 So, in the next few modules you want to talk about semiconductor nanocrystals and their ultrafast processes. So, as you know nanomaterials are strange things they are neither like atoms or molecules nor really like bulk material. As we have studied maybe even in class 11 and 12 in atoms we talk about orbitals discrete energy levels. And then in the first approximation we use them as linear combinations either in hybridized or unhybridized state to generate what are called molecular orbitals when we want to talk about molecules. Once again in molecules we have discrete energy levels. Now the issue is that more the number of atomic orbitals that is used to generate the molecular orbitals smaller is the energy gap between the bonding and anti-bonding levels produced or even if there are several bonding levels for bonding molecular orbitals energy gap between them would decrease energy gap between the anti-bonding orbitals would decrease. So if you keep on increasing the number of atoms like this what would happen is that the energy gaps between the bonding molecular orbitals would keep decreasing to practically zero energy gaps between anti-bonding orbitals would keep decreasing to practically zero. So you would end up getting no longer discrete states but continuous bands and band theory is what is used to discuss solids bulk solids may they be conductors or nonconductors, semiconductors whatever whether it is conductor or semiconductor or nonconductor depends on what the band gap is. Now if you think of the reverse process if you think of taking a solid and breaking it down into smaller and smaller and smaller particles what will happen there will come a time when these continuous bands are no longer going to be continuous rather they will break down into discrete levels. So there is still not be molecules but you will have band structure that looks something like this. So you can also think that there is a sort of a band gap but within the band you have discrete energy levels and this phenomenon is especially observed when we work with moderate band gaps that is for semiconductor nano crystals. And these energy levels are designated for further discussion of designation of energy levels one can follow this paper by Klimov in annual review of physical chemistry 2007. So this energy levels are designated like this the lower ones the valence band the energy levels are labeled 1s whole 1p whole 1d whole and so on and so forth. In the upper level they are designated 1s electron 1p electron 1d electrons 1 and so forth. What does this mean what it means is that when there is a transition what happens an electron goes to a higher energy level and it leaves behind a vacancy this is something that we know from our understanding of solid state chemistry or solid state physics. So the conduction band is always occupied by electrons and the vacancies they leave behind which are called holes are in the lower levels. So what happens when an electron goes to a higher energy level of course the energy increases and that transition would require light of higher energy shorter wavelength. Now what is the meaning of this whole energy level that means the separation charge separation has started in you can think from a lower energy level. So the hole is in a lower energy level means that the energy is more you can think if you want to think of it as a molecule you can think that the transition has taken place from a lower energy orbital rather from rather than from the highest occupied orbital. And this spd these are essentially like term symbols and they stand for the total angular momentum of the particle we are talking about we are not going to enter a very detailed discussion of that because I mean once again semiconductor nano crystals can be half a course by itself we really do not have so much time anymore we are slowly approaching the end of this course. So let us just take it axiomatically that in the conduction band we have 1s 1p 1d in increasing order of energy of electron and in the valence band you have 1s 1p 1d in decreasing order of energy but then do not forget that lower the whole occupying a lower energy level actually means greater energy because that is where the energy has started from. So if you think of transition between these levels so you can have something like this say this one is 1s half once again we will not get into a discussion of what half is but if one knows this total angular momentum and z component angular momentum it is not very difficult to figure out. So this transition what has happened is that the electron has gone from here to here and the whole is in this level so this is 1s half whole and this is 1s electron right. So this transition would be the lowest energy transition in the energy levels that we have drawn here is that quite clear. If you have some other transition let us say 2p 3 by 2 whole 1p electron if these two energy levels are involved see it is a longer arrow energy gap is bigger. So if you look at the absorption spectrum the lowest energy absorption is and today we are going to discuss cadmium selenide and nothing else many semiconductors are there many studies are there we will only focus on few representative studies on cadmium selenide. So the lowest energy transition is the 1s whole 1s electron these two energy levels are involved and this is how one writes it 1s e dash 1s 3 by 2h you do not read the dash just call it 1s e 1s 3 by 2 whole okay that is the designation of the band the second one is 1s e 2s 3 by 2 whole which means the electron is still in the lowest energy state in the conduction band but the whole is somewhere here. So of course energy gap is larger that is why it shows up at a higher energy band as a higher energy band in the absorption spectrum and so on and so forth you can see all you can designate all these bands it is a little difficult because as you go higher up in the energy there is also a scattering profile and the degenerate transitions so it becomes sort of smooth but the lowest energy bands are usually visible and this lowest energy band is called the bandage absorption it is called bandage because this is what it is right one end of the lower energy band one end of the higher energy band that is the meaning of bandage the lowest energy transition is the bandage transition. Again it is also well known that if one changes the size of the nanoparticle then the color of absorption and emission also change so this is a very common figure whenever anybody does any work on nanocrystals semiconductor nanocrystals you would see this photograph of vials with changing color the material is the same only the size of the particle is different and you see the absorption spectra have also moved towards lower energy as have emission spectra as the size has become larger or smaller what happens when size