1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high quality educational resources for free. 5 00:00:10,660 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,320 at ocw.mit.edu. 8 00:00:25,033 --> 00:00:26,450 JEFFREY GROSSMAN: What we're going 9 00:00:26,450 --> 00:00:29,060 to do today is now, as I mentioned, shift gears. 10 00:00:29,060 --> 00:00:32,540 And we're going to go from atoms and molecules to solids. 11 00:00:32,540 --> 00:00:38,150 So the outline for today is, well, OK, so here we are. 12 00:00:38,150 --> 00:00:40,190 But remember, last time we didn't really 13 00:00:40,190 --> 00:00:41,380 talk about hydrogen storage. 14 00:00:41,380 --> 00:00:45,020 And some people-- well, like two people-- 15 00:00:45,020 --> 00:00:47,450 wanted to hear about hydrogen storage. 16 00:00:47,450 --> 00:00:51,350 So I thought what I'll do is I will just briefly discuss 17 00:00:51,350 --> 00:00:52,130 this problem-- 18 00:00:52,130 --> 00:00:55,040 this energy problem, which is a very interesting problem 19 00:00:55,040 --> 00:00:58,130 and really, as I'll show at the very end, 20 00:00:58,130 --> 00:01:01,070 is another one of these huge phase space 21 00:01:01,070 --> 00:01:03,350 problems where quantum mechanical modeling is 22 00:01:03,350 --> 00:01:04,440 going to be important. 23 00:01:04,440 --> 00:01:05,090 OK. 24 00:01:05,090 --> 00:01:08,270 And then we'll move on to understanding what happens 25 00:01:08,270 --> 00:01:09,560 when you have a crystal. 26 00:01:09,560 --> 00:01:13,790 When you go from these atoms and molecules to an ordered array 27 00:01:13,790 --> 00:01:16,970 of atoms and molecules, which is a periodic crystal, 28 00:01:16,970 --> 00:01:17,517 what changes? 29 00:01:17,517 --> 00:01:19,100 And there's something really important 30 00:01:19,100 --> 00:01:22,190 that changes in this wave function that we've 31 00:01:22,190 --> 00:01:23,403 been caring so much about. 32 00:01:23,403 --> 00:01:25,820 And that's due to something called Bloch's function, which 33 00:01:25,820 --> 00:01:26,660 we'll talk about. 34 00:01:26,660 --> 00:01:33,400 And it results in the orbitals becoming curvy. 35 00:01:33,400 --> 00:01:33,900 OK. 36 00:01:33,900 --> 00:01:35,390 So that's kind of the punch line. 37 00:01:35,390 --> 00:01:37,015 That's what we're going to build up to. 38 00:01:37,015 --> 00:01:39,440 And then next week we'll pick up from that 39 00:01:39,440 --> 00:01:41,220 and talk about what that means. 40 00:01:41,220 --> 00:01:41,840 OK. 41 00:01:41,840 --> 00:01:43,860 Any questions at all? 42 00:01:43,860 --> 00:01:46,180 Thoughts? 43 00:01:46,180 --> 00:01:48,170 Hopes? 44 00:01:48,170 --> 00:01:48,900 Fears? 45 00:01:48,900 --> 00:01:50,040 Aspirations? 46 00:01:50,040 --> 00:01:51,470 OK. 47 00:01:51,470 --> 00:01:52,130 Right. 48 00:01:52,130 --> 00:01:56,930 That's my hydrogen storage introduction slide. 49 00:01:56,930 --> 00:01:59,360 And what are the two problems for hydrogen storage? 50 00:01:59,360 --> 00:02:00,350 Two challenges? 51 00:02:00,350 --> 00:02:02,510 Who can tell me? 52 00:02:02,510 --> 00:02:03,777 Yeah? 53 00:02:03,777 --> 00:02:04,860 AUDIENCE: Making hydrogen. 54 00:02:04,860 --> 00:02:05,270 JEFFREY GROSSMAN: Right. 55 00:02:05,270 --> 00:02:06,350 AUDIENCE: And then storing it. 56 00:02:06,350 --> 00:02:08,174 JEFFREY GROSSMAN: Why do we need to make hydrogen? 57 00:02:08,174 --> 00:02:09,882 AUDIENCE: The Earth's gravitational field 58 00:02:09,882 --> 00:02:12,082 is [INAUDIBLE] so it all evaporates. 59 00:02:12,082 --> 00:02:13,040 JEFFREY GROSSMAN: Yeah. 60 00:02:13,040 --> 00:02:13,880 Well, not all of it. 61 00:02:13,880 --> 00:02:15,480 There's still some of it. 62 00:02:15,480 --> 00:02:18,170 There's some of it with which we make from. 63 00:02:18,170 --> 00:02:20,880 But where is it? 64 00:02:20,880 --> 00:02:22,320 It's tied up. 65 00:02:22,320 --> 00:02:23,920 It's busy. 66 00:02:23,920 --> 00:02:24,490 You know? 67 00:02:24,490 --> 00:02:27,340 And you got to say, hey, I need you. 68 00:02:27,340 --> 00:02:29,410 Get off of that methane molecule or get off 69 00:02:29,410 --> 00:02:32,830 of this other molecule and come over here and be 70 00:02:32,830 --> 00:02:34,510 an energy carrier for me. 71 00:02:34,510 --> 00:02:40,360 That takes energy to make the hydrogen be available to use 72 00:02:40,360 --> 00:02:43,300 as an energy carrier. 73 00:02:43,300 --> 00:02:45,854 And then the other problem is what? 74 00:02:45,854 --> 00:02:46,730 AUDIENCE: Storing it. 75 00:02:46,730 --> 00:02:47,938 JEFFREY GROSSMAN: Storing it. 76 00:02:47,938 --> 00:02:48,582 Right. 77 00:02:48,582 --> 00:02:49,540 Because it's very hard. 78 00:02:49,540 --> 00:02:52,970 So I'm going to talk about the second problem 79 00:02:52,970 --> 00:02:57,920 because, again, there's sort of a molecular solution that's 80 00:02:57,920 --> 00:03:00,080 really exciting where computation can play a role. 81 00:03:00,080 --> 00:03:00,580 OK. 82 00:03:00,580 --> 00:03:02,662 So I'm just going to spend the first 15, 83 00:03:02,662 --> 00:03:04,120 20 minutes here talking about this. 84 00:03:04,120 --> 00:03:07,505 I think this is-- 85 00:03:07,505 --> 00:03:11,070 there are high opinions about hydrogen-- gesundheit-- 86 00:03:11,070 --> 00:03:12,150 storage. 87 00:03:12,150 --> 00:03:14,280 And the hydrogen economy-- 88 00:03:14,280 --> 00:03:16,650 the hydrogen economy is this sort of vision 89 00:03:16,650 --> 00:03:20,100 that President Bush put forth. 90 00:03:20,100 --> 00:03:23,310 And he really kickstarted a large research program 91 00:03:23,310 --> 00:03:27,810 in this vision where you would be able to-- 92 00:03:27,810 --> 00:03:30,510 a simple chemical reaction between hydrogen and oxygen 93 00:03:30,510 --> 00:03:31,800 generates energy. 94 00:03:31,800 --> 00:03:32,850 What's that called? 95 00:03:36,280 --> 00:03:38,510 There's a device that does that. 96 00:03:38,510 --> 00:03:39,603 That's a fuel cell. 97 00:03:39,603 --> 00:03:41,270 We're not going to talk about that part. 98 00:03:41,270 --> 00:03:43,485 We're not talking about what you do with the hydrogen 99 00:03:43,485 --> 00:03:45,110 because there's all kinds of challenges 100 00:03:45,110 --> 00:03:46,650 there too, by the way. 101 00:03:46,650 --> 00:03:47,570 OK. 102 00:03:47,570 --> 00:03:49,340 So there's really great research going 103 00:03:49,340 --> 00:03:52,820 on in fuel cells, which are what you put the hydrogen into 104 00:03:52,820 --> 00:03:54,255 to make electricity. 105 00:03:57,950 --> 00:03:59,660 There is a new national commitment. 106 00:03:59,660 --> 00:04:01,310 And there's going to be-- 107 00:04:01,310 --> 00:04:03,170 first car driven by a child born today 108 00:04:03,170 --> 00:04:06,110 could be powered by hydrogen and pollution free. 109 00:04:06,110 --> 00:04:06,950 Yes. 110 00:04:06,950 --> 00:04:09,730 For $10 million per car. 111 00:04:09,730 --> 00:04:11,590 OK. 112 00:04:11,590 --> 00:04:14,320 I mean, I think a lot of people were excited about this. 113 00:04:14,320 --> 00:04:18,100 But this is a very large investment 114 00:04:18,100 --> 00:04:22,420 for government research, which is very little money. 115 00:04:22,420 --> 00:04:25,480 And in a technology that clearly has 116 00:04:25,480 --> 00:04:28,330 at least 50 years of research that's 117 00:04:28,330 --> 00:04:31,840 needed before you're going to see a trial. 118 00:04:31,840 --> 00:04:36,520 So maybe this should have said a child's future grandchild 119 00:04:36,520 --> 00:04:37,720 born today. 120 00:04:37,720 --> 00:04:39,140 And that would be cool. 121 00:04:39,140 --> 00:04:40,390 OK. 122 00:04:40,390 --> 00:04:43,930 And yet, there are some really great programs going on 123 00:04:43,930 --> 00:04:47,445 to test the hydrogen economy. 124 00:04:47,445 --> 00:04:48,820 And those are the kinds of things 125 00:04:48,820 --> 00:04:52,113 we need to do now to lay the groundwork for all 126 00:04:52,113 --> 00:04:53,780 the challenges that have to be overcome. 127 00:04:53,780 --> 00:04:56,287 Again, I'm just going to talk about one 128 00:04:56,287 --> 00:04:57,370 of them, which is storage. 129 00:04:57,370 --> 00:05:00,100 But so I don't-- 130 00:05:00,100 --> 00:05:03,190 now, then Obama came in and he slashed the budget. 131 00:05:03,190 --> 00:05:05,980 Actually, he tried to kill it-- 132 00:05:05,980 --> 00:05:09,460 the hydrogen research in this country to 0, which I thought 133 00:05:09,460 --> 00:05:13,720 was interesting because in that administration 134 00:05:13,720 --> 00:05:17,492 the sense was well, it's a great technology. 135 00:05:17,492 --> 00:05:18,700 But it's a very long way off. 136 00:05:18,700 --> 00:05:20,290 Let's try to work on-- 137 00:05:20,290 --> 00:05:23,770 reshuffle our investment a little into other things. 138 00:05:23,770 --> 00:05:26,940 Things that might give a shorter return 139 00:05:26,940 --> 00:05:31,570 in terms of technologies that are needed in clean energy 140 00:05:31,570 --> 00:05:33,240 more rapidly. 141 00:05:33,240 --> 00:05:34,660 OK. 142 00:05:34,660 --> 00:05:35,680 I've already shown this. 143 00:05:35,680 --> 00:05:39,610 I love this history of hydrogen. When you have a moment, just 144 00:05:39,610 --> 00:05:41,710 read some Jules Verne where he said-- 145 00:05:41,710 --> 00:05:43,450 I think I already read this quote. 146 00:05:43,450 --> 00:05:46,390 But he said, "I believe that one day hydrogen and oxygen, which 147 00:05:46,390 --> 00:05:49,060 together form water will be used either alone 148 00:05:49,060 --> 00:05:52,300 or together as an inexhaustible source of heat and light." 149 00:05:52,300 --> 00:05:55,840 I think that's really cool that was in the 1800s. 150 00:05:55,840 --> 00:05:58,420 And then there are some things that 151 00:05:58,420 --> 00:06:05,560 happened along the way like that blimp thing that I showed you. 152 00:06:05,560 --> 00:06:09,310 But see, now that has caused a lot of misunderstanding 153 00:06:09,310 --> 00:06:14,680 about hydrogen in that it's, yes, it can combust. 154 00:06:14,680 --> 00:06:16,730 But how dangerous is it? 155 00:06:16,730 --> 00:06:20,470 So I'll show you a slide on that. 156 00:06:20,470 --> 00:06:20,970 Right. 157 00:06:20,970 --> 00:06:25,920 So the problem in the storage is that the low volumetric density 158 00:06:25,920 --> 00:06:28,110 of anything that's a gas requires 159 00:06:28,110 --> 00:06:29,610 something that can densify it. 160 00:06:29,610 --> 00:06:32,130 So you got to compress it basically. 161 00:06:32,130 --> 00:06:34,770 Here's your metric gasoline. 162 00:06:34,770 --> 00:06:39,240 Remember, gasoline is this beautiful high energy density 163 00:06:39,240 --> 00:06:41,460 way of storing energy. 164 00:06:41,460 --> 00:06:45,510 32 megajoules per liter compared with liquid hydrogen-- 165 00:06:45,510 --> 00:06:47,250 if you can liquefy it, it's 8. 166 00:06:47,250 --> 00:06:49,350 And gas under very high pressure-- 167 00:06:49,350 --> 00:06:53,070 5,000 PSI is only 3. 168 00:06:53,070 --> 00:06:56,670 So even if I squeeze this hydrogen gas a ton, 169 00:06:56,670 --> 00:07:00,450 it's still only a tenth of the energy density of gasoline. 170 00:07:00,450 --> 00:07:00,950 OK. 171 00:07:00,950 --> 00:07:05,180 So that's why storing enough hydrogen on board is hard. 172 00:07:05,180 --> 00:07:09,950 And it adds a lot of weight and volume. 173 00:07:09,950 --> 00:07:13,220 And this would be how much-- this is a little less. 174 00:07:13,220 --> 00:07:15,290 This is about 3,000 PSI-- 175 00:07:15,290 --> 00:07:18,900 200 bar-- so that would be a gas tank of hydrogen, 176 00:07:18,900 --> 00:07:22,910 which is currently how some car manufacturers are making 177 00:07:22,910 --> 00:07:24,320 hydrogen powered cars-- 178 00:07:24,320 --> 00:07:28,820 again, $10, $20 million per vehicle-- 179 00:07:28,820 --> 00:07:30,600 that store hydrogen this way. 180 00:07:30,600 --> 00:07:32,060 OK. 181 00:07:32,060 --> 00:07:34,538 If you could liquefy it, you can get down to smaller tanks. 182 00:07:34,538 --> 00:07:36,830 What I want to tell you about is something that I think 183 00:07:36,830 --> 00:07:41,340 is really exciting here, which is a chemical storage approach. 184 00:07:41,340 --> 00:07:44,250 OK. 185 00:07:44,250 --> 00:07:47,470 Now, right. 186 00:07:47,470 --> 00:07:48,292 OK. 187 00:07:48,292 --> 00:07:50,500 Of course, you can imagine that when you store things 188 00:07:50,500 --> 00:07:54,970 at this high pressure or at low temperatures and high pressures 189 00:07:54,970 --> 00:07:58,900 you have very challenging engineering and packaging 190 00:07:58,900 --> 00:07:59,710 issues. 191 00:07:59,710 --> 00:08:02,350 And that's one of the biggest concerns. 192 00:08:02,350 --> 00:08:05,315 And because there's also a lot of misunderstanding 193 00:08:05,315 --> 00:08:06,440 of the safety of hydrogen-- 194 00:08:06,440 --> 00:08:09,590 I'm not saying it's completely safe. 195 00:08:09,590 --> 00:08:12,820 But people actually are looking at the safety 196 00:08:12,820 --> 00:08:15,540 of having something under that much pressure. 197 00:08:15,540 --> 00:08:17,010 OK. 198 00:08:17,010 --> 00:08:19,470 And so actually, when you have a hydrogen powered car, 199 00:08:19,470 --> 00:08:24,280 you tend to do a drop test, which isn't really true. 200 00:08:24,280 --> 00:08:26,500 They just are doing drop tests on the containers 201 00:08:26,500 --> 00:08:28,250 that they're developing. 202 00:08:28,250 --> 00:08:28,750 And 203 00:08:28,750 --> 00:08:31,000 They drop it so that it's going to impact 204 00:08:31,000 --> 00:08:34,030 the ground at 52 miles an hour. 205 00:08:34,030 --> 00:08:39,039 And I guarantee you that one exploding drop test, 206 00:08:39,039 --> 00:08:45,520 and the market is going to have some serious issues whatever 207 00:08:45,520 --> 00:08:47,620 the cause of it is. 208 00:08:47,620 --> 00:08:50,800 But this is a real concern is the safety-- 209 00:08:50,800 --> 00:08:52,480 forget about whether it's hydrogen. 210 00:08:52,480 --> 00:08:55,270 But if you store anything under that high pressure, 211 00:08:55,270 --> 00:08:57,880 there are serious safety concerns and real packaging 212 00:08:57,880 --> 00:08:58,620 challenges. 