ISSN:
1573-7357
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
We describe a new formulation of the theory of inhomogeneous superconductors, applicable to all cases in which the Bogoliubov equations reduce to ordinary differential equations, and intermediate between the Bogoliubov, or wave function, and Gor'kov, or Green's function, formulations; the method is illustrated for the simple case of zero magnetic field, but a generalization appears straightforward. The method is based on the former formulation, and hence is well suited for the investigation of boundary effects; neither the wave functions nor the energy eigenvalues are required, however, for either the free energy or the self-consistency equation. We show that the free energy can be expressed in terms of a single function ℰ, derive integral and differential equations for ℰ, and give a thorough discussion of its properties. We also show that the self-consistency equation for the order parameter Δ(x) can be expressed in terms of ℰ and a generalization of it. We apply the new formulation in two investigations of the question of anomalous terms in the free energy near the transition temperature. First we calculate the free energy for arbitrary Δ(x), but only to terms quadratic in Δ(x) and its derivatives; we show that anomalous terms occur if, and only if, either some odd-numbered derivative of Δ(x) fails to vanish at a free surface, or Δ(x) (or one of its derivatives) is discontinuous. Second, we calculate the free energy to all orders in Δ(x) for four models in which ℰ can be obtained analytically, illustrating the conclusions of the first investigation, and resolving a contradiction in the literature. Among these analytically soluble models is the previously unknown case% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca% qGOaGaaeiEaiaabMcacaqG9aGaeuiLdq0aaSbaaSqaaGqadiaa-jea% caWFdbGaa83uaaqabaGccaqGOaGaaeymaiaabccacaqGRaGaaeiiai% aabwgadaahaaWcbeqaaiaab2cacqaHXoqycaqG4baaaOGaaeykamaa% CaaaleqabaGaaeylaiaabgdaaaGccaGGUaaaaa!4C6D!\[\Delta {\text{(x) = }}\Delta _{BCS} {\text{(1 + e}}^{{\text{ - }}\alpha {\text{x}}} {\text{)}}^{{\text{ - 1}}} .\]
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00120865
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