WEBVTT 00:00:00.000 --> 00:00:03.180 align:middle line:84% JUDY HOYT: Where we were-- where we are. 00:00:03.180 --> 00:00:05.250 align:middle line:84% I've been out last week, I went to a conference. 00:00:05.250 --> 00:00:07.310 align:middle line:90% So you had two-- 00:00:07.310 --> 00:00:11.270 align:middle line:84% your first two lectures on dopant diffusion 00:00:11.270 --> 00:00:12.740 align:middle line:90% given by the TA. 00:00:12.740 --> 00:00:17.390 align:middle line:84% And here we are, Lecture Number 10, 00:00:17.390 --> 00:00:19.940 align:middle line:84% we're going to talk about some more advanced models 00:00:19.940 --> 00:00:21.110 align:middle line:90% for dopant diffusion. 00:00:21.110 --> 00:00:25.312 align:middle line:84% And last week, you also had your homeworks Number 3 handed out. 00:00:25.312 --> 00:00:27.020 align:middle line:84% So hopefully, you've all started on that. 00:00:27.020 --> 00:00:30.180 align:middle line:84% You're using SUPREM for your homework. 00:00:30.180 --> 00:00:33.800 align:middle line:84% Get used to running SUPREM because that homework is due 00:00:33.800 --> 00:00:34.910 align:middle line:90% this Thursday. 00:00:34.910 --> 00:00:37.880 align:middle line:84% And then next Tuesday, you'll have another homework, homework 00:00:37.880 --> 00:00:39.770 align:middle line:84% Number 4 going out, which is also 00:00:39.770 --> 00:00:42.410 align:middle line:84% going to use the process SUPREM IV, 00:00:42.410 --> 00:00:47.600 align:middle line:90% and more sophisticated models. 00:00:47.600 --> 00:00:48.860 align:middle line:90% OK. 00:00:48.860 --> 00:00:53.390 align:middle line:84% So as I mentioned, today's lecture 00:00:53.390 --> 00:00:56.060 align:middle line:84% is essentially the third lecture out of a number 00:00:56.060 --> 00:01:00.470 align:middle line:84% that are on dopant diffusion and profile measurement. 00:01:00.470 --> 00:01:03.350 align:middle line:84% Last couple of lectures, Maggie talked 00:01:03.350 --> 00:01:06.860 align:middle line:84% about the relatively, quote, unquote, "simple cases 00:01:06.860 --> 00:01:08.540 align:middle line:90% of constant diffusivity." 00:01:08.540 --> 00:01:11.420 align:middle line:84% Diffusivity does not change in space, basically, 00:01:11.420 --> 00:01:13.460 align:middle line:90% throughout the sample. 00:01:13.460 --> 00:01:16.310 align:middle line:84% That applies when you have a low dopant concentration. 00:01:16.310 --> 00:01:19.160 align:middle line:84% We'll talk today about what happens when that assumption is 00:01:19.160 --> 00:01:22.520 align:middle line:84% broken, such as, when you're diffusing the well of the CMOS. 00:01:22.520 --> 00:01:24.770 align:middle line:84% So it's relatively low dopant concentration, 00:01:24.770 --> 00:01:28.010 align:middle line:90% say, 10 to the 18th or less. 00:01:28.010 --> 00:01:30.350 align:middle line:84% And we also saw-- you saw last time 00:01:30.350 --> 00:01:33.322 align:middle line:84% that you can design a diffuse layer based on a sheet 00:01:33.322 --> 00:01:34.280 align:middle line:90% resistance requirement. 00:01:34.280 --> 00:01:36.050 align:middle line:84% As a device engineer, you might be 00:01:36.050 --> 00:01:38.630 align:middle line:84% told you need a diffuse layer, such and such a thickness, 00:01:38.630 --> 00:01:42.925 align:middle line:84% and it should have so many ohms per square of sheet resistance. 00:01:42.925 --> 00:01:45.050 align:middle line:84% You saw how you could use some of the urban curves, 00:01:45.050 --> 00:01:46.508 align:middle line:84% or you can use numerical techniques 00:01:46.508 --> 00:01:50.180 align:middle line:90% to design a diffuse layer. 00:01:50.180 --> 00:01:52.250 align:middle line:84% Last time, also on Thursday, Maggie 00:01:52.250 --> 00:01:54.920 align:middle line:84% introduced a relatively simple but very powerful 00:01:54.920 --> 00:01:56.510 align:middle line:84% method for doing numerical solutions. 00:01:56.510 --> 00:01:59.660 align:middle line:84% There's only about three or four cases for diffusion 00:01:59.660 --> 00:02:01.110 align:middle line:90% that you can solve analytically. 00:02:01.110 --> 00:02:02.220 align:middle line:84% And of course, we try to give you 00:02:02.220 --> 00:02:03.030 align:middle line:90% some of those in the homework. 00:02:03.030 --> 00:02:05.197 align:middle line:84% But most of the time, you have to do it numerically. 00:02:05.197 --> 00:02:07.750 align:middle line:84% And this is a very simple technique. 00:02:07.750 --> 00:02:10.410 align:middle line:84% It's called the finite difference algorithm. 00:02:10.410 --> 00:02:13.430 align:middle line:84% It's slow, it's not actually what's used in SUPREM IV. 00:02:13.430 --> 00:02:14.450 align:middle line:90% SUPREM IV is the-- 00:02:14.450 --> 00:02:15.940 align:middle line:84% was more sophisticated than that. 00:02:15.940 --> 00:02:18.510 align:middle line:84% But if you were stuck on a desert island and all 00:02:18.510 --> 00:02:20.010 align:middle line:84% you had was a computer with you, you 00:02:20.010 --> 00:02:22.010 align:middle line:84% could write your own finite difference technique 00:02:22.010 --> 00:02:24.660 align:middle line:84% in about three or four minutes, and you could solve a diffusion 00:02:24.660 --> 00:02:26.580 align:middle line:84% problem like that numerically, even 00:02:26.580 --> 00:02:30.120 align:middle line:90% if you didn't have SUPREM IV. 00:02:30.120 --> 00:02:31.620 align:middle line:84% Maggie also talked about that you 00:02:31.620 --> 00:02:33.450 align:middle line:84% need to modify Fick's law for something 00:02:33.450 --> 00:02:35.550 align:middle line:84% called electric field effects, so electric fields 00:02:35.550 --> 00:02:37.110 align:middle line:90% that are induced. 00:02:37.110 --> 00:02:39.270 align:middle line:84% These can enhance the effective diffusivity 00:02:39.270 --> 00:02:40.737 align:middle line:90% of a high concentration species. 00:02:40.737 --> 00:02:42.570 align:middle line:84% So you have a high concentration of arsenic, 00:02:42.570 --> 00:02:45.720 align:middle line:84% let's say diffusing, it can actually, 00:02:45.720 --> 00:02:47.890 align:middle line:84% in a low concentration of boron background, 00:02:47.890 --> 00:02:51.390 align:middle line:84% its diffusivity can be enhanced by up to a factor of 2 00:02:51.390 --> 00:02:53.200 align:middle line:84% due to this electric field effect. 00:02:53.200 --> 00:02:55.200 align:middle line:84% But even more of an enhancement can 00:02:55.200 --> 00:02:56.910 align:middle line:84% be obtained, more than a factor of 2, 00:02:56.910 --> 00:03:00.970 align:middle line:84% very dramatic diffusion of those species of lower concentration. 00:03:00.970 --> 00:03:04.290 align:middle line:84% So she showed an example with a low boron concentration that 00:03:04.290 --> 00:03:07.140 align:middle line:84% was constant at the beginning of the diffuse profile 00:03:07.140 --> 00:03:08.760 align:middle line:84% and then diffusing arsenic into it. 00:03:08.760 --> 00:03:11.070 align:middle line:84% And then after the diffusion, it actually 00:03:11.070 --> 00:03:13.500 align:middle line:84% hadn't moved, even though the profile was flat. 00:03:13.500 --> 00:03:15.630 align:middle line:84% That cannot be-- that's non Fickian diffusion. 00:03:15.630 --> 00:03:18.250 align:middle line:84% Fick's law would say, once the profile is flat, no diffusion. 00:03:18.250 --> 00:03:21.487 align:middle line:84% So that's due to the electric field effect. 00:03:21.487 --> 00:03:23.820 align:middle line:84% So that's kind of what we went over-- what you went over 00:03:23.820 --> 00:03:24.920 align:middle line:90% last week. 00:03:24.920 --> 00:03:27.990 align:middle line:84% Today, I want to cover three items-- 00:03:27.990 --> 00:03:29.640 align:middle line:84% concentration dependent diffusion, 00:03:29.640 --> 00:03:33.420 align:middle line:84% because this is really prevalent in Silicon, especially, 00:03:33.420 --> 00:03:35.040 align:middle line:90% in making MOSFETs-- 00:03:35.040 --> 00:03:37.470 align:middle line:84% segregation interfacial dopant pileup, 00:03:37.470 --> 00:03:40.320 align:middle line:84% and starting to look at an atomic scale 00:03:40.320 --> 00:03:43.780 align:middle line:90% model of dopant diffusion. 00:03:43.780 --> 00:03:44.280 align:middle line:90% OK. 00:03:44.280 --> 00:03:46.320 align:middle line:90% Let's go on to slide number 2. 00:03:46.320 --> 00:03:50.310 align:middle line:84% And this plot introduces so-called Fermi level effects, 00:03:50.310 --> 00:03:52.350 align:middle line:84% or concentration dependent diffusion. 00:03:52.350 --> 00:03:55.440 align:middle line:84% And it's a cartoon plot, but it shows concentration 00:03:55.440 --> 00:03:59.700 align:middle line:84% on a log scale, on the y-axis, versus depth. 00:03:59.700 --> 00:04:02.110 align:middle line:84% And there are a couple of solutions here. 00:04:02.110 --> 00:04:04.710 align:middle line:84% Look at this one starting at the surface concentration of 10 00:04:04.710 --> 00:04:07.270 align:middle line:84% to the 18th going down here in red. 00:04:07.270 --> 00:04:08.970 align:middle line:84% This is a complementary error function, 00:04:08.970 --> 00:04:11.400 align:middle line:84% just like we learned a couple of lectures 00:04:11.400 --> 00:04:14.040 align:middle line:84% ago when you have a constant surface concentration. 00:04:14.040 --> 00:04:16.200 align:middle line:84% If you have a higher surface concentration, 00:04:16.200 --> 00:04:19.709 align:middle line:84% say, up in mid 10 of the 20, this red curve here, 00:04:19.709 --> 00:04:22.350 align:middle line:84% dotted curve, shows what a complementary error function 00:04:22.350 --> 00:04:23.890 align:middle line:90% would look like. 00:04:23.890 --> 00:04:25.680 align:middle line:84% Well, what we actually find when we 00:04:25.680 --> 00:04:27.900 align:middle line:84% do diffusions of a number of dopants in Silicon, 00:04:27.900 --> 00:04:29.830 align:middle line:84% that's not what they look like in high concentrations. 00:04:29.830 --> 00:04:31.330 align:middle line:84% They don't look like this red curve. 00:04:31.330 --> 00:04:35.190 align:middle line:84% They look more like the blue curve, or the green one. 00:04:35.190 --> 00:04:37.560 align:middle line:84% They have a much flatter top, and they have 00:04:37.560 --> 00:04:40.560 align:middle line:90% a much more abrupt drop-off. 00:04:40.560 --> 00:04:43.930 align:middle line:84% And this tends to occur when the dopant concentration is 00:04:43.930 --> 00:04:45.550 align:middle line:90% greater than ni. 00:04:45.550 --> 00:04:47.200 align:middle line:90% So n much, much greater than ni. 00:04:47.200 --> 00:04:49.810 align:middle line:84% So here's n, the concentration of arsenic-- 00:04:49.810 --> 00:04:50.950 align:middle line:90% let's say this is arsenic-- 00:04:50.950 --> 00:04:53.350 align:middle line:84% is 2 10 to the 20 up here at this point. 00:04:53.350 --> 00:04:56.470 align:middle line:90% And ni is a few 10 to the 18th. 00:04:56.470 --> 00:04:58.795 align:middle line:84% So orders of magnitude larger than ni, 00:04:58.795 --> 00:05:00.910 align:middle line:84% you see this that the complementary error 00:05:00.910 --> 00:05:03.610 align:middle line:84% function does not yield the correct profile, 00:05:03.610 --> 00:05:06.200 align:middle line:84% has more box-like than it would be. 00:05:06.200 --> 00:05:09.550 align:middle line:84% And by the way here, this little dashed line, horizontal line, 00:05:09.550 --> 00:05:10.750 align:middle line:90% shows you where ni is. 00:05:10.750 --> 00:05:13.960 align:middle line:84% So clearly over this portion of the profile, 00:05:13.960 --> 00:05:16.930 align:middle line:90% n is much, much greater than ni. 00:05:16.930 --> 00:05:18.850 align:middle line:84% So at high dopant concentrations we 00:05:18.850 --> 00:05:20.920 align:middle line:84% observe for a lot of the dopants in Silicon 00:05:20.920 --> 00:05:23.830 align:middle line:84% that the diffusivity appears to be larger than it 00:05:23.830 --> 00:05:25.790 align:middle line:90% is at low concentrations. 00:05:25.790 --> 00:05:30.160 align:middle line:84% And so what this means is that fixed equations have 00:05:30.160 --> 00:05:32.770 align:middle line:84% to be solved numerically since, basically what 00:05:32.770 --> 00:05:34.840 align:middle line:84% we're finding, what we see, is that diffusivity 00:05:34.840 --> 00:05:38.240 align:middle line:84% is no longer a constant throughout the sample. 00:05:38.240 --> 00:05:41.350 align:middle line:84% In fact, it depends upon the local dopant concentration 00:05:41.350 --> 00:05:42.230 align:middle line:90% at each point. 00:05:42.230 --> 00:05:44.900 align:middle line:84% So at each point along this profile from here, here, 00:05:44.900 --> 00:05:47.380 align:middle line:84% here, here, and here, the diffusivity 00:05:47.380 --> 00:05:49.760 align:middle line:90% is a slightly different number. 00:05:49.760 --> 00:05:53.440 align:middle line:84% And in fact, according to this, for the blue curve 00:05:53.440 --> 00:05:56.380 align:middle line:84% is proportional to the ratio of n over ni. 00:05:56.380 --> 00:05:58.240 align:middle line:84% And of course, n over ni is changing 00:05:58.240 --> 00:06:00.050 align:middle line:90% as I walk down this profile. 00:06:00.050 --> 00:06:02.050 align:middle line:84% So there's no way I can do an analytic solution. 00:06:02.050 --> 00:06:04.990 align:middle line:84% Every point in space I need to apply a different diffusion 00:06:04.990 --> 00:06:06.280 align:middle line:90% coefficient. 00:06:06.280 --> 00:06:09.140 align:middle line:84% That's what we mean by not equals to a constant. 00:06:09.140 --> 00:06:13.810 align:middle line:84% So in that case, remember this was Fick's second law, 00:06:13.810 --> 00:06:17.960 align:middle line:84% when I cannot pull the d out of the partial derivative. 00:06:17.960 --> 00:06:22.100 align:middle line:84% So partial d, partial t is equal to the first derivative 00:06:22.100 --> 00:06:24.170 align:middle line:84% with respect to x of this product. 00:06:24.170 --> 00:06:26.800 align:middle line:84% The diffusivity of the dopant, the effective diffusivity times 00:06:26.800 --> 00:06:28.120 align:middle line:90% partial c, partial x. 00:06:28.120 --> 00:06:29.955 align:middle line:90% Where now da effective-- 00:06:29.955 --> 00:06:31.330 align:middle line:84% I should write here-- is actually 00:06:31.330 --> 00:06:34.900 align:middle line:84% a function of x in this high concentration case. 00:06:34.900 --> 00:06:37.990 align:middle line:84% So I cannot pull it out of the derivative. 00:06:37.990 --> 00:06:42.330 align:middle line:84% I have to solve it most likely numerically. 00:06:42.330 --> 00:06:42.830 align:middle line:90% OK. 00:06:42.830 --> 00:06:45.440 align:middle line:84% So that's an introduction to what people see. 00:06:45.440 --> 00:06:47.100 align:middle line:90% How do we explain this effect? 00:06:47.100 --> 00:06:50.420 align:middle line:84% Well, as you might imagine, people 00:06:50.420 --> 00:06:52.820 align:middle line:84% explain this dependence on the Fermi level. 00:06:52.820 --> 00:06:56.840 align:middle line:84% Basically, it's the dependence on dopant, or Fermi level, 00:06:56.840 --> 00:06:59.420 align:middle line:84% based on concentrations of point defects. 00:06:59.420 --> 00:07:03.170 align:middle line:84% We saw several chapters ago if I move the Fermi level 00:07:03.170 --> 00:07:07.310 align:middle line:84% up and down the band gap, so if I increase the dopant above NI, 00:07:07.310 --> 00:07:09.180 align:middle line:90% that I change dramatically. 00:07:09.180 --> 00:07:11.660 align:middle line:84% In fact, you had a homework problem that was due last week. 00:07:11.660 --> 00:07:13.910 align:middle line:84% You saw when you went to a high concentration 00:07:13.910 --> 00:07:15.680 align:middle line:84% that the total concentration of vacancies 00:07:15.680 --> 00:07:19.340 align:middle line:84% went up because some of the charged vacancies went up. 00:07:19.340 --> 00:07:19.940 align:middle line:90% OK. 00:07:19.940 --> 00:07:22.670 align:middle line:84% More vacancies, if you need vacancies for diffusion, 00:07:22.670 --> 00:07:24.750 align:middle line:84% make sense the diffusivity would go up. 00:07:24.750 --> 00:07:28.910 align:middle line:84% So this-- basically, we saw that charge point defects 00:07:28.910 --> 00:07:31.730 align:middle line:84% obey the same statistics as shallow donors and acceptors. 00:07:31.730 --> 00:07:34.550 align:middle line:84% In fact, we wrote equations in Chapter 3, 00:07:34.550 --> 00:07:38.150 align:middle line:84% we wrote down these equations for the concentration 00:07:38.150 --> 00:07:39.860 align:middle line:90% of any charge point defect here. 00:07:39.860 --> 00:07:42.150 align:middle line:84% Here's the concentration of vacancies that 00:07:42.150 --> 00:07:43.710 align:middle line:90% are single negatively charged. 00:07:43.710 --> 00:07:46.050 align:middle line:84% We could write it in terms of the neutral vacancy 00:07:46.050 --> 00:07:48.870 align:middle line:84% concentration, which is only a function of temperature 00:07:48.870 --> 00:07:53.610 align:middle line:84% times this exponential of e Fermi minus ev minus, OK. 00:07:53.610 --> 00:07:55.980 align:middle line:84% So it's the distance between the Fermi level here 00:07:55.980 --> 00:07:59.250 align:middle line:90% and this energy-- 00:07:59.250 --> 00:08:02.520 align:middle line:84% defect energy level ev minus, that distance, 00:08:02.520 --> 00:08:04.410 align:middle line:90% that whole thing to the kt. 00:08:04.410 --> 00:08:07.080 align:middle line:84% So as I move the Fermi level up and down, 00:08:07.080 --> 00:08:10.360 align:middle line:84% I can exponentially increase or decrease. 00:08:10.360 --> 00:08:13.140 align:middle line:84% And we can write the same kind of equations for interstitials. 00:08:13.140 --> 00:08:14.880 align:middle line:84% So as you can imagine then, we're 00:08:14.880 --> 00:08:19.050 align:middle line:84% going to use this idea to say that the diffusivity, which 00:08:19.050 --> 00:08:21.420 align:middle line:84% depends directly on these concentrations, 00:08:21.420 --> 00:08:23.250 align:middle line:84% if you're diffusing with point defects, 00:08:23.250 --> 00:08:25.950 align:middle line:84% must then depend on the Fermi level, 00:08:25.950 --> 00:08:30.180 align:middle line:84% and use that to explain the concentration dependence. 00:08:30.180 --> 00:08:32.220 align:middle line:84% So I just want to derive now here 00:08:32.220 --> 00:08:35.669 align:middle line:84% on-- shown on slide 4 of your handouts 00:08:35.669 --> 00:08:39.240 align:middle line:84% an explicit expression for the charge defect concentration. 00:08:39.240 --> 00:08:41.820 align:middle line:84% This time in terms of carrier concentration. 00:08:41.820 --> 00:08:44.220 align:middle line:84% On the last slide, if we just go back one second, 00:08:44.220 --> 00:08:47.452 align:middle line:90% here it's implicit in here. 00:08:47.452 --> 00:08:49.410 align:middle line:84% I'm going to show you the carrier concentration 00:08:49.410 --> 00:08:51.240 align:middle line:90% is embedded in here. 00:08:51.240 --> 00:08:53.670 align:middle line:84% Right now, you don't see it explicitly, 00:08:53.670 --> 00:08:56.130 align:middle line:84% but I want to make an expression where 00:08:56.130 --> 00:09:00.000 align:middle line:84% we can see the dependence on n over ni directly. 00:09:00.000 --> 00:09:01.862 align:middle line:84% And we'll do this for a particular case 00:09:01.862 --> 00:09:03.070 align:middle line:90% and then we'll generalize it. 00:09:03.070 --> 00:09:04.937 align:middle line:84% So this is a little bit hard to see, 00:09:04.937 --> 00:09:06.270 align:middle line:90% the font size is a little small. 00:09:06.270 --> 00:09:09.450 align:middle line:84% But what this is saying is from the last slide we just saw, 00:09:09.450 --> 00:09:14.370 align:middle line:84% that the concentration of cv plus 00:09:14.370 --> 00:09:17.042 align:middle line:84% is equal to the neutral concentration of vacancy. 00:09:17.042 --> 00:09:18.750 align:middle line:84% So the concentration of neutral vacancies 00:09:18.750 --> 00:09:23.490 align:middle line:84% times this exponential ev plus minus e Fermi over kt. 00:09:23.490 --> 00:09:27.390 align:middle line:84% So that's-- and this is our band diagram. 