pdae models of integrated circuits and index analysis

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PDAE models of integrated circuits andindex analysisMartin Bodestedt a & Caren Tischendorf aa Institut für Mathematik, Technische Universität Berlin , Strassedes 17. Juni 136, 10623, Berlin, GermanyPublished online: 30 Jan 2007.

To cite this article: Martin Bodestedt & Caren Tischendorf (2007) PDAE models of integratedcircuits and index analysis, Mathematical and Computer Modelling of Dynamical Systems:Methods, Tools and Applications in Engineering and Related Sciences, 13:1, 1-17, DOI:10.1080/13873950600557329

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PDAE models of integrated circuits and index analysis

MARTIN BODESTEDT* and CAREN TISCHENDORF

Institut fur Mathematik, Technische Universitat Berlin, Strasse des 17. Juni 136,10623, Berlin, Germany

A coupled system modelling an electric circuit containing semiconductors is presented. Themodified nodal analysis leads to a differential algebraic equation (DAE) describing the electricnetwork. The nonlinear behaviour of the semiconductors is modelled by the drift diffusionequations. Coupling relations are defined and a generalization of the tractability index to systems ofinfinite dimensions is presented and applied to the resulting partial differential algebraic equation(PDAE). The PDAE turns out to have the same index as the electrical network equations.

Keywords: Semiconductor; Partial differential algebraic equation; Index analysis; Coupledsystem

1. Introduction

In the development of integrated memory circuits, the modelling of semiconductorswith equivalent models is getting more and more cumbersome. The decreasing spatialscales and higher frequencies lead to larger equivalent models requiring an extensivetuning effort. Therefore, it is worthwhile to replace them with partial differentialequations (PDEs).

A model coupling the stationary drift diffusion equations with an electrical networkdifferential algebraic equation (DAE) is presented.

A DAE is an implicit differential equation

fðx0; x; tÞ ¼ 0; x0 2 Rm; x 2 Dx � R

m; t 2 I � R;

with fv(v, x, t) singular for t 2 J � I. The sensitivity with respect to perturbations ofthe data can, for DAEs, depend not only on the perturbations themselves but also theirderivatives as the following simple example shows:

x01 þ x2 ¼ 0;

x1 ¼ 0;

*Corresponding author. Email: [email protected]

Mathematical and Computer Modelling of Dynamical SystemsVol. 13, No. 1, February 2007, 1 – 17

Mathematical and Computer Modelling of Dynamical SystemsISSN 1387-3954 print/ISSN 1744-5051 online � 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/13873950600557329

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has the solution x¼ 0. If we perturb the right-hand side with (d1, d2)T we get the

solution ðx�1; x�2Þ ¼ ðd2; d1 � d02Þ. In contrast to the ODE case we see that the deviationof the solution of the unperturbed problem from the solution of the perturbed problemdepends on the derivatives of the perturbation

kx� x�k � Cðkdk þ kd0kÞ:

The perturbation index of a problem is one plus the order of the highest derivative thatappears in such an estimate as above (for a less heuristic definition, see [1]).Since the perturbation index requires knowledge of the exact solution it is not always

possible to determine it. Therefore, several other index concepts have been proposed.The tractability index is a projector based approach that utilizes a matrix chain to giveinformation on the DAE’s sensitivity with respect to perturbations without knowledgeof the exact solution. For finite dimensional DAEs it has been shown that thetractability index and the perturbation index coincide, under some assumptions [2].A generalization of the tractability index to infinite dimensional systems [3] is applied

to the network DAE coupled with the drift diffusion equations. We show that if thedynamical behaviour of the pn-junctions is modelled by a capacitor, then the coupledsystem has the same generalized index (1 or 2) as the network DAE as long as theapplied voltages are low.In Section 2 we model the electric network, present the drift diffusion equations and

derive boundary conditions. The coupled system and the generalization of thetractability index are presented in sections 3 and 4. The greater part of the index proofconsists of an existence proof for the linearized drift diffusion equations contained insection 5. Thereafter, we summarize our results and present some simulation results.

2. Modelling integrated circuits with semiconductor PDEs

We consider an RLC network with one semiconductor modelled by the stationary driftdiffusion equations. Generalization to several semiconductors is straightforward. Thecircuit has nþ 1 nodes and contains semiconductors, resistors, inductors, capacitors,and independent voltage and current sources denoted {S, R, L, C, V, Ii}, respectively.By kE2N with E from the index set above we mean the number of the element E in thecircuit. Additionally, we also consider kIc voltage controlled current sources Icconnected parallel to capacitors. These appear in, for instance, diode equivalent circuitsfor pn-junctions.The topology of the network is defined through the incidence matrices AE2Rn6 kE.

