Nonequilibrium Quasiparticle Distribution Research Articles

Nonequilibrium quasiparticle distribution in superconducting resonators: Effect of pair-breaking photons

Many superconducting devices rely on the finite gap in the excitation spectrum of a superconductor: thanks to this gap, at temperatures much smaller than the critical one the number of excitations (quasiparticles) that can impact the device’s behavior is exponentially small. Nevertheless, experiments at low temperature usually find a finite, non-negligible density of quasiparticles whose origin has been attributed to various non-equilibrium phenomena. Here, we investigate the role of photons with energy exceeding the pair-breaking threshold 2\Delta2Δ as a possible source for these quasiparticles in superconducting resonators. Modeling the interacting system of quasiparticles, phonons, sub-gap and pair-breaking photons using a kinetic equation approach, we find analytical expressions for the quasiparticles’ density and their energy distribution. Applying our theory to measurements of quality factor as function of temperature and for various readout powers, we find they could be explained by assuming a small number of photons above the pair-breaking threshold. We also show that frequency shift data can give evidence of quasiparticle heating.

Field, frequency, and temperature dependencies of the surface resistance of nitrogen diffused niobium superconducting radio frequency cavities

We investigate the rf performance of several single-cell superconducting radio-frequency cavities subjected to low temperature heat treatment in nitrogen environment. The cavities were treated at temperature 120–165 °C for an extended period of time (24–48 h) either in high vacuum or in a low partial pressure of ultrapure nitrogen. The improvement in Q0 with a Q rise was observed when nitrogen gas was injected at ∼300 °C during the cavity cooldown from 800 °C and held at 165 °C, without any degradation in accelerating gradient over the baseline performance. The treatment was applied to several elliptical cavities with frequency ranging from 0.75 to 3.0 GHz, showing an improved quality factor as a result of low temperature nitrogen treatments. The Q rise feature is similar to that achieved by nitrogen alloying Nb cavities at higher temperature, followed by material removal by electropolishing. The surface modification was confirmed by the change in electronic mean free path and tuned with the temperature and duration of heat treatment. The decrease of the temperature-dependent surface resistance with increasing rf field, resulting in a Q rise, becomes stronger with increasing frequency and decreasing temperature. The data suggest a crossover frequency of ∼0.95 GHz above that the Q rise phenomenon occurs at 2 K. Some of these results can be explained qualitatively with an existing model of intrinsic field-dependence of the surface resistance with both equilibrium and nonequilibrium quasiparticle distribution functions. The change in the Q slope below 0.95 GHz may result from masking contribution of trapped magnetic flux to the residual surface resistance. Published by the American Physical Society 2024

Nonequilibrium Quasiparticle Distribution in Superconducting Resonators: An Analytical Approach

In the superconducting state, the presence of a finite gap in the excitation spectrum implies that the number of excitations (quasiparticles) is exponentially small at temperatures well below the critical one. Conversely, minute perturbations can significantly impact both the distribution in energy and number of quasiparticles. Typically, the interaction with the electromagnetic environment is the main perturbation source driving quasiparticles out of thermal equilibrium, while a phonon bath is responsible for restoration of equilibrium. Here we derive approximate analytical solutions for the quasiparticle distribution function in superconducting resonators and explore the impact of nonequilibrium on two measurable quantities: the resonator's quality factor and its resonant frequency. Applying our results to experimental data, we conclude that while at intermediate temperatures there is clear evidence for the nonequilibrium effects due to heating of the quasiparticles by photons, the low-temperature measurements are not explained by this mechanism.

