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 --> 00:00:18,332 at ocw.mit.edu. 8 00:00:25,556 --> 00:00:30,710 JEFFREY GROSSMAN: So we're here, but don't forget about this. 9 00:00:30,710 --> 00:00:34,750 And so the quiz is next Thursday. 10 00:00:34,750 --> 00:00:36,500 I'm going to send an email out with office 11 00:00:36,500 --> 00:00:42,050 hours for extra time. 12 00:00:42,050 --> 00:00:44,000 But if anyone wants to meet anytime, 13 00:00:44,000 --> 00:00:46,370 just send me an email, please. 14 00:00:46,370 --> 00:00:49,520 This PSET 2, we decided is going to be due 15 00:00:49,520 --> 00:00:51,230 the day before on May 9. 16 00:00:51,230 --> 00:00:55,550 And that's when your PSET 3 will be posted as well. 17 00:00:55,550 --> 00:00:58,610 And PSET 3 will have to do with the things we've been talking 18 00:00:58,610 --> 00:01:02,510 about the last two lectures and today, which 19 00:01:02,510 --> 00:01:05,010 is what you can do with solids. 20 00:01:08,660 --> 00:01:11,890 Any questions about the plan, the schedule? 21 00:01:14,620 --> 00:01:16,675 OK. 22 00:01:16,675 --> 00:01:17,175 Right. 23 00:01:19,830 --> 00:01:21,810 Now, there are other practice problems. 24 00:01:24,870 --> 00:01:27,100 I'll throw one out from last year's quiz. 25 00:01:27,100 --> 00:01:33,270 So that's a practice quiz that I gave out 26 00:01:33,270 --> 00:01:35,303 maybe a year ago or two. 27 00:01:35,303 --> 00:01:36,720 And then there's last year's quiz, 28 00:01:36,720 --> 00:01:38,670 and I'll pull a problem or two out from that, 29 00:01:38,670 --> 00:01:40,800 and we'll do one today, and then maybe we'll 30 00:01:40,800 --> 00:01:42,600 look at another one on Tuesday. 31 00:01:47,580 --> 00:01:50,310 What I want to cover in terms of the lecture material today 32 00:01:50,310 --> 00:01:54,270 is, I want to briefly start with that band structure 33 00:01:54,270 --> 00:01:58,320 and caspase, and make sure we're feeling good about it, 34 00:01:58,320 --> 00:02:01,810 and then tell you a few of the things you can do. 35 00:02:01,810 --> 00:02:03,270 We talked about the band structure, 36 00:02:03,270 --> 00:02:07,461 we've talked about how it has so much information. 37 00:02:07,461 --> 00:02:09,419 Well, what other kinds of information is there? 38 00:02:09,419 --> 00:02:12,750 What can you do with these sorts of properties of solids? 39 00:02:17,280 --> 00:02:18,360 I love this cartoon. 40 00:02:18,360 --> 00:02:20,340 "At some point, his theory becomes so abstract, 41 00:02:20,340 --> 00:02:22,830 it can only be conveyed using interpretive dance." 42 00:02:22,830 --> 00:02:24,450 And mostly it's because of the visuals 43 00:02:24,450 --> 00:02:30,350 that I get on my colleagues here at MIT just dancing. 44 00:02:30,350 --> 00:02:34,380 But we're not going to get this complicated, right? 45 00:02:34,380 --> 00:02:38,730 So the point isn't for me to do a lecture on transport theory, 46 00:02:38,730 --> 00:02:44,122 and tell you all about phonons and go into great detail 47 00:02:44,122 --> 00:02:45,330 about these different things. 48 00:02:45,330 --> 00:02:48,040 That is, again, not been our point here. 49 00:02:48,040 --> 00:02:52,515 The point is to keep relevant and keep pretty applied 50 00:02:52,515 --> 00:02:54,090 in this class. 51 00:02:54,090 --> 00:02:58,890 So I'm going to give you just a little hint as to what 52 00:02:58,890 --> 00:03:02,730 you can do with the DOS and the band structure. 53 00:03:02,730 --> 00:03:04,750 But we're not going to go into great detail. 54 00:03:04,750 --> 00:03:08,850 But you can find those details in many places online 55 00:03:08,850 --> 00:03:09,430 and in books. 56 00:03:09,430 --> 00:03:13,140 And if you come to me, I'm happy to talk about these concepts 57 00:03:13,140 --> 00:03:15,785 that we'll talk about. 58 00:03:15,785 --> 00:03:17,160 So the plan is, we're going to do 59 00:03:17,160 --> 00:03:19,560 a little bit of review, caspase, feeling good 60 00:03:19,560 --> 00:03:22,270 about oneness, all that. 61 00:03:22,270 --> 00:03:27,073 And then I want to show you that problem. 62 00:03:27,073 --> 00:03:28,740 We'll talk about just one band structure 63 00:03:28,740 --> 00:03:31,840 problem that I happened to give last year on the exam. 64 00:03:31,840 --> 00:03:37,530 And then we'll talk a little bit about some properties, 65 00:03:37,530 --> 00:03:39,193 electron transport a couple of minutes. 66 00:03:39,193 --> 00:03:40,860 And then we'll talk about magnetization, 67 00:03:40,860 --> 00:03:42,960 and then we'll do some calculations together again, 68 00:03:42,960 --> 00:03:44,835 because I think that's been really enjoyable. 69 00:03:44,835 --> 00:03:49,470 Actually, seriously, I always enjoy calculating together. 70 00:03:49,470 --> 00:03:52,620 But how many of you feel like that's useful, 71 00:03:52,620 --> 00:03:56,320 to do some simulations here in class? 72 00:03:56,320 --> 00:03:58,360 One, two, more? 73 00:03:58,360 --> 00:03:59,620 OK. 74 00:03:59,620 --> 00:04:02,320 Because please ask questions. 75 00:04:02,320 --> 00:04:03,250 This is the time. 76 00:04:03,250 --> 00:04:08,110 Because this is where we can, if you have any questions, please 77 00:04:08,110 --> 00:04:09,040 stop. 78 00:04:09,040 --> 00:04:13,060 And say, I got no idea what you're talking about. 79 00:04:13,060 --> 00:04:14,150 That's OK. 80 00:04:14,150 --> 00:04:15,770 I love those. 81 00:04:15,770 --> 00:04:16,930 That's what I want to hear. 82 00:04:16,930 --> 00:04:19,222 [LAUGHS] Well, that's actually not what I want to hear. 83 00:04:19,222 --> 00:04:21,490 But if I hear it, it's fine. 84 00:04:21,490 --> 00:04:24,760 That helps me to know where we need 85 00:04:24,760 --> 00:04:28,170 to have a little discussion. 86 00:04:28,170 --> 00:04:33,060 Now, this is the concept that we have 87 00:04:33,060 --> 00:04:35,190 to be really good in our understanding. 88 00:04:35,190 --> 00:04:39,090 But this is really the key of this second part 89 00:04:39,090 --> 00:04:41,650 of the second part, if you want, of the class. 90 00:04:41,650 --> 00:04:46,290 Which is, we did quantum, we did quantum for atoms, 91 00:04:46,290 --> 00:04:49,980 we did quantum for molecules, and now it's all about solids. 92 00:04:49,980 --> 00:04:54,600 And the big difference is that when you have a solid, 93 00:04:54,600 --> 00:04:58,350 you have over and over and over again, 94 00:04:58,350 --> 00:05:02,850 the same atom, or same couple of atoms in your basis. 95 00:05:02,850 --> 00:05:05,820 And therefore, the same potential that the electron 96 00:05:05,820 --> 00:05:07,380 feels. 97 00:05:07,380 --> 00:05:10,430 And over and over again, how many times? 98 00:05:10,430 --> 00:05:14,350 AUDIENCE: [INAUDIBLE] 2? 99 00:05:14,350 --> 00:05:16,481 JEFFREY GROSSMAN: If I had to give a number. 100 00:05:16,481 --> 00:05:19,710 AUDIENCE: Oh my god, there's a couple thousand. 101 00:05:19,710 --> 00:05:22,070 JEFFREY GROSSMAN: Well, I'd say more like a couple of 10 102 00:05:22,070 --> 00:05:23,150 to the 23rd thousands. 103 00:05:23,150 --> 00:05:24,500 AUDIENCE: [LAUGHTER] 104 00:05:24,500 --> 00:05:25,917 JEFFREY GROSSMAN: 10 to the 23rds. 105 00:05:28,730 --> 00:05:35,120 If I take a piece of material, this big, this big, even 106 00:05:35,120 --> 00:05:35,930 this big. 107 00:05:35,930 --> 00:05:37,040 Even really tiny. 108 00:05:37,040 --> 00:05:41,060 You got like, 10 23rd repeats of this potential. 109 00:05:41,060 --> 00:05:41,990 That's a lot. 110 00:05:41,990 --> 00:05:45,650 So it's almost infinite. 111 00:05:45,650 --> 00:05:48,770 And when you have that periodically repeating 112 00:05:48,770 --> 00:05:53,950 well that the electron feels, something happens. 113 00:05:53,950 --> 00:05:56,120 And the something that happens was 114 00:05:56,120 --> 00:05:59,960 this mathematical thing that, who 115 00:05:59,960 --> 00:06:02,990 can remember what it's called? 116 00:06:02,990 --> 00:06:04,940 It was a theorem, I believe. 117 00:06:04,940 --> 00:06:06,440 AUDIENCE: Bloch's theorem? 118 00:06:06,440 --> 00:06:08,960 JEFFREY GROSSMAN: Bloch's theorem. 119 00:06:08,960 --> 00:06:12,620 And we don't need to know much about Bloch's theorem 120 00:06:12,620 --> 00:06:16,130 except the endgame. 121 00:06:16,130 --> 00:06:18,020 The key is what it means. 122 00:06:18,020 --> 00:06:24,050 And it's a mathematical consequence. 123 00:06:24,050 --> 00:06:30,620 It's a consequence of having an electron, and therefore 124 00:06:30,620 --> 00:06:33,620 its solution being a wave function that solves 125 00:06:33,620 --> 00:06:34,790 the Schrodinger equation. 126 00:06:34,790 --> 00:06:37,430 It's a consequence of having that 127 00:06:37,430 --> 00:06:40,730 in the context of a periodic potential. 128 00:06:40,730 --> 00:06:43,010 You can't get around it. 129 00:06:43,010 --> 00:06:43,970 It's a theorem. 130 00:06:43,970 --> 00:06:50,500 So because of Bloch's theorem, you 131 00:06:50,500 --> 00:06:52,523 got to have this requirement, if you 132 00:06:52,523 --> 00:06:54,190 want to think about it as a requirement, 133 00:06:54,190 --> 00:06:58,090 that when the wave function repeats, 134 00:06:58,090 --> 00:06:59,440 the properties just repeat. 135 00:06:59,440 --> 00:07:01,180 The density's the same here as it is here 136 00:07:01,180 --> 00:07:03,190 as it is here, 10 to the 23rd out. 137 00:07:03,190 --> 00:07:04,960 But when the wave function repeats, 138 00:07:04,960 --> 00:07:06,260 it picks up this phase. 139 00:07:08,800 --> 00:07:14,770 And that phase is a number that lives in k-space, that 140 00:07:14,770 --> 00:07:15,898 lives in the inverse space. 141 00:07:15,898 --> 00:07:17,440 And that's why we spent all that time 142 00:07:17,440 --> 00:07:21,790 talking about what inverse space is, what reciprocal space is, 143 00:07:21,790 --> 00:07:23,680 what k-space is. 144 00:07:23,680 --> 00:07:30,490 And that phase is, you can think about it as another quantum 145 00:07:30,490 --> 00:07:31,970 number. 146 00:07:31,970 --> 00:07:36,940 So the result of solving the Schrodinger equation 147 00:07:36,940 --> 00:07:38,920 and getting psi in a periodic potential 148 00:07:38,920 --> 00:07:42,190 gives you a new number. 149 00:07:42,190 --> 00:07:45,850 But it's very different than these numbers. 150 00:07:45,850 --> 00:07:48,880 How is it different? 151 00:07:48,880 --> 00:07:49,380 Yeah. 152 00:07:49,380 --> 00:07:50,490 AUDIENCE: The vector. 153 00:07:50,490 --> 00:07:51,970 JEFFREY GROSSMAN: OK, yeah. 154 00:07:51,970 --> 00:07:52,470 Right. 155 00:07:52,470 --> 00:07:59,790 These are like 1 or 2 or 1, 2, 0, minus 1, 156 00:07:59,790 --> 00:08:03,570 whereas this is a place in space. 157 00:08:03,570 --> 00:08:05,820 And what else is really interesting about that vector? 158 00:08:05,820 --> 00:08:08,920 AUDIENCE: It's in reciprocal space. 159 00:08:08,920 --> 00:08:11,230 JEFFREY GROSSMAN: It's a reciprocal space. 160 00:08:11,230 --> 00:08:12,670 Absolutely. 161 00:08:12,670 --> 00:08:13,810 Nice job. 162 00:08:13,810 --> 00:08:15,550 You guys did that together, totally. 163 00:08:15,550 --> 00:08:16,510 Team effort, man. 164 00:08:16,510 --> 00:08:17,380 I love it. 165 00:08:17,380 --> 00:08:22,840 And what else? 166 00:08:22,840 --> 00:08:24,290 It's a vector in reciprocal space. 167 00:08:24,290 --> 00:08:24,790 What else? 168 00:08:24,790 --> 00:08:25,981 AUDIENCE: Kind of in the Brillouin zone. 169 00:08:25,981 --> 00:08:28,030 JEFFREY GROSSMAN: OK, well, it doesn't have to be. 170 00:08:28,030 --> 00:08:29,980 But we said it doesn't really matter if you go outside of it, 171 00:08:29,980 --> 00:08:31,540 because you just loop back. 172 00:08:31,540 --> 00:08:35,780 But what about its variation in the Brillouin zone? 173 00:08:35,780 --> 00:08:37,680 How can it vary? 174 00:08:37,680 --> 00:08:38,590 AUDIENCE: Anything. 175 00:08:38,590 --> 00:08:40,360 JEFFREY GROSSMAN: Yeah. 176 00:08:40,360 --> 00:08:43,590 This is a vector that can be anything. 177 00:08:43,590 --> 00:08:45,210 This thing can be anything. 178 00:08:45,210 --> 00:08:47,100 These things, can they be anything? 179 00:08:47,100 --> 00:08:48,733 Can l be pi? 180 00:08:48,733 --> 00:08:49,660 AUDIENCE: No. 181 00:08:49,660 --> 00:08:52,540 JEFFREY GROSSMAN: Or square root of 2, right? 182 00:08:52,540 --> 00:08:57,560 So it's a different kind of quantum number. 183 00:08:57,560 --> 00:09:00,125 But you see, it has the same impact. 184 00:09:03,060 --> 00:09:06,560 Because, what do these things do? 185 00:09:06,560 --> 00:09:08,670 What does this do, anyway? 186 00:09:08,670 --> 00:09:11,130 What's one of the things that this does? 187 00:09:11,130 --> 00:09:11,820 Or this? 188 00:09:11,820 --> 00:09:12,570 What does l do? 189 00:09:12,570 --> 00:09:17,700 When I change l from 0 to 1, what's 190 00:09:17,700 --> 00:09:20,250 one of the things that happens? 191 00:09:20,250 --> 00:09:23,505 AUDIENCE: Moving from, like, s to p? 192 00:09:23,505 --> 00:09:24,880 JEFFREY GROSSMAN: Yeah, yeah, OK. 193 00:09:24,880 --> 00:09:28,750 So let's write that down. 194 00:09:28,750 --> 00:09:33,760 Well, let's change n up to here to s, 1s, 2s. 195 00:09:33,760 --> 00:09:37,480 And now you just said I'm moving to p. 196 00:09:37,480 --> 00:09:41,660 Now, what is this that I'm always just assuming you know, 197 00:09:41,660 --> 00:09:43,130 is what that vertical axis is? 198 00:09:43,130 --> 00:09:43,630 What is it? 199 00:09:43,630 --> 00:09:44,785 AUDIENCE: Fill energy from lowest to highest. 200 00:09:44,785 --> 00:09:45,790 JEFFREY GROSSMAN: Yeah. 201 00:09:45,790 --> 00:09:49,060 So this is the energy going from lowest to highest, exactly. 202 00:09:49,060 --> 00:09:53,820 And so what did changing the quantum number, l, from 0 to 1 203 00:09:53,820 --> 00:09:54,320 do? 204 00:09:54,320 --> 00:09:55,655 AUDIENCE: It moved from s to p. 205 00:09:55,655 --> 00:09:59,990 JEFFREY GROSSMAN: Yeah, it moves it around. 206 00:09:59,990 --> 00:10:02,570 It moves it around. 207 00:10:02,570 --> 00:10:05,150 OK, so you can change the energy, 208 00:10:05,150 --> 00:10:07,460 and actually, you can also not change the energy. 209 00:10:07,460 --> 00:10:09,570 m doesn't change the energy. 210 00:10:09,570 --> 00:10:10,660 It gives degeneracy. 211 00:10:13,730 --> 00:10:15,500 But certainly you can change the energy 212 00:10:15,500 --> 00:10:16,925 by changing these quantum numbers. 213 00:10:19,890 --> 00:10:22,910 That's what you can do with k. 214 00:10:22,910 --> 00:10:24,590 That's what k does. 215 00:10:24,590 --> 00:10:29,840 k, which is a result of living in a periodically repeating 216 00:10:29,840 --> 00:10:33,890 world, k changes the levels. 