mottphysics 1talk

Mott physics

E. Bascones

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

Mott physics. Course Outline

Metals and Insulators. Basic concepts: Fermi liquids, Mott

insulators, Slater insulators, nature of magnetism

The Mott transition: Mott-Hubbard vs Brinkmann-Rice

transition, DMFT description. Charge-transfer vs Mott insulators.

Finite temperatures.

Doping a Mott insulator. The case of cuprates.

Single-orbital systems

Multi-orbital systems

Mott physics in Multi-orbital systems (at & away half filling)

- Degenerate bands. Effect of Hund’s coupling. Hund’s metals

- Non degenerate bands:Orbital selective Mott transition. Hund

- Spin-orbital Mott insulators (iridates)

Mott physics in iron superconductors

1st Talk: Basic concepts

Independent electron & Fermi liquid descriptions

Mott transition: Breakdown of independent electron picture.

Itinerant versus atomic description

Magnetic exchange. Slater versus Mott insulators

Bloch theory for Fermi gas:

A(k, ) ( - (k))

A(k, ):

Band states are eigenstates,

i.e. infinite lifetime

Electron spectral function

Probability that an electron has

momentum k and energy

Band energy

States filled up to the Fermi level

Fermi surface in metals

Metals and Insulators. Independent electrons

Metals and Insulators. Independent electrons

Metallicity

in clean systems

Bands crossing

the Fermi level

(finite DOS)

Fig: Calderón et al, PRB, 80, 094531 (2009)

Insulating behaviour

in clean systems

Bands below

Fermi level filled

Fig: Hess & Serene, PRB 59, 15167 (1999)

Metals and Insulators. Independent electrons

Spin degeneracy:

Each band can hold 2 electrons per unit cell

Even number

of electrons

per unit cell

Insulating

Metallic (in case

of band overlap)

Odd number

of electrons

per unit cell

Metallic

Weakly correlated metals: Fermi liquid description

Band theory based on kinetic energy of electrons in presence of a lattice

but electrons interact!

Why does an independent electron theory works at all?

Weakly correlated metals: Fermi liquid description

Band theory based on kinetic energy of electrons in presence of a lattice

but electrons interact

Why does an independent electron theory works at all?

Fermi liquid theory (effective theory to describe small energy excited states):

- Elementary excitations: quasiparticles with charge e and spin ½

-The quasiparticles are not electrons but there is a one-to-one correspondence

with an electron

Mattuck

Bloch theory for Fermi gas:

A(k, ) ( - (k))

A(k, ):

Band states are eigenstates,

i.e. infinite lifetime

Electron spectral function

Probability that an electron has

momentum k and energy

Band energy

States filled up to the Fermi level

Fermi surface in metals

Weakly correlated metals: Fermi liquid description

Fermi liquid:

There is a Fermi surface.

Close to the Fermi surface the

elementary excitations are

quasiparticles with renormalized

energy *(k) and finite lifetime 1/

Spectral function is broadened

and peaks at *(k)

Weakly correlated metals: Fermi liquid description

A(k, ): Electron spectral function

Probability that an electron has momentum k and energy

Fig: Damascelli, Hussain, Shen, RMP 75, 473 (2003)

Fermi liquid description

Fig: Lu et al, Nature 455, 81 (2008)

Angle Resolved Photoemission

Experiments (ARPES) would

show energy bands

Weakly correlated metals: Fermi liquid description

There is a Fermi surface. Quasiparticles with renormalized

energy *(k) and finite lifetime 1/

Spectral function is broadened

and peaks at *(k) A quasiparticle is well defined if

F

Zero T: quasiparticles at the Fermi Surface have infinite lifetime

~A ( - *F)2 + B T2

Temperature

In Fermi liquid (phase space arguments)

Close to the Fermi surface

quasiparticles are well defined

2

Weakly correlated metals: Fermi liquid description

Renormalized mass m*=m/Z

electrons become heavier

Renormalized band energy (k)

Z: quasiparticle weight 0 Z 1

smaller Z : larger effect of interactions

Z=0 there are no quasiparticles

Z also gives the quasiparticle

peak height in the spectral function

Fermi liquid description

Fig: Lu et al, Nature 455, 81 (2008)

Angle Resolved Photoemission

Experiments (ARPES) would

show energy bands but with a

renormalized bandwidth

How well defined it is the band and how much reduced is the bandwidth

give an idea of the value of Z.

