Long range Coulomb interactionsat

low densities of polarons

S. Fratini - Grenoble

P. Quémerais (LEPES-CNRS Grenoble)

G. Rastelli, S. Ciuchi (Università dell’Aquila, Italy)

G. Chuev (Institute of theoretical & experimental biophysicsRussian Academy of Science, Pushchino, Russia)

J.-L. Raimbault (Ecole Polytechnique, France)

Subtitle:

on the role played by the long-range polarization (large polarons) in metal-insulator transitions

- metal-insulator transition scenarios

- what happens in a polarizable medium? Wigner crystal of polarons: melting & instability

- 2 examples, 1 analogy: cuprates and metal-amonia solutions

Outline

Anderson localization (monoelectronic model)

Disorder can localize a free particle(no interaction between carriers)

Peierls transition (2k F susceptibility in 1D)

formation of a CDW, a gap opens at the Fermi level

Mott transition in semiconductors (long-range Coulomb interactions & screening)

instability of the metallic phase against the formationof localized states around charged dopants

(ex: nonpolar doped semiconductors)

Wigner crystallization (long-range Coulomb interactions)

minimization of electrostatic energy at low concentrations (ex: electrons on liquid He, semiconductor heterostroctures)

Polarization catastrophe (long-range Coulomb interactions)

polarizability of individual units diverges due to collective effects (local field, cf. Clausius-Mossotti)

Metal-Insulator Transitions (a selection)

Mott-Hubbard transition (short range interaction)

half-filled band --> should be metallic, often insulating(ex: transition metal oxides, organic salts...)

The Mott Transition in semiconductors (1949)

Long-range Coulomb interactions / screening

Donor atoms in usual semiconductors (Si,Ge)Non-polar: κ= ε

s= ε

∞

Bound statearound the donor

Free electronscreening

No more bound state when

Mott Criterion

nc1/3a

0*=0.26

- Instability of the metallic phase- carrier density jumps at the transition- works in a number of doped nonpolar semiconductors (later understood as Anderson localization)

− e2

r

Impurity band

conduction band

Impurity levels

[Thomas, J Phys Chem 88, 3749 (84)]

Polarization catastrophe (1927)

(predicts if a solid is a metal or an insulator from the polarizability of the individual atoms, before Bloch theory...)

Dilute gas of polarizable units, each with characteristic frequency ω

0responds to external field

Dense system (solid, liquid): units respondto external field + macroscopic polarization

ω0

Polarizability

e2

α = m ω

02

p = dipole

(transverse)

Transverse frequency reduced by Lorenz local field factor (dipole-dipole interactions)

ωcoll

2 = ω02 - ω

P2/3

The restoring force for the electrons vanishes at critical nc ---> metal

Metals: R>M/d Insulators: R<M/d~1/n

Wigner crystallization (1934)

At low enough densities, electrons in a compensating jellium of positive charge crystallize --> insulating phase (at T=0K)

Kinetic energyper particle behaves as:

1/rs2

Coulomb interaction ~

1/rs

At low densities, the minimizationof energy yields a crystallized state

3D --> BCC2D --> Hexagonal

Dimensionless density parameter rs=R

s /a

0

n : density of electrons

H=Hkin

+Hint

jellium sphereelectron

The system is separatedinto neutral spheres of radius Rscontaining one electron -e and a compensating charge +e

Wigner decomposition

{Ri} Bravais lattice sites

ri = R

i + u

i Quadratic expansion of

+

€

u i

Dipole-dipole interactions

Neutral spheres do not interact, E=0

+

€

u i

+

€

u j

Mean Field (Wigner)

Beyond mean field

Warning !! : not all crystallized structures are MECHANICALLY stable.

Ex: Simple Cubic <0

Mean field(Wigner 1933)

Dipole-dipole interactions (bcc)(Carr 1961)

BUT: Comparison of energies is not accurate enough (the liquid is strongly interacting at such low densities)

=> Phenomenological Lindeman criterion, originally for thermal melting of solids

The crystal melts when the oscillations of the localized particles attain a fraction of the interparticle distance [Pines & Nozières 1958]

QMC:

2D

3D

Melting towards electron liquid: Eliq

<EWC

On the formation of (large) polaronsand their crystallization

Ionic Crystal Dipolar Liquid

Two sources of polarization

+ -

+-

+ -

+-+

-+-

+-

+ -+-+

- + -+ -

+ -+ -

+ -

+ -

+-

+-

+-

+-

+-

+-

1) Electronic polarizability (core electrons)

2) Ionic distortion, molecular orientation

+ + -- ++ -- + -

+ + -- ++ -- + -

+ + -- +

fast dynamics(follows instantaneously the motion ofadded charge)

slow dynamics(leads to polaron formation)

Polarizable medium

Dielectric constant depends on frequency

NH3

€

ε s

€

ε∞

≈2≈5

≈22 ≈30

(isotropic)

La2CuO4

Ionic crystalChen & al. 91

Formation of Fröhlich polarons + + -- +

+ -- +-

+ + -- ++

-- + -+ + -- +

+ -- + -

Ionic distortion LO phonons create a polarization to screen the electron charge

localizesthe electronic wave function

self-consistentproblem

Hydrogenic bound state phonon energy α 2=

Single polaron: already a nontrivial problem

Weak coupling: perturbation theory

Strong coupling:static polarization field

Electron localized in astatic potential well

All coupling theory: path integrals[R. Feynman 1958]

NOT quadratic!

From weak to strong coupling: Feynman Path-Integral approach

What choice for the trial model?

the polarization is replaced by a fictitious particle

- not fixed, does not break translational invariance !