213 00:08:58,620 --> 00:09:01,280 OK. 214 00:09:01,280 --> 00:09:03,370 Remember, it's not just how the package operates 215 00:09:03,370 --> 00:09:04,960 when you buy it, it's how it operates 216 00:09:04,960 --> 00:09:08,940 after 10 years of corrosion and operation. 217 00:09:08,940 --> 00:09:09,480 OK. 218 00:09:09,480 --> 00:09:13,390 So I think chemical storage is a really interesting way to go. 219 00:09:13,390 --> 00:09:17,380 It does not involve high pressures. 220 00:09:17,380 --> 00:09:17,880 OK. 221 00:09:17,880 --> 00:09:21,780 Basically, in chemical storage, what you have is 222 00:09:21,780 --> 00:09:23,460 you have essentially a sponge. 223 00:09:23,460 --> 00:09:27,870 You have a material that soaks hydrogen up. 224 00:09:27,870 --> 00:09:29,730 So it literally just absorbs hydrogen 225 00:09:29,730 --> 00:09:33,130 and it changes into a hydride. 226 00:09:33,130 --> 00:09:35,710 And then you pull the hydrogen back out 227 00:09:35,710 --> 00:09:37,280 and it's just a sponge. 228 00:09:37,280 --> 00:09:39,520 OK. 229 00:09:39,520 --> 00:09:41,840 Now the problem is that-- 230 00:09:41,840 --> 00:09:43,815 and there are lots of references. 231 00:09:43,815 --> 00:09:45,940 And there's lots of great places to read about some 232 00:09:45,940 --> 00:09:46,523 of this stuff. 233 00:09:46,523 --> 00:09:49,070 And I've put them here-- put some of them here. 234 00:09:49,070 --> 00:09:51,430 This is a really nice review. 235 00:09:51,430 --> 00:09:54,470 And the problem is if you look at the periodic table, 236 00:09:54,470 --> 00:09:57,605 you see a lot of times we're excited about materials design. 237 00:09:57,605 --> 00:09:59,230 And we look at the whole periodic table 238 00:09:59,230 --> 00:10:01,190 and say, what should we do today? 239 00:10:01,190 --> 00:10:01,840 Right? 240 00:10:01,840 --> 00:10:03,100 Don't you all feel that way? 241 00:10:03,100 --> 00:10:04,005 That's how I feel. 242 00:10:04,005 --> 00:10:06,130 Every morning-- like what am I going to make today? 243 00:10:06,130 --> 00:10:07,920 Polonium! 244 00:10:07,920 --> 00:10:10,400 No. 245 00:10:10,400 --> 00:10:12,230 Ciborium. 246 00:10:12,230 --> 00:10:14,360 Where's ciborium? 247 00:10:14,360 --> 00:10:15,860 There it is. 248 00:10:15,860 --> 00:10:16,850 OK. 249 00:10:16,850 --> 00:10:17,990 That's a neat material. 250 00:10:17,990 --> 00:10:20,660 Ciborium carbide is higher than diamond. 251 00:10:20,660 --> 00:10:22,530 It's harder than diamond. 252 00:10:22,530 --> 00:10:23,030 Right? 253 00:10:23,030 --> 00:10:24,980 Now it's a problem that see ciborium only 254 00:10:24,980 --> 00:10:28,220 exists for 10 to the minus-- 255 00:10:28,220 --> 00:10:30,650 I don't know-- 5 seconds or something. 256 00:10:30,650 --> 00:10:32,600 But still, for that short of a time 257 00:10:32,600 --> 00:10:34,820 you can make really hard materials with it. 258 00:10:34,820 --> 00:10:36,200 But see-- now-- OK-- 259 00:10:36,200 --> 00:10:37,670 it's a little tangent. 260 00:10:37,670 --> 00:10:42,480 For hydrogen storage, you got a big problem, which is density. 261 00:10:42,480 --> 00:10:42,980 Right? 262 00:10:42,980 --> 00:10:44,870 So if you're going to use a sponge 263 00:10:44,870 --> 00:10:48,890 to soak hydrogen up and then release it when you need it 264 00:10:48,890 --> 00:10:51,440 and then soak it back up, the sponge can't 265 00:10:51,440 --> 00:10:56,070 weigh 1,000 times more than the hydrogen. That's a problem. 266 00:10:56,070 --> 00:10:56,570 Right? 267 00:10:56,570 --> 00:10:59,390 Because then you're just going to add thousands of pounds 268 00:10:59,390 --> 00:11:01,310 of weight to your car. 269 00:11:01,310 --> 00:11:03,720 So that really limits what you can use. 270 00:11:03,720 --> 00:11:06,350 And so only the light elements are really 271 00:11:06,350 --> 00:11:07,520 going to be useful here. 272 00:11:07,520 --> 00:11:08,900 And then you have a short list. 273 00:11:08,900 --> 00:11:10,370 And then you say, I don't want them 274 00:11:10,370 --> 00:11:12,180 to be really that poisonous. 275 00:11:12,180 --> 00:11:14,700 It might not be a good thing in the back of your car. 276 00:11:14,700 --> 00:11:16,040 Right? 277 00:11:16,040 --> 00:11:18,620 And so then the list-- you can essentially 278 00:11:18,620 --> 00:11:21,800 argue that the list is about eight elements that you 279 00:11:21,800 --> 00:11:22,850 might store. 280 00:11:22,850 --> 00:11:25,100 And that's just not a lot of room to play. 281 00:11:25,100 --> 00:11:27,510 That's why this is challenging. 282 00:11:27,510 --> 00:11:29,610 As I'm going to show you, there actually 283 00:11:29,610 --> 00:11:31,740 is more room than you think. 284 00:11:31,740 --> 00:11:34,390 More and more room as research goes on. 285 00:11:34,390 --> 00:11:35,610 OK. 286 00:11:35,610 --> 00:11:38,250 And so there are a lot of materials that do fall 287 00:11:38,250 --> 00:11:40,510 into-- when you combine these eight elements together, 288 00:11:40,510 --> 00:11:42,780 you can make many different materials. 289 00:11:42,780 --> 00:11:47,460 And they do fall into a range of categories-- 290 00:11:47,460 --> 00:11:49,800 kinds of structures, kinds of elements, 291 00:11:49,800 --> 00:11:52,085 and so forth that can pick up hydrogen. Yeah? 292 00:11:52,085 --> 00:11:53,460 AUDIENCE: I'm absolutely curious. 293 00:11:53,460 --> 00:11:56,730 What is metallic hydrogen or how is it made? 294 00:11:56,730 --> 00:11:58,870 JEFFREY GROSSMAN: Well, that's-- 295 00:11:58,870 --> 00:11:59,370 Yeah. 296 00:11:59,370 --> 00:12:01,930 Let me get to the metal hydrides. 297 00:12:01,930 --> 00:12:02,430 OK. 298 00:12:06,410 --> 00:12:11,210 So these are some examples of the densities you can get to. 299 00:12:11,210 --> 00:12:12,880 OK. 300 00:12:12,880 --> 00:12:15,140 Look at the density of methane. 301 00:12:15,140 --> 00:12:17,860 That's a great hydrogen storage material. 302 00:12:17,860 --> 00:12:19,990 Why is it totally useless? 303 00:12:19,990 --> 00:12:21,752 Somebody tell me. 304 00:12:21,752 --> 00:12:23,362 AUDIENCE: [INAUDIBLE] carbon. 305 00:12:23,362 --> 00:12:25,070 JEFFREY GROSSMAN: Well, carbon isn't bad. 306 00:12:27,700 --> 00:12:29,420 What? 307 00:12:29,420 --> 00:12:30,350 Not so toxic. 308 00:12:30,350 --> 00:12:31,020 Not a problem. 309 00:12:31,020 --> 00:12:32,100 We can handle that. 310 00:12:32,100 --> 00:12:33,020 AUDIENCE: Explosive. 311 00:12:33,020 --> 00:12:33,895 JEFFREY GROSSMAN: OK. 312 00:12:33,895 --> 00:12:34,490 Explosive. 313 00:12:34,490 --> 00:12:36,590 That could be a little bit of an issue. 314 00:12:36,590 --> 00:12:38,660 We actually like that it's explosive-- the way 315 00:12:38,660 --> 00:12:40,190 we use it today. 316 00:12:40,190 --> 00:12:42,365 AUDIENCE: [INAUDIBLE] 317 00:12:42,365 --> 00:12:43,990 JEFFREY GROSSMAN: It's hard to densify. 318 00:12:43,990 --> 00:12:45,930 You sort of have the same problem. 319 00:12:45,930 --> 00:12:46,950 All right. 320 00:12:46,950 --> 00:12:49,050 What if I could make liquid methane 321 00:12:49,050 --> 00:12:50,790 at really high densities? 322 00:12:50,790 --> 00:12:54,060 Why would that still be a really challenging hydrogen storage 323 00:12:54,060 --> 00:12:55,170 material? 324 00:12:55,170 --> 00:12:57,880 It's such a great weight percent density. 325 00:13:02,630 --> 00:13:03,890 Exactly. 326 00:13:03,890 --> 00:13:08,120 You see, a sponge that's going to-- a chemical sponge that's 327 00:13:08,120 --> 00:13:10,340 going to absorb hydrogen and give it back out when 328 00:13:10,340 --> 00:13:16,130 I want it, it's got to be just the right amount of binding. 329 00:13:16,130 --> 00:13:21,510 It's got to want the hydrogen just enough to hold onto it 330 00:13:21,510 --> 00:13:24,080 but not so much that it won't let it go. 331 00:13:24,080 --> 00:13:24,800 Right? 332 00:13:24,800 --> 00:13:27,110 And it turns out that's really hard. 333 00:13:27,110 --> 00:13:31,280 That window of binding is really hard to tune. 334 00:13:31,280 --> 00:13:33,760 And here it's way too hard, way too high. 335 00:13:33,760 --> 00:13:36,320 So methane-- the carbon is holding on to those hydrogens. 336 00:13:36,320 --> 00:13:40,440 And you need to go to thousands of degrees to pull them off. 337 00:13:40,440 --> 00:13:44,570 You're not going to put a 1,000 degree Celsius 338 00:13:44,570 --> 00:13:46,070 furnace in your car-- 339 00:13:46,070 --> 00:13:48,800 heating system-- to pull off the hydrogen. First of all, 340 00:13:48,800 --> 00:13:53,200 you just explode your gas probably. 341 00:13:53,200 --> 00:13:55,030 But also because it takes so much energy. 342 00:13:55,030 --> 00:13:55,280 Right? 343 00:13:55,280 --> 00:13:56,230 So it's the kinetics. 344 00:13:56,230 --> 00:13:57,847 It's getting this hydrogen in and out 345 00:13:57,847 --> 00:13:58,930 that has to be just right. 346 00:13:58,930 --> 00:14:05,540 And that's where these metal hydrides are quite interesting. 347 00:14:05,540 --> 00:14:06,880 They have lower densities. 348 00:14:06,880 --> 00:14:08,560 Higher though than the targets set 349 00:14:08,560 --> 00:14:12,280 by DOE, which was 6% at least some years ago. 350 00:14:12,280 --> 00:14:14,990 I forget what their target is today. 351 00:14:14,990 --> 00:14:19,870 So they're in a pretty good range to be interesting. 352 00:14:19,870 --> 00:14:21,310 And there's been a lot of work. 353 00:14:21,310 --> 00:14:24,530 And I just put a few kind of papers and examples here. 354 00:14:24,530 --> 00:14:27,310 There's been a lot of work on these different materials. 355 00:14:27,310 --> 00:14:30,670 And in particular, functionalizing materials 356 00:14:30,670 --> 00:14:34,990 in different ways to see how they pick up-- 357 00:14:34,990 --> 00:14:37,270 how these kinetics change-- how they pick up hydrogen. 358 00:14:37,270 --> 00:14:41,011 In this case, you're still just looking at the weight percent. 359 00:14:41,011 --> 00:14:42,320 OK. 360 00:14:42,320 --> 00:14:42,820 Yeah. 361 00:14:42,820 --> 00:14:43,737 And same in this case. 362 00:14:43,737 --> 00:14:46,190 I'll get to the kinetics in a few minutes. 363 00:14:46,190 --> 00:14:47,140 These are the sponges. 364 00:14:47,140 --> 00:14:48,160 Look at this. 365 00:14:48,160 --> 00:14:53,750 Sodium alanate doped with a little bit of titanium 366 00:14:53,750 --> 00:15:00,280 can give you up to 3.7% or 1.8%, depending on how you make it. 367 00:15:00,280 --> 00:15:01,530 And that's what it looks like. 368 00:15:01,530 --> 00:15:05,070 There it is before you fill it up with your fuel. 369 00:15:05,070 --> 00:15:06,905 And there it is when it's filled up. 370 00:15:06,905 --> 00:15:08,780 And then it's just going to go back and forth 371 00:15:08,780 --> 00:15:09,972 and back and forth. 372 00:15:09,972 --> 00:15:11,180 That's the concept, at least. 373 00:15:17,390 --> 00:15:20,090 So that's basically just restating what I said. 374 00:15:20,090 --> 00:15:23,130 The metal soaks up and releases the hydrogen like a sponge. 375 00:15:23,130 --> 00:15:25,460 It becomes a part of the chemical structure 376 00:15:25,460 --> 00:15:27,890 of the material, so it doesn't require 377 00:15:27,890 --> 00:15:30,560 high pressures or cryogenic temperatures for operation. 378 00:15:30,560 --> 00:15:32,730 But it's the irreversibility that's the problem. 379 00:15:32,730 --> 00:15:35,270 How do you get to the right kinetics? 380 00:15:35,270 --> 00:15:39,090 Now, here's a chart that I'm not going to go through. 381 00:15:39,090 --> 00:15:41,150 But it's, again, just more data for you 382 00:15:41,150 --> 00:15:42,510 to look at, if you want. 383 00:15:42,510 --> 00:15:46,010 And you can see that there's a lot of work going on here. 384 00:15:46,010 --> 00:15:48,260 There's a lot of really interesting research going 385 00:15:48,260 --> 00:15:52,160 on in this area for looking at materials 386 00:15:52,160 --> 00:15:55,760 that can form a hydride or not. 387 00:15:55,760 --> 00:15:59,540 So they can-- like sodium alanate 388 00:15:59,540 --> 00:16:04,490 can be sodium alanate, just sodium aluminum, the metal. 389 00:16:04,490 --> 00:16:09,770 Or it can soak up the hydrogen. And this table gives you 390 00:16:09,770 --> 00:16:12,380 what weight you can expect. 391 00:16:12,380 --> 00:16:14,750 And it gives you things like what temperature 392 00:16:14,750 --> 00:16:18,560 are you going to need to go to get the hydrogen out, 393 00:16:18,560 --> 00:16:20,520 because that's a really important parameter. 394 00:16:20,520 --> 00:16:23,150 And so I'm going to finally come to-- oh. 395 00:16:23,150 --> 00:16:26,690 Now, I have a few sides here that I won't go through. 396 00:16:26,690 --> 00:16:28,880 But they're just here for you to look at, 397 00:16:28,880 --> 00:16:31,310 just looking at a few cases of different materials 398 00:16:31,310 --> 00:16:35,420 that have gotten a lot of attention in the last 10 years. 399 00:16:35,420 --> 00:16:39,320 And what I do is I tell you why they're interesting 400 00:16:39,320 --> 00:16:42,830 and what their challenges are. 401 00:16:42,830 --> 00:16:44,540 So we go through a few. 402 00:16:44,540 --> 00:16:46,070 There's aluminum hydride. 403 00:16:46,070 --> 00:16:48,650 You go from the pure aluminum to aluminum hydride, 404 00:16:48,650 --> 00:16:50,120 back and forth. 405 00:16:50,120 --> 00:16:53,120 You can get up to 10% weight. 406 00:16:53,120 --> 00:16:58,070 But it's almost completely irreversible. 407 00:16:58,070 --> 00:17:00,320 Now, this is the interesting thing 408 00:17:00,320 --> 00:17:02,000 that I want to tell you about. 409 00:17:02,000 --> 00:17:04,640 And this is where you start to get really 410 00:17:04,640 --> 00:17:07,420 cool phase space opening up. 411 00:17:07,420 --> 00:17:10,069 Again, that's what I get really excited about. 412 00:17:10,069 --> 00:17:11,990 You look at a problem in energy. 413 00:17:11,990 --> 00:17:16,069 And you try to find new ways of opening up material's design, 414 00:17:16,069 --> 00:17:17,660 axes for designing materials. 415 00:17:17,660 --> 00:17:18,990 And this is one of them. 416 00:17:18,990 --> 00:17:20,569 It turns out that-- 417 00:17:20,569 --> 00:17:23,000 you see in-- let's take magnesium. 