Physica D 159 (2001) 71–90 Electron self-trapping in a discrete two-dimensional lattice L. Brizhik a , A. Eremko a , B. Piette b , W.J. Zakrzewski b,∗ a Bogolyubov Institute for Theoretical Physics, 03143 Kiev, Ukraine b Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK Received 25 June 2001; received in revised form 2 August 2001; accepted 2 August 2001 Communicated by F.H. Busse Abstract We study analytically and numerically the electron–phonon interaction in an isotropic two-dimensional lattice. We show that the properties of the system depend crucially on the electron–phonon coupling constant and that the system admits stationary soliton-like solutions when the coupling constant takes numerical values within some finite interval. We predict the lower critical value of the coupling constant and study some properties of the corresponding solutions. We estimate the period of oscillation of the slightly excited field configurations. We also prove that above the upper critical value of the coupling constant the regime of strong localisation (small polaron) takes place. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear schr ¨ odinger; Electron self-trapping; Solitons; Lattice 1. Introduction A wide class of organic and inorganic materials was recently synthesised. Some of them possess strongly anisotropic physical characteristics and reveal many unusual properties, due to which some of these compounds find numerous applications in modern technologies and attract an ever increasing scientific interest. Many applica- tions in nanotechnologies, including quantum computation, information processing, sensing, etc., are based on the discoveries of charge and spin density waves, high-temperature superconductivity in perovskites, superconductivity in low-dimensional (LD) organic and inorganic materials, including the recently discovered superconductivity in magnesium diboride, MgB 2 [1] and in organic polymer films [2]. Generally speaking, the study of electrical and optical properties in LD materials has revealed the important role of collective excitations due to the electron–phonon interaction (EPI) which can result in the creation of lo- calised modes or self-trapped states. As was predicted by Davydov and Kislukha [3], in one-dimensional (1D) molecular chains such self-trapped excitations, e.g. an Amide-I excitation in -helical proteins, possess soliton properties. The radiative life-time of such soliton-like excitations is much higher than that of excitons [4]. The self-trapping is described by the nonlinear Schr¨ odinger equation (NLSE) where the nonlinearity arises from the ∗ Corresponding author. E-mail addresses: brizhik@bitp.kiev.ua (L. Brizhik), eremko@bitp.kiev.ua (A. Eremko), b.m.a.g.piette@durham.ac.uk (B. Piette), w.j.zakrzewski@durham.ac.uk (W.J. Zakrzewski). 0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII:S0167-2789(01)00332-3