Figure 93: Uniform tunnel barrier;
Random tunnel barriers, some with resistance above the resistance quantum.
make a transition to a Mott insulating anti-ferromagnet. And for a anti-ferromagnet the resistivity
goes up as the temperature goes down.
Another example for a Mott insulator is shown in fig. 94. It shows the conductivity of Si doped with
P, which is a donor in Si, at low temperature. If the doping is above∼ 4 · 1018 cm−^3 it’s acting like a
metal and it’s a so-called degenerate semiconductor but if the doping is below∼ 4 · 1018 cm−^3 it acts
like an insulator because the donors freeze out. This transition is also a Mott transition.
12.3 Electronic phase transitions
12.3.1 Mott Transition^12
The simple models that we use to describe metals or semiconductors neglect the electron-electron
interactions. And it often happens that materials that supposed to be a metal (based on the inde-
pendent electron models) are actually insulators. Mott showed that electron-electron interactions can
cause an insulating state where electrons can be better described as localized than extended over the
whole crystal. The Mott transition is a transition from a metallic to an insulating state that takes
place because of the electron-electron interaction. The electron screening is related to the Mott tran-
sition. If the screening length is very short, in some sense the electron electron interactions are not
that important. But if it is long, the electron-electron interactions become more important. This also
means that metals with a low electron density go more likely through a Mott transition than metals
with a high electron density.
A simple model to describe a Mott insulator is showed in fig. 95. The material is considered to
be consisting of coupled metal crystals connected by tunnel junctions. For strong coupling, if the
crystals are close together, the material is a metal. For weak enough coupling the material is an
insulator. By changing the tunnel coupling you can go from a metal to an insulator. The tunnel
coupling is characterized by a resistance. If crystals are arranged in a two dimensional square array
the critical resistanceRequals~/e^2 = 25. 5 kΩ. If the resistance of the tunnel coupling is below this
value, the material acts like a metal at low temperatures, if it is above, it acts like an insulator (see
fig. 96). If the coupling is strong and the material is cooled down the crystals come closer together
(^12) Nevill Francis Mott won the Nobel prize in physics 1977