WEBVTT 00:00:00.000 --> 00:00:00.875 align:middle line:90% JUDY HOYT: All right. 00:00:00.875 --> 00:00:04.950 align:middle line:84% We're beginning the lecture on handout number 15. 00:00:04.950 --> 00:00:09.470 align:middle line:84% This is the second lecture on dopant diffusion and profile 00:00:09.470 --> 00:00:10.310 align:middle line:90% measurement. 00:00:10.310 --> 00:00:14.790 align:middle line:84% And we're working on chapter 7 of your text. 00:00:14.790 --> 00:00:17.540 align:middle line:84% So last time, we introduced some of the basics 00:00:17.540 --> 00:00:18.860 align:middle line:90% of dopant diffusion. 00:00:18.860 --> 00:00:22.280 align:middle line:84% We talked about basic definition of sheet resistance, 00:00:22.280 --> 00:00:25.700 align:middle line:84% and the scaling requires from us that we pointed out 00:00:25.700 --> 00:00:28.442 align:middle line:84% there's a conflict between the need for smaller series 00:00:28.442 --> 00:00:29.900 align:middle line:84% resistance and the need to decrease 00:00:29.900 --> 00:00:33.930 align:middle line:84% junction depths to improve control of the channel charge. 00:00:33.930 --> 00:00:35.870 align:middle line:84% We also talked about short channel effects 00:00:35.870 --> 00:00:41.060 align:middle line:84% and how dr decreases as l goes down. 00:00:41.060 --> 00:00:44.560 align:middle line:84% We mentioned the use of predeposition, either in the gas 00:00:44.560 --> 00:00:46.330 align:middle line:90% phase, or by ion implantation. 00:00:46.330 --> 00:00:48.850 align:middle line:84% And then the use of subsequent driving of dopants 00:00:48.850 --> 00:00:51.580 align:middle line:90% to create a doped regions. 00:00:51.580 --> 00:00:53.578 align:middle line:84% We spoke about macroscopic diffusion 00:00:53.578 --> 00:00:55.870 align:middle line:84% from the point of view of Fick's first and second laws, 00:00:55.870 --> 00:00:57.042 align:middle line:90% and gave a few cases. 00:00:57.042 --> 00:00:58.750 align:middle line:84% There were only about three or four cases 00:00:58.750 --> 00:01:01.870 align:middle line:84% where there are simple analytic solutions to the diffusion 00:01:01.870 --> 00:01:03.890 align:middle line:90% equation. 00:01:03.890 --> 00:01:06.530 align:middle line:84% Today, I want to just briefly review those analytic solution 00:01:06.530 --> 00:01:11.390 align:middle line:84% examples, and then talk about a brief introduction 00:01:11.390 --> 00:01:14.580 align:middle line:84% to numerical solutions to the diffusion equation. 00:01:14.580 --> 00:01:17.930 align:middle line:84% Then we'll talk about design of diffuse layers based 00:01:17.930 --> 00:01:19.820 align:middle line:84% upon certain device requirements, 00:01:19.820 --> 00:01:21.530 align:middle line:90% such as the sheet resistance. 00:01:21.530 --> 00:01:25.160 align:middle line:84% And our first deviation from Fickian 00:01:25.160 --> 00:01:28.190 align:middle line:84% diffusion, we'll discuss electric field effects 00:01:28.190 --> 00:01:31.000 align:middle line:90% on dopant diffusion. 00:01:31.000 --> 00:01:35.520 align:middle line:84% So let's go on to slide number 2 and just review from last time 00:01:35.520 --> 00:01:39.750 align:middle line:84% this pictorial view of Fick's second law. 00:01:39.750 --> 00:01:43.110 align:middle line:84% What I'm showing here is a volume element 00:01:43.110 --> 00:01:47.700 align:middle line:84% that is shown by the blue rectangular element. 00:01:47.700 --> 00:01:52.020 align:middle line:84% And there is a flux f sub in going in the left-hand face, 00:01:52.020 --> 00:01:56.920 align:middle line:84% and a flux f sub out going out the right-hand face. 00:01:56.920 --> 00:01:59.970 align:middle line:84% Remember again, our basic definition of flux 00:01:59.970 --> 00:02:04.020 align:middle line:84% is, it's a number of atoms or particles per unit time per unit 00:02:04.020 --> 00:02:05.290 align:middle line:90% area. 00:02:05.290 --> 00:02:09.360 align:middle line:84% So looking at in this cube, we can 00:02:09.360 --> 00:02:13.110 align:middle line:84% calculate the net flow of atoms into this volume, 00:02:13.110 --> 00:02:14.310 align:middle line:90% into that small volume. 00:02:14.310 --> 00:02:19.890 align:middle line:84% As shown in the upper left corner is equal to the area 00:02:19.890 --> 00:02:23.370 align:middle line:84% a of one of those faces times the difference 00:02:23.370 --> 00:02:26.100 align:middle line:90% between f out minus f in. 00:02:26.100 --> 00:02:27.690 align:middle line:90% So that's the net flow of atoms. 00:02:27.690 --> 00:02:30.820 align:middle line:84% And we can write that as a times-- 00:02:30.820 --> 00:02:34.630 align:middle line:84% and just looking at the geometry, it's delta 00:02:34.630 --> 00:02:38.830 align:middle line:84% x times the change in concentration, 00:02:38.830 --> 00:02:42.918 align:middle line:84% delta c, divided by delta time, delta t. 00:02:42.918 --> 00:02:44.710 align:middle line:84% So we can just rearrange this a little bit. 00:02:44.710 --> 00:02:46.270 align:middle line:84% The a's go out, and what we see is 00:02:46.270 --> 00:02:50.530 align:middle line:84% that the first derivative of the flux, delta f over delta x, 00:02:50.530 --> 00:02:56.020 align:middle line:84% is equal to the time change, delta c over delta t. 00:02:56.020 --> 00:02:59.560 align:middle line:84% So mathematically, if we want to write this, 00:02:59.560 --> 00:03:01.510 align:middle line:84% not in terms of deltas or differences, 00:03:01.510 --> 00:03:04.540 align:middle line:84% but in terms of differentials, what 00:03:04.540 --> 00:03:09.640 align:middle line:84% we see is a time rate of change of the concentration in the box, 00:03:09.640 --> 00:03:12.490 align:middle line:84% partial c, partial t is equal to the first derivative 00:03:12.490 --> 00:03:15.880 align:middle line:84% of the flux, partial f, partial x. 00:03:15.880 --> 00:03:18.430 align:middle line:84% And then we can substitute in the equation on the center 00:03:18.430 --> 00:03:19.900 align:middle line:90% of the page for the flux. 00:03:19.900 --> 00:03:21.490 align:middle line:84% We know from Fick's first law that f 00:03:21.490 --> 00:03:24.070 align:middle line:84% is equal to d, partial c, partial x. 00:03:24.070 --> 00:03:30.040 align:middle line:84% So that is, indeed, a way of deriving just bookkeeping 00:03:30.040 --> 00:03:32.720 align:middle line:90% and deriving Fick's second law. 00:03:32.720 --> 00:03:35.150 align:middle line:84% Now, the equation in the center of the page 00:03:35.150 --> 00:03:38.650 align:middle line:84% can be simplified somewhat if we are in the special case 00:03:38.650 --> 00:03:40.400 align:middle line:84% where the diffusivity is a constant, where 00:03:40.400 --> 00:03:42.710 align:middle line:90% it doesn't vary in space. 00:03:42.710 --> 00:03:45.320 align:middle line:84% I can pull the d out of the partial derivative 00:03:45.320 --> 00:03:49.280 align:middle line:84% with respect to x, and we just get partial c, partial t. 00:03:49.280 --> 00:03:52.490 align:middle line:84% So the time rate of change of the concentration 00:03:52.490 --> 00:03:54.350 align:middle line:84% is equal to a constant diffusivity. 00:03:54.350 --> 00:03:57.830 align:middle line:84% It's a number times the second derivative of the concentration 00:03:57.830 --> 00:03:59.530 align:middle line:90% with respect to x. 00:03:59.530 --> 00:04:02.920 align:middle line:84% So that is-- at the bottom of the page would be then 00:04:02.920 --> 00:04:05.140 align:middle line:84% the special case of Fick's second law 00:04:05.140 --> 00:04:08.870 align:middle line:84% when the diffusivity is a constant. 00:04:08.870 --> 00:04:11.230 align:middle line:84% So let's go on to slide number 3. 00:04:11.230 --> 00:04:14.800 align:middle line:84% And these are a couple of cases we solved last time where 00:04:14.800 --> 00:04:18.670 align:middle line:84% we wrote down the solutions last time for two cases which 00:04:18.670 --> 00:04:21.880 align:middle line:84% are commonly encountered and for which analytic solutions are 00:04:21.880 --> 00:04:22.540 align:middle line:90% available. 00:04:22.540 --> 00:04:24.130 align:middle line:84% On the left-hand side, I'm reviewing 00:04:24.130 --> 00:04:26.080 align:middle line:90% the case of a predeposition. 00:04:26.080 --> 00:04:29.810 align:middle line:84% And again, by definition in this case, 00:04:29.810 --> 00:04:34.660 align:middle line:84% we are assuming that there is a supply of dopant, or atoms that 00:04:34.660 --> 00:04:37.840 align:middle line:84% are diffusing, that holds the surface concentration 00:04:37.840 --> 00:04:40.536 align:middle line:90% at a constant value c sub s. 00:04:40.536 --> 00:04:42.020 align:middle line:90% That number is fixed. 00:04:42.020 --> 00:04:46.040 align:middle line:84% And we said in this case that the shape of the profile, c, 00:04:46.040 --> 00:04:48.350 align:middle line:84% as a function of depth x and time 00:04:48.350 --> 00:04:51.110 align:middle line:84% is given by this surface concentration c sub 00:04:51.110 --> 00:04:55.100 align:middle line:84% s times the complementary error function of x over 2 square root 00:04:55.100 --> 00:04:55.860 align:middle line:90% et. 00:04:55.860 --> 00:04:59.600 align:middle line:84% So that is the actual analytic solution to Fick's second law 00:04:59.600 --> 00:05:01.460 align:middle line:90% for a constant diffusivity. 00:05:01.460 --> 00:05:03.860 align:middle line:84% And we also know that the integral 00:05:03.860 --> 00:05:09.350 align:middle line:84% of that function integrated over our space, 00:05:09.350 --> 00:05:12.470 align:middle line:84% or throughout the silicon, has a value q. 00:05:12.470 --> 00:05:14.690 align:middle line:84% That's the dose, so the area under the curve. 00:05:14.690 --> 00:05:17.000 align:middle line:84% And that can be solved for, and it turns out 00:05:17.000 --> 00:05:22.210 align:middle line:84% to be 2 c sub s over square root of pi times square root of dt. 00:05:22.210 --> 00:05:23.960 align:middle line:84% So this is a particular case and, in fact, 00:05:23.960 --> 00:05:26.660 align:middle line:84% showing a plot on the vertical axis 00:05:26.660 --> 00:05:29.900 align:middle line:84% is c over cs as a function of depth. 00:05:29.900 --> 00:05:33.470 align:middle line:84% For several different times, the three 00:05:33.470 --> 00:05:36.050 align:middle line:84% curves here shown where the square root of dt-- 00:05:36.050 --> 00:05:37.760 align:middle line:84% or 2 times the square root of dt is 00:05:37.760 --> 00:05:41.050 align:middle line:84% being varied from 0.1 micron all the way up to 1 micron. 00:05:41.050 --> 00:05:44.400 align:middle line:84% And you get a feel for how the shape of the complementary error 00:05:44.400 --> 00:05:48.360 align:middle line:90% function evolves with time. 00:05:48.360 --> 00:05:49.860 align:middle line:84% In contrast, on the right-hand side, 00:05:49.860 --> 00:05:52.235 align:middle line:84% we're showing the case of a deriving that would typically 00:05:52.235 --> 00:05:53.870 align:middle line:90% be done after Pre-Dep. 00:05:53.870 --> 00:05:57.320 align:middle line:84% And deriving takes place usually under the assumption 00:05:57.320 --> 00:06:01.160 align:middle line:84% of a constant dose q, so the constant area under the curve. 00:06:01.160 --> 00:06:06.090 align:middle line:84% And what's happening as a function of time then is 00:06:06.090 --> 00:06:08.660 align:middle line:84% the shape of the curve is changing, and it's-- 00:06:08.660 --> 00:06:11.630 align:middle line:84% we know the solution is a Gaussian, the solution 00:06:11.630 --> 00:06:14.030 align:middle line:90% to Fick's second law. 00:06:14.030 --> 00:06:16.850 align:middle line:84% And in fact, if you look at this equation as a function of time, 00:06:16.850 --> 00:06:19.460 align:middle line:84% you'll see that the concentration of the surface 00:06:19.460 --> 00:06:21.380 align:middle line:84% is dropping as a function of time 00:06:21.380 --> 00:06:24.410 align:middle line:84% according to 1 over the square root of dt. 00:06:24.410 --> 00:06:27.710 align:middle line:84% And the profile is broadening its width 00:06:27.710 --> 00:06:32.210 align:middle line:84% or its standard deviation is increasing according to dt. 00:06:32.210 --> 00:06:34.340 align:middle line:84% And there are a couple of different cases 00:06:34.340 --> 00:06:36.300 align:middle line:90% shown on the right. 00:06:36.300 --> 00:06:40.760 align:middle line:84% So these solutions are valid under this particular case where 00:06:40.760 --> 00:06:43.940 align:middle line:84% the concentration of a diffusing dopant is less than n sub i. 00:06:43.940 --> 00:06:47.520 align:middle line:84% And that implies in that case that the diffusivity 00:06:47.520 --> 00:06:48.130 align:middle line:90% is a constant. 00:06:48.130 --> 00:06:49.300 align:middle line:90% It doesn't vary in space. 00:06:49.300 --> 00:06:51.930 align:middle line:84% We'll talk in subsequent lectures about the case 00:06:51.930 --> 00:06:56.890 align:middle line:84% when the concentration is quite high, or higher than n sub y. 00:06:56.890 --> 00:06:59.370 align:middle line:84% So let's go on to slide number 4. 00:06:59.370 --> 00:07:01.710 align:middle line:84% And this review is from last time. 00:07:01.710 --> 00:07:03.870 align:middle line:84% What we showed is an Arrhenius plot 00:07:03.870 --> 00:07:05.790 align:middle line:84% of the intrinsic diffusion coefficients 00:07:05.790 --> 00:07:07.980 align:middle line:84% of some common dopants in silicon. 00:07:07.980 --> 00:07:11.670 align:middle line:84% And what we're showing on the left axis on a log scale 00:07:11.670 --> 00:07:15.820 align:middle line:84% is a diffusivity and units of centimeters squared per second. 00:07:15.820 --> 00:07:21.540 align:middle line:84% And the x-axis is 1,000 over temperature in Kelvin. 00:07:21.540 --> 00:07:23.430 align:middle line:84% The lower x-axis, the upper x-axis, 00:07:23.430 --> 00:07:27.490 align:middle line:84% actually has the temperatures actually indicated. 