2nd talk. The Mott transition

icmm.csic.es

2nd talk. The Mott transition

Mott physics

2nd Talk

E. Bascones

Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC)


Summary I

Independent electrons: Odd number of electrons/unit cell = metal

Interactions in many metals can be described following Fermi liquid

theory:

Description in k-space. Fermi surface and energy bands are

meaningful quantities. Rigid band shift

There are elementary excitations called quasiparticles with

charge e and spin ½

Quasiparticle have finite lifetime & renormalized energy

dispersion (heavier mass). Better defined close to Fermi level & low T

Quasiparticle weight Z , it also gives mass renormalization m*

Increasing correlations: smaller Z. m* (and Z) can be estimated

from ARPES bandwidth, resistivity, specific heat and susceptibility

~ 0 + A T 2

A ~ m* 2

C ~ T ~

~ m* ~ m*


Summary I-b

Interactions are more important in f and d electrons and decrease

with increasing principal number (U 3d > U 4d …) .

With interactions energy states depend on occupancy: non-rigid

band shift

In one orbital systems with one electron per atom (half-filling) onsite

interactions can induce a metal insulator transition : Mott

transition.

In Mott insulators : description in real space (opposed to k-space)

Mott insulators are associated with avoiding double occupancy not

with magnetism (Slater insulators)

Magnetism:

Weakly correlated metals: Fermi surface instability

Mott insulators: Magnetic exchange (t 2 /U). Spin models


Outline II: The Mott transition in single band systems

The Mott-Hubbard transtion. Hubbard bands. Mott and

charge transfer insulators

The correlated metallic state. Brinkman-Rice transition

The DMFT description of the Mott transition

Finite temperatures


The Mott transition. Paramagnetic state

Paramagnetic

Mott

Insulator

Metal-Insulator

transition with

decreasing pressure

Antiferromagnetism

Increasing Pressure: decreasing U/W

McWhan et al, PRB 7, 1920 (1973)


The Mott transition. Paramagnetic state

Atomic lattice single orbital per site and average occupancy 1: half filling

Hopping

saves energy t

Double occupancy

costs energy U

Small U/t

Metal

Increasing U/t

Mott transition

Large U/t

Insulator

W

Double

occupancy

Single

occupancy

U


Hubbard model. Kinetic and On-site interaction Energy

Hopping

saves energy t

Kinetic energy

Intra-orbital

repulsion

Double occupancy

costs energy U

Atomic lattice with a single orbital per site and average occupancy 1 (half filling)

E

Hopping restricted to first nearest neighbors: Electron-hole symmetry

Tight-binding (hopping)

Intra-orbital

repulsion


Mott insulator. Paramagnetic state: Hubbard bands

Double

electron

occupancy

Single

electron

occupancy

U


Mott insulator. Paramagnetic state: Hubbard bands

Double

electron

occupancy

U

Single

electron

occupancy

Remove an electron

(as in photoemission)

Empty

state


Mott insulator. Paramagnetic state: Hubbard bands

Double

electron

occupancy

U

Single

electron

occupancy

Remove an electron

(as in photoemission)

Empty

state

Empty state is free to move


Mott insulator. Paramagnetic state: Hubbard bands

t

t


Mott insulator. Paramagnetic state: Hubbard bands

t

t

Double

occupancy

U

U

Single

occupancy

W

Lower Hubbard band


Mott insulator. Paramagnetic state: Hubbard bands

t

t

Upper Hubbard Band

W

U

Lower Hubbard Band

W


Mott insulator. Paramagnetic state: Hubbard bands

Doubly occupied states

Upper Hubbard Band

W

U

Lower Hubbard Band

W

Non-degenerate bands

Singly occupied states


The Mott-Hubbard transition. Paramagnetic state

Increasing U

U=0

W

W

Double

degenerate

band (spin)

Gap

U- W

W

U

Non-degenerate

bands


The Mott-Hubbard transition. Paramagnetic state

Increasing U

U=0

W

Double

degenerate

band (spin)

W

W

Mott transition

Uc= W

Gap

U- W

W

W

U

Non-degenerate

bands

Gap opens at the Fermi level at Uc


Mott vs charge transfer insulators

3d oxides

U=0

4s band

3d narrow band

2p oxygen band


Mott vs charge transfer insulators

3d oxides

U=0

4s band

W

U

3d narrow band

W

2p oxygen band


Mott vs charge transfer insulators

3d oxides

U=0

Mott insulator

4s band

Lowest excitation

energy d-type (Mott)

3d narrow band

Charge transfer

insulator

2p oxygen band

Lowest excitation

energy p-type


Mott vs charge transfer insulators

Cuprates are

charge transfer insulators


The Brinkman-Rice transition from the metallic state.

The uncorrelated metallic state: The Fermi sea |FS>

W

Spin degenerate

Energy states are filled

according to their kinetic energy.

States are well defined in k-space


The Brinkman-Rice transition from the metallic state.

