We find a ln(/) correction because the echo is proportional to 1/t and is integrated up to the time when inelastic scattering processes destroy the phase coherence of the two complementary waves and the echo disappears. As a consequence the conductance is reduced and the resistance is increased, both by the same fraction . A similar argument can be made in real space where the coherent back-scattering yields the same correction to the diffusion constant D. And since the conductance is proportional to the diffusion constant one obtains the same conductance corrections G

This is not the end of the story, nature is even more generous. If we apply a magnetic field perpendicular to the film then our electron wave picks up a magnetic phase shift. For a closed loop this phase shift is equal to e/ where is the enclosed magnetic flux, and e the charge of the electron. Since our two complementary waves (along the closed loop) surround the flux in opposite direction their magnetic phase shifts have different signs and the two amplitudes have no longer the same phase. When their combined phase shift is of the order of one they no longer interfere constructively.

Let l be the mean free path of the electrons and the time between two collisions. Now we consider an electron on a closed loop with n scattering events. This electron returns after the time t=n to its starting point. (With t we denote the time which an electron needs to diffuse along a closed loop). Of course, there are many different closed loops with n scattering events. Each of these closed loops encloses an area F. The size of F for the different closed loops lies between -n² and +n² where n²=2Dt. (In two dimensions the diffusion constant has the value D=²/2.) In a magnetic field H the resulting phase shift between two complementary waves has a Gaussian distribution with a width of 2eFB/ 4DBt. For t>1/4DB there are closed loops with constructive and others with destructive interference. In an average the interference cancels out. For t<1/4DB, on the other hand, the phase shift between complementary waves is sufficiently small that the interference remains constructive. This means that in a given magnetic field we roughly accumulate the echo up to a time t=1/4DB. If we plot now the resistance as a function of 1/B then the 1/B axis is equivalent to a time axis.


In Fig.4 we have plotted the resistance of a Mg film (upper curve) as a function of 1/B in a semi-logarithmic plot. The corresponding time scale is indicated at the top of the figure. In our experiments the field B=1T is equivalent to a time of 0.3 ps. The ordinate, i.e. the resistance, represents (besides a constant) the integrated echo (corresponding to the number of sailors who returned up to a time t to the bar). On a logarithmic time scale the integrated echo is a linear function. For B<0.01T (which corresponds to t>30ps) the curve becomes flat. Here the diffusion time is so large that electron phonon processes destroy the coherence of the two complementary waves. The length of the coherent echo can be extended by lowering the temperature of the film sample.

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