1 00:00:00,050 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,350 Your support will help MIT OpenCourseWare 4 00:00:06,350 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,209 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,209 --> 00:00:17,834 at ocw.mit.edu. 8 00:00:25,620 --> 00:00:27,870 PROFESSOR: We'll be talking about the diode and charge 9 00:00:27,870 --> 00:00:29,830 separation inside of a solar cell. 10 00:00:29,830 --> 00:00:31,540 So at the end of last class, we ended 11 00:00:31,540 --> 00:00:33,680 with this little demonstration right here, 12 00:00:33,680 --> 00:00:36,690 where after much discussion about band 13 00:00:36,690 --> 00:00:38,990 gaps and light absorption, we agreed 14 00:00:38,990 --> 00:00:40,610 that that little piece of silicon 15 00:00:40,610 --> 00:00:42,080 there should be absorbing the light 16 00:00:42,080 --> 00:00:46,420 and free charges should be generated inside the silicon. 17 00:00:46,420 --> 00:00:48,800 But without some fixed fields, some 18 00:00:48,800 --> 00:00:51,810 built in field inside of the silicon material, 19 00:00:51,810 --> 00:00:54,110 it wasn't possible to measure any current output 20 00:00:54,110 --> 00:00:55,380 once we shown light on it. 21 00:00:55,380 --> 00:00:56,620 And that made sense. 22 00:00:56,620 --> 00:00:58,490 There were free charges being generated, 23 00:00:58,490 --> 00:01:02,090 but due to Brownian motion, there was no net charge flow. 24 00:01:02,090 --> 00:01:04,267 They were just moving around with no net movement. 25 00:01:04,267 --> 00:01:05,850 And as a result, there was no current. 26 00:01:05,850 --> 00:01:08,580 Now if we attach the batteries in series, 27 00:01:08,580 --> 00:01:11,160 the batteries apply to field across this little piece 28 00:01:11,160 --> 00:01:15,110 of silicon resulting in current flow-- more 29 00:01:15,110 --> 00:01:18,630 precisely, drift current across that piece of silicon. 30 00:01:18,630 --> 00:01:20,580 And today we're going to be talking 31 00:01:20,580 --> 00:01:26,060 about how the solar cell device has a built in electric field. 32 00:01:26,060 --> 00:01:28,407 So obviously we don't want an external power source. 33 00:01:28,407 --> 00:01:30,490 We don't want to have a set of batteries attached. 34 00:01:30,490 --> 00:01:34,130 That defeats the purpose of an autonomous energy generation 35 00:01:34,130 --> 00:01:34,986 device. 36 00:01:34,986 --> 00:01:36,860 We'll be talking about how the built in field 37 00:01:36,860 --> 00:01:38,670 inside of a solar cell comes into being. 38 00:01:41,340 --> 00:01:43,910 As we've followed in the last few classes, 39 00:01:43,910 --> 00:01:47,384 we will be discussing physics when it is necessary, 40 00:01:47,384 --> 00:01:48,800 taking a very engineering approach 41 00:01:48,800 --> 00:01:51,802 to this so as to keep everybody from a business background, 42 00:01:51,802 --> 00:01:53,510 from a mechanical engineering background, 43 00:01:53,510 --> 00:01:56,420 and from a material science background on the same page. 44 00:01:56,420 --> 00:02:00,330 We will have, in the lecture, I think n plus 2. 45 00:02:00,330 --> 00:02:02,970 Very physics-y lecture, and we'll 46 00:02:02,970 --> 00:02:05,360 be getting to the nitty-gritty of semiconductor physics 47 00:02:05,360 --> 00:02:06,940 and how it relates to solar cell devices. 48 00:02:06,940 --> 00:02:08,731 So for those of you who are already experts 49 00:02:08,731 --> 00:02:10,199 in this material, bear with me. 50 00:02:10,199 --> 00:02:12,816 And again, try to relate to the solar cell device. 51 00:02:12,816 --> 00:02:15,190 You may not have had that connection in previous lectures 52 00:02:15,190 --> 00:02:17,040 in previous classes. 53 00:02:17,040 --> 00:02:18,230 So we'll jump right on in. 54 00:02:18,230 --> 00:02:20,640 To remind everybody, you're here in the fundamentals. 55 00:02:20,640 --> 00:02:22,140 We'll be getting to the technologies 56 00:02:22,140 --> 00:02:24,040 and cross cutting themes after we really 57 00:02:24,040 --> 00:02:26,980 have a good, solid understanding of how a solar cell works. 58 00:02:26,980 --> 00:02:29,975 The conversion efficiency is the output energy versus the input. 59 00:02:29,975 --> 00:02:32,600 We have our inputs in the solar spectrum, the outputs in charge 60 00:02:32,600 --> 00:02:34,820 collection, and we've been steadily making progress 61 00:02:34,820 --> 00:02:37,020 down toward the outputs here. 62 00:02:37,020 --> 00:02:38,580 We discussed the solar spectrum, then 63 00:02:38,580 --> 00:02:40,270 light absorption, the charge excitation, 64 00:02:40,270 --> 00:02:45,330 and finally now we're on to charge drift and diffusion. 65 00:02:45,330 --> 00:02:49,310 So we have the total solar cell efficiency 66 00:02:49,310 --> 00:02:51,390 as a product of all the individual processes. 67 00:02:51,390 --> 00:02:53,480 And any one of these processes can kill 68 00:02:53,480 --> 00:02:54,784 the efficiency of the device. 69 00:02:54,784 --> 00:02:56,950 That's why it's important to think about your device 70 00:02:56,950 --> 00:02:57,970 like so. 71 00:02:57,970 --> 00:03:00,530 And just not to make this introduction so boring, 72 00:03:00,530 --> 00:03:02,190 I want to really emphasize this point. 73 00:03:02,190 --> 00:03:04,090 So I'm making it over and over again. 74 00:03:04,090 --> 00:03:07,530 But I wanted to relate this to other engineering devices 75 00:03:07,530 --> 00:03:08,440 as well. 76 00:03:08,440 --> 00:03:10,730 Namely, today it's a Toyota Prius. 77 00:03:10,730 --> 00:03:12,770 These are all the components inside of a Prius 78 00:03:12,770 --> 00:03:15,870 that have to work well for the car to function. 79 00:03:15,870 --> 00:03:18,320 If one of these components, let's say 80 00:03:18,320 --> 00:03:22,750 this one, the inverter, is broken or not functioning well, 81 00:03:22,750 --> 00:03:26,240 you're not going to have the car in an autonomous mode. 82 00:03:26,240 --> 00:03:28,100 So in a similar manner, in a solar cell 83 00:03:28,100 --> 00:03:30,100 we have to have all the different pieces working 84 00:03:30,100 --> 00:03:32,490 well together. 85 00:03:32,490 --> 00:03:34,180 So the essence of charge separation, 86 00:03:34,180 --> 00:03:36,920 we're going to begin our exploration of charge 87 00:03:36,920 --> 00:03:40,210 separation using the diode analogy. 88 00:03:40,210 --> 00:03:43,170 And just to situate everybody, I brought 89 00:03:43,170 --> 00:03:47,090 in a number of small discrete components, small diodes. 90 00:03:47,090 --> 00:03:49,890 Those are the ones that are orange-ish 91 00:03:49,890 --> 00:03:51,460 that have two leads coming out. 92 00:03:51,460 --> 00:03:53,030 There are a bunch of transistors in here as well. 93 00:03:53,030 --> 00:03:54,739 Those have three little leads coming out. 94 00:03:54,739 --> 00:03:56,196 You'll be able to distinguish them. 95 00:03:56,196 --> 00:03:58,320 But just to situate ourselves, these are diodes. 96 00:03:58,320 --> 00:04:02,660 The essence of a diode is that you have a dissimilar material 97 00:04:02,660 --> 00:04:05,340 on either side of the device. 98 00:04:05,340 --> 00:04:07,510 That's why I've worn these dissimilar colors today 99 00:04:07,510 --> 00:04:09,927 to denote that on different sides. 100 00:04:09,927 --> 00:04:12,260 And of course, you have this mixed region in the middle. 101 00:04:12,260 --> 00:04:15,680 We have here an n and a p on either side. 102 00:04:15,680 --> 00:04:18,040 And if current is attempting to flow in one direction, 103 00:04:18,040 --> 00:04:18,640 it will be barred. 104 00:04:18,640 --> 00:04:20,723 But if it attempts to flow in the other direction, 105 00:04:20,723 --> 00:04:22,360 it will go through rather easily. 106 00:04:22,360 --> 00:04:25,240 That's the essence of what a diode is. 107 00:04:25,240 --> 00:04:26,640 How is it made? 108 00:04:26,640 --> 00:04:29,150 Well, we're manufacturing materials 109 00:04:29,150 --> 00:04:31,660 of the same base element. 110 00:04:31,660 --> 00:04:32,930 Let's say, silicon. 111 00:04:32,930 --> 00:04:34,630 But we're doing something special 112 00:04:34,630 --> 00:04:37,650 to the material to add particular types of charges 113 00:04:37,650 --> 00:04:38,300 on either side. 114 00:04:38,300 --> 00:04:39,150 We call doping. 115 00:04:39,150 --> 00:04:41,210 And we'll get to that in a few slides. 116 00:04:41,210 --> 00:04:42,430 Why do we care about diodes? 117 00:04:42,430 --> 00:04:45,520 Well, this is the essence of charge separation. 118 00:04:45,520 --> 00:04:48,160 And this is what drives the voltage inside of a solar cell 119 00:04:48,160 --> 00:04:48,660 device. 120 00:04:48,660 --> 00:04:49,750 That's why we care. 121 00:04:49,750 --> 00:04:52,170 It's pretty important for at least understanding 122 00:04:52,170 --> 00:04:55,810 the traditional semiconductor-based solar cells 123 00:04:55,810 --> 00:04:58,550 like this one here. 124 00:04:58,550 --> 00:05:00,840 So for those history folks, I figured 125 00:05:00,840 --> 00:05:03,350 I would add a quick at description. 126 00:05:03,350 --> 00:05:05,120 You typically see the diode represented 127 00:05:05,120 --> 00:05:06,650 like this in an equivalent circuit 128 00:05:06,650 --> 00:05:11,630 diagram, a little triangle and a line orthogonal 129 00:05:11,630 --> 00:05:13,480 to the direction of the current path. 130 00:05:13,480 --> 00:05:18,220 And does anybody know where that comes from? 131 00:05:18,220 --> 00:05:19,620 No? 132 00:05:19,620 --> 00:05:20,280 All right. 133 00:05:20,280 --> 00:05:25,030 So back in the day of that, we used 134 00:05:25,030 --> 00:05:28,151 to have vacuum tube components for a lot of our-- I 135 00:05:28,151 --> 00:05:31,660 would say the discrete components within our circuits. 136 00:05:31,660 --> 00:05:35,590 In the case of a diode, you could envision a very simple 137 00:05:35,590 --> 00:05:39,290 one where you have a filament that by thermionic emission 138 00:05:39,290 --> 00:05:41,630 heats electrons off of the filament, 139 00:05:41,630 --> 00:05:46,050 and then they're collected by this other collector up top. 140 00:05:46,050 --> 00:05:50,400 If you remember our etymology in Greek, ana is above, 141 00:05:50,400 --> 00:05:51,510 cata is below. 142 00:05:51,510 --> 00:05:52,470 Catatombs, right? 143 00:05:52,470 --> 00:05:53,950 Catatonic, below. 144 00:05:53,950 --> 00:05:56,520 So the anode up above is collecting these electrons, 145 00:05:56,520 --> 00:05:58,510 and then the electron flow is moving 146 00:05:58,510 --> 00:06:01,090 like this, which means that our current flow, defined 147 00:06:01,090 --> 00:06:02,810 as the flow of positive charge, is 148 00:06:02,810 --> 00:06:04,460 moving the opposite direction. 149 00:06:04,460 --> 00:06:07,950 And that's why we have a similar representation right here 150 00:06:07,950 --> 00:06:09,260 from our vacuum technology. 151 00:06:09,260 --> 00:06:11,080 You could envision also a point where 152 00:06:11,080 --> 00:06:13,100 the field is concentrated at that tip 153 00:06:13,100 --> 00:06:15,910 and electrons are spreading off. 154 00:06:15,910 --> 00:06:17,437 So learning objectives. 155 00:06:17,437 --> 00:06:19,520 Today is going to be rather intense, rather dense. 156 00:06:19,520 --> 00:06:24,650 And so we will be trying to hold as much as possible in our RAM, 157 00:06:24,650 --> 00:06:27,840 so that by the end of class we can really truly understand 158 00:06:27,840 --> 00:06:29,650 the solar cell as an entirety. 159 00:06:29,650 --> 00:06:32,190 If you get lost along the way, come back 160 00:06:32,190 --> 00:06:33,510 to these learning objectives. 161 00:06:33,510 --> 00:06:35,780 They're like flag posts along the way. 162 00:06:35,780 --> 00:06:38,284 So you can find yourself again. 163 00:06:38,284 --> 00:06:39,950 What we're going to do first is describe 164 00:06:39,950 --> 00:06:41,832 how the conductivity of a semiconductor 165 00:06:41,832 --> 00:06:43,790 can be modified by the intentional introduction 166 00:06:43,790 --> 00:06:44,960 of dopants. 167 00:06:44,960 --> 00:06:46,630 What this means is we're going to learn 168 00:06:46,630 --> 00:06:49,170 how to create the n and the p over here. 169 00:06:52,380 --> 00:06:56,840 So if we look at a semiconductor such as silicon-- 170 00:06:56,840 --> 00:06:58,730 this is a little piece of silicon right here. 171 00:06:58,730 --> 00:07:00,610 Here's a silicon-based solar cell. 172 00:07:00,610 --> 00:07:02,992 It's a fairly easy semiconductor to understand. 173 00:07:02,992 --> 00:07:04,700 It's what's called a unary semiconductor. 174 00:07:04,700 --> 00:07:06,650 It means it's comprised of one element. 175 00:07:06,650 --> 00:07:08,440 Silicon right here in the periodic table. 176 00:07:08,440 --> 00:07:10,940 This is just an excerpt from the rightmost side 177 00:07:10,940 --> 00:07:12,590 of the periodic table. 178 00:07:12,590 --> 00:07:15,300 And silicon has 4 valence electrons. 179 00:07:15,300 --> 00:07:18,340 So it forms what are called sp3 hybridized orbitals when 180 00:07:18,340 --> 00:07:19,510 it's in a crystal structure. 181 00:07:19,510 --> 00:07:23,460 It has four bonds with its nearest neighbors, 182 00:07:23,460 --> 00:07:24,820 all of equivalent type. 183 00:07:24,820 --> 00:07:26,740 And those are covalent bonds. 184 00:07:26,740 --> 00:07:29,740 So you can envision that if you were to introduce an atom 185 00:07:29,740 --> 00:07:32,200 and substitute out one silicon atom in that lattice 186 00:07:32,200 --> 00:07:35,430 for something else that has 5 valence electrons, 187 00:07:35,430 --> 00:07:39,450 such as a phosphorus, it's almost of equivalent size. 188 00:07:39,450 --> 00:07:41,300 So those material scientists in the room 189 00:07:41,300 --> 00:07:42,950 from the Hume-Rothery rules, you should 190 00:07:42,950 --> 00:07:45,450 be able to estimate that the miscibility is rather large. 191 00:07:45,450 --> 00:07:48,140 In other words, that you could mix in a high concentration 192 00:07:48,140 --> 00:07:51,670 of phosphorus into your silicon given the similar size 193 00:07:51,670 --> 00:07:54,420 and similar atomic structure. 194 00:07:54,420 --> 00:07:56,050 Electronic structure, rather. 195 00:07:56,050 --> 00:07:59,440 So you substitute here a group 5 element. 196 00:07:59,440 --> 00:08:03,400 Silicon is a group 4 element, as is carbon, germanium, and tin. 197 00:08:03,400 --> 00:08:05,480 So we substitute in a group 5 element. 198 00:08:05,480 --> 00:08:08,670 Let's say a phosphorus atom in for one of our silicon atoms 199 00:08:08,670 --> 00:08:09,810 right here. 200 00:08:09,810 --> 00:08:11,530 And this phosphorus atom will have 201 00:08:11,530 --> 00:08:13,880 4 plus 1 valence electrons. 202 00:08:13,880 --> 00:08:15,660 So those 4 valence electrons will 203 00:08:15,660 --> 00:08:18,300 bond to the silicon atoms, the nearest neighbors. 204 00:08:18,300 --> 00:08:21,170 And that one extra valence electron will be left over. 205 00:08:21,170 --> 00:08:22,745 It will have nobody to bond to. 206 00:08:22,745 --> 00:08:24,540 And we'll describe mathematically 207 00:08:24,540 --> 00:08:26,710 what happens to that electron in a minute. 208 00:08:26,710 --> 00:08:28,550 But in principle, just intuitively you 209 00:08:28,550 --> 00:08:30,800 should be able to understand that that electron should 210 00:08:30,800 --> 00:08:32,258 be able to be removed rather easily 211 00:08:32,258 --> 00:08:34,920 and move around the lattice with relative ease. 212 00:08:34,920 --> 00:08:37,419 Likewise, instead of moving one to the right, if we move one 213 00:08:37,419 --> 00:08:39,600 to the left into our group 3 column here, 214 00:08:39,600 --> 00:08:43,470 we can dope our material with a boron atom, let's say. 215 00:08:43,470 --> 00:08:46,030 Substituting one silicon atom for a boron atom. 216 00:08:46,030 --> 00:08:47,980 Boron has 3 valence electrons. 217 00:08:47,980 --> 00:08:51,830 Now, those 3 will form bonds with the 3 silicon atoms. 218 00:08:51,830 --> 00:08:53,420 And then you'll have a missing bond, 219 00:08:53,420 --> 00:08:55,600 a missing electron if you will. 220 00:08:55,600 --> 00:08:57,590 That will be a hole. 221 00:08:57,590 --> 00:09:00,240 So with phosphorus, or group 5 elements in silicon, 222 00:09:00,240 --> 00:09:03,290 we can dope the material with electrons. 223 00:09:03,290 --> 00:09:05,390 And with group 3 elements in silicon, 224 00:09:05,390 --> 00:09:08,200 we can dope the material with holes. 225 00:09:08,200 --> 00:09:10,180 And this is interesting because now we 226 00:09:10,180 --> 00:09:13,890 can arbitrarily change the density of charge carriers 227 00:09:13,890 --> 00:09:15,370 in our bands. 228 00:09:15,370 --> 00:09:17,530 As long as these holes and electrons 229 00:09:17,530 --> 00:09:20,320 can become dissociated from the dopant atoms 230 00:09:20,320 --> 00:09:21,980 and move freely around the lattice. 231 00:09:21,980 --> 00:09:23,490 So that's a big question, how easy 232 00:09:23,490 --> 00:09:27,300 is it to remove that electron around the phosphorus atom? 233 00:09:27,300 --> 00:09:29,950 If you want to think about this electron just 234 00:09:29,950 --> 00:09:32,950 from a quantum mechanical sense, you 235 00:09:32,950 --> 00:09:36,319 could almost envision that this electron is attracted 236 00:09:36,319 --> 00:09:38,360 to the phosphorus atom because the phosphorus has 237 00:09:38,360 --> 00:09:41,440 one extra proton than the silicon, right? 