WEBVTT 00:00:00.000 --> 00:00:00.890 align:middle line:90% 00:00:00.890 --> 00:00:06.800 align:middle line:84% JUDY HOYT: We're going to begin this lecture on handout number 00:00:06.800 --> 00:00:07.770 align:middle line:90% 14. 00:00:07.770 --> 00:00:10.820 align:middle line:84% We'll be moving now to chapter 7. 00:00:10.820 --> 00:00:14.060 align:middle line:84% This will be our first lecture on chapter 7 00:00:14.060 --> 00:00:17.520 align:middle line:84% on the topic of dopant diffusion and profile measurement. 00:00:17.520 --> 00:00:21.390 align:middle line:84% So far, we've discussed a number of major topics, 00:00:21.390 --> 00:00:26.310 align:middle line:84% including the fabrication of wafers themselves and cleaning, 00:00:26.310 --> 00:00:28.250 align:middle line:90% point defects in silicon. 00:00:28.250 --> 00:00:30.260 align:middle line:84% And the last couple of lectures, we've 00:00:30.260 --> 00:00:33.110 align:middle line:84% been talking about the details of silicon thermal oxidation, 00:00:33.110 --> 00:00:37.100 align:middle line:84% including two-dimensional and stress effects. 00:00:37.100 --> 00:00:40.040 align:middle line:84% During the next few lectures, including this one, 00:00:40.040 --> 00:00:42.830 align:middle line:84% we're going to discuss the accurate control and placement 00:00:42.830 --> 00:00:45.980 align:middle line:84% of active dopant regions through a process called 00:00:45.980 --> 00:00:48.260 align:middle line:90% dopant diffusion. 00:00:48.260 --> 00:00:50.630 align:middle line:84% Today I'm going to give an overall introduction 00:00:50.630 --> 00:00:54.440 align:middle line:84% to diffusion in silicon from chapter 7. 00:00:54.440 --> 00:00:57.080 align:middle line:84% So let's go on to slide number 2. 00:00:57.080 --> 00:01:00.740 align:middle line:84% And here I'd like to give an introduction 00:01:00.740 --> 00:01:03.740 align:middle line:84% to the basic concepts of why we care 00:01:03.740 --> 00:01:06.620 align:middle line:84% about the details of diffusion in silicon. 00:01:06.620 --> 00:01:10.400 align:middle line:84% And what's being shown here is obviously a silicon MOSFET. 00:01:10.400 --> 00:01:13.850 align:middle line:84% We see the source, drain, and gate regions. 00:01:13.850 --> 00:01:18.230 align:middle line:84% And each one of these regions has a certain resistance 00:01:18.230 --> 00:01:20.840 align:middle line:90% associated with it. 00:01:20.840 --> 00:01:23.810 align:middle line:84% And that resistance turns out to be dramatically dependent 00:01:23.810 --> 00:01:26.600 align:middle line:84% on the placement of the atoms themselves 00:01:26.600 --> 00:01:28.070 align:middle line:90% and of the doped regions. 00:01:28.070 --> 00:01:31.520 align:middle line:84% And not only that, but the placement of those 00:01:31.520 --> 00:01:34.040 align:middle line:84% regions determines many of the so-called short channel 00:01:34.040 --> 00:01:38.550 align:middle line:84% characteristics of MOSFETs that we'll talk about. 00:01:38.550 --> 00:01:43.460 align:middle line:84% So as the device shrinks by some scale factor, say k-- 00:01:43.460 --> 00:01:45.770 align:middle line:84% we make some dimensions smaller-- 00:01:45.770 --> 00:01:49.730 align:middle line:84% the junction depths should also scale by k 00:01:49.730 --> 00:01:52.550 align:middle line:84% to maintain the same electric field 00:01:52.550 --> 00:01:55.290 align:middle line:84% patterns in the lateral and vertical dimensions. 00:01:55.290 --> 00:01:58.380 align:middle line:90% 00:01:58.380 --> 00:02:00.480 align:middle line:84% And so that's an important region the reason 00:02:00.480 --> 00:02:03.030 align:middle line:84% we need to control the doping profiles. 00:02:03.030 --> 00:02:05.910 align:middle line:84% And finally, the doping of other materials, not just 00:02:05.910 --> 00:02:08.490 align:middle line:84% the silicon itself, but of the polysilicon gate 00:02:08.490 --> 00:02:10.560 align:middle line:84% affects things like gate depletion 00:02:10.560 --> 00:02:14.190 align:middle line:84% and limits how well the gate voltage controls the channel 00:02:14.190 --> 00:02:15.830 align:middle line:90% potential. 00:02:15.830 --> 00:02:20.890 align:middle line:84% So we really need to understand the placement of these atoms. 00:02:20.890 --> 00:02:24.100 align:middle line:90% Let's go on to slide 3. 00:02:24.100 --> 00:02:26.615 align:middle line:84% Since we just talked about the idea 00:02:26.615 --> 00:02:28.490 align:middle line:84% of the resistance of these different regions, 00:02:28.490 --> 00:02:33.100 align:middle line:84% I just want to remind people what the general form 00:02:33.100 --> 00:02:35.290 align:middle line:84% for how one calculates the resistance 00:02:35.290 --> 00:02:38.710 align:middle line:90% of either a bar or a sheet. 00:02:38.710 --> 00:02:40.930 align:middle line:84% We know that the resistance in ohms 00:02:40.930 --> 00:02:43.720 align:middle line:84% can be calculated by the product of rho, 00:02:43.720 --> 00:02:46.390 align:middle line:84% the resistivity of the material, which has the units of ohm 00:02:46.390 --> 00:02:51.160 align:middle line:84% centimeters, times the length of the resistor bar divided 00:02:51.160 --> 00:02:53.500 align:middle line:84% by the cross-sectional area through which the current 00:02:53.500 --> 00:02:54.440 align:middle line:90% flows. 00:02:54.440 --> 00:02:58.330 align:middle line:84% So the resistance in rho L over a. 00:02:58.330 --> 00:03:03.200 align:middle line:84% Shown on the left is a picture of a cube. 00:03:03.200 --> 00:03:07.270 align:middle line:84% So let's say we have a cube, and we can calculate its resistance 00:03:07.270 --> 00:03:10.090 align:middle line:84% as just being the resistive times the length 00:03:10.090 --> 00:03:14.060 align:middle line:84% of the cube divided by this cross-sectional area. 00:03:14.060 --> 00:03:18.250 align:middle line:84% This resistivity, where the resistivity of the cube 00:03:18.250 --> 00:03:22.840 align:middle line:84% is given by, essentially, the electric field 00:03:22.840 --> 00:03:26.020 align:middle line:90% divided by the current density. 00:03:26.020 --> 00:03:28.440 align:middle line:84% So now, if we look on the right, and instead 00:03:28.440 --> 00:03:31.698 align:middle line:84% of having a cube-- in semiconductors or in silicon, 00:03:31.698 --> 00:03:33.990 align:middle line:84% we typically don't have a cube or a chunk of material-- 00:03:33.990 --> 00:03:37.230 align:middle line:84% we're usually measuring the resistance of a thin sheet 00:03:37.230 --> 00:03:39.720 align:middle line:90% in the near surface region. 00:03:39.720 --> 00:03:43.440 align:middle line:84% And that sheet generally has dimensions 00:03:43.440 --> 00:03:48.670 align:middle line:84% of L length and width that are much larger than its thickness. 00:03:48.670 --> 00:03:50.370 align:middle line:84% So if we have a shallow junction, 00:03:50.370 --> 00:03:54.990 align:middle line:84% as pictured on the right, we can calculate its resistance 00:03:54.990 --> 00:03:57.600 align:middle line:84% as follows, again, using the same formula 00:03:57.600 --> 00:03:59.170 align:middle line:90% at the top of the page. 00:03:59.170 --> 00:04:03.430 align:middle line:84% It's the resistivity in ohm centimeters times the length 00:04:03.430 --> 00:04:05.400 align:middle line:90% now divided by the area. 00:04:05.400 --> 00:04:09.240 align:middle line:84% But now we're assuming here that we have a square sheet 00:04:09.240 --> 00:04:13.540 align:middle line:84% so that the length and the width of the square are equal-- 00:04:13.540 --> 00:04:15.390 align:middle line:90% in this case, equal to W. 00:04:15.390 --> 00:04:21.720 align:middle line:84% So in that case, the W's cancel out because we have the length 00:04:21.720 --> 00:04:22.560 align:middle line:90% equal to W's. 00:04:22.560 --> 00:04:25.890 align:middle line:84% And when we divide by the cross-sectional area 00:04:25.890 --> 00:04:29.160 align:middle line:84% through which the current flows, it's equal to W times xj. 00:04:29.160 --> 00:04:30.550 align:middle line:90% The W's cancel out. 00:04:30.550 --> 00:04:33.540 align:middle line:90% So we end up at the resistance. 00:04:33.540 --> 00:04:37.680 align:middle line:84% The sheet resistance is just given by rho 00:04:37.680 --> 00:04:40.260 align:middle line:90% over the resistivity over xj. 00:04:40.260 --> 00:04:44.280 align:middle line:84% So that's a simple way and a convenient way 00:04:44.280 --> 00:04:46.740 align:middle line:84% of calculating resistance of various structures 00:04:46.740 --> 00:04:50.130 align:middle line:90% in semiconductor devices. 00:04:50.130 --> 00:04:54.590 align:middle line:84% So let's go on to slide number 4. 00:04:54.590 --> 00:04:56.660 align:middle line:84% What we just talked about was for a sheet 00:04:56.660 --> 00:05:00.470 align:middle line:84% that was uniformly doped to a given doping concentration. 00:05:00.470 --> 00:05:03.290 align:middle line:84% If the doping concentration throughout the sheet 00:05:03.290 --> 00:05:06.590 align:middle line:84% is non-uniformly doped in depth, then we 00:05:06.590 --> 00:05:08.420 align:middle line:84% can still calculate the sheet resistance, 00:05:08.420 --> 00:05:11.150 align:middle line:90% but we need to do an integral. 00:05:11.150 --> 00:05:15.170 align:middle line:84% It's equal to-- the sheet resistance is just rho over xj. 00:05:15.170 --> 00:05:18.840 align:middle line:84% But at this point, we need to integrate. 00:05:18.840 --> 00:05:24.200 align:middle line:84% So we integrate 1 over the resistivity. 00:05:24.200 --> 00:05:30.410 align:middle line:84% So it's 1 over the integral of the carrier concentration N 00:05:30.410 --> 00:05:32.930 align:middle line:84% minus the background doping concentration NB-- 00:05:32.930 --> 00:05:35.900 align:middle line:84% so that's just the net doping concentration-- 00:05:35.900 --> 00:05:38.690 align:middle line:84% times the mobility, which is generally 00:05:38.690 --> 00:05:41.240 align:middle line:84% a function of the doping concentration dx. 00:05:41.240 --> 00:05:44.870 align:middle line:84% We integrate that over from 0 to xj. 00:05:44.870 --> 00:05:47.600 align:middle line:84% You can do that numerically, essentially, 00:05:47.600 --> 00:05:51.035 align:middle line:90% for an arbitrary doping profile. 00:05:51.035 --> 00:05:52.410 align:middle line:84% The equation has actually already 00:05:52.410 --> 00:05:55.920 align:middle line:84% been integrated numerically for certain special analytic 00:05:55.920 --> 00:06:01.030 align:middle line:84% profiles, and we'll talk about those results later. 00:06:01.030 --> 00:06:04.410 align:middle line:84% So that's basically how we get from the doping 00:06:04.410 --> 00:06:07.290 align:middle line:84% profile to the electrical properties such as the sheet 00:06:07.290 --> 00:06:10.080 align:middle line:84% resistance or the resistance of a region. 00:06:10.080 --> 00:06:12.960 align:middle line:84% Experimentally, we measure the sheet resistance using 00:06:12.960 --> 00:06:14.580 align:middle line:90% a four-point probe setup-- 00:06:14.580 --> 00:06:17.490 align:middle line:84% I think we discussed this a couple of lectures ago-- 00:06:17.490 --> 00:06:23.080 align:middle line:84% or a Van der Pauw structure, as discussed in your text. 00:06:23.080 --> 00:06:25.240 align:middle line:90% Let's go on to slide number 5. 00:06:25.240 --> 00:06:28.780 align:middle line:84% Here, again, I'm picturing that same MOSFET structure 00:06:28.780 --> 00:06:31.810 align:middle line:84% with the resistance of the different regions. 00:06:31.810 --> 00:06:36.650 align:middle line:84% And in general, as a rule of thumb, of course, 00:06:36.650 --> 00:06:38.740 align:middle line:84% we'd like the resistance of the regions that 00:06:38.740 --> 00:06:43.480 align:middle line:84% are extrinsic to the device, such as the contact resistance, 00:06:43.480 --> 00:06:46.150 align:middle line:84% the source and drain resistance, and the resistance 00:06:46.150 --> 00:06:48.730 align:middle line:84% of these extension regions, hopefully should 00:06:48.730 --> 00:06:52.060 align:middle line:84% be no more than about 10% of the channel resistance. 00:06:52.060 --> 00:06:54.880 align:middle line:84% That is, we'd like to have the intrinsic resistance 00:06:54.880 --> 00:06:57.190 align:middle line:84% of the channel dominate the overall resistance 00:06:57.190 --> 00:07:01.520 align:middle line:84% of the device because that's what the gate has control over. 00:07:01.520 --> 00:07:04.120 align:middle line:84% So if we apply this 10% criterion, 00:07:04.120 --> 00:07:09.010 align:middle line:84% you can write the equation that's shown on slide 5 00:07:09.010 --> 00:07:09.760 align:middle line:90% as follows. 00:07:09.760 --> 00:07:12.260 align:middle line:84% That 2 times the contact resistance-- 00:07:12.260 --> 00:07:15.220 align:middle line:84% so that's the resistance of the metal contact-- 00:07:15.220 --> 00:07:19.000 align:middle line:84% plus the resistance of the source region, as shown here, 00:07:19.000 --> 00:07:21.500 align:middle line:84% plus the resistance of the drain, 00:07:21.500 --> 00:07:25.310 align:middle line:84% plus 2 times the resistance of the source drain extensions. 00:07:25.310 --> 00:07:27.050 align:middle line:84% Remember, these shallow extensions 00:07:27.050 --> 00:07:33.350 align:middle line:84% are the shallow xj regions that attach the source and drains, 00:07:33.350 --> 00:07:35.690 align:middle line:84% essentially, to the channel region. 00:07:35.690 --> 00:07:38.780 align:middle line:84% We'd like the sum of those terms to be less than 00:07:38.780 --> 00:07:43.410 align:middle line:84% or equal to 1/10 the channel resistance. 