WEBVTT
Kind: captions
Language: en
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I have actually checked on the website the
Amazon India has this book, this is the Ad
00:00:23.420 --> 00:00:29.310
for that. I am, if you have not purchased
it so far, or could not, this is what recent
00:00:29.310 --> 00:00:40.860
I saw yesterday. It is 718 rupees or whatever
possible and this is second edition, 1st January
00:00:40.860 --> 00:00:49.390
2009. This is available at website www.Amazon.in
Silicon VLSI Technology. This is written here,
00:00:49.390 --> 00:00:55.230
you can take, they say five days or five to
eight business days.
00:00:55.230 --> 00:01:00.070
So in case you have not purchased or do not
want to purchase, is yours. But in case you
00:01:00.070 --> 00:01:07.311
want to, this you can note down or maybe you
can go on Google and say Amazon India. So
00:01:07.311 --> 00:01:14.980
whichever way it is, so do not tell me that
book is not available or something, something.
00:01:14.980 --> 00:01:22.979
Is that okay? Coming back, and those who want
to write that website, maybe you can. In case
00:01:22.979 --> 00:01:25.439
you feel that is what I am marking.
00:01:25.439 --> 00:01:34.280
We were other day looking for Bruce Deal model,
Bruce Deal and Andrew Grove model for oxidation.
00:01:34.280 --> 00:01:40.289
The first paper published in this area way
back in 65 and this model is very popular
00:01:40.289 --> 00:01:47.859
in the literature called Deal-Grove model.
And I just told you other day, Grove was the
00:01:47.859 --> 00:01:56.829
one of the CEO or Chief of Intel from say
1992 to 1997. He is the scientist or CEO of
00:01:56.829 --> 00:02:04.689
Intel. Andy Grove also has a book on physics
and technology of semiconductors or rather
00:02:04.689 --> 00:02:07.299
technology of semiconductor which is old book.
00:02:07.299 --> 00:02:13.300
But many things remain same irrespective whether
today or yesterday. So one can also look for
00:02:13.300 --> 00:02:20.750
Grove’s book in the library. There is another
book on VLSI technology by Wolf, which is
00:02:20.750 --> 00:02:27.570
also very good which is rather recent compared
to Plummer of course is older. But around
00:02:27.570 --> 00:02:34.440
2006 Wolf has also published. So any of these
books are good enough, please do not say that
00:02:34.440 --> 00:02:38.560
the books are not there, because of that whatever
happened.
00:02:38.560 --> 00:02:46.360
At least do not blame books. So the assumption
what he did was that initially there is a
00:02:46.360 --> 00:02:53.410
finite thin oxide and at t is equal to 0 minus,
this is the oxide available and t is equal
00:02:53.410 --> 00:03:02.280
to 0 plus, oxidant attacks this silicon layer,
silicon dioxide layer and starts the oxidation.
00:03:02.280 --> 00:03:06.450
We did last time, I am just trying to recapitulate.
00:03:06.450 --> 00:03:12.120
Then the oxidants we see then, reacts with,
diffuses through the thin oxide, reaches to
00:03:12.120 --> 00:03:19.440
the silicon-silicon dioxide interphase, reacts
with silicon and forms SiO2. And we have already
00:03:19.440 --> 00:03:25.850
last time said there are three fluxes, F1
in the gas phase, F2 in the oxide, and F3
00:03:25.850 --> 00:03:32.580
near the interphase of silicon-silicon dioxide.
And we also discussed other day that in steady
00:03:32.580 --> 00:03:40.950
state if the oxide, oxidant is coming, diffusing,
reacting, the system has to be in steady state.
00:03:40.950 --> 00:03:47.110
And in steady state all fluxes must be equal.
This we have done last time, I am just trying
00:03:47.110 --> 00:03:49.570
to push it again.
00:03:49.570 --> 00:03:56.290
And we say F1 is equal to F2 equal to F3 and
that we call it F. And we will show you this
00:03:56.290 --> 00:04:01.700
process again but this is just to show you
how oxidation proceeds. Inside a furnace where
00:04:01.700 --> 00:04:06.390
temperature is kept 800 to 1200, wafers are
vertically start on quartz rack. Oxygen is
00:04:06.390 --> 00:04:16.840
entering here and oxidizes the silicon. So
let us start with the model, this is what
00:04:16.840 --> 00:04:21.030
we did last time. So now we actually look
for the model.
00:04:21.030 --> 00:04:28.760
Here is the model, same thing what I wrote
but there are now few more definitions. Let
00:04:28.760 --> 00:04:37.060
us say CG is the oxidant gas or oxidant concentration
in the gas stream. That is not near the surface
00:04:37.060 --> 00:04:45.380
but inside the whole tube actually. This is
called CG, then corresponding to this CG we
00:04:45.380 --> 00:04:55.030
have a oxidant concentration at oxide interface
is CS or rather C0. Or sorry C0, forget it,
00:04:55.030 --> 00:05:02.150
CS we have, CS is the concentration here.
So there is a gradient because there is no
00:05:02.150 --> 00:05:08.389
enough oxygen here or oxidant here, so there
is a gradient from CG to CS.
00:05:08.389 --> 00:05:15.550
Now these are called gas phase concentrations.
So equilibrium concentration in, at the solid
00:05:15.550 --> 00:05:22.249
also can be found and we define C star as
the equilibrium oxidant concentration in solid
00:05:22.249 --> 00:05:30.569
related to this CG. CG is the gas phase concentration
and C star is the equivalent of that in the
00:05:30.569 --> 00:05:39.120
solid phase. Whereas C0 is similarly is equilibrium
constant, oxidant concentration in solid related
00:05:39.120 --> 00:05:44.740
to CS. So this relates to here and this relates
to here.
00:05:44.740 --> 00:05:50.629
This is, then we say initial oxide thickness
is CI, initial concentration at the interface
00:05:50.629 --> 00:06:00.440
is CI. And we say already xo oxide thickness
exist or xi rather but xo is the oxide thickness
00:06:00.440 --> 00:06:07.250
which we want to find at a given time and
temperature. So let us look at the fluxes
00:06:07.250 --> 00:06:16.240
now. According to Deal-Grove model flux F1
which is in the gas phase from the ambient
00:06:16.240 --> 00:06:20.509
to the surface, we know the flux will be proportional
to the gradient.
00:06:20.509 --> 00:06:27.120
If CG is the gas concentration in the bulk,
CS is the gas concentration near the surface,
00:06:27.120 --> 00:06:33.199
then F1 must be proportional to CG minus CS.
Whatever at the surface and what is available
00:06:33.199 --> 00:06:39.750
in the stream, the gradient is set and that
difference between the two is the flux F1
00:06:39.750 --> 00:06:50.490
and the proportionality constant is called
mass transfer coefficient hG. So F1 is hG
00:06:50.490 --> 00:06:56.310
times CG minus CS. This is the first flux
which we are looking into, that is the gas
00:06:56.310 --> 00:06:57.830
phase flux.
00:06:57.830 --> 00:07:04.639
However we are more interested to know there
are terms which probably you should learn
00:07:04.639 --> 00:07:10.050
from thermodynamics but there are words called
pressure, total pressure and there are words
00:07:10.050 --> 00:07:15.480
called partial pressures. Partial pressure
is defined as the pressure of a gas inside
00:07:15.480 --> 00:07:25.460
a stream in a given temperature in a given
area. So if I know F1 is hG, this I want to
00:07:25.460 --> 00:07:31.249
replace this CG and CS which we are not known
to me. CG and CS cannot be monitored, so I
00:07:31.249 --> 00:07:36.520
will try to know is there equivalence of these
in terms of partial pressures.
00:07:36.520 --> 00:07:42.270
And if I know the partial pressures, then
I also know correspondingly to that partial
00:07:42.270 --> 00:07:46.370
pressure, using Hertz equation we will find
out what is the solid phase concentration
00:07:46.370 --> 00:07:52.341
for this partial pressures. So at the end
I am interested to correlate CG, CS with C
00:07:52.341 --> 00:07:58.379
star and C0. This is what I am looking for.
So the first thing I do is after I write hG
00:07:58.379 --> 00:08:05.740
into CG minus CS as the first flux and whereas
I repeat hG is called mass transfer coefficients.
00:08:05.740 --> 00:08:14.080
We define CG by ideal gas law, P is equal
to RT, there is nothing great about. This
00:08:14.080 --> 00:08:21.389
is concentration, so P is equal to RT. So
CG is equal to PG by kT. N is number one,
00:08:21.389 --> 00:08:28.009
molecular this, so we, RkT is, NkT can be
written as N is equal to 1. Therefore RT is
00:08:28.009 --> 00:08:38.979
kT. So C, sorry CG is PG by kT. CS is PS by
kT where PG, PS are the partial pressures
00:08:38.979 --> 00:08:46.730
of the oxidant at gas ambient and oxide surface.
Again same, PG in the gas stream and PS is
00:08:46.730 --> 00:08:52.820
the at near the interface of oxide and gas
stream.
00:08:52.820 --> 00:09:01.981
So if I substitute CG and CS in the last equation
by PG and PS term, then I get F1 is equal
00:09:01.981 --> 00:09:12.810
to hG PG by kT minus PS by kT or is equal
to hG by kT into PG minus PS. Please remember
00:09:12.810 --> 00:09:21.320
if I fix temperature and if I fix the gas
flows, total gas flows, hG is the constant
00:09:21.320 --> 00:09:27.130
which is the mass transfer coefficient proportional
to the total pressures. We will see in case
00:09:27.130 --> 00:09:31.779
of CVD. We invoke Henry’s law for gases
and fluids.
00:09:31.779 --> 00:09:38.350
There is Henry law which actually relates
the partial pressure to solid state concentration.
