Superconductivity

Prerequisites: Symmetry Breaking, Canonical Quantum Mechanics. Full path (20 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract

Why is it interesting?

A superconductor is simply a material in which electromagnetic gauge invariance is spontaneously broken.

Weinberg QFT Vol 1

Intuitive

• For a nice laymen introduction to superconductivity, see Draw me a picture of a Cooper pair by Brian Skinner

First I will explain how superconductivity arises, and then turn to the profound, and as it transpired inspirational, importance of hidden symmetry in this case.

An electron moving through a lattice of positively charged ions experiences an electrical attraction, which causes a slight distortion of the lattice.As a bell continues to ring after having been struck, so the lattice’s distortion may persist for a short while after the electron has passed. A second electron coming through finds a distorted lattice, and interacts with it. If the timing, speed, and spinning motions are right, the two interactions with the lattice cause the electrons to attract one another magnetically.They act cooperatively, like a single particle where the two spins, or individual magnetism, of the constituent electrons have canceled out.

American Leon Cooper was the first to realize this possibility, in 1956,and since then these twins have been called “Cooper Pairs.”8 Their con stituent electrons are fermions—particles with half-integer spins, whichin quantum mechanics act like cuckoos, where two in the same nest are forbidden. In a Cooper pair, the duo collectively has integer spin and actslike a boson.9 Bosons, by contrast, are like penguins, where large numbers cooperate as a colony. Bosons can collect together into the lowest-possible energy state—an effect known as Bose-Einstein condensation, after the scientists whose work led to this phenomenon being understood. It is manifested in weird phenomena such as the “superfluid” ability of liquid helium to flow through narrow openings without friction and of super-conductivity. In superconductivity, the Cooper Pairs act like bosons,which in concert make a “Bose condensate.” An analogy of the difference between conventional conductors, which have resistance to the flow of current and become hot, and superconduc tors, which offer no measurable resistance at all, is dancing in a wild night-club in contrast to what happens when a professional troupe performs a routine onstage.10 On a packed floor at a nightclub, hundreds of individual dancers are vigorously jiving, waving their arms, rocking from side to side,and bumping into one another. This state is like that of electrons in a metal. To model the electric field, imagine the dance floor is tilted to one side, as on a cruise ship, which is listing slightly. The force of gravity will push the dancers gently toward one side of the room. As they drift across,they continue dancing, the whole ensemble behaving quite chaotically.The more collisions you have, the more energy you waste.

This is how electrons behave in a warm metal when an electric fieldpushes them in one direction. The electrons move in the direction dic-tated by the field, meanwhile bumping into one another, losing energy asheat. An overall movement of dancers—in this case electric charge—en-sues; electric current flows, but there is a lot of resistance along the way.

The Cooper Pairs in a superconductor are like professional ballroomdancers who are performing as a troupe, rather than as individual pairs.However, in this particular routine, your partner is not dancing with you cheek to cheek, but instead is far across the room, their motion mirroring your own precisely. A large number of dancers may be between you and your partner, each of them in turn being paired with another, somewhere in the crowd. The entire company performs as a coherent whole, a sens eof order existing throughout the ballroom.

Any disturbance that would hinder a single dancer in the first example would have to affect the full ensemble of performers in the second case.The collective power of the organized troupe enables it to continue unimpeded; their motion—the electric current for the real case of electrons—loses no energy.

All that is required for this to happen is the existence of the organized pairs. The dynamics of the choreography will determine precisely how they go about it, but the concept itself depends only on the ability of the electrons to pair off. Today we know that this powerful pair bonding is an example of spontaneous symmetry breaking; the symmetry that has be-come hidden in the real case of electrons in a superconductor is gauge in-variance of electromagnetism. This work would lead to two Nobel Prizes.

Nambu appears to have been the first to recognize that gauge invariance does hold true in the BCS Theory but has become hidden. He had identified a profound truth: When the temperature gets cold enough, the fun damental patterns of electromagnetism—gauge invariance —may be hidden, as a result of which strange things happen, such as the appearance of the boson like Cooper Pairs. …. In a superconductor, the ground state contains Cooper Pairs. It costs energy to break up any pair, liberating individual electrons. Once liberated,the electrons have higher energy, the difference from their original bond ing in pairs being called the “energy gap.” The freed electrons receive this energy, which via E=mc2 makes them appear to have gained mass. This gave Nambu an idea: If the universe itself was like a superconductor, could the masses of particles arise by some analogous mechanism? from "The Infinity Puzzle" by Frank Close

Concrete

Below a critical temperature $T_c$ a superconductor shows two defining effects: zero DC resistance, and the expulsion of magnetic fields from its interior (the Meissner effect). The second one is the more fundamental, since a perfect conductor with zero resistance alone would only freeze in whatever field it had when it was cooled. We use SI units.