418 00:17:23,000 --> 00:17:24,859 So I showed you aluminum, sodium. 419 00:17:24,859 --> 00:17:27,050 Here's magnesium, so another example 420 00:17:27,050 --> 00:17:31,400 of one of these sponges, these metal sponges. 421 00:17:31,400 --> 00:17:37,050 And when it's sponged, no, soaked, it looks like this. 422 00:17:37,050 --> 00:17:39,720 It's got hydrogen in it. 423 00:17:39,720 --> 00:17:42,007 And it's actually a pretty decent weight percent, 424 00:17:42,007 --> 00:17:44,060 7.7 weight percent. 425 00:17:44,060 --> 00:17:46,970 The problem is that this is the amount of energy it 426 00:17:46,970 --> 00:17:49,820 takes to get the hydrogen out. 427 00:17:49,820 --> 00:17:52,220 That's too much. 428 00:17:52,220 --> 00:17:55,520 What you need for an onboard hydrogen storage is you 429 00:17:55,520 --> 00:18:00,890 need that delta H to be somewhere 430 00:18:00,890 --> 00:18:04,100 in the range of 20 to 50 kilojoules per mole. 431 00:18:04,100 --> 00:18:05,900 And what is so cool is that-- 432 00:18:05,900 --> 00:18:09,750 what's been shown-- and this is just a bunch of data. 433 00:18:09,750 --> 00:18:11,990 But I want you to think size going 434 00:18:11,990 --> 00:18:15,710 this way and the tuning, that delta H parameter, 435 00:18:15,710 --> 00:18:17,270 on this axis. 436 00:18:17,270 --> 00:18:20,120 What's been shown is that, when it's a bulk material-- 437 00:18:20,120 --> 00:18:22,760 it's out here at 75-- 438 00:18:22,760 --> 00:18:27,512 and as you go down to smaller sizes, that gets tuned. 439 00:18:27,512 --> 00:18:28,220 And look at that. 440 00:18:28,220 --> 00:18:31,970 It tunes all the way down into the point where it's completely 441 00:18:31,970 --> 00:18:34,200 reversed the stability even. 442 00:18:34,200 --> 00:18:36,740 And so what you see is that, just by changing the size, 443 00:18:36,740 --> 00:18:40,610 just by nanostructuring the material, what you're doing 444 00:18:40,610 --> 00:18:42,830 is you're changing the kinetics. 445 00:18:42,830 --> 00:18:45,200 You're changing the desorbtion energy 446 00:18:45,200 --> 00:18:49,080 by changing the size of the material. 447 00:18:49,080 --> 00:18:49,970 So that's an axis. 448 00:18:49,970 --> 00:18:51,890 That's a tunability axis. 449 00:18:51,890 --> 00:18:55,020 Now, why is this interesting-- 450 00:18:55,020 --> 00:18:55,520 to what? 451 00:18:55,520 --> 00:18:56,360 To our class. 452 00:18:56,360 --> 00:18:59,550 Why is this interesting to quantum mechanics? 453 00:18:59,550 --> 00:19:02,690 Well, because these are all quantum 454 00:19:02,690 --> 00:19:04,230 mechanical calculations. 455 00:19:04,230 --> 00:19:07,280 And you can see that, even just the different levels theory, 456 00:19:07,280 --> 00:19:09,650 oh, you now know what some of these mean, because these 457 00:19:09,650 --> 00:19:11,060 are different DFT functionals. 458 00:19:11,060 --> 00:19:14,390 Just look at those, three different DFT functionals. 459 00:19:14,390 --> 00:19:16,115 Well, they give you similar trends. 460 00:19:18,487 --> 00:19:20,570 But you can see that there's still some variation. 461 00:19:20,570 --> 00:19:26,210 If I go to classical potentials, it's very, very hard to know, 462 00:19:26,210 --> 00:19:28,910 when I engineer something into a new domain like this, 463 00:19:28,910 --> 00:19:31,250 if it's right. 464 00:19:31,250 --> 00:19:33,500 So this is really a problem that's 465 00:19:33,500 --> 00:19:38,895 best suited for accurate quantum mechanical methods, 466 00:19:38,895 --> 00:19:41,270 because you're going to make a classical potential that's 467 00:19:41,270 --> 00:19:42,860 fit here, basically. 468 00:19:42,860 --> 00:19:45,710 But all of the interesting stuff happens as you go down 469 00:19:45,710 --> 00:19:47,300 to smaller sizes. 470 00:19:47,300 --> 00:19:49,140 So it goes down into this window. 471 00:19:49,140 --> 00:19:52,483 So notice that I might want it in this range. 472 00:19:52,483 --> 00:19:53,900 I might want the desorbtion energy 473 00:19:53,900 --> 00:19:57,170 to be in this range, which means I have a very narrow window 474 00:19:57,170 --> 00:20:00,090 of size that I can pick. 475 00:20:00,090 --> 00:20:02,870 I better get that right, that size right, 476 00:20:02,870 --> 00:20:05,510 if I'm going to try to make predictions of new materials. 477 00:20:05,510 --> 00:20:08,330 And that's where-- again, and then the phase space opens up 478 00:20:08,330 --> 00:20:12,260 even more as you talk about-- 479 00:20:12,260 --> 00:20:15,600 this is nanoscaling phase space that allows you to tune delta 480 00:20:15,600 --> 00:20:18,810 H. But you can also tune it by alloying different materials. 481 00:20:18,810 --> 00:20:21,710 So now, you have this really cool playground. 482 00:20:21,710 --> 00:20:25,220 I have a bunch of different metals, sodium, aluminum, 483 00:20:25,220 --> 00:20:26,660 magnesium. 484 00:20:26,660 --> 00:20:28,490 I have size. 485 00:20:28,490 --> 00:20:30,860 And I have composition. 486 00:20:30,860 --> 00:20:33,260 What is going to give you the best fuel? 487 00:20:33,260 --> 00:20:35,870 What's going to give you the best kinetics? 488 00:20:35,870 --> 00:20:39,200 Perfect problem for computation and, really, a problem 489 00:20:39,200 --> 00:20:42,110 that quantum mechanics is needed for. 490 00:20:42,110 --> 00:20:50,280 So that's my 20-minute hydrogen storage ramble. 491 00:20:50,280 --> 00:20:53,947 Any questions about it before we move on? 492 00:20:53,947 --> 00:20:56,280 And I'd be happy to talk more with people about hydrogen 493 00:20:56,280 --> 00:20:57,238 storage, if you'd like. 494 00:21:00,210 --> 00:21:02,860 It's cool problems. 495 00:21:02,860 --> 00:21:03,360 OK. 496 00:21:06,510 --> 00:21:09,390 Now, I want to turn to this goal of going from atoms 497 00:21:09,390 --> 00:21:13,780 to crystals, to solids. 498 00:21:13,780 --> 00:21:15,990 Now, what we're going to do today 499 00:21:15,990 --> 00:21:24,320 is just lay some of the basic theory that we need to know. 500 00:21:24,320 --> 00:21:27,690 And then next week, we're going to start looking 501 00:21:27,690 --> 00:21:30,330 at the ramifications of that. 502 00:21:30,330 --> 00:21:33,720 We'll end with a band structure, which is really exciting. 503 00:21:33,720 --> 00:21:36,090 How many of you have seen a band structure? 504 00:21:36,090 --> 00:21:37,170 Yeah. 505 00:21:37,170 --> 00:21:38,760 Is it fun to look at band structures? 506 00:21:38,760 --> 00:21:40,350 Have you gotten to that point? 507 00:21:40,350 --> 00:21:42,660 Yeah, it's really so-so. 508 00:21:42,660 --> 00:21:43,930 We'll get there. 509 00:21:43,930 --> 00:21:46,770 We're going to feel really excited about band structures, 510 00:21:46,770 --> 00:21:48,750 because it's just it's really cool stuff. 511 00:21:48,750 --> 00:21:52,325 And it happens because you go to that. 512 00:21:52,325 --> 00:21:53,700 So now, we're at this point where 513 00:21:53,700 --> 00:21:56,010 we want to know about crystals and not 514 00:21:56,010 --> 00:21:58,950 just atoms and molecules. 515 00:21:58,950 --> 00:22:04,710 So what happens that we talked about so far is this. 516 00:22:04,710 --> 00:22:07,110 We talked about this on Tuesday. 517 00:22:07,110 --> 00:22:11,040 I have my atomic levels, say, S and P. 518 00:22:11,040 --> 00:22:16,020 And when I bring them together to form, say, 519 00:22:16,020 --> 00:22:18,990 a molecule, a dimer, like H2, well, 520 00:22:18,990 --> 00:22:22,290 then they can form combined-- they 521 00:22:22,290 --> 00:22:24,750 can form molecular orbitals. 522 00:22:24,750 --> 00:22:26,940 So that's the molecule there. 523 00:22:26,940 --> 00:22:29,190 And we talked about how you can get bonding status 524 00:22:29,190 --> 00:22:30,857 and antibonding states, because you have 525 00:22:30,857 --> 00:22:33,310 wiggles that are overlapping. 526 00:22:33,310 --> 00:22:35,580 And so they can be in phase and out of phase and so 527 00:22:35,580 --> 00:22:36,900 forth in the wave functions. 528 00:22:36,900 --> 00:22:38,850 And then you square it to find out 529 00:22:38,850 --> 00:22:40,620 the probability distributions. 530 00:22:40,620 --> 00:22:42,780 But now, when I go to a crystal, you 531 00:22:42,780 --> 00:22:45,090 see you're taking this to the next level, 532 00:22:45,090 --> 00:22:47,750 to the next infinite level. 533 00:22:47,750 --> 00:22:50,090 So now, you don't just have another atom there, 534 00:22:50,090 --> 00:22:53,100 you've got an atom everywhere, all the way out to infinity. 535 00:22:53,100 --> 00:22:57,300 And what ends up happening is that, as we'll see, 536 00:22:57,300 --> 00:23:01,280 is that, instead of getting just these combined molecular 537 00:23:01,280 --> 00:23:05,270 orbitals that overlap a wave function with another wave 538 00:23:05,270 --> 00:23:07,670 function, you get these bands. 539 00:23:07,670 --> 00:23:12,170 You get these bands of energy states. 540 00:23:12,170 --> 00:23:16,670 And the structure of those bands is the band structure. 541 00:23:16,670 --> 00:23:19,190 And they're very-- these don't just look like this. 542 00:23:19,190 --> 00:23:23,390 They actually wiggle, which is what's really cool about that. 543 00:23:23,390 --> 00:23:25,160 Oh, there's the wiggle. 544 00:23:25,160 --> 00:23:29,180 Yeah, they wiggle. 545 00:23:29,180 --> 00:23:34,400 I mean, that-- I just went from all doing all this, 546 00:23:34,400 --> 00:23:36,920 drawing this a whole bunch of times. 547 00:23:36,920 --> 00:23:40,160 And now, I'm saying they're wiggling. 548 00:23:40,160 --> 00:23:41,240 So they look like this. 549 00:23:44,620 --> 00:23:46,820 How is that possible? 550 00:23:46,820 --> 00:23:49,240 How can an energy level wiggle? 551 00:23:49,240 --> 00:23:51,972 What's it wiggling into? 552 00:23:51,972 --> 00:23:53,180 That's what we're getting to. 553 00:23:53,180 --> 00:23:56,258 That's called reciprocal space or k-space. 554 00:23:56,258 --> 00:23:57,800 So that's what we're going to get to. 555 00:24:01,090 --> 00:24:05,390 And I think you all know this, but, of course, 556 00:24:05,390 --> 00:24:07,790 when you talk about a molecule, you don't usually 557 00:24:07,790 --> 00:24:10,458 talk about a metal. 558 00:24:10,458 --> 00:24:12,750 I mean, you talk about the spin states of the molecule. 559 00:24:12,750 --> 00:24:14,890 You can ask whether it's magnetic or not. 560 00:24:14,890 --> 00:24:19,300 But you don't usually say it's a metal. 561 00:24:19,300 --> 00:24:21,910 When you have a solid, well, we know these terms. 562 00:24:21,910 --> 00:24:24,160 It can be an insulator, which would mean 563 00:24:24,160 --> 00:24:25,960 that you have bands filled. 564 00:24:25,960 --> 00:24:29,410 And we fill them, just like in the atoms and molecules. 565 00:24:29,410 --> 00:24:30,880 We get our bands. 566 00:24:30,880 --> 00:24:33,280 We get our wiggles states now. 567 00:24:33,280 --> 00:24:34,880 And we fill those with electrons, 568 00:24:34,880 --> 00:24:37,550 just like we did with the atoms and molecules. 569 00:24:37,550 --> 00:24:42,340 And if you fill them up, and you're in the middle of a band 570 00:24:42,340 --> 00:24:44,570 when you stop, you're a metal. 571 00:24:44,570 --> 00:24:47,440 If you're here, and you stop at the top of a band, 572 00:24:47,440 --> 00:24:50,740 and there's a big jump to the next band, it's an insulator. 573 00:24:50,740 --> 00:24:54,430 If it's a little jump, it's a semiconductor. 574 00:24:54,430 --> 00:24:58,240 Anybody know what the gap of a semiconductor is, typically? 575 00:24:58,240 --> 00:24:59,428 AUDIENCE: [INAUDIBLE] 576 00:24:59,428 --> 00:25:00,970 JEFFREY GROSSMAN: One point what now? 577 00:25:00,970 --> 00:25:02,344 Say that again. 578 00:25:02,344 --> 00:25:04,510 AUDIENCE: [INAUDIBLE] 579 00:25:04,510 --> 00:25:07,180 JEFFREY GROSSMAN: Yeah, that sounds about right, 1 to 3. 580 00:25:07,180 --> 00:25:10,680 And what's an insulator? 581 00:25:10,680 --> 00:25:12,180 AUDIENCE: Anything higher than the-- 582 00:25:12,180 --> 00:25:13,400 JEFFREY GROSSMAN: Anything higher, maybe? 583 00:25:13,400 --> 00:25:14,410 Yeah, 5, maybe. 584 00:25:14,410 --> 00:25:16,100 Yeah, I love it. 585 00:25:16,100 --> 00:25:20,960 And then there's the term wide bandgap semiconductor, 586 00:25:20,960 --> 00:25:23,840 which is really a term that's used when you really 587 00:25:23,840 --> 00:25:26,150 wish your material were a semiconductor. 588 00:25:26,150 --> 00:25:27,818 But it just has a really big gap. 589 00:25:27,818 --> 00:25:29,610 And most people would call it an insulator. 590 00:25:29,610 --> 00:25:32,780 We say, it's a wide bandgap semiconductor, 591 00:25:32,780 --> 00:25:35,010 so I hope it could be useful for electronics. 592 00:25:38,520 --> 00:25:42,930 Now, I think most of you have seen crystal symmetries. 593 00:25:42,930 --> 00:25:45,360 I mean, you certainly saw it if you took 3012. 594 00:25:45,360 --> 00:25:47,775 How many of you have not seen crystals? 595 00:25:50,440 --> 00:25:54,580 So this is not about having a deep understanding 596 00:25:54,580 --> 00:25:56,878 of crystal symmetries. 597 00:25:56,878 --> 00:25:58,420 But there is something important that 598 00:25:58,420 --> 00:26:02,410 comes about from having periodic potential, which is the focus. 599 00:26:02,410 --> 00:26:03,340 So I just have-- 600 00:26:03,340 --> 00:26:06,350 I didn't want to go into details here. 601 00:26:06,350 --> 00:26:09,340 But I will just give you the very basic elements, again, 602 00:26:09,340 --> 00:26:11,770 in a couple of minutes. 603 00:26:11,770 --> 00:26:15,020 When you have a repeating pattern, 604 00:26:15,020 --> 00:26:18,860 you can identify what repeats. 605 00:26:18,860 --> 00:26:20,890 And that's the unit cell in the crystal. 606 00:26:23,500 --> 00:26:28,790 It's a very powerful way of picturing an infinite material. 607 00:26:28,790 --> 00:26:31,385 Because how many atoms do I need to represent silicon? 608 00:26:34,520 --> 00:26:36,410 How many of you know? 609 00:26:36,410 --> 00:26:37,910 What's the structure of silicon? 610 00:26:40,600 --> 00:26:41,762 Anybody know? 611 00:26:41,762 --> 00:26:43,410 AUDIENCE: Cubic. 612 00:26:43,410 --> 00:26:44,870 JEFFREY GROSSMAN: Cubic, OK. 613 00:26:44,870 --> 00:26:46,042 AUDIENCE: Tetrahedral? 