 Hello again, in the previous session of this class we dealt with the Pauli exclusion principle and therefore, we also got introduced to fermions and bosons. Then we went on to deal with quantum tunneling as well as the problem of a particle in a box both of which are examples of the time independent Schrodinger equation and we learned that the energy levels in the particle in a box are quantized. So, what we will do today is to deal with interacting potentials which leads for us to consider the formation of bands of energy and then we deal with the dimensionality of quantum systems that is go from 3 to 2 to 1 to 0 dimensions and introduce ourselves to the concept of the density of states in quantum systems. Now, one of the things that we dealt with in the last class was the quantum well of finite depth, but we started with a quantum well of infinite depth where the energy levels are quantized as we see here with wave functions that correspond to sinusoidal waves. Now, notice that the energy levels are quantized and the energy gap between successive levels increases as the quantum number increases from 1 to 2 to 3 to 4 etcetera on to infinity. What we also noticed is that if you had a potential well of finite depth as opposed to infinitely deep potential wells, then the concept of tunneling leads us to notice that the wave function no longer is confined to within the well, but actually spills out into the outside, outside the box. Within the box the potential is 0, outside the potential is V which is less than infinite. In such case the tunneling phenomenon of quantum mechanics leads to the possibility a finite non-zero possibility of the particles being outside the box even though the energy is less than that of the height of the well namely V. And the shallower the depth of the well the greater the extent of leakage or spilling over of the wave function beyond the walls or the well and the number of levels of energy quantized energy that a particle in such a shallow box or such a finite box can undertake the number of levels is greater as the depth of the well increases. The shallower the well the smaller the number of levels that it can accommodate for a given well width. We also learned that in a finite potential as you increase the width in a finite potential well as you increase the width of the well keeping the depth of the well constant then the narrower the well the fewer the energy levels it can accommodate. And this is a direct result of the Heisenberg uncertainty principle whereby if delta x is smaller delta P x is greater leading to higher energies in a more severely constrained system. So, as the well width is increased the energy levels actually gradually are reduced compared to this very narrow well and more and more number of levels are accommodated within the finite height of the well. So, far we have dealt with single potential wells that is a particle moving within a single potential well the under the influence of a single potential well. While that is interesting in itself because quantum wells are actually realized in practice as we shall see later in the session. While these single wells are interesting in themselves what is really interesting also is a particle that moves in under the influence of more than one such well each of them adjacent to the other. So, let us consider the case of two potential wells adjoining each other. So, think of a particle that is moving here in the first well on the left so to speak, but as we saw earlier the wave function of any of these states which the particle can undertake over here wave function leaks into because of quantum tunneling into the adjoining cell as well adjoining well as well. Therefore, you can consider these as interacting potential wells the particle that is moving in one of these boxes or one of these wells is also affected by the potential that is next door. So, what I want you to see here is that I have shown here in this drawing two sets of adjoining potential wells one in which they are close to each other and the other one in which they are slightly more separated from each other. So, these are two identical wells adjoining each other very close to each other these are two identical wells that are slightly more separated from each other. What I want you to see is that while we had a single earlier while we had a single energy level corresponding to each of these wave functions over here what we now have are split energies that is now we have two energy levels closely spaced to each other for each of these states that used to be a single level in the single quantum well. So, what we have now have therefore are two solutions to the Schrodinger equation one is the so called symmetric wave function that you can see here sinusoidal mostly sinusoidal symmetric wave function and the other one that is the anti symmetric wave function you would have come across these things in your earlier classes. So, you have two solutions one is symmetric and the other one is anti symmetric and the two have different energy levels the anti symmetric wave function always has a higher energy level than the symmetric wave function. So, each of the levels that was a single one earlier in a single quantum well is now split into two and this separation between the symmetric and anti symmetric states the separation in energy in energy becomes greater and greater as the quantum number increases from 1, 2, 2, to 