Stationary and dynamical properties of polarons in the anharmonic Holstein model N. K. Voulgarakis and G. P. Tsironis Department of Physics, University of Crete and Foundation for Research and Technology-Hellas, P. O. Box 2208, 71003 Heraklion, Crete, Greece ~Received 17 June 2000; published 11 December 2000! We study the semiclassical Holstein model with a hard nonlinear on-site potential in one, two, and three dimensions. Using stationary solutions, we obtain the ground-state phase diagram as a function of the param- eter characterizing the nonlinearity as well as the electron-phonon coupling. The basic result is that in the presence of the nonlinear on-site potential medium and large polaron formation is also possible in two and three dimensions. Linearizing the equations of motion around the stationary solution, we calculate the normal modes and study their stability. Normal mode analysis shows the existence of a low-frequency pinning mode not only in one dimension ~1D! but also in 2D and 3D. This result enables us to construct numerically one- and two-dimensional moving polarons and study their mobility properties. The dramatic reduction of the polaron effective mass is the most characteristic effect of the lattice nonlinearity. DOI: 10.1103/PhysRevB.63.014302 PACS number~s!: 63.20.Kr, 63.20.Ry, 71.38.2k I. INTRODUCTION Properties of electron self-localization in crystal lattices due to interaction with phonons constitutes one of the funda- mental problems in condensed matter physics. This question has long history starting almost seven decades ago, when Landau 1 first introduced the concept of polaron, i.e, the qua- siparticle formed by an electron accompanied by its own lattice distortion. In the preceding decades many theoretical works 2–13 have been devoted to the phenomenon of polaron formation. In general, the analytical studies are based on the adiabatic treatment of continuous or discrete models in the weak or strong electron-phonon coupling, following either variational or perturbation methods. Analytical results are in good agreement with the numerical calculations of finite- cluster exact diagonalizations 12 or quantum Monte Carlo methods. 13 The general result is that while in one dimension ~1D! the polaron ~small or large! is always the ground state at any nonvanishing coupling strength, in higher dimensions only small polaron formation is possible and a minimum coupling is required for this purpose. Thus, due to their small size two and three dimensional polarons are expected to be pinned rather than mobile. In one-dimensional large polarons on the other hand, where the continuum limit is valid, Davydov 14 has shown that solitary states are formed that can propagate uniformly in molecular chains. Furthermore, over the last two decades, real-time numerical simulations 15–17 in one-dimensional models gave additional insight to the ques- tion of polaron formation and its dynamical properties. It should be mentioned that all the above studies are re- stricted in the special case where the phonon on-site potential is quadratic. This assumption is sufficient to describe the main aspects of the polaron problem in the weak coupling limit, where the presence of the electron does not displace substantially the oscillators from their equilibrium position and the linear approximation is adequate. In the opposite limit of strong electron-lattice interaction, the oscillators are greatly perturbed from the equilibrium position and higher terms of the potential expansion should be taken into ac- count. Although it would be interesting physically to con- sider such nonlinear effects, this aspect has not been ad- dressed sufficiently and only in low dimensions. 18–22 The main aim of this work is the study of the Holstein model 4 including on-site lattice anharmonicity in one, two, and three dimensions. We have assumed only hard lattice nonlinearity since, as explained below, it results in more in- teresting dynamical behavior of the polaron. The organiza- tion of this paper is the following: In Sec. II we introduce the anharmonic Holstein model and derive the semiclassical equations of motion. In Sec. III we numerically calculate the adiabatic ground state and discuss its modifications effected by the lattice nonlinearity. The following two sections are devoted to the polaron dynamical properties. More particu- larly, in Sec. IV, we consider small oscillations around the adiabatic polaron solution and calculate its normal modes focusing our interest in the lower frequencies. The linear stability of the polaron solution is checked and the investi- gation of the possibility of having mobile polarons is per- formed in Sec. V. Finally, in Sec. VI we summarize and discuss our results. II. THE MODEL In order to study the effects of lattice nonlinearity in the Holstein model we consider the following Hamiltonian: H 5H el 1H lat 1H int , ~1! The first term describes an electron in the tight binding ap- proximation H el 52J ( i 51 N ( n i c i ² c n i , ~2! where c i ² and c i are the electron creation and annihilation operators, respectively, at site i, and J is the hopping integral between the site i and its nearest neighbors n i . The second term is the Hamiltonian of N identical anharmonic Einstein oscillators with mass M: PHYSICAL REVIEW B, VOLUME 63, 014302 0163-1829/2000/63~1!/014302~7!/$15.00 ©2000 The American Physical Society 63 014302-1