00:07:27.490 --> 00:07:31.180 align:middle line:90% So we are intrinsic diffusivity. 00:07:31.180 --> 00:07:33.030 align:middle line:84% So they apply when in the carrier 00:07:33.030 --> 00:07:37.071 align:middle line:84% concentrate of the concentration of the dopant is less than ni 00:07:37.071 --> 00:07:39.330 align:middle line:90% at the diffusion temperature. 00:07:39.330 --> 00:07:41.250 align:middle line:84% One thing you can notice right off the bat 00:07:41.250 --> 00:07:43.800 align:middle line:84% is that there are some fast diffusers in silicon, 00:07:43.800 --> 00:07:46.470 align:middle line:84% such as boron, phosphorus, and indium. 00:07:46.470 --> 00:07:48.570 align:middle line:90% They're the upper three curves. 00:07:48.570 --> 00:07:51.390 align:middle line:84% And the slower diffusers are arsenic 00:07:51.390 --> 00:07:53.640 align:middle line:90% and antimony and silicon. 00:07:53.640 --> 00:07:55.620 align:middle line:84% It turns out just for practical reasons 00:07:55.620 --> 00:07:57.720 align:middle line:84% because of their high electrical solubility, 00:07:57.720 --> 00:08:01.380 align:middle line:84% that boron is often the p type opener of choice. 00:08:01.380 --> 00:08:02.520 align:middle line:90% It pretty much is. 00:08:02.520 --> 00:08:05.035 align:middle line:84% And arsenic in n type dope as you 00:08:05.035 --> 00:08:06.410 align:middle line:84% can choose arsenic or phosphorus, 00:08:06.410 --> 00:08:09.540 align:middle line:84% but arsenic is usually chosen because of its higher 00:08:09.540 --> 00:08:13.600 align:middle line:84% electrical solubility, and also sometimes its lower diffusion 00:08:13.600 --> 00:08:14.100 align:middle line:90% coefficient. 00:08:14.100 --> 00:08:17.610 align:middle line:90% It's easier to control profiles. 00:08:17.610 --> 00:08:20.240 align:middle line:84% So what-- typically, we're going to have issues 00:08:20.240 --> 00:08:23.630 align:middle line:84% with controlling diffusion in p type regions 00:08:23.630 --> 00:08:26.630 align:middle line:84% because boron is diffusing at any given temperature, 00:08:26.630 --> 00:08:30.410 align:middle line:84% almost an order of magnitude of higher diffusivity 00:08:30.410 --> 00:08:33.270 align:middle line:90% than the case of arsenic. 00:08:33.270 --> 00:08:36.450 align:middle line:84% All right, let's go on to slide number 5. 00:08:36.450 --> 00:08:39.870 align:middle line:84% And here what we are trying to talk about 00:08:39.870 --> 00:08:44.010 align:middle line:84% is the effect of successive diffusions in, say, 00:08:44.010 --> 00:08:49.110 align:middle line:84% a Moss process flow on the doping profiles. 00:08:49.110 --> 00:08:51.400 align:middle line:84% You remember from the first lecture of this course, 00:08:51.400 --> 00:08:52.230 align:middle line:90% we talked-- 00:08:52.230 --> 00:08:55.650 align:middle line:84% we kind of walked through an example of an integrated circuit 00:08:55.650 --> 00:08:57.210 align:middle line:90% fabrication process. 00:08:57.210 --> 00:09:00.000 align:middle line:84% And you know that they involve a large number of different steps 00:09:00.000 --> 00:09:02.040 align:middle line:90% at various temperatures. 00:09:02.040 --> 00:09:05.580 align:middle line:84% We don't just simply do a Pre-Dep and then a drive-in 00:09:05.580 --> 00:09:07.770 align:middle line:90% and then it's done. 00:09:07.770 --> 00:09:11.850 align:middle line:84% That could be oxidations, it could be other high temperature 00:09:11.850 --> 00:09:15.670 align:middle line:84% steps in which the dopants are going to move. 00:09:15.670 --> 00:09:19.860 align:middle line:84% So if you have a whole series of steps, what's the question is, 00:09:19.860 --> 00:09:22.500 align:middle line:84% what is the actual final profile, 00:09:22.500 --> 00:09:24.100 align:middle line:90% and how can we approximate it? 00:09:24.100 --> 00:09:27.690 align:middle line:84% Well, it turns out in the case of Gaussian diffusion 00:09:27.690 --> 00:09:32.492 align:middle line:84% that we can write the total effective d2 product 00:09:32.492 --> 00:09:34.450 align:middle line:84% as being a measure of the thermal budget that's 00:09:34.450 --> 00:09:35.810 align:middle line:90% used in that process. 00:09:35.810 --> 00:09:37.990 align:middle line:84% And for Gaussian diffusion, we can actually 00:09:37.990 --> 00:09:42.550 align:middle line:84% sum up the products of the diffusivity in any given time 00:09:42.550 --> 00:09:46.420 align:middle line:84% step times the length of that time step, d times t. 00:09:46.420 --> 00:09:49.720 align:middle line:84% We can sum up those dt products and add them up, 00:09:49.720 --> 00:09:54.520 align:middle line:84% and end up with an effective dt for the entire process. 00:09:54.520 --> 00:09:57.500 align:middle line:84% Again, assuming that in each case, 00:09:57.500 --> 00:10:02.830 align:middle line:84% the diffusion satisfies the Gaussian assumptions. 00:10:02.830 --> 00:10:05.350 align:middle line:84% What we'll typically find as you go through a whole process 00:10:05.350 --> 00:10:09.140 align:middle line:84% is that some of the processing steps may be negligible. 00:10:09.140 --> 00:10:11.860 align:middle line:84% And that's because the diffusion coefficients in this formula 00:10:11.860 --> 00:10:15.370 align:middle line:84% is exponentially activated, or it varies 00:10:15.370 --> 00:10:17.240 align:middle line:90% with temperature exponential. 00:10:17.240 --> 00:10:19.870 align:middle line:84% So the highest temperature steps in the process 00:10:19.870 --> 00:10:22.037 align:middle line:90% typically dominate. 00:10:22.037 --> 00:10:23.620 align:middle line:84% It's not always the case, for example, 00:10:23.620 --> 00:10:26.095 align:middle line:84% when we talk about transient enhanced diffusion. 00:10:26.095 --> 00:10:27.220 align:middle line:90% We see that's not the case. 00:10:27.220 --> 00:10:30.370 align:middle line:84% But for normal diffusion, you expect the highest temperature 00:10:30.370 --> 00:10:31.820 align:middle line:90% steps to dominate this equation. 00:10:31.820 --> 00:10:33.490 align:middle line:84% And so you can usually zoom in on those 00:10:33.490 --> 00:10:37.150 align:middle line:84% as being the ones that primarily determine the dt product 00:10:37.150 --> 00:10:40.080 align:middle line:84% and, therefore, the shape of the profile. 00:10:40.080 --> 00:10:41.580 align:middle line:84% One thing you have to be aware of 00:10:41.580 --> 00:10:45.270 align:middle line:84% is that this Gaussian solution we've been talking about only 00:10:45.270 --> 00:10:48.120 align:middle line:84% holds if the delta function approximation 00:10:48.120 --> 00:10:49.990 align:middle line:84% for the initial profile is reasonable. 00:10:49.990 --> 00:10:51.610 align:middle line:90% We talked about this last time. 00:10:51.610 --> 00:10:56.250 align:middle line:84% So it holds in the case where the width of the profile 00:10:56.250 --> 00:10:58.620 align:middle line:84% initially when you start is small 00:10:58.620 --> 00:11:02.100 align:middle line:84% compared to the final width, or compared to the total dt. 00:11:02.100 --> 00:11:05.250 align:middle line:84% If that assumption is not valid then, 00:11:05.250 --> 00:11:11.590 align:middle line:84% and we don't have a simple Gaussian solution, necessarily. 00:11:11.590 --> 00:11:14.140 align:middle line:84% OK, let's go on to slide number 6. 00:11:14.140 --> 00:11:20.230 align:middle line:84% And here, we're going to go through some principles 00:11:20.230 --> 00:11:22.760 align:middle line:84% and examples of the design of diffuse layers. 00:11:22.760 --> 00:11:27.370 align:middle line:84% So what we want to do is, as an engineer, a process engineer, 00:11:27.370 --> 00:11:28.960 align:middle line:84% or working with device engineers, 00:11:28.960 --> 00:11:34.570 align:middle line:84% we'll want to typically design a diffusion process, or a Pre-Dep 00:11:34.570 --> 00:11:36.470 align:middle line:84% and diffusion and a deriving process 00:11:36.470 --> 00:11:39.250 align:middle line:84% so that we end up with a certain sheet resistance that's required 00:11:39.250 --> 00:11:44.140 align:middle line:90% electrically by the device. 00:11:44.140 --> 00:11:48.160 align:middle line:84% So that's the goal of this section. 00:11:48.160 --> 00:11:50.650 align:middle line:84% And I'm showing on the left-hand side just 00:11:50.650 --> 00:11:56.140 align:middle line:84% the example of a cube of silicon where the top region is colored 00:11:56.140 --> 00:11:59.740 align:middle line:84% in orange, and that is the little sheet through which we're 00:11:59.740 --> 00:12:03.490 align:middle line:84% going to be passing a current and measuring the voltage. 00:12:03.490 --> 00:12:08.710 align:middle line:84% And this little sheet has a particular geometry. 00:12:08.710 --> 00:12:10.420 align:middle line:84% What we're going to consider is the case 00:12:10.420 --> 00:12:16.120 align:middle line:84% where the length of this resistor 00:12:16.120 --> 00:12:18.160 align:middle line:90% is actually equal to its width. 00:12:18.160 --> 00:12:24.000 align:middle line:84% So it's a square, and it has dimension w by w. 00:12:24.000 --> 00:12:27.060 align:middle line:84% And so we're going to define the sheet 00:12:27.060 --> 00:12:29.280 align:middle line:90% resistance of this junction. 00:12:29.280 --> 00:12:33.872 align:middle line:84% Last time we talked about it as being the resistivity, which 00:12:33.872 --> 00:12:35.580 align:middle line:84% is an intrinsic property of the material, 00:12:35.580 --> 00:12:38.010 align:middle line:84% depending on how it's doped and things, divided 00:12:38.010 --> 00:12:40.380 align:middle line:90% by xj at the junction depth. 00:12:40.380 --> 00:12:43.020 align:middle line:84% We can see xj marked on the left-hand side. 00:12:43.020 --> 00:12:45.690 align:middle line:84% And the units of that sheet resistance 00:12:45.690 --> 00:12:51.570 align:middle line:84% then are given it to be ohms per square. 00:12:51.570 --> 00:12:53.250 align:middle line:84% So the sheet resistance is, indeed, 00:12:53.250 --> 00:12:58.140 align:middle line:84% the resistance r that you would get in a resistor made 00:12:58.140 --> 00:13:00.810 align:middle line:90% of 1 square of this material. 00:13:00.810 --> 00:13:07.950 align:middle line:84% And we can either write it as Roman r sub s, or rho, Greek-- 00:13:07.950 --> 00:13:09.600 align:middle line:90% the Greek letter rho sub s. 00:13:09.600 --> 00:13:13.680 align:middle line:84% And the s indicating it's for a sheet. 00:13:13.680 --> 00:13:16.440 align:middle line:84% The resistivity rho, remember, is given by 1 00:13:16.440 --> 00:13:19.320 align:middle line:84% over a cube with electronic charge times the carrier 00:13:19.320 --> 00:13:21.570 align:middle line:84% concentration n times mobility, which 00:13:21.570 --> 00:13:23.040 align:middle line:90% is in general, a function of n. 00:13:23.040 --> 00:13:26.880 align:middle line:84% And the resistivity has units along centimeter, xj has units 00:13:26.880 --> 00:13:28.437 align:middle line:90% of centimeter, they cancel out. 00:13:28.437 --> 00:13:30.645 align:middle line:84% We end up with a sheet resistance in ohms per square. 00:13:30.645 --> 00:13:33.390 align:middle line:90% 00:13:33.390 --> 00:13:37.470 align:middle line:84% That's all reasonably intuitive assuming 00:13:37.470 --> 00:13:41.850 align:middle line:84% you have a constantly doped-- a constant doped layer. 00:13:41.850 --> 00:13:44.340 align:middle line:84% For if a layer is non-uniformly doped, 00:13:44.340 --> 00:13:50.328 align:middle line:84% you need to integrate the doping profile times the mobility 00:13:50.328 --> 00:13:52.120 align:middle line:84% in order to calculate the sheet resistance. 00:13:52.120 --> 00:13:55.260 align:middle line:84% So the equation shown near the bottom of the slide 00:13:55.260 --> 00:13:58.080 align:middle line:84% says, sheet resistance can be obtained 00:13:58.080 --> 00:14:01.830 align:middle line:84% by 1 over the integral q times the integral from 0 00:14:01.830 --> 00:14:04.170 align:middle line:84% to the junction depth of the doping-- of the carrier 00:14:04.170 --> 00:14:09.990 align:middle line:84% concentration profile, and of x, or the doping profile minus nb. 00:14:09.990 --> 00:14:11.760 align:middle line:84% Or nb is the background concentration 00:14:11.760 --> 00:14:16.890 align:middle line:84% that tells you where you cut off the electron concentration. 00:14:16.890 --> 00:14:20.370 align:middle line:84% That quantity times the mobility, which is, 00:14:20.370 --> 00:14:25.300 align:middle line:84% in general, a function of ndx integrated. 00:14:25.300 --> 00:14:27.000 align:middle line:84% So that's what you would do in the case 00:14:27.000 --> 00:14:31.050 align:middle line:84% of a non-uniformly doped layer where the doping concentration 00:14:31.050 --> 00:14:33.550 align:middle line:90% is varying in thickness. 00:14:33.550 --> 00:14:36.450 align:middle line:84% Now, it turns out this equation you can solve it yourself. 00:14:36.450 --> 00:14:39.180 align:middle line:84% But this equation has already been numerically integrated 00:14:39.180 --> 00:14:43.140 align:middle line:84% by Irvin for a couple of different analytical profiles. 00:14:43.140 --> 00:14:45.180 align:middle line:84% In particular, the case of the Gaussian 00:14:45.180 --> 00:14:47.460 align:middle line:84% and the complementary error function profiles. 00:14:47.460 --> 00:14:50.040 align:middle line:84% And the solutions to those integrals 00:14:50.040 --> 00:14:52.680 align:middle line:84% are given in your text, either in Chapter 7, 00:14:52.680 --> 00:14:56.650 align:middle line:84% or there's also an appendix where the solutions are given. 00:14:56.650 --> 00:15:00.325 align:middle line:84% So let's go on to slide number 7. 