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 00:00:26,660 --> 00:00:28,720 JEFFREY C. GROSSMAN: How's everybody doing? 9 00:00:28,720 --> 00:00:29,740 Good? 10 00:00:29,740 --> 00:00:30,880 Yeah. 11 00:00:30,880 --> 00:00:35,260 I wanted to start with some inspiration. 12 00:00:35,260 --> 00:00:42,970 And so because I love videos, and I think 13 00:00:42,970 --> 00:00:44,955 that's a way of inspiring. 14 00:00:44,955 --> 00:00:47,080 And sometimes I think if you feel really passionate 15 00:00:47,080 --> 00:00:51,880 about something, you can use song and video 16 00:00:51,880 --> 00:00:54,160 as a way of conveying your ideas. 17 00:00:54,160 --> 00:00:58,840 I have a good friend, actually, who's been doing that. 18 00:00:58,840 --> 00:01:02,350 So I want to show you-- anybody heard of Fog and Smog? 19 00:01:02,350 --> 00:01:03,220 OK. 20 00:01:03,220 --> 00:01:11,590 Let's show you this little video, just a minute of it. 21 00:01:11,590 --> 00:01:14,410 DJ DAVE: What's crazy to me is like nobody ever 22 00:01:14,410 --> 00:01:16,570 talks in person anymore. 23 00:01:16,570 --> 00:01:19,580 Like you got Twitter and Facebook and everybody 24 00:01:19,580 --> 00:01:21,250 sending text messages. 25 00:01:21,250 --> 00:01:23,920 Like nobody's ever sitting down like this and just talking. 26 00:01:23,920 --> 00:01:25,000 You know? 27 00:01:25,000 --> 00:01:27,700 For all these technologies, we're actually further apart. 28 00:01:27,700 --> 00:01:29,710 You know what I'm saying? 29 00:01:29,710 --> 00:01:30,340 Hello? 30 00:01:30,340 --> 00:01:32,140 JEFFREY C. GROSSMAN: That's my good friend DJ Dave. 31 00:01:32,140 --> 00:01:34,060 DJ DAVE: (RAPPING) Would you please put your phone down? 32 00:01:34,060 --> 00:01:35,368 You're not walking straight. 33 00:01:35,368 --> 00:01:36,790 You're stumbling around. 34 00:01:36,790 --> 00:01:37,738 You're in public, man. 35 00:01:37,738 --> 00:01:40,420 You're being kinda rude, and text messages 36 00:01:40,420 --> 00:01:41,860 are not that important. 37 00:01:41,860 --> 00:01:43,318 Excuse me, lady. 38 00:01:43,318 --> 00:01:44,860 Could you please put your phone down? 39 00:01:44,860 --> 00:01:47,300 You're talking too loud, like nobody's around. 40 00:01:47,300 --> 00:01:50,380 I'm sure you'll find a way to hear what you have to say. 41 00:01:50,380 --> 00:01:52,702 I don't want to know that much about your day. 42 00:01:52,702 --> 00:01:54,960 I'm standing on line at my favorite cafe, 43 00:01:54,960 --> 00:01:58,415 trying to get my coffee, like I do every single day. 44 00:01:58,415 --> 00:02:00,790 This dude in front of me is texting up a storm. 45 00:02:00,790 --> 00:02:03,950 Thumbs moving like bees in a damn swarm. 46 00:02:03,950 --> 00:02:06,610 I mean, it's none of my business how you spend your time, 47 00:02:06,610 --> 00:02:08,050 but pay attention, dude. 48 00:02:08,050 --> 00:02:09,370 You're like next in line. 49 00:02:09,370 --> 00:02:12,210 He finally gets to the counter and is like, um. 50 00:02:12,210 --> 00:02:13,810 I'm like, man, you're hella dumb. 51 00:02:13,810 --> 00:02:15,102 JEFFREY C. GROSSMAN: All right. 52 00:02:15,102 --> 00:02:15,602 Anyway. 53 00:02:15,602 --> 00:02:18,340 DJ DAVE: (RAPPING) Could you you please put your phone down. 54 00:02:18,340 --> 00:02:18,700 JEFFREY C. GROSSMAN: All right. 55 00:02:18,700 --> 00:02:20,320 So I won't show all the thing. 56 00:02:20,320 --> 00:02:22,990 That's actually a good family friend, DJ Dave. 57 00:02:22,990 --> 00:02:27,850 I lived in his room for a year, awesome guy, 58 00:02:27,850 --> 00:02:31,960 and he's made some really nice videos. 59 00:02:31,960 --> 00:02:35,530 The point is-- this is not quantum mechanics related-- 60 00:02:35,530 --> 00:02:38,650 it is a topic near and dear to my heart as some of you know. 61 00:02:38,650 --> 00:02:40,840 And I'm sure you're all closing your laptops 62 00:02:40,840 --> 00:02:45,010 and putting your phones down, especially after seeing that. 63 00:02:45,010 --> 00:02:47,350 But the point is, he's passionate. 64 00:02:47,350 --> 00:02:48,860 So what does he do? 65 00:02:48,860 --> 00:02:51,460 He conveys his passion in song. 66 00:02:51,460 --> 00:02:53,560 Now, if any of you-- 67 00:02:53,560 --> 00:02:56,200 so this is related to the class-- if any of you 68 00:02:56,200 --> 00:02:58,150 have an idea that you're passionate about, 69 00:02:58,150 --> 00:02:59,980 like say for your final project, and you'd 70 00:02:59,980 --> 00:03:02,740 like to do something like that, I would really 71 00:03:02,740 --> 00:03:03,880 be fully supportive. 72 00:03:03,880 --> 00:03:07,750 I tried to-- some of you may remember my attempts 73 00:03:07,750 --> 00:03:13,060 to get you to do that in 3012, and now guess how many people 74 00:03:13,060 --> 00:03:16,280 actually did that in 3012? 75 00:03:16,280 --> 00:03:17,470 Right, not so many. 76 00:03:17,470 --> 00:03:22,000 But if somebody wants to take a crack at conveying 77 00:03:22,000 --> 00:03:29,050 their project in this format, there's 78 00:03:29,050 --> 00:03:31,570 got to be a bonus point or two in that just alone. 79 00:03:31,570 --> 00:03:34,000 Right? 80 00:03:34,000 --> 00:03:36,520 Are any of you are juniors, junior lab? 81 00:03:36,520 --> 00:03:38,380 Are you doing that in Van Vliet's class? 82 00:03:38,380 --> 00:03:39,430 AUDIENCE: Yeah, we did it last summer. 83 00:03:39,430 --> 00:03:40,513 JEFFREY C. GROSSMAN: Yeah. 84 00:03:40,513 --> 00:03:45,670 So I saw some of those, and I'm like, man, I want that. 85 00:03:45,670 --> 00:03:48,010 I think that's really creative and cool. 86 00:03:48,010 --> 00:03:49,800 So I see a lot of really cool stuff. 87 00:03:49,800 --> 00:03:50,800 I saw a couple of those. 88 00:03:50,800 --> 00:03:51,980 They were really great. 89 00:03:51,980 --> 00:03:57,460 So if anybody wants to do that, let me know, and I'd 90 00:03:57,460 --> 00:03:59,110 be happy to support however I can, 91 00:03:59,110 --> 00:04:03,000 put you in touch with Dave if you want or whatever. 92 00:04:03,000 --> 00:04:03,710 OK. 93 00:04:03,710 --> 00:04:04,210 All right. 94 00:04:04,210 --> 00:04:06,400 Any questions about class products? 95 00:04:06,400 --> 00:04:10,030 So I've been talking to some of you about the projects, 96 00:04:10,030 --> 00:04:16,450 and if anyone is still interested, it's not too late. 97 00:04:16,450 --> 00:04:18,610 But we're getting close to being too late, 98 00:04:18,610 --> 00:04:20,162 so please, come talk to me. 99 00:04:20,162 --> 00:04:21,579 If you still want to do a project, 100 00:04:21,579 --> 00:04:25,780 this would be like the last week I think to get something going. 101 00:04:28,690 --> 00:04:32,020 What I want to do today is-- well, before I start, 102 00:04:32,020 --> 00:04:32,870 here's where we are. 103 00:04:32,870 --> 00:04:33,370 OK? 104 00:04:33,370 --> 00:04:36,160 So what I want to do is continue on our march. 105 00:04:36,160 --> 00:04:38,200 We talked about atoms to solids, laid a little 106 00:04:38,200 --> 00:04:38,950 of the groundwork. 107 00:04:38,950 --> 00:04:45,520 We'll review some of that and then go onto some of the things 108 00:04:45,520 --> 00:04:48,070 you can do once you model solids. 109 00:04:48,070 --> 00:04:50,020 I want to do some calculations, assuming 110 00:04:50,020 --> 00:04:54,880 that it works, using the nanoHUB interactively together. 111 00:04:54,880 --> 00:05:00,610 Because again, there is nothing like calculating the behavior 112 00:05:00,610 --> 00:05:02,300 of electrons with friends. 113 00:05:02,300 --> 00:05:06,460 There's no better way to create a good group dynamics 114 00:05:06,460 --> 00:05:07,510 and interactions. 115 00:05:11,330 --> 00:05:17,850 Computing has the nuts and bolts, 116 00:05:17,850 --> 00:05:21,110 a lot of which we're talking about, some of which were not, 117 00:05:21,110 --> 00:05:24,530 and then it has sort of the philosophy. 118 00:05:24,530 --> 00:05:25,190 Right? 119 00:05:25,190 --> 00:05:28,580 And I really tend to agree with some of the things 120 00:05:28,580 --> 00:05:33,500 that people have said over the years on how to use computing. 121 00:05:33,500 --> 00:05:35,000 And so I just want to bring this in, 122 00:05:35,000 --> 00:05:38,720 and I'll come back to this at the end of the term 123 00:05:38,720 --> 00:05:39,930 a little bit more. 124 00:05:39,930 --> 00:05:42,830 But Richard Hamming had a number of very interesting things 125 00:05:42,830 --> 00:05:46,100 to say about research and about computing. 126 00:05:46,100 --> 00:05:49,110 The purpose of computing is insight not numbers. 127 00:05:49,110 --> 00:05:49,610 Right? 128 00:05:49,610 --> 00:05:54,560 So the computation is only as good as your creativity 129 00:05:54,560 --> 00:05:57,920 and your thinking on both sides, what you put into the computer 130 00:05:57,920 --> 00:05:59,290 and what you get out. 131 00:05:59,290 --> 00:06:01,837 He also said what are the most important problems 132 00:06:01,837 --> 00:06:02,420 in your field? 133 00:06:02,420 --> 00:06:03,800 Are you working on one of them? 134 00:06:03,800 --> 00:06:05,150 Why not? 135 00:06:05,150 --> 00:06:12,222 I love that, and that's very much an MIT ecosystem way, 136 00:06:12,222 --> 00:06:13,430 but it's not true everywhere. 137 00:06:13,430 --> 00:06:15,140 But we should always remind ourselves, 138 00:06:15,140 --> 00:06:17,410 even if I solve this problem-- 139 00:06:17,410 --> 00:06:19,160 so I'd say, my problem is I want to change 140 00:06:19,160 --> 00:06:22,530 the bandgap of silicon from indirect to direct. 141 00:06:22,530 --> 00:06:23,030 OK? 142 00:06:23,030 --> 00:06:25,010 I can do that now, because we've learned how 143 00:06:25,010 --> 00:06:26,380 to do that using simulation. 144 00:06:26,380 --> 00:06:29,120 So I can probe materials using a computer. 145 00:06:29,120 --> 00:06:32,210 You should still say, why? 146 00:06:32,210 --> 00:06:34,490 The point isn't-- you can't just say, why? 147 00:06:34,490 --> 00:06:36,380 Well, because it would be useful. 148 00:06:36,380 --> 00:06:38,770 Or why, well, because it can make a better solar cell. 149 00:06:38,770 --> 00:06:39,570 That's not enough. 150 00:06:39,570 --> 00:06:40,070 Right? 151 00:06:40,070 --> 00:06:43,490 It's really how much impact can you have? 152 00:06:43,490 --> 00:06:46,400 What is this going to do to the technology you're working on 153 00:06:46,400 --> 00:06:49,070 or the science problem you're trying to answer? 154 00:06:49,070 --> 00:06:51,500 And taking it those extra steps goes a long way, 155 00:06:51,500 --> 00:06:53,390 and that applies to computing, just 156 00:06:53,390 --> 00:06:56,022 like any other area of research. 157 00:06:56,022 --> 00:06:57,980 Better to solve the right problem the wrong way 158 00:06:57,980 --> 00:07:00,370 than the wrong problem the right way. 159 00:07:00,370 --> 00:07:02,830 So choosing the problem that's a related statement. 160 00:07:02,830 --> 00:07:04,760 In research, if you know what you're doing, 161 00:07:04,760 --> 00:07:06,460 then you shouldn't be doing it. 162 00:07:06,460 --> 00:07:12,160 I like that a lot, and that's where 163 00:07:12,160 --> 00:07:15,100 computation is a real enabler. 164 00:07:15,100 --> 00:07:21,610 I love experiments too, but I love the world of computation, 165 00:07:21,610 --> 00:07:23,200 especially at this scale. 166 00:07:23,200 --> 00:07:25,150 Where you can design-- 167 00:07:25,150 --> 00:07:26,860 within a limited number of atoms-- 168 00:07:26,860 --> 00:07:28,510 you can design anything. 169 00:07:28,510 --> 00:07:31,090 You can think up anything and try it. 170 00:07:31,090 --> 00:07:31,660 Right? 171 00:07:31,660 --> 00:07:35,260 So there's really the phase space is just enormous, 172 00:07:35,260 --> 00:07:37,840 and you can really play around and try 173 00:07:37,840 --> 00:07:41,630 crazy ideas, random thoughts, and tangents out as you go. 174 00:07:41,630 --> 00:07:44,530 Those are often where the best creativity happens. 175 00:07:44,530 --> 00:07:45,680 And then this is it. 176 00:07:45,680 --> 00:07:46,180 Right? 177 00:07:46,180 --> 00:07:47,200 Machines should work. 178 00:07:47,200 --> 00:07:48,910 People should think. 179 00:07:48,910 --> 00:07:52,803 So as you go after this class, and you come out, 180 00:07:52,803 --> 00:07:54,970 one of the most important things to come out of this 181 00:07:54,970 --> 00:07:57,970 is it's always a balance. 182 00:07:57,970 --> 00:08:00,460 When you do computation, it's a balance 183 00:08:00,460 --> 00:08:04,150 between your time and the computer's time. 184 00:08:04,150 --> 00:08:07,630 And what you'd like to do is create 185 00:08:07,630 --> 00:08:09,340 a balance, such that the computer is 186 00:08:09,340 --> 00:08:12,460 solving things for you. 187 00:08:12,460 --> 00:08:15,580 And you're spending most of your time not solving those things, 188 00:08:15,580 --> 00:08:19,330 not wasting your time on things that the computer could do, 189 00:08:19,330 --> 00:08:21,550 but rather on thinking about the kind of problem 190 00:08:21,550 --> 00:08:23,140 that you're going to make it do. 191 00:08:23,140 --> 00:08:27,250 And you'd be surprised how often in computation this balance 192 00:08:27,250 --> 00:08:28,840 is not well kept. 193 00:08:28,840 --> 00:08:29,640 OK? 194 00:08:29,640 --> 00:08:33,100 And how often you might spend a lot of your time 195 00:08:33,100 --> 00:08:36,730 actually spinning your wheels on something 196 00:08:36,730 --> 00:08:38,620 that the machine could actually handle. 197 00:08:38,620 --> 00:08:40,480 Right? 198 00:08:40,480 --> 00:08:42,580 And how much you might think the machine 199 00:08:42,580 --> 00:08:44,500 is actually going to somehow solve 200 00:08:44,500 --> 00:08:48,640 a problem that actually you need to think about to solve. 201 00:08:48,640 --> 00:08:51,130 Or how much more quickly you could solve it, if you just 202 00:08:51,130 --> 00:08:52,330 thought for an hour. 203 00:08:52,330 --> 00:08:55,640 You might have saved a million hours of computer time. 204 00:08:55,640 --> 00:08:56,250 Right? 205 00:08:56,250 --> 00:09:00,350 Seriously, so that balance always has to be kept in mind. 206 00:09:00,350 --> 00:09:03,890 And then the one I like to add is Spider Man's uncle, 207 00:09:03,890 --> 00:09:06,590 with great power comes great responsibility, 208 00:09:06,590 --> 00:09:09,610 because you have now power. 209 00:09:09,610 --> 00:09:10,930 You're gaining power. 210 00:09:10,930 --> 00:09:13,120 That's what this class is doing to you. 211 00:09:13,120 --> 00:09:16,180 You're gaining the power to calculate things 212 00:09:16,180 --> 00:09:17,950 like band structures. 213 00:09:17,950 --> 00:09:18,700 Right? 214 00:09:18,700 --> 00:09:21,850 And the responsibility is that I want 215 00:09:21,850 --> 00:09:24,460 you to know what you're doing. 216 00:09:24,460 --> 00:09:28,065 In the sense that you don't just say 217 00:09:28,065 --> 00:09:29,440 I'm going to do a bang structure. 218 00:09:29,440 --> 00:09:30,490 Oh, here it is. 219 00:09:30,490 --> 00:09:33,440 You check, and you're careful about it. 220 00:09:33,440 --> 00:09:33,940 Right? 221 00:09:33,940 --> 00:09:36,820 And you understand how to validate the calculations 222 00:09:36,820 --> 00:09:40,720 that you're doing against experiment, 223 00:09:40,720 --> 00:09:44,020 and you understand how to converge the calculations. 224 00:09:44,020 --> 00:09:46,670 You understand how to make sure you understand what they mean. 225 00:09:46,670 --> 00:09:48,250 That's the responsibility. 226 00:09:48,250 --> 00:09:49,240 OK? 227 00:09:49,240 --> 00:09:52,480 So a little bit of philosophy there. 228 00:09:52,480 --> 00:09:54,232 Anyway, OK. 229 00:09:54,232 --> 00:09:55,690 We're going to review a little bit, 230 00:09:55,690 --> 00:09:57,550 and we'll talk about some properties. 