1 00:00:00,000 --> 00:00:01,375 PROFESSOR: --called case studies. 2 00:00:01,375 --> 00:00:04,860 That is, we really look at how some of the tools 3 00:00:04,860 --> 00:00:06,700 that we have learned during this class-- 4 00:00:06,700 --> 00:00:09,480 some of the energy tools, like density functional theory, 5 00:00:09,480 --> 00:00:12,990 some of the statistical tools, like thermodynamic 6 00:00:12,990 --> 00:00:15,180 integration-- come together and are actually 7 00:00:15,180 --> 00:00:19,320 used to predict the properties of materials and structures 8 00:00:19,320 --> 00:00:22,090 of real interest. 9 00:00:22,090 --> 00:00:23,580 This is-- I called it lecture 20, 10 00:00:23,580 --> 00:00:27,240 but I think I have been slightly inconsistent with the numbering 11 00:00:27,240 --> 00:00:28,810 of my lectures. 12 00:00:28,810 --> 00:00:30,030 We count also the labs. 13 00:00:30,030 --> 00:00:33,360 Anyhow, everything is on the web. 14 00:00:33,360 --> 00:00:36,780 So what will you learn today is actually 15 00:00:36,780 --> 00:00:40,380 how to become an expert in two of these jobs 16 00:00:40,380 --> 00:00:42,100 for the 21st century. 17 00:00:42,100 --> 00:00:45,210 One is the virtual alchemist, and the one that is fancier, 18 00:00:45,210 --> 00:00:47,880 I think, is the nanotechnologist. 19 00:00:47,880 --> 00:00:51,360 There was probably an ad for GE a few months 20 00:00:51,360 --> 00:00:54,780 ago on TV in which there was a nanotechnologist marrying 21 00:00:54,780 --> 00:00:55,590 a supermodel. 22 00:00:55,590 --> 00:00:58,350 It was sort of fairly intriguing for all of us 23 00:00:58,350 --> 00:01:00,630 doing the initial calculations. 24 00:01:00,630 --> 00:01:04,800 But more seriously, what we are really going to focus on 25 00:01:04,800 --> 00:01:10,340 is how can we break the n-cube barrier of electronic structure 26 00:01:10,340 --> 00:01:11,340 calculation. 27 00:01:11,340 --> 00:01:13,890 You have learned that, say, when we study a system using 28 00:01:13,890 --> 00:01:17,850 density functional theory, the computational cost scales 29 00:01:17,850 --> 00:01:20,700 as the cube of the number of atoms. 30 00:01:20,700 --> 00:01:23,700 And that basically scales very rapidly 31 00:01:23,700 --> 00:01:25,930 the size of the system that you can do. 32 00:01:25,930 --> 00:01:28,710 I mean, even on a supercomputer nowadays, 33 00:01:28,710 --> 00:01:31,740 we can't really do more than 500 atoms, say. 34 00:01:31,740 --> 00:01:36,420 And on a regular cluster, we are more likely close to 100 atoms. 35 00:01:36,420 --> 00:01:40,200 And the reason of the cubic scaling in DFT 36 00:01:40,200 --> 00:01:43,920 is ultimately due to the orthonormality constraints. 37 00:01:43,920 --> 00:01:46,860 Every single particle orbital needs 38 00:01:46,860 --> 00:01:51,090 to be orthogonal to every other single particle orbital. 39 00:01:51,090 --> 00:01:53,700 So if you double the size of your system, 40 00:01:53,700 --> 00:01:55,380 you double the number of electrons. 41 00:01:55,380 --> 00:01:57,870 That is, you have twice as many orbitals 42 00:01:57,870 --> 00:02:02,460 that need to be orthogonal with twice as many other orbitals. 43 00:02:02,460 --> 00:02:05,910 So you have four times more matrix elements 44 00:02:05,910 --> 00:02:07,200 that you need to calculate. 45 00:02:07,200 --> 00:02:09,300 But also, those matrix elements are 46 00:02:09,300 --> 00:02:12,790 calculated in a unit cell that is twice as large. 47 00:02:12,790 --> 00:02:14,340 So there is another factor of two, 48 00:02:14,340 --> 00:02:16,650 and that gives us a factor of eight. 49 00:02:16,650 --> 00:02:19,200 And of course, there is a lot of effort in the community 50 00:02:19,200 --> 00:02:24,000 nowadays to go towards linear scaling algorithm, algorithm 51 00:02:24,000 --> 00:02:26,970 that just double in computational cost 52 00:02:26,970 --> 00:02:29,280 when you double the size of your unit cell. 53 00:02:29,280 --> 00:02:32,940 And they are all based on the sort of physical principle 54 00:02:32,940 --> 00:02:36,150 that, in many ways, nature is local. 55 00:02:36,150 --> 00:02:40,500 So your electron needs only to know what's happening nearby 56 00:02:40,500 --> 00:02:43,230 and doesn't need to be really orthogonal to electrons 57 00:02:43,230 --> 00:02:45,060 that are very, very far away. 58 00:02:45,060 --> 00:02:48,660 And you see an example of sort of where this linear scaling 59 00:02:48,660 --> 00:02:52,920 direction are heading. 60 00:02:52,920 --> 00:02:56,100 Other methods scale even worse than cubic. 61 00:02:56,100 --> 00:02:59,230 We have sort of discussed that sort of standard Hartree-Fock 62 00:02:59,230 --> 00:03:03,330 scales as the full power, and highly correlated 63 00:03:03,330 --> 00:03:06,630 quantum chemistry methods scale as the fifth, sixth, seventh 64 00:03:06,630 --> 00:03:08,280 power. 65 00:03:08,280 --> 00:03:11,370 An interesting exception, I think, 66 00:03:11,370 --> 00:03:15,270 among the methods that try to go beyond DFT in accuracy 67 00:03:15,270 --> 00:03:18,480 is what people call quantum Monte Carlo approaches. 68 00:03:18,480 --> 00:03:21,810 They tend to be approaches that are very accurate but based 69 00:03:21,810 --> 00:03:24,730 on a stochastic sampling of the wave function, 70 00:03:24,730 --> 00:03:27,720 so really, a stochastic solution for the wave 71 00:03:27,720 --> 00:03:29,670 function in the Schrodinger equation. 72 00:03:29,670 --> 00:03:32,140 And those in principle scale as the cube. 73 00:03:32,140 --> 00:03:34,230 And there have been sort of recent progress 74 00:03:34,230 --> 00:03:36,310 in making them truly linear scaling. 75 00:03:36,310 --> 00:03:38,460 So I think they are a very promising avenue 76 00:03:38,460 --> 00:03:39,750 for the future. 77 00:03:39,750 --> 00:03:43,200 But in this class, we'll sort of confine ourselves to DFT. 78 00:03:43,200 --> 00:03:47,400 And in particular, we'll look at two examples in which we 79 00:03:47,400 --> 00:03:52,140 use electronic structure DFT calculation to actually obtain 80 00:03:52,140 --> 00:03:55,440 the relevant parameters for a Hamiltonian 81 00:03:55,440 --> 00:04:00,420 that basically models the important degrees of freedom 82 00:04:00,420 --> 00:04:01,660 for our system. 83 00:04:01,660 --> 00:04:05,160 So we'll actually look at semiconductor alloys, where 84 00:04:05,160 --> 00:04:07,110 you'll see, very naturally, one can sort of 85 00:04:07,110 --> 00:04:10,950 solve the problem of substitutional disorder 86 00:04:10,950 --> 00:04:15,390 on that alloy using an Ising-like Hamiltonian, so 87 00:04:15,390 --> 00:04:17,880 something that you have seen in the previous classes. 88 00:04:17,880 --> 00:04:19,560 And then we'll look at another example 89 00:04:19,560 --> 00:04:22,470 to functionalize nanotubes, that is 90 00:04:22,470 --> 00:04:27,660 carbon nanotubes, to which organic ligands have 91 00:04:27,660 --> 00:04:29,520 been covalently attached. 92 00:04:29,520 --> 00:04:33,510 And you'll see how we sort of develop a linear scaling 93 00:04:33,510 --> 00:04:35,130 tight binding-like procedure. 94 00:04:35,130 --> 00:04:36,660 We'll explain this. 95 00:04:36,660 --> 00:04:38,580 That allows us actually to study, 96 00:04:38,580 --> 00:04:42,600 with full DFT precision, systems that have tens of thousands 97 00:04:42,600 --> 00:04:45,490 of atoms in the units cell. 98 00:04:45,490 --> 00:04:48,840 So let's start with the first example, that 99 00:04:48,840 --> 00:04:50,400 of semiconductor alloys. 100 00:04:50,400 --> 00:04:53,140 Semiconductor alloys have been of central interest 101 00:04:53,140 --> 00:04:56,220 since, I would say, the late 80s, when 102 00:04:56,220 --> 00:04:59,890 people started to realize that there wasn't only silicon. 103 00:04:59,890 --> 00:05:03,550 And in particular, it was very important to tune 104 00:05:03,550 --> 00:05:06,010 out the physical properties of your semiconductor, 105 00:05:06,010 --> 00:05:09,280 and alloying is a very simple way of doing that. 106 00:05:09,280 --> 00:05:13,330 If you, say, mix together something like gallium arsenide 107 00:05:13,330 --> 00:05:17,110 and aluminum arsenide, depending on the concentration, 108 00:05:17,110 --> 00:05:20,590 you have a substitutional alloy with a bandgap 109 00:05:20,590 --> 00:05:21,810 that is variable. 110 00:05:21,810 --> 00:05:24,040 So it can be used very efficiently, say, 111 00:05:24,040 --> 00:05:28,150 to tune the emission frequency of your laser light, 112 00:05:28,150 --> 00:05:31,750 if this semiconductor is your active layer. 113 00:05:31,750 --> 00:05:34,250 And this is something which, in our department, 114 00:05:34,250 --> 00:05:36,520 in the material science course, there 115 00:05:36,520 --> 00:05:38,230 is a lot of ongoing effort. 116 00:05:38,230 --> 00:05:40,690 In particular, Professor Fitzgerald 117 00:05:40,690 --> 00:05:42,880 works a lot on some of these alloys 118 00:05:42,880 --> 00:05:46,160 to actually improve mobility of the carriers. 119 00:05:46,160 --> 00:05:48,670 And I had here some kind of beautiful picture 120 00:05:48,670 --> 00:05:53,140 of actually an array of vertical KVT lasers 121 00:05:53,140 --> 00:05:57,760 in which sort of a number of vertical pillars 122 00:05:57,760 --> 00:06:00,940 were grown on a surface with a composition 123 00:06:00,940 --> 00:06:03,020 gradient for that active layer. 124 00:06:03,020 --> 00:06:04,930 So what you achieve on a single breadboard 125 00:06:04,930 --> 00:06:08,200 is a very high density of emitting lasers. 126 00:06:08,200 --> 00:06:12,070 And each one emits at a slightly different wavelength, 127 00:06:12,070 --> 00:06:15,970 because the bandgap is tuned by the compositional gradient 128 00:06:15,970 --> 00:06:17,950 in that array. 129 00:06:17,950 --> 00:06:22,480 And another set of reasons that sort of intrigued and prompted 130 00:06:22,480 --> 00:06:24,880 the study of the semiconductor alloys 131 00:06:24,880 --> 00:06:27,340 was actually what is called the ordering, 132 00:06:27,340 --> 00:06:29,650 multilayer ordering, superlattice 133 00:06:29,650 --> 00:06:31,210 ordering in those alloys. 134 00:06:31,210 --> 00:06:33,760 That is, when doing X-ray diffraction 135 00:06:33,760 --> 00:06:39,220 of a standard semiconductor, you would get a set of broad pixel. 136 00:06:39,220 --> 00:06:42,220 And what people note is that under certain growth 137 00:06:42,220 --> 00:06:46,960 circumstances, they would find naturally in the diffraction 138 00:06:46,960 --> 00:06:50,860 figures some intermediate peaks. 139 00:06:50,860 --> 00:06:55,240 So in this case, some peaks are closer to the origin, 140 00:06:55,240 --> 00:06:57,490 or in this case, sort of the green dots. 141 00:06:57,490 --> 00:07:00,490 So yeah, midway between the two red dots. 142 00:07:00,490 --> 00:07:02,710 What it means is that there is actually 143 00:07:02,710 --> 00:07:05,560 some kind of ordering that has periodicity that 144 00:07:05,560 --> 00:07:07,780 is in that particular crystallography direction, 145 00:07:07,780 --> 00:07:10,420 has a wavelength that is twice as large 146 00:07:10,420 --> 00:07:12,760 or a wave vector that is one half. 147 00:07:12,760 --> 00:07:16,185 And so it sort of hints at some kind of ordering. 148 00:07:16,185 --> 00:07:17,560 So there were a number of reasons 149 00:07:17,560 --> 00:07:19,270 why these were interesting. 150 00:07:19,270 --> 00:07:22,810 And I'll show you sort of one of the ways one could actually 151 00:07:22,810 --> 00:07:27,490 tackle the problem of studying these systems. 152 00:07:27,490 --> 00:07:28,950 I had another figure here. 153 00:07:28,950 --> 00:07:31,510 They just didn't come-- they didn't come out-- 154 00:07:31,510 --> 00:07:33,790 in which, again, I was sort of showing you 155 00:07:33,790 --> 00:07:38,440 basically the whole set of semiconductors 156 00:07:38,440 --> 00:07:43,270 and plotting that as a function of lattice parameter 157 00:07:43,270 --> 00:07:46,090 and bandgap. 158 00:07:46,090 --> 00:07:49,030 And basically showing that, if you look at all, say, 159 00:07:49,030 --> 00:07:54,040 the III-V semiconductor, they sit in a certain region 160 00:07:54,040 --> 00:07:56,890 of this diagram. 161 00:07:56,890 --> 00:07:58,780 But in reality, there isn't anything 162 00:07:58,780 --> 00:08:04,210 that is really useful that-- say, as the bandgap of silicon. 163 00:08:04,210 --> 00:08:08,920 But it meets at the wavelength for which conventional fiber 164 00:08:08,920 --> 00:08:11,230 optics cables are transparent. 165 00:08:11,230 --> 00:08:15,040 And so a lot of effort in trying to engineer alloys 166 00:08:15,040 --> 00:08:19,450 or new materials that go towards that soft spot in which you 167 00:08:19,450 --> 00:08:21,820 can grow things that seem more thick on silicon. 168 00:08:21,820 --> 00:08:24,340 So you can use all your silicon technology, 169 00:08:24,340 --> 00:08:27,400 but they have the right direct bandgap 170 00:08:27,400 --> 00:08:32,919 that is appropriate to optical fibers, 1.5 microns. 171 00:08:32,919 --> 00:08:36,669 OK, so from the conceptual or computational point of view, 172 00:08:36,669 --> 00:08:39,370 our problem is really trying to figure out 173 00:08:39,370 --> 00:08:45,176 a way to calculate efficiently the energetics and the phase 174 00:08:45,176 --> 00:08:48,460 ability of a semiconductor alloy. 175 00:08:48,460 --> 00:08:51,420 And I sort of shown you here an example. 176 00:08:51,420 --> 00:08:55,120 Remember, semiconductors have the diamond or the zinc blende 177 00:08:55,120 --> 00:08:56,030 structure. 178 00:08:56,030 --> 00:09:00,520 So they are made by two, say, interpenetrating lattices. 179 00:09:00,520 --> 00:09:03,280 And what I'm showing here is just one 180 00:09:03,280 --> 00:09:08,000 of the two FCC sub-lattices of a typical zinc blende structure. 181 00:09:08,000 --> 00:09:11,170 Say, suppose that you are studying gallium aluminum 182 00:09:11,170 --> 00:09:12,280 arsenide. 183 00:09:12,280 --> 00:09:14,650 Arsenic is the group V semiconductor. 184 00:09:14,650 --> 00:09:16,990 Gallium and aluminum are group III. 185 00:09:16,990 --> 00:09:19,630 So you would have a sub-lattice where 186 00:09:19,630 --> 00:09:21,910 gallium and aluminum sit, and this 187 00:09:21,910 --> 00:09:23,350 is what you are looking at. 188 00:09:23,350 --> 00:09:26,210 It's an FCC sub-lattice-- and where there 189 00:09:26,210 --> 00:09:29,080 can be substitutional disorder. 