00:15:00.325 --> 00:15:01.700 align:middle line:84% And at the top of the slide we're 00:15:01.700 --> 00:15:03.710 align:middle line:84% showing an example of Irvin's curves 00:15:03.710 --> 00:15:10.130 align:middle line:84% for the particular case of p type Gaussian profiles in an n 00:15:10.130 --> 00:15:11.210 align:middle line:90% type background. 00:15:11.210 --> 00:15:16.400 align:middle line:84% And what you see in these curves on the y-axis 00:15:16.400 --> 00:15:17.850 align:middle line:90% is the surface concentration. 00:15:17.850 --> 00:15:22.200 align:middle line:84% So that would be c sub s, and atoms per cubic centimeter. 00:15:22.200 --> 00:15:25.670 align:middle line:84% The x-axis now is effective conductivity, 00:15:25.670 --> 00:15:29.600 align:middle line:84% and it has units of ohm centimeter to the minus 1. 00:15:29.600 --> 00:15:32.068 align:middle line:90% 00:15:32.068 --> 00:15:33.610 align:middle line:84% So let's see if x-axis-- you can just 00:15:33.610 --> 00:15:36.640 align:middle line:84% go to the bottom of the slide, thinking about it, that 00:15:36.640 --> 00:15:39.430 align:middle line:84% is the effective average conductivity, so 00:15:39.430 --> 00:15:41.410 align:middle line:84% the inverse of the average resistivity. 00:15:41.410 --> 00:15:44.110 align:middle line:84% So we can write this as a signal bar 00:15:44.110 --> 00:15:50.920 align:middle line:84% and it's equal to a 1 over the product of rho sub s times xj. 00:15:50.920 --> 00:15:53.875 align:middle line:84% So it's 1 over the sheet resistance times the junction 00:15:53.875 --> 00:15:55.480 align:middle line:90% depth. 00:15:55.480 --> 00:15:59.980 align:middle line:84% So there are three key parameters we need to identify 00:15:59.980 --> 00:16:01.220 align:middle line:90% and using these curves. 00:16:01.220 --> 00:16:02.803 align:middle line:84% There's a surface concentration, which 00:16:02.803 --> 00:16:06.200 align:middle line:90% is the y-axis in the profile. 00:16:06.200 --> 00:16:08.200 align:middle line:84% There is-- again, this is for a particular type. 00:16:08.200 --> 00:16:09.700 align:middle line:90% This is for a Gaussian profile. 00:16:09.700 --> 00:16:13.810 align:middle line:84% There is the sheet resistance, which 00:16:13.810 --> 00:16:16.420 align:middle line:84% is coming into this effective conductivity. 00:16:16.420 --> 00:16:18.350 align:middle line:84% That's really part of the x-axis. 00:16:18.350 --> 00:16:20.200 align:middle line:90% And there's the junction depth. 00:16:20.200 --> 00:16:23.740 align:middle line:84% So the x-axis has sheet resistance and junction depth 00:16:23.740 --> 00:16:26.860 align:middle line:90% built into it. 00:16:26.860 --> 00:16:28.540 align:middle line:84% And now the metallurgical junction 00:16:28.540 --> 00:16:31.340 align:middle line:84% is the place, the point in depth, 00:16:31.340 --> 00:16:35.660 align:middle line:84% where the chemical concentration in the diffuse region 00:16:35.660 --> 00:16:38.960 align:middle line:84% of the fused dopant equals the background concentration cb. 00:16:38.960 --> 00:16:42.500 align:middle line:84% And we'll do an example shortly after this where you can 00:16:42.500 --> 00:16:44.258 align:middle line:90% see how these curves are used. 00:16:44.258 --> 00:16:45.800 align:middle line:84% But each curve in the different color 00:16:45.800 --> 00:16:49.020 align:middle line:84% is for a different background concentration. 00:16:49.020 --> 00:16:52.070 align:middle line:84% So this, basically, tells you that you can take these three 00:16:52.070 --> 00:16:56.270 align:middle line:84% key parameters, and they have a unique relationship 00:16:56.270 --> 00:16:58.203 align:middle line:84% among themselves for the particular case 00:16:58.203 --> 00:16:59.870 align:middle line:84% where the shape of the profile is known. 00:16:59.870 --> 00:17:03.080 align:middle line:90% In this case, it's Gaussian. 00:17:03.080 --> 00:17:05.390 align:middle line:84% So in fact, let's go on to slide number 8. 00:17:05.390 --> 00:17:06.619 align:middle line:90% We're going to walk through-- 00:17:06.619 --> 00:17:08.960 align:middle line:84% the best way to understand these Irvin's curves 00:17:08.960 --> 00:17:11.760 align:middle line:90% is to go through an example. 00:17:11.760 --> 00:17:14.150 align:middle line:84% And what we're asked to do in this example 00:17:14.150 --> 00:17:16.280 align:middle line:84% is to calculate the deriving conditions, which 00:17:16.280 --> 00:17:23.660 align:middle line:84% means the temperature, and the time to produce a CMOS p-well. 00:17:23.660 --> 00:17:26.270 align:middle line:84% So it's a p type, a legion, and it's 00:17:26.270 --> 00:17:28.380 align:middle line:84% called a p-well of a certain thickness. 00:17:28.380 --> 00:17:32.510 align:middle line:84% And what we want to do, we're told 00:17:32.510 --> 00:17:35.480 align:middle line:84% that the constraint is that the sheet resistance of that p-well 00:17:35.480 --> 00:17:39.540 align:middle line:84% shouldn't be 900 ohms per square. 00:17:39.540 --> 00:17:40.200 align:middle line:90% OK. 00:17:40.200 --> 00:17:41.880 align:middle line:84% And it has to have a certain junction 00:17:41.880 --> 00:17:44.910 align:middle line:84% depth, which is shown schematically in the figure xj 00:17:44.910 --> 00:17:46.780 align:middle line:90% to be 3 microns. 00:17:46.780 --> 00:17:49.950 align:middle line:84% And that xj, again, is the point where the boron concentration 00:17:49.950 --> 00:17:53.550 align:middle line:84% then will reach the background concentration, which 00:17:53.550 --> 00:17:59.140 align:middle line:84% is given by 10 to the 15th per cubic centimeter. 00:17:59.140 --> 00:18:04.140 align:middle line:84% So again, we're informing a p-well by ion and planning bond 00:18:04.140 --> 00:18:07.440 align:middle line:90% into the region, onto the wafer. 00:18:07.440 --> 00:18:09.970 align:middle line:84% And the region on the left is blocked so we're just 00:18:09.970 --> 00:18:11.100 align:middle line:90% putting it in one region. 00:18:11.100 --> 00:18:15.390 align:middle line:84% And typically, you would implant boron at a certain dose, 00:18:15.390 --> 00:18:17.910 align:middle line:90% say, 10 to the 13th, and energy. 00:18:17.910 --> 00:18:19.822 align:middle line:84% And then we want to drive that in. 00:18:19.822 --> 00:18:21.780 align:middle line:84% And we're interested in the driving conditions, 00:18:21.780 --> 00:18:23.340 align:middle line:90% what temperature. 00:18:23.340 --> 00:18:25.890 align:middle line:84% In time, what we need to get it to go that far 00:18:25.890 --> 00:18:31.090 align:middle line:84% and to have a particular sheet resistance. 00:18:31.090 --> 00:18:33.170 align:middle line:84% And we're going to use Irvin's curves to do this. 00:18:33.170 --> 00:18:37.540 align:middle line:90% So let's look on slide number 9. 00:18:37.540 --> 00:18:40.237 align:middle line:84% So these are the constraints again repeated. 00:18:40.237 --> 00:18:42.070 align:middle line:84% The sheet resistance is 900 ohms per square. 00:18:42.070 --> 00:18:44.950 align:middle line:84% The junction depth has to be 3 microns, and the background 00:18:44.950 --> 00:18:49.330 align:middle line:84% concentration, either nb or a substrate concentration, 00:18:49.330 --> 00:18:52.810 align:middle line:90% maybe nbc is 10 to the 15th. 00:18:52.810 --> 00:18:56.950 align:middle line:84% So given the requirements that we have been told, 00:18:56.950 --> 00:18:59.860 align:middle line:84% I can actually immediately find the x-axis on the Irvine curve 00:18:59.860 --> 00:19:01.720 align:middle line:84% with the average conductivity layer. 00:19:01.720 --> 00:19:05.470 align:middle line:84% It's just 1 over the sheet resistance times xj. 00:19:05.470 --> 00:19:07.540 align:middle line:84% So the average conductivity calculates out 00:19:07.540 --> 00:19:12.470 align:middle line:84% to be 3.7 ohms centimeter to minus 1. 00:19:12.470 --> 00:19:15.350 align:middle line:84% We can now from Irvin's curve, we can obtain, 00:19:15.350 --> 00:19:17.355 align:middle line:84% assuming this is the Gaussian, that's 00:19:17.355 --> 00:19:20.030 align:middle line:84% an assumption we need to make at this point, which we'll show 00:19:20.030 --> 00:19:23.000 align:middle line:84% is reasonable later, we can then obtain the surface 00:19:23.000 --> 00:19:24.240 align:middle line:90% concentration. 00:19:24.240 --> 00:19:27.350 align:middle line:84% So in order for me knowing the shape of the Gaussian, 00:19:27.350 --> 00:19:29.480 align:middle line:84% basically, Irvin's intervals tells us 00:19:29.480 --> 00:19:32.690 align:middle line:84% that the surface concentration has to be about 4 times 10 00:19:32.690 --> 00:19:35.900 align:middle line:84% to the 17th per cubic centimeter in order 00:19:35.900 --> 00:19:38.510 align:middle line:84% to achieve an average conductivity that 00:19:38.510 --> 00:19:42.290 align:middle line:84% has been specified as 3.7, assuming 00:19:42.290 --> 00:19:43.920 align:middle line:90% the junction depth 3 microns. 00:19:43.920 --> 00:19:47.240 align:middle line:84% So that's-- and if you can-- we want to see how we actually do 00:19:47.240 --> 00:19:50.030 align:middle line:84% that, we can go on to slide number 10, 00:19:50.030 --> 00:19:54.110 align:middle line:84% and this actually shows how the curves were used. 00:19:54.110 --> 00:19:56.660 align:middle line:84% Again, this is for a p type Gaussian profile. 00:19:56.660 --> 00:20:00.200 align:middle line:84% And all we've done is on the x-axis 00:20:00.200 --> 00:20:03.590 align:middle line:84% picked out-- it's a little hard because this is a long log plot. 00:20:03.590 --> 00:20:08.150 align:middle line:84% And the x-axis picked out the average conductivity to be 3.7. 00:20:08.150 --> 00:20:12.950 align:middle line:84% And that's a little tricky to find that, but you can. 00:20:12.950 --> 00:20:15.680 align:middle line:84% And then we see-- we read off the y-axis 00:20:15.680 --> 00:20:18.830 align:middle line:84% as shown by that horizontal line service 00:20:18.830 --> 00:20:21.290 align:middle line:84% concentration about 4 times 10 to the 17th. 00:20:21.290 --> 00:20:23.960 align:middle line:84% And again, we have multiple curves we can choose. 00:20:23.960 --> 00:20:26.870 align:middle line:84% I choose to use the orange curve, which 00:20:26.870 --> 00:20:29.930 align:middle line:84% is for the case of a background concentration, in this example, 00:20:29.930 --> 00:20:33.140 align:middle line:90% of 10 to the 15th. 00:20:33.140 --> 00:20:34.820 align:middle line:84% If the background concentration, say, 00:20:34.820 --> 00:20:39.620 align:middle line:84% had been a little bit higher, say, 10 to the 16th, or 10 00:20:39.620 --> 00:20:42.860 align:middle line:84% to the 17th, what you see would have happened 00:20:42.860 --> 00:20:46.160 align:middle line:84% is that the surface concentration 00:20:46.160 --> 00:20:49.100 align:middle line:84% would have actually moved down a little bit on the y-axis. 00:20:49.100 --> 00:20:52.970 align:middle line:84% So it would be intersect a slightly different curve. 00:20:52.970 --> 00:20:58.720 align:middle line:84% So that's how we come up with the surface concentration. 00:20:58.720 --> 00:21:01.680 align:middle line:90% So let's go on to slide 11. 00:21:01.680 --> 00:21:04.020 align:middle line:84% And thinking about that surface concentration, 00:21:04.020 --> 00:21:07.750 align:middle line:84% c sub s, we know it's well below the solid solubility of boron 00:21:07.750 --> 00:21:08.250 align:middle line:90% and silicon. 00:21:08.250 --> 00:21:10.290 align:middle line:90% It's only 4 10 to the 17th. 00:21:10.290 --> 00:21:14.032 align:middle line:84% And it's probably likely below the value of n sub i 00:21:14.032 --> 00:21:15.240 align:middle line:90% at the diffusion temperature. 00:21:15.240 --> 00:21:17.490 align:middle line:84% Again, at 1,000 degrees, which is probably 00:21:17.490 --> 00:21:20.340 align:middle line:84% somewhere near or at a minimum temperature 00:21:20.340 --> 00:21:23.640 align:middle line:84% you'll be using to drive this in, n sub i is about mid 10 00:21:23.640 --> 00:21:24.900 align:middle line:90% to the 18th. 00:21:24.900 --> 00:21:27.430 align:middle line:84% And so mid 10 to the 17th is well below that. 00:21:27.430 --> 00:21:30.992 align:middle line:84% So it's probably valid then to say 00:21:30.992 --> 00:21:32.200 align:middle line:90% that the profile is Gaussian. 00:21:32.200 --> 00:21:37.660 align:middle line:84% It's probably valid to say that it is a constant diffusivity d. 00:21:37.660 --> 00:21:41.590 align:middle line:84% And, therefore, we can assume this Gaussian type of solution 00:21:41.590 --> 00:21:42.640 align:middle line:90% because it's a deriving. 00:21:42.640 --> 00:21:45.820 align:middle line:84% So we can write down by inspection 00:21:45.820 --> 00:21:48.190 align:middle line:84% the equation in the center of the slide, which 00:21:48.190 --> 00:21:49.315 align:middle line:90% is the Gaussian equation. 00:21:49.315 --> 00:21:52.270 align:middle line:90% 00:21:52.270 --> 00:21:58.420 align:middle line:84% And then we know that at the point where 00:21:58.420 --> 00:22:03.490 align:middle line:84% x equals xj, the carrier concentration, or the doping 00:22:03.490 --> 00:22:07.573 align:middle line:84% concentration, c of the boron reaches that of the background. 00:22:07.573 --> 00:22:08.740 align:middle line:90% That's the definition of xj. 00:22:08.740 --> 00:22:10.490 align:middle line:84% So if you look in the middle of the slide, 00:22:10.490 --> 00:22:13.030 align:middle line:84% we're saying the background doping-- 00:22:13.030 --> 00:22:15.970 align:middle line:84% background concentration cbc is equal to the surface 00:22:15.970 --> 00:22:17.990 align:middle line:84% concentration, c sub s, which we just located, 00:22:17.990 --> 00:22:21.340 align:middle line:84% which is found times exponential to the minus xj squared 00:22:21.340 --> 00:22:22.120 align:middle line:90% over 4 dt. 00:22:22.120 --> 00:22:24.460 align:middle line:90% That's the definition of xj. 00:22:24.460 --> 00:22:27.037 align:middle line:84% So we know xj in this equation we just found, c sub s. 00:22:27.037 --> 00:22:28.870 align:middle line:84% We know the background doping concentration. 