WEBVTT 00:00:00.030 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.810 Commons license. 00:00:03.810 --> 00:00:06.850 Your support will help MIT OpenCourseWare to continue to 00:00:06.850 --> 00:00:10.510 offer high-quality educational resources for free. 00:00:10.510 --> 00:00:13.390 To make a donation or view additional material from 00:00:13.390 --> 00:00:15.830 hundreds of MIT courses, visit ocw.mit.edu. 00:00:20.960 --> 00:00:22.290 PROFESSOR: OK. 00:00:22.290 --> 00:00:27.140 Folks, I may not look like Don Sadoway, but for today, I am 00:00:27.140 --> 00:00:29.040 Don Sadoway. 00:00:29.040 --> 00:00:31.810 So let's-- 00:00:31.810 --> 00:00:34.760 I guess you can hear me pretty well, right? 00:00:34.760 --> 00:00:36.210 OK. 00:00:36.210 --> 00:00:41.470 Professor Sadoway is in a faraway, terrible place. 00:00:41.470 --> 00:00:43.770 It's called Vienna. 00:00:43.770 --> 00:00:44.800 Yeah, I know. 00:00:44.800 --> 00:00:46.260 We're here. 00:00:46.260 --> 00:00:47.840 What can I say? 00:00:47.840 --> 00:00:52.450 And he'll be back next Tuesday. 00:00:52.450 --> 00:00:55.100 And so my name is Ron Ballinger, and 00:00:55.100 --> 00:00:58.490 I'm taking his place. 00:00:58.490 --> 00:01:00.380 Put a muffler on that. 00:01:00.380 --> 00:01:01.920 OK, great. 00:01:01.920 --> 00:01:03.960 I'm sure you're all anxious, before we get 00:01:03.960 --> 00:01:09.080 started, to see that. 00:01:09.080 --> 00:01:10.330 Right. 00:01:10.330 --> 00:01:11.380 The average-- 00:01:11.380 --> 00:01:17.520 it was about 66, which is about 10 points lower than the 00:01:17.520 --> 00:01:22.490 last five or 10 quiz ones. 00:01:22.490 --> 00:01:23.870 So it's a little bit lower. 00:01:23.870 --> 00:01:28.440 It was a little bit harder test. I'm sure Professor 00:01:28.440 --> 00:01:32.330 Sadoway will mention this when he comes back, but for those 00:01:32.330 --> 00:01:38.350 of you who are below the 50 mark, it's time to do 00:01:38.350 --> 00:01:41.750 something about it, and when I say do something about it, I'm 00:01:41.750 --> 00:01:45.540 sure your recitation instructors will be very happy 00:01:45.540 --> 00:01:49.070 to help out in any way possible, but there are other 00:01:49.070 --> 00:01:52.150 options, not the least of which is a tutor. 00:01:52.150 --> 00:01:55.590 And so if you think you need one and you need to be kind of 00:01:55.590 --> 00:01:59.560 ruthless about examining yourself-- 00:01:59.560 --> 00:02:02.990 if you think you need one, go down and see Hilary and there 00:02:02.990 --> 00:02:08.430 are tutors which are available for individual instruction. 00:02:08.430 --> 00:02:12.350 So I think it's a great time to take it take advantage of 00:02:12.350 --> 00:02:15.200 that because I can guarantee you that test number two will 00:02:15.200 --> 00:02:16.660 not be any easier. 00:02:16.660 --> 00:02:20.980 In fact, it'll be considerably harder because it'll be pretty 00:02:20.980 --> 00:02:23.010 much new material. 00:02:23.010 --> 00:02:23.380 OK. 00:02:23.380 --> 00:02:28.790 So that's enough for that. 00:02:28.790 --> 00:02:31.980 Last day, we were-- 00:02:31.980 --> 00:02:32.530 I wasn't here. 00:02:32.530 --> 00:02:34.920 I was in Washington. 00:02:34.920 --> 00:02:36.860 They're actually trying to build what's 00:02:36.860 --> 00:02:39.970 called an exoflop computer. 00:02:39.970 --> 00:02:42.250 Does anybody know what exo means? 00:02:42.250 --> 00:02:44.190 10 to the 18th. 00:02:44.190 --> 00:02:49.090 10 to the 18th floating point operations per second. 00:02:49.090 --> 00:02:50.830 Why do they need that? 00:02:50.830 --> 00:02:54.070 Because when you go beyond 3091 then starting modeling 00:02:54.070 --> 00:02:55.470 these atoms-- 00:02:55.470 --> 00:02:58.430 especially F-block atoms-- you need a supercomputer that 00:02:58.430 --> 00:03:07.360 large and to do one run on a petaflop machine takes 100,000 00:03:07.360 --> 00:03:12.100 CPUs 10 hours for one atom. 00:03:12.100 --> 00:03:15.540 So solving the Schrodinger equation's a bit tricky. 00:03:15.540 --> 00:03:16.820 OK. 00:03:16.820 --> 00:03:23.210 Remember last time, we talked about metallic bonding and 00:03:23.210 --> 00:03:25.030 we're sort of sneaking up on it. 00:03:25.030 --> 00:03:28.080 Remember, Paul Drude-- 00:03:28.080 --> 00:03:29.940 well, let's look up here. 00:03:29.940 --> 00:03:31.980 These are the characteristics of metallic solids. 00:03:31.980 --> 00:03:34.680 They have high electrical conductivity. 00:03:34.680 --> 00:03:36.640 They have high thermal conductivity. 00:03:36.640 --> 00:03:39.870 They shine and they have ductility. 00:03:39.870 --> 00:03:44.000 So we're going to deal with the first three today, and 00:03:44.000 --> 00:03:48.150 later on in the course, we'll talk about the ductility part. 00:03:48.150 --> 00:03:54.610 But recall that Paul Drude modeled the solid as a series 00:03:54.610 --> 00:04:07.150 of cations, which amounted to the nucleus plus the inner 00:04:07.150 --> 00:04:11.420 core electrons and then allowed the electrons-- 00:04:11.420 --> 00:04:13.690 the remaining, the outer shell electrons, the valence 00:04:13.690 --> 00:04:20.290 electrons, to float around, called it a free electron gas. 00:04:20.290 --> 00:04:24.440 And that model explained the temperature dependence of heat 00:04:24.440 --> 00:04:29.500 capacity, but it was not so good when it came to 00:04:29.500 --> 00:04:30.560 electrical conductivity. 00:04:30.560 --> 00:04:34.020 It explained the fact that you get electrical conductivity, 00:04:34.020 --> 00:04:38.440 but not the fact that some materials are conductors and 00:04:38.440 --> 00:04:40.570 some materials are insulators. 00:04:40.570 --> 00:04:44.770 So we need to deal with that and for that, we need to talk 00:04:44.770 --> 00:04:46.620 to two sets of folks. 00:04:46.620 --> 00:04:58.040 The first one is Felix Bloch and he was a 1928-- 00:04:58.040 --> 00:04:59.220 he was a-- 00:04:59.220 --> 00:05:02.080 he got his PhD under Heisenberg. 00:05:02.080 --> 00:05:04.290 Boy, it would've been nice back in those days, working 00:05:04.290 --> 00:05:06.140 for those great folks. 00:05:06.140 --> 00:05:12.680 And he applied quantum mechanics to solids. 