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Physical Review

Superconductivity of a Charged Ideal 2-Dimensional Bose Gas

Robert Mccredie May

Published Jul 15, 1959• 45 citations
Publisher

Abstract

It is well-known that the 2-dimensional Bose gas does not condense, and that no transition occurs in its thermodynamic behavior. Despite this, we find that in a magnetic field a 2-dimensional gas of charged, ideal bosons has a nearly sharp transition temperature, below which it displays an "imperfect" Meissner-Ochsenfeld effect, practically indistinguishable from the perfect London one. This fact is relevant to the application of the Bose gas model of superconductivity to thin films.The main interest of the model is, however, that it exhibits an (essentially perfect) Meissner-Ochsenfeld effect without any irregularity in ${C}_{V}(T)$.

DOI
10.1103/physrev.115.254

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It is well-known that the 2-dimensional Bose gas does not condense, and that no transition occurs in its thermodynamic behavior. Despite this, we find that in a magnetic field a 2-dimensional gas of charged, ideal bosons has a nearly sharp transition temperature, below which it displays an "imperfect" Meissner-Ochsenfeld effect, practically indistinguishable from the perfect London one. This fact is relevant to the application of the Bose gas model of superconductivity to thin films.The main interest of the model is, however, that it exhibits an (essentially perfect) Meissner-Ochsenfeld effect without any irregularity in CV(T){C}_{V}(T)CV​(T).
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​.
We present closed-form, analytical expressions for the thermodynamic properties of an ideal, two-dimensional (2D) charged Fermi or Bose gas in the presence of a uniform magnetic field of arbitrary strength. We consider both the homogeneous quantum gas (in which case our expressions are exact) and the inhomogeneous gas within the local-density approximation. Our results for the Fermi gas are relevant to the current-density-functional theory of low-dimensional electronic systems in magnetic fields. For a 2D charged Bose gas (CBG) in a homogeneous magnetic field, we show that the uniform system undergoes a sharp transition at a critical temperature ${T}_{c}^{\ensuremath{\star}},$ below which there is a macroscopic occupation of the lowest Landau level. An examination of the one-body density matrix, however, reveals the absence of long-range order, thereby indicating that the transition cannot be interpreted to a Bose-Einstein condensate. Nevertheless, for $T<{T}_{c}^{\ensuremath{\star}}$ and weak magnetic fields, the system still exhibits magnetic properties which are practically indistinguishable from those of a condensed, superconducting CBG. We therefore conclude that while a condensate is a sufficient condition for the ideal CBG to exhibit a superconducting state, it may not be a necessary condition.
The breakdown of the Meissner-Ochsenfeld effect in the three-dimensional Bose gas as the applied field passes through its critical value is an entropy-driven weakly first-order transition, rather than the second-order transition usually ascribed to the system. The transition is second order at the usual Bose condensation temperature Tc{T}_{c}Tc​ as well as at T=0, with a line of first-order transitions connecting these critical points. The first-order transitions make the Bose gas resemble familiar superconductors, and a Landau-Ginzburg analysis indicates that the Bose gas is always a type-I superconductor.
We consider the mathematical model of an ideal gas of charged bosons or fermions in an n-dimensional space, treating n as a continuous variable. The investigation shows the extent to which the magnetic behavior depends on the dimensionality of the system. In particular, the charged Bose gas in a homogeneous magnetic field does not condense unless n > 4, in contrast to the field-free gas which condenses for n > 2: however so long as n > 2 and T < Tc this system completely expels homogeneous magnetic fields weaker than a certain critical field, Hc. (Tc is the field free transition temperature.) This field expulsion comes explicitly from the condensed ground-state bosons for n > 4; but it is still present, and Hc has the same form, for 4 > n > 2 where there is no condensation.