Physics · Atomic and quantum physics · Year 3

Superconductivity

Why does a disc cooled in liquid nitrogen float above a magnet?

Open the experiment

The situation

In 1911, in a laboratory in Leiden, Heike Kamerlingh Onnes's team measured the electrical resistance of mercury cooled in liquid helium and saw the resistance vanish suddenly at about 4.2 K. Today at exhibitions a black ceramic disc cooled in liquid nitrogen floats in the air above a magnet, and the MRI scanners in hospitals create a strong field with a persistent current in a superconducting coil. How can the resistance be exactly zero, and why does the cold disc shun the magnet and “freeze” above it?

Topic

Superconductivity. A disc cooled in liquid nitrogen levitates above a magnet (Meissner effect)

Key ideas

• Meissner effect
• Cooper pairs
• Critical temperature

Formulas

Critical field: in a stronger field superconductivity is destroyed.

Rutgers formula: the jump in heat capacity at $T_c$ ($V$ is the molar volume).

Solution of the London equation: the field decays at a depth ${\lambda }_L$ from the surface.

BCS theory: the energy needed to break a Cooper pair.

Josephson effects and the magnetic flux quantum.

Theory

Resistance vanishes: the main experimental facts

Onnes saw the resistance of mercury drop abruptly to zero at $T_c\approx 4.15$ K. Lower the temperature in the cryostat and repeat this on the $R(T)$ graph: above that point the resistance falls because lattice vibrations (phonons) decrease, and at $T_c$ it vanishes abruptly. Copper, by contrast, has never become superconducting at the lowest temperatures reached — a residual resistance remains in it. A current set up in a superconducting ring does not decay ($\tau =L/R\to \infty$). Superconductivity is destroyed if the critical temperature, the critical field $B_c$ or the critical current is exceeded. In an ordinary metal the same electrons carry both heat and current: the Wiedemann–Franz law $\kappa /(\sigma T)=L_0\approx 2.44\cdot {10}^{-8}$ W·Ω/K² (details — in the “Electric current in metals” experiment). In a superconductor $\sigma \to \infty$ but the thermal conductivity does not increase: the pairs carry no entropy — the law breaks down.

Thermodynamics of the phase transition: the Rutgers formula

The free energy of the superconducting state is lower than that of the normal state by $B_c^2/2{\mu }_0$ per unit volume (the condensation energy). The transition without a field is a second-order phase transition: there is no latent heat, but the heat capacity jumps. Thermodynamics links this jump to the slope of the $B_c(T)$ curve at $T_c$ — the Rutgers formula; for lead it gives ≈ 52 mJ/(mol·K). The two-fluid model on the $C(T)$ graph reproduces exactly this jump; BCS theory predicts $\mathrm{\Delta }C=1.43\gamma T_c$. At the boundary between the normal and superconducting regions there is a surface energy, roughly $(\xi -\lambda )B_c^2/2{\mu }_0$ per unit area. For $\xi >\lambda$ it is positive — boundaries are “expensive”, this is type I; for $\kappa =\lambda /\xi >1/\sqrt{2}$ it is negative — the field enters as thin vortices, this is type II.

The Meissner effect and the London equations

In 1933 Meissner and Ochsenfeld found that a superconductor pushes the field out of its interior: it is not simply an ideal conductor with $R=0$ — a type I disc cooled in a field also expels the field and rises. The London brothers (1935) proposed the equation $\nabla \times {𝐣}_s=-(n_s{e}^2/m)𝐁$: the field decays from the surface inwards as $B_0{e}^{-x/{\lambda }_L}$ (${\lambda }_L\approx 53$ nm for $n_s={10}^{28}$ m⁻³). The persistent currents at the surface repel the magnet. The force is calculated by the mirror dipole method: behind the superconductor there is taken to be an oppositely directed “image” magnet, $F=3{\mu }_0{m}^2/(32\pi h^4)$. In type II superconductors (YBCO, Nb) a field stronger than $B_{c1}$ enters as Abrikosov vortices, each carrying one ${\mathrm{\Phi }}_0$; vortices “pinned” on defects hold the disc in place — which is why a disc cooled in a field even hangs below the magnet.

