1 00:00:00,660 --> 00:00:04,950 Welcome back. This is now the 17th lecture of the condensed matter, of course. 2 00:00:05,220 --> 00:00:12,090 When we left off last time, we were talking about band structure and we're talking about electrons in conduction bands and holes in valence bands. 3 00:00:12,270 --> 00:00:19,380 And what we're going to do today is we're going to start by thinking about how electrons and conduction bands and holes in valence bands move around. 4 00:00:19,680 --> 00:00:23,190 So let's start by drawing the bottom of a conduction band. 5 00:00:23,190 --> 00:00:29,970 So we have some E and some K and there's going to be some quadratic minimum of a conduction band. 6 00:00:30,210 --> 00:00:36,950 Let's call this point here came in where the bottom occurs and this point here, we'll call it E, 7 00:00:36,960 --> 00:00:43,050 not the energy of the bottom of the conduction band very generally already over here, very generally, 8 00:00:43,050 --> 00:00:49,590 we can we can write for an electron near the bottom of the band that E is approximately e 9 00:00:49,590 --> 00:00:59,910 nought plus which is the Taylor expansion D squared e d squared times k minus came in squared, 10 00:01:00,540 --> 00:01:06,390 maybe other terms which will ignore. And given this, we will define an effective mass. 11 00:01:06,780 --> 00:01:11,399 An electron effective mass. Electron effective. 12 00:01:11,400 --> 00:01:17,010 Mass effective. Mass. 13 00:01:18,150 --> 00:01:25,320 I'm a star. I'm stars of electron via one over am star electron. 14 00:01:25,650 --> 00:01:29,070 Equals one over a squared d squared. 15 00:01:29,070 --> 00:01:45,770 E d k squared. Such that the energy is not plus the usual h bar squared k minus came in a squared over two an electron star. 16 00:01:46,050 --> 00:01:52,980 Now this looks very much like a free electron, except the momentum, the the wave vector should be measured with respect to the wherever. 17 00:01:52,980 --> 00:01:57,330 The minimum in the momentum is is the usual free electron parabola. 18 00:01:57,660 --> 00:02:06,990 Now given this, we can also calculate actually given any form of the energy of the of the electron in this, 19 00:02:07,260 --> 00:02:15,870 in this band we can write a group velocity V group very generally is d omega decay if it's a general expression for a group velocity, 20 00:02:16,290 --> 00:02:24,120 since we can replace omega by one over h bar times in times in energy d decay. 21 00:02:24,660 --> 00:02:29,969 This is the general relationship for the group velocity of an electron in a band. 22 00:02:29,970 --> 00:02:41,250 In our case it will be using this dispersion up here it will be h bar k minus came in over m star of the electron. 23 00:02:41,640 --> 00:02:43,950 So it means that if you have this picture over here, 24 00:02:44,160 --> 00:02:50,790 if you if you move the electron to the right of the minimum, then the electron is moving to the right. 25 00:02:50,790 --> 00:02:55,080 If you move the electron to the left of the minimum, then the electron is moving to the left. 26 00:02:55,350 --> 00:02:58,740 This is true independent of where in the bronze down the minimum occurs. 27 00:02:58,980 --> 00:03:02,910 So really the issue is where your position with respect to the minimum, 28 00:03:02,910 --> 00:03:09,210 if you want to figure out which direction your group velocity is now, there are a couple of caveats here. 29 00:03:09,570 --> 00:03:17,700 Caveat, one are generally effective masses, right? 30 00:03:17,700 --> 00:03:32,609 This masses may be anisotropic, masses can be anisotropic, meaning that the energy is it may be parabolic in all directions, 31 00:03:32,610 --> 00:03:36,900 but depending on which crystal direction you choose, the curvature may be different. 32 00:03:37,140 --> 00:03:42,890 This is sort of natural. If you have something like a tetrahedral crystal where the axes are different lengths in different directions, 33 00:03:42,900 --> 00:03:50,010 you might expect that the curvature of the energy in the different displays in K in the different directions might be different. 34 00:03:50,010 --> 00:03:56,639 And indeed it often is. So one has to worry about the fact that depending on which way you're going, your effect of mass may seem different. 35 00:03:56,640 --> 00:04:04,890 We're going to ignore this generally, so ignore this complication, but more generally you should be aware that it does occur. 36 00:04:05,310 --> 00:04:13,560 The second complication that you should be aware and it occurs is you can have situations if you plot E versus K, 37 00:04:13,890 --> 00:04:19,620 you can have a situation where there are two minima that occur and exactly the same energy. 38 00:04:21,220 --> 00:04:29,650 This E so that the minimum of the conduction band can occur at two places in the burn zone, K1 and K2. 39 00:04:30,070 --> 00:04:35,020 These are known as valleys. Valleys for obvious reason. 40 00:04:36,940 --> 00:04:40,719 This may look like it is something that is very unusual to have happen, 41 00:04:40,720 --> 00:04:47,290 that you have just such a coincidence that the minima that the same energy occurs at the minimum in two places in the bronze zone. 42 00:04:47,530 --> 00:04:52,149 But in fact it's actually quite common to have this be due to the symmetries of the crystal. 43 00:04:52,150 --> 00:04:56,020 So generally this will happen. If the crystal is particularly symmetric, 44 00:04:56,020 --> 00:04:59,589 you'll have to minimum in the conduction band at two different points in the bronze 45 00:04:59,590 --> 00:05:03,790 zone or even four different points or even six different points of of the barren zone. 46 00:05:03,970 --> 00:05:08,170 So you have to keep track of whether the electron is in this minimum or this minimum. 47 00:05:08,410 --> 00:05:13,719 But the general rule applies that if you displace yourself in the minimum to the right, you're moving to the right. 48 00:05:13,720 --> 00:05:18,370 If you replace yourself from to the minimum are to the left of the minimum, then you're moving to the left. 49 00:05:18,460 --> 00:05:26,860 Okay. Now, the final point. And generally, we're actually we're going to ignore this complication and also ignore this as well. 50 00:05:27,940 --> 00:05:33,250 Just be aware that it sometimes occurs. The final complication is perhaps a much more important complication. 51 00:05:33,490 --> 00:05:36,670 If you have a band structure, here's a broken zone. 52 00:05:36,670 --> 00:05:41,830 If you have a band structure like the tight binding band structures we drew before, it looks like this. 53 00:05:42,220 --> 00:05:48,280 The generally the effective mass is given by the second derivative of the energy with respect to K. 54 00:05:48,610 --> 00:05:53,739 So that would mean that the star of the electron here is greater than zero. 55 00:05:53,740 --> 00:06:06,290 But up here m star of the electron would be less than zero, which seems rather odd to have a negative mass for a particle, but we'll see in a moment. 56 00:06:06,290 --> 00:06:11,860 That actually makes quite a bit of sense. And really we should think of the mass of the electron as being negative. 57 00:06:12,100 --> 00:06:18,730 The general definition that now so the electron one over h by square squared e.g. case where will always hold. 58 00:06:18,940 --> 00:06:22,390 And that is what we should always think of as the effect of mass. 59 00:06:23,230 --> 00:06:30,700 So far so good. This was the easy part. The hard part is when we start thinking about holes at the top of of a valence band. 60 00:06:32,470 --> 00:06:38,440 So the reason it gets hard is because minus signs will come back and bite you one way or the other. 