00:09:38.350 --> 00:09:45.269
This is called Henry’s law. According to
Henry's law the C star which is the equilibrium
00:09:45.269 --> 00:09:52.320
oxide concentration in solid state is proportional
to the partial pressure in gas stream. Similarly
00:09:52.320 --> 00:09:58.290
C0 which is at the surface of SiO2, C0 is
proportional to partial pressure at the surface
00:09:58.290 --> 00:10:03.660
which is PS. And there is a constant associated
with this equilibrium, equivalent to this
00:10:03.660 --> 00:10:07.130
which is called Henry's constant.
00:10:07.130 --> 00:10:19.490
So C star is H times PG, C0 is H time PS where
H is called Henry's constant. So now we can
00:10:19.490 --> 00:10:26.120
see I have relationship with C star and C0
in terms of PG and PS through Henry's constant,
00:10:26.120 --> 00:10:34.420
so I can use these two equations to go back
into 3 and find the flux F1. What is the method?
00:10:34.420 --> 00:10:42.019
First I converted CG and CS into equivalent
partial pressures by mass transfer coefficients.
00:10:42.019 --> 00:10:46.589
The partial pressures are related to their
equivalent solid state concentrations by using
00:10:46.589 --> 00:10:50.800
Henry's law or Henry, H is the Henry’s constant.
00:10:50.800 --> 00:10:58.630
So now I have C star and C0 which are at the
surface. Please remember C star and C0 are
00:10:58.630 --> 00:11:03.470
at the surface of SiO2 which are in solid
phase, that is the oxidant we are going to
00:11:03.470 --> 00:11:13.970
actually diffuse through. So F1 therefore,
is that okay? I repeat first we relate CG
00:11:13.970 --> 00:11:20.660
to PS, PG. CS, CG, PS, PG and then we wrote
this equation. Then we say okay, using, invoking
00:11:20.660 --> 00:11:27.529
the Henry’s law for fluids C star is proportional
to PG, C0 is proportional to PS. Then C star
00:11:27.529 --> 00:11:38.240
is H time, H is H PG and C0 with H time PS.
And now I substitute PG, PS from here in this
00:11:38.240 --> 00:11:39.389
equation 3.
00:11:39.389 --> 00:11:52.959
If I do that, F1 is hG by kT, C star by H
minus C0 by H. Or F1 is hG by HkT C star minus
00:11:52.959 --> 00:12:00.899
C0. And I can redefine this hG upon HkT which
is all constant at a temperature and marks
00:12:00.899 --> 00:12:10.459
total flow and pressure. Then h is hG upon
HkT is redefined as term h, which of course
00:12:10.459 --> 00:12:20.829
is proportional to mass flow. So F1 is h times
C star minus C0. So first flux in the gas
00:12:20.829 --> 00:12:28.040
phase is related to solid phase concentration
difference, h times C star minus C0, so this
00:12:28.040 --> 00:12:29.540
is the first flux we obtained.
00:12:29.540 --> 00:12:38.459
Now as I say what is our game is to find F2,
find F3 and then what do we do? We write F1
00:12:38.459 --> 00:12:45.899
is equal to F2 equal to F3 and then solve.
Ultimately what I am really trying to do is
00:12:45.899 --> 00:12:53.899
I want to find dx0 by dt. What is dx0 by dt?
Rate of oxide growth, that is what I am interested
00:12:53.899 --> 00:12:59.089
in, rate of oxide growth. So if I know the
time, I know how much oxide, I will be growing
00:12:59.089 --> 00:13:07.020
in case of oxidation. So that is my purpose.
So first flux, F1 I got. Now secondly if we
00:13:07.020 --> 00:13:12.260
look at our figure again, just a minute.
00:13:12.260 --> 00:13:20.470
If we look at this figure, F1 we have found
in terms of C star, C0. F2 is the flux which
00:13:20.470 --> 00:13:28.600
is diffusing inside thin oxide. F2 is the
flux of oxidant passing through the oxide
00:13:28.600 --> 00:13:30.380
thickness.
00:13:30.380 --> 00:13:40.430
And that is equal to C0 minus CI. CI as I
say is concentration of oxidant at silicon-silicon
00:13:40.430 --> 00:13:41.430
dioxide interface.
00:13:41.430 --> 00:13:46.720
CI, so there is a gradient diffusion.
00:13:46.720 --> 00:13:55.410
So F2 is then proportional to C0 minus CI
by x0, this is gradient. C0 minus CI by x0
00:13:55.410 --> 00:14:02.939
is essentially a gradient. And what is the
gradient constant should be? Since it is diffusing
00:14:02.939 --> 00:14:08.939
what should be the constant of proportionality
here? Diffusion coefficients of that oxidant
00:14:08.939 --> 00:14:14.560
in the oxide. Please remember diffusion coefficient
is for the material oxide or whatever specie
00:14:14.560 --> 00:14:20.379
going in that material. Diffusivity is different
in different material for different gas flows
00:14:20.379 --> 00:14:25.839
going in. So specifically you have to, earlier
we have talked about impurities in silicon.
00:14:25.839 --> 00:14:33.380
Now we are talking of equivalent solids concentration
of gases in the oxide and how do they diffuse
00:14:33.380 --> 00:14:34.380
through.
00:14:34.380 --> 00:14:42.620
So if I write that, I write F2 is minus D
effective. Minus is because I am subtracting
00:14:42.620 --> 00:14:52.410
CI minus C0 by x0, where D, why I did this?
Because final concentration minus the initial
00:14:52.410 --> 00:14:59.329
concentration divided by x is actually slope.
It is essentially how we define, that is going
00:14:59.329 --> 00:15:05.759
down. So that is the final concentration minus
the initial concentration divided by x0 is
00:15:05.759 --> 00:15:12.600
the gradient. So minus D effective CI minus
C0 by this where D effective, diffusion coefficient
00:15:12.600 --> 00:15:16.629
of oxidant in oxide.
00:15:16.629 --> 00:15:24.600
This is flux, F2. Now this is now made available
to react with silicon, oxidation, oxidant
00:15:24.600 --> 00:15:29.170
is now diffusing. Please remember what is
the process we are saying. The gas, from the
00:15:29.170 --> 00:15:36.339
gas phase oxygen is coming, equivalently it
gets to the silicon oxide surface, then diffuses
00:15:36.339 --> 00:15:42.579
through and then reacts with silicon to form
fresh oxide. Is that clear? That is what Deal-Grove
00:15:42.579 --> 00:15:49.749
model is suggesting. Gas phase through oxide
and to react with silicon, that is the process
00:15:49.749 --> 00:15:51.480
we have defined.
00:15:51.480 --> 00:15:56.069
So we want to know what is the flux 3 which
is (propo) and we know flux 3 is essentially
00:15:56.069 --> 00:16:02.879
as many silicon atoms are available or concentration
of silicon available, CI at that place, that
00:16:02.879 --> 00:16:07.870
is the only one which it can react with. Oxygen
cannot react more than CI because whatever
00:16:07.870 --> 00:16:14.689
available only can react. So we say the flux
F3 is proportional to CI and the proportionality
00:16:14.689 --> 00:16:19.889
constant is called reaction rate constant
k, ks.
00:16:19.889 --> 00:16:27.879
Reaction, ks is called reaction rate constant.
So F3 is ks CI. So now I have three fluxes.
00:16:27.879 --> 00:16:37.029
F1 is from gas phase to the silicon dioxide
surface. From silicon dioxide surface to silicon
00:16:37.029 --> 00:16:45.149
surface diffuse and then reacts with silicon
to form fresh oxide. And as I say if I assume
00:16:45.149 --> 00:16:50.029
it is in steady state, which will happen,
I put the wafer in, I have constant temperature,
00:16:50.029 --> 00:16:56.439
constant flow, system will go into steady
state. In that case the flux F is F1 equal
00:16:56.439 --> 00:16:58.189
to F2 equal to F3.
00:16:58.189 --> 00:17:01.310
C star remove kar diya?
00:17:01.310 --> 00:17:08.439
No, what we are available concentration in
the solid phase from the bulk is C star, of
00:17:08.439 --> 00:17:13.390
which only C0 is really going to diffuse because
that is at the surface.
00:17:13.390 --> 00:17:16.870
Cs is available.
00:17:16.870 --> 00:17:22.020
Cs is available which correspond to C0 in
solid.
00:17:22.020 --> 00:17:23.030
Means?
00:17:23.030 --> 00:17:29.590
Correspond I just now said, gas concentration
equivalent in solid concentration is related.
00:17:29.590 --> 00:17:36.620
I just now said through the mass transfer
coefficient. And through that to partial pressure,
00:17:36.620 --> 00:17:42.360
through that to Henry’s constant to the
actual concentrations. So we are saying the
00:17:42.360 --> 00:17:48.340
gas is coming, equivalently how much gas is
available overall, of which how much will
00:17:48.340 --> 00:17:54.070
diffuse from the surface is C0. It will go
to CI because there is a gradient, this is
00:17:54.070 --> 00:17:59.380
very few oxidant here and very large oxidant
here, so it will diffuse and when it reaches
00:17:59.380 --> 00:18:03.309
here, it will start reacting with the silicon
surface and form oxides.
00:18:03.309 --> 00:18:11.059
So we do little maths, there is nothing very
serious maths. Since we have already done,
00:18:11.059 --> 00:18:18.780
F1, F2, F3 are equal, we say, first we equate
F1 equal to F2 and next time we will equate
00:18:18.780 --> 00:18:26.590
F2 equal to F3. So if I write F1 equal to
F2, I get h C star minus C0, minus D effective
00:18:26.590 --> 00:18:36.600
CI minus C0 by x0. And if I write F2 equal
to F3, I get minus D effective CI minus C0
00:18:36.600 --> 00:18:44.730
by x0 is ks CI. So I have equation 9 and 10
which are, I got it from equating F1 equal
00:18:44.730 --> 00:18:48.640
to F2 and F2 equal to F3.
00:18:48.640 --> 00:18:57.159
Two equations to announce, I am interested
to know C0. I am interested to know CI in
00:18:57.159 --> 00:19:05.880
terms of C star available gas phase concentration.
