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arXiv:2001.02540 · New Journal of Physics

Nonlinear effects in superconducting thin film microwave resonators

C N Thomas, S Withington, Z Sun, T Skyrme, D J Goldie

Published May 29, 2020• 21 citations
arXivHTMLPDF

Abstract

Abstract We discuss how reactive and dissipative nonlinearities affect the intrinsic response of superconducting thin-film resonators. We explain how most, if not all, of the complex phenomena commonly seen can be described by a model in which the underlying resonance is a single-pole Lorentzian, but whose centre frequency and quality factor change as external parameters, such as readout power and frequency, are varied. What is seen during a vector-network-analyser measurement is series of samples taken from an ideal Lorentzian that is shifting and spreading as the readout frequency is changed. According to this model, it is perfectly proper to refer to, and measure, the resonant frequency and quality factor of the underlying resonance, even though the swept-frequency curves appear highly distorted and hysteretic. In those cases where the resonance curve is highly distorted, the specific shape of the trajectory in the Argand plane gives valuable insights into the second-order physical processes present. We discuss the formulation and consequences of this approach in the case of nonlinear kinetic inductance, two-level-system loss, quasiparticle generation, and a generic model based on a power-law form. The generic model captures the key features of specific dissipative nonlinearities, but additionally leads to insights into how general dissipative processes create characteristic forms in the Argand plane. We provide detailed formulations in each case, and indicate how they lead to the wide variety of phenomena commonly seen in experimental data. We also explain how the properties of the underlying resonance can be extracted from this data. Overall, our paper provides a self-contained compendium of behaviour that will help practitioners interpret and determine important parameters from distorted swept-frequency measurements.

