# Industrial Mathematics

Chairmen: A. M. Anile and A. Fasano
________ Centro de Matemática Aplicada
Instituto Superior Técnico
June 7-12, Lisbon (Portugal)

1. Mathematical Modeling of Semicondutor Devices and Circuits. Industrial Motivation
Technology CAD is one of the main activities in many R&D centers of microelectronics companies. Numerical simulation is routinely used in the design of microelectronics devices such as bipolar transistors, MOSFET, etc..and in the ensuing design of electric circuits. The performance of the simulation chain device-circuits simulation is essential for the future developments in the microelectronics industry.

2. Mathematical Modeling of Industrial Processes for the Production of Advanced Materials
In the last decades these has been an impressive acceleration in the production of new materials with particular mechanical properties. A remarkable development has been observed in the sectors of polymers and of composite materials, which have become fundamental branches of modern technology. Both the chemistry and the physics involved are a remarkable sources of challenging problems. Such a complexity is of course present also at the level of mathematical modeling, which is a crucial step in the investigation of the technical processes and a basic tool for numerical simulation. The aim of these lectures is to illustrate the most relevant mathematical aspects of such highly sofisticated technological problems.

### Introduction to Hydrodynamical Models of Carrier Transport in Semiconductor Devices

A.M. Anile (University of Catania, Italy)

Device simulation at the industrial level is based on a mathematical model for the charge carrier transport in semiconductors which is based on the drift-diffusion equations, a non linear parabolic system of PDE's describing the motion of electrons in the conduction band of a semiconductor as due to the action of both diffusion (under a density gradient, such as junctions, interfaces, etc.due to inhomogeneous doping) and drift (under an external and self-consistent electric field).

This system of equations, called the Van Roesbrok equations is at the core of all the device simulators currently used in an industrial context. It has been the subject of deep mathematical studies concerning existence, uniqueness and well posedeness of initial and boundary value problems. The present technology push towards integration leads, on one side to miniaturized devices (used as basic components for microprocessors), on the other to many more functions on a single chip (e.g. sensors and actuators for control) and to more stringent requirements on performance (such as a higher voltage breakdown for power devices) . As a consequence the standard drift-diffusion model is no longer adequate in order to describe reliably modern devices.

Hydrodynamical models have been introduced,as a bridge between the drift-diffusion equations and the full kinetic treatment based on the Boltzmann equation (which presents very difficult computational problems).These models are obtained from the moment equations by ad hoc closure assumptions and heuristic reasoning. They are implemented in several TCAD simulation tools. However they are still unsatisfactory in practical use because of the large number of free parameters and because the numerical algorithms are not sufficiently robust. Several groups have started to build more satisfactory hydrodynamical model/ or energy models using more rigourous approaches and better numerical algorithm. The aim of this first part of the course is to give an introduction to hydrodynamical models for describing carrier transport in semiconductor devices.

Lecture 1 :
Introduction to electron transport in semiconductors. Band structure. The semiclassical description. Statistical properties.
Lecture 2 :
Heuristic derivation of the semiclassical Boltzmann transport equation for electrons and holes in semiconductors. Main properties. The drift-diffusion system.
Lecture 3 :
Entropy based moment equations. Closures and maximum entropy. Hyperbolicity. Relationship with extended thermodynamics.
Lecture 4 :
Extended hydrodynamical models for electron transport in semiconductors. Parabolic and non parabolic bands. Maximum entropy closures and comparison with Monte Carlo simulations.
Lecture 5 :
Higher order numerical schemes for hyperbolic systems of conservation laws. Numerical solutions of extended extended hydrodynamical models . Comparison with Monte Carlo simulations.

