Computational Modeling in Electronics and Biomathematicsricsac/LectureNotesCMEBM.pdf · 4.Part IV applies models and computational methods adopted for ion transport in cellular biology

Notes of the Course

Computational Modeling inElectronics and Biomathematics

Master of Science in Mathematical EngineeringDipartimento di Matematica Politecnico di Milano, Italy

Prof. Riccardo Sacco

Dipartimento di Matematica Politecnico di Milano

Piazza Leonardo da Vinci 32 20133 Milano, Italy

E-mail: [email protected]

Home Page: http://www1.mate.polimi.it/~ricsac/

Dr. Aurelio Giancarlo Mauri

Dipartimento di Matematica Politecnico di Milano

Piazza Leonardo da Vinci 32 20133 Milano, Italy

E-mail: [email protected]

Contents

I. Advection-Diffusion Conservation Laws in 1D:

Models and Numerical Approximation 3

1. 1D conservation laws 5

1.1. The 1D conservation law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2. Conservation for the continuous problem . . . . . . . . . . . . . . . . . . . 7

1.3. Time semidiscretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4. The continuous maximum principle . . . . . . . . . . . . . . . . . . . . . . 9

1.5. The weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6. Finite element approximation . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6.1. Finite element spaces . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6.2. Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6.3. Finite element formulation . . . . . . . . . . . . . . . . . . . . . . 14

1.6.4. Connection between the PM method and the standard FE method 15

1.7. Conservation for the discrete problem . . . . . . . . . . . . . . . . . . . . 17

1.8. The discrete maximum principle . . . . . . . . . . . . . . . . . . . . . . . 17

1.9. FE approximation of 1D model problems . . . . . . . . . . . . . . . . . . 18

1.9.1. The diffusion BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.9.2. The reaction-diffusion BVP . . . . . . . . . . . . . . . . . . . . . . 19

1.9.3. The advection-diffusion BVP . . . . . . . . . . . . . . . . . . . . . 20

1.10. Stabilization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.10.1. Stabilization of the reaction-diffusion BVP . . . . . . . . . . . . . 22

1.10.2. Stabilization for the advection-diffusion BVP . . . . . . . . . . . . 24

1.11. Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.12. The time-dependent 1D conservation law . . . . . . . . . . . . . . . . . . 28

1.13. Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.13.1. The stationary 1D conservation law . . . . . . . . . . . . . . . . . 30

1.13.1.1. Diffusion problem . . . . . . . . . . . . . . . . . . . . . . 30

1.13.1.2. Reaction-diffusion problem . . . . . . . . . . . . . . . . . 31

1.13.1.3. Advection-diffusion problem . . . . . . . . . . . . . . . . 34

1.13.2. The time-dependent 1D conservation law . . . . . . . . . . . . . . 36

1.14. Application: heating of a room . . . . . . . . . . . . . . . . . . . . . . . . 38

1.14.0.1. Test 1. Analysis as a function of external temperature . . 40

1.14.0.2. Test 2. Analysis as a function of the initial temperature . 43

1.14.0.3. Test 3. Analysis as a function of glass thermal diffusivity 44

II Contents

II. Mathematical Models in Cellular Biology and Electrophysiology 47

2. Introduction to Cellular Biology and Ion Transport 49

2.1. Cells: structure, membrane and ion channels . . . . . . . . . . . . . . . . 50

2.1.1. The cell membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.2. Ionic channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.2. Transport of charged particles . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2.1. Concentrations and fluxes . . . . . . . . . . . . . . . . . . . . . . . 53

2.3. Charge transport in ionic fluids . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3.1. The Nernst-Planck equation . . . . . . . . . . . . . . . . . . . . . . 57

3. ODE Models in Cellular Electrophysiology 59

3.1. Reduced order modeling of membrane electrophysiology . . . . . . . . . . 60

3.2. General form of transmembrane ionic current densities . . . . . . . . . . . 65

3.3. The ODE model of ion transport . . . . . . . . . . . . . . . . . . . . . . . 65

3.4. Transmembrane current models . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.1. The linear resistor model . . . . . . . . . . . . . . . . . . . . . . . 67

3.4.2. The Goldman-Hodgkin-Katz model . . . . . . . . . . . . . . . . . . 67

3.4.3. The Hodgkin-Huxley model . . . . . . . . . . . . . . . . . . . . . . 70

3.5. Thermal equilibrium of a system of monovalent ions . . . . . . . . . . . . 71

4. PDE Models in Cellular Electrophysiology 73

4.1. The cable equation model . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.1. The geometrical setting . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.2. Electrical equivalent representation . . . . . . . . . . . . . . . . . . 74

4.1.3. The cable equation model system . . . . . . . . . . . . . . . . . . . 77

4.2. A PDE model of ion flow in a fluid medium . . . . . . . . . . . . . . . . . 77

4.2.1. The geometrical setting . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.2. Balance of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.3. Balance of momentum . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2.4. The Velocity-Extended Poisson-Nernst-Planck model . . . . . . . . 81

4.2.5. The electrochemical potential . . . . . . . . . . . . . . . . . . . . . 82

5. The Poisson-Nernst-Planck model in one spatial dimension 85

5.1. Geometrical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2. Biophysical assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3. Boundary conditions at channel openings . . . . . . . . . . . . . . . . . . 88

5.4. A graphical representation of the PNP model in 1D . . . . . . . . . . . . 90

5.5. The PNP model in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

III. Solution Maps and Functional Techniques 95

6. Solution Maps 97

Contents III

IV. Modeling and Simulation of Semiconductor Devices for Application in

Electronics 125

7. Modeling and Simulation of Semiconductor Devices 127

7.1. A Short Introduction to Integrated Circuits . . . . . . . . . . . . . . . . . 128

7.2. Multiscale Structure of Integrated Circuits . . . . . . . . . . . . . . . . . . 129

7.3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.4. Conductivity in a Semiconductor Material . . . . . . . . . . . . . . . . . . 131

7.5. Equilibrium and Non-Equilibrium Statistics . . . . . . . . . . . . . . . . . 133

7.6. The Drift-Diffusion Transport Equations . . . . . . . . . . . . . . . . . . . 138

7.7. The Continuity Equations and Thermal Equilibrium . . . . . . . . . . . . 139

7.8. The Drift-Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.9. The linear resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.9.1. Physical principles of the linear resistor . . . . . . . . . . . . . . . 141

7.9.2. Boundary Conditions at Ohmic Contacts . . . . . . . . . . . . . . 143

7.9.3. The Drift-Diffusion model of the linear resistor . . . . . . . . . . . 145

7.10. The p-n junction diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.10.1. Energy Band Diagram of Isolated Materials . . . . . . . . . . . . . 146

7.10.2. The p-n Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.10.3. The p-n Junction at Thermal Equilibrium . . . . . . . . . . . . . . 148

7.10.3.1. Mathematical Model at Thermal Equilibrium . . . . . . . 150

7.10.3.2. Functional Iteration and Numerical Approximation at Ther-

mal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 151

7.11. Recombination/Generation Mechanisms . . . . . . . . . . . . . . . . . . . 152

7.11.1. Shockley-Hall-Read (SHR) R/G . . . . . . . . . . . . . . . . . . . 153

7.11.2. Auger (Au) R/G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.11.3. The Impact Ionization Generation Process . . . . . . . . . . . . . . 155

7.11.4. Total R/G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.12. The p-n Junction in Non-Equilibrium Conditions . . . . . . . . . . . . . . 157

7.12.1. p-n Junction in Non-Equilibrium . . . . . . . . . . . . . . . . . . . 158

7.12.2. The Gummel Map in Non-Equilibrium . . . . . . . . . . . . . . . . 160

7.12.2.1. The NLP Equation . . . . . . . . . . . . . . . . . . . . . 160

7.12.2.2. The SG Method . . . . . . . . . . . . . . . . . . . . . . . 161

7.12.2.3. The Stabilized PM Method . . . . . . . . . . . . . . . . . 162

7.12.2.4. The SG Method . . . . . . . . . . . . . . . . . . . . . . . 163

7.13. Metal-Semiconductor Contacts: Physical Principles, Modeling and Nu-

merical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.13.1. Energy Band Diagram of Isolated Materials . . . . . . . . . . . . . 165

7.13.2. Energy Band Diagram of MS Contact . . . . . . . . . . . . . . . . 166

7.13.2.1. MS Contact at Thermal Equilibrium . . . . . . . . . . . . 166

7.13.2.2. MS Contact in Non-Equilibrium Conditions . . . . . . . . 168

7.13.3. Current Flow in a MS Junction . . . . . . . . . . . . . . . . . . . . 169

7.13.4. Mathematical Model of a MS Contact . . . . . . . . . . . . . . . . 171

7.13.5. Functional Iteration and Numerical Approximation . . . . . . . . . 172

IV Indice

7.14. The MOS Structure: Physical Principles, 1D Models and Numerical Ap-

proximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Indice 1

Introduction

These notes represent the supporting material for the course entitled

Computational Modeling in Electronics and Biomathematics

that is held within the MSc programme in Mathematical Engineering at Politecnico di

Milano Italy. For further information on this MSc programme, refer to:

http://www.mate.polimi.it/

The course has a duration of 56 hours (classes) plus a number of 28 hours devoted

to laboratory activity in a computer room. The scope of the course is to introduce the

students to the mathematical modeling of classes of problems that can be encountered in

Electronics and Biology, focusing on the several common features shared by these classes,

and to the computational techniques that can be used for their successful simulation.

These notes are divided into 4 distinct parts:

1. Part I introduces the reader to the mathematical modeling and finite element ap-

proximation of time-dependent differential conservation laws in one spatial dimen-

sion. This part of the material is quite general and is propedeutic to the numerical

study of the various models object of the course.

2. Part II gives an introduction to cellular biology and electrophysiology by illustrating

a hierarchy of increasingly complex mathematical models of ion transport across a

cellular membrane based on the use of systems of ordinary and partial differential

equations.

3. Part III describes the functional iteration techniques used to decouple the equation

systems and gives a result of existence and uniqueness for each decoupled equation

in the iteration. The issue of existence and uniqueness of the fixed point of the

iteration is also addressed.

4. Part IV applies models and computational methods adopted for ion transport in

cellular biology to the simulation of a number of semiconductor devices that are

commonly employed in micro and nanoelectronics industry.

All numerical examples and computations illustrated in these notes have been per-

formed with the 1D finite element code bio1d-1.3.0 written using Matlab by Aurelio

Giancarlo Mauri and Riccardo Sacco. The lecture notes, the Matlab files and the sup-

porting material of the computer laboratory sessions can be downloaded at the link

https://beep.metid.polimi.it

upon authorized credentials.

Part I.

Advection-Diffusion Conservation

Laws in 1D:

Models and Numerical

Approximation

1. 1D Conservation Laws: Models,

Properties and Finite Element

Approximation

In this chapter, we consider the numerical approximation of a conservation law in the

one-dimensional (1D) case. The model describes the detailed balance among the time

rate of change of a scalar variable, the flux of such variable due to advective and dif-

fusive transport mechanisms and the effect of a net production rate. The temporal

semidiscretization of the conservation law is performed using the θ-method, which in

the particular choice θ = 1 becomes the Backward Euler (BE) method. The numerical

approximation of the sequence of problems obtained from time discretization is carried

out by means of the Primal Mixed Finite Element Method (PM-FEM) using piecewise

linear continuous and piecewise constant discontinuous finite elements for the discrete

particle density and flux, respectively. We conclude the presentation with a discussion of

a series of simple examples that aim to assess the accuracy and the stability of the PM

formulation, especially in the presence of possibly dominating reactive and/or advective

terms, and the role played by the choice of θ in the numerical stability of the PM-FEM

scheme with respect to the time variable.

6 1. 1D conservation laws

1.1. The 1D conservation law

Let x and t denote the spatial coordinate on the real line and the time variable, respec-

tively. We also denote from now by Ω an open bounded interval (a, b) and by ∂Ω = a, bits boundary on which an outward unit normal n is defined, in such a way that n(a) = −1

and n(b) = +1 (see Fig. 1.1). Unless otherwise explicitely stated, we shall always assume

a = 0 and b = L for a given L > 0. We indicate by t0 a given initial time and by T > 0

the observational time in such a way that IT := (t0, t0 + T ) represents the time interval

of the problem at hand. Unless otherwise stated, we shall always assume t0 = 0. Finally,

we associate with Ω and IT the space-time cylinder QT := Ω× IT .

Figure 1.1.: Computational domain in 1D.

The conservation law in 1D is mathematically represented by the following initial

value/boundary value problem (IVBVP) for the dependent variable u = u(x, t):

∂u

∂t+∂ J

∂ x= P ∀(x, t) ∈ QT (1.1a)

J = v u− µ∂ u∂ x

∀(x, t) ∈ QT (1.1b)

u(x, 0) = u0(x) ∀x ∈ Ω (1.1c)

γ(x, t)J · n(x, t) = α(x, t)u(x, t)− β(x, t) ∀(x, t) ∈ ∂Ω× (0, T ). (1.1d)

Equation (1.1a) has the physical meaning of a conservation equation for the variable u.

A relevant instance of (1.1a) is given by the case where µ = P = 0 and v is the velocity

of a fluid having mass density equal to u. In such a case (1.1a) is referred to as the

continuity equation or fluid mass conservation law.

The function J is the flux density associated with u and accounts for an advective

contribution vu, v = v(x, t) being a given advective field, a diffusive contribution −µ∂ u∂ x

,

according to Fick’s law, µ being the diffusion coefficient of u. The diffusion coefficient

µ is a strictly positive and bounded function in QT . P = P (x, t) is a given function

representing the net production rate.

The given function u0 is the initial datum of the IVBVP whereas eq. (1.1d) expresses

the boundary conditions. In particular, we assume α : (∂Ω× IT )→ R+, whereas γ(x, t)

may be equal to 1 or 0 at x = 0 and x = L for all t ∈ IT . If γ = 1, the corresponding

relation (1.1d) is known as Robin condition, while if γ = 0, relation (1.1d) degenerates

into the so-called Dirichlet condition. In the case of a Robin condition, if α = 0 then the

(outward) flux density J is given at the boundary and equal to −β, and relation (1.1d)

takes the name of Neumann condition. In the case of a Dirichlet condition the value of

the unknown u is given at the boundary and is equal to β/α.

1.2. Conservation for the continuous problem 7

1.2. Conservation for the continuous problem

Let x be the spatial coordinate of a point belonging to the domain Ω = (a, b), let t be

a time instant, and let Vx ∈ R3 denote an arbitrary time-invariant volume centered at x

whose cross-sectional area in the plane y-z is denoted by S (see Fig. 1.2).

Vx

xx

yz

x x

S− +

Figure 1.2.: Volume Vx for the calculation of particle number.

Let us assume that u has the physical meaning of a concentration (number of parti-

cles/unit volume) of a chemical substance. By definition, the number of particles con-

tained in the volume Vx at time t is

Nxu (t) =

∫Vx

u(x, t) dxdydz, (1.2a)

whereas the volumetric net production term is

Px(t) :=

∫Vx

P (x, t) dxdydz. (1.2b)

Theorem 1.2.1 (Conservation of mass). Let x− and x+ denote the abscissae of the

boundary of the volume Vx that surrounds point x (see figure 1.2). Then, we have

dNxu (t)

dt=

∫Vx

P (x, t) dxdydz − S(J(x+, t)− J(x−, t)

). (1.2c)

Proof. The result follows by integrating (1.1a) over the volume Vx and using the funda-

mental theorem of Calculus.

According to Thm. 1.2.1, the IVBVP (1.1) expresses the physical fact that, in sta-

tionary conditions, the net production rate of u inside the volume exactly balances the

net flux of u across the volume boundary. Then, in absence of net production inside the

volume (P = 0), the flux of u crossing the volume is constant (SJ(x+, t) = SJ(x−, t)).

This is, for instance, what happens in the case of an ideal non-reacting fluid (for which

the density is a constant in both time and space). If P > 0 then the outflow flux is

greater than the inflow flux (SJ(x+, t) > SJ(x−, t)), while the opposite situation occurs

if P < 0.

These three cases are summarized in Fig. 1.3 where the black dot represents point

x, each black arrow represents a nodal (particle) current I := J · S while each red

arrow represents Px. In equivalent electrical terms, we see that the mass conservation

principle 1.2.1 is nothing but the classical Kirchhoff’s current law at node x.


Iin

Iout

Iin

Iout

Iin

Px

Px

Iout

Figure 1.3.: Electrical interpretation of flux balance at point x. The quantities Iin and

Iout represent the nodal (particle) current entering and leaving node x. Left:

Px = 0; center: Px > 0; right: Px < 0.

Assumption 1.2.1 (The net production rate). Throughout this chapter we assume that

the net production rate P takes the following form

P (x, t) = g(x, t)− κ(x, t)u(x, t) ∀(x, t) ∈ QT (1.3)

where g and κ are nonnegative given functions of position and time. Relation (1.3) is the

mathematical model of a first-order chemical reaction. The function g represents the pro-

duction mechanism of u whereas the function κu represents the consumption mechanism

of u occurring at a rate κ (units s−1).

1.3. Time semidiscretization

Let us divide the time interval IT into NT ≥ 1 subintervals τk = (tk, tk+1), k = 0, . . . , NT .

For sake of simplicity we assume from now on that the width of each time subinterval is

the same and equal to ∆t := T/NT . For any function Φ = Φ(x, t) we set Φk := Φ(x, tk).

Then, for each θ ∈ [0, 1] we introduce the θ-method for the temporal semidiscretization

of the IVBVP (1.1):

given u0, for each k = 0, . . . , NT − 1 solve:

uk+1 − uk

∆t+ θ

∂ Jk+1

∂ x+ (1− θ)∂ J

k

∂ x+ θκk+1uk+1 = θgk+1 + (1− θ)(gk − κkuk) ∀x ∈ Ω

(1.4a)

where

Jq = vq uq − µq ∂ uq

∂ x∀x ∈ Ω, q = 0, . . . , NT (1.4b)

and the boundary conditions are defined as

γqJq · n = αquq − βq ∀x ∈ ∂Ω, q = 0, . . . , NT . (1.4c)

If θ = 0 the time discretization is called Forward Euler method (FE), if θ = 1 the

time discretization is called Backward Euler method (BE) whereas if θ = 1/2 the time

discretization is called Crank-Nicolson method (CN). This latter scheme is also well-

known as trapezoidal integration method (TI)). The FE method is an explicit method

whereas all the schemes obtained for θ ∈ (0, 1] are implicit methods. We refer to Sect. 1.12

and Sects. 1.13.2, 1.14 for the theoretical and computational analysis of the performance

of the θ-method for specific values of the parameter θ. Unless otherwise stated, we always

1.4. The continuous maximum principle 9

assume θ = 1 (BE method) so that at each time level tk+1, k = 0, . . . , NT − 1 we are led

to solving the following Diffusion-Advection-Reaction (DAR) two-point BVP:

∂ J

∂ x+ σu = f ∀x ∈ Ω (1.5a)

J = v u− µ∂ u∂ x

∀x ∈ Ω (1.5b)

γ(x)J · n(x) = α(x)u(x)− β(x) ∀x ∈ ∂Ω (1.5c)

where, recalling (1.3), we have σ := 1/∆t + κ and f := uk/∆t + gk+1. In writing (1.5)

we are using the same symbol u (without temporal superscript). The function σ = σ(x)

is called reaction term whereas the function f = f(x) is called production term.

Assumption 1.3.1. From now on, otherwise differently stated, we assume that σ(x) ≥ 0

for all x ∈ Ω and f(x) ≥ 0 for all x ∈ Ω. We also assume that

0 < µ0 ≤ µ(x) ≤ µ1 < +∞ ∀x ∈ Ω. (1.5d)

1.4. The continuous maximum principle

To examine some important properties associated with the BVP (1.5), we introduce the

second-order linear differential operator

Lu :=∂ J

∂ x+ σu =

∂

∂ x

(v u− µ∂ u

∂ x

)+ σu : Ω→ R. (1.6)

Theorem 1.4.1 (Inverse-monotonicity). Let u ∈ C2(Ω) ∩ C0(Ω) be such that:

Lu(x) ≥ 0 for all x ∈ Ω (1.7a)

u(x) ≥ 0 for all x ∈ ∂Ω. (1.7b)

Then, we have

u(x) ≥ 0 for all x ∈ Ω. (1.7c)

In such an event, we say that L is inverse-monotone or, equivalently, positivity-preserving.

Unless otherwise stated, we shall always assume throughout the remainder of the text

that L is inverse monotone.

Remark 1.4.1 (Physical interpretation of inverse monotonicity). Inverse monotonicity

has an important physical significance, because it mathematically expresses the obvious

fact that the dependent variable of the problem, say, a concentration, a temperature or a

mass density, cannot take negative values.

An important consequence of the inverse-monotonicity property is the following result.

Theorem 1.4.2 (Comparison principle). Suppose that there exists a function φ ∈ C2(Ω)∩C0(Ω) such that

Lu(x) ≤ Lφ(x) ∀x ∈ Ω (1.8a)

u(x) ≤ φ(x) ∀x ∈ ∂Ω. (1.8b)


Then, we have

u(x) ≤ φ(x) ∀x ∈ Ω. (1.8c)

In such an event, we say that φ is a barrier function for u.

Proof. We only need to apply Thm. 1.4.1 to the function w(x) := φ(x)− u(x) and then

we get (1.8c) using (1.7c).

Combining (1.7c) and (1.8c), we obtain the following result which is a very useful

tool in the approximation process of the BVP (1.5).

Theorem 1.4.3 (Maximum principle). Suppose that L is inverse-monotone and that

the comparison principle holds for a suitable barrier function φ. Then, setting Mφ :=

maxx∈Ω

φ(x), we have

0 ≤ u(x) ≤Mφ ∀x ∈ Ω (1.9)

and we say that u satisfies a maximum principle (MP).

1.5. The weak formulation

Let us consider the boundary conditions (1.5c) in the special case where γ = β = 0

and α = 1 on ∂Ω. This corresponds to the homogeneous Dirichlet boundary condition

u(0) = u(L) = 0. Then, we introduce the following pair of functional spaces:

Q := L2(Ω) =

q : Ω→ R |

∫Ωq2 dx < +∞

(1.10a)

V := H10 (Ω) =

w : Ω→ R |w ∈ L2(Ω),

∂ w

∂ x∈ L2(Ω), w(0) = w(L) = 0

. (1.10b)

Finally, we assume that f ∈ L2(Ω), and that µ, v, and σ belong to L∞(Ω), the space

of essentially superiorly bounded functions in the domain Ω. To construct an integral

formulation of problem (1.5), known as the weak formulation, we write Eq. (1.5b) as

µ−1 (J − vu) +∂ u

∂ x= 0,

we multiply this latter relation by an arbitrary function q in Q and integrate the resulting

expression over Ω, to obtain∫Ωµ−1 (J − vu) q dx+

∫Ωq∂ u

∂ xdx = 0 ∀q ∈ Q. (1.10c)

We proceed in an analogous manner with Eq. (1.5a) and multiply it by an arbitrary

function w in V , to obtain∫Ωw∂ J

∂ xdx+

∫Ωσuw dx =

∫Ωfw dx ∀w ∈ V. (1.10d)

Since we are going to search for J into the space Q, we need to integrate by parts the

first term in (1.10d) because of lack of regularity of J . By doing so, Eq. (1.10d) becomes∫Ω

∂

∂ x(Jw) dx−

∫ΩJ∂ w

∂ xdx+

∫Ωσuw dx =

∫Ωfw dx ∀w ∈ V. (1.10e)

1.6. Finite element approximation 11

Now, the use of Green’s theorem in the first integral of (1.10e) yields∫Ω

∂

∂ x(Jw) dx =

∫∂ΩwJ · n d(∂Ω) = w(0)J(0) · n(0) + w(L)J(L) · n(L) = 0

because of the fact that w ∈ V .

Collecting the above relations, the weak (or distributional) solution of (1.5) with

homogeneous Dirichlet boundary conditions is the function pair (J, u) ∈ (Q × V ) such

that: ∫ L

0µ−1 (J − v u) q dx+

∫ L

0q∂ u

∂ xdx = 0 ∀q ∈ Q (1.11a)

−∫ L

0J∂ w

∂ xdx+

∫ L

0σuw dx =

∫ L

0f w dx ∀w ∈ V. (1.11b)

The following result can be proved (cf. [RT91] and [QV97], Chapt. 5).

Theorem 1.5.1 (Existence and uniqueness). Assume that f ∈ L2(Ω), and that µ, v,

and σ belong to L∞(Ω). Assume also that ∂ v/∂ x belongs to L∞(Ω), with

σ(x) +1

2

∂ v(x)

∂ x≥ 0 in Ω.

Then the solution pair (J, u) ∈ (Q× V ) of system (1.11) exists and is unique.

1.6. Finite element approximation

In this section, we are going to study the numerical approximation of the model BVP (1.5),

in the simple case of homogeneous Dirichlet boundary conditions, using the so-called Fi-

nite Element Method (FEM). The main idea of the FEM is to project the weak form (1.11)

over a pair of suitable chosen finite dimensional subspaces Qh ⊂ Q and Vh ⊂ V , h > 0

being the so-called discretization parameter (see Fig. 1.4). In this way, the integral for-

mulation (1.11) is transformed into a linear algebraic system whose solution yields the

approximation pair (Jh, uh) ∈ (Qh × Vh).

Figure 1.4.: The FEM as a projection scheme. The infinite dimensional spaces V and Q

are denoted by circles. The finite dimensional subspaces Vh ⊂ V and Qh ⊂ Qare denoted as arrows.


1.6.1. Finite element spaces

We start by introducing a partition Th of Ω into a number M = M(h) ≥ 2 of 1-simplex

(intervals) Ki = [xi, xi+1], i = 1, . . . ,M , in such a way that x1 := a and xM+1 := b. We

denote by hi := xi+1 − xi the length of each interval and set h := maxTh

hi. The partition

Th takes the name of triangulation of the domain Ω. Each Ki is an element of the

triangulation, while the quantities xi, i = 1, . . . , N are the vertices of the triangulation,

having set N := M+1. The same terminology is adopted when Ω is a bounded set of Rd,d ≥ 2, and in such a case 2-simplices are triangular elements (d = 2) while 3-simplices

are tetrahedral elements (d = 3). We associate with Th the following pair of function

spaces:

Qh := qh ∈ Q : qh|Ki ∈ P0(Ki) ∀Ki ∈ Th (1.12a)

Vh :=wh ∈ C0(Ω) ⊂ V : wh|Ki ∈ P1(Ki) ∀Ki ∈ Th

. (1.12b)

Qh is the vector space of piecewise constant polynomials defined over Ω, while Vh is the

vector space of piecewise linear continuous polynomials over Ω. Qh and Vh are the finite

element spaces of degree 0 and 1 associated with Th. The dimension of Qh is simply

equal to the number of elements M , while the dimension N of Vh is equal to the number

of vertices minus the two endopints x = 0 and x = L where, because of boundary

conditions, the solution is known, so that N = M + 1 − 2 = M − 1. Fig. 1.5 shows an

example of Th (with M = 4) and of functions qh and wh.

Figure 1.5.: Triangulation in 1D and finite element functions. Black bullets denote nodal

values, while black ticks and black squares are the spatial positions of the

degrees of freedom of Vh and Qh, respectively.


1.6.2. Basis functions

By definition of vector space, any function qh ∈ Qh and wh ∈ Vh can be written in the

form:

qh(x) =M∑k=1

qkψk(x) (1.13a)

wh(x) =N∑j=1

wjϕj(x) (1.13b)

where ψk, k = 1, . . . ,M and ϕj , j = 1, . . . , N are the basis functions of Qh and Vh, while

the real numbers qk and wj are degrees of freedom of qh and wh, that is, the coordinates

of qh and wh with respect to the bases ψk and ϕj, respectively. A particularly

interesting choice for the basis of Vh is represented by the so-called Lagrangian basis

functions, such that

ϕi(xj) = δij i, j = 1, . . . , N. (1.14)

Using (1.14) in the evaluation of (1.13b) at a generic node x = xi yields

wh(xi) =N∑j=1

wjϕj(x) =N∑j=1

wjδji = wi i = 1, . . . , N.

This result allows us to interpret the quantities wi as the nodal values of wh at the nodes

of Th. The Lagrangian property in the case of the basis functions of Qh simply consists

of taking

ψk(x) = χk(x) :=

1 if x ∈ Kk

0 elsewhere(1.15)

with k = 1, . . . ,M . The function χk is the characteristic function of Kk. A basis function

of Qh and two basis functions of Vh associated with the triangulation Th of Fig. 1.5 are

plotted in Fig. 1.6.

Figure 1.6.: Basis functions of Qh and Vh. We have dim(Qh) = 4 and dim(Vh) = 3.


1.6.3. Finite element formulation

Given a function g : Ω→ R such that∫

Ω |g(x)|dx < +∞, we indicate by g|k the restriction

of g over the element Kk, k = 1, . . . ,M , and we set

g|k :=

∫Kk

g|k(x) dx

hk. (1.16a)

The primal mixed FE approximation of the BVP (1.5) reads:

Find (Jh, uh) ∈ (Qh × Vh) such that:∫ L

0µ−1 (Jh − v uh) qh dx+

∫ L

0

∂ uh∂ x

qh dx = 0 ∀qh ∈ Qh (1.16b)

−∫ L

0Jh∂ wh∂ x

dx+

∫ L

0σuhwh dx =

∫ L

0f wh dx ∀wh ∈ Vh. (1.16c)

By definition of basis of a vector space, and using the expansions (1.13), system (1.16)

becomes:

Jk

∫Kk

(µ|k)−1 dx−N∑j=1

uj

∫Kk

(µ|k)−1v|kϕj dx+

N∑j=1

uj

∫Kk

∂ ϕj∂ x

dx = 0 k = 1, . . . ,M

(1.17a)

−M∑k=1

Jk

∫ L

0ψk∂ ϕi∂ x

dx+N∑j=1

uj

∫ L

0σϕjϕi dx =

∫ L

0f ϕi dx i = 1, . . . , N.

(1.17b)

The M finite element equations (1.17a) constitute a diagonal system for the approxi-

mate flux densities Jk, k = 1, . . . ,M , so that they can be solved independently over each

mesh element Kk. To do this, we replace v|k with its integral mean value v|k to obtain

Jk = v|kuk + uk+1

2−Hk(µ)

uk+1 − ukhk

k = 1, . . . ,M (1.18)

where

Hk(µ) := ((µ|k)−1)−1 (1.19)

is the harmonic average of µ over the element Kk. Comparing (1.18) with the exact

expression (1.5b), we see that the following approximations have been introduced over

the element Kk:

µ(x)|k ⇒ Hk(µ), (1.20a)

∂ u(x)

∂ x|k ⇒ uk+1 − uk

hk, (1.20b)

(v(x)u(x))|k ⇒ v|kuk + uk+1

2≡ v|k uh|k. (1.20c)

Approximation (1.20a) states that the exact diffusion coefficient µ has been replaced

by its harmonic average. Approximation (1.20b) states that the exact diffusive flux

∂ u(x)/∂ x has been replaced by the first-order incremental ratio (uk+1 − uk)/hk. Ap-

proximation (1.20c) states that the exact advective flux v(x)u(x) has been replaced by

its average v|k uh|k.