190 00:09:29,080 --> 00:09:32,140 On each side, there is a certain probability 191 00:09:32,140 --> 00:09:35,230 of finding either a gallium or an aluminum that 192 00:09:35,230 --> 00:09:37,550 is basically proportional to the concentration. 193 00:09:37,550 --> 00:09:39,580 So if you have a 50/50 alloy, you 194 00:09:39,580 --> 00:09:42,610 will have 50% probability of finding gallium 195 00:09:42,610 --> 00:09:43,660 in a certain side. 196 00:09:43,660 --> 00:09:45,580 And then there is a second sub-lattice 197 00:09:45,580 --> 00:09:47,680 that is almost invisible that would 198 00:09:47,680 --> 00:09:51,760 be given by the anion or the arsenic side. 199 00:09:51,760 --> 00:09:55,270 Of course, one could sort of study this system 200 00:09:55,270 --> 00:09:58,480 with a sort of brute force approach. 201 00:09:58,480 --> 00:10:02,050 We could really look at large supercell, 202 00:10:02,050 --> 00:10:05,740 and perform many calculations, and find out 203 00:10:05,740 --> 00:10:11,650 which arrangement of atoms give us lower energy configuration, 204 00:10:11,650 --> 00:10:14,110 figure out what, say, would be the bandgap 205 00:10:14,110 --> 00:10:16,120 or what would be the lattice parameter 206 00:10:16,120 --> 00:10:17,890 of certain configuration, and try 207 00:10:17,890 --> 00:10:20,650 to figure out what are the macroscopic properties. 208 00:10:20,650 --> 00:10:24,610 But statistics at this stage will always be limited. 209 00:10:24,610 --> 00:10:28,070 If you want to study a supercell that is large enough, 210 00:10:28,070 --> 00:10:30,760 you probably need, again, a cell in which 211 00:10:30,760 --> 00:10:34,060 the correlation between faraway atoms is lost, 212 00:10:34,060 --> 00:10:37,540 and you might easily need something like 100 atoms. 213 00:10:37,540 --> 00:10:39,340 But then in particular, you need a lot 214 00:10:39,340 --> 00:10:44,800 of statistics that are true to the 100 ways of arranging atoms 215 00:10:44,800 --> 00:10:46,210 on that lattice. 216 00:10:46,210 --> 00:10:49,510 And you can only calculate a few of these. 217 00:10:49,510 --> 00:10:53,950 And so you become very limited in analyzing the thermodynamics 218 00:10:53,950 --> 00:10:55,180 of your problem. 219 00:10:55,180 --> 00:10:58,060 At the end, what you really care about 220 00:10:58,060 --> 00:11:03,370 is understanding what is the energy of a configuration. 221 00:11:03,370 --> 00:11:06,670 That is, what is the energy as a function of-- you have seen 222 00:11:06,670 --> 00:11:07,480 this before-- 223 00:11:07,480 --> 00:11:09,610 configuration of variable sigma. 224 00:11:09,610 --> 00:11:13,780 We use the nomenclature in which sigma is equal to 1, 225 00:11:13,780 --> 00:11:16,000 if, say, we have a gallium atom here. 226 00:11:16,000 --> 00:11:17,710 This example specifically is actually 227 00:11:17,710 --> 00:11:20,860 for gallium indium phosphide, so we'll have, 228 00:11:20,860 --> 00:11:22,780 again, group III indium. 229 00:11:22,780 --> 00:11:24,520 And sigma equal to minus 1. 230 00:11:24,520 --> 00:11:26,620 That means that there is indium on one side. 231 00:11:26,620 --> 00:11:30,370 This will make actually the connection with the Ising model 232 00:11:30,370 --> 00:11:32,500 very, very apparent. 233 00:11:32,500 --> 00:11:36,100 And so what you want to do is calculate the energetics 234 00:11:36,100 --> 00:11:38,810 of a lot of configuration and then extract 235 00:11:38,810 --> 00:11:40,780 the thermodynamic properties. 236 00:11:40,780 --> 00:11:43,293 And you have seen already in Professor 237 00:11:43,293 --> 00:11:44,710 [? Seder's ?] lectures, and you'll 238 00:11:44,710 --> 00:11:47,620 see again partially in this lecture and the next lecture 239 00:11:47,620 --> 00:11:50,230 how you can use actually Monte Carlo approaches 240 00:11:50,230 --> 00:11:55,150 to extract basically what are the average ensemble 241 00:11:55,150 --> 00:11:55,990 properties. 242 00:11:55,990 --> 00:11:58,630 But our first goal for us is actually 243 00:11:58,630 --> 00:12:02,710 finding out an inexpensive way to calculate 244 00:12:02,710 --> 00:12:06,760 the energy of an arbitrary configuration of atoms, 245 00:12:06,760 --> 00:12:10,060 without having to do the explicit first principle 246 00:12:10,060 --> 00:12:10,960 calculation. 247 00:12:10,960 --> 00:12:13,480 And this is where the fundamental concept 248 00:12:13,480 --> 00:12:15,520 that you'll see in this lecture comes about. 249 00:12:15,520 --> 00:12:20,390 That is, how we actually devise from the initial calculation, 250 00:12:20,390 --> 00:12:24,250 how we extract from the initial calculation, the parameters 251 00:12:24,250 --> 00:12:27,010 for the model Hamiltonian that describes 252 00:12:27,010 --> 00:12:31,330 the energetic of my system for any arbitrary configuration. 253 00:12:31,330 --> 00:12:34,390 And as usual, you need to put ingenuity 254 00:12:34,390 --> 00:12:36,640 in solving a difficult problem. 255 00:12:36,640 --> 00:12:38,860 And this is one of the possible solution. 256 00:12:38,860 --> 00:12:41,740 You'll actually see some of different alternative 257 00:12:41,740 --> 00:12:43,910 approaches in one of the next lecture. 258 00:12:43,910 --> 00:12:46,450 But what we are doing here is, we are actually 259 00:12:46,450 --> 00:12:50,290 sort of exploiting the fact that, say, if you are looking 260 00:12:50,290 --> 00:12:53,950 at a substitutional alloy, you have substitutions 261 00:12:53,950 --> 00:12:57,670 on a sub-lattice between atoms that are really chemically very 262 00:12:57,670 --> 00:12:58,510 similar. 263 00:12:58,510 --> 00:13:01,390 If you are studying, say, gallium indium phosphide, 264 00:13:01,390 --> 00:13:04,600 gallium and indium are going to be chemically very similar. 265 00:13:04,600 --> 00:13:05,920 They are both group III. 266 00:13:05,920 --> 00:13:09,010 They are just sort of in a slightly different in shape. 267 00:13:09,010 --> 00:13:12,400 And of course, their orbitals will be sort of slightly 268 00:13:12,400 --> 00:13:14,440 different, and they will correspond 269 00:13:14,440 --> 00:13:17,110 to a different period in the periodic table. 270 00:13:17,110 --> 00:13:21,490 But their chemical properties are still fairly similar. 271 00:13:21,490 --> 00:13:24,850 And so for a moment, we could actually think-- 272 00:13:24,850 --> 00:13:27,700 and I'll show you how it works mathematically-- 273 00:13:27,700 --> 00:13:32,580 that a real configuration of a crystal-- 274 00:13:32,580 --> 00:13:35,260 and remember, from the point of view of density function 275 00:13:35,260 --> 00:13:38,230 theory, having a certain arrangement 276 00:13:38,230 --> 00:13:40,810 of red and blue atoms means that there 277 00:13:40,810 --> 00:13:43,630 are very specific pseudo potentials 278 00:13:43,630 --> 00:13:45,940 sitting on each one of the sides. 279 00:13:45,940 --> 00:13:48,370 But one of these arrangements could actually 280 00:13:48,370 --> 00:13:53,290 be sort of considered as a perturbation on what we 281 00:13:53,290 --> 00:13:56,050 call a virtual crystal, a crystal that has 282 00:13:56,050 --> 00:13:59,450 full translational symmetry. 283 00:13:59,450 --> 00:14:01,720 So the sort of idea that I'm trying to give you-- 284 00:14:01,720 --> 00:14:03,670 and we'll see it mathematically in a way-- 285 00:14:03,670 --> 00:14:07,930 is that we could think at the real configuration of a system 286 00:14:07,930 --> 00:14:11,920 as a perturbation in which we take a pseudo atom that 287 00:14:11,920 --> 00:14:14,830 doesn't exist but is somehow very 288 00:14:14,830 --> 00:14:17,890 similar to both gallium and indium. 289 00:14:17,890 --> 00:14:21,700 And then we turn that with a small change 290 00:14:21,700 --> 00:14:25,330 in the pseudo potential in either gallium or indium. 291 00:14:25,330 --> 00:14:27,610 And if we can do that, well, then we 292 00:14:27,610 --> 00:14:31,180 can study this problem with density functional theory, 293 00:14:31,180 --> 00:14:33,130 because this is actually a system that 294 00:14:33,130 --> 00:14:35,050 has two atoms per unit cell. 295 00:14:35,050 --> 00:14:37,900 And then we can try and use perturbation theory 296 00:14:37,900 --> 00:14:42,130 to figure out what would be the energy of our configuration 297 00:14:42,130 --> 00:14:43,240 over there. 298 00:14:43,240 --> 00:14:44,890 And this is actually fairly trivial 299 00:14:44,890 --> 00:14:47,080 to sort of see it in mathematical form, 300 00:14:47,080 --> 00:14:49,900 when you actually think again at the fact 301 00:14:49,900 --> 00:14:51,970 that in density functional theory, 302 00:14:51,970 --> 00:14:54,370 the potential acting on the electron 303 00:14:54,370 --> 00:14:58,420 is really given by this array of pseudo potential 304 00:14:58,420 --> 00:15:01,470 sitting on lattice sides. 305 00:15:01,470 --> 00:15:05,010 And so if we have a real configuration-- remember, 306 00:15:05,010 --> 00:15:07,500 the red and blue crystal on the left-- 307 00:15:07,500 --> 00:15:10,830 well, to that, it will correspond 308 00:15:10,830 --> 00:15:16,140 an external potential that is an array of red and blue pseudo 309 00:15:16,140 --> 00:15:19,980 potential, that can actually be rewritten 310 00:15:19,980 --> 00:15:24,750 in terms of the semi-sum and the semi-difference of pseudo 311 00:15:24,750 --> 00:15:25,990 potential. 312 00:15:25,990 --> 00:15:28,440 So this would be the exact expression. 313 00:15:28,440 --> 00:15:30,690 We have the true pseudo potential 314 00:15:30,690 --> 00:15:35,220 on each sort of FCC sub-lattice site. 315 00:15:35,220 --> 00:15:39,750 But we write, say, gallium pseudo potential 316 00:15:39,750 --> 00:15:47,060 as being given by the sum of the average between the gallium 317 00:15:47,060 --> 00:15:51,140 and the indium pseudo potential, say, plus one 318 00:15:51,140 --> 00:15:52,790 half the semi-difference. 319 00:15:52,790 --> 00:15:55,280 You see, if I'm sitting on a site, 320 00:15:55,280 --> 00:15:58,790 and I'm telling you that the pseudo potential acting 321 00:15:58,790 --> 00:16:02,810 on that side is going to be the sum between the average 322 00:16:02,810 --> 00:16:08,210 between gallium and indium plus one half the semi-difference, 323 00:16:08,210 --> 00:16:12,140 you see that the indium cancels out, and the gallium sums up. 324 00:16:12,140 --> 00:16:15,080 And what you have is the gallium pseudo potential there. 325 00:16:15,080 --> 00:16:20,630 And so a sigma variable equal to plus 1 at the site R 326 00:16:20,630 --> 00:16:25,130 really gives you a gallium pseudo potential in that site. 327 00:16:25,130 --> 00:16:27,830 And a sigma variable equal to minus 1 328 00:16:27,830 --> 00:16:31,230 gives you an indium pseudo potential in that site. 329 00:16:31,230 --> 00:16:35,660 But in this form, and in the limit in which the two pseudo 330 00:16:35,660 --> 00:16:38,240 potential are fairly similar, you 331 00:16:38,240 --> 00:16:41,450 can see that this term is actually small. 332 00:16:41,450 --> 00:16:43,460 The more similar gallium and indium 333 00:16:43,460 --> 00:16:46,580 are, the smaller this system is. 334 00:16:46,580 --> 00:16:50,270 But this term here overall is the only one 335 00:16:50,270 --> 00:16:53,520 where the configurational variables are present. 336 00:16:53,520 --> 00:16:55,760 There are no configurational variable here. 337 00:16:55,760 --> 00:16:59,750 And so this represents, really, the green average virtual 338 00:16:59,750 --> 00:17:02,810 crystal to which at every site, we 339 00:17:02,810 --> 00:17:06,140 add or subtract depending on the sign of sigma 340 00:17:06,140 --> 00:17:09,560 R, the semi-difference of the two pseudo potential. 341 00:17:09,560 --> 00:17:13,790 And this term really turns a green atom 342 00:17:13,790 --> 00:17:15,859 in a red or blue atom. 343 00:17:15,859 --> 00:17:19,160 And for the specific case of semiconductor alloys, 344 00:17:19,160 --> 00:17:22,160 this term is smaller, because disorder 345 00:17:22,160 --> 00:17:26,490 takes place between atoms that are very small. 346 00:17:26,490 --> 00:17:29,540 And so what we are going to do is use density function theory 347 00:17:29,540 --> 00:17:33,350 to study this crystal then use perturbation theory 348 00:17:33,350 --> 00:17:37,490 to understand what is going on when we do this perturbation. 349 00:17:37,490 --> 00:17:40,100 And so this is also sort of a chance 350 00:17:40,100 --> 00:17:44,570 for me to at least hint at how we actually 351 00:17:44,570 --> 00:17:49,220 perform perturbation theory in density functional theory 352 00:17:49,220 --> 00:17:50,250 itself. 353 00:17:50,250 --> 00:17:51,920 And that's a very useful tool. 354 00:17:51,920 --> 00:17:54,170 Say, it's used, and you'll see it, 355 00:17:54,170 --> 00:17:57,260 and you have seen it before, to calculate things 356 00:17:57,260 --> 00:17:58,790 like phonon dispersions. 357 00:17:58,790 --> 00:18:02,180 That is, what is the force or what 358 00:18:02,180 --> 00:18:06,680 is the frequency of perturbation in the atomic position 359 00:18:06,680 --> 00:18:08,570 with a certain wavelength that is really 360 00:18:08,570 --> 00:18:10,310 the normal mode of the crystal. 361 00:18:10,310 --> 00:18:13,860 Or it can be used in a variety of problems, 362 00:18:13,860 --> 00:18:16,730 say, to study what is the response of a system 363 00:18:16,730 --> 00:18:18,980 to an electric field and then calculate 364 00:18:18,980 --> 00:18:20,750 Raman or infrared spectra. 365 00:18:20,750 --> 00:18:23,270 It can be used to calculate the response of a system 366 00:18:23,270 --> 00:18:26,930 to a magnetic field and then calculate the NMR spectra, 367 00:18:26,930 --> 00:18:30,740 or it can even be used to calculate higher order 368 00:18:30,740 --> 00:18:32,360 important physical quantities. 369 00:18:32,360 --> 00:18:35,750 If we go to sort of higher order response, 370 00:18:35,750 --> 00:18:38,540 we can start to calculate what is the third order 371 00:18:38,540 --> 00:18:40,910 Hamiltonian in the phonons, so what are the-- 372 00:18:40,910 --> 00:18:43,910 in the phonons, what are the lifetime of phonons. 373 00:18:43,910 --> 00:18:49,430 Or we can calculate how an electronic state couples 374 00:18:49,430 --> 00:18:51,560 with a phonon state. 375 00:18:51,560 --> 00:18:53,990 And so we can actually get out the parameters, 376 00:18:53,990 --> 00:18:55,790 say, for superconductivity. 377 00:18:55,790 --> 00:18:59,600 That is, when electrons travel through a crystal paired 378 00:18:59,600 --> 00:19:03,020 because of the interaction with the phonon degrees of freedom. 379 00:19:03,020 --> 00:19:06,890 Or we could calculate, say, resonant Raman scattering 380 00:19:06,890 --> 00:19:10,500 that, again, is sort of a strong electron-phonon component. 