WEBVTT 00:00:00.000 --> 00:00:02.420 align:middle line:84% The following content is provided under a Creative 00:00:02.420 --> 00:00:03.600 align:middle line:90% Commons license. 00:00:03.600 --> 00:00:06.050 align:middle line:84% Your support will help MIT OpenCourseWare 00:00:06.050 --> 00:00:09.650 align:middle line:84% continue to offer high quality educational resources for free. 00:00:09.650 --> 00:00:12.620 align:middle line:84% To make a donation, or to view additional materials 00:00:12.620 --> 00:00:16.520 align:middle line:84% from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.520 --> 00:00:19.915 align:middle line:90% at ocw.mit.edu. 00:00:19.915 --> 00:00:22.490 align:middle line:84% GANG CHEN: How about we start by reviewing what 00:00:22.490 --> 00:00:25.460 align:middle line:90% we talked in the last lecture. 00:00:25.460 --> 00:00:30.290 align:middle line:84% We introduced the thermoelectric effects, 00:00:30.290 --> 00:00:33.170 align:middle line:90% including the Seebeck effect. 00:00:33.170 --> 00:00:36.050 align:middle line:84% When we have a temperature difference, 00:00:36.050 --> 00:00:38.000 align:middle line:90% a voltage is generated. 00:00:38.000 --> 00:00:39.950 align:middle line:90% That's the phenomenon. 00:00:39.950 --> 00:00:44.270 align:middle line:84% And that will give a microscopic picture 00:00:44.270 --> 00:00:49.280 align:middle line:84% of what happens with the diffusion of charge 00:00:49.280 --> 00:00:53.840 align:middle line:84% on one side of the path, the charge to the other side. 00:00:53.840 --> 00:00:58.160 align:middle line:84% And the total effect is when we have 00:00:58.160 --> 00:01:01.980 align:middle line:84% a junction that made up two different materials, we 00:01:01.980 --> 00:01:04.230 align:middle line:90% pass current through it. 00:01:04.230 --> 00:01:08.250 align:middle line:84% You will have either cooling or heating effect, 00:01:08.250 --> 00:01:11.770 align:middle line:84% depends on the direction of the current. 00:01:11.770 --> 00:01:16.720 align:middle line:84% And this is because the charge itself also carry heat. 00:01:16.720 --> 00:01:20.560 align:middle line:84% So when the charge flow from one material to another material, 00:01:20.560 --> 00:01:22.660 align:middle line:84% if there are more heat flows out, 00:01:22.660 --> 00:01:25.060 align:middle line:90% then the junction cools down. 00:01:25.060 --> 00:01:29.170 align:middle line:84% And if there are more heat flows in the junction itself and you 00:01:29.170 --> 00:01:31.330 align:middle line:84% reverse the current direction, you 00:01:31.330 --> 00:01:34.730 align:middle line:84% can see that the effect is reversed. 00:01:34.730 --> 00:01:42.350 align:middle line:84% And Thomson effect is when we have a uniform conductor, 00:01:42.350 --> 00:01:48.680 align:middle line:84% or semi-conductor, we have a different imposed time 00:01:48.680 --> 00:01:52.730 align:middle line:84% and the flow of current through, then along the material 00:01:52.730 --> 00:01:57.500 align:middle line:84% there is uniform cooling, there is a distributed cooling, 00:01:57.500 --> 00:02:00.900 align:middle line:84% or heating, that happens along the material. 00:02:00.900 --> 00:02:06.380 align:middle line:84% And this is because, really, at each section of the material, 00:02:06.380 --> 00:02:08.389 align:middle line:84% because the temperature difference, 00:02:08.389 --> 00:02:14.090 align:middle line:84% the charge, the electrons, carry different amount of heat. 00:02:14.090 --> 00:02:17.120 align:middle line:90% So that's the Thomson effect. 00:02:17.120 --> 00:02:21.720 align:middle line:84% And then we did the analysis, simple analysis of a device. 00:02:21.720 --> 00:02:23.730 align:middle line:90% We have a real device. 00:02:23.730 --> 00:02:25.550 align:middle line:90% We pass current through it. 00:02:25.550 --> 00:02:28.340 align:middle line:84% There is a thermoelectric effect. 00:02:28.340 --> 00:02:33.110 align:middle line:84% There also undesirable side effects. 00:02:33.110 --> 00:02:36.780 align:middle line:84% The side effect is one, there's [INAUDIBLE] heating. 00:02:36.780 --> 00:02:41.010 align:middle line:84% And two, there is a reverse heat flow conduction, 00:02:41.010 --> 00:02:45.190 align:middle line:84% and this diminished degree the device performance. 00:02:45.190 --> 00:02:48.900 align:middle line:84% And if we do the device analysis and find out 00:02:48.900 --> 00:02:54.420 align:middle line:84% that the efficiency, or the coefficient of performance, 00:02:54.420 --> 00:02:57.900 align:middle line:84% we found that those depends on, eventually, 00:02:57.900 --> 00:03:00.270 align:middle line:84% the material property which you will 00:03:00.270 --> 00:03:03.510 align:middle line:90% call the figure of merit ZT. 00:03:03.510 --> 00:03:05.650 align:middle line:84% And, of course, those are heat engines. 00:03:05.650 --> 00:03:09.000 align:middle line:84% So the pre-factor and the caudal factor, 00:03:09.000 --> 00:03:11.730 align:middle line:84% hence, something that's less than 1. 00:03:11.730 --> 00:03:14.370 align:middle line:84% If it's less than 1, it's dependent 00:03:14.370 --> 00:03:17.550 align:middle line:84% on the material property, and that 1 to be 00:03:17.550 --> 00:03:20.250 align:middle line:90% larger, as large as possible. 00:03:20.250 --> 00:03:24.570 align:middle line:84% And then we discuss applications. 00:03:24.570 --> 00:03:28.290 align:middle line:84% I mentioned that at this stage, the materials is not 00:03:28.290 --> 00:03:34.050 align:middle line:84% competitive to replace the main refrigerators, or main power 00:03:34.050 --> 00:03:36.210 align:middle line:84% generator, like an internal combustion engine. 00:03:36.210 --> 00:03:39.420 align:middle line:84% So there are two directions to go. 00:03:39.420 --> 00:03:41.760 align:middle line:84% One, if you're an engineer, think 00:03:41.760 --> 00:03:44.550 align:middle line:90% about really using those there. 00:03:44.550 --> 00:03:48.480 align:middle line:84% You have to look at the system and the innovation 00:03:48.480 --> 00:03:53.580 align:middle line:84% where it makes sense from both the efficiency 00:03:53.580 --> 00:03:57.900 align:middle line:84% and, particularly, cost commercial product on the view. 00:03:57.900 --> 00:04:01.650 align:middle line:84% And the other direction is go to improve the materials. 00:04:01.650 --> 00:04:06.990 align:middle line:84% So what we'll be talking in the next today's lecture 00:04:06.990 --> 00:04:11.460 align:middle line:84% and next lecture is on understanding 00:04:11.460 --> 00:04:15.540 align:middle line:84% what really determines the microscopic aspects 00:04:15.540 --> 00:04:19.140 align:middle line:84% of the materials, and what are the directions which you 00:04:19.140 --> 00:04:22.530 align:middle line:90% go to improve the materials. 00:04:22.530 --> 00:04:23.370 align:middle line:90% What are the ideas? 00:04:23.370 --> 00:04:26.120 align:middle line:90% 00:04:26.120 --> 00:04:32.840 align:middle line:84% So let me, again, start with this figure of merit ZT. 00:04:32.840 --> 00:04:34.580 align:middle line:84% I said that when you do thermoelectric, 00:04:34.580 --> 00:04:36.840 align:middle line:84% so people say I work on thermoelectric. 00:04:36.840 --> 00:04:40.250 align:middle line:84% So when people ask you what's your ZT, right? 00:04:40.250 --> 00:04:42.290 align:middle line:84% That shows you know something about it. 00:04:42.290 --> 00:04:45.710 align:middle line:90% And I have three parameters-- 00:04:45.710 --> 00:04:50.840 align:middle line:84% the electrical conductivity, the Seebeck coefficient, 00:04:50.840 --> 00:04:53.730 align:middle line:90% and the thermal conductivity. 00:04:53.730 --> 00:04:57.420 align:middle line:84% And the thermal conductivity, the heat conduction 00:04:57.420 --> 00:05:00.640 align:middle line:84% in the material consists of two parts. 00:05:00.640 --> 00:05:03.390 align:middle line:90% One is the topical vibration. 00:05:03.390 --> 00:05:06.930 align:middle line:84% The lattice in the material, the atoms in the material, 00:05:06.930 --> 00:05:08.940 align:middle line:90% vibrate and conduct heat. 00:05:08.940 --> 00:05:12.420 align:middle line:84% And the other is when charge itself, 00:05:12.420 --> 00:05:15.210 align:middle line:84% it carries states also conducted. 00:05:15.210 --> 00:05:20.280 align:middle line:84% So here, we decompose this k into [INAUDIBLE] 00:05:20.280 --> 00:05:22.200 align:middle line:90% and the electron. 00:05:22.200 --> 00:05:27.720 align:middle line:84% So those are when you think about ZT, 00:05:27.720 --> 00:05:31.170 align:middle line:84% you look at those parameters-- electric conductivity, Seebeck 00:05:31.170 --> 00:05:34.110 align:middle line:84% coefficient, and the [INAUDIBLE] contribution 00:05:34.110 --> 00:05:36.570 align:middle line:84% to thermal conductivity, electron contribution, 00:05:36.570 --> 00:05:38.520 align:middle line:90% thermal conductivity. 00:05:38.520 --> 00:05:43.290 align:middle line:84% And what I want to do next is to give you-- go back 00:05:43.290 --> 00:05:45.420 align:middle line:90% into more details. 00:05:45.420 --> 00:05:48.120 align:middle line:84% So what are the microscopic pictures 00:05:48.120 --> 00:05:51.120 align:middle line:90% determine those properties? 00:05:51.120 --> 00:05:56.070 align:middle line:84% And what are the physical more together 00:05:56.070 --> 00:05:58.320 align:middle line:84% with some mathematical formulation 00:05:58.320 --> 00:06:01.950 align:middle line:90% for those properties? 00:06:01.950 --> 00:06:04.500 align:middle line:90% So today, we hope-- 00:06:04.500 --> 00:06:09.060 align:middle line:84% I hope you all power those topics. 00:06:09.060 --> 00:06:15.420 align:middle line:84% We will first review some basic concepts in solid state. 00:06:15.420 --> 00:06:20.460 align:middle line:84% And then using the simplest picture, that's the same picture 00:06:20.460 --> 00:06:23.490 align:middle line:84% I talked in the first lecture how we use a three step. 00:06:23.490 --> 00:06:25.500 align:middle line:90% I derive [INAUDIBLE], right? 00:06:25.500 --> 00:06:29.550 align:middle line:84% It's a really popular going from left and minus particle, 00:06:29.550 --> 00:06:33.420 align:middle line:84% from left to right, from right to left. 00:06:33.420 --> 00:06:36.210 align:middle line:84% So with that imbalance, you will see 00:06:36.210 --> 00:06:40.980 align:middle line:84% we can come up with expressions for the electric conductivity, 00:06:40.980 --> 00:06:43.260 align:middle line:84% Seebeck, thermal conductivity, all those. 00:06:43.260 --> 00:06:46.960 align:middle line:84% And then I will do a leap of faith. 00:06:46.960 --> 00:06:51.040 align:middle line:84% I'm not going to the detailed derivation from this. 00:06:51.040 --> 00:06:55.140 align:middle line:84% You know we go-- we can go to more formal transfer theory 00:06:55.140 --> 00:07:00.360 align:middle line:84% and have the more rigorous expressions for the transport 00:07:00.360 --> 00:07:03.390 align:middle line:90% coefficients. 00:07:03.390 --> 00:07:06.510 align:middle line:90% So let me start. 00:07:06.510 --> 00:07:12.150 align:middle line:84% Like I said, this is a review, but I know for most of you 00:07:12.150 --> 00:07:13.440 align:middle line:90% it's new. 00:07:13.440 --> 00:07:16.710 align:middle line:84% Because if you are, for example, from mechanical engineering, 00:07:16.710 --> 00:07:18.060 align:middle line:90% you have not seen this. 00:07:18.060 --> 00:07:21.090 align:middle line:90% So don't worry about it. 00:07:21.090 --> 00:07:23.710 align:middle line:84% You just follow-- let's follow through. 00:07:23.710 --> 00:07:27.990 align:middle line:84% Some of you will take this other result from wavelengths. 00:07:27.990 --> 00:07:31.110 align:middle line:84% Others, I hope the picture is simple enough for you 00:07:31.110 --> 00:07:35.520 align:middle line:90% to appreciate those pictures. 00:07:35.520 --> 00:07:38.490 align:middle line:84% So let's start with the basic concept. 00:07:38.490 --> 00:07:43.170 align:middle line:84% First, in terms of the atomic vibration, 00:07:43.170 --> 00:07:48.300 align:middle line:84% because atomic vibration in the solid carries [INAUDIBLE], 00:07:48.300 --> 00:07:48.850 align:middle line:90% right? 00:07:48.850 --> 00:07:52.980 align:middle line:84% So now, I think about the bonding between atoms. 00:07:52.980 --> 00:07:55.860 align:middle line:84% You can think of this either solid or even just a hydrogen 00:07:55.860 --> 00:07:57.000 align:middle line:90% molecule. 00:07:57.000 --> 00:07:59.760 align:middle line:84% The bonding between the two atoms, 00:07:59.760 --> 00:08:02.280 align:middle line:84% two hydrogen atoms in the molecule, 00:08:02.280 --> 00:08:06.040 align:middle line:84% when they are far apart, they are attractive. 00:08:06.040 --> 00:08:08.470 align:middle line:84% And when they are too close, they 00:08:08.470 --> 00:08:11.560 align:middle line:84% repel each other because the electrons start to overlap, 00:08:11.560 --> 00:08:13.210 align:middle line:90% and they-- 00:08:13.210 --> 00:08:17.530 align:middle line:84% all the mechanics will say the basic principle is the exclusion 00:08:17.530 --> 00:08:18.110 align:middle line:90% principle. 00:08:18.110 --> 00:08:21.310 align:middle line:84% It doesn't allow the electron to take this [INAUDIBLE]. 00:08:21.310 --> 00:08:27.940 align:middle line:84% So we have a typical potential curve in the far separation 00:08:27.940 --> 00:08:29.800 align:middle line:90% between the atoms. 00:08:29.800 --> 00:08:31.690 align:middle line:90% It's attractive. 00:08:31.690 --> 00:08:35.860 align:middle line:84% And when they close, it's repulsive 00:08:35.860 --> 00:08:40.990 align:middle line:84% and the atoms sits in the equilibrium position. 00:08:40.990 --> 00:08:44.740 align:middle line:84% So when we have many atoms, they will 00:08:44.740 --> 00:08:47.030 align:middle line:90% have constraints on both sides. 00:08:47.030 --> 00:08:49.220 align:middle line:84% So they have the most stable position. 00:08:49.220 --> 00:08:54.220 align:middle line:84% That's where the energy is the minimum, right? 00:08:54.220 --> 00:08:56.870 align:middle line:84% Now, because this curve, the energy is a minimum. 00:08:56.870 --> 00:08:59.740 align:middle line:84% If you look at the mathematics, when 00:08:59.740 --> 00:09:02.050 align:middle line:84% I have a curve, minimum or maximum, 00:09:02.050 --> 00:09:04.120 align:middle line:90% the first derivative is zero. 00:09:04.120 --> 00:09:11.350 align:middle line:84% So when I look at the expansion, I can expand this potential 00:09:11.350 --> 00:09:15.680 align:middle line:84% into the minimum point, plus, the first derivative is zero. 00:09:15.680 --> 00:09:17.560 align:middle line:84% Then I go to the second derivative. 00:09:17.560 --> 00:09:20.030 align:middle line:90% So you can go to higher order. 00:09:20.030 --> 00:09:24.280 align:middle line:84% But for the understanding here, we just go to second order, 00:09:24.280 --> 00:09:28.120 align:middle line:84% and then the force between the two atoms 00:09:28.120 --> 00:09:32.380 align:middle line:84% is just the derivative of the potential. 00:09:32.380 --> 00:09:35.570 align:middle line:84% So F equals the potential derivative. 