 So, we are in the last lecture on this solid state physics part, which is an additional lecture, as I told you yesterday. Since we could not cover some of the important aspects which I wanted to do, so I added this lecture. So, this should finish the basic solid state physics part which we are planning. So, if you remember what we have been talking about in the last couple of lectures on solid state physics, we mainly we were worried about the description of the quantum mechanical description of a solid, especially metallic solid. We started with the classical picture. We saw some success and many failures. Then we went to the quantum mechanical description again treating the free electron model. Some more modifications were needed or some of these discrepancies between the theory and the experiment led to the further development which actually we started fearing in the last class. So, one of the main issues associated with this whole topic as it stands now is that the mean free path that the electron experiences in a metal, whether you take the positive ions into account, their potentials into account or not as we are going to see, the mean free paths are much larger than what is expected. The experimentally observed mean free paths are much more than what is expected in a periodic array of potential bills. So, this is an issue which has to be bothered about. The large mean free path is something which is not able to be comprehended just because we know that there are many scattering centers and hence one needs to really understand what is really happening in the case of a periodic lattice where you have this electron movement. So, in that respect we started modifying the free electron picture, the quantum mechanical free electron picture namely the Somerfield model. So, we will slowly move away from the free electron concept. So, we will take the positive ions into account in the form of these potentials associated with that. But as a first step what we are going to do is instead of really worrying about the actual potentials that will be taken later, first we will worry about the translational invariance or the periodicity into account. The wave function that we have been writing down, the plane wave solutions that we have been writing down that is e to the power of ikr functions will not be acceptable now. So, we will try to modify it. This modification was actually done by Bloch and that gives you the Bloch's theorem where we have seen that the wave function now gets modulated. The plane wave actually is modulated by a function which actually is having the same periodicity as the lattice potential or the periodicity of the lattice. So, the wave function phi x is no longer e to the power of ikx type, but it is multiplied by a function which actually is periodic in the lattice. So, this is what the difference is. We have started with a free electron picture. Now, it is actually modulated. This part actually tells you that it actually is taken the periodicity of the lattice is taken into account. So, we talked about this in the last class. So, the condition that has to be satisfied here is that you cannot take any function as u, but you have to satisfy this condition that u x plus n a equal to u x where a is so called the lattice constant. So, this is the periodicity long range translational invariance, long range order whatever you are calling. So, one can actually show as I mentioned yesterday whether I when I take x plus n a that means I go multiple distances of multiple of this lattice constant. I see that the charge density which actually is determining all the physical properties remain unchanged that is what you are seeing here. This condition can be satisfied when you have this substitution here. It is self-explanatory. I will not go into the details of it. So, the implications are because of this when I take k any particular value of k and if I take this condition 2 m phi divided by a something like the periodic boundary condition which we talked about yesterday when your m is plus or minus 1 plus or minus 2 even this is there. So, whether it is k or anything of this kind you are actually going to have the same result. So, which means that you are remember this is a not l, l was the one which was actually used for the periodic boundary condition when where l was the total length of the crystal. Here a is the lattice constant which is a separation between two adjacent neighboring atoms. So, what does it mean? One can show that when you have this condition satisfied there are many k values that are acceptable with all our conditions. So, there is no uniqueness as far as k is concerned earlier in the case of a free electron picture your k was unique. But now if I take a k value and if I either add or subtract this fraction it is not going to change the physical properties it is not going to change any of the properties that we are interested in. So, that means very important difference has come into a picture now that the uniqueness of k has gone. k has always identified as a momentum because h bar k is a momentum. So, k is important physically because it is relating it was relating to the momentum. But now because of this there is some rethink that is needed because the uniqueness of k has gone because I can always add or subtract which means that k can be kind of anything. So, that is what is actually shown here. So, this of course is one can really show that when I take any of these e to the power of i 2 m pi by a one can actually show that 2x plus na is ux or this condition is satisfied. So, the point is you are actually getting into a large number of k values which will essentially satisfy the same physical requirement. So, that is k prime is given by k plus 2 m pi by a this condition takes away the uniqueness of k values not only that because of this you can have unique values of k only in the range between of course, in the range of 2 pi by a. So, which can be taking into account the symmetry I can tell that the distinguishable unique values of k can be taken only in the range from minus pi by a to plus pi by a. Now remember a is coming in the denominator pi by a represents the k. So, what we are actually doing we are in the k space as I was mentioning yesterday this is nothing but the reciprocal space reciprocal lattice we are talking we have been talking about some questions were they are regarding this. So, what it means is that in the reciprocal space or in the k space only physically distinguishable points are there only in the region between minus pi by a to plus pi by a. So, this region of the reciprocal lattice or the k space becomes little different or more important compared to other regions of this one because this is because any other k value if you have you can always bring back to this region you can either subtracting with this number or adding with this factor one can always make sure that the corresponding k you have a corresponding k value within this range. Remember this is actually giving 2 pi by a that is what is seen here. So, the physically distinguishable values of k are only in the range between minus pi by a to plus pi by a this is what is shown in this picture. So, one can have at various points these are these are what you are seeing here this is an one-dimensional reciprocal lattice or the k space and we can actually have this parabola the free electron kind of parabola that we have been talking about earlier we had only one parabola because it was unique k value but now because of this 2 pi m by a business you have many parabolas one can construct but physically distinguishable k values are only in the range between these two blue lines this is from minus pi by a to plus pi by a because this is 2 pi by a. So, this is a region which actually is physically interesting because it gives rise to the physically distinguishable k values. So, if you see these are all now we will call it as n equal to 0, n equal to 1 the number index which was coming in the expression. So, depending on these values one can actually have various parabola remember again everything is a parabola and you can actually have a large number of infinite number of such parabola and one can get the information of all these parabola whatever information these parabola contain one can actually get them back to the region of minus pi by a to plus pi by a by doing this business. If you see this part which actually is in the corresponding to n equal to 0 this is like our original parabola which is centered at the 0 k value which was used in the case of a free electron picture that is what is shown here and when it goes out of this one I will not worry that is why it is stopped here. Then for the second one that is corresponding to n equal to 1 the part which is coming here is this one and this one. So, that is what is shown here. Similarly, I can always take the portion of this parabola in appearing in the range between minus pi by a to plus pi by a if I put it I will get these kind of sections. These are all sections belonging to the parabola this is a lowest one then these two this is continuous and then it comes like this then this is the third one fourth one like that it goes. So, this is obtained by this side if you take this was actually subtracted. So, you bring it here from here you add