00:07:43.410 --> 00:07:46.820 align:middle line:84% So, in general, as we scale devices, 00:07:46.820 --> 00:07:48.620 align:middle line:84% the channel length becomes shorter 00:07:48.620 --> 00:07:50.580 align:middle line:84% and the channel resistance goes down. 00:07:50.580 --> 00:07:53.330 align:middle line:84% So we similarly need to scale down 00:07:53.330 --> 00:07:56.030 align:middle line:84% these extrinsic resistances of the source drain 00:07:56.030 --> 00:07:59.070 align:middle line:90% and the extension regions. 00:07:59.070 --> 00:08:02.660 align:middle line:84% So to reduce those resistances-- those parasitic regions-- 00:08:02.660 --> 00:08:08.510 align:middle line:84% we would like to increase, in general, the junction depth xj. 00:08:08.510 --> 00:08:11.540 align:middle line:84% However, there's a problem that if we make the junctions 00:08:11.540 --> 00:08:15.440 align:middle line:84% deeper, it will make it easier for voltages at the drain 00:08:15.440 --> 00:08:19.300 align:middle line:84% to affect the current flow in the channel because of the way 00:08:19.300 --> 00:08:21.390 align:middle line:84% the field patterns are established. 00:08:21.390 --> 00:08:23.450 align:middle line:84% So this two-dimensional spreading 00:08:23.450 --> 00:08:25.910 align:middle line:84% of the electric field from the drain 00:08:25.910 --> 00:08:29.570 align:middle line:84% can attract carriers from the source, even when the device is 00:08:29.570 --> 00:08:31.170 align:middle line:90% supposedly in the off state. 00:08:31.170 --> 00:08:33.710 align:middle line:84% So we end up with something called drain-induced barrier 00:08:33.710 --> 00:08:39.049 align:middle line:84% lowering if the junctions are too deep. 00:08:39.049 --> 00:08:41.450 align:middle line:84% So this results in, for device designers, 00:08:41.450 --> 00:08:45.380 align:middle line:84% kind of a fundamental design tradeoff for MOSFETs 00:08:45.380 --> 00:08:49.670 align:middle line:84% that is a design trade off between the series resistance 00:08:49.670 --> 00:08:53.150 align:middle line:84% versus the DIBL or the ability of the gate 00:08:53.150 --> 00:08:58.620 align:middle line:84% to control the current and turn the device on and off. 00:08:58.620 --> 00:09:02.353 align:middle line:84% So if we go on to slide number 6, 00:09:02.353 --> 00:09:03.770 align:middle line:84% essentially what we're saying here 00:09:03.770 --> 00:09:05.720 align:middle line:84% is that there is a major challenge 00:09:05.720 --> 00:09:07.880 align:middle line:84% that we need to keep the junctions shallow 00:09:07.880 --> 00:09:11.060 align:middle line:84% so that DIBL and short-channel effects are 00:09:11.060 --> 00:09:13.760 align:middle line:90% reduced as we scale. 00:09:13.760 --> 00:09:15.530 align:middle line:84% But at the same time, we need to keep 00:09:15.530 --> 00:09:17.570 align:middle line:84% the resistance of the source-drain region 00:09:17.570 --> 00:09:20.640 align:middle line:84% small so that we can maximize the current drive get 00:09:20.640 --> 00:09:22.640 align:middle line:84% the maximum amount of current out of the device. 00:09:22.640 --> 00:09:25.990 align:middle line:84% And these are conflicting requirements. 00:09:25.990 --> 00:09:28.770 align:middle line:84% And you can see the effect of these conflicting requirements 00:09:28.770 --> 00:09:32.100 align:middle line:84% to a certain extent by examining this chart. 00:09:32.100 --> 00:09:34.660 align:middle line:84% This is a chart that I took out of the text. 00:09:34.660 --> 00:09:36.720 align:middle line:84% It's a little bit dated in the sense 00:09:36.720 --> 00:09:42.000 align:middle line:84% it's from the 1997 Technology Roadmap for Semiconductors. 00:09:42.000 --> 00:09:44.340 align:middle line:84% But since it's consistent with what's in your text, 00:09:44.340 --> 00:09:45.840 align:middle line:90% we'll go ahead and look at it. 00:09:45.840 --> 00:09:50.560 align:middle line:84% If you look at, say, the last three rows in the chart, 00:09:50.560 --> 00:09:56.175 align:middle line:84% you'll see the contact region junction depth contact 00:09:56.175 --> 00:10:00.780 align:middle line:84% xj from, say, the year 2000 being in the range 00:10:00.780 --> 00:10:02.710 align:middle line:90% of 50 to 100 nanometers. 00:10:02.710 --> 00:10:06.600 align:middle line:84% And as we go smaller and smaller channel lengths out further 00:10:06.600 --> 00:10:11.160 align:middle line:84% in time, that's scaling down to shallower and shallower 00:10:11.160 --> 00:10:14.070 align:middle line:90% junction depths at the channel. 00:10:14.070 --> 00:10:17.670 align:middle line:84% So right near the channel or in the source-drain extension 00:10:17.670 --> 00:10:20.700 align:middle line:84% regions, you can see that's even much thinner, 00:10:20.700 --> 00:10:26.080 align:middle line:84% and that also scales down with time to thicknesses by 2009 00:10:26.080 --> 00:10:30.020 align:middle line:84% on the order of 15 to 30 nanometers-- quite shallow. 00:10:30.020 --> 00:10:33.400 align:middle line:84% And at the same time as we're scaling these junctions 00:10:33.400 --> 00:10:37.330 align:middle line:84% to maintain the good electrostatic control, 00:10:37.330 --> 00:10:39.590 align:middle line:84% the drain extension concentration-- 00:10:39.590 --> 00:10:42.790 align:middle line:84% so that's the doping concentration in the extension 00:10:42.790 --> 00:10:46.090 align:middle line:84% regions-- is going up dramatically, say, 00:10:46.090 --> 00:10:50.020 align:middle line:84% from 10 to the 19th up to 10 to the 20 or perhaps even higher. 00:10:50.020 --> 00:10:53.800 align:middle line:84% And that higher doping requirement 00:10:53.800 --> 00:10:56.800 align:middle line:84% is arising from the fact that we're 00:10:56.800 --> 00:10:59.140 align:middle line:90% making the sheet shallower. 00:10:59.140 --> 00:11:03.730 align:middle line:84% The junction depth is smaller to compensate for that 00:11:03.730 --> 00:11:06.160 align:middle line:90% and have the resistance go down. 00:11:06.160 --> 00:11:08.993 align:middle line:84% We really need to up the dopant concentration. 00:11:08.993 --> 00:11:10.660 align:middle line:84% And there's a fundamental physical limit 00:11:10.660 --> 00:11:15.100 align:middle line:84% on how much dopant we can put in the silicon 00:11:15.100 --> 00:11:17.000 align:middle line:84% and how much it will be electrically active. 00:11:17.000 --> 00:11:20.060 align:middle line:84% So this is becoming a real problem. 00:11:20.060 --> 00:11:23.320 align:middle line:84% We need to find new ways to activate dopants to higher 00:11:23.320 --> 00:11:26.800 align:middle line:84% levels if we're going to be able to manage this design tradeoff. 00:11:26.800 --> 00:11:29.660 align:middle line:90% 00:11:29.660 --> 00:11:32.760 align:middle line:84% So let's go on to slide 7, again, 00:11:32.760 --> 00:11:34.970 align:middle line:84% which is that same picture, to remind you. 00:11:34.970 --> 00:11:39.590 align:middle line:84% So, basically, the ITRS requirements in the future 00:11:39.590 --> 00:11:44.810 align:middle line:84% really require or dictate that we know the dopant positions 00:11:44.810 --> 00:11:48.710 align:middle line:84% in the device with almost atomic-scale accuracy 00:11:48.710 --> 00:11:51.390 align:middle line:84% in both two-dimensional and three-dimensional profiles. 00:11:51.390 --> 00:11:54.170 align:middle line:84% So being able to scale the device really 00:11:54.170 --> 00:11:57.920 align:middle line:84% amounts to, in the front-end processing to a large extent, 00:11:57.920 --> 00:12:00.110 align:middle line:84% to being able to control very precisely 00:12:00.110 --> 00:12:03.980 align:middle line:84% the shape of the doping profiles where the dopants end up. 00:12:03.980 --> 00:12:08.240 align:middle line:84% And what I'm going to spend some time in the next few slides 00:12:08.240 --> 00:12:11.720 align:middle line:84% is giving you examples from the present literature 00:12:11.720 --> 00:12:16.520 align:middle line:84% on device scaling, which emphasize or give you 00:12:16.520 --> 00:12:20.240 align:middle line:84% some sense of how these doping profiles need to be controlled 00:12:20.240 --> 00:12:23.130 align:middle line:84% and what their impact is on device performance. 00:12:23.130 --> 00:12:27.533 align:middle line:84% So, again, perhaps you won't understand the detailed device 00:12:27.533 --> 00:12:28.950 align:middle line:84% physics, but it's just to give you 00:12:28.950 --> 00:12:33.900 align:middle line:84% a flavor for why studying dopant diffusion 00:12:33.900 --> 00:12:36.495 align:middle line:90% is such an important topic. 00:12:36.495 --> 00:12:39.040 align:middle line:90% 00:12:39.040 --> 00:12:44.520 align:middle line:84% So let's go on to slide number 8 and talk about a topic called 00:12:44.520 --> 00:12:46.680 align:middle line:90% the short-channel effect. 00:12:46.680 --> 00:12:52.410 align:middle line:84% And this basically takes place when the distance 00:12:52.410 --> 00:12:53.760 align:middle line:90% between the source and drain-- 00:12:53.760 --> 00:12:56.370 align:middle line:90% that is the channel length L-- 00:12:56.370 --> 00:13:00.720 align:middle line:84% becomes comparable to the MOS depletion width 00:13:00.720 --> 00:13:04.060 align:middle line:90% in the vertical direction. 00:13:04.060 --> 00:13:08.370 align:middle line:84% And then that the source-drain potentials themselves 00:13:08.370 --> 00:13:10.170 align:middle line:84% from the source and drain regions 00:13:10.170 --> 00:13:13.200 align:middle line:84% end up having a strong effect on the control of the current 00:13:13.200 --> 00:13:14.260 align:middle line:90% in the device. 00:13:14.260 --> 00:13:16.890 align:middle line:84% So in words, that's a way of expressing 00:13:16.890 --> 00:13:18.810 align:middle line:90% the short-channel effect. 00:13:18.810 --> 00:13:23.160 align:middle line:84% And what I'm showing the top equation on slide number 8 00:13:23.160 --> 00:13:25.872 align:middle line:84% is the equation we looked at briefly-- 00:13:25.872 --> 00:13:26.580 align:middle line:90% we didn't derive. 00:13:26.580 --> 00:13:30.540 align:middle line:84% We just wrote it down in class several lectures ago-- 00:13:30.540 --> 00:13:32.340 align:middle line:90% for the threshold voltage. 00:13:32.340 --> 00:13:34.050 align:middle line:84% And this is for the threshold voltage 00:13:34.050 --> 00:13:38.070 align:middle line:84% in a MOSFET that has a constant doping in the channel-- a very 00:13:38.070 --> 00:13:38.910 align:middle line:90% simple profile. 00:13:38.910 --> 00:13:39.870 align:middle line:90% It's just constant. 00:13:39.870 --> 00:13:42.240 align:middle line:84% And it's a relatively long channel device. 00:13:42.240 --> 00:13:44.490 align:middle line:84% So you will not see this short-channel effect. 00:13:44.490 --> 00:13:47.130 align:middle line:84% The gate length L is much, much longer 00:13:47.130 --> 00:13:49.020 align:middle line:90% than the depletion widths. 00:13:49.020 --> 00:13:51.690 align:middle line:84% And we said we can write down these three terms, roughly, 00:13:51.690 --> 00:13:53.832 align:middle line:84% to calculate the threshold voltage. 00:13:53.832 --> 00:13:55.290 align:middle line:84% And remember, the threshold voltage 00:13:55.290 --> 00:13:58.050 align:middle line:84% is an important parameter as far as the device and circuit 00:13:58.050 --> 00:14:00.450 align:middle line:90% designers are concerned. 00:14:00.450 --> 00:14:04.800 align:middle line:84% Now, when we get to the short-channel case, which 00:14:04.800 --> 00:14:08.610 align:middle line:84% is shown below, that threshold voltage equation 00:14:08.610 --> 00:14:11.760 align:middle line:84% has to be modified to a certain extent. 00:14:11.760 --> 00:14:15.900 align:middle line:84% And, in fact, the third term, which 00:14:15.900 --> 00:14:19.800 align:middle line:84% is represented on the second equation by the bulk charge 00:14:19.800 --> 00:14:24.350 align:middle line:90% QB prime over WLC ox. 00:14:24.350 --> 00:14:27.980 align:middle line:84% That term ends up being smaller than it would 00:14:27.980 --> 00:14:32.180 align:middle line:90% be in the long-channel case. 00:14:32.180 --> 00:14:36.170 align:middle line:84% And this QB prime is smaller, and that ends up 00:14:36.170 --> 00:14:39.830 align:middle line:84% affecting-- that third term being smaller affects the Vt. 00:14:39.830 --> 00:14:42.750 align:middle line:84% Threshold voltage actually goes down. 00:14:42.750 --> 00:14:45.890 align:middle line:84% And this diagram on the lower left 00:14:45.890 --> 00:14:52.340 align:middle line:84% is an explanation that people have proposed to explain this. 00:14:52.340 --> 00:14:55.490 align:middle line:84% And it's essentially because charge sharing-- 00:14:55.490 --> 00:14:58.880 align:middle line:84% that some of the charge under the gate 00:14:58.880 --> 00:15:03.350 align:middle line:84% is balanced by the charge from the source-drain regions. 00:15:03.350 --> 00:15:07.760 align:middle line:84% So that effectively gives you an effective channel length. 00:15:07.760 --> 00:15:12.060 align:middle line:84% The L prime is actually shorter than the actual gate length. 00:15:12.060 --> 00:15:17.510 align:middle line:84% So there's this tendency as we scale devices for the threshold 00:15:17.510 --> 00:15:20.282 align:middle line:84% voltage itself to drop or to roll off. 00:15:20.282 --> 00:15:21.990 align:middle line:84% And if you want to learn more about this, 00:15:21.990 --> 00:15:23.910 align:middle line:84% I've got a reference on the bottom. 00:15:23.910 --> 00:15:27.750 align:middle line:84% There are a number of books, but this is one by Taur and Ning 00:15:27.750 --> 00:15:32.490 align:middle line:84% where you can look at this so-called Vt roll-off effect. 00:15:32.490 --> 00:15:34.080 align:middle line:84% If we just go on to slide number 9, 00:15:34.080 --> 00:15:38.970 align:middle line:84% you can actually see some data on threshold voltage roll-off. 00:15:38.970 --> 00:15:41.100 align:middle line:90% The upper plot is for nMOSFETs. 00:15:41.100 --> 00:15:43.500 align:middle line:84% And you can see on the vertical axis 00:15:43.500 --> 00:15:47.220 align:middle line:84% is the threshold voltage Vt as a function of channel length. 