Module 2,3 & 4

N.B.Balamurugan

Associate Professor

ECE Department

Thiagarajar College of Engineering

Madurai-15 (nbbalamurugan @tce.edu)

VLSI DEVICE MODELING

AND SIMULATION

UNIT II NOISE MODELING 9

Noise sources in MOSFET, Flicker noise modeling, Thermal noise modeling, model for

accurate distortion analysis, nonlinearities in CMOS devices and modeling, calculation

ofdistortion in analog CMOS circuit

UNIT III BSIM4 MOSFET MODELING 9

Gate dielectric model, Enhanced model for effective DC and AC channel length and

width, Threshold voltage model, Channel charge model, mobility model, Source/drain

resistance model, I-V model, gate tunneling current model, substrate current models,

Capacitance models, High speed model, RF model, noise model, junction diode

models, Layout-dependent parasitics model.

UNIT IV OTHER MOSFET MODELS 9

The EKV model, model features, long channel drain current model, modeling second

order effects of the drain current, modeling of charge storage effects, Non-quasi-static

modeling, noise model temperature effects, MOS model 9, MOSAI model)

UNIT V MODELLING OF PROCESS VARIATION AND QUALITY 9

ASSURANCE

Influence of process variation, modeling of device mismatch for Analog/RF

Applications, Benchmark circuits for quality assurance, Automation of the tests

OLD SYALLABUS OF SSDMAS

24 October 2013 2

OUTLINEUNIT 2: MESH, NODE, MODIFIED NODAL ANALYSIS

Solving Linear Networks : Sparse Matrix

Solving Non linear Networks using NEWTON

RAPHSON Method.

UNIT 3 Stiffness and MULTISTEP Method

UNIT 4 Finite Difference Solution to

Poisson Equation, Continuity equation,

Schodinger Equation, drift diffusion Equation

HydroDynamic Equation

24 October 2013 3

• Introduction

Circuit Simulations

Circuit Simulators – SPICE

• Device Equations

• Network Equations

Elements

Passive and Active Elements

Equivalent circuit Model

Networks

Mesh, Nodal, Modified Nodal, Hybrid

• Solution of Linear Circuit Equation

• Solution of Nonlinear Circuit Equation

• Solution of Differential Circuit Equation

• Solution of Partial Differential Circuit Equation24 October 2013 4

CIRCUIT SIMULATION

• It is a technique for checking and verifying the design

of electrical and electronic circuits and systems.

• Circuit simulation combines

Mathematical modeling of the circuit elements or devices.

Formulation of circuit / Network Equation.

Techniques for solution of these equations.

24 October 2013 5

Circuit Simulation

Simulator:

Solve numerically

Input and setup Circuit

Output

Types of analysis:

– DC Analysis

– DC Transfer curves

– Transient Analysis

– AC Analysis, Noise, Distortions, Sensitivity

)()(

)()()(

)(

tFUtDXY

tBUtGXdt

tdXCXf

dt

tdXCXf

)()(

24 October 2013 6

Program Structure (a closer look)

Numerical Techniques:

– Formulation of circuit equations

– Solution of ordinary differential equations

– Solution of nonlinear equations

– Solution of linear equations

Input and setup Models

Output24 October 2013 7

CIRCUIT SIMULATORS

•They use a Detailed (so called circuit level /

Transistor Level) description of the circuit and perform

relatively accurate simulation.

• Typically a simulation uses physical models of the circuit

elements,solves the resuliting differential and algebraic

equations and generates time waveforms of node

Voltages and element Currents.

• Early techniques for circuit simulation using computer

were introduced in 1950 and 1960s.

24 October 2013 8

SPICE

• It is Universal standard simulator used to simulate

the operation of various electric circuits and devices.

• It was developed by L.W.Nagel at the Univ. of California,

Berkeley in 1968.

• The first simulator was named as CANCER (Computer Analysis

of Non-Linear Circuits Excluding Radiation).

It could not handle more components or circuit nodes.

1970 Improvements in CANCER continued

1971 an improved version of CANCER named SPICE 1 was

released.

1975 SPICE2 was introduced.

1983 SPICE 2G.6 version was released. All these version

were written in FORTRAN source code. Later it was

rewriten in C. 24 October 2013 9

SPICE 3 New C version of the program

Many SPICE like simulators in the market.

Meta-software’s HSPICE

Intusoft’s IS-SPICE

Spectrum Software’s MICRO-CAP

Microsim’s PSPICE

Texas instruments TINA SPICE

National Instruments LT SPICE

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DEVICE EQUATIONS

A device is a any simulation elements described by

means of a current voltage relationship.

Ex: Resistor

V = I R

I is found from V.

Linear Element

When the element equation contains no terms with

powers of 2 or higher, the element is said to be linear

Element circuit circuit.

Linear Circuit

A network of linear elements is said to be a linear circuit.

24 October 2013 13

Branch Constitutive Equations

(BCE)Ideal elements

Element Branch Eqn Variable parameter

Resistor v = R·i -

Capacitor i = C·dv/dt -

Inductor v = L·di/dt -

Voltage Source v = vs i = ?

Current Source i = is v = ?

VCVS vs = AV · vc i = ?

VCCS is = GT · vc v = ?

CCVS vs = RT · ic i = ?

CCCS is = AI · ic v = ?

24 October 2013 14

NETWORK EQUATIONS

ELEMENTS and NETWORKS

•An element is a two terminal electrical device.

•An electrical Network or Circuit is a system consisting of

a set of elements and a set of nodes, where every element

terminal is unique node,and every node is identified

with atleast the element terminal.

• A network is completely connected, i.e there is always

atleast one path from one node to another.

24 October 2013 15

Passive Elements

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NETWORKThey are two basic important techniques used in finding solutions

for a network

MESH ANALYSIS

• If a network has a large number of voltage sources, it is useful to use MESH analysis.

• It is only applicable only for planar networks.

• 1. Check whether the circuit is applicable.

• 2. To select mesh current

• 3. Writing Kirchhoff’s voltage law equations in terms of unknowns and solving them to the final solution

NODE ANALYSIS

• If a network has more

current sources, Node

analysis is more useful.

