 You can follow along with this presentation using printed slides from the Nanohub. Visit www.nanohub.org and download the PDF file containing the slides for this presentation. Print them out and turn each page when you hear the following sound. Enjoy the show. Recombination generation, this is lecture 13. Now, actually we have in last 12 lectures, we have done a very important thing. The last lecture was very important because until last lecture we talked about equilibrium system. That means take a chunk of silicon, put it on your desk, in a dark room let us say nothing happening and the system is in equilibrium with the room, same temperature and everything and we knew how to calculate the number of electrons and holes in that material, where the electrons sit, how they are distributed in terms of farm media statistics for electrons. Now that is all good of course, but a chunk of silicon sitting on your desk is a good paper width, but it is not very useful and what we will start from today is part of being this equilibrium of this semiconductor so that electron and hole will flow and we will get something useful out of it. So if you remember very quickly going back to the beginning that the problem we are interested in is a piece of semiconductor perhaps uniformly doped, but most likely non-uniformly doped, contacted between two contacts and we want to find out how much current flows and one thing we immediately realize from the very beginning that this will depend on what material it is, how the atoms are arranged and so that is the first thing we discussed in professor chapter one and two and then we immediately said that the number of electrons and the current flow is proportional to the number of electrons and the velocity and the number of electrons, how do we calculate that? Well that is what we did in the last 12 lectures sort of because what we did was use quantum mechanics and this is very important equilibrium statistical mechanics that is we derived farm media statistics without assuming any electric field or potential and in that case we were able to calculate how many electrons do I have for variable doping and for variable materials for different materials we are able to calculate it. Now this should be clear to you as you do the homework it will become more clear. The one point I want to make very clearly that although we have discussed quantum mechanics and statistical mechanics these are vast subjects there are courses for it many courses in Purdue for this for example and every other university but what we will take out from that at least for this course is the only thing is effective masses which essentially takes the curvature of the ek diagram for each band and we just had two effective masses because only two or two bands sort of on the top were involved in conduction process so two effective masses that's my quantum mechanics sort of and a bandgap of course bandgap came from quantum mechanics and from statistical mechanics well just the Fermi Dirac statistics and the temperature that's all I take and if I know that I can calculate electron concentration for arbitrary material very powerful concept actually. Now of course next we'll have to calculate the velocity when we apply an electric field or a density gradient if we can somehow calculate the velocity finally I have the current. Now it turns out calculating current you need a little bit more because when you want to calculate a current the semiconductor has to be contacted by two contacts it will part up the electron and hold density in a particular way and the velocity there will be a corresponding change in the velocity. Now that is a little bit more complicated that we'll do in a few class down but today we'll sort of think about a device that has one contact sort of and that is a problem in which you have a chunk of silicon and you put a flashlight on it right and turn off the flashlight after a little bit and look at how the electron and hold that were generated in response of the light how they decay as a function of time because that tells me that when I have a perturbation remember on the left side by applying a bias I'm part of being the equilibrium and the question is it tries to restore equilibrium it wants to go back where it was and how it does so these two problems have many analogies so we'll start from a sort of a quote unquote single contact problem in the sense that my flashlight and the photon stream that is coming in is like a single contact and then I will go to the double and triple and other contacts a little bit later. Now when you have a semiconductor and you shine light on it or you perturb it some way then what happens that the electron and hold numbers changes right in the equilibrium let's say you have 10 to the power 10 per centimeter cube of number at a given temperature no light and nothing now in this case all in a sudden if you perturb it the electron and hold number is going to change now if you shut the light off then the electrons and holds are gradually going to find each other and then they are going to recombine each other because they don't want to be at the higher carrier concentration now one way they could recombine is an electron in the conduction band you see I have shown here and conduction and a valence band an electron red electron in the conduction band finds a blank space the whole white hole in the valence band and then recombine and when they recombine they give off a photon now this is a direct recombination that occurs only in a certain subset of material I will explain in a second but this is called a direct material a direct recombination that happens in direct band gap materials generally now one thing you have to notice here that in real space how it look this is in case space energy case space in the real space what you will see and red electron is moving around and a white a like a hole is moving around they find each other and then they recombine a photon comes out the excess energy the electron had that comes out as a photon but this process