00:15:47.220 --> 00:15:52.320 align:middle line:84% And these curves-- one is for in the linear regime 00:15:52.320 --> 00:15:55.020 align:middle line:90% with a low drain-to-source bias. 00:15:55.020 --> 00:15:58.560 align:middle line:84% The triangle is for the saturation regime 00:15:58.560 --> 00:16:00.940 align:middle line:84% with a source-drain bias of 3 volts. 00:16:00.940 --> 00:16:03.450 align:middle line:84% And you can see, indeed, the threshold voltage is dropping 00:16:03.450 --> 00:16:07.200 align:middle line:84% or rolling off going towards 0 as we decrease the channel 00:16:07.200 --> 00:16:08.050 align:middle line:90% length-- 00:16:08.050 --> 00:16:11.370 align:middle line:90% same trend for the pMOSFET. 00:16:11.370 --> 00:16:14.310 align:middle line:84% And that's an effect that needs to be controlled 00:16:14.310 --> 00:16:15.330 align:middle line:90% as we scale the devices. 00:16:15.330 --> 00:16:18.330 align:middle line:84% And the way we partly control that is 00:16:18.330 --> 00:16:24.220 align:middle line:84% by controlling the dopant profiles underneath the gate. 00:16:24.220 --> 00:16:27.690 align:middle line:84% So if we go on to slide number 10, 00:16:27.690 --> 00:16:29.820 align:middle line:84% it's discussing something called channel 00:16:29.820 --> 00:16:31.560 align:middle line:90% doping profile engineering. 00:16:31.560 --> 00:16:35.610 align:middle line:84% Essentially, this refers to optimizing the final doping 00:16:35.610 --> 00:16:41.940 align:middle line:84% profiles in of the p-type regions in the nMOS that 00:16:41.940 --> 00:16:44.610 align:middle line:84% is under the gate or the n-type regions 00:16:44.610 --> 00:16:49.840 align:middle line:84% under the gate in the pMOS in the channel region. 00:16:49.840 --> 00:16:55.500 align:middle line:84% So this is just a schematic cartoon picture 00:16:55.500 --> 00:16:57.360 align:middle line:90% of cross-section of a MOSFET. 00:16:57.360 --> 00:17:00.330 align:middle line:84% And this channel doping profile engineering 00:17:00.330 --> 00:17:04.050 align:middle line:84% is referring to these red regions-- 00:17:04.050 --> 00:17:08.160 align:middle line:84% both these dark red so-called halo regions 00:17:08.160 --> 00:17:11.700 align:middle line:84% that are marked here that go around the perimeter 00:17:11.700 --> 00:17:16.020 align:middle line:84% of the source-drain extensions and this lighter red 00:17:16.020 --> 00:17:17.880 align:middle line:90% retrograde well regions. 00:17:17.880 --> 00:17:19.920 align:middle line:84% These two-dimensional dopant profiles 00:17:19.920 --> 00:17:23.790 align:middle line:84% are engineered or designed to minimize 00:17:23.790 --> 00:17:25.289 align:middle line:90% these short-channel effects. 00:17:25.289 --> 00:17:27.990 align:middle line:90% 00:17:27.990 --> 00:17:31.500 align:middle line:84% So let's go on to page 11 and give you an example 00:17:31.500 --> 00:17:35.910 align:middle line:90% on page 11 taking in depth-- 00:17:35.910 --> 00:17:39.540 align:middle line:84% going right through the center of the channel-- 00:17:39.540 --> 00:17:43.470 align:middle line:84% the profile concentration of the channel 00:17:43.470 --> 00:17:45.690 align:middle line:84% of the doping as a function of depth. 00:17:45.690 --> 00:17:48.480 align:middle line:84% And there are two different doping profiles 00:17:48.480 --> 00:17:49.540 align:middle line:90% that are shown here. 00:17:49.540 --> 00:17:53.970 align:middle line:84% One is for a uniform well-- uniformly doped phosphorus. 00:17:53.970 --> 00:17:56.100 align:middle line:84% It's reasonably constant doping around 00:17:56.100 --> 00:17:59.070 align:middle line:90% the [? 2a17 ?] at the surface. 00:17:59.070 --> 00:18:03.300 align:middle line:90% And the other-- the blue line-- 00:18:03.300 --> 00:18:07.410 align:middle line:84% is the so-called super steep retrograde well, 00:18:07.410 --> 00:18:09.900 align:middle line:84% where you have certain well doping. 00:18:09.900 --> 00:18:12.000 align:middle line:84% And then near the surface, the profile 00:18:12.000 --> 00:18:16.330 align:middle line:84% falls off in doping very rapidly. 00:18:16.330 --> 00:18:20.610 align:middle line:84% These two profiles, as it turns out, in the channel region 00:18:20.610 --> 00:18:23.520 align:middle line:84% give very closely the same threshold voltage 00:18:23.520 --> 00:18:26.070 align:middle line:90% for a given device. 00:18:26.070 --> 00:18:29.650 align:middle line:84% But you get better leakage current control. 00:18:29.650 --> 00:18:34.460 align:middle line:84% So you can scale the device to a smaller L effective. 00:18:34.460 --> 00:18:36.100 align:middle line:84% So, again, this is an example of how 00:18:36.100 --> 00:18:40.690 align:middle line:84% you need to control the doping profiles in order 00:18:40.690 --> 00:18:42.670 align:middle line:90% to optimize the device design. 00:18:42.670 --> 00:18:45.280 align:middle line:90% 00:18:45.280 --> 00:18:48.070 align:middle line:84% Let's go on now to slide number 12. 00:18:48.070 --> 00:18:53.650 align:middle line:84% And this is sort of an extreme case of scaling, where we're 00:18:53.650 --> 00:18:56.020 align:middle line:84% trying to scale the MOSFET gate length down 00:18:56.020 --> 00:18:59.530 align:middle line:90% to 25 nanometers dimensions. 00:18:59.530 --> 00:19:04.120 align:middle line:84% And in order to do this, if you look back in the literature, 00:19:04.120 --> 00:19:08.860 align:middle line:84% there's a paper by Taur, Wann and Frank in 1998-- 00:19:08.860 --> 00:19:12.400 align:middle line:84% the so-called the super-halo profile. 00:19:12.400 --> 00:19:14.780 align:middle line:84% And what the super-halo profile is 00:19:14.780 --> 00:19:18.820 align:middle line:84% is a fairly sophisticated ion-implanted and then 00:19:18.820 --> 00:19:23.530 align:middle line:84% diffused profile, say, for an nMOSFET of boron doping 00:19:23.530 --> 00:19:26.200 align:middle line:84% that creates a fairly complicated 00:19:26.200 --> 00:19:28.540 align:middle line:84% two-dimensional boron doping profile. 00:19:28.540 --> 00:19:30.760 align:middle line:84% And what you're looking at here is the MOSFET. 00:19:30.760 --> 00:19:34.510 align:middle line:84% And in the central region here underneath the channel, 00:19:34.510 --> 00:19:37.150 align:middle line:84% you see these p-type doping contours. 00:19:37.150 --> 00:19:39.880 align:middle line:84% It looks sort of like butterfly shaped. 00:19:39.880 --> 00:19:43.090 align:middle line:84% There's one contour here shown that around a doping of 10 00:19:43.090 --> 00:19:48.460 align:middle line:84% to the 19th and another contour shown at about 5 10 the 18th. 00:19:48.460 --> 00:19:54.190 align:middle line:84% And you can see this is designed to put reasonably high boron 00:19:54.190 --> 00:20:00.100 align:middle line:84% doping against the source-drain regions 00:20:00.100 --> 00:20:05.050 align:middle line:84% to help stand off the field in this 00:20:05.050 --> 00:20:08.720 align:middle line:90% from the source-drain regions. 00:20:08.720 --> 00:20:14.740 align:middle line:84% And so that's refers to this halo design-- 00:20:14.740 --> 00:20:15.940 align:middle line:90% very sophisticated. 00:20:15.940 --> 00:20:19.780 align:middle line:84% And there's also the doping of the source and drain regions 00:20:19.780 --> 00:20:22.990 align:middle line:84% themselves designed to be quite abrupt in both the vertical 00:20:22.990 --> 00:20:25.510 align:middle line:90% and the lateral dimensions. 00:20:25.510 --> 00:20:28.090 align:middle line:90% 00:20:28.090 --> 00:20:31.540 align:middle line:84% So what exactly are the effect of these doping profiles 00:20:31.540 --> 00:20:33.280 align:middle line:90% on electrical performance? 00:20:33.280 --> 00:20:37.090 align:middle line:84% Let's go on to slide number 13, which 00:20:37.090 --> 00:20:41.500 align:middle line:84% shows the short-channel threshold 00:20:41.500 --> 00:20:45.460 align:middle line:84% voltage roll-off and basically how the Vt varies 00:20:45.460 --> 00:20:46.720 align:middle line:90% with channel length. 00:20:46.720 --> 00:20:48.880 align:middle line:84% So on the y-axis is the threshold voltage 00:20:48.880 --> 00:20:49.990 align:middle line:90% of the device. 00:20:49.990 --> 00:20:53.713 align:middle line:84% The x-axis is the channel length in nanometers. 00:20:53.713 --> 00:20:55.630 align:middle line:84% And there's a couple of different designs here 00:20:55.630 --> 00:20:57.220 align:middle line:90% that are shown. 00:20:57.220 --> 00:21:00.340 align:middle line:84% The dashed line with the open squares 00:21:00.340 --> 00:21:01.780 align:middle line:90% refers to the retrograde. 00:21:01.780 --> 00:21:05.200 align:middle line:84% So that's not the super halo, but a more classical 00:21:05.200 --> 00:21:08.020 align:middle line:90% super steep retrograde profile. 00:21:08.020 --> 00:21:10.360 align:middle line:84% You can see it's threshold voltage rolls 00:21:10.360 --> 00:21:13.690 align:middle line:84% off quite rapidly as we shrink the channel 00:21:13.690 --> 00:21:15.320 align:middle line:90% length below 30 nanometers. 00:21:15.320 --> 00:21:17.260 align:middle line:90% So that's not going to work. 00:21:17.260 --> 00:21:21.280 align:middle line:84% But the super-halo profile shown by the diamonds and the stars 00:21:21.280 --> 00:21:24.470 align:middle line:84% have a much flatter, nearly flat short channel 00:21:24.470 --> 00:21:27.610 align:middle line:90% and Vt roll-off characteristics. 00:21:27.610 --> 00:21:29.683 align:middle line:84% And furthermore, the Vt roll-off is not 00:21:29.683 --> 00:21:31.600 align:middle line:84% that sensitive to the vertical junction depth, 00:21:31.600 --> 00:21:35.780 align:middle line:84% as you can see by comparing the diamonds to the stars. 00:21:35.780 --> 00:21:39.040 align:middle line:84% So this lower variation of the Vt 00:21:39.040 --> 00:21:41.980 align:middle line:84% with L effective or with channel length 00:21:41.980 --> 00:21:45.160 align:middle line:84% allows a larger design window, which 00:21:45.160 --> 00:21:46.660 align:middle line:84% we need because there's always going 00:21:46.660 --> 00:21:49.300 align:middle line:84% to be some process variations in the channel 00:21:49.300 --> 00:21:52.300 align:middle line:90% length across the wafer. 00:21:52.300 --> 00:21:54.850 align:middle line:84% And this enables the technologists 00:21:54.850 --> 00:21:57.920 align:middle line:84% to push the channel length down to smaller dimensions. 00:21:57.920 --> 00:21:59.800 align:middle line:84% So it's not so much a fundamental improvement 00:21:59.800 --> 00:22:01.570 align:middle line:84% in device performance, but it really 00:22:01.570 --> 00:22:06.400 align:middle line:84% enables you to manufacture circuits with these shorter 00:22:06.400 --> 00:22:07.630 align:middle line:90% channel devices. 00:22:07.630 --> 00:22:10.120 align:middle line:84% And again, a lot of it boils down 00:22:10.120 --> 00:22:12.940 align:middle line:84% to controlling and detailed doping 00:22:12.940 --> 00:22:17.980 align:middle line:84% profiles in the source train and under in the channel regions. 00:22:17.980 --> 00:22:20.490 align:middle line:90% 00:22:20.490 --> 00:22:22.250 align:middle line:84% Let's take a look at slide 14, again, 00:22:22.250 --> 00:22:25.230 align:middle line:84% is another example of the fact that not only do 00:22:25.230 --> 00:22:28.260 align:middle line:84% we care about the doping in the vertical direction, of course, 00:22:28.260 --> 00:22:33.090 align:middle line:84% but the dopant profiles for the source-grain dopants themselves 00:22:33.090 --> 00:22:36.750 align:middle line:84% in the lateral direction are very important because it's not 00:22:36.750 --> 00:22:40.560 align:middle line:84% just the junction depth, but it's the lateral gradient 00:22:40.560 --> 00:22:42.340 align:middle line:90% of the source-grain doping. 00:22:42.340 --> 00:22:44.610 align:middle line:84% So this is a plot of the threshold voltage 00:22:44.610 --> 00:22:46.200 align:middle line:90% versus channel length again. 00:22:46.200 --> 00:22:47.790 align:middle line:84% But this time, the different curves 00:22:47.790 --> 00:22:52.480 align:middle line:84% refer to different lateral source-drain gradients. 00:22:52.480 --> 00:22:57.160 align:middle line:84% So you've got the top curve refers to a lateral gradient-- 00:22:57.160 --> 00:23:01.830 align:middle line:84% so how quickly the arsenic doping profile rolls off 00:23:01.830 --> 00:23:04.020 align:middle line:84% at about 2 nanometers per decade. 00:23:04.020 --> 00:23:07.890 align:middle line:84% We've got a curve at 4, 8 and 16. 00:23:07.890 --> 00:23:12.060 align:middle line:84% And you can see that for lateral gradients larger 00:23:12.060 --> 00:23:14.860 align:middle line:84% than about 4 nanometers per decade, 00:23:14.860 --> 00:23:17.790 align:middle line:84% the Vt roll-off is just too large. 00:23:17.790 --> 00:23:20.860 align:middle line:84% The threshold voltage is approaching 0. 00:23:20.860 --> 00:23:24.310 align:middle line:84% You wouldn't be able to make a 25 nanometer MOSFET-- 00:23:24.310 --> 00:23:25.990 align:middle line:84% so, again, illustrates the importance 00:23:25.990 --> 00:23:28.520 align:middle line:84% of controlling the lateral doping profile 00:23:28.520 --> 00:23:33.800 align:middle line:84% and of controlling diffusion processes themselves. 00:23:33.800 --> 00:23:39.580 align:middle line:84% So given that brief introduction to the electrical effects, 00:23:39.580 --> 00:23:43.300 align:middle line:84% let me go on now on slide number 15 00:23:43.300 --> 00:23:46.347 align:middle line:84% and talk about dopant diffusion fundamentals. 00:23:46.347 --> 00:23:48.430 align:middle line:84% So I've tried to make the point that understanding 00:23:48.430 --> 00:23:50.145 align:middle line:84% these profiles in detail is important. 