00:09:35.570 --> 00:09:42.220 align:middle line:84% So I get a force equals the spring constant between the-- 00:09:42.220 --> 00:09:45.970 align:middle line:84% So now, we can take this two atoms vibration as a spring. 00:09:45.970 --> 00:09:49.740 align:middle line:90% So it's a mass spring system. 00:09:49.740 --> 00:09:53.190 align:middle line:84% And for mass spring system, we all 00:09:53.190 --> 00:09:56.790 align:middle line:84% know the simplest picture is mass print. 00:09:56.790 --> 00:10:02.600 align:middle line:84% And there is a natural frequency for this vibration. 00:10:02.600 --> 00:10:08.180 align:middle line:84% The vibration frequency is the spring constant and divided 00:10:08.180 --> 00:10:12.300 align:middle line:90% by the mass of the atom. 00:10:12.300 --> 00:10:15.060 align:middle line:84% This is very relevant when you think 00:10:15.060 --> 00:10:22.330 align:middle line:84% about what material you want to choose for thermoelectric. 00:10:22.330 --> 00:10:27.250 align:middle line:84% And typically, the larger is the spring constant 00:10:27.250 --> 00:10:28.570 align:middle line:90% between the atoms. 00:10:28.570 --> 00:10:33.550 align:middle line:84% You look at this, the higher is the frequency, the more 00:10:33.550 --> 00:10:35.800 align:middle line:90% vibrational energy, the more-- 00:10:35.800 --> 00:10:38.100 align:middle line:84% the bonding of the atoms is stronger. 00:10:38.100 --> 00:10:39.910 align:middle line:84% There is a larger spring constant. 00:10:39.910 --> 00:10:43.800 align:middle line:84% And this will lead to higher thermal conductivity. 00:10:43.800 --> 00:10:47.220 align:middle line:84% And diamond, for example, has the highest thermal conductivity 00:10:47.220 --> 00:10:50.700 align:middle line:84% among the materials, and that's because the bonding 00:10:50.700 --> 00:10:53.820 align:middle line:84% between carbon in diamond is very strong, 00:10:53.820 --> 00:10:57.480 align:middle line:84% and the carbon atom is very light. 00:10:57.480 --> 00:10:57.980 align:middle line:90% OK. 00:10:57.980 --> 00:11:00.950 align:middle line:90% So that's just a-- 00:11:00.950 --> 00:11:05.810 align:middle line:84% it has implications for this, even though it's very simple. 00:11:05.810 --> 00:11:09.860 align:middle line:84% And this is what we learn in classical mechanics 00:11:09.860 --> 00:11:14.580 align:middle line:84% dynamics, the natural frequency of vibration. 00:11:14.580 --> 00:11:17.910 align:middle line:84% And if you go to quantum mechanics, 00:11:17.910 --> 00:11:21.488 align:middle line:84% what it tells you is just a small addition, 00:11:21.488 --> 00:11:22.530 align:middle line:90% you know what I'm saying? 00:11:22.530 --> 00:11:24.410 align:middle line:90% The frequency is still the same. 00:11:24.410 --> 00:11:26.130 align:middle line:90% New is still the same. 00:11:26.130 --> 00:11:30.040 align:middle line:84% But what it tells you is that energy cannot be any value. 00:11:30.040 --> 00:11:32.190 align:middle line:84% We all know quantum mechanics, the basics, 00:11:32.190 --> 00:11:34.410 align:middle line:90% the energy is complex. 00:11:34.410 --> 00:11:38.880 align:middle line:84% So if your oscillation is either this frequency 00:11:38.880 --> 00:11:42.510 align:middle line:84% in classical mechanics, the oscillation amplitude, 00:11:42.510 --> 00:11:44.640 align:middle line:90% the velocity, can be any value. 00:11:44.640 --> 00:11:49.650 align:middle line:84% And in quantum mechanics, we say the energy must 00:11:49.650 --> 00:11:54.790 align:middle line:90% be integer of this [INAUDIBLE]. 00:11:54.790 --> 00:11:55.450 align:middle line:90% OK. 00:11:55.450 --> 00:12:00.910 align:middle line:84% So that's the new thing coming from quantum mechanics. 00:12:00.910 --> 00:12:04.240 align:middle line:84% So it must be multiples of the Ishmael. 00:12:04.240 --> 00:12:08.890 align:middle line:84% And so this Ishmael, you can think of this as a basic energy 00:12:08.890 --> 00:12:11.900 align:middle line:90% quantum, and that's-- 00:12:11.900 --> 00:12:14.470 align:middle line:90% a solid is called a phonon. 00:12:14.470 --> 00:12:16.810 align:middle line:90% And this one-half is also-- 00:12:16.810 --> 00:12:18.970 align:middle line:84% is related to the so-called uncertainty 00:12:18.970 --> 00:12:21.100 align:middle line:90% principle in quantum mechanics. 00:12:21.100 --> 00:12:23.650 align:middle line:84% It's also called the zero point energy. 00:12:23.650 --> 00:12:29.170 align:middle line:84% So if you recall before I talk about the photon, 00:12:29.170 --> 00:12:31.720 align:middle line:84% right, when we do electromagnetic waves, 00:12:31.720 --> 00:12:36.430 align:middle line:84% and this atomic vibration is analogous 00:12:36.430 --> 00:12:39.250 align:middle line:90% to the electromagnetic wave. 00:12:39.250 --> 00:12:41.710 align:middle line:90% So those are all waves. 00:12:41.710 --> 00:12:45.020 align:middle line:84% We'll extend this more to waves in the next step. 00:12:45.020 --> 00:12:47.200 align:middle line:84% So that's where the name coming from, 00:12:47.200 --> 00:12:52.390 align:middle line:84% photon for electromagnetic wave, and the phonon for lattice 00:12:52.390 --> 00:12:53.590 align:middle line:90% vibration. 00:12:53.590 --> 00:12:58.380 align:middle line:84% And that's the basic energy quanta of vibration. 00:12:58.380 --> 00:12:58.880 align:middle line:90% OK. 00:12:58.880 --> 00:12:59.840 align:middle line:90% So this is a-- 00:12:59.840 --> 00:13:03.440 align:middle line:84% we're talking about the individual oscillator. 00:13:03.440 --> 00:13:09.800 align:middle line:84% And now, I look at the atomic chain. 00:13:09.800 --> 00:13:15.890 align:middle line:84% So when I think about solid, the most simple picture 00:13:15.890 --> 00:13:17.330 align:middle line:90% is a crystal. 00:13:17.330 --> 00:13:20.640 align:middle line:84% Crystal, a periodic arrangement of atoms. 00:13:20.640 --> 00:13:25.940 align:middle line:84% So now, I started with one dimension of atomic chain, 00:13:25.940 --> 00:13:30.950 align:middle line:84% and the distance between the atoms 00:13:30.950 --> 00:13:35.540 align:middle line:84% is the lattice constant, A. So we have N atoms. 00:13:35.540 --> 00:13:39.550 align:middle line:84% So the length of this chain is [INAUDIBLE] A. Now, 00:13:39.550 --> 00:13:43.700 align:middle line:84% again, you recall in the electromagnetic wave, 00:13:43.700 --> 00:13:48.010 align:middle line:84% the last lecture, I derived the Planck blackbody 00:13:48.010 --> 00:13:51.520 align:middle line:90% radiation [INAUDIBLE]. 00:13:51.520 --> 00:13:55.450 align:middle line:84% I say that when we consider a cavity in the length direction, 00:13:55.450 --> 00:13:57.610 align:middle line:90% I have standing width. 00:13:57.610 --> 00:14:00.140 align:middle line:84% I'm going to make the same argument. 00:14:00.140 --> 00:14:06.010 align:middle line:84% If I clamp the two ends of this atomic chain, do not move. 00:14:06.010 --> 00:14:10.870 align:middle line:84% Again, take a rubber band as an example. 00:14:10.870 --> 00:14:15.010 align:middle line:84% You fix this end, you oscillate it. 00:14:15.010 --> 00:14:17.930 align:middle line:84% You see the waves, different waves. 00:14:17.930 --> 00:14:21.650 align:middle line:84% And the violin is another example. 00:14:21.650 --> 00:14:24.130 align:middle line:90% So the length gave me-- 00:14:24.130 --> 00:14:26.830 align:middle line:84% those are the wave of the constraint. 00:14:26.830 --> 00:14:30.760 align:middle line:84% The wave has to be multiples of lambda, 00:14:30.760 --> 00:14:32.970 align:middle line:90% which is the wavelength over 2. 00:14:32.970 --> 00:14:38.450 align:middle line:84% So I have-- the length must be more than 2. 00:14:38.450 --> 00:14:41.820 align:middle line:90% It could be any integer. 00:14:41.820 --> 00:14:43.920 align:middle line:90% And then the weight vector. 00:14:43.920 --> 00:14:46.630 align:middle line:84% Again, I'm talking about because of the weight, 00:14:46.630 --> 00:14:51.690 align:middle line:84% so the inverse of the wavelength is the weight vector 00:14:51.690 --> 00:14:53.140 align:middle line:90% is 2 pi over lambda. 00:14:53.140 --> 00:14:54.510 align:middle line:90% That's the definition. 00:14:54.510 --> 00:15:00.240 align:middle line:84% And the fundamentally is like the Fourier transform. 00:15:00.240 --> 00:15:03.720 align:middle line:84% If you're an electrical engineer, the time signal, 00:15:03.720 --> 00:15:06.270 align:middle line:90% or sine omega t signal. 00:15:06.270 --> 00:15:09.850 align:middle line:84% And here is key and frequency is omega. 00:15:09.850 --> 00:15:14.580 align:middle line:84% The relation is omega equals 2 pi over t, the same. 00:15:14.580 --> 00:15:18.840 align:middle line:84% This is a spatially periodic vibration, 00:15:18.840 --> 00:15:22.230 align:middle line:84% so the weight vector and wavelength relation 00:15:22.230 --> 00:15:24.700 align:middle line:90% is 2 pi over lambda. 00:15:24.700 --> 00:15:26.950 align:middle line:90% So that's a Fourier transform. 00:15:26.950 --> 00:15:30.730 align:middle line:90% And so this is the same picture. 00:15:30.730 --> 00:15:35.050 align:middle line:84% I clamped the two ends of the [INAUDIBLE]. 00:15:35.050 --> 00:15:40.510 align:middle line:84% I have those other [INAUDIBLE] wave vectors. 00:15:40.510 --> 00:15:45.010 align:middle line:84% And I went this pretty fast in the last time I 00:15:45.010 --> 00:15:46.870 align:middle line:90% talk about electromagnetic wave. 00:15:46.870 --> 00:15:49.970 align:middle line:90% But I did say the standing wave. 00:15:49.970 --> 00:15:52.010 align:middle line:90% I did write the mathematics. 00:15:52.010 --> 00:15:52.840 align:middle line:90% You go to check. 00:15:52.840 --> 00:15:55.660 align:middle line:84% When I wrote the mathematics, I say 00:15:55.660 --> 00:15:59.230 align:middle line:84% I can do the summation of two traveling waves. 00:15:59.230 --> 00:16:04.300 align:middle line:84% One goes from left to right, the others go from right to left. 00:16:04.300 --> 00:16:08.180 align:middle line:90% So here k is 2 pi over lambda. 00:16:08.180 --> 00:16:09.400 align:middle line:90% Those are the values. 00:16:09.400 --> 00:16:16.150 align:middle line:90% So now, I do the traveling wave. 00:16:16.150 --> 00:16:18.280 align:middle line:84% So rather than clamp the two ends, 00:16:18.280 --> 00:16:20.920 align:middle line:84% I say, OK, I don't have that constraint. 00:16:20.920 --> 00:16:23.490 align:middle line:84% The waves can go both positive direction 00:16:23.490 --> 00:16:26.110 align:middle line:90% and negative direction. 00:16:26.110 --> 00:16:26.650 align:middle line:90% OK. 00:16:26.650 --> 00:16:31.840 align:middle line:84% So in this case, the weight vector is pi over-- or minus pi 00:16:31.840 --> 00:16:34.540 align:middle line:90% over 8, 2 plus pi over 8. 00:16:34.540 --> 00:16:37.680 align:middle line:84% And two traveling wave will make a standing wave. 00:16:37.680 --> 00:16:41.740 align:middle line:84% So if you have the same wave lengths there. 00:16:41.740 --> 00:16:48.110 align:middle line:84% But also, if you compare what I wrote before, OK, 00:16:48.110 --> 00:16:51.030 align:middle line:84% I didn't put the [INAUDIBLE] limit, right? 00:16:51.030 --> 00:16:55.730 align:middle line:84% | this wave every vector can have any value. 00:16:55.730 --> 00:17:01.420 align:middle line:84% On the other hand, when I think about very logically vector, 00:17:01.420 --> 00:17:06.760 align:middle line:84% what it means in a crystal, because the two atoms, 00:17:06.760 --> 00:17:09.040 align:middle line:90% the smallest distance is here. 00:17:09.040 --> 00:17:12.130 align:middle line:84% Now, if you have wavelengths shorter than that, 00:17:12.130 --> 00:17:14.270 align:middle line:90% it doesn't mean much. 00:17:14.270 --> 00:17:21.180 align:middle line:84% So that's why I say my wavelength is limited to here. 00:17:21.180 --> 00:17:23.359 align:middle line:84% If you look at this, this is a two wave. 00:17:23.359 --> 00:17:27.290 align:middle line:84% The two-- one atomic distance is A, 00:17:27.290 --> 00:17:32.180 align:middle line:84% so this is the minimum you can have in terms of wavelengths. 00:17:32.180 --> 00:17:36.290 align:middle line:84% So those are the region of my weight 00:17:36.290 --> 00:17:43.800 align:middle line:84% vector going from negative pi over A 2 plus pi over A. 00:17:43.800 --> 00:17:48.120 align:middle line:84% Now, you think about the frequency. 00:17:48.120 --> 00:17:51.750 align:middle line:84% When I talk about the one atom vibrating, 00:17:51.750 --> 00:17:58.770 align:middle line:84% the frequency is K or M, square root of K over M. Now, 00:17:58.770 --> 00:18:02.460 align:middle line:84% I have the whole lattice, different wave vector. 00:18:02.460 --> 00:18:08.390 align:middle line:84% So each wavelength corresponding to one mode of vibration. 00:18:08.390 --> 00:18:12.020 align:middle line:84% And the thing about, if my whole atomic chain 00:18:12.020 --> 00:18:18.500 align:middle line:84% is one way, one way with pure, one pure, 00:18:18.500 --> 00:18:21.110 align:middle line:84% the frequency is higher, or lower, 00:18:21.110 --> 00:18:26.400 align:middle line:84% or same as my individual K over M square root. 00:18:26.400 --> 00:18:35.510 align:middle line:90% 00:18:35.510 --> 00:18:40.380 align:middle line:90% Anybody is making a guess? 00:18:40.380 --> 00:18:44.510 align:middle line:84% AUDIENCE: If an atom was moving fast, yes, I'd 00:18:44.510 --> 00:18:47.815 align:middle line:84% say it's the same as the [INAUDIBLE]. 00:18:47.815 --> 00:18:49.820 align:middle line:90% GANG CHEN: Well, let's suppose-- 00:18:49.820 --> 00:18:53.690 align:middle line:84% again, let me take this as right. 00:18:53.690 --> 00:19:00.230 align:middle line:84% Now, all the atoms moving just one pure, all the moves. 00:19:00.230 --> 00:19:07.670 align:middle line:84% And then let's say is a one pure, and more pure 00:19:07.670 --> 00:19:09.140 align:middle line:90% since the frequency. 00:19:09.140 --> 00:19:12.560 align:middle line:90% Do they change, or they don't? 00:19:12.560 --> 00:19:19.670 align:middle line:90% 00:19:19.670 --> 00:19:21.770 align:middle line:84% You probably can go back and do an experiment 00:19:21.770 --> 00:19:25.040 align:middle line:84% and put some beads or a rubber band, 00:19:25.040 --> 00:19:37.460 align:middle line:84% and then you'll find out the relation between the frequency 00:19:37.460 --> 00:19:40.500 align:middle line:90% and the wavelength. 00:19:40.500 --> 00:19:43.620 align:middle line:84% In this case, it's changing, right? 00:19:43.620 --> 00:19:47.640 align:middle line:84% When all the atoms move in unison, 00:19:47.640 --> 00:19:52.560 align:middle line:84% when, say, only one pure, it looks like the whole atoms-- 00:19:52.560 --> 00:19:55.690 align:middle line:84% all the atoms in the chain is moving same time. 00:19:55.690 --> 00:19:58.620 align:middle line:90% So it's slower. 00:19:58.620 --> 00:20:00.120 align:middle line:90% The frequency is smaller. 00:20:00.120 --> 00:20:02.737 align:middle line:84% So that's-- in the long wavelengths, 00:20:02.737 --> 00:20:03.820 align:middle line:90% this is the weight vector. 