238 00:09:41,440 --> 00:09:44,730 So it has a positive charge here in the nucleus. 239 00:09:44,730 --> 00:09:46,180 And here's a negative charge. 240 00:09:46,180 --> 00:09:47,810 In net, they balance out. 241 00:09:47,810 --> 00:09:49,929 But if the electron goes too far away, 242 00:09:49,929 --> 00:09:51,720 it will feel that attractive potential back 243 00:09:51,720 --> 00:09:53,840 and be drawn back towards the phosphorus atom. 244 00:09:53,840 --> 00:09:55,923 So the big question is, what is the binding energy 245 00:09:55,923 --> 00:09:57,834 of that electron to the phosphorus atom? 246 00:09:57,834 --> 00:09:59,250 And a simple way to think about it 247 00:09:59,250 --> 00:10:02,010 is through a hydrogenic model. 248 00:10:02,010 --> 00:10:05,570 If you consider a hydrogen atom to have an electron surrounding 249 00:10:05,570 --> 00:10:10,010 it, then that binding energy is well-defined. 250 00:10:10,010 --> 00:10:13,230 But in the case of a phosphorus atom within a silicon lattice, 251 00:10:13,230 --> 00:10:14,510 it's a little trickier. 252 00:10:14,510 --> 00:10:19,380 Because now we have to worry about the electron 253 00:10:19,380 --> 00:10:22,120 screening coming from the other elements in our lattice. 254 00:10:22,120 --> 00:10:25,420 And we also have to worry about a property of the electron 255 00:10:25,420 --> 00:10:29,610 in a crystal, which changes the mobility of that electron 256 00:10:29,610 --> 00:10:30,910 through the crystal. 257 00:10:30,910 --> 00:10:34,610 So what we do is we treat this as a hydrogen 258 00:10:34,610 --> 00:10:38,814 atom, 13.6 eV binding energy. 259 00:10:38,814 --> 00:10:40,230 But then we do a couple of things. 260 00:10:40,230 --> 00:10:41,604 We account for electron screening 261 00:10:41,604 --> 00:10:44,134 and we account for what we call effective mass. 262 00:10:44,134 --> 00:10:45,800 We'll get back to this in a few lectures 263 00:10:45,800 --> 00:10:48,212 and describe exactly what this is here. 264 00:10:48,212 --> 00:10:50,170 We need a little bit more semiconductor physics 265 00:10:50,170 --> 00:10:51,850 to understand it in its entirety. 266 00:10:51,850 --> 00:10:54,380 But think about an electron moving in a crystal 267 00:10:54,380 --> 00:10:55,850 as moving differently than moving 268 00:10:55,850 --> 00:10:57,310 in free space in a vacuum. 269 00:10:57,310 --> 00:10:59,340 That's what this term here is about. 270 00:10:59,340 --> 00:11:01,740 This other term, the epsilon, is rather straightforward. 271 00:11:01,740 --> 00:11:02,470 It's the electron screening. 272 00:11:02,470 --> 00:11:04,595 It's the fact that you have so many other electrons 273 00:11:04,595 --> 00:11:06,730 in your system here that screening the charges, 274 00:11:06,730 --> 00:11:10,420 screening the electron from the extra positive charge 275 00:11:10,420 --> 00:11:12,110 in the nucleus here. 276 00:11:12,110 --> 00:11:15,140 And so it becomes easier for that electron to move away. 277 00:11:15,140 --> 00:11:19,140 This is about a factor of 0.1, the effective mass. 278 00:11:19,140 --> 00:11:21,150 And that's about 1/100. 279 00:11:21,150 --> 00:11:23,210 And so overall, we're reducing the binding energy 280 00:11:23,210 --> 00:11:28,490 by about a factor of 1,000 down to about 10 m eV in practice. 281 00:11:28,490 --> 00:11:30,870 So if we were to run through the calculation for silicon, 282 00:11:30,870 --> 00:11:33,520 the binding energy of that electron around the phosphorus 283 00:11:33,520 --> 00:11:35,730 atom in the silicon lattice would be around 10 m 284 00:11:35,730 --> 00:11:37,470 eV on that order. 285 00:11:37,470 --> 00:11:40,590 And that is very, very small compared 286 00:11:40,590 --> 00:11:44,160 to the thermal energy in your system, just kt, Boltzmann's 287 00:11:44,160 --> 00:11:46,334 constant times the temperature in Kelvin, 288 00:11:46,334 --> 00:11:47,750 we have a thermal energy somewhere 289 00:11:47,750 --> 00:11:49,540 on the order of 26 m eV. 290 00:11:49,540 --> 00:11:51,666 And that's enough to dissociate that electron 291 00:11:51,666 --> 00:11:53,790 and allow it to move freely throughout the lattice. 292 00:11:53,790 --> 00:11:54,625 Question? 293 00:11:54,625 --> 00:11:56,445 AUDIENCE: Is that electron screening 294 00:11:56,445 --> 00:11:58,630 related to the [INAUDIBLE] constant at all? 295 00:11:58,630 --> 00:12:00,410 PROFESSOR: Yes, absolutely. 296 00:12:00,410 --> 00:12:02,201 AUDIENCE: I just had a [INAUDIBLE] question 297 00:12:02,201 --> 00:12:03,500 about the holes. 298 00:12:03,500 --> 00:12:07,560 So I understand that-- I can imagine an electron moving 299 00:12:07,560 --> 00:12:09,340 around in the lattice. 300 00:12:09,340 --> 00:12:11,930 But I imagine that a hole moving would 301 00:12:11,930 --> 00:12:17,668 be forming a bond-- breaking and forming bonds over and over. 302 00:12:17,668 --> 00:12:19,858 Does that actually happen, or are the electrons 303 00:12:19,858 --> 00:12:21,924 more free to move? 304 00:12:21,924 --> 00:12:22,590 PROFESSOR: Yeah. 305 00:12:22,590 --> 00:12:24,160 This is an excellent question. 306 00:12:24,160 --> 00:12:25,830 The question, if I may paraphrase, 307 00:12:25,830 --> 00:12:28,240 relates to the hole and electron mobilities. 308 00:12:28,240 --> 00:12:30,449 How easy is it for them to move around the lattice? 309 00:12:30,449 --> 00:12:32,490 And typically, the hole mobility is about a third 310 00:12:32,490 --> 00:12:34,200 of the electron mobility in silicon. 311 00:12:34,200 --> 00:12:37,060 So absolutely, it is a little bit more difficult 312 00:12:37,060 --> 00:12:38,500 for those holes to move around. 313 00:12:38,500 --> 00:12:42,130 Holes, you may recall, are quasi-particles. 314 00:12:42,130 --> 00:12:45,653 OK, so we have an understanding of doping, 315 00:12:45,653 --> 00:12:47,870 a rough understanding at this point. 316 00:12:47,870 --> 00:12:51,420 We understand that if we dope a certain type of material, let's 317 00:12:51,420 --> 00:12:54,940 say our n-type material, with phosphorus, 318 00:12:54,940 --> 00:12:57,000 we have an excess of electrons. 319 00:12:57,000 --> 00:12:59,120 And if we dope another type of material 320 00:12:59,120 --> 00:13:03,160 with an excess of, say, boron, we would then 321 00:13:03,160 --> 00:13:04,900 introduce a large number of holes 322 00:13:04,900 --> 00:13:08,350 and the material might have a majority conductivity 323 00:13:08,350 --> 00:13:09,100 through our holes. 324 00:13:09,100 --> 00:13:11,080 Hence, we call it p-type. 325 00:13:11,080 --> 00:13:13,920 n and p comes from negative and positive. 326 00:13:13,920 --> 00:13:15,790 Negative being the electron charge 327 00:13:15,790 --> 00:13:17,660 and positive being the hole charge. 328 00:13:17,660 --> 00:13:20,291 So that's where the n- and p-type come from. 329 00:13:20,291 --> 00:13:21,340 Great. 330 00:13:21,340 --> 00:13:25,200 So now, pictorially we're going to draw 331 00:13:25,200 --> 00:13:30,800 a pn-junction or a junction with our fixed and mobile charges. 332 00:13:30,800 --> 00:13:37,010 OK, so let's start out with a simple review of Gauss' law. 333 00:13:37,010 --> 00:13:39,810 Spatially variant fixed charge creates an electric field. 334 00:13:39,810 --> 00:13:41,130 What do I mean by that? 335 00:13:41,130 --> 00:13:43,870 I mean that if you have a high concentration 336 00:13:43,870 --> 00:13:46,370 of fixed charge over here and a lower concentration of fixed 337 00:13:46,370 --> 00:13:49,209 charge over here, an electric field will develop. 338 00:13:49,209 --> 00:13:50,750 And that's the expression right there 339 00:13:50,750 --> 00:13:55,190 that relates-- that describes this in mathematical terms 340 00:13:55,190 --> 00:13:58,330 where we're using psi here as our electric field. 341 00:13:58,330 --> 00:14:00,200 We're using psi instead of e. 342 00:14:00,200 --> 00:14:02,060 You probably remember your physics textbooks 343 00:14:02,060 --> 00:14:04,060 using e as the electric field because we're 344 00:14:04,060 --> 00:14:07,100 going to reserve that variable for the electron energy. 345 00:14:07,100 --> 00:14:10,500 That will come in several slides forward. 346 00:14:10,500 --> 00:14:12,580 So the charge density will be given 347 00:14:12,580 --> 00:14:16,640 by that rho and the material permittivity by our epsilon. 348 00:14:16,640 --> 00:14:18,690 And an example is a capacitor here. 349 00:14:18,690 --> 00:14:21,710 All I've done is expanded the dimensionality 350 00:14:21,710 --> 00:14:23,692 of the derivative here. 351 00:14:23,692 --> 00:14:24,900 I'm looking at the capacitor. 352 00:14:24,900 --> 00:14:26,490 I have fixed charge on either side. 353 00:14:26,490 --> 00:14:29,850 And I have electric field in between. 354 00:14:29,850 --> 00:14:35,010 Now, this right here, obviously in the case of a capacity, 355 00:14:35,010 --> 00:14:36,670 you're under high vacuum. 356 00:14:36,670 --> 00:14:40,560 Just as a very quick aside, this is-- I can't resist. 357 00:14:40,560 --> 00:14:43,690 This is really an interesting historical note. 358 00:14:43,690 --> 00:14:45,979 This space right here in the diode as we recall, 359 00:14:45,979 --> 00:14:48,520 where you have the thermionic emission of the electrons going 360 00:14:48,520 --> 00:14:50,853 off to the other plate, this space right instead of here 361 00:14:50,853 --> 00:14:54,070 is called the space charge region. 362 00:14:54,070 --> 00:14:55,290 Keep that name in mind. 363 00:14:55,290 --> 00:14:58,230 We're going to come back to it in several slides. 364 00:14:58,230 --> 00:14:59,920 So a bit of history there. 365 00:14:59,920 --> 00:15:03,060 So we have the capacitor as our prime example. 366 00:15:03,060 --> 00:15:05,310 We have fixed charge on either side and electric field 367 00:15:05,310 --> 00:15:06,310 in between. 368 00:15:06,310 --> 00:15:08,770 And note that the charge will move parallel 369 00:15:08,770 --> 00:15:10,100 to the electric field. 370 00:15:10,100 --> 00:15:12,150 If there's no electric field-- in other words, 371 00:15:12,150 --> 00:15:15,036 if we're just exciting charge inside of a bare piece 372 00:15:15,036 --> 00:15:16,410 of silicon with no electric field 373 00:15:16,410 --> 00:15:18,310 applied-- imagine our batteries here 374 00:15:18,310 --> 00:15:20,090 are taken out of the circuit. 375 00:15:20,090 --> 00:15:21,970 We'll have simple Brownian motion 376 00:15:21,970 --> 00:15:24,850 of those photo-excited electrons until they decay back down 377 00:15:24,850 --> 00:15:26,200 into their ground state. 378 00:15:26,200 --> 00:15:28,187 However, if we apply an electric field, 379 00:15:28,187 --> 00:15:29,520 now we have the Brownian motion. 380 00:15:29,520 --> 00:15:32,420 But superimposed on top of it, a certain drift 381 00:15:32,420 --> 00:15:35,470 of the electrons in response to that electric field. 382 00:15:35,470 --> 00:15:37,700 And we can describe this mathematically as well. 383 00:15:37,700 --> 00:15:40,600 The drift currents for holes and electrons being 384 00:15:40,600 --> 00:15:42,739 described by the charge q. 385 00:15:42,739 --> 00:15:44,280 Note that the charge q is going to be 386 00:15:44,280 --> 00:15:47,660 different for the electrons and the holes-- the same value, 387 00:15:47,660 --> 00:15:49,060 but negative or positive. 388 00:15:49,060 --> 00:15:53,430 The mobility of holes and mobility of electrons. 389 00:15:53,430 --> 00:15:55,500 The density of holes or electrons. 390 00:15:55,500 --> 00:15:58,380 And then finally, the electric field right here. 391 00:15:58,380 --> 00:16:00,810 And the reason in this equation you typically 392 00:16:00,810 --> 00:16:04,340 see the q's being the same value here, 393 00:16:04,340 --> 00:16:06,932 although no minus sign is put in front, 394 00:16:06,932 --> 00:16:08,640 is because under the same electric field, 395 00:16:08,640 --> 00:16:11,056 you'll have electrons drifting in this direction and holes 396 00:16:11,056 --> 00:16:13,980 drifting in the opposite direction. 397 00:16:13,980 --> 00:16:17,600 So it's important to keep these signs straight in your mind. 398 00:16:17,600 --> 00:16:18,675 Yeah, question? 399 00:16:18,675 --> 00:16:19,830 AUDIENCE: [INAUDIBLE]. 400 00:16:19,830 --> 00:16:20,746 PROFESSOR: Absolutely. 401 00:16:20,746 --> 00:16:22,460 So p and n denote the concentrations 402 00:16:22,460 --> 00:16:24,870 of free electrons or free hole-- actually, 403 00:16:24,870 --> 00:16:27,950 free holes for free electrons, respectively. 404 00:16:27,950 --> 00:16:31,670 So you can almost think of p as being synonymous 405 00:16:31,670 --> 00:16:36,710 with the density of boron atoms. 406 00:16:36,710 --> 00:16:39,450 If every hole is ionized, meaning 407 00:16:39,450 --> 00:16:42,460 dissociated from the dopant atom and free to move around 408 00:16:42,460 --> 00:16:45,256 the crystal, then the density of holes 409 00:16:45,256 --> 00:16:46,630 and the density of boron atoms is 410 00:16:46,630 --> 00:16:48,530 going to be almost identical. 411 00:16:48,530 --> 00:16:50,800 Likewise, for phosphorus atoms and/or 412 00:16:50,800 --> 00:16:53,360 the donor atoms and the electrons, 413 00:16:53,360 --> 00:16:56,860 you'll have almost an equal number. 414 00:16:56,860 --> 00:16:58,710 So that's right here. 415 00:16:58,710 --> 00:17:00,962 The n is related to the number of electrons 416 00:17:00,962 --> 00:17:02,670 that are free to move around the crystal. 417 00:17:02,670 --> 00:17:06,630 The p, the density of holes that are free to move around 418 00:17:06,630 --> 00:17:07,440 in the crystal. 419 00:17:07,440 --> 00:17:12,910 And these are given in units of number of particles charge 420 00:17:12,910 --> 00:17:14,740 carriers per unit volume. 421 00:17:14,740 --> 00:17:16,619 So per centimeter cubed, let's say. 422 00:17:16,619 --> 00:17:17,483 Question. 423 00:17:17,483 --> 00:17:21,347 AUDIENCE: [INAUDIBLE] you're creating more holes. 424 00:17:21,347 --> 00:17:23,659 So is that not on the same order of magnitude 425 00:17:23,659 --> 00:17:25,700 as the number of holes introduced by [INAUDIBLE]? 426 00:17:25,700 --> 00:17:26,866 PROFESSOR: Yeah, absolutely. 427 00:17:26,866 --> 00:17:31,170 So the density of holes and electrons here in the dark 428 00:17:31,170 --> 00:17:34,190 is dictated by the dopant density. 429 00:17:34,190 --> 00:17:34,870 Where are we? 430 00:17:34,870 --> 00:17:36,810 Right here, for instance, the dopant density. 431 00:17:36,810 --> 00:17:40,620 If we add light, that's a complicating factor. 432 00:17:40,620 --> 00:17:44,564 The intrinsic population of electrons and holes 433 00:17:44,564 --> 00:17:46,730 and the dopant concentration of electrons and holes, 434 00:17:46,730 --> 00:17:49,400 we are adding a photo-generated population as well. 435 00:17:49,400 --> 00:17:51,700 Let's leave that for next lecture 436 00:17:51,700 --> 00:17:52,980 and try to simplify things. 437 00:17:52,980 --> 00:17:55,410 So we just discussed the diode in the dark. 438 00:17:55,410 --> 00:17:59,640 Perhaps we turn the light off here just for effect. 439 00:17:59,640 --> 00:18:02,312 So we're trying to take this piece by piece, so that we 440 00:18:02,312 --> 00:18:04,270 can really construct everything and not add too 441 00:18:04,270 --> 00:18:07,390 much into the pot at one time. 442 00:18:07,390 --> 00:18:10,890 I wanted to discuss the notion of electron drift. 443 00:18:10,890 --> 00:18:12,530 And I also wanted to review the concept 444 00:18:12,530 --> 00:18:15,459 of diffusion, which everyone should be familiar with. 445 00:18:15,459 --> 00:18:17,500 This is the reason why we're breathing right now, 446 00:18:17,500 --> 00:18:20,670 and why all the air molecules aren't crowded 447 00:18:20,670 --> 00:18:22,100 into that corner of the room. 448 00:18:22,100 --> 00:18:25,590 It's because of Fick's law, the process of diffusion. 449 00:18:25,590 --> 00:18:29,630 If we have a concentration of a particular species in one 450 00:18:29,630 --> 00:18:33,850 spatial location, the natural tendency through random motion 451 00:18:33,850 --> 00:18:35,880 is for that concentration to distribute 452 00:18:35,880 --> 00:18:38,210 itself equally throughout. 453 00:18:38,210 --> 00:18:40,960 And this is described very nicely by Fick's law. 454 00:18:40,960 --> 00:18:44,040 If we want to describe a current to it, 455 00:18:44,040 --> 00:18:46,160 we would then describe the current of holes 456 00:18:46,160 --> 00:18:50,000 and the current of electrons by, again, the charge. 457 00:18:50,000 --> 00:18:54,160 The diffusivity, this is a quantity typically given 458 00:18:54,160 --> 00:18:56,920 in units of centimeters squared per second. 459 00:18:56,920 --> 00:18:58,080 And the gradient. 460 00:18:58,080 --> 00:19:00,750 In other words, the concentration gradient. 461 00:19:00,750 --> 00:19:02,720 In this case, in one dimension. 462 00:19:02,720 --> 00:19:06,360 And that, again, should be review for most folks 463 00:19:06,360 --> 00:19:07,600 from physics. 464 00:19:07,600 --> 00:19:09,720 And so we can see here two different methods 465 00:19:09,720 --> 00:19:12,597 of currents, two different methods of current flow inside 466 00:19:12,597 --> 00:19:13,680 of a semiconductor device. 