it so that it comes here the addition or subtraction is by a factor of 2 pi by a times whatever n you have to do. So, that is what is giving rise to this. So, this is actually what you are seeing this particular scheme is called peter zone scheme because there I am going to call them as zones now later on we will change it and this gives rise to what is known as a reduced zone everything is reduced to this zone or this range of k values which actually is between minus pi by a and plus pi by a. This physically important regime or region or the zone is nothing but the first Brillouin zone. The first Brillouin zone actually is the smallest in this case the length that is there in the reciprocal space. So, in fact, in the direct lattice there is a concept of what is known as a Wichner Seyd cell which actually is a smallest volume cell. A corresponding thing in the case of a reciprocal lattice or the k space is called a first Brillouin zone you have different Brillouin zones in this picture all the k values that are possible for the electron. Again remember we are not talking about the real motion of the electron in the real lattice we are only worried about what are the k values physically distinguishable k values physically meaningful k values of electrons. So, these values only we are worried about. So, we directly talk in terms of this reciprocal k values which means that we are in the reciprocal lattice already. So, the reciprocal lattice one-dimensional case the minimum size is between minus pi by a and plus pi by a and that is called a first Brillouin zone. Similarly, you have second, third and higher Brillouin zone. So, here all the different parts of these zones are brought into the first Brillouin zone this is actually called a reduced zone scheme which actually is a very important thing in many of the understanding later. So, what is the difference between this picture and our free electron picture that we have been talking about in the last class? The difference what you are seeing is in the case of a completely free electron picture where there is no periodicity taken into account. I had only one parabola there was no this kind of constraints it was simply going this parabola was going all the way up to plus infinity here minus infinity here. Now, what you are seeing here is because of this periodicity being taken into account there is a difference very important difference all these things have kind of become pieces and these parabola different parabola are possible that is number one and physically meaningful region takes into account certain pieces of these parabola different parabola. So, that the picture is complete because in this case and in this case the picture is complete in the sense that all physically meaningful k values have been represented in this one equally they are represented in this picture also. So, this picture is very important. But, let us look at what was our motivation one of the motivations was to find out what is the role of this lattice potential into this electron motion that has not been taken into account. But, what you have taken into account is a translation the symmetry has been taken into account the discreteness the symmetry that has been taken into account of course, for an one dimensional situation which can be extended to three dimensions. Therefore, a part of the problem has already been incorporated into this already. What is not taken into account is the actual positive ion potentials that it will take up to sometime. So, and you can see as I mentioned earlier in the case of free electron picture there is no question of seeing an energy band or energy band gaps. If you ask whether are you able to see that here also we are not able to see here in this case again you do not have energy bands you cannot see energy bands or if you want to tell that energy bands are there you should get some kind of a gaps the energy gaps that picture again is not coming. So, they nearly taking the lattice points or the symmetry into account it is not really giving you the actual information that we are looking for. And what we are not getting we know the energy band gaps or energy bands you are not able to see. And we know also that we are not taken into account certain important factor namely the lattice potential which means that lattice potentials must be having a role in determining the energy band gaps. Because this is missing and we know that this is not taken into account other factor. So, definitely there must be a connection between the two. So, this is our guess at this point of time that the lattice potentials unless they are actually taken into account physically into the problem you are not going to encounter energy gaps. Because of this reason this particular picture can actually be called because this is different from the free electron picture and definitely it is not exactly which actually takes into account the actual lattice potentials. So, this actually is called what is known as an empty lattice model what we are doing is we are taking the lattice into account, but the positive ions are actually taken in the limit of 0. So, you have the lattice. So, lattice properties are taken into account, but actual people who are sitting at the lattice points that is a positive ion core that is that effect actually is not taken into account. So, this is kind of a theoretical kind of a picture where you call the lattice is there, but it is empty. So, this is the empty lattice model which actually does not give you the energy bands or the energy band gaps which we are looking for. But this is definitely an improvement over the completely free electron picture where you have only one parabola. Now, you have many parabola and or we have all the relevant pieces of all the parabola into a physically important region of k space namely the first Briello-Einstone that extends from minus pi by a to plus pi by a. Now, what I am going to do is I am not going to give a very rigorous proof of any of these things, but some of the things somewhat in a qualitative or a classical or a semi-classical sense, but that idea is very important. Remember I am repeating it, the main one of the main issues in front of us is how to explain very large mean free paths encountered by electrons when they are actually moving in this solid. This is again I have already mentioned in the case of free electron theory wave vector was unique which and momentum that because of that was unique. Now, your k is not unique, your k can actually have you tell some k, I can always change it without any with change in the physical properties because I can always add or subtract certain fraction. Because of this reason the physical significance of p equal to h bar k as a momentum is gone and hence it is still dimensional it is alright. So, this is called what is known as crystal momentum. So, in the block picture h bar k is not a real momentum of the electron, but it is only the crystal momentum because of the crystal properties this is happening into account and it has the momentum properties momentum dimensions. So, it is called a crystal momentum. We are going to see the effects of this. Now, speed of the electron here I am not really giving up an actual derivation speed of the electron in metal is actually given by 1 by h bar d e by d k. This is nothing to be difficult because you know what is a group velocity expression that was d omega by d k. So, if you see this is e and h bar it is nothing but omega. So, this is essentially the group velocity expression meaning wise that is the thing. So, this let us assume such a thing actually is true or valid for own dimensions. So, what does it mean? The real speed of an electron in the real lattice is actually related to the first derivative of the energy versus k diagram e versus k diagram the slope of which will give you the velocity of the electron having a particular k value. So, if you want to find out what is the speed what you need to know is you have the parabola and you have to find out what is the value of this slope and that we will give you the speed of this one at corresponding to that particular value. So, this is a very important the slide. The state of the electron is given by a particular value of k in real lattice whatever is the electron the correspondingly it has a particular value of k. The speed corresponding to that particular k is nothing but the first derivative of this parabola calculated at that particular k value. So, your e versus k diagram is there and you know that your k is some value let us say k prime what you have to find out is you simply find out the what is the slope at that point that will give you the speed the actual speed of the electron. So, if k changes the velocity also can change because from one point to other as you know the slope changes and this changes. But what is important is if your k remains constant your derivative will always be a constant that means the speed is not going to change. That means if you start with an electron with a certain k value irrespective of what is there in the solid what is the lattice and so on. If the k is not changing your speed is going to be unaffected. This is a very very important point compared to what we have been thinking. We thought positive ions are there all these things