00:05:16.280 --> 00:05:25.500 He said, OK, let's consider that in a solid, the atoms are 00:05:25.500 --> 00:05:32.960 arranged in an array. 00:05:32.960 --> 00:05:33.650 What's an array? 00:05:33.650 --> 00:05:35.970 Well, it's actually called a crystal. 00:05:40.640 --> 00:05:43.650 It's an ordered array of atoms. We're going to say a 00:05:43.650 --> 00:05:48.040 lot more about that as we go along, but-- 00:05:48.040 --> 00:06:06.350 and then he said, well, then let's apply the Schrodinger 00:06:06.350 --> 00:06:08.120 equation to this system. 00:06:08.120 --> 00:06:10.600 Now it's a big difference between the Schrodinger 00:06:10.600 --> 00:06:14.390 equation for a single atom in a gas and 00:06:14.390 --> 00:06:16.306 multiple atoms in a solid. 00:06:16.306 --> 00:06:19.270 It's a different set of boundary conditions, different 00:06:19.270 --> 00:06:23.890 set of conditions all together. 00:06:23.890 --> 00:06:30.160 And when you do that, you get a set of solutions for the 00:06:30.160 --> 00:06:38.670 valence electrons which is, as you might 00:06:38.670 --> 00:06:40.035 expect, it's periodic. 00:06:42.820 --> 00:06:57.340 It's periodic and it invokes wave-like properties of the 00:06:57.340 --> 00:07:05.890 electron and you end up with a set of values of the 00:07:05.890 --> 00:07:11.130 wavelengths for the electron that are such that it allows 00:07:11.130 --> 00:07:16.930 mobility, which is, after all, what we're after. 00:07:16.930 --> 00:07:19.220 These electrons got to move through the solid if we're 00:07:19.220 --> 00:07:23.540 going to have conductivity. 00:07:23.540 --> 00:07:37.280 And this is an example where classical physics wouldn't 00:07:37.280 --> 00:07:38.900 allow that. 00:07:38.900 --> 00:07:43.790 So you can't get this kind of behavior 00:07:43.790 --> 00:07:45.810 with classical physics. 00:07:45.810 --> 00:07:48.050 So that's one piece. 00:07:48.050 --> 00:07:54.360 And the second group, two guys. 00:07:54.360 --> 00:08:06.380 Walter Heitler and Fritz London. 00:08:06.380 --> 00:08:11.510 We know Fritz London from London dispersion force-- 00:08:11.510 --> 00:08:12.850 the same guy. 00:08:12.850 --> 00:08:16.430 These guys were very, very productive. 00:08:16.430 --> 00:08:19.290 And he did a post-doc-- 00:08:22.260 --> 00:08:24.650 that's another word for slave labor, by the way. 00:08:27.180 --> 00:08:29.070 Folks will know that here. 00:08:29.070 --> 00:08:30.060 For guess who? 00:08:30.060 --> 00:08:31.440 Well, he did it for Schrodinger. 00:08:39.920 --> 00:08:41.930 Interesting story, I guess. 00:08:41.930 --> 00:08:46.380 These guys showed up for their slave labor at Schrodinger's 00:08:46.380 --> 00:08:50.190 lab, only to discover that Schrodinger had taken a 00:08:50.190 --> 00:08:52.330 position at a different university. 00:08:52.330 --> 00:08:56.720 So these guys showed up and he said, goodbye. 00:08:56.720 --> 00:09:00.060 Anyway, that's a bummer. 00:09:00.060 --> 00:09:00.960 OK. 00:09:00.960 --> 00:09:03.220 And they sat down and they said, well, OK. 00:09:03.220 --> 00:09:08.270 Let's see if we can go at this from an energy point of view 00:09:08.270 --> 00:09:11.980 as opposed from the quantum mechanical point of view and 00:09:11.980 --> 00:09:17.560 let's see if we can apply LCAOMO-- 00:09:20.420 --> 00:09:22.810 now don't break out in hives because of the test-- 00:09:26.120 --> 00:09:33.070 to a solid, to large aggregates. 00:09:33.070 --> 00:09:34.320 Large what? 00:09:37.480 --> 00:09:40.860 Of atoms. How big is large? 00:09:40.860 --> 00:09:42.880 Well, I don't know. 00:09:42.880 --> 00:09:46.670 Dream up a number-- say, 10 to the 23rd. 00:09:46.670 --> 00:09:50.190 Lots of atoms. And let's see what happens. 00:09:50.190 --> 00:09:54.860 Well, we've already done a little of this in the past. We 00:09:54.860 --> 00:10:01.020 looked at the energetics of the stability of hydrogen or 00:10:01.020 --> 00:10:02.130 the stability of helium. 00:10:02.130 --> 00:10:05.710 So we can take that and we can kind of add up and we'll see 00:10:05.710 --> 00:10:06.940 what happens. 00:10:06.940 --> 00:10:13.360 Well, remember, we had atom A and now let's only deal with 00:10:13.360 --> 00:10:17.550 the valence electrons. 00:10:17.550 --> 00:10:21.985 And so this would be 0 and there's another atom A-- 00:10:24.880 --> 00:10:25.620 0-- 00:10:25.620 --> 00:10:30.830 and this would be A2 and we've been through this before. 00:10:30.830 --> 00:10:31.810 Let's just take-- 00:10:31.810 --> 00:10:37.940 we know that we get splitting and we get a sigma bonding 00:10:37.940 --> 00:10:42.820 orbital and a sigma star anti-bonding orbital. 00:10:42.820 --> 00:10:48.990 Well, let's start adding some atoms here, additional atoms. 00:10:48.990 --> 00:10:50.230 What do we get? 00:10:50.230 --> 00:10:54.680 Well, let's just try for A sub N-- 00:10:54.680 --> 00:10:56.436 lots of atoms. What do we get? 00:11:02.160 --> 00:11:07.640 We end up with lots of states. 00:11:07.640 --> 00:11:09.460 Remember, we have to have conservation of states. 00:11:09.460 --> 00:11:11.840 So for every atom we add-- 00:11:11.840 --> 00:11:16.410 let's say this is copper, for example, which 00:11:16.410 --> 00:11:19.960 has an s1, one s. 00:11:19.960 --> 00:11:21.880 It's an odd number of electrons. 00:11:21.880 --> 00:11:26.990 So every time I add a copper to this, I add extra states. 00:11:26.990 --> 00:11:27.870 And then what do I do? 00:11:27.870 --> 00:11:33.510 I start filling them using Aufbau, just like we've done 00:11:33.510 --> 00:11:40.140 in the past. Well, if we keep going, and by the way, not to 00:11:40.140 --> 00:11:49.030 scale, it starts looking like a whole bunch of states. 00:11:49.030 --> 00:11:52.370 And this energy level here, these energies, what? 00:11:52.370 --> 00:11:54.180 We know for hydrogen, this is what? 00:11:54.180 --> 00:11:58.560 Minus 13.6 electron volts. 00:11:58.560 --> 00:12:02.110 So what are we dealing with here in terms of state 00:12:02.110 --> 00:12:03.330 differences? 00:12:03.330 --> 00:12:09.550 Well, let's take 10 to the 23rd atoms and let's see if we 00:12:09.550 --> 00:12:10.800 can calculate energy. 