467 00:19:13,680 --> 00:19:18,670 We can have diffusion or we can have drift. 468 00:19:18,670 --> 00:19:20,660 Drift occurs when there's an electric field 469 00:19:20,660 --> 00:19:26,500 present and diffusion occurs when-- well, in this case, 470 00:19:26,500 --> 00:19:28,900 diffusion can occur in the absence of an electric field 471 00:19:28,900 --> 00:19:32,350 simply when there is a concentration gradient present. 472 00:19:32,350 --> 00:19:34,840 So we can envision, just quickly, 473 00:19:34,840 --> 00:19:36,970 that if we have an electric field confined 474 00:19:36,970 --> 00:19:39,200 to a certain region of our solar cell device, 475 00:19:39,200 --> 00:19:42,380 the electric field will be dominant in one portion. 476 00:19:42,380 --> 00:19:44,710 And if the rest of our device we don't 477 00:19:44,710 --> 00:19:47,930 have a strong electric field, but the electrons nevertheless 478 00:19:47,930 --> 00:19:49,530 can sense the electric field far away 479 00:19:49,530 --> 00:19:51,370 because of the concentration gradient, 480 00:19:51,370 --> 00:19:53,870 diffusion can drive those electrons-- 481 00:19:53,870 --> 00:19:57,300 can drive those electron toward the electric field. 482 00:19:57,300 --> 00:20:01,190 So in our solar cell device, the ratio of drift and diffusion 483 00:20:01,190 --> 00:20:03,650 current will change as a function of distance 484 00:20:03,650 --> 00:20:08,900 from the pn-junction, from the built-in field. 485 00:20:08,900 --> 00:20:12,600 OK, so I want everybody to upload into their RAM 486 00:20:12,600 --> 00:20:15,180 the checkerboard example that we went through. 487 00:20:15,180 --> 00:20:18,570 Because this will be important for understanding the next few 488 00:20:18,570 --> 00:20:20,510 slides. 489 00:20:20,510 --> 00:20:22,560 So let's imagine these n-and p-type materials are 490 00:20:22,560 --> 00:20:24,490 in contact, but there's this imaginary barrier 491 00:20:24,490 --> 00:20:25,281 right between them. 492 00:20:27,680 --> 00:20:31,560 Here are p-type materials, here are n-type materials. 493 00:20:31,560 --> 00:20:33,690 And let's parse through this figure. 494 00:20:33,690 --> 00:20:35,530 Let's dwell, just a minute. 495 00:20:35,530 --> 00:20:37,010 So we have this imaginary boundary 496 00:20:37,010 --> 00:20:39,650 between p-type material over here and n-type material. 497 00:20:39,650 --> 00:20:41,720 The boundary is right here in the middle. 498 00:20:41,720 --> 00:20:44,800 The blue dots here are representing 499 00:20:44,800 --> 00:20:47,590 mobile holes, holes that are free to move around 500 00:20:47,590 --> 00:20:48,870 the material. 501 00:20:48,870 --> 00:20:51,800 And while there's this imaginary barrier here, 502 00:20:51,800 --> 00:20:53,940 these holes are just moving in Brownian motion. 503 00:20:53,940 --> 00:20:55,680 There's no net current. 504 00:20:55,680 --> 00:20:58,100 There's no net charge flow. 505 00:20:58,100 --> 00:21:01,710 These minus signs that are left behind are the boron atoms. 506 00:21:01,710 --> 00:21:05,160 The boron atoms have one less proton in their nucleus 507 00:21:05,160 --> 00:21:08,000 than the silicon atoms do. 508 00:21:08,000 --> 00:21:10,970 And so the hole plus the negative charge 509 00:21:10,970 --> 00:21:14,050 that's within the boron atom core if you will, 510 00:21:14,050 --> 00:21:16,280 that combination is neutral. 511 00:21:16,280 --> 00:21:20,100 So we haven't perturbed the net charge of the entire system. 512 00:21:20,100 --> 00:21:22,630 But locally, if we draw a little circle around these two 513 00:21:22,630 --> 00:21:25,360 for instance, the boron atom that's fixed, 514 00:21:25,360 --> 00:21:26,300 that's not moving. 515 00:21:26,300 --> 00:21:28,100 The boron atom is bound to the neighboring silicon atoms. 516 00:21:28,100 --> 00:21:29,420 It's not moving around. 517 00:21:29,420 --> 00:21:32,410 And the hole, which is free to move around the material, 518 00:21:32,410 --> 00:21:34,790 now we have net charge neutrality. 519 00:21:34,790 --> 00:21:36,380 But if that hole moves too far away, 520 00:21:36,380 --> 00:21:40,940 then you could have a field building up. 521 00:21:40,940 --> 00:21:42,680 On the other side, very similar. 522 00:21:42,680 --> 00:21:45,250 Here we have phosphorus atoms embedded within our silicon 523 00:21:45,250 --> 00:21:46,070 lattice. 524 00:21:46,070 --> 00:21:48,677 The phosphorus atoms have one extra proton in their nucleus 525 00:21:48,677 --> 00:21:49,760 than the silicon atoms do. 526 00:21:49,760 --> 00:21:53,980 Hence, we denote that as a positive charge right here. 527 00:21:53,980 --> 00:21:57,530 And there's an extra electron associated with the phosphorus 528 00:21:57,530 --> 00:21:58,220 atoms. 529 00:21:58,220 --> 00:22:01,180 Now as those electrons move around, when it's in isolation, 530 00:22:01,180 --> 00:22:02,960 again, there's no net charge flow. 531 00:22:02,960 --> 00:22:04,300 There's no current. 532 00:22:04,300 --> 00:22:09,840 And we have a situation of net charge neutrality. 533 00:22:09,840 --> 00:22:11,880 Question is, what happens if we remove 534 00:22:11,880 --> 00:22:14,260 the barrier in between the two? 535 00:22:14,260 --> 00:22:15,990 Let me ask you just a basic question. 536 00:22:15,990 --> 00:22:19,130 Would the phosphorus atoms diffuse over to this side? 537 00:22:19,130 --> 00:22:19,970 AUDIENCE: No. 538 00:22:19,970 --> 00:22:22,345 PROFESSOR: Probably not, because the phosphorus atoms are 539 00:22:22,345 --> 00:22:24,280 bound to 4 silicon neighbors. 540 00:22:24,280 --> 00:22:26,540 Those bonds are really tough and it's probably not 541 00:22:26,540 --> 00:22:28,270 going to break. 542 00:22:28,270 --> 00:22:30,990 Unless you heat it up to a pretty high temperature. 543 00:22:30,990 --> 00:22:32,809 But the electrons are free to move around. 544 00:22:32,809 --> 00:22:34,100 They're in the conduction band. 545 00:22:34,100 --> 00:22:35,910 They're able to move. 546 00:22:35,910 --> 00:22:38,290 And because the binding energy to the phosphorus atoms 547 00:22:38,290 --> 00:22:41,849 is on that order of 10 m EV on that order, 548 00:22:41,849 --> 00:22:43,640 the thermal energy here at room temperature 549 00:22:43,640 --> 00:22:45,780 is enough to dissociate them, and to allow 550 00:22:45,780 --> 00:22:49,140 those electrons to move away from the phosphorus atom thanks 551 00:22:49,140 --> 00:22:51,400 to the screening potential of the surrounding lattice 552 00:22:51,400 --> 00:22:54,370 and thanks to the fact that the electrons inside of a crystal 553 00:22:54,370 --> 00:22:58,450 are more easily-- can more easily move than in vacuum. 554 00:22:58,450 --> 00:23:01,180 So when we remove that boundary in between, 555 00:23:01,180 --> 00:23:02,950 what begins to happen? 556 00:23:02,950 --> 00:23:04,220 So we've removed the boundary. 557 00:23:04,220 --> 00:23:06,610 We have these two materials in direct contact. 558 00:23:06,610 --> 00:23:09,400 The nuclei are not moving. 559 00:23:09,400 --> 00:23:12,140 The fixed charge associated with the boron atoms 560 00:23:12,140 --> 00:23:14,830 in this side and the phosphorus atoms on that side 561 00:23:14,830 --> 00:23:16,160 are not moving. 562 00:23:16,160 --> 00:23:20,250 However, the instant you remove this boundary in between, 563 00:23:20,250 --> 00:23:22,430 there is a high concentration of electrons over here 564 00:23:22,430 --> 00:23:24,970 and a very low concentration of electrons over there, which 565 00:23:24,970 --> 00:23:28,929 means that you will have a diffusion process. 566 00:23:28,929 --> 00:23:31,220 So the concentration gradient drives the electrons over 567 00:23:31,220 --> 00:23:32,510 to this side. 568 00:23:32,510 --> 00:23:35,340 Likewise, holes are being driven from here over to here. 569 00:23:35,340 --> 00:23:36,580 Now, notice what happens. 570 00:23:36,580 --> 00:23:38,940 Holes carry a positive charge. 571 00:23:38,940 --> 00:23:41,450 The fixed charge from the phosphorus is also positive. 572 00:23:41,450 --> 00:23:43,430 So you have a buildup of net positive charge 573 00:23:43,430 --> 00:23:45,620 on this side of the junction. 574 00:23:45,620 --> 00:23:47,570 Electrons are negative charge carriers. 575 00:23:47,570 --> 00:23:49,880 The boron atoms have a net negative charge here. 576 00:23:49,880 --> 00:23:51,421 And so you have a net negative charge 577 00:23:51,421 --> 00:23:52,770 building up on this side. 578 00:23:52,770 --> 00:23:54,640 And now you have a field beginning 579 00:23:54,640 --> 00:23:57,150 to develop, an electric field. 580 00:23:57,150 --> 00:24:00,220 And at some point, this built-in electric field 581 00:24:00,220 --> 00:24:02,640 will counteract the diffusion process. 582 00:24:02,640 --> 00:24:05,760 The diffusion current is driving electrons in this direction, 583 00:24:05,760 --> 00:24:08,720 but the field will be driving them back the other way. 584 00:24:08,720 --> 00:24:12,360 And it's that equilibrium that establishes the pn-junction. 585 00:24:12,360 --> 00:24:14,200 It's that equilibrium that defines 586 00:24:14,200 --> 00:24:18,820 the width of this region here, this transition region, which 587 00:24:18,820 --> 00:24:20,390 comes in a variety of names. 588 00:24:20,390 --> 00:24:22,930 The transition region is also called the depletion region. 589 00:24:22,930 --> 00:24:26,740 And it's also called the space charge region. 590 00:24:26,740 --> 00:24:31,610 Interesting how these words and terms come back into use. 591 00:24:31,610 --> 00:24:37,200 So that's the essence of the formation of this junction. 592 00:24:37,200 --> 00:24:39,100 We have this diffusion process that 593 00:24:39,100 --> 00:24:43,380 drives the carriers from this side into this side. 594 00:24:43,380 --> 00:24:45,330 And then as we have the electrons moving over 595 00:24:45,330 --> 00:24:51,032 here, and summing to the boron nuclei that 596 00:24:51,032 --> 00:24:52,740 are left behind once the holes, likewise, 597 00:24:52,740 --> 00:24:54,156 have moved over to the other side. 598 00:24:54,156 --> 00:24:56,610 We have a buildup of net negative charge on this side, 599 00:24:56,610 --> 00:24:59,040 a buildup of net positive charge on that side, 600 00:24:59,040 --> 00:25:04,810 and the establishment of a built-in electric field. 601 00:25:04,810 --> 00:25:06,120 That's cool. 602 00:25:06,120 --> 00:25:08,430 It takes a while to really get it. 603 00:25:08,430 --> 00:25:10,460 And so the chessboard example was brilliant. 604 00:25:10,460 --> 00:25:12,100 I thank Joe for that. 605 00:25:12,100 --> 00:25:14,000 And hopefully, this as well reinforces 606 00:25:14,000 --> 00:25:16,250 several of those concepts. 607 00:25:16,250 --> 00:25:19,250 And it might take some time studying it on your own 608 00:25:19,250 --> 00:25:21,340 or setting up special office hours with Joe 609 00:25:21,340 --> 00:25:23,380 or coming to my office hours on Mondays. 610 00:25:23,380 --> 00:25:26,220 But whatever it takes, make sure that you understand 611 00:25:26,220 --> 00:25:30,040 this concept well-- the fundamental-- 612 00:25:30,040 --> 00:25:31,975 gaining an intuition about the concept. 613 00:25:31,975 --> 00:25:33,710 We'll be getting into the math soon. 614 00:25:33,710 --> 00:25:35,751 And if you don't have a good, solid understanding 615 00:25:35,751 --> 00:25:39,920 of the intuition of where charges are moving around, 616 00:25:39,920 --> 00:25:44,760 it becomes less easy, let's say, to really get the math. 617 00:25:44,760 --> 00:25:48,690 So we have a buildup of net charge 618 00:25:48,690 --> 00:25:50,040 on either side of the junction. 619 00:25:50,040 --> 00:25:51,790 We have a buildup of net negative charge 620 00:25:51,790 --> 00:25:53,499 on one side of the junction and a buildup 621 00:25:53,499 --> 00:25:55,873 of net positive charge on the other side of the junction. 622 00:25:55,873 --> 00:25:57,400 And this line right here is meant 623 00:25:57,400 --> 00:26:00,990 to represent that dividing line where the two materials 624 00:26:00,990 --> 00:26:02,640 initially came together. 625 00:26:02,640 --> 00:26:06,420 So the dashed line is meant to represent the real charge 626 00:26:06,420 --> 00:26:11,050 distribution and the boxed colored rectangles, 627 00:26:11,050 --> 00:26:14,340 the rectilinear boxes, are meant to represent an approximation 628 00:26:14,340 --> 00:26:17,140 that we'll use for the rest of today's class. 629 00:26:17,140 --> 00:26:19,190 We're making successive approximations 630 00:26:19,190 --> 00:26:23,070 here because we want the mathematics to be manageable. 631 00:26:23,070 --> 00:26:29,830 You can, of course, discretize this entire material 632 00:26:29,830 --> 00:26:32,590 in one dimension, for instance, and do a finite element 633 00:26:32,590 --> 00:26:33,290 solution. 634 00:26:33,290 --> 00:26:36,000 We'll show you the equations that you can use to do that 635 00:26:36,000 --> 00:26:37,960 later on in today's class. 636 00:26:37,960 --> 00:26:40,080 So we have the net charge distribution 637 00:26:40,080 --> 00:26:44,150 on either side of the junction shown here. 638 00:26:44,150 --> 00:26:49,310 We also have, as a result of that net charge, 639 00:26:49,310 --> 00:26:50,460 an electric field. 640 00:26:50,460 --> 00:26:55,130 And this goes back to Gauss' law, or the derivative form 641 00:26:55,130 --> 00:26:55,670 there of. 642 00:26:55,670 --> 00:27:00,990 And so we have here a built-in electric field as you can see. 643 00:27:00,990 --> 00:27:04,314 And the field reaches a maximum right here in the middle. 644 00:27:04,314 --> 00:27:05,480 So that kind of makes sense. 645 00:27:05,480 --> 00:27:07,490 If you're too far away from that junction, 646 00:27:07,490 --> 00:27:09,740 you're not going to-- if you're a charge carrier, 647 00:27:09,740 --> 00:27:11,220 you're not going to see the field. 648 00:27:11,220 --> 00:27:13,700 But if you're right there in the middle of that junction, 649 00:27:13,700 --> 00:27:14,950 that field is going to be very strong. 650 00:27:14,950 --> 00:27:16,783 You'll be swept out of there pretty quickly. 651 00:27:16,783 --> 00:27:19,030 That's why the field reaches a maximum right here 652 00:27:19,030 --> 00:27:19,730 in the middle. 653 00:27:19,730 --> 00:27:22,220 So that intuitively makes a lot of sense. 654 00:27:22,220 --> 00:27:27,260 Now, if we take one further integral of the electric field, 655 00:27:27,260 --> 00:27:29,340 here the electric field, arc psi. 656 00:27:29,340 --> 00:27:34,020 If we integrate our arc psi, we will get the potential. 657 00:27:34,020 --> 00:27:37,500 And this potential right here essentially follows 658 00:27:37,500 --> 00:27:40,830 this wave-like curve, where you have 659 00:27:40,830 --> 00:27:42,970 a lower potential on this side and a higher 660 00:27:42,970 --> 00:27:44,460 potential on that side. 661 00:27:44,460 --> 00:27:47,570 And if we take this potential and translate it into something 662 00:27:47,570 --> 00:27:50,580 that we can understand, which is electron energy, 663 00:27:50,580 --> 00:27:52,790 by multiplying the potential by q. 664 00:27:52,790 --> 00:27:55,150 And q in the case of an electron is a negative number. 665 00:27:55,150 --> 00:27:57,614 That's why we're flipping this. 666 00:27:57,614 --> 00:27:59,030 So we're flipping this upside down 667 00:27:59,030 --> 00:28:00,738 because we're multiplying this value here 668 00:28:00,738 --> 00:28:02,180 by a negative number. 669 00:28:02,180 --> 00:28:04,200 Now, what we can see is that there's 670 00:28:04,200 --> 00:28:07,590 an energy gain for the electron by going from the p-side 671 00:28:07,590 --> 00:28:08,380 to the n-side. 672 00:28:10,920 --> 00:28:12,870 This is the electron energy. 673 00:28:12,870 --> 00:28:15,780 Obviously, if you give them an opportunity, 674 00:28:15,780 --> 00:28:18,310 all the electrons would want to come down to this side. 675 00:28:18,310 --> 00:28:21,944 But why wouldn't all of them come down over here? 676 00:28:21,944 --> 00:28:23,282 AUDIENCE: Because of diffusion. 677 00:28:23,282 --> 00:28:24,615 PROFESSOR: Because of diffusion. 678 00:28:24,615 --> 00:28:26,670 So there's that balance of the two effects. 679 00:28:26,670 --> 00:28:29,360 So there is a net energy gain for the electrons, 680 00:28:29,360 --> 00:28:32,426 but diffusion is what's driving some of the electrons 681 00:28:32,426 --> 00:28:34,300 back across the other side of the junction is 682 00:28:34,300 --> 00:28:39,340 that equilibrium, which is establishing the final energy 683 00:28:39,340 --> 00:28:42,660 band diagram of the pn-junction, which is really looking 684 00:28:42,660 --> 00:28:44,950 something very much like that. 685 00:28:44,950 --> 00:28:49,220 So this right here describes for you how you go from atoms 686 00:28:49,220 --> 00:28:51,240 and charge carriers up at the top 687 00:28:51,240 --> 00:28:55,450 to charge distribution, electric field, potential, and finally 688 00:28:55,450 --> 00:28:56,980 electron energy. 689 00:28:56,980 --> 00:28:58,006 Yes. 690 00:28:58,006 --> 00:28:59,683 AUDIENCE: Are these four diagrams 691 00:28:59,683 --> 00:29:01,615 specifically for electrons? 