The inputs of the system are the functions is(�)2RkI and vs(�)2RkV describing thebehaviour of the independent sources Ii and V.In the modified nodal analysis (MNA) [4], Kirchoff’s laws and the specific relations

describing the network elements are combined in a differential algebraic equation(DAE). The unknowns are reduced to a vector x(t)¼ (e(t), iL(t), iV(t))

T2Rnþ kLþ kV

containing the node potentials, the currents through the inductors and the currentsthrough the voltage sources,

0 ¼ACqCðATCe; tÞ

0 þ ARgðATRe; tÞ þ ALiLðtÞ þ AViV þ AIc iIðAT

Ice; tÞ þ AIi iSþ ASjS;

0 ¼FðiLðtÞ; tÞ0 � ATLe;

0 ¼ATVe� vsðtÞ: ð2:1Þ

2 M. Bodestedt and C. Tischendorf

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The functions qC and F are the electric charges and electromagnetic fluxes, respectively,and g and iI describe the voltage dependence of resistances and controlled currentsources, respectively.

We assume that the resistors, inductors and capacitors are locally passive, i.e. thatthe matrices

GðtÞ ¼ @gðw; tÞ@w

; LðtÞ ¼ @Fðw; tÞ@w

; CðtÞ ¼ @qCðt;wÞ@w

ð2:2Þ

are weakly (not necessarily symmetric) positive definite.Before we turn to the determination of the semiconductor current we explain why we

consider the MNA equations to model the dynamics of the electric circuit. The MNAequations have a relatively small number of unknowns and can be set upautomatically – two important features for the development of digital memory circuitshaving up to 107 network components. Furthermore, the MNA equations lead toDAEs of at most index 2 [5] if all capacitances and inductances are passive andcontrolled sources satisfy weak topological assumptions concerning their controllingvoltages and currents. The higher index variables depend only linearly on the othernetwork variables. It implies the weak instability known for higher index DAEs to beharmless in case of network DAEs formulated by MNA. Finally, we want to remarkthat per se it is not always an advantage to use an ODE formulation instead of a higherindex DAE. This is well known, for instance, for mechanical systems where the ODEsolution leads to a so-called drift-off effect due to the fact that information onconstraints gets lost in the index reduction process [6].

2.1 The stationary drift diffusion equations

A semiconductor occupying a region O2R3 with some doping profile N(�)2H1(O) isconsidered.Todetermine the semiconductor current jSwe solve thePoisson equation for theelectrostatic potential c together with the stationary continuity equations for the currentdensities, Jn, Jp. The current densities in turn depend on the charge carrier densities n, p.

EDc ¼ qðn� p�NÞ; ð2:3aÞ

divJn ¼ qRðn; pÞ; ð2:3bÞ

divJp ¼ �qRðn; pÞ; ð2:3cÞ

Jn ¼ qmnðUTrn� nrcÞ; ð2:3dÞ

Jp ¼ �qmpðUTrpþ prcÞ: ð2:3eÞ

The functions c, n, p, Jn and Jp depend on space, y2O, and time, t2 [0, ?). Theimplicit time dependence cannot be omitted since we are to apply time-dependentboundary conditions determined by the surrounding circuit.

Close to the thermal equilibrium we can assume constant temperature. As aconsequence the thermal potential UT is constant. Temperature-dependent models ofintegrated circuits are studied in [7]. Other constants are the elementary charge q andthe electric permittivity constant E.

Generally, the mobilities of the charge carriers, mn and mp, depend on the dopingprofile and the electric field; but for low applied voltages it is sufficient to model themas only space(doping)-dependent non-negative functions [8].

PDAE models of integrated circuits and index analysis 3

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The function R models the recombination and generation of charge carriers. We usethe Shockley –Read –Hall recombination,

Rðn; pÞ ¼ np� n2itpðnþ niÞ þ tnðpþ niÞ

; ð2:4Þ

for two particle transition which is sufficient close to the thermal equilibrium [8]. Theconstants ni, tn and tp denote the intrinsic carrier concentration in the material and theaverage times between generation and recombination for the charge carriers.We consider a one-port semiconductor with metal – semiconductor contacts. The

device geometry is a region O \ R3 with a boundary consisting of two non-emptyclosed parts �O¼�O1

[�O2with Dirichlet conditions and one non-empty insulating

part �I with Neumann conditions. We also assume �O1\�O2

¼ {;}. This assumption isneeded in the construction of solution spaces independent of time and it only excludessemiconductor geometries with a short-circuit between the metal – semiconductorcontacts.

Remark 2.1 The stationary drift diffusion equations do not take the capacitive behaviour

of the pn-junctions into account. To compensate for this we always assume that a small

capacitor (C� 10714F) has been incorporated into the electrical network described by (2.1)

between the semiconductor contacts �O1and �O2

.