Nonlocal measurement of quasiparticle charge and energy relaxation in proximitized semiconductor nanowires using quantum dots

The lowest-energy excitations of superconductors do not carry an electric charge, as their wave function is equally electron-like and hole-like. This fundamental property is not easy to study in electrical measurements that rely on the charge to generate an observable signal. The ability of a quantum dot to act as a charge filter enables us to solve this problem and measure the quasiparticle charge in superconducting-semiconducting hybrid nanowire heterostructures. We report measurements on a three-terminal circuit, in which an injection lead excites a non-equilibrium quasiparticle distribution in the hybrid system, and the electron or hole component of the resulting quasiparticles is detected using a quantum dot as a tunable charge and energy filter. The results verify the chargeless nature of the quasiparticles at the gap edge and reveal the complete relaxation of injected charge and energy in a proximitized nanowire, resolving open questions in previous three-terminal experiments.

Josephson junctions based on amorphous MoGe: prospects for use in superconducting electronics

We have fabricated and characterized all-MoGe Josephson junctions with a very thin Al/AlO x /(Al) barrier, where the amorphous MoGe films exhibit superconducting transition temperatures up to 7 K. Due to the uniformity of the surface morphology of the MoGe films, the junctions demonstrate high uniformity of their tunneling properties. The experimental data on the temperature dependence of the subgap current agree well with theoretical calculations. The results obtained imply that Josephson tunnel junctions based on amorphous superconductors are promising candidates for use in superconducting electronics, especially in applications requiring multiple stacked junctions or the creation of a nonequilibrium quasiparticle distribution.

Controllable supercurrent in mesoscopic superconductor-normal metal-ferromagnet crosslike Josephson structures

A nonmonotonic dependence of the critical Josephson supercurrent on the injection current through a normal metal/ferromagnet weak link from a single domain ferromagnetic strip has been observed experimentally in nanofabricated planar crosslike S-N/F-S Josephson structures. This behavior is explained by 0–π and π–0 transitions, which can be caused by the suppression and Zeeman splitting of the induced superconductivity due to interaction between N and F layers, and the injection of spin-polarized current into the weak link. A model considering both effects has been developed. It shows the qualitative agreement between the experimental results and the theoretical model in terms of spectral supercurrent-carrying density of states of S-N/F-S structures and the spin-dependent double-step nonequilibrium quasiparticle distribution.

Voltage control of superconducting exchange interaction and anomalous Josephson effect

Exerting control of the magnetic exchange interaction in heterostructures is of both basic interest and has potential for use in spin-based applications relying on quantum effects. We here show that the sign of the exchange interaction in a spin-valve, determining whether the ferro- or antiferromagnetic configuration is favored, can be controlled via an electric voltage. This occurs due to an interplay between a nonequilibrium quasiparticle distribution and the presence of spin-polarized Cooper pairs. Additionally, we show that a voltage-induced distribution controls the anomalous supercurrent that occurs in magnetic Josephson junctions, obviating the challenging task to manipulate the magnetic texture of the system. This demonstrates that two key phenomena in superconducting spintronics, the magnetic exchange interaction and the phase shift generating the anomalous Josephson effect, can be controlled electrically. Our findings are of relevance for spin-based superconducting devices which in practice most likely have to be operated precisely by nonequilibrium effects.

Voltage-induced metal-insulator transition in a one-dimensional charge density wave

We present a theoretical investigation of the voltage-driven metal insulator transition based on solving coupled Boltzmann and Hartree-Fock equations to determine the insulating gap and the electron distribution in a model system -- a one dimensional charge density wave. Electric fields that are parametrically small relative to energy gaps can shift the electron distribution away from the momentum-space region where interband relaxation is efficient, leading to a highly non-equilibrium quasiparticle distribution even in the absence of Zener tunneling. The gap equation is found to have regions of multistability; a non-equilibrium analog of the free energy is constructed and used to determine which phase is preferred.