00:20:03.820 --> 00:20:05.760 align:middle line:84% Long wavelength frequency is smaller. 00:20:05.760 --> 00:20:10.230 align:middle line:84% And when you go to larger wavelengths at the end, 00:20:10.230 --> 00:20:11.170 align:middle line:90% it's closed. 00:20:11.170 --> 00:20:13.650 align:middle line:84% Coming back to here, you can see, 00:20:13.650 --> 00:20:17.420 align:middle line:90% here is a K over M square root. 00:20:17.420 --> 00:20:23.460 align:middle line:84% So come back to individual atom and high frequency. 00:20:23.460 --> 00:20:24.060 align:middle line:90% OK. 00:20:24.060 --> 00:20:29.580 align:middle line:84% So this is the relation between the wave vector. 00:20:29.580 --> 00:20:33.900 align:middle line:84% And the frequency is, again, the dispersion. 00:20:33.900 --> 00:20:38.220 align:middle line:84% Remember, for photon, for the electromagnetic wave, 00:20:38.220 --> 00:20:40.020 align:middle line:90% omega equals [INAUDIBLE] k. 00:20:40.020 --> 00:20:42.270 align:middle line:90% That's a linear dispersion. 00:20:42.270 --> 00:20:50.580 align:middle line:84% And here, the real one we have is approximately half of a sine. 00:20:50.580 --> 00:20:52.710 align:middle line:90% This is like a sine function. 00:20:52.710 --> 00:20:58.530 align:middle line:84% And often, we do a linear approximation, take the slope. 00:20:58.530 --> 00:21:03.720 align:middle line:84% So the speed of sound is that that's-- the slope is 00:21:03.720 --> 00:21:05.170 align:middle line:90% for the longitudinal wave. 00:21:05.170 --> 00:21:06.810 align:middle line:90% That's the speed of sound. 00:21:06.810 --> 00:21:12.780 align:middle line:84% So in solid, you lock one end, propagate to the other end. 00:21:12.780 --> 00:21:15.060 align:middle line:90% That's the speed of sound. 00:21:15.060 --> 00:21:19.092 align:middle line:90% And that's this slope here. 00:21:19.092 --> 00:21:21.840 align:middle line:84% And at a very high frequency, when 00:21:21.840 --> 00:21:27.670 align:middle line:84% you go to look at the mass of the atoms, 00:21:27.670 --> 00:21:31.320 align:middle line:84% you can see this frequency can go as high as terahertz 00:21:31.320 --> 00:21:33.060 align:middle line:90% or tens of terahertz. 00:21:33.060 --> 00:21:34.710 align:middle line:90% So very high frequency. 00:21:34.710 --> 00:21:37.680 align:middle line:84% Of course, the acoustic wave that we can hear 00:21:37.680 --> 00:21:41.190 align:middle line:90% is only a few 100 hertz. 00:21:41.190 --> 00:21:43.980 align:middle line:90% So this is a simple one-- 00:21:43.980 --> 00:21:46.440 align:middle line:90% monatomic atomic chain. 00:21:46.440 --> 00:21:50.460 align:middle line:84% And now if I extend this a little bit-- 00:21:50.460 --> 00:21:53.510 align:middle line:84% suppose my chain has two different atoms. 00:21:53.510 --> 00:21:56.060 align:middle line:90% 00:21:56.060 --> 00:21:58.720 align:middle line:90% Gallium arsenide has a crystal. 00:21:58.720 --> 00:22:00.220 align:middle line:90% One atom is gallium. 00:22:00.220 --> 00:22:02.646 align:middle line:90% The other atom is arsenide. 00:22:02.646 --> 00:22:05.590 align:middle line:90% 00:22:05.590 --> 00:22:09.880 align:middle line:84% So in this case, this is a white one here. 00:22:09.880 --> 00:22:12.820 align:middle line:90% So from the 22 atoms-- 00:22:12.820 --> 00:22:15.586 align:middle line:90% the same atom-- that's one here. 00:22:15.586 --> 00:22:17.820 align:middle line:84% Of course, this one simple picture 00:22:17.820 --> 00:22:21.430 align:middle line:84% is-- the outer atom is moving in unison with this atom. 00:22:21.430 --> 00:22:30.100 align:middle line:84% So they form the same face into two sets of subatom lattice. 00:22:30.100 --> 00:22:33.490 align:middle line:84% And in this case, the dispersion-- 00:22:33.490 --> 00:22:36.010 align:middle line:84% the relation between the wavelength-- 00:22:36.010 --> 00:22:38.470 align:middle line:90% wave vector and frequency-- 00:22:38.470 --> 00:22:42.280 align:middle line:84% is similar to what we drew before. 00:22:42.280 --> 00:22:49.250 align:middle line:84% And in the other case, we could have this set of atoms moving up 00:22:49.250 --> 00:22:52.400 align:middle line:84% and the other set of atoms moving down. 00:22:52.400 --> 00:22:54.575 align:middle line:90% So it's a much harder stretch. 00:22:54.575 --> 00:22:57.580 align:middle line:90% 00:22:57.580 --> 00:22:59.060 align:middle line:90% We are very close. 00:22:59.060 --> 00:23:04.510 align:middle line:84% You are forcing them in terms of a different phase. 00:23:04.510 --> 00:23:09.700 align:middle line:84% It's much harder [INAUDIBLE] than the higher frequency. 00:23:09.700 --> 00:23:14.650 align:middle line:84% So in this case, we have two type-- 00:23:14.650 --> 00:23:17.380 align:middle line:90% I'm going to have two atoms. 00:23:17.380 --> 00:23:19.940 align:middle line:90% One is the acoustic branch. 00:23:19.940 --> 00:23:21.310 align:middle line:90% That's the bottom part. 00:23:21.310 --> 00:23:25.250 align:middle line:84% That's this picture here, roughly. 00:23:25.250 --> 00:23:29.000 align:middle line:90% The other is the other phase. 00:23:29.000 --> 00:23:30.580 align:middle line:90% That's the [INAUDIBLE] approach. 00:23:30.580 --> 00:23:33.360 align:middle line:90% 00:23:33.360 --> 00:23:40.640 align:middle line:84% So that's the atomic vibration in one dimension. 00:23:40.640 --> 00:23:46.430 align:middle line:84% And of course, a crystal is not the just one dimension. 00:23:46.430 --> 00:23:48.740 align:middle line:90% A solid is just one dimension. 00:23:48.740 --> 00:23:50.640 align:middle line:90% It's three dimension. 00:23:50.640 --> 00:23:54.210 align:middle line:84% So any direction you look, it look 00:23:54.210 --> 00:23:57.270 align:middle line:84% like you got a one-dimensional periodical arrangement. 00:23:57.270 --> 00:24:01.350 align:middle line:84% It's just that space in between the atoms will be different. 00:24:01.350 --> 00:24:02.980 align:middle line:90% So how we express it? 00:24:02.980 --> 00:24:06.048 align:middle line:90% 00:24:06.048 --> 00:24:12.350 align:middle line:84% So here, we'll go back to what I said before-- 00:24:12.350 --> 00:24:14.330 align:middle line:90% Fourier transform. 00:24:14.330 --> 00:24:16.110 align:middle line:90% And the periodic signal-- 00:24:16.110 --> 00:24:24.190 align:middle line:84% if the period is t, We see the frequency is the 2 pi over t. 00:24:24.190 --> 00:24:28.060 align:middle line:84% And in terms of wave, we want to mention 00:24:28.060 --> 00:24:30.580 align:middle line:90% wavelengths of the lambda. 00:24:30.580 --> 00:24:34.810 align:middle line:84% And the Fourier transform is 2 pi lambda. 00:24:34.810 --> 00:24:39.900 align:middle line:84% But in three dimension, each direction has a lambda. 00:24:39.900 --> 00:24:44.580 align:middle line:84% If I look at the crystal, a cube-- take a simple cube. 00:24:44.580 --> 00:24:46.320 align:middle line:90% Put them periodically arranged. 00:24:46.320 --> 00:24:48.280 align:middle line:90% You look at the other direction. 00:24:48.280 --> 00:24:51.150 align:middle line:90% The period is cubic 3-- 00:24:51.150 --> 00:24:54.870 align:middle line:84% no square root of 3 times the lattice constant a. 00:24:54.870 --> 00:24:57.870 align:middle line:90% So that's your periodicity. 00:24:57.870 --> 00:25:01.080 align:middle line:84% You look at, say, a different [INAUDIBLE] 00:25:01.080 --> 00:25:03.060 align:middle line:90% you have different periodicity. 00:25:03.060 --> 00:25:09.560 align:middle line:84% So what I do is I transform this periodicity in space. 00:25:09.560 --> 00:25:13.580 align:middle line:90% Let's say I take a crystal-- 00:25:13.580 --> 00:25:17.430 align:middle line:84% and say, here is a, say, face on the structure-- 00:25:17.430 --> 00:25:18.380 align:middle line:90% face on the cube. 00:25:18.380 --> 00:25:20.670 align:middle line:84% And we add a face on the surface. 00:25:20.670 --> 00:25:22.790 align:middle line:90% There is also an atom. 00:25:22.790 --> 00:25:29.090 align:middle line:84% And I do a Fourier transform of this space in three dimensions. 00:25:29.090 --> 00:25:30.660 align:middle line:90% That's what it looks like. 00:25:30.660 --> 00:25:33.740 align:middle line:90% Now it looks a little bit ugly. 00:25:33.740 --> 00:25:37.910 align:middle line:84% But I can say each crystallographic direction here 00:25:37.910 --> 00:25:44.570 align:middle line:84% is a p direction, And the shortest distance, L-- 00:25:44.570 --> 00:25:48.830 align:middle line:84% this point-- is the longest real space 00:25:48.830 --> 00:25:55.930 align:middle line:84% because the k is 2 pi over lambda. 00:25:55.930 --> 00:26:00.430 align:middle line:84% So if I think about this from center 00:26:00.430 --> 00:26:04.030 align:middle line:84% to this point, the shortest, I know for a cube, 00:26:04.030 --> 00:26:06.760 align:middle line:90% that's a 111 direction. 00:26:06.760 --> 00:26:10.800 align:middle line:84% It's the universe of the atomic space. 00:26:10.800 --> 00:26:15.000 align:middle line:84% So this is a way that the physicists like 00:26:15.000 --> 00:26:19.500 align:middle line:90% to use the reciprocal space. 00:26:19.500 --> 00:26:20.470 align:middle line:90% I joke with them-- 00:26:20.470 --> 00:26:23.370 align:middle line:84% I say, you guys don't live in the real world. 00:26:23.370 --> 00:26:27.780 align:middle line:84% They like to express things in the reciprocal space. 00:26:27.780 --> 00:26:31.530 align:middle line:90% So now let's look at a simple-- 00:26:31.530 --> 00:26:34.560 align:middle line:84% in terms of mathematics-- expression of this, 00:26:34.560 --> 00:26:36.150 align:middle line:90% You go to electrical engineers. 00:26:36.150 --> 00:26:40.020 align:middle line:84% They like to do signal analysis and frequency. 00:26:40.020 --> 00:26:42.480 align:middle line:90% They don't do time signal. 00:26:42.480 --> 00:26:45.010 align:middle line:84% Time signal-- you have to plot the time. 00:26:45.010 --> 00:26:49.980 align:middle line:84% You do a hard spectral analysis, you can use frequency. 00:26:49.980 --> 00:26:51.935 align:middle line:84% It's a much simpler mathematical language. 00:26:51.935 --> 00:26:54.840 align:middle line:90% 00:26:54.840 --> 00:26:58.950 align:middle line:84% So now you get to the real world because that-- 00:26:58.950 --> 00:27:01.170 align:middle line:90% before, it looked very simple. 00:27:01.170 --> 00:27:03.570 align:middle line:84% Now you can start to look at a string. 00:27:03.570 --> 00:27:06.330 align:middle line:84% For most mechanical engineers, you look at this, 00:27:06.330 --> 00:27:07.890 align:middle line:90% and you'll get the-- 00:27:07.890 --> 00:27:09.660 align:middle line:90% what is that, they'll say. 00:27:09.660 --> 00:27:12.490 align:middle line:84% It's really looking at the different crystallographic 00:27:12.490 --> 00:27:16.430 align:middle line:84% direction, the vibration frequency. 00:27:16.430 --> 00:27:22.970 align:middle line:84% And if you look at this all silicon gallium arsenide 00:27:22.970 --> 00:27:26.330 align:middle line:84% or phase-centered cubic, as I do before, 00:27:26.330 --> 00:27:28.580 align:middle line:84% different crystal has different-- 00:27:28.580 --> 00:27:30.470 align:middle line:84% those different directions-- different 00:27:30.470 --> 00:27:34.190 align:middle line:84% reciprocal-- the shape of this basic unit 00:27:34.190 --> 00:27:36.020 align:middle line:90% cell in reciprocal space. 00:27:36.020 --> 00:27:39.950 align:middle line:84% So I look at the gamma direction. 00:27:39.950 --> 00:27:42.920 align:middle line:90% 00:27:42.920 --> 00:27:46.700 align:middle line:84% So this is diagonal, from this to this-- diagonal direction. 00:27:46.700 --> 00:27:50.240 align:middle line:84% So if I look at that direction, what's the vibrational situation 00:27:50.240 --> 00:27:50.920 align:middle line:90% spectrum? 00:27:50.920 --> 00:27:53.480 align:middle line:84% So if I have the lattice wave propagating 00:27:53.480 --> 00:27:59.920 align:middle line:84% in the diagonal direction-- so that's from gamma to L. 00:27:59.920 --> 00:28:05.020 align:middle line:84% And those are the optical phonons. 00:28:05.020 --> 00:28:07.270 align:middle line:90% Those are the acoustic phonons. 00:28:07.270 --> 00:28:09.560 align:middle line:84% Acoustic phonons start from 0 here. 00:28:09.560 --> 00:28:15.580 align:middle line:84% Optical is from the top of the acoustic. 00:28:15.580 --> 00:28:19.300 align:middle line:84% And why I have different branches-- 00:28:19.300 --> 00:28:23.350 align:middle line:84% I have different branches because the vibration-- 00:28:23.350 --> 00:28:26.680 align:middle line:84% the sound wave you receive is longitudinal. 00:28:26.680 --> 00:28:29.890 align:middle line:90% Here, is longitudinal wave only. 00:28:29.890 --> 00:28:33.730 align:middle line:84% But in solid, you can have longitudinal as well as 00:28:33.730 --> 00:28:35.560 align:middle line:90% transverse. 00:28:35.560 --> 00:28:38.170 align:middle line:90% You can sustain the shear. 00:28:38.170 --> 00:28:44.620 align:middle line:84% So I have a longitudinal wave, transverse wave. 00:28:44.620 --> 00:28:46.690 align:middle line:84% And really, transverse-- if you look 00:28:46.690 --> 00:28:48.640 align:middle line:84% at the electromagnetic wave, I said 00:28:48.640 --> 00:28:51.980 align:middle line:90% there are two polarizations. 00:28:51.980 --> 00:28:55.370 align:middle line:84% See, acoustic wave transfers also two. 00:28:55.370 --> 00:29:00.920 align:middle line:84% But in highly symmetric direction, like from gamma to L, 00:29:00.920 --> 00:29:03.210 align:middle line:84% in this case, this is [INAUDIBLE] symmetric. 00:29:03.210 --> 00:29:07.250 align:middle line:84% So the two directions-- this direction and this direction-- 00:29:07.250 --> 00:29:08.570 align:middle line:90% the same wave vector-- 00:29:08.570 --> 00:29:10.810 align:middle line:90% they have same frequency. 00:29:10.810 --> 00:29:14.710 align:middle line:84% So they become degenerate just for [INAUDIBLE] one. 00:29:14.710 --> 00:29:16.460 align:middle line:90% But in some other direction. 00:29:16.460 --> 00:29:17.810 align:middle line:90% It's not highly symmetric. 00:29:17.810 --> 00:29:21.550 align:middle line:84% You can say there are three polarizations-- 00:29:21.550 --> 00:29:25.250 align:middle line:84% longitudinal, one longitudinal, two latitudinal. 00:29:25.250 --> 00:29:31.260 align:middle line:84% So acoustic branches, optical the branches. 00:29:31.260 --> 00:29:37.470 align:middle line:84% That's the station from 1D to 3D. 00:29:37.470 --> 00:29:41.300 align:middle line:84% AUDIENCE: What is the spacing between the different wavelength 00:29:41.300 --> 00:29:41.900 align:middle line:90% vectors? 00:29:41.900 --> 00:29:43.590 align:middle line:84% GANG CHEN: Different wave vector-- 00:29:43.590 --> 00:29:46.630 align:middle line:84% the spacing-- those are experiments. 00:29:46.630 --> 00:29:50.730 align:middle line:84% The data is comparing experiment. 00:29:50.730 --> 00:29:55.140 align:middle line:84% AUDIENCE: The frequency should be in hertz or [INAUDIBLE] 00:29:55.140 --> 00:29:57.240 align:middle line:90% frequency is in the hertz? 