Technical report, IDE1066, February 2011
Model of MOSFET in Delphi
Master’s Thesis in Microelectronics and Photonics
Andrey Prokhorov
Olesya Gerzheva
School of Information Science, Computer and Electrical Engineering
2
Acknowledgment
We want to thank our supervisors: Docent Ying Fu from the Royal institute of Technology
for his valuable suggestions which helped us greatly during the process of preparing and writing
this thesis; and we also want to thank Doctor Lars Landin who supervised us in the beginning of
work.
Thanks to Håkan Pettersson, the Head of the Dept. of Mathematics, Physics and Electrical
Engineering, and main teacher in semiconductor physics, for his help and advice.
Thanks to all members of our families for their support and encouragement.
3
Abstract
In modern times the increasing complexity of transistors and their constant decreasing
size require more effective techniques to display and interpret the processes that are inside of
devices.
In this work, we are modeling a two‐dimensional n‐MOSFET with a long channel
and uniformly doped substrate. We assume that this device is a large geometry device so that
short‐channel and narrow‐width effects can be neglected.
As a result of the thesis, a demonstration program was built. In this executable file, the
user can choose parameters of the MOSFET‐model: drain and gate voltage, and different
geometrical parameters of the device (junction depth and effective channel length). In the
advanced regime of the program, the user can also specify the model re‐calculation parameter,
doping concentration in n
+
and bulk regions. The program shows the channel between the
source and drain region with surface diagrams of carrier density and potential energy as an
output. It is possible to save all calculated results to a file and process it in any other program,
for example, plot graphics in Matlab or Matematica.
The model can be used in lectures that are related to semiconductor physics in order to
explain the basic working mechanisms of MOSFETs as well as for further detailed analysis of the
processes in MOSFETs. It is possible to use our modeling techniques to rebuild the model in
another computer language, or even to build other models of transistors, performing similar
calculations and approximations.
It is possible to download the executable file of the model here:
http://studentdevelop.com/projects/MOSFET_model.zip
Keywords
MOSFET, semiconductor, model, Delphi, channel, carrier concentration, potential energy.
4
Table of Contents
Acknowledgment .......................................................................................................................................... 2
Abstract ......................................................................................................................................................... 3
1. Basic theory of semiconductors ................................................................................................................ 5
1.1 Energy bands ....................................................................................................................................... 5
1.2 Intrinsic semiconductor ....................................................................................................................... 5
1.2 Carrier concentration and Fermi level ................................................................................................ 6
2. The MOSFET .............................................................................................................................................. 8
2.1 Introduction......................................................................................................................................... 8
2.2 Energy band diagrams ......................................................................................................................... 9
2.3 The MOSFET ...................................................................................................................................... 12
2.4 Characteristics of the MOSFET .......................................................................................................... 13
2.5 Operating regions of the MOSFET ..................................................................................................... 14
3. Modeling of MOSFET ............................................................................................................................... 17
3.1 Numerical methods ........................................................................................................................... 17
3.2 Description of the model................................................................................................................... 19
3.2.1 Programming model ................................................................................................................... 19
3.2.2 Mathematical model .................................................................................................................. 21
4. Development environment Delphi .......................................................................................................... 23
5. Results ..................................................................................................................................................... 24
6. Conclusion ............................................................................................................................................... 29
References ................................................................................................................................................... 30
Appendix ..................................................................................................................................................... 31
Program code of the model in Delphi ..................................................................................................... 31
Download the model ............................................................................................................................... 35
5
1. Basic theory of semiconductors
1.1 Energy bands
Semiconductor materials are the basis of modern electronics. Different kinds of transistors,
diodes and solar cells are made of semiconductors. The most popular semiconductor material is
silicon. Atoms in a silicon crystal have four valence electrons to share with four nearest
neighbors. Electrons of an isolated atom may occupy only certain discrete energy levels. If two
atoms in a semiconductor move closer to each other then energy levels split to accommodate all
electrons in the system.
Usually, the system has a large number of atoms and the higher energy levels tend to unite
into two separate bands of allowed energies, called the Conduction band and the Valence band,
respectively [1]. The Conduction band – is the upper band, where energy levels are almost
empty. Energy level E
c
– is the bottom of the conduction band. The Valence band – is the lower
band where energy levels are full. Energy level
E
V
– is the top of the valence band. The
difference between two of these levels is called the bandgap energy (
g
E
)[2].
The bandgap energy of most semiconductors decreases as the temperature increases
(Eq.1.1) [1].
⎪
⎩
⎪
⎨
⎧
≤⋅⋅−
>>⋅⋅−
≥⋅−
=
−
−
−
K) 170 T(for T106.05  1006.117.1
K) 170 T300K (for T103.05  1003.9179.1
K) 250 T(for 1073.2206.1
)(
275
275
4
T
T
T
TE
g
, (1.1)
where
T
– is the temperature, (K).
1.2 Intrinsic semiconductor
A pure semiconductor without dopant species added is called undoped or
intrinsic
semiconductor.
Intrinsic semiconductor has the same number of electrons in the conduction band
as the number of holes in the valence band, at a given temperature.