If Z vanishes the band is not well defined. Smaller Z: narrower band

Fermi liquid behaviour

Metal (Fermi liquid)

Resistivity increases with temperature

~ 0 + A T2

A ~ m*2

Specific heat linear with temperature

C ~ T ~ m*

Magnetic susceptibility

does not depend on temperature

~ ~ m*

Experimental measurements

help to identify the strength

of interactions in metals

Not always easy to probe (phonons , …)

Metals and Insulators. Mott insulators

Fig: Pickett, RMP 61, 433 (1989)

Electron counting

La2CuO4: 2 La (57x2)+Cu (29) + 4 O (4x8)=175 electrons

Metallic behavior

expected

Breakdown of independent electron picture

Fig: Pickett, RMP 61, 433 (1989)

Metallic behavior

expected

Insulating behavior is found

Breakdown of independent electron picture

Mott insulators

Fig: Pickett, RMP 61, 433 (1989)

Metallic behavior

expected

Insulating behavior is found

Mott insulator:

Insulating behavior due to electron-electron interactions

Do not be confused with Anderson localization which is due to disorder

Kinetic energy. Delocalizing effect

Fig: Calderón et al, PRB, 80, 094531 (2009)

atomic site (ij) Atomic

orbital

spin

Adding

electrons

Filling bands

(rigid band shift)

Kinetic energy

Going from one atom to another

Delocalizing effect

Interaction energy

1 Atomic level.

Tight-binding (hopping) Intra-orbital repulsion

E

Consider 1 atom with a single orbital

Two electrons in the same

atom repel each other

1 electron (two possible states)

E =0

2 electron (the energy changes)

To add a second electron

to single filled orbital

costs energy U

Energy states depend

on the occupancy

(non-rigid band shift)

Kinetic and 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)

Kinetic and 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

Mott insulators

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

For U >> t electrons localize: Mott insulator

The Mott transition

Atomic lattice with a single orbital per site and average occupancy 1

half filling

Hopping

saves energy t

Double occupancy

costs energy U

For U >> t electrons localize: Mott insulator

Small U/t

Metal

Large U/t

Insulator

Increasing U/t

Mott transition

The Bandwidth

Increasing coordination number increases kinetic energy gain and bandwidth

1 dimension: hops to two neighbors

2 dimensions square lattice:

hops to four neighbors

2 dimensions triangular lattice:

hops to six neighbors

Bandwidth: (half bandwidth) D, bandwidth W

Parameter controlling Mott transition U/D or U/W

Itinerant vs localized electrons

Fig: Calderón et al, PRB, 80, 094531 (2009)

Metal: Electrons delocalized in real space,

localized in k-space.

Description in terms of electronic

bands

Mott Insulator: Electrons localized in real space,

delocalized in k-space.

Spin models. Description as localized

spins is meaningful

Itinerant vs localized electrons

Metal (Fermi liquid) Mott insulator

Resistivity increases with temperature Resistivity decreases with temperature

~ 0 + A T2

A ~ m*2

Specific heat linear with temperature

C ~ T ~ m*

Magnetic susceptibility

does not depend on temperature

~ ~ m*

Specific heat activated like behavior

Magnetic susceptibility inversely

proportional to temperature

~ + C’/(T+ )

Itinerant vs localized electrons

s & p

electrons

generally

delocalized

3d: competition between

kinetic energy & interaction

Interaction strength decreases

in 4d & overall in 5d

4f electrons are localized, 5f are also expected to be quite localized

Metals and Insulators. Independent electrons

Spin degeneracy:

Each band can hold 2 electrons per unit cell

Even number

of electrons

per unit cell

Insulating

Metallic (in case

of band overlap)

Odd number

of electrons

per unit cell

Metallic

Slater vs Mott insulators

Antiferromagnetism doubles the unit cell

1 electron per site

2 electrons per unit cell

(even number of electrons/unit cell)

Slater insulators: Insulating behavior due to unit cell doubling

(Antiferromagnetism)

The shape of the Fermi can lead to an antiferromagnetic instability

Slater vs Mott insulators

Antiferromagnetism doubles the unit cell

1 electron per site

2 electrons per unit cell

(even number of electrons/unit cell)

Slater insulators: Insulating behavior due to unit cell doubling

(Antiferromagnetism)

Mott insulators: Insulating behavior does not require AF

The shape of the Fermi can lead to an antiferromagnetic instability

Slater vs Mott insulators

Paramagnetic

Mott

Insulator

Metal-Insulator

transition with

decreasing pressure

Increasing Pressure: decreasing U/W Antiferromagnetism

McWhan et al, PRB 7, 1920 (1973)

Large U limit. The Insulator. Magnetic exchange

Mott insulator:

Avoid double occupancy

(no constraint on spin ordering)

Large U limit. The Insulator. Magnetic exchange

Virtual transition

t2/U

Mott insulator:

Avoid double occupancy

(no constraint on spin ordering)

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

Nature of antiferromagnetism

Fermi surface instability Antiferromagnetic exchange

- Delocalized electrons. Energy

bands in k-space and Fermi surface

good starting point to describe

the system.

-The shape of the Fermi surface

presents a special feature (nesting)

-In the presence of small

interactions antiferromagnetic

ordering appears.

- Ordering can be incommensurate

Spin Density Wave

Magnetism driven by interactions

- Localized electrons. Spins localized in

real space

-Kinetic energy favors virtual hopping

of electrons (t2/ E ~ t2/ E ).

-Virtual hopping results in interactions

between the spins. Magnetic Exchange

Spin models

- Magnetic ordering appears if frustration

(lattice, hopping, …) does not avoid it.

- Commesurate ordering

Magnetism driven by kinetic energy

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 to avoiding double occupancy not

with magnetism (Slater insulators)

Magnetism:

Weakly correlated metals: Fermi surface instability

Mott insulators: Magnetic exchange (t2/U). Spin models