00:22:28.870 --> 00:22:30.340 align:middle line:84% The only thing we don't know is dt. 00:22:30.340 --> 00:22:32.320 align:middle line:84% So we can invert that equation and solve 00:22:32.320 --> 00:22:38.070 align:middle line:84% for dt, plug in all the numbers and you get a dt product. 00:22:38.070 --> 00:22:43.370 align:middle line:84% In order to obtain this Gaussian of 3.7 times 10 to the minus 9 00:22:43.370 --> 00:22:45.330 align:middle line:90% centimeters squared. 00:22:45.330 --> 00:22:47.040 align:middle line:84% That's the dt product, the so-called-- 00:22:47.040 --> 00:22:49.710 align:middle line:84% if you would like to use the term thermal budget 00:22:49.710 --> 00:22:52.060 align:middle line:90% for the deriving process. 00:22:52.060 --> 00:22:53.440 align:middle line:90% So that gives us an idea. 00:22:53.440 --> 00:22:55.590 align:middle line:84% And we don't know what d is, or what t is, 00:22:55.590 --> 00:22:59.650 align:middle line:84% but we're going to design a certain temperature in time. 00:22:59.650 --> 00:23:02.490 align:middle line:84% So we constrain to the fact that the dt product 00:23:02.490 --> 00:23:05.180 align:middle line:90% will be this number. 00:23:05.180 --> 00:23:07.870 align:middle line:84% So let's go on to slide number 12. 00:23:07.870 --> 00:23:09.730 align:middle line:84% And again, I'm repeating the dt product 00:23:09.730 --> 00:23:14.020 align:middle line:84% from the top, 3.9 times 10 to the minus 9, 3.7. 00:23:14.020 --> 00:23:17.407 align:middle line:84% Let's just assume a temperature and see what happens, 00:23:17.407 --> 00:23:18.990 align:middle line:84% see what time is associated with that. 00:23:18.990 --> 00:23:20.657 align:middle line:84% And if it's a reasonable amount of time. 00:23:20.657 --> 00:23:23.260 align:middle line:84% So let's say we know we're diffusing pretty deep 00:23:23.260 --> 00:23:24.860 align:middle line:90% down to 3 microns. 00:23:24.860 --> 00:23:26.620 align:middle line:84% So let's just say we assume we drive in 00:23:26.620 --> 00:23:29.060 align:middle line:90% at 1100 degrees centigrade. 00:23:29.060 --> 00:23:32.440 align:middle line:84% And then you can look up from your plots, 00:23:32.440 --> 00:23:35.245 align:middle line:84% or you can calculate diffusivity of boron at 1100, 00:23:35.245 --> 00:23:39.130 align:middle line:84% it's about 1.5 times 10 to the minus 13 centimeters squared 00:23:39.130 --> 00:23:40.250 align:middle line:90% per second. 00:23:40.250 --> 00:23:43.150 align:middle line:84% And so then you can solve for time, the drive in time, 00:23:43.150 --> 00:23:45.260 align:middle line:84% calculate that out, you get about 6.7 hours. 00:23:45.260 --> 00:23:46.510 align:middle line:90% That's plenty long. 00:23:46.510 --> 00:23:49.640 align:middle line:84% That tells you probably from a practical point of view, 00:23:49.640 --> 00:23:52.720 align:middle line:84% you would not want to do this drive in any lower temperature 00:23:52.720 --> 00:23:56.340 align:middle line:84% because it would just take too darn long. 00:23:56.340 --> 00:23:56.840 align:middle line:90% OK. 00:23:56.840 --> 00:24:00.980 align:middle line:84% So now one thing we'd like to calculate, 00:24:00.980 --> 00:24:03.740 align:middle line:84% we have figured out the surface concentration 00:24:03.740 --> 00:24:05.160 align:middle line:90% from Irvin's curves. 00:24:05.160 --> 00:24:07.310 align:middle line:90% And we have the dt product. 00:24:07.310 --> 00:24:10.670 align:middle line:84% It will be interesting to calculate the initial dose, 00:24:10.670 --> 00:24:13.475 align:middle line:84% or the dose, the area under the curve, for the Gaussian profile. 00:24:13.475 --> 00:24:14.600 align:middle line:90% We have everything we need. 00:24:14.600 --> 00:24:18.440 align:middle line:84% Again, we know the dose q is the surface concentration c sub s 00:24:18.440 --> 00:24:20.270 align:middle line:90% times the square root of pi dt. 00:24:20.270 --> 00:24:22.310 align:middle line:84% And we have all those quantities. 00:24:22.310 --> 00:24:25.610 align:middle line:84% We can plug them in and we get q equal to 4.3 times 10 00:24:25.610 --> 00:24:27.230 align:middle line:90% to the 13th. 00:24:27.230 --> 00:24:30.470 align:middle line:84% Because the integral under the Gaussian. 00:24:30.470 --> 00:24:34.730 align:middle line:84% It's a relatively low dose compared to our implant range. 00:24:34.730 --> 00:24:36.950 align:middle line:90% So it can be implanted. 00:24:36.950 --> 00:24:38.810 align:middle line:84% So this dose can easily be implanted 00:24:38.810 --> 00:24:43.250 align:middle line:84% into a very narrow layer close to the surface, which 00:24:43.250 --> 00:24:46.430 align:middle line:84% justifies implicitly that narrow layer might only 00:24:46.430 --> 00:24:48.530 align:middle line:90% be 1,000 angstroms deep. 00:24:48.530 --> 00:24:51.170 align:middle line:84% And that justifies our implicit assumption 00:24:51.170 --> 00:24:55.160 align:middle line:84% in using a Gaussian profile that the initial distribution would 00:24:55.160 --> 00:24:57.820 align:middle line:84% be a delta function, maybe only again, 00:24:57.820 --> 00:24:59.070 align:middle line:90% a few thousand angstroms deep. 00:24:59.070 --> 00:25:01.290 align:middle line:84% And you're going to a final junction 00:25:01.290 --> 00:25:04.740 align:middle line:90% depth that is 3 microns. 00:25:04.740 --> 00:25:10.920 align:middle line:84% And certainly, that seems justified 00:25:10.920 --> 00:25:14.310 align:middle line:90% using the Gaussian profile. 00:25:14.310 --> 00:25:16.980 align:middle line:84% OK, let's go on to slide number 13. 00:25:16.980 --> 00:25:19.860 align:middle line:84% And now, that was the case we just talked about something 00:25:19.860 --> 00:25:21.480 align:middle line:90% we had implanted, the dose. 00:25:21.480 --> 00:25:24.420 align:middle line:84% Let's just take the case where if we used 00:25:24.420 --> 00:25:32.310 align:middle line:84% a gas phase Pre-Dep step as an example to get the boron 00:25:32.310 --> 00:25:34.570 align:middle line:90% initially into the wafer. 00:25:34.570 --> 00:25:36.780 align:middle line:90% And so that was done at 9:50. 00:25:36.780 --> 00:25:40.110 align:middle line:84% You can then use the charts in your book, or in the handouts, 00:25:40.110 --> 00:25:42.840 align:middle line:84% to look up the solid solubility of boron at 9:50, 00:25:42.840 --> 00:25:47.188 align:middle line:84% and you'll see it's pretty high, it's 2.5 10 to the 20. 00:25:47.188 --> 00:25:49.730 align:middle line:84% And at that temperature, we can also look up the diffusivity. 00:25:49.730 --> 00:25:52.430 align:middle line:90% It's about 4 10 to the minus 15. 00:25:52.430 --> 00:25:56.300 align:middle line:84% So you can solve for the dose in the complementary error function 00:25:56.300 --> 00:25:57.680 align:middle line:90% profile q. 00:25:57.680 --> 00:26:00.575 align:middle line:84% You have everything you need in that equation 2 c 00:26:00.575 --> 00:26:04.130 align:middle line:84% sub s over the square root of pi square root of dt. 00:26:04.130 --> 00:26:06.030 align:middle line:84% And so the time that you would require 00:26:06.030 --> 00:26:10.670 align:middle line:84% to deposit that dose into the wafer t of the Pre-Dep 00:26:10.670 --> 00:26:14.630 align:middle line:84% can be solved and you find about 5.5 seconds. 00:26:14.630 --> 00:26:21.470 align:middle line:84% So you now see that the dt product of the Pre-Dep, which 00:26:21.470 --> 00:26:24.530 align:middle line:84% you can calculate, it's 5.5 seconds times the diffusivity 00:26:24.530 --> 00:26:28.130 align:middle line:84% at 9:50, which we see from above is 4 10 to the minus 15th. 00:26:28.130 --> 00:26:32.370 align:middle line:84% So that dt Pre-Dep is about 2 times 10 to the minus 14th. 00:26:32.370 --> 00:26:34.620 align:middle line:84% And comparing that to the dt of the drive in, which we 00:26:34.620 --> 00:26:37.160 align:middle line:84% solved for earlier, which is about 1 and 1/2 times 10 00:26:37.160 --> 00:26:37.910 align:middle line:90% to the minus 13th. 00:26:37.910 --> 00:26:41.540 align:middle line:84% So in an order of magnitude difference there. 00:26:41.540 --> 00:26:45.350 align:middle line:84% Again, this tells us that the assumption of Gaussian diffusion 00:26:45.350 --> 00:26:46.970 align:middle line:90% is justified, whether-- 00:26:46.970 --> 00:26:49.070 align:middle line:84% even if we've done a solid phase Pre-Dep. 00:26:49.070 --> 00:26:53.660 align:middle line:90% 00:26:53.660 --> 00:26:56.520 align:middle line:84% OK, so that's the end of slide 13. 00:26:56.520 --> 00:26:59.870 align:middle line:84% That certainly gives you an idea of how these Irvin curves can 00:26:59.870 --> 00:27:02.240 align:middle line:90% be used to design a process. 00:27:02.240 --> 00:27:06.140 align:middle line:84% The alternative you could use is to read off. 00:27:06.140 --> 00:27:09.518 align:middle line:84% If you go back to just momentarily slide 10, 00:27:09.518 --> 00:27:11.810 align:middle line:84% you might-- the problem might be specified differently. 00:27:11.810 --> 00:27:18.290 align:middle line:84% You might be given some kind of surface concentration, 00:27:18.290 --> 00:27:21.020 align:middle line:84% and background concentration, and have 00:27:21.020 --> 00:27:26.090 align:middle line:84% to back out what the effect of conductivity of the layer 00:27:26.090 --> 00:27:26.780 align:middle line:90% might be. 00:27:26.780 --> 00:27:31.500 align:middle line:84% So different ways of getting those curves. 00:27:31.500 --> 00:27:34.640 align:middle line:84% So let's go on now to slide number 14. 00:27:34.640 --> 00:27:38.120 align:middle line:84% And I want to shift gears a little bit because we 00:27:38.120 --> 00:27:41.060 align:middle line:84% have more or less covered a few cases 00:27:41.060 --> 00:27:43.910 align:middle line:84% for which the analytic solutions can be obtained. 00:27:43.910 --> 00:27:46.190 align:middle line:84% And those are important cases but, primarily, 00:27:46.190 --> 00:27:48.080 align:middle line:84% when the dopant concentration is low, 00:27:48.080 --> 00:27:54.190 align:middle line:84% or satisfying certain special conditions during Pre-Deps, 00:27:54.190 --> 00:27:57.070 align:middle line:84% it'll turn out there's most cases we cannot solve 00:27:57.070 --> 00:27:58.270 align:middle line:90% analytically. 00:27:58.270 --> 00:28:01.870 align:middle line:84% So these solutions have to be obtained by using computers 00:28:01.870 --> 00:28:03.340 align:middle line:90% using numerical methods. 00:28:03.340 --> 00:28:08.290 align:middle line:84% And so I wanted to introduce you in the next few slides to some 00:28:08.290 --> 00:28:11.500 align:middle line:84% of the simplest types of numerical methods 00:28:11.500 --> 00:28:14.650 align:middle line:84% so you have an idea, or a feel, for how you yourself 00:28:14.650 --> 00:28:16.660 align:middle line:84% could implement these numerical techniques, 00:28:16.660 --> 00:28:17.800 align:middle line:90% if you need to do it. 00:28:17.800 --> 00:28:23.690 align:middle line:84% And there are textbooks on this topic, in particular, 00:28:23.690 --> 00:28:25.750 align:middle line:84% if you want to look at more detailed information 00:28:25.750 --> 00:28:28.210 align:middle line:84% on numerical solutions of diffusion equation. 00:28:28.210 --> 00:28:31.420 align:middle line:84% There's a book by J Crank called, 00:28:31.420 --> 00:28:33.070 align:middle line:84% The mathematics of diffusion, has 00:28:33.070 --> 00:28:37.745 align:middle line:84% a chapter on numerical methods for this particular equation. 00:28:37.745 --> 00:28:39.370 align:middle line:84% The nice thing about numerical methods, 00:28:39.370 --> 00:28:41.260 align:middle line:84% they can be used to solve arbitrary 00:28:41.260 --> 00:28:42.490 align:middle line:90% initial starting profiles. 00:28:42.490 --> 00:28:45.460 align:middle line:84% So you don't have to make any detailed assumptions-- arbitrary 00:28:45.460 --> 00:28:47.110 align:middle line:84% boundary conditions, and in the case 00:28:47.110 --> 00:28:49.700 align:middle line:84% where the diffusivity varies in space. 00:28:49.700 --> 00:28:52.310 align:middle line:84% So if you have a concentration dependent diffusivity, 00:28:52.310 --> 00:28:55.750 align:middle line:84% which will be the case we'll talk about in later 00:28:55.750 --> 00:29:00.010 align:middle line:84% lectures when the dopant concentration is quite high 00:29:00.010 --> 00:29:03.140 align:middle line:84% and you can use numerical methods. 00:29:03.140 --> 00:29:06.190 align:middle line:84% So what I'm picturing here on slide number 14 00:29:06.190 --> 00:29:10.930 align:middle line:84% is a very schematic picture of atoms moving around 00:29:10.930 --> 00:29:12.760 align:middle line:90% between planes in a lattice. 00:29:12.760 --> 00:29:16.270 align:middle line:84% And so let's use this picture to get some physical insight 00:29:16.270 --> 00:29:18.080 align:middle line:90% into the diffusion process. 00:29:18.080 --> 00:29:22.390 align:middle line:84% So each plane here is indicated by a vertical bar. 00:29:22.390 --> 00:29:24.820 align:middle line:90% And 0, 1, 2, 3, 4. 00:29:24.820 --> 00:29:27.550 align:middle line:84% And the distance between these planes 00:29:27.550 --> 00:29:31.960 align:middle line:84% is given by the distance delta x as units of length. 00:29:31.960 --> 00:29:36.310 align:middle line:84% And in sub i here, so n0, n1, and 2 00:29:36.310 --> 00:29:40.780 align:middle line:84% is the planar atomic density of the i-th point. 00:29:40.780 --> 00:29:45.760 align:middle line:84% So it's the number of atoms per square centimeter in a given 00:29:45.760 --> 00:29:46.840 align:middle line:90% plane. 00:29:46.840 --> 00:29:52.300 align:middle line:84% And c sub i has a volume concentration, 00:29:52.300 --> 00:29:54.752 align:middle line:84% and it's the average volume concentration at any point. 