00:29:57.240 --> 00:29:58.920 align:middle line:90% GANG CHEN: Frequency is-- 00:29:58.920 --> 00:30:04.230 align:middle line:84% if it's omega-- so this is another thing that, very often, 00:30:04.230 --> 00:30:07.680 align:middle line:84% people get really confused because it depends 00:30:07.680 --> 00:30:09.700 align:middle line:90% on who likes to plot out what. 00:30:09.700 --> 00:30:15.820 align:middle line:90% 00:30:15.820 --> 00:30:20.220 align:middle line:90% And sometimes-- here is k-- 00:30:20.220 --> 00:30:21.270 align:middle line:90% omega. 00:30:21.270 --> 00:30:24.510 align:middle line:84% And sometimes you'll see this one is not [INAUDIBLE] 00:30:24.510 --> 00:30:25.770 align:middle line:90% terms of the frequency. 00:30:25.770 --> 00:30:28.690 align:middle line:90% This is centimeter inverse. 00:30:28.690 --> 00:30:31.770 align:middle line:84% Well, that's not the frequency unit. 00:30:31.770 --> 00:30:34.530 align:middle line:90% So what they mean? 00:30:34.530 --> 00:30:39.150 align:middle line:84% And in this case, if you want to look at-- the first frequency 00:30:39.150 --> 00:30:43.260 align:middle line:90% is 2 pi mil angular frequency. 00:30:43.260 --> 00:30:46.470 align:middle line:84% You have to say what exactly is a better plot, the angular 00:30:46.470 --> 00:30:48.270 align:middle line:90% frequency or frequency? 00:30:48.270 --> 00:30:54.510 align:middle line:84% And mil is-- if we practice wave vector inverse, 00:30:54.510 --> 00:30:56.765 align:middle line:90% we plotted 1 over lambda. 00:30:56.765 --> 00:31:00.010 align:middle line:90% 00:31:00.010 --> 00:31:05.500 align:middle line:84% So you have to have to convert the frequency equals to c. 00:31:05.500 --> 00:31:06.250 align:middle line:90% What about lambda? 00:31:06.250 --> 00:31:10.460 align:middle line:90% 00:31:10.460 --> 00:31:12.810 align:middle line:90% So people have different habits. 00:31:12.810 --> 00:31:13.310 align:middle line:90% Yes. 00:31:13.310 --> 00:31:15.546 align:middle line:84% AUDIENCE: This is speed of sound or speed of light? 00:31:15.546 --> 00:31:16.400 align:middle line:90% GANG CHEN: Huh? 00:31:16.400 --> 00:31:18.470 align:middle line:84% AUDIENCE: Speed of sound or speed of light? 00:31:18.470 --> 00:31:21.042 align:middle line:84% GANG CHEN: This is the speed of sound. 00:31:21.042 --> 00:31:22.020 align:middle line:90% Oh, I'm sorry. 00:31:22.020 --> 00:31:22.520 align:middle line:90% I'm sorry. 00:31:22.520 --> 00:31:23.430 align:middle line:90% Speed of light. 00:31:23.430 --> 00:31:24.110 align:middle line:90% AUDIENCE: OK. 00:31:24.110 --> 00:31:24.950 align:middle line:90% GANG CHEN: Yeah. 00:31:24.950 --> 00:31:26.360 align:middle line:90% Not speed of sound. 00:31:26.360 --> 00:31:28.520 align:middle line:84% When they do this kind of expression, 00:31:28.520 --> 00:31:30.530 align:middle line:84% they always go the speed of light. 00:31:30.530 --> 00:31:33.710 align:middle line:90% 00:31:33.710 --> 00:31:38.130 align:middle line:84% Speed of sound is different for different materials. 00:31:38.130 --> 00:31:46.050 align:middle line:84% So you're asking, what are the wave vector numbers? 00:31:46.050 --> 00:31:51.240 align:middle line:84% So this is where the number of waves-- 00:31:51.240 --> 00:31:53.470 align:middle line:90% when I go from here to here-- 00:31:53.470 --> 00:31:56.700 align:middle line:84% well, this is called from pi or a minus pi 00:31:56.700 --> 00:31:58.140 align:middle line:90% over a to plus pi over a. 00:31:58.140 --> 00:32:00.390 align:middle line:84% That's called the first [INAUDIBLE]. 00:32:00.390 --> 00:32:03.180 align:middle line:84% And the number of wave vectors, k, 00:32:03.180 --> 00:32:06.910 align:middle line:84% equals the number of atoms you have. 00:32:06.910 --> 00:32:10.000 align:middle line:90% 00:32:10.000 --> 00:32:13.380 align:middle line:84% So these points-- from here to here-- 00:32:13.380 --> 00:32:14.530 align:middle line:90% the number is n. 00:32:14.530 --> 00:32:18.510 align:middle line:90% 00:32:18.510 --> 00:32:21.390 align:middle line:90% That's actually important later. 00:32:21.390 --> 00:32:25.170 align:middle line:84% This number is more important when you do counting. 00:32:25.170 --> 00:32:29.960 align:middle line:84% Once you go to microscopic picture, you do the counting. 00:32:29.960 --> 00:32:33.060 align:middle line:84% AUDIENCE: In silicon dispersion relation, 00:32:33.060 --> 00:32:39.110 align:middle line:84% we have a mode that seemingly correspond to the optical mode 00:32:39.110 --> 00:32:40.490 align:middle line:90% in gallium arsenide. 00:32:40.490 --> 00:32:43.130 align:middle line:84% Then what will be-- the mode will 00:32:43.130 --> 00:32:47.035 align:middle line:84% look like for the silicon at the top of it? 00:32:47.035 --> 00:32:49.880 align:middle line:84% GANG CHEN: Oh, they'll be optical modes. 00:32:49.880 --> 00:32:57.420 align:middle line:84% Well, it turns out silicon is a one lattice point. 00:32:57.420 --> 00:32:59.700 align:middle line:90% How many atoms? 00:32:59.700 --> 00:33:03.980 align:middle line:90% And the silicon is-- 00:33:03.980 --> 00:33:07.520 align:middle line:84% in one lattice point, you have two atoms. 00:33:07.520 --> 00:33:09.680 align:middle line:84% So there is another atom-- see that 00:33:09.680 --> 00:33:11.630 align:middle line:84% the one quarter, one quarter, one quarter. 00:33:11.630 --> 00:33:15.530 align:middle line:84% If you think this is one, there is another-- this is a lattice-- 00:33:15.530 --> 00:33:16.640 align:middle line:90% this is the point-- 00:33:16.640 --> 00:33:17.720 align:middle line:90% a lattice point. 00:33:17.720 --> 00:33:20.708 align:middle line:84% So there is another atom at the one quarter, 00:33:20.708 --> 00:33:21.750 align:middle line:90% one quarter, one quarter. 00:33:21.750 --> 00:33:25.600 align:middle line:84% And it's not-- it's an FCC, but there's two atoms. 00:33:25.600 --> 00:33:27.770 align:middle line:90% If there are two atoms, that-- 00:33:27.770 --> 00:33:31.590 align:middle line:84% say, in the lattice point, if there are two atoms, 00:33:31.590 --> 00:33:35.550 align:middle line:84% then there's three acoustic branch-- 00:33:35.550 --> 00:33:37.500 align:middle line:90% three acoustic branch. 00:33:37.500 --> 00:33:40.245 align:middle line:90% And then there are another-- 00:33:40.245 --> 00:33:44.160 align:middle line:84% so two atoms, n minus 1-- so times 3. 00:33:44.160 --> 00:33:46.920 align:middle line:84% So there are three optical branch. 00:33:46.920 --> 00:33:51.030 align:middle line:84% If you have three atoms, you have three acoustic [INAUDIBLE] 00:33:51.030 --> 00:33:54.870 align:middle line:84% branch, then you have six optical branch. 00:33:54.870 --> 00:33:59.190 align:middle line:90% So the total is 3 times n. 00:33:59.190 --> 00:34:00.120 align:middle line:90% That's n. 00:34:00.120 --> 00:34:04.650 align:middle line:84% At every lattice point on an atom [INAUDIBLE]. 00:34:04.650 --> 00:34:10.139 align:middle line:84% AUDIENCE: What's the direction of the two transverse wave? 00:34:10.139 --> 00:34:12.610 align:middle line:84% GANG CHEN: Roughly, it's perpendicular. 00:34:12.610 --> 00:34:15.270 align:middle line:84% But when you actually do it, you have 00:34:15.270 --> 00:34:19.469 align:middle line:90% to solve the dynamic matrix. 00:34:19.469 --> 00:34:21.360 align:middle line:84% You have to solve the matrix to find out 00:34:21.360 --> 00:34:26.550 align:middle line:84% the exact direction of the vibration, so the eigenvectors. 00:34:26.550 --> 00:34:29.400 align:middle line:90% You can always decompose. 00:34:29.400 --> 00:34:31.469 align:middle line:84% But there is an eigenvector that you 00:34:31.469 --> 00:34:34.600 align:middle line:84% should go to solve the equation of motion. 00:34:34.600 --> 00:34:37.530 align:middle line:84% You can solve the Newtonian equation of motion 00:34:37.530 --> 00:34:39.190 align:middle line:90% to get those eigenvectors. 00:34:39.190 --> 00:34:42.710 align:middle line:90% 00:34:42.710 --> 00:34:44.780 align:middle line:90% OK. 00:34:44.780 --> 00:34:49.239 align:middle line:84% That's all about crystal and vibration. 00:34:49.239 --> 00:34:55.480 align:middle line:90% And now let's see the electrons. 00:34:55.480 --> 00:34:59.650 align:middle line:84% I started with crystal vibration by hydrogen molecule. 00:34:59.650 --> 00:35:06.310 align:middle line:84% I have the two atoms and the separation spreads, [INAUDIBLE]. 00:35:06.310 --> 00:35:10.360 align:middle line:84% And when talking about electron, let's also start with hydrogen. 00:35:10.360 --> 00:35:14.664 align:middle line:84% This time, I'm starting the hydrogen atom. 00:35:14.664 --> 00:35:16.920 align:middle line:90% And the hydrogen atom-- 00:35:16.920 --> 00:35:18.940 align:middle line:84% again, you know quantum mechanics-- 00:35:18.940 --> 00:35:23.780 align:middle line:84% this is the energy levels of a hydrogen atom-- 00:35:23.780 --> 00:35:27.180 align:middle line:84% electrons, in this case, I'm talking. 00:35:27.180 --> 00:35:31.190 align:middle line:84% And the electron-- the energy level-- 00:35:31.190 --> 00:35:34.210 align:middle line:90% 00:35:34.210 --> 00:35:35.600 align:middle line:90% I didn't even write it down. 00:35:35.600 --> 00:35:43.635 align:middle line:90% 00:35:43.635 --> 00:35:45.460 align:middle line:90% You can see I wrote-- 00:35:45.460 --> 00:35:49.130 align:middle line:90% 00:35:49.130 --> 00:35:56.370 align:middle line:84% so E of the electron is dependent on the n-- 00:35:56.370 --> 00:36:01.205 align:middle line:84% is minus 13.6 electron volts n square. 00:36:01.205 --> 00:36:05.820 align:middle line:90% 00:36:05.820 --> 00:36:10.445 align:middle line:84% So what is the one-- so let me say-- 00:36:10.445 --> 00:36:15.000 align:middle line:84% if my electron volt is one electron mode in-- 00:36:15.000 --> 00:36:19.170 align:middle line:84% one charge of electron in one mode and one voltage-- 00:36:19.170 --> 00:36:21.150 align:middle line:90% for electric field, one voltage. 00:36:21.150 --> 00:36:23.730 align:middle line:84% And that's the energy of the electron. 00:36:23.730 --> 00:36:30.150 align:middle line:84% And this is the inverse, 1.6 to the minus 19 charge. 00:36:30.150 --> 00:36:41.070 align:middle line:84% So that's the energy of the electrons in the hydrogen atom. 00:36:41.070 --> 00:36:47.180 align:middle line:84% And its electrons are moving in three dimensions. 00:36:47.180 --> 00:36:52.470 align:middle line:84% And so when I solve the Schrödinger equation, 00:36:52.470 --> 00:36:53.820 align:middle line:90% I'll find the wave function. 00:36:53.820 --> 00:36:56.350 align:middle line:84% Again, this is going in quantum mechanics. 00:36:56.350 --> 00:36:58.540 align:middle line:84% We're not going to do any quantum-- 00:36:58.540 --> 00:37:02.760 align:middle line:84% I'm just introducing the basic concept. 00:37:02.760 --> 00:37:06.450 align:middle line:84% And it turns out because we have a three dimension, 00:37:06.450 --> 00:37:09.700 align:middle line:84% r theta phi is in three-dimensional space. 00:37:09.700 --> 00:37:15.800 align:middle line:84% So there is three quantum numbers, n, l, m. 00:37:15.800 --> 00:37:18.230 align:middle line:84% Those are all results of quantum mechanics, 00:37:18.230 --> 00:37:21.710 align:middle line:84% except that in s, you have to go through even 00:37:21.710 --> 00:37:24.200 align:middle line:90% relativistic quantum mechanics. 00:37:24.200 --> 00:37:29.200 align:middle line:84% And those n-- you can see now-- all integers. 00:37:29.200 --> 00:37:33.320 align:middle line:84% The quantum number-- n is [INAUDIBLE] 1 to 3. 00:37:33.320 --> 00:37:34.970 align:middle line:90% All those integers-- 00:37:34.970 --> 00:37:36.815 align:middle line:90% L must be less than n. 00:37:36.815 --> 00:37:41.450 align:middle line:84% m must be less than L. This is all mathematics-- coming 00:37:41.450 --> 00:37:43.610 align:middle line:90% from mathematics. 00:37:43.610 --> 00:37:47.660 align:middle line:84% But what it really means is I wrote down that-- 00:37:47.660 --> 00:37:49.280 align:middle line:90% I forgot to write the energy. 00:37:49.280 --> 00:37:52.400 align:middle line:84% Energy is minus 13.6 electron volts 00:37:52.400 --> 00:37:58.560 align:middle line:84% divided by n squared, so energy doesn't depend on L and n. 00:37:58.560 --> 00:38:01.740 align:middle line:90% It depends only on n. 00:38:01.740 --> 00:38:06.105 align:middle line:90% So when n equals 1, I have-- 00:38:06.105 --> 00:38:08.610 align:middle line:90% 00:38:08.610 --> 00:38:10.290 align:middle line:90% L has-- you look at it. 00:38:10.290 --> 00:38:14.540 align:middle line:90% L has to be 0 L has to be 0. 00:38:14.540 --> 00:38:17.870 align:middle line:84% And the S could be plus or minus 1/2. 00:38:17.870 --> 00:38:19.910 align:middle line:90% So there are two quantum states. 00:38:19.910 --> 00:38:22.422 align:middle line:84% So here, I have two quantum states. 00:38:22.422 --> 00:38:24.380 align:middle line:84% So if you look at the theoretical table, that's 00:38:24.380 --> 00:38:28.430 align:middle line:84% why you have hydrogen, helium, failed. 00:38:28.430 --> 00:38:29.960 align:middle line:90% Helium is stabled. 00:38:29.960 --> 00:38:31.730 align:middle line:90% Two failed. 00:38:31.730 --> 00:38:36.140 align:middle line:84% And the next quantum level, you go to a plus 2. 00:38:36.140 --> 00:38:40.950 align:middle line:84% It's pretty far away from the first level. 00:38:40.950 --> 00:38:43.160 align:middle line:84% So you look at the periodic table again. 00:38:43.160 --> 00:38:43.880 align:middle line:90% You fail. 00:38:43.880 --> 00:38:46.800 align:middle line:90% This is the one quantum state. 00:38:46.800 --> 00:38:53.290 align:middle line:84% One quantum state can only have one electron, [INAUDIBLE]. 00:38:53.290 --> 00:38:58.180 align:middle line:84% That's the one I've seen as a different state. 00:38:58.180 --> 00:39:04.510 align:middle line:90% So hydrogen, helium, lithium. 00:39:04.510 --> 00:39:10.240 align:middle line:90% Lithium is [INAUDIBLE] 2s now. 00:39:10.240 --> 00:39:13.480 align:middle line:84% So when you go to continue to look at the periodic table, 00:39:13.480 --> 00:39:14.350 align:middle line:90% you fill in. 00:39:14.350 --> 00:39:17.890 align:middle line:90% Here, each energy level is 2n. 00:39:17.890 --> 00:39:21.070 align:middle line:90% 00:39:21.070 --> 00:39:25.090 align:middle line:84% Once you go to n equals 2, the number of-- total number 00:39:25.090 --> 00:39:31.360 align:middle line:84% of electrons you can have, up to filled all the orbital-- 00:39:31.360 --> 00:39:34.820 align:middle line:84% the quantum state is n plus 2 is what? 00:39:34.820 --> 00:39:39.260 align:middle line:90% 00:39:39.260 --> 00:39:40.530 align:middle line:90% 8, right? 00:39:40.530 --> 00:39:41.030 align:middle line:90% 11. 00:39:41.030 --> 00:39:44.700 align:middle line:90% 00:39:44.700 --> 00:39:47.440 align:middle line:90% So that's the hydrogen atom. 00:39:47.440 --> 00:39:50.430 align:middle line:84% And this is a basic model for the periodic table. 00:39:50.430 --> 00:39:51.810 align:middle line:90% Look at the periodic table. 00:39:51.810 --> 00:39:55.980 align:middle line:84% You can use this and pretty much interpret 00:39:55.980 --> 00:40:01.050 align:middle line:84% a lot of those first 30 atoms of [INAUDIBLE]. 