nestfsss

Non-equilibrium superconducting thin film steady-state simulations

This repository contains MATLAB code to solve for steady-state solutions to the Chang & Scalapino equations, including photon and phonon injection. These kinetic equations describe quasiparticle, phonon and photon interactions in superconductors. The self-consistent steady-state quasiparticle and phonon distributions can then be used to calculate surface impedance and other transport properties, and can also be used to estimate pair-breaking efficiencies ($\eta_{pb}$) for different types, intensities and energies of absorbed power. Absorbed photons (light) or phonons of any frequency (sub-gap or pair-breaking) can be included. This information is useful for designing superconducting devices and electronics such as detectors, microwave resonators, and qubit elements.

Theory

Full theoretical details and derivations are available in (PhD thesis):
Guruswamy, T. (2018) "Nonequilibrium behaviour and quasiparticle heating in thin film superconducting microwave resonators". doi:10.17863/CAM.24510.

Other relevant references:
Chang, J.-J. & Scalapino, D. J. Kinetic-equation approach to nonequilibrium superconductivity. Physical Review B 15, 2651–2670 (1977)
Goldie, D. J. & Withington, S. Non-equilibrium superconductivity in quantum-sensing superconducting resonators. Superconductor Science and Technology 26, 015004 (2013)
Guruswamy, T., Goldie, D. J. & Withington, S. Quasiparticle generation efficiency in superconducting thin films. Superconductor Science and Technology 27, 055012 (2014)
Guruswamy, T., Goldie, D. J. & Withington, S. Nonequilibrium superconducting thin films with sub-gap and pair-breaking photon illumination. Superconductor Science and Technology 28, 054002 (2015)
de Visser, P. J. et al. The non-equilibrium response of a superconductor to pair-breaking radiation measured over a broad frequency band. Applied Physics Letters 106, 252602 (2015)

Installation

Add the top-level directory to your MATLAB pathdef. Most recently tested with MATLAB R2022a.

Operation

• Instantiate a Superconductor object
  • contains material parameters as well as the arrays representing energy, quasiparticle distribution, and phonon distribution
  • preconfigured materials are available: Sc_Aluminum, Sc_Niobium, etc.
• Instantiate a ThinFilm object around the Superconductor
  • contains parameters defining signal (above-gap pair-breaking photons), probe (sub-gap microwave/readout photons), and phonon injection
• Instantiate an Iterator object around the ThinFilm
  • contains parameters related to the solution convergence, number of iterations, etc.
• Run methods of the Iterator object to solve for the steady-state solution given the parameters set.
  • it.main_iteration() returns a new object with (hopefully) converged distributions, available via the internal Superconductor object.
  • it.with_without_signal() returns two objects, with solutions with the phonon and signal terms disabled, and with the phonon and signal terms enabled.

See simple_test.m for an example which calculates and then plots $f(E)$.

For a sanity check, try turning off all absorbed power and ensure all distributions converge to thermal distributions at the bath temperature $T_B$.

Convergence and solution validity notes

• Solutions may fail to converge:
  • if the absorbed power is very high
  • if the phonon trapping factor is very high
  • if the starting distribution is very strange.
• Many assumptions detailed in the theory references above
  • only considers redistribution of quasiparticles, not changes in the density of states¹
  • assumes 3-D, clean-limit, BCS superconductors
• No geometry is included. This approach assumes uniform photon/phonon absorption and uniform quasiparticle/phonon response. Consider calculating the quasiparticle diffusion length and time is in your system to understand the spatial scale over which this approach might be valid.

Footnotes

1. Semenov, A. V., Devyatov, I. A., de Visser, P. J. & Klapwijk, T. M. Coherent excited states in superconductors due to a microwave field. Physical Review Letters 117, 047002 (2016). ↩