Enhanced quasiparticle lifetime in a superconductor by selective blocking of recombination phonons with a phononic crystal

When quasiparticles in a BCS superconductor recombine into Cooper pairs, phonons are emitted within a narrow band of energies above the pairing energy at 2$\Delta$. We show that a phonon bandgap restricting the escape of recombination phonons from a superconductor can increase the quasiparticle recombination lifetime by more than an order of magnitude. A phonon bandgap can be realized and matched to the recombination energy with a phononic crystal, a periodically-patterned dielectric membrane. We discuss in detail the non-equilibrium quasiparticle and phonon distributions that arise in a superconductor due to a phonon bandgap and a pair-breaking photon signal. Although intrinsically a non-equilibrium effect, the lifetime enhancement in the small-signal regime is remarkably similar to an estimate from an equilibrium formulation. The equilibrium estimate closely follows $\exp(\Omega_{bg}/k_BT_b)$, where $\Omega_{bg}$ is the phonon bandgap energy bandwidth above 2$\Delta$, and $T_b$ is the phonon bath temperature of the coupled electron-phonon system. We discuss the impact of a phononic bandgap on the performance of superconducting devices, and propose a superconducting microwave resonator circuit to measure the enhancement in the quasiparticle lifetime.

Relaxation of photoexcitations in polaron-induced magnetic microstructures

We investigate the evolution of a photoexcitation in correlated materials over a wide range of time scales. The system studied is a one-dimensional model of a manganite with correlated electron, spin, orbital, and lattice degrees of freedom, which we relate to the three-dimensional material Pr$_{1-x}$Ca$_{x}$MnO$_3$. The ground-state phases for the entire composition range are determined and rationalized by a coarse-grained polaron model. At half-doping a pattern of antiferromagnetically coupled Zener polarons is realized. Using time-dependent density-matrix renormalization group (tDMRG), we treat the electronic quantum dynamics following the excitation. The emergence of quasiparticles is addressed, and the relaxation of the nonequilibrium quasiparticle distribution is investigated via a linearized quantum-Boltzmann equation. Our approach shows that the magnetic microstructure caused by the Zener polarons leads to an increase of the relaxation times of the excitation.

Interplay of the Inverse Proximity Effect and Magnetic Field in Out-of-Equilibrium Single-Electron Devices

The magnetic field is shown to affect significantly non-equilibrium quasiparticle (QP) distributions under conditions of inverse proximity effect on the remarkable example of a single-electron hybrid turnstile. This effect suppresses the gap in the superconducting leads in the vicinity of turnstile junctions with a Coulomb blockaded island, thus, trapping hot QPs in this region. Applied magnetic field creates additional QP traps in the form of vortices or regions with reduced superconducting gap in the leads resulting in release of QPs away from junctions. We present clear experimental evidence of such interplay of the inverse proximity effect and a magnetic field revealing itself in the superconducting gap enhancement in a magnetic field as well as in significant improvement of the turnstile characteristics. The observed interplay of the inverse proximity effect and external magnetic field, and its theoretical explanation in the context of QP overheating are important for various superconducting and hybrid nanoelectronic devices, which find applications in quantum computation, photon detection and quantum metrology.

The non-equilibrium response of a superconductor to pair-breaking radiation measured over a broad frequency band

We have measured the absorption of terahertz radiation in a BCS superconductor over a broad range of frequencies from 200 GHz to 1.1 THz, using a broadband antenna-lens system and a tantalum microwave resonator. From low frequencies, the response of the resonator rises rapidly to a maximum at the gap edge of the superconductor. From there on, the response drops to half the maximum response at twice the pair-breaking energy. At higher frequencies, the response rises again due to trapping of pair-breaking phonons in the superconductor. In practice, this is a measurement of the frequency dependence of the quasiparticle creation efficiency due to pair-breaking in a superconductor. The efficiency, calculated from the different non-equilibrium quasiparticle distribution functions at each frequency, is in agreement with the measurements.

