Recombination-limited energy relaxation in a Bardeen-Cooper-Schrieffer superconductor

Bipolar thermoelectric Josephson engine

Thermoelectric effects in metals are typically small due to the nearly perfect particle-hole symmetry around their Fermi surface. Furthermore, thermo-phase effects and linear thermoelectricity in superconducting systems have been identified only when particle-hole symmetry is explicitly broken, since thermoelectric effects were considered impossible in pristine superconductors. Here, we experimentally demonstrate that superconducting tunnel junctions develop a very large bipolar thermoelectricity in the presence of a sizable thermal gradient thanks to spontaneous particle-hole symmetry breaking. Our junctions show Seebeck coefficients of up to ±300 μV K^-1, which is comparable with quantum dots and roughly 10⁵ times larger than the value expected for normal metals at subkelvin temperatures. Moreover, by integrating our junctions into a Josephson interferometer, we realize a bipolar thermoelectric Josephson engine generating phase-tunable electric powers of up to ~140 nW mm^-2. Notably, our device implements also the prototype for a persistent thermoelectric memory cell, written or erased by current injection. We expect that our findings will lead to applications in superconducting quantum technologies.

Gate-Controlled Suspended Titanium Nanobridge Supercurrent Transistor

Under standard conditions, the electrostatic field-effect is negligible in conventional metals and was expected to be completely ineffective also in superconducting metals. This common belief was recently put under question by a family of experiments that displayed full gate-voltage-induced suppression of critical current in superconducting all-metallic gated nanotransistors. To date, the microscopic origin of this phenomenon is under debate, and trivial explanations based on heating effects given by the negligible electron leakage from the gates should be excluded. Here, we demonstrate the control of the supercurrent in fully suspended superconducting nanobridges. Our advanced nanofabrication methods allow us to build suspended superconducting Ti-based supercurrent transistors which show ambipolar and monotonic full suppression of the critical current for gate voltages of V_G^C ≃ 18 V and for temperatures up to ∼80% of the critical temperature. The suspended device architecture minimizes the electron-phonon interaction between the superconducting nanobridge and the substrate, and therefore, it rules out any possible contribution stemming from charge injection into the insulating substrate. Besides, our finite element method simulations of vacuum electron tunneling from the gate to the bridge and thermal considerations rule out the cold-electron field emission as a possible driving mechanism for the observed phenomenology. Our findings promise a better understanding of the field effect in superconducting metals.

Thermionic junction devices utilizing phonon blocking

Electrothermal elements are used in various energy harvesters, coolers, and radiation detectors. The optimal operation of these elements relies on mastering two competing boundary conditions: the maximization of the electrothermal response and the blockade of lattice (phonon) thermal conduction. In this work, we propose and demonstrate that efficient electrothermal operation and phonon blocking can be achieved in solid-state thermionic junctions, paving the way for new phonon-engineered high-efficiency refrigerators and sensors. Our experimental demonstration uses semiconductor-superconductor (Sm-S) junctions where the electrothermal response arises from the superconducting energy gap and the phonon blocking results from the acoustic transmission bottleneck at the junction. We demonstrate a cooling platform where a silicon chip, suspended only from the Sm-S junctions, is cooled by ~40% from the bath temperature. We also show how the observed effect can be used in radiation detectors and multistage electronic refrigerators suitable for cooling of quantum technology devices.

Thermodynamics of a Phase-Driven Proximity Josephson Junction

We study the thermodynamic properties of a superconductor/normal metal/superconductor Josephson junction in the short limit. Owing to the proximity effect, such a junction constitutes a thermodynamic system where phase difference, supercurrent, temperature and entropy are thermodynamical variables connected by equations of state. These allow conceiving quasi-static processes that we characterize in terms of heat and work exchanged. Finally, we combine such processes to construct a Josephson-based Otto and Stirling cycles. We study the related performance in both engine and refrigerator operating mode.

Thermodynamic cycles in Josephson junctions

A superconductor/normal metal/superconductor Josephson junction is a coherent electron system where the thermodynamic entropy depends on temperature and difference of phase across the weak-link. Here, exploiting the phase-temperature thermodynamic diagram of a thermally isolated system, we argue that a cooling effect can be achieved when the phase drop across the junction is brought from 0 to π in a iso-entropic process. We show that iso-entropic cooling can be enhanced with proper choice of geometrical and electrical parameters of the junction, i.e. by increasing the ratio between supercurrent and total junction volume. We present extensive numerical calculations using quasi-classical Green function methods for a short junction and we compare them with analytical results. Interestingly, we demonstrate that phase-coherent thermodynamic cycles can be implemented by combining iso-entropic and iso-phasic processes acting on the weak-link, thereby engineering the coherent version of thermal machines such as engines and cooling systems. We therefore evaluate their performances and the minimum temperature achievable in a cooling cycle.