•

24 October 2013 25

DC MESH Analysis

BRANCH AND MESH CURRENTS:

Applying KVL around the left loop

Applying KVL around the right loop

MATRICES AND MESH CURRENTS:• The n simultaneous equations of an n-mesh network can be written in

matrix form

When KVL is applied to the three mesh network

24 October 2013 26

Basic CircuitsMesh Analysis: Example 7.1.

Write the mesh equations and solve for the currents I1, and I2.

+

_10V

4 2

6 7

2V20V

I1 I2+

+_

_

Figure 7.2: Circuit for Example 7.1.

Mesh 1 4I1 + 6(I1 – I2) = 10 - 2

Mesh 2 6(I2 – I1) + 2I2 + 7I2 = 2 + 20

Eq (7.9)

Eq (7.10)

24 October 2013 27

Basic Circuits

Mesh Analysis: Example 7.1, continued.

Simplifying Eq (7.9) and (7.10) gives,

10I1 – 6I2 = 8

-6I1 + 15I2 = 22

Eq (7.11)

Eq (7.12)

» % A MATLAB Solution

»

» R = [10 -6;-6 15];

»

» V = [8;22];

»

» I = inv(R)*V

I =

2.2105

2.3509

I1 = 2.2105

I2 = 2.3509

24 October 2013 28

Basic CircuitsMesh Analysis: Example 7.2

+

_

6

10

9

11

3

4

20V 10V

8V

12V

I1 I2

I3

+

+

__

_

_

+

+_

Mesh 1: 6I1 + 10(I1 – I3) + 4(I1 – I2) = 20 + 10

Mesh 2: 4(I2 – I1) + 11(I2 – I3) + 3I2 = - 10 - 8

Mesh 3: 9I3 + 11(I3 – I2) + 10(I3 – I1) = 12 + 8

Eq (1)

Eq (2)

Eq (3)24 October 2013 29

Basic Circuits

Mesh Analysis:

Clearing Equations (1), (2) and (3) gives,

20I1 – 4I2 – 10I3 = 30

-4I1 + 18I2 – 11I3 = -18

-10I1 – 11I2 + 30I3 = 20

In matrix form:

20

18

30

3

2

1

301110

11184

10420

I

I

I

WE NOW MAKE AN IMPORTANT

OBSERVATION!!

Standard Equation form

24 October 2013 30

Basic CircuitsMesh Analysis: Standard form for mesh equations

Consider the following:

R11 =

of resistance around mesh 1, common to mesh 1 current I1.

R22 = of resistance around mesh 2, common to mesh 2 current I2.

R33 = of resistance around mesh 3, common to mesh 3 current I3.

)3(

)2(

)1(

3

2

1

333231

232221

131211

emfs

emfs

emfs

I

I

I

RRR

RRR

RRR

24 October 2013 31

NODE VOLTAGE METHOD

• In a N node circuit one of the nodes is chosen as

reference, then it is possible to write N-1 nodal

equations by assuming N-1 node voltages.

• Applying KCL at node 1,2,3

24 October 2013 32

Nodal Analysis

24 October 2013 33

STEADY STATE AC ANALYSIS

• Two analysis have applied it only resistive circuits.

• These concepts can also be used for sinusoidal steady

state condition.

• In the sinusoidal steady state analysis,

we use voltage phasors, current Phasors,

impedances,admittances, to write branch

equations, KVL and KCL equations.

• For ac circuits, the method of writing loop equation is

• modified slightly. If the impedances are complex,

the sum of their voltages is found by vector addition.

24 October 2013 34

Modified Nodal Approach

• It is another Method of writing circuit equations that is less demanding of space and almost as flexible.

• It allows a broad range of elements to be modeled.

• we will consider networks with two terminals elements only.

• We will label the elements as either being current controlled or voltage controlled.

• R , Inductor,and Independent voltage source will be considered current controlled.

• Conductors , Capacitors and independent currnet source will

be considered voltage controlled.

24 October 2013 35

• If the branch relation is written

i = f(v)

The element is considered voltage controlled.

• If the branch relation is written

v = g(i)

The element is considered current controlled.

• This generates a set of equations with more

unknowns than equations

24 October 2013 36

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38

Advantages and problems of MNA

24 October 2013

39

Modified Nodal Analysis (MNA)

24 October 2013

40

Modified Nodal Analysis (2)

0

6

0

0

0

001077

000110

1011

00

0111

00

0100111

0000111

5

7

6

4

3

2

1

88

88

433

3

2

3

2

1

ES

i

i

i

e

e

e

e

EE

RR

RR

RRR

RG

RG

R

s

24 October 2013

SOLVING OF NETWORK

EQUATIONS

USING SPARSE MATRICESSPARSE MATRIX

A sparse matrix is a matrix in which the great majority of the

elements are zeros. Such matrices are used to solve network equations

and other engineering disciplines.

SPARSITY IN A CIRCUIT

Consider a complex linear circuit, containing large number of nodes.

By writing the node equations we came to know that the matrix have

large number of zeroes since each node is connected to only a few other

nodes. Sparsity is a key feature of large scale circuits such as VLSI digital

circuits or electric networks.

Sparse matrix

x x 0 0 0 0 0 0

0 x x 0 0 0 0 0

0 0 x x 0 0 0 0 An Sparse Matrix of order 8x8

0 0 0 x x 0 0 0 x-non-zero elements

0 0 0 0 x x 0 0

0 0 0 0 0 x x 0

0 0 0 0 0 0 x x

0 0 0 0 0 0 0 x

A circuit’s node equations when represented in matrix will also have same

matrix like the above one.

Storing of all entries need more space and harder to manipulate. Hence we

can store only the non zero elements and it will be easy to manipulate.

Solving sparse matrices

In network theory, big problem is solving a large system of linear

equations.

Ax=B

Sparse matrix can be solved using LU factorization.

Solution procedure

• Rewrite A=LU,where L is Lower triangular matrix,U is upper triangular

matrix.

• Solve Lz=b for z.

• Solve Ux=z for b.

5 1 2 1 0 0 5 1 2

A= 1 4 1 = 0.2 1 0 0 3.8 0.6

2 2 5 0.4 0.42 1 0 0 3.94

L U

Fill-in

When a sparse matrix is factorize into L and U ,it contains non zeroelements in places whereas the same place in sparse matrix contains zeros.This unwanted non zero elements are called as fill-ins.

Fill-in is a major problem in large matrices. Hence it should be reduced.

