Coupled Mathematical Models for Physical and Biological Nanoscale Systems and Their Applications

We show that the nonlinear bifurcations found by simulations in single quantum wells in the terahertz regime [5, 6, 25] also occur in semiconductor superlattices (SSLs) in the gigahertz range [2, 3, 11]. The only exception is the second Hopf bifurcation to quasi-periodic orbits on a torus. The advantage of experiments on SSLs in the gigahertz range is that the experiments can be conducted at room temperature and a chaotic oscillator due to the random dressing of a period two-orbit has already been measured [31, 32]. We determine [42] that the route to chaos for SSLs in the sequential tunneling regime is the period doubling cascade. Shorter (10-period) superlattices are observed to exhibit faster oscillations compared with longer (50-period) ones. Two plateaus are observed as functions of the voltage bias, and intrinsically chaotic dynamics on the second plateau are possible only for shorter SSLs, while the dynamics in the first plateau contain intrinsic chaos only for longer (N > 50) SSLs [21].

Electrons move along potential or thermal gradients. In the presence of a global gradient, applied e.g. to the two terminals of a conductor, this induces electric charge and heat currents. They can also flow between two equilibrated terminals (at the same voltage and temperature) if detailed balance is broken in some part of the system. A minimal model involving two metallic islands in series is introduced whose internal potential and temperatures can be externally modulated. The conditions for a finite electric flow are discussed.

Non-perturbative approaches in nanoscience are discussed. Traditional applications of these approaches cover description of charge transport and optical phenomena in nano-scale systems. We focus on finite-size effects in spin systems near the critical point, based on Monte Carlo (MC) method and some analytical arguments. We have performed MC simulations of the 3D Ising model for small, as well as large linear lattice sizes up to \(L=2560\), providing a numerical evidence for a recent challenging prediction, according to which the asymptotic decay of corrections to finite-size scaling is remarkably slower than it was expected before. Our approach along with several other non-perturbative approaches, like, e.g., the non-perturbative nonequilibrium Greens functions (NEGF) method, reveals a potential application of non-perturbative methods to nanoscience and nanotechnology through condensed matter physics, including semiconductor physics and physics of disordered systems like spin glasses.

A 3D continuum model is used to find optical and acoustic phonon fields in zincblende GaAs. We start out using continuum elastic differential equations and the Maxwell-Poisson equation to describe dynamic lattice strain and internal strain effects accounting for the full crystal symmetry of zincblende GaAs. The analytical model is derived in detail in a first-principles analysis. Our results reveal that for a slab of crystal GaAs grown along the [001] direction the mechanical displacements along the x and z directions \(u_x, u_z\) couple while the mechanical displacement \(u_y\) couple solely to the electric field E by virtue of piezoelectricity. As a consequence optical and acoustic phonon fields are inherently coupled due to piezoelectricity and acoustic and optical phonon modes must be found by solving simultaneously the full elastic solid and electric governing equations and the relevant (elastic and electric) boundary conditions. We then derive phonon dispersion curves for a GaAs slab and compare cases with and without anisotropy and piezoelectricity and show that neglecting the latter in the description of both acoustic and optical modes of GaAs, as is done in many classical descriptions, is a too crude approximation. We finally discuss two novel results: (i) confined coupled acousto-optical \(u_y-u_\phi \) modes cannot exist in piezoelectric media except at certain discrete \(q_x\) wavenumber values, and (ii) piezoelectricity prohibits the existence of optical phonon fields at the LO phonon frequency. The model presented is general and can be applied to other materials and other crystal structures.

Angiogenesis is a complex multiscale process by which diffusing vessel endothelial growth factors induce sprouting of blood vessels that carry oxygen and nutrients to hypoxic tissue. There is strong coupling between the kinetic parameters of the relevant branching—growth—anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. A hybrid mesoscale tip cell model involves stochastic branching, fusion (anastomosis) and extension of active vessel tip cells with reaction-diffusion growth factor fields. Anastomosis prevents indefinite proliferation of active vessel tips, precludes a self-averaging stochastic process and ensures that a deterministic description of the density of active tips holds only for ensemble averages over replicas of the stochastic process. Evolution of active tips from a primary vessel to a tumor adopts the form of an advancing soliton that can be characterized by ordinary differential equations for its position, velocity and a size parameter. A short review of other angiogenesis models and possible implications of our work is also given.

From multicellular tissues to bacterial colonies, three dimensional cellular structures arise through the interaction of cellular activities and mechanical forces. Simple bacterial communities provide model systems for analyzing such interaction. Biofilms are bacterial aggregates attached to wet surfaces and encased in a self-produced polymeric matrix. Biofilms in flows form filamentary structures that contrast with the wrinkled layers observed on air/solid interfaces. We are able to reproduce both types of shapes through elastic rod and plate models that incorporate information from the biomass production and differentiation processes, such as growth rates, growth tensors or inner stresses, as well as constraints imposed by the interaction with environment.