217 00:10:33,890 --> 00:10:35,570 It's the same idea. 218 00:10:35,570 --> 00:10:39,120 It's very different, but it has the same impact. 219 00:10:39,120 --> 00:10:41,540 It changes these levels around. 220 00:10:41,540 --> 00:10:47,900 And it's very different because it can do it continuously, 221 00:10:47,900 --> 00:10:55,220 as opposed to these big leaps that we have in the atom. 222 00:10:55,220 --> 00:10:57,110 Or in the molecule. 223 00:10:57,110 --> 00:11:00,560 In the molecule, there are also discrete levels. 224 00:11:00,560 --> 00:11:06,050 And so we got this picture that we came to last time, 225 00:11:06,050 --> 00:11:10,100 where you see, k can be a vector. 226 00:11:10,100 --> 00:11:13,265 And it can live anywhere in this inverse space. 227 00:11:17,350 --> 00:11:22,390 And anytime you change it, you may change the energy levels. 228 00:11:22,390 --> 00:11:25,350 You may change them. 229 00:11:25,350 --> 00:11:28,940 So it's a really important thing when you have a solid. 230 00:11:28,940 --> 00:11:39,380 Now, again, if I have a periodically repeating atom, 231 00:11:39,380 --> 00:11:43,080 but the distance between them is like, 100 angstroms. 232 00:11:46,140 --> 00:11:49,010 What's the impact of k going to be? 233 00:11:49,010 --> 00:11:50,887 We did it in class. 234 00:11:50,887 --> 00:11:52,720 AUDIENCE: It's going to be relatively small. 235 00:11:52,720 --> 00:11:54,540 JEFFREY GROSSMAN: Relatively small. 236 00:11:54,540 --> 00:11:56,670 Like, almost nothing. 237 00:11:56,670 --> 00:12:01,230 Because the whole effect comes from the fact 238 00:12:01,230 --> 00:12:08,490 that you have an electron in a lattice of periodic potentials, 239 00:12:08,490 --> 00:12:11,250 where they're sort of interacting, 240 00:12:11,250 --> 00:12:14,190 where they're close enough that there's some interaction. 241 00:12:14,190 --> 00:12:17,350 If there's no interaction, and your atoms are really, 242 00:12:17,350 --> 00:12:23,270 really far apart, then it's basically 243 00:12:23,270 --> 00:12:25,730 like you just have isolated atoms, 244 00:12:25,730 --> 00:12:28,520 sitting in their potentials. 245 00:12:28,520 --> 00:12:30,960 And a whole lot of empty space in between. 246 00:12:30,960 --> 00:12:33,260 And then the levels will just look flat 247 00:12:33,260 --> 00:12:34,670 as a function of k-space. 248 00:12:34,670 --> 00:12:36,440 Doesn't mean you can't vary k. 249 00:12:36,440 --> 00:12:37,910 You could still vary k. 250 00:12:37,910 --> 00:12:41,600 But the bands that you get will be completely flat. 251 00:12:41,600 --> 00:12:45,290 They'll go back to your molecular level description. 252 00:12:45,290 --> 00:12:46,970 So does everybody see that? 253 00:12:49,740 --> 00:12:51,720 All right. 254 00:12:51,720 --> 00:12:54,420 And we talked about that. 255 00:12:54,420 --> 00:12:59,220 And so then the question was, well, what playground of k 256 00:12:59,220 --> 00:13:01,530 matters? 257 00:13:01,530 --> 00:13:06,307 And this is something we talked about, 258 00:13:06,307 --> 00:13:07,390 but it's really important. 259 00:13:07,390 --> 00:13:09,900 So I want to review it here. 260 00:13:09,900 --> 00:13:14,670 So we just said that k is this new index that 261 00:13:14,670 --> 00:13:17,700 is a continuous variable, that changes the energy levels. 262 00:13:17,700 --> 00:13:20,300 It moves them around. 263 00:13:20,300 --> 00:13:23,970 OK, so what does that mean for a calculation, right? 264 00:13:23,970 --> 00:13:28,400 Well, one thing that it means is that when 265 00:13:28,400 --> 00:13:43,820 I do a calculation, if an energy level can now do this, 266 00:13:43,820 --> 00:13:46,520 as a function of k, then that means 267 00:13:46,520 --> 00:13:48,380 I need to actually do my calculation 268 00:13:48,380 --> 00:13:50,660 with a mesh of k-points. 269 00:13:50,660 --> 00:13:52,160 How fine a mesh do I need? 270 00:13:56,100 --> 00:13:57,397 What's the answer to that? 271 00:13:57,397 --> 00:13:58,151 AUDIENCE: Depends. 272 00:13:58,151 --> 00:13:59,640 JEFFREY GROSSMAN: Thank you. 273 00:13:59,640 --> 00:14:00,646 On what? 274 00:14:00,646 --> 00:14:03,130 AUDIENCE: How curvy the bands are. 275 00:14:03,130 --> 00:14:04,630 JEFFREY GROSSMAN: Yeah. 276 00:14:04,630 --> 00:14:06,343 How curvy are my bands? 277 00:14:06,343 --> 00:14:07,665 AUDIENCE: Pretty curvy. 278 00:14:07,665 --> 00:14:09,290 JEFFREY GROSSMAN: This is pretty curvy. 279 00:14:09,290 --> 00:14:13,760 So maybe I need a lot of k-points to capture this. 280 00:14:13,760 --> 00:14:15,170 You got a grid. 281 00:14:15,170 --> 00:14:18,710 I'm solving for this problem on a grid of k-points, basically. 282 00:14:18,710 --> 00:14:23,420 And if I only have a k-point here and here, well, 283 00:14:23,420 --> 00:14:27,140 let's say I have it actually even worse, I have it here 284 00:14:27,140 --> 00:14:32,900 and here, I'm not going to really capture 285 00:14:32,900 --> 00:14:35,480 the right physics here. 286 00:14:35,480 --> 00:14:38,090 So the density of k-points has to be such 287 00:14:38,090 --> 00:14:42,770 that you capture the features, in terms of how 288 00:14:42,770 --> 00:14:44,790 much these energy levels vary. 289 00:14:44,790 --> 00:14:47,870 So let's go back to these two atoms that are very far apart. 290 00:14:47,870 --> 00:14:49,760 How many k-points do I need for those? 291 00:14:53,050 --> 00:14:54,460 AUDIENCE: Two. 292 00:14:54,460 --> 00:14:56,602 JEFFREY GROSSMAN: What if they're really far apart? 293 00:14:56,602 --> 00:14:57,780 AUDIENCE: One. 294 00:14:57,780 --> 00:14:58,720 JEFFREY GROSSMAN: One. 295 00:14:58,720 --> 00:14:59,220 Whoa. 296 00:14:59,220 --> 00:15:02,520 All the calculations you've been doing, until we did the solids, 297 00:15:02,520 --> 00:15:04,950 everything before that was actually 1 k-point. 298 00:15:04,950 --> 00:15:06,780 It was at 000. 299 00:15:06,780 --> 00:15:08,370 We never varied it. 300 00:15:08,370 --> 00:15:11,670 We never worried about it. 301 00:15:11,670 --> 00:15:13,620 It was never an index. 302 00:15:13,620 --> 00:15:17,760 OK, so now, if the bands are flat, 303 00:15:17,760 --> 00:15:19,890 you could just have one k-point, and you've 304 00:15:19,890 --> 00:15:25,120 got the variation of the bands nailed, since they don't vary. 305 00:15:25,120 --> 00:15:26,400 So you just need the one. 306 00:15:26,400 --> 00:15:29,580 But if they vary a lot, like if I bring atoms together, 307 00:15:29,580 --> 00:15:32,340 and they vary a lot, and there's a lot of interaction, 308 00:15:32,340 --> 00:15:35,730 then I just need more and more k-points to represent that 309 00:15:35,730 --> 00:15:36,690 correctly. 310 00:15:36,690 --> 00:15:40,470 OK, so the k-point is something that's 311 00:15:40,470 --> 00:15:44,830 now an essential part of your calculation. 312 00:15:44,830 --> 00:15:48,310 Now, that's talking about what you 313 00:15:48,310 --> 00:15:50,380 need to get the density right, which 314 00:15:50,380 --> 00:15:52,270 is to get the wave function right, which is 315 00:15:52,270 --> 00:15:55,550 to get the energy levels right. 316 00:15:55,550 --> 00:16:01,100 That's a new component of a quantum mechanical simulation. 317 00:16:01,100 --> 00:16:02,900 But then there was this other thing 318 00:16:02,900 --> 00:16:08,150 that's actually a little bit separate in a way, which 319 00:16:08,150 --> 00:16:12,980 is to visualize that variation. 320 00:16:12,980 --> 00:16:17,760 Now, visualizing this variation leads to these plots, 321 00:16:17,760 --> 00:16:19,883 which are called 322 00:16:19,883 --> 00:16:21,690 AUDIENCE: Band structures. 323 00:16:21,690 --> 00:16:23,230 JEFFREY GROSSMAN: Band structures. 324 00:16:23,230 --> 00:16:25,050 This is the band structure of the material. 325 00:16:28,410 --> 00:16:32,800 Ooh, it's another kind of phase map, 326 00:16:32,800 --> 00:16:37,410 for those of you who felt the excitement of phase diagrams 327 00:16:37,410 --> 00:16:38,400 in 301-2. 328 00:16:38,400 --> 00:16:40,260 This is like a phase map for that electron. 329 00:16:43,020 --> 00:16:50,160 It's a really important description 330 00:16:50,160 --> 00:16:52,290 of how electrons behave in materials. 331 00:16:58,948 --> 00:17:00,240 Everything going OK over there? 332 00:17:00,240 --> 00:17:00,865 AUDIENCE: Yeah. 333 00:17:03,003 --> 00:17:04,920 JEFFREY GROSSMAN: It looked like you got hurt. 334 00:17:04,920 --> 00:17:06,190 No, you're good. 335 00:17:06,190 --> 00:17:06,690 OK. 336 00:17:06,690 --> 00:17:09,440 All right, I got a little concerned, but we're over it. 337 00:17:12,904 --> 00:17:16,410 [LAUGHS] Sorry. 338 00:17:16,410 --> 00:17:23,750 The phase map is a walkthrough k-space. 339 00:17:23,750 --> 00:17:27,530 Now, I just mentioned that you have a grid that you just 340 00:17:27,530 --> 00:17:31,220 have to put on here to do the calculation 341 00:17:31,220 --> 00:17:33,467 to get the wiggles that may or may not be there. 342 00:17:33,467 --> 00:17:35,300 Well, you should take a guess it whether you 343 00:17:35,300 --> 00:17:37,190 think they're there, and then you should see, 344 00:17:37,190 --> 00:17:40,010 and then you should converge it. 345 00:17:40,010 --> 00:17:45,797 But now I'm taking a walk around in that zone. 346 00:17:45,797 --> 00:17:47,630 And the question is, where am I going to go? 347 00:17:53,080 --> 00:17:56,330 Well, I'm using 4 by 4 by 4 k-points. 348 00:17:56,330 --> 00:17:59,080 So I'm using a grid of 64 points. 349 00:17:59,080 --> 00:18:02,860 And I'm plopping it down into this Brillouin zone. 350 00:18:02,860 --> 00:18:04,510 Should I just look at the energies 351 00:18:04,510 --> 00:18:06,290 at each of those points? 352 00:18:06,290 --> 00:18:10,270 Well, you could do that, right? 353 00:18:10,270 --> 00:18:12,970 Ideally, you'd like to be able to visualize 354 00:18:12,970 --> 00:18:19,490 this variation all over k-space, at every single point. 355 00:18:19,490 --> 00:18:21,800 But why don't we do that? 356 00:18:21,800 --> 00:18:23,480 Tell me why we don't usually do that. 357 00:18:29,264 --> 00:18:32,496 AUDIENCE: Would we be more interested in isosurfaces? 358 00:18:32,496 --> 00:18:35,460 JEFFREY GROSSMAN: Well, that is true, 359 00:18:35,460 --> 00:18:37,960 but that's more of a density thing. 360 00:18:37,960 --> 00:18:40,710 So that's where you'd want to know the isosurfaces 361 00:18:40,710 --> 00:18:42,880 of the charged density. 362 00:18:42,880 --> 00:18:43,380 Yeah? 363 00:18:43,380 --> 00:18:44,691 AUDIENCE: Once you know a certain amount, 364 00:18:44,691 --> 00:18:46,441 it's repetitive throughout this direction. 365 00:18:46,441 --> 00:18:48,372 JEFFREY GROSSMAN: OK, that's definitely true. 366 00:18:48,372 --> 00:18:49,830 But let's say you didn't know that. 367 00:18:56,170 --> 00:18:58,450 I'd like to do the most accurate calculation possible. 368 00:18:58,450 --> 00:19:02,440 Let's say I put a grid over this k-space of a million 369 00:19:02,440 --> 00:19:03,760 by a million by a million. 370 00:19:03,760 --> 00:19:05,500 Don't do that on the nanoHUB. 371 00:19:05,500 --> 00:19:06,880 We'll get calls. 372 00:19:06,880 --> 00:19:11,410 [LAUGHS] But let's say I did that. 373 00:19:11,410 --> 00:19:13,120 Now, I've got a million by a million 374 00:19:13,120 --> 00:19:15,040 by a million points in here, where 375 00:19:15,040 --> 00:19:20,110 each point has a new set of energy eigenvalues. 376 00:19:20,110 --> 00:19:21,910 Why can't I just visualize that? 377 00:19:25,690 --> 00:19:26,190 Yeah. 378 00:19:26,190 --> 00:19:29,451 AUDIENCE: Would it just take too long to simulate? 379 00:19:29,451 --> 00:19:31,980 JEFFREY GROSSMAN: Well, it would take a whole lot of time 380 00:19:31,980 --> 00:19:33,330 to simulate. 381 00:19:33,330 --> 00:19:35,070 What about visualizing it, though? 382 00:19:35,070 --> 00:19:37,310 How would you visualize that? 383 00:19:37,310 --> 00:19:40,160 AUDIENCE: [INAUDIBLE] 3D [INAUDIBLE],, right? 384 00:19:40,160 --> 00:19:44,540 JEFFREY GROSSMAN: But see, it's not even really just 3D, right? 385 00:19:44,540 --> 00:19:46,310 Because at each one of these points, 386 00:19:46,310 --> 00:19:48,215 I've got a spectrum of energies. 387 00:19:51,380 --> 00:19:58,730 So it would be making your eigenvalues, 388 00:19:58,730 --> 00:20:01,640 each one into a 3D curve. 389 00:20:01,640 --> 00:20:04,850 And I want to look at them all. 390 00:20:04,850 --> 00:20:08,040 You could do that. 391 00:20:08,040 --> 00:20:13,730 But you got to remember, if I go down to here, 392 00:20:13,730 --> 00:20:17,360 you can't just do it within this volume element, right? 393 00:20:17,360 --> 00:20:24,470 My eigenvalues might shift up, down in different ways, 394 00:20:24,470 --> 00:20:25,950 depending on where I am. 395 00:20:25,950 --> 00:20:29,060 How am I actually going to see that? 396 00:20:29,060 --> 00:20:33,470 So in a way, what we have is actually a strong visualization 397 00:20:33,470 --> 00:20:34,310 challenge. 398 00:20:39,070 --> 00:20:40,780 There's sort of a good story here, 399 00:20:40,780 --> 00:20:44,440 because luckily, we can just look 400 00:20:44,440 --> 00:20:49,440 at a 2D map of this variation and get all the information we 401 00:20:49,440 --> 00:20:54,920 need, for almost all materials. 402 00:20:54,920 --> 00:20:57,510 And so, how do you get it a 2D map? 403 00:20:57,510 --> 00:21:01,760 Well then, instead of doing the whole of volume, 404 00:21:01,760 --> 00:21:04,820 you just walk through the volume along lines. 405 00:21:04,820 --> 00:21:08,840 You literally just walk along this line, and then this line, 406 00:21:08,840 --> 00:21:11,510 and then down, and then back, and then up. 407 00:21:11,510 --> 00:21:14,900 And as you're doing this walk through k-space, 408 00:21:14,900 --> 00:21:18,590 you plot the energy bands, and you connect them together. 409 00:21:18,590 --> 00:21:21,870 And you draw lines between those connections. 410 00:21:21,870 --> 00:21:23,480 This is the same energy level just 411 00:21:23,480 --> 00:21:25,730 moving around, along that walk. 412 00:21:29,330 --> 00:21:34,370 So that seems like an oversimplification, 413 00:21:34,370 --> 00:21:37,910 because I'm not going through the whole Brillouin zone 414 00:21:37,910 --> 00:21:38,780 by any means. 415 00:21:38,780 --> 00:21:43,110 I'm just going through very particular lines in there. 416 00:21:43,110 --> 00:21:45,510 And yet it's all we need. 417 00:21:45,510 --> 00:21:49,970 And the reason is that most of the things 418 00:21:49,970 --> 00:21:53,070 I want to know about a material, most of the things 419 00:21:53,070 --> 00:21:55,430 I want to know, are going to come 420 00:21:55,430 --> 00:22:00,950 from the variation in these bands between high symmetry 421 00:22:00,950 --> 00:22:04,670 points in k-space. 