B.Ya. Yavidov, P.Yu. Tsyba, S.M. Zholdasova,... .

B.Ya. Yavidov*, P.Yu. Tsyba, S.М. Zholdasova**, R. Myrzakulov, S. Dzhumanov*Optical condactivity of small Frohlich polarons

(*Phase Transitions Physics Laboratory, Institute of Nuclear Physics, Tashkent c.)(**Physics Department, Aktobe State University named after K.Zhubanov, Aktobe c.)

(L.N. Gumilyov Eurasian Nationality University, Astana c.)

Optical conductivity of small Frohlich polarons is calculated and compared with those of small Holstein polarons. Opticalabsorption of the cuprates is briefly discussed within small Frohlich polaron model.

Polarons have been extensively studied since a seminal paper of Landau [2]. They are dividedinto small and large polarons in accordance with the size of their wave function. In the first casea carrier is coupled to intramolecular vibrations and self-trapped on a single site. Its size is thesame as the size of the phonon cloud, both are about the lattice constant (so-called small Holsteinpolaron (SHP)) [2]. In the case of large polarons the size of polaron is also the same as the size ofthe phonon cloud, but the polaron extends over many lattice constants [3]. In Ref. [4] polarons witha very different internal structure were introduced. They were called small Frohlich polarons (SFP).SFP size is about the lattice constant, but its phonon cloud spreads over the whole crystal. Themodel Hamiltonian of Ref. [4] is

H = He +Hph +He−ph (1)

where

He = −t∑~n

(c+~n c~n+~a + h.c.

)(2)

is the electron hopping energy,

Hph =∑~m,α

(− ~

2

2M∂2

∂uα+Mω2u2α

2

)(3)

is the Hamiltonian of the vibrating ions,

He−ph =∑~m,~n,α

f~m,α (~n) c+~n c~nu~m,α (4)

describes an interaction between the electron that belongs to a lower chain and the ions of anupper chain (Fig.1(b)). Here c+~n (c~n) is a creation (destruction) operator of an electron on a cite ~n,u~m,α is the α = z, y-polarized displacement of the ~m-th ion and f~m,α (~n) is an interacting density-displacement type force between an electron on a site ~n and the α polarized vibration of the ~m-thion. M is the mass of the vibrating ions and ω is their frequency. An explicit form of y – and z –coordinates of the interacting force are:

f~m,z (~n) =kzb(

|~n− ~m|2 + b2)3/2 (5)

f~m,y (~n) =ky |~n− ~m|(

|~n− ~m|2 + b2)3/2 (6)

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Л.Н. Гумилев атындағы ЕҰУ Хабаршы - Вестник ЕНУ им.Л.Н.Гумилева, 2010, №4

In the SHP model an electron performs hopping motion in a one-dimensional chain of the molecules andinteracts with the single site intramolecular vibrations. (b) In the SFP model an electron hops on a lower

chain and interacts with the ions vibrations of an upper infinite chain via a density-displacement type forcef~m,α (n) . The distances between the chains and between the ions are assumed equal to 1.

where k are some force constant. The distance along the chain |~n− ~m| is measured in the units ofa lattice constant a = 1. The distance between the chains is b = 1 too. A case of an electron coupledto a single site intramolecular vibrations represents the canonical Holstein model (Fig.1(a)) in whichan electron-phonon interacting force is defined as f~m,α (~n) = kαδ~m,~n. The model has direct relevanceto cuprates and to polaronic (bipolaronic) mechanism of High-TC superconductivity as Hamiltonian(1) mimics an interaction of a hole on the CuO2 plane (filled circles) with apical oxygen vibrations(open circles) of cuprates. An important role of a such type interaction is now overwhelming (seefor example [5]).

Optical conductivity of both small and large polarons have been studied extensively [6]. Anoptical absorption of small polarons is nearly adiabatic process so one can apply the familiar Franck-Condon principle. According the small polaron theory a general formula for the optical conductivityof small polarons at T = 0 is written as [7]

σ (ν) =σ0t̃

2

~ν√

2Ea~ωexp

[− (ν − 4Ea)

2(2√

2Ea~ω)2]

(7)

where σ0 is a constant, ν is the photon frequency and Ea is an activation energy for hopping process.The main difference between polarons with the Holstein and the Frohlich interactions is that in theformer case electron deforms only the site where it seats, while in the second case it deforms alsomany neighboring sites. This difference can be seen (i) in diagonal transitions of polaron from siteto site which ensures a lighter polaron in the Frohlich model [?, ?] and (ii) in the optical absorptionspectra. Due to the photon absorption SHP hops to an undeformed site, and Ea = Ep/2 [?]. HoweverSFP hops to a deformed neighboring site, so that Ea = γEp/2 [9,10], where

EP =1

2Mω2∑~m,α

f2~m,α (~n) (8)

and

γ = 1−

∑~m,α

f~m,α (~n) f~m,α (~n+ ~a)∑~m,α

f2~m,α (~n)(9)

SFP model enables one take into account real crystal lattice structure and type of electron-latticeinteraction (through parameter γ). When an electron on Cu−O chain (or CuO2 plane) coupled to the

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B.Ya. Yavidov, P.Yu. Tsyba, S.M. Zholdasova,... .

only z – polarized vibrations of an apical oxygen ions one finds γ = 0.286779. Calculation of γ withtwo-dimensional and three-dimensional isotropic vibrations of an upper chain ions yields 0.392008and 0.727797, respectively. We have calculated optical conductivity of SFP for a chain model ofcuprates (Fig.2 and Fig.3) at different values of γ and EP . As one can see that SPF model enablesus reproduce an anomalous midinfrared optical absorption of cuprates with the band maximumenergies from 0.1 eV up to 0.5 eV [11-15]. Further improvement of agreement to the experimentsmay be achieved by calculating an actual value of γ taking into account more complex structure ofcuprates and other relevant interactions as well as an influence of disorder.

