Andreev Bound States Research Articles

Direct Microwave Spectroscopy of Andreev Bound States in Planar Ge Josephson Junctions

We demonstrate microwave measurements of the Andreev-bound-state (ABS) spectrum in planar Josephson junctions (JJs) defined in Ge high-mobility two-dimensional hole gases contacted by superconducting platinum germanosilicide (PtSiGe). The JJs and readout circuitry are located on separate chips and inductively coupled via flip-chip bonding. For a device with 350-nm junction length, the spectroscopic signatures were consistent with the short-junction limit, with an induced superconducting gap Δ∗≈48μeV and transmission τ≈0.94. The interaction between the highest-transmission ABS and the resonator was well described by a Jaynes-Cummings model with a vacuum Rabi splitting of approximately 6 MHz. A device with a junction length of 1μm showed an ABS spectrum consistent with a long-junction model. Time-resolved monitoring of the readout resonator in the dispersive regime revealed gate-voltage tunable-junction parity fluctuations on the time scale of seconds. Our work indicates a viable path toward hybrid quantum devices based on planar Ge. Published by the American Physical Society 2024

Josephson effect in a Fibonacci quasicrystal

Quasiperiodicity has recently been proposed to enhance superconductivity and its proximity effect. Simultaneously, there has been significant experimental progress in the fabrication of quasiperiodic structures, including in reduced dimensions. Motivated by these developments, we use microscopic tight-binding theory to investigate the DC Josephson effect through a ballistic Fibonacci chain attached to two superconducting leads. The Fibonacci chain is one of the most-studied examples of quasicrystals, hosting a rich multifractal spectrum, containing topological gaps with different winding numbers. We study how the Andreev-bound states (ABS), current-phase relation, and the critical current depend on the quasiperiodic degrees of freedom, from short to long junctions. While the current-phase relation shows a traditional 2π sinusoidal or sawtooth profile, we find that the ABS develop quasiperiodic oscillations and that the Andreev reflection is qualitatively altered, leading to quasiperiodic oscillations in the critical current as a function of junction length. Surprisingly, despite earlier proposals of quasiperiodicity enhancing superconductivity compared to crystalline junctions, we do not, in general, find that it enhances the critical current. However, we find significant current enhancement for reduced interface transparency because of the modified Andreev reflection. Furthermore, by varying the chemical potential, e.g., by an applied gate voltage, we find a fractal oscillation between superconductor-normal metal-superconductor (SNS) and superconductor-insulator-superconductor (SIS) behavior. Finally, we show that the winding of the subgap states leads to an equivalent winding in the critical current, such that the winding numbers, and thus the topological invariant, can be determined. Published by the American Physical Society 2024

Spin-Degeneracy Breaking and Parity Transitions in Three-Terminal Josephson Junctions

Hybrid Josephson junctions (JJs) realized in superconductor-semiconductor heterostructures host fermionic modes known as Andreev bound states (ABSs). In these structures, a promising and yet unexplored avenue for harnessing spin and parity degrees of freedom is offered by JJs with three or more superconducting terminals, where phase-induced spin polarization and transitions of the ground state to an odd parity were predicted to arise. Here we spectroscopically probe the two-dimensional band structure of ABSs in a phase-controlled InAs/Al three-terminal JJ. Andreev bands show signatures of spin-degeneracy breaking, with level splitting in excess of ∼9 GHz, and zero-energy crossings associated to ground state fermion parity transitions. Spin splitting and parity transitions are enabled and controlled by locally applied magnetic fluxes, in the absence of Zeeman effect or Coulomb blockade. Our results underscore the potential of multiterminal hybrid devices for phase engineering ABSs, with significant implications for spin- and parity-based quantum devices. Published by the American Physical Society 2024

Spectral and Transport Properties of Andreev Molecule Coupled to Majorana Wire

AbstractThe quasiparticle spectrum and transport properties of the double quantum dot (DQD) deposited on a superconducting substrate (Andreev molecule) and side‐coupled to a nanowire hosting Majorana zero modes (MZMs) are studied. Placing a DQD on the superconducting substrate induces the trivial Andreev‐bound states (ABSs) in quantum dots. However, coupling of DQD with a nanowire causes the leakage of the MZM from the topological nanowire into quantum dots. The relationship between the Andreev states and the Majorana mode for different values of the coupling parameters is analyzed. Additionally, it is shown that the connection point of a metallic tip, treated as an scanning tunneling microscope (STM) tip, affects the measured results of the differential conductance.