3, to 4 etcetera. Now, notice that the separation between the levels of the symmetric and anti symmetric states anti symmetric wave functions is greater in the case of the two wells that are close to each other than in the case of the well that is slightly separated from slight with a greater separation between each other that is the interaction between these two potentials that is the interaction that this particle in one well feels due to the potential next door this interaction is reduced when the two wells are slightly more separated accordingly because of the reduced interaction you can see that the separation of these levels separation between the symmetric and anti symmetric states is now reduced. So, the point to take here is that when you have this interacting interaction between two adjacent potentials then you have splitting of the energy levels between the symmetric and anti symmetric states and the split between these symmetric and anti symmetric states is enhanced when the potentials are brought closer to each other so that they interact more strongly. Now, let us extend this and go to a number of potentials identical potentials that are in line which is the each other so to speak that is we subject our particle say an electron to the potential of a series of equally spaced potentials along the x axis these are in one dimensional potentials to make the problem easier. So, what we have here here is a single potential well of a finite height 2 of them 3 of them 4 of them and 5 of them equally spaced. So, what you see here is that this single energy level corresponding to n equal to 1 only n equal to 1 is shown for the sake of simplicity so that we avoid clutter. So, this energy level split into 2 then when you have 3 adjacent levels 3 interacting potentials then it is split into 3 and into 4 and into 5 and so on. So, this energy level which was unique in the case of a single potential is then split into as many levels as there are interacting potentials that are equally spaced along this x axis. So, we can think of this as a periodic potential you are all probably familiar from your early classes of the effects of periodic potential in crystalline solids. So, what we have here is a simulation of a simple case of a one dimensional potential that is periodic and illustrating illustrating clearly that energy levels are split when you have a particle that is subject to the potentials of subject to the influence of interacting potentials especially in a periodic array. So, this is similar to the periodic potential in crystalline solid. So, what this is illustrating is that bands of energy you can call these bands of energy. So, think of this as a potential due to an atom due to an ion in a solid. So, you have a another ion which is in a periodic array in that solid in the crystalline solid. So, what we can think of this as a simplified illustration of a periodic potential in a crystalline solid the periodic potential is simplified as a square well potential to make the problem easier. And what I want you to remember what I said last time I want to repeat that and that is that these are rigorous computer simulated solutions of the Schrodinger problem Schrodinger equation for one dimensional potentials by using appropriate boundary conditions as we have discussed before. So, this is showing for our understanding of the formation of bands this is showing that if you had an electron in a crystalline solid for example that is subject to the potentials of adjoining ions in a periodic array then it has energies that are in a band. We can extend this further by noting the following. So, what we have shown here in the in the previous case we had shown simply square well potentials. So, called square well potentials that as you can imagine is really an idealization real potentials have more complex form to illustrate the complexity of the real potentials in a crystalline solid what is shown here are these so called saw tooth wave potentials. So, we begin with a square well potential and then we modulate that as shown here with a saw tooth with increasing amplitude for the saw tooth that is you are going from a square well potential gradually from here to here to here to here we are replacing the square well potential with a saw tooth potential which is a bit more realistic than the square well potential in representing the potentials in real solids. So, while there are details to be observed here regarding the preservation of symmetry and anti symmetry and so on as we go from a square well to this saw tooth configuration the point to take away once again is that each of these levels notice that we have four periodic potentials here and therefore, this level over here at the bottom is split into four levels in the band or in the incipient band as we might say therefore, this is a more rigorous illustration of what happens in a periodic potential of a slightly more realistic potential function that approaches that of a solid or crystalline solid. Now you would have learnt earlier in the earlier part of the course you have dealt with band gaps and so on and certainly you are familiar with the energy bands in silicon and the band gap and so on. So, what this is shown what this shows here is how the atomic states of silicon atomic energy states of silicon this represents the outer electrons the valence electron shell of the silicon atom how these levels split into bands