381 00:19:10,500 --> 00:19:13,100 But in general, the problem is just this. 382 00:19:13,100 --> 00:19:15,980 Perturbation theory looks at what 383 00:19:15,980 --> 00:19:20,840 happens to a certain system that has an external potential v0 384 00:19:20,840 --> 00:19:23,420 when you add a perturbation. 385 00:19:23,420 --> 00:19:26,630 And the perturbation is sort of parameterized 386 00:19:26,630 --> 00:19:28,670 by a certain strength lambda. 387 00:19:28,670 --> 00:19:32,400 So for small lambda, the perturbation is weak. 388 00:19:32,400 --> 00:19:35,690 So we are sort of thinking at the limit of what happens 389 00:19:35,690 --> 00:19:37,830 when the perturbation is weak. 390 00:19:37,830 --> 00:19:43,310 So you add a lambda delta v term to your external potential. 391 00:19:43,310 --> 00:19:46,730 And in this specific case, that delta v term 392 00:19:46,730 --> 00:19:48,440 is going to be that semi-difference 393 00:19:48,440 --> 00:19:50,220 between the pseudo potential. 394 00:19:50,220 --> 00:19:53,090 So it's the term that really transform a virtual atom 395 00:19:53,090 --> 00:19:55,790 in a real gallium and in a real indium. 396 00:19:55,790 --> 00:20:00,380 What happens when you apply this small perturbation is that, 397 00:20:00,380 --> 00:20:05,330 of course, your new self-consistent charge density 398 00:20:05,330 --> 00:20:06,690 will change. 399 00:20:06,690 --> 00:20:10,220 So if, for a given external potential v0, 400 00:20:10,220 --> 00:20:17,280 you add a ground state charge density n0 for a perturbation 401 00:20:17,280 --> 00:20:19,910 in which you add the term lambda delta v, 402 00:20:19,910 --> 00:20:22,370 your new ground state charge density 403 00:20:22,370 --> 00:20:25,430 is going to be a certain charge density that 404 00:20:25,430 --> 00:20:28,160 will define us n lambda. 405 00:20:28,160 --> 00:20:31,070 But we can think of this n lambda 406 00:20:31,070 --> 00:20:38,310 as being sort of expanded in a power series of power lambdas. 407 00:20:38,310 --> 00:20:42,020 So the new n lambda is going to be given by the original ground 408 00:20:42,020 --> 00:20:44,580 state charge density plus a term that we 409 00:20:44,580 --> 00:20:49,350 call the linear response that is a first order in lambda, 410 00:20:49,350 --> 00:20:52,030 plus a term that is a second-order response, 411 00:20:52,030 --> 00:20:54,870 plus a term that is third order and fourth order 412 00:20:54,870 --> 00:20:56,470 and so on and so forth. 413 00:20:56,470 --> 00:21:00,690 And the smaller the lambda is, the less relevant 414 00:21:00,690 --> 00:21:02,880 will be the higher order terms. 415 00:21:02,880 --> 00:21:07,650 And what is really important is just this first order response 416 00:21:07,650 --> 00:21:09,640 in the charge density. 417 00:21:09,640 --> 00:21:13,650 And this actually sort of gives us a lot of power. 418 00:21:13,650 --> 00:21:18,090 That is, just knowing the first order response in the charge 419 00:21:18,090 --> 00:21:23,190 density allows us to calculate the energy of a system 420 00:21:23,190 --> 00:21:25,230 in the presence of this perturbation 421 00:21:25,230 --> 00:21:27,390 up to the second order. 422 00:21:27,390 --> 00:21:30,870 And this basically can be seen by using 423 00:21:30,870 --> 00:21:34,230 Hellmann-Feynman theorem here. 424 00:21:34,230 --> 00:21:38,160 This is-- you have seen it sort of when we discussed forces, 425 00:21:38,160 --> 00:21:41,820 and it's really a sort of direct consequence of the ground state 426 00:21:41,820 --> 00:21:45,120 properties of the orbitals in the Schrodinger equation. 427 00:21:45,120 --> 00:21:47,550 But basically, what does Hellmann-Feynman theorem 428 00:21:47,550 --> 00:21:48,330 tells us? 429 00:21:48,330 --> 00:21:49,920 Well, it tells us that if we need 430 00:21:49,920 --> 00:21:53,370 to look at the change in energy with respect 431 00:21:53,370 --> 00:21:56,070 to a certain perturbation, the only thing 432 00:21:56,070 --> 00:21:58,320 that we need to calculate is actually 433 00:21:58,320 --> 00:22:05,720 the expectation value of the orbitals with respect 434 00:22:05,720 --> 00:22:07,000 to the perturbation. 435 00:22:07,000 --> 00:22:10,220 So this is really Hellmann-Feynman theorem. 436 00:22:10,220 --> 00:22:11,240 You might remember this. 437 00:22:11,240 --> 00:22:12,260 We have discussed it. 438 00:22:12,260 --> 00:22:15,350 We don't need to take into account the derivative 439 00:22:15,350 --> 00:22:16,460 of, say, lambda-- 440 00:22:16,460 --> 00:22:20,630 sorry, I'm using too many lambdas here. 441 00:22:20,630 --> 00:22:23,900 But let's call it lambda again here. 442 00:22:23,900 --> 00:22:27,620 We don't need to take into account the changes in the wave 443 00:22:27,620 --> 00:22:28,970 functions themselves. 444 00:22:28,970 --> 00:22:32,010 We just calculate the expectation value. 445 00:22:32,010 --> 00:22:34,700 So for any given arbitrary lambda, 446 00:22:34,700 --> 00:22:36,915 the derivative of the energy, the change 447 00:22:36,915 --> 00:22:38,600 in the energy with respect to lambda, 448 00:22:38,600 --> 00:22:41,540 is just given by this simple integral. 449 00:22:41,540 --> 00:22:47,180 And so you see that if we then say that we actually 450 00:22:47,180 --> 00:22:53,180 expand the energy up to the first-order term, what we have 451 00:22:53,180 --> 00:22:58,835 is that the energy is equal to e0 plus lambda d 452 00:22:58,835 --> 00:23:00,950 e over d lambda. 453 00:23:00,950 --> 00:23:04,460 And then there would be a one half lambda square term. 454 00:23:04,460 --> 00:23:07,045 But then in itself, we can expand-- 455 00:23:07,045 --> 00:23:08,750 it's a little bit father, so I'm sort 456 00:23:08,750 --> 00:23:12,470 of skipping some of the passages-- we can expand this n 457 00:23:12,470 --> 00:23:15,320 lambda in powers of lambda. 458 00:23:15,320 --> 00:23:19,250 And so what you see is that up to the second order 459 00:23:19,250 --> 00:23:22,280 in lambda, the only terms that are 460 00:23:22,280 --> 00:23:24,740 relevant in the expansion of your energy-- 461 00:23:24,740 --> 00:23:27,140 and this is what is important-- 462 00:23:27,140 --> 00:23:31,130 are, for the first order term, just the integral 463 00:23:31,130 --> 00:23:34,550 of the perturbation with respect to the ground state's charge 464 00:23:34,550 --> 00:23:37,430 density, and for the second-order term, 465 00:23:37,430 --> 00:23:40,580 the integral of the perturbation with the first order 466 00:23:40,580 --> 00:23:42,830 change in the charge density. 467 00:23:42,830 --> 00:23:47,190 So perturbation induces a change. 468 00:23:47,190 --> 00:23:51,440 And if we are able to calculate the linear response to that, 469 00:23:51,440 --> 00:23:54,260 this n1, we actually get correction 470 00:23:54,260 --> 00:23:57,860 to the energy up to the second-order term. 471 00:23:57,860 --> 00:24:00,900 And so there is a whole branch of density functional theory 472 00:24:00,900 --> 00:24:04,190 that's called density functional perturbation theory that 473 00:24:04,190 --> 00:24:10,700 really deals on how to calculate this linear order perturbation. 474 00:24:10,700 --> 00:24:14,240 And I really won't go into the detail. 475 00:24:14,240 --> 00:24:16,460 The sort of other difficulty with respect 476 00:24:16,460 --> 00:24:19,790 to a standard set of quantum mechanical perturbation theory 477 00:24:19,790 --> 00:24:21,590 is that in density functional theory, 478 00:24:21,590 --> 00:24:24,980 the Hamiltonian is in itself self-consistent. 479 00:24:24,980 --> 00:24:27,320 And so you have an added layer of complexity, 480 00:24:27,320 --> 00:24:30,780 because it depends from the charge density itself. 481 00:24:30,780 --> 00:24:33,020 And so when you have got a perturbation 482 00:24:33,020 --> 00:24:35,750 in your external potential, the perturbation 483 00:24:35,750 --> 00:24:39,740 in your Kohn-Sham potential contains also term 484 00:24:39,740 --> 00:24:43,130 in the Hartree and in the exchange correlation term. 485 00:24:43,130 --> 00:24:45,380 So we won't really go into these details, 486 00:24:45,380 --> 00:24:48,380 but there is, again, an iterative self-consistent 487 00:24:48,380 --> 00:24:52,520 procedure to obtain the self-consistent linear order 488 00:24:52,520 --> 00:24:54,920 response to an external perturbation. 489 00:24:54,920 --> 00:24:59,100 And all of this is actually implemented in the PWSCF code. 490 00:24:59,100 --> 00:25:01,250 So some of you might at certain point 491 00:25:01,250 --> 00:25:04,010 need to use that, again, to calculate quantities 492 00:25:04,010 --> 00:25:05,930 like the phonon dispersions. 493 00:25:05,930 --> 00:25:08,450 And there are actually examples in the distribution 494 00:25:08,450 --> 00:25:10,920 that tell you how to do that. 495 00:25:10,920 --> 00:25:15,350 And so if we go back to our original problem, all of this 496 00:25:15,350 --> 00:25:17,810 was actually to convince you that if we 497 00:25:17,810 --> 00:25:20,750 are able to calculate this linear response, 498 00:25:20,750 --> 00:25:24,260 the first order change in the charge density induced 499 00:25:24,260 --> 00:25:26,720 by a certain perturbation, we actually 500 00:25:26,720 --> 00:25:31,400 have the energy of a system up to the second order 501 00:25:31,400 --> 00:25:34,520 in the perturbation, where the first order term, again, 502 00:25:34,520 --> 00:25:37,490 is just given by the integral of the ground state charge 503 00:25:37,490 --> 00:25:39,560 density in the perturbation. 504 00:25:39,560 --> 00:25:41,390 And the second order term is given 505 00:25:41,390 --> 00:25:45,020 by the integral of the linear response 506 00:25:45,020 --> 00:25:46,850 times the perturbation. 507 00:25:46,850 --> 00:25:50,240 But you see, what is now the fundamental step 508 00:25:50,240 --> 00:25:53,570 is that these quantities don't depend 509 00:25:53,570 --> 00:25:56,330 on the configurational variables. 510 00:25:56,330 --> 00:25:59,780 These are quantities that are calculated on the ground 511 00:25:59,780 --> 00:26:02,180 state-- that is, the green crystal-- 512 00:26:02,180 --> 00:26:05,150 something that doesn't depend on configuration. 513 00:26:05,150 --> 00:26:09,890 So this can be calculated once and for all just 514 00:26:09,890 --> 00:26:13,920 on the system that contains two atoms per unit cell. 515 00:26:13,920 --> 00:26:16,790 And once you have calculated this, 516 00:26:16,790 --> 00:26:20,210 this will give you the energy up to the second order 517 00:26:20,210 --> 00:26:22,470 in the configurational variables. 518 00:26:22,470 --> 00:26:25,100 So doing this calculation will take some time, 519 00:26:25,100 --> 00:26:27,620 but it's really on a two-atom unit cell. 520 00:26:27,620 --> 00:26:30,350 But once you have this, what you have 521 00:26:30,350 --> 00:26:35,420 is really a analytical and extremely inexpensive 522 00:26:35,420 --> 00:26:39,200 expression to calculate the energy up to the second order. 523 00:26:39,200 --> 00:26:43,730 So as long as your disorder is weak, 524 00:26:43,730 --> 00:26:46,760 you will get a very good approximation of the energy 525 00:26:46,760 --> 00:26:50,180 just by doing this set of sums that a computer will 526 00:26:50,180 --> 00:26:55,140 take probably a nanosecond or something like this. 527 00:26:55,140 --> 00:26:59,720 And again, this really looks like an Ising model 528 00:26:59,720 --> 00:27:00,980 Hamiltonian. 529 00:27:00,980 --> 00:27:03,600 If you remember what you had in the Ising model-- 530 00:27:03,600 --> 00:27:09,240 is that you had a sort of coupling between nearby spins 531 00:27:09,240 --> 00:27:13,290 that was parameterized by a certain exchange parameter 532 00:27:13,290 --> 00:27:14,410 here. 533 00:27:14,410 --> 00:27:16,790 The only difference is that, being these 534 00:27:16,790 --> 00:27:20,640 are semiconductor alloys, interactions 535 00:27:20,640 --> 00:27:22,450 tend to be long range. 536 00:27:22,450 --> 00:27:28,350 And so it's not only atoms or sigma variables sitting nearby 537 00:27:28,350 --> 00:27:31,830 that sort of interact, but the interaction sort of 538 00:27:31,830 --> 00:27:34,930 goes on farther and farther. 539 00:27:34,930 --> 00:27:39,060 And so it's a sort of long range Ising Hamiltonian. 540 00:27:39,060 --> 00:27:43,440 But once you have this, you can do all the thermodynamics 541 00:27:43,440 --> 00:27:47,420 that you want to your problem and very inexpensively. 542 00:27:50,440 --> 00:27:55,360 I'll sort of go very quickly through this. 543 00:27:55,360 --> 00:28:00,850 But again, what we have seen up to now is really this sort 544 00:28:00,850 --> 00:28:04,750 of fundamental concept that the electronic structure 545 00:28:04,750 --> 00:28:08,620 calculation or our energy calculation allows us 546 00:28:08,620 --> 00:28:11,060 to extract parameters-- in this case, 547 00:28:11,060 --> 00:28:13,960 the J interaction constants-- 548 00:28:13,960 --> 00:28:16,720 that gives us the energy of our system. 549 00:28:16,720 --> 00:28:19,450 And as usual, when you really need 550 00:28:19,450 --> 00:28:22,700 to find out what are the parameters to extract, 551 00:28:22,700 --> 00:28:25,540 you need to think at what are your relevant degrees 552 00:28:25,540 --> 00:28:26,190 of freedom. 553 00:28:26,190 --> 00:28:29,210 So you need to think at the physics of your problem. 554 00:28:29,210 --> 00:28:31,900 And if you have a semiconductor alloy, 555 00:28:31,900 --> 00:28:34,840 the first degree of freedom that comes to mind correctly 556 00:28:34,840 --> 00:28:36,940 is the configurational degree of freedom. 557 00:28:36,940 --> 00:28:41,530 That is, we can have on a certain site one atom, gallium, 558 00:28:41,530 --> 00:28:43,510 or another atom, indium. 559 00:28:43,510 --> 00:28:47,830 But actually, it's not the only degrees of freedom that is 560 00:28:47,830 --> 00:28:53,410 relevant, because depending on which atom you put on a certain 561 00:28:53,410 --> 00:28:57,550 side, what you have is that, depending on also what is 562 00:28:57,550 --> 00:29:01,540 the surrounding environment-- if it's a gallium between gallium 563 00:29:01,540 --> 00:29:04,720 atoms or if it's a gallium between indium atoms-- 564 00:29:04,720 --> 00:29:07,690 that atom will see a broken symmetry. 