Functoriality and duality in Morse-Conley-Floer homology

In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known~\cite{abbondandoloschwarz, aizenbudzapolski,audindamian, kronheimermrowka, schwarz}, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of isolating map and flow map allows the results to generalize to local Morse homology and Morse-Conley-Floer homology. We prove Poincar\'e type duality statements for local Morse homology and Morse-Conley-Floer homology.Comment: To appear in the Journal of Fixed Point theory and its Application

Morse-Conley-Floer Homology

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue of the Conley index and the Morse-Conley relations. The approach will be referred to as Morse-Conley-Floer homology

Continuation Sheaves in Dynamics: Sheaf Cohomology and Bifurcation

Continuation of algebraic structures in families of dynamical systems is described using category theory, sheaves, and lattice algebras. Well-known concepts in dynamics, such as attractors or invariant sets, are formulated as functors on appropriate categories of dynamical systems mapping to categories of lattices, posets, rings or abelian groups. Sheaves are constructed from such functors, which encode data about the continuation of structure as system parameters vary. Similarly, morphisms for the sheaves in question arise from natural transformations. This framework is applied to a variety of lattice algebras and ring structures associated to dynamical systems, whose algebraic properties carry over to their respective sheaves. Furthermore, the cohomology of these sheaves are algebraic invariants which contain information about bifurcations of the parametrized systems

Scanning spreading resistance microscopy of two-dimensional diffusion of boron implanted in free-standing silicon nanostructures

B implants of 1keV, 1×10¹⁵at.cm⁻² into 125-nm-wide, free-standing Si nanostructures have been characterized using scanning spreading resistancemicroscopy following a 0s, 1050°Canneal in N₂. A curved diffusion front has been observed. B in the center of the ridge diffuses further than at the sides. A similar effect has been observed in SUPREM-IV simulations. It is attributed to a reduction in transient enhanced diffusion close to the vertical surfaces due to recombination of ion-implantation-induced excess Si self-interstitials

Electrical Characterization of Submicrometer Silicon Devices by Cross-Sectional Contact Mode Atomic Force Microscopy

Two contact mode atomic force microscopic (AFM) techniques under ambient conditions are presented for the electrical evaluation of cross sectioned silicon devices. In the first technique, a conductive AFM tip is used as a voltage probe to determine the local potential distribution on the cross section of a silicon device under operation. The electrical potential is measured simultaneously with the surface topography with nanometer resolution and mV accuracy, offering an easy way of correlating topographic and electrical features. A second method, nanometer spreading resistance profiling (nano-SRP), performs localized spreading resistance measurements to determine the spatial distribution of charge carriers in silicon structures. The conversion of the resistance profiles into charge carrier profiles as well as the applied correction factors are discussed in more detail. Both methods are used to map electrical characteristics of state-of-the-art silicon structures

Thickness dependence of the resistivity of Platinum group metal thin films

We report on the thin film resistivity of several platinum-group metals (Ru, Pd, Ir, Pt). Platinum-group thin films show comparable or lower resistivities than Cu for film thicknesses below about 5\,nm due to a weaker thickness dependence of the resistivity. Based on experimentally determined mean linear distances between grain boundaries as well as ab initio calculations of the electron mean free path, the data for Ru, Ir, and Cu were modeled within the semiclassical Mayadas--Shatzkes model [Phys. Rev. B 1, 1382 (1970)] to assess the combined contributions of surface and grain boundary scattering to the resistivity. For Ru, the modeling results indicated that surface scattering was strongly dependent on the surrounding material with nearly specular scattering at interfaces with SiO2 or air but with diffuse scattering at interfaces with TaN. The dependence of the thin film resistivity on the mean free path is also discussed within the Mayadas--Shatzkes model in consideration of the experimental findings.Comment: 28 pages, 9 figure