,
i
nnp
=
=
(1.2)
Where
p
– is the free hole concentration, (cm
3
);
n
– is the free electron concentration, (cm
3
);
i
n
– is the intrinsic carrier concentration, (cm
3
).
The formula 1.3 shows the intrinsic carrier concentration as a function of the temperature.
6
]
2
)(
2
)(
exp[))(()(
0
0
2/3
0
0
kT
TE
kT
TE
T
T
TnTn
gg
ii
+−=, (cm
3
) (1.3)
where
T
0
– is the nominal temperature (
T
0
= 300
K
) [1].
1.2 Carrier concentration and Fermi level
If we consider an intrinsic case without impurities added to the semiconductor, then the
number of electrons (occupied conduction band levels) is given by the total number of states
)(EN multiplied by the occupancy )(EF, integrated over the conduction band:
∫∫
∞∞
==
CC
EE
dEEFENdEEnn,)()()(
(1.4)
)(En
– is the electron density [(cm
3
eV)
1
],
)(EN –
is the density of allowed energy states [(cm
3
eV)
1
],
)(EF
– is the FermiDirac distribution function.
C
n
EE
h
m
EN −=
2
3
2
)
2
(4)( π
(1.5)
where
n
m – is the effective mass of the electrons;
h
– is the Planck constant;
C
E
‐
is the conduction band edge.
The FermiDirac distribution function )(EF shows the probability of electron occupation
of an electronic state with energy E.
kTEE
F
e
EF
/)(
1
1
)(
−
+
=
(1.6)
The energy at which the probability of occupation by an electron is
2
1
called the Fermi
energy (
F
E
).
Large numbers of states are allowed in the conduction and valence band. However, there
would not be many electrons in the conduction band for an intrinsic semiconductor because
electrons prefer states with lower energy that are in the valence band. Therefore, an electron can
occupy one of these upper states with a very small probability. Most of the allowed states in the
valence band will be occupied by electrons. Hence, the electron can occupy one of these states
with a probability near one. Unoccupied electron states in the valence band are referred to as
holes. Substituting Eq. 1.5 and Eq. 1.6 into Eq. 1.7 we can find the carrier concentration [2].
7
∫
∞
−
+
−
=
C
F
E
kTEE
C
n
e
dEEE
h
m
n
/)(
2
3
2
1
)
2
(4π
(1.7)
If kTEEx
C
/)(
−≡
, where
x
– is the carrier energy in units of
kT
,
and kTEE
CF
/)(
−
≡
η
, where
η
ₖ猠瑨攠䙥rm椠汥癥氠楮⁵n楴猠潦=
kT
, then Eq.7 becomes
∫
∞
+−
⋅
=
0
2
1
2
3
2
1)exp(
)
2
(4
η
π
x
dxx
h
kTm
n
n
. (1.8)
By collecting up parameters, we can express the electron concentration as:
),(
2
2/1
η
π
FNn
C
⋅=
(1.9)
where
2/3
2
)
2
(2
h
kTm
N
n
C
π
≡
(1.10)
C
N – is the effective density of states in the conduction band;
)(
2/1
η
F – is the Fermi – Dirac integral of the order of
2
1
.
We can find the Fermi – Dirac integral for the two cases, when
1−<<
η
and
1>>
η
:
⎪
⎪
⎩
⎪
⎪
⎨
⎧
>>+−++
−<<+−
=
−
−
1...)4267.0125.01(
3
2
1...])exp()21(exp[
2
)(
22
2
3
2
3
2
1
ηηηπη
ηηη
π
η
if
if
F
(1.11).
The approximate FermiDirac integral of the order of
2
1
for the value of
η
Ⱐ潶敲⁴桥慮ge=
(
∞−
to
∞+
) will be [7]:
)exp(43
2
)(
8
3
2
1
ηπ
π
η
−+
≈
−
a
F
, (1.12)
where
(
)
㔰ΦΦ1(ㄷ.0數瀨㘸.016.㌳
24
++−−+= ηηη
a. (1.13)
8
2. The MOSFET
2.1 Introduction
A MOS (
M
etal
O
xide
S
emiconductor) diode is a structure where a thin layer of oxide is
grown on top of semiconductor substrate, and after that, a metal layer is deposited on the oxide,
as is shown in Fig. 2.1.
Fig.2.1 Crosssection of a MOS diode
In the MOS diode the voltage applied to the gate controls the state of the Sisurface
underneath. There are two states of the MOS diode that can be used to make a voltagecontrolled
switch – accumulation and inversion. The MOS diode is in an accumulation state when a
negative gate voltage is applied, that attracts the holes from the ptype silicon to the surface; and
in the inversion state when a positive voltage (larger than the threshold voltage) is applied,
creating an inverted layer of electrons at the surface. The threshold voltage is the gate voltage
when the channel just starts to form at the oxidesubstrate interface.
There are two modes of the switch: on and off that correspond to the existence or absence
of the electron layer (the channel). When the gate voltage is below the threshold voltage there is
no channel and the source and drain n
+
regions are isolated by the ptype substrate. This is the
offmode of the switch. When the gate voltage is higher than the threshold voltage (onmode) the
current flows through the surface and the channel appears [4].