565 00:29:07,690 --> 00:29:12,370 And so it will want to relax from the ideal sort 566 00:29:12,370 --> 00:29:14,710 of periodic Bravais lattice position 567 00:29:14,710 --> 00:29:17,860 and will likely like to move around, 568 00:29:17,860 --> 00:29:20,770 either to make space for larger atoms that 569 00:29:20,770 --> 00:29:24,460 are nearby or sort of to make space for itself if there 570 00:29:24,460 --> 00:29:26,210 are smaller atoms. 571 00:29:26,210 --> 00:29:30,340 And so this is sort of where intuition becomes important. 572 00:29:30,340 --> 00:29:34,690 And so you need to decide what are the important degrees 573 00:29:34,690 --> 00:29:35,510 of freedom. 574 00:29:35,510 --> 00:29:37,120 And in this case, what we are saying 575 00:29:37,120 --> 00:29:40,990 is that there are not only configurational variables that 576 00:29:40,990 --> 00:29:42,760 are important degrees of freedom, 577 00:29:42,760 --> 00:29:47,710 but there are also displacements of the atoms in response 578 00:29:47,710 --> 00:29:51,070 to a certain configuration away from equilibrium. 579 00:29:51,070 --> 00:29:53,080 And so once you have identified that there 580 00:29:53,080 --> 00:29:55,880 is this second set of degrees of freedom, 581 00:29:55,880 --> 00:29:58,420 well, what you are really saying is that we really 582 00:29:58,420 --> 00:30:02,800 need to consider, in the expansion of the energy 583 00:30:02,800 --> 00:30:05,020 over all my degrees of freedom, not 584 00:30:05,020 --> 00:30:07,990 only the configurational variables but also 585 00:30:07,990 --> 00:30:11,020 the displacement of an atom, say, 586 00:30:11,020 --> 00:30:13,130 from its equilibrium position. 587 00:30:13,130 --> 00:30:16,450 And so we need to expand their energy up 588 00:30:16,450 --> 00:30:20,770 to the second order, not only in configuration but also 589 00:30:20,770 --> 00:30:22,600 in displacements. 590 00:30:22,600 --> 00:30:25,540 And because we are expanding up to the second order 591 00:30:25,540 --> 00:30:27,910 in configuration and in displacements, 592 00:30:27,910 --> 00:30:31,000 there is going to be also a mixed term that 593 00:30:31,000 --> 00:30:34,210 is half configuration and half displacement. 594 00:30:34,210 --> 00:30:37,900 If you want this term, think of F 595 00:30:37,900 --> 00:30:41,260 as being the second derivative of the energy with respect 596 00:30:41,260 --> 00:30:43,810 to u and with respect towards sigma. 597 00:30:43,810 --> 00:30:46,780 So this term here is telling us what 598 00:30:46,780 --> 00:30:51,910 is the force that is induced on the atom R 599 00:30:51,910 --> 00:30:57,190 when, on the side R prime, there is a certain atom given 600 00:30:57,190 --> 00:31:01,060 by the configurational variable sigma. 601 00:31:01,060 --> 00:31:05,200 So we really need to expand in all our degrees of freedom. 602 00:31:05,200 --> 00:31:08,300 But then we can do something more. 603 00:31:08,300 --> 00:31:12,190 And we can even sort of apply an equilibrium condition. 604 00:31:12,190 --> 00:31:15,580 Because at the end, what we want to do for the system is not 605 00:31:15,580 --> 00:31:19,000 really the dynamics, but what we really want to do 606 00:31:19,000 --> 00:31:23,060 is the thermodynamics of the different configurations. 607 00:31:23,060 --> 00:31:25,090 So what we are saying is that, well, 608 00:31:25,090 --> 00:31:29,300 suppose that we have chosen a certain configuration of atoms. 609 00:31:29,300 --> 00:31:31,840 Now, what we really need to take into account 610 00:31:31,840 --> 00:31:35,110 is not only how the energy changes but also 611 00:31:35,110 --> 00:31:39,250 how the atom moves around to accommodate the configuration 612 00:31:39,250 --> 00:31:40,840 as best as possible. 613 00:31:40,840 --> 00:31:46,150 But that really means that the atoms will move in order 614 00:31:46,150 --> 00:31:49,270 to minimize the energy, or if you want, 615 00:31:49,270 --> 00:31:52,960 in order to have really zero force at a certain point. 616 00:31:52,960 --> 00:31:55,150 If you start from the periodic lattice 617 00:31:55,150 --> 00:31:58,810 and you sort of decide a certain configuration, on those sites, 618 00:31:58,810 --> 00:31:59,950 there will be forces. 619 00:31:59,950 --> 00:32:02,870 Atoms move a little bit away from that site, 620 00:32:02,870 --> 00:32:04,570 so that their force is 0. 621 00:32:04,570 --> 00:32:07,810 So what we have is here the sort of physical embodiment 622 00:32:07,810 --> 00:32:09,310 of the degrees of freedom. 623 00:32:09,310 --> 00:32:12,010 The energy is expanded up to the second order. 624 00:32:12,010 --> 00:32:14,860 But then we also apply an equilibrium condition. 625 00:32:14,860 --> 00:32:18,940 That is, the force on the ions for a given configuration sigma 626 00:32:18,940 --> 00:32:20,050 must be 0. 627 00:32:20,050 --> 00:32:22,390 So the derivative, the first order derivative 628 00:32:22,390 --> 00:32:25,690 of the energy with displacement, that is the force, must be 0. 629 00:32:25,690 --> 00:32:29,560 And if you apply that at this and you work the algebra, 630 00:32:29,560 --> 00:32:33,580 you really obtain, for a given set 631 00:32:33,580 --> 00:32:37,780 of configurational variable, what are the displacements away 632 00:32:37,780 --> 00:32:39,850 from equilibrium that are induced 633 00:32:39,850 --> 00:32:42,020 by this configurational variable. 634 00:32:42,020 --> 00:32:46,600 And of course, they are mediated by this response function 635 00:32:46,600 --> 00:32:50,830 of our system, the force on an atom induced 636 00:32:50,830 --> 00:32:54,400 by our configurational variable, and really, 637 00:32:54,400 --> 00:32:57,010 the second-order derivative, the stiffness 638 00:32:57,010 --> 00:32:59,710 of our potential energy surface with respect 639 00:32:59,710 --> 00:33:01,420 to atomic displacements. 640 00:33:01,420 --> 00:33:04,090 And so we use this equilibrium condition. 641 00:33:04,090 --> 00:33:06,190 That is, for a given configuration, 642 00:33:06,190 --> 00:33:08,290 the displacement are really given. 643 00:33:08,290 --> 00:33:11,680 And we take it, and we stick it back inside. 644 00:33:11,680 --> 00:33:17,920 So we substitute, for the green u of R, this expression here. 645 00:33:17,920 --> 00:33:20,440 Those are all sort of tensorial quantities 646 00:33:20,440 --> 00:33:23,440 which I sort of removed the sums. 647 00:33:23,440 --> 00:33:30,280 But so we are back to our spin Ising Hamiltonian 648 00:33:30,280 --> 00:33:32,770 in which the only thing that is relevant 649 00:33:32,770 --> 00:33:34,570 is the configurational variables, 650 00:33:34,570 --> 00:33:37,210 because at the end, that's the only thing that can happen. 651 00:33:37,210 --> 00:33:39,890 We can have a certain atom in a certain place, 652 00:33:39,890 --> 00:33:45,070 but we take into account the fact that the ions sort of 653 00:33:45,070 --> 00:33:47,080 go back, move to equilibrium. 654 00:33:47,080 --> 00:33:51,130 And we take that into account by a renormalization 655 00:33:51,130 --> 00:33:55,090 of the interaction constant j by, if you want, 656 00:33:55,090 --> 00:34:01,690 the displacement term or what we would call the elastic terms. 657 00:34:01,690 --> 00:34:09,489 OK, so we have all the tools to calculate this regional j, this 658 00:34:09,489 --> 00:34:12,310 that would be nothing else than the dynamical force 659 00:34:12,310 --> 00:34:14,500 constant matrix that you diagonalize 660 00:34:14,500 --> 00:34:17,350 in your final calculation or this mixed term that 661 00:34:17,350 --> 00:34:19,840 in the literature are called Kanzaki forces. 662 00:34:19,840 --> 00:34:21,800 We do this on the virtual crystal. 663 00:34:21,800 --> 00:34:25,510 And then we really have our spin Hamiltonian, 664 00:34:25,510 --> 00:34:32,139 including the energetics effects of atomic displacements. 665 00:34:32,139 --> 00:34:36,880 And so we have really the Hamiltonian of our system, 666 00:34:36,880 --> 00:34:40,300 and we can start doing the thermodynamics of it, 667 00:34:40,300 --> 00:34:45,130 with a small caveat that what we have done up to now 668 00:34:45,130 --> 00:34:46,449 is really this step. 669 00:34:46,449 --> 00:34:48,070 We have been able to calculate, if you 670 00:34:48,070 --> 00:34:52,929 want, the energy of a certain configuration, 671 00:34:52,929 --> 00:34:55,179 starting from a virtual crystal. 672 00:34:55,179 --> 00:34:57,880 But we haven't really-- and this is a subtlety that I won't go 673 00:34:57,880 --> 00:34:58,810 into the detail-- 674 00:34:58,810 --> 00:35:00,580 we haven't really taken into account 675 00:35:00,580 --> 00:35:04,390 that our sort of end original products 676 00:35:04,390 --> 00:35:07,010 have actually different lattice parameters. 677 00:35:07,010 --> 00:35:09,550 So in itself, at the end, if we need 678 00:35:09,550 --> 00:35:11,590 to understand phased ability, we need 679 00:35:11,590 --> 00:35:16,120 to understand what is the lower Gibbs 680 00:35:16,120 --> 00:35:18,340 free energy for our problem, that is, 681 00:35:18,340 --> 00:35:21,010 if entropy drives this configuration 682 00:35:21,010 --> 00:35:24,610 to be stable, or, if you want, if enthalpy and elastic 683 00:35:24,610 --> 00:35:27,250 interaction drive segregation in two 684 00:35:27,250 --> 00:35:30,310 compounds with different lattice parameters 685 00:35:30,310 --> 00:35:31,990 to be the stable state. 686 00:35:31,990 --> 00:35:36,130 And of course, the closer the lattice parameters 687 00:35:36,130 --> 00:35:40,090 are, the more irrelevant this term is going to be 688 00:35:40,090 --> 00:35:42,370 and the more relevant term in which 689 00:35:42,370 --> 00:35:44,170 the disorder atoms will be. 690 00:35:44,170 --> 00:35:47,930 And temperature is always going to favor this. 691 00:35:47,930 --> 00:35:50,470 But this might become very expensive 692 00:35:50,470 --> 00:35:53,500 if the original compounds have very different lattice 693 00:35:53,500 --> 00:35:54,940 parameters, because we really need 694 00:35:54,940 --> 00:35:57,940 to stretch things to start having 695 00:35:57,940 --> 00:36:00,640 the possibility of mixing them together. 696 00:36:00,640 --> 00:36:04,000 And all sort of-- the phase diagonal stability 697 00:36:04,000 --> 00:36:08,530 of this system is really driven by these two competing sort 698 00:36:08,530 --> 00:36:11,470 of forces, the temperature that want 699 00:36:11,470 --> 00:36:15,100 to mix up things around but chemistry 700 00:36:15,100 --> 00:36:17,140 that, in order to mix things around, 701 00:36:17,140 --> 00:36:19,930 you need to have atoms that are chemically very similar. 702 00:36:19,930 --> 00:36:22,000 And you need to have lattice parameters that 703 00:36:22,000 --> 00:36:25,960 are fairly similar, so that you don't build up a lot of strain. 704 00:36:28,600 --> 00:36:31,225 And so, again, without going into detail, 705 00:36:31,225 --> 00:36:33,440 it is actually very easy to calculate 706 00:36:33,440 --> 00:36:36,100 some of those elastic terms. 707 00:36:36,100 --> 00:36:39,820 And so when we do that, what we really have is a way 708 00:36:39,820 --> 00:36:43,660 to calculate how much energy it takes to stretch things so they 709 00:36:43,660 --> 00:36:48,250 have the same sort of unit cell and then how much-- 710 00:36:48,250 --> 00:36:50,290 and that was the tricky part-- 711 00:36:50,290 --> 00:36:52,870 it cost to mix things around, it cost 712 00:36:52,870 --> 00:36:55,750 to have a certain configuration. 713 00:36:55,750 --> 00:36:57,610 And the very nice thing is that now, we 714 00:36:57,610 --> 00:37:01,540 can sort of test our perturbation theory approach 715 00:37:01,540 --> 00:37:05,620 and compare it with full blown density functional theory 716 00:37:05,620 --> 00:37:08,170 calculations for cells that are small 717 00:37:08,170 --> 00:37:11,410 enough so that you can actually do a density functional theory 718 00:37:11,410 --> 00:37:12,220 calculation. 719 00:37:12,220 --> 00:37:13,750 These are older results. 720 00:37:13,750 --> 00:37:15,850 Now we could do much larger cells. 721 00:37:15,850 --> 00:37:18,160 But the concept stands. 722 00:37:18,160 --> 00:37:22,060 We can do a full self-consistent calculation, say, 723 00:37:22,060 --> 00:37:26,050 for a super lattice of gallium planes and indium 724 00:37:26,050 --> 00:37:30,360 planes in the 111 direction. 725 00:37:30,360 --> 00:37:33,020 And so we can calculate what is the lattice 726 00:37:33,020 --> 00:37:35,630 parameter of this structure and what 727 00:37:35,630 --> 00:37:38,000 is actually the sort of Gibbs free energy 728 00:37:38,000 --> 00:37:41,310 of mixing for this system. 729 00:37:41,310 --> 00:37:45,530 And we can do that with a full explicit calculation, 730 00:37:45,530 --> 00:37:50,090 and we can do it with a set of perturbation approach. 731 00:37:50,090 --> 00:37:53,330 And you see, we can compare the equilibrium lattice parameters 732 00:37:53,330 --> 00:37:54,410 in both cases. 733 00:37:54,410 --> 00:37:56,810 And basically, the difference-- this is atomic units. 734 00:37:56,810 --> 00:37:59,900 This is PWSCF-- is basically negligible. 735 00:37:59,900 --> 00:38:03,200 And the energies are also very good. 736 00:38:03,200 --> 00:38:06,560 This sort of error that the perturbation theory makes 737 00:38:06,560 --> 00:38:11,000 is actually here, just 1 millielectron volt per atom. 738 00:38:11,000 --> 00:38:12,740 And as you sort of move around, you 739 00:38:12,740 --> 00:38:16,025 see that there are lot of cases in which the error is larger, 740 00:38:16,025 --> 00:38:19,800 4 millielectron volt. And 4 millielectron 741 00:38:19,800 --> 00:38:21,770 volt is very small, but it's still 742 00:38:21,770 --> 00:38:25,320 10% of this mixing free energy. 743 00:38:25,320 --> 00:38:27,260 So it's sort of somehow relevant. 744 00:38:27,260 --> 00:38:30,920 But we, I would say, still do very well. 745 00:38:30,920 --> 00:38:34,250 And calculating this energy of mixing 746 00:38:34,250 --> 00:38:36,890 is absolutely inexpensive. 747 00:38:36,890 --> 00:38:39,200 It takes a nanosecond on your computer. 748 00:38:39,200 --> 00:38:42,410 You just basically sum with the appropriate configurational 749 00:38:42,410 --> 00:38:44,780 variables the term with your interaction constants. 750 00:38:44,780 --> 00:38:48,780 This requires a full self-consistent calculation. 751 00:38:48,780 --> 00:38:51,770 So with this approach, you can calculate the energies 752 00:38:51,770 --> 00:38:54,140 of millions of structures that are 753 00:38:54,140 --> 00:38:56,810 very large, while with this, you can only 754 00:38:56,810 --> 00:39:00,890 calculate tens of structures that are reasonably small. 