1 quick and dirty introduction to mott insulators introduction to solid state physics...

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Quick and Dirty Introduction to Mott Insulators

Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html

PHYS 624: Quick and dirty introduction to Mott Insulators 2

Weakly correlated electron liquid: Coulomb interaction effects

( ) ( ) ( )Fn eD U r r

assume: ( )

( , 0) ( )F

F

e U

f T

r

When local perturbation potential is switched on, some electrons will leave this region in order to ensure constant (chemical potential is a thermodynamic potential; therefore, in equilibrium it must be homogeneous throughout the crystal).

( )U r

F


Thomas –Fermi screening

•Except in the immediate vicinity of the perturbation charge, assume that is caused by the induced space charge → Poisson equation:

2

0

( )( )

e nU

r

r

/2 2

2

02

0

1( )

( )

in vacuum: ( ) 0, ( )4

TFr r

TFF

F

er U

r r r r

re D

qD U

r

r

2 1/3

2 /3 2 /3 1/32 2 2 22 2

0

3 1 2 4( ) 3 , 3 3

2 2 2F F TFF

n m nD n n r

m a

1/ 6 20

03 20

23 3

41,

2

8.5 10 , 0.55Å

TF

CuCu TF

nr a

a me

n cm r

( )U r


Mott Metal-Insulator transition

•Below critical electron concentration, the potential well of the screened field extends far enough for a bound state to be formed → screening length increases so that free electrons become localized → Mott Insulators (e.g., transition metal oxides, glasses, amorphous semiconductors)!

2 2001/3

1/30

1

4

4

TF

ar a

n

n a


Metal vs. Insulator

T

T

Fundamental requirements for electron transport in Fermi systems:

• quantum-mechanical states for electron-hole excitations must be available at energies immediately above the ground state since the external field provides vanishingly small energy

• these excitations must describe delocalized charges that can contribute to transport over the macroscopic sample sizes.


Metal-Insulator Transitions

From weakly correlated Fermi liquid to strongly correlated Mott insulators

nc2nc

n

STRONG CORRELATION WEAK CORRELATION

INSULATOR STRANGE METAL F. L. METAL

Mott Insulator: A solid in which strong repulsion between the particles impedes their flow → simplest cartoon is a system with a classical ground state in which there is one particle on each site of a crystalline lattice and such a large repulsion between two particles on the same site that fluctuations involving the motion of a particle from one site to the next are suppressed.


Energy band theory

Electron in a periodic potential (crystal) energy band ( : 1-D tight-binding band)

N = 1 N = 2 N = 4 N = 8 N = 16 N =

EF

kinetic energy gain

( ) 2 cos( )k t ka


Band (Bloch-Wilson) insulator

Wilson’s rule 1931: partially filled energy band metal otherwise insulator

metal insulatorsemimetal

Counter example: transition-metal oxides, halides, chalcogenides Fe: metal with 3d6(4sp)2

FeO: insulator with 3d6


Mott gedanken experiment (1949)

energy cost U

electron transfer integral tt

Competition between W(=2zt) and U Metal-Insulator Transition

e.g.: V2O3, Ni(S,Se)2

d atomic distance

d (atomic limit: no kinetic energy gain): insulatord 0 : possible metal as seen in alkali metals


Mott vs. Bloch-Wilson insulators

•Band insulator, including familiar semiconductors, is state produced by a subtle quantum interference effects which arise from the fact that electrons are fermions.

•Nevertheless one generally accounts band insulators to be “simple” because the band theory of solids successfully accounts for their properties

•Generally speaking, states with charge gaps (including both Mott and Bloch-Wilson insulators) occur in crystalline systems at isolated “occupation numbers” where is the number of particles per unit cell.

•Although the physical origin of a Mott insulator is understandable to any child, other properties, especially the response to doping are only partially understood.

•Mott state, in addition to being insulating, can be characterized by: presence or absence of spontaneously broken symmetry (e.g., spin antigerromagnetism); gapped or gapless low energy neutral particle excitations; presence or absence of topological order and charge fractionalization.

* *

*


Theoretical modeling: Hubbard Hamiltonian

Hubbard Hamiltonian 1960s: on-site Coulomb interaction is most dominant

Hubbard’s solution by the Green’s function decoupling method insulator for all finite U value

Lieb and Wu’s exact solution for the ground state of the 1-D Hubbard model (PRL 68) insulator for all finite U value

e.g.: U ~ 5 eV, W ~ 3 eV for most 3d transition-metal oxide such as MnO, FeO, CoO, NiO : Mott insulator

band structure correlation


Trend in the Periodic Table

U

U


Solving Hubbard model in dimensions •In -D, spatial fluctuation can be neglected. → mean-field solution becomes exact.•Hubbard model → single-impurity Anderson model in a mean-field bath.•Solve exactly in the time domain → “dynamical” mean-field theory


From non-Fermi liquid metal to Mott insulator

Model: Mobile spin-up electrons interact with frozen spin-down electrons.

NOTE: DOS defined even though there are no fermionic quasiparticles.


Experiment: Photoemission Spectroscopy

h (K,) > We- (Ek,k,)

N-particle (N1)-particle

P(| i | f )

Sudden approximation

Einstein’s photoelectric effect

Photoemission current is given by:

EiN

EfN 1

fi

Ni

Nfr

TkE EEiTfeZ

A BNi

,

12/ )(||1

)(


Mott Insulating Material: V2O3

a = 4.95 Å

c = 14.0 Å

–(1012) cleavage plane

Vanadium

Oxygen

surface-layer thickness =

side view

2.44Å

top view


Phase diagram of V2O3


Bosonic Mott insulator in optical lattices

•Superfluid state with coherence, Mott Insulator without coherence, and superfluid state after restoring the coherence.

1 quick and dirty introduction to mott insulators introduction to solid state physics...

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