614 00:26:46,042 --> 00:26:47,500 JEFFREY GROSSMAN: Tetrahedral, yes. 615 00:26:47,500 --> 00:26:50,190 What's the symmetry? 616 00:26:50,190 --> 00:26:51,460 Come back to that. 617 00:26:51,460 --> 00:26:56,370 AUDIENCE: [INAUDIBLE] 618 00:26:56,370 --> 00:26:58,200 JEFFREY GROSSMAN: Any others? 619 00:26:58,200 --> 00:27:00,480 We'll come back to it. 620 00:27:00,480 --> 00:27:06,100 But-- oh, yeah, you have unit cells. 621 00:27:06,100 --> 00:27:09,840 And they get repeated-- 622 00:27:09,840 --> 00:27:11,820 the cell is repeated-- 623 00:27:11,820 --> 00:27:13,180 on the lattice. 624 00:27:13,180 --> 00:27:19,630 The lattice defines the repeatness, the repeatiness, 625 00:27:19,630 --> 00:27:20,440 repetition. 626 00:27:20,440 --> 00:27:22,900 No, none of those are good words. 627 00:27:22,900 --> 00:27:26,110 But the lattice defines how it repeats. 628 00:27:26,110 --> 00:27:31,330 Is it going to repeat like this, in a rectangular way? 629 00:27:31,330 --> 00:27:34,150 Is it going to repeat in a square way? 630 00:27:34,150 --> 00:27:35,980 And you fill all of space. 631 00:27:35,980 --> 00:27:39,640 If you fill all of space, then it's 632 00:27:39,640 --> 00:27:41,140 a particular kind of lattice. 633 00:27:43,870 --> 00:27:47,100 Now, because a crystal is periodic, 634 00:27:47,100 --> 00:27:51,570 the whole point here is we don't have to model all the atoms. 635 00:27:51,570 --> 00:27:53,290 Because if we did, we'd need to model 10 636 00:27:53,290 --> 00:27:56,100 to the 23rd, which would be a bummer. 637 00:27:56,100 --> 00:27:58,590 And so what's cool is that, since it's periodic, 638 00:27:58,590 --> 00:28:04,720 we can just model the ones that periodicalize, or repeat. 639 00:28:04,720 --> 00:28:06,220 And these are the crystal symmetries 640 00:28:06,220 --> 00:28:08,080 that fill all of space. 641 00:28:08,080 --> 00:28:11,590 So this is all, for those of you who had 3012, 642 00:28:11,590 --> 00:28:12,790 I know this is just review. 643 00:28:17,230 --> 00:28:21,130 You have, basically, seven symmetry classes, 644 00:28:21,130 --> 00:28:23,920 crystal classes, sorry. 645 00:28:23,920 --> 00:28:25,815 Some of you talked about cubic. 646 00:28:25,815 --> 00:28:27,190 And then you have different types 647 00:28:27,190 --> 00:28:30,190 of unit cells, which, basically, is just whether you have, 648 00:28:30,190 --> 00:28:33,910 in the repeat cell, atoms in every corner, 649 00:28:33,910 --> 00:28:37,300 in every corner in the center, in the faces, all faces, 650 00:28:37,300 --> 00:28:39,100 some faces. 651 00:28:39,100 --> 00:28:41,032 Those are four options. 652 00:28:41,032 --> 00:28:42,490 And when you bring it all together, 653 00:28:42,490 --> 00:28:45,280 you have 14 of these kinds of lattices, 654 00:28:45,280 --> 00:28:49,410 which are the ones that can fill all of space. 655 00:28:49,410 --> 00:28:50,820 Those are Bravais lattices. 656 00:28:50,820 --> 00:28:54,470 This is all mostly review, right? 657 00:28:54,470 --> 00:28:57,390 And it's not critical. 658 00:28:57,390 --> 00:28:59,650 But, yeah, is this feeling-- 659 00:28:59,650 --> 00:29:01,230 are you feeling good about this? 660 00:29:01,230 --> 00:29:02,280 Yeah. 661 00:29:02,280 --> 00:29:05,070 You like it. 662 00:29:05,070 --> 00:29:07,770 So here's two different kinds of crystals. 663 00:29:07,770 --> 00:29:09,630 And then you have the crystal lattice, which 664 00:29:09,630 --> 00:29:10,955 is determined by these dots. 665 00:29:10,955 --> 00:29:13,080 And then you have the basis, which can be anything. 666 00:29:13,080 --> 00:29:14,850 You can put a basis on the dot. 667 00:29:14,850 --> 00:29:17,910 Now, this is how we're going to talk to our code, by the way. 668 00:29:17,910 --> 00:29:20,640 When we model a solid, this is what 669 00:29:20,640 --> 00:29:23,850 we're going to tell it, because it needs to know. 670 00:29:23,850 --> 00:29:25,170 It's going to need to know. 671 00:29:25,170 --> 00:29:27,450 When you run your SIESTA code-- and we'll do that-- 672 00:29:27,450 --> 00:29:30,390 we'll run some SIESTA calculations next week-- 673 00:29:30,390 --> 00:29:33,570 it's going to want to know what your lattices, well, 674 00:29:33,570 --> 00:29:35,130 what symmetry lattice you have. 675 00:29:35,130 --> 00:29:36,510 Is it FCC? 676 00:29:36,510 --> 00:29:38,970 Is it simple cubic? 677 00:29:38,970 --> 00:29:41,610 And then it's going to want to know the lattice spacing. 678 00:29:41,610 --> 00:29:44,400 And then it's going to want to know everything in your basis. 679 00:29:44,400 --> 00:29:47,280 Is it one atom in your basis? 680 00:29:47,280 --> 00:29:50,587 Is it like FCC, with two atoms in your basis, 681 00:29:50,587 --> 00:29:52,170 which sounds a lot like silicon to me? 682 00:29:54,858 --> 00:29:56,650 Or is it something like, with Mickey Mouse, 683 00:29:56,650 --> 00:30:00,190 which cannot be entered into a computer? 684 00:30:00,190 --> 00:30:02,970 So that's how you're going to input structures. 685 00:30:02,970 --> 00:30:06,670 Now, here's where it gets really interesting. 686 00:30:06,670 --> 00:30:08,380 And this is where-- 687 00:30:08,380 --> 00:30:11,500 this is really the key of today's lecture 688 00:30:11,500 --> 00:30:16,840 is the next five, six lines, next bit. 689 00:30:21,420 --> 00:30:25,080 There is something called an inverse lattice, which is also 690 00:30:25,080 --> 00:30:27,100 called reciprocal space. 691 00:30:27,100 --> 00:30:28,170 And I think because-- 692 00:30:28,170 --> 00:30:31,800 I think you did learn all this, if you had 3012, yes? 693 00:30:31,800 --> 00:30:32,790 So you all feel-- 694 00:30:32,790 --> 00:30:36,400 do you feel your oneness with reciprocal space? 695 00:30:36,400 --> 00:30:38,520 Yeah, I'm seeing some strong nods 696 00:30:38,520 --> 00:30:41,580 and some not-so-strong nods. 697 00:30:41,580 --> 00:30:45,420 So this is a really important concept 698 00:30:45,420 --> 00:30:47,370 that, essentially, gives this. 699 00:30:47,370 --> 00:30:49,800 It gives us the wiggles. 700 00:30:49,800 --> 00:30:53,400 This reciprocal space is the language 701 00:30:53,400 --> 00:30:56,310 we need to describe the wiggles. 702 00:30:56,310 --> 00:30:59,670 So just to review for most of you, 703 00:30:59,670 --> 00:31:02,700 with each real space lattice, like the square that I 704 00:31:02,700 --> 00:31:05,400 showed you, you can make a reciprocal lattice. 705 00:31:05,400 --> 00:31:07,890 And it's the set of vectors that are 706 00:31:07,890 --> 00:31:09,600 commensurate with the real space lattice. 707 00:31:09,600 --> 00:31:14,280 And you can define them very, very ingeniously. 708 00:31:14,280 --> 00:31:18,714 If the real space lattice is vectors a, b, and c-- 709 00:31:18,714 --> 00:31:21,850 oh, like here, it's a 2D lattice. 710 00:31:21,850 --> 00:31:25,780 So this would be like a. 711 00:31:25,780 --> 00:31:26,990 That would be vector a. 712 00:31:26,990 --> 00:31:29,606 And maybe, this would be vector b. 713 00:31:29,606 --> 00:31:30,730 This would be vector b. 714 00:31:30,730 --> 00:31:32,230 Which one? 715 00:31:32,230 --> 00:31:34,730 Any thoughts? 716 00:31:34,730 --> 00:31:37,160 We're just brushing up here. 717 00:31:37,160 --> 00:31:41,330 But you call them a star, b star, and c star. 718 00:31:41,330 --> 00:31:45,200 And the way they're defined is in a very important way. 719 00:31:45,200 --> 00:31:47,360 First of all, a star-- 720 00:31:47,360 --> 00:31:50,270 that's the reciprocal vector-- times a is 1. 721 00:31:50,270 --> 00:31:53,110 So there's a normalization. 722 00:31:53,110 --> 00:31:55,610 And secondly, there's something a little more mathematically 723 00:31:55,610 --> 00:32:00,410 rigorous, which is that it's the cross product of the other two. 724 00:32:00,410 --> 00:32:04,630 So a star is the cross product of b and c. 725 00:32:04,630 --> 00:32:08,050 b star is the cross product of c and a. 726 00:32:08,050 --> 00:32:10,013 So you can write down-- 727 00:32:10,013 --> 00:32:11,680 basically, when you have your real space 728 00:32:11,680 --> 00:32:13,480 lattice and-- hello there. 729 00:32:13,480 --> 00:32:16,560 [DOOR CLOSES] 730 00:32:16,560 --> 00:32:21,340 Why do I have those little n's there in front of those vectors 731 00:32:21,340 --> 00:32:24,640 to define my real space lattice? 732 00:32:24,640 --> 00:32:28,500 This is what we use to talk about lattice? 733 00:32:28,500 --> 00:32:29,630 Why are those n's there? 734 00:32:33,526 --> 00:32:35,480 AUDIENCE: Signify the twilight zone. 735 00:32:35,480 --> 00:32:38,370 JEFFREY GROSSMAN: The twilight zone? 736 00:32:38,370 --> 00:32:39,240 Very good guess. 737 00:32:41,750 --> 00:32:45,684 What does n usually mean when I put it in an equation? 738 00:32:45,684 --> 00:32:46,680 AUDIENCE: [INAUDIBLE] 739 00:32:46,680 --> 00:32:48,260 JEFFREY GROSSMAN: A what number? 740 00:32:48,260 --> 00:32:48,650 AUDIENCE: Integer. 741 00:32:48,650 --> 00:32:49,940 JEFFREY GROSSMAN: Some integer, yeah. 742 00:32:49,940 --> 00:32:50,570 Why? 743 00:32:50,570 --> 00:32:52,010 I don't know. 744 00:32:52,010 --> 00:32:54,150 We just like n for integer. 745 00:32:54,150 --> 00:32:56,780 It just happened. 746 00:32:56,780 --> 00:32:59,330 So why would those be integers? 747 00:32:59,330 --> 00:33:03,630 a1, a2, a3 are the vectors of the lattice. 748 00:33:03,630 --> 00:33:05,120 And now, I have integers there. 749 00:33:05,120 --> 00:33:06,370 Why? 750 00:33:06,370 --> 00:33:08,865 AUDIENCE: [INAUDIBLE] 751 00:33:08,865 --> 00:33:10,990 JEFFREY GROSSMAN: Yeah, because this is my lattice. 752 00:33:10,990 --> 00:33:12,400 This is my lattice. 753 00:33:12,400 --> 00:33:15,490 It goes off to infinity. 754 00:33:15,490 --> 00:33:22,630 so it's n is 1 to get here, 2 get there, 3 to get there. 755 00:33:22,630 --> 00:33:24,610 That's why it's n. 756 00:33:24,610 --> 00:33:26,770 It's an integer times the lattice factor. 757 00:33:26,770 --> 00:33:30,700 And it gets me to any other lattice point, 758 00:33:30,700 --> 00:33:33,310 depending on what my n's are. 759 00:33:33,310 --> 00:33:38,890 Now, the same-- in reciprocal space, 760 00:33:38,890 --> 00:33:41,890 you construct another lattice, which 761 00:33:41,890 --> 00:33:46,540 is given by these mathematical expressions, as I said, 762 00:33:46,540 --> 00:33:48,880 just cross products of the first lattice. 763 00:33:48,880 --> 00:33:50,830 And you get a reciprocal lattice, 764 00:33:50,830 --> 00:33:55,780 which we like to call G. We say that those have 765 00:33:55,780 --> 00:34:01,000 their own lattice vectors and their own integers in front 766 00:34:01,000 --> 00:34:06,070 that take you anywhere you want in the reciprocal lattice. 767 00:34:06,070 --> 00:34:08,670 Now, a couple of important things, one 768 00:34:08,670 --> 00:34:13,800 is, if I take e to the iG.R, where 769 00:34:13,800 --> 00:34:18,570 R is a lattice vector in real space, 770 00:34:18,570 --> 00:34:23,290 and G is a lattice vector in reciprocal space, I get 1. 771 00:34:23,290 --> 00:34:24,670 That's important. 772 00:34:24,670 --> 00:34:26,969 You'll see in a minute why. 773 00:34:26,969 --> 00:34:32,810 But also, if I happen to write a function that 774 00:34:32,810 --> 00:34:36,050 looks like something like this, where I have it as an exponent, 775 00:34:36,050 --> 00:34:39,110 e to the i some reciprocal lattice vector 776 00:34:39,110 --> 00:34:42,830 times R, where R can now be anywhere in real space-- 777 00:34:42,830 --> 00:34:44,780 R is just a vector-- 778 00:34:44,780 --> 00:34:46,040 it's not a lattice vector-- 779 00:34:46,040 --> 00:34:48,380 then that function will automatically 780 00:34:48,380 --> 00:34:52,370 be periodic in the real space. 781 00:34:52,370 --> 00:34:55,150 Isn't that cool? 782 00:34:55,150 --> 00:34:57,492 That function will automatically be periodic. 783 00:34:57,492 --> 00:34:59,200 That sounds like something we might want. 784 00:35:03,630 --> 00:35:07,790 Now, here's a simple picture. 785 00:35:11,010 --> 00:35:15,730 This is a real space BCC lattice. 786 00:35:15,730 --> 00:35:18,670 And when I construct the inverse lattice using 787 00:35:18,670 --> 00:35:21,250 the mathematical rules I just showed you, 788 00:35:21,250 --> 00:35:24,040 I get this, which is cool. 789 00:35:24,040 --> 00:35:26,930 BCC turns into FCC. 790 00:35:26,930 --> 00:35:28,460 When you do the inverse lattice, you 791 00:35:28,460 --> 00:35:32,870 get a new lattice, which happens to be FCC. 792 00:35:32,870 --> 00:35:37,670 So it's just a new space, but it's reciprocal space. 793 00:35:37,670 --> 00:35:38,930 Now, why is this so important? 794 00:35:38,930 --> 00:35:39,800 We're getting there. 795 00:35:44,060 --> 00:35:49,660 Within this reciprocal space, you can define a zone. 796 00:35:49,660 --> 00:35:52,110 And you can define it very simply. 797 00:35:52,110 --> 00:35:56,340 You just draw-- you just go out to your-- you 798 00:35:56,340 --> 00:36:00,480 just go out to the next point here. 799 00:36:00,480 --> 00:36:02,380 And you draw your bisection lines. 800 00:36:02,380 --> 00:36:03,630 So you go out to these points. 801 00:36:03,630 --> 00:36:06,690 And you draw lines. 802 00:36:06,690 --> 00:36:08,230 In this case, it's a 2D lattice. 803 00:36:08,230 --> 00:36:11,310 So the area that's carved out is a zone. 804 00:36:11,310 --> 00:36:15,075 And that, when you do it with just to the first neighbors, 805 00:36:15,075 --> 00:36:18,630 when you do it like that, and you draw the lines, 806 00:36:18,630 --> 00:36:21,120 you get the Brillouin zone. 807 00:36:21,120 --> 00:36:24,900 Did you guys talk about Brillouin zone in the fall? 808 00:36:24,900 --> 00:36:28,350 It's just this very first zone, first-- 809 00:36:31,080 --> 00:36:33,250 I don't want to get too technical here. 810 00:36:33,250 --> 00:36:37,680 But it's the zone we care about in the calculations. 811 00:36:37,680 --> 00:36:40,500 This part of the reciprocal space 812 00:36:40,500 --> 00:36:44,160 is all we need, this Brillouin zone. 813 00:36:49,730 --> 00:36:50,570 He was a cool guy. 814 00:36:53,270 --> 00:36:57,260 And you can go out-- you can keep doing this 815 00:36:57,260 --> 00:36:58,760 and go to the next nearest neighbors 816 00:36:58,760 --> 00:37:02,720 and drop bisecting lines and create the next volume. 