It can be either reduced by permuting or more easier elimination graphs. Alsoin elimination graphs minimal degree ordering is the best way to reduce

fill-ins.

example

for the resistor circuit given ,we can

Apply sparsity since each node is

connected to few other nodes. While

Solving,a matrix of order 12x12 will be

obtained with large number of zeros,

Which is a sparse matrix.

Newton’s Method,

Solving NonLinear Equation

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

History

• Discovered by Isaac Newton and published in his Method of Fluxions, 1736

• Joseph Raphson described the method in Analysis Aequationum in 1690

• Method of Fluxions was written earlier in 1671

• Today it is used in a wide variety of subjects, including Computer Vision and Artificial Intelligence

October 24, 201351

Review: Classification of

Equations• Linear: independent variable appears to the

first power only, either alone or multiplied by a

constant

• Nonlinear:

– Polynomial: independent variable appears raised to

powers of positive integers only

– General non-linear: all other equations

October 24, 201352

Review: Solution Methods

• Linear: Easily solved analytically

• Polynomials: Some can be solved analytically

(such as by quadratic formula), but most will

require numerical solution

• General non-linear: unless very simple, will

require numerical solution

October 24, 201353

Newton’s Method

• Newton’s Method (also know as the Newton-Rapshon Method) is another widely-used algorithm for finding roots of equations

• In this method, the slope (derivative) of the function is calculated at the initial guess value and projected to the x-axis

• The corresponding x-value becomes the new guess value

• The steps are repeated until the answer is obtained to a specified tolerance

October 24, 201354

Newton’s Method

Initial guess xi

y(xi)Tangent Line: Slope = y’(xi)

October 24, 201355

Newton Raphson

i

iii

ii

ii

xf

xfxx

rearrange

xx

xfxf

fdx

dygenttan

'

0'

'

1

1

f(xi)

xi

tangent

xi+1

If Initial guess at the root is xi,

a tangent can be extended from

the point [xi,f(xi)]. The point where

this tangent crosses the x axis usually

represents an improved estimate

of the root.f(xi) -

0

Slope =

ixf '

ixf '

October 24, 201356

Newton’s Method

xi

y(xi)

y(xi)/y’(xi)

New guess for x:xn = xi - y(xi)/y’(xi)

October 24, 201357

Newton’s Method Example

• Find a root of this equation:

• The first derivative is:

• Initial guess value: x = 10

October 24, 201358


• For x = 10:

• This is the new value of x

October 24, 201359


Initial guess =

10

y = 1350y’ = 580

New x-value = 7.6724

October 24, 201360


• For x = 7.6724:

October 24, 201361


Initial value = 7.6724

y = 368.494y’ = 279.621

New x-value = 6.3546

October 24, 201362


• Continue iterations:

Method quickly converges to this root

October 24, 201363

Newton’s Method - Comments

• Usually converges to a root much faster than the

bisection method

• In some cases, the method will not converge

(discontinuous derivative, initial slope = 0, etc.)

• In most cases, however, if the initial guess is

relatively close to the actual root, the method will

converge

• Don’t necessarily need to calculate the derivative:

can approximate the slope from two points close

to the x-value. This is called the Secant Method

October 24, 201364

October 24, 201365

Need to Solve

( 1) 0d

t

VV

d sI I e

Nonlinear Problems - Example

0

1

IrI1 Id

0)1(1

0

11

1

1

IeIeR

III

tV

e

s

dr

11)( Ieg

October 24, 201366

Nonlinear Equations

• Given g(V)=I

• It can be expressed as: f(V)=g(V)-I

Solve g(V)=I equivalent to solve f(V)=0

Hard to find analytical solution for f(x)=0

Solve iteratively

October 24, 201367

Nonlinear Equations – Iterative Methods

• Start from an initial value x0

• Generate a sequence of iterate xn-1, xn, xn+1

which hopefully converges to the solution x*

• Iterates are generated according to an iteration

function F: xn+1=F(xn)

October 24, 201368

Newton-Raphson (NR) Method

Consists of linearizing the system.

Want to solve f(x)=0 Replace f(x) with its linearized version

and solve.

Note: at each step need to evaluate f and f’

functionIterationxfdx

xdfxx

xxdx

xdfxfxf

iesTaylor Serxxdx

xdfxfxf

kk

kk

kkk

kk

)()(

)()(

)()(

...)*()(

)(*)(

1

1

11

Solving Nonlinear Equations

• nonlinear elements when simulating integrated

circuits.

• For a simple nonlinear equation

f(x)=0

• the usual method of solution is some variant on

the Newton-Raphson (NR) procedure.

October 24, 201369

• In order to understand the NR procedure we

must realize that the entire procedure is based

upon a linear approximation to the function at

the current point, xc.

• Thus, if we expand the function in a Taylor

series about the current point we have

• if we further assume that the x+ is the solution,

i.e.,

(which is the basic NR step.)

October 24, 201370

Newton-Raphson example.

October 24, 201371

Newton-Raphson Applied to Circuit

Simulation

• The first difficulty is that nonlinear circuit equations are not just a

function of a single variable. Further, there is a set of equations and not

just a single one. This problem is easily handled by extending the 1D

NR Scheme to multidimensions using vector calculus.

• The update formula now becomes

• were ∂f/∂x is known is the Jacobian of the set of circuit equations and

is the matrix of partial derivatives.

October 24, 201372

• Rewriting this set of equations we have

• This is precisely in the form

The final difficulty we face that is how to generate the equations in the first place

and how to take the partial derivative of this large set of equations.

First individual elements can be linearised and the equation written for this network

And we still have the same set of equations required for NR.

October 24, 201373

Example

• We will assume a simple form for the diode equation

• The linearized form of this becomes

• If we rewrite this grouping the known and unknown quantities we have

We will consider the network in 8.49

October 24, 201374

Nonlinear network used for NR

October 24, 201375

October 24, 201376

October 24, 201377

Adv and disadv of NR method

ADV:

• convergence rate for Newton’s method is very high.

• Error estimates are very good.

• NR method can find complex roots.

DIS ADV:

• If the local min/max is selected as an initial Guess,the slope does not interset

with X axis.

• The formula for xi will lead to an infinite value.