692 00:29:04,513 --> 00:29:07,411 Would the hole ones be [INAUDIBLE] 693 00:29:07,411 --> 00:29:08,860 the reverse of that? 694 00:29:08,860 --> 00:29:13,150 PROFESSOR: So if you wanted to look in terms of hole energy, 695 00:29:13,150 --> 00:29:15,690 this equation here would be equally valid, 696 00:29:15,690 --> 00:29:17,441 but your charge would be a positive value. 697 00:29:17,441 --> 00:29:20,023 And so you'd have a curve that looked very similar to this one 698 00:29:20,023 --> 00:29:22,560 right here, where there'd be a net energy gain for the holes 699 00:29:22,560 --> 00:29:24,360 to go on to the other side. 700 00:29:24,360 --> 00:29:27,980 And that's what establishes the separation of charge. 701 00:29:27,980 --> 00:29:30,890 AUDIENCE: So the bottom one is the only one that would change 702 00:29:30,890 --> 00:29:31,547 [INAUDIBLE]? 703 00:29:31,547 --> 00:29:32,630 PROFESSOR: That's correct. 704 00:29:32,630 --> 00:29:33,250 Yep. 705 00:29:33,250 --> 00:29:35,569 Everything else identical. 706 00:29:35,569 --> 00:29:37,360 So this is important, this electron energy. 707 00:29:37,360 --> 00:29:40,380 And this jump right here confuses a lot of folks. 708 00:29:40,380 --> 00:29:43,290 Because in books, they don't often describe very clearly 709 00:29:43,290 --> 00:29:46,030 what is potential and what is energy, 710 00:29:46,030 --> 00:29:48,230 almost using those two terms interchangeably. 711 00:29:48,230 --> 00:29:50,850 But it's that cue, the fact that the electron has 712 00:29:50,850 --> 00:29:54,560 a negative charge that sets things right. 713 00:29:54,560 --> 00:29:56,640 OK, this is pretty important. 714 00:29:56,640 --> 00:29:58,330 This is the foundation, the fundamental, 715 00:29:58,330 --> 00:30:01,440 of how a pn-junction comes into being. 716 00:30:01,440 --> 00:30:03,920 And the summary of our understanding 717 00:30:03,920 --> 00:30:07,550 so far is that when the light-- here we go. 718 00:30:07,550 --> 00:30:09,850 When light creates an electron-hole pair, 719 00:30:09,850 --> 00:30:13,360 a pn-junction can separate the positive and negative charges 720 00:30:13,360 --> 00:30:15,660 because of the built-in electric field. 721 00:30:15,660 --> 00:30:17,332 Let me repeat that one more time. 722 00:30:17,332 --> 00:30:19,040 When light creates an electron-hole pair, 723 00:30:19,040 --> 00:30:20,675 a pn-junction potentially can separate 724 00:30:20,675 --> 00:30:22,050 the positive and negative charges 725 00:30:22,050 --> 00:30:24,150 because of that built-in electric field. 726 00:30:24,150 --> 00:30:28,040 For very small light intensities, 727 00:30:28,040 --> 00:30:29,870 very small light intensities, such 728 00:30:29,870 --> 00:30:33,770 that you can think of that photo-generated carrier 729 00:30:33,770 --> 00:30:36,330 as a perturbation to the system, not fundamentally 730 00:30:36,330 --> 00:30:39,526 altering the energy levels yet-- it does eventually happen. 731 00:30:39,526 --> 00:30:41,150 But if you have, say, one photon coming 732 00:30:41,150 --> 00:30:42,880 into your material exciting electron 733 00:30:42,880 --> 00:30:45,690 over here, that electron can move down, 734 00:30:45,690 --> 00:30:48,010 can be swept out of the device, because 735 00:30:48,010 --> 00:30:50,370 of that built-in electric field. 736 00:30:50,370 --> 00:30:54,600 Again, assuming that we have a very small impact 737 00:30:54,600 --> 00:30:57,260 on the electrostatic [INAUDIBLE] system 738 00:30:57,260 --> 00:30:58,880 that it's just a small perturbation, 739 00:30:58,880 --> 00:31:00,840 maybe one single electron in our system 740 00:31:00,840 --> 00:31:03,330 is not going to affect you much because we have 10 741 00:31:03,330 --> 00:31:07,560 to the 23 or so atoms per cubic centimeter. 742 00:31:07,560 --> 00:31:09,980 We have a very small perturbation to our system, 743 00:31:09,980 --> 00:31:13,496 that charge carrier will be swept out very quickly. 744 00:31:13,496 --> 00:31:15,245 The built-in electric field is established 745 00:31:15,245 --> 00:31:19,310 at a pn-junction because of the balance of electron and hole 746 00:31:19,310 --> 00:31:21,330 drift and diffusion currents. 747 00:31:21,330 --> 00:31:24,000 Let me be more precise about this one point right here. 748 00:31:24,000 --> 00:31:25,010 Because the built-in electric field 749 00:31:25,010 --> 00:31:26,240 is established at a pn-junction because 750 00:31:26,240 --> 00:31:28,198 of the balance of electron drift and diffusion. 751 00:31:28,198 --> 00:31:29,290 We talked about that. 752 00:31:29,290 --> 00:31:31,500 But hole drift and diffusion as well. 753 00:31:31,500 --> 00:31:36,040 So if you go back over to here, notice how the electron drift 754 00:31:36,040 --> 00:31:40,970 and diffusion are opposed and equal and opposite to the hole 755 00:31:40,970 --> 00:31:42,570 diffusion and drift. 756 00:31:42,570 --> 00:31:44,960 So let's think about it from this perspective 757 00:31:44,960 --> 00:31:47,190 right back here. 758 00:31:47,190 --> 00:31:48,890 This is clear, our electron diffusion 759 00:31:48,890 --> 00:31:50,600 is going in that direction because we 760 00:31:50,600 --> 00:31:52,183 have a high concentration of electrons 761 00:31:52,183 --> 00:31:54,550 that tend to move toward the low-concentration regime. 762 00:31:54,550 --> 00:31:56,770 So our electron diffusion current 763 00:31:56,770 --> 00:31:58,840 is pointing toward the left. 764 00:31:58,840 --> 00:32:02,290 Our hole diffusion current is pointing to the right. 765 00:32:02,290 --> 00:32:06,510 And the hole drift current, because of this built-in field, 766 00:32:06,510 --> 00:32:08,570 is pointing opposite the hole diffusion. 767 00:32:08,570 --> 00:32:11,400 So the hole drift current will be pointing to the left. 768 00:32:11,400 --> 00:32:13,330 Whereas, the electron drift current 769 00:32:13,330 --> 00:32:15,010 would be pointing to the right. 770 00:32:15,010 --> 00:32:17,101 So we have, in total, four currents here. 771 00:32:17,101 --> 00:32:19,350 The way to think about it would be, for example, let's 772 00:32:19,350 --> 00:32:20,660 take the electrons first. 773 00:32:20,660 --> 00:32:24,850 Electron drift or electron diffusion and drift counteract. 774 00:32:24,850 --> 00:32:26,660 And then you can take the holes. 775 00:32:26,660 --> 00:32:30,190 Hole diffusion and drift counteract. 776 00:32:30,190 --> 00:32:31,480 They're all balanced. 777 00:32:31,480 --> 00:32:35,400 And this is resulting in zero net current flow when 778 00:32:35,400 --> 00:32:38,930 you have no illumination and no external field applied 779 00:32:38,930 --> 00:32:41,308 to that device. 780 00:32:41,308 --> 00:32:44,275 That's pretty nifty. 781 00:32:44,275 --> 00:32:46,150 So the built-in electric field is established 782 00:32:46,150 --> 00:32:48,420 at the pn-junction because of the balance of drift 783 00:32:48,420 --> 00:32:50,820 and diffusion current for both electrons and holes. 784 00:32:54,910 --> 00:32:56,840 So we have a small in-class exercise. 785 00:32:56,840 --> 00:32:58,780 I'd like you to take out these. 786 00:32:58,780 --> 00:33:00,756 And I'd like you to work in pairs. 787 00:33:00,756 --> 00:33:02,880 So that if anybody reaches a small stumbling block, 788 00:33:02,880 --> 00:33:06,440 you can help each other work through the problems. 789 00:33:06,440 --> 00:33:10,510 So first off, let's focus on this upper left-hand portion 790 00:33:10,510 --> 00:33:12,700 right here as shown right there. 791 00:33:12,700 --> 00:33:14,700 This is a replica of the sheet that you 792 00:33:14,700 --> 00:33:16,550 should have in front of you. 793 00:33:16,550 --> 00:33:19,720 So what we're doing is we're dividing-- we're considering 794 00:33:19,720 --> 00:33:21,990 the pn-junction under no bias. 795 00:33:21,990 --> 00:33:24,730 That means that we're not applying a battery 796 00:33:24,730 --> 00:33:26,200 to our pn-junction. 797 00:33:26,200 --> 00:33:28,920 So we have a pn-junction and we don't attach a battery 798 00:33:28,920 --> 00:33:29,420 in series. 799 00:33:29,420 --> 00:33:32,690 We just have the pn-junction under zero bias conditions. 800 00:33:32,690 --> 00:33:36,390 So the way we'd represent that in that equivalent or model 801 00:33:36,390 --> 00:33:38,990 circuit diagram right here is notice 802 00:33:38,990 --> 00:33:41,710 how we have these little lines extending 803 00:33:41,710 --> 00:33:44,230 from the p- and the n-regions and going above. 804 00:33:44,230 --> 00:33:46,246 Those represent the external circuit outside 805 00:33:46,246 --> 00:33:47,120 of the actual device. 806 00:33:47,120 --> 00:33:49,510 The device would be this one right here 807 00:33:49,510 --> 00:33:51,490 where we have p and our n. 808 00:33:51,490 --> 00:33:53,490 We have our positive and negative charges 809 00:33:53,490 --> 00:33:57,340 here representing the charges that have built up 810 00:33:57,340 --> 00:33:59,380 in that space charge region that are creating 811 00:33:59,380 --> 00:34:00,707 the built-in electric field. 812 00:34:00,707 --> 00:34:03,040 And under no bias conditions-- I'll give you the answer. 813 00:34:03,040 --> 00:34:04,340 It's pretty straightforward. 814 00:34:04,340 --> 00:34:05,820 We would just draw a straight line 815 00:34:05,820 --> 00:34:08,810 across right there bridging that external circuit because there 816 00:34:08,810 --> 00:34:10,139 is no external bias. 817 00:34:10,139 --> 00:34:12,300 There is no battery applied. 818 00:34:12,300 --> 00:34:16,650 And so from an energy band diagram, e versus x, for e 819 00:34:16,650 --> 00:34:19,420 being the electron energy in this case. 820 00:34:19,420 --> 00:34:21,170 And our p-type material being on this side 821 00:34:21,170 --> 00:34:23,280 and our n-type material being on that side, 822 00:34:23,280 --> 00:34:26,630 let's draw the energy band diagram, just like we've done 823 00:34:26,630 --> 00:34:28,949 or we've alluded to so far. 824 00:34:28,949 --> 00:34:32,120 Sketch out the energy band diagram for this pn-junction. 825 00:34:32,120 --> 00:34:33,440 And I'll give you a hint. 826 00:34:33,440 --> 00:34:36,320 It's going to look something very similar to that right 827 00:34:36,320 --> 00:34:38,960 there, except that you'll have to take into account both 828 00:34:38,960 --> 00:34:41,280 the conduction and valence bands, 829 00:34:41,280 --> 00:34:45,219 because they're both affected in a similar manner to that. 830 00:34:45,219 --> 00:34:48,030 So why don't you go ahead and give it your best? 831 00:34:48,030 --> 00:34:53,270 Draw the energy band diagram in terms of the electron energy 832 00:34:53,270 --> 00:34:54,590 and position. 833 00:34:54,590 --> 00:34:56,840 Notice your p-type on one side, n-type on the other. 834 00:34:56,840 --> 00:34:59,120 You can assume that the point of division 835 00:34:59,120 --> 00:35:01,599 is like right here in the middle. 836 00:35:01,599 --> 00:35:03,140 And for those who are a little quick, 837 00:35:03,140 --> 00:35:05,080 you can go on and draw the relative magnitudes 838 00:35:05,080 --> 00:35:06,802 of electron drift and diffusion currents. 839 00:35:06,802 --> 00:35:08,760 So why don't we give that a quick, little shot? 840 00:35:08,760 --> 00:35:12,840 Maybe 30 seconds to think about it and another 30 seconds 841 00:35:12,840 --> 00:35:15,110 to draw and chat over it with your colleagues. 842 00:35:22,050 --> 00:35:23,650 So we want to draw the band diagram, 843 00:35:23,650 --> 00:35:25,880 just to make sure everybody's in the same page. 844 00:35:25,880 --> 00:35:27,255 So we're going to be starting out 845 00:35:27,255 --> 00:35:32,820 with our valence band and our conduction band over here. 846 00:35:32,820 --> 00:35:35,550 And as we move toward our space charge region, 847 00:35:35,550 --> 00:35:38,700 as we move toward our depletion region, our transition region 848 00:35:38,700 --> 00:35:42,780 here in the middle, we're going to see some effect. 849 00:35:42,780 --> 00:35:45,910 We're going to see some bending of the bands, if you will. 850 00:35:45,910 --> 00:35:48,120 So your question is to figure out, 851 00:35:48,120 --> 00:35:49,800 do the bands do something like this? 852 00:35:49,800 --> 00:35:52,480 Or do they do something like that? 853 00:35:52,480 --> 00:35:55,210 Do the bands go up or do they bend down? 854 00:35:55,210 --> 00:35:57,660 And that should be a relatively easy copy 855 00:35:57,660 --> 00:35:59,500 and paste from two slides prior. 856 00:36:06,370 --> 00:36:08,490 The correct answer is, indeed, we 857 00:36:08,490 --> 00:36:11,820 have the band bending down like this. 858 00:36:11,820 --> 00:36:14,510 So we would have the bands higher on one side, lower 859 00:36:14,510 --> 00:36:15,740 on the other. 860 00:36:15,740 --> 00:36:18,570 The electron energy is higher on the p-type side 861 00:36:18,570 --> 00:36:19,580 than the n-type side. 862 00:36:19,580 --> 00:36:21,246 So if we add one electron to our system, 863 00:36:21,246 --> 00:36:23,350 for example a photo-generated carrier 864 00:36:23,350 --> 00:36:26,317 from one photon coming into our device, 865 00:36:26,317 --> 00:36:28,150 that electron will have the natural tendency 866 00:36:28,150 --> 00:36:29,460 to go onto this side. 867 00:36:29,460 --> 00:36:32,000 It will be swept down. 868 00:36:32,000 --> 00:36:34,470 Note that the net current flow inside 869 00:36:34,470 --> 00:36:37,050 of the device, in the absence of an external excitation, 870 00:36:37,050 --> 00:36:41,410 like light or a battery pack over here, an external bias 871 00:36:41,410 --> 00:36:44,340 voltage source, the net current flow is zero. 872 00:36:44,340 --> 00:36:45,940 Because electron diffusion is pointed 873 00:36:45,940 --> 00:36:47,840 in that direction and electron drift 874 00:36:47,840 --> 00:36:49,175 pointed in that direction. 875 00:36:51,960 --> 00:36:53,330 All right, good. 876 00:36:53,330 --> 00:37:00,240 So now that we've done one example, I wanted to-- oh, 877 00:37:00,240 --> 00:37:01,150 one last thing. 878 00:37:01,150 --> 00:37:02,167 So hole diffusion. 879 00:37:02,167 --> 00:37:03,750 So this is electron diffusion pointing 880 00:37:03,750 --> 00:37:06,050 to the left and electron drift pointing to the right. 881 00:37:06,050 --> 00:37:08,210 Hole diffusion is pointing to the right and hole 882 00:37:08,210 --> 00:37:12,500 drift pointing to the left as one might expect. 883 00:37:12,500 --> 00:37:17,060 Let's try these two right now, forward bias and reverse bias. 884 00:37:17,060 --> 00:37:19,380 And your trick is to figure out, if you 885 00:37:19,380 --> 00:37:22,920 have a battery for instance, if the battery looks like this. 886 00:37:22,920 --> 00:37:25,460 This would be positive terminal and negative terminal. 887 00:37:25,460 --> 00:37:28,800 If you have a battery, which way would you 888 00:37:28,800 --> 00:37:33,970 align the battery to induce a forward bias or reverse bias? 889 00:37:33,970 --> 00:37:37,220 It's perhaps a little bit beyond my expectation 890 00:37:37,220 --> 00:37:38,440 of what you would get. 891 00:37:38,440 --> 00:37:39,440 Let me add that for you. 892 00:37:39,440 --> 00:37:41,070 Let me just give that to you right here. 893 00:37:41,070 --> 00:37:42,653 The forward bias would look like that. 894 00:37:42,653 --> 00:37:44,590 The reverse bias would look like that. 895 00:37:44,590 --> 00:37:45,180 OK. 896 00:37:45,180 --> 00:37:47,720 And now the big question is, what do the band diagrams 897 00:37:47,720 --> 00:37:48,220 look like? 898 00:37:48,220 --> 00:37:49,710 How would these look? 899 00:37:53,480 --> 00:37:59,270 If we forward bias, if we forward bias our device 900 00:37:59,270 --> 00:38:01,929 and we inject electrons into this side right here, 901 00:38:01,929 --> 00:38:03,220 what would happen to our bands? 902 00:38:03,220 --> 00:38:04,678 What would you expect would happen? 903 00:38:07,470 --> 00:38:11,560 We'll prove this out in the next few slides, 904 00:38:11,560 --> 00:38:14,029 but I want to see how people's intuition is 905 00:38:14,029 --> 00:38:14,820 doing this morning. 906 00:38:23,480 --> 00:38:26,410 So as you bias a device, you're shifting one level 907 00:38:26,410 --> 00:38:27,332 relative to the other. 908 00:38:27,332 --> 00:38:29,540 And so the basic question that you have to answer is, 909 00:38:29,540 --> 00:38:33,370 should it shift up or should it shift down under 910 00:38:33,370 --> 00:38:36,410 Forward and reverse bias conditions? 911 00:38:36,410 --> 00:38:37,134 Spend a minute. 912 00:38:37,134 --> 00:38:38,050 Talk to your neighbor. 913 00:38:38,050 --> 00:38:40,630 Discuss it. 914 00:38:40,630 --> 00:38:41,620 All right, folks. 915 00:38:41,620 --> 00:38:43,814 Why don't we tie it in? 916 00:38:47,559 --> 00:38:49,350 We should be able to get band diagrams that 917 00:38:49,350 --> 00:38:50,750 look something like that. 918 00:38:50,750 --> 00:38:53,260 I saw several of you have already begun 919 00:38:53,260 --> 00:38:55,520 reaching this consensus here. 920 00:38:55,520 --> 00:38:56,702 Rationale? 921 00:38:56,702 --> 00:38:58,160 An easy way to think about this is, 922 00:38:58,160 --> 00:39:00,618 if our battery is aligned with our pn-junction in that way, 923 00:39:00,618 --> 00:39:02,740 and we're injecting electrons into one side, 924 00:39:02,740 --> 00:39:05,140 we then have a large electron diffusion current. 