WEBVTT
Kind: captions
Language: en
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Okay. So let us go we were looking of impurity
contribute to resistivity of the seen last
00:00:23.039 --> 00:00:25.640
time. This is what we were doing it.
00:00:25.640 --> 00:00:34.030
We also calculated, I just want to quickly
go through what we did last time. We calculated
00:00:34.030 --> 00:00:42.040
the available substitutional impurities or
interstitial impurities by putting this thermodynamics.
00:00:42.040 --> 00:00:48.379
And we then said that these are available
to the N e to the power minus Es by kT where
00:00:48.379 --> 00:00:52.420
N is number of atoms available in the lattice
per cc.
00:00:52.420 --> 00:00:58.949
And then we also said if it is interstitial,
we declared it NI0. If it is vacancies, it
00:00:58.949 --> 00:01:07.049
is Nv0 and we derived an expression which
NI0 is 27, 10 to power 27 times this. Nv0,
00:01:07.049 --> 00:01:08.970
this is all that we did last time.
00:01:08.970 --> 00:01:13.970
Today we start with quickly something more
about it. There is also possibility as I said
00:01:13.970 --> 00:01:20.040
last time, Frenkel defects are also available
and they can also be created at given temperature.
00:01:20.040 --> 00:01:26.479
If N is the number of atoms in crystal per
cc or per volume. N dash is number of available
00:01:26.479 --> 00:01:35.460
interstitial sides, sorry it is sites per
volume and nf is number of Frenkel defects
00:01:35.460 --> 00:01:41.570
per volume. And Ef is the activation energy.
Then by similar arguments we have entropy
00:01:41.570 --> 00:01:45.909
is equal to k times Boltzmann's constant time
ln of Cnf.
00:01:45.909 --> 00:01:53.510
Please remember I am now looking for vacancy
interstitial, so both these are we have taken
00:01:53.510 --> 00:02:01.189
into care. And by same argument I can show
you that by similar d, we can write S by into
00:02:01.189 --> 00:02:08.259
kT and T delta G. Delta G, by same method
we can calculate. The available Frenkel pair
00:02:08.259 --> 00:02:16.260
will be N into N dash e to the power Ef by
2kT where silicon, Ef and silicon is 1.1 electron
00:02:16.260 --> 00:02:21.860
volt. Please remember this energy is smaller,
so Frenkel pair creations are not smaller.
00:02:21.860 --> 00:02:29.750
Frenkel pair creations are not smaller simply
because on interstitial side vacancy can pair
00:02:29.750 --> 00:02:34.450
very easily and move together. One jumps there,
then the vacancy is created on the backside.
00:02:34.450 --> 00:02:41.600
A slight electron volt transport. So this
of course is not so important. So maths you
00:02:41.600 --> 00:02:46.680
can do again, I have not done it. I just wrote
down the final answer. The method is same
00:02:46.680 --> 00:02:49.520
as I we did for interstitial and vacancies.
00:02:49.520 --> 00:02:58.280
Okay, once we say the, we know the defects
at a given temperature, we like to know how
00:02:58.280 --> 00:03:03.730
these impurities move inside a crystal and
we are not looking impurities concentration
00:03:03.730 --> 00:03:10.180
in amorphous semiconductors and in polycrystalline
semiconductor. Though we are interested at
00:03:10.180 --> 00:03:15.370
least in the case of solar cell, doping of
amorphous materials and we are interested
00:03:15.370 --> 00:03:20.150
in the case of CMOS, poly gates or silicon
gate devices, poly diffusions.
00:03:20.150 --> 00:03:28.540
However as we say we will first look into
crystalline structure in which impurities
00:03:28.540 --> 00:03:36.150
are entering. Now these impurities if they
sit only on the substitutional sides as I
00:03:36.150 --> 00:03:43.160
say other day, then only they can contribute
to resistivities. Otherwise they will sit
00:03:43.160 --> 00:03:52.590
into interstitial side and do nothing except
creating strain. If you have written down,
00:03:52.590 --> 00:03:59.570
I may move further. Please remember this is
available on Google sites. If you wish to
00:03:59.570 --> 00:04:06.130
read sometimes, find time if not very much
busy with other more important activity like
00:04:06.130 --> 00:04:14.760
TV, mobile and or Internet. Look for this,
maybe interesting.
00:04:14.760 --> 00:04:19.019
This is a diffusion process has nothing to
do with electrical engineering or something,
00:04:19.019 --> 00:04:24.440
it is a diffusion in anything. So basically
it can be available on chemical engineering
00:04:24.440 --> 00:04:29.990
sites, chemistry sites, material science sites.
Many places you can get same expressions because
00:04:29.990 --> 00:04:36.849
it is a thermodynamics related situation.
Is that okay? So let us start how impurities
00:04:36.849 --> 00:04:37.849
move.
00:04:37.849 --> 00:04:45.420
Please remember silicon has a primitive cell
which is shown here which is one silicon atom
00:04:45.420 --> 00:04:51.520
is bonded to nearest neighbor by four atoms.
And this is called primitive cell. If you
00:04:51.520 --> 00:04:57.349
have seen our unit cell, there was a primitive
cell colored shown by. The corner one, the
00:04:57.349 --> 00:05:03.699
one which is just interposed from 1 by 4,
1 by 4 site and three other this. So these
00:05:03.699 --> 00:05:08.210
are essentially is called primitive cell.
The minimum amount of bonding which makes
00:05:08.210 --> 00:05:13.910
silicon atoms go is, silicon lattice go is
this cell, primitive cell.
00:05:13.910 --> 00:05:20.960
Now there are if you see, primitive cell,
there are five voids, they are arranged in
00:05:20.960 --> 00:05:29.710
tetrahedrally, 1, 2, 3, 4 and back one side,
so 5 sides. Here are some are occupied but
00:05:29.710 --> 00:05:33.719
most are available sites for impurities not
all voids are substitutional by something.
00:05:33.719 --> 00:05:42.349
But they are voids available. So one of the
possible mechanism is how interstitial diffuses
00:05:42.349 --> 00:05:50.240
through interstitial. So let us say an impurity
sits a position 1, it can hop through to 2,
00:05:50.240 --> 00:05:56.110
to another interstitial site, it can hop to
3, another (interst). Of course this is random,
00:05:56.110 --> 00:06:01.600
this is only shown one method, one place but
can have any random placements. 3 can go to
00:06:01.600 --> 00:06:05.080
4, 4, 5, 6 and maybe further ahead.
00:06:05.080 --> 00:06:11.010
So this is called interstitial diffusion.
Impurities hop from one void to the other,
00:06:11.010 --> 00:06:17.289
this is called (substitu) interstitial diffusion.
Even now though it is doing, this process
00:06:17.289 --> 00:06:22.690
may not contribute to resistivity but it has
more importance because as it releases, the
00:06:22.690 --> 00:06:27.749
void strain releases and there is a possibility
of silicon moves from here. It can occupy
00:06:27.749 --> 00:06:33.699
another void and release a vacancy down. So
the whole purpose is how vacancies also can
00:06:33.699 --> 00:06:45.159
be moving with interstitial motion. Now in
case of silicon, this is of course few lines,
00:06:45.159 --> 00:06:46.629
so you can always see.
00:06:46.629 --> 00:06:52.180
Silicon diameter of interstitial void is 2.36
Armstrongs. What is the void essentially I
00:06:52.180 --> 00:06:53.180
am saying?
00:06:53.180 --> 00:06:59.259
Between this the circle which touches all
four or rather backside one as well, is called
00:06:59.259 --> 00:07:00.259
sphere there.
00:07:00.259 --> 00:07:08.029
And that is circle which is shown here, has
a dia of 2.36 Armstrongs which means radius
00:07:08.029 --> 00:07:17.979
is 1.18 Armstrong. This is called tetrahedral
radius, 1.18 Armstrong is called tetrahedral
00:07:17.979 --> 00:07:25.139
radius. And of course between the two constriction,
between the two atoms on the constriction
00:07:25.139 --> 00:07:33.729
side, the gap is 2.10 Armstrong. Since lattice
vibrates at any temperature, there is a lattice
00:07:33.729 --> 00:07:39.400
vibrations, the lattice keeps on rocking as
well as stretching. More physics, some day.
00:07:39.400 --> 00:07:44.160
And it has some frequency which is called
jump frequency or frequency of oscillations
00:07:44.160 --> 00:07:53.070
or vibration. And typically it is 10 to power
13 to 10 to power 14 per second in different
00:07:53.070 --> 00:07:59.479
lattices. This is typical monitored number
derived from and measured from atomic spectroscopies.
00:07:59.479 --> 00:08:05.129
If interstitial impurity has to jump from
one side to the other, it has to overcome
00:08:05.129 --> 00:08:11.099
energy barrier. It cannot just go, it has
to cross some barrier. Now this barrier which
00:08:11.099 --> 00:08:21.499
is 700 to 1200, thermal vibrations occur with
frequency new which is given by 4 mu 0 e to
00:08:21.499 --> 00:08:22.499
the power Em.
00:08:22.499 --> 00:08:28.300
Em is the barrier energy, it has to cross
this much energy to come out. It should get
00:08:28.300 --> 00:08:34.510
excited enough, pass through barrier and jump
to the next side. This 4 of course is called
00:08:34.510 --> 00:08:41.830
degeneracy, from where it comes? It is a random,
it can go this side, it can go this side,
00:08:41.830 --> 00:08:47.029
it can go this side, it can go this side,
so it has four possibility of motion. So it
00:08:47.029 --> 00:08:52.990
is called degeneracy factor. So typically
the jump frequency is 4 mu 0 e to the power
00:08:52.990 --> 00:08:57.610
minus Em by kT.
00:08:57.610 --> 00:09:01.760
Typically Em for this substitutional sides,
the barrier is something shown here. So if
00:09:01.760 --> 00:09:08.350
atom has to go from this side to this side,
it must cross this much energy. This is equivalent
00:09:08.350 --> 00:09:14.540
model of energy. I hope in second year or
maybe earlier you might have done chronic
00:09:14.540 --> 00:09:20.170
pain model, this is replication of that to
some extent. So you have one atom here and
00:09:20.170 --> 00:09:27.160
it has to occupy this. It must cross the barrier
of energy, Em to reach to the next side.