422 00:22:04,670 --> 00:22:08,390 So I can go to other points in the Brillouin zone, 423 00:22:08,390 --> 00:22:09,860 and I'll get other levels. 424 00:22:09,860 --> 00:22:13,970 I'll get a different set of levels. 425 00:22:13,970 --> 00:22:15,680 But it's probably not going to add 426 00:22:15,680 --> 00:22:19,960 to my knowledge of the material and the properties 427 00:22:19,960 --> 00:22:21,220 that I can get. 428 00:22:21,220 --> 00:22:24,850 So for example, I'm very interested in this 429 00:22:24,850 --> 00:22:28,300 where this has a maximum, the valence band, and where 430 00:22:28,300 --> 00:22:31,240 the conduction band has a minimum, right? 431 00:22:31,240 --> 00:22:33,670 And you'll notice, the minimum for silicon 432 00:22:33,670 --> 00:22:36,730 is actually not at this high symmetry point x. 433 00:22:36,730 --> 00:22:40,420 But it does come along the path to x from another high symmetry 434 00:22:40,420 --> 00:22:42,110 point, gamma. 435 00:22:42,110 --> 00:22:43,470 It came along that path. 436 00:22:43,470 --> 00:22:46,850 And there's no place in k-space that gives me a lower point, 437 00:22:46,850 --> 00:22:47,570 you see. 438 00:22:47,570 --> 00:22:50,540 So I nailed the thing I needed I nailed it 439 00:22:50,540 --> 00:22:53,010 by going from one high symmetry point to another. 440 00:22:53,010 --> 00:22:55,325 And that's what a band diagram is. 441 00:22:55,325 --> 00:22:58,730 It's a walk through k-space along lines 442 00:22:58,730 --> 00:23:00,260 between high symmetry points. 443 00:23:00,260 --> 00:23:02,390 And that gives you the map that you need. 444 00:23:02,390 --> 00:23:07,220 And these sometimes look really complicated, but just think, 445 00:23:07,220 --> 00:23:10,070 it's saving you a whole lot of trouble 446 00:23:10,070 --> 00:23:13,310 compared to all the data you actually could have. 447 00:23:13,310 --> 00:23:15,080 But it's giving you what you need. 448 00:23:15,080 --> 00:23:20,780 Are there any questions about that? 449 00:23:20,780 --> 00:23:25,546 That's sort of the key point here from this band structure. 450 00:23:28,880 --> 00:23:37,000 And we've talked about this, how you get your bands, 451 00:23:37,000 --> 00:23:39,940 just like in an atom or a molecule. 452 00:23:39,940 --> 00:23:42,370 You get your energy levels from solving the Schrodinger 453 00:23:42,370 --> 00:23:42,980 equation. 454 00:23:42,980 --> 00:23:48,830 And then you fill them up, using the Pauli exclusion rule, 455 00:23:48,830 --> 00:23:50,810 right? 456 00:23:50,810 --> 00:23:53,070 Add a little bit of Hund's rule to it, 457 00:23:53,070 --> 00:23:54,440 although that can be violated. 458 00:23:54,440 --> 00:23:55,250 Don't tell anyone. 459 00:23:57,850 --> 00:23:59,630 And you fill them up and then you're done. 460 00:23:59,630 --> 00:24:01,820 And when you're done, you've got your Fermi energy. 461 00:24:04,580 --> 00:24:08,510 Now, it's the same exact thing here in the band structure. 462 00:24:08,510 --> 00:24:12,470 You fill your bands up, until you've filled them 463 00:24:12,470 --> 00:24:14,780 up with the number of electrons that you 464 00:24:14,780 --> 00:24:17,210 had in your simulation. 465 00:24:17,210 --> 00:24:19,520 And then you've got your Fermi energy, 466 00:24:19,520 --> 00:24:24,380 and then if there's a small gap there, and no bands cross it, 467 00:24:24,380 --> 00:24:25,730 it cannot be a what? 468 00:24:28,700 --> 00:24:30,087 It cannot be a-- 469 00:24:30,087 --> 00:24:31,037 AUDIENCE: Conductor. 470 00:24:31,037 --> 00:24:32,120 JEFFREY GROSSMAN: A metal. 471 00:24:32,120 --> 00:24:33,432 It can be a conductor. 472 00:24:33,432 --> 00:24:34,580 AUDIENCE: I don't know. 473 00:24:34,580 --> 00:24:37,070 JEFFREY GROSSMAN: It can be a conductor, a semiconductor, 474 00:24:37,070 --> 00:24:42,740 or a really bad insulating conductor, also known 475 00:24:42,740 --> 00:24:45,180 as an insulator. 476 00:24:45,180 --> 00:24:47,790 But it can't be a metal. 477 00:24:47,790 --> 00:24:51,480 Because if no bands cross, if there's a gap here 478 00:24:51,480 --> 00:24:55,650 between where you filled to and the next place, 479 00:24:55,650 --> 00:24:58,770 then it's not a metal. 480 00:24:58,770 --> 00:25:02,230 So we talked about that. 481 00:25:02,230 --> 00:25:05,438 And that's, certainly if you have 482 00:25:05,438 --> 00:25:07,480 an odd number of electrons, it has to be a metal. 483 00:25:10,720 --> 00:25:13,570 And there's a calculation, I showed this last time 484 00:25:13,570 --> 00:25:14,470 of an insulator. 485 00:25:14,470 --> 00:25:22,240 And we also talked about how this density of states 486 00:25:22,240 --> 00:25:24,680 is another really important thing here. 487 00:25:24,680 --> 00:25:27,340 And since this is in your homework as well, 488 00:25:27,340 --> 00:25:32,560 in your current PSET, I'll get to transport in a sec. 489 00:25:32,560 --> 00:25:34,810 I want to make sure we know what density of states is. 490 00:25:42,280 --> 00:25:48,085 Somebody tell me what the density of states of the carbon 491 00:25:48,085 --> 00:25:48,960 atom would look like. 492 00:25:51,810 --> 00:25:53,235 AUDIENCE: [INAUDIBLE] 493 00:25:53,235 --> 00:25:55,090 JEFFREY GROSSMAN: OK, so let's draw. 494 00:25:55,090 --> 00:26:01,930 So we had the carbon atom would have 1s and 2s and 2p. 495 00:26:01,930 --> 00:26:03,460 And how many would I put here? 496 00:26:03,460 --> 00:26:04,287 AUDIENCE: 2. 497 00:26:04,287 --> 00:26:06,370 JEFFREY GROSSMAN: OK, and what about the next one? 498 00:26:06,370 --> 00:26:06,745 AUDIENCE: 2. 499 00:26:06,745 --> 00:26:08,350 JEFFREY GROSSMAN: OK, I'm liking it. 500 00:26:08,350 --> 00:26:10,036 And here? 501 00:26:10,036 --> 00:26:10,900 AUDIENCE: 2. 502 00:26:10,900 --> 00:26:11,920 JEFFREY GROSSMAN: 2? 503 00:26:11,920 --> 00:26:12,700 OK, like that. 504 00:26:12,700 --> 00:26:13,330 AUDIENCE: Aw. 505 00:26:13,330 --> 00:26:14,830 JEFFREY GROSSMAN: Oh, wait, hold on. 506 00:26:14,830 --> 00:26:18,280 OK, thank you, like this. 507 00:26:18,280 --> 00:26:19,750 Oh, sorry. 508 00:26:19,750 --> 00:26:20,860 All right. 509 00:26:20,860 --> 00:26:23,300 That would be an open shell singlet. 510 00:26:23,300 --> 00:26:24,610 Thank you very much. 511 00:26:24,610 --> 00:26:26,770 OK, now. 512 00:26:26,770 --> 00:26:29,450 OK, so what's my density of states over this? 513 00:26:29,450 --> 00:26:32,030 How do I do it? 514 00:26:32,030 --> 00:26:32,760 OK, yeah. 515 00:26:32,760 --> 00:26:35,320 So I'm seeing people go like this. 516 00:26:35,320 --> 00:26:37,990 So this was energy here. 517 00:26:37,990 --> 00:26:40,060 And now this is going to be energy here. 518 00:26:43,920 --> 00:26:47,107 OK, what's my next step? 519 00:26:47,107 --> 00:26:48,190 AUDIENCE: Drawing a curve. 520 00:26:48,190 --> 00:26:49,060 JEFFREY GROSSMAN: Drawing a curve. 521 00:26:49,060 --> 00:26:50,650 OK, I'm going to draw a curve. 522 00:26:50,650 --> 00:26:51,700 What is that? 523 00:26:51,700 --> 00:26:53,047 AUDIENCE: 1s. 524 00:26:53,047 --> 00:26:54,130 JEFFREY GROSSMAN: It's 1s. 525 00:26:54,130 --> 00:26:55,750 Why not? 526 00:26:55,750 --> 00:26:57,070 I've just broadened it. 527 00:26:57,070 --> 00:26:58,720 We'll make that 1s. 528 00:26:58,720 --> 00:27:01,330 And what's my next curve I want to draw? 529 00:27:01,330 --> 00:27:02,410 OK, sure. 530 00:27:02,410 --> 00:27:08,140 And it should be spaced such that it 531 00:27:08,140 --> 00:27:09,910 should be the right energy spacing 532 00:27:09,910 --> 00:27:13,360 that I get from my quantum mechanical simulations. 533 00:27:13,360 --> 00:27:14,680 OK. 534 00:27:14,680 --> 00:27:17,110 And what's next? 535 00:27:17,110 --> 00:27:19,000 2p. 536 00:27:19,000 --> 00:27:21,640 Let's make this go out further. 537 00:27:21,640 --> 00:27:22,810 OK, now 2p. 538 00:27:22,810 --> 00:27:24,127 Now, how should I do this? 539 00:27:24,127 --> 00:27:25,210 Should I make it the same? 540 00:27:28,521 --> 00:27:31,370 AUDIENCE: Or shift it down [INAUDIBLE] energy. 541 00:27:31,370 --> 00:27:33,160 JEFFREY GROSSMAN: Shift it down. 542 00:27:33,160 --> 00:27:33,785 AUDIENCE: Same. 543 00:27:33,785 --> 00:27:35,250 JEFFREY GROSSMAN: Same. 544 00:27:35,250 --> 00:27:37,110 Is it different than 2s? 545 00:27:37,110 --> 00:27:39,990 AUDIENCE: [INAUDIBLE] 546 00:27:39,990 --> 00:27:41,670 JEFFREY GROSSMAN: Well, let's see. 547 00:27:41,670 --> 00:27:44,400 I had two electrons in each of these. 548 00:27:44,400 --> 00:27:46,686 How many electrons could go into 2p? 549 00:27:46,686 --> 00:27:47,500 AUDIENCE: 6. 550 00:27:47,500 --> 00:27:49,030 JEFFREY GROSSMAN: 6. 551 00:27:49,030 --> 00:27:51,190 So I actually have more states there. 552 00:27:51,190 --> 00:27:54,520 I have 3 times as many as here. 553 00:27:54,520 --> 00:27:56,770 So in a way, if you're thinking about these 554 00:27:56,770 --> 00:28:03,910 as sort of a vertical line, that is spread out 555 00:28:03,910 --> 00:28:06,310 by some smoothing function, just to make it 556 00:28:06,310 --> 00:28:10,600 into a nice, pretty curve, if it's an atom, if it's really 557 00:28:10,600 --> 00:28:12,730 an atom, then it's really just a line. 558 00:28:12,730 --> 00:28:13,760 It's a delta function. 559 00:28:16,630 --> 00:28:19,870 But then this should have three lines all on top of each other, 560 00:28:19,870 --> 00:28:21,420 right? 561 00:28:21,420 --> 00:28:23,430 And so if I look at that, it's going 562 00:28:23,430 --> 00:28:32,390 to lead to a bigger peak, that has more area under it. 563 00:28:32,390 --> 00:28:33,473 Three times as much, yeah. 564 00:28:33,473 --> 00:28:34,348 AUDIENCE: [INAUDIBLE] 565 00:28:34,348 --> 00:28:35,360 JEFFREY GROSSMAN: OK. 566 00:28:35,360 --> 00:28:38,880 So you can fit 6 electrons into this peak. 567 00:28:38,880 --> 00:28:41,720 This is what the density of states is, right? 568 00:28:41,720 --> 00:28:46,340 It is the density of states. 569 00:28:46,340 --> 00:28:51,950 [LAUGHS] It is these levels. 570 00:28:51,950 --> 00:28:54,520 It's these levels. 571 00:28:54,520 --> 00:28:55,020 OK. 572 00:28:55,020 --> 00:28:56,020 Does everybody see that? 573 00:28:56,020 --> 00:29:00,290 Now, where's my Fermi energy for the carbon atom? 574 00:29:03,748 --> 00:29:06,453 AUDIENCE: [INAUDIBLE] 575 00:29:06,453 --> 00:29:07,870 JEFFREY GROSSMAN: Is it over here? 576 00:29:07,870 --> 00:29:09,690 AUDIENCE: No. 577 00:29:09,690 --> 00:29:11,877 [INAUDIBLE] 578 00:29:11,877 --> 00:29:12,835 JEFFREY GROSSMAN: Here? 579 00:29:12,835 --> 00:29:14,735 AUDIENCE: No. 580 00:29:14,735 --> 00:29:17,630 JEFFREY GROSSMAN: It feels like it should be here, right? 581 00:29:17,630 --> 00:29:19,260 We like Fermi energies between peaks, 582 00:29:19,260 --> 00:29:21,240 but that's just because we like semiconductors. 583 00:29:21,240 --> 00:29:24,030 No, we like metals too, and insulators. 584 00:29:24,030 --> 00:29:25,863 They're all good. 585 00:29:25,863 --> 00:29:26,780 So where should it be? 586 00:29:29,405 --> 00:29:30,280 Does this sound good? 587 00:29:30,280 --> 00:29:31,495 AUDIENCE: No. 588 00:29:31,495 --> 00:29:34,294 JEFFREY GROSSMAN: Over here? 589 00:29:34,294 --> 00:29:35,710 Hmm? 590 00:29:35,710 --> 00:29:38,200 Somebody tell me when to stop. 591 00:29:38,200 --> 00:29:40,000 Somebody say stop. 592 00:29:40,000 --> 00:29:40,752 AUDIENCE: Stop. 593 00:29:40,752 --> 00:29:41,710 JEFFREY GROSSMAN: Yeah. 594 00:29:41,710 --> 00:29:44,080 OK, now why am I stopping here? 595 00:29:44,080 --> 00:29:48,530 So that's my Fermi energy for the carbon atom. 596 00:29:48,530 --> 00:29:49,688 Why is it there? 597 00:29:49,688 --> 00:29:53,104 AUDIENCE: [INAUDIBLE] 598 00:29:53,104 --> 00:29:54,730 JEFFREY GROSSMAN: Well, it's an atom. 599 00:29:54,730 --> 00:29:59,170 So let's not think about it necessarily as conducting. 600 00:29:59,170 --> 00:30:01,650 AUDIENCE: It's where you stop filling. 601 00:30:01,650 --> 00:30:04,740 JEFFREY GROSSMAN: That's the definition, right? 602 00:30:04,740 --> 00:30:07,755 Here I filled and I stopped. 603 00:30:07,755 --> 00:30:08,630 AUDIENCE: [INAUDIBLE] 604 00:30:08,630 --> 00:30:09,588 JEFFREY GROSSMAN: Yeah. 605 00:30:09,588 --> 00:30:11,777 So here I filled and I stopped, and here I filled 606 00:30:11,777 --> 00:30:12,360 and I stopped. 607 00:30:12,360 --> 00:30:15,270 And what the gentleman over there said is, 608 00:30:15,270 --> 00:30:17,760 it's where you stop filling, right? 609 00:30:17,760 --> 00:30:20,040 And so here is where I stop filling, 610 00:30:20,040 --> 00:30:23,850 because I had this peak of states, right? 611 00:30:23,850 --> 00:30:28,310 It's a peak of states, the density of states, that is only 612 00:30:28,310 --> 00:30:32,100 filled by 2 electrons, right? 613 00:30:32,100 --> 00:30:34,650 Does everybody see that? 614 00:30:34,650 --> 00:30:41,610 So if I had something else, like say, neon, 615 00:30:41,610 --> 00:30:44,456 would my Fermi energy be inside the peak? 616 00:30:44,456 --> 00:30:45,345 AUDIENCE: No. 617 00:30:45,345 --> 00:30:46,220 JEFFREY GROSSMAN: No. 618 00:30:46,220 --> 00:30:46,920 AUDIENCE: It would be just out. 619 00:30:46,920 --> 00:30:49,040 JEFFREY GROSSMAN: Just outside of it, because I 620 00:30:49,040 --> 00:30:50,010 would have filled it, right? 621 00:30:50,010 --> 00:30:50,510 Yeah. 622 00:30:50,510 --> 00:30:53,000 AUDIENCE: Besides the change in the location of the Fermi 623 00:30:53,000 --> 00:30:58,358 energy, would the density states look the same for nitrogen? 624 00:30:58,358 --> 00:30:59,900 JEFFREY GROSSMAN: Very good question. 625 00:30:59,900 --> 00:31:00,770 It would, actually. 626 00:31:00,770 --> 00:31:05,630 In the sense that if you're looking at atoms, 627 00:31:05,630 --> 00:31:07,250 a density of states is a little bit 628 00:31:07,250 --> 00:31:09,080 of an odd thing for an atom. 629 00:31:09,080 --> 00:31:12,650 But I'm trying to explain it in the context of atoms, 630 00:31:12,650 --> 00:31:15,650 because I really want you guys to see the connection, 631 00:31:15,650 --> 00:31:18,330 and really what it is. 632 00:31:18,330 --> 00:31:20,090 But if you look at atoms, they're 633 00:31:20,090 --> 00:31:21,770 nothing more than levels. 634 00:31:21,770 --> 00:31:27,200 And those discrete levels become little shapes 635 00:31:27,200 --> 00:31:28,520 that you broaden a little. 636 00:31:28,520 --> 00:31:30,290 So it would look the same as nitrogen, 637 00:31:30,290 --> 00:31:33,080 because in nitrogen, you would have that. 638 00:31:33,080 --> 00:31:34,620 Now, what does that do? 639 00:31:34,620 --> 00:31:38,390 Well, it'll change these energy level positions. 