you see is very interesting because these two electrons and I will show you in a second the electron and hole must be massed in wavelength so electron and hole as soon as they see each other they don't recombine in fact they see each other every 10 to the power minus 13 seconds about 100 femtoseconds or so they see each other out of a million encounter they are bouncing off each other buffeting against each other all the time but one in a million perhaps on that order that is when the wavelength of the red electron is exactly similar very close to the wavelength of the white hole and only then they can recombine you see that makes it very less probable it is like listening to a radio station you see unless you have the right tuner you don't really get the receive the signal so similarly the wavelength has to match same is the case here without this and that is what makes life difficult here and that's why this recombination direct recombination has a certain rate we will calculate that a little bit later now many materials gallium arsenide indium phosphide indium antimonyde these do have direct band gap and a significant amount of direct recombination and lasers and leds these are all work and it can only work because of this direct recombination can occur now there is something else which doesn't require as is not as restrictive in terms of matching the wave wavelength of the electrons and holes and that involves something called an excitonic recombination so this is another way electron and hole can disappear with each other now these unlike gallium arsenide indium phosphide i just talked about these generally occur in mostly one-dimensional solid remember we talked about one-dimensional semiconductor two-dimensional semiconductor and three five bravais lattices and 2d right do you remember those so in mostly one-dimensional one like a organic transistor or carbon nanotube in these cases what happens the electrons and holes they have strong coulomb attraction so when an electron moves around there is a bunch of places there is a around it there is a space in which it has this coulomb attraction and as soon as a hole gets into this even though it doesn't have the same wavelength they begin to circle each other and eventually they recombine so this is a property of a material that has strong coulomb interaction among the electron and hole mostly in 1d so what happens that once they begin to circle each other they lose a little bit of energy even though initial wavelength might not have been perfectly maxed and then they give out a photon with wavelength which is a little bit less why because they lost a little bit energy as they began to circle each other and in real space what would it mean that after the electron and hole have been captured around each other it's like a binary star you know binary that they begin to circle each other and eventually when they have the right wavelength in right place they recombine so this process is one-dimensional semiconductor very effective and is an important problem these days solar cell is an important topic for research and in solar cells is this is a very important recombination mechanism what other way can a semiconductor can electron and hole disappear the third process will be called an indirect recombination or trap assisted recombination now these occurs in indirect band gap semiconductor what are indirect band gap semiconductors silicon germanium this type of thing and this is trap assisted and you can see that in the energy band diagram I have drawn a blue line and the blue line is supposed to indicate the level of traps now what is the distinction between a donor and a acceptor in a trap well no distinction really but I will show later the ones that are close to the mid gap only those can help in this process if you are close to the conduction band like a donor or if you are close to the valence band like an acceptor then this process it they will give away electrons you know donors and acceptors but they cannot help you in terms of recombination so I will call trap a material that gives a band gap level in the middle of the gap so for example copper or gold in silicon which has a mid gap state and that's what would be a called trap that can help in this recombination process you know that's why you will see in modern microprocessors all the interconnects are made of copper right they copper interconnect they have multiple layers of interconnections now if you look at it them any picture go to the website and look at them in picture then you will see that they are encapsulated by a very thin pinhole free covering each of the copper interconnect the reason is even a tiny fraction of the copper if it could escape and go down in the transistor underneath your whole IC will die right there so not even a tiny fraction of coppers are allowed to skip down because this will immediately kill your transistor because of this recombination process right gold and copper in silicon but other materials will have different and the idea is this that the red electron comes in gets around the orbit of the of the trap right that's a blue one in real space you can see start circling each other in this case again wavelength need not be matched and it recombines with the holes because holes does the same thing and in the process it releases energy now compared to the direct recombination process do you think that this rate will be more or less because in one case you see it now will require the help of a trap not only have they do they have to find each other they will also have to find a third-party mediator which can help them to recombine so therefore this rate is actually very small on the order of maybe in modern transistors 10 to the power minus 3 seconds to 10 to the power minus 6 seconds and this is why many of the DRAM memory that you have in computer can actually work and can only be made through indirect band gap material not through gallium arsenide you cannot make a memory out of gallium arsenide per se we'll explain that later on I'm just