WEBVTT 00:00:00.000 --> 00:00:01.472 align:middle line:90% [SQUEAKING] 00:00:01.472 --> 00:00:03.416 align:middle line:90% [RUSTLING] 00:00:03.416 --> 00:00:04.874 align:middle line:90% [CLICKING] 00:00:04.874 --> 00:00:16.860 align:middle line:90% 00:00:16.860 --> 00:00:18.280 align:middle line:90% JUEJUN HU: OK, so let's start. 00:00:18.280 --> 00:00:20.828 align:middle line:84% So starting in this lecture, we're 00:00:20.828 --> 00:00:22.620 align:middle line:84% going to talk about a different topic here. 00:00:22.620 --> 00:00:25.160 align:middle line:84% We are going to discuss about electrical and transport 00:00:25.160 --> 00:00:28.600 align:middle line:84% properties in amorphous materials. 00:00:28.600 --> 00:00:33.200 align:middle line:84% The reading materials, they cover ionic transport. 00:00:33.200 --> 00:00:36.160 align:middle line:84% There's relatively little coverage on, essentially, 00:00:36.160 --> 00:00:40.080 align:middle line:84% electronic structures and electronic transport behavior. 00:00:40.080 --> 00:00:42.680 align:middle line:84% You basically just have to more or less rely on the slides, 00:00:42.680 --> 00:00:46.700 align:middle line:84% or go to ChatGPT and search for relevant materials. 00:00:46.700 --> 00:00:47.200 align:middle line:90% OK. 00:00:47.200 --> 00:00:50.680 align:middle line:84% So to start with, this slide summarizes 00:00:50.680 --> 00:00:53.320 align:middle line:84% some of the very basics of electrical conduction. 00:00:53.320 --> 00:00:55.560 align:middle line:84% So electrical conduction happens when you apply 00:00:55.560 --> 00:00:57.100 align:middle line:90% electric field in a material. 00:00:57.100 --> 00:01:00.500 align:middle line:84% The charged particles start to move around. 00:01:00.500 --> 00:01:04.640 align:middle line:84% You can have charged particles two different opposite charges. 00:01:04.640 --> 00:01:07.260 align:middle line:84% You can have positively-charged ions, negatively-charged ions, 00:01:07.260 --> 00:01:08.460 align:middle line:90% or electrons. 00:01:08.460 --> 00:01:11.180 align:middle line:84% They will move along different directions in electric field, 00:01:11.180 --> 00:01:13.700 align:middle line:84% but both of them will contribute to positive electrical 00:01:13.700 --> 00:01:14.700 align:middle line:90% conductivity. 00:01:14.700 --> 00:01:16.860 align:middle line:84% So we can see that from the equation here. 00:01:16.860 --> 00:01:18.660 align:middle line:84% This is something you probably have already 00:01:18.660 --> 00:01:20.080 align:middle line:90% learned from general physics. 00:01:20.080 --> 00:01:23.220 align:middle line:90% 00:01:23.220 --> 00:01:25.740 align:middle line:84% Basically, if you look at the electrical conductivity 00:01:25.740 --> 00:01:28.980 align:middle line:84% expression here, n will be the concentration 00:01:28.980 --> 00:01:31.500 align:middle line:84% of the charged carriers, Z is the charge 00:01:31.500 --> 00:01:35.000 align:middle line:84% number, e is the elementary charge, 00:01:35.000 --> 00:01:37.600 align:middle line:84% and mu is basically the mobility. 00:01:37.600 --> 00:01:43.780 align:middle line:90% 00:01:43.780 --> 00:01:50.260 align:middle line:84% Now, one thing to keep in mind is that regardless 00:01:50.260 --> 00:01:53.580 align:middle line:84% of the sign of the charge, whether it's 00:01:53.580 --> 00:01:55.980 align:middle line:84% a positive charge or negative charge, 00:01:55.980 --> 00:01:57.768 align:middle line:84% you always get a positive contribution 00:01:57.768 --> 00:01:59.060 align:middle line:90% to the electrical conductivity. 00:01:59.060 --> 00:02:00.040 align:middle line:90% Which makes sense. 