00:12:10.800 --> 00:12:12.910 Well, let's take for copper-- 00:12:12.910 --> 00:12:16.110 the molar volume of copper is what? 00:12:16.110 --> 00:12:22.220 7.11 centimeter cube for mole. 00:12:22.220 --> 00:12:22.790 All right. 00:12:22.790 --> 00:12:25.510 And that's N to the 23rd-- 00:12:25.510 --> 00:12:32.310 actually, 6.02 times 10 to the 23rd atoms. 00:12:32.310 --> 00:12:36.930 And let's just ask ourselves, what's the sort of range here? 00:12:36.930 --> 00:12:39.720 Well, we know it's 13.6 here. 00:12:39.720 --> 00:12:42.470 We know that's 0. 00:12:42.470 --> 00:12:43.820 We're all friends. 00:12:43.820 --> 00:12:47.880 So let's call it 10 ev, just for grins. 00:12:47.880 --> 00:12:50.240 Not a bad number. 00:12:50.240 --> 00:12:54.600 And let's ask ourselves, well, if we've got 10 to the 23rd 00:12:54.600 --> 00:13:01.340 atoms and 10 ev-- let's take 10 ev over 10 to the 23rd and 00:13:01.340 --> 00:13:02.000 you get what? 00:13:02.000 --> 00:13:14.940 Well, 10 to the minus 22 ev per state. 00:13:14.940 --> 00:13:16.370 That's small. 00:13:16.370 --> 00:13:20.120 That's really, really, really, really small and if you 00:13:20.120 --> 00:13:23.800 convert that to joules, you end up with 10 to 00:13:23.800 --> 00:13:27.690 the minus 41 joules. 00:13:27.690 --> 00:13:29.990 That's really small. 00:13:29.990 --> 00:13:31.160 So what are we saying? 00:13:31.160 --> 00:13:37.310 We're saying that this organization here-- 00:13:37.310 --> 00:13:41.850 when we and put a lot of atoms in there, we end up with what 00:13:41.850 --> 00:13:45.060 amounts to a band. 00:13:48.380 --> 00:13:49.925 A band of states. 00:13:52.540 --> 00:13:54.150 And what do we do? 00:13:58.170 --> 00:14:02.790 We populate this band just like we did using the Aufbau 00:14:02.790 --> 00:14:07.910 principle, but what does it really look like? 00:14:07.910 --> 00:14:12.550 Well, in the case of copper, you remember, 00:14:12.550 --> 00:14:19.810 copper has a 1s electron. 00:14:19.810 --> 00:14:28.660 Copper is 3d10s1. 00:14:28.660 --> 00:14:35.490 If we start filling start this band, we're going to get-- 00:14:39.860 --> 00:14:47.900 and so we start filling and we get up here and we find that 00:14:47.900 --> 00:14:53.890 we end up with a half-filled band, because there are states 00:14:53.890 --> 00:14:55.916 that are not occupied. 00:14:58.540 --> 00:15:03.290 But remember, the distance between this guy and this guy, 00:15:03.290 --> 00:15:06.680 an occupied and unoccupied state, is only 10 to 00:15:06.680 --> 00:15:09.020 the minus 41 joules. 00:15:09.020 --> 00:15:12.650 So that's pretty small. 00:15:12.650 --> 00:15:15.980 10 to the minus 22 electron volts. 00:15:15.980 --> 00:15:21.530 To give you an example, 0.025 electron volts, which is, if 00:15:21.530 --> 00:15:25.120 you want to convert that to temperature, that's about 300 00:15:25.120 --> 00:15:27.160 degrees Kelvin. 00:15:27.160 --> 00:15:30.350 So very, very small energy. 00:15:30.350 --> 00:15:35.640 So that means if I were to take and if I were to apply a 00:15:35.640 --> 00:15:39.750 potential here, then what happens? 00:15:39.750 --> 00:15:41.960 Well, I add a little energy-- and I don't 00:15:41.960 --> 00:15:43.250 have to add much energy-- 00:15:43.250 --> 00:15:48.120 and it's very easy for me to promote one or more of these 00:15:48.120 --> 00:15:51.600 electrons up into the above here and 00:15:51.600 --> 00:15:55.560 then I can get migration. 00:15:55.560 --> 00:16:00.200 So the endpoint is that if we apply a potential, we end up 00:16:00.200 --> 00:16:09.830 with conductivity, which is what we were after. 00:16:09.830 --> 00:16:14.100 Moreover, we can say a few more things. 00:16:14.100 --> 00:16:16.590 If I shine-- 00:16:16.590 --> 00:16:18.560 now this is a sort of mixed metaphor here. 00:16:18.560 --> 00:16:21.180 This is an energy diagram and this some plates on an 00:16:21.180 --> 00:16:24.270 electrode so be a little bit careful. 00:16:24.270 --> 00:16:26.370 We shine photons on this thing. 00:16:26.370 --> 00:16:27.130 What happens? 00:16:27.130 --> 00:16:29.560 What's the energy of the photons, of light? 00:16:29.560 --> 00:16:33.190 Between 200 or 400 and 700 nanometers. 00:16:33.190 --> 00:16:35.330 It's about one electron volt. 00:16:35.330 --> 00:16:37.890 So there's plenty of electron volts here. 00:16:37.890 --> 00:16:40.500 I'm going to-- with a metal, not only will I have 00:16:40.500 --> 00:16:44.270 conductivity, but there'll be enough energy here to promote 00:16:44.270 --> 00:16:50.010 electrons and those electrons will move around and we'll end 00:16:50.010 --> 00:16:54.710 up with re-emission and we end up with opaqueness. 00:16:54.710 --> 00:16:59.660 In other words, we absorb light and readmit it and so we 00:16:59.660 --> 00:17:05.930 end up with, in the case of a metal, luster. 00:17:05.930 --> 00:17:07.400 So there's a couple of things. 00:17:07.400 --> 00:17:10.530 So if we recall back here-- 00:17:10.530 --> 00:17:12.680 we're talking about high electrical conductivity. 00:17:12.680 --> 00:17:14.250 We had-- we need to say a lot more about that, 00:17:14.250 --> 00:17:15.500 but we're OK here. 00:17:15.500 --> 00:17:17.940 But Drude, it was OK as well. 00:17:17.940 --> 00:17:21.000 High thermal conductivity, Drude did that. 00:17:21.000 --> 00:17:22.100 That was fine. 00:17:22.100 --> 00:17:23.100 Luster. 00:17:23.100 --> 00:17:28.590 OK, So it works for copper, seems to be OK. 00:17:28.590 --> 00:17:31.180 But what about another one? 00:17:31.180 --> 00:17:34.830 Like, say, beryllium. 00:17:34.830 --> 00:17:35.850 What's with beryllium? 00:17:35.850 --> 00:17:37.900 Well, beryllium is 2s2s2. 00:17:40.940 --> 00:17:46.270 So now let's draw the thing for beryllium. 00:17:46.270 --> 00:17:47.290 We have beryllium. 00:17:47.290 --> 00:17:56.900 We have-- must be a 2s and now it's 2p and we populate this. 00:17:56.900 --> 00:18:01.040 And now we want to add a beryllium-- 00:18:01.040 --> 00:18:03.530 n beryllium atoms, all right? 00:18:03.530 --> 00:18:04.980 What's happening? 00:18:04.980 --> 00:18:06.550 well, it'll be a little bit. 00:18:06.550 --> 00:18:11.270 We got this band that we've talked about earlier. 00:18:11.270 --> 00:18:13.680 Now we go to fill it. 00:18:13.680 --> 00:18:17.390 So we're filling these guys and we fill it, and lo and 00:18:17.390 --> 00:18:20.740 behold, we find out that we fill it all the way to the top 00:18:20.740 --> 00:18:24.220 and so we're screwed. 