9
2.2 Energy band diagrams
The energy band diagrams of the three separate components (metal, oxide and
semiconductor) of the MOS diode are shown in the Fig. 2.2, where:
0
E – is the vacuum energy
level of free electrons. The parameter
ox
E
– is the bandgap of
2
SiO
, typically )0.90.8(
−
=
ox
E
eV
.
m
q
Φ⋅
– is the work function of the metal.
Figure 2.2 The energy band diagrams of MOS diode components
The work function is the energy that must be given to an electron to pass over the surface
energy barrier, in other words, across the energy difference between the vacuum level
0
E and
the Fermi energy of the metal
fm
E
:
fmm
EEq
−
=
Φ
⋅
0
. (2.1)
For aluminum the work function is: 1.4
=
Φ
⋅
m
q
eV
.
Electron affinity (
χ
⋅q
) – is the energy difference between the vacuum level
0
E and the
conduction band edge
c
E at the surface:
c
EEq
−
=
⋅
0
χ
††††††††††††††††††††††††⠲⸲⤠
10
The affinity is a property of a material and it is not affected by the presence of impurities or
imperfections.
s
q
χ
⋅
– is the electron affinity in the semiconductor, its quantity varies as a
function of doping. For silicon, the affinity is 05.4
=
⋅
s
q
χ
eV
.
s
q
Φ⋅
– is the work function of the semiconductor:
p
g
ss
q
E
qq
ϕχ ++⋅=Φ⋅
2
(eV) for ptype, (2.3)
where
p
ϕ
– is the Fermi potential for ptype silicon.
n
g
ss
q
E
qq
ϕχ ++⋅=Φ⋅
2
(eV) for ntype, (2.4)
Where:
n
ϕ
– is Fermi potential for ntype silicon. For the same doping concentration:
ϕϕϕ ==
np
(2.5)
)ln(
i
b
t
n
N
V=ϕ
(V), (2.6)
where:
q
kT
V
t
=
– is the thermal voltage,
b
N – is the substrate doping concentration.
For ptype silicon with an acceptor concentration
315
10
−
=
cmN
b
and 29.0=⋅
ϕ
q
eV
, the
work function will be 90.4=Φ⋅
s
q
eV
. When three components of the MOS structure are
connected, the work function can be determined only by the difference between metal and
semiconductor parts [1].
The energy band diagram of an ideal ptype MOS diode at V=0 is shown in Fig. 2.3. At
zero bias, the energy difference between the metal work function
m
q Φ
⋅
and the work function
s
q Φ⋅ is zero or, in other words, the work function difference
ms
q
Φ
⋅
is zero.
0)
2
()( =⋅++⋅−Φ⋅=Φ⋅−Φ⋅≡Φ⋅
b
g
msmms
q
E
qqqqq ψχ, (2.7)
where the sum of the three variables in the brackets is equal to
s
q Φ⋅.
11
qΦ
s
E
g
/2
qχ
s
VACUUM
LEVEL
METAL
ALUMINIUM
E
f
qΦ
m
E
f
E
c
E
i
INSULATOR
(SILICON DIOXIDE)
SEMICONDUCTOR
(SILICON pTYPE)
d
qψ
B
Figure 2.3 Energy band diagram of an ideal MOS diode at V=0
12
2.3 The MOSFET
A MOSFET –
M
etal
O
xide
S
emiconductor
F
ield
E
ffect
T
ransistor (Fig. 2.4) is a transistor
based on the MOS diode. On the top of the oxide, a gate electrode is deposited (a conducting
layer of metal). Under the oxide and inside the substrate there are two heavily doped regions:
source and drain. The sourcetodrain electrodes are equivalent to two pn junctions that are
situated backtoback. The central MOS diode with the inverted channel connects the source and
drain junctions. The flow of charge carriers in the channel region between the source and the
drain is thus controlled by an electric field, hence the name MOSFET, created by a voltage
g
V
applied to the gate electrode.
The MOSFET may be nchannel or pchannel depending on the type of carriers in the
channel region [1]. In our MOSFET model, the channel contains electrons (nchannel), the
source and drain regions are heavily n
+
doped and the substrate is ptype.
Figure 2.4 Nchannel MOSFET diagram
13
When there is no voltage applied to the gate and there is no conduction channel between
the drain and source regions, the MOSFET is referred to as a normallyoff device (an
enhancementmode device). A certain minimum voltage – called a threshold (turnon) voltage
V
th
, should be applied to the gate to induce a conduction channel.
If a conduction channel exists between the source and the drain regions even at zero gate
voltage (normallyon device) – then it is called a depletionmode device. In this case, the current
flow is not exactly at the surface, some carriers are in the bulk of the silicon.
2.4 Characteristics of the MOSFET
Under normal operating conditions, the drain and source voltages should be applied in a
way that the source and draintosubstrate pn junctions will be reverse biased (i.e. a negative
voltage is applied to the pside with respect to the nside). There will be no significant current
until the voltage reaches the critical value called “the junction breakdown voltage” after which
the current dramatically increases.
In case when the source and bulk regions are grounded (V
b
=V
s
=V
sb
=0), depletion regions
are formed around the n
+
source and drain region (even when the gate voltage is zero) due to n
+

p junctions formed with the ptype substrate of concentration N
b
(cm
3
).
