Mixed finite element methods for elliptic convection dominated problems arising in semiconductor physics [Elektronische Ressource] / Stefan Holst

Stefan Holst

Sujets

• Mathematics
• Physik

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2005
• Nombre de lectures: 41
• Langue: Deutsch
• Poids de l'ouvrage: 2 Mo

Extrait

Mixed Finite-Element Methods for Elliptic
Convection-dominated Problems Arising in
Semiconductor Physics
Dissertation
zur Erlangung des Grades
Doktor der Naturwissenschaften
Am Fachbereich Physik, Mathematik und Informatik
der Johannes Gutenberg-Universit at Mainz
Stefan Holst,
geboren in Berlin
Mainz, 2005Tag der mundlic hen Prufung: 13. Januar 2006
D77 - Mainzer DissertationAbstract
In dieser Arbeit wird ein adaptives numerisches Verfahren zur Simulation
einer Klasse von makroskopischen Halbleitermodellen vorgestellt und analy-
siert. Dazu wird zun achst in die mathematische Modellierung von Halb-
leitern eingefuhrt. Dies dient zur Einordnung der im weiteren Verlauf nu-
merisch in 2D genauer untersuchten Energie{Transport Modelle. Diese
Modellklasse beschreibt den Flu von geladenen Teilchen, d.h. von negativ
geladenen Elektronen und sogenannten L ochern, das sind Pseudoteilchen mit
positiver Ladung, und deren Energieverteilung in einem Halbleiterkristall
anhand eines Systems von nichtlinearen gekoppelten partiellen Di eren tial-
gleichungen.
Eine wesentliche Schwierigkeit in der numerischen Behandlung dieser Gle-
ichungen stellen einerseits die nichtlineare Kopplung und die nur teilweise
durch die Daten absch atzbaren lokalen Ph anomene, sogenannter \hot elec-
tron e ects", dieser teils konvektionsdominanten Gleichungen dar. Die pri-
m aren Gr o en der Modelle sind in der hier fur die Simulationen verwen-
deten Formulierung Teilchen- und Energiedichten. Weiterhin entscheidend
ist fur den Anwender die Gr o e des Strom usses durch Teile des Randes,
sogenannte Kontakte. Das hier betrachtete numerische Verfahren verwen-
det gemischte Finite Elemente als Ansatzraum fur die diskrete L osung. Die
stetige Diskretisierung der Normalkomponente der Stromdichte ist aus Sicht
der Anwendung der entscheidende Vorteil dieser Elemente. Es wird gezeigt,
da im Laufe des Algorithmus unter bestimmten Bedingungen an die Tri-
angulierung sichergestellt ist, da die Teilchendichten positiv bleiben. In
diesem Zusammenhang wird ebenfalls eine a priori Fehlerabsch atzung fur die
diskrete L osung einer linearen Konvektions-Di usions-Gleic hung bewiesen.
Die lokalen Ph anomene im Halbleiter werden durch adaptive Verfahren, die
auf a posteriori Fehlersch atzern beruhen, geeignet aufgel ost. Es ndet an
dieser Stelle ein Vergleich verschiedener Fehlersch atzer statt.
Au erdem wird ein Verfahren zur Fehlersch atzung in von der L osung ab-
geleiteten Gr o en, sogenannten ‘functional outputs’, auf die Diskretisierung
mit gemischten Finiten Elementen ub ertragen. An einem Beispielproblem
wird dargestellt, wie dieses Verfahren noch erfolgversprechend angewendet
werden kann, wenn Standardfehlersch atzer keine Reduktion des Fehlers im
Zuge iterativer Gitterverfeinerung erzielen.Contents
1. Introduction 5
1.1 Semiconductor Device Modeling . . . . . . . . . . . . . . . . . 8
1.1.1 Semi-Classical Picture of Quantum Mechanics . . . . . 8
1.1.2 Macroscopic Models . . . . . . . . . . . . . . . . . . . 17
1.1.3 Energy-transport Models { An Overview . . . . . . . . 22
1.2 A Mixed Finite{Element Framework . . . . . . . . . . . . . . 25
1.2.1 Hybridization . . . . . . . . . . . . . . . . . . . . . . . 30
1.2.2 Basic Adaptive Algorithm . . . . . . . . . . . . . . . . 32
2. A Hybridized Mixed-FEM for Convection-Di usion Problems 35
2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 A priori Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 Proofs of technical lemmas . . . . . . . . . . . . . . . 48
2.3 A Posteriori Error Estimation . . . . . . . . . . . . . . . . . . 51
22.3.1 An Embedded Estimator Controlling the L -Error . . 52
2.3.2 Benchmark Problems . . . . . . . . . . . . . . . . . . 53
2.3.3 Error Control Based on the Current Density . . . . . 56
2.4 The DWR-Estimation for Mesh Re nemen t Control . . . . . 61
2.4.1 Methodology for Standard Finite{Element Methods . 62
2.4.2 DWR-Estimator for Mixedtds . 65
2.4.3 A Problem of the SIAM 100-Digit Challenge . . . . . 68
3. The Semiconductor Application 75
3.1 The Complete Physical Model . . . . . . . . . . . . . . . . . . 75
3.2 Thermal equilibrium . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Global Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 81
34 Contents
3.3.1 Re nemen t Strategy . . . . . . . . . . . . . . . . . . . 83
3.4 Semiconductor Devices . . . . . . . . . . . . . . . . . . . . . . 85
3.4.1 A Ballistic Diode . . . . . . . . . . . . . . . . . . . . . 85