Controllable Andreev Bound States in Bilayer Graphene Josephson Junctions from Short to Long Junction Limits.

We demonstrate that the mode number of Andreev bound states in bilayer graphene Josephson junctions can be modulated by controlling the superconducting coherence length insitu. By exploiting the quadratic band dispersion of bilayer graphene, we control the Fermi velocity and thus the coherence length via the application of electrostatic gating. Tunneling spectroscopy of the Andreev bound states reveals a crossover from short to long Josephson junction regimes as we approach the charge neutral point of the bilayer graphene. Furthermore, analysis of different mode numbers of the Andreev energy spectrum allows us to estimate the phase-dependent Josephson current quantitatively. Our Letter provides a new way for studying multimode Andreev levels by tuning the Fermi velocity.

Transport signatures of Bogoliubov Fermi surfaces in normal metal/time-reversal symmetry broken d-wave superconductor junctions

In recent times, Bogoliubov Fermi surfaces (BFSs) in superconductors (SCs) have drawn significant attention due to a substantial population of Bogoliubov quasiparticles (BQPs) together with Cooper pairs (CPs) in them. The BQPs as zero energy excitations give rise to captivating and intricate charge dynamics within the BFSs. In this theoretical study, we propose to reveal the unique signatures of the topologically protected BFSs in bulk d-wave SCs using normal metal/time-reversal symmetry (TRS) broken d-wave SC hybrid setup, in terms of the differential conductance and Fano factor (FF). Orientation of crystal a axis with respect to junction normal, quantified by the parameter α, is crucial for transport properties in these hybrid devices. For α = 0, an enhancement in zero-bias conductance (ZBC) can be identified as a key signature of BFSs. However, for α≠0 , this feature does not replicate due to the presence of the localized Andreev bound state (ABS) at the interface. The interplay of ABS and BFSs gives rise to an anomalous behavior in ZBC compared to the α = 0 case. This behavior remains qualitatively similar even at finite temperatures. Finally, we explain this anomalous behavior by analyzing the effective charge of the carriers in terms of the FF. The simplicity of our setup based on d-wave SC makes our proposal persuasive.

Supercurrent noise in a phase-biased superconductor-normal ring in thermal equilibrium

In superconductor-normal-superconductor (SNS) junctions, supercurrent is mediated via Andreev bound states (ABSs) controlled by the phase difference between the two superconductors. Theory has long predicted significant noise of such supercurrent in equilibrium, due to thermal excitation between the ABSs. Via the fluctuation-dissipation theorem (FDT), this leads to a finite dissipative conductance that coexists with the supercurrent but is hidden in dc measurement. Here, we directly measure the supercurrent noise at radio frequency in a phase-biased SNS ring inductively coupled to a superconducting resonator. We also measure the admittance of the same system whose real part is the dissipative conductance, and quantitatively verify the FDT relation of the SNS ring. The dissipative conductance shows a 1/T temperature dependence, in contrast to the Drude conductance of an unproximitized metal. Using linear response theory, we attribute this behavior to the enhanced current correlations due to the electron-hole symmetry imposed by the proximity effect. Our results reveal the commonly overlooked role of thermal fluctuations in superconducting hybrid systems and other mesoscopic or quantum material systems with large orbital susceptibility. Published by the American Physical Society 2024

Subgap spectroscopy along hybrid nanowires by nm-thick tunnel barriers

Tunneling spectroscopy is widely used to examine the subgap spectra in semiconductor-superconductor nanostructures when searching for Majorana zero modes (MZMs). Typically, semiconductor sections controlled by local gates at the ends of hybrids serve as tunnel barriers. Besides detecting states only at the hybrid ends, such gate-defined tunnel probes can cause the formation of non-topological subgap states that mimic MZMs. Here, we develop an alternative type of tunnel probes to overcome these limitations. After the growth of an InSb-Al hybrid nanowire, a precisely controlled in-situ oxidation of the Al shell is performed to yield a nm-thick AlOx layer. In such thin isolating layer, tunnel probes can be arbitrarily defined at any position along the hybrid nanowire by shadow-wall angle-deposition of metallic leads. In this work, we make multiple tunnel probes along single nanowire hybrids and successfully identify Andreev bound states (ABSs) of various spatial extension residing along the hybrids.