Nonequilibrium superconducting thin films with sub-gap and pair-breaking photon illumination

We calculate nonequilibrium quasiparticle and phonon distributions for a number of widely-used low transition temperature thin-film superconductors under constant, uniform illumination by sub-gap probe and pair-breaking signal photons simultaneously. From these distributions we calculate material-characteristic parameters that allow rapid evaluation of an effective quasiparticle temperature using a simple analytical expression, for all materials studied (Mo, Al, Ta, Nb, and NbN) for all photon energies. We also explore the temperature and energy-dependence of the low-energy quasiparticle generation efficiency η by pair-breaking signal photons finding in the limit of thick films at low bath temperatures that is material-independent. Taking the energy distribution of excess quasiparticles into account, we find as the bath temperature approaches the transition temperature in agreement with the assumption of the two-temperature model of the nonequilibrium response that is appropriate in that regime. The behaviour of η with signal frequency scaled by the superconducting energy gap is also shown to be material-independent, and is in qualitative agreement with recent experimental results. An enhancement of η in the presence of sub-gap (probe) photons is shown to be most significant at signal frequencies near the superconducting gap frequency and arises due to multiple photon absorption events that increase the average energy of excess quasiparticles above that in the absence of the probe.

Recovering of superconductivity in S/F bilayers under spin-dependent nonequilibrium quasiparticle distribution

We study theoretically the influence of spin accumulation on superconductivity in a superconductor/ferromagnet bilayer. It is well-known that the superconductivity in S/F bilayers is suppressed by the proximity to a ferromagnet. The spin accumulation by itself is also a depairing factor. But here we show that creation of the spin accumulation on top of effective exchange depairing, caused by the proximity to a ferromagnet, can lead to an opposite result. The superconductivity can be partially recovered by spin-dependent quasiparticle distribution. The systems with realistic parameters are considered and the possible experimental setup is proposed.

Long-range spin imbalance in mesoscopic superconductors under Zeeman splitting

We develop a theory of spin relaxation in Zeeman-splitted superconducting films at low temperatures. A new mechanism of spin relaxation, specific only for Zeeman-splitted superconductors is proposed. It can explain the extremely high spin relaxation lengths, experimentally observed in Zeeman-splitted superconductors, and their strong growth with the magnetic field. In the framework of this mechanism the observed spin signal is formed by the spin-independent nonequilibrium quasiparticle distribution weighted by the spin-split DOS. We demonstrate that the relaxation length of such a spin signal is determined by the energy relaxation length at energies of the order of the superconducting gap.

Long-range triplet Josephson current driven by the bias voltage

We study the long-range triplet Josephson current in a clean junction composed of two s-wave superconductors and a normal-metal/ferromagnet/normal-metal trilayer. Through applying the bias voltages on the metal regions by two antiparallel half-metal electrodes, we show that the amplitude and direction of this long-range current can be controlled easily and flexibly. Such current arises from the fact that the applied voltage can produce a nonequilibrium spin-dependent quasiparticle distribution in the metal regions so that the Cooper pairs acquire an extra momenta, which will lead to a spin-flip processes in the metal regions. This processes can produce the parallel spin triplet pairs in the central ferromagnet layer. In particular, if the voltage is applied only on one metal region, we further find that the recently discovered long-range superharmonic Josephson current will appear because of the transport of an even number of parallel spin triplet pairs.

Enhancing of the critical temperature of an in-plane FFLO state in heterostructures by the orbital effect of the magnetic field

It is well-known that the orbital effect of the magnetic field suppresses superconducting $T_c$. We show that for a system, which is in the Larkin-Ovchinnikov-Fulde-Ferrell (FFLO) state at zero external magnetic field, the orbital effect of an applied magnetic field can lead to the enhancement of the critical temperature higher than $T_c$ at zero field. We concentrate on two systems, where the in-plane FFLO-state was predicted recently. These are equilibrium S/F bilayers and S/N bilayers under nonequilibrium quasiparticle distribution. However, it is suggested that such an effect can take place for any plane superconducting heterostructure, which is in the in-plane FFLO-state (or is close enough to it) at zero applied field.

