Understanding Recombination in Semiconductors

Okay welcome back, so if you remember in the last class we had introduced first we have you know derived Einstein relation between drift and diffusion mobility and diffusion coefficient and then we had introduced the concept of recombination, traps you know how electrons are captured, electrons are emitted, holes are captured, holes are emitted there was an eventually we came to an expression which gives the total recombination rate of electrons or holes because of these traps. This recombination rate is very important whether you are shining light, you are not shining light in the presence of traps, the operation of devices like LEDs and photo detectors and so on, this is very important things because this recombination leads to a current called recombination current we will see later or generation recombination current that has real impact on how the device performs. So we should definitely try to understand this recombination very well, okay. So we will come to the white board now, we will see where we stopped in the last class. There was a typo in my last slide here, you know there is a type typing mistake, this actually is not. you know I define tau as the lifetime, it is actually given by 1 by sigma thermal velocity times the total trap density, not n the total trap density, there is a typing mistake there last time. So I told you that, let me come to a new slide here, I told you that electron cross section is assumed to be whole cross section, is assumed to be sigma, the thermal velocity of electron and whole is assumed to be the same and another thing is that, If this is your conduction band, this is your valence band, another thing that has been assumed actually in all the mathematics that were solved which I did not show it here is that your trap energy level, actually I am not drawing it well, your trap energy level E t is assumed to be at E i which is at mid gap, okay. If the trap energy level is at mid gap or E i then your inner recombination rate becomes maximum. So So that is why we assume it here and that helps in the mathematics otherwise some cos hyperbolic term comes there. Anyways when you do this then you get a major recombination this is your total recombination rate, recombination rate okay. This is equal to 1 by tau into Pn minus ni square by P plus n plus 2 ni, ni is the intrinsic carrier concentration and tau as I told you is given by 1 by sigma Vth. So, if your tau is large you know if your tau is large which is a large lifetime then your recombination rate is also small it will recombine small and if your tau is very small say nanosecond or picosecond very small then your recombination rate is faster it will recombine too fast okay. This sort of recombination and you remember this. Pn and Ni square they are not equal here because it is a non equilibrium you are shining light for example the excess carriers that are coming out because of this traps emitting in cross section and so on this is not an ideal situation here. So this actually is driving the recombination this is actually giving the resistance to the recombination here okay. So this has 3 assumptions one is that cross section of electron and hole traps are same thermal velocity is same and the third assumption is that the trap energy level is supposed to be exactly at mid gap. in which case this is valid, okay. This is the total recombination rate, the unit is per centimeter cube per second here, okay. So suppose I take a moderately you know n type dope semiconductor, moderately n type dope semiconductor I am shining light generating some carriers, if I am generating some carriers, so in equilibrium, in equilibrium the whole concentration will be say P naught in equilibrium which is ni square by n. Okay, n i square by n, okay, right. So, now I am going to shine light and generate some excess carriers and I am going to understand how the recombination rate depends on that, okay, how the recombination rate depends on that. So, u look at u here 1 by tau p n minus n i square by, sorry. Pn minus Ni square by P plus n plus 2 Ni. Now because it is a non-equilibrium, we are shining light generating some excess carrier, there are large numbers, you know, these are large numbers that are generating compared to Ni, much much larger, okay. So you can conveniently ignore Ni in the top, in the bottom at least, here you can ignore bottom. So what you can do is that you can ignore 2ni here okay, you can ignore 2ni here and then I told you that you have n, it is an n-type doped semiconductor predominantly, it is an n-type doped semiconductor and you are shining light for example, you are generating excess carriers, excess electrons and holes because it is an n-type the excess electron that are generating, see excess electrons that you generate is suppose delta n. Because you are shining light that will always be equal to excess holes that you are generating but because it is n type doped this quantity is much smaller than the background n type doping so you cannot this is can be neglected in term in comparison to n type but because the p type is minority and that is given by ni square by n because p type is minority the delta p is actually larger than p okay. So for n type semiconductor when you generate excess carrier the excess holes actually matter more than excess electrons okay. because p is minority here, okay. So I can neglect the p also here, okay. I can neglect the p here. So what I can do is that 1 by tau p n by n minus n i square by n. So this is given by 1 by tau p. You remember n i square by n actually is the equilibrium hole concentration. This is your equilibrium hole concentration without any shining light. equilibrium hole concentration, okay. And this is your total hole concentration that you are coming, total hole concentration because of shining light. The difference of total hole concentration minus the equilibrium hole concentration is called the excess carrier that you are generating, excess hole concentration that you are generating which is equal to delta P. See you have a baseline P naught which is the equilibrium concentration. On top of that you are shining light and generating some excess carrier delta P. The total is P the total whole concentration. So P minus P naught gives you delta P. So in other words