Abstract
We have seen how the conductivity of a metal is limited by collisions, which determine the electron mean free path. After a collision, an electron completely loses all memory of its quantum state as specified by its quasi-momentum k. It is thus impossible to follow a Bloch state over any distance much greater than the mean free path. To understand the microscopic origin of these collisions, one had to measure the conductivity of very pure metals at low temperatures. This was made possible by the work of the physicist Kammerlingh Onnes, who specialised in the liquefaction of gases and opened the way to the use of cryogenic fluids. In 1911, he succeeded in liquefying 4He, at a temperature of 4.2 K. He then suggested using the low temperatures created in this way to study the low temperature conductivity of pure metals. Quite unexpectedly, he discovered that the conductivity of mercury increased by several orders of magnitude at temperatures below 4.18 K. This discovery of superconductivity remained a mystery for more than 40 years. With hindsight, it is clear that the prerequisites of quantum mechanics had not yet been established. Of course, this did not prevent the physicists of the first half of the twentieth century from gradually getting a hold on the fundamental manifestations of this phenomenon through ideas based entirely on experimental observations.
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- 1.
We shall see in Sect. 5.2 that changes in B nevertheless occur within the superconducting loop.
- 2.
We shall see that in reality currents only flow at the surface of the wire, over some thickness λ (see Sect. 5.2). The critical current density is thus in this surface layer \(\textbf{j}_{\textrm{c}}\approx H_{\textrm{c}}/\lambda\).
- 3.
It is worth noting how important empirical models can be for scientific progress. A full formulation is often only possible much later, when many conceptual obstacles have been removed. And even though today the BCS theory provides an explicit understanding of the observed relationships, the simplified representation due to the London brothers remains an extremely useful guide to understanding many experimental observations.
- 4.
Can the quantities \(\phi(\textbf{r})\) and \(\theta(\textbf{r})\) be defined at the same point of space? In the case of a one-particle wave function, the state and energy are defined, but the phase cannot be simultaneously measured. However, in a macroscopic system, the density and phase can be defined simultaneously, because the relevant uncertainty relation is \(\varDelta n_{\textrm{s}}\varDelta\theta> 1\). Since n s is macroscopic, the two quantities can be defined accurately enough, and can thus be treated semi-classically.
- 5.
In an almost perfect metal with very low resistivity, the mean free path can be long enough to ensure that the electron wave functions are coherent on a macroscopic scale. Effects associated with this quantum coherence can then be detected, appearing as periodic oscillations in certain macroscopic quantities, such as the magnetic susceptibility or the resistance of the ring. These observations bear no relation to a quantisation of the magnetic flux.
- 6.
Note that, for a connected sample, Eq. (5.26) resumes into
$$\lambda^{2}\mu_{0}\textbf{j}=-\textbf{A}$$and we directly obtain then an elegant representation of the two London equations, which are recovered by differentiating with respect to time or by taking the curl.
- 7.
Some features can be understood by attempting Problem 11: Direct and Alternating Josephson Effects in Zero Magnetic Field.
- 8.
See Problem 12: Josephson Junction in a Magnetic Field.
References
Rose-Innes, A.C., Rhoderick, E.H.: Introduction to Superconductivity. International Series in Solid State Physics, Pergamon Press, Oxford (1978)
Basdevant, J.L., Dalibard, J.: Quantum Mechanics. Springer-Verlag, Berlin, Heidelberg (2002)
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Alloul, H. (2011). Introduction to Superconductivity. In: Introduction to the Physics of Electrons in Solids. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13565-1_5
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DOI: https://doi.org/10.1007/978-3-642-13565-1_5
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