00:29:54.752 --> 00:29:56.960 align:middle line:84% So you can calculate and you can get between the two. 00:29:56.960 --> 00:29:59.270 align:middle line:84% c sub i is just n sub i over delta x. 00:29:59.270 --> 00:30:03.430 align:middle line:84% Or n sub i is just c sub i times delta x. 00:30:03.430 --> 00:30:06.070 align:middle line:90% So let's go on to slide 15. 00:30:06.070 --> 00:30:08.360 align:middle line:84% At the top, I've just reproduced that picture. 00:30:08.360 --> 00:30:11.170 align:middle line:84% It's only a smaller version of it. 00:30:11.170 --> 00:30:13.660 align:middle line:84% And just reminding ourselves of the relationship 00:30:13.660 --> 00:30:16.222 align:middle line:90% between ci and ni. 00:30:16.222 --> 00:30:17.890 align:middle line:84% Now, as we're looking at this lattice, 00:30:17.890 --> 00:30:21.850 align:middle line:84% we know that the atoms are relatively fixed in the lattice. 00:30:21.850 --> 00:30:26.710 align:middle line:84% But they do vibrate about their average position in a plane, 00:30:26.710 --> 00:30:28.270 align:middle line:90% where average position. 00:30:28.270 --> 00:30:30.010 align:middle line:84% And they're vibrating at some frequency, 00:30:30.010 --> 00:30:33.550 align:middle line:84% b sub d, which is called the Debye frequency. 00:30:33.550 --> 00:30:37.810 align:middle line:84% And that's-- that number is typically on the order of about 00:30:37.810 --> 00:30:39.647 align:middle line:90% 10 to the 13th per second. 00:30:39.647 --> 00:30:40.980 align:middle line:90% So it's a pretty high frequency. 00:30:40.980 --> 00:30:44.380 align:middle line:84% You're vibrating about their average lattice positions 00:30:44.380 --> 00:30:46.610 align:middle line:90% at a finite temperature. 00:30:46.610 --> 00:30:48.820 align:middle line:84% Now sometimes and during the vibration, 00:30:48.820 --> 00:30:52.310 align:middle line:84% an atom will actually be able to surmount the energy barrier that 00:30:52.310 --> 00:30:55.160 align:middle line:84% exists in going from one lattice position to the next, 00:30:55.160 --> 00:30:56.760 align:middle line:90% or one plane to the next. 00:30:56.760 --> 00:31:00.470 align:middle line:84% So it'll actually be able to hop to an adjacent plane. 00:31:00.470 --> 00:31:03.710 align:middle line:84% And there's an energy barrier, which we'll call an e sub b, 00:31:03.710 --> 00:31:06.110 align:middle line:84% to get from one plane to the next. 00:31:06.110 --> 00:31:11.090 align:middle line:84% And this hopping frequency we call 00:31:11.090 --> 00:31:14.300 align:middle line:84% b sub b, which is just a Debye frequency, 00:31:14.300 --> 00:31:17.000 align:middle line:84% vd-- that's the vibration frequency-- times the Boltzmann 00:31:17.000 --> 00:31:22.985 align:middle line:84% factor exponential, the minus eb is the energy barrier over kt. 00:31:22.985 --> 00:31:26.810 align:middle line:84% So this v sub b, small v sub b, gives us 00:31:26.810 --> 00:31:29.690 align:middle line:84% an idea of the frequency of hopping from atoms 00:31:29.690 --> 00:31:31.500 align:middle line:90% from one plane to the next. 00:31:31.500 --> 00:31:33.230 align:middle line:84% And what we assume in this derivation 00:31:33.230 --> 00:31:35.000 align:middle line:84% is that there's equal probability 00:31:35.000 --> 00:31:39.750 align:middle line:84% of jumping right or left because this is a random process. 00:31:39.750 --> 00:31:42.110 align:middle line:84% So if we look at any given plane, 00:31:42.110 --> 00:31:44.930 align:middle line:84% we can say the number of atoms jumping 00:31:44.930 --> 00:31:47.570 align:middle line:84% to the right of that plane per unit time 00:31:47.570 --> 00:31:53.012 align:middle line:84% is just vb over 2 times the number of atoms in that plane n. 00:31:53.012 --> 00:31:55.950 align:middle line:84% Path 1 go right, number jumping left, per unit time 00:31:55.950 --> 00:31:59.400 align:middle line:84% is just bv over 2 times the number in the plane n. 00:31:59.400 --> 00:32:02.380 align:middle line:90% Those two are equal. 00:32:02.380 --> 00:32:04.990 align:middle line:84% Now, if we do the bookkeeping and say, 00:32:04.990 --> 00:32:07.570 align:middle line:84% look at the number of atoms crossing a plane, 00:32:07.570 --> 00:32:10.990 align:middle line:84% the particular plane at i equals 2 in the above diagram. 00:32:10.990 --> 00:32:14.670 align:middle line:84% So the plane label 2, the number of atoms crossing 00:32:14.670 --> 00:32:16.080 align:middle line:90% that plane per unit time. 00:32:16.080 --> 00:32:20.630 align:middle line:84% Well, we can just write that as a flux. 00:32:20.630 --> 00:32:22.480 align:middle line:84% And f is an equation on the bottom. 00:32:22.480 --> 00:32:26.230 align:middle line:84% Well, that's just going to be the number of atoms, 00:32:26.230 --> 00:32:32.740 align:middle line:84% basically, per unit time jumping to the right minus those jumping 00:32:32.740 --> 00:32:34.150 align:middle line:90% to the left. 00:32:34.150 --> 00:32:41.950 align:middle line:84% So you write that as v minus vb over 2, n2 minus n1. 00:32:41.950 --> 00:32:47.350 align:middle line:84% Again, assuming equal probability going either way. 00:32:47.350 --> 00:32:49.420 align:middle line:84% And then you can convert the difference 00:32:49.420 --> 00:32:53.890 align:middle line:84% between atomic density and number 00:32:53.890 --> 00:32:57.040 align:middle line:84% of atoms per cubic centimeter just by multiplying, using 00:32:57.040 --> 00:33:01.100 align:middle line:84% the definition of n is equal to delta x times the concentration. 00:33:01.100 --> 00:33:04.840 align:middle line:84% You have then expression for this flux 00:33:04.840 --> 00:33:07.900 align:middle line:90% crossing plane i equals 2. 00:33:07.900 --> 00:33:10.990 align:middle line:84% And then you multiply and divide by delta x and you end up with 00:33:10.990 --> 00:33:13.400 align:middle line:84% this equation that this flux is equal to-- looking 00:33:13.400 --> 00:33:14.650 align:middle line:90% in the right-hand side-- 00:33:14.650 --> 00:33:21.310 align:middle line:84% minus bb over 2 delta x squared times the delta p over delta 00:33:21.310 --> 00:33:25.250 align:middle line:84% x, something that's starting to look like a slope. 00:33:25.250 --> 00:33:28.090 align:middle line:84% So immediately see some kind of thick law-- 00:33:28.090 --> 00:33:31.630 align:middle line:84% Fick's law here where a flux is related to a slope times 00:33:31.630 --> 00:33:33.430 align:middle line:84% something out in front that is a constant. 00:33:33.430 --> 00:33:35.940 align:middle line:90% 00:33:35.940 --> 00:33:37.440 align:middle line:84% So that's at the bottom of slide 15. 00:33:37.440 --> 00:33:39.330 align:middle line:90% Let's go on to slide 16. 00:33:39.330 --> 00:33:42.810 align:middle line:84% And I've just repeated that equation that we just had, 00:33:42.810 --> 00:33:47.800 align:middle line:84% but I added one more thing on the right, one more equality. 00:33:47.800 --> 00:33:50.880 align:middle line:84% And we set that flux equal to a diffusivity, a number 00:33:50.880 --> 00:33:53.250 align:middle line:84% d, times the slope delta c over delta 00:33:53.250 --> 00:33:55.740 align:middle line:84% x, where we define the diffusivity 00:33:55.740 --> 00:33:57.900 align:middle line:84% in this particular equation to be 00:33:57.900 --> 00:34:04.040 align:middle line:84% equal to the vb over 2 times delta x squared. 00:34:04.040 --> 00:34:06.730 align:middle line:90% So there is Fick's first law. 00:34:06.730 --> 00:34:09.880 align:middle line:84% The flux is equal to minus a diffusivity, a constant number, 00:34:09.880 --> 00:34:12.070 align:middle line:90% times delta c over delta x. 00:34:12.070 --> 00:34:17.560 align:middle line:84% And then the diffusivity has some atomic scale mechanisms 00:34:17.560 --> 00:34:18.560 align:middle line:90% of u to it. 00:34:18.560 --> 00:34:24.100 align:middle line:84% In fact, it's proportional to the jump frequency, v sub d, 00:34:24.100 --> 00:34:27.400 align:middle line:84% and therefore, proportional be related to the energy barrier 00:34:27.400 --> 00:34:32.480 align:middle line:84% to hop between adjacent positions, or adjacent clients. 00:34:32.480 --> 00:34:34.210 align:middle line:84% So it gives you some idea of what-- 00:34:34.210 --> 00:34:36.460 align:middle line:84% at the time of scale, what goes into determining 00:34:36.460 --> 00:34:37.489 align:middle line:90% the diffusivity d. 00:34:37.489 --> 00:34:39.929 align:middle line:90% 00:34:39.929 --> 00:34:40.429 align:middle line:90% OK. 00:34:40.429 --> 00:34:42.449 align:middle line:84% In addition to giving us an atomic scale view, 00:34:42.449 --> 00:34:44.630 align:middle line:84% we can actually use this type of formalism 00:34:44.630 --> 00:34:50.600 align:middle line:84% to go ahead and derive a numerical solution to a fix 00:34:50.600 --> 00:34:51.163 align:middle line:90% equation. 00:34:51.163 --> 00:34:52.580 align:middle line:84% So let's go on-- to do that, we'll 00:34:52.580 --> 00:34:57.830 align:middle line:84% go on to slide number 17, which is a slightly different sketch, 00:34:57.830 --> 00:34:59.660 align:middle line:84% but the exact same idea is that what we're 00:34:59.660 --> 00:35:01.790 align:middle line:90% going to be going through. 00:35:01.790 --> 00:35:05.090 align:middle line:84% And what I'm showing on the left-hand side in the bottom 00:35:05.090 --> 00:35:09.890 align:middle line:84% is a plot of concentration as a function of x, so distance. 00:35:09.890 --> 00:35:14.810 align:middle line:84% And let's say it has some shape, this dark, dark decreasing black 00:35:14.810 --> 00:35:15.350 align:middle line:90% line. 00:35:15.350 --> 00:35:19.430 align:middle line:84% And it's decreasing from left to right. 00:35:19.430 --> 00:35:22.480 align:middle line:84% And what we'll then do is we're going to discretize this, 00:35:22.480 --> 00:35:26.180 align:middle line:84% and then look at discrete positions in space x. 00:35:26.180 --> 00:35:30.200 align:middle line:84% And we'll look at concentration C0, C1, and C2, 00:35:30.200 --> 00:35:32.060 align:middle line:84% and the spacing between those planes 00:35:32.060 --> 00:35:37.220 align:middle line:84% is going to be a constant, just like we used before, delta x. 00:35:37.220 --> 00:35:39.770 align:middle line:84% And so what we're going to look at 00:35:39.770 --> 00:35:42.650 align:middle line:84% is in the time interval, delta t, a certain time interval, 00:35:42.650 --> 00:35:46.490 align:middle line:84% delta t, what is the number of atoms crossing 00:35:46.490 --> 00:35:48.320 align:middle line:90% a certain plane per unit area? 00:35:48.320 --> 00:35:52.340 align:middle line:84% And we call that to the plane r, which we had labeled r above, 00:35:52.340 --> 00:35:53.750 align:middle line:90% q sub r. 00:35:53.750 --> 00:35:57.830 align:middle line:84% It's just the flux, f sub r times delta t. 00:35:57.830 --> 00:35:59.990 align:middle line:84% That's the number of atoms crossing the unit 00:35:59.990 --> 00:36:02.480 align:middle line:90% area in a given time interval. 00:36:02.480 --> 00:36:04.730 align:middle line:84% That's just basically-- this comes from the definition 00:36:04.730 --> 00:36:06.530 align:middle line:90% of what flux is. 00:36:06.530 --> 00:36:11.810 align:middle line:84% And we then can write in for the flux. 00:36:11.810 --> 00:36:15.440 align:middle line:84% The flux is just the diffusivity d times the concentration 00:36:15.440 --> 00:36:18.830 align:middle line:84% difference across that plane, across plane r. 00:36:18.830 --> 00:36:21.290 align:middle line:84% Well, the concentration difference across plane r 00:36:21.290 --> 00:36:24.890 align:middle line:84% is C1 minus C0 divided by delta x. 00:36:24.890 --> 00:36:30.590 align:middle line:84% So we can approximate it that way, the flux crossing plane r. 00:36:30.590 --> 00:36:32.270 align:middle line:84% What's the flux crossing plane s? 00:36:32.270 --> 00:36:35.390 align:middle line:84% Well, we can do the exact same thing, or the number of atoms 00:36:35.390 --> 00:36:36.395 align:middle line:90% crossing plane s. 00:36:36.395 --> 00:36:40.040 align:middle line:84% It's just the diffusivity times delta t times this time. 00:36:40.040 --> 00:36:45.360 align:middle line:84% We subtract c2 minus c1 over delta x. 00:36:45.360 --> 00:36:47.580 align:middle line:84% So we can then do some bookkeeping 00:36:47.580 --> 00:36:50.640 align:middle line:84% and calculate the net number of atoms 00:36:50.640 --> 00:36:53.500 align:middle line:84% accumulated in that shaded region between plane r and plane 00:36:53.500 --> 00:36:54.000 align:middle line:90% s. 00:36:54.000 --> 00:36:59.790 align:middle line:84% And the time interval delta t is just qr minus qs. 00:36:59.790 --> 00:37:01.590 align:middle line:84% And we just subtract these two expressions 00:37:01.590 --> 00:37:03.450 align:middle line:84% and you see what you end up with is 00:37:03.450 --> 00:37:06.900 align:middle line:84% equal to a diffusivity d times the time interval delta 00:37:06.900 --> 00:37:11.550 align:middle line:84% t, times, in parentheses, the c0 minus 2c1 00:37:11.550 --> 00:37:14.595 align:middle line:84% plus c2, that whole thing, quantity over delta x. 00:37:14.595 --> 00:37:17.830 align:middle line:90% 00:37:17.830 --> 00:37:21.730 align:middle line:84% Now, if we divide that net gain in the number of atoms divided 00:37:21.730 --> 00:37:24.430 align:middle line:84% by delta x is shown below, that is the net gain 00:37:24.430 --> 00:37:28.660 align:middle line:84% in the concentration in that shaded region that is delta c. 00:37:28.660 --> 00:37:32.020 align:middle line:84% And that's just qr minus qs, that whole quantity 00:37:32.020 --> 00:37:34.390 align:middle line:90% divided by delta x. 00:37:34.390 --> 00:37:37.420 align:middle line:84% So I can substituting in qr minus qs 00:37:37.420 --> 00:37:40.210 align:middle line:84% and you end up with the equation near the bottom, which just says 00:37:40.210 --> 00:37:43.030 align:middle line:84% the net gain in the concentration, delta c 00:37:43.030 --> 00:37:46.870 align:middle line:84% in this shaded region, is just the diffusivity d times delta 00:37:46.870 --> 00:37:51.130 align:middle line:84% t, times this quantity in parentheses, which is related 00:37:51.130 --> 00:37:53.470 align:middle line:84% to the concentrations in those three regions, 00:37:53.470 --> 00:37:56.900 align:middle line:90% divided by delta x squared. 