640 00:31:38,390 --> 00:31:39,770 It'll shift them around. 641 00:31:39,770 --> 00:31:40,850 This may even change. 642 00:31:40,850 --> 00:31:43,820 The 1s may go down or up, this may change a little. 643 00:31:43,820 --> 00:31:47,240 And then here, you're going to change not the peak, 644 00:31:47,240 --> 00:31:49,400 because you still got 6 states, and you still 645 00:31:49,400 --> 00:31:52,010 got 6 electrons you can put in there. 646 00:31:52,010 --> 00:31:54,410 But you will change the Fermi energy within the peak, 647 00:31:54,410 --> 00:31:56,100 because you can fill it by one more. 648 00:32:05,170 --> 00:32:09,130 And just like going all the way back to our first 649 00:32:09,130 --> 00:32:14,470 or second lecture, where we had the orbitals of the hydrogen 650 00:32:14,470 --> 00:32:15,700 atom. 651 00:32:15,700 --> 00:32:20,710 So we had the nucleus and we had this electron 652 00:32:20,710 --> 00:32:23,140 that does not orbit. 653 00:32:23,140 --> 00:32:25,060 But there it is orbiting. 654 00:32:25,060 --> 00:32:29,110 And then we had the 1s, and then we had the 2s. 655 00:32:29,110 --> 00:32:31,090 And remember how we talked about, 656 00:32:31,090 --> 00:32:33,250 it can actually make transitions between the two 657 00:32:33,250 --> 00:32:36,798 without actually existing between the two. 658 00:32:36,798 --> 00:32:38,590 There are some times it has to cross paths, 659 00:32:38,590 --> 00:32:40,600 which are totally forbidden. 660 00:32:40,600 --> 00:32:42,700 It's sort of exciting, it's like some sort 661 00:32:42,700 --> 00:32:44,960 of interesting fiction. 662 00:32:44,960 --> 00:32:47,270 That this electron is going on an adventure. 663 00:32:47,270 --> 00:32:51,470 And well, that's what this is, right? 664 00:32:51,470 --> 00:32:54,940 There is no state in here, in this energy. 665 00:32:54,940 --> 00:32:57,670 So the DOS tells you that too. 666 00:32:57,670 --> 00:32:58,840 That is what's in the DOS. 667 00:32:58,840 --> 00:33:00,820 That information is contained in the DOS. 668 00:33:00,820 --> 00:33:08,230 There cannot be an electron with an energy where there's no DOS, 669 00:33:08,230 --> 00:33:12,470 there's no states to support it there. 670 00:33:12,470 --> 00:33:17,570 So for an atom, like the hydrogen atom or the carbon 671 00:33:17,570 --> 00:33:19,217 atom as well, that's fairly obvious, 672 00:33:19,217 --> 00:33:21,050 because you had your levels and you knew it. 673 00:33:21,050 --> 00:33:22,710 Couldn't be in between those levels. 674 00:33:22,710 --> 00:33:25,340 That's the quantization stuff we talked about. 675 00:33:25,340 --> 00:33:31,920 But now when you get to solids, and it's a more messy picture. 676 00:33:31,920 --> 00:33:35,070 But it's the same understanding that you get out of it. 677 00:33:35,070 --> 00:33:37,630 It's the same exact understanding. 678 00:33:37,630 --> 00:33:43,680 So what you do for atom, molecule, or solid 679 00:33:43,680 --> 00:33:49,380 is you take your levels and you turn them over, 680 00:33:49,380 --> 00:33:52,470 and you count up the number of states, 681 00:33:52,470 --> 00:33:56,050 and that has to be the same. 682 00:33:56,050 --> 00:33:58,440 And you get shapes to the DOS. 683 00:33:58,440 --> 00:34:00,720 If it was an atom, I just had these little lines 684 00:34:00,720 --> 00:34:02,820 here from the levels, and I broaden them. 685 00:34:02,820 --> 00:34:06,530 Now, the levels are no longer flat. 686 00:34:06,530 --> 00:34:09,590 If it's a solid, they can vary. 687 00:34:09,590 --> 00:34:12,739 And when you turn that on its side, 688 00:34:12,739 --> 00:34:16,860 it can be really different, right? 689 00:34:16,860 --> 00:34:18,523 But it's telling you the same thing. 690 00:34:18,523 --> 00:34:20,190 It's telling you exactly the same thing. 691 00:34:20,190 --> 00:34:23,083 You're filling your electrons up through here. 692 00:34:23,083 --> 00:34:24,750 And you're stopping, well, I don't know, 693 00:34:24,750 --> 00:34:25,792 is that the Fermi energy? 694 00:34:25,792 --> 00:34:27,239 I didn't tell you. 695 00:34:27,239 --> 00:34:29,639 But if it is, then you're stopping there. 696 00:34:29,639 --> 00:34:32,745 And those are your unoccupied bands, and so forth. 697 00:34:32,745 --> 00:34:34,320 Does everybody see that? 698 00:34:34,320 --> 00:34:37,710 It's the same idea. 699 00:34:37,710 --> 00:34:42,830 But it's more complicated when you have these curvinesses. 700 00:34:42,830 --> 00:34:44,330 Yeah. 701 00:34:44,330 --> 00:34:46,045 Right? 702 00:34:46,045 --> 00:34:48,409 We good? 703 00:34:48,409 --> 00:34:49,310 Sort of? 704 00:34:49,310 --> 00:34:52,219 Any questions? 705 00:34:52,219 --> 00:34:54,420 Thoughts? 706 00:34:54,420 --> 00:34:54,920 Yeah. 707 00:34:54,920 --> 00:34:58,418 AUDIENCE: So when you're looking at allotropic systems-- 708 00:34:58,418 --> 00:34:59,780 so in this case, you have done-- 709 00:34:59,780 --> 00:35:01,380 I'm assuming that the graphite, this 710 00:35:01,380 --> 00:35:02,630 would be completely different. 711 00:35:02,630 --> 00:35:04,260 JEFFREY GROSSMAN: Completely different. 712 00:35:04,260 --> 00:35:06,411 AUDIENCE: And so how does that tie 713 00:35:06,411 --> 00:35:08,248 to solving the density of states? 714 00:35:08,248 --> 00:35:10,040 JEFFREY GROSSMAN: You can write the density 715 00:35:10,040 --> 00:35:12,110 of states for any material. 716 00:35:12,110 --> 00:35:16,130 And the same concept always holds, right? 717 00:35:16,130 --> 00:35:18,950 That you have these accessible places where 718 00:35:18,950 --> 00:35:24,050 the electron can be, with some probability 719 00:35:24,050 --> 00:35:26,960 of them being there, right? 720 00:35:26,960 --> 00:35:29,510 In other words, it's all about how many 721 00:35:29,510 --> 00:35:33,460 possible energy levels you have as a function of energy. 722 00:35:33,460 --> 00:35:34,550 That's really what it is. 723 00:35:34,550 --> 00:35:39,590 And so, whatever material you have, the density of states 724 00:35:39,590 --> 00:35:43,310 simply has the same meaning, but it will look very different. 725 00:35:43,310 --> 00:35:45,600 And in diamond, you're going to get a big bandgap, 726 00:35:45,600 --> 00:35:47,780 and in graphite, well, it's a metal. 727 00:35:51,290 --> 00:35:54,350 In graphene, you have a crossing at one 728 00:35:54,350 --> 00:35:57,560 part in the band structure, goes right through what 729 00:35:57,560 --> 00:35:59,510 would have been the gap. 730 00:35:59,510 --> 00:36:00,010 OK? 731 00:36:05,850 --> 00:36:11,865 And that's why when you do your homework, when you do PSET 5, 732 00:36:11,865 --> 00:36:15,030 there are two ways I want you to think about how 733 00:36:15,030 --> 00:36:16,920 these molecules absorb light. 734 00:36:16,920 --> 00:36:21,450 One way is that they just have some cutoff, 735 00:36:21,450 --> 00:36:25,090 below which in energy or above which in wavelength, 736 00:36:25,090 --> 00:36:26,740 they cannot absorb light. 737 00:36:26,740 --> 00:36:28,840 So that's just a single cutoff. 738 00:36:28,840 --> 00:36:31,020 But the other way is to actually use 739 00:36:31,020 --> 00:36:37,380 the features in this plot of levels, 740 00:36:37,380 --> 00:36:39,780 of accessible states for the electrons, 741 00:36:39,780 --> 00:36:44,460 to use this information and weight the absorption 742 00:36:44,460 --> 00:36:45,480 of the sun by this. 743 00:36:45,480 --> 00:36:50,220 Because that's actually what's really going on, is that 744 00:36:50,220 --> 00:36:54,720 the molecule cannot absorb the same at all the wavelengths 745 00:36:54,720 --> 00:36:55,680 of light. 746 00:36:55,680 --> 00:36:59,730 Because in some places it can't absorb any light. 747 00:36:59,730 --> 00:37:02,890 In some parts of the DOS, there's no probability. 748 00:37:02,890 --> 00:37:06,780 So that turns out to have a really big impact. 749 00:37:09,290 --> 00:37:11,000 OK. 750 00:37:11,000 --> 00:37:13,310 Questions, thoughts? 751 00:37:13,310 --> 00:37:17,765 Now, let me, before I go on, it's OK. 752 00:37:17,765 --> 00:37:22,970 I thought maybe we'd have time to look at Intro to Solar, 753 00:37:22,970 --> 00:37:25,340 but that would have been ahead anyway. 754 00:37:25,340 --> 00:37:27,770 Oh wait, no. 755 00:37:27,770 --> 00:37:32,695 Where is the view? 756 00:37:38,458 --> 00:37:40,600 Huh. 757 00:37:40,600 --> 00:37:41,950 That's not looking very good. 758 00:37:46,080 --> 00:37:54,585 I had what I thought was a nice example of a problem. 759 00:37:54,585 --> 00:37:55,085 No. 760 00:38:01,080 --> 00:38:02,240 Hang on one second. 761 00:38:11,080 --> 00:38:11,980 Oh boy. 762 00:38:11,980 --> 00:38:15,160 Never show anyone your desktop in a presentation. 763 00:38:15,160 --> 00:38:16,375 That's a bad idea. 764 00:38:19,310 --> 00:38:22,160 But anyway, oh, final quiz. 765 00:38:22,160 --> 00:38:23,483 AUDIENCE: That's last year. 766 00:38:23,483 --> 00:38:25,400 JEFFREY GROSSMAN: Yeah, that's from last year. 767 00:38:25,400 --> 00:38:26,735 Yeah, don't worry. 768 00:38:26,735 --> 00:38:29,360 It's in a very safe location for this year. 769 00:38:29,360 --> 00:38:30,712 AUDIENCE: [LAUGHS] 770 00:38:30,712 --> 00:38:33,170 JEFFREY GROSSMAN: Actually, it's very unsafe, it's up here. 771 00:38:33,170 --> 00:38:34,850 Anyway, wait. 772 00:38:34,850 --> 00:38:35,845 Where did it go? 773 00:38:35,845 --> 00:38:37,570 Ah. 774 00:38:37,570 --> 00:38:38,500 Why can't I see it? 775 00:38:38,500 --> 00:38:39,530 AUDIENCE: [LAUGHS] 776 00:38:39,530 --> 00:38:41,203 JEFFREY GROSSMAN: Here we go. 777 00:38:41,203 --> 00:38:43,120 OK, that's the problem I wanted to talk about. 778 00:38:43,120 --> 00:38:45,250 OK, so this was a problem from last year's quiz. 779 00:38:45,250 --> 00:38:50,261 AUDIENCE: [LAUGHS] 780 00:38:50,261 --> 00:38:54,810 JEFFREY GROSSMAN: OK, so your advisor has just spilled coffee 781 00:38:54,810 --> 00:38:57,210 on the culmination of your entire summer UROP work. 782 00:38:59,750 --> 00:39:02,090 This is not a true story. 783 00:39:02,090 --> 00:39:03,590 The prediction of the band structure 784 00:39:03,590 --> 00:39:05,710 and of an amazing new material for solar cells. 785 00:39:05,710 --> 00:39:08,290 Unfortunately, in your excitement to show your work, 786 00:39:08,290 --> 00:39:10,757 you forgot to back up your data. 787 00:39:10,757 --> 00:39:12,340 You dropped your computer in a hot tub 788 00:39:12,340 --> 00:39:14,050 and you forgot everything you did. 789 00:39:14,050 --> 00:39:15,003 AUDIENCE: [LAUGHS] 790 00:39:15,003 --> 00:39:17,170 JEFFREY GROSSMAN: So you only have that one printout 791 00:39:17,170 --> 00:39:20,290 now stained with coffee. 792 00:39:20,290 --> 00:39:21,880 Can you believe it? 793 00:39:21,880 --> 00:39:22,960 This is what you got. 794 00:39:22,960 --> 00:39:25,510 Nothing else. 795 00:39:25,510 --> 00:39:28,230 OK? 796 00:39:28,230 --> 00:39:30,750 Was this material-- you want to put it back together, 797 00:39:30,750 --> 00:39:33,330 because you know-- it was 4:00 in the morning, 798 00:39:33,330 --> 00:39:35,255 you had had a lot of NoDoz. 799 00:39:35,255 --> 00:39:36,630 But still, you're pretty sure you 800 00:39:36,630 --> 00:39:41,160 remember it was going to change the whole field of solar, OK? 801 00:39:41,160 --> 00:39:44,490 Now, from this, was this material metal, 802 00:39:44,490 --> 00:39:46,796 a semiconductor, or an insulator? 803 00:39:46,796 --> 00:39:50,246 AUDIENCE: Semiconductor. 804 00:39:50,246 --> 00:39:52,720 JEFFREY GROSSMAN: Tell me why. 805 00:39:52,720 --> 00:39:54,130 Really? 806 00:39:54,130 --> 00:39:56,180 Seriously? 807 00:39:56,180 --> 00:39:56,680 Yeah. 808 00:39:56,680 --> 00:39:59,650 AUDIENCE: Doesn't look like The bands are crossing. 809 00:39:59,650 --> 00:40:02,600 Looks like the tallest bandgap is just under the 0. 810 00:40:02,600 --> 00:40:03,670 I can't really tell. 811 00:40:03,670 --> 00:40:06,412 And then it doesn't seem like the bandgap 812 00:40:06,412 --> 00:40:08,710 for the bands is crossing, either. 813 00:40:08,710 --> 00:40:11,215 JEFFREY GROSSMAN: It doesn't look like it? 814 00:40:11,215 --> 00:40:12,415 Can you be certain? 815 00:40:12,415 --> 00:40:12,970 AUDIENCE: No. 816 00:40:15,850 --> 00:40:20,290 It's also a fuller cell. 817 00:40:20,290 --> 00:40:22,040 JEFFREY GROSSMAN: Well, you do need metal, 818 00:40:22,040 --> 00:40:25,700 and maybe it was a revolution in the metal contact. 819 00:40:28,400 --> 00:40:32,600 That wouldn't revolutionize solar PV, but still. 820 00:40:32,600 --> 00:40:34,850 Again, see, we're inclined to want 821 00:40:34,850 --> 00:40:37,630 to connect these things up. 822 00:40:37,630 --> 00:40:39,670 We're sort of used to staring at silicon, 823 00:40:39,670 --> 00:40:41,570 and we think that's going to do that. 824 00:40:41,570 --> 00:40:47,110 But I don't know that you have enough information here. 825 00:40:50,250 --> 00:40:53,070 I'm not sure you know enough about this 826 00:40:53,070 --> 00:40:56,040 to know that this band didn't actually split, and one of them 827 00:40:56,040 --> 00:40:59,190 crossed up and went this way. 828 00:40:59,190 --> 00:41:00,830 How do you know? 829 00:41:00,830 --> 00:41:03,200 There is one thing you know it's not. 830 00:41:03,200 --> 00:41:05,407 There's one of those three you know it's not. 831 00:41:05,407 --> 00:41:06,740 AUDIENCE: It's not an insulator. 832 00:41:06,740 --> 00:41:08,198 JEFFREY GROSSMAN: Not an insulator. 833 00:41:08,198 --> 00:41:09,038 Why not? 834 00:41:09,038 --> 00:41:10,160 AUDIENCE: [INAUDIBLE] 835 00:41:10,160 --> 00:41:11,910 JEFFREY GROSSMAN: Because you can make out 836 00:41:11,910 --> 00:41:14,730 at least that much, that these bands, 837 00:41:14,730 --> 00:41:16,800 they're about an EV or two apart. 838 00:41:16,800 --> 00:41:18,990 They may be closer, they may overlap. 839 00:41:18,990 --> 00:41:21,970 But it is not going to be an insulator, right? 840 00:41:21,970 --> 00:41:23,275 But you can't tell. 841 00:41:23,275 --> 00:41:27,550 So this a little bit of a tricky question. 842 00:41:27,550 --> 00:41:30,030 You seem to remember that the material would absorb light 843 00:41:30,030 --> 00:41:30,780 very efficiently. 844 00:41:30,780 --> 00:41:32,340 Reconstruct the band structure plot 845 00:41:32,340 --> 00:41:34,575 that would correspond to such a material. 846 00:41:37,435 --> 00:41:38,810 What are you supposed to do here? 847 00:41:43,880 --> 00:41:44,705 Yeah. 848 00:41:44,705 --> 00:41:47,690 AUDIENCE: [INAUDIBLE] associated with some wavelength 849 00:41:47,690 --> 00:41:49,552 or a region of wavelengths alike. 