231 00:09:57,550 --> 00:09:59,230 We'll come back to this band structure 232 00:09:59,230 --> 00:10:05,037 which is so critical to this part of the class. 233 00:10:05,037 --> 00:10:07,120 And if I have time, I'll talk about magnetization, 234 00:10:07,120 --> 00:10:09,120 but probably, I'll talk about magnetization 235 00:10:09,120 --> 00:10:10,570 on Thursday a little. 236 00:10:10,570 --> 00:10:12,480 All right? 237 00:10:12,480 --> 00:10:14,130 Any questions? 238 00:10:14,130 --> 00:10:19,490 So we've had a rap video, some philosophy, 239 00:10:19,490 --> 00:10:20,750 a little bit of an outline. 240 00:10:20,750 --> 00:10:21,660 Where else can we go? 241 00:10:21,660 --> 00:10:22,160 Right? 242 00:10:22,160 --> 00:10:24,260 Things are good. 243 00:10:24,260 --> 00:10:26,540 Rondo, what's up with that? 244 00:10:26,540 --> 00:10:28,400 Was that a good decision? 245 00:10:28,400 --> 00:10:29,270 No? 246 00:10:29,270 --> 00:10:30,500 How many people saw that? 247 00:10:30,500 --> 00:10:31,590 I just want to know. 248 00:10:31,590 --> 00:10:33,290 OK, so a little more people are-- 249 00:10:33,290 --> 00:10:35,130 But you only saw it on a highlight. 250 00:10:35,130 --> 00:10:36,360 Oh, you watched the game. 251 00:10:36,360 --> 00:10:36,860 All right. 252 00:10:36,860 --> 00:10:38,890 Good. 253 00:10:38,890 --> 00:10:39,650 Yeah. 254 00:10:39,650 --> 00:10:41,720 But they're going to win without Rondo tonight. 255 00:10:41,720 --> 00:10:43,200 Anyway, OK. 256 00:10:43,200 --> 00:10:43,700 Yeah. 257 00:10:43,700 --> 00:10:45,230 You bump a ref on purpose? 258 00:10:45,230 --> 00:10:46,010 Come on. 259 00:10:46,010 --> 00:10:47,270 Come on, man. 260 00:10:47,270 --> 00:10:50,070 Anyway, I can't spend more time on it. 261 00:10:50,070 --> 00:10:52,380 I don't want to do that. 262 00:10:52,380 --> 00:10:55,850 Let's take a walk through memory lane for just a moment, 263 00:10:55,850 --> 00:10:59,810 because I want to make sure we all know where we are. 264 00:10:59,810 --> 00:11:00,650 Right? 265 00:11:00,650 --> 00:11:01,790 We started with this. 266 00:11:01,790 --> 00:11:03,470 There were some strange observations 267 00:11:03,470 --> 00:11:05,330 by some very smart people. 268 00:11:05,330 --> 00:11:07,240 Tell me what those are. 269 00:11:07,240 --> 00:11:09,850 We could ask questions like about these on the quiz. 270 00:11:09,850 --> 00:11:10,810 AUDIENCE: [INAUDIBLE] 271 00:11:10,810 --> 00:11:11,740 JEFFREY C. GROSSMAN: Those are electrons. 272 00:11:11,740 --> 00:11:12,240 Yeah. 273 00:11:12,240 --> 00:11:13,210 That's an electron. 274 00:11:13,210 --> 00:11:14,043 Those are electrons. 275 00:11:14,043 --> 00:11:15,132 What's this experiment? 276 00:11:15,132 --> 00:11:16,820 AUDIENCE: [INAUDIBLE] 277 00:11:16,820 --> 00:11:18,200 JEFFREY C. GROSSMAN: This one is? 278 00:11:18,200 --> 00:11:19,190 What's this one? 279 00:11:19,190 --> 00:11:21,080 AUDIENCE: Collapsing if that happens. 280 00:11:21,080 --> 00:11:22,830 JEFFREY C. GROSSMAN: Why does it collapse? 281 00:11:22,830 --> 00:11:25,160 AUDIENCE: [INAUDIBLE] classical. 282 00:11:25,160 --> 00:11:29,285 JEFFREY C. GROSSMAN: Classical, then it's got to be what? 283 00:11:29,285 --> 00:11:30,377 AUDIENCE: [INAUDIBLE] 284 00:11:30,377 --> 00:11:31,460 JEFFREY C. GROSSMAN: Yeah. 285 00:11:31,460 --> 00:11:32,960 It's giving off energy, and it's not 286 00:11:32,960 --> 00:11:35,090 going to hang into its orbit. 287 00:11:35,090 --> 00:11:36,830 But then quantum mechanics came along 288 00:11:36,830 --> 00:11:38,660 and said, no, that's not true. 289 00:11:38,660 --> 00:11:39,290 Right? 290 00:11:39,290 --> 00:11:40,040 Because what? 291 00:11:45,000 --> 00:11:47,430 It just is. 292 00:11:47,430 --> 00:11:50,040 It's the 'It just is" answer. 293 00:11:50,040 --> 00:11:52,290 No, because what it said is that the levels 294 00:11:52,290 --> 00:11:54,780 of a quantum mechanical object are quantized, 295 00:11:54,780 --> 00:11:56,460 and they have to be in those places. 296 00:11:56,460 --> 00:11:59,700 And there's no in between, and that's what this is. 297 00:11:59,700 --> 00:12:00,756 What's this? 298 00:12:00,756 --> 00:12:02,580 AUDIENCE: [INAUDIBLE] 299 00:12:02,580 --> 00:12:05,400 JEFFREY C. GROSSMAN: Which comes from that same observation. 300 00:12:05,400 --> 00:12:08,160 You'd think that the universe, the hydrogen 301 00:12:08,160 --> 00:12:11,407 might emit colors in any frequency, but instead only 302 00:12:11,407 --> 00:12:12,240 certain frequencies. 303 00:12:12,240 --> 00:12:15,090 And it comes from that quantization of those electron 304 00:12:15,090 --> 00:12:18,120 levels, and this, somebody already said this. 305 00:12:18,120 --> 00:12:19,540 Right? 306 00:12:19,540 --> 00:12:20,510 Photoelectric effect. 307 00:12:20,510 --> 00:12:20,890 OK? 308 00:12:20,890 --> 00:12:22,265 Now, those are very smart people, 309 00:12:22,265 --> 00:12:23,500 and the weirdness kept going. 310 00:12:23,500 --> 00:12:26,470 Now, is that guy ordering two drinks at the bar? 311 00:12:26,470 --> 00:12:29,770 Why does he have two? 312 00:12:29,770 --> 00:12:33,610 Double slit experiment, that was a cool one, and the weirdness 313 00:12:33,610 --> 00:12:34,850 just kept going. 314 00:12:34,850 --> 00:12:36,670 Right? 315 00:12:36,670 --> 00:12:41,710 Now, it became very clear that matter behaved like waves. 316 00:12:41,710 --> 00:12:46,900 If we took this little 10-slide, where we are, 317 00:12:46,900 --> 00:12:49,840 quick, what road if we come on, and we turned it 318 00:12:49,840 --> 00:12:51,670 into a music video. 319 00:12:51,670 --> 00:12:53,830 Right? 320 00:12:53,830 --> 00:12:55,510 Anybody want to do that? 321 00:12:55,510 --> 00:12:58,810 That's exciting. 322 00:12:58,810 --> 00:13:00,720 OK. 323 00:13:00,720 --> 00:13:03,970 It became clear that matter behave like waves and vice 324 00:13:03,970 --> 00:13:04,890 versa. 325 00:13:04,890 --> 00:13:06,370 OK? 326 00:13:06,370 --> 00:13:09,220 And that we had to lose our classical concepts 327 00:13:09,220 --> 00:13:11,470 of absolute position of momentum and instead 328 00:13:11,470 --> 00:13:15,040 consider a particle as a wave whose square is the probability 329 00:13:15,040 --> 00:13:17,110 of finding it, and there it is. 330 00:13:17,110 --> 00:13:21,590 And this became our goal, our task of finding that. 331 00:13:21,590 --> 00:13:22,090 Right? 332 00:13:22,090 --> 00:13:25,630 That's what we needed to do on a computer, 333 00:13:25,630 --> 00:13:28,710 and there's a picture of a wave. 334 00:13:28,710 --> 00:13:30,530 How do we do this? 335 00:13:30,530 --> 00:13:31,620 How do we describe this? 336 00:13:31,620 --> 00:13:35,080 Well, we needed our F equals ma. 337 00:13:35,080 --> 00:13:36,270 Right? 338 00:13:36,270 --> 00:13:40,360 So when matter is just a particle, we have F equals ma, 339 00:13:40,360 --> 00:13:43,810 but now that it's a wave, we needed something different. 340 00:13:43,810 --> 00:13:46,500 And from the elements and the bits 341 00:13:46,500 --> 00:13:49,350 of pieces of quantum mechanics, which 342 00:13:49,350 --> 00:13:52,410 people were putting together, out comes 343 00:13:52,410 --> 00:13:53,970 the Schrodinger equation. 344 00:13:53,970 --> 00:13:56,940 And that's our F equals ma, which tells us 345 00:13:56,940 --> 00:14:00,750 how to describe the behavior of waves, 346 00:14:00,750 --> 00:14:03,870 if they're quantum mechanical, and it was wonderful. 347 00:14:03,870 --> 00:14:05,660 It explained many things. 348 00:14:05,660 --> 00:14:07,490 It gave us atomic orbitals. 349 00:14:07,490 --> 00:14:09,710 It predicted the energy levels of hydrogen. 350 00:14:09,710 --> 00:14:12,560 Remember how wonderful that was? 351 00:14:12,560 --> 00:14:14,870 It gave us, basically, the means to understand 352 00:14:14,870 --> 00:14:16,920 a whole lot of this. 353 00:14:16,920 --> 00:14:17,420 Right? 354 00:14:22,490 --> 00:14:27,390 But nature does more than one electron, 355 00:14:27,390 --> 00:14:29,162 and it was impossible to solve for more 356 00:14:29,162 --> 00:14:30,120 than a single electron. 357 00:14:30,120 --> 00:14:32,890 So here's where computational quantum mechanics comes in. 358 00:14:32,890 --> 00:14:33,390 OK? 359 00:14:37,113 --> 00:14:39,030 But there was still a problem, and the problem 360 00:14:39,030 --> 00:14:41,430 is that we don't have the age of the universe 361 00:14:41,430 --> 00:14:43,200 to do our calculations. 362 00:14:43,200 --> 00:14:46,980 And so that's how long it would take to currently solve 363 00:14:46,980 --> 00:14:49,530 the Schrodinger equation exactly on a computer 364 00:14:49,530 --> 00:14:53,220 for like only a handful of particles. 365 00:14:53,220 --> 00:14:55,008 Right? 366 00:14:55,008 --> 00:14:56,550 Obviously, you could solve it for one 367 00:14:56,550 --> 00:15:00,380 or two particles exactly in a lot less time than that, 368 00:15:00,380 --> 00:15:03,610 but once you get up to 10, you're done. 369 00:15:03,610 --> 00:15:06,750 And so we started to try to take some inspiration, 370 00:15:06,750 --> 00:15:10,960 and we looked at this, a guy's back. 371 00:15:10,960 --> 00:15:14,410 And we said, OK, well, maybe that's telling us something. 372 00:15:14,410 --> 00:15:17,740 It's telling us a lot, but it's telling us 373 00:15:17,740 --> 00:15:18,790 some interesting physics. 374 00:15:18,790 --> 00:15:21,510 It's telling us that the ions are a whole lot heavier 375 00:15:21,510 --> 00:15:22,760 and slower than the electrons. 376 00:15:22,760 --> 00:15:23,890 So we can freeze them out. 377 00:15:23,890 --> 00:15:27,070 So we started to peel away at the Schrodinger equation, 378 00:15:27,070 --> 00:15:28,470 and we made it simpler. 379 00:15:28,470 --> 00:15:28,970 OK? 380 00:15:28,970 --> 00:15:35,090 We made some approximations, and there are basically 381 00:15:35,090 --> 00:15:39,170 the two paths that I talked about in those approximations. 382 00:15:39,170 --> 00:15:45,560 At the time, it's about 50 years ago, maybe 60, 383 00:15:45,560 --> 00:15:49,130 these two paths were fairly distinct. 384 00:15:49,130 --> 00:15:51,960 And the chemists mostly worked on saying, well, here's 385 00:15:51,960 --> 00:15:53,210 the equation we have to solve. 386 00:15:53,210 --> 00:15:53,990 We can't solve it. 387 00:15:53,990 --> 00:15:56,180 Let's make psi something simpler. 388 00:15:56,180 --> 00:15:57,590 Let's simplify psi. 389 00:15:57,590 --> 00:16:00,380 And the physicists mostly said, well, 390 00:16:00,380 --> 00:16:02,020 let's not think about psi. 391 00:16:02,020 --> 00:16:03,620 Let's just change the Hamiltonian. 392 00:16:03,620 --> 00:16:05,360 That's h there. 393 00:16:05,360 --> 00:16:06,590 Right? 394 00:16:06,590 --> 00:16:09,320 And those are two approaches that 395 00:16:09,320 --> 00:16:13,130 led to many different models, many different theories, 396 00:16:13,130 --> 00:16:16,040 many different theories. 397 00:16:16,040 --> 00:16:19,040 In the theories, in both sides, the effect 398 00:16:19,040 --> 00:16:23,600 of these simplifications led to the picture 399 00:16:23,600 --> 00:16:25,020 that we call a mean field picture. 400 00:16:25,020 --> 00:16:27,350 Which is that instead of having to worry about all 401 00:16:27,350 --> 00:16:29,960 these quantum mechanical electrons interacting 402 00:16:29,960 --> 00:16:31,670 with each other, we just worry about them 403 00:16:31,670 --> 00:16:34,720 as if it's each one is in the field of the others, 404 00:16:34,720 --> 00:16:36,470 an average field of the others, and that's 405 00:16:36,470 --> 00:16:38,280 called the mean field method. 406 00:16:38,280 --> 00:16:39,020 OK? 407 00:16:39,020 --> 00:16:41,150 Those are the methods we're using in this class. 408 00:16:41,150 --> 00:16:43,800 That's the method we're using. 409 00:16:43,800 --> 00:16:48,440 And then we talked about density functional theory, 410 00:16:48,440 --> 00:16:51,950 and the breakthrough there is that trying 411 00:16:51,950 --> 00:16:54,590 to keep track of a wave function, let's 412 00:16:54,590 --> 00:17:01,970 say, on a grid of a 2 by 2 grid or 3 by 3 or whatever. 413 00:17:01,970 --> 00:17:04,190 Well, this is for actually a 2 by 2 by 2 grid, 414 00:17:04,190 --> 00:17:07,099 and you have N electrons. 415 00:17:07,099 --> 00:17:12,950 Well, if you have N is 1, so 1 electron, 416 00:17:12,950 --> 00:17:16,640 you have 8 points you need to keep track of. 417 00:17:16,640 --> 00:17:19,430 But if you had just 10 electrons, 418 00:17:19,430 --> 00:17:23,790 then you see that you get 10 to the 9th points. 419 00:17:23,790 --> 00:17:24,290 OK? 420 00:17:24,290 --> 00:17:27,750 This is the problem of using grids to represent things. 421 00:17:27,750 --> 00:17:32,270 And so if you use the density instead, it's just a constant. 422 00:17:32,270 --> 00:17:35,310 Tracking the density is much easier, 423 00:17:35,310 --> 00:17:37,820 and the crux of density functional theory, 424 00:17:37,820 --> 00:17:41,030 which was worked out in the '60s, is that there is a 1 to 1 425 00:17:41,030 --> 00:17:43,580 correspondence between the exact density of the system 426 00:17:43,580 --> 00:17:44,540 and the wave function. 427 00:17:44,540 --> 00:17:46,502 OK? 428 00:17:46,502 --> 00:17:47,960 And we talked about this last time, 429 00:17:47,960 --> 00:17:49,670 about how we sit here, because it's 430 00:17:49,670 --> 00:17:54,260 a nice compromise between accuracy and applicability. 431 00:17:54,260 --> 00:17:56,720 But I want to come back to this, because this is not-- 432 00:17:56,720 --> 00:17:59,660 part of the reason I wanted to remind you of this is that it's 433 00:17:59,660 --> 00:18:02,300 just always fun to take these walks through memory lane, 434 00:18:02,300 --> 00:18:03,920 and it only took us five minutes. 435 00:18:03,920 --> 00:18:07,160 But another reason is that I want 436 00:18:07,160 --> 00:18:10,190 to remind you that we are not solving for quantum mechanics 437 00:18:10,190 --> 00:18:11,450 exactly. 438 00:18:11,450 --> 00:18:12,980 We're not. 439 00:18:12,980 --> 00:18:14,210 We cannot. 440 00:18:14,210 --> 00:18:14,710 OK? 441 00:18:20,060 --> 00:18:24,150 This is what I mean by the responsibility part. 442 00:18:24,150 --> 00:18:25,190 OK? 443 00:18:25,190 --> 00:18:30,170 There are some things that DFT can give you, 444 00:18:30,170 --> 00:18:33,230 can calculate about those electrons, that are typically 445 00:18:33,230 --> 00:18:35,300 very accurate, but there are some things 446 00:18:35,300 --> 00:18:38,270 where it's actually quite off. 447 00:18:38,270 --> 00:18:39,890 It's an approximation. 448 00:18:39,890 --> 00:18:43,280 Does anybody remember where it tends to do well? 449 00:18:43,280 --> 00:18:46,460 I did talk about this in the third or fourth lecture 450 00:18:46,460 --> 00:18:48,380 and where it tends to do badly. 451 00:18:48,380 --> 00:18:49,670 Yeah. 452 00:18:49,670 --> 00:18:51,230 Or one of those two. 453 00:18:51,230 --> 00:18:53,362 AUDIENCE: [INAUDIBLE] solids, periodic structures. 454 00:18:54,040 --> 00:18:55,040 JEFFREY C. GROSSMAN: OK. 455 00:18:55,040 --> 00:18:58,640 What about the solids, what properties? 456 00:18:58,640 --> 00:19:00,323 AUDIENCE: Bandgaps. 457 00:19:00,323 --> 00:19:01,740 JEFFREY C. GROSSMAN: That would be 458 00:19:01,740 --> 00:19:06,310 a case where it falls into the not so well category. 459 00:19:06,310 --> 00:19:08,440 OK? 460 00:19:08,440 --> 00:19:10,680 So this is pretty important. 461 00:19:10,680 --> 00:19:11,945 AUDIENCE: Lattice parameters? 462 00:19:11,945 --> 00:19:13,820 JEFFREY C. GROSSMAN: Lattice parameters, yes. 463 00:19:13,820 --> 00:19:16,320 It tends to do very well for lattice parameters. 464 00:19:16,320 --> 00:19:18,260 So how far apart should atoms be, 465 00:19:18,260 --> 00:19:21,800 if they come together in a molecule or solid or a liquid? 