12. Band Theory of Solids | CSIR-NET Physics & GATE | Nearly Free Electron Model & Bloch Theorem

Added: 2026-05-10

[00:00:05]Right.

[00:00:11]Ready.

[00:00:13]>> So, good morning all of you.

[00:00:14]>> Good morning, sir.

[00:00:16]>> The summary of the last lecture we have discussed this uh huh right free electron theory of metals. So we have successfully completed this uh free electron theory which can explain the specific heat of solids. Am I right?

[00:00:37]Good. Now take a fresh face and put a heading band theory of solids.

[00:00:44]Band theory of solids.

[00:00:52]This is a very important definite question. No doubt about it. either band theory of solids or free electron theory of metals definitely you'll get a question very important listen carefully so what is this band theory of solids first why they have introduced this that means what is the purpose of introducing this uh band theory of solids study ready so till now we have taken free electron theory say yes or no >> yeah free electron theory till now we have discussed what is the meaning of free electron elect that means you are assuming the electron is moving freely inside the latice that is nothing but a free electron theory am I right whenever it is a free the potential energy equal to zero and the energy is only kinetic energy non-relativistic that's why h cross² k² by 2 m so finally if you treat the electron is moving inside the latis as a free particle its energy is a proportional to water k² we can say the energy is a proportional to k² cool now I'm taking the graph energy versus k this is the energy and this is the k I have taken so one thing is very clear energy is a proportional to k square means the curve is parabola say sr no this is a curve we What if you treat the electron as a free particle? Is it a discrete spectrum or continuous spectrum?

[00:02:35]>> That's good. So we got continuous spectrum.

[00:02:44]Very nice. So these are these are the results I mean till now what we have done in the free electron theory. Then what is the problem here? Let's see.

[00:02:53]Actually this a free electron theory a free electron theory can explain can explain thermal conductivity thermal conductivity I think we didn't explain this thermal conductivity and specific heats this one we explained clearly specific heats of solids specific heats of solids uh successfully.

[00:03:29]It's not a failure theory. It's a successful theory only by which we can explain the thermal conductivity as well as specific heats of solids. But unfortunately, but this could not explain this could not explain explain the behavior of the behavior of the behavior of solids.

[00:04:02]What behaviors are? Yes, some solids are behaving like a conductors, some are semiconductors, some are insulators.

[00:04:12]How these solids become insulator or conductor or a semiconductor on what basis? So this classification came this was not explained by a free electron theory. I hope you understand what is the failure of a free electron theory.

[00:04:27]This could not explain the behavior of solids. What is that conducting behavior?

[00:04:34]How conductors we got? Or next one in sorry semiconductors anything. Okay.

[00:04:41]Semiconductors and insulators.

[00:04:48]Oho fine.

[00:04:51]So why this solid? Ironizer solid. This is also solid. Why this is bea be behaving like insulator? Why iron is behaving like a conductor? What is the reason for that? But this was not explained by free electron.

[00:05:07]So then they started what is the reason behind this?

[00:05:12]Then they thought that actually here we have treated electron is a completely free. But if you observe carefully, if you observe carefully, this is the latis point. Listen, this is the latis point.

[00:05:27]I mean lat these are the latice points and at each latice point you have a ion core say yes S or no that two positive ion core that's why plus J nucleus nucleus plus JD E positive ion core means nucleus plus J E plus J E something like this you have at each lattice point you have a positive ion core which is a fixed to the latice. Now what you are saying your electron is moving in the latice but if you observe carefully the electron is moving in the periodic latice or it is moving in the latis where we are having positive ion coursees. So the electron is coming close to the positive ion core again going away again coming close again going away like that. That means this electron is not at all a free particle.

[00:06:25]It is moving in the potential interaction or it is moving with the potential interaction of nucleus or ion core. Free means what? Potential is a zero. Whenever the electron is moving or clo coming close to the nucleus, what is the potential here? The potential at the nucleus we can say J E by 4 pi epsilon KN.

[00:06:53]Yeah, that potential is there. Then why you are saying your potential is completely zero.

[00:07:00]Okay. So now we are going to treat the electron is not at all free. The electron is moving in the periodic potential.

[00:07:11]>> Periodic potential >> potential is repeating. See observe carefully maybe increase maybe decrease but whatever it is it is repeating after certain intervals s why that intervals came your latice structure your latice structure is having a periodic behavior that's why your potential is also having a periodic behavior that's good so finally what you understand actually electron is not free electron is not free.

[00:07:44]Okay. So, electron is moving is moving in the periodic potential in the periodic potential of the latis of the latis.

[00:08:02]1 second.

[00:08:05]Okay. So if I assume this just I'm taking one one nucleus. This is the potential V of R potential energy and this is R. Okay. So everybody know what is the potential? The potential equal to potential energy. I'm talking about energy. This is the potential and electron is moving in that potential.

[00:08:26]That's why minus JD E² by 4 pi epsilon KN I hope so very this side no response at all and here we have the nucleus plus J and the electron is coming close to this nucleus then we are having this interaction say yes or no then the graph will be attractive that's why negative like this so this will be the potential curve at the nucleus.

[00:08:59]So, but it's a latice is a combination of so many positive ion coursees. That's why we have to assume this is the potential energy everyone.

[00:09:13]This is R. I hope you are observing this is the positive ion core plus JD and again this is also plus JD and this is also plus JD this is also the positive ion core something like this.

[00:09:40]Now if I if you consider the entire latis in one dimension we can't draw three dimension in one dimension it's look like this. Now your electron is moving in this potential.

[00:09:53]So what are the potentials very I mean what are the potentials it is experiencing. See are you ready? So this is the nucleus. This is also nuclear site. This is also nucleus. Vicinity of the nucleus.

[00:10:09]Ready? So just now I have drawn for one nucleus quite similar to that remaining also you can draw. So this is the nucleus.

[00:10:18]I hope you are observing say no this is the nucleus. Okay that's why this is the nucleus at every nucleus just I'm drawing this graph.

[00:10:34]I hope you are observing this is the nucleus this side and this side. This is the nucleus this side and this side.

[00:10:47]Okay. So if you just combine this just like this.

[00:10:53]Now what happened here at the nucleus in the vicinity of the nucleus you got a potential well and after that you got a potential barrier and again potential well and barrier like this.

[00:11:12]So this is the vicinity of the nucleus.