850 00:41:49,552 --> 00:41:51,635 JEFFREY GROSSMAN: OK, you could certainly do that. 851 00:41:54,770 --> 00:42:02,170 And in a way, any band structure above the bandgap 852 00:42:02,170 --> 00:42:05,050 is associated with the region of light. 853 00:42:05,050 --> 00:42:09,400 Because except for the bandgap, below which you cannot absorb 854 00:42:09,400 --> 00:42:13,090 light, as soon as you start having states, 855 00:42:13,090 --> 00:42:14,740 you can absorb light. 856 00:42:14,740 --> 00:42:16,000 But the key is here. 857 00:42:16,000 --> 00:42:18,050 It absorbs light very efficiently. 858 00:42:18,050 --> 00:42:19,980 So what do I need to do in here? 859 00:42:19,980 --> 00:42:21,300 AUDIENCE: Put a direct bandgap. 860 00:42:21,300 --> 00:42:22,258 JEFFREY GROSSMAN: Yeah. 861 00:42:22,258 --> 00:42:25,860 You got to put a direct bandgap in there somehow. 862 00:42:25,860 --> 00:42:27,610 That's the key. 863 00:42:27,610 --> 00:42:29,350 So you got to reconstruct this. 864 00:42:29,350 --> 00:42:31,540 And it didn't matter to me where. 865 00:42:31,540 --> 00:42:33,880 And there were all kinds of really creative curves 866 00:42:33,880 --> 00:42:36,100 people drew. 867 00:42:36,100 --> 00:42:37,630 But somewhere in here, maybe this 868 00:42:37,630 --> 00:42:41,440 is going to come down a little and then go down here. 869 00:42:41,440 --> 00:42:46,810 And then maybe come in like here, where you can't see. 870 00:42:46,810 --> 00:42:50,500 But somehow, you got to try to make this a direct bandgap 871 00:42:50,500 --> 00:42:52,210 material. 872 00:42:52,210 --> 00:42:54,100 It's actually pretty hard, because the stain 873 00:42:54,100 --> 00:42:56,670 doesn't give you a lot of room. 874 00:42:56,670 --> 00:42:58,910 But it's actually, that mirrors how hard it 875 00:42:58,910 --> 00:43:01,490 is to do that for silicon in real life. 876 00:43:01,490 --> 00:43:07,362 The stain and silicon are deeply connected. 877 00:43:07,362 --> 00:43:09,320 One minute later, you realize that actually, it 878 00:43:09,320 --> 00:43:11,780 didn't absorb light efficiently at all, 879 00:43:11,780 --> 00:43:14,210 but instead it was a revolutionary metal electrode. 880 00:43:14,210 --> 00:43:15,250 Now what would you draw? 881 00:43:15,250 --> 00:43:16,180 AUDIENCE: Overlapping. 882 00:43:16,180 --> 00:43:17,180 JEFFREY GROSSMAN: Right. 883 00:43:17,180 --> 00:43:19,340 You'd cross it up. 884 00:43:19,340 --> 00:43:21,620 Maybe you'd try to get a lot of wiggles in here 885 00:43:21,620 --> 00:43:24,770 if you could, or just certainly several crossings, 886 00:43:24,770 --> 00:43:30,140 so that the electrons don't really see any gap. 887 00:43:30,140 --> 00:43:31,970 You can't see a gap. 888 00:43:31,970 --> 00:43:37,490 So that's kind of a nice problem. 889 00:43:37,490 --> 00:43:40,670 And you can see, there's not a very long problem. 890 00:43:40,670 --> 00:43:43,520 But you've got to know a few things to answer the question 891 00:43:43,520 --> 00:43:46,310 about these sorts of topics. 892 00:43:46,310 --> 00:43:49,400 OK, so we'll do another one of those next Tuesday, or two. 893 00:43:53,770 --> 00:43:55,040 Wait, where am I here? 894 00:43:59,640 --> 00:44:01,860 Let's talk about a few other properties 895 00:44:01,860 --> 00:44:04,780 that you can calculate. 896 00:44:04,780 --> 00:44:07,480 Again, I'm not going to go into the details of transport. 897 00:44:07,480 --> 00:44:14,340 But this is sort of basic double E or basic physics stuff. 898 00:44:14,340 --> 00:44:16,560 If you put an electron in a field, 899 00:44:16,560 --> 00:44:18,370 it's going to push it with a force. 900 00:44:18,370 --> 00:44:19,960 And so the F equals ma of that. 901 00:44:19,960 --> 00:44:20,460 Yeah? 902 00:44:20,460 --> 00:44:21,790 AUDIENCE: I have a question. 903 00:44:21,790 --> 00:44:22,290 [INAUDIBLE] 904 00:44:22,290 --> 00:44:23,332 JEFFREY GROSSMAN: Please. 905 00:44:23,332 --> 00:44:26,180 AUDIENCE: [INAUDIBLE] absorb things that are greater here. 906 00:44:26,180 --> 00:44:28,550 It absorbs light that has energy greater than the gap. 907 00:44:28,550 --> 00:44:28,710 JEFFREY GROSSMAN: Greater than the gap. 908 00:44:28,710 --> 00:44:31,500 AUDIENCE: But don't you want a very small gap 909 00:44:31,500 --> 00:44:33,000 for a semiconductor? 910 00:44:33,000 --> 00:44:35,160 JEFFREY GROSSMAN: For what application? 911 00:44:35,160 --> 00:44:36,000 For solar cells? 912 00:44:36,000 --> 00:44:37,550 AUDIENCE: Oh yeah, sure. 913 00:44:37,550 --> 00:44:40,230 JEFFREY GROSSMAN: So that's a very good point. 914 00:44:40,230 --> 00:44:46,600 And who can tell me why you can't have too low of a gap? 915 00:44:46,600 --> 00:44:50,477 And I'll talk about this as we move into solar PV. 916 00:44:50,477 --> 00:44:52,810 AUDIENCE: Because I think the barrier's not high enough, 917 00:44:52,810 --> 00:44:54,820 and they keep hopping over. 918 00:44:54,820 --> 00:44:55,990 [INAUDIBLE] 919 00:44:55,990 --> 00:44:58,460 JEFFREY GROSSMAN: That wouldn't necessarily be a problem. 920 00:44:58,460 --> 00:44:58,960 Yeah. 921 00:44:58,960 --> 00:45:02,942 AUDIENCE: It would probably get saturated with lower energy. 922 00:45:02,942 --> 00:45:06,336 JEFFREY GROSSMAN: And what's wrong with that, though? 923 00:45:06,336 --> 00:45:08,232 AUDIENCE: You're not going to be able to use 924 00:45:08,232 --> 00:45:09,760 the full potential of the cell. 925 00:45:09,760 --> 00:45:12,150 JEFFREY GROSSMAN: Well, that's what it is, is it 926 00:45:12,150 --> 00:45:14,320 actually is what sets the potential of the cell. 927 00:45:14,320 --> 00:45:17,940 So this goes back to, and it's a very good question, right? 928 00:45:17,940 --> 00:45:20,700 Why not just use the lowest gap possible? 929 00:45:20,700 --> 00:45:23,280 And then you'll absorb all the light, or most of the light. 930 00:45:23,280 --> 00:45:26,130 Well, the reason is that it goes back to this picture 931 00:45:26,130 --> 00:45:30,420 that when you're taking photons and making them into electrons, 932 00:45:30,420 --> 00:45:33,630 what you're really doing is it's again, 933 00:45:33,630 --> 00:45:35,190 it's that pumping water analogy. 934 00:45:35,190 --> 00:45:36,840 You're giving the electrons energy 935 00:45:36,840 --> 00:45:38,830 and pushing them up a hill. 936 00:45:38,830 --> 00:45:42,120 Now, in the solar fuel's case, we're 937 00:45:42,120 --> 00:45:45,420 using that push to change the molecule's configuration 938 00:45:45,420 --> 00:45:46,830 to store energy. 939 00:45:46,830 --> 00:45:50,340 In the solar cell case, we're using that push 940 00:45:50,340 --> 00:45:54,830 to get just energy, voltage out, right? 941 00:45:58,100 --> 00:46:02,010 So I'd like to not get just a teeny, teeny bit of voltage. 942 00:46:02,010 --> 00:46:04,930 I'd like to get it up a hill. 943 00:46:04,930 --> 00:46:12,800 Now, if I make my band gap such that I can only push it really, 944 00:46:12,800 --> 00:46:14,980 really high, you might think you want that. 945 00:46:14,980 --> 00:46:15,897 But then what happens? 946 00:46:15,897 --> 00:46:17,355 AUDIENCE: Doesn't absorb any light. 947 00:46:17,355 --> 00:46:19,310 JEFFREY GROSSMAN: Doesn't absorb any light. 948 00:46:19,310 --> 00:46:21,320 So now you have these two competing effects. 949 00:46:21,320 --> 00:46:22,840 AUDIENCE: Small bandgap. 950 00:46:22,840 --> 00:46:24,830 JEFFREY GROSSMAN: To absorb the light. 951 00:46:24,830 --> 00:46:28,610 Larger bandgap to give you more potential for yourself. 952 00:46:28,610 --> 00:46:32,925 AUDIENCE: And large bandgap, more potential, small bandgap, 953 00:46:32,925 --> 00:46:34,140 small potential. 954 00:46:34,140 --> 00:46:35,520 JEFFREY GROSSMAN: But more light. 955 00:46:35,520 --> 00:46:38,700 And there's a sweet spot. 956 00:46:38,700 --> 00:46:40,500 And I'll talk about that. 957 00:46:40,500 --> 00:46:43,440 I'll talk about that a little bit on Tuesday. 958 00:46:43,440 --> 00:46:49,020 That that is two competing attributes of solar PV 959 00:46:49,020 --> 00:46:54,060 that limit how much energy you can get from the sun. 960 00:46:58,470 --> 00:47:00,390 So if you put an electron in a field, 961 00:47:00,390 --> 00:47:05,430 it's going to move until you hit this equilibrium point where 962 00:47:05,430 --> 00:47:11,880 the acceleration is 0, because the force it feels 963 00:47:11,880 --> 00:47:13,710 is balanced by the scattering. 964 00:47:13,710 --> 00:47:18,870 That's sort of basic transport, electron transport equations. 965 00:47:18,870 --> 00:47:22,410 And so you can get an equation for the drift velocity 966 00:47:22,410 --> 00:47:23,190 of the electron. 967 00:47:23,190 --> 00:47:25,050 Or the thing that we care about a lot 968 00:47:25,050 --> 00:47:28,740 is this electrical conductivity, something that you can measure, 969 00:47:28,740 --> 00:47:31,430 which is nothing more than the drift velocity times the number 970 00:47:31,430 --> 00:47:32,565 density. 971 00:47:32,565 --> 00:47:34,440 That's actually a hard thing to separate out, 972 00:47:34,440 --> 00:47:37,680 but it can be done in experiment. 973 00:47:37,680 --> 00:47:40,530 And so the current is that, and it's 974 00:47:40,530 --> 00:47:43,720 equal to the electrical connectivity times the field. 975 00:47:43,720 --> 00:47:45,540 Now, here's the thing. 976 00:47:49,590 --> 00:47:53,040 That equation, then, if you just look at these very simple F 977 00:47:53,040 --> 00:47:55,020 equals ma for an electron in a field, 978 00:47:55,020 --> 00:47:58,060 that very simply comes down to this equation. 979 00:47:58,060 --> 00:48:00,450 And here's the thing. 980 00:48:00,450 --> 00:48:05,290 That mass, what we want now is, well, 981 00:48:05,290 --> 00:48:10,080 what's the electron conductivity in the solid? 982 00:48:10,080 --> 00:48:12,180 What's the electron conductivity in the solid? 983 00:48:12,180 --> 00:48:14,790 And it turns out that that mass that's 984 00:48:14,790 --> 00:48:19,530 on the previous page, that m, it becomes 985 00:48:19,530 --> 00:48:25,890 a kind of effective mass of the electron that 986 00:48:25,890 --> 00:48:28,955 can be different than just the mass 987 00:48:28,955 --> 00:48:31,080 of the electron, the standard mass of the electron. 988 00:48:31,080 --> 00:48:33,550 The reason is that it's in this periodic crystal. 989 00:48:33,550 --> 00:48:37,920 Remember, the electron is going to feel 990 00:48:37,920 --> 00:48:41,595 this very strange environment when 991 00:48:41,595 --> 00:48:42,720 it's in a periodic crystal. 992 00:48:42,720 --> 00:48:45,180 And it's going to feel differences in its energy band. 993 00:48:45,180 --> 00:48:47,820 And what that does is it changes, 994 00:48:47,820 --> 00:48:49,800 actually the effective mass of the electron, 995 00:48:49,800 --> 00:48:52,180 as it's in different parts of k-space. 996 00:48:54,880 --> 00:48:58,090 Well, it turns out that that effective mass is directly 997 00:48:58,090 --> 00:49:01,960 related to this curvature. 998 00:49:04,570 --> 00:49:06,400 I'm just rewriting the previous equation, 999 00:49:06,400 --> 00:49:08,770 but now I'm putting in the band structure here. 1000 00:49:08,770 --> 00:49:13,690 These little v's are nothing more than the slope of E 1001 00:49:13,690 --> 00:49:16,180 in the band structure, the slope of the energy 1002 00:49:16,180 --> 00:49:17,730 in the band structure versus k. 1003 00:49:23,252 --> 00:49:25,210 So this is called effective mass approximation. 1004 00:49:25,210 --> 00:49:28,540 It's an approximation, but it's a pretty good one. 1005 00:49:28,540 --> 00:49:33,197 The electron feels a different weight to it. 1006 00:49:33,197 --> 00:49:34,030 It's heavy or light. 1007 00:49:34,030 --> 00:49:35,410 Think about it that way. 1008 00:49:35,410 --> 00:49:39,130 Depending on whether it's in a flat part of the band, where 1009 00:49:39,130 --> 00:49:41,720 it's really heavy, or whether it's 1010 00:49:41,720 --> 00:49:44,420 in a curvy part of the band, where it's 1011 00:49:44,420 --> 00:49:47,060 feeling more light and bouncy. 1012 00:49:47,060 --> 00:49:51,830 And that changes the connectivity 1013 00:49:51,830 --> 00:49:53,220 that can be measured. 1014 00:49:53,220 --> 00:49:56,755 So just from the curvature of these bands, 1015 00:49:56,755 --> 00:49:58,130 it's a really nice way of getting 1016 00:49:58,130 --> 00:50:01,910 a sense of how mobile an electron or hole is 1017 00:50:01,910 --> 00:50:03,340 in a material. 1018 00:50:03,340 --> 00:50:06,380 All you have to do is you just take 1019 00:50:06,380 --> 00:50:15,230 an integral over the derivative of the slope, 1020 00:50:15,230 --> 00:50:19,850 basically, as you go along k-space there. 1021 00:50:19,850 --> 00:50:22,040 So does anybody know what a Fermi function is? 1022 00:50:22,040 --> 00:50:23,420 It's just a wait. 1023 00:50:23,420 --> 00:50:26,060 But-- yeah? 1024 00:50:26,060 --> 00:50:29,070 How many of you don't know what a Fermi function is? 1025 00:50:29,070 --> 00:50:29,570 OK. 1026 00:50:29,570 --> 00:50:33,350 So this comes from Fermi-Dirac statistics. 1027 00:50:33,350 --> 00:50:36,770 It's not in any way critical here. 1028 00:50:36,770 --> 00:50:39,470 But it's an occupation function. 1029 00:50:39,470 --> 00:50:42,110 It's saying that at a given temperature, 1030 00:50:42,110 --> 00:50:50,080 you're likely to have electrons or not at a certain energy. 1031 00:50:50,080 --> 00:50:54,540 And so the Fermi function at 0 temperature, 1032 00:50:54,540 --> 00:50:56,040 F is going to look like this. 1033 00:50:59,300 --> 00:51:03,620 Where this would be 1, and this would be 0. 1034 00:51:09,050 --> 00:51:14,560 And the Fermi function is 1 over e 1035 00:51:14,560 --> 00:51:26,030 to the E minus EF divided by KT, plus 1. 1036 00:51:26,030 --> 00:51:27,770 Where e is where you are in the band 1037 00:51:27,770 --> 00:51:30,110 structure. e is where you are in the band structure, 1038 00:51:30,110 --> 00:51:34,310 and you're looking at this as you go above the Fermi energy. 1039 00:51:34,310 --> 00:51:35,570 Can electrons be there? 1040 00:51:35,570 --> 00:51:38,900 Well, at 0 temperature, no. 1041 00:51:38,900 --> 00:51:42,440 But actually, when you go to some finite temperature, 1042 00:51:42,440 --> 00:51:45,980 just temperature alone can kick electrons up. 1043 00:51:48,630 --> 00:51:51,620 OK, that's what the Fermi function is in that equation 1044 00:51:51,620 --> 00:51:52,790 for. 1045 00:51:52,790 --> 00:51:55,580 Because at some temperature, you can actually 1046 00:51:55,580 --> 00:51:58,250 have a distribution that looks more like this. 1047 00:52:00,930 --> 00:52:05,790 T equals 0, T equals something. 