925 00:39:05,140 --> 00:39:07,450 We have the bands shifting in this direction. 926 00:39:07,450 --> 00:39:10,600 Now the barrier, the energy barrier for the electrons 927 00:39:10,600 --> 00:39:13,810 to overcome, to go from the region of high concentration 928 00:39:13,810 --> 00:39:16,680 to the region of low concentration, is smaller. 929 00:39:16,680 --> 00:39:18,930 And you'll get a larger diffusion current as a result. 930 00:39:18,930 --> 00:39:21,480 More electrons will be moving over that n-type silicon 931 00:39:21,480 --> 00:39:23,110 into the p-type silicon. 932 00:39:23,110 --> 00:39:26,090 And the drift current will be smaller. 933 00:39:26,090 --> 00:39:29,600 Notice here, the balance of the two, if you add them together, 934 00:39:29,600 --> 00:39:32,020 you have a net electron flow in that direction, which 935 00:39:32,020 --> 00:39:33,910 means you have a net current flow 936 00:39:33,910 --> 00:39:35,870 as defined by the flow of positive charges 937 00:39:35,870 --> 00:39:37,346 in the other direction. 938 00:39:37,346 --> 00:39:39,380 A lot of definitions to keep straight. 939 00:39:39,380 --> 00:39:42,770 Whereas, under reverse bias here, you're shifting the bands 940 00:39:42,770 --> 00:39:44,490 like so. 941 00:39:44,490 --> 00:39:46,240 Because you have the large electric field, 942 00:39:46,240 --> 00:39:49,720 the drift current will be increased, but only slightly. 943 00:39:49,720 --> 00:39:54,080 Because there's a limited number of carriers here 944 00:39:54,080 --> 00:39:57,070 that you can pull from the p-type material. 945 00:39:57,070 --> 00:39:59,750 And the diffusion current is going to be drastically reduced 946 00:39:59,750 --> 00:40:04,070 because now the energy barrier is so large that the electrons 947 00:40:04,070 --> 00:40:06,470 have a very hard time getting from the n-type material 948 00:40:06,470 --> 00:40:08,010 into the p-type material. 949 00:40:08,010 --> 00:40:10,910 And then that net current flow is in the opposite direction. 950 00:40:13,374 --> 00:40:15,040 Does this make intuitive sense to folks? 951 00:40:15,040 --> 00:40:15,540 Ashley. 952 00:40:15,540 --> 00:40:21,655 AUDIENCE: So when you touch the battery [INAUDIBLE], 953 00:40:21,655 --> 00:40:26,027 is the minus side injecting electrons [INAUDIBLE]? 954 00:40:26,027 --> 00:40:26,610 PROFESSOR: OK. 955 00:40:26,610 --> 00:40:29,900 So in between these two, the easiest way 956 00:40:29,900 --> 00:40:32,920 to go from battery to electron energy band diagram 957 00:40:32,920 --> 00:40:37,841 is to follow the same sequence that we did over here. 958 00:40:37,841 --> 00:40:39,590 And if you're thinking battery, and you're 959 00:40:39,590 --> 00:40:41,173 thinking positive and negative, you're 960 00:40:41,173 --> 00:40:45,160 shifting the potential of one side relative to the other. 961 00:40:45,160 --> 00:40:49,336 Then, take that potential shift and flip it for the electron. 962 00:40:49,336 --> 00:40:53,380 AUDIENCE: I follow why the diagrams are the way they are. 963 00:40:53,380 --> 00:40:58,750 But is it conceptually accurate to think of the negative side 964 00:40:58,750 --> 00:41:01,628 as putting more electrons into the n-side? 965 00:41:01,628 --> 00:41:02,530 PROFESSOR: One could. 966 00:41:02,530 --> 00:41:03,750 AUDIENCE: Like injecting carriers? 967 00:41:03,750 --> 00:41:04,390 PROFESSOR: The way I prefer to think 968 00:41:04,390 --> 00:41:06,230 about it is that you're changing the energy 969 00:41:06,230 --> 00:41:08,500 levels, the potential one relative to the other. 970 00:41:08,500 --> 00:41:10,780 And as a consequence, you have more electrons flowing 971 00:41:10,780 --> 00:41:13,420 from that n-side to the p-side. 972 00:41:13,420 --> 00:41:16,050 And that's what's pulling the current through the circuit. 973 00:41:19,930 --> 00:41:23,100 It's almost a tail wagging dog, dog wagging tail. 974 00:41:23,100 --> 00:41:25,250 But I would take a slight preference 975 00:41:25,250 --> 00:41:28,360 toward understanding it from the perspective of if you change 976 00:41:28,360 --> 00:41:30,206 the potential one relative to the other, 977 00:41:30,206 --> 00:41:31,580 and allow for a greater diffusion 978 00:41:31,580 --> 00:41:35,250 current, that motion of electrons, the electron moving 979 00:41:35,250 --> 00:41:37,880 from here to here, will pull current 980 00:41:37,880 --> 00:41:39,490 through the entire circuit. 981 00:41:39,490 --> 00:41:42,110 And that's what you'll be measuring with your external-- 982 00:41:42,110 --> 00:41:44,110 essentially, all of the electrons in the circuit 983 00:41:44,110 --> 00:41:46,708 are moving through. 984 00:41:46,708 --> 00:41:50,398 AUDIENCE: [INAUDIBLE] so we've opposed the field that 985 00:41:50,398 --> 00:41:51,469 was set up [INAUDIBLE]. 986 00:41:51,469 --> 00:41:52,260 PROFESSOR: Exactly. 987 00:41:52,260 --> 00:41:53,968 We are opposing the field that was set up 988 00:41:53,968 --> 00:41:56,900 by the balance of the drift and diffusion currents initially. 989 00:41:56,900 --> 00:41:57,872 Yeah. 990 00:41:57,872 --> 00:41:59,573 AUDIENCE: Under the first bias, how 991 00:41:59,573 --> 00:42:02,732 come the many electrons in the anti-material 992 00:42:02,732 --> 00:42:04,676 aren't [INAUDIBLE] through the entire circuit 993 00:42:04,676 --> 00:42:06,150 and injected into the [INAUDIBLE]. 994 00:42:06,150 --> 00:42:07,025 PROFESSOR: Certainly. 995 00:42:07,025 --> 00:42:11,480 So the reverse bias is such that the battery 996 00:42:11,480 --> 00:42:12,720 is aligned to prevent that. 997 00:42:15,977 --> 00:42:17,810 perhaps one simple way of thinking about it. 998 00:42:20,029 --> 00:42:22,320 Yeah, I think that would be the way I would describe it 999 00:42:22,320 --> 00:42:22,986 most accurately. 1000 00:42:22,986 --> 00:42:26,150 You could-- and you are dragging some of the electrons 1001 00:42:26,150 --> 00:42:27,770 out through the external circuit. 1002 00:42:27,770 --> 00:42:32,940 But what is driving the current is mostly the drift current. 1003 00:42:32,940 --> 00:42:34,790 It's pulling carriers from-- essentially, 1004 00:42:34,790 --> 00:42:35,998 through your external circle. 1005 00:42:35,998 --> 00:42:38,770 Because carriers are moving from the p-type to the n-type 1006 00:42:38,770 --> 00:42:40,360 and creating that charge imbalance 1007 00:42:40,360 --> 00:42:43,860 resulting in the drift current. 1008 00:42:43,860 --> 00:42:46,730 Why don't we continue moving on just a little bit 1009 00:42:46,730 --> 00:42:49,960 because we have some nice demos we'd like to get to. 1010 00:42:49,960 --> 00:42:51,920 I do know this is really, really important. 1011 00:42:51,920 --> 00:42:54,030 And I would welcome you, actually urge you, 1012 00:42:54,030 --> 00:42:56,390 encourage you to come to some of the office hours 1013 00:42:56,390 --> 00:43:00,160 and our recitation as well. 1014 00:43:00,160 --> 00:43:02,890 I'd like to continue moving forward 1015 00:43:02,890 --> 00:43:05,680 for the benefit of several of those who may have seen 1016 00:43:05,680 --> 00:43:07,510 similar material in the past. 1017 00:43:07,510 --> 00:43:10,820 But let me, before we move too far ahead, 1018 00:43:10,820 --> 00:43:12,690 on our I-V characteristics, we still 1019 00:43:12,690 --> 00:43:16,560 don't know how to map an I-V characteristic of that device. 1020 00:43:16,560 --> 00:43:19,589 We don't know how the current voltage, I current V voltage, 1021 00:43:19,589 --> 00:43:21,130 we don't know how the current voltage 1022 00:43:21,130 --> 00:43:24,080 response is going to look like for that particular device. 1023 00:43:24,080 --> 00:43:27,060 But we do know that under no bias conditions 1024 00:43:27,060 --> 00:43:29,080 we're at bias voltage equals 0. 1025 00:43:29,080 --> 00:43:30,450 So V is at 0. 1026 00:43:30,450 --> 00:43:32,710 And we know that we have no net current flow. 1027 00:43:32,710 --> 00:43:34,760 And so our current is going to be at 0 as well. 1028 00:43:34,760 --> 00:43:35,850 And so we know that we're going to have 1029 00:43:35,850 --> 00:43:37,360 one point on our I-V curve that's 1030 00:43:37,360 --> 00:43:39,890 going to be situated at 0, 0. 1031 00:43:39,890 --> 00:43:41,990 Likewise, in this case over here, 1032 00:43:41,990 --> 00:43:44,872 we have electron diffusion driving the circuit. 1033 00:43:44,872 --> 00:43:46,830 And in the other, we have electron drift, which 1034 00:43:46,830 --> 00:43:48,440 is in the opposite direction. 1035 00:43:48,440 --> 00:43:51,140 And so you'd expect that the current flow 1036 00:43:51,140 --> 00:43:53,840 in I under forward bias conditions-- forward bias is 1037 00:43:53,840 --> 00:43:55,990 positive-- would be somewhere up here. 1038 00:43:55,990 --> 00:43:57,970 And likewise, under reverse bias, 1039 00:43:57,970 --> 00:44:00,970 that our current would be opposite the sign. 1040 00:44:00,970 --> 00:44:03,110 So whereas we had positive current over here, 1041 00:44:03,110 --> 00:44:06,290 we would expect negative current over there. 1042 00:44:06,290 --> 00:44:08,110 So our I-V characteristic is going 1043 00:44:08,110 --> 00:44:11,450 to look something like a combination of one point 1044 00:44:11,450 --> 00:44:14,730 right here and some function up in this quadrant 1045 00:44:14,730 --> 00:44:16,940 and some function down in this quadrant, 1046 00:44:16,940 --> 00:44:20,090 just by glancing at the net current flows 1047 00:44:20,090 --> 00:44:25,715 and the type of bias that we've applied to our diode. 1048 00:44:25,715 --> 00:44:28,920 We'll come back to this at the end of class 1049 00:44:28,920 --> 00:44:33,100 and confirm it using those small apparatus over there. 1050 00:44:33,100 --> 00:44:34,630 So current flow in a pn-junction. 1051 00:44:34,630 --> 00:44:36,455 We're going to describe the nature of the drift, diffusion, 1052 00:44:36,455 --> 00:44:38,270 and illumination currents in a diode. 1053 00:44:38,270 --> 00:44:40,317 Show the direction and magnitude in the dark. 1054 00:44:40,317 --> 00:44:42,150 Eventually, we'll do this under illumination 1055 00:44:42,150 --> 00:44:44,050 as well, but let's just focus in the dark. 1056 00:44:44,050 --> 00:44:45,800 In fact, we've already done this. 1057 00:44:45,800 --> 00:44:47,510 I've already-- I've already tricked you 1058 00:44:47,510 --> 00:44:50,370 into doing learning objective number 3 on your own, 1059 00:44:50,370 --> 00:44:52,339 before even describing the math. 1060 00:44:52,339 --> 00:44:54,630 But I will show you the mathematics as well so that you 1061 00:44:54,630 --> 00:44:56,480 have the complete picture. 1062 00:44:56,480 --> 00:44:58,080 We have the diffusion current. 1063 00:44:58,080 --> 00:44:59,800 This is the essence of diffusion current. 1064 00:44:59,800 --> 00:45:01,820 We talked about Fick's law as well for holes 1065 00:45:01,820 --> 00:45:03,270 and for electrons. 1066 00:45:03,270 --> 00:45:04,740 We have our drift current. 1067 00:45:04,740 --> 00:45:06,710 We spoke about the nature of the drift current 1068 00:45:06,710 --> 00:45:09,100 as well for holes and electrons. 1069 00:45:09,100 --> 00:45:11,440 And we know that by combining the two, 1070 00:45:11,440 --> 00:45:14,170 we can describe the net current flow across a pn-junction. 1071 00:45:14,170 --> 00:45:17,830 So by combining the two, by combining drift and diffusion-- 1072 00:45:17,830 --> 00:45:20,870 drift and diffusion for electrons and holes-- 1073 00:45:20,870 --> 00:45:25,220 we can describe the current flow across that pn-junction. 1074 00:45:25,220 --> 00:45:28,350 So when the electric field is large, 1075 00:45:28,350 --> 00:45:30,370 the drift current term is going to dominate. 1076 00:45:30,370 --> 00:45:31,960 If you have a large field, that's 1077 00:45:31,960 --> 00:45:36,170 going to be the dominant current source if you will. 1078 00:45:36,170 --> 00:45:38,500 And when the electric field is small, 1079 00:45:38,500 --> 00:45:40,560 diffusion is going to dominate. 1080 00:45:40,560 --> 00:45:43,380 So if we have a device like this right here. 1081 00:45:43,380 --> 00:45:46,380 And if I told you that the built-in electric field was 1082 00:45:46,380 --> 00:45:49,075 located pushed up toward the top 1/100 of the device. 1083 00:45:49,075 --> 00:45:51,830 So it was located very close to the surface. 1084 00:45:51,830 --> 00:45:55,230 You might surmise that the drift current is going 1085 00:45:55,230 --> 00:45:57,300 to be having a large influence. 1086 00:45:57,300 --> 00:45:59,830 If you're a single electron, you'll 1087 00:45:59,830 --> 00:46:01,790 feel the influence of the electric field 1088 00:46:01,790 --> 00:46:03,440 near the front surface of the device. 1089 00:46:03,440 --> 00:46:06,480 And deep within the bulk, the diffusion current 1090 00:46:06,480 --> 00:46:11,830 is going to be driving the carriers toward that junction. 1091 00:46:11,830 --> 00:46:13,700 There's a relationship between the mobility 1092 00:46:13,700 --> 00:46:15,380 and the diffusivity of carriers. 1093 00:46:15,380 --> 00:46:17,870 It's known as the Einstein relation. 1094 00:46:17,870 --> 00:46:19,390 And that's given to you over there 1095 00:46:19,390 --> 00:46:22,230 on the right-hand side for a band conductor. 1096 00:46:22,230 --> 00:46:24,610 Note that if you have-- this is an advanced concept. 1097 00:46:24,610 --> 00:46:28,430 But if you have a dispersive hopping mechanism, 1098 00:46:28,430 --> 00:46:30,810 for instance, you may need to become a little bit more 1099 00:46:30,810 --> 00:46:33,310 sophisticated in how you relate the diffusivity of a carrier 1100 00:46:33,310 --> 00:46:34,180 to your mobility. 1101 00:46:34,180 --> 00:46:36,970 But let's assume that is valid for nice, well-behaved 1102 00:46:36,970 --> 00:46:38,260 semiconductors. 1103 00:46:38,260 --> 00:46:41,312 It's just a warning for those who work on organic materials. 1104 00:46:41,312 --> 00:46:44,040 I'd be happy to talk to you more about that in a bit. 1105 00:46:44,040 --> 00:46:46,756 So again, we have Gauss' law. 1106 00:46:46,756 --> 00:46:48,630 The question is, what is that electric field? 1107 00:46:48,630 --> 00:46:49,540 What is arc psi? 1108 00:46:49,540 --> 00:46:51,109 What is our built-in electric field? 1109 00:46:51,109 --> 00:46:52,650 Well, we know that the charge density 1110 00:46:52,650 --> 00:46:56,340 is going to be comprised of the free hole 1111 00:46:56,340 --> 00:46:59,940 density, the free electron density, the fixed 1112 00:46:59,940 --> 00:47:02,854 density of our donor atoms. 1113 00:47:02,854 --> 00:47:04,645 You could think of these as the phosphorus, 1114 00:47:04,645 --> 00:47:07,890 the ionized phosphorus atoms, and our fixed density 1115 00:47:07,890 --> 00:47:09,650 of ionized acceptor atoms. 1116 00:47:09,650 --> 00:47:12,455 We can also think of this as our boron atoms and silicon. 1117 00:47:12,455 --> 00:47:15,750 And so that is our fixed-- actually, not our fixed. 1118 00:47:15,750 --> 00:47:16,760 These are fixed charges. 1119 00:47:16,760 --> 00:47:17,610 These are mobile charges. 1120 00:47:17,610 --> 00:47:19,900 But under equilibrium, they've distributed themselves 1121 00:47:19,900 --> 00:47:20,850 in a certain way. 1122 00:47:20,850 --> 00:47:22,950 That is our charge density, our rho. 1123 00:47:22,950 --> 00:47:24,750 And that will figure into Gauss' law 1124 00:47:24,750 --> 00:47:27,300 and allow us to calculate arc psi. 1125 00:47:27,300 --> 00:47:32,390 And so in summa, we have here arc psi as a function of dx. 1126 00:47:32,390 --> 00:47:35,270 We have an expression that relates the electric field 1127 00:47:35,270 --> 00:47:38,725 to the density of dopant atoms inside of our material. 1128 00:47:38,725 --> 00:47:41,100 Note that we made one critical assumption going from here 1129 00:47:41,100 --> 00:47:44,690 to here, where we went from the ionized donor and acceptor 1130 00:47:44,690 --> 00:47:47,570 concentration to the total dopant concentration. 1131 00:47:47,570 --> 00:47:49,320 We're assuming that everything is ionized. 1132 00:47:49,320 --> 00:47:52,130 Again, because the ionization energy, 1133 00:47:52,130 --> 00:47:54,220 because the binding energy of those free carriers 1134 00:47:54,220 --> 00:47:57,190 to their dopant atoms is so small 1135 00:47:57,190 --> 00:48:01,400 that under kt, under thermal energy at room temperature 1136 00:48:01,400 --> 00:48:05,500 it's enough to dissociate the charge. 1137 00:48:05,500 --> 00:48:07,280 OK, so we've gotten to here. 1138 00:48:07,280 --> 00:48:10,440 We can describe quantitatively using this approximation, 1139 00:48:10,440 --> 00:48:12,170 using the box function approximation. 1140 00:48:12,170 --> 00:48:14,870 We can describe the electric field quantitatively. 1141 00:48:14,870 --> 00:48:16,710 And now we have to get to the potential 1142 00:48:16,710 --> 00:48:19,530 and find the electron energy as well. 1143 00:48:19,530 --> 00:48:21,930 So we do a little bit of accounting. 