Two-site model for a small polaron: Mass renormalization and optical conductivity
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LECTRONIC PROPERTIES OF SOLID
TwoSite Model for a Small Polaron: Mass Renormalization and Optical Conductivity1 B. Yavidov Institute of Nuclear Physics, Academy of Sciences of Uzbekistan, Tashkent, 100214 Uzbekistan email: [email protected] Received February 2, 2009
Abstract—The renormalization of the mass of an electron interacting with many ions of a lattice via the long range (Fröhlich) electron–phonon interaction and optical absorption of electrons are studied at zero tem perature. Ions are assumed to be isotropic threedimensional oscillators. The optical conductivity and the renormalized mass of small adiabatic Fröhlich polarons are calculated and compared with those of small adi abatic Holstein polarons. PACS numbers: 63.20.kd, 63.20.kg, 71.38.Ht, 71.38.Cn, 78.67.n DOI: 10.1134/S1063776109060132 1
1. INTRODUCTION Polarons have been extensively studied since a sem inal paper of Landau [1]. They are divided into small and large polarons in accordance with the size of their wave function. In the first case, a carrier is coupled to intramolecular vibrations and selftrapped on a single site. Its size is the same as the size of the phonon cloud, both are about the lattice constant (socalled small Holstein polaron, SHP). In the case of large polarons, the size of the polaron is also the same as the size of the phonon cloud, but the polaron extends over the dis tance of many lattice constants. Polarons with a very different internal structure were introduced in [2, 3]. They were called small Fröhlich polarons (SFP). An SFP size is about the lat tice constant, but its phonon cloud spreads over the whole crystal. Within the model in [3], the renormal ized mass appears to be much smaller compared with that in the canonical Holstein model [4]. Recently [5], this model has been extended to the adiabatic limit and it was found that the SFP mass is renormalized much less than the mass of the SHP in this limit. An electron interacting with vibrations of a chain of ions polarized perpendicular to the chain was considered in [5]. The model was introduced in order to mimic high Tc cuprates, where inplane CuO2 carriers are strongly coupled to the caxis polarized vibrations of apical oxygen ions [6]. However, apical ions vibrate in all directions. To describe a more realistic case, we con sider an electron hopping between two sites and inter acting with threedimensional (3D) vibrations of ions of the chain, as shown in Fig. 1a. In addition, we cal culate the optical conductivity of the system to show that the longrange electron–phonon interaction
qualitatively changes the polaron hopping and the optical conductivity compared with those in the Hol stein model, Fig. 1b. 2. MASS RENORMALIZATION We first derive an analytic expression for the renor malized hopping integral of an SFP in the nonadia batic limit and in the adiabatic limit in order to eluci date the effect of ion’s longitudinal vibrations in the renormalized hopping integral. The Hamiltonian of the model is [2–5] H = H e + H ph + H e–ph ,
(1)
whe
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