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PippardCorrections

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clean vs. dirty superconductors

The conservation of the total momentum upon application of an external magnetic field is actually a deep consequence of the quantum nature of the macroscopic superconducting ground state, and the London fully understood this implication.

Historically however there was a problem: the values of the penetration depth {$\lambda$} obtained by Eq. 8 are too small for Al, In and other known superconductors, if one assumes that {$n$} is the full number density of conduction electrons and {$m$} their effective mass.

The answer proposed by Pippard is that electrons interact with fields averaging their own response on some local scale, therefore the interaction between fields and currents become non-local. The length scale {$\xi$} controlling this non-local interaction, called coherence length, has an intrinsic limit, due to the quantum nature of superconductivity

{$\qquad\qquad \xi_0=\frac {\hbar v_F}{\pi\Delta}\propto \frac {\hbar v_F}{k_B T_c}$}

where {$v_F$} is the Fermi velocity and the middle expression is from BCS, with the superconducting gap {$\Delta$}. The last expression was deduced by Pippard by an uncertainty principle argument. Assuming that {$\xi_0$} is the minimum space to define the electron wave function {$\xi_0\Delta p\ge \hbar$} implies that only electrons with momentum within a crust {$k_BT_c$} of Fermi energy can contribute, i.e. just a fraction, not all the {$n$} conduction electrons.

Typically {$\xi_0$} is (much) greater than {$\lambda$} for type I superconductors, e.g. a clean superconducting simple metals^[1]. However the extrinsic electron mean free path, {$\cal l$} cannot be neglected, since it limits the effective coherence length by

{$\qquad\qquad \frac 1 \xi \,=\, \frac 1 {\xi_0}\, +\, \frac 1 {\cal l}$}

This leads to an increased penetration depth for dirty superconductors, by replacing {$n$} with the correct fraction. The calculation is not straightforward. One limit which is of interest to µSR is for dirty (short {$l$}) local (short {$\xi_0$}, i.e. type II) superconductors. In this case

{$ (1) \qquad\qquad \lambda=\sqrt{\frac {m} {\mu_0 ne^2}(1+\xi_0/l)} $}

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