817 00:37:02,720 --> 00:37:05,370 And here are just a few examples. 818 00:37:05,370 --> 00:37:07,067 The first Brillouin zone is what we're 819 00:37:07,067 --> 00:37:08,150 going to be interested in. 820 00:37:11,690 --> 00:37:15,020 This is what it looks like for BCC and FCC lattices. 821 00:37:15,020 --> 00:37:17,690 And then you can go out to further and further points. 822 00:37:17,690 --> 00:37:20,910 And you get some really interesting volumes here. 823 00:37:23,450 --> 00:37:25,790 Again, I'm not going to go too much into this. 824 00:37:25,790 --> 00:37:28,340 But we do care about this Brillouin zone. 825 00:37:28,340 --> 00:37:31,685 And I'll tell you why when we get to the band structure. 826 00:37:36,570 --> 00:37:40,200 No, I'll tell you why right now. 827 00:37:40,200 --> 00:37:42,150 You see-- and then I'll come back to it, 828 00:37:42,150 --> 00:37:44,240 because it might not make total sense right now-- 829 00:37:46,770 --> 00:37:50,130 you see, the Brillouin zone is an area of-- 830 00:37:50,130 --> 00:37:55,770 it's a volume, in 3D, of space, of reciprocal space, not 831 00:37:55,770 --> 00:37:57,180 real space, reciprocal space. 832 00:37:59,700 --> 00:38:03,110 And what ends up happening-- so again, 833 00:38:03,110 --> 00:38:05,060 I'm jumping to the punchline here-- 834 00:38:05,060 --> 00:38:07,190 but I want to tell to you while you're thinking 835 00:38:07,190 --> 00:38:09,800 about this Brillouin zone-- 836 00:38:09,800 --> 00:38:14,420 is that, you see, when you move around in reciprocal space 837 00:38:14,420 --> 00:38:18,500 within this first Brillouin zone, what ends up happening 838 00:38:18,500 --> 00:38:23,760 is that the wave function can change, depending 839 00:38:23,760 --> 00:38:27,410 on where it is in that zone. 840 00:38:27,410 --> 00:38:32,087 Now, that's a result of having a periodic potential 841 00:38:32,087 --> 00:38:33,920 and this thing called Bloch's Theorem, which 842 00:38:33,920 --> 00:38:36,890 we're going to talk about next. 843 00:38:36,890 --> 00:38:42,320 But that is what leads the variation in the energies 844 00:38:42,320 --> 00:38:44,390 is what leads to the curviness. 845 00:38:44,390 --> 00:38:49,400 This curviness, if you want to think about it, 846 00:38:49,400 --> 00:38:59,320 is a tour through the Brillouin zone. 847 00:38:59,320 --> 00:39:02,530 I take a tour through this special zone 848 00:39:02,530 --> 00:39:04,220 in reciprocal space. 849 00:39:04,220 --> 00:39:09,150 And I find that the wave function varies. 850 00:39:09,150 --> 00:39:12,900 So I may take a tour by going from the middle of the zone 851 00:39:12,900 --> 00:39:15,370 out to one of these edges. 852 00:39:15,370 --> 00:39:18,750 Maybe I'll go up to the middle of the plane here of the phase 853 00:39:18,750 --> 00:39:20,970 and back to the edge. 854 00:39:20,970 --> 00:39:23,790 And you cruise around this Brillouin zone. 855 00:39:23,790 --> 00:39:25,800 And that's what gives you wiggles. 856 00:39:25,800 --> 00:39:27,870 That's how you get the wiggles. 857 00:39:27,870 --> 00:39:30,690 I mean, the crystal is what gives you the wiggles. 858 00:39:30,690 --> 00:39:33,450 The way you calculate them is by cruising around 859 00:39:33,450 --> 00:39:35,370 in reciprocal space in the code. 860 00:39:41,690 --> 00:39:47,060 By the way, in this part of the class, 861 00:39:47,060 --> 00:39:50,450 I'm very purposefully not going into technical details 862 00:39:50,450 --> 00:39:51,200 of the codes. 863 00:39:51,200 --> 00:39:54,500 However, if anyone is interested, 864 00:39:54,500 --> 00:39:56,690 I would be very happy to work with you 865 00:39:56,690 --> 00:40:00,200 or share with you more details about how the codes are doing 866 00:40:00,200 --> 00:40:04,250 these calculations, what's the math under the hood, what kind 867 00:40:04,250 --> 00:40:05,960 of loops are happening. 868 00:40:05,960 --> 00:40:09,800 And you're certainly welcome to download codes. 869 00:40:09,800 --> 00:40:12,320 There are free ones that-- 870 00:40:12,320 --> 00:40:18,930 for solids, you can download, for free, SIESTA, which 871 00:40:18,930 --> 00:40:20,410 is what's running on the hub. 872 00:40:20,410 --> 00:40:21,690 You can download PWSCF. 873 00:40:24,270 --> 00:40:26,770 You can download ABINIT. 874 00:40:26,770 --> 00:40:29,880 These are just a few that come to mind. 875 00:40:29,880 --> 00:40:30,523 They're free. 876 00:40:30,523 --> 00:40:31,440 You can download them. 877 00:40:31,440 --> 00:40:33,570 Most of them, I think you can run on your laptop. 878 00:40:33,570 --> 00:40:35,160 And if you're interested in going more 879 00:40:35,160 --> 00:40:37,743 under the hood in the codes, I'd be very happy to talk to you. 880 00:40:37,743 --> 00:40:40,500 But that's not the focus of this part of the class, 881 00:40:40,500 --> 00:40:41,880 as I've said. 882 00:40:41,880 --> 00:40:46,200 For molecules-- as I've said, there are many codes here-- 883 00:40:46,200 --> 00:40:48,270 this is just a very short list-- but there's 884 00:40:48,270 --> 00:40:58,390 GAMES and NWCHEM, which are a few that come to mind, 885 00:40:58,390 --> 00:40:59,320 both free. 886 00:40:59,320 --> 00:41:01,420 And then there are bunch that you pay for. 887 00:41:01,420 --> 00:41:04,270 If you want to spend some money, if you 888 00:41:04,270 --> 00:41:09,570 got 5, 10 grand to spare, first, come see me and-- 889 00:41:09,570 --> 00:41:10,644 [LAUGHTER] 890 00:41:10,644 --> 00:41:12,880 --I'm kidding. 891 00:41:12,880 --> 00:41:16,150 But you can buy VASP. 892 00:41:16,150 --> 00:41:21,370 Or you could buy a code called Gaussian, 893 00:41:21,370 --> 00:41:23,890 which are both-- we heavily use VASP in my group. 894 00:41:23,890 --> 00:41:25,390 These are very good codes, actually. 895 00:41:25,390 --> 00:41:26,683 But they cost money. 896 00:41:26,683 --> 00:41:28,600 They have some-- these different codes are all 897 00:41:28,600 --> 00:41:31,300 doing the same thing. 898 00:41:31,300 --> 00:41:32,550 They're all doing Schrodinger. 899 00:41:32,550 --> 00:41:35,010 And they're all using their approximations we talked about. 900 00:41:35,010 --> 00:41:37,840 And they have different advantages and disadvantages, 901 00:41:37,840 --> 00:41:40,200 which, again, I'd be happy to talk with people about, 902 00:41:40,200 --> 00:41:43,770 if they'd like to dig a little deeper. 903 00:41:43,770 --> 00:41:47,430 Now, all of this stuff about the Brillouin zone, 904 00:41:47,430 --> 00:41:50,730 which comes from the symmetries of the crystal, 905 00:41:50,730 --> 00:41:55,170 comes into play because we have periodic potentials. 906 00:41:55,170 --> 00:41:57,090 That's it. 907 00:41:57,090 --> 00:41:57,930 That's the point. 908 00:41:57,930 --> 00:42:11,600 So I now went from a potential that looks like this. 909 00:42:11,600 --> 00:42:14,360 Well, let me try to-- 910 00:42:14,360 --> 00:42:18,440 if there's an atom here, there's a potential well 911 00:42:18,440 --> 00:42:21,860 that an electron might feel as it gets attracted to the atom. 912 00:42:25,540 --> 00:42:30,750 And I went from putting, maybe, two atoms together to now 913 00:42:30,750 --> 00:42:33,788 having an array of them that go out to infinity. 914 00:42:38,000 --> 00:42:41,050 Now, that has repercussions. 915 00:42:41,050 --> 00:42:44,590 And so the potential, now, can be written. 916 00:42:44,590 --> 00:42:46,690 So you could try to write the whole thing out 917 00:42:46,690 --> 00:42:48,130 as a function of all of space. 918 00:42:48,130 --> 00:42:51,130 Or you can just say, hey, for every plus R, 919 00:42:51,130 --> 00:42:54,520 I get back to V, because that's basically what it is. 920 00:42:54,520 --> 00:42:57,290 It's a periodic potential in the lattice. 921 00:42:57,290 --> 00:43:00,640 So the potential now, every R, which is a lattice vector, 922 00:43:00,640 --> 00:43:02,800 is the same. 923 00:43:02,800 --> 00:43:06,440 Now, it turns out this is actually really hard to solve. 924 00:43:06,440 --> 00:43:09,490 And in textbooks, in quantum textbooks, 925 00:43:09,490 --> 00:43:14,740 like if you take a solid state physics class, what you'll see 926 00:43:14,740 --> 00:43:19,030 is that they'll show this picture, especially 927 00:43:19,030 --> 00:43:20,767 from a long time ago-- 928 00:43:20,767 --> 00:43:22,600 there's some really good solid state physics 929 00:43:22,600 --> 00:43:23,590 texts from the '70s. 930 00:43:25,812 --> 00:43:27,520 And you look at some of those older ones, 931 00:43:27,520 --> 00:43:29,895 they'll show those pictures, say, it's too hard to solve. 932 00:43:29,895 --> 00:43:31,840 And I love that, because it's not anymore, 933 00:43:31,840 --> 00:43:34,130 because we can use computers to do it, 934 00:43:34,130 --> 00:43:36,700 which is what we're going to do in this class. 935 00:43:36,700 --> 00:43:39,058 But back then, and even longer before, 936 00:43:39,058 --> 00:43:40,100 it was too hard to solve. 937 00:43:40,100 --> 00:43:48,880 And so people would basically model a crystal going back 938 00:43:48,880 --> 00:43:54,150 exactly to the problem of an atom and the potentials 939 00:43:54,150 --> 00:43:55,650 being so hard to solve analytically 940 00:43:55,650 --> 00:43:56,730 unless they're simple. 941 00:43:56,730 --> 00:43:58,050 They simplified this. 942 00:43:58,050 --> 00:43:59,700 And if you make it a periodically 943 00:43:59,700 --> 00:44:01,740 repeating square well like that, you 944 00:44:01,740 --> 00:44:04,400 could solve it analytically. 945 00:44:04,400 --> 00:44:07,080 And for those of you who are interested, 946 00:44:07,080 --> 00:44:10,610 this is called the Kronig-Penney model. 947 00:44:10,610 --> 00:44:12,960 And you can look up-- 948 00:44:12,960 --> 00:44:14,460 I think that's how you spell Penney. 949 00:44:14,460 --> 00:44:19,040 You could look up the solution to that online, if you want. 950 00:44:19,040 --> 00:44:22,850 It won't happen-- we're not going to do it in the class. 951 00:44:22,850 --> 00:44:24,780 But beyond simple things like this, 952 00:44:24,780 --> 00:44:27,830 which is not a very realistic crystal, although you 953 00:44:27,830 --> 00:44:32,750 do get some key physical meaning out of this, beyond that, 954 00:44:32,750 --> 00:44:39,210 you need a computer, which I think is pretty cool. 955 00:44:39,210 --> 00:44:40,890 So now, we get to Bloch's Theorem, 956 00:44:40,890 --> 00:44:47,340 which is what gets us to this dependence, this wiggle, 957 00:44:47,340 --> 00:44:48,570 this wiggle. 958 00:44:48,570 --> 00:44:51,060 So we're setting it up. 959 00:44:51,060 --> 00:44:52,650 We have reciprocal space. 960 00:44:52,650 --> 00:44:57,000 We defined this first zone in reciprocal space. 961 00:44:57,000 --> 00:44:59,700 We said that, if you go through the zone, things change. 962 00:44:59,700 --> 00:45:01,650 Bloch's Theorem is the reason. 963 00:45:01,650 --> 00:45:03,540 It's the reason why things change 964 00:45:03,540 --> 00:45:04,950 as you tour through this zone. 965 00:45:07,558 --> 00:45:09,850 There are different ways to talk about Bloch's Theorem. 966 00:45:09,850 --> 00:45:11,650 Here's a fairly simple way to look at it. 967 00:45:14,340 --> 00:45:19,620 So if you write the reciprocal lattice vector like this, 968 00:45:19,620 --> 00:45:22,320 2 pi-- and the 2's are just there 969 00:45:22,320 --> 00:45:24,630 to make math a little easier, so you 970 00:45:24,630 --> 00:45:27,070 can write it however you want. 971 00:45:27,070 --> 00:45:28,170 This is that integer. 972 00:45:28,170 --> 00:45:30,690 There's the reciprocal lattice vector. 973 00:45:30,690 --> 00:45:34,950 If you write it like that, then, if you take this exponential-- 974 00:45:34,950 --> 00:45:36,420 [SNEEZE] 975 00:45:36,420 --> 00:45:42,020 --gesundheit-- which I've already alluded to before, 976 00:45:42,020 --> 00:45:45,260 and you do that, actually, explicitly out. 977 00:45:45,260 --> 00:45:48,230 So R is a vector in real space. 978 00:45:50,980 --> 00:45:52,540 R is a vector in real space. 979 00:45:52,540 --> 00:45:57,880 Now, if I dot R with G, which is a reciprocal lattice vector, 980 00:45:57,880 --> 00:46:01,520 then I get the reciprocal lattice vector dot that. 981 00:46:01,520 --> 00:46:04,660 And that's just alpha, beta, gamma, or just the coefficients 982 00:46:04,660 --> 00:46:06,370 of the real space lattice vector, 983 00:46:06,370 --> 00:46:08,538 because R will be somewhere in real space. 984 00:46:08,538 --> 00:46:11,080 And so I can represent it, in terms of the real space lattice 985 00:46:11,080 --> 00:46:14,710 vectors time some coefficients. 986 00:46:14,710 --> 00:46:17,770 And what you can see when you do this, it's very simple math. 987 00:46:17,770 --> 00:46:19,870 What you can see is that, as you vary where 988 00:46:19,870 --> 00:46:23,440 you are in real space, then, well, these things 989 00:46:23,440 --> 00:46:24,340 are going to change. 990 00:46:24,340 --> 00:46:27,040 Alpha, beta, and gamma would change between, say, 0 and 1. 991 00:46:30,030 --> 00:46:33,350 And the function e to the iG.r changes as well. 992 00:46:36,530 --> 00:46:40,100 But you can see from this little derivation 993 00:46:40,100 --> 00:46:44,720 that, since these are integers, this function is always going 994 00:46:44,720 --> 00:46:45,990 to vary with the periodicity. 995 00:46:45,990 --> 00:46:47,000 So this is what I said. 996 00:46:47,000 --> 00:46:48,530 Here's the math behind it. 997 00:46:48,530 --> 00:46:50,090 I said this before. 998 00:46:50,090 --> 00:46:54,020 So you now have a function, e to the iG.R 999 00:46:54,020 --> 00:46:56,343 And that's the reason why. 1000 00:46:56,343 --> 00:46:58,260 If you just spend a few minutes staring at it, 1001 00:46:58,260 --> 00:47:00,980 you'll see why this is true. 1002 00:47:00,980 --> 00:47:03,745 You have a way of making a function automatically 1003 00:47:03,745 --> 00:47:04,245 periodic. 1004 00:47:07,070 --> 00:47:11,030 Now, why do I care? 1005 00:47:11,030 --> 00:47:14,410 Well, because if I have this periodicity, 1006 00:47:14,410 --> 00:47:17,020 if I have this ordering-- 1007 00:47:17,020 --> 00:47:18,700 here, it's drawn is an approximation. 