Combining the expressions for the current densities, (2.3d), (2.3e), with thecontinuity equations, (2.3b), (2.3c), we get the so-called drift diffusion equations forsemiconductors. We will use the quite common convention to also include the Laplaceequation in what we denote the drift diffusion equations.At the metal – semiconductor contacts the Dirichlet boundary condition for the

carrier densities depend on the doping N and the intrinsic carrier concentration ni.

nðyÞ ¼ 1

2NðyÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNðyÞ2 þ 4n2i

q� �; y 2 �O; ð2:5aÞ

pðyÞ ¼ 1

2�NðyÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNðyÞ2 þ 4n2i

q� �; y 2 �O: ð2:5bÞ

For the electrostatic potential we have time-dependent Dirichlet conditions. Thecontact at �O1

is connected to node i and the contact at �O2is connected to node j in

the network and we have

cðy; tÞ ¼ cbiðyÞ þ eiðtÞ; y 2 �O1; ð2:6aÞ

cðy; tÞ ¼ cbiðyÞ þ ejðtÞ; y 2 �O2: ð2:6bÞ

The built-in potential is doping dependent and defined in such a way that the device isin the thermal equilibrium when the externally applied potentials are zero [9].To be able to formulate solution spaces independent of time, we homogenize the

electrostatic potential.Let h2H2(O) be a solution of the Laplace equation in O with the boundary

conditions

hðyÞ ¼ 1; y 2 �O1; hðyÞ ¼ 0; y 2 �O2

; ð2:7aÞ

rhðyÞ � nðyÞ ¼ 0; y 2 �I: ð2:7bÞ


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The vector n is the exterior unit normal on the boundary @O. Now, we merge (2.6) in

cðy; tÞ ¼ cbiðyÞ þ hðyÞATSeðtÞ þ ejðtÞ; y 2 �O: ð2:8Þ

The condition �O1\�O2

¼ {;} is fulfilled for all standard semiconductor geometries andensures the existence of the function h.

The potential is by (2.3) only determined up to an additive constant. Therefore, wecan neglect the term ej in the boundary conditions. Then,

u0ðy; tÞ :¼ cðy; tÞ � ðcbiðyÞ þ cEðy; tÞÞ; ð2:9Þ

satisfies homogeneous Dirichlet conditions at the metal – semiconductor contacts. Toshorten the notation we have introduced cE ¼ hðyÞAT

SeðtÞ.For analytical reasons it is useful to obtain a divergence structure on the PDE system

via transformation of carrier densities into the Slotboom variables U1, U2;

n ¼ ni exp ðc=UTÞU1; ð2:10aÞ

p ¼ ni exp ð�c=UTÞU2: ð2:10bÞ

In [9] it has been shown that the charge carrier densities are positive. Thisproperty is invariant under the variable change (2.10). Now, the current densityrelations read

Jn ¼ qmnUTni exp ðc=UTÞrU1; ð2:11aÞ

Jp ¼ �qmpUTni exp ð�c=UTÞrU2: ð2:11bÞ

The Slotboom variables have time-dependent Dirichlet conditions

U1ðy; tÞ ¼ exp �cEðy; tÞUT

� �; y 2 �O; ð2:12aÞ

U2ðy; tÞ ¼ expcEðy; tÞUT

� �; y 2 �O ð2:12bÞ

and are homogenized

u1ðy; tÞ :¼ U1ðy; tÞ � exp �cEðy; tÞUT

� �; ð2:13aÞ

u2ðy; tÞ :¼ U2ðy; tÞ � expcEðy; tÞUT

� �: ð2:13bÞ

On the insulating boundary we assume a vanishing outward electric field andvanishing outward current densities,

rcðyÞ � nðyÞ ¼ JnðyÞ � nðyÞ ¼ JpðyÞ � nðyÞ ¼ 0; y 2 �I: ð2:14Þ

Close to the insulating boundary the doping profiles of most semiconductors areconstant in the boundary normal direction, and therefore we can assume

rcbiðyÞ � nðyÞ ¼ 0; y 2 �I: ð2:15Þ


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From (2.7b), (2.14) and (2.15), homogeneous Neumann conditions on the insulatingboundary are obtained for the electrostatic potential. By (2.11), the vanishing outwardcurrent densities (2.14), the condition (2.7b) and the fact that the charge carriermobilities are positive, we have homogeneous Neumann conditions for thehomogenized Slotboom variables u1, u2.