00:23:50.145 --> 00:23:51.520 align:middle line:84% And that's what we're going to do 00:23:51.520 --> 00:23:53.470 align:middle line:90% in the next several lectures. 00:23:53.470 --> 00:23:54.647 align:middle line:90% So what is diffusion? 00:23:54.647 --> 00:23:56.230 align:middle line:84% Diffusion is really the redistribution 00:23:56.230 --> 00:24:00.430 align:middle line:84% of atoms from regions where they exist in high concentration 00:24:00.430 --> 00:24:03.550 align:middle line:90% to regions of low concentration. 00:24:03.550 --> 00:24:05.740 align:middle line:84% Diffusion occurs essentially at all temperatures. 00:24:05.740 --> 00:24:09.280 align:middle line:84% But the diffusivity-- or the diffusion coefficient-- 00:24:09.280 --> 00:24:11.630 align:middle line:84% has an exponential dependence on temperature. 00:24:11.630 --> 00:24:14.920 align:middle line:84% So above a certain temperature is when the diffusion rates 00:24:14.920 --> 00:24:18.760 align:middle line:90% really become large. 00:24:18.760 --> 00:24:22.900 align:middle line:84% In silicon IC processing, there are two different steps 00:24:22.900 --> 00:24:25.720 align:middle line:84% that we refer to in diffusion historically. 00:24:25.720 --> 00:24:29.050 align:middle line:84% The first step was so-called predeposition. 00:24:29.050 --> 00:24:32.500 align:middle line:90% 00:24:32.500 --> 00:24:34.630 align:middle line:84% And what this refers to is that you 00:24:34.630 --> 00:24:38.410 align:middle line:84% had an initial step in which the dopants were introduced 00:24:38.410 --> 00:24:41.710 align:middle line:84% into the silicon wafer with a required integrated 00:24:41.710 --> 00:24:44.590 align:middle line:90% dose into the substrate. 00:24:44.590 --> 00:24:47.980 align:middle line:84% Originally, in the early days, this predeposition step 00:24:47.980 --> 00:24:49.540 align:middle line:84% to introduce the dopants was done 00:24:49.540 --> 00:24:55.510 align:middle line:84% by diffusion of the dopants into the wafer from a doped glass 00:24:55.510 --> 00:24:57.130 align:middle line:84% or by introducing into the silicon 00:24:57.130 --> 00:25:00.550 align:middle line:84% by heating in a doped gas ambient. 00:25:00.550 --> 00:25:04.210 align:middle line:84% The pre-dep is rarely done these days in that particular way. 00:25:04.210 --> 00:25:06.190 align:middle line:84% In the more modern technology, it's 00:25:06.190 --> 00:25:08.560 align:middle line:84% usually done by ion implantation, which 00:25:08.560 --> 00:25:11.180 align:middle line:84% is a process that we're going to discuss later 00:25:11.180 --> 00:25:15.040 align:middle line:84% and is covered in detail in chapter 8. 00:25:15.040 --> 00:25:19.300 align:middle line:84% Let's go on to slide number 16, which 00:25:19.300 --> 00:25:24.580 align:middle line:84% very pictorially illustrates the predeposition 00:25:24.580 --> 00:25:25.730 align:middle line:90% and the drive-in process. 00:25:25.730 --> 00:25:28.390 align:middle line:84% The second process in creating a region 00:25:28.390 --> 00:25:29.870 align:middle line:84% of the wafer with a certain doping 00:25:29.870 --> 00:25:33.040 align:middle line:84% is what is typically referred to as the drive-in. 00:25:33.040 --> 00:25:37.120 align:middle line:84% This is a subsequent anneal after the pre-dep that 00:25:37.120 --> 00:25:39.820 align:middle line:84% then diffuses and redistributes the dopant, 00:25:39.820 --> 00:25:42.190 align:middle line:84% giving the required junction depth 00:25:42.190 --> 00:25:43.990 align:middle line:84% that you need to get the right resistance 00:25:43.990 --> 00:25:46.990 align:middle line:84% and giving you the right profile or surface concentration, 00:25:46.990 --> 00:25:47.920 align:middle line:90% hopefully. 00:25:47.920 --> 00:25:52.010 align:middle line:84% So, again, schematically we have two processes going on here. 00:25:52.010 --> 00:25:55.780 align:middle line:84% The first one would be the ion implant step or the pre-dep, 00:25:55.780 --> 00:26:01.030 align:middle line:84% which would result in the bright orange or the red region. 00:26:01.030 --> 00:26:04.660 align:middle line:84% And that has introduced a controlled integrated dose 00:26:04.660 --> 00:26:08.080 align:middle line:84% of atoms per square centimeter into the silicon. 00:26:08.080 --> 00:26:11.110 align:middle line:84% And then without introducing any additional atoms but keeping 00:26:11.110 --> 00:26:15.010 align:middle line:84% the dose constant, we then diffuse in-- or drive in-- 00:26:15.010 --> 00:26:16.270 align:middle line:90% that profile. 00:26:16.270 --> 00:26:18.430 align:middle line:84% And that would then give us the final junction 00:26:18.430 --> 00:26:21.850 align:middle line:84% depth represented by the lighter orange region. 00:26:21.850 --> 00:26:24.470 align:middle line:90% 00:26:24.470 --> 00:26:26.440 align:middle line:84% So let's go on now to slide number 17. 00:26:26.440 --> 00:26:30.640 align:middle line:84% And here I'm comparing somewhat qualitatively 00:26:30.640 --> 00:26:34.450 align:middle line:84% these two different methods of doing predeposition. 00:26:34.450 --> 00:26:36.850 align:middle line:84% On the left hand column, we're talking about some 00:26:36.850 --> 00:26:42.230 align:middle line:84% of the characteristics of doing ion implantation of the atoms. 00:26:42.230 --> 00:26:44.320 align:middle line:84% And on the right hand column we're 00:26:44.320 --> 00:26:47.410 align:middle line:84% talking about doing this more old-fashioned solid or gas 00:26:47.410 --> 00:26:51.850 align:middle line:84% phase in diffusion in order to do the pre-dep. 00:26:51.850 --> 00:26:53.930 align:middle line:90% Now, so ion implantation-- 00:26:53.930 --> 00:26:56.410 align:middle line:90% What are some of its advantages? 00:26:56.410 --> 00:27:01.428 align:middle line:84% Which we'll see when we talk about in chapter 8. 00:27:01.428 --> 00:27:03.220 align:middle line:84% It's done at room temperature, essentially, 00:27:03.220 --> 00:27:06.910 align:middle line:84% so you can mask it with simple materials like photoresist. 00:27:06.910 --> 00:27:09.350 align:middle line:84% It gives you, probably most importantly, 00:27:09.350 --> 00:27:11.740 align:middle line:84% a very precise control of the number 00:27:11.740 --> 00:27:14.020 align:middle line:84% of atoms that are introduced per square centimeter 00:27:14.020 --> 00:27:16.060 align:middle line:90% into the substrate. 00:27:16.060 --> 00:27:18.320 align:middle line:84% It also gives you very accurate depth control. 00:27:18.320 --> 00:27:22.240 align:middle line:90% So those are key advantages. 00:27:22.240 --> 00:27:25.790 align:middle line:84% The problems with it, of course, is as the implant process 00:27:25.790 --> 00:27:30.487 align:middle line:84% occurs at high energies, it actually damages the crystal 00:27:30.487 --> 00:27:31.320 align:middle line:90% to a certain extent. 00:27:31.320 --> 00:27:34.350 align:middle line:84% And we have to heal this by an annealing process. 00:27:34.350 --> 00:27:36.230 align:middle line:84% But unfortunately, the damage itself 00:27:36.230 --> 00:27:37.670 align:middle line:90% can enhance the diffusion rate. 00:27:37.670 --> 00:27:39.110 align:middle line:84% And we're going to spend some time 00:27:39.110 --> 00:27:44.310 align:middle line:84% in this course talking about the transient-enhanced diffusion. 00:27:44.310 --> 00:27:47.820 align:middle line:84% The dislocations or extended defects 00:27:47.820 --> 00:27:51.510 align:middle line:84% associated with the damage can lead to junction leakage, which 00:27:51.510 --> 00:27:53.140 align:middle line:90% is not desirable. 00:27:53.140 --> 00:27:55.505 align:middle line:84% And you may have some channeling of the implant. 00:27:55.505 --> 00:27:56.880 align:middle line:84% We'll talk about how that affects 00:27:56.880 --> 00:27:58.622 align:middle line:90% the profiles in chapter 8. 00:27:58.622 --> 00:28:03.120 align:middle line:90% 00:28:03.120 --> 00:28:06.420 align:middle line:84% The advantages of the pre-dep by gas phase 00:28:06.420 --> 00:28:09.870 align:middle line:84% is there's no damage created by this process. 00:28:09.870 --> 00:28:11.400 align:middle line:90% And it can be done in batches. 00:28:11.400 --> 00:28:14.460 align:middle line:84% But it has some serious limitations. 00:28:14.460 --> 00:28:17.670 align:middle line:84% Usually, you are limited to introducing the dopant 00:28:17.670 --> 00:28:19.770 align:middle line:84% at the surface at a high concentration 00:28:19.770 --> 00:28:22.290 align:middle line:90% of the solid solubility. 00:28:22.290 --> 00:28:25.470 align:middle line:84% It's very hard to achieve low surface concentrations 00:28:25.470 --> 00:28:27.730 align:middle line:90% without a long drive-in step. 00:28:27.730 --> 00:28:30.060 align:middle line:84% And so it's hard to control the shape of the profile 00:28:30.060 --> 00:28:31.860 align:middle line:90% to certain types of shapes. 00:28:31.860 --> 00:28:35.580 align:middle line:84% And low-dose predepositions are very difficult, 00:28:35.580 --> 00:28:37.890 align:middle line:90% and that's a major limiter. 00:28:37.890 --> 00:28:41.800 align:middle line:84% So, as I said, except in very special cases, 00:28:41.800 --> 00:28:43.510 align:middle line:84% people typically use ion implantation 00:28:43.510 --> 00:28:46.600 align:middle line:90% for the predeposition step. 00:28:46.600 --> 00:28:48.070 align:middle line:90% Let's go on to slide number 18. 00:28:48.070 --> 00:28:52.910 align:middle line:84% Again, if we are talking about predeposition, in that case 00:28:52.910 --> 00:28:55.450 align:middle line:84% the dopants are typically introduced 00:28:55.450 --> 00:28:57.580 align:middle line:90% at their solid solubility limit. 00:28:57.580 --> 00:29:00.940 align:middle line:84% So you have some atmosphere of gas, 00:29:00.940 --> 00:29:04.570 align:middle line:84% say, that you introduce the wafers at high temperature 00:29:04.570 --> 00:29:07.090 align:middle line:90% into this gas atmosphere. 00:29:07.090 --> 00:29:08.740 align:middle line:84% And at the surface, you would end up 00:29:08.740 --> 00:29:12.280 align:middle line:84% with the solid solubility of that dopant. 00:29:12.280 --> 00:29:13.780 align:middle line:84% And just to give you an example here 00:29:13.780 --> 00:29:17.890 align:middle line:84% of what some of the solubilities are showing in this plot, 00:29:17.890 --> 00:29:20.470 align:middle line:84% solid solubility as a function of temperature. 00:29:20.470 --> 00:29:24.550 align:middle line:84% And say, if you are doping something with boron, 00:29:24.550 --> 00:29:28.150 align:middle line:84% for example, and you're heating away for up to 1,000 degrees, 00:29:28.150 --> 00:29:29.710 align:middle line:84% the solid solubility is somewhere 00:29:29.710 --> 00:29:34.320 align:middle line:84% in the range of 2 to 3 times 10 to the 20. 00:29:34.320 --> 00:29:37.740 align:middle line:84% So they are soluble in bulk silicon up to that value. 00:29:37.740 --> 00:29:41.820 align:middle line:84% And above that, presumably, they start to precipitate out 00:29:41.820 --> 00:29:45.533 align:middle line:90% into another phase. 00:29:45.533 --> 00:29:47.700 align:middle line:84% So it gives you an idea of the surface concentration 00:29:47.700 --> 00:29:52.590 align:middle line:84% you might get of these dopants if you did a predeposition 00:29:52.590 --> 00:29:55.210 align:middle line:90% at that temperature. 00:29:55.210 --> 00:29:59.700 align:middle line:84% Now, if you go on to slide 19, it 00:29:59.700 --> 00:30:01.380 align:middle line:84% turns out there's a subtle difference. 00:30:01.380 --> 00:30:03.360 align:middle line:84% And a point we want to make in this course 00:30:03.360 --> 00:30:08.400 align:middle line:84% is that dopants also have what's called an electrical solubility 00:30:08.400 --> 00:30:10.650 align:middle line:84% that is different from the solid solubility 00:30:10.650 --> 00:30:14.460 align:middle line:84% that we defined according to precipitation. 00:30:14.460 --> 00:30:18.060 align:middle line:84% The electrical solubility refers to the maximum doping 00:30:18.060 --> 00:30:19.980 align:middle line:84% concentration in terms of electron 00:30:19.980 --> 00:30:22.230 align:middle line:84% density per cubic centimeter that you 00:30:22.230 --> 00:30:24.772 align:middle line:90% can achieve with that dopant. 00:30:24.772 --> 00:30:26.730 align:middle line:84% And that generally varies with the temperature, 00:30:26.730 --> 00:30:27.480 align:middle line:90% as we see here. 00:30:27.480 --> 00:30:32.740 align:middle line:84% This is a plot for the maximum electron concentration doping 00:30:32.740 --> 00:30:35.552 align:middle line:84% you can achieve using arsenic in silicon 00:30:35.552 --> 00:30:36.760 align:middle line:90% as a function of temperature. 00:30:36.760 --> 00:30:38.770 align:middle line:84% And there's a lot of different points 00:30:38.770 --> 00:30:42.400 align:middle line:84% on this curve from different measurements in the literature. 00:30:42.400 --> 00:30:46.880 align:middle line:84% But they generally follow this roughly this straight line. 00:30:46.880 --> 00:30:49.030 align:middle line:84% And so, again, what this is saying is that 00:30:49.030 --> 00:30:52.610 align:middle line:84% at 1,000 degrees, if you look at the curve, 00:30:52.610 --> 00:30:55.420 align:middle line:84% you can get something like 3 to 4 times 10 00:30:55.420 --> 00:30:59.440 align:middle line:84% to the 20 electrons per cubic centimeter 00:30:59.440 --> 00:31:01.750 align:middle line:84% by introducing arsenic into the lattice. 00:31:01.750 --> 00:31:05.590 align:middle line:84% If you introduce more arsenic than that, 00:31:05.590 --> 00:31:09.160 align:middle line:84% it may still be below the solid solubility, 00:31:09.160 --> 00:31:10.730 align:middle line:84% but you won't get any more electrons. 00:31:10.730 --> 00:31:13.423 align:middle line:90% It's not electrically active. 