---
title: "Superconductivity: Phenomenology and BCS"
module: Superconductivity
moduleNumber: 10
lessonNumber: 1
order: 1001
summary: >
  Below a critical temperature some materials lose all resistance and expel
  magnetic flux — the Meissner effect that defines the state. The isotope effect
  points to lattice vibrations, and BCS theory binds electrons into Cooper pairs
  through phonon exchange. The paired condensate opens an energy gap, quantizes
  magnetic flux, and drives the Josephson effects.
topics: [Superconductivity]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 10 — Solid State Physics; §10-9 Superconductivity, Flux Quantization, Josephson Junction"
---

In 1911, while probing the properties of metals at liquid-helium temperatures,
Kamerlingh Onnes found that the resistance of mercury dropped abruptly to zero
below a **critical temperature** $T_c = 4.2\ \text{K}$. Below $T_c$ the
resistivity is not merely small but exactly zero: currents set up in
superconducting rings have persisted for years with no measurable decay.

$$
% caption: Onnes's measurement: the resistance of mercury falls sharply to zero
% at the critical temperature T_c = 4.2 K, marking the onset of superconductivity.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.6,0) node[below] {$T$ (K)};
  \draw[->, black] (0,0) -- (0,3.4) node[left] {$R$};
  % flat above Tc then sudden drop at x corresponding to 4.2
  \draw[acc, very thick] (5.8,2.4) -- (3.4,2.15) -- (3.15,2.0);
  \draw[acc, very thick] (3.15,2.0) -- (3.1,0.05) -- (0.3,0.03);
  \draw[black, dashed] (3.12,0) -- (3.12,2.1);
  \node[anchor=north, font=\scriptsize] at (3.12,0) {$T_c = 4.2$};
  \node[anchor=west, font=\scriptsize] at (4.2,2.5) {normal metal};
  \node[anchor=west, font=\scriptsize] at (0.4,0.35) {superconducting};
\end{tikzpicture}
$$

Critical temperatures range from below $0.1\ \text{K}$ (hafnium, iridium) to
$9.25\ \text{K}$ (niobium) for elements. An applied magnetic field lowers $T_c$;
above a **critical field** $B_c$ superconductivity vanishes at any temperature.
The metal compound $\text{MgB}_2$ superconducts at $39\ \text{K}$, and the cuprate
ceramics — beginning with the 1986 discovery and reaching $\text{YBa}_2\text{Cu}_3
\text{O}_7$ at $92\ \text{K}$ — superconduct above the $77\ \text{K}$ boiling
point of inexpensive liquid nitrogen.

## The Meissner effect

Zero resistance is not the defining property. Consider cooling a material through
$T_c$ in a small external field. Zero resistance alone (via Faraday's law) would
only freeze the field already inside. What is observed is stronger: the field is
actively **expelled**, and the interior field becomes exactly zero.

$$
% caption: The Meissner effect. Above T_c the field threads the sample; cooled
% below T_c, screening supercurrents on the surface expel the field so B = 0
% inside — the superconductor is a perfect diamagnet.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- above Tc: field passes through ---
  \begin{scope}
    \draw[black] (-1.0,0) circle (0.9);
    \foreach \y in {-0.6,-0.2,0.2,0.6}
      \draw[black, ->] (-2.4,\y) -- (0.6,\y);
    \node[anchor=north] at (-1.0,-1.15) {$T > T_c$};
  \end{scope}
  % --- below Tc: field expelled ---
  \begin{scope}[xshift=5.6cm]
    \draw[acc, thick, fill=acc!10] (-1.0,0) circle (0.9);
    % field lines curving around
    \draw[black, ->] (-2.4,0.6) .. controls (-1.4,0.6) and (-1.05,1.2) .. (-1.0,1.35);
    \draw[black, ->] (-2.4,-0.6) .. controls (-1.4,-0.6) and (-1.05,-1.2) .. (-1.0,-1.35);
    \draw[black, ->] (-2.4,0.2) .. controls (-1.7,0.2) and (-1.75,1.0) .. (-0.2,1.15);
    \draw[black, ->] (-2.4,-0.2) .. controls (-1.7,-0.2) and (-1.75,-1.0) .. (-0.2,-1.15);
    \node[acc, font=\scriptsize] at (-1.0,0) {$B = 0$};
    \node[anchor=north] at (-1.0,-1.55) {$T < T_c$};
  \end{scope}
\end{tikzpicture}
$$

> **Definition (Meissner effect).** The expulsion of magnetic flux from a material
> as it is cooled below $T_c$, leaving zero field inside. It, not zero
> resistance, is the criterion for superconductivity. The expelling **screening
> currents** flow on the surface and make the superconductor a perfect diamagnet.