We analyse the unfolding pathway of biomolecules comprising several independent modules in pulling experiments. In a recently proposed model, a critical velocity \(v_{c}\) has been predicted, such that for pulling speeds \(v>v_{c}\) it is the module at the pulled end that opens first, whereas for \(v<v_{c}\) it is the weakest. Here, we introduce a variant of the model that is closer to the experimental setup, and discuss the robustness of the emergence of the critical velocity and its dependence on the model parameters. We also propose a possible experiment to test the theoretical predictions of the model, which seems feasible with state-of-art molecular engineering techniques.

This paper presents a heuristic derivation of a geometric minimum action method that can be used to determine most-probable transition paths in noise-driven dynamical systems. Particular attention is focused on systems that violate detailed balance, and the role of the stochastic vorticity tensor is emphasized. The general method is explored through a detailed study of a two-dimensional quadratic shear flow which exhibits bifurcating most-probable transition pathways.

Thin-film modules of all technologies often suffer from performance degradation over time. Some of the performance changes are reversible and some are not, which makes deployment, testing, and energy-yield prediction more challenging. Manufacturers devote significant empirical efforts to study these phenomena and to improve semiconductor device stability. Still, understanding the underlying reasons of these instabilities remains clouded due to the lack of ability to characterize materials at atomistic levels and the lack of interpretation from the most fundamental material science. The most commonly alleged causes of metastability in CdTe device, such as “migration of Cu,” have been investigated rigorously over the past fifteen years. Still, the discussion often ended prematurely with stating observed correlations between stress conditions and changes in atomic profiles of impurities or CV doping concentration. Multiple hypotheses suggesting degradation of CdTe solar cell devices due to interaction and evolution of point defects and complexes were proposed, and none of them received strong theoretical or experimental confirmation. It should be noted that atomic impurity profiles in CdTe provide very little intelligence on active doping concentrations. The same elements could form different energy states, which could be either donors or acceptors, depending on their position in crystalline lattice. Defects interact with other extrinsic and intrinsic defects; for example, changing the state of an impurity from an interstitial donor to a substitutional acceptor often is accompanied by generation of a compensating intrinsic interstitial donor defect. Moreover, all defects, intrinsic and extrinsic, interact with the electrical potential and free carriers so that charged defects may drift in the electric field and the local electrical potential affects the formation energy of the point defects. Such complexity of interactions in CdTe makes understanding of temporal changes in device performance even more challenging and a closed solution that can treat the entire system and its interactions is required. In this book chapter we first present validation of the tool that is used to analyze Cu migration in single crystal (sx) CdTe bulk. Since the usual diffusion analysis has limited validity, our simulation approach presented here provides more accurate concentration profiles of different Cu defects that lead to better understanding of the limited incorporation and self-compensation mechanisms of Cu in CdTe. Finally, simulations are presented that study Cu ion’s role in light soaking experiments of CdTe solar cells under zero-bias and forward-bias stress conditions.

In this work, we have developed a multiscale coupling method between the multiscale micromorphic molecular dynamics (MMMD) and the nonlinear finite element method. The multiscale micromorphic molecular dynamics (MMMD) is a three-scale non-equilibrium molecular dynamics that span from microscale to mesoscale and to macroscale. A multiscale computational algorithm is formulated to couple the continuum scale equations of motion in terms of finite element method (FEM) with the coarse scale molecular dynamics of MMMD. To validate the computational formulation, we apply the multiscale coupling method to simulate nano-indentation of silicon crystals.

We present a general method for the electronic characterization of aperiodic 2D materials using ab-initio tight binding models. Specifically studied is the subclass of twisted, stacked heterostructures, but the formalism provided can be implemented for any 2D system without long-range interactions. This new method provides a multi-scale approach for dealing with the ab-initio calculation of electronic transport properties in stacked nanomaterials, allowing for fast and efficient simulation of multi-layered stacks in the presence of twist angles, magnetic field, and defects. We calculate the electronic density of states in twisted bilayer systems of graphene and MX\(_2\) transition metal dichalcogenides (TMDCs). We comment on the interesting features of their density of states as a function of twist-angle and local configuration and how these features are experimentally observable. These results support the bilayer twist-angle as a new variable for controlling electronic properties in artificial nanomaterials (“Twistronics”).

The main objective of this review chapter is to give the reader a practical toolbox for applications in quantitative biology and computational drug discovery. The computational technique of molecular dynamics is discussed, with special attention to force fields for protein simulations and methods for the calculation of solvation free energies. Additionally, computational methods aimed at characterizing and identifying ligand binding pockets on protein surfaces are discussed. Practical information about available databases and software of use in drug design and discovery is provided.

In this chapter, we present the formation and modeling techniques of two important sensing devices built out of engineered artificial membranes: the Ion Channel Switch (ICS) biosensor, and the Electroporation Measurement Platform (EMP). The ICS biosensor can be used to detect femto-molar concentrations of analyte species in an electrolyte solution, and the EMP is used to study the dynamics of electroporation in engineered membranes. The engineered membrane in the ICS and EMP are design to mimic the electrophysiological properties of real cell membranes. Common to both platforms is the bioelectronic interface for performing electrical measurements. Experimental measurements of the two platforms are performed by estimating the current response of the engineered membrane which depends on the charging dynamics at the bioelectronic interface and membrane, as well as dynamics of aqueous pores and conducting ion-channels in the membrane.