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Parker, W. H. (1975). Modified heating theory of nonequilibrium superconductors. Phys. Rev. B, 12(9), 3667–3672.
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Perrin, N., & Vanneste, C. (1983). Response of superconducting films to a periodic optical irradiation. Phys. Rev. B, 28(9), 5150–5159.
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Smolyaninov, I. I., Zayats, A. V., Stanishevsky, A., & Davis, C. C. (2002). Optical control of photon tunneling through an array of nanometer-scale cylindrical channels. Phys. Rev. B, 66(20), 205414_1–205414_5.
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Su, M. Y., Carter, S. G., & Sherwin, M. S. (2003). Strong-field terahertz optical mixing in excitons. Phys. Rev. B, 67(12).
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Ciulin, V., Carter, S. G., & Sherwin, M. S. (2004). Terahertz optical mixing in biased GaAs single quantum wells. Phys. Rev. B, 70(11), 115312–(1–6).
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Kerman, A. J., Yang, J. K. W., Molnar, R. J., Dauler, E. A., & Berggren, K. K. (2009). Electrothermal feedback in superconducting nanowire single-photon detectors. Phys. Rev. B, 79(10), 4.
Abstract: We investigate the role of electrothermal feedback in the operation of superconducting nanowire single-photon detectors (SNSPDs). It is found that the desired mode of operation for SNSPDs is only achieved if this feedback is unstable, which happens naturally through the slow electrical response associated with their relatively large kinetic inductance. If this response is sped up in an effort to increase the device count rate, the electrothermal feedback becomes stable and results in an effect known as latching, where the device is locked in a resistive state and can no longer detect photons. We present a set of experiments which elucidate this effect and a simple model which quantitatively explains the results.
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Bulaevskii, L. N., Graf, M. J., Batista, C. D., & Kogan, V. G. (2011). Vortex-induced dissipation in narrow current-biased thin-film superconducting strips. Phys. Rev. B, 83(14), 9.
Abstract: A vortex crossing a thin-film superconducting strip from one edge to the other, perpendicular to the bias current, is the dominant mechanism of dissipation for films of thickness d on the order of the coherence length ξ and of width w much narrower than the Pearl length Λâ‰<ab>wâ‰<ab>ξ. At high bias currents I*<I<Ic the heat released by the crossing of a single vortex suffices to create a belt-like normal-state region across the strip, resulting in a detectable voltage pulse. Here Ic is the critical current at which the energy barrier vanishes for a single vortex crossing. The belt forms along the vortex path and causes a transition of the entire strip into the normal state. We estimate I* to be roughly Ic/3. Furthermore, we argue that such “hot†vortex crossings are the origin of dark counts in photon detectors, which operate in the regime of metastable superconductivity at currents between I* and Ic. We estimate the rate of vortex crossings and compare it with recent experimental data for dark counts. For currents below I*, that is, in the stable superconducting but resistive regime, we estimate the amplitude and duration of voltage pulses induced by a single vortex crossing.
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Bulaevskii, L. N., Graf, M. J., & Kogan, V. G. (2012). Vortex-assisted photon counts and their magnetic field dependence in single-photon superconducting detectors. Phys. Rev. B, 85(1), 9.
Abstract: We argue that photon counts in a superconducting nanowire single-photon detector (SNSPD) are caused by the transition from a current-biased metastable superconducting state to the normal state. Such a transition is triggered by vortices crossing the thin and narrow superconducting strip from one edge to another due to the Lorentz force. Detector counts in SNSPDs may be caused by three processes: (a) a single incident photon with sufficient energy to break enough Cooper pairs to create a normal-state belt across the entire width of the strip (direct photon count), (b) thermally induced single-vortex crossing in the absence of photons (dark count), which at high-bias currents releases the energy sufficient to trigger the transition to the normal state in a belt across the whole width of the strip, and (c) a single incident photon of insufficient energy to create a normal-state belt but initiating a subsequent single-vortex crossing, which provides the rest of the energy needed to create the normal-state belt (vortex-assisted single-photon count). We derive the current dependence of the rate of vortex-assisted photon counts. The resulting photon count rate has a plateau at high currents close to the critical current and drops as a power law with high exponent at lower currents. While the magnetic field perpendicular to the film plane does not affect the formation of hot spots by photons, it causes the rate of vortex crossings (with or without photons) to increase. We show that by applying a magnetic field one may characterize the energy barrier for vortex crossings and identify the origin of dark counts and vortex-assisted photon counts.