61 00:06:38,710 --> 00:06:44,980 So it is really hard to keep track of the minus signs. And to make matters worse, some books put the minus signs in different places. 62 00:06:45,340 --> 00:06:50,950 So this is one of these key situations where as long as you know what you're doing, you can move the minus sign around. 63 00:06:50,970 --> 00:06:54,430 You have the even number of minus signs. At the end of the day, you got the right answer. 64 00:06:54,760 --> 00:07:01,270 But what I am going to tell you is what I think is the most natural way to think of where you put the minus signs. 65 00:07:01,270 --> 00:07:04,540 And I think most books agree with me on this, but not all. 66 00:07:04,540 --> 00:07:10,480 So be warned about that. So this is the situation with holes at the top of valence bands. 67 00:07:10,930 --> 00:07:22,299 Top of bands. So so that is means up here where we found this odd result that the mass of the electron is 68 00:07:22,300 --> 00:07:27,250 negative when we're thinking about the top of a band like over there where we found this result, 69 00:07:27,430 --> 00:07:33,190 we're usually thinking when we're at the top of the band about a nearly filled band and nearly filled valence band. 70 00:07:33,520 --> 00:07:39,790 So we draw again E versus K, and we'll imagine that we have the top of some down, 71 00:07:40,060 --> 00:07:51,910 which is mostly filled and we'll think about, okay, we'll put this at K max and okay, and we will call this E e not also here. 72 00:07:52,480 --> 00:07:59,500 We imagine if we put an electron up at the top of this band somewhere, its energy is going to be okay. 73 00:07:59,500 --> 00:08:09,190 So for the case of Paul's top of band, top of band, the energy of an electron, e electron. 74 00:08:10,840 --> 00:08:16,959 It's really good to keep keeping your head track of whether you're talking about an electron energy or a whole energy, 75 00:08:16,960 --> 00:08:19,810 because you'll see that they're going to get confused in a second. 76 00:08:20,200 --> 00:08:28,870 So the electron energy, if you put an electron near the top of the band, that will be some E0 plus one half D squared, 77 00:08:28,870 --> 00:08:35,559 e d k squared times k minus k max squared exactly the same way as it was before. 78 00:08:35,560 --> 00:08:38,980 But now this quantity here is negative. So this is less than zero. 79 00:08:39,580 --> 00:08:44,559 And this is why we should think of the mass of the electron here as being a negative quantity, 80 00:08:44,560 --> 00:08:48,490 because the energy actually goes down as you go away from the maximum. 81 00:08:49,000 --> 00:08:59,520 So here, what we're going to do to make our life a little bit simpler is to define a mass of a whole define mass 82 00:08:59,530 --> 00:09:16,780 of whole to be h bar squared over m hall r star is minus d squared e for the electron decay squared. 83 00:09:17,260 --> 00:09:20,650 So this thing here is greater than zero. 84 00:09:20,770 --> 00:09:30,190 At top of that top of band. And you'll notice from comparing this definition to the above definition, 85 00:09:30,550 --> 00:09:38,110 that the star of the whole is actually equal to minus the star of the electron. 86 00:09:40,600 --> 00:09:47,800 Okay, so up here where the mass of the electron appears to be negative, we've defined the mass of the hall to be minus of that. 87 00:09:48,010 --> 00:09:54,760 Now, why is it that this makes sense to do? Why is it we should define the mass of a hall to be negative, the mass of electron? 88 00:09:54,760 --> 00:10:00,030 And why should it be the mass of the hall? Be positive at the top of the Valence Band. 89 00:10:00,040 --> 00:10:04,509 So let's think about this for a second. So here we have the same top of the Valence band. 90 00:10:04,510 --> 00:10:10,870 We imagine the whole Valence band is filled except for one hall, which we're putting at the top of the band here. 91 00:10:12,130 --> 00:10:22,540 So the energy of the hall e hall be some constant in our Twitter and then we want to find 92 00:10:22,540 --> 00:10:27,760 out how does the energy of this hole change as we displace it from the top of the band? 93 00:10:28,180 --> 00:10:34,050 So let's imagine this point here, just so we don't have to carry around these K axes and K means anymore. 94 00:10:34,300 --> 00:10:40,090 Let us decide this point. Here is cable zero as we displace K from. 95 00:10:40,090 --> 00:10:42,579 This is just for simplicity. It doesn't have to be a capable zero. 96 00:10:42,580 --> 00:10:50,350 But let's imagine is let us ask what happens to the energy as you displace the hall from the top of the band? 97 00:10:50,590 --> 00:10:52,930 So we're going to move the hall from here to here. 98 00:10:54,100 --> 00:10:59,950 Well, when you do that, the energy actually has gone up from this place in the hall from the top of the band. 99 00:10:59,950 --> 00:11:05,380 Why is that as squared to star hall times? 100 00:11:05,560 --> 00:11:08,950 I guess k squared here we're displacing it from Capel zero. 101 00:11:09,310 --> 00:11:13,240 The reason that the energy has gone up when we displace the hall from the top of 102 00:11:13,240 --> 00:11:18,280 the band is because the electrons are trying to go to the lowest energy possible. 103 00:11:18,460 --> 00:11:21,760 The holes try to bubble up to the highest energy possible. 104 00:11:21,760 --> 00:11:27,219 It's like is like an air pocket or a bubble or a like pushing a balloon underwater. 105 00:11:27,220 --> 00:11:31,450 If you take this balloon here and you push it underwater, that costs you energy. 106 00:11:31,450 --> 00:11:38,290 It costs you positive energy to take this hall and push it underneath the electrons because the electrons want to go to the lowest energy possible. 107 00:11:38,590 --> 00:11:44,499 That clear, that is the intuition of why it is that you should think of the mass of the hall as being positive, 108 00:11:44,500 --> 00:11:50,560 because when you displace it from the top of the band, it cost you positive energy to move away from the top of the band. 109 00:11:50,740 --> 00:11:53,950 So far, so good. Okay, good. All right. 110 00:11:54,820 --> 00:11:58,900 So that's the easy sign to keep track of, but the signs get more complicated. 111 00:12:01,030 --> 00:12:05,640 Let us ask about the momentum of a whole. What is the momentum of a hall? 112 00:12:08,280 --> 00:12:24,329 Momentum of a whole. And a good way to think about this is to realise that a filled band, filled band has no momentum. 113 00:12:24,330 --> 00:12:28,350 Has no crystal momentum. No crystal momentum. 114 00:12:31,930 --> 00:12:36,430 Carrying no current inert has no momentum at all. 115 00:12:36,820 --> 00:12:44,850 So if we want to know what the momentum is of a hole in the top of the band, let's imagine, here's our band. 116 00:12:44,860 --> 00:12:50,920 We fill the entire band like this, except for one state which is missing out of a state like this. 117 00:12:51,250 --> 00:12:55,480 We imagine taking this hole, this missing particle from the band. 118 00:12:55,720 --> 00:13:01,450 If we add it to a band, it's empty. Except for that one electron placed in. 119 00:13:01,840 --> 00:13:09,520 Then this plus this equals a fill down equals filled van filled with file filled band. 120 00:13:11,170 --> 00:13:17,770 So momentum. So total momentum equals zero. Total h party equals zero. 121 00:13:18,070 --> 00:13:26,410 Okay, so the sum of these two put together the fill band with the absence of the whole, plus the empty band with this one electron. 