noc19-ph02 Lecture 65-Tight Binding Model- II

[Music] [Music] what we have done in the previous lecture is introduced the idea of tight binding approximation which is nothing but equivalent to linear combination of atomic orbitals or what is known as LC AO and the idea is to combine them in such a way that Bloch's theorem is satisfied and therefore we write sorry okay and I'm first working in one dimension so it's ikx equals summation and arrays to I K and a Phi X minus and a intense goes from minus infinity to plus infinity so what we are doing is we are taking this crystal lattice where atoms have distance a between them and we're starting from some point and an equals minus infinity to plus infinity e raise to I KN a and we also showed that if I take sorry okay X plus a it turns out to be equal to e raised to i ka sorry K X so this form of the wave function satisfies the block's condition what we want to do now is calculate energy E as a function of K using the tight binding wave function and this will involve some approximations which will spell out as we go along the way to calculate the energy EK is to calculate the expectation value so I K X the Hamiltonian sorry K X where so I K X is normalized and what does that mean recall my earlier discussion of normalizing the block wave function which was related to normalization of plane waves so the way we do normalization these wave functions is by integrating size star K X sorry KX DX over length L where L is length of n cells and since I K X equals E raised to I KX u KX we normalize in two steps number one step number one is u KX is normalized over a cell so what that means is that integration UK x star UK x DX over a cell is equal to one and then I can write this normalization integration psych a star x sy k X DX over length over length L or equivalently over n cells as equal to integration he raised to minus I KX u K X star times e raised to I KX u K X DX and this is equal to supposed to be 1 right so let me write a normalization constant that I want to determine in front of the wave function so I have this integration I raised to minus ikx and e raise to K is give me 1 and I have UK star X UK x CN mod Square where CN is the normalization constant this should be 1 this integral I can write summation over n cells times integration over 1 cell and this integration over 1 cell is that of UK star X UK X DX so this gives me summation over n times 1 which is n all this implies that mod CN square times n is equal to 1 or the normalization Cohn's constant is 1 over root n so a block wave function is normalized like this sorry K X is written as 1 over root n e raised to I KX u KX where it is understood that u KX is normalized over sin now let's see how I'm going to normalize the tight binding wave function and I have soy KX equals summation N equals minus infinity to infinity e raised to I K n a Phi X minus n a I have this wave function and I put in front of this a normalization constant C n CN is for n cells so when I take size star KX sorry KX this comes out to be mod CN square summation M summation n raised to I K minus M plus n a Phi of X minus M a star Phi of X minus n a DX integral integral DX now let's write this as again clearly sine square CN square summation M summation n arrays to I K minus M plus n a can be taken outside the integral and I have integral Phi star X minus M a Phi X minus n a DX let me show the approximation that we are going to make now so what we are doing is we are taking this atomic chain and here is my let's say the wave function out here let this be the NH site and let this be the Emeth site and here is the wave function Phi which is concentrated near MH site and here is a Phi concentrated near the NH site and what you can see is there's hardly any overlap between two so what I can write is that if I have Phi X minus ma star multiplying phi x minus na it's almost equal to 0 for m not equal to n so even if i take the site right next to the NS side the overlap is nearly zero and for M equals n I'm going to have mod Phi X minus na square DX and this I can take to be 1 because I'm taking fire to be the atomic wave function which is normalized so if I look at this integration so I K star X sy KX DX it comes out to be CN mod square summation m summation and arrays to I K minus M plus n a times Delta MN where Delta MN is a function which is 0 if M is not equal to 1 and M equal to 1 it is equal to 1 so this gives me C n mod square summation only one of the indices of M or n will be left times 1 which is equal to CN square times n exactly like what we did for the free electron wave function and this should be one and this gives me that mod CN is equal to one over root n therefore I can say that neglecting overlap between the atomic wave functions on two different atoms and this is an approximation I'm neglecting the overlap between two atomic wave functions there on two different atoms I have the wave function tight binding wave function so I KX equals 1 over root n summation N equals minus infinity to infinity e raise to I ke na Phi X minus na this is my normalized tight binding wave function when I've normalized it next I calculate ek by taking the expectation value Phi K X H so I K X which will be equal to 1 over N summation M summation n I raise to I K minus M plus n a integration of Phi X minus ma expectation value of H with respect to that so that's why I said integration for I X - na keep in mind now that this term is independent of the Hamiltonian and therefore it has been taken out of the expectation value and what we now are to focus on is calculation of this term here and again we will be making certain approximations when we calculate this so let us do that now so we wish to calculate e K equals 1 over N summation M summation n I raised to I K minus M plus n a Phi X minus ma H the Hamiltonian Phi X minus na where the Hamiltonian h equals minus H Cross's square over 2 m d2 by DX square plus VX VX is periodic so to facilitate this summation over all the sides what I'm going to do is change what I'm going to do is divide this sum into the same site M equals N and M not equal to n so what I'm going to right now is e okay and there's a purpose for this 1 over N let's take the same side so I'll just sum over m e raised to minus K MN and now the same so that