The uncorrelated metallic state: The Fermi sea |FS>

W

Spin degenerate

Energy states are filled

according to their kinetic energy.

States are well defined in k-space

Probability in real space: ¼ per the 4 possible states (half filling)

Cost in interaction energy per particle

=U/4

Kinetic energy gain per particle

(constant DOS)

=W/4=D/2


The Brinkman-Rice transition from the metallic state.

The uncorrelated metallic state: The Fermi sea 1FS>

E=K+U


=U/4



=D/2


The Brinkman-Rice transition from the metallic state.

The correlated metallic state: Gutzwiller wave function

| >= j [ 1-(1- )n j n j ]1FS>

Variational Parameter

=1 U=0

Uncorrelated

=0 U=

uniformly diminishes

the concentration of

doubly occupied sites

Correlated

Gutzwiller Approximation. Constant DOS


The Brinkman-Rice transition from the metallic state.

The correlated metallic state: Gutzwiller wave function

Uncorrelated

Correlated


The Brinkman-Rice transition from the metallic state.

The correlated metallic state: Gutzwiller wave function

uncorrelated

correlated

Average potential energy

reduced due to reduced

double occupancy

correlated

uncorrelated

Kinetic Energy

is reduced


The Brinkman-Rice transition

Correlated metallic state

W

Heavy quasiparticle

(reduced Kinetic Energy)

Quasiparticle disappears

U


The Brinkman-Rice transition

Correlated metallic state. Fermi liquid like aproach

W

Heavy quasiparticle

(reduced Kinetic Energy)

Quasiparticle disappears

Reduced

quasiparticle residue

Quasiparticle disappears

at the Mott transition


Mott-Hubbard vs Brinkman-Rice transition

The Mott-Hubbard transition (insulator) Uc=W

W

W

W

Gap

U- W

U

The Brinkman-Rice transition (metallic) Uc=2W

W

Heavy quasiparticle

(reduced K.E.)

Reduced quasiparticle residue

F* ~Z F

Quasiparticle disappears


The Brinkman-Rice transition from the metallic state.

The correlated metallic state: Gutzwiller wave function

uncorrelated

correlated

Transition happens when

double occupancy

dissapears

correlated

uncorrelated

Energy of

independent

localized electrons


Large U limit. The Insulator. Magnetic exchange

Antiferromagnetic interactions

between the localized spins

(not always ordering)

Effective exchange interactions

J ~t 2 /U

Antiferromagnetic correlations/ordering can reduce the energy

of the localized spins

Double occupancy is not zero


Transition between correlated metal and insulator

The correlated metallic state: Gutzwiller wave function

Uncorrelated

insulator

Uncorrelated

Metal

Correlated

Insulator

t 2 /U

Correlated

Metal

Transition happens

with non vanishing

double occupancy


Mott-Hubbard vs Brinkman-Rice transition

The Mott-Hubbard transition (insulator)

W

W

W

Gap

U- W

U

The Brinkman-Rice transition (metallic)

W

Heavy quasiparticle

(reduced K.E.)

Reduced quasiparticle residue

F* ~Z F

Quasiparticle disappears


Mott-Hubbard + Brinkman-Rice transition

Heavy quasiparticle which disappears when

F* vanishes at U c2 > U c1

U

F* ~Z F

Gap U- W

between the

Hubbard bands

opens at

U c1 =W=2D

- Density of States: Quasiparticle and Hubbard

Bands three peak structure.

- Two energy scales: F* and the gap between

the Hubbard bands


Mott transition. Paramagnetic state. DMFT picture

Infinite dimensions

Georges et al , RMP 68, 13 (1996)

U/D=1

Three peak structure

U/D=2

U/D=2.5

F*

Heavy quasiparticles

(coherent)

U/D=3

U/D=4

Hubbard bands

(incoherent)

Two energy scales: F* and the gap between the Hubbard bands

F* Fermi liquid,

F* Non-Fermi liquid


Mott transition. Paramagnetic state. DMFT picture

Infinite dimensions

U/D=1

U/D=2

Transfer of spectral weight

from the quasiparticle peak

to the Hubbard bands

U/D=2.5

U/D=3

U/D=4

The gap between the

Hubbard bands

opens in the metallic state

Quasiparticles disappear at the Mott transition

Georges et al , RMP 68, 13 (1996)


The Mott transition.

Quasiparticle weight vanishes

at the Mott transition

Best order parameter for the

transition

Georges et al , RMP 68, 13 (1996)

Quasiparticle weight : Step at Fermi surface

Luttinger theorem

(original version):

At the Mott transition

the Fermi surface disappears

Fermi surface volume

proportional

to carrier density

Localization

In real space

Delocalization

in momentum space


The Mott transition. Paramagnetic state. DMFT picture

Quasiparticle weight vanishes at the Mott transition

but double occupancy does not

Georges et al , RMP 68, 13 (1996)

DMFT numerical results can depend on the a

approximation used to solve the impurity problem


The Mott transition. Paramagnetic state.

