the junction. We find that the proximitized defect states invariably have a singlet ground state, manifest in the Zeeman splitting of the subgap excitation. We also find, in some cases, bound states that converge to zero energy and remain there. We discuss the role of the spin-orbit term, present both in the barrier and the superconductor, in the realization of such topologically trivial zero-energy states.

The Anderson-Josephson quantum dot-a theory perspective

Recent progress in nanoscale manufacturing has allowed to experimentally investigate quantum dots coupled to two superconducting leads in controlled and tunable setups. The equilibrium Josephson current was measured in on-chip superconducting quantum interference devices, and subgap states were investigated using weakly coupled metallic leads for spectroscopy. This has reinstated two 'classic' problems on the agenda of theoretical condensed matter physics: (1) the Josephson effect and (2) quantum spins in superconductors. The relevance of the former is obvious as the barrier, which separates the two superconductors in a standard Josephson junction, is merely replaced by the quantum dot with well separated energy levels. For odd filling of the dot it acts as a quantum mechanical spin-1/2 and thereby the relevance of the latter becomes apparent also. For normal conducting leads and at odd dot filling the Kondo effect strongly modifies the transport properties as can, e.g. be studied within the Anderson model. One can expect the same for superconducting leads, and in certain parameter regimes remnants of Kondo physics, i.e. strong electronic correlations, will affect the Josephson current. In this topical review, we discuss the status of the theoretical understanding of the Anderson-Josephson quantum dot in equilibrium, mainly focusing on the Josephson current. We introduce a minimal model consisting of a dot which can only host a single spin-up and a single spin-down electron repelling each other by a local Coulomb interaction. The dot is tunnel-coupled to two superconducting leads as described by the Bardeen-Cooper-Schrieffer Hamiltonian. This model was investigated using a variety of methods, some capturing aspects of Kondo physics, while others failing in this respect. We briefly review this. The model shows a first order level-crossing quantum phase transition when varying any parameter, provided that the others are within appropriate ranges. At vanishing temperature it leads to a jump of the Josephson current. To study the qualitative behavior of the phase diagram, or the Josephson current, several of the methods can be used. However, for a quantitative description, elaborate quantum many-body methods must be employed. We show that a quantitative agreement between accurate results obtained for the simple model and measurements of the current can be reached. This confirms that the experiments reveal the finite temperature signatures of the zero temperature transition. In addition, we consider two examples of more complex dot geometries, which might be experimentally realized in the near future. The first is characterized by the interplay of the above level-crossing physics and the Fano effect, and the second by the interplay of superconductivity and almost degenerate singlet and triplet two-body states.

Transport signatures of an Andreev molecule in a quantum dot-superconductor-quantum dot setup

Hybrid devices combining quantum dots with superconductors are important building blocks of conventional and topological quantum-information experiments. A requirement for the success of such experiments is to understand the various tunneling-induced non-local interaction mechanisms that are present in the devices, namely crossed Andreev reflection, elastic co-tunneling, and direct interdot tunneling. Here, we provide a theoretical study of a simple device that consists of two quantum dots and a superconductor tunnel-coupled to the dots, often called a Cooper-pair splitter. We study the three special cases where one of the three non-local mechanisms dominates, and calculate measurable ground-state properties, as well as the zero-bias and finite-bias differential conductance characterizing electron transport through this device. We describe how each non-local mechanism controls the measurable quantities, and thereby find experimental fingerprints that allow one to identify and quantify the dominant non-local mechanism using experimental data. Finally, we study the triplet blockade effect and the associated negative differential conductance in the Cooper-pair splitter, and show that they can arise regardless of the nature of the dominant non-local coupling mechanism. Our results should facilitate the characterization of hybrid devices, and their optimization for various quantum-information-related experiments and applications.

Supercurrent in a Double Quantum Dot

We demonstrate the Josephson effect in a serial double quantum dot defined in a nanowire with epitaxial superconducting leads. The supercurrent stability diagram adopts a honeycomb pattern. We observe sharp discontinuities in the magnitude of the critical current, I_{c}, as a function of dot occupation, related to doublet to singlet ground state transitions. Detuning of the energy levels offers a tuning knob for I_{c}, which attains a maximum at zero detuning. The consistency between experiment and theory indicates that our device is a faithful realization of the two-impurity Anderson model.