00:40:01.050 --> 00:40:05.120 align:middle line:90% Now, I'm going to, again, say-- 00:40:05.120 --> 00:40:10.130 align:middle line:84% starting from here, I'm going into a [INAUDIBLE] chain 00:40:10.130 --> 00:40:14.590 align:middle line:84% because I want to understand electrons in solid. 00:40:14.590 --> 00:40:18.510 align:middle line:90% 00:40:18.510 --> 00:40:20.916 align:middle line:90% So a [INAUDIBLE] chain-- 00:40:20.916 --> 00:40:26.250 align:middle line:84% I'm thinking about hydrogen as a simple example, 00:40:26.250 --> 00:40:28.500 align:middle line:84% but the hydrogen [INAUDIBLE] gas, 00:40:28.500 --> 00:40:31.200 align:middle line:84% and then thinking about the solid. 00:40:31.200 --> 00:40:38.920 align:middle line:84% And so electrons-- they are waves in quantum mechanics. 00:40:38.920 --> 00:40:42.990 align:middle line:84% So both waves and particle characteristics. 00:40:42.990 --> 00:40:47.280 align:middle line:84% So waves-- each electron has those wave function as part 00:40:47.280 --> 00:40:47.970 align:middle line:90% of [INAUDIBLE]. 00:40:47.970 --> 00:40:50.550 align:middle line:90% This is the wave function. 00:40:50.550 --> 00:40:53.580 align:middle line:84% Quantum mechanics will give you this function. 00:40:53.580 --> 00:40:58.210 align:middle line:90% And now they are waves. 00:40:58.210 --> 00:41:03.550 align:middle line:84% So let's plot a plus 1, approximately. 00:41:03.550 --> 00:41:05.560 align:middle line:90% And they spread out-- 00:41:05.560 --> 00:41:10.270 align:middle line:84% not just confined to one electron. 00:41:10.270 --> 00:41:15.160 align:middle line:84% And now, this is the next atom, a plus 1. 00:41:15.160 --> 00:41:18.930 align:middle line:84% They start to overlap with each other. 00:41:18.930 --> 00:41:21.980 align:middle line:90% What happens? 00:41:21.980 --> 00:41:25.640 align:middle line:84% And these two electrons from the two atoms 00:41:25.640 --> 00:41:28.400 align:middle line:84% both will have the same states because once 00:41:28.400 --> 00:41:29.960 align:middle line:90% they start to overlap-- 00:41:29.960 --> 00:41:34.130 align:middle line:84% once they start overlapping, the state of the two atoms 00:41:34.130 --> 00:41:36.500 align:middle line:90% are the identical states. 00:41:36.500 --> 00:41:39.270 align:middle line:84% And the basic quantum mechanics doesn't allow it. 00:41:39.270 --> 00:41:41.891 align:middle line:90% 00:41:41.891 --> 00:41:48.570 align:middle line:84% So the state will start to adjust a little bit 00:41:48.570 --> 00:41:52.350 align:middle line:90% and form back. 00:41:52.350 --> 00:41:55.710 align:middle line:90% So now n equals 1-- 00:41:55.710 --> 00:41:59.320 align:middle line:84% this is originally n equals 2, 3, 4. 00:41:59.320 --> 00:42:02.910 align:middle line:84% Think about-- they started forming bands. 00:42:02.910 --> 00:42:08.280 align:middle line:84% So there are-- if you have n atoms, again, in the chain, 00:42:08.280 --> 00:42:12.560 align:middle line:84% between here and here, I have an n point. 00:42:12.560 --> 00:42:15.920 align:middle line:84% And each point now-- they start to-- 00:42:15.920 --> 00:42:21.230 align:middle line:84% you can see here [INAUDIBLE] different and those same 00:42:21.230 --> 00:42:22.650 align:middle line:90% allowable weight vector-- 00:42:22.650 --> 00:42:26.125 align:middle line:84% now electron is a wave, extended through the whole lattice. 00:42:26.125 --> 00:42:29.580 align:middle line:90% 00:42:29.580 --> 00:42:31.800 align:middle line:84% This is one quantum mechanical state. 00:42:31.800 --> 00:42:35.610 align:middle line:84% So the waves can have different vectors, different period, 00:42:35.610 --> 00:42:42.960 align:middle line:84% same as we argued for the lattice. 00:42:42.960 --> 00:42:49.700 align:middle line:84% So that's-- a basic quantum mechanics dictate. 00:42:49.700 --> 00:42:56.130 align:middle line:84% The energy has to split a little bit, forming, now, a band. 00:42:56.130 --> 00:42:58.830 align:middle line:84% It's quasi continuous with [INAUDIBLE] continuous 00:42:58.830 --> 00:43:00.660 align:middle line:90% because that k-- 00:43:00.660 --> 00:43:03.150 align:middle line:90% there are many k points. 00:43:03.150 --> 00:43:06.180 align:middle line:90% But they still return here. 00:43:06.180 --> 00:43:10.770 align:middle line:84% So here, the endpoint depends on the n for your number 00:43:10.770 --> 00:43:13.760 align:middle line:90% of atoms in a chain. 00:43:13.760 --> 00:43:18.490 align:middle line:90% This is very interesting. 00:43:18.490 --> 00:43:20.860 align:middle line:84% because even this simple picture-- 00:43:20.860 --> 00:43:25.000 align:middle line:84% I said the hydrogen picture tells you the periodic table. 00:43:25.000 --> 00:43:27.070 align:middle line:84% What this picture tells you-- whether you 00:43:27.070 --> 00:43:30.630 align:middle line:90% have an insulator or conductor. 00:43:30.630 --> 00:43:33.760 align:middle line:90% Let's think about-- again. 00:43:33.760 --> 00:43:39.480 align:middle line:84% If I think of hydrogen as a simple extreme, 00:43:39.480 --> 00:43:45.240 align:middle line:84% each hydrogen has one electron, So when the electron field 00:43:45.240 --> 00:43:51.960 align:middle line:84% go to the energy, the lowest energy level fails first. 00:43:51.960 --> 00:43:52.505 align:middle line:90% They're lazy. 00:43:52.505 --> 00:43:55.510 align:middle line:90% 00:43:55.510 --> 00:44:00.280 align:middle line:90% Now, thermodynamics tells you-- 00:44:00.280 --> 00:44:04.570 align:middle line:84% everybody wants to go to minimum energy space. 00:44:04.570 --> 00:44:12.110 align:middle line:84% So if I have n atoms, I have-- between here and here-- 00:44:12.110 --> 00:44:13.970 align:middle line:90% n point. 00:44:13.970 --> 00:44:18.070 align:middle line:84% Each point has a spin [INAUDIBLE]. 00:44:18.070 --> 00:44:21.590 align:middle line:84% The spin is not reflected in this curve. 00:44:21.590 --> 00:44:25.170 align:middle line:90% 00:44:25.170 --> 00:44:27.230 align:middle line:90% So each point has two states-- 00:44:27.230 --> 00:44:31.580 align:middle line:84% to spins, s equals plus minus half. 00:44:31.580 --> 00:44:34.710 align:middle line:84% So now I start to fill the bottom electrons-- 00:44:34.710 --> 00:44:39.930 align:middle line:84% if I have hydrogen, There are really two end quantum states. 00:44:39.930 --> 00:44:41.270 align:middle line:90% I have only an electron. 00:44:41.270 --> 00:44:44.220 align:middle line:90% Where it goes? 00:44:44.220 --> 00:44:48.100 align:middle line:90% It fills about half, right? 00:44:48.100 --> 00:44:51.580 align:middle line:84% So the electron maximum [INAUDIBLE] energy 00:44:51.580 --> 00:44:55.100 align:middle line:84% will be somewhere in between the band. 00:44:55.100 --> 00:44:58.448 align:middle line:90% 00:44:58.448 --> 00:45:00.360 align:middle line:90% So this is where-- 00:45:00.360 --> 00:45:03.600 align:middle line:84% it depends on [INAUDIBLE] field here 00:45:03.600 --> 00:45:11.350 align:middle line:84% depends on how many electrons you can fill to within the band. 00:45:11.350 --> 00:45:15.190 align:middle line:90% And it's either 0 temperature-- 00:45:15.190 --> 00:45:18.790 align:middle line:90% 0 Kelvin, this point-- 00:45:18.790 --> 00:45:24.230 align:middle line:84% where [INAUDIBLE] just think about the water-filled bucket. 00:45:24.230 --> 00:45:30.620 align:middle line:84% The [INAUDIBLE] energy level is [INAUDIBLE]. 00:45:30.620 --> 00:45:36.560 align:middle line:84% And so if it's a non-zero temperature, 00:45:36.560 --> 00:45:39.150 align:middle line:90% those electrons do not move. 00:45:39.150 --> 00:45:43.170 align:middle line:84% They need a lot of energy to move because there is no quantum 00:45:43.170 --> 00:45:46.560 align:middle line:84% state in this, here, quantum mechanical-- 00:45:46.560 --> 00:45:49.320 align:middle line:84% no state allows energy to move to-- 00:45:49.320 --> 00:45:51.570 align:middle line:84% when they move, they change state. 00:45:51.570 --> 00:45:55.260 align:middle line:84% The state has to be adjacent, allowable in the solvent. 00:45:55.260 --> 00:45:57.815 align:middle line:84% So only electrons in this region are movable. 00:45:57.815 --> 00:46:01.350 align:middle line:90% 00:46:01.350 --> 00:46:06.540 align:middle line:84% And in metal-- because the next state-- 00:46:06.540 --> 00:46:09.030 align:middle line:90% just close to this point-- 00:46:09.030 --> 00:46:09.980 align:middle line:90% they're very close. 00:46:09.980 --> 00:46:12.250 align:middle line:84% The energy difference is very close. 00:46:12.250 --> 00:46:15.690 align:middle line:84% So the thermal energy is large enough 00:46:15.690 --> 00:46:19.180 align:middle line:84% to allow them to go to the next state. 00:46:19.180 --> 00:46:24.860 align:middle line:84% So only when your electrons fill to level their empty state 00:46:24.860 --> 00:46:29.440 align:middle line:90% in the band it's a conductor. 00:46:29.440 --> 00:46:30.280 align:middle line:90% It's [INAUDIBLE]. 00:46:30.280 --> 00:46:33.400 align:middle line:90% That's [INAUDIBLE]. 00:46:33.400 --> 00:46:39.330 align:middle line:84% And what happens if my number of electron filled here? 00:46:39.330 --> 00:46:43.510 align:middle line:90% 00:46:43.510 --> 00:46:45.750 align:middle line:90% Then it needs a lot of energy. 00:46:45.750 --> 00:46:52.220 align:middle line:84% Depends on the difference of the energy here and here. 00:46:52.220 --> 00:46:58.220 align:middle line:84% If this difference is very large, you've got the insulator. 00:46:58.220 --> 00:47:01.960 align:middle line:90% Electron cannot move. 00:47:01.960 --> 00:47:14.070 align:middle line:84% So this-- from the [INAUDIBLE] of electrons to the empty 00:47:14.070 --> 00:47:15.060 align:middle line:90% state-- 00:47:15.060 --> 00:47:16.300 align:middle line:90% that's the band gap. 00:47:16.300 --> 00:47:20.415 align:middle line:90% 00:47:20.415 --> 00:47:23.750 align:middle line:84% And if this band gap is very large, that's an insulator. 00:47:23.750 --> 00:47:24.960 align:middle line:90% How large is large? 00:47:24.960 --> 00:47:27.500 align:middle line:90% A few electron volts. 00:47:27.500 --> 00:47:31.800 align:middle line:84% And this is because at room temperature, 00:47:31.800 --> 00:47:37.950 align:middle line:84% we said that, in the first class, the energy is about kdt. 00:47:37.950 --> 00:47:40.560 align:middle line:84% k is the Boltzmann constant times temperature, 00:47:40.560 --> 00:47:42.670 align:middle line:90% thermal energy. 00:47:42.670 --> 00:47:45.180 align:middle line:84% And that's about 26 milli-electron volts. 00:47:45.180 --> 00:47:48.370 align:middle line:90% 00:47:48.370 --> 00:47:52.930 align:middle line:84% And so if this one is a few electron 00:47:52.930 --> 00:47:56.470 align:middle line:90% volts, much larger than kdt. 00:47:56.470 --> 00:47:59.590 align:middle line:84% So very, very few chance electron 00:47:59.590 --> 00:48:03.190 align:middle line:84% can come over here due to thermal. 00:48:03.190 --> 00:48:05.920 align:middle line:90% And that's insulator. 00:48:05.920 --> 00:48:08.150 align:middle line:84% But then for this gap, it's not large. 00:48:08.150 --> 00:48:13.060 align:middle line:84% Plus, for example, this is an insulator. 00:48:13.060 --> 00:48:18.400 align:middle line:84% And the gap is a view of about 4 to 5 electron volts. 00:48:18.400 --> 00:48:24.200 align:middle line:84% Now, silicon-- that gap is about 1.12 electron volts. 00:48:24.200 --> 00:48:30.460 align:middle line:84% [INAUDIBLE] So silicon becomes a semiconductor 00:48:30.460 --> 00:48:34.000 align:middle line:84% because this gap is not very large. 00:48:34.000 --> 00:48:36.400 align:middle line:84% And if you cool silicon to 0 Kelvin, 00:48:36.400 --> 00:48:42.320 align:middle line:84% it should be an insulator because there's not 00:48:42.320 --> 00:48:48.080 align:middle line:84% enough thermal energy to kick the electrons from here to here. 00:48:48.080 --> 00:48:53.570 align:middle line:84% But at room temperature, the thermal energy, kdt-- 00:48:53.570 --> 00:48:55.430 align:middle line:84% because of the thermal energy some electrons 00:48:55.430 --> 00:49:00.130 align:middle line:90% are kicked from here to there. 00:49:00.130 --> 00:49:05.650 align:middle line:84% And if they are kicked from here to here, what's left behind 00:49:05.650 --> 00:49:07.690 align:middle line:90% are some empty states. 00:49:07.690 --> 00:49:11.590 align:middle line:84% Now, those electrons-- those other electrons are happy now. 00:49:11.590 --> 00:49:16.380 align:middle line:84% There are some states they can move. 00:49:16.380 --> 00:49:19.590 align:middle line:84% And these outer electrons that kicked up 00:49:19.590 --> 00:49:23.812 align:middle line:84% is also happy because there are also empty states they can move. 00:49:23.812 --> 00:49:26.770 align:middle line:90% 00:49:26.770 --> 00:49:30.340 align:middle line:84% So the motion of those electrons-- 00:49:30.340 --> 00:49:34.600 align:middle line:84% those electrons-- it's much easier. 00:49:34.600 --> 00:49:36.040 align:middle line:90% So you can think of this-- 00:49:36.040 --> 00:49:39.060 align:middle line:84% the thing that's really moving is the electron moving 00:49:39.060 --> 00:49:41.040 align:middle line:90% because there are empty states. 00:49:41.040 --> 00:49:45.910 align:middle line:84% But it's much easier to think them as positive charge. 00:49:45.910 --> 00:49:47.700 align:middle line:84% So don't think those are electrons. 00:49:47.700 --> 00:49:50.520 align:middle line:84% Just think of those empty states as holes. 00:49:50.520 --> 00:49:52.950 align:middle line:90% There's no hole. 00:49:52.950 --> 00:49:56.110 align:middle line:90% No positive charge. 00:49:56.110 --> 00:49:58.650 align:middle line:90% It's electron that's mobile. 00:49:58.650 --> 00:50:03.970 align:middle line:84% But we just think this as a whole positive charge, 00:50:03.970 --> 00:50:06.980 align:middle line:90% and those are electrons. 00:50:06.980 --> 00:50:08.720 align:middle line:90% And you can even put-- 00:50:08.720 --> 00:50:12.170 align:middle line:84% so that's an intrinsic silicon [INAUDIBLE], right? 00:50:12.170 --> 00:50:16.970 align:middle line:84% But you could even put the impurities into silicon, 00:50:16.970 --> 00:50:23.240 align:middle line:84% into semiconductors to add more electrons or add more holes. 00:50:23.240 --> 00:50:28.190 align:middle line:84% So when I add-- for example, if I put a phosphor in silicon, 00:50:28.190 --> 00:50:30.740 align:middle line:90% phosphor has 5 charge. 00:50:30.740 --> 00:50:33.410 align:middle line:90% Silicon is 4 in the outer shell. 00:50:33.410 --> 00:50:36.410 align:middle line:84% So there's one extra pair of silicon. 00:50:36.410 --> 00:50:39.380 align:middle line:84% So that extra-- because it's very close. 00:50:39.380 --> 00:50:42.210 align:middle line:84% The energy level of phosphor very close to here. 00:50:42.210 --> 00:50:47.300 align:middle line:84% So that electron can easily be excited to this region. 00:50:47.300 --> 00:50:52.060 align:middle line:90% So it's an n-type semiconductor. 00:50:52.060 --> 00:50:59.220 align:middle line:84% And on the other hand, if I put boron into silicon, 00:50:59.220 --> 00:51:02.435 align:middle line:90% the outer has three. 