Having eliminated the variables Jk, k = 1, . . . ,M , in favor of the sole variables uj ,

j = 1, . . . , N , we can now replace the former variables into the finite element equa-

tions (1.17b). By doing so, we obtain a tridiagonal system for the N = M − 1 unknowns

ui, i = 1, . . . , N , that can be written in matrix form as

Ku = b (1.21)

where K ∈ RN×N is the stiffness matrix, u ∈ RN is the column vector of nodal unknowns

and b ∈ RN is the (column) load vector. The stiffness matrix can be expressed as the

sum of four matrices

K = Kd + Ka + Kr

where Kd is the diffusion matrix, Ka is the advection matrix and Kr is the reaction

matrix.

1.6.4. Connection between the PM method and the standard FE method

It is interesting (and important) to compare the PM finite element scheme with the

standard FE method for the approximation of the DAR problem (1.5). To this purpose

we consider the simplified example, well known as the Generalized Poisson Problem, to

be solved in the open interval Ω = (0, L):

find u = u(x) such that:

∂

∂x

(−µ∂u

∂x

)= f ∀x ∈ Ω (1.22a)

u(0) = u(L) = 0, (1.22b)

where µ is a given function satisfying (1.5d) and f is a given function in L2(Ω). The

BVP (1.22) is a special case of (1.5) having set σ = γ = β = 0 and α = 1.

The standard Galerkin FE method of degree 1 for the numerical approximation

of (1.22) consists of the following discrete problem:

Find usth ∈ Vh such that:∫ L

0µ∂ usth∂ x

∂ wh∂ x

dx =

∫ L

0f wh dx ∀wh ∈ Vh. (1.23a)

Using the expansion (1.13b) we easily see that (1.23a) is equivalent to solving the linear

algebraic system

KstUst = b, (1.23b)

where the right-hand side b is the same as in (1.21), the vector Ust ∈ RN contains

the nodal values of the unknown function usth whereas the stiffness matrix Kst has the

following expression

(Kst)ij =

∫ L

0µ∂ ϕj(x)

∂ x

∂ ϕi(x)

∂ xdx, i, j = 1, . . . , N. (1.23c)

Noting that ∂ ϕi(x)/∂ x and ∂ ϕj(x)/∂ x are constant over each mesh element, and using

the definition of integral average (1.16a) we can compute for each row i = 1, . . . , N the


three nonzero entries of the stiffness matrix Kst as:

(Kst)i,i−1 =

∫ xi

xi−1

µ|i−1∂ ϕi−1(x)

∂ x

∂ ϕi(x)

∂ xdx = −µ|i−1

hi−1, (1.23d)

(Kst)i,i =

∫ xi

xi−1

µ|i−1∂ ϕi(x)

∂ x

∂ ϕi(x)

∂ xdx

+

∫ xi+1

xi

µ|i∂ ϕi(x)

∂ x

∂ ϕi(x)

∂ xdx =

µ|i−1

hi−1+µ|ihi

(1.23e)

(Kst)i,i+1 =

∫ xi+1

xi

µ|i∂ ϕi+1(x)

∂ x

∂ ϕi(x)

∂ xdx = −µ|i

hi. (1.23f)

The use of the PM method for the finite element approximation of (1.22) leads to solving

the following linear algebraic system

KPMUPM = b, (1.23g)

where, for each row i = 1, . . . , N , the three nonzero entries of the stiffness matrix KPM

acting on the vector of the nodal unknowns UPM are given by:

(KPM )i,i−1 =

∫ xi

xi−1

((µ|i−1)−1)−1∂ ϕi−1(x)

∂ x

∂ ϕi(x)

∂ xdx = −Hi−1(µ)

hi−1, (1.23h)

(KPM )i,i =

∫ xi

xi−1

((µ|i−1)−1)−1∂ ϕi(x)

∂ x

∂ ϕi(x)

∂ xdx

+

∫ xi+1

xi

((µ|i)−1)−1∂ ϕi(x)

∂ x

∂ ϕi(x)

∂ xdx =

Hi−1(µ)

hi−1+Hi(µ)

hi(1.23i)

(KPM )i,i+1 =

∫ xi+1

xi

((µ|i)−1)−1∂ ϕi+1(x)

∂ x

∂ ϕi(x)

∂ xdx = −Hi(µ)

hi. (1.23j)

Comparing the two sets of expressions of the stiffness matrix entries for the standard

Galerkin FE method of degree 1 and for the PM FE method we immediately see that

the difference between the two schemes consists in the different average of the diffusion

coefficient over each mesh element. In particular, the standard Galerkin FE method

approximates the diffusivity with its standard integral average whereas the PM method

approximates the diffusivity with its harmonic average. If µ is constant or mildly varying

with x the two approaches behave substantially in the same manner. However, if µ

experiences large variations over the spatial coordinate, it is expected that the solution

computed by the two methods is strongly different. The theoretical and computational

study carried out in [BO83] advocates a clear superiority of the harmonic average over

the standard average, in terms of nodal accuracy of the computed solution uh, in the

case where µ is a function with a finite jump of increasing size. A similar performance

of the harmonic average approach can be verified in multi-dimensional simulations of

semiconductor devices using the Drift-Diffusion model (see [BMP87, BMP89b, BMP89a,

BMM+05, BMM+06]).

From the above analysis, we can also conclude that the standard Galerkin FE method

of degree 1 can be interpreted a posteriori as a PM finite element method in which the

approximate flux Jh has the following expression over each element Kk ∈ Th

Jstk = −µk∂usth∂x

(1.23k)

1.7. Conservation for the discrete problem 17

whereas the corresponding approximate flux computed by the PM method is

JPMk = −Hk(µ)∂uPMh∂x

. (1.23l)

1.7. Conservation for the discrete problem

Let us consider the discretized equilibrium equation (1.17b) at each internal node xi,

i = 2, . . . , N − 1. We get the following system of nodal conservation laws

Ji − Ji−1 = (fi − σui)(hi−1 + hi

2

)i = 2, . . . , N − 1.

The above equation expresses the fact that at each internal node of the partition the

output flux Ji is equal to the sum of the input flux Ji−1 plus the nodal net production

term Pi := (fi − σui)(hi−1 + hi)/2, in complete analogy with the conclusions of the

previous section. In particular, if f = σ = 0, we get strong flux conservation at each

internal node xi, i = 2, . . . , N − 1, otherwise flux is conserved in a weak sense, i.e., its

jump at the node xi is balanced by the nodal net production rate. In conclusion, the

PM finite element scheme reproduces in a remarkable manner on the discrete level the

conservation properties enjoyed by the solution of the BVP (1.5) at the continuous level:

exact current conservation across a control volume in the case of no production, weak

current conservation across a control volume in the case of non-zero net production.

1.8. The discrete maximum principle

In this section, we consider the discrete counterpart of the maximum principle introduced

in Sect. 1.4. For this, we assume that the solution u of the BVP (1.5) satisfies the a

priori estimate (1.9), and we characterize the conditions under which the same estimate

is satisfied also by the FE approximation uh computed by solving the linear algebraic

system (1.21). Should this occur, we say that uh satisfies a discrete maximum principle

(DMP). With this scope, the following definitions turn out to be useful (cf. [RST96]).

Definition 1.8.1 (Inverse monotone matrix). An invertible square matrix A of size n

is said to be inverse monotone if

A−1 ≥ 0 (1.24)

the inequality being understood in the element-wise sense.

The following result is a consequence of Def. 1.8.1.

Proposition 1.8.1. Assume that A is inverse-monotone. Then

Aw ≤ Az ⇒ w ≤ z

(always in the element-wise sense), w, z being two vectors of Rn.

Proof. Let v := A(z−w). By assumption, we have v ≥ 0, so that, multiplying by A−1

and using (1.24), we still have A−1v ≥ 0, i.e., z−w ≥ 0.


Remark 1.8.1. Assume that x is the solution of the linear system Ax = b, with b ≥ 0

and A satisfying Def. 1.8.1. Then every component of x is nonnegative, because of

Prop. 1.8.1. This property is of particular importance if x has the physical meaning of a

concentration.

A special class of monotone matrices is that introduced below (cf. [Var62]).

Definition 1.8.2 (M-matrix). An invertible square matrix A is an M-matrix if:

• A is inverse-monotone;

• Aij ≤ 0 for i 6= j.

To verify the property of being an M-matrix using directly Def. 1.8.2 is, in general,

prohibitive. The following (necessary and sufficient) condition is useful (cf. [RST96]).

Theorem 1.8.1 (Discrete comparison principle). Let A be an invertible matrix with

non-positive off diagonal entries (Aij ≤ 0 for i 6= j). Then, A is an M-matrix if and

only if there exists a positive vector e such that Ae ≥ 0 (in the component-wise sense),

with at least one row index i∗ such that (Ae)i∗ > 0.

Remark 1.8.2. A first choice to try for the test vector in Thm. 1.8.1 is

e = [1, 1, . . . , 1]T ∈ Rn.

Thus, computing the matrix-vector product Ae amounts to computing the row sum for

each row of A.

The property of K to be an M-matrix yields the following important result.

Theorem 1.8.2 (Sufficient condition for DMP). Assume that the stiffness matrix K of

the FEM system (1.21) is an M-matrix. Then uh satisfies the DMP, i.e.

0 ≤ uh(x) ≤Mφ ∀x ∈ Ω. (1.25)

1.9. FE approximation of 1D model problems

In the following sections, we apply Thms. 1.8.2- 1.8.1 to the study of three particular

subproblems deriving from (1.5). For ease of presentation, we set L = 1 and assume that

the finite element partition is uniform, with mesh size h = 1/M . We also assume that

the coefficients µ, v, σ and f are constant, with f = 1. We set γ = 0, α = 1 and β = 0,

so that u1 = β(0) = 0 and uN = β(1) = 0, respectively. The linear system (1.21) has

size N = M − 1.

1.9. FE approximation of 1D model problems 19

1.9.1. The diffusion BVP

In this case we have σ = v = 0, so that

K = Kd =µ

h

2 −1 0 . . . 0

−1 2 −1 . . . 0

0. . .

. . .. . . 0

0 . . . −1 2 −1

0 . . . 0 −1 2

. (1.26)

The above matrix is symmetric and positive definite. Applying the discrete comparison

principle 1.8.1 with e = [1, . . . , 1]T ∈ Rn yields

Ke =µ

h[1, 0, . . . , 0, 1]T

which allows to conclude that K is an M-matrix. Using Thm. 1.8.2 with φ(x) = x(1 −x)/(2µ)(≡ u(x)!), we obtain that uh satisfies the DMP with Mφ = 1/(8µ).

1.9.2. The reaction-diffusion BVP

In this case we have v = 0, so that K = Kd + Kr where Kd is given by (1.26) while the

reaction matrix is

Kr =σh

6

4 1 . . . 0

1 4 1 . . . 0

0. . .

. . .. . . 0

0 . . . 1 4 1

0 . . . 0 1 4

.

Matrix K is symmetric and positive definite. The off-diagonal entries Kij are < 0 (to be

more conservative) if and only if the following condition holds

Perdloc < 1 (1.27)

where

Perdloc :=σh2

6µ(1.28)

is the local Peclet number associated with the reaction-diffusion problem. Thus, provided

that the mesh size h is sufficiently small to satisfy (1.27), we can use Thm. 1.8.2 with

φ(x) = 1/σ, to conclude that uh satisfies the DMP with Mφ = 1/σ. To see the role of

condition (1.27), we study the reaction-diffusion BVP in the case µ = 10−4 and σ = 1 on

a grid with M = 10 (h = 0.1). The corresponding value of the Peclet number is equal to

16.7 so that (1.27) is violated. The effect of this is illustrated in Fig. 1.7 where the graph

of the computed solution pair (uh, Jh) is superposed to the exact solution pair (u, J).

Spurious (unphysical) oscillations arise in the neighbourhood of x = 0 and x = 1.

These latter can be removed by reducing the size of the grid. For instance, taking M = 41

(h = 1/41) we obtain the result shown in Fig. 1.8.

Oscillations have disappeared, and it can be checked that uh is uniformly bounded

from above by the function φ(x) = 1/σ over [0, 1] so that the DMP is satisfied.


x

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2 uh

uex

x

0 0.2 0.4 0.6 0.8 1-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Jh

Jex

Figure 1.7.: Reaction-diffusion problem: exact and approximate solutions using M = 10.

x

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

uh

uex

x

0 0.2 0.4 0.6 0.8 1-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Jh

Jex

Figure 1.8.: Reaction-diffusion problem: exact and approximate solutions computed us-

ing M = 41.

1.9.3. The advection-diffusion BVP

In this case we have σ = 0 and we take v > 0, so that K = Kd + Ka where Kd is given

by (1.26) while the advection matrix is

Ka =v

2

0 +1 . . . 0

−1 0 +1 . . . 0

0. . .

. . .. . . 0

0 . . . −1 0 +1

0 . . . 0 −1 0

.

1.10. Stabilization techniques 21

Matrix K is positive definite. The off-diagonal entries Kij are < 0 if and only if the

following condition holds

Peadloc < 1 (1.29)

where

Peadloc :=|v|h2µ

(1.30)

is the local Peclet number associated with the advection-diffusion problem. Thus, pro-

vided that the mesh size h is sufficiently small to satisfy (1.29), we can use Thm. 1.8.2

with φ(x) = x/v, to conclude that uh satisfies the DMP with Mφ = 1/v. To see the role

of condition (1.29), we study the advection-diffusion BVP in the case µ = 5 · 10−3 and

v = 1 on a grid with M = 10 (h = 0.1). The corresponding value of the Peclet number

is equal to 10 so that (1.29) is violated. The effect of this is illustrated in Fig. 1.9 where

the computed solution (uh, Jh) is superposed to the exact solution pair (u, J).

x

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

uh

uex

x

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Jh

Jex

Figure 1.9.: Advection-diffusion problem: exact and approximate solutions using M =

10.

Spurious (unphysical) oscillations arise in the neighbourhood of x = 1 and propagate

back throughout the entire domain [0, 1] polluting the overall quality of the computed

solution. The oscillations can be removed by reducing the size of the grid. For instance,

taking M = 100 (h = 1/100) we obtain the result shown in Fig. 1.10.

Oscillations have disappeared, and it can be checked that uh is uniformly bounded

from above by the barrier function φ(x) = x/v over [0, 1] so that the DMP is satisfied.

1.10. Stabilization techniques

The multi-dimensional analogues of conditions (1.27) or (1.29) may lead to the solution

of linear algebraic systems that are computationally very demanding because of the dra-

matic increase of the total number of dofs. As a matter of fact, denoting by Nmin the

minimum number of nodes required to satisfy (1.27) or (1.29), the corresponding number

of dofs in 2D and 3D is O(N2min) and O(N3

min), respectively. An alternative to the above


x

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

uh

uex

x

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Jh

Jex


100.

difficulties is represented by the use of a stabilized FEM. This latter method is charac-

terized by seeking an approximate solution uh ∈ Vh and a corresponding approximate

flux density Jh ∈ Qh, which in general do not coincide with the solution pair uh and Jh

computed by solving system (1.21), in such a way that the DMP is satisfied at a much

cheaper cost than required by a naive (uniform) reduction of the grid size. In matrix

terms, a stabilized FEM can be written as

Ku = b (1.31)

where K and b are the modified stiffness matrix and load vector, respectively, while u is

the vector of nodal unknowns. In the following sections, we illustrate how to select the

modified matrix and load vectors in such a way to satisfy the DMP without restrictions

on the mesh size.

1.10.1. Stabilization of the reaction-diffusion BVP

Given a continuous function ζ = ζ(x), we consider the problem of computing the integral

of ζ over the element Kk

IKk(ζ) =

∫Kk

ζ(x)dx. (1.32a)

A possible approximate manner for solving the above problem is to use the trapezoidal

quadrature rule

IKk(ζ) ' hk

2(ζk + ζk+1). (1.32b)

Assuming that ζ ∈ C2(Kk), the quadrature error associated with (1.32b) can be estimated

as

|IKk(ζ)− hk

2(ζk + ζk+1)| ≤ M

12h3k (1.32c)


whereM := maxx∈Kk

|ζ ′′(x)|. Relation (1.32c) shows that the use of the trapezoidal integra-

tion formula introduces a quadrature error of the order of h3k. In the case of the reaction-

diffusion BVP the stabilization approach consists of using the trapezoidal quadrature

rule (1.32b) to compute the entries Krij and bi, i, j = 1, . . . , n. By doing so, b ≡ b be-

cause the trapezoidal rule is exact if f is constant, while K = Kd+Kr, where Kr = σhI,

I being the identity matrix of order n. Applying the discrete comparison principle 1.8.1

with e = [1, . . . , 1]T ∈ Rn yields

Ke =[µh

+ σh, σh, . . . , σh,µ

h+ σh

]T> 0

which allows to conclude that K is an M-matrix for all h > 0 and uh satisfies the DMP

with Mφ = 1/σ.

x

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

uh

uex

x

0 0.2 0.4 0.6 0.8 1-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Jh

Jex

Figure 1.11.: Reaction-diffusion problem. Exact and approximate solutions computed

using M = 10. The lumping stabilization has been used.

The effect of the use of the trapezoidal numerical quadrature can be seen in Fig. 1.11,

which shows the computed solution pair (uh, Jh) superposed to the exact solution pair

(u, J) over the same mesh as in Fig. 1.7. Oscillations have disappeared, and it can be

checked that maxx∈[0,1]

uh(x) = 1/σ = 1 so that the DMP is satisfied.

Remark 1.10.1 (Reduced integration and lumping). The use of trapezoidal quadrature

is called reduced integration, because instead of computing exactly the entries Krij, we

are deliberately introducing a quadrature error. The advantage of reduced integration is

that the reaction matrix becomes diagonal so that the off-diagonal entries of K are ≤ 0

for all h > 0. The diagonalization of the reaction matrix is often called lumping because

the entries Krii can be interpreted as obtained by summing by row the matrix Kr. This is

equivalent to “lump” the weight of the reaction term into each mesh node xi.


1.10.2. Stabilization for the advection-diffusion BVP

In the case of the advection-diffusion BVP the stabilization approach consists of replacing

the diffusion coefficient µ with the following modified expression

µ := µ(

1 + Φ(Peadloc))

(1.33)

where Φ : R+ → R+ is a suitable stabilization function to be chosen in such a way that:

Φ(t) ≥ 0 if t ≥ 0; (1.34a)

limt→0+

Φ(t) = 0. (1.34b)

The introduction of the artificial diffusion µΦ(Peadloc) is equivalent to solving the modified

advection-diffusion two-point BVP:

∂ J

∂ x= f in Ω = (0, 1) (1.35a)

J = v u− µ∂ u∂ x

in Ω = (0, 1) (1.35b)

u(0) = u(1) = 0. (1.35c)

Definition 1.10.1 (Peclet number of the modified advection-diffusion problem). The

Peclet number associated with (1.35) is

Pead

loc :=|v|h2µ

=|v|h

2µ(1 + Φ(Peadloc))=

Peadloc1 + Φ(Peadloc)

. (1.36)

The following condition for the proper choice of the stabilization function is an im-

mediate consequence of Def. 1.10.1.

Proposition 1.10.1 (Choice of Φ).

Pead

loc < 1 ⇔ Φ(Peadloc) > Peadloc − 1. (1.37)

Thus, provided that Φ is chosen in such a way to satisfy (1.37), we can use the same

arguments as in Sect. 1.9.1 to conclude that uh satisfies the DMP with Mφ = 1/v. Two

special choices of Φ satisfying (1.37) are:

• Upwind (UP) stabilization function

ΦUP (t) := t t ≥ 0 (1.38)

• Scharfetter-Gummel (SG) stabilization function

ΦSG(t) := t− 1 + B(2t) t ≥ 0 (1.39)

where

B(X) :=

X

eX − 1X 6= 0

1 X = 0

(1.40)

is the inverse of the Bernoulli function. A plot of B(X) and B(−X) is reported in

Fig. 1.12.


Figure 1.12.: Plot of B(X) (solid line) and of B(−X) (dashed line).

The function B enjoys the following properties:

B(X) > 0 ∀X ∈ R; (1.41)

B(0) = 1; (1.42)

eXB(X) = B(−X) = X + B(X); (1.43)

limX→+∞

B(X) = 0+, limX→−∞

B(X) = −X; (1.44)

limX→+∞

B(−X) = X, limX→−∞

B(X) = 0+. (1.45)

The effect of the use of UP and SG artificial diffusion stabilizations can be seen in

Fig. 1.13 and Fig. 1.14, respectively, where the computed solution pair (uh, Jh) is plotted

superposed to the exact solution pair (u, J) over the same mesh as in Fig. 1.9.

x

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9u

h

uex

x

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Jh

Jex


10 and the UP stabilization.

No oscillations affect the computed solutions, and it can be checked that in both cases

uh is uniformly bounded from above by the barrier function φ(x) = x/v over [0, 1] so


x

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9u

h

uex

x

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Jh

Jex


10 and the SG stabilization.

that the DMP is satisfied.

1.11. Convergence analysis

We conclude the overview of the FEM applied to the numerical solution of the 1D con-

servation law (1.5) by providing a quantitative measure of the accuracy of the computed

solution pair (Jh, uh) ∈ (Qh×Vh) as a function of h in the case of homogeneous Dirichlet

boundary conditions u(0) = u(L) = 0, corresponding to setting γ = β = 0 and α = 1

on ∂Ω. The function spaces Q and V defined in (1.10) are both Hilbert spaces endowed

with the norms:

‖q‖Q :=

(∫Ωq2(x) dx

)1/2

q ∈ Q (1.46a)

‖w‖V = ‖w‖H10 (Ω) :=

(∥∥∥∂ w∂ x

∥∥∥2

L2(Ω)

)1/2w ∈ V. (1.46b)

In the remainder of the discussion, we denote by C a positive constant independent of

h, but possibly depending on u, J and on problem data, whose value is not necessarily

the same at each occurrence.

Definition 1.11.1 (Convergence). The FEM is said to be convergent if

limh→0

(‖u− uh‖V + ‖J − Jh‖Q) = 0. (1.46c)

Moreover, the FEM is said to be convergent with order p > 0, if there exists a positive

constant C independent of h such that

‖u− uh‖V + ‖J − Jh‖Q ≤ Chp. (1.46d)

The following convergence results can be proved to hold (cf. [RT91]).

1.11. Convergence analysis 27

Theorem 1.11.1 (Convergence of the primal-mixed FE approximation). Assume that

the solution pair of the BVP (1.5) with homogeneous Dirichlet boundary conditions is

such that u ∈ (H2(Ω) ∩ H10 (Ω)) and J ∈ H1(Ω), and let h be sufficiently small. Then,

there exists a constant C > 0 independent of h such that

‖u− uh‖V + ‖J − Jh‖Q ≤ Ch(‖u‖H2(0,L) + ‖J‖H1(0,L)). (1.46e)

Thus, the FEM (in non-stabilized and stabilized forms) is convergent with order 1 ac-

cording to Def. 1.11.1.

Theorem 1.11.2 (Convergence of uh to u in the L2 norm). From (1.46e) we immediately

have

‖u− uh‖L2 < ‖u− uh‖V + ‖J − Jh‖Q ≤ Ch(‖u‖H2(0,L) + ‖J‖H1(0,L)). (1.46f)

Thus, the piecewise linear approximation uh computed by the PM-FEM converges to u

(only) with order 1 according to Def. 1.11.1. The following (much better) second-order

convergence estimate can be proved using duality arguments (cf. [QV97]) in the case of

the non-stabilized method and of the method with lumping and/or SG stabilization.

‖u− uh‖L2 ≤ Ch2. (1.46g)

Estimates (1.46f) and (1.46g) indicate that the amount of extra-diffusion introduced

by the upwind method significantly spoils the accuracy of uh compared to the approxi-

mation computed by the non-stabilized method or by the method with SG stabilization.

For any continuous function η : [0, L]→ R, we introduce the so-called discrete maxi-

mum norm associated with the triangulation Th and defined as

‖η‖∞,h := maxxi∈Th

|η(xi)|. (1.46h)

Then, the following results can be proved to hold (cf. [RST96]).

Theorem 1.11.3 (Uniform convergence for the reaction-diffusion problem). Assume that

µ ∈ C1([0, L]) and that σ and f are given continuous functions on [0, 1], and let h be

sufficiently small. Then, the FEM (in non-stabilized and stabilized forms) is convergent

in the discrete maximum norm and we have

‖u− uh‖∞,h ≤ Ch2. (1.46i)

Theorem 1.11.4 (Uniform convergence for the advection-diffusion problem). Assume

that µ ∈ C1([0, L]), v ∈ C1([0, L]) and f is given a continuous function on [0, 1], and let

h be sufficiently small. Then, the non-stabilized, the UP and SG stabilized FEMs are all

convergent in the discrete maximum norm and we have:

‖u− uh‖∞,h ≤ Ch2 Non stabilized FEM (1.46ja)

‖u− uh‖∞,h ≤ Ch Upwind stabilization (1.46jb)

‖u− uh‖∞,h ≤ Ch2 SG stabilization. (1.46jc)


Estimate (1.46jb) shows again that the upwind stabilization is far less accurate than

the SG stabilization as h becomes asymptotically small. An even better result can be

proved for the SG method in the special case of an advection-diffusion BVP with constant

coefficients.

Theorem 1.11.5 (Nodal exactness of the SG scheme). Assume that µ > 0, v and f are

given constants (with f = 1), and that Th is a uniform partition of Ω. Then the solution

uh of the SG FEM satisfies the following relation

uh(xi) = u(xi) =1

v

[xi −

evxi/µ − 1

ev/µ − 1

]i = 1, . . . , N. (1.46k)

Therefore, the SG scheme is nodally exact in the special case of constant coefficients and

uniform grid.

The nodal exactness of the SG stabilized method is a particular case of a more general

property of the FEM, referred to as superconvergence. The occurrence of this property

is visible in Fig. 1.14 (left panel).

1.12. The time-dependent 1D conservation law

In this section we address the study of the 1D conservation law (1.1) with respect to

the temporal coordinate. To simplify the presentation, we set γ = β = 0 and α = 1,

corresponding to homogeneous Dirichlet boundary conditions at the endpoints of Ω.

Therefore, our considered IVBVP reads: find u(x, t) for all (x, t) ∈ QT such that:

∂u

∂t+∂ J

∂ x+ κu = g ∀(x, t) ∈ QT (1.11a)

J = vu− µ∂ u∂ x

∀(x, t) ∈ QT (1.11b)

u(x, 0) = u0(x) ∀x ∈ Ω (1.11c)

u(x, t) = 0 ∀(x, t) ∈ ∂Ω× (0, T ). (1.11d)

The IVBVP (1.11) is the heat equation in conservation form and represents the mathe-

matical model of the spatial and temporal diffusion and passive transport of heat in a

medium of length L with thermal conductivity µ and heat transport velocity v, subject

to a heat production g, a heat consumption rate κ, an initial thermal distribution u0 and

boundary temperature equal to zero for all t ∈ (0, T ).

Property 1.12.1. Set κ = g = 0 in (1.11a). Then, we have

limt→∞

u(x, t) = 0 ∀x ∈ Ω. (1.11e)

Relation (1.11e) expresses the fact that the IVBVP (1.11) is asymptotically stable.

Asymptotic stability of the IVBVP problem is a consequence of the fact that µ is

strictly positive and expresses the physical property that the system modeled by (1.11)

is dissipative.

1.12. The time-dependent 1D conservation law 29

Let us now use the θ-method introduced in Sect. 1.3 for the discretization of the

IVBVP (1.11) with respect to spatial and temporal variables. In the following, we denote

by C a positive constant independent of h and ∆t whose value may not be the same at

each occurrence. Using the notation of Sect. 1.3 and Sect. 1.6.3, the linear algebraic

system that has to be solved at each time level tk, k = 0, . . . , NT − 1, reads

Aθuk+1 = fk+1

θ (1.11f)

where:

Aθ = θ(Kd + Ka + Kr) +1

∆tM (1.11g)

fk+1θ = −(1− θ)(Kd + Ka + Kr)uk +

1

∆tMuk + θgk+1 + (1− θ)gk, (1.11h)

having introduced the mass matrix

Mij =

∫ L

0ϕj(x)ϕi(x) dx i, j = 1, . . . , N. (1.11i)

In the above relations, θ is a parameter to be chosen in the interval [0, 1]. In particular,

we notice that:

1. If θ = 0, relation (1.11f) becomes an explicit formula to compute the solution at

time level tk+1 using information only at the level tk. The corresponding scheme is

called Forward Euler or Explicit Euler method.

2. If θ > 0, the family of schemes associated with (1.11f) takes the name of implicit

methods because to compute uk+1 we need to solve a linear system using informa-

tion at both levels tk and tk+1.

Definition 1.12.1 (Stability). Set κ = g = 0 in (1.11a). The θ-method (1.11f) is said

to be stable (or also, asymptotically stable) if the sequenceuk

is such that

limk→∞

uk = 0. (1.11j)

Condition (1.11j) is the discrete counterpart of (1.11e) and is not automatically in-

herited by the θ-method. In this sense, the following condition needs be satisfied (for the

proof, see [QV97], Chapt. 11 and [QSS07], Chapt. 13).

Theorem 1.12.1 (Stability of the θ-method). If θ ≥ 1/2 the θ-method verifies (1.11j)

for every value of ∆t. In this case we say that the θ-method is unconditionally stable. If

0 ≤ θ < 1/2 the θ-method verifies (1.11j) only if

∆t ≤ Ch2

µ0(1− 2θ). (1.11k)

In this case we say that the θ-method is conditionally stable.

Conditional stability can be a severe computational constraint for the θ-method be-

cause for a small value of h (which is typically needed to achieve a reasonable spatial

accuracy of the approximation) the corresponding value of ∆t to make the θ-method

stable turns out to be even smaller. Therefore, the use of an implicit method is in order

thanks to its unconditional stability. The choice of such a method is governed by the

next important result.


Theorem 1.12.2 (Convergence of the θ-method). Under suitable regularity assumptions

on the data in (1.1) we have

‖uk − ukh‖L2(Ω) ≤ ‖u0 − u0h‖L2(Ω) + C(u0, f, u)

[h2 + ∆tp(θ)

]∀k ∈ [1, NT ] (1.11l)

where p(1/2) = 2 and p(θ) = 1 for all θ 6= 1/2.

Thm. 1.12.2 tells us that:

1. if limh→0‖u0 − u0

h‖L2(Ω) = 0 then the θ-method is convergent ;

2. the θ-method is convergent with order 2 with respect to both spatial and time co-

ordinates only in the case of the Crank-Nicolson scheme. All the other implicit

schemes with θ > 1/2 converge linearly with respect to ∆t. Quadratic convergence

with respect to the spatial coordinate is a consequence of (1.46g).

1.13. Numerical tests

In this section, we verify the performance of the numerical schemes illustrated in this

chapter. In Sect. 1.13.1 we study the DAR problem in the stationary case whereas in

Sect. 1.13.2 we study the 1D conservation law in the time-dependent regime.

1.13.1. The stationary 1D conservation law

We study three BVPs of the form (1.5), the first being a purely diffusion problem, the

second being a reaction-diffusion problem, the third being an advection-diffusion problem.