Tuning the Coupling of an Individual Magnetic Impurity to a Superconductor: Quantum Phase Transition and Transport

The exchange scattering at magnetic adsorbates on superconductors gives rise to Yu-Shiba-Rusinov (YSR) bound states. Depending on the strength of the exchange coupling, the magnetic moment perturbs the Cooper pair condensate only weakly, resulting in a free-spin ground state, or binds a quasiparticle in its vicinity, leading to a (partially) screened spin state. Here, we use the flexibility of Fe-porphin (FeP) molecules adsorbed on a Pb(111) surface to reversibly and continuously tune between these distinct ground states. We find that the FeP moment is screened in the pristine adsorption state. Approaching the tip of a scanning tunneling microscope, we exert a sufficiently strong attractive force to tune the molecule through the quantum phase transition into the free-spin state. We ascertain and characterize the transition by investigating the transport processes as function of tip-molecule distance, exciting the YSR states by single-electron tunneling as well as (multiple) Andreev reflections.

Spin-Orbit Interaction and Induced Superconductivity in a One-Dimensional Hole Gas

Low dimensional semiconducting structures with strong spin-orbit interaction (SOI) and induced superconductivity attracted great interest in the search for topological superconductors. Both the strong SOI and hard superconducting gap are directly related to the topological protection of the predicted Majorana bound states. Here we explore the one-dimensional hole gas in germanium silicon (Ge-Si) core-shell nanowires (NWs) as a new material candidate for creating a topological superconductor. Fitting multiple Andreev reflection measurements shows that the NW has two transport channels only, underlining its one-dimensionality. Furthermore, we find anisotropy of the Landé g-factor that, combined with band structure calculations, provides us qualitative evidence for the direct Rashba SOI and a strong orbital effect of the magnetic field. Finally, a hard superconducting gap is found in the tunneling regime and the open regime, where we use the Kondo peak as a new tool to gauge the quality of the superconducting gap.

Mirage Andreev Spectra Generated by Mesoscopic Leads in Nanowire Quantum Dots

We study transport mediated by Andreev bound states formed in InSb nanowire quantum dots. Two kinds of superconducting source and drain contacts are used: epitaxial Al/InSb devices exhibit a doubling of tunneling resonances, while, in NbTiN/InSb devices, Andreev spectra of the dot appear to be replicated multiple times at increasing source-drain bias voltages. In both devices, a mirage of a crowded spectrum is created. To describe the observations a model is developed that combines the effects of a soft induced gap and of additional Andreev bound states both in the quantum dot and in the finite regions of the nanowire adjacent to the quantum dot. Understanding of Andreev spectroscopy is important for the correct interpretation of Majorana experiments done on the same structures.

Yu-Shiba-Rusinov screening of spins in double quantum dots

A magnetic impurity coupled to a superconductor gives rise to a Yu–Shiba–Rusinov (YSR) state inside the superconducting energy gap. With increasing exchange coupling the excitation energy of this state eventually crosses zero and the system switches to a YSR ground state with bound quasiparticles screening the impurity spin by ħ/2. Here we explore indium arsenide (InAs) nanowire double quantum dots tunnel coupled to a superconductor and demonstrate YSR screening of spin-1/2 and spin-1 states. Gating the double dot through nine different charge states, we show that the honeycomb pattern of zero-bias conductance peaks, archetypal of double dots coupled to normal leads, is replaced by lines of zero-energy YSR states. These enclose regions of YSR-screened dot spins displaying distinctive spectral features, and their characteristic shape and topology change markedly with tunnel coupling strengths. We find excellent agreement with a simple zero-bandwidth approximation, and with numerical renormalization group calculations for the two-orbital Anderson model.

Coupling superconductors to mesoscopic systems leads to unusual effects that could be exploited in new devices including topological quantum computers. Here the authors present a double quantum dot with a Yu–Shiba–Rusinov ground state arising from the interplay of Coulomb interactions and superconductivity.