00:18:24.220 --> 00:18:25.470 We have no conductivity. 00:18:28.530 --> 00:18:31.540 What I can imagine now is there's 100 computers in here. 00:18:31.540 --> 00:18:33.910 50 of them have Skype. 00:18:33.910 --> 00:18:34.750 Guess what's going to happen? 00:18:34.750 --> 00:18:38.570 By the end of the day, there'll be 00:18:38.570 --> 00:18:41.410 50 things up there. 00:18:41.410 --> 00:18:44.910 So we know beryllium's metal and it has conductivity. 00:18:44.910 --> 00:18:46.410 So what's the deal? 00:18:46.410 --> 00:18:52.560 Well, it turns out that while beryllium has the 2s band 00:18:52.560 --> 00:18:57.280 full, the 2p orbitals still are there. 00:18:57.280 --> 00:18:58.745 They're still there and so there's going 00:18:58.745 --> 00:19:06.410 to be a band unfilled. 00:19:06.410 --> 00:19:11.210 There's going to be a band for the 2p orbitals. 00:19:11.210 --> 00:19:13.690 Well, guess what? 00:19:13.690 --> 00:19:24.650 It turns out that the way I've drawn it, the 2p band overlaps 00:19:24.650 --> 00:19:26.180 the 2s band. 00:19:26.180 --> 00:19:33.110 And so what that means is that I can promote into the 2p band 00:19:33.110 --> 00:19:36.940 and I can achieve my conductivity. 00:19:36.940 --> 00:19:41.850 So that's another-- so we solved the problem of the 00:19:41.850 --> 00:19:44.940 filled s-orbitals, and we've got 00:19:44.940 --> 00:19:47.800 conductivity in both cases. 00:19:47.800 --> 00:19:55.760 So we're OK so far, but Drude wasn't far behind. 00:20:05.500 --> 00:20:06.750 What can we say about insulators? 00:20:15.630 --> 00:20:18.170 It looks like we have a problem. 00:20:18.170 --> 00:20:21.210 Well, let's take a look at another example. 00:20:21.210 --> 00:20:22.650 And let's try carbon. 00:20:26.290 --> 00:20:28.170 Well, we know carbon is what? 00:20:28.170 --> 00:20:29.420 2s2p2. 00:20:32.870 --> 00:20:34.650 And so we can go-- 00:20:34.650 --> 00:20:38.470 and we know, by the way, that most of the time carbon will 00:20:38.470 --> 00:20:42.100 hybridize and so we'll end up with 2sp3. 00:20:45.360 --> 00:20:49.940 So n equals 2sp3 hybridized. 00:20:49.940 --> 00:20:51.820 So we see that happen. 00:20:51.820 --> 00:20:57.440 And we also know that carbon's not a metal and that we have 00:20:57.440 --> 00:21:01.470 strong bonds. 00:21:04.040 --> 00:21:05.720 In the structure that we're going to talk about, they're 00:21:05.720 --> 00:21:09.440 covalent bonds and so they're very strong bonds. 00:21:09.440 --> 00:21:10.690 So let's do this. 00:21:10.690 --> 00:21:14.720 Carbon in the gas phase-- 00:21:14.720 --> 00:21:21.160 we have 2s and 2p. 00:21:21.160 --> 00:21:25.540 We know that what happens is we end up with hybridization 00:21:25.540 --> 00:21:29.030 and so we ends up and we fill these guys. 00:21:33.940 --> 00:21:35.870 So that's what we get. 00:21:35.870 --> 00:21:46.620 Now this would be for diamond in the gas phase. 00:21:49.550 --> 00:21:52.780 So with this hybridization, we get bands. 00:21:52.780 --> 00:21:57.220 We start adding carbon atoms to this and what do we get? 00:21:57.220 --> 00:22:04.280 Well, we get a band that's the 2p band. 00:22:04.280 --> 00:22:08.820 The sp3 band is different looking than the bands for 00:22:08.820 --> 00:22:14.430 magnesium or copper or beryllium in that there's a 00:22:14.430 --> 00:22:19.080 separation between the sigma and the sigma-star 00:22:19.080 --> 00:22:20.500 anti-bonding orbitals. 00:22:23.420 --> 00:22:31.860 So we start doing these guys up using Aufbau, and we 00:22:31.860 --> 00:22:40.650 discover that there's an energy gap e sub g between the 00:22:40.650 --> 00:22:45.180 sigma bonding orbital part of the band and 00:22:45.180 --> 00:22:48.030 the sigma-star band. 00:22:48.030 --> 00:22:49.150 So what's happening? 00:22:49.150 --> 00:22:52.480 Well, that's actually pretty good sized. 00:22:52.480 --> 00:22:58.220 It's about 5.4 electron volts. 00:22:58.220 --> 00:23:00.020 Now compare that with what? 00:23:00.020 --> 00:23:07.250 Visible light is around one electron volt. 00:23:07.250 --> 00:23:12.110 So now we have a very, very different situation and 00:23:12.110 --> 00:23:15.010 there's some terminology we need to have here. 00:23:15.010 --> 00:23:18.870 This is the so-called conduction band and this is 00:23:18.870 --> 00:23:23.530 the so-called valence. 00:23:23.530 --> 00:23:26.210 band. 00:23:26.210 --> 00:23:28.210 Valence band and conduction band. 00:23:28.210 --> 00:23:33.820 So in order for us to get electrical conductivity, we 00:23:33.820 --> 00:23:39.500 have to somehow promote an electron across the 5.4 00:23:39.500 --> 00:23:43.090 electron-volt band gap. 00:23:43.090 --> 00:23:50.500 This is the so-called band gap. 00:23:50.500 --> 00:23:54.540 Now we know that visible light is about 1 electron volt so we 00:23:54.540 --> 00:23:56.910 know that's not going to do it. 00:23:56.910 --> 00:24:00.870 And in fact, we might expect that even a diamond is what? 00:24:00.870 --> 00:24:02.480 Transparent to visible light. 00:24:02.480 --> 00:24:05.710 So if the photons come in, if there's no promotion, there's 00:24:05.710 --> 00:24:10.870 no mission, transparency to visible light. 00:24:10.870 --> 00:24:15.390 So that's a big number. 00:24:15.390 --> 00:24:17.740 Let's try another one. 00:24:17.740 --> 00:24:19.710 Let's try silicon. 00:24:22.520 --> 00:24:30.970 Now I'm going down group four, where silicon is going to be 00:24:30.970 --> 00:24:35.690 also sp3 hybridization. 00:24:35.690 --> 00:24:38.450 So we can do that, only this time over 00:24:38.450 --> 00:24:41.800 here, carbon is what? 00:24:41.800 --> 00:24:46.040 n equals 2. 00:24:46.040 --> 00:24:53.980 For silicon, n equals 3 and so we can do the same thing. 00:24:53.980 --> 00:24:55.950 Here's the 3s. 00:24:55.950 --> 00:24:57.200 Here's the 3p. 00:24:59.560 --> 00:25:06.270 And we hybridize and we end up with before and then we end up 00:25:06.270 --> 00:25:12.300 with a band, same kind of band structure where this is a 00:25:12.300 --> 00:25:20.980 sigma-star, star, this is e sub g, this is the valence 00:25:20.980 --> 00:25:24.200 band, conduction band. 00:25:24.200 --> 00:25:28.270 Now we have states up here, but in this 00:25:28.270 --> 00:25:29.930 case, what do you figure? 