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arXiv:2411.08391

A Simple Model of Superconductors: Insights from Free Fermion and Boson Gases

Mi-Ra Hwang, Eylee Jung, MuSeong Kim, DaeKil Park

Published Nov 13, 2024• 0 citations
arXivHTMLPDF

Abstract

Superconductors at temperatures below the critical temperature $T_c$ can be modeled as a mixture of Fermi and Bose gases, where the Fermi gas consists of conduction electrons and the Bose gas comprises Cooper pairs. This simple model enables the computation of the temperature dependence of $2 r(T) / N$, where $N$ is the total number of conduction electrons and $r(T)$ is the number of Cooper pairs at temperature $T$. Analyzing $2 r(T) / N$ across various superconductors may provide significant insights into the mechanisms behind high-temperature superconductivity, especially regarding coherence in Cooper pairs.

DOI
10.48550/arxiv.2411.08391

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Superconductors at temperatures below the critical temperature TcT_cTc​ can be modeled as a mixture of Fermi and Bose gases, where the Fermi gas consists of conduction electrons and the Bose gas comprises Cooper pairs. This simple model enables the computation of the temperature dependence of 2r(T)/N2 r(T) / N2r(T)/N, where NNN is the total number of conduction electrons and r(T)r(T)r(T) is the number of Cooper pairs at temperature TTT. Analyzing 2r(T)/N2 r(T) / N2r(T)/N across various superconductors may provide significant insights into the mechanisms behind high-temperature superconductivity, especially regarding coherence in Cooper pairs.
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.
We review efforts to unify both the Bardeen, Cooper and Schrieffer (BCS) and Bose-Einstein condensation (BEC) pictures of superconductivity. We have finally achieved this in terms of a "\textit{complete} boson-fermion (BF) model" (CBFM) that reduces in special cases to all the main continuum (as opposed to "spin") statistical theories of superconductivity. Our BF model is "complete" in the sense that not only two-electron (2e) but also two-hole (2h) Cooper pairs (CPs) are allowed in arbitrary proportions. In contrast, BCS-Bogoliubov theory--which can also be considered as the theory of a mixture of kinematically independent electrons, 2e- and 2h-CPs--allows only equal, 50%-50%, mixtures of the two kinds of CPs. This is obvious from the perfect symmetry about μμμ, the electron chemical potential, of the well-known Bogoliubov v2(ε)v^{2}(ε)v2(ε) and u2(ε)u^{2}(ε)u2(ε) coefficients, where εεε is the electron energy. The CBFM is then applied to see: a) whether the BCS model interaction for the electron-phonon dynamical mechanism is sufficient to predict the unusually high values of TcT_{c}Tc​ (in units of the Fermi temperature) of ≃0.01−0.1\simeq 0.01-0.1≃0.01−0.1 exhibited by the so-called ``exotic'' superconductors \cite{Brandow} in both 2D and 3D--relative to the low values of ≲10−3\lesssim 10^{-3}≲10−3 more or less correctly predicted by BCS theory for conventional, elemental superconductors; and b) whether it can at least suggest, if not explain, why "hole superconductors" have higher TcT_{c}Tc​'s.
Abstract The superfluid density ns(T)n_{s}(T)ns​(T) n s ( T ) of a superconductor is calculated based on the generalized Bose–Einstein condensation (GBEC) theory that addresses a fully-interacting ternary boson-fermion gas mixture of free electrons as fermions, plus two-electron Cooper pairs (2eCPs) and also, explicitly, two-hole Cooper pairs (2hCPs), both as bosons. Here we consider two special cases (i) 100%–0% (i.e., with no condensed 2hCPs) and (ii) 0%–100% (i.e., with no condensed 2eCPs). Subsumed in GBEC are the Bardeen–Cooper–Schrieffer (BCS) and Bose–Einstein condensation (BEC) theories along with the BCS-BEC crossover theory extended with 2hCPs. We find that in the weak-coupling regime ns(0)n_{s}(0)ns​(0) n s ( 0 ) agrees with data from the Uemura et al. (2004) graph for several elemental SCs by taking in 3D with a quadratic energy-dispersion relation while in 2D with a linear relation are much too far below the data. In the strong-coupling regime the linear behavior of critical temperature TcT_{c}Tc​ T c vs ns(0)n_{s}(0)ns​(0) n s ( 0 ) obtained here is just as Božović et al. (2016) found. However, in 2D with a linear relation accounting for 0%–100%, ns(T)/ns(0)n_{s}(T)/n_{s}(0)ns​(T)/ns​(0) n s ( T ) / n s ( 0 ) compares well with some high- TcT_{c}Tc​ T c -cuprate SC data between the two coupling regimes.