00:09:27.160 --> 00:09:34.720
This essentially is what this expression shows.
Typical jump rate or frequency is 1 per minute
00:09:34.720 --> 00:09:41.529
at around between 700 to 1200 date, varies
little bit but it is around 1 jump per minute
00:09:41.529 --> 00:09:49.980
is what the rate with which substitutional
this, sorry interstitial actually jump. This
00:09:49.980 --> 00:09:56.060
is important because how many atoms are where
at a given instant is relative to, at a given
00:09:56.060 --> 00:10:01.639
temperature is some way relevant to know how
many impurities are available where.
00:10:01.639 --> 00:10:06.520
I introduce some impurity in silicon where
they will lie. So I like to know where they
00:10:06.520 --> 00:10:12.160
can at best go and how will they go. So this
is physics telling that there is a possibility.
00:10:12.160 --> 00:10:21.149
Now these jumps, impurities can come and occupy
that void, so they are entering in. Now this
00:10:21.149 --> 00:10:27.300
jump frequency, why I have got it? There is
something this equation has to do with diffusion
00:10:27.300 --> 00:10:33.880
coefficient but maybe we will see this later.
This relationship which I am drawing, I need
00:10:33.880 --> 00:10:40.790
to have for creation of constants, diffusion
constants or diffusion coefficients. Is that
00:10:40.790 --> 00:10:47.880
okay? Everyone? Okay. So this is something
called interstitial.
00:10:47.880 --> 00:10:53.519
The most important transfer of impurities
inside a material is through substitutional
00:10:53.519 --> 00:11:01.430
sides. Substitutional means wherever silicon
atom is there and if there is a vacancy, an
00:11:01.430 --> 00:11:07.889
impurity atom can occupy that vacancy and
sit there. But it can actually jump from say
00:11:07.889 --> 00:11:13.709
let position 1, if it finds a vacancy here,
so it actually may jump here. If it finds
00:11:13.709 --> 00:11:18.769
a vacancy here, it may jump here and keep
jumping wherever it finds another vacancy.
00:11:18.769 --> 00:11:26.509
So impurity can move from one vacancy side
to the other vacancy side by process called
00:11:26.509 --> 00:11:27.509
substitutional diffusion.
00:11:27.509 --> 00:11:34.561
This is most important diffusion, how impurities
actually travel inside silicon. That is what
00:11:34.561 --> 00:11:40.209
our aim. Why are we doing all this? Maybe
it should be very clear to you that my interest
00:11:40.209 --> 00:11:44.740
in doing all this is not just because I want
to understand physics alone or maybe I am
00:11:44.740 --> 00:11:50.860
interested x of you, I am maybe interested.
But the interest part is how much resistance
00:11:50.860 --> 00:11:56.180
it finally offers because of the profile it
gives. Because current is related to that,
00:11:56.180 --> 00:12:01.500
there are some wear, I am very keen to know
how much is the current I can get if I apply
00:12:01.500 --> 00:12:03.860
x voltage.
00:12:03.860 --> 00:12:10.230
My interest in electrical engineering is only
IV and CV. If anyone hurts me on IV or CV,
00:12:10.230 --> 00:12:15.830
I am going to look into you. Why cannot I
do better? So I look into physics, I look
00:12:15.830 --> 00:12:20.529
into material, I look everything because at
the end of the day circuit must function the
00:12:20.529 --> 00:12:26.839
way I thought I have designed. And to do this,
I must understand everything around that which
00:12:26.839 --> 00:12:31.990
helps me to improve. So please do not think
this technology course is only of this side.
00:12:31.990 --> 00:12:38.569
This is essentially covering those areas which
normally we do not cover anywhere. Normally,
00:12:38.569 --> 00:12:40.269
I do not say…..
00:12:40.269 --> 00:12:46.370
Please remember generally vacancies are fair
compared to voids. So essentially you can
00:12:46.370 --> 00:12:51.420
say the jump rate of diffusion process will
be smaller because available sites are smaller
00:12:51.420 --> 00:12:57.089
compared to interstitials. Interstitial side
almost everywhere. Vacancies are need to be
00:12:57.089 --> 00:13:01.980
created by some energy, e to the power Es
by kT. So you have now two energy, one Es
00:13:01.980 --> 00:13:07.220
you have to first create a vacancy and then
allow it move. So that will be now new energies
00:13:07.220 --> 00:13:13.440
will be, will have actually energy creation
for vacancy plus energy (creati) energy for
00:13:13.440 --> 00:13:14.630
barrier to cross.
00:13:14.630 --> 00:13:20.290
So actually it will be two energy sum now
which will be required for vacancy motion.
00:13:20.290 --> 00:13:26.319
Is that point clear? Voids are not to be if
they are there. But in case of, first vacancy
00:13:26.319 --> 00:13:31.910
have to be created. Once vacancy, it can jump
only if it crosses a barrier.
00:13:31.910 --> 00:13:39.709
So based on the same, similar analysis, one
can write mu is now 4 mu 0, e to the power
00:13:39.709 --> 00:13:48.420
En plus Es where Es is, the En, Es are binding.
Es is binding energy and En is the vacancy
00:13:48.420 --> 00:13:54.100
creation energy. Is that okay? I just now
said. To create a vacancy, I need En energy.
00:13:54.100 --> 00:14:02.279
To move it I need barrier to cross which is
Es. Now silicon-silicon bond, binding energy
00:14:02.279 --> 00:14:08.399
is larger than silicon impurity bond. Is that
word clear? Two silicon atoms, okay maybe
00:14:08.399 --> 00:14:09.480
first you write down.
00:14:09.480 --> 00:14:16.589
This part is most important to show that substitutional
impurity diffusions has much lesser chance
00:14:16.589 --> 00:14:22.779
compared to void ones because there are enough
voids, so many many of them can actually enter
00:14:22.779 --> 00:14:29.110
which is very contrasting. People believe
that impurities must be sitting on first vacancy.
00:14:29.110 --> 00:14:34.519
No, first they may likely to sit to the voids
itself, interstitial sides and then maybe
00:14:34.519 --> 00:14:39.420
move for seeing a vacancy around.
00:14:39.420 --> 00:14:45.370
This very interesting part, silicon is also,
atoms are also moving because of thermal energy.
00:14:45.370 --> 00:14:50.910
Some bonds are broken anyway but silicon-silicon
bond is very strong, very very strong in case
00:14:50.910 --> 00:14:56.750
of bond strength. That is law if you apply.
The bond and force is very high. Compared
00:14:56.750 --> 00:15:02.100
to silicon impurity atoms, the binding energy
for silicon-silicon is much higher. So it
00:15:02.100 --> 00:15:06.779
is unlikely that silicon bonds maybe keep
breaking every now and then unless you heat
00:15:06.779 --> 00:15:08.189
it very heavily.
00:15:08.189 --> 00:15:14.620
Give lot of thermal energy, otherwise silicon
does not break, bonds does not, do not break
00:15:14.620 --> 00:15:18.779
very easily. Therefore silicon self-diffusion
going from one side to the other is lesser
00:15:18.779 --> 00:15:26.790
event, not that it will not happen but lesser
chance of moving one silicon atom to the other
00:15:26.790 --> 00:15:33.411
silicon side is very smaller probability event
compared to impurities getting inside the
00:15:33.411 --> 00:15:38.660
crystal. Because after all you are giving
temperature, enough thermal energy is provided.
00:15:38.660 --> 00:15:44.040
Every possible mechanism can happen. So we
want to eliminated saying that okay, silicon-silicon
00:15:44.040 --> 00:15:50.970
self-diffusion is smallest among all of them,
does not mean 0, finite, small compared to
00:15:50.970 --> 00:15:53.410
the interstitial and substitutional.
00:15:53.410 --> 00:15:59.279
There is one more (possi), maybe two more
possibility that there which also are very
00:15:59.279 --> 00:16:08.089
low, maybe this one. There is a possibility
that atoms may move either vacancies are at
00:16:08.089 --> 00:16:15.770
silicon atoms and they can actually keep moving
around nearer sides. This is very interesting
00:16:15.770 --> 00:16:21.130
which is very very small probability that
this atom of impurity or silicon jumps to
00:16:21.130 --> 00:16:24.610
the next side, next side, next side and come
apart.
00:16:24.610 --> 00:16:29.170
This has a very very small, this is called
interchange diffusion, very very unlikely
00:16:29.170 --> 00:16:37.129
event but can occur. One in billion or even
lower probable but can occur. But at given
00:16:37.129 --> 00:16:45.670
temperatures higher than it may occur as well
to some extent. However which is the best
00:16:45.670 --> 00:16:51.709
possible diffusion therefore? Substitutional
and interstitial together is very possible.
00:16:51.709 --> 00:16:56.550
Atom first come to interstitial side, jumps
to vacancy, allows another impurity come to
00:16:56.550 --> 00:16:59.209
interstitial side, jump to vacancy.
00:16:59.209 --> 00:17:04.980
This vacancy atom may move to another vacancy,
it may create another void there from where
00:17:04.980 --> 00:17:09.780
interstitial move, another vacancy is brought
in and this is called cooperative diffusion.
00:17:09.780 --> 00:17:15.900
This is the most likely diffusion in which
interstitials and vacancies go together to
00:17:15.900 --> 00:17:25.740
push atoms inside. Is that okay? They both
can help each other to actually get more and
00:17:25.740 --> 00:17:30.470
more impurities getting diffused inside the
crystal. This is most likely event and has
00:17:30.470 --> 00:17:38.730
the largest probability.
00:17:38.730 --> 00:17:45.450
I can do some quick calculation for this as
well. Is that okay? Two possibly, so there
00:17:45.450 --> 00:17:51.570
are four possible mechanism in which impurities
can get inside. Most likely is the last one
00:17:51.570 --> 00:17:57.560
but the first two actually tell you that together
they will help in the fourth case. Is that
00:17:57.560 --> 00:18:01.289
okay? Everyone?