122 00:13:26,830 --> 00:13:31,569 Sorry, the filled band with the absence of this electron plus the empty band with the presence of this electron. 123 00:13:31,570 --> 00:13:35,260 When you put them together, their total momentum is going to be zero. 124 00:13:35,680 --> 00:13:50,260 So what we have then is h bar K for the hall over here plus h bar K for the electron should which is over here should give us a total of zero. 125 00:13:50,860 --> 00:14:02,110 So far so good. So that means that h pa k for a hall must be minus h bar k for the missing electron. 126 00:14:05,930 --> 00:14:11,300 Okay. So that is a good way to keep track of what the momentum of a horse should be. 127 00:14:12,050 --> 00:14:15,350 There is one of these confusing minus signs. 128 00:14:15,690 --> 00:14:21,450 Now let's think about a given. This what the velocity is of a hole at the top of are. 129 00:14:21,860 --> 00:14:27,680 These things are actually going to be end up being important because you need to know which way the electrons or the holes are moving. 130 00:14:28,010 --> 00:14:31,340 Keeping track of it for an electron at the bottom of the band, everything seems very natural, 131 00:14:31,550 --> 00:14:35,720 but hole at the top of the band, everything starts getting confused. 132 00:14:35,720 --> 00:14:43,490 So it is important to keep track of these correctly. So let's think about velocity of hall at top of band. 133 00:14:44,540 --> 00:14:54,800 Top of band. So again, we start with the energy for the electron at the top of the band. 134 00:14:55,400 --> 00:15:03,320 So that is E not minus H bar squared, k squared K electron squared over two and Mr. Hall. 135 00:15:04,100 --> 00:15:08,900 Okay. Now why did I draw it this way? So Mr. Hall is greater than zero. 136 00:15:09,110 --> 00:15:17,060 The energy for the electron decreases as you increase the momentum of the electron away from zero. 137 00:15:17,500 --> 00:15:28,620 The band is curving down, so the energy for the electron is decreasing the group velocity then which is always one over h bar D. 138 00:15:28,670 --> 00:15:30,380 So this is group velocity for the electron. 139 00:15:30,740 --> 00:15:44,690 The electron decay electron is then minus h for k electron over Mr. Hall or since Mr. Hall is minus Mr. of the electron, 140 00:15:44,690 --> 00:15:49,490 we can write this is h bar k e over m star electron. 141 00:15:49,670 --> 00:15:53,180 But Mr. Electron is is less than zero. 142 00:15:53,660 --> 00:16:02,090 So what this means is if I have a, if I had an empty band like this and I had an electron here near the top of the band, 143 00:16:02,330 --> 00:16:05,660 the velocity of this electron is actually going this way. 144 00:16:06,290 --> 00:16:08,660 The group is going this way, right. 145 00:16:09,110 --> 00:16:14,180 The MC because you can think of that as the electron mass is going negative is negative because of the negative curvature. 146 00:16:14,330 --> 00:16:18,980 So the group velocity is actually in the opposite direction as the wave vector. 147 00:16:19,770 --> 00:16:27,080 Okay, I know it's I know it's confusing, but but if you just keep track of the science carefully, everything comes out correct. 148 00:16:27,870 --> 00:16:33,710 Let's compare that to the energy situation if we think about it in terms of holes. 149 00:16:34,070 --> 00:16:46,070 So the energy of a hole near the top of the band is some constant plus h bar squared K hole squared over two and a star hole. 150 00:16:46,880 --> 00:16:51,440 So the energy of the hole increases as you increase its momentum away from the top of the band. 151 00:16:51,950 --> 00:16:56,660 So the group velocity, the group hole equals one over h bar. 152 00:16:57,200 --> 00:17:13,880 The hall decay hall equals h bar k hall over Mr. Hall and K Hall, we just decided is the same thing as minus K for the electron. 153 00:17:14,120 --> 00:17:25,430 So this is the same thing as minus k, h, bar h parquet electron over Mr. HALL Which is in fact the same thing we just calculated up here. 154 00:17:25,790 --> 00:17:37,010 So the point here is that if we have a filled band like this and one empty state here, the group velocity of that is going in this direction as well. 155 00:17:37,190 --> 00:17:44,210 This is an extremely important principle that will come back many times or at least several times, that the velocity of a state, 156 00:17:44,570 --> 00:17:59,270 the velocity of an orbital of an orbital orbital is independent of whether that orbital is filled. 157 00:17:59,930 --> 00:18:08,420 Whether the orbital is filled. 158 00:18:12,650 --> 00:18:17,150 So you can think of it as you have a bunch of orbitals in your system and the orbitals have some velocity, 159 00:18:17,330 --> 00:18:21,560 you can put an electron in that orbital and the electron will move that out with that velocity. 160 00:18:21,710 --> 00:18:27,050 If you remove the electron from that orbital, it is still the orbital is still moving with the same velocity it is. 161 00:18:27,050 --> 00:18:36,080 There is no electron in it. Okay. So this is kind of an important principle which sort of actually makes a very good amount of sense. 162 00:18:36,080 --> 00:18:43,069 Even when we think about the we think about the hydrogen atom or something, we think about these orbitals being there, 163 00:18:43,070 --> 00:18:49,309 whether or not that there's an electron in the orbital and the orbitals can be moving with some velocity and 164 00:18:49,310 --> 00:18:54,080 they're still there and moving independent of whether we put the electron in or we didn't put the electron in. 165 00:18:55,820 --> 00:19:04,390 All right. So let us now actually try to take what we've learned and calculate some dynamics. 166 00:19:05,650 --> 00:19:14,410 Dynamics of electrons and holes. Dynamics of electrons and holes, channels and halls. 167 00:19:17,710 --> 00:19:25,630 And the simplest way to handle dynamics is what we started the term with, which is to think about the theory. 168 00:19:27,790 --> 00:19:37,269 So we're back to Judith Theory after many weeks. And so let's recall that the equation that we derived some p dot equals F just Newton's 169 00:19:37,270 --> 00:19:43,930 equations with a drag term p over tau and we can write that out for electrons. 170 00:19:44,590 --> 00:19:51,520 For electrons we will have the mass of the electron. 171 00:19:51,520 --> 00:19:53,950 I'm going to put a star here. Times V Dot. 172 00:19:53,950 --> 00:19:59,679 So the only thing different from what we did at the beginning of the term is I'm now putting a star next to the mass. 173 00:19:59,680 --> 00:20:03,520 So it's the affective mass, the curvature at the bottom of the band that matters, 174 00:20:03,520 --> 00:20:09,070 not the bear electron mass, which is not an important quantity in a semiconductor or an insulator. 175 00:20:09,520 --> 00:20:18,460 We will have the force, the usual, the usual Lorentz force, which is always with you as a another joke. 176 00:20:18,730 --> 00:20:25,540 I never mind the Tao for the electron, the scattering time for the electron. 177 00:20:25,750 --> 00:20:31,450 So this should look fairly familiar. This is the Duda transport equation that we had before. 178 00:20:31,720 --> 00:20:34,780 There is some acceleration and mass times acceleration. 