Historical Introduction

[…] something unexpected occurred. The disappearance did not take place gradually but abruptly. […] Thus the mercury at $4.2K$ has entered a new state, which, owing to its particular electrical properties, can be called the state of superconductivity. – Kamerlingh Onnes, Nobel Lecture, December 11, 1913 [1]

Checklist

Check Steve Simon: SS Basic for history of conductors –when was Fermi surface concept introduced?
Add some history to the Landau theory –how did he come up with it. Generalistions...
Explain Type I and Type II
Finish Bardeen-Cooper-Schrieffer section
Add Gauge theory section
Add section on time-reversal symmetry breaking (to appendices, perhaps?)
Add GL geometric formulation to appendices.
Josephson effect
Nambu-Goldstone
High-temperature superconductors
Unconventional superconductors

Conductors

Drude model of conductors

Drude assumed that conduction electrons in a solid experience two interactions: an externally applied force $\mathbf{F}$ which accelerates them (whose origin, Drude takes to be a field gradient $\mathbf{F}\propto\mathbf{E}$ on the electric charges $q$ ), and a collision probability $1/\tau$ with the nuclei –a deceleration proportional to the momentum $\mathbf{p}$ . The average acceleration the electrons experience is therefore

\dot{\mathbf{p}}=-\frac{1}{\tau}\mathbf{p}+\mathbf{F}.

In terms of the current density $\mathbf{J}=nq\mathbf{p}$ , where $n$ is the number of charge carriers, the Drude model becomes

\dot{\mathbf{J}}=-\frac{1}{\tau}\mathbf{J}+\frac{nq²}{m}\mathbf{E}, \tag{1}

where $m$ is the mass of the carriers. In the steady-state, the Drude model captures Ohm’s Law

\mathbf{J}=σ\mathbf{E}, \tag{2}

with Ohm’s constant derived $σ≡\frac{nq²\tau}{m}$ [2] .

Equation 2

Equations 2, 1

Landau theory of phase transitions

The Landau theory describes continuous phase transitions due to a loss of symmetry[3] , one would begin by writing a free energy expansion in terms of an order parameter $η$ , such that

\begin{aligned} η = 0, \quad & T>T_C, \\ η\neq 0, \quad & T<T_C, \end{aligned}

where, here, an additional first-order transition is parametrised by temperature $T$ , $T_C$ is the critical temperature. The continuity of the free energy varying the parameter $η$ across the phase transition is enforced by the smoothness of the free energy, which must be a second-order differential function of the parameter. For example, in ferroelectrics, the spontaneous polarisation plays the role of the continuous parameter $η$ , and in ferromagnetism, it is the spontaneous magnetisation[4] .

Conventional superconductivity

Brief history

In 1911, investigations of matter approaching absolute zero shook our understanding of Physics. Kamerlingh Onnes had founded a cryogenics laboratory at the University of Leiden, with the ambition to investigate electron motion. Two schools of thought existed in the scientific community –those who imaged electrons flowing through a conductor as free and therefore freezing at absolute zero –in other words, the resistance would diverge; and others who expected the ceasing lattice vibrations to allow the electrons to flow unimpeded, thus the resistance would steadily decrease to zero. Work by Augustus Matthiessen had shown that metal conductivity increases with temperature. [5, 6]

Bringing the philosophical divergence into the realm of science, Kamerlingh Onnes measured the electrical conductivity of pure metals, starting with mecury, and later tin and lead. On the 8th of April 1911, working with mercury wire immersed in liquid helium, he discovered the resistance went to zero at 4.2K. The 1913 Nobel Prize in Physics was awarded to him for, “his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium”.

The second, equally-surprising characteristic of superconductivity, was discovered by Miessner and Ochsenfeld in 1933 [7] : the total expulsion of magnetic fields below the critical temperature. Thus, superconductivity became defined as a phase of matter characterised by two phenomena: zero resistance to current, and the total expulsion of magnetic fields.

For further history of superconductivity, see [8, 9]

London theory of the Meissner effect

Theoretical advances came in the 1935 hydrodynamic theory of the London brothers[10] . The theory almost follows from the Drude model of conductivity and Maxwell’s equations.

A perfect conductor is described by the Drude model in the collisionless limit $\tau→0$ as

\dot{\mathbf{J}}=\frac{nq²}{m}\mathbf{E}

Taking the curl of this equation, and applying Faraday’s law to relate the curl of the electric field with the magnetic flux, we have

References

H. Kamerlingh Onnes, Investigations into the properties of substances at low temperatures, which have led, amongst other things, to the preparation of liquid helium, NobelPrize.org, Dec. 11, 1913. [Online]. Available: https://www.nobelprize.org/prizes/physics/1913/onnes/lecture/ [Accessed: Jul. 4, 2023]. (↩︎)
S. Simon, The oxford solid state basics, CERN Document Server; Oxford Univ. Press, 2013. [Online]. Available: https://cds.cern.ch/record/1581455 [Accessed: Jul. 6, 2023]. (↩︎)
L. Landau,
1. ON THE THEORY OF SUPERCONDUCTIVITY (p.540)
,
in Collected Papers Of L. D. Landau, , 1965[Online]. Available: http://archive.org/details/d.-ter-haar-collected-papers-of-l.-d.-landau [Accessed: Jul. 6, 2023]. (↩︎)
L. Landau and E. Lifshitz, CHAPTER XIV - PHASE TRANSITIONS OF THE SECOND KIND AND CRITICAL PHENOMENA, in Statistical Physics (Third Edition), L. Landau and E. Lifshitz, Eds. Butterworth-Heinemann, 1980, pp. 446–516.doi:10.1016/B978-0-08-057046-4.50021-X (↩︎)
On the influence of temperature on the electric conducting power of metals, Philosophical Transactions of the Royal Society of London, vol. 152, pp. 1–27, 1862. [Online]. Available: https://www.jstor.org/stable/108819 [Accessed: Jul. 4, 2023]. (↩︎)
A. Matthiessen and A. Vogt, IV. On the influence of temperature on the electric conducting-power of alloys, Philosophical Transactions of the Royal Society of London, vol. 154, pp. 167–200, 1997. doi:10.1098/rstl.1864.0004 (↩︎)
W. Meissner and R. Ochsenfeld, Ein neuer Effekt bei Eintritt der Supraleitfähigkeit, Naturwissenschaften, vol. 21, no. 44, pp. 787–788, 1933. doi:10.1007/BF01504252 (↩︎)
P. Saunders G. A., The rise of the superconductors. CRC Press, 2004. doi:10.1201/9780203646311 (↩︎)
D. Van Delft and P. Kes, The discovery of superconductivity, Physics Today, vol. 63, no. 9, pp. 38–43, 2010. doi:10.1063/1.3490499 (↩︎)
F. London, H. London, and F. Lindemann, The electromagnetic equations of the supraconductor, Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences, vol. 149, no. 866, pp. 71–88, 1997. doi:10.1098/rspa.1935.0048 (↩︎)