[00:11:14]The electron is bounded to the potential level. Okay. And after that we are having a barrier and it is crossing the barrier and again going to the >> potential well. So it is moving in a either potential well or a barrier. So what we are assuming is we are assuming let in the vicinity of the nucleus >> in the vicinity of the nucleus >> electron is free >> and the barrier outside height I'm taking it as v not how much >> that's it because this is the barrier height and I'm assuming this electron potential energy is zero that's why whatever you are studying this is called nearly free electron theory.

[00:12:03]>> Electron theory >> sir why this word is taken because we are assuming again in the vicinity of the nucleus the electron is free that is assumption we are taking to study without any assumption we can't proceed ideal case always we will study after that we can implement is that clear so I hope everybody understand what exactly happening in the latis so this is the free electron And the energy we got continuous. This is wrong. Okay. If it is a continuous all the solids must be conductors. All the solids must be conductors. But it's not correct. But some solids are conductors. Some are semiconductors. Some are insulators.

[00:12:48]So what's happening here? Maybe some energy bands are forming. What is the reason for that? Then whatever you assumed is wrong. The actually electron is not at all free completely. So it is moving in the periodic potential of the latis like this and that to one more assumption we made in the vicinity of the nucleus again we assumed it as a free particle that means just like a free particle in a 1D box clear come on That's why band theory of solids is also called as nearly free electron theory.

[00:14:05]Not completely free.

[00:14:59]So in the vicinity of the nucleus we got the potential well and after that we have potential barrier. So this is the potential well and this is the barrier.

[00:15:27]So finally we are going to show that how there exist a discontinuous in the energy bands. Okay, it's not at all continuous. There must be some discontinuous.

[00:15:39]Let us see.

[00:17:02]So one more assumption we made here in the vicinity of the nucleus electron is assumed to be free.

[00:17:25]That's why it is called a nearly free electron theory.

[00:19:45]So finally in the band theory of solids we are assuming the electron is moving in the periodic potential of the latice.

[00:19:52]How that potential is varying? Zero in the vicinity of the nucleus outside the vicinity V not some potential barrier.

[00:20:00]So 0 V 0 V not like that. So periodic potential let us see right stop.

[00:20:31]Okay. So now this is the assumption in the band theory of solids. We are assuming the electron in the vicinity of the nucleus is free but outside the nucleus it is not free bound bound particle laser. So this is our latice okay now I'm considering this is the nucleus of course. So here we have the potential well say no. Okay listen babu very important I'm taking the potential well and after that we have a potential barrier oh you have to take a little bit far okay so now I'm taking in the vicinity of the nucleus what you are having a potential well say no this is the vicinity of the nucleus this here somewhere. So again beyond the nucleus I mean outside the nucleus we have potential barrier and again potential well in the nucleus vicinity potential barrier well something like this it's a keep on varying cool now I'm assuming uh this is the potential well anything you can take shall I take this one yeah potential well I'm taking this is the potential well that means here we have the nucleus vicinity I'm taking this is a zero and a that means I have taken width of the well as a and this is the barrier listen carefully barrier potential V I'm taking this width as B 0 minus of minus B B Understand what is the width of the barrier.

[00:22:30]>> What is the width of the potential?

[00:22:32]Well, >> that's good. If this is a 0 to a, this is what barrier. Barrier height is what?

[00:22:41]>> 1 second. Barrier width must be b.

[00:22:45]>> This is already a that's why.

[00:22:49]>> Understand? Next one. This is the potential well.

[00:22:53]>> Potential.

[00:22:54]>> Potential.

[00:22:56]Now potential well with >> that's why it is already a plus b that's why everybody understand this point like this cool now carefully observe your potential is varying periodically so what is the periodicity first find this potential potential is varying is varying periodically.

[00:23:29]Now what is the periodicity?

[00:23:37]Now carefully see all of you.

[00:23:40]Is it a well?

[00:23:43]>> What is this side?

[00:23:45]Huh?

[00:23:46]>> Now again the first end of the well is here. Say yes or no.

[00:23:55]Okay. So your potential and one more this end second end is at >> and second end of the well exist again at >> that means your well potential zero v =0 exist from 0 to a 0 to a and again and again exist from a + b to a + Your potential zero exists from 0 to A to >> I mean 0 to A and >> A + B to >> A >> that means ZERO BECOME >> A become a b >> after what interval it's a repeating value >> that's it I hope everybody understand so the periodicity is nothing but how much >> that's So we can say one thing this is the periodic potential with the periodicity.

[00:25:01]What is the periodicity?

[00:25:03]>> This is definitely again equal to what?

[00:25:08]>> So your electron in the latice is moving in the periodic potential of periodicity.

[00:25:15]>> That's good.

[00:25:22]One minute.

[00:25:34]Okay.

[00:25:35]So fine, very good. But the problem with uh this model is what we have to find the wave function of the electron. We have only one choice that is the shinger equation. We can't find shinger equation each and every reason. How many showing this equation you will write? 10 to the^ of 23 you will write. No, it's not possible. That's why here what we are going to do. Are you ready? Right now we have to take the showing users equation.

[00:26:08]I hope everybody understand this diagram. What is happening here? This is the first end of the potential well and again repeating at this point that means after a + b. So every x is repeating after x + a + b. I hope everybody understand this point. Am I right? Good.

[00:26:31]That's why your potential is a periodic potential with the periodicity a + b.

[00:26:38]Now I have to take the shinger equation.

[00:26:40]Let us start shinger's equation.

[00:26:45]Before going to enter into this periodic potential, let us go back to the free particle and again we'll discuss that.

[00:26:52]Wait now I am assuming free particle free electron. What is the potential?

[00:26:58]Everybody say now come on what is the shinger equation? do ² by dx² + 2 m e by h +² = good now I assume this thing is nothing but what k² am I right that's why we will write s of x is nothing but is it a plane wave say yes or no good it's a plane wave Okay. The solution of Shinger equation of a free particle. What is the solution? Plane wave. What is that plane wave?

[00:27:47]>> Now but actually our electron is moving is moving in a periodic potential.

[00:27:59]Am I right or wrong? Periodic potential.