755 00:39:00,890 --> 00:39:06,650 So it's very appropriate to use the linear response term 756 00:39:06,650 --> 00:39:11,030 to then do-- to use one of these statistical sampling methods 757 00:39:11,030 --> 00:39:13,700 that you have seen with Professor [? Seder. ?] 758 00:39:13,700 --> 00:39:16,520 And again, I won't go into detail, because you 759 00:39:16,520 --> 00:39:18,380 have seen them partially, and you'll 760 00:39:18,380 --> 00:39:20,730 see them over and over again. 761 00:39:20,730 --> 00:39:23,420 But just to remind you sort of the concepts, 762 00:39:23,420 --> 00:39:25,520 if you want to understand what is 763 00:39:25,520 --> 00:39:29,240 the stability of a certain structure, what you really 764 00:39:29,240 --> 00:39:32,160 need to compare are their Gibbs free energy. 765 00:39:32,160 --> 00:39:35,690 So you want to operate into the thermodynamic ensemble in which 766 00:39:35,690 --> 00:39:38,300 you have constant pressure and constant temperature, 767 00:39:38,300 --> 00:39:40,410 because that's really what your experiment is. 768 00:39:40,410 --> 00:39:44,670 You're at a certain temperature, and you let the system relax. 769 00:39:44,670 --> 00:39:47,900 And as usual, you can't calculate-- 770 00:39:47,900 --> 00:39:50,270 explicitly, there is not a formula 771 00:39:50,270 --> 00:39:52,470 that gives you free energy. 772 00:39:52,470 --> 00:39:54,780 It's a logarithm of a partitioned function. 773 00:39:54,780 --> 00:39:57,530 And so there are sort of statistical ways 774 00:39:57,530 --> 00:40:00,080 by which that can be calculated. 775 00:40:00,080 --> 00:40:03,320 And one of the most powerful approaches 776 00:40:03,320 --> 00:40:06,440 is that given by thermodynamic integration. 777 00:40:06,440 --> 00:40:11,210 That really is based on the fact that while we can't calculate 778 00:40:11,210 --> 00:40:13,790 free energy, we can actually calculate 779 00:40:13,790 --> 00:40:16,500 the derivative of free energy. 780 00:40:16,500 --> 00:40:20,000 So the derivative of the free energy with a certain 781 00:40:20,000 --> 00:40:23,070 parameter, like the derivative of the free energy, 782 00:40:23,070 --> 00:40:25,610 the Gibbs free energy with respect to the number 783 00:40:25,610 --> 00:40:30,560 of particles, is actually going to be given by the ensemble 784 00:40:30,560 --> 00:40:33,290 average-- that's when we use the sort of brackets, 785 00:40:33,290 --> 00:40:35,480 the bracket sort of symbol-- 786 00:40:35,480 --> 00:40:39,260 by the ensemble average of the derivative of the energy 787 00:40:39,260 --> 00:40:41,500 with respect to the number of particles. 788 00:40:41,500 --> 00:40:46,370 So that allows us to calculate derivatives of free energy 789 00:40:46,370 --> 00:40:49,070 in the appropriate thermodynamic ensemble. 790 00:40:49,070 --> 00:40:52,070 And so what we can do is calculate free energy 791 00:40:52,070 --> 00:40:56,930 differences by integrating the derivative of free energy 792 00:40:56,930 --> 00:41:02,630 from state A and state B. And obviously, this integral 793 00:41:02,630 --> 00:41:07,250 will be done by step-wise calculating these derivatives 794 00:41:07,250 --> 00:41:09,090 at different points. 795 00:41:09,090 --> 00:41:11,450 But these derivatives, that in the case of the Gibbs 796 00:41:11,450 --> 00:41:14,700 free energy with respect to the number of particles, nothing 797 00:41:14,700 --> 00:41:17,120 less than the chemical potential as a function 798 00:41:17,120 --> 00:41:20,990 of the concentration, needs to be evaluated 799 00:41:20,990 --> 00:41:22,490 for every concentration. 800 00:41:22,490 --> 00:41:25,910 Say, in going from concentration A to concentration B, 801 00:41:25,910 --> 00:41:27,740 what we really want to figure out 802 00:41:27,740 --> 00:41:31,070 is what is the free energy difference between these two 803 00:41:31,070 --> 00:41:32,160 concentration. 804 00:41:32,160 --> 00:41:35,570 So each of these points in this integral 805 00:41:35,570 --> 00:41:40,070 requires an entire statistical sampling and then 806 00:41:40,070 --> 00:41:42,710 an entire statistical simulation. 807 00:41:42,710 --> 00:41:49,480 And so this is what you would actually do. 808 00:41:49,480 --> 00:41:54,790 You choose a temperature, say, 1,000 kelvin, 809 00:41:54,790 --> 00:41:58,690 and you choose a pressure, say, 0 pressure or 1 atmosphere, 810 00:41:58,690 --> 00:42:01,930 that for all practical purposes is 0. 811 00:42:01,930 --> 00:42:05,470 And then under these conditions, you 812 00:42:05,470 --> 00:42:10,600 have your total energy expressed in configurational variables. 813 00:42:10,600 --> 00:42:14,320 And you decide what is your chemical potential, that is, 814 00:42:14,320 --> 00:42:18,160 what is sort of the additional cost of switching 815 00:42:18,160 --> 00:42:20,560 an atom from gallium to indium. 816 00:42:20,560 --> 00:42:23,020 And then you start with your Monte Carlo sampling. 817 00:42:23,020 --> 00:42:25,600 You keep switching atoms from gallium to indium 818 00:42:25,600 --> 00:42:27,670 or from indium to gallium. 819 00:42:27,670 --> 00:42:30,430 Each of these switches will have a certain energy 820 00:42:30,430 --> 00:42:34,840 cost that is weighed with the Metropolis factor. 821 00:42:34,840 --> 00:42:39,310 And in this simulation, at a given pressure 822 00:42:39,310 --> 00:42:41,800 and at a given chemical potential, 823 00:42:41,800 --> 00:42:46,630 after of millions of sweeps and millions of trials in which you 824 00:42:46,630 --> 00:42:51,070 try to change a spin variable from plus 1 or minus 1, 825 00:42:51,070 --> 00:42:54,850 you do this over and over again, and you end up with a point 826 00:42:54,850 --> 00:43:00,130 there that tells you what is the concentration of, say, 827 00:43:00,130 --> 00:43:05,140 gallium 0.9 for a given chemical potential. 828 00:43:05,140 --> 00:43:07,780 And you repeat this over and over again. 829 00:43:07,780 --> 00:43:12,070 That is, you sweep all your possible chemical potential. 830 00:43:12,070 --> 00:43:15,490 And for each choice of chemical potential, 831 00:43:15,490 --> 00:43:18,340 you get a certain average concentration. 832 00:43:18,340 --> 00:43:20,930 And so you do this over and over again. 833 00:43:20,930 --> 00:43:24,250 And so you see, what you get is a curve 834 00:43:24,250 --> 00:43:28,480 that tells you what is the concentration as a function 835 00:43:28,480 --> 00:43:30,220 of the chemical potential. 836 00:43:30,220 --> 00:43:34,060 And you get one of these curves for each temperature 837 00:43:34,060 --> 00:43:36,080 at which you do the simulation. 838 00:43:36,080 --> 00:43:40,270 So you get concentration as a function of chemical potential. 839 00:43:40,270 --> 00:43:43,480 And if we go back for a moment to the previous slide, 840 00:43:43,480 --> 00:43:46,580 this is really the inverse of what you need. 841 00:43:46,580 --> 00:43:48,580 You need the thermodynamic integration 842 00:43:48,580 --> 00:43:50,710 to calculate the free energy difference between two 843 00:43:50,710 --> 00:43:56,080 concentration, the integral of the average chemical potential 844 00:43:56,080 --> 00:43:58,160 for a given concentration. 845 00:43:58,160 --> 00:44:00,550 So the only thing that you have to do 846 00:44:00,550 --> 00:44:11,200 is, really, look at this curve, turn it around if you want to, 847 00:44:11,200 --> 00:44:15,300 so that you have x as a function of mu. 848 00:44:15,300 --> 00:44:19,860 And say, for the 1,000 kelvin case, 849 00:44:19,860 --> 00:44:22,690 you'll have something like this. 850 00:44:22,690 --> 00:44:25,650 So you have sort of inverted-- 851 00:44:25,650 --> 00:44:28,500 your simulations have given you concentration as a function 852 00:44:28,500 --> 00:44:29,760 of chemical potential. 853 00:44:29,760 --> 00:44:32,670 You invert that, and you integrate it, 854 00:44:32,670 --> 00:44:36,480 and you get a free energy curve. 855 00:44:36,480 --> 00:44:39,000 And you see, things get interesting 856 00:44:39,000 --> 00:44:41,960 when you start being below a critical temperature. 857 00:44:41,960 --> 00:44:45,030 So you see, we got 820 kelvin. 858 00:44:45,030 --> 00:44:48,960 And you really see that the curve is still continuous, 859 00:44:48,960 --> 00:44:53,550 but it sort of starts to have a slope in the middle that 860 00:44:53,550 --> 00:44:55,960 is going towards infinity. 861 00:44:55,960 --> 00:45:02,190 And if we go below, at 650 kelvin, what happens 862 00:45:02,190 --> 00:45:05,610 is that we have opened a miscibility gap. 863 00:45:05,610 --> 00:45:12,450 That is, there is a region of concentrations from 0.1 to 0.8 864 00:45:12,450 --> 00:45:14,490 that never appear. 865 00:45:14,490 --> 00:45:19,080 So at 650 kelvin, this system doesn't 866 00:45:19,080 --> 00:45:25,680 want to mix in the concentration regime between 0.8 and 0.1. 867 00:45:25,680 --> 00:45:29,260 And this is typical of model alloys. 868 00:45:29,260 --> 00:45:30,990 Miscibility gap has opened. 869 00:45:30,990 --> 00:45:32,820 The two things don't mix. 870 00:45:32,820 --> 00:45:38,460 And if you think at sort of how your curve could look like, 871 00:45:38,460 --> 00:45:41,430 we could sort of somehow interpolate 872 00:45:41,430 --> 00:45:46,740 in some arbitrary way this curve and then invert it. 873 00:45:46,740 --> 00:45:48,510 And when you integrate the data-- 874 00:45:52,310 --> 00:45:57,220 it seems that the computer has gone mad. 875 00:45:57,220 --> 00:46:01,080 And let's see. 876 00:46:01,080 --> 00:46:03,330 OK, we'll keep the green forever. 877 00:46:03,330 --> 00:46:06,600 It's now in the realm of the computer. 878 00:46:06,600 --> 00:46:08,700 But this is good, because when you invert this 879 00:46:08,700 --> 00:46:11,280 and you integrate, what you see is 880 00:46:11,280 --> 00:46:15,090 that for your free energy as a function of the concentration-- 881 00:46:15,090 --> 00:46:16,920 is that what you would really obtain 882 00:46:16,920 --> 00:46:23,130 is real free energies only from 0 to 0.1 or from between 1 883 00:46:23,130 --> 00:46:26,370 and something like 0.8, 0.85. 884 00:46:26,370 --> 00:46:29,820 And in the middle, you have this famous miscibility 885 00:46:29,820 --> 00:46:34,530 gap in which, really, the free energy of the solid solution 886 00:46:34,530 --> 00:46:38,640 is higher than the free energy of the n compounds. 887 00:46:38,640 --> 00:46:41,280 And so you have the max error and the tangent. 888 00:46:41,280 --> 00:46:45,490 And so what you have is the composition of your system. 889 00:46:45,490 --> 00:46:47,520 And so you can calculate your phase diagram. 890 00:46:47,520 --> 00:46:51,090 And at the end, for this system, it would really look like this. 891 00:46:51,090 --> 00:46:55,470 Below 820 kelvin, you open a miscibility gap 892 00:46:55,470 --> 00:46:57,150 in which, really, the two systems 893 00:46:57,150 --> 00:46:58,700 don't want to mix together. 894 00:46:58,700 --> 00:47:01,260 And so in many ways, semiconductor alloys 895 00:47:01,260 --> 00:47:04,500 are really model solutions. 896 00:47:04,500 --> 00:47:07,890 And below the critical temperature, 897 00:47:07,890 --> 00:47:09,750 they start to segregate. 898 00:47:09,750 --> 00:47:14,250 And of course, you can do a lot of other intriguing things. 899 00:47:14,250 --> 00:47:18,000 That is, in your simulation, you figure out, say, 900 00:47:18,000 --> 00:47:21,270 what is the distributions of, say, 901 00:47:21,270 --> 00:47:25,320 gallium-phosphorus distance or indium-phosphorus distances 902 00:47:25,320 --> 00:47:28,410 at any given concentrations. 903 00:47:28,410 --> 00:47:30,420 And what you actually discover is 904 00:47:30,420 --> 00:47:34,200 that there is a sort of very bimodal distribution, 905 00:47:34,200 --> 00:47:37,230 that there are short distances with the phosphorus, 906 00:47:37,230 --> 00:47:39,660 and there are long distance with the phosphorus-- that 907 00:47:39,660 --> 00:47:42,240 are, of course, due to the fact that these 908 00:47:42,240 --> 00:47:44,790 are gallium-phosphorus distances and these are 909 00:47:44,790 --> 00:47:46,530 indium-phosphorus distances. 910 00:47:46,530 --> 00:47:49,680 But they sort of depends with the concentration. 911 00:47:49,680 --> 00:47:52,300 But those are things that can be measured. 912 00:47:52,300 --> 00:47:54,930 And so in this sort of graph, I've 913 00:47:54,930 --> 00:48:00,150 put an example of what are the numbers that we 914 00:48:00,150 --> 00:48:03,450 get from the Monte Carlo simulations 915 00:48:03,450 --> 00:48:07,710 on the Ising Hamiltonian and what are actually the EXAES 916 00:48:07,710 --> 00:48:10,440 results that people measure. 917 00:48:10,440 --> 00:48:12,880 And you see, there is really a very good agreement. 918 00:48:12,880 --> 00:48:16,380 Say, these are the short gallium phosphide distances 919 00:48:16,380 --> 00:48:18,900 as a function of concentration between what 920 00:48:18,900 --> 00:48:24,330 the experiment measure and what your thermodynamical DFT is 921 00:48:24,330 --> 00:48:26,160 actually predicting. 922 00:48:26,160 --> 00:48:30,630 And I think I'll skip this. 923 00:48:30,630 --> 00:48:33,420 There were some other details about epitaxial system. 924 00:48:33,420 --> 00:48:35,220 But otherwise, I'm worried I won't 925 00:48:35,220 --> 00:48:37,860 be able to do the second part. 926 00:48:37,860 --> 00:48:42,180 So this was the first example. 927 00:48:42,180 --> 00:48:45,150 Again, what is important there is that we 928 00:48:45,150 --> 00:48:47,220 have a physical problem. 929 00:48:47,220 --> 00:48:51,330 We have identified what are the important degrees of freedom, 930 00:48:51,330 --> 00:48:54,900 and we have calculated those degrees of freedom 931 00:48:54,900 --> 00:48:57,750 from calculations that are actually manageable. 932 00:48:57,750 --> 00:49:00,180 But those degrees of freedom in those calculation 933 00:49:00,180 --> 00:49:04,260 have given us the parameter of a Hamiltonian that, in this case, 934 00:49:04,260 --> 00:49:07,920 was a Ising-like long range Hamiltonian in which we 935 00:49:07,920 --> 00:49:12,720 can do sort of much more complex statistical mechanics analysis. 936 00:49:12,720 --> 00:49:16,850 And so we can actually obtain phase diagrams. 937 00:49:16,850 --> 00:49:18,780 That was one example. 938 00:49:18,780 --> 00:49:20,630 The other example that I want to show you 939 00:49:20,630 --> 00:49:24,360 here is instead how we sort of repeat 940 00:49:24,360 --> 00:49:27,290 the same process in order instead 941 00:49:27,290 --> 00:49:33,050 to calculate Hamiltonians that have a linear scaling cost. 942 00:49:33,050 --> 00:49:35,750 That is, that allow us to calculate 943 00:49:35,750 --> 00:49:37,760 the electronic structure and energies 944 00:49:37,760 --> 00:49:39,800 of systems that are really large, 945 00:49:39,800 --> 00:49:44,450 that are sort of thousands or tens of thousands of atoms. 