will create lot of problem scattering will be there and hence the speed will naturally decrease. This is our usual guess. But this statement tells you that when your k is not changing and as we are going to see very soon if there is no external field whatever be the field let us not worry about it if there is no external field k remains constant. When k remains constant the speed of the electron remains constant. So, this is something which is very important. So, this is actually a kind of a hint for getting an answer to the fact that the mean free paths are actually very large. If this statement is true one can explain very large mean free path that we know experimentally. So, this is what I have been telling in spite of the presence of this core potentials the lattice potentials speed of the electron for a particular wave vector is constant and is time independent provided there is no external field. This is as I mentioned is a very important point how do we prove that I am just giving a very semi classical kind of a picture here we assume that the electric field is someone which is applied externally. So, the force is nothing but minus e times the electric field one can actually find out what is the rate of absorption is the kind of mechanics derivation the power absorbed is nothing but the absorption of energy which actually is given by d by dt which is given by the mechanics definition is f dot v where f is the external field and v is the speed. So, this can be substituted in this manner. So, I can write rate of change of energy absorption and can modify this as I have given here these are all all the steps are here which actually tells you that my h bar dk divided by dt is minus e times the the force that has been applied. So, this tells you that if my external force is 0 my dk by dt is 0 which means that my k is constant this was a statement that was mentioned earlier. So, if you do not have an external influence on the system if you are not applying an external field the electron continues to have the same k value and as long as its k value does not change the speed also is going to be a constant. Irrespective of what is there in the material irrespective of the scattering centers irrespective of the lattice potentials all these things become irrelevant as far as the field is concerned provided there is no external field that is applied. This is a very important breakthrough when you discuss the free electron model and it comes out. This is equation if you see is I told you that h bar k is a momentum. So, this is basically rate of change of momentum and this is a force. So, this has some resemblance with the Newton's law, but too much importance should not be given to this one because f is not the real force in this case not only force and also as I mentioned earlier this momentum calling this h bar k as a momentum is not a good idea because it is only the crystal momentum not the real momentum of the electron. So, this also has to be kept in mind in spite of the similarity of this expression with our usual equation of dp by dt is equal to the force that is Newton's second law. So, do not give too much importance to this one. So, the effect of external force either in the form of an electric field or in some other way is to change the wave vector and the change in the wave vector will cause a change in the speed electron speed that is the information that we get from this semi classical treatment. So, now let us look at let us look at a very very important manifestation of these ideas. As k increases assume that there is an external force that is given that means your k will increase you have a parabola part of the parabola your k increases k increases when the k increases beyond this plus or minus pi by a because that is our limit of the first Brillouin zone it will go to the second Brillouin zone part. But the second Brillouin zone does not have any physical extra physical significance one can always reduce or add so that they can be brought back to this minus pi by a to plus pi by a region we can do that that means when k goes out of this when it k is out of the thing on one side it is equivalent to taking that same thing on the left hand side within the pi minus pi by a to plus pi by a region pi by a plus delta is exactly nothing but minus pi by a plus delta on the other side. So, because it will always come back to a corresponding value can always be found out for any value that goes out of this range minus pi by a to plus pi by a range if it goes either way in the opposite side it will appear a corresponding value. That means when you keep on you apply a continuous electric field for example from outside what is going to happen is you are continuously changing your k as dictated by the our Newton's second law kind of an expression the k keeps on changing when k goes out of the first Brillouin zone it will come back this again gets changed it goes to the right or left it comes back and forth. So, that means the k value actually oscillates between the two that is between minus pi by a and plus pi by a the k keeps on changing oscillating one under the influence of the external field. k oscillates means the corresponding speed the real speed of the electron also oscillates. This kind of an oscillation is actually called the block oscillations block oscillations are very important theoretically from the picture that we have been talking about. So, which actually prevents an electron. So, in the real lattice we remember V is the actual real speed of the electron in the real lattice what is the speed of this electron in the metallic piece for example. Because of these arguments the V actually keeps oscillating the direction keeps oscillating and hence there is no the electron cannot really move in the forward direction according to this picture fortunately then there is no conduction at all in spite of all our attempts. What happens is the advantage is or the good point of this is that the usual frequency of oscillation the block oscillation is something like 10 to the power of 5 hertz that means the speed also oscillates at with this frequency of 10 to the power of 5 hertz. Similarly, in the k space the k also oscillates between the with the frequency of 10 to the power of 5 that that frequency is the same frequency of k oscillation is nothing but the frequency in the the speed of the real electron in the real lattice. However, in an actual material the collisions and other things are there that frequency will be 10 to the power of 14. So, you will not be able to see the block oscillation in the real lattice because of this large amount of collisions that are generally present in a really prepared material. So, it is extremely extremely difficult to observe these block oscillations in an actual experiment even though the theoretically this year people have been trying various things some people have come with some success in actually observing this in a very carefully prepared samples and so on. But the theoretical idea at this point of time for me is to tell you that the physical significance of this minus pi by a to plus pi by a or the first Briello and so on is very very important in the understanding of this electron dynamics in solids that is a my main point. But in that process this also tells you the connection between whatever happens in k space there is a connection with what is going to happen to the speed of the electron in the real lattice. If k oscillates the velocity in the real space also is going to oscillate that is what is seen. So, these block oscillations are they theoretically that is a very important concept. A related issue we have been talking about velocity so far the speed of the electron so far. Now, we will go one step further we want to see what happens to acceleration acceleration is given by again a semi classical treatment is dv by dt. So, this is as we have seen we have the expression it is 1 by h dE by dk. So, I take one more derivative with respect to time this gets this thing this can be rearranged as 1 over h bar h bar d square E by dk square dk by dt. This can be written I will write looking at this picture I can write this has a particular form because dk by dt is something like force. So, I can write this as because I know this expression f is h bar dk by dt already I have shown you. So, because of this I can write this expression as f by m star because I should have something which has dimensions of mass so that this equation is satisfied provided I am ready to accept that m star whatever I am defining is nothing but h bar square because I need h bar here and there is an h bar. So, that becomes h bar square divided by the second derivative of energy with respect to k. So, again it becomes very important that the E versus k diagram which actually gives you the slope and now it is giving you the curvature second derivative is related to the curvature. So, for both these things in the case of speed it was a slope now it is a curvature which is important and in both these cases what is needed is a parabola I mean the parabola or the E curve diagram. So, E k diagram is very important. So, E k diagram from this picture I can actually get an expression for the mass of this electron m star I am going to to call because it is not the not the normal mass but