Analogy between Mott transition & liquid-gas transition

Metal: liquid

First order phase transition

(some exception could exist)

Insulator: gas

(larger entropy)

The particles in the gas

are the doubly occupied

sites. Density is smaller

in the insulator (gas)


The Mott transition. Finite temperatures. DMFT

Georges et al , RMP 68, 13 (1996)

In the region between the dotted lines both

a metallic and an insulator solution exist

A gap between

Hubbard bands

opens at U c1

Mott transition

The quasiparticle peak

disappears at U c2

At zero temperature the Mott transition happens at U c2

when the quasiparticle peak disappears


The Mott transition. Finite temperatures

The system becomes

insulating with

increasing temperature

First order transition

Georges et al , RMP 68, 13 (1996)


The Mott transition. Finite temperatures

The system becomes

insulating with

increasing temperature

First order transition

Georges et al , RMP 68, 13 (1996) McWhan et al, PRB 7, 1920 (1973)


The Mott transition. Finite temperatures

Also in liquid gas transition

Critical point:

No distinction of

what it is a metal

and what an insulator

at higher temperatures


The Mott transition. Finite temperatures

Critical point

Histeresis

First order

Limelette et al, Science 302, 89 (2003)


The Mott transition. Finite temperatures

U/D=2.5

T=0.05 D

T=0.03 D

T=0.08 D

T=0.10 D

The quasiparticle weight Z decreases with increasing temperature


The Mott transition. Finite temperatures

U/D=2.4

Resistivity decreases

with temperature

(insulator)

Resistivity increases

with temperature

(metal)

Change from metallic to insulating

like behavior at a given temperature

Georges et al, J. de Physique IV 114, 165 (2004), arXiv:0311520


The Mott transition. Finite temperatures

Not so clear distinction between a metal and an insulator at finite temperatures

Georges et al, J. de Physique IV 114, 165 (2004), arXiv:0311520


The Mott transition. Finite temperatures

2.85

3

2.65 2.45

2.25

2

The slope of the linear T

dependence increases

with interactions

3.1

Fermi liquid: Specific heat

linear with temperature

C ~ T ~ m*

Mass enhanced

with interactions

DMFT Georges et al , RMP 68, 13 (1996)


The Mott transition. Finite temperatures

2.85

3

2.65

2.45

2.25

2

U/D=1

The slope of the linear T

dependence increases

with interactions

3.1

Fermi liquid: Specific heat

linear with temperature

C ~ T ~ m*

Linearity is lost at a temperature which

decreases with increasing interactions

DMFT

Mass enhanced

with interactions


The Mott transition. Finite temperatures

T-linear dependence

at low temperatures

(Metallic)

U/D=4

Change to insulating

Like behavior at high

temperatures

U/D=2

Activated behavior at low temperatures

(Insulating)

DMFT Georges et al , RMP 68, 13 (1996)


Summary II: The Mott transition.

Half-filling. Zero T . Paramagnetic state

At half filling and zero temperature. Hubbard model (only on-site

interactions) Mott transtion: Metal-insulator transition at a given U/W

Mott-Hubbard approach: Insulator as starting point. A hole or a

doubly occupied state is able to move. Non-degenerate lower and

upper Hubbard bands (width W). Gap U-W. Transition Uc=W

U=0 Degenerate

W

W

W

Gap

U- W

U

Non-degenerate

Charge transfer insulators: Lowest excitation with different orbital

character than the one which opens the gap


Summary II-b: The Mott transition.

Half-filling. Zero T . Paramagnetic state

Brinkmann-Rice approach: Metal as starting point. The correlated

metal avoids double occupancy (Gutzwiller). Quasiparticles with

larger mass, renormalized Fermi energy, reduced quasiparticle weight

Z. Transition U ~2 W when Z=0

W

Heavy quasiparticle

(reduced K.E.)

Reduced quasiparticle residue

F* ~Z F

Quasiparticle disappears

Z as an order parameter for the transition


Summary II-c: The Mott transition.

Half-filling. Zero T . Paramagnetic state

U/D=1

U/D=2

DMFT:

3-peak spectral function Hubbard

bands+ quasiparticle peak

U/D=2.5

U/D=3

U/D=4

2 energy scales: * F

Gap: U-W

Z dies at the transition, Gap

opens at smaller U

Similarity with liquid-gas

transition: number of particles in

the gas is the number of doubly

occupied states


Summary II-d: The Mott transition. Finite temperatures

First order transition & critical point

U/D=4

U/D=2

For intermediate U/t

The metallic character decreases with temperature and eventually can become

insulator. Change from Fermi liquid behavior at low temperature to insulating

behavior at higher temperatures

T=0.03 D

Incoherence increases with increasing

temperature & quasiparticles can

disappear

T=0.08 D

T=0.05 D

T=0.10 D

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