946 00:49:44,450 --> 00:49:47,220 And you know, this is because this comes from some research 947 00:49:47,220 --> 00:49:48,320 example. 948 00:49:48,320 --> 00:49:50,900 We were interested in sort of studying 949 00:49:50,900 --> 00:49:55,010 what is the effect on a carbon nanotube 950 00:49:55,010 --> 00:49:57,200 of organic functionalization. 951 00:49:57,200 --> 00:50:00,950 That is, what happens if we take a ligand-- 952 00:50:00,950 --> 00:50:02,950 this would be-- it's a benzene ring. 953 00:50:02,950 --> 00:50:05,900 So again, let's kind of call it phenyl moiety. 954 00:50:05,900 --> 00:50:10,460 And this sort of ring is attached with a covalent bond 955 00:50:10,460 --> 00:50:13,130 to an infinitely longer carbon nanotube. 956 00:50:13,130 --> 00:50:14,673 And people have started doing this. 957 00:50:14,673 --> 00:50:16,340 And of course, when it happens, what you 958 00:50:16,340 --> 00:50:20,540 have is a nanotube that has a random array 959 00:50:20,540 --> 00:50:21,770 of these funny rings. 960 00:50:21,770 --> 00:50:25,760 So again, we have the thermodynamical problem 961 00:50:25,760 --> 00:50:29,930 of finding out what is the effect of this configuration 962 00:50:29,930 --> 00:50:32,840 of this order on some of the intriguing properties 963 00:50:32,840 --> 00:50:34,280 of the carbon nanotube. 964 00:50:34,280 --> 00:50:37,160 Like, there are sort of ballistic electrical 965 00:50:37,160 --> 00:50:40,190 conductivity, that is electrical conductivity, that 966 00:50:40,190 --> 00:50:44,180 is independent of length, so very different 967 00:50:44,180 --> 00:50:45,980 from an ohmic conductor-- 968 00:50:45,980 --> 00:50:49,250 or what is their thermal conductivity. 969 00:50:49,250 --> 00:50:53,090 But again, we really don't have the computational resources, 970 00:50:53,090 --> 00:50:55,550 and no one has the computational resources 971 00:50:55,550 --> 00:50:58,190 to study, with density functional theory, a system 972 00:50:58,190 --> 00:51:01,520 with tens of thousands of atoms. 973 00:51:01,520 --> 00:51:05,030 And you'll see this in sort of the last part of the class, 974 00:51:05,030 --> 00:51:05,720 really. 975 00:51:05,720 --> 00:51:08,850 It's not just a matter of waiting a few more years. 976 00:51:08,850 --> 00:51:13,410 I mean, the scaling cost kills us in many ways. 977 00:51:13,410 --> 00:51:16,610 So we really need to use sort of ingenuity. 978 00:51:16,610 --> 00:51:21,440 And what we are going to use here is this idea. 979 00:51:21,440 --> 00:51:24,950 We want to break away from this orthonormalization 980 00:51:24,950 --> 00:51:27,740 constraint that is really what drives 981 00:51:27,740 --> 00:51:30,170 the cubic scaling of a system. 982 00:51:30,170 --> 00:51:34,010 That is, what we want to do is using a system 983 00:51:34,010 --> 00:51:37,250 as sort of picture in which we really 984 00:51:37,250 --> 00:51:41,780 look at Kohn-Sham orbitals that are localized, 985 00:51:41,780 --> 00:51:47,000 so we don't have to worry about orthonormality with orbitals 986 00:51:47,000 --> 00:51:48,230 that are far away. 987 00:51:48,230 --> 00:51:51,920 Because if two orbitals are far away, and both of them 988 00:51:51,920 --> 00:51:55,850 are localized, their overlap is going to be 0. 989 00:51:55,850 --> 00:51:59,000 And so they are orthogonal by default, 990 00:51:59,000 --> 00:52:04,820 without us having to sort of impose that. 991 00:52:04,820 --> 00:52:06,680 And the way we are going to do this 992 00:52:06,680 --> 00:52:11,200 is, again, a mapping from the full density functional theory 993 00:52:11,200 --> 00:52:12,450 calculation. 994 00:52:12,450 --> 00:52:13,460 So we start-- 995 00:52:13,460 --> 00:52:16,880 I'm giving you here the example of silicon. 996 00:52:16,880 --> 00:52:20,360 If you look at the band structure of silicon, 997 00:52:20,360 --> 00:52:24,500 again, what you have are very localized 998 00:52:24,500 --> 00:52:27,680 core orbitals down there that don't really have 999 00:52:27,680 --> 00:52:28,930 any effect on the chemistry. 1000 00:52:28,930 --> 00:52:30,800 They corresponds to flat bands. 1001 00:52:30,800 --> 00:52:34,280 But what is interesting is what the valence electrons do, 1002 00:52:34,280 --> 00:52:36,890 and this is sort of the very celebrated band 1003 00:52:36,890 --> 00:52:38,480 structure of silicon. 1004 00:52:38,480 --> 00:52:42,650 This in particular is the gamma point, so at 0, 0, 0. 1005 00:52:42,650 --> 00:52:45,350 And so because silicon has two atoms 1006 00:52:45,350 --> 00:52:48,860 per unit cell, four valence electrons per atom, 1007 00:52:48,860 --> 00:52:54,560 we have eight valence electrons. 1008 00:52:54,560 --> 00:52:55,670 Spin the generator. 1009 00:52:55,670 --> 00:52:58,880 So we have bands that contain two electrons each. 1010 00:53:02,910 --> 00:53:03,410 Sorry. 1011 00:53:06,980 --> 00:53:11,300 So what we have is four valence bands. 1012 00:53:11,300 --> 00:53:13,220 And you see, 1, 2, 3, 4. 1013 00:53:13,220 --> 00:53:15,380 As we move around the Brillouin zone, 1014 00:53:15,380 --> 00:53:17,300 they can become the generator. 1015 00:53:17,300 --> 00:53:21,110 And those that are studying semiconductor physics, 1016 00:53:21,110 --> 00:53:24,530 they will know that this is the top of the valence band, 1017 00:53:24,530 --> 00:53:27,230 and we have heavy holes and light holes 1018 00:53:27,230 --> 00:53:31,010 depending on the curvature up here. 1019 00:53:31,010 --> 00:53:33,620 But so what is the electronic structure of our problem? 1020 00:53:33,620 --> 00:53:40,640 Well, remember, the band index and the quasimomentum k 1021 00:53:40,640 --> 00:53:42,500 across the Brillouin zone are really 1022 00:53:42,500 --> 00:53:46,160 the quantum numbers for the crystal, for my problem. 1023 00:53:46,160 --> 00:53:48,860 If I study a molecule, and I have 1024 00:53:48,860 --> 00:53:53,420 a molecule with eight electrons, I will have only four-- 1025 00:53:53,420 --> 00:53:55,730 in the case of spin degeneracy-- 1026 00:53:55,730 --> 00:53:58,440 eigenstates that are occupied. 1027 00:53:58,440 --> 00:54:01,520 If, instead of studying a molecule with eight electrons, 1028 00:54:01,520 --> 00:54:05,150 I study a crystal with eight electrons per unit cell, 1029 00:54:05,150 --> 00:54:09,740 instead of having four states, I have four bands. 1030 00:54:09,740 --> 00:54:13,100 And the energy of each eigenstate, 1031 00:54:13,100 --> 00:54:15,860 thanks to Bloch theorem, can be cataloged 1032 00:54:15,860 --> 00:54:20,840 depending on this band index and depending on the quasimomentum. 1033 00:54:20,840 --> 00:54:23,330 And the quasimomentum really sort of tells me 1034 00:54:23,330 --> 00:54:27,590 how the phase of things changes when we go from one unit cell 1035 00:54:27,590 --> 00:54:28,580 to the other. 1036 00:54:28,580 --> 00:54:31,550 The same concept takes place also not in-- 1037 00:54:31,550 --> 00:54:35,150 for the electronic eigenstates but for the normal modes, 1038 00:54:35,150 --> 00:54:36,890 for the phonon modes of a crystal. 1039 00:54:36,890 --> 00:54:40,190 We classify them according to wavelength 1040 00:54:40,190 --> 00:54:43,040 and according to band index. 1041 00:54:43,040 --> 00:54:48,200 And so your infinite number of electrons in the crystal 1042 00:54:48,200 --> 00:54:52,550 can all sort of be summarized in this picture, in which we 1043 00:54:52,550 --> 00:54:55,940 have a continuum of states. 1044 00:54:55,940 --> 00:54:57,740 So if you want, we have mapped the fact 1045 00:54:57,740 --> 00:55:01,280 that we had eight electrons repeated infinitely 1046 00:55:01,280 --> 00:55:05,750 into a problem in which we have eight electrons, that is four 1047 00:55:05,750 --> 00:55:09,020 bands, with a sort of continuum variation, 1048 00:55:09,020 --> 00:55:12,810 depending on which wavelength we are looking at. 1049 00:55:12,810 --> 00:55:15,830 And so this picture sort of represents 1050 00:55:15,830 --> 00:55:21,020 all the possible occupied states in your system 1051 00:55:21,020 --> 00:55:25,790 that are infinite but are classified in a group of four 1052 00:55:25,790 --> 00:55:28,850 with a continuum index k. 1053 00:55:28,850 --> 00:55:32,210 Now, the interesting things about quantum mechanics 1054 00:55:32,210 --> 00:55:35,780 is that you can change your representation 1055 00:55:35,780 --> 00:55:37,950 and still look at the same problem. 1056 00:55:37,950 --> 00:55:40,010 It's the same thing in classical mechanics. 1057 00:55:40,010 --> 00:55:42,890 If you are sort of studying something in three dimension, 1058 00:55:42,890 --> 00:55:45,420 you can rotate your coordinate system. 1059 00:55:45,420 --> 00:55:47,420 And so your equation of motion will change. 1060 00:55:47,420 --> 00:55:49,040 Your trajectory will change. 1061 00:55:49,040 --> 00:55:52,760 But really, you are just moving everything accordingly, 1062 00:55:52,760 --> 00:55:54,680 and you are looking at the same problem 1063 00:55:54,680 --> 00:55:58,040 in a different coordinate system. 1064 00:55:58,040 --> 00:56:01,460 That is even more true in quantum mechanics, in which you 1065 00:56:01,460 --> 00:56:05,900 can perform a unitary transformation between all 1066 00:56:05,900 --> 00:56:09,800 your eigenstates and still look at the same problem. 1067 00:56:09,800 --> 00:56:12,500 Usually, what we sort of choose to be 1068 00:56:12,500 --> 00:56:14,180 is into a representation in which 1069 00:56:14,180 --> 00:56:17,090 we are looking at eigenstates of the Hamiltonian. 1070 00:56:17,090 --> 00:56:20,450 So each of these states here is exactly 1071 00:56:20,450 --> 00:56:23,720 the characteristic of being an eigenstate of the Hamiltonian. 1072 00:56:23,720 --> 00:56:26,060 You apply the Hamiltonian to that state. 1073 00:56:26,060 --> 00:56:28,310 What you obtain is a constant that 1074 00:56:28,310 --> 00:56:31,500 is the energy eigenvalue times the state itself. 1075 00:56:31,500 --> 00:56:34,520 But we can mix this around, and we can still 1076 00:56:34,520 --> 00:56:38,870 look at the same problem, just in a different representation. 1077 00:56:38,870 --> 00:56:41,660 Exactly as in classical mechanics, 1078 00:56:41,660 --> 00:56:43,700 we can do an orthogonal transformation 1079 00:56:43,700 --> 00:56:46,830 of our coordinate system and look at the same problem. 1080 00:56:46,830 --> 00:56:52,130 So what we can do is actually do a unitary transformation, 1081 00:56:52,130 --> 00:56:55,040 a unitary rotation, of our orbitals 1082 00:56:55,040 --> 00:56:57,770 and still look at the same problem. 1083 00:56:57,770 --> 00:57:03,230 And again, without really going into the details of what it is 1084 00:57:03,230 --> 00:57:05,180 and what quantum mechanics is, it's 1085 00:57:05,180 --> 00:57:07,160 something that would look like this. 1086 00:57:07,160 --> 00:57:13,130 That is, if we start from our eigenstates, at each k point-- 1087 00:57:13,130 --> 00:57:15,830 that is, at each point in the Brillouin zone-- 1088 00:57:15,830 --> 00:57:20,750 if we have those four bands, we can mix them together 1089 00:57:20,750 --> 00:57:22,580 with a unitary transformation. 1090 00:57:22,580 --> 00:57:27,260 But not only that, we can sort of mix those mixed bands 1091 00:57:27,260 --> 00:57:30,890 with all the other bands at different k points 1092 00:57:30,890 --> 00:57:32,840 with the appropriate unitary factor. 1093 00:57:32,840 --> 00:57:37,310 And so what we can obtain is a new set of orbitals that 1094 00:57:37,310 --> 00:57:41,270 are, again, infinite and that really represent 1095 00:57:41,270 --> 00:57:43,820 the same quantum mechanical problem, 1096 00:57:43,820 --> 00:57:46,280 just sort of turned it around. 1097 00:57:46,280 --> 00:57:48,740 And our quantum numbers that before, 1098 00:57:48,740 --> 00:57:53,330 where the band index and the quasimomentum 1099 00:57:53,330 --> 00:58:00,090 become some new quantum numbers, that will be-- again, 1100 00:58:00,090 --> 00:58:02,880 one of them will be related to the bands. 1101 00:58:02,880 --> 00:58:07,080 So for silicon, we'll go from 1 to 4. 1102 00:58:07,080 --> 00:58:10,560 And the other one will count all the sort 1103 00:58:10,560 --> 00:58:12,510 of infinite number of electrons that we 1104 00:58:12,510 --> 00:58:16,050 have when we go from one isolated unit cell to one unit 1105 00:58:16,050 --> 00:58:18,060 cell periodically repeated. 1106 00:58:18,060 --> 00:58:21,600 And before, what we had was a quasicontinuum, 1107 00:58:21,600 --> 00:58:23,490 basically, of wavelengths. 1108 00:58:23,490 --> 00:58:26,160 And in this new representation, what we 1109 00:58:26,160 --> 00:58:30,090 have is a quasicontinuum of localization vectors. 1110 00:58:30,090 --> 00:58:34,650 So we go from extended Bloch orbitals 1111 00:58:34,650 --> 00:58:36,960 that are eigenstates of the Hamiltonian 1112 00:58:36,960 --> 00:58:42,570 to localized orbitals in which the electron is sitting in each 1113 00:58:42,570 --> 00:58:44,640 of the different unit cells. 1114 00:58:44,640 --> 00:58:48,390 And really, this is an arbitrary transformation. 1115 00:58:48,390 --> 00:58:52,020 And these matrices are sort of our parameters, 1116 00:58:52,020 --> 00:58:55,740 the sort of quantum mechanical equivalent of the three Euler 1117 00:58:55,740 --> 00:58:59,220 angles that rotate the Cartesian system from one 1118 00:58:59,220 --> 00:59:01,090 representation to another. 1119 00:59:01,090 --> 00:59:05,860 And so we can choose, say, this set of transformation, 1120 00:59:05,860 --> 00:59:10,740 so that what we obtain is the same quantum mechanical picture 1121 00:59:10,740 --> 00:59:15,480 but in a representation that is as localized as possible. 1122 00:59:15,480 --> 00:59:19,650 And how we do that sort of requires a lot of algebra. 1123 00:59:19,650 --> 00:59:23,280 But this is the sort of very intriguing result 1124 00:59:23,280 --> 00:59:26,500 that takes place when we actually make it in practice. 1125 00:59:26,500 --> 00:59:30,960 We take-- again, all these states and each of them 1126 00:59:30,960 --> 00:59:32,220 is delocalized. 1127 00:59:32,220 --> 00:59:35,040 It's Bloch orbital energy eigenstates 1128 00:59:35,040 --> 00:59:37,920 that is extended over the whole crystal. 