00:37:56.900 --> 00:38:02.040 align:middle line:84% So what it says is, at a given point c1 in this curve, 00:38:02.040 --> 00:38:04.470 align:middle line:84% if I know the concentrations of the neighboring points, 00:38:04.470 --> 00:38:07.590 align:middle line:84% so I know its neighbors, I know the concentration at c zero 00:38:07.590 --> 00:38:11.340 align:middle line:84% and at c2, if you calculate a new concentration 00:38:11.340 --> 00:38:14.850 align:middle line:84% after a certain time step, delta t at c1, 00:38:14.850 --> 00:38:17.970 align:middle line:84% just by knowing its neighbors at this time 00:38:17.970 --> 00:38:21.070 align:middle line:84% step, the concentration of this neighbors. 00:38:21.070 --> 00:38:24.700 align:middle line:84% So if we go on to slide number 18, 00:38:24.700 --> 00:38:27.630 align:middle line:84% we can then use that information to come up 00:38:27.630 --> 00:38:30.150 align:middle line:84% with something called an explicit finite difference 00:38:30.150 --> 00:38:31.470 align:middle line:90% formula. 00:38:31.470 --> 00:38:33.000 align:middle line:84% That's what the equation being shown 00:38:33.000 --> 00:38:35.940 align:middle line:90% near the top of the slide 18. 00:38:35.940 --> 00:38:39.060 align:middle line:84% What you're saying is that the concentration at any given point 00:38:39.060 --> 00:38:44.970 align:middle line:84% i in one of these slabs, at a new time, t plus delta t. 00:38:44.970 --> 00:38:48.970 align:middle line:84% So we're going to evolve this profile over time. 00:38:48.970 --> 00:38:53.490 align:middle line:84% So c sub i plus on the left-hand side 00:38:53.490 --> 00:38:55.410 align:middle line:84% is equal to the concentration at time t. 00:38:55.410 --> 00:38:59.670 align:middle line:84% So one interval before that, whatever it was before, plus we 00:38:59.670 --> 00:39:04.320 align:middle line:84% add in this difference, this term 00:39:04.320 --> 00:39:06.360 align:middle line:84% on the right that gives the change 00:39:06.360 --> 00:39:10.080 align:middle line:84% in the concentration in the i-th slab after a time step delta t. 00:39:10.080 --> 00:39:12.540 align:middle line:84% Then that term is just d, the diffusivity, 00:39:12.540 --> 00:39:15.720 align:middle line:84% delta t divided by delta x squared times the quantity 00:39:15.720 --> 00:39:16.963 align:middle line:90% in parentheses. 00:39:16.963 --> 00:39:18.630 align:middle line:84% Where again, the quantity of parentheses 00:39:18.630 --> 00:39:21.360 align:middle line:84% in order evaluated at any point ci, all you need to know 00:39:21.360 --> 00:39:22.920 align:middle line:84% is the concentration at that point, 00:39:22.920 --> 00:39:27.330 align:middle line:84% and the concentration of its two nearest neighbors in whatever 00:39:27.330 --> 00:39:28.500 align:middle line:90% way that we've discretized. 00:39:28.500 --> 00:39:31.710 align:middle line:84% It's something we've discretized this continuous function 00:39:31.710 --> 00:39:36.290 align:middle line:90% into little intervals, delta x. 00:39:36.290 --> 00:39:39.110 align:middle line:84% So that is an explicit formula that you 00:39:39.110 --> 00:39:41.660 align:middle line:84% can use to evolve a concentration 00:39:41.660 --> 00:39:44.790 align:middle line:90% profile in shape over time. 00:39:44.790 --> 00:39:46.970 align:middle line:84% And if you look at it for a few minutes, 00:39:46.970 --> 00:39:50.540 align:middle line:84% you can notice the similarity of that difference formula 00:39:50.540 --> 00:39:54.790 align:middle line:84% to the differential equation for diffusion. 00:39:54.790 --> 00:39:59.920 align:middle line:84% Which if you take the partial t on this equation shown 00:39:59.920 --> 00:40:02.870 align:middle line:84% in the center, and move it over to the right-hand side, 00:40:02.870 --> 00:40:06.790 align:middle line:84% you can see that partial c is equal to some constant d 00:40:06.790 --> 00:40:11.620 align:middle line:84% times a second derivative of concentration with respect to x. 00:40:11.620 --> 00:40:16.990 align:middle line:84% And in fact, this partial second derivative with respect to x, 00:40:16.990 --> 00:40:20.740 align:middle line:84% it can be numerically evaluated as the term 00:40:20.740 --> 00:40:25.210 align:middle line:84% in the first equation on slide 18 on the right-hand side. 00:40:25.210 --> 00:40:30.760 align:middle line:84% That is the numerical solution to that differential equation. 00:40:30.760 --> 00:40:33.190 align:middle line:90% And I won't derive it here. 00:40:33.190 --> 00:40:36.400 align:middle line:84% If you go back to Crank's book, or any of the books 00:40:36.400 --> 00:40:41.290 align:middle line:84% on numerical methods and you find that the method 00:40:41.290 --> 00:40:42.550 align:middle line:90% is numerically stable. 00:40:42.550 --> 00:40:45.580 align:middle line:84% So it works from a numerical point of view. 00:40:45.580 --> 00:40:49.940 align:middle line:84% It gives reasonable answers for a particular condition. 00:40:49.940 --> 00:40:51.770 align:middle line:84% We have-- when we're doing the solution, 00:40:51.770 --> 00:40:53.990 align:middle line:84% we have a condition on a quantity r 00:40:53.990 --> 00:40:59.360 align:middle line:84% where this quantity r is what multiplies the concentration 00:40:59.360 --> 00:41:03.050 align:middle line:84% change in a slab at a certain time step delta t. 00:41:03.050 --> 00:41:04.910 align:middle line:84% That indeed, delta t over delta x 00:41:04.910 --> 00:41:07.370 align:middle line:84% squared, that multiplier has to be less than 00:41:07.370 --> 00:41:09.530 align:middle line:90% or equal to one-half. 00:41:09.530 --> 00:41:11.160 align:middle line:84% If that multiplier gets too large, 00:41:11.160 --> 00:41:13.760 align:middle line:84% then that's-- the whole method breaks down, become unstable, 00:41:13.760 --> 00:41:16.340 align:middle line:84% and you get nonsense, absolutely, 00:41:16.340 --> 00:41:18.890 align:middle line:90% totally unmeaningful solutions. 00:41:18.890 --> 00:41:19.910 align:middle line:90% So you need to keep-- 00:41:19.910 --> 00:41:22.670 align:middle line:84% you need to adjust if you're doing this numerical, make 00:41:22.670 --> 00:41:25.460 align:middle line:84% sure you adjust your time steps delta t 00:41:25.460 --> 00:41:29.270 align:middle line:84% to be small enough so that this holds. 00:41:29.270 --> 00:41:35.840 align:middle line:84% And your delta x to be of appropriate size with respect 00:41:35.840 --> 00:41:41.900 align:middle line:84% to delta t so that you don't get yourself out of the physically-- 00:41:41.900 --> 00:41:44.238 align:middle line:84% out of the numerically stable solution. 00:41:44.238 --> 00:41:45.530 align:middle line:90% It's a very powerful technique. 00:41:45.530 --> 00:41:46.690 align:middle line:90% It's very simple. 00:41:46.690 --> 00:41:49.050 align:middle line:84% That's one of the nice things about it. 00:41:49.050 --> 00:41:50.730 align:middle line:84% And it's certainly going to be required 00:41:50.730 --> 00:41:54.630 align:middle line:84% to use some method, numerical method, when the diffusivity is 00:41:54.630 --> 00:41:57.900 align:middle line:90% not a constant. 00:41:57.900 --> 00:41:59.400 align:middle line:84% If you were to sit down, you could 00:41:59.400 --> 00:42:01.620 align:middle line:84% write your own simple simulator using 00:42:01.620 --> 00:42:07.800 align:middle line:84% this very relatively easy to use finite difference formula. 00:42:07.800 --> 00:42:11.640 align:middle line:84% It is not the particular type of numerical method that's been-- 00:42:11.640 --> 00:42:13.410 align:middle line:90% that is being used in SUPREM IV. 00:42:13.410 --> 00:42:16.200 align:middle line:84% SUPREM IV has a more sophisticated numerical solution 00:42:16.200 --> 00:42:18.517 align:middle line:84% method for the diffusion equation. 00:42:18.517 --> 00:42:20.850 align:middle line:84% This is much more simplified, and this particular method 00:42:20.850 --> 00:42:21.598 align:middle line:90% can be slow. 00:42:21.598 --> 00:42:22.890 align:middle line:90% It has all of its own problems. 00:42:22.890 --> 00:42:24.990 align:middle line:84% But the beauty of it is the simplicity. 00:42:24.990 --> 00:42:29.580 align:middle line:84% And just to give you an idea of how one might use 00:42:29.580 --> 00:42:32.910 align:middle line:90% numerical solutions to obtain-- 00:42:32.910 --> 00:42:35.190 align:middle line:84% numerical methods to obtain solutions 00:42:35.190 --> 00:42:38.120 align:middle line:90% of the diffusion equation. 00:42:38.120 --> 00:42:39.500 align:middle line:90% OK. 00:42:39.500 --> 00:42:45.020 align:middle line:84% Let's go on now to slide number 19. 00:42:45.020 --> 00:42:48.850 align:middle line:84% What I want to talk about at this point are from now on, 00:42:48.850 --> 00:42:51.670 align:middle line:84% and this is the rest of this lecture and the next couple 00:42:51.670 --> 00:42:55.630 align:middle line:84% of lectures, will be the so-called modifications 00:42:55.630 --> 00:42:56.440 align:middle line:90% of Fick's laws. 00:42:56.440 --> 00:42:59.740 align:middle line:84% Fick's laws are introduced in a lot of basic courses-- 00:42:59.740 --> 00:43:03.010 align:middle line:90% material science or physics. 00:43:03.010 --> 00:43:07.270 align:middle line:84% And when we are diffusing dopants 00:43:07.270 --> 00:43:10.120 align:middle line:84% in semiconductors in practical cases, 00:43:10.120 --> 00:43:14.140 align:middle line:84% there are a few number of cases where Fick's laws need 00:43:14.140 --> 00:43:17.750 align:middle line:84% to be modified to take into account the things that 00:43:17.750 --> 00:43:19.750 align:middle line:84% are really going on in semiconductor processing. 00:43:19.750 --> 00:43:23.470 align:middle line:84% And the first thing that we'll talk about here on slide 19 00:43:23.470 --> 00:43:25.060 align:middle line:84% is so-called electric field effect. 00:43:25.060 --> 00:43:27.600 align:middle line:90% 00:43:27.600 --> 00:43:35.140 align:middle line:84% And this occurs in the case when the doping concentration is 00:43:35.140 --> 00:43:37.240 align:middle line:90% higher than n sub i. 00:43:37.240 --> 00:43:41.440 align:middle line:84% So n sub i is intrinsic carrier concentration at a diffusion 00:43:41.440 --> 00:43:43.660 align:middle line:84% temperature, again, of 1,000. n sub 00:43:43.660 --> 00:43:47.710 align:middle line:84% i is close to mid 10 to the 18th or so. 00:43:47.710 --> 00:43:49.990 align:middle line:84% If the doping concentration gets higher than that, 00:43:49.990 --> 00:43:56.193 align:middle line:84% you can start to have a non-uniformity in the charge 00:43:56.193 --> 00:43:57.610 align:middle line:84% distribution in the semiconductor, 00:43:57.610 --> 00:44:00.400 align:middle line:84% and electric field effects can become important. 00:44:00.400 --> 00:44:03.070 align:middle line:84% And this is a schematic demonstration 00:44:03.070 --> 00:44:08.050 align:middle line:84% of how electric fields can arise when dopants 00:44:08.050 --> 00:44:09.330 align:middle line:90% are diffusing and silicon. 00:44:09.330 --> 00:44:12.670 align:middle line:84% It's a plot of concentration schematically versus depth, 00:44:12.670 --> 00:44:19.180 align:middle line:84% and the red line is supposed to represent the profile 00:44:19.180 --> 00:44:23.920 align:middle line:84% at any given point in time for, say, arsenic that's diffusing. 00:44:23.920 --> 00:44:25.600 align:middle line:90% And then arsenic is a donor. 00:44:25.600 --> 00:44:31.450 align:middle line:84% It has one extra electron floating around it. 00:44:31.450 --> 00:44:34.250 align:middle line:84% But the atom itself in the absence 00:44:34.250 --> 00:44:38.720 align:middle line:84% of this electron, this extra valence electron, the donor 00:44:38.720 --> 00:44:40.160 align:middle line:90% itself is positively charged. 00:44:40.160 --> 00:44:41.630 align:middle line:90% It's an ion. 00:44:41.630 --> 00:44:44.660 align:middle line:84% This arsenic's diffusing and there 00:44:44.660 --> 00:44:48.350 align:middle line:84% are-- the electrons associated with this heavily doped arsenic 00:44:48.350 --> 00:44:51.770 align:middle line:84% region can diffuse ahead of the dopant. 00:44:51.770 --> 00:44:54.560 align:middle line:84% And so this blue line is meant to represent 00:44:54.560 --> 00:44:57.330 align:middle line:84% the distribution or the diffusion profile of electrons, 00:44:57.330 --> 00:44:58.880 align:middle line:90% let's say. 00:44:58.880 --> 00:45:03.530 align:middle line:84% So the electrons diffuse, and holes diffuse more rapidly 00:45:03.530 --> 00:45:06.380 align:middle line:84% than their associated donors and acceptors. 00:45:06.380 --> 00:45:11.190 align:middle line:84% Because of that more rapid diffusion, they'll diffuse-- 00:45:11.190 --> 00:45:13.820 align:middle line:84% the profile of electrons will be a little bit deeper 00:45:13.820 --> 00:45:16.820 align:middle line:84% and diffuse ahead of the dopant that they're 00:45:16.820 --> 00:45:19.370 align:middle line:84% associated with until they reach some steady state 00:45:19.370 --> 00:45:26.770 align:middle line:84% condition where the drift flux from the internal electric field 00:45:26.770 --> 00:45:30.880 align:middle line:84% is going to balance the diffusion, the diffusion flux. 00:45:30.880 --> 00:45:33.490 align:middle line:84% And so what we look at here is there, 00:45:33.490 --> 00:45:36.170 align:middle line:84% if could just look at this schematic picture, 00:45:36.170 --> 00:45:39.957 align:middle line:84% you'll see there is indeed from your elementary electronics, 00:45:39.957 --> 00:45:42.040 align:middle line:84% or solid state physics, there is an electric field 00:45:42.040 --> 00:45:45.640 align:middle line:84% induced by the fact that there's a net positive charge 00:45:45.640 --> 00:45:47.350 align:middle line:90% on the left region. 