466 00:19:21,800 --> 00:19:23,840 DFT can do very well. 467 00:19:23,840 --> 00:19:26,960 It does very well for vibrational properties, 468 00:19:26,960 --> 00:19:31,160 vibrational spectrum, the phonons in a material. 469 00:19:31,160 --> 00:19:34,040 It tends to do worse for excited states, 470 00:19:34,040 --> 00:19:36,290 like looking at, say, the optical gap 471 00:19:36,290 --> 00:19:39,680 or the electronic gap of material. 472 00:19:39,680 --> 00:19:44,000 It tends to the approximations that we 473 00:19:44,000 --> 00:19:47,840 had to use to get a method that's workable, all of these 474 00:19:47,840 --> 00:19:49,310 have approximations, except this. 475 00:19:49,310 --> 00:19:49,970 See? 476 00:19:49,970 --> 00:19:51,710 We're up to two electrons here. 477 00:19:54,230 --> 00:20:01,820 Those approximations lead to DFT not doing so well for the gap. 478 00:20:01,820 --> 00:20:05,865 However, as you may have noticed in your homeworks, that's 479 00:20:05,865 --> 00:20:07,490 one of the things you have to calculate 480 00:20:07,490 --> 00:20:08,900 in your second homework. 481 00:20:08,900 --> 00:20:12,410 So here's the point I want to make. 482 00:20:12,410 --> 00:20:15,440 These methods have advantages and disadvantages. 483 00:20:15,440 --> 00:20:17,120 They all are approximate. 484 00:20:17,120 --> 00:20:21,230 Some quantum mechanical methods can describe some things 485 00:20:21,230 --> 00:20:24,950 very accurately and some things not as accurately, 486 00:20:24,950 --> 00:20:27,320 and others may have different balances. 487 00:20:27,320 --> 00:20:29,750 And as you go down to more accurate methods that 488 00:20:29,750 --> 00:20:32,180 can describe more things very accurately, 489 00:20:32,180 --> 00:20:36,650 you tend to be able to do less and less numbers of electrons. 490 00:20:36,650 --> 00:20:37,250 Right? 491 00:20:37,250 --> 00:20:42,050 Limit your phase space to 12, 20 electrons, 492 00:20:42,050 --> 00:20:45,980 and you're pretty limited in how many materials you can explore. 493 00:20:45,980 --> 00:20:46,480 Right? 494 00:20:49,370 --> 00:20:52,670 Now, why can we still calculate the gap? 495 00:20:52,670 --> 00:20:55,520 Well, and this is, actually, there's 496 00:20:55,520 --> 00:20:57,530 going to be an addendum to your homework 497 00:20:57,530 --> 00:21:00,650 which I'm going to have to-- 498 00:21:00,650 --> 00:21:04,513 I forgot to put this in there, but if this 499 00:21:04,513 --> 00:21:05,555 is the density of states. 500 00:21:13,380 --> 00:21:13,880 OK. 501 00:21:13,880 --> 00:21:17,300 Let's say that's the density of states. 502 00:21:17,300 --> 00:21:19,130 By the way, if I asked you what's 503 00:21:19,130 --> 00:21:22,760 the gap of this material, what would you say? 504 00:21:25,380 --> 00:21:26,212 Yeah? 505 00:21:26,212 --> 00:21:28,572 AUDIENCE: Those are between the first density of states 506 00:21:28,572 --> 00:21:29,520 and second density of states. 507 00:21:29,520 --> 00:21:31,853 JEFFREY C. GROSSMAN: Well, that would be true but I need 508 00:21:31,853 --> 00:21:34,800 to know one thing very importantly or I cannot answer 509 00:21:34,800 --> 00:21:36,590 whether that's the case. 510 00:21:36,590 --> 00:21:38,730 What do I need to know? 511 00:21:38,730 --> 00:21:41,317 I need to know the position of something here. 512 00:21:41,317 --> 00:21:42,382 AUDIENCE: [INAUDIBLE] 513 00:21:42,382 --> 00:21:44,590 JEFFREY C. GROSSMAN: The Fermi energy, who said that? 514 00:21:44,590 --> 00:21:45,600 Very good. 515 00:21:45,600 --> 00:21:47,800 Fermi energy, came from over there. 516 00:21:47,800 --> 00:21:48,300 OK? 517 00:21:48,300 --> 00:21:51,310 Now, if I tell you that the Fermi energy is here. 518 00:21:51,310 --> 00:21:52,560 Then, you're absolutely right. 519 00:21:52,560 --> 00:21:55,320 It's the difference-- and in the DOS plots 520 00:21:55,320 --> 00:21:58,860 you get on the nanoHUB, these are essentially 521 00:21:58,860 --> 00:22:01,440 levels that come out. 522 00:22:01,440 --> 00:22:03,600 If it's a molecule, they're levels that come out, 523 00:22:03,600 --> 00:22:05,308 and then they're spread out a little bit. 524 00:22:05,308 --> 00:22:07,830 So it's very likely this peak height versus that 525 00:22:07,830 --> 00:22:10,740 is going to give you the gap. 526 00:22:10,740 --> 00:22:13,350 And in the nanoHUB tool, the Fermi energy 527 00:22:13,350 --> 00:22:15,820 is always set to 0 for you. 528 00:22:15,820 --> 00:22:16,320 OK? 529 00:22:16,320 --> 00:22:20,670 But if I don't specify it, you may not know. 530 00:22:20,670 --> 00:22:23,280 Now, is that a semiconductor or an insulator? 531 00:22:23,280 --> 00:22:23,910 Is it a metal? 532 00:22:27,570 --> 00:22:28,890 Semiconductor? 533 00:22:28,890 --> 00:22:30,618 How do you know? 534 00:22:30,618 --> 00:22:31,410 You have a feeling? 535 00:22:31,410 --> 00:22:32,290 AUDIENCE: [INAUDIBLE] between the two bands. 536 00:22:32,290 --> 00:22:32,700 JEFFREY C. GROSSMAN: OK. 537 00:22:32,700 --> 00:22:33,950 Well, let's say I put it here. 538 00:22:37,380 --> 00:22:38,610 Semiconductor? 539 00:22:38,610 --> 00:22:40,830 Insulator? 540 00:22:40,830 --> 00:22:43,710 Wait, why can I not answer that question? 541 00:22:43,710 --> 00:22:45,172 What don't you know? 542 00:22:45,172 --> 00:22:47,220 AUDIENCE: You don't know the scale. 543 00:22:47,220 --> 00:22:50,310 JEFFREY C. GROSSMAN: What if I told you that this was 0.1, 544 00:22:50,310 --> 00:22:56,040 and this was 0.2, and the scale is in eV, then what is it? 545 00:22:58,860 --> 00:23:01,490 It's a 0.1 eV gap semiconductor. 546 00:23:01,490 --> 00:23:02,710 Right? 547 00:23:02,710 --> 00:23:03,320 OK? 548 00:23:03,320 --> 00:23:06,050 So well, that's like almost a metal, 549 00:23:06,050 --> 00:23:07,310 but it's still semiconductor. 550 00:23:07,310 --> 00:23:09,890 It's a very low bandgap semiconductor, 551 00:23:09,890 --> 00:23:12,260 where actually electrons will probably 552 00:23:12,260 --> 00:23:16,400 populate the conduction band just thermally 553 00:23:16,400 --> 00:23:20,300 to an interesting probability. 554 00:23:20,300 --> 00:23:26,840 Now, if I told you this was 10 eV, what would it be? 555 00:23:26,840 --> 00:23:27,710 AUDIENCE: Insulator? 556 00:23:27,710 --> 00:23:28,640 JEFFREY C. GROSSMAN: Insulator. 557 00:23:28,640 --> 00:23:29,140 OK? 558 00:23:29,140 --> 00:23:32,420 Now, I'm going to go on. 559 00:23:32,420 --> 00:23:36,810 Don't worry, but the last point is DFT, 560 00:23:36,810 --> 00:23:40,610 I'm not going to go into the details of why, 561 00:23:40,610 --> 00:23:44,820 but DFT tends to underestimate the bandgap. 562 00:23:44,820 --> 00:23:45,630 OK? 563 00:23:45,630 --> 00:23:50,040 And so there are now sophisticated ways 564 00:23:50,040 --> 00:23:52,920 of correcting that and going beyond DFT that we're not going 565 00:23:52,920 --> 00:23:54,450 to talk about in this class. 566 00:23:54,450 --> 00:23:56,130 But what we are going to do is we're 567 00:23:56,130 --> 00:23:57,660 going to use an approximation that 568 00:23:57,660 --> 00:24:00,630 was used for 20 years in the scientific literature 569 00:24:00,630 --> 00:24:03,820 and community to correct for this problem. 570 00:24:03,820 --> 00:24:06,300 And that is simply called a scissor shift, 571 00:24:06,300 --> 00:24:10,320 where basically, you see, DFT gets the bandgap wrong, 572 00:24:10,320 --> 00:24:15,110 but it gets qualitatively very often the picture right. 573 00:24:15,110 --> 00:24:17,240 It just gets the bandgap too small. 574 00:24:17,240 --> 00:24:22,730 So very often, we just shift the conduction band states 575 00:24:22,730 --> 00:24:27,087 by some correction. 576 00:24:30,500 --> 00:24:31,340 OK? 577 00:24:31,340 --> 00:24:32,960 I will give you that correction. 578 00:24:32,960 --> 00:24:35,240 That will be the addendum in the homework, 579 00:24:35,240 --> 00:24:36,900 because you'll need that. 580 00:24:36,900 --> 00:24:38,810 So the DOS plots that you calculate 581 00:24:38,810 --> 00:24:43,370 for these solar fuels, think of them as pretty good, 582 00:24:43,370 --> 00:24:45,500 except you have to shift these states up. 583 00:24:45,500 --> 00:24:49,010 So qualitatively, these states are not bad, 584 00:24:49,010 --> 00:24:51,290 but they're just too low in energy 585 00:24:51,290 --> 00:24:53,840 compared to the occupied states. 586 00:24:53,840 --> 00:24:54,530 OK? 587 00:24:54,530 --> 00:25:01,250 So that's a method that is not really used very often anymore, 588 00:25:01,250 --> 00:25:03,710 but it was used heavily in the '70s and '80s, 589 00:25:03,710 --> 00:25:09,680 to use DFT to look at DOS and band structures. 590 00:25:09,680 --> 00:25:11,480 Nowadays, we do have methods that 591 00:25:11,480 --> 00:25:14,870 go further that allow us to actually calculate 592 00:25:14,870 --> 00:25:18,080 these things without some rigid shift approximation, 593 00:25:18,080 --> 00:25:20,370 but calculate them more from first principles. 594 00:25:20,370 --> 00:25:21,870 I won't go into those in this class. 595 00:25:21,870 --> 00:25:22,370 OK? 596 00:25:22,370 --> 00:25:23,970 Any questions? 597 00:25:23,970 --> 00:25:25,640 So you'll have to download your DOS plot 598 00:25:25,640 --> 00:25:28,690 and do the shift that I give you to the data. 599 00:25:28,690 --> 00:25:30,260 All right? 600 00:25:30,260 --> 00:25:35,570 Seems like a cheat, and it actually is a cheat. 601 00:25:35,570 --> 00:25:39,080 But like I said the qualitative nature of these states 602 00:25:39,080 --> 00:25:41,120 is still very good often. 603 00:25:41,120 --> 00:25:42,110 OK? 604 00:25:42,110 --> 00:25:46,040 So that's a property DFT does not do so well on. 605 00:25:46,040 --> 00:25:46,550 OK. 606 00:25:46,550 --> 00:25:49,430 I won't go through the self-consistent cycle. 607 00:25:49,430 --> 00:25:53,460 What we did on Tuesday is we talked about solids. 608 00:25:53,460 --> 00:25:53,960 OK? 609 00:25:53,960 --> 00:25:55,820 So we talked about the periodic cell, 610 00:25:55,820 --> 00:25:57,710 and we talked about crystal symmetries. 611 00:25:57,710 --> 00:25:59,810 And this was all review, I think, for most of you, 612 00:25:59,810 --> 00:26:01,880 and most symmetries are-- 613 00:26:01,880 --> 00:26:04,700 most common lattices are cubic or hexagonal, close back. 614 00:26:04,700 --> 00:26:06,590 Anybody know why? 615 00:26:06,590 --> 00:26:07,320 Yeah. 616 00:26:07,320 --> 00:26:08,278 AUDIENCE: Lower energy. 617 00:26:08,278 --> 00:26:09,362 JEFFREY C. GROSSMAN: Yeah. 618 00:26:09,362 --> 00:26:10,212 Because of what? 619 00:26:10,212 --> 00:26:11,367 AUDIENCE: More atoms. 620 00:26:11,367 --> 00:26:12,950 JEFFREY C. GROSSMAN: You pack them in. 621 00:26:12,950 --> 00:26:14,000 Makes sense. 622 00:26:14,000 --> 00:26:15,890 OK? 623 00:26:15,890 --> 00:26:17,708 And then, and this is sort of where 624 00:26:17,708 --> 00:26:19,250 we spent some time Tuesday and I want 625 00:26:19,250 --> 00:26:24,920 to make sure we feel our oneness with this k-space thing. 626 00:26:24,920 --> 00:26:26,760 So associated with each real space lattice, 627 00:26:26,760 --> 00:26:29,360 there exists something we call reciprocal lattice. 628 00:26:29,360 --> 00:26:31,222 It's a set of wave vectors which are 629 00:26:31,222 --> 00:26:32,930 commensurate with the real space lattice. 630 00:26:32,930 --> 00:26:37,140 Sometimes, we call it G. Why not? 631 00:26:37,140 --> 00:26:39,310 Those are the lattice vectors in reciprocal space. 632 00:26:39,310 --> 00:26:43,290 And we define them as a star, b star, c star. 633 00:26:43,290 --> 00:26:49,080 That's a lattice in real space, BCC, BCC lattice. 634 00:26:49,080 --> 00:26:51,690 And there's the reciprocal space lattice. 635 00:26:51,690 --> 00:26:52,740 It's a real lattice. 636 00:26:52,740 --> 00:26:55,560 It has distances in lattice world, 637 00:26:55,560 --> 00:26:57,810 in reciprocal lattice world. 638 00:26:57,810 --> 00:26:59,040 It just looks different. 639 00:26:59,040 --> 00:27:01,740 And actually, the inverse of BCC is FCC. 640 00:27:01,740 --> 00:27:03,300 So we talked about all this. 641 00:27:03,300 --> 00:27:05,310 And then we talked about how you can define 642 00:27:05,310 --> 00:27:07,660 a zone within reciprocal space. 643 00:27:07,660 --> 00:27:09,990 And the first-- if you take-- 644 00:27:09,990 --> 00:27:13,860 if you draw lines between neighbors and nearest neighbors 645 00:27:13,860 --> 00:27:16,950 and you draw bisections, then you define that zone. 646 00:27:16,950 --> 00:27:18,570 And that's called the Brillouin zone. 647 00:27:18,570 --> 00:27:22,350 And so you see here in this square lattice, it's a square. 648 00:27:22,350 --> 00:27:25,080 In here, it's going to be a hexagon. 649 00:27:25,080 --> 00:27:28,260 And if we define that zone in three dimensions-- 650 00:27:28,260 --> 00:27:32,680 well, here, it's squares and hexagons on the edges. 651 00:27:32,680 --> 00:27:35,470 But it's a volume. 652 00:27:35,470 --> 00:27:39,450 Does everybody-- so we're just defining space, still. 653 00:27:39,450 --> 00:27:42,910 And then came the Bloch's theorem. 654 00:27:42,910 --> 00:27:47,160 And in Bloch's theorem, what we said 655 00:27:47,160 --> 00:27:50,460 is that values of a function, like the density, 656 00:27:50,460 --> 00:27:53,730 should be the same because it's a crystal. 657 00:27:53,730 --> 00:27:56,100 But the wave function is periodic only when 658 00:27:56,100 --> 00:27:57,480 multiplied by a phase factor. 659 00:27:57,480 --> 00:28:00,150 And that's-- I didn't go into a lot of detail. 660 00:28:00,150 --> 00:28:01,740 But that's Bloch's theorem. 661 00:28:01,740 --> 00:28:04,020 So it says that the wave function picks up 662 00:28:04,020 --> 00:28:09,870 a phase factor, which depends on this new quantum number. 663 00:28:09,870 --> 00:28:15,390 That's all review from last week. 664 00:28:15,390 --> 00:28:18,040 And you can see that if the wave function looks like that-- 665 00:28:18,040 --> 00:28:19,500 so if you add-- 666 00:28:19,500 --> 00:28:20,910 the periodicity of the lattice is 667 00:28:20,910 --> 00:28:23,768 big R. R is just anywhere in real space. 668 00:28:23,768 --> 00:28:25,560 Then you're going to get this phase factor. 669 00:28:25,560 --> 00:28:27,560 And if you square them, say, to get the density, 670 00:28:27,560 --> 00:28:30,780 then you can show that the densities are the same. 671 00:28:30,780 --> 00:28:35,100 Also, and this is important, the wave function 672 00:28:35,100 --> 00:28:37,347 picks up a phase if you just go by the periodicity 673 00:28:37,347 --> 00:28:38,430 of the real space lattice. 674 00:28:38,430 --> 00:28:44,970 But if I increase k, if I add an inverse lattice vector to k, 675 00:28:44,970 --> 00:28:47,740 then they're exactly the same. 676 00:28:47,740 --> 00:28:50,880 So in k-space, I can also translate around. 677 00:28:53,640 --> 00:28:56,600 And the energies are the same. 678 00:28:56,600 --> 00:28:59,330 We're going somewhere here. 679 00:28:59,330 --> 00:29:00,260 We're going somewhere. 680 00:29:00,260 --> 00:29:01,385 Give me a minute. 681 00:29:01,385 --> 00:29:02,510 And then I showed you this. 682 00:29:02,510 --> 00:29:03,950 And does anybody have questions about this? 683 00:29:03,950 --> 00:29:06,590 The parallel here, I think, is pretty self-explanatory. 684 00:29:09,960 --> 00:29:13,387 In the hydrogen atom, we impose this spherical symmetry. 685 00:29:13,387 --> 00:29:14,220 AUDIENCE: [SNEEZING] 686 00:29:14,220 --> 00:29:15,940 JEFFREY C. GROSSMAN: Gesundheit. 687 00:29:15,940 --> 00:29:18,610 And we got out of that quantization-- we solved. 688 00:29:18,610 --> 00:29:21,070 And here, we impose a translational symmetry. 689 00:29:21,070 --> 00:29:26,200 And we get out of that k-vectorization. 690 00:29:26,200 --> 00:29:27,760 We-- but k is not a-- 691 00:29:27,760 --> 00:29:29,170 k is a quantum number. 