00:18:01.289 --> 00:18:08.490
If Ns and NI are concentration of available
substitutional and interstitial sites at temperature
00:18:08.490 --> 00:18:16.650
T, then the effective jump frequency can be
given by available substitutional site into
00:18:16.650 --> 00:18:25.630
total sites into jump frequency for substitutional
plus available interstitial site to the total
00:18:25.630 --> 00:18:32.980
sites into jump frequency for interstitial.
This is very standard average method. Available
00:18:32.980 --> 00:18:39.370
ones to the total with a jump rate, available
one with ratio to the total into that jump
00:18:39.370 --> 00:18:44.980
rate. This is the net possibility. If it is
only substitutional, what will happen?
00:18:44.980 --> 00:18:52.059
NI will be much smaller than Ns. Then you
say this term will be negligible. Only substitutional
00:18:52.059 --> 00:18:57.150
impurities may move. If NIs are much larger
than Ns, we say only interstitial diffusion.
00:18:57.150 --> 00:19:03.750
But in a given this, both are together and
Ns, Ni also concentration keep changing as
00:19:03.750 --> 00:19:08.720
numbers start getting more and more inside.
So relative to, for considering both defects
00:19:08.720 --> 00:19:14.360
relative each other is effective new. Please
remember this is most important thing which
00:19:14.360 --> 00:19:16.530
you should understand.
00:19:16.530 --> 00:19:21.100
This expression is only trying to say both
together can happen. And this expression I
00:19:21.100 --> 00:19:26.220
have derived from the average method, one
among so much into this, second among so much
00:19:26.220 --> 00:19:32.580
into this. However it is important to note
that natural random jump events may not be
00:19:32.580 --> 00:19:39.720
very large. These are called natural random
jumps. Most of the impurities actually travel
00:19:39.720 --> 00:19:46.799
because of concentration gradient. Larger
comparative is here, smaller impurity is here
00:19:46.799 --> 00:19:51.779
and they try to diffuse through to make equalization.
00:19:51.779 --> 00:19:59.010
So question arises and why do we argue this
if we know it is only gradient dependent?
00:19:59.010 --> 00:20:07.080
No, some of the effects what I should call
them, anomalous effect. I see some profile,
00:20:07.080 --> 00:20:15.000
none of the standard expression fit to that.
Then I come and see is this material at this
00:20:15.000 --> 00:20:20.450
time has some other diffusivity going on.
So I look into which is the other mechanism
00:20:20.450 --> 00:20:26.371
which might have added or reduced it so that
the profile should have gone up but it is
00:20:26.371 --> 00:20:28.640
moving down.
00:20:28.640 --> 00:20:33.679
So to understand the actual profile which
I will get in real diffusion, I will have
00:20:33.679 --> 00:20:39.850
to model it. And to model it, I should know
from where these possibilities can occur.
00:20:39.850 --> 00:20:46.010
So it is not that these are very strong forces
there but in case the profiles do not match
00:20:46.010 --> 00:20:50.940
as in the case boron it does not match, phosphorus
it does not, mostly it matches with arsenic
00:20:50.940 --> 00:20:55.980
in normal this. But all other impurities show
what is called as anomalous effects.
00:20:55.980 --> 00:21:02.300
Anomalous means from standard diffusion theory
it does not match my profile. So I say why.
00:21:02.300 --> 00:21:08.659
This is more important if you are actually
looking into very very highly doped crystals
00:21:08.659 --> 00:21:14.770
areas or very very low dope area. In between
the diffusion coefficient, diffusivity is
00:21:14.770 --> 00:21:21.899
essentially governed by gradients. Most devices
are in that range but source drain, 10 to
00:21:21.899 --> 00:21:29.510
power 20. Some kind of PI diodes, 10 to power
13. So on those areas, these diffusion techniques
00:21:29.510 --> 00:21:32.320
are very very important, other ones.
00:21:32.320 --> 00:21:41.909
We have two hours, so now we will start with
this how the diffusion starts and I want to
00:21:41.909 --> 00:21:48.690
find at the end of the day, maybe I will show
you one figure what is my ultimate aim of
00:21:48.690 --> 00:21:54.660
doing all this. Why I am so much worried?
00:21:54.660 --> 00:22:10.000
By some way this is my crystal. Maybe silicon
right now and this is my surface I call it.
00:22:10.000 --> 00:22:21.899
And I introduce impurities. This
is what I will do. Technique of doing it,
00:22:21.899 --> 00:22:27.750
we will discuss in techniques, how impurities
are introduced actually. Now if they are getting,
00:22:27.750 --> 00:22:34.320
since there is a large concentration of impurities
at the surface compared to what silicon it
00:22:34.320 --> 00:22:38.649
has, it can be p-type, n-type, smaller doping,
higher doping but difference.
00:22:38.649 --> 00:22:46.169
So there is a gradient. These impurities tend
to enter the lattice and keep moving downwards
00:22:46.169 --> 00:22:52.279
because of the gradient it created. If I,
let us say this is x is equal to 0, then if
00:22:52.279 --> 00:23:04.279
I plot concentration of impurities Nx versus
x, impurities getting down, this is x down.
00:23:04.279 --> 00:23:11.620
Then I may say there are number of profiles.
One possible profile is this. Other possible
00:23:11.620 --> 00:23:21.990
profile is, sorry exponential. This is Gaussian,
this is exponential and one more we shall,
00:23:21.990 --> 00:23:24.570
that is what we will start with.
00:23:24.570 --> 00:23:37.530
So this Nx is what I am interested to know,
what is the profile? That means from the surface
00:23:37.530 --> 00:23:42.640
the concentration will be normally very high
but as you go down the concentration starts
00:23:42.640 --> 00:23:49.289
reducing. Now this profile decides some kind
of resistance available to you because carrier
00:23:49.289 --> 00:23:54.990
available will be proportional to x now. Normally
we say dopants are equal to the electrons
00:23:54.990 --> 00:23:56.690
or holes depending on the what dopant.
00:23:56.690 --> 00:24:01.510
If it is n-type, all the doping impurities
actually are equal to available electrons,
00:24:01.510 --> 00:24:10.580
not everyone of it but mostly. We say n is
equal to Nd plus ionized one, p is equal to
00:24:10.580 --> 00:24:17.090
Na plus ionized ones. They are almost equal.
So resistivity decided by n and p, so that
00:24:17.090 --> 00:24:24.190
that means Nd and Nes. But if Nd, Ne they
are not constant, that means there is a profile.
00:24:24.190 --> 00:24:34.769
So n and p are also functions of x which means
if I calculate the resistivity or resistance,
00:24:34.769 --> 00:24:37.480
I will find it is a function of x.
00:24:37.480 --> 00:24:42.630
In many cases the way I do it, I take average
of that value, integral over that range and
00:24:42.630 --> 00:24:48.649
say okay average resistivity is this. But
in real life resistivity will higher have
00:24:48.649 --> 00:24:54.299
up, sorry, resistivity will be smaller up
or conductivity higher up and will go down.
00:24:54.299 --> 00:25:00.179
Now this matters lot in smaller devices now
because your channel length is of the order
00:25:00.179 --> 00:25:05.929
of 14, 10 nanometers. Your diffusions or substrate
are 10 to power 19 kind of things, very thin
00:25:05.929 --> 00:25:08.179
layer of channel is going to be created.
00:25:08.179 --> 00:25:13.929
Of course there are other physics effect called
quantum effects but some other day. Now there
00:25:13.929 --> 00:25:19.990
how many real items are available to us? How
much really Nx I have? Do I have a profile
00:25:19.990 --> 00:25:27.100
say uniform? All those issues will finally
decide IV of the MOS transistor or BJT transistor.
00:25:27.100 --> 00:25:33.399
So this number is very crucial for me. How
many actually? And this how many makes me
00:25:33.399 --> 00:25:39.390
actually go through all of it. Since I am
interested to know Nx, Px for my electrical
00:25:39.390 --> 00:25:44.970
properties, I am now looking into how are
impurities going to contribute to this Nx
00:25:44.970 --> 00:25:46.200
and Px profiles.
00:25:46.200 --> 00:25:51.710
Is that clear? So that is the purpose of doing
all this. I am not just doing this physics
00:25:51.710 --> 00:25:57.430
because I like. Maybe I like it, as I keep
saying I really like it. Over the years you
00:25:57.430 --> 00:26:04.919
will also like it. When there is no stress
of exam, you also start liking, I mean this
00:26:04.919 --> 00:26:11.710
is natural. So there are two laws in which
impurities, diffusion can be modeled. First
00:26:11.710 --> 00:26:19.090
is called Fick’s first law and second is
Fick’s second law. I think what I will do
00:26:19.090 --> 00:26:24.440
is I will leave it Fick’s first and second
law for a while.
00:26:24.440 --> 00:26:29.380
I will first finish the impurities because
someone say lab you want some numbers. So
00:26:29.380 --> 00:26:34.660
I will show you actually how diffusion is
done there. So what we are essentially doing
00:26:34.660 --> 00:26:41.850
here is the impurity which are coming inside
the silicon, how they get inside and what
00:26:41.850 --> 00:26:48.120
is the profile it creates, is what is our
major interest of doing all of it. So I start
00:26:48.120 --> 00:26:52.710
with something which is related to that and
as I say time permitting I will come back
00:26:52.710 --> 00:26:53.710
to that.
00:26:53.710 --> 00:26:59.580
This is very interesting. Since our diffusion
is solid impurities are getting into solid
00:26:59.580 --> 00:27:06.260
material, if it is liquid-liquid system which
we have solid bits getting into liquid, both
00:27:06.260 --> 00:27:13.110
forming liquid crystal growth. There we can
as if stir it and uniformly dope it but that
00:27:13.110 --> 00:27:19.710
is not so in the case of diffusion in normal
sense. So we are interested in, if I start
00:27:19.710 --> 00:27:27.669
these impurities diffusion from the top let
us say of the wafer, what is the maximum concentration
00:27:27.669 --> 00:27:36.260
I can get at the surface. For your kind information
maybe, just a minute I will just say numbers
00:27:36.260 --> 00:27:38.250
and come back to this.