179 00:20:34,780 --> 00:20:40,090 There's some force, there's a charge of minus E on the electron, there's Lorentz force and there's a drag force here. 180 00:20:40,390 --> 00:20:47,230 This equation holds true whether we're talking about an electron near the bottom, a band, or an electron near the top of the band. 181 00:20:47,410 --> 00:20:54,310 However, near the top of the band, the sign of the mass gets to be negative, and that seems a little bit weird, 182 00:20:54,550 --> 00:21:04,360 but if we think about it, near top of band, near top of a band, we can consider the equation for holes. 183 00:21:04,960 --> 00:21:10,960 So we would have an m star whole times, the velocity of the whole same equation. 184 00:21:10,960 --> 00:21:15,040 But the charge on the whole is now plus e right? 185 00:21:15,040 --> 00:21:18,370 This again equals plus e, the charge on the whole is plus E. 186 00:21:18,730 --> 00:21:31,570 Then we have E plus the Crosby Ren's force and then minus m star hole velocity of the hole over some scattering time for the hall as well. 187 00:21:31,990 --> 00:21:36,730 So we have a equation for the holes at the top of the band as well. 188 00:21:37,240 --> 00:21:41,350 Now remember that the we have two things here. 189 00:21:41,920 --> 00:21:44,980 We have two equations, one for the electrons, one for the holes. 190 00:21:45,310 --> 00:21:50,950 And the difference in the two equations is that the sign of the charge is different for the electrons in the holes. 191 00:21:51,160 --> 00:21:57,250 But also Mr. Hall was defined to be minus the star of the electron. 192 00:21:57,250 --> 00:22:03,640 So in fact these two equations are exactly the same if this equation has been multiplied by an overall minus one 193 00:22:03,820 --> 00:22:10,120 and Star Hall is minus M star of the electron and so is actually just repeating exactly the same JUDA equation. 194 00:22:10,960 --> 00:22:15,220 So this is actually quite important. It's exactly the same statement that I just wrote over here. 195 00:22:15,460 --> 00:22:18,310 The dynamics of the state are exactly the same. 196 00:22:18,310 --> 00:22:23,170 Whether you're talking about an electron moving the state or a whole moving the state, the absence of an electron, 197 00:22:23,170 --> 00:22:30,460 the orbital still moves with the same dynamics, whether or not the orbital is filled with an electron gun. 198 00:22:31,120 --> 00:22:37,180 So the thing that is funny about the equation in the semiconductors is if you're up at the top of the band, 199 00:22:37,420 --> 00:22:40,840 you have a choice of two things that both seem a little bit weird. 200 00:22:41,020 --> 00:22:45,429 One possibility is that you have to think about a negative mass for the electrons. 201 00:22:45,430 --> 00:22:47,080 If you're thinking about electrons in the state, 202 00:22:47,320 --> 00:22:53,170 the other possibility is you think about positive mass holes, but now these are positively charged particles. 203 00:22:53,410 --> 00:23:01,000 Now this resolves one of our all puzzles, all puzzle from the second day or the third day of term of the puzzle, 204 00:23:02,470 --> 00:23:12,460 which was why is sign of hall coefficient hall coefficient. 205 00:23:16,800 --> 00:23:17,910 Sometimes positive. 206 00:23:23,100 --> 00:23:31,889 So you'll recall that we derived due to theory that the whole coefficient is one over n times the charge on the particle moving around. 207 00:23:31,890 --> 00:23:38,370 This is the charge on the electron. So if you have a band which is mostly empty and there are just a couple of electrons moving around, 208 00:23:38,370 --> 00:23:41,040 then you have the theory for these electrons moving around, 209 00:23:41,040 --> 00:23:45,569 just like we thought about in Sommerfeld Theory or in Judith Theory at the beginning of the term. 210 00:23:45,570 --> 00:23:47,820 And you'll get a whole coefficient which is negative. 211 00:23:48,060 --> 00:23:54,180 But if you have a a band which is mostly filled and you just have a couple of holes in the top of the band running around, 212 00:23:54,510 --> 00:23:59,850 then either way you look at it, what you think about it as holes running around with a positive charge, 213 00:24:00,060 --> 00:24:05,100 or you think about it as electrons running around, but now they're massive electrons negative. 214 00:24:05,250 --> 00:24:11,819 What you'll get is you'll get the opposite sign because now the things carrying the charge have the have the wrong sign. 215 00:24:11,820 --> 00:24:23,160 They have positive sign instead of negative sign. So this is what you might have to do it way back in 1901 that it appeared that for some materials, 216 00:24:23,400 --> 00:24:26,580 for some reason, the charge carrier had the wrong sign. 217 00:24:26,580 --> 00:24:32,070 And indeed that is what's going on in some materials. The charge is carried by holes rather than electrons. 218 00:24:32,340 --> 00:24:38,190 So if you have a semiconductor and you doped with electrons, you always get this sign of the hall coefficient. 219 00:24:38,190 --> 00:24:41,490 If you do it with holes, you'll get the opposite sign of the hall coefficient. 220 00:24:41,790 --> 00:24:47,400 Now for metals, real, real metals like calcium or beryllium or manganese or something like that, 221 00:24:47,730 --> 00:24:52,320 which are not just lightly doped where they, you know, we have some complicated band structure. 222 00:24:52,650 --> 00:24:55,770 It is less clear what the sign of this is supposed to come out as, 223 00:24:55,980 --> 00:24:59,850 because you can have a situation like we discussed with calcium in the last 224 00:24:59,850 --> 00:25:03,809 lecture where you have a partially filled lowest band in a partially filled, 225 00:25:03,810 --> 00:25:11,520 highest band. So you can think of it as having some holes in the first bronze and some holes in the first band, but some electrons in the second band. 226 00:25:11,760 --> 00:25:14,240 So the holes are trying to give you a positive hall coefficient. 227 00:25:14,430 --> 00:25:18,659 The electrons in the second band are trying to give you a negative coefficient of the two of them compete 228 00:25:18,660 --> 00:25:22,680 with each other and then you have to start worrying about which one has a greater scattering time. 229 00:25:22,890 --> 00:25:30,030 Okay, so it gets a little bit more complicated for real, for a genuine real metals like beryllium or or manganese or calcium. 230 00:25:30,690 --> 00:25:34,020 But in the case of a doped semiconductor, it comes out very clearly. 231 00:25:34,230 --> 00:25:38,219 If you have basically electron transport, you get this sign of the hall coefficient. 232 00:25:38,220 --> 00:25:41,970 If you have basically hole transport, you have the opposite side of the hall coefficient. 233 00:25:42,000 --> 00:25:44,700 So far so good. So we resolve this one puzzle. 234 00:25:45,000 --> 00:25:49,530 There is something else that I mentioned when we talked about due to theory all the way back at the beginning of the term, 235 00:25:49,830 --> 00:25:56,430 we remember that due to theory, works well, works well for semiconductors. 236 00:25:57,330 --> 00:26:08,129 For semiconductors. Why is that? Besides the fact that it can get the sign intelligently and the reason why the theory works 237 00:26:08,130 --> 00:26:12,180 particularly well for semiconductors is because usually when you're talking about semiconductors, 238 00:26:12,180 --> 00:26:14,730 you're talking about a very low density of charge carriers. 