00:51:02.435 --> 00:51:08.430 align:middle line:84% So the born energy level is somewhere here, 00:51:08.430 --> 00:51:12.818 align:middle line:84% close to the bottom of this band. 00:51:12.818 --> 00:51:13.860 align:middle line:90% This is the valence band. 00:51:13.860 --> 00:51:15.210 align:middle line:90% This is the conduction band. 00:51:15.210 --> 00:51:17.530 align:middle line:90% So the energy levels here-- 00:51:17.530 --> 00:51:21.070 align:middle line:84% those electrons are much happier to go here, 00:51:21.070 --> 00:51:24.550 align:middle line:90% so left behind holes. 00:51:24.550 --> 00:51:29.980 align:middle line:84% So boron becomes an acceptor, entrap the electron. 00:51:29.980 --> 00:51:34.480 align:middle line:84% And they create the holes in those bands, 00:51:34.480 --> 00:51:36.300 align:middle line:90% anticipating in these bands-- 00:51:36.300 --> 00:51:38.680 align:middle line:90% that the electrons [INAUDIBLE]. 00:51:38.680 --> 00:51:43.780 align:middle line:84% So that's how you do doping in semiconductor. 00:51:43.780 --> 00:51:45.250 align:middle line:90% You create the energy states. 00:51:45.250 --> 00:51:48.280 align:middle line:84% Sometimes even impurity-- even frequency 00:51:48.280 --> 00:51:50.420 align:middle line:84% could create those additional states. 00:51:50.420 --> 00:51:53.150 align:middle line:90% 00:51:53.150 --> 00:51:56.230 align:middle line:84% In silicon, we do artificial doping, 00:51:56.230 --> 00:51:59.680 align:middle line:84% when we do thermoelectric material's weakened state, 00:51:59.680 --> 00:52:03.190 align:middle line:84% a missing atom could create those kind of situation. 00:52:03.190 --> 00:52:08.200 align:middle line:84% So the doping in material-- in thermoelectrics, doping 00:52:08.200 --> 00:52:09.400 align:middle line:90% is very important. 00:52:09.400 --> 00:52:15.560 align:middle line:84% Of course, this is true for all semiconductor device. 00:52:15.560 --> 00:52:17.877 align:middle line:90% Any questions? 00:52:17.877 --> 00:52:22.330 align:middle line:90% 00:52:22.330 --> 00:52:25.520 align:middle line:90% Let's move on. 00:52:25.520 --> 00:52:29.710 align:middle line:90% Look at the real material now. 00:52:29.710 --> 00:52:32.200 align:middle line:84% That's a one-dimensional picture. 00:52:32.200 --> 00:52:34.600 align:middle line:84% And I look at, again, three dimensions. 00:52:34.600 --> 00:52:39.740 align:middle line:84% Go to the virtual reciprocal space. 00:52:39.740 --> 00:52:43.730 align:middle line:84% And 111 is the gamma 2 L point direction. 00:52:43.730 --> 00:52:49.000 align:middle line:84% 111-- I only do one half of it because it's symmetric. 00:52:49.000 --> 00:52:51.820 align:middle line:84% Positive wave vector, lengthy wave vector symmetric-- 00:52:51.820 --> 00:52:55.540 align:middle line:90% I draw half to save space. 00:52:55.540 --> 00:53:00.460 align:middle line:84% So this is a gamma to x that went 00 direction. 00:53:00.460 --> 00:53:03.760 align:middle line:90% And it turns out the silicon-- 00:53:03.760 --> 00:53:08.440 align:middle line:84% the maximum point where this electron fill to-- 00:53:08.440 --> 00:53:10.600 align:middle line:90% at the valence band. 00:53:10.600 --> 00:53:12.920 align:middle line:90% And the minimum point-- 00:53:12.920 --> 00:53:15.010 align:middle line:90% that's the conduction band. 00:53:15.010 --> 00:53:18.570 align:middle line:84% They do not agree in terms of the wave vector. 00:53:18.570 --> 00:53:22.090 align:middle line:84% So this is an indirect semiconductor. 00:53:22.090 --> 00:53:24.310 align:middle line:84% And gallium arsenide, on the other hand, 00:53:24.310 --> 00:53:28.060 align:middle line:90% is a direct [INAUDIBLE]. 00:53:28.060 --> 00:53:33.620 align:middle line:84% There's a big difference in these two materials, direct 00:53:33.620 --> 00:53:34.640 align:middle line:90% and indirect. 00:53:34.640 --> 00:53:40.810 align:middle line:84% Microelectronics industry is established on silicon. 00:53:40.810 --> 00:53:45.310 align:middle line:84% But the lasers-- semiconductor lasers-- 00:53:45.310 --> 00:53:48.610 align:middle line:84% nobody can use silicone to make semiconductors. 00:53:48.610 --> 00:53:50.890 align:middle line:90% People are making progress. 00:53:50.890 --> 00:53:55.510 align:middle line:84% It's much more difficult to make a laser out of silicon. 00:53:55.510 --> 00:53:57.270 align:middle line:90% It's a direct gap. 00:53:57.270 --> 00:53:59.210 align:middle line:90% Semiconductor is much easier. 00:53:59.210 --> 00:54:02.130 align:middle line:84% The reason is that in the gallium arsenide 00:54:02.130 --> 00:54:05.400 align:middle line:84% device, microelectronic device, you reject the electron here. 00:54:05.400 --> 00:54:07.140 align:middle line:90% You eject hole. 00:54:07.140 --> 00:54:08.382 align:middle line:90% That's by design. 00:54:08.382 --> 00:54:09.840 align:middle line:84% You put a current, and you actually 00:54:09.840 --> 00:54:12.060 align:middle line:90% inject it into the material. 00:54:12.060 --> 00:54:18.270 align:middle line:84% And then the electrons flow off to the bottom of-- 00:54:18.270 --> 00:54:23.592 align:middle line:84% recombine from here and here-- give off a photon. 00:54:23.592 --> 00:54:27.100 align:middle line:84% It's much easier to do that in direct gap 00:54:27.100 --> 00:54:30.970 align:middle line:84% because there's no wave vector mismatch. 00:54:30.970 --> 00:54:34.870 align:middle line:84% And here it's very difficult for electron recombine 00:54:34.870 --> 00:54:37.670 align:middle line:90% with the hole. 00:54:37.670 --> 00:54:41.630 align:middle line:84% Well, you see that's good for solar cells, Solar cells-- you 00:54:41.630 --> 00:54:44.470 align:middle line:84% don't want recombination, really, 00:54:44.470 --> 00:54:47.930 align:middle line:90% if you want to get a charge. 00:54:47.930 --> 00:54:50.820 align:middle line:90% But there are various points. 00:54:50.820 --> 00:54:54.350 align:middle line:84% This one-- solar cell is not good. 00:54:54.350 --> 00:54:56.090 align:middle line:90% It doesn't absorb well. 00:54:56.090 --> 00:55:01.730 align:middle line:84% You can't create-- when photon comes in, a photon-- 00:55:01.730 --> 00:55:07.390 align:middle line:84% The wave vector-- the wavelength is about 1 micron. 00:55:07.390 --> 00:55:10.330 align:middle line:84% Wavelengths of photon-- a half micron. 00:55:10.330 --> 00:55:14.060 align:middle line:84% You think about the visible light-- half micron. 00:55:14.060 --> 00:55:17.060 align:middle line:84% And the wave vector of photon is 2 pi 00:55:17.060 --> 00:55:22.670 align:middle line:84% over lambda, 2 pi over a half micron. 00:55:22.670 --> 00:55:26.515 align:middle line:84% And here is pi over aa is lattice constant, 00:55:26.515 --> 00:55:27.190 align:middle line:90% a few angstrom. 00:55:27.190 --> 00:55:31.360 align:middle line:90% 00:55:31.360 --> 00:55:33.490 align:middle line:90% So this is a few angstrom. 00:55:33.490 --> 00:55:38.710 align:middle line:84% If it's a 2 pi over a half micron, that's exactly-- 00:55:38.710 --> 00:55:42.450 align:middle line:90% almost right at the center. 00:55:42.450 --> 00:55:45.190 align:middle line:84% The wave vector of photon, very small. 00:55:45.190 --> 00:55:50.350 align:middle line:84% Wave vector hk-- h over lambda is momentum. 00:55:50.350 --> 00:55:53.220 align:middle line:84% Momentum of photon is very small. 00:55:53.220 --> 00:55:58.270 align:middle line:84% So in gallium arsenide, the photon comes in and go-- 00:55:58.270 --> 00:56:00.170 align:middle line:90% you go from here to-- 00:56:00.170 --> 00:56:01.790 align:middle line:84% if the energy is large enough, you 00:56:01.790 --> 00:56:05.660 align:middle line:84% can lift the electron from here to here. 00:56:05.660 --> 00:56:07.590 align:middle line:90% That was all. 00:56:07.590 --> 00:56:11.370 align:middle line:84% But in silicon, because they interact-- the semiconductor, 00:56:11.370 --> 00:56:14.130 align:middle line:90% the photon momentum is here. 00:56:14.130 --> 00:56:18.200 align:middle line:84% You cannot leave the electron from here to here. 00:56:18.200 --> 00:56:19.605 align:middle line:90% Doesn't have enough momentum. 00:56:19.605 --> 00:56:20.480 align:middle line:90% It has enough energy. 00:56:20.480 --> 00:56:23.920 align:middle line:90% 00:56:23.920 --> 00:56:25.290 align:middle line:90% What can solve that problem? 00:56:25.290 --> 00:56:29.236 align:middle line:90% 00:56:29.236 --> 00:56:30.865 align:middle line:90% Some method. 00:56:30.865 --> 00:56:35.780 align:middle line:90% You solve a huge problem. 00:56:35.780 --> 00:56:37.360 align:middle line:90% See, this is a really-- 00:56:37.360 --> 00:56:39.910 align:middle line:84% even in the microscopic world, it's 00:56:39.910 --> 00:56:42.580 align:middle line:84% the energy conservation, the momentum conservation. 00:56:42.580 --> 00:56:46.100 align:middle line:84% You learn either in continuum or in discrete. 00:56:46.100 --> 00:56:48.830 align:middle line:90% That always [INAUDIBLE]. 00:56:48.830 --> 00:56:51.342 align:middle line:90% So silicon don't absorb well. 00:56:51.342 --> 00:56:53.870 align:middle line:90% Gallium arsenide absorb well. 00:56:53.870 --> 00:56:56.420 align:middle line:84% If you look at a gallium arsenide solar cell, 00:56:56.420 --> 00:56:57.860 align:middle line:90% people use the same field. 00:56:57.860 --> 00:56:58.910 align:middle line:90% They absorb well. 00:56:58.910 --> 00:57:01.790 align:middle line:84% Silicon-- it's very hard to use the same field, 00:57:01.790 --> 00:57:03.920 align:middle line:90% because they don't absorb. 00:57:03.920 --> 00:57:06.890 align:middle line:84% So your compromise is make thicker 00:57:06.890 --> 00:57:09.106 align:middle line:90% for that chance to happen. 00:57:09.106 --> 00:57:12.034 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:57:12.034 --> 00:57:13.190 align:middle line:90% GANG CHEN: Ah, yes. 00:57:13.190 --> 00:57:17.840 align:middle line:84% But that's exactly how absorb happens. 00:57:17.840 --> 00:57:20.570 align:middle line:90% You need a phonon. 00:57:20.570 --> 00:57:23.550 align:middle line:90% Phonon has that wave vector. 00:57:23.550 --> 00:57:25.590 align:middle line:84% Phonon doesn't have enough energy. 00:57:25.590 --> 00:57:28.890 align:middle line:90% The omega is too small. 00:57:28.890 --> 00:57:33.780 align:middle line:90% So phonon plus the photon-- 00:57:33.780 --> 00:57:36.360 align:middle line:90% phonon gave you the vector. 00:57:36.360 --> 00:57:37.600 align:middle line:90% Photon gave you the energy. 00:57:37.600 --> 00:57:39.690 align:middle line:90% Can lift it from here to here. 00:57:39.690 --> 00:57:42.310 align:middle line:84% But you've got three particles to work together, 00:57:42.310 --> 00:57:45.520 align:middle line:84% Think about your coordination at MIT. 00:57:45.520 --> 00:57:47.850 align:middle line:84% You get three professor together. 00:57:47.850 --> 00:57:49.890 align:middle line:90% Much less chance. 00:57:49.890 --> 00:57:51.490 align:middle line:90% That's the probability. 00:57:51.490 --> 00:57:54.350 align:middle line:90% 00:57:54.350 --> 00:57:56.197 align:middle line:90% We'll have that experience. 00:57:56.197 --> 00:58:00.180 align:middle line:90% 00:58:00.180 --> 00:58:02.430 align:middle line:84% Now we're going to do a little bit more. 00:58:02.430 --> 00:58:06.690 align:middle line:84% That's the basic idea of the energy levels, 00:58:06.690 --> 00:58:12.270 align:middle line:84% vibrational energy level, and the electronic energy 00:58:12.270 --> 00:58:14.660 align:middle line:90% level in solids. 00:58:14.660 --> 00:58:18.390 align:middle line:84% And I'm going to do a little more. 00:58:18.390 --> 00:58:22.650 align:middle line:84% And I approximate-- first, I think about the free electron. 00:58:22.650 --> 00:58:26.860 align:middle line:84% The kinetic energy-- 1/2 mv square. 00:58:26.860 --> 00:58:33.580 align:middle line:84% And mv square-- quantum mechanics say-- 00:58:33.580 --> 00:58:36.140 align:middle line:84% first, m comes away as the momentum. 00:58:36.140 --> 00:58:37.870 align:middle line:90% So p squared 2m-- 00:58:37.870 --> 00:58:42.970 align:middle line:84% that's the relation from p as the m wave. 00:58:42.970 --> 00:58:47.320 align:middle line:84% And in quantum mechanics, p is hk. 00:58:47.320 --> 00:58:49.520 align:middle line:90% Oh, I got h squared. 00:58:49.520 --> 00:58:50.020 align:middle line:90% Sorry. 00:58:50.020 --> 00:58:53.440 align:middle line:84% It should be h squared k squared over 2m. 00:58:53.440 --> 00:58:54.933 align:middle line:90% So that's a free electron. 00:58:54.933 --> 00:59:02.630 align:middle line:90% 00:59:02.630 --> 00:59:10.420 align:middle line:84% For the electron in the solid, I'm going to do a similar thing. 00:59:10.420 --> 00:59:12.380 align:middle line:84% Because this-- here is a maximum. 00:59:12.380 --> 00:59:14.310 align:middle line:90% Here is a minimum. 00:59:14.310 --> 00:59:16.320 align:middle line:84% Again, the first-order derivative 00:59:16.320 --> 00:59:19.470 align:middle line:84% is 0, unless this is a very sharp. 00:59:19.470 --> 00:59:22.060 align:middle line:90% There are some cases it's not-- 00:59:22.060 --> 00:59:25.190 align:middle line:84% the derivative is not continuous. 00:59:25.190 --> 00:59:26.960 align:middle line:84% So the band is not a part of-- what 00:59:26.960 --> 00:59:31.190 align:middle line:84% I say is it's not a continuous curvature here. 00:59:31.190 --> 00:59:32.490 align:middle line:90% But see, most of-- 00:59:32.490 --> 00:59:35.730 align:middle line:84% this maximum-minimum-- the first derivative is 0. 00:59:35.730 --> 00:59:38.066 align:middle line:84% So I have second-order [INAUDIBLE]. 00:59:38.066 --> 00:59:40.754 align:middle line:90% 00:59:40.754 --> 00:59:43.230 align:middle line:90% Mathematics I extension. 00:59:43.230 --> 00:59:47.610 align:middle line:84% So here, if I, for example, take a [INAUDIBLE], 00:59:47.610 --> 00:59:48.990 align:middle line:90% of course I put a k. 00:59:48.990 --> 00:59:50.940 align:middle line:90% I didn't put a k. 00:59:50.940 --> 00:59:57.860 align:middle line:84% This is not 0, so I should put kx minus k0 square. 00:59:57.860 --> 01:00:00.800 align:middle line:84% Here, I put the center at the center. 01:00:00.800 --> 01:00:06.530 align:middle line:84% So this is the sigma general, not the most general form 01:00:06.530 --> 01:00:12.090 align:middle line:90% of doing the mathematics, 01:00:12.090 --> 01:00:15.590 align:middle line:84% And here, it looks very similar to free electron. 01:00:15.590 --> 01:00:21.140 align:middle line:84% This is just a second-order derivative of the curvature. 01:00:21.140 --> 01:00:25.465 align:middle line:84% Now, if I treat that just as a mass, equivalent of mass-- 01:00:25.465 --> 01:00:30.020 align:middle line:84% if I call it effective mass, then it's a free electron. 01:00:30.020 --> 01:00:33.220 align:middle line:84% I can do the rest of my work as a free electron. 01:00:33.220 --> 01:00:36.130 align:middle line:90% So mathematical simplification. 01:00:36.130 --> 01:00:44.630 align:middle line:84% Here are the effective mass of the electron in the crystal. 01:00:44.630 --> 01:00:48.860 align:middle line:84% If I take the effective mass, I write this pretty 01:00:48.860 --> 01:00:50.430 align:middle line:90% much like a free electron. 