Mottphysics 1talk
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Transcript of Mottphysics 1talk
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Mott physics
E. Bascones
Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC)
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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
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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
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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
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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)
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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
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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?
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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
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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)
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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)
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Fermi liquid description
Fig: Lu et al, Nature 455, 81 (2008)
Angle Resolved Photoemission
Experiments (ARPES) would
show energy bands
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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
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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
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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
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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 , …)
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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
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Breakdown of independent electron picture
Fig: Pickett, RMP 61, 433 (1989)
Metallic behavior
expected
Insulating behavior is found
Breakdown of independent electron picture
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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
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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
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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)
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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)
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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
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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
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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
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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
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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
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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+ )
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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
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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
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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
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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
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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)
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Large U limit. The Insulator. Magnetic exchange
Mott insulator:
Avoid double occupancy
(no constraint on spin ordering)
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Large U limit. The Insulator. Magnetic exchange
Virtual transition
t2/U
Mott insulator:
Avoid double occupancy
(no constraint on spin ordering)
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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
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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
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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*
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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