00:25:29.930 --> 00:25:32.040 n equals 3. 00:25:32.040 --> 00:25:36.290 So those valence electrons are hanging out further away from 00:25:36.290 --> 00:25:37.860 the nucleus. 00:25:37.860 --> 00:25:42.300 And we know generally that the energy drops off, the valence 00:25:42.300 --> 00:25:44.840 electron energy drops off as we get further 00:25:44.840 --> 00:25:46.760 away from the nucleus. 00:25:46.760 --> 00:25:51.040 So you might expect that the energy of this gap would be a 00:25:51.040 --> 00:25:53.360 little bit smaller and indeed. 00:25:53.360 --> 00:25:53.760 it is. 00:25:53.760 --> 00:25:58.090 Very fortunate for us, it's 1.1 electron volts. 00:25:58.090 --> 00:25:59.340 Now we're getting close. 00:26:04.010 --> 00:26:15.830 And so this is 1/4 or even 1/5 of that for carbon and 00:26:15.830 --> 00:26:20.850 remember, visible light is on the order of 00:26:20.850 --> 00:26:21.700 one electron volt. 00:26:21.700 --> 00:26:24.700 So one, one and half electron volts. 00:26:24.700 --> 00:26:26.190 So what happens? 00:26:26.190 --> 00:26:32.560 This is close enough so that we get semi-conduction. 00:26:42.210 --> 00:26:47.410 It's called a semiconductor and we'll make a definition. 00:26:47.410 --> 00:26:55.030 If the band gap is greater than 3 electron volts, we call 00:26:55.030 --> 00:27:00.160 it an insulator. 00:27:00.160 --> 00:27:05.976 If the band gap is less than-- well, let's give it a range. 00:27:16.140 --> 00:27:24.880 1 to 3 electron volts, we call it a semiconductor and of 00:27:24.880 --> 00:27:28.280 course, if the band gap is equal to 0, 00:27:28.280 --> 00:27:32.340 we call it a metal. 00:27:32.340 --> 00:27:39.040 So that's a distinction, which is a little bit arbitrary, but 00:27:39.040 --> 00:27:41.450 pretty good. 00:27:41.450 --> 00:27:44.830 So now, if I take a look at silicon, what do I see? 00:27:44.830 --> 00:27:48.340 Well, if I had a piece of silicon here as opposed to 00:27:48.340 --> 00:27:51.200 diamond and I looked at it, it would be gray. 00:27:51.200 --> 00:27:54.680 We would have color and the reason it would have color is 00:27:54.680 --> 00:27:59.210 because I get some promotion here and I get re-emission and 00:27:59.210 --> 00:28:00.730 so now I get color. 00:28:00.730 --> 00:28:04.835 So it's not transparent to visible light. 00:28:16.640 --> 00:28:17.100 OK. 00:28:17.100 --> 00:28:21.520 Let's take a look in general at what we-- 00:28:21.520 --> 00:28:23.230 what we're talking about is photoexcitation. 00:28:31.760 --> 00:28:35.120 Now this is a diagram which is-- 00:28:35.120 --> 00:28:36.710 you should have in your aid sheet. 00:28:36.710 --> 00:28:39.680 Sooner or later, you should have it in your aid sheet. 00:28:39.680 --> 00:28:40.110 OK. 00:28:40.110 --> 00:28:43.180 So let's draw a general band. 00:28:46.340 --> 00:28:47.410 Here's our two bands. 00:28:47.410 --> 00:28:49.500 This is the valence band. 00:28:49.500 --> 00:28:56.630 This is the conduction band and we have a band gap. 00:28:59.530 --> 00:29:07.480 And now what happens if I were a photon here? 00:29:07.480 --> 00:29:09.910 Well, I guess it depends. 00:29:09.910 --> 00:29:11.380 If the photon-- 00:29:11.380 --> 00:29:17.740 if e photon greater than e band gap, then what happens? 00:29:17.740 --> 00:29:24.570 I get promotion of an electron up from the valence band to 00:29:24.570 --> 00:29:26.630 the conduction band. 00:29:26.630 --> 00:29:27.940 OK. 00:29:27.940 --> 00:29:29.210 So then what happens? 00:29:29.210 --> 00:29:32.865 Well, the photon is quantized. 00:29:32.865 --> 00:29:34.610 It's one shot. 00:29:34.610 --> 00:29:35.570 The photon's gone. 00:29:35.570 --> 00:29:38.070 Now I guess we need to settle one little thing. 00:29:38.070 --> 00:29:40.520 What happens if the photon is really a lot greater 00:29:40.520 --> 00:29:41.460 than the band gap? 00:29:41.460 --> 00:29:45.030 In another words, let's say it's a million electron volts 00:29:45.030 --> 00:29:46.010 or something like that. 00:29:46.010 --> 00:29:50.260 Well, for purposes of 3.091, what we're going to assume is 00:29:50.260 --> 00:29:58.820 that any excess energy goes to heat. 00:29:58.820 --> 00:30:03.290 Let's not worry about what happens for other things. 00:30:03.290 --> 00:30:07.850 In fact, that's not far off. 00:30:07.850 --> 00:30:11.990 Some of these LEDs that you see, they get warm. 00:30:11.990 --> 00:30:14.600 So there is some heat. 00:30:14.600 --> 00:30:16.500 So the photon-- 00:30:16.500 --> 00:30:18.450 once it goes away, I get-- 00:30:20.960 --> 00:30:22.890 the electron falls back down. 00:30:22.890 --> 00:30:27.410 Well, if the electron falls back down, then what do I get? 00:30:27.410 --> 00:30:32.250 I get another photon out, but now I've basically built 00:30:32.250 --> 00:30:35.900 myself a diode or some kind of device. 00:30:35.900 --> 00:30:37.150 That photon-- 00:30:41.090 --> 00:30:44.610 the energy of the photon is equal to what? 00:30:44.610 --> 00:30:47.320 It's equal to the band gap, which is 00:30:47.320 --> 00:30:50.160 equal to HC over lambda. 00:30:50.160 --> 00:30:55.780 So I can adjust the wavelength here if I can 00:30:55.780 --> 00:30:57.790 adjust the band gap. 00:31:00.350 --> 00:31:03.720 See where we're going with this? 00:31:03.720 --> 00:31:06.510 Well, what happens if I-- 00:31:06.510 --> 00:31:18.700 let's say I hook this up to a resistor and I draw a current. 00:31:18.700 --> 00:31:22.230 Well, as long as I keep the light shining on here, 00:31:22.230 --> 00:31:26.650 photons, then I can keep promoting these guys, and I 00:31:26.650 --> 00:31:28.820 can keep drawing current. 00:31:28.820 --> 00:31:31.020 So what do I have? 00:31:31.020 --> 00:31:34.190 I have something that I can generate 00:31:34.190 --> 00:31:39.500 electricity with light. 00:31:39.500 --> 00:31:42.600 And so that's sort of solar-powered 00:31:42.600 --> 00:31:43.390 something, isn't it? 00:31:43.390 --> 00:31:44.760 We can generate current. 