1144 00:48:21,930 --> 00:48:24,500 If we want to account for all of the electrons flowing 1145 00:48:24,500 --> 00:48:26,750 through our system, we would have 1146 00:48:26,750 --> 00:48:30,430 to consider as well, in any voxel of our material, 1147 00:48:30,430 --> 00:48:32,580 in any unit volume of our material, 1148 00:48:32,580 --> 00:48:35,420 we have electrons going in, electrons coming out. 1149 00:48:35,420 --> 00:48:38,500 And within it, we could have electrons being generated, 1150 00:48:38,500 --> 00:48:41,270 say by light, or electrons recombining. 1151 00:48:41,270 --> 00:48:43,060 Meaning they're losing their energy 1152 00:48:43,060 --> 00:48:45,750 and falling back into a ground state where they're bound. 1153 00:48:45,750 --> 00:48:47,772 And that's this U-term right here. 1154 00:48:47,772 --> 00:48:49,730 And so the continuity equations are essentially 1155 00:48:49,730 --> 00:48:52,080 a set of accounting equations that 1156 00:48:52,080 --> 00:48:55,430 just describe the difference, the net change, 1157 00:48:55,430 --> 00:48:56,290 going in and out. 1158 00:48:56,290 --> 00:48:58,460 If something changes on either surface, 1159 00:48:58,460 --> 00:49:01,110 it's because your either added carriers or took them away. 1160 00:49:01,110 --> 00:49:03,880 And the way you add is by shining light on your sample 1161 00:49:03,880 --> 00:49:05,120 and generating more carriers. 1162 00:49:05,120 --> 00:49:06,480 G, for instance. 1163 00:49:06,480 --> 00:49:09,590 That's one way of generating carriers by adding light. 1164 00:49:09,590 --> 00:49:11,020 Or recombination. 1165 00:49:11,020 --> 00:49:13,606 And one way of recombining is, for example, 1166 00:49:13,606 --> 00:49:14,980 through a defect in our material, 1167 00:49:14,980 --> 00:49:17,070 where the electron is moving along. 1168 00:49:17,070 --> 00:49:19,844 It finds a defect and it falls back into a ground state. 1169 00:49:19,844 --> 00:49:22,260 So the continuity equations are nothing more, nothing less 1170 00:49:22,260 --> 00:49:24,310 than housekeeping, bookkeeping. 1171 00:49:24,310 --> 00:49:26,630 But it's very important as you'll see in the next slide 1172 00:49:26,630 --> 00:49:29,670 right here where we combine five equations that 1173 00:49:29,670 --> 00:49:33,240 describe current transport in a pn-junction. 1174 00:49:33,240 --> 00:49:35,400 We have drift and diffusion up here. 1175 00:49:35,400 --> 00:49:37,120 Those are the equations we just saw. 1176 00:49:37,120 --> 00:49:39,520 We have the equation describing the electric field. 1177 00:49:39,520 --> 00:49:42,460 And we have our accounting equations down here. 1178 00:49:42,460 --> 00:49:45,010 And if we look at these equations in unison, 1179 00:49:45,010 --> 00:49:50,550 we'll notice that, oh, well, we have our hole current appearing 1180 00:49:50,550 --> 00:49:51,530 twice here. 1181 00:49:51,530 --> 00:49:54,980 We have our electric field appearing in two cases. 1182 00:49:54,980 --> 00:49:55,530 Sorry. 1183 00:49:55,530 --> 00:49:57,180 Back. 1184 00:49:57,180 --> 00:49:58,620 Back. 1185 00:49:58,620 --> 00:49:59,380 There. 1186 00:49:59,380 --> 00:50:01,800 We have our electric field appearing in those three 1187 00:50:01,800 --> 00:50:03,070 equations right here. 1188 00:50:03,070 --> 00:50:05,510 And we have a system of non-linear equations 1189 00:50:05,510 --> 00:50:10,200 as a result. It is not possible to solve these analytically 1190 00:50:10,200 --> 00:50:10,980 in their entirety. 1191 00:50:10,980 --> 00:50:12,730 We'll have to make a series of assumptions 1192 00:50:12,730 --> 00:50:16,210 to really get to the full solution. 1193 00:50:16,210 --> 00:50:19,140 But it's possible to solve them numerically 1194 00:50:19,140 --> 00:50:21,520 using computer simulations. 1195 00:50:21,520 --> 00:50:22,930 This is important because we want 1196 00:50:22,930 --> 00:50:25,540 to be able to define the current coming out of our solar cell 1197 00:50:25,540 --> 00:50:26,090 device. 1198 00:50:26,090 --> 00:50:27,840 Without solving these equations, we really 1199 00:50:27,840 --> 00:50:30,130 don't understand what the current is out 1200 00:50:30,130 --> 00:50:33,110 of a pn-junction, out of a solar cell under different bias 1201 00:50:33,110 --> 00:50:33,670 conditions. 1202 00:50:33,670 --> 00:50:35,836 That's why these equations right here are important. 1203 00:50:46,540 --> 00:50:49,260 What I want to do is not to just dwell on the current. 1204 00:50:49,260 --> 00:50:51,110 I want to emphasize that to calculate 1205 00:50:51,110 --> 00:50:53,230 the voltage across the semiconductor-- sorry, 1206 00:50:53,230 --> 00:50:56,910 a semiconductor device, a pn-junction-based solar cell. 1207 00:50:56,910 --> 00:50:59,670 To calculate the voltage across a solar cell, 1208 00:50:59,670 --> 00:51:02,660 we have to understand a little bit more. 1209 00:51:02,660 --> 00:51:07,810 So far, we've been assuming that each electron is 1210 00:51:07,810 --> 00:51:09,120 a unique individual. 1211 00:51:09,120 --> 00:51:11,980 That an electron over here is a small perturbation 1212 00:51:11,980 --> 00:51:14,580 to a larger system in that it will flow down 1213 00:51:14,580 --> 00:51:16,440 based on that electric field. 1214 00:51:16,440 --> 00:51:18,150 For a voltage, we're really talking 1215 00:51:18,150 --> 00:51:19,722 about an ensemble of electrons. 1216 00:51:19,722 --> 00:51:21,180 All of the electrons in the system. 1217 00:51:21,180 --> 00:51:23,970 And so we have to define some chemical potential 1218 00:51:23,970 --> 00:51:24,920 for our system. 1219 00:51:24,920 --> 00:51:27,510 We have to define some-- one could 1220 00:51:27,510 --> 00:51:30,090 think of it as an average energy for our system. 1221 00:51:30,090 --> 00:51:32,930 So let me define chemical potential for you 1222 00:51:32,930 --> 00:51:35,930 in a semiconductor. 1223 00:51:35,930 --> 00:51:38,100 If we describe the semiconductor-- here's 1224 00:51:38,100 --> 00:51:40,110 our valence band and our conduction band. 1225 00:51:40,110 --> 00:51:43,270 This is our band gap right here in between the two bands 1226 00:51:43,270 --> 00:51:45,982 as described by e sub g, our conduction band, valence band. 1227 00:51:45,982 --> 00:51:47,940 And these little red dots here are representing 1228 00:51:47,940 --> 00:51:50,350 our covalently bonded electrons in the ground 1229 00:51:50,350 --> 00:51:53,260 state at 0 Kelvin. 1230 00:51:53,260 --> 00:51:55,955 If we heat this material up just a little bit, 1231 00:51:55,955 --> 00:51:57,580 there will be some fraction of carriers 1232 00:51:57,580 --> 00:52:00,910 moving across into the conduction band just 1233 00:52:00,910 --> 00:52:02,680 through simple probability. 1234 00:52:02,680 --> 00:52:06,330 And we'll get to the precise equation that 1235 00:52:06,330 --> 00:52:08,100 describes the probability distribution 1236 00:52:08,100 --> 00:52:10,020 function in a few lectures. 1237 00:52:10,020 --> 00:52:12,440 But for now, let's just assume that a certain number 1238 00:52:12,440 --> 00:52:14,600 through thermal processes are able to be 1239 00:52:14,600 --> 00:52:16,580 excited across the band gap. 1240 00:52:16,580 --> 00:52:18,920 So these are thermally-excited electrons. 1241 00:52:18,920 --> 00:52:22,270 And we call these-- this is the intrinsic carrier 1242 00:52:22,270 --> 00:52:23,270 concentration. 1243 00:52:23,270 --> 00:52:25,420 So these carriers up here are free to move around 1244 00:52:25,420 --> 00:52:26,140 the crystal. 1245 00:52:26,140 --> 00:52:27,400 They can conduct electricity. 1246 00:52:27,400 --> 00:52:28,654 They can conduct charge. 1247 00:52:28,654 --> 00:52:30,070 We could measure a current flowing 1248 00:52:30,070 --> 00:52:33,130 through if we applied a bias voltage across material 1249 00:52:33,130 --> 00:52:33,970 like this. 1250 00:52:33,970 --> 00:52:35,230 Those are our charge carriers. 1251 00:52:35,230 --> 00:52:36,500 These holes as well. 1252 00:52:36,500 --> 00:52:40,516 And so this is the intrinsic carrier concentration. 1253 00:52:40,516 --> 00:52:41,890 To give you an order of magnitude 1254 00:52:41,890 --> 00:52:44,210 of the intrinsic carrier concentration in silicon, 1255 00:52:44,210 --> 00:52:48,290 it's around 10 to the 10 carriers per cubic centimeter. 1256 00:52:48,290 --> 00:52:55,180 That's 1 over 10 to the 12 per atom. 1257 00:52:55,180 --> 00:52:58,680 So each atom of silicon is generating roughly 1 over 10 1258 00:52:58,680 --> 00:53:01,080 to the 12 free carriers due to this process. 1259 00:53:01,080 --> 00:53:03,520 Very small concentration of free carriers in the material. 1260 00:53:03,520 --> 00:53:06,440 Around 5/10 into the 22 atoms per cubic centimeter 1261 00:53:06,440 --> 00:53:09,660 and only 10 to the 10 free carriers per cubic centimeter 1262 00:53:09,660 --> 00:53:11,770 due to this thermal process at room temperature. 1263 00:53:11,770 --> 00:53:15,410 So it's a very small effect in silicon. 1264 00:53:15,410 --> 00:53:18,579 As you shrink your band gap, the energy 1265 00:53:18,579 --> 00:53:20,120 necessary to excite across a band gap 1266 00:53:20,120 --> 00:53:24,630 becomes less since you have more thermal carriers. 1267 00:53:24,630 --> 00:53:27,610 So the chemical potential is describing the average energy 1268 00:53:27,610 --> 00:53:30,950 necessary to remove an infinitesimally small quantity 1269 00:53:30,950 --> 00:53:33,200 of electrons to the system. 1270 00:53:33,200 --> 00:53:35,640 And again, infinitesimally small quantity meaning it's 1271 00:53:35,640 --> 00:53:36,479 a mere perturbation. 1272 00:53:36,479 --> 00:53:37,770 You're not changing the volume. 1273 00:53:37,770 --> 00:53:39,680 You're not changing the electrostatics of the system. 1274 00:53:39,680 --> 00:53:41,630 You're just adding one electron to the system, 1275 00:53:41,630 --> 00:53:43,046 or taking it away, and determining 1276 00:53:43,046 --> 00:53:45,340 how much energy was required. 1277 00:53:45,340 --> 00:53:48,030 And so we describe that chemical potential, 1278 00:53:48,030 --> 00:53:49,980 we call it by a different name. 1279 00:53:49,980 --> 00:53:52,770 We call it Fermi level, but it is the exact same thing. 1280 00:53:52,770 --> 00:53:55,950 It's the energy necessary to remove that electron 1281 00:53:55,950 --> 00:53:57,930 from the material. 1282 00:53:57,930 --> 00:53:59,740 So you can envision that it's going 1283 00:53:59,740 --> 00:54:03,320 to be the Fermi level on either side of the pn-junction that 1284 00:54:03,320 --> 00:54:05,480 will determine the voltage output of our solar cell 1285 00:54:05,480 --> 00:54:05,980 device. 1286 00:54:05,980 --> 00:54:08,146 And that's why we want to be able to calculate this. 1287 00:54:10,420 --> 00:54:13,530 So let's look, first off what happens to our Fermi level 1288 00:54:13,530 --> 00:54:16,850 when we dope our material, when we intentionally 1289 00:54:16,850 --> 00:54:20,340 add, in this case, boron atoms to make it p-type 1290 00:54:20,340 --> 00:54:22,980 or phosphorus atoms to make the material n-type. 1291 00:54:22,980 --> 00:54:23,810 What happens? 1292 00:54:23,810 --> 00:54:26,790 Well, we end up shifting our Fermi level 1293 00:54:26,790 --> 00:54:29,240 toward the valence band or shifting the Fermi 1294 00:54:29,240 --> 00:54:32,220 level toward the conduction band by the addition of holes 1295 00:54:32,220 --> 00:54:33,536 or electrons. 1296 00:54:33,536 --> 00:54:35,410 An easy way to think about it without getting 1297 00:54:35,410 --> 00:54:38,120 into complicated semiconductor math is the following. 1298 00:54:38,120 --> 00:54:41,920 If I'm adding more electrons to my system, by doping at n-type, 1299 00:54:41,920 --> 00:54:44,507 essentially I'm shifting the energy level up. 1300 00:54:44,507 --> 00:54:45,965 It's an easy way to think about it. 1301 00:54:45,965 --> 00:54:47,880 If I'm removing the electrons from the system 1302 00:54:47,880 --> 00:54:50,710 by adding holes down here, then I'm 1303 00:54:50,710 --> 00:54:52,630 shifting the Fermi level down. 1304 00:54:52,630 --> 00:54:53,830 Easy way to think about it. 1305 00:54:53,830 --> 00:54:55,110 We'll get to the math. 1306 00:54:55,110 --> 00:54:56,610 Actually, if you're really curious, 1307 00:54:56,610 --> 00:54:59,080 I think I included one slide of the derivation 1308 00:54:59,080 --> 00:55:00,880 of the precise chemical potential 1309 00:55:00,880 --> 00:55:03,850 so you can walk through that as well. 1310 00:55:03,850 --> 00:55:05,990 So voltage across a pn-junction. 1311 00:55:05,990 --> 00:55:09,070 Under zero bias conditions, this is in the dark 1312 00:55:09,070 --> 00:55:10,820 and without a battery pack attached to it. 1313 00:55:10,820 --> 00:55:12,790 So this is an unperturbed pn-junction 1314 00:55:12,790 --> 00:55:19,090 just in the dark without any external bias. 1315 00:55:19,090 --> 00:55:23,130 We have the Fermi level constant throughout. 1316 00:55:23,130 --> 00:55:25,920 Both the p-type and the n-type flat. 1317 00:55:25,920 --> 00:55:28,780 Notice the bands are bending, so the distance 1318 00:55:28,780 --> 00:55:30,960 of the Fermi level from the valence band 1319 00:55:30,960 --> 00:55:33,640 here is smaller than the distance from the Fermi level 1320 00:55:33,640 --> 00:55:35,530 to the valence band in the n-type material. 1321 00:55:35,530 --> 00:55:36,690 That's because in the n-type material 1322 00:55:36,690 --> 00:55:38,064 we have more electrons, and we're 1323 00:55:38,064 --> 00:55:40,570 pushing the Fermi level higher. 1324 00:55:40,570 --> 00:55:43,850 The vacuum level follows these bands. 1325 00:55:43,850 --> 00:55:45,722 It just happens to be way up there. 1326 00:55:45,722 --> 00:55:47,680 And so the amount of energy necessary to remove 1327 00:55:47,680 --> 00:55:50,480 the electron from the system, to move it to the vacuum level, 1328 00:55:50,480 --> 00:55:52,680 changes from the p-type to the n-type. 1329 00:55:52,680 --> 00:55:54,480 But for the purposes of our diagram right 1330 00:55:54,480 --> 00:55:58,140 here, we're drawing the Fermi level constant throughout. 1331 00:55:58,140 --> 00:56:02,030 Now, we have that transition region, the depletion region, 1332 00:56:02,030 --> 00:56:04,950 the space charge region-- all anonymous, same thing. 1333 00:56:04,950 --> 00:56:08,100 We have that transition region here in the middle. 1334 00:56:08,100 --> 00:56:11,300 So if we begin quantifying the different parameters here, 1335 00:56:11,300 --> 00:56:15,350 we have a Fermi level distance from the valence band. 1336 00:56:15,350 --> 00:56:16,987 So that's this distance right here. 1337 00:56:16,987 --> 00:56:18,820 We have the distance between the Fermi level 1338 00:56:18,820 --> 00:56:21,130 and the conduction band in the n-type material. 1339 00:56:21,130 --> 00:56:24,040 And then we have our built-in potential. 1340 00:56:24,040 --> 00:56:27,220 And when we multiply our built-in potential by q, 1341 00:56:27,220 --> 00:56:28,085 we get an energy. 1342 00:56:28,085 --> 00:56:32,600 Essentially, an energy gain across that pn-junction. 1343 00:56:32,600 --> 00:56:36,801 So that quantity here is equal to the band gap minus these two 1344 00:56:36,801 --> 00:56:37,300 parameters. 1345 00:56:37,300 --> 00:56:40,230 Minus this, minus that. 1346 00:56:40,230 --> 00:56:42,200 And so you can see that the built-in potential 1347 00:56:42,200 --> 00:56:45,392 across the junction is benefited by higher doping. 1348 00:56:45,392 --> 00:56:47,100 The higher we dope our material, the more 1349 00:56:47,100 --> 00:56:49,680 we shift our Fermi level toward either band. 1350 00:56:49,680 --> 00:56:52,630 And the greater the separation we get, 1351 00:56:52,630 --> 00:56:55,750 the smaller this quantity is, the smaller that quantity is, 1352 00:56:55,750 --> 00:56:57,900 and the more our built-in potential approximates 1353 00:56:57,900 --> 00:57:00,240 the actual band gap of the material. 1354 00:57:00,240 --> 00:57:02,524 This built-in potential will relate to-- it 1355 00:57:02,524 --> 00:57:03,940 wont' be identical to, but it will 1356 00:57:03,940 --> 00:57:06,760 be associated with our maximum voltage 1357 00:57:06,760 --> 00:57:08,737 that we can get out of the device. 1358 00:57:08,737 --> 00:57:11,070 And so we'll want to engineer our material in such a way 1359 00:57:11,070 --> 00:57:13,180 so as to maximize that quantity. 1360 00:57:13,180 --> 00:57:15,750 That's important. 1361 00:57:15,750 --> 00:57:20,010 The relation between our built-in potential and the dope 1362 00:57:20,010 --> 00:57:21,530 intensity is shown here. 1363 00:57:21,530 --> 00:57:23,430 For now, you'll have to take my word for it. 1364 00:57:23,430 --> 00:57:26,170 I'm sparing you a lot of semiconductor physics. 1365 00:57:26,170 --> 00:57:28,170 It's written on the next slide right here if you 1366 00:57:28,170 --> 00:57:30,130 care to look in some detail. 1367 00:57:30,130 --> 00:57:32,000 It's also described in, I believe, Chapter 2 1368 00:57:32,000 --> 00:57:34,290 in Martin Green's textbook. 1369 00:57:34,290 --> 00:57:37,015 But for now, let's assume that the built-in junction potential 1370 00:57:37,015 --> 00:57:39,170 is a function of the dopant concentrations 1371 00:57:39,170 --> 00:57:40,499 just by our intuition. 1372 00:57:40,499 --> 00:57:42,290 If we're adding more electrons to one side, 1373 00:57:42,290 --> 00:57:45,960 we're going to be shifting the Fermi energy up. 1374 00:57:45,960 --> 00:57:49,760 OK, so the voltage across a pn-junction. 1375 00:57:49,760 --> 00:57:51,320 Now, let's bias our device. 