00:09:27.390 --> 00:09:32.130 align:middle line:84% Remember, e sub v here is the valence band position. 00:09:32.130 --> 00:09:33.450 align:middle line:90% ec is the conduction band. 00:09:33.450 --> 00:09:35.190 align:middle line:84% This-- unfortunately, the notation 00:09:35.190 --> 00:09:36.900 align:middle line:90% is a little bit unfortunate. 00:09:36.900 --> 00:09:37.890 align:middle line:90% It's awfully similar. 00:09:37.890 --> 00:09:40.800 align:middle line:84% But ev plus is the energy level in the band 00:09:40.800 --> 00:09:43.020 align:middle line:84% gap of the single positively charged 00:09:43.020 --> 00:09:44.907 align:middle line:90% vacancy, OK, it's ev plus. 00:09:44.907 --> 00:09:45.990 align:middle line:90% It's not the valence band. 00:09:45.990 --> 00:09:47.670 align:middle line:84% The valence band energy is just E sub b 00:09:47.670 --> 00:09:50.040 align:middle line:90% for the valence band energy. 00:09:50.040 --> 00:09:53.070 align:middle line:84% The mid-gap position here is called ei, which 00:09:53.070 --> 00:09:55.530 align:middle line:90% is shown by this dashed line. 00:09:55.530 --> 00:09:57.480 align:middle line:84% The Fermi level in this particular example 00:09:57.480 --> 00:09:59.550 align:middle line:90% is given right here by ef. 00:09:59.550 --> 00:10:01.750 align:middle line:84% So those are all the relevant energies. 00:10:01.750 --> 00:10:04.590 align:middle line:84% So I want to now take this expression, this ev 00:10:04.590 --> 00:10:07.290 align:middle line:84% plus minus e Fermi and expand it out 00:10:07.290 --> 00:10:12.310 align:middle line:84% in terms of some other relevant quantities. 00:10:12.310 --> 00:10:14.850 align:middle line:84% So this ev plus minus e Fermi, that's 00:10:14.850 --> 00:10:16.980 align:middle line:84% actually just some distance between here and here, 00:10:16.980 --> 00:10:18.400 align:middle line:90% from this point here to here. 00:10:18.400 --> 00:10:22.080 align:middle line:84% So it's equal to this vector, this a value. 00:10:22.080 --> 00:10:27.810 align:middle line:84% But you can also write a as just the sum of d plus c minus b. 00:10:27.810 --> 00:10:30.240 align:middle line:84% Just add up d plus c, subtract b, that's 00:10:30.240 --> 00:10:33.730 align:middle line:90% equal to the energy distance a. 00:10:33.730 --> 00:10:36.900 align:middle line:84% And so you can take each one of these terms, d, c, and b, 00:10:36.900 --> 00:10:39.270 align:middle line:84% and write down quantities for them. 00:10:39.270 --> 00:10:42.240 align:middle line:84% For example, d, this distance right here, 00:10:42.240 --> 00:10:45.390 align:middle line:90% is just ei minus e Fermi, OK? 00:10:45.390 --> 00:10:47.850 align:middle line:90% So I can write d like that. 00:10:47.850 --> 00:10:52.200 align:middle line:84% d is just the valence band energy ev minus ei, 00:10:52.200 --> 00:10:53.400 align:middle line:90% so we've written that. 00:10:53.400 --> 00:10:59.610 align:middle line:84% And subtracting off b, b is just ev minus-- 00:10:59.610 --> 00:11:03.390 align:middle line:84% ev, the valence band energy minus ev plus. 00:11:03.390 --> 00:11:06.250 align:middle line:84% So if you want to look at it in terms of those distances, 00:11:06.250 --> 00:11:07.110 align:middle line:90% I think that helps. 00:11:07.110 --> 00:11:09.000 align:middle line:84% You can also recognize that mathematically 00:11:09.000 --> 00:11:10.710 align:middle line:84% all we really did in this equation 00:11:10.710 --> 00:11:14.560 align:middle line:84% is we added and subtracted quantities from the same side. 00:11:14.560 --> 00:11:16.620 align:middle line:84% So we haven't changed anything, but we've 00:11:16.620 --> 00:11:20.040 align:middle line:84% rearranged the terms in a way that's going to become useful. 00:11:20.040 --> 00:11:23.340 align:middle line:84% Then you can just rearrange this here 00:11:23.340 --> 00:11:29.220 align:middle line:84% where you substituted in here an eg over 2 for ei. 00:11:29.220 --> 00:11:31.495 align:middle line:90% So I've just rearranged. 00:11:31.495 --> 00:11:33.370 align:middle line:84% So it's the same quantity as we talked about. 00:11:33.370 --> 00:11:36.720 align:middle line:84% So if I substitute this in, this expression right here, 00:11:36.720 --> 00:11:40.290 align:middle line:84% into the argument on the top of the numerator 00:11:40.290 --> 00:11:42.460 align:middle line:84% of this exponential, this is what we get. 00:11:42.460 --> 00:11:44.250 align:middle line:84% We get something that looks like this. 00:11:44.250 --> 00:11:46.320 align:middle line:84% cv plus is the neutral concentration 00:11:46.320 --> 00:11:48.820 align:middle line:84% of vacancies-- the concentration of neutral vacancies 00:11:48.820 --> 00:11:52.270 align:middle line:84% times this exponential, times another somewhat 00:11:52.270 --> 00:11:54.110 align:middle line:84% more complicated looking exponential. 00:11:54.110 --> 00:11:58.540 align:middle line:84% So we've just rearranged these energy quantities. 00:11:58.540 --> 00:12:00.440 align:middle line:84% But I've done it in a particular way. 00:12:00.440 --> 00:12:04.660 align:middle line:84% And the reason we've done it, we factored out this ei minus ef 00:12:04.660 --> 00:12:07.810 align:middle line:84% is because that is directly-- that exponential of that is 00:12:07.810 --> 00:12:09.975 align:middle line:90% directly related to n over ni. 00:12:09.975 --> 00:12:12.100 align:middle line:84% That's why we wrote it in terms of this expression. 00:12:12.100 --> 00:12:17.020 align:middle line:84% Because, in fact, you know that the electron concentration 00:12:17.020 --> 00:12:19.990 align:middle line:84% divided by ni is just exponentially 00:12:19.990 --> 00:12:23.120 align:middle line:84% dependent on the distance between the Fermi level nei. 00:12:23.120 --> 00:12:25.960 align:middle line:84% That's something that we learned from Chapter 1. 00:12:25.960 --> 00:12:30.010 align:middle line:84% So that over kt, the exponential of that over kt, that's just 00:12:30.010 --> 00:12:31.315 align:middle line:90% n over ni. 00:12:31.315 --> 00:12:32.890 align:middle line:84% So I have something in this equation 00:12:32.890 --> 00:12:34.010 align:middle line:90% that looks just like that. 00:12:34.010 --> 00:12:36.670 align:middle line:84% So in fact, if I invert this, ni over n 00:12:36.670 --> 00:12:40.870 align:middle line:84% is just the ei minus e Fermi over kt, just inverting that. 00:12:40.870 --> 00:12:43.600 align:middle line:84% So this expression right here, this exponential, 00:12:43.600 --> 00:12:47.590 align:middle line:84% I'm going to be able to put in a term ni over n, replace Boris. 00:12:47.590 --> 00:12:50.250 align:middle line:90% 00:12:50.250 --> 00:12:52.980 align:middle line:84% In fact, that's exactly what we do here. 00:12:52.980 --> 00:12:55.410 align:middle line:84% In this equation that's in the red box, 00:12:55.410 --> 00:12:57.750 align:middle line:84% we can write then the concentration of v 00:12:57.750 --> 00:13:01.800 align:middle line:84% plus as ni over n times the concentration 00:13:01.800 --> 00:13:05.340 align:middle line:84% of neutral vacancies times 1 more exponential factor, 00:13:05.340 --> 00:13:07.770 align:middle line:84% this e to the minus, and then this everything 00:13:07.770 --> 00:13:11.580 align:middle line:90% in curly brackets. 00:13:11.580 --> 00:13:13.085 align:middle line:84% Well, let's look at the numerator 00:13:13.085 --> 00:13:14.460 align:middle line:84% here of what's in curly brackets. 00:13:14.460 --> 00:13:18.240 align:middle line:84% This eg over 2, if I expand this out, 00:13:18.240 --> 00:13:19.920 align:middle line:90% there's a negative sign in here. 00:13:19.920 --> 00:13:23.880 align:middle line:84% I've used this simple mathematical expression here, 00:13:23.880 --> 00:13:28.530 align:middle line:84% eg over 2, plus ev plus, minus the valence band energy. 00:13:28.530 --> 00:13:33.390 align:middle line:84% Well, just rearranging, that's ev plus minus this quantity 00:13:33.390 --> 00:13:35.750 align:middle line:84% here, the valence band energy plus eg over 2. 00:13:35.750 --> 00:13:39.750 align:middle line:84% Well, that is just ei, what's in parentheses, right? 00:13:39.750 --> 00:13:42.510 align:middle line:84% What's right here is just the definition of the mid-gap point 00:13:42.510 --> 00:13:46.000 align:middle line:90% ei, the intrinsic energy. 00:13:46.000 --> 00:13:49.840 align:middle line:84% So this numerator up here in this exponential 00:13:49.840 --> 00:13:53.740 align:middle line:84% can be written as ev plus minus ei. 00:13:53.740 --> 00:13:58.960 align:middle line:84% Well, then the exponential of that whole thing over kt, 00:13:58.960 --> 00:14:02.440 align:middle line:84% that's just equal to, essentially, 00:14:02.440 --> 00:14:05.020 align:middle line:90% the concentration of-- 00:14:05.020 --> 00:14:08.530 align:middle line:84% intrinsic concentration of v plus, 00:14:08.530 --> 00:14:11.540 align:middle line:90% of positively charged vacancies. 00:14:11.540 --> 00:14:14.710 align:middle line:84% So basically what happens is, this expression 00:14:14.710 --> 00:14:17.020 align:middle line:84% in the rectangular red box shows me 00:14:17.020 --> 00:14:19.330 align:middle line:84% that I can write the concentration of v 00:14:19.330 --> 00:14:22.510 align:middle line:90% plus under extrinsic conditions. 00:14:22.510 --> 00:14:25.720 align:middle line:84% I can write it as the concentration of v 00:14:25.720 --> 00:14:30.610 align:middle line:84% plus under intrinsic conditions times ni over n. 00:14:30.610 --> 00:14:34.060 align:middle line:84% So I immediately have factored out the dopant dependence. 00:14:34.060 --> 00:14:37.250 align:middle line:84% And so if I substitute in here-- well, it's simple. 00:14:37.250 --> 00:14:41.620 align:middle line:84% If I substitute in here n equals ni, this term just goes to 1, 00:14:41.620 --> 00:14:43.780 align:middle line:84% and then the concentration is just 00:14:43.780 --> 00:14:45.500 align:middle line:90% the intrinsic concentration. 00:14:45.500 --> 00:14:49.160 align:middle line:84% But if I were to pump ni way up, or n way up, 00:14:49.160 --> 00:14:52.330 align:middle line:84% let's say I make n very large, this number becomes very small 00:14:52.330 --> 00:14:54.850 align:middle line:84% and the concentration of v plus goes down. 00:14:54.850 --> 00:14:57.670 align:middle line:84% Alternatively, I can make n very small by going 00:14:57.670 --> 00:14:59.410 align:middle line:90% to very heavy p type material. 00:14:59.410 --> 00:15:02.200 align:middle line:84% So I make n's very small, that pumps up this number-- 00:15:02.200 --> 00:15:04.062 align:middle line:90% this whole quantity. 00:15:04.062 --> 00:15:05.770 align:middle line:84% So in p type material, this concentration 00:15:05.770 --> 00:15:06.812 align:middle line:90% is going to be very high. 00:15:06.812 --> 00:15:08.320 align:middle line:84% It's the exact same equation we just 00:15:08.320 --> 00:15:10.195 align:middle line:84% learned that you did in your homework problem 00:15:10.195 --> 00:15:13.270 align:middle line:84% where you were manipulating the energy levels 00:15:13.270 --> 00:15:17.020 align:middle line:84% and subtracting all these energy differences in the band gap. 00:15:17.020 --> 00:15:19.510 align:middle line:84% The only difference is, now we have a more convenient way 00:15:19.510 --> 00:15:20.980 align:middle line:84% of remembering it and of writing it 00:15:20.980 --> 00:15:23.800 align:middle line:84% in terms of the ratio of the dopant concentration 00:15:23.800 --> 00:15:26.530 align:middle line:90% to the intrinsic concentration. 00:15:26.530 --> 00:15:29.200 align:middle line:84% And that's very convenient for thinking 00:15:29.200 --> 00:15:33.470 align:middle line:84% about these concentration dependent effects. 00:15:33.470 --> 00:15:36.970 align:middle line:84% So if we go on to slide number 5, 00:15:36.970 --> 00:15:40.120 align:middle line:84% then basically what I've just written down 00:15:40.120 --> 00:15:43.840 align:middle line:84% is that we can write cv plus as ni 00:15:43.840 --> 00:15:47.290 align:middle line:84% over n times the concentration under intrinsic conditions. 00:15:47.290 --> 00:15:49.600 align:middle line:84% Or if you want to invert this, it's a little bit easier 00:15:49.600 --> 00:15:52.630 align:middle line:84% to think it either in terms of p over ni, or n over ni, 00:15:52.630 --> 00:15:54.610 align:middle line:90% you can just write ni over n. 00:15:54.610 --> 00:15:58.460 align:middle line:90% Well, you know what n is, right? 00:15:58.460 --> 00:16:02.030 align:middle line:84% pn product is always equal to ni squared. 00:16:02.030 --> 00:16:07.870 align:middle line:84% So you can write n as being equal to ni squared over p. 00:16:07.870 --> 00:16:10.090 align:middle line:84% So if we want to write it in terms of the majority 00:16:10.090 --> 00:16:12.250 align:middle line:84% carrier given this expression, then 00:16:12.250 --> 00:16:15.190 align:middle line:84% I can substitute for this lower n in the denominator, 00:16:15.190 --> 00:16:16.420 align:middle line:90% ni squared over p. 00:16:16.420 --> 00:16:19.940 align:middle line:84% And what I end up with is p over ni times that concentration. 00:16:19.940 --> 00:16:22.390 align:middle line:84% So clearly, the concentration-- cv plus 00:16:22.390 --> 00:16:24.977 align:middle line:84% goes up in heavily p type material. 00:16:24.977 --> 00:16:27.310 align:middle line:84% And that's kind of what we know is saying mathematically 00:16:27.310 --> 00:16:29.230 align:middle line:90% what we knew intuitively. 00:16:29.230 --> 00:16:30.970 align:middle line:90% ev plus is down here. 00:16:30.970 --> 00:16:33.310 align:middle line:84% When I make the material very heavily p type, 00:16:33.310 --> 00:16:35.470 align:middle line:84% I bring the Fermi level down towards that. 00:16:35.470 --> 00:16:37.510 align:middle line:84% And as that distance decreases, I'm 00:16:37.510 --> 00:16:40.390 align:middle line:84% going to pump up the concentration of v plus 00:16:40.390 --> 00:16:43.760 align:middle line:90% vacancies. 00:16:43.760 --> 00:16:46.940 align:middle line:84% And in fact, it's directly proportional to p over ni. 00:16:46.940 --> 00:16:50.660 align:middle line:84% Similarly, for the double positively charged, 00:16:50.660 --> 00:16:55.610 align:middle line:84% I can write it as cv plus plus is equal to p over ni squared. 00:16:55.610 --> 00:16:58.430 align:middle line:84% It turns out there's a square quantity in there. 00:16:58.430 --> 00:17:02.000 align:middle line:84% cv minus is what we just derived is n over ni times 00:17:02.000 --> 00:17:06.530 align:middle line:84% the intrinsic concentration of cv minus, and cv double minus, 00:17:06.530 --> 00:17:10.319 align:middle line:84% depends similarly with the n over ni squared. 00:17:10.319 --> 00:17:14.630 align:middle line:84% So in general, you could write a general using this derivation, 00:17:14.630 --> 00:17:18.800 align:middle line:84% sort of a general rule that the concentration of any vacancy 00:17:18.800 --> 00:17:22.579 align:middle line:84% in any charged state r, r could be zero for neutral, it could 00:17:22.579 --> 00:17:25.550 align:middle line:84% be plus 1 minus 1, that concentration 00:17:25.550 --> 00:17:29.270 align:middle line:84% under extrinsic conditions is just n over ni to the minus 00:17:29.270 --> 00:17:32.720 align:middle line:84% r times the concentration of that species 00:17:32.720 --> 00:17:34.950 align:middle line:90% under intrinsic conditions. 00:17:34.950 --> 00:17:38.660 align:middle line:84% So all this is saying, that's a generalized term, 00:17:38.660 --> 00:17:42.240 align:middle line:84% is that the concentrations of charge point defects 00:17:42.240 --> 00:17:45.360 align:middle line:84% and, of course, the total point defect populations increase 00:17:45.360 --> 00:17:49.650 align:middle line:84% or decrease directly proportional to n over ni. 00:17:49.650 --> 00:17:53.155 align:middle line:84% So with all this mathematical manipulation, 00:17:53.155 --> 00:17:54.530 align:middle line:84% what it boils down to physically, 00:17:54.530 --> 00:17:58.310 align:middle line:84% if the dopants diffuse using these point defects, 00:17:58.310 --> 00:18:01.100 align:middle line:84% the vacant charge vacancies are interstitial. 00:18:01.100 --> 00:18:03.260 align:middle line:84% Then the diffusivity of the pair that 00:18:03.260 --> 00:18:07.520 align:middle line:84% is of the dopant and the charged vacancy, or interstitial, 00:18:07.520 --> 00:18:11.190 align:middle line:84% is proportional to the point defect concentration. 00:18:11.190 --> 00:18:14.090 align:middle line:84% Then the total diffusivity will follow these same trends. 00:18:14.090 --> 00:18:17.750 align:middle line:84% So if I-- as the concentration of vacancies 00:18:17.750 --> 00:18:21.180 align:middle line:84% goes up because I'm moving the Fermi level up very high, 00:18:21.180 --> 00:18:23.720 align:middle line:84% so I get-- let's say I get a lot more of these cv minuses, 00:18:23.720 --> 00:18:25.850 align:middle line:84% total concentration of vacancies goes up. 00:18:25.850 --> 00:18:29.180 align:middle line:84% If I have a diffuser like antimony 00:18:29.180 --> 00:18:31.310 align:middle line:84% that diffuses with vacancies, then you 00:18:31.310 --> 00:18:34.100 align:middle line:84% expect its diffusivity to be enhanced 00:18:34.100 --> 00:18:36.710 align:middle line:84% because it has more vacancies around it to diffuse with. 00:18:36.710 --> 00:18:38.360 align:middle line:84% And it should be enhanced according 00:18:38.360 --> 00:18:41.130 align:middle line:90% to this n over ni expression. 00:18:41.130 --> 00:18:44.520 align:middle line:84% So the higher I make n over ni, the more of these vacancies, 00:18:44.520 --> 00:18:46.520 align:middle line:84% the more the diffusion coefficient should go up. 00:18:46.520 --> 00:18:47.687 align:middle line:90% That's the general argument. 00:18:47.687 --> 00:18:50.960 align:middle line:90% 00:18:50.960 --> 00:18:53.200 align:middle line:84% So what evidence do we have of this? 00:18:53.200 --> 00:18:55.600 align:middle line:84% Let me show you some experimental data which 00:18:55.600 --> 00:18:57.310 align:middle line:84% indicates, although it doesn't tell you 00:18:57.310 --> 00:18:59.435 align:middle line:84% that there are vacancies or interstitials involved, 00:18:59.435 --> 00:19:03.730 align:middle line:84% but at least indicates these dependencies make some sense-- 00:19:03.730 --> 00:19:06.370 align:middle line:84% the Fermi level dependence, that is. 00:19:06.370 --> 00:19:09.640 align:middle line:84% And here's some experiments that are called ISO concentration 00:19:09.640 --> 00:19:12.310 align:middle line:84% experiments-- and you'll see in a moment why we call them ISO 00:19:12.310 --> 00:19:14.170 align:middle line:90% concentration-- 00:19:14.170 --> 00:19:17.230 align:middle line:84% indicating the dependence of the diffusion coefficient 00:19:17.230 --> 00:19:18.950 align:middle line:90% on the concentration. 00:19:18.950 --> 00:19:25.390 align:middle line:84% So you might have boron 10 diffusing in a boron 11 00:19:25.390 --> 00:19:27.350 align:middle line:90% background, for example. 00:19:27.350 --> 00:19:29.950 align:middle line:84% And so just-- let's take a look at what we mean by this. 00:19:29.950 --> 00:19:32.410 align:middle line:84% This is an ISO concentration diffusion experiment. 00:19:32.410 --> 00:19:35.800 align:middle line:84% So if we were to plot concentration of a species 00:19:35.800 --> 00:19:38.590 align:middle line:84% as a function of depth, what we do 00:19:38.590 --> 00:19:40.450 align:middle line:84% is we put two species in the sample. 00:19:40.450 --> 00:19:43.030 align:middle line:84% The first one is a background species, which I've shown here 00:19:43.030 --> 00:19:47.290 align:middle line:84% by this orange box, or this constant concentration profile 00:19:47.290 --> 00:19:48.320 align:middle line:90% in this region. 00:19:48.320 --> 00:19:51.610 align:middle line:84% So this could be my B11, so it's boron 11. 