3. The coupled system

The network DAE and the semiconductor PDEs are coupled in two ways. First, thenode potentials in the network appear in the boundary conditions for the drift diffusionvariables. Second, the current flowing over the metal – semiconductor boundaries jSmust be taken into account in Kirchoff’s Current Law for the network. Summing (2.3b)and (2.3c) and applying the divergence theorem one can deduce that it is only necessaryto evaluate jS at �O1

,

jSðtÞ ¼ �Z

�O1

ðJn þ JpÞ � n ds: ð3:1Þ

We have now derived boundary conditions and coupling relations for an electric circuitcontaining a one-port semiconductor. The resulting PDAE is of the form

0 ¼ACqC�AT

Ce; t�0þ ARg

�AT

Re; t�þ ALiL þ AViV þ AIc iI

�AT

Ice; t�þ AIi iSþ ASjS; ð3:2aÞ

0 ¼ FðiLðtÞ; tÞ0 � ATLe ð3:2bÞ

0 ¼ ATVe� vsðtÞ ð3:2cÞ

0 ¼ jS �Z

�O1

niUT mn expu0 þ cbi þ cE

UT

� �r u1 þ exp

�cE

UT

� ��

� mp exp�u0 � cbi � cE

UT

� �r u2 þ exp

cE

UT

� �� nds; ð3:2dÞ

0 ¼ EqD u0 þ cbi þ cEð Þ � ni exp

u0 þ cbi þ cE

UT

� �u1 þ exp

�cE

UT

� ��

þ ni exp�u0 � cbi � cE

UT

� �u2 þ exp

cE

UT

� �� þN; ð3:2eÞ

0 ¼ div mnniUT expu0 þ cbi þ cE

UT

� �r u1 þ exp

�cE

UT

� �� Sðu0; u1; u2; eÞ; ð3:2fÞ

0 ¼ div mpniUT exp�u0 � cbi � cE

UT

� �r u2 þ exp

cE

UT

� �� Sðu0; u1; u2; eÞ; ð3:2gÞ

and is subject to boundary conditions

ujðy; tÞ ¼ 0; y 2 �O; 8t 2 ½0;1Þ; ð3:3aÞ

rujðy; tÞ � nðyÞ ¼ 0; y 2 �I; 8t 2 ½0;1Þ; ð3:3bÞ


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for j¼ 0, 1, 2. The function S is the corresponding recombination – generation term forthe transformed homogenized variables

R u0 þ cbi þ cE; nieu0þcbiþcE

UT

u1 þ e

�cEUT

; nie

�u0þcbiþcEUT ðu2 þ e

cEUTÞ

� �¼: Sðu0; u1; u2; eÞ:

We define

X0 ¼ fv 2 H1ðOÞ j vðyÞ ¼ 0; y 2 �Og

X1 ¼ X0 \H2ðOÞ; ð3:4Þ

and seek solutions ðe; iL; iV; jS; u0; u1; u2Þ 2 X ¼ Rn � R

kL � RkV � R� X0 � X2

1 subjectto (3.3b).

To evaluate the boundary integral (3.2d) we need ru1, ru22L2(@O), i.e. u1,u22H1(@O). For the trace operator we have [10]

B : HmðOÞ ! Hm�1ð@OÞ:

Hence, if we take u1, u22H2(O) the boundary integral (3.2d) is well defined.

Remark 3.1 A standard way to weaken the regularity assumptions for u1, u2 is to multiply

the integrand in (3.2d) by a smooth function, apply the Gaussian Theorem and obtain an

integral over O. Then, in some sense, Jn, Jp2L2(O) and thereby u1, u22H1(O) would suffice.

In the next two sections we turn to the index analysis of this model.

4. Abstract differential algebraic equations

For finite dimensional semi-linear DAEs the connection between the tractability indexand the sensitivity with respect to time-dependent perturbations has been wellestablished [2]. In [5,11] it was shown that the tractability index of a circuit DAEderived with the modified nodal analysis can be determined from the topology of thenetwork.

The topology of an electric network is described by a connected graph with nodesand directed branches. By a tree we mean a set of branches that form a base in thebranch space. A loop is a set of l branches joining l nodes. A cutset is a set ofbranches with the properties: the network remains connected if they are not allremoved, but if they are all removed, the network is split into two non-connectedparts.

Theorem 4.1 ([5,11]) Consider a network without (PDE-modelled) semiconductors (nS¼ 0).

Let the nodes of the voltage controlled current sources Ic be connected by capacitive paths.

Assume that the capacitors, resistors and inductors are passive (see (2.2)). Then the MNA

network equations (2.1a) has tractability index

0, when the network contains a capacitive tree and no voltage sources,1, when the network does not contain any CV-loops with at least one voltage source nor

LI-cutsets, and2, otherwise.