1129 00:59:37,920 --> 00:59:41,760 We mix all these things together with a criterion 1130 00:59:41,760 --> 00:59:43,470 that those mixing transformation are 1131 00:59:43,470 --> 00:59:46,980 chosen so that the new states are as localized as possible. 1132 00:59:46,980 --> 00:59:50,010 And what we obtain at the end is fairly beautiful. 1133 00:59:50,010 --> 00:59:53,610 If we look at, say, something like silicon, again, 1134 00:59:53,610 --> 00:59:56,820 sort of a perennial diamond structure, 1135 00:59:56,820 --> 00:59:59,850 one silicon atom fourfold coordinated 1136 00:59:59,850 --> 01:00:02,130 with the other silicon atoms-- 1137 01:00:02,130 --> 01:00:11,060 and it's a bit difficult to see, but what we would have 1138 01:00:11,060 --> 01:00:14,930 is that one silicon atom, in red, 1139 01:00:14,930 --> 01:00:20,210 is connected to four silicon atoms. 1140 01:00:20,210 --> 01:00:22,895 And this blue over blue silicon atom 1141 01:00:22,895 --> 01:00:27,510 is connected to other four silicon atoms. 1142 01:00:31,400 --> 01:00:35,150 And the localized orbital that you obtain 1143 01:00:35,150 --> 01:00:42,050 is really what the chemist would have told us probably in 1930. 1144 01:00:42,050 --> 01:00:47,480 It's really a covalent bond between the two orbitals. 1145 01:00:47,480 --> 01:00:50,870 So what we have is that these two atoms here 1146 01:00:50,870 --> 01:00:54,050 are connected by an orbital that has, 1147 01:00:54,050 --> 01:00:59,750 really, this pile up of covalent charge between those two atoms. 1148 01:00:59,750 --> 01:01:04,400 And there is a certain amount of sort of back-bonding 1149 01:01:04,400 --> 01:01:08,180 between each of those two atoms and the remaining three 1150 01:01:08,180 --> 01:01:09,290 neighbors. 1151 01:01:09,290 --> 01:01:13,820 So this is how the localized orbital looks like. 1152 01:01:13,820 --> 01:01:17,720 And that's been just obtained by a unitary transformation. 1153 01:01:17,720 --> 01:01:20,370 But when we have this player here, 1154 01:01:20,370 --> 01:01:23,120 it becomes very easy now to construct 1155 01:01:23,120 --> 01:01:25,640 the electronic structure of a system that 1156 01:01:25,640 --> 01:01:29,420 might have some amount of configurational disorder. 1157 01:01:29,420 --> 01:01:32,300 Because, if instead of having somewhere, 1158 01:01:32,300 --> 01:01:35,420 say, silicon-silicon in an infinite silicon matrix, 1159 01:01:35,420 --> 01:01:38,930 we have maybe gallium and arsenic for a moment locally. 1160 01:01:38,930 --> 01:01:42,600 Well, maybe what we can do, and I'll show you how we'll do it, 1161 01:01:42,600 --> 01:01:46,550 maybe we can put there the exact orbital corresponding 1162 01:01:46,550 --> 01:01:48,890 to the gallium arsenic bond. 1163 01:01:48,890 --> 01:01:52,860 And if you look at it, it's actually going to be-- again, 1164 01:01:52,860 --> 01:01:54,950 because things are similar, it's going 1165 01:01:54,950 --> 01:01:58,370 to be very similar to the silicon orbital. 1166 01:01:58,370 --> 01:02:00,740 The gallium arsenic covalent bonds 1167 01:02:00,740 --> 01:02:03,140 is just, because of the difference 1168 01:02:03,140 --> 01:02:07,370 of electronic activity, sort of tilted more 1169 01:02:07,370 --> 01:02:12,500 towards the electronegative arsenic in this problem. 1170 01:02:12,500 --> 01:02:15,440 Or say, again, to give you the feeling, 1171 01:02:15,440 --> 01:02:19,700 it's nothing less than the sort of standard chemical picture. 1172 01:02:19,700 --> 01:02:22,820 If we have a system that is still silicon, 1173 01:02:22,820 --> 01:02:24,560 but it's amorphous silicon-- 1174 01:02:24,560 --> 01:02:26,900 and I've got here atoms around. 1175 01:02:26,900 --> 01:02:28,640 But amorphous silicon is something 1176 01:02:28,640 --> 01:02:31,970 in which a tetrahedral short range holder is still 1177 01:02:31,970 --> 01:02:36,530 preserved on average but long range holder is lost. 1178 01:02:36,530 --> 01:02:40,220 Well, atoms that are tetrahedrally coordinated 1179 01:02:40,220 --> 01:02:43,850 won't have a perfect 109 degrees angles, 1180 01:02:43,850 --> 01:02:48,780 but their bonds still look very much like those. 1181 01:02:48,780 --> 01:02:52,020 And sometimes, you have atoms that are five-fold coordinated. 1182 01:02:52,020 --> 01:02:54,200 And so they have their own bond. 1183 01:02:54,200 --> 01:02:55,940 And if you want, these are really 1184 01:02:55,940 --> 01:02:59,270 what we call the LEGO bricks of electronic structure. 1185 01:02:59,270 --> 01:03:01,730 That is, in a very complex system, 1186 01:03:01,730 --> 01:03:03,350 the electronic structure of the system 1187 01:03:03,350 --> 01:03:07,940 is going to be made up locally by the appropriate orbitals 1188 01:03:07,940 --> 01:03:10,700 that corresponds to the local environment. 1189 01:03:10,700 --> 01:03:14,150 And the nature and physics is inherently local, 1190 01:03:14,150 --> 01:03:17,360 so there is a large degree of transferability. 1191 01:03:17,360 --> 01:03:20,390 And again, to give you a sort of more feeling for this, 1192 01:03:20,390 --> 01:03:23,900 suppose that instead of looking at a crystal, 1193 01:03:23,900 --> 01:03:25,340 we look at a molecule. 1194 01:03:25,340 --> 01:03:28,100 We look at something like a benzene ring. 1195 01:03:28,100 --> 01:03:30,620 And you go to a representation in which states, instead 1196 01:03:30,620 --> 01:03:33,050 of being eigenstates of the Hamiltonian, 1197 01:03:33,050 --> 01:03:34,910 are localized states. 1198 01:03:34,910 --> 01:03:38,600 What you see are electronic states that are really 1199 01:03:38,600 --> 01:03:42,110 carbon-hydrogen single bonds, or what 1200 01:03:42,110 --> 01:03:47,390 you see are carbon-carbon single bonds, or what you see 1201 01:03:47,390 --> 01:03:50,610 are carbon-carbon double bonds, in which 1202 01:03:50,610 --> 01:03:54,530 you have both this orbital and this reflection 1203 01:03:54,530 --> 01:03:58,800 across the symmetry plane of the benzene molecule. 1204 01:03:58,800 --> 01:04:02,210 So really, what you are seeing is very intuitive chemical 1205 01:04:02,210 --> 01:04:03,110 picture. 1206 01:04:03,110 --> 01:04:07,190 But the way it's been obtained is actually 1207 01:04:07,190 --> 01:04:12,530 from an exact transformation of our quantum mechanical problem, 1208 01:04:12,530 --> 01:04:14,570 solving density functional theory, that is, 1209 01:04:14,570 --> 01:04:17,300 with a certain degree of accuracy. 1210 01:04:17,300 --> 01:04:20,030 And we'll see now how we apply this 1211 01:04:20,030 --> 01:04:23,540 to the case of carbon nanotubes, because what we really 1212 01:04:23,540 --> 01:04:27,500 want to do is study chemical disorder on these carbon 1213 01:04:27,500 --> 01:04:29,300 nanotubes. 1214 01:04:29,300 --> 01:04:33,800 And carbon nanotube is nothing else than a sheet of graphite-- 1215 01:04:33,800 --> 01:04:36,650 that is what is called the graphene sheet-- 1216 01:04:36,650 --> 01:04:40,770 that has been rolled over together to create a tube. 1217 01:04:40,770 --> 01:04:43,490 So if you take this hexagonal lattice, 1218 01:04:43,490 --> 01:04:48,020 there are sort of discrete infinite way of rolling things 1219 01:04:48,020 --> 01:04:48,980 together. 1220 01:04:48,980 --> 01:04:51,710 And the way the carbon nanotube community 1221 01:04:51,710 --> 01:04:54,620 has agreed to sort of use a nomenclature 1222 01:04:54,620 --> 01:04:59,090 is by defining what is called a chiral vector, that, 1223 01:04:59,090 --> 01:05:03,930 in this case here, would be this CH vector. 1224 01:05:03,930 --> 01:05:08,900 So if, on a graphene sheet, you define a vector, 1225 01:05:08,900 --> 01:05:11,630 that is really just an integer number 1226 01:05:11,630 --> 01:05:17,330 of times the two a1 and a2 primitive lattice vectors-- 1227 01:05:17,330 --> 01:05:20,780 that is, once you have defined this green CH, 1228 01:05:20,780 --> 01:05:23,720 you have defined your nanotube, by saying 1229 01:05:23,720 --> 01:05:26,720 that you are cutting the graphene sheet 1230 01:05:26,720 --> 01:05:31,110 perpendicular to this vector and rolling what you have cut, 1231 01:05:31,110 --> 01:05:34,650 so that the top of the vector touches the bottom. 1232 01:05:34,650 --> 01:05:38,280 And so you have sort of a perfect cylinder. 1233 01:05:38,280 --> 01:05:40,380 And then there is a nomenclature. 1234 01:05:40,380 --> 01:05:44,070 And you can sort of-- a generic nanotube 1235 01:05:44,070 --> 01:05:48,750 will be called armchair, zig-zag, or chiral nanotube, 1236 01:05:48,750 --> 01:05:52,080 depending on how you have rolled things around. 1237 01:05:52,080 --> 01:05:57,520 And so if your chiral vector is an n number of a1s 1238 01:05:57,520 --> 01:06:01,680 and an n number of a2, equal, your chiral vector 1239 01:06:01,680 --> 01:06:04,240 will be this blue vector here. 1240 01:06:04,240 --> 01:06:06,420 And so what you are going to do is, 1241 01:06:06,420 --> 01:06:09,180 you are going to roll the nanotube so 1242 01:06:09,180 --> 01:06:12,540 that perpendicular to the direction of the tube, 1243 01:06:12,540 --> 01:06:17,070 you can clearly see this sort of zig-zag pattern. 1244 01:06:17,070 --> 01:06:18,750 And so an armchair-- 1245 01:06:18,750 --> 01:06:24,450 sorry, a zig-zag-- oh, sorry. 1246 01:06:24,450 --> 01:06:28,540 A zig-zag nanotube is the one in which 1247 01:06:28,540 --> 01:06:30,910 you have a chiral vector that is made 1248 01:06:30,910 --> 01:06:34,510 by an integer number of times a1 and zero 1249 01:06:34,510 --> 01:06:36,890 number a2 or vice versa. 1250 01:06:36,890 --> 01:06:40,090 So in the form n, 0 or 0, m. 1251 01:06:40,090 --> 01:06:44,500 And so you see, this blue vector for the zig-zag 1252 01:06:44,500 --> 01:06:48,430 is going to be 1, 2, 3, 4, 5 times a1. 1253 01:06:48,430 --> 01:06:51,700 So this will be the zig-zag nanotube, 1254 01:06:51,700 --> 01:06:53,950 and you see this zig-zag pattern. 1255 01:06:53,950 --> 01:07:01,390 If, instead, you choose n for a1 equal to the n for a2, what 1256 01:07:01,390 --> 01:07:04,900 you have is a vector that goes in this direction, 1257 01:07:04,900 --> 01:07:06,860 like the red vector here. 1258 01:07:06,860 --> 01:07:09,550 And so what you are doing is actually 1259 01:07:09,550 --> 01:07:12,760 sort of cutting the nanotube like this 1260 01:07:12,760 --> 01:07:14,650 and rolling it together. 1261 01:07:14,650 --> 01:07:19,090 And again, now perpendicular to the axis of the nanotube, 1262 01:07:19,090 --> 01:07:20,950 you see this pattern. 1263 01:07:20,950 --> 01:07:23,470 That is called an armchair pattern. 1264 01:07:23,470 --> 01:07:27,500 And so an armchair nanotube is identified by these two quantum 1265 01:07:27,500 --> 01:07:28,000 numbers. 1266 01:07:28,000 --> 01:07:32,010 And then identical zig-zag is n, 0 or 0, n. 1267 01:07:32,010 --> 01:07:36,070 And everything else has a sort of more ordered chirality 1268 01:07:36,070 --> 01:07:40,960 and is called, generically speaking, a chiral nanotube. 1269 01:07:40,960 --> 01:07:43,060 OK, so this is how they look like. 1270 01:07:43,060 --> 01:07:46,100 You see, a zig-zag, very long in this direction. 1271 01:07:46,100 --> 01:07:48,370 We cut it, we see the zig-zag pattern. 1272 01:07:48,370 --> 01:07:52,570 An armchair, we cut, and we see that sort of hexagonal pattern 1273 01:07:52,570 --> 01:07:56,380 that, for some reason, is called an armchair. 1274 01:07:56,380 --> 01:07:58,960 And now the interesting bit comes 1275 01:07:58,960 --> 01:08:03,910 when we look at the band structure of this system. 1276 01:08:03,910 --> 01:08:07,780 We could have started, say, from the band structure of graphite 1277 01:08:07,780 --> 01:08:11,200 or graphene, in which, again, the Brillouin zone-- 1278 01:08:11,200 --> 01:08:14,560 the extended Brillouin zone is this hexagon. 1279 01:08:14,560 --> 01:08:18,040 And what happens is a bit difficult to visualize. 1280 01:08:18,040 --> 01:08:20,140 But actually, what happens is that there 1281 01:08:20,140 --> 01:08:22,510 are only six points-- 1282 01:08:22,510 --> 01:08:24,640 so the black dots there-- 1283 01:08:24,640 --> 01:08:30,040 at which they are unoccupied states 1284 01:08:30,040 --> 01:08:32,420 touch the occupied states. 1285 01:08:32,420 --> 01:08:36,880 So usually-- and we have seen it over and over again-- 1286 01:08:36,880 --> 01:08:39,340 in something like a metal, we will 1287 01:08:39,340 --> 01:08:41,640 have maybe a band like this. 1288 01:08:41,640 --> 01:08:48,550 And then we'll have Fermi energy that cuts the occupied states 1289 01:08:48,550 --> 01:08:50,350 from the unoccupied states. 1290 01:08:50,350 --> 01:08:54,790 In graphite, the band dispersion is much more exotic. 1291 01:08:54,790 --> 01:08:56,770 It actually looks like this. 1292 01:08:56,770 --> 01:08:58,930 In certain points of the Brillouin zone, 1293 01:08:58,930 --> 01:09:02,380 it's actually sort of straight lines. 1294 01:09:02,380 --> 01:09:04,010 You'll see it in a moment. 1295 01:09:04,010 --> 01:09:07,750 And the Fermi surface is exactly here. 1296 01:09:07,750 --> 01:09:10,300 So there is only-- the Fermi energy is here. 1297 01:09:10,300 --> 01:09:12,609 The Fermi surface is on a point. 1298 01:09:12,609 --> 01:09:15,880 And seen from above, if you want, 1299 01:09:15,880 --> 01:09:21,130 these are where these combs are that I've sort of cut there 1300 01:09:21,130 --> 01:09:22,479 in a profile. 1301 01:09:22,479 --> 01:09:26,680 What happens when you go from graphene to a nanotube? 1302 01:09:26,680 --> 01:09:28,330 Well, instead of having a system that 1303 01:09:28,330 --> 01:09:33,189 is infinitely B-dimensional, you get a system in which, 1304 01:09:33,189 --> 01:09:36,130 by rolling one of the dimensions, what 1305 01:09:36,130 --> 01:09:39,790 you do actually is you quantize your quantum 1306 01:09:39,790 --> 01:09:41,583 numbers in one direction. 1307 01:09:41,583 --> 01:09:43,000 So you still have a Brillouin zone 1308 01:09:43,000 --> 01:09:46,540 that is a continuum in the direction of the tube axis. 1309 01:09:46,540 --> 01:09:50,229 But because you have rolled it, maybe you have only 10 hexagons 1310 01:09:50,229 --> 01:09:53,439 instead of an infinite number of hexagons in the direction 1311 01:09:53,439 --> 01:09:55,150 perpendicular to the axis. 1312 01:09:55,150 --> 01:09:57,280 Now, what you are effectively doing 1313 01:09:57,280 --> 01:10:01,210 is really looking at the band structure only along 1314 01:10:01,210 --> 01:10:05,290 a certain number of cuts of this Brillouin zone. 1315 01:10:05,290 --> 01:10:09,460 And so depending on how you roll your system, what you can have 1316 01:10:09,460 --> 01:10:14,530 is that this subset of lines that corresponds 1317 01:10:14,530 --> 01:10:17,890 to having a continuum index in one direction 1318 01:10:17,890 --> 01:10:21,250 become a discrete index, because in one direction, 1319 01:10:21,250 --> 01:10:24,940 we go from infinite hexagon to a finite number of hexagons. 