1.13.1.1. Diffusion problem

This example serves as a verification of the convergence analysis of Sect. 1.11. We set

a = 0, b = L = 5, µ = 1, σ = v = 0 and f = e−x(x2 − (4 + L)x + 2(L + 1)), in such a

way that the exact solution is the pair

u(x) = xe−x(L− x), J(x) = −e−x(x2 − (2 + L)x+ L),

such that u(0) = u(L) = 0. The graphs of u and J are shown in Fig. 1.15.

In the numerical approximation of the problem, the mesh size is uniform and equal

to h = 5/M , M = [10, 20, 40, 80, 160, 320]T . The load vector entries are computed using

the trapezoidal rule (1.32b).

>>

||u-u_h||_inf ||u-u_h||_L^2 ||u-u_h||_H^1 ||j-j_h||_L^2

1.66291e-01 3.08380e-01 9.20279e-01 8.67073e-01

4.30457e-02 7.96382e-02 4.50240e-01 4.43141e-01

1.08192e-02 2.00721e-02 2.23670e-01 2.22768e-01

2.70842e-03 5.02825e-03 1.11647e-01 1.11533e-01

6.77428e-04 1.25770e-03 5.57995e-02 5.57853e-02

1.69371e-04 3.14465e-04 2.78968e-02 2.78950e-02

1.13. Numerical tests 31

x

0 1 2 3 4 50

0.5

1

1.5

u(x)

x

0 1 2 3 4 5-5

-4

-3

-2

-1

0

J(x)

Figure 1.15.: Diffusion problem. Exact solution pair. Left panel: u(x). Right panel:

J(x).

Convergence orders:

1.9498 1.9532 1.0314 0.9684

1.9923 1.9883 1.0093 0.9922

1.9981 1.9971 1.0024 0.9981

1.9993 1.9993 1.0006 0.9995

1.9999 1.9998 1.0002 0.9999

>>

The above tabulated values illustrate the convergence history of the method by re-

porting the discrete maximum norm, the L2 norm and the H1 norm of u − uh and the

L2 norm of J − Jh as a function of the number of mesh elements M . Fig. 1.16, instead,

displays the same information under the form of log-log curves (i.e., where logarithmic

scale is used on both x and y axes).

Results clearly confirm the theoretical conclusions of Sect. 1.11. In particular, we can

see that ‖u−uh‖H1 and ‖J −Jh‖L2 reduce by a factor a two after each mesh refinement,

this indicating a linear convergence rate, while ‖u− uh‖∞,h and ‖u− uh‖L2 reduce by a

factor of 4 at each refinement, this indicating quadratic convergence of the approximation.

1.13.1.2. Reaction-diffusion problem

This example serves to analyze to effect of using uniform and non-uniform grids on

the stability and accuracy of the computed approximate solution of a reaction-diffusion

problem. We set a = 0, b = L = 1, µ = 10−4, σ = 1, v = 0 and f = 1, in such a way that

the exact solution is the pair of functions:

u(x) =1

σ

(1 +

eα − 1

e−α − eα(eα(x−1) + e−αx)

),

J(x) = −µασ

eα − 1

e−α − eα(eα(x−1) − e−αx)


h10

-210

-110

010

-4

10-2

100

|| u - uh ||

h,inf

h10

-210

-110

010

-4

10-2

100

|| u - uh ||

L2

h10

-210

-110

010

-2

10-1

100

|| u - uh ||

H1

h10

-210

-110

010

-2

10-1

100

|| J - Jh ||

L2

p=2 p=2

p=1 p=1

Figure 1.16.: Diffusion problem: log-log error curves. Upper left panel: ‖u − uh‖∞,h.

Upper right panel: ‖u − uh‖L2 . Bottom left panel: ‖u − uh‖H1 . Bottom

right panel: ‖J − Jh‖L2 .

where α :=√σ/µ = 100. Again, we have u(0) = u(1) = 0. We compare the results of

three simulations, all performed using the same number of subdivisions, M = 20. In the

first case, we use a uniform grid, h = 1/20, so that the Peclet number is Perdloc = 4.1667.

x

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

uh

uex

x

0 0.2 0.4 0.6 0.8 1-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Jh

Jex

Figure 1.17.: Reaction-diffusion problem. Uniform mesh, no lumping of the reaction

matrix.

>> rdbvp


1.81566e-01 9.61332e-02 5.60700e+00 5.60617e-04


Fig. 1.17 shows uh and Jh, with the expected oscillations close to the endpoints. In

particular, from the above reported output values of the simulation we see that the

maximum nodal error eu is equal to 1.81566e-01 and the error in H1 is 5.60700e+00.

In the subsequent computation, we use a different collocation of the grid nodes,

according to the following law

xk =L

2(1− cos(kπ/M)) k = 0, . . . ,M, (1.12)

which amounts to taking a more refined mesh size close to the endpoints where oscillations

occur.

x

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

uh

uex

x

0 0.2 0.4 0.6 0.8 1-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Jh

Jex

Figure 1.18.: Reaction-diffusion problem. Nonuniform mesh, no lumping of the reaction

matrix.

>> rdbvp


3.66961e-02 7.45903e-03 2.80895e+00 2.80894e-04

The effect of choosing a non-uniform mesh can be seen in Fig. 1.18. Oscillations have

been suppressed, with a maximum nodal error eu equal to 3.66961e-02 and an error in

H1 equal to 2.80895e+00.

In the final simulation, we go back to a uniform mesh, but adopt the stabilization

technique based on the use of trapezoidal rule for the computation of the reaction matrix.

>> rdbvp


3.03500e-02 1.12081e-01 5.20262e+00 5.20141e-04

Fig. 1.19 indicates that the computed solution satisfies the discrete maximum princi-

ple, unlike the previous two cases. The maximum nodal error eu is equal to 3.03500e-02


x

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

uh

uex

x

0 0.2 0.4 0.6 0.8 1-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Jh

Jex

Figure 1.19.: Reaction-diffusion problem. Uniform mesh, lumping of the reaction matrix.

(smaller than in the non-uniform case) but the error in H1 is 5.20262e+00, quite larger

than in the non-uniform case because the steep gradient of the exact solution in the

neighbourhood of the endpoints is not sufficiently well resolved with a mesh size of 1/20.

1.13.1.3. Advection-diffusion problem

This example serves to analyze to effect of using different stabilizations on the accuracy of

the computed approximate solution of an advection-diffusion problem on a non-uniform

mesh. We set a = 0, b = L = 1, µ = 5 · 10−3, σ = 0, v = 1 and f = 1, in such a way that

exact solution is the pair

u(x) =1

v

(x− eE x − 1

eE − 1

), J(x) = x+ CE

where E := 1/µ and CE := (B(E) − 1)/E. The number of mesh elements is M =

[10, 20, 40, 80, 160, 320, 640, 1280]T , and the grid nodes are collocated according to (1.12).

For each triangulation Th,i, i = 1, . . . , 8, we set h := maxhk, k = 1, . . . ,M(i).

h10

-410

-310

-210

-110

-4

10-2

100

102

|| u - uh ||

h,inf

h10

-410

-310

-210

-110

-5

100

|| u - uh ||

L2

h10

-410

-310

-210

-110

-1

100

101

102

|| u - uh ||

H1

h10

-410

-310

-210

-110

-4

10-2

100

|| J - Jh ||

L2

p=2 p=2

p=1 p=1

Figure 1.20.: Advection-diffusion problem. Uniform mesh. No stabilization.


h10

-410

-310

-210

-110

-2

10-1

100

|| u - uh ||

h,inf

h10

-410

-310

-210

-110

-3

10-2

10-1

100

|| u - uh ||

L2

h10

-410

-310

-210

-1

1

2

3

4

5

|| u - uh ||

H1

h10

-410

-310

-210

-110

-4

10-3

10-2

10-1

|| J - Jh ||

L2

p=1p=1

p=1

p=1

Figure 1.21.: Advection-diffusion problem. Uniform mesh. Upwind stabilization.

h10

-410

-310

-210

-110

-16

10-14

10-12

|| u - uh ||

h,inf

h10

-410

-310

-210

-110

-4

10-2

100

|| u - uh ||

L2

h10

-410

-310

-210

-1

100

|| u - uh ||

H1

h10

-410

-310

-210

-110

-4

10-3

10-2

10-1

|| J - Jh ||

L2

p=2

p=1

p=1

Figure 1.22.: Advection-diffusion problem. Uniform mesh. SG stabilization.

Fig. 1.20 illustrates the convergence history of the FE discretization as a function

of h, in the case of the formulation without including the stabilization terms. Log-log

plots are used to allow an easy interpretation of the convergence order of the method.

Results clearly show that the asymptotical rates predicted by the theoretical estimates

of Sect. 1.11 are verified. In particular, convergence is linear for u−uh and J −Jh in the

V and Q norms, respectively, while is quadratic for u − uh in the L2 and ∞, h norms.

This in particular means that, despite the onset of spurious oscillations when the mesh

size is not sufficiently small, in the limit h → 0 the FE approximation becomes stable

and accurate.

The behaviour of the numerical method changes when stabilization terms are in-

cluded. Fig. 1.21 illustrates the convergence history of the FE discretization as a func-

tion of h, in the case of the formulation with upwind artificial diffusion. Comparing the

accuracy of uh in the L2 and ∞, h norms with the corresponding function computed by

the non-stabilized method, we see that the accuracy is degraded from quadratic to linear


convergence rate. This is the price to be paid for ensuring the DMP even for a coarse

grid size.

Things get better when the SG stabilization terms is included. Fig. 1.22 illustrates the

convergence history of the FE discretization as a function of h. Comparing the accuracy

of uh in the L2 and ∞, h norms with the corresponding function computed by the non-

stabilized method, we see that second-order accuracy is recovered as h → 0. Notice in

the upper left panel of Fig. 1.22 the onset of round-off error in the evaluation of the

discretization error in the discrete maximum norm. Since the SG scheme is in this case

nodally exact (cf. Thm. 1.11.5) the maximum nodal error should be equal to zero. This

is not possible using finite precision arithmetic as the plot actually shows. Moreover, it

is also interesting to see that the numerical stability of computation worsens as h → 0

because of the increase of the ill-conditioning of the stiffness matrix.

1.13.2. The time-dependent 1D conservation law

In this example we study the IVBVP (1.11) using the θ-method in the case where µ = 1,

v = κ = 0, Ω = (0, 1), IT = [0, 1], u0(x) = sin(2πx) and g is such that the exact solution

is u(x, t) = sin(2πx) cos(2πt).

∆ t

0 0.02 0.04 0.06 0.08 0.110

-6

10-5

10-4

10-3

10-2

10-1

|| u(tfin

) - uh(t

fin) ||

L2(0,1)

BE

CN

Figure 1.23.: The time-dependent problem. Convergence of the θ-method as a function

of ∆t. The spatial mesh size is h = 1/500. The temporal mesh size is

∆t = (10 · k)−1, k = 1, . . . , 20. BE method: solid line. CN method: dashed

line. Logarithmic scale is used on the y-axis.

Figs. 1.23 and 1.24 illustrate the results obtained by solving the IVBVP with the BE

and CN methods. Comparison between the two schemes is carried out by computing the

quantity ‖uNT −uNTh ‖L2(Ω). Fig. 1.23 addresses the study of convergence with respect to

∆t having fixed h = 1/500 whereas the time grid size is allowed to vary in the interval

[1/10, 1/200] by setting ∆t = (10 ·k)−1, k = 1, . . . , 20. Conversely, Fig. 1.24 addresses the

study of convergence with respect to h having fixed ∆t = 1/100 whereas the spatial grid

size is allowed to vary in the interval [1/10, 1/200] by setting h = (10 ·k)−1, k = 1, . . . , 20.


h

0 0.02 0.04 0.06 0.08 0.110

-6

10-5

10-4

10-3

10-2

|| u(tfin

) - uh(t

fin) ||

L2(0,1)

BE

CN

Figure 1.24.: The time-dependent problem. Convergence of the θ-method as a function

of h. The temporal mesh size is ∆t = 1/100. The spatial mesh size is

h = (10 · k)−1, k = 1, . . . , 20. BE method: solid line. CN method: dashed

line. Logarithmic scale is used on the y-axis.

The two convergence studies clearly indicate that the CN method is far more accurate

than the BE method with respect to both h and ∆t.

Let us now address the issue of the conditional stability of the FE method (θ = 0).

In the considered case condition (1.11k) amounts to requiring that

NT ≥ NT,min = CM2, (1.13)

NT,min denoting the minimum number of time intervals needed to prevent numerical

instabilities to arise. We set C = 2 and M = 30, so that we expect the FE method to be

asymptotically stable if NT,min = 1800.

x

0 0.2 0.4 0.6 0.8 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

uh(x,t)

x

0 0.2 0.4 0.6 0.8 1

-6

-4

-2

0

2

4

6

Jh(x,t)

Figure 1.25.: The time-dependent problem. Stability of the FE method. M = 30, NT =

NT,min = 1800. The method is asymptotically stable.


x

0 0.2 0.4 0.6 0.8 1

×10303

-6

-4

-2

0

2

4

6

uh(x,t)

x

0 0.2 0.4 0.6 0.8 1

×106

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Jh(x,t)

Figure 1.26.: The time-dependent problem. Stability of the FE method. M = 30, NT =

NT,min = 900. The mathod fails to be asymptotically stable.

Fig. 1.25 illustrates the spatial distribution of the computed solution as a function of

time in the case NT = NT,min = 1800. As we can see, the solution uh is asymptotically

stable whereas the computed flux Jh is accurate as it can be, compatibly with the (rather)

coarse choice of h.

Fig. 1.26 illustrates the spatial distribution of the computed solution as a function

of time in the case NT = NT,min/2 = 900. As we can see, this time the solution pair

(uh, Jh) fails to be asymptotically stable and unphysical (unbounded) oscillations pollute

the plot.

1.14. Application: heating of a room

In this concluding section we use the IVBVP (1.1) as a simplified 1D model of the heat

transfer design problem schematically illustrated in Fig. 1.27. The geometry consists of

a room of length L whose left side is a wall set at a fixed temperature T = Twall and

whose right side is exposed to sunlight through a window glass of thickness δ L. The

outside temperature is equal to a fixed value Tout which is in general not in equilibrium

with T . The object of the analysis is to study the evolution of the room temperature T

over the time period [0, tfinal], starting from an initial uniform temperature T 0, under

the assumption of neglecting heat diffusion along the vertical direction y and along the

direction perpendicular to the x− y plane, so that we can write T = T (x, t).

The mathematical model of the heated room reads: Given the initial temperature

T 0, determine the room temperature T = T (x, t) for all x ∈ Ω = (0, L) and for all

1.14. Application: heating of a room 39

T(x,0) = T0

Twall

WallT = T(x,t)

Room

x=0 x=Lx

δ

Window

T = Tout

Figure 1.27.: Geometrical model of a room with a window on the right side x = L and a

wall at fixed temperature at the left side x = 0.

t ∈ (0, tfinal) such that:

∂T

∂t+∂ Q

∂ x= 0 ∀(x, t) ∈ QT (1.14a)

Q = −µroom∂ T

∂ x∀(x, t) ∈ QT (1.14b)

T (x, 0) = T 0 ∀x ∈ Ω (1.14c)

T (0, t) = Twall ∀t ∈ (0, tfinal) (1.14d)

Q · n(L, t) = v (T (L, t)− Tout) ∀t ∈ (0, tfinal). (1.14e)

The variable Q is the heat flux and physically represents the mechanism of heat transfer

between two points x1 and x2 at a velocity that is proportional to the ratio between the

(-)gradient of the temperature field across the two points (according to Fick’s law of dif-

fusion) and the average temperature that automatically tends to arise to drive the system

back to local equilibrium. The coefficient µroom is the thermal diffusivity of the air in

the room. It should be noted that each medium represented in Fig. 1.27 is characterized

by its own value of µ which is thus, in general, a discontinuous function of space. The

temperature values in the initial condition (1.14c) and in the boundary condition (1.14d)

are not necessarily in equilibrium, i.e., in general, we have T 0 6= Twall. Indeed, it is

the jump associated with these two values to act as initializing thermal source inside the

room. Then, heat flows by thermal diffusion between hot and cold regions until it reaches

the interface at x = L where the window is located. The physical process occurring there

is represented by the Newton-Fourier condition (1.14e). This latter relation is a Robin

boundary condition that expresses a local dynamical thermal balance between the tem-

perature immediately inside the room (x = L−) and the temperature immediately outside

the room (x = (L+ δ)+). Clearly, thermal exchange across the window depends strongly

on the thermal transfer velocity v that determines the thermal isolation property of the


window glass. If the glass is adiabatic, v is very small and heat flow cannot leave the room

whereas if v is large, heat is rapidly allowed to leave the room (or viceversa) depending

on whether T (L−, t) is larger or smaller than Tout. When a dynamical equilibrium is

reached, ∂T/∂t becomes equal to zero and the heat flow becomes constant throughout

the whole room because ∂Q/∂x = 0. In conclusion, in equilibrium conditions a finite

temperature jump across the window thickness can be sustained depending on the value

of the thermal exchange velocity v. This latter parameter can be roughly estimated as

v =µglassδ

,

µglass denoting the thermal diffusivity of the material constituting the window. The

thermotecnical engineer should conduct a trial-and-error design of the window through a

trade-off between the need of saving material (reducing δ) and the need of ensuring a good

isolation performance in both winter and summer seasons (reducing v). In the remainder

of the section, we try to simulate such an optimization analysis by solving numerically

system (1.14) in different working conditions. In all subsequent computations, except

otherwise indicated, we set tfinal = 60s and use the BE method for time discretization

with NT = 60, so that ∆t = 1s. We also set Twall = 300K and M = 100 so that

h = 5 · 10−3m. The length L is set equal to 0.5m so that our analysis is confined in the

close proximity of the window. Finally, the glass thickness δ is equal to 10−2m.

1.14.0.1. Test 1. Analysis as a function of external temperature

In this section we set µroom = 10−2m2s−1 and µglass = 10−3m2s−1 in such a way that

the thermal exchange velocity is equal to v = 0.1ms−1.

x

0 0.1 0.2 0.3 0.4 0.5

300

300

300

300

300

300

300

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

×10-12

-2

-1.5

-1

-0.5

0

0.5

1

Qh(x,t)

Figure 1.28.: Test case 1a. Left panel: room temperature (look at the scale on the y axis).

Right panel: room heat flux. µroom = 10−2m2s−1, µglass = 10−3m2s−1,

δ = 10−2m, T 0 = Tout = Twall = 300K.

In the first simulation test (1a) we set T 0 = Tout = Twall = 300K, so that we expect

that the model returns as solution a constant temperature distribution in the room which

is in thermal equilibrium with the external environment. This conjecture is confirmed

by the results shown in Fig. 1.28. Th (left panel) is substantially flat with a maximum


x

0 0.1 0.2 0.3 0.4 0.5

292

293

294

295

296

297

298

299

300

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Qh(x,t)

Figure 1.29.: Test case 1b. Left panel: room temperature. Right panel: room heat flux.

µroom = 10−2m2s−1, µglass = 10−3m2s−1, δ = 10−2m, T 0 = Twall = 300K,

Tout = 290K.

x

0 0.1 0.2 0.3 0.4 0.5300

301

302

303

304

305

306

307

308

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

Qh(x,t)

Figure 1.30.: Test case 1c. Left panel: room temperature. Right panel: room heat flux.


Tout = 310K.

excursion of 1.7735e−11 and a similar trend is obtained in the computation of Qh (right

panel) where the onset of round-off is neatly visible because of fluctuations around 10−12.

In the second simulation test (1b) we set T 0 = Twall = 300K and Tout = Twall − 10 =

290K, so that we expect the heat to flow from the room towards the external environment.

This conjecture is confirmed by the results shown in Fig. 1.29. Th tends to a linear profile

after an initial transient (left panel) and correspondingly the heat flux tends to a constant

value throughout the room (right panel). We notice that because of the finite (positive)

value of v, a finite temperature jump of about 2K occurs in the temperature across the

window glass, the room temperature being greater than the external temperature.

In the third simulation test (1c) we set T 0 = Twall = 300K and Tout = Twall +


x

0 0.1 0.2 0.3 0.4 0.5

×1091

-6

-4

-2

0

2

4

6

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

×1092

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Qh(x,t)



Tout = 310K. In this case the FE method is used instead of the BE method

as in Fig. 1.30, with NT < NT,min so that condition (1.13) is not satisfied.

x

0 0.1 0.2 0.3 0.4 0.5300

301

302

303

304

305

306

307

308

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Qh(x,t)



Tout = 310K. In this case the FE method is used instead of the BE method

as in Fig. 1.30, with NT > NT,min so that condition (1.13) is satisfied.

10 = 310K, so that, this time, we expect the heat to flow into the room from the

external environment. This conjecture is confirmed by the results shown in Fig. 1.30.

Th tends to a linear profile after an initial transient (left panel) and correspondingly

the heat flux tends to a constant value throughout the room (right panel). As in the

previous case, a finite temperature jump of about 2K occurs in the temperature across

the window glass, the room temperature being lower than the external temperature. We

show in Fig. 1.31 the results on the same simulation test (1c) obtained using the FE time


discretization method instead of the BE method. We notice that the computed solution

pair (Th, Qh) exhibits a markedly oscillatory behaviour and tends to become unbounded

as time increases. The numerical instabilities that are clearly visible in the plots are

due to the fact that the FE method is conditionally absolutely stable whereas the BE

method is unconditionally absolutely stable. With the data used in test case (1c) we have

NT = 60 and NT,min = 48000 so that (1.13) is manifestly violated and oscillations occur

as in Fig. 1.31. If we increase the spatial mesh size, using only M = 10 elements, so

that h = 5 · 10−2, we obtain NT,min = 480. Taking NT = 500 we obtain the results

shown in Fig. 1.32 where we can see that oscillations are disappeared at the expense of

a reduced spatial accuracy of the approximation of the solution pair (T,Q) compared to

what obtained with the BE method.

1.14.0.2. Test 2. Analysis as a function of the initial temperature

In this section we use the same values of µroom and µglass as in the previous section and

we set Tout = Twall − 10 = 290K.

x

0 0.1 0.2 0.3 0.4 0.5290

291

292

293

294

295

296

297

298

299

300

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Qh(x,t)

Figure 1.33.: Test case 2a. Left panel: room temperature. Right panel: room heat flux.

µroom = 10−2m2s−1, µglass = 10−3m2s−1, δ = 10−2m, T 0 = Tout = 290K,

Twall = 300K.

In the first simulation test (2a) we set T 0 = Tout = Twall − 10 = 290K so that the

initial temperature is lower than the wall temperature at x = 0. This jump discontinuity

is expected to be smoothed out during time evolution until a stationary condition similar

to that of case 1b is reached. This conjecture is confirmed by the results shown in

Fig. 1.33.

In the second simulation test (2b) we set T 0 = Twall + 10 = 310K so that the initial

temperature is greater than the wall temperature at x = 0. Also in this case, the jump

initial discontinuity should be smoothed out during time evolution until a stationary

condition similar to that of case 1b is reached. This conjecture is confirmed by the

results shown in Fig. 1.34.


x

0 0.1 0.2 0.3 0.4 0.5

292

294

296

298

300

302

304

306

308

310

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Qh(x,t)

Figure 1.34.: Test case 2b. Left panel: room temperature. Right panel: room heat

flux. µroom = 10−2m2s−1, µglass = 10−3m2s−1, δ = 10−2m, T 0 = 310K,

Tout = 290K, Twall = 300K.

1.14.0.3. Test 3. Analysis as a function of glass thermal diffusivity

In this section we use the same values of µroom and Tout as in the previous section, whereas

we modify the value of the thermal diffusivity of the material constituting the window

glass. We consider the four decreasing values µglass = 10−4, 10−5, 10−6, 10−7m2s−1. A

first qualitative indication that is expected from the results of this section is that the finite

temperature jump across the window should be smaller than in case 2b because of the

reduced heat transmission coefficient. A second qualitative indication is that in the limit

of vanishing thermal diffusivity of the window glass, the Robin boundary condition (1.14e)

tends to an homogeneous Neumann condition for the heat flux. However, the steady-

state temperature distribution is not in general expected to be constant throughout the

room because of the initial jump discontinuity although there should be not a significant

variation for very small values of µglass. The a priori qualitative analysis is confirmed by

the simulation results shown in Fig. 1.35, Fig. 1.36, Fig. 1.37 and Fig. 1.38.


x

0 0.1 0.2 0.3 0.4 0.5

298

300

302

304

306

308

310

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Qh(x,t)

Figure 1.35.: Test case 3a. Left panel: room temperature. Right panel: room heat flux.

µroom = 10−2m2s−1, µglass = 10−4m2s−1, δ = 10−2m, T 0 = Twall + 10 =

310K, Tout = Twall − 10 = 290K, Twall = 300K.

x

0 0.1 0.2 0.3 0.4 0.5300

301

302

303

304

305

306

307

308

309

310

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Qh(x,t)

Figure 1.36.: Test case 3b. Left panel: room temperature. Right panel: room heat flux.




x

0 0.1 0.2 0.3 0.4 0.5300

301

302

303

304

305

306

307

308

309

310

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Qh(x,t)




x

0 0.1 0.2 0.3 0.4 0.5300

301

302

303

304

305

306

307

308

309

310

Th(x,t)

x

0 0.1 0.2 0.3 0.4 0.5

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

Qh(x,t)

Figure 1.38.: Test case 3d. Left panel: room temperature. Right panel: room heat flux.



Part II.

Mathematical Models in Cellular

Biology and Electrophysiology

2. Introduction to Cellular Biology and Ion

Transport

In this chapter, we give an introduction to the biological setting of the course. We

start by illustrating the structure of a cell, of its contents and its membrane. Then,

we introduce the ionic channels and the mobile and fixed charges that contribute to

ion flow throughout the cellular membrane. Finally, we describe the basic biophysical

mechanisms that determine ion transport, the diffusion and the drift mechanisms, and

we provide their mathematical characterization expressed by the Nernst-Planck equation.

For more detailed information, we refer to [Hil01, GH06, BF01].

50 2. Introduction to Cellular Biology and Ion Transport

2.1. Cells: structure, membrane and ion channels

The basic living unit of the body is the cell : each organ is an aggregate of many different

cells held together by intercellular supporting structures. The entire body contains about

100 trillion cells (∼ 1014 cells). Although the many cells of the body often differ markedly

from one another (since each type of cell is specifically targeted to perform one or a few

particular functions), all of them have certain basic characteristics that are alike: a

typical cell, as seen by the light microscope, is shown in Fig. 2.1. Its two major parts

are the nucleus and the cytoplasm. The nucleus is separated from the cytoplasm by a

nuclear membrane, and the cytoplasm is separated from the surrounding fluids by a cell

membrane. The principal fluid medium of the cell is water, which is present in most

cells, except for fat cells, in a concentration of 70 to 85 %. Many organelles (such as

mithocondria or sarcoplasmic reticulum) as well as cellular chemicals are dissolved in the

water. Others are suspended in the water as solid particulates. Chemical reactions take

place among the dissolved chemicals or at the surfaces of the suspended particles and

organelles.

Figure 2.1.: Structure of a cell (reprinted from www.catalog.flatworldknowledge.com).

2.1. Cells: structure, membrane and ion channels 51

2.1.1. The cell membrane

The cell membrane (also called the plasma membrane), which envelops the cell, is a thin,

pliable, elastic structure with a thickness tm of only 7.5 to 10 nanometers. The diameter

dc of cells varies from 7.5 to 150 micrometers . Thus, the ratio tm/dc varies in the range

50× 10−6 ÷ 1.3× 10−3. Despite of its relative small dimension, the cell membrane plays

a fundamental role for the life of the cell because:

• it preserves the integrity of the cell separating the intracellular fluid from the

extracellular fluid ;

• it regulates the passage of substances from the outside to the inside of the cell, and

viceversa.

The cell membrane consists almost entirely of a lipid bilayer, but it also contains large

numbers of protein molecules in the lipid, many of which penetrate all the way through

the membrane, as shown in Fig. 2.2. Most of these penetrating proteins constitute a

pathway through the cell membrane. Some proteins, called channel proteins, allow free

movement of water as well as selected ions or molecules. The channel proteins are usually

highly selective with respect to the types of molecules or ions that are allowed to cross

the membrane.

Figure 2.2.: Cellular membrane bilayer (reprinted from 2012books.lardbucket.org).

Ions provide inorganic chemicals for cellular reactions. Also, they are necessary for

operation of some of the cellular control mechanisms. For instance, ions acting at the cell

membrane are required for transmission of electrical impulses in nerve and muscle fibers.

The most important ions in the cell are potassium K+, magnesium Mg++, phosphate,

sulfate SO−−4 , bicarbonate HCO−3 , and smaller quantities of sodium Na+, chloride Cl−,

and calcium Ca++. An ion is called cation if positively charged, anion if negatively

charged.


2.1.2. Ionic channels

Certain cells, commonly called excitable cells, are unique because of their ability to

generate electrical signals. Some examples are neuron cells, muscle cells, and touch

receptor cells. Like all cells, an excitable cell maintains a different concentration of ions

in its cytoplasm than in its extracellular environment. Together, these concentration

differences create a small drop of electrical potential across the plasma membrane defined

as

ϕm := ϕ(in) − ϕ(out). (2.1)

The quantities ϕ(in) and ϕ(out) are the values of the electric potential ϕ at the two sides

of the membrane, measured with respect to a common reference potential (the ground

potential, for simplicity). The voltage drop across the channel is usually referred to as

membrane potential and plays a relevant role in determining ion flow in a channel. As a

matter of fact, when ϕm reaches a threshold value, typically around 55mV, specialized

channels in the plasma membrane, called ion channels, open and allow rapid ion move-

ment into or out of the cell, and this movement creates an electrical signal. All of these

processes characterize the so-called cellular electrical activity (CEA) that represents the

way ion channels can generate an electrical current flowing between a cell and another,

thus providing a fundamental aspect in the life of every biological system.

Ionic channels are large proteins that reside in the membrane of cells (Fig. 2.3) and

conduct ions through a narrow tunnel of fixed charge formed by the amino acid residues

of the protein.

Channels are ideally placed across the membrane in series with the intracellular en-

vironment to control the cellular biological function [SBT02]. Ion channels should be

viewed as natural nanotubes that relate the electrolyte solutions in and outside the cell

to the ion concentration gradient and to the electric field that is established across the cell

membrane, synthetically, to the electrochemical potential gradient across the membrane.

Figure 2.3.: Ion channels function as pores to permit the flux of ions down their electro-

chemical potential gradient (reprinted from www.nature.com).

2.2. Transport of charged particles 53

Channels are responsible for signaling in the nervous system, coordination of muscle

contraction, and transport in all tissues. Channels are obvious targets for drugs and

disease [Sch96]: as a matter of fact, many of the drugs used in clinical medicine act

directly or indirectly through channels.

Assumption 2.1.1 (Homogeneization of ionic fluid). In these notes we take the point

of view of Continuum Fluid-Mechanics. This means that ions are treated as compressible

fluids, characterized by a density that is, in general, not constant neither in space, nor in

time. An alternative approach, that is not addressed in these notes, consists in treating

ions as spherical particles, with a radius, a mass and a charge. This atomistic approach is

the foundation of Molecular Dynamics (for more information about this latter approach,

see [KBA05]).