00:19:51.610 --> 00:19:55.330 align:middle line:84% It sets the Fermi level to a constant value in this one 00:19:55.330 --> 00:19:58.330 align:middle line:84% region, right, because it's a high concentration of dopant 00:19:58.330 --> 00:20:01.420 align:middle line:84% that's p-type, so it defines p over ni. 00:20:01.420 --> 00:20:05.170 align:middle line:84% And then underneath that at a smaller concentration, 00:20:05.170 --> 00:20:08.950 align:middle line:84% I could put boron 11, which is the dopant 00:20:08.950 --> 00:20:10.150 align:middle line:90% that I want to study. 00:20:10.150 --> 00:20:12.920 align:middle line:84% I want to study its diffusion coefficient. 00:20:12.920 --> 00:20:14.920 align:middle line:84% So I do a diffusion experiment where 00:20:14.920 --> 00:20:17.730 align:middle line:84% I start with an initial profile, a Gaussian profile of boron, 00:20:17.730 --> 00:20:19.000 align:middle line:90% I let it diffuse. 00:20:19.000 --> 00:20:21.160 align:middle line:84% All of it in a background concentration 00:20:21.160 --> 00:20:24.910 align:middle line:90% of high concentration boron 10. 00:20:24.910 --> 00:20:26.125 align:middle line:90% OK. 00:20:26.125 --> 00:20:27.250 align:middle line:90% I measured the diffusivity. 00:20:27.250 --> 00:20:29.140 align:middle line:84% Now, I go again and I take another sample. 00:20:29.140 --> 00:20:31.120 align:middle line:84% And this time I put a higher concentration 00:20:31.120 --> 00:20:34.360 align:middle line:84% of B10, again, constant in space but higher 00:20:34.360 --> 00:20:36.130 align:middle line:84% than it was in the previous sample. 00:20:36.130 --> 00:20:38.290 align:middle line:84% And what you see is that the diffusion 00:20:38.290 --> 00:20:40.420 align:middle line:84% of the dopant of low concentration dopant 00:20:40.420 --> 00:20:42.190 align:middle line:90% is more enhanced now. 00:20:42.190 --> 00:20:44.860 align:middle line:84% And you keep doing this ISO concentration, 00:20:44.860 --> 00:20:46.990 align:middle line:84% you keep putting in this box like profile 00:20:46.990 --> 00:20:50.260 align:middle line:84% as the background at higher and higher concentrations, 00:20:50.260 --> 00:20:54.830 align:middle line:84% and you measure each sample and you measure its diffusivity. 00:20:54.830 --> 00:20:57.670 align:middle line:84% So the nice thing about that is the profiles 00:20:57.670 --> 00:21:00.970 align:middle line:84% remain Gaussian because in space, the diffusion 00:21:00.970 --> 00:21:04.420 align:middle line:84% of the dopant in any one of these regions 00:21:04.420 --> 00:21:07.990 align:middle line:84% is a constant now, as long as you stay inside the orange box. 00:21:07.990 --> 00:21:09.760 align:middle line:84% It's just higher than it would be 00:21:09.760 --> 00:21:12.040 align:middle line:84% in the absence of the background dopant. 00:21:12.040 --> 00:21:14.990 align:middle line:84% So it makes analysis of the profiles a lot easier. 00:21:14.990 --> 00:21:17.720 align:middle line:84% You can still get Gaussian diffusion. 00:21:17.720 --> 00:21:20.595 align:middle line:84% So people have done these type of background concentration 00:21:20.595 --> 00:21:21.970 align:middle line:84% studies where they move the Fermi 00:21:21.970 --> 00:21:25.110 align:middle line:84% level up and down with another species. 00:21:25.110 --> 00:21:29.120 align:middle line:84% In fact, here is some data I took from a literature on slide 00:21:29.120 --> 00:21:34.350 align:middle line:84% number 7 from Marc Law's work published back in 1993-- 00:21:34.350 --> 00:21:36.170 align:middle line:90% so it's about 10 years old now-- 00:21:36.170 --> 00:21:39.470 align:middle line:90% where they did just that. 00:21:39.470 --> 00:21:43.550 align:middle line:84% They had a background concentration of anti-dopant 00:21:43.550 --> 00:21:47.240 align:middle line:84% that they created with, say, arsenic. 00:21:47.240 --> 00:21:49.340 align:middle line:84% And then they looked at diffusion of phosphorus 00:21:49.340 --> 00:21:51.420 align:middle line:90% underneath it. 00:21:51.420 --> 00:21:55.040 align:middle line:84% The Fermi level was constant in each one of these samples. 00:21:55.040 --> 00:21:57.200 align:middle line:84% They got this triangle, and they got, 00:21:57.200 --> 00:22:00.500 align:middle line:84% say, the electron concentration was 10 of the 19th here, 00:22:00.500 --> 00:22:03.380 align:middle line:84% going all the way up to low 10 of the 20. 00:22:03.380 --> 00:22:07.010 align:middle line:84% And then they extracted the diffusivity of the phosphorus 00:22:07.010 --> 00:22:11.270 align:middle line:84% as a function of the background electron concentration. 00:22:11.270 --> 00:22:14.360 align:middle line:84% And in fact, you see this is the experimental values, 00:22:14.360 --> 00:22:16.880 align:middle line:84% diffusivity going up very rapidly. 00:22:16.880 --> 00:22:21.440 align:middle line:84% When you get above about, oh, say, mid 10 to the 18th, 00:22:21.440 --> 00:22:24.490 align:middle line:90% this was at 1,000 degrees. 00:22:24.490 --> 00:22:27.400 align:middle line:84% Well, what is ni equal to at 1,000? 00:22:27.400 --> 00:22:31.120 align:middle line:84% Maybe you remember from the homework roughly? 00:22:31.120 --> 00:22:33.157 align:middle line:84% About mid 10 to the 18th, in that range. 00:22:33.157 --> 00:22:34.990 align:middle line:84% So you see, that's exactly what's happening, 00:22:34.990 --> 00:22:36.157 align:middle line:90% the passivity is a constant. 00:22:36.157 --> 00:22:38.000 align:middle line:90% This was the calculated. 00:22:38.000 --> 00:22:42.880 align:middle line:84% This is the calculated theoretical line where they 00:22:42.880 --> 00:22:45.940 align:middle line:90% use this type of expression. 00:22:45.940 --> 00:22:47.850 align:middle line:84% So the effect of diffusivity was given 00:22:47.850 --> 00:22:52.950 align:middle line:84% by a constant, d0 plus d minus times n over ni, 00:22:52.950 --> 00:22:55.180 align:middle line:84% plus d double minus times n over ni squared. 00:22:55.180 --> 00:22:57.120 align:middle line:90% So there's three terms here. 00:22:57.120 --> 00:22:59.250 align:middle line:84% And what this equation tells you is, well, 00:22:59.250 --> 00:23:03.960 align:middle line:84% if I plug in n equals ni, then it becomes a constant, OK? 00:23:03.960 --> 00:23:07.090 align:middle line:84% At ni or below, this whole thing goes to a constant number. 00:23:07.090 --> 00:23:09.300 align:middle line:84% And so if you go below ni, indeed, the diffusivity 00:23:09.300 --> 00:23:10.760 align:middle line:90% is approaching a constant here. 00:23:10.760 --> 00:23:13.050 align:middle line:84% It just looks like it's about 1 and 1/2 times 10 00:23:13.050 --> 00:23:15.380 align:middle line:90% to the minus 14th. 00:23:15.380 --> 00:23:18.380 align:middle line:84% As I crank up n over ni, this time right 00:23:18.380 --> 00:23:20.990 align:middle line:84% here starts to take over and over ni. 00:23:20.990 --> 00:23:22.580 align:middle line:90% And you see an increase. 00:23:22.580 --> 00:23:24.590 align:middle line:84% And I crank up n over and ni even higher, 00:23:24.590 --> 00:23:28.040 align:middle line:84% see here I get a factor of 10 or 20. 00:23:28.040 --> 00:23:30.860 align:middle line:84% This n over ni squared kicks in and, indeed, you 00:23:30.860 --> 00:23:34.317 align:middle line:90% see this square dependence. 00:23:34.317 --> 00:23:35.900 align:middle line:84% So this is based on experimental data, 00:23:35.900 --> 00:23:39.590 align:middle line:84% and it fits this type of empirical fit quite well. 00:23:39.590 --> 00:23:41.840 align:middle line:90% Now, what are these numbers? 00:23:41.840 --> 00:23:44.778 align:middle line:84% And what is the meaning of d zero, d minus, or d 00:23:44.778 --> 00:23:45.320 align:middle line:90% double minus? 00:23:45.320 --> 00:23:47.030 align:middle line:84% Well, we don't really know the meaning. 00:23:47.030 --> 00:23:49.930 align:middle line:84% But what we assign the meaning to is, 00:23:49.930 --> 00:23:53.540 align:middle line:84% we presume that this coefficient d minus 00:23:53.540 --> 00:23:56.660 align:middle line:84% has something to do with the diffusivity of the pair 00:23:56.660 --> 00:24:00.500 align:middle line:84% of the dopant with the single negatively charged point 00:24:00.500 --> 00:24:02.030 align:middle line:90% defect, whatever it should be. 00:24:02.030 --> 00:24:04.430 align:middle line:84% So it could be that the pair of the phosphorus 00:24:04.430 --> 00:24:06.620 align:middle line:90% pairing with d minus. 00:24:06.620 --> 00:24:10.880 align:middle line:84% So that diffusivity value has to do with that pair diffusivity. 00:24:10.880 --> 00:24:15.330 align:middle line:84% This term here has to do with the pairing of the dopant 00:24:15.330 --> 00:24:17.190 align:middle line:90% with v double minus. 00:24:17.190 --> 00:24:18.990 align:middle line:84% Or if you believe in interstitials, 00:24:18.990 --> 00:24:20.240 align:middle line:90% the I double minus, whichever. 00:24:20.240 --> 00:24:22.115 align:middle line:84% These experiments don't tell you whether it's 00:24:22.115 --> 00:24:24.720 align:middle line:84% a dop-- whether it's a vacancy or interstitial enhancing it. 00:24:24.720 --> 00:24:28.500 align:middle line:84% All it says is that at some point defect it has-- 00:24:28.500 --> 00:24:31.200 align:middle line:84% the concentration of that point defect 00:24:31.200 --> 00:24:33.240 align:middle line:84% depends on the Fermi level because it's charged. 00:24:33.240 --> 00:24:36.450 align:middle line:84% So as I move the Fermi level up by increasing n over ni, 00:24:36.450 --> 00:24:40.680 align:middle line:84% the diffusivity overall goes up according to n over ni. 00:24:40.680 --> 00:24:43.230 align:middle line:84% That's basically what it tells us. 00:24:43.230 --> 00:24:44.890 align:middle line:90% So those are the types-- 00:24:44.890 --> 00:24:46.410 align:middle line:84% one type of experiment people have 00:24:46.410 --> 00:24:51.390 align:middle line:84% used to observe these so-called Fermi level effects. 00:24:51.390 --> 00:24:54.050 align:middle line:84% So what we do based on this experiment showing 00:24:54.050 --> 00:24:58.040 align:middle line:84% on slide number 8, we write the diffusivity 00:24:58.040 --> 00:25:00.050 align:middle line:84% in terms of the local carrier concentration 00:25:00.050 --> 00:25:03.240 align:middle line:84% at each point in space in the sample. 00:25:03.240 --> 00:25:04.310 align:middle line:90% So here's an example. 00:25:04.310 --> 00:25:07.280 align:middle line:84% Equation 2 shows you the case for an n type dopant, 00:25:07.280 --> 00:25:11.000 align:middle line:84% a convenient way to write it as the effective diffusivity of a. 00:25:11.000 --> 00:25:14.420 align:middle line:84% a could be the arsenic, or phosphorous, 00:25:14.420 --> 00:25:19.160 align:middle line:84% or whatever, is a constant d0, plus d minus times n over ni, 00:25:19.160 --> 00:25:20.910 align:middle line:84% plus d double minus n over ni squared. 00:25:20.910 --> 00:25:23.090 align:middle line:84% So we write that type of expression. 00:25:23.090 --> 00:25:25.610 align:middle line:84% And we write a very much analogous type of expression 00:25:25.610 --> 00:25:26.570 align:middle line:90% for p type dopants. 00:25:26.570 --> 00:25:30.020 align:middle line:84% But now, the dependency is on p over ni. 00:25:30.020 --> 00:25:33.830 align:middle line:84% So presumably, this term here, this d plus, 00:25:33.830 --> 00:25:36.830 align:middle line:84% refers to the diffusion of the dopant, maybe boron, 00:25:36.830 --> 00:25:40.560 align:middle line:84% with a single positively charged vacancy, for example. 00:25:40.560 --> 00:25:43.610 align:middle line:84% So as I increase p over ni, that v concentration 00:25:43.610 --> 00:25:46.460 align:middle line:84% goes way up and so does the effective diffusivity. 00:25:46.460 --> 00:25:49.150 align:middle line:90% 00:25:49.150 --> 00:25:51.600 align:middle line:84% So again, we talked about what each one of these d's 00:25:51.600 --> 00:25:53.890 align:middle line:90% corresponds to. 00:25:53.890 --> 00:25:57.000 align:middle line:84% So again, if I'm under intrinsic conditions 00:25:57.000 --> 00:25:59.760 align:middle line:84% and I have an n type dopant where intrinsic means 00:25:59.760 --> 00:26:04.950 align:middle line:84% p equals n equals ni, that's the definition of intrinsic. 00:26:04.950 --> 00:26:07.770 align:middle line:84% But I just substitute in here for n, ni, 00:26:07.770 --> 00:26:10.180 align:middle line:84% and you see that the diffusivity is indeed a constant. 00:26:10.180 --> 00:26:14.250 align:middle line:84% It's the sum of three numbers, but it's a constant number. 00:26:14.250 --> 00:26:18.510 align:middle line:84% It just depends-- doesn't depend on the local concentration 00:26:18.510 --> 00:26:19.770 align:middle line:90% when you're intrinsic. 00:26:19.770 --> 00:26:23.220 align:middle line:90% 00:26:23.220 --> 00:26:26.970 align:middle line:84% And each individual diffusivity, this d to the r power, 00:26:26.970 --> 00:26:29.410 align:middle line:84% or if you want to call it-- it's not really to a power, 00:26:29.410 --> 00:26:32.040 align:middle line:84% it's just a symbol, a superscripted symbol that 00:26:32.040 --> 00:26:33.930 align:middle line:84% tells you the charge of the point defect, 00:26:33.930 --> 00:26:37.770 align:middle line:84% each one of these diffusivities is exponentially activated. 00:26:37.770 --> 00:26:40.350 align:middle line:84% You can write it in Arrhenius type fashion. 00:26:40.350 --> 00:26:42.780 align:middle line:90% So d is equal to d dot zero. 00:26:42.780 --> 00:26:44.250 align:middle line:90% This is using the SUPREM-- 00:26:44.250 --> 00:26:45.750 align:middle line:84% you should learn this because you'll 00:26:45.750 --> 00:26:48.660 align:middle line:84% use it in your homework-- the SUPREM sort of notation. 00:26:48.660 --> 00:26:50.220 align:middle line:90% Any diffusivity is d dot zero. 00:26:50.220 --> 00:26:54.840 align:middle line:84% That's the pre-exponential times e to the minus d dot e. 00:26:54.840 --> 00:26:59.170 align:middle line:84% That's the activation energy divided by kt. 00:26:59.170 --> 00:27:04.690 align:middle line:84% So each one of these terms has this exponentially activated. 00:27:04.690 --> 00:27:08.530 align:middle line:84% So sometimes this equation, equation number two here, 00:27:08.530 --> 00:27:12.130 align:middle line:84% is rewritten in a slightly different fashion. 00:27:12.130 --> 00:27:16.540 align:middle line:84% People like to talk about these parameters beta and gamma. 00:27:16.540 --> 00:27:20.920 align:middle line:84% Beta is defined as the ratio of d minus to d zero. 00:27:20.920 --> 00:27:22.360 align:middle line:90% So it's that ratio. 00:27:22.360 --> 00:27:24.640 align:middle line:84% And gamma is defined as the ratio of d double 00:27:24.640 --> 00:27:25.690 align:middle line:90% minus to d zero. 00:27:25.690 --> 00:27:27.850 align:middle line:84% So when I make that definition, then I 00:27:27.850 --> 00:27:31.970 align:middle line:84% can rewrite equation 2 in this fashion. 00:27:31.970 --> 00:27:34.910 align:middle line:84% The effective diffusivity is da star, 00:27:34.910 --> 00:27:38.210 align:middle line:84% where this is under intrinsic conditions, 00:27:38.210 --> 00:27:42.710 align:middle line:84% and you factor that out, and it's 1 plus beta n over ni 00:27:42.710 --> 00:27:44.840 align:middle line:84% plus gamma n over ni squared divided 00:27:44.840 --> 00:27:47.010 align:middle line:90% by 1 plus beta plus gamma. 00:27:47.010 --> 00:27:49.755 align:middle line:84% So it's just another mathematical formulation. 00:27:49.755 --> 00:27:52.130 align:middle line:84% People sometimes like to think about these beta and gamma 00:27:52.130 --> 00:27:54.260 align:middle line:84% terms, the ratios of the diffusivities, 00:27:54.260 --> 00:27:57.710 align:middle line:84% rather than the absolute numbers. 00:27:57.710 --> 00:27:59.720 align:middle line:84% For a p type dopant in this equation, 00:27:59.720 --> 00:28:03.470 align:middle line:84% you would just replace the n by p, basically, 00:28:03.470 --> 00:28:04.610 align:middle line:90% for a p type dopant. 00:28:04.610 --> 00:28:07.430 align:middle line:84% And then the beta and gamma are redefined 00:28:07.430 --> 00:28:11.210 align:middle line:84% according to the appropriate positively charged 00:28:11.210 --> 00:28:12.260 align:middle line:90% diffusivities. 00:28:12.260 --> 00:28:15.020 align:middle line:84% So again, this is the, quote, unquote, "Fermi model." 00:28:15.020 --> 00:28:17.960 align:middle line:84% When you run SUPREM IV, as you'll 00:28:17.960 --> 00:28:21.260 align:middle line:84% do for your next homework set, and you use a Fermi model, 00:28:21.260 --> 00:28:24.650 align:middle line:84% you're invoking this type of concentration dependence 00:28:24.650 --> 00:28:26.750 align:middle line:84% on the diffusivity for the dopant. 00:28:26.750 --> 00:28:30.120 align:middle line:90% 00:28:30.120 --> 00:28:32.960 align:middle line:84% So if we go on to slide 11, just to show you, 00:28:32.960 --> 00:28:37.200 align:middle line:84% I've taken from Tables 7-5 in the text. 00:28:37.200 --> 00:28:38.422 align:middle line:90% These are the quantities. 00:28:38.422 --> 00:28:40.130 align:middle line:84% This is the way we write the diffusivity, 00:28:40.130 --> 00:28:42.422 align:middle line:84% and these are the quantities that are in this equation. 00:28:42.422 --> 00:28:44.750 align:middle line:84% There's the pre-exponential factor, d dot zero, 00:28:44.750 --> 00:28:48.330 align:middle line:84% and then there's the activation energy. 00:28:48.330 --> 00:28:51.060 align:middle line:90% So just here are some examples. 00:28:51.060 --> 00:28:55.490 align:middle line:84% The first two rows refer to the d zero 00:28:55.490 --> 00:29:00.330 align:middle line:84% term, both the pre-exponential and the activation energy. 00:29:00.330 --> 00:29:03.530 align:middle line:84% And the last row, a couple of rows, 00:29:03.530 --> 00:29:06.710 align:middle line:90% refer to the double minus term. 00:29:06.710 --> 00:29:10.910 align:middle line:84% So just from looking at this chart, 00:29:10.910 --> 00:29:12.680 align:middle line:84% let's say we take the case of arsenic. 00:29:12.680 --> 00:29:15.050 align:middle line:84% And these are the numbers I took out of your text, 00:29:15.050 --> 00:29:17.510 align:middle line:84% and some of these are also used in SUPREM. 00:29:17.510 --> 00:29:20.690 align:middle line:84% Based on looking at this chart, people 00:29:20.690 --> 00:29:22.220 align:middle line:90% have fit data to arsenic. 00:29:22.220 --> 00:29:24.980 align:middle line:84% And what can you say about the different types 00:29:24.980 --> 00:29:27.980 align:middle line:84% of point defects that arsenic supposedly diffuses with? 00:29:27.980 --> 00:29:31.980 align:middle line:84% What are they based on-- if there's no number in the chart, 00:29:31.980 --> 00:29:34.980 align:middle line:84% then there's nothing been observed for that. 00:29:34.980 --> 00:29:39.100 align:middle line:84% So arsenic diffuses with what types of point defects? 00:29:39.100 --> 00:29:43.180 align:middle line:84% Single negatively charged and neutral, 00:29:43.180 --> 00:29:46.300 align:middle line:84% just because there's nothing else filled in in the chart. 00:29:46.300 --> 00:29:49.900 align:middle line:84% So people have observed for arsenic, primarily, 00:29:49.900 --> 00:29:51.880 align:middle line:90% an n over ni dependence. 00:29:51.880 --> 00:29:53.530 align:middle line:84% For phosphorus, how about phosphorus? 00:29:53.530 --> 00:29:57.950 align:middle line:84% What is it diffused with based on this chart? 00:29:57.950 --> 00:30:02.392 align:middle line:84% Got neutral, single, and double negatively charged. 00:30:02.392 --> 00:30:03.350 align:middle line:90% So there's three terms. 00:30:03.350 --> 00:30:06.080 align:middle line:84% And I showed you that when we saw Mark Law's data. 00:30:06.080 --> 00:30:08.210 align:middle line:90% He had those three terms. 00:30:08.210 --> 00:30:10.040 align:middle line:90% He fit the arsenic-- 00:30:10.040 --> 00:30:12.200 align:middle line:84% the phosphorus data back on slide 7. 00:30:12.200 --> 00:30:14.670 align:middle line:84% He fit this to a three term type of model. 