[00:28:04]of periodicity a + b then what to do here we are going to introduce blocks theorem what is that blocks theorem so what is the use of a blocks theorem if you know the solution in a particular region you can get the solution after the periodicity >> after the periodicity >> for example you Know the potential sorry you know the solution from 0 to a after the periodicity which reason you are getting this solution you can find by using blocks theorem. So blocks theorem is a simply gives you the shinger equation solution of a particle moving in periodic potential 1 minute. This gives the solution of shinger's equation solution of shinger equation of a particle of a particle moving moving in periodic potential.

[00:29:21]So what is this blocks theorem? Are you ready? The blocks theorem is given by is given by s of x = what is that? x= >> this is the solution of shinger equation of a particle moving in periodic potential. Clear?

[00:29:40]Now I am assuming because this is the theorem not the problem. I'm assuming periodicity is one a periodicity is how much I'm taking a >> a that means come on say v of x + a = v of x I'm not doing the problem I'm explaining the blocks theorem I'm assuming the particle is moving in this potential with the periodicity a then what is THE SOLUTION ARE YOU READY S OF X = u k of X of X >> into E RA to I uh I sorry 1 minute s of X = U K ofX into E RA to I K X 1 minute u K ofX = uh E R to I Kx X one minute. So what is this? This is called block function.

[00:30:49]>> Block function.

[00:30:50]>> Wait, what is this?

[00:30:55]>> Solution. Earlier earlier free particle solution something solution. This is the solution of uh uh showing us equation.

[00:31:03]Listen carefully. Listen carefully. So C of X equal to U K of X into EA to I K X.

[00:31:12]Now this block function u k of x having periodicity having periodicity of of the uh latice or potential latis or potential.

[00:31:35]So that means if you substitute uk what is the periodicity you have taken a uk of x + a equal to uk of x.

[00:31:48]So if you want to find the solution after a that means s of x + a = u k of x + a into e ra to i k x + a. This is having periodic property. Again it is coming uk of x into e ra to i kx into e to i k a. But what is uk of x into e to i kx your s of x. Finally what happened?

[00:32:22]S of x + a = s of x into e to i k a i once again. See this is the blocks theorem. Bloxs theorem use is simply if you know the solution in one region you can get the solution in another reg reason after a periodicity >> you know s of x I want to find s of x at x + a clear once again see just remember this is not derive we can't derive all these things okay just remember the statement what is the blocks theorem x = x 1 second. So this is the plane wave earlier whatever we have assumed if it is a free. So that is a free sol free wave uh plane wave that is a free particle solution. So if a particle is moving in a periodic potential they introduced a small function called block function. What is that block function?

[00:33:29]>> That block function is having >> or periodic properties of the latice.

[00:33:34]What is that? UK of >> UK.

[00:33:39]>> Wait, what I'm saying is this is my solution at a particular region. Now I want to find the solution after a periodicity a come on s of >> i + split it e to i kx into Now u of x + a equal to >> then uk of x into i to ikx is nothing but >> that's it. So s of x + a after a periodicity a = s of x into e to i k a.

[00:34:26]Now come to this. What is your periodicity?

[00:34:30]>> Now I know the solution side. I want to find the solution after a periodicity a + b. Tell me s of x into e ra to i >> and they over that's it. That's the blocks theorem. It simply tells you if you know the solution in one region after a periodicity what could be the solution of shooting equation.

[00:34:59]Understand? Still not understanding I'll explain. Come on right now.

[00:35:11]Always remember one thing in physics.

[00:35:14]Whenever you are discussing a new thing, always they will modify the old one.

[00:35:20]This is the free particle wave function.

[00:35:22]They multiplied with the block function.

[00:35:24]What is your doubt? Stand up. Why you are asking him?

[00:35:28]Don't have manners at all. What is your doubt? Ask. He knows. He came to teach here.

[00:35:38]Ask.

[00:35:44]So we get the quantum mechanics. We got solutions.

[00:35:51]But we are getting.

[00:35:54]>> Did I solve anything till now? Then why you are asking? Did I apply shing equation till now? Then why you are asking?

[00:36:03]I didn't apply any shinger equation to this. Did I apply? Sit down.

[00:36:14]I have not taken any shinger equation for this. Then why you are talking about that?

[00:36:19]This is I have written free.

[00:36:22]Okay. This we know. Now we want to get the solution of a particle moving in a periodic potential. That is I'm explaining I clearly said he won't listen properly. I am not doing the problem. This is what I said without listening properly. Simply asking on that day also I clearly tell in the textbooks they will write multiplied but I am taking double because of it is a single electron. Alkaly atoms we are studying.

[00:36:53]No you have to listen properly.

[00:36:57]Maximum if you are listening properly each and every sentence you won't get the doubt if you are getting a doubt I'm waste that means I didn't explain properly that's why each and every point I will explain while I'm you people won't listen properly what we will do mind absent I clearly said I'm not doing the problem we are explaining the blocks theorem Come.

[00:37:48]I don't want get any student doubt while explanation is going on because I want to explain what is this, what is this.

[00:37:54]I'm clearly explaining in the order and you are jumping to the next step. What I will do it's not correct.

[00:38:02]Wait after the task.

[00:38:26]So the blocks theorem is just an idea to get the solution of shinger equation of a particle moving in a periodic potential because we know the solution of a free particle that is modified with block function that block function is having the property of periodicity.

[00:39:08]Focus on what I am explaining in your mind. If you are thinking something else ways to sit in the class what I'm explaining, focus that's enough.

[00:40:11]very very very important. blocks theorem gives the solution of uh shinga's equation moving equation of a particle of a particle moving in a periodic potential. potential in the theorem I am assuming this potential I'm not doing the problem again I will come back to the problem just wait to understand blocks theorem again I have taken some potential of periodicity a we are searching what theorem will explain the sol the solution if you know the solution I want to find solution here here here everywhere but I can't solve these many times and similarly if you know the solution here you can get here you can get here like that so what is a block theorem it's just modification of the free particle solution with a blocks function that blocks function is having a property of periodicity. Whatever the periodicity of latice or potential periodicity of potential Slow, slow, slowly. Understand.

[00:42:30]Ready?

[00:42:33]Ready?

[00:42:59]Right. Once again, listen.

[00:43:03]We don't have any shoringer equation solution readily with us. If you know the solution here after repetition of space after repetition of space and time of periodicity again we can't solve the shinger equation like that if you want to solve infinite shinger equations you have to solve that's why we are searching for any other thing we have a block theorem.