In fig. 2.5, a crosssection of an nchannel MOSFET is illustrated, where X
sd
and X
dd
are
the widths of the depletion region under the source and drain, respectively.
Figure 2.5 Crosssection of nchannel MOSFET
14
b
ii
ddsd
qN
XX
ϕεε
0
2
==
(cm) at V
ds
=V
bs
=0, (2.8)
where
i
ϕ
– is the builtin potential between the source/drain to substrate pn junction.
)ln(
i
bsd
ti
n
NN
V=ϕ
(V), (2.9)
where
q
kT
V
t
= –
is the thermal voltage,
N
sd
– is the concentration in the n
+
regions (cm
3
)
With positive voltage V
ds
applied to the drain contact, and positive gate voltage less than
the threshold voltage (V
gs
< V
th
), applied to the gate, the ptype region will be depleted under the
gate oxide. Holes are pushed away from the surface, therefore immobile negative bulk charge Q
b
appears at the silicon substrate.
If V
gs
> V
th
is applied to the gate, then a conduction channel with mobile negative charge Q
i
is formed at the surface.
This channel is called inversion layer because the surface layer is
inverted from ptype to ntype after formation of the channel. The thickness of this inversion
layer depends on the applied bias.
When V
gs
= V
th
, the concentration of the minority carriers (electrons) at the surface equals
that of the majority carriers (holes) in the bulk but the higher the V
gs
is
than V
th
the higher will be
minority charge density Q
i
(inversion charge) will be.
Q
g
=Q
i
+Q
b,,
(2.10)
where Q
g
– is the gate charge.
If there is a voltage difference between the source and the drain, then the current I
ds
will
flow through the channel, due to the drift of carriers from the source to the drain [1].
2.5 Operating regions of the MOSFET
Linear region
The linear region is a region where I
ds
increases linearly with V
ds
for a given V
gs
, which is
higher than V
th
. If a small drain voltage is applied, electrons will flow from the source to the
drain, therefore current will flow in the reverse direction (from the drain to the source) through
the conduction channel (see Fig. 2.6 a).
15
Saturation region
This is a region where I
ds
no longer increases with V
ds
, I
ds
is saturated. When the drain
voltage increases, eventually it reaches V
Dsat
; the thickness of the inversion layer x
i
near y=L is
reduced to zero. This is called the Pinchoff point (see Fig. 2.6 b). At this point, the drain current
remains the same, because for V
d
>V
Dsat
, at point P the voltage V
Dsat
remains the same. Therefore,
the number of carriers arriving to point P from the source remains the same (this is the same with
current arriving from the drain) [2]. The pinchedoff portion of the channel moves towards the
source end due to the widening of the drain depletion region. If the voltage V
Dsat
increases
beyond pinchoff, the pinchoff region between the channel pinchoff point and drain region
causes the effective channel length to decrease from L to L’ (see Fig. 2.6 c).
Breakdown region
When V
ds
increases even more, the transistor enters a region where I
ds
suddenly increases
until breakdown of the draintosubstrate pn junction occurs. The breakdown is caused by the
high electric field in the drain end. In short devices, this is called hotcarrier effect, due to the
high electric field at the drain end, and it can also result in device breakdown. [1].
Cutoff region
This is the region where V
gs
< V
th
for which no channel exist between the source and the
drain, and I
ds
=0.
16
Figure 2.6 Operating regions in the MOSFET
a) Linear region
b) Pinchoff point
c) Saturation region
17
3. Modeling of MOSFET
In order to describe the model that was built, we need to explain it from two different
points of view – practical and theoretical. That will be done in the following paragraphs
Programming model and Mathematical model respectively. Since all the mathematical
calculations were made by a program, we used numerical methods in our model. Numerical
methods will be described in the next section.
3.1 Numerical methods
First derivative
The derivative of a function f(x) is the limit where an increment of the function is divided
by an independent variable that goes to zero (vanish):
0 ,lim)( →Δ
Δ
Δ
=
∂
∂
=
′
x
x
y
x
y
xf (3.1)
We should replace the ratio
x
y
Δ
Δ
of infinitely small increments by the ratio of finite
differences in order to perform numerical solution of the derivative. Then, the smaller increment
of the argument we take, the more precise numerical value of the derivative we will get.
In the twopoint method for calculating derivatives two points are used that are
obtained by adding and subtracting Δx from the desirable point x, where the derivative should be
determined.
Fig. 3.1 Geometrical illustration of first derivative
18
According to Figure 3.1, we can write:
x
xxyxxy
x
y
x
y
Δ
Δ
−
−
Δ
+
=
Δ
Δ
≈
∂
∂
2
)()(
(3.2)
Let us write a formula for the case of an array
A
that has
n
elements [0..n]. The
values of
A
are written in such a way that the difference between indexes (x) and (x+1) is Δx, so
the formula (1) will look like:
2
)1()1(
−
−
+
≈
∂
∂
xAxA
x
A
(3.3)
Example:
If we take function y=x
2
and ∆x = 0.001, the first derivative will be:
2001.0
)001.0()001.0(
⋅
−
−
+
=
Δ
Δ
≈
∂
∂
=
′
xyxy
x
y
x
y
y,
If we want to find derivative of this function at the point x = 1, then:
2
002.0
)999.0()001.1(
)1(
2
2
=
−
=
Δ
Δ
≈
∂
∂
=
′
x
y
x
y
y
That could be proved by analytical solving: 212)1( ;2)(
2
=⋅=
′
=
′
=
′
yxxy
Second derivative
The second derivative is calculated as a derivative of the first derivative.