00:31:13.423 --> 00:31:16.090 align:middle line:84% It may not precipitate until you get up into the 10 to 21 range. 00:31:16.090 --> 00:31:17.920 align:middle line:84% But so there's this intermediate range, 00:31:17.920 --> 00:31:21.520 align:middle line:84% say, at 1,000 degrees between 310 to the 20 and 10 00:31:21.520 --> 00:31:25.510 align:middle line:84% to the 21 in which you may not see 00:31:25.510 --> 00:31:29.710 align:middle line:84% silicon-arsenic precipitates, but you do not have any higher 00:31:29.710 --> 00:31:31.823 align:middle line:90% electron concentration. 00:31:31.823 --> 00:31:33.490 align:middle line:84% That turned out to be a bit of a mystery 00:31:33.490 --> 00:31:34.690 align:middle line:90% to people for a number of years. 00:31:34.690 --> 00:31:35.980 align:middle line:84% They couldn't see precipitates, but they 00:31:35.980 --> 00:31:37.938 align:middle line:84% knew they wouldn't weren't getting the electron 00:31:37.938 --> 00:31:40.630 align:middle line:84% concentration above a certain number. 00:31:40.630 --> 00:31:44.200 align:middle line:84% And it was subject to a lot of discussions in the literature. 00:31:44.200 --> 00:31:48.730 align:middle line:84% In fact, if you go on to slide number 20-- 00:31:48.730 --> 00:31:50.800 align:middle line:90% I took this from your text. 00:31:50.800 --> 00:31:54.430 align:middle line:84% People eventually came up with a number of models 00:31:54.430 --> 00:31:57.100 align:middle line:84% to try to explain how it is you can 00:31:57.100 --> 00:32:01.420 align:middle line:84% get more arsenic in the lattice if it does not precipitate. 00:32:01.420 --> 00:32:05.350 align:middle line:84% And yet it doesn't contribute electrons 00:32:05.350 --> 00:32:07.390 align:middle line:90% to the doping in this range. 00:32:07.390 --> 00:32:10.450 align:middle line:84% And here on the left, I'm imagining 00:32:10.450 --> 00:32:12.010 align:middle line:90% arsenic in the lattice-- 00:32:12.010 --> 00:32:14.620 align:middle line:84% the pink atom in the silicon lattice-- 00:32:14.620 --> 00:32:18.220 align:middle line:84% a relatively lightly doped sample, say, in the 10 00:32:18.220 --> 00:32:22.190 align:middle line:84% to the 20 range or so-- heavily doped, but not too high. 00:32:22.190 --> 00:32:24.580 align:middle line:84% And what you see is arsenic generally surrounded 00:32:24.580 --> 00:32:26.170 align:middle line:90% by four silicon atoms. 00:32:26.170 --> 00:32:29.380 align:middle line:84% And it donates its extra fifth electron 00:32:29.380 --> 00:32:32.620 align:middle line:84% that is not conveniently bonded to the conduction band, 00:32:32.620 --> 00:32:35.220 align:middle line:90% and you can get a free electron. 00:32:35.220 --> 00:32:38.640 align:middle line:84% Now, on the right side is shown a hypothetical case where 00:32:38.640 --> 00:32:41.970 align:middle line:84% you have, say, a lot of arsenic on the lattice. 00:32:41.970 --> 00:32:46.620 align:middle line:84% So you might have 8 times 10 to the 20 or 10 to the 21 00:32:46.620 --> 00:32:50.460 align:middle line:84% or something in which people still 00:32:50.460 --> 00:32:52.590 align:middle line:90% didn't see a second phase. 00:32:52.590 --> 00:32:54.420 align:middle line:84% They didn't see precipitation happening. 00:32:54.420 --> 00:33:00.780 align:middle line:84% What you might have, say, four arsenic atoms near each other-- 00:33:00.780 --> 00:33:03.420 align:middle line:84% these four arsenic atoms surrounding, say, a vacancy-- 00:33:03.420 --> 00:33:04.870 align:middle line:90% a silicon vacancy. 00:33:04.870 --> 00:33:09.840 align:middle line:84% So this AS 4 and V is a complex that people have hypothesized 00:33:09.840 --> 00:33:14.458 align:middle line:84% which would enable you to introduce four arsenic atoms 00:33:14.458 --> 00:33:15.750 align:middle line:90% in the vicinity of the vacancy. 00:33:15.750 --> 00:33:19.200 align:middle line:84% And yet no electrons will be donated-- 00:33:19.200 --> 00:33:20.800 align:middle line:90% no free electrons. 00:33:20.800 --> 00:33:22.530 align:middle line:84% So these four arsenic atoms would 00:33:22.530 --> 00:33:27.450 align:middle line:84% be essentially electrically inactive, and yet essentially 00:33:27.450 --> 00:33:32.020 align:middle line:84% on substitutional lattice sites at high arsenic concentration. 00:33:32.020 --> 00:33:34.870 align:middle line:84% So this might be one way of accounting for the fact 00:33:34.870 --> 00:33:36.870 align:middle line:84% that the electrical solubility is a little lower 00:33:36.870 --> 00:33:38.620 align:middle line:90% than the solid solubility. 00:33:38.620 --> 00:33:41.880 align:middle line:84% So, again, we just point this out for a couple of reasons, 00:33:41.880 --> 00:33:44.670 align:middle line:84% mainly because it points to the fact 00:33:44.670 --> 00:33:47.430 align:middle line:84% that as we increase doping, we don't always 00:33:47.430 --> 00:33:50.220 align:middle line:84% get an increase in the electron concentration, 00:33:50.220 --> 00:33:55.680 align:middle line:84% and therefore a decrease in resistance. 00:33:55.680 --> 00:33:58.470 align:middle line:84% OK, so let's go on to slide number 21. 00:33:58.470 --> 00:34:01.590 align:middle line:84% And we're going to consider macroscopic first-- 00:34:01.590 --> 00:34:03.900 align:middle line:84% macroscopic models for diffusion. 00:34:03.900 --> 00:34:07.320 align:middle line:84% Later on, we'll talk about the more atomistic 00:34:07.320 --> 00:34:10.380 align:middle line:84% diffusion mechanisms and effects. 00:34:10.380 --> 00:34:12.870 align:middle line:84% And hopefully, you've read part of chapter 7, 00:34:12.870 --> 00:34:16.920 align:middle line:84% and you know that macroscopic dopant diffusion is described 00:34:16.920 --> 00:34:19.530 align:middle line:84% by Fick's first law, which describes 00:34:19.530 --> 00:34:21.929 align:middle line:84% how the flux or the flow of dopant 00:34:21.929 --> 00:34:24.389 align:middle line:84% depends upon the doping gradient. 00:34:24.389 --> 00:34:29.100 align:middle line:84% And I'm showing here a schematic sketch of concentration 00:34:29.100 --> 00:34:31.320 align:middle line:90% as a function of distance. 00:34:31.320 --> 00:34:33.150 align:middle line:90% And one of these curves-- 00:34:33.150 --> 00:34:36.000 align:middle line:84% the one with a higher peak concentration-- 00:34:36.000 --> 00:34:40.020 align:middle line:84% occurs at some time called t1, an earlier time. 00:34:40.020 --> 00:34:42.840 align:middle line:84% And then the second curve that has moved out 00:34:42.840 --> 00:34:45.870 align:middle line:84% a further distance corresponds to time t2. 00:34:45.870 --> 00:34:50.880 align:middle line:84% And essentially, what we see is that the flux F-- 00:34:50.880 --> 00:34:52.800 align:middle line:84% moving in this case to the right-- 00:34:52.800 --> 00:34:56.460 align:middle line:84% is equal to minus a constant called 00:34:56.460 --> 00:35:00.780 align:middle line:84% the diffusion constant times the gradient dC by dx. 00:35:00.780 --> 00:35:05.190 align:middle line:84% So as you take the slope along that curve, wherever 00:35:05.190 --> 00:35:08.400 align:middle line:84% the slope is very steep you get a large flux, and therefore 00:35:08.400 --> 00:35:11.125 align:middle line:90% a large movement in the profile. 00:35:11.125 --> 00:35:12.750 align:middle line:84% As you get near the top of the profile, 00:35:12.750 --> 00:35:15.523 align:middle line:84% the concentration gradient is getting small, 00:35:15.523 --> 00:35:17.190 align:middle line:84% and then the flux is a little bit lower. 00:35:17.190 --> 00:35:19.530 align:middle line:84% But at the flux at any given point 00:35:19.530 --> 00:35:22.320 align:middle line:90% can be given by this equation. 00:35:22.320 --> 00:35:25.260 align:middle line:84% When the concentration gradient goes to 0, 00:35:25.260 --> 00:35:28.770 align:middle line:84% essentially, the dopant or the atoms 00:35:28.770 --> 00:35:31.260 align:middle line:84% are uniformly distributed, say, in the solid, 00:35:31.260 --> 00:35:33.645 align:middle line:84% and the flow would stop according to Fick's first law. 00:35:33.645 --> 00:35:36.400 align:middle line:90% 00:35:36.400 --> 00:35:38.310 align:middle line:84% So let's go on to slide number 22. 00:35:38.310 --> 00:35:43.230 align:middle line:84% Again this F equals minus D the partial C by dx. 00:35:43.230 --> 00:35:46.800 align:middle line:84% This is a general sort of flux law 00:35:46.800 --> 00:35:50.010 align:middle line:84% which hopefully may be familiar to you. 00:35:50.010 --> 00:35:53.160 align:middle line:84% It's similar to Fourier's law or analogous to Fourier's law 00:35:53.160 --> 00:35:58.750 align:middle line:84% of heat conduction or to Ohm's law for current flow. 00:35:58.750 --> 00:36:01.120 align:middle line:84% In this case, the proportionality constant 00:36:01.120 --> 00:36:03.640 align:middle line:84% is called the diffusivity D. It has 00:36:03.640 --> 00:36:07.750 align:middle line:84% units of length squared per time or centimeter squared 00:36:07.750 --> 00:36:10.070 align:middle line:90% per second. 00:36:10.070 --> 00:36:11.770 align:middle line:84% And as it turns out, we'll see that D 00:36:11.770 --> 00:36:16.510 align:middle line:84% is related to the atomic hop rate or the jump frequency 00:36:16.510 --> 00:36:19.318 align:middle line:84% over some kind of energy barrier. 00:36:19.318 --> 00:36:20.860 align:middle line:84% And this energy barrier is associated 00:36:20.860 --> 00:36:24.970 align:middle line:84% with the formation of migration energies of mobile species. 00:36:24.970 --> 00:36:27.350 align:middle line:84% D is generally exponentially activated. 00:36:27.350 --> 00:36:30.190 align:middle line:84% So it's dependent on temperature in exponential fashion. 00:36:30.190 --> 00:36:33.520 align:middle line:84% And in the silicon lattice, by symmetry D is isotropic. 00:36:33.520 --> 00:36:37.570 align:middle line:84% So it doesn't depend on which direction you're diffusing. 00:36:37.570 --> 00:36:41.110 align:middle line:84% And of course, the negative sign in this Fick's first law 00:36:41.110 --> 00:36:44.780 align:middle line:84% indicates that the flow is down the concentration gradient. 00:36:44.780 --> 00:36:48.740 align:middle line:84% So just by drawing yourself a profile, 00:36:48.740 --> 00:36:52.520 align:middle line:90% you see that dC/dx is negative. 00:36:52.520 --> 00:36:54.220 align:middle line:84% So in order to get flux to the right, 00:36:54.220 --> 00:36:58.490 align:middle line:84% you need to have the negative sign 00:36:58.490 --> 00:37:00.890 align:middle line:90% Let's go on to slide number 23. 00:37:00.890 --> 00:37:07.190 align:middle line:84% This illustrates a derivation of Fick's second law, which 00:37:07.190 --> 00:37:10.850 align:middle line:84% describes how the change in concentration in a small volume 00:37:10.850 --> 00:37:13.700 align:middle line:84% element is determined by the change in the fluxes 00:37:13.700 --> 00:37:17.790 align:middle line:90% into and out of that volume. 00:37:17.790 --> 00:37:22.040 align:middle line:84% So if you take a look at this volume element, 00:37:22.040 --> 00:37:25.260 align:middle line:84% which has a certain length to it-- 00:37:25.260 --> 00:37:26.510 align:middle line:90% this delta x. 00:37:26.510 --> 00:37:30.900 align:middle line:84% And there's a certain concentration change 00:37:30.900 --> 00:37:33.470 align:middle line:84% in a certain time period, delta t, 00:37:33.470 --> 00:37:37.280 align:middle line:84% in that volume element, which we'll call in delta C. 00:37:37.280 --> 00:37:39.890 align:middle line:84% There's a flux in coming from the left phase, 00:37:39.890 --> 00:37:43.340 align:middle line:84% and there's a flux out going out the right phase. 00:37:43.340 --> 00:37:46.742 align:middle line:84% And just by a bookkeeping on the upper left, 00:37:46.742 --> 00:37:48.200 align:middle line:84% we can write a simple equation that 00:37:48.200 --> 00:37:50.420 align:middle line:84% says the change in the concentration delta 00:37:50.420 --> 00:37:56.510 align:middle line:84% C in the time period delta t is just equal to the difference 00:37:56.510 --> 00:37:57.710 align:middle line:90% in these two fluxes-- 00:37:57.710 --> 00:38:00.980 align:middle line:84% the incoming flux minus the outgoing flux-- 00:38:00.980 --> 00:38:03.185 align:middle line:84% divided by the distance, delta x. 00:38:03.185 --> 00:38:06.310 align:middle line:90% 00:38:06.310 --> 00:38:07.930 align:middle line:84% Mathematically, instead of writing it 00:38:07.930 --> 00:38:12.340 align:middle line:84% in terms of these differences, we can write it mathematically 00:38:12.340 --> 00:38:15.370 align:middle line:90% as partial derivatives. 00:38:15.370 --> 00:38:21.670 align:middle line:84% We can write delta C by delta t as partial C by partial t. 00:38:21.670 --> 00:38:24.670 align:middle line:84% So the time rate of change of the concentration 00:38:24.670 --> 00:38:29.290 align:middle line:84% is equal to the partial of the flux divided by x. 00:38:29.290 --> 00:38:32.740 align:middle line:84% So it's the gradient of the flux. 00:38:32.740 --> 00:38:35.400 align:middle line:84% That's what we're saying with Fick's second law. 00:38:35.400 --> 00:38:39.640 align:middle line:84% And now for the flux F we can substitute in Fick's first law. 00:38:39.640 --> 00:38:43.560 align:middle line:84% It's just F is equal to D partial C by partial x. 00:38:43.560 --> 00:38:45.210 align:middle line:84% So we can substitute that in, and we 00:38:45.210 --> 00:38:48.780 align:middle line:84% get this equation in the middle of the slide, which essentially 00:38:48.780 --> 00:38:54.540 align:middle line:84% is the differential form of Fick's second law. 00:38:54.540 --> 00:38:58.080 align:middle line:84% Now, what we do is we make the assumption at this point 00:38:58.080 --> 00:38:59.950 align:middle line:84% that the diffusivity is constant. 