1008 00:47:18,700 --> 00:47:20,840 Here, it's drawn as maybe more realistic, 1009 00:47:20,840 --> 00:47:22,150 what my potentials look like. 1010 00:47:27,640 --> 00:47:32,740 Then the values of some function, like the density, 1011 00:47:32,740 --> 00:47:36,020 should be what at each lattice site? 1012 00:47:36,020 --> 00:47:39,320 Well, I have it there in blue. 1013 00:47:39,320 --> 00:47:41,283 AUDIENCE: They'd be identical [INAUDIBLE] 1014 00:47:41,283 --> 00:47:42,450 JEFFREY GROSSMAN: Thank you. 1015 00:47:42,450 --> 00:47:45,240 Thank you very much. 1016 00:47:45,240 --> 00:47:50,910 Now, that seems obvious. 1017 00:47:50,910 --> 00:47:52,980 I mean, I've got, basically, an array 1018 00:47:52,980 --> 00:47:56,460 that goes on forever that's the same every lattice vector. 1019 00:47:56,460 --> 00:47:58,030 It's just the same thing. 1020 00:47:58,030 --> 00:48:00,390 This is a pure, perfect crystal. 1021 00:48:00,390 --> 00:48:03,480 There are no defects. 1022 00:48:03,480 --> 00:48:07,200 It's completely unrealistic, because, as you all know, 1023 00:48:07,200 --> 00:48:08,820 there are defects in reality. 1024 00:48:08,820 --> 00:48:11,382 But we can put those into the simulation as well. 1025 00:48:11,382 --> 00:48:13,215 But for this derivation, there's no defects. 1026 00:48:13,215 --> 00:48:15,640 So you have infinite crystal. 1027 00:48:15,640 --> 00:48:18,780 And what that means is that, everywhere, the properties 1028 00:48:18,780 --> 00:48:19,600 should be the same. 1029 00:48:19,600 --> 00:48:22,300 If I sit on this atom or this atom or in between these two 1030 00:48:22,300 --> 00:48:24,175 and then in between those two, the properties 1031 00:48:24,175 --> 00:48:27,000 should be the same, if I go by a lattice vector. 1032 00:48:27,000 --> 00:48:28,230 Here's the kicker, though. 1033 00:48:28,230 --> 00:48:30,720 The wave function is not the same. 1034 00:48:33,770 --> 00:48:35,150 And that's really the key. 1035 00:48:35,150 --> 00:48:38,030 The wave function is not the same. 1036 00:48:38,030 --> 00:48:39,800 It's almost the same, but it's not. 1037 00:48:42,320 --> 00:48:43,430 It's periodic. 1038 00:48:43,430 --> 00:48:45,320 The wave function is periodic but only 1039 00:48:45,320 --> 00:48:49,160 when it's multiplied by a phase factor. 1040 00:48:49,160 --> 00:48:50,710 And that is Bloch's Theorem. 1041 00:48:50,710 --> 00:48:52,460 And there are three or four different ways 1042 00:48:52,460 --> 00:48:53,450 that it can be derived. 1043 00:48:53,450 --> 00:48:55,670 I will post them onto the Stellar website. 1044 00:48:55,670 --> 00:48:59,150 They're not something you need to know for quiz or homework. 1045 00:48:59,150 --> 00:49:00,650 But that is Bloch's Theorem. 1046 00:49:00,650 --> 00:49:04,280 It's that the wave function in a infinitely repeating 1047 00:49:04,280 --> 00:49:12,800 periodic crystal is going to equal some function that 1048 00:49:12,800 --> 00:49:16,450 is periodic, some function that's 1049 00:49:16,450 --> 00:49:19,810 periodic that has the periodicity of the lattice, 1050 00:49:19,810 --> 00:49:23,620 some function, u, times the phase. 1051 00:49:23,620 --> 00:49:25,570 And that, this is that function u. 1052 00:49:25,570 --> 00:49:27,430 You see, it's periodic. 1053 00:49:27,430 --> 00:49:32,900 It equals it, if you go plus big R, plus one lattice vector. 1054 00:49:32,900 --> 00:49:34,817 So if you move over by a whole-- 1055 00:49:34,817 --> 00:49:36,400 when I see lattice vector, by the way, 1056 00:49:36,400 --> 00:49:37,780 let's bring it to reality. 1057 00:49:37,780 --> 00:49:41,860 I just mean if silicon were cubic, 1058 00:49:41,860 --> 00:49:44,790 I'd just be going from one silicon atom to another. 1059 00:49:44,790 --> 00:49:45,790 That's a lattice vector. 1060 00:49:50,230 --> 00:49:54,570 And simple cubic, I'm thinking. 1061 00:49:54,570 --> 00:49:57,600 And so you have a function that also 1062 00:49:57,600 --> 00:50:00,980 is the same in that lattice distance, 1063 00:50:00,980 --> 00:50:04,040 as you go from, say, one atom to another in the simplest case. 1064 00:50:04,040 --> 00:50:06,140 But the wave function is that times the phase. 1065 00:50:06,140 --> 00:50:07,940 And that's really important, because that 1066 00:50:07,940 --> 00:50:10,620 brings in a new variable. 1067 00:50:10,620 --> 00:50:12,980 In fact, this is a new quantum number. 1068 00:50:12,980 --> 00:50:14,480 That brings in a new quantum number. 1069 00:50:18,250 --> 00:50:22,840 This is a quantum number, k, that lives in inverse space. 1070 00:50:27,950 --> 00:50:38,500 So Bloch's Theorem says that, if I go from one point in space 1071 00:50:38,500 --> 00:50:42,220 to another point that's translated by just a lattice 1072 00:50:42,220 --> 00:50:48,430 factor, then I equal my original wave function time this phase. 1073 00:50:48,430 --> 00:50:49,750 And you can show-- 1074 00:50:49,750 --> 00:50:51,280 I mean, this is actually trivial, 1075 00:50:51,280 --> 00:50:53,360 because this is a complex number. 1076 00:50:53,360 --> 00:50:55,570 So when you square it, you can show 1077 00:50:55,570 --> 00:50:58,390 that the squares are the same, which is really important. 1078 00:50:58,390 --> 00:51:02,290 The charge density in the lattice is periodic. 1079 00:51:02,290 --> 00:51:04,930 Does everybody see that? 1080 00:51:04,930 --> 00:51:07,150 The charge density in a crystal should be periodic. 1081 00:51:07,150 --> 00:51:09,880 Between any two silicon atoms, the bond 1082 00:51:09,880 --> 00:51:14,510 is the same for as long as you can see, 10 to 23rd bonds. 1083 00:51:14,510 --> 00:51:16,580 It's a whole lot of sameness. 1084 00:51:16,580 --> 00:51:18,860 And you can see that the equations show that it is. 1085 00:51:18,860 --> 00:51:23,960 But the wave function can vary by this phase. 1086 00:51:23,960 --> 00:51:29,630 You can also show that if psi k, this thing that I'm 1087 00:51:29,630 --> 00:51:33,530 going to talk about more, is a solution to the Schrodinger 1088 00:51:33,530 --> 00:51:37,913 equation, then psi k plus any reciprocal-- 1089 00:51:37,913 --> 00:51:39,080 this is a reciprocal number. 1090 00:51:39,080 --> 00:51:40,370 Remember, this is a number. 1091 00:51:40,370 --> 00:51:44,570 k is a vector in reciprocal space. 1092 00:51:44,570 --> 00:51:46,610 That's the Bloch's Theorem. 1093 00:51:46,610 --> 00:51:50,450 That's a reciprocal vector that lives in reciprocal space 1094 00:51:50,450 --> 00:51:54,860 dotted with the lattice vector. 1095 00:51:54,860 --> 00:51:58,610 And if I add the reciprocal lattice vector to it, 1096 00:51:58,610 --> 00:52:00,630 then that also must be a solution. 1097 00:52:00,630 --> 00:52:03,470 So that's also another consequence of Bloch's Theorem. 1098 00:52:03,470 --> 00:52:06,215 So here's the-- 1099 00:52:06,215 --> 00:52:08,480 I know some of you might not quite 1100 00:52:08,480 --> 00:52:10,580 be feeling your oneness with this. 1101 00:52:10,580 --> 00:52:12,020 That's fine. 1102 00:52:12,020 --> 00:52:15,830 We'll talk about this a little bit more next week. 1103 00:52:15,830 --> 00:52:20,480 I don't need you to go into great fundamental depth 1104 00:52:20,480 --> 00:52:23,030 in Bloch's Theorem and how it's derived. 1105 00:52:23,030 --> 00:52:25,490 And I'll show you, you can basically-- 1106 00:52:25,490 --> 00:52:33,200 because proper derivations require some more knowledge 1107 00:52:33,200 --> 00:52:34,950 of quantum mechanics. 1108 00:52:34,950 --> 00:52:38,590 And I'm trying to avoid that. 1109 00:52:38,590 --> 00:52:40,950 But basically, one of the easiest ways to look at it 1110 00:52:40,950 --> 00:52:44,910 mathematically is that the translation vector 1111 00:52:44,910 --> 00:52:48,150 associated with the lattice commutes with the Hamiltonian. 1112 00:52:48,150 --> 00:52:51,060 Anybody know what that means? 1113 00:52:51,060 --> 00:52:51,560 Yeah? 1114 00:52:51,560 --> 00:52:55,027 AUDIENCE: Is that like how we take ab equals ba? 1115 00:52:55,027 --> 00:52:56,360 JEFFREY GROSSMAN: Yeah, kind of. 1116 00:52:56,360 --> 00:52:59,840 Yeah, so you can switch the order of the operations. 1117 00:52:59,840 --> 00:53:01,670 But that's, again, not-- 1118 00:53:01,670 --> 00:53:03,590 we don't need to know that. 1119 00:53:03,590 --> 00:53:05,270 What we need to know is this. 1120 00:53:05,270 --> 00:53:07,160 This is what we need to know. 1121 00:53:07,160 --> 00:53:09,300 This is the punchline of it all. 1122 00:53:09,300 --> 00:53:12,860 So we've gone through some derivation. 1123 00:53:12,860 --> 00:53:15,005 We won't do more much more derivation. 1124 00:53:15,005 --> 00:53:16,130 But this is the punch line. 1125 00:53:16,130 --> 00:53:17,838 This is what you got to take out of this, 1126 00:53:17,838 --> 00:53:20,900 so please do take this away. 1127 00:53:20,900 --> 00:53:25,310 When we had hydrogen atoms, when we solved the Schrodinger 1128 00:53:25,310 --> 00:53:29,030 equation for a hydrogen atom, we saw 1129 00:53:29,030 --> 00:53:32,550 that the way we did it was we imposed spherical symmetry. 1130 00:53:32,550 --> 00:53:34,070 So we had a symmetry. 1131 00:53:34,070 --> 00:53:35,570 We did have a symmetry. 1132 00:53:35,570 --> 00:53:39,200 It wasn't a crystal symmetry, which I showed you, 1133 00:53:39,200 --> 00:53:40,340 14 Bravais lattices. 1134 00:53:40,340 --> 00:53:42,200 But it was a symmetry, spherical. 1135 00:53:42,200 --> 00:53:46,550 And when I imposed that, we saw that there are-- these quantum 1136 00:53:46,550 --> 00:53:48,650 numbers came out, remember? 1137 00:53:48,650 --> 00:53:49,350 We did that. 1138 00:53:49,350 --> 00:53:50,930 And that was so cool, because it just 1139 00:53:50,930 --> 00:53:54,140 solved so many of the issues of the time, 1140 00:53:54,140 --> 00:53:56,780 just the fact that electrons were quantized 1141 00:53:56,780 --> 00:53:58,400 was so important. 1142 00:53:58,400 --> 00:53:59,180 All right. 1143 00:53:59,180 --> 00:54:03,070 Now, we have a new kind of symmetry. 1144 00:54:03,070 --> 00:54:05,185 We have something called translational symmetry. 1145 00:54:05,185 --> 00:54:07,060 We're still solving the Schrodinger equation, 1146 00:54:07,060 --> 00:54:08,590 but now it's a periodic solid. 1147 00:54:08,590 --> 00:54:11,320 And when this kind of symmetry is in the game, 1148 00:54:11,320 --> 00:54:15,680 we have a different kind of quantum number. 1149 00:54:15,680 --> 00:54:16,970 That's what matters most here. 1150 00:54:16,970 --> 00:54:18,800 We have a different kind of quantum number. 1151 00:54:18,800 --> 00:54:25,340 We have a quantum number that is a reciprocal lattice vector. 1152 00:54:25,340 --> 00:54:28,850 It's a vector in reciprocal space. 1153 00:54:28,850 --> 00:54:31,560 That's the key. 1154 00:54:31,560 --> 00:54:33,260 And because we have-- you see, just 1155 00:54:33,260 --> 00:54:37,100 like-- so these l's and m's had their rules. 1156 00:54:37,100 --> 00:54:39,860 This could be plus or minus that and so forth and so on. 1157 00:54:39,860 --> 00:54:41,210 And so you could-- 1158 00:54:41,210 --> 00:54:44,960 they lived within I don't know. 1159 00:54:44,960 --> 00:54:47,180 They lived within in the counting 1160 00:54:47,180 --> 00:54:50,610 of the principal quantum number. 1161 00:54:50,610 --> 00:54:52,150 Now, we have a different situation. 1162 00:54:52,150 --> 00:54:55,260 We have a quantum number that can vary. 1163 00:54:55,260 --> 00:54:57,330 It doesn't have to just be 0, 1. 1164 00:54:57,330 --> 00:55:00,750 Or it's not necessarily tied to the principle quantum number. 1165 00:55:00,750 --> 00:55:02,040 But it can vary. 1166 00:55:02,040 --> 00:55:07,170 And its domain of varying is that Brillouin zone. 1167 00:55:07,170 --> 00:55:07,830 That's it. 1168 00:55:07,830 --> 00:55:09,900 So that's where it can play. 1169 00:55:09,900 --> 00:55:14,717 These guys have variations dictated 1170 00:55:14,717 --> 00:55:16,050 by the principal quantum number. 1171 00:55:16,050 --> 00:55:19,660 This has variations dictated by the Brillouin zone. 1172 00:55:19,660 --> 00:55:23,480 That's really the key here, the Brillouin zone, 1173 00:55:23,480 --> 00:55:26,805 which comes about from reciprocal space. 1174 00:55:26,805 --> 00:55:27,680 Is everybody with me? 1175 00:55:34,840 --> 00:55:37,500 Well, here's just a drawing. 1176 00:55:37,500 --> 00:55:38,740 I don't even know where-- 1177 00:55:38,740 --> 00:55:40,840 I should really have a reference here. 1178 00:55:40,840 --> 00:55:44,428 This is one of my Google image searches. 1179 00:55:44,428 --> 00:55:45,220 And there are many. 1180 00:55:45,220 --> 00:55:47,140 And you can look at drawings online. 1181 00:55:47,140 --> 00:55:51,970 But if you have a very simple wave function-- 1182 00:55:51,970 --> 00:55:53,680 that's the real and imaginary parts-- 1183 00:55:53,680 --> 00:55:56,800 and you have some periodically repeating function-- 1184 00:55:56,800 --> 00:56:00,160 so that's my psi, and that's my periodic function-- 1185 00:56:00,160 --> 00:56:02,830 then you can get a sense of this variation 1186 00:56:02,830 --> 00:56:04,750 that I'm talking about. 1187 00:56:04,750 --> 00:56:08,390 This is a very simple one-dimensional example. 1188 00:56:08,390 --> 00:56:11,120 So I have a one-dimensional wave function. 1189 00:56:11,120 --> 00:56:13,830 And I have a periodically repeating function. 1190 00:56:13,830 --> 00:56:15,830 And remember, what Bloch's Theorem says 1191 00:56:15,830 --> 00:56:18,350 is that that is now how I'm going 1192 00:56:18,350 --> 00:56:20,480 to write a solution to the Schrodinger equation. 1193 00:56:20,480 --> 00:56:26,210 It's going to be a combination of a plane wave like that, 1194 00:56:26,210 --> 00:56:28,440 where k is in the reciprocal space, 1195 00:56:28,440 --> 00:56:30,440 and some periodically repeating function. 1196 00:56:30,440 --> 00:56:36,110 And just to illustrate that you can get variation now, check 1197 00:56:36,110 --> 00:56:40,310 out what happens when I change k from, say, something not 1198 00:56:40,310 --> 00:56:43,560 equal to 0 to something like pi over a, 1199 00:56:43,560 --> 00:56:46,850 you can see that these functions change dramatically. 1200 00:56:46,850 --> 00:56:52,010 The wave function of the crystal changes. 