giving you some references and here the the the trap will give away its energy by vibrating the excess energy it has by vibrating quite a bit in the beginning and gradually giving away its excess energy to the other atoms nearby and that's called a phonon now as I said this is a very important process we'll explain next two classes in fact just thinking about this this particular process very important but the theory is a little bit complicated so we'll see now there is something called third thing called n or one of the recombination is called oj recombination now oj recombination is slightly different remember the problem my problem is my electron and hole has to have the same wavelength before they can recombine or they need a trap but if the density of electrons is very high then there is an extra mechanism in which the two electrons from the conduction band shown here as number two the red and the blue which is difficult to see one one is blue and then what they do is they bump against each other and one electron goes down in the valence band right and finds empty hole recombines there and the other electron goes up so in this process you can see even though the two initial electrons do not have exactly maxed wave vector to the hole but they can scatter against each other eventually forcing one to arbitrary high value and the other one into the hole this is oj recombination in fact anytime you have a laser semiconductor laser indium phosphide for example the telecommunication laser that you have this is the most important recombination process because in that doping is so high that electron concentration is extremely high so they bump against each other all the time and that forces the recombination so this oj recombination and what will happen that the extra electron which went up will gradually lose this energy down to the ground level by sending out heat waves or phonons as a result again the process in real space will be the green one and the red one will scatter against each other and push one the red one let's say into an empty space the hole and the electron other electron goes up way up in the conduction band so this is the oj process very important for many many materials spatially lasers now there is a final one it's called an impact ionization which is the reverse of oj you can so call it an inverse oj process in some way because instead of a recombination this is like a generation mechanism consider the electron one red on the top and consider a band energy band diagram that is under electric field do you remember that generally we draw it flat the ec and ev we generally draw it flat but as soon as we put a battery on it did i show you before that it changes the potential and therefore the electron and hole conduction bands and valence bands they will change so here what i'm showing here is an extreme electric field so you see although the number one starting out has no energy no kinetic energy it has a lot of potential energy now if it starts moving then it will gain a lot of kinetic energy right and lose potential energy in the process but eventually what will happen that if that red electron can scatter of another electron this time number two from the valence band and once they scatter it will kick an electron out from the valence band into the conduction band and in the process one will lose its energy become three right because it has to kick somebody out it had to give that energy and so one becomes three and the electron that was in the valence band that went to number four went up so now i have more electrons in the conduction band and a little bit extra hole in the valence band where did electron come in the valence band is it not full of holes not really right valence band is actually full of electrons with very tiny amount of holes around right remember it's below the Fermi level below the Fermi level is full of electrons and only a one minus f there's a hole only a tiny bit of hole so in this process you increase the number of electrons and you increase the number of holes right so this is an impact ionization process many time you can call it an inverse OJ process because two electrons collided and they eventually went to different places so therefore this process is again very important as we will see now let me explain to you this statement strange statement i made that the electrons and holes have to have the same wavelength before they have to recombine i mean that's a strange statement why before we do that let me remind you that there are these two types of semiconductors right indirect and direct band gap semiconductor you know that that any time the the top of the valence band and the bottom of the conduction band lowest point are aligned at a particular k need not be k equals zero at any particular k then that's a direct band gap material like gallium arsenide on the right hand panel and any time it is not they are not aligned and they have a different value of k then they will be on indirect band gap material like silicon like germanium and you know that right this this point you have seen many times now i'm going to focus on gallium arsenide direct band gap material consider my flash lamp is exciting an electron from the valence band to the conduction band in gallium arsenide and therefore i have shown a ek diagram that is essentially centered around the same point you can see let's say k equals zero you remember the end of the brilliance on is two pi over a for silicon generally it's pi over a but for silicon for the cubic structure this is two pi over a and let's say a photon is coming in with a wave vector k photon this thing is coming in it kicks the electron up in the conduction band the red electron up in the conduction band leaves behind the hole now one thing it has to do is to conserve energy right energy conservation that means whatever was the energy of the electron in the valence band ev plus photon whatever energy it gave it that must be equal to the electron in the energy of the electron in the conduction band e sub c okay and