Optical and transport properties of small polarons from Dynamical Mean-Field Theory

Optical and transport properties of small polarons from Dynamical Mean-Field Theory S. 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: Optical absorption: Photoemission: activated behavior (hopping barrier) IR broad peak (transitions within the polaron potential well) 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)

Holstein model (1959) Solid = lattice of deformable molecules, (electronic level if occupied) E 0 E P tight binding electrons, bandwidth 2D~2zt Einstein bosons (phonons, excitons...): 0 a + i a i local interaction: g (a + i +a i ) c+ i c i Polaron energy E P =g²/ 0 ~ 0.1-0.5 ev -> Define 2 dimensionless parameters

Polaron formation, d>1 Adiabaticity: = 0 /D g<<1 : adiabatic, slow phonons (ordinary metals, most oxides) g>>1 antiadiabatic, extremely narrow bands (molecular solids, AF background...) Interaction strength: small polarons if =E P /D >1, adiabatic (bound state out of band) ²=E P / 0 >1, antiadiabatic (# phonons in polaron cloud) Adiabatic Antiadiabatic weak-coupling crossover crossover polarons polarons

Simplest picture 2sites, adiabatic limit (Polder) Study uncorrelated hops between 2 molecules Integrate out fast electrons U(q) Double well phonon potential: +/- =electronic state U(q) =Aq 2 ±Bq q Elastic energy Electronic gain + Review: Austin & Mott, Adv.Phys 18, 41 (1969)

Photoemission Spectrum inside individual molecule, => r(k, ) distribution of peaks with gaussian envelope In a solid, the distribution gets smeared 0 q + Shen et al. PRL 93, 267002 (2004) Ca 2-x Na x CuO 2 Cl 2

Optical conductivity: IR absorption peak Optical excitation is fast, lattice cannot relax ( )~exp [ -( -E opt ) 2 /s 2 ] Reik, Z. Phys. 203, 346 (1967) E opt =4E a Ea q + Franck-Condon Broadening Kudinov - Sov.Phys.Sol.St. (1970) - Ti0 2 Note: such simple picture is valid only if s>>d (see below)

Transport: activated mobility carrier mobility is activated Holstein, Ann. Phys. 8, 343 (1959) ~e Ea/kT Ea q + polaron trapped on a site: incoherent hopping energy barrier E a 1000/T Morin - Phys.Rev. (1954) - NiO

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

Dynamical mean field theory (DMFT) mean field (ordinary): isolate a particle, the rest of the system is described by an effective field h to be determined self-consistently average on space AND time h mean field (dynamical): idem, but h(t) is time dependent, local fluctuations average 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)

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, )

Calculation of conductivity (Kubo formula) units: ohm cm Boltzmann statistics current vertex 2 approximations: - dynamical mean-field, convolution of spectral functions, neglects current vertex corrections - independent polarons, valid at low density BUT spectral function (cf. ARPES) 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

Antiadiabatic regime, D<< 0 2 2 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. )

Adiabatic regime, D>> 0 Weak coupling, up to c : - edge at 0, single-phonon excitation - fine structure at multiples of 0 - washed out as T 0 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

Lineshapes in limiting cases Weak coupling: [1] [1] [2] Polaronic regime: s>>d, ''Reik'' gaussian lineshape, [3] [2] - Franck-Condon line broadened by phonon fluctuations - strong coupling, any d (lattice dimensionality) [3] [3] s<<d, sharp ''photoionization'' threshold, - transitions from localized level to electron continuum - any coupling, any d... analytical description of intermediate case? DMFT [2]

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

Transport: 3 regimes I II I. coherent motion T< 0 tunneling with large eff. mass r exp ( 0 /kt) III. residual scattering T>Ep r T³ / ² II. activated behavior 0 <T<Ep r exp (E a /kt)

Transport: 3 regimes I II Analytical formula valid in nonadiabatic regime: y=t/ 0 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

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 [1] [2] [3] 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

Non-uniform peak width

Results in 1D (worst case) Adiabatic, s/d=0 Alexandrov, Kabanov, Ray, Physica C224 (1994) Adiabatic, intermediate s/d Schubert et al., PRB72, 104204 (2005)

Optical absorption in NiO

Spectral density Weak e-ph coupling Single phonon processes (metals) [cf. Engelsberg & Schrieffer] - bandwidth ~ 2D - weakly renormalized spectrum, only around 0 KINKS! DOS= k r(k, ) 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, CDW NiCuS2, NiSSe, (TaSe 4 ) 2 I...

Photoemission Smearing of Poisson distr: 1 molecule solid Spectrum inside individual molecule, => r(k, ) distribution of peaks with gaussian envelope 0 q + Perfetti et al., PRL 87, 216404 (2001) (TaSe 4 ) 2 I