00:38:59.950 --> 00:39:02.520 align:middle line:90% So it doesn't depend on x. 00:39:02.520 --> 00:39:04.570 align:middle line:84% In that case, and only in that case, 00:39:04.570 --> 00:39:07.680 align:middle line:84% then we can pull the D out of the derivative. 00:39:07.680 --> 00:39:13.713 align:middle line:84% The partial by dx, if we do the chain rule, is 0. 00:39:13.713 --> 00:39:15.630 align:middle line:84% And then we can get the equation at the bottom 00:39:15.630 --> 00:39:18.870 align:middle line:84% of the page, which says that the time rate of change 00:39:18.870 --> 00:39:22.080 align:middle line:84% of the concentration at any given point 00:39:22.080 --> 00:39:26.760 align:middle line:84% is a constant D times the second derivative of the concentration 00:39:26.760 --> 00:39:28.380 align:middle line:90% with respect to x. 00:39:28.380 --> 00:39:31.380 align:middle line:90% And that is Fick's second law. 00:39:31.380 --> 00:39:33.700 align:middle line:84% And again, this particular formulation 00:39:33.700 --> 00:39:37.620 align:middle line:84% only applies if the diffusion diffusivity is a constant. 00:39:37.620 --> 00:39:38.590 align:middle line:90% It doesn't depend on x. 00:39:38.590 --> 00:39:41.640 align:middle line:90% 00:39:41.640 --> 00:39:44.490 align:middle line:90% So let's go on to slide 24 now. 00:39:44.490 --> 00:39:47.880 align:middle line:84% And there are a handful of cases, maybe three or so 00:39:47.880 --> 00:39:52.530 align:middle line:84% or four, in which it's possible to write down 00:39:52.530 --> 00:39:56.730 align:middle line:84% or to derive relatively simple analytic solutions 00:39:56.730 --> 00:39:58.260 align:middle line:90% to the diffusion equation. 00:39:58.260 --> 00:40:01.470 align:middle line:84% In all the other cases and most of the cases 00:40:01.470 --> 00:40:02.990 align:middle line:84% we'll end up using in this course, 00:40:02.990 --> 00:40:04.532 align:middle line:84% we'll have to do numerical solutions. 00:40:04.532 --> 00:40:07.600 align:middle line:84% And we'll talk next time about how numerical solutions work. 00:40:07.600 --> 00:40:10.620 align:middle line:84% But for now, let's look at a couple of special cases 00:40:10.620 --> 00:40:14.250 align:middle line:84% where we can solve this equation by hand. 00:40:14.250 --> 00:40:17.670 align:middle line:84% The first case is pictured on slide 24, which 00:40:17.670 --> 00:40:19.530 align:middle line:90% is called the steady state. 00:40:19.530 --> 00:40:23.940 align:middle line:84% And what that refers to is that, in fact, we have a profile that 00:40:23.940 --> 00:40:28.030 align:middle line:84% is not varying in time, in which case 00:40:28.030 --> 00:40:31.570 align:middle line:84% we write partial C partial t equals 0. 00:40:31.570 --> 00:40:34.750 align:middle line:84% And so then we have a relatively simple equation 00:40:34.750 --> 00:40:37.510 align:middle line:84% that D is equal to the second derivative of C with respect 00:40:37.510 --> 00:40:38.860 align:middle line:90% to x equals 0. 00:40:38.860 --> 00:40:40.570 align:middle line:90% D times that is equal to 0. 00:40:40.570 --> 00:40:44.110 align:middle line:84% So we can just simply integrate this equation twice, 00:40:44.110 --> 00:40:46.670 align:middle line:84% and we end up with a linear profile over distance. 00:40:46.670 --> 00:40:49.900 align:middle line:84% So if you just take the equation that the concentration C equals 00:40:49.900 --> 00:40:56.830 align:middle line:84% a plus bx, and we differentiate that twice, indeed we get 0. 00:40:56.830 --> 00:40:59.140 align:middle line:84% So this says that in a steady state 00:40:59.140 --> 00:41:04.510 align:middle line:84% solution to the diffusion equation is a linear equation. 00:41:04.510 --> 00:41:07.050 align:middle line:84% In fact, when we did the solution 00:41:07.050 --> 00:41:09.780 align:middle line:84% of the diffusion of the oxidant through the oxide 00:41:09.780 --> 00:41:12.240 align:middle line:84% during thermal oxidation, this is the equation 00:41:12.240 --> 00:41:13.200 align:middle line:90% that we actually use. 00:41:13.200 --> 00:41:17.700 align:middle line:84% This is implicitly the solution that we were assuming. 00:41:17.700 --> 00:41:19.440 align:middle line:90% And I think you'll recognize-- 00:41:19.440 --> 00:41:20.910 align:middle line:84% from the last few lectures, you'll 00:41:20.910 --> 00:41:28.350 align:middle line:84% recognize on the lower pictures this steady state solution is 00:41:28.350 --> 00:41:32.340 align:middle line:84% either in the left side-- being in the thin oxide regime, 00:41:32.340 --> 00:41:34.890 align:middle line:84% we get a straight line that's just 00:41:34.890 --> 00:41:38.430 align:middle line:84% a constant number of the diffusion through the oxide. 00:41:38.430 --> 00:41:40.900 align:middle line:84% Or on the right hand side, we get, again, 00:41:40.900 --> 00:41:46.478 align:middle line:84% a linear profile of the oxygen through the oxide 00:41:46.478 --> 00:41:47.520 align:middle line:90% during thermal oxidation. 00:41:47.520 --> 00:41:50.040 align:middle line:84% And again, in that case it's a concentration profile 00:41:50.040 --> 00:41:51.630 align:middle line:90% that's not changing with time. 00:41:51.630 --> 00:41:54.990 align:middle line:90% 00:41:54.990 --> 00:41:56.570 align:middle line:84% So let's go on to slide number 25 00:41:56.570 --> 00:42:01.760 align:middle line:84% and do the first solution of a case that's 00:42:01.760 --> 00:42:04.430 align:middle line:84% a little more complicated than that, which is 00:42:04.430 --> 00:42:05.960 align:middle line:90% called the limited source case. 00:42:05.960 --> 00:42:12.110 align:middle line:84% So we consider that we have the dopant in this region, 00:42:12.110 --> 00:42:16.490 align:middle line:84% and it has a fixed dose Q, so a fixed number of atoms 00:42:16.490 --> 00:42:17.810 align:middle line:90% per square centimeter. 00:42:17.810 --> 00:42:19.940 align:middle line:84% And we're going to introduce it as a delta function 00:42:19.940 --> 00:42:22.420 align:middle line:90% at the origin. 00:42:22.420 --> 00:42:26.770 align:middle line:84% And then we're going to let it diffuse and diffuse out. 00:42:26.770 --> 00:42:31.640 align:middle line:84% And as it turns out, the C-- if diffusivity is a constant, 00:42:31.640 --> 00:42:36.620 align:middle line:84% it diffuses into the shape of a Gaussian profile. 00:42:36.620 --> 00:42:40.930 align:middle line:84% And so, basically, the boundary conditions 00:42:40.930 --> 00:42:45.550 align:middle line:84% are that we have essentially a delta function at time t equals 00:42:45.550 --> 00:42:49.420 align:middle line:90% 0 and that the dose-- 00:42:49.420 --> 00:42:52.120 align:middle line:84% the integral of this delta function-- 00:42:52.120 --> 00:42:55.800 align:middle line:90% is a constant. 00:42:55.800 --> 00:42:57.590 align:middle line:90% Let's go on to slide number 26. 00:42:57.590 --> 00:42:59.630 align:middle line:84% And we find that the solution that 00:42:59.630 --> 00:43:03.570 align:middle line:84% satisfies Fick's second law is written down by this equation. 00:43:03.570 --> 00:43:07.370 align:middle line:84% And in fact, it's a Gaussian distribution. 00:43:07.370 --> 00:43:09.500 align:middle line:84% The concentration is a function of x and time 00:43:09.500 --> 00:43:13.880 align:middle line:84% can be given by that constant dose Q divided by 2 times 00:43:13.880 --> 00:43:19.040 align:middle line:84% the square root of pi Dt times the exponential of minus 00:43:19.040 --> 00:43:20.960 align:middle line:90% x squared over 4 Dt. 00:43:20.960 --> 00:43:26.108 align:middle line:84% So that's what's known as a Gaussian profile. 00:43:26.108 --> 00:43:27.650 align:middle line:84% And the important consequence of this 00:43:27.650 --> 00:43:32.270 align:middle line:84% are that one, of course, the dose Q remains constant. 00:43:32.270 --> 00:43:34.620 align:middle line:84% That means then that the peak concentration-- 00:43:34.620 --> 00:43:36.410 align:middle line:84% so the concentration at the origin-- 00:43:36.410 --> 00:43:40.970 align:middle line:84% is going to decrease according to the square root of Dt 00:43:40.970 --> 00:43:42.900 align:middle line:90% over time. 00:43:42.900 --> 00:43:44.930 align:middle line:84% So the peak concentration goes down. 00:43:44.930 --> 00:43:47.990 align:middle line:84% And the width of the profile or the diffusion distance 00:43:47.990 --> 00:43:52.490 align:middle line:84% from the origin is going to increase 00:43:52.490 --> 00:43:56.700 align:middle line:84% according to 2 times the square root of Dt. 00:43:56.700 --> 00:44:00.630 align:middle line:84% You see that just by looking at the argument of the exponential 00:44:00.630 --> 00:44:02.400 align:middle line:90% there. 00:44:02.400 --> 00:44:05.542 align:middle line:84% So at this distance-- this distance equal to 2 times 00:44:05.542 --> 00:44:06.500 align:middle line:90% the square root of Dt-- 00:44:06.500 --> 00:44:12.140 align:middle line:84% the doping concentration will fall off by 1 over e. 00:44:12.140 --> 00:44:15.380 align:middle line:84% And in fact, what we do is we often call this distance-- 00:44:15.380 --> 00:44:16.520 align:middle line:90% we give it a special name. 00:44:16.520 --> 00:44:20.930 align:middle line:84% We call the diffusion length L. We typically write it 00:44:20.930 --> 00:44:23.510 align:middle line:84% as twice a square root Dt or sometimes 00:44:23.510 --> 00:44:25.610 align:middle line:90% just the square root of Dt. 00:44:25.610 --> 00:44:28.970 align:middle line:84% It gives us an idea of the width or the broadening 00:44:28.970 --> 00:44:29.735 align:middle line:90% of the profile. 00:44:29.735 --> 00:44:34.240 align:middle line:90% 00:44:34.240 --> 00:44:37.330 align:middle line:84% So let's go on to slide number 27 which shows pictorially 00:44:37.330 --> 00:44:41.290 align:middle line:84% the time evolution of a Gaussian profile. 00:44:41.290 --> 00:44:47.230 align:middle line:84% The left hand plot is on linear y-axis and the right hand plot 00:44:47.230 --> 00:44:48.940 align:middle line:90% is on a logarithmic scale-- 00:44:48.940 --> 00:44:51.860 align:middle line:90% give you a little better view. 00:44:51.860 --> 00:44:53.890 align:middle line:84% So, first, let's look at the left. 00:44:53.890 --> 00:44:56.560 align:middle line:84% And we there are three curves shown here. 00:44:56.560 --> 00:45:02.620 align:middle line:84% The red is for time t equals some time t 0. 00:45:02.620 --> 00:45:04.930 align:middle line:84% And the y-axis is plotting the concentration 00:45:04.930 --> 00:45:07.330 align:middle line:90% in a normalized fashion. 00:45:07.330 --> 00:45:10.990 align:middle line:84% And the x-axis is in units of diffusion distance. 00:45:10.990 --> 00:45:17.680 align:middle line:84% So its units are units of 2 times square root of Dt0. 00:45:17.680 --> 00:45:22.090 align:middle line:84% And you can see in going from the red to the blue curve, 00:45:22.090 --> 00:45:25.990 align:middle line:84% indeed the concentration has dropped by 1 00:45:25.990 --> 00:45:28.930 align:middle line:84% over the square root of t because, basically, 00:45:28.930 --> 00:45:31.960 align:middle line:90% the blue curve is 4 times t0. 00:45:31.960 --> 00:45:34.010 align:middle line:84% So it's dropped by a factor of 2. 00:45:34.010 --> 00:45:36.470 align:middle line:90% And it's broadened. 00:45:36.470 --> 00:45:39.920 align:middle line:84% And same thing by looking at the green curve. 00:45:39.920 --> 00:45:42.830 align:middle line:84% You see the same type of phenomenon. 00:45:42.830 --> 00:45:45.470 align:middle line:84% And on the right on the semi-log plot, 00:45:45.470 --> 00:45:47.525 align:middle line:84% you just get to see a little bit more detail. 00:45:47.525 --> 00:45:50.000 align:middle line:84% You get your eye calibrated for what a Gaussian looks 00:45:50.000 --> 00:45:51.960 align:middle line:90% like on a logarithmic scale. 00:45:51.960 --> 00:45:53.900 align:middle line:84% So you can see many orders of magnitude 00:45:53.900 --> 00:45:58.490 align:middle line:84% down below the peak what the broadening actually looks like. 00:45:58.490 --> 00:46:00.290 align:middle line:84% Because in semiconductor processing, 00:46:00.290 --> 00:46:02.930 align:middle line:84% linear scales for dopants are not all that useful 00:46:02.930 --> 00:46:06.890 align:middle line:84% because, in fact, we often care about how the dopant falls off 00:46:06.890 --> 00:46:10.050 align:middle line:84% over many, many orders of magnitude of concentration. 00:46:10.050 --> 00:46:12.530 align:middle line:84% So we typically like to use a semi-log type plot. 00:46:12.530 --> 00:46:15.080 align:middle line:90% 00:46:15.080 --> 00:46:17.750 align:middle line:84% OK, so we have one solution for one case. 00:46:17.750 --> 00:46:22.070 align:middle line:84% Let's go on to slide number 28 and talk 00:46:22.070 --> 00:46:27.170 align:middle line:84% about the second case, which is a fixed dose Q, just like we 00:46:27.170 --> 00:46:29.130 align:middle line:90% talk about, constant in time. 00:46:29.130 --> 00:46:31.550 align:middle line:84% But now we're diffusing near a surface. 00:46:31.550 --> 00:46:34.910 align:middle line:84% Before we have the origin, and we assume the silicon 00:46:34.910 --> 00:46:37.350 align:middle line:84% was semi-infinite in both directions. 00:46:37.350 --> 00:46:40.040 align:middle line:84% Let's say we had this delta function of dose Q 00:46:40.040 --> 00:46:41.300 align:middle line:90% as the initial profile. 00:46:41.300 --> 00:46:46.280 align:middle line:84% But we're right near the surface of the silicon wafer. 00:46:46.280 --> 00:46:53.720 align:middle line:84% Well, there's a relatively simple trick for solving this. 00:46:53.720 --> 00:46:57.260 align:middle line:84% If we can assume that there's no dopant loss through evaporation 00:46:57.260 --> 00:46:59.810 align:middle line:84% or segregation at the surface-- that the dopant is contained 00:46:59.810 --> 00:47:00.740 align:middle line:90% in the surface-- 00:47:00.740 --> 00:47:02.810 align:middle line:84% if there's evaporation, all bets are off, 00:47:02.810 --> 00:47:05.820 align:middle line:84% and we have to solve it differently. 