it is given by this kind of a complicated expression which I never thought in the beginning. So, it is actually related to what is going to happen to the shape of this one. Now, I have shown you here as long as it is a parabola a given parabola unlike the slope slope changes from point to point the curvature is fixed. So, for a given parabola this is a constant but if I have two parabola like this one is like this here it is like this this has got more curvature and hence where because of the denominator here this is going to have a small mass. On the other hand if my parabola is like this it has a smaller curvature and the mass is going to be larger. So, this is something which is different compared to our usual thinking electron I know what is a mass and from relativity we know what is going to happen but this is a totally different thing we are not in the relativistic speeds only thing is we are actually having is the electron is moving in a periodic system that is all. So, this idea of effective mass so this is called effective mass because this is not a real mass this is a very very important aspect of this electron dynamics in solids. So, let us see further in all the above expressions I have already mentioned e versus k diagram is critical in determining the dynamic electron motion this relationship is actually called and because of this importance this is called the band structure as we are going to see even though you are not able to see the bands so far this e versus k diagram generally gives what is known as a band structure of the solid and band structure gives you everything. For a completely filled electron case when e is proportional to k square you can see that the first derivative is proportional to k as I mentioned so the speed keeps changing from one point together of the parabola that means if you take two electrons having two different k values they must have different speeds the actual speeds whereas the effective masses are concerned they must be the same because your second derivative is actually a constant speed depends on k but the mass is not but now comes a very important issue this is true only if my parabola is fixed suppose due to whatever reason if my parabola gets distorted at some point of k values for some k values if my parabola is not really the original parabola it actually distorts not parabolic throughout if that is the situation my e k diagram is not parabolic throughout in that case you can expect not only the change in the speed because of the change in the first derivative that of course is there in even in the case of a good parabola undistorted parabola but when it is distorted you can expect different masses corresponding to different k values provided your parabola is not parabola throughout or it is a distorted parabola so if your band structure is represented by a distorted parabola you can expect electrons of different k values to have different masses electrons of different k values means electrons at one region of the band structure compared that with the electrons of a different region of the band structure they can have different masses electrons remember they are electrons but they can have different masses provided that the band structure representing the electrons actually is a distorted parabola this is a again a very very important breakthrough big break from what we have been talking about in fact you can actually get into the situation where your mass can be negative because mass in this case is critically determined by the second derivative second derivative can actually be negative in some some cases so you should be prepared to accept even negative masses for electron this is something which is very strange what are the consequences of this so people have found theoretically and experimentally effective masses can have a very wide range it can actually go from 10 to the power of minus 2 the rest mass of the electron but what the mass which we with which we are familiar and it can actually be sometimes 1000 times so which means that in some solids and some cases the electron behaves as if they are very very light and in some cases they are extreme behaving as if they are extremely heavy 1000 times the actual mass of the electron so such systems are actually the very heavy situation is called heavy fermions electrons are fermions such materials are called heavy fermions I will not be able to go into those details but just wanted to tell you that you have this concept of heavy fermions systems many of the heavy fermions systems are actually super conducting for a variety of reasons I will not be able to talk about that but this is from the solid state point of view it is a very important aspect what are the heavy fermions basically the electrons in that system will have masses which are much larger than the usual rest mass of the electron as I told you earlier you should be ready to accept even negative masses electrons with negative masses are actually called holes so you have two concepts coming there you have very wide range of masses possible for the electron when it is in a in the part of a periodic system and you are actually able to get something what is known as a hole not just merely telling that hole is a vacancy a vacant electron you have much more things to add to that before we proceed further we can also see I mean our original problem was that how you can have a very large mean free paths now you see that this picture the semi classical treatment given by this picture is able to explain very long mean free paths because we have this condition if there are no external fields if the k values are not changed if your parabola is undistorted your speed of the electron will remain the same irrespective of what is there in the solid so this is a very important point as far as a wave propagation because electron in this case is a wave it is a quantum mechanical system a wave when it is passing through a discrete set of points the scattering is much less than what we expect and hence this is able to explain large mean free paths so the picture that is emerging is this one this is not only true for electrons in the case of solids it is in general true for any wave that is passing through a medium where the lattice spacing is comparable to the wavelength that we have so now as a consequence of this what is shown here is I am showing you instead of showing a real parabola good parabola I am showing you a distorted parabola my parabola is distorted towards the band just this is pi pi by a and minus pi by a so you can see this is parabolic for most of the ranges of k values but very close to the band here this I am going to call it as a banded there the parabola is actually flattening out instead of going up like this it is actually flattening what is going to happen let us take a electron which is having a k value somewhere here in the parabolic region no problem it has an electron it is a usual electron that we know and it has got an effective mass which is positive because the second derivative here is positive I take another electron which can be having a k value in this range very close to the band edge either plus pi by a here or it can be here what is going to happen if I calculate the second derivative here is a negative my these are all hand drawn things it may not show but what is meant is having a negative actually there is a second derivative negative second derivative is negative means effective mass actually it must be negative because effective mass is related to the second derivative but having a concept of mass being negative is something which we are not very familiar with so to avoid this negative mass thing negative effective mass what one can do is in all these things we are worried about the dynamics means we are interested in the force or the acceleration acceleration in all these cases most of the time supplied electric field it is given by the force divided by the mass force is essentially e times e divided by m so the factor that is coming into picture is e by m and what we are telling is that we have electron that is a minus charge in minus c and you have a minus m in this case so if you are having a minus charge and a minus m the dynamics is essentially determined by the factor minus e divided by minus m which is actually positive so when the electron is somewhere here in this region of the band the dynamics is determined by this positive quantity e by m because it is minus e divided by minus m the same physics can be mentioned with respect to another thing I want a positive e by m what how can I do that I can do get still get a positive e by m if I take this minus e to be plus e and then it becomes e by m which is again positive so if I do not want to use this negative mass the physics does not change even if I take a positive mass provided I am ready to accept that electronic charge becomes positive so an electron with a positive charge that is showed here is nothing but the hole so hole is nothing but an electron with a negative effective mass as far as the dynamics is concerned an electron with a negative effective mass that means an electron close to the band edges will behave as if they are holes and they will have exactly same properties as that of an electron with a negative effective mass so this minus