01:00:50.430 --> 01:00:54.950 align:middle line:84% So I'm going to do free electron, most of the time, 01:00:54.950 --> 01:00:57.590 align:middle line:90% except replacing the mass. 01:00:57.590 --> 01:01:01.580 align:middle line:84% So this is, again, my dispersion relation. 01:01:01.580 --> 01:01:06.470 align:middle line:84% Before, I said dispersion relation between frequency 01:01:06.470 --> 01:01:08.270 align:middle line:90% and wave vector. 01:01:08.270 --> 01:01:11.815 align:middle line:84% And now, if you think about energy relate to frequency, 01:01:11.815 --> 01:01:13.400 align:middle line:90% that's new. 01:01:13.400 --> 01:01:15.410 align:middle line:90% So frequency, wave vector. 01:01:15.410 --> 01:01:18.570 align:middle line:84% Except for electron, people don't talk to a frequency. 01:01:18.570 --> 01:01:22.170 align:middle line:84% People like to talk about energy. 01:01:22.170 --> 01:01:26.070 align:middle line:84% Close relation between the vector frequency or the vector-- 01:01:26.070 --> 01:01:29.443 align:middle line:90% and energy dispersion relations. 01:01:29.443 --> 01:01:30.860 align:middle line:84% AUDIENCE: So the effective mass is 01:01:30.860 --> 01:01:34.330 align:middle line:90% different for every direction? 01:01:34.330 --> 01:01:37.240 align:middle line:84% GANG CHEN: It could be different for every direction. 01:01:37.240 --> 01:01:41.830 align:middle line:84% In some case, it could be isotropic. 01:01:41.830 --> 01:01:43.780 align:middle line:90% Next one-- let me show you. 01:01:43.780 --> 01:01:47.050 align:middle line:84% If we take the energy as constant, I do this. 01:01:47.050 --> 01:01:49.630 align:middle line:84% So if you look at this, this is-- 01:01:49.630 --> 01:01:52.750 align:middle line:84% OK, what's the constant energy surface looking like? 01:01:52.750 --> 01:01:56.260 align:middle line:84% In 3D, If these are the same, it's a sphere. 01:01:56.260 --> 01:01:57.320 align:middle line:90% They are different. 01:01:57.320 --> 01:01:58.400 align:middle line:90% They are ellipsoid. 01:01:58.400 --> 01:02:02.770 align:middle line:90% 01:02:02.770 --> 01:02:07.790 align:middle line:84% So this is what's in gallium arsenide. 01:02:07.790 --> 01:02:08.970 align:middle line:90% This is a silicon electron. 01:02:08.970 --> 01:02:12.995 align:middle line:90% 01:02:12.995 --> 01:02:13.905 align:middle line:90% End of story. 01:02:13.905 --> 01:02:16.400 align:middle line:84% So in this case, we say, OK, there, you 01:02:16.400 --> 01:02:21.670 align:middle line:84% got six pair of pockets, different directions. 01:02:21.670 --> 01:02:28.680 align:middle line:90% 01:02:28.680 --> 01:02:31.420 align:middle line:90% And this is just a review. 01:02:31.420 --> 01:02:33.070 align:middle line:90% I showed this before. 01:02:33.070 --> 01:02:37.300 align:middle line:84% Now we know what is phonon, what is electron. 01:02:37.300 --> 01:02:42.280 align:middle line:84% And Fermi-Dirac distribution is for electron. 01:02:42.280 --> 01:02:45.243 align:middle line:84% Bose-Einstein distribution for phonon and photon. 01:02:45.243 --> 01:02:46.660 align:middle line:84% Most of the time, this [INAUDIBLE] 01:02:46.660 --> 01:02:51.980 align:middle line:84% is 0 because phonon and photon are not real particles. 01:02:51.980 --> 01:02:55.000 align:middle line:84% That's another way they are not real. 01:02:55.000 --> 01:02:57.670 align:middle line:90% They can disappear. 01:02:57.670 --> 01:03:02.010 align:middle line:90% Electron also disappear. 01:03:02.010 --> 01:03:03.300 align:middle line:90% But the most of the time-- 01:03:03.300 --> 01:03:04.200 align:middle line:90% and not disappear. 01:03:04.200 --> 01:03:05.670 align:middle line:90% Electron cannot disappear. 01:03:05.670 --> 01:03:07.110 align:middle line:90% Electron is the real part. 01:03:07.110 --> 01:03:12.210 align:middle line:84% As we go from conduction band to valence band, we combine-- 01:03:12.210 --> 01:03:15.070 align:middle line:84% just [INAUDIBLE] the electron still is there. 01:03:15.070 --> 01:03:18.690 align:middle line:90% 01:03:18.690 --> 01:03:20.990 align:middle line:90% OK, now I'm going to do math. 01:03:20.990 --> 01:03:23.400 align:middle line:90% The constant. 01:03:23.400 --> 01:03:27.360 align:middle line:84% I'm going into more detail constant. 01:03:27.360 --> 01:03:30.440 align:middle line:84% I say, OK, I know about semiconductor. 01:03:30.440 --> 01:03:30.940 align:middle line:90% Yes. 01:03:30.940 --> 01:03:31.607 align:middle line:90% AUDIENCE: Sorry. 01:03:31.607 --> 01:03:33.550 align:middle line:84% Is there any [INAUDIBLE] mass conservation 01:03:33.550 --> 01:03:38.160 align:middle line:84% or effective mass conservation [INAUDIBLE]? 01:03:38.160 --> 01:03:41.562 align:middle line:84% GANG CHEN: Mass conservation for-- 01:03:41.562 --> 01:03:42.145 align:middle line:90% good question. 01:03:42.145 --> 01:03:44.850 align:middle line:90% 01:03:44.850 --> 01:03:52.350 align:middle line:84% I don't think we can talk about mass conservation for-- 01:03:52.350 --> 01:03:54.750 align:middle line:84% under the effective mass picture. 01:03:54.750 --> 01:03:57.810 align:middle line:90% I'm sure this is the only from-- 01:03:57.810 --> 01:04:00.690 align:middle line:90% I think mostly from [INAUDIBLE]. 01:04:00.690 --> 01:04:02.850 align:middle line:84% Electron-- of course, the true math 01:04:02.850 --> 01:04:06.810 align:middle line:84% is still the [INAUDIBLE] 0.1 to 0.31. 01:04:06.810 --> 01:04:10.140 align:middle line:84% But in the solid, when you look at the transport, 01:04:10.140 --> 01:04:11.850 align:middle line:90% I see the free electron. 01:04:11.850 --> 01:04:15.000 align:middle line:84% Then usually fill the [INAUDIBLE] 01:04:15.000 --> 01:04:18.030 align:middle line:90% of the background, lattice. 01:04:18.030 --> 01:04:21.060 align:middle line:84% So now they feel either heavier or lighter 01:04:21.060 --> 01:04:24.540 align:middle line:84% than the free electron mass, which 01:04:24.540 --> 01:04:28.340 align:middle line:90% is 0.1 to 0.31 [INAUDIBLE]. 01:04:28.340 --> 01:04:29.052 align:middle line:90% Yes. 01:04:29.052 --> 01:04:32.840 align:middle line:84% AUDIENCE: When people do the [INAUDIBLE] structure 01:04:32.840 --> 01:04:36.170 align:middle line:84% calculation for electrons, do they 01:04:36.170 --> 01:04:41.480 align:middle line:84% assume independent national approximation? 01:04:41.480 --> 01:04:44.255 align:middle line:90% If they include [INAUDIBLE] 01:04:44.255 --> 01:04:49.160 align:middle line:84% GANG CHEN: I think they should include the electron-- 01:04:49.160 --> 01:04:53.000 align:middle line:84% for example, you do a density functional theory calculation. 01:04:53.000 --> 01:04:57.670 align:middle line:84% They will include the outer electron effect. 01:04:57.670 --> 01:04:58.960 align:middle line:90% And you don't do-- 01:04:58.960 --> 01:05:01.060 align:middle line:90% just an independent electron. 01:05:01.060 --> 01:05:05.570 align:middle line:84% You have electron influence the other electron in the same atom. 01:05:05.570 --> 01:05:10.130 align:middle line:90% 01:05:10.130 --> 01:05:12.630 align:middle line:84% So now I'm going to do a spherical-- 01:05:12.630 --> 01:05:15.500 align:middle line:84% I'm not going to do parabolic or elliptical, 01:05:15.500 --> 01:05:19.160 align:middle line:84% but I'm going to do a spherical [INAUDIBLE]. 01:05:19.160 --> 01:05:22.640 align:middle line:84% And it's the same in all directions. 01:05:22.640 --> 01:05:26.120 align:middle line:84% So that's a really like a free electron, 01:05:26.120 --> 01:05:28.640 align:middle line:84% except that this mass could be different. 01:05:28.640 --> 01:05:33.380 align:middle line:84% Like I said, it's good for gallium arsenide, 01:05:33.380 --> 01:05:36.050 align:middle line:84% Let me suppose-- I want to find out-- 01:05:36.050 --> 01:05:40.365 align:middle line:84% if I dope the material, how many electrons 01:05:40.365 --> 01:05:43.300 align:middle line:90% I have in this conduction band? 01:05:43.300 --> 01:05:46.990 align:middle line:90% 01:05:46.990 --> 01:05:51.340 align:middle line:90% And I'm going to do counting. 01:05:51.340 --> 01:05:56.140 align:middle line:90% This is a strategy that I use. 01:05:56.140 --> 01:06:02.800 align:middle line:84% And I use before same for photon. 01:06:02.800 --> 01:06:04.920 align:middle line:90% I did that photon in cavity. 01:06:04.920 --> 01:06:08.080 align:middle line:84% I count how many photons in the cavity. 01:06:08.080 --> 01:06:10.750 align:middle line:90% 01:06:10.750 --> 01:06:15.760 align:middle line:84% So now, if you remember, each quantum mechanical state 01:06:15.760 --> 01:06:19.940 align:middle line:90% can have one electron. 01:06:19.940 --> 01:06:24.920 align:middle line:84% And in the state, what's the average number of electrons 01:06:24.920 --> 01:06:27.110 align:middle line:90% is given by Fermi-Dirac? 01:06:27.110 --> 01:06:29.362 align:middle line:90% Maximum one electron, right? 01:06:29.362 --> 01:06:31.490 align:middle line:90% We said in the-- 01:06:31.490 --> 01:06:36.660 align:middle line:84% this is the average given a certain temperature, quantum 01:06:36.660 --> 01:06:41.670 align:middle line:84% potential, average number of electrons in one quantum state. 01:06:41.670 --> 01:06:47.030 align:middle line:84% Now I just need to see how many quantum states I have, right? 01:06:47.030 --> 01:06:52.550 align:middle line:84% One quantum state, this is the average number of electrons. 01:06:52.550 --> 01:06:54.980 align:middle line:84% So I just need to-- really, all those counting 01:06:54.980 --> 01:06:59.840 align:middle line:84% looking very complicated, it's just counting quantum states. 01:06:59.840 --> 01:07:01.940 align:middle line:84% And before, when I do electromagnetic wave, 01:07:01.940 --> 01:07:06.350 align:middle line:84% I was counting electromagnetic modes. 01:07:06.350 --> 01:07:07.885 align:middle line:90% Same way. 01:07:07.885 --> 01:07:10.820 align:middle line:84% So all that counting-- counting how many modes, 01:07:10.820 --> 01:07:14.810 align:middle line:84% the modes is essentially a state. 01:07:14.810 --> 01:07:16.280 align:middle line:90% So I'm counting. 01:07:16.280 --> 01:07:18.830 align:middle line:90% So in the x-- so it's 3D-- 01:07:18.830 --> 01:07:21.570 align:middle line:90% kx, ky, kz. 01:07:21.570 --> 01:07:27.450 align:middle line:84% And kx, I have nx number, n atoms in the x direction; 01:07:27.450 --> 01:07:31.291 align:middle line:90% ky, ny number; kz, nz number. 01:07:31.291 --> 01:07:35.100 align:middle line:90% 01:07:35.100 --> 01:07:42.090 align:middle line:84% And the factor of 2 is because I have spin, electron spin 1/2, 01:07:42.090 --> 01:07:43.710 align:middle line:90% positive/minus-- 01:07:43.710 --> 01:07:45.630 align:middle line:90% negative. 01:07:45.630 --> 01:07:49.860 align:middle line:84% Electromagnetic wave is a factor of 2. 01:07:49.860 --> 01:07:52.545 align:middle line:84% If I want to do full arm, maybe it's a factor of 3 01:07:52.545 --> 01:07:55.450 align:middle line:84% because my [INAUDIBLE] if they are all the same. 01:07:55.450 --> 01:07:58.976 align:middle line:90% 01:07:58.976 --> 01:08:03.640 align:middle line:84% And so this is the number of quantum states. 01:08:03.640 --> 01:08:05.990 align:middle line:84% And this is, at each quantum state, 01:08:05.990 --> 01:08:09.360 align:middle line:84% what's the average number of electrons. 01:08:09.360 --> 01:08:12.040 align:middle line:84% You can think that for a phonon or for a photon, 01:08:12.040 --> 01:08:16.308 align:middle line:84% it's all the same way, all right? 01:08:16.308 --> 01:08:18.350 align:middle line:84% AUDIENCE: Gang, so it's counting the wave vector, 01:08:18.350 --> 01:08:20.024 align:middle line:90% it's counting the number? 01:08:20.024 --> 01:08:20.899 align:middle line:90% GANG CHEN: Ah, well-- 01:08:20.899 --> 01:08:24.080 align:middle line:84% AUDIENCE: Why is it number minus the nz over 2? 01:08:24.080 --> 01:08:27.689 align:middle line:90% GANG CHEN: This is a-- 01:08:27.689 --> 01:08:28.709 align:middle line:90% yes. 01:08:28.709 --> 01:08:31.080 align:middle line:90% It's counting the number. 01:08:31.080 --> 01:08:36.035 align:middle line:84% But see, then I do the integration conversion. 01:08:36.035 --> 01:08:36.660 align:middle line:90% AUDIENCE: Yeah. 01:08:36.660 --> 01:08:39.047 align:middle line:84% I see you're counting the wave vector. 01:08:39.047 --> 01:08:39.630 align:middle line:90% GANG CHEN: OK. 01:08:39.630 --> 01:08:44.580 align:middle line:90% Here is-- I'm counting n. 01:08:44.580 --> 01:08:49.940 align:middle line:84% These are the discrete states, right? 01:08:49.940 --> 01:08:53.380 align:middle line:84% And then this is a standing wave picture. 01:08:53.380 --> 01:08:56.130 align:middle line:90% So everything's positive. 01:08:56.130 --> 01:09:00.180 align:middle line:84% And when I do traveling with two directions, 01:09:00.180 --> 01:09:03.766 align:middle line:84% I said that these are equivalent, OK? 01:09:03.766 --> 01:09:08.819 align:middle line:84% So that's why I go from negative n 1/2 to positive n 1/2. 01:09:08.819 --> 01:09:12.660 align:middle line:90% It's just equivalent. 01:09:12.660 --> 01:09:14.069 align:middle line:90% OK? 01:09:14.069 --> 01:09:15.859 align:middle line:90% Why I count the numbers? 01:09:15.859 --> 01:09:19.583 align:middle line:84% Because my next step is convert the number into integration. 01:09:19.583 --> 01:09:21.125 align:middle line:84% Because we're counting states, right? 01:09:21.125 --> 01:09:25.410 align:middle line:84% It's better to do integrated states rather than 01:09:25.410 --> 01:09:28.050 align:middle line:90% fractional number. 01:09:28.050 --> 01:09:30.600 align:middle line:90% This is the number space. 01:09:30.600 --> 01:09:33.689 align:middle line:90% That's where I convert this-- 01:09:33.689 --> 01:09:38.069 align:middle line:84% if I think about wave vector, k is a continuous function. 01:09:38.069 --> 01:09:44.939 align:middle line:84% And between two states, that delta k is 2 pi over l. 01:09:44.939 --> 01:09:50.706 align:middle line:84% So this is converting this integer summation into-- 01:09:50.706 --> 01:09:58.042 align:middle line:84% no, this summation into integration, right? 01:09:58.042 --> 01:10:03.640 align:middle line:84% I did that same thing when I do electromagnetic modes. 01:10:03.640 --> 01:10:08.740 align:middle line:84% So in the x-direction, if this is the length, 01:10:08.740 --> 01:10:12.153 align:middle line:90% but the next two adjacent k-- 01:10:12.153 --> 01:10:15.370 align:middle line:84% n is 2 pi over [INAUDIBLE] x, so this 01:10:15.370 --> 01:10:19.320 align:middle line:84% is the number converted into here, [INAUDIBLE] down here. 01:10:19.320 --> 01:10:22.750 align:middle line:84% So here, I have the integration kx, ky, kz. 01:10:22.750 --> 01:10:26.080 align:middle line:90% 01:10:26.080 --> 01:10:31.600 align:middle line:84% Those are all mathematics converting-- 01:10:31.600 --> 01:10:34.930 align:middle line:84% so the conversion and the counting. 01:10:34.930 --> 01:10:37.480 align:middle line:84% And then the reason I want to do this now 01:10:37.480 --> 01:10:41.320 align:middle line:84% is I have an integration dkx dky dkz. 01:10:41.320 --> 01:10:46.540 align:middle line:84% And 2 pi now, I have is an 8 pi cubic. 