00:30:14.670 --> 00:30:18.140 align:middle line:84% So these three terms come from experiments like the one 00:30:18.140 --> 00:30:18.770 align:middle line:90% that he did. 00:30:18.770 --> 00:30:21.530 align:middle line:90% 00:30:21.530 --> 00:30:23.420 align:middle line:90% OK. 00:30:23.420 --> 00:30:27.090 align:middle line:84% So these are the Fermi models for extrinsic diffusion. 00:30:27.090 --> 00:30:30.890 align:middle line:84% So when n is greater than ni, or when p is greater than ni, 00:30:30.890 --> 00:30:33.420 align:middle line:90% we use these types of equations. 00:30:33.420 --> 00:30:36.740 align:middle line:84% And this is what's used in your process simulator SUPREM IV. 00:30:36.740 --> 00:30:40.270 align:middle line:90% 00:30:40.270 --> 00:30:40.960 align:middle line:90% OK. 00:30:40.960 --> 00:30:42.550 align:middle line:90% Let's go on to page number 10. 00:30:42.550 --> 00:30:44.590 align:middle line:84% So on slide number 10, I just want 00:30:44.590 --> 00:30:47.440 align:middle line:84% to go over a little example just to give you a feel for how 00:30:47.440 --> 00:30:49.700 align:middle line:90% these numbers work. 00:30:49.700 --> 00:30:52.130 align:middle line:84% An example asks you to calculate the effective diffusion 00:30:52.130 --> 00:30:55.070 align:middle line:84% coefficient at 1,000 degrees for two 00:30:55.070 --> 00:30:57.185 align:middle line:84% different box-shaped arsenic profiles. 00:30:57.185 --> 00:30:59.060 align:middle line:84% So they're going to give you an easy profile. 00:30:59.060 --> 00:31:01.160 align:middle line:90% It'll be a box like thing. 00:31:01.160 --> 00:31:03.510 align:middle line:84% And there are two different ones. 00:31:03.510 --> 00:31:06.980 align:middle line:84% One is doped at 10 to the 18th, and the other profile 00:31:06.980 --> 00:31:09.538 align:middle line:90% is doped at 10 to the 20. 00:31:09.538 --> 00:31:12.080 align:middle line:84% So the first thing you need to do when you're calculating any 00:31:12.080 --> 00:31:14.450 align:middle line:84% of these is figure out for the temperature you're at, 00:31:14.450 --> 00:31:16.700 align:middle line:84% what is ni, because everything varies. 00:31:16.700 --> 00:31:18.890 align:middle line:84% If you're less than ni, ni or below, 00:31:18.890 --> 00:31:20.750 align:middle line:84% then you have a constant diffusivity 00:31:20.750 --> 00:31:25.100 align:middle line:84% and you just plug in for n, n equals ni, right? 00:31:25.100 --> 00:31:30.203 align:middle line:84% If you're above ni, then the n is equal to the-- 00:31:30.203 --> 00:31:32.120 align:middle line:84% essentially, whatever the dopant concentration 00:31:32.120 --> 00:31:34.310 align:middle line:84% is because it's being controlled extrinsically 00:31:34.310 --> 00:31:35.720 align:middle line:90% by the dopant concentration. 00:31:35.720 --> 00:31:37.820 align:middle line:84% The carriers come primarily from the dopant. 00:31:37.820 --> 00:31:43.610 align:middle line:84% So in this example, we calculate ni to be about 710 to the 18th. 00:31:43.610 --> 00:31:46.460 align:middle line:84% So if I have a dopant profile that's at 1 E18, 00:31:46.460 --> 00:31:48.710 align:middle line:90% well, that's much less than ni. 00:31:48.710 --> 00:31:51.690 align:middle line:84% Then what is n? n is equal to ni, right, 00:31:51.690 --> 00:31:54.780 align:middle line:84% the intrinsic carrier concentration. 00:31:54.780 --> 00:31:57.840 align:middle line:84% Because that's just due to the thermal activation 00:31:57.840 --> 00:31:59.440 align:middle line:90% of the breaking of bonds. 00:31:59.440 --> 00:32:03.480 align:middle line:84% So if you just substitute in here d0, which 00:32:03.480 --> 00:32:04.750 align:middle line:90% we got from the prior-- 00:32:04.750 --> 00:32:06.900 align:middle line:84% this is the numbers we get from the prior table 00:32:06.900 --> 00:32:09.450 align:middle line:90% on the prior slide, and d minus. 00:32:09.450 --> 00:32:11.910 align:middle line:84% And again, n over ni is 1, so this term 00:32:11.910 --> 00:32:13.500 align:middle line:90% is just being multiplied by 1. 00:32:13.500 --> 00:32:15.000 align:middle line:90% There's a 1 here in front of it. 00:32:15.000 --> 00:32:18.630 align:middle line:84% Add those up and you get a number that's about four-- 00:32:18.630 --> 00:32:23.110 align:middle line:84% 1.4 times 10 to the minus 15th centimeters squared per second. 00:32:23.110 --> 00:32:25.360 align:middle line:84% Again, we should be aware of the units. 00:32:25.360 --> 00:32:27.630 align:middle line:84% So you say, OK, when I use the table 00:32:27.630 --> 00:32:30.420 align:middle line:84% and I use the two term model, this is what I get. 00:32:30.420 --> 00:32:32.400 align:middle line:84% Now, a sanity check on that value 00:32:32.400 --> 00:32:36.990 align:middle line:84% might be when we were talking earlier in Chapter 7, 00:32:36.990 --> 00:32:38.850 align:middle line:84% we talked about intrinsic diffusion, 00:32:38.850 --> 00:32:40.560 align:middle line:84% and we didn't give a two term model. 00:32:40.560 --> 00:32:42.870 align:middle line:84% We just gave a constant diffusivity 00:32:42.870 --> 00:32:46.110 align:middle line:84% number, which was obtained by fitting a single activation 00:32:46.110 --> 00:32:48.310 align:middle line:90% energy to those expressions. 00:32:48.310 --> 00:32:50.700 align:middle line:84% So the question is, when I use this two term model, 00:32:50.700 --> 00:32:52.860 align:middle line:84% how does my number compare to the case 00:32:52.860 --> 00:32:58.080 align:middle line:84% when I use the single term model that's shown in Table 7-3? 00:32:58.080 --> 00:33:01.950 align:middle line:84% And in Table 7-3, you were given this number here, 00:33:01.950 --> 00:33:04.560 align:middle line:90% this exponential dependence. 00:33:04.560 --> 00:33:06.217 align:middle line:84% The activation energy, if we want 00:33:06.217 --> 00:33:08.550 align:middle line:84% to average out these activation energies, you were given 00:33:08.550 --> 00:33:11.850 align:middle line:84% was 3.99, and the pre-exponential was 9.17. 00:33:11.850 --> 00:33:13.830 align:middle line:84% When you calculate that out at 1,000, 00:33:13.830 --> 00:33:16.260 align:middle line:84% you get about 1.48 times 10 to the minus 15th, 00:33:16.260 --> 00:33:17.160 align:middle line:90% very, very close. 00:33:17.160 --> 00:33:19.590 align:middle line:84% So again, at a sanity check, this two term model 00:33:19.590 --> 00:33:22.680 align:middle line:84% is not screwing up your intrinsic diffusion 00:33:22.680 --> 00:33:23.280 align:middle line:90% coefficient. 00:33:23.280 --> 00:33:25.655 align:middle line:84% You're getting exactly what you would have gotten had you 00:33:25.655 --> 00:33:28.560 align:middle line:84% gone back to the simpler table and the simpler calculation 00:33:28.560 --> 00:33:30.030 align:middle line:90% in Table 7-3. 00:33:30.030 --> 00:33:33.240 align:middle line:84% But now if we do the next part, how about for the case where 00:33:33.240 --> 00:33:34.920 align:middle line:90% the profile is still 1 E20. 00:33:34.920 --> 00:33:38.750 align:middle line:90% 00:33:38.750 --> 00:33:40.700 align:middle line:84% Well, at 1 E20, what do you know? 00:33:40.700 --> 00:33:43.910 align:middle line:84% Well, you know n is much, much greater than ni. 00:33:43.910 --> 00:33:48.410 align:middle line:84% So n over ni is large, that means the concentration 00:33:48.410 --> 00:33:51.230 align:middle line:84% of single negatively charged point defect, 00:33:51.230 --> 00:33:53.460 align:middle line:90% cv minus goes way up. 00:33:53.460 --> 00:33:55.050 align:middle line:84% So the diffusivity should be enhanced. 00:33:55.050 --> 00:33:58.560 align:middle line:84% So now solved, right, at that same equation, 00:33:58.560 --> 00:33:59.690 align:middle line:90% here's the first term. 00:33:59.690 --> 00:34:01.910 align:middle line:84% The d zero term doesn't change, right, 00:34:01.910 --> 00:34:03.740 align:middle line:84% because it's not multiplied by anything. 00:34:03.740 --> 00:34:06.110 align:middle line:90% This is the d minus term. 00:34:06.110 --> 00:34:07.700 align:middle line:84% But what it's multiplied by here is 00:34:07.700 --> 00:34:11.090 align:middle line:84% n over ni, which is a big number, 10 to the 20 over 710 00:34:11.090 --> 00:34:11.600 align:middle line:90% to the 18th. 00:34:11.600 --> 00:34:15.050 align:middle line:84% That's more than an order of magnitude. 00:34:15.050 --> 00:34:17.159 align:middle line:90% That's two orders of magnitude. 00:34:17.159 --> 00:34:19.050 align:middle line:84% So you're really pumping up this term. 00:34:19.050 --> 00:34:20.090 align:middle line:90% So now what do you get? 00:34:20.090 --> 00:34:23.699 align:middle line:90% 1.6 times 10 to the minus 14th. 00:34:23.699 --> 00:34:27.570 align:middle line:84% So the highly doped layer has a 10-fold higher diffusivity 00:34:27.570 --> 00:34:30.239 align:middle line:84% diffusion coefficient in the extrinsic material. 00:34:30.239 --> 00:34:32.250 align:middle line:84% So it'll be diffusing with a diffusivity that's 00:34:32.250 --> 00:34:34.170 align:middle line:90% 10 times greater. 00:34:34.170 --> 00:34:38.280 align:middle line:84% And that's exactly how these calculations go. 00:34:38.280 --> 00:34:40.949 align:middle line:84% Now, if you have the concentration changing in space 00:34:40.949 --> 00:34:43.139 align:middle line:84% at every single point along the profile, 00:34:43.139 --> 00:34:45.989 align:middle line:84% the computer has to keep track of the diffusivity value 00:34:45.989 --> 00:34:48.480 align:middle line:84% at every single point and apply that correct diffusivity 00:34:48.480 --> 00:34:53.995 align:middle line:84% value when it's doing the calculation of a diffusion 00:34:53.995 --> 00:34:54.495 align:middle line:90% profile. 00:34:54.495 --> 00:34:57.650 align:middle line:90% 00:34:57.650 --> 00:34:58.150 align:middle line:90% OK. 00:34:58.150 --> 00:35:00.220 align:middle line:90% So let's go on to slide 12. 00:35:00.220 --> 00:35:02.140 align:middle line:84% So basically, the consequence of this 00:35:02.140 --> 00:35:04.690 align:middle line:84% from a practical point of view is that the profiles 00:35:04.690 --> 00:35:06.040 align:middle line:90% are very steep. 00:35:06.040 --> 00:35:10.060 align:middle line:84% They have-- they're flat topped and they fall off rapidly. 00:35:10.060 --> 00:35:11.590 align:middle line:90% And why is that? 00:35:11.590 --> 00:35:15.460 align:middle line:84% Well, if I'm working-- walking my way along this profile, 00:35:15.460 --> 00:35:17.590 align:middle line:84% first of all, I'm much, much greater than ni 00:35:17.590 --> 00:35:19.810 align:middle line:90% by over a factor of 10. 00:35:19.810 --> 00:35:21.950 align:middle line:84% And the diffusivity here is-- it's a large number. 00:35:21.950 --> 00:35:23.800 align:middle line:84% So that means it's going to-- when you're high, 00:35:23.800 --> 00:35:25.342 align:middle line:84% when you're above and over ni, you're 00:35:25.342 --> 00:35:26.710 align:middle line:90% going to be diffusing very fast. 00:35:26.710 --> 00:35:29.480 align:middle line:84% So these guys get over to the right very quickly. 00:35:29.480 --> 00:35:33.100 align:middle line:84% But as I start to go to-- my concentration starts 00:35:33.100 --> 00:35:35.200 align:middle line:84% to drop at the diffusion front, when 00:35:35.200 --> 00:35:37.720 align:middle line:84% I get close to n equals ni, the diffusivity 00:35:37.720 --> 00:35:39.340 align:middle line:84% is dropping like a rocket, right? 00:35:39.340 --> 00:35:42.520 align:middle line:84% Because n over ni is going down as I move down this profile. 00:35:42.520 --> 00:35:44.800 align:middle line:84% In fact, n over ni is only 1 here. 00:35:44.800 --> 00:35:48.280 align:middle line:84% So what's happening is the diffusivity 00:35:48.280 --> 00:35:50.560 align:middle line:84% is falling off very rapidly at the diffusion front. 00:35:50.560 --> 00:35:54.790 align:middle line:84% So it tends to make a very sharp, abrupt profile there, 00:35:54.790 --> 00:35:57.440 align:middle line:84% because as you walk down here, your diffusion coefficient is 00:35:57.440 --> 00:35:58.950 align:middle line:84% going way down, it's slowing down. 00:35:58.950 --> 00:36:01.970 align:middle line:84% So you get these box-like profiles. 00:36:01.970 --> 00:36:05.090 align:middle line:84% A box-like profile originating from-- 00:36:05.090 --> 00:36:07.940 align:middle line:84% and again, this assumption was that you have a constant source 00:36:07.940 --> 00:36:12.050 align:middle line:84% surface concentration of arsenic, or whatever so 00:36:12.050 --> 00:36:14.480 align:middle line:84% that the surface concentration is about 5, 10 to the 20. 00:36:14.480 --> 00:36:16.230 align:middle line:84% Instead of getting the complementary error 00:36:16.230 --> 00:36:20.720 align:middle line:84% function, which is here, you get a more box-like profile. 00:36:20.720 --> 00:36:24.350 align:middle line:84% And that's exactly because of the concentration dependence 00:36:24.350 --> 00:36:25.707 align:middle line:90% to the diffusivity. 00:36:25.707 --> 00:36:28.040 align:middle line:84% And people observe-- you observe these all the time when 00:36:28.040 --> 00:36:29.513 align:middle line:90% you do arsenic source strains. 00:36:29.513 --> 00:36:32.180 align:middle line:84% It's never-- it never looks like a complementary error function. 00:36:32.180 --> 00:36:34.295 align:middle line:84% It's always very box like, and that's why. 00:36:34.295 --> 00:36:38.530 align:middle line:90% 00:36:38.530 --> 00:36:41.260 align:middle line:84% Besides source drains, actually, you can see this-- 00:36:41.260 --> 00:36:42.790 align:middle line:90% we always talk about Moss beds. 00:36:42.790 --> 00:36:46.930 align:middle line:84% But if you're doing bipolar technology making NPN 00:36:46.930 --> 00:36:48.430 align:middle line:90% structures, you see the exact-- 00:36:48.430 --> 00:36:51.280 align:middle line:90% the same issues. 00:36:51.280 --> 00:36:53.470 align:middle line:90% And here's an example. 00:36:53.470 --> 00:36:56.440 align:middle line:84% On the left, I'm showing a starting structure 00:36:56.440 --> 00:36:58.760 align:middle line:90% for an NPN bipolar transistor. 00:36:58.760 --> 00:37:01.180 align:middle line:84% So this is a plot on a semi-log scale 00:37:01.180 --> 00:37:05.440 align:middle line:84% of concentration as a function of distance into the device. 00:37:05.440 --> 00:37:10.270 align:middle line:84% And what it is is that the collector down here is lightly 00:37:10.270 --> 00:37:12.820 align:middle line:84% doped around 10 of the 16th n-type with phosphorus. 00:37:12.820 --> 00:37:13.900 align:middle line:90% Here's the base. 00:37:13.900 --> 00:37:16.320 align:middle line:84% We're assuming the base was grown by epitaxial crystal 00:37:16.320 --> 00:37:16.820 align:middle line:90% growth. 00:37:16.820 --> 00:37:18.195 align:middle line:84% And we'll talk later in the class 00:37:18.195 --> 00:37:19.780 align:middle line:90% what that is in the course. 00:37:19.780 --> 00:37:23.290 align:middle line:84% But it has a very abrupt, constant doping profile. 00:37:23.290 --> 00:37:25.210 align:middle line:84% Like this is about 1,000 angstroms. 00:37:25.210 --> 00:37:29.350 align:middle line:84% And then you deposit a layer of n plus polysilicon on top, 00:37:29.350 --> 00:37:32.680 align:middle line:84% and that gives you this high arsenic concentration. 00:37:32.680 --> 00:37:35.950 align:middle line:84% Now, what you're interested in is annealing this 00:37:35.950 --> 00:37:39.380 align:middle line:84% to drive in the arsenic a little ways into the epi, 00:37:39.380 --> 00:37:45.050 align:middle line:84% into the base for 1,000 degrees for 30 minutes. 00:37:45.050 --> 00:37:47.240 align:middle line:84% And you want to see what the profile looks like. 00:37:47.240 --> 00:37:51.920 align:middle line:84% Well, on the right, I'm showing the SUPREM simulation that 00:37:51.920 --> 00:37:54.440 align:middle line:84% includes both the electric field effects that Maggie talked 00:37:54.440 --> 00:37:57.590 align:middle line:84% about last time, as well as this concentration, or Fermi level 00:37:57.590 --> 00:37:58.238 align:middle line:90% effect. 00:37:58.238 --> 00:37:59.780 align:middle line:84% And you can see what that looks like. 00:37:59.780 --> 00:38:02.180 align:middle line:84% Here is, again, concentration versus distance. 00:38:02.180 --> 00:38:04.490 align:middle line:90% And look at the arsenic profile. 00:38:04.490 --> 00:38:06.790 align:middle line:84% Here it is in the poly relatively constant. 00:38:06.790 --> 00:38:08.893 align:middle line:90% It is indeed very box like. 00:38:08.893 --> 00:38:11.060 align:middle line:84% It doesn't look like a complementary error function. 00:38:11.060 --> 00:38:12.950 align:middle line:84% It's almost constant, and then shuts off 00:38:12.950 --> 00:38:14.480 align:middle line:90% very quickly right here. 00:38:14.480 --> 00:38:18.800 align:middle line:84% That's due to the concentration dependent diffusivity. 00:38:18.800 --> 00:38:21.200 align:middle line:90% Here's the boron profile. 00:38:21.200 --> 00:38:24.110 align:middle line:84% It kind of Gaussian-ish, sort of, 00:38:24.110 --> 00:38:26.960 align:middle line:84% but it has this little dip in it right here, 00:38:26.960 --> 00:38:29.420 align:middle line:84% this divot, right where the arsenic-- 00:38:29.420 --> 00:38:33.330 align:middle line:84% right where the pn junction, the metallurgical junction happens. 00:38:33.330 --> 00:38:36.740 align:middle line:84% And in fact, this little divot is due to the electric field 00:38:36.740 --> 00:38:40.175 align:middle line:84% effect, which significantly impacts the profile, the boron 00:38:40.175 --> 00:38:41.300 align:middle line:90% profile, near the junction. 00:38:41.300 --> 00:38:43.850 align:middle line:84% Remember, the electric field here is being generated 00:38:43.850 --> 00:38:47.090 align:middle line:84% by-- mostly by this rapidly decreasing arsenic 00:38:47.090 --> 00:38:50.450 align:middle line:84% concentration, and the boron feels that and gets-- 00:38:50.450 --> 00:38:52.790 align:middle line:90% its diffusion gets modified. 00:38:52.790 --> 00:38:55.110 align:middle line:84% There's no way you would calculate this by hand, 00:38:55.110 --> 00:38:56.860 align:middle line:84% you would come up with anything like that. 00:38:56.860 --> 00:38:58.460 align:middle line:84% But this is exactly what is simulated 00:38:58.460 --> 00:38:59.585 align:middle line:90% in the numerical simulator. 00:38:59.585 --> 00:39:03.230 align:middle line:84% And this is what people measure by SIMS and things. 00:39:03.230 --> 00:39:05.210 align:middle line:84% So you really need numerical simulation 00:39:05.210 --> 00:39:08.570 align:middle line:84% to accurately model modern devices bipolar, be 00:39:08.570 --> 00:39:12.380 align:middle line:84% they MOSFETs or any other type of device 00:39:12.380 --> 00:39:14.465 align:middle line:84% because of these high concentration effects. 00:39:14.465 --> 00:39:17.790 align:middle line:90% 00:39:17.790 --> 00:39:18.290 align:middle line:90% OK. 00:39:18.290 --> 00:39:19.707 align:middle line:84% So let me go on to talk-- so we've 00:39:19.707 --> 00:39:21.510 align:middle line:84% talked about the electric field effect 00:39:21.510 --> 00:39:24.650 align:middle line:84% and the high concentration and what happens. 00:39:24.650 --> 00:39:26.780 align:middle line:84% There's another effect when we get to an interface 00:39:26.780 --> 00:39:30.110 align:middle line:84% that we're going to need to take into account 00:39:30.110 --> 00:39:33.030 align:middle line:84% that's going to be important in determining practical profiles. 00:39:33.030 --> 00:39:36.200 align:middle line:90% And this is called segregation. 00:39:36.200 --> 00:39:38.240 align:middle line:84% We know that dopants have different solubilities 00:39:38.240 --> 00:39:39.000 align:middle line:90% in different materials. 00:39:39.000 --> 00:39:40.958 align:middle line:84% So let's say I have a dopant and it's coming up 00:39:40.958 --> 00:39:43.210 align:middle line:90% against an oxide layer. 00:39:43.210 --> 00:39:47.360 align:middle line:84% It's in Silicon and there's an oxide layer right next door, 00:39:47.360 --> 00:39:47.992 align:middle line:90% next to it. 00:39:47.992 --> 00:39:49.700 align:middle line:84% It could have two layers right next door, 00:39:49.700 --> 00:39:53.870 align:middle line:84% could be silicon and nitride, or whatever-- silicon nitride. 