 So, here we are, we are back and we are trying to understand how the amplitudes were dealt with in this study by Wies and Koworkers. As discussed they said lambda Wies to be 5 centimeter nanometer where the assumption was that there is no contribution from the electron. So A n Wies the normalized amplitude is equal to chi n, the entire thing is because of hole of electron. And then we are looking for another wavelength which is designated A n i r there is no reason to call it n i r actually then it could be a visible wavelength also for all you know. But it is called n i r because it is obtained from the we are working with these amplitudes remember and these amplitudes came from the n i r region okay. So this A n i r yeah n i r is probe even Wies is probe. So what is the wavelength where there is 50-50 contribution as you will see we will derive something and from there we will arrive at what lambda n i r is as well okay. So start with this A n at n i r is equal to A n at n i r contribution of that from hole plus contribution from electron that is very simply put and then what we are looking for is this A n n i r equal to 0.5 chi n e plus 0.5 chi n h 50-50 contribution okay. Now see you have two equations A n Wies equal to chi n e and A n n i r equal to 0.5 chi n e plus 0.5 chi n h. Now remember what is the definition of chi chi is the contribution of electron or hole right to the nth mode of decay right. So irrespective of the wavelength chi will remain same for a given value of n this is important to understand. In fact when I read the paper for the first time I did not understand this. So it required some time for me this to sink in are we clear about that it does not matter which wavelength we take as long as we are working with the same n the respective chi should be same chi n should be constant chi n e should be constant chi n h should also be another constant clear. Now see now if you simplify this we already have a workable formula for chi n e is not because this amplitude A n Wies is an experimental quantity can we get an expression for chi n h in terms of the amplitudes what will it be it is quite simple take this value of chi n e and plug it in here. So you will get A n n i r is equal to 0.5 A n Wies plus 0.5 chi n h. So what will be the expression for chi n h yeah yes but I am proving the same n that is what we are saying. So what we are saying is that a particular n value stands for a particular decay mechanism. Now that shows up in all wavelengths all probe wavelengths at least many probe wavelengths we are looking at the contribution of the same mechanism of decay in lambda Wies as well as lambda n i r. See how did we get all these time constants by a global fit across the wavelengths right. So the basic premise of the work is that there are certain fixed decay mechanisms and effects of this can be seen across the probe spectrum. So as long as you work with a particular value of n chi n e is the same irrespective of the probe wavelength chi n h is the same irrespective of the probe wavelength. That is the basic premise of this work and that is the part that may not be very easy to understand when we start are we convinced otherwise you cannot say that 4.5 picosecond is a particular decay mechanism 4.5 you get from all the wavelengths right as you saw earlier it shows up in TRPL it shows up in transient absorption visible it shows up in nir as well that is exactly the point this is a comparison between time resolved photoluminescence and transient absorption fitted independently and that is the beginning of the story fitted independently they get a very good match first of all where did 6 time constants come from 3 came from one experiment 3 came from another experiment in PL itself. Now when they do transient absorption again picosecond and nanosecond they get actually the same time constants this mismatch is not much and what is very prominently absent in transient absorption is 0.73. So then that is why they got encouraged and they looked in nir and when they did global analysis of the nir data then they got the same time constant of course they would get the same time constant because they fix the lifetimes but the reason why they are justified fitting the lifetimes is that in any case everything is coming the same the only problem is that in this time constant if you take 5 instead of 6 this 0.73 and 0.45 you get this sum over I eta y kind of thing and the time constant becomes less than 4.5 as you go from higher to lower energy probe okay. So the basic premise is that the time constants are the same irrespective of the experiment irrespective of the probe wavelength otherwise this analysis cannot be done yeah this is where we were we said we are going to put n nir equal to 0.5 nvis plus 0.5 chi nh. So what is the expression for chi nh chi nh equal to yeah no but then that has to be divided by 0.5 also a n nir minus 0.5 chi n e divided by 0.5 so you get this chi nh turns out to be 2 a nir minus a nvis alright. Now using this what one wants to work out is eta n e so eta n e turns out to be so you understand what we are doing right we have got the expression for chi n e we have got the expression for chi nh in terms of measurable quantities that is amplitudes normalized amplitudes. So just plug in the values in the numerator for this expression well you get a nvis instead of chi n e in the denominator you get a nvis plus 2 what is this 2 a n nir minus a nvis so the denominator a nvis minus a nvis cancel each other you are left with 2 a nvis a n nir sorry. So finally you get the expression eta n e is equal to 0.5 a nvis by a nr alright. Now what we want is this lambda nir how will you find that lambda nir in fact I have given you the answer already it is 1170 nanometer how will that be obtained we just look at the amplitudes basically plot the ratios of a nvis and a nnir what we are looking for is eta n e to be equal to half when will eta n e be equal to half when this ratio of amplitudes is equal to 1. So what they did is they plotted this ratio of a nvis and a nnir and they found that it becomes 1. So you understand what it will be right a nnir well a nvis is basically the same throughout and this is a nnir you see there is an inflection here basically the inflection point is the 50-50 point in fact you do not even have to do the