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Santhanam, P., Wind, S., & Prober, D. E. (1987). Localization, superconducting fluctuations, and superconductivity in thin films and narrow wires of aluminum. Phys. Rev. B, 35(7), 3188–3206.
Abstract: We report a comprehensive set of experiments on wide and narrow thin-film strips of aluminum which test the predictions of recent localization theory. The experiments on wide films in the two-dimensional regime confirm the theoretical predictions and also yield insight into inelastic mechanisms and spin-orbit scattering rates. Our extension of the existing theory for one-dimensional systems to include spin-orbit scattering and Maki-Thompson superconducting fluctuations is verified by the experiments. We find clear evidence for one-dimensional localization, with inferred inelastic rates identical to those in two-dimensional films. The prediction of the localization theory for a dimensional crossover from two-dimensional to one-dimensional behavior is also confirmed. We have reanalyzed the results of some previous experiments on thin films and narrow wires in light of these results.
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Vercruyssen, N., Verhagen, T. G. A., Flokstra, M. G., Pekola, J. P., & Klapwijk, T. M. (2012). Evanescent states and nonequilibrium in driven superconducting nanowires. Phys. Rev. B, 85, 224503(1–10).
Abstract: We study the nonlinear response of current transport in a superconducting diffusive nanowire between normal reservoirs. We demonstrate theoretically and experimentally the existence of two different superconducting states appearing when the wire is driven out of equilibrium by an applied bias, called the global and bimodal superconducting states. The different states are identified by using two-probe measurements of the wire, and measurements of the local density of states with tunneling probes. The analysis is performed within the framework of the quasiclassical kinetic equations for diffusive superconductors.
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Semenov, A. D., Nebosis, R. S., Gousev, Y. P., Heusinger, M. A., & Renk, K. F. (1995). Analysis of the nonequilibrium photoresponse of superconducting films to pulsed radiation by use of a two-temperature model. Phys. Rev. B, 52(1), 581–590.
Abstract: Photoresponse of a superconducting film in the resistive state to pulsed radiation has been studied in the framework of a model assuming that two different effective temperatures can be assigned to the quasiparticle and phonon nonequilibrium distributions. The coupled electron-phonon-substrate system is described by a system of time-dependent energy-balance differential equations for effective temperatures. An analytical solution of the system is given and calculated voltage transients are compared with experimental photoresponse signals taking into account the radiation pulse shape and the time resolution of the readout electronics. It is supposed that a resistive state (vortices, fluxons, network of intergrain junctions, hot spots, phase slip centers) provides an ultrafast connection between electron temperature changes and changes of the film resistance and thus plays a minor role in the temporal evolution of the response. In accordance with experimental observations a two-component response was revealed from simulations. The slower component corresponds to a bolometric mechanism while the fast component is connected with the relaxation of the electron temperature. Calculated photoresponse transients are presented for different ratios of the electron and phonon specific heat, radiation pulse durations and fluences, and frequency band passes of registration electronics. From the amplitude of the bolometric component we determine the radiation energy absorbed in a film. This enables us to reveal an intrinsic electron-phonon scattering time even if it is much shorter than the time resolution of readout electronics. We analyze experimental voltage transients for NbN, YBa2Cu3O7, and TlBa2Ca2Cu3O9 superconducting films and find the electron-phonon interaction times at the transition temperatures of 17, 2.5, and 1.8 ps, respectively. The values are in reasonable agreement with data of other experiments.
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Tinkham, M., Free, J. U., Lau, C. N., & Markovic, N. (2003). Hysteretic I–V curves of superconducting nanowires. Phys. Rev. B, 68, 134515(1 to 7).
