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arXiv:1002.2289 · Europhysics Letters (EPL)

Multi-gapped states in BCS superconductors

Eldad Bettelheim

Published Jun 1, 2010• 3 citations
arXivHTMLPDF

Abstract

A BCS (Bardeen-Cooper-Schrieffer) superconductor, which is placed out of equilibrium, can develop quantum instabilities, which manifest themselves in oscillations of the superconductor's order parameter (pairing amplitude Δ). These instabilities are manifestations of the Cooper instability. Inelastic collisions are essential in resolving those instabilities. Incorporating the quantum instabilities and collisions in a unified approach based on Richardson's exact solution of the pairing Hamiltonian, we find that a BCS superconductor may end up in a state in which the spectrum has more than one gap.

DOI
10.1209/0295-5075/90/67002

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A BCS (Bardeen-Cooper-Schrieffer) superconductor, which is placed out of equilibrium, can develop quantum instabilities, which manifest themselves in oscillations of the superconductor's order parameter (pairing amplitude Δ). These instabilities are manifestations of the Cooper instability. Inelastic collisions are essential in resolving those instabilities. Incorporating the quantum instabilities and collisions in a unified approach based on Richardson's exact solution of the pairing Hamiltonian, we find that a BCS superconductor may end up in a state in which the spectrum has more than one gap.
I consider relaxation of the pairing amplitude in a disordered Bardeen-Cooper-Schrieffer (BCS) superconductor in the absence of the two-particle collisions. My main assumption is that nonmagnetic and magnetic disorder scattering rates are much smaller than the value of the superconducting pairing gap ${\mathrm{\ensuremath{\Delta}}}_{0}$. I derive a system of nonlinear equations which describe the collisionless relaxation of the pairing amplitude following a quench of the pairing strength. I find that in a superconductor in which scattering on paramagnetic impurities is dominant, the pairing amplitude in a steady state varies periodically with time even for small deviations from equilibrium. It is shown that such a steady state emerges due to scattering on paramagnetic impurities which leads to a decrease in the value of the resonant frequency of the amplitude mode below $2{\mathrm{\ensuremath{\Delta}}}_{0}$.
Recent experimental progress in the fields of cold quantum gases and ultrafast optical spectroscopy of quantum materials allows us to controllably induce and probe nonadiabatic dynamics of superconductors and superfluids. The time evolution of the gap function before relaxation with the lattice is determined by the superposition of coherently evolving individual Cooper pairs within the manifold of the Bardeen-Cooper-Schrieffer (BCS) wave function. While dynamics following an abrupt quench of the pairing interaction strength in the single-band BCS model has been exactly solved due to the integrability of the model, the dynamics of postquench multiband superconductors remain under scrutiny. Here, we develop a generalization of the Volkov-Kogan Laplace-space perturbative method that allows us to determine the nonadiabatic gap dynamics of two-band fully gapped superconductors for a wide range of quench amplitudes. Our approach expands the long-time dynamics around the steady-state asymptotic value of the gap, which is self-consistently determined, rather than around the equilibrium value of the gap. We explicitly demonstrate that this method recovers the exact solution of the long-time gap dynamics in the single-band case and perfectly agrees with a numerical solution of the two-band model. We discover that dephasing of Cooper pairs from different bands leads to faster collisionless relaxation of the gap oscillation with a power law of ${t}^{\ensuremath{-}3/2}$ instead of the well-known ${t}^{\ensuremath{-}1/2}$ behavior found in the single-band case. Furthermore, the gap oscillations display beating patterns arising from the existence of two different asymptotic gap values. Our results have important implications to a variety of two-band superconductors driven out of equilibrium, such as iron-based superconductors, MgB2{\mathrm{MgB}}_{2}MgB2​, and SrTiO3{\mathrm{SrTiO}}_{3}SrTiO3​.
BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).
The dynamics of BCS (Bardeen-Cooper-Schrieffer) superconductors is fairly well understood due to the availability of a mean-field solution for the pairing Hamiltonian, a solution which gives the quantum state of a superconductor as a state of almost-free fermions interacting only with a condensate. As a result, transition probabilities may be computed, and expressed in terms of matrix elements of electron creation and annihilation operators between approximate eigenstates. These matrix elements are also called ``coherence factors.'' Mean-field theory is however not sufficient to describe all eigenstates of a superconductor, a deficiency which is hardly important in (or very close to) equilibrium, but one that becomes relevant in certain out-of-equilibrium situations. We report here on a computation of matrix elements (coherence factors) for the pairing Hamiltonian between any ``two-arc'' eigenstates in the thermodynamic limit.
Analytic expressions for the Bardeen–Cooper–Schrieffer (BCS) gap of a many-electron system with the BCS model interaction are obtained in one, two, and three dimensions in the weak coupling limit, but for arbitrary interaction width ν≡ℏωD/EF, 0<ν<∞. Here ℏωD is the maximum energy of a force-mediating boson, and EF is the Fermi energy (which is fixed by the electronic density). Our results address both phononic (ν≪1) as well as nonphononic (e.g., exciton, magnon, plasmon, etc.) pairing mechanisms where the mediating boson energies are not small compared with EF, provided weak electron–boson coupling prevails. The essential singularity in coupling, sometimes erroneously attributed to the two-dimensional character of the BCS model interaction with ν≪1, is shown to appear in one, two, and three dimensions before the limit ν→0 is taken.
A generic Hamiltonian, which incorporates the effect of the orbital contraction on the hopping amplitude between the nearest sites, is studied both analytically at the weak coupling limit and numerically at the intermediate and strong coupling regimes for finite atomic cluster. The effect of the orbital contraction due to hole localization at atomic sites is specified with two coupling parameters V and W (multiplicative and additive contraction terms). The singularity of the vertex part of the two-particle Green’s function determines the critical temperature Tc and the relaxation rate Γ(T) of the order parameter at temperature above Tc. Unlike in conventional BCS superconductors, Γ has a non-zero imaginary part which may influence the fluctuation conductivity of superconductor above Tc. We compute the ground state energy as a function of the particle number and magnetic flux through the cluster, and show the existence of the parity gap ∆ appearing at the range of system parameters consistent with the appearance of Cooper instability. Numeric