Mottphysics 2talk

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 T2

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 (U3d > U4d …) .

With interactions energy states depend on occupancy: non-rigid

band shift

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

site 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 (t2/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

Increasing Pressure: decreasing U/W Antiferromagnetism

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

Large U/t

Insulator

Increasing U/t

Mott transition

W

Single

occupancy

Double

occupancy

U

Hubbard model. Kinetic and On-site interaction Energy

Tight-binding (hopping) Intra-orbital repulsion

Kinetic energy Intra-orbital repulsion

E

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

Hopping

saves energy t Double occupancy

costs energy U

Hopping restricted to first nearest neighbors: Electron-hole symmetry

Single

electron

occupancy

Double

electron

occupancy

U

Mott insulator. Paramagnetic state: Hubbard bands

U

Remove an electron

(as in photoemission)


Empty

state

Single

electron

occupancy

Double

electron

occupancy


U

Empty

state

Empty state is free to move

Remove an electron

(as in photoemission)

Single

electron

occupancy

Double

electron

occupancy

t

t


Single

occupancy

Double

occupancy

U

t

t

U

W

Lower Hubbard band


t

t

U

W

W Lower Hubbard Band

Upper Hubbard Band



U

W

W Lower Hubbard Band

Upper Hubbard Band

Singly occupied states

Doubly occupied states

Non-degenerate bands

The Mott-Hubbard transition. Paramagnetic state

U

W

W

W

Double

degenerate

band (spin)

Increasing U

U=0

Non-degenerate

bands

Gap

U- W

The Mott-Hubbard transition. Paramagnetic state

U

W

W

W

Double

degenerate

band (spin)

Increasing U

W

W

U=0

Non-degenerate

bands

Gap

U- W

Mott transition

Uc= W

Gap opens at the Fermi level at Uc

Mott vs charge transfer insulators

U=0

3d oxides

3d narrow band

2p oxygen band

4s band


U=0

3d oxides

3d narrow band

2p oxygen band

4s band

U

W

W


U=0

3d oxides

3d narrow band

2p oxygen band

4s band

Lowest excitation

energy p-type

Lowest excitation

energy d-type (Mott)

Mott insulator

Charge transfer

insulator


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 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

Cost in interaction energy per particle

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

Kinetic energy gain per particle

(constant DOS)

=U/4

<K>=W/4=D/2


The uncorrelated metallic state: The Fermi sea 1FS>

<U/D>

<K/D>

E=K+U

<E/D>

=U/4

<K>=D/2


The correlated metallic state: Gutzwiller wave function

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

Variational Parameter

=1 U=0

=0 U=

Gutzwiller Approximation. Constant DOS

uniformly diminishes

the concentration of

doubly occupied sites

Uncorrelated

Correlated



Correlated

Uncorrelated



<K>uncorrelated

<K>correlated

correlated

uncorrelated

Kinetic Energy

is reduced

Average potential energy

reduced due to reduced

double occupancy


The Brinkman-Rice transition

W

Heavy quasiparticle

(reduced Kinetic Energy) Quasiparticle disappears

Correlated metallic state

U

The Brinkman-Rice transition

W

Heavy quasiparticle

(reduced Kinetic Energy) Quasiparticle disappears

Correlated metallic state. Fermi liquid like aproach

Reduced

quasiparticle residue

Quasiparticle disappears

at the Mott transition

Mott-Hubbard vs Brinkman-Rice transition

U W W

W

Gap

U- W

The Mott-Hubbard transition (insulator) Uc=W

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

W

Heavy quasiparticle

(reduced K.E.)

Reduced quasiparticle residue


F* ~Z F


<K>uncorrelated

<K>correlated

correlated

uncorrelated Transition happens when

double occupancy

dissapears


Energy of

independent

localized electrons

Large U limit. The Insulator. Magnetic exchange

Antiferromagnetic interactions

between the localized spins

(not always ordering)

J ~t2/U

Effective exchange interactions

Antiferromagnetic correlations/ordering can reduce the energy

of the localized spins

Double occupancy is not zero


Correlated

Metal

Uncorrelated

Metal Correlated

Insulator

Uncorrelated

insulator

Transition between correlated metal and insulator

t2/U

Transition happens

with non vanishing

double occupancy

Mott-Hubbard vs Brinkman-Rice transition

U W W

W

Gap

U- W

The Mott-Hubbard transition (insulator)

The Brinkman-Rice transition (metallic)

W

Heavy quasiparticle

(reduced K.E.)