if I write this equation again better here it means that U the total recombination rate is equal to 1 by tau times delta P right. So this is delta P by tau. This is a very important fundamental very important quantity okay. It is given by centimeter cube per second. It tells you that the recombination happens as delta excess carriers by tau. How many excess carriers are there and how they are decaying, okay. So if you see if your lifetime is very short, your recombination is very fast and if you have more holes then of course your recombination also will be very fast. This is a very important, a very very important relation. If you are told that you know your recombination rate is say 10 to the power 16 per centimeter per cube and your lifetime is say 1 nanosecond, what is the excess carrier generating? The excess carrier you are generating is delta p is equal to u times tau which is 10 to the power 16 times 10 to the power minus 9 second which is equal to 10 to the power 7 per centimeter cube of excess electron holes you are generating. So this is a very powerful relation also for electrons it will hold true in a moderately p-type dope semiconductor excess electron by that lifetime that you are doing. This is your total recombination rate the rate at which electrons or holes are recombining. The rate at which electrons or holes are recombining is the rate at which electrons or holes are recombining. And in case I did not tell you before this entire premise this entire expression here that we have learnt here you know the entire expression that we have learnt here this recombination okay this is this is called SRH recombination statistics what does SRH stand for SRH stands for Shockley Reed these are three people actually Reed and Hall. Shockley-Reed Hall statistics gives you this recombination, this is a trap related recombination, okay. This is a trap related recombination and I am showing you a shining light example to show how the trap related recombination can actually you know take part here. So now we know the recombination, we know also generation when you shine light you might generate carriers for that the light has to be absorbed by the semiconductor and you typically shine light that can be absorbed by the semiconductor. So the light energy has to be equal to or more than the energy of the semiconductor if you remember that. So suppose I have a semiconductor and the band gap is EG. So the light that I have to shine the energy of the light for it to get absorbed the energy of the light is h nu which is equal to hc by lambda that has to be at least greater than or equal to EG. If it is less than the band gap then it will pass through it will not get absorbed it has to be at least the band gap of the energy or more than the band gap of the energy of the band gap. So from here I can say that lambda has to be less than equal to hc by EG. Okay, and this is simplified as you know 1242 by EG in electron volt will give you in nanometer. What I mean is that lambda, suppose the band gap of a semiconductor is 1.2 electron volt, then the lambda minimum lambda you know is lambda less than equal to 1242 by 1.2 EV will be almost equal to how much? Almost equal to 1000, so 1000 nanometer. So, lambda less than or equal to 1000 nanometer will be absorbed in the material. So, 900 nanometer will be absorbed, 950 nanometer will be absorbed but 1050 will not be absorbed okay, because 1050 nanometer is lower in energy than this okay. So, that we should keep that in mind just in case okay. Now we know generation when you shine light you can have generation or even you can have more thermal generation in other case we have recombination. which is because of say traps and then it is called Shockley-Hall-Reed recombination. There can be also direct band to band recombination that might emit light. That is a different thing. That is during your LED that we will talk about. We will not talk about that right now. What is important next logically is continuity equation. Continuity. This continuity equation will allow you to derive the solution for any kind of semiconductor situation. Okay what does continuity equation actually mean? Suppose I have a semiconductor and here I am going to take I have a semiconductor like this and I am gonna go and take a slice of this I am gonna take a slice of this okay I am going to go. I am going to take a slice of this which is a very small thickness, this is the thickness okay. I am going to take a slice of this, the slice thickness is dx and the area uniform area is A. There is current that is flowing J of n, this point is x, this point is x plus dx. So the current at this point is J of f of x, the current that is coming out is J x plus dx. Now you might say it will not it will violate Kirchhoff's law because the current has to be continuous everywhere true but here there will be some recombination taking place within this volume that is why the current the particle rate has to be conserved not the current okay because there will be some recombination generation of carriers that will come out here in this volume if you see this volume well. So some current is flowing here Jn of x some current is coming out here Jn x plus dx. So Within this volume the red shaded volume that I am giving the area is a ok within this volume you have a generation of gl per centimeter cube per second maybe optically it is your shining light within this volume there is also recombination the recombination is given by you know if I am talking about electron then delta n by tau. This is also centimeter cube per centimeter cube by second, so this is what at rate at which the electrons are recombining, this is the rate at which electrons are generating, electrons are flowing in, there is a current, electrons are flowing out, there is a current. So the total rate of particle flow within this block, the total rate of particle flow, electron flow I mean, the total rate of particle flow, okay, the total rate of particle flow is given by d n. by DT the rate of flow of particle okay actually it is a position of both X and T okay the rate at which particles are flowing here and what will that rate be so actually there is a current here flow in there is a current out here flow out there is generation here here there is recombination lost also currents are particles