And correspondingly I derive, solve these
00:19:05.880 --> 00:19:15.510
two equations 9 and 10. How do I eliminate
CI out of this? Substitute here and find C0
00:19:15.510 --> 00:19:24.320
or get C0 from here, substitute here to get
CI. This maths little longer, so I just wrote
00:19:24.320 --> 00:19:30.360
down the final equation. I repeat, solve 9
and 10 to get CI and C0. Substitute one of
00:19:30.360 --> 00:19:36.390
them to the other, you will get the other
value or substitute how much C0 into the second
00:19:36.390 --> 00:19:41.760
equation, you will get CI first, whichever
way you do.
00:19:41.760 --> 00:19:55.559
So I get CI is equal to C star upon 1 plus
ks by h plus ks x0 by D effective. Then C0
00:19:55.559 --> 00:20:01.570
is 1 plus ks x0 by D effective times C star
and the denominator is same. Please remember
00:20:01.570 --> 00:20:13.059
denominator is same, 1 plus ks x0 by this,
ks by h. So I have now CI and C0. I may not
00:20:13.059 --> 00:20:18.020
be very, I do not need to know C0 very much
but I am interested to know CI, why? Because
00:20:18.020 --> 00:20:23.630
that is the concentration is going to react
with silicon. So there I figured out both
00:20:23.630 --> 00:20:31.320
can be calculated but I am more interested
to know CI in terms of C star which I got.
00:20:31.320 --> 00:20:36.779
So what Deal-Grove model says now, what is
the oxidation rate. Can anyone suggest what
00:20:36.779 --> 00:20:46.410
is the oxidation rate? If F is the flux and
N is the number which is available for reaction,
00:20:46.410 --> 00:20:59.050
so F by N is essentially equal to dx0 by dt.
If, what is dx by dt? The available flux divided
00:20:59.050 --> 00:21:07.559
by available concentration where it can react,
the ratio of that is essentially dx0 by dt.
00:21:07.559 --> 00:21:15.780
So that is Grove-Deal’s model which says
if N1 is the concentration of oxidant molecules,
00:21:15.780 --> 00:21:23.070
then dx0 by dt is F by N1 and F is equal to
F1 equal to F2 equal to F3. So F3 is the smallest
00:21:23.070 --> 00:21:28.740
term, Ks CI, so I used F3 instead of, I can
write anyone of them. But then there will
00:21:28.740 --> 00:21:34.400
be two variables coming, so I just want to
remove, I have got CI value anyway. So I got
00:21:34.400 --> 00:21:45.840
ks CI by N1. Please remember this N1 for oxygen
is 2.22, 10 to power 22, N1 for H2O HOH molecule
00:21:45.840 --> 00:21:48.559
is 4 point double of that.
00:21:48.559 --> 00:21:57.919
OH, OH double the concentration whereas pure
oxygen has N1 as 2.22, 10 to power 22. This
00:21:57.919 --> 00:22:04.020
has been chemically monitored, measured by
many experiments, by atomic mass spectroscopy
00:22:04.020 --> 00:22:11.669
to FTIR everything may, one is monitoring
that. So if I have dx0 by Ks CI but this I
00:22:11.669 --> 00:22:18.820
substitute CI from the last equation. I just
derived expression for CI, so substitute CI
00:22:18.820 --> 00:22:30.060
here, so it gets k upon C star upon N1, 1
upon, 1 plus ks by h, k sorry, ks x0 by D
00:22:30.060 --> 00:22:39.220
effective. So how does it look like? We will
do some simple looking expression for this,
00:22:39.220 --> 00:22:43.650
first you note down this. Yes.
00:22:43.650 --> 00:22:55.710
So I get dx0 by dt is ks C star by N1, 1 upon
1 plus ks by h plus ks x0 by D effective.
00:22:55.710 --> 00:23:01.570
So this equation now I know, I can modify
this equation to suit some kind of good-looking
00:23:01.570 --> 00:23:08.370
expressions. What is good-looking? Some A,
B constant, so it looks very simple equation,
00:23:08.370 --> 00:23:14.640
algebraic. Of course not algebraic, first
order differential equation. Is that noted?
00:23:14.640 --> 00:23:29.770
This one, dx0? So I have now the oxidation
rate and I will modify exactly in form, that
00:23:29.770 --> 00:23:34.580
is what I am going to do. Is it okay?
00:23:34.580 --> 00:23:41.840
Okay, so dx0 by dt is 2 C star D effective
by N1. I actually multiply it, 2 D effective
00:23:41.840 --> 00:23:55.230
by ks both sides, this and here and I get
this expression. Readjust the terms, 2 D effective,
00:23:55.230 --> 00:24:04.710
1 plus ks by, 1 by ks plus 1 upon h plus 2x0,
into this constant. And now I define some
00:24:04.710 --> 00:24:16.000
terms. I define term A as 2 D effective 1
upon ks plus 1 upon h as A. And 2 D effective
00:24:16.000 --> 00:24:30.600
C star by N1, I define as B. Okay? Same expression,
this I have defined as B and this I defined
00:24:30.600 --> 00:24:37.559
as A. So I get a very nice looking simple
differential equation first order, dx0 by
00:24:37.559 --> 00:24:40.270
dt is B upon A plus 2x0.
00:24:40.270 --> 00:24:47.049
Please remember ks is a function of temperature,
h is a function of mass transfer coefficients
00:24:47.049 --> 00:24:56.580
or mass flows. In normal case h is much higher
than ks but let us see how much. dx0 by dt,
00:24:56.580 --> 00:25:04.549
B upon A plus 2x0 is the simplest equation
we get, with good-looking. Assuming that B
00:25:04.549 --> 00:25:13.080
and A are constants at a given temperature
for a given mass flow. If you change that,
00:25:13.080 --> 00:25:22.900
these terms will, B, and A values will also
correspondingly change. Is it okay?
00:25:22.900 --> 00:25:29.970
So that is a very trivial looking this but
it is important. Why are we doing this again?
00:25:29.970 --> 00:25:36.390
Because at the end when I monitor the oxide,
I have the furnace, I actually oxidize the
00:25:36.390 --> 00:25:42.669
wafers and then I may do some characterization
to know the oxide thickness. But if probably
00:25:42.669 --> 00:25:48.850
I am doing something process on a system where
there is no furnace and there is nothing to
00:25:48.850 --> 00:25:54.640
this. I must be able to get relative oxide
thicknesses every now and then for change
00:25:54.640 --> 00:26:00.630
in oxide thickness or whatever I am doing
in process which is running on a CAD tool.
00:26:00.630 --> 00:26:05.440
That is called T-CAD for technology tools
are available, earlier ones we used to have
00:26:05.440 --> 00:26:08.269
program from called Stanford called Supreme
process stimulator.
00:26:08.269 --> 00:26:14.710
There are now many, one is Intra, other is
. There are many such program, this is which
00:26:14.710 --> 00:26:19.580
is device plus process simulator. Since we
have many process simulators now, we like
00:26:19.580 --> 00:26:26.250
to know what models they use because they
will also find out what is the oxide capacitance
00:26:26.250 --> 00:26:32.519
every now and then. So they need to know oxide,
how much grown. So you may specify temperature,
00:26:32.519 --> 00:26:37.570
you may specify gas flows, you may specify
but then at the end software has to do something
00:26:37.570 --> 00:26:44.429
to evaluate that. We need models to do that,
so all this effort is to see a model which
00:26:44.429 --> 00:26:47.090
is going into a T-CAD tool.
00:26:47.090 --> 00:26:51.980
So whenever you are using central or others,
you probably do not even look at the models,
00:26:51.980 --> 00:26:57.320
you just, it adds data and you just substitute
and you just ahh, good! But actually what
00:26:57.320 --> 00:27:02.580
I have gone through is this and maybe tomorrow
for thin oxide, better models will be required.
00:27:02.580 --> 00:27:08.169
So you must know how do we actually derive
models, what is the physics behind, chemistry
00:27:08.169 --> 00:27:15.580
behind, materials science behind and you again
behind.
00:27:15.580 --> 00:27:26.210
So I rewrite the same term, A dx0 by dt. So
if I solve the, if I see this equation, this
00:27:26.210 --> 00:27:33.000
is the quadratic equation. So we put an initial
condition to solve this differential equation.
00:27:33.000 --> 00:27:39.500
Sorry, not it is a simple differential equation.
I say at t is equal to 0, according to Deal-Grove’s
00:27:39.500 --> 00:27:46.030
model there is initial oxide x0 is xi. That
is how we started with. There is an initial
00:27:46.030 --> 00:27:54.809
oxide. So x0 is xi and corresponding to this
time, xi, if I use, okay, so we substitute
00:27:54.809 --> 00:27:55.809
here.
00:27:55.809 --> 00:28:03.169
Let us say tau is the time taken to create
this xi. Let us say tau is the time taken,
00:28:03.169 --> 00:28:08.669
equivalently actually it is existing. We do
not know what time we did, we did not do anything.
00:28:08.669 --> 00:28:13.419
So we say okay, equivalently if we have to
go this much oxide, how much time? So that
00:28:13.419 --> 00:28:24.830
time I declared as tau. So I say now Ax0,
if I put at this, A 2x0 dx0 is Bt. And I integrate
00:28:24.830 --> 00:28:31.500
this, then I get Ax0, 2x0, so I have Bt, plus
initial condition I will put it at x is equal
00:28:31.500 --> 00:28:34.610
to xi, tau is the time taken.
00:28:34.610 --> 00:28:42.419
So I rewrite this term, Ax0 by 2 x0 square,
Bt plus B tau. Tau is the time taken to grow
00:28:42.419 --> 00:28:47.370
oxide thickness of xi and there is, this is
only a fictitious number. Why fictitious?
00:28:47.370 --> 00:28:53.110
Because initial oxide is already there, we
are just trying to equate it into a time frame.
00:28:53.110 --> 00:29:02.630
Okay, or to say tau is xi square plus A xi
upon B. Axi, xi square Axi by B is essentially
00:29:02.630 --> 00:29:08.710
this. xi also I do not know in fact. So we
assume normally tau should be very small because
00:29:08.710 --> 00:29:09.970
thin oxide is there.
