Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figure

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

Phase formation and thermal stability of ultrathin nickel-silicides on Si(100)

The solid-state reaction and agglomeration of thin nickel-silicide films was investigated from sputter deposited nickel films (1-10 nm) on silicon-on-insulator (100) substrates. For typical anneals at a ramp rate of 3 degrees C/s, 5-10 nm Ni films react with silicon and form NiSi, which agglomerates at 550-650 degrees C, whereas films with a thickness of 3.7 nm of less were found to form an epitaxylike nickel-silicide layer. The resulting films show an increased thermal stability with a low electrical resistivity up to 800 degrees C

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

Understanding the effect of confinement in scanning spreading resistance microscopy measurements

Scanning spreading resistance microscopy (SSRM) is a powerful technique for quantitative two-and three-dimensional carrier profiling of semiconductor devices with sub-nm spatial resolution. However, considering the sub-10 nm dimensions of advanced devices and the introduction of three-dimensional architectures like fin field effect transistor (FinFET) and nanowires, the measured spreading resistance is easily impacted by parasitic series resistances present in the system. The limited amount of material, the presence of multiple interfaces, and confined current paths may increase the total resistance measured by SSRM beyond the expected spreading resistance, which can ultimately lead to an inaccurate carrier quantification. Here, we report a simulation assisted experimental study to identify the different parameters affecting the SSRM measurements in confined volumes. Experimentally, the two-dimensional current confinement is obtained by progressively thinning down uniformly doped blanket silicon on insulator wafers using scalpel SSRM. The concomitant SSRM provides detailed electrical information as a function of depth up to oxide interface. We show that the resistance is most affected by the interface traps in case of a heterogeneous sample, followed by the intrinsic resistance of the current carrying paths. Furthermore, we show that accurate carrier quantification is ensured for typical back contact distances of 1 μm if the region of interest is at least nine times larger than the probe radius. © 2020 Author(s)

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