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arXiv:1806.09449 · Physica C Superconductivity

Superconducting properties of a nonideal bipolaron gas

Victor Dmitrievich Lakhno

Published Jan 1, 2018• 20 citations
arXivHTMLPDF

Abstract

The properties of a Bose gas of translation-invariant (TI) bipolarons analogous to Cooper pairs are considered. As in the BCS theory, the description of a TI-bipolaron gas is based on the electron-phonon interaction and Froehlich Hamiltonian. As distinct from the BCS theory, when the correlation length greatly exceeds the mean distance between the pairs, here we deal with the opposite case when the correlation length is much less than the distance between the pairs. We calculate the critical temperature of the transition of a TI-bipolaron Bose-gas into the superconducting state, its energy, heat capacity and heat of the transition. The results obtained are used to explain the experiments on high-temperature superconductors. Possible ways of raising the critical temperature of high-temperature superconductors are discussed.

DOI
10.1016/j.physc.2018.10.009

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The properties of a Bose gas of translation-invariant (TI) bipolarons analogous to Cooper pairs are considered. As in the BCS theory, the description of a TI-bipolaron gas is based on the electron-phonon interaction and Froehlich Hamiltonian. As distinct from the BCS theory, when the correlation length greatly exceeds the mean distance between the pairs, here we deal with the opposite case when the correlation length is much less than the distance between the pairs. We calculate the critical temperature of the transition of a TI-bipolaron Bose-gas into the superconducting state, its energy, heat capacity and heat of the transition. The results obtained are used to explain the experiments on high-temperature superconductors. Possible ways of raising the critical temperature of high-temperature superconductors are discussed.
The monograph presents the theory of translation-invariant polarons and bipolarons based on the theory of squeezed vacuum wave functions. It is shown that the Tulub ansatz, which establishes a connection between the generalized Bogolyubov transformation with the unitary squeezed operator gives a solution to the spectral problem for a bipolaron. The solutions obtained are used to construct a theory of superconductivity based on the Froehlich Hamiltonian with a strong electron-phonon interaction. The role of Cooper pairs in it is played by TI bipolarons of spatially delocalized electrons with a small correlation length. The theory developed explains a large number of experiments on the thermodynamic, spectroscopic and transport characteristics of high-temperature superconductors, Josephson tunneling, angle-resolved photoemission spectroscopy, neutron scattering, etc. The book is intended for physicists and mathematicians who work in the field of the theory of condensed matter, as well as graduate students and senior students of universities.
Various BCS-like pairing theories are still invariably used to describe the unusual superconducting/superfluid states and properties of condensed matters without clarifying the fermionic or bosonic nature of Cooper pairs and the relevant mechanisms of superconductivity and superfluidity. We show that the bosonization of Cooper pairs and the novel superconductivity and superfluidity resulting from the condensation of an attractive Bose gas of Cooper pairs into a superfluid state in high-Tc cuprates and other exotic systems occur under the condition εF≲2εA (where εF is the Fermi energy, εA is the energy of the effective attraction between fermions). We argue that the simple BCS criterion (i.e. formation of an energy gap on the Fermi surface) is insufficient for the occurrence of unconventional superconductivity (superfluidity) and the superconducting/superfluid transition in attractive Bose systems is more λ-like than the usual Bose–Einstein condensation. The new and specific criteria for unconventional superconductivity and superfluidity are formulated. Mean-field theory of the three- and two-dimensional superfluid Bose condensates of Cooper pairs describes reasonably well the novel superconducting states (i.e., two distinct superconducting phases below the bulk superconducting transition temperature Tc and a vortex-like state above Tc) and properties (such as λ-like transition at Tc, first-order phase transition and gapless bosonic excitations somewhat below Tc or even far below Tc) of high-Tc cuprates in full agreement with the experimental findings. The unusual superconducting/superfluid states and properties of heavy-fermion and organic compounds, Sr2RuO4, 3He, 4He and atomic Fermi gases are explained more clearly by this theory.
