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Cambridge University Press eBooks

Superconductivity of an Ideal Charged Boson System

T.D. Lee

Published Oct 25, 1991• 1 citation
Publisher

Abstract

Recently Friedberg, Lee and Ren have pointed out that at low density the ideal charged boson system turns out not to be a superconductor, but becomes a type II superconductor at high density. This conclusion differs from the well-known Schafroth solution of superconductivity at any density for the same problem. Schafroth's analysis is found to contain a mistake due to the neglect of the electrostatic exchange energy E ex . Based on the Schafroth solution, E ex is shown to be +∞ in the normal phase, but 0 in the condensed phase (at T = 0). Of course, the correct solution has to give a finite E ex . This research was supported in part by the U.S. Department of Energy. SCHAFROTH'S SOLUTION Schafroth's superconductivity solution of an ideal charged boson system published 35 years ago has always been considered to be the definitive work, comparable in depth to the analysis made by Landau on the diamagnetism of an ideal charged fermion system. However, recently it was found that the Schafroth solution contains a serious mistake due to the neglect of the electrostatic exchange energy E ex . It turns out that based on the Schafroth solution, E ex is +∞ in the normal phase, but 0 in the condensed phase (at T = 0). Of course, the correct solution has to give a finite E ex . For ideal charged particles, bosons or fermions, there is only the electromagnetic interaction.

DOI
10.1017/cbo9780511563980.012

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Dielectric approach to a dense charged boson gas at T=0

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Physical review. B, Solid state•Oct 1, 1975•14 citations
Recently Friedberg, Lee and Ren have pointed out that at low density the ideal charged boson system turns out not to be a superconductor, but becomes a type II superconductor at high density. This conclusion differs from the well-known Schafroth solution of superconductivity at any density for the same problem. Schafroth's analysis is found to contain a mistake due to the neglect of the electrostatic exchange energy E ex . Based on the Schafroth solution, E ex is shown to be +∞ in the normal phase, but 0 in the condensed phase (at T = 0). Of course, the correct solution has to give a finite E ex . This research was supported in part by the U.S. Department of Energy. SCHAFROTH'S SOLUTION Schafroth's superconductivity solution of an ideal charged boson system published 35 years ago has always been considered to be the definitive work, comparable in depth to the analysis made by Landau on the diamagnetism of an ideal charged fermion system. However, recently it was found that the Schafroth solution contains a serious mistake due to the neglect of the electrostatic exchange energy E ex . It turns out that based on the Schafroth solution, E ex is +∞ in the normal phase, but 0 in the condensed phase (at T = 0). Of course, the correct solution has to give a finite E ex . For ideal charged particles, bosons or fermions, there is only the electromagnetic interaction.
It is shown that an ideal gas of charged bosons exhibits the essential equilibrium features of a superconductor. The onset of Bose-Einstein condensation marks the transition temperature Tc{T}_{c}Tc​. Below Tc{T}_{c}Tc​ a Meissner-Ochsenfeld effect is exhibited which is described in a very good approximation by London's equation.The singular nature of the condensed ideal Bose gas exhibits itself in a space dependence of the London constant $\ensuremath{\lambda}$, determined by the boundary conditions on the wave function. It is shown that the electrostatic repulsion between the bosons compensates this effect and leads to a spatially constant $\ensuremath{\lambda}$, independently of the boundary conditions.The critical field Hc(T){H}_{c}(T)Hc​(T) is determined and found to be related to the penetration depth d(T)d(T)d(T) by ${H}_{c}=\frac{\ensuremath{\hbar}c}{2e{d}^{2}}$ (eee being the boson charge).The B(H)B(H)B(H) law is different from the one usually assumed for actual superconductors. Corresponding changes occur in the thermodynamical relation.A comparison with superconducting metals is made. The main conclusion is that if superconductivity in metals is due to the concurrence of bosons, then the number of these bosons must be strongly temperaturedependent below Tc{T}_{c}Tc​.
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).
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.
Abstract The method of canonical transformations proposed by one of the authors ten years ago in connection with a microscopic theory of superfluidity for Bose systems, is generalized here to Fermi systems, and forms the basis of a method for investigating the problem of superconductivity. Starting from Fröhlich's Hamiltonian, the energy of the superconducting ground state and the one‐Fermion and collective excitations corresponding to this state are obtained. It turns out that the final formulae for the ground state and one‐Fermion excitations recently obtained by Bardeen, Cooper and Schrieffer are correct in the first approximation. The physical picture appears to be closer to the one proposed by Schafroth, Butler and Blatt. The effect on superconductivity of the Coulomb interaction between the electrons is analyzed in detail. A criterion for the superfluidity of a Fermi system with a four‐line vertex Hamiltonian is established.
The generalized dielectric constant is calculated to the order next to the Bogoliubov approximation. This is done by using the analogy between the condensed boson system and a fictitious fermion system with spin degeneracy equal to the total number of particles (instead of two). From the zero of the dielectric constant, we have calculated the first-order correction to the Bogoliubov plasmon energy and the half linewidth of the plasmon states. The real part of the solution is examined from a graphical view point to show that up to the specified order of approximation there exists only one mode of elementary excitation. The screening of the system to a static impurity charge is shown to be exponential at a long distance. The response of the system to a static transverse vector potential shows a perfect Meissner effect at long- and short-wavelength limits. We have also examined the diagrammatic structure for the number of particles in condensate. We show that the series contains only terms of integer powers of rs34{r}_{s}^{\frac{3}{4}}rs43​​. The entire treatment is fully number conserving.