### Mathematical Foundations of Electrical Network Analysis

P. Rentrop, M. Guenther
(Technical University, Darmstadt, Germany)

In electric circuits simulation, the transient behaviour of input signals and the output signals of a circuit are studied. The simulations can be performed in the time domain or in the frequency domain (harmonic balance).
The basis for the mathematical model is the Modified Nodal Analysis (MNA).
Application of Kirchoff's laws leads to large systems of implicit ordinary differential equations or to special structured differential-algebraic systems. For the MOSFET substitute circuits or companion models on several levels are used whose data fit the behaviour of the real device. In an advanced state the data are taken from the device simulation or are assembled in table models.
The circuits equations are generated automatically by packages like SPICE or in the advanced package TITAN of the SIEMENS company.
From the point of view of numerical simulation, the circuits can be roughly divided into three classes :

• small circuits with oscillating behaviour,
• middle-sized circuits (up to several hundreds elements) without pronounced oscillating behaviour
• highly integrated circuits with more than thousand elements.
These classes generate different numerical difficulties. In case of oscillating circuits efficient methods for the limit cycle computation with an unknown period are necessary. Middle-sized circuits usually suffer from hidden singularities, which are described by the index concept. In very large circuits less than five percent of the elements are active, most parts of the circuits are latent. This is reflected in multirate strategies, where the components of a differential equatio system are integrated with respect to their inherent local time constants. The aim of this second part of the course is to provide an introduction to the relevant advanced mathematical and computational techniques to deal with electric circuits simulation.

Lecture 1 :
Mathematical modeling of electrical networks. SPICE approach:
Charge Oriented MNA, Kirchoff's laws, constitutive element relations; companion models for semiconductor devices; special circuits.
Lecture 2:
Mathematical foundations of ordinary differential equations.
Existence, Uniqueness, parameter dependence; initial value problem; linear and nonlinear systems; matrix exponential.
Lecture 3 :
Stiff and implicit systems of differential equations.
Stiffness: A-stability; differential-algebraic equations, index concepts; special circuits revisited.
Lecture 4:
Numerical simulation of non oscillatory circuits. BDF approach and Newton-Raphson procedure in SPICE; higher index systems; integration schemes for higher index systems.
Lecture 5:
Numerical simulation of oscillatory circuits. Harmonic balance vs.shooting methods; discussion; Colpitts and ring oscillator; shooting and monodromy matrix; stability problems due to Chua; RF applications and PDAE approach for multiple signals.

### Mathematical Modeling in Polymer Science

A. Fasano (University of Firenze, Italy)

A polymerization process consists in the aggregation of relatively simple molecules (monomers) into chains which can contains thousands of elements. The molecular structure of the resulting product is responsible for its peculiar thermodynamical and mechanical complex. For a given polymer in the solid state the mechanical properties depend in a crucial way on its thermal history, which determines the crystal volume fraction and the size distribution of crystals. Therefore it is quite important to study the solidification process. Here it is necessary to describe the two basic stages of crystallization: nucleation and crystal growth, which both take place in a relatively large temperature interval.

Some of the models proposed in the literature will be discussed, along with the general features of phase change processes occurring over a temperature interval. Then the specific case of the solidification of a molten sample under prescribed pressure will be dealt with, taking into account the flow induced by thermal shrinking.
The complex question of polymerization will also be addresses for the specific procedure of the Ziegler-Natta process.

Lecture 1.
Mathematical models for polymer solidification: structure of polymers crystals, the phenomenon of impingement, the Kolmogorov-Avrami model, other models.
Lecture 2.
A thermodynamic approach to nucleation (Ziabicki's theory). Solving a model for a cooling process: theoretical and numerical results.
Lecture 3.
General features of phase change processes over finite temperature intervals: order parameter, additivity rules, travelling waves.
Lecture 4.
Isobaric solidification of polymers with thermally induced flows: comparison with experiments.
Lecture 5.
Modeling the Ziegler-Natta process for the polymerization of gaseous monomers.

There will an illustration with some numerical results given by A. Mancini (Univ. Firenze) in the end of Lecture 3.

### Mathematical Modeling of Composite Materials Manufacturing Processes

L. Preziosi (Politecnico di Torino, Italy)

Composite materials are obtained by injecting polymerizing resins into suitably designed porous materials. They have remarkable mechanical properties that justify their rapidly expanding employment e.g. in automotive and aeronautical industry.

The injection process differs considerably from the standard penetration of a liquid through a porous medium due to the presence of several factors: variable thermal fields induced by strongly exothermic polymerization (both processes influence viscosity to a great extent), and mechanical deformations.

Lecture 1. General features of the technological problem. Mass balances.
Lecture 2. Momentum and energy balances.
Lecture 3. Incompressible flows in deformable porous media.
Lecture 4. Application to compression moulding processes.
Lecture 5. Application to resin injection moulding processes.