00:47:05.820 --> 00:47:08.478 align:middle line:84% So we assume that there's no segregation or evaporation. 00:47:08.478 --> 00:47:10.520 align:middle line:84% And we also assume that the annealing takes place 00:47:10.520 --> 00:47:14.490 align:middle line:84% over a long time so that the initial profile is reasonably 00:47:14.490 --> 00:47:17.430 align:middle line:84% can be reasonably approximated by a delta function compared 00:47:17.430 --> 00:47:19.290 align:middle line:90% to the final profile. 00:47:19.290 --> 00:47:21.900 align:middle line:84% If those two assumptions are hold, 00:47:21.900 --> 00:47:26.520 align:middle line:84% then we can essentially solve it by assuming that we 00:47:26.520 --> 00:47:29.040 align:middle line:90% have virtual diffusion-- 00:47:29.040 --> 00:47:31.590 align:middle line:84% that we have a symmetric diffusion 00:47:31.590 --> 00:47:35.250 align:middle line:84% with an imaginary delta function of equal dose 00:47:35.250 --> 00:47:39.460 align:middle line:90% Q on the left-hand side. 00:47:39.460 --> 00:47:42.630 align:middle line:84% So we can solve it by using, essentially, 00:47:42.630 --> 00:47:47.490 align:middle line:84% the prior solution, pretending that the medium is 00:47:47.490 --> 00:47:49.570 align:middle line:90% semi-infinite. 00:47:49.570 --> 00:47:52.450 align:middle line:84% So, in fact, if we go on to slide number 27, 00:47:52.450 --> 00:47:55.900 align:middle line:84% that same the graph is shown at the top. 00:47:55.900 --> 00:47:57.580 align:middle line:84% Effectively, what this means is that we 00:47:57.580 --> 00:48:03.310 align:middle line:84% have a dose of 2Q introduced into a virtual infinite medium 00:48:03.310 --> 00:48:07.600 align:middle line:84% by symmetry so that the concentration profile is 00:48:07.600 --> 00:48:11.290 align:middle line:84% given, again, just by the same Gaussian we had last time. 00:48:11.290 --> 00:48:13.960 align:middle line:84% But instead in that formula wherever we had Q, 00:48:13.960 --> 00:48:18.130 align:middle line:84% we replace it with 2Q from the prior solution. 00:48:18.130 --> 00:48:21.420 align:middle line:84% So the surface concentration now goes Q 00:48:21.420 --> 00:48:24.600 align:middle line:90% over the square root of pi Dt. 00:48:24.600 --> 00:48:28.870 align:middle line:84% But it's very similar to what we had last time. 00:48:28.870 --> 00:48:31.710 align:middle line:84% And so this is what the equation looks 00:48:31.710 --> 00:48:35.160 align:middle line:84% like for diffusing with a fixed dose 00:48:35.160 --> 00:48:40.270 align:middle line:84% into a surface where we have no loss from the surface. 00:48:40.270 --> 00:48:44.200 align:middle line:90% Again, it's a Gaussian profile. 00:48:44.200 --> 00:48:46.240 align:middle line:84% So let's go on to slide number 30. 00:48:46.240 --> 00:48:52.270 align:middle line:84% And the third case, essentially, that we can solve analytically 00:48:52.270 --> 00:48:55.960 align:middle line:84% is called the case of an infinite source. 00:48:55.960 --> 00:48:59.710 align:middle line:84% And what this is essentially an infinite source 00:48:59.710 --> 00:49:05.920 align:middle line:84% of dopant which is made up of small slices, 00:49:05.920 --> 00:49:08.140 align:middle line:84% essentially each diffusing as a Gaussian. 00:49:08.140 --> 00:49:12.670 align:middle line:84% So what we look at in this plot looking at the black line-- we 00:49:12.670 --> 00:49:16.240 align:middle line:84% have a concentration C being constant everywhere 00:49:16.240 --> 00:49:18.760 align:middle line:90% x is less than the origin. 00:49:18.760 --> 00:49:21.310 align:middle line:84% And then there's a step function at the origin. 00:49:21.310 --> 00:49:25.520 align:middle line:84% And then it's 0 everywhere greater than the origin. 00:49:25.520 --> 00:49:28.090 align:middle line:84% So we have the step function at x equals 0. 00:49:28.090 --> 00:49:30.970 align:middle line:84% And that's the initial profile in black. 00:49:30.970 --> 00:49:34.630 align:middle line:84% And what we're going to find is that the diffuse profile 00:49:34.630 --> 00:49:35.890 align:middle line:90% looks like the red. 00:49:35.890 --> 00:49:39.280 align:middle line:84% The step function gets rounded to the left of 0. 00:49:39.280 --> 00:49:42.220 align:middle line:84% And some of that dopant then has diffused 00:49:42.220 --> 00:49:45.647 align:middle line:84% into the right-hand side at x greater than 0. 00:49:45.647 --> 00:49:47.230 align:middle line:84% And it gives us that diffused profile. 00:49:47.230 --> 00:49:49.487 align:middle line:90% 00:49:49.487 --> 00:49:51.070 align:middle line:84% So how are we going to solve for this? 00:49:51.070 --> 00:49:52.630 align:middle line:84% Well, actually, we do it by using 00:49:52.630 --> 00:49:57.040 align:middle line:84% the solution we had obtained previously and essentially 00:49:57.040 --> 00:49:59.440 align:middle line:84% by a linear superposition of solutions 00:49:59.440 --> 00:50:00.910 align:middle line:90% for each of these thin slices. 00:50:00.910 --> 00:50:05.920 align:middle line:84% So we break up this infinite source on the left-hand half 00:50:05.920 --> 00:50:13.030 align:middle line:84% plane into a series of very, very small thin slices, each 00:50:13.030 --> 00:50:14.620 align:middle line:90% of which has a certain dose. 00:50:14.620 --> 00:50:17.200 align:middle line:84% And its dose, by definition, is just 00:50:17.200 --> 00:50:19.400 align:middle line:84% the concentration C times delta x-- 00:50:19.400 --> 00:50:23.970 align:middle line:84% the width of that little thin slice. 00:50:23.970 --> 00:50:26.910 align:middle line:84% And in fact, after some time t, we 00:50:26.910 --> 00:50:29.370 align:middle line:84% know how to write down for that little slice what 00:50:29.370 --> 00:50:30.490 align:middle line:90% the profile looks like. 00:50:30.490 --> 00:50:33.420 align:middle line:90% In fact, it's a Gaussian. 00:50:33.420 --> 00:50:37.710 align:middle line:84% So, now, if each slice then of all these slices 00:50:37.710 --> 00:50:41.430 align:middle line:84% can be added up, their Gaussians associated 00:50:41.430 --> 00:50:46.710 align:middle line:84% with their diffusion, we can then find the diffused profile. 00:50:46.710 --> 00:50:49.240 align:middle line:84% And that's exactly what we're doing here. 00:50:49.240 --> 00:50:51.630 align:middle line:84% So that equation at the bottom of slide 30 00:50:51.630 --> 00:50:56.790 align:middle line:84% shows that the concentration in this infinite source case 00:50:56.790 --> 00:50:59.520 align:middle line:84% can be given by the sum of all those Gaussians. 00:50:59.520 --> 00:51:03.780 align:middle line:84% So we go for the sum from i equals 1 to n, 00:51:03.780 --> 00:51:09.900 align:middle line:84% where n is some large number, of this delta xi comes 00:51:09.900 --> 00:51:14.070 align:middle line:84% from the dose-- remember, there was a Q front of the Gaussian-- 00:51:14.070 --> 00:51:17.250 align:middle line:84% times this exponential of x minus xi 00:51:17.250 --> 00:51:20.700 align:middle line:84% because we're sliding this the position 00:51:20.700 --> 00:51:25.680 align:middle line:84% of this particular thin slice along the x-axis. 00:51:25.680 --> 00:51:28.320 align:middle line:84% That exponential squared over 4 Dt. 00:51:28.320 --> 00:51:30.720 align:middle line:84% So we're summing up all these Gaussians 00:51:30.720 --> 00:51:32.310 align:middle line:90% at the bottom of slide 30. 00:51:32.310 --> 00:51:35.200 align:middle line:90% 00:51:35.200 --> 00:51:39.490 align:middle line:84% So, in fact, analytically, or in an exact sense, 00:51:39.490 --> 00:51:42.430 align:middle line:84% the solution which satisfies Fick's second law 00:51:42.430 --> 00:51:46.180 align:middle line:84% is written down at the top of slide 31. 00:51:46.180 --> 00:51:52.120 align:middle line:84% The concentration is actually equal to concentration C 00:51:52.120 --> 00:51:57.070 align:middle line:84% prime over 2 times the quantity in square brackets 00:51:57.070 --> 00:52:01.690 align:middle line:84% 1 minus the error function of the argument 00:52:01.690 --> 00:52:05.530 align:middle line:84% x over 2 times the square root of Dt. 00:52:05.530 --> 00:52:12.430 align:middle line:84% And we can write this as C sub s times the complementary error 00:52:12.430 --> 00:52:14.800 align:middle line:84% function of x over 2 times the square root of Dt, 00:52:14.800 --> 00:52:19.090 align:middle line:84% where the second equation and third gives you 00:52:19.090 --> 00:52:22.150 align:middle line:84% the definition of what we mean by the error function. 00:52:22.150 --> 00:52:25.090 align:middle line:84% Error function of z is just equal to 2 00:52:25.090 --> 00:52:28.660 align:middle line:84% over the square root of pi times the integral 00:52:28.660 --> 00:52:31.840 align:middle line:84% from 0 to z of this exponential-- 00:52:31.840 --> 00:52:34.520 align:middle line:84% it's integral of a Gaussian, basically. 00:52:34.520 --> 00:52:36.950 align:middle line:84% So the error function is the integral of the Gaussian. 00:52:36.950 --> 00:52:40.110 align:middle line:84% The complementary error function is 1 minus that. 00:52:40.110 --> 00:52:42.560 align:middle line:84% And then these error functions and complementary error 00:52:42.560 --> 00:52:44.240 align:middle line:90% functions have been tabulated. 00:52:44.240 --> 00:52:49.580 align:middle line:84% So in that sense, it can be calculated analytically. 00:52:49.580 --> 00:52:51.770 align:middle line:84% So we know that the complementary error function 00:52:51.770 --> 00:52:54.980 align:middle line:84% is what the shape of this profile 00:52:54.980 --> 00:52:56.900 align:middle line:90% can be calculated according to. 00:52:56.900 --> 00:52:59.430 align:middle line:90% 00:52:59.430 --> 00:53:03.510 align:middle line:84% So let's look at slide number 32, 00:53:03.510 --> 00:53:06.240 align:middle line:84% which, again, the error function solutions 00:53:06.240 --> 00:53:10.350 align:middle line:84% are made up of a sum of Gaussian delta function solutions. 00:53:10.350 --> 00:53:15.990 align:middle line:84% And what you see is that here in this plot, 00:53:15.990 --> 00:53:19.240 align:middle line:84% the initial profile is shown in the dashed line in green 00:53:19.240 --> 00:53:26.870 align:middle line:84% and that you have the subsequent profiles 00:53:26.870 --> 00:53:33.840 align:middle line:84% are time t equals t0 in black, 4t0, in blue, and 9t0 in red. 00:53:33.840 --> 00:53:38.610 align:middle line:84% And that the dose beyond x equals 0 00:53:38.610 --> 00:53:41.760 align:middle line:84% continues to increase with annealing time 00:53:41.760 --> 00:53:45.765 align:middle line:84% in this infinite source sort of solution. 00:53:45.765 --> 00:53:48.860 align:middle line:90% 00:53:48.860 --> 00:53:52.540 align:middle line:90% So let's go on to slide 33. 00:53:52.540 --> 00:53:55.360 align:middle line:90% 00:53:55.360 --> 00:54:00.220 align:middle line:84% We can take this as another special case, 00:54:00.220 --> 00:54:04.240 align:middle line:84% in fact, by just looking at the plot in the upper part 00:54:04.240 --> 00:54:06.010 align:middle line:90% of the slide 33. 00:54:06.010 --> 00:54:08.770 align:middle line:84% What we see is that at x equals 0, 00:54:08.770 --> 00:54:13.250 align:middle line:84% the concentration is actually held fixed. 00:54:13.250 --> 00:54:18.430 align:middle line:84% So if we have a situation where we have a constant surface 00:54:18.430 --> 00:54:21.760 align:middle line:84% concentration, then, in fact, the solution 00:54:21.760 --> 00:54:24.040 align:middle line:84% to the diffusion equation is just the right-hand side 00:54:24.040 --> 00:54:25.840 align:middle line:90% of the above figure. 00:54:25.840 --> 00:54:28.300 align:middle line:84% An example of this might be the case 00:54:28.300 --> 00:54:32.110 align:middle line:84% where we're doing diffusion from a gas ambient into the solid 00:54:32.110 --> 00:54:34.587 align:middle line:84% where the gas concentration above the solid solubility 00:54:34.587 --> 00:54:35.170 align:middle line:90% of the dopant. 00:54:35.170 --> 00:54:38.470 align:middle line:84% Then, in that case, at the surface of the silicon wafer 00:54:38.470 --> 00:54:40.750 align:middle line:84% the concentration of the dopant is 00:54:40.750 --> 00:54:42.410 align:middle line:90% fixed at the solid solubility. 00:54:42.410 --> 00:54:44.510 align:middle line:90% So it's constant. 00:54:44.510 --> 00:54:47.630 align:middle line:84% So if we take just that right-hand solution 00:54:47.630 --> 00:54:49.640 align:middle line:84% in the above figure, we can write it down 00:54:49.640 --> 00:54:52.220 align:middle line:90% from the previous solution. 00:54:52.220 --> 00:54:54.890 align:middle line:84% Concentration is just C sub s, which 00:54:54.890 --> 00:54:56.630 align:middle line:84% is the surface concentration, which 00:54:56.630 --> 00:55:01.080 align:middle line:84% is a constant, times the complementary error function. 00:55:01.080 --> 00:55:04.710 align:middle line:84% And in fact, you can integrate this equation 00:55:04.710 --> 00:55:07.230 align:middle line:84% to find the dose on the right-hand side. 00:55:07.230 --> 00:55:10.240 align:middle line:84% And the dose is given by this integral, 00:55:10.240 --> 00:55:13.800 align:middle line:84% which can be done integrating from 0 to infinity. 