increases does absorption the bandage absorption move to higher energy or lower energy why because the energy is given by a sort of a particle in a box kind of model particle in a box because you have 1 by R square here but it is a little more complicated than that a particle in a box works if you want to just compare different energies. So EG effective for a nanoparticle of radius R is given by EG infinity means the band gap of the bulk solid plus h square pi square by 2 R square multiplied by 1 by Me plus 1 by MH of course when we have something like this what is this this is the reciprocal of the effective mass minus 1 by 0.8 E square by epsilon R so this is the additional R dependence that comes in over and above the simple particle in a box model of course one can ask what is the particle what is the box and there is some confusion often in people's mind that sometimes you understand what the particle is but why does a box model work after all we have a spherical nanoparticle at least that is what we are talking about right now we are not talking about rods or anything well the particle is the exciton a an electron whole pair that moves together tightly bound electron whole pair and what is the box the size of the box is given by the diameter of the nanoparticle because the way it works the one dimensional box model is that see outside the electron and hole considering them to be a particle the particle always has to be in the nanoparticle it cannot remain outside so the situation is exactly the same as the particle in a box model that we study in quantum mechanics so it basically means v equal to infinity outside the particle and it does not matter in which direction there is no theta 5 dependence so outside the particle v equal to infinity particle cannot exist inside the particle what you assume here is that the potential energy is 0 why is potential energy is 0 because right now we are talking about only one particle only one exciton in nanoparticle so there is nothing that it has to interact with the idea is that interaction with the lattice and all is completely neglected so we are pretending as if once you form the exciton remember there is no question of interaction between electron and hole separately that is already accounted for because electron and hole together forms exciton that is why v can be taken to be 0 and that is why this model more or less works so let us just explain this terms this is band gap in bulk as I said already this capital R is the radius of the nanoparticle and this is the effective mass of electron and hole now there is another point of confusion that is often there what is the meaning of effective mass of hole hole is just a gap but effective mass of hole can actually be calculated and that has been done long before people started even talking about nanoparticles so that is something that is very well established the standard formula by which the effective mass of hole can be calculated right and what I have in this all this discussion is not from any textbook so whatever papers we have used the references are given on every slide great now let us go on and talk about ultrafast time resolved PL study of cadmium selenite this is a paper by Rosenthal and co workers so here you see well let us neglect the fact that absorbance arbitrary unit is written the solid line denotes the absorption spectrum the dashed line denotes the fluorescence spectrum Rosenthal has actually called it fluorescence and not photo luminescence because the model that they used involved as we are going to see singlet and triplet states that is not accepted universally people are more comfortable calling this simply photo luminescence because spin states and all may not be easy to incorporate in a system like this but for now the important thing for us to notice is that all the emission that is there is the emission spectrum is a mirror image of the bandage absorption which means that it is completely bandage emission now when see exciton formation essentially is generation of the what is called a carrier right the carrier is exciton which is tightly bound electron whole pair now when they recombine that excess energy has to be given out that energy can be given out in a radiative manner or in a non radiative manner both actually happen and we are going to discuss that in a little more detail. So there are several things that can happen before this electron whole recombination is complete first of all it can just recombine that is one thing secondly what can happen is that the surface of the nanoparticles they always defects usually there are defects it is not impossible to make completely defect free nanoparticles but it is not easy and you have capping agents and all just to pacify the surface so very often what happens is that either the electron or the whole is trapped by the surface some dangling bond at the surface or even sometimes the protecting the capping group capping agent when that happens of course now see either the electron or the whole is not even there in the nanoparticle it is separated somewhere else it is in another energy manifold. So it cannot recombine so easily what will happen then is that you will get a long lifetime and you will get usually if it is a radiative trap state then you will get a redshifted long lifetime emission more often than not trapping is just a non radiative process you get non radiative traps and you do not see anything the other thing that can happen is Auger recombination. Auger recombination means this energy is transferred to third particle and once again that is going to show up in the dynamics of recombination a third thing that can happen pretty much like what happened in gold is that you can have exchange of energy with the surroundings it is excess energy can go into the lattice non radiatively and that would cause a severe decrease in lifetime as well as your quantum yield. So what Rosenthal did was that she looked at nanoparticles of different size very simple kind of experiment to start with. So these are nanoparticles of diameter 25 angstrom to 60 angstrom so relatively large nanoparticle not really 2 nanometer, 3 nanometer, 5 nanometer not like that and I do not know if you see it in this decade actually you do right what happens in 25 angstroms do you see a first component and that first component is not there so much when you go to longer lifetimes right. So this is a summary of amplitudes I have not shown the second amplitude amplitudes and time constants associated with this nanoparticles of different size. So what we see is that as a size increases from this diameter not radius 25 angstrom to 60 angstrom well even before going there all these decays are fit to bi exponential functions there have been more complicated treatment of this kind of data as we will discuss in the next