2.2. Transport of charged particles

In this section we start to introduce the basic concepts of ion charges and ion motion in

a biological channel.

2.2.1. Concentrations and fluxes

Consistently with Assumption 2.1.1, we denote henceforth by Cion the concentration of

a certain ionic species having chemical valence zion with zion > 0 for cations, zion < 0 for

anions while neutral species have zion = 0. The quantity Cion has usually several different

units according to the specific context in which is treated:

• if Cion is a number density we use the symbol cion and

[cion] =# of ions

m3;

• if Cion is a molar density we use the symbol Cion and

[Cion] =mol

m3= 10−3 mol

dm3= 10−3 mol

l= 10−3M = 1mM

where 1M is 1 molar.

Conversion between molar and number density is regulated by the following relation

cion := NAvCion (2.2)

where NAv = 6.023 · 1023 mol−1 is the Avogadro constant.

Let vion denote the velocity of the ions. This velocity is the result of the action of

several forces, including electrical, chemical, thermal and fluid forces. The molar flux

density f ion associated with the motion of the ion particles is defined as

f ion := Cionvion (2.3)

while

J ion := qzioncionvion (2.4)


J

n

ion

S

x

Figure 2.4.: Electrical current flowing across a surface S.

is the current density associated with the motion of the ion particles, q = 1.602 · 10−19 C

being the elementary charge. The units of f ion are mol m−2 s−1 while the units of J ion

are C m−2 s−1 = A m−2.

Let us denote by S the cross-sectional area of the channel and by n the outward unit

normal vector on S. The electrical current (units A = C s−1) associated with the flow of

the ionic species of number density nion through the channel is defined as (see Fig. 2.4)

Iion :=

∫SJ ion · n dS. (2.5)

The ionic electrical current is therefore the amount of ionic charge that crosses a given

surface in a unit of time. Analogously, the ionic molar flux (units mol s−1), defined as

Fion :=

∫Sf ion · n dS, (2.6)

is the amount of ionic moles that crosses a given surface in a unit of time.

2.3. Charge transport in ionic fluids

When dealing with charge transport in a biological fluid we need to take care of the fact

that the number density cion of a chemical species may experience very large variations

inside a biological fluid. This is the case, for instance, of the concentration of calcium

Ca2+ inside and outside a smooth muscle cell in the human body. Denoting by cin

and cext the values of calcium concentration in the intracellular and extracellular sites,

respectively, we typically have cin = 10−4mM and cext = 5mM. These numbers show that

there is a large gradient of calcium molar density across the membrane thickness, of the

order of 109mM m−1. According to the basic theory of molecular diffusion, associated

with such concentration gradient, there is a diffusion flux density (units mol m−2 s−1)

defined as

fdiffc = −Dc∇ c (2.7a)

where c is the molar concentration of the considered chemical species and Dc its diffusion

coefficient (units m2 s−1). Eq. (2.7a) expresses the intuitive physical principle that the

2.3. Charge transport in ionic fluids 55

flux moves in the opposite direction with respect to the gradient in such a way that regions

at higher concentration tend to migrate towards regions at lower concentration to restore

dynamical equilibrium. This principle is commonly known as Fick’s law of diffusion. In

presence of an electric field E, each ion particle is subject also to the electrical (or drift)

force

Felc = qzcE. (2.7b)

zc >0

zc >0

E

+

_

Fc

el

Fcel

Figure 2.5.: Electrical force.

The corresponding act of motion of the particle, according to the sign of its chem-

ical valence, is represented schematically in Fig. 2.5. The hydrodynamical resistance

exerted by the fluid to the motion of ion particles due to the application of the electrical

force (2.7b) is expressed by the relation

Fmechc = −ηcvelc (2.7c)

where ηc is the ion viscosity (units Kg s−1) and velc is the drift velocity of the particles.

Using the fact that the electrical force and the hydrodynamical force satisfy the action-

reaction principle

Felc + Fmechc = 0,

we have

qzcE − ηcvelc = 0,

from which it follows that

velc = qzcµmechc E, (2.7d)

where

µmechc :=1

ηc(2.7e)

is the mechanical mobility, a measure of the attitude to drift motion in the medium (units

Kg−1 s). In a similar manner, it is convenient to define the electrical mobility

µc :=|velc ||E|

(2.7f)

from which, taking the module of (2.7d), we get

|velc ||E|

= µc = q|zc|µmechc ⇒ µmechc =µcq|zc|

. (2.7g)


Using this latter relation in (2.7d), we finally get

velc := µczc|zc|

E. (2.7h)

This latter relation shows that the ion drift velocity is directly proportional to the applied

electric field through a positive coefficient, the electrical mobility (units m2V−1s−1).

According to the basic theory of hydrodynamics, associated with the above drift velocity,

there is a drift flux density (units mol m−2 s−1) defined as

fdriftc = cvelc = µczc|zc|

cE. (2.7i)

Of course, since the species c has a chemical valence zc, in correspondance of fdiffc we

also have a diffusion current density (units A m−2) defined as

Jdiffc = qzcfdiffc = −qzcDc∇ c. (2.7j)

Clearly, in the case of a perfect conductor, Jdiffc is equal to zero because the concen-

tration is constant and current transport is determined by the sole drift mechanism,

mathematically expressed by Ohm’s law

Jc = σcE (2.7k)

where σc denotes the conductivity of the conducting medium (units S m−1).

In the case of ions in motion throughout the biological fluid the ohmic current density

(also often referred to as drift current density) is defined as

Jdriftc = qzcfdriftc = q|zc|µccE. (2.7l)

Assuming, for the moment, to neglect other driving mechanisms to ion flow (such as ther-

mal forces and/or fluid-mechanical forces), application of linear superposition of effects

yields the following drift-diffusion model of ionic charge transport in a fluid

Jc = Jdriftc + Jdiffc = q|zc|µccE − qzcDc∇ c = σcE − qzcDc∇ c (2.7m)

where

σc := q|zc|µcc (2.7n)

is the ion electrical conductivity (units S m−1). The transport relation (2.7m) is the

natural extension of Ohm’s law (2.7k) to the case where a concentration gradient is

present. For this reason, Eq. (2.7m) is referred to as the generalized Ohm’s law.

Remark 2.3.1 (Thermal equilibrium). The special (and important) case where Jdriftc =

−Jdiffc corresponds to Jc = 0 and is commonly referred to as thermal equilibrium.In

this situation, the net current flow across a section is null although the separate drift

and diffusion contributions are non-zero.

Remark 2.3.2 (The Drift-Diffusion model for semiconductors). Eq. (2.7l) is the funda-

mental relation at the basis of the charge transport model that is most widely used in the

simulation and design of modern electronic devices, the so-called Drift-Diffusion (DD)

2.3. Charge transport in ionic fluids 57

system. In this specific application two ion species must be considered, anions (electrons,

zc = −1) and cations (holes, zc = +1). These two species strongly mutually interact

because of quantum-mechanical effects, and also exchange energy with the surrounding

environment that is, unlike the case of biological fluids, a periodic crystalline medium at

rest, typically made of silicon atoms. For a thourough description of the model and a de-

tailed analysis of its properties we refer to [Sel84], [Mar86] and [Jer96]. A technological

presentation of the DD system is provided in the book [MK02].

2.3.1. The Nernst-Planck equation

Let us consider the cellular membrane represented in Fig. 2.2 and let us denote by x

the spatial coordinate aligned with the channel axis. The following relation between

diffusivity and mobility holds

Dc =KBT

q|zc|µc, (2.8)

where KB is Boltzmann’s constant (units J K−1) and T is the absolute temperature of the

environment (units K). Formula (2.8) is well known as Einstein’s relation. Combining

relations (2.7a) and (2.7i), and using (2.8), we see that the number flux density of ions

through the membrane along the direction x that is driven by the simultaneous effect of

a concentration gradient and an electric field is expressed by the relation

fc =qzcKBT

DccE︸︷︷︸Drift

−Dc∂c

∂x︸︷︷︸Fick diffusion

(2.9)

We notice that the expression (2.9) is of the same mathematical form as (1.1b) upon

making the following identifications: u→ c, µ→ Dc, v → qzcDcE/(KBT ) and J → fc.

The units of the flux are

[fc] = [Dc] ·m−4 = m2 s−1 ·m−4 = m−2 s−1.

Multiplying this latter expression by qzc yields

Jc = qzc × fc = −qzcDc

(∂c

∂x+zcq

KTc∂ϕ

∂x

). (2.10)

The units of Jc are

[Jc] = [q]× [fc] = Cm−2 s−1 = A m−2

from which we conclude that Jc is a current density. Equation (2.10) is known as Nernst-

Planck relation and is the most widely used modeling tool in Electrophysiology to describe

the motion of ions under the effect of diffusive and drift forces.

We conclude this section with some concepts and definitions that will play an impor-

tant role in the discussion to follow.

Definition 2.3.1 (Nernst potential). The Nernst potential (units V) associated with the

ionic species c is defined as

Ec =KBT

zcqln

(c(out)

c(in)

)=RT

zcFln

(c(out)

c(in)

)(2.11)

where R is the universal gas constant (units J mol−1 K−1) and F is Faraday’s constant

(units C mol−1).


Definition 2.3.2 (Thermal equilibrium).

Thermal equilibrium ⇔ Jc = 0 ⇔ ϕm = Ec. (2.12)

It is important to notice that thermal equilibrium is a dynamical state in which, at

the microscopic level, particles manifest continuously the tendency to diffuse and drift

across the cellular membrane, for instance, because of thermal agitation. Correspond-

ingly, thermal equilibrium is indispensable to introduce the Nernst potential that is a

fundamental quantity in the analysis of the electrical activity of every excitable cell.

3. ODE Models in Cellular Electrophysiology

As anticipated in Chapter 2, the cell membrane is a biological interface that separates

the interior of every cell from the outside environment. The cell membrane is selectively

permeable to ions and organic molecules and controls the movement of substances in

and out of the cell. The basic function of cell membrane is to protect the cell from

its surroundings, but it is also involved in a variety of cellular processes such as cell

adhesion, ion conductivity and cell signaling. In this chapter, we focus on membrane

electrical activity by developing a biophysically sound description of the ionic currents

through the membrane. The resulting mathematical model is based on an equivalent

electrical representation of the cell and is constituted by a system of Ordinary Differential

Equations (ODEs) derived from the application of Kirchhoff’s current law. The system

must be solved at each spatial point of the membrane surface and at each time level of

the temporal evolution of the biophysical problem under given initial conditions for the

membrane potential and the ionic concentrations in the intra- and extracellular sites.

60 3. ODE Models in Cellular Electrophysiology

3.1. Reduced order modeling of membrane electrophysiology

In this section we derive the electrical equivalent model of the electrophysiological be-

haviour of the cellular membrane. To this purpose, we start recalling the fundamental

set of Maxwell’s equations that govern the propagation of an electromagnetic signal in a

medium:

rotH = J +∂D

∂ tAmpere Law (3.1a)

rotE = −∂B∂ t

Faraday Law (3.1b)

divD = ρ Gauss Law (3.1c)

divB = 0 absence of magnetic charge (3.1d)

where:

• E: electric field, units: Vm−1;

• D: electric displacement, units: Cm−2;

• H: magnetic field, units: Am−1;

• B: magnetic induction, units: Vsm−2;

• J : total conduction current density, given by the sum of the ionic currents flowing

in the fluid, units: Am−2;

• ρ: electric charge density, units: Cm−3.

The Maxwell system is complemented by the following set of constitutive laws that

characterize the electromagnetic properties of the medium:

D = εE (3.2a)

B = µmH (3.2b)

where:

• ε: dielectric permittivity, units Fm−1;

• µm: magnetic permeability, units s2F−1m−1 ≡ Hm−1.

Assumption 3.1.1 (Slow time variation of B). Throughout the text we assume that

the magnetic induction B varies slowly in time compared to the spatial variation of the

electric field. This amounts to approximating the Faraday law (3.1b) by

rotE = 0. (3.3)

The consequence of (3.3) is that there exists a scalar potential ϕ = ϕ(x, t) (called electric

potential) such that

E = −∇ϕ. (3.4)

3.1. Reduced order modeling of membrane electrophysiology 61

Remark 3.1.1 (Validity of Assumption 3.1.1). Assumption 3.1.1 decouples the electric

field dynamics from that of the magnetic field and is valid until the characteristic scale

length of the problem at hand is sufficiently small with respect to the wavelength associated

with the maximum frequency of the signal that is propagating in the physical system. Such

condition is largely verified even at the frequency of Ghz.

Taking the divergence of (3.1a) and using (3.1c), we get the continuity equation

∂ ρ

∂ t+ divJ = 0 (3.5)

which expresses the physical fact that the time rate of change of the charge accumulated

in a volume is balanced by the net flux of the total conduction current across the volume

surface.

nout

nin

membrane

extracellular

region

mt

channel

Σout

intracellularregion

Σin

Figure 3.1.: Three-dimensional detail of the cellular membrane.

Let us now use the continuity equation (3.5) to describe current flow across the

ion channel schematically illustrated in the 3D picture of Fig. 3.1. With this aim, we

consider the inner surface Σin of area S and we construct the 3D cylindrical control

volume T = Σin × h, h being an arbitrarily small quantity, shown in Fig. 3.2.

Integration of (3.5) in the volume T , having assumed that, because of symmetry, no

current density flows out of the lateral surface of the cylinder, yields

∂ ρ

∂ tSh+ J totin · ninS + J tmin · noutS = 0,

where J totin and J tmin denote the total conduction current density injected by the intra-

cellular region and the transmembrane current density that reaches the channel region,

respectively, while the current density J totin is the sum of the ionic currents flowing from

the cytoplasm. Dividing out the previous relation by S and defining the surface charge

density distributed on Σin (units Cm−2)

σin := ρh,

we getdσindt− J totin + J tmin = 0 at Σin. (3.6a)


Σin

J . n= 0

Jin

x

intracellular region

channel

h

T

tot

Jin

tm

Figure 3.2.: Three-dimensional control volume across Σin.

Proceeding in a completely similar manner at Σout we obtain

dσoutdt− J tmout + J totout = 0 at Σout (3.6b)

where σout is the surface charge density distributed on Σout. Since no production or

reaction mechanisms occur in the channel region, we have

J tmout = J tmin (3.6c)

and, similarly, we also have

J totout = J totin . (3.6d)

Summing (3.6a) and (3.6b) and using (3.6c) and (3.6d) we conclude that the following

relation holds

σin(x, t) + σout(x, t) = K(x) (3.6e)

at each point x belonging to the external surface of the cell Σcell and for all times t,

where K(x), at each fixed point x, is a constant.

Assuming that K(x) = 0 at the initial time t0, we conclude that having separately

enforced across the cellular membrane:

• the continuity of the total current (Eq. (3.6d)) and

• the continuity of the transmembrane current (Eq. (3.6c)),

implies the instantaneous formation of a charge double layer across the membrane thick-

ness. This double layer is constituted by two time-varying surface charge densities,

distributed over the internal and external membrane walls, equal in module and opposite

in sign. The resulting partition of the total current flowing across the membrane into

3.1. Reduced order modeling of membrane electrophysiology 63

Itot

out

Itot

in

+

_

ϕm

Itm

cap Itm

cond

_ _ _ _

+ + + +

Figure 3.3.: Electrodynamical picture of current flow across the cellular membrane

through the use of an electrical equivalent representation. The model is

a consequence of the current continuity equation.

two distinct contributions is schematically represented by the electric equivalent circuit

shown in Fig. 3.3. The transversal arrow superposed to the two circuital elements indi-

cates that in general the considered electrical parameter is a nonlinear function of the

system state variable, the membrane potential ϕm = ϕm(x, t) = ϕ(in)(x, t)− ϕ(out)(x, t)

defined in (2.1).

The various symbols introduced in Fig. 3.3 have the following definitions and meaning:

• Itotin := J totin × S: total conduction current flowing across the channel mouth at Σin

from the intracellular region (units A);

• Itotout := J totout×S: total conduction current flowing out of the channel mouth at Σout

(units A);

• ϕm: membrane potential across the channel (units V);

• Itmcap :=dσmdt× S =

dQmdt

: transmembrane capacitive current (units A), where σm

is the surface charge density and Qm = σm × S is the total charge accumulated on

each side of the membrane (unit C);

• Itmcond = J tmcond × S: transmembrane conduction current (units A).

The capacitive current Itmcap flows across the (generally nonlinear) capacitance in the

left branch of the circuit while the transmembrane conduction current Itmcond flows across

the (generally nonlinear) conductance in the right branch of the circuit. From a mathe-

matical perspective, the equivalent electrical scheme of Fig. 3.3 is a reduced order model

of membrane electrophysiology because the complex effort required by the solution of

the partial differential equation (PDE) (3.5), in correspondance of every point x ∈ Σcell,

is reduced to the solution of the following ordinary differential equation (ODE) at every


point x ∈ Σcell:

Itmcap(x, t) + Itmcond(x, t)− Itotout(x, t) = 0 t > t0 (3.6f)

Itmcap(x, t) =dQm(x, t)

dt(3.6g)

Itmcond(x, t) = Itmcond(ϕm(x, t)) (3.6h)

where t0 is the initial time of the electrophysiological analysis and Itmcond is a (generally)

nonlinear function of the membrane potential ϕm. Eq. (3.6f) is the classic Kirchhoff

Current Law (KCL) of circuit theory.

In the linear case, we have:

Qm = Cmϕm (3.6i)

Cm =εmtmS (3.6j)

Itmcond = Gtotm ϕm (3.6k)

Gtotm = gtotm S (3.6l)

where εm is the dielectric permittivity of the membrane (units Fm−1) and gtotm is the total

specific conductance of the membrane (units Sm−2). In principle, both Cm and Gm may

depend on the spatial position x on the cell surface and on time t. Assuming for ease of

presentation that they are constant quantities, the KCL takes the form of the following

linear initial value problem (IVP):

Cmdϕm(t)

dt+Gmϕm(t) = Itotout(t) t > t0 (3.6m)

ϕm(t0) = ϕm,0 (3.6n)

where ϕm,0 is a given initial datum and Itotout is a given external current.

Example 3.1.1. Verify that the solution of the IVP (3.6m)- (3.6n) is

ϕm(t) = ϕm,0 exp

(− t− t0

τm

)+

∫ t

t0

Itotout(s)

Cmexp

(− t− s

τm

)ds t ≥ t0

where

τm :=CmGm

is the time constant of the circuit (units s).

Definition 3.1.1 (0D - Lumped parameter model). A model like that schematically rep-

resented in Fig. 3.3 will be henceforth referred to as a 0D model, because the dependence

of the unknown ϕm on the spatial variable is neglected in Eq. (3.6f) once the point x is

fixed on Σcell. Using the language of electrical engineering, the 0D model is also called

lumped parameter model, because the complex bio-physical behaviour of the cell mem-

brane with respect to ion transport is synthetically represented by the lumped electrical

parameters of Fig. 3.3. A more general mathematical formulations based on the use of a

distributed parameter approach will be considered in Chapter 4.

3.2. General form of transmembrane ionic current densities 65

3.2. General form of transmembrane ionic current densities

In this section we address the delicate issue of characterizing in mathematical terms

the function Itmcond in (3.6h). This function is a transmembrane ionic current that flows

through ion channels, transporters, or pumps that are located within the cell mem-

brane. To define each of these currents we adopt the formalism of Hodgkin and Hux-

ley [HH52a, HH52b], generalized here to allow for nonlinear instantaneous current-voltage

relations and ion concentration effects [Mor06]. In the remainder of these notes we always

assume, otherwise differently stated, that in the considered electrolyte solution a number

of Mion ≥ 1 ionic species is flowing. Each ion has a concentration ci and a ionic valence

zi in such a way that the amount of charge carried by the ionic species per unit volume

is qzici, q being the elementary charge of the electron equal to 1.602 · 10−19 Coulomb. In

accordance with (2.5), we introduce the quantity S that represents an arbitrarily chosen

cross-sectional area of the membrane surface across which the considered transmembrane

current Itmi is flowing. Then, the transmembrane current density (units A m−2) associ-

ated with the i-th ionic species has the following expression:

J tmi = J tmi (x, t; PH−H(x, t)) i = 1, . . . ,Mion, (3.7)

PH−H(x, t) =y(x, t), ϕm(x, t), c(in)(x, t), c(out)(x, t)

. (3.8)

In the above relation, x denotes the spatial position vector along the membrane, t is the

time variable while PH−H is a collection of biophysical parameters defined as follows:

• y(x, t) = (y1, . . . , yNg) is a vector of gating variables where Ng is the total number of

gating variables in all of the channel types that arise in our system. The individual

components of y are dimensionless variables in the range [0, 1] and describe the

time-dependent activation or inactivation profile of the channel;

• c(in)(x, t) = (c(in)1 , . . . , c

(in)Mion

) (and similarly c(out)) is the vector of ion concentra-

tions in the intracellular (respectively extracellular) space. By including the whole

vector of ion concentrations, we allow for the possibility that the current density

carried by the i-th species of ion is influenced by the concentrations of other ionic

species on the two sides of the membrane.

The functional relation (3.7) expresses the bio-physical fact that the current density

of the i-th ion may be influenced by the transmembrane current density of the other

channels (possibly of more than one type) that carry the i-th species of ion across the

membrane separating the intracellular and the extracellular space. Finally, the explicit

dependence of J tmi on x reflects the possible inhomogeneity of the membrane, because

the density of channels may vary from one location to another.

3.3. The ODE model of ion transport

Let us assume from now on that (3.6i) and (3.6j) always hold, and define the specific

membrane capacitance (units F m−2)

cm :=CmS. (3.9a)


Assume also that Itotin = Itotout = 0 in such a way that the circuit of Fig. 3.3 is in open

circuit conditions as illustrated in Fig. 3.4. This choice amounts to investigating the

electrical behaviour of the cell without the influence of external sources but only as the

effect of the transmembrane currents.

ϕm

Itm

cap

+

_Itm

cond

_ _ _ _++++

I=0

Figure 3.4.: Membrane equivalent electrical circuit in open circuit conditions.

Then, setting Itotout(x, t) = 0 into (3.6f) and replacing the general expression of the trans-

membrane current density (3.7), we obtain the following ODE to be solved at each time

level t > t0 in correspondance of each spatial position x on the membrane:

cmdϕmdt

(x, t) = −Mion∑i=1

J tmi (x, t; PH−H(x, t)) (3.9b)

ϕ(x, 0) = ϕ0(x). (3.9c)

In (3.9), ϕ0(x) is the initial value of the membrane potential at each point of the

membrane. The concentrations c(in)(x, t) and c(out)(x, t) of the Mion ionic species are

assumed to be (biophysically suitable) given functions. We shall see that this is not

the case with the more general PDE-based model treated in Chapter 4. The gating

variables y may be given functions or the output of a suitable ODE model, as illustrated

in Sect. 3.4.3. Under these assumptions, (3.9) is a system of Cauchy problems for the

membrane potential ϕm to be solved (in principle) at each spatial position x of the

membrane. In practice, this cannot be done and the ODE system (3.9) is solved only at

a finite number of points xk suitable selected over the membrane surface.

3.4. Transmembrane current models

In this section we present the models for the transmembrane current densities that are

most commonly used in the theoretical and computational description of cellular electrical

activity. For a fully detailed treatment of this complex subject, we refer to [KS98, Hil01,

ET10].

3.4. Transmembrane current models 67

3.4.1. The linear resistor model

This is the simplest current-voltage relationship because the ionic current density Ji of

the i-th ion can be expressed as

J tmi = gi (ϕm − Ei) = gi

(ϕ(in) − ϕ(out) − KBT

zqln

(c

(out)i

c(in)i

)), (3.10)

gi being the specific conductance associated with the ionic species ci. The units of gi are

[gi] = A V−1 m−2 = S m−2.

Figure 3.5.: Characteristic curve for the linear resistor.

By inspection, we see that (3.10) is consistent with the thermal equilibrium condi-

tion (2.3.2). The graphical representation of the current-voltage relationship (3.10) in

the (ϕm, J) plane, shown in Fig. 3.5, is a straight line whose slope is equal to gi. Despite

its simplicity, the linear resistor model proves to be quite accurate in many cases and it

is used for instance in [Fro03].

3.4.2. The Goldman-Hodgkin-Katz model

The Goldman-Hodgkin-Katz (GHK) model is a first, significant, example of improvement

of the linear resistor formulation of the previous section. To derive a realistic model of

fluxes and currents that flow across the cellular membrane it is necessary to make some

simplifying assumptions (see also [KS98, Rub90]):

1. equation (2.10) holds across the membrane;

2. the electric field is constant across the membrane;

3. the current density is constant across the membrane.

Referring to Fig. 3.6 for the notation and indicating by x the spatial coordinate parallel to

the channel (black arrow in the figure) and by l the membrane thickness, the application

of assumption 2. and the use of (3.4) and of (2.1) yield

E = −∂ϕ∂x

= constant = −ϕ(out) − ϕ(in)

l=ϕml. (3.11)


Figure 3.6.: Schematics of ion electrodiffusion across the cell membrane.

Using assumption 1. and using (3.11) in (2.10) we get

J tmi = −qziDi

(∂ci∂x− ziq

KBTciϕml

). (3.12)

Then, applying assumption 3. to (3.12) we obtain

J tmi = constant =⇒ ∂J tmi∂x

= 0 =⇒ ∂2ci∂x2

− ziq

KBT· ϕml· ∂ci∂x

= 0.

The solution of the above differential equation is given by

ci(x) = A+B exp

(ziq

KBTϕm

x

l

)(3.13)

where A and B are arbitrary constants that can be found by imposing the following

boundary conditions at the two sides of the channel:

ci(0) = c(in)i

ci(l) = c(out)i

=⇒

A+B = c

(in)i

A+Beziq

KBTϕm = c

(out)i

=⇒

A = −

(c(out)i − c(in)

i eziq

KBTϕm)

eziq

KBTϕm − 1

B =c

(out)i − c(in)

i

eziq

KBTϕm − 1

.

For notational simplicity, we introduce the dimensionless variable

X := ziϕm

KBT/q

which has the physical meaning of a normalized electric potential. Then, to compute the

constant current density throughout the channel we replace (3.13) into (3.12) to obtain

J tmi = −qziDi

(BX

lexp

(Xx

l

)− X

lA−BX

lexp

(Xx

l

)).

The first and third term in the braces at the right-hand side mutually cancel out, and

we are left with the constant current density

J tmi = qziDiX

lA = −qziDi

1

l

[X

eX − 1c

(out)i − XeX

eX − 1c

(in)i

].

3.4. Transmembrane current models 69

Using definition (1.40) and property (1.43), the constant current density can be writ-

ten as

J tmi = −qziDi1

l

[B(

ziϕmKBT/q

)c

(out)i − B

(− ziϕmKBT/q

)c

(in)i

](3.14)

where B is the inverse of the Bernoulli function defined in (1.40). Relation (3.14) is the

celebrated Goldman-Hodgkin-Katz (GHK) equation for the current density associated

with the i-th ion.

Let us check that (3.14) satisfies the thermal equilibrium condition (2.3.2). Using

property (1.43), we have

J tmi = 0 =⇒ c(out)i − c(in)

i exp

(ziqϕmKBT

)= 0

from which we get

ϕm =KBT

ziqln

(c

(out)i

c(in)i

)≡ Ei

that is, the membrane potential coincides with the Nernst potential as required at thermal

equilibrium. Conversely, setting ϕm = Ei into (3.14) yields

J tmi = −qziDi1

l

qziKBT

· KBT

qziln

(c

(out)i

c(in)i

)c

(out)i

c(in)i

− 1

c(out)i −

− qziKBT

· KBT

qziln

(c

(out)i

c(in)i

)c

(in)i

c(out)i

− 1

c(in)i

= 0,

which is the thermal equilibrium condition.

The GHK current density enjoys other interesting properties. Assume that ϕm = 0

(i.e., the intra- and extracellular potentials have the same value). Then, using prop-

erty (1.42) the GHK current density degenerates in

J tmi = −qziDic

(out)i − c(in)

i

l.

This formula corresponds to a pure diffusion ion flow across the membrane in agreement

with the Nernst-Planck relation (3.12) in absence of electric field.

Conversely, assume that c(out)i = c

(in)i = ci (i.e., the intra- and extracellular ion

concentrations have the same value ci). Then, using property (1.43) the GHK current

density degenerates in

J tmi = −qziDicil

(−zi

ϕmKBT/q

).

Using (2.8) and (3.11), the previous relation becomes

J tmi = q|zi|µiciE.

This formula corresponds to a pure drift ion flow across the membrane in agreement with

the Nernst-Planck relation (3.12) in absence of a concentration gradient.

The above analysis shows that the GHK expression of the ion current density au-

tomatically adapts itself to all possible transport regimes. This makes it amenable to

numerical computations and explains the reason of its wide success and implementation

in contemporary simulation tools.


3.4.3. The Hodgkin-Huxley model

In a series of historical papers [HH52a, HH52b] that owned them the Nobel Prize in

Medicine in 1963, Hodgkin and Huxley proposed and analyzed a mathematical model

for ion membrane current conduction that extends the simple linear resistor theory by

accounting for the mechanism of voltage-gating which in turn permits the simulation of

the propagation of an action potential.

The Hodgkin-Huxley (HH) model, as illustrated in [HH52b], includes three ionic

species, Na+, K+ and Cl− in the mathematical description of electrodiffusion because

they are responsible for the majority of the ionic current in a cellular action poten-

tial [KS98, Hil01, ET10].

The HH model is a special instance of the general formulation (3.9b)- (3.9c) and

consists of the following nonlinear (stiff) ODE system:

cmdvmdt

= −(J tmK + J tmNa + J tmL ) (3.15a)

J tmK = n4 gK(vm − vK) (3.15b)

J tmNa = hm3 gNa(vm − vNa) (3.15c)

J tmL = gL(vm − vL) (3.15d)

∂m

∂t= αm(vm)(1−m)− βm(vm)m (3.15e)

∂n

∂t= αn(vm)(1− n)− βn(vm)n (3.15f)

∂h

∂t= αh(vm)(1− h)− βh(vm)h. (3.15g)

Comparing the expressions for the transmembrane current densities J tmK , J tmNa and J tmLwith the general expression (3.7), we see that

y = [m,n, h]T

is the vector of the gating variables while the dependent variable vm is defined as

vm := ϕm − Ec,m (3.15h)

ϕm being, as usual, the membrane potential while Ec,m is the membrane resting potential,

that is, the equilibrium potential of the membrane. The variable vm biophysically repre-

sents the displacement of the membrane potential from the resting state of the membrane,

which corresponds to the situation of thermal equilibrium introduced in Definition 2.3.2

in the case of a single ionic species. Correspondingly, the quantities vK , vNa and vL

represent the deviations of the Nernst potentials of potassium, sodium and chloride from

the membrane resting potential, and are treated in [HH52b] as model fitting parameters

(so-called “adjusted potentials”). All potentials vm, vK , vNa and vL are expressed in

mV, and, precisely, the shifted reversal potentials are set equal to

vK = −12 mV, vNa = 115 mV, vL = 10.613 mV,

3.5. Thermal equilibrium of a system of monovalent ions 71

The specific membrane capacitance cm is equal to 1µFcm−2 while the constant specific

conductances gK , gNa and gL are expressed in mS cm−2 and are set equal to

gK = 36 mS cm−2, vNa = 120 mS cm−2, vL = 0.3 mS cm−2.