Thermal transport of Josephson junction based on two-dimensional electron gas

We study the phase-dependent thermal transport of a short temperature-biased Josephson junction based on two-dimensional electron gas (2DEG) with both Rashba and Dresselhaus couplings. Except for thermal equilibrium temperature T, characters of thermal transport can also be manipulated by interaction parameter h0 and the parameter [Formula: see text] . A larger value and a sharper switching behavior of thermal conductance can be obtained if h0 takes suitable values and [Formula: see text] is larger. Finally, we propose a possible experimental setup based on the discussed Josephson junction and find that the temperature of the right superconducting electrode TR is influenced by the same three parameters in a similar way with thermal conductance. This setup may provide a valid method to select moderately-doped 2DEG materials and superconducting electrodes to control the change of temperature and obtain an efficient temperature regulator.

Phase-coherent solitonic Josephson heat oscillator

Since its recent foundation, phase-coherent caloritronics has sparkled continuous interest giving rise to numerous concrete applications. This research field deals with the coherent manipulation of heat currents in mesoscopic superconducting devices by mastering the Josephson phase difference. Here, we introduce a new generation of devices for fast caloritronics able to control local heat power and temperature through manipulation of Josephson vortices, i.e., solitons. Although most salient features concerning Josephson vortices in long Josephson junctions were comprehensively hitherto explored, little is known about soliton-sustained coherent thermal transport. We demonstrate that the soliton configuration determines the temperature profile in the junction, so that, in correspondence of each magnetically induced soliton, both the flowing thermal power and the temperature significantly enhance. Finally, we thoroughly discuss a fast solitonic Josephson heat oscillator, whose frequency is in tune with the oscillation frequency of the magnetic drive. Notably, the proposed heat oscillator can effectively find application as a tunable thermal source for nanoscale heat engines and coherent thermal machines.

Phase-Tunable Josephson Thermal Router

A fundamental aspect of electronics is the ability to distribute a charge current among different terminals. On the other hand, despite the great interest in dissipation, storage, and conversion of heat in solid state structures, the control of thermal currents at the nanoscale is still in its infancy. Here, we show the experimental realization of a phase-tunable thermal router able to control the spatial distribution of an incoming heat current, thus providing the possibility of tuning the electronic temperatures of two output terminals. This ability is obtained thanks to a direct current superconducting quantum interference device (dc SQUID), which can tune the coherent component of the electronic heat currents flowing through its Josephson junctions. By varying the external magnetic flux and the bath temperature, the SQUID allows us to regulate the size and the direction of the thermal gradient between two drain electrodes. Our results offer new opportunities for all microcircuits requiring an accurate energy management, including electronic coolers, quantum information architectures, and thermal logic components.

Towards phase-coherent caloritronics in superconducting circuits

The emerging field of phase-coherent caloritronics (from the Latin word calor, heat) is based on the possibility of controlling heat currents by using the phase difference of the superconducting order parameter. The goal is to design and implement thermal devices that can control energy transfer with a degree of accuracy approaching that reached for charge transport by contemporary electronic components. This can be done by making use of the macroscopic quantum coherence intrinsic to superconducting condensates, which manifests itself through the Josephson effect and the proximity effect. Here, we review recent experimental results obtained in the realization of heat interferometers and thermal rectifiers, and discuss a few proposals for exotic nonlinear phase-coherent caloritronic devices, such as thermal transistors, solid-state memories, phase-coherent heat splitters, microwave refrigerators, thermal engines and heat valves. Besides being attractive from the fundamental physics point of view, these systems are expected to have a vast impact on many cryogenic microcircuits requiring energy management, and possibly lay the first stone for the foundation of electronic thermal logic.