00:31:44.760 --> 00:31:48.850 What happens if I take-- 00:31:48.850 --> 00:31:52.320 and now instead of doing that, I hook up a battery to this 00:31:52.320 --> 00:31:57.070 thing and now I pump current in here? 00:31:57.070 --> 00:31:57.980 I pump current in here. 00:31:57.980 --> 00:32:03.520 In that case, I can force the electrons up here. 00:32:03.520 --> 00:32:10.480 So I can force electrons to go in the reverse and I could 00:32:10.480 --> 00:32:15.410 make a photosensor or I could force the electrons up here 00:32:15.410 --> 00:32:19.530 and let them come back down and I know this wavelength 00:32:19.530 --> 00:32:24.350 here, I can make a light-emitting diode. 00:32:24.350 --> 00:32:28.470 So just this one little concept here, which is very 00:32:28.470 --> 00:32:38.830 simplified, gives us the basis for photovoltaics. 00:32:41.360 --> 00:32:46.000 And that's exactly the way it works. 00:32:48.980 --> 00:32:49.950 OK. 00:32:49.950 --> 00:32:52.120 Let's put this in another way. 00:32:55.810 --> 00:33:01.380 Let's plot the percent absorption. 00:33:01.380 --> 00:33:03.490 In other words, if the energy's 00:33:03.490 --> 00:33:05.870 high enough, I absorb-- 00:33:05.870 --> 00:33:12.730 versus wavelength this way and since energy is inversely 00:33:12.730 --> 00:33:17.840 proportional to wavelength, we have energy going this way. 00:33:21.420 --> 00:33:24.570 Let's put some numbers in here. 00:33:24.570 --> 00:33:30.370 Let's say this is 400, this is 700 and this is Professor 00:33:30.370 --> 00:33:31.620 Sadoway's dreaded nanometers. 00:33:34.480 --> 00:33:42.970 So this is visible light and let's put carbon on there. 00:33:42.970 --> 00:33:46.500 Well, if you do the calculation, convert iy, it 00:33:46.500 --> 00:33:53.860 turns out that for carbon, with this band gap, you end up 00:33:53.860 --> 00:33:58.160 with behavior where if the energy is above 00:33:58.160 --> 00:34:00.480 the band gap, I get-- 00:34:00.480 --> 00:34:02.800 well, let's call this 100. 00:34:02.800 --> 00:34:04.430 That's 0, right? 00:34:04.430 --> 00:34:07.110 I get 100% absorption. 00:34:07.110 --> 00:34:09.320 When I get to the band gap, below the 00:34:09.320 --> 00:34:10.420 band gap, what happens? 00:34:10.420 --> 00:34:14.380 Well, I drop off and it goes like that. 00:34:14.380 --> 00:34:18.960 In the case of carbon, this wavelength, right, which is, 00:34:18.960 --> 00:34:25.565 by the way, called the absorption edge-- 00:34:28.200 --> 00:34:35.910 that number comes out to be about 229. 00:34:35.910 --> 00:34:47.210 If I try it with silicon, that number-- 00:34:47.210 --> 00:34:53.930 so this would be carbon, this would be silicon and carbon-- 00:34:57.520 --> 00:34:59.660 1125. 00:34:59.660 --> 00:35:00.470 All right. 00:35:00.470 --> 00:35:05.740 So the absorption edge for silicon is 1125, which means 00:35:05.740 --> 00:35:10.930 it absorbs in the visible range and so that's the way we 00:35:10.930 --> 00:35:12.960 get the luster. 00:35:12.960 --> 00:35:14.470 And this would be in the what? 00:35:14.470 --> 00:35:15.720 Infrared. 00:35:18.380 --> 00:35:23.320 And this would be in the UV region, if you wanted to-- 00:35:23.320 --> 00:35:28.780 which would be far hard UV and this would be far infrared. 00:35:33.840 --> 00:35:34.110 OK. 00:35:34.110 --> 00:35:37.840 This is the paper-- 00:35:37.840 --> 00:35:39.440 Heitler and London's paper. 00:35:39.440 --> 00:35:44.720 It's in German and you can see in Zurich. 00:35:44.720 --> 00:35:46.010 And this is their original paper. 00:35:46.010 --> 00:35:49.620 You can go down to the library and you can get this original 00:35:49.620 --> 00:35:52.790 paper and if you know how to read German, it's-- 00:35:52.790 --> 00:35:55.040 we have it. 00:35:55.040 --> 00:35:56.470 This is some of the original paper. 00:35:56.470 --> 00:36:00.180 You notice the Schrodinger equation up there. 00:36:00.180 --> 00:36:01.130 Nasty-- 00:36:01.130 --> 00:36:03.950 really nasty, but this is the calculations. 00:36:03.950 --> 00:36:04.700 This is radius. 00:36:04.700 --> 00:36:09.500 This is energy and you can see a point here where you get an 00:36:09.500 --> 00:36:13.520 energy minimum and that's where the bands operate. 00:36:13.520 --> 00:36:15.870 This is another way to look at it. 00:36:19.510 --> 00:36:22.580 This would be the bottom of the bonding orbital or the top 00:36:22.580 --> 00:36:27.160 of the anti-bonding orbitals. 00:36:27.160 --> 00:36:32.020 This is another way to look at what we've drawn so far. 00:36:32.020 --> 00:36:34.060 Somebody is talking. 00:36:34.060 --> 00:36:36.070 Something you guys ought to know, this room is 00:36:36.070 --> 00:36:38.350 acoustically perfect. 00:36:38.350 --> 00:36:43.470 If you pass gas in the back row, I'll hear it, OK? 00:36:43.470 --> 00:36:47.070 So if you're talking up there, I'll hear it. 00:36:47.070 --> 00:36:47.870 OK. 00:36:47.870 --> 00:36:49.790 So you can see what's going on here. 00:36:49.790 --> 00:36:53.780 We've drawn that and this another way to look at it. 00:36:53.780 --> 00:36:56.380 By the way, there's a mistake in your book that 00:36:56.380 --> 00:36:57.320 doesn't make any sense. 00:36:57.320 --> 00:36:58.570 2s3p-- 00:37:01.070 --> 00:37:04.770 it's 2p in the description for beryllium. 00:37:04.770 --> 00:37:09.350 OK, this is another way to look at it. 00:37:09.350 --> 00:37:10.650 This is in the archival notes. 00:37:10.650 --> 00:37:12.640 It's one of the few pieces of the archival notes which I 00:37:12.640 --> 00:37:14.930 don't understand. 00:37:14.930 --> 00:37:17.200 So don't worry about that. 00:37:17.200 --> 00:37:18.370 OK. 00:37:18.370 --> 00:37:21.280 Here's what happens when you-- 00:37:21.280 --> 00:37:23.940 where we illustrated this where you apply a potential. 00:37:23.940 --> 00:37:27.200 What happens is you depopulate the bonding orbitals, and you 00:37:27.200 --> 00:37:30.520 populate the antibonding orbitals rules, and you get 00:37:30.520 --> 00:37:33.320 conduction. 00:37:33.320 --> 00:37:34.500 This is another-- 00:37:34.500 --> 00:37:35.750 this is the right way to look at it. 00:37:38.320 --> 00:37:40.370 This sort of illustrates the things we've gone through the 00:37:40.370 --> 00:37:41.270 whole time. 00:37:41.270 --> 00:37:45.680 A metal has no band gap. 00:37:45.680 --> 00:37:51.150 That no band gap is achieved either by half-filled set of 00:37:51.150 --> 00:37:55.190 orbitals or overlap between one or 00:37:55.190 --> 00:37:56.920 between different levels. 