becomes e raised to 0 which is 1 Phi X minus ma H Phi X minus ma plus 1 more term that includes both m and n but M is not equal to N and now I have a raise to I K minus M plus n a Phi X minus ma H Phi X minus n a I will take both these contributions one by one and see how they work out so let's first focus on the first term which is 1 over N summation M Phi X minus ma the Hamiltonian Phi X minus na now recall that Phi X is the atomic orbital at each side because what we have done is made the linear combination of atomic orbitals satisfying blocks theorem so each Phi is an atomic orbital let us look at the H the Hamiltonian the Hamiltonian consists of two parts the kinetic energy part plus VX if you look at VX this is how it looks so let me make this chain in one dimension of atoms vx is nothing but the sum of these atomic potentials so atomic potential would have been something like what I show in green for each site and when I add these the VX potential that I get I'm going to show it in red it's pretty much Adam like near the site and in between it becomes a little lower because we are adding negative potentials so it differs a bit but mainly in the interstitial region if I were to plot the difference Delta V X between the two potentials Delta V X will be something like this almost zero near the atomic sites and quite large in between so keep this picture in mind because this will help us calculate these matrix elements so this is Delta V X the one shown in dark blue it is quite large in between and very small near the sides now let's go back to the calculation of this matrix element one over N summation M integration Phi of X minus ma the Hamiltonian minus H cross square over 2 m d2 by DX square plus 3 X 5x minus ma DX DX runs from 0 to the length over which we are applying the periodic boundary conditions and the left term should be Phi star now I can say that this Hamiltonian acting on Phi X minus ma will give me about e atomic atomic orbital times Phi X minus ma I'm saying this about because H is not the atomic Hamiltonian 1 over N summation M integration 0 to L Phi star X minus ma H Phi X minus ma D X now this integral will contribute only for M such that M a is less than or equal to L so M will give you a count of n integral 0 to L Phi star X minus ma H Phi X minus M ay DX will be about e atomic as I argued earlier and sum over m of this integral 0 to L then is going to give me n times Epsilon atomic it is understood that epsilon atomic is not exactly the orbital eigen value of the atom but very close to it Ketil calls it minus alpha alpha being greater than zero and therefore this whole expression comes out to be 1 over n times n times Epsilon atomic and I can cancel this N and n so this is equal to epsilon atomic let us now look at the second term the second term which is 1 over N summation M summation n am not equal to n e raised to I K minus M plus n a times the expectation value Phi X minus ma H Phi X minus n a let us look at this term now M is not equal to n so let me again just to visualize make that atomic chain and the potential that we drew earlier is something like this and symbolically I have the wavefunction right on each site this is my Phi now when I calculate this term I am actually talking about a sigh at some site which I'm showing by shaded by this purple color and sighs some other site at some other site here could be right the next site or could be farther out when I write this expectation value this is going to be just the expectation value term it's going to be integration Phi star X minus ma H is minus H cross square over 2 m d2 by DX square plus V X Phi X minus na and M is not equal to n most of the time the atomic orbitals are you know exponentially decaying so even this term D 2 Phi by DX square if you go to the region between the atoms is going to be proportional to Phi itself that you can check if Phi is exponentially decaying maybe alpha X so for the first term what you will see is that the overlap in between out here is negligible if M is not equal to n then the first term should give me roughly 0 this also makes sense according to the approximation that we have been making in normalizing the wave function we have been neglecting the overlap of wave functions from sites which are different so the only term which will contribute in this is the V X term so let's just slowly understand so I conclude by looking at this picture that this H cross square over 2m the minus sign d2 by DX square sorry X minus na DX is roughly equal to 0 let me see if I put the minus sign is roughly equal to 0 and for the second term if I look at this Phi so there should be a star here Phi star X minus ma V X 5x minus na DX M is not equal to n let me look at these atoms again this is the potential it has some value in between remember I had plotted it earlier so the the Delta VX is quite large in between and here are the wave functions so although we have neglected right let me just point out we have neglected Phi X minus ma Phi X minus na when M is not equal to n now you see this product yes when it's multiplied by Delta V X is going to be large only from this intermediate region so this part cannot be neglected if I did that the only value for the expectation value of the Hamiltonian I will get is the atomic atomic orbitals and the band will be flat so we are not going to neglect this term and what we are going to do is make what is known as the nearest neighbor approximation and by looking at the picture you can make out it's when we look at the atomic orbitals next to each other then this term is appreciable there otherwise it becomes negligible because if I look at this site I'm now taking and the site 1 removed from the next site you will see that the overlap is going to be very small so a nearest neighbor approximation what I'm going to say is that the integral Y star X minus ma V X Phi X minus na DX is not equal to 0 for N equals M plus minus 1 that means right next to each other for others is going to be 0 so if I look at that term again the term I'm after is 1 over N summation M summation n arrays to minus or plus I K minus M plus n a Phi X minus ma H Phi X minus n a so what we are going to say is that this term Phi X minus M a H Phi X minus n a which is equal to integral 0 to L Phi star X minus M ay H Phi X minus