Effect of Lattice Disorder on the Superconducting Transition Temperature

Abstract

A theory is presented to explain the dependence of the superconducting transition temperature T c on the changes in the phonon frequency spectrum and electronic density of states which result from lattice disorder. Numerical calculations of T c are presented for films composed of crystalline granules, for films composed of amorphous granules, and for homogeneous amorphous metals. The calculations are in good agreement with experimental values of T c .Experiments 1 ' 2 on disordered films of a variety of metals and on disordered dilute Sn-Cu alloys 3 have shown that the superconducting transition temperature T c increases with increasing lattice disorder. None of the explanations proposed 4 ' 5 for the enhancement of T c in superconducting films has been applicable to all experimental situations discussed in this paper. Although no calculations are presented for the case of very thin films, 4 the theory proposed here is applicable to films composed of small metallic crystallites, 1 ' 6 films composed of amorphous metallic granules, 1 ' 2 ' 6 and homogeneous amorphous superconducting alloys. In particular, a two-parameter model for granular films of nearly-free-electron metals yields numerical values for the enhancement of T c which are in good agreement with the experimental values found by Buckel and Hilsch 1 and von Minnigerode. 2 The theory proposed here is based on the assumption that the average amplitude of ionic vibrations is larger in a disordered lattice than a perfect crystal. For disordered films which consist of small crystalline granules, 6 this increase in the average amplitude of ionic vibrations results from the many ions which are in lattice positions of reduced symmetry near the surface of the granules. Since these ions are held in place by weaker ionic forces than those found in a bulk crystal, they undergo localized ionic vibrations of larger amplitude and lower frequency than those found in a bulk crystal. As a result, the formation of granules (1) increases the average amplitude of ionic vibrations and decreases the average phonon frequency (^ph)> anc * (2) broadens the peaks in the phonon density of states D(w). For homogeneous amorphous metals, the increased average amplitude of ionic vibrations results from the weakened forces acting on all of the ions. In this case, the first effect is usually more important than the second. Both the effects found in homogeneous amorphous metals and those found in crystalline granular films occur in the case of disordered films which consist of amorphous granules.The increase in the average amplitude of the ionic vibrations increases the average electronphonon coupling constant. This increases both T c and the phonon contribution 7 X to the electron mass renormalization constant Z(0) = (1 + X)Z e . Here, Z e is the Coulomb mass renormalization constant. 8 Note that X is approximately given by 7where N(Q) is the electronic band density of states at the Fermi level, M is the ionic mass, (oOp^2) is the average squared phonon frequency, and (J 2 ) is the aver...

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