00:37:56.920 --> 00:38:02.950 An insulator has a large band gap and a semiconductor has a 00:38:02.950 --> 00:38:06.360 sort of intermediate band gap. 00:38:06.360 --> 00:38:08.930 Let's get a little-- 00:38:08.930 --> 00:38:12.010 sort of wet your whistle for your reading for next Tuesday. 00:38:12.010 --> 00:38:15.560 I think we have Monday lecture on Tuesday. 00:38:15.560 --> 00:38:18.570 If you go over to the reactor-- 00:38:18.570 --> 00:38:21.130 they have a reactor here at MIT-- 00:38:21.130 --> 00:38:26.650 once a week, a truck backs up and it's full of silicon logs. 00:38:26.650 --> 00:38:28.780 These things are about 12 inches in diameter and they're 00:38:28.780 --> 00:38:34.140 about this tall and they bring them into the reactor and 00:38:34.140 --> 00:38:36.460 they're irradiated with neutrons. 00:38:36.460 --> 00:38:37.390 Well, what do you think happens? 00:38:37.390 --> 00:38:46.250 Since everybody in here knows the Periodic Table by heart, 00:38:46.250 --> 00:38:48.130 what's the next atom-- what's the next 00:38:48.130 --> 00:38:51.030 element over from silicon? 00:38:51.030 --> 00:38:52.160 Phosphorus. 00:38:52.160 --> 00:38:52.670 OK. 00:38:52.670 --> 00:38:53.860 So guess what? 00:38:53.860 --> 00:38:56.850 You take and you irradiate the silicon. 00:38:56.850 --> 00:38:58.440 You absorb a neutron. 00:38:58.440 --> 00:39:00.690 It adds one to z, doesn't it? 00:39:00.690 --> 00:39:02.890 And it becomes phosphorus. 00:39:02.890 --> 00:39:09.560 Well, phosphorus has one more electron than silicon. 00:39:09.560 --> 00:39:17.630 So I have implanted phosphorus into silicon by transmutation. 00:39:17.630 --> 00:39:19.860 And there's one more electron that's in there. 00:39:19.860 --> 00:39:20.610 Now where's it come from? 00:39:20.610 --> 00:39:21.350 I don't know. 00:39:21.350 --> 00:39:23.170 The electron bank, right? 00:39:23.170 --> 00:39:27.000 But what do you figure the energy of that guy is? 00:39:27.000 --> 00:39:29.550 Well, it's a lot higher than the electrons in silicon. 00:39:32.400 --> 00:39:33.600 And so guess what? 00:39:33.600 --> 00:39:38.000 That electron, we'll find out, doesn't reside down here. 00:39:38.000 --> 00:39:42.880 It resides up here or very close to that. 00:39:42.880 --> 00:39:45.830 So that's what's called-- 00:39:45.830 --> 00:39:47.780 and we'll talk about it next time-- it's called doping, 00:39:47.780 --> 00:39:52.850 only now we're doping it in a special kind of way. 00:39:52.850 --> 00:39:53.510 OK. 00:39:53.510 --> 00:39:54.870 We've got-- let's keep going. 00:40:04.160 --> 00:40:07.800 I want to get this one up there. 00:40:07.800 --> 00:40:09.050 There we go. 00:40:12.740 --> 00:40:13.990 OK. 00:40:17.050 --> 00:40:19.900 Now-- so far we've talked about photons. 00:40:19.900 --> 00:40:22.050 They're a one shot deal. 00:40:22.050 --> 00:40:33.430 What about thermal excitation? 00:40:33.430 --> 00:40:36.700 Well, we can make our-- the same drawing we have in the 00:40:36.700 --> 00:40:41.850 past. We got our material here where we have a valence band, 00:40:41.850 --> 00:40:46.140 conduction band, and now we put-- 00:40:49.410 --> 00:40:54.780 we add thermal energy. 00:40:54.780 --> 00:40:58.340 By the way, what's the one electron volt in temperature? 00:40:58.340 --> 00:41:00.040 Just to give you a feel-- 00:41:00.040 --> 00:41:04.060 one electron volt is 11,600 degrees Kelvin if you convert 00:41:04.060 --> 00:41:05.420 that to temperature. 00:41:05.420 --> 00:41:08.650 So one electron volt seems like a small number, but in 00:41:08.650 --> 00:41:10.830 temperature, is pretty warm. 00:41:10.830 --> 00:41:11.460 OK. 00:41:11.460 --> 00:41:19.480 So now the thermal energy is what? 00:41:19.480 --> 00:41:20.730 It's constant. 00:41:24.040 --> 00:41:27.950 Unlike the photons, which are one off, right? 00:41:27.950 --> 00:41:28.980 So it's constant. 00:41:28.980 --> 00:41:30.530 So what happens? 00:41:30.530 --> 00:41:39.762 Now I take an electron from the valence band and I promote 00:41:39.762 --> 00:41:44.010 it up here. 00:41:44.010 --> 00:41:46.020 So it looks sort of like-- 00:41:46.020 --> 00:41:48.360 so far up there with photons. 00:41:48.360 --> 00:41:51.080 But there's one critical difference. 00:41:51.080 --> 00:41:53.730 It stays up there. 00:41:53.730 --> 00:41:57.290 It stays up there because the energy is constant. 00:41:57.290 --> 00:41:59.290 So what does that mean? 00:41:59.290 --> 00:42:03.225 Well, it leaves behind a broken bond. 00:42:10.080 --> 00:42:16.240 And that broken bond has a lot of energy, and in fact, it 00:42:16.240 --> 00:42:17.900 doesn't like to stick around. 00:42:17.900 --> 00:42:19.500 So given a chance, it'll move. 00:42:19.500 --> 00:42:22.850 It's like a hot potato and so that broken 00:42:22.850 --> 00:42:27.040 bond is called a hole. 00:42:31.050 --> 00:42:32.170 We're not too imaginative. 00:42:32.170 --> 00:42:34.610 It's a hole in the lattice. 00:42:34.610 --> 00:42:37.200 But it has high energy. 00:42:37.200 --> 00:42:38.770 So what's the deal with this hole? 00:42:38.770 --> 00:42:42.050 Well, this is-- 00:42:42.050 --> 00:42:51.580 a hole is a 0 in a land of minus 1. 00:42:51.580 --> 00:42:52.380 So what does it mean? 00:42:52.380 --> 00:42:55.915 It's actually net positive. 00:42:58.860 --> 00:43:03.570 So now in the case of thermal excitation, I end up with an 00:43:03.570 --> 00:43:09.070 electron in the conduction band and a hole 00:43:09.070 --> 00:43:10.200 in the valence band. 00:43:10.200 --> 00:43:16.160 So I end up with an electron in the conduction band and a 00:43:16.160 --> 00:43:21.800 hole and the electrical engineers call this p. 00:43:21.800 --> 00:43:25.550 They're not very imaginative either-- positive in the 00:43:25.550 --> 00:43:26.590 valence band. 00:43:26.590 --> 00:43:30.720 So we get 241. 00:43:30.720 --> 00:43:34.940 We get two charge carriers for every event. 00:43:34.940 --> 00:43:36.420 That, we get one. 00:43:36.420 --> 00:43:40.035 This, we get two charge carriers for every event. 