Effects of finite electron mean free path on the attenuation,electromagnetic generation,and detection of ultrasonic shear waves in superconductors

Use of the general many-body formalism allows us to derive a system of equations describing the propagation of electromagnetic shear waves coupled to ultrasonic shear waves in normal and superconducting metals with arbitrary electron mean free paths. From this system of equations we derive general expressions in terms of correlation functions for the attenuation coefficient as well as for the generation and detection (radiation) efficiency of ultrasonic shear waves, assuming specular reflection. It is shown that, for arbitrary mean free path, generation and radiation efficiency are equal. Furthermore, they are related in a simple way to the residual surface resistance caused by ultrasound generation. In order to incorporate the frictional force between electrons and lattice, a reactive part of which remains finite even at T = 0, one has to introduce the current-stress tensor correlation function. The calculation of this correlation function for an impure BCS superconductor is more complicated than the calculation of the conductivity from the current-current correlation function. If suitably normalized, the electromagnetic contribution to the attenuation coefficient depends on mean free path only through the conductivity. For sufficiently short mean free path the generation and radiation efficiency in a superconductor can become equal to and even larger than the efficiency in the normal metal because the frictional force, which opposes the electromagnetic field driving the ions, is less effective in the superconducting than in the normal state. It is shown that the interpretation of experiments on the radiation efficiency in superconducting tin has to be modified when the finite electron mean free path is taken into account. Agreement between theory and experiment can still be achieved by a suitable choice of the Fermi velocity and the mean free path. The London penetration depth calculated from this Fermi velocity turns out to be smaller than generally accepted values.