Subgap states and quantum phase transitions in one-dimensional superconductor–ferromagnetic insulator heterostructures

We theoretically study the spectral properties of a one dimensional semiconductor-superconductor-ferromagnetic insulator (SE-SU-FMI) hybrid nanostructure, motivated by recent experiments where such devices have been fabricated using epitaxial growing techniques. We model the hybrid structure as a one-dimensional single-channel semiconductor nanowire under the simultaneous effect of two proximity-induced interactions: superconducting pairing and a (spatially inhomogeneous) Zeeman exchange field. The coexistence of these competing mechanisms generates a rich quantum phase diagram and a complex subgap Andreev bound state (ABS) spectrum. By exploiting the symmetries of the problem, we classify the solutions of the Bogoliubov-de Gennes equations into even and odd ABS with respect to the spatial inversion symmetry $x\ensuremath{\rightarrow}\ensuremath{-}x$. We find the ABS spectrum of the device as a function of the different parameters of the model: the length $L$ of the coexisting SU-FMI region, the induced Zeeman exchange field ${h}_{0}$, and the induced superconducting coherence length $\ensuremath{\xi}$. In particular we analyze the evolution of the subgap spectrum as a function of the length $L$. Interestingly, we generically find spin-polarized ABS emerging in the subgap region, which, depending on the ratio ${h}_{0}/\mathrm{\ensuremath{\Delta}}$, can eventually cross below the Fermi energy at certain critical values ${L}_{c}$, and induce spin- and fermion parity-changing quantum phase transitions. We argue that this type of device constitute a promising highly-tunable platform to engineer subgap ABS.

Multiple Andreev reflections in topological Josephson junctions with chiral Majorana modes

Andreev bound states (ABS) occur in Josephson junctions when the total phase of the Andreev and normal reflections is a multiple of $2\pi$. In ballistic junctions with an applied voltage bias, a quasi-particle undergoes multiple Andreev reflections before entering the leads, resulting in peaks in the current-voltage $I(V)$ curve. Here we present a general model for Josephson junctions with spin-active interlayers i.e., magnetic or topological materials with broken time-reversal symmetry. We investigate how ABS change the peak positions and shape of $I(V)$, which becomes asymmetric for a single incident angle. We show how the angle-resolved $I(V)$ curve becomes a spectroscopic tool for the chirality and degeneracy of ABS.

Revisiting Andreev processes in superconductor-graphene-superconductor (SGS) Josephson junctions: comparison with experimental results

In view of the recent progress in experiments on charge transport through various Josephson junctions made out of graphene, we have made a careful comparison between the theory and some of the available experimental results. Within the framework of a transfer matrix approach, we have first analytically derived the spectrum of Andreev bound states (ABS) in a superconductor -graphene-superconductor (SGS) junction for a wide range of experimentally relevant parameters. We have particularly considered the case of monolayer graphene (MLG). The theoretical results can account for both the retro Andreev reflection (RAR) and the specular Andreev reflection (SAR) in the relevant parameter range. Using the ABS spectrum we have evaluated the current through such junctions and the junction conductance from the analytically derived expressions at different bias voltages for a range of other system parameters directly taken from the experimental works. These theoretical results have then been compared with experimental results. Evaluated current and the conductance show scaling behaviour with change in the junction length and agree well with the experimental results. In the relevant parameter regime where the SAR process is dominant, the calculated values of the current and the conductivity have been found much lower than the corresponding values observed when the RAR process is dominant.