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In this work we investigate how these index criteria transfer to integrated circuitmodels of the form (3.2), i.e. when nS¼ 1.First we review the generalization of the tractability index to infinite dimensions [3],

then, we apply it to (3.2).We consider PDAEs as abstract DAEs in the form

Aðdðwðy; tÞ; tÞÞ0 þ bðwðy; tÞÞ ¼ 0; t 2 J � Rþ; y 2 O; ð4:1Þ

where A : Z! Y, d(�, t) : X ! Z and b(�) : X ! Y are maps between real Hilbertspaces. Solutions w : J ! X are paths in X.As in the finite case [13], the index analysis is done via linearization. Along the path

w*2X, at this point not necessarily a solution, we define

DðtÞ ¼ d 0wðw�; tÞ; B0 ¼ b0wðw�Þ; ð4:2Þ

in the Frechet sense.

Definition 4.2 ([3]) The leading term of (4.1) is said to be properly stated if

kerA imDðtÞ ¼ Z ð4:3Þ

for all t 2 J and these spaces are spanned by continuously differentiable bases.

Due to the passivity assumption (2.2) we can choose d(�, t) such that DðtÞ has aconstant nullspace. In [14,15] is was shown that this well-matched decomposition ofthe leading term improves the stability properties of numerical schemes such as BDFand Runge –Kutta.Now, we need an operator chain. Let Lb(X) be the set of linear bounded operators on

the Hilbert space X. For i 0, let

G0ðtÞ ¼ ADðtÞ;

Qi ¼ LbðXÞ; Qi ¼ Q2i ; imQi ¼ kerGi;

Pi ¼ I�Qi;

Giþ1ðtÞ ¼ GiðtÞ þ BiQi;

Biþ1 ¼ BiPi:

Definition 4.3 ([3]) The Abstract Differential Algebraic System (4.1) with properly stated

leading term has ADAS index m2N when

(i) dimðker GiÞ r 2 Nþ for all i � m� 1

(ii) the operator Gm is injective and GmðXÞ ¼ Y.

We remark that the tractability index is a special case of the ADAS index.The PDAE (3.2) can be cast in the form (4.1) with

A ¼ ATC 0 0 0 0 0 00 IkL 0 0 0 0 0

� �T

; dðwðtÞ; tÞ ¼ qCðATCe; tÞ

FðiL; tÞ

� �; ð4:4Þ


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and the Hilbert spaces

X ¼ Rn � R

kL � RkV � R� X0 � X1 � X1;

Y ¼ Rn � R

kL � RkV � R� X�0 � L2ðOÞ � L2ðOÞ;

Z ¼ RnC � R

kL :

The matrix IkL is the identity matrix with dimension kL. We have

D ¼ CðtÞATC 0 0 0 0 0 0

0 LðtÞ 0 0 0 0 0

� �:

Obviously, G0 ¼ AD is not injective and we have a first result.

Lemma 4.4 If the circuit contains at least one semiconductor modelled by stationary PDEs

the index is at least 1.

Therefore, we calculate

and

Q0 ¼

QC 0 0 0 0 0 00 0 0 0 0 0 00 0 IkV 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

0BBBBBBBB@

1CCCCCCCCA:

The linear functionals b23(�) : X0!R, and b24(�), b25(�) : X1!R, have the actions

b23ðoÞ ¼Z

�O1

mnni expU0�

UT

� �rU1� � noþ mpni exp

�U0�

UT

� �rU2� � no

� �ds;

b24ðoÞ ¼Z

�O1

mnniUT expU0�

UT

� �n � ro ds; ð4:5Þ

b25ðoÞ ¼ �Z

�O1

mpniUT exp�U0�UT

� �n � ro ds:


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The path w*¼ (e*, iL*, iV*, jS*, u0*, u1*, u2*)T is the linearization point and

cE� ¼ hATSe�; U1� ¼ u1� þ exp

�cE�UT

� �;

U0� ¼ u0� þ cE� þ cbi; U2� ¼ u2� þ expcE�UT

� �:

The matrix QC is a projector onto kerATC and Ikv the identity matrix with dimension

kV. We are not going to specify the functionals b21 – b51 for reasons soon to beclarified. The evaluation of F(w*) is the left-hand side of the linearized semiconductorequations (3.2e) – (3.2g).To find the index 1 criteria we construct

Here, we have used the condition that the nodes of the semiconductors (see Remark2.1) and voltage controlled current sources must be connected by capacitivebranches, i.e.

ATSQC ¼ 0 and AT

IcQC ¼ 0: ð4:7Þ

If the surrounding circuit, with the semiconductor branches omitted, is describedby index 1 MNA equations then the upper left block in G1 is a non-singular matrix [11].

Lemma 4.5 If the surrounding circuit is modelled by MNA equations of index 1 and

the block operator F(w*) is injective and densely surjective then the PDAE (3.2) has ADAS

index 1.

Due to the block structure of G1 it is clear that the matrix chain for the MNAequations is realized in the upper left block. Therefore, the index 2 result followsstraightforwardly in accordance with [11].

Lemma 4.6 If the surrounding circuit is modelled by MNA equations of index 2 and the

block operator F(w*) is injective and densely surjective then the PDAE (3.2) has ADAS

index 2.