m is avoided by checking actually checking at sign change so I make this minus m to plus m provided I am ready to take this minus e to plus c so that the ratio is still positive because of this you have this you are actually now encountering a plus c situation that is nothing but the hole so remember I can have a similar thing here then all whatever we are telling will cancel out and there is no contribution but just to give you an idea about this I am taking one electron here and trying to see if I calculate the dynamics of this particular electron and if I compare with this one that is why I have written here clearly this electron is an electron with normal electron with a positive effective mass so this is not a hole because I can explain everything with the positive mass I do not have to worry about it whereas when it comes here my mass is becoming negative to overcome that or to avoid such a possibility of using a negative effective mass what I can do is that I can think that this is an electron with a positive charge that is called a hole and that is what is written here electron with a negative effective mass is treated as a hole so basically a hole has a positive charge exactly having the same magnitude as the electric charge so this is a very elementary idea of a hole that generally we encounter in many situations of semiconductors I will give you some information as we go along so why is this effective mass you know that electron is electron as I mentioned it is not the relativistic effect that we know where the mass can change here why you are having effective mass is you are actually trying to describe the motion of this electron purely in terms of the applied force you are completely forgetting a force that is there a periodic force which is produced by the positive ion you are not real that I am going to call it as a lattice force fl and you are not worried about that you are not really taking into account you want to explain the motion in terms of only one part of the force knowing very well that both the forces are actually contributing to the motion naturally you will get these kind of things if you are having a mechanism where you are actually take the real force into account of full force into account including the lattice force your masses will work out all right you will get a normal mass that we expect for a electron so that is why you get into this concept of effective mass you have to use effective mass because knowledge about this lattice force and properly taking them into account is not going to be easy so we always tend to be happy with our concept of effective mass so using this idea of the so-called empty lattice model I talked about something which actually is then regarding the distorted parabola but if you see my parabola is not really distorted the parabola has various sections in the reduced zone see but nothing is distorted if you want to see the distortion which of course I assumed to explain the effective mass and things like that strictly speaking one has to really see how the distortion can happen the distortion happens as I mentioned because when you really take into account the periodic potentials then it also explains the band gaps so this part is what I am going to show where we are actually trying to solve or see the real band model of metals real band model if a realistic metal is going to be a complex issue it is actually a research problem we will not be worried about that I will give you a very very simple flavor of this by giving the simplest model for the band picture that is what is known as the chronic penny model chronic penny model there is another reason for me to talk about here because this is something connected with what you have seen in the case of quantum mechanics so in the case of quantum mechanics you started seeing the you started seeing the infinite square well potential the effects you also have seen the finite square well potential what we know in a case of a solid at least one dimensional solid is something simple one dimensional solid can be approximated something like this you have this positive ions whose periodicity was taken into account even in the empty lattice model but the actual potentials or the potential will the attractive part of the potential so we are ignored the actual form of the potential wells we do not know but what we can do is that we can approximate these potential wells to be finite of course we know but we will kind of make it finite square wells like the way we have shown so when an electron comes in the region of this one it will see a well here and similarly next one there is a well there is a well there is a well this is an idealized picture this is not a realistic picture that because the actual form of the potential is not like this but to start with chronic penny what they did was they approximated like this so what you have to do is you have to do this problem of not just one finite square well you have to do an infinite number of such things arranged periodically so compared to the quantum mechanics problem that you have done for a finite square well what is extra only thing that is extra here is the periodicity also must be guaranteed otherwise it is just matching the wave functions the derivatives at the boundaries and so on I am not going to do this problem it is straight forward except that there is a periodicity here what is the periodicity here the periodicity is a plus b where you can see a is the region this is the this part the width of the well at something like your a l in the case of a particle in a box and this part this width is actually from minus b to 0 that is so the periodicity is always a plus b this is the only thing that is extra compared to what you know in the case of a finite square well so what one can do is one can proceed I will to save some time I will not do this quantum mechanics again you have not just one square well you have infinite amount of finite square wells extra condition that is needed usual boundary conditions of matching the boundary the wave function and the first derivative definitely there plus you have the periodicity taken so this condition is very important you bring in the blocks theorem for this periodicity that is actually done here so this is something which is extra so when we do this I will straight away go to the expression you get into what is the determinant of this kind because you get various conditions using all these things block theorem some conditions then you have these conditions of demanded by the wave function the Schrodinger equation conditions all these things will give you this kind of a determinant and for a non trivial solution of all these equations there are four equations this must be 0 this actually reduces to this kind of a expression this is not difficult all these things are written here I will come to the physics of this so what I will do is all this kind this this things can be I mean made in the simple form I bring in a new constant p which actually takes care of this potential well that is depth and the width of this thing the v naught a and then you have the b and of course m is a mass of the electron and dash bar square so I get an expression of this kind so you can see the k 1 of course is determining the energy of the electron because we have already defined what is k 1 as shown here k 1 is given by this one so what does it give so what I am going to do is I am actually trying to plot this function I know that right hand side is very simple because it is a cos function so cos function if I write k it has to be from minus 1 to plus 1 so this is the range that is possible when I try to plot this one I will not worry about it when I plot I will give you the results straight away the left hand side is plotted and it comes something like this it oscillates because there is a sin term here a cos term here it will oscillate it is as a function of k 1 it actually oscillates something like this but since this equality has to hold you can only take the solutions which are actually appearing between this 1 and minus 1 and if I do that I can see that in this region there is one part which actually is has to be the solution and then there is some part there is this solution cannot be acceptable and then again there is an acceptable part unacceptable acceptable like that so these are the allowed values of k which means these are the allowed k values are possible for the electron acceptable k values of this one so this red boxes essentially tell you this is the range of k values which are acceptable as far as the wave function the k values of these electrons so they are called energy bands now I am called this as energy band 1 energy band 2 energy band 3 so this is the first time we are able to see the energy bands separated by these gaps where the solution is unacceptable because of this cos 1 minus 1 problem so we have seen the free the classical model no energy gaps no bands we have seen the summer field model quantum mechanical there is a parabola was going all the way from 0 to infinity or minus infinity to plus infinity no concept of energy bands energy band gaps we have seen the empty lattice model which again did not show this kind of a picture only after taking into account the real potentials in the form of this