In-Class Exercise

• Draw a flow chart of Newton’s Method

• Write the MATLAB code to apply Newton’s

Method to the previous example:

October 24, 201378

Define converge tolerance tol

while abs(f(x)) > tol

Input initial guess xCalculate f(x)

Calculate slope fpr(x)x = x – f(x)/fpr(x)

Calculate f(x)

Output root x

October 24, 201379

MATLAB Code

• MATLAB functions defining the function and

its derivative:

October 24, 201380

MATLAB Code

October 24, 201381

MATLAB Results

>> Newton

Enter initial guess

10

Root found: 5.6577

>> Newton

Enter initial guess

0

Root found: 1.4187

>> Newton

Enter initial guess

-10

Root found: -2.0764

October 24, 201382

Excel and MATLAB Tools

• General non-linear equations:

– Excel: Goal Seek, Solver

– MATLAB: fzero

• Polynomials:

– MATLAB: roots

• Graphing tools are also important to locate

roots approximately

October 24, 201383

roots Example

• For polynomials, the MATLAB function roots

finds all of the roots, including complex roots

• The argument of roots is an array containing

the coefficients of the equation

• For example, for the equation

the coefficient array is [3, -15, -20, 50]

October 24, 201384

roots Example

>> A = [3, -15, -20, 50];

>> roots(A)

ans =

5.6577

-2.0764

1.4187

October 24, 201385

roots Example

• Now find roots of

>> B = [3, -5, -20, 50];

>> roots(B)

ans =

-2.8120

2.2393 + 0.9553i

2.2393 - 0.9553i

Two complex roots

October 24, 201386

Summary

• The bisection method and Newton’s method

(or secant method) are widely-used algorithms

for finding the roots of equations

• When using any tool to find the roots of non-

linear equations, remember that multiple roots

may exist

• The initial guess value will affect which root is

found

October 24, 201387

88

Ordinary Differential Equations

• Equations which are composed of an unknown

function and its derivatives are called differential

equations.

• Differential equations play a fundamental role in

engineering because many physical phenomena are

best formulated mathematically in terms of their rate

of change.

v- dependent variable

t- independent variablev

m

cg

dt

dv

89

• When a function involves one dependent variable, the equation is called an ordinary differential equation (or ODE). A partial differential equation (or PDE) involves two or more independent variables.

• Differential equations are also classified as to their order.

– A first order equation includes a first derivative as its highest derivative.

– A second order equation includes a second derivative.

• Higher order equations can be reduced to a system of first order equations, by redefining a variable.

90

Figure PT7.2

UNIT 3:Stiffness and

Multistep Methods

91

Stiff problem:1. Natural time constants2. Input time constants3. Interval of interest

If these are widely separated, then the problem is stiff

92

Stiff system are both individual and systems of ODEs

that have both fast and slow components to their

solution.

We introduce the idea of an implicit solution technique

as one commonly used remedy for this problem

93

Stiffness and Multistep Methods

• Two areas are covered:

– Stiff ODEs will be described - ODEs that have both

fast and slow components to their solution.

– Implicit solution technique and multistep methods

will be described.

94

STIFFNESS

In many cases, the rapidly varying components are transient that die away

quickly,after which the solution becomes dominated by the slowly varying

components. Although the transient phenomena exist for only a short part of

the integration interval, they can dictate the time step for the entire solution.

95

STIFFNESS

•Definition

•Proof – one problem

•Solving the problem by Numerically

Explicit Euler’s Method

Implicit Euler’s Method

Step size

•Literature survey

96

Stiffness• A stiff system is the one involving rapidly changing

components together with slowly changing ones.

• Both individual and systems of ODEs can be stiff:

• If y(0)=0, the analytical solution is developed as:

teydt

dy 200030001000

tt eey 002.2998.03 1000

Suppose that lemda1 = 1000 and lemda2 =1 are widely

sepatared and giving a stiff system. Note that the first

exponential waveform dies out in 5 micro secs.

To get an accurate solution of the fast waveform we need

a small time step and to have an efficient simulatin of the

slow waveform we eed a large time step. The obvious

97

Figure 26.1

98

99

100

• If Euler’s method is used to solve the problem

numerically:

The stability of this formula depends on the step size

h:

)1(or 11

1

ahyyhayyy

hdt

dyyy

iiiii

iii

iyah

ah

i as/2

11

101

• Thus, for transient part of the equation, the step size

must be <2/1000=0.002 to maintain stability.

• While this criterion maintains stability, an even

smaller step size would be required to obtain an

accurate solution.

• Rather than using explicit approaches, implicit

methods offer an alternative remedy.

• An implicit form of Euler’s method can be developed

by evaluating the derivative at a future time.

102

103

• Insight into the step size required for stability of such a

solution can be gained by examining the homogeneous

part of the ODE:

• The solution starts at y(0)=y0 and asymptotically

approaches zero.

ateyy

aydt

dy

0 is the solution.

104

• If Euler’s method is used to solve the problem

numerically:

The stability of this formula depends on the step size

h:

)1(or 11

1

ahyyhayyy

hdt

dyyy

iiiii

iii

iyah

ah

i as/2

11

105

ah

yy

hayyy

hdt

dyyy

ii

iii

iii

11

11

11

iasyi 0

• Backward or implicit Euler’s method

The approach is called unconditionally stable.

Regardless of the step size:

aydt

dy

106

107

MULTI STEP METHODS

• Definition

• Non self staring Heun method

Euler’s method as Predictor

Trapezoidals rule as a corrector

• Slightly Modified Predictor and corrector

• Derivation and Error analysis of Predictor- corrector Formulas

• Error Estimates

• Modifiers

One step methods give information at a single xi to predict a value of

the dependent variable yi+1 at a future point xi+1.

Multistep gives valuable information from previous points. The curvature of

the lines connecting these previous values that provides information regarding

the trajectory of the solution.

At each point, tn, we are going to compute an approximation, xn, to

the exact solution, x(tn). A large number of methods are available for

108Graphical depiction of the fundamental difference between (a) one step and

(b) Multistep methods for solving ODEs.

109

Multistep Methods

The Non-Self-Starting Heun Method

• It is characteristic of most multistep methods.

• It use an open integration formula (the mid point method) to

make an initial estimate.

• Huen method uses Euler’s method as a predictor and

trapezoidal rule as a corrector.

• Predictor is the weak link in the method because it has the

greatest error, O(h2).