00:39:53.870 --> 00:39:56.030 align:middle line:84% But they have different solubilities 00:39:56.030 --> 00:39:59.960 align:middle line:84% and so they tend to redistribute across an interface between two 00:39:59.960 --> 00:40:03.260 align:middle line:84% materials until something called the chemical potential is 00:40:03.260 --> 00:40:05.870 align:middle line:90% the same. 00:40:05.870 --> 00:40:09.590 align:middle line:84% And basically, the ratio of the equilibrium dopant 00:40:09.590 --> 00:40:11.660 align:middle line:84% concentration on each side of the interface 00:40:11.660 --> 00:40:13.530 align:middle line:90% is the segregation coefficient. 00:40:13.530 --> 00:40:16.040 align:middle line:84% So we've already seen segregation coefficients. 00:40:16.040 --> 00:40:19.680 align:middle line:84% We talked about it in Chapter 3 on crystal growth. 00:40:19.680 --> 00:40:23.370 align:middle line:84% We defined a segregation coefficient k0 with respect 00:40:23.370 --> 00:40:25.140 align:middle line:84% to crystal growth to be the concentration 00:40:25.140 --> 00:40:28.260 align:middle line:84% in the solid of the dopant, say, the boron or the arsenic, 00:40:28.260 --> 00:40:30.382 align:middle line:84% divided by the concentration in the liquid phase. 00:40:30.382 --> 00:40:31.590 align:middle line:90% So there we had an interface. 00:40:31.590 --> 00:40:34.027 align:middle line:84% One was the same material, it was silicon. 00:40:34.027 --> 00:40:35.610 align:middle line:84% It's just that one case it was liquid, 00:40:35.610 --> 00:40:37.440 align:middle line:90% the other case it wasn't solid. 00:40:37.440 --> 00:40:40.288 align:middle line:84% We can do this exact same thing, we 00:40:40.288 --> 00:40:42.330 align:middle line:84% can define a segregation coefficient, in general, 00:40:42.330 --> 00:40:45.870 align:middle line:84% between two materials, material A and material B 00:40:45.870 --> 00:40:50.730 align:middle line:84% in this segregation coefficient k and k0, or it may be-- 00:40:50.730 --> 00:40:53.130 align:middle line:84% the subscript may indicate which two materials may 00:40:53.130 --> 00:40:55.920 align:middle line:84% be from silicon to silicon dioxide, whatever, 00:40:55.920 --> 00:40:58.740 align:middle line:84% is in general-- is the concentration in material B 00:40:58.740 --> 00:41:02.980 align:middle line:84% divided by the concentration of material A, just the ratio. 00:41:02.980 --> 00:41:05.790 align:middle line:84% So given long enough time, if you 00:41:05.790 --> 00:41:07.590 align:middle line:84% were to put some dopant in this material 00:41:07.590 --> 00:41:09.900 align:middle line:84% and let it-- and heat it up and let it move around, 00:41:09.900 --> 00:41:13.062 align:middle line:84% it's going to arrange itself so that the ratio 00:41:13.062 --> 00:41:15.270 align:middle line:84% of the concentrations on either side of the interface 00:41:15.270 --> 00:41:18.180 align:middle line:90% is exactly k, k0. 00:41:18.180 --> 00:41:20.280 align:middle line:84% If k0 is 1, then it will arrange itself 00:41:20.280 --> 00:41:22.290 align:middle line:84% to have the same concentration at that interface 00:41:22.290 --> 00:41:23.230 align:middle line:90% on either side. 00:41:23.230 --> 00:41:24.960 align:middle line:84% If k0 is 10, it's going to want to have 00:41:24.960 --> 00:41:27.480 align:middle line:84% a factor of 10 difference in concentration going 00:41:27.480 --> 00:41:30.390 align:middle line:84% from material B up to material A. 00:41:30.390 --> 00:41:33.030 align:middle line:84% So there is-- this different solubility 00:41:33.030 --> 00:41:39.430 align:middle line:84% causes this equilibrium segregation. 00:41:39.430 --> 00:41:39.930 align:middle line:90% OK. 00:41:39.930 --> 00:41:41.760 align:middle line:84% When I'm calculating in SUPREM, when 00:41:41.760 --> 00:41:43.500 align:middle line:84% I'm calculating the interface flux, 00:41:43.500 --> 00:41:47.250 align:middle line:84% what's the boundary condition at the interface? 00:41:47.250 --> 00:41:50.130 align:middle line:84% Well, you saw-- if you're inside a given material, 00:41:50.130 --> 00:41:52.650 align:middle line:84% you saw last time in the finite difference case, 00:41:52.650 --> 00:41:54.810 align:middle line:84% if you're inside silicon one given material, 00:41:54.810 --> 00:41:56.790 align:middle line:84% you could use your neighboring concentrations 00:41:56.790 --> 00:41:59.410 align:middle line:84% to figure out a flux to either side. 00:41:59.410 --> 00:42:01.960 align:middle line:84% It's a little bit different if you have an interface flux. 00:42:01.960 --> 00:42:03.502 align:middle line:84% In fact, what you write is that right 00:42:03.502 --> 00:42:07.630 align:middle line:84% at this interface between A and B, the flux F to the right 00:42:07.630 --> 00:42:09.970 align:middle line:84% is equal to-- now, we don't use a diffusion coefficient, 00:42:09.970 --> 00:42:13.030 align:middle line:84% it's equal to a transport, interface transport coefficient 00:42:13.030 --> 00:42:18.250 align:middle line:84% h, which has units of length per unit time in, for example, 00:42:18.250 --> 00:42:20.290 align:middle line:90% centimeters per second. 00:42:20.290 --> 00:42:22.720 align:middle line:84% It gives you an idea how fast this thing 00:42:22.720 --> 00:42:25.570 align:middle line:84% is going to reach its equilibrium concentration. 00:42:25.570 --> 00:42:28.600 align:middle line:84% It's h times this ratio times this quantity, 00:42:28.600 --> 00:42:31.540 align:middle line:90% ca minus cb over k naught. 00:42:31.540 --> 00:42:35.060 align:middle line:84% Where again, k0 is defined according to this. 00:42:35.060 --> 00:42:37.880 align:middle line:84% So if you just look at this flux equation for a moment, 00:42:37.880 --> 00:42:38.990 align:middle line:90% what is it saying? 00:42:38.990 --> 00:42:43.540 align:middle line:84% Well, it says if, let's say, h is some reasonable value, if ca 00:42:43.540 --> 00:42:46.060 align:middle line:84% is much, much different, the concentration on this side, 00:42:46.060 --> 00:42:50.770 align:middle line:84% then cb over k0, this difference is going to be a large number. 00:42:50.770 --> 00:42:52.710 align:middle line:90% There's going to be a big flux. 00:42:52.710 --> 00:42:55.900 align:middle line:84% It's going to allow flux to flow, basically, 00:42:55.900 --> 00:42:59.080 align:middle line:90% until ca approaches cb over k0. 00:42:59.080 --> 00:43:01.510 align:middle line:90% And then the flux goes to zero. 00:43:01.510 --> 00:43:04.240 align:middle line:84% So it's going to force then the concentration 00:43:04.240 --> 00:43:06.670 align:middle line:84% profile across that interface to be pegged 00:43:06.670 --> 00:43:09.790 align:middle line:84% to have a difference, a ratio, that's equal to k0, 00:43:09.790 --> 00:43:12.487 align:middle line:84% and how rapidly it approaches that equilibrium. 00:43:12.487 --> 00:43:14.320 align:middle line:84% Well, that will be given to a certain extent 00:43:14.320 --> 00:43:16.630 align:middle line:84% by what h value you use for that. 00:43:16.630 --> 00:43:18.490 align:middle line:84% Because at each time step, you have 00:43:18.490 --> 00:43:22.690 align:middle line:84% a flux that's equal to ca minus cb over k0 times the h value. 00:43:22.690 --> 00:43:25.420 align:middle line:84% If h is very small, it'll take a long time 00:43:25.420 --> 00:43:28.780 align:middle line:84% to reach the equilibrium segregated condition. 00:43:28.780 --> 00:43:30.250 align:middle line:90% If h is very rapid-- 00:43:30.250 --> 00:43:33.830 align:middle line:84% high, then it reaches that very quickly. 00:43:33.830 --> 00:43:37.960 align:middle line:84% So h is a measure of how easily the species is transported 00:43:37.960 --> 00:43:41.670 align:middle line:90% across the interface. 00:43:41.670 --> 00:43:43.620 align:middle line:84% So a very common thing that you will 00:43:43.620 --> 00:43:47.580 align:middle line:84% see happening when you anneal wafers, 00:43:47.580 --> 00:43:49.890 align:middle line:84% or when you grow oxides, is segregation 00:43:49.890 --> 00:43:52.260 align:middle line:84% at an interface between silicon and silicon dioxide. 00:43:52.260 --> 00:43:55.340 align:middle line:90% 00:43:55.340 --> 00:43:57.370 align:middle line:90% And so let's look at that case. 00:43:57.370 --> 00:43:59.860 align:middle line:84% By the way, I should say that segregation coefficients, 00:43:59.860 --> 00:44:04.240 align:middle line:84% the ratio of some species on one side of an interface to another 00:44:04.240 --> 00:44:06.980 align:middle line:84% material, the other side of the interface, it sounds trivial, 00:44:06.980 --> 00:44:09.100 align:middle line:84% but it's-- actually, in practice, it's very hard. 00:44:09.100 --> 00:44:11.440 align:middle line:84% You might say, well, why don't you just use SIMS? 00:44:11.440 --> 00:44:14.840 align:middle line:84% Just profile through the sample and measure the concentration 00:44:14.840 --> 00:44:16.840 align:middle line:84% on this side and the concentration on that side. 00:44:16.840 --> 00:44:20.200 align:middle line:84% Well, a lot of techniques like SIMS near an interface, 00:44:20.200 --> 00:44:21.640 align:middle line:90% they don't perform very well. 00:44:21.640 --> 00:44:23.470 align:middle line:84% Because as you get close to the interface, 00:44:23.470 --> 00:44:25.630 align:middle line:84% you're changing the material and the sputter rate 00:44:25.630 --> 00:44:26.680 align:middle line:90% starts to change. 00:44:26.680 --> 00:44:29.260 align:middle line:84% All the assumptions that you need to make in SIMS 00:44:29.260 --> 00:44:32.060 align:middle line:84% tend to be degraded, to a certain extent. 00:44:32.060 --> 00:44:34.920 align:middle line:84% So you would think we have perfect numbers for this but we 00:44:34.920 --> 00:44:35.420 align:middle line:90% don't. 00:44:35.420 --> 00:44:36.790 align:middle line:90% We have rough numbers. 00:44:36.790 --> 00:44:39.910 align:middle line:84% And these are the rough numbers that are in SUPREM. 00:44:39.910 --> 00:44:43.000 align:middle line:84% They are-- the k values, of course, are adjustable, 00:44:43.000 --> 00:44:45.340 align:middle line:84% and you can adjust them to fit your-- 00:44:45.340 --> 00:44:46.780 align:middle line:90% whatever your experiments show. 00:44:46.780 --> 00:44:49.690 align:middle line:84% But k0, so the ratio of the concentration 00:44:49.690 --> 00:44:53.680 align:middle line:84% of the dopant in silicon-- to that in silicon dioxide. 00:44:53.680 --> 00:44:57.050 align:middle line:84% For boron, it's less than 1, it's 0.3. 00:44:57.050 --> 00:44:58.900 align:middle line:84% So what that means is that boron wants 00:44:58.900 --> 00:45:02.080 align:middle line:84% to go into the oxide layers, concentration in the silicon 00:45:02.080 --> 00:45:04.040 align:middle line:90% will be lower. 00:45:04.040 --> 00:45:08.770 align:middle line:84% So boron tends to be depleted in the silicon and higher 00:45:08.770 --> 00:45:10.090 align:middle line:90% in the oxide. 00:45:10.090 --> 00:45:12.800 align:middle line:84% Arsenic, the n-type dopants are just the opposite. 00:45:12.800 --> 00:45:14.720 align:middle line:84% They tend to segregate into the silicon. 00:45:14.720 --> 00:45:17.980 align:middle line:84% So they pile up at the silicon, and they're lower in the oxide. 00:45:17.980 --> 00:45:19.870 align:middle line:84% And this is giving them a ratio of 10. 00:45:19.870 --> 00:45:21.860 align:middle line:84% Of course, it also depends on temperature. 00:45:21.860 --> 00:45:25.890 align:middle line:84% So you need to take that into account. 00:45:25.890 --> 00:45:29.240 align:middle line:84% So in fact, if we go to slide 16, 00:45:29.240 --> 00:45:30.860 align:middle line:90% we can see an example of this. 00:45:30.860 --> 00:45:33.050 align:middle line:84% These are some SUPREM simulations. 00:45:33.050 --> 00:45:35.940 align:middle line:84% And what's been done here in the upper left, 00:45:35.940 --> 00:45:38.840 align:middle line:84% you're seeing the simulation of the oxidation 00:45:38.840 --> 00:45:40.580 align:middle line:90% of a uniformly doped substrate. 00:45:40.580 --> 00:45:43.220 align:middle line:84% So initially, it was uniformly doped. 00:45:43.220 --> 00:45:45.920 align:middle line:84% So the concentration of boron at the surface 00:45:45.920 --> 00:45:50.180 align:middle line:84% was all the same, say, 10 to the 18th, 00:45:50.180 --> 00:45:52.130 align:middle line:84% throughout the entire wafer from the surface 00:45:52.130 --> 00:45:53.000 align:middle line:90% all the way through. 00:45:53.000 --> 00:45:54.875 align:middle line:84% And you take that and you put it in a furnace 00:45:54.875 --> 00:45:56.380 align:middle line:90% and you oxidize it. 00:45:56.380 --> 00:45:59.180 align:middle line:84% Well, you can see where the oxide was grown. 00:45:59.180 --> 00:46:02.060 align:middle line:84% All of a sudden, the boron has a profile to it. 00:46:02.060 --> 00:46:03.830 align:middle line:84% And it's a little hard to tell the profile 00:46:03.830 --> 00:46:04.940 align:middle line:90% from these contours. 00:46:04.940 --> 00:46:07.970 align:middle line:84% Each contour, each color represents a different boron 00:46:07.970 --> 00:46:08.940 align:middle line:90% concentration. 00:46:08.940 --> 00:46:12.680 align:middle line:84% But if you want to take a cut right through the center, 00:46:12.680 --> 00:46:14.810 align:middle line:84% this is what the boron looks like. 00:46:14.810 --> 00:46:18.200 align:middle line:84% There's a certain concentration of boron in the oxide 00:46:18.200 --> 00:46:21.620 align:middle line:84% here, around 6 or 7 times 10 to the 17th. 00:46:21.620 --> 00:46:23.630 align:middle line:90% Then there's a drop. 00:46:23.630 --> 00:46:25.970 align:middle line:84% There's a factor of 3 drop because, again, we 00:46:25.970 --> 00:46:28.940 align:middle line:84% said the segregation coefficient was about 0.3. 00:46:28.940 --> 00:46:31.290 align:middle line:84% And it's depleted somewhat in the silicon. 00:46:31.290 --> 00:46:33.563 align:middle line:84% So it's come down in the silicon. 00:46:33.563 --> 00:46:35.480 align:middle line:84% That's because there's been an interface flux. 00:46:35.480 --> 00:46:38.510 align:middle line:84% The concentration originally in the oxide was 0. 00:46:38.510 --> 00:46:40.610 align:middle line:84% The concentration in here was about 10 00:46:40.610 --> 00:46:42.020 align:middle line:90% to the 18th in the bulk. 00:46:42.020 --> 00:46:46.130 align:middle line:84% And there's been a flux from the silicon to the oxide 00:46:46.130 --> 00:46:48.950 align:middle line:84% because in equilibrium, it wants to set this ratio 00:46:48.950 --> 00:46:50.720 align:middle line:90% to be equal to 3. 00:46:50.720 --> 00:46:54.660 align:middle line:84% And so you see this depleted boron concentration 00:46:54.660 --> 00:46:56.030 align:middle line:90% in the silicon. 00:46:56.030 --> 00:46:57.530 align:middle line:90% How about n-type dopants? 00:46:57.530 --> 00:47:00.140 align:middle line:84% Well, arsenic and phosphorus we said their segregation 00:47:00.140 --> 00:47:01.350 align:middle line:90% coefficient is 10. 00:47:01.350 --> 00:47:03.030 align:middle line:90% So it's a positive number. 00:47:03.030 --> 00:47:05.990 align:middle line:84% So they actually tend to pile up at the interface, 00:47:05.990 --> 00:47:09.480 align:middle line:84% and they're low concentration in the oxide. 00:47:09.480 --> 00:47:12.650 align:middle line:84% So they tend to segregate into the silicon. 00:47:12.650 --> 00:47:17.210 align:middle line:84% This arsenic profile looks a little steeper than phosphorus 00:47:17.210 --> 00:47:18.960 align:middle line:84% because it has a slower diffusivity. 00:47:18.960 --> 00:47:22.070 align:middle line:84% So it's not getting to the interface. 00:47:22.070 --> 00:47:24.810 align:middle line:84% It's not able to diffuse across the interface, 00:47:24.810 --> 00:47:27.540 align:middle line:84% or transport across it as rapidly 00:47:27.540 --> 00:47:29.670 align:middle line:84% because the delivery of the arsenic from the bulk 00:47:29.670 --> 00:47:32.590 align:middle line:90% is a little bit slower. 00:47:32.590 --> 00:47:35.280 align:middle line:84% So there's an example of segregation 00:47:35.280 --> 00:47:38.730 align:middle line:84% that happens during an oxidation process. 00:47:38.730 --> 00:47:41.758 align:middle line:90% 00:47:41.758 --> 00:47:43.800 align:middle line:84% So that's something that's incorporated in SUPREM 00:47:43.800 --> 00:47:47.220 align:middle line:90% and is very commonly observed. 00:47:47.220 --> 00:47:51.630 align:middle line:84% The next interfacial effect that I'll talk about, 00:47:51.630 --> 00:47:53.910 align:middle line:90% interfacial dopant pileup. 00:47:53.910 --> 00:47:56.310 align:middle line:84% Be careful, this is not equilibrium segregation. 00:47:56.310 --> 00:47:59.160 align:middle line:84% This gets a little tricky because we just talked about 00:47:59.160 --> 00:48:01.930 align:middle line:84% cross an interface, the doping concentration being different. 00:48:01.930 --> 00:48:04.598 align:middle line:84% This is actually piling up right at the interface. 00:48:04.598 --> 00:48:05.890 align:middle line:90% So it's a little bit different. 00:48:05.890 --> 00:48:09.480 align:middle line:90% It's a monolayer type effect. 00:48:09.480 --> 00:48:11.497 align:middle line:84% And particularly as junctions become shallow, 00:48:11.497 --> 00:48:13.080 align:middle line:84% it's observed that some of the dopants 00:48:13.080 --> 00:48:16.620 align:middle line:84% pile up in this very narrow interfacial layer 00:48:16.620 --> 00:48:19.690 align:middle line:84% at the interface between oxide and silicon. 00:48:19.690 --> 00:48:22.020 align:middle line:84% This pileup is separate from and is 00:48:22.020 --> 00:48:25.050 align:middle line:84% larger than the normal equilibrium segregation that 00:48:25.050 --> 00:48:26.620 align:middle line:90% might occur during annealing. 00:48:26.620 --> 00:48:29.850 align:middle line:84% So just as an example, if this is the oxide to the left, 00:48:29.850 --> 00:48:32.910 align:middle line:84% the silicon to the right, normal segregation 00:48:32.910 --> 00:48:35.480 align:middle line:84% would give you a little height difference like this. 00:48:35.480 --> 00:48:38.100 align:middle line:84% This interfacial pileup gives you 00:48:38.100 --> 00:48:39.970 align:middle line:84% a height difference that's even larger, 00:48:39.970 --> 00:48:45.590 align:middle line:84% and it's just at the interface, and it integrates to some dose. 00:48:45.590 --> 00:48:47.860 align:middle line:84% It may be as thin as a monolayer, 00:48:47.860 --> 00:48:49.480 align:middle line:90% but it can actually-- 00:48:49.480 --> 00:48:52.900 align:middle line:84% the interface essentially acting as a sink for the dopants. 00:48:52.900 --> 00:48:57.140 align:middle line:84% And it can trap up to about 10 to the 15th per square at that 00:48:57.140 --> 00:48:57.640 align:middle line:90% interface. 00:48:57.640 --> 00:48:58.765 align:middle line:90% That's a pretty big number. 00:48:58.765 --> 00:49:01.300 align:middle line:84% If I consider that I might only be implanting 00:49:01.300 --> 00:49:03.770 align:middle line:84% 2 times 10 to the 15th arsenic, and I implant 00:49:03.770 --> 00:49:05.770 align:middle line:84% it right near the surface, a lot of that arsenic 00:49:05.770 --> 00:49:07.750 align:middle line:84% gets sucked up right into that interface 00:49:07.750 --> 00:49:10.160 align:middle line:84% and never become electrically active. 00:49:10.160 --> 00:49:12.718 align:middle line:84% So when people were implanting the arsenic really deep 00:49:12.718 --> 00:49:14.260 align:middle line:84% into the substrate, nobody noticed it 00:49:14.260 --> 00:49:16.760 align:middle line:84% because the concentration near the surface was low. 00:49:16.760 --> 00:49:19.147 align:middle line:84% But now that we have shallow implants, a lot of people 00:49:19.147 --> 00:49:20.980 align:middle line:84% are finding that their dopants are sucked up 00:49:20.980 --> 00:49:25.690 align:middle line:84% into that interface and so they lose a lot of effective dose 00:49:25.690 --> 00:49:27.320 align:middle line:84% that they would have expected to have. 00:49:27.320 --> 00:49:29.770 align:middle line:84% So people-- SUPREM IV now includes 00:49:29.770 --> 00:49:35.230 align:middle line:84% an interfacial segregation module to model this effect. 