3.4.2 A MESFET Device . . . . . . . . . . . . . . . . . . . . 87
3.4.3 A Double-Gate MESFET . . . . . . . . . . . . . . . . 90
3.4.4 A Deep Submicron MOSFET . . . . . . . . . . . . . . 93
Bibliography 101Chapter 1
Introduction
Telecommunication was one of the major factors driving the development
of modern societies in the last century. This development is closely related
to the evolution of semiconductor technology. A technology that started in
1947 when Bardon, Brattain and Schockley presented the rst semiconduc-
tor device (a germanium transistor) for which they were awarded the Nobel
Prize in 1956.
Semiconductor devices as a replacement for the electronic devices used at
that time were much smaller. The combination of several transistors and
other structures in an electronic circuit on a single semiconductor crystal led
to the so-called integrated circuits (J. Kilby of Texas Instruments, R. Noyce
of Fairchild Semiconductor) which was the beginning of a process leading to
2todays chips containing millions of transistors on an area of about 1cm . In
the industry the term \scaling" refers to the continued reduction of the size
of the structures on a chip.
The characteristic length scales reached dimensions of 90nm in June 2004
in Intel’s Pentium 4 and the process to build structures with 65nm is al-
ready in development. In these dimensions quantum e ects cannot always
be neglected. However, in the energy-transport model that we will use for
the device simulation it is looked at the movement of electrons in a semi-
conductor in a continuum sense. Similar to uid dynamics a hierarchy of
models exists to describe this \electron gas". The quantum physical struc-
ture of the crystal is incorporated only via special predetermined param-
eters in the most detailed so called microscopic model in this hierarchy,
the semi{classical semiconductor Boltzmann equation. The least complex
56 1. Introduction
and best understood macroscopic model, the drift-di usion model, consists
of the mass conservation equation and a constitutive equation for the cur-
rent density only. It has been derived from phenomenological considerations
[102], a rigorous derivation is due to Poupaud [97]. We will clarify the po-
sition of the energy-transport model in the hierarchy of these semi{classical
semiconductor models.
Numerical simulations of chips with microscopic models are impossible.
Even macroscopic models are only applied to simulate very few connected
devices. In the past the behaviour of a device in an integrated circuit was
described with a rather small set of parameters that had been extracted from
precedingly-performed device simulations. The scaling of device dimensions
has led to a larger set of necessary parameters to describe the device reac-
tion on external inputs. For techniques to automatically identify signi can t
parameters see [70]. There is a demand for more detailed information on
the interaction of the device with a connected circuit. Therefore methods to
directly couple device and circuit simulation have been developed, see [109].
The active region of a device is modeled by macroscopic device equations
and the remaining circuit is described by di eren tial algebraic equations. A
very interesting question in this coupling in present devices is for instance
the device temperature.
In both of these elds the drift-di usion model is not accurate enough to
capture the additional features. Re ned models were proposed by physi-
cists and later mathematically analyzed, namely the hydrodynamic model
introduced by Bl tekj r [18] and Baccarani and Wordemann [7] and the
energy-transport model introduced by Stratton [111].
Theort models are more complex than the drift-di usion equa-
tions but keep their parabolic nature. For the numerical solution of the hy-
drodynamic model, which contains hyperbolic modes, special (e cien t) al-
gorithms are necessary (see, e.g.,[61]). This intermediate complexity makes
the energy-transport system appropriate for fast and accurate semiconduc-
tor simulations. Extensions of recently developed algorithmic device design
optimization techniques [71] for the drift-di usion equations seem to be
possible.
To solve this model with a exible and robust numerical method that can
automatically adapt to the local behaviour of solutions to semiconductor
device simulations is the main goal of this work. In [49] it was shown
that the energy-transport equations can be written in a drift-di usion form.
The unknowns in this formulation are the electron number and the electron
energy density which should remain positive to be physically meaningful.
Under this premise the discretization of drift dominated problems is not7
straight forward, for instance, stand