692 00:29:29,170 --> 00:29:31,240 But it can be continuous. 693 00:29:31,240 --> 00:29:33,530 It can be continuous. 694 00:29:33,530 --> 00:29:37,350 And you look at it in the first Brillouin zone. 695 00:29:37,350 --> 00:29:43,300 Now, somebody tell me, given this, why would 696 00:29:43,300 --> 00:29:46,605 I worry about k only-- 697 00:29:46,605 --> 00:29:47,980 AUDIENCE: Sorry to interrupt you. 698 00:29:47,980 --> 00:29:48,580 JEFFREY C. GROSSMAN: No problem. 699 00:29:48,580 --> 00:29:49,210 AUDIENCE: I'm just videotaping. 700 00:29:49,210 --> 00:29:51,420 And we're just hearing the mic move a lot. 701 00:29:51,420 --> 00:29:52,212 I'm just going to-- 702 00:29:52,212 --> 00:29:53,362 JEFFREY C. GROSSMAN: Oh. 703 00:29:53,362 --> 00:29:54,320 AUDIENCE: There you go. 704 00:29:54,320 --> 00:29:55,945 JEFFREY C. GROSSMAN: So I should stop-- 705 00:29:55,945 --> 00:29:56,680 I kind of-- no. 706 00:29:56,680 --> 00:29:57,790 AUDIENCE: [LAUGHTER] 707 00:29:58,300 --> 00:30:01,570 JEFFREY C. GROSSMAN: That rap got me a little bit excited. 708 00:30:01,570 --> 00:30:04,620 So let's go back to slide 1. 709 00:30:04,620 --> 00:30:06,342 AUDIENCE: [LAUGHTER] 710 00:30:09,320 --> 00:30:12,770 JEFFREY C. GROSSMAN: So I'm adding an inverse lattice 711 00:30:12,770 --> 00:30:16,340 vector, a full vector, that gets me to another lattice 712 00:30:16,340 --> 00:30:17,570 point in inverse space. 713 00:30:17,570 --> 00:30:20,330 And I say my wave function is the same. 714 00:30:20,330 --> 00:30:21,560 Now, why is it-- 715 00:30:21,560 --> 00:30:23,000 I also defined a Brillouin zone. 716 00:30:26,370 --> 00:30:28,770 And you can see my definition of the Brillouin zone 717 00:30:28,770 --> 00:30:31,400 is, in fact, that it's a zone that-- 718 00:30:31,400 --> 00:30:32,785 it's a lattice vector. 719 00:30:32,785 --> 00:30:34,410 You see, those are the lattice vectors. 720 00:30:34,410 --> 00:30:37,260 Why is it that I only now care about the variation of k 721 00:30:37,260 --> 00:30:38,760 within that first Brillouin zone? 722 00:30:38,760 --> 00:30:39,420 AUDIENCE: [INAUDIBLE] 723 00:30:39,420 --> 00:30:40,503 JEFFREY C. GROSSMAN: Yeah? 724 00:30:40,503 --> 00:30:42,090 AUDIENCE: Same thing. 725 00:30:42,090 --> 00:30:44,732 JEFFREY C. GROSSMAN: The same. 726 00:30:44,732 --> 00:30:46,940 That's why we care so much about that Brillouin zone. 727 00:30:46,940 --> 00:30:48,340 What is it? 728 00:30:48,340 --> 00:30:52,760 Well, it's the boundaries of the variability 729 00:30:52,760 --> 00:30:58,340 of the wave function in terms of how it depends on k. 730 00:30:58,340 --> 00:31:02,660 So I only need to vary k in this first zone. 731 00:31:02,660 --> 00:31:05,070 But I do need to vary it in that zone. 732 00:31:05,070 --> 00:31:08,240 I have to vary it in that zone because there will be variation 733 00:31:08,240 --> 00:31:11,050 if it's a solid. 734 00:31:11,050 --> 00:31:14,470 But if I vary it and then go out here, well, it's just-- 735 00:31:14,470 --> 00:31:15,490 I can translate it back. 736 00:31:15,490 --> 00:31:17,350 And it's the same thing, you see? 737 00:31:17,350 --> 00:31:23,140 So I need to move k around in this first zone. 738 00:31:23,140 --> 00:31:26,410 And when I do, and this is a slide I've showed before-- 739 00:31:26,410 --> 00:31:28,690 when I do, what you get is, as you 740 00:31:28,690 --> 00:31:32,170 go-- as you think about a system going from atoms to molecules 741 00:31:32,170 --> 00:31:35,410 to bands in your solid, you see this 742 00:31:35,410 --> 00:31:37,920 is the energy of that band. 743 00:31:37,920 --> 00:31:38,910 It wiggles. 744 00:31:38,910 --> 00:31:41,550 That's what happens in that first zone. 745 00:31:41,550 --> 00:31:42,930 You have a new quantum number. 746 00:31:42,930 --> 00:31:44,370 You change it. 747 00:31:44,370 --> 00:31:46,530 And you vary the energy. 748 00:31:46,530 --> 00:31:47,220 It wiggles. 749 00:31:47,220 --> 00:31:50,310 So now these orbitals are wiggly. 750 00:31:50,310 --> 00:31:52,080 And that's the big-- 751 00:31:52,080 --> 00:31:53,280 the most important thing. 752 00:31:53,280 --> 00:31:54,460 And you can see here-- 753 00:31:54,460 --> 00:31:57,060 so here's my k-space. 754 00:31:57,060 --> 00:32:01,900 And I can vary kx, ky, kz if it's three-dimensional solid. 755 00:32:01,900 --> 00:32:03,730 And at each point in k-space, you 756 00:32:03,730 --> 00:32:08,207 are going to get a set of levels, 757 00:32:08,207 --> 00:32:10,040 just like you did for that atom or molecule. 758 00:32:10,040 --> 00:32:11,540 You're going to get a set of levels. 759 00:32:11,540 --> 00:32:12,840 And they will vary. 760 00:32:12,840 --> 00:32:17,310 But they will be the same if I translate 761 00:32:17,310 --> 00:32:21,120 by the inverse lattice vector. 762 00:32:21,120 --> 00:32:24,210 So if I go from here out some G vector, 763 00:32:24,210 --> 00:32:25,920 then I get exactly the same levels. 764 00:32:25,920 --> 00:32:28,675 But if I'm inside this Brillouin zone, they're going to vary. 765 00:32:32,740 --> 00:32:34,360 Any questions? 766 00:32:34,360 --> 00:32:35,995 K-space is really important for solids. 767 00:32:41,080 --> 00:32:43,380 Mm-hmm. 768 00:32:43,380 --> 00:32:44,320 And so what do we do? 769 00:32:44,320 --> 00:32:46,020 Well-- so the inverse lattice-- 770 00:32:46,020 --> 00:32:47,700 it's related to the real space lattice. 771 00:32:47,700 --> 00:32:53,160 If I double the real space lattice so it's periodic, 772 00:32:53,160 --> 00:32:55,770 but with more atoms in the cell-- 773 00:32:55,770 --> 00:32:58,200 so it's periodic over a larger domain-- 774 00:32:58,200 --> 00:33:02,490 then, basically, I'm-- it's like a Fourier series. 775 00:33:02,490 --> 00:33:06,150 So if I go to larger real space lattice, then 776 00:33:06,150 --> 00:33:09,790 I have finer and finer k-space. 777 00:33:09,790 --> 00:33:13,570 And so what that means is that you 778 00:33:13,570 --> 00:33:14,950 can choose a certain k-mesh. 779 00:33:14,950 --> 00:33:19,090 This is a variable in the input now for solids. 780 00:33:19,090 --> 00:33:22,060 For solids, this is a new critical variable 781 00:33:22,060 --> 00:33:25,540 in your calculations of solids. 782 00:33:25,540 --> 00:33:26,917 You need a k-mesh. 783 00:33:26,917 --> 00:33:28,000 And this is just a number. 784 00:33:28,000 --> 00:33:29,800 You'll enter 8 or 2 or 4. 785 00:33:29,800 --> 00:33:31,900 And it'll just do a cube because it's simple. 786 00:33:31,900 --> 00:33:33,233 But you don't have to do a cube. 787 00:33:33,233 --> 00:33:35,650 But the code does a cube on the nanoHUB. 788 00:33:35,650 --> 00:33:40,180 So if I enter 8 by 8 by 8, I get 512 k-points on a mesh. 789 00:33:40,180 --> 00:33:45,760 And what that does is it allows me to-- 790 00:33:45,760 --> 00:33:51,280 it is-- it, effectively, is like increasing how much real space 791 00:33:51,280 --> 00:33:53,880 I had because they're related. 792 00:33:53,880 --> 00:33:56,910 If I had a larger unit cell with more real space, 793 00:33:56,910 --> 00:34:00,220 I'd have finer and finer k-space. 794 00:34:00,220 --> 00:34:02,310 And so you can think about these things 795 00:34:02,310 --> 00:34:03,900 as being able to go back and forth. 796 00:34:08,400 --> 00:34:12,449 Now, the reason that you don't go to-- and now, well, 797 00:34:12,449 --> 00:34:14,710 how many k-points do you need? 798 00:34:14,710 --> 00:34:17,340 Well, you need to have enough k-points-- 799 00:34:17,340 --> 00:34:21,780 you need to distribute k-points in your Brillouin zone such 800 00:34:21,780 --> 00:34:22,420 that-- 801 00:34:22,420 --> 00:34:22,920 what? 802 00:34:22,920 --> 00:34:23,670 What's the answer? 803 00:34:23,670 --> 00:34:25,679 How many k-points do I need? 804 00:34:25,679 --> 00:34:27,775 It's my favorite answer. 805 00:34:27,775 --> 00:34:28,650 AUDIENCE: It depends. 806 00:34:28,650 --> 00:34:30,840 JEFFREY C. GROSSMAN: It depends. 807 00:34:30,840 --> 00:34:32,010 What does it depend on? 808 00:34:32,010 --> 00:34:33,600 Did that come from the back? 809 00:34:33,600 --> 00:34:34,982 I love it. 810 00:34:34,982 --> 00:34:35,940 What does it depend on? 811 00:34:39,510 --> 00:34:41,052 AUDIENCE: The electrons [INAUDIBLE].. 812 00:34:41,052 --> 00:34:43,790 JEFFREY C. GROSSMAN: It might depend on the electrons. 813 00:34:43,790 --> 00:34:45,409 What else might it depend on? 814 00:34:45,409 --> 00:34:47,546 How many k-points do I need? 815 00:34:47,546 --> 00:34:50,360 AUDIENCE: [INAUDIBLE] 816 00:34:50,360 --> 00:34:51,500 JEFFREY C. GROSSMAN: Yes. 817 00:34:51,500 --> 00:34:52,969 Yes. 818 00:34:52,969 --> 00:34:54,659 Look at this picture. 819 00:34:54,659 --> 00:34:56,420 Let's look back at this picture. 820 00:34:56,420 --> 00:34:57,470 Think about this. 821 00:34:57,470 --> 00:35:00,170 And now try to think about how many k-points do I need. 822 00:35:03,570 --> 00:35:06,142 AUDIENCE: Enough [INAUDIBLE]. 823 00:35:06,142 --> 00:35:07,350 JEFFREY C. GROSSMAN: Exactly. 824 00:35:07,350 --> 00:35:09,330 I need enough. 825 00:35:09,330 --> 00:35:10,440 Now, what's enough? 826 00:35:10,440 --> 00:35:13,920 Well, enough so that you can resolve how 827 00:35:13,920 --> 00:35:15,260 the energy is going to change. 828 00:35:18,280 --> 00:35:28,210 So now tell me, if the wiggles are slow and not so much, 829 00:35:28,210 --> 00:35:32,560 if the wiggles are like this, or maybe even they're really flat, 830 00:35:32,560 --> 00:35:36,930 do I need a lot of k-points or a little bit of k-points? 831 00:35:36,930 --> 00:35:38,010 How many? 832 00:35:38,010 --> 00:35:40,390 A lot? 833 00:35:40,390 --> 00:35:42,060 Just a little bit because there's just 834 00:35:42,060 --> 00:35:43,780 not that much variation. 835 00:35:43,780 --> 00:35:47,040 But if the wiggles looked like this, 836 00:35:47,040 --> 00:35:49,040 would I need a lot or a little? 837 00:35:49,040 --> 00:35:51,960 And so this is the behavior of bands 838 00:35:51,960 --> 00:35:55,650 that look like this-- are more sort of metallicy. 839 00:35:55,650 --> 00:35:58,320 And the behavior of bands that look like this, the flat ones, 840 00:35:58,320 --> 00:36:00,240 are more "insulatorey." 841 00:36:02,750 --> 00:36:04,730 And in between-- and we'll see-- on Thursday, 842 00:36:04,730 --> 00:36:08,210 we'll talk about how you can actually calculate the mobility 843 00:36:08,210 --> 00:36:09,320 from these bands from this 844 00:36:09,320 --> 00:36:09,820 Wiggle. 845 00:36:12,200 --> 00:36:14,420 But you can-- how many do you need? 846 00:36:14,420 --> 00:36:15,300 Well, it depends. 847 00:36:15,300 --> 00:36:17,540 It depends on the nature of the material. 848 00:36:17,540 --> 00:36:19,010 But you have to converge that. 849 00:36:19,010 --> 00:36:21,080 That's part of that responsibility. 850 00:36:21,080 --> 00:36:25,620 Spiderman's uncle knew all about it. 851 00:36:25,620 --> 00:36:29,080 And so this is where we ended. 852 00:36:29,080 --> 00:36:30,950 Oh, man. 853 00:36:30,950 --> 00:36:31,580 Come on. 854 00:36:31,580 --> 00:36:33,142 There we go. 855 00:36:33,142 --> 00:36:34,100 This is where we ended. 856 00:36:34,100 --> 00:36:39,800 And so you have energy levels in the Brillouin zone. 857 00:36:39,800 --> 00:36:42,620 k is a continuous variable. 858 00:36:42,620 --> 00:36:45,393 But you don't actually continue it everywhere 859 00:36:45,393 --> 00:36:46,310 because it's too hard. 860 00:36:46,310 --> 00:36:47,840 So you just do-- what the computer 861 00:36:47,840 --> 00:36:50,750 does is a bunch of calculations at different k-points 862 00:36:50,750 --> 00:36:53,810 on the grid you give it. 863 00:36:53,810 --> 00:36:55,670 And what you're doing then is you're 864 00:36:55,670 --> 00:36:58,040 moving through-- when you give it different k-points, 865 00:36:58,040 --> 00:36:59,360 literally, this is what you do. 866 00:36:59,360 --> 00:37:01,490 You say, compute the-- 867 00:37:01,490 --> 00:37:04,010 what the computer is doing is it's saying, 868 00:37:04,010 --> 00:37:07,460 compute the energy levels at k-point 0, 0, 0. 869 00:37:07,460 --> 00:37:12,350 Now compute the energy levels at k-points point 1, point 1, 870 00:37:12,350 --> 00:37:13,820 point 1. 871 00:37:13,820 --> 00:37:16,370 And you literally just give it coordinates of k-points 872 00:37:16,370 --> 00:37:17,990 within the Brillouin zone. 873 00:37:17,990 --> 00:37:20,870 And for each of those k-points, it does the calculations. 874 00:37:20,870 --> 00:37:25,800 And it finds a set of energy levels, which are different. 875 00:37:25,800 --> 00:37:28,800 And those-- and then you fill that in. 876 00:37:28,800 --> 00:37:30,135 And you get your curvy bands. 877 00:37:33,090 --> 00:37:33,810 Who's with me? 878 00:37:33,810 --> 00:37:35,190 Who's got questions? 879 00:37:35,190 --> 00:37:37,830 Anyone have questions? 880 00:37:37,830 --> 00:37:40,620 That's the new-- yeah? 881 00:37:40,620 --> 00:37:42,727 AUDIENCE: So at each point, is a distinct number 882 00:37:42,727 --> 00:37:44,420 of energy states within the electrons? 883 00:37:45,807 --> 00:37:46,890 JEFFREY C. GROSSMAN: Yeah. 884 00:37:46,890 --> 00:37:50,370 Well, each k-point is the same number of energy states. 885 00:37:50,370 --> 00:37:54,320 But they can move around depending on where 886 00:37:54,320 --> 00:37:57,530 you are in that Brillouin zone. 887 00:37:57,530 --> 00:38:00,320 And their "moving-aroundness" is what gives you the band 888 00:38:00,320 --> 00:38:01,358 structure. 889 00:38:04,850 --> 00:38:07,482 So in the code, and I did this on the previous slide-- 890 00:38:07,482 --> 00:38:08,440 I didn't talk about it. 891 00:38:08,440 --> 00:38:09,070 It was the second point. 892 00:38:09,070 --> 00:38:11,200 But on the code, you-- each one of these you 893 00:38:11,200 --> 00:38:14,320 can think of as occupied by electrons 894 00:38:14,320 --> 00:38:19,270 because it's looking at this k-point and then this k-point. 895 00:38:19,270 --> 00:38:21,040 But when you think about the full band 896 00:38:21,040 --> 00:38:24,490 structure of a material, that is not how you look at it. 897 00:38:24,490 --> 00:38:27,132 You don't have-- you would-- 898 00:38:27,132 --> 00:38:28,840 as you keep increasing your k-point mesh, 899 00:38:28,840 --> 00:38:31,750 you would have infinite number of electrons in a band. 900 00:38:31,750 --> 00:38:32,750 That is not the case. 901 00:38:32,750 --> 00:38:35,320 What you have is the same as before, 902 00:38:35,320 --> 00:38:38,890 the same exact thing as molecular states, 903 00:38:38,890 --> 00:38:40,690 which is that you can have-- one band 904 00:38:40,690 --> 00:38:44,180 can hold two electrons-- one up, one down. 905 00:38:44,180 --> 00:38:48,600 That's exactly the same as in a molecular state. 906 00:38:53,810 --> 00:38:56,810 But depending on where you are in the Brillouin zone, 907 00:38:56,810 --> 00:38:59,540 the energy of that band now can change dramatically. 908 00:38:59,540 --> 00:39:01,580 And that changes everything in terms 909 00:39:01,580 --> 00:39:02,810 of properties of material. 910 00:39:02,810 --> 00:39:04,610 In particular, what we're focusing on here 911 00:39:04,610 --> 00:39:07,190 is optical properties and electronic properties 912 00:39:07,190 --> 00:39:13,060 because those are very quantum mechanically, which we like. 913 00:39:13,060 --> 00:39:16,780 Now-- and the Fermi energy is in the code-- 914 00:39:16,780 --> 00:39:20,080 we talked about the Fermi energy a little bit for molecules. 915 00:39:20,080 --> 00:39:25,060 It's sort of the energy at which you stop filling. 916 00:39:25,060 --> 00:39:28,417 It's the energy-- so you fill your bands up 917 00:39:28,417 --> 00:39:29,250 to the Fermi energy. 918 00:39:29,250 --> 00:39:31,210 You can just think about it that way. 919 00:39:31,210 --> 00:39:33,947 Don't worry about-- so much about whether it's in the gap 920 00:39:33,947 --> 00:39:35,030 or where it is in the gap. 921 00:39:35,030 --> 00:39:38,050 We don't need to worry about that in this class. 