00:02:00.040 --> 00:02:02.397 align:middle line:84% Essentially, what happens is if you apply 00:02:02.397 --> 00:02:04.980 align:middle line:84% electric fields, the positive- and negative-charged particles, 00:02:04.980 --> 00:02:06.680 align:middle line:84% they will move in opposite directions, 00:02:06.680 --> 00:02:09.600 align:middle line:84% but the current direction always aligns with the electric field 00:02:09.600 --> 00:02:10.180 align:middle line:90% direction. 00:02:10.180 --> 00:02:12.080 align:middle line:84% So this is something to keep in mind when 00:02:12.080 --> 00:02:14.920 align:middle line:84% you do the summation to get the total electrical conductivity 00:02:14.920 --> 00:02:16.020 align:middle line:90% in solid material. 00:02:16.020 --> 00:02:19.040 align:middle line:84% The other thing I also want to mention about this equation 00:02:19.040 --> 00:02:20.800 align:middle line:90% here is about this letter here. 00:02:20.800 --> 00:02:26.160 align:middle line:84% So for this equation here, v is the velocity. 00:02:26.160 --> 00:02:30.440 align:middle line:84% It's basically a product of mobility times electric field. 00:02:30.440 --> 00:02:36.720 align:middle line:84% Unfortunately, v looks exactly the same as the Greek letter nu. 00:02:36.720 --> 00:02:40.360 align:middle line:84% There's no way to tell the difference on PowerPoint. 00:02:40.360 --> 00:02:45.560 align:middle line:84% So that's why any letter v that's in the bold form, 00:02:45.560 --> 00:02:47.140 align:middle line:90% this is representing a vector. 00:02:47.140 --> 00:02:49.120 align:middle line:90% This is the velocity. 00:02:49.120 --> 00:02:50.280 align:middle line:90% OK? 00:02:50.280 --> 00:02:53.600 align:middle line:90% Nu is without the bold. 00:02:53.600 --> 00:02:56.120 align:middle line:84% On the board, I'm going to write v with a vector sign. 00:02:56.120 --> 00:02:56.620 align:middle line:90% OK? 00:02:56.620 --> 00:03:01.860 align:middle line:90% Basically, this is our velocity. 00:03:01.860 --> 00:03:04.180 align:middle line:90% This is nu. 00:03:04.180 --> 00:03:05.840 align:middle line:90% This is v. This is our velocity. 00:03:05.840 --> 00:03:06.340 align:middle line:90% OK? 00:03:06.340 --> 00:03:09.860 align:middle line:90% 00:03:09.860 --> 00:03:10.500 align:middle line:90% OK. 00:03:10.500 --> 00:03:12.760 align:middle line:84% So the other thing I also want to highlight here, 00:03:12.760 --> 00:03:13.772 align:middle line:90% this is also something-- 00:03:13.772 --> 00:03:15.980 align:middle line:84% I assume you probably have seen the Einstein relation 00:03:15.980 --> 00:03:18.740 align:middle line:90% in kinetics class, yes? 00:03:18.740 --> 00:03:19.260 align:middle line:90% Yeah. 00:03:19.260 --> 00:03:22.660 align:middle line:84% Essentially, it links two kinds of different kinetic parameters, 00:03:22.660 --> 00:03:23.160 align:middle line:90% right? 00:03:23.160 --> 00:03:27.220 align:middle line:84% One is diffusion coefficient D, the other is mobility. 00:03:27.220 --> 00:03:29.020 align:middle line:84% And the two are naturally linked together 00:03:29.020 --> 00:03:31.140 align:middle line:84% because they both describe how these charged 00:03:31.140 --> 00:03:33.540 align:middle line:84% particles will actually move under the action 00:03:33.540 --> 00:03:35.580 align:middle line:90% of external forces. 00:03:35.580 --> 00:03:40.240 align:middle line:84% In the case of D, it's moving due to a concentration gradient. 00:03:40.240 --> 00:03:43.140 align:middle line:84% So essentially, due to a chemical potential gradient. 00:03:43.140 --> 00:03:46.460 align:middle line:84% In the case of mu, it's moving under the action 00:03:46.460 --> 00:03:48.180 align:middle line:90% of electric field. 00:03:48.180 --> 00:03:53.092 align:middle line:84% So both should be related to how mobile the particle is 00:03:53.092 --> 00:03:54.300 align:middle line:90% and the other external force. 00:03:54.300 --> 00:03:56.217 align:middle line:84% That's why they are naturally linked together. 