In the next section we prove sufficient conditions for F(w*) to be injective and denselysurjective.

: (4.6)


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5. Unique solvability of the drift diffusion equations

This section is entirely devoted to the proof of the following lemma.

Lemma 5.1 If the linearization point w* is a solution of (3.2) and the voltage applied over the

Dirichlet boundaries is small, then the operator F(w*) is injective and densely surjective.

The introduction of the Slotboom variables allows a functional analytical approachfor systems of elliptic equations to prove Lemma 5.1. The equation

Fðw�Þu ¼ b ð5:1Þ

can therefore be written in divergence form

�divðBruÞ þ Cu ¼ b; ð5:2Þ

where B is a positive definite matrix for all t 2 J and y2O. By multiplying withv 2 X0 � X2

1 and integrating by parts we obtain the corresponding weak formulationZOðrvÞTBrudyþ

ZOvTCudy ¼

ZOvTbdy: ð5:3Þ

The left-hand side is a bilinear form a¼ a(u, v), and for each b 2 X�0 � ½L2ðOÞ�2 theright-hand side is the evaluation of a linear functional lð�Þ : X0 � X2

1 ! R. Therefore,equation (5.3) is equivalent to

aðv; uÞ ¼ lðvÞ: ð5:4Þ

A Fredholm alternative for Garding forms will yield the result.

Definition 5.2 ([10]) A bounded, bilinear form g : X6X!R is a Garding form if the

embedding X!Y is continuous and the Garding inequality

gðu; uÞ ckuk2X � dkuk2Y ð5:5Þholds for all u2X with constants c4 0 and d. Moreover, if the embedding X!Y is

compact, then the Garding form is regular.

Throughout this section the space X is to be understood as the solution space for thedrift diffusion equations, X ¼ X0 � X2

1, and not as the entire solution space for thePDAE as defined in section 3. The space X consists of functions that are zero on asubset of the boundary with surface measure greater than zero, namely �O1

[�O2.

Hence, we can use the canonical norm for H10ðOÞ

3,

kuk1;2;0 ¼ZO

X2i¼0jruij2dy

!1=2

; ð5:6Þ

on it [12]. We need to show that F(u*) is densely surjective and since L2(O) is dense inX�0 we take L2(O)

3 with the usual norm

kuk2 ¼ZO

X2i¼0juij2dy

!1=2

: ð5:7Þ


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as the space Y in Definition 5.2. The space X is both continuously and compactlyembedded in Y [10].We apply Young’s inequality to (5.3) with v¼ u and obtainZ

OðruÞTBrudyþ

ZOuTCudy

c

ZOjru0j2 þ jru1j2 þ jru2j2

dy� d

ZOju0j2 þ ju1j2 þ ju2j2

dy

¼ ckuk21;2;0 � dkuk22;

with c,d4 0. Hence, a fulfills the Garding inequality and is a regular Garding form.We apply a Fredholm alternative for Garding forms.

Theorem 5.3 ([10]) Consider equation (5.4). Let a : X6X!R be a regular Garding form on

the real Hilbert space X and let l2X*. If the homogeneous equation (5.4) with l¼ 0 has the

trivial solution u¼ 0 only, then, for each l2X*, the inhomogeneous equation (5.4) has a unique

solution.

We prove the uniqueness under the assumption that the point of linearization is asolution of the original system (3.2). This allows us to replace the divergence terms inthe first column of the matrix function C by the recombination – generation term S*.The existence and local uniqueness of solutions of (3.2) for O2R was shown in [16].We let v¼ (du0, u1, u2)2X, with d4 0 a small parameter, and have

ZOr

du0u1

u2

0B@

1CA

T Eq 0 0

0 nimnUTeU0�=UT

� �0

0 0 nimpUTe�U0�=UT

� �0BB@

1CCAr

u0

u1

u2

0B@

1CA

þdu0u1

u2

0B@

1CA

T nieU0�=UTU1� þ nie

�U0�=UTU2� nie�U0�=UT � nie

U0�=UT

S�UTþ @S�

@u0

� S�UT� @S�

@u0

@S�@u1

@S�@u2

@S�@u1

@S�@u2

0BB@

1CCA

u0

u1

u2

0B@

1CAdy ¼ 0:

We put a¼ ni exp(U0*/UT) and b¼ ni exp(7U0*/UT), keep the divergence terms to theleft and move the rest to the right-hand side. By applying the Poincare – Friedrichinequality on the remaining left-hand side we obtainZ