the momentum has also to match right so hkv is the momentum of the hole hk photon well photon wave vector and photon momentum and you can see the other one is the electron so they must balance momentum balance and energy balance now let's calculate the photon wave vector not very complicated right because wave vector is given by two pi divided by lambda why in micron because i chose so my calculation will be easier so lambda wavelength in micron let's say now do i know how to calculate the wavelength of a photon you have done it in chapter 2 that generally it is 1.21 divided by energy of the photon right and that's in ev if you express things in ev that's in that's in ev now what is the energy of the photon required for an electron to go from the conduction to the valence band equal to the band gap right on the average band gap and band gap is how much one or two ev let's assume about a 1.21 ev just for the sake of argument if i assume that band gap is 1.21 ev photon energy is 1.21 ev and so my k photon is 2 pi per micron 6 6 per micron now think about it compare that to the end of the Brillouin zone end of the Brillouin zone is 2 pi over a and how much is a do you remember from the Bravais lattices how far apart are the atoms about 4 5 angstrom right so you put 5 angstrom n and do you see how big it is it is about 10 000 divided by 5 about 2000 and that's 6 so the end of the Brillouin zone is 10 000 right per micron let's say and that's really like on the order of one 1 in 10 000 so if you compare to the end of the Brillouin zone this will look like a vertical transition do you see this because the energy of the photon is so large in this case so therefore their wave vector this photon wave vector is actually tiny compared to the Brillouin zone so therefore it looks vertical it is not vertical but anytime it looks vertical that means it is negligible that says that the momentum of the valence electron h k v must be approximately equal to the momentum of the valence electron us conduction band electron h k c and therefore since k v is equal to k c their wavelength must match so anytime a photon is involved a photon is involved then the wave numbers or wave lengths of the electrons and holes must be matched and that's what makes it so difficult for recombination you see they bump into each other but most of the time the wavelength don't match and as a result they don't recombine they just bump and there is a scattering and then they go away without recombining direct band gap recombination you see now this is a very important point you should remember this again once you understand you'll not forget but make sure that you do now what happens in an indirect band gap material do I have no hope well I really have no hope of direct recombination because photon is not bringing in much wave vector into the game it gives you a lot of energy but not much wave vector so therefore there is no way that it is going to take an electron from the valence band in indirect band gap material and pump it into the conduction band because the wave vectors are so different right do you remember in silicon that the bottom of the conduction band is almost to the age of the brillo in zone and the bottom of the valence band well in the gamma point in the zone center so therefore so different no hope that a photon will typically do it again but in this case a phonon can do it phonon is a lattice vibration so it can do it again I'll write the energy conservation and the momentum conservation again I try to calculate the wave vector for the phonon I will not explain this very much but you can see the wavelength of phonon it is the wavelength of a sound wave right because phonon essentially sound wave moves by moving the atoms around right and so so essentially you can calculate the wavelength lambda by the velocity of the sound and the phonon energy phonon energy is small but the velocity of sound is also very small compared to velocity of light how much is it you know it's about a thousand meter per second right is that right velocity of sound approximately and what is the velocity of light 10 to the power eight or so right so huge difference five orders of magnitude difference so you can see why phonon wavelength phonon essentially this atoms move up and down bob up and down so you can see the wavelength that you can fit can be huge and as a result that is actually almost equal to the Brillouin zone edge and so when a phonon is involved it doesn't have much energy but it can give you a lot of wave vector change this wave vector and in the process it can help recombine recombination process so it's indirect band gap recombination by the way i'm writing this word bz uh in the bottom bz stands for Brillouin zone so it takes a little bit to write so remember that this word means end of the Brillouin zone 2 pi over a now how does physically how does traps help well if you think about a trap trap has a dimension of a trap is a impurity atom right foreign atom you have a bunch of green deep green silicon and you have one light green impurity atom sitting there and what it does that the wave vector of the trap when the trap is moving up and down is on the order of 2 pi over a and what it helps is that the trap level is right there shown here in the energy band diagram as a small red line and what the trap does it provides a lot of wave vector so that the electron from that bottom of the conduction band can move to the trap right and once it is there it is perfectly aligned with the valence band and so now when the right hole comes in it just drops in and then in the process it recombines so that is how the indirect band gap recombination occurs you can immediately see physical why indirect band gap recombination is so difficult right it needs to be first captured by a trap if you don't have a trap it's not going to recombine at all once it has a trap then it has to wait for a hole to come along and so therefore this is on the order of a millisecond to a microsecond for this recombination to occur how would you increase