0-π phase-controllable thermal Josephson junction

Two superconductors coupled by a weak link support an equilibrium Josephson electrical current that depends on the phase difference ϕ between the superconducting condensates. Yet, when a temperature gradient is imposed across the junction, the Josephson effect manifests itself through a coherent component of the heat current that flows opposite to the thermal gradient for |ϕ| < π/2 (refs 2-4). The direction of both the Josephson charge and heat currents can be inverted by adding a π shift to ϕ. In the static electrical case, this effect has been obtained in a few systems, for example via a ferromagnetic coupling or a non-equilibrium distribution in the weak link. These structures opened new possibilities for superconducting quantum logic and ultralow-power superconducting computers. Here, we report the first experimental realization of a thermal Josephson junction whose phase bias can be controlled from 0 to π. This is obtained thanks to a superconducting quantum interferometer that allows full control of the direction of the coherent energy transfer through the junction. This possibility, in conjunction with the completely superconducting nature of our system, provides temperature modulations with an unprecedented amplitude of ∼100 mK and transfer coefficients exceeding 1 K per flux quantum at 25 mK. Then, this quantum structure represents a fundamental step towards the realization of caloritronic logic components such as thermal transistors, switches and memory devices. These elements, combined with heat interferometers and diodes, would complete the thermal conversion of the most important phase-coherent electronic devices and benefit cryogenic microcircuits requiring energy management, such as quantum computing architectures and radiation sensors.

Quantum-limited heat conduction over macroscopic distances

The emerging quantum technological apparatuses1, 2, such as the quantum computer3-6, call for extreme performance in thermal engineering7. Cold distant heat sinks are needed for the quantized electric degrees of freedom due to the increasing packaging density and heat dissipation. Importantly, quantum mechanics sets a fundamental upper limit for the flow of information and heat, which is quantified by the quantum of thermal conductance8-10. However, the short distance between the heat-exchanging bodies in the previous experiments11-14 hinders their applicability in quantum technology. Here, we present experimental observations of quantum-limited heat conduction over macroscopic distances extending to a metre. We achieved this improvement of four orders of magnitude in the distance by utilizing microwave photons travelling in superconducting transmission lines. Thus, it seems that quantum-limited heat conduction has no fundamental distance cutoff. This work establishes the integration of normal-metal components into the framework of circuit quantum electrodynamics15-17 which provides a basis for the superconducting quantum computer18-21. Especially, our results facilitate remote cooling of nanoelectronic devices using far-away in-situ-tunable heat sinks22, 23. Furthermore, quantum-limited heat conduction is important in contemporary thermodynamics24, 25. Here, the long distance may lead to ultimately efficient mesoscopic heat engines with promising practical applications26.

Tunable quasiparticle trapping in Meissner and vortex states of mesoscopic superconductors

Nowadays, superconductors serve in numerous applications, from high-field magnets to ultrasensitive detectors of radiation. Mesoscopic superconducting devices, referring to those with nanoscale dimensions, are in a special position as they are easily driven out of equilibrium under typical operating conditions. The out-of-equilibrium superconductors are characterized by non-equilibrium quasiparticles. These extra excitations can compromise the performance of mesoscopic devices by introducing, for example, leakage currents or decreased coherence time in quantum devices. By applying an external magnetic field, one can conveniently suppress or redistribute the population of excess quasiparticles. In this article, we present an experimental demonstration and a theoretical analysis of such effective control of quasiparticles, resulting in electron cooling both in the Meissner and vortex states of a mesoscopic superconductor. We introduce a theoretical model of quasiparticle dynamics, which is in quantitative agreement with the experimental data.

Excessive excitation induced by overheating may deteriorate the resistance-free operation of superconductor-based devices. Here, Taupin et al. propose an effective control of excess quasiparticles and their spatial distribution in a mesoscopic superconducting disc by applying a small magnetic field.

Nanoscale phase engineering of thermal transport with a Josephson heat modulator

Macroscopic quantum phase coherence has one of its pivotal expressions in the Josephson effect, which manifests itself both in charge and energy transport. The ability to master the amount of heat transferred through two tunnel-coupled superconductors by tuning their phase difference is the core of coherent caloritronics, and is expected to be a key tool in a number of nanoscience fields, including solid-state cooling, thermal isolation, radiation detection, quantum information and thermal logic. Here, we show the realization of the first balanced Josephson heat modulator designed to offer full control at the nanoscale over the phase-coherent component of thermal currents. Our device provides magnetic-flux-dependent temperature modulations up to 40 mK in amplitude with a maximum of the flux-to-temperature transfer coefficient reaching 200 mK per flux quantum at a bath temperature of 25 mK. Foremost, it demonstrates the exact correspondence in the phase engineering of charge and heat currents, breaking ground for advanced caloritronic nanodevices such as thermal splitters, heat pumps and time-dependent electronic engines.