00:03:56.217 --> 00:03:58.740 align:middle line:90% 00:03:58.740 --> 00:04:00.980 align:middle line:84% And so if you plug everything into the equation here, 00:04:00.980 --> 00:04:03.920 align:middle line:90% this is what you end up getting. 00:04:03.920 --> 00:04:07.800 align:middle line:84% If you want to get the total electrical conductivity 00:04:07.800 --> 00:04:10.040 align:middle line:84% in a material, you just sum up the contribution 00:04:10.040 --> 00:04:12.680 align:middle line:84% from each individual charged species. 00:04:12.680 --> 00:04:16.292 align:middle line:84% And as I said, because this term here, Z, is squared. 00:04:16.292 --> 00:04:18.000 align:middle line:84% And that's why, regardless of whether you 00:04:18.000 --> 00:04:20.140 align:middle line:84% have a positive-charge or negative-charge species, 00:04:20.140 --> 00:04:22.040 align:middle line:84% you always get a positive contribution 00:04:22.040 --> 00:04:24.560 align:middle line:90% to the electrical conductivity. 00:04:24.560 --> 00:04:25.060 align:middle line:90% OK? 00:04:25.060 --> 00:04:26.460 align:middle line:90% Because the Z here is squared. 00:04:26.460 --> 00:04:29.040 align:middle line:90% 00:04:29.040 --> 00:04:29.960 align:middle line:90% Make sense? 00:04:29.960 --> 00:04:30.680 align:middle line:90% OK? 00:04:30.680 --> 00:04:31.800 align:middle line:90% All right. 00:04:31.800 --> 00:04:34.560 align:middle line:84% So what we will do next is let's take a look at now-- 00:04:34.560 --> 00:04:36.540 align:middle line:84% essentially, play divide and conquer. 00:04:36.540 --> 00:04:39.560 align:middle line:84% Let's look at the contribution to electrical conductivity 00:04:39.560 --> 00:04:42.600 align:middle line:84% from both ionic species and the electronic species. 00:04:42.600 --> 00:04:46.775 align:middle line:84% Starting with ionic species, you also probably 00:04:46.775 --> 00:04:48.400 align:middle line:84% have learned about the ionic conduction 00:04:48.400 --> 00:04:51.160 align:middle line:84% mechanism in crystalline materials in kinetics class. 00:04:51.160 --> 00:04:52.760 align:middle line:84% There are two kinds of mechanisms. 00:04:52.760 --> 00:04:54.900 align:middle line:84% You can have a vacancy mechanism, 00:04:54.900 --> 00:04:57.140 align:middle line:84% where essentially, the ions swap position 00:04:57.140 --> 00:04:59.077 align:middle line:84% with neighboring vacancy, and as a result, 00:04:59.077 --> 00:05:00.660 align:middle line:84% this will lead to an electric current, 00:05:00.660 --> 00:05:05.200 align:middle line:84% or the ions can be staying in interstitial sites, 00:05:05.200 --> 00:05:06.700 align:middle line:84% and they can essentially move around 00:05:06.700 --> 00:05:09.580 align:middle line:84% from one interstitial moving to a neighboring one, 00:05:09.580 --> 00:05:11.080 align:middle line:84% and therefore conduct electricity. 00:05:11.080 --> 00:05:16.340 align:middle line:84% So in amorphous materials, it becomes a little harder 00:05:16.340 --> 00:05:20.620 align:middle line:84% to precisely define a lattice site, as well as the vacancy 00:05:20.620 --> 00:05:24.260 align:middle line:84% site, because we do have this kind 00:05:24.260 --> 00:05:27.420 align:middle line:84% of regular crystalline lattice structure. 00:05:27.420 --> 00:05:31.340 align:middle line:84% So instead, the language we want to use here 00:05:31.340 --> 00:05:34.420 align:middle line:84% is that there are actually preferred locations or preferred 00:05:34.420 --> 00:05:36.820 align:middle line:90% sites. 00:05:36.820 --> 00:05:40.620 align:middle line:84% This size gives, essentially, a low energy 00:05:40.620 --> 00:05:42.900 align:middle line:84% when ions are actually populating it. 00:05:42.900 --> 00:05:47.300 align:middle line:84% So then, the electrical conduction process becomes, 00:05:47.300 --> 00:05:49.100 align:middle line:84% essentially, an ion-hopping process 00:05:49.100 --> 00:05:51.850 align:middle line:90% between these low-energy sites. 