First-principles study of the pentacene/Cu(1 1 1) interface: Adsorption states and vacuum level shifts

We have studied the interaction of pentacene with a Cu(1 1 1) surface using density functional theory (DFT) within a generalized gradient approximation (GGA) and the van der Waals density functional [vdW-DF, M. Dion, H. Rydberg, E. Schröder, D.C. Langreth, B.I. Lundqvist, Phys. Rev. Lett. 92 (2004) 246401]. The adsorption energy is accurately predicted by vdW-DF, while the equilibrium distances between pentacene and the metal substrate (Z_C) are overestimated by both GGA and vdW-DF. The work function changes depend significantly on Z_C. The experimental work function change can be successfully reproduced by GGA if the experimentally reported adsorption geometry is used, whereas the magnitude of the work function change is underestimated if calculated adsorption geometries are applied. We examined the IDIS model [H. Vázquez, R. Qszwaldowski, P. Pou, J. Ortega, R. Pérez, F. Flores, A. Kahn, Europhys. Lett. 65 (2004) 802] to compare it with the GGA results. The interface dipoles estimated by the IDIS model fairly agree with the GGA results, provided that the adsorption distance is large. On the other hand, they tend to deviate from the GGA results as the adsorption distance becomes smaller, where back donation from the metal surface to the adsorbate occurs. Our analysis reveals that at experimentally reported metal–organic distance, back donation is significant enough to induce polarization of pentacene molecules perpendicular to the surface, which leads to a reduction of the work function. Thus, at the experimentally reported metal–organic distance, the work function change estimated by a simple IDIS model deviates from that calculated by self-consistent GGA calculations. We also found that at the experimentally reported metal–organic distance, the transferred electrons create weak chemical bonds between pentacene and the Cu(1 1 1) surface, illustrating the reactive nature of pentacene.     

Dynamical analysis of Mathieu equation with two kinds of van der Pol fractional-order terms

In this paper the dynamics of Mathieu equation with two kinds of van der Pol (VDP) fractional-order terms is investigated. The approximately analytical solution is obtained by the averaging method. The steady-state solution, existence conditions and stability condition for the steady-state solution are presented, and it is found that the two kinds of VDP fractional coefficients and fractional orders remarkably affect the steady-state solution, which is characterized by the additional damping coefficient (ADC) and additional stiffness coefficient (ASC). The comparisons between the analytical and numerical solutions verify the correctness and satisfactory precision of the approximately analytical solution. The presented typical amplitude–frequency curves illustrate the important effects of two kinds of VDP fractional-order terms on system dynamics. The application of two VDP fractional-order terms in vibration control is discussed. At last, the detailed results are summarized and the conclusions are made.