are lost here. So, dn by dt the rate at which it is going is equal to what is equal to actually the you know if your current is flowing jn. and the current that is flowing out is Jnx plus dx this is Jnx then the rate of particle flow is actually given by 1 by Q Jnx plus dx minus Jnx which can be written as which can be written as what 1 by Q d Jnx by dx into delta x of course this is the current density So, if you want to get normalize current the total current you have to multiply the area A okay you have to multiply the area A. Similarly this is the total particle flow in flow out minus flow in okay flow out minus flow in this is the total current flow in and flow out okay and G L is the optical generation rate. So, this also has to be multiplied by delta x times A. So, this gives you the total generation rate and delta n by tau okay. is the recombination the carrier cell losing because of recombination times delta X into A gives you the total recombination current here the rate recombination rate. So if you look at the previous slide the total rate of particle flow is given by particle flow out minus particle flow in plus generation minus recombination. Okay so the total rate of particle flow is given by flow out minus flow in which is given by this quantity plus generation because you are generating carrier which is given by this minus recombination. which is given by this. You see what is that flow out minus flow in that is the generation rate plus optical generation rate or any other generation rate minus recombination rate gives you the total rate of particle flow. So now I can rephrase this equation and write it very well okay. I can write dn by dt of course there will be A into delta x is equal to dn 1 by Q dNX by dx into delta X into A plus GL minus delta N by tau delta X by A. So of course this quantity can go away, this will be cancelled out everywhere, okay. If you remember I mean this derivative of the current how does it come if you recall actually this quantity comes because of this. It is like you know if you recall in the last class about the diffusion kindle side set the same thing f of x plus delta x minus f of x is given by df x by dx into delta x that is what I am doing here. So I can do that dn by dt is equal to 1 by q d Jn x dn by dt by dx plus gl minus delta n by tau this is called continuity equation I will make it simplified also again little bit more this is called continuity equation that tells you how currents will actually flow. Now you look at this equation this term carefully current look at the term current jn it has two component electron current one is diffusion current one is drift current. Diffusion current is given by Q dn dn by dx drift current is given by Q mu n field. Suppose there is no field then this component goes okay then what remains is Jn of x will be equal to Q dn dn by dx. You substitute that in the continuity equation if you remember the continuity equation dn by dt is equal to 1 by Q of x of this which is q dn dn by dx plus gl minus delta n by tau what it will be dn by dt is equal to dn sorry here I will write dn you have two derivative now this is one there is already one so this square N x actually it will be N of x comma t, but I will just write it as N for example by d x squared. plus g L minus delta N by tau. If it is in steady state, if we talk about steady state, in steady state there is no time variation, there is no time variation which means this quantity is 0 in steady state this quantity will be 0 then what will happen there then you will have dn d square n by dx square plus gl minus delta n by t is equal to 0. Now this n is actually the background concentration n0 I mean the equilibrium concentration n0 which is constant this is constant plus the excess carrier concentration that you are generating whatever means it is and dn by dx. will be equal to d of this but this is constant right so the derivative will also be only delta nx by dx. So I can write this expression again as dn d square delta nx by dx square plus gl minus delta nx by tau is equal to 0. Okay this is a simplified form of continuity equation assuming there is no field because if there is a field there is a drift component that I have ignored here okay. If there is no optical generation then this quantity will go to 0 if there is no optical generation then your equation will look like dn d square delta n by delta x square is equal to delta n by tau. You see this is a second order differential equation. You can write it as del square n by d square equal to delta n by dn into tau n and this quantity can be written as ln square in other words I can write ln is equal to dn by half it is called the diffusion mean diffusion length mean free path of diffusion you can say it is or you can just say it is a diffusion length actually that is the average length over which electrons and holes or electrons will diffuse okay. So I can write it as d square delta n by d square x equal to delta n by ln square. This is it is like you know d square y by dx square equal to y by some constant. How do you solve it? Actually it has a generic solution which is delta n x will be equal to some a1 e to the power x by ln plus a2 e to the power minus x by ln. You have to take some boundary condition and do things like that, but we can do two approximation one is there can be two situations not approximation two situations where we can approximate one is that number one is that your the semiconductor you are talking about the length of the semiconductor okay the length of the semiconductor is much larger than the diffusion length ln and in one case the length of the semiconductor is much shorter than the diffusion length ln okay. Because the length of the semiconductor will come because you are talking about a finite block of semiconductor in which you are going to recombine the things. So, in one case the semiconductor can be much larger than the diffusion length, in one case the semiconductor length can be much smaller than the diffusion length, in which case this can be a actually this solution this solution will give you a hyperbolic equation, a hyperbolic sine cosine equation ok. We do not have to memorize or you know derive that directly here, but you can simplify that actually when you have. Two unique situation when either the length of the