00:27:38.250 --> 00:27:50.659
The maximum silicon concentration at a normal
temperature is 5 into 10 power to 22 atoms
00:27:50.659 --> 00:27:57.480
per cc. This is the maximum carrier concentration
or maximum doping, maximum atoms per cc available
00:27:57.480 --> 00:28:03.100
to you. Based on this our density recalculations
we perform or rather from the major densities
00:28:03.100 --> 00:28:10.480
this number has been found. Okay, for the
lattice of 8 atoms in per cell. So this is
00:28:10.480 --> 00:28:14.769
the maximum atoms, so how much doping we can
do? Certainly not 5 into 10 to power 22.
00:28:14.769 --> 00:28:22.990
Because if it replaces all the atoms, then
there is nothing silicon. So obviously it
00:28:22.990 --> 00:28:29.029
has to be less than 5 into 10 power 22. So
this is a natural limit up to which silicon
00:28:29.029 --> 00:28:38.010
can get into, impurities can get into silicon.
That at a given temperature is different and
00:28:38.010 --> 00:28:46.360
this term is called solid solubility. Solubility
is essentially between liquid and solid but
00:28:46.360 --> 00:28:52.900
here is the case we are talking, terming it
as solid solubility. Is that clear? Atoms,
00:28:52.900 --> 00:28:59.220
how many atoms per cc are inside at the surface,
we will like to know how maximum can reach
00:28:59.220 --> 00:29:02.779
there at the surface. That is called solid
solubility.
00:29:02.779 --> 00:29:12.899
Here is some type of impurities you know which
contribute to electrons and holes. There are
00:29:12.899 --> 00:29:18.880
some story about if there is why do not you
buy, put impurities from the second group
00:29:18.880 --> 00:29:24.860
or even the first group. Then it can create
two electrons or two holes, three electrons
00:29:24.860 --> 00:29:32.340
or three holes. Why only one kind of thing,
three or five we take. There is answer later
00:29:32.340 --> 00:29:37.000
when I calculate diffusion coefficients. What
will happen if there is a double such system
00:29:37.000 --> 00:29:38.000
appears?
00:29:38.000 --> 00:29:46.260
However as of now arsenic, phosphorus, antimony
are standard n-type impurities and boron,
00:29:46.260 --> 00:29:54.820
aluminum, gallium are mostly p-type impurities.
99 percent now we use arsenic as n-type dopant
00:29:54.820 --> 00:30:02.139
and boron even now is the only good dopant
for p-type. Yeah, there are some devices in
00:30:02.139 --> 00:30:09.200
which boron plus aluminum has been tried but
it is not great success. There is some new
00:30:09.200 --> 00:30:13.019
methods are being tried there, some other
day about this.
00:30:13.019 --> 00:30:18.200
P-type impurity concentrations, difficult
to get into the much numbers. We will see
00:30:18.200 --> 00:30:25.130
this number soon. So here is some graph shown
here. This is temperature versus solubility.
00:30:25.130 --> 00:30:37.170
I can say this is my 10 to power 22 or maybe
5 into 10 to power 22 is the limit of silicon.
00:30:37.170 --> 00:30:42.650
So nothing much can go below. Also the graph
shown here, arsenic, phosphorus, phosphorus
00:30:42.650 --> 00:30:48.289
deep at higher temperature. We will explain,
this is anomalous. At high temperature phosphorus,
00:30:48.289 --> 00:30:49.429
why it comes out?
00:30:49.429 --> 00:30:57.250
So it actually reduces the concentration.
Boron of course at 1200 does not do much.
00:30:57.250 --> 00:31:04.230
Now one can see from here the numbers were,
roughly it is 4 into 20 for arsenic, boron
00:31:04.230 --> 00:31:11.800
but arsenic can go up to 10 to power 21, 4
into 10 to power 21. What does that means?
00:31:11.800 --> 00:31:17.919
Arsenic is much easier to get in compared
to boron. So what is the problem? Why solid
00:31:17.919 --> 00:31:24.919
solubility of arsenic is highest compared
to boron? So here is some numbers which, a
00:31:24.919 --> 00:31:31.000
sheet which last time I forgot, so maybe taken
from Plummer yesterday and hand-written on
00:31:31.000 --> 00:31:32.000
something.
00:31:32.000 --> 00:31:37.080
If you have taken down the graph, these graphs
are also available in Plummer, so nothing
00:31:37.080 --> 00:31:41.680
great about. Of course they, if you see their
book, I do not know, yesterday I wanted but
00:31:41.680 --> 00:31:48.830
there will be also two such graphs. One is
this and the other is dotted lines for both
00:31:48.830 --> 00:31:54.399
arsenic, phosphorus. Can you think what could
be they? This will be slightly lower than
00:31:54.399 --> 00:32:03.230
all of them. Say arsenic, 4 into 21, maybe
this is let us say 5 into 21 but actually
00:32:03.230 --> 00:32:08.559
dotted curve will be slightly below, 3 into
21 or 4 into 21. Why these dots come lower
00:32:08.559 --> 00:32:11.440
then, is that clear to you? Something?
00:32:11.440 --> 00:32:18.090
Not all atoms actually get ionized, they may
come in only at the substitutional sides.
00:32:18.090 --> 00:32:24.990
If they sit, they ionize which essentially
means this is the actual number which have
00:32:24.990 --> 00:32:31.690
come in. But the actual ionized atoms will
be slightly lower than the available. So actually
00:32:31.690 --> 00:32:37.190
our interest is in that number not even this.
But for theory let us look into solubility
00:32:37.190 --> 00:32:42.460
at given temperature. So please look at the
books, they will show you three graphs for
00:32:42.460 --> 00:32:49.100
solid solubility, three graphs solid solubility
with activated impurity. Activated means ionized,
00:32:49.100 --> 00:32:52.210
they are actually sitting at substitutional
side.
00:32:52.210 --> 00:32:58.710
In this everything which is getting into lattice
without straining it, is possible. Is that
00:32:58.710 --> 00:33:03.060
okay? So this fact has to be understood why
that number is slightly smaller and in other
00:33:03.060 --> 00:33:08.429
calculation when I will give you graph, I
will give both, I mean the full graph. We
00:33:08.429 --> 00:33:14.950
will have to always choose the dotted pairs
which are activated numbers for the concentration.
00:33:14.950 --> 00:33:21.070
Is that point clear? You must use dotted curves
instead of the hard ones because dotted are
00:33:21.070 --> 00:33:26.520
activated impurities. Activated means these
are the ones only contributing to resistivity
00:33:26.520 --> 00:33:32.710
and we are only interested in currents and
voltages, no more. So we will say okay, they
00:33:32.710 --> 00:33:35.470
are dotted. So we will see that graph later.
00:33:35.470 --> 00:33:42.020
Whenever impurities try to get into a lattice,
it try to, even if we say strain-free, it
00:33:42.020 --> 00:33:47.669
does strain the lattice. For example, when
the, even if interstitial side between the
00:33:47.669 --> 00:33:53.840
two atoms, when there is a constriction there,
so it has to pump in, it has to push in. Now
00:33:53.840 --> 00:33:58.429
that means some of the earlier atoms may not
retrace back to its original position, maybe
00:33:58.429 --> 00:34:04.649
slightly moved. So there is this partial strain
which always exists even if we do room temperature.
00:34:04.649 --> 00:34:08.159
The problem is we recover it by some other
technique that is another issue.
00:34:08.159 --> 00:34:15.370
For a silicon we have actually discussed other
day or just now that there is a tetrahedral
00:34:15.370 --> 00:34:24.200
radius which is 2.36 Armstrong was dia, so
1.18 Armstrong was the radius. For each such
00:34:24.200 --> 00:34:30.040
atom size of phosphorus, arsenic, antimony,
I think there are given boron, aluminum, gallium,
00:34:30.040 --> 00:34:34.240
indium, gold, silver, all impurities we have
actually found out the tetrahedral radius
00:34:34.240 --> 00:34:42.880
for their lattices. Now one of the these impurities
may come into silicon. This is, is that clear?
00:34:42.880 --> 00:34:47.909
They are impurity atoms, they also have a
crystalline structure, many of them, not necessary
00:34:47.909 --> 00:34:49.040
all of them.
00:34:49.040 --> 00:34:55.540
So if their tetrahedral radius is known and
I know silicon tetrahedral radius. Now we
00:34:55.540 --> 00:35:03.900
say if the tetrahedral radius of silicon matches
with tetrahedral radius of that impurity,
00:35:03.900 --> 00:35:10.040
then the maximum will come because they will
not strain anything. Same size, if the size
00:35:10.040 --> 00:35:15.400
is bigger, the smaller atoms and one is bigger
sitting there, so it will strain the lattice
00:35:15.400 --> 00:35:21.740
anyway. If it is smaller, it will be inverse
stress. So if the size tetrahedral radius
00:35:21.740 --> 00:35:28.369
of an impurity is not identical to silicon
or tetrahedral radius, then it strains the
00:35:28.369 --> 00:35:35.400
lattice, marginally but it does. This is essentially
called replaced by, this is explained by term
00:35:35.400 --> 00:35:43.540
called misfit factor. It is called misfit
factor, so maybe I write down in that sheet.
00:35:43.540 --> 00:35:47.820
I come back and this number maybe read out
for you so that you know.
00:35:47.820 --> 00:36:02.700
For silicon R0 is the tetrahedral radius,
1.18 Armstrong. Phosphorus, it is 1.10, okay.
00:36:02.700 --> 00:36:19.740
Arsenic, it is 1.18. Boron, it is 0.88. Please
note, when I say please note down. Silicon,
00:36:19.740 --> 00:36:29.430
1.18; phosphorus, 1.10; boron, arsenic, 1.18;
boron, 0.88; aluminum, 1.026; gallium, 1.26;
00:36:29.430 --> 00:36:40.060
aluminum, 1.026, in gallium, 1.26; indium,
1.44 largest atom around. But gold and this
00:36:40.060 --> 00:36:49.260
is even higher. Gold is 1.5 Armstrong, silver
is 1.52 Armstrongs. Is that okay?