Cooper pairs and BCS theory

According to the theory of Bardeen, Cooper and Schrieffer (1957), a moving electron slightly attracts the positive ions; the positive “trail” left behind (a phonon) attracts a second electron. Electrons with opposite momenta and spins form a Cooper pair. All the pairs are in a single quantum state: to scatter one of them takes at least energy $2\mathrm{\Delta }$, and at low temperature there is no such energy — the resistance is zero. BCS: $2\mathrm{\Delta }(0)=3.53k_B{T}_c$, and $\mathrm{\Delta }(T)$ falls to zero at $T_c$ (an approximate formula on the graph). The energy gap is seen in tunnel spectroscopy: the $dI/dV$ graph of a normal metal–insulator–superconductor junction has a peak at $eV=\pm \mathrm{\Delta }$. The isotope effect ($T_c\propto M^{-1/2}$, 1950) proved the role of the lattice. The size of a pair — the coherence length ${\xi }_0\approx \hslash v_F/\pi \mathrm{\Delta }$ — is from tens of nanometres to a micrometre: inside one pair there are millions of other pairs, so the animation is schematic.

Josephson effects and the SQUID

Pairs tunnel through a very thin insulator between two superconductors (Josephson, 1962). Stationary (DC) effect: current flows even without voltage, $I=I_c\sin \phi$, where $\phi$ is the difference between the phases of the wave functions on the two sides. Non-stationary (AC) effect: at a voltage $V$ the phase grows continuously and the current oscillates at the frequency $f=2eV/h$ — 483.6 MHz for every microvolt; the modern volt standard is based on this effect. A loop with two junctions is a SQUID: its critical current depends periodically on the flux through the loop, $I_c=2I_0|\cos (\pi \mathrm{\Phi }/{\mathrm{\Phi }}_0)|$, with a period equal to the flux quantum ${\mathrm{\Phi }}_0=h/2e$. The “$2e$” in it is direct proof that the current carriers are pairs. A SQUID senses even the very weak fields produced by the brain and heart. Model: the loop inductance is neglected, and the junctions are identical and strongly damped.

High-temperature superconductors and superfluidity

In 1986 Bednorz and Müller found superconductivity at about 35 K in a copper-oxide ceramic, and in 1987 YBa₂Cu₃O₇ (YBCO), with $T_c\approx 92$ K, crossed the boundary of liquid nitrogen (77 K). Such cuprates are type II; the pairing mechanism in them is still not fully explained, and their normal-state resistance depends almost linearly on temperature. MgB₂ (2001) — 39 K. Most MRI scanners, however, use a coil of NbTi wire and work in liquid helium (4.2 K). Liquid ⁴He becomes superfluid below 2.17 K (the λ point; Kapitsa; Allen and Misener, 1938): its viscosity vanishes, and its thermal conductivity is so great that bubbles have no time to form — boiling stops at once. A thin film creeps up the wall of the vessel and drips off outside. Simplifications of the model: cooling is sped up; mercury freezes at 234 K — $R(T)$ is for solid mercury (the Bloch–Grüneisen formula), and the sample resistance is taken as $R(300\text{K})=10\mathrm{\Omega }$.

Did you know?

• Heike Kamerlingh Onnes was the first to liquefy helium, in 1908, and discovered superconductivity in mercury in 1911; he received the Nobel Prize in Physics in 1913.
• YBa₂Cu₃O₇ (YBCO) was discovered in 1987: its $T_c\approx 92$ K makes it the first superconductor above the boiling point of liquid nitrogen (77 K). And liquid nitrogen is much cheaper than liquid helium.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

• Many-electron atoms — Year 1
• Bands in solids — Year 1
• Schrödinger and quantum tunnelling — Year 3
• Quantum dots and the potential well — Year 3
• The molecule: bonding and spectra — Year 3