as the silicon atoms condense into a crystalline solid. So, these levels represent energy states of silicon atom that is silicon atoms are far away from one another. So, that they do not influence one another. So, these are atomic energy states undisturbed atomic energy states as you bring the silicon atoms large numbers of them into a crystalline solid as it condenses into a crystalline solid because of the interaction between the potentials of the adjacent silicon atoms as we have seen in the simulation just shown because of that these atomic energy states which are distinct bore like energy states these split into bands. So, what we see here is the widening of these energy levels into bands as shown here. So, here you have the upper electron energy level these are actually the anti symmetric states these are symmetric states. So, there is a lower energy over here for this band of silicon. So, in the equilibrium configuration that is at the equilibrium distance. So, this x axis represents the separation between silicon atoms at the equilibrium separation between silicon atoms then a band gap occurs between the upper band and the lower band. So, the lower band is the valence band the upper band is the conduction band and there is a gap E g which is 1.1 E v for silicon that develops as silicon atoms are condensed into a bulk crystal. So, here is a more schematic diagram of energy bands in a metal silicon is a semiconductor. So, there is a band gap as we have just learned. Now, in a metal something similar happens namely the free atom energy levels spread into bands these represent the inner core electrons and therefore, they spread into narrower bands because the interaction between inner core electrons adjacent on adjacent atoms is very minimal. Therefore, because of the reduced interaction then the spreading of these levels into a band is a narrow spread this is a narrow spread. So, as you go to outer and outer electrons these bands get wider and wider representing a greater degree of interaction between electrons in the outer orbits of an atom. In the case of a metal as you know as you have already learned in some earlier class the outer band the conduction band in this case is really not full unlike in the case of silicon where the valence band is full and the conduction band is empty here the conduction band is not full let us say it is half full. Therefore, this is a representation of the energy bands of a crystalline metallic solid. Now, what we have seen here is the formation of bands and what I want you to note is that atomic energy levels are spreading into bands of energy. Now, consider a general case of band formation in a bulk crystalline solid let us forget about whether it is a metal or a semiconductor atoms that are far apart from one another notionally at infinite separation they have atomic levels e 1, e 2, e 3 etcetera. When you condense them condense large numbers of them into a crystalline solid then these levels spread into bands as we have learned. Now, one thing that we must remember over here is that this spreading this splitting of these energy levels as they come together into a crystalline solid is influenced by the Pauli exclusion principle which if you recall says that two electrons cannot have the same wave function they cannot be in the same state. Therefore, as they crowd into a solid crystalline solid these electrons have to spread into bands because of the Pauli exclusion principle being valid. Now, what I have shown here is a notional band gap between the valence band and the conduction band. So, as you condense this set of atoms into a solid crystalline solid a band gap develops. Now, this is a bulk crystalline solid what I mean by that is that it is of a macroscopic size of the order of you know contains of the order of 10 to the 23 or Avogadro number of atoms. So, we are talking about a large number of atoms where there are large number of electrons. So, the bands is spread into relatively wide the energy levels spread into relatively wide bands and therefore, you have a gap between the valence band and the conduction band as denoted here. Let us consider band formation in a nano crystal that is we go from a bulk crystal where as I just said we have approximately an Avogadro number of atoms into a nano crystal where you may have or may be 10,000 atoms or 100,000 atoms something of that order. Now, as you bring these atoms together these are the energy levels of the individual atoms non interacting atoms at infinite distance we bring them together. Now, notice that we have a fewer number of atoms within this nano crystal much fewer than what the case is in a bulk solid bulk crystal. So, as a result the spread of these atomic energy levels into bands will now correspond to the relatively small number of electrons that are present relatively small number of atoms that are there in these nano crystals. Let us say we have 10 to the 5 atoms in this nano crystal 10 to the 6 of that order then the width of these bands the width of the each of these bands will be less than what it would be for the same material in a bulk crystal. That is we go back to this let us say this material and this material are the same in one case you have the bulk crystal with about 10 to the 23 atoms and in the other case you have a nano crystal with about 10 to the 5 or 10 to the 6 atoms correspondingly the width of the bands representing even e 2 e 3 