00:29:09.970 --> 00:29:16.139
But if I somehow figure out how much was xi,
then I should be able to know how much time
00:29:16.139 --> 00:29:23.360
equivalently it would have been. Okay, so
this is my equation and this looks what? A
00:29:23.360 --> 00:29:29.570
simple quadratic equation and therefore I
can find x0 terms. I repeat the term which
00:29:29.570 --> 00:29:35.679
I am getting is Ax0 plus 2 x0 square. 2, 2
of course will cancel. x0 square is equal
00:29:35.679 --> 00:29:38.070
to Bt plus tau.
00:29:38.070 --> 00:29:45.590
So this is the expression you will get, x0
square plus Ax0 is Bt plus tau. For thicker
00:29:45.590 --> 00:29:51.560
oxide growths tau can be neglected, why? Because
t will be much larger than tau. Tau is very
00:29:51.560 --> 00:29:57.529
small but it is existing. But for thinner
oxide that may be comparable, so we must figure
00:29:57.529 --> 00:30:02.940
out how much is actually thin oxide during
initial time must be actually evaluated. I
00:30:02.940 --> 00:30:10.080
will show you how. So if this quadratic equation
can we have solution of minus A plus minus
00:30:10.080 --> 00:30:18.429
A square plus 4Bt plus tau by 2, minus A by
2 plus minus A by 2, 1 plus t plus tau upon
00:30:18.429 --> 00:30:20.860
A square by 4B.
00:30:20.860 --> 00:30:27.440
And of course negative solution is neglected.
Why? Because there is nothing called negative
00:30:27.440 --> 00:30:35.340
oxide growths. So we say it is only minus
A by 2 plus A by 2 terms and assumption is,
00:30:35.340 --> 00:30:40.840
and time should be such that this term should,
because this is 1 plus, so obviously this
00:30:40.840 --> 00:30:47.039
term will be larger than A by 2 and therefore
positive growths are expected. Is that clear?
00:30:47.039 --> 00:30:55.860
I repeat if this term is, even if it is point
something, 0.1 plus something is there, which
00:30:55.860 --> 00:31:01.799
means A by 2 times this will be always larger
than A by 2. And therefore positive growths
00:31:01.799 --> 00:31:03.450
are expected.
00:31:03.450 --> 00:31:11.899
So if I write only plus sign, I get x0 is
and I put minus inside, so A by 2 into 1 plus
00:31:11.899 --> 00:31:21.870
t by tau, A square by B to the power half
under root minus 1. Now we define some whatever
00:31:21.870 --> 00:31:29.240
we aim, we define, we actually give some nomenclature
to them. And why, we will see soon. We call
00:31:29.240 --> 00:31:40.509
B as a parabolic rate constant, B is called
parabolic rate constant. And B by A is called
00:31:40.509 --> 00:31:47.659
linear rate constant. This is definition namewise
and why we name linear and parabolic, will
00:31:47.659 --> 00:31:51.210
be soon seen when we will take the two cases.
00:31:51.210 --> 00:32:02.680
I repeat B is defined as parabolic rate constant
and B by A. In many models, this is given
00:32:02.680 --> 00:32:12.500
kp and this is given kl, parabolic kp and
kl, so most of the earlier CAD tools may be
00:32:12.500 --> 00:32:20.919
using or even central uses Kp means parabolic
rate constant. Kl subscript all is essentially
00:32:20.919 --> 00:32:27.640
is linear rate constant which is same as what
I have been using. This is Deal-Grove model,
00:32:27.640 --> 00:32:33.100
I cannot change, k is there because that is
Deal’s model. If it is my model, I can do
00:32:33.100 --> 00:32:39.230
any other names but this is Deal’s model
and that is how they are defined and way back
00:32:39.230 --> 00:32:40.340
in 65.
00:32:40.340 --> 00:32:49.049
Two limiting cases, okay, just look at these
terms.
00:32:49.049 --> 00:32:55.750
This term is smaller or larger, depends on
this, t plus tau is larger than this, t plus
00:32:55.750 --> 00:32:58.539
tau is smaller than this, two extreme cases.
00:32:58.539 --> 00:33:06.779
So first we say t plus tau is much smaller
than A square by 4B. So we can then expand
00:33:06.779 --> 00:33:15.480
it by binomial term, so 1 plus x to the power
half. If x is less than 1, 1 upon half x.
00:33:15.480 --> 00:33:24.919
So this 1 plus half t plus tau by A square
by B minus 1. 1,1, cancels. So x0 is B by
00:33:24.919 --> 00:33:38.980
A t plus tau. For a given temperature, given
gas flow, B and A both are constant. How is
00:33:38.980 --> 00:33:49.690
x0 related to time? Linear. B is, x0 is proportional
to tau, tau is very small. So x0 is proportional
00:33:49.690 --> 00:33:57.049
to time, what is this growth is? Linear. It
is linearly increasing. So now we understood
00:33:57.049 --> 00:34:04.889
why I named B by A as linear rate constant,
because x0 is B by A times t and therefore
00:34:04.889 --> 00:34:11.830
B by A is named as linear rate constant.
00:34:11.830 --> 00:34:17.080
So initially what will happen that what does
that mean? If t is smaller, what does that
00:34:17.080 --> 00:34:25.630
mean? Initially oxide will grow linearly with
time and as time increases we believe it will
00:34:25.630 --> 00:34:31.159
become parabolic. And let us see how that
can happen. Is that clear? So initial growth
00:34:31.159 --> 00:34:37.629
of silicon dioxide is linear with time and
then starts, it reduces the rate. Can you
00:34:37.629 --> 00:34:44.889
think why it will reduce the rate? As the
time increases oxide thickness will increase,
00:34:44.889 --> 00:34:48.190
so the gradient will decrease. Is that clear
to you?
00:34:48.190 --> 00:34:55.200
So if the flux available is less, whatever
is available may react but available is less.
00:34:55.200 --> 00:35:01.890
So the oxidation rate will go down. So that
is the essential as you go. Thicker oxide,
00:35:01.890 --> 00:35:07.790
it will, much smaller impurities I mean oxidant
will reach there, so smaller thickness relative
00:35:07.790 --> 00:35:15.830
to the time will wave off. Is that point clear
why this is happening? Before we come to parabolic,
00:35:15.830 --> 00:35:19.650
these are few terms to be explained.
00:35:19.650 --> 00:35:27.480
B by A is 2, now we substitute B and A, 2
Defective C star by N1, 2 Defective ks plus
00:35:27.480 --> 00:35:35.900
this can be rewritten as C star upon N1, ks
h by ks plus. In general for a given mass
00:35:35.900 --> 00:35:43.420
flow in oxidation furnace and for the temperatures
which we normally use from 800 to 1200, h
00:35:43.420 --> 00:35:51.319
is much larger than ks. ks is e to the power
minus e by kT kind, minus 4, minus 5. Whereas
00:35:51.319 --> 00:35:57.810
B will be or h, sorry, h will be order of
few centimeter per seconds which is much higher.
00:35:57.810 --> 00:36:04.609
So what happens? In most normal cases h is,
so which is it limited by? h or ks? Smaller
00:36:04.609 --> 00:36:12.170
the one limits it. Larger one does not limit
it. So h is, I do not say every time, in CVD
00:36:12.170 --> 00:36:18.560
we will say no, it is mass limited but in
this particular case we say h is much larger
00:36:18.560 --> 00:36:28.390
than ks. So if I do this, I can neglect h,
1 upon h it can goes. So we get B by A is
00:36:28.390 --> 00:36:38.210
C star by N1 into ks. Is that okay? If h is
larger than ks, this can be neglected. 1 upon
00:36:38.210 --> 00:36:46.061
h is smaller compared to 1 upon ks and therefore
we neglect h and we get C star upon N1 into
00:36:46.061 --> 00:36:47.061
ks.
00:36:47.061 --> 00:36:57.780
And as I say ks, how do we define ks? Reaction
rate constant. What is reaction rate? Available
00:36:57.780 --> 00:37:03.980
oxidant reacting with silicon. The rate with
which that reaction takes place, it is a chemical
00:37:03.980 --> 00:37:10.410
process which essentially means which is essentially
e to the power minus temperature dependent
00:37:10.410 --> 00:37:18.310
term. B by A is also linear rate constant
which also follows ks dependence of temperature.
00:37:18.310 --> 00:37:25.619
Is that clear? Bs, B by A is a function of
ks, ks follows temperature dependence e to
00:37:25.619 --> 00:37:31.069
the power. So B by A will also follow temperature
dependence same as ks.
00:37:31.069 --> 00:37:40.550
So we write then, this is the x0 C star by
N1, ks t plus tau. This is called, so initial
00:37:40.550 --> 00:37:48.900
oxide growth is related to with time, linear
growth and it is also limited by available
00:37:48.900 --> 00:37:54.960
reaction rate at CI. How much is available
only can, please remember it is availability
00:37:54.960 --> 00:38:02.089
of CI I mean is the silicon atoms which can
it react. I may have enough oxidant but not
00:38:02.089 --> 00:38:09.089
all is possible to react. Initially enough
oxidant has come but there are not enough
00:38:09.089 --> 00:38:15.710
bonds where oxygen can, SiO-Si bond can be
formed, so the growth rate is limited by available
00:38:15.710 --> 00:38:17.310
bonds.
00:38:17.310 --> 00:38:23.061
So it is essentially decided by ks which is
temperature dependent. Typically we will show
00:38:23.061 --> 00:38:30.970
you the term but I just now want to clarify.
Please remember oxidant is reaching enough
00:38:30.970 --> 00:38:35.380
amount, thickness is very small, much of it
diffuses through. Whatever is available, most
00:38:35.380 --> 00:38:40.310
of it will reach CI. But available oxidation
will be limited by the reaction there which
00:38:40.310 --> 00:38:48.500
is temperature dependent which is essentially
how many atoms can react. This is the case
00:38:48.500 --> 00:38:56.400
I, what is that case we talked? Time is very
low, smaller times, t plus tau.
