Metal -Insulator Transitions

Electron - Electron Interactions

Electron-electron interactions

Including electron-electron interactions into the description of solids is very, very difficult.

One of the simplest approximation is to say that the electron-electron interactions screen the nuclei-electron interactions.

2 22 2 22 2

, 0 0 02 2 4 4 4A A B

i Ai A i A i j A Be A iA ij AB

Z e Z Z eeHm m r r r

Screening = Abschirmung

Electron screening (Abschirmung)

2

0

( )e rV

If a charge is put in a metal, the other charges will move

The Helmholtz equation in 3-d

0

exp4

se k r rV

r r

0

( )e rE

E V

2

0 0

( ) inde rV

2 2

0

( ) .se rV k V

If ind is proportional to -V,

Poisson equation04

eVr r

2

0

indsk V

Thomas-Fermi screening

Thomas - Fermi screening length2 1/ 3 2 1/ 3

22 4 / 3

0 0

3 32s

F

e n me nkE

exp4

se k r rV

r r

Fn D E eV E

x

EF

32F

F

nD EE

2ind Fe n e D E V

22

0 0 0 0

( ) ( ) 32

ind

F

e r e r e nV VE

22

0 0

3 ( )2 F

e n e rV VE

Electron screening

Thomas - Fermi screening length

2 1/ 3 2 1/ 32

2 4 / 30 0

3 32s

F

e n me nkE

exp4

sk r rV

r r

Screening length depends on the electron density

Kittel

2 1/ 3sk n

Around a point defect

Friedel oscillations

Only wave vectors k < kF can contribute to the screening

sin( )Fk rr

Friedel oscillations or Rudermann-Kittel oscillations

Copper Surface

http://www.almaden.ibm.com/vis/stm/atomo.html

Metal-insulator transition

Atoms far apart: insulator

Atoms close together: metal

Mott transition

The number of bound states in a finite potential well depends on the width of the well. There is a critical width below which the valence electrons are no longer bound.

Mott transition

For low electron densities the screening is weak. The electrons are bound and the material is an insulator.

For high electron densities the screening is strong, the valence electrons are not bound and the material is a metal. The 1s state of a screened Coulomb potential becomes unbound at ks = 1.19a0.

Kittel

There are bound state solutions to the screened potential for ks < 1.19/a0

Mott transition (low electron density)

Bohr radius

High-temperature oxide superconductors / antiferromagnets

There are bound state solutions to the unscreened potential (hydrogen atom)

1/ 32

0

4 3s

nka

Nevill Francis Mott Nobel prize 1977

Mott argued that the transition should be sharp.

degenerate semiconductor

Kittel

P in Si

Semiconductor conductivity at low temperature

The kinetic energy can be minimized by allowing the electron states to spread out over the whole system giving them the lowest values of k and p. This leads to a higher potential energy.

Wigner crystal

At low electron densities, electrons moving in a uniform positive background should form a crystal.

Eugene Wigner

The potential energy is lowest if the electrons are at fixed positions as far apart as possible.

For low electron densities, the total energy is lowest for a crystal of electrons.

Metal-insulator transition

Clusters far apart: insulator

Clusters close together: metal

Quantum dots

Consider a solid formed by weakly coupling quantum dots together

The molecules are weakly coupled in the crystals and act like quantum dots.

Charging effects

The motion of electrons through a single quantum dot is correlated.

quantum dot

e e

After screening, the next most simple approach to describing electron-electron interactions are charging effects.

2

mV

nA

0.50

1

Single electron transistor