[00:43:26]So what this blocks theorem is saying laser according to the block theorem the wave function of the shinger equation s of x of a particle moving in a periodic potential >> moving in a periodic potential >> equal to they have taken plane wave this is the free particle are you observing this is the free particle solution free particle solution and just it is MODIFIED WITH FUNCTION CALLED BLOCK function and this is a free particle solution e ra to ikx example we are taking if it is moving in a potential v you know how to solve that you know that energy greater than v energy less than I'm not entering into the problem please listen okay and this is the block function still if anybody didn't understand please listen block function this block function sir where you have introduced the periodicity. This block function is having the periodicity. What is that? U K of X + A = UK.

[00:44:37]>> Is that clear? Now see okay for example I would like to find the solution after a periodicity A. That means X gives I mean X ts to X + A. Then uh s of x + a = u k of x + a e to >> x + a. This is what I'm saying. Listen.

[00:45:08]Okay. This is already having the periodic property.

[00:45:13]Now it is a e to >> this is the wave function after a periodicity a this is nothing but so this is the solution in a particular region after periodicity and this is the solution known solution known solution into e to i k a where a is what >> that's it and this is the wave function after wave function in a region in a region after periodicity after periodicity uh a and this is the known own solution.

[00:46:14]Now come to our problem. This is BLOCKS THEOREM. EVERYBODY UNDERSTAND? YES SIR.

[00:46:19]>> SURE. YES SIR.

[00:46:20]>> GOOD.

[00:46:23]Now for example I know the solution s of x after a periodicity b. What is the solution? S of x and that s of x is the known solution which you solved by using shinger equation. Everybody any doubts m now come to the problem babu babu okay it's very clear we have potential barrier and a potential well again barrier potential well like this this is the zero this is a v knot this is a v knot now we are studying our electron is moving definitely ly its energy is less than barrier potential because the nuclear potential is very high. So whatever the electron is moving with some kindinetic energy uh definitely less. How do you know sir? Beta decay.

[00:47:23]In the beta decay the electron which is coming out of this atom is how much? Uh 7 to 8 m.

[00:47:32]7 to 8 m. Okay. So 7 to 8 mu kinetic energy which is very very very very less compared to the nuclear potential energy. So that's why the energy of the particle is less than v. Here we have taken potential is zero. Come on. This is a zero a minus b a a + b. Am I right?

[00:47:59]Okay enough. Now first I'm considering this. Now I'm applying shinger equation to the problem to the problem. Now first one 0 less than x less than a I'm going to find the solution.

[00:48:15]Now here the potential is come on shinger equation.

[00:48:23]/= >> now I am assuming this as alpha square >> that's why shall I write the solution plus or minus >> here >> here plus or minus then the solution sinosidal exponential = a e to alpha x + b e 2 - alpha x i kx minus i kx here we have taken as alpha square over that's good second one I'm taking minus b less than x less than zero now this is the reason we are studying our potential is v kn but unfortunately less than e energy of the electron so showing us equation do² i I dx² + 2 m e - v but this is a negative that's why what we will do minus directly I'm writing because you know already this is not quantum mechanics >> v not greater than e comma 2 m Huh? V - E by H cross² S = Z.

[00:49:59]Now I'm assuming this entire thing as beta².

[00:50:05]Okay. Then the solution exponential or sinosidal?

[00:50:10]>> C E R to beta X + D E R to minus beta.

[00:50:17]>> Now your doubt is clear. This is a solution. Now tell me >> huh?

[00:50:27]>> Where?

[00:50:48]>> How it will be? friendships, relations.

[00:50:52]As long as you like any person, although he's cold, although she's cold, we won't mind. Okay? Impression. That is called impression. Your best friend, you fighted. Next day, what you will do again? You will because you like them so much. If you don't like one person, whatever he say, you won't like. Huh?

[00:51:13]It's a true relations, anything. If you are sitting in the class with a negative mindset, it will be like this only.

[00:51:26]I don't know how many of you observe my intention while I'm explaining no student should get the doubt in the explanation. Yes, I don't know how many of you observe while I'm explaining in the meanwhile I will ask shall I ask a doubt? Shall I ask a doubt? All those doubts are asked by the earlier students earlier batches because while I'm explaining one student got a doubt means maybe next batch also some students will get the same doubt.

[00:51:51]If it is a valid I will explain if it is a not valid we won't explain.

[00:52:04]If you don't like a person, although he didn't say anything, he didn't scold but you feel he scolded us. That's the point.

[00:52:14]That means those people sit on a branch and they will cut their own branch. They will understand after few years.

[00:52:26]Come on.

[00:52:28]This is the just showing us equation solution of the given potential barrier or given periodic potential. And one more thing our periodic potential what is this? V of X + A B equal to V of X.

[00:53:07]If you listen with 100% concentration, I will explain each and every point. Maybe I won't give the notes but I will explain maximum while you are solving the problems only you will get the doubt.

[00:53:27]after explanation also if you are getting it out means I didn't explain properly one more time I'm reminding you whatever we are doing is what the nearly free electron that means what is the assumption we made in the vicinity of the nucleus electron is assumed to be free. Please remember that free mind and sit in the class. If you are thinking something different, you won't understand what I'm explaining.

[00:54:56]Sit down. Sit down. First, make your mind free and sit in the class. After that, read. Sit down. Sit down.

[00:55:32]Right.

[00:57:02]One more time please see that blocks function I mean blocks theorem. Bloxs theorem will tell you if you know the solution of shinger equation in a particular region. If anybody ask what is the blocks theorem don't write s of x equal to u k ux sorry u k of x e ra to ikx please tell them if we know the solution of shinger equation in a particular region after the periodicity of the potential >> periodicity of the potential >> or a region after the periodicity of the potential we can say the solution simply earlier solution into e to i k periodicity T now please listen stop writing a mark please listen please listen please. Okay cool.

[00:57:55]So here potential zero we know the solution sinosidal potential V not of course energy of the electron is definitely less than V not and we got this solution that is a exponential.

[00:58:09]Now everyone see I would like to find the potential in this region.

[00:58:16]Okay. So here the potential is V and again repeated V after interval of - B a that means A minus of minus A + B otherwise 0 A + B a + b minus 0.