• One way to improve Heun’s method is to develop a predictor

that has a local error of O(h3).

hyxfyy iiii 2),(1

0

1

110

Multistep Methods

Corrector formula

and you iterate until a maximum number of iteratios is reached.

h

yxfyxfyy

j

iiiii

j

i2

),(),( 1

111

111

112

113

114

115

116

117

Integration Formulas/

Newton-Cotes Formulas.

Open Formulas.

Closed Formulas.

dxxfyyi

ni

x

x

nnii )(1

1

fn(x) is an nth order interpolating

polynomial.

dxxfyyi

ni

x

x

nnii )(1

1

11

118

Adams Formulas (Adams-Bashforth).

Open Formulas.

• The Adams formulas can be derived in a variety of

ways. One way is to write a forward Taylor series

expansion around xi. A second order open Adams

formula:

Closed Formulas.

• A backward Taylor series around xi+1 can be written:

)(12

5

2

1

2

3 43

11 hOfhffhyy iiiii

)( 11

0

11

nn

k

kikii hOfhyy

Listed in Table

26.2

119

Higher-Order multistep Methods/

Milne’s Method.

• Uses the three point Newton-Cotes open formula as a predictor and three point Newton-Cotes closed formula as a corrector.

Fourth-Order Adams Method.

• Based on the Adams integration formulas. Uses the fourth-order Adams-Bashforth formula as the predictor and fourth-order Adams-Moulton formula as the corrector.

Finite Difference Method

Schrodinger equation

Need of Scaling (Moore’s Law )

Gordon Moore, a physical chemist working in electronics, made a prediction in 1965, the number of transistors on an integrated chip, would double every 18 months.

Dr. Gordon Moore

15/3/2012121

Karunya

Device Modeling Enable System Transformations

15/3/2012122

Karunya

• Smaller• Faster• Cheaper

Miniaturisation: why?

15/3/2012123

Karunya

1980 → 1985 → 1988 → 1991 → 1994 → 1999 → 2003 …

LGATE = 10 μm 5 μm 2.5 μm 1.3 μm 0.63 μm 0.25 μm 130 nm ….

1980 → 1985 → 1988 → 1991 → 1994 → 1999 → 2003 …


1980 → 1985 → 1988 → 1991 → 1994 → 1999 → 2003 …


1980 → 1985 → 1988 → 1991 → 1994 → 1999 → 2003 …


1980 → 1985 → 1988 → 1991 → 1994 → 1999 → 2003 …


1980 → 1985 → 1988 → 1991 → 1994 → 1999 → 2003 …

LGATE = 10 μm 5 μm 2.5 μm 1.3 μm 0.63 μm 0.25 μm 130 nm …. 1980 → 1985 → 1988 → 1991 → 1994 → 1999 → 2011 …


Scaling• Shrink dimensions maintaining aspect-ratio• Must shrink electrostatic features as well (depletion regions→ doping

level and profiles)

15/3/2012124

Karunya

In Classical systems,

The particle will take a path betweenthe two positions.

In Quantum systems,

Instead, the particle is awave… and it doesn’ttake one path from theinitial position to the finalposition, it takes allpossible paths.

•DOS- Density of statesDescribes the number of states at each energy levels that are available to be occupied by charge carriers.

• is the Momentum of the charge carriers.

p

Typically, near the silicon surface, the inversion layer charges are confined to a

potential well formed by:

Oxide barrier

bend Si-conduction band at the surface due to the applied voltage, Vg

Due to the confinement of inversion layer e-:

e- energy levels are grouped in discrete sub-bands of energy, Ej

Energy Wavefunction Particle Reduced Potential

mass PlankConstant

If the potential is dependent of x, then the solution is based on numerical solution

x

xfxxflim)x('fx

0

Continuity Equation

Derivation of Continuity

Equation• Consider carrier-flux into/out-of an infinitesimal volume:

Jn(x) Jn(x+dx)

dx

Area A, volume Adx

Adxn

AdxxJAxJqt

nAdx

n

nn

)()(

1

EE130/230M Spring 2013 Lecture 6, Slide 139


n

n n

x

xJ

qt

n

)(1

L

p

pG

p

x

xJ

qt

p

)(1

Continuity

Equations:

dxx

xJxJdxxJ n

nn

)()()(

L

n

n Gn

x

xJ

qt

n

)(1

Derivation of Minority Carrier Diffusion

Equations• The minority carrier diffusion equations are derived from

the general continuity equations, and are applicable only

for minority carriers.

• Simplifying assumptions:

1. The electric field is small, such that

in p-type

material

in n-type material

2. n0 and p0 are independent of x (i.e. uniform doping)

3. low-level injection conditions prevail

x

nqD

x

nqDnqJ nnnn

x

pqD

x

pqDpqJ pppp



• Starting with the continuity equation for electrons:

L

n

n Gn

x

nnqD

xqt

nn

1 00

L

n

n Gn

x

nD

t

n

2

2

L

n

n Gn

x

xJ

qt

n

)(1

Carrier Concentration Notation


• The subscript “n” or “p” is used to explicitly denote n-type or p-type material, e.g.

pn is the hole (minority-carrier) concentration in n-type mat’l

np is the electron (minority-carrier) concentration in n-type mat’l

L

p

nnp

n

L

n

pp

n

p

Gp

x

pD

t

p

Gn

x

nD

t

n

2

2

2

2

• Thus the minority carrier diffusion equations are

Summary

• The continuity equations are established based on conservation of carriers, and therefore hold generally:

• The minority carrier diffusion equations are derived from the continuity equations, specifically for minority carriers under certain conditions (small E-field, low-level injection, uniform doping profile):

L

p

nL

n

n Gp

x

xJ

qt

pG

n

x

xJ

qt

n

)(1

)(1

L

p

nnP

nL

n

pp

N

pG

p

x

pD

t

pG

n

x

nD

t

n

2

2

2

2


Drift-Diffusion Modeling

Outline of the Lecture

• Classification of PDEs

• Why Numerical Analysis?

• Numerical Solution Sequence

• Flow-Chart of Equilibrium Poisson Equation Solver

• Discretization of the Continuity Equation

• Numerical Solution Techniques for Sparse Matrices

• Flow-Chart of 1D Drift-Diffusion Simulator

Classification of PDEs

Classification of PDEs

Different mathematical and physicalbehaviors: Elliptic Type Parabolic Type Hyperbolic Type

System of coupled equations for several variables: Time : first-derivative (second-derivative for wave

equation) Space: first- and second-derivatives

Classification of PDEs (cont.)