SuperConga: An open-source framework for mesoscopic superconductivity

We present SuperConga, an open-source framework for simulating equilibrium properties of unconventional and ballistic singlet superconductors, confined to two-dimensional (2D) mesoscopic grains in a perpendicular external magnetic field, at arbitrary low temperatures. It aims at being both fast and easy to use, enabling research without access to a computer cluster, and visualization in real-time with OpenGL. The core is written in C++ and CUDA, exploiting the embarrassingly parallel nature of the quasiclassical theory of superconductivity by utilizing the parallel computational power of modern graphics processing units. The framework self-consistently computes both the superconducting order-parameter and the induced vector potential and finds the current density, free energy, induced flux density, local density of states (LDOS), and the magnetic moment. A user-friendly Python frontend is provided, enabling simulation parameters to be defined via intuitive configuration files, or via the command-line interface, without requiring a deep understanding of implementation details. For example, complicated geometries can be created with relative ease. The framework ships with simple tools for analyzing and visualizing the results, including an interactive plotter for spectroscopy. An overview of the theory is presented, as well as examples showcasing the framework's capabilities and ease of use. The framework is free to download from https://gitlab.com/superconga/superconga, which also links to the extensive user manual, containing even more examples, tutorials, and guides. To demonstrate and benchmark SuperConga, we study the magnetostatics, thermodynamics, and spectroscopy of various phenomena. In particular, we study flux quantization in solenoids, vortex physics, surface Andreev bound-states, and a “phase crystal.” We compare our numeric results with analytics and present experimental observables, e.g., the magnetic moment and LDOS, measurable with, for example, scanning probes, STM, and magnetometry.

Probing Majorana bound states through an inhomogeneous Andreev double dot interferometer

In modern experiments with hybrid superconducting (SC)/semiconducting nanowires the presence of zero-energy Andreev bound states (ABSs), characterized by a partial overlap of the Majorana wave functions, is a common problem that significantly complicates the detection of a genuine Majorana bound state (MBS). In this article, taking into account spatial inhomogeneity of experimentally investigated nanowire samples, we study interference transport features of a curved heterostructure in which two normal wires (or arms) are separated by a superconducting wire. Since Andreev reflection on the two N/S interfaces with smoothly changing electrostatic and SC pairing potentials results in the emergence of bound states, the low-energy interference transport is described in the framework of the model of two noninteracting Andreev levels or the Andreev double quantum dot. A set of limiting cases is analyzed allowing us to highlight the interference properties that are unique for the different types of ABSs, such as bulk ABS, inhomogeneous ABS and MBS. In particular, considerable attention is paid to the features of the Aharonov-Bohm (AB) effect. It is shown that the response of each state can be recognized analyzing both AB period and extrema positions of the conductance oscillations which take place without any fine tuning of the system parameters.

Signatures of the long-range phase transition in topological Josephson junctions

The extended Kitaev models spark a recent surge of interest due to its unprecedented topological phase with fractional invariants. In this extended model of topological superconductors, long-range pairing interactions can induce a pair of subgap massive edge modes (MEMs), posing novel challenges to the unambiguous detection of Majorana bound states (MBSs) sustained by the same model in the short-range limit. Here we investigate the effect of a power-law decayed long-range pairing on the dc and ac Josephson currents between two Majorana nanowires. By varying the magnetic field and long-range pairing, the nanowires can be driven into different phases hosting MEMs, MBSs, or trivial Andreev bound states (ABSs), which considerably modulate the Josephson current. For weakly linked Josephson junctions, we show that the MEMs emerging in the long-range phases can induce a zigzag profile in the dc current-phase relation featured by two sign reversals deviating from the point where the superconducting phase difference $\ensuremath{\phi}=\ensuremath{\pi}$. In contrast, the Josephson current indicates a sharp sign reversal at $\ensuremath{\phi}=\ensuremath{\pi}$ in the short-range phase that hosts MBSs, while a smooth sinelike dc Josephson current appears in the presence of ABSs. In the nonequilibrium junction biased by a voltage, we identify a $4\ensuremath{\pi}$ ac Josephson current in the short-range phases, indicating a MBSs-mediated tunneling process dominating the supercurrents. Differently, when the system enters MEMs-hosted long-range phases, the ordinary ac Josephson current with $2\ensuremath{\pi}$ periodicity can be restored. In addition, we find the ac Josephson current induced by MEMs is significantly suppressed under the strong long-range pairing interactions. These signatures can serve as probes to discriminate MBSs from MEMs and ABSs.