[00:58:35]Everybody understand? So this is a region having the periodicity of x + >> say s or no that means v of x + a + b equal to >> x according to blocks theorem blocks theorem the wave function after the periodicity x + a + b = s of x. This is the known known into e to i k periodicity.

[00:59:19]So what would be the solution in this region? in this region. So in this region solution s of x equal to known solution understand in this region you know the solution in this region we are finding the solution known solution into e to i k of so that is your uh solution after The periodicity a + b understand clear right any doubt?

[01:00:01]Ah s of x written a huh that is the in this region. Okay I will give the numbers just wait. Okay fine uh >> I bet k not beta k plane wave plane wave k here beta we defined something else.

[01:00:22]Okay. No, K only. K only. Ready?

[01:00:26]It's not uh beta. It's not alpha. It is a K only. Or maybe alpha. Maybe alpha but not at all beta. When it is alpha, we'll tell you please wait. Ready? Ah, stop. So his doubt is okay. Valid doubt.

[01:00:40]He got a doubt here. Is it a beta or K?

[01:00:44]So it's a K only. K means what? Under root 2me by HR square. It may be alpha.

[01:00:52]When it will be alpha, just wait. Just wait. But it is not at all beta because blocks theorem contains the solution of a free particle. Free particle from that only we derived. Is that clear? Good. So it's a k only not at all beta or not at all alpha. Is that clear? Right. Now once again I'm taking at x =0 at x =0 x equal to0 means this one >> this is my s1 and this is my s2 I'm taking the boundary conditions now first one s1 equal to s2 where >> s1 you know s2 you Come on. Say a + b. Put x =0 in s1 and s2 + >> c plus. Next second one differentiation do s1 by dox = >> very few people are saying at x = 0. Now once you differentiate this I alpha. Am I right? Huh? I alpha A - B and this is beta C.

[01:02:14]>> See, >> that's good.

[01:02:18]So, a - b = - iota beta by alpha c - d equation one and equation two. You can rearrange this uh I mean once you uh okay you'll get four equations a plus b equal to c plus d and here a minus b equal to this one like that one more boundary you have to take shall I take x= to a x = a ready x = a x = a this is my s2 let us say this is my s3 now Okay. Ready? Start. S2 at x= a. x = a h a e to alpha a plus b e >> good equal to just s just s c e ra to beta a plus d e^ - beta a into >> plus. Next one. Second condition, differentiation. Shall I do it? If you do the differentiation, I ala >> come on. A E R to I alpha A minus B E R to - I alpha A slowly. Now this is a beta C E R to beta A minus D E R to minus beta A into E R to I K.

[01:04:05]That's it. We got four equations 1 2 and three and a four.

[01:04:13]Okay fine. H then what to do next?

[01:04:18]We have to solve these equations. Solve these equations means trivial solution is all coefficients a b c d all are zero. So non-trivial solution non trivial solution that means what I will take coefficients of determinant are a b c d equal to zero.

[01:04:45]Just take the coefficients of a b c d and take the determinant is equal to zero. Four equations are there. Now take the coefficient of a. For example, one uh next >> one one c also this one. This one 1 >> - one - 1 by alpha c minus d. Is it okay? But what about that?

[01:05:12]Where you will write that?

[01:05:16]You have to take the coefficients of a, b, c, determinate zero. You have four equations where you will write that is my doubt.

[01:05:26]>> Huh?

[01:05:29]>> Two determin if you add this what you will get a in terms of c and d. If you subtract what you'll get B. If you add that there also you'll get uh no nothing you won't get.

[01:05:53]Huh?

[01:05:58]Oh 4x4 2x2 4x4 matrix 2x2 SS 4x4 4x4 no 2x2 only I think okay come on right now how we will take we are not solving this we are not solving just I'm saying coefficients of uh uh a b c d how we will Take we have to add once subtract once add once subtract once then you will get here a b and there also you'll get AB in terms of CD. You'll get AB in terms of CD. Then what you will do?

[01:06:58]Yeah, here also you'll get you can bring I alpha this side.

[01:07:02]In the quantum mechanics also we'll do the same thing. We'll bring that and we will add once and subtract once.

[01:07:09]Any textbook is do you have text book?

[01:07:11]Please check. He didn't solve. He has written that uh determinant.

[01:07:17]We are not solving actually.

[01:07:22]4x4 written. Ah. Oh, 4x4.

[01:07:26]Sorry.

[01:07:28]4x4. Correct.

[01:07:36]Okay.

[01:07:41]So, four equations. Uh, A, B, C, D.

[01:07:43]Four. Um, yes. A1 B1 C1 D1 A2 B2 C2 D2 I thought that it's only 2x2. No, sorry.

[01:07:56]C3 D3 A4 B4 C4 D4 Please check the answer. Alpha square + beta square by 2 alpha beta sin hyperbolic alpha a cos alpha square beta square minus alpha square by 2 alpha beta what he assumed alpha and beta check that also maybe I'm not following that book same final final expression will be there alpha square + beta square by to alpha beta sin hyperbolic alpha a something hyperbolic in terms of hyperbolic sign not there.

[01:08:52]Okay. Okay. Leave it right.

[01:09:00]Huh?

[01:09:04]Alpha square + beta square by 2 alpha beta sin hyperbolic they will take it as a p that alpha square plus beta square by 2 alpha beta they will take it as a p finally a graph will be there okay come Okay Bob that's uh here 1 one minus one minus one and the same thing one and like that. So take the coefficients of ABCD abcd it's a 4x4 matrix determinant somebody said sum of two det 2x2 determinance a 4x4 sorry no in my notebook I have written 2x2 that's getting We are not solving the determinant.

[01:10:23]Okay, don't worry. After finding the determinant the value I will tell you So the non-trivial solution determinant is equal to zero. Determinant of coefficients equal to zero.

[01:12:54]H why that confusion? Four equations, four coefficients. 4x4 Ready right stop. So we are not solving the determinant. Finally I'm writing what is the expression you will get after finding the determinant and equate to zero. Everyone listen.

[01:14:51]So once you have taken the determinant you'll get like this all this is also no need to remember don't get tension see one more time my intention is to simply show that how we are getting energy uh band gaps that means uh one continuous band will be there after that forbidden again continuous band will be there again forbidden that is what I want to show okay like alpha squar + beta squar by 2 alpha beta into sin hyperbolic I think sin hyperbolic beta yes sin hyperbolic beta b cos alpha a plus cos hyperbolic beta b beta b beta b into cos alpha a equal to cos k a Not exactly cos kos k of a plus b I think so this is the expression you'll get if you want derivation see the textbook textbook also I think they have written determinant and after that they have written I think okay fine no problem so this is the expression everyone please stop and see the board now once again alpha square + beta square by to alpha beta. Here is a sign hyperbolic cos alpha cos hyperbolic cos alpha cos ka.