General form of second-order PDEs ( 2 variables)

PDE Model Problems

Hyperbolic (Propagation)

• Advection equation (First-order linear)

• Wave equation (Second-order linear )

PDE Model Problems (cont.)

Parabolic (Time- or space-marching)

• Burger’s equation (Second-order nonlinear)

• Fourier equation (Second-order linear )

(Diffusion / dispersion)

PDE Model Problems (cont.)

Elliptic (Diffusion, equilibrium problems)

• Laplace/Poisson (second-order linear)

• Helmholtz equation

Boundary and InitialConditions

R

s

n

R

Initial conditions: starting point for propagation problems

Boundary conditions: specified on domain boundaries to provide the interior solution in computational domain

Numerical Methods

Complex geometryComplex equations (nonlinear, coupled)Complex initial / boundary conditions

No analytic solutionsNumerical methods needed !!

Why Numerical Analysis?

Coupling of Transport Equations to

Poisson and Band-Structure Solvers

D. Vasileska and S.M. Goodnick, Computational

Electronics, published by Morgan & Claypool , 2006.

Drift-Diffusion Approach

Constitutive Equations

• Poisson

• Continuity Equations

• Current Density Equations

1

1

J

J

n n

p p

nU

t q

pU

t q

D AV p n N N

( ) ( )

( ) ( )

n n n

p p p

dnJ qn x E x qD

dx

dnJ qp x E x qD

dx

S. Selberherr: "Analysis and

Simulation of Semiconductor

Devices“, Springer, 1984.

Poisson/Laplace Equation

SolutionPoisson/Laplace Equation

No knowledge of solving of PDEs

Method of images

With knowledge for solving of PDEs

Theoretical Approaches

Numerical Methods:

finite difference

finite elements

Poisson

Green’s function method

Laplace

Method of separation of variables

(Fourier analysis)

Numerical Solution Sequence

Numerical Solution Details

Governing

Equations

ICS/BCS

Discretization

System of

Algebraic

Equations

Equation

(Matrix)

Solver

Approximate

Solution

Continu

ous

Solution

s

Finite-

Difference

Finite-

Volume

Finite-

Element

Spectral

Discret

e Nodal

Values

Tridiago

nal

SOR

Gauss-

Seidel

Krylov

Multigrid

φi (x,y,z,t)

p (x,y,z,t)

n (x,y,z,t)

D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes,

Arizona State University, Tempe, AZ.

What is next?

• MESH

• Finite Difference Discretization

• Boundary Conditions

MESH TYPE

The course of action taken in three steps is dictated by the nature of the problem

being solved, the solution region, and the boundary conditions. The most commonly

used grid patterns for two-dimensional problems are

Common grid patterns: (a) rectangular grid, (b) skew grid, (c) triangular grid, (d)

circular grid.

Finite Difference Schemes

Before finding the finite difference solutions to specific PDEs, we will look at how

one constructs finite difference approximations from a given differential equation. This

essentially involves estimating derivatives numerically. Let’s assume f(x) shown below:

Estimates for the derivative of f (x) at P using forward, backward, and central

differences.


We can approximate derivative of f(x), slope or the tangent at P by the slope of the arc

PB, giving the forward-difference formula,

x

xfxxfxf

)()()( 00

0

or the slope of the arc AP, yielding the backward-difference formula,

x

xxfxfxf

)()()( 00

0

or the slope of the arc AB, resulting in the central-difference formula,

x

xxfxxfxf

2

)()()( 00

0


We can also estimate the second derivative of f (x) at P as

x

xxfxf

x

xfxxf

x

x

xxfxxfxf

)()()()(1

)2/()2/()(

0000

000

or

2

0000

)()(2)()(

x

xxfxfxxfxf

Any approximation of a derivative in terms of values at a discrete set of points is

called finite difference approximation.


)()(!3

1)()(

!2

1)()()( 0

3

0

2

000 xfxxfxxfxxfxxf

The approach used above in obtaining finite difference approximations is rather

intuitive. A more general approach is using Taylor’s series. According to the wellknown

expansion,

)()(!3

1)()(

!2

1)()()( 0

3

0

2

000 xfxxfxxfxxfxxf

and

Upon adding these expansions,

4

0

2

000 )()()()(2)()( xxfxxfxxfxxf

where O(x)4 is the error introduced by truncating the series. We say that this error is of the

order (x)4 or simply O(x)4. Therefore, O(x)4 represents terms that are not greater than (

x)4. Assuming that these terms are negligible,

2

0000

)()(2)()(

x

xxfxfxxfxf


)()(!3

1)()(

!2

1)()()( 0

3

0

2

000 xfxxfxxfxxfxxf

)()(!3

1)()(

!2

1)()()( 0

3

0

2

000 xfxxfxxfxxfxxf

3

000 )()()(2)()( xxfxxxfxxf

Subtracting

from

We obtain

and neglecting terms of the order (x)3 yields

x

xxfxxfxf

2

)()()( 00

0

This shows that the leading errors of the order (x)2. Similarly, the forward and backward

difference formula have truncation errors of O(x).

Poisson Equation

• The Poisson equation is of the following general form:

It accounts for Coulomb carrier-carrier interactions in the

Hartree approximation

It is always coupled with some form of transport simulator

except when equilibrium conditions apply

It has to be frequently solved during the simulation procedure

to properly account for the fields driving the carriers in the

transport part

There are numerous ways to numerically solve this equation

that can be categorized into direct and iterative methods

)()(2 rfr

Poisson Equation Linearization

• The 1D Poisson equation is of the form:

2

2

exp exp( / )

exp exp( / )

D A

F ii i T

B

i Fi i T

B

d ep n N N

dx

E En n n V

k T

E Ep n n V

k T

Finite Difference Representation

1 1 1

1 12 2 2

1 2 1

( ) ( )

n n n

i i i i i

n

i i i i i i

n p

p n C p n

Equilibrium:

exp( ), exp( )n n

i i i in p

Non-Equilibrium:

n calculated using PM coupling and p still calculated as in equilibrium case (quasi-equilibrium approximation)

Boundary Conditions

• There are three types of boundary conditions that are specified during the discretization process of the Poisson equation:

Dirichlet (this is a boundary condition on the potential)

Neumann (this is a boundary condition on the derivative of the potential, i.e. the electric field)

Mixed boundary condition (combination of Dirichlet and Neumann boundary conditions)

• Note that when applying the boundary conditions for a particular structure of interest, at least one point MUST have Dirichlet boundary conditions specified on it to get the connection to the real world.