ratio from this plot itself you can see this inflection you see the inflection right so that is where it is and we can work with that safely because it is only from cold relaxation alright. So they determined that lambda nir equal to 1170 what is the next step we have all this we have this 6 time constants we have the expression for eta n e we have the expression for eta n h and we already have said that 0.73 picosecond is ultrafast cold relaxation we know the lambda v's and lambda nir's now what remains to be done is work out the ratios of the amplitudes okay of v's and nir and then get this percent electron to percent whole ratio well you can get percent electron or percent whole here for whatever reason they have written it as percent electron to percent whole which is a little strange because you get numbers like 0 that is okay but as you will see you will also get a number that is infinity so I do not know why they wanted to take the ratio they could have just written it separately but then it is their paper not mine what can I do. So you understand what is going on here right see lambda v's lambda nir are specific wavelengths using them what we work out is a wavelength independent contribution of electron and of whole to each and every relaxation process we say that relaxation process number 1.73 for that electron contribution that I can say without doing anything percentage electron contribution is 0 percentage whole contribution is 1 that is what we have started with that is the whole relaxation and then as you go from 1, 2, 3, 4, 5, 6 you will see that percentage contribution of electron will increase percentage contribution of whole will decrease until at the end you will have no contribution from the whole at all are we clear. What is the need of working out this 50-50 point the need of working out 50-50 point is to arrive at this formula and this is the formula that takes us to chi n e which is wavelength independent and from there well that is basically what we work out so this is the result and here you might notice that instead of fi basics only one is taken this is just a sum of the two only one is taken because first of all that contribution is very small and they mean more or less the same kind of thing look at the last line and neglect this strange notation let us read the numerators first and then the denominators percentage electron contribution for the first pathway is 0 for the second pathway is 11% for the third pathway 23 4th 85 and 5th and 6th 100 and whole contribution is 100 for pathway number 1 89 for pathway number 2 77 for pathway number 3 15 you see the dramatic change here from 77 it goes to 15 when you go from 48 picosecond to about 700 picosecond 700 picosecond is of the order of nanosecond so at the moment of well at the point of transition from ultrafast dynamics to fast dynamics you see there is a significant changeover of relative contributions of electron and whole and then this one is electron all the way okay. So to explain this what they considered and you will see why they considered this they considered 3 kinds of populations of nano crystals and 3 kinds of populations means one population with one kind of distribution of electron and whole trap second with another kind third with another we will see where all that comes from and why okay another thing if I forget please remind me at the end to say what more could have been done or what more should have been done in this paper and I want you to find out whether that has been done in the 9 years that have passed between publication of this paper and today alright. So this is what you have the time constants and the contribution what is numerator what is denominator very easy to remember because if you remember the first component is only due to whole. So denominator is whole numerator is electron okay let us go one by one let us look at the first component first component we do not even have to think it is whole relaxation. So what would be the meaning of whole relaxation this what it essentially means is that there is a population I mean there is a large population where the whole is actually not on the highest level somewhere down below that means the electron has been taken out from a lower energy level while exciting and that whole floats up that is a relaxation okay. So to start with we should write that tau 1 as 1 by k r 1 plus 1 by k nr 1 yeah but I hope you will agree with me if I erase one of this translated denominator can I erase k r or k nr what can I erase k r right here only whole is involved. So whole is going from one level to another level there is no question of emission of light okay it is just relaxation. So I can just erase this so actually it is 1 by k r 1 now the convention that is used throughout is k r or k nr is written and then you write the a number in sequence of appearance of that nothing else. So this 1 2 3 4 that we write here is not necessarily correlated with this 1 2 3 4 and that is another point where we can get confused okay one needs to be careful about that in the first one there is no problem because both are 1 but later on you will see things will get jumbled up a little bit okay. So first one is accounted for whole relaxation what about the second one here what is the contribution of electron what is the contribution of whole percentage 11% electron 89% whole okay so you can safely say 11% electron means whatever is in excess electron or whole that is being trapped you can think like that you have 11 electrons you have 89 holes this is what is involved in the relaxation pathway. So what will happen the model that is being used is that 11 electrons will radiatively recombine with 11 holes and the remaining 18-11 78 holes will get trapped am I clear yeah 11% electron 89% whole means 100 excitons handling here well 100 excitons 100 carriers 11 electrons are there 89 holes are there that are relaxing through this pathway so this 11 electrons will radiatively recombine with 11 holes and the remaining 78 78 right yeah 78 holes are going to get trapped thus that is the only way they can get relaxed by themselves okay this is definitely questionable because who has said that out of the 11 electrons 4 do not get trapped right so this is one problem with trying to do so much of detailed analysis but it is still it is a commendable