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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
24 October 2013 10
24 October 2013 11
24 October 2013 12
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
24 October 2013 16
24 October 2013 17
24 October 2013 18
24 October 2013 19
24 October 2013 20
24 October 2013 21
24 October 2013 22
24 October 2013 23
24 October 2013 24
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
24 October 2013 37
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
Newton’s Method Example
• For x = 10:
• This is the new value of x
October 24, 201359
Newton’s Method Example
Initial guess =
10
y = 1350y’ = 580
New x-value = 7.6724
October 24, 201360
Newton’s Method Example
• For x = 7.6724:
October 24, 201361
Newton’s Method Example
Initial value = 7.6724
y = 368.494y’ = 279.621
New x-value = 6.3546
October 24, 201362
Newton’s Method Example
• 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 …
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 …
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 …
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 …
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 …
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 …
LGATE = 10 μm 5 μm 2.5 μm 1.3 μm 0.63 μm 0.25 μm 22 nm ….
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
EE130/230M Spring 2013 Lecture 6, Slide 140
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
EE130/230M Spring 2013 Lecture 6, Slide 141
EE130/230M Spring 2013 Lecture 6, Slide 142
• 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
EE130/230M Spring 2013 Lecture 6, Slide 143
• 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
EE130/230M Spring 2013 Lecture 6, Slide 144
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.
Finite Difference Schemes
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
Finite Difference Schemes
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.
Finite Difference Schemes
)()(!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
Finite Difference Schemes
)()(!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:
- Central coefficient of the linearized Poisson Equation
- Forcing function
Solve for the updated potential
given the forcing function using LU decomposition
Test maximum
absolute error update
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
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:
- Central coefficient of the linearized Poisson Equation
- Forcing function
Solve for the updated potential
given the forcing function using LU decomposition
Test maximum
absolute error update
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.
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:
0
0
)(
0)(
ww
ew
wwet
we
t
w
dw
wdcoll
d
vE
vEvE
K. Tomizawa, Numerical Simulation Of
Submicron Semiconductor Devices.