01:10:46.540 --> 01:10:51.280 align:middle line:84% lx ly lz is the volume, factor of 2. 01:10:51.280 --> 01:10:55.275 align:middle line:90% And now the f-- 01:10:55.275 --> 01:10:58.620 align:middle line:90% I'm going to just ignore that 1. 01:10:58.620 --> 01:11:02.100 align:middle line:84% Plus 1, that's Boltzmann approximation. 01:11:02.100 --> 01:11:04.360 align:middle line:90% So the Fermi-Dirac distribution. 01:11:04.360 --> 01:11:07.130 align:middle line:90% 01:11:07.130 --> 01:11:10.610 align:middle line:84% OK, so I'm plugging this f already here. 01:11:10.610 --> 01:11:16.040 align:middle line:84% So that's my number of electrons in this region, 01:11:16.040 --> 01:11:17.790 align:middle line:90% in the conduction band. 01:11:17.790 --> 01:11:20.670 align:middle line:90% 01:11:20.670 --> 01:11:22.620 align:middle line:90% OK. 01:11:22.620 --> 01:11:25.590 align:middle line:84% I encountered this classical problem. 01:11:25.590 --> 01:11:29.980 align:middle line:84% My integration is over a wave vector. 01:11:29.980 --> 01:11:34.185 align:middle line:84% My function is over energy, right? 01:11:34.185 --> 01:11:37.140 align:middle line:84% Same thing in the electromagnetic wave. 01:11:37.140 --> 01:11:41.910 align:middle line:84% My function, my integration is over kx, ky, kz. 01:11:41.910 --> 01:11:44.580 align:middle line:90% My function is over energy. 01:11:44.580 --> 01:11:47.880 align:middle line:84% And I have to choose either energy or wave 01:11:47.880 --> 01:11:52.260 align:middle line:84% vector as my variable to do this integration. 01:11:52.260 --> 01:11:55.840 align:middle line:84% I already have the relation between E and k. 01:11:55.840 --> 01:12:05.045 align:middle line:84% That's my problem here, E and k, right? 01:12:05.045 --> 01:12:07.740 align:middle line:84% So I'm going to do that conversion. 01:12:07.740 --> 01:12:12.510 align:middle line:84% In this case, it's much easier for me to convert dkx, 01:12:12.510 --> 01:12:16.440 align:middle line:84% dky, dkz into E because you can say 01:12:16.440 --> 01:12:17.790 align:middle line:90% that's a Cartesian coordinate. 01:12:17.790 --> 01:12:21.210 align:middle line:84% I can convert it into spherical coordinate. 01:12:21.210 --> 01:12:27.240 align:middle line:84% And so I'm doing this conversion. 01:12:27.240 --> 01:12:31.380 align:middle line:84% Now I do per unit volume number density. 01:12:31.380 --> 01:12:35.940 align:middle line:90% And I do that dkx, dky, dkz. 01:12:35.940 --> 01:12:40.725 align:middle line:84% In spherical coordinate, k would be magnitude. 01:12:40.725 --> 01:12:42.670 align:middle line:90% So it's 4 pi k squared dk. 01:12:42.670 --> 01:12:45.892 align:middle line:90% 01:12:45.892 --> 01:12:51.386 align:middle line:84% So that's from Cartesian to spherical function and variant. 01:12:51.386 --> 01:12:57.450 align:middle line:90% 01:12:57.450 --> 01:13:00.980 align:middle line:84% Now I replace because I know the k, 01:13:00.980 --> 01:13:05.000 align:middle line:84% k square is 2 me minus Ecx squared, right? 01:13:05.000 --> 01:13:07.840 align:middle line:90% That's the expression. 01:13:07.840 --> 01:13:11.680 align:middle line:84% This is a ac squared times k squared. 01:13:11.680 --> 01:13:16.870 align:middle line:84% Now, k is my radius, like r, right? 01:13:16.870 --> 01:13:19.300 align:middle line:90% So I replace this k square. 01:13:19.300 --> 01:13:22.240 align:middle line:90% 01:13:22.240 --> 01:13:25.910 align:middle line:84% k square equals 2 pi [INAUDIBLE] minus c over x squared. 01:13:25.910 --> 01:13:28.990 align:middle line:90% dk becomes this d. 01:13:28.990 --> 01:13:32.400 align:middle line:84% And everything is in [INAUDIBLE]. 01:13:32.400 --> 01:13:37.080 align:middle line:84% This will be same thing you do all the time in counting 01:13:37.080 --> 01:13:39.930 align:middle line:90% the numbers in crystal. 01:13:39.930 --> 01:13:41.670 align:middle line:90% You convert the states-- 01:13:41.670 --> 01:13:44.650 align:middle line:84% number of states summation into integration. 01:13:44.650 --> 01:13:47.370 align:middle line:84% You choose your integration variable, 01:13:47.370 --> 01:13:48.890 align:middle line:90% either in energy or wave vector. 01:13:48.890 --> 01:13:51.690 align:middle line:90% 01:13:51.690 --> 01:13:58.710 align:middle line:84% And if I group everything all together, this is my f. 01:13:58.710 --> 01:14:04.680 align:middle line:84% And this one, I call it the density of states. 01:14:04.680 --> 01:14:06.111 align:middle line:90% OK. 01:14:06.111 --> 01:14:08.400 align:middle line:90% It uses-- it's mathematics. 01:14:08.400 --> 01:14:14.630 align:middle line:84% [INAUDIBLE] nothing-- physics wise, 01:14:14.630 --> 01:14:16.340 align:middle line:90% you don't have much physics. 01:14:16.340 --> 01:14:21.680 align:middle line:84% It's just a mathematical way of converting my summation 01:14:21.680 --> 01:14:22.790 align:middle line:90% into integration. 01:14:22.790 --> 01:14:26.600 align:middle line:84% So this is the number of quantum states per unit volume 01:14:26.600 --> 01:14:28.760 align:middle line:90% because I normalize the volume. 01:14:28.760 --> 01:14:31.430 align:middle line:84% And per energy interval, because here 01:14:31.430 --> 01:14:40.170 align:middle line:84% is d, that's a physical interpretation of my derivation. 01:14:40.170 --> 01:14:43.710 align:middle line:84% And in fact, if you go to look at this one, 01:14:43.710 --> 01:14:45.480 align:middle line:84% you can do the integration [INAUDIBLE]. 01:14:45.480 --> 01:14:48.000 align:middle line:84% That's why I use the Boltzmann distribution. 01:14:48.000 --> 01:14:49.650 align:middle line:90% If I put a Fermi-Dirac-- 01:14:49.650 --> 01:14:54.960 align:middle line:84% if I put a, say, 1 over this, not this, plus 1, 01:14:54.960 --> 01:14:58.161 align:middle line:90% I can't do this integration, OK? 01:14:58.161 --> 01:15:00.760 align:middle line:84% So the reason I use the Boltzmann distribution 01:15:00.760 --> 01:15:03.070 align:middle line:84% is that I can do this integration, 01:15:03.070 --> 01:15:06.640 align:middle line:90% and this integration gives me-- 01:15:06.640 --> 01:15:13.390 align:middle line:84% here is a kpt, and here is E minus c maybe exponent. 01:15:13.390 --> 01:15:19.060 align:middle line:84% So how I determine the chemical potential mu, if you give me 01:15:19.060 --> 01:15:21.670 align:middle line:90% the number of dopant-- 01:15:21.670 --> 01:15:25.300 align:middle line:84% so I put a phosphor in the semiconductor. 01:15:25.300 --> 01:15:27.370 align:middle line:90% Per unit volume, I put how much. 01:15:27.370 --> 01:15:28.920 align:middle line:90% I know that. 01:15:28.920 --> 01:15:35.410 align:middle line:84% And if I assume each phosphor donates one electron-- 01:15:35.410 --> 01:15:37.120 align:middle line:90% assumption, right? 01:15:37.120 --> 01:15:39.790 align:middle line:84% And there are more rigorous statistical ways 01:15:39.790 --> 01:15:42.280 align:middle line:84% to say how many electrons are activated. 01:15:42.280 --> 01:15:45.640 align:middle line:84% But because phosphor energy is so close to silicon, 01:15:45.640 --> 01:15:48.010 align:middle line:84% so you can roughly approximate this. 01:15:48.010 --> 01:15:52.450 align:middle line:84% So every atom that you put in there donates one electron. 01:15:52.450 --> 01:15:55.960 align:middle line:84% So with that, I know the left-hand side. 01:15:55.960 --> 01:15:58.000 align:middle line:90% I can do the inversion. 01:15:58.000 --> 01:16:00.610 align:middle line:84% This is where my chemical potential is. 01:16:00.610 --> 01:16:06.010 align:middle line:84% So chemical potential is determined by the number 01:16:06.010 --> 01:16:09.270 align:middle line:90% of carriers you put in. 01:16:09.270 --> 01:16:11.380 align:middle line:84% And then you will see thermoelectrically, 01:16:11.380 --> 01:16:14.040 align:middle line:84% you have to optimize this carrier density 01:16:14.040 --> 01:16:18.900 align:middle line:90% to get the maximum performance. 01:16:18.900 --> 01:16:20.286 align:middle line:90% OK. 01:16:20.286 --> 01:16:26.200 align:middle line:90% So let me summarize this, right? 01:16:26.200 --> 01:16:30.490 align:middle line:90% This way of doing the-- 01:16:30.490 --> 01:16:33.520 align:middle line:84% converting it to the number counting, 01:16:33.520 --> 01:16:36.820 align:middle line:84% you can always start with that summation, I said. 01:16:36.820 --> 01:16:39.560 align:middle line:84% But at the end, the most times you 01:16:39.560 --> 01:16:42.070 align:middle line:84% will end up with this density of states. 01:16:42.070 --> 01:16:44.080 align:middle line:84% It's just that you derive [INAUDIBLE] state. 01:16:44.080 --> 01:16:46.240 align:middle line:90% Every time. 01:16:46.240 --> 01:16:49.540 align:middle line:84% Once you become comfortable, you can say, OK, this, 01:16:49.540 --> 01:16:51.790 align:middle line:84% I can write it directly the number 01:16:51.790 --> 01:16:54.190 align:middle line:90% of electrons per quantum state. 01:16:54.190 --> 01:16:58.600 align:middle line:84% This is the number of quantum state per unit volume per energy 01:16:58.600 --> 01:16:59.830 align:middle line:90% interval times [INAUDIBLE]. 01:16:59.830 --> 01:17:02.800 align:middle line:84% That's the number of [INAUDIBLE] quantum states there are. 01:17:02.800 --> 01:17:07.690 align:middle line:84% And that gives me the number of electrons near energy E. 01:17:07.690 --> 01:17:10.855 align:middle line:84% And I integrate over all possible energy states. 01:17:10.855 --> 01:17:12.605 align:middle line:84% That should give me the number of charges. 01:17:12.605 --> 01:17:15.550 align:middle line:90% 01:17:15.550 --> 01:17:18.520 align:middle line:84% So once you gone through this a few times, 01:17:18.520 --> 01:17:22.580 align:middle line:84% you can just start from the [INAUDIBLE]. 01:17:22.580 --> 01:17:26.720 align:middle line:84% And I wrote down for Boltzmann-- so then this is just 01:17:26.720 --> 01:17:29.750 align:middle line:90% under Boltzmann distribution. 01:17:29.750 --> 01:17:35.180 align:middle line:84% I can do this integration, and that's what we have. 01:17:35.180 --> 01:17:43.960 align:middle line:84% And what it means is if I gave a number of dopants 01:17:43.960 --> 01:17:49.400 align:middle line:84% to the phosphor, I can see where my potential is, [INAUDIBLE] 01:17:49.400 --> 01:17:53.600 align:middle line:84% potential, from here given n [INAUDIBLE] inverse, 01:17:53.600 --> 01:17:56.690 align:middle line:90% I know E matches c. 01:17:56.690 --> 01:18:00.430 align:middle line:84% And the Boltzmann-- so this is only valid 01:18:00.430 --> 01:18:05.680 align:middle line:90% when this mu is far below Ec. 01:18:05.680 --> 01:18:09.070 align:middle line:84% Once the mu gets into here, you have 01:18:09.070 --> 01:18:12.790 align:middle line:84% to use the Fermi-Dirac statistics. 01:18:12.790 --> 01:18:15.690 align:middle line:84% That's a degenerate semiconductor. 01:18:15.690 --> 01:18:18.420 align:middle line:90% This is the only approximation. 01:18:18.420 --> 01:18:20.910 align:middle line:90% But here, it's general. 01:18:20.910 --> 01:18:26.020 align:middle line:84% You can always do that math to inverse the given number. 01:18:26.020 --> 01:18:27.794 align:middle line:90% You can get the [INAUDIBLE]. 01:18:27.794 --> 01:18:31.280 align:middle line:90% 01:18:31.280 --> 01:18:36.170 align:middle line:84% And the density of states is a mathematical concept. 01:18:36.170 --> 01:18:38.780 align:middle line:90% But see, it's very convenient. 01:18:38.780 --> 01:18:43.280 align:middle line:84% And really, it's the number of quantum states 01:18:43.280 --> 01:18:45.500 align:middle line:90% per energy interval that we-- 01:18:45.500 --> 01:18:49.080 align:middle line:84% and in the parabolic band, it's a problem. 01:18:49.080 --> 01:18:51.300 align:middle line:90% It's a square root relation. 01:18:51.300 --> 01:18:55.010 align:middle line:84% It's obviously increasing as square root. 01:18:55.010 --> 01:18:58.160 align:middle line:84% And of course, below Ec, this is where Ec is. 01:18:58.160 --> 01:19:01.220 align:middle line:84% At the bottom here, there's no states in this region, 01:19:01.220 --> 01:19:04.190 align:middle line:90% in the forbidden band gap. 01:19:04.190 --> 01:19:08.150 align:middle line:84% So that's a typical bulk material [INAUDIBLE] substrates. 01:19:08.150 --> 01:19:11.870 align:middle line:84% And then when you fill, it turns out that most of the time, 01:19:11.870 --> 01:19:16.070 align:middle line:84% there's only a small region because here is kt. 01:19:16.070 --> 01:19:20.170 align:middle line:84% So the electrons are mostly in this region 01:19:20.170 --> 01:19:23.120 align:middle line:90% when you do the measurement. 01:19:23.120 --> 01:19:26.080 align:middle line:84% Once nE becomes very large, that is 01:19:26.080 --> 01:19:28.130 align:middle line:90% [INAUDIBLE] factor very small. 01:19:28.130 --> 01:19:30.270 align:middle line:84% Much larger than kt, this is a very small. 01:19:30.270 --> 01:19:33.420 align:middle line:90% 01:19:33.420 --> 01:19:34.731 align:middle line:90% OK? 01:19:34.731 --> 01:19:38.920 align:middle line:90% So that's what we went through. 01:19:38.920 --> 01:19:44.920 align:middle line:84% And I saw that I was going much faster. 01:19:44.920 --> 01:19:46.120 align:middle line:90% I will do that next time. 01:19:46.120 --> 01:19:48.365 align:middle line:84% This is-- actually, just write down my [INAUDIBLE] 01:19:48.365 --> 01:19:50.740 align:middle line:90% in the original syllabus. 01:19:50.740 --> 01:19:54.370 align:middle line:90% Any questions? 01:19:54.370 --> 01:19:57.670 align:middle line:84% Now, if you're a mechanical engineer, 01:19:57.670 --> 01:20:01.890 align:middle line:84% again, I'm sure this is a [INAUDIBLE] normal mechanic 01:20:01.890 --> 01:20:03.520 align:middle line:90% [INAUDIBLE] talk about this. 01:20:03.520 --> 01:20:06.940 align:middle line:84% And I don't expect you to completely master it, 01:20:06.940 --> 01:20:11.200 align:middle line:84% but I think what you'll get is say, oh, I 01:20:11.200 --> 01:20:13.750 align:middle line:90% know the periodic table. 01:20:13.750 --> 01:20:20.296 align:middle line:84% I know the metal and dielectric difference. 01:20:20.296 --> 01:20:22.840 align:middle line:84% And I know that [INAUDIBLE] vibration 01:20:22.840 --> 01:20:27.585 align:middle line:84% in the crystal quantize and the basic unit of that quantization 01:20:27.585 --> 01:20:29.830 align:middle line:90% of the phonon. 01:20:29.830 --> 01:20:32.800 align:middle line:84% Now you know electron, you know phonon. 01:20:32.800 --> 01:20:35.330 align:middle line:84% And then the thermal conductivity, remember, 01:20:35.330 --> 01:20:37.550 align:middle line:90% comes from electron phonon. 01:20:37.550 --> 01:20:41.130 align:middle line:84% And in thermal electrics, the s squared comes from electron. 01:20:41.130 --> 01:20:45.554 align:middle line:84% So you have to really get this basic picture [INAUDIBLE]. 01:20:45.554 --> 01:20:48.300 align:middle line:90% 01:20:48.300 --> 01:20:49.675 align:middle line:90% Any other questions? 01:20:49.675 --> 01:20:52.930 align:middle line:90% 01:20:52.930 --> 01:20:54.480 align:middle line:90% OK. 01:20:54.480 --> 01:20:55.000 align:middle line:90%