00:38:56.400 --> 00:39:03.680
However if t is much larger than tau, and
t is much larger than A square by B, longer
00:39:03.680 --> 00:39:08.830
time oxidation is performed. t will be always
larger than tau, tau is very small. So if
00:39:08.830 --> 00:39:17.730
t is much larger time, t plus tau is t. And
then we say t is larger than A square by 4B.
00:39:17.730 --> 00:39:26.109
Then we can neglect 1 there and rewrite the
term x0 as 2 root t B by A into A by 2 minus
00:39:26.109 --> 00:39:33.040
A by 2. If I readjust these terms, this is
just substitute into the equation which we
00:39:33.040 --> 00:39:36.120
wrote earlier in quadratic solution.
00:39:36.120 --> 00:39:42.369
Substitute, t is much greater than this, 1
can be neglected there. Therefore under root
00:39:42.369 --> 00:39:51.880
of that is root t B and into A by 2 minus
A by 2. And if I do this, it essentially comes
00:39:51.880 --> 00:40:02.839
to be equal to root Bt. We are neglecting
small terms, so compared to this everything
00:40:02.839 --> 00:40:08.500
is small. This term is smaller than this,
we already said through this. So we always
00:40:08.500 --> 00:40:18.690
say x0 is root of Bt which means x0 square
is Bt. What is this law? Parabola.
00:40:18.690 --> 00:40:27.250
This x0 square is Bt is parabola. So x0 growth
is now under root of Bt means it reduces the
00:40:27.250 --> 00:40:36.109
oxidation rate as, therefore oxide thickness
as time is larger and larger initially. So
00:40:36.109 --> 00:40:43.700
initially linear, and then parabolic growth
starts in the case of oxidation. So this is
00:40:43.700 --> 00:40:52.020
the, and since B is parabolic constant, means
is giving a constant of parabolicity, we call
00:40:52.020 --> 00:41:02.290
B as parabolic rate constant. B is parabolic
rate constant. I told you this A square by
00:41:02.290 --> 00:41:10.260
4B is much smaller compared to t, so A is
much smaller. Obviously this term is much
00:41:10.260 --> 00:41:13.910
larger than this, so that is neglected.
00:41:13.910 --> 00:41:23.550
In actual model you do not neglect anything
but it is 100 minus 0.5, think of it whether
00:41:23.550 --> 00:41:29.160
you want to retain 0.5. If you wish, fine.
99.95 is the answer or 100, your choice. What
00:41:29.160 --> 00:41:35.980
decimal accuracy you want, 61 precision, 128
bits of precision, computer can do any basic.
00:41:35.980 --> 00:41:49.510
Here is some experiments are performed just
to compare those values which I am talking
00:41:49.510 --> 00:41:56.431
about. I have done an oxidation, experiment
means actually experiment was performed. I
00:41:56.431 --> 00:42:05.280
have grown an oxide or silicon at 920 degree
centigrade and assumed that tau given to me
00:42:05.280 --> 00:42:16.820
is 50 seconds. Now we monitor x0 at different
times. I keep growing 5 minute, say time in
00:42:16.820 --> 00:42:23.660
hours, so this is some 6 minutes and this
is 18 minutes, this is 24 minutes, this is
00:42:23.660 --> 00:42:28.230
30 minutes, this is 60, 40 minutes. Roughly
36 minutes.
00:42:28.230 --> 00:42:33.880
So I have different oxide thickness or oxide
times, oxidation times. This is in hours,
00:42:33.880 --> 00:42:39.849
please remember this is in hours. And I have
monitored by some way the oxide thickness.
00:42:39.849 --> 00:42:52.339
We will see how to monitor oxide thickness.
And I monitored, measured them. For 0.11 hour,
00:42:52.339 --> 00:42:58.790
it is 0.041 micron. 0.3, 0.10; 0.4, 0.128;
0.5, 0.153; 0.6, 0.177. Since you did not
00:42:58.790 --> 00:43:04.550
want to leave, I used it to show I can take
care. How do you get that value, also is as
00:43:04.550 --> 00:43:05.760
per you used to.
00:43:05.760 --> 00:43:14.790
Since x0 square is Bt plus tau minus Ax0,
so I get x0 is Bt plus tau by x0 minus A.
00:43:14.790 --> 00:43:21.490
What is this equation looks like? This is
y is equal to Nx plus C kind of equation,
00:43:21.490 --> 00:43:32.230
linear. So that if I plot t plus tau by x0,
this term versus x0, so some way at 0, that
00:43:32.230 --> 00:43:40.900
is t plus tau, at tau by x0 at this point,
whatever is the constant is minus A. So you
00:43:40.900 --> 00:43:47.500
can monitor that minus A, how much is minus
A. So in actual I just want to make parabolic,
00:43:47.500 --> 00:43:54.210
I said neglected but in calculation I have,
I really take care because I need to know
00:43:54.210 --> 00:43:55.210
A.
00:43:55.210 --> 00:44:05.910
So by extrapolating this curve, I will get
minus A. And the slope is B, so I could get
00:44:05.910 --> 00:44:14.349
B and BA and therefore B and B by A, that
is kl and kp are monitored if I know actual
00:44:14.349 --> 00:44:21.990
oxide thicknesses at different growth temperatures
at different growth times. Assumption everywhere
00:44:21.990 --> 00:44:33.470
is temperature is constant and also the mass
flows are constant. So if, we can see there
00:44:33.470 --> 00:44:41.550
that if I do experiment, I will be able to
measure B and B by A. And then what do I have
00:44:41.550 --> 00:44:51.410
to do? This I have to repeat for many temperatures
to get B and B by A dependence with temperature.
00:44:51.410 --> 00:44:55.880
And then I figured out later, maybe I do not
know whether I have graph, I have, okay I
00:44:55.880 --> 00:45:02.130
will show you. It shows that it is actually
following the physics that is ks and diffusivity,
00:45:02.130 --> 00:45:07.280
is essentially whatever their temperature
dependence is, same happens to B and B by
00:45:07.280 --> 00:45:13.490
A as well. We will see this little later.
So is that okay? So I can monitor B and B
00:45:13.490 --> 00:45:20.630
by A by actually monitoring the oxide thicknesses
at different times of oxidations. Is that
00:45:20.630 --> 00:45:22.020
point clear?
00:45:22.020 --> 00:45:32.970
So let us say by typical experiment which
I did or rather someone else has done, of
00:45:32.970 --> 00:45:40.520
course I have calculated but this data was
taken from our lab many years ago. So B is
00:45:40.520 --> 00:45:50.240
0.2 micron square per hour, A is 0.5 micron
and B by A therefore is 0.4 microns per hour,
00:45:50.240 --> 00:45:59.160
0.4 micron per hour. So I just now said if
I know the data, I will be able to plot time
00:45:59.160 --> 00:46:07.069
versus x0 by this. And if I know this, I will
be able to evaluate B as well as B by A.
00:46:07.069 --> 00:46:17.710
Now as I said you I will repeat this experiment,
what do I do? At different temperatures. Again
00:46:17.710 --> 00:46:25.990
oxide thicknesses for, only thing is now I
may do it for a given time, I mean same time
00:46:25.990 --> 00:46:30.800
so that, but even if you do different time,
graph will show the slopes. So it does not
00:46:30.800 --> 00:46:35.530
really matter as long as you, but preferably
you do same so that you know where the slopes
00:46:35.530 --> 00:46:44.690
are moving, just to see them. Why we want
to do this? Because I want to know whether
00:46:44.690 --> 00:46:50.920
B and B by A really follows D and ks. That
is what we are doing from the experiment,
00:46:50.920 --> 00:46:57.970
from the theory we are looking, that B by
A is following ks and B is following D effective.
00:46:57.970 --> 00:47:05.359
So can, does that temperature dependence appears,
so we like to see that. So we actually do
00:47:05.359 --> 00:47:12.730
repeated experiment at 1000, 1100 and 1200,
monitor oxide thicknesses at same times and
00:47:12.730 --> 00:47:24.680
replot B and, replot x0 versus time and get
B and A for all of them, for different temperatures.
00:47:24.680 --> 00:47:32.540
This is for 920, repeated for 1000, repeated
for 1100, repeated for 1200.
00:47:32.540 --> 00:47:44.720
If I do this, the data I get is 920 is this
and 1200 is this. This is B by A, please remember
00:47:44.720 --> 00:47:50.109
units. B is expressed as micron square per
hour and B by A is expressed as micron per
00:47:50.109 --> 00:48:01.390
hour. Linear and parabolic words square. So
if I plot now B versus 1 upon T for dry oxidation,
00:48:01.390 --> 00:48:08.210
of course I have not talked about this but
we will come, dry means only pure oxygen is
00:48:08.210 --> 00:48:19.800
passed. So for the dry oxide and I plot B
versus T, 1 upon T, I see a straight line.
00:48:19.800 --> 00:48:27.450
And its slope is 1.23 electron volt. If I
plot D versus temperature, 1 by T in fact,
00:48:27.450 --> 00:48:40.400
and I actually see for dry oxide case, the
slope is 1.23 electron volt. If I repeat the
00:48:40.400 --> 00:48:48.910
same graphical this for B by A versus temperature,
I plot B by A versus 1 upon T and I get slope
00:48:48.910 --> 00:49:01.120
of 2 electron volt. This is experimental because
what I did, I actually went in and I did oxidation,
00:49:01.120 --> 00:49:05.670
monitored the thicknesses at different temperature
for different times and plotted them to get
00:49:05.670 --> 00:49:08.210
B and B by A at different temperatures.
00:49:08.210 --> 00:49:16.290
So this is no model, this is essentially what
I can measure. And now I want to prove that
00:49:16.290 --> 00:49:23.380
what I said in a model probably fits to what
I monitored. Therefore Grove-Deal model is
00:49:23.380 --> 00:49:31.460
reasonably good. Is that point clear to you?