it if you wanted to shorten it just increase the number of traps right and we'll see a transistor or a diode that diode will show how it this increase actually help the performance of a diode later on now let me give you some so that's all these are various processes by which extra electron can recombine with extra hole this is not rocket science very simple now let me explain a concept which is a very important concept in preparation of derivation of the recombination generation formula the overall formula that we'll do in the next class I want to describe a few concepts and this you need to understand very clearly steady state and transient response consider that we have a room the yellow room and let's say this room this looks like yellow here and whatever the temperature of that room is and my computer and that is the supposed to be a thermometer on the right hand side it has a certain temperature and my computer and rain is sitting there in equilibrium with the room now if I change the temperature of the room all on a sudden so let's say the temperature of the room is changing then the electron device has to go to now a new equilibrium point do you remember in the last class we said as you increase the temperature the number of electrons changes from the freeze out to intrinsic to the extrinsic region do you remember that means that every temperature the electron concentration is actually different so all on a sudden if you increase the temperature then the number of electrons might change right from one value to another so what is going to happen that let's say in the equilibrium case this was your electron and hole concentration as you increase the temperature all on a sudden it will go to a new value electron and hole will go to a new value so this is a transient response but eventually if you keep the temperature the same eventually it will reach 20 steady state and how is it different from transient well this is the here you allowed it to reach to a equilibrium value but if you turn the temperature on and off the thermostat on and off all the time then of course what will happen that it will never be able to reach a level it will go up and then it will go down go up and go down so that would be a transient response now I want you to understand this concept very well because I will show you through a series of cartoons that what this concepts exactly mean now one thing I want to first point out is a notion of equilibrium and the notion of a detailed balance equilibrium what is equilibrium many times a bunch of silicon a chunk of silicon sitting on your desk it looks like nothing is happening not really inside the semiconductor a violent storm is raising electrons are bouncing off each other every hundred femtoseconds going in their way and they are redistributing energy redistributing wavelength they are in a violent motion everywhere however if you look at individual processes the number of electrons at a given time in the conduction band you will see two electrons going to the valence band two electrons coming from the valence band in the conduction band so if you just look at the number of electrons at a given time it looks like it's 10 to the power 10 but this is not the same 10 to the power 10 if you could tag the electrons not the same 10 to the power 10 that it was there two seconds ago constantly changing from valence band to conduction band redistributing its energy but the rate is the same going in and out is the same and therefore that is called equilibrium equilibrium doesn't mean that everybody is sitting in their place if everybody sat in their place there will be no fermi dirac statistics remember fermi dirac statistics they had to change their position go through all the combination that you could allow now if you just freeze them in a given place the number of permutation will be one no fermi dirac statistics fermi dirac statistics implies that electrons are violently moving around each other and redistributing their energy all the time okay so equilibrium is a really a very active place and one way to think about equilibrium think about usa here to be the device and the china mexico and india to be various other energy states this is equilibrium it means from mexico to usa let's say two person coming in and from usa to mexico two person going out so if you look at usa it's not the same person same set of people static but the number is the same and you see between any pair of states the rates are exactly balanced so from india four person per day let's say in and out so this is called detailed balance detailed balance means individually each pair of states balanced among each other you don't even have to look to other other things this is detailed balance and there is a property of equilibrium and equilibrium only okay so this i have a few words here you can read it later on but the point is as i said fermi dirac statistics bosch Einstein distribution relies on this notion of detail balance okay now what is steady state well steady state is the following consider the figure on the right hand side and let me walk you through the number of people coming from mexico to usa is three but the other way around is two so they are not individually balanced you see they are not individually balanced and that's let's say the same for all other things you see that the rates in and out are not individually balanced that means detailed balance has been broken right but look at the sum three people coming in from mexico six or four from china let's say and five from india how many total 12 how many going out i also have two six and four 12 going out so if i was just focusing on the usa that energy level then i will see no change in number although two individual states the numbers are changing the rates are very very different this is steady state it means for the relevant energy state that i'm interested in no net change in number but that doesn't mean that every rate is individually balanced no detail balance here in steady state so if you look at this side by side this