922 00:39:38,050 --> 00:39:41,050 But it is the energy at where-- at which you stop filling. 923 00:39:41,050 --> 00:39:45,300 And so you can see that if you-- 924 00:39:45,300 --> 00:39:47,020 I have some properties. 925 00:39:47,020 --> 00:39:48,640 I thought I had somebody else. 926 00:39:48,640 --> 00:39:58,650 But if you were to fill these and the bands-- 927 00:39:58,650 --> 00:40:02,230 and these bands didn't stop nicely, like this, 928 00:40:02,230 --> 00:40:05,598 but they just crossed over it, what would you have? 929 00:40:05,598 --> 00:40:07,140 What kind of material would you have? 930 00:40:12,047 --> 00:40:13,450 AUDIENCE: [INAUDIBLE] 931 00:40:13,450 --> 00:40:15,650 JEFFREY C. GROSSMAN: Yeah. 932 00:40:15,650 --> 00:40:17,690 It's a metal. 933 00:40:17,690 --> 00:40:19,700 It's a metal. 934 00:40:19,700 --> 00:40:23,720 And if you have the bands stop where you stop filling 935 00:40:23,720 --> 00:40:27,060 and then there's a gap, it's a semiconductor or an insulator. 936 00:40:30,360 --> 00:40:33,690 Let's do a few calculations. 937 00:40:33,690 --> 00:40:37,110 And then I'll come back to structural properties. 938 00:40:37,110 --> 00:40:41,775 We are on our favorite place, the HUB of nano. 939 00:40:46,770 --> 00:40:48,580 Come on. 940 00:40:48,580 --> 00:40:49,430 Yes! 941 00:40:49,430 --> 00:40:50,365 We got to that stage. 942 00:40:52,870 --> 00:40:55,940 Launch Tool-- it sounds like a good idea. 943 00:40:55,940 --> 00:40:59,800 And this is where we had some issues last time-- 944 00:40:59,800 --> 00:41:01,210 not going to kid you. 945 00:41:01,210 --> 00:41:03,730 There it is. 946 00:41:03,730 --> 00:41:06,400 And we go to here. 947 00:41:06,400 --> 00:41:08,600 Now, let's do a few things. 948 00:41:12,010 --> 00:41:17,450 And so this is the same as-- 949 00:41:17,450 --> 00:41:18,950 so it's not letting me scroll again. 950 00:41:18,950 --> 00:41:20,143 That's wonderful. 951 00:41:26,630 --> 00:41:28,250 There it is. 952 00:41:28,250 --> 00:41:34,400 Now, this is the same as the tool 953 00:41:34,400 --> 00:41:38,200 that you're using for the homework number 2 954 00:41:38,200 --> 00:41:39,770 to look at molecules. 955 00:41:39,770 --> 00:41:44,420 And in looking at molecules, you just select Molecule. 956 00:41:44,420 --> 00:41:48,380 But all the code is doing when you look at molecules-- 957 00:41:48,380 --> 00:41:51,470 all the code is doing is it's saying, 958 00:41:51,470 --> 00:41:55,370 well, whatever you give me, whatever sort of thing 959 00:41:55,370 --> 00:42:00,800 you give me, I'm going to put space in my periodic cell. 960 00:42:04,900 --> 00:42:06,940 I'm going to put enough space so that it 961 00:42:06,940 --> 00:42:08,200 doesn't see its own image. 962 00:42:11,930 --> 00:42:12,980 That's all it's doing. 963 00:42:12,980 --> 00:42:15,260 It's still a periodic calculation. 964 00:42:15,260 --> 00:42:18,500 It's just putting a whole bunch of space in there. 965 00:42:18,500 --> 00:42:22,940 You could choose Solid and do molecule calculations 966 00:42:22,940 --> 00:42:28,800 and just make sure that there's space between what you put in. 967 00:42:28,800 --> 00:42:34,020 However, when you choose Solid, then you 968 00:42:34,020 --> 00:42:36,390 can see that it's actually entering the coordinates 969 00:42:36,390 --> 00:42:37,520 in a different way. 970 00:42:37,520 --> 00:42:40,740 It's using a slightly different language. 971 00:42:40,740 --> 00:42:46,850 Has anybody seen this beautiful non-scrolling thing? 972 00:42:46,850 --> 00:42:48,770 Man. 973 00:42:48,770 --> 00:42:49,520 Oh, yeah. 974 00:42:49,520 --> 00:42:51,830 Well, it's good to be challenged. 975 00:42:51,830 --> 00:42:52,580 That's fun. 976 00:42:55,490 --> 00:43:00,050 So-- you gotta be kidding me. 977 00:43:02,780 --> 00:43:04,632 Did you see that coming? 978 00:43:04,632 --> 00:43:05,840 Did somebody see that coming? 979 00:43:08,720 --> 00:43:10,520 Should we go back to the-- 980 00:43:10,520 --> 00:43:11,660 to DJ Dave? 981 00:43:11,660 --> 00:43:12,160 Wait. 982 00:43:15,080 --> 00:43:16,805 What if I did Firefox? 983 00:43:19,610 --> 00:43:21,910 Is that better? 984 00:43:21,910 --> 00:43:23,440 It's the same. 985 00:43:23,440 --> 00:43:24,520 That's beautiful. 986 00:43:24,520 --> 00:43:25,480 You get the same thing? 987 00:43:29,280 --> 00:43:31,440 It did remember me. 988 00:43:31,440 --> 00:43:33,860 Now let's close that. 989 00:43:33,860 --> 00:43:35,944 How often does that happen? 990 00:43:35,944 --> 00:43:38,180 AUDIENCE: [INAUDIBLE] 991 00:43:38,180 --> 00:43:41,102 JEFFREY C. GROSSMAN: But what percentage, would you say? 992 00:43:41,102 --> 00:43:43,787 AUDIENCE: [INAUDIBLE] 993 00:43:43,787 --> 00:43:44,870 AUDIENCE: 60% of the time. 994 00:43:44,870 --> 00:43:47,820 JEFFREY C. GROSSMAN: 60% of the time? 995 00:43:47,820 --> 00:43:48,553 60-- 996 00:43:48,553 --> 00:43:51,803 AUDIENCE: I feel the nanoHub [INAUDIBLE] are generally not 997 00:43:51,803 --> 00:43:55,070 stable for the Macintosh machines and for the Unix-- 998 00:43:55,070 --> 00:43:56,570 JEFFREY C. GROSSMAN: Oh, no, no, no. 999 00:43:56,570 --> 00:43:58,550 You can't be blaming this on a Mac. 1000 00:43:58,550 --> 00:44:00,010 This is-- because it's not a Mac. 1001 00:44:00,010 --> 00:44:01,250 AUDIENCE: Not a Macintosh. 1002 00:44:01,250 --> 00:44:01,700 JEFFREY C. GROSSMAN: Thank you. 1003 00:44:01,700 --> 00:44:02,400 AUDIENCE: I mean the software. 1004 00:44:02,400 --> 00:44:03,817 JEFFREY C. GROSSMAN: That must be. 1005 00:44:03,817 --> 00:44:07,910 AUDIENCE: I think the software is unfriendly for Windows. 1006 00:44:07,910 --> 00:44:11,315 JEFFREY C. GROSSMAN: Well, that's a big problem. 1007 00:44:11,315 --> 00:44:11,940 So here we are. 1008 00:44:11,940 --> 00:44:13,250 This looks a little bit better. 1009 00:44:13,250 --> 00:44:15,350 Now, when you choose Solid-- 1010 00:44:15,350 --> 00:44:16,910 so now we have-- look at this. 1011 00:44:16,910 --> 00:44:17,772 [GASPING] 1012 00:44:19,100 --> 00:44:22,078 You can choose your crystal symmetry because it's a solid. 1013 00:44:22,078 --> 00:44:23,870 And remember those symmetries that you saw? 1014 00:44:23,870 --> 00:44:26,270 Well, these are those symmetries except now 1015 00:44:26,270 --> 00:44:28,670 you have to choose a lattice constant. 1016 00:44:28,670 --> 00:44:30,890 See, when you did the molecule, the lattice constant 1017 00:44:30,890 --> 00:44:34,190 was simply big. 1018 00:44:34,190 --> 00:44:36,200 It was just big so that the molecule 1019 00:44:36,200 --> 00:44:38,245 didn't fill its periodic image. 1020 00:44:38,245 --> 00:44:39,620 But now that we're doing a solid, 1021 00:44:39,620 --> 00:44:41,390 the lattice constant has to be, well, 1022 00:44:41,390 --> 00:44:44,960 like the lattice constant of the material. 1023 00:44:44,960 --> 00:44:47,540 And depending on the symmetry of the material, 1024 00:44:47,540 --> 00:44:49,610 you're going to get different lattice constants. 1025 00:44:49,610 --> 00:44:52,370 So if I-- now-- and have any of you 1026 00:44:52,370 --> 00:44:54,938 seen fractional coordinates before? 1027 00:44:54,938 --> 00:44:56,230 Does anybody know what this is? 1028 00:44:58,810 --> 00:45:03,360 So this is actually an important language of solids where, 1029 00:45:03,360 --> 00:45:04,590 oftentimes-- 1030 00:45:04,590 --> 00:45:08,280 so with a solid, you have a periodically repeating space 1031 00:45:08,280 --> 00:45:10,560 that is set by a lattice constant. 1032 00:45:10,560 --> 00:45:14,820 And so often, it's simpler to think about the coordinates 1033 00:45:14,820 --> 00:45:17,580 that go inside that space-- the basis, if you want-- 1034 00:45:17,580 --> 00:45:21,390 as fractions of the lattice. 1035 00:45:21,390 --> 00:45:24,720 So for silicon, actually-- 1036 00:45:24,720 --> 00:45:30,180 for silicon, it's an FCC crystal with a basis 1037 00:45:30,180 --> 00:45:34,500 at one corner of the FCC and another atom 1038 00:45:34,500 --> 00:45:38,140 at a quarter-quarter-quarter out in terms of the lattice. 1039 00:45:38,140 --> 00:45:40,350 This is not at a quarter-quarter-quarter 1040 00:45:40,350 --> 00:45:42,210 angstroms. 1041 00:45:42,210 --> 00:45:44,880 This is a quarter-quarter-quarter scaled 1042 00:45:44,880 --> 00:45:46,820 by that lattice-- 1043 00:45:46,820 --> 00:45:48,400 does everybody see that-- 1044 00:45:48,400 --> 00:45:51,995 which is a cubic face-centered lattice. 1045 00:45:51,995 --> 00:45:53,370 So that's fractional coordinates. 1046 00:45:53,370 --> 00:45:55,550 You can go back and forth. 1047 00:45:55,550 --> 00:45:58,980 You can write a 10-line Python script to go back and forth 1048 00:45:58,980 --> 00:45:59,700 between the two. 1049 00:45:59,700 --> 00:46:03,120 But in solids, often we like to think about positions in terms 1050 00:46:03,120 --> 00:46:04,420 of the symmetry of the lattice. 1051 00:46:04,420 --> 00:46:07,320 And so that's how we write them. 1052 00:46:07,320 --> 00:46:11,430 Now, in this case, I have two silicon atoms. 1053 00:46:11,430 --> 00:46:20,580 And we're going to ignore everything and just run it. 1054 00:46:20,580 --> 00:46:24,030 Is that how you guys do it, anyway, you just simulate? 1055 00:46:24,030 --> 00:46:28,070 It's fun to do that, just simulate, 1056 00:46:28,070 --> 00:46:32,490 except that you need to know what you did. 1057 00:46:32,490 --> 00:46:36,820 Now, that's what I want to ask you about-- is what did I do? 1058 00:46:36,820 --> 00:46:39,150 Oh, look at that. 1059 00:46:39,150 --> 00:46:40,410 This is solid. 1060 00:46:40,410 --> 00:46:42,090 I only gave it two atoms. 1061 00:46:42,090 --> 00:46:43,790 And yet, because it has the symmetry 1062 00:46:43,790 --> 00:46:46,415 in the lattice and all that, it knows how to construct a solid. 1063 00:46:46,415 --> 00:46:48,173 And that's what it looks like. 1064 00:46:48,173 --> 00:46:49,590 When the computer thinks about it, 1065 00:46:49,590 --> 00:46:52,660 it thinks about it like that. 1066 00:46:52,660 --> 00:46:53,160 Uh-oh. 1067 00:46:56,100 --> 00:46:59,950 Where's the-- oh, here we go. 1068 00:46:59,950 --> 00:47:07,020 Now, key outputs-- it's telling you that it has-- well, these 1069 00:47:07,020 --> 00:47:08,760 should really be 4. 1070 00:47:08,760 --> 00:47:13,330 It has four up electrons and four down electrons. 1071 00:47:13,330 --> 00:47:15,350 That doesn't sound like that many electrons. 1072 00:47:15,350 --> 00:47:18,060 How come I only have eight electrons in my simulation 1073 00:47:18,060 --> 00:47:21,700 and I'm simulating an infinite crystal? 1074 00:47:21,700 --> 00:47:24,687 AUDIENCE: [INAUDIBLE] 1075 00:47:24,687 --> 00:47:25,770 JEFFREY C. GROSSMAN: Yeah. 1076 00:47:25,770 --> 00:47:26,790 So very good. 1077 00:47:26,790 --> 00:47:28,562 But how come-- so it's only the valence 1078 00:47:28,562 --> 00:47:30,520 because that's where all the chemistry happens. 1079 00:47:30,520 --> 00:47:31,530 We talked about that. 1080 00:47:31,530 --> 00:47:35,130 But why are the only two atoms? 1081 00:47:35,130 --> 00:47:38,130 How can I simulate a solid with only two atoms? 1082 00:47:38,130 --> 00:47:39,990 AUDIENCE: [INAUDIBLE] 1083 00:47:39,990 --> 00:47:41,850 AUDIENCE: Because it's symmetric. 1084 00:47:41,850 --> 00:47:44,100 JEFFREY C. GROSSMAN: Because it's symmetric. 1085 00:47:44,100 --> 00:47:48,640 However, because I only have two atoms, especially, 1086 00:47:48,640 --> 00:47:50,860 and it's symmetric and it repeats, 1087 00:47:50,860 --> 00:47:54,930 I need to worry about the convergence of what, which 1088 00:47:54,930 --> 00:47:56,686 we just talked about? 1089 00:47:56,686 --> 00:47:59,130 AUDIENCE: [INAUDIBLE] 1090 00:47:59,130 --> 00:48:01,760 JEFFREY C. GROSSMAN: And so you saw-- the density 1091 00:48:01,760 --> 00:48:03,333 of states we talked about. 1092 00:48:03,333 --> 00:48:04,250 Oh, this is beautiful. 1093 00:48:04,250 --> 00:48:04,940 Look at this. 1094 00:48:04,940 --> 00:48:06,320 It's beautiful. 1095 00:48:06,320 --> 00:48:09,770 Those are the peaks of the solid of crystalline silicon. 1096 00:48:09,770 --> 00:48:11,510 It's just like in the molecules. 1097 00:48:11,510 --> 00:48:13,580 In the molecules, we had the levels. 1098 00:48:13,580 --> 00:48:16,290 And we turned them on their side. 1099 00:48:16,290 --> 00:48:17,790 We turned the levels on their side. 1100 00:48:17,790 --> 00:48:19,740 And we said that's the same as the DoS. 1101 00:48:19,740 --> 00:48:23,520 But in the solid, it's the band structure we turn on its side. 1102 00:48:23,520 --> 00:48:24,998 Here's the band structure. 1103 00:48:27,670 --> 00:48:29,980 So those are my levels now. 1104 00:48:29,980 --> 00:48:30,850 Those are my levels. 1105 00:48:30,850 --> 00:48:34,210 They're wiggly in reciprocal space. 1106 00:48:34,210 --> 00:48:34,990 But if I turn-- 1107 00:48:34,990 --> 00:48:36,580 I can still turn them on their side. 1108 00:48:36,580 --> 00:48:40,870 And when I do that, I get the density of states, 1109 00:48:40,870 --> 00:48:44,770 directly comparable to the situation of the molecules. 1110 00:48:44,770 --> 00:48:47,110 But in the molecules, what I turned on its side 1111 00:48:47,110 --> 00:48:48,760 didn't wiggle. 1112 00:48:48,760 --> 00:48:50,990 Does everybody see that? 1113 00:48:50,990 --> 00:48:52,870 So there's the band structure. 1114 00:48:52,870 --> 00:48:56,980 Now, this is-- the nanoHUB has issues with putting 1115 00:48:56,980 --> 00:48:58,060 Greek letters on plots. 1116 00:48:58,060 --> 00:49:00,790 So this is the band structure with the Greek letters. 1117 00:49:00,790 --> 00:49:03,400 And I'll talk about-- well, actually, I don't need to wait. 1118 00:49:03,400 --> 00:49:05,500 These just correspond to different points 1119 00:49:05,500 --> 00:49:07,750 in that Brillouin zone. 1120 00:49:07,750 --> 00:49:10,660 You cruise through phase space in the Brillouin zone. 1121 00:49:10,660 --> 00:49:12,460 And you plot the energies-- 1122 00:49:12,460 --> 00:49:16,700 the code is plotting the energies along that path. 1123 00:49:16,700 --> 00:49:19,450 So you get a sense of the variation. 1124 00:49:19,450 --> 00:49:23,220 Could these energies vary differently 1125 00:49:23,220 --> 00:49:26,360 than I've plotted here? 1126 00:49:26,360 --> 00:49:27,905 Could they vary differently? 1127 00:49:33,950 --> 00:49:35,390 How could they vary differently? 1128 00:49:35,390 --> 00:49:36,140 The answer is yes. 1129 00:49:44,080 --> 00:49:46,200 This is a cruise through phase space. 1130 00:49:46,200 --> 00:49:48,000 Gamma is always the origin. 1131 00:49:48,000 --> 00:49:50,640 Gamma's at k-point 0, 0, 0. 1132 00:49:50,640 --> 00:49:53,170 And then I went to some other k-point. 1133 00:49:53,170 --> 00:49:54,910 And we called it X. We like to give 1134 00:49:54,910 --> 00:49:57,460 high-symmetry points in the Brillouin zone 1135 00:49:57,460 --> 00:50:03,250 a letter, a Greek letter, or not. 1136 00:50:03,250 --> 00:50:04,780 And then I went back to gamma. 1137 00:50:04,780 --> 00:50:08,440 Here, I went to a point we called L. And I-- 1138 00:50:08,440 --> 00:50:10,570 the way you do this is you just, basically, 1139 00:50:10,570 --> 00:50:13,600 go in a line from this point to that point. 1140 00:50:13,600 --> 00:50:17,470 And you plot the change in the energy of each band 1141 00:50:17,470 --> 00:50:20,770 as you go from point to point in the Brillouin zone. 1142 00:50:20,770 --> 00:50:23,770 So now, tell me, are those the only variations 1143 00:50:23,770 --> 00:50:25,380 that you could have? 1144 00:50:25,380 --> 00:50:25,890 Why not? 1145 00:50:30,990 --> 00:50:33,108 Tell me why not. 1146 00:50:33,108 --> 00:50:35,400 AUDIENCE: If you choose a different path, [INAUDIBLE].. 