The total transmembrane current density J tmm is the sum of a membrane capacitance

contribution cmdϕmdt

and of three ionic current densities. The first current density

is associated with potassium, the second current density with sodium while the third

contribution is a leakage current density J tmL associated with Cl−. This latter term is

usually small compared to the other two ion current densities.

The variablesm, n and h are called gating variables, because they describe the opening

state of the channel. These variables typically vary between 0 and 1, and, at each point

of the membrane, are governed by the ODEs (3.15e)-(3.15g), where the time rates αs and

βs, s = m,n, h, are experimentally determined functions expressed in ms−1. Hodgkin

and Huxley [HH52b] used the following expressions for them:

αm(vm) = B((25− vm)/10) (3.15i)

βm(vm) = 4 exp(−vm/18) (3.15j)

αn(vm) = 0.1B((10− vm)/10) (3.15k)

βn(vm) = 0.125 exp(−vm/80) (3.15l)

αh(vm) = 0.07 exp(−vm/20) (3.15m)

βh(vm) = C((30− vm)/10) (3.15n)

where C(t) := 1/(et + 1). A complete mathematical analysis of the ODE system (3.15)

goes beyond the scope of these notes and can be found in [KS98] and [ET10]. It is worth

noting that accounting for each considered type of ion channel would lead to a system

of increased complexity compared to the linear model (3.10) and the GHK model (3.14)

in terms of the number of state variables. Since each of these state variables is a local

property of the membrane interface, the overall complexity introduced by the HH model

is quite significant and is numerically investigated in the various examples treated in the

laboratories of the course that are specifically devoted to the subject.

3.5. Thermal equilibrium of a system of monovalent ions

In Sect. 2.3.1 we have addressed the important issue of thermal equilibrium for the

electrodiffusive flow of a single ionic species flowing across the cell membrane. In this

section we consider the more general case of a system of Mion monovalent ions (i.e.,

zi = ±1) that are moving in a biological fluid environment, in such a way that

Mion = M+ +M−

where M+ is the number of ions with zi = +1 and M− the number of ions with zi = −1.

To extend Defns. 2.3.1 and 2.3.2 to this (more realistic) situation we can profitably use

the GHK theory of Sect. 3.4.2 and write the following generalization of Def. 2.3.2 as


Definition 3.5.1 (Thermal equilibrium for a system of Mion ≥ 1 monovalent ions).

Thermal equilibrium ⇔ J tmtot,Mion= 0 ⇔ ϕm = Ec,Mion (3.16a)

where

J tmtot,Mion=

Mtot∑i=1

J tmi (3.16b)

J tmi being given by (3.14), and

Ec,Mion =RT

Fln

M+∑i=1

P+i c

(+,out)i +

M−∑i=1

P−i c(−,out)i

M+∑i=1

P+i c

(+,in)i +

M−∑i=1

P−i c(−,in)i

=KBT

qln

M+∑i=1

P+i c

(+,out)i +

M−∑i=1

P−i c(−,out)i

M+∑i=1

P+i c

(+,in)i +

M−∑i=1

P−i c(−,in)i

. (3.16c)

Formula (3.16c) is the so-called Goldman equation and the quantity P±i := D±i /l is the

membrane permeability with respect to the i-th ion, i = 1, . . . ,Mion. The quantity Ec,Mion

is the reversal (or equilibrium) potential of the whole system of ions and is also called

Goldman potential.

Remark 3.5.1. It is important to notice that Def. 3.5.1 does not require the single

ion current density to be equal to zero in thermal equilibrium, but only the (weaker)

condition that the total current density sums up to zero. It is clear that in the case

where the cell is much more permeable to a specific ion than to all the others (highly

selective membrane), the Goldman relation tends to the classical Nernst potential (2.11)

and thermal equilibrium of the whole system is regulated by the most permeant species.

4. PDE Models in Cellular Electrophysiology

In this chapter we address the extension of the ODE model illustrated in Chapt. 3.

The formulation is based on the generalization of the infinite set of initial-value prob-

lems (3.9b)- (3.9c) that must be solved at each spatial position x on the cell membrane

and at each time level t. The extension of this latter model is based on the adoption of a

continuum approach for ion transport where both diffusion and drift forces determine the

spatial and time evolution of the variables affecting the electrical activity of the biological

system. We start by presenting the cable equation model that is the most widely used

method to describe the propagation of an electrical signal along a nerve fiber. Then, we

consider the Poisson-Nernst-Planck (PNP) model in which the effect of a translational

fluid velocity is included to determine transmembrane ion current flow. Finally, the PNP

model is illustrated in detail in the case of a simplified 1D geometrical setting.

74 4. PDE Models in Cellular Electrophysiology

4.1. The cable equation model

In this section we mathematically describe the biophysical mechanism of intercellular

communication through the propagation of an action potential along an axon fiber. Our

presentation is closely based on the material discussed in [ET10], Chapt. 1., to which we

refer for further details.

4.1.1. The geometrical setting

Let us consider the schematical representation of a biophysical network shown in Fig. 4.1.

V_

in

(x,t)ϕm

Rout

jionm

cablex

cell

0 L

ϕm,out

2a

mc

Figure 4.1.: Schematic view of a biophysical connection between an excitable cell (for

example, a neuron) and a terminal output (indicated by a resistive load RL).

The connecting cable is the simplified representation of a nerve fiber (an

axon).

The excitable cell (a neuron, for example) is subject to an external stimulus (the

applied voltage signal V in). The signal gives rise to a modification of the membrane

potential ϕm which is transmitted instantaneously to the output receiving terminal (the

resistive load Rout) giving rise to the output voltage drop ϕm,out. The transmission is

operated by the “biological cable” represented in the scheme by a cylinder of radius a

and length L. In the realistic case, such a cable is a nerve fiber that constitutes the

protrusion of the neuron’s body (soma) towards the environment surrounding the cell.

Such nerve fiber has the same physiological content as that of the mother cell and has the

fundamental role of transmitting a nervous signal (in the form of an electrical impulse) to

a receiving terminal. In what follows, we indicate by S = πa2 the cross section of the axon

fiber, a being the radius of the axon. In typical cases we have a L, although exceptions

may occur (as in the case of the giant squid axon analyzed in the work [HH52b]).

4.1.2. Electrical equivalent representation

The electrical parameters that are used in the equivalent circuit of Fig. 4.1 have the same

biophysical meaning as those introduced in Chapter 3. In particular:

• cm (units Fm−2) is the membrane capacitance per unit area (of the cell and of the

cylindrical axon). It is assumed to be a given positive quantity;

4.1. The cable equation model 75

• jionm (units Am−2) is the membrane ionic current per unit area and is in general a

nonlinear function of the membrane potential and of the ion concentrations flowing

throughout the cell membrane. It can be described with any of the mathematical

models discussed in Sect. 3.4.

Using the above electrical lumped parameters it is possible to account for the spatial

propagation of the input signal V in = V in(t) as illustrated below.

We make the following assumptions:

1. the intra and extracellular regions are characterized by spatially constant ion con-

centrations;

2. the extracellular region is equipotential (for convenience we set its potential equal

to zero);

3. the intracellular region has a uniform electrical resistivity ρin (units Ωm).

Assumption 3. implies that the resistance of an infinitesimal piece dx of axon is equal

to

dR(x) =ρindx

S=ρindx

πa2. (4.1a)

(x,t)m

ϕ (x+dx,t)m

ϕ

x x+dx

dϕm(x,t)

Iin(x,t)

dR(x)x

Figure 4.2.: Ohm’s law for an infinitesimal piece of axon of length dx.

As a consequence, applying Ohm’s law (2.7k) between two points along the axis of

the cable at x and x + dx, respectively, with the convention on the sign of the current

Iin as in Fig. 4.2, we get

dϕm(x, t) := ϕm(x, t)− ϕm(x+ dx, t) = −(ϕm(x+ dx, t)− ϕm(x, t))

= dR(x)Iin(x, t) =ρindx

πa2Iin(x, t)

from which, letting dx tend to zero, we obtain the following differential relation for

the longitudinal electrical current flowing along the axon axis due to the intracellular

longitudinal gradient of the membrane potential

Iin(x, t) = −πa2

ρin

∂ ϕm∂ x

(x, t). (4.1b)

Having identified the expression of ohmic current flow in the direction of the axon

axis, we need now to complete the cable transmission model with the balance of currents


flowing in and out across a volumetric differential element of the cable because capacitive

and ohmic effects occur also in the transversal direction of the fiber. To account for these

latter electrodiffusive phenomena we consider the scheme shown in Fig. 4.3.

Iin

(x,t) Iin

(x+dx,t)

x

dR(x) dR(x)x+dx/2

+ + +_ _ _

dQcap

m(x+dx/2,t)

dt____

dt____dQ

ion

m (x+dx/2,t)

dx

x+dx2a

Figure 4.3.: Kirchhoff current law for an infinitesimal volume of axon fiber of length dx

and lateral surface 2πa.

The balance of currents at x+ dx/2 yields

Iin(x, t) = Iin(x+ dx, t) +dQcapmdt

(x+ dx/2, t) +dQionmdt

(x+ dx/2, t) (4.1c)

where:

• Iin(x, t) and Iin(x + dx, t) are the longitudinal currents entering and leaving the

infinitesimal volume, respectively;

•

dQcapmdt

(x+ dx/2, t) = cm × 2πadx∂ ϕm∂ t

(x+ dx/2, t) (4.1d)

is the time rate of change of the infinitesimal amount of charge stored at each side

of the membrane and distributed over a surface differential area equal to 2πadx;

•

dQionmdt

(x+ dx/2, t) = jionm (x+ dx/2, t; PH−H(x+ dx/2, t))× 2πadx (4.1e)

is the time rate of change of the ionic charge that is flowing across the surface

differential area equal to 2πadx, where jionm is the ion current density expressed

according to the formalism of the Generalized Hodgkin-Huxley model of Sect. 3.2.

Replacing (4.1d) and (4.1e) into (4.1c), dividing out both sides by 2πadx and letting

dx tend to zero, we obtain the following balance of longitudinal and transversal current

densities at point x and time instant t

− 1

2πa

∂ Iin∂ x

(x, t) = cm∂ ϕm∂ t

(x, t) + jionm (x, t; PH−H(x, t)) (4.1f)

where Iin(x, t) is given by (4.1b).

4.2. A PDE model of ion flow in a fluid medium 77

4.1.3. The cable equation model system

Collecting the relations obtained in Sect. 4.1.1, the cable equation system of PDEs de-

scribing the propagation of an electrical impulse along the axis of a nerve fiber consists of

the following equations, to be solved in the space-time cylinder QTfin := (0, L)×(0, Tfin),

Tfin being the length of the simulation time:

cm∂ ϕm∂ t

(x, t) = − 1

2πa

∂ Iin∂ x

(x, t)− jionm (x, t; PH−H(x, t)) (4.2a)

Iin(x, t) = −πa2

ρin

∂ ϕm∂ x

(x, t) (4.2b)

where the ionic current density jionm is modeled in Sect. 3.2.

The boundary conditions corresponding to the configuration shown in Fig. 4.1 are:

ϕm(0, t) = V in(t) t ∈ (0, Tfin) (4.2c)

− Iin(L, t) +1

Routϕm(L, t) = 0 t ∈ (0, Tfin) (4.2d)

while the initial condition is

ϕm(x, 0) = ϕ0m(x) x ∈ (0, L). (4.2e)

We note that if the longitudinal resistivity of the cable ρin → +∞ then the cable equation

model degenerates into the ODE formulation studied in Chapt. 3.

4.2. A PDE model of ion flow in a fluid medium

In this section we carry out a refinement of the cable equation model illustrated in

Sect. 4.1 by including biophysical phenomena that were neglected in the cable equation

formulation. This extension consists of a phenomenological derivation of a model for ion

flow in a fluid medium based on a system of partial differential equations (PDEs) that

include electrical, chemical and fluid driving forces. Our presentation is closely based on

the material discussed in [Sch09] and [MSV15] to which we refer for further details on

mathematical and numerical aspects.

4.2.1. The geometrical setting

Fig. 4.4 illustrates a schematical picture of the problem at hand. A system of M ≥ 1

chemical species is flowing in the fluid medium Ω ⊂ R3 under the application of external

electrical, chemical, thermal and mechanical forces. The spatial position of a point in the

fluid is indicated by x while the time coordinate is t. Every function and/or parameter is

assumed to depend on x and t and such dependence is left understood, except otherwise

stated. The fluid is assumed to be incompressible and its motion is governed by the time-

dependent incompressible Navier-Stokes equations (see [QV97] for a general theory and

numerical approximation). The velocity of the fluid is denoted by u while the number

density of each species i, i = 1, . . . ,M , is denoted by ni, the corresponding chemical

valence is zi and the particle average velocity is vi. For ease of notation we omit the

subscript i, except where needed. We also denote by Ti and Tf the absolute temperatures

of ions and fluid, while Tsys indicates a given (constant) temperature.


Figure 4.4.: Ion flow in a fluid medium: computational domain and physical phenomena.

4.2.2. Balance of mass

Conservation of the number of particles is expressed by the following equation

∂n

∂t+ divf = P (4.3a)

where

f = nv (4.3b)

is the ion flux density and P is the net production rate (units m−3 s−1).

Tx

xΣx

f

n

P

Figure 4.5.: Conservation law: integral interpretation.

Remark 4.2.1 (Conservation law). Eq. (4.3a) is the basic example of a conservation law

in differential form and can be regarded as the multi-dimensional extension of (1.1a). To

explain the physical meaning of (4.3a) we proceed as in Thm. 1.2.1, and we fix arbitrarily

a point x ∈ Ω and pick around x a control volume Tx (see Fig. 4.5). Then, we integrate


the conservation law over Tx and apply the divergence theorem obtaining

dNTx(t)

dt= pTx(t)−

∫Σx

f(x, t) · n dσ. (4.3c)

The quantity NTx(t) is the total number of ion particles contained in the control volume

Tx at time t while pTx(t) is the number of particles produced or consumed in Tx per unit

time. Thus, Eq. (4.3a) tells us that the time rate of change of the number of ion particles

in Tx equals the balance between the net production rate of particles in Tx and the net

number of particles that crosses the surface Σx of Tx per unit time. In steady-state

conditions this latter balance is identically equal to zero so that

dNTx(t)

dt= 0 (4.3d)

and the total number of particles in the control volume remains constant.

4.2.3. Balance of momentum

Conservation of momentum is expressed by Newton’s second law of dynamics

mnDv

Dt= F tot (4.4a)

where m is the mass of the particle (units Kg), F tot is the total force per unit volume

(units N m−3) that is acting on n, while

Dv

Dt=∂v

∂t+ (v ·∇ )v (4.4b)

is the material derivative (also called Lagrangian derivative) of v. The k-th component

of the second term at the right-hand side of (4.4b) is given by

((v ·∇ )v)k = v ·∇ vk =

3∑j=1

vj∂vk∂xj

k = 1, . . . , 3.

The above expression is a nonlinear function of v and makes the solution of (4.4a) a

nontrivial task.

Assumption 4.2.1. We assume

Dv

Dt= 0. (4.4c)

This amounts to neglecting inertial forces and reduces the momentum balance equa-

tion (4.4a) to the simpler force equilibrium relation

F tot = 0. (4.4d)

We also assume

Ti = Tf = Tsys. (4.4e)

This amounts to neglecting thermal energy exchange among the components of the mul-

tiphase system at hand.


Rh

viscousdrag

−P −P

−P

v

Figure 4.6.: Ion particle in motion in the fluid (cyan), the viscosity effect (green) and the

hydrostatic pressure (red).

The total force density acting on each ion species is

F tot = qznE − 6πηRhnv + div (−PI) (4.4f)

having denoted by I the identity tensor of dimension 3.

The first term at the right-hand side of (4.4f) is the Coulomb force.

The second term is the viscous drag exchanged between fluid and particles according

to the so-called Stokes approximation. In this latter theory particles are assumed to have

a spherical shape whose hydrodynamic radius is given by

Rh =KBTsys6πηD

(4.4g)

where η is the fluid viscosity while D is the molecular diffusion coefficient of the moving

ion particle.

The third term at the right-hand side of (4.4f) is the hydrostatic pressure exerted by

the other surrounding particles. Using the theory of ideal gases, we have

P = nKBTsys. (4.4h)

The geometrical representation of particle motion throughout the fluid under the

action of the forces exerted on the considered particle by the fluid and by the surrounding

particles is schematically depicted in Fig. 4.6.

Replacing (4.4f), (4.4g) and (4.4h) into the simplified momentum balance equa-

tion (4.4d) and dividing both sides by n, we get

v =qz

KBTsysDE −D∇n

n=

z

|z|q|z|

KBTsysDE −D∇n

n=

z

|z|µE −D∇n

n

where we have defined the electrical mobility of the ion particle as

µ :=q|z|DKBTsys

. (4.4i)


4.2.4. The Velocity-Extended Poisson-Nernst-Planck model

Including the mechanism of passive transport due to electrolyte fluid velocity u, the ion

charge transport model is constituted by the following system of PDEs:

∂ni∂t

+ divf i = Pi i = 1, . . . ,M (4.5a)

f i = nivi =z

|z|µiniE −Di∇ni + niu (4.5b)

Di = µiKBTsysq|zi|

. (4.5c)

Eq. (4.5c) is well known as Einstein’s relation. The boxed term at the right-hand side

in (4.5b) is an additional translational contribution to ion flow due to fluid motion and

represents the coupling between electrolyte fluid motion and electrodiffusive ion trans-

port.

Using a completely similar approach to express the balance of mass and momentum

for the electrolyte fluid, the following system of PDEs is obtained to describe the motion

of the fluid:

div u = 0 (4.6a)

ρf∂u

∂t= div σ(u, p) + q

M∑i=1

ziniE (4.6b)

σ(u, p) = 2µfε(u)− pδ (4.6c)

ε(u) =1

2(∇u + (∇u)T ). (4.6d)

The equation system (4.6) is referred to as the time-dependent Stokes system to describe

the motion of an incompressible and viscous fluid with a constant density ρf [Kg m−3]

(see [QV97] for a complete mathematical and numerical treatment). In the electrolyte

fluid model, u is the fluid velocity [ms−1], p is the fluid pressure [Pa], µf the fluid shear

viscosity [Kg m−1s−1], σ the stress tensor [Pa] and ε the strain rate tensor [s−1]. The

symbol δ is the identity tensor of dimension 3. Notice that in accordance with the

assumption of slow fluid motion, the quadratic convective term in the inertial forces

has been neglected in the momentum balance equation (4.6b). The boxed term at the

right-hand side of equation (4.6b) physically corresponds to the electric pressure exerted

by the ionic charge on the electrolyte fluid, and mathematically represents the coupling

between electrodiffusive ion transport and electrolyte fluid motion.

To complete the electrochemical-fluid picture of ion transport in the moving fluid we

need determine the electric field E. This is done by solving at each time level t the

Poisson equation

divD = ρfix + ρmob (4.7a)

where:

D = εE (4.7b)

E = −∇ϕ. (4.7c)


The variable ϕ is the electric potential (units V), the vector field D is the electric dis-

placement (units Cm−2) and ε (units Fm−1) is the dielectric permittivity of the channel

fluid medium. Relation (4.7c) is the consequence of Assumption 3.1.1 and allows to ex-

press the electric field E (units Vm−1) as a gradient field in terms of the electric potential.

Eq. (4.7a) is referred to as the Poisson equation and represents Gauss’ law in differential

form. The quantity ρfix is a given function of space and time and biophysically repre-

sents the volumetric charge due to the presence of fixed ions in the channel membrane.

Its mathematical characterization is carried out in Sect. 5.1. The quantity ρmob, defined

as

ρmob = q

M∑i=1

zini, (4.7d)

is the mobile charge density and biophysically represents the total charge density of

mobile ions that are flowing inside the channel medium.

The system of PDEs (4.5), (4.6) and (4.7) in the scalar unknowns ni, i = 1, . . . ,M ,

p and ϕ, and for the vector-valued variables f i, i = 1, . . . ,M , u and E is called

velocity-extended Poisson-Nernst-Planck (VE-PNP) model. For a mathematical anal-

ysis of the VE-PNP system, we refer to the classical book [Rub90] and to the more

recent works [Jer02, JS09].

4.2.5. The electrochemical potential

We can write in an equivalent manner the VE-PNP system by replacing (4.7c) and (4.5c)

in the first two terms at the right-hand side of (4.5b), to obtain

− zi|zi|

µini(−∇ϕ)−Di∇n = − zi|zi|

µini

(∇ϕ+

KBTsysqzi

∇ ln

(ninref

))nref being a reference concentration. Therefore, introducing the electrochemical potential

ϕeci := ϕ+KBTsysqzi

ln

(ninref

)i = 1, . . . ,M (4.8a)

we can write the ion flux density in the compact form

f i = nivi = − zi|zi|

µini∇ϕeci + niu. (4.8b)

Notice that, inverting (4.8a), we obtain the well-known Maxwell-Boltzmann statistics

ni = nref exp

(qzi

ϕeci − ϕKBTsys

)i = 1, . . . ,M. (4.8c)

The remarkable conclusion of this analysis is that the average ion velocity can be ex-

pressed as

vi =zi|zi|

µiEeci + u i = 1, . . . ,M, (4.8d)

where

Eeci := −∇ϕeci i = 1, . . . ,M, (4.8e)


is the electrochemical field acting on the i-th ion species. Using (2.4), the ion current

density is given by

J i = qzinivi = σiEeci + qziniu i = 1, . . . ,M (4.8f)

having defined the ion electrical conductivity

σi := qµi|zi|ni i = 1, . . . ,M. (4.8g)

In the case of a perfect conductor, the ionic density is spatially uniform so that from (4.8c)

we have ϕeci = ϕ + K, K being a constant, so that Eeci = E and Ohm’s law (2.7k) is

automatically recovered from (4.8f). However, in the case of a general ionic fluid in which

the concentration is not spatially (and temporally) uniform, the use of the generalized

Ohm’s law (4.8f) allows to account for both drift and diffusion contributions to ion flow

and provides the right modeling view to ionic transport in the fluid.

5. The Poisson-Nernst-Planck model in one

spatial dimension

In this chapter we consider the PNP model illustrated in Sect. 4.2 in the case of a one-

dimensional (1D) channel geometry. In particular, we derive a reduced boundary-value

initial-value differential problem starting from a multi-domain partition of the intra and

extracellular environments, for which appropriate boundary conditions are provided at

the entrance and outlet sections of the channel. Our presentation is based on [CE93b]

to which we refer for all the biophysical details. To be adherent to this latter reference,

and to simplify the exposition, we neglect throughout the chapter the contribution of

the electrolyte fluid velocity to the translational motion of the ions in the channel. This

amounts to setting u = 0 in (4.8f).

5.1. Geometrical model

Following the presentation of [CE93b], we illustrate in Fig. 5.1 the schematic representa-

tion of a cross-section of a biological channel, assuming rotational invariance around the

channel axis (x axis).

+++ + + +

++

0_

0 d

_

+

_

+

+

_

_

+

+

+

_

_

region

L R

Intracellular

region

ExtracellularMembrane

σ

d+a x

Figure 5.1.: Schematic view of the biophysical problem. Five regions can be distin-

guished. From left to right: intracellular bathing solution, antichamber,

channel region, antichamber, bathing solution in the extracellular side. At

the endpoints, x = 0 and x = L, terminal contacts are marked in red color.

The red bullets at the endpoints of the domain, x = L and x = R, are the terminals

86 5. The Poisson-Nernst-Planck model in one spatial dimension

of a Cellular Electrophysiology equipment, and represent the physical places where the

electrical potential is fixed and the solution intra and extra-cellular concentrations are

accessible to experimental measurements. Then, five spatial regions can be distinguished,

ordered from left to right:

1. Region 1, L ≤ x ≤ 0−: this is the bathing solution in the intracellular side.

2. Region 2, 0− < x ≤ 0: this is the channel antichamber (or access region) from the

intracellular side.

3. Region 3, 0 ≤ x < d: this is the channel region.

4. Region 4, d ≤ x < d+: this is the channel antichamber from the extracellular side.

5. Region 5, d+ ≤ x ≤ R: this is the bathing solution in the extracellular side.

Finally, two kinds of charges can be identified in Fig. 5.1:

• a fixed charge density σ (units Cm−2) uniformly distributed over the surface of the

channel;

• a mobile charge density ρmob (units Cm−3) of cations and anions that are flowing

through the channel fluid under the action of electrodiffusive forces.

The mobile charge density accounts for the ions that are flowing throughout the

membrane channel and is defined in (4.7d). The surface charge density accounts for ions

trapped within the bilayer (porous) lipid structure constituting the cellular membrane,

and has the effect to modify the motion of cations and anions throughout the channel

because if σ < 0 cations are attracted while anions are repelled and the opposite occurs

if σ > 0.

The surface charge density σ can be accounted for in the term ρfix at the right-hand

side of the differential Gauss law (4.7a) as follows. Let x and x + dx two points along

the channel axis and let dΣlat(x) = 2πa(x)dx denote the infinitesimal lateral surface of

the channel, a(x) being the value at x of the radius of the channel circular cross-section.

The infinitesimal fixed charge distributed over dΣlat(x) is

dQσ(x) = σ(x)2πa(x)dx.

The volumetric fixed charge ρfix(x) contained in the infinitesimal volume dV (x) =

πa(x)2dx is

dQV (x) = ρfix(x)πa(x)2dx

so that, equating the two infinitesimal charges we find

ρfix(x) =dQσ(x)

πa2(x)dx=

2σ(x)

a(x).

Therefore, the volumetric concentration P (x) (units m−3) of fixed ions in the channel

electrostatically equivalent to the surface concentration σ(x)/q (units m−2) of fixed ions

located on the membrane lipid bilayer is

P (x) =ρfix(x)

q=

2σ(x)

qa(x). (5.1)

5.2. Biophysical assumptions 87

5.2. Biophysical assumptions

The endpoints of the domain, x = L and x = R, are located sufficiently far from the

antichamber and channel regions, in such a way that appropriate equilibrium conditions

can be applied. More importantly, the endpoints are assumed to be the physical place

where the solution intra and extra-cellular electrochemical conditions are accessible to

experimental measurements. Because of this, following [CE93a, CE93b], we assume that

at x = L and x = R:

(A1) the electric potential ϕ is a known given quantity of time, so that:

ϕ(L, t) = ϕL(t) (5.2a)

ϕ(R, t) = ϕR(t); (5.2b)

(A2) the ion concentrations ni are known given quantities of time, so that:

ni(L, t) = ni,L(t), i = 1, . . . ,M (5.2c)

ni(R, t) = ni,R(t), i = 1, . . . ,M (5.2d)

where M ≥ 1 is the number of ions flowing in the cellular solution. For each

i = 1, . . . ,M , we set nmaxi,L = maxt∈[0,T ]

ni,L(t) and nmaxi,R = maxt∈[0,T ]

ni,R(t). The boundary

values for the ion concentrations satisfy the electroneutrality constraint at each

time level t:

M∑i=1

zini,L(t) = 0 (5.2e)

M∑i=1

zini,R(t) = 0 (5.2f)

where zi is the charge number associated with each ion species ni (zi > 0 for cations,

zi < 0 for anions and zi = 0 for neutral species).

We close the characterization of the bathing regions by assuming that:

(A3) the ion flux densities vanish inside the baths for each time level t:

Ji(x, t) = 0 L ≤ x ≤ 0−, i = 1, . . . ,M (5.2g)

Ji(x, t) = 0 d+ ≤ x ≤ R, i = 1, . . . ,M. (5.2h)

Continuing to move from the periphery of the domain towards the channel region, we

encounter the antichamber openings, at x = 0− and x = d+, respectively. We make the

following assumptions within the antichamber regions:

(A4) electroneutrality holds at the channel mouth entrances:

qP (0−, t) + q

M∑j=1

zjnj(0−, t) = 0 (5.2i)

qP (d+, t) + qM∑j=1

zjnj(d+, t) = 0; (5.2j)


(A5) the electric potential is spatially constant in the antichamber regions:

ϕ(x, t) = ϕ(0−, t) 0− ≤ x ≤ 0 (5.2k)

ϕ(x, t) = ϕ(d+, t) d− ≤ x ≤ d+; (5.2l)

(A6) the ion concentrations are spatially constant in the antichamber regions:

ni(x, t) = ni(0−, t) 0− ≤ x ≤ 0 i = 1, . . . ,M (5.2m)

ni(x, t) = ni(d+, t) d ≤ x ≤ d+ i = 1, . . . ,M. (5.2n)

5.3. Boundary conditions at channel openings

In this section we use the geometrical multi-domain representation of the problem of

Sect. 5.1 and the assumptions made in Sect. 5.2 to derive the boundary conditions to be

supplied to the PNP Model at x = 0 and x = d (channel openings). Using (4.8f) in one

spatial dimension and having set u = 0, we can write the PNP current density as

Ji = −q |zi|µini∂ϕeci∂x

i = 1, . . . ,M (5.3a)

where

ϕeci := ϕ+1

ziVth ln

(ninref

)i = 1, . . . ,M (5.3b)

is the electrochemical potential and

nref := maxi=1,...,M

nmaxi,L ;nmaxi,R

is a reference concentration. Inverting (5.3b) we obtain the well-known Maxwell-Boltzmann

(MB) statistics for the ion densities

ni = nref exp

(ziϕeci − ϕVth

)i = 1, . . . ,M. (5.3c)

Using (A3) at each time level t, we get

ϕeci (x, t) = const =: ϕeci,L(t) L ≤ x ≤ 0− i = 1, . . . ,M (5.3d)

ϕeci (x, t) = const =: ϕeci,R(t) d+ ≤ x ≤ R i = 1, . . . ,M (5.3e)

where ϕeci,L(t) and ϕeci,R(t), i = 1, . . . ,M , are yet to be determined. Using (5.3d) and (5.3e)

and assumptions (A5) and (A6) into the MB statistics (5.3c), we find:

ni(x, t) = nref exp

zi(ϕeci,L(t)− ϕ(x, t)

)Vth

L ≤ x ≤ 0 i = 1, . . . ,M

ni(x) = nref exp

zi(ϕeci,R(t)− ϕ(x, t)

)Vth

d ≤ x ≤ R i = 1, . . . ,M.