approach that one can learn from okay so this is one problem in fact I wanted you to tell me I have told you the answer so the problem is this here after doing so much of calculation finally you are working within the ambit of some approximation and the approximation is that if you have a smaller number of electron or smaller number of holes all of them relax by radiative recombination that actually may not be true justification for using that is that you have 78 holes that are getting trapped as against 3 electrons getting trapped to neglect the 3 electrons okay so this is definitely an approximation that is being used so do not get the impression that it is an absolute perfect approach lots of approximations are actually involved okay so I can draw it like this let us see what I mean here first of all this hole relaxation has already taken place in the first 730 second now from here 2 pathways are there one is radiative you understand what this means right these 2 lines at the bottom these are energy levels in the valence band this one is the lowest energy level of the conduction band this here is a hole trap this here is an electron trap and we are working with shallow traps that is another approximation why do I say we are working with shallow traps because there is no redshifted emission right if there are deep traps would not you expect PL that is redshifted it will be significant change right but if there are shallow traps electron or hole then whatever whenever those trapped electron holes recombine the energy involved will not be very different from the bandage recombination energy yeah that is why these are all shallow traps that is that comes from the steady state spectrum so this is one pathway the radiative pathway now recombination has taken place that is why you do not see electron or hole electron is designated as a field circle hole is designated as an empty circle okay so this one is given the name KR2 and this is where the deviation from this notation up here begins well not really because this is true that is also 2 we will see where the deviation comes you understand KR2 is the rate constant associated with the electron hole recombination in this kind of a situation so this is bandage electron hole recombination okay and that is associated with what kind of time constant something like 4.5 because second time constant right the other thing that is there is KNR2 where the hole gets trapped alright so that is one thing and this is labeled fast hole trapping and the need for using the adjective fast will be apparent in a few minutes have you understood this diagram yeah this is the diagram that is easy to understand later on things get a little messed up little bit of hand waving is there okay now let us go to the next one 48 picosecond time constant electron contribution 23 hole contribution 77 so if I go by the previous treatment what does this mean it means 23 electrons radiatively recombine with 23 holes and 77-23 54 holes relax by themselves right so what kind of diagram will I get similar what I drew earlier what will change instead of writing KR1 KR2 I have to write KR3 yes so this same thing what we write is that tau 3 is equal to 1 by KR2 plus KNR3 and this is where the deviation begins actually so see the here it is KR2 here also it is KR2 why because in both the cases the radiative process is electron hole recombination at bandage there is no reason why we should use a different rate constant there however the non radiative rate constant is definitely different right trapping is definitely different because time constants are different by an order of magnitude in the first case we are working with a 4.5 picosecond time constant now we are working with a 48 picosecond time constant 10 times more and we are saying that the radiative rate constant is the same in both the cases so obviously non radiative rate constant is going to be different will it be larger or will it be smaller for population 2 will so we have the names you can use the names KNR3 is it larger than or smaller than KNR2 yeah smaller smaller than KNR2 smaller rate constant is associated with a longer time constant okay so smaller rate constant means what slower yeah basic chemical kinetics if rate constant is small the process is slow so here this is called this hole trapping is called slow hole trapping why would we have a fast hole trapping and a slow hole trapping what could the reasons be this I think you can tell first of all it could be different traps different kinds of traps where the hole is getting trapped or it could be different density of traps you can have in population 1 perhaps there is a large number of hole traps in population 2 there may be a smaller number of hole traps right that is why it is called population 1 and population 2 yeah KNR2 is smaller KNR is not slower or faster time constant is slower or faster you can say well everything is smaller and larger what I am saying is tau 3 is large larger than tau 2 and if you look at the expression in the denominator tau 2 has KR2 tau 3 also has KR2 so now my question was KNR3 and KNR2 which one is larger since tau 3 is larger KNR2 has to be smaller it is in the denominator that is what I am saying smaller smaller rate constant is not faster smaller rate constant is associated with a slower reaction rate equal to K multiplied by concentration so if K is small rate will be small so slower process will be slower yeah Is it really okay to take that large KR3 instead of KR4 because the time scale here is 40 yes it is almost in transport then tau yeah so even if the KNR3 is smaller than KNR2 or something it can be I mean there is no restriction on what the values of KNR2 and 3 are right so that is not the model that is being used the model is that whenever you have bandage recombination bandage recombination is independent of number to be honest bandage recombination is one electron one whole forget everything else that kind of a situation so bandage recombination have to have the same lifetime when trapping takes place it is not so easy for the electrons and holes to recombine radiatively that is why it slows down that is why when say copper is introduced or manganese is introduced lifetime goes from picosecond to hundreds of nanoseconds because they are like physically separated right in different