1147 00:50:35,400 --> 00:50:37,800 JEFFREY C. GROSSMAN: If you choose a different path. 1148 00:50:37,800 --> 00:50:43,530 This is just one path going from L to gamma to X. You see, 1149 00:50:43,530 --> 00:50:45,210 here we are. 1150 00:50:45,210 --> 00:50:48,720 I went-- you see, here they are for FCC, 1151 00:50:48,720 --> 00:50:50,070 I think, or maybe it's BCC. 1152 00:50:50,070 --> 00:50:53,700 I went from gamma to L to X. I went just 1153 00:50:53,700 --> 00:50:55,800 along a line in this Brillouin zone. 1154 00:50:55,800 --> 00:51:01,160 But what if I had gone like this or what if I'd done like this? 1155 00:51:01,160 --> 00:51:03,500 Would I get different curvatures? 1156 00:51:03,500 --> 00:51:06,830 You better believe it. 1157 00:51:06,830 --> 00:51:10,520 You are taking paths through the Brillouin zone 1158 00:51:10,520 --> 00:51:11,900 to get the band structure. 1159 00:51:11,900 --> 00:51:16,100 That's how you get the band structure. 1160 00:51:16,100 --> 00:51:25,000 Now, what ends up happening is that the interesting variations 1161 00:51:25,000 --> 00:51:28,390 in these energy bands, the interesting ones, 1162 00:51:28,390 --> 00:51:30,670 the ones that are important for optical and electronic 1163 00:51:30,670 --> 00:51:33,640 properties, tend to happen between the high-symmetry 1164 00:51:33,640 --> 00:51:35,200 points in the Brillouin zone. 1165 00:51:35,200 --> 00:51:38,200 They tend to happen between the origin and the center 1166 00:51:38,200 --> 00:51:41,770 of this face and the center of this edge 1167 00:51:41,770 --> 00:51:43,870 and going back to an origin. 1168 00:51:43,870 --> 00:51:48,640 They tend to happen between the path of one high-symmetry point 1169 00:51:48,640 --> 00:51:49,760 to another. 1170 00:51:49,760 --> 00:51:54,250 And so, de facto, we just don't care 1171 00:51:54,250 --> 00:51:56,800 much about everything else in the Brillouin zone 1172 00:51:56,800 --> 00:52:00,440 because it doesn't really give us extra information. 1173 00:52:00,440 --> 00:52:02,720 It would give us a different band structure. 1174 00:52:02,720 --> 00:52:04,550 But it's not that-- 1175 00:52:04,550 --> 00:52:08,660 it's not giving us something that's 1176 00:52:08,660 --> 00:52:11,540 as relevant to the key fundamental properties 1177 00:52:11,540 --> 00:52:14,190 we care about. 1178 00:52:14,190 --> 00:52:16,170 So that's one thing. 1179 00:52:16,170 --> 00:52:17,930 Now-- so that's where those-- 1180 00:52:17,930 --> 00:52:20,150 now, the code just picks the paths. 1181 00:52:20,150 --> 00:52:23,170 But in all these codes, you can enter whatever path you want. 1182 00:52:23,170 --> 00:52:24,920 So if you want to cruise around in k-space 1183 00:52:24,920 --> 00:52:27,560 and get energy variations, you can do it. 1184 00:52:27,560 --> 00:52:30,650 You are not limited. 1185 00:52:30,650 --> 00:52:33,680 Now, is this the right band structure for silicon? 1186 00:52:44,328 --> 00:52:45,808 AUDIENCE: [INAUDIBLE] 1187 00:52:45,808 --> 00:52:47,600 JEFFREY C. GROSSMAN: Let's go back to our-- 1188 00:52:47,600 --> 00:52:50,330 not bad-- looks pretty good. 1189 00:52:50,330 --> 00:52:51,620 I like that. 1190 00:52:51,620 --> 00:52:55,860 That's why I want to-- now, if we go back to the input, 1191 00:52:55,860 --> 00:52:57,470 what is it that-- 1192 00:52:57,470 --> 00:53:01,770 of these parameters, what is it that I might need to check? 1193 00:53:01,770 --> 00:53:03,470 AUDIENCE: [INAUDIBLE] 1194 00:53:03,470 --> 00:53:03,880 JEFFREY C. GROSSMAN: What is that? 1195 00:53:03,880 --> 00:53:04,505 Say that again. 1196 00:53:04,505 --> 00:53:05,915 AUDIENCE: [INAUDIBLE] 1197 00:53:05,915 --> 00:53:07,290 JEFFREY C. GROSSMAN: So I might-- 1198 00:53:07,290 --> 00:53:09,630 I have a 4-by-4 grid. 1199 00:53:09,630 --> 00:53:12,690 Would I get a-- would I get curvy band structures if I do 1200 00:53:12,690 --> 00:53:13,740 a 1-by-1-by-1? 1201 00:53:16,450 --> 00:53:17,140 Let's see. 1202 00:53:17,140 --> 00:53:20,340 Let's try it. 1203 00:53:20,340 --> 00:53:21,000 I'll do 14. 1204 00:53:26,920 --> 00:53:30,340 This seems to be stable for now. 1205 00:53:30,340 --> 00:53:33,670 Uh-oh, where's my Simulate button? 1206 00:53:45,710 --> 00:53:47,340 There it is. 1207 00:53:47,340 --> 00:53:54,290 So I'm doing a 1-by-1-by-1 k-point. 1208 00:53:54,290 --> 00:53:56,310 This is another key concept I want 1209 00:53:56,310 --> 00:53:59,000 to make sure we understand. 1210 00:53:59,000 --> 00:53:59,630 There it is. 1211 00:53:59,630 --> 00:54:00,740 There's silicon. 1212 00:54:00,740 --> 00:54:02,690 I didn't change anything about the real space. 1213 00:54:02,690 --> 00:54:04,482 I didn't change anything about the k-space. 1214 00:54:04,482 --> 00:54:09,020 I just only evaluated the density at one point. 1215 00:54:09,020 --> 00:54:12,800 Now, here's what's going to perhaps surprise 1216 00:54:12,800 --> 00:54:15,200 you, which is that the band structure still curves. 1217 00:54:18,410 --> 00:54:23,510 Now-- and it curves differently, actually, 1218 00:54:23,510 --> 00:54:27,410 than it did when I had 4-by-4-by-4. 1219 00:54:27,410 --> 00:54:28,610 Why would it still curve? 1220 00:54:32,730 --> 00:54:35,400 This is actually something I didn't talk about yet. 1221 00:54:35,400 --> 00:54:38,950 But it is an aside that I was going to talk about. 1222 00:54:38,950 --> 00:54:43,053 But since I have it here, I'll just talk about it now. 1223 00:54:43,053 --> 00:54:44,970 I probably could have done all that internally 1224 00:54:44,970 --> 00:54:46,980 without voicing it. 1225 00:54:46,980 --> 00:54:50,220 But here we are. 1226 00:54:50,220 --> 00:54:56,970 So you see, when you calculate the band structure, 1227 00:54:56,970 --> 00:55:00,610 what you do is you do two things. 1228 00:55:00,610 --> 00:55:04,620 First, you converge the wave function in density. 1229 00:55:04,620 --> 00:55:08,640 First, you converge the density at the k-points I input 1230 00:55:08,640 --> 00:55:11,920 into the input. 1231 00:55:11,920 --> 00:55:15,130 So if I have a 1-by-1-by-1 k-point density, well, 1232 00:55:15,130 --> 00:55:17,897 that gives me some grid. 1233 00:55:17,897 --> 00:55:19,980 That gives me some density and some wave function. 1234 00:55:22,510 --> 00:55:25,900 Now, then to calculate the band structure, what the code does 1235 00:55:25,900 --> 00:55:27,400 is it does a second step. 1236 00:55:27,400 --> 00:55:29,410 It takes that converged density. 1237 00:55:29,410 --> 00:55:31,750 And it cruises around in k-space. 1238 00:55:31,750 --> 00:55:33,880 And that's how it plots this out. 1239 00:55:33,880 --> 00:55:35,830 But it doesn't change the density anymore. 1240 00:55:35,830 --> 00:55:37,747 It doesn't change the converged wave function. 1241 00:55:37,747 --> 00:55:39,430 It just evaluates it at all kinds 1242 00:55:39,430 --> 00:55:41,540 of different points in k-space. 1243 00:55:41,540 --> 00:55:43,500 So there's two steps. 1244 00:55:43,500 --> 00:55:48,020 First, you converge the density at some number of k-points. 1245 00:55:48,020 --> 00:55:50,870 Then you cruise around in the Brillouin zone 1246 00:55:50,870 --> 00:55:52,490 with that density. 1247 00:55:52,490 --> 00:55:54,710 And you evaluate your functions. 1248 00:55:54,710 --> 00:55:59,060 And sure, the wrong density will still 1249 00:55:59,060 --> 00:56:01,820 wiggle as you move around in k-space. 1250 00:56:01,820 --> 00:56:03,140 It'll still wiggle. 1251 00:56:03,140 --> 00:56:04,700 But it's the wrong density because I 1252 00:56:04,700 --> 00:56:07,990 didn't have enough k-points in the first place to converge it. 1253 00:56:07,990 --> 00:56:08,990 Does everybody see that? 1254 00:56:12,510 --> 00:56:14,500 So now, how many k-points do I need? 1255 00:56:14,500 --> 00:56:16,320 Well, the number of k-points I need 1256 00:56:16,320 --> 00:56:20,110 is when this converges and doesn't change anymore 1257 00:56:20,110 --> 00:56:22,450 if the band structure-- if properties of the band 1258 00:56:22,450 --> 00:56:24,040 structure are what I want. 1259 00:56:24,040 --> 00:56:28,050 That's how many k-points I need. 1260 00:56:28,050 --> 00:56:29,610 Does everybody see that? 1261 00:56:29,610 --> 00:56:30,600 That's the difference. 1262 00:56:30,600 --> 00:56:36,240 Now-- no. 1263 00:56:36,240 --> 00:56:37,560 That wasn't good. 1264 00:56:37,560 --> 00:56:39,310 Here we go. 1265 00:56:39,310 --> 00:56:41,410 Oh, yeah. 1266 00:56:41,410 --> 00:56:43,390 These so-called web browsers-- 1267 00:56:43,390 --> 00:56:46,605 Marc Andreessen was my student, by the way, that-- 1268 00:56:46,605 --> 00:56:48,770 does anybody know who he is? 1269 00:56:48,770 --> 00:56:50,432 No one knows who he is. 1270 00:56:50,432 --> 00:56:51,800 I'm dated. 1271 00:56:51,800 --> 00:56:53,120 No. 1272 00:56:53,120 --> 00:56:55,100 He was at Illinois, where I did my PhD. 1273 00:56:55,100 --> 00:56:56,240 And I was his TA. 1274 00:56:56,240 --> 00:57:01,300 And he went and founded Netscape-- 1275 00:57:01,300 --> 00:57:01,800 good guy. 1276 00:57:04,485 --> 00:57:06,360 That's not one of my browser options anymore. 1277 00:57:06,360 --> 00:57:08,880 So what's up with that? 1278 00:57:08,880 --> 00:57:13,320 Now, what I want to do now is the following. 1279 00:57:13,320 --> 00:57:16,340 I'm going to go to one atom. 1280 00:57:20,430 --> 00:57:22,957 And what I want to know is-- 1281 00:57:22,957 --> 00:57:23,790 and you can do that. 1282 00:57:23,790 --> 00:57:27,450 Just like before, you can mess around. 1283 00:57:27,450 --> 00:57:32,247 One atom-- we'll just leave it at one k-point. 1284 00:57:32,247 --> 00:57:34,830 And what I want to know is, am I going to get a band structure 1285 00:57:34,830 --> 00:57:37,560 with one atom in the unit cell? 1286 00:57:37,560 --> 00:57:38,640 Oh, there it is. 1287 00:57:38,640 --> 00:57:41,082 They're kind of far apart. 1288 00:57:41,082 --> 00:57:42,040 What do you guys think? 1289 00:57:45,060 --> 00:57:47,520 That's one atom in the corner of an FCC cell. 1290 00:57:55,130 --> 00:57:57,740 Will I get a band structure? 1291 00:57:57,740 --> 00:58:00,173 Just say yes or no. 1292 00:58:00,173 --> 00:58:01,650 AUDIENCE: Yes. 1293 00:58:01,650 --> 00:58:03,410 JEFFREY C. GROSSMAN: There it is. 1294 00:58:03,410 --> 00:58:06,578 Now, why is it curving? 1295 00:58:06,578 --> 00:58:07,870 Let me ask it a different way. 1296 00:58:07,870 --> 00:58:10,900 How could I get these bands to stop curving? 1297 00:58:10,900 --> 00:58:12,732 Just stop curving already! 1298 00:58:12,732 --> 00:58:14,090 AUDIENCE: [INAUDIBLE] 1299 00:58:14,090 --> 00:58:14,840 JEFFREY C. GROSSMAN: What is that? 1300 00:58:14,840 --> 00:58:16,250 AUDIENCE: Larger k-mesh space? 1301 00:58:16,250 --> 00:58:19,473 JEFFREY C. GROSSMAN: Larger k-mesh space or other space? 1302 00:58:19,473 --> 00:58:20,640 AUDIENCE: [INAUDIBLE] space. 1303 00:58:20,640 --> 00:58:24,290 JEFFREY C. GROSSMAN: Why are they curving again? 1304 00:58:24,290 --> 00:58:25,700 Why are these things curving? 1305 00:58:30,947 --> 00:58:34,286 AUDIENCE: [INAUDIBLE] 1306 00:58:35,217 --> 00:58:36,300 JEFFREY C. GROSSMAN: Yeah. 1307 00:58:36,300 --> 00:58:38,860 It has to do with that. 1308 00:58:38,860 --> 00:58:41,880 But why do you get-- we talked a lot in the last half-hour 1309 00:58:41,880 --> 00:58:45,360 about how you get this variation in k-space of the band. 1310 00:58:45,360 --> 00:58:46,490 Why do you get that, again? 1311 00:58:50,210 --> 00:58:52,287 Why do you get that? 1312 00:58:52,287 --> 00:58:53,870 You get it because of Bloch's theorem. 1313 00:58:53,870 --> 00:58:54,662 I'll tell you that. 1314 00:58:54,662 --> 00:58:56,516 But why does that matter? 1315 00:58:56,516 --> 00:58:57,740 AUDIENCE: It's periodic. 1316 00:58:57,740 --> 00:59:00,290 JEFFREY C. GROSSMAN: It's periodic. 1317 00:59:00,290 --> 00:59:02,760 You got a periodic potential. 1318 00:59:02,760 --> 00:59:05,000 And so if it's-- 1319 00:59:05,000 --> 00:59:08,810 so when would you expect the curviness to stop curving? 1320 00:59:11,740 --> 00:59:17,860 Well-- or if you don't really have a crystal anymore. 1321 00:59:17,860 --> 00:59:19,780 This looks a little less curvy. 1322 00:59:19,780 --> 00:59:22,420 What if I actually made the spacing-- 1323 00:59:22,420 --> 00:59:25,650 if I changed the lattice constant to 10 1324 00:59:25,650 --> 00:59:27,333 and then simulated? 1325 00:59:27,333 --> 00:59:28,500 What's going to happen then? 1326 00:59:32,140 --> 00:59:33,640 That was pretty fast. 1327 00:59:33,640 --> 00:59:34,390 And look at that. 1328 00:59:34,390 --> 00:59:36,670 Now they're far apart. 1329 00:59:36,670 --> 00:59:37,840 They sort of look like-- 1330 00:59:41,530 --> 00:59:42,085 let's see. 1331 00:59:42,085 --> 00:59:43,210 There's the band structure. 1332 00:59:43,210 --> 00:59:45,490 Ooh! 1333 00:59:45,490 --> 00:59:47,690 I love that reaction there. 1334 00:59:47,690 --> 00:59:50,590 That was-- that had some meaning in it. 1335 00:59:50,590 --> 00:59:55,330 Whoever that was, thank you, because I felt a connection. 1336 00:59:55,330 --> 00:59:57,080 What just happened? 1337 00:59:57,080 --> 00:59:58,660 What is this? 1338 00:59:58,660 --> 00:59:59,320 What are these? 1339 01:00:02,962 --> 01:00:03,920 AUDIENCE: Energy bands. 1340 01:00:03,920 --> 01:00:04,970 JEFFREY C. GROSSMAN: Energy bands. 1341 01:00:04,970 --> 01:00:06,320 They are still energy bands-- 1342 01:00:06,320 --> 01:00:07,940 varying k-space. 1343 01:00:07,940 --> 01:00:09,410 But they don't vary in k-space. 1344 01:00:09,410 --> 01:00:11,285 But you-- doesn't mean you can't compute them 1345 01:00:11,285 --> 01:00:13,340 at different k-points, which the code did. 1346 01:00:13,340 --> 01:00:18,340 But they're flat, which means they're not really 1347 01:00:18,340 --> 01:00:20,830 interacting in any sort of periodic way anymore. 1348 01:00:20,830 --> 01:00:23,020 They're so far apart that the periodic potentials 1349 01:00:23,020 --> 01:00:24,740 don't really overlap. 1350 01:00:24,740 --> 01:00:26,270 They don't feel each other. 1351 01:00:26,270 --> 01:00:27,730 So what do I have left then? 1352 01:00:27,730 --> 01:00:28,930 What is it? 1353 01:00:28,930 --> 01:00:30,654 What am I staring at? 1354 01:00:30,654 --> 01:00:31,593 AUDIENCE: Silicon. 1355 01:00:31,593 --> 01:00:33,010 JEFFREY C. GROSSMAN: Silicon atom. 1356 01:00:33,010 --> 01:00:34,635 That's all that-- those are the states. 1357 01:00:34,635 --> 01:00:37,624 That's-- which state is that? 1358 01:00:37,624 --> 01:00:39,757 AUDIENCE: [INAUDIBLE] 1359 01:00:39,757 --> 01:00:40,840 JEFFREY C. GROSSMAN: Yeah. 1360 01:00:40,840 --> 01:00:42,130 But which state is that? 1361 01:00:45,677 --> 01:00:46,308 AUDIENCE: 4s? 1362 01:00:46,308 --> 01:00:48,850 JEFFREY C. GROSSMAN: It would be if we had all the electrons. 1363 01:00:48,850 --> 01:00:49,850 So it is a little tough. 1364 01:00:49,850 --> 01:00:51,720 But we only have the outer four. 1365 01:00:51,720 --> 01:00:54,380 So it's actually the 3s. 1366 01:00:54,380 --> 01:00:58,660 3s, 3p-- our friend, silicon atom, is back. 1367 01:00:58,660 --> 01:01:00,750 We went from atoms to molecules to solids. 1368 01:01:00,750 --> 01:01:04,350 And now we just did an atom. 1369 01:01:04,350 --> 01:01:06,480 And you see, as I bring those atoms, 1370 01:01:06,480 --> 01:01:11,190 as I-- so those are the atomic levels of silicon. 1371 01:01:11,190 --> 01:01:12,450 They don't vary in k-space. 1372 01:01:12,450 --> 01:01:15,330 A molecule of silicon wouldn't vary in k-space. 1373 01:01:15,330 --> 01:01:18,570 But as I start to make this thing feel 1374 01:01:18,570 --> 01:01:26,310 a "repeatiness," then Bloch's theorem kicks in 1375 01:01:26,310 --> 01:01:28,920 and energies depend on k. 1376 01:01:28,920 --> 01:01:32,640 And that changes the properties of the material. 