1376 00:57:51,320 --> 00:57:55,400 We have under zero bias conditions this expression. 1377 00:57:55,400 --> 00:57:57,550 Sorry, there we go. 1378 00:57:57,550 --> 00:58:00,200 Under zero bias conditions, we have that expression there. 1379 00:58:00,200 --> 00:58:02,020 And under biased conditions, now we 1380 00:58:02,020 --> 00:58:04,700 have an applied biased voltage, our V sub a. 1381 00:58:04,700 --> 00:58:06,070 We're applying a bias voltage. 1382 00:58:06,070 --> 00:58:08,130 We've effectively shifted-- notice here, 1383 00:58:08,130 --> 00:58:10,870 we've shifted our bands when relative to the other. 1384 00:58:10,870 --> 00:58:15,340 So we've shifted this side up by a certain V sub a. 1385 00:58:15,340 --> 00:58:20,510 And now if you'll notice the quantities here, 1386 00:58:20,510 --> 00:58:24,320 we have a separation of the Fermi energy 1387 00:58:24,320 --> 00:58:25,620 on one side to the other. 1388 00:58:25,620 --> 00:58:28,610 There is a bias now across this device. 1389 00:58:28,610 --> 00:58:31,270 There is a driving force for electrons 1390 00:58:31,270 --> 00:58:34,230 to go and complete an external circuit, to travel 1391 00:58:34,230 --> 00:58:35,570 through the external circuit. 1392 00:58:35,570 --> 00:58:37,028 Because the electrons that are over 1393 00:58:37,028 --> 00:58:40,390 here have a higher energy net than the electrons over there-- 1394 00:58:40,390 --> 00:58:43,640 the ensemble, on average. 1395 00:58:43,640 --> 00:58:47,529 The transition region, likewise, will get smaller. 1396 00:58:47,529 --> 00:58:48,570 Sorry for that animation. 1397 00:58:48,570 --> 00:58:51,460 But if we keep track here, the transition region 1398 00:58:51,460 --> 00:58:53,800 becomes smaller under forward bias. 1399 00:58:53,800 --> 00:58:58,049 Because we're depleting-- we're removing the amount of charge 1400 00:58:58,049 --> 00:58:58,840 that was over here. 1401 00:58:58,840 --> 00:59:00,130 We're squeezing it back. 1402 00:59:00,130 --> 00:59:03,030 We're reducing that barrier height. 1403 00:59:03,030 --> 00:59:06,490 And so over here, if we go back to this diagram, 1404 00:59:06,490 --> 00:59:09,140 you can now draw in what's written in these red circles. 1405 00:59:09,140 --> 00:59:12,620 You can draw in the actual depletion 1406 00:59:12,620 --> 00:59:17,040 width, the width of the space charge region on your diagrams, 1407 00:59:17,040 --> 00:59:19,630 just like that. 1408 00:59:19,630 --> 00:59:23,920 I'll give you just a second to complete that diagram there. 1409 00:59:23,920 --> 00:59:26,840 Under reverse bias, likewise, the width of the depletion 1410 00:59:26,840 --> 00:59:27,715 region will increase. 1411 00:59:36,310 --> 00:59:38,390 And the depletion region is increasing, 1412 00:59:38,390 --> 00:59:39,950 the built-in charge is increasing, 1413 00:59:39,950 --> 00:59:41,815 the amount of band bending is increasing, 1414 00:59:41,815 --> 00:59:44,450 and the amount of drift current also increasing. 1415 00:59:44,450 --> 00:59:45,610 So it all fits together. 1416 00:59:45,610 --> 00:59:48,490 It's beginning to really come together nicely in one 1417 00:59:48,490 --> 00:59:50,720 nice picture in our minds. 1418 00:59:50,720 --> 00:59:55,028 So yes, one question. 1419 00:59:55,028 --> 00:59:57,134 AUDIENCE: So I understand that for our solar cell, 1420 00:59:57,134 --> 01:00:01,019 we wouldn't want to actually use a battery to drive current. 1421 01:00:01,019 --> 01:00:03,310 PROFESSOR: Let's get to illuminated current next class. 1422 01:00:03,310 --> 01:00:05,580 For now, we'll just focus on the battery. 1423 01:00:05,580 --> 01:00:06,080 Yeah. 1424 01:00:06,080 --> 01:00:10,500 AUDIENCE: So should we not quite yet understand 1425 01:00:10,500 --> 01:00:14,129 why forward bias and reverse bias applies to [INAUDIBLE]? 1426 01:00:14,129 --> 01:00:15,920 PROFESSOR: Let's leave that for next class. 1427 01:00:15,920 --> 01:00:19,407 For now, let's assume that the illumination current-- if you 1428 01:00:19,407 --> 01:00:20,990 really want to satisfy your curiosity, 1429 01:00:20,990 --> 01:00:22,656 your illumination current is going to be 1430 01:00:22,656 --> 01:00:24,070 one additional arrow to this. 1431 01:00:24,070 --> 01:00:26,320 It's going to be in addition to everything else that's 1432 01:00:26,320 --> 01:00:27,220 going on. 1433 01:00:27,220 --> 01:00:28,900 But the majority of the field is going 1434 01:00:28,900 --> 01:00:33,280 to be created by what is doped into the material. 1435 01:00:33,280 --> 01:00:35,490 So think of the illumination for now 1436 01:00:35,490 --> 01:00:37,492 as a small perturbation to the system. 1437 01:00:37,492 --> 01:00:39,200 That's the easiest way to think about it. 1438 01:00:39,200 --> 01:00:40,800 To justify to yourself why we need 1439 01:00:40,800 --> 01:00:43,760 to understand first the solar cell in the dark, 1440 01:00:43,760 --> 01:00:46,090 and then because of that small perturbation, 1441 01:00:46,090 --> 01:00:49,590 we can treat it as a linear superposition of effects. 1442 01:00:49,590 --> 01:00:51,850 And we'll add the illumination next class. 1443 01:00:51,850 --> 01:00:53,969 But bear with me in the dark first. 1444 01:00:53,969 --> 01:00:55,760 Because if we really don't understand this, 1445 01:00:55,760 --> 01:00:57,218 we're not going to understand fully 1446 01:00:57,218 --> 01:00:59,670 how the solar cell operates in the light either. 1447 01:00:59,670 --> 01:01:02,590 So next, we'll draw the I-V response. 1448 01:01:02,590 --> 01:01:04,400 We'll want to really get to this last point 1449 01:01:04,400 --> 01:01:07,860 right here where we can draw the current voltage response. 1450 01:01:07,860 --> 01:01:10,130 And we want to recognize that minority carrier flux is 1451 01:01:10,130 --> 01:01:12,070 what's regulating the current. 1452 01:01:12,070 --> 01:01:15,050 So to do this well, to do this properly, 1453 01:01:15,050 --> 01:01:18,690 we have to shift our focus from here 1454 01:01:18,690 --> 01:01:20,480 where we were talking about ensembles 1455 01:01:20,480 --> 01:01:24,090 and individual particles. 1456 01:01:24,090 --> 01:01:25,870 Here we're going to be discussing 1457 01:01:25,870 --> 01:01:30,160 in terms of carrier densities on either side of the junction. 1458 01:01:30,160 --> 01:01:32,440 Densities of electrons and holes. 1459 01:01:32,440 --> 01:01:33,940 And unfortunately, I'm going to have 1460 01:01:33,940 --> 01:01:35,880 to move rather quickly through these slides. 1461 01:01:35,880 --> 01:01:40,360 This is the essence of why a solar cell behaves 1462 01:01:40,360 --> 01:01:41,440 like a diode. 1463 01:01:41,440 --> 01:01:43,180 And it's really something that is best 1464 01:01:43,180 --> 01:01:45,830 done by studying on your own. 1465 01:01:45,830 --> 01:01:48,440 I'm happy to walk you through the most salient points, 1466 01:01:48,440 --> 01:01:50,610 the most important approximations 1467 01:01:50,610 --> 01:01:53,200 that we have along the way, but this is best 1468 01:01:53,200 --> 01:01:55,460 understood by going home, looking up 1469 01:01:55,460 --> 01:01:58,590 the readings on stellar, and walking through the derivation 1470 01:01:58,590 --> 01:01:59,290 yourself. 1471 01:01:59,290 --> 01:02:00,910 It's not something that very easily I 1472 01:02:00,910 --> 01:02:05,160 can convey a series of equations in the class. 1473 01:02:05,160 --> 01:02:09,930 So before we go into detail into current flows, 1474 01:02:09,930 --> 01:02:13,074 I wanted to touch on this with the space charge region. 1475 01:02:13,074 --> 01:02:15,240 We can describe the width of the space charge region 1476 01:02:15,240 --> 01:02:19,410 now by the built-in bias across the junction. 1477 01:02:19,410 --> 01:02:22,560 The applied bias right here and some fundamental material 1478 01:02:22,560 --> 01:02:24,400 properties as well. 1479 01:02:24,400 --> 01:02:26,230 And this little epsilon right here 1480 01:02:26,230 --> 01:02:31,340 is essentially the dielectric constant and the vacuum 1481 01:02:31,340 --> 01:02:32,822 permittivity, a constant. 1482 01:02:32,822 --> 01:02:34,530 And so you want to take that into account 1483 01:02:34,530 --> 01:02:36,610 when you're running your actual calculations. 1484 01:02:36,610 --> 01:02:39,540 It's very easy to be off on the width of the space charge 1485 01:02:39,540 --> 01:02:41,470 region by several orders of magnitude 1486 01:02:41,470 --> 01:02:46,060 if you don't do the proper accounting for those fixed 1487 01:02:46,060 --> 01:02:47,030 variables. 1488 01:02:47,030 --> 01:02:49,110 And so if you walk through the equations 1489 01:02:49,110 --> 01:02:51,550 to describe the width of the space charge region, 1490 01:02:51,550 --> 01:02:54,140 you'll find that in a typical solar cell device, 1491 01:02:54,140 --> 01:02:55,422 it's on the order of a micron. 1492 01:02:55,422 --> 01:02:57,630 And this is related to one of your homework problems, 1493 01:02:57,630 --> 01:03:00,230 so stash that away somewhere in your brain. 1494 01:03:00,230 --> 01:03:01,569 It's on the order of a micron. 1495 01:03:01,569 --> 01:03:03,110 The width of that space charge region 1496 01:03:03,110 --> 01:03:05,380 typically could be on that order. 1497 01:03:05,380 --> 01:03:07,965 So it's going to be some multiple. 1498 01:03:07,965 --> 01:03:09,340 And the reason that's interesting 1499 01:03:09,340 --> 01:03:11,280 is because the entire solar cell device 1500 01:03:11,280 --> 01:03:14,110 is about 100 to 200 microns. 1501 01:03:14,110 --> 01:03:17,590 That's for a crystalline silicon device. 1502 01:03:17,590 --> 01:03:20,590 That means that in a crystalline silicon device, 1503 01:03:20,590 --> 01:03:23,090 the diffusion current is what's driving 1504 01:03:23,090 --> 01:03:25,730 most of the current flow inside of the solar cell. 1505 01:03:25,730 --> 01:03:27,740 If our solar cell is much, much, much thinner, 1506 01:03:27,740 --> 01:03:30,470 say in the order of a micron, then our drift current 1507 01:03:30,470 --> 01:03:32,026 would be dominating. 1508 01:03:32,026 --> 01:03:33,650 This is a more advanced topic and we'll 1509 01:03:33,650 --> 01:03:35,608 return to that when we describe the differences 1510 01:03:35,608 --> 01:03:38,400 between thin film operation and crystalline silicon solar cell 1511 01:03:38,400 --> 01:03:39,760 operation. 1512 01:03:39,760 --> 01:03:41,060 Device capacitance. 1513 01:03:41,060 --> 01:03:45,320 For those who are running experimental measurements 1514 01:03:45,320 --> 01:03:48,890 and want to determine the RC time constant 1515 01:03:48,890 --> 01:03:52,920 necessary to have the device settle into a measurable state, 1516 01:03:52,920 --> 01:03:54,480 that's there. 1517 01:03:54,480 --> 01:03:57,060 And the capacitance is described, again, 1518 01:03:57,060 --> 01:04:00,500 by the various properties as well as 1519 01:04:00,500 --> 01:04:04,140 the built-in potential which is related to the dopant density. 1520 01:04:04,140 --> 01:04:07,810 OK, so under zero bias, we have a concentration 1521 01:04:07,810 --> 01:04:10,292 of holes on one side of our junction that's high. 1522 01:04:10,292 --> 01:04:12,750 In the p-type material, the concentration of holes is high. 1523 01:04:12,750 --> 01:04:15,690 And it's approximately equal to the acceptor concentration. 1524 01:04:15,690 --> 01:04:17,620 On the n-type side of our junction, 1525 01:04:17,620 --> 01:04:20,730 the whole population is drastically reduced. 1526 01:04:20,730 --> 01:04:25,480 And the flip side, our electron concentration is very high 1527 01:04:25,480 --> 01:04:27,877 and it drops into the p-type side. 1528 01:04:27,877 --> 01:04:29,710 So we have predominately holes on this side. 1529 01:04:29,710 --> 01:04:31,470 Predominately electrons in that side. 1530 01:04:31,470 --> 01:04:35,330 But a small concentration of holes on the n-type side 1531 01:04:35,330 --> 01:04:38,220 and a small concentration of electrons in the p-type side. 1532 01:04:38,220 --> 01:04:40,730 And we call this the minority carrier. 1533 01:04:40,730 --> 01:04:42,340 And we call that the majority carrier. 1534 01:04:42,340 --> 01:04:44,100 The majority carrier is in the majority. 1535 01:04:44,100 --> 01:04:46,220 The minority carrier in the minority. 1536 01:04:46,220 --> 01:04:48,890 And an interesting thing happens when we bias our device. 1537 01:04:48,890 --> 01:04:51,580 When we bias our solar cell device, 1538 01:04:51,580 --> 01:04:53,360 populations of both carriers increase. 1539 01:04:53,360 --> 01:04:56,070 But because this is on a log scale, 1540 01:04:56,070 --> 01:05:00,510 we're increasing this-- shall we say the minority carrier 1541 01:05:00,510 --> 01:05:04,610 concentration in an absolute sense by a lot. 1542 01:05:04,610 --> 01:05:08,240 And it's because of this drastic uptick in the minority carrier 1543 01:05:08,240 --> 01:05:11,060 concentration right at the edge of the space charge region 1544 01:05:11,060 --> 01:05:14,380 that we have current flow across that junction. 1545 01:05:14,380 --> 01:05:19,520 And that's described by a series of equations here. 1546 01:05:19,520 --> 01:05:22,420 Let's see, the way to walk through the derivation, 1547 01:05:22,420 --> 01:05:24,860 there's a series of approximations to make. 1548 01:05:24,860 --> 01:05:29,010 Think of it in terms of electron and hole fluxes 1549 01:05:29,010 --> 01:05:30,590 on either side of the junction. 1550 01:05:30,590 --> 01:05:33,090 You can make a series of assumptions as to what currents 1551 01:05:33,090 --> 01:05:34,574 matter and which don't. 1552 01:05:34,574 --> 01:05:36,240 You can, of course, make the assumptions 1553 01:05:36,240 --> 01:05:38,060 as to the charge distribution as well, 1554 01:05:38,060 --> 01:05:39,940 fixed using that box potential. 1555 01:05:39,940 --> 01:05:44,830 And you can consider the cases in which the minority carrier 1556 01:05:44,830 --> 01:05:49,430 concentration here is dominating to be the regions of interest 1557 01:05:49,430 --> 01:05:52,380 for a current generation inside of our device. 1558 01:05:52,380 --> 01:05:56,500 And so if we walk through, again the diffusion equation, 1559 01:05:56,500 --> 01:05:59,360 the diffusion of carriers at the edge of the space 1560 01:05:59,360 --> 01:06:00,700 charge region. 1561 01:06:00,700 --> 01:06:03,520 And from the previous slides here, 1562 01:06:03,520 --> 01:06:05,810 add our approximations in, we will ultimately 1563 01:06:05,810 --> 01:06:08,760 derive an expression that has an exponential relation 1564 01:06:08,760 --> 01:06:11,590 between current and our voltage. 1565 01:06:11,590 --> 01:06:16,200 And if we continue through the series of calculations, 1566 01:06:16,200 --> 01:06:19,100 including the continuity equation for accountability, 1567 01:06:19,100 --> 01:06:20,950 we wind up with an expression that 1568 01:06:20,950 --> 01:06:23,670 looks like this right here, where the total current flowing 1569 01:06:23,670 --> 01:06:27,466 through the device will be equal to the electron current 1570 01:06:27,466 --> 01:06:29,340 at the edge of our space charge region coming 1571 01:06:29,340 --> 01:06:31,370 from the p-type side. 1572 01:06:31,370 --> 01:06:34,000 The hole concentration coming from the n-type side. 1573 01:06:34,000 --> 01:06:35,660 The addition of the two together. 1574 01:06:35,660 --> 01:06:38,700 And that's effectively, this equation right here, 1575 01:06:38,700 --> 01:06:41,880 where we have an exponential relation between current 1576 01:06:41,880 --> 01:06:44,590 and voltage. 1577 01:06:44,590 --> 01:06:50,610 So again, this we really need to go home and study. 1578 01:06:50,610 --> 01:06:52,894 And if you'd like to do this pictorially, 1579 01:06:52,894 --> 01:06:54,560 I provide you with that link right there 1580 01:06:54,560 --> 01:06:55,930 so you can see it visually. 1581 01:06:55,930 --> 01:07:00,120 If you're more of a math type, Chapters 3 and 4-- actually, 1582 01:07:00,120 --> 01:07:03,580 Chapter 4 in Martin Green is probably the best place to go. 1583 01:07:03,580 --> 01:07:05,400 And the important thing to keep in mind 1584 01:07:05,400 --> 01:07:08,784 is that in the pn-junction, the current flow 1585 01:07:08,784 --> 01:07:10,200 across that junction is determined 1586 01:07:10,200 --> 01:07:13,830 by the minority carrier current flow at the edge of the space 1587 01:07:13,830 --> 01:07:15,380 charge region. 1588 01:07:15,380 --> 01:07:18,460 And that's why we wind up with the exponential relation 1589 01:07:18,460 --> 01:07:21,060 between voltage and current. 1590 01:07:21,060 --> 01:07:23,000 I decided the best way to emphasize 1591 01:07:23,000 --> 01:07:26,540 this voltage-current relation is actually to measure it, 1592 01:07:26,540 --> 01:07:27,710 to run it. 1593 01:07:27,710 --> 01:07:28,700 To do it. 1594 01:07:28,700 --> 01:07:31,330 And we have around 10 minutes left before the end of class. 1595 01:07:31,330 --> 01:07:33,110 Do you think that is time for our demo? 1596 01:07:33,110 --> 01:07:33,960 Cutting it kind of short. 1597 01:07:33,960 --> 01:07:35,220 AUDIENCE: Probably wait for Tuesday. 1598 01:07:35,220 --> 01:07:37,261 PROFESSOR: Probably wait for Tuesday on the demo. 1599 01:07:37,261 --> 01:07:38,400 AUDIENCE: [INAUDIBLE]. 