potential well infinite number we are able to see these boxes and their separations these boxes the red boxes represent the bands because band is nothing but allowed k values and not allowed k values disallowed k values that is separating them so this is the first picture of our energy bands and the gaps so this is something very important this is a very very elementary theory as far as energy bands are concerned the textbook material area elementary solid state physics book will have this thing but this is a starting point so one can actually see instead of having a rigorous proof of this what we can do is that we can actually try to see the meaning how meaningful we can check various conditions which we know like for example what happens when we my v0 b is very large what is going to happen when my v0 is v0 b is 0 that means there is no potential I should get a free electron kind of picture that is what is shown here when it has to be s bar square k square by 2 m the usual summer field is k square the picture so all these conditions which we know earlier all these things are verified here also in these limiting conditions which means that this particular method is alright so the concept of energy bands the way we derive now straight away if I go back I this is a picture I got bands and band gaps the actual picture the height varies like this and so on this is something which I related to what I talked about yesterday this is a kind of a band the lowest energy band this is almost like a energy level as you go to higher and higher energies you can see that band width is increasing this is what happens between 3d and 4f 3d is the case where the energy is more because they are outside you will see that the width is more whereas the 4f they are inside they will be having smaller width this has got importance in the magnetism part we have seen talking about it so and as you go to lower and lower it is essentially like a core level the core levels the atomic levels so if I translate this information into the ek diagram that parabola what you are seeing is this parabola gets distorted like this this is a distortion which I use to explain free effective mass and the related ideas so it is not like that the bands are not the parabola is not continuous like this the parabola actually goes like this this is in the full parabola scheme but if I take everything in the minus pi by a plus pi by 8c scheme I will see that this parabola earlier this parabola point was touching with this one but now you can see there is a gap there is a flattening out there is a flattening out here and then there is a gap here this is the energy gap so this is a first band second band third band and so on so just give you an example of this assume that my band structure in a given case is given by this e is given by a minus b cos k a I can find out the velocity by taking the first derivative one can see that velocity corresponding to k and velocity corresponding to minus k are opposite that is why the complete thing in equilibrium the net velocity will be zero so that is why in the absence of an electric field here also the band picture also this picture also gives you zero current the second derivative is this one which means that my effective mass is effective mass is not constant it is dependent on k as you can see and it is cos k a dependence inverse of course all one can see that v is in this range and so on this is an expression which is simply coming out of that very important information so we have some idea about the bands and band gaps now we want to see how you can get a metal how can get an insulator and how can get a semiconductor so for that let us take the length of the Brillouin zone the first Brillouin zone is all that we need the length is minus pi by a to plus pi by a that is something like 2 pi by a is a length from the periodic boundary condition this is another reason why I talked about it the range of k values the separation between the first one k value and the next k value is delta k is nothing but 2 pi by l we have seen yesterday so the number of states in this range is one so total number of states possible in the Brillouin zone is actually this divided by this so this is the whole length and the length between two adjacent things is 2 pi by l this is this that gives you l by a and do not worry about whether it is exactly n minus 1 a or something like that for the length this is nothing but n times a where n is the number of atoms so n is the number of atoms or the number of unit cells the correct thing is number of unit cells so what does it mean number of states in a band is nothing but a number of unit cells crystallographic information so number so you have various bands as we have seen here how many are here how many are here how many are here is determined by how many unit cells are present so this information is very important number of unit cells is given by this so this is true for all the bands taking the spin degeneracy so now I have n electrons can be accommodated n unit cells are possible but taking the account of spin degeneracy plus up spin and down spin I can actually take 2n electron now let us look at this thing for a monovalent metal that means there are n electrons that have to be accommodated so which means that the first band can take maximum of 2n electrons if it is anything more than 2n if it is 2n plus 1 then it has to go to the second band but my monovalent metal I have only one per atom and if I take a one mole of the substance I have to essentially worry about n unit cells that means n electrons must be accommodated so n I have a space for 2n but I will be complete when I have n filled that means it is here so that means I have half filled and half empty so the band the first band I am talking about the first band itself there are spaces available here where it can go so you can see that if I apply an electric fuel or if I am ready to give energy these things can actually get excited and they can go here without really encountering an energy gap this whole thing is the first band the band gap will come here only I have not reached here the highest energy electron is far away from the band it does not see the band gap because it is far below so within the band itself there are vacancies available and hence they are ready to accept the energy without encountering the energy gap so this becomes a better this is a metal sodium is a good metal now we have seen earlier that we expected that the conductivity should go up when I becomes divalent but that is not really true divalent or trivalent things were actually becoming less conducting compared to monovalent that was something which was very difficult for us to comprehend at that time now let us look at it now in this picture if I have a divalent thing when I have number of unit cells to be n is actually 2n that 2n means I have to use this space completely and I will have to fill up to this point when I am filling up to this point any excitation that is possible has to encounter this energy gap so that it can go to the next level only because in this level in this band everything is gone you have to go to the next you have to depend on the next band so naturally it whenever it encounters an energy gap the conductivity is going to come down and hence a divalent metal is less conducting in this picture it is a very simplified picture but it gives you this idea that a divalent thing is expected to give a less conductivity compared to the monovalent this was related to the whole coefficient also I am coming to that because anything close to the band edge you remember when it is here in the case of sodium it is here it is very much in the parabolic region of the band the positive curvature effective mass is positive we expect normal things in the case of verilium kind of divalent things we found that the whole effect was giving a wrong sign at that time we thought it is a wrong sign why is it happening because you are in this region the top of the band is a band where it is flattening out the negative curvature negative effective mass you expect a sign change for everything so the hall effective so it is not an electron which actually is really doing the job we can tell that it is a negative mass electron is doing the job or the hole is doing the job and hence a sign change positive hole is positively charged so a positive charge is expected to give a sign change in the case of hall so that is why hall effect is giving hall coefficient is having a different sign for verilium compared to something like sodium so this is again consistent with this very very simple elementary model so using this I can construct various things a metal is something where the highest occupied band has still vacant spaces in the same band itself so so that these things can be used for excitation by when the system takes energy either from the electric field or from thermal and whatever if it is an insulator situation is something like if anything happens it has to go to the next band crossing this energy gap of each then if this energy gap is small then it is you call it as a semiconductor it is comparable to kbt you call it a semiconductor otherwise you call it as a so the idea many people have this idea that there are no energy gaps in the case of a metal that