5.3. Boundary conditions at channel openings 89

Using (A2) in the equations above at z = L and z = R, respectively, we obtain, for each

time level t, the boundary values for the electrochemical potential:

ϕeci,L(t) = ϕL(t) +Vthzi

ln

(ni,L(t)

nref

)i = 1, . . . ,M (5.3f)

ϕeci,R(t) = ϕR(t) +Vthzi

ln

(ni,R(t)

nref

)i = 1, . . . ,M. (5.3g)

From the previous discussion, we see that the electrochemical potentials are spatially

constant in the bathing regions, but this does not necessarily mean that also the electric

potential and the ion concentrations are constant in those regions. On the contrary, it is

expected that these quantities experience a spatial variation in such a way that diffusive

and drift current densities mutually cancel out each other to ensure the equilibrium

conditions (5.2g)-(5.2h). This means that it makes sense to introduce the voltage drops

occurring across the bathing solution regions, the so-called built-in potentials:

ϕbi(0−, t) := ϕ(L, t)− ϕ(0−, t) (5.4a)

ϕbi(d+, t) := ϕ(d+, t)− ϕ(R, t). (5.4b)

To determine the built-in potentials we enforce the charge neutrality conditions (5.2i)

and (5.2j) obtaining, at x = 0−:

P (0−) +M∑i=1

zini(0−, t) = P (0−) +

M∑i=1

zinref exp

(zi

(ϕeci (0−, t)− ϕ(0−, t))

Vth

)

= P (0−) +M∑i=1

zinref exp

(zi

(ϕeci,L(t)− ϕ(L, t))

Vth

)exp

(zi

(ϕ(L, t)− ϕ(0−, t))

Vth

)

= P (0−) +

M∑i=1

zini,L(t) exp

(ziϕbi(0

−, t)

Vth

)= 0 (5.4c)

and similarly at x = d+:

P (d+) +M∑i=1

zini(d+, t) = P (d+) +

M∑i=1

zinref exp

(zi

(ϕeci (d+, t)− ϕ(d+, t))

Vth

)

= P (d+) +

M∑i=1

zinref exp

(zi

(ϕeci,R(t)− ϕ(R, t))

Vth

)exp

(zi

(ϕ(R, t)− ϕ(d+, t))

Vth

)

= P (d+) +M∑i=1

zini,R(t) exp

(−zi

ϕbi(d+, t)

Vth

)= 0. (5.4d)

Remark 5.3.1. In the particular case of the KCl solution (z1 = +1, z2 = −1) considered

in [CE93a], equations (5.4c) and (5.4d) can be explicitly solved, to yield:

ϕbi(0−, t) = Vth ln

(−P (0−) +

√(P (0−))2 + 4(n1,L(t))2

2n1,L(t)

)

ϕbi(d+, t) = Vth ln

(P (d+) +

√(P (d+))2 + 4(n1,R(t))2

2n1,R(t)

)


where we notice that n1,L(t) = n2,L(t) and n1,R(t) = n2,R(t), at each time level t, because

of the electroneutrality constraints (5.2e) and (5.2f). It can be easily checked that the

above expressions can be written in the equivalent form:

ϕbi(0−, t) = −Vth sinh−1

(P (0−)

2n1,L(t)

)(5.4e)

ϕbi(d+, t) = +Vth sinh−1

(P (d+)

2n1,R(t)

). (5.4f)

In order to complete the characterization of the boundary values for the dependent

variables of the problem we need the value of the electric potential and of the ion con-

centrations at the two endpoints of the channel. The potential is determined using (5.4)

and (A5), which yield at each time level t:

ϕ(0, t) = ϕ(0−, t) = ϕL(t)− ϕbi(0−, t) (5.5a)

ϕ(d, t) = ϕ(d+, t) = ϕR(t) + ϕbi(d+, t). (5.5b)

To determine the ion concentrations we use (5.3f) and (5.4a) at x = 0 to obtain:

ni(0, t) = ni(0−, t) = nref exp

(zi

(ϕeci (0−, t)− ϕ(0−, t))

Vth

)= nref exp

(zi

(ϕeci,L(t)− ϕ(0−, t))

Vth

)

= nref exp

(zi

(ϕeci,L(t)− ϕ(L, t))

Vth

)exp

(zi

(ϕ(L, t)− ϕ(0−, t))

Vth

)= ni,L(t) exp

(ziϕbi(0

−, t)

Vth

)i = 1, . . . ,M (5.6a)

and, similarly, (5.3g) and (5.4b) at x = d to obtain:

ni(d, t) = ni(d+, t) = nref exp

(zi

(ϕeci (d+, t)− ϕ(d+, t))

Vth

)= nref exp

(zi

(ϕeci,R(t)− ϕ(d+, t))

Vth

)

= nref exp

(zi

(ϕeci,R(t)− ϕ(R, t))

Vth

)exp

(zi

(ϕ(R, t)− ϕ(d+, t))

Vth

)= ni,R(t) exp

(−zi

ϕbi(d+, t)

Vth

)i = 1, . . . ,M. (5.6b)

5.4. A graphical representation of the PNP model in 1D

In this section we give a pictorial representation of the various biophysical assumptions

introduced in Section 5.2 and the corresponding graphs of the dependent variables of

the problem, i.e., the electric and electrochemical potentials, the ion number and current

densities.

Fig. 5.2 schematically illustrates the spatial distributions of the dependent variables

of the PNP model at a fixed time instant t. The top panel of the figure shows the ion

number density n and the ion current density J whereas the bottom panel of the figure

5.4. A graphical representation of the PNP model in 1D 91

0_

d+

nL n

R

0_

d+

ϕec(x,t) ϕ(x,t)

ϕ ec

L

ϕ ec

RϕL

ϕR

xL 0 d R

n(x,t)

L x0 d R

J(x,t)

Figure 5.2.: Spatial distributions of the dependent variables of the PNP model in 1D. Top

panel: n(x, t), J(x, t). Bottom panel: ϕ(x, t), ϕec(x, t). Solid lines indicate

that the variable is a solution output of the PNP model. Dashed lines indicate

that the plotted value is the result of an assumption of Section 5.2.

shows the electric potential ϕ and the ion electrochemical potential ϕec. The accessible

boundary values for the ion number density are denoted by nL and nR whereas those

for the electric potential are denoted by ϕL and ϕL. The boundary values for the elec-

trochemical potential are determined using (5.3f) and (5.3g) and are denoted by ϕecL and

ϕecR , respectively. The solid line in the various plots indicates that the plotted quantity

is the computed output of the PNP model system. The dashed line, instead, indicates

that the plotted quantity is the result of one of the assumptions made in Sect. 5.2.

Specifically, looking at the plot of the current density J , we see that J is equal to

zero in the bath regions in accordance with Assumption (A3). Instead, the fact that J is

still equal to zero also in the antichamber regions is the result of Assumptions (A5) and

(A6). Finally, in the channel region the current density is not spatially constant unless

steady-state conditions are reached. As a matter of fact, solving the ion mass balance

equation at time t we get

J(x, t) = J(0, t)− qz∫ x

0

∂n

∂t(ξ, t)dξ ∀x ∈ (0, d), ∀t > 0,

so that we obtain

J(x,+∞) = J(0,+∞) ∀x ∈ (0, d).

Since the current density J is zero in the union of bathing and antichamber regions,


in such regions the electrochemical potential ϕec turns out to be constant, equal to

the boundary values ϕecL and ϕecR , respectively. The plots of ion number density and

electric potential in the antichamber regions are the result of Assumptions (A5) and (A6).

The constant values of n and ϕ in these regions are determined using (5.5) and (5.6),

respectively.

5.5. The PNP model in 1D

Let Tfinal > 0 denote the temporal duration of ion electrodiffusion into the channel.

Then, the equations for the PNP model to be solved in the space-time cylinder QTfinal:=

(0, d)× (0, Tfinal) read:

qzi∂ni∂t

+∂Ji∂x

= 0 i = 1, . . . ,M (5.7a)

Ji = qµi|zi|niE − qziDi∂ni∂x

i = 1, . . . ,M (5.7b)

∂E

∂x=q

ε

(M∑i=1

zini + P

)(5.7c)

E = −∂ϕ∂x

(5.7d)

where diffusivities and electrical mobilities are related by

Di =µiVth|zi|

i = 1, . . . ,M. (5.8)

For each time t ∈ (0, Tfinal), the Dirichlet boundary conditions at x = 0 are:

ϕ(0, t) = ϕL(t)− ϕbi(0−, t) (5.9a)

ni(0, t) = ni,L(t) exp

(ziϕbi(0

−, t)

Vth

)i = 1, . . . ,M, (5.9b)

where the built-in potential at x = 0− is determined by solving the nonlinear algebraic

equation

P (0−, t) +M∑i=1

zini,L(t) exp

(ziϕbi(0

−, t)

Vth

)= 0 (5.9c)

whereas the boundary conditions at x = d are:

ϕ(d, t) = ϕR(t) + ϕbi(d+, t) (5.9d)

ni(d, t) = ni,R(t) exp

(−zi

ϕbi(d+, t)

Vth

)i = 1, . . . ,M, (5.9e)

where the built-in potential at x = d+ is determined by solving the nonlinear algebraic

equation

P (d+) +

M∑i=1

zini,R(t) exp

(−zi

ϕbi(d+, t)

Vth

)= 0. (5.9f)

(5.9g)

5.5. The PNP model in 1D 93

In the above relations, the boundary data at x = 0 and at x = d for ϕ and ni are given,

experimentally accessible, functions of time. Moreover, at time t = 0 the following initial

conditions are prescribed for the ion concentrations in the interior of the channel

ni(x, 0) = n0i (x) i = 1, . . . ,M (5.9h)

where the initial data n0i are given positive bounded functions in (0, d).

Part III.

Solution Maps and Functional

Techniques

6. Solution Maps

Abstract

In this chapter we illustrate and analyze a family of solution maps for the decoupled

treatment of the various sets of equations that constitute the PNP and Cable Equation

models introduced in Chapter 4. The solution maps are based on the classic Gum-

mel’s Decoupled Algorithm originally proposed in [Gum64] for the Drift-Diffusion model

in semiconductors and consist of two main steps: (1) time semidiscretization with the

Backward Euler method; (2) iterative solution of the nonlinear system of PDEs at each

discrete time level by a fixed point method. The theoretical analysis conducted in this

chapter is mainly based on [Jer96] and on [PJ97].

Part IV.

Modeling and Simulation of

Semiconductor Devices for

Application in Electronics

7. Modeling and Simulation of

Semiconductor Devices

In this chapter, we describe a number of semiconductor devices used in electronics indus-

try and the corresponding mathematical models for their functioning. The attention is

focused on the most widely used mathematical instrument for the simulation of modern

electronics components, the so-called Drift-Diffusion (DD) equation model, that consists

of a nonlinearly coupled set of PDEs in conservation form, completely similar to the

PNP equations discussed in Chapt. 4, that describe charge transport and electric field

distribution inside a semiconductor device. For each considered device we illustrate, in

stationary conditions, the basic principles of its functioning and the associated mathe-

matical formulation, referring to Chapt. 1 for the numerical discretization.

128 7. Modeling and Simulation of Semiconductor Devices

7.1. A Short Introduction to Integrated Circuits

Fig. 7.1 shows the three scientists of Bell Telephone Labs that mainly contributed to the

advancement of modern electronics: John Bardeen, William Shockley and Walter Brat-

tain. They developed the ideas and experiments that led to the invention and fabrication

of the first transistor device in 1947 (Fig. 7.2). Because of this fundamental result, they

were awarded the Nobel Prize in Physics in 1956.

Figure 7.1.: The inventors of transistor.

Figure 7.2.: Replica of the first transistor developed at Bell Labs in 1947.

The next important step toward modern semiconductor device technology was taken

in 1958, with the introduction of Integrated Circuits (IC), an example of which is shown

in Fig. 7.3.

From that time until present, an ever-increasing progress has continued, according

to the indication of ”Moore’s Law”, formulated by Gordon Moore (co-founder of Intel

Corporation) in 1965 and stating that the number of transistors on integrated circuits

doubles approximately every two years. The predictive power of Moore’s law is very

well demonstrated in Fig. 7.4 which illustrates the trend of the increase of the number

of transistors/chip NTr as a function of the fabrication year, over the period 1970-2010.

7.2. Multiscale Structure of Integrated Circuits 129

Figure 7.3.: Integrated circuit with related packaging and terminal connections.

Figure 7.4.: Moore’s law. On the y-axis: number of transistor/chip. On the x-axis:

fabrication year. Circles denote experimental data, while the solid line is a

representation of Moore’s law.

From the curve (solid line in Fig. 7.4) we see that Ntr moves from 2000 in 1971 to 200

millions in 2010, with an expected value of 2 billions in 2015. These numbers give an idea

of the enormously high level of integration characterizing nowadays device technology,

which undergoes the name of Ultra Large Scale Integration (ULSI).

7.2. Multiscale Structure of Integrated Circuits

To better become familiar with integrated device technology, we provide in Fig. 7.5 a

schematic view of the structure of a semiconductor device that is typically encountered

in modern electronics industry.

Proceeding from top to bottom, and from left to right in a counterclockwise orienta-

tion, we can distinguish four spatial scales, each one being characterized by a different

order of magnitude of the characteristic length `:

1. macroscale: it is the scale of the silicon wafer constituting the substrate of any


Figure 7.5.: Schematic multiscale structure of an electronic device. Top left: silicon wafer

(macroscale); top right: an IC (mesoscale). Bottom right: a Metal-Oxide-

Semiconductor (MOS) transistor (micro/nanoscale); bottom left: atomic

structure of silicon lattice with the presence of a doping impurity of a P

(phosphorus) atom (atomic scale).

electronic device, ` = O(10−2) m;

2. mesoscale: it is the scale of the integrated circuit (IC), ` = O(10−3) m;

3. micro/nanoscale: it is the scale of the single component of the IC, ` = O(10−6÷10−9) m;

4. atomic scale: it is the scale of the material lattice, ` = O(10−10) m.

Each scale has its appropriate mathematical model:

• Atomic scale: quantum picture, based on the single-electron Schroedinger equa-

tion.

• Micro/nano scale: semi-classical/quantum picture, based on a two-family hier-

archy: (1) semi-classical hierarchy, composed of: the Boltzmann Transport Equa-

tion (BTE), the Hydrodynamic Model (HD) and the Drift-Diffusion Model (DD);

(2) quantum hierarchy, composed of: the Wigner Transport Equation (WTE),

the Quantum-Hydrodynamic Model (QHD) and the Quantum-Corrected Drift-

Diffusion Model (QCDD).

• Meso/macro scale: lumped parameter circuit modeling, coupled with Maxwell

equations to account for electromagnetic propagation throughout the IC and ther-

mal models to describe heat power consumption and dissipation.

The computational effort required to numerically solve the equation system at each

above mentioned scale varies considerably, with a typical (dramatic) increase going from

7.3. Notation 131

macroscale to atomic scale. An excellent trade-off between machine time cost and phys-

ical accuracy is represented by the DD model, which is by far the most widely used

mathematical tool for industrial simulation of semiconductor devices. For this reason,

we focus our discussion in the remainder of the text to this mathematical description

and refer to the specialized literature for further details about the various models in the

multiscale hierarchy introduced in this section.

7.3. Notation

In the remainder of this chapter we apply the PNP equation system illustrated in Sect. 4.2

to the mathematical study of semiconductor devices in stationary conditions. This is

equivalent to considering a binary mixture of monovalent anions and cations, i.e., such

that their chemical valence is equal to ±1 as in the case of the KCl solution or the NaCl

solution. We use the symbol p (positive) and n (negative) to refer to the concentrations

of holes and electrons, respectively. In our analysis we follow [Man13, SMJ15], the series

of references [Jer96, BRGC86, RGQ93] in the context of semiconductor device modeling,

and the theory of [Rub90, Jer02, JS09, Sch09]. For the numerical simulation examples

of each considered device, we refer the reader to the material of laboratory sessions that

can be downloaded at

https://beep.metid.polimi.it/

upon authorized credentials.

7.4. Conductivity in a Semiconductor Material

In a semiconducting medium like silicon, unlike the case of a conductor like copper or

aluminum, there exist forbidden energy states for charge carriers that act as an obstacle to

current flow in the material. Such forbidden states belong to the energy interval (Ev, Ec)

of width Egap = Ec−Ev = 1.12 eV (at room temperature) represented in Fig. 7.6, where

Ec and Ev denote the energies of the bottom of the conduction band and the top of the

valence band, respectively, while Ei is the intrinsic Fermi level of the material (situated

at the mid-gap energy (Ev + Ec)/2).

According to the above physical picture, electrical charge transport is due to the con-

tribution of two kinds of charge carriers, electrons (negatively charged) in the conduction

band and holes (positively charged) in the valence band. Denoting by n = n(x, t) and

p = p(x, t) the number densities of electron and hole particles per unit volume, at a

spatial position x and at a time instant t, the electrical conductivity is given by

σSie = qµnn+ qµpp, (7.1)

where µn and µp are the mobilities of electron and hole particles, respectively.

To give an indication of the order of magnitude of σSie , we need to observe that in

intrinsic silicon (i.e., silicon without the presence of any external impurity in its lattice

structure) we have

n = p = ni (7.2)


Figure 7.6.: Energy band structure for intrinsic silicon in the presence of an external elec-

tric field. The arrows indicate current flow of electrons and holes according

to Ohm’s law applied in the conduction and valence bands, respectively.

where ni is the so-called intrinsic concentration which at room temperature is approxi-

mately equal to 1016m−3 (to be compared with 8.5 · 1028m−3 in copper), while

µn ' 0.14m2V−1s−1, µp ' 0.048m2V−1s−1

(to be compared with 4.4 · 10−3m2V−1s−1 in copper), so that the resulting value of

electrical conductivity is σSie ' 2 · 10−7Sm−1, that is, 14 orders of magnitude smaller

than that of copper!! This explains the difference between a conductor (like copper or

aluminum) and a semiconductor (or even worse, an insulator, like diamond or silicon

dioxide) in terms of their attitude in allowing an easy conduction of current.

As a matter of fact, intrinsic silicon does not appreciably allow current flow. This

is the reason for introducing from the external environment impurities (called dopants)

in the intrinsic silicon periodic structure in such a way to modify artificially the (low)

conductivity of the material.

Figure 7.7.: Periodic lattice structure with local dopant impurities.

Specifically, as shown in Fig. 7.7, atoms of third and fifth groups of the periodic table

7.5. Equilibrium and Non-Equilibrium Statistics 133

(Boron and Phosphorus, for example) are added with given concentrations in ionized

form equal to N−A and N+D respectively (acceptor and donors), so that the electric charge

density in the doped material is given by

ρ = qp− qn+ qN+D − qN

−A in Cm−3. (7.3)

The first two contributions at the right-hand-side are associated with the mobile charge

carriers (those which actually transport current flow), while the second two contributions

are given by fixed ionized impurities. From a mathematical point of view, N−A and N+D

must be considered as given functions of x only, because they are assumed to be time-

invariant (∂ N−A /∂ t = ∂ N+D/∂ t = 0).

7.5. Equilibrium and Non-Equilibrium Statistics

In this section, we briefly review some basic aspects concerning the relation between the

energy band structure of a semiconductor and its electrical conductive properties.

The first important concept that we need introduce is that of thermal equilibrium.

Definition 7.5.1 (Thermal equilibrium (1)). A semiconductor material is at thermal

equilibrium conditions if no external electric, chemical or mechanical forces are applied

on it. In such conditions current flow of electrons and holes is identically zero at each

point of the material and at each time level, in mathematical terms

Jn(x, t) = Jp(x, t) = 0 ∀x ∈ Ω ∀t. (7.4)

Let us assume to be at thermal equilibrium conditions. Then, we introduce the

Fermi-Dirac distribution function

fD(E) =1

1 + exp

(E − EFKBT

) (7.5)

which gives the probability that an energy state E (in eV) is occupied by an electron at

a temperature T . The quantity EF is the Fermi level and represents a reference energy

at which fD(Ef ) =1

2. Fig. 7.8 shows the graph of fD at various temperatures.

We see that all energies above EF are unoccupied at T = 0K, while at higher

temperatures thermal energy is sufficient for populating higher energy states. Unless

otherwise specified, we assume from now on T = 300K (room temperature).

Definition 7.5.2 (Thermal equilibrium (2)). In a semiconductor at thermal equilibrium

the Fermi level is constant at each point of the material and at each time level, in math-

ematical terms

EF (x, t) = EF ∀x ∈ Ω ∀t (7.6)

for a suitable constant Ef .

At a given temperature, the total number of electrons (holes) in the conduction

(valence) band is given by:

n = Nc exp

(−(Ec − EF )

KBT

)(7.7a)

p = Nv exp

(−(EF − Ev)

KBT

)(7.7b)


Figure 7.8.: Fermi-Dirac distribution function for three values of the temperature and for

EF = 0.56 eV.

where Nc and Nv are the effective density of states in the conduction and valence bands,

respectively. From the above expressions, we are therefore able to determine the number

of carriers available for current conduction once we know the position of the Fermi level

inside the energy gap.

Remark 7.5.1 (Intrinsic semiconductor). In the special case where no doping impurities

are present (N+D = N−A = 0), the semiconductor is said to be intrinsic and we have

n(x, t) = p(x, t) = ni ∀x, ∀t > 0

where ni is the so-called intrinsic concentration which at room temperature is approxi-

mately equal to 1016m−3. Using (7.7) we obtain

Nc exp

(−(Ec − EF )

KBT

)= Nv exp

(−(EF − Ev)

KBT

)

⇒ EF =Ec + Ev

2+KBT

2ln

(Nv

Nc

).

(7.8)

This relation shows that the Fermi level in an intrinsic semiconductor is situated close to

the midgap level (Ev + Ec)/2 = Eg/2, except for the energy amount KBT/2 ln(Nv/Nc).

In the case of silicon, at T = 300 K, the value of the energy gap is Eg = 1.12 eV and the

value of the effective density of states is Nc = 2.8 · 1025 m−3 and Nv = 1.04 · 1025 m−3

(the difference to being ascribed to the fact that the effective mass for electrons and holes

is not the same), from which relation (7.8) yields

EF = 0.56− 0.02589

2· 0.9904 = (0.56− 0.0128) eV.

The energy shift from midgap is thus about 2.3% so that in the case of an intrinsic

semiconductor made of silicon it is a reasonable approximation to set

EF := Ei =Eg2

(7.9)


where Ei is called, for notational coherence, intrinsic Fermi level.

In order to determine the carrier number density ni in an intrinsic semiconductor we

use (7.7) with n = ni and p = ni, to obtain

pn = nini = n2i = NcNv exp

(− EgKBT

)from which we get

ni =√NcNv exp

(− Eg

2KBT

). (7.10)

This relation shows that the intrinsic concentration is a characteristic property of the

semiconductor material and, for a given temperature, depends only on its energy band

parameters.

Using the notion of intrinsic concentration and intrinsic Fermi level, we can express

the number density of electrons as a function of the sole difference between the actual

position of the Fermi level EF and that of the corresponding intrinsic energy level Ei.

As a matter of fact, setting EF = Ei into (7.7a) we get

ni = Nc exp

(−(Ec − Ei)

KBT

)= Nc exp

(Ei − EcKBT

)from which it follows, using again (7.7a)

n = Nc exp

(EF − EcKBT

)= Nc exp

(Ei − EcKBT

)exp

(EF − EiKBT

)

= ni exp

(EF − EiKBT

).

(7.11)

Proceeding similarly with holes, setting EF = Ei into (7.7b) we get

ni = Nv exp

(−(Ei − Ev)

KBT

)= Nv exp

(Ev − EiKBT

)from which it follows, using again (7.7b)

p = Nv exp

(Ev − EFKBT

)= Nv exp

(Ev − EiKBT

)exp

(Ei − EFKBT

)

= ni exp

(Ei − EFKBT

).

(7.12)

Relations (7.11) and (7.12) are well-known as the Maxwell-Boltzmann (MB) statistics

for electron and hole charge carrier densities in a semiconductor material at thermal

equilibrium. This form of carrier statistics has a very relevant practical use because it

allows to measure directly the excess (or decrease) of free charge carriers with respect

to the undoped material. In particular, if EF is positioned well above the intrinsic level

(but below Ec), the material is of n-type (see Fig. 7.9(a)) because EF − Ei > 0 so that

p ni, while the reverse situation occurs if EF is positioned well below the intrinsic

level (but above Ev, see Fig. 7.9(b)).


(a) n-type silicon (b) p-type silicon

Figure 7.9.: Band structure of n-type and p-type doped materials.

Recalling that the relation between potential ϕ and energy E is E = −ϕ/q, the MB

relations at thermal equilibrium can be written as:

n = ni exp

(ϕ− ϕFVth

)(7.13a)

p = ni exp

(ϕF − ϕVth

)(7.13b)

where ϕF = −EF /q is the Fermi potential, constant in all the material due to Def. 7.5.2.

Recalling Def. 7.5.1, it is natural (and mathematically convenient) to set EF = 0eV,

or, equivalently, ϕF = 0V. In this manner, the MB statistics in thermal equilibrium

conditions can be written as:

n = ni exp

(ϕ

Vth

)(7.13c)

p = ni exp

(− ϕ

Vth

). (7.13d)

Theorem 7.5.1 (Thermal equilibrium (3)). In a semiconductor at thermal equilibrium,

the product between the number of electrons and the number of holes per unit volume is

constant at each point of the material and at each time level, in mathematical terms

p(x, t)n(x, t) = n2i ∀x ∈ Ω ∀t (mass-action law). (7.14)

Proof. Compute pn using (7.13).

Example 7.5.1 (Thermal equilibrium statistics in doped silicon). In this example, we

use the MB thermal equilibrium statistics to determine the values of n and p and the

position of EF (with respect to the intrinsic level) under the assumption that silicon

contains 8 · 1022 m−3 atoms of arsenic and 2 · 1022 m−3 atoms of boron. Arsenic belongs

to the V group of table of elements, therefore it correponds to a dopant concentration of

donors N+D , while boron belongs to the III group of table of elements, so that it correponds

to a dopant concentration of acceptors N−A . The resulting net doping concentration is

N+D −N

−A = +6 ·1022 m−3 ≡ ∆N which indicates that silicon is of n-type. The number of

electrons available for current conduction in the conduction band equals the excess donor

concentration, so that

n = ∆N = 6 · 1022 m−3


while using Thm. 7.5.1 we obtain

p =n2i

n=

1032

6 · 1022= 1.7 · 109 m−3

which is 13 orders of magnitude less than the electron concentration, consistently with

fact that the material is of n-type. To find the location of the Fermi level with respect to

the intrinsic level, we use (7.11) to obtain

EF − Ei = KBT log

(n

ni

)= +0.4036 eV.

The resulting band diagram is shown in Fig. 7.10.

Figure 7.10.: Band structure of an n-type material.

The application of an external electric force to the semiconductor material drives the

system away from thermal equilibrium, and the situation is referred to as non-equilibrium

condition. In such a case, it is not possible to introduce a unique Fermi level in the

material, rather, two distinct reference levels must be considered, one for each carrier

type. With this purpose, relations (7.13) can be generalized into the following non-

equilibrium MB statistics:

n = ni exp

(ϕ− ϕnVth

)(7.15a)

p = ni exp

(ϕp − ϕVth

)(7.15b)

where ϕn and ϕp can be defined, in a very natural manner, as the quasi Fermi poten-

tials for electrons and holes, generalizing the notion of Fermi potential used in (7.13).

Clearly, (7.15) and (7.13) coincide at thermal equilibrium, while in the general case of

non-equilibrium conditions we have the following generalization of (7.14)

p(x, t)n(x, t) = n2i exp

(ϕp(x, t)− ϕn(x, t)

Vth

)∀x ∈ Ω ∀t. (7.16)

Thus, the relative distance between the quasi Fermi levels in the semiconductor measures

how far the system is from thermal equilibrium regime. Fig. 7.11 shows the ratio pn/n2i

as a function of ϕp − ϕn in the interval [−1, 1] V (which corresponds to (ϕp − ϕn)/Vth

in the interval [−40, 40]). We see that the variation of the excess charge carrier density

with respect to the intrinsic condition is huge, ranging from 10−20 to 1020, due to its

exponential dependance on the distance between the quasi Fermi potentials.


Figure 7.11.: Variation of pn/n2i as a function of the distance between the quasi Fermi

potentials in the range [−1, 1] V.

7.6. The Drift-Diffusion Transport Equations

The phenomenological form of the current density of electrons and holes is expressed by

the following relations:

Jn = −qnvn Electron current density (7.17a)

Jp = +qpvp Hole current density (7.17b)

where vn and vp are the drift velocities of electrons and holes in the semiconductor,

respectively. Notice that (7.17) are special instances of (2.4) when zion = −1 (electrons)

and zion = +1 (holes), respectively. The principal difference with respect to the case

of a perfect conducting material is that the number densities n and p may experience

very large variations inside the material, according to the fact that this latter is doped

with an excess of donor ionized impurities (N+D N−A ) or with an excess of acceptor

ionized impurities (N−A N+D ). In the former case, the material is of n-type material and

electrical conductivity is mainly due to electrons, while in the latter case the material is of

p-type and electrical conductivity is mainly due to holes. In general, we cannot conclude,

as in the case of a perfect conducting material, that in a (doped) semiconductor the

number densities of free carriers are uniform in space (and time). Rather, they may

experience large variations across the interface region between a p-type material and a

n-type material (the so-called p-n junction). To describe this new physical behaviour

compared to the case of perfect conductors, we assume to model charge current flow in

a semiconductor as the sum of two principal mechanisms:

• diffusion current, according to Fick’s law;

• drift current, according to Ohm’s law.

7.7. The Continuity Equations and Thermal Equilibrium 139

In mathematical terms, diffusion current densities are given by:

Jdiffn = −Dn(−q∇n) (7.18a)

Jdiffp = −Dp(+q∇p) (7.18b)

while drift current densities are given by:

Jdriftn = qµnnE (7.19a)

Jdriftp = qµppE (7.19b)

in such a way that electron and hole current densities are expressed through the celebrated

Drift-Diffusion Transport Equations:

Jn = Jdriftn + Jdiff

n = qµnnE + qDn∇n (7.20a)

Jp = Jdriftp + Jdiff

p = qµppE − qDp∇p (7.20b)

where Dn and Dp are the diffusivities of electrons and holes (in m2s−1).

We notice that in regions where one (or both) of the two charge carrier densities are

nearly constant the corresponding current densities coincide with the classical Ohm’s law,

and the material behaves electrically as a linear resistor. However, wherever n and/or p

are not uniformly distributed, the diffusion current contribution may become significant,

deviating the electrical behaviour of the semiconductor from being of ohmic type. In

the limit, low-field regions with large carrier gradients may experience a non-negligible

current flow due to a Fick’s diffusion mechanism. These facts will be documented in the

study of current flow in a p-n junction.

7.7. The Continuity Equations and Thermal Equilibrium

The total conduction current density J can be written as

J = Jn + Jp (7.21)

so that the continuity equation (3.5) becomes

q∂ p

∂ t− q∂ n

∂ t︸︷︷︸time rate of change of electric charge density

+ divJn + divJp︸︷︷︸current sources/sinks

= 0.

The above relation can be conveniently splitted into the following system of two continuity

equations:

−q∂ n∂ t

+ divJn = qR Electron continuity equation (7.22a)

q∂ p

∂ t+ divJn = −qR Hole continuity equation (7.22b)

upon introducing the (yet unspecified) function R which has the meaning of net recom-

bination rate. This function can be conveniently written as

R := R−G (7.22c)


where the two functions R and G are nonnegative and have the role of describing the

physical processes, occurring in the material at each spatial position and at each time

level, that drive the system locally away from thermal equilibrium conditions. In particu-

lar, this latter condition corresponds toR = 0, i.e., R = G, whereasR > 0 corresponds to

a local predominance of recombination in the material (R > G) while R < 0 corresponds

to a local predominance of generation in the material (G > R). In the former case, there

is a local excess of free charge carriers with respect to equilibrium, so that recombination

mechanisms (for instance, electron-hole neutralization) come into play to restore local

equilibrium. In the latter case, there is a local defect of free charge carriers with respect

to equilibrium, so that charge generation mechanisms (for instance, thermal generation)

come into play to increase carrier population thus restoring local equilibrium. A detailed

discussion and study of the form of the function R will be addressed in Sect. 7.11.