The rule is
2
v
uvvu
v
u
′
−
′
=
′
⎟
⎠
⎞
⎜
⎝
⎛
(3.4).
( )
2
2
2
4
))()(()2(2)()(
2
)()(
x
xxyxxyxxxxyxxy
x
xxyxxy
x
y
x
y
Δ
Δ−−Δ+⋅
′
Δ−Δ⋅
′
Δ−−Δ+
=
=
′
⎟
⎠
⎞
⎜
⎝
⎛
Δ
Δ−−Δ+
=
′
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
∂
∂
Finally we get the second derivative:
22
2
)()(2)(
x
xxyxyxxy
x
y
Δ
Δ−+⋅−Δ+
=
∂
∂
. (3.5)
19
3.2 Description of the model
3.2.1 Programming model
The programming model of the MOSFET is represented by several matrixes – 2D arrays of
main parameters of the MOSFET – Fermilevel
E
f
, carrier concentration N and potential
energy φ. The size of one matrix (number of elements) is limited by the performance of the CPU
and the memory size of the computer. We were working with the computer that has following
parameters: CPU 2300 Mhz and 2 Gb RAM, that allowed us to operate with matrixes
2000 x 1000 elements.
By trying to calculate values with different sizes of matrixes, we determined that the
optimal size of array for a demonstration model is 200 x 100 elements, since it is time of
working is more important that accuracy.
In every matrix, the following zones are described: source, drain, gate, their contacts and
oxide layer. Schematically, it is shown in the figure 3.2.
Figure 3.2. Model of MOSFET with geometrical parameters
In our model, every zone of the MOSFET will be represented by an area of cells in a two
dimensional array (figure 3.3). For example, the source contact will be described as rectangular
with coordinates (0, 0, 400, 100) – top left corner (x
1
,y
1
) and bottom right corner (x
2
,y
2
).
20
Figure 3.3. Representation of the model in computer memory
Each element of the matrix (a cell) represents a small unit inside the MOSFET. For
example, for 200 nmwidth transistor – the width of one unit will correspond to: 200 nm divided
by 2000 – which is 0.1 nm. For each element, the mathematic model is applied, i.e. in every
point carrier concentration n and electric field E are calculated.
The real MOSFET, as it is shown in the figure 3.4, is fabricated with much bigger contacts
and thicker oxide layer than was described in the model. However, for the model, the size of
contact and oxide layer do not affect any calculations, since in all points of oxide the carrier
density is determined as 0 (no charges) and in all points of metal contacts primitively ε is much
larger than in other regions (ε =1000 for metal and ε =12 for nregions).
Figure 3.4. Real ( fabrication) MOSFET structure
21
3.2.2 Mathematical model
The mathematical model of the MOSFET is described by formulas used to calculate the
main parameters: E
f
, n and φ. The Fermi level can be calculated from the doping concentration
N
D
that is given by the user through the program interface (in the bulk region and in nregions).
From the Fermi level, it is possible to calculate the carrier concentration n and from that
the potential energy.
To calculate the Fermi level, we used an approximation of the FermiDirac integral of the
order of
2
1
(see Eq. 1.12 and Eq. 1.13 in chapter 1):
)exp(43
2
)(
8
3
2
1
ηπ
π
η
−+
≈
−
a
F, where
[
]
{
}
㔰Φ1(ㄷ.0數e㘸.016.㌳
24
++−−+= ηηηa and
kTEE
CF
/)( −≡
η
.
Then, from the Fermi level, we can calculate the carrier concentration using Eq. 1.8 from
chapter 1
).(
2
2/1
η
π
FNn
C
⋅=
The potential energy can be found from the Poisson equation. It cannot be solved
analytically hence numerical methods should be used. In order to calculate the potential energy
( eE−=
ϕ
), the Laplace operator needs to be calculated:
2
2
2
2
2
2
2
z
E
y
E
x
E
EE
δ
δ
δ
δ
δ
δ
++=∇=Δ
. (3.6)
If we want to solve Poisson equation in a point (x,y), then we need E(x,y) at this point and
calculate the Laplace operator:
y
E
x
E
yxEyxE
2
2
2
2
2
),(),(
∂
∂
+
∂
∂
=∇=Δ
(3.7)
That means we need to find two second derivatives of
E
by x and by y. If we use the
formula (2) for second derivative, we can rewrite the equation (3):
222
2
2
2
)()(2)()()(2)(
y
yyEyEyyE
x
xxExExxE
y
E
x
E
Δ
Δ++⋅−Δ+
+
Δ
Δ−+⋅−Δ+
=
∂
∂
+
∂
∂
(3.8)
If we have a matrix a(i,j) where each element is a value of
),(
2
yxE∇
, and ∆x and ∆y is a
small value
Δ
, we finally we get:
22
),(
),(),(2),(
),(),(2),(
),(
2
22
2
2
2
2
jinA
jiEjiEjiE
jiEjiEjiE
j
E
i
E
jiE
⋅=
Δ
Δ−+⋅−Δ+
+
Δ
Δ−+⋅−Δ+
=
∂
∂
+
∂
∂
=∇
, (3.9)
where
0
2
εε
Δ
=
e
A .