Establishing the screening current costs an energy density $B^2/2\mu_0$; when $B >
B_c$ there is not enough energy available and the material reverts to normal.
Superconductors come in two kinds:

- **Type I** ("soft"), mostly pure metal elements, show the complete Meissner
  effect up to a single $B_c$, which is too small for useful magnets.
- **Type II** ("hard"), usually alloys, have two critical fields. Below $B_{c1}$
  they fully expel flux; between $B_{c1}$ and $B_{c2}$ the field penetrates as
  quantized **flux tubes** (vortices) of normal material threading the
  superconductor; above $B_{c2}$ they go normal. Because $B_{c2}$ can reach tens
  of tesla, type II materials build high-field magnets.

$$
% caption: Magnetization versus applied field. A type I superconductor expels
% flux completely up to B_c; a type II superconductor expels fully to B_c1, then
% admits flux tubes up to the much larger B_c2 before going normal.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- type I ---
  \begin{scope}
    \draw[->, black] (0,0) -- (3.2,0) node[below] {$B_{\text{app}}$};
    \draw[->, black] (0,-2.0) -- (0,0.4) node[left, font=\scriptsize] {magnetization};
    \draw[acc, very thick] (0,0) -- (1.8,-1.6);
    \draw[acc, very thick] (1.8,-1.6) -- (1.85,0) -- (3.0,0);
    \draw[black, dashed] (1.8,0) -- (1.8,-1.6);
    \node[anchor=south, font=\scriptsize] at (1.9,0.02) {$B_c$};
    \node[anchor=north] at (1.6,-2.0) {type I};
  \end{scope}
  % --- type II ---
  \begin{scope}[xshift=5.2cm]
    \draw[->, black] (0,0) -- (3.4,0) node[below] {$B_{\text{app}}$};
    \draw[->, black] (0,-2.0) -- (0,0.4) node[left, font=\scriptsize] {magnetization};
    \draw[acc, very thick] (0,0) -- (1.0,-1.4);
    \draw[acc, very thick] (1.0,-1.4) .. controls (2.0,-0.7) and (2.6,-0.15) .. (3.0,0);
    \draw[black, dashed] (1.0,0) -- (1.0,-1.4);
    \node[anchor=south, font=\scriptsize] at (0.95,0.02) {$B_{c1}$};
    \node[anchor=south, font=\scriptsize] at (2.95,0.02) {$B_{c2}$};
    \node[anchor=north] at (1.6,-2.0) {type II};
  \end{scope}
\end{tikzpicture}
$$

The flux threading a superconducting loop is **quantized**. Because no emf can
exist in the loop, the enclosed flux is frozen, and quantum mechanics sharpens
this to

$$
\Phi_m = n\,\frac{h}{2e}, \qquad \Phi_0 = \frac{h}{2e} = 2.07\times10^{-15}\ \text{T}\cdot\text{m}^2.
$$

The quantum of flux $\Phi_0$ is a **fluxoid**; each flux tube in a type II
superconductor carries exactly one. The factor of $2e$, not $e$, is the first
hint that the charge carriers are electron _pairs_.

## BCS theory and Cooper pairs

The mechanism was found in 1957 by Bardeen, Cooper, and Schrieffer. The decisive
clue was the **isotope effect** (1950): the critical temperature depends on the
average isotopic mass $M$ as

$$
M^\alpha T_c = \text{constant},
$$

with $\alpha \approx 0.5$ for many superconductors. A dependence on the *mass of
the ions* means the lattice vibrations — phonons — are essential. If the ions
were infinitely heavy (fixed lattice), $T_c$ would be zero.

The pairing works through the lattice. An electron moving through the lattice
attracts the nearby positive ions, displacing them slightly and leaving a region
of enhanced positive charge behind it — a propagating lattice distortion, i.e. a
phonon. A second electron is attracted to that positive region. The net effect is
an **attraction** between the two electrons, mediated by the lattice.

$$
% caption: Cooper pairing: a moving electron draws the positive ions inward,
% leaving a trailing region of excess positive charge (a phonon) that attracts a
% second electron; the two bind into a Cooper pair below T_c.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ion lattice, pulled inward near center
  \foreach \i in {0,1,2,3,4,5}{
    \foreach \j in {0,1,2}{
      \fill[black] (\i*1.0,\j*0.95) circle (2pt);
    }
  }
  % distortion region (denser positive)
  \draw[acc, dashed, thick] (2.5,0.95) ellipse (0.7 and 0.55);
  \node[acc, font=\scriptsize, anchor=south] at (2.5,1.55) {excess positive};
  % first electron leading
  \fill[acc] (3.6,0.95) circle (3pt);
  \draw[acc, ->, thick] (3.7,0.95) -- (4.5,0.95);
  \node[acc, anchor=south, font=\scriptsize] at (3.9,1.15) {electron 1};
  % second electron trailing, attracted
  \fill[acc] (1.4,0.95) circle (3pt);
  \draw[acc, ->, thick] (1.5,0.95) -- (2.1,0.95);
  \node[acc, anchor=north, font=\scriptsize] at (1.4,0.7) {electron 2};
\end{tikzpicture}
$$

Below $T_c$ this attraction can exceed the Coulomb repulsion, and the electrons
form a bound **Cooper pair** with opposite spins and opposite momenta — total
spin zero, total momentum zero.

