Superconductivity, BCS theory and Mott-Hubbard localisation

A superconductor has no resistance and pushes magnetic fields out

Intuition

Normal wires waste energy as heat because electrons keep bumping into things. A superconductor is a wire where nothing gets in the way, below a certain cold temperature. It also acts like a force field against magnets.

In 1908 Kamerlingh Onnes liquefied helium (boiling point −269 °C, 4.2 K) by cooling to 1.5 K. In 1911 the resistance of a solid Hg wire in liquid He fell to zero at 4.2 K. He called this superconductivity. Lead and tin followed in 1913, at 6 K and 4 K, and he won the 1913 Nobel Prize in Physics. The Meissner effect was observed in January 1914.

Below the critical temperature $T_c$ two things hold. The resistivity is zero. The material completely expels magnetic fields, which is perfect diamagnetism. The Meissner effect is this expulsion on cooling through $T_c$.

The slide on “room-temperature superconductivity” gives the first superconductor (1911) as $T_c = 4.2$ K and a record for “high-temperature” superconductivity of 151 K (−122 °C), reported on 9 March 2026, beating a record that stood for more than 30 years. It adds that claims of room-temperature superconductivity have not survived testing, citing the failed LK-99 replication.

BCS theory ties Tc to the density of states at the Fermi level

Quantum mechanics explains superconductivity as electrons forming pairs, called Cooper pairs, at low temperature. The pairs flow through the lattice without scattering, so there is zero resistance. BCS proposed this in 1957 and won the 1972 Nobel Prize in Physics.

Plain words first. $T_c$ is a competition. The pairing attraction between electrons pushes $T_c$ up. Having lots of electron seats right at the Fermi level also pushes it up. Then the exponential turns that into a temperature.

$k_B T_c = \hbar\omega\,\exp\!\left(\frac{-1}{V\,N(E_f)}\right)$

$N(E_f)$ is the density of states at the Fermi energy. $V$ is the strength of the attractive interaction between electrons. $\hbar\omega/k_B$ is the Debye temperature, which represents the collective or acoustic vibration of the lattice.

Exam detail

Know the symbols of the BCS equation and the direction of each effect on $T_c$. The lecture states the formula without a worked number, so do not expect to be given a quantity from the slides.

Chemists use this to design materials. $N(E_f)$ is related to the width of the band in energy. Band width is set by orbital overlap: more overlap gives more orbital splitting and a wider band. A wide band spreads the same electrons over many energies, so $N(E_f)$ is small and $T_c$ is low. A narrow band packs them in, so $N(E_f)$ is large and $T_c$ is high.

Fullerene intercalates show the trade between overlap and Tc

Small cations let the C₆₀ molecules sit close together, giving a small cell edge $a$, strong overlap, a large band width, a small $N(E_f)$ and a lower $T_c$. Large cations push them apart, giving weak overlap, a small band width, a large $N(E_f)$ and a higher $T_c$.

Compare

Small cation

C60 close, small $a$. Strong overlap. Wide band. Small $N(E_f)$. Lower $T_c$.

Large cation

C60 far apart, large $a$. Weak overlap. Narrow band. Large $N(E_f)$. Higher $T_c$.

Pull atoms far enough apart and the band picture fails

Take sodium. At normal spacing it is a metal with a band. As the Na-Na distance grows, overlap shrinks and the band model breaks down. Electrons stay on individual atoms or molecules. This is Mott-Hubbard localisation, and the result is a Mott-Hubbard insulator. The slide plots the calculated band structure of Na against Na-Na distance, going from crystal orbitals (band conductor) to atomic orbitals.

Lecture 8 closes the loop with ammonia. Intercalating NH₃ into A₃C₆₀ separates the C₆₀ molecules further. The result is increased $T_c$ or Mott-Hubbard localisation, in the compound $(\text{NH}_3)_x\text{A}_3\text{C}_{60}$. The slide does not say which of the two happens for which composition.

Pitfall

More separation raises $T_c$ only up to a point. Past that point the lecture says the material localises and becomes a Mott-Hubbard insulator instead.

Recall

Why do large cations in A3C60 tend to give a higher Tc?

They push the C60 molecules apart. Less orbital overlap means a narrower band, so there are more states at the Fermi level (large N(E_f)), and the BCS formula gives a higher Tc.