00:05:51.850 --> 00:05:54.890 align:middle line:84% And this low-energy site, obviously, is not periodic. 00:05:54.890 --> 00:05:57.250 align:middle line:84% If you look at this plot I'm showing you 00:05:57.250 --> 00:06:00.090 align:middle line:84% on the left-hand side here, what I'm showing, essentially, 00:06:00.090 --> 00:06:04.570 align:middle line:84% in yellow color, this labels the low-energy sites 00:06:04.570 --> 00:06:06.650 align:middle line:84% where the lithium ion, in this case, 00:06:06.650 --> 00:06:09.650 align:middle line:84% preferentially sits in the silicate glass. 00:06:09.650 --> 00:06:12.210 align:middle line:84% So you can see that as we increase the lithium ion 00:06:12.210 --> 00:06:16.490 align:middle line:84% concentration, the density of this kind of low-energy site 00:06:16.490 --> 00:06:19.050 align:middle line:84% actually increases in the glass material. 00:06:19.050 --> 00:06:24.490 align:middle line:84% But even for this very high lithium ion concentration, 00:06:24.490 --> 00:06:26.210 align:middle line:84% you notice that these sites, still they 00:06:26.210 --> 00:06:27.530 align:middle line:90% are separated from each other. 00:06:27.530 --> 00:06:29.490 align:middle line:84% They are not completely connected. 00:06:29.490 --> 00:06:31.650 align:middle line:84% So then if you want to have the lithium ion 00:06:31.650 --> 00:06:33.090 align:middle line:84% move from one end of the material 00:06:33.090 --> 00:06:35.465 align:middle line:84% to the other end of material during electrical conduction 00:06:35.465 --> 00:06:38.290 align:middle line:84% process, inevitably, it involves the process 00:06:38.290 --> 00:06:41.010 align:middle line:84% of the lithium ion hopping from one low-energy site 00:06:41.010 --> 00:06:43.170 align:middle line:90% to another low-energy site. 00:06:43.170 --> 00:06:46.730 align:middle line:84% And in that process, they have to go through, essentially, 00:06:46.730 --> 00:06:48.990 align:middle line:90% barriers in between them. 00:06:48.990 --> 00:06:52.290 align:middle line:84% So you can think of the ionic conduction process 00:06:52.290 --> 00:06:56.285 align:middle line:84% also as a thermally-activated process, where the ions will 00:06:56.285 --> 00:06:57.910 align:middle line:84% have to essentially overcome the energy 00:06:57.910 --> 00:06:59.950 align:middle line:84% barrier between these low-energy sites 00:06:59.950 --> 00:07:03.810 align:middle line:90% in order to conduct the current. 00:07:03.810 --> 00:07:04.310 align:middle line:90% OK. 00:07:04.310 --> 00:07:08.090 align:middle line:90% 00:07:08.090 --> 00:07:09.990 align:middle line:84% So what we will do next is that we 00:07:09.990 --> 00:07:12.270 align:middle line:84% will try to essentially develop a model 00:07:12.270 --> 00:07:15.797 align:middle line:84% to capture what is the ionic conductivity in solid materials, 00:07:15.797 --> 00:07:17.130 align:middle line:90% by looking at this picture here. 00:07:17.130 --> 00:07:18.830 align:middle line:84% What we are assuming is we're basically 00:07:18.830 --> 00:07:21.830 align:middle line:84% looking at two neighboring low-energy sites. 00:07:21.830 --> 00:07:24.790 align:middle line:84% There is some energy barrier in between, that E, 00:07:24.790 --> 00:07:26.790 align:middle line:90% this is our energy barrier. 00:07:26.790 --> 00:07:29.150 align:middle line:84% And the ion will have to overcome the energy barrier 00:07:29.150 --> 00:07:32.670 align:middle line:84% in order to transport the electric current. 