1D Discretization

• The resultant finite difference equations can be represented in a

matrix form Au= f, where:

x0 x1 x2 x3 x4

)(

2;

2;

)(

2

where

10

0

0

2

,),,,,,(,),,,,,(

444

333

222

111

00

543210543210

ei

wi

ei

iwi

ei

iei

wi

wi

idxdxdx

cdxdx

bdxdxdx

a

cba

cba

cba

cba

cb

fffff

A

fu

Neumann Dirichletx5

2D Discretization

• In 2D, the finite-difference discretization of the Poisson equation

leads to a five point stencil:

N=5,M=4

Dirichlet: 0,4,5,9,10,14,15,19

Neuman: 1,2,3,16,17,18

xi-1 xi xi+1

xi-N

xi+N

widx

eidx

sidx

nidx

xi-1 xi xi+1

xi-N

xi+N

widx

eidx

sidx

nidx

0 1 2 3 4

5 6 7 8 9

10 11 12 13 14

15 16 17 18 19

Va

0 1 2 3 4

5 6 7 8 95 6 7 8 9

10 11 12 13 1410 11 12 13 14

15 16 17 18 1915 16 17 18 19

Va

2D Discretization (cont’d)

Dirichlet: 0,4,5,9,10,14,15,19

Neuman: 1,2,3,16,17,18

0

0

0

0

1

2

2

2

1

1

1

1

1

1

2

2

2

1

18

17

16

13

12

11

8

7

6

3

2

1

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

18181818

17171717

16161616

1313131313

1212121212

1111111111

88888

77777

66666

3333

2222

1111

f

f

f

V

f

f

f

V

f

f

f

V

f

f

f

V

dcba

dcba

dcba

edcba

edcba

edcba

edcba

edcba

edcba

edcb

edcb

edcb

a

a

a

a

Flow-Chart of Equilibrium

Poisson Equation Solver

Initialize parameters:

-Mesh size

-Discretization coefficients

-Doping density

-Potential based on charge neutrality

Solve for the updated potential

given the forcing function using LU decomposition

Update:

- Central coefficient of the linearized Poisson Equation

- Forcing function

Test maximum

absolute error update

Equilibrium solver

VA = VA+V

Calculate coefficients for:

- Electron continuity equation

- Hole continuity equation

- Update generation recombination rate

Solve electron continuity equation using LU decomposition

Solve hole continuity equation using LU decomposition

Update:


- Forcing function



Test maximum


Maximum voltage exceeded?

Calculate current

STOP yes

no

> tolerance

< tolerance

Non-Equilibrium solver

> tolerance

< tolerance

V is a fraction of the thermal voltage VT

Discretization of the Continuity

Equation

Sharfetter-Gummel Discretization Scheme

• The discretization of the continuity equation in conservative form requires the knowledge of the current densities

on the mid-points of the mesh lines connecting neighboring grid nodes. Since solutions are available only on the grid nodes, interpolation schemes are needed to determine the solutions.

• There are two schemes that one can use:

(a)Linearized scheme: V, n, p, and D vary linearly between neighboring mesh points

(b) Scharfetter-Gummel scheme: electron and hole densities follow exponential variation between mesh points

peDExepxJneDExenxJ

ppp

nnn

)()()()(

• Within the linearized scheme, one has that

• This scheme can lead to substantial errors in regions of high electric fields and highly doped devices.

2/12/11

2/12/12/1

iii

iiiii neD

a

VVenJ

21 ii nn

i

ii

a

nn 1

i

i

i

iiii

i

i

i

iiiii

a

eD

a

VVen

a

eD

a

VVenJ

2/112/1

2/112/112/1

2

2

(a) Linearized Scheme

(b) Sharfetter-Gummel Scheme• One solves the electron current density equation

for n(V), subject to the boundary conditions

• The solution of this first-order differential equation leads to

x

V

V

neD

a

VVen

x

neD

a

VVenJ

ii

iii

ii

iiii

2/11

2/1

2/11

2/12/1

11

)(and)(

iiii

nVnnVn

Vt

VVBn

Vt

VVBn

a

eDJ

e

eVgVgnVgnVn

iii

iii

i

ii

VtVV

VtVV

iiii

i

111

2/12/1

/)(

/)(

11

1)(),()(1)(

1

1)(

xe

xxB is the Bernouli function

Solution of the Coupled DD

Equations

There are two schemes that are used in solving the

coupled set of equations which comprises the Drift-

Diffusion model:

– Gummel’s method

– Newton’s method

Gummel’s relaxation method, which solves the equations with

the decoupled procedure, is used in the case of weak coupling:

• Low current densities (leakage currents, subthreshold regime),

where the concentration dependent diffusion term in the current

continuity equation is dominant

• The electric field strength is lower than the avalanche threshold,

so that the generation term is independent of V

• The mobility is nearly independent of E

The computational cost of the Gummel’s iteration is one matrix

solution for each carrier type plus one iterative solution for the

linearization of the Poisson Equation

Gummel’s Method

Gummel’s Method (cont’d)The solution strategy when using Gummel’s relaxation

scheme is the following one:

• Find the equilibrium solution of the linearized Poisson

equation

• After the solution in equilibrium is obtained, the applied

voltage is increased in steps V VT

• Now the scaled Poisson equation becomes:

i

DAi

i

n

NNVV

N

n

xd

Vd

VVVN

n

xd

Vd

expexp

expexp

2

2

2

2

i

DApn

i

n

NNVV

N

n

xd

Vdexp)exp(exp)exp(

2

2

Gummel’s Method (cont’d)The 1D discretized electron current continuity equation (as

long as Einstein’s relations are valid) is:

For holes, one can obtain analogous equations by

substituting:

02

1

11111

21

11121

iiiiiiiiii

i

iiiiiii

i

RGaaVVBnVVBna

D

VVBnVVBna

D

/

/

pnVV ,

The decoupled iteration scheme goes as follows:

(1) Solve the Poisson equation with a guess for the quasi-

Fermi levels (use the applied voltage as initial guess)

(2) The potential is used to update the Bernouli functions

(3) The above equations are solved to provide an update for

the quasi-Fermi levels, that enter into the Poisson

equation

Gummel’s Method (cont’d)The criterion for convergence is:

In the case of strong coupling, one can use the extended

Gummel’s scheme

kTT

kk

kTT

kk

kk

p

pVV

V

Vpp

n

nVV

V

Vnn

VVV

lnexp

lnexp

1

1

1

1

1

111

k

k

Tk

k

Tkk

p

pV

n

nVVV lnmax,lnmax,max

Gummel’s Method (cont’d)

initial guess

of the solution

solve

Poisson’s eq.