is not really true what you see here is this one you have energy gaps but what happens is as far as the highest occupied band is concerned the highest energy electrons they do not encounter any energy gap for excitation because excitation within the same band is possible they are possible with so it does not actually encounter any energy gap that is something which is different that is what makes this different from this one of this one here any excitation means they have to cross the gap here the excitation does not mean that you have to cross the gap without crossing the gap they can actually take the energy they can get excited they can participate in the conduction this is the difference between the three categories of materials in general this is possible only after the block picture taking the periodicity into account the lattice potential into account I can tell that the simplest model that can give you this picture is the chrony pendulum so I can proceed further I think I have some time I can proceed there are certain symmetries associated with the bands I will only give the results you showed you something earlier so if I take the speed at k and if I calculate the speed at minus k this as a negative sign here whereas here effective mass is same for both energy has this particular symmetry relationship these are all connected with the actual symmetry of the crystal systems the direct lattice symmetry get translated into the all these symmetries are in the context of k space symmetry the brillo and zone symmetry these symmetries are actually related to the direct lattice symmetry so that is why again the direct lattice crystal structure symmetry that is very important in determining all these things including the band symmetry that one has to worry about this is a picture I gave you earlier so the band edges as I was mentioning this band edges when the electrons are at the top of the band these things are happening and you can expect a difference in the behavior compared to the case of an electron which is situated very well into the parabolic region so near the bottom of the band middle of the band things are all right when you are closer to the edges the flattened part of the parabola things are not all right and you are actually going to get negative sign inversions in many of the physical properties we are coming to a very important section again electrons in a band velocity will be zero at the bottom of the band at the top of the band because the curvature with the first derivative is good to be zero effective mass would be positive at the bottom and negative at the top I already told now can we give a simple picture very simple picture why this is happening why this band gaps are coming this is a very very simple picture if you see the whole problem is happening towards a band edges band edges out of the regions where k equal to n pi by a minus or plus this in matter so k is nothing but 2 pi by lambda where lambda is kind of the wavelength of an electron I am using a I mean going back to the kind of a classical or a semi classical idea but it is just to give a get an idea it is okay not very rigorous taking the k k is of course electron wave vector and if I write if I can get a lambda and I can write k equal to 2 pi by lambda this gives me 2 a equal to n lambda where a is a lattice constant so which means this expression that resembles our usual Bragg reflection what are we looking at we are looking for the free movement of electrons one point to the left to right let us say this is a condition which resembles the Bragg reflection it is a reflection when I take this theta to be actually pi it is coming back exactly so that is what actually is happening pi by 2 actually this is exactly coming back so what does it mean that when this condition is satisfied an electron is actually not able to move in the forward direction it actually gets reflected back it is completely reflected back your theta is pi by 2 so that your angle of reflection is actually pi it actually goes back in the same direction from where it came so it is completely reflected back so when an electron is not in a position to move forward because of the reflection complete reflection into the original side you are telling that is the region or that is the wavelength that is corresponding to the problem case the problem region that is exactly what is happening here so when you are finding out the so called the bad region the bandage what is one can think that the electron wave is not in a position to move forward but because these lattice forces because of the lattice periodicity that is present there is reflecting that force is reflecting it back something like the Bragg reflection that happens for electron waves or x-rays so it just goes back and the k for us is nothing but the k of the wave that is moving and since that is not possible once it is reflected we will tell that you have a band gap because those k values are not physically possible so you have a band gap that is happening whenever this condition is satisfied or this condition is satisfied which in turn talks about this condition which is nothing but the bandage so it is a very simple kind of a crude picture just to visualize what can happen when at the boundaries at the zone boundaries this is extending further what happens in the case of case when you have no electric field but in the band picture so we talked about the electrons in the classical picture what happens there is no electric field no current in the case of somer field picture no electric field no current in the case of a block we did not really discuss but we have some idea but in the band picture let us see what happens when the electric field is applied zero and what is going to happen in this case also a completely filled band is not contributing because we on this side and we on this side because it is completely filled both sides we will cancel each other that means there is no current in the absence of electric field here also no current provided the band is completely filled same thing happens for the band which is with some empty states also so a band whether it is completely filled or not filled you are not going to get any conductivity when the applied field is zero this is exactly like the free electron picture quantum mechanical or even otherwise let us not compare the classical picture when non-zero field is applied completely filled band what is going to happen the highest electrons as I mentioned they are not able to if they want to receive this energy they have to cross the barrier so the current due to these electrons will not be there and so as long as the completely filled bands are considered this is difference compared to the free electron picture even when the applied field is not is there even when the applied field is there the completely filled bands cannot come contribute to the conductivity in a significant manner because they are completely filled and any receiving of energy will need to jump the energy gap which is generally very large and hence it is not possible so the contribution from the filled band even when e applied is non-zero is minimally this is quite different from the free electron picture free electron picture electric field is a absolutely you are getting some conductivity because there there is band picture was not there so every electron was in the same band one can say however the current due to lower electrons cancel in pairs so there is no current now electrical conductivity when e applied is non-zero if it is having some empty states if there are empty states something like this one can apply an electric field and you can see that these are the states which are like the holes because they are close to the band edges they are kind of holes and hence they move in the direction of the applied field as you can see here electrical conductivity then you have just one electron is absent here what the difference is here that states that means this one and this one are absent this is a hole this is a hole here what I am talking about is I am having just one electron is missing that means this electron is there corresponding electron at a minus k1 is not there this is missing so a hole I can tell this is exactly like a hole at k equal to minus 1 so this picture is the actual hole that we know that an absence absent electron is called a hole here so what is going to happen in this case when there is no applied electric field here one can show that we know that the total contribution for a completely filled shell is 0 so j is minus e times the summation of these velocities for from 1 to n all the electrons is a completely filled shell it is 0 we know now we have one we can see so that is a hole at k equal to minus k1 so that should give rise to a current that is j prime it is given by minus e and one hole is there so the net contribution will be 1 to n minus 1 because one electron is not there times exactly same expression this will not be 0 because there is this is not n minus 1 but it is n this can be written in this form minus e of course will be there 1 to n v1 minus whatever is not there that is at the side k1 this is not there so I write minus 1 so this is but the first term is 0 because of this expression a filled band will not contribute so this is 