1048 00:52:10,320 --> 00:52:12,120 Now, what's important here? 1049 00:52:12,120 --> 00:52:15,360 Well, what's important is that when 1050 00:52:15,360 --> 00:52:22,410 you think about conductivity, you want to know, you see. 1051 00:52:22,410 --> 00:52:28,320 The difference here between drift velocity and current 1052 00:52:28,320 --> 00:52:31,290 or connectivity is that you got to know something about how 1053 00:52:31,290 --> 00:52:32,640 many carriers you have. 1054 00:52:36,630 --> 00:52:40,680 And that's going to be temperature-dependent. 1055 00:52:40,680 --> 00:52:41,872 Does everybody see that? 1056 00:52:41,872 --> 00:52:43,080 That's why this is important. 1057 00:52:43,080 --> 00:52:45,480 Now, but what if my Fermi function is like this? 1058 00:52:45,480 --> 00:52:47,550 And OK, let's say I have silicon, 1059 00:52:47,550 --> 00:52:50,390 and I'm going to now plot on top of this, 1060 00:52:50,390 --> 00:52:55,170 I'm going to use a different color, and plot on top of this 1061 00:52:55,170 --> 00:52:56,640 the density of states. 1062 00:52:56,640 --> 00:52:59,610 And let's say my density of states was like this. 1063 00:52:59,610 --> 00:53:02,130 OK, I don't know if it is or not. 1064 00:53:02,130 --> 00:53:07,220 And then here's the bandgap. 1065 00:53:07,220 --> 00:53:09,630 This is my DOS. 1066 00:53:09,630 --> 00:53:11,630 That doesn't look like a different color at all. 1067 00:53:11,630 --> 00:53:12,380 AUDIENCE: [LAUGHS] 1068 00:53:12,380 --> 00:53:13,970 JEFFREY GROSSMAN: It's not even close. 1069 00:53:13,970 --> 00:53:17,180 It's like they painted it yellow on the outside. 1070 00:53:17,180 --> 00:53:19,130 And seriously, it's totally white. 1071 00:53:19,130 --> 00:53:20,000 All right, anyway. 1072 00:53:23,870 --> 00:53:26,120 So here's my Fermi function. 1073 00:53:26,120 --> 00:53:28,460 I'm at room temperature, and this is 1074 00:53:28,460 --> 00:53:30,120 my distribution of electrons. 1075 00:53:34,640 --> 00:53:35,690 This is the gap. 1076 00:53:39,682 --> 00:53:40,890 Tell me what's going on here. 1077 00:53:43,570 --> 00:53:45,150 Somebody explain this to me. 1078 00:53:48,780 --> 00:53:49,900 Yeah, Sam? 1079 00:53:49,900 --> 00:53:52,505 AUDIENCE: Even though there's probability 1080 00:53:52,505 --> 00:53:55,060 that you can have electrons in that space, because there's 1081 00:53:55,060 --> 00:53:56,646 no space for them to go, you still [INAUDIBLE].. 1082 00:53:56,646 --> 00:53:57,930 JEFFREY GROSSMAN: I love it. 1083 00:53:57,930 --> 00:53:59,700 That's absolutely right. 1084 00:53:59,700 --> 00:54:02,010 You cannot have electrons there. 1085 00:54:02,010 --> 00:54:04,770 The Fermi function says I can. 1086 00:54:04,770 --> 00:54:07,980 In your face, Fermi function, says the DOS. 1087 00:54:07,980 --> 00:54:09,240 I didn't say that. 1088 00:54:09,240 --> 00:54:11,160 I would never say that to a Fermi function. 1089 00:54:11,160 --> 00:54:13,020 But the DOS said it. 1090 00:54:13,020 --> 00:54:14,902 The DOS said, uh-uh! 1091 00:54:14,902 --> 00:54:17,741 AUDIENCE: [LAUGHS] 1092 00:54:17,741 --> 00:54:20,070 JEFFREY GROSSMAN: We got zero probability here, man! 1093 00:54:20,070 --> 00:54:22,150 You can't be here. 1094 00:54:22,150 --> 00:54:26,550 So the DOS said, no states, no deal. 1095 00:54:26,550 --> 00:54:27,570 Go away. 1096 00:54:27,570 --> 00:54:28,860 So the Fermi function tried. 1097 00:54:28,860 --> 00:54:31,053 It said, hey, I got enough energy from temperature 1098 00:54:31,053 --> 00:54:32,220 to give you a few electrons. 1099 00:54:32,220 --> 00:54:33,720 And the DOS says, I don't want them. 1100 00:54:33,720 --> 00:54:36,510 Because they're not at the energy where I've got 1101 00:54:36,510 --> 00:54:39,120 any states. 1102 00:54:39,120 --> 00:54:42,390 Now, what can I do, then, to populate 1103 00:54:42,390 --> 00:54:45,010 the conduction band of silicon? 1104 00:54:45,010 --> 00:54:47,975 What can I do? 1105 00:54:47,975 --> 00:54:49,780 AUDIENCE: More thermal energy. 1106 00:54:49,780 --> 00:54:51,450 JEFFREY GROSSMAN: More thermal energy. 1107 00:54:51,450 --> 00:54:55,810 I'm going to run my solar cell at 10,000 degrees. 1108 00:54:55,810 --> 00:54:59,740 Well, you'll get more and more. 1109 00:54:59,740 --> 00:55:02,320 And you will start to get electrons in there. 1110 00:55:02,320 --> 00:55:04,570 But actually, that's not always practical. 1111 00:55:04,570 --> 00:55:05,836 AUDIENCE: [LAUGHS] 1112 00:55:05,836 --> 00:55:10,617 JEFFREY GROSSMAN: So how do we get electrons into there? 1113 00:55:10,617 --> 00:55:12,200 Then there are other reasons electrons 1114 00:55:12,200 --> 00:55:18,830 come into those states, not just from thermal. 1115 00:55:18,830 --> 00:55:23,460 But what's one way that we can get electrons in there? 1116 00:55:23,460 --> 00:55:24,390 AUDIENCE: Doping. 1117 00:55:24,390 --> 00:55:25,348 JEFFREY GROSSMAN: Yeah. 1118 00:55:25,348 --> 00:55:26,476 What does doping do? 1119 00:55:26,476 --> 00:55:29,385 AUDIENCE: [INAUDIBLE] changes it from energy. 1120 00:55:29,385 --> 00:55:31,760 JEFFREY GROSSMAN: Well, it can change it from the energy. 1121 00:55:31,760 --> 00:55:35,571 But what might I dope silicon with? 1122 00:55:35,571 --> 00:55:37,375 AUDIENCE: Phosphorus. 1123 00:55:37,375 --> 00:55:38,610 JEFFREY GROSSMAN: Phosphorus. 1124 00:55:38,610 --> 00:55:39,547 Why? 1125 00:55:39,547 --> 00:55:40,673 AUDIENCE: Excess electron. 1126 00:55:40,673 --> 00:55:42,090 JEFFREY GROSSMAN: Excess electron, 1127 00:55:42,090 --> 00:55:44,400 but compared to silicon. 1128 00:55:44,400 --> 00:55:51,360 Now, if I have an excess electron sitting in my crystal, 1129 00:55:51,360 --> 00:55:55,093 but really the crystal is still kind of silicony, 1130 00:55:55,093 --> 00:55:57,260 because I don't have many of these excess electrons. 1131 00:55:57,260 --> 00:55:59,720 I just have one phosphorus atom every 1000. 1132 00:55:59,720 --> 00:56:01,880 So it's really still silicon. 1133 00:56:01,880 --> 00:56:04,850 But I filled all my electrons up to here. 1134 00:56:04,850 --> 00:56:06,200 And now I've got one extra. 1135 00:56:06,200 --> 00:56:07,204 Where does it go? 1136 00:56:07,204 --> 00:56:08,820 AUDIENCE: It goes in there. 1137 00:56:08,820 --> 00:56:10,580 JEFFREY GROSSMAN: Goes in there. 1138 00:56:10,580 --> 00:56:15,200 And now I'm populating it without thermal energy, right? 1139 00:56:15,200 --> 00:56:17,990 I don't need to work with my Fermi function, 1140 00:56:17,990 --> 00:56:20,180 I did it chemically. 1141 00:56:20,180 --> 00:56:23,750 I put an atom in there that had a spare electron, right? 1142 00:56:23,750 --> 00:56:26,030 And that is actually, in fact, exactly what 1143 00:56:26,030 --> 00:56:29,630 happens in real materials. 1144 00:56:29,630 --> 00:56:32,670 Who knows what it's called when you have an extra electron, 1145 00:56:32,670 --> 00:56:34,410 or an excess of electrons? 1146 00:56:34,410 --> 00:56:35,490 AUDIENCE: N-type? 1147 00:56:35,490 --> 00:56:37,690 JEFFREY GROSSMAN: N-type. 1148 00:56:37,690 --> 00:56:39,272 Now, what if I wanted holes? 1149 00:56:39,272 --> 00:56:40,480 What if I wanted extra holes? 1150 00:56:40,480 --> 00:56:41,466 AUDIENCE: B-type. 1151 00:56:41,466 --> 00:56:43,880 JEFFREY GROSSMAN: B-type. 1152 00:56:43,880 --> 00:56:45,436 And what would I put in there? 1153 00:56:45,436 --> 00:56:46,744 AUDIENCE: Boron. 1154 00:56:46,744 --> 00:56:47,810 JEFFREY GROSSMAN: Boron. 1155 00:56:47,810 --> 00:56:49,364 Because? 1156 00:56:49,364 --> 00:56:50,915 AUDIENCE: It's an electron. 1157 00:56:50,915 --> 00:56:52,373 JEFFREY GROSSMAN: It's an electron. 1158 00:56:52,373 --> 00:56:54,320 Now, you could dope it with many other things. 1159 00:56:54,320 --> 00:56:59,540 You don't have to dope silicon with just phosphorus or boron. 1160 00:56:59,540 --> 00:57:02,210 It turns out that those are easy and there 1161 00:57:02,210 --> 00:57:04,490 are some advantages in terms of their processing, 1162 00:57:04,490 --> 00:57:06,890 and chemically, they can be happy. 1163 00:57:06,890 --> 00:57:09,530 But there are many ways of doping materials. 1164 00:57:09,530 --> 00:57:12,150 Many, many ways of doping materials. 1165 00:57:12,150 --> 00:57:14,272 And one of the things you do with doping, 1166 00:57:14,272 --> 00:57:15,980 there are many things you do with doping. 1167 00:57:15,980 --> 00:57:19,940 But one of the things you do is you get electrons or holes 1168 00:57:19,940 --> 00:57:21,530 into these bands. 1169 00:57:21,530 --> 00:57:23,750 You populate them, right? 1170 00:57:23,750 --> 00:57:25,460 But this expression is ignoring that. 1171 00:57:25,460 --> 00:57:28,010 This is just saying, well, we're going 1172 00:57:28,010 --> 00:57:31,460 to see if there's some thermal energy there, 1173 00:57:31,460 --> 00:57:33,100 and we'll use that. 1174 00:57:33,100 --> 00:57:34,620 OK? 1175 00:57:34,620 --> 00:57:35,120 Yeah. 1176 00:57:37,930 --> 00:57:40,000 Any questions about that? 1177 00:57:40,000 --> 00:57:42,170 Now, we talked about this a lot. 1178 00:57:42,170 --> 00:57:44,790 So I'm not going to go over this again, 1179 00:57:44,790 --> 00:57:49,570 that the direct transitions are easy for photons. 1180 00:57:49,570 --> 00:57:53,710 The indirect ones where you shift where you are in k-space 1181 00:57:53,710 --> 00:57:56,860 are hard, OK? 1182 00:57:56,860 --> 00:58:01,870 And that is why silicon is an expensive solar cell. 1183 00:58:01,870 --> 00:58:03,370 So we talked about this, and I don't 1184 00:58:03,370 --> 00:58:05,530 want to belabor the point, because I 1185 00:58:05,530 --> 00:58:09,010 want to try to get to some simulations, if possible. 1186 00:58:09,010 --> 00:58:11,500 Any questions about that point, though, the optical 1187 00:58:11,500 --> 00:58:12,250 transitions? 1188 00:58:15,575 --> 00:58:16,302 All right. 1189 00:58:16,302 --> 00:58:17,760 So the kinds of things you can get. 1190 00:58:17,760 --> 00:58:20,310 Well, you can get optical properties, right? 1191 00:58:23,370 --> 00:58:25,950 And you can also get magnetic properties 1192 00:58:25,950 --> 00:58:27,440 from these densities of states. 1193 00:58:27,440 --> 00:58:29,190 So we talked about the optical properties. 1194 00:58:29,190 --> 00:58:35,340 And the optical properties is really the context of your PSET 1195 00:58:35,340 --> 00:58:36,720 5, but it's for a molecule. 1196 00:58:39,800 --> 00:58:41,250 We haven't talked about megnetism, 1197 00:58:41,250 --> 00:58:43,430 and I want to come back to it, because I did mention 1198 00:58:43,430 --> 00:58:45,140 that electrons have spin. 1199 00:58:45,140 --> 00:58:49,220 And therefore, they also have a magnetic moment. 1200 00:58:49,220 --> 00:58:52,790 And we call this, just think of it as some constant 1201 00:58:52,790 --> 00:58:54,530 that we call the Bohr magneton. 1202 00:58:54,530 --> 00:58:55,350 It has a value. 1203 00:58:58,970 --> 00:59:02,900 Well, and if you think about an atom or a molecule, 1204 00:59:02,900 --> 00:59:06,710 you could just get that by knowing how many electrons 1205 00:59:06,710 --> 00:59:12,740 are spinning in one direction more than the other, right? 1206 00:59:12,740 --> 00:59:15,170 So you can just count how many spin up 1207 00:59:15,170 --> 00:59:18,945 electrons minus how many spin down electrons you have, 1208 00:59:18,945 --> 00:59:20,570 and multiply that by the Bohr magneton. 1209 00:59:20,570 --> 00:59:22,320 And that will give you the magnetic moment 1210 00:59:22,320 --> 00:59:24,530 of the atom or molecule. 1211 00:59:24,530 --> 00:59:25,910 But what would I do for a solid? 1212 00:59:30,000 --> 00:59:31,770 Do I still have it? 1213 00:59:31,770 --> 00:59:32,910 Yeah. 1214 00:59:32,910 --> 00:59:35,820 See, there's the atomic picture where I could just count. 1215 00:59:35,820 --> 00:59:38,700 And in this case, the difference is 3. 1216 00:59:38,700 --> 00:59:39,330 Right? 1217 00:59:39,330 --> 00:59:41,050 But now, I've got a solid. 1218 00:59:41,050 --> 00:59:42,050 So what do I want to do? 1219 00:59:47,545 --> 00:59:49,455 What do I want to look at for a solid? 1220 00:59:49,455 --> 00:59:50,640 AUDIENCE: Density of states. 1221 00:59:50,640 --> 00:59:52,200 JEFFREY GROSSMAN: Density of states. 1222 00:59:52,200 --> 00:59:54,390 And it turns out that you can plot 1223 00:59:54,390 --> 00:59:58,380 the density of states for spin up and spin down electrons 1224 00:59:58,380 --> 00:59:59,930 separately. 1225 00:59:59,930 --> 01:00:01,840 And they can be different. 1226 01:00:01,840 --> 01:00:05,660 They can occupy the two spins differently 1227 01:00:05,660 --> 01:00:09,920 in a more complicated way, but very much same 1228 01:00:09,920 --> 01:00:13,010 idea as what can happen in and out of them. 1229 01:00:13,010 --> 01:00:14,630 But it's more complicated as you get 1230 01:00:14,630 --> 01:00:17,930 these curvy bands and all kinds of other things 1231 01:00:17,930 --> 01:00:19,470 that can happen in a solid. 1232 01:00:19,470 --> 01:00:24,230 So now, you look at the separate density of states for electrons 1233 01:00:24,230 --> 01:00:26,210 with spin up versus spin down. 1234 01:00:26,210 --> 01:00:28,400 And that is something that I didn't talk about yet. 1235 01:00:28,400 --> 01:00:32,300 It's an input parameter in SIESTA 1236 01:00:32,300 --> 01:00:35,660 that is checked yes by default. And it's 1237 01:00:35,660 --> 01:00:38,910 called spin polarized, question mark, 1238 01:00:38,910 --> 01:00:40,170 I think is what it's called. 1239 01:00:40,170 --> 01:00:41,400 Spin polarized? 1240 01:00:41,400 --> 01:00:44,340 I should give the right intonation. 1241 01:00:44,340 --> 01:00:45,840 And you say yes or no. 1242 01:00:45,840 --> 01:00:47,683 And you say yes, because that's the default. 1243 01:00:47,683 --> 01:00:49,350 But what are you doing when you say yes? 1244 01:00:49,350 --> 01:00:51,210 Well, what you're doing is you're 1245 01:00:51,210 --> 01:00:56,430 telling the calculation to allow the up and down electrons 1246 01:00:56,430 --> 01:01:00,000 to have freedom from one another. 1247 01:01:00,000 --> 01:01:01,510 That's actually critical. 1248 01:01:01,510 --> 01:01:04,380 So if you didn't do that, you would 1249 01:01:04,380 --> 01:01:08,820 be simulating the material with the constraint 1250 01:01:08,820 --> 01:01:11,530 that the up and down electrons are exactly 1251 01:01:11,530 --> 01:01:12,405 at the same energies. 1252 01:01:15,030 --> 01:01:19,560 So spin polarized, the reason you 1253 01:01:19,560 --> 01:01:22,470 do calculations that are not spin polarized 1254 01:01:22,470 --> 01:01:24,960 is because many systems don't have 1255 01:01:24,960 --> 01:01:29,400 a difference between up and down electrons, right? 1256 01:01:29,400 --> 01:01:32,160 And they are actually all the same, right? 