entities but not in this case it is in the same particle in the same particle when you have an electron and a hole it does not matter what it what the situation is in another particle do not forget lesser number of holes means what what we are saying is that we have done one photon excitation so in a single particle there is only one electron hole pair all we are saying is that the distribution is different for a particular relaxation pathway total number in any case is same over the ensemble but how how many holes are combining with how many electrons in a particular pathway that is being discussed and the case of a trap is different because in one nanoparticle there are many trap states so you can talk about in a single nanoparticle you can talk about a greater density or lesser density of traps so even for one exciton number of traps is going to alter the rate but bandage emission recombination of an electron and a hole in valence band and your conduction band that is constant constant means when that has its own distribution that is why you will have a lifetime yeah that is independent of anything else that is why the radiative rate constants have been taken to the with the same are we clear now this is not trap if it was a trap emission then what you are saying would be definitely correct alright so this is population 2 and in fact population 3 is very much like population 2 when I show it you will see but let us go to this nanosecond time constant now so we are going to picosecond to nanosecond now there will be a more fundamental change what happens here 85 electrons 15 holes what does that mean electron trap right so what they did and it is not very clear why they did not want to do it from another population why they did not use a fourth population what they did here is that well there is another reason why they did it I will tell you what they did is this but just now just go with me and see what they said then we will come back to that they said that this tau 4 originates not here but here tau 4 originates in a in nanoparticles where the hole is already trapped and the reason why they say it is that now here the radiative constant is different because second to nanosecond right now we come to a situation where you are referring to just before this here for a nanosecond decay radiative constant cannot be associated with a bandage electron hole pair it must involve a situation where one of the carriers is trapped so here you see a different radiative rate constant is used here okay so here 3 is much smaller as the lifetime is time constant associated is much larger so 1 by kr 3 plus 1 by knr 4 so what will happen one thing that can happen is that first of all these 2 can recombine and this recombination is different from the recombination in the earlier stage because it involves a trapped hole that is why the time constant is much bigger it constant is much smaller the other thing that can happen is this electron can get trapped so you end up with nanoparticles where electron is trapped hole is also trapped that is bound to happen so now you have created a situation where both are trapped so if there is any ultra long lifetime in PL that will come from here okay because if these traps are completely isolated they would not even recombine they have to sort of come back to the core and then only recombination can take place okay. Similarly the next one was associated with population 2 exactly same kind of situation and then population 3 so what they are saying is that there are different kinds so basic processes are only of 3 kinds okay but there are different kinds of population with different density of traps that is what shows up in the different time constants in nanosecond time scale because second is justified by the fundamental trapping and bandage electron hole recombination processes okay so this is a this is what they did now can you tell me what you honestly think about this analysis do you think it makes sense do you think it is all rubbish or do you think it is somewhere in between so what I think is that this model is it requires further verification it is a good model to start with but it may not be complete how can one try to complete this study what I would think is this is a situation where one can take resort to simulation when I say simulation I do not mean protein folding and all that nothing to do with that what I am saying is that in this kind of a numerical you can do it with things like math cad and now I am sure there are better software for this what one can do is you basically you want to be able to fit the data just like fitting is already done what you need to do is you need to guess values of tau 1, tau 2, tau 3, tau 4, tau 5, tau 6 right and then vary them until you get a good fit look at them and see whether they make sense not all good fit will give you time constants that make any sense or rate constants so this is something that should be done in cases like this actually the story is incomplete did the story get completed in the last 8 years I leave that to you to read up and see and easy way of doing it is to see which papers have cited this paper go through them and see whether this has been complete also these people actually published a corrigan done later on I like you to read that up as well read this paper it is a good example of how one can handle very complicated ultrafast time resolved absorption and emission data that way it is instructive and it is not necessarily that the understanding generated from the data analysis is restricted to nanoparticles it can actually be extrapolated to other systems molecular systems with comparable or more or less complexity okay so that is what we wanted to say here so far we have talked about exciton dynamics in semiconductor nanoparticles that is only the beginning of the story so in the next two or three modules we are going to talk about dynamics of multi excitons and maybe just touch upon a very recent paper that has been published on other kinds of carriers that are there polaroons trions and so on and so forth perhaps we will not go into too much of detail of that but at least we will mention what they are and what has been seen but multi exciton dynamics is something that you definitely want to talk about and that is essentially a single paper by Clemoff