## Mott Transition

A Mott transition is a metal-insulator transition that occurs when the electron density of a material is changed. Usually a sequence of samples are prepared each with slightly different chemical compositions so that they have different electron densities. The samples with a large electron density are metallic and those with a low electron density are insulating. Sometimes it is possible to induce a Mott transition by applying a pressure or changing the temperature. This will only work if the material is close to a critical electron density.

The valence electrons of an isolated metal atom are bound to the atom. If other metal atoms are brought nearby, the electron clouds will overlap and electron screening will modify the potential that the valence electrons see. To understand the consequence of this screening, consider a particle confined to a finite potential well. The Schrödinger equation for such a system has only a finite number of bound states represented by the blue lines in the diagram. The energy of the bound states increase when the potential is made narrower and at some point the bound states pop out of the potential well and become unbound. Adjust the width of the potential well to see the energy levels pop out.

 $E$ eV $x$ Å

$V_0=$ 5 [eV]

$L=$ 10 [Å]

A similar effect occurs as metal atoms are brought close to each other. Due to electron screening, the potential that an electron sees gets narrower and at some point the valence electron states become unbound. For low electron densities the screening effect is weak, all electrons are bound to the atoms and the material is an insulator. For high electron densities the screening is much stronger. In that case, the valence electrons are unbound and the material is a metal.

Fig. 1 Hydrogen atoms with a large lattice constant (left, insulator) and a small lattice constant (right, metal)

Mott origially considered a hypothetical crystal of hydrogen atoms where the distance between the atoms could be adjusted. When the hydrogen atoms are far apart from each other, the electrons are bound to their respective protons. In this case, the application of an electric field will cause the atoms polarize, but there will be no electrical conduction through the crystal. The crystal is an electrical insulator.

When the hydrogen atoms move closer together, the wave functions of the electrons start to overlap and the electrons from one atom start to screen the Coulomb potential of the neighboring atom. Therefore the potential well gets narrower, and at a critical electron density, the electrons become unbound and the insulating crystal turns into a metal. The critical electron density is for hydrogen was calculated by the Mott to be,

$$n^{1/3}a_0 = 0.19,$$

where $a_0$ is the Bohr radius. Mott argued that the transition from insulator to metal is quite abrupt as the valence electrons become unbound.

### Mott transition of a semiconductor

Although it is not experimentally possible to move hydrogen atoms closer together to observe a Mott transition, a similar experiment can be performed with doped semiconductors. A semiconductor like silicon is an insulator at low temperatures. If silicon is doped with phosphorus donor atoms, one valence electron will be weakly bound to each donor atom at low temperature. If the concentration of the donor atoms is increased, at one point their wavefunctions start to overlap and the electrons become unbound. Heavily doped silicon is a metal even at low temperatures.

### References

State of references: 16.03.2021