Enhanced Quasiparticle Lifetime in a Superconductor by Selective Blocking of Recombination Phonons with a Phononic Crystal

When quasiparticles in a BCS (Bardeen-Cooper-Schrieffer) 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 band gap restricting the escape of recombination phonons from a superconductor can increase the quasiparticle recombination lifetime by more than an order of magnitude. A phonon band gap can be realized and matched to the recombination energy with a phononic crystal, a periodically patterned dielectric membrane. We discuss in detail the nonequilibrium quasiparticle and phonon distributions that arise in a superconductor due to a phonon band gap and a pair-breaking photon signal. Although intrinsically a nonequilibrium 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 (sub bg) divided by k (sub B)T (sub b)), where Omega (sub bg) is the phonon band gap energy bandwidth above 2 and T (sub b) is the phonon bath temperature of the coupled electron-phonon system. We discuss the impact of a phononic band gap on the performance of a superconducting circuit element and propose a microwave resonator to measure the enhancement in the quasiparticle lifetime.