Effects of anisotropy of the superconducting energy gap

Clem, John Richard

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https://hdl.handle.net/2142/23968

Description

Title

Effects of anisotropy of the superconducting energy gap

Author(s)

Clem, John Richard

Issue Date

1965

Doctoral Committee Chair(s)

Bardeen, John

Kadanoff, L.P.

Department of Study

Physics

Discipline

Physics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

anisotropy

superconducting energy gap

thermodynamic properties

single-crystal superconductors

BCS model

Language

en

Abstract

Various effects of anisotropy of the superconducting
energy gap are theoretically considered. In order to estimate
the effects of anisotropy upon the thermodynamic properties
of pure, single-crystal superconductors, a factorable BCSlike
model for the effective electron-electron matrix
element Vp p • = [1 + a. (p) ] V [1 +a(p')] is used. The IN\ W\
effects of anisotropy upon the temperature dependence of
the gap parameter 6p ' the critical field, and the specific
heat near the critical temperature are shown to be small and
proportional to the mean-squared anisotropy < a2 ), which
.is of the order of 0.02 for typical superconductors.
A number of processes, such as those involved in lowtemperature
specific heat, nuclear spin-lattice re1axati9n
time, tunneling, surface resistance, and longitudinal
ultrasonic attenuation are more sensitive to the details
of the anisotropy than are the above thermodynamic properties.
Theoretical expressions which explicitly include the anisotropy
of a general gap parameter are given for these processes.
In order to simplify the description of these effects, two
quantities, the anisotropy distribution function P(a) and
the reduced, an~sotropic, effective density of states, are
introduced.
The influence of the presence of non-magnetic impurities
upon the anisotropy of the superconducting energy gap parameter
is consideredo Using the above BeS-like model within the
context of an earlier theory by Markowitz and Kadanoff, it
is shown that when impurities are present the wavevectordependent
gap parameter Ap is replaced by a complex, wavevectorand
energy-dependent gap parameter A(p,(I) =
VV\
A. «I) + a (p) Aa (CD) •
~ Vo.A
The behavior of Ai(m) and Aa(ro} is extensively examined as
a function of impurity concentration: it is found, for
example, that the magnitude of the anisotropic part Aa(ro)
of the gap parameter tends to zero in the limit of large impurity
concentration. A model calculation, assuming a
rectangular shape for the anisotropy distribution function
p'(a) I illustrates the behavior for small and moderate impurity
concentrations. The behavior for large impurity concentrations
is found to depend, to lowest order, only upon
< a2 >. The behavior of th~.effective density of states is
also examined and is shown to become isotropic as the impurity
concentration increases. The precise shape of the effective
density of states for energies near the gap is obtained for
the large impurity concentration limit.
The experimental manifestations of the reduction of the
anisotropy by impurity scattering are considered. The
thermodynamic properties are discussed in terms of the behavior
of quantities of the order of the mean-squared
anisotropy; the more sensitive experimental processes are
discussed in terms of the impurity concentration-dependent
behavior of the effective density of states.

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