London equations. Assume a density $n_s$ of superconducting carriers with charge $e$ and mass $m$ that move without friction. Newton's law $m\,\dot{\vec v} = e\vec E$ for the supercurrent $\vec J_s = n_s e \vec v$ gives the first London equation, and London's further assumption gives the second: $$\frac{\partial \vec J_s}{\partial t} = \frac{n_s e^2}{m}\vec E, \qquad \nabla \times \vec J_s = -\frac{n_s e^2}{m}\vec B .$$ Combining the second with Ampère's law $\nabla \times \vec B = \mu_0 \vec J_s$ and $\nabla\cdot\vec B = 0$ gives $$\nabla^2 \vec B = \frac{1}{\lambda_L^2}\vec B, \qquad \lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}} .$$ For a superconductor filling $x > 0$ with an applied field parallel to the surface, the solution is $B(x) = B(0)\, e^{-x/\lambda_L}$. The field decays over the London penetration depth, typically tens to hundreds of nanometres. That is the Meissner effect. In the London gauge the two equations are summarised by $\vec J_s = -\frac{n_s e^2}{m}\vec A$.

Ginzburg-Landau theory. Near $T_c$ the superconducting state is described by a complex order parameter $\psi(\vec x)$, with $|\psi|^2$ proportional to the density of superconducting carriers. The free energy is $$F = \int d^3x \left[ \alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*}\left|\left(-i\hbar\nabla - q\vec A\right)\psi\right|^2 + \frac{\vec B^2}{2\mu_0} \right],$$ with $\beta > 0$ and $\alpha \propto (T - T_c)$. Above $T_c$ the minimum is $\psi = 0$. Below $T_c$, $\alpha < 0$ and the minimum is at $|\psi|^2 = -\alpha/\beta$, with an arbitrary phase. This is the symmetry breaking discussed in the Abstract layer. Microscopic theory shows that the carriers are pairs of electrons, so $q = 2e$, and one conventionally sets $m^* = 2m_e$. Varying $F$ gives two length scales: the penetration depth $\lambda$ and the coherence length $\xi$ over which $|\psi|$ can vary. Their ratio $\kappa = \lambda/\xi$ separates type I superconductors ($\kappa < 1/\sqrt 2$), which expel the field completely until superconductivity breaks down, from type II ($\kappa > 1/\sqrt 2$), where above a lower critical field the field enters as a lattice of quantised vortices.

Flux quantisation. Write $\psi = |\psi| e^{i\theta}$. Deep inside a superconducting ring the current vanishes, which requires $\hbar\nabla\theta = q\vec A$. Integrating around the ring and using that $\psi$ is single valued, so $\theta$ changes by $2\pi n$, $$\Phi = \oint \vec A\cdot d\vec l = \frac{2\pi\hbar}{q}\, n = n\,\frac{h}{2e}, \qquad \Phi_0 = \frac{h}{2e} \approx 2.07\times 10^{-15}\,\text{Wb}.$$ The measured flux quantum with $2e$ instead of $e$ was direct evidence for pairing.

BCS theory. In conventional superconductors an effective attraction between electrons near the Fermi surface, mediated by phonons, binds them into Cooper pairs. The single-particle excitations then have energies $$E_{\vec k} = \sqrt{\xi_{\vec k}^2 + \Delta^2},$$ where $\xi_{\vec k}$ is the normal-state energy measured from the Fermi level and $\Delta$ is the energy gap. In the weak-coupling limit, with $N(0)$ the density of states at the Fermi level, $V$ the pairing interaction and $\omega_D$ the Debye frequency, $$k_B T_c \approx 1.13\,\hbar\omega_D\, e^{-1/(N(0)V)}, \qquad 2\Delta(0) \approx 3.53\, k_B T_c .$$ Breaking a pair costs at least $2\Delta$, so at temperatures well below $T_c$ the number of thermally excited quasiparticles is suppressed by a factor of order $e^{-\Delta/k_B T}$.

Abstract

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1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Group Theory
6. Hilbert Space
7. Legendre Transformation
8. Newton's Second Law
9. Vector Calculus
10. Newtonian Mechanics
11. Symmetry
12. Variational Calculus
13. Configuration Space
14. Euler-Lagrange Equations
15. Lagrangian Formalism
16. Phase Space
17. Hamiltonian Formalism
18. Symmetry Breaking
19. Schrödinger Equation
20. Canonical Quantum Mechanics