00:49:31.460 --> 00:49:41.520
So B and B by A for dry, we do wet oxidations
in 95 degree water wafers. So I did it, we
00:49:41.520 --> 00:49:52.910
did same experiment for wet oxides and I find
for B the slope is 0.78ev and B by A has slope
00:49:52.910 --> 00:50:02.700
of 2.05ev. Let us look at this term again,
B and B by A. Okay, if you have noted down,
00:50:02.700 --> 00:50:09.990
as I say B by A is also called KL and B is
also called Kp in many simulators. And that
00:50:09.990 --> 00:50:16.960
can be written as C1 exponential minus E1
by kT, C2 exponential minus E2 by, this is
00:50:16.960 --> 00:50:21.329
the model they have used.
00:50:21.329 --> 00:50:26.020
And if you see here, these are the yields,
E1 and E2. This is the temperatures, slope
00:50:26.020 --> 00:50:33.260
you have got it. C star into ks by N1 is,
that is what we have just derived. B by A
00:50:33.260 --> 00:50:44.200
is C star ks by N1. B is D effective by C
star. ks, C is constant. N1 is constant. So
00:50:44.200 --> 00:50:52.869
if this temperature dependence has to come,
ks must have similar relations. If this relation
00:50:52.869 --> 00:50:59.619
has to be followed in this, D effective must
be for, I mean E2 must be following the activation
00:50:59.619 --> 00:51:03.359
energy associated with D effective. Is that
point clear?
00:51:03.359 --> 00:51:11.109
If these are equals from the graphs if I say,
then the since here they are constants, only
00:51:11.109 --> 00:51:18.070
D effective is temperature dependent, ks is
temperature dependent. So obviously this E1
00:51:18.070 --> 00:51:27.809
must confirm to ks and E2 must conform to
D effective. And yes, this experiment was
00:51:27.809 --> 00:51:36.630
further extended and once we did this, we
found that they did which essentially means
00:51:36.630 --> 00:51:43.799
what? The linear rate constant B by A essentially
is governed by reaction rate constant which
00:51:43.799 --> 00:51:52.890
has temperature dependence of E to the power
E1 by kT where E1 is 1.23 for dry oxidation,
00:51:52.890 --> 00:51:56.910
0.78 for wet oxidation.
00:51:56.910 --> 00:52:05.920
So ks actually follows what the growth is,
this. Diffusivity of oxidant for dry or wet,
00:52:05.920 --> 00:52:13.400
you can see from here, sorry, this was ks
and this one. For B we find it is 1.23 and
00:52:13.400 --> 00:52:19.520
0.78 and we figure out D effective have the
same energy of oxidant in oxide as such we
00:52:19.520 --> 00:52:24.430
know and same is reaction rates we know about
in real life. We actually monitor by different
00:52:24.430 --> 00:52:32.349
methods. So we figured out that this process
is reaction rate limited and this process
00:52:32.349 --> 00:52:37.200
is diffusion limited and which is obvious.
00:52:37.200 --> 00:52:43.809
Initially when the oxide thickness is small,
the available oxidant is enough, it is the
00:52:43.809 --> 00:52:51.079
reaction possible at a given temperature to
convert silicon into silicon dioxide. Si plus
00:52:51.079 --> 00:52:57.940
2O2, SiO-SiO bond has to formed. Now this
at a given temperature is, this is the reaction,
00:52:57.940 --> 00:53:03.670
so it has to, this reaction will be temperature
dependent. That is what exactly we did experiment
00:53:03.670 --> 00:53:06.270
and we found yes, it does depends on this.
00:53:06.270 --> 00:53:13.530
However when I increase the time, that means
I have larger time, by then already oxide
00:53:13.530 --> 00:53:20.040
thickness has grown. So the available oxidant
now at the interface of silicon, silicon oxide
00:53:20.040 --> 00:53:27.490
is smaller and it is this as much as you can
diffuse through is going to available for
00:53:27.490 --> 00:53:34.050
oxidation. Any amount coming I will oxidize
but available is only what you will supply.
00:53:34.050 --> 00:53:40.329
So it is the diffusivity which decided how
much oxidant I can provide, initially everything
00:53:40.329 --> 00:53:42.730
available, so reaction is following.
00:53:42.730 --> 00:53:49.660
In next time reaction can be done any amount
but available kitna, so that means the reaction
00:53:49.660 --> 00:53:55.600
rate constant is dominant in linear initial
times and parabolic rate constants are dominant
00:53:55.600 --> 00:54:01.770
in thicker oxide times.
00:54:01.770 --> 00:54:19.819
This is called Hu's paper way back in 71.
Very famous person, now we also we figured
00:54:19.819 --> 00:54:27.880
out that the oxide thickness for 111 plane
is different from 100. We have shown the last
00:54:27.880 --> 00:54:33.799
time that oh, maybe we have to show, sorry
we have not shown. So why do you think that
00:54:33.799 --> 00:54:45.869
C1 constant is larger for compared to this?
The reason why 111 shows oxide thickness thicker,
00:54:45.869 --> 00:54:52.880
can anyone suggest why it cannot be thicker?
Because more atoms on that plane is, the question
00:54:52.880 --> 00:54:55.660
asked was exactly this Miller planes.
00:54:55.660 --> 00:55:02.289
Miller plane stain how many atoms are available
on that to react. So 111 shows the maximum
00:55:02.289 --> 00:55:07.880
silicon atoms, 4 of them in fact are four
corners and 3 inside. Since they are the largest
00:55:07.880 --> 00:55:14.020
number, the growth rate is highest along the
plane. This is exactly what is data has been
00:55:14.020 --> 00:55:16.799
shown, taken from .
00:55:16.799 --> 00:55:25.510
Based on this I can again show you this, same
time once again and again. The slope activation
00:55:25.510 --> 00:55:33.320
in ks and activation energy related with the
diffusion is found identical to what is measured
00:55:33.320 --> 00:55:42.380
for B by A and B which verifies B by A is
ks limited and B is D effective limited, identical.
00:55:42.380 --> 00:55:49.740
The energy associated with ks and 1 upon T
and D effective 1 upon T, their slope essentially
00:55:49.740 --> 00:55:57.829
matches with the actual data measured by B
and B by A which means the Grove-Deal model
00:55:57.829 --> 00:56:08.190
to a great extent is valid, except for the
assumptions which we may have to modify as
00:56:08.190 --> 00:56:10.690
things go.
00:56:10.690 --> 00:56:17.040
But for a thicker oxide less than say, thicker
oxide around 100 Armstrongs or above, Grove-Deal
00:56:17.040 --> 00:56:25.039
model fits very well. Anything below 100 is
not true. For D effective and ks, for whom,
00:56:25.039 --> 00:56:31.240
ks will be had different activation energy
which is essentially what B by A we got. B
00:56:31.240 --> 00:56:37.609
by A is proportional to ks. So if I monitor
ks by, not by B by A method, by actual reaction
00:56:37.609 --> 00:56:43.539
by thermodynamics, so if I evaluate thermodynamically
this equation, I get whatever activation energy
00:56:43.539 --> 00:56:48.860
associated. I figured out that is same as
what B by A I got by experiment, so which
00:56:48.860 --> 00:56:52.600
means B by A is case limited.
00:56:52.600 --> 00:57:00.160
I did same thing for diffusivity experiments
from the chemical point of view and whatever
00:57:00.160 --> 00:57:05.430
energies I found for both wet and this, I
matched it with B values and I finally I got
00:57:05.430 --> 00:57:08.530
the same within errors, within experimental
errors.
00:57:08.530 --> 00:57:12.460
Which energy should they have in experiment?
00:57:12.460 --> 00:57:18.089
Activation energy is the energy required to
react, is essentially, it is enthalpy. See
00:57:18.089 --> 00:57:27.230
the, yeah, any reaction whether binding or
dissociation or enthalpy formation is essentially
00:57:27.230 --> 00:57:35.530
related to energy. If A plus B has to go to
C plus D, then the reaction is favored forward
00:57:35.530 --> 00:57:42.760
if the Gibbs energy is plus, that is enthalpy
minus entropy, T delta S is positive. If enthalpy
00:57:42.760 --> 00:57:47.440
minus T delta is negative, dissociation B,
C plus D will go back to A plus B.
00:57:47.440 --> 00:57:53.780
We will see in CVD. This is where we adjust
the growth. If I want growth, what should
00:57:53.780 --> 00:58:00.510
I do? A plus B should be stronger and delta
G must be positive. So A plus B will react.
00:58:00.510 --> 00:58:06.740
When I want to etch, what should I do? I do
not want reaction, I want etching. So I see
00:58:06.740 --> 00:58:12.059
to it, C plus D goes back to A plus B. That
is exactly what we do to depositions and etching
00:58:12.059 --> 00:58:18.391
are identical, the reaction are favored or
not favored. Is that point, if something is
00:58:18.391 --> 00:58:24.700
formed, I reverse it, do not form. Remove
it. So that is exactly what I am trying to
00:58:24.700 --> 00:58:33.020
say. So what essentially I am trying to say
you, that Grove-Deal model to a great extent
00:58:33.020 --> 00:58:43.910
is a good model except for as I say very thin
oxides.
00:58:43.910 --> 00:58:51.720
If I see orientation, as I already said B
by, orientation is only, afterwards it is
00:58:51.720 --> 00:58:57.819
the diffusivity. In thick oxide, how much
is available is going to decide. Only in thinner
00:58:57.819 --> 00:59:04.849
oxide, available bonds will decide available
reaction. So we know B by A is 100, 111 is
00:59:04.849 --> 00:59:12.369
at least 1.7 times B by A of 100 constants,
I already given C1, C2, you can see value.
00:59:12.369 --> 00:59:21.470
And this is also essentially the ratio of
bonds of 111 to 100. So initial oxide will
00:59:21.470 --> 00:59:26.930
be thicker for 111 compared to 100 by 1.7
times.
