Dielectric function spectra from a nondegenerate polaron gas

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Dielectric function spectra from a nondegenerate polaron gas

C.A. Barboza

a

_{, E.L. Albuquerque}

a,∗

_{, C. Chesman}

a

_{, V.N. Freire}

b

a_{Departamento de Física, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil} b_{Departamento de Física, Universidade Federal do Ceará, 60455-970 Fortaleza-CE, Brazil}

Received 6 January 2007; accepted 26 January 2007 Available online 7 February 2007

Communicated by R. Wu

Abstract

A Green function formalism is employed to perform an analytical calculation of the dielectric function due to an electron–phonon interaction within a Hartree–Fock approximation. Our theoretical approach is based on the polar or Fröhlich type of electron–phonon interaction, with the electronic hopping between the first-neighbor, to characterize the polaron model. Our numerical results for the dielectric function profile show resonant peaks, all of them at lower frequencies than the main one, attributed to the polaron crossover regime between the quasi-free-electrons and the small polarons.

PACS: 71.38.-k; 71.45.Gm; 71.38.Ht; 77.22.Ch

Keywords: Small polarons; Dielectric response function; Electron–phonon interaction; Linear response function

Polaronic phenomena have been studied in conducting polymers for decades (for a review see Ref.[1]). Although there are many ways of addressing the polaron dynamics[2,3], the dielectric function measurements can provide important information because a hopping process of small polarons (strong electron–phonon coupling) has a high probability of involving a dielectric relaxation process[4]. Furthermore, the dielectric function (ω), measuring the response to an external applied electric field, is an important tool for the understanding of many-body effects in the normal-state polaron system[5]. It is therefore natural to investigate its behavior due to a realistic electron–phonon coupling in one-dimensional system (like conducting polymers), in which no adiabatic separation between the electronic and phonon degrees of freedom is made[6,7].

Even considering that exact diagonalization of vibrating molecules coupled with electrons[8]revealed an excellent agreement with analytical result using the Holstein molecular crystal model[9], which is the simplest model to describe polarons, we have chosen our theoretical approach for studying such system based on the polar or Fröhlich type of electron–phonon interaction[10], namely

(1) H= He+ Hph+ He_−ph,

where Heis the electronic Hamiltonian

(2) He= n ωencn†cn+ n,m (n_=m) Vnmc†ncm,

Hphis the Hamiltonian of the phonon bath,

(3) Hph= k ωkb†_kbk, * _{Corresponding author.}

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and He_−phis the electron–phonon interaction, (4) He−ph= nk γnkcn†cn b_k†+ bk.

Here c†n (cn)creates (annihilates) an electron in the nth electronic state with energy ωen, while b†k (bk)creates (annihilates) a phonon in mode k with energy ωk. Also, Vnmis the hopping potential, and the sum over m is limited to the nearest neighbors. This is a general Hamiltonian, describing the polaronic states beyond the Holstein polaron model. Troughout this Letter we use the units in which_{¯h = 1.}

The electronic Hamiltonian can describe a scattering process in which the electron starts in one continuous manifold and ends in another, and the states n belong to the target that causes the scattering process. These states may be the eigenstates of the target Hamiltonian. The phonon Hamiltonian represents the thermal environment as a harmonic-phonon bath, with the states k taken to be different manifolds of continuous-scattering states. The coupling between the electronic system and this bath is assumed in Eq.(4)

to originate from a target-state dependent shift in the equilibrium position of each phonon mode. An exact solution to this scattering problem can thus be obtained for the particular case where the target is represented by a single state n and the phonon bath contains only one oscillator of frequency ω, which means a single state k. In this situation the Hamiltonian(1)takes the simpler form

(5) H= ωec†_ncn+ V m=n c†_ncm+ ωkb_k†bk+ γnkcn†cn b_k†+ bk,

which will be consider from now on.

