Microscopic insight into the pump–probe relaxation dynamics of superconductors: Model study of MgB₂ relaxation within nonlinear response theory

Abstract

ReceivedHere we present a quantum-statistical formulation of relaxation dynamics of superconductors related to pump-probe (PP) spectroscopy. The method is based on the perturbation expansion of the non-equilibrium density matrix for calculation of multi-time correlation functions, and the corresponding response function. As a model for our study, the high-temperature superconductor MgB 2 has been selected. Knowledge of the electronic structure and of the corresponding Eliashberg function of electron-phonon (EP) interactions enabled us to distinguish nonequilibrium processes in a sequence of interactions with laser pulses on a microscopic level. Temperature-dependent relaxation dynamics as a function of delay time between the pump and probe pulses have been derived. We have shown that an abrupt increase of the relaxation time at T c is the direct consequence of sudden changes in the character of EP coupling in transition from an adiabatic to a stabilized superconducting anti-adiabatic state, as it predicts the anti-adiabatic theory of electron-vibration interactions. The BCS model, which is based on adiabatic character of EP coupling, is basically unable to reflect the enormous sudden increase of the relaxation time at T c . Differences in the optical pump-optical probe and the optical pump-terahertz probe settings of PP relaxation dynamics are discussed within diagrammatic perturbation theory.Keywords: pump-probe spectroscopy; relaxation dynamics of superconductors; electron-phonon coupling; theory of superconductivity; anti-adiabatic theory of superconductivity IntroductionPump-probe (PP) spectroscopy, a method of nonlinear optics, is an important experimental tool in the study of ultrafast relaxation dynamics of excited states in condensed matter systems [1,2]. The interpretation of experimental results relies on various types of phenomenological equations of motion for the relaxation of "hot electrons" and/or high-energy phonons in coupling to bosonic degrees of freedom [3][4][5][6][7]. The assumption of quasi-equilibrium, which is often behind these methods, is discussed in [8], where the authors present an analytic solution of excited state relaxation dynamics within the Boltzmann kinetic equation, the method of which deals with electrons and phonons out of equilibrium.In general, however, phenomenological treatments not only ignore nonlinear aspects of processes running in PP experiments, but ignore also the microscopic character of nonequilibrium states -coherences and/or populations, which can evolve in the studied system over a sequence of different time periods via the interaction of matter with the incident fields. In these circumstances, mainly in the case of superconductors, it is difficult to disclose the relationship between the derived relaxation parameters, i.e. relaxation times, or electron-phonon (EP) coupling constants (), and the microscopic mechanism of the superconducting state transition. The theory of high-temperature superconductors (e.g. cuprates, pnictides) is far from being settled,...

Aspects of strong electron–phonon coupling in superconductivity of compressed metal hydrides MH6 (M = Mg, Ca, Sc, Y) with Im-3m structure

“…More specific arrangement of the theory toward superconductivity is available in the form of open access papers [52][53][54] in physical journals. On semiempirical level, the theory was applied to study superconductivity 48,[55][56][57][58][59][60][61][62] in anti-adiabatic state for different groups of superconductors or to study some spectral properties of superconductors, e.g. ARPES 55,56 or pump-probe spectroscopy of relaxation processes in superconductors 62 .…”

“…On semiempirical level, the theory was applied to study superconductivity 48,[55][56][57][58][59][60][61][62] in anti-adiabatic state for different groups of superconductors or to study some spectral properties of superconductors, e.g. ARPES 55,56 or pump-probe spectroscopy of relaxation processes in superconductors 62 . It has been shown that, no matter if EPC is strong or weak, adiabatic → anti-adiabatic state transition is the effect common for superconductors of different groups, e.g.…”

“…Electronic band structure (BS) of studied hydrides in fixed nuclear geometries has been calculated by a computer code 62 Solid2000. This code is not based on Kohn-Sham DFT method but on the Hartree-Fock SCF (HF-SCF) method for infinite 3D-periodic cyclic cluster (CC) 62 with semiempirical quasi-relativistic INDO Hamiltonian 63 and parameterization based on results of atomic Dirac-Fock calculations 64 . This method of BS calculation has many advantages and some drawbacks as well.…”

“…For MH 6 it is 1331 k-points. The HF-SCF CC procedure is performed for each k-point of the grid with the INDO Hamiltonian matrix elements that obey the boundary conditions of the cyclic cluster 62 . The Pyykko-Lohr quasirelativistic basis set of the valence electron atomic orbitals AO (s,p-AO for Mg, Ca ; s,p,d-AO for Sc, Y and 1s for H atom) has been used.…”

“…In general, the precision of the results of band structure calculation increases with increasing dimension of the basic cluster. It has been shown 62,[65][66][67][68] , however, that there is an effect of saturation, a bulk limit beyond which the effect of increasing dimension on e.g. total electronic energy, orbital energies, HOMO-LUMO difference…, is negligible.…”