Starting from the Friedberg-TD Lee Hamiltonian describing a coexisting and dynamically interacting many-particle binary boson-fermion gas mixture with a coupling $(\ensuremath{\lambda})$-dependent gap $2\ensuremath{\Delta}(\ensuremath{\lambda})$ in the boson dispersion relation for the sss-wave Cooper or BCS model interaction, we deduce several observed characteristic features of high-temperature superconductors at the simplest level. Analytic expressions for both the unpaired-fermion and boson number densities, as well for the fermion chemical potential $\ensuremath{\mu}(\ensuremath{\lambda},T)$, all of which vary with the degree of bosonization and with temperature TTT, are derived in detail using two-time, finite-temperature Green function techniques. Simple implicit formulas are then obtained for both two and three dimensions for the pseudogap T∗{T}^{*}T∗ and Bose-Einstein condensation Tc{T}_{c}Tc​ temperatures in terms of $\ensuremath{\mu}(\ensuremath{\lambda},T)$ and $2\ensuremath{\Delta}(\ensuremath{\lambda})$. In particular, even at the sss-wave level we find a self-consistent description of the generic phase diagram observed in cuprates, including the appearance of a pseudogap and a dome-shaped Tc{T}_{c}Tc​ vs doping behavior both of which hinge on the gapped boson spectrum.
The crucial demonstration that superfluidity was linked to Bose particles and Bose-Einstein condensation came after the discovery of He3 in 1949, which failed to show the characteristic A transition within a reasonable wide temperature interval around the critical temperature for the onset of superfluidity in He4. The idea of the bosonization of electrons due to their pairing, introduced by Ogg (1946) became very attractive as an explanation for superconductivity after Schafroth (1955) showed that a gas of charged bosons with charge 2e would indeed show a Meissner effect, which is one of the most crucial features of superconductors (see Chapter 1). However this phenomenological Bose-gas picture was condemned and later on practically forgotten because it was unable to account quantitatively for the critical parameters of the classical (low-temperature) superconductors. Using for Tc the Bose-Einstein condensation temperature Eq.(l.S) one obtains the utterly meaningless result Tc = lOOOO.R&s; with the atomic density of electron pairs ~ 1022cm~3 and with the effective mass m** = 2me. The failure of the Schafroth picture stems from a very large size £ of pairs in classical superconductors, which can be estimated from the uncertainty principle: The uncertainty of the momentum 8p is estimated from the uncertainty of the kinetic energy 8E ~ vpSp, which should be of the order of Tc, the only characteristic energy of the superconducting state. Therefore and turns out to be more than one micron for simple metals with Tc ~ IK and the Fermi velocity VF — I0scm/s. That means that pairs in classical superconductors overlap strongly and the Shafroth model of real space pairs is wrong. The finally successful theory for classical superconductors was formulated by Bardeen, Cooper and Schrieffer (1957). In this chapter we present the main points of the BCS theory, using the Bogoliubov transformation (1947,1958) for the weak-coupling regime and the Green&s;s function technique for the intermediate coupling in Chapter 3. More details will be found in the excellent books by Schrieffer(1965), De Gennes (1989) and Abrikosov, Gor&s;kov and Dzyaloshinskii (1963).
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.