Models in CM physics uses and misuses

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Models in CM physics uses and misuses

• George Sawatzky
• ubc

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Although we know the exact theory i.e. all
interactions and elementary particles of
importance in CM physics the many body nature of
the problem makes a solution impossible and we
resort to models to try to get an understanding
of the diversity of physical properties. This is
not only because of curiosity but also because we
would like to optimize properties
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Solids exhibit a Wide diversity of properties
Take for example only the transition metal oxides

• Metals CrO2, Fe3O4 Tgt120K
• Insulators Cr2O3, SrTiO3,CoO
• Semiconductors Cu2O
• Semiconductor metal VO2,V2O3, Ti4O7
• Superconductors La(Sr)2CuO4, LiTiO4,NaxCoO2
• Piezo and Ferroelectric BaTiO3
• Catalysts Fe,Co,Ni Oxides
• Ferro and Ferri magnets CrO2, gammaFe2O3
• Antiferromagnets alfa Fe2O3, MnO,NiO ---
• Properties depend in detail on composition and
  structure

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Atoms in a periodic array in solids
Bloch Wilson 1937
We are interested in the potential Produced by
the nuclei and the inner electrons on the
outermost Valence electrons
K2p/wave length
Ef is the Fermi level up to which Each k state
is filled with 2 electrons
ONLY METALS !!
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More atomic like states for atoms in solids with
large inter-atomic spacing compared to orbital
radius
Electrons can quantum mechanically Tunnel from
atom to atom forming again Waves and bands of
states but now the Bands are finite in width. If
such a band is full ( 2 electrons per atom for S
orbitals the material will be an insulator
Because of a forbidden gap to the next band of
states INSULATOR OR SEMICONDUCTOR
Still rather boring since we have no magnetism
and systems With an odd number of electrons per
atom would all be metallic
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One electron band theory

• Electrons are in delocalized states labeled by a
  wave vector k forming bands
• There are two electrons per k state ( spin up and
  down) (non magnetic)
• An even number of electrons per unit cell could
  yield either an insulating or metallic state but
  an odd number would always yield a metal
• Bloch Wilson theory of 1937 already falsified in
  1937 Verwey and de Boer ( CoO is an insulator)
  and explained by Peierls ( stay at home principle
  for the d electrons coined by Herring )

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Surely a lattice of H atoms separated by say 1 cm
would not behave like a metal

• What have we forgotten ?
• The electron electron repulsive interaction

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Simplest model single band Hubbard
Row of H atoms 1s orbitals only
Largest coulomb Interaction is on site U
The hole can freely Propagate leading to A width
The electron can freely Propagate leading to a
width
E gap 12.9eV-W
The actual motion of the Particles will turn out
to be more complicated
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For large UgtgtW

• One electron per site ----Insulator
• Low energy scale physics contains no charge
  fluctuations
• Spin fluctuations determine the low energy scale
  properties
• Can we project out the high energy scale?

Heisenberg Spin Hamiltonian
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Doping a Mott Hubbard system
U
N
N
EF
PES
PES
(1-x)/2x
2
2
EF
EF
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x0.0
x0.1
x0.2
x0.3
x0.4
x0.5
x0.6
x0.7
x0.8
These states would be visible in a two particle
addition spectral function
x0.9
Meinders et al, PRB 48, 3916 (1993)
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These particles block 2 or more states
Bosons block 0 states Fermions block 1 state
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What would a mean field theory give you?
Elfimov unpublished
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Sometimes we get so involved in the beauty and
complexity of the model that we forget what the
validating conditions were and use them outside
of the range of validity
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Remember that Transition metal compounds

• Consist of atoms on a lattice not a jelium
• The charge carriers and spins live on atoms
• The atoms or ions can be strongly polarizable
• Polarizability is very non uniform i.e. O2- is
  highly polarizable Cu2 is not
• We cannot use conventional screening models to
  screen short range interactions

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Hossain et al., Nature Physics 4, 527 (2008)
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Correlated Electrons in a Solid
U
dn dn ? dn-1 dn1
Cu (d9)
O (p6)
?
p6 dn ? p5 dn1
U EITM EATM - Epol
? EIO EATM - Epol dEM
If ? lt (Ww)/2 ? Self doped metal
Epol depends on surroundings!!!
EI ionization energyEA electron affinity
energyEM Madelung energy

• J.Hubbard, Proc. Roy. Soc. London A 276, 238
  (1963)
• ZSA, PRL 55, 418 (1985)

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Cu2 (d9) Impurity in CuO2 lattice Eskes PRL
61,1475 (1988)
Other symmetry States at about 0.4 eV Below ZR
Zhang rice singlet Forms the lowest energy
band for a lattice of impurities
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Is single band Hubbard justified for Cuprates?
Zhang Rice PRB 1988 37,3759
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Problem with ZR singlets

• The combination of O 2p states is not compatible
  with a band structure state
• The wave functions are non orthogonal

From ZR PRL 37,3759
Note it goes to infinity at k0, should we see
it at Gamma in ARPES? Luckly it goes to 1 for K
Pi/2,Pi/2 and along the antiferromagnetic zone
boundary where the doped holes go at low doping
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Problems with ZR singlets

• As we dope the system the integrety of the ZR
  states disappears
• As we dope the system the ZR states strongly
  overlap forbidden by Pauli so they must change.

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Effective Hamiltonians can be misleading

• Hubbard like models are based on the assumption
  that longer range coulomb interactions are
  screened and the short range on site interactions
  remain
• However U for the atom is about 20 eV but U as
  measured in the solid is only of order 5 eV
• HOW IS THIS POSSIBLE?