00:49:35.230 --> 00:49:40.960 align:middle line:84% In fact, on slide 18 shows some data of what happens. 00:49:40.960 --> 00:49:46.300 align:middle line:84% Here on the left, this is from Cazenave, IEDM, 1998. 00:49:46.300 --> 00:49:49.900 align:middle line:84% This is concentration of arsenic as a function of depth, 00:49:49.900 --> 00:49:53.740 align:middle line:84% and the "as implanted" is shown here in blue with the boxes. 00:49:53.740 --> 00:49:57.580 align:middle line:84% And this is annealed, the rapid thermal anneal, 00:49:57.580 --> 00:49:58.960 align:middle line:90% 1,050 for 30 seconds. 00:49:58.960 --> 00:50:01.930 align:middle line:84% There's a thin oxide layer, by the way, 00:50:01.930 --> 00:50:03.040 align:middle line:90% that was on the surface. 00:50:03.040 --> 00:50:05.440 align:middle line:84% He was using as a cap when he did this anneal. 00:50:05.440 --> 00:50:08.170 align:middle line:84% And then he stripped the oxide before doing the SIMS. 00:50:08.170 --> 00:50:10.690 align:middle line:84% And this is the red layer, the red profile. 00:50:10.690 --> 00:50:13.150 align:middle line:84% If you integrate it, its area, or its dose, 00:50:13.150 --> 00:50:15.260 align:middle line:84% is only-- is 6 times 10 to the 14th. 00:50:15.260 --> 00:50:17.185 align:middle line:90% So 30% of the arsenic was lost. 00:50:17.185 --> 00:50:20.220 align:middle line:90% 00:50:20.220 --> 00:50:21.810 align:middle line:84% Now, he had an oxide cap there, hoping 00:50:21.810 --> 00:50:23.227 align:middle line:84% to keep the arsenic in the sample, 00:50:23.227 --> 00:50:25.080 align:middle line:84% hoping to keep the arsenic from evaporating. 00:50:25.080 --> 00:50:27.667 align:middle line:84% It didn't evaporate, it got stuck at the interface 00:50:27.667 --> 00:50:29.250 align:middle line:84% and then was stripped off with the HF. 00:50:29.250 --> 00:50:33.820 align:middle line:84% So he prevented evaporation, but it is a problem. 00:50:33.820 --> 00:50:37.320 align:middle line:84% And in fact, if you go on the SIMS [INAUDIBLE] at the right, 00:50:37.320 --> 00:50:39.630 align:middle line:84% he tried to understand what was going on. 00:50:39.630 --> 00:50:42.480 align:middle line:84% What he saw, if he did not do the HF dip 00:50:42.480 --> 00:50:45.870 align:middle line:84% after the rapid thermal anneal, he saw this red profile. 00:50:45.870 --> 00:50:49.133 align:middle line:84% So indeed, in the oxide right near the interface-- and again, 00:50:49.133 --> 00:50:50.550 align:middle line:84% SIMS doesn't resolve the interface 00:50:50.550 --> 00:50:55.240 align:middle line:84% very well-- he saw a pileup in that interfacial region. 00:50:55.240 --> 00:50:57.120 align:middle line:84% And that's where all that extra dose went. 00:50:57.120 --> 00:51:00.840 align:middle line:84% Then when you strip the oxide, of course, everything piled up, 00:51:00.840 --> 00:51:02.610 align:middle line:84% but the interface is gone and you end up 00:51:02.610 --> 00:51:05.380 align:middle line:90% with a much lower dose. 00:51:05.380 --> 00:51:06.960 align:middle line:84% So it's not very well understood. 00:51:06.960 --> 00:51:08.760 align:middle line:84% But you can imagine, this is very important 00:51:08.760 --> 00:51:10.590 align:middle line:90% for source drains. 00:51:10.590 --> 00:51:14.500 align:middle line:84% If we go on to slide number 19, in fact, 00:51:14.500 --> 00:51:16.440 align:middle line:84% if we look at right near the channel, 00:51:16.440 --> 00:51:19.020 align:middle line:84% we implant these very shallow junctions 00:51:19.020 --> 00:51:23.507 align:middle line:84% called the source and drain extensions, 00:51:23.507 --> 00:51:24.590 align:middle line:90% sometimes called the tips. 00:51:24.590 --> 00:51:28.220 align:middle line:84% These junctions today are on the order of 500 angstroms or less, 00:51:28.220 --> 00:51:29.070 align:middle line:90% maybe 300 angstroms. 00:51:29.070 --> 00:51:30.320 align:middle line:90% So they're very, very shallow. 00:51:30.320 --> 00:51:32.900 align:middle line:84% So you need a very shallow implant right near the surface. 00:51:32.900 --> 00:51:35.730 align:middle line:84% And lo and behold, what's above that implant, of course, 00:51:35.730 --> 00:51:36.800 align:middle line:90% is an oxide. 00:51:36.800 --> 00:51:39.200 align:middle line:84% So you have an interface between silicon and oxide, 00:51:39.200 --> 00:51:41.570 align:middle line:84% and a lot of dopant right underneath there. 00:51:41.570 --> 00:51:45.350 align:middle line:84% So you will go ahead, you will design your source train 00:51:45.350 --> 00:51:48.290 align:middle line:84% extension to give you the right sheet resistance, 00:51:48.290 --> 00:51:50.720 align:middle line:84% and then you go measure it and you find out it's half-- 00:51:50.720 --> 00:51:53.240 align:middle line:84% or it's twice what the sheet resistance you would design. 00:51:53.240 --> 00:51:55.190 align:middle line:84% Because half your dose has segregated 00:51:55.190 --> 00:51:58.070 align:middle line:84% to that interface where it's not electrically active. 00:51:58.070 --> 00:52:00.000 align:middle line:84% So it sounds like a subtle effect, 00:52:00.000 --> 00:52:02.960 align:middle line:84% but it's actually extremely important, particularly, 00:52:02.960 --> 00:52:05.180 align:middle line:84% when you need low sheet resistance 00:52:05.180 --> 00:52:07.618 align:middle line:90% contacts to the channel. 00:52:07.618 --> 00:52:09.410 align:middle line:84% It's very annoying that half of your dopant 00:52:09.410 --> 00:52:10.950 align:middle line:90% gets sucked up by that oxide. 00:52:10.950 --> 00:52:12.740 align:middle line:84% So it's just a fact of life and something 00:52:12.740 --> 00:52:14.840 align:middle line:90% you need to take into account. 00:52:14.840 --> 00:52:16.550 align:middle line:84% SUPREM does have empirical models. 00:52:16.550 --> 00:52:19.160 align:middle line:84% Of course, you have to adjust all the coefficients 00:52:19.160 --> 00:52:20.795 align:middle line:90% to fit your particular data. 00:52:20.795 --> 00:52:23.530 align:middle line:90% 00:52:23.530 --> 00:52:24.030 align:middle line:90% OK. 00:52:24.030 --> 00:52:25.500 align:middle line:84% So those are some special effects 00:52:25.500 --> 00:52:26.875 align:middle line:84% that I wanted to include and when 00:52:26.875 --> 00:52:29.370 align:middle line:90% we talk about dopant diffusion. 00:52:29.370 --> 00:52:32.010 align:middle line:84% And now, I want to go on and talk about atomic scale model. 00:52:32.010 --> 00:52:33.510 align:middle line:84% Everything we've talked about so far 00:52:33.510 --> 00:52:36.330 align:middle line:84% has been-- except for several lectures ago-- 00:52:36.330 --> 00:52:39.840 align:middle line:84% it's been, really, at the macroscopic diffusion. 00:52:39.840 --> 00:52:43.290 align:middle line:84% We defined Fick's law, you added electric field effects 00:52:43.290 --> 00:52:45.720 align:middle line:90% and Fermi level effects. 00:52:45.720 --> 00:52:47.490 align:middle line:84% But a lot of effects, especially, 00:52:47.490 --> 00:52:50.130 align:middle line:84% those like OED, oxidation enhanced diffusion 00:52:50.130 --> 00:52:53.070 align:middle line:84% and TED, transient enhanced diffusion, 00:52:53.070 --> 00:52:55.350 align:middle line:90% they're action at a distance. 00:52:55.350 --> 00:52:56.970 align:middle line:84% They're very important experimentally, 00:52:56.970 --> 00:52:59.520 align:middle line:84% but they cannot be explained by these simple macroscopic 00:52:59.520 --> 00:53:00.750 align:middle line:90% models. 00:53:00.750 --> 00:53:02.940 align:middle line:84% So we really want to look at dopant diffusion 00:53:02.940 --> 00:53:08.400 align:middle line:84% as best as we can at the level of the atomic scale. 00:53:08.400 --> 00:53:13.380 align:middle line:84% And here is a way of doing that, showing you on slide 21. 00:53:13.380 --> 00:53:14.850 align:middle line:84% There are two different mechanisms 00:53:14.850 --> 00:53:16.530 align:middle line:90% pictured on this slide. 00:53:16.530 --> 00:53:18.575 align:middle line:84% One that we've talked about, we haven't really 00:53:18.575 --> 00:53:19.950 align:middle line:84% gone through the specifics of it. 00:53:19.950 --> 00:53:24.040 align:middle line:84% But we've hinted at this vacancy assisted mechanism. 00:53:24.040 --> 00:53:25.680 align:middle line:84% So imagine this chemical equation. 00:53:25.680 --> 00:53:29.520 align:middle line:84% I have a dopant A, it could be arsenic, boron, whatever. 00:53:29.520 --> 00:53:33.420 align:middle line:84% And it gets-- it's paired with a vacancy right nearby. 00:53:33.420 --> 00:53:36.770 align:middle line:84% And it goes into a pair, an av pair. 00:53:36.770 --> 00:53:38.790 align:middle line:84% av pair, what I mean by pair, well, they 00:53:38.790 --> 00:53:42.280 align:middle line:84% maintain within a lattice constant are two of each other. 00:53:42.280 --> 00:53:44.700 align:middle line:84% There's this pair and they move as a pair 00:53:44.700 --> 00:53:46.170 align:middle line:90% throughout the lattice. 00:53:46.170 --> 00:53:47.560 align:middle line:90% So how can that happen? 00:53:47.560 --> 00:53:50.608 align:middle line:84% Well, you can imagine this-- let's 00:53:50.608 --> 00:53:51.900 align:middle line:90% say this is my first time step. 00:53:51.900 --> 00:53:53.280 align:middle line:90% Here's a vacancy right here. 00:53:53.280 --> 00:53:55.680 align:middle line:84% Here's a dopant atom pictured in the light. 00:53:55.680 --> 00:53:58.500 align:middle line:84% And the silicon atoms are all the dark black. 00:53:58.500 --> 00:54:02.400 align:middle line:84% So this dopant here may exchange site with the vacancy, OK? 00:54:02.400 --> 00:54:04.050 align:middle line:84% The dopant moves to this vacancy site, 00:54:04.050 --> 00:54:05.280 align:middle line:90% the vacancy moves up there. 00:54:05.280 --> 00:54:06.030 align:middle line:90% All right, fine. 00:54:06.030 --> 00:54:08.530 align:middle line:84% Well, if they just switch back, they haven't moved anywhere. 00:54:08.530 --> 00:54:09.450 align:middle line:84% They're just switching back and forth. 00:54:09.450 --> 00:54:10.658 align:middle line:90% That doesn't do you any good. 00:54:10.658 --> 00:54:13.800 align:middle line:84% But imagine I move the dopant to this point and the vacancy 00:54:13.800 --> 00:54:14.820 align:middle line:90% moves up there. 00:54:14.820 --> 00:54:17.490 align:middle line:84% Now, all of a sudden, this vacancy is sitting here. 00:54:17.490 --> 00:54:19.770 align:middle line:84% It can move independently for a second 00:54:19.770 --> 00:54:22.600 align:middle line:84% of the dopant and exchange with its neighboring silicon atom. 00:54:22.600 --> 00:54:25.470 align:middle line:84% So now, I have a vacancy here and a silicon atom here. 00:54:25.470 --> 00:54:27.400 align:middle line:84% And the vacancy can move over here. 00:54:27.400 --> 00:54:29.070 align:middle line:84% So all of a sudden now, the vacancy 00:54:29.070 --> 00:54:31.890 align:middle line:84% is sitting over here now, again, next to that dopant atom. 00:54:31.890 --> 00:54:33.520 align:middle line:90% Now, it can exchange with it. 00:54:33.520 --> 00:54:37.980 align:middle line:84% So just by moving the vacancy around in a circular motion 00:54:37.980 --> 00:54:42.180 align:middle line:84% around just open atom and exchanging sites with it, 00:54:42.180 --> 00:54:45.030 align:middle line:84% the vacancy and the silicon can essentially move as a pair 00:54:45.030 --> 00:54:46.470 align:middle line:90% gradually through the lattice. 00:54:46.470 --> 00:54:48.940 align:middle line:84% And it's much easier to do that than if you imagine 00:54:48.940 --> 00:54:51.970 align:middle line:84% there was no vacancy there, and every time the dopant had 00:54:51.970 --> 00:54:54.230 align:middle line:84% to move, it had to break the bonds. 00:54:54.230 --> 00:54:56.810 align:middle line:84% You don't need to do that when you have a vacancy. 00:54:56.810 --> 00:55:00.412 align:middle line:84% So this pairing of the vacancies is 00:55:00.412 --> 00:55:02.620 align:middle line:84% believed to be a very efficient mechanism for dopants 00:55:02.620 --> 00:55:04.900 align:middle line:90% to move in silicon. 00:55:04.900 --> 00:55:07.240 align:middle line:84% You can do something similar with interstitials, 00:55:07.240 --> 00:55:09.920 align:middle line:84% or interstitial c assisted mechanisms. 00:55:09.920 --> 00:55:12.700 align:middle line:84% So here I have a chemical equation, dopant a, 00:55:12.700 --> 00:55:16.555 align:middle line:84% plus an interstitial forms, an ai pair. 00:55:16.555 --> 00:55:18.590 align:middle line:84% And it will help in this assist diffusion. 00:55:18.590 --> 00:55:20.860 align:middle line:84% Well, here again, my pink atom here is the dopant. 00:55:20.860 --> 00:55:22.690 align:middle line:90% Here is an interstitial. 00:55:22.690 --> 00:55:25.960 align:middle line:84% It can come along and kick out the dopant off a lattice site, 00:55:25.960 --> 00:55:30.220 align:middle line:84% make it interstitial, and help it get moving that way. 00:55:30.220 --> 00:55:32.830 align:middle line:84% Or here's an interstitial c, what is that? 00:55:32.830 --> 00:55:36.040 align:middle line:84% Here is a silicon atom sharing a lattice site 00:55:36.040 --> 00:55:37.750 align:middle line:90% with another silicon atom. 00:55:37.750 --> 00:55:40.000 align:middle line:84% It can then start to share with the dopant atom. 00:55:40.000 --> 00:55:41.950 align:middle line:84% And then the dopant atom can move along 00:55:41.950 --> 00:55:44.810 align:middle line:84% and share with the next bonded silicon atom. 00:55:44.810 --> 00:55:46.660 align:middle line:84% And so it can move along, perhaps, 00:55:46.660 --> 00:55:49.150 align:middle line:84% the bond direction as an interstitial c 00:55:49.150 --> 00:55:53.230 align:middle line:84% as two objects sort of sharing the same lattice site. 00:55:53.230 --> 00:55:56.800 align:middle line:84% So either interstitials excess hanging around, 00:55:56.800 --> 00:56:00.520 align:middle line:84% or vacancies hanging around, either one, these point defects 00:56:00.520 --> 00:56:03.040 align:middle line:84% can assist with the motion of the dopant in the lattice. 00:56:03.040 --> 00:56:06.630 align:middle line:90% 00:56:06.630 --> 00:56:10.140 align:middle line:84% So we're going to make some inferences about mechanisms. 00:56:10.140 --> 00:56:12.690 align:middle line:84% I think we talked about this a little bit last time, 00:56:12.690 --> 00:56:15.180 align:middle line:84% or a couple of lectures ago when we talked about stacking 00:56:15.180 --> 00:56:16.980 align:middle line:90% faults and oxidation. 00:56:16.980 --> 00:56:19.020 align:middle line:84% This is a picture of local oxidation. 00:56:19.020 --> 00:56:21.360 align:middle line:84% So over on the right, I'm having oxidation take place. 00:56:21.360 --> 00:56:23.530 align:middle line:84% On the left, I'm underneath a nitride 00:56:23.530 --> 00:56:25.740 align:middle line:90% so there's no oxidation. 00:56:25.740 --> 00:56:29.220 align:middle line:84% And what people see is that deep in the substrate 00:56:29.220 --> 00:56:31.140 align:middle line:84% underneath where you're oxidizing, 00:56:31.140 --> 00:56:33.420 align:middle line:84% you see that oxidation induced stacking faults. 00:56:33.420 --> 00:56:35.940 align:middle line:84% Remember, we said they grow underneath the region 00:56:35.940 --> 00:56:37.410 align:middle line:84% where there's oxidizing underneath. 00:56:37.410 --> 00:56:38.993 align:middle line:84% The region where there's no oxidation, 00:56:38.993 --> 00:56:41.740 align:middle line:90% they don't, they stay constant. 00:56:41.740 --> 00:56:45.150 align:middle line:84% So it's believed that oxidation injects from the surface, 00:56:45.150 --> 00:56:48.270 align:middle line:84% injects interstitials into the bulk, which 00:56:48.270 --> 00:56:50.130 align:middle line:84% aids the growth of stacking faults, 00:56:50.130 --> 00:56:53.310 align:middle line:84% and also can enhance the diffusion of dopants 00:56:53.310 --> 00:56:55.740 align:middle line:90% like boron. 00:56:55.740 --> 00:56:58.170 align:middle line:84% Now similarly, it's also been found, 00:56:58.170 --> 00:57:00.270 align:middle line:84% if I took this starting wafer and instead 00:57:00.270 --> 00:57:02.730 align:middle line:84% of putting in an oxidizing furnace and growing an oxide, 00:57:02.730 --> 00:57:05.470 align:middle line:84% I put it in a furnace with ammonia. 00:57:05.470 --> 00:57:08.170 align:middle line:84% And I grow over here in this region on the right, 00:57:08.170 --> 00:57:10.430 align:middle line:90% I would grow silicon nitride. 00:57:10.430 --> 00:57:12.310 align:middle line:90% So I'm nitriding. 00:57:12.310 --> 00:57:13.540 align:middle line:90% I'm thermally nitriding. 00:57:13.540 --> 00:57:16.030 align:middle line:84% I'm reacting silicon with ammonia. 00:57:16.030 --> 00:57:17.530 align:middle line:84% People have found that this actually 00:57:17.530 --> 00:57:20.920 align:middle line:84% has the opposite effect, that boron diffusion is retarded, 00:57:20.920 --> 00:57:23.260 align:middle line:84% and stacking faults actually shrink. 00:57:23.260 --> 00:57:28.700 align:middle line:84% So people believe that thermal nitration injects vacancies 00:57:28.700 --> 00:57:29.770 align:middle line:90% so-- 00:57:29.770 --> 00:57:32.560 align:middle line:84% by these inferences by observation 00:57:32.560 --> 00:57:34.160 align:middle line:90% of stacking faults and things. 00:57:34.160 --> 00:57:37.000 align:middle line:84% So the nice thing now is, if I do an experiment, 00:57:37.000 --> 00:57:40.840 align:middle line:84% I can inject interstitials by oxidizing in one region. 00:57:40.840 --> 00:57:43.270 align:middle line:84% I can tend to take the wafer, or another wafer, 00:57:43.270 --> 00:57:46.000 align:middle line:84% and put it in a furnace with ammonia and inject vacancies. 00:57:46.000 --> 00:57:49.120 align:middle line:84% And I can see what happens to the dopant profiles 00:57:49.120 --> 00:57:51.100 align:middle line:84% under these different injection conditions. 00:57:51.100 --> 00:57:54.850 align:middle line:84% And then, therefore, decide does the dopant diffuse faster 00:57:54.850 --> 00:57:56.020 align:middle line:90% with interstitials? 00:57:56.020 --> 00:57:58.150 align:middle line:84% Well, then, it must tend to diffuse 00:57:58.150 --> 00:57:59.938 align:middle line:90% with interstitial pairs. 00:57:59.938 --> 00:58:01.480 align:middle line:84% So there's a way to make an inference 00:58:01.480 --> 00:58:03.970 align:middle line:84% about the diffusion mechanism using 00:58:03.970 --> 00:58:07.515 align:middle line:90% oxidation and nitridation. 00:58:07.515 --> 00:58:09.140 align:middle line:84% So there's been a number of experiments 00:58:09.140 --> 00:58:11.910 align:middle line:84% that have been done over the years, the last 20 years, 00:58:11.910 --> 00:58:16.160 align:middle line:84% and some of them are shown here on the results on slide 23. 00:58:16.160 --> 00:58:19.670 align:middle line:84% And what people have seen over the years is that boron 00:58:19.670 --> 00:58:23.990 align:middle line:84% and phosphorus, and to a little bit, a certain extent arsenic, 00:58:23.990 --> 00:58:27.530 align:middle line:84% they have enhanced diffusion coefficients 00:58:27.530 --> 00:58:30.090 align:middle line:84% under the influence of thermal oxidation. 00:58:30.090 --> 00:58:32.180 align:middle line:84% So during a thermal oxidation, the boron, 00:58:32.180 --> 00:58:34.940 align:middle line:84% the phosphorus diffusivity tend to go up. 00:58:34.940 --> 00:58:36.420 align:middle line:90% Antimony is just the opposite. 00:58:36.420 --> 00:58:39.140 align:middle line:84% It slows down compared to inert when you're 00:58:39.140 --> 00:58:41.700 align:middle line:90% in an oxidizing condition. 00:58:41.700 --> 00:58:45.290 align:middle line:84% So here's just an example of a plot. 00:58:45.290 --> 00:58:48.470 align:middle line:84% On the left axis is concentration versus depth. 