We can simplify the equation (3.9):
2
),(),(),(),(4),(
),(
Δ
Δ−+
Δ
+
+
Δ
−
+
⋅
−
Δ+
=⋅
jiEjiEjiEjiEjiE
jinA
(3.10)
We can continue:
2
),()),(),(),(),((),(4 Δ⋅⋅−Δ−+Δ++Δ−+Δ+=⋅ jinAjiEjiEjiEjiEjiE
4
),(
4
),(),(),(),(
),(
2
jinAjiEjiEjiEjiE
jiE
Δ⋅
−
Δ−+Δ++Δ−+Δ+
=
)],(),(),(),([),(
0
2
yxNyxNyxnyxp
q
yxE
aD
−+−=∇
εε
In the bulk region: p(x,y) – N
a
(x,y) = 0
Eventually:
)],(),([),(
0
2
yxnyxN
q
yxE
D
−=∇
εε
In this step, we can find the electric field E(x,y) from the carrier concentration n.
0
2
4
)),(),((
4
),(),(),(),(
),(
εε
⋅
−Δ⋅
−
Δ−+Δ++Δ−+Δ+
=
jinjiNA
jiEjiEjiEjiE
jiE
D
(3.11)
Since the potential is:
][][),(
0
22
DD
Nn
e
NnAAeeEjiV −
Δ−
=−⋅−=−==
εε
ϕ,
0
22
εε
Δ−
=
e
AA
.
Then, finally, the formula to calculate the potential will be:
0
22
4
)),(),((
4
),(),(),(),(
),(
εε⋅
−Δ⋅
−
Δ−+Δ++Δ−+Δ+
=
jinjiNq
jiVjiVjiVjiV
jiV
D
.
23
4. Development environment Delphi
To build our MOSFET model we used the Borland Delphi 7 program. Delphi is a
programming language that originally was developed from the objectoriented Turbo Pascal
language.
Borland Delphi is a so called development environment, where a programmer can
comfortably make Windowsbased applications. The main window of Borland Delphi is shown
in figure 4.1.
Figure 4.1 Interface of Borland Delphi
We chose this development environment because we have had experience with this
program before and it is easy to build applications with it: the user just needs to drag elements to
a form. For example, we used lists, buttons, and comboboxed elements in our MOSFET model.
24
5. Results
The main window of our program is shown in the figure 5.1.
Figure 5.1 Main window of MOSFET model program
The program has two regimes: demonstration and advanced. The demonstration regime can
be used in lectures, for explanation of the basic working mechanism of the MOSFET. The
advanced regime can be used for detailed investigations of the MOSFET model. In the regime
the user can change many parameters of the model (figure 5.2).
Figure 5.2 Window of the MOSFET model program in the advanced regime
25
In the figures 5.3 – 5.5, the results of our calculations are shown for the donor
concentration N
D
, and carrier concentration n and potential energy φ, when n and φ parameters
were not recalculated (the first approximation).
Figure 5.3 Donor concentration N
D
In figure 5.3, the doping profile is shown for all regions of the MOSFET as a surface
diagram. It is clearly seen there that the three contacts have the highest doping concentration (N
D
= 10
20
cm
3
), the highly doped source and drain have N
D
=10
18
cm
3
, and the bulk region
(substrate) – N
D
=10
15
cm
3
. In the oxide part, the donor concentration goes down to zero, since it
is undoped.
26
Figure 5.4. First approximation of carrier concentration
Figure 5.5 First approximation of potential energy
The first approximation of the carrier concentration and the potential energy is shown in
figures 5.4 and 5.5. In these figures the source and the drain region have rectangular form. This
is the initial guess for the model which will be improved later. After 5 000 loops of re
calculations (iterations), when neighboring cells are considered, the form of the regions becomes
smooth, as shown in figures 5.6 and 5.7.
27
Figure 5.6 Potential energy after 5000 iterations
In figure 5.6, the result of the potential energy calculation is shown as a surface diagram,
with drain voltage V
D
= 0.15 V and gate voltage V
G
= 0.2 V. The potential energy at the drain is
lower than the energy at the source because of the additional voltage applied to drain.
Figure 5.7 Carrier concentration after 5000 iterations
28
After 5000 recalculations of the initial guess of the carrier concentration, the depletion region
appears; there were no depletion region in the first approximation. The result is shown in figure
5.7. The gate voltage moves holes away from the gate region. If the gateinducted vertical
electric field is strong enough, a channel between these two nregions will appear. We define in
the model, that the channel appears in the place where the concentration of electrons is greater
than 10
17
cm
3
, since the donor concentration in the bulk region is N
D
=10
15
cm
3
and n=10
18
cm
3
in the nregions.