239 00:26:14,940 --> 00:26:28,620 You're talking about just a few electrons in the conduction band or just a few holes in the valence band because because density is low is low. 240 00:26:28,800 --> 00:26:35,220 Whether we're talking about electrons in the conduction band or holes in the valence band, the density is fairly low. 241 00:26:35,220 --> 00:26:36,330 And why does that help you? 242 00:26:36,660 --> 00:26:43,739 The reason that the theory failed back when we were studying it for the first time is because we neglected Fermi statistics. 243 00:26:43,740 --> 00:26:51,629 We neglected the fact that the electrons were fermions and that made us mess up all of these things, like the the heat capacity. 244 00:26:51,630 --> 00:26:54,840 We got that completely wrong because we neglected Fermi statistics. 245 00:26:55,050 --> 00:26:59,820 But if the density is really low, you're actually entitled to neglect Fermi statistics. 246 00:26:59,820 --> 00:27:00,450 Why is that? 247 00:27:00,690 --> 00:27:07,710 Well, if the probability of having two electrons ever sit in the same orbital is extremely low because the density of the electrons is very low, 248 00:27:08,040 --> 00:27:10,829 then you can just treat these as classical particles. 249 00:27:10,830 --> 00:27:16,710 Maximum Boltzmann particles can, because it's very unlikely they're going to sit in the same orbital in the first place. 250 00:27:17,320 --> 00:27:21,450 A certain better defined way of saying that is the Fermi energy. 251 00:27:22,290 --> 00:27:25,919 Given this density, the Fermi energy increases with density. 252 00:27:25,920 --> 00:27:31,860 If the density is low, the Fermi energy is low, if the Fermi energy is much less than the temperature, which would be true. 253 00:27:31,860 --> 00:27:37,079 If the density is low, then you don't have to worry about Fermi statistics. 254 00:27:37,080 --> 00:27:40,200 You can treat the particles as being classical. So. 255 00:27:40,680 --> 00:27:55,589 So particles are classical. Our classical and you can consider just regular classical due to theory classical Boltzmann 256 00:27:55,590 --> 00:27:59,190 dynamics maximal Boltzmann distributions without worrying about Fermi distributions. 257 00:27:59,460 --> 00:28:02,730 So this is yes, it is similar based. 258 00:28:04,950 --> 00:28:09,599 The whole yes world will come to that in a moment. But also you obey for me, statistics. 259 00:28:09,600 --> 00:28:14,880 You can only put if you think about it for a second. You can only put one hall in a particular case state. 260 00:28:15,120 --> 00:28:19,500 If you start with the Valence band filled, you can take it out once, but you can't take it out again. 261 00:28:20,200 --> 00:28:23,760 Okay. Yes. Holes of the Fermi statistics, they are fermions. 262 00:28:24,330 --> 00:28:32,940 All right. So this is actually bringing us to the to the realm of statistical mechanics or thermodynamics of our particles in in semiconductors. 263 00:28:33,860 --> 00:28:41,960 So we have to start thinking about things like how many are how many electrons get excited from the Valence band to the conduction band? 264 00:28:42,230 --> 00:28:46,910 How did they get there and what are they doing once they're there? What is the effect of temperature? 265 00:28:48,200 --> 00:28:55,129 So one thing that I told you from the in the last lecture that I should be a little bit more honest about. 266 00:28:55,130 --> 00:29:02,090 We're going to go back to think about doping for a second and how doping is affected by temperature, doping and temperature. 267 00:29:06,640 --> 00:29:21,100 Okay. So let us recall that when we took a and opened and opened like, like a phosphorus box in silicon and they opened, 268 00:29:21,400 --> 00:29:28,570 we realised that phosphorus equals a silicon plus one proton plus one electron. 269 00:29:28,960 --> 00:29:37,420 And I told you the electron goes into the conduction band, into conduction band band. 270 00:29:37,420 --> 00:29:41,350 And I told you to ignore the fact that there's an extra proton ignore. 271 00:29:42,130 --> 00:29:47,830 Now this issue of being able to ignore the extra proton that we put in, in addition, 272 00:29:48,250 --> 00:29:53,110 that is something that is actually we have to be a little bit more careful about and we have 273 00:29:53,110 --> 00:29:56,860 to worry about temperature and things like that to ask whether it's it's okay to ignore it. 274 00:29:56,860 --> 00:30:01,930 So think about this a little bit more carefully. We put in a proton and we put in an electron. 275 00:30:02,140 --> 00:30:05,080 So what is an electron and a proton do when you put them together? 276 00:30:05,290 --> 00:30:11,740 Which is really going to happen is you are going to form form a hydrogen atom or a hydrogen atom. 277 00:30:12,370 --> 00:30:20,170 Hydrogen atom. We put the electron in the conduction man, but we also put a positive charge behind. 278 00:30:20,350 --> 00:30:25,329 And the electron is going to bind to that positive charge is attracted to that positive charge. 279 00:30:25,330 --> 00:30:29,830 By Coulomb interactions, you form a hydrogen atom, but it is not a regular hydrogen atom. 280 00:30:30,130 --> 00:30:40,300 It is slightly different from a hydrogen atom because one m star not m not the mass of the electron. 281 00:30:40,390 --> 00:30:43,570 It is the effective mass of the electron, not the massive electron. 282 00:30:43,570 --> 00:30:49,000 The dynamics of the electron running around in the conduction man is given by M Star, not the bear mass of the electron. 283 00:30:49,000 --> 00:30:55,719 So that is the difference. One. The second difference is something you learned in A&M in the Coulomb interaction. 284 00:30:55,720 --> 00:31:06,940 The attraction back to the proton is is squared over for pi epsilon, not epsilon relative times r r epsilon, 285 00:31:07,020 --> 00:31:19,990 a relative dielectric constant relative dielectric constant, not the usual e squared over four pi epsilon not R. 286 00:31:21,340 --> 00:31:30,430 Okay, so the, the Coulomb interaction is, is reduced by the relative dielectric constant because you're living in a semiconductor, 287 00:31:30,430 --> 00:31:40,090 not living in the vacuum of outer space. So if you recall, we are regular hydrogen atom, regular hydrogen, regular hydrogen atom. 288 00:31:41,410 --> 00:31:49,660 There's a binding energy binding which is well, okay, you probably well, 289 00:31:49,690 --> 00:31:57,190 maybe do remember this formula is eight epsilon not squared Planck's constant squared, 290 00:31:57,580 --> 00:32:02,860 but you probably remember the magic number 13.6 EV, which is a red berg. 291 00:32:04,060 --> 00:32:07,660 So this is the binding energy of a, of a regular hydrogen atom. 292 00:32:07,900 --> 00:32:18,370 But in a semiconductor the binding energy e binding of the electron back to the proton. 293 00:32:18,370 --> 00:32:23,560 The additional charge that I put in with the doped is now m star electron. 294 00:32:23,890 --> 00:32:36,670 It is a fourth over eight epsilon relative epsilon not squared over h squared which I can then write as r m 295 00:32:36,670 --> 00:32:47,170 star of the electron divided by the mass of the electron times one over epsilon relative squared times 13.65. 296 00:32:48,580 --> 00:32:55,270 Now this expression can be a whole lot different from 13.6, and generically it's going to be a lot smaller. 297 00:32:55,540 --> 00:32:59,739 Typically effect of masses of electrons and semiconductors can be a third of the 298 00:32:59,740 --> 00:33:03,460 mass of an electronic can be even a 10th of the mass with electron or even smaller. 