
Industrial MathematicsChairmen: A. M. Anile and A. Fasano 
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Centro de Matemática Aplicada Instituto Superior Técnico June 712, Lisbon (Portugal) 
Introduction to Hydrodynamical Models of Carrier Transport in Semiconductor DevicesA.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 driftdiffusion 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 selfconsistent 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 driftdiffusion model is no longer adequate in order to describe reliably modern devices.
Hydrodynamical models have been introduced,as a bridge between the driftdiffusion 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 driftdiffusion 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 AnalysisP. 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 differentialalgebraic
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 :
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: Astability; differentialalgebraic equations, index
concepts; special circuits revisited.
Lecture 4:
Numerical simulation of non oscillatory circuits. BDF approach
and NewtonRaphson 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 ScienceA. 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 ZieglerNatta process.
Lecture 1.
Mathematical models for polymer solidification: structure
of polymers crystals, the phenomenon of impingement, the
KolmogorovAvrami 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 ZieglerNatta 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 ProcessesL. 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.