Definition 7.7.1 (Thermal equilibrium (4)). In a semiconductor at thermal equilibrium

net recombination rate is equal to zero at each point of the material and at each time

level, in mathematical terms

R(x, t) = 0 ∀x ∈ Ω ∀t. (7.23)

From Def. 7.7.1, it follows that if the local difference ϕp−ϕn is positive, then charge

carrier number densities are much larger than in equilibrium so that R > 0 is expected,

and recombination restores equilibrium by charge mutual neutralization. If ϕp − ϕn is

negative, then charge carrier number densities are much lower than in equilibrium so that

R < 0 is expected, and charge production under the form of thermal generation enters

the scene to drive back system to equilibrium.

7.8. The Drift-Diffusion Model

Collecting together the transport equations (7.20), the carrier continuity equations (7.22),

the Poisson equation (4.7a) with the constitutive laws for the electric displacement vec-

tor (4.7b) and for the space charge density (7.3), we obtain the well-known Drift-Diffusion

(DD) Model for Semiconductor Devices:

− q∂ n∂ t

+ divJn = q(R−G) Electron continuity equation (7.24a)

q∂ p

∂ t+ divJn = −q(R−G) Hole continuity equation (7.24b)

divD = qp− qn+ qN+D − qN

−A Poisson equation (7.24c)

Jn = qµnnE + qDn∇n Electron current density (7.24d)

Jp = qµppE − qDp∇p Hole current density (7.24e)

D = εE Electric displacement (7.24f)

E = −∇ϕ. Electric field (7.24g)

7.9. The linear resistor 141

The equation system (7.24) is to be solved in a bounded domain Ω ⊂ Rd, d = 1, 2, 3

(typically a polyhedron) whose boundary Γ is typically divided into disjoint subsets

ΓD and ΓN , on which Dirichlet or Neumann boundary conditions are enforced for the

dependent variables ϕ, n and p or the normal components of their associated vector-

valued functions D, Jn and Jp. In the time-dependent regime, suitable initial conditions

n0 = n0(x) : Ω → R+, p0 = p0(x) : Ω → R+ must be supplied for the electron and hole

concentrations. The associated initial datum ϕ0 for the electric potential is the solution

of the Poisson equation (7.24c) with n = n0 and p = p0. In the next sections we apply the

DD model to mathematically describe charge transport in several devices of importance

in the modern technology of nanoelectronics. Only the static solution of the DD system

is considered, i.e., we always assume henceforth∂ n

∂ t=∂ p

∂ t= 0 in (7.24).

7.9. The linear resistor

We start illustrating the basic principles underlying the functioning of the device and

then we write the associated mathematical model using the DD system.

7.9.1. Physical principles of the linear resistor

Let us describe conduction current transport in a solid subject to the action of an exter-

nally applied electric force as shown in Fig. 7.12(a).

(a) Macroscopic scale (b) Atomic scale

Figure 7.12.: Current transport in a solid. Left: macroscopic scale perspective. Right:

atomic scale perspective.

Fig. 7.12(b) shows the lattice structure of the solid, with lattice constant a (typically

of the order of nm), where the atoms of the solid (black bullets) are located at the

vertices of the lattice cube. Throughout the structure, an electron charge density −qNeis flowing (z = −1), Ne being the number density of flowing electrons (in m−3). The

mass of the lattice atoms is assumed to be much larger than electron mass me. For

instance, in the case of copper (symbol: Cu) the molar mass is 63.546 grams per mole,

that is, mCu = 63.546/NAv = 1.055 · 10−25 Kg while the effective electron mass me,Cu '


1845/1660 ·m0 = 1.01 ·10−30 Kg, that is, 100000 times smaller, m0 = 9.1 ·10−31 Kg being

the electron mass in vacuum.

Electron charge flow is due to the action of an externally applied electric field

E = −VaLi,

i being the unit vector of the x-axis, Va and L being the applied voltage across the

material and the length of the device, typically of the order of µm, respectively. Using

Newton’s law of dynamics, the force acting on each charge is

−qE = meae

from which we get that the acceleration acting on each particle is

ae = −qEme

.

During their motion throughout the lattice structure, electrons travel at an average speed

defined as

vD = aeτe = −qτeme

E (7.25)

where τe is the average time of flight of the electron bewteen two consecutive interactions

(elastic collisions) with the atoms of the lattice. The quantity vD is the drift velocity

of the electron particles. Therefore, using (2.4), the current density associated with the

number charge density −qNe is

Je = −qNevD = qNeqτeme

E = qNeµeE = σeE (7.26)

where:

µe :=qτeme

Electron mobility in m2V−1s−1 (7.27a)

σe := qµeNe Electron conductivity in Sm−1. (7.27b)

Relation (7.26) expresses the well-known Ohm’s law (cf. (2.7k)) stating that in a con-

ducting material the current density is directly proportional to the applied electric field.

Introducing the specific conductance

ge :=σeL

measured in Sm−2, we obtain the classical Ohm’s law for a linear resistor device

Je = ge Va, (7.28)

Je denoting the module of the vector Je. To obtain a current relation from (7.28), we

only need to multiply both sides by the cross-sectional area S of the resistor, which yields

Ie = Ge Va, (7.29)

where Ge := geS is the conductance of the resistor measured in S. In the case of copper,

the electrical conductivity is very large because Ne is huge while the mobility is rather

small due to the many collisions experienced by particles in the lattice. In particular,


Figure 7.13.: J − V characteristic of a linear resistor.

at T = 293.16 K, we have Ne = 8.5 · 1028m−3 and µe = 4.4 · 10−3m2V−1s−1 so that

σCue = 6 · 107Sm−1.

To compare a conductor like copper with a semiconductor material like silicon (the

most used in electronic device technology), at T = 293.16 K we have Ne = 1016m−3 (the

so-called intrinsic concentration) and µe = 0.14m2V−1s−1 so that σSie = 2 · 10−7Sm−1

which is 14 orders of magnitude smaller than σCue . This huge difference in the electrical

conductivity is the reason why silicon is not employed in its natural form in device

technology but, instead, is subject to a sophisticated process (called doping) that consists

of substituting the atoms of the silicon lattice with atoms of chemical elements of the third

and fifth groups of the periodic table, to increase artificially (and in a very controlled

manner) the local value of σSie . The J − V characteristic corresponding to (7.28) is

schematically illustrated in Fig. 7.13 in the case of a linear resistor made of copper and

having length L = 1µm.

7.9.2. Boundary Conditions at Ohmic Contacts

Ohmic contacts are assumed to be ideal, i.e. they are equipotential surfaces and no

voltage drop occurs at the interface between the contact and the neighbouring material.

The corresponding boundary conditions are of Dirichlet type and take the following

general form:

ϕ = ϕD on ΓD (7.30a)

n = nD on ΓD (7.30b)

p = pD on ΓD (7.30c)

where ΓD is the portion of the device boundary Γ occupied by the ohmic contacts, ϕD is

the boundary datum for the electric potential at the contact, possibly depending on time

t, while nD and pD are positive boundary data for electron and hole number densities.

Conditions (7.30) reduce to the sole (7.30a) in the case where the neighbouring material

is a perfect insulator.

Below, we consider the case where the contact is made of a metal lead as happens at

the two device terminals in Fig. 7.12(a). Thermodynamical equilibrium and charge neu-

trality are assumed to be verified at each ohmic contact. These two conditions correspond


to the following algebraic system for nD and pD:pDnD = n2

i

pD − nD +N+D −N

−A = 0

whose solution yields:

nD =DopD +

√Dop2

D + 4n2i

2(7.31a)

pD =−DopD +

√Dop2

D + 4n2i

2(7.31b)

having set DopD := (N+D −N

−A )|ΓD

.

Figure 7.14.: Band diagram in a metal-semiconductor contact at thermal equilibrium (n-

type material). Notice that energy bands are flat at the contact.

To determine the boundary datum for the electric potential at the contact, we exploit

the assumption of local thermal equilibrium at the metal-semiconductor interface. This

implies in particular that there exists a unique Fermi level at the contact, which, for

convenience, is fixed equal to the externally applied voltage Vext|ΓD(see Fig. 7.14).

The above assumption corresponds to setting

ϕn = ϕp = Vext|ΓDon ΓD (7.32)

from which, using MB statistics (7.13) (with ϕF = Vext|ΓD, we get

ϕD = Vext|ΓD+ Vbi|ΓD

(7.33)

where:

Vbi|ΓD= Vth log

(nDni

)n-type doping (7.34a)

Vbi|ΓD= −Vth log

(pDni

)p-type doping. (7.34b)

Remark 7.9.1 (n-type and p-type doping). Realistic doping profiles used in semicon-

ductor technology are chacterized by the fact that one of the two ionized dopant impurities


locally dominates the other. In such a case, the expressions (7.31) for the evaluation of

the local boundary carrier concentrations simplifies considerably.

In particular, if N+D N−A (strongly local n-type material), then (7.31) can be well

approximated by:

nD ' N+D |ΓD

(7.35a)

pD 'n2i

N+D |ΓD

. (7.35b)

If N−A N+D (strongly local p-type material), then (7.31) can be well approximated

by:

nD 'n2i

N−A |ΓD

(7.36a)

pD ' N−A |ΓD. (7.36b)

Consistently, the expressions (7.34) for the built-in potentials become:

Vbi|ΓD' Vth log

(N+D |ΓD

ni

)n-type doping (7.37a)

Vbi|ΓD= −Vth log

(N−A |ΓD

ni

)p-type doping. (7.37b)

7.9.3. The Drift-Diffusion model of the linear resistor

Let us consider the linear resistor of length L depicted in Fig. 7.12(a). Let us assume

that the device is uniformly n-doped so that N−A (x) = 0 for all x ∈ [0, L] whereas

N+D (x) = N

+D > 0 for all x ∈ [0, L]. Let us also assume that mobilities are constant in Ω

and that recombination-generation mechanisms can be neglected, so that R = 0 in the

continuity equations (7.22). The left contact is grounded whereas at the right contact a

bias voltage equal to Va is externally applied. The DD model system (7.24) in the case

of a linear uniformly doped n-type resistor reads:

−∂ Jn∂ x

= 0 (7.38a)

Jn = qµnnE + qDn∂ n

∂ x(7.38b)

∂ Jp∂ x

= 0 (7.38c)

Jp = qµppE − qDp∂ p

∂ x(7.38d)

∂ E

∂ x=

q

ε0εr(p− n+N

+D) (7.38e)

E = −∂ ϕ∂ x

. (7.38f)


Equations (7.38) are supplied by the following boundary conditions:

ϕ = ϕbi,0 = Vth log

(N

+D

ni

)at x = 0 (7.39a)

ϕ = Va + ϕbi,L = Va + Vth log

(N

+D

ni

)at x = L (7.39b)

n = N+D at x = 0 (7.39c)

n = N+D at x = L (7.39d)

p =n2i

N+D

at x = 0 (7.39e)

p =n2i

N+D

at x = L. (7.39f)

The numerical solution of system (7.38)-(7.39) proceeds along the lines illustrated in

Chapt. 6. Computational experience reveals that the decoupled Gummel algorithm con-

verges very slowly in non-equilibrium conditions (Va 6= 0) whereas the fully coupled

Newton iteration is very rapid and robust with respect to the choice of the initial guess[ϕ(0), n(0), p(0)

]T.

7.10. The p-n junction diode

Most electronic devices contain the presence of semiconductor regions, characterized by

different doping types and separated by a contact surface called junction. When two

neighbouring semiconductor regions are of p and n-type, respectively, and a voltage drop

is applied across them, the resulting device is called p-n junction. In this chapter, we

apply the semiconductor device model equations to the study of the p-n junction. The

basic principles underlying the physical and electrical behaviour of the contact are first

illustrated, and then, the appropriate set of boundary conditions and the FE numerical

approximation of the p-n junction model are discussed in detail.

7.10.1. Energy Band Diagram of Isolated Materials

Let us assume from now on that the material constituting the p-type region is located

on the left, while the n-type region is located on the right. The concentration of fixed

ionized dopant impurities is equal to N−A (acceptors) in the p region and to N+D (donors)

in the n-region. The band diagram of the two materials, mutually separated for the

moment, is schematically represented in Fig. 7.15.

The energy levels EFp and EFn denote the Fermi levels in the p and n regions,

respectively, while the energies qΦSp and qΦSn are the work functions of the two semi-

conductor regions. Using (7.12) in the p region, with p = N−A , and (7.11) in the n region,

7.10. The p-n junction diode 147

Figure 7.15.: Band diagram for the p-type region (left) and the n-type region (right).

with n = N+D , we get:

Ei|p − EFp = KBT ln

(N−Ani

), (7.40a)

EFn − Ei|n = KBT ln

(N+D

ni

), (7.40b)

where Ei|p and Ei|n are the intrinsic levels in the p-type and n-type materials, respec-

tively, so that from Fig. 7.15 we obtain:

qΦSp = qχS +EGap

2+KBT ln

(N−Ani

)= 4.61eV + ln

(N−Ani

), (7.40c)

qΦSn = qχS +EGap

2−KBT ln

(N+D

ni

)= 4.61eV −KBT ln

(N+D

ni

). (7.40d)

The two above relations show that the value of the work function for the two semiconduc-

tor regions depends on the value of the doping concentration in the two regions. Using

realistic data, if N+D = 1022 m−3 and N−A = 3 · 1022 m−3, for example, then we have

qΦSp = 4.61 + 0.39 = 5 eV, while qΦSn = 4.61− 0.37 = 4.24 eV.

7.10.2. The p-n Junction

In this section, we abandon the assumption of having two isolated pieces of (doped)

semiconductor and consider the case where the two semiconductor regions are put in

intimate contact under the action of an externally applied voltage Va. The origin of

the x-axis is located in correspondance with the contact at the p-side of the junction,

in such a way that the p-side occupies the region [0, xj) while the n-side occupies the

portion [xj , L], L and xj being the total length of the device and the junction coordinate,

respectively. Without loss of generality, we assume xj = L/2. The terminal at x = L is

set to ground voltage, so that Va is the value of the external voltage at the p-contact at

x = 0 (see Fig. 7.16).


Figure 7.16.: Electrical scheme of the p-n junction.

The doping profile D is described by the piecewise constant function

D(x) :=

−N−A x ∈ [0, xj)

+N+D x ∈ [xj , L].

(7.41)

The p-n junction corresponding to this form of doping is called abrupt junction. If

N+D = N−A , the junction is called symmetric otherwise it is called asymmetric.

7.10.3. The p-n Junction at Thermal Equilibrium

In this section, we set Va = 0 V. This corresponds to the case where no electrical external

forces are applied to the device, so that current flow throughout the device is identically

zero.

The first physical observation is that at thermal equilibrium the Fermi level EF is

constant throughout the entire length of the device (cf. Def. 7.5.2). Since the externally

applied voltage is equal to zero we have EF (x) = 0 eV for all x ∈ [0, L]. Then, far from

the junction coordinate xj , the two materials are in the same local equilibrium conditions

as in the separate situation, so that the distances between EF and Ev and EF and Ec

are exactly equal to those in Fig. 7.15 (left and right sides, respectively).

The second physical observation is that the energy gap Egap separating Ev and Ec

is a constant quantity, characteristic of the semiconducting material, and the energies

Ev = Ev(x), Ec = Ec(x) and Ei = Ei(x) are continuous functions of the spatial position

x ∈ [0, L]. Thus, the conduction and valence bands must necessarily bend in order to

maintain their distance Ec(x)−Ev(x) constant and equal to Egap. Correspondingly, also

the intrinsic Fermi level Ei = Ei(x) which is located at mid-gap within the forbidden

energy interval, has to bend.

The resulting band diagram of the system is shown in Fig. 7.17.

The hole number density p = p(x) in the p-side region far from the junction is equal to

the equilibrium value peq,p := N−A , while its value in the n-side region far from the junction

is equal to peq,n := n2i /N

+D . As peq,p peq,n, we use to say that peq,p is the equilibrium

majoritary concentration while peq,n is the equilibrium minoritary concentration. In an

analogous manner, the electron number density n = n(x) in the n-side region far from

the junction is equal to the majoritary equilibrium value neq,n := N+D while in the p-

side region far from the junction it is equal to the minoritary equilibrium concentration


Figure 7.17.: Band diagram for the p-n junction at thermal equilibrium.

neq,p := n2i /N

−A . As before, we have neq,n neq,p. In the remaining transition region

around the junction, the concentration profile of holes must rapidly decrease from peq,p

to peq,n, and the concentration profile of electrons must rapidly increase from neq,p to

neq,n.

Based on the above analysis of carrier number densities in the various regions of

the device, it turns out that the majoritary electrons in the n-side should have a natural

tendency to flow by diffusion towards the p-side where they are minoritary carriers, and a

similar tendency should hold for the majoritary holes from the p-side to the n-side. Such

a natural tendency is contrasted by the presence of the built-in energy barrier Φbi that

automatically forms after the two semiconductor material regions are joined together.

To determine the height of the built-in barrier, we see from Fig. 7.17 that

Φbi = Egap − (EF − Ev)|x=0 − (Ec − EF )|x=L

= Egap − ((Ei − Ev)− (Ei − EF ))|x=0 − ((Ec − Ei)− (EF − Ei))|x=L

= Egap −Egap

2− Egap

2+ (Ei − EF )|x=0 + (EF − Ei)|x=L,

from which, using the equilibrium MB statistics (7.11) and (7.12), with n = N+D and

p = N−A , we get

Φbi = KBT ln

(N−Ani

)+KBT ln

(N+D

ni

)= KBT ln

(N−AN

+D

n2i

). (7.42)

Therefore, majoritary electron and hole diffusive flow into the opposite minoritary regions

leaves the n and p sides of the junction depleted of the respective mobile carrier densities.

In terms of electrical charge, this amounts to the formation of a double layer of fixed

charge across the junction, which is constituted by “non-compensated” ionized dopant


impurities. This double layer is represented by the ”+” and ”-” charges visible in the

neighbourhood of x = xj in Fig. 7.17. The effect of the double layer is to give rise to the

onset of an intrinsic electric field E directed from right to left, whose action is to push

electrons from left to right back towards the n region and holes from right to left back

towards the p region, respectively.

The net balance between the diffusive current densities Jdiffn = qDn∂ n/∂ x, Jdiff

p =

−qDp∂ p/∂ x with the respective drift current densities Jdriftn = qµnnE, Jdrift

p = qµppE,

is equal to zero, because of the thermal equilibrium condition. This fact is expressed in

Fig. 7.18, which displays all the contributions to current flow across the junction.

Figure 7.18.: Current contributions across a p-n junction at thermal equilibrium.

7.10.3.1. Mathematical Model at Thermal Equilibrium

Let us denote by Ω = (0, L) the computational domain and by xj = L/2 the coordinate

of the junction separating the p and n regions. At thermal equilibrium, current flow of

both carrier types is identically null and the same holds for the net recombination rate.

Therefore, the (stationary) DD dystem (7.24) reduces to the following nonlinear Poisson

equation that describes the electric potential distribution in the p-n junction at thermal

equilibrium:

−∂2ϕ

∂ x2=

q

εs

[ni exp

(− ϕ

Vth

)− ni exp

(ϕ

Vth

)+D

](7.43a)

ϕ(0) = ϕ0, ϕ(L) = ϕL (7.43b)

where εs := ε0εr is the semiconductor dielectric constant and the boundary values for

the electric potential can be computed using the MB statistics (7.13a) and (7.13b) (with


ϕF = 0) at x = 0 and x = L respectively, to obtain:

ϕ0 = −Vth log

(N−Ani

)(7.44a)

ϕL = +Vth log

(N+D

ni

). (7.44b)

Once the BVP (7.43) is solved, the free carrier number densities can be determined by

computing for all x ∈ [0, L] the expressions:

p(x) = ni exp

(−ϕ(x)

Vth

)(7.45a)

n(x) = ni exp

(+ϕ(x)

Vth

). (7.45b)

Exercise 7.10.1. Prove that the current densities Jn and Jp associated with the equilib-

rium number densities (7.45) are identically equal to zero.

Exercise 7.10.2. Compute the built-in potential

ϕbi := ϕ0 − ϕL (7.45c)

and check how it is related to the built-in energy barrier (7.42).

7.10.3.2. Functional Iteration and Numerical Approximation at Thermal

Equilibrium

To treat the nonlinear character of the Poisson equation (7.43a), we use the Newton

iterative process illustrated in Chapt. 6 with the following identifications:

U := ϕ

F (ϕ) := −∂2ϕ

∂ x2+

q

εs

[ni exp

(ϕ

Vth

)− ni exp

(− ϕ

Vth

)−D

].

To compute the Frechet derivative of F , we set D ≡ 1 and

f(ϕ) :=q

εs

[ni exp

(ϕ

Vth

)− ni exp

(− ϕ

Vth

)−D

]to obtain

F ′(ϕ)w = −∂2w

∂ x2+∂ f

∂ ϕ(ϕ)w = −∂

2w

∂ x2+

2qniεsVth

cosh

(ϕ

Vth

)w

where ϕ is an arbitrary function belonging to H1(0, L) and satisfying the boundary

conditions (7.44), while w is an arbitrary function belonging to V := H10 (0, L).

Then, given ϕ(0), at each step k ≥ 0 until convergence, we have to solve the following

linear BVP:

−∂2W

∂ x2+

2qniεsVth

cosh

(ϕ(k)

Vth

)W = −F (ϕ(k)) in Ω = (0, L) (7.46a)

W (0) = W (L) = 0, (7.46b)


and then set

ϕ(k+1) = ϕ(k) + τkW (7.47)

where τk is a damping parameter, suitably chosen in the interval (0, 1], introduced to

avoid excessively large increments w between consecutive Newton steps, particularly

during the very first iterations where the available iterate ϕ(k) may be quite far from the

exact solution.

The equation system (7.46) is a reaction-diffusion BVP of the form (1.5) where

µ := 1, v := 0, σ :=2qniεsVth

cosh

(ϕ(k)

Vth

), f := −F (ϕ(k)).

Therefore, for its approximation we use the FE formulation illustrated in Sect. 1.6 with

the lumping stabilization of the reaction term illustrated in Sect. 1.10.1 to avoid the onset

of possible spurious oscillations in the computed electric potential.

7.11. Recombination/Generation Mechanisms

Let us consider a piece of semiconductor material in thermal equilibrium conditions.

According to Defn. 7.5.1 and Thm. 7.5.1, current flow in the material is identically

equal to zero and the mass-action law establishes that the product pn is constant in

all the material. However, this does not mean that nothing is happening at all in the

semiconductor. Indeed, at a microscopic scale, it could be seen that electrons and holes

are in continuous fluctuation due to their thermal energy, but the macroscopic result of

such a process is that the net recombination rate R is identically zero at each point and

at each time level.

Having dealt so far with thermal equilibrium, it is now of our interest to better analyze

a deviation from such a condition, i.e., we assume that the carrier densities are subject

to a change from n to n′ = n + δn and from p to p′ = p + δp, δn and δp being small

perturbations with respect to the equilibrium state. Accordingly, we have

p′n′ = pn+ nδp+ pδn+ δnδp = n2i + nδp+ pδn+ δnδp.

Neglecting higher order terms, we can conclude that if δn > 0 and δp > 0 we have

p′n′ > n2i so that an excess of charge carriers has been produced in the semiconductor

material, and a recombination event (neutralization of excess charge) must occur in order

to drive the system back to the original equilibrium condition. If δn < 0 and δp < 0

we have p′n′ < n2i so that an excess of charge carriers has been removed from the

semiconductor material and a generation event (due to thermal agitation or an external

input source) must occur in order to drive the system back to the original equilibrium

condition.

In general mathematical terms, an appropriate ”phenomenological model” for the net

recombination rate R is given by

R(p, n) = (pn− n2i )F (p, n) (7.48a)

7.11. Recombination/Generation Mechanisms 153

where F is a function modeling the specific recombination/generation (R/G) event taking

place in the material to restore equilibrium. Comparing (7.48a) with (7.22c) we see that:

R(p, n) = p nF (p, n) (7.48b)

G(p, n) = n2i F (p, n). (7.48c)

7.11.1. Shockley-Hall-Read (SHR) R/G

SHR recombination/generation is a two-particle process (the electron-hole pair which is

recombining or generating), which mathematically expresses the probability that:

(RSHR) an electron in the conduction band neutralizes a hole at the top of the valence

band through the mediation of an unoccupied trapping energy level Et located

somewhere in the semiconductor energy gap (Fig. 7.19);

Figure 7.19.: Left: SHR recombination (the electron is represented by a black bullet, the

hole by a red bullet while the unoccupied trap is a white bullet).

(GSHR) an electron is emitted from the top of the valence band to the bottom of the

conduction band, through the mediation of an unoccupied trapping energy level Et

located somewhere in the semiconductor energy gap, leaving a hole in the valence

band (Fig. 7.20).

Figure 7.20.: Left: SHR generation (the electron is represented by a black bullet, the

hole by a red bullet while the unoccupied trap and energy level are a white

bullet).


Process RSHR is a recombination event, while process GSHR is a generation event.

From the energetical point of view the SHR process is an indirect recombination/gen-

eration mechanism because its occurrence requires the assistance of an unoccupied trap

level. The following expression is usually employed for the modulating function F

FSHR(p, n) =1

τn(p+ ni) + τp(n+ ni). (7.49)

The quantities τn and τp (expressed in s) are called carrier lifetimes and are physically

defined as the reciprocals of the capture rates per single carrier associated with the energy

trap distribution within the semiconductor energy gap. Their typical order of magnitude

lies in the range 10−3µs÷1µs. Denoting by τn,0 and τp,0 the reference values of τn and τp

in the considered semiconductor material, it can be verified that carrier lifetimes actually

depend on the total number of dopant impurities located at each spatial position x in

the lattice. A phenomenological model to describe such a dependence is

τν(x) =τν,0

Fτ (Ntot(x))ν = n, p (7.50)

where Fτ is a function accounting for lifetime reduction as a consequence of the increase

of total ionized impurities defined as

Fτ (Ntot) = 1 +Ntot

Nref(7.51)

where Nref is a fitting parameter of the order of 5 · 1022 m−3. From (7.50)-(7.51), we

see that in a heavily doped semiconductor carrier lifetimes may be significantly reduced

with respect to their reference values. This may affect the contribution of R/G currents

to the total current flow in the device.

7.11.2. Auger (Au) R/G

Auger recombination/generation is a three-particle process (the electron-hole pair which

is recombining or generating, plus a third carrier which receives or provides the energy

excess), which mathematically expresses the probability that:

(R2n,1pAU ) a high-energy electron in the conduction band moves to the valence band where

it neutralizes a hole, transmitting the excess energy to another electron in the

conduction band;

(G2n,1pAU ) an electron in the valence band moves to the conduction band by taking the

energy from a high-energy electron in the conduction band and leaves a hole in the

valence band.

The above two processes involve three carriers: two electrons and one hole (see Fig. 7.21).

In an analogous manner, the other three-particle events involving two holes and one

electron are (see Fig. 7.22):

(R1n,2pAU ) an electron in the conduction band moves to the valence band where it neutral-

izes a hole, transmitting the excess energy to another hole in the valence band;

7.11. Recombination/Generation Mechanisms 155

Figure 7.21.: Auger recombination/generation with two electrons and one hole (the elec-

trons are represented by a black and white bullet, the hole by a red bullet).

(G1n,2pAU ) an electron in the valence band moves to the conduction band by taking the

energy from a high-energy hole in the valence band and leaves a hole in the valence

band.

Figure 7.22.: Auger recombination/generation with one electron and two holes (the elec-

tron is represented by a black bullet, the holes by a red and yellow bullet).

The following expression is usually employed for the modulating function F

FAu(p, n) = Cn n+ Cp p (7.52)

where the quantities Cn and Cp (in m6s−1) are the so-called Auger capture coefficients

typically of the order of magnitude of 10−43m6s−1. The small values of the Auger co-

efficients express the physical fact that Auger R/G is relevant only when both carrier

densities attain high values, that is, typically, in high-injection regime as is the case with

forward-biased p-n junctions (see Chapt. 7.10).

7.11.3. The Impact Ionization Generation Process

The Impact Ionization (II) mechanism is a generation process characterized by the same

microscopic events as in the case of Auger generation, and plays an important role in

determining the electric behaviour of a p-n junction in highly reverse bias conditions.

The main difference between Auger generation and II is that in the former case the high-

energy carrier generation is initiated by a local high number density of carriers of both


polarities, as is typically the case with p-n junctions in high-injection conditions, while

in the latter case the high-energy carrier generation is triggered by the presence of very

high electric fields, as is typically the case with strongly reverse-biased p-n junctions (see

Chapt. 7.10). Fig. 7.23 shows a schematic picture of the II mechanism.

Figure 7.23.: The II generation process.

The process is initiated by an incident electron (indicated with a black bullet, left)

which travels from left to right in the depleted region between the p and n sides of the

junction. The travelling electron gains enough energy from the electric field to excite

an electron-hole pair out of a silicon-silicon lattice bond (indicated by a black and red

bullets, right and left below in the figure, respectively). This collision-production event is

called impact ionization and is such that the total number of carriers after its occurrence

is equal to three, the initial electron plus the generated electron-hole pair. Then, the

process can be repeated, until an avalanche of generated carriers is produced within the

region.

A widely used model for the II generation process is that proposed by Chynoweth

and takes the following form

GII = GII(Jn,Jp;E) =1

q(αn(E)|Jn|+ αp(E)|Jp|) (7.53)

where the ionization coefficients αν , ν = n, p (in m−1) are expressed by the phenomeno-

logical relation

αν = αν,∞ exp

(−Ecrit,ν

E‖,ν

)ν = n, p.

In the previous formula, αν,∞ is the value of the ionization rate at very high fields, Ecrit,ν

is a threshold value of the field under which no II occurs, while

E‖,ν :=E · Jν|Jν |

ν = n, p

is the component of the electric field in the direction of current flow. The above relations

indicate, consistently with physical expectation, that the effective electric field partici-

pating to the ionization process is not the electric field vector, but only the component

of the field directed along the current streamline. This important distinction allows to

avoid overestimation of the II contribution in multi-dimensional simulations, such as, for

instance, the study of a 2 or 3D MOS transistor where the current flowing in the channel

from Source to Drain contacts is almost perpendicular to the electric field directed from

the Gate contact toward the substrate.