00:36:49.260 --> 00:36:58.109
And the difference between R0 to this tetrahedral
radius are that it is called misfit factor
00:36:58.109 --> 00:37:03.220
epsilon. Is that clear to you? The difference
between silicon tetrahedral radius and impurity
00:37:03.220 --> 00:37:07.599
tetrahedral radius is called misfit factor.
It can be plus or minus. If the impurity atom
00:37:07.599 --> 00:37:16.890
is larger tetrahedral radius, it will be minus.
If it is smaller, it is positive. So what
00:37:16.890 --> 00:37:21.950
I define, you have noted these numbers, you
noted down what I wrote.
00:37:21.950 --> 00:37:33.490
What I am now saying you that I have a formulation
which says r is equal to r0, 1 plus minus
00:37:33.490 --> 00:37:46.320
epsilon where epsilon is called misfit factor.
You subtract 1.18 from each impurity atoms,
00:37:46.320 --> 00:37:54.220
each impurity atom tetrahedral radius and
find out what is epsilon for each. r0 is for
00:37:54.220 --> 00:38:02.920
silicon, r is the tetrahedral radius for impurities.
If epsilon is 0, when this can occur? When
00:38:02.920 --> 00:38:13.900
r is r0. If r is r0, epsilon is 0. Which impurities
according to you has? Arsenic.
00:38:13.900 --> 00:38:20.880
Since arsenic has the least misfit factor,
the possibility of number of atoms of arsenic
00:38:20.880 --> 00:38:26.450
getting in silicon is the highest at solid
solubility. Is that point clear? Because they
00:38:26.450 --> 00:38:34.380
do not strain the lattice compared to others.
Next will be which? Depending on of course,
00:38:34.380 --> 00:38:41.780
0.68 is for phosphorus. So next best may be
phosphorus. Boron has 0.254 as epsilon difference,
00:38:41.780 --> 00:38:48.750
so it will have a smaller number. So do you
get the point that graph which I showed you
00:38:48.750 --> 00:38:55.300
is someway related to misfit factor. Is that
point clear? Why those graph, arsenic shows
00:38:55.300 --> 00:38:58.270
highest concentration followed by phosphorus,
followed by boron.
00:38:58.270 --> 00:39:05.280
And if you plot for all the impurities, correspondingly
you get solid solubility curves for all impurities.
00:39:05.280 --> 00:39:12.280
But since we are only interested into silicon
IC process, I am restricting only these three.
00:39:12.280 --> 00:39:19.310
But please remember in case I needed, I get
those values for my other impurities as well.
00:39:19.310 --> 00:39:24.950
So this fact that misfit factor decide solid
solubility should be understood that why people
00:39:24.950 --> 00:39:30.340
actually say that arsenic is the best n-type
dopant or best dopant in silicon.
00:39:30.340 --> 00:39:37.819
But if the gallium arsenide lattice, it may
have different kinds of misfit factors for
00:39:37.819 --> 00:39:42.780
different impurities there. Of course there
is no, by the way which is the easiest doping
00:39:42.780 --> 00:39:53.590
this in the case of gallium arsenide? Silicon.
Okay, some other time. Of course, I am not
00:39:53.590 --> 00:39:59.849
teaching gallium arsenide but my own PhD work
for some gallium arsenide, 35 years, 40 years
00:39:59.849 --> 00:40:02.339
ago. So I still enjoy that.
00:40:02.339 --> 00:40:08.700
And as I said those days when I shifted to
silicon, my guide was saying that oh, this
00:40:08.700 --> 00:40:15.170
is the area of future, why are you shifting?
Then I told him ki sir, woh future never comes.
00:40:15.170 --> 00:40:23.480
So I would prefer to be in present, so I shift
to silicon. So my choice was not bad as far
00:40:23.480 --> 00:40:28.359
as technology goes but if I would have continued,
I would have published many more poor papers.
00:40:28.359 --> 00:40:33.160
For silicon there were 20 lakh people working,
gallium arsenide 100. So I should have been
00:40:33.160 --> 00:40:39.069
there but did not realize that I will become
teacher. Maybe I should have become the otherwise.
00:40:39.069 --> 00:40:46.000
Though I did try work in industry as well
as R&D labs, before I became teacher.
00:40:46.000 --> 00:40:51.349
That is why we say technology was my first
jobs where I did for 15 years. So I understand
00:40:51.349 --> 00:40:55.930
more of technology compared to many not because
they are smarter, they are, they know more
00:40:55.930 --> 00:41:01.079
knowledgeable maybe. Many of them I have been
taught by me, so maybe more better than me
00:41:01.079 --> 00:41:08.050
but simply because I enjoy. And after great
fight with my head, you are the sufferers
00:41:08.050 --> 00:41:12.890
for that but I forced them to give me this
course for the last time because I said I
00:41:12.890 --> 00:41:15.010
want to record it once.
00:41:15.010 --> 00:41:20.920
Many of my old student who learnt technology
from me, 20 years, 15 years, they kept on
00:41:20.920 --> 00:41:26.569
saying sir, aapka yeh course web par nahi
hai. Mein bola okay, last time I will record
00:41:26.569 --> 00:41:32.940
it. Now this was the old time case that students
ka thinking has changed, student attitudes
00:41:32.940 --> 00:41:39.960
has changed. Maybe I have also changed. So
things may not be as good or as bad, as they
00:41:39.960 --> 00:41:44.690
were earlier. But I am known as technology
scene earlier phase of my career and suddenly
00:41:44.690 --> 00:41:47.819
I became designer for no good reason.
00:41:47.819 --> 00:41:56.079
These days I am working on meta materials
and antennas and something else. There are
00:41:56.079 --> 00:42:03.570
two laws of diffusion. One is Fick’s first
law, and the other Fick’s second law. As
00:42:03.570 --> 00:42:09.809
I said we will derive them or I will leave
it to the posting. What is Fick’s first
00:42:09.809 --> 00:42:18.790
law says? Let us say there is a semiconductor
bar, it has two planes I have made, A by root
00:42:18.790 --> 00:42:25.460
3, A by root 3 are the two planes. This Fick's
law statement, I am just showing a figure.
00:42:25.460 --> 00:42:31.500
This is called cross-sectional area. Impurities
are coming inside this area and getting inside.
00:42:31.500 --> 00:42:38.599
This is my x direction. This is my y direction
and this is my z direction in a crystal. Now
00:42:38.599 --> 00:42:45.770
it can be found by not going too detail on
this, this we derive again. I just wrote down
00:42:45.770 --> 00:42:52.750
there, maybe I said, j is called flux density.
This plus, please us this word j which is
00:42:52.750 --> 00:42:57.830
current density but here I am using a flux
density because over the years I have been
00:42:57.830 --> 00:43:01.840
using it. Some other books may give something
else.
00:43:01.840 --> 00:43:10.190
What is flux density? Number as, someone asked
you what is the definition of flux density,
00:43:10.190 --> 00:43:19.450
number of atoms or number of particles moving
per unit area per second is essentially called
00:43:19.450 --> 00:43:27.640
flux density. So when someone asked us many
years ago that why IIT Bombay and many other
00:43:27.640 --> 00:43:33.079
IITs only have electrical engineering department
and not electronic as communication instrument,
00:43:33.079 --> 00:43:37.960
I have some mixture of n of them. So I said,
my statement was simple.
00:43:37.960 --> 00:43:43.250
After all in all electrical engineering we
are interested in the electron transport and
00:43:43.250 --> 00:43:49.280
nothing else. Maybe hole is additional feature
and it is only the flux density matters. If
00:43:49.280 --> 00:43:55.000
it is very high flux density, we say it is
a power area, very large flux, large amount
00:43:55.000 --> 00:43:59.890
of currents amps, tens of amp, fifties of
or hundreds of amps. Flux density is very
00:43:59.890 --> 00:44:05.920
high actually. If it is very very small, we
say nano. In between if the signal you need
00:44:05.920 --> 00:44:11.150
moderately, flux density requires for electrons
motion.
00:44:11.150 --> 00:44:18.980
So all areas are covered essentially by number
per cc per second. So electrical engineering
00:44:18.980 --> 00:44:26.540
is only electron transport and nothing more.
So we keep working only on electron transport.
00:44:26.540 --> 00:44:32.270
So this flux density is per unit area, so
it is dn by dt. This is the statement. We
00:44:32.270 --> 00:44:39.450
will derive this later, 1 upon A, dn by dt
where n is the number which essentially we
00:44:39.450 --> 00:44:47.690
are saying per unit volume actually. Impurities
are coming and going from this plane in one
00:44:47.690 --> 00:44:52.540
or one into two and there is a diffusion process
remains.
00:44:52.540 --> 00:44:58.460
Some numbers can go from 2 to 1 but there
will be net numbers going from 1 to 2 if there
00:44:58.460 --> 00:45:04.660
is a gradient set. Now this gradient is let
us say if N2 is the number here and N1 is
00:45:04.660 --> 00:45:12.320
the number here per cc. So dn by dx is N2
minus N1 upon, if this plane distance is a
00:45:12.320 --> 00:45:18.500
root 3, so N2 minus N1 by a root 3, what is
this a root 3? The distance between the plane,
00:45:18.500 --> 00:45:24.609
which distance I am talking? Is the Miller
distance, we will see next time.
00:45:24.609 --> 00:45:30.380
You have the planes, so Miller planes, so
we will see what is the minimum distance they
00:45:30.380 --> 00:45:37.670
have along 100, 111, other planes. So if I
do this which I have done there again, please
00:45:37.670 --> 00:45:42.220
just note down, do not note down because I
am going to post this. I just wanted to rewrite
00:45:42.220 --> 00:45:44.230
because to show you.
00:45:44.230 --> 00:45:50.650
And then we define this mu a square by 6,
whatever term is coming here as diffusion
00:45:50.650 --> 00:45:57.910
coefficient or diffusion constant D. Then
I write 1 upon A dn by dt is minus D dn by
00:45:57.910 --> 00:46:06.829
dx or j which is this minus dn by dx. This
is Fick’s first law, that the flux density
00:46:06.829 --> 00:46:14.200
is related to gradient, proportional to gradient.