00:07:32.670 --> 00:07:34.790 align:middle line:84% Now, one thing I do want to mention here 00:07:34.790 --> 00:07:36.630 align:middle line:84% is that because of the statistical nature 00:07:36.630 --> 00:07:39.070 align:middle line:84% of glass structure, when we are referring 00:07:39.070 --> 00:07:42.110 align:middle line:84% to this parameters of the low-energy sites, 00:07:42.110 --> 00:07:45.030 align:middle line:84% such as depths of the well, or the energy barrier separating 00:07:45.030 --> 00:07:47.530 align:middle line:84% two neighboring sites, all those numbers 00:07:47.530 --> 00:07:49.370 align:middle line:84% should be considered as an ensemble 00:07:49.370 --> 00:07:52.118 align:middle line:90% average across solid material. 00:07:52.118 --> 00:07:53.910 align:middle line:84% They are not-- unlike crystalline material, 00:07:53.910 --> 00:07:56.130 align:middle line:90% they are not well-defined. 00:07:56.130 --> 00:07:58.970 align:middle line:84% OK, so now let's consider how we actually 00:07:58.970 --> 00:08:01.970 align:middle line:84% derive the ionic conductivity in this picture here. 00:08:01.970 --> 00:08:06.370 align:middle line:84% We will start out essentially with a random walk theory. 00:08:06.370 --> 00:08:09.610 align:middle line:84% If you consider an ion will be moving around 00:08:09.610 --> 00:08:14.530 align:middle line:84% between neighboring lesser sites, 00:08:14.530 --> 00:08:17.210 align:middle line:84% the average distance it will move, 00:08:17.210 --> 00:08:23.030 align:middle line:84% r, correlates with the distance for every hop. 00:08:23.030 --> 00:08:25.450 align:middle line:84% So d will be the separation between neighboring 00:08:25.450 --> 00:08:26.850 align:middle line:90% low-energy sites. 00:08:26.850 --> 00:08:30.690 align:middle line:84% So it's essentially the distance where for one atomic hop. 00:08:30.690 --> 00:08:31.370 align:middle line:90% OK? 00:08:31.370 --> 00:08:35.049 align:middle line:84% And M is the total number of atomic hops. 00:08:35.049 --> 00:08:43.730 align:middle line:84% So from the random walk theory, the total distance within scales 00:08:43.730 --> 00:08:46.510 align:middle line:84% quadratically with the number of hops, 00:08:46.510 --> 00:08:49.110 align:middle line:84% or square root of number of hops. 00:08:49.110 --> 00:08:50.610 align:middle line:90% OK? 00:08:50.610 --> 00:08:53.900 align:middle line:90% 00:08:53.900 --> 00:08:57.390 align:middle line:84% Now, the other equation that you probably also 00:08:57.390 --> 00:09:00.670 align:middle line:84% have seen in your kinetic class is that the average diffusion 00:09:00.670 --> 00:09:03.470 align:middle line:84% distance, this is correlated with the-- this 00:09:03.470 --> 00:09:05.370 align:middle line:84% is given by this expression here, 00:09:05.370 --> 00:09:09.390 align:middle line:84% where tau is the time of the diffusion, 00:09:09.390 --> 00:09:11.750 align:middle line:84% and D is our diffusion coefficient. 00:09:11.750 --> 00:09:16.150 align:middle line:90% 00:09:16.150 --> 00:09:17.830 align:middle line:84% So this is basically the definition 00:09:17.830 --> 00:09:19.590 align:middle line:90% of the diffusion length. 00:09:19.590 --> 00:09:22.950 align:middle line:84% For 1-D, the coefficient is 2, for 3-D, the coefficient is 6, 00:09:22.950 --> 00:09:26.190 align:middle line:84% because you have to diffuse along three different 00:09:26.190 --> 00:09:27.230 align:middle line:90% directions. 00:09:27.230 --> 00:09:29.870 align:middle line:84% So now, if you combine this equation here, 00:09:29.870 --> 00:09:32.850 align:middle line:84% these two equations here, for example, 00:09:32.850 --> 00:09:38.870 align:middle line:84% if you take the simple 1-D case, so you have two equations, 00:09:38.870 --> 00:09:44.730 align:middle line:84% basically r equals to d square root M, 00:09:44.730 --> 00:09:51.690 align:middle line:84% we know that r also equals to square root 2 tau D for the 1-D 00:09:51.690 --> 00:09:54.050 align:middle line:90% case, right? 