Fig. 3: Different regimes of quasiparticle recombination as a function of temperature and disorder.

Each regime (indicated in a shade of gray) is characterized by a different temperature dependence of the quasiparticle (QP) relaxation time. The x-axis is chosen such that the dotted vertical line, ql = 1 for 2Δ₀ phonons, is approximately on the same point for different superconductors. For simplicity, we take ql for 2Δ₀ phonons only in this sketch, such that on the right side of the ql = 1 dotted line, both recombination and scattering interactions are disordered. The solid line gives the values at which the thermal quasiparticle number equals the number of localized quasiparticle states, which we estimate as \({\tilde{n}}_{\,{\mbox{qp}}}^{{\mbox{loc}}\,}=3/(4\pi {\xi }^{3})\). Below this solid line, quasiparticles will localize and recombine on-site. Below the dashed line, the scattering time, Equation (3), is longer than the recombination time, Equation (1), at a quasiparticle density equal to \({\tilde{n}}_{\,{\mbox{qp}}}^{{\mbox{loc}}\,}\), i.e., \({\tau }_{{{{\rm{rec}}}}}({\tilde{n}}_{\,{\mbox{qp}}}^{{\mbox{loc}}\,})\)^27,29. In that regime, delocalization via phonon absorption limits quasiparticle recombination (Fig. 1d), which is indicated in light gray. For ql < 1 (the right side of the dotted line), phonon absorption is always slower than recombination at \({\tilde{n}}_{\,{\mbox{qp}}}^{{\mbox{loc}}\,}\). The colored areas give the experimental conditions of refs. ^21,32,47 and Fig. 2. The fade at lower temperatures for refs. ^21,47 are added because these are measurements via a pulsed excitation, which does not give information on the quasiparticle temperature. In ref. ²¹, disorder is increased by implanting ions. There is however no data on how much \(1/\sqrt{\xi \Delta }\) increased in these films, which we indicated by an additional fade to higher disorder. The recombination regimes are consistent with the observed temperature dependence of the relaxation time in the respective experiments.