Noise of a Chargeless Fermi Liquid

We construct a Fermi liquid theory to describe transport in a superconductor-quantum dot-normal metal junction close to the singlet-doublet (parity changing) transition of the dot. Though quasiparticles do not have a definite charge in this chargeless Fermi liquid, in the case of particle-hole symmetry, a mapping to the Anderson model unveils a hidden U(1) symmetry and a corresponding pseudocharge. In contrast to other correlated Fermi liquids, the back scattering noise reveals an effective charge equal to the charge of Cooper pairs, e^{*}=2e. In addition, we find a strong suppression of noise when the linear conductance is unitary, even for its nonlinear part.

Scaling of Yu-Shiba-Rusinov energies in the weak-coupling Kondo regime

The competition of the free-spin state of a paramagnetic impurity on a superconductor with its screened counterpart is characterized by the energy scale of Kondo screening compared to the superconducting pairing energy Δ. When the experimental temperature suppresses Kondo screening, but preserves superconductivity, i.e., when Δ/k_B > T > T_K (k_B is Boltzmann’s constant and T_K the Kondo temperature), this description fails. Here, we explore this temperature range in a set of manganese phthalocyanine molecules decorated with ammonia on Pb(111). We show that these molecules suffice the required energy conditions by exhibiting weak-coupling Kondo resonances. We correlate the Yu-Shiba-Rusinov bound states energy inside the superconducting gap with the intensity of the Kondo resonance. The observed correlation follows the expectations for a classical spin on a superconductor. This finding is important in view of many theoretical predictions using a classical spin model, in particular for the description of Majorana bound states in magnetic nanostructures on superconducting substrates.

The description of a paramagnetic impurity on a superconductor remains elusive in the weak-coupling Kondo regime. Here, Hatter et al. correlate the energy of the Yu-Shiba-Rusinov bound states with the intensity of the Kondo resonances in such a regime, revealing a behavior well described by classical spin models.

Higher-order spin and charge dynamics in a quantum dot-lead hybrid system

Understanding the dynamics of open quantum systems is important and challenging in basic physics and applications for quantum devices and quantum computing. Semiconductor quantum dots offer a good platform to explore the physics of open quantum systems because we can tune parameters including the coupling to the environment or leads. Here, we apply the fast single-shot measurement techniques from spin qubit experiments to explore the spin and charge dynamics due to tunnel coupling to a lead in a quantum dot-lead hybrid system. We experimentally observe both spin and charge time evolution via first- and second-order tunneling processes, and reveal the dynamics of the spin-flip through the intermediate state. These results enable and stimulate the exploration of spin dynamics in dot-lead hybrid systems, and may offer useful resources for spin manipulation and simulation of open quantum systems.

Andreev molecules in semiconductor nanowire double quantum dots

Chains of quantum dots coupled to superconductors are promising for the realization of the Kitaev model of a topological superconductor. While individual superconducting quantum dots have been explored, control of longer chains requires understanding of interdot coupling. Here, double quantum dots are defined by gate voltages in indium antimonide nanowires. High transparency superconducting niobium titanium nitride contacts are made to each of the dots in order to induce superconductivity, as well as probe electron transport. Andreev bound states induced on each of dots hybridize to define Andreev molecular states. The evolution of these states is studied as a function of charge parity on the dots, and in magnetic field. The experiments are found in agreement with a numerical model.

Quantum dots in a nanowire are one possible approach to creating a solid-state quantum simulator. Here, the authors demonstrate the coupling of electronic states in a double quantum dot to form Andreev molecule states; a potential building block for longer chains suitable for quantum simulation.

Shiba Bound States across the Mobility Edge in Doped InAs Nanowires

We present a study of Andreev quantum dots fabricated with small-diameter (30 nm) Si-doped InAs nanowires where the Fermi level can be tuned across a mobility edge separating localized states from delocalized states. The transition to the insulating phase is identified by a drop in the amplitude and width of the excited levels and is found to have remarkable consequences on the spectrum of superconducting subgap resonances. While at deeply localized levels only quasiparticle cotunneling is observed, for slightly delocalized levels Shiba bound states form and a parity-changing quantum phase transition is identified by a crossing of the bound states at zero energy. Finally, in the metallic regime, single Andreev resonances are observed.