1201 00:56:52,010 --> 00:56:52,850 Everybody see that? 1202 00:56:55,820 --> 00:56:57,893 So that's really important. 1203 00:56:57,893 --> 00:56:59,810 And the reason that's important and the reason 1204 00:56:59,810 --> 00:57:03,050 this is all going to come home for us as we move on 1205 00:57:03,050 --> 00:57:09,800 to seeing why this all matters is that that means you have 1206 00:57:09,800 --> 00:57:16,380 energies that also depend on k. 1207 00:57:16,380 --> 00:57:22,580 The energies also depend on this variation in reciprocal space. 1208 00:57:22,580 --> 00:57:23,420 All right. 1209 00:57:23,420 --> 00:57:25,230 So now, if I'm in reciprocal space, 1210 00:57:25,230 --> 00:57:27,380 and I move from the origin out to the edge 1211 00:57:27,380 --> 00:57:30,410 of the Brillouin zone, not only am I changing my function, 1212 00:57:30,410 --> 00:57:32,210 I'm changing this. 1213 00:57:32,210 --> 00:57:38,360 I'm changing-- the position of this level can vary now. 1214 00:57:38,360 --> 00:57:39,260 It now can vary. 1215 00:57:39,260 --> 00:57:42,770 It can move up or down. 1216 00:57:42,770 --> 00:57:49,535 And that's the key result, the one that's 1217 00:57:49,535 --> 00:57:50,660 going to matter most to us. 1218 00:57:57,070 --> 00:58:00,870 Is everybody-- any questions? 1219 00:58:00,870 --> 00:58:01,563 This is-- yeah? 1220 00:58:01,563 --> 00:58:03,230 AUDIENCE: I was just wondering, is the k 1221 00:58:03,230 --> 00:58:04,688 right here that we're dealing with, 1222 00:58:04,688 --> 00:58:06,865 is that the same as the crystal momentum? 1223 00:58:06,865 --> 00:58:08,240 JEFFREY GROSSMAN: Very good, yes, 1224 00:58:08,240 --> 00:58:12,860 the crystal momentum would be that times h bar, yes, 1225 00:58:12,860 --> 00:58:17,220 which is like momentum, kind of, but not really. 1226 00:58:17,220 --> 00:58:19,032 But it is, sort of. 1227 00:58:19,032 --> 00:58:19,740 AUDIENCE: So it-- 1228 00:58:19,740 --> 00:58:23,745 JEFFREY GROSSMAN: Quasi is a good word, momentum. 1229 00:58:23,745 --> 00:58:26,872 AUDIENCE: So it it an inverse lattice vector or-- 1230 00:58:26,872 --> 00:58:27,830 JEFFREY GROSSMAN: Yeah. 1231 00:58:27,830 --> 00:58:29,606 AUDIENCE: [INAUDIBLE] 1232 00:58:29,606 --> 00:58:31,790 JEFFREY GROSSMAN: Ah, well, that is exactly it, 1233 00:58:31,790 --> 00:58:34,740 yeah, yeah, yeah. 1234 00:58:37,470 --> 00:58:44,100 And you can get a feel for-- 1235 00:58:44,100 --> 00:58:50,970 as you vary k, so then things happen, like the momentum 1236 00:58:50,970 --> 00:58:53,970 can change. 1237 00:58:53,970 --> 00:59:00,140 And that means the velocity can change. 1238 00:59:00,140 --> 00:59:02,330 And if the velocity can change-- 1239 00:59:02,330 --> 00:59:06,290 well, actually, the mass can change as well. 1240 00:59:06,290 --> 00:59:08,390 And if those things can change-- 1241 00:59:08,390 --> 00:59:09,890 of course, it's not really changing. 1242 00:59:09,890 --> 00:59:12,020 The mass of the electron isn't really changing. 1243 00:59:12,020 --> 00:59:16,910 But the effective mass, the mass that it feels in the crystal, 1244 00:59:16,910 --> 00:59:18,460 is changing. 1245 00:59:18,460 --> 00:59:19,940 And, boy, is that important. 1246 00:59:19,940 --> 00:59:24,910 And we'll talk about that, because, from the way that-- 1247 00:59:24,910 --> 00:59:27,670 here's my levels in an atom or molecule. 1248 00:59:27,670 --> 00:59:30,340 And now, remember, I said that they do this. 1249 00:59:30,340 --> 00:59:35,420 From the way they do this, how heavy an electron feels itself 1250 00:59:35,420 --> 00:59:38,930 to be in the crystal varies. 1251 00:59:38,930 --> 00:59:44,830 It has an effective mass that's dependent on the wiggles here. 1252 00:59:44,830 --> 00:59:47,610 And because of that, that's directly related 1253 00:59:47,610 --> 00:59:51,090 to how fast an electron can cruise through the crystal. 1254 00:59:51,090 --> 00:59:53,620 It's directly related to the mobility. 1255 00:59:53,620 --> 00:59:55,530 So the transport properties are deeply 1256 00:59:55,530 --> 00:59:58,110 dependent on this wiggling. 1257 01:00:01,050 --> 01:00:03,000 So Bloch's Theorem is a formal theorem 1258 01:00:03,000 --> 01:00:06,060 that tells you that you have to write wave functions in such 1259 01:00:06,060 --> 01:00:10,050 a way that you have a dependence on the inverse lattice. 1260 01:00:10,050 --> 01:00:16,680 And then how that dependence enters into a system 1261 01:00:16,680 --> 01:00:19,050 or a material is very complex. 1262 01:00:19,050 --> 01:00:21,300 And it depends on solving the Schrodinger equation 1263 01:00:21,300 --> 01:00:22,008 for the material. 1264 01:00:26,180 --> 01:00:30,020 But you see, you can think about it like this, if you want. 1265 01:00:30,020 --> 01:00:31,730 Here's my inverse space. 1266 01:00:31,730 --> 01:00:36,030 And I'm going to change where I am in reciprocal space. 1267 01:00:36,030 --> 01:00:38,180 Remember, again, I only have reciprocal space 1268 01:00:38,180 --> 01:00:39,870 because I have a crystal. 1269 01:00:39,870 --> 01:00:41,660 I have a lattice. 1270 01:00:41,660 --> 01:00:43,160 So I have an inverse space. 1271 01:00:43,160 --> 01:00:45,380 And if I change where I am in inverse space, 1272 01:00:45,380 --> 01:00:48,620 these would be my new energy levels. 1273 01:00:48,620 --> 01:00:50,880 You get a new set of energy levels, 1274 01:00:50,880 --> 01:00:55,160 those e's that I just showed you, for every point in space, 1275 01:00:55,160 --> 01:00:58,050 in k-space, every point in k-space. 1276 01:00:58,050 --> 01:01:00,870 So now, this is just blown up. 1277 01:01:00,870 --> 01:01:05,440 It used to be the good old days, last Thursday. 1278 01:01:05,440 --> 01:01:07,120 We just had these levels. 1279 01:01:07,120 --> 01:01:10,330 We just got through talking about how you get these from-- 1280 01:01:10,330 --> 01:01:12,490 you can build up two atoms, three atoms. 1281 01:01:12,490 --> 01:01:13,780 And you get these levels. 1282 01:01:13,780 --> 01:01:15,140 And they didn't go anywhere. 1283 01:01:15,140 --> 01:01:17,890 And now, they've blown up, because anywhere you go 1284 01:01:17,890 --> 01:01:21,270 in this reciprocal space, they can change. 1285 01:01:21,270 --> 01:01:23,030 They can change. 1286 01:01:23,030 --> 01:01:26,170 So now, you need to compute those same levels 1287 01:01:26,170 --> 01:01:30,310 in the computer for different points in k-space. 1288 01:01:30,310 --> 01:01:33,670 And that's what a computer does when it solves for a solid. 1289 01:01:33,670 --> 01:01:34,720 That's what it does. 1290 01:01:34,720 --> 01:01:37,390 That's basically the difference, is you now 1291 01:01:37,390 --> 01:01:41,680 have an extra degree of freedom that you didn't have 1292 01:01:41,680 --> 01:01:46,330 in your atoms and molecules or an extra quantum number 1293 01:01:46,330 --> 01:01:49,830 that you need to explore, k. 1294 01:01:49,830 --> 01:01:51,390 So what are my levels at 0? 1295 01:01:51,390 --> 01:01:52,650 Well, there are these. 1296 01:01:52,650 --> 01:01:53,440 I don't know. 1297 01:01:53,440 --> 01:01:54,480 Well, they're not here. 1298 01:01:54,480 --> 01:01:55,320 What are my levels here? 1299 01:01:55,320 --> 01:01:56,070 They look like this. 1300 01:01:56,070 --> 01:01:57,070 Here, they look at this. 1301 01:01:57,070 --> 01:01:58,740 And that's how the computer does it. 1302 01:01:58,740 --> 01:02:00,240 It computes the same thing. 1303 01:02:00,240 --> 01:02:02,970 It computes the energy levels, fills them up. 1304 01:02:02,970 --> 01:02:05,850 But at each point in k-space, they can be different. 1305 01:02:05,850 --> 01:02:08,850 And this is exactly that difference. 1306 01:02:08,850 --> 01:02:10,650 You change where you are in k-space. 1307 01:02:10,650 --> 01:02:11,460 And look at that. 1308 01:02:11,460 --> 01:02:13,680 This level starts going down. 1309 01:02:13,680 --> 01:02:15,650 This level goes up. 1310 01:02:15,650 --> 01:02:18,040 I don't know what's going on here. 1311 01:02:18,040 --> 01:02:19,270 Maybe that's this one. 1312 01:02:19,270 --> 01:02:20,230 That one's going up. 1313 01:02:20,230 --> 01:02:24,360 All kinds of things can happen as you cruise around 1314 01:02:24,360 --> 01:02:27,890 phase space, in k-space. 1315 01:02:27,890 --> 01:02:30,420 Those energy levels will change. 1316 01:02:30,420 --> 01:02:32,947 And that is what leads to a band structure. 1317 01:02:32,947 --> 01:02:34,280 That's what a band structure is. 1318 01:02:41,000 --> 01:02:44,960 It's just a continuous change of energy levels 1319 01:02:44,960 --> 01:02:46,220 as a function of k-space. 1320 01:02:49,650 --> 01:02:54,690 And each of these bands is still just an energy level. 1321 01:02:54,690 --> 01:02:57,150 So how many electrons go into it? 1322 01:02:57,150 --> 01:02:59,210 Still just an energy level, it's just 1323 01:02:59,210 --> 01:03:02,600 a weird one that wiggles depending on where you 1324 01:03:02,600 --> 01:03:04,310 are in the reciprocal lattice. 1325 01:03:04,310 --> 01:03:06,590 But how many electrons do I put into this band 1326 01:03:06,590 --> 01:03:09,310 as I fill them up? 1327 01:03:09,310 --> 01:03:13,510 Two, I still just put two electrons in here. 1328 01:03:13,510 --> 01:03:15,160 And then I put two electrons in here. 1329 01:03:15,160 --> 01:03:16,577 And then I put two electrons here. 1330 01:03:16,577 --> 01:03:17,500 Oh, but look at that. 1331 01:03:17,500 --> 01:03:19,420 It was actually two bands. 1332 01:03:19,420 --> 01:03:21,350 That's tricky. 1333 01:03:21,350 --> 01:03:23,890 So you get these really interesting behaviors, 1334 01:03:23,890 --> 01:03:26,320 where bands can merge and become degenerate. 1335 01:03:26,320 --> 01:03:30,880 Ah, but that's not really that different, is it, 1336 01:03:30,880 --> 01:03:33,040 then the idea that you could have 1337 01:03:33,040 --> 01:03:39,700 states like this, where I had-- this is just the atom, 1S 2S, 1338 01:03:39,700 --> 01:03:41,200 2P. 1339 01:03:41,200 --> 01:03:46,360 I had three energy bands that had the same energy. 1340 01:03:46,360 --> 01:03:47,980 That's called degeneracy. 1341 01:03:47,980 --> 01:03:50,080 Here it is in the hydrogen atom. 1342 01:03:50,080 --> 01:03:52,360 There it is in a very complex crystal. 1343 01:03:52,360 --> 01:03:55,490 Well, it's actually a simple crystal. 1344 01:03:55,490 --> 01:03:57,470 So you have degeneracies, where-- 1345 01:03:57,470 --> 01:04:01,190 these would be like two levels that are the same energy. 1346 01:04:01,190 --> 01:04:03,860 But now, I'm cruising around in k-space. 1347 01:04:03,860 --> 01:04:07,100 And I get to a part of k-space where they no longer 1348 01:04:07,100 --> 01:04:07,680 are the same. 1349 01:04:07,680 --> 01:04:08,720 They actually split. 1350 01:04:08,720 --> 01:04:09,650 This one goes down. 1351 01:04:09,650 --> 01:04:13,100 And these two go up, let's say. 1352 01:04:13,100 --> 01:04:16,260 And so their degeneracy splits. 1353 01:04:16,260 --> 01:04:19,740 And you see that all the time in band structures. 1354 01:04:19,740 --> 01:04:21,930 But there are a certain number of total levels here. 1355 01:04:21,930 --> 01:04:23,347 And you fill them up, just like we 1356 01:04:23,347 --> 01:04:26,190 did for an atom or a molecule. 1357 01:04:26,190 --> 01:04:30,570 Now, somebody tell me what-- so that's a band structure. 1358 01:04:30,570 --> 01:04:35,090 We'll talk about the band structure in all of its glory 1359 01:04:35,090 --> 01:04:35,810 next week. 1360 01:04:35,810 --> 01:04:36,980 But we're not done. 1361 01:04:36,980 --> 01:04:41,380 Somebody tell me-- by the way, this is a beautiful thing. 1362 01:04:41,380 --> 01:04:42,230 Are we feeling that? 1363 01:04:44,970 --> 01:04:47,250 I mean, is it a little emotional moment for you 1364 01:04:47,250 --> 01:04:49,170 guys, a little bit? 1365 01:04:49,170 --> 01:04:50,160 Yeah? 1366 01:04:50,160 --> 01:04:53,100 I felt that. 1367 01:04:53,100 --> 01:04:55,710 There's so much in the band structure. 1368 01:04:55,710 --> 01:04:59,310 There is so much material science, so much 1369 01:04:59,310 --> 01:05:01,570 physics and chemistry, in this band structure, 1370 01:05:01,570 --> 01:05:04,170 so much to learn about a material, just from this. 1371 01:05:09,690 --> 01:05:11,940 So we still have occupied and unoccupied. 1372 01:05:14,907 --> 01:05:16,240 That wasn't supposed to do that. 1373 01:05:20,600 --> 01:05:22,500 Oh, why is it doing that? 1374 01:05:22,500 --> 01:05:24,055 OK, there we go. 1375 01:05:24,055 --> 01:05:26,180 So you still-- that's what I was saying-- you still 1376 01:05:26,180 --> 01:05:27,560 will fill them up. 1377 01:05:27,560 --> 01:05:28,880 And then you'll have-- 1378 01:05:28,880 --> 01:05:29,950 what is that? 1379 01:05:29,950 --> 01:05:30,700 AUDIENCE: Bandgap. 1380 01:05:30,700 --> 01:05:33,130 JEFFREY GROSSMAN: That's the bandgap. 1381 01:05:33,130 --> 01:05:36,470 But what's interesting about this bandgap? 1382 01:05:36,470 --> 01:05:36,970 Yes? 1383 01:05:36,970 --> 01:05:37,970 AUDIENCE: It's indirect. 1384 01:05:37,970 --> 01:05:39,670 JEFFREY GROSSMAN: It's indirect. 1385 01:05:39,670 --> 01:05:44,500 And now, that's what indirect means, you see. 1386 01:05:44,500 --> 01:05:48,320 Can I get an indirect-- remember, we had the molecules. 1387 01:05:48,320 --> 01:05:49,980 We had the molecules. 1388 01:05:52,860 --> 01:05:54,400 This is a molecule. 1389 01:05:54,400 --> 01:05:56,890 Those are my energy levels of a molecule. 1390 01:05:56,890 --> 01:05:57,810 And this was the gap. 1391 01:06:01,680 --> 01:06:04,592 Can I have an indirect bandgap in a molecule? 1392 01:06:08,450 --> 01:06:11,024 What do you think? 1393 01:06:11,024 --> 01:06:13,940 AUDIENCE: If it's a molecule of silicon. 1394 01:06:13,940 --> 01:06:17,260 JEFFREY GROSSMAN: Well, even if it's a molecule silicon, 1395 01:06:17,260 --> 01:06:18,925 can I have an indirect bandgap? 1396 01:06:22,250 --> 01:06:24,149 Why not? 1397 01:06:24,149 --> 01:06:25,570 AUDIENCE: It's not periodic. 1398 01:06:25,570 --> 01:06:26,470 JEFFREY GROSSMAN: It's not periodic. 1399 01:06:26,470 --> 01:06:27,010 Therefore? 