[01:16:25]Otherwise there are some websites who will give you the determinants of anything any matrix on values on vectors. Just open that and give these values and check what is the determinant you are getting if you are interested.

[01:16:39]But whatever the book it is given I have taken. Now let me find this. I hope we have defined alpha square = 2 m e by h cross² and beta² = 2 m v - e by h cross² once you add e e gets cancel I think that alpha squar + beta squared uh will be equal to how much 2 m v by h cross² yes or no Okay. And uh after that what we can do is we are taking an approximation.

[01:17:23]We are taking an approximation here.

[01:17:26]Something is missing.

[01:17:38]Correct. 2 alpha beta. Okay. Here we are taking an approximation. What is that approximation? Listen. Earlier we have taken this is the potential barrier. Potential barrier height V not and the length you have taken B. Say no. Here we are taking an approximation again. B tends to zero.

[01:18:02]>> B t >> V not ts to infinity. B >> of course such that B V is again finite because this is the barrier height I'm assuming again it is of infinite height because electron energy is very less compared to the barrier potential and this B is what tending to zero that's why this BV kn is very very small is that clear good so once you have taken B ts to zero all of you please observe that B tends to Then sin hyperbolic I hope you are observing sin hyperbolic if theta tends to zero sin theta is also nearly equal to theta sin hyperbolic is also nearly equal to theta now what is the answer next beta ts to0 we are taking this approximation beta b is nearly equal to 1 not 1 - theta² by 2 we are taking it as 1. If you take it and substitute alpha² m v by uh h cross² into 2 alpha beta slowly slowly sin hyperbolic beta uh beta b come on.

[01:19:23]Next cos this is taken as >> into one uh cos >> alpha >> in equal to cos k a >> b ts to zero. Now here with your permission I will multiply with okay beta beta gets cancel okay so this time I multiplying with a here and uh uh here also I will get uh a finally 2 m v what is this a b 2 h cross² 22 gets cancel if you Huh? H cross² >> alpha >> A >> alpha A don't cancel A. Y are we they are taking some a using for some other purpose just keep it as it is into cos alpha a plus cos alpha a equal to cos ka shall I take cos cos alpha a common don't take I'm assuming this has some parameter p because m constant v not constant this one constant alpha whatever it is fixed fixed value. A is the width of the barrier. AB is the product of width and two widths. So let us suppose P = MV A B by H cross² alpha A then finally I what I get comma P cos alpha A plus no sign s part is missing.

[01:21:34]Is it cos alpha a sin alpha a?

[01:21:42]I have given wrong.

[01:21:55]H sin alpha a not cos alpha a once again alpha square + beta square by 2 alpha beta sin hyperbolic sin alpha a plus cos hyperbolic cos alpha a equal to this one it's not s cosma it's a sign that's good yes now I understood why they have taken this.

[01:22:25]It's okay. So now this MV AB by H cross² let this P equal to uh MV A by H cross² that implies yes I remember the final expression I should get sin theta by theta that is not coming so maybe I have given here wrong so please correct it P sin alpha A by alpha A plus cos alpha A equal to cos K H ah thanks.

[01:23:00]This expression will tell you where we will get the energy band where we will get the forbidden gap. How it explain? I will explain. Just wait. Come on.

[01:23:18]Don't expect the complete derivation.

[01:23:20]Leave it. Just write the expression final. This is the sin hyperbolic sign cos hyperbolic cos after that this is the approximation we are taking b ts to0 now just take it here small confusion by mistake I have taken cos alpha a sorry please correct Sin hyperbolic theta is theta. Cos hyperbolic theta take it as a 1.

[01:24:39]And then we got the final approximation.

[01:24:42]P sin alpha by alpha A by alpha A plus cos alpha A equal to cos K A shall ask one one question here everyone please see V not you have taken infinity B you have taken 10 into zero that's Why this P value is a small or large?

[01:25:17]Okay, just remember that you have already defined B tends to zero, V tends to infinity but B V not is finite.

[01:25:38]Huh? You you asked earlier read that last equation >> put P equal to zero tending to zero.

[01:25:57]>> So in that case only your K will become alpha but K never be equal to beta.

[01:26:39]Don't bother about this derivation. No need, no use. Okay. But approximation remember from that approximation we got this final expression.

[01:26:52]But always remember that P is very small.

[01:27:14]I try my level west that graph. It's bit difficult to draw but I try. I'll try.

[01:27:48]Ready?

[01:27:52]Stop. Listen carefully. Maximum try to understand. If not understanding, read the book. Maybe then you will understand. First let me explain, please. So this is the final expression.

[01:28:04]Everyone please listen.

[01:28:07]P sin alpha A by alpha A plus cos alpha A equal to cos KL. Simple math not at all difficult. Listen. SL please focus on this right hand side what is that cos k a cos theta am I right cos theta this one what is the minimum value minus one maximum value + oh your right hand side must be line between minus1 and + one So if the right hand side is line maximum and minimum values you fix it maximum value + one minimum value that's why left hand side total value also line between >> wait this value also total value total value try to understand line between line between minus1 and + one So it indicates that you are input you are giving values of what alpha a whatever the alpha a you are giving that means whatever alpha a allowed >> such that this entire value lie in between >> only those values of alpha a are allowed and the remaining alpha a values are forbidden.

[01:29:46]So wherever alpha a value alpha a value of this expression is giving minus1 below and + one above. All those values are forbidden.

[01:30:03]So you are having the spectrum only between >> or only in a region of alpha a values of >> a values >> in which in which or for which we will get this entire expression value >> if you didn't understand this statement just once again read the book now see now I'm taking the graph this is the y-axis I'm taking B sin alpha A by alpha A plus cos alpha A I'm taking that value is it clear because this value definitely lie in between + one and now I'm taking uh somewhere zero or is it now this is my alpha a x-axis. Okay. So now I'm giving the input is alpha a x-axis and the value is what? P sin alpha by alpha a by alpha a cos alpha a.