00:55:13.800 --> 00:55:15.400 align:middle line:90% And you can do that. 00:55:15.400 --> 00:55:18.390 align:middle line:84% And it turns out that dose Q is equal to 2 00:55:18.390 --> 00:55:23.310 align:middle line:84% C sub s over the square root of pi times the square root of Dt. 00:55:23.310 --> 00:55:25.920 align:middle line:84% So, again, now we see that this dose or the number 00:55:25.920 --> 00:55:29.340 align:middle line:84% of the integral of these curves on the right-hand side 00:55:29.340 --> 00:55:31.890 align:middle line:84% is increasing with time according 00:55:31.890 --> 00:55:33.480 align:middle line:90% to the square root of Dt. 00:55:33.480 --> 00:55:38.130 align:middle line:84% So we're getting a higher and higher dose into the sample. 00:55:38.130 --> 00:55:41.420 align:middle line:84% So let's go on to slide number 34. 00:55:41.420 --> 00:55:43.910 align:middle line:84% And here we are graphically comparing 00:55:43.910 --> 00:55:47.420 align:middle line:84% on the left and the right-hand side the two different types 00:55:47.420 --> 00:55:49.340 align:middle line:84% of classical processes that we talked 00:55:49.340 --> 00:55:52.850 align:middle line:84% about in terms of their diffusion profile shapes. 00:55:52.850 --> 00:55:56.240 align:middle line:84% On the left is the predeposition case where 00:55:56.240 --> 00:55:58.640 align:middle line:84% we have, say, a constant surface concentration, 00:55:58.640 --> 00:56:00.410 align:middle line:84% assuming the pre-dep was being done 00:56:00.410 --> 00:56:03.150 align:middle line:90% by a gas phase in diffusion. 00:56:03.150 --> 00:56:05.690 align:middle line:84% And there are two plots shown on the left. 00:56:05.690 --> 00:56:09.080 align:middle line:84% In the upper left is a plot on a linear scale. 00:56:09.080 --> 00:56:13.220 align:middle line:84% The lower left is a plot on a semi-log plot. 00:56:13.220 --> 00:56:15.530 align:middle line:84% But in either case, what you're looking at 00:56:15.530 --> 00:56:18.180 align:middle line:84% is a complementary error function. 00:56:18.180 --> 00:56:20.270 align:middle line:84% And you can see that the surface concentration 00:56:20.270 --> 00:56:24.200 align:middle line:84% is a constant normalized at C over C s equals 1. 00:56:24.200 --> 00:56:29.000 align:middle line:84% And that at the different times, you 00:56:29.000 --> 00:56:35.840 align:middle line:84% can see the twice square root of Dt is 0.1, 0.5, and 1 micron. 00:56:35.840 --> 00:56:38.640 align:middle line:84% The area under these curves is increasing. 00:56:38.640 --> 00:56:42.810 align:middle line:84% And fact, it's increasing by this factor square root of Dt. 00:56:42.810 --> 00:56:44.520 align:middle line:84% So the longer you would do the pre-dep, 00:56:44.520 --> 00:56:50.310 align:middle line:84% the more dose you would deposit into this silicon surface. 00:56:50.310 --> 00:56:53.140 align:middle line:84% On the right hand side, we see instead their Gaussian profiles 00:56:53.140 --> 00:56:53.640 align:middle line:90% . 00:56:53.640 --> 00:56:55.770 align:middle line:84% This would be the case for a drive-in, which 00:56:55.770 --> 00:56:57.220 align:middle line:90% has a constant dose. 00:56:57.220 --> 00:57:01.950 align:middle line:84% And so you see what's happening over time at the shorter time. 00:57:01.950 --> 00:57:03.990 align:middle line:84% We have a certain peak concentration. 00:57:03.990 --> 00:57:07.080 align:middle line:84% That peak concentration is then falling or dropping 00:57:07.080 --> 00:57:08.550 align:middle line:90% for the second profile. 00:57:08.550 --> 00:57:11.430 align:middle line:84% And the profile is broadening, and then it falls again, 00:57:11.430 --> 00:57:13.600 align:middle line:84% and the profile broadens further. 00:57:13.600 --> 00:57:16.770 align:middle line:84% So that's for Q equals a constant, integral is constant, 00:57:16.770 --> 00:57:20.400 align:middle line:84% and the left-hand side is for the surface concentration 00:57:20.400 --> 00:57:21.820 align:middle line:90% is a constant. 00:57:21.820 --> 00:57:23.700 align:middle line:84% That's just to get your eyes calibrated 00:57:23.700 --> 00:57:27.655 align:middle line:84% for complementary error function versus a Gaussian type 00:57:27.655 --> 00:57:28.155 align:middle line:90% of solution. 00:57:28.155 --> 00:57:31.730 align:middle line:90% 00:57:31.730 --> 00:57:36.840 align:middle line:84% Let's go on to slide number 35 and talk a little bit 00:57:36.840 --> 00:57:42.320 align:middle line:84% about dopant diffusion coefficients themselves. 00:57:42.320 --> 00:57:44.180 align:middle line:84% And we're just going to talk first 00:57:44.180 --> 00:57:47.900 align:middle line:84% about what we call intrinsic dopant diffusion, which 00:57:47.900 --> 00:57:53.150 align:middle line:84% happens in the case when the dopant concentration is 00:57:53.150 --> 00:57:55.640 align:middle line:90% less than n sub i. 00:57:55.640 --> 00:57:59.360 align:middle line:84% So the semiconductor is considered to be intrinsic. 00:57:59.360 --> 00:58:02.480 align:middle line:84% And generally, we can write these intrinsic diffusion 00:58:02.480 --> 00:58:05.960 align:middle line:84% coefficients in an Arrhenius-type relationship 00:58:05.960 --> 00:58:07.940 align:middle line:84% for the diffusion coefficient is just 00:58:07.940 --> 00:58:12.470 align:middle line:84% some constant D 0 written here times exponential 00:58:12.470 --> 00:58:15.140 align:middle line:90% minus EA over kT. 00:58:15.140 --> 00:58:17.960 align:middle line:84% And this chart, which is taken from your textbook, 00:58:17.960 --> 00:58:20.780 align:middle line:84% shows some rough numbers for what 00:58:20.780 --> 00:58:24.690 align:middle line:84% D 0 looks like in units of centimeters squared per second 00:58:24.690 --> 00:58:26.210 align:middle line:84% and what the activation energy looks 00:58:26.210 --> 00:58:29.330 align:middle line:84% like for a couple of different species 00:58:29.330 --> 00:58:32.940 align:middle line:84% and some of the pot that are dopants in silicon. 00:58:32.940 --> 00:58:35.530 align:middle line:84% So, for example, if you look at boron, 00:58:35.530 --> 00:58:39.360 align:middle line:84% it has an activation energy here of 3.5 electron volts. 00:58:39.360 --> 00:58:43.710 align:middle line:90% And the prefactor is about 1. 00:58:43.710 --> 00:58:47.460 align:middle line:84% Thing to note is that n sub i-- the intrinsic carrier 00:58:47.460 --> 00:58:50.680 align:middle line:84% concentration-- is very large at process temperatures. 00:58:50.680 --> 00:58:53.880 align:middle line:84% So, actually, intrinsic diffusion conditions 00:58:53.880 --> 00:58:58.930 align:middle line:84% apply under many different conditions or cases. 00:58:58.930 --> 00:59:01.800 align:middle line:84% So, for example, at 1,000 degrees C, 00:59:01.800 --> 00:59:05.460 align:middle line:84% n sub i is roughly equal to about 7 times 10 to the 18th. 00:59:05.460 --> 00:59:08.340 align:middle line:84% So this diffusion coefficient written here 00:59:08.340 --> 00:59:11.790 align:middle line:84% in this simple chart would apply anytime the concentration 00:59:11.790 --> 00:59:15.540 align:middle line:84% of that dopant at 1,000 degrees is less than about 7 10 00:59:15.540 --> 00:59:17.310 align:middle line:90% to the 18th. 00:59:17.310 --> 00:59:20.160 align:middle line:84% You can use this constant diffusion coefficient. 00:59:20.160 --> 00:59:22.770 align:middle line:84% When we get above that-- when the doping concentration is 00:59:22.770 --> 00:59:24.593 align:middle line:90% larger than n sub i-- 00:59:24.593 --> 00:59:26.010 align:middle line:84% we'll talk about next time there's 00:59:26.010 --> 00:59:27.660 align:middle line:84% some interesting Fermi level effects 00:59:27.660 --> 00:59:31.440 align:middle line:84% that come into play as the point defect concentrations become 00:59:31.440 --> 00:59:35.710 align:middle line:84% modulated by the carrier concentrations themselves 00:59:35.710 --> 00:59:36.910 align:middle line:90% of the diffusion species. 00:59:36.910 --> 00:59:39.690 align:middle line:90% 00:59:39.690 --> 00:59:43.370 align:middle line:84% So we'd go on to slide number 36. 00:59:43.370 --> 00:59:45.790 align:middle line:84% This is just a graphical representation. 00:59:45.790 --> 00:59:49.460 align:middle line:84% It's a plot of the diffusivity in centimeters squared 00:59:49.460 --> 00:59:53.030 align:middle line:84% per second as a function of 1,000 over T. 00:59:53.030 --> 00:59:55.790 align:middle line:90% So it's Arrhenius-type plot. 00:59:55.790 --> 00:59:57.860 align:middle line:84% In the upper y-axis, you can read the temperature 00:59:57.860 --> 00:59:59.310 align:middle line:90% if you prefer that. 00:59:59.310 --> 01:00:02.570 align:middle line:84% And what you can see for these dopants right off the bat 01:00:02.570 --> 01:00:06.710 align:middle line:84% looking at them is that there's a pretty large difference 01:00:06.710 --> 01:00:10.910 align:middle line:84% or discrepancy between so-called fast diffusers in silicon 01:00:10.910 --> 01:00:12.860 align:middle line:90% and the slower ones-- 01:00:12.860 --> 01:00:17.390 align:middle line:84% say, the fast diffusers being boron and phosphorus 01:00:17.390 --> 01:00:19.850 align:middle line:90% among the common dopants. 01:00:19.850 --> 01:00:24.320 align:middle line:84% Boron, again, the only really available p-type dopant, 01:00:24.320 --> 01:00:26.030 align:middle line:90% is relatively fast. 01:00:26.030 --> 01:00:30.650 align:middle line:84% It can be up to a factor of 10 or 20 or 30 faster diffusion 01:00:30.650 --> 01:00:33.680 align:middle line:84% coefficient than the slower diffusers such as arsenic 01:00:33.680 --> 01:00:35.540 align:middle line:90% or antimony. 01:00:35.540 --> 01:00:37.550 align:middle line:84% So this gives you a rough idea of, when 01:00:37.550 --> 01:00:39.810 align:middle line:84% we talk about doping diffusion, what we're 01:00:39.810 --> 01:00:42.012 align:middle line:84% going to have to worry about a little bit more 01:00:42.012 --> 01:00:43.720 align:middle line:84% would be the fast diffusers-- say, boron. 01:00:43.720 --> 01:00:45.990 align:middle line:90% 01:00:45.990 --> 01:00:47.990 align:middle line:84% The other thing I want to point out with respect 01:00:47.990 --> 01:00:50.480 align:middle line:84% to this plot on slide number 36 is 01:00:50.480 --> 01:00:53.120 align:middle line:84% that earlier versions of the text 01:00:53.120 --> 01:00:56.750 align:middle line:84% had an error in this corresponding figure. 01:00:56.750 --> 01:01:00.380 align:middle line:84% And so on the website, we had posted the errata. 01:01:00.380 --> 01:01:03.590 align:middle line:84% And you'll be able to see that to make sure that you're using, 01:01:03.590 --> 01:01:05.390 align:middle line:84% if you're reading curves off the plot-- 01:01:05.390 --> 01:01:07.648 align:middle line:90% reading values off the plot-- 01:01:07.648 --> 01:01:08.940 align:middle line:90% that you have the right values. 01:01:08.940 --> 01:01:10.315 align:middle line:84% One way to check that, of course, 01:01:10.315 --> 01:01:12.590 align:middle line:84% is to back to slide 35 and actually 01:01:12.590 --> 01:01:15.230 align:middle line:84% compute directly with a calculator 01:01:15.230 --> 01:01:16.880 align:middle line:90% the diffusion coefficients. 01:01:16.880 --> 01:01:19.530 align:middle line:90% 01:01:19.530 --> 01:01:22.730 align:middle line:90% So let's go on to slide 37. 01:01:22.730 --> 01:01:26.300 align:middle line:84% And I'd like to summarize this introduction 01:01:26.300 --> 01:01:29.090 align:middle line:90% to dopant diffusion. 01:01:29.090 --> 01:01:31.700 align:middle line:84% We've talked about that the placement of dope regions 01:01:31.700 --> 01:01:33.260 align:middle line:84% is critical because it determines 01:01:33.260 --> 01:01:37.370 align:middle line:84% many of the characteristics of short-channel MOSFETs. 01:01:37.370 --> 01:01:42.290 align:middle line:84% That's why we spent so much time calculating in great detail 01:01:42.290 --> 01:01:44.750 align:middle line:84% dopant diffusion, as we'll do over the next three or four 01:01:44.750 --> 01:01:46.940 align:middle line:90% lectures. 01:01:46.940 --> 01:01:49.670 align:middle line:84% Turns out there's a design tradeoff between the series 01:01:49.670 --> 01:01:52.520 align:middle line:84% resistance of a MOSFET, which means 01:01:52.520 --> 01:01:55.490 align:middle line:84% you would like to have deeper source-drains to minimize 01:01:55.490 --> 01:01:58.880 align:middle line:84% the series resistance and the short-channel effects, 01:01:58.880 --> 01:02:02.180 align:middle line:84% such as the control of Vt, would dictate that you 01:02:02.180 --> 01:02:04.670 align:middle line:90% have a shallower source-drain. 01:02:04.670 --> 01:02:06.770 align:middle line:84% So this is a fundamental tradeoff. 01:02:06.770 --> 01:02:11.390 align:middle line:84% And therefore, the channel doping profile engineering 01:02:11.390 --> 01:02:15.840 align:middle line:84% is a way of compromising that design tradeoff. 01:02:15.840 --> 01:02:17.900 align:middle line:84% And so channel doping profiles need 01:02:17.900 --> 01:02:21.360 align:middle line:84% to be controlled very accurately. 01:02:21.360 --> 01:02:23.130 align:middle line:84% Beyond that motivation, we talked 01:02:23.130 --> 01:02:25.890 align:middle line:84% about some simple analytic solutions 01:02:25.890 --> 01:02:29.640 align:middle line:84% that the time evolution of a doping profile, 01:02:29.640 --> 01:02:33.660 align:middle line:84% if the case is simple, is governed by a fixed loss-- 01:02:33.660 --> 01:02:36.615 align:middle line:84% the so-called diffusion equation. 01:02:36.615 --> 01:02:37.990 align:middle line:84% There are a couple of cases where 01:02:37.990 --> 01:02:39.910 align:middle line:90% there are analytic solutions. 01:02:39.910 --> 01:02:42.520 align:middle line:84% We talked about the diffusion of a Gaussian profile 01:02:42.520 --> 01:02:46.060 align:middle line:84% with a fixed dose or diffusion of a complementary error 01:02:46.060 --> 01:02:50.260 align:middle line:84% function, which we apply for a constant surface concentration. 01:02:50.260 --> 01:02:52.950 align:middle line:90% And finally--