00:45:47.350 --> 00:45:49.890 align:middle line:84% That is, if you subtract all the arsenic donors, 00:45:49.890 --> 00:45:51.640 align:middle line:84% you subtract electron concentration there, 00:45:51.640 --> 00:45:53.630 align:middle line:84% there's a net positive charge on the left. 00:45:53.630 --> 00:45:55.422 align:middle line:84% There's a little bit of net negative charge 00:45:55.422 --> 00:45:58.060 align:middle line:84% on the right that produces an electric field pointing 00:45:58.060 --> 00:46:00.980 align:middle line:90% from left to right. 00:46:00.980 --> 00:46:04.150 align:middle line:84% So that electric field then tends 00:46:04.150 --> 00:46:07.510 align:middle line:84% to cause electrons to drift from right to left, right? 00:46:07.510 --> 00:46:12.117 align:middle line:84% Because they drift against the field. 00:46:12.117 --> 00:46:14.200 align:middle line:84% So they're going to be drifting from right to left 00:46:14.200 --> 00:46:16.810 align:middle line:84% while they're diffusing from left to right, because there's 00:46:16.810 --> 00:46:18.170 align:middle line:90% a concentration gradient. 00:46:18.170 --> 00:46:21.260 align:middle line:84% So the electric field will build up to the point 00:46:21.260 --> 00:46:25.400 align:middle line:84% where there's steady state where that drift flux just 00:46:25.400 --> 00:46:31.000 align:middle line:84% balances the diffusion flux for the electrons. 00:46:31.000 --> 00:46:33.700 align:middle line:84% And again, the physical origin of this electric field 00:46:33.700 --> 00:46:37.270 align:middle line:84% is the fact that, well, electrons are charged in holes, 00:46:37.270 --> 00:46:39.327 align:middle line:84% and they have somewhat higher mobility. 00:46:39.327 --> 00:46:41.410 align:middle line:84% They can move around the lattice much more readily 00:46:41.410 --> 00:46:44.690 align:middle line:90% than the dopant ions themselves. 00:46:44.690 --> 00:46:45.190 align:middle line:90% OK. 00:46:45.190 --> 00:46:47.773 align:middle line:84% And again, this is only going to happen to the particular case 00:46:47.773 --> 00:46:50.980 align:middle line:84% where the concentration of the dopant is higher than n sub i. 00:46:50.980 --> 00:46:54.907 align:middle line:84% Otherwise, the background electron concentration 00:46:54.907 --> 00:46:55.990 align:middle line:90% is set by the temperature. 00:46:55.990 --> 00:46:59.650 align:middle line:84% It's just a constant to throughout space. 00:46:59.650 --> 00:47:02.350 align:middle line:84% So the electric field would not develop right 00:47:02.350 --> 00:47:05.560 align:middle line:84% at the edge of this diffusion profile. 00:47:05.560 --> 00:47:07.270 align:middle line:90% Let's go on to slide number 20. 00:47:07.270 --> 00:47:11.350 align:middle line:84% And so what we do is, we modify Fick's first law, 00:47:11.350 --> 00:47:14.710 align:middle line:84% and we modify it in a way that's very similar and pretty 00:47:14.710 --> 00:47:16.630 align:middle line:84% much almost identical to what we do 00:47:16.630 --> 00:47:20.470 align:middle line:84% in our electronics, or solid state physics classes. 00:47:20.470 --> 00:47:22.090 align:middle line:84% We say that the total impurity atom 00:47:22.090 --> 00:47:24.940 align:middle line:84% flux in the presence of both a concentration 00:47:24.940 --> 00:47:27.790 align:middle line:84% gradient and an electric field can be written down 00:47:27.790 --> 00:47:29.590 align:middle line:90% as a sum of two fluxes. 00:47:29.590 --> 00:47:33.020 align:middle line:84% One is f, which is the normal Fickian diffusion flux, which 00:47:33.020 --> 00:47:38.060 align:middle line:84% is given by minus d, partial c, partial x, plus f prime, 00:47:38.060 --> 00:47:42.050 align:middle line:84% where f prime is now the drift flux due to the electric field. 00:47:42.050 --> 00:47:44.270 align:middle line:84% And for the drift flux, I wrote down 00:47:44.270 --> 00:47:47.160 align:middle line:84% z, which is going to get the charge. 00:47:47.160 --> 00:47:50.870 align:middle line:84% The charge number z would be 1 for electron, 00:47:50.870 --> 00:47:53.930 align:middle line:84% depends on the number of electronic charges associated 00:47:53.930 --> 00:47:59.360 align:middle line:84% with whatever is drifting, times the mobility mu, times 00:47:59.360 --> 00:48:01.820 align:middle line:84% the concentration c, times the electric field. 00:48:01.820 --> 00:48:03.987 align:middle line:90% That is the flux due to drift. 00:48:03.987 --> 00:48:05.570 align:middle line:84% Again, it's not a diffusion processes, 00:48:05.570 --> 00:48:08.220 align:middle line:84% that's the drift process, a different physical process. 00:48:08.220 --> 00:48:10.700 align:middle line:84% You can add these two together and write down 00:48:10.700 --> 00:48:12.120 align:middle line:90% as the total flux. 00:48:12.120 --> 00:48:15.500 align:middle line:84% So we have a non-Fickian term on the right. 00:48:15.500 --> 00:48:18.800 align:middle line:84% But because of the Einstein relation, we can actually, 00:48:18.800 --> 00:48:21.680 align:middle line:84% when we're talking about the diffusion of carriers, 00:48:21.680 --> 00:48:23.690 align:middle line:84% like electrons and holes, we know 00:48:23.690 --> 00:48:27.290 align:middle line:84% that their mobility is directly proportional to the diffusivity 00:48:27.290 --> 00:48:29.840 align:middle line:84% and the proportionality constant is q over kt. 00:48:29.840 --> 00:48:34.520 align:middle line:84% So mu is equal to d times q over kt. 00:48:34.520 --> 00:48:37.460 align:middle line:84% So we can substitute in for the mobility 00:48:37.460 --> 00:48:40.940 align:middle line:84% in the upper right-hand equation with a substitute 00:48:40.940 --> 00:48:44.363 align:middle line:90% in q over t times d. 00:48:44.363 --> 00:48:45.780 align:middle line:84% We can also substitute in the fact 00:48:45.780 --> 00:48:49.250 align:middle line:84% that electric field e is the gradient of the potential where 00:48:49.250 --> 00:48:52.460 align:middle line:84% the potential psi can be related to the carrier 00:48:52.460 --> 00:48:57.020 align:middle line:84% concentration in as minus kt over q times the ln of n 00:48:57.020 --> 00:48:58.920 align:middle line:90% over ni. 00:48:58.920 --> 00:49:03.930 align:middle line:84% So basically by substituting in the expression 00:49:03.930 --> 00:49:07.650 align:middle line:84% proportional diffusivity for mu, and substituting in 00:49:07.650 --> 00:49:10.560 align:middle line:84% for the electric field, how it relates to the potential 00:49:10.560 --> 00:49:14.550 align:middle line:84% and, therefore, to n over ni, we get the equation 00:49:14.550 --> 00:49:17.040 align:middle line:84% near the bottom for the flux f, which 00:49:17.040 --> 00:49:20.820 align:middle line:84% is the usual Fickian minus the times partial z, partial x, 00:49:20.820 --> 00:49:24.570 align:middle line:84% times minus d, times the concentration c, 00:49:24.570 --> 00:49:28.620 align:middle line:84% times now we've come up with this psi 00:49:28.620 --> 00:49:35.700 align:middle line:84% term, partial partial x of mu potential within n over ni. 00:49:35.700 --> 00:49:39.210 align:middle line:90% 00:49:39.210 --> 00:49:43.680 align:middle line:84% So now this is the new flux equation, 00:49:43.680 --> 00:49:47.665 align:middle line:84% and we can-- now we use for the carrier concentration. 00:49:47.665 --> 00:49:48.790 align:middle line:90% And you have to be careful. 00:49:48.790 --> 00:49:51.270 align:middle line:84% It's not just equal to nd because ni 00:49:51.270 --> 00:49:52.960 align:middle line:90% can be of order nd in this. 00:49:52.960 --> 00:49:56.230 align:middle line:84% So we need to make sure we use the full expression 00:49:56.230 --> 00:49:57.480 align:middle line:90% for the carrier concentration. 00:49:57.480 --> 00:50:01.140 align:middle line:84% And it's one-half-- the bottom equation here 00:50:01.140 --> 00:50:05.100 align:middle line:84% is one-half times the quantity, square root of nd square plus r 00:50:05.100 --> 00:50:07.440 align:middle line:90% and i square plus nd. 00:50:07.440 --> 00:50:11.230 align:middle line:84% That comes from charge neutrality. 00:50:11.230 --> 00:50:13.930 align:middle line:84% Given that, we-- therefore, there's 00:50:13.930 --> 00:50:16.720 align:middle line:84% a relationship between end, the carrier 00:50:16.720 --> 00:50:21.310 align:middle line:84% concentration, and the concentration of the dopant c. 00:50:21.310 --> 00:50:29.150 align:middle line:84% Because again, the concentration c is just nd minus na. 00:50:29.150 --> 00:50:30.660 align:middle line:90% So it's the net concentration. 00:50:30.660 --> 00:50:34.610 align:middle line:84% So we have a relationship between n and c. 00:50:34.610 --> 00:50:37.580 align:middle line:84% We're going to use that to make this flux equation 00:50:37.580 --> 00:50:41.260 align:middle line:84% f to be written only in terms of the concentration c. 00:50:41.260 --> 00:50:44.770 align:middle line:84% So we go on to the next page, slide 21. 00:50:44.770 --> 00:50:47.560 align:middle line:84% What we find is that we can do this compression 00:50:47.560 --> 00:50:51.820 align:middle line:84% in this equation, and we find that the total flux now f 00:50:51.820 --> 00:50:56.320 align:middle line:84% is written as something called minus h times the diffusivity 00:50:56.320 --> 00:51:00.400 align:middle line:84% of a dopant d sub a, partial c, partial x. 00:51:00.400 --> 00:51:02.920 align:middle line:84% So we've been able to actually write down 00:51:02.920 --> 00:51:06.310 align:middle line:84% an equation that looks just like Fick's first law, 00:51:06.310 --> 00:51:09.100 align:middle line:84% but there's an h factor multiplying it. 00:51:09.100 --> 00:51:10.990 align:middle line:84% And this h factor, it turns out to be 00:51:10.990 --> 00:51:15.640 align:middle line:84% equal to 1 plus c divided by the quantity square root of c 00:51:15.640 --> 00:51:17.185 align:middle line:90% squared plus 4 ni squared. 00:51:17.185 --> 00:51:21.220 align:middle line:90% 00:51:21.220 --> 00:51:25.390 align:middle line:90% So basically, it looks-- 00:51:25.390 --> 00:51:26.950 align:middle line:84% in the presence of an electric field, 00:51:26.950 --> 00:51:29.020 align:middle line:84% it looks just like we have an enhancement 00:51:29.020 --> 00:51:32.800 align:middle line:84% in the diffusivity of the dopant by a factor called h. 00:51:32.800 --> 00:51:35.170 align:middle line:84% And you can-- we just worked that out. 00:51:35.170 --> 00:51:39.760 align:middle line:84% So interestingly, we see that it-- electric field effect 00:51:39.760 --> 00:51:43.250 align:middle line:84% on a dopant that is creating the field on that dopant itself, 00:51:43.250 --> 00:51:45.140 align:middle line:84% it enhances the diffusivity, and it turns out 00:51:45.140 --> 00:51:48.350 align:middle line:90% h has an upper bound of t. 00:51:48.350 --> 00:51:50.960 align:middle line:84% So h is not going to be any larger than 2, 00:51:50.960 --> 00:51:53.480 align:middle line:84% and that means that the electric field enhancement 00:51:53.480 --> 00:51:56.150 align:middle line:84% term can enhance the diffusivity of the dopant that 00:51:56.150 --> 00:52:01.220 align:middle line:84% causes the field by as much as, or a maximum of a factor of 2. 00:52:01.220 --> 00:52:04.880 align:middle line:84% And this happens when the doping concentration term, or c 00:52:04.880 --> 00:52:07.340 align:middle line:90% is much, much greater than ni. 00:52:07.340 --> 00:52:11.450 align:middle line:84% And you can see that by substituting in and calculating 00:52:11.450 --> 00:52:13.460 align:middle line:84% out what h is in certain conditions. 00:52:13.460 --> 00:52:16.160 align:middle line:90% 00:52:16.160 --> 00:52:23.480 align:middle line:84% In the special case where c is ni, then h goes back to being 1. 00:52:23.480 --> 00:52:25.940 align:middle line:84% So how do we get to this equation 00:52:25.940 --> 00:52:27.630 align:middle line:90% where the simplified h factor? 00:52:27.630 --> 00:52:31.340 align:middle line:84% Well, if you go on to slide number 22-- 00:52:31.340 --> 00:52:33.710 align:middle line:84% actually, I'm not going to go through all the algebra. 00:52:33.710 --> 00:52:38.640 align:middle line:84% But basically, what we did was, we used a particular fact, 00:52:38.640 --> 00:52:40.880 align:middle line:84% which is that-- along the top of slide 22-- 00:52:40.880 --> 00:52:43.880 align:middle line:84% partial partial x on the logarithm of x 00:52:43.880 --> 00:52:45.350 align:middle line:90% is equal to 1 over x. 00:52:45.350 --> 00:52:49.340 align:middle line:84% And what we needed to find in that earlier 00:52:49.340 --> 00:52:52.250 align:middle line:84% diffusion and that earlier drift term, 00:52:52.250 --> 00:52:58.970 align:middle line:84% there was a term that went partial partial x of ln of n 00:52:58.970 --> 00:53:01.190 align:middle line:90% over ni. 00:53:01.190 --> 00:53:05.420 align:middle line:84% And so just by going through the algebra shown on-- 00:53:05.420 --> 00:53:07.040 align:middle line:84% we can substitute-- on slide 22, we 00:53:07.040 --> 00:53:09.740 align:middle line:84% can substitute for the carrier concentration n, 00:53:09.740 --> 00:53:15.050 align:middle line:84% and we can find an expression for partial partial x of Lin 00:53:15.050 --> 00:53:16.190 align:middle line:90% of n over ni. 00:53:16.190 --> 00:53:19.550 align:middle line:84% And you see that in the middle of the slide. 