DOI
10.1088/1367-2630/ab97e8

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Abstract We discuss how reactive and dissipative nonlinearities affect the intrinsic response of superconducting thin-film resonators. We explain how most, if not all, of the complex phenomena commonly seen can be described by a model in which the underlying resonance is a single-pole Lorentzian, but whose centre frequency and quality factor change as external parameters, such as readout power and frequency, are varied. What is seen during a vector-network-analyser measurement is series of samples taken from an ideal Lorentzian that is shifting and spreading as the readout frequency is changed. According to this model, it is perfectly proper to refer to, and measure, the resonant frequency and quality factor of the underlying resonance, even though the swept-frequency curves appear highly distorted and hysteretic. In those cases where the resonance curve is highly distorted, the specific shape of the trajectory in the Argand plane gives valuable insights into the second-order physical processes present. We discuss the formulation and consequences of this approach in the case of nonlinear kinetic inductance, two-level-system loss, quasiparticle generation, and a generic model based on a power-law form. The generic model captures the key features of specific dissipative nonlinearities, but additionally leads to insights into how general dissipative processes create characteristic forms in the Argand plane. We provide detailed formulations in each case, and indicate how they lead to the wide variety of phenomena commonly seen in experimental data. We also explain how the properties of the underlying resonance can be extracted from this data. Overall, our paper provides a self-contained compendium of behaviour that will help practitioners interpret and determine important parameters from distorted swept-frequency measurements.
We discuss how reactive and dissipative nonlinearities affect the intrinsic response of superconducting thin-film resonators. We explain how most, if not all, of the complex phenomena commonly seen can be described by a model in which the underlying resonance is a single-pole Lorentzian, but whose centre frequency and quality factor change as external parameters, such as readout power and frequency, are varied. What is seen during a vector-network-analyser measurement is series of samples taken from an ideal Lorentzian that is shifting and spreading as the readout frequency is changed. According to this model, it is perfectly proper to refer to, and measure, the resonant frequency and quality factor of the underlying resonance, even though the swept-frequency curves appear highly distorted and hysteretic. In those cases where the resonance curve is highly distorted, the specific shape of the trajectory in the Argand plane gives valuable insights into the second-order physical processes present. We discuss the formulation and consequences of this approach in the case of nonlinear kinetic inductance, two-level-system loss, quasiparticle generation, and a generic model based on a power-law form. The generic model captures the key features of specific dissipative nonlinearities, but additionally leads to insights into how general dissipative processes create characteristic forms in the Argand plane. We provide detailed formulations in each case, and indicate how they lead to the wide variety of phenomena commonly seen in experimental data. We also explain how the properties of the underlying resonance can be extracted from this data. Overall, our paper provides a self-contained compendium of behaviour that will help practitioners interpret and determine important parameters from distorted swept-frequency measurements.
In this thesis I describe work on developing theoretical and numerical models of supercon- ducting thin-film microwave resonators. Superconducting resonators are used in a variety of applications, one of which is as kinetic inductance detectors (KIDs). KIDs are ultra-low noise, highly sensitive, multiplexable detectors, with uses in a wide variety of fields including astrophysics, medical imaging, and particle physics. Resonators are also crucial for supercon- ducting qubit readout, superconducting mixers and parametric amplifiers, and as multiplexers for other devices. The results described in this thesis apply to all thin film resonators, but are primarily described in the context of KIDs. I develop models of how constant absorbed power affects superconducting thin films, driving them out of thermal equilibrium with the bath. Using the Chang & Scalapino equations, I calculate numerically the steady-state quasiparticle and phonon distributions for various sources of power (sub-gap and above-gap photons and phonons, single frequency and broadband). Many new results emerge, a few of which are: the quasiparticle heating effects of microwave power, explaining the experimentally measured saturation of resonator performance; the frequency dependence of the quasiparticle generation efficiency in the sub-mm wavelength range; and the importance of phonon trapping in thin films at low temperatures. I then use these nonequilibrium results in higher-level device models. Starting from a generic framework for resonators, I present and implement a model of a KID with variable thermal isolation. I calculate the effects of quasiparticle heating on both the large-signal and small-signal behaviour, highlighting the effects of electrothermal feedback caused by readout power. Electrothermal feedback is shown to be able to increase or decrease the magnitude and bandwidth of the responsivity and noise, depending on the operating point. Finally, I propose an experimental measurement of quasiparticle heating effects in a new device – a four-port ring resonator. In such a device, unlike in the standard two-port resonator, the device can be heated and read out independently. I develop detailed models for the thermal and electrical behaviour of these devices, and suggest a scheme by which the quasiparticle heating effects of microwave power, predicted in this thesis, can be measured.
This experiment is designed to compare the feedback from an altered superconductor resonator with a conventional, unaltered superconductor resonator. Superconducting transmission lines are patterned on a lanthanum aluminium substrate chip with a geometry that resonates within the microwave range. Nonlinearities, distortion of the signals in the superconductor, can be engineered into the transmission line using an ion beam or other artificial source of radiation. The sample in this experiment, thin-film yttrium barium copper oxide: YBCO was sent to a facility in Italy to be engineered with a nonlinearity and then characterized in a low temperature (77 K) cryostat. This means that the power dependence of a traditional resonator will differ greatly from this sample. Two key figures of merit of superconducting resonators are their critical power, where the power input to the resonator is most absorbed, and critical frequency of the sample, the frequency at which the critical power occurs. Both were found to vary with temperature, and at high incident power the frequency response of the resonator begins to experience deformation. The change of the power response as frequency is swept through (bending) and the hysteresis in the curve (bifurcation) form the core disparities between traditional superconducting resonators and those with engineered nonlinearities.
We studied the effect of these quasiparticles on the temperature and microwave power dependence of the characteristics of superconducting thin-film resonator, assuming that there are a significant number of quasiparticles in the superconductor even atT≪TcT \ll T_cT≪Tc​. It is demonstrated that the temperature dependence of both the internal quality factor and resonance frequency are well described by that of the normal conductivity of quasiparticles including the contributions from Kondo effect, phonon, and electron-electron scatterings. It also found that the readout microwave power dependence of the internal quality factor is well explained by the change of quasiparticle density due to the energy re-distribution of quasiparticles under microwave field in the resonator.
We investigate the nonlinear behavior of a superconducting microwave resonator subjected to a dissipative mechanism where the associated quality factor (Q factor) decreases with increasing dissipated power, leading to a dissipative feedback effect. By modifying the Rothwarf-Taylor equations, we establish a macroscopic quasiparticle heating (QPH) model that directly links the quality factor to the microwave readout power. The key finding is the identification of a distinctive oval-shaped distortion in the resonance circle in the complex plane. This distortion serves as a practical experimental signature for identifying the readout power regime in which QPH dominates the loss, under conditions where other nonlinear mechanisms are sufficiently weak. To validate the model, we design and fabricate a niobium (Nb) half-wavelength coplanar waveguide (CPW) resonator and conduct systematic bath temperature and readout power sweeps. The model provides a well fit to the observed oval-shaped resonance circle distortion across a wide range of operating conditions, confirming the QPH mechanism as the primary source of the dissipative non-linearity in the parameter space investigated.
We consider the effects of quasiparticle heating on the dynamical behaviour of superconducting microresonators operating with high readout powers. It is seen that resonance-curve distortion and hysteretic switching are an inevitable consequence of the forms of the quasiparticle heating and cooling functions. The model uses a diagrammatic representation of dynamical behaviour to gain an insight into how the instantaneous operating point moves in response to step changes in readout power, signal power, bath temperature, and frequency. The coupling quality factor has a marked effect on non-linear behaviour. The model is used to calculate the waveforms generated when a device switches between hysteretic states. The work suggests that measured response times need not be representative of intrinsic quasiparticle relaxation rates.