# Superconductivity

I wrote this note in the process of learning Josephson junctions. “It would be a good way to learn this from the bottom-up”, I thought, and then looked for various YouTube videos, textbooks, articles and lecture notes on superconductivity. I had watched YouTube videos about superconductivity and floating magnets (and prototypes of floating trains) before, but was no where close to understanding them. I started likewise – watched few YouTube videos about Meissner effect, spin of electron, Josephson effect etc., and then when it was not enough I started looking for textbooks and articles. After learning some bits, I began to appreciate the findings of Meissner and Ochsenfeld, more so because it was discovered at a time when it was important to incite rapid interest in a field so crucial to our understanding of matter. It sort of opened new doors to peek into the superconductive world. Fritz London and Heinz London then came with a mathematical formulation to describe the Meissner effect. The London equations are first hand derivation of the Maxwell’s equation and the current density equation (eyebrows raised), and therefore were not difficult to grasp. I understood the nitty-gritty details of the Meissner effect and realized that “External magnetic field cannot penetrate a superconductor!”, was not a true statement (not completely). The magnetic field did penetrate the superconductor, but the depth to which it penetrated was small enough to treat superconductors as something that expelled external magnetic fields altogether. Maybe this statement made a strong impression that was needed to catch eyes of listeners, or may be not, but letting go of this deeply held belief was not easy (it never is I thought and moved on).

My curiosity led me to study the macroscopic theory and how/why everything was tied to the phase of the wavefunction of superconducting particles. I think it was natural to go into Landau theory and his phenomenological model of second order phase transition to understand the origin. This phase transition i.e. solid to gas or say ferromagnetism to anti-ferromagnetism did not have a direct relation with the phase of the exponential function I was trying to understand, but helped develop some understanding of the superconducting domain. The superconducting domain started looking like a ball of spaghetti (a lake full of it) where everything was tangled up so nicely that understanding one would unlock all other. I didn’t know how much of this made sense and served my learning purpose, but I did realize that I had lost track. The allure of this novel field however kept me diverted and I soon ran into the dark alleys of condensed matter physics and began wandering from “broken symmetry” to “mean-field theory” to the concept of “second quantization”. I learned bits and pieces of each, unable to form a clear idea (not knowing how to relate to my old bits) I called time and finally decided to whip my mind-horse back to track. I then focused towards understanding the making of the superconducting wavefunction, the concept of cooper pairs and the BCS theory, few thermodynamical models and things easily available in the Tinkham book. My plan here is to grind all those pieces of information learned from various sources, and come up with a sweet and sour sauce based on my taste, palate and perception. I hope it is edible.

## Introduction:

Kamerlingh Onnes work on cryogenics made possible the liquefaction of Helium. Able to reach temperature as low as 2K with his liquid Helium apparatus, he conducted series of experiments on various metals to test their electrical properties at extremely low temperatures. It was with Mercury that he noticed a significant drop in resistance just below a critical temperature, and discovered superconductivity. A result driven by curiosity to probe with a new toy just revealed to mankind, this fascinating discovery allured the human mind with all its profuse glory and possibilities, which then led to even more interesting developments. The Meissner effect was one of the early ones that helped stretch our ignorance further, and amuse scientists still trying to wrap their heads around superconductivity and the cause of this something spectacular. Nonetheless, it showed us a way to peek into this new state of matter with the help of our beloved Maxwell’s equations. Let me write down the four Maxwell’s equations before getting into the superconducting stuff:

$\nabla.\,\vec{D} = \rho_V$
$\nabla.\, \vec{B} = 0$