I repeat the flux density is proportional
00:46:14.200 --> 00:46:23.881
to gradient, this is the Fick’s first. Why
this minus sign? Gradient down, okay, minus
00:46:23.881 --> 00:46:24.881
sign.
00:46:24.881 --> 00:46:31.560
D is the diffusion coefficient or diffusion
constant and we know diffusion coefficient
00:46:31.560 --> 00:46:39.300
can be rewritten as 4 mu a square by 6, exponential
En plus Es by kT for vacancy transport. So
00:46:39.300 --> 00:46:46.829
D is equal to D0, this term is called D0 exponential
minus En plus Es by kT. So first Fick’s
00:46:46.829 --> 00:46:55.650
law says j is equal to minus dn by dx, gradient,
that is our first thing. So if the impurity
00:46:55.650 --> 00:47:01.410
concentration is higher here and lower here,
impurities will move towards the lower side,
00:47:01.410 --> 00:47:03.910
it is like a potential difference.
00:47:03.910 --> 00:47:10.030
Unless there is a potential difference, energy
does not move. The only difference there is
00:47:10.030 --> 00:47:17.030
one can say it is not a random motion. In
this case the way it is as I say probability
00:47:17.030 --> 00:47:21.320
wise, 50 percent chance going ahead, 50 percent,
but keep going plus minus, plus minus, at
00:47:21.320 --> 00:47:27.280
some number if there is a gradient, you will
be further away from the starting point. So
00:47:27.280 --> 00:47:34.440
this is essentially statement of Fick’s
first law. That the amount of impurities per
00:47:34.440 --> 00:47:41.150
unit area, per unit time at the surface of
silicon or rather when they enter silicon,
00:47:41.150 --> 00:47:44.210
it will be proportional to the gradient it
has set in.
00:47:44.210 --> 00:47:50.920
Now that is the term we want to calculate.
So we must first get j value somewhere and
00:47:50.920 --> 00:47:58.040
must get dn by dx relationship with that later.
If I can calculate Nx, that is what all that
00:47:58.040 --> 00:48:05.370
my interest is. Is that okay? So for you this
of course, you need not have written but I
00:48:05.370 --> 00:48:10.230
have written there again. But just to repeat
in case I do not, then I feel I will show
00:48:10.230 --> 00:48:11.849
where from Fick’s first law is coming.
00:48:11.849 --> 00:48:17.180
The Fick’s second law is essentially a statement,
I do not know how many of you have done your
00:48:17.180 --> 00:48:24.000
devices well but hopefully. So but any other’s
devices we bound where we talk lot of it but
00:48:24.000 --> 00:48:29.349
continuity equation has nothing to device.
Continuity of transport of any fluid, solid,
00:48:29.349 --> 00:48:40.740
gas, anything is continuity. And according
to the divergence theorem, the dn by dt, dj
00:48:40.740 --> 00:48:50.550
by dx that is the flux density gradient is
equal to minus dn by dt. This we will discuss,
00:48:50.550 --> 00:48:56.910
this is called continuity equations. So time
dependent term is related to space dependent
00:48:56.910 --> 00:49:02.041
term. This is called continuity equation,
we will derive this next time or as I say
00:49:02.041 --> 00:49:03.690
may post it.
00:49:03.690 --> 00:49:11.670
And if I use dj by dx is minus dn by dt, use
this j here, differentiate j here, so I get
00:49:11.670 --> 00:49:19.599
this equation. dn by dt is D d2n by dx square.
This is called diffusion equation. This is
00:49:19.599 --> 00:49:26.619
what we want to solve. At the end this is
what we want to solve, why? For a given time
00:49:26.619 --> 00:49:33.640
impurities are coming in and also moving in,
is that point clear why this equation is relevant?
00:49:33.640 --> 00:49:40.359
Impurities are coming inside, going with time
but also moving in space. So I am not interested
00:49:40.359 --> 00:49:46.841
in only Nx but I am also interested in Nxt.
But if t is known, I know I will only get
00:49:46.841 --> 00:49:50.170
Nx profile at the end of t. Is that clear?
00:49:50.170 --> 00:49:57.990
That is what I want to do. This equation is
my precursor of finding Nx functions. This
00:49:57.990 --> 00:50:03.420
equation is what we are going to solve now.
And once we solve this, only thing catch word
00:50:03.420 --> 00:50:12.010
in this maybe I have said it, okay. Here I
assume D is a function of nothing, that is
00:50:12.010 --> 00:50:19.780
constant. D is independent of everything.
But in real life that is not so. D is a function
00:50:19.780 --> 00:50:24.230
of concentration itself. You can understand
some way. If there are larger atom, the other
00:50:24.230 --> 00:50:31.030
impurities will require more effort to get
in. So it is a gradient dependent term. So
00:50:31.030 --> 00:50:37.490
if Nx are present, that means D will be get
affected by N itself. N is larger, D will
00:50:37.490 --> 00:50:43.170
be smaller you take from me. Because they
will be stopped by some other people. It is
00:50:43.170 --> 00:50:44.170
a crowd business.
00:50:44.170 --> 00:50:52.410
Since D is a function of Nx, D is a function
of x, so if I differentiate J, D dn by dx,
00:50:52.410 --> 00:51:00.830
then I must write this as a function of x.
And if I then differentiate, I will get two
00:51:00.830 --> 00:51:06.950
terms. One related to Dd by dx, the other
related to d2n by dx square. Now this term
00:51:06.950 --> 00:51:16.290
many cases, this equation is not linear equation,
it is a non-linear equation. And therefore
00:51:16.290 --> 00:51:20.970
analytically cannot be solved easily. Some
people can do by linearization. If you are
00:51:20.970 --> 00:51:26.900
expert in maths, there are certain condition
in which you can linearize it.
00:51:26.900 --> 00:51:33.260
If you cannot, what is the easiest way? Own
a system and solve numerically this equation.
00:51:33.260 --> 00:51:39.831
Any non-linear second order differential equation
can be solved by N methods. Whichever method
00:51:39.831 --> 00:51:46.099
you prefer, you can solve. Linearize it also,
by then come to Gauss- Seidel, Gauss–Euler,
00:51:46.099 --> 00:51:52.630
whichever method, you can choose methods.
N methods are solving second order non-linear
00:51:52.630 --> 00:51:58.880
differential equations. I will assume right
now linearity for analytical purpose. But
00:51:58.880 --> 00:52:05.900
in real life the models which I will substitute
in software for process simulation, I will
00:52:05.900 --> 00:52:08.579
use D as a function of N itself.
00:52:08.579 --> 00:52:14.191
And let it take, because there is a grid,
it will find what N and find D there. Why
00:52:14.191 --> 00:52:21.470
should I care for it? But why I care many
times? Even if in a software when I write,
00:52:21.470 --> 00:52:28.690
what is the criteria I normally put for writing
good software? Time taken to solve is the
00:52:28.690 --> 00:52:34.549
major criterion writing a good software. You
may have very interesting software written
00:52:34.549 --> 00:52:38.150
but if it takes ages to solve, then there
is no point in using that.
00:52:38.150 --> 00:52:44.801
So as much as simplicity you can create, so
put some small model inside which may not
00:52:44.801 --> 00:52:50.530
be accurate but enough for that and partly
linearize it. So there are tricks in all modeling
00:52:50.530 --> 00:52:57.601
people, they keep using some tricks and then
say oh, so fast it works. It works fast because
00:52:57.601 --> 00:53:04.520
you have assumed some few things. If you do
not, it takes hours or ages. So please remember
00:53:04.520 --> 00:53:11.049
you will do only linear system because that
is easy to solve analytically. Real life,
00:53:11.049 --> 00:53:18.780
since this is not very strong term, D by dx,
Dd by dx, so for first order this term can
00:53:18.780 --> 00:53:24.299
be neglected and you can use only first ones.
00:53:24.299 --> 00:53:32.160
The first thing we start is now looking into
profiles, that is our ultimate, that is what
00:53:32.160 --> 00:53:41.200
we are going to do, work at. So we start with
profiles. First, let me say and then draw.
00:53:41.200 --> 00:53:46.339
This is silicon surface and as I say this
is the depth in silicon surface, shown x,
00:53:46.339 --> 00:53:58.680
this is the silicon surface. This is silicon
surface, this is silicon wafer of thickness
00:53:58.680 --> 00:54:06.250
t and t is very large compared to anything.
t is, for a boundary condition what will say?
00:54:06.250 --> 00:54:11.180
t equal to infinite, means our thickness is
infinite.
00:54:11.180 --> 00:54:19.859
So I have impurities introduced from surface
side. Essentially wafers sit in a rack like
00:54:19.859 --> 00:54:30.510
this and source of impurities are impinging
on it in any technique. I assume and that
00:54:30.510 --> 00:54:39.240
is very important, in time frame, okay maybe
we will come back to this later. Is that model
00:54:39.240 --> 00:54:45.839
clear what I am saying? Impurities are impinging
at x is equal to 0 at which is silicon surface
00:54:45.839 --> 00:54:53.960
and they will get inside silicon along the
x axis. The assumption is it is isotropic
00:54:53.960 --> 00:54:59.700
diffusion, means y and z do not play. It is
not true, actually I should do Nxyz or delta
00:54:59.700 --> 00:55:02.190
as a term we should solve for.
00:55:02.190 --> 00:55:10.510
But most cases this is good enough. dn by
dt is D d2n and this is our diffusion equation.
00:55:10.510 --> 00:55:17.470
Just now we wrote, Fick’s second law. We
take, of course this we will come back. This
00:55:17.470 --> 00:55:24.410
is the condition I am putting because I need
to solve, so initial condition I create. I
00:55:24.410 --> 00:55:29.049
take a Laplace transform, I hope 99.9999 people
know Laplace transforms. If not, what you
00:55:29.049 --> 00:55:35.250
did? At least communication people if they
do not know Laplace transform, 4-year transform,
00:55:35.250 --> 00:55:40.730
they will not be within communication, next
day bahar khada kar denge.