00:09:54.050 --> 00:09:57.250 align:middle line:84% So then you can then, essentially, 00:09:57.250 --> 00:10:02.450 align:middle line:84% from that you can then calculate D. 00:10:02.450 --> 00:10:07.410 align:middle line:84% This would equal to, you do a square on both sides, 00:10:07.410 --> 00:10:14.810 align:middle line:84% becomes 1/2 M divided by tau, and then d squared. 00:10:14.810 --> 00:10:16.530 align:middle line:90% Right? 00:10:16.530 --> 00:10:21.410 align:middle line:84% Now, M is the number of hops, and tau 00:10:21.410 --> 00:10:24.130 align:middle line:90% is the duration, all the time. 00:10:24.130 --> 00:10:27.450 align:middle line:84% So M divided by tau essentially gives you the frequency 00:10:27.450 --> 00:10:29.730 align:middle line:90% of this kind of atomic hops. 00:10:29.730 --> 00:10:33.730 align:middle line:84% So that's where we're now introducing our Greek letter 00:10:33.730 --> 00:10:36.970 align:middle line:90% nu d squared. 00:10:36.970 --> 00:10:42.590 align:middle line:84% So nu will be the frequency of this successive ionic hops 00:10:42.590 --> 00:10:44.730 align:middle line:84% between the neighboring low-energy sites. 00:10:44.730 --> 00:10:48.070 align:middle line:90% 00:10:48.070 --> 00:10:50.450 align:middle line:84% OK, so if you look at this equation here, 00:10:50.450 --> 00:10:55.110 align:middle line:84% we are essentially simplifying the diffusion coefficient 00:10:55.110 --> 00:10:58.750 align:middle line:84% into this hopping frequency, nu, here. 00:10:58.750 --> 00:11:02.110 align:middle line:84% So all we need to do in order to derive ionic conductivity, 00:11:02.110 --> 00:11:05.242 align:middle line:84% if you look at the expression here, if you know D, 00:11:05.242 --> 00:11:06.950 align:middle line:84% you can then use it to actually calculate 00:11:06.950 --> 00:11:08.870 align:middle line:90% the ionic conductivity. 00:11:08.870 --> 00:11:09.410 align:middle line:90% OK? 00:11:09.410 --> 00:11:12.190 align:middle line:84% So what we need to do is then to predict 00:11:12.190 --> 00:11:15.430 align:middle line:84% what is the frequency of successive hops 00:11:15.430 --> 00:11:17.510 align:middle line:84% between the neighboring low-energy sites. 00:11:17.510 --> 00:11:19.420 align:middle line:90% OK. 00:11:19.420 --> 00:11:22.990 align:middle line:84% Now, this is actually quite straightforward 00:11:22.990 --> 00:11:25.710 align:middle line:84% because this is essentially a process where it's thermally 00:11:25.710 --> 00:11:27.470 align:middle line:90% activated. 00:11:27.470 --> 00:11:29.110 align:middle line:90% Essentially, you have an ion. 00:11:29.110 --> 00:11:31.550 align:middle line:84% It's constantly vibrating at this low-energy site 00:11:31.550 --> 00:11:34.750 align:middle line:84% with a vibrational frequency of nu 0. 00:11:34.750 --> 00:11:37.510 align:middle line:84% Now, every time it vibrates, you attempt 00:11:37.510 --> 00:11:41.290 align:middle line:84% to overcome the energy barrier and hop to a neighboring site. 00:11:41.290 --> 00:11:44.290 align:middle line:84% Now, this kind of attempt is not always successful. 00:11:44.290 --> 00:11:47.850 align:middle line:84% So the probability of having enough energy for the ion 00:11:47.850 --> 00:11:51.290 align:middle line:84% to overcome the energy barrier will be given by the Boltzmann 00:11:51.290 --> 00:11:52.810 align:middle line:90% factor here. 00:11:52.810 --> 00:11:55.770 align:middle line:84% So according to this Arrhenius law, 00:11:55.770 --> 00:11:58.650 align:middle line:84% basically it says that the frequency of successful hops 00:11:58.650 --> 00:12:01.690 align:middle line:84% is going to be given by attempt frequency 00:12:01.690 --> 00:13:30.060 align:middle line:84% or the vibrational frequency times the [AUDIO OUT] 00:13:30.060 --> 01:13:44.000 align:middle line:90%