00:58:48.470 --> 00:58:52.280 align:middle line:84% And so here's an example of antimony. 00:58:52.280 --> 00:58:55.550 align:middle line:84% Antimony was diffused, and this profile 00:58:55.550 --> 00:58:59.520 align:middle line:84% here without any dots on it is the inert case. 00:58:59.520 --> 00:59:01.610 align:middle line:84% And if you look at the junction depth for the case 00:59:01.610 --> 00:59:03.650 align:middle line:84% where it was diffusing under oxidation, 00:59:03.650 --> 00:59:04.940 align:middle line:90% it's actually shallower. 00:59:04.940 --> 00:59:07.920 align:middle line:84% So there was less diffusion of the antimony when you diffused 00:59:07.920 --> 00:59:09.670 align:middle line:84% it in the furnace-- the same temperature, 00:59:09.670 --> 00:59:12.385 align:middle line:90% but under oxidizing ambient. 00:59:12.385 --> 00:59:13.510 align:middle line:90% Boron is just the opposite. 00:59:13.510 --> 00:59:14.130 align:middle line:90% Look at boron. 00:59:14.130 --> 00:59:16.290 align:middle line:84% This is the inert case for boron. 00:59:16.290 --> 00:59:19.050 align:middle line:84% The junction depth here is about 0.4 microns. 00:59:19.050 --> 00:59:22.320 align:middle line:84% When you diffuse it with oxidation going on 00:59:22.320 --> 00:59:24.510 align:middle line:84% above it, again, it's not touching the boron, 00:59:24.510 --> 00:59:26.910 align:middle line:84% the oxidation is taking place up high in the sample. 00:59:26.910 --> 00:59:31.000 align:middle line:84% The boron has a junction depth about 0.8. 00:59:31.000 --> 00:59:34.230 align:middle line:90% So it's dramatically enhanced. 00:59:34.230 --> 00:59:36.900 align:middle line:84% The idea is that oxidation increases 00:59:36.900 --> 00:59:38.460 align:middle line:84% the concentration of interstitials, 00:59:38.460 --> 00:59:41.330 align:middle line:90% silicon interstitials, ci. 00:59:41.330 --> 00:59:45.520 align:middle line:84% Now, it decreases cv from their equilibrium values. 00:59:45.520 --> 00:59:48.690 align:middle line:84% So from this, I would conclude that boron diffuses with i, 00:59:48.690 --> 00:59:51.390 align:middle line:84% with interstitials, because I put more i in the sample, 00:59:51.390 --> 00:59:52.290 align:middle line:90% it goes faster. 00:59:52.290 --> 00:59:55.470 align:middle line:84% And antimony diffuses probably more with vacancies. 00:59:55.470 --> 00:59:58.080 align:middle line:84% Because when I decrease the vacancy population 00:59:58.080 --> 01:00:00.210 align:middle line:84% by injecting a lot of excess interstitials, 01:00:00.210 --> 01:00:01.710 align:middle line:90% antimony slows down. 01:00:01.710 --> 01:00:05.950 align:middle line:84% So antimony must favor diffusion with vacancies. 01:00:05.950 --> 01:00:07.650 align:middle line:84% So it's by these types of experiments, 01:00:07.650 --> 01:00:10.980 align:middle line:84% injecting interstitials with oxide, 01:00:10.980 --> 01:00:13.410 align:middle line:84% oxidation, or thermal nitridation 01:00:13.410 --> 01:00:15.180 align:middle line:84% to inject vacancies, that people try 01:00:15.180 --> 01:00:18.780 align:middle line:84% to figure out what is the mechanism 01:00:18.780 --> 01:00:22.480 align:middle line:90% of the dopant diffusion. 01:00:22.480 --> 01:00:24.690 align:middle line:90% So let's go on to slide 24. 01:00:24.690 --> 01:00:27.210 align:middle line:84% And in fact, the interesting thing-- 01:00:27.210 --> 01:00:30.240 align:middle line:84% the injected interstitial level depends on the generation rate 01:00:30.240 --> 01:00:34.470 align:middle line:84% at the very interface between the oxide and the silicon, 01:00:34.470 --> 01:00:36.463 align:middle line:84% and the recombination rate at that interface. 01:00:36.463 --> 01:00:38.880 align:middle line:84% So there's certain generation rate of these interstitials, 01:00:38.880 --> 01:00:40.650 align:middle line:90% and certain number combine-- 01:00:40.650 --> 01:00:41.520 align:middle line:90% recombine. 01:00:41.520 --> 01:00:43.710 align:middle line:90% Those adult go into the bulk. 01:00:43.710 --> 01:00:48.540 align:middle line:84% And in fact, the concentration of excess of interstitials 01:00:48.540 --> 01:00:51.520 align:middle line:90% depends on the oxidation rate. 01:00:51.520 --> 01:00:54.300 align:middle line:84% That's what people find, which is interesting. 01:00:54.300 --> 01:00:57.210 align:middle line:84% So if I oxidize faster at a given temperature, 01:00:57.210 --> 01:01:00.930 align:middle line:84% I'm going to get a higher oxidation, 01:01:00.930 --> 01:01:04.570 align:middle line:90% or interstitial concentration. 01:01:04.570 --> 01:01:08.430 align:middle line:84% So for example, if I plot the interstitial supersaturation 01:01:08.430 --> 01:01:11.280 align:middle line:84% ratio-- and this is the ratio of ci 01:01:11.280 --> 01:01:13.980 align:middle line:84% divided by ci star, where ci star means 01:01:13.980 --> 01:01:16.150 align:middle line:84% the concentration of interstitials in equilibrium. 01:01:16.150 --> 01:01:19.290 align:middle line:84% So that's in a neutral ambient without any perturbing 01:01:19.290 --> 01:01:21.510 align:middle line:90% due to oxidation. 01:01:21.510 --> 01:01:25.800 align:middle line:84% And I look at this ratio, the dashed line is for wet 002, 01:01:25.800 --> 01:01:28.190 align:middle line:90% and the solid is for dry. 01:01:28.190 --> 01:01:31.370 align:middle line:84% Well, we know the oxidation rate in water, in 1 O2 01:01:31.370 --> 01:01:32.390 align:middle line:90% is a lot faster. 01:01:32.390 --> 01:01:35.580 align:middle line:84% You see, the whole dashed line is higher. 01:01:35.580 --> 01:01:38.930 align:middle line:84% So if I really, really wanted to enhance the diffusion of boron, 01:01:38.930 --> 01:01:40.070 align:middle line:90% what would I do? 01:01:40.070 --> 01:01:41.900 align:middle line:84% I could put it in the substrate, and I 01:01:41.900 --> 01:01:44.540 align:middle line:84% would subject the substrate to wet oxidation 01:01:44.540 --> 01:01:45.650 align:middle line:90% at a given temperature. 01:01:45.650 --> 01:01:46.970 align:middle line:90% And that would really boost up. 01:01:46.970 --> 01:01:48.803 align:middle line:84% I could make the boron diffuse a lot faster. 01:01:48.803 --> 01:01:52.400 align:middle line:84% And generally, you want to slow things down. 01:01:52.400 --> 01:01:56.162 align:middle line:84% So this-- as you can see, this depends primarily 01:01:56.162 --> 01:01:56.870 align:middle line:90% upon temperature. 01:01:56.870 --> 01:02:00.410 align:middle line:84% There's a little influence of the wet versus dry. 01:02:00.410 --> 01:02:02.840 align:middle line:84% But the big influence here is on temperature. 01:02:02.840 --> 01:02:05.390 align:middle line:84% And the interstitial supersaturation ratio 01:02:05.390 --> 01:02:09.440 align:middle line:84% is much larger at low oxidation temperatures. 01:02:09.440 --> 01:02:13.460 align:middle line:84% That's because ci star is going down rapidly while you continue 01:02:13.460 --> 01:02:16.880 align:middle line:84% to inject a lot of interstitial ci. 01:02:16.880 --> 01:02:20.870 align:middle line:84% So we expect the enhancement in diffusion-- or diffusivity 01:02:20.870 --> 01:02:22.430 align:middle line:84% to be small at high temperatures, 01:02:22.430 --> 01:02:26.090 align:middle line:84% like, at 1,200, where the supersaturation ratio is only 01:02:26.090 --> 01:02:27.260 align:middle line:90% a factor of 2. 01:02:27.260 --> 01:02:29.990 align:middle line:84% But to be large at low temperature, say, 800, 01:02:29.990 --> 01:02:33.100 align:middle line:84% very large, where you can get ratios of 10, 01:02:33.100 --> 01:02:35.760 align:middle line:84% or 100, 100 times faster diffusion 01:02:35.760 --> 01:02:38.490 align:middle line:84% than you would get under equilibrium non-oxidizing 01:02:38.490 --> 01:02:39.400 align:middle line:90% conditions. 01:02:39.400 --> 01:02:41.750 align:middle line:84% So this tells us where OED, where 01:02:41.750 --> 01:02:45.870 align:middle line:84% ORD is going to be most prevalent at low temperatures. 01:02:45.870 --> 01:02:48.390 align:middle line:90% 01:02:48.390 --> 01:02:50.850 align:middle line:84% So how do people model the interstitial and vacancy 01:02:50.850 --> 01:02:53.500 align:middle line:90% components of diffusion? 01:02:53.500 --> 01:02:55.800 align:middle line:84% Well, here's-- again, I just want to show you some 01:02:55.800 --> 01:03:01.050 align:middle line:84% experimental data, some SIMS plots with both arsenic 01:03:01.050 --> 01:03:04.410 align:middle line:84% and antimony in the sample at the same time. 01:03:04.410 --> 01:03:09.730 align:middle line:84% So the red here is shown under inert conditions. 01:03:09.730 --> 01:03:10.690 align:middle line:90% So no oxidation. 01:03:10.690 --> 01:03:12.150 align:middle line:84% You can see the arsenic is abrupt. 01:03:12.150 --> 01:03:16.070 align:middle line:84% If you oxidize it, the arsenic diffuses a little faster. 01:03:16.070 --> 01:03:18.190 align:middle line:84% So it's being influenced by the interstitials. 01:03:18.190 --> 01:03:22.480 align:middle line:84% Antimony, at the same time under inert conditions, 01:03:22.480 --> 01:03:23.920 align:middle line:90% is a little broader. 01:03:23.920 --> 01:03:28.670 align:middle line:84% But if you oxidize it, it maintains-- it diffuses less. 01:03:28.670 --> 01:03:31.750 align:middle line:84% So here's OED, oxidation enhanced diffusion 01:03:31.750 --> 01:03:34.360 align:middle line:84% of arsenic taking place at the same time as ORD, 01:03:34.360 --> 01:03:38.000 align:middle line:84% oxidation retarded diffusion, of antimony. 01:03:38.000 --> 01:03:41.560 align:middle line:84% So both types of point defects, interstitials and vacancies, 01:03:41.560 --> 01:03:44.725 align:middle line:84% are important in diffusing in silicon. 01:03:44.725 --> 01:03:47.650 align:middle line:90% 01:03:47.650 --> 01:03:50.560 align:middle line:84% So what people do is-- it's somewhat empirical but it 01:03:50.560 --> 01:03:51.160 align:middle line:90% works-- 01:03:51.160 --> 01:03:52.990 align:middle line:84% is you say that the dopants diffuse 01:03:52.990 --> 01:03:56.320 align:middle line:84% with a certain fraction, f sub i, of interstitial type 01:03:56.320 --> 01:03:58.750 align:middle line:84% diffusion, and a certain fraction 01:03:58.750 --> 01:04:03.320 align:middle line:84% f sub c-- f sub b, which is just 1 minus fi, of vacancy type. 01:04:03.320 --> 01:04:04.870 align:middle line:84% So we're just going to apportion-- 01:04:04.870 --> 01:04:06.680 align:middle line:84% for any given dopant we're going to say, 01:04:06.680 --> 01:04:11.710 align:middle line:84% well, X percent, or x fraction, f sub i fraction, 01:04:11.710 --> 01:04:15.100 align:middle line:84% is associated with it moving with interstitials. 01:04:15.100 --> 01:04:17.650 align:middle line:84% So we write in a very generalized form 01:04:17.650 --> 01:04:22.730 align:middle line:84% the diffusivity of any dope, da, is da star, where da 01:04:22.730 --> 01:04:25.820 align:middle line:84% star is the normal equilibrium diffusivity measured 01:04:25.820 --> 01:04:28.370 align:middle line:84% under inert conditions-- no oxidation, 01:04:28.370 --> 01:04:29.840 align:middle line:90% no thermal nitridation. 01:04:29.840 --> 01:04:31.820 align:middle line:84% We're not perturbing the surface in any way. 01:04:31.820 --> 01:04:35.750 align:middle line:84% So that's da star times this quantity, f sub 01:04:35.750 --> 01:04:37.970 align:middle line:84% i, which is a number between zero and 1, 01:04:37.970 --> 01:04:43.670 align:middle line:84% times ci over ci star, plus f sub v, times cv over ci star-- 01:04:43.670 --> 01:04:44.840 align:middle line:90% cv star. 01:04:44.840 --> 01:04:47.390 align:middle line:84% Again, the star means equilibrium, no oxidation 01:04:47.390 --> 01:04:50.450 align:middle line:90% or nitridation. 01:04:50.450 --> 01:04:53.240 align:middle line:84% So you can see I can enhance the diffusivity just 01:04:53.240 --> 01:04:57.370 align:middle line:84% by enhancing ci over ci star, assuming 01:04:57.370 --> 01:04:59.110 align:middle line:90% f sub i is greater than zero. 01:04:59.110 --> 01:05:01.060 align:middle line:84% So if I have a dopant like boron, 01:05:01.060 --> 01:05:03.190 align:middle line:84% people believe that f sub i is 1. 01:05:03.190 --> 01:05:06.010 align:middle line:84% If I pump up ci over ci star, then I 01:05:06.010 --> 01:05:07.880 align:middle line:84% get a great enhancement proportional 01:05:07.880 --> 01:05:11.540 align:middle line:84% to ci or ci star in the diffusivity. 01:05:11.540 --> 01:05:13.750 align:middle line:84% So again, oxidation injects interstitials, 01:05:13.750 --> 01:05:17.540 align:middle line:84% so it's going to raise ci star, and it reduces vacancy. 01:05:17.540 --> 01:05:20.290 align:middle line:84% So this goes down, cv over cv star 01:05:20.290 --> 01:05:22.720 align:middle line:90% by a recombination mechanism. 01:05:22.720 --> 01:05:24.770 align:middle line:84% And nitridation does exactly the opposite. 01:05:24.770 --> 01:05:26.630 align:middle line:84% So this is the mathematical formulation 01:05:26.630 --> 01:05:31.000 align:middle line:84% we can use to express these observations. 01:05:31.000 --> 01:05:32.920 align:middle line:84% And in fact, you go on to slide 27, 01:05:32.920 --> 01:05:34.580 align:middle line:90% people have tried to measure-- 01:05:34.580 --> 01:05:36.820 align:middle line:84% they have measured the enhancement of the diffusion, 01:05:36.820 --> 01:05:39.730 align:middle line:84% or the retarded diffusion, under different conditions. 01:05:39.730 --> 01:05:43.630 align:middle line:84% And these are the f sub i and f sub v values that people-- 01:05:43.630 --> 01:05:45.910 align:middle line:90% that are roughly in SUPREM. 01:05:45.910 --> 01:05:47.180 align:middle line:90% So what do we see? 01:05:47.180 --> 01:05:49.000 align:middle line:90% Well, for boron, f sub i is 1. 01:05:49.000 --> 01:05:53.470 align:middle line:84% So they're saying it diffuses entirely by interstitials. 01:05:53.470 --> 01:05:55.120 align:middle line:84% So that's roughly what people believe. 01:05:55.120 --> 01:05:56.600 align:middle line:90% Phosphorous is close to 1. 01:05:56.600 --> 01:05:58.910 align:middle line:84% Arsenic, which is our most popular n-type dopant, 01:05:58.910 --> 01:06:00.470 align:middle line:90% unfortunately, is mixed. 01:06:00.470 --> 01:06:03.440 align:middle line:84% It diffuses both by interstitial mechanism 01:06:03.440 --> 01:06:05.030 align:middle line:90% and by vacancy mechanism. 01:06:05.030 --> 01:06:09.240 align:middle line:84% Antimony is just the opposite, entirely by vacancies. 01:06:09.240 --> 01:06:10.520 align:middle line:90% So these [INAUDIBLE] numbers. 01:06:10.520 --> 01:06:13.400 align:middle line:84% Of course, you can modify them at will in the simulator, 01:06:13.400 --> 01:06:17.480 align:middle line:84% but these are the ones that are programmed in into SUPREM IV. 01:06:17.480 --> 01:06:20.090 align:middle line:90% 01:06:20.090 --> 01:06:22.610 align:middle line:90% So let's go on to slide 28. 01:06:22.610 --> 01:06:24.230 align:middle line:84% Again, this is a general formulation 01:06:24.230 --> 01:06:29.300 align:middle line:84% for how we write the diffusivity in terms of si, fi and fv. 01:06:29.300 --> 01:06:31.850 align:middle line:84% How does this actually relate to our previous description? 01:06:31.850 --> 01:06:35.090 align:middle line:84% We keep making-- now, that I've made the model more atomistic, 01:06:35.090 --> 01:06:38.210 align:middle line:84% how does it actually relate back to the more macroscopic 01:06:38.210 --> 01:06:38.810 align:middle line:90% description? 01:06:38.810 --> 01:06:41.420 align:middle line:84% Well, this is the macroscopic way we wrote it, right? 01:06:41.420 --> 01:06:47.060 align:middle line:84% We said da effective is just da zero, some number 01:06:47.060 --> 01:06:50.940 align:middle line:90% times e to the minus e over kt. 01:06:50.940 --> 01:06:53.340 align:middle line:84% Well, I can rewrite this expression on top 01:06:53.340 --> 01:06:54.750 align:middle line:90% under inert conditions. 01:06:54.750 --> 01:06:58.470 align:middle line:84% Inert meaning, ci over ci star is 1. 01:06:58.470 --> 01:07:00.210 align:middle line:90% cv over cv star is 1. 01:07:00.210 --> 01:07:02.040 align:middle line:84% So there's no oxidation or nitrogen. 01:07:02.040 --> 01:07:05.380 align:middle line:84% Then da is just sum of two terms. 01:07:05.380 --> 01:07:07.500 align:middle line:84% The diffusivity of the paired species 01:07:07.500 --> 01:07:12.090 align:middle line:84% ai, plus the diffusivity of the paired species av. 01:07:12.090 --> 01:07:13.620 align:middle line:84% And in fact, I can break this down 01:07:13.620 --> 01:07:17.190 align:middle line:84% even further where I write this diffusivity of the paired 01:07:17.190 --> 01:07:24.770 align:middle line:84% ai as a little d, diffusivity of ai times the concentration of c 01:07:24.770 --> 01:07:28.230 align:middle line:84% of these ai pairs, divided by ca, 01:07:28.230 --> 01:07:33.460 align:middle line:84% plus a comparable expression analogous for vacancies. 01:07:33.460 --> 01:07:36.750 align:middle line:84% So what this is saying is, I can sum up the diffusivity. 01:07:36.750 --> 01:07:38.490 align:middle line:84% And if I just look at this one term, 01:07:38.490 --> 01:07:41.730 align:middle line:84% it's looking like the diffusivity of the pair, 01:07:41.730 --> 01:07:44.070 align:middle line:84% say, the dopant paired with the interstitials, 01:07:44.070 --> 01:07:46.980 align:middle line:84% times the ratio of its concentration, 01:07:46.980 --> 01:07:49.740 align:middle line:84% to the total concentration of the arsenic, 01:07:49.740 --> 01:07:51.180 align:middle line:90% or whatever the dopant is. 01:07:51.180 --> 01:07:54.100 align:middle line:84% So if I make this go up, then this will go up. 01:07:54.100 --> 01:07:58.800 align:middle line:84% So at an atomistic level, we can decompose 01:07:58.800 --> 01:08:02.660 align:middle line:84% this effect of diffusivity into these two different mechanisms. 01:08:02.660 --> 01:08:07.270 align:middle line:90% 01:08:07.270 --> 01:08:08.610 align:middle line:90% So let's go on to slide 29. 01:08:08.610 --> 01:08:10.560 align:middle line:84% And, say, there's another way people 01:08:10.560 --> 01:08:15.690 align:middle line:84% can look at this atomistic scale reactions and diffusions. 01:08:15.690 --> 01:08:18.053 align:middle line:84% And people do it through a chemical reaction. 01:08:18.053 --> 01:08:19.470 align:middle line:84% If you're familiar with chemistry, 01:08:19.470 --> 01:08:20.470 align:middle line:90% this makes sense to you. 01:08:20.470 --> 01:08:23.069 align:middle line:84% If you're not, you think, why am I going to all this effort 01:08:23.069 --> 01:08:24.848 align:middle line:84% when I can express it mathematically 01:08:24.848 --> 01:08:25.890 align:middle line:90% a little bit differently? 01:08:25.890 --> 01:08:30.000 align:middle line:84% But what we say is, the reaction where a substitutional dopant 01:08:30.000 --> 01:08:33.850 align:middle line:84% atom a interacts with an interstitial silicon atom 01:08:33.850 --> 01:08:36.200 align:middle line:90% to form a mobile species. 01:08:36.200 --> 01:08:40.689 align:middle line:84% So the simple reaction says that a, substitutional, plus i, 01:08:40.689 --> 01:08:42.040 align:middle line:90% goes to ai. 01:08:42.040 --> 01:08:44.710 align:middle line:84% Now, the important thing to realize about this equation 01:08:44.710 --> 01:08:49.000 align:middle line:84% is that on the left-hand side, a and i are both immobile. 01:08:49.000 --> 01:08:51.310 align:middle line:90% So a is immobile by itself. 01:08:51.310 --> 01:08:54.939 align:middle line:84% We're assuming that anytime the atom, the arsenic or the boron, 01:08:54.939 --> 01:08:57.560 align:middle line:84% is substitutional on the lattice at high temperature, 01:08:57.560 --> 01:08:58.779 align:middle line:90% it can't move. 01:08:58.779 --> 01:09:00.910 align:middle line:84% That the only way it can become mobilized 01:09:00.910 --> 01:09:04.479 align:middle line:84% is when it's in a pair form with an a right next to an i, 01:09:04.479 --> 01:09:07.779 align:middle line:84% or next to a vacancy if you want to do it in terms of vacancies. 