is the middle one is transient so first one is you see first one is detail balance means steady state right or means equilibrium equilibrium the second one is transient and let's compare it to the third one and let me so the last one is steady state where 12 in 12 out population is same and independent of time but think about the one in the middle in the transient state i have two three and five 10 coming in three five and four 12 going out that means in the transient phase neither detail balance is valid not integrated flux is valid none of them are valid because this is changing all around now if you take the whole thing as a whole meaning take the magenta mexico china india usa all thing together since electron can neither be created or destroyed the whole thing globally of course is in equilibrium in steady state the whole thing globally but if you look at individually they are not so these three things has to be cleanly understood because we'll be using many of this in later derivation okay let we'll get started today on the recombination generation formula but we will actually do the dirty work in the next class but let me explain to you and we will focus on this one this is the most difficult one the others really require one line to derive so we'll do the hard work here but let's get started i'm talking about trap assisted indirect recombination silicon and in germanium right and then we will see assume that i have a few traps here the total number of traps mid-gap levels is let's say here six blue plus red blue ones are the ones that has already captured an electron it's full the red ones are ones that is empty so the trap is empty remember just like donor sometimes it can be empty sometimes it can be full similarly here i'm saying anytime they are full this is n small n sub t and anytime they are empty then that's the red ones that's the that's the empty traps now let's think about an electron the black one on the left is an electron and white open circle is a hole so first thing is that if you allow if you start here start watching the movie from here you will see that the electron is sort of bounced back and forth and it will then get captured in one of the empty traps right okay now why is the electron doing that who is actually bouncing it around do you remember this green thing that in the background that i have drawn is actually not a uniform material these are all the atoms are sitting there all the atoms are sitting there i have drawn it as a flat space but the atoms are sitting there and remember they are actually jiggling in temperature and so when the black electron goes around when it meets a right atom it might be scattered and goes in another direction and it scatters in various ways eventually meeting the red red trap which is the empty trap okay so if you have that after that electron has been captured how many traps do you have now well i still have six because i'm not destroying any trap i just got one trap which was initially empty red one converted into in the right hand top side into a blue circle which means i have that i have that full so therefore now i have four full traps and two empty traps that's it so but still it's six number has changed between n t and p t right a little bit later let's say the hole gets captured by one that had an electron so because this white hole got captured in a blue trap that had an extra electron so it got captured so therefore i am returning to the same space i was before three electron and three hole but in the process look at the what the fate of the electron and hole they have disappeared from the picture so you see what the trap did was assist in the process of finding electron and hole temporarily sort of storing the electron so that a hole can come later on and find it and disappear in the process so this is the mechanics of indirect band gap transition okay now to get started on this we have to calculate the rate at which the electrons and holes disappear so let me just get started on that derivation let's say i'm thinking about electrons d and d t per unit time how many electrons disappear now do you agree with the formula on the right hand side the figure on the left hand side the green figure is a projection one one dimensional space actually it's a three dimensional material so on the right hand side i have recopied it but this time drawn it in three dimension and you can see again i have drawn the three red circles and the three blue circles now this electron the black electron essentially is going moving at the velocity let's say v thermal and i'll explain what that in a second and in a time t in a time t it will go v thermal multiplied by t that's one side of the box the other side of the box is a the cross sectional area so a multiplied by vthd this is the volume of the box the first two term is the volume of the box now if i didn't have any trap or empty trap to capture this electron it doesn't matter how big the boxes so the piece of t or piece of t is the number of red red empty traps available for capture and sigma n says how big the red is if the red is tiny right it will not be able to capture much but if the red is a big circle and i'll explain what that means then it will be able to capture the electrons very easily and i have to divide a by t because i want to and by the way the whole process is proportional to the number of electrons of course if i have more electrons more recombination no rocket science again very simple and i can most of the other things are constant so i can call it and capture constant c sub n put it in here and i'll explain what it means in a second and you can see it's proportional to the number of empty traps and proportional to the number of electrons what is the solution of this equation the entity this is an exponential right do you see immediately that if you shine light and then turn off your torch light then as a function of time the extra electron and hole will gradually decay of almost exponentially of course not exactly exponential because piece of t as the