O

C1dEqju0j2 þ C1C2 ju1j2 þ ju2j2

� �dy � ð5:8aÞ

ZO

C1dEqju0j2 þ C1mnaju1j

2 þ C1mpbju2j2

� �dy � ð5:8bÞ

�ZO

�dðaU1� þ bU2�Þu20 þ dbu0u1 � dau0u2

þ S�UTþ @S�@u0

� �u0u1 þ

@S�@u1

u21 þ@S�@u2

u1u2

� S�uTþ @S�@u0

� �u0u2 þ

@S�@u1

u1u2 þ@S�@u2

u22

�dy ð5:8cÞ


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with C2¼min(infO{mna}, infO {mp b})4 0 and C1 being the Poincare – Friedrichconstant. We apply Young’s inequality again and obtain

ð5:8aÞ �ZO

��dðaU1� þ bU2�Þ þ dþ S�

UTþ @S�@u0

��

� �u20

þ 1

2db2 þ S�

UTþ @S�@u0

�� @S�@u1

þ @S�@u2

� �u21

þ 1

2da2 þ S�

UTþ @S�@u0

��þ @S�@u1

� @S�@u2

� �u22

�dy: ð5:9Þ

Now, if

�dðaU1� þ bU2�Þ þ dþ S�UTþ @S�@u0

�� C1dE

2qð5:10aÞ

db2 þ S�UTþ @S�@u0

�� @S�@u1

þ @S�@u2� C1C2 ð5:10bÞ

da2 þ S�UTþ @S�@u0

��þ @S�@u1

� @S�@u2� C1C2 ð5:10cÞ

for all y2O, we are done. Since U1* and U2* always remain non-negative, theinequality (5.10a) is fulfilled if

S�UTþ @S�@u0

�� 1

2

C1Eq� 2

� �d: ð5:11aÞ

The inequalities (5.10b) and (5.10c) are satisfied if

@S�@u2� @S�@u1

�� 1

3C1C2 ð5:11bÞ

db2 � 1

3C1C2 ð5:11cÞ

S�UTþ @S�@u0

�� 1

3C1C2 ð5:11dÞ

da2 � 1

3C1C2 ð5:11eÞ

hold for all y2O. Now, we choose a d such that

d � C1

3

minðinfOðmnaÞ; infOðmpbÞÞmaxðsupO a2; supO b2Þ ð5:12Þ

for all y2O. By differentiating the recombination – generation S one sees that (5.11b)and (5.11d) can be satisfied by choosing solutions close to the thermal equilibrium.


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Since we are primarily interested in solutions close to equilibrium due to the lowvoltages applied in integrated memory circuits, this restriction is not a severe one.We have (5.11), fulfilled for all y2O, which implies

C

ZO½ju0j2 þ ju1j2 þ ju2j2�dy � 0 ð5:13Þ

with the constant

C ¼ 1

4lmin

dEq;minfinf

OðmnaÞ; inf

OðmpbÞg

�

and, hence, u¼ 0. Theorem 5.3 now ensures the unique solvability of (5.1), and therebyLemma 5.1 is proven.In the next section we summarize Lemma 4.4 –Lemma 4.6 and Lemma 5.1 in a

theorem.

6. Results

Theorem 6.1 Assume that the abstract DAE (3.2) fulfil the following:

. the capacitors, resistors and inductors are passive;

. the nodes of the semiconductor and the controlled current sources are connected bycapacitive branches;

. the linearization point is a solution of (3.2); and

. the voltage applied over the semiconductor contacts is small.

Then (3.2) has the ADAS index

0, if the network has a capacitive tree, no voltage source and no semiconductor;1, if the network does not contain a LI-cutset nor a CV-loop with at least one voltage

source;2, otherwise.

It is not necessary to linearize in the solution of (3.2); however, one needs to definethe distance between the linearization point and the solution.The corresponding results for a PDAE with the non-stationary drift – diffusion

equation were proven in [17].It is probably possible to relax the assumption on capacitors being parallel to the

semiconductor by altering the assumption on the function h(�) used in thehomogenization; however, then the dynamics of the pn-junctions are neglected whichobviously is unacceptable. The condition AT

SQC ¼ 0 should not be seen as a restrictionbut a sensible way of modelling the junctions time-behaviour.A more challenging problem is to relax the restriction on the applied voltages. It is

known that the stationary drift diffusion equations do not have a unique solution forarbitrary Dirichlet conditions, and it is possible that this fact will lead to higherindex for some applied voltages. For the proof one needs to find the projectionoperator Q1, which includes characterization of the kernel of the linearized driftdiffusion equations.


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Another interesting task is to investigate the relation between the ADAS index andthe perturbation index, and at some point also consider perturbations in space.

Finally we remark that analysis, implementation and simulations of the correspond-ing space semi-discretized system are ongoing [18].