1257 01:01:35,040 --> 01:01:38,910 What do I have if I fill this up? 1258 01:01:38,910 --> 01:01:39,810 What atom is that? 1259 01:01:39,810 --> 01:01:40,790 AUDIENCE: Neon. 1260 01:01:40,790 --> 01:01:43,060 JEFFREY GROSSMAN: It's neon. 1261 01:01:43,060 --> 01:01:46,720 And here, every up electron is the same as every down electron 1262 01:01:46,720 --> 01:01:49,330 for a given state. 1263 01:01:49,330 --> 01:01:51,200 Everything is the same here. 1264 01:01:51,200 --> 01:01:56,050 So it's not like this up electron of this p orbital 1265 01:01:56,050 --> 01:01:59,590 wanted to be there, and the down one wanted to be there. 1266 01:01:59,590 --> 01:02:01,450 That doesn't happen for neon. 1267 01:02:01,450 --> 01:02:04,390 And it doesn't happen for many systems and many materials. 1268 01:02:04,390 --> 01:02:07,330 But certainly in some cases, and certainly 1269 01:02:07,330 --> 01:02:10,630 for magnetic materials, that's what causes magnetism. 1270 01:02:13,450 --> 01:02:17,500 It's that you have a difference between spin up and spin down. 1271 01:02:17,500 --> 01:02:21,130 And that as you occupy the density of states, 1272 01:02:21,130 --> 01:02:26,680 literally you get a difference in the number, right? 1273 01:02:26,680 --> 01:02:29,920 So that's what causes the magnetism. 1274 01:02:29,920 --> 01:02:34,950 So for many materials, there is no spin polarization. 1275 01:02:34,950 --> 01:02:37,595 Or in other words, there is no difference. 1276 01:02:37,595 --> 01:02:39,220 Spin up and spin down is the same here, 1277 01:02:39,220 --> 01:02:40,990 same here, same here. 1278 01:02:40,990 --> 01:02:45,100 But if it's a magnetic material, what you'll find 1279 01:02:45,100 --> 01:02:47,380 is that if you plot the density of states 1280 01:02:47,380 --> 01:02:49,900 separately for spin up and spin down 1281 01:02:49,900 --> 01:02:54,100 electrons, like in this case for iron, there's a difference. 1282 01:02:54,100 --> 01:02:57,670 And in fact, the magnetic moment of that material 1283 01:02:57,670 --> 01:03:01,840 is nothing more than the integral of that difference. 1284 01:03:01,840 --> 01:03:04,660 Again, you're integrating the DOS. 1285 01:03:04,660 --> 01:03:09,710 And you're finding something really cool about the material. 1286 01:03:09,710 --> 01:03:13,265 So that's the same thing as just counting 1287 01:03:13,265 --> 01:03:15,140 number of up a number of down, but now you're 1288 01:03:15,140 --> 01:03:19,740 counting it in this very complex filling of states 1289 01:03:19,740 --> 01:03:20,870 that happens in a material. 1290 01:03:23,900 --> 01:03:26,300 Any questions? 1291 01:03:26,300 --> 01:03:32,510 So if I did a calculation of iron with spin polarized 1292 01:03:32,510 --> 01:03:38,070 checked no, then you see, these would be identical. 1293 01:03:38,070 --> 01:03:41,037 And I'd get no magnetism. 1294 01:03:41,037 --> 01:03:41,870 And you can do that. 1295 01:03:41,870 --> 01:03:45,260 You can actually do the calculation of iron 1296 01:03:45,260 --> 01:03:48,343 with spin polarized checked no, and it's completely wrong. 1297 01:03:48,343 --> 01:03:49,760 It's not the ground state of iron. 1298 01:03:53,510 --> 01:03:55,940 And if you do it with spin polarized on, 1299 01:03:55,940 --> 01:03:57,920 then it allows those up and down electrons 1300 01:03:57,920 --> 01:04:00,140 to move relative to each other. 1301 01:04:00,140 --> 01:04:07,065 And the outputs, you'll see that in the plots, 1302 01:04:07,065 --> 01:04:09,440 there's something called the projected density of states. 1303 01:04:09,440 --> 01:04:13,400 And it's projected for one spin and another spin. 1304 01:04:13,400 --> 01:04:16,280 And that would allow you to calculate 1305 01:04:16,280 --> 01:04:18,140 the magnetic properties of materials 1306 01:04:18,140 --> 01:04:19,940 using the nanoHUB tool. 1307 01:04:19,940 --> 01:04:23,330 And I think last year we might have done a homework on that. 1308 01:04:23,330 --> 01:04:24,960 But this year, we won't. 1309 01:04:24,960 --> 01:04:28,380 But I want you to know about it, because it's kind of cool. 1310 01:04:28,380 --> 01:04:29,870 OK? 1311 01:04:29,870 --> 01:04:30,770 Any questions? 1312 01:04:34,410 --> 01:04:34,910 Mm-hm. 1313 01:04:37,995 --> 01:04:39,370 A couple other things you can do, 1314 01:04:39,370 --> 01:04:46,270 and first principles, quantum molecular dynamics, 1315 01:04:46,270 --> 01:04:48,210 you can do it. 1316 01:04:48,210 --> 01:04:52,920 You can do MD, Just like you did MD in the first part. 1317 01:04:52,920 --> 01:04:59,910 You're just getting f from the Hellmann-Feynman theorem, which 1318 01:04:59,910 --> 01:05:03,990 I referred to a couple lectures ago. 1319 01:05:03,990 --> 01:05:09,030 But it's basically just a way of getting the forces 1320 01:05:09,030 --> 01:05:11,010 using quantum mechanics. 1321 01:05:11,010 --> 01:05:13,560 And I don't really want to go into it in detail, 1322 01:05:13,560 --> 01:05:15,060 but I just want to tell you, you can 1323 01:05:15,060 --> 01:05:19,500 get the force on these atoms from these quantum mechanical 1324 01:05:19,500 --> 01:05:20,280 simulations. 1325 01:05:20,280 --> 01:05:21,330 And you can do dynamics. 1326 01:05:21,330 --> 01:05:24,090 Here's one I did a long time ago. 1327 01:05:24,090 --> 01:05:28,120 And here the problem was that, how do nanotubes grow, 1328 01:05:28,120 --> 01:05:30,150 which was a big question, OK? 1329 01:05:30,150 --> 01:05:36,730 And so what you have is, you have a piece of iron here. 1330 01:05:36,730 --> 01:05:39,970 And this is what happens when you 1331 01:05:39,970 --> 01:05:42,880 grow nanotubes, is you have these seed particles, 1332 01:05:42,880 --> 01:05:43,540 like iron. 1333 01:05:43,540 --> 01:05:47,440 It can be other things, nickel, other things will work. 1334 01:05:47,440 --> 01:05:49,460 Not gold. 1335 01:05:49,460 --> 01:05:51,610 And then you flow some kind of gas over it 1336 01:05:51,610 --> 01:05:55,200 that dumps carbon onto the catalyst. 1337 01:05:55,200 --> 01:05:59,620 Now, the problem is that understanding how that then 1338 01:05:59,620 --> 01:06:03,230 leads to the growth of a tube is still an open question, 1339 01:06:03,230 --> 01:06:03,730 by the way. 1340 01:06:03,730 --> 01:06:04,810 Still an open question. 1341 01:06:04,810 --> 01:06:07,570 Nobody actually really knows the full answer to that. 1342 01:06:07,570 --> 01:06:09,340 But certainly, people were just debating, 1343 01:06:09,340 --> 01:06:13,120 does the carbon go inside the catalyst or not? 1344 01:06:13,120 --> 01:06:15,670 Does it stay on the surface, does it saturate the catalyst 1345 01:06:15,670 --> 01:06:20,920 first, and then come out from the bulk, from the particle? 1346 01:06:20,920 --> 01:06:24,370 And that's a perfect problem for quantum, 1347 01:06:24,370 --> 01:06:27,050 because there's so many different bonding environments, 1348 01:06:27,050 --> 01:06:30,160 and so many possibilities here, that classical potentials will 1349 01:06:30,160 --> 01:06:31,380 fail. 1350 01:06:31,380 --> 01:06:36,190 They're not going to capture all of the different complexities 1351 01:06:36,190 --> 01:06:40,507 in how carbon needs to bond to iron, inside and out. 1352 01:06:40,507 --> 01:06:42,590 And so it's a great problem for quantum mechanics. 1353 01:06:42,590 --> 01:06:46,930 And we studied this, and we fed the simulation carbon, 1354 01:06:46,930 --> 01:06:52,660 and we just watched how the tube winds up making its way out 1355 01:06:52,660 --> 01:06:53,450 of the system. 1356 01:06:53,450 --> 01:06:55,492 And we showed that it doesn't saturate, actually, 1357 01:06:55,492 --> 01:07:00,287 inside the iron, but rather stays on the surface. 1358 01:07:00,287 --> 01:07:01,495 And it makes for cool movies. 1359 01:07:05,230 --> 01:07:07,510 Somebody tell me something you notice about this. 1360 01:07:07,510 --> 01:07:09,790 Besides just, it's kind of a fun movie. 1361 01:07:09,790 --> 01:07:14,650 But do you notice anything? 1362 01:07:14,650 --> 01:07:17,900 There's the initial nanotube growth. 1363 01:07:17,900 --> 01:07:20,170 And that's where it ended. 1364 01:07:20,170 --> 01:07:22,270 So I notice two things. 1365 01:07:22,270 --> 01:07:22,945 Yeah. 1366 01:07:22,945 --> 01:07:24,820 AUDIENCE: Some rings are more than 6 carbons. 1367 01:07:24,820 --> 01:07:25,903 JEFFREY GROSSMAN: Oh yeah. 1368 01:07:25,903 --> 01:07:27,520 Actually, that's a really good point. 1369 01:07:27,520 --> 01:07:31,380 So that was a very interesting part of the analysis is, 1370 01:07:31,380 --> 01:07:33,280 what kind of carbon you get. 1371 01:07:33,280 --> 01:07:37,150 When you catalyze carbon on a surface, what does it do? 1372 01:07:37,150 --> 01:07:39,640 Does it form nice graphene and then curve it? 1373 01:07:39,640 --> 01:07:42,550 Or does it form the defects so that you can curve it? 1374 01:07:42,550 --> 01:07:45,130 Where are the penalties in energy coming from? 1375 01:07:45,130 --> 01:07:49,463 But actually, I was going to say some negative things, two 1376 01:07:49,463 --> 01:07:50,130 negative things. 1377 01:07:50,130 --> 01:07:54,200 They're not negatives, but they're limitations. 1378 01:07:54,200 --> 01:07:55,700 Tell me something obvious about this 1379 01:07:55,700 --> 01:07:57,418 compared to, say, the protein simulations 1380 01:07:57,418 --> 01:07:58,460 we did in the first part. 1381 01:07:58,460 --> 01:07:59,910 AUDIENCE: Very small. 1382 01:07:59,910 --> 01:08:00,870 JEFFREY GROSSMAN: Yeah. 1383 01:08:00,870 --> 01:08:02,970 This is really tiny. 1384 01:08:02,970 --> 01:08:05,632 And how long do you think this went? 1385 01:08:05,632 --> 01:08:06,848 AUDIENCE: [INAUDIBLE] 1386 01:08:06,848 --> 01:08:08,140 JEFFREY GROSSMAN: What is that? 1387 01:08:08,140 --> 01:08:09,515 AUDIENCE: When was it found? 1388 01:08:09,515 --> 01:08:10,015 What year? 1389 01:08:10,015 --> 01:08:13,150 JEFFREY GROSSMAN: I'm not going to tell you, because that 1390 01:08:13,150 --> 01:08:15,550 would make me feel old. 1391 01:08:15,550 --> 01:08:16,569 A while ago. 1392 01:08:16,569 --> 01:08:22,359 [LAUGHS] But anyway, how long can you 1393 01:08:22,359 --> 01:08:25,120 run a molecular dynamics simulation having to do 1394 01:08:25,120 --> 01:08:26,800 quantum mechanics at each step? 1395 01:08:26,800 --> 01:08:28,240 AUDIENCE: Very short period. 1396 01:08:28,240 --> 01:08:29,200 JEFFREY GROSSMAN: Yeah. 1397 01:08:29,200 --> 01:08:30,960 AUDIENCE: [INAUDIBLE] 1398 01:08:30,960 --> 01:08:32,710 JEFFREY GROSSMAN: Well, how long could you 1399 01:08:32,710 --> 01:08:35,050 do proteins with classical MD? 1400 01:08:35,050 --> 01:08:36,370 Huge systems. 1401 01:08:36,370 --> 01:08:38,109 Could you do them for seconds? 1402 01:08:38,109 --> 01:08:38,770 No. 1403 01:08:38,770 --> 01:08:40,621 But how long? 1404 01:08:40,621 --> 01:08:41,794 AUDIENCE: Milliseconds? 1405 01:08:41,794 --> 01:08:44,229 JEFFREY GROSSMAN: Yeah, actually even longer. 1406 01:08:44,229 --> 01:08:47,410 With classical MD, you can get up to microseconds now. 1407 01:08:47,410 --> 01:08:52,420 Certainly nanoseconds, just 10, 100 nanoseconds, no problem. 1408 01:08:52,420 --> 01:08:55,600 Here, we're at 10 picoseconds, right? 1409 01:08:55,600 --> 01:08:57,340 Even 2 picoseconds. 1410 01:08:57,340 --> 01:09:01,517 Nowadays, you can push 50, 100 picoseconds, 1411 01:09:01,517 --> 01:09:03,100 but you wouldn't go further than that. 1412 01:09:03,100 --> 01:09:04,760 It's really hard. 1413 01:09:04,760 --> 01:09:07,910 So the time and size are really limited. 1414 01:09:07,910 --> 01:09:10,390 And that's why with quantum MD, you 1415 01:09:10,390 --> 01:09:13,000 study small systems for short timescales. 1416 01:09:13,000 --> 01:09:14,319 This was now cluster growth. 1417 01:09:17,709 --> 01:09:21,520 I think quantum, these are just other things you can do. 1418 01:09:21,520 --> 01:09:24,430 This is supposed to be just making 1419 01:09:24,430 --> 01:09:27,702 sure you see that it's not just the band structure in DOS. 1420 01:09:27,702 --> 01:09:28,660 There are other things. 1421 01:09:28,660 --> 01:09:32,740 You can do a lot of things with quantum simulations. 1422 01:09:32,740 --> 01:09:36,340 Water is a really cool problem. 1423 01:09:36,340 --> 01:09:39,130 And I love Cavendish. 1424 01:09:39,130 --> 01:09:40,540 I love reading about Cavendish. 1425 01:09:40,540 --> 01:09:42,880 Has anybody read about Cavendish? 1426 01:09:42,880 --> 01:09:46,180 He's such a fascinating guy. 1427 01:09:46,180 --> 01:09:47,770 He's a university dropout. 1428 01:09:47,770 --> 01:09:51,130 He compared the connectivities of electrolytes 1429 01:09:51,130 --> 01:09:55,570 and expressed a version of Ohm's law way ahead of time. 1430 01:09:55,570 --> 01:10:00,790 And he measured the gravitational constant. 1431 01:10:00,790 --> 01:10:03,790 And he came up with really water, which at that point 1432 01:10:03,790 --> 01:10:10,360 was phlogiston and dephlogisticated air, which 1433 01:10:10,360 --> 01:10:12,850 I love those names. 1434 01:10:12,850 --> 01:10:14,650 Which is hydrogen and oxygen. 1435 01:10:14,650 --> 01:10:19,225 But anyway, water has these really interesting properties. 1436 01:10:19,225 --> 01:10:22,068 This is a wonderful website to read about water. 1437 01:10:22,068 --> 01:10:23,110 I'll give you the answer. 1438 01:10:23,110 --> 01:10:26,030 It actually looks like this. 1439 01:10:26,030 --> 01:10:29,310 Water sees water like this, not like this. 1440 01:10:29,310 --> 01:10:33,320 So it really sees it much more spherical 1441 01:10:33,320 --> 01:10:35,150 than we usually draw it. 1442 01:10:35,150 --> 01:10:39,020 And yet, it does have these very definite bondings 1443 01:10:39,020 --> 01:10:39,920 that it likes to do. 1444 01:10:39,920 --> 01:10:43,820 And the point I want to make is, when you go into an MD code, 1445 01:10:43,820 --> 01:10:45,920 and you put water in there, did you use water 1446 01:10:45,920 --> 01:10:47,600 in your protein simulations? 1447 01:10:47,600 --> 01:10:48,335 Yeah. 1448 01:10:48,335 --> 01:10:49,460 Which water did you pick? 1449 01:10:53,210 --> 01:10:54,710 You must have picked a potential, 1450 01:10:54,710 --> 01:10:57,300 or the code picked a potential. 1451 01:10:57,300 --> 01:10:58,000 Right? 1452 01:10:58,000 --> 01:10:59,620 Tip 3P to 4P. 1453 01:10:59,620 --> 01:11:02,120 Well, there's hundreds of them. 1454 01:11:02,120 --> 01:11:04,430 These are just a few water potentials. 1455 01:11:04,430 --> 01:11:05,510 Why are there so many? 1456 01:11:05,510 --> 01:11:08,450 Well, because some of them reproduce some things better 1457 01:11:08,450 --> 01:11:09,860 than others, right? 