Solve electron eq.

Solve hole eq.

nconverged?

converged?n

y

y

initial guess

of the solution

solve

Poisson’s eq.

Solve electron eq.

Solve hole eq.

nconverged?

converged?n

y

y

initial guess

of the solution

Solve Poisson’s eq.

Electron eq.

Hole eq.

Update

generation rate

nconverged?

converged?n

y

y

initial guess

of the solution

Solve Poisson’s eq.

Electron eq.

Hole eq.

Update

generation rate

nconverged?

converged?n

y

y

Original Gummel’s scheme Modified Gummel’s scheme

• The three equations that constitute the DD model, written in residual form are:

• Starting from an initial guess, the corrections are calculated by solving:

0),,( 0),,( 0),,( pnvFpnvFpnvF pnv

p

n

v

ppp

nnn

vvv

F

F

F

p

n

v

p

F

n

F

v

Fp

F

n

F

v

Fp

F

n

F

v

F

kkk

kkk

kkk

ppp

nnn

VVV

1

1

1

Newton’s Method

Newton’s Method (cont’d)

p

n

V

n

Fp

F

n

F

F

F

F

p

n

V

p

F

n

F

V

Fn

F

V

FV

F

n

vv

p

n

v

ppp

nn

v

000

00

0

0

00

• The method can be simplified by the following iterative scheme:

111

11

1

kpkpp

kp

kknn

kv

kvkvv

kv

nn

FV

V

FFp

p

F

pp

FnV

V

FFn

V

F

pp

Fn

n

FFV

V

F

k+1 k

Flow-Chart of 1D Drift-

Diffusion Simulator

Initialize parameters:

-Mesh size

-Discretization coefficients

-Doping density

-Potential based on charge neutrality



Update:


- Forcing function

Test maximum


Equilibrium solver

VA = VA+V

Calculate coefficients for:

- Electron continuity equation

- Hole continuity equation

- Update generation recombination rate

Solve electron continuity equation using LU decomposition

Solve hole continuity equation using LU decomposition

Update:


- Forcing function



Test maximum


Maximum voltage exceeded?

Calculate current

STOP yes

no

> tolerance

< tolerance

Non-Equilibrium solver

> tolerance

< tolerance

V is a fraction of the thermal voltage VT

Hydrodynamic Modeling

• In small devices there exists non-stationary transport and carriers are moving through the device with velocity larger than the saturation velocity

– In Si devices non-stationary transport occurs because of the different order of magnitude of the carrier momentum and energy relaxation times

– In GaAs devices velocity overshoot occurs due to intervalley transfer

T. Grasser (ed.): "Advanced Device Modeling and Simulation“, World Scientific Publishing

Co., 2003, ISBN: 9-812-38607-6 M.

M. Lundstrom, Fundamentals of Carrier Transport, 1990.

http://www.iue.tuwien.ac.at/books/adv_dev_mod

Velocity Overshoot in Silicon

-5x106

0

5x106

1x107

1.5x107

2x107

2.5x107

0 0.5 1 1.5 2 2.5 3 3.5 4

1 kV/cm5 kV/cm10 kV/cm50 kV/cm

time [ps]

Dri

ft v

elo

city

[cm

/s]

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3 3.5 4

1 kV/cm5 kV/cm10 kV/cm50 kV/cm

Ene

rgy [e

V]

time [ps]

Scattering mechanisms:

• Acoustic deformation potential scattering

• Zero-order intervalley scattering (f and g-

phonons)

• First-order intervalley scattering (f and g-

phonons)

g

f

kz

kx

ky

g

f

g

ff

kz

kx

ky

X. He, MS Thesis, ASU, 2000.

How is the Velocity Overshoot Accounted

For?

• In Hydrodynamic/Energy balance modeling

the velocity overshoot effect is accounted for

through the addition of an energy conservation

equation in addition to:

– Particle Conservation (Continuity Equation)

– Momentum (mass) Conservation Equation

Hydrodynamic Model due to

BlotakjerConstitutive Equations: Poisson +• More convenient set of balance equations is in terms of n, vd

and w:

colld

d

Bdd

coll

d

dddd

colld

t

we

vmw

kn

nw

t

w

tm

e

vnmnwnm

mmt

t

nn

t

n

)(

2

*

3

2

)(

*

*2

1

*3

2)*(

*

)(

2

2

vE

vv

vE

vvv

v

Closure

• To have a closed set of equations, one either:

(a) ignores the heat flux altogether

(b) uses a simple recipe for the calculation of the heat flux:

)(*2

5,

2

wvm

nTkTn B q

• Substituting T with the density of the carrier energy, the

momentum and energy balance equations become:

colld

dB

dd

coll

dddd

d

t

nwen

vmwk

nwnt

nw

t

nenvnmnwn

t

n

)(

*2

1

3

2

)(*

2

1

3

2)(

2

2

vE

vv

pEpv

p

Momentum Relaxation Rate

• The momentum rate is determined by a steady-state MC

calculation in a bulk semiconductor under a uniform bias

electric field, for which:

dp

dpcoll

dd

vm

eEw

wm

e

tm

e

t

*)(

0)(**

v

EvEv

K. Tomizawa, Numerical Simulation Of

Submicron Semiconductor Devices.

Energy Relaxation Rate

• The emsemble energy relaxation rate is also determined by a

steady-state MC calculation in a bulk semiconductor under a

uniform bias electric field, for which:

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K. Tomizawa, Numerical Simulation Of

Submicron Semiconductor Devices.

Date post:	26-Nov-2015
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Module 2,3 & 4

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