The magnetic transition in moderately small superconductors, and the influence of elastic strain

Abstract

Measurements have been made of the onset of the superconducting phase transition of tin whiskers (single crystals of diameter 1—2 μm and length several millimetres) as a function of temperature T , magnetic field H , and elastic strain ε up to 2%. For samples of this size (denoted ‘moderately small ’ since they are larger than λ(0)), there is a range of temperatures, approximately 20-30 mK below the transition temperature T e (ε), for which the transition is of second order in the Ehrenfest sense. Below the temperature denoted T ' (ε) the transition is of first order and may exhibit hysteresis. The phase diagram at constant strain is derived from the equation Δ G = − A ψ 2 + 1 2 B ψ 4 + 1 3 C ψ 6 , where Δ G is the free energy difference between the superconducting and normal states, ψ is the wave function of the superconducting electrons, and the coefficients A , B and C each comprise two terms, of which one is field-dependent, being proportional to H 2 . The other, field-independent, term is Ginzburg’s (1958) expression for the zero-field energy difference, so that A contains a term proportional to ( T c — T ), and B is independent of T . Coefficient C contains a field-independent term, assumed independent of T , which we introduce for consistency. The condition A = 0 describes both the secondorder transition and limiting supercooling, while the transition at thermodynamic equilibrium in the first order region and limiting superheating are described by B 2 = − 16 3 A C and B 2 = — 4 AC respectively. The Landau critical point ( H ', T ') is given by B — 0, A = 0. If the limiting metastable transitions for a cylinder in parallel field are included on a phase diagram, then the supercooling curve is a continuation of the second order curve while the curve for thermodynamic equilibrium branches from it tangentially if C ( H ', T ') > 0, or at a slope which is 1.32 times greater than this if C ( H ', T ’) = 0. The case C ( H ', T ') < 0 is discussed elsewhere (Nabarro & Bibby 1974, following paper in this volume). The last case arises because our observations indicate that the field-independent term in C is negative. Estimates of the sample size were made by using the present theory and were in fair agreement with estimates made electron-microscopically. Expressions for the change at the superconducting transition of the specific heat and other second derivatives of the Gibbs free energy above and below T ' are derived. It is shown theoretically that the transition remains of second order when the sample is strained elastically. Some of the Ehrenfest relations describing a second order transition with two independent variables are experimentally verified from our data.