1320 01:10:24,940 --> 01:10:27,790 And so depending on how many and what 1321 01:10:27,790 --> 01:10:30,130 is the orientation of the cut, we 1322 01:10:30,130 --> 01:10:34,730 can have that these lines cross or do not cross these points. 1323 01:10:34,730 --> 01:10:38,590 And if they cross those points, we'll have a metallic nanotube. 1324 01:10:38,590 --> 01:10:41,740 If they don't, we'll have a semiconducting nanotube. 1325 01:10:41,740 --> 01:10:44,950 So just a very slight change in orientation 1326 01:10:44,950 --> 01:10:48,640 in the chirality vector might allow us to miss this point 1327 01:10:48,640 --> 01:10:51,910 and make something that was metallic into a semiconductor. 1328 01:10:51,910 --> 01:10:54,460 And the rule to remember this is trivial. 1329 01:10:54,460 --> 01:10:56,740 If the difference between the indices 1330 01:10:56,740 --> 01:11:00,040 is 0 or a multiple of three, your nanotube 1331 01:11:00,040 --> 01:11:02,080 is going to be metallic. 1332 01:11:02,080 --> 01:11:03,790 Otherwise, it's semiconducting. 1333 01:11:03,790 --> 01:11:07,240 So say, all armchair nanotubes are metallic, 1334 01:11:07,240 --> 01:11:11,290 because m is equal to n, so the difference is 0. 1335 01:11:11,290 --> 01:11:13,330 So they are metallic. 1336 01:11:13,330 --> 01:11:17,290 Zig-zag nanotubes, well, only 1 out of 3 1337 01:11:17,290 --> 01:11:19,420 is going to be metallic. 1338 01:11:19,420 --> 01:11:21,190 6, 0 is going to be metallic. 1339 01:11:21,190 --> 01:11:23,350 7, 0, 8, 0 are semiconducting. 1340 01:11:23,350 --> 01:11:26,570 9, 0 is metallic, and so on and so forth. 1341 01:11:26,570 --> 01:11:29,470 And this sort of only breaks down 1342 01:11:29,470 --> 01:11:31,930 when we have very narrow nanotubes 1343 01:11:31,930 --> 01:11:35,680 in which this sort of model picture of what is called band 1344 01:11:35,680 --> 01:11:39,380 folding starts to break apart, because the way we have thought 1345 01:11:39,380 --> 01:11:43,750 of the nanotube is as being identical to a perfect sheet 1346 01:11:43,750 --> 01:11:48,400 of graphene that has been quantized in one direction. 1347 01:11:48,400 --> 01:11:50,120 But really, there is curvature. 1348 01:11:50,120 --> 01:11:53,000 And if the nanotube is narrow, there is a lot of curvature. 1349 01:11:53,000 --> 01:11:56,620 And so that sort of might change things around a little bit. 1350 01:11:56,620 --> 01:11:58,060 But so here is an example. 1351 01:11:58,060 --> 01:12:00,790 You see, if you have a metallic nanotube, 1352 01:12:00,790 --> 01:12:04,720 like in this case, when you quantize it in one direction, 1353 01:12:04,720 --> 01:12:06,850 you are rolling it, you actually end up 1354 01:12:06,850 --> 01:12:10,430 with lines that go through the black points. 1355 01:12:10,430 --> 01:12:14,170 And so what you have is that your band structure there 1356 01:12:14,170 --> 01:12:19,120 sort of sees these cones of the graphite system. 1357 01:12:19,120 --> 01:12:22,750 If you have, say, in this case a semiconducting a0 system, when 1358 01:12:22,750 --> 01:12:25,600 you quantize anything in that perpendicular direction, 1359 01:12:25,600 --> 01:12:28,510 your lines actually do not go through the black points. 1360 01:12:28,510 --> 01:12:32,020 And so you never encounter a corner, 1361 01:12:32,020 --> 01:12:34,180 and you never find that gap. 1362 01:12:34,180 --> 01:12:36,580 And again, a very slight change in orientation 1363 01:12:36,580 --> 01:12:40,210 can give you a very variable gap and can bring you 1364 01:12:40,210 --> 01:12:43,750 to be metallic very quickly. 1365 01:12:43,750 --> 01:12:46,770 And so we start really going again into the details. 1366 01:12:46,770 --> 01:12:52,650 We really need to take all our occupied states-- 1367 01:12:52,650 --> 01:12:55,590 that is, the one below here that I'm not showing, 1368 01:12:55,590 --> 01:12:58,590 everything below the Fermi energy 1369 01:12:58,590 --> 01:13:01,110 that is here for this metallic nanotube-- 1370 01:13:01,110 --> 01:13:04,620 and we actually want to take a little bit of the states 1371 01:13:04,620 --> 01:13:08,220 above, because those are also important. 1372 01:13:08,220 --> 01:13:12,450 And with all these states, the red, the blue, and the black-- 1373 01:13:12,450 --> 01:13:15,540 so all of the occupied states and a bit of the unoccupied 1374 01:13:15,540 --> 01:13:16,110 one-- 1375 01:13:16,110 --> 01:13:19,290 we want to sort of mix them together 1376 01:13:19,290 --> 01:13:24,870 and find what are the localized orbitals coming from this. 1377 01:13:24,870 --> 01:13:30,660 And I won't go into the details on how we sort of extracted 1378 01:13:30,660 --> 01:13:34,740 those when you have some of the discontinuity that comes about 1379 01:13:34,740 --> 01:13:35,730 in a metal. 1380 01:13:35,730 --> 01:13:37,890 But so basically, what's happening 1381 01:13:37,890 --> 01:13:40,350 is that we start from our electronic structure 1382 01:13:40,350 --> 01:13:43,170 calculation, maybe of the carbon nanotube 1383 01:13:43,170 --> 01:13:46,680 with a functionalized ligand. 1384 01:13:46,680 --> 01:13:49,830 And in a new quantum mechanical representation, 1385 01:13:49,830 --> 01:13:53,130 what we have obtained are sort of the building blocks 1386 01:13:53,130 --> 01:13:55,890 of the pristine nanotube, and the building 1387 01:13:55,890 --> 01:14:01,080 blocks of the ligand, and things like what is exactly 1388 01:14:01,080 --> 01:14:03,510 the orbital that connects a carbon 1389 01:14:03,510 --> 01:14:06,900 atom in the ligand to a carbon atom in the sidewall, 1390 01:14:06,900 --> 01:14:10,270 with sort of full first principle precision. 1391 01:14:10,270 --> 01:14:12,690 And so with these blocks now, we can actually 1392 01:14:12,690 --> 01:14:16,590 construct not only the band structure 1393 01:14:16,590 --> 01:14:19,410 and the electronic structure of this system periodically 1394 01:14:19,410 --> 01:14:24,030 repeated, but also a system in which the ligand is first here, 1395 01:14:24,030 --> 01:14:27,630 here, and then here, with all the chemical disorder 1396 01:14:27,630 --> 01:14:28,900 that we want. 1397 01:14:28,900 --> 01:14:32,340 And because the orbitals do not overlap with each other, 1398 01:14:32,340 --> 01:14:38,190 the computational cost of doing this is very, very reduced. 1399 01:14:38,190 --> 01:14:40,860 Because basically, in our new representation, 1400 01:14:40,860 --> 01:14:45,270 orbitals that are far away are orthogonal with each other, 1401 01:14:45,270 --> 01:14:48,210 when we actually represent the Hamiltonian-- 1402 01:14:48,210 --> 01:14:51,210 remember, in a lot of your PWSCF calculation, 1403 01:14:51,210 --> 01:14:53,190 you have represented the Hamiltonian 1404 01:14:53,190 --> 01:14:55,890 in a plane wave basis set. 1405 01:14:55,890 --> 01:14:58,050 And so in a plane wave basis set, 1406 01:14:58,050 --> 01:15:01,880 your Hamiltonian has the diagonally dominant part, 1407 01:15:01,880 --> 01:15:03,720 that is g squared over 2. 1408 01:15:03,720 --> 01:15:07,080 But then it also terms all over the place 1409 01:15:07,080 --> 01:15:10,110 that corresponds to the Fourier transform of the potential. 1410 01:15:10,110 --> 01:15:14,370 When we actually represent the Hamiltonian 1411 01:15:14,370 --> 01:15:18,150 on this basis of localized orbitals, what you have 1412 01:15:18,150 --> 01:15:21,510 is that this is truly diagonally dominant. 1413 01:15:21,510 --> 01:15:24,690 And as soon as we consider orbitals 1414 01:15:24,690 --> 01:15:27,070 that are far away from each other, 1415 01:15:27,070 --> 01:15:30,220 there is no Hamiltonian matrix between them. 1416 01:15:30,220 --> 01:15:32,910 If we calculate the matrix element between this, 1417 01:15:32,910 --> 01:15:34,920 the Hamiltonian, and this green one, 1418 01:15:34,920 --> 01:15:38,620 it's going to be 0, because they really do not overlap. 1419 01:15:38,620 --> 01:15:42,660 So this is what is called a sparse diagonally dominant 1420 01:15:42,660 --> 01:15:43,620 Hamiltonian. 1421 01:15:43,620 --> 01:15:46,380 And diagonalizing this is something 1422 01:15:46,380 --> 01:15:49,500 that actually doesn't require cubic scaling 1423 01:15:49,500 --> 01:15:51,500 but only requires linear scaling. 1424 01:15:51,500 --> 01:15:55,800 So one can really sort of study very large systems. 1425 01:15:55,800 --> 01:15:59,430 And to show you an enlargement of a cutout 1426 01:15:59,430 --> 01:16:03,130 from a perfect nanotube, you see, this is what you get. 1427 01:16:03,130 --> 01:16:08,550 You get orbitals that look either as beautiful sort 1428 01:16:08,550 --> 01:16:12,210 of bonding combination of sp2 orbitals-- 1429 01:16:12,210 --> 01:16:14,220 if you think, we know from chemistry 1430 01:16:14,220 --> 01:16:17,280 that the carbon is sp2 hybridized. 1431 01:16:17,280 --> 01:16:19,800 So it means that the graphitic backbone 1432 01:16:19,800 --> 01:16:22,170 is given by a bonding combination 1433 01:16:22,170 --> 01:16:26,160 of an sp2 orbital sitting here and an sp2 sitting here. 1434 01:16:26,160 --> 01:16:28,050 And the same happens here. 1435 01:16:28,050 --> 01:16:33,730 And then there is a set of pz orbitals sitting on each tube. 1436 01:16:33,730 --> 01:16:35,580 And when those mix together, they 1437 01:16:35,580 --> 01:16:39,490 give rise to the metallic graphitic-like bonds. 1438 01:16:39,490 --> 01:16:42,360 And again, this looks a lot like a pz orbital, 1439 01:16:42,360 --> 01:16:47,070 but it is nowhere similar to the pz orbital of a carbon atom. 1440 01:16:47,070 --> 01:16:51,220 It's the pz orbital specific to a carbon nanotube. 1441 01:16:51,220 --> 01:16:54,390 So with this, we can-- as previously 1442 01:16:54,390 --> 01:16:58,290 in the case of the alloys, we can first validate our system. 1443 01:16:58,290 --> 01:17:00,540 That is, we can start constructing 1444 01:17:00,540 --> 01:17:02,550 the electronic structure of something 1445 01:17:02,550 --> 01:17:05,730 that we can also fully calculate from first principle. 1446 01:17:05,730 --> 01:17:09,060 And we can compare them and these continuous lines 1447 01:17:09,060 --> 01:17:13,380 from the models and dots from the full calculation. 1448 01:17:13,380 --> 01:17:16,230 And you see, the model works perfectly 1449 01:17:16,230 --> 01:17:19,920 and doesn't describe the higher energy 1450 01:17:19,920 --> 01:17:21,900 bonds that have not been included 1451 01:17:21,900 --> 01:17:25,020 in our sort of orbital construction procedure. 1452 01:17:25,020 --> 01:17:27,780 But for all the bonds that we wanted to consider, 1453 01:17:27,780 --> 01:17:29,280 this is perfect. 1454 01:17:29,280 --> 01:17:33,390 And once we sort of feel confident that this works, 1455 01:17:33,390 --> 01:17:36,511 we can actually use it to calculate the-- 1456 01:17:39,160 --> 01:17:41,860 let me actually go farther-- 1457 01:17:41,860 --> 01:17:44,050 we can actually use it to calculate 1458 01:17:44,050 --> 01:17:47,470 the electronic structure of systems 1459 01:17:47,470 --> 01:17:52,010 that are too large to attack directly, something like this. 1460 01:17:52,010 --> 01:17:56,320 This has just a conducting region of 3,000 atoms 1461 01:17:56,320 --> 01:17:58,870 and a very long sort of semi-infinite leads 1462 01:17:58,870 --> 01:17:59,680 on the side. 1463 01:17:59,680 --> 01:18:02,455 And we can figure out a lot of what happens, 1464 01:18:02,455 --> 01:18:05,650 say, to the ballistic conductivity of this system. 1465 01:18:05,650 --> 01:18:08,420 And I'm not really going into the details. 1466 01:18:08,420 --> 01:18:12,730 I wanted to go before concluding to a previous slide 1467 01:18:12,730 --> 01:18:17,920 here, in which I was actually showing how, 1468 01:18:17,920 --> 01:18:21,550 in practice, the matrix element between a localized 1469 01:18:21,550 --> 01:18:24,280 orbital and another localized orbital 1470 01:18:24,280 --> 01:18:26,730 changes as a function of the distance, 1471 01:18:26,730 --> 01:18:30,550 say, as we sit on a carbon atom, and we 1472 01:18:30,550 --> 01:18:33,550 look at the overlap between the pz orbital here 1473 01:18:33,550 --> 01:18:36,400 and we move away from the nanotube. 1474 01:18:36,400 --> 01:18:39,940 And you see, in a semi-logarithmic scale, 1475 01:18:39,940 --> 01:18:43,000 we have very good exponential decay. 1476 01:18:43,000 --> 01:18:45,760 Certain point goes into the numerical noise. 1477 01:18:45,760 --> 01:18:47,920 And we can really look at-- 1478 01:18:47,920 --> 01:18:53,860 validate what are the effects of our truncation by cutting 1479 01:18:53,860 --> 01:18:58,000 shorter and shorter Hamiltonians, by neglecting 1480 01:18:58,000 --> 01:19:00,820 a large number of terms that are far away. 1481 01:19:00,820 --> 01:19:03,775 And you see, there is almost an invisible effect 1482 01:19:03,775 --> 01:19:05,040 on the band structure. 1483 01:19:05,040 --> 01:19:09,100 So that really sort of tells us how much localized things are, 1484 01:19:09,100 --> 01:19:11,320 how much they overlap, and if you want, 1485 01:19:11,320 --> 01:19:15,790 tells us how long range or short range chemistry is. 1486 01:19:15,790 --> 01:19:18,700 That is very important when we consider 1487 01:19:18,700 --> 01:19:20,930 one of this configuration. 1488 01:19:20,930 --> 01:19:22,870 So with this, I'll conclude. 1489 01:19:22,870 --> 01:19:25,570 Again, the two important messages 1490 01:19:25,570 --> 01:19:28,330 that you want to remember from this class 1491 01:19:28,330 --> 01:19:31,490 is how you take a physical problem, 1492 01:19:31,490 --> 01:19:33,970 and you can use density functional calculation 1493 01:19:33,970 --> 01:19:36,610 to find out what are the important parameters 1494 01:19:36,610 --> 01:19:39,320 and construct a model for your energy. 1495 01:19:39,320 --> 01:19:42,400 The first case where configurational variables 1496 01:19:42,400 --> 01:19:47,890 inducing relaxation and gives us Ising-like Hamiltonian, 1497 01:19:47,890 --> 01:19:50,260 in this case, we have localized orbitals. 1498 01:19:50,260 --> 01:19:54,040 And once we determine those, from a small calculation 1499 01:19:54,040 --> 01:19:55,900 or from a fragment, we can actually 1500 01:19:55,900 --> 01:19:58,622 calculate the very large system.