Models in CM physics uses and misuses George Sawatzky ubc.

Models in CM physics uses and misuses

George Sawatzky ubc

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

Solids exhibit a Wide diversity of properties

• Metals: CrO2, Fe3O4 T>120K• 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

Take for example only the transition metal oxides

Atoms in a periodic array in solids

We are interested in the potentialProduced by the nuclei and the inner electrons on the outermost “Valence” electrons

K2π/wave length

Ef is the Fermi level up to which Each k state is filled with 2 electrons

ONLY METALS !!

Bloch Wilson 1937

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 againWaves 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

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 )

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

The hole can freely Propagate leading to A width

The electron can freely Propagate leading to a width

Largest coulombInteraction is on site U

Simplest model single band HubbardRow of H atoms1s orbitals only

E gap = 12.9eV-W

The actual motion of the Particles will turn out to be more complicated

For large U>>W

• 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?

ji

jiSJSH,

UtJ /4 2

Heisenberg Spin Hamiltonian

N N

EFPES PES

U

EF

N-1 N-12

EF

N+1N-1

2

Doping a Mott – Hubbard system

(1-x)/2x

x=0.0

x=0.1

x=0.2

x=0.3

x=0.4

x=0.5

x=0.6

x=0.7

x=0.8

x=0.9

Meinders et al, PRB 48, 3916 (1993)

These states would be visible in a two particle addition spectral function

These particles block 2 or more states

Bosons – block 0 statesFermions – block 1 state

Num

ber

of h

oles

LDA+U potential correction

SC Hydrogen

a =2.7 ÅU=12eV

LDA+U DOS

0.0 0.2 0.4 0.6 0.8 1.0-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

LDA

+U

cor

rect

ion

(eV

)

Number of holes

spin up spin down

0.0

0.2

0.4

0.0

0.2

0.4

0.0

0.2

0.4

DO

S (

stat

es/e

V c

ell)

0.0

0.2

0.4

0.0

0.2

0.4

0.5

0.4

0.3

0.2

0.1

-6 -4 -2 0 2 4 6 8 10 12 140.0

0.2

0.4

Energy (eV)

0.0

0.2

0.4

0.0

0.2

0.4

DO

S (

stat

es/e

V c

ell)

0.0

0.2

0.4

0.0

0.2

0.4

0.99

0.9

0.8

0.7

0.6

Elfimov unpublished What would a mean field theory give you?

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

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

Hossain et al., Nature Physics 4, 527 (2008)

Correlated Electrons in a Solid

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

If Δ < (W+w)/2 Self doped metal

dn dn dn-1 dn+1U :

p6 dn p5 dn+1Δ :

U = EITM – EA

TM - Epol

Δ = EIO – EA

TM - Epol + δEM

EI ionization energyEA electron affinity energyEM Madelung energy

Cu (d9)

O (p6)

Epol depends on surroundings!!! 4

2

p R

αzeE

Cu2+ (d9) Impurity in CuO2 lattice Eskes PRL 61,1475 (1988)

Zhang rice singlet Forms the lowest energy band for a lattice of “impurities”

Other symmetryStates at about 0.4 eVBelow ZR

Is single band Hubbard justified for Cuprates?

Zhang Rice PRB 198837,3759

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 k=0, 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

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.

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?

Coulomb interactions in solids

How large is U ?How are short range interactions

screened in solids?

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

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)

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

il

alliii

innPnnzPUH

,2int 2)2(

So the reduction of the Hubbard U in a polarizable medium like this introduces a strong nextnn repulsive interaction. This changes our model!!

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

Homogeneous Maxwell Equations

(r,r’) —> (r – r’) —> (q)

Ok if polarizability is uniform

ε(q)

(q)VV(q)

0

In most correlated electron systems and molecular solids the polarizability is actuallyVery NONUNIFORM

In many solids the plarizability is very non uniform

• Short range interactions cannot be described in terms of Є(r-r’) but rather of Є(r,r’) and so we cannot use Є(q) to screen

• Rather than working with Є 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

+

eћ ћ

e

—

PES (EI) IPES (EA)

Full polarization can develop provided that Dynamic Response Time of the polarizable medium is faster than

hopping time of the charge

E (polarizability) > W ; E MO energy splitting in molecules, plasma frequency in metals-----

A Picture of Solvation of ions in a polarizable medium

Reduction of U due to polarizability of O2- (SOLVATION)

U = EITM – EA

TM -2Epol

EI ionization energyEA electron affinity energy

i

Epol2

1

Epol = 2 For 6 nn of O2- ~ 13eVFor 4 nn As3- ~17eV

ELECTONIC POLARON

What about intersite interaction V?

For pnictides the Fe-As-Fe nn bond angle is ~70 degrees Therefore the contribution to V is attractive ~4 eV

Fro the cuprates the Cu-O-Cu bond angle is 180 degrees therefor the repulsive interaction is enhanced.

Polarization cloud For Two charges on Neighboring Fe “ELECTRONIC

BIPOLARON

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 P=αE

• For Z=1 and 1 electron restoring force =

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

Single band model is only valid at low energy scales i.e. less than .5 eV!!! In

doped systems

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 interactionsMeinders 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)

arXiv:0808.1390 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

U for C60Gas phase :

SmalleyI = 7.6 eVA = 2.65 eVE = 1.6 eV

T1u-Hu

U = I – A – E = 3.4 eV U [‘atomic’] = 3.4 eV

Solid Screening ---Solvation

4

2

p R

αzeE Z=12 [FCC] but smaller at surface

EI = EI0 – Ep

EA = EA0+Ep

effect:reduction Iincrease A

U [‘solid’] = 1.6 eVNow:

Compares well with our experiments !

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)

arXiv:0808.1390 Heavy anion solvation of polarity fluctuations in Pnictides G.A. Sawatzky, I.S. Elfimov, J. van den Brink, J. Zaanen

arXiv:08110214v Electronic polarons and bipolarons in Fe-based superconductorsMona Berciu, Ilya Elfimov and George A. Sawatzky