1377 01:01:32,640 --> 01:01:36,340 So that's all really important. 1378 01:01:36,340 --> 01:01:40,360 Now, one more thing while we have the tool up-- 1379 01:01:42,982 --> 01:01:44,440 well, so now we're actually-- we're 1380 01:01:44,440 --> 01:01:46,992 pretty well-equipped to try a whole bunch of stuff. 1381 01:01:46,992 --> 01:01:48,950 And we're going to understand what we're doing. 1382 01:01:48,950 --> 01:01:51,140 Let me just ask you one more thing. 1383 01:01:51,140 --> 01:01:52,930 If I look at these inputs-- 1384 01:01:52,930 --> 01:01:57,150 let's pull up alpha-quartz, let's say. 1385 01:01:57,150 --> 01:01:59,940 There's alpha-quartz. 1386 01:01:59,940 --> 01:02:03,540 Now, if I look at these inputs, what else might 1387 01:02:03,540 --> 01:02:05,040 you want to think about converging? 1388 01:02:05,040 --> 01:02:08,610 We talked about the k-point mesh, something 1389 01:02:08,610 --> 01:02:10,808 not as bad as alpha-quartz. 1390 01:02:14,020 --> 01:02:19,270 Aluminum-- well, that's nice, one atom. 1391 01:02:19,270 --> 01:02:20,950 What else besides the k-point mesh, 1392 01:02:20,950 --> 01:02:25,160 and you better use more k-points than one for aluminum-- 1393 01:02:25,160 --> 01:02:28,260 besides that, what else might-- ooh, I just pointed to it-- 1394 01:02:28,260 --> 01:02:31,650 might I want to converge or make sure is right-- 1395 01:02:31,650 --> 01:02:32,910 the lattice constant. 1396 01:02:32,910 --> 01:02:39,270 You see, I'm just giving this to you because it's there. 1397 01:02:39,270 --> 01:02:40,630 Where did it come from? 1398 01:02:40,630 --> 01:02:43,500 Is it the right lattice constant for this choice 1399 01:02:43,500 --> 01:02:47,950 of theory and basis set? 1400 01:02:47,950 --> 01:02:49,060 Is it the right one? 1401 01:02:49,060 --> 01:02:53,110 Well, you should make sure there are no forces in the system. 1402 01:02:53,110 --> 01:02:54,420 You should relax your system. 1403 01:02:54,420 --> 01:02:56,170 And that gets to the point I want to make. 1404 01:02:56,170 --> 01:02:58,030 How do you relax a solid, because there's 1405 01:02:58,030 --> 01:03:01,557 now two things, not one, that you need to care about? 1406 01:03:01,557 --> 01:03:02,890 Tell me what the two things are. 1407 01:03:05,500 --> 01:03:07,220 Well, how do you relax a molecule? 1408 01:03:07,220 --> 01:03:09,650 When you say optimize structure, what's it doing? 1409 01:03:12,452 --> 01:03:14,210 AUDIENCE: Minimizing free energy. 1410 01:03:14,210 --> 01:03:17,143 JEFFREY C. GROSSMAN: By doing what? 1411 01:03:17,143 --> 01:03:18,060 What's it calculating? 1412 01:03:18,060 --> 01:03:18,768 AUDIENCE: Energy. 1413 01:03:21,142 --> 01:03:23,100 JEFFREY C. GROSSMAN: Which is also called what? 1414 01:03:23,100 --> 01:03:23,767 AUDIENCE: Force. 1415 01:03:23,767 --> 01:03:26,250 JEFFREY C. GROSSMAN: The force on each atom. 1416 01:03:26,250 --> 01:03:27,780 And it pushes them along the force 1417 01:03:27,780 --> 01:03:30,240 until they get to some minimum. 1418 01:03:30,240 --> 01:03:32,550 Now, when I have a solid, I have two things. 1419 01:03:32,550 --> 01:03:36,210 I have the force on each atom, same as the molecule, 1420 01:03:36,210 --> 01:03:39,150 the force on each atom in the basis, as well as what? 1421 01:03:42,608 --> 01:03:43,353 AUDIENCE: Volume. 1422 01:03:43,353 --> 01:03:45,770 JEFFREY C. GROSSMAN: The volume, which is set by the what? 1423 01:03:45,770 --> 01:03:46,510 AUDIENCE: Lattice constant. 1424 01:03:46,510 --> 01:03:48,700 JEFFREY C. GROSSMAN: By the lattice constant. 1425 01:03:48,700 --> 01:03:53,260 So you need to converge both of those. 1426 01:03:53,260 --> 01:03:57,340 And, well, in aluminum, there's only one atom in the basis 1427 01:03:57,340 --> 01:04:02,390 because, you see, aluminum is an FCC metal. 1428 01:04:02,390 --> 01:04:06,190 So all I need is one atom in the corner of an FCC unit cell. 1429 01:04:06,190 --> 01:04:07,360 And I can simulate it. 1430 01:04:07,360 --> 01:04:09,940 But there is the lattice constant 1431 01:04:09,940 --> 01:04:11,050 that I need to play with. 1432 01:04:11,050 --> 01:04:13,660 Now, if I had a more complex solid, 1433 01:04:13,660 --> 01:04:17,110 like alpha-quartz, which has a number of atoms in the basis, 1434 01:04:17,110 --> 01:04:20,733 then the relative positions of those atoms may be important. 1435 01:04:23,900 --> 01:04:25,520 And you may need to relax that. 1436 01:04:25,520 --> 01:04:26,818 You may not. 1437 01:04:26,818 --> 01:04:28,610 But it's something you need to be aware of. 1438 01:04:28,610 --> 01:04:29,390 And you can go in. 1439 01:04:29,390 --> 01:04:31,473 And you can look at the forces on these materials. 1440 01:04:31,473 --> 01:04:35,060 And you definitely have to know whether you're 1441 01:04:35,060 --> 01:04:38,360 at the minimum in terms of the lattice constant. 1442 01:04:38,360 --> 01:04:41,210 So let's do aluminum just because it's here. 1443 01:04:41,210 --> 01:04:45,440 We'll do four k-points, which is not enough to look at a metal. 1444 01:04:45,440 --> 01:04:48,515 But we'll do it because it should be fairly fast. 1445 01:04:52,613 --> 01:04:53,405 There's the energy. 1446 01:04:58,290 --> 01:05:05,700 And-- oh, look at that. 1447 01:05:05,700 --> 01:05:07,010 Isn't that beautiful? 1448 01:05:07,010 --> 01:05:09,530 It knew to change the color and everything-- 1449 01:05:09,530 --> 01:05:12,440 very smart, smart tool. 1450 01:05:12,440 --> 01:05:14,102 And here we go back. 1451 01:05:14,102 --> 01:05:15,560 And tell me-- somebody tell me what 1452 01:05:15,560 --> 01:05:17,143 they think the band structure is going 1453 01:05:17,143 --> 01:05:19,220 to look like for aluminum. 1454 01:05:19,220 --> 01:05:20,540 AUDIENCE: [INAUDIBLE] 1455 01:05:20,540 --> 01:05:22,373 JEFFREY C. GROSSMAN: Is it going to be what? 1456 01:05:22,373 --> 01:05:23,120 Flat? 1457 01:05:23,120 --> 01:05:23,480 AUDIENCE: Curved. 1458 01:05:23,480 --> 01:05:24,647 JEFFREY C. GROSSMAN: Curved. 1459 01:05:24,647 --> 01:05:26,130 How many say curved? 1460 01:05:26,130 --> 01:05:26,930 How many say flat? 1461 01:05:29,460 --> 01:05:31,180 Good. 1462 01:05:31,180 --> 01:05:31,810 Look at that. 1463 01:05:31,810 --> 01:05:33,500 That's curved. 1464 01:05:33,500 --> 01:05:34,690 That's seriously curved. 1465 01:05:37,210 --> 01:05:40,870 And if we include-- if we looked at different energies, 1466 01:05:40,870 --> 01:05:46,510 we'd see even more spaghetti, lots of curvy-- 1467 01:05:46,510 --> 01:05:48,340 Oh, and look at this. 1468 01:05:48,340 --> 01:05:50,020 Oh, let's get the DoS. 1469 01:05:50,020 --> 01:05:51,250 This is so exciting. 1470 01:05:51,250 --> 01:05:53,980 I forgot about the DoS. 1471 01:05:53,980 --> 01:05:57,580 There's the DoS near the Fermi energy. 1472 01:05:57,580 --> 01:05:59,290 Fermi energy is right here. 1473 01:05:59,290 --> 01:06:01,570 Is there a gap in this material? 1474 01:06:01,570 --> 01:06:02,470 No. 1475 01:06:02,470 --> 01:06:04,330 Should there have been? 1476 01:06:04,330 --> 01:06:07,210 No because, remember, in a metal, 1477 01:06:07,210 --> 01:06:11,110 you've got states coming in all over the place 1478 01:06:11,110 --> 01:06:14,790 from below the Fermi energy and above-- continues across it. 1479 01:06:14,790 --> 01:06:18,610 That's what tells you it's a metal. 1480 01:06:18,610 --> 01:06:20,950 Now, is alpha-quartz-- let's-- 1481 01:06:20,950 --> 01:06:23,330 that's a nice material. 1482 01:06:23,330 --> 01:06:26,170 Mostly, I want to simulate it because I 1483 01:06:26,170 --> 01:06:29,480 think it'll be pretty-looking. 1484 01:06:29,480 --> 01:06:31,910 And there it is. 1485 01:06:31,910 --> 01:06:39,580 Now, is alpha-quartz-- I'm going to go down to 2 by 2. 1486 01:06:42,870 --> 01:06:48,494 Is alpha-quartz a metal or a semiconductor, or what is it? 1487 01:06:48,494 --> 01:06:49,790 AUDIENCE: Insulator? 1488 01:06:49,790 --> 01:06:51,570 JEFFREY C. GROSSMAN: Insulator? 1489 01:06:51,570 --> 01:06:52,070 Yeah. 1490 01:06:52,070 --> 01:06:56,120 What's one clue to the fact that it's an insulator? 1491 01:06:56,120 --> 01:06:56,790 What is that? 1492 01:06:56,790 --> 01:06:58,796 AUDIENCE: The inflection in your voice. 1493 01:06:58,796 --> 01:07:00,500 AUDIENCE: [LAUGHTER] 1494 01:07:00,500 --> 01:07:02,420 JEFFREY C. GROSSMAN: I'm that easy to read? 1495 01:07:02,420 --> 01:07:04,820 Man, I got to work on that. 1496 01:07:04,820 --> 01:07:09,150 Is it an insulator or a metal? 1497 01:07:09,150 --> 01:07:09,950 I'll try. 1498 01:07:09,950 --> 01:07:12,890 Well, besides the inflection in my voice, 1499 01:07:12,890 --> 01:07:14,880 what else could give you a clue? 1500 01:07:18,670 --> 01:07:20,710 Well, look at it. 1501 01:07:20,710 --> 01:07:21,955 Is it a transparent material? 1502 01:07:21,955 --> 01:07:23,080 Well, you can't tell there. 1503 01:07:23,080 --> 01:07:23,990 Ooh, it is pretty. 1504 01:07:23,990 --> 01:07:25,440 Look at this. 1505 01:07:25,440 --> 01:07:26,580 Isn't that cool? 1506 01:07:26,580 --> 01:07:29,550 See, this is a periodically repeating structure. 1507 01:07:29,550 --> 01:07:31,910 But the basis has nine atoms in it. 1508 01:07:31,910 --> 01:07:33,745 It's a more complex structure. 1509 01:07:37,460 --> 01:07:38,240 And let's look. 1510 01:07:38,240 --> 01:07:40,520 Now, it's an insulator. 1511 01:07:40,520 --> 01:07:44,190 I'm going to totally agree with that. 1512 01:07:44,190 --> 01:07:46,590 You see right through it. 1513 01:07:46,590 --> 01:07:49,680 So you know it's not what? 1514 01:07:49,680 --> 01:07:53,700 Absorbing in the visible, at least. 1515 01:07:53,700 --> 01:07:57,720 But since it's an insulator and it has a large band gap, 1516 01:07:57,720 --> 01:07:59,730 even though it's a solid, what do 1517 01:07:59,730 --> 01:08:02,520 you think the bands are going to look like, curvy or not 1518 01:08:02,520 --> 01:08:04,894 so curvy? 1519 01:08:04,894 --> 01:08:07,230 AUDIENCE: Not so curvy. 1520 01:08:07,230 --> 01:08:08,760 JEFFREY C. GROSSMAN: Not as curvy. 1521 01:08:08,760 --> 01:08:10,800 They're not usually as curvy. 1522 01:08:10,800 --> 01:08:12,100 So here's the DoS. 1523 01:08:12,100 --> 01:08:15,360 Look at that beautiful band gap. 1524 01:08:15,360 --> 01:08:19,439 This is a band gap only a mother could-- 1525 01:08:19,439 --> 01:08:20,970 a father or mother-- 1526 01:08:20,970 --> 01:08:24,000 that's the beautiful-- and look at this. 1527 01:08:27,120 --> 01:08:29,970 There's the band structure. 1528 01:08:29,970 --> 01:08:33,160 And you can see that-- look at all those bands. 1529 01:08:33,160 --> 01:08:34,319 Look at all those levels. 1530 01:08:34,319 --> 01:08:35,399 And they do curve. 1531 01:08:35,399 --> 01:08:36,359 They do curve. 1532 01:08:36,359 --> 01:08:37,986 But they're not as crazy-- 1533 01:08:37,986 --> 01:08:39,569 well, they are kind of spaghetti-like. 1534 01:08:39,569 --> 01:08:42,569 But they're not spaghetti-like near the Fermi energy 1535 01:08:42,569 --> 01:08:44,100 or near the gap. 1536 01:08:44,100 --> 01:08:45,120 So you look at that. 1537 01:08:45,120 --> 01:08:47,910 And you're like, insulator, first of all, 1538 01:08:47,910 --> 01:08:49,649 just because of how big the gap is. 1539 01:08:49,649 --> 01:08:51,359 And then what we're going to do-- 1540 01:08:51,359 --> 01:08:53,310 and I still have just 10 minutes for today. 1541 01:08:53,310 --> 01:08:55,200 But on Thursday, what we're going to do 1542 01:08:55,200 --> 01:08:57,490 is we're going to show how, when you look at this, 1543 01:08:57,490 --> 01:09:00,899 you can say, not a very good conductor, certainly 1544 01:09:00,899 --> 01:09:03,271 not a good hole conductor. 1545 01:09:03,271 --> 01:09:04,729 This is where the holes would move. 1546 01:09:04,729 --> 01:09:05,689 This is where the electrons would 1547 01:09:05,689 --> 01:09:08,050 move because we're going to get to that a little bit 1548 01:09:08,050 --> 01:09:08,550 on Thursday. 1549 01:09:08,550 --> 01:09:12,260 Band structure is so rich with information. 1550 01:09:12,260 --> 01:09:14,279 It's a beautiful thing. 1551 01:09:14,279 --> 01:09:16,120 It's emotional. 1552 01:09:16,120 --> 01:09:19,194 And if any of you are feeling that, it's OK. 1553 01:09:19,194 --> 01:09:20,319 I'm touching the mic again. 1554 01:09:20,319 --> 01:09:21,279 Sorry. 1555 01:09:21,279 --> 01:09:21,819 It's OK. 1556 01:09:21,819 --> 01:09:23,540 We're good. 1557 01:09:23,540 --> 01:09:24,400 Check that. 1558 01:09:24,400 --> 01:09:24,900 Roger. 1559 01:09:28,300 --> 01:09:33,069 So we're simulating solids. 1560 01:09:33,069 --> 01:09:34,660 You understand about band structures. 1561 01:09:34,660 --> 01:09:41,620 We've talked about silicon and occupied, unoccupied-- 1562 01:09:41,620 --> 01:09:45,170 you can get the densities, just like you can in molecules. 1563 01:09:45,170 --> 01:09:51,439 And as I said, you can calculate the-- 1564 01:09:51,439 --> 01:09:51,939 you what? 1565 01:09:51,939 --> 01:09:54,580 You need to find the equilibrium lattice constant. 1566 01:09:54,580 --> 01:09:58,437 And so if you just do the calculation 1567 01:09:58,437 --> 01:10:00,020 at different lattice constants, you'll 1568 01:10:00,020 --> 01:10:02,860 get a variation in energy. 1569 01:10:02,860 --> 01:10:03,860 That's what will happen. 1570 01:10:03,860 --> 01:10:05,420 And you'll get a minimum. 1571 01:10:05,420 --> 01:10:07,340 And that'll be the lattice constant 1572 01:10:07,340 --> 01:10:09,800 that you should run at for your simulations. 1573 01:10:09,800 --> 01:10:11,035 And that's pretty important. 1574 01:10:13,760 --> 01:10:15,830 But if you had that curve already, well, 1575 01:10:15,830 --> 01:10:19,530 then you've also got the bulk modulus, which is pretty cool. 1576 01:10:19,530 --> 01:10:22,040 Now, I think that you probably did this calculation 1577 01:10:22,040 --> 01:10:24,470 using classical potentials. 1578 01:10:24,470 --> 01:10:25,990 Is that right? 1579 01:10:25,990 --> 01:10:28,130 You calculated the bulk modulus? 1580 01:10:28,130 --> 01:10:29,920 So this isn't any different. 1581 01:10:29,920 --> 01:10:33,250 It's just the-- did you do-- 1582 01:10:33,250 --> 01:10:35,200 you did periodically repeating structures, 1583 01:10:35,200 --> 01:10:38,840 right, using classical force fields? 1584 01:10:43,410 --> 01:10:46,570 I'm seeing a whole lot of "I'm not sure." 1585 01:10:46,570 --> 01:10:48,630 Yeah? 1586 01:10:48,630 --> 01:10:50,695 It's a very "I'm not sure" kind of moment. 1587 01:10:50,695 --> 01:10:52,392 It's OK. 1588 01:10:52,392 --> 01:10:53,620 It's OK. 1589 01:10:53,620 --> 01:10:58,426 Is it-- did you look at solid materials in your homeworks? 1590 01:10:58,426 --> 01:11:01,270 It's still no, not so sure. 1591 01:11:01,270 --> 01:11:03,340 Silicon? 1592 01:11:03,340 --> 01:11:04,510 Carbon? 1593 01:11:04,510 --> 01:11:07,068 Not so much. 1594 01:11:07,068 --> 01:11:08,500 AUDIENCE: [INAUDIBLE] proteins. 1595 01:11:08,500 --> 01:11:09,750 JEFFREY C. GROSSMAN: Proteins? 1596 01:11:09,750 --> 01:11:15,810 So proteins are a bit more complex of a beast, 1597 01:11:15,810 --> 01:11:18,390 by the way, which you can't really do using-- 1598 01:11:18,390 --> 01:11:21,520 you can do parts of proteins using quantum mechanics. 1599 01:11:21,520 --> 01:11:22,510 But it's just too hard. 1600 01:11:22,510 --> 01:11:24,420 It's too big. 1601 01:11:24,420 --> 01:11:29,950 You can't do that many atoms to do most full-sized proteins. 1602 01:11:29,950 --> 01:11:35,400 So that's a great problem to study with classical models. 1603 01:11:35,400 --> 01:11:38,220 For crystals, like the two-atom silicon, 1604 01:11:38,220 --> 01:11:40,110 eight electrons-- are all those crystals 1605 01:11:40,110 --> 01:11:43,140 in there-- and not to say 1,000 electrons. 1606 01:11:43,140 --> 01:11:46,290 You can do things like calculate the change in the lattice 1607 01:11:46,290 --> 01:11:46,883 with-- 1608 01:11:46,883 --> 01:11:48,300 change in energy with the lattice. 