00:53:19.550 --> 00:53:22.580 align:middle line:84% And then we substitute in that the dopant concentration nd 00:53:22.580 --> 00:53:23.900 align:middle line:90% is equal to c. 00:53:23.900 --> 00:53:26.840 align:middle line:84% And we're able to see then that the flux, the total flux is 00:53:26.840 --> 00:53:30.680 align:middle line:84% the usual thickening term d minus d times partial 00:53:30.680 --> 00:53:34.910 align:middle line:84% c partial x, minus this other term, which ends up 00:53:34.910 --> 00:53:38.720 align:middle line:84% being able to be written just like partial c, partial x 00:53:38.720 --> 00:53:43.860 align:middle line:84% in terms of partial c, partial x, times the h factor 00:53:43.860 --> 00:53:47.595 align:middle line:84% where h is equal to, as I show on the bottom, 1 plus c 00:53:47.595 --> 00:53:49.720 align:middle line:84% over the square root of c squared plus 4ni squared. 00:53:49.720 --> 00:53:53.400 align:middle line:84% So that's just justifies the derivation of the h factor 00:53:53.400 --> 00:53:56.790 align:middle line:90% on slide 22. 00:53:56.790 --> 00:53:59.550 align:middle line:84% OK, so just go back to slide 21 for one second 00:53:59.550 --> 00:54:01.080 align:middle line:84% and just want to remind ourselves 00:54:01.080 --> 00:54:06.000 align:middle line:84% that the total flux f now in the presence of a high carrier 00:54:06.000 --> 00:54:08.850 align:middle line:84% concentration can be written just like the Ficking diffusion, 00:54:08.850 --> 00:54:11.310 align:middle line:84% but the h, there's a multiplier h 00:54:11.310 --> 00:54:14.960 align:middle line:84% from the diffusivity of the dopant atom a. 00:54:14.960 --> 00:54:17.750 align:middle line:90% Let's skip on to slide 23 now. 00:54:17.750 --> 00:54:19.790 align:middle line:84% However, so that's what we're talking 00:54:19.790 --> 00:54:22.880 align:middle line:84% about the amount of enhancement you 00:54:22.880 --> 00:54:26.570 align:middle line:84% might get in the high concentration species. 00:54:26.570 --> 00:54:28.730 align:middle line:84% But you may have two different species diffusing. 00:54:28.730 --> 00:54:31.280 align:middle line:84% And in the case where the species 00:54:31.280 --> 00:54:34.490 align:middle line:84% are at different concentrations, the electric field term 00:54:34.490 --> 00:54:36.860 align:middle line:84% can actually cause an even larger change 00:54:36.860 --> 00:54:40.190 align:middle line:84% in the diffusivity of the low concentration [INAUDIBLE].. 00:54:40.190 --> 00:54:44.210 align:middle line:84% So let's look at a particular example shown on slide 23. 00:54:44.210 --> 00:54:47.360 align:middle line:84% And what we're showing here is concentration 00:54:47.360 --> 00:54:50.960 align:middle line:84% as a function of depth, and the red curve in high concentration, 00:54:50.960 --> 00:54:55.580 align:middle line:84% let's say the high concentration arsenic profile. 00:54:55.580 --> 00:54:57.080 align:middle line:84% The initial arsenic profile looks-- 00:54:57.080 --> 00:54:59.840 align:middle line:84% it was on a planet or something it looks like this as shown 00:54:59.840 --> 00:55:02.120 align:middle line:90% in the solid red curve. 00:55:02.120 --> 00:55:04.460 align:middle line:84% The background doping is initially all constant, 00:55:04.460 --> 00:55:04.970 align:middle line:90% it's p-type. 00:55:04.970 --> 00:55:10.840 align:middle line:84% So this boron is shown by the solid blue line. 00:55:10.840 --> 00:55:13.320 align:middle line:84% Now, we're going to simulate including electric field 00:55:13.320 --> 00:55:15.990 align:middle line:84% effects at 1,000 degrees, the diffusion 00:55:15.990 --> 00:55:18.360 align:middle line:90% of the arsenic and the boron. 00:55:18.360 --> 00:55:21.000 align:middle line:84% Well, it turns out the arsenic does indeed diffuse, 00:55:21.000 --> 00:55:22.470 align:middle line:90% as shown by the dashed line. 00:55:22.470 --> 00:55:26.640 align:middle line:84% But that the electric field that it produces right near its-- 00:55:26.640 --> 00:55:29.160 align:middle line:84% the slope right near the edge of its profile, 00:55:29.160 --> 00:55:30.930 align:middle line:84% actually, that electric field can 00:55:30.930 --> 00:55:34.830 align:middle line:84% cause a flux of boron ions, which can cause 00:55:34.830 --> 00:55:37.270 align:middle line:90% the bond to itself to move. 00:55:37.270 --> 00:55:40.800 align:middle line:84% So the final boron profile now is 00:55:40.800 --> 00:55:44.460 align:middle line:84% shown by the blue dashed line, has this little hump 00:55:44.460 --> 00:55:45.660 align:middle line:90% in it and then a dip. 00:55:45.660 --> 00:55:48.180 align:middle line:84% And that little dip occurs right near 00:55:48.180 --> 00:55:51.750 align:middle line:84% where the junction is, right where the boron-- 00:55:51.750 --> 00:55:53.370 align:middle line:90% the arsenic in the boron cross. 00:55:53.370 --> 00:55:55.620 align:middle line:84% And there's no way by Fickian diffusion 00:55:55.620 --> 00:55:58.800 align:middle line:84% you would ever predict that a constant profile of boron 00:55:58.800 --> 00:56:00.150 align:middle line:90% would end up being-- 00:56:00.150 --> 00:56:02.700 align:middle line:84% looking the way it does in this particular plot. 00:56:02.700 --> 00:56:05.220 align:middle line:84% Again, that's because of the second term, the electric field 00:56:05.220 --> 00:56:07.320 align:middle line:90% term, non-Fickian term. 00:56:07.320 --> 00:56:10.960 align:middle line:84% So when you want to think about that, how that occurred, 00:56:10.960 --> 00:56:16.120 align:middle line:84% look at the electric field arrow drawn from left to right. 00:56:16.120 --> 00:56:18.040 align:middle line:84% Again, drawing from positive, net positive 00:56:18.040 --> 00:56:22.540 align:middle line:84% to net negative charge, that direction of e pulls the bond-- 00:56:22.540 --> 00:56:24.220 align:middle line:84% and again, remember, the boron ions, 00:56:24.220 --> 00:56:27.550 align:middle line:84% as they diffuse, they're b minus, these acceptors 00:56:27.550 --> 00:56:28.670 align:middle line:90% and that negative charge. 00:56:28.670 --> 00:56:30.045 align:middle line:84% So they're going to be pulled in, 00:56:30.045 --> 00:56:32.680 align:middle line:84% they're going to move to the left by the electric field. 00:56:32.680 --> 00:56:34.840 align:middle line:84% They'll be pulled into the n plus region 00:56:34.840 --> 00:56:39.160 align:middle line:84% and deplete some of the boron in the region past the junction. 00:56:39.160 --> 00:56:42.040 align:middle line:84% And so you end up with this sort of clump and dip, 00:56:42.040 --> 00:56:44.620 align:middle line:84% which is a non-Fickian type of motion, 00:56:44.620 --> 00:56:47.830 align:middle line:84% but induced by the electric field created by the higher 00:56:47.830 --> 00:56:51.530 align:middle line:90% concentration arsenic species. 00:56:51.530 --> 00:56:55.690 align:middle line:84% So this can have quite a bit of an effect on the diffusion 00:56:55.690 --> 00:56:57.280 align:middle line:90% profiles near a junction. 00:56:57.280 --> 00:56:59.750 align:middle line:84% And we see this all the time in our devices. 00:56:59.750 --> 00:57:02.900 align:middle line:84% In fact, you go on to slide number 23, 00:57:02.900 --> 00:57:07.370 align:middle line:84% showing a simulation example of an MOSFET. 00:57:07.370 --> 00:57:08.960 align:middle line:84% These are two SUPREM simulations, 00:57:08.960 --> 00:57:11.480 align:middle line:90% both around 1,000 degrees. 00:57:11.480 --> 00:57:14.960 align:middle line:84% For MOSFETs, you'll recognize the polysilicon gate here 00:57:14.960 --> 00:57:17.180 align:middle line:90% in magenta. 00:57:17.180 --> 00:57:21.020 align:middle line:84% And the source and drain regions are arsenic, 00:57:21.020 --> 00:57:24.090 align:middle line:90% and they are shown in black. 00:57:24.090 --> 00:57:26.780 align:middle line:84% And if you look first at the left 00:57:26.780 --> 00:57:29.120 align:middle line:84% in the case of an electric field, 00:57:29.120 --> 00:57:31.190 align:middle line:84% you see these different contours, 00:57:31.190 --> 00:57:35.720 align:middle line:84% the green, yellow, sort of light orange, and then a dark orange, 00:57:35.720 --> 00:57:37.490 align:middle line:90% or red. 00:57:37.490 --> 00:57:40.940 align:middle line:84% We see the boron uniformly doped, 00:57:40.940 --> 00:57:46.140 align:middle line:84% and its contours are shown by the different colors. 00:57:46.140 --> 00:57:48.320 align:middle line:84% And now what happens on the right 00:57:48.320 --> 00:57:50.360 align:middle line:84% in the presence of an electric field, 00:57:50.360 --> 00:57:53.810 align:middle line:84% instead of these uniform lateral doping controls for the boron, 00:57:53.810 --> 00:57:58.490 align:middle line:84% you see there's an electric field that acts on the boron 00:57:58.490 --> 00:58:02.088 align:middle line:84% and it changes the doping distribution in its shape. 00:58:02.088 --> 00:58:03.630 align:middle line:84% In fact, they dominate-- it dominates 00:58:03.630 --> 00:58:06.000 align:middle line:84% the shape of the bond distribution, particularly, 00:58:06.000 --> 00:58:09.570 align:middle line:84% near the source drain junctions of the MOSFET device. 00:58:09.570 --> 00:58:13.050 align:middle line:84% The boron basically gets pulled in to underneath the source 00:58:13.050 --> 00:58:16.230 align:middle line:84% and drains, and there's quite a bit of loss of boron 00:58:16.230 --> 00:58:20.205 align:middle line:84% from the channel region into the source trains that 00:58:20.205 --> 00:58:22.080 align:middle line:84% do electric field effect that's not predicted 00:58:22.080 --> 00:58:24.550 align:middle line:90% on the left-hand side. 00:58:24.550 --> 00:58:27.600 align:middle line:84% So even though h, the h factor has been in the arsenic, 00:58:27.600 --> 00:58:31.200 align:middle line:84% diffusivity is limited to, at most, a factor of 2. 00:58:31.200 --> 00:58:34.110 align:middle line:84% So there's not too much effect on the junction of the arsenic, 00:58:34.110 --> 00:58:36.940 align:middle line:90% if it's in the black junctions. 00:58:36.940 --> 00:58:39.870 align:middle line:84% The regions don't change that significantly. 00:58:39.870 --> 00:58:44.400 align:middle line:84% But it dominates the diffusion of the boron underneath, 00:58:44.400 --> 00:58:45.780 align:middle line:90% or at a lower concentration. 00:58:45.780 --> 00:58:49.830 align:middle line:90% 00:58:49.830 --> 00:58:51.560 align:middle line:84% So let's go on to slide number 25. 00:58:51.560 --> 00:58:55.670 align:middle line:84% I just want to summarize this second lecture on diffusion. 00:58:55.670 --> 00:59:00.770 align:middle line:84% We reviewed a few cases of simple constant diffusivity. 00:59:00.770 --> 00:59:04.370 align:middle line:84% These generally apply well to the case of low doping 00:59:04.370 --> 00:59:07.280 align:middle line:84% concentrations in a seamless process flow 00:59:07.280 --> 00:59:10.190 align:middle line:84% that usually means the diffusion of the seamless wells, which 00:59:10.190 --> 00:59:15.560 align:middle line:84% are fairly lightly doped, say, below mid 10 of the 17th, 00:59:15.560 --> 00:59:17.663 align:middle line:90% or below 10 to the 18th. 00:59:17.663 --> 00:59:19.830 align:middle line:84% They're usually diffuse in at very high temperatures 00:59:19.830 --> 00:59:21.380 align:middle line:90% so n sub i is large. 00:59:21.380 --> 00:59:25.080 align:middle line:84% And they satisfy the fact that the dopant concentration 00:59:25.080 --> 00:59:26.060 align:middle line:90% was less than ni. 00:59:26.060 --> 00:59:28.670 align:middle line:84% So you have constant diffusivities. 00:59:28.670 --> 00:59:32.240 align:middle line:84% But beyond that case, turns out most diffusion problems 00:59:32.240 --> 00:59:36.050 align:middle line:84% in devices need to be solved numerically. 00:59:36.050 --> 00:59:39.740 align:middle line:84% And we introduced a very simple but powerful method 00:59:39.740 --> 00:59:42.020 align:middle line:84% for numerical solution of the diffusion equation. 00:59:42.020 --> 00:59:45.860 align:middle line:84% And it actually was a finite difference formula. 00:59:45.860 --> 00:59:49.740 align:middle line:84% It's not that sophisticated, and it's not 00:59:49.740 --> 00:59:52.342 align:middle line:84% actually the formulation that is used in SUPREM IV, 00:59:52.342 --> 00:59:54.300 align:middle line:84% but it gives you an idea if you needed to write 00:59:54.300 --> 00:59:55.815 align:middle line:90% your own simple method. 00:59:55.815 --> 00:59:59.530 align:middle line:90% 00:59:59.530 --> 01:00:02.180 align:middle line:84% We talked about the first type of modification 01:00:02.180 --> 01:00:04.450 align:middle line:84% we need to make to Fick's first law, 01:00:04.450 --> 01:00:06.760 align:middle line:84% and that was for electric field effects, which 01:00:06.760 --> 01:00:10.480 align:middle line:84% can enhance the diffusivity of a high concentration species 01:00:10.480 --> 01:00:13.030 align:middle line:84% up to a factor of 2, but can dramatically 01:00:13.030 --> 01:00:17.590 align:middle line:84% impact the motion of species at lower concentrations. 01:00:17.590 --> 01:00:19.720 align:middle line:84% And what we're going to talk about next time 01:00:19.720 --> 01:00:24.833 align:middle line:84% are further corrections to the simple Ficking 01:00:24.833 --> 01:00:26.500 align:middle line:84% diffusion that we've been talking about, 01:00:26.500 --> 01:00:28.910 align:middle line:84% and more realistic dopant profiles, 01:00:28.910 --> 01:00:32.090 align:middle line:84% such as the case when we have concentration 01:00:32.090 --> 01:00:33.820 align:middle line:90% dependent diffusivity. 01:00:33.820 --> 01:00:38.940 align:middle line:84% OK, that's all I have for today, and thanks very much. 01:00:38.940 --> 01:00:42.000 align:middle line:90%