Fano-Andreev effect in a T-shape double quantum dot in the Kondo regime

In the present work, we investigate the electronic transport through a T-shape double quantum dot system coupled to two normal leads and to one superconducting lead. We explore the interplay between Kondo and Andreev states due to proximity effects. We find that Kondo resonance is modified by the Andreev bound states, which manifest through Fano antiresonances in the local density of states of the embedded quantum dot and normal transmission. This means that there is a correlation between Andreev bound states and Fano resonances that is robust under the influence of high electronic correlation. We have also found that the dominant couplings at the quantum dots are characterized by a crossover region that defines the range where the Fano-Kondo and the Andreev-Kondo effect prevail in each quantum dot. Likewise, we find that the interaction between Kondo and Andreev bound states has a notable influence on the Andreev transport.

Non-Fermi-liquid behavior in quantum impurity models with superconducting channels

We study how the non-Fermi-liquid nature of the overscreened multi-channel Kondo impurity model affects the response to a BCS pairing term that, in the absence of the impurity, opens a gap Δ. We find that the low-energy spectrum in the limit Δ → 0 actually does not correspond to the spectrum strictly at Δ = 0. In particular, in the two-channel Kondo model the Δ → 0 ground state is an orbitally degenerate spin-singlet, while it is an orbital singlet with a residual spin degeneracy at Δ = 0. In addition, there are fractionalized spin-1/2 sub-gap excitations whose energy in units of Δ tends towards a finite and universal value when Δ → 0; as if the universality of the anomalous power-law exponents that characterise the overscreened Kondo effect turned into universal energy ratios when the scale invariance is broken by Δ ≠ 0. This intriguing phenomenon can be explained by the renormalisation flow towards the overscreened fixed point and the gap cutting off the orthogonality catastrophe singularities. We also find other non-Fermi liquid features at finite Δ: the local density of states lacks coherence peaks, the states in the continuum above the gap are unconventional, and the boundary entropy is a non-monotonic function of temperature. The persistent sub-gap excitations are characteristic of the non-Fermi-liquid fixed-point of the model, and thus depend on the impurity spin and the number of screening channels.

Robust Majorana Conductance Peaks for a Superconducting Lead

Experimental evidence for Majorana bound states largely relies on measurements of the tunneling conductance. While the conductance into a Majorana state is in principle quantized to 2e^{2}/h, observation of this quantization has been elusive, presumably due to temperature broadening in the normal-metal lead. Here, we propose to use a superconducting lead instead, whose gap strongly suppresses thermal excitations. For a wide range of tunneling strengths and temperatures, a Majorana state is then signaled by symmetric conductance peaks at eV=±Δ of a universal height G=(4-π)2e(2)/h. For a superconducting scanning tunneling microscope tip, Majorana states appear as spatial conductance plateaus while the conductance varies with the local wave function for trivial Andreev bound states. We discuss effects of nonresonant (bulk) Andreev reflections and quasiparticle poisoning.

Magnetic anisotropy in Shiba bound states across a quantum phase transition

The exchange coupling between magnetic adsorbates and a superconducting substrate leads to Shiba states inside the superconducting energy gap and a Kondo resonance outside the gap. The exchange coupling strength determines whether the quantum many-body ground state is a Kondo singlet or a singlet of the paired superconducting quasiparticles. Here we use scanning tunnelling spectroscopy to identify the different quantum ground states of manganese phthalocyanine on Pb(111). We observe Shiba states, which are split into triplets by magnetocrystalline anisotropy. Their characteristic spectral weight yields an unambiguous proof of the nature of the quantum ground state. Our results provide experimental insights into the phase diagram of a magnetic impurity on a superconducting host and shine light on the effects induced by magnetic anisotropy on many-body interactions.

The exchange coupling strength between magnetic adsorbates and a superconducting surface determines the nature of the system's quantum ground state. Here, the authors use scanning tunnelling microscopy to explore the ground state and excited state properties of manganese phthalocyanine adsorbed on a Pb(111) surface.