The problem of the evolution from BCS theory with cooperative Cooper pairing to the formation and condensation of composite bosons has attracted considerable attention for the past several decades. It has gained renewed impetus in the mid-eighties with the discovery of the high-Tc superconductors, which have a coherence length comparable to the interparticle spacing. More recently, this subject has spurred a great deal of research activity in connection with experiments involving dilute atomic gases of fermionic atoms. The initial objective of this work will be to use functional integral techniques to analyze the low-temperature BCS-to-BEC evolution of d-wave superconductors within the saddle point (mean field) approximation for a continuum model. Then, the same mathematical formalism will be applied to the problem of the BCS-to-BEC evolution of fully spin-polarized p-wave Fermi gases in two dimensions. We find that a quantum phase transition occurs for both systems as they are driven from the BCS-like regime of weakly interacting fermionic pairs to the opposite BEC-like regime of strongly interacting bosonic molecules. This is in contrast to the smooth crossover predicted and observed in systems that exhibit s-wave pairing symmetry. We calculate several spectroscopic and thermodynamic properties that signal the occurrence of this phase transition, and suggest some possible experimental realizations. Finally, fluctuations about the saddle point solution are included in the calculations, and the effects of such correction are analyzed in the low (T~0) and high (T~Tc) temperature limits. We conclude that, at high temperatures, the bosonic degrees of freedom that arise from two-particle bound states become essential to describe the strong coupling limit,\nas the saddle point approximation alone becomes unreliable.
Using the Boson-Fermion formalism of superconductivity we calculate the condensation energy for several superconductors ranging from conventional to unconventional, or high temperature superconductors. It is calculated as the difference between the Helmholtz free energies of the superconducting and the normal state, which is a gas of N attractive electron gas, while the superconducting state is formed by the condensed Cooper pairs taken as composite bosons, coming from a fraction of the electrons inside the Debye shell, plus those electrons inside and outside the Debye shell that are unable to pair. After giving the analytic expressions for the internal energy U and the entropy S we obtain the Helmholtz free energy F = U -TS for both the superconducting and the normal states as functions of temperature. In the search for universalities, we calculate the ratio of the condensation energy at T=0 to the Sommerfeld constant (the normal state electronic specific heat over the temperature when T it's almost zero) using two different methods: the Boson-Fermion formalism developed here, as well as an analytical expression deduced from a combination of the BCS and Ginzburg-Landau theories. We find for the Boson-Fermion formalism Econd/γ0=0.252 Tc1.997E_{cond}/γ_0= 0.252\,T_{c}^{1.997}Econd​/γ0​=0.252Tc1.997​, which is the same behavior described by the experimental fit of Kim et al Econd/γ0=0.2 Tc2.06E_{cond}/γ_0= 0.2\,T_{c}^{2.06}Econd​/γ0​=0.2Tc2.06​ and by the one recently reported by Tallon et al for overdoped cuprate superconductors; while for the Ginzburg-Landau-BCS we get the expression Econd/γ0=0.236 Tc2E_{cond}/γ_0= 0.236\,T_{c}^{2}Econd​/γ0​=0.236Tc2​, also in very good agreement with the partifirst method.
Observed crossover events between different power functions of absolute temperature occurring below about ~1 K in the temperature dependence of the heat capacity or the thermal conductivity of the conventional superconductors are identified as transitions from Maxwell-Boltzmann to Bose-Einstein (BE) statistics of the Cooper-pairs. Because of the low mass of the Cooper pairs of 2me (with me as mass of the electron) and their high density, the BE-condensation temperature, TBE, of the Cooper-pairs is about five orders of magnitude higher than for the dilute alkali atom condensates. The condensation temperature TBE turns out to be proportional to the superconducting transition temperature TSC. From the observed TBE-values it is possible to calculate the density of the Cooper pairs. Assuming that the Cooper pairs form a dense gas of bosons, the diameter of the Cooper-pair orbitals turns out to be equal to the London penetration depth. As a conclusion, due to the large orbital diamagnetism of the Cooper-pairs pairs, only one layer of Cooper-pairs, next to the inner surface of the sample, is sufficient to shield an applied external magnetic field completely.