Optical and transport properties of small polarons

from Dynamical Mean-Field TheoryS. Fratini, S. Ciuchi

Outline:

● Historical overview

● DMFT for Holstein polaron

● Optical conductivity

● Transport

Polarons: historical overview

● Landau (1933) “self-trapping” phenomenological model : the electron polarizes the medium, that deforms and traps it

● Fröhlich (1954) microscopic model, long range e-ph interaction with LO modes=> large polarons (hydrogenic states, Rp>>a)

● Holstein (1959) What if the polaron size Rp~a ?experimental problem: resistivity of NiO, CoO, MnO Yamashita & Kurosawa (1954), Heikes & Johnston (1957)

thermally activated behavior, but fixed number of carriers(≠semiconductors, activated n)

Small Polarons: experiments

● Transport: activated behavior (hopping barrier)

● Optical absorption: IR broad peak (transitions within the polaron potential well)

● Photoemission: broad peak (multi-phonon shakeoff)

... should be cross-checked!

PROBLEMS:

1) textbook formulas are only valid in limiting cases 2) textbook formulas are only valid for independent polarons (in real systems at finite density, interplay with electronic correlations)

... here we address 1)

● tight binding electrons, bandwidth 2D~2zt ● Einstein bosons (phonons, excitons...): 0 a

+iai

● local interaction: g (a+i+ai) c

+ici

Solid = lattice of deformable molecules, (electronic level if occupied)

E0

EP

Holstein model (1959)

Polaron energyEP=g²/0 

 ~ 0.1-0.5 eV

-> Define 2 dimensionless parameters

Interaction strength:small polarons if

=EP/D >1 , adiabatic

(bound state out of band)

²=EP/0>1 , antiadiabatic

(# phonons in polaron cloud)

Adiabaticity: =0/D

g<<1 : adiabatic, slow phonons(ordinary metals, most oxides)

g>>1 antiadiabatic, extremely narrow bands(molecular solids, AF background...)

polarons

crossover

polaronswe

ak-c

oupl

ing crossover

Adia

bati

c A

nti

adia

bati

c

Polaron formation, d>1

Double well phonon potential:+/- =electronic state

U(q) =Aq2±Bq

Electronic gain

+ –

Simplest picture – 2sites, adiabatic limit (Polder)

●Study uncorrelated hops between 2 molecules

●Integrate out “fast” electrons

Elastic energy

U(q)

q

Review: Austin & Mott, Adv.Phys 18, 41 (1969)

q

+ –

Photoemission

0

Shen et al. – PRL 93, 267002 (2004) Ca2-xNaxCuO2Cl2

Spectrum inside individual “molecule”,=> r(k,) distribution of peaks with gaussian envelope

In a solid, the distributiongets smeared

Optical conductivity: IR absorption peak

q

+ –

Eopt=4Ea

Franck-Condon Broadening

Optical excitation is fast,lattice cannot relax

()~exp [ -(-Eopt)2/s2 ]

Reik, Z. Phys. 203, 346 (1967)

Kudinov - Sov.Phys.Sol.St. (1970) - Ti02

Ea

Note: such simple picture is valid only if s>>D (see below)

q

+ –

Transport: activated mobility

~e­Ea/kT

carrier mobility is activatedHolstein, Ann. Phys. 8, 343 (1959)

polaron trapped on a site:incoherent hopping energy barrier Ea

Ea

Morin - Phys.Rev. (1954) - NiO

1000/T

DMFT results

●Method and single particle solution

●Optical conductivity

●Transport

References:S. Ciuchi, F. de Pasquale, S. Fratini, D. Feinberg, PRB 56, 4494 (1996)S. Fratini, F. de Pasquale, S. Ciuchi, PRB 63, 153101 (2001)S. Fratini, S. Ciuchi, PRL 91, 256403 (2003)S. Fratini, S. Ciuchi, cond-mat/0512202

●Method and single particle solution

●Optical conductivity

●Transport

References:S. Ciuchi, F. de Pasquale, S. Fratini, D. Feinberg, PRB 56, 4494 (1996)S. Fratini, F. de Pasquale, S. Ciuchi, PRB 63, 153101 (2001)S. Fratini, S. Ciuchi, PRL 91, 256403 (2003)S. Fratini, S. Ciuchi, cond-mat/0512202

DMFT results

Dynamical mean field theory (DMFT)

● mean field (dynamical):idem, but h(t) is time dependent, local fluctuationsaverage on space NOT time -> local self energy Georges, Kotliar, Krauth, Rozenberg, RMP 68, 13 (1996)

- becomes exact if d→∞ - excellent approximation at finite d for local phenomena (Holstein model: OK) avoid “small” parameter - analytical solution for single polaron Ciuchi, Feinberg, Fratini, De Pasquale PRB (1996)

h(t)

● mean field (ordinary):isolate a particle, the rest of the system is described by an effective field h to be determined self-consistentlyaverage on space AND time

h

Typical spectral density – Strong e-ph coupling

Multi-phonon “shakeoff” processes - spectral weight is redistributed, width >2D- coexistence of narrow antiadiabatic features + broad adiabatic continuum

“polaron” subband(exponentially narrowed)

narrow peaks, cf. molecular spectra (increasing width) high energy broad

incoherent continuum(gaussian envelope)

DOS=k r(k,)

DMFT results

●Method and single particle solution

●Optical conductivity

●Transport

References:S. Ciuchi, F. de Pasquale, S. Fratini, D. Feinberg, PRB 56, 4494 (1996)