Novel non-local effects in three-terminal hybrid devices with quantum dot

We predict non-local effect in the three-terminal hybrid device consisting of the quantum dot (QD) tunnel coupled to two normal and one superconducting reservoirs. It manifests itself as the negative non-local resistance and results from the competition between the ballistic electron transfer (ET) and the crossed Andreev scattering (CAR). The effect is robust both in the linear and non-linear regimes. In the latter case the screening of charges and the long-range interactions play significant role. We show that sign change of the non-local conductance depends on the subgap Shiba/Andreev states, and it takes place even in absence of the Coulomb interactions. The effect is large and can be experimentally verified using the four probe setup. Since the induced non-local voltage changes sign and magnitude upon varying the gate potential and/or coupling of the quantum dot to the superconducting lead, such measurement could hence provide a controlled and precise method to determine the positions of the Shiba/Andreev states. Our predictions ought to be contrasted with non-local effects observed hitherto in the three-terminal planar junctions where the residual negative non-local conductance has been observed at large voltages, related to the Thouless energy of quasiparticles tunneling through the superconducting slab.

Hard gap in epitaxial semiconductor-superconductor nanowires

Many present and future applications of superconductivity would benefit from electrostatic control of carrier density and tunnelling rates, the hallmark of semiconductor devices. One particularly exciting application is the realization of topological superconductivity as a basis for quantum information processing. Proposals in this direction based on the proximity effect in semiconductor nanowires are appealing because the key ingredients are currently in hand. However, previous instances of proximitized semiconductors show significant tunnelling conductance below the superconducting gap, suggesting a continuum of subgap states--a situation that nullifies topological protection. Here, we report a hard superconducting gap induced by the proximity effect in a semiconductor, using epitaxial InAs-Al semiconductor-superconductor nanowires. The hard gap, together with favourable material properties and gate-tunability, makes this new hybrid system attractive for a number of applications, as well as fundamental studies of mesoscopic superconductivity.

Andreev bound-state dynamics in quantum-dot Josephson junctions: a washing out of the 0-π transition

We consider a Josephson junction formed by a quantum dot connected to two bulk superconductors in the presence of Coulomb interaction and coupling to both an electromagnetic environment and a finite density of electronic quasiparticles. In the limit of a large superconducting gap we obtain a Born-Markov description of the relevant Andreev bound-states dynamics. We calculate the current-phase relation and we find that the experimentally unavoidable presence of quasiparticles can dramatically modify the 0-π standard transition picture. We show that photon-assisted quasiparticle absorption allows the dynamic switching from the 0 to the π state and vice versa, washing out the 0-π transition predicted by purely thermodynamic arguments. Spectroscopic signatures of Andreev bound-states broadening are investigated by considering microwave irradiation.

Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures

The physics and operating principles of hybrid superconductor-semiconductor devices rest ultimately on the magnetic properties of their elementary subgap excitations, usually called Andreev levels. Here we report a direct measurement of the Zeeman effect on the Andreev levels of a semiconductor quantum dot with large electron g-factor, strongly coupled to a conventional superconductor with a large critical magnetic field. This material combination allows spin degeneracy to be lifted without destroying superconductivity. We show that a spin-split Andreev level crossing the Fermi energy results in a quantum phase transition to a spin-polarized state, which implies a change in the fermionic parity of the system. This crossing manifests itself as a zero-bias conductance anomaly at finite magnetic field with properties that resemble those expected for Majorana modes in a topological superconductor. Although this resemblance is understood without evoking topological superconductivity, the observed parity transitions could be regarded as precursors of Majorana modes in the long-wire limit.

In-gap states of a quantum dot coupled between a normal and a superconducting lead

We study the in-gap states of a quantum dot hybridized with one conducting and another superconducting electrode. The proximity effect suppresses the electronic states in the entire subgap regime |ω| < Δ, where Δ denotes the energy gap of the superconductor. The Andreev scattering mechanism can induce, however, some in-gap states whose line-broadening (inverse life-time) is controlled by the hybridization of the quantum dot with the normal electrode. We show that the number of such Andreev bound states is substantially dependent on the competition between the Coulomb repulsion and the induced on-dot pairing. We discuss the signatures of these in-gap states in the tunneling conductance, especially in a low-bias regime.

Tunneling spectroscopy of quasiparticle bound states in a spinful Josephson junction

The spectrum of a segment of InAs nanowire, confined between two superconducting leads, was measured as function of gate voltage and superconducting phase difference using a third normal-metal tunnel probe. Subgap resonances for odd electron occupancy-interpreted as bound states involving a confined electron and a quasiparticle from the superconducting leads, reminiscent of Yu-Shiba-Rusinov states-evolve into Kondo-related resonances at higher magnetic fields. An additional zero-bias peak of unknown origin is observed to coexist with the quasiparticle bound states.