Non-equilibrium superconductivity in quantum-sensing superconducting resonators

Low temperature microwave superconducting resonators (SRs) are attractive candidates for producing quantum-sensitive, arrayable energy or power detectors for astrophysical and other precision measurement applications. Their readout uses a microwave probe signal with quanta of energy well below the threshold for pair-breaking in the superconductor. We have calculated the non-equilibrium quasiparticle and phonon distributions generated by the photons of the probe signal of a resonator operating well below its superconducting transition temperature Tc as the absorbed probe power was changed using the coupled kinetic equations described by Chang and Scalapino. The calculations give insight into a rate equation estimate which suggests that the quasiparticle distributions can be driven far from their thermal equilibrium value for typical readout powers. From the driven quasiparticle distribution functions, the driven quasiparticle number densities and lifetimes were calculated. An effective temperature to describe the driven quasiparticles was defined. The non-equilibrium lifetimes were compared to the distribution-averaged thermal lifetimes at the effective temperature and good agreement was found typically within a few per cent. We used the non-equilibrium quasiparticle distribution to model a representative SR. The complex conductivity and hence the frequency dependence of the experimentally measured forward scattering parameter S21 of the SR as a function of absorbed power were found. The non-equilibrium S21 cannot be accurately modeled by a thermal distribution even at its own elevated temperature, having a higher quality factor in all cases studied, although for low absorbed powers the two effective temperatures are similar. From the non-equilibrium lifetimes and number densities we determined the achievable noise equivalent power (NEP) of the resonator used as a power detector as a function of absorbed microwave power. Simpler expressions to evaluate the effective quasiparticle temperature as a function of absorbed power have also been derived. We conclude that multiple photon absorption from the microwave probe increases the quasiparticle number above the thermal background and ultimately limits the achievable NEP of the resonator at temperatures well below Tc.

Long-Range Proximity Effect for Opposite-Spin Pairs in Superconductor-Ferromagnet Heterostructures Under Nonequilibrium Quasiparticle Distribution

By now it is known that in a singlet superconductor-ferromagnet (S-F) structure the superconducting correlations carried by opposite-spin pairs penetrate into the ferromagnet over a short distance of the order of magnetic coherence length. The long-range proximity effect (LRPE), taking place on the length scale of the normal metal coherence length, can only be maintained by equal-spin pairs, which can be generated by magnetic inhomogeneities in the system. In this Letter, we have predicted a new type of LRPE, which can take place in S-F heterostructures under nonequilibrium conditions. The superconducting correlations in the F region are generated by opposite-spin Cooper pairs and equal-spin pairs are not involved. The possibility for an opposite-spin pair to penetrate into the ferromagnet over a large distance is provided by creation of the proper nonequilibrium quasiparticle distribution there. This leads to a sharp increase (up to a few orders of magnitude) of the critical Josephson current through a S-F-S junction at some values of the voltage controlling the nonequilibrium distribution in the F interlayer.

Coherent nonlocal correlations in Andreev interferometers

Andreev interferometers, hybrid normal–superconducting loops, are well suited for studying phase-coherent effects, as they combine the robust quantum phase of a superconductor with a finite resistance normal section where phase-dependent transport properties can be easily measured. While they have previously been used to demonstrate local, phase-coherent properties, they may also be integrated into structures where nonlocal phase coherence occurs. In this paper, we use Andreev interferometer devices to experimentally establish the existence of nonlocal phase coherence between two normal metals linked by a superconductor. The generation of phase-coherent nonlocal signals is brought about by producing a nonequilibrium quasiparticle distribution in the normal section of the interferometer and tuning the phase with an external magnetic flux through the loop. Quasiclassical modeling of our experiment shows that the nonequilibrium distribution, coupled with the flux, leads to an induced supercurrent that is not constant along the length of the interferometer's normal section. The supercurrent–quasiparticle current conversion that occurs in this section is manifested in the production of flux-dependent nonlocal voltages through the mechanisms of crossed Andreev reflection and elastic cotunneling.