7. Simulation results

A small index 1 circuit containing only a voltage source, resistor, an artificial capacitorand a PDE-modelled diode was simulated (see figure 1). The circuit was supplied with asinusoidal voltage, vS¼ 5 sin(2p 108t) V, the resistor had a resistance of 100O and thecapacitance was set to 1.0 * 10

713 F. The diode was discretized by a finite volumescheme in 22 non-equidistant intervals. For the time integration a Runge –Kuttamethod for DAEs with properly stated leading terms was applied. In figure 2 theswitching behaviour of the diode is clearly visible.

Figure 1. A simple test circuit.

Figure 2. The switching behaviour of a 1D diode.


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For DAEs of index higher than 1 it is well known that small but high oscillatingperturbations lead to large deviations from the numerical solution of the unperturbedsystem. Since the circuit is index 1 this effect should not appear. As a norm for the errorwe used the maximum norm taken over 5000 time steps. The perturbation frequenciesused were 100 MHz and 10 GHz. In figure 3 we see that the error does not change eventhough the frequency of the perturbation was increased 100 times.

Acknowledgements

This work was supported by the German Federal Ministry of Education and Researchunder the registration number 03TIMB3.The authors are grateful to Prof. Ansgar Jngel for a valuable discussion concerning

the proof of the theorem and to Dr. Monica Selva for the help with the simulations.The corresponding author also thanks Steffen Voigtmann for his preparatory work onthe simplified model [19].

References

[1] Harier, E. and Wanner, G., 1989, Solving Ordinary Differential Equations, II. (Berlin: Springer-Verlag).[2] Marz, R., 2003, Nonlinear Differential-Algebraic Equations with Properly Formulated Leading Term.

Technical Report, Humboldt University of Berlin.[3] Lamour, R., Marz, R. and Tischendorf, C., 2001, PDAEs and Further Mixed Systems as Abstract

Differential Algebraic Systems. Technical Report, Humboldt University of Berlin.

Figure 3. The error for two perturbation frequencies.


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[4] Feldman, U. and Gunther, M., 1993, The DAE-Index in Electric Circuit Simulation. Technical Report,Technical University Munich.

[5] Estevez Schwarz, D. and Tischendorf, C., 2000, Structural Analysis of Electric Circuits andConsequences for MNA. Int. J. Circ. Theor. Appl., 28, 131 – 162.

[6] Fuhrer, C. and Leimkuhler, B.J., 1991, Numerical Solution of Differential-Algebraic Equations forConstrained Mechanical Motion. Numer. Math., 59, 55 – 69.

[7] Bartel, A., First-Order Thermal PDAE Models in Electric Circuit Design. Paper presented at the 4thMATHMOD, Vienna, Austria, 5 – 7 February, 2003.

[8] Sze, S.M., 1981, Physics of Semiconductor Devices (New York: John Wiley).[9] Markowich, P.A., 1986, The Stationary Semiconductor Device Equations (Vienna: Springer).[10] Zeidler, E., 1990, Nonlinear Functional Analysis and its Applications, II/A (New York: Springer).[11] Tischendorf, C., 1999, Topological Index Calculation of Differential-Algebraic Equations in Circuit

Simulation. Surveys Math. Indust., 8, 187 – 199.[12] Troianiello, G.M., 1987, Elliptic Differential Equations and Obstacle Problems (New York: Plenum

Press).[13] Marz, R., 2003, Differential Algebraic Systems with Properly Stated Leading Term and MNA

Equations. In: K. Antreich, R. Bulirsch, A. Gilg and P. Rentrop (Eds), Mathematical Modeling,Simulation and Optimization of Integrated Electrical Circuits, Vol. 146, (Basel: Birkhauser), pp. 135 – 141.

[14] Higueras, I., Marz, R. and Tischendorf, C., 2000, Stability Preserving Integration of Index-1 DAEs.Appl. Numer. Math., 45, 175 – 200.

[15] Higueras, I., Marz, R. and Tischendorf, C., 2003, Stability Preserving Integration of Index-2 DAEs.Appl. Numer. Math., 45, 201 – 229.

[16] Alı, G., Bartel, A., Gunther, M. and Tischendorf, C., 2003, Elliptic Partial Differential-AlgebraicMultiphysics Models in Electrical Network Design. Mathematical Models & Methods in AppliedSciences, 13(9), 1261 – 1278.

[17] Tischendorf, C., 2004.Coupled Systems of Differential Algebraic and Partial Differential Equations inCircuit and Device Simulation. Modeling and Numerical Analysis. Habilitation Thesis, HumboldtUniversity of Berlin.

[18] Selva Soto, M. and Tischendorf, C., 2004, Numerical Analysis of DAEs from Coupled Circuit andSemiconductor Simulation, Technical Report, Humboldt University of Berlin.

[19] Schulz, S., 2002, Ein PDAE-Netzwerkmodell als Abstraktes Differential-Algebraisches System. DiplomaThesis, Humboldt University of Berlin.


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pdae models of integrated circuits and index analysis

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