1400 01:06:27,010 --> 01:06:28,730 AUDIENCE: [INAUDIBLE] 1401 01:06:28,730 --> 01:06:31,550 JEFFREY GROSSMAN: Yeah, there is nothing to vary. 1402 01:06:31,550 --> 01:06:35,830 K-space does nothing for me here. 1403 01:06:38,910 --> 01:06:39,720 Yeah? 1404 01:06:39,720 --> 01:06:42,012 AUDIENCE: Can you explain what an indirect bandgap is? 1405 01:06:42,012 --> 01:06:43,720 JEFFREY GROSSMAN: Yeah, yeah, absolutely. 1406 01:06:43,720 --> 01:06:45,137 And I'm going to talk about a lot. 1407 01:06:45,137 --> 01:06:46,940 Yeah, sorry, I jumped a little ahead. 1408 01:06:46,940 --> 01:06:48,220 I get a little excited. 1409 01:06:48,220 --> 01:06:50,510 Indirect bandgap is such a cool thing. 1410 01:06:50,510 --> 01:06:54,560 And it's such a pain point in silicon. 1411 01:06:54,560 --> 01:06:56,660 You see, if I want-- 1412 01:06:56,660 --> 01:06:58,250 what you're going to do in homework 2, 1413 01:06:58,250 --> 01:07:00,620 and what we talked about last Thursday, is that, 1414 01:07:00,620 --> 01:07:04,190 if I shine light onto this material-- remember, 1415 01:07:04,190 --> 01:07:06,020 these are all occupied-- 1416 01:07:06,020 --> 01:07:09,260 what I can do is kick an electron up in energy. 1417 01:07:09,260 --> 01:07:11,210 That's a photoexcitation. 1418 01:07:11,210 --> 01:07:13,010 And that's what allows me to store energy 1419 01:07:13,010 --> 01:07:14,510 from the sun in the molecules you're 1420 01:07:14,510 --> 01:07:18,180 going to work on in your next problem set. 1421 01:07:18,180 --> 01:07:21,630 But you see, that's just a direct bandgap. 1422 01:07:21,630 --> 01:07:22,500 Why is it direct? 1423 01:07:22,500 --> 01:07:25,620 Well, because it's just the bandgap. 1424 01:07:25,620 --> 01:07:27,540 In a molecule, there's no direct or indirect. 1425 01:07:27,540 --> 01:07:29,542 But this is direct, because you just 1426 01:07:29,542 --> 01:07:30,750 have the energy from the sun. 1427 01:07:30,750 --> 01:07:34,260 And that's the energy that it needs to go up. 1428 01:07:34,260 --> 01:07:40,280 Now, when you have k-space, things are different. 1429 01:07:40,280 --> 01:07:44,000 Because now, these things, these energies, they wiggle. 1430 01:07:48,200 --> 01:07:50,800 And here's the kicker. 1431 01:07:50,800 --> 01:07:55,090 You see, what's the gap now of this material? 1432 01:07:55,090 --> 01:07:57,757 Well, here, they wiggle. 1433 01:07:57,757 --> 01:07:59,090 Don't look at that for a second. 1434 01:07:59,090 --> 01:08:00,970 Just look at this. 1435 01:08:00,970 --> 01:08:02,590 I now don't have a molecule. 1436 01:08:02,590 --> 01:08:06,280 I have a periodically repeating solid. 1437 01:08:06,280 --> 01:08:10,510 And therefore, I get these wiggles of each energy band. 1438 01:08:10,510 --> 01:08:12,190 And now, I shine light on it. 1439 01:08:12,190 --> 01:08:13,480 Where does this electron go? 1440 01:08:13,480 --> 01:08:16,180 Is this the gap of the material? 1441 01:08:16,180 --> 01:08:17,109 That's the key. 1442 01:08:17,109 --> 01:08:17,979 Is that the gap? 1443 01:08:22,240 --> 01:08:26,170 Well, it's the gap at this point in k-space. 1444 01:08:26,170 --> 01:08:28,240 But here's the real important part. 1445 01:08:28,240 --> 01:08:32,700 There is a lower energy gap in this material. 1446 01:08:32,700 --> 01:08:36,479 If I'm able to go from this point k-space 1447 01:08:36,479 --> 01:08:40,319 to this point in k-space, then the gap is only this big. 1448 01:08:46,109 --> 01:08:49,580 This would be called a direct gap. 1449 01:08:49,580 --> 01:08:52,220 And this would be called an indirect gap. 1450 01:08:58,189 --> 01:09:04,893 And that is why silicon solar cells are expensive. 1451 01:09:04,893 --> 01:09:06,310 We will talk about that next week. 1452 01:09:06,310 --> 01:09:10,420 But that's why silicon solar cells are expensive. 1453 01:09:10,420 --> 01:09:18,880 You see, when you have to go to a different part of k-space 1454 01:09:18,880 --> 01:09:23,750 to create your excitation, well, you need help. 1455 01:09:23,750 --> 01:09:25,880 You can't do it with just light. 1456 01:09:25,880 --> 01:09:27,560 You need some assistance. 1457 01:09:27,560 --> 01:09:29,729 You've got to call it in. 1458 01:09:29,729 --> 01:09:33,017 And that means it's a lot harder to do 1459 01:09:33,017 --> 01:09:34,809 and, therefore, a very inefficient process. 1460 01:09:37,390 --> 01:09:39,130 And as we talked about last week, 1461 01:09:39,130 --> 01:09:40,600 and as you'll do in your homework, 1462 01:09:40,600 --> 01:09:43,840 you want to absorb as much of that sunlight from the sun 1463 01:09:43,840 --> 01:09:44,460 as you can. 1464 01:09:47,170 --> 01:09:50,050 So if your gap-- 1465 01:09:50,050 --> 01:09:52,698 our next application-- not Tuesday but probably next 1466 01:09:52,698 --> 01:09:54,490 Thursday, we might start to talk about it-- 1467 01:09:54,490 --> 01:09:56,830 will be solar cells. 1468 01:09:56,830 --> 01:09:58,690 And we'll talk about solar cells then. 1469 01:09:58,690 --> 01:10:03,460 But you don't want your bandgap to be too high, because if it's 1470 01:10:03,460 --> 01:10:05,097 really high, remember, you're not 1471 01:10:05,097 --> 01:10:06,430 going to absorb much of the sun. 1472 01:10:10,410 --> 01:10:12,600 Now, it also can't be 0 or close to 0 1473 01:10:12,600 --> 01:10:18,180 for a similar reason of the solar fuels but different. 1474 01:10:18,180 --> 01:10:19,480 But it's related. 1475 01:10:19,480 --> 01:10:21,340 And we'll get to that later. 1476 01:10:21,340 --> 01:10:27,840 But the point is that this gap is very high for a solar cell. 1477 01:10:27,840 --> 01:10:28,830 This is a big gap. 1478 01:10:28,830 --> 01:10:30,990 You want it to have a small gap. 1479 01:10:30,990 --> 01:10:32,280 It does have a small gap. 1480 01:10:32,280 --> 01:10:36,720 Silicon does have a small gap from here to here, 1.1, 1481 01:10:36,720 --> 01:10:37,710 beautiful EV. 1482 01:10:40,430 --> 01:10:43,610 But it needs to call it in. 1483 01:10:43,610 --> 01:10:45,050 It needs help. 1484 01:10:45,050 --> 01:10:47,780 It can't just get that from the light alone. 1485 01:10:47,780 --> 01:10:51,830 And therefore, that's a very low efficiency process and a very 1486 01:10:51,830 --> 01:10:53,520 low probability process. 1487 01:10:53,520 --> 01:10:56,180 So yes, it can absorb light, if it has 1488 01:10:56,180 --> 01:10:58,430 help at those lower energies. 1489 01:10:58,430 --> 01:11:00,620 But it's very infrequent. 1490 01:11:00,620 --> 01:11:03,140 It's very improbable. 1491 01:11:06,080 --> 01:11:10,550 And that indirect bandgap is the reason why silicon is not 1492 01:11:10,550 --> 01:11:12,560 a good optical absorber, does not 1493 01:11:12,560 --> 01:11:15,500 absorb light efficiently in the parts of the spectrum 1494 01:11:15,500 --> 01:11:17,500 that you'd like it to. 1495 01:11:17,500 --> 01:11:19,810 It can absorb light, if you make it thick enough 1496 01:11:19,810 --> 01:11:21,610 in that part of the spectrum. 1497 01:11:21,610 --> 01:11:23,990 But that's not what you want to do. 1498 01:11:23,990 --> 01:11:25,120 But you got to do it. 1499 01:11:25,120 --> 01:11:26,842 So that's the indirect bandgap. 1500 01:11:26,842 --> 01:11:28,300 Now, does everybody see-- but, see, 1501 01:11:28,300 --> 01:11:30,440 it has a direct gap at gamma. 1502 01:11:30,440 --> 01:11:31,440 Does everybody see this? 1503 01:11:31,440 --> 01:11:32,725 It's a really cool important-- 1504 01:11:32,725 --> 01:11:33,872 [PHONE RINGING] 1505 01:11:33,872 --> 01:11:36,590 --result. Tell them you'll call them back 1506 01:11:36,590 --> 01:11:40,730 unless it's about the band structure of silicon, whoever's 1507 01:11:40,730 --> 01:11:43,580 phone that was. 1508 01:11:43,580 --> 01:11:46,790 Now, somebody tell me what would happen 1509 01:11:46,790 --> 01:11:50,400 if I spaced my molecules-- 1510 01:11:50,400 --> 01:11:53,030 this is a silicon solid. 1511 01:11:53,030 --> 01:11:54,280 Now, what would happen-- 1512 01:11:54,280 --> 01:11:57,290 oh, let me, in the last few minutes of class-- 1513 01:11:57,290 --> 01:12:00,020 uh-oh, what's it doing? 1514 01:12:00,020 --> 01:12:01,710 Yeah, that doesn't look good, does it? 1515 01:12:04,360 --> 01:12:07,150 Somebody tell me what would happen 1516 01:12:07,150 --> 01:12:09,130 if I spaced my atoms in the crystal 1517 01:12:09,130 --> 01:12:10,230 further and further apart. 1518 01:12:14,540 --> 01:12:16,700 What's going to happen to that band structure 1519 01:12:16,700 --> 01:12:19,322 if I make them go further and further apart? 1520 01:12:19,322 --> 01:12:20,940 AUDIENCE: You'll smooth it out. 1521 01:12:20,940 --> 01:12:21,380 JEFFREY GROSSMAN: What is that? 1522 01:12:21,380 --> 01:12:22,950 AUDIENCE: You'll smooth it out [INAUDIBLE] 1523 01:12:22,950 --> 01:12:23,820 JEFFREY GROSSMAN: You'll smooth it out. 1524 01:12:23,820 --> 01:12:25,440 What will happen eventually? 1525 01:12:25,440 --> 01:12:26,580 AUDIENCE: [INAUDIBLE] 1526 01:12:26,580 --> 01:12:28,740 JEFFREY GROSSMAN: Yeah, exactly. 1527 01:12:28,740 --> 01:12:34,050 So I can plot the band structure of an atom. 1528 01:12:34,050 --> 01:12:36,700 And what would I get? 1529 01:12:36,700 --> 01:12:38,380 My variation in case based will be? 1530 01:12:41,160 --> 01:12:42,660 What would the band structure-- what 1531 01:12:42,660 --> 01:12:45,880 would the variation in k-space for an atom be, 1532 01:12:45,880 --> 01:12:47,200 just an isolated atom? 1533 01:12:47,200 --> 01:12:48,195 AUDIENCE: Zero. 1534 01:12:48,195 --> 01:12:49,570 JEFFREY GROSSMAN: Zero variation, 1535 01:12:49,570 --> 01:12:51,820 exactly, everything's flat. 1536 01:12:51,820 --> 01:12:53,682 You're the same everywhere in k-space. 1537 01:12:53,682 --> 01:12:56,140 You're just back to the picture that we've been working in. 1538 01:12:59,110 --> 01:12:59,920 So let's just see. 1539 01:13:02,540 --> 01:13:07,180 Now, you'll use this to do molecule calculations 1540 01:13:07,180 --> 01:13:08,150 for your next homework. 1541 01:13:08,150 --> 01:13:10,070 But we'll do some calculations. 1542 01:13:13,530 --> 01:13:15,670 I just want to show one last point. 1543 01:13:15,670 --> 01:13:18,420 And we'll do some calculations also next week. 1544 01:13:18,420 --> 01:13:20,580 Oh, it's loading. 1545 01:13:20,580 --> 01:13:23,610 The nanoHUB toolkit is loading. 1546 01:13:27,290 --> 01:13:29,990 How do I-- do you think-- 1547 01:13:29,990 --> 01:13:31,490 tell me, while this is loading-- 1548 01:13:31,490 --> 01:13:32,900 if it ever finishes-- 1549 01:13:32,900 --> 01:13:38,960 tell me what you think is going to be an additional convergence 1550 01:13:38,960 --> 01:13:40,280 parameter for solids. 1551 01:13:44,380 --> 01:13:47,270 Think about this picture. 1552 01:13:47,270 --> 01:13:50,610 It looks like that may never load. 1553 01:13:50,610 --> 01:13:57,610 Think about this picture here and tell me what you think 1554 01:13:57,610 --> 01:13:59,440 is going to be an important convergence 1555 01:13:59,440 --> 01:14:00,700 parameter for solids. 1556 01:14:00,700 --> 01:14:01,466 Yeah? 1557 01:14:01,466 --> 01:14:03,090 AUDIENCE: [INAUDIBLE] 1558 01:14:03,090 --> 01:14:06,180 JEFFREY GROSSMAN: Yeah, it's exactly right. 1559 01:14:06,180 --> 01:14:08,280 I mean, because now, how am I going 1560 01:14:08,280 --> 01:14:11,730 to know if the variation is correct? 1561 01:14:11,730 --> 01:14:13,860 This variation I'm telling you is extremely 1562 01:14:13,860 --> 01:14:15,360 important for crystals. 1563 01:14:15,360 --> 01:14:17,198 This is what goes to a band structure. 1564 01:14:17,198 --> 01:14:18,990 This is what gives you your band structure. 1565 01:14:18,990 --> 01:14:20,220 Is it right? 1566 01:14:20,220 --> 01:14:23,700 Well, did you do enough of these energy level 1567 01:14:23,700 --> 01:14:27,630 calculations at enough points in k-space? 1568 01:14:27,630 --> 01:14:32,400 The code will now have that as an input, you see. 1569 01:14:32,400 --> 01:14:34,680 Ah-ha, there it is, beautiful. 1570 01:14:38,760 --> 01:14:44,280 I think this is a bust here. 1571 01:14:44,280 --> 01:14:48,150 Ah, oh, this is exciting. 1572 01:14:48,150 --> 01:14:53,240 This reminds me of the days of dial-up modem. 1573 01:14:53,240 --> 01:14:54,272 I just read about them. 1574 01:14:54,272 --> 01:14:55,730 I don't know about them, of course. 1575 01:14:58,290 --> 01:14:58,970 So here you go. 1576 01:14:58,970 --> 01:15:00,590 This is SIESTA. 1577 01:15:00,590 --> 01:15:05,390 It's not letting me scroll down again, beautiful-- 1578 01:15:05,390 --> 01:15:07,190 run JML viewer, solid. 1579 01:15:10,740 --> 01:15:12,320 It's not responding. 1580 01:15:12,320 --> 01:15:13,980 That's also interesting. 1581 01:15:13,980 --> 01:15:17,390 Oh, there's the scrollbar. 1582 01:15:17,390 --> 01:15:20,210 Can I run it? 1583 01:15:20,210 --> 01:15:24,145 It's really not-- yeah, I can't even click this. 1584 01:15:24,145 --> 01:15:25,300 Ah-ha, oh, yeah-- 1585 01:15:25,300 --> 01:15:26,710 AUDIENCE: [INAUDIBLE] 1586 01:15:26,710 --> 01:15:28,158 JEFFREY GROSSMAN: Right. 1587 01:15:28,158 --> 01:15:29,950 That's the first time that's ever happened. 1588 01:15:29,950 --> 01:15:31,450 AUDIENCE: I got that last night. 1589 01:15:31,450 --> 01:15:32,590 JEFFREY GROSSMAN: Yeah. 1590 01:15:32,590 --> 01:15:36,280 Well, so anyway, that would have been 1591 01:15:36,280 --> 01:15:37,600 a band structure of silicon. 1592 01:15:37,600 --> 01:15:39,370 But what you can see here, look at this. 1593 01:15:39,370 --> 01:15:40,840 Oh, but look under here. 1594 01:15:40,840 --> 01:15:42,610 You see, don't pay attention to that. 1595 01:15:42,610 --> 01:15:46,110 Look at this, k-point density. 1596 01:15:46,110 --> 01:15:51,490 That is a new convergence parameter for solids. 1597 01:15:51,490 --> 01:15:54,520 Can I get a wiggly band structure for silicon 1598 01:15:54,520 --> 01:15:56,500 with just one k-point in my simulation? 1599 01:15:56,500 --> 01:15:57,430 You can, actually. 1600 01:15:57,430 --> 01:15:58,810 And I'll tell you why. 1601 01:15:58,810 --> 01:15:59,630 Will it be right? 1602 01:15:59,630 --> 01:16:01,850 No. 1603 01:16:01,850 --> 01:16:04,450 And I'm going to pick up on that point 1604 01:16:04,450 --> 01:16:07,480 to make sure we understand this convergence parameter 1605 01:16:07,480 --> 01:16:09,780 when we start next week.