- K, M are variational parameters

- many-body --> two-body

- after minimization, the trial model itself gives a good representation of the polaron

Trial model

Take a system of polarons at low density and let them interact through the Coulomb repulsion

OR

Take a Wigner Crystal of electrons and put it in a host polar material

jellium

electron

Polarizable host

Two competing effects arise from the ionic polarization

Static screening of LR repulsion:reduces interaction energy

tends to DESTABILIZE the WC

Polaron formation: Decrease of kinetic energy

MP>>m

tends to STABILIZE the WC

Who wins?

η= ε∞

/εs

Polarizability ratio(no bipolaron formation if η>0.1)

Polaronic Wigner crystal (PWC)

- Fröhlich e-ph interaction + Coulomb e-e repulsion (long-range) - Integrate out phonons, expand for small oscillations

Beyond mean field: polaron-polaron correlations

Mean field: polaron in a harmonic potential: average effect of other localized polarons

Feynman approach to the PWC

Mean field: polaron in a harmonic potential

Melting of PWC:

2 degrees of freedom => 2 Lindeman criteria

Simple picture based on trial model:the polaron is a composite particle,with 2 independent d.o.f.

(i) center of mass (ii) relative motion

Sketch of absorption spectrum

Electron-electron interactions

Melting of PWC: two scenarios

(i) (ii)

Mott

Electrons are “dressed” by the phonons

=> The PWC melts as an ordinary WC, but r

c is renormalized by Mp and ε

s

Weak and intermediate coupling, melting driven by center of mass fluctuation

strong coupling, fluctuations of internal d.o.f.

the polarons have a large mass and cannot delocalize

=> polaron dissociation

~ Mott criterion !

Beyond mean field: collective modes

- trial model: 2 particles/site, in 3D => 6 branches

- dispersion caused by dipole-dipole interactions

Wigner crystal of electrons Wigner crystal of polarons

Polaronpeak in σ(ω)

Remember polarization catastrophe?

Optcond, figura articolo

Softening of polaroniccollective mode

+transfer of spectral weightto low freq. collective mode

ALL DUE TO DIPOLE-DIPOLEINTERACTIONS

alternative to screening,cf. [Cataudella et al.] [Devreese et al.]

Increasing density

Polaronpeak

Softening of transverse collective frequencyis a precursor to the polarization catastrophe

ωpol

2=ω02-ω

p2/3ε

∞

Characterist frequency of isolated polarons

Softening due to collective effects

Dielectric properties: ε (k,w)

It can overscreen the coulomb repulsion between 2 test charges [cf. talk by Takada]

moves the ions away from equilibrium positions

Static limit:

<0

Dynamical response:

Becomes generally negative when approaching the polarization catastrophe

Could induce pairing of free electrons coexisting with a PWC

The collective bosonic excitations of the PWC could mediate an attractive interaction

BUT: hard to justify theoretically

e-phattraction

PWCattraction

Can all this be of some relevance in real compounds?

[Lupi et al. PRL 99]

Cuprates: softening of MIR peak

[Lucarelli et al. IJMP 03]

STM: hole crystallization?

Measure tunneling I vs. bias V with atomic resolution

=> conductance dI/dV= g(r,V) ~ Local DOS

FT of local DOS => spots in reciprocal space

- q varies with bias => qp interference, g(q,V) ~ A(k,V) A(k+q,V)

- dispersionless q, 4a0x4a0 modulations, 30-80 A corr. length

=> charge ordering

Several interpretations (crystal of superconducting pairs, condensate with

spatial modulations, stripes, Wigner crystal of holes...)

A Wigner crystal with periodicity 4x4 can be stabilized provided

that the holes are subject to an additional localizing effect whose

characteristic energy scale is ~ 150meV, the same that is observed in σ(ω)

candidates => polaron energy Ep, exchange energy

[G.R., S.F. & P.Q.: Eur. Phys. Jour. B, 42, 305 (2004)]

Na-CCOC, non SC (low doping) [Hanaguri et al., Nature 430, 1001 (2004)

NON-Dispersive features in PG regions

Bi-2212, superconducting [McElroy, PRL94 197005 (2005)]

Inhomogeneous gap maps, from OD to UD

Metal-Ammonia solutions: Na-NH3

Solvated electrons=polarons

Optical absorptionpeak at 0.8 eV

σ(ω)

Phase diagram

Polarons are blue !

Bronze metallic phase (polarons have dissociated)

[JC. Wasse, website]

[Edwards 2001]

Softening of polaron peak

[Schlauf et al]

“

”

A possible analogy?

region with negativedielectric constant ?

A suggestion:

If the counter charges are mobileas in liquid NH3, phase separation

If counter charges are frozenas in the cuprates, superconductivity

Conclusion

There is a lot of interesting physics in

long range interactions, why restrict to short range?

References

P.Q.: Mod. Phys. Lett., 9, 1665 (1995)

P.Q & S. F: Mod. Phys. Lett., 11, 1303 (1997)

P.Q. & S.F: Int. Jour. Mod. Phys B, 12, 3131 (1998)first prediction of the red shift

S.F. &P.Q.: Mod. Phys. Lett. 12, 1003 (1998)

S.F & P.Q.: Eur. Phys. Jour. B, 14, 99 (2000)

P.Q. & S.F.: Physica C, 341-348, 225 (2000)

S.F. & P.Q.: Eur. Phys. Jour. B, 29, 41 (2002)

P.Q., J.L.R. & S.F,: J. Phys. IV France 12, pr9-227 (2002)on the analogy with metal-amonia solutions

G.R., S.F. & P.Q.: Eur. Phys. Jour. B, 42, 305 (2004) on the modulations observed by STM

G.R and S.C.: Phys. Rev. B 71, 184303 (2005)

G.R. and S.F.: in preparationon the stabilization of a WC in low-dimensional solids