[01:31:16]Clear? Are you understanding?

[01:31:19]This entire value lie between + one and >> + one and minus one.

[01:31:35]Try please try please try whether you understand or not just listen first let me explain that's it okay cool h ready start alpha a is nothing but the angle that's why I'm taking this is a zero and somewhere I'm taking uh p<unk> by2 somewhere<unk> by2 2 somewhere pi next 3 p<unk> by 2 next 2 pi next this side >> -<unk> by 2 >> -<unk> - 3<unk> by 2 something like this is it okay now carefully observe if alpha a is tending to zero Then this value equal to 1 that is nothing but P still we have taken P is very small say S or no that is what we are discussing till now P is tending to zero that's why which is the dominating value so whatever the allowed region it will be decided by cos alpha A cool now ready one more time I'm saying as alpha A is tending to 0.

[01:33:00]Okay, this value is 1. So that's why P + cos alpha A I'm taking that P is tending to zero. What is the reason I already explained there tending to zero very small that's why the right left hand sorry left hand side value is dominated by now ready carefully observe now these are my 0 by 2 pi 3 p<unk> by 2 2 pi something now this is forget about this only dominating is what cos alpha okay we are not finding exact values approximations now ready. So cos pi alpha a = pi cos pi minus 1 or + 1 - observe observe next so cos alpha is tending to pi by 2 then this value will be zero and alpha not exactly at pi close to pi understand but I'm taking like that and again it is coming close to zero coming close to zero somewhere.

[01:34:13]somewhere cos alpha a zero how much it is say yes or no somewhere okay that means + one again it become zero again it become minus1 and around at pi it is becoming minus1 close to zero it is going to + one understand everybody Similarly, Similarly, left side near to zero it is going to + one and exactly not pi. Uh nearly at pi it is becoming minus one.

[01:35:00]Is it okay? Now many times I told you we will first explain only first build one jone. If you observe carefully where it is going to minus one means it is a pi by pi by a alpha a then alpha is what >> and this is also >> so minus pi and plus pi is which region that also don't didn't understand alpha is pi means alpha is what pi by Wait.

[01:35:36]Okay. Once again, what happened here? As alpha a coming close to zero, it is going to + one. As alpha is going towards pi, not exactly at pi 9 nearly coming close to pi that is a jone boundary you are getting minus one. So definitely your energy spectrum lie in between + one and okay slowly slowly. Next. Ready. One more. One more we will take.

[01:36:18]Uh ready. So 2 pi 2 pi cos 2 pi forget about this. This is nearly equal to zero. Huh? Cos 2 pi 1 + 1 and nearly coming towards 3 p<unk> by 2 cos 3 p<unk> by 2.

[01:36:35]>> Huh?

[01:36:36]>> Zero. Is it okay? Next.

[01:36:42]Uhhuh. Come on.

[01:36:45]>> Again. Coming towards pi it will be minus one. Now only we are having this is allowed. That means if you extend this carefully observe here this is the allowed energy band this is the allowed energy band okay but here we are getting beyond minus1 is it allowed that's why this is what forbidden Similarly from here to here allowed from here to here allowed so that's why again I'm taking ah sorry now beyond this what is happening above + one is it allowed okay still not understanding in the same way if you Proceed like this. Somewhere here nearly minus pi you will get minus1 and again here what you are getting a forbidden gap like this. So what is happening here? So this is origin of course. Okay. Somewhere very close to the origin we are having this energy band. This is the allowed energy band. And nearly at the jone boundary please please see this because exactly we won't get it's impossible nearly to the jone boundary we are getting minus1 that means beyond minus1 no energy is allowed because these values are not allowed that means in this region what we are having forbidden these values are forbidden and this gives you a forbidden gap and this is The allowed allowed and this is also allowed and again here we have forbidden.

[01:38:55]So this is the structure what we are getting in this diagram. In this diagram means we have taken x-axis is alpha a and yaxis is this. But still we are not getting a clear picture. That's why we will take this diagram and draw in the reciprocal latice and there we will see what will happen. There you will understand. So how the energy bands are getting and at jone boundaries what we are getting from this diagram my intention is to to say that near to the jone boundaries we are having forbidden energy gaps. This equation is telling you near to the jone boundaries we are getting forbidden gaps because only particular values of LA sorry only particular values of alpha a only allowed and after beyond that that means this is the reason this is the reason where alpha a where alpha a gives this value beyond minus1 that's why this reason is forbidden similarly this region is forbidden. Similarly, this region is forbidden. Like that we are getting allowed energy forbidden.

[01:40:05]Allowed energy forbidden. But here we are not clearly understanding how that structure will be. That's why again we will take this and represent in the reciprocal latis. First write down.

[01:40:27]I will give some points here. Maybe it will be useful. Still not understanding, please write those points.

[01:40:40]Huh? One thing I want to say here uh the energy spectrum how it is alternate energy uh allowed bands that means allowed energy band and forbidden and allowed forbidden like that. Earlier we got what continuous So now it is discrete and there exist a forbidden gap between two allowed bands.

[01:41:04]That's important.

[01:41:27]Sir how we we will draw this just to forget this term then where this term is becoming +1 and minus1 minus1 is becoming at pi that's why very close to pi as it is approaching 2 pi it should go to minus next where it will become + one zero so it is approaching to zero it will go to + this is called jone boundary.

[01:42:06]So at jone boundary we are getting forbidden gap.

[01:42:14]Slowly slowly slowly draw we won't get any question but we should know now at least how we are getting that what is the mathematics there at least Oh. Although derivation is not understanding just in the beginning how we proceed what is the approximations we have taken potential well how we got periodic and blocks theorem you must remember the blocks theorem because the question will come on blocks theorem or blocks function so don't forget that and this derivation no one will ask this is only helping you at the jone boundaries why we are getting forbidden gaps Somewhere I explained in the X-ray draction or somewhere at the brillwan jone boundaries we will get the draction phenomena and the wave become standing wave and we have shown that group velocity is equal to zero there. There I explained it become a standing wave that means it is not transmitting any energy and I I said at that boundary you'll get a forbidden gaps. We will understand why we are getting in the why we are getting at the jone boundaries. We will explain in the band theory of solids. This is what I said on that day. So this is the derivation near to the jone boundary we are getting a forbidden gap. Please please observe. Of course, this is origin.

