Polaron Theory

8/10/2019 Polaron Theory

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Activated Conductivity

=ne/m=A exp (- /kT)

1) semiconducting gap:carrier density is activated

2) interactions:carrier mobility is activated

Ep

polaron trapped on a site:hopping transportuntrapping energy Ep

Morin (1954)


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Small polaron transport has since been reported in:

almost every transition metal oxide:NiO, MnO, CoO, CuO, ZnO, LaCoO3 ...Fe3TiO4, TiO2, SrLaTiO3, SrLaVO3 ... (titanates, vanadates)LaCaMnO3, Tl2Mn2O7... (manganites, pyrochlores)

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

biological and organic compounds:DNA, TCNQ, oligoacenes (pentacene, anthracene, etc.) ...

otherNiCuS2, NiSSe ...


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Holstein model (1959)

tight binding electrons : - t c +ic j optical bosons (phonons, excitons...): 0 a+iailocal interaction: g (a +i+a i) c

+ici

2 dimensionless parameters =0 /t =g/ 0t or =(g/ 0)

interaction strength :

polarons if >1 ( >1)

+ density (take n independent polarons)+ temperature...

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

E0E1

adiabaticity : 1 fast phonons, slow electrons(narrow band solids)


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we are left with:

2 (adiabatic/nonadiabatic)x 2 (strong/weak coupling)x 3 (low, room, high temperature)

------------------------------------------------------= 12 different regimes ! i.e. basically 12 different theoretical approaches...

but:people are mainly interested in the strong coupling regime, at RT=> 2 formulas are sufficient (adiabatic/nonadiabatic)=> both appear in the original paper by Holstein (1959)


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Open questions (an arbitrary selection)

many polaron theory -> individual single polarons

transport, 2 main experimental problems:

1) simple Arrhenius law always fails at low T2) Holstein's theory fails to explain self- consistently

d.c. transport and optical absorption data:microscopic parameters extracted from d.c.resistivity are inconsistent with ()


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examples: 1) failure of pure Arrhenius behaviour

the data fit nicely to eq. (1) can be fitted according to eq. (2) LaSrMn2O7, Chatterjee PRB2003 LaTiO 3.41 , Kuntscher PRB2003

some easier transport channel appears at low T

VRH ? exp (T -

) coherent (band) motion of small polarons?often, magnetic transitions at low T


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Coherent band motion at low T

Instead of hopping incoherently, the electron can tunnel coherently to any quantum state of equivalent energy, and form a band (need quantum phonons )

no phonon scattering up to 0 , exp (- 0 /kT)

in principle dominates at low temperature

BUT t* is exponentially reduced , very sensitive to disorder

There are only a few reports of this low T regime in thesmall polaron literature (but very common for large polarons):- LaTiO 3.41 , Kuntscher PRB2003 (could be LP)- LaCaMnO 3, Zhao PRL2000 (within FM phase, t enhanced)

- naphthalene, Schein PR78 (+ Schn...)- NiS 2-xSe x, Kwizera PRB80

1/T


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more examples:

Pr2CuO4, Homes PRB 2002 LaTiO3, Lunkenheimer PRB 2003


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more examples:

La 0.66 Ca 0.33 MnO 3 films, Jaime PRB 96 La 1.2Sr1.8Mn 2O7, Chen PRB2003

Tc=238KTc=125K


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examples: 2) inconsistency with optics

BaTiO3, Iguchi PRB91 BaTiO3, Berglund PR67

d.c.=68meV

opt=450- 600 meV

Standard theory predicts: = 4 here~

8(d.c.)=Ep/2(opt)=2Ep

(d.c.)(opt)


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Dynamical mean field theory (DMFT)

mean field (dynamical):idem, but h(t) is time dependent (average on space only)

- becomes exact infinite dimensions- excellent approximation d=2,3 systems for local phenomena

(Holstein polaron: OK)- analytical solutionif n independent polarons(Ciuchi, Feinberg, Fratini, De Pasquale 1996)

h(t)

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

h


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G( )


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The DMFT loop

impurity propagator = lattice propagator

Numerical (QMC,ED, NRG)Analytical


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Single polaron DOS & damping

Weak coupling

weakly renormalized DOS distributionlow energy coherent parthigh energy weakly incoherent part


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Calculation of resistivity (Kubo formula)

2 approximations:- dynamical mean field- independent polarons

no small parameter :- valid for any t, 0, g- at all temperatures T

BUT

spectral function (cf. ARPES)

Boltzmann statistics

units: ohm cm


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Pros:

one can treat both- phonon quantum fluctuations (00): access low T regime- electronic dispersion (t0): no need for ad hoc regularization

(+ 1 theory instead of 12 different approximations)

Cons:

Analytical method but final integral is numeric,no direct formula for (T)

(still searching a general formula, found one in nonadiabatic regime...)


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DMFT results: 3 regimes

I. coherent motion T


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Analytical formula in nonadiabatic regime

At large =0 /t the resistivity obeys:

The DMFT data are well described by:

y=T/ 0

Activation gap: sensibly reduced: d.c=1/2 Ep 3/8 Ep (solve mismatch with optics ?)temperature dependent ! (downturn of Arrhenius plots at low T)

Onset of phonon quantum fluctuations:a ROBUST precursor to the low T coherent motion (not destroyed by disorder).


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Tentative application: titanates

fit yieldsEp=200meV

polaron absorption should beat 2Ep = 400meV = 3200cm+ independent estimate of 0=130cm


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Tentative application: vanadates

Tokura, PRB95

peak position according to DMFT fitpeak position according to Arrhenius fit


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Concluding remarksSmall polarons are quite common in oxides (but not only)

Standard theory by Holstein often fails to describe self- consistentlyd.c. transport and optical absorption experiments

The mismatch is due to inconsistencies in the standardapproximations (ad- hoc regularization, two-site model)

DMFT seems to go in the right direction:- enhancement of conductivity at low T , deviations from Arrhenius

related to phonon quantum fluctuations- reduction of polaron energy extracted from experimental data related to weak dispersion of optical phonons (local)

restore agreement with ()

work in progress (long term): systematic reanalysis of experiments(contact with experimentalists needed...)


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Note:

The theoretical inconsistencies pointed out here were clearly statedin Holstein's original work in 1959.

The scientific community does not seem to be aware of suchinconsistencies.

Holstein's paper has been cited 1496 times to date.

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Polaron Theory