Studies of static and driven lattice systems with the phase field crystal model

Study of static and dynamical properties of two dimensional lattice systems has become an important topic in nanoscience. Competition between the intrinsic ordering of adsorbed layers and the underlying substrate often leads to the appearance of spatially modulated structures. Important examples include spin density waves, charge density waves, vortex lattices in superconducting films with pinning centers and weakly adsorbed monolayers. In addition to static properties of such systems, their dynamics under external driving force is of great importance in tribology to understand the microscopic origins of friction. This Thesis deals with the static and dynamic properties of adsorbed layers under the influence of a driving force using an extended version of the Phase Field Crystal Model. In the case of periodic potentials, the various commensurate phases are described in detail and the complete phase diagram is mapped out as a function of pinning strength and lattice mismatch. For finite temperatures, Monte Carlo simulations are used to determine the phase diagram as a function of temperature. For one of the commensurate phases behavior is found consistent with the Ising universality class. The results concerning the effect of disordered pinning potential are also presented. The nonlinear response to a driving force on an initially pinned commensurate phase is then studied via overdamped dynamic equations of motion for different values of mismatch and pinning strengths. For large pinning strength the driven depinning transitions are continuous, and the sliding velocity varies with the force from the threshold with power-law exponents in agreement with analytical predictions. Transverse depinning transitions in the moving state are also found in two dimensions. Surprisingly, for sufficiently weak pinning potential we find a discontinuous depinning transition with hysteresis even in one dimension under overdamped dynamics. Also, structural changes of the system are characterized in some detail close to the depinning transitions