00:59:26.930 --> 00:59:32.349
Afterwards why they will, it will not this
because availability of oxidant is going to
00:59:32.349 --> 00:59:38.809
decide and not the rate. So then 111 and 100
will have same thing as diffusion limitations.
00:59:38.809 --> 00:59:45.530
But initially available bonds to react at
a given temperature will be decided by which
00:59:45.530 --> 00:59:56.700
kind of planes you have. We also have in real
life polysilicon as gate and we will see that
00:59:56.700 --> 01:00:03.740
later, the growth of oxide is different from
poly compared to crystallines. We are not
01:00:03.740 --> 01:00:10.530
done poly so far, so I do not want to preamp
but just for the heck of it, I may show you
01:00:10.530 --> 01:00:11.980
something which is of…..
01:00:11.980 --> 01:00:24.880
See poly crystals, say let us say this is
crystalline, but its orientation is different
01:00:24.880 --> 01:00:32.829
at different x, y, z. There are crystallites
but there are many of them, poly, large numbers.
01:00:32.829 --> 01:00:37.770
So in this part they may be crystalline but
their orientation will be different compared
01:00:37.770 --> 01:00:43.430
to orientations here and here. Some may be
larger crystallites, some will be smaller.
01:00:43.430 --> 01:00:49.089
The line between two such crystallites is
called grain boundary.
01:00:49.089 --> 01:01:00.890
Each crystallite is called grain, so these
are grain boundaries. That is two crystallites
01:01:00.890 --> 01:01:06.690
are meeting at that point, that is why it
is called grain boundaries. Now grain boundaries
01:01:06.690 --> 01:01:14.440
do not have crystals there. So it is like
a void in the system. So if you push something,
01:01:14.440 --> 01:01:20.490
it may not go through crystallite but may
actually go through the grain boundaries.
01:01:20.490 --> 01:01:26.280
So more and more atoms may be possible to
be oxidized because much of the oxidant now
01:01:26.280 --> 01:01:31.829
will not necessarily go through silicon or
silicon dioxide layer on the top but through
01:01:31.829 --> 01:01:38.020
grain boundaries and react. So do you expect
thicker oxide for poly? Because more deeper,
01:01:38.020 --> 01:01:42.650
at least thicker oxide because it will, oxygen
can go deeper in the polycrystallines. Polyoxide
01:01:42.650 --> 01:01:53.380
is mini-crystallite and hence many grain boundaries.
01:01:53.380 --> 01:01:59.510
Oxidant diffuses faster through the grain
boundaries leading to enhanced oxidation rate
01:01:59.510 --> 01:02:06.140
and the model which is suggested is a t to
the power n where a is fit constant, n is
01:02:06.140 --> 01:02:12.380
also fit constant. Kyaa karenge? Iska oxide
lenge, time se plot karenge aur iss function
01:02:12.380 --> 01:02:21.500
ko fit kar denge. So we say okay, this is
the law polyoxides follow. It does not follow
01:02:21.500 --> 01:02:28.460
Deal-Grove model. It follows another law.
Is that okay? Why it is so? Because we have
01:02:28.460 --> 01:02:32.329
not yet studied very strongly the diffusivity
through grain boundaries.
01:02:32.329 --> 01:02:37.440
There are many micro crystalline theories,
some other day, if you are really working
01:02:37.440 --> 01:02:44.250
PhD for that, then I will show you how even
this model is not correct. But as of now we
01:02:44.250 --> 01:02:52.789
will not discuss. Is it okay? So we say polycrystallite,
crystal, poly when oxidize, it oxidizes thicker
01:02:52.789 --> 01:03:03.940
compared, faster therefore compared to normal
or silicon. And the formula it fits into is
01:03:03.940 --> 01:03:10.099
x0 a t to the power n, n is typically more
than half which essentially is closer to parabolic.
01:03:10.099 --> 01:03:20.700
n, greater than half is parabola. So it is
slightly higher than parabola, faster growth.
01:03:20.700 --> 01:03:29.819
This is like Bt, okay, so as root Bt. So essentially
it is saying if you do that, it is essentially
01:03:29.819 --> 01:03:36.430
parabolic growth because thicker oxide is
growing. But n is not half, but it is larger
01:03:36.430 --> 01:03:41.730
than half. Is that clear? Thicker. And there
is no thinner oxide, because by then grain,
01:03:41.730 --> 01:03:47.029
enough oxidant will be made available and
it will be decided only by this available
01:03:47.029 --> 01:03:50.250
this.
01:03:50.250 --> 01:03:56.540
The next effect for us, under pressure the
crystals are stressed leading to enhanced
01:03:56.540 --> 01:04:03.980
effect. Whenever you put a wafer inside pressures
and there is an experiment done for water
01:04:03.980 --> 01:04:10.260
wafer or water solution and we actually put
high pressure on that and their oxidation
01:04:10.260 --> 01:04:15.299
rate, Jayraman’s experiment very popular
one. This pressure effects are coming back
01:04:15.299 --> 01:04:22.280
in some other way now in FETs as well as MOSFETs.
For the pressure we are talking, strains.
01:04:22.280 --> 01:04:28.680
Pressure is essentially stress proportional
to strain. So when the lattice is not matched,
01:04:28.680 --> 01:04:33.940
then it is a strain which is essentially due
to the pressure of, that is stress. Stress
01:04:33.940 --> 01:04:40.990
is pressure. So force per unit area is pressure.
So strain is essentially now coming back once
01:04:40.990 --> 01:04:47.650
again, to improve your motilities. But in
oxide, this is not crystalline, it is oxide.
01:04:47.650 --> 01:04:57.270
In oxide more bonds are available because
under pressure lattice is heavily pressed
01:04:57.270 --> 01:04:58.500
and it breaks actually.
01:04:58.500 --> 01:05:05.230
Lattice porosity is not 100 percent, so more
oxidation is possible. So formula was figured
01:05:05.230 --> 01:05:11.490
out in linear growth regime, B by A at any
pressure is B by A at atmospheric pressure
01:05:11.490 --> 01:05:19.340
times p to the power n. Typically the pressure
which we used by Jayraman earlier was 1 to
01:05:19.340 --> 01:05:28.830
3 atmospheric pressure and n was found to
be 0.7 to 0.8, again fit function. So all
01:05:28.830 --> 01:05:36.440
modeling people very happy, they fit. Experimental
data fit.
01:05:36.440 --> 01:05:41.500
Then you can say that I could have fitted
something, p should be some numbers, n should
01:05:41.500 --> 01:05:47.309
be some, I can fit any in such combination.
But I know roughly that how much pressure
01:05:47.309 --> 01:05:54.109
I can put actually before it actually breaks.
So I figured out up to 3 atmospheric it sustains.
01:05:54.109 --> 01:05:58.920
So I fixed those value in between and for
those values how much n it can fit to the
01:05:58.920 --> 01:06:04.760
data? So some physics was brought afterwards.
After I really see what is happening, I say
01:06:04.760 --> 01:06:09.029
okay, isko funda bhi maar dete hein, which
is not true because there are n combinations
01:06:09.029 --> 01:06:17.700
I can make out of it. But that is all modelers
do including person like me.
01:06:17.700 --> 01:06:21.720
Then there is an oxidation which many times
we will have to do which is called oxidation
01:06:21.720 --> 01:06:28.750
due to impurities present in the oxide. Doped
have impurities present in like, heavily doped
01:06:28.750 --> 01:06:35.260
silicon has different oxidation rate compared
to lightly doped silicon. So there is a difference
01:06:35.260 --> 01:06:43.630
in growth rates and typically the fitting
function. For doping greater than 10 to power
01:06:43.630 --> 01:06:51.539
9 per, 19 per cc. Where do you think these
numbers will appear? Which area of MOSFET?
01:06:51.539 --> 01:06:52.539
Source-drain.
01:06:52.539 --> 01:06:58.680
So in source-drain the field oxide growths
will be thicker naturally because of B by
01:06:58.680 --> 01:07:05.920
A is 20 times B by A undoped, 2 times B by
A for p-type. Beta for B is doped, is twice
01:07:05.920 --> 01:07:12.651
B type undoped. And B doped is 0.04 times
undoped for n and p-type. This is the fit
01:07:12.651 --> 01:07:19.370
data again. These numbers do not actually
matter but close to these numbers can be fitted.
01:07:19.370 --> 01:07:26.260
In real life it may become 0.30, 0.0385 but
it is okay to say 0.04.
01:07:26.260 --> 01:07:32.950
The real data may not be exactly this numbers
but this is good enough because most of the
01:07:32.950 --> 01:07:38.570
times we are not very keen, we not actually
just do by, we actually monitor. So I know
01:07:38.570 --> 01:07:44.599
how much oxide thickness I have there. So
I may not make mistake in future this but
01:07:44.599 --> 01:07:51.700
to model I cannot do very very accurate model.
If I have to, then whatever, every time I
01:07:51.700 --> 01:07:57.869
get experiment data for the lab, I really
try to fit in this and actually get the fit
01:07:57.869 --> 01:07:58.869
models.
01:07:58.869 --> 01:08:03.349
For this lab or these furnaces, these everything
I know this data which may not be universal,
01:08:03.349 --> 01:08:11.109
only for this people. This is how all people
do. We get spice parameters in the circuit
01:08:11.109 --> 01:08:20.190
simulator, all kinds of effects are taken
care, all physics is introduced by fitting.
01:08:20.190 --> 01:08:29.670
What do you mean by fitting? I mean the physics
cannot be fit, but that is what it happens.
01:08:29.670 --> 01:08:32.800
As long as IV, IDS, VDS, characteristics appears,
circuit performance appears, who cares whether
01:08:32.800 --> 01:08:36.340
it is k to the power or not to the power.
01:08:36.340 --> 01:08:41.830
This function fits, that is what we want.
But if I have to understand what is happening,
01:08:41.830 --> 01:08:47.560
then I, is the mobility dependency is how
much is correct, is the oxide thickness is
01:08:47.560 --> 01:08:53.830
varying, how much. Then I will start Bt is
constant or not constant, how much available
01:08:53.830 --> 01:08:58.900
there. So I will now put physics and since
it does not fit, I will put some constants,
01:08:58.900 --> 01:09:06.120
exponential constants so that it fits to the
lab data for a given technology.