module. But here Rosenthal group had fitted to a bi exponential function and then what they got is they got a very short lifetime 3 to 5 picosecond and relatively longer but still short enough lifetime that went from 43.8 to 90.6 picosecond. And the percent amplitude of the ultra short component the picosecond component decreased from 69.2% to 31.1% so naturally the amplitude of the relatively longer component increased accordingly. So all this is shown nicely in these two graphs the only thing I do not know is that can you see the difference in colors which one is which color okay bravo I cannot see it. So what we see is that both the lifetimes actually increase as the size of the nanoparticle increases naturally this is the the two axes are actually different this is a shorter lifetime this is a longer lifetime and this is what we have discussed already amplitude of the short component decreases as a diameter increases amplitude of the relatively longer component increases with increase in diameter okay so these are the qualitative trends. The most important inference of this paper comes from the next figure that we are going to see what they did was they plotted reciprocal of lifetime against nanoparticle size why would one be interested in reciprocal of lifetime yeah it is a rate constant right and that is what by the time we are done with this discussion we will remind ourselves of something that is actually obvious but something that we tend to forget many times. So this is the plot and this is the plot for tau 1 okay the shorter component 1 by tau 1 plotted against diameter that of course would decrease because tau itself increases and this is fit to a polynomial function of diameter and the best fit is obtained for a function like this a minus b by d plus c by d cube nothing in d square d to the power 4 is not required and this is more profound than what it might look like at first glance first of all a what is a what happens when d equal to infinity second and third terms vanish yeah even then this lifetime will be there that is what it means rate constant will not become 0 right rate constant will not become 0 lifetime will not become infinity finite lifetime will be there that is all it means and that would be the limit of bulk material what is the significance of b by d what is the significance of c by d cube these are actually important terms because b by d is proportional to the density of trap states what happens when traps when a size increases what happens to a very fundamental property by which nanoparticles are characterized well many things happen it is not so easy for you to read my mind so I will tell you but it is quite obvious also a size increases the surface to volume ratio decreases right that is something that we all know and one of the claims to fame of nanoparticles is that for there there are very high surface to volume ratio lot of the material is exposed and that is what makes them good catalysts and stuff like that okay so that is one thing surface to volume ratio decreases as d increases so if surface to volume ratio decreases then naturally density of trap states will also decrease so this is something that we may not realize if you think only of the surface we might think that as the surface increases number of surface traps will also increase number of surface strap does increase but density decreases right so this first one is density of trap states and what is c by d cube c by d cube is something that is proportional to oscillator strength what is the oscillator strength what does it give us information about yeah so how strong a transition is what kind of transition actually this as well as that what what kind of transition radiative transition right absorption can only be only involve photon I guess but deexcitation can be radiative or non radiative so once again let us not forget that this oscillator strength is directly proportional to the square of transition moment integral which again is proportional to Einstein's B coefficient Einstein's a coefficient is also proportional to Einstein's B coefficient Einstein's B coefficient is first stimulated emission a coefficient is first spontaneous emission so basically the second term has got to do with oscillator strength that is radiative transition so what we see is that when the size becomes larger then first of all trapping is less probable because density of trap states decreases and you have a greater radiative rate okay and this is what is important this fit a term for non radiative process as well as a term for radiative process so let us not forget that tau is equal to 1 by kr plus knr so 1 by tau equal to kr plus knr so the trait constant that we get by taking a simple reciprocal is really a sum of a radiative rate constant and a non radiative rate constant just because it is 1 tau we should not think that it is either trapping process or radiative process or something like that so the point we are trying to make here is that every rate constant that we get every component that we get has a radiative part and a non radiative part and this is going to be extremely useful in the discussion that we are going to perform in the next module or the next and what they had done is they had assigned this tens of picosecond component to relaxation from a triplet state to ground state as I said this is not accepted by everybody there are other explanations of slower recombination so this is one part of the story the other part of the story is that as you said you can have trapping and when you have trapping very often you have long long lived trap states and this is an example of how long lived it can be in fact the lifetime can be microsecond so here in this spectrum as you will understand this is the bandage emission this is a trap emission so when you record the lifetime here it is going to be ultra fast with maybe a nanosecond component as we will see in the next module if you record the photoluminescence decay here you will actually get a long component here you see the PL is not over even in almost 3 microsecond it is almost 1 microsecond time constant if it is single exponential it never is but one thing that we need to remember is that total intensity what is the contribution in a multi exponential decay what is the contribution of each component to the steady state intensity ai tau i right ai is important here the issue is the number of trap states of course is much less so that is why the total intensity from trap states is very small even though the lifetime is very large intensity is less because the amplitudes are very often in 1% 2% 5% which 10% is huge right so that is another aspect that we should remember okay so we will close this discussion and we will come back for the next module which will be a little short one and then another one which will be a long one.