S. Fratini, F. de Pasquale, S. Ciuchi, PRB 63, 153101 (2001)

S. Fratini, S. Ciuchi, PRL 91, 256403 (2003)

S. Fratini, S. Ciuchi, cond-mat/0512202

Calculation of conductivity (Kubo formula)

2 approximations:- dynamical mean-field, convolution

of spectral functions, neglects current vertex corrections

- independent polarons, valid at low density

no « small parameter »:- valid for any D, 0, g,T

- treats electron dispersion and phonon quantum fluctuations on the same footing- no analytic continuation-

d.c.= (=0) ,

most sensitive to quantum effects

BUT

spectral function (cf. ARPES)

Boltzmann statisticsunits: ohm cm

current vertex

Antiadiabatic regime, D<<0

Weak coupling:- peak at

0 , single-phonon excitation

+ few replicas - all have same width 4D- asymmetric and sharp edge at T<<4D- peaks shrink as T>

0,

agrees with common wisdom based on D→0 expansion

Strong coupling, ~ independent molecules- multi-phonon peaks, gaussian distribution

(envelope ~ Reik) - peak width is not uniform: lowest ~ exp(-2), otherwise -p - peaks broaden as T>

0 ,

finite D effects, beyond Holstein decoupling(also affects

d.c. )

2 2

Adiabatic regime, D>>0

Weak coupling, up to c:

- edge at 0 , single-phonon excitation

- fine structure at multiples of 0

- washed out as T0

Strong coupling, c:

- broad, slightly asymmetric peak - fine structure if finite 0 - peak position

max<2Ep

- lineshape depends on ratio s/D = phonon broadening/

electron bandwidth

[1] [2]

[3]

Lineshapes in limiting casesWeak coupling:

Polaronic regime:

● s>>D, ''Reik'' gaussian lineshape,

- Franck-Condon line broadened by phonon fluctuations- strong coupling, any d (lattice dimensionality)

● s<<D, sharp ''photoionization'' threshold,

- transitions from localized level to electron continuum- any coupling, any d

... analytical description of intermediate case?

[1]

[2]

[3]

[2]

[3]

DMFT

Adiabatic regime, intermediate coupling

(narrow) polaron crossover region, c:

- reentrant behavior governed by W<0

(renormalized polaronic bandwidth): T<W, weak coupling;

T>W, polaronic;- Polaron Interband Transitions,

thermally activated resonances with nonmonotonic T dependence

Adiabatic regime, intermediate coupling

(narrow) polaron crossover region, c:

- reentrant behavior governed by W<0

(renormalized polaronic bandwidth): T<W, weak coupling;

T>W, polaronic;- Polaron Interband Transitions,

thermally activated resonances with nonmonotonic T dependence

DMFT results

●Method and single particle solution

●Optical conductivity

●Transport

References:S. Ciuchi, F. de Pasquale, S. Fratini, D. Feinberg, PRB 56, 4494 (1996)

S. Fratini, F. de Pasquale, S. Ciuchi, PRB 63, 153101 (2001)

S. Fratini, S. Ciuchi, PRL 91, 256403 (2003)

S. Fratini, S. Ciuchi, cond-mat/0512202

Transport: 3 regimes

I. coherent motion T<0

 tunneling with large eff. mass

r exp (­ 0/kT)

II. activated behavior 0<T<Ep        r exp (Ea/kT)

III. residual scattering T>Ep rT³/²

I

II

Transport: 3 regimes

I

II

Arrhenius plot:activation energy Ea and absolute value of resistivity much less than “textbook” results obtained assuming D-->0 Conduction is “enhanced” by finite bandwidth effects

Analytical formula valid in nonadiabatic regime:

y=T/0

Concluding remarks

● The present DMFT results do not rely on “small parameters” and give access to the evolution of the optical and transport properties of (few) small polarons beyond the usual textbook limiting cases

● Phonon quantum fluctuations 0≠0 and electron dispersion D≠0 are treated on the same footing. Some results based on the assumption D→0 that predict a thermal narrowing of the polaron-band are apparently invalidated

● Main open question: how does this apply to real systems with a finite density of electrons? interplay with e-e repulsion, Hubbard-Holstein model

[1] [2]

[3]

Non-uniform peak width

Results in 1D (worst case)

Alexandrov, Kabanov, Ray, Physica C224 (1994)

Schubert et al., PRB72, 104204 (2005)

Adiabatic, s/D=0

Adiabatic, intermediate s/D

Optical absorption in NiO

Spectral density – Weak e-ph coupling

DOS=k r(k,)

Single phonon processes (metals) [cf. Engelsberg & Schrieffer]

- bandwidth ~ 2D- weakly renormalized spectrum, only around 0 KINKS!

low energy, coherent (Im =0)

high energy, weakly incoherent (Im 0)

Small polarons have been reported in:

● almost every transition metal oxide: NiO, MnO, CoO, CuO, ZnO, LaCoO3 ...Fe3O4, Fe3TiO4, TiO2, SrLaTiO3, SrLaVO3 LaCaMnO3, Tl2Mn2O7...

● atomic and molecular solids:Ne, Ar, Kr, Xe... N2,O2,CO...

● biological and organic compounds:DNA, TCNQ, anthracene...

● Other, CDWNiCuS2, NiSSe, (TaSe4)2I ...

q

+ –

Photoemission

0Spectrum inside individual “molecule”,=> r(k,) distribution of peaks with gaussian envelope

Perfetti et al., PRL 87, 216404 (2001) (TaSe4)2I

Smearing of Poisson distr:1 molecule solid