electrons are recombining the piece of t itself might change and therefore may not be exactly an exponential but you can see this is what the result is eventually going to be what is that v thermal well v thermal is the average energy by which the electrons move around and you know half mv squared is three halves kt this is from the statistical mechanics we know that that's the case if it is three dimensional solid three halves kt one dimensional solid half kt and two dimensional solid graphene kt right so from that we should be able to calculate the thermal velocity thermal velocity is on the order of 10 to the power 7 centimeter per second so that's all semiconductors have about the same number 10 to the power 7 centimeter per second now this capture cross section i will discuss this and end there the capture cross section is a geometrical thing assume that you have a sphere and somebody is throwing dart at your sphere right so there is a sphere sitting somebody is throwing dart now what is the probability that the dart will hit the sphere depends on what how big the radius is right and not general radius not 4 by 3 pi r cube because for the person who is throwing the dart it is the cross section that matters pi r naught square now what is the typical diameter of a trap you know trap replaced one of the atoms in the lattice and the atom spacing was how much about five angstrom right and so r naught without doing any calculation you can estimate on the order of five angstrom i have a little circle the atom is about to capture electrons okay so if i put r naught equals on the order of 5 to 10 nanometer uh 5 to 10 angstrom then you can immediately realize that this value is going to be on the order of 10 to the power minus 14 centimeter square do you see that 10 angstrom 10 to the power minus 7 centimeter r square 10 to the power minus 14 centimeters right okay so in fact if you if electron is trying to come in and get captured in one of the red uh empty traps the capture cross section is going to be on the order of on the order of 10 to the power 14 to 10 to the power minus 16 depends on the exact material but on that order okay now this is zinc capture model in silicon that means how zinc behaves copper will have a slightly different value and gold will have a slightly different value so the electron comes in and they get captured in this one but once they get captured it doesn't want a second electron to be captured in that anymore right because it already has one now if you want a second electron to get into the first trap which is already has one it doesn't want anymore you will really have to give it a lot of energy or if its effective cross section is much smaller now because it doesn't want to capture anymore that is 10 to the power 18 minus 18 and if it already has two if you try to put a third one in there or it's not buying anymore it is it is full so it's captured it's sort of trying to disappear itself from the view of the electrons it doesn't want anymore and so therefore it becomes smaller and smaller cross section and they do not participate in the recombination process you see so they disappear on the other hand as an electron let's say it has captured a certain number of electrons for the hole it is very attractive because it is a charged thing and therefore a hole can easily capture so look at this number so as soon as it has captured an electron for the second electron e2 the cross section is small it doesn't want a second electron but it wants a hole and that's in that case for the first hole you can see that it is 10 to the power minus 15 it's much larger now assume that it has for some reason captured two electron now then it is even better for the holes because it has two electrons it really wants a hole and so therefore the hole will come in and essentially it will have a huge cross section for the hole and this is a zinc it's a mid gap state in silicon so that's the model for gold and copper you will have similar types of models it's a done very well in your textbook take a look at this particular picture it's done in a form of a table try to understand the table because remembering these things you know it's not very helpful you have to understand them physically so that it stays with you regardless of where you go independent of the textbook textbook you use and this has a name called cascade model for capture it was first done in 1968 or so and the person who did it lacks i actually worked with him for a few years when i was in bell labs beautiful beautiful theory actually it's i cannot do that here it's a introductory course but in some other course maybe okay let me end so the first thing i wanted to mean say was that once you part of equilibrium restoration of the equilibrium can happen through many channels direct recombination oj right oj recombination trap assisted recombination some are more important in some solid and others in other solids other materials so we'll have to really know them all but apply them carefully now the one i think i also mentioned spend some time the direct recombination is photon assisted photon has actually very little wave vector that's why when you stand in front of your mirror in the morning the mirror doesn't start to rotate hopefully because although the photons are bouncing off the mirror but hopefully they contain such a small amount of momentum that in order to see the momentum you'll have to have a tiny mirrors suspended at a very very low temperature and then you can see that of course it jiggles around right the photons photon does have a wave vector very small phone on on the other hand is a huge wave vector no energy so that's the distinction indirect recombination and i try to explain to you the notion of an equilibrium lots of things are happening in equilibrium it's not sitting there and that is the origin of Fermi drug statistics and steady state and transient dynamics you need to understand this clearly so that when we apply them you have no questions about the relevance or appropriateness okay thank you