29
6. Conclusion
We have built a model of the MOSFET that is possible to use as a demonstration program
for teaching and for further analysis. In the model it is possible to save the results of the
calculation to a file as a twodimensional array, where parameters are calculated in every point
of the device.
Large mathematical analysis was made and surface diagrams of the carrier concentration,
the potential energy and the donor concentration distributions were presented.
The channel distribution in the MOSFET was calculated and plotted. All calculation
processes are described in sufficient details to allow reconstitution of the model in another
computer language or building other models of transistors performing similar calculations and
approximations.
30
References
[1]
Narain Arora, MOSFET Modeling for VLSI Simulation: Theory and Practice, ISBN
13: 9789812568625, ISBN10: 981256862X, World Scientific Publishing Co. Pte.
Ltd., Singapore, p. 632, 2007.
[2]
S.M. Sze, Semiconductor devices: Physics and technology, SBN/ISSN: 0471333727,
2 Uppl., Wiley, New York viii, p. 564. , 2002.
[3]
S. M. Sze, Kwok K. Ng, Physics of Semiconductor Devices, ISBNI 3: 978047 11 4323
9, ISBN10: 0471143235, John Wiley & Sons, Inc., Hoboken, New Jersey, p. 5, 2007.
[4]
Sima Dimitrijev, Principles of Semiconductor Devices, ISBN10: 0195161130, ISBN
13: 9780195161137, Oxford University Press, USA, p. 578, 2005.
[5]
Umesh K. Mishra, Jasprit, Singh Semiconductor device physics and design, ISBN 9781
402064807, Dordrecht, The Netherlands: Springer, p. 560, 2008.
[6]
E. J. Farrell, S. E. Laux, P. L. Corson, E. M. Buturla Animation and 3D color display of
multiplevariable data: Application to semiconductor design.
[7]
Ying Fu, M. Willander Physical models of semiconductor quantum devices, ISBN 07923
84571, USA:
Springer, p. 266, 1999.
31
Appendix
Program code of the model in Delphi
program MOSFET;
uses
SysUtils, Math;
(* Comments:
V: the potential energy (eV); ND: doping profile (cm3)
W: dielectric constant, F: Fermi energy
cell size=0.1 nm, DM: carrier effective mass
*)
const q=1.6E19; (* C *)
delta =1E7; (* cm, step: 1 nm *)
x = 0.1; (* update parameter *)
mu= 1450; (* cm^2/Vs *)
eps0=8.85E14; (* permittivity, F/cm *)
k = 1.38E23; (* Boltzmann constant, J/K *)
h = 6.62E34; (* Planck constant , J*s *)
m0 = 9.1095E31; (* effective mass, kg *)
var
m :integer; //scaleparameter of matrix
IX:integer; //width of matrix
IY:integer; //height of matrix
Leff, Xj: string;
IX1,IX2,recount : integer;
ff,ff1,ff2,ff3:Text;
ND_n,NB : Real;
eps : array [1..2000,1..1000] of integer; //dialectic constant (permittivity)
V,ND,P,n : array [1..2000,1..1000] of real;
//Vthe potential energy (eV), NDdoping concentration (cm3),
//P  temp array, ncarrier concentration (cm3)
A,B,DM,F,T, min_v, breakdown_v : real;
VD, VG : Real; //VD  Drain voltage (V) and VG  Gate voltage (V)
k_channel : Integer = 0; //number of point in channel array
channel:array[1..2, 0..1000]of integer;
procedure FERMI(ND,DM,T:real; var F:Real);
var S,X,Y,Z: Extended;
j : LongInt;
begin
T:=300; (* temperature [K] *)
T:=T/1.1604E4; (* eV *)
Z:=0;
F:=0.5; //start value
X:=ND/(power((T*DM),1.5)*6.037E21);
S:=1.0E3/T;
F:=F/T;
F:=FS;
S:=S*0.5;
for j:=1 to 500000 do
32
begin
F:=F+S;
Y:=0.0;
if not(F>90.0) then
begin
Y:=Power(F,4)+33.6*F*(1.00.68*exp(0.17*power((1.0+F),2)));
Y:=1.0/(exp(F)+1.32934038675/power((50.0+Y),0.375))X;
if (J=1) then Z:=Y;
end;
if (Y*Z<=0) then break;
end;
F:=(F+0.5*S)*T;
end;
(* CC  *)
procedure CC(var A,DM,T,nu: real); //calc carrier concentration
var aa,n,FD : real;
begin
nu:=(nuA)/T;
n:=0;
if (nu <= 90) then
begin