na DX is not equal to 0 for n equals M plus minus 1 and 0 for all other terms and in this also what we said earlier is that the kinetic energy term right Phi X minus ma minus H cross square over 2 m d2 by DX square Phi X minus na is going to be almost zero and it's the second term that contributes and that is Phi X minus ma V X Phi X minus na is not equal to zero and in this VX also as I showed in the previous slide actually is the Delta X Delta VX term I'm putting a little Delta V X here I'm emphasizing okay let me just encircle this this term is the Delta V X term which is large between the two atomic sites that's the main contributor this nonzero term so it's because of the Delta V X that the bands arise if this was not there the bands would not arise it suppose Delta V X was 0 each atomic site was independent all I would have gotten is e atomic as the expectation value of the Hamiltonian and the total band would be flat energy will be equal to a atomic for each okay but now this is what we have so we are calculating this term integration 0 to L Phi star X minus ma the Hamiltonian minus H cross square over 2 m d2 by DX square plus V X Phi of X minus na DX and we are saying this is nonzero for N equals M plus or minus 1 which is I'm calling the nearest approximation so you see again that the main contribution to this term comes from that Delta V X term that I talked about earlier and what I'm now going to call this as 0 to L Phi star X minus ma H Phi X minus na DX I'm going to call is equal to gamma since I'm integrating from 0 to L and this is nonzero that is equal to gamma for m equal to 0 to n minus 1 or 1 to n because that's where we are integrating from 0 to L if the orbit lies or the atom lies outside it its contribution within the region is going to be very small so this term summation 1 over n m e raised to I K minus M plus n a Phi X minus ma H Phi X minus n a the summation over n also n equals M plus minus 1 now can be written as 1 over N summation m and for the other term I'm going to have Y raise to I K minus M let's take an equals M plus 1 M plus 1 a times gamma plus e raised to I K minus M plus M minus 1 a gamma and that's it so this will turn out to be 1 over n times sum over m now each term is the same is going to give me a factor of n times 2 gamma cosine of K so recall what we were calculating we were calculating EK equals 1 over n summation m summation n raised to I K minus M plus n a Phi X minus ma H Phi X minus n a and by splitting it into M equals n term and not equal to n term and making the nearest neighbor approximation we have gotten this to be about epsilon atomic plus 2 gamma cosine of ka that's your EK notice that this gamma term arises because of that delta VX mainly so if i were to plot it EK what's this game he atomic is some number that your book it l cos alpha there's some epsilon atomic and i'm plotting the blows own the band in the first zone and if gamma were negative this is how it's going to look this is being plotted between minus PI by a to PI by a this is a cosine curve the lowest value here is going to be e atomic plus 2 gamma oh I'm taking 4 here gamma less than zero the overlap integral is giving me zero and the width of the band from his bottom to the top is for gamma the top value out here is epsilon atomic minus 2 gamma I should put this for gamma as gamma mod because I'm taking gamma to be negative if on the other hand gamma was positive then the curve would look like this this will be for gamma greater than zero so you see similar kind of bands that we obtained earlier in the other extreme limit of nearly free electron model is obtained when we take lcao the linear combination of atomic orbitals and make appropriate combinations to make it satisfy Bloch's theorem so this what we have done is calculation tight binding calculation for bands in 1d now we have focus so far on one dimension calculations and we have taken sy KX to be of the form summation N equals 0 to n minus 1 or n equals 1 to n the same thing arrays to I K and a Phi X minus n a now this is easily extended to two or three dimensions so in two or three dimensions I'm going to have psy K vector our vector is equal to summation N equals 0 to n minus 1 e raise to I K dot R n where RN is the position of the NSL phi r minus RN this is going to be the form of the wave function and rest of the calculations go exactly in the same way as we did for one-dimensional calculations except that when you take two dimensions for example if I take a square lattice and the atoms are at the corners number of nearest neighbors becomes larger for example in 2d for the central atom which I'll encircle I have one two three four nearest neighbors and RN the net vector is going to be RN which is some M a X plus some n a Y similarly when I go to three dimensions suppose I take a cubic lattice and here are the atoms at the corners for each atom there are going to be six nearest neighbors and our n is going to be of the form some M a X plus L a y+ n a Z and K is also going to have three components KX x plus KY y plus K Zed Zed and it will require a little more bookkeeping but procedure is exactly the same I'll be giving you problems related to this and I argue to read this from little Spock tight binding approximation so this winds up our different ways of looking at bands so I conclude this lecture by noticing that we looked at band structure or energy bands from the point of view of nearly free electron in which I applied the perturbative approach then I looked at it from the mixing of plane waves where we mix many many plane waves to get the wave function in a periodic structure and third we have looked at it from the tight binding approximation point of view and if you look at it the first one the nearly free electrons model and the tight binding approximation these are the two extremes of band structure calculations in one we have electrons moving freely and then we apply a very weak perturbation and see how bands arise in the tight binding case we have electrons tight bound to atoms and then we D localize them a bit so that the electrons interact with only neighboring potentials in the next lecture I'll be talking about how particles move in a band when a force is applied thank you [Music] 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