etcetera are now reduced. So, if there is a gap between the conduction band and the valence band with each of them of a smaller width than before it means that the band gap between the top of the valence band and the bottom of the conduction band that band gap is now increased. So, narrower bands in the case of smaller crystals nano crystals and therefore, the band gap becomes wider the band gap representing the frequency of the light that might be emitted when there is a transition between the conduction band and the valence band. E G represents the frequency E G represents the band gap and therefore, the frequency of light that would be emitted if there was a transition between the conduction band and the valence band. If this becomes greater if the band gap becomes greater than the frequency of the light that is emitted is greater. And because blue light has a higher frequency than red light such a change in the wavelength of light from a lower frequency to a higher frequency is called a blue shift. And this phenomenon that leads to such an increase in band gap and therefore, an increase in the frequency of light that might be emitted in a transition like this. This phenomenon is the so called quantum confinement phenomenon. Actually quantum confinement takes place in any solid because the electrons cannot move out of the solid. But here in any solid you would not notice the difference between what happens in let us say a crystal of 1 millimeter versus a crystal of 1 centimeter in size. Whereas when you come to the nanometric regime you begin to see the differences between what would be the case in the bulk material and the nano material and a nano crystal the same material. So, this is the blue shift. This can be seen dramatically in the case of semiconductors like cadmium selenide. So, what is shown here is the famous picture of the light emission from under fluorescence from cadmium selenide nanocrystals with increasing particle size from left to right. It turns out that it is possible to prepare cadmium selenide nanocrystals with highly controlled diameter all the way from let us say about 8 nanometers on the right to about 2 nanometers on the left. So, in the size controlled crystals of cadmium selenide one can observe the progressive change in the band gap as we reduce the size from about 8 nanometers to about 2 nanometers where the band gap is such that blue light is emitted. So, this is a vivid illustration of how the band gap can be changed by controlling the size of crystalline semiconductors. Actually this phenomenon has very important practical applications which we shall probably return to later in the segment of this later in this segment of the course. Now such quantum confinement or the blue shift is not confined to cadmium selenide. Cadmium selenide is a so called compound semiconductor. Silicon is the prototypical semiconductor which we are all familiar with, but silicon is an indirect band gap semiconductor which you know and therefore it is a poor emitter of light. The optical efficiency is very low because it is an indirect band gap semiconductor. Therefore, such dramatic effects as we see here in the case of cadmium selenide due to size variation cannot be seen in the case of silicon. Nevertheless, measurements have been made of the band gap of silicon in silicon nanowires that is by controlling the diameters of nanowires of silicon which can be grown through a method known as the VLS method which we shall return to later. As a function of the diameter of nanowires as you go from about 7 or 8 nanometers for the diameter of nanowires and that is steadily reduced to about 1 nanometer you can see that the familiar 1.1 EV band gap for silicon present at about 7 or 8 nanometers of diameter for the nanowire that increases steadily to go beyond 3 electron volts from 1.1 to 3 electron volts when the wire diameter is reduced wire act these are actually single crystalline wires. So, when the wire diameter is reduced to about 1 nanometer the energy band gap shoots up all the way to about 3 electron volts. This is borne out these are actually some of these data points are from calculations, but it is borne out by experimental work too that there is a indeed a blue shift in the silicon semiconductor as well. Now, in the very first class of this segment we learned about how properties and materials scale with size and we talked about the smooth scaling with size that is as the size is reduced. There is a smooth scaling of the melting point the cohesive energy and so forth. Now, it turns out that that set of properties where there is smooth scaling is different from such a variation in the band gap semiconductors. So, the variation in the band gap or the enhancement of band gap when the size is reduced is not one of those that scales smoothly with size as a size reduced and enhanced. So, this is a this part of a second set of properties where scaling with size is not smooth. Let us return to the potential well or the particle in a box earlier we dealt with the infinite one dimensional potential well, but it is easy to extend this to a three to three dimensions that is a potential well that extends along the x y and z axis. So, if you recall the treatment there we had an extension of l along the x axis for the potential well. So, you can have l x l x l x l y and l z in a three dimensional potential well in which case the energy Eigen functions in analogy with