00:55:40.730 --> 00:55:48.369
Microwave terms mein chal jata hai. loag match
kartehi nahi udhar, so udhar chal jate hai.
00:55:48.369 --> 00:55:57.280
Of course, this is a trivial maths, any network
person must know it. S NxS is equal to the
00:55:57.280 --> 00:56:04.380
Laplace transform or this is S Nx S, minus
the initial condition, Nxt equal to 0, equal
00:56:04.380 --> 00:56:11.310
to D times d2Nx S upon dx square. This is
the Laplace transform or diffusion equation.
00:56:11.310 --> 00:56:18.710
I can rearrange these diffusion equations
slightly, is it okay? Nahi, kisiko doubt hai
00:56:18.710 --> 00:56:24.540
toh bol dena, there is nothing wrong with
this. Laplace transform sikh lijiye baba,
00:56:24.540 --> 00:56:31.859
nahi hota toh aisa nahi chalega. Life is very
tough without transforms. Is it okay?
00:56:31.859 --> 00:56:41.869
I rearrange this, that equation again. I write
d2Nx S by dx square is S by D, please remember
00:56:41.869 --> 00:56:48.559
Laplace transform is only for time, x does
not change. So d2Nx S upon dx square is S
00:56:48.559 --> 00:57:00.400
by D Nx S minus Nxt0 by D which is why initial
condition term. Now here is my to solve this
00:57:00.400 --> 00:57:06.410
equation, this is very easy to solve if I
know this. This equation I can solve, if I
00:57:06.410 --> 00:57:12.040
know this, that means I must know my initial
condition. So I have conditions which I impose
00:57:12.040 --> 00:57:16.010
myself and say this is my initial condition.
00:57:16.010 --> 00:57:23.579
So let us assume, if you have written down
the formula which is trivial, let us assume
00:57:23.579 --> 00:57:29.460
impurity source provides impurities at the
surface which is x is equal to 0 at t is equal
00:57:29.460 --> 00:57:35.770
to 0. So that means, what does that mean?
Prior to t equal to 0, there are no impurities.
00:57:35.770 --> 00:57:43.880
At t is equal to 0, source starts. Is that
clear? Prior to t equal to 0, there is but
00:57:43.880 --> 00:57:50.980
once it starts, it never ends. Certain number
of atoms per cc are constantly available to
00:57:50.980 --> 00:57:53.359
me infinite times.
00:57:53.359 --> 00:58:02.400
Just a minute before I show this. At t is
equal to, less than equal to 0, no impurities
00:58:02.400 --> 00:58:08.760
are impinging. At t is equal to 0, the available
concentration is in N0 which remains constant
00:58:08.760 --> 00:58:14.800
for all times to come. This is my initial
condition which I start with and this is real
00:58:14.800 --> 00:58:20.090
life condition, that is why I did it. Is that
okay?
00:58:20.090 --> 00:58:24.570
At t is equal to 0, no impurities, we start
the source at t is equal to 0 and some fixed
00:58:24.570 --> 00:58:33.520
number appears for all time to come. Which
means Nxt0 is less than equal to 0. However
00:58:33.520 --> 00:58:40.530
t equal to 0, plus what I said, I have written
again. Source of impurities at the surface,
00:58:40.530 --> 00:58:49.359
thus impurity source is unit step function
in time as shown. Our next assumption, yeh
00:58:49.359 --> 00:58:57.460
to ekk condition lag gai, what is the other
assumption I need or other conditions I need?
00:58:57.460 --> 00:59:02.530
x ki condition chahiye naa, t ka toh dikha
diya, ab x ka bhi chhahiye. How many conditions
00:59:02.530 --> 00:59:08.780
you need, boundary conditions? Second order
equation need two BCs, so let us see which
00:59:08.780 --> 00:59:15.000
are the two boundary conditions we have. Our
next, first assumption is it is unit source.
00:59:15.000 --> 00:59:23.720
t is equal to 0, then it starts, I have constant
source available or also called infinite source,
00:59:23.720 --> 00:59:29.360
why? It keeps coming, there is no stopping
on that. So either it is called infinite source
00:59:29.360 --> 00:59:36.940
diffusion or called constant source diffusion.
All the time in infinite or constantly available
00:59:36.940 --> 00:59:39.420
for all times.
00:59:39.420 --> 00:59:46.440
The second BC or secondly we want to see BCs,
so we say our next assumption is that impurity
00:59:46.440 --> 00:59:53.140
source keeps constant impurity concentration
at x is equal to 0. At the surface we always
00:59:53.140 --> 01:00:00.539
get N0 whatever number all the time. At x
is equal to 0, this number is fixed, how much?
01:00:00.539 --> 01:00:08.750
N0. And how much will be N0 roughly? Solid
solubility because at that temperature the
01:00:08.750 --> 01:00:16.170
maximum available to enter there is so much.
So N0 will be actually you pick from solid
01:00:16.170 --> 01:00:21.160
solubility graph, is that correct? N0 will
be picked up from solid solubility graph.
01:00:21.160 --> 01:00:28.410
Because we know at that temperature how much
N0 can reach at the just below surface of
01:00:28.410 --> 01:00:30.480
the silicon.
01:00:30.480 --> 01:00:36.079
Okay, This value is defined as N0 because
that is the number which we were constantly
01:00:36.079 --> 01:00:43.190
pushing. So we said x is equal to 0, this
number is fixed. So the first boundary condition
01:00:43.190 --> 01:00:49.040
therefore say, Nxt0 plus onwards is N0 which
is constant.
01:00:49.040 --> 01:01:00.799
Now once I know my initial conditions, I know
the equations which are wrote, has equation
01:01:00.799 --> 01:01:09.140
analytical equation given by, solution is
AS exponential under root of S by D into x
01:01:09.140 --> 01:01:16.619
plus BS, exponential minus S by D to the power
half x. This is the solution of second order
01:01:16.619 --> 01:01:24.109
differential equation which I have used. This
is very simple, second order differential
01:01:24.109 --> 01:01:28.680
equation ka simplest solution ye hai. Now
the first boundary condition you have said
01:01:28.680 --> 01:01:34.480
here, the second boundary condition, boundary
is what? First is x is equal to 0, where is
01:01:34.480 --> 01:01:38.160
the second boundary? Far away.
01:01:38.160 --> 01:01:44.549
x is equal to infinite. If the impurities
are coming in with the constant this, why
01:01:44.549 --> 01:01:51.380
should it become 0? All impurities will go
there naa. Why should it go 0? So there is
01:01:51.380 --> 01:01:55.589
an issue, means this is what you are saying.
If infinite source is there, the infinite
01:01:55.589 --> 01:02:01.920
N it will reach because there is no stopping
it, it just goes there. Now the problem is
01:02:01.920 --> 01:02:10.150
if I put first x is equal to infinity in this
term, what does that mean? This is positive,
01:02:10.150 --> 01:02:16.960
S by D is constant, x is positive, x is infinite,
what does that mean?
01:02:16.960 --> 01:02:24.869
This Nx will become infinite but I already
told you diffusion is always gradient based.
01:02:24.869 --> 01:02:32.750
So obviously impurity concentration cannot
reach from N0 to infinite, that is not possible.
01:02:32.750 --> 01:02:41.490
So what should be happening? AS must be unequivocally
0, the first constant pre-exponent must be
01:02:41.490 --> 01:02:48.810
unequivocally 0. Is that point clear? If x
is equal to infinity, the first term will
01:02:48.810 --> 01:02:53.309
go to infinite, that means concentration will
reach infinite which is never possible because
01:02:53.309 --> 01:02:59.410
impurity will diffuse down with the gradient
which essentially means the first term must
01:02:59.410 --> 01:03:05.810
vanish, which means AS must be guaranteedly
0.
01:03:05.810 --> 01:03:13.329
If that is so, the second term has removed
the first term itself, second boundary condition.
01:03:13.329 --> 01:03:20.140
Then I get NxS is BS, exponential minus S
by D to the power half into x. So this is
01:03:20.140 --> 01:03:29.089
the solution but what is still not known to
me? BS, first AS, because of the second boundary
01:03:29.089 --> 01:03:34.819
I just removed that. But now I must know BS,
so I look into the real life situation. Let
01:03:34.819 --> 01:03:42.570
us see what it happens. Is that okay? Please
note down. This is the solution. If x goes
01:03:42.570 --> 01:03:47.549
to infinity, this term blows and since this
term blows, it is again the principle of diffusion
01:03:47.549 --> 01:03:55.809
and therefore AS must be 0. So the actual
solution in this specific case is BS times
01:03:55.809 --> 01:04:01.510
exponential minus S by D to the power half,
is that okay everyone?
01:04:01.510 --> 01:04:07.380
The first of course initial condition I already
showed, a step function of source I have introduced.
01:04:07.380 --> 01:04:14.359
The second is I say at the surface the concentration
is fixed, that is what I said, is N0, solid
01:04:14.359 --> 01:04:18.790
solubility limit. That is the number which
is available at x is equal to 0. If you are
01:04:18.790 --> 01:04:27.349
not very satisfied, say x is 0 plus because
at the surface we do not know outside, but
01:04:27.349 --> 01:04:34.050
just below surface you can say or at the surface
we say the concentration is N0.
01:04:34.050 --> 01:04:42.820
If I know this boundary condition, x is Nx,
so we can see. From below that, we do not
01:04:42.820 --> 01:04:49.980
know, in the silicon there is no concentration.
x minus, there is no concentration. But just
01:04:49.980 --> 01:05:00.940
at x is equal to 0, it becomes N0. Now this
essentially means Nx0 at all times, your impurity
01:05:00.940 --> 01:05:09.910
source are coming anyway, all times is N0
ut. Is that correct? It is a step function.
01:05:09.910 --> 01:05:17.349
So if this is my second boundary or rather
first boundary condition, that at x is equal
01:05:17.349 --> 01:05:24.891
to 0, concentration is N0 ut. Why this ut
has to be done? Because step, so I have to
01:05:24.891 --> 01:05:31.940
because in Laplace transform it will give
something. What will it give? What is Laplace
01:05:31.940 --> 01:05:41.039
transform, N0 ut? N0 by S. So we have to take
care of that ut term, constant by S. ut is
01:05:41.039 --> 01:05:45.410
only giving me that constancy.