In the presence of an external electric field E, linear response theory[11,12]yields a dielectric function (ω) given by: (6) (ω)= 1 +P (ω) E = ∞− π Z_e2 ωe c+ c†,c+ c†_ω, or (7) (ω)= ∞−π Z 2 e ωe c; c† ω+ c; c†_−ω,

where the last step uses a standard identity[13]. Here,· · ·ωis the Fourier transform of the retarded commutator Green function of the arguments shown, Zeis the electron’s oscillator strength, and _∞is the high frequency dielectric constant. We have neglected the interaction of the radiation field with the phonon states (2nd order dipole term).

The Green functionc; c†ωcan be determined using its equations of motion approach[14], yielding:

(8) id dt cn; c_n†_ω= (1/2π) + ωecn, cn† ω+ V cn−1; c†n ω+ cn+1; cn† ω + γnkcnbk; c†n ω+ cnb†k; c † n ω . On the other hand,

(9) id dt cn−1; c†n ω= (1/2π) + ωe cn; c_n†_ω+ Vcn−1; c†n ω+ cn+1; cn† ω + γnkcnbk; c†n ω+ cnb†_k; c†n ω , (10) id dt cn+1; c†n ω= δ(t − t)+ ωe cn; c_n† ω+ V cn−1; c†n ω+ cn+1; cn† ω + γnkcn bk+ b†_k ; c† n ω, (11) id dt cnbk; cn† ω= (ωe+ ωk) cnbk; cn† ω+ V (cn−1+ cn)bk; c†n ω + γnkcnc†ncn; c † n ω+ cn bk+ bk† bk; c†_n_ω, (12) id dt cnb†k; c † n ω= (ωe− ωk) cnb†k; c † n ω+ V cn−1b†k; c † n ω+ cn+1b†k; c † n ω + γnkcn bk+ b†_k bk; c_n†_ω−cnc†ncn; c†n ω .

As it is discussed by Zubarev[15], the simplest way to decouple the above Green functions is by using the Hartree–Fock method, in which the corresponding approximation for these triple Green functions leads to the generalized Fock method[16]. This approx-imation, although fairly coarse, is however more appropriated than the canonical transformation one, related to the evaluation of the average energy from which one can afterwards reduce the thermodynamic functions[17]. Of course our method is not suitable if one consider correlation between electrons leading to superconducting state, but it reflects a number of interesting properties of a system of interacting particles, as those considered in this work. Therefore, taking into account the Hartree–Fock decoupling[15]

(13) cn bk+ b†_k bk; c†n ω= b†_kbk ω cn; c†_n_ω, (14) cnc†ncn; cn† ω= 1−c†_ncn ω cn; c†_n_ω, (15) cn bk+ b†k b_k†; c†_n_ω=1+b_k†bk ω cn; c†_n_ω,

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Fig. 1. The dielectric function profile, as expressed by the function f (ω), plotted versus the reduced frequency ω/ωe, considering the following values of the

electron–phonon coupling: (a) γnk/ωe= 0; (b) γnk/ωe= 0.5; (c) γnk/ωe= 1.0.

we find, after a bit of algebra

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cn; c_n†_ω=1+ 2π(Ξ + Υ ) κ− η − μ , with Ξ, Υ, κ, η, μ defined inAppendix A.

By substitution of Eq.(16)into Eq.(7), it is easy to get the contribution of the electron–phonon coupling to the dielectric function by means of the function f (ω)= (πZ2e)−1[(ω) − ∞], which is plotted versus the reduced frequency ω/ωe inFig. 1. Unlike the usual strong-coupling situation, in which the polaron absorption is mainly in a frequency region much higher than the phonon frequency ωk= 1.2 THz [18], we will focus our attention on the intermediate coupling regime, near the adiabatic limit, where interesting properties can be expected in the optical spectra. Therefore, the following values of the electron–phonon coupling is considered here: (a) no electron–phonon coupling, i.e., γnk/ωe= 0; (b) γnk/ωe= 0.5; (c) γnk/ωe= 1.0. We have taken the hopping potential V /ωeand the average electrons’ occupational numberc_k†ck equal to 1, with ωe= 4.0 THz. Besides, expecting that our results are relevant at low enough density, i.e., a dilute nondegenerate polaron gas, where exchange effects can be neglected, we assume here that the average occupation number of the lattice phonon is given by the Planck distribution

(17) b_k†bk =exp(ωk/kBT )− 1 ₋₁ .