> **Definition (Cooper pair).** A bound pair of electrons with opposite spin and
> opposite momentum, held together by phonon exchange. Having zero net spin, a
> pair behaves as a boson and does **not** obey the exclusion principle, so any
> number of pairs occupy the same lowest-energy state — a coherent condensate.

Because all the pairs occupy one quantum state and act together, a single pair
cannot be scattered by a lattice ion without breaking it. Breaking a pair costs
the **superconducting energy gap** $E_g$, predicted by BCS to be

$$
E_g = 3.5\,k T_c
$$

at $T = 0$. For cadmium ($T_c = 0.517\ \text{K}$) this gives $E_g = 1.56\times
10^{-4}\ \text{eV}$, within 4% of the measured value — some four orders of
magnitude smaller than a semiconductor gap. When a pair carries net momentum, the
whole condensate carries current, and since scattering that changes one pair's
momentum is forbidden below the gap, the current flows without resistance.

$$
% caption: The superconducting energy gap shrinks from 3.5 kT_c at T = 0 to zero
% at T_c as thermal excitations break Cooper pairs, following the BCS curve.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[below] {$\frac{T}{T_c}$};
  \draw[->, black] (0,0) -- (0,3.4) node[left, font=\scriptsize] {$\frac{E_g(T)}{E_g(0)}$};
  \draw[acc, very thick, domain=0:0.985, samples=90, variable=\x]
    plot ({\x*6.0},{3.0*sqrt(1-\x*\x)});
  \draw[black, dashed] (6.0,0) -- (6.0,-0.05) node[below, font=\scriptsize] {$1$};
  \draw[black, dashed] (0,3.0) -- (0.1,3.0);
  \node[anchor=east, font=\scriptsize] at (-0.05,3.0) {$1$};
\end{tikzpicture}
$$

The gap closes as $T \to T_c$, where $E_g \to 0$ and pairs break freely. The
critical field follows a similar law, $B_c(T) = B_c(0)\bigl[1 - (T/T_c)^2\bigr]$.
Cooper pairs are large: estimating the pair size from the uncertainty principle
gives $\Delta x \sim 10^3\ \text{nm}$, about ten thousand atomic diameters, so the
pairs overlap heavily and the condensate is a single coherent quantum state on a
macroscopic scale.

## The Josephson effects

That macroscopic coherence is visible when two superconductors are separated by a
thin insulating barrier — a **Josephson junction**. Cooper pairs can
[tunnel](/quantum-mechanics/wave-mechanics-1d/barrier-penetration-and-quantum-tunneling)
across it with no resistance. Josephson predicted in 1962 that with no applied
voltage a supercurrent flows,

$$
I = I_{\max}\sin(\phi_2 - \phi_1),
$$

set by the phase difference of the pair wave functions on the two sides (the **dc
Josephson effect**). With a dc voltage $V$ across the junction, the current
instead alternates at

$$
f = \frac{2eV}{h}
$$

(the **ac Josephson effect**), again showing the pair charge $2e$.

$$
% caption: A Josephson junction: two superconductors separated by a thin
% insulating barrier through which Cooper pairs tunnel, carrying a supercurrent
% set by the phase difference of the two condensates.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[acc, thick, fill=acc!10] (0,0) rectangle (2.4,1.4);
  \node[acc, align=center] at (1.2,0.7) {superconductor 1\\phase $p_1$};
  \draw[black, thick, fill=black!6] (2.4,0) rectangle (3.0,1.4);
  \node[black, font=\scriptsize, rotate=90] at (2.7,0.7) {barrier};
  \draw[acc, thick, fill=acc!10] (3.0,0) rectangle (5.4,1.4);
  \node[acc, align=center] at (4.2,0.7) {superconductor 2\\phase $p_2$};
  % tunneling arrow
  \draw[acc, ->, thick] (2.2,1.75) .. controls (2.7,2.1) .. (3.2,1.75);
  \node[acc, anchor=south, font=\scriptsize] at (2.7,2.0) {pair tunneling};
\end{tikzpicture}
$$

Because frequency is measurable to extreme precision, the ac Josephson effect
fixes the ratio $e/h$ and sets voltage standards. The same coherence underlies
SQUID magnetometers sensitive to a single flux quantum. Superconductivity closes
this module: it is the point where the quantum statistics of
[Cooper-pair bosons](/statistical-mechanics/bose-systems/bose-einstein-condensation-and-the-fermion-gas), the
[tunneling](/quantum-mechanics/wave-mechanics-1d/barrier-penetration-and-quantum-tunneling)
of the Schrödinger equation, and the lattice physics of solids combine into a
single macroscopic quantum state.