the one dimensional case would then be the product of three sinusoidal functions each with a quantum number n 1 n 2 and n 3 which are all integers, but n 1 and n 2 and n 3 correspond to the length l x l y and l z for the boxes along the respective directions. The Eigen values of energy for such a case of a three dimensional potential well would have three quantum numbers e n 1 n 2 n 3 with the expression for the energy being h bar square over 2 m l square into n 1 square plus n 2 square plus n 3 square, where each of them is an integer and we have simplified this case by making this into l x equal to l y equal to l z equal to l. So, the algebra of this is simplified when all the l's are equal. So, the energy Eigen values then would be h bar square over 2 m l square n 1 square plus n 2 square plus n 3 square. Now, we have three quantum numbers and you can observe from the formation of this from the nature of this expression that you can have different solutions that yield the same value of energy. You can have n 1 n 2 n 3 with certain numbers, you can have n 2 n 3 n 1 with the same certain numbers, but distributed differently and so forth. So, you can have multiple solutions for which the energy is the same. That is the case of degeneracy different combinations of n 1 and n 2 and n 3 giving the same value of energy. That is many different states of a particle in a three dimensional box would have the same energy and the states are essentially indistinguishable. So, such degeneracy which I suppose you have come across earlier is a common feature among quantum mechanical systems and it is a result of the symmetry namely l x l y and l z being the same. So, in this case it provides cubic symmetry. So, because of the symmetry then the degeneracy is enhanced and this is a general phenomenon in quantum mechanics that this degeneracy is broken by breaking the symmetry that is if you had l x l y and l z different from one another then the symmetry is broken and that then you would have a smaller extent of degeneracy. In a previous session I introduce you to the density of states. The concept of the density of states that is how many states of energy that a particle can assume in a given energy interval in a quantum mechanical system. Consider a three dimensional infinitely deep potential well as we just did. So, then you have E n is equal to h bar square there is an error here I am sorry it is not 8 is 2 h bar square over 2 m l square into n x square plus n y square plus n z square. Now, if you consider l to be equal to 1 centimeter that is we consider a macroscopic box not a quantum box in the usual sense, but a macroscopic box of extension 1 centimeter. So, we substitute l equal to 1 centimeter when we do that and we consider the value of the Planck's constant E n E n then becomes 3.7 into 10 to the power of minus 15 into n x square plus n y square plus n z square. So, many electron volts. So, by plugging in the values of different quantum numbers one can calculate the energy of the corresponding states. As we said earlier the difference between successive levels or neighboring levels increases as the quantum numbers increase. So, for higher quantum numbers succeeding levels are separated by a greater interval of energy. Now, let us assume a metallic sample which is a crystalline sample with a periodic potential as we have said earlier. Let us assume a metallic sample in which case experiments show that the maximum value of E n is of the order of 5 E v is so called Fermi energy. So, experiments show that the maximum value of E n is about 5 E v. Now, if you put that 5 E v in into this expression and we take n x equal to n y equal to n z for simplicity then the quantum number corresponding to the highest energy level namely 5 electron volts is about 1.8 into 10 to the power of 7. So, I can imagine that these numbers you know for bulk materials quantum numbers are very large. Now, this is the top most level we can get the energy level just below the maximum level just below the top most level by substituting for one of the quantum numbers instead of n x you would have n x minus 1 that is 10 to the power of 7 minus 1. If we do that arithmetic will tell us that the energy difference between the top most level and the one just below that is of the order of 1.25 times 10 to the minus 7 E v which is indeed a very small energy interval. So, what this is telling us is that in a macroscopic sample levels of energy that an electron can assume are very closely spaced that is the interval of energy d e between successive levels which are all discrete by the way you know these is important to remember that all these energies are discrete because these numbers are discrete the energy interval between successive levels is very small 10 to the power of minus 7 E v. Therefore, it makes sense to talk about the by the way these are called quasi-continuous states they are actually not continuous they are discrete but they are quasi-continuous because there are such large number of levels in an interval of let us say 1 electron volt the separation between them being 10 to the power of minus 7 in a 1 electron energy 1 electron volt interval there would be a large number of energy levels possible. Therefore, one can speak of the concept of the density of states this is where the concept of the density of states comes in defined as d n divided by d e or the number of energy levels per unit energy