01:05:45.410 --> 01:05:51.280
This condition is also, the kind of boundary
condition initially when I used is also called
01:05:51.280 --> 01:05:56.731
infinite source condition or constant source
condition. If you have a constant source,
01:05:56.731 --> 01:06:05.690
so we have Nx0 S is N0 by S taking Laplace
transform the second boundary condition, rather
01:06:05.690 --> 01:06:14.461
first boundary condition. So we write Nx0
S is N0 by S equal to BS exponential 0. x
01:06:14.461 --> 01:06:23.690
is equal to 0, so exponential 0, so we get
BS is equal to N0 by S. Is that correct? Second
01:06:23.690 --> 01:06:27.900
boundary condition, first boundary condition
removed AS.
01:06:27.900 --> 01:06:35.220
The first boundary condition give me the BS
is equal to N0 by S. This is just substitution
01:06:35.220 --> 01:06:45.460
of x is equal to 0 in the equation. And therefore
the infinite source or a constant source case,
01:06:45.460 --> 01:06:55.890
the solution of diffusion equation is NxS
is N0 by S, exponential under root of minus
01:06:55.890 --> 01:07:06.220
S by D times x. This is the solution of diffusion
inside silicon when starts with constant source
01:07:06.220 --> 01:07:14.010
at x0, constantly N0. This is the equation
you get for it. Is that okay, solution? So
01:07:14.010 --> 01:07:19.260
now I have the diffusion equations solving
done for profile, this is my profile which,
01:07:19.260 --> 01:07:26.650
this is in what this, this is explained and
I want to come back to time frame.
01:07:26.650 --> 01:07:33.849
So what should I do? Take inverse Laplace
transform of this, kay hoga? Anyway it is
01:07:33.849 --> 01:07:38.559
not so easy for you and you have not seen
that function, so do not try unless you have
01:07:38.559 --> 01:07:41.770
done a course somewhere or done something.
01:07:41.770 --> 01:07:48.839
Taking inverse Laplace transform, Nxt is N01
minus error function of x upon 2 root Dt.
01:07:48.839 --> 01:07:56.319
That is very important. I will explain the
error function soon quickly before we leave.
01:07:56.319 --> 01:08:06.570
Can anyone tell me what is the unit of this?
D is always defined as centimeter square per
01:08:06.570 --> 01:08:13.900
second. Is that correct? Into time, under
root of that means it is centimeter if everything
01:08:13.900 --> 01:08:24.130
is, so Dt is, root Dt is essentially distance,
is that correct? Root Dt is essentially a
01:08:24.130 --> 01:08:30.839
distance, we will see D is a function of temperature
T, some temperature dependence.
01:08:30.839 --> 01:08:39.700
So what does that mean? If Dt term which is
temperature dependent and time dependent,
01:08:39.700 --> 01:08:48.270
so for a given temperature for a given time,
I have fixed root Dt, 2 root Dt. Is that clear?
01:08:48.270 --> 01:08:55.870
So now I know where the impurities are going
for this time and temperature at every x.
01:08:55.870 --> 01:09:01.890
Is that point clear? D is fixed for a given
temperature, T I have fixed, okay I will do
01:09:01.890 --> 01:09:08.050
1 hour diffusion, so I know the time. Please
remember everywhere we do seconds, so 1 hour
01:09:08.050 --> 01:09:10.680
yaani 3600 seconds. Is that okay to you?
01:09:10.680 --> 01:09:20.750
So please 3600, do not mischief 1 there. So
if I plot this function, normally Nx0 by N0
01:09:20.750 --> 01:09:29.290
versus y, y I define x by 2 root Dt, I define
y to plot, nothing this. So if I say between
01:09:29.290 --> 01:09:39.000
plus y and minus y, this is symmetric function.
It is initial value is this, and as time proceeds,
01:09:39.000 --> 01:09:47.320
and at where it will all go finally? Infinite.
It will go to 0 and infinite. Some way this
01:09:47.320 --> 01:09:52.850
has reached to 1, actually it is isotropic,
it should reach to 1 isotropically. Now this
01:09:52.850 --> 01:10:03.440
function is called error function. Now for
those last slide for the day, this is please
01:10:03.440 --> 01:10:04.720
note down this.
01:10:04.720 --> 01:10:11.120
Some data about error functions. And since
I am going to use this constant source diffusion
01:10:11.120 --> 01:10:18.160
very often, I first want to give little expression
for error function because those will be used
01:10:18.160 --> 01:10:25.470
directly by me in my solving the actual profile
evaluations. Is that okay? Everyone, noted
01:10:25.470 --> 01:10:31.150
down? Those who wish to, everyone does not
but those who wish to.
01:10:31.150 --> 01:10:42.740
Error function x is, has a definition in maths,
2 upon root pi, it is an integral 0, 2 upon
01:10:42.740 --> 01:10:48.070
root pi, 0 to x, e to the power minus alpha
square. Alpha is any other parameter, variable.
01:10:48.070 --> 01:10:57.020
So you can write y, you can write z, any parameter.
So e to the power minus alpha square d alpha.
01:10:57.020 --> 01:11:02.520
This is the definition of error function,
plots I have already shown you. If you see
01:11:02.520 --> 01:11:09.900
this integral which is shown bottom, please
see the last line, 0 to z, e minus y square
01:11:09.900 --> 01:11:11.250
dy.
01:11:11.250 --> 01:11:19.160
If I expand them in series, it will be y minus
y cube by 3 into 1 factorial, y5, plus. 5
01:11:19.160 --> 01:11:26.500
into 3 factorial minus y to the power 7, 7
into 3 factorial and so on and so forth. So
01:11:26.500 --> 01:11:34.000
this series is essentially e to the power
minus y square dy, 0 to z, whichever it is.
01:11:34.000 --> 01:11:46.020
I should not say z, I should say y only. So
this expression is a series but I know this
01:11:46.020 --> 01:11:52.950
is the expression I need and this is 2 upon
root pi, 0 to x, e to the power, this is called
01:11:52.950 --> 01:12:01.260
error function. Error function at 0, you can
take from me. When it is 0, all ys are 0,
01:12:01.260 --> 01:12:04.250
so what is the sum? 0.
01:12:04.250 --> 01:12:11.750
So one can say error function 0 is 0. You
say from the series, everyone has x term,
01:12:11.750 --> 01:12:21.040
so x, 0, 0, 0, 0, 0, 0, everywhere 0. So error
function x is equal to 0 is always 0. Error
01:12:21.040 --> 01:12:27.970
function x equal to infinite is very important.
2 upon root pi, 0 to infinite, e to the power
01:12:27.970 --> 01:12:34.810
minus alpha square d alpha, now this integral
0 to infinity e to the power minus alpha square
01:12:34.810 --> 01:12:43.021
d alpha, mathematically can be derived as
for series some of this is root pi by 2, is
01:12:43.021 --> 01:12:50.620
root, this is a slightly diverging series
and difficult to sum up. But 1 minus t kind
01:12:50.620 --> 01:12:53.800
of equivalence can be done and you can sum
it up.
01:12:53.800 --> 01:13:01.680
So it gives you root pi by 2. 0 to infinite,
e to the power minus alpha square d alpha
01:13:01.680 --> 01:13:12.340
is root pi by 2. So if I put here this 2 by
root pi into root pi by 2, means exponential,
01:13:12.340 --> 01:13:21.900
sorry error function infinity is 1. And that
graph was shown to the 1, it will go to the
01:13:21.900 --> 01:13:22.950
1 always, maximum.
01:13:22.950 --> 01:13:29.810
Okay, there is few more terms. We actually
our profile which we are going to get is 1
01:13:29.810 --> 01:13:35.770
minus error functions and that is called since
1 is the infinite part, so you subtract rest
01:13:35.770 --> 01:13:43.560
is compliment to that. So complimentary error
function, erscx, this is remember, error,
01:13:43.560 --> 01:13:51.890
infinity minus error function x, error function
infinity is 1, this term. So 1 minus error
01:13:51.890 --> 01:13:58.510
function x, is called complimentary error.
You can think like this, in integral 0 to
01:13:58.510 --> 01:14:03.750
x, x to infinity. So that is essentially doing
the same job.
01:14:03.750 --> 01:14:09.730
If I differentiate error function, it is 2
upon root pi, exponential minus x, this is
01:14:09.730 --> 01:14:16.140
the most important differential really because
it is essentially, which is this term coming?
01:14:16.140 --> 01:14:23.000
Exponential minus x square kya define karata
hein? It is a normalized x as of now. We have
01:14:23.000 --> 01:14:29.490
defined, it is a Gaussian profile. So aap
error function se Gaussian mein jayenge, that
01:14:29.490 --> 01:14:35.600
is what exactly we are going to see this in
the next time. Similarly if I take second
01:14:35.600 --> 01:14:41.370
order differential, then error function as
x is, iska differential karo. Minus 4 upon
01:14:41.370 --> 01:14:44.850
pi, x e to the power minus 2 x square.
01:14:44.850 --> 01:14:52.140
These are the error function terms which you
note down because I will be assuming that
01:14:52.140 --> 01:14:58.330
you know error function algebra, so we substitute
whenever any differential second order, first
01:14:58.330 --> 01:15:04.250
order comes or infinite 0 comes. We can just
substitute there as it is. Is that okay?
01:15:04.250 --> 01:15:11.840
So we have found from our diffusion equation
before we quit that Nxt is N0, complimentary
01:15:11.840 --> 01:15:18.661
error function of x upon 2 root Dt. This is
the diffusion profile which I got, for which
01:15:18.661 --> 01:15:25.750
case? Which is the case I discussed today?
Constant source or infinite source whichever
01:15:25.750 --> 01:15:31.620
book we are using, some use infinite source,
some use constant source. This will give always
01:15:31.620 --> 01:15:34.170
complimentary error function profiles.