Resolving thermoelectric "paradox" in superconductors

For almost a century, thermoelectricity in superconductors has been one of the most intriguing topics in physics. During its early stages in the 1920s, the mere existence of thermoelectric effects in superconductors was questioned. In 1944, it was demonstrated that the effects may occur in inhomogeneous superconductors. Theoretical breakthrough followed in the 1970s, when the generation of a measurable thermoelectric magnetic flux in superconducting loops was predicted; however, a major crisis developed when experiments showed puzzling discrepancies with the theory. Moreover, different experiments were inconsistent with each other. This led to a stalemate in bringing theory and experiment into agreement. With this work, we resolve this stalemate, thus solving this long-standing "paradox," and open prospects for exploration of novel thermoelectric phenomena predicted recently.

Rectification of electronic heat current by a hybrid thermal diode

Thermal diodes--devices that allow heat to flow preferentially in one direction--are one of the key tools for the implementation of solid-state thermal circuits. These would find application in many fields of nanoscience, including cooling, energy harvesting, thermal isolation, radiation detection and quantum information, or in emerging fields such as phononics and coherent caloritronics. However, both in terms of phononic and electronic heat conduction (the latter being the focus of this work), their experimental realization remains very challenging. A highly efficient thermal diode should provide a difference of at least one order of magnitude between the heat current transmitted in the forward temperature (T) bias configuration (Jfw) and that generated with T-bias reversal (Jrev), leading to ℛ = Jfw/Jrev ≫ 1 or ≪ 1. So far, ℛ ≈ 1.07-1.4 has been reported in phononic devices, and ℛ ≈ 1.1 has been obtained with a quantum-dot electronic thermal rectifier at cryogenic temperatures. Here, we show that unprecedentedly high ratios of ℛ ≈ 140 can be achieved in a hybrid device combining normal metals tunnel-coupled to superconductors. Our approach provides a high-performance realization of a thermal diode for electronic heat current that could be successfully implemented in true low-temperature solid-state thermal circuits.

Reduction of dissipative nonlinear conductivity of superconductors by static and microwave magnetic fields

A theory of dissipative nonlinear conductivity, σ(1)(ω,H), of s-wave superconductors under strong electromagnetic fields at low temperatures is proposed. Closed-form expressions for σ(1)(H) and the surface resistance R(s)(ω,H) are obtained in the nonequilibrium dirty limit for which σ(1)(H) has a significant minimum as a function of a low-frequency (ħω ≪ k(B)T) magnetic field H. The calculated microwave suppression of R(s)(H) is in good agreement with recent experiments on alloyed Nb resonator cavities. It is shown that superimposed dc and ac fields, H = H(0) + H(a)cosωt, can be used to reduce ac dissipation in thin film nanostructures by tuning σ(1)(H(0)) with the dc field.

A quantum diffractor for thermal flux

Macroscopic phase coherence between weakly coupled superconductors leads to peculiar interference phenomena. Among these, magnetic flux-driven diffraction might be produced, in full analogy to light diffraction through a rectangular slit. This can be experimentally revealed by the electric current and, notably, also by the heat current transmitted through the circuit. The former was observed more than 50 years ago and represented the first experimental evidence of the phase-coherent nature of the Josephson effect, whereas the second one was still lacking. Here we demonstrate the existence of heat diffraction by measuring the modulation of the electronic temperature of a small metallic electrode nearby-contacted to a thermally biased short Josephson junction subjected to an in-plane magnetic field. The observed temperature dependence exhibits symmetry under magnetic flux reversal, and clear resemblance with a Fraunhofer-like modulation pattern. Our approach, joined to widespread methods for phase-biasing superconducting circuits, might represent an effective tool for controlling the thermal flux in nanoscale devices.

Excitation of single quasiparticles in a small superconducting Al island connected to normal-metal leads by tunnel junctions

We investigate the dynamics of individual quasiparticle excitations on a small superconducting aluminum island connected to normal metallic leads by tunnel junctions. We find the island to be free of excitations within the measurement resolution. This allows us to show that the residual heating, which typically limits experiments on superconductors, has an ultralow value of less than 0.1 aW. By injecting electrons with a periodic gate voltage, we probe electron-phonon interaction and relaxation down to a single quasiparticle excitation pair, with a measured recombination rate of 16 kHz. Our experiment yields a strong test of BCS theory in aluminum as the results are consistent with it without free parameters.