01:09:06.120 --> 01:09:12.310
Then I have a circuit simulator with modified
model card, no one starts to be putting my
01:09:12.310 --> 01:09:18.100
model card, so I will put my model card. That
will give me the accurate result, this is
01:09:18.100 --> 01:09:25.320
how all designers do. Whatever actuality I
may, they will fit to a value which fits to
01:09:25.320 --> 01:09:32.350
the data and then of course many of us will
like to physics. So force physics on it but
01:09:32.350 --> 01:09:35.540
that is more interesting to see.
01:09:35.540 --> 01:09:43.640
Before we quit last this, all these models
since it starts with thin oxide assumption,
01:09:43.640 --> 01:09:52.440
if you are going thin oxide, then what is
the model? Since we are scaling down the technologies
01:09:52.440 --> 01:10:01.120
at least below 90 and even 65, the gate oxide
thickness is now reducing below 40 Armstrongs
01:10:01.120 --> 01:10:07.680
and in the 28 nanometer or 22 nanometer, it
is less than 5 Armstrongs, it is even close
01:10:07.680 --> 01:10:12.750
to 1 Armstrong now. 1 Armstrong of course
you cannot grow and that was the question
01:10:12.750 --> 01:10:15.000
asked in your quiz.
01:10:15.000 --> 01:10:20.890
If you scale down technologies, the oxide
thickness are becoming less than 5 Armstrongs
01:10:20.890 --> 01:10:26.200
and then there is one monolayer which you
cannot grow. And therefore you need high k,
01:10:26.200 --> 01:10:32.620
so that capacitance remains constant, epsilon
A by t. So increase 1 epsilon, increase t,
01:10:32.620 --> 01:10:40.880
the ratio remains same. So use high k, here
we are not talking of high k. We are just
01:10:40.880 --> 01:10:48.110
saying it is reducing. Till very late we were
working on thin, 65 even there is SiO2 thickness,
01:10:48.110 --> 01:10:50.610
gate insulators have been used.
01:10:50.610 --> 01:10:59.220
Below 14, 4 nanometers, I started using nano
because these days everyone must talk nano,
01:10:59.220 --> 01:11:05.500
not Armstrong. So I also wrote 4 nanometers.
The kinetics of thin oxide growth was not
01:11:05.500 --> 01:11:11.730
exactly as predicted by Deal and Grove. So
for example, thicker it may still work because
01:11:11.730 --> 01:11:17.720
then it is not thin oxide. So we are not interested
to know something about parabolic because
01:11:17.720 --> 01:11:23.930
by then it is already thick. Where is the
worry? The linear portion because there the
01:11:23.930 --> 01:11:26.890
thickness is smaller.
01:11:26.890 --> 01:11:34.390
So we figured out that if I see the Grove-Deal
model and if I see the actual thickness, it
01:11:34.390 --> 01:11:42.390
is somewhere different slope which essentially
means thinner oxides do not follow exactly
01:11:42.390 --> 01:11:50.720
in the very thin oxide regime the Deal-Grove
model. To reduce the oxide thickness what
01:11:50.720 --> 01:11:56.890
should I reduce? One is of course time, but
time cannot be 1 second. I mean you cannot
01:11:56.890 --> 01:12:03.100
push and take out, you can do, so you will
put some minutes. So what can I reduce? Temperature.
01:12:03.100 --> 01:12:09.520
So I actually reduce the temperature, I do
not go oxide at 900, 1000, 1200 but I go at
01:12:09.520 --> 01:12:14.750
800 possible. You can say why not 600, there
is no reaction. The reaction minimum temperature
01:12:14.750 --> 01:12:24.250
for SiO bond to form is 800. So the thinnest
of oxide can be grown around 800 degree centigrade
01:12:24.250 --> 01:12:35.720
which is around 1073 degrees kelvin. Please
convert everywhere into kelvins. So there
01:12:35.720 --> 01:12:45.490
are models which are available in market.
Quickly we will show you and…..
01:12:45.490 --> 01:12:53.720
So is that clear to you? For thinner oxide
regime, this is what we see, this is what
01:12:53.720 --> 01:13:05.540
Grove-Deal predicts. One of my PhD student
way back in 80s did this work which model
01:13:05.540 --> 01:13:06.950
I will show you later.
01:13:06.950 --> 01:13:12.270
There is first model which appeared was Reisman’s
model which has some modification they did.
01:13:12.270 --> 01:13:20.350
x0 is a t plus t in to the power this. t in
is a fit parameter which is essentially xi
01:13:20.350 --> 01:13:29.290
by a. a and b are constants. As a fit parameter
xi is also fit parameter. I get the data and
01:13:29.290 --> 01:13:37.390
fit it into this model and we say okay, Reisman
says if you grow this and fit into this, it
01:13:37.390 --> 01:13:45.860
will fit. So fit get a, b and xi from the
actual data and use this model for your lab,
01:13:45.860 --> 01:13:48.160
for your CAD tool.
01:13:48.160 --> 01:13:55.790
You can see everywhere what we are doing is
fit. So someone should say is it b by a or
01:13:55.790 --> 01:14:03.870
is it what? No, I do not know what is there.
It is a and b. Then there was Han and Helms
01:14:03.870 --> 01:14:11.970
model which, he say there are two parallel
reactions are going on in this. So he says
01:14:11.970 --> 01:14:22.310
B1 upon 2x0 by A1 and B2 upon 2x0 by A2. Now
this you have this B1, B2, A1, A2 as fit parameters
01:14:22.310 --> 01:14:29.480
and try to adjust x0 versus time using these
two parallel terms.
01:14:29.480 --> 01:14:34.940
So if this does not fit the data, you add
from here some term and see it may be minus
01:14:34.940 --> 01:14:40.180
as well, sign B2 may be minus, depends on
the fit you want. And you see that it fits
01:14:40.180 --> 01:14:51.900
to available data. This is called Han and
Helms model. The most important model which
01:14:51.900 --> 01:14:58.890
was accepted for quite some time was from
Plummer and his student Massoud and Massoud
01:14:58.890 --> 01:15:07.720
has the best of data as well in way back in
90s. Massoud published much of the experimental
01:15:07.720 --> 01:15:14.870
data for thin oxides as well to thick oxides
but and his data has been taken as most standard
01:15:14.870 --> 01:15:25.790
data by almost all industries. This was work
done at Stanford. Yeh doh term hain, yeh aapne
01:15:25.790 --> 01:15:31.010
pehle likha naa, ekk aur term add kardi meine.
01:15:31.010 --> 01:15:36.050
Then Massoud and Plummer has a model, they
say okay, the second term instead of B2 by
01:15:36.050 --> 01:15:43.750
something, they I added a term called e to
the power exponential x0 by L, again to, and
01:15:43.750 --> 01:15:48.810
they expected that the L value which they
will use should be less than 70 Armstrongs
01:15:48.810 --> 01:15:56.780
beyond which this is not valid, thicker oxides.
So they added another parameter here C and
01:15:56.780 --> 01:16:05.910
L to fit the data and that is, that was accepted
for many years or rather even now the first
01:16:05.910 --> 01:16:15.080
attempt is to use Massoud’s data and Plummer’s
model. We also did some more kinetics on that
01:16:15.080 --> 01:16:25.620
for thin oxide. Our work was published in
JAP 1989, myself, Vasi, and More, my PhD student.
01:16:25.620 --> 01:16:31.510
So we suggested some equations, that S is
called the available site.
01:16:31.510 --> 01:16:39.120
There is a term which we created called site,
so we say oxidant plus site may form oxygen
01:16:39.120 --> 01:16:47.570
site combination. Oxygen’s O2S plus another
site may form 2OS at, with a constant, I mean
01:16:47.570 --> 01:16:53.961
proportionality constant K2, reaction rate
constant. Then we say Si plus, Si-Si bond
01:16:53.961 --> 01:17:00.490
will react with OS to form Si-O-Si bond and
create another site. That is how oxide will
01:17:00.490 --> 01:17:02.700
keep growing, thin oxide will keep.
01:17:02.700 --> 01:17:09.140
We also introduced many constant in this expression,
P is some constant, omega into silicon-silicon
01:17:09.140 --> 01:17:20.310
bond times n K3 into K1, K2. These are called
constants of reactions. Q is K1 and R is K1,
01:17:20.310 --> 01:17:28.940
K2. This of course those who are very keen
can see our paper of 1989. We fitted this
01:17:28.940 --> 01:17:36.810
data for 800 degree and 900 degree centigrade
with Massoud experimented data, which he published
01:17:36.810 --> 01:17:42.230
for different pressures but 0.1 atmospheric
pressure is the best result they claim.
01:17:42.230 --> 01:17:50.330
So we fitted our model to this and by making
proper choices of, K1, K2 also we derived
01:17:50.330 --> 01:18:00.750
what values we should have. And based on our
analysis of this we could get P, Q, R values
01:18:00.750 --> 01:18:06.930
which fits into thin oxide. So we could fit
data up to 20 Armstrongs of oxide thickness.
01:18:06.930 --> 01:18:15.140
Below 20 of course our model also did not
fit. 20 to 40 Armstrongs or 60 Armstrongs
01:18:15.140 --> 01:18:23.740
our model fitted very well with the experimental
data then known. So if tomorrow someone wants
01:18:23.740 --> 01:18:28.210
another model, you can always try something.
01:18:28.210 --> 01:18:33.140
So you should be able to find some reactions,
what should be the real materials going on,
01:18:33.140 --> 01:18:38.300
what bonds can it create, what is the binding
energies available, what is the space charge
01:18:38.300 --> 01:18:46.100
around. So there are many things which you
can think and add on to a model. Is that okay?
01:18:46.100 --> 01:18:53.000
So this finishes the modeling part, next time
we will do oxidation techniques and characterization.