299 00:33:03,670 --> 00:33:06,850 The dielectric constant in semiconductors can be numbers like ten. 300 00:33:07,120 --> 00:33:08,319 So that means that this number, 301 00:33:08,320 --> 00:33:15,760 the actual binding energy of the electron to the proton in the semiconductor can be as much as a factor of a thousand less than this. 302 00:33:15,760 --> 00:33:31,990 For example, in silicon in silicon e binding of the electron back to its doped nucleus is 0.04 for electron volts, a much, much, much smaller number. 303 00:33:32,470 --> 00:33:35,530 Now, why is that important? Okay. Oh, look at this. 304 00:33:37,810 --> 00:33:42,040 I need. That can be. With this in here, I'm going to draw. 305 00:33:43,370 --> 00:33:47,240 So here is the bottom of the conduction band. Here's the top of a valence band. 306 00:33:48,370 --> 00:33:56,910 So when you put a dope in, when you put that phosphorus opened in, you form a bound state, which is a very small distance. 307 00:33:57,100 --> 00:34:02,860 Binding. A binding below the bottom of the conduction band. 308 00:34:03,130 --> 00:34:08,830 So you gain a tiny bit of energy by binding. Instead of just sitting with cake with zero in the conduction band, 309 00:34:09,010 --> 00:34:13,030 you gain a little bit of energy by binding to the positive nucleus that you came from. 310 00:34:13,420 --> 00:34:20,920 However, this energy here is very, very small and at room temperature, at room temperature, write this down. 311 00:34:23,770 --> 00:34:29,960 A tea room. The electron breaks free. 312 00:34:32,870 --> 00:34:36,470 Breaks free. And goes off into the conduction band. 313 00:34:37,310 --> 00:34:44,090 Breaks free and goes into conduction band. 314 00:34:44,780 --> 00:34:47,840 Is thermally excited into the conduction band. 315 00:34:48,530 --> 00:34:57,050 And this is why this is why it is okay to think about when we add our phosphorus doping to silicon. 316 00:34:57,200 --> 00:35:00,170 We really should just think a room temperature or higher temperature. 317 00:35:00,380 --> 00:35:03,980 We should think about this as just directly adding electrons into the conduction band. 318 00:35:04,400 --> 00:35:08,240 If you cool it down, if you cool this, maybe I'll write that down. 319 00:35:08,570 --> 00:35:18,680 So doping adds electrons adds directly to two conduction band. 320 00:35:24,070 --> 00:35:28,090 Or similarly, if you do the same thing with holes, when you add in the hole, 321 00:35:28,330 --> 00:35:35,800 it would form a a bound state to the to the negative charge on the nucleus that we left behind. 322 00:35:36,370 --> 00:35:38,780 We are putting in a boron or an aluminium or something. 323 00:35:39,610 --> 00:35:46,570 And at a finite temperature, that hole breaks free of its nucleus and runs off into the valence band. 324 00:35:47,350 --> 00:35:57,790 However, at low temperature, at low temperature, the electron becomes becomes bound, becomes bound again. 325 00:35:57,940 --> 00:36:02,230 Bound to the nucleus. To the open nucleus. 326 00:36:03,940 --> 00:36:16,320 Nucleus again. So if you are measuring the density of electrons in the conduction band, if you cool the system down a lot, 327 00:36:16,500 --> 00:36:20,430 all of a sudden you would discover that the density of electrons in conduction band was dropping 328 00:36:20,640 --> 00:36:25,350 because all of these electrons were getting frozen back onto their the nucleus that they came from. 329 00:36:25,590 --> 00:36:31,620 This is known as carrier freeze out. Carrier freeze out. 330 00:36:36,000 --> 00:36:40,560 Okay. All right. So, so far, so good. 331 00:36:40,650 --> 00:36:48,870 Everyone happy with this? All right, so now it's time to do some serious statistical mechanics thinking about how many electrons 332 00:36:48,870 --> 00:36:53,130 we should have in the conduction band and how many holes we should have in the Valence Band. 333 00:36:53,160 --> 00:36:58,860 This is a rather important calculation that actually shows up on exams very frequently. 334 00:36:59,220 --> 00:37:03,180 So it's a good thing to go through several times. 335 00:37:03,660 --> 00:37:11,700 All right. So let's start with this generic picture of the bottom of a conduction band, the top of a valence band. 336 00:37:11,880 --> 00:37:16,530 They do not have to have the same curvature. In general, they could have a different effect of masses of John. 337 00:37:16,770 --> 00:37:19,920 The effect of masses being slightly different. They don't have to be a line. 338 00:37:20,580 --> 00:37:25,770 It could be an indirect gap at the top of the Valence band. Could be in different place than the bottom of the conduction band. 339 00:37:26,070 --> 00:37:30,209 We'll call this energy conduction easy for in conduction. 340 00:37:30,210 --> 00:37:33,600 We call this energy even for a valence. 341 00:37:34,050 --> 00:37:41,280 This energy gap is a D and e gap equals e conduction minus e valence. 342 00:37:41,790 --> 00:37:47,280 And we'll imagine that the chemical potential is somewhere in the middle of the band and we'll try to figure out where it is. 343 00:37:49,500 --> 00:37:49,950 All right. 344 00:37:50,430 --> 00:37:57,900 And so the we start with the Valence band being completely filled and we're going to try to figure out what happens when we add temperature. 345 00:37:58,320 --> 00:38:02,430 So a couple of things we're going to need. One is what the spectra are. 346 00:38:02,700 --> 00:38:06,120 So for energy greater than energy conduction, 347 00:38:06,420 --> 00:38:19,690 we have energy as a function of K is energy conduction plus H bar squared minus K mean squared over to Mr. Electron. 348 00:38:20,220 --> 00:38:25,590 And then for energy less than E valence, 349 00:38:26,790 --> 00:38:41,280 we have the upside down parabola e k equals E valence minus h bar squared k minus k max squared over two m star whole. 350 00:38:41,700 --> 00:38:46,470 And Mr. Hall, remember, is positive. So we have a negative curvature of the top of the band. 351 00:38:47,560 --> 00:38:51,760 And then we have to think about counting states. So counting states. 352 00:38:53,470 --> 00:38:58,540 And we'll think back to when we did Sommerfeld theory for free electrons. 353 00:38:59,440 --> 00:39:11,920 For free electrons. We had our well, our energy was e equals H per squared K squared over two M and the density of states. 354 00:39:12,370 --> 00:39:21,790 So the density of states per unit volume while g heavy was given by this expression which we calculated to mass. 355 00:39:22,360 --> 00:39:31,930 So three halves are over two pi squared h bar cubed epsilon to the one half. 356 00:39:31,960 --> 00:39:35,770 We calculated this, I think, on the second or third day of the term. 357 00:39:36,130 --> 00:39:41,620 This is the density of states. Of states per unit volume. 358 00:39:42,610 --> 00:39:54,290 States per volume. So we'd like to use exactly the same calculation, except for the conduction band and the Valence band here. 359 00:39:54,620 --> 00:39:59,179 So without actually doing the work, it's pretty easy to see what the result is going to come out. 360 00:39:59,180 --> 00:40:07,759 As the density of states in the conduction band for E greater than E conduction is going to be too massive. 361 00:40:07,760 --> 00:40:19,549 The electrons star to the three halves divided by the same two pi squared h bar cubed and it is the same e to the two one half, 362 00:40:19,550 --> 00:40:25,190 but it's e minus e conduction band. So you're measuring the energy with respect to the bottom of the conduction band. 363 00:40:25,190 --> 00:40:30,020 The parabola starts at the bottom of the conduction band, but otherwise it's exactly the same calculation. 