Transport measurement of Andreev bound states in a Kondo-correlated quantum dot

We report nonequilibrium transport measurements of gate-tunable Andreev bound states in a carbon nanotube quantum dot coupled to two superconducting leads. In particular, we observe clear features of two types of Kondo ridges, which can be understood in terms of the interplay between the Kondo effect and superconductivity. In the first type (type I), the coupling is strong and the Kondo effect is dominant. Levels of the Andreev bound states display anticrossing in the middle of the ridge. On the other hand, crossing of the two Andreev bound states is shown in the second type (type II) together with the 0-π transition of the Josephson junction. Our scenario is well understood in terms of only a single dimensionless parameter, k(B)T(K)(min)/Δ, where T(K)(min) and Δ are the minimum Kondo temperature of a ridge and the superconducting order parameter, respectively. Our observation is consistent with measurements of the critical current, and is supported by numerical renormalization group calculations.

Zero-bias anomaly in a nanowire quantum dot coupled to superconductors

We studied the low-energy states of spin-1/2 quantum dots defined in InAs/InP nanowires and coupled to aluminum superconducting leads. By varying the superconducting gap Δ with a magnetic field B we investigated the transition from strong coupling Δ << T(K) to weak-coupling Δ >> T(K), where T(K) is the Kondo temperature. Below the critical field, we observe a persisting zero-bias Kondo resonance that vanishes only for low B or higher temperatures, leaving the room to more robust subgap structures at bias voltages between Δ and 2Δ. For strong and approximately symmetric tunnel couplings, a Josephson supercurrent is observed in addition to the Kondo peak. We ascribe the coexistence of a Kondo resonance and a superconducting gap to a significant density of intragap quasiparticle states, and the finite-bias subgap structures to tunneling through Shiba states. Our results, supported by numerical calculations, own relevance also in relation to tunnel-spectroscopy experiments aiming at the observation of Majorana fermions in hybrid nanostructures.

Josephson current through interacting double quantum dots with spin-orbit coupling

We study the effect of Rashba spin-orbit interaction on the Josephson current through a double quantum dot in the presence of Coulomb repulsion. In particular, we describe the characteristic effects on the magnetic field-induced singlet-triplet transition in the molecular regime. Exploring the whole parameter space, we analyze the effects of the device asymmetry, the orientation of the applied magnetic field with respect to the spin-orbit interaction, and finite temperatures. We find that at finite temperatures the orthogonal component of the spin-orbit interaction exhibits a similar effect to the Coulomb interaction inducing the occurrence of a π-phase at particle-hole symmetry. This provides a new route to the experimental observability of the π-phase in multi-level quantum dots.

Thermoelectric transport in the three terminal quantum dot

The thermoelectric transport in the system composed of a quantum dot in contact with superconducting, ferromagnetic and normal metal electrodes has been studied. Such a system can support pure spin current in the normal electrode. In the limit of a large superconducting gap and weak coupling between the dot and the electrodes we investigate the sub-gap charge and spin transport via Andreev mechanism using the standard master equation technique, which is known to be valid in the sequential tunnelling regime. The Zeeman splitting of the dot level induces pure spin current in the ferromagnetic electrode under an appropriate bias. This opens a novel possibility to switch the spin current between two electrodes by electric means. The calculated spin and charge thermopower coefficients attain very large values, of the order of a few hundreds μV K(-1), and show similar dependences on the position of the on-dot energy level and temperature.

Hybrid superconductor-quantum dot devices

Advances in nanofabrication techniques have made it possible to make devices in which superconducting electrodes are connected to non-superconducting nanostructures such as quantum dots. The properties of these hybrid devices result from a combination of a macroscopic quantum phenomenon involving large numbers of electrons (superconductivity) and the ability to control single electrons, offered by quantum dots. Here we review research into electron transport and other fundamental processes that have been studied in these devices. We also describe potential applications, such as a transistor in which the direction of a supercurrent can be reversed by adding just one electron to a quantum dot.