Controlling Andreev Bound States with the Magnetic Vector Potential.

Tunneling spectroscopy measurements are often used to probe the energy spectrum of Andreev bound states (ABSs) in semiconductor-superconductor hybrids. Recently, this spectroscopy technique has been incorporated into planar Josephson junctions (JJs) formed in two-dimensional electron gases, a potential platform to engineer phase-controlled topological superconductivity. Here, we perform ABS spectroscopy at the two ends of planar JJs and study the effects of the magnetic vector potential on the ABS spectrum. We show that the local superconducting phase difference arising from the vector potential is equal in magnitude and opposite in sign at the two ends, in agreement with a model that assumes localized ABSs near the tunnel barriers. Complemented with microscopic simulations, our experiments demonstrate that the local phase difference can be used to estimate the relative position of localized ABSs separated by a few hundred nanometers.

Determination of spin polarization of metals and gap of superconductors by spin-polarized Andreev reflection spectroscopy

Andreev reflection spectroscopy (ARS) is a unique technique that can determine the spin polarization of a metal as well as the gap of a superconductor but d-wave superconductors are seldomly used in ARS due to the lack of quantitative model to extract parameters from experiments. Here we calculate spin-polarized ARS based on the modified Blonder-Tinkham-Klapwijk model for both s- and d-wave superconductors of a three-dimensional (3D) interface with a point angle. We show that the 3D interface and the point angle can drastically affect the spectra for both s- and d-wave ARS. For s-wave, the 3D effect is similar to that of an interfacial barrier but it can generate a variety of rich features in d-wave ARS and induce Andreev bound states (ABS). The spin polarization can linearly suppress both s- and d-wave spectra including ABS. We show that the ABS states can be controlled by the spin polarization, the point angle and the interfacial barrier but very high polarization is required to completely suppress the ABS. Furthermore, the Fermi wavevector mismatch as well as the carrier effective mass mismatch can cause effects very similar to that of an interfacial barrier for both s- and d-wave ARS.

Nonlocal signatures of hybridization between quantum dot and Andreev bound states

We investigate local and nonlocal signatures of hybridization between a quantum dot state and an extended Andreev bound state (ABS) in a gate-defined InAs nanowire with multiple side probes. When a quantum dot in one of the side probes was hybridized with an ABS in the nanowire, a characteristic spectroscopic pattern was observed both locally, i.e., in the probe with the quantum dot, and nonlocally, in the tunnel conductance of a remote probe. Nonlocal signatures of hybridization reveal the extended nature of the ABS.

Probing electron-hole weights of an Andreev bound state by transient currents

Andreev bound states (ABSs) are localized quantum states that contain both electron and hole components. They ubiquitously reside in inhomogeneous superconducting systems. Following theoretical analysis, we propose to probe the electron-hole weights of an ABS via a local tunneling measurement that detects the transient current under a steplike pulse bias. With our protocol, the ABS energy level can also be obtained from peaks of the Fourier spectrum of the transient current. Our protocol can be applied to detect robust zero-energy Majorana bound states (MBSs), which have equal electron-hole weights, in candidate platforms where local tunneling spectroscopy measurement is possible. In the one-dimensional Majorana nanowire model, we numerically calculate the electron-hole weights for different types of low-energy bound states, including ABSs, quasi-MBSs, and MBSs.

Distinguishing Majorana and Andreev bound states in a topological superconducting nanowire with a potential barrier

We propose a scheme to distinguish the Majorana bound states (MBSs) and the trivial Andreev bound states (ABSs) in a topological superconducting nanowire by applying a sharp local potential barrier on one part of the wire. The position and strength of the local potential barrier could be modulated by a gate voltage applied to the nanowire. It is found that a new pair of MBSs appear at both sides of the potential in the presence of the local potential barrier, as a manifestation of the topological nature. Differently, the local potential barrier does not induce extra ABSs. Such a difference can be traced back to the evolution of the conductance spectrum with the Zeeman field or the local potential strength. When modulating the local potential strength, the zero-bias conductance peak induced by MBSs can be split into two-, three-, or four-peak structures, while the zero-bias peak by ABSs is only split into two-peak form.

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.