1609 01:11:48,300 --> 01:11:51,440 And from that-- well, this is the volume. 1610 01:11:51,440 --> 01:11:54,780 But that's just the same as the change in the lattice. 1611 01:11:54,780 --> 01:11:56,280 And from that second derivative, you 1612 01:11:56,280 --> 01:11:58,822 can get the bulk modulus, which is a very important property. 1613 01:12:02,380 --> 01:12:04,130 And this is something I just went through. 1614 01:12:04,130 --> 01:12:05,797 This was that slide that I talked about. 1615 01:12:05,797 --> 01:12:10,460 But it was-- really should have been an internal discussion. 1616 01:12:10,460 --> 01:12:14,840 And where you find the converged ground state density 1617 01:12:14,840 --> 01:12:17,960 and potential with some k-point mesh that you gave it-- 1618 01:12:17,960 --> 01:12:19,340 that's what you input-- 1619 01:12:19,340 --> 01:12:22,250 is the k-points that go into that calculation. 1620 01:12:22,250 --> 01:12:25,700 And then for that, you calculate the energies at k-points 1621 01:12:25,700 --> 01:12:27,110 along all the lines. 1622 01:12:27,110 --> 01:12:30,380 And some software-- well, step 3 is not really a step. 1623 01:12:30,380 --> 01:12:33,440 But use some software to plot it. 1624 01:12:33,440 --> 01:12:35,875 And you can download the data for this from the nanoHUB. 1625 01:12:35,875 --> 01:12:37,250 There's a-- I've been showing you 1626 01:12:37,250 --> 01:12:39,750 the version of the plot that gives you the Greek characters, 1627 01:12:39,750 --> 01:12:40,550 which is an image. 1628 01:12:40,550 --> 01:12:42,883 But there's another version that gives you the raw data. 1629 01:12:42,883 --> 01:12:44,822 It just doesn't put the characters on it. 1630 01:12:44,822 --> 01:12:47,030 But that will allow you to compare one run to another 1631 01:12:47,030 --> 01:12:50,570 and download the raw data. 1632 01:12:50,570 --> 01:12:57,030 And then calculating the DoS is also a two-step process. 1633 01:12:57,030 --> 01:12:59,030 You converge the ground state density potential. 1634 01:12:59,030 --> 01:13:01,510 And then you use that potential to calculate the energies. 1635 01:13:01,510 --> 01:13:04,570 And usually, you want to do this at a dense k-mesh 1636 01:13:04,570 --> 01:13:10,840 because the subtle variations-- see, in a molecule, oftentimes, 1637 01:13:10,840 --> 01:13:15,520 the DoS is pretty clear. 1638 01:13:15,520 --> 01:13:22,340 If I turn-- well, I guess I could've just done this. 1639 01:13:26,000 --> 01:13:30,260 If I turn, say, an atom or a molecule on its side, 1640 01:13:30,260 --> 01:13:32,100 then I have this. 1641 01:13:32,100 --> 01:13:35,540 And you use some kind of smearing function. 1642 01:13:35,540 --> 01:13:37,970 And so your-- well, that was a bad centering. 1643 01:13:37,970 --> 01:13:40,405 And so your DoS would look like that. 1644 01:13:40,405 --> 01:13:41,780 But there'd be bigger peaks where 1645 01:13:41,780 --> 01:13:43,910 you have lots of states and smaller peaks 1646 01:13:43,910 --> 01:13:45,330 where you don't have many. 1647 01:13:45,330 --> 01:13:49,760 But, you see, now, I'm turning a band structure, 1648 01:13:49,760 --> 01:13:54,770 and you saw alpha-quartz-- so now I'm turning-- 1649 01:13:54,770 --> 01:13:58,080 see if I can just show it one more time-- 1650 01:13:58,080 --> 01:13:59,050 oh, man. 1651 01:13:59,050 --> 01:13:59,550 Seriously? 1652 01:13:59,550 --> 01:14:00,690 Oh, it was in another-- 1653 01:14:03,430 --> 01:14:06,010 I'm turning this on its side now. 1654 01:14:06,010 --> 01:14:08,920 And that's not as easy. 1655 01:14:08,920 --> 01:14:09,820 That's trickier. 1656 01:14:09,820 --> 01:14:12,460 So there's different weights of a band 1657 01:14:12,460 --> 01:14:15,140 at different parts in energy. 1658 01:14:15,140 --> 01:14:19,660 And when I turn this on this side, you can get DoS shapes, 1659 01:14:19,660 --> 01:14:23,170 Density of States, that are-- that have a-- 1660 01:14:23,170 --> 01:14:25,360 more features in them. 1661 01:14:25,360 --> 01:14:29,200 And to get all those features just right, 1662 01:14:29,200 --> 01:14:31,210 you need to use a dense k-point mesh, which 1663 01:14:31,210 --> 01:14:34,240 is what the code does. 1664 01:14:34,240 --> 01:14:35,150 This can be measured. 1665 01:14:35,150 --> 01:14:38,770 The DoS can be measured experimentally-- 1666 01:14:38,770 --> 01:14:42,220 can use STM to actually directly measure the DoS. 1667 01:14:45,130 --> 01:14:47,020 And we talked about this already. 1668 01:14:47,020 --> 01:14:50,320 You fill up your electrons to the Fermi level. 1669 01:14:50,320 --> 01:14:52,280 If there's any bands crossing, it's a metal. 1670 01:14:52,280 --> 01:14:54,580 If not, it's an insulator or a semiconductor. 1671 01:14:54,580 --> 01:14:58,930 If you have an odd number of electrons in the unit cell, 1672 01:14:58,930 --> 01:15:01,740 it has to be a metal because the Fermi level is 1673 01:15:01,740 --> 01:15:04,590 going to come in in the middle. 1674 01:15:04,590 --> 01:15:07,030 Each band takes two electrons. 1675 01:15:07,030 --> 01:15:08,650 So if you fill up to the Fermi level 1676 01:15:08,650 --> 01:15:11,070 and the Fermi level is in the middle of a band, 1677 01:15:11,070 --> 01:15:12,750 by definition, it's a metal-- 1678 01:15:17,530 --> 01:15:20,270 sodium. 1679 01:15:20,270 --> 01:15:23,540 And then here is a picture of diamond. 1680 01:15:23,540 --> 01:15:26,270 And the thing that I want to, again, emphasize 1681 01:15:26,270 --> 01:15:33,590 is that from DFT, these characters of these wiggles 1682 01:15:33,590 --> 01:15:36,740 and of these features are pretty darn good, 1683 01:15:36,740 --> 01:15:40,680 but this distance is not. 1684 01:15:40,680 --> 01:15:45,000 And so what we-- what we're going to do in this class, 1685 01:15:45,000 --> 01:15:47,030 and we're going to be proud about it, 1686 01:15:47,030 --> 01:15:49,190 is we're going to just shift these bands up 1687 01:15:49,190 --> 01:15:51,975 by a rigid amount to fix the gap. 1688 01:15:51,975 --> 01:15:52,850 AUDIENCE: [INAUDIBLE] 1689 01:15:52,850 --> 01:15:55,058 JEFFREY C. GROSSMAN: That was probably electronvolts. 1690 01:15:55,058 --> 01:15:56,990 Yeah, it says it here, electronvolts. 1691 01:15:56,990 --> 01:16:02,440 And here, it looks like-- what do we have, like, 1692 01:16:02,440 --> 01:16:06,270 a 3-eV gap from DFT? 1693 01:16:06,270 --> 01:16:08,040 And so that's about half. 1694 01:16:08,040 --> 01:16:12,780 Very often, DFT gives you a gap that's about half 1695 01:16:12,780 --> 01:16:16,230 of the experimental gap, very often, not at-- 1696 01:16:16,230 --> 01:16:18,750 always, but very often. 1697 01:16:18,750 --> 01:16:22,920 For germanium, it makes it a semimetal, which it's not. 1698 01:16:22,920 --> 01:16:25,134 Does anybody know what a semimetal is? 1699 01:16:25,134 --> 01:16:26,398 AUDIENCE: [INAUDIBLE] 1700 01:16:26,398 --> 01:16:27,690 JEFFREY C. GROSSMAN: Not quite. 1701 01:16:30,540 --> 01:16:32,280 Here's an indirect band gap. 1702 01:16:32,280 --> 01:16:34,350 A semimetal is a cool material. 1703 01:16:34,350 --> 01:16:37,080 Germanium is not a semimetal, but DFT 1704 01:16:37,080 --> 01:16:40,650 will tell you it is because of the gap problem. 1705 01:16:40,650 --> 01:16:42,570 Here's an indirect band gap material. 1706 01:16:42,570 --> 01:16:43,980 Which one is it? 1707 01:16:43,980 --> 01:16:44,730 AUDIENCE: Silicon. 1708 01:16:44,730 --> 01:16:45,120 JEFFREY C. GROSSMAN: Yeah. 1709 01:16:45,120 --> 01:16:45,620 Sure. 1710 01:16:45,620 --> 01:16:47,880 Why not? 1711 01:16:47,880 --> 01:16:52,770 And if it's silicon, this will be 1.1 eV in experiment 1712 01:16:52,770 --> 01:16:55,995 and about 0.6 eV in DFT. 1713 01:16:59,750 --> 01:17:01,760 And that's the band gap error. 1714 01:17:01,760 --> 01:17:02,983 But again, you can shift it. 1715 01:17:02,983 --> 01:17:04,400 And that's what we're going to do. 1716 01:17:04,400 --> 01:17:07,130 We're going to be OK with that. 1717 01:17:07,130 --> 01:17:10,430 In-- if you switch to germanium, experiment says it's about 1718 01:17:10,430 --> 01:17:13,550 a 0.6- or 0.7-eV gap, something around there. 1719 01:17:13,550 --> 01:17:17,720 But DFT-- that's silicon. 1720 01:17:17,720 --> 01:17:20,210 But DFT will put it here. 1721 01:17:20,210 --> 01:17:22,295 It'll put this conduction band below. 1722 01:17:25,330 --> 01:17:26,650 It has to actually-- 1723 01:17:26,650 --> 01:17:29,440 it's-- why didn't I draw that right? 1724 01:17:29,440 --> 01:17:31,390 It's actually-- comes below. 1725 01:17:31,390 --> 01:17:39,250 So that is a semimetal when you have, 1726 01:17:39,250 --> 01:17:43,630 basically, its crossings across the Fermi level, 1727 01:17:43,630 --> 01:17:46,660 but in an indirect-- from indirect states. 1728 01:17:46,660 --> 01:17:47,890 That's called a semimetal. 1729 01:17:47,890 --> 01:17:50,980 And DFT will tell you germanium is that, which is not-- 1730 01:17:50,980 --> 01:17:55,510 but you can still shift all those states up, which we do. 1731 01:17:55,510 --> 01:17:56,350 Oh, there's a metal. 1732 01:17:56,350 --> 01:17:57,790 Oh, look at that, spaghetti. 1733 01:17:57,790 --> 01:17:59,530 You now know-- oh, and there it is. 1734 01:17:59,530 --> 01:18:02,477 See, often in papers, you'll see in the literature-- 1735 01:18:02,477 --> 01:18:04,060 so you're now becoming sort of experts 1736 01:18:04,060 --> 01:18:09,220 at seeing a band structure and seeing it, really seeing it. 1737 01:18:09,220 --> 01:18:11,350 And it's a powerful thing. 1738 01:18:11,350 --> 01:18:13,263 And there it is with the DoS next to it, 1739 01:18:13,263 --> 01:18:15,430 you see, because that's just this on the side, which 1740 01:18:15,430 --> 01:18:22,190 is why you then plot the DoS sideways, often, 1741 01:18:22,190 --> 01:18:24,290 except that doesn't quite look right. 1742 01:18:24,290 --> 01:18:28,100 But anyway-- oh, maybe this-- 1743 01:18:28,100 --> 01:18:30,528 yeah. 1744 01:18:30,528 --> 01:18:31,570 That's what it should be. 1745 01:18:34,740 --> 01:18:37,080 And then we've talked about this. 1746 01:18:37,080 --> 01:18:38,538 And we're going to talk about this. 1747 01:18:38,538 --> 01:18:40,413 We're going to come back to this a little bit 1748 01:18:40,413 --> 01:18:42,390 when we talk about solar cells. 1749 01:18:42,390 --> 01:18:44,070 But I have mentioned this a bunch. 1750 01:18:44,070 --> 01:18:47,670 Certainly, this would be a good third homework set-- 1751 01:18:47,670 --> 01:18:51,750 is to think about this indirect versus direct band gap problem. 1752 01:18:51,750 --> 01:18:54,720 Somebody tell me, again, why silicon solar cells 1753 01:18:54,720 --> 01:18:57,690 are expensive just so I can go home today and feel 1754 01:18:57,690 --> 01:19:01,240 really good or really bad because it's bad news? 1755 01:19:01,240 --> 01:19:02,490 AUDIENCE: [INAUDIBLE] bandgap. 1756 01:19:02,490 --> 01:19:02,730 JEFFREY C. GROSSMAN: Yeah. 1757 01:19:02,730 --> 01:19:05,280 But why is that-- why does that make silicon solar cells 1758 01:19:05,280 --> 01:19:07,083 expensive? 1759 01:19:07,083 --> 01:19:08,683 AUDIENCE: You have to make them thick. 1760 01:19:08,683 --> 01:19:11,016 JEFFREY C. GROSSMAN: Why do you have to make them thick? 1761 01:19:11,016 --> 01:19:14,502 AUDIENCE: [INAUDIBLE] 1762 01:19:19,000 --> 01:19:21,910 JEFFREY C. GROSSMAN: Yeah, because you've got to get-- 1763 01:19:21,910 --> 01:19:23,160 you see, you got to get-- 1764 01:19:23,160 --> 01:19:24,710 I think I had it shown here. 1765 01:19:24,710 --> 01:19:25,240 Yeah. 1766 01:19:25,240 --> 01:19:27,867 See, that's easy for a photon to do. 1767 01:19:27,867 --> 01:19:28,700 I talked about this. 1768 01:19:28,700 --> 01:19:30,310 Now, here it is in the cartoon. 1769 01:19:30,310 --> 01:19:34,120 That's easy for a photon to do. 1770 01:19:34,120 --> 01:19:37,990 That's easy for a photon to do at different places in energy. 1771 01:19:37,990 --> 01:19:40,000 And this would be absorbing light. 1772 01:19:40,000 --> 01:19:41,500 These blue arrows would be the light 1773 01:19:41,500 --> 01:19:43,540 that come from the solar flux that you're 1774 01:19:43,540 --> 01:19:46,270 downloading in your homework. 1775 01:19:46,270 --> 01:19:48,820 That's where these blue arrows come. 1776 01:19:48,820 --> 01:19:50,980 And it's easy for those blue arrows to do this. 1777 01:19:50,980 --> 01:19:55,540 But to see-- the thing that matters is this minimum amount 1778 01:19:55,540 --> 01:19:59,470 of energy, the band gap of the material that is the lowest 1779 01:19:59,470 --> 01:20:02,290 amount of energy that it would take to get an electron up 1780 01:20:02,290 --> 01:20:04,630 from the valence into the conduction bands. 1781 01:20:04,630 --> 01:20:08,080 And that, in this case, is not vertical. 1782 01:20:08,080 --> 01:20:08,890 It's not vertical. 1783 01:20:08,890 --> 01:20:10,577 It's an indirect gap. 1784 01:20:10,577 --> 01:20:11,410 Did I have it there? 1785 01:20:14,200 --> 01:20:17,460 And because of that, it's not very good at absorbing light. 1786 01:20:17,460 --> 01:20:21,360 Those photons, those blue lines-- 1787 01:20:21,360 --> 01:20:23,130 let's get them back here. 1788 01:20:23,130 --> 01:20:26,300 Those blue lines-- they need help. 1789 01:20:26,300 --> 01:20:28,120 They can do this. 1790 01:20:28,120 --> 01:20:30,160 But they can't do this. 1791 01:20:30,160 --> 01:20:32,770 And to do this, you need help. 1792 01:20:32,770 --> 01:20:35,710 So you need some kind of kick. 1793 01:20:35,710 --> 01:20:38,620 And that means it's a much less efficient process. 1794 01:20:38,620 --> 01:20:43,420 And absorbing light in that whole region of the spectrum 1795 01:20:43,420 --> 01:20:46,150 where a photon would need to do this-- that whole region 1796 01:20:46,150 --> 01:20:47,260 is very inefficient. 1797 01:20:47,260 --> 01:20:48,910 And yet that's where the sun is. 1798 01:20:48,910 --> 01:20:50,230 That's where the sun is. 1799 01:20:50,230 --> 01:20:52,510 And so you've got to make them really thick. 1800 01:20:52,510 --> 01:20:54,320 It's inefficient, but not zero. 1801 01:20:54,320 --> 01:20:55,570 So you make them really thick. 1802 01:20:55,570 --> 01:20:57,310 So you get that region of the sun. 1803 01:20:57,310 --> 01:20:59,980 But to make them really thick, you see, 1804 01:20:59,980 --> 01:21:02,763 you're generating electrons and holes. 1805 01:21:02,763 --> 01:21:05,180 And those-- that generation of electrons and holes-- well, 1806 01:21:05,180 --> 01:21:08,450 that-- then you've got to get them out. 1807 01:21:08,450 --> 01:21:10,690 You can't just let them stay there. 1808 01:21:10,690 --> 01:21:14,970 And to get them out, well, you need a good material. 1809 01:21:14,970 --> 01:21:19,080 And if they have to go a long ways to get out, 1810 01:21:19,080 --> 01:21:22,230 which they do because you have to make it so thick because it 1811 01:21:22,230 --> 01:21:25,520 doesn't absorb light well because of the band structure, 1812 01:21:25,520 --> 01:21:26,978 then-- 1813 01:21:26,978 --> 01:21:27,770 where was I, again? 1814 01:21:27,770 --> 01:21:28,395 No. 1815 01:21:28,395 --> 01:21:32,030 Then you need it to be a really pure material. 1816 01:21:32,030 --> 01:21:34,530 That costs money. 1817 01:21:34,530 --> 01:21:37,900 Otherwise, they'll run into traps and other things. 1818 01:21:37,900 --> 01:21:40,410 So what we're going to do is, on Thursday, 1819 01:21:40,410 --> 01:21:43,660 we'll talk a little bit more about some other properties. 1820 01:21:43,660 --> 01:21:46,470 We'll talk about magnetism, very briefly 1821 01:21:46,470 --> 01:21:49,303 show you that you can do molecular dynamics, 1822 01:21:49,303 --> 01:21:50,220 a couple other things. 1823 01:21:50,220 --> 01:21:53,690 And then we'll start turning our attention to solar cells.