 We are discussing radiative and non-radiative relaxation pathways of semiconductor nano crystals. As we said there is a vast body of literature on this and there are differing opinions as well. So the limited scope of this course will not be able to discuss everything that I leave to the interest of few people but what we are discussing now is one single paper published in ACS nano in 2011 by Wies and her group where they had taken a very rigorous approach towards understanding these rate constants or amplitudes also obtained in time resource studies of cadmium selenide nano crystals in particular. Now what I would suggest is that one should try and see go through the papers that I have cited her paper because if you do that you will see that there have been people who have supported this approach there have been people who have opposed this approach we are talking about science that is evolving now not science that is established 100 years ago. So to be honest the jury is out but the reason why we still discuss this paper is that this is a I would say it is a little bold approach but a thorough approach never the less to try to understand the temporal parameters obtained in ultrafast studies. So we have discussed this already in the previous module what they did is they worked with 5 nanometer cadmium selenide particles they did work with a little bigger particles as well just to verify whether whatever they say is applicable for another set or not and it turned out that they were applicable this as we discussed is absorption spectrum and we have already talked about what these bands mean this is the most prominent band edge absorption that we get and the photo luminescence that we get is more or less mirror image of the band edge absorption quantum yield is only 5% and then the unique approach of this paper is that when we talked about earlier about these data fitting and all we have said that if you use a sufficient number of exponentials you can fit even an elephant. So here they have used a large number of exponential functions and as we will see they have tried to make use of all the time constants that they get make sense of all the time constant that they get. So first of all they performed an ultrafast experiment you can see the full scale here is about nanosecond and from there they obtained 3 time constants 0.73 picosecond, 4.5 picosecond and 48 picosecond and from TCSPC experiments they obtained 1.4 nanosecond, 13 nanosecond, 45 nanosecond. Point to note is that there have there has been a lot of work on semiconductor nanoprystals where people have done only TCSPC and have happily lived with these 3 time constants. But as is very obvious in this work it is not enough to do just TCSPC. One should look at ultrafast time scales as well because as you see the photo luminescence is decayed to about 10% of what it was at the beginning in a matter of a nanosecond. In 500 picosecond it goes down to maybe 20%. So there is a very very prominent ultrafast component that one cannot neglect. Of course when you look at photo luminescence perhaps it is a nanosecond components that dominate at least in the steady state spectrum. Because PL that gets over in tens and hundreds of picosecond do not contribute sufficiently. If you remember contribution to steady state intensity is ai tau i where ai is the normalized amplitude. So it is not only ai tau i also contributes. So if you have 2 tau i's one is 10 picosecond and one is 100 nanosecond that itself is a factor of 100,000 by 10, 10,000. And let us say we have a picosecond component of 90% of the decay is in 10 picosecond and well time constant is 10 picosecond. Amplitude is 0.9. What is amplitude multiplied by tau here? 0.9 into 10 picosecond. So to bring everything to picosecond. So that is 0.9 multiplied by 10, 9. And 10% 0.1 multiplied by 100 nanosecond. How much is that? 10 nanosecond that is 10,000 picosecond. So 10,000 divided by 9 is how much? Let us say 10,000 divided by 10 is 1000. So contribution of the long component will actually be 1000 times that of the ultra short component in the steady state spectrum even though its amplitude is only 10%. You can have other examples where you can tweak these numbers and see that this thing actually holds but we take this a little bit. So 3 time constants from ultrafast experiment, 3 time constants from the T6 PC experiment and then they had done a visible transient absorption as well in picosecond time scale as well as nanosecond time scale. And from there they got these time constants the only difference was 1.4 and 0.7. This was the only component that was different otherwise more or less everything was same and that is remarkable. And that is why they got encouraged to probe this further. And as you see this 0.7 picosecond component does not even show up in transient absorption using visible probe. It does have indications of showing up when the probe is NIR. I think we have discussed all this in the previous module. In NIR we see that as one goes from probe wavelength of 900nm to 1400nm decays get faster and faster and it is very clear that the long time constant decays are all tail matched. Long time decays are all tail matched. In short time the difference is there as you go to the radar side you get the decays becoming faster. And when fitted just as such this is the kind of data that we had shown you. This 4.7 becomes 2.5. But then that does not explain anything. In fact what they also try to do is they try to use things like state exponential model. But what they found is that even by using state exponential model the number of terms does not decrease. So state exponential model in this case does not teach us anything. That is why they embarked upon the analysis that we are discussing now. So another thing that was known from earlier work of Wies and coworkers as well as Klimowen coworkers is that if you look at this NIR probe transient absorption this is ascribed to bandage to higher energy state relaxation. Well higher energy to bandage relaxation or rather I can say that this absorption is from bandage to higher energy. How does one go from bandage to higher energy? One thing is that electrons can go higher up. I mean electrons might have gone higher up or might go higher up when you excite by the probe or the hole can keep going down or both can happen. So what is it actually? To understand that one thing that is known is that this higher energy side is dominated by electrons lower energy side is dominated by holes. That was established in the papers that I have shown. So this seems to be mostly a hole contribution because it goes up there. So what they did was they had done a global analysis but not really global analysis because they have fixed the lifetimes. They fixed the lifetimes because they already see a very good match between PL and transient absorption data for 5 of the time constants. And this 0.7 because second time constant was also well resolved. That is why they fixed it but this is one step that can be questioned. So did this global analysis and this is what they got. All this time constants 1, 2, 3, 4, 5, 6 and then they worked with the amplitudes. They plotted the amplitudes and you get this kind of a plot. So look at which one is C1 out of this. You can see the color I hope. C1 means the amplitude associated with tau 1. This is also called A1 in later discussion. So which one is C1 out of this? This one right. And see what happens to C1? It is not there is 0 and then it goes up in the NIR range and goes to saturation. Why does that happen? Because this 0.73 is associated with