00:43:45.260 --> 00:43:49.770 So now we have what? 00:43:49.770 --> 00:43:52.770 Let's ask ourselves, what's the-- 00:43:55.420 --> 00:43:57.340 can we do a little math on this? 00:43:57.340 --> 00:44:03.940 And the broken bond, by the way, is a hole and so the 00:44:03.940 --> 00:44:13.660 number of electrons is also equal to the number of holes 00:44:13.660 --> 00:44:17.210 because it's a one for one, and in the electrical 00:44:17.210 --> 00:44:20.060 engineering world, they say n equals p. 00:44:23.930 --> 00:44:27.000 And so we can now go back and say something about this 00:44:27.000 --> 00:44:27.830 conductivity. 00:44:27.830 --> 00:44:36.570 We can do some calculations here and we can calculate the 00:44:36.570 --> 00:44:42.730 conductivity due to the electrical connectivity, and 00:44:42.730 --> 00:44:47.920 it's really the sum over i of n of i, which would be the 00:44:47.920 --> 00:45:00.430 number of the population of carriers times-- 00:45:00.430 --> 00:45:04.510 that's the value of the charge on the carrier-- 00:45:04.510 --> 00:45:07.110 times something called the mobility. 00:45:14.210 --> 00:45:16.570 OK, so what's the mobility? 00:45:16.570 --> 00:45:22.510 Well, you could imagine that these electrons have to move 00:45:22.510 --> 00:45:25.500 back and forth in the lattice and in a metal versus a 00:45:25.500 --> 00:45:27.500 covalence solid or something like that, it might be a 00:45:27.500 --> 00:45:29.350 little bit different. 00:45:29.350 --> 00:45:31.340 The resistance might be different and so-- 00:45:31.340 --> 00:45:32.890 in fact, it is. 00:45:32.890 --> 00:45:40.660 mu i is equal to the velocity of the charge carrier divided 00:45:40.660 --> 00:45:43.440 by the electric field. 00:45:46.870 --> 00:45:47.840 OK. 00:45:47.840 --> 00:45:49.090 So we're almost there. 00:45:54.540 --> 00:45:59.880 Now we need to-- and we can simplify this because of the n 00:45:59.880 --> 00:46:04.590 equals p business, and we can say that the electrical 00:46:04.590 --> 00:46:06.790 conductivity is equal to what? 00:46:06.790 --> 00:46:12.760 It's equal to n sub e times e, which is the electronic charge 00:46:12.760 --> 00:46:20.760 times mu sub e plus n sub h, which would be the holes, the 00:46:20.760 --> 00:46:24.690 electronic charge, times mu sub h. 00:46:24.690 --> 00:46:30.610 But since n is equal to p, we end up with n sub e times the 00:46:30.610 --> 00:46:38.870 electronic charge times mu sub e plus u sub h. 00:46:38.870 --> 00:46:40.490 OK. 00:46:40.490 --> 00:46:43.600 And so that-- what we really need to get is this. 00:46:43.600 --> 00:46:47.450 Well, we don't have time to go through that, but from a 00:46:47.450 --> 00:46:50.930 quantum mechanical calculation, we can get that. 00:46:50.930 --> 00:46:52.740 n sub e is equal to what? 00:46:52.740 --> 00:46:58.700 It's equal to some constant times T to 3/2 times the 00:46:58.700 --> 00:47:10.070 exponent of minus E sub g over 2k sub B times T. 00:47:10.070 --> 00:47:11.650 So what is all this stuff? 00:47:11.650 --> 00:47:12.790 Well, this is a constant. 00:47:12.790 --> 00:47:15.370 This is the temperature. 00:47:15.370 --> 00:47:16.370 This is the band gap. 00:47:16.370 --> 00:47:20.340 So it's a representation of the binding. 00:47:26.020 --> 00:47:27.090 What's down here? 00:47:27.090 --> 00:47:29.840 Well, k sub B is Boltzmann's constant. 00:47:29.840 --> 00:47:36.020 This is absolute temperature and so this represents the 00:47:36.020 --> 00:47:40.895 disruptive force. 00:47:44.800 --> 00:47:49.860 So it's a balance between how tightly they're bound and how 00:47:49.860 --> 00:47:52.810 much energy I've got to do this. 00:47:52.810 --> 00:47:54.115 Well, let's-- 00:47:54.115 --> 00:47:56.000 we've got one minute. 00:47:56.000 --> 00:48:00.090 Let's do a quick calculations for silicon, just to 00:48:00.090 --> 00:48:01.960 close the loop here. 00:48:01.960 --> 00:48:08.540 For silicon at room temperature, that number comes 00:48:08.540 --> 00:48:15.560 out 1.3 times 10 to the 10th per centimeter cubed. 00:48:15.560 --> 00:48:17.310 Say well, that's billions. 00:48:17.310 --> 00:48:19.260 That's a big deal. 00:48:19.260 --> 00:48:23.980 What about copper at room temperature? 00:48:23.980 --> 00:48:31.800 Well, that number comes out about 8.5 times 10 to the 22. 00:48:31.800 --> 00:48:33.950 Now we're talking big numbers. 00:48:33.950 --> 00:48:38.190 So there's a 10 to the 12th difference between these two 00:48:38.190 --> 00:48:42.630 Well, let's take a quick look before we finish up. 00:48:42.630 --> 00:48:42.790 OK. 00:48:42.790 --> 00:48:45.580 This is the band gap as a function of position in the 00:48:45.580 --> 00:48:46.120 Periodic Table. 00:48:46.120 --> 00:48:49.690 Now you guys ought to be able to rationalize this. 00:48:49.690 --> 00:48:52.460 Lead is a metal, bigger atoms you can see. 00:48:52.460 --> 00:48:54.690 10 actually comes in two flavors. 00:48:54.690 --> 00:48:56.320 Gray and white and one of them-- 00:48:56.320 --> 00:48:57.550 I think it's white-- 00:48:57.550 --> 00:49:01.250 actually forms covalent bonds and they use 00:49:01.250 --> 00:49:03.880 tin for night vision. 00:49:03.880 --> 00:49:07.860 Think about where that band gap is. 00:49:07.860 --> 00:49:09.980 But here's the punch line. 00:49:09.980 --> 00:49:12.390 We're trying to rationalize this large difference in 00:49:12.390 --> 00:49:13.420 conductivity. 00:49:13.420 --> 00:49:16.630 Well, there's copper up there at 10 to the 7th, there 00:49:16.630 --> 00:49:20.820 silicon at 10 to the minus 4 and ten to the minus 4, 10 to 00:49:20.820 --> 00:49:23.060 the minus-- about 10 to the 12th or thereabouts-- 00:49:23.060 --> 00:49:26.490 and so, lo and behold, this-- 00:49:26.490 --> 00:49:30.660 we were able to rationalize with these fairly simple 00:49:30.660 --> 00:49:33.880 models, the range of conductivities that go all the 00:49:33.880 --> 00:49:36.060 way from metals to 00:49:36.060 --> 00:49:38.150 semiconductors and even diamond. 00:49:38.150 --> 00:49:38.730 Look at diamond-- 00:49:38.730 --> 00:49:40.630 10 to the minus 11. 00:49:40.630 --> 00:49:42.660 You can rationalize that the band gap-- 00:49:42.660 --> 00:49:43.620 where's the band gap? 00:49:43.620 --> 00:49:46.050 5.4 electron volts. 00:49:46.050 --> 00:49:48.550 OK, we're out of time.