1458 01:11:09,860 --> 01:11:13,130 And that gets to be a really serious problem 1459 01:11:13,130 --> 01:11:16,580 if you want to study some of the fundamental problems of water. 1460 01:11:16,580 --> 01:11:20,510 And there's so many mysteries that we still don't know, 1461 01:11:20,510 --> 01:11:24,560 in terms of what happens when water evaporates, freezes. 1462 01:11:24,560 --> 01:11:26,480 And so, which one is best? 1463 01:11:26,480 --> 01:11:29,090 Well, this is a perfect problem for quantum mechanics. 1464 01:11:29,090 --> 01:11:31,790 And I won't go through the details. 1465 01:11:31,790 --> 01:11:38,000 But you can do water with quantum. 1466 01:11:38,000 --> 01:11:39,530 And this is a good way to validate 1467 01:11:39,530 --> 01:11:43,550 which potential you use in your classical simulations. 1468 01:11:43,550 --> 01:11:45,500 You can also calculate phonons. 1469 01:11:45,500 --> 01:11:51,170 You can just displace atoms and calculate energy differences, 1470 01:11:51,170 --> 01:11:52,640 and get frequencies. 1471 01:11:52,640 --> 01:11:54,050 So you can calculate phonons. 1472 01:11:54,050 --> 01:11:58,040 And when you calculate photons in a crystal, 1473 01:11:58,040 --> 01:12:00,860 that's different than in a molecule. 1474 01:12:03,600 --> 01:12:05,670 When you calculate phonons in a crystal, 1475 01:12:05,670 --> 01:12:08,610 you're getting the, waves the sound waves. 1476 01:12:08,610 --> 01:12:12,520 You're going to get the acoustic waves in the material, right? 1477 01:12:16,475 --> 01:12:17,550 You're just using water. 1478 01:12:17,550 --> 01:12:21,720 In a water molecule, how many phonon modes do I have? 1479 01:12:25,060 --> 01:12:27,520 In a water molecule? 1480 01:12:27,520 --> 01:12:29,850 Who can tell me? 1481 01:12:29,850 --> 01:12:30,510 Take a guess. 1482 01:12:30,510 --> 01:12:31,080 AUDIENCE: 6. 1483 01:12:31,080 --> 01:12:32,580 JEFFREY GROSSMAN: It's a good guess. 1484 01:12:32,580 --> 01:12:37,020 Let's get rid of translational and rotational. 1485 01:12:37,020 --> 01:12:40,510 How many waves can the water molecules stretch? 1486 01:12:40,510 --> 01:12:41,965 AUDIENCE: Two. 1487 01:12:41,965 --> 01:12:44,760 JEFFREY GROSSMAN: Yeah, there's actually three, right? 1488 01:12:44,760 --> 01:12:49,830 Because the OHs can stretch, and the bond can stretch. 1489 01:12:49,830 --> 01:12:54,690 But then there's two different ways that it can go, right? 1490 01:12:54,690 --> 01:13:01,840 And so those are your phonon modes for the molecule. 1491 01:13:01,840 --> 01:13:03,720 It's just stretch frequencies, right? 1492 01:13:03,720 --> 01:13:05,370 And they can be measured. 1493 01:13:05,370 --> 01:13:09,300 In a solid, you calculate your phonon modes 1494 01:13:09,300 --> 01:13:12,640 by moving atoms around in the lattice. 1495 01:13:12,640 --> 01:13:18,815 And that actually gives you the behavior 1496 01:13:18,815 --> 01:13:20,940 of the sound waves in the material, which tells you 1497 01:13:20,940 --> 01:13:23,400 something about how sound travels in the solid, 1498 01:13:23,400 --> 01:13:25,380 and you can calculate the sound velocity, 1499 01:13:25,380 --> 01:13:28,740 you can calculate thermal transport. 1500 01:13:28,740 --> 01:13:31,020 You can calculate the heat capacity of the material. 1501 01:13:31,020 --> 01:13:34,350 All of this are things that we do with quantum mechanics 1502 01:13:34,350 --> 01:13:37,260 that we're not going to do in any of our problems, 1503 01:13:37,260 --> 01:13:37,860 or anymore. 1504 01:13:37,860 --> 01:13:40,152 But I just wanted you to be aware of some of the things 1505 01:13:40,152 --> 01:13:43,260 that people do in research, and that 1506 01:13:43,260 --> 01:13:47,970 are really a big part of quantum mechanical simulations. 1507 01:13:51,450 --> 01:13:52,950 So those are some of the properties. 1508 01:13:52,950 --> 01:13:54,450 So that was sort of the point here. 1509 01:13:54,450 --> 01:13:57,360 You can do a wide calculator, a wide range of properties. 1510 01:13:57,360 --> 01:14:01,783 This has been our focus, these two. 1511 01:14:01,783 --> 01:14:03,450 And we're going to stay with that focus. 1512 01:14:03,450 --> 01:14:07,860 But there's a very large range of things you can do. 1513 01:14:07,860 --> 01:14:10,110 Now, I want to just end. 1514 01:14:10,110 --> 01:14:12,420 I got 5 minutes, OK? 1515 01:14:12,420 --> 01:14:17,740 And we'll start solar with this on Tuesday. 1516 01:14:17,740 --> 01:14:19,740 But for five minutes, I want to do a simulation. 1517 01:14:23,910 --> 01:14:25,800 So we'll start solar on Tuesday. 1518 01:14:25,800 --> 01:14:28,050 But here we are. 1519 01:14:28,050 --> 01:14:29,580 You know which tool it is. 1520 01:14:29,580 --> 01:14:30,510 It's our favorite. 1521 01:14:30,510 --> 01:14:33,260 AUDIENCE: Are we [INAUDIBLE] Tuesday, or no? 1522 01:14:33,260 --> 01:14:36,680 JEFFREY GROSSMAN: And we'll do another problem or two 1523 01:14:36,680 --> 01:14:37,880 in class. 1524 01:14:37,880 --> 01:14:40,200 And I can do a little review. 1525 01:14:40,200 --> 01:14:43,050 I sort of did that on Tuesday of this week. 1526 01:14:43,050 --> 01:14:47,120 Does anybody want to do more review next week? 1527 01:14:47,120 --> 01:14:48,470 What do you guys think? 1528 01:14:48,470 --> 01:14:49,670 A little bit? 1529 01:14:49,670 --> 01:14:50,630 Yeah? 1530 01:14:50,630 --> 01:14:52,820 OK. 1531 01:14:52,820 --> 01:14:54,890 What I'll do next Tuesday is we'll 1532 01:14:54,890 --> 01:15:03,830 do another quiz problem from last year like we did today. 1533 01:15:03,830 --> 01:15:06,560 I'll ask if you have any questions about this practice 1534 01:15:06,560 --> 01:15:09,410 quiz but, I'll also post the solutions. 1535 01:15:09,410 --> 01:15:15,560 And I'll go over some key things to keep in mind for the quiz. 1536 01:15:15,560 --> 01:15:17,870 So we'll spend the first half of class on that, then. 1537 01:15:17,870 --> 01:15:20,510 But let me just spend the last few minutes here. 1538 01:15:20,510 --> 01:15:21,590 Let's do a simulation. 1539 01:15:23,985 --> 01:15:25,360 AUDIENCE: It's open notes, right? 1540 01:15:25,360 --> 01:15:27,500 JEFFREY GROSSMAN: It's open notes. 1541 01:15:27,500 --> 01:15:28,450 Open notes, yeah. 1542 01:15:28,450 --> 01:15:30,875 AUDIENCE: Does that mean computer is open? 1543 01:15:30,875 --> 01:15:34,570 JEFFREY GROSSMAN: I think no. 1544 01:15:34,570 --> 01:15:40,265 Because that's like, everything, right? 1545 01:15:40,265 --> 01:15:41,890 What did you guys do in the first part? 1546 01:15:41,890 --> 01:15:43,080 AUDIENCE: We had the computers. 1547 01:15:43,080 --> 01:15:43,755 JEFFREY GROSSMAN: You had the computers? 1548 01:15:43,755 --> 01:15:45,672 AUDIENCE: We had the computers, but we weren't 1549 01:15:45,672 --> 01:15:46,938 allowed to use the internet. 1550 01:15:46,938 --> 01:15:51,160 So I don't know how they were checking that. 1551 01:15:51,160 --> 01:15:58,127 But I [INAUDIBLE] my computer, so I don't know. 1552 01:15:58,127 --> 01:15:59,960 JEFFREY GROSSMAN: OK, let's put it this way. 1553 01:15:59,960 --> 01:16:02,570 If we do open computers, then I'll make the problems harder. 1554 01:16:02,570 --> 01:16:03,425 AUDIENCE: No. 1555 01:16:03,425 --> 01:16:04,300 JEFFREY GROSSMAN: OK. 1556 01:16:04,300 --> 01:16:05,940 [LAUGHS] All right. 1557 01:16:08,500 --> 01:16:11,950 Because Wikipedia's pretty powerful, right? 1558 01:16:11,950 --> 01:16:13,000 And Google. 1559 01:16:13,000 --> 01:16:18,747 And so if you have that, you can find a lot of close answers 1560 01:16:18,747 --> 01:16:19,580 with a quick search. 1561 01:16:19,580 --> 01:16:23,740 So I'd have to be a little more creative. 1562 01:16:23,740 --> 01:16:25,570 But you won't need your computers. 1563 01:16:25,570 --> 01:16:27,230 And it's full open notes, right? 1564 01:16:29,920 --> 01:16:30,820 OK. 1565 01:16:30,820 --> 01:16:34,990 Now here's what I wanted to do. 1566 01:16:38,295 --> 01:16:38,795 OK. 1567 01:16:42,120 --> 01:16:45,600 Let's go to our favorite thing here, SIESTA. 1568 01:16:45,600 --> 01:16:46,380 There it is. 1569 01:16:51,030 --> 01:16:57,150 So I wanted to show you first of all, this feature here, 1570 01:16:57,150 --> 01:16:58,790 which is that spin polarized. 1571 01:16:58,790 --> 01:16:59,590 Do you see? 1572 01:16:59,590 --> 01:17:00,470 Yes? 1573 01:17:00,470 --> 01:17:01,710 No. 1574 01:17:01,710 --> 01:17:02,270 OK? 1575 01:17:02,270 --> 01:17:08,965 Now, when it's yes, and I calculate silicon, 1576 01:17:08,965 --> 01:17:12,625 here's the calculation of the density of states of silicon, 1577 01:17:12,625 --> 01:17:13,750 is what I want to show you. 1578 01:17:18,360 --> 01:17:22,280 What you are going to find. 1579 01:17:22,280 --> 01:17:24,620 Oh, I didn't fill out the structure. 1580 01:17:24,620 --> 01:17:27,740 Is that, OK, so this is just rounding out 1581 01:17:27,740 --> 01:17:31,625 our SIESTA understanding of what the tool does. 1582 01:17:31,625 --> 01:17:33,500 Because there were just these few things that 1583 01:17:33,500 --> 01:17:36,240 we hadn't talked about. 1584 01:17:36,240 --> 01:17:37,490 This is the density of states. 1585 01:17:37,490 --> 01:17:40,150 And you now really feel your oneness with this. 1586 01:17:40,150 --> 01:17:41,150 You know what these are. 1587 01:17:41,150 --> 01:17:47,690 You look at this, and you get emotional, which is OK. 1588 01:17:47,690 --> 01:17:48,530 But look at this. 1589 01:17:48,530 --> 01:17:54,920 I also have here, because I did, let's see 1590 01:17:54,920 --> 01:17:57,950 if it does this right. 1591 01:17:57,950 --> 01:18:00,350 Total, and there. 1592 01:18:00,350 --> 01:18:01,910 Look at that, OK. 1593 01:18:01,910 --> 01:18:04,463 You see, it's automatically doing 1594 01:18:04,463 --> 01:18:06,380 what's called the projected density of states. 1595 01:18:06,380 --> 01:18:08,900 And what a projected density of states is, is it's 1596 01:18:08,900 --> 01:18:12,620 the density of states, but only for a certain part 1597 01:18:12,620 --> 01:18:14,360 of the system. 1598 01:18:14,360 --> 01:18:16,070 Which turns out to be really interesting. 1599 01:18:16,070 --> 01:18:18,070 Like sometimes you just want to know the density 1600 01:18:18,070 --> 01:18:19,670 of states for one type of atom. 1601 01:18:19,670 --> 01:18:23,390 Where is that atom contributing to the states? 1602 01:18:23,390 --> 01:18:26,270 Or in this case, because it has been polarized, 1603 01:18:26,270 --> 01:18:29,420 it's giving me the density of states for spin down and spin 1604 01:18:29,420 --> 01:18:30,410 up separately. 1605 01:18:30,410 --> 01:18:31,340 And look at this. 1606 01:18:31,340 --> 01:18:37,320 They are, while spin down, spin down, you can't see it. 1607 01:18:40,230 --> 01:18:41,850 There's some spin up in there. 1608 01:18:41,850 --> 01:18:44,760 They're exactly on top of each other. 1609 01:18:44,760 --> 01:18:46,774 What does that mean? 1610 01:18:46,774 --> 01:18:48,615 AUDIENCE: Same energy. 1611 01:18:48,615 --> 01:18:49,990 JEFFREY GROSSMAN: All the spin up 1612 01:18:49,990 --> 01:18:52,940 electrons when they're paired with spin down, 1613 01:18:52,940 --> 01:18:57,100 they all have the same energy. 1614 01:18:57,100 --> 01:18:59,140 They have the exact same position on this. 1615 01:18:59,140 --> 01:19:01,360 And they lead to this total DOS here. 1616 01:19:01,360 --> 01:19:04,300 They just add up. 1617 01:19:04,300 --> 01:19:08,230 So if I did silicon with spin polarized off, 1618 01:19:08,230 --> 01:19:12,665 then the answer would be exactly the same. 1619 01:19:12,665 --> 01:19:15,040 So there was really no need to do it with spin polarized, 1620 01:19:15,040 --> 01:19:20,470 but it's the default. And now if I do iron, you see? 1621 01:19:20,470 --> 01:19:22,930 And I press simulate. 1622 01:19:22,930 --> 01:19:29,428 Then what you'll find is that you should see a difference. 1623 01:19:29,428 --> 01:19:30,220 Let's see if we do. 1624 01:19:34,140 --> 01:19:36,450 Here is iron. 1625 01:19:36,450 --> 01:19:39,160 And now you have the total density of states. 1626 01:19:39,160 --> 01:19:40,800 Oh, what is this? 1627 01:19:40,800 --> 01:19:43,923 I forget, is this an insulator? 1628 01:19:43,923 --> 01:19:45,690 AUDIENCE: No, it's a [INAUDIBLE].. 1629 01:19:45,690 --> 01:19:47,023 JEFFREY GROSSMAN: How do I know? 1630 01:19:47,023 --> 01:19:48,880 AUDIENCE: [INAUDIBLE] 1631 01:19:48,880 --> 01:19:52,920 JEFFREY GROSSMAN: Because I got some DOS here, right? 1632 01:19:52,920 --> 01:19:54,000 At the Fermi energy. 1633 01:19:57,450 --> 01:20:01,810 Oh, we got to do it for iron. 1634 01:20:01,810 --> 01:20:02,900 And now look at this. 1635 01:20:02,900 --> 01:20:03,760 This is beautiful. 1636 01:20:07,600 --> 01:20:12,220 OK, so this is the total, the green. 1637 01:20:12,220 --> 01:20:13,810 That's the total. 1638 01:20:13,810 --> 01:20:14,590 And look at this. 1639 01:20:14,590 --> 01:20:18,340 The red is the spin down. 1640 01:20:18,340 --> 01:20:20,680 And the blue is the spin up. 1641 01:20:20,680 --> 01:20:23,730 And look at those differences. 1642 01:20:23,730 --> 01:20:24,900 What does it mean? 1643 01:20:24,900 --> 01:20:25,620 Whoa! 1644 01:20:25,620 --> 01:20:27,698 I mean, you've got to integrate it to know. 1645 01:20:27,698 --> 01:20:28,740 But there's a difference. 1646 01:20:28,740 --> 01:20:30,870 There's a big difference in the spin up 1647 01:20:30,870 --> 01:20:32,460 and spin down electrons and where 1648 01:20:32,460 --> 01:20:35,370 they sit in energy space in the magnetic material. 1649 01:20:35,370 --> 01:20:36,240 There can be. 1650 01:20:36,240 --> 01:20:38,670 That's what makes it a magnet. 1651 01:20:38,670 --> 01:20:42,180 And so if you integrate that difference up to here, 1652 01:20:42,180 --> 01:20:45,810 well then, you're going to get this magnetization. 1653 01:20:45,810 --> 01:20:51,280 If I simulated iron with spin polarized turned off, 1654 01:20:51,280 --> 01:20:53,670 it would force those to be exactly the same, 1655 01:20:53,670 --> 01:20:55,810 and they would lie exactly on top of each other, 1656 01:20:55,810 --> 01:20:58,500 just like they just did in silicon. 1657 01:20:58,500 --> 01:21:01,260 But that would be not iron. 1658 01:21:01,260 --> 01:21:05,550 So you got to be aware of what this spin thing is, 1659 01:21:05,550 --> 01:21:07,440 and when you need it, when it's important. 1660 01:21:07,440 --> 01:21:09,215 Keeping it on is safe. 1661 01:21:09,215 --> 01:21:10,840 But I wanted you to know what it means. 1662 01:21:10,840 --> 01:21:12,330 So OK. 1663 01:21:12,330 --> 01:21:13,530 Very good. 1664 01:21:13,530 --> 01:21:15,500 Have a good weekend.