Abstract: Experimental I–V curves of superconducting MoGe nanowires show hysteresis for the thicker wires and none for the thinner wires. A rather quantitative account of these data for representative wires is obtained by numerically solving the one-dimensional heat flow equation to find a self-consistent distribution of temperature and local resistivity along the wire, using the measured linear resistance R(T) as input. This suggests that the retrapping current in the hysteretic I–V curves is primarily determined by heating effects, and not by the dynamics of phase motion in a tilted washboard potential as often assumed. Heating effects and thermal fluctuations from the low-resistance state to a high-resistance, quasinormal regime appear to set independent upper bounds for the switching current.
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Pekker, D., Shah, N., Sahu, M., Bezryadin, A., & Goldbart, P. M. (2009). Stochastic dynamics of phase-slip trains and superconductive-resistive switching in current-biased nanowires. Phys. Rev. B, 80, 214525 (1 to 17).
Abstract: Superconducting nanowires fabricated via carbon-nanotube templating can be used to realize and study quasi-one-dimensional superconductors. However, measurement of the linear resistance of these nanowires have been inconclusive in determining the low-temperature behavior of phase-slip fluctuations, both quantal and thermal. Thus, we are motivated to study the nonlinear current-voltage characteristics in current-biased nanowires and the stochastic dynamics of superconductive-resistive switching, as a way of probing phase-slip events. In particular, we address the question: can a single phase-slip event occurring somewhere along the wire—during which the order-parameter fluctuates to zero—induce switching, via the local heating it causes? We explore this and related issues by constructing a stochastic model for the time evolution of the temperature in a nanowire whose ends are maintained at a fixed temperature. We derive the corresponding master equation as a tool for evaluating and analyzing the mean switching time at a given value of current (smaller than the depairing critical current). The model indicates that although, in general, several phase-slip events are necessary to induce switching via a thermal runaway, there is indeed a regime of temperatures and currents in which a single event is sufficient. We carry out a detailed comparison of the results of the model with experimental measurements of the distribution of switching currents, and provide an explanation for the rather counterintuitive broadening of the distribution width that is observed upon lowering the temperature. Moreover, we identify a regime in which the experiments are probing individual phase-slip events, and thus offer a way of unearthing and exploring the physics of nanoscale quantum tunneling of the one-dimensional collective quantum field associated with the superconducting order parameter.
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Romijn, J., Klapwijk, T. M., Renne, M. J., & Mooij, J. E. (1982). Critical pair-breaking current in superconducting aluminum strips far below Tc. Phys. Rev. B, 26(7), 3648–3655.
Abstract: Critical currents of narrow, thin aluminum strips have been measured as a function of temperature. For the smallest samples uniformity of the current density is obtained over a large temperature range. Hence the intrinsic limit on the currentcarrying capacity of the superconductor was measured outside the Ginzburg-Landau -regime. The experimental values are compared with recent theoretical predictions by Kupriyanov and Lukichev. An approximate method of solving their equations is given, the results of which agree with the exact solution to within 1%. Experimental data are in excellent agreement with theoretical predictions. The absolute values agree if one assumes a Ïl value of 4×10–16 Ωm2 with vF=1.3×106 m/s. This value for Ïl is the same as that found from measurements of the anomalous skin effect but differs from values extracted from size-effect-limited resistivity.
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Huard, B., Pothier, H., Esteve, D., & Nagaev, K. E. (2007). Electron heating in metallic resistors at sub-Kelvin temperature. Phys. Rev. B, 76, 165426(1–9).
Abstract: In the presence of Joule heating, the electronic temperature in a metallic resistor placed at sub-Kelvin temperatures can significantly exceed the phonon temperature. Electron cooling proceeds mainly through two processes: electronic diffusion to and from the connecting wires and electron-phonon coupling. The goal of this paper is to present a general solution of the problem in a form that can easily be used in practical situations. As an application, we compute two quantities that depend on the electronic temperature profile: the second and the third cumulant of the current noise at zero frequency, as a function of the voltage across the resistor. We also consider time-dependent heating, an issue relevant for experiments in which current pulses are used, for instance, in time-resolved calorimetry experiments.
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