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Physical Review B

Erosion of nodal Fermi spheres in nonequilibrium d -wave superconductors

Michael F. Smith, P. Pairor

Published Dec 29, 2005• 1 citation
Publisher

Abstract

The relaxation of a $d$-wave superconductor that has been driven out of equilibrium by an optical pulse is investigated. We consider a simple model for the low-energy nonequilibrium state, one in which the unpaired quasiparticles form Fermi spheres near the gap nodes, and calculate the decay rate of the quasiparticle population due to phonon-assisted recombination. In the high density limit for the photoinjected quasiparticles, the decay rate $dn∕dt$ is found to vary as ${n}^{5∕2}$, which differs from the ${n}^{2}$ form commonly adopted in phenomenological models of the relaxation dynamics. In the low density limit, the decay is exponential. From numerical estimates, we determine that phonon-assisted recombination could play an important role over the picosecond time scales of current interest. We compare our results to pump-probe optical experiments on high ${T}_{C}$ cuprates and find reasonable agreement for the decay rate in underdoped YBCO for low laser intensity.

DOI
10.1103/physrevb.72.212513

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The relaxation of a ddd-wave superconductor that has been driven out of equilibrium by an optical pulse is investigated. We consider a simple model for the low-energy nonequilibrium state, one in which the unpaired quasiparticles form Fermi spheres near the gap nodes, and calculate the decay rate of the quasiparticle population due to phonon-assisted recombination. In the high density limit for the photoinjected quasiparticles, the decay rate dn∕dtdn∕dtdn∕dt is found to vary as n5∕2{n}^{5∕2}n5∕2, which differs from the n2{n}^{2}n2 form commonly adopted in phenomenological models of the relaxation dynamics. In the low density limit, the decay is exponential. From numerical estimates, we determine that phonon-assisted recombination could play an important role over the picosecond time scales of current interest. We compare our results to pump-probe optical experiments on high TC{T}_{C}TC​ cuprates and find reasonable agreement for the decay rate in underdoped YBCO for low laser intensity.
In the transient state of d-wave superconductors, we investigate the temporal variation of photoinduced changes in the superfluid weight. We derive the formula that relates the nonlinear response function to the nonequilibrium distribution function. The latter qunatity is obtained by solving the kinetic equation with the electron-electron and the electron-phonon interaction included. By numerical calculations, a nonexponential decay is found at low temperatures in contrast to the usual exponential decay at high temperatures. The nonexponential decay originates from the nonmonotonous temporal variation of the nonequilibrium distribution function at low energies. The main physical process that causes this behavior is not the recombination of quasiparticles as previous phenomenological studies suggested, but the absorption of phonons.
Motivated by recent ultrafast pump-probe experiments on high-temperature superconductors, we discuss the transient dynamics of a d-wave BCS model after a quantum quench of the interaction parameter. We find that the existence of gap nodes, with the associated nodal quasiparticles, introduces a decay channel which makes the dynamics much faster than in the conventional s-wave model. For every value of the quench parameter, the superconducting gap rapidly converges to a stationary value smaller than the one at equilibrium. Using a sudden approximation for the gap dynamics, we find an analytical expression for the reduction of spectral weight close to the nodes, which is in qualitative agreement with recent experiments.
We develop a computationally inexpensive model to examine the dynamics of boson-assisted electron relaxation in solids, studying nonequilibrium dynamics in a metal, in a nodal superconductor with a stationary density of states, and in a nodal superconductor where the gap dynamically opens. In the metallic system, the electron population resembles a thermal population at all times, but the presence of even a fixed nodal gap both invalidates a purely thermal treatment and sharply curtails relaxation rates. For a gap that is allowed to open as electron relaxation proceeds, effects are even more pronounced, and gap dynamics become coupled to the dynamics of the electron population. Comparisons to experiments reveal that phase-space restrictions in the presence of a gap are likely to play a significant role in the widespread observation of coexisting femtosecond and picosecond dynamics in the cuprate high-temperature superconductors.
We use density-matrix theory to calculate the ultrafast dynamics of unconventional superconductors from a microscopic viewpoint. We calculate the time evolution of the optical conductivity as well as pump-probe spectra for a ddd-wave order parameter. Three regimes can be distinguished in the spectra. The Drude response at low photon energies is the only one of those which has been experimentally measured so far. At higher energies, we predict two more regimes: the pair-breaking peak, which is reduced as Cooper pairs are broken up by the exciting pulse, and a suppression above the pair-breaking peak due to nonequilibrium quasiparticles. Furthermore, we consider the influence of the electron-phonon coupling and derive rate equations that have been widely used so far.
We theoretically analyze the process of charge recombination in the planar Mott-Hubbard insulators with the aim to explain the short picosecond-range lifetimes of photoexcited carriers, experimentally studied via pump-probe experiments on the undoped cuprates. The recombination mechanism consists of two essential ingredients: the formation of a metastable sss-type bound holon-doublon pair, i.e., the Mott exciton, and the decay of such an excitonic state via the multimagnon emission. In spite of the large gap that requires many bosons to be emitted, the latter process is fast due to a large exchange scale and strong charge-spin coupling in planar systems. As the starting microscopic model we consider the single-band Hubbard model and then a more realistic three-band model for cuprates, both leading to the same minimal one. The decay rate of the exciton is evaluated numerically via the Fermi golden rule, having consistency also with the direct time-evolution calculation. The decay rate reveals exponential dependence on the ratio of the Mott-Hubbard gap and the exchange coupling, the result qualitatively reproduced also within a toy exciton-boson model.
A comparative analysis has been done of the formerly established two self-consistent solutions for the density of quasiparticle states in doped d-wave superconductors, having strikingly different and disputed behavior near the Fermi energy. One of them (1) remains finite in this limit, while the other (2) tends to zero. To resolve this discrepancy, the known Ioffe-Regel criterion for band states, widely used for doped semiconductors, was applied to these solutions. It is shown that both them are valid in limited and different energy regions, where the corresponding quasiparticles are weakly damped. In particular, density of states of nodal quasiparticles near the Fermi level is provided by the (2) solution, while the (1) only applies far enough from this level.