364 00:40:30,360 --> 00:40:38,870 People happy with that OC again. So G Valence may be a little bit more complicated, but not too much more complicated. 365 00:40:39,370 --> 00:40:54,979 R is two star hall r three has over two pi squared h bar cubed and here we measure are e minus e valence minus E to the one half. 366 00:40:54,980 --> 00:41:01,070 And we're measuring going in the downward direction while we need this thing to the one half to be a positive number. 367 00:41:01,070 --> 00:41:04,370 So it has to be e valence minus E we measure going down. 368 00:41:04,370 --> 00:41:08,600 We get an increasing number of states as we go down to lower energy. 369 00:41:09,080 --> 00:41:14,660 Good. All right. So then we can write in conduction band. 370 00:41:18,200 --> 00:41:31,580 We just use our statistical mechanics. The density of electrons, density N of electrons of E's is given by N as a function of T, 371 00:41:31,640 --> 00:41:36,650 that's an n equals the integral from the bottom of the conduction band up to energy infinity, 372 00:41:36,980 --> 00:41:45,980 the energy density of states as a function of energy and the Fermi factor theta energy minus mu. 373 00:41:46,790 --> 00:41:53,780 Okay, so the integral of the density of states is basically a sum over all eigen states. 374 00:41:54,020 --> 00:41:57,560 And the Fermi factor is the probability that that eigen state is going to be filled. 375 00:41:57,980 --> 00:42:04,100 So this is just counting up all the states that are filled at a given temperature, but we need to know what the chemical potential is. 376 00:42:06,000 --> 00:42:11,520 So now we're going to. Well, let's write out the Fermi function here. 377 00:42:11,970 --> 00:42:18,540 The Fermi function is one over e to the beta energy minus mu plus one. 378 00:42:18,960 --> 00:42:27,870 Now if, if we're at a fairly high temperature or if the density is low or strictly speaking, 379 00:42:28,080 --> 00:42:32,730 what we need is beta energy minus mu is much greater than one. 380 00:42:32,970 --> 00:42:44,370 This means that NU is much below, much below, below ac meaning that are below by at least the amount of the temperature. 381 00:42:44,700 --> 00:42:49,139 So, you know, fairly well below the bottom of the conduction band. 382 00:42:49,140 --> 00:42:52,890 And that is pretty easy because the gaps between the conduction band and the Valence band, 383 00:42:52,890 --> 00:42:56,760 there can be a couple of electron volts, much bigger than typical temperatures. 384 00:42:57,120 --> 00:43:07,620 That means that an F can be approximated by well, you basically you can just drop the plus one and you get the just the regular Boltzmann factor. 385 00:43:07,920 --> 00:43:12,630 This is just Boltzmann. Maxwell Boltzmann. Maxwell Boltzmann. 386 00:43:17,550 --> 00:43:26,070 So let us put that in and we have and of t then is going to be integral to energy zero. 387 00:43:26,530 --> 00:43:31,739 Very easy from the bottom of the conduction band up to infinity of g of energy. 388 00:43:31,740 --> 00:43:41,010 And then the maximal Boltzmann factor here, the minus beta energy minus mu making that approximation and this temperature. 389 00:43:42,870 --> 00:43:49,570 Then how do we do this integral? Well, first thing we're going to do is we are going to well, what are we going to do? 390 00:43:49,590 --> 00:43:52,850 We're going to write out what this is. 391 00:43:53,370 --> 00:44:03,150 So this has a to m star of the electron to the three halves and it has a two pi squared h bar cubed and is integral 392 00:44:03,870 --> 00:44:11,459 the energy and conduction band to infinity times e minus e conduction to the one half and then E to the minus. 393 00:44:11,460 --> 00:44:19,160 Beta energy is mu. So this is the one half that we have up there and we have to do this integral. 394 00:44:19,160 --> 00:44:29,700 And the way we do this integral is by shifting out a factor of E to the minus beta c minus mu out frontal. 395 00:44:29,700 --> 00:44:40,170 Then we're left with an integral D energy from AC to infinity D minus easy to the one half, and then the minus beta E minus sc. 396 00:44:41,580 --> 00:44:44,920 Which is nice. We can then write if we want to. 397 00:44:45,300 --> 00:44:56,460 Well, x equals you want us to see if we want. So this becomes integral of the X from zero to infinity x to the one half, even minus beta x. 398 00:44:57,360 --> 00:45:02,050 Then we can do that integral by defining y squared equals x. 399 00:45:02,070 --> 00:45:06,120 This becomes a simple Gaussian integral, and we get better than minus three. 400 00:45:06,120 --> 00:45:09,420 Half times square root of pi over two. 401 00:45:10,730 --> 00:45:22,160 Okay. So we put that all together and we end up getting the density of electrons in the conduction band is a function of t is then putting together 402 00:45:22,160 --> 00:45:39,590 all the factors one quarter two mass of the electron star kb t over pi h bar squared r to the three half e to the minus beta energy. 403 00:45:39,590 --> 00:45:43,310 The conduction band minus mu. So this is our expression. 404 00:45:43,490 --> 00:45:49,160 We now know how many electrons there are in the conduction band is a function of temperature and is a function of MU. 405 00:45:49,400 --> 00:45:53,420 The important thing to take away. There are some pretty factors here which aren't that crucial. 406 00:45:53,630 --> 00:46:03,650 The important thing here to take away is this basically activated that you have to activate the energy up from the chemical 407 00:46:03,650 --> 00:46:09,500 potential into the conduction band in order to get an electron into the conduction band as you drop the temperature. 408 00:46:09,740 --> 00:46:12,920 The number of electrons in the conduction band is going to drop exponentially. 409 00:46:13,560 --> 00:46:17,970 Okay. All right. So far, so good. Now we have to do the same thing for holes. 410 00:46:18,000 --> 00:46:22,790 I think we can still do this in time. So the number of holes in the valence band. 411 00:46:24,020 --> 00:46:28,850 Well, okay, let's write that. That's going to be integral from minus infinity to the top of the valence band. 412 00:46:29,270 --> 00:46:33,290 The energy, the density of states in the Valence Band. 413 00:46:33,710 --> 00:46:39,920 And then I'll write one minus the Fermi factor in the Valence Band. 414 00:46:39,920 --> 00:46:43,130 Beta Epsilon minus new, minus me. Why is that? 415 00:46:43,160 --> 00:46:49,370 Well, if the probability of an electron being a state is the Fermi function, then the probability of a hole being the state. 416 00:46:49,370 --> 00:46:52,460 The absence of electron is one minus the Fermi function. 417 00:46:54,880 --> 00:47:00,540 All right. So this factor here, we're going to basically go through the same manipulations. 418 00:47:00,540 --> 00:47:11,530 So this is we can write this as a one minus one over E to the beta, minus new plus one that becomes, Oh, I'm running out of space, aren't I? 419 00:47:12,460 --> 00:47:22,420 All right, well, this thing becomes, ah, this whole thing here becomes one over one plus either the minus beta, 420 00:47:23,050 --> 00:47:28,390 a minus mu that maybe took two steps, but but you can check algebraically. 421 00:47:28,390 --> 00:47:36,010 That's true. And again, we are going to assume something analogous to well, do I have more space somewhere? 