This average occupation number is approximately equal to one when ωk/kBT = 1[19], and this value is considered here for the numerical behavior of the dielectric function, neglecting any temperature effect.

When there is no electron–phonon coupling,Fig. 1(a) shows a typical pattern of the dielectric function appropriated to describe an ionic crystal, with a resonant peak at ω/ωe= 6, as expected. Considering an intermediate electron–phonon coupling regime, more resonant peaks take place, all of them at lower frequencies than the main resonant peak at ω/ωe= 6, tending to zero at higher frequencies (seeFigs. 1(b) and 1(c)). This behavior can be attributed to the polaron crossover regime between the quasi-free-electrons and the small polarons, which takes place at intermediate values of the electron–phonon coupling strength. Indeed, if we restrict our focus to the adiabatic regime, this crossover (which takes place when the polaron energy is of the order of the free electron bandwidth) is characterized by the successive opening of energy gaps in the low-frequency part of the spectral density as the electron–phonon coupling increases. As a consequence there are multiple resonant peaks, as shown by the dielectric function spectra depicted inFig. 1(b) (for γnk/ωe= 0.5) andFig. 1(c) (for γnk/ωe= 1.0), respectively.

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In summary, we have studied here the behavior of the dielectric function, as a response to an applied electric field, in the presence of an electron–phonon interaction. The method of calculation used a Green function equation of motion formalism, in which the electron–phonon coupling were considered taking into account a simpler form derived from the Fröhlich Hamiltonian, and decoupled using a Hartree–Fock approximation. The dielectric function spectra show resonant peaks, all of them at lower frequencies than the main one, attributed to the polaron crossover regime between the quasi-free-electrons and the small polarons. Experimental investigation of the results presented here can be done in BaO–TiO2–SnO2ternary system by using terahertz time-domain spectroscopy, far-infrared (FIR) reflectivity, and Raman-scattering measurements[20]. By instance, the numerical analysis of the far-infrared reflection spectra can be made using a fitting procedure based on the factorized form of the dielectric function, as those described in this work[21].

Acknowledgements

We would like to thank partial financial support from the Brazilian Research Agencies CNPq-Rede Nanobioestruturas and FINEP-CTInfra. Appendix A (A.1) Ξ= yV+ γnkV[1 − V + γnk(N−+ N+)](1 + b † kbk) (ω− V ) − [γnkV (M₊+ M₋)+ γ_nk2V (N₋+ N₊)](1 − c†ncn) , (A.2) Υ= γnk1−c†_ncn  y(M₊+ M−) ω− V + γnkV (γnk+ ω/2π)(N−+ N+) ω2_{− 2ωV} D, (A.3) κ= ω − ωen− yV ωen (ω− V ) − γnkV[M₊+ M₋+ γnk(N₋+ N₊)](1 − cn†cn) , (A.4) η=1+b†_kbk 1− V + γnkV (N₋+ N₊)+ γnk yωen(M++ M−)(1− cn†cn) ω− V , (A.5) μ=γnkV ωωen(N−+ N+)(1− c † ncn) (ω2_{− 2ωV )D} , where (A.6) y= ω(ω− V ) (ω2_{− 2ωV )}+ 1+ V ω(ω− V ) (ω− V )(ω − V )2_{− V}2 , (A.7) N_±= 1

(ω± ωk− V )[ω ± ωk+ ωen] − V ωen[(1 + V A−)+ A−(ω± ωk− V )],

(A.8)

M_±= V B∓(ω− ωk− V ± V )

(ω∓ ωk− V )(ω ∓ ωk± ωen)− [V ωen(1+ V A_∓)+ V ωenA_∓(ω∓ ωk− V )],

(A.9) D= 1− γnk1−c†_ncn y(M₊+ M−) ω− V + γnkV (N−+ N+) ω2_{− 2ωV} , (A.10) A_±= 1+ V ω± ωk− V ± V2, (A.11) B_±=γnkA± 1+ V . References

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