aa:=Power(nu,4)+33.6*nu*(1.00.68*EXP(0.17*power((1.0+nu),2)));
FD:=1.0/(exp(F)+1.32934038675/power((50.0+aa),0.375));
n:=FD*power((T*DM),1.5)*6.037E21;
end;
A:=n;
end;
procedure main;
var x,y,i,j,k, model:integer; z:real;
begin
T:=300; (* temperature [K] *)
T:=T/1.1604E4; (* eV *)
DM:=1.08; //effective mass for silicon
FERMI(NB,DM,T,A);
FERMI(ND_n,DM,T,B);
breakdown_v := B;
for Y:=1 to IY do
for X:=1 to IX do
begin
(* oxide part *)
eps[X,Y]:=4;
ND[X,Y]:=0;
(* nregion *)
if (Y < IY10*m) then
eps[X,Y]:=12;
ND[X,Y]:=1.0E15;
V[X,Y]:=A;
(* source *)
if (Y <= IY10*m) and (Y >= IY33*m) and (X >= 0) and (X <= 67*m) then
begin
eps[X,Y]:=12;
ND[X,Y]:=1.0E18;
V[X,Y]:=B;
end;
(* source contact *)
33
if (Y >= IY10*m) and (X >= 0) and (X <= 32*m) then
begin
eps[X,Y]:=1000;
V[X,Y]:=B;
ND[X,Y]:=1.0E20;
end;
(* drain *)
if (Y <= IY10*m) and (Y >= IY33*m) and (X >= 133*m) then
begin
eps[X,Y]:=12;
ND[X,Y]:=1.0E18;
V[X,Y]:=BVD;
end;
(* drain contact *)
if (Y >= IY10*m) and (X >= 168*m) then
begin
eps[X,Y]:=1000;
V[X,Y]:=BVD;
ND[X,Y]:=1.0E20;
end;
(* gate contact *)
if (Y > IY5*m) and (X > 62*m) and (X < 137*m) then
begin
eps[X,Y]:=500;
V[X,Y]:=VG;
ND[X,Y]:=1.0E20;
end;
end;
for K:=1 to recount do
begin
// 
for j:=1 to IY do //do 300 Y=2,IY1
for i:=1 to IX do //do 300 X=2,IX1
begin
if (eps[i,j] = 1000) then n[i,j]:=1E20;
if (eps[i,j] = 4) then
begin
n[i,j]:=0; //no charges in the oxide
ND[i,j]:=0;
end;
end;
for j:=2 to IY1 do
begin
//search for IX1  max V from left to right
for I:=2 to IX1 do
if (V[I,j] > 0) then
begin
IX1:=I;
break;
end;
// search for IX2  max V from right to left
for I:=IX1 downto 2 do //do 401 I=2,IX
begin
if (V[I,j] > VD) then
begin
IX2:=I;
break;
end;
end;
34
// 
for i:=2 to IX1 do
begin
P[i,j]:=V[i,j];
//calc n
if (eps[i,j] = 12) then // for Source, Drain and pregion
begin
A:=V[i,j];
F:=0;
if (i <= IX1) then F:=0;
if (i > IX1) and (i < IX2) then F:=(iIX1)*(VD/(IX2IX1));
if (i >= IX2) then F:=VD;
CC(A,DM,T,F);
n[i,j]:=A; //n  elect density
if (n[i,j] < 1E15) then n[i,j]:=1E15;
end;
//calc potential
if (eps[i,j] <= 20) then //all regions except contacts where eps=1000
begin
B:=0.25*(V[i,j+1]+V[i,j1]+V[i+1,j]+V[i1,j]
(ND[i,j]n[i,j]) * 9.05E23 * 1.0E2/ (eps[i,j]));
P[i,j]:=B;
end;
end;
end;
// 
//UPDATE Potential V
for j:=2 to IY1 do //do 310 Y=2,IY1
for i:=2 to IX1 do //do 310 X=2,IX1
if (eps[i,j] <= 20) then //if current point is part of MOSFET
begin
if k < Round(recount*0.8) then
V[i,j]:=0.1*P[i,j]+0.9*V[i,j]
else
V[i,j]:=0.01*P[i,j]+0.99*V[i,j];
end;
// edges definition 
j:=1;
for i:=2 to IX1 do //do X=2,IX1
if eps[i,j]<=100 then V[i,j]:= V[i,j+1];
j:=IY;
for i:=2 to IX1 do //do X=2,IX1
if eps[i,j]<=100 then V[i,j]:=V[i,j1];
i:=1;
for j:=1 to IY do //do Y=1,IY
if eps[i,j]<=100 then V[i,j]:=V[i+1,j];
i:=IX;
for j:=1 to IY do //do Y=1,IY
if eps[i,j]<=100 then V[i,j]:=V[i1,j];
35
end;
//                        
//checking for device breakdown
min_v := 1000;
i:=1;
for j:=1 to IY do //do 310 X=2,IX1
if (eps[i,j]=12) and (min_v > V[i,j]) then min_v := V[i,j];
if (breakdown_v > min_v ) then
begin
writeln('The device has breakdown!');
end;
//                        
// CHANNEL
k_channel:=0;
for i:=1 to IX do
for j:=1 to IY do
begin
if (n[i,j]>=1E17) and (n[i,j]<1E20) and (i>=75*m) and (i<=135*m) then
begin
inc(k_channel);
channel[1,k_channel] := i;
channel[2,k_channel] := j;
break;
end;
end;
// writing results to files could be here
end;
end.
//MAIN PROGRAM
begin
Main;
write('finished.');
readln;
end.
Download the model
It is possible to download the executable file of the model here:
http://studentdevelop.com/projects/MOSFET_model.zip
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