Optical conductivity of SFP and SHP as a function of photon energy ~ν for different values of γ. Frequency ofan apical oxygen ions is equal to 0.075 eV [5] and polaron shift EP = 0.5eV.

Optical conductivity of SFP and SHP as a function of photon energy ~ν for different values of γ. Frequency ofan apical oxygen ions is equal to 0.075 eV [5] and polaron shift EP = 1eV.

It is worthwhile to note that optical curves of SHP and SFP are different. An optical curve ofSFP has a more asymmetric gaussian shape. It is also different from optical conductivity of thelarge Frohlich polarons studied using the effective mass approximation and ignoring detailed crystalstructure. The optical conductivity of large polarons has an asymmetric shape with a threshold atthe optical phonon frequency ωOL [16]. This shape also depends on the many-body (polaron-polaron)interactions [17] and on the magnitude of polaron coupling constant α = (e2/2~ωε̃)

√2m∗ω/~, where

m∗ – is effective mass of an electron, ε̃−1 = ε−1∞ −ε−10 , ε∞ and ε0 high frequency and static dielectric

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Л.Н. Гумилев атындағы ЕҰУ Хабаршы - Вестник ЕНУ им.Л.Н.Гумилева, 2010, №4

constants of the lattice. It was established that optical conductivity spectra of large Frohlich polaronsexhibit relaxed state peaks at moderate and large values of α. However, recently quantum MonteCarlo simulations is showed that at strong-coupling regime peaks due to relaxed excited states are"washed out" by large broadening of these states [18] and optical conductivity spectra of largepolarons obtained in [19] is restricted to the region of α < 6. While the optical conductivity of ourdiscrete model is different, its gross features are more reminiscent of the canonical shape of largepolaron optical conductivity [16].

REFERENCES

1.Landau L.D., Phys. Z. Sowjetunion J. Phys. USSR, v.3, p.664 (1933).2.Holstein T., Ann. Phys. v.8, p.325 (1959); v.8, p.343 (1959).3.Frohlich H., Adv. Phys. v.3, p.325 (1954).4.Alexandrov A.S. and Kornilovitch P.E., Phys. Rev. Lett. v.82, p.807 (1999).5. Timusk T., Homes C.C., and Reichardt W., in Anharmonic Properties of High-TC Cuprates,edited by D.Mihailové et al. (Wolrd Scientific, Singapore, 1995), p.1716.Devreese T. Jozef and Alexandrov S. Alexandre, Rep. Prog. Phys. v.72, p.066501 (2009) andreferences therein.7.Mahan G.D., Many-Particle Physics (Kluwer Academic/Plenum Publishers, 2000).8.Alexandrov A.S. and Yavidov B.Ya., Phys. Rev. B, v.69, p.073101 (2004)9.Alexandrov A.S. and Bratkovsky A.M., J. Phys. Condens. Matter, v.11, p.L531(1999).10.Yavidov B., Zh.Eksp.Teor.Fiz., v.135, No.6, p.1173 (2009).11.Timusk T. and Tanner D.B., in The Physical Properties of High Temperature Superconductors,edited by D.M.Ginsberg (World Scientific, 1989).12.Orenstein J. et al., Phys. Rev. B, v. 42, p.6342 (1990).13.Uchida S. et al., Phys. Rev. B, v.43, p.7942 (1991).14.Schlesinger Z. et al., Phys. Rev. B, v.41, p.11237 (1990).15.Xiang-Xin Bi and Eklund C. Peter, Phys. Rev. Lett. v.70, p.2625 (1993).16.Kartheuser E., Evrard R. and Devreese J.T., Phys. Rev. Lett. v.22, p.94(1969).17.Tempere J. and Devreese J.T., Phys. Bev. B, v.64, p.104504 (2001).18.Mishchenko A.S., et al. Phys. Rev. Lett. v.91, p.236401 (2003).19.Devreese J., Sitter J. De and Goovaerts M., Phys. Rev. B, v.5, p.2367 (1972).

Явидов Б.Я., Цыба П.Ю., Жолдасова С.М., Мырзакулов Р., Джуманов С.Фрелих кiшi полярондарының оптикалық откiзгiштiгiФрелих кiшi полярондарының оптикалық откiзгiштiгi есептелiп, Холстейн кiшi полярондарының шеңберiнде куп-

раттардың оптикалық абсорбциясы қысқаша талқыланған.

Явидов Б.Я., Цыба П.Ю., Жолдасова С.М., Мырзакулов Р., Джуманов С.Оптическая проводимость малых поляронов ФрелихаОптическая проводимость малых поляронов Фрелиха вычислена и сравнена с оптической проводимостью малых

поляронов Холстейна. Кратко обсуждается оптическая абсорбция купратов в рамках модели малых поляронов Фрелиха.

Поступила в редакцию 12.03.10Рекомендована к печати 17.05.10

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