[01:47:06]Please read about blocks theorem. Don't neglect. Right on. First point, the energy spectrum of electron here. Here here first point. The energy spectrum of the electron consist of consist of alternate regions alternate regions.

[01:47:31]alternate regions of allowed energy bands.

[01:47:37]Of allowed energy bands of allowed energy bands alternate uh regions of allowed energy bands. Next, second point.

[01:48:03]The boundaries of the boundaries of the allowed energy levels.

[01:48:12]The boundaries of the allowed energy levels corresponds to corresponds to corresponds to cos k a = + r -1 implies implies implies k = + r - n pi by a in bracket a jone boundaries jone boundaries because cos ka must take only plus r minus one maximum and minimum. Next point.

[01:48:59]As a P ts to zero, as a P tends to zero, the potential barrier becomes as P ts to 0, the potential barrier becomes 0 0.

[01:49:21]And the electron treated as sorry potential barrier becomes weak.

[01:49:29]Potential barrier becomes weak. Weak and electron treated as free. Electron treated as free.

[01:49:44]Next point.

[01:49:48]As a p value increases, as a p value increases, as a p value increases, the area of the potential well, the area of the potential well increases increases and the electron is treated.

[01:50:15]And the electron is treated.

[01:50:18]And the electron is treated bound to the electron is treated as a bound particle. Electron electron is treated as a bound particle. That's enough.

[01:50:38]Now listen. So finally what do you understand from that graph? We are getting the discontinuity discontinuities in the energy spectrum where at the jone boundary. 1 minute. 1 minute. All of you please listen last very important. So earlier we have taken electron as a free free electron theory.

[01:51:02]If electron is a free completely what we get already I have discussed it's energy equal to h cross² k² by 2m are you listening that implies your energy is a proportional to k² so we got this spectrum this is e and uh this is what kind of spectrum It is continuous.

[01:51:35]Wait. Now till now today what we have taken is electron as a nearly free electron.

[01:51:44]Nearly a free electron. This is what we have assumed. Say yes or no. And finally what happened? This is my reciprocal space. Everyone please observe this is the reciprocal space. Now I'm going to represent that in the energy versus K diagram. That is not energy versus K diagram. Where we are getting the discontinuities that is only telling where at the jone boundaries. This is the K and I'm taking this E. Everyone please listen. This is a zero.

[01:52:24]by a -<unk> by a 2<unk> by a - 2<unk> by a something like this. Now everybody know this is my jone boundary.

[01:52:43]This is the jone boundary.

[01:52:47]So first let us this is also a jone boundary.

[01:52:52]So according to that diagram if you observe carefully up to the jone boundary what you are having a continuous spectrum say no.

[01:53:05]Okay.

[01:53:08]So this is the origin and this is the jone boundary and the spectrum is latis and here what we are getting >> forbidden >> from that I hope everybody understand near the jone boundary we are getting forbidden and again co that means uh I hope you are observing that we have drawn some lines okay so white color lines earlier Here sir why you are carefully drawing the flat region there because at that point we are getting a standing wave whose slope must be zero there that's why it should become flat and again I hope you are observing and sir why you have taken concave here convex here.

[01:54:06]Just wait tomorrow. Ready? This one.

[01:54:23]Okay. Again forbidden gap.

[01:54:30]Again forbidden gap. Something like this. So here we are having forbidden forbidden in this region and this is allow allowed like this. So alternate allowed forbidden allowed forbidden. This is what we understand from that graph. And if the same thing if you represent with the energy versus K the free electron it is a continuous spectrum. But in this nearly free electron theory what we got is alternate energy bands exist or alternate forbidden bands exist or alternate allowed energy bands exist. And tomorrow we'll see so what it will gives the information and what is the effective mass and what is the velocity of the electron why the velocity of the electron is negative or why it is decreasing all these things we will discuss with the help of this uh graph only this graph will tell you the remaining part. Is that clear right now?

[01:55:43]If you observe carefully bottom and top parabola this is a gate question which of the following is correct incorrect something like this. So energy versus K in the band theory of solids it is a parabola in sorry parabola at bottom and top of the energy band but the curves curvatures are opposite direction. Curvatures are opposite direction. Please remember and here what you will get minimum what you'll get maximum in between what you'll get point of inflection that point of inflection will tell you the firmy level and that fermy level will decide you whether it behaves like a conductor or semi semiconductor or insulator everything will come from this graph only okay write I think I need not to say all these things. Slope here. What is the slope? D by DK. Slope how much? Zero. At the maximum slope zero. That means D by DK equal to zero. But this is a minimum.

[01:57:01]Minimum means first derivative. D² E by DK². Again tomorrow I will explain just writing greater than zero. This is the maximum that's why d² e by dk² is negative don't forget and somewhere in the middle there exist a point of inflection point of inflection I explained all these things okay point of inflection here also d by dk0 but d² e by dk² square = 0. This is very crucial and again we will discuss that graph I mean that curve tomorrow.

[01:57:54]What is the name given for this diagram in the note uh in the textbooks?

[01:57:58]Extended jone scheme.

[01:58:01]Extended jone scheme means you need not to bother about all those things. You just only study about this first brillan the remaining nature is repeated repeated. So this is called extended jone boundary scheme.

[01:59:19]But my request once read the book and leave it then if any points I didn't explain properly you that may fulfill The free electron we got a parabola But in nearly free electron top and bottom are having uh concave and convex is something like that. Okay. So curvatures in the opposite direction. That's why one is a concave another is the convex.

[02:00:10]That convex part is the maximum. That concave part is the minimum. In the middle there exists a point of inflection. What are the significance of all those things? Tomorrow we'll explain beautifully and tomorrow class is very crucial. Effective mass everything will come here.

[02:00:43]Maximum tomorrow it will be complete problems also and after tomorrow we have to start superc conductivity.

[02:01:03]by Wednesday we have to close this otherwise we'll be in trouble Thursday onwards we have to use this morning session for either SM or for digital otherwise it won't be complete Please read the book. Please read the book. He explained properly. Please read the wahab book and leave it once. read and leave it because before the interview if you read you feel unfamiliar. Now once you read and left while you are revising for before interviews you feel okay I know this something earlier some conference will be there otherwise what will happen generally student psychology if it is unfamiliar what we will do we will leave it that's it no first read and again we can revise later don't neglect please