interval that interval being electron volts let us remember 10 to the power of 1 electron volt is about 10 to the power of minus 19 joules. So, the density of levels then talks about the number of energy levels that a quantum system has per electron volt energy interval the number of discrete energy levels of a quantum mechanical particle derives from its confinement to a region of space even if that confinement is not necessarily even if that confined region is not very small there is a confinement and confinement invariably results in quantization of energy levels. Now, such confinement can take place not only in the familiar 3 dimensional objects that we can have, but such confinement can be in 1 dimension in 2 dimensions 3 dimensions and even in 0 dimensions. So, in the world of nanomaterials nanotechnology not only do you have confinement on all along all 3 dimensions which is referred to as a quantum dot. So, a quantum dot in principle has no extension. So, in all dimensions it is a very small object all 3 dimensions. So, that is a quantum dot or a 0 dimensional object or you can have a 1 dimensional object such as for example, the carbon nanotube you can think of that as a 1 dimensional object in an object or a 2 dimensional quantum object the prototypical example of that is the graphene sheet which I think we will shall return to later. So, you can have confinement in 1 dimension 2 dimensions 3 dimensions and even in 0 dimensions. Now, the number of energy states available to electrons in a quantum mechanical system must therefore, really depend on the dimensionality of the system. This is sort of intuitively obvious that this number of energy states should be dependent on this because the number of atoms and the number of electrons present would be dictated by the fact whether it is a 1 dimensional, 2 dimensional, 3 dimensional, 0 dimensional object. Now, if you recall the solution to the time independent Schrodinger equation for the infinite potential well we had the reciprocal vector k. So, we can think of the so called k space when you have confinement the region of k space available to a particle is limited for example, to a line in 1 dimension to an area in 2 dimensions and to a volume in 3 dimensions. Therefore, we can see that this density of states should really behave differently should have different kinds of functional dependence on the number of dimensions in the object. Simple analysis which we shall not go through here shows that the density of states has the following dependence on the number of dimensions in a crystal. If it is bulk material, 3 dimensional bulk material there is nearly no confinement. If you take a 1 centimeter object as I was saying a moment ago in such case for all intents and purposes there is no confinement of the electron. 1 centimeter is very much larger than the mean free path of an electron and therefore, this is really not confined. In such a case the functional dependence of the density of states that is d n d e number of states per unit energy interval this varies as e to the power of half. If you have a quantum well where there is a 1 degree of confinement then the density of states is independent of energy d n d e is equal to constant. If you have a quantum wire so that an electron for example, is confined to move along a line. So, there are 2 degrees of confinement then the density of states varies as e to the minus half. If it is a quantum dot where an electron for example, is confined in all dimensions then the density of states is a direct delta function that is it spikes at specific values of energy but it is 0 on either side of such a value of energy. So, graphically what this shows is the density of states d e as a function of energy. So, what I just said earlier about the functional dependence of the density of state functional dependence of density of states and energy is shown here graphically in 3 dimensions there is a parabolic dependence of the density of states energy on e and then you have 1 over e to the power of half dependence in 1 dimension and in 2 dimensions the density of states is independent of energy. In a slightly different representation what I have shown here on the right hand side is the density of states as a function of energy in 3 dimensions over here. So, you have this parabolic dependence we are thinking of a nano system here and in a quantum well you have a sort of a staircase over here and in the case of a quantum wire you have this spike dependence where there is a sharp fall off on either side of a certain value of energy whereas, in the case of a quantum dot you have the density of states 0 except at specific values of energy as shown here. So, you have spikes of density of states in the case of a quantum dot which is the reason why in the illustration of cadmium selenide what we saw was sharply blue and sharply green in other words emission of light monochromatic light from particles of a fixed wavelength fixed size. So, that is an illustration of the density of states being very sharply defined in the case of a quantum dot system. Now, continuing with the concept of the density of states one usually considers energy states of electrons or holes in material systems to understand its behavior. Therefore, the density of states usually refers to the density of electron energy states say in a semiconductor or a metal as you might expect although we are not really gone