F* ~Z F

U

Gap U- W

between the

Hubbard bands

opens at

Uc1=W=2D

F* ~Z F

Heavy quasiparticle which disappears when

F* vanishes at Uc2 > Uc1

Mott-Hubbard + Brinkman-Rice transition

- Density of States: Quasiparticle and Hubbard

Bands three peak structure.

- Two energy scales: F* and the gap between

the Hubbard bands

Hubbard bands

(incoherent)

Heavy quasiparticles

(coherent)

Georges et al , RMP 68, 13 (1996) Infinite dimensions

U/D=1

U/D=2

U/D=2.5

U/D=3

U/D=4

Three peak structure

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

F*

Mott transition. Paramagnetic state. DMFT picture

F* Fermi liquid, F* Non-Fermi liquid

Mott transition. Paramagnetic state. DMFT picture

Georges et al , RMP 68, 13 (1996)

Infinite dimensions

U/D=1

U/D=2

U/D=2.5

U/D=3

U/D=4

Transfer of spectral weight

from the quasiparticle peak

to the Hubbard bands

Quasiparticles disappear at the Mott transition

The gap between the

Hubbard bands

opens in the metallic state

The Mott transition.

Quasiparticle weight vanishes

at the Mott transition

Best order parameter for the

transition


Quasiparticle weight : Step at Fermi surface

At the Mott transition

the Fermi surface disappears

Localization

In real space

Delocalization

in momentum space

Luttinger theorem

(original version):

Fermi surface volume

proportional

to carrier density

The Mott transition. Paramagnetic state. DMFT picture


DMFT numerical results can depend on the a

approximation used to solve the impurity problem

Quasiparticle weight vanishes at the Mott transition

but double occupancy does not

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

In the region between the dotted lines both

a metallic and an insulator solution exist

A gap between

Hubbard bands

opens at Uc1

The quasiparticle peak

disappears at Uc2


Mott transition

At zero temperature the Mott transition happens at Uc2

when the quasiparticle peak disappears

The Mott transition. Finite temperatures

First order transition

The system becomes

insulating with

increasing temperature



First order transition

The system becomes

insulating with

increasing temperature

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


Critical point:

No distinction of

what it is a metal

and what an insulator

at higher temperatures

Also in liquid gas transition


Histeresis

First order

Critical point

Limelette et al, Science 302, 89 (2003)


T=0.03 D

T=0.05 D

T=0.08 D T=0.10 D

The quasiparticle weight Z decreases with increasing temperature

U/D=2.5


U/D=2.4

Change from metallic to insulating

like behavior at a given temperature Resistivity increases

with temperature

(metal)

Resistivity decreases

with temperature

(insulator)

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

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 slope of the linear T

dependence increases

with interactions

C ~ T ~ m*

Fermi liquid: Specific heat

linear with temperature

Mass enhanced

with interactions

3.1

3

2.85

2.65 2.45

2.25

2

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


The slope of the linear T

dependence increases

with interactions

C ~ T ~ m*

Fermi liquid: Specific heat

linear with temperature

Mass enhanced

with interactions Linearity is lost at a temperature which

decreases with increasing interactions

U/D=1 2

2.25

2.45 2.65

2.85

3

3.1

DMFT


U/D=4

U/D=2

Activated behavior at low temperatures

(Insulating)

T-linear dependence

at low temperatures

(Metallic)

Change to insulating

Like behavior at high

temperatures

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

Charge transfer insulators: Lowest excitation with different orbital

character than the one which opens the gap

U W W

W

Gap

U- W

U=0 Degenerate

Non-degenerate

Summary II-b: The Mott transition.


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

Z as an order parameter for the transition

W

Heavy quasiparticle

(reduced K.E.)



F* ~Z F

Summary II-c: The Mott transition.


U/D=1

U/D=2

U/D=2.5

U/D=3

U/D=4

DMFT:

3-peak spectral function Hubbard

bands+ quasiparticle peak

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

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

Incoherence increases with increasing

temperature & quasiparticles can

disappear

T=0.03 D

T=0.05 D

T=0.08 D

T=0.10 D

For intermediate U/t

U/D=4

U/D=2