The Josephson heat interferometer

The Josephson effect is perhaps the prototypical manifestation of macroscopic phase coherence, and forms the basis of a widely used electronic interferometer--the superconducting quantum interference device (SQUID). In 1965, Maki and Griffin predicted that the thermal current through a temperature-biased Josephson tunnel junction coupling two superconductors should be a stationary periodic function of the quantum phase difference between the superconductors: a temperature-biased SQUID should therefore allow heat currents to interfere, resulting in a thermal version of the electric Josephson interferometer. This phase-dependent mechanism of thermal transport has been the subject of much discussion but, surprisingly, has yet to be realized experimentally. Here we investigate heat exchange between two normal metal electrodes kept at different temperatures and tunnel-coupled to each other through a thermal 'modulator' (ref. 5) in the form of a direct-current SQUID. We find that heat transport in the system is phase dependent, in agreement with the original prediction. Our Josephson heat interferometer yields magnetic-flux-dependent temperature oscillations of up to 21 millikelvin in amplitude, and provides a flux-to-temperature transfer coefficient exceeding 60 millikelvin per flux quantum at 235 millikelvin. In addition to confirming the existence of a phase-dependent thermal current unique to Josephson junctions, our results point the way towards the phase-coherent manipulation of heat in solid-state nanocircuits.

Thermally induced parametric instability in a back-action evading measurement of a micromechanical quadrature near the zero-point level

We report the results of back-action evading experiments utilizing a tightly coupled electro-mechanical system formed by a radio frequency micromechanical resonator parametrically coupled to a NbTiN superconducting microwave resonator. Due to excess dissipation in the microwave resonator, we observe a parametric instability induced by a thermal shift of the mechanical resonance frequency. In light of these measurements, we discuss the constraints on microwave dissipation needed to perform BAE measurements far below the zero-point level.

Micrometre-scale refrigerators

A superconductor with a gap in the density of states or a quantum dot with discrete energy levels is a central building block in realizing an electronic on-chip cooler. They can work as energy filters, allowing only hot quasiparticles to tunnel out from the electrode to be cooled. This principle has been employed experimentally since the early 1990s in investigations and demonstrations of micrometre-scale coolers at sub-kelvin temperatures. In this paper, we review the basic experimental conditions in realizing the coolers and the main practical issues that are known to limit their performance. We give an update of experiments performed on cryogenic micrometre-scale coolers in the past five years.

Thermal conductance by the inverse proximity effect in a superconductor

We study heat transport in hybrid lateral normal-metal-superconductor-normal-metal structures. We find the thermal conductance of a short superconducting wire to be strongly enhanced beyond the BCS value due to the inverse proximity effect, resulting from contributions of elastic cotunneling and crossed Andreev reflection of quasiparticles. Our measurements agree with a model based on the quasiclassical theory of inhomogeneous superconductivity in the diffusive limit.

Hierarchical energy relaxation in mesoscopic tunnel junctions: effect of a nonequilibrium environment on low-temperature transport

We develop a theory of far from the equilibrium transport in arrays of tunnel junctions. We find that if the rate of the electron-electron interactions exceeds the rate of the electron-phonon energy exchange, the energy relaxation ensuring the charge transfer may occur sequentially. In particular, cotunneling transport in arrays of junctions is dominated by the relaxation via the intermediate bosonic environment, the electron-hole excitations, rather than by the electron-phonon mechanism. The current-voltage characteristics are highly sensitive to the spectrum of the environmental modes and to the applied bias, which sets the lower bound for the effective temperature. We demonstrate that the energy gap in the electron-hole spectrum which opens below some critical temperature T* due to long-range Coulomb interactions gives rise to the suppression of the tunneling current.

Electronic refrigeration at the quantum limit

We demonstrate quantum-limited electronic refrigeration of a metallic island in a low-temperature microcircuit. We show that matching the impedance of the circuit enables refrigeration at a distance, of about 50 microm in our case, through superconducting leads with a cooling power determined by the quantum of thermal conductance. In a reference sample with a mismatched circuit this effect is absent. Our results are consistent with the concept of electromagnetic heat transport. We observe and analyze the crossover between electromagnetic and quasiparticle heat flux in a superconductor.