through the treatment the density of states determines the various electronic optical and other properties of the material because the density of states determines the actual energy configuration of a system. One of the things I want to point out is that the density of states is really the density of states available for electrons to occupy the actual occupation of these levels depends on the statistics of the system. For example, the Fermi direct statistics for electrons the Fermi direct statistics determine the probability of occupation of a given energy level. So, the actual energy configuration that is the energy levels actually occupied by a population of electrons in let us say semiconductor is given by the product of the density of energy states and the probability of occupancy of a given energy state which is derived from the statistics. Therefore, the density of states is not the density of electron occupation the density of states is the density of states available for occupation and that has to be multiplied by the probability of occupancy of a given state to get the actual configuration of occupation of different levels of energy. Now, because of the way this is defined it is clear that the dimensionality of the material namely the nano structures that we are dealing with alters the properties in a democratic way because the density of states would be different in these objects of different dimensionality. The dimensionality and the extent of confinement affect the properties in a pronounced fashion. So, this is one of the fundamental aspects of nano materials nano crystals where the behavior is determined by the fact that they may have fewer dimensions than in a typical bulk sample. Now, one more thing to extend this concept of density of states though it has been applied to an ensemble of electrons or holes we have basically referred to that so far. The concept is extendable to other quantized forms of energy for example, phonons which are the quant of vibration energy. So, this density of states is extendable to all so called quasi particles phonons, magnons, polarons and so on. And the corresponding properties that is the phonon density of states then determines for example, the mechanical properties of a solid material. It is useful to be familiar with the density of states in a bulk material versus a nano material nano metric sample. As you may expect intuitively the density of states depends on the number of atoms in a sample because this number determines the number of electrons in that sample if you are dealing with electronic properties of the material. Therefore, the density of states depends on the number of atoms in the sample, the size of the sample and therefore, the number in the density of the energy states is determined by the size of the sample. In a metal one can show that the in a metal of let us say 1 centimeter cubed volume 1 centimeter by 1 centimeter by 1 centimeter which is the sample we considered a while ago. One can show that the density of states is of the order of 10 to the 22 per electron volt that is that is the essentially we have an avagadro number of energy states possible in a bulk metal sample of 1 centimeter in extension. By contrast if you consider a nano metric sample of silicon measuring let us say 100 nanometers by 100 nanometers by 10 nanometers then one can show that the density of states in the conduction band you know remember these silicon sample has a band gap and you have the valence band that is full and the conduction band that is empty at zero temperature. The density of states in the conduction band is of the order of 10 to the 5 per electron volt that is you can see the enormous difference in the density of states between difference between the density of states in a bulk sample versus the same quantity in a nano metric sample. Therefore, you can appreciate that quite apart from the dimensionality of the sample namely 0 d, 1 d, 2 d and 3 d the density of states is also dictated by the small number of atoms relatively small number of atoms present in such a structure and therefore, the density of states which affects the properties of such ensembles is significantly smaller than in a bulk sample. I think we will conclude there for this session. We will come back in the next session to deal with real quantum wells. So, what we have done today is to work with quantum wells and arrays of quantum wells to lead to the concept of bands of energy and how these bands of energy are formed in bulk material of macroscopic extension versus nano materials where the bands are narrower than corresponding bands of energy in bulk material leading to the concept of enhancement of band gap when the size of a sample is size of a given material is greatly reduced to nano metric dimensions. So, that is the so called blue shift and then we have considered the concept of the density of states density of quantum energy states in a confined material and we have shown that at least we have reviewed not really established that the density of states depends different differently an energy in objects of 0, 1, 2 and 3 dimensions and we have said that the density of states is a very important quantity that determines the electronic and other properties of material. So, we will continue with the realization or illustrations of the realization of actual quantum wells and the devices that come out of that in the next session. Thank you.