ultrafast hole relaxation. Remember as you go higher energy to lower energy side in the NIR domain, NIR probe domain you get signal that is more and more predominated by holes. What all things can happen? So as you go from higher energy to lower energy side it is dominated by holes right. What all things can happen? First of all it is important to have this very clear because we will need it in the subsequent discussion. So you have an electron and you have a hole. You have created an exciton. Now first of all they can recombine right and go back to the ground state. And that recombination is the only process that contributes to emission of light. You can perhaps have non-radiative recombinations also. But when an electron hole recombines that is the only time when you get light out of the system. That is is that clear? Other things can also happen. You can have the electron going up the energy ladder. You can have the hole going down which also the hole going down means essentially increase in energy right. That can happen independent of each other. The electron can go up higher in energy irrespective of the hole right or let us put this I think we will all understand. Electron can be say trapped or hole can be trapped independent of the other carrier yeah. So these are the different things that can happen. In trapping process electron and hole one of these are one of these is usually affected. But in recombination both are involved and that is the process that is that gives you light. Remember this will come back to this okay. So this 0.3 picosecond component is ascribed to ultrafast hole relaxation. And we will see by the time we have done what that means. You understand what hole relaxation means. It essentially means that initially the hole is in one of these lower levels it floats up to the highest level in conduction in valence band. That is hole relaxation. Electron relaxation means electron is in a higher energy level it sinks to the lowest band lowest energy level in the conduction band. That is electron relaxation this is hole relaxation. So this 0.3 picosecond is associated with ultrafast hole relaxation. Why because in NIR in the lower energy side this 0.7 picosecond is observed to a greater extent. And the lower energy side as shown is dominated by hole relaxation. Then another important thing to remember and we said it earlier is that each component is really 1 by kr plus knr. Every tau every time constant is equal to 1 by kr plus knr. There can be a situation where for a particular component kr is much much larger than knr. So you can neglect knr. We can have cases and we will have a case at least where knr is so much larger that you can neglect kr. But those are special cases. The general case is that every component every tau is 1 by kr plus knr. This holds for everything not just nanoparticles. And this is something that somehow we have already talked about this knr can be associated with electron trapping, hole trapping, auxiliary combination right. And kr can be associated with radiative electron hole recombination. So but then how to make any sense of this. Now we start discussion of this rather rigorous and sometimes and possibly questionable treatment of the amplitudes that this group had done. And I want to tell you by the time we had done what you think of this analysis. And when I say I want you to tell me what you think of this analysis I do not mean I want you to tell me what I think of this analysis. You should feel free to say that this analysis is rubbish if you feel so. Right. So this is what they did. First of all they define well there is not much to define here it is quite straight forward. The fraction of the total normalized amplitude of a given component that is accounted for by electron decay they defined to be eta ne equal to chi ne by chi ne plus chi nh where the chi's are the fraction of total population of electron and hole respectively in bandages that decay via cn. It is important to understand this statement. You understand what they are trying to say. What does n denote of cn what is n? What is c? Normalize coefficient amplitude okay. And it has also been used as a okay that might be a little misleading. What is n? What does n tell you? Which component you are working with? So the 0.73 picosecond component is n equal to 1 component. 4.5 picosecond component for that n equal to 2. 48 picosecond n equal to 3 and so on and so forth okay. So this subscript of the tau's that are given at the top the subscripts are the n. Now understand what they are saying. chi ne and chi nh are the fractions of total population of electrons and holes. E for electron H for hole in the bandages. So in this study we complete neglect things that are not in the bandage for now at least in the initial definition that decay via n right. You have say avocado number of electron avocado number of hole right. Different electron hole pairs will do different things right. We can only work with statistics. We can only work with fractions that decay by this component or that that is all they are saying. And eta ne is given by chi ne divided by chi ne plus chi ny that is very straightforward. And of course then the fraction attributed to hole decay would be 1- eta ne. The 2 eta's have to add up to 1 okay. What is eta? It is the fraction of the total normalized amplitude that is accounted for by electron or hole depending on what subscript you are using. Are we clear? Then they started trying to simplify this problem. What happens for so one more parameter is there which we have not really talked about here and that is wavelength. They are doing transient absorption all right. So different wavelengths are there. So is there can you think of a wavelength where there is no contribution from either electron or hole. If you can do that then the problem becomes a little simpler. Can you think of a wavelength from what we have seen already? Probe wavelength where the hole does not contribute. I will make things easy yeah. Higher energy side of course. Actually let us remind ourselves where do we see the hole dynamics very prominently in NIR right. So if we just go to that visible transient absorption there we will reach a situation where hole dynamics is not even there. So what they have done is they have taken this position of absorption maximum right or transient absorption minimum 572 nanometer. That position they said there is no contribution from holes okay. So a and v is equal to chi ne that is the starting point. They have defined this wavelength to be lambda v's okay. Of course there is many other lambda v's why 572 even 500 is lambda v's. But here they are defined is you can treat it as a proper name not as a generic name. They are defining 570 nanometer to be lambda v's okay. So a and v's is equal to chi ne. So what they are saying is chi h chi n h equal to 0. For which n? For which n? For all n? All n 1 2 3 4 5 6. For all the n's for probe wavelength 572 nanometer which is given the name lambda v's there is no contribution of the hole. That is what they are saying. And what is the basis of saying that the hole dynamics is seen in this NIR okay. Then what they do is in fact you can now take this and any other wavelength where you know what is the contribution of electron what is the contribution of hole. So just to keep the calculation simple they have looked for a wavelength where there is 50-50 contribution from electron and hole. How does one find it? That is what we will learn now okay. But perhaps it is better if you take a break and discuss this in the next module from here onwards.