422 00:47:36,080 --> 00:47:42,400 Now I maybe I don't. So we're going to assume something analogous to what we assumed. 423 00:47:42,670 --> 00:47:54,129 But the electrons will assume that that chemical potential is not too close to close to close, 424 00:47:54,130 --> 00:48:00,600 meaning within the within the temperature, within the range of cavity of the top of the valence band E.V. 425 00:48:01,270 --> 00:48:11,889 In which case this thing can be approximated as this whole thing here can be approximately equal to the Boltzmann factor. 426 00:48:11,890 --> 00:48:15,400 Either the beta epsilon minus mu, much less than one. 427 00:48:17,360 --> 00:48:22,099 Okay. So it is, again, more or less the same, the same story. 428 00:48:22,100 --> 00:48:27,170 But here you have to realise that energy is less than you and you're activating a whole 429 00:48:27,380 --> 00:48:32,570 downwards into the valence band rather than a electron up into the conduction band. 430 00:48:33,290 --> 00:48:39,950 So plugging in the expression for the density of states in the valence band, 431 00:48:40,280 --> 00:48:49,399 we get to our whole to the three halves to pi squared h bar cubed integral the energy 432 00:48:49,400 --> 00:48:58,880 minus infinity to valence e valence minus E the one half in the beta epsilon minus mu. 433 00:49:00,140 --> 00:49:08,150 And exactly the same manipulations are with maybe a couple of minus signs put in various places. 434 00:49:09,170 --> 00:49:13,160 I think you're supposed to do it for homework, so maybe I won't go through it. And also I'm running out of time. 435 00:49:13,790 --> 00:49:25,340 So this whole thing becomes, at the end of the day, a very analogous expression to what we found above is one quarter to Mr. Hall capped 436 00:49:26,450 --> 00:49:36,470 over pi h by a squared all to three halves e to the minus beta mu minus e valence. 437 00:49:36,920 --> 00:49:41,260 And again. It is the same story. You have some free factors here, 438 00:49:41,560 --> 00:49:46,810 but the important part is that you have to activate a whole down from the chemical potential 439 00:49:47,080 --> 00:49:52,899 into the valence band because it takes energy to push and hold down to the Valence band. 440 00:49:52,900 --> 00:49:58,270 The holes want to go up as much as possible. So if we knew the chemical potential and we knew the temperature, 441 00:49:58,420 --> 00:50:04,960 we can figure out the density of electrons in the conduction man and the density of holes in the valence band. 442 00:50:05,230 --> 00:50:12,400 But now there's a trick. And the trick is to look at the product of energy, times p of t let us do that. 443 00:50:13,480 --> 00:50:24,280 We get a bunch of factors. So one half out front kb t over a pi squared cubed. 444 00:50:24,740 --> 00:50:30,190 Then there's a mass of an electron star, mass of whole star to three halves. 445 00:50:30,430 --> 00:50:32,410 And then if you look at the exponential factors, 446 00:50:32,620 --> 00:50:38,620 you'll notice that the exponential factors in the minus beta news and the the plus beta new will cancel. 447 00:50:38,830 --> 00:50:46,630 So the only thing that's left over is even the minus beta e conduction minus e valence e conduction minus valence is egad. 448 00:50:48,430 --> 00:50:51,340 So if you know the gap energy and you know the temperature, 449 00:50:51,490 --> 00:50:58,149 you know the product of the number of electrons in the conduction band and the number of holes in the valence band. 450 00:50:58,150 --> 00:51:04,480 And this, this law, very important law is known as that. 451 00:51:04,780 --> 00:51:07,780 That is known as the law of mass action. 452 00:51:08,440 --> 00:51:15,490 Law of mass action. If you talk to chemists, they know this. 453 00:51:16,090 --> 00:51:20,200 They have a law of mass action. And the nomenclature comes from this, from chemistry. 454 00:51:20,200 --> 00:51:25,750 Actually, I won't maybe go through why it is it's called the same thing is a law of mass action in chemistry. 455 00:51:25,960 --> 00:51:30,250 But the important thing here to realise is that the product is independent of the 456 00:51:30,250 --> 00:51:34,390 chemical potential and that means it is independent of how we have doped the system. 457 00:51:34,660 --> 00:51:39,550 So that is a couple more things. If we have an intrinsic, oh, did I use that word? 458 00:51:39,970 --> 00:51:46,510 I never used that word before. So intrinsic means undocked. 459 00:51:48,740 --> 00:51:54,080 So if I have an intrinsic semiconductor, I should have probably used that word in the last lecture. 460 00:51:54,380 --> 00:52:01,190 For an intrinsic semiconductor, the density of electrons in the conduction band equals the density of holes in the valence band. 461 00:52:01,190 --> 00:52:03,769 Because each time you take an electron out of the valence manning, 462 00:52:03,770 --> 00:52:07,400 you put it up in the conduction band, you created one hole and you created one electron, 463 00:52:07,670 --> 00:52:14,660 in which case you can just take the square root of the law of mass action and you can tell what and in PR at any temperature. 464 00:52:14,720 --> 00:52:21,620 So you are good there. But for a doped system, doped then is more complicated. 465 00:52:22,580 --> 00:52:27,320 Then we add doping and are. 466 00:52:28,850 --> 00:52:33,079 We have the number of electrons in the conduction band, 467 00:52:33,080 --> 00:52:45,500 minus the number of holes in the valence band equals the number of donor impurities impurities minus the number of acceptor impurities. 468 00:52:51,180 --> 00:52:59,940 Because basically each time you add a donor impurity, you add one electron into the conduction band without adding a hole into the valence band. 469 00:53:00,240 --> 00:53:05,460 I recently had an acceptor. You add a hole into the Valence band without adding an electron into the conduction band. 470 00:53:05,740 --> 00:53:14,550 And this is true. As long as as long as T is greater than T frees out. 471 00:53:17,590 --> 00:53:27,310 Where the electrons in the conduction man get get down to their donors and the holes in the in the valence band get bound to their acceptors. 472 00:53:27,580 --> 00:53:35,110 So the key thing to realise here is that the product of LMP is fixed independent of the doping. 473 00:53:35,110 --> 00:53:45,040 The difference of N.A. depends on the doping. This means that as you add electrons, as you add donor impurities, the chemical potential goes up. 474 00:53:45,340 --> 00:53:51,190 So you get more electrons and fewer holes. But the product of the number of electrons in the hole stays fixed. 475 00:53:51,400 --> 00:53:57,219 As you add more acceptors, the chemical potential goes down, you get more holes than electrons. 476 00:53:57,220 --> 00:54:05,980 But the product of the Tuesdays suspects maybe even right that before calling it quits, adding donors. 477 00:54:08,140 --> 00:54:19,540 New goes up. New goes up where adding acceptors, acceptors mew goes down. 478 00:54:20,320 --> 00:54:25,570 But because of the law of mass action, the product of those two is always going to stay fixed. 479 00:54:25,930 --> 00:54:27,309 All right, so next time, 480 00:54:27,310 --> 00:54:32,920 we'll pick up and actually talk about some real semiconductor devices and the devices that change the world, like the transistor. 481 00:54:33,190 --> 00:54:36,520 And I'll see you I think Wednesday. Wednesday. I think it's Wednesday. 482 00:54:36,720 --> 00:54:37,480 Okay. Have a good weekend.