01:09:07.779 --> 01:09:10.479 align:middle line:84% So the substitutional species themselves 01:09:10.479 --> 01:09:13.149 align:middle line:84% are immobile, only as a pair are they immobile. 01:09:13.149 --> 01:09:16.939 align:middle line:84% And this is what the model is saying. 01:09:16.939 --> 01:09:18.729 align:middle line:84% So we can actually be used to explain 01:09:18.729 --> 01:09:20.590 align:middle line:84% a lot of different phenomena at a distance 01:09:20.590 --> 01:09:22.720 align:middle line:90% that people have observed. 01:09:22.720 --> 01:09:26.210 align:middle line:84% For example, if you have an interstitial supersaturation. 01:09:26.210 --> 01:09:28.640 align:middle line:90% So you pump up i a lot. 01:09:28.640 --> 01:09:30.069 align:middle line:84% This is going to drive more dopant 01:09:30.069 --> 01:09:31.837 align:middle line:84% atoms into the mobile state by shifting 01:09:31.837 --> 01:09:34.420 align:middle line:84% the equation to the right, and enhance the dopant diffusivity. 01:09:34.420 --> 01:09:35.050 align:middle line:90% And that's OED. 01:09:35.050 --> 01:09:37.510 align:middle line:84% So example, if you're a chemist and I 01:09:37.510 --> 01:09:39.760 align:middle line:90% tell you I flood the reactor-- 01:09:39.760 --> 01:09:41.979 align:middle line:84% I flood the silicon with a lot of i, 01:09:41.979 --> 01:09:44.585 align:middle line:84% well, this reaction tends to get-- if we add more i, 01:09:44.585 --> 01:09:46.210 align:middle line:84% this reaction gets pushed to the right. 01:09:46.210 --> 01:09:47.830 align:middle line:90% So I form more pairs. 01:09:47.830 --> 01:09:51.340 align:middle line:84% If I have more pairs, arsenic can diffuse more readily-- 01:09:51.340 --> 01:09:54.550 align:middle line:90% or the dopant, and you get OED. 01:09:54.550 --> 01:09:58.510 align:middle line:84% Now, the interesting thing is that there's 01:09:58.510 --> 01:10:01.900 align:middle line:84% can-- this equation also predicts some effects even 01:10:01.900 --> 01:10:03.590 align:middle line:90% under inert conditions. 01:10:03.590 --> 01:10:09.070 align:middle line:84% So this can indicate that the interior of the silicon 01:10:09.070 --> 01:10:14.170 align:middle line:84% will be injected by this mobile ai species as it diffuses in. 01:10:14.170 --> 01:10:16.630 align:middle line:84% And that's going to drive the equation to the left, 01:10:16.630 --> 01:10:19.180 align:middle line:84% to this way, releasing interstitials 01:10:19.180 --> 01:10:21.370 align:middle line:84% in the interior when the dopant regains 01:10:21.370 --> 01:10:23.620 align:middle line:90% its substitutional position. 01:10:23.620 --> 01:10:26.002 align:middle line:84% So interestingly, this chemical reaction 01:10:26.002 --> 01:10:27.460 align:middle line:84% tells us that silicon interstitials 01:10:27.460 --> 01:10:30.940 align:middle line:84% can be pumped into the interior of the sample 01:10:30.940 --> 01:10:32.382 align:middle line:90% by dopant diffusion. 01:10:32.382 --> 01:10:34.090 align:middle line:84% So for example, let's say I have arsenic, 01:10:34.090 --> 01:10:37.420 align:middle line:84% it diffuses in by pairs, it finds its way 01:10:37.420 --> 01:10:39.130 align:middle line:84% and settles into a certain position 01:10:39.130 --> 01:10:40.690 align:middle line:90% where it's now substitutional. 01:10:40.690 --> 01:10:43.060 align:middle line:84% And then what happens, these interstitials are released. 01:10:43.060 --> 01:10:44.650 align:middle line:84% So all of a sudden, the arsenic is 01:10:44.650 --> 01:10:46.800 align:middle line:84% carried in with it, all these excess interstitials, 01:10:46.800 --> 01:10:48.550 align:middle line:84% there's a bunch of interstitials released, 01:10:48.550 --> 01:10:50.560 align:middle line:84% they could then impact the dopant-- 01:10:50.560 --> 01:10:52.658 align:middle line:84% the diffusion of a dopant nearby. 01:10:52.658 --> 01:10:54.700 align:middle line:84% And in fact, that's exactly-- that's exactly what 01:10:54.700 --> 01:11:00.670 align:middle line:84% happens in this profile on Page 30 of phosphorus. 01:11:00.670 --> 01:11:04.030 align:middle line:84% For years, it was observed that high concentration phosphorus 01:11:04.030 --> 01:11:06.930 align:middle line:90% had a kink and a tail. 01:11:06.930 --> 01:11:08.250 align:middle line:90% It was kinked region here. 01:11:08.250 --> 01:11:10.200 align:middle line:84% And then at lower concentrations, 01:11:10.200 --> 01:11:11.580 align:middle line:90% it had a long tail. 01:11:11.580 --> 01:11:13.950 align:middle line:90% It was into the substrate. 01:11:13.950 --> 01:11:16.890 align:middle line:84% And people thought of all kinds of mechanisms to explain this. 01:11:16.890 --> 01:11:20.340 align:middle line:84% Well, one mechanism in terms of this chemical equilibrium 01:11:20.340 --> 01:11:22.590 align:middle line:84% formulation that we're talking about 01:11:22.590 --> 01:11:24.900 align:middle line:84% today is to say, all right, the phosphorus 01:11:24.900 --> 01:11:29.070 align:middle line:84% diffuses with interstitials so they diffuse as a pair. 01:11:29.070 --> 01:11:32.190 align:middle line:84% And then, eventually, phosphorus finds a substitutional site. 01:11:32.190 --> 01:11:33.107 align:middle line:90% It stays there. 01:11:33.107 --> 01:11:35.190 align:middle line:84% And then it's going to release them into the bulk. 01:11:35.190 --> 01:11:38.910 align:middle line:84% So all of a sudden, I have a flux here, 01:11:38.910 --> 01:11:41.080 align:middle line:90% I have a flux of a pair. 01:11:41.080 --> 01:11:43.500 align:middle line:84% And then I get a flux of extra interstitials. 01:11:43.500 --> 01:11:46.320 align:middle line:84% This, in turn, enhances the tail diffusivity 01:11:46.320 --> 01:11:48.030 align:middle line:90% of the phosphorus profile. 01:11:48.030 --> 01:11:50.820 align:middle line:84% So the reason the tail is enhanced, people believe, 01:11:50.820 --> 01:11:52.802 align:middle line:84% is because the phosphorus itself is 01:11:52.802 --> 01:11:54.510 align:middle line:84% pumping in a whole bunch of interstitials 01:11:54.510 --> 01:11:58.590 align:middle line:84% and then releasing them somewhere down in this depth. 01:11:58.590 --> 01:12:03.410 align:middle line:84% So that's a way to use this to explain qualitatively 01:12:03.410 --> 01:12:07.290 align:middle line:84% the tail region of phosphorus diffusion. 01:12:07.290 --> 01:12:09.080 align:middle line:84% On slide 31, there's a famous effect 01:12:09.080 --> 01:12:11.508 align:middle line:90% that people call emitter push. 01:12:11.508 --> 01:12:13.800 align:middle line:84% Again, they thought of lots of reasons to explain this. 01:12:13.800 --> 01:12:14.820 align:middle line:90% What is emitter push? 01:12:14.820 --> 01:12:17.130 align:middle line:84% Well, when you're making a bipolar transistor, 01:12:17.130 --> 01:12:18.780 align:middle line:84% you have a high concentration emitter-- 01:12:18.780 --> 01:12:20.490 align:middle line:84% it could be phosphorus or arsenic. 01:12:20.490 --> 01:12:23.460 align:middle line:84% And then you have a more lightly dope base. 01:12:23.460 --> 01:12:25.680 align:middle line:84% And then you have a more lightly dope collector. 01:12:25.680 --> 01:12:27.960 align:middle line:84% What people observed is, wherever 01:12:27.960 --> 01:12:30.120 align:middle line:84% the emitter, the phosphorus was being diffused, 01:12:30.120 --> 01:12:32.250 align:middle line:84% right underneath it, the boron base 01:12:32.250 --> 01:12:36.750 align:middle line:84% was pushed out, almost like the emitter was pushing the boron 01:12:36.750 --> 01:12:38.310 align:middle line:90% faster, to diffuse faster. 01:12:38.310 --> 01:12:40.290 align:middle line:84% Far away from the emitter over here, 01:12:40.290 --> 01:12:42.330 align:middle line:90% it didn't diffuse quite so much. 01:12:42.330 --> 01:12:45.175 align:middle line:84% But underneath it, it diffused quite a bit. 01:12:45.175 --> 01:12:47.550 align:middle line:84% And again, there were a lot of models people came up with 01:12:47.550 --> 01:12:48.217 align:middle line:90% to explain this. 01:12:48.217 --> 01:12:51.420 align:middle line:84% Well, again, people could say that this high concentration 01:12:51.420 --> 01:12:53.340 align:middle line:84% of phosphorus is pumping interstitials 01:12:53.340 --> 01:12:55.262 align:middle line:84% because p and i are diffusing together. 01:12:55.262 --> 01:12:56.970 align:middle line:84% So the interstitials themselves are being 01:12:56.970 --> 01:13:00.030 align:middle line:90% carried in towards the base. 01:13:00.030 --> 01:13:02.460 align:middle line:84% We get a high supersaturation of these interstitials. 01:13:02.460 --> 01:13:06.600 align:middle line:84% They get released when the phosphorus stops diffusing, 01:13:06.600 --> 01:13:10.420 align:middle line:84% and they're released into the boron base. 01:13:10.420 --> 01:13:13.840 align:middle line:84% This excess interstitials then enhances the boron diffusivity 01:13:13.840 --> 01:13:17.110 align:middle line:84% and causes it to push in, because we know that we're-- 01:13:17.110 --> 01:13:19.900 align:middle line:90% boron has an f sub i of 1. 01:13:19.900 --> 01:13:21.610 align:middle line:84% So this is an interesting effect. 01:13:21.610 --> 01:13:24.190 align:middle line:84% We said we could inject interstitials by oxidation 01:13:24.190 --> 01:13:25.480 align:middle line:90% and enhance boron. 01:13:25.480 --> 01:13:28.000 align:middle line:84% We can also inject interstitials by other processes 01:13:28.000 --> 01:13:31.360 align:middle line:84% just by the presence nearby of a high concentration 01:13:31.360 --> 01:13:33.790 align:middle line:84% diffusion of another species like phosphorus. 01:13:33.790 --> 01:13:36.810 align:middle line:90% So this is called full coupling. 01:13:36.810 --> 01:13:40.440 align:middle line:84% Full coupling means that the diffusion of the dopants 01:13:40.440 --> 01:13:41.850 align:middle line:84% is affected by the interstitials. 01:13:41.850 --> 01:13:43.980 align:middle line:84% And likewise, the diffusion of the interstitials 01:13:43.980 --> 01:13:45.695 align:middle line:84% is affected by the presence of dopants. 01:13:45.695 --> 01:13:47.820 align:middle line:84% And where those interstitials end up in your wafer, 01:13:47.820 --> 01:13:50.350 align:middle line:84% they could impact some other process. 01:13:50.350 --> 01:13:53.310 align:middle line:84% So that's what they mean by full coupling in the SUPREM model. 01:13:53.310 --> 01:13:56.590 align:middle line:90% 01:13:56.590 --> 01:14:00.210 align:middle line:84% So if we go on to slide 32, again, this 01:14:00.210 --> 01:14:02.080 align:middle line:84% is that same equation I showed before. 01:14:02.080 --> 01:14:05.040 align:middle line:84% And if you're a chemist and you assume chemical equilibrium 01:14:05.040 --> 01:14:08.150 align:middle line:84% between these dopants a and the defects i, 01:14:08.150 --> 01:14:10.320 align:middle line:84% you can write a law of mass action 01:14:10.320 --> 01:14:14.280 align:middle line:84% that says the concentration of the products, c of ai, 01:14:14.280 --> 01:14:18.720 align:middle line:84% is just a constant at any given temperature of the multiply, 01:14:18.720 --> 01:14:23.520 align:middle line:84% or the product of the reactants ca times ci. 01:14:23.520 --> 01:14:27.450 align:middle line:84% So you can actually write this in a chemical equation. 01:14:27.450 --> 01:14:29.640 align:middle line:84% And then the neat thing is, given this relationship 01:14:29.640 --> 01:14:31.740 align:middle line:84% that it's just the product of ca times ci, 01:14:31.740 --> 01:14:35.740 align:middle line:84% I can apply Fick's first law to this mobile species. 01:14:35.740 --> 01:14:38.290 align:middle line:84% So if I want to differentiate cai 01:14:38.290 --> 01:14:41.250 align:middle line:84% by dx, so this is Fick's first law, 01:14:41.250 --> 01:14:43.450 align:middle line:84% it says the flux of the mobile pair 01:14:43.450 --> 01:14:46.270 align:middle line:84% is just some constant times the concentration 01:14:46.270 --> 01:14:48.610 align:middle line:90% gradient of the mobile pair. 01:14:48.610 --> 01:14:51.970 align:middle line:84% Well, I can now apply the chain rule to this. 01:14:51.970 --> 01:14:56.290 align:middle line:84% And applying the chain rule, I can see that the flux of ai 01:14:56.290 --> 01:14:58.990 align:middle line:84% depends on some diffusivity times the term that 01:14:58.990 --> 01:15:02.680 align:middle line:84% goes like the gradient of the arsenic, or the dopant, 01:15:02.680 --> 01:15:06.490 align:middle line:84% plus a term that goes like the gradient in the interstitials. 01:15:06.490 --> 01:15:10.180 align:middle line:84% So what this is saying is that grains in the defects, 01:15:10.180 --> 01:15:14.110 align:middle line:84% as well as gradients in the dopants cause dopant diffusion. 01:15:14.110 --> 01:15:15.790 align:middle line:84% When we talked about Fick's law earlier, 01:15:15.790 --> 01:15:17.665 align:middle line:84% we said, well, we have a gradient of arsenic, 01:15:17.665 --> 01:15:20.287 align:middle line:84% and that's gradient of arsenic is what drives the diffusion. 01:15:20.287 --> 01:15:22.120 align:middle line:84% Well, not only will the gradient of arsenic, 01:15:22.120 --> 01:15:24.700 align:middle line:84% but somehow if you create by some other mechanism 01:15:24.700 --> 01:15:28.580 align:middle line:84% a gradient of interstitials, that will also drive diffusion. 01:15:28.580 --> 01:15:30.050 align:middle line:90% So there's a hidden term. 01:15:30.050 --> 01:15:32.350 align:middle line:84% And that's because we're doing a pair model. 01:15:32.350 --> 01:15:34.750 align:middle line:84% We're saying that arsenic, or boron, or whatever, 01:15:34.750 --> 01:15:37.510 align:middle line:84% has to diffuse by means of pairing 01:15:37.510 --> 01:15:39.050 align:middle line:84% and so it gives you this extra term. 01:15:39.050 --> 01:15:40.675 align:middle line:84% And there are a lot of interesting ways 01:15:40.675 --> 01:15:43.270 align:middle line:84% you can accidentally create gradients of interstitials, 01:15:43.270 --> 01:15:46.240 align:middle line:84% not even realizing it, and then end up driving dopant 01:15:46.240 --> 01:15:47.665 align:middle line:90% diffusion at a faster rate. 01:15:47.665 --> 01:15:50.390 align:middle line:90% 01:15:50.390 --> 01:15:52.947 align:middle line:84% So on Page 33, this is actually-- 01:15:52.947 --> 01:15:54.530 align:middle line:84% I'm not going to derive this, but this 01:15:54.530 --> 01:15:58.850 align:middle line:84% is the actual overall flux equation that SUPREM uses, 01:15:58.850 --> 01:16:00.752 align:middle line:90% and it's discussed in your text. 01:16:00.752 --> 01:16:02.210 align:middle line:84% It's a fairly complicated-- you say 01:16:02.210 --> 01:16:04.670 align:middle line:84% that the total flux of boron interstitial pairs 01:16:04.670 --> 01:16:07.887 align:middle line:84% is the product of all these terms. 01:16:07.887 --> 01:16:09.720 align:middle line:84% Well, there are a-- we can look at the terms 01:16:09.720 --> 01:16:11.100 align:middle line:90% and make sense out of them. 01:16:11.100 --> 01:16:14.430 align:middle line:84% This dvi star, that's just the inner-- the star indicates 01:16:14.430 --> 01:16:17.130 align:middle line:84% its inert low concentration diffusion 01:16:17.130 --> 01:16:19.590 align:middle line:84% driven by the dopant gradient, the usual good 01:16:19.590 --> 01:16:21.120 align:middle line:90% old Fick's diffusion. 01:16:21.120 --> 01:16:23.250 align:middle line:84% And then all the rest are correct in factors. 01:16:23.250 --> 01:16:27.360 align:middle line:84% In these large parentheses, we see the high concentration 01:16:27.360 --> 01:16:28.900 align:middle line:90% effects due to the Fermi level. 01:16:28.900 --> 01:16:32.340 align:middle line:84% So that's what this beta time is due to. 01:16:32.340 --> 01:16:35.160 align:middle line:84% The interstitial supersaturation, 01:16:35.160 --> 01:16:38.460 align:middle line:84% ci over ci star, again, if I inject interstitials, 01:16:38.460 --> 01:16:42.330 align:middle line:84% we know that will cause an enhancement in diffusivity. 01:16:42.330 --> 01:16:44.400 align:middle line:84% And this term over here at the very end, 01:16:44.400 --> 01:16:47.640 align:middle line:84% partial partial x of the ln of p over ni, that's 01:16:47.640 --> 01:16:48.850 align:middle line:90% the electric field effect. 01:16:48.850 --> 01:16:50.730 align:middle line:84% So all of these are lumped together in order 01:16:50.730 --> 01:16:52.500 align:middle line:90% to calculate the total flux. 01:16:52.500 --> 01:16:54.960 align:middle line:90% 01:16:54.960 --> 01:16:55.460 align:middle line:90% OK. 01:16:55.460 --> 01:16:57.390 align:middle line:90% So let me just summarize. 01:16:57.390 --> 01:16:59.180 align:middle line:84% We talked about Fermi level effects. 01:16:59.180 --> 01:17:02.240 align:middle line:84% They apply, they're important when the carrier concentration 01:17:02.240 --> 01:17:04.490 align:middle line:90% is greater than ni. 01:17:04.490 --> 01:17:07.160 align:middle line:84% The diffusivity is dependent upon the local carrier 01:17:07.160 --> 01:17:08.240 align:middle line:90% concentration. 01:17:08.240 --> 01:17:11.060 align:middle line:84% It's either determined by the diffusion species itself, 01:17:11.060 --> 01:17:13.830 align:middle line:84% or by the background doping, whichever is higher. 01:17:13.830 --> 01:17:19.670 align:middle line:84% And we tend to get diffusivities of this form, this formulation. 01:17:19.670 --> 01:17:23.060 align:middle line:84% This leads to very box-like profiles. 01:17:23.060 --> 01:17:25.580 align:middle line:84% We talked about segregation at the oxide interface. 01:17:25.580 --> 01:17:27.470 align:middle line:84% It determines the boundary conditions. 01:17:27.470 --> 01:17:30.770 align:middle line:84% Boron segregates into the oxide, it's depleted from the silicon. 01:17:30.770 --> 01:17:33.140 align:middle line:90% Arsenic, and phosphorus pile up. 01:17:33.140 --> 01:17:36.320 align:middle line:84% They go out of the oxide and they go into the silicon. 01:17:36.320 --> 01:17:38.610 align:middle line:84% There's also interfacial dopant pileup, 01:17:38.610 --> 01:17:40.220 align:middle line:84% which is different from segregation 01:17:40.220 --> 01:17:41.780 align:middle line:90% at the oxide silicon interface. 01:17:41.780 --> 01:17:44.180 align:middle line:84% And this results in dramatic dose loss, particularly, 01:17:44.180 --> 01:17:47.540 align:middle line:90% for shallow source drains. 01:17:47.540 --> 01:17:51.500 align:middle line:84% We know from OE-- we know OED, ORD, and growth and shrinkage 01:17:51.500 --> 01:17:54.330 align:middle line:84% of stacking faults can be explained by this atomic scale 01:17:54.330 --> 01:17:55.260 align:middle line:90% diffusion picture. 01:17:55.260 --> 01:17:58.230 align:middle line:90% 01:17:58.230 --> 01:17:59.790 align:middle line:84% We said the boron and [? phosph ?] 01:17:59.790 --> 01:18:02.330 align:middle line:84% diffuse primarily with interstitials and ammonia, 01:18:02.330 --> 01:18:03.380 align:middle line:90% primarily, by vacancy. 01:18:03.380 --> 01:18:06.660 align:middle line:84% This has been determined by a lot of experiments. 01:18:06.660 --> 01:18:08.750 align:middle line:84% And if we use this chemical equation 01:18:08.750 --> 01:18:11.060 align:middle line:84% formula for dopant defect interaction, 01:18:11.060 --> 01:18:15.080 align:middle line:84% it can explain a lot of action at a distance effects, like, 01:18:15.080 --> 01:18:18.320 align:middle line:84% OED, phosphorus tail, emitter push, things 01:18:18.320 --> 01:18:21.720 align:middle line:84% that people had a hard time explaining for many years. 01:18:21.720 --> 01:18:23.490 align:middle line:84% So that's about all I have to say today. 01:18:23.490 --> 01:18:25.760 align:middle line:84% I know it's a pretty dense lecture. 01:18:25.760 --> 01:18:29.900 align:middle line:84% But we'll finish up Chapter 7 on Thursday. 01:18:29.900 --> 01:18:33.040 align:middle line:84% And Thursday, remember, your homework is due. 01:18:33.040 --> 01:18:38.000 align:middle line:90%