7.12. The p-n Junction in Non-Equilibrium Conditions 157

7.11.4. Total R/G

Having introduced the three above R/G mechanisms, for the purpose of device modeling,

the simplest and most common approach is to add each single controbution in a linear

manner, so that

R = R(p, n,E) = (pn− n2i )(F

SHR(p, n) + FAU (p, n))−GII(Jn,Jp;E). (7.54)

7.12. The p-n Junction in Non-Equilibrium Conditions

The next case we are going to examine is that corresponding to the application of a

non-zero external electric force across the device terminals at x = 0 and x = L. In

particular, we assume that the contact at x = L is kept fixed at ground potential while

the terminal at x = 0 is set to a voltage Va, as depicted in Fig. 7.16. This means that,

due to the assumption of thermal equilibrium at ohmic contacts, the values of the quasi

Fermi potentials ϕn and ϕp are enforced to be coinciding at both x = 0 and x = L, and

equal to:

ϕn(0) = ϕp(0) = Va (7.55a)

ϕn(L) = ϕp(L) = 0. (7.55b)

Depending on the sign of Va, we distinguish between: Va > 0: forward bias conditions

and Va < 0: reverse bias conditions. These two conditions correspond to quite a different

electrical behaviour of the p-n junction device. Roughly speaking, in the forward bias

regime the built-in energy barrier Φbi in Fig. 7.17 is reduced by the application of the

external voltage Va, so that current flow throughout the system is highly increased. In

technical parlance, the device is said to be in ”on” conditions. Conversely, in the reverse

bias regime the built-in barrier Φbi in Fig. 7.17 is furtherly enhanced, compared to thermal

equilibrium, by the application of the external voltage Va, so that current flow throughout

the system, due to carrier thermal generation, is almost negligible and the device is in

”off” conditions.


7.12.1. Mathematical Model of a p-n Junction in Non-Equilibrium

Conditions

The DD model describing a p-n junction under non-equilibrium conditions reads:

−∂ Jn∂ x

= −q(R−G) (7.56a)


∂ x(7.56b)

∂ Jp∂ x

= −q(R−G) (7.56c)

Jp = qµppE − qDp∂ p

∂ x(7.56d)

∂ E

∂ x=

q

εrε0(p− n+D) (7.56e)

E = −∂ ϕ∂ x

. (7.56f)


ϕ = ϕ0 + Va = −Vth log

(N−Ani

)+ Va at x = 0 (7.57a)

ϕ = ϕL = +Vth log

(N+D

ni

)at x = 0 (7.57b)

n = n0 at x = 0 (7.57c)

n = nL at x = L (7.57d)

p = p0 at x = 0 (7.57e)

p = pL at x = L. (7.57f)

The above relations express thermal equilibrium and charge neutrality at each device

terminal, which is thus assumed to be an ideal ohmic contact. Replacing (7.57a), (7.57b)

and (7.55) into the MB statistics (7.15), the boundary values for the number densities

are given by:

n0 = ni exp

(ϕ(0)− ϕn(0)

Vth

)=

n2i

N−Aat x = 0 (7.58a)

nL = ni exp

(ϕ(L)− ϕn(L)

Vth

)= N+

D at x = L (7.58b)

p0 = ni exp

(ϕp(0)− ϕ(0)

Vth

)= N−A at x = 0 (7.58c)

pL = ni exp

(ϕp(L)− ϕ(L)

Vth

)=

n2i

N+D

at x = L. (7.58d)

To complete the description of current transport in a p-n junction in non-equilibrium

conditions, we need specify the models for carrier mobilities µn and µp and for the re-

combination and generation rates R and G. In view of the subsequent analysis, we assume


that mobilities are constant and equal to their low-field values µ0n and µ0

p, respectively,

and that R and G account only for the Shockley-Hall-Read (SHR) recombination/gener-

ation mechanism (see Sect. 7.11), in such a way that

R(n, p) = (pn− n2i )FSHR(p, n) (7.59a)

FSHR(p, n) = [τn (p+ ni) + τp (n+ ni)]−1 (7.59b)

τn and τp being the electron and hole lifetimes, respectively. Eliminating from sys-

tem (7.56) the variables Jn, Jp and E as functions of ϕ, n and p, and using (7.59a), we

obtain the following reduced model:

− ∂

∂ x

(qDn

∂ n

∂ x− qµnn

∂ ϕ

∂ x

)= −q(pn− n2

i )FSHR(p, n) (7.60a)

− ∂

∂ x

(qDp

∂ p

∂ x+ qµpp

∂ ϕ

∂ x

)= −q(pn− n2

i )FSHR(p, n) (7.60b)

−∂2ϕ

∂ x2=

q

εrε0(p− n+D) (7.60c)

to be supplied by the boundary conditions (7.57).

Exercise 7.12.1. Compute the total voltage drop ∆V across the p-n junction defined as

∆V := ϕ(0)− ϕ(L) (7.60d)

as a function of the externally applied voltage Va and of the built-in potential (7.45c).


7.12.2. The Gummel Map in Non-Equilibrium

The Gummel algorithm for the iterative solution of system (7.60)- (7.57) reads:

given [u(0)n , u

(0)p ]T , for every k ≥ 0 until convergence:

Solve the Nonlinear Poisson Equation (NLP):

−∂2ϕ

∂ x2+

q

εrε0

(−u(k)

p exp

(− ϕ

Vth

)+ u(k)

n exp

(ϕ

Vth

)−D

)= 0 (7.61a)

Set ϕ(k+1) = ϕ (7.61b)

Compute intermediate electron and hole number densities:

n(k) = u(k)n eϕ

(k+1)/Vth = nie(ϕ(k+1)−ϕ(k)

n )/Vth

p(k) = u(k)p e−ϕ

(k+1)/Vth = nie(ϕ

(k)p −ϕ(k+1))/Vth

Solve the Linear Electron Continuity Equation (LEC):

− ∂

∂ x

(qDn

∂ n

∂ x− qµnn

∂ ϕ(k+1)

∂ x

)+ qp(k)FSHR(p(k), n(k))n

= qn2iFSHR(p(k), n(k)) (7.61c)

Set n(k+1) = n and set u(k+1)n = n(k+1) exp

(−ϕ

(k+1)

Vth

)(7.61d)

Solve the Linear Hole Continuity Equation (LHC):

− ∂

∂ x

(qDp

∂ p

∂ x+ qµpp

∂ ϕ(k+1)

∂ x

)+ qn(k+1)FSHR(p(k), n(k+1))p

= qn2iFSHR(p(k), n(k+1)) (7.61e)

Set p(k+1) = p and set u(k+1)p = p(k+1) exp

(ϕ(k+1)

Vth

). (7.61f)

In the following sections we describe the functional iteration approach and the cor-

responding numerical schemes used to treat each subproblem in the decoupled algo-

rithm (7.61).

7.12.2.1. The NLP Equation in Non-Equilibrium Conditions

To treat the nonlinear character of the Poisson equation (7.61a), we proceed in the same

manner as in Sect. 7.10.3.2, i.e., we use the Newton iterative process with the following

identifications:

U := ϕ

F (ϕ) := −∂2ϕ

∂ x2+

q

εs

[ni exp

(ϕ− ϕ(k)

n

Vth

)− ni exp

(ϕ

(k)p − ϕVth

)−D

]


where k is the counter of the outer iterations and

un := ni exp

(− ϕnVth

)(7.62)

up := ni exp

(ϕpVth

)(7.63)

are the Slotboom variables associated with electrons and holes, respectively. To compute

the Frechet derivative of F , we proceed as done in Sect. 7.10.3.2, to obtain

F ′(ϕ)w = −∂2w

∂ x2+

qniεsVth

(ϕ− ϕ(k)

n

Vth+ϕ

(k)p − ϕVth

)w

where ϕ is an arbitrary function belonging to H1(0, L) and satisfying the boundary

conditions (7.57a)- (7.57b), while w is an arbitrary function belonging to V := H10 (0, L).

Then, given ϕ(0), at each step j ≥ 0 of Newton’s inner iteration until convergence,

we have to solve the following linear BVP:

−∂2W

∂ x2+

qniεsVth

[exp

(ϕ(j) − ϕ(k)

n

Vth

)+ exp

(ϕ

(k)p − ϕ(j)

Vth

)]W

= −F (ϕ(j)) in Ω = (0, L)

(7.64a)

W (0) = W (L) = 0, (7.64b)

and then set

ϕ(j+1) = ϕ(j) + τjW (7.65)

where τj ∈ (0, 1] is a suitable damping parameter.

The equation system (7.64) is a reaction-diffusion BVP of the form (1.5) where

µ := 1, v := 0,

σ :=qniεsVth

[exp

(ϕ(j) − ϕ(k)

n

Vth

)+ exp

(ϕ

(k)p − ϕ(j)

Vth

)], f := −F (ϕ(j)).

Therefore, for its approximation we use the FE formulation illustrated in Sect. 1.6 with

the lumping stabilization of the reaction term illustrated in Sect. 1.10.1 to avoid the onset

of possible spurious oscillations in the computed electric potential.

Remark 7.12.1 (Slotboom variables and Fermi pseudopotentials). The choice of using

the (exponentials) of the quasi Fermi potentials ϕn and ϕp instead of the Slotboom vari-

ables un and up is made to avoid the possible onset of overflow problems in the evaluation

of e±ϕ/Vth due to the huge dynamical range of these latter terms (cf. Fig. 7.11).

7.12.2.2. The Linear Continuity Equations in Non-Equilibrium Conditions

Concerning with the solution of the linear electron and hole continuity equations, a special

treatment is introduced in the Gummel Decoupled Algorithm (7.61) to separate the

contributions due to recombination and generation in the SHR expression (7.59a). In the


case of the electron continuity equation, the numerator of the recombination mechanism

is brought at the left-hand side of the equation by “freezing” the term proportional to

p at its (available) value at the previous iteration k, while the denominator of the SHR

formula (7.59a) is computed using the (available) values of n and p at the previous step

k. A slightly different strategy is adopted to solve the hole continuity equation, in which

the numerator of the recombination mechanism is brought at the left-hand side of the

equation by “freezing” the term proportional to n at its value n(k+1) just determined

by the solution of the electron equation (7.61c), while the denominator of the SHR

formula (7.59a) is computed using the values p(k) and n(k+1). This splitting of the R/G

term is called lagging approach and corresponds to extending to the nonlinear case the

classical Gauss-Seidel method for the iterative solution of linear algebraic systems. An

alternative approach would be to use in both electron and hole equations the values n(k),

p(k) at the previous iteration to compute the splitted SRH R/G term. In such a case,

the lagging method corresponds to extending to the nonlinear case the classical Jacobi

method for the iterative solution of linear algebraic systems.

7.12.2.3. The Stabilized PM Method

Both linearized electron and hole continuity equations in the Gummel algorithm (7.61)

are diffusion-advection-reaction BVPs of the form (1.5) with:

µ := qDn, v := qµn∂ ϕ(k+1)

∂ x,

σ := qp(k)FSHR(p(k), n(k)), f := qn2iFSHR(p(k), n(k)),

in the case of the electron equation, and:

µ := qDp, v := −qµp∂ ϕ(k+1)

∂ x,

σ := qn(k+1)FSHR(p(k), n(k+1)), f := qn2iFSHR(p(k), n(k+1)),

in the case of the hole equation. The use of the primal-mixed FE formulation with the

aid of stabilization terms (lumping for the zero-th order term and artificial diffusion for

the convective term) proceeds as described in detail in Sect. 1.6.

We assume for simplicity to adopt a uniform partition of the device domain [0, L] into

M ≥ 2 subintervals Ki := [xi, xi+1] of length h = L/M , with x1 := 0 and xM+1 := L.

Moreover, we let U := p and denote by Ui, i = 1, . . . ,M + 1 ≡ N , the nodal values of

the finite element approximant Uh ∈ Vh, such that U1 = p0 and UN = pL, respectively.

Moreover, we recall that the mobility µp is constant and equal to µ0p, so that Dp = µ0

pVth

(due to Einstein’s relation), and that the electric potential ϕ(k+1) is a piecewise linear

continuous function over [0, L] in accordance with the FE approximation described in

Sect. 7.12.2.1. For each element Ki, i = 1, . . . ,M , we set ∆ϕ(k+1)i := ϕ

(k+1)i+1 −ϕ(k+1)

i and

we denote by

Ei := −∆ϕ

(k+1)i

h(7.66)

the constant value of the electric field.


Then, the (stabilized) primal-mixed FE discrete equations (1.21) are given by:

−Ji−1 + Ji + qn(k+1)FSHR(p(k), n(k+1))Ui h

= qn2iFSHR(p(k), n(k+1))h i = 2, . . . ,M

(7.67)

where the approximate (stabilized) flux on each element Ki is given by

Ji = qµ0pEi

Ui + Ui+1

2− qDp

(1 + Φ(Peadloc)

) Ui+1 − Uih

i = 1, . . . ,M, (7.68)

having defined the local Peclet number

Peadloc =|v|h2µ

=qµ0

p|Ei|h2qµ0

pVth=|Ei|h2Vth

and where Φ is the stabilization function introduced in Sect. 1.10.2.

According to the electrical equivalent interpretation based carried out at the end of

Sect. 1.1, each one of the discrete equations (7.67) can be considered as the following

Kirchhoff Current Law at node xi

− Ip,i−1 + Ip,i + qRiAh = 0 i = 1, . . . ,M, (7.69)

where A is the two-dimensional cross-section of the device and I = J ·A is a nodal current

(expressed in A). Eq. (7.69) expresses the fact the net hole current crossing the node xi

is balanced by the recombination/generation current qRiAh associated with the volume

Vxi = A · h surrounding node xi.

7.12.2.4. The Scharfetter-Gummel Method

In this section, we wish to further analyze the structure of the discrete equations (7.67)

in the case where the SG stabilization (1.39) is adopted. For ease of presentation we

refer to the hole continuity equation (7.61e) supplied by the Dirichlet boundary condi-

tions (7.57e)- (7.57f). The analysis of the electron continuity equation (7.61c) proceeds

in a completely analogous manner.

Replacing the expression (1.39) for Φ into (7.68) and setting for brevityX := ∆ϕ(k+1)i /Vth,

the current density over the element Ki can be written as

Ji = −qDp

h

[(|X|2

+ B(|X|) +X

2

)Ui+1 −

(|X|2

+ B(|X|)− X

2

)Ui

]i = 1, . . . ,M.

Let us consider the coefficient multiplying Ui+1. If X > 0 we have

|X|2

+ B(|X|) +X

2= X +

X

eX − 1=

XeX

eX − 1= B(−X),

and the same holds if X < 0. With a similar argument we can see that the coefficient

multiplying Ui is such that

|X|2

+ B(|X|)− X

2= B(X),

so that, going back to the original physical variable p, the current density over the element

Ki can be written in the following form

Ji = −qDp

h

[B

(−

∆ϕ(k+1)i

Vth

)pi+1 − B

(∆ϕ

(k+1)i

Vth

)pi

]i = 1, . . . ,M. (7.70)


The above formula is well-known in the semiconductor community as the Scharfetter-

Gummel equation for the hole current density over an element of size h. It was proposed

in 1969 by D. Scharfetter and H.K. Gummel (two scientists of Bell Labs) and from

that time on it has become the ”workhorse” of every computational algorithm for the

numerical simulation of the DD equation system.

Figure 7.24.: Upwinding effect associated by the use of the SG formula.

The SG formula (7.70) has several appealing properties:

1. if ϕ(k+1)i = ϕ

(k+1)i+1 (i.e., the electric field is equal to zero over Ki), relation (7.70)

becomes

Ji = −qDppi+1 − pi

hon Ki (7.71)

since B(0) = 1. Equation (7.71) is the first-order approximation of a pure diffusion

current density;

2. if (ϕ(k+1)i+1 −ϕ(k+1)

i )/Vth 0 (i.e., the electric field is directed from right to left over

Ki), relation (7.70) becomes

Ji = −qDp

h· pi+1 ·

ϕ(k+1)i+1 − ϕ(k+1)

i

Vth= qµ0

ppi+1Ei on Ki. (7.72)

Equation (7.72) is a pure drift current density transporting a charge density equal

to qpi+1;

3. if (ϕ(k+1)i+1 −ϕ(k+1)

i )/Vth 0 (i.e., the electric field is directed from left to right over

Ki), relation (7.70) becomes

Ji = −qDn

h· (−pi) ·

(−ϕ

(k+1)i+1 − ϕ(k+1)

i

Vth

)= qµ0

ppiEi on Ki. (7.73)

Equation (7.73) is a pure drift current density transporting a charge density equal

to qpi.

The three limiting cases discussed above are significant instances of any electronic

application. The automatic adaptive behaviour of the SG formula (7.70) to deal with each

of them is the result of the approximation properties provided by the Bernoulli weights

7.13. Metal-Semiconductor Contacts: Physical Principles, Modeling andNumerical Approximation 165

B(±∆ϕ(k+1)i /Vth). In particular, the two previous examples show that an automatic

upwinding effect is obtained if current flow is dominated by the drift mechanism, because

in such a case the transported (positive) charge is that corresponding to the node of the

element located ”up” with respect to the ”wind” (the electric field) as shown in Fig. 7.24.

7.13. Metal-Semiconductor Contacts: Physical Principles,

Modeling and Numerical Approximation

Every electronic device is basically made by a semiconductor region and metal strips,

put in intimate contact with the semiconductor region, through which the device is

electrically connected with the surrounding environment. In this section, we apply the

semiconductor device model equations to the study of the metal-semiconductor (MS)

contact. The basic principles underlying the physical and electrical behaviour of the

contact are first illustrated, and then, the appropriate set of boundary conditions and

the FE numerical approximation of the MS contact model are discussed in detail.

7.13.1. Energy Band Diagram of Isolated Materials

Let us assume that the material constituting the metal contact region is gold (Au) and

that the semiconductor region is made by silicon with a concentration of ionized donor

dopant impurities equal to N+D . The band diagram of the two materials, mutually sepa-

rated for the moment, is schematically represented in Fig. 7.25. The spatial coordinate

is denoted by x, while, as usual, energy is measured in eV, and increases from bottom to

top.

Figure 7.25.: Band diagram for the metal (left) and the semiconductor (right).

Several energy levels can be distinguished in Fig. 7.25. E0 is the vacuum level,

while EFM and EFS denote the energies of the Fermi level in the metal and in the

semiconductor, respectively. The energies qΦM and qχS are the metal work function

and the semiconductor electron affinity, respectively. The metal work function is the

energy required to move an electron from the metal Fermi level EFM up the vacuum

level E0, while the semiconductor electron affinity is the energy required to move an

electron from the the bottom of the conduction band EC up to the vacuum level E0.


Thus, the two energies have the same physical meaning, but are referred to a different

starting energy, EFM in the case of the metal, EC in the case of the semiconductor. In

any event, qΦM and qχS are characteristic constants of each material and do not depend

on dopant impurity concentration or on temperature. Their values are qΦM = 4.75 eV

and qχS = 4.05 eV, respectively. This is not the case with qΦS , that represents the work

function of the semiconductor material. As a matter of fact, from the band diagram in

Fig. 7.25, we have

qΦS = qχS + EC − EFS = qχS + (EC − Ei)− (EFS − Ei)

= qχS +EGap

2− (EFS − Ei).

Using (7.11) with n = N+D , we get

qΦS = qχS +EGap

2−KBT ln

(N+D

ni

), (7.74)

so that we see that the value of the work function for the semiconductor depends on the

value of the doping concentration. Using realistic data, if N+D = 1023 m−3, for example,

then we have qΦS = 4.61− 0.43 = 4.18 eV.

7.13.2. Energy Band Diagram of MS Contact

In this section, we discuss the case where metal and semiconductor are put in intimate

contact under the action of an externally applied voltage Va. The origin of the x-axis is

located at the metal-semiconductor junction, in such a way that the metal occupies the

region x < 0 while the semiconductor occupies the portion x > 0 of the real line. The

terminal at the end of the semiconductor region is set to ground voltage, so that Va is

the value of the external voltage at the MS contact (see Fig. 7.26).

Figure 7.26.: Electrical scheme of the MS contact.

In what follows, we always make the simplifying assumption that the hole number

density p in the semiconductor material is identically equal to zero, and that the electrical

properties of the metal are not affected by the presence of the semiconductor.

7.13.2.1. MS Contact at Thermal Equilibrium

In this case, Va = 0 V. Then, we recall the three following fundamental physical properties

of the system under investigation:


1. the Fermi level is constant at every point of the device because of Def. 7.5.1 and

Def. 7.5.2;

2. the vacuum energy level E0 is a continuous function of position;

3. the electron affinity is constant at every point in the semiconductor.

Using the above properties, we obtain the energy band diagram of the system shown in

Fig. 7.27.

Figure 7.27.: Band diagram of the MS contact at thermal equilibrium. The MS junc-

tion is located at x = 0. The metal (Au) is on the left, while the n-type

semiconductor is on the right.

The principal information that we can draw from the energy level distribution is

summarized below:

• Electrons close to x = 0 in the semiconductor side have an energy larger than those

located far from the interface. This is a consequence of the fact that electrons in the

semiconductor conduction band before contacting the two materials had a larger

energy than those in the metal (cf. Fig. 7.25), so that a transfer of negative charge

naturally occurs from the semiconductor toward the metal once the two materials

are put in contact.

• Such negative charge transfer leaves the semiconductor region close to x = 0 almost

depleted of free electrons, while hole number density is in any case assumed to

be negligible. As a result, a reasonable model of the MS junction device, is the

complete depletion approximation (CDA) already introduced in Sect. ?? for the

p-n junction, which consists of setting n = p = 0 for x ∈ (0, xd], xd being yet to be

determined, and n = N+D for x > xd.

• Electrons at x = 0− (in the metal) experience an energy barrier equal to qΦB that

they have to overcome in order to be injected from metal to semiconductor.

• Electrons at x = x+d (in the semiconductor) experience an energy barrier equal

to qΦi, Φi being called the built-in voltage of the MS junction, that they have to

overcome in order to be injected from semiconductor to metal.


From Fig. 7.27, it is immediate to see that

qΦB = q(ΦM − χS) (7.75)

which is independent of the dopant impurity concentration in the semiconductor. As far

as the barrier for electrons in the semiconductor, we have

qΦi = qΦB − (Ec − EF )|x≥xd = q(ΦM − χS)− q(ΦS − χS) = q(ΦM − ΦS). (7.76)

Thus, unlike the case for electrons in the metal side of the MS junction, the barrier for

electrons in the semiconductor to flow toward the metal depends on the value of dopant

concentration.

Exercise 7.13.1. Using the CDA, draw a plot of the space charge density ρ(x) = q(p(x)−n(x)+N+

D (x)) in the MS junction. What is the space charge accumulated at the interface

in the metal region?

Exercise 7.13.2. Using Gauss’ theorem, show that the width xd of the depleted region

at thermal equilibrium is given by

xd =

√2Φiε

qN+D

(7.77)

where Φi is expressed in V.

7.13.2.2. MS Contact in Non-Equilibrium Conditions

In this case Va 6= 0. Since the metal is not able to sustain a voltage drop, we assume

that Va falls entirely across the depleted region inside the semiconductor region. Then,

we distinguish between:

• Va > 0 (forward bias conditions);

• Va < 0 (reverse bias conditions).

(a) Va > 0 (b) Va < 0

Figure 7.28.: Electron current flow in a MS contact in non-equilibrium conditions.

When the MS junction is forwardly biased, the metal is at a higher potential than

the semiconductor bulk, so that electrons in the semiconductor material are attracted

by the positive polarity and an appreciable electron flux flows from the n-doped region

towards the metal (Fig. 7.28(a)). In terms of built-in voltage, the barrier opposing to


electron flow from semiconductor to metal becomes q(Φi−Va), qΦi being given by (7.76).

Therefore, the magnitude of the barrier is reduced in the forward bias regime compared

to that in thermal equilibrium.

Conversely, when the MS junction is reversedly biased, the metal is at a more negative

potential than the semiconductor bulk, so that electrons in the semiconductor material

are repelled from the contact region and, consequently, the depleted region widens and

extends more deeply into the bulk substrate along the x-axis (Fig. 7.28(b)). In terms

of built-in voltage, the barrier opposing to electron flow from semiconductor to metal is

still equal to q(Φi − Va) = q(Φi + |Va|), so that its magnitude is increased in the reverse

bias regime compared to that in thermal equilibrium.

Exercise 7.13.3. Draw schematically the energy band diagram of the MS contact in both

forward and reverse bias conditions.

7.13.3. Current Flow in a MS Junction

In this section, we address the evaluation of the net current density J flowing across the

MS junction at x = 0. For this, we switch off all the R/G mechanisms occurring in the

semiconductor region, consistently with the previous assumption p = 0. We also denote

by nMS = +1 and nSM = −1 the two normals directed outwardly with respect to metal

and semiconductor, respectively, and by JMS and JSM the electron current densities from

metal to semiconductor and from semiconductor to metal, respectively. Fig. 7.29 shows

the various contributions to current flow throughout the MS junction.

Figure 7.29.: Net electron current flow (in red) across the MS contact. The interface is

represented by a blue rectangle of infinitesimal width.

Current balance across the MS junction is expressed by the following relation

J = JMS · nMS + JSM · nSM = JMS − JSM . (7.78)

Recalling (7.17a), we can write (7.78) as

J = −qnMvMS + qnSvSM (7.79)

where nM and nS are the electron number densities at x = 0 on the metal and semi-

conductor sides, while vMS and vSM are the injection velocities across the junction from

metal to semiconductor and viceversa.

At thermal equilibrium, J = 0, so that

|JMS | = |JSM | = qneqveq ≡ Jeq (7.80)


where neq and veq denote the equilibrium values of the electron number density and elec-

tron velocity at the interface, respectively. The value of neq can be computed using (7.7a)

to obtain

neq = Nc exp

(−(Ec − EF )|x=0

KBT

)= Nc exp

(− qΦB

KBT

). (7.81)

This quantity can be expressed more conveniently in terms of the semiconductor bulk

doping density N+D as

neq = Nc exp

(− qΦB

KBT

)exp

(−(Ec − EF )|x≥xd

KBT

)exp

(+

(Ec − EF )|x≥xdKBT

)= N+

D exp

(− qΦB

KBT

)exp

(+

(Ec − EF )|x≥xdKBT

)= N+

D exp

(− qΦi

KBT

) (7.82)

where charge neutrality in the n region of the device far from the MS contact and the

first identity in (7.76) have been used.

The value of veq can be determined using the theory of Richardson thermionic emission

of an electron from a material at a temperature T > 0 which predicts the following

expression of the electron current density spontaneously flowing out of the metal

Jth−ionic = A∗nT2 exp

(− qΦe

KBT

)(7.83)

where A∗n (measured in Am−2K−2) is the effective Richardson constant for electrons

and qΦe is the energy barrier experienced by electrons in the metal. Replacing (7.83)

into (7.80) with Φe = ΦB and using (7.81), gives

veq =A∗nT

2

qNc. (7.84)

In non-equilibrium conditions, electron current from metal to semiconductor is not

affected by the presence of the semiconductor, so that the current balance at the MS

junction becomes

J = −qneqveq + qnv

where n and v are the values of the electron number density and of the injection velocity

at the MS interface, respectively. Assuming v = veq (which amounts to assuming that the

semiconductor always acts as a thermionic emitter irrespective of the biasing conditions),

current balance across the MS junction is given by the following relation

J = q(n− neq)veq. (7.85)

According to (7.85), current flow and electron number density at the MS interface are not

given quantities, rather they have to obey a detailed balance principle which expresses

the opposite tendency of electrons to flow from the metal and from the semiconductor

under the action of electrical and thermal forces.

Remark 7.13.1 (Robin condition at the MS contact). In mathematical terms, rela-

tion (7.85) is a Robin boundary condition (for the dependent variable n) of the form (1.5c),

upon defining (in the notation of the advection-diffusion-reaction BVP (1.5)) u := −qnand setting µ := Dn, v := −µnE, α := veq and β := ueq veq, with ueq := −qneq.


Remark 7.13.2 (J-V characteristic of a MS junction). If we assume that all the applied

voltage drop Va falls across the depleted region [0, xd], then the value of the electron

number density n at x = 0 can be expressed as a function of the equilibrium electron

number density neq as

n|x=0 = neq exp

(qVaKBT

). (7.86)

The above relation tells us that the electron concentration in the semiconductor side of the

MS contact is larger than the bulk doping number density if the junction is in forward bias

regime because the built-in barrier to electron flow from the semiconductor towards the

metal is lowered by the application of a positive external voltage Va. In the opposite case

of a reverse bias condition (Va < 0), electrons are furtherly swept away from the junction

by the externally applied electric field so that the interface region is more depleted than

at thermal equilibrium and the electron number density decreases according to a factor

e−|Va|/Vth with respect to the equilibrium value neq.

Replacing (7.86) into (7.85) we get

J = J0

[exp

(VaVth

)− 1

](7.87)

where J0 := qneqveq is the saturation current density of the MS contact.

7.13.4. Mathematical Model of a MS Contact

In this section, we summarize the simplified DD system to be solved in stationary con-

ditions for the simulation of a MS contact in one spatial dimension. The formulation is

based on the following assumptions:

• hole number density is neglected (p = 0);

• the metal region is not affected by the presence of the semiconductor and acts only

as a boundary condition for the semiconductor material.

Assuming that the MS contact is at x = 0 and denoting by L the length of the semicon-

ductor, the model describing electron and electric potential distributions in the semicon-

ductor region Ω = (0, L) of the MS contact reads:

−∂ Jn∂ x

= 0 (7.88a)


∂ x(≡ −qnvn) (7.88b)

∂ E

∂ x=

q

εrε0(N+

D − n) (7.88c)

E = −∂ ϕ∂ x

. (7.88d)



ϕ = Vext + Vth log

(neqni

)at x = 0 (7.89a)

−Jn · nSM = q(n− neq)veq at x = 0 (7.89b)

ϕ = Vth log

(N+D

ni

)at x = L (7.89c)

n = N+D at x = L. (7.89d)

We notice that the boundary condition (7.89a) follows from the MB relation (7.15a)

upon setting ϕn|x=0 = Va (quasi-Fermi level imposed by the metal and equal to applied

voltage) and n|x=0 = neq, neq being given by (7.81) (or, equivalently, (7.82)).

7.13.5. Functional Iteration and Numerical Approximation

The functional iteration to solve the nonlinear system of PDEs (7.88) and the correspond-

ing FE discretization proceed in exactly the same manner as described in Chapt. 7.10 for

the p-n junction device in thermal equilibrium, forward bias and reverse bias conditions.

The principal difference consists in the boundary condition at the metal-semiconductor

interface at x = 0, which is, in the present case, of Robin type while for the p-n junction

the corresponding condition was of Dirichlet type because in that case the contact was

assumed to be ideal and ohmic, so that veq = +∞ (infinite recombination velocity) and

n = neq.

7.14. The MOS Structure: Physical Principles, 1D Models

and Numerical Approximation

In this section, we apply the semiconductor device model equations to the study of a fun-

damental component in semiconductor device technology: the Metal-Oxide-Semiconductor

(MOS) structure. The complexity of the physical principles underlying the functioning

of a MOS device suggest introducing a simpler description based on 1D approximations

of the structure. To this purpose, we consider two particular electronic substructures

obtained by suitably cutting the MOS device along the y and x axes, perpendicular and

parallel to the oxide-semiconductor interface, respectively. The obtained substructure are

denoted as the MOS capacitor and the MOS channel. The appropriate set of boundary

conditions for each substructure and the numerical implementation using the FEM are

discussed in detail.

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