0 this becomes plus e p at minus k1 so what is this this e is a positive charge that is a hole this is a current and remember v minus k1 is in the negative direction so there is a positive charge moving in the negative x direction that means there is a current along the negative x direction this but we can also write e is minus k1 can this v minus k1 is minus v k1 so this minus comes here I can write this is minus e v k plus 1 so minus e is an electron which actually is having a positive velocity but electron having a positive velocity that means a current again in the same direction as in the whole picture so whether you are talking in terms of an electron when you have a completely filled shell except for one electron you can use all the electrons that is n minus 1 electron and describe the motion or you can describe the motion purely in terms of one hole that is the advantage of this whole concept the whole concept of hole becomes important or useful because of this one you do not have to worry about all the entire n minus 1 electrons simply worry about the vacant electron and call it as a hole whatever that hole is going to move whichever direction whatever current that hole is going to contribute to the current the exactly same thing is going to happen when you do the same thing with an n the remaining n minus 1 electron so you do not have to worry about n minus 1 electron purely you can write everything in terms of your hole one hole that is advantage so this is a case there is a current even when there is no applied field as I have written here there is one electron missing but applied electric field is 0 here even when there is no applied electric field the ideal case but even when there is an applied electric field 0 if there is a vacancy of just one electron in the band you can get a very small current over and above this if I apply an electric this is what actually shown here this is a current that will be coming out of this and over and above if I apply an electric field this will increase the current because the velocity gets modified by the electric field as given by this these expressions are all self-explanatory everything is in terms of effective mass of course it you will see you see that the originally the single electron the uncompensated electron that corresponding hole gives rise to a current this current gets magnified or it increased supplemented by the electric field and you get an additional increase in the current here so depending on if you have a very interesting case if you just have one electron short of completeness you can have a current a small current even in the absence of an electric field now if you apply an electric field you see that there is a further increase in the current given by these expressions everything in terms of your m star because we are close to the conduction at the band edge okay I forgot to mention this thing here the what we do in semiconductor since I am not talking about semiconductor language so the highest occupied band this band we call it as the valence band and because of thermal energy being comparable to this gap some of these things get excited here so this becomes the conduction band so this is called a conduction band this is called a valence band for a semiconductor and these excitations will happen and when they go from here for example suppose you are taking one electron from here to here you will describe the motion here as n-1 electrons or I say mentioned just one hole is contributing plus there is an free electron here at the bottom of the band a normal electron so that will contribute a current here so you have an electronic current here a hole current here in the case of a semiconductor which has got this electron band structure like this so this is what I was mentioning so you have when you have excitations creating some hole in the highest occupied band and some very small number of electrons in the unoccupied the next band you will have contributions of hole from the lower band and electrons from the higher band that is why in semiconductors you always talk in terms of both for example if I straight away I am going from metals to intrinsic semiconductors you see that there are three holes correspondingly they have three electrons because number is again conserved at t not equal to 0 eg is comparable to your kbt so the current density instead of writing simply sigma e you have a hole contribution and a electron contribution like this one hole coefficient one can show that it is actually given by the electronic contribution and the hole contribution exactly this one so this can be further simplified right in terms of mobility is where mobility is e tau by m where tau is the relaxation time as we have defined earlier one can write in this one so this is basically telling you that unlike the case of metals we are we read too much we do not worry about the hole concept in the case of intrinsic semiconductors this is an issue and you have to really contribute I mean find out the contributions of holes and electrons holes from the valence band the highest occupied band and the excited electrons which are actually creating the holes those electrons which go to the so called the conduction band here this is the valence band and this is a conduction band you have to worry about a electrical conduction from these electrons this is electronic path this is a hole path that is what is this is hole conductivity electronic conductivity similarly electronic part of the hole coefficient and the hole coefficient of the hole effect so that is what is shown here so this gives you a kind of a cute summary of things which are there in the case of atoms I mean electrons which are moving in mostly I talked about metals a very little extension to the case of semiconductors that is what I am doing I will finish this I will be ready to answer some questions whatever you will have so with this I am forced to finish super semi conducting or the solid state part in all every sense solid state is complete because crystal structure is over this is also over we say that at a temperature higher than 0 some of the electrons lying close to the fermi energy level they rise in the energy states above that yeah so does the level of the fermi energy level goes down as the temperature is increased yes yes yes that is what I in fact one of the slides that expression was there there is a small temperature dependence for the fermi energy it actually comes down but what I told you is that that dependence is very small generally we ignore so if you go through the slides on the free electron model this information was there so e f has a small temperature dependence but generally we ignore so actually it comes down that is true yes definitely it is actually the chemical potential it is called a chemical potential chemical potential has a temperature dependence when you tell that the chemical potential is at t equal to 0 you call it as a fermi energy so actual name the correct name should be chemical potential only for t equal to 0 we call it as a fermi energy but since we are kind of diluting it we told the fermi energy throughout we can explain the periodic force again in detail periodic force yeah yeah it is not possible in a I mean in a question and succession because see if we know everything about the periodic force then you could have actually incorporated in and all these effective mass problem I mean you could have solved it since that is not easy to work it out that is why we write only in terms of the external force and bring in this concept of effective mass because it depends on the actual arrangement of atoms lattice and things like that now it is not an easy thing to write why do we need it actually we are able to explain everything with the concept of effective mass which is an actually a consequence of the fact that we do not know I mean we are not able to properly account for the lattice forces okay but somebody is actually interested in that particular problem let us say lattice dynamics related issue they will worry about it but to understand the physical properties in general we do not have to really look at that aspect and get bothered we can simply write everything in terms of the effective mass effective mass will take care everything is consistent you do not have to really worry about it that is the issue one one really wants to do one has to only attack that problem and try to solve it maybe like a lattice dynamics kind of a thing sir when the electron is moving through the periodic potential yeah the mass of the electron varies from 10 power minus 2 times of rest mass of electron to 10 power 3 times of rest mass of electron yeah so I cannot understand about the negative mass 10 power minus 2 times of mass of electron yeah so first of all there is a clarification this is not in the same material the range is not from the same material from material to material if you take certain materials it can be very low certain materials it can be very large number one number two because the curvature is the one which actually determines effective mass in some cases using that formula it works out to be negative when that happens only and the band edges and band edge is a problem as I showed you so that problem is overcome by calling that particular case