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Coulomb interactions in solids

• How large is U ?
• How are short range interactions screened in
  solids?

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I will show that

• The polarizability of anions results in a strong
  reduction of the Hubbard on site U
• The charged carriers living on transition metal
  ions are dressed by virtual electron hole
  excitations on the anions resulting in electronic
  polarons
• The nearest neighbor coulomb interactions can be
  either screened or antiscreened depending on the
  details of the structure

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polarizability in TM compounds is very non
uniform
The dielectric constant is a function of r,r,w
and not only r-r,w and so Is a function of
q,q,w
Strong local field corrections for short range
interactions
Meinders et al PRB 52, 2484 (1995) Van den Brink
et al PRL 75, 4658 (1995)
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Reduction of onsite interactions and changing the
nearest neighbor interactions with polarizable
ions in a lattice
We assume that the hole and electron move slowly
compared to the response time of the
polarizability of the atoms. Note the oppositely
polarized atoms next to the hole and extra
electron
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So the reduction of the Hubbard U in a
polarizable medium like this introduces a strong
nextnn repulsive interaction. This changes our
model!!
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Note short range interactions are reduced
screened and intermediate range interactions
are enhanced or antiscreened-quite opposite to
conventional wisdom in solid state physics
Jeroen van den Brink Thesis U of Groningen 1997
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Homogeneous Maxwell Equations
?(r,r) gt ?(r r) gt ?(q)
Ok if polarizability is uniform
In most correlated electron systems and
molecular solids the polarizability is
actually Very NONUNIFORM
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In many solids the plarizability is very non
uniform

• Short range interactions cannot be described in
  terms of ?(r-r) but rather of ?(r,r) and so we
  cannot use ?(q) to screen
• Rather than working with ? go back work in real
  space with polarizability
• Atomic plarizabilities are high frequency i.e. of
  order 5 or more eV. Most correlated systems
  involve narrow bands i.e. less than 2 eV and so
  the response of atomic polarizability to the
  motion of a charge in a narrow band is
  instantaneous.
• i.e Electrons are dressed by the polarizable
  medium and move like heavier polarons

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A Picture of Solvation of ions in a polarizable
medium
PES (EI)
IPES (EA)
? ?
? ?
e
e


Full polarization can develop provided that
Dynamic Response Time of the polarizable medium
is faster than hopping time of the charge ?E
(polarizability) gt W ?E ? MO energy
splitting in molecules, plasma frequency in
metals-----
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Reduction of U due to polarizability of O2-
(SOLVATION)
U EITM EATM -2Epol
EI ionization energyEA electron affinity
energy
Epol 2
For 6 nn of O2- 13eV For 4 nn As3- 17eV
ELECTONIC POLARON
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What about intersite interaction V?
Fro the cuprates the Cu-O-Cu bond angle is 180
degrees therefor the repulsive interaction is
enhanced.
For pnictides the Fe-As-Fe nn bond angle is 70
degrees Therefore the contribution to V is
attractive 4 eV
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Polarization cloud For Two charges on Neighboring
Fe ELECTRONIC BIPOLARON
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Rough estimateAtomic or ionic polarizability
volume

• Consider atom nucleus at the center of a
  uniformly charge sphere of electrons
• In a field E a dipole moment is induced PaE
• For Z1 and 1 electron restoring force

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Concluding remarks

• Models are great but on applying them to real
  systems one should be aware of the approximations
  made to get to them
• In testing models one has to remain within the
  energy range excluding contributions from other
  states not included.
• Non uniform polarizabilities can introduce
  surprises with regard to short range coulomb
  interactions
• We would all be dead if it was not for solvation
  and so would weakly correlated electron systems

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Single band model is only valid at low energy
scales i.e. less than .5 eV!!! In doped systems
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polarizability in TM compounds is very
inhomogeneous
The dielectric constant is a function of r,r,w
and not only r-r,w and so Is a function of
q,q,w
Strong local field corrections for short range
interactions
Meinders et al PRB 52, 2484 (1995) Van den Brink
et al PRL 75, 4658 (1995) J. van den Brink and
G.A. Sawatzky Non conventional screening of the
Coulomb interaction In low dimensional and finite
size systems. Europhysics Letters 50, 447 (2000)
arXiv0808.1390 Heavy anion solvation of
polarity fluctuations G.A. Sawatzky, I.S.
Elfimov, J. van den Brink, J. Zaanen arXiv
0811.0214v1 Electronic polarons and bipolarons
Mona Berciu, Ilya Elfimov and George A sawatzky
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U for C60
Gas phase
Smalley
U atomic 3.4 eV
U I A E 3.4 eV
Solid ? Screening ---Solvation
EI EI0 Ep EA EA0Ep
Z12 FCC but smaller at surface
Now
U solid 1.6 eV
Compares well with our experiments !
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polarizability in TM compounds is very non
uniform
The dielectric constant is a function of r,r,w
and not only r-r,w and so Is a function of
q,q,w
Strong local field corrections for short range
interactions
Meinders et al PRB 52, 2484 (1995) Van den Brink
et al PRL 75, 4658 (1995)
arXiv0808.1390 Heavy anion solvation of
polarity fluctuations in Pnictides G.A.
Sawatzky, I.S. Elfimov, J. van den Brink, J.
Zaanen arXiv08110214v Electronic polarons and
bipolarons in Fe-based superconductors Mona
Berciu, Ilya Elfimov and George A. Sawatzky