1600 01:07:38,400 --> 01:07:40,650 PROFESSOR: So we're going to run the demos on Tuesday, 1601 01:07:40,650 --> 01:07:41,629 I suppose. 1602 01:07:41,629 --> 01:07:43,670 Because we're running a little bit short on time. 1603 01:07:43,670 --> 01:07:45,128 I know David is really disappointed 1604 01:07:45,128 --> 01:07:46,830 because he's been working hard to get 1605 01:07:46,830 --> 01:07:48,980 these in perfect ship-shape condition. 1606 01:07:48,980 --> 01:07:50,756 But we'll have another couple of days 1607 01:07:50,756 --> 01:07:52,630 to work out some of the bugs in the software, 1608 01:07:52,630 --> 01:07:54,463 so we don't need to restart it every time we 1609 01:07:54,463 --> 01:07:57,224 take a new I-V measurement. 1610 01:07:57,224 --> 01:07:58,640 But by and large, what we'll do is 1611 01:07:58,640 --> 01:08:01,490 we'll measure the current voltage relation 1612 01:08:01,490 --> 01:08:03,380 for a real solar cell device. 1613 01:08:03,380 --> 01:08:05,660 And we'll see that it definitely does follow, 1614 01:08:05,660 --> 01:08:08,150 at least under forward bias conditions, 1615 01:08:08,150 --> 01:08:09,600 this exponential relation. 1616 01:08:09,600 --> 01:08:13,280 So as v goes forward, as v-- this v over here 1617 01:08:13,280 --> 01:08:15,490 is our applied bias condition. 1618 01:08:15,490 --> 01:08:19,890 As our v increases, we have that exponential current output 1619 01:08:19,890 --> 01:08:21,120 from our device. 1620 01:08:21,120 --> 01:08:23,700 As v goes towards a negative number, 1621 01:08:23,700 --> 01:08:25,779 we have pretty much a flat lining of our current. 1622 01:08:25,779 --> 01:08:30,314 It goes into a slightly negative condition and a flat lining 1623 01:08:30,314 --> 01:08:30,939 of the current. 1624 01:08:30,939 --> 01:08:37,109 So if we look at this again right here, our I versus V, 1625 01:08:37,109 --> 01:08:40,160 now we can plot the same equation. 1626 01:08:40,160 --> 01:08:43,460 Or essentially, the same curve on all three of these. 1627 01:08:43,460 --> 01:08:47,180 We'll want to plot the same curve-- this one over here. 1628 01:08:47,180 --> 01:08:49,520 And for now, let's assume that this big lump over here 1629 01:08:49,520 --> 01:08:51,517 is a constant. 1630 01:08:51,517 --> 01:08:54,100 And so we just need to plot an exponential function, something 1631 01:08:54,100 --> 01:08:56,580 similar to this right here. 1632 01:08:56,580 --> 01:09:00,399 And under zero bias conditions, we'll be right there at 0, 0. 1633 01:09:00,399 --> 01:09:02,729 Under forward bias conditions will be further up. 1634 01:09:02,729 --> 01:09:06,040 And the reverse bias conditions will be down there. 1635 01:09:06,040 --> 01:09:11,370 And let me give you a second just to draw this down. 1636 01:09:11,370 --> 01:09:12,410 And please do. 1637 01:09:12,410 --> 01:09:15,090 We'll be validating it to ourselves on Tuesday. 1638 01:09:15,090 --> 01:09:17,939 My sincere apologies for not having enough time to really 1639 01:09:17,939 --> 01:09:21,120 go into the demos as well. 1640 01:09:21,120 --> 01:09:23,479 But suffice to say the following. 1641 01:09:23,479 --> 01:09:25,189 Under forward bias conditions, when 1642 01:09:25,189 --> 01:09:28,260 we reduce the barrier height between the n- 1643 01:09:28,260 --> 01:09:31,710 and the p-type side, we're now allowing that diffusion current 1644 01:09:31,710 --> 01:09:34,590 from the n to the p to take over. 1645 01:09:34,590 --> 01:09:37,470 And as we reduce the barrier more and more and more, 1646 01:09:37,470 --> 01:09:40,770 we have this exponentially increasing density of electrons 1647 01:09:40,770 --> 01:09:43,760 flowing through our system. 1648 01:09:43,760 --> 01:09:46,390 And that's why we have this exponentially increasing 1649 01:09:46,390 --> 01:09:48,109 curve here. 1650 01:09:48,109 --> 01:09:51,700 As we reverse bias-- our device, we're 1651 01:09:51,700 --> 01:09:55,323 increasing the barrier height for the electrons to pass over. 1652 01:09:55,323 --> 01:10:00,080 So our diffusion current practically vanishes. 1653 01:10:00,080 --> 01:10:02,570 And our drift current stays, more or less, 1654 01:10:02,570 --> 01:10:05,300 constant all the way throughout because there's only 1655 01:10:05,300 --> 01:10:08,100 a finite carrier density, a finite minority carrier 1656 01:10:08,100 --> 01:10:10,510 concentration inside of the p-type silicon. 1657 01:10:10,510 --> 01:10:12,534 The thermally excited carriers for instance, 1658 01:10:12,534 --> 01:10:14,700 that are finding their way to the junction and being 1659 01:10:14,700 --> 01:10:15,830 drifted across. 1660 01:10:15,830 --> 01:10:18,370 And that concentration is virtually finite. 1661 01:10:18,370 --> 01:10:21,560 At some point, you'll reverse bias this so much 1662 01:10:21,560 --> 01:10:24,850 that the carriers will begin tunneling across 1663 01:10:24,850 --> 01:10:27,740 from the valence band into the conduction band over here 1664 01:10:27,740 --> 01:10:30,640 and you'll have a catastrophic failure of your device, 1665 01:10:30,640 --> 01:10:33,237 but that's more of an advanced point. 1666 01:10:33,237 --> 01:10:33,737 Yeah. 1667 01:10:33,737 --> 01:10:37,550 AUDIENCE: So does the x indicate the coordinate of I and V 1668 01:10:37,550 --> 01:10:38,360 at steady state? 1669 01:10:38,360 --> 01:10:41,390 PROFESSOR: So the entire line represents 1670 01:10:41,390 --> 01:10:44,860 the I-V characteristic under all biased conditions. 1671 01:10:44,860 --> 01:10:49,150 The x represents the current and voltage 1672 01:10:49,150 --> 01:10:52,920 for this particular operating point on the solar cell. 1673 01:10:52,920 --> 01:10:54,190 On the I-V characteristic. 1674 01:10:54,190 --> 01:10:55,980 So if you're under forward bias condition. 1675 01:10:55,980 --> 01:10:58,110 In other words, your v is positive, 1676 01:10:58,110 --> 01:11:01,340 you will have a positive current flow through your device. 1677 01:11:01,340 --> 01:11:04,490 Positive current flow, electrons flowing that way. 1678 01:11:04,490 --> 01:11:07,580 And under reverse bias, notice the sign of the current 1679 01:11:07,580 --> 01:11:09,925 is flipped because now the electrons used 1680 01:11:09,925 --> 01:11:10,800 to be going that way. 1681 01:11:10,800 --> 01:11:13,100 The electrons are predominately moving the other direction. 1682 01:11:13,100 --> 01:11:15,280 The net current flow is in the opposite direction. 1683 01:11:15,280 --> 01:11:18,570 And the voltage is also changing sign. 1684 01:11:18,570 --> 01:11:20,212 AUDIENCE: You can say, OK, I'm going 1685 01:11:20,212 --> 01:11:21,779 to put myself in this voltage? 1686 01:11:21,779 --> 01:11:22,570 PROFESSOR: Exactly. 1687 01:11:22,570 --> 01:11:24,150 And that's important because we want 1688 01:11:24,150 --> 01:11:27,876 to be able to describe what our current voltage 1689 01:11:27,876 --> 01:11:29,250 characteristic is of a solar cell 1690 01:11:29,250 --> 01:11:32,000 because the product of the two is the power. 1691 01:11:32,000 --> 01:11:34,200 For efficiency, we want to know what the power out 1692 01:11:34,200 --> 01:11:35,429 of our solar cell is. 1693 01:11:35,429 --> 01:11:37,220 And it's the product of current and voltage 1694 01:11:37,220 --> 01:11:41,560 that will give us the power of our solar cell device. 1695 01:11:41,560 --> 01:11:43,830 Time for a few questions, actually. 1696 01:11:43,830 --> 01:11:47,940 About five minutes of Q&A since we had to forgo the demo. 1697 01:12:02,170 --> 01:12:04,020 Promise me you'll do one thing. 1698 01:12:04,020 --> 01:12:05,620 Promise me you'll go home. 1699 01:12:05,620 --> 01:12:07,750 And sometime between now and Tuesday, you're 1700 01:12:07,750 --> 01:12:09,620 going to read through these chapters 1701 01:12:09,620 --> 01:12:12,630 and do the derivations that will work 1702 01:12:12,630 --> 01:12:15,280 through to this ideal diode equation right here. 1703 01:12:15,280 --> 01:12:16,800 Because this is pretty important. 1704 01:12:16,800 --> 01:12:17,820 And you have to convince yourself 1705 01:12:17,820 --> 01:12:19,180 that that's indeed the case. 1706 01:12:19,180 --> 01:12:20,970 I can wave my hands and phenomenologically 1707 01:12:20,970 --> 01:12:23,680 describe, OK, we have an energy barrier. 1708 01:12:23,680 --> 01:12:24,430 I kind of get it. 1709 01:12:24,430 --> 01:12:26,950 I should have an exponential probability 1710 01:12:26,950 --> 01:12:28,595 of passing over an energy barrier. 1711 01:12:28,595 --> 01:12:30,970 At least from quantum mechanics that kind of makes sense. 1712 01:12:30,970 --> 01:12:33,350 So I kind of get how if I forward bias my device, 1713 01:12:33,350 --> 01:12:38,630 I'll have an exponentially increasing diffusion current. 1714 01:12:38,630 --> 01:12:40,750 I get that if I go into reverse bias conditions, 1715 01:12:40,750 --> 01:12:44,430 there's just a finite density of carriers, minority carriers, 1716 01:12:44,430 --> 01:12:45,730 in my p-type material. 1717 01:12:45,730 --> 01:12:48,050 If I'm not, say, shining a light on the material, 1718 01:12:48,050 --> 01:12:50,480 I'm not changing that concentration by much. 1719 01:12:50,480 --> 01:12:53,080 And I'm going to flat line. 1720 01:12:53,080 --> 01:12:55,415 Yes, they'll be drifting across, but they'll 1721 01:12:55,415 --> 01:12:59,660 be limited more by their ability to reach the junction. 1722 01:12:59,660 --> 01:13:03,650 You can hand wave all you want, but this is really-- whoopsie-- 1723 01:13:03,650 --> 01:13:07,800 the equations that describe the ideal diode equation. 1724 01:13:07,800 --> 01:13:09,634 Those are really where it's at. 1725 01:13:09,634 --> 01:13:11,050 With that equation, you can really 1726 01:13:11,050 --> 01:13:13,540 understand how material properties will 1727 01:13:13,540 --> 01:13:15,670 impact solar cell performance, let's say. 1728 01:13:15,670 --> 01:13:16,705 Yes, question? 1729 01:13:16,705 --> 01:13:19,675 AUDIENCE: I have a question about the band diagrams. 1730 01:13:23,140 --> 01:13:27,100 For all three diagrams, the energy levels [INAUDIBLE] 1731 01:13:27,100 --> 01:13:28,090 the same. 1732 01:13:28,090 --> 01:13:31,060 We're just changing where the n-type side is. 1733 01:13:31,060 --> 01:13:33,040 Both of them [INAUDIBLE]? 1734 01:13:33,040 --> 01:13:34,040 PROFESSOR: Exactly. 1735 01:13:34,040 --> 01:13:35,968 Yep, absolutely. 1736 01:13:35,968 --> 01:13:41,170 The point was here, what we've done for simplicity is shifting 1737 01:13:41,170 --> 01:13:42,470 one side relative to the other. 1738 01:13:45,600 --> 01:13:46,100 Yeah. 1739 01:13:46,100 --> 01:13:49,180 In energy-- see, energy is a funny thing 1740 01:13:49,180 --> 01:13:52,970 because you can redefine your zero. 1741 01:13:52,970 --> 01:13:53,470 Yeah. 1742 01:13:53,470 --> 01:13:56,370 So depending on where you're defining your zero point at, 1743 01:13:56,370 --> 01:13:58,220 you can move things relative to each other. 1744 01:13:58,220 --> 01:14:01,220 But let's assume that you're changing 1745 01:14:01,220 --> 01:14:03,420 both sides relative to some universal ground 1746 01:14:03,420 --> 01:14:06,830 potential that's off on an external component 1747 01:14:06,830 --> 01:14:08,606 of your circuit. 1748 01:14:08,606 --> 01:14:09,522 AUDIENCE: [INAUDIBLE]. 1749 01:14:27,072 --> 01:14:28,530 PROFESSOR: That's a great question. 1750 01:14:28,530 --> 01:14:31,519 So the question was, when you dope your material, what 1751 01:14:31,519 --> 01:14:33,060 are the experimental methods that you 1752 01:14:33,060 --> 01:14:35,620 can use to really determine whether or not 1753 01:14:35,620 --> 01:14:40,110 the atoms have occupied a substitutional position? 1754 01:14:40,110 --> 01:14:43,930 One of the methods is called Rutherford backscattering 1755 01:14:43,930 --> 01:14:47,260 or channeling, IN channeling, where you introduce ions 1756 01:14:47,260 --> 01:14:48,320 down your lattice. 1757 01:14:48,320 --> 01:14:50,360 And if there is an atom here, for example, 1758 01:14:50,360 --> 01:14:52,330 an interstitial site, it will scatter 1759 01:14:52,330 --> 01:14:54,680 some fraction of those introduced ions back 1760 01:14:54,680 --> 01:14:56,590 at the detector. 1761 01:14:56,590 --> 01:15:01,460 And so channeling is one characterization method 1762 01:15:01,460 --> 01:15:06,310 for determining the interstitial to substitutional ratio inside 1763 01:15:06,310 --> 01:15:09,420 of a semiconducting material. 1764 01:15:09,420 --> 01:15:13,350 Another way that we have very strong evidence 1765 01:15:13,350 --> 01:15:15,942 that boron is occupying a substitutional site and not 1766 01:15:15,942 --> 01:15:21,000 an interstitial site is we can calculate the binding energy 1767 01:15:21,000 --> 01:15:22,890 of this hole to the boron atom here 1768 01:15:22,890 --> 01:15:24,620 on the substitutional site. 1769 01:15:24,620 --> 01:15:26,610 In principle, if the interstitial was also-- 1770 01:15:26,610 --> 01:15:28,840 if it was a charge to defect, we could 1771 01:15:28,840 --> 01:15:33,190 calculate the binding energy of the charge to that defect. 1772 01:15:33,190 --> 01:15:35,840 And the density functional theory 1773 01:15:35,840 --> 01:15:37,770 could tell you which is the more likely 1774 01:15:37,770 --> 01:15:39,750 given your experimental observation that you 1775 01:15:39,750 --> 01:15:42,280 get about one hole for every one boron anatomy inside 1776 01:15:42,280 --> 01:15:43,359 of your sample. 1777 01:15:43,359 --> 01:15:45,150 Boron has been studied to death in silicon. 1778 01:15:45,150 --> 01:15:47,150 It's pretty well-known that it's substitutional. 1779 01:15:47,150 --> 01:15:49,580 But if you're working with a new semiconducting material, 1780 01:15:49,580 --> 01:15:50,640 you never know. 1781 01:15:50,640 --> 01:15:51,973 You have to run the experiments. 1782 01:15:55,060 --> 01:15:56,978 Yeah, question. 1783 01:15:56,978 --> 01:16:01,768 AUDIENCE: Under bias [INAUDIBLE] diagrams, are the p- 1784 01:16:01,768 --> 01:16:04,642 and n-sides far away from the [INAUDIBLE]. 1785 01:16:04,642 --> 01:16:07,050 Are those bands actually-- do they actually 1786 01:16:07,050 --> 01:16:10,584 have a slope to them because of the applied field? 1787 01:16:10,584 --> 01:16:11,460 PROFESSOR: Yeah. 1788 01:16:11,460 --> 01:16:15,880 So the question was far away from the space charge region 1789 01:16:15,880 --> 01:16:17,950 here, do these bands have a slope? 1790 01:16:17,950 --> 01:16:21,730 My argument would be that if we go back to the way 1791 01:16:21,730 --> 01:16:24,830 we derive-- there we go. 1792 01:16:24,830 --> 01:16:27,100 The way we derive this band to begin with, 1793 01:16:27,100 --> 01:16:29,790 it's predicated upon an electric field, which is predicated 1794 01:16:29,790 --> 01:16:31,430 upon a charge distribution. 1795 01:16:31,430 --> 01:16:34,390 So the biggest question is, what does this real charge 1796 01:16:34,390 --> 01:16:35,970 distribution look like? 1797 01:16:35,970 --> 01:16:37,370 Most of the time, it's relatively 1798 01:16:37,370 --> 01:16:41,980 concentrated to the near space charge region. 1799 01:16:41,980 --> 01:16:46,280 If you get too far away, OK, yes, it decays exponentially. 1800 01:16:46,280 --> 01:16:48,970 But it falls below the background intrinsic carrier 1801 01:16:48,970 --> 01:16:51,464 concentration and effectively doesn't matter. 1802 01:16:51,464 --> 01:16:54,370 AUDIENCE: Under applied bias, [INAUDIBLE]? 1803 01:16:54,370 --> 01:16:55,142 PROFESSOR: Yes. 1804 01:16:55,142 --> 01:16:57,100 AUDIENCE: Usually they're just drawing it flat, 1805 01:16:57,100 --> 01:16:59,500 but is it such a small effect they're just 1806 01:16:59,500 --> 01:17:00,950 drawing it flat [INAUDIBLE]? 1807 01:17:00,950 --> 01:17:01,220 PROFESSOR: Yeah. 1808 01:17:01,220 --> 01:17:02,530 So under applied bias, what happens 1809 01:17:02,530 --> 01:17:03,670 to these charge distributions? 1810 01:17:03,670 --> 01:17:05,170 Well, if you reverse bias it, you're 1811 01:17:05,170 --> 01:17:07,400 essentially increasing the amount of charge 1812 01:17:07,400 --> 01:17:08,240 on either side of the junction. 1813 01:17:08,240 --> 01:17:09,906 If you forward bias, you're reducing it. 1814 01:17:09,906 --> 01:17:11,570 So under forward bias, probably you 1815 01:17:11,570 --> 01:17:13,320 wouldn't see the effect you're describing. 1816 01:17:13,320 --> 01:17:17,490 Under reverse bias, the width of the space charge region, 1817 01:17:17,490 --> 01:17:21,960 let's say, increases from, let's call it 1 micron to 3 microns. 1818 01:17:21,960 --> 01:17:24,940 And the thickness of the entire device is 200 microns. 1819 01:17:24,940 --> 01:17:27,740 So still, I think it would be a relatively small effect 1820 01:17:27,740 --> 01:17:30,507 several 10's of microns away from the junction. 1821 01:17:30,507 --> 01:17:32,090 It really depends on the length scale, 1822 01:17:32,090 --> 01:17:35,850 the geometry of your device relative to the length scale 1823 01:17:35,850 --> 01:17:39,490 of your space charge region. 1824 01:17:39,490 --> 01:17:40,162 Good? 1825 01:17:40,162 --> 01:17:42,620 Well, to give you time to reach your next class, thank you. 1826 01:17:42,620 --> 01:17:45,203 We'll come back on Tuesday, and we'll have a great demo set up 1827 01:17:45,203 --> 01:17:46,180 for you.