semiconductor is much larger than the diffusion length and or the length of the semiconductor is much smaller than the diffusion length. So if your length of the semiconductor is much smaller than the is much larger than the diffusion length then you know this is a solution delta n x will have an exponential decay it will basically it will decay exponentially as e to the power minus x by ln which means this is suppose delta n. Excess carrier concentration at x equal to 0 it will be here over distance x it will decay exponentially like this. This is excess carrier remember this is excess carrier not the total carrier. The total carrier will be the excess carrier plus the baseline equilibrium concentration okay. This excess carrier will decay exponentially like this as e to the power minus x by ln okay e to the power minus x by ln. And if the semiconductor length is smaller than the diffusion length then If the semiconductor length is smaller than the diffusion length then this excess carrier delta n of x at x it will decay linearly it will decay linearly. So it will be like n of if this is point is sorry if this point is a delta n of 0 then it will be something like delta n of 0 1 minus it is something like the x by the length of the semiconductor l for example. So it will decay linearly. And if it is a short diode, it is a short sort of a material or very short, we call it a short diode or something. It is a long diode or long semiconductor. It will decay exponentially. It will decay exponentially. And this has a lot of important consequences for device. If the delta x decays like that, then the current, the diffusion current will also go as J dn into d delta n by dx. And if this decays exponentially then this also will decay exponentially which means the diffusion current also will decay exponentially, the diffusion current also will decay exponentially in this case it is a straight line the derivative of this is constant so the diffusion current also will be constant. And in this case the diffusion current will decay exponentially. So they are very important things actually that you know this how the decay has a lot of role to play actually whether it is a long diode or it is a short diode and this decay of excess carriers comes from solving the continuity equation. What are the assumptions we made? We made some very big assumptions. One of them is that there is no field and that is why you are able to do Derive this equation if there is a field then you know there will be drift component that we are neglecting and also we are neglecting that optical generation rate. Only in that case we are able to put this equation up, the second order differential equation that we are solving to get this decaying profile. So we will see in real devices actually when you inject carriers you know you will have this excess carriers will decay. And you are injecting electron somehow, you are injecting electron, so this electron will decay exponentially like this with position or if it is a long one, if it is a short one it will decay linearly, right because the electrons will decay exponentially like this the derivative also will decay exponentially which means the diffusion current also will decay exponentially. So there are important things actually, important implications for real device applications that will come how you know when we do it, for example if you have a P type semiconductor you are shining light here. We will do this all in the next class of course, but if you shine light here you will generate excess carriers here, this excess carriers also will diffuse. So that is one thing, you might shine light everywhere uniformly. if you shine light everywhere uniformly then everywhere it will be generated then nothing will diffuse because everywhere the concentration will be same. Remember in this case there is no field it is only the diffusion of excess carriers that is happening. So continuity equation is very important and you have seen that continuity equation we have conveniently neglected the drift current if you can if you want to take the drift current then this expression will have an extra term because if you recall that the. The way we had derived it is this we had neglected, so if you take that then in this expression you have to also take that expression. So then it will become a little longer expression and that can be done of course but. not we should not do it right now if and when the necessity arises then we can definitely do that okay. So the things that we have learned right now okay the things that we have learned right now I can tabulate it again here we have learned about the sorry we have learned about the Shockley-Ried Hall statistics and how recombination takes place I told you the total recombination rate delta n by tau there is a tau is a lifetime this is the excess carrier you have this is the recombination rate you have. you might shine light and generate some carriers which is optical generation rate. In the presence of all of this you can actually solve and get the continuity equation and this continuity equation will be our very important starting point in many of the semiconductor devices. This continuity equation tells you that you know how you solve for the carrier movement or the carrier distribution or the current flow. It will basically depend on dn, d square delta nx for n type, for p type it will be delta p. plus GL optical minus delta N by tau is equal to 0 in steady state, in steady state. So similarly for holes you will get this for P type and this is without consideration of field and in any case you can always solve this equation to get the carrier profile delta N. Once you get the carrier profile delta N you can get QDND of this you will get the diffusion current okay and not only that if you get delta N by multiplying mu times field times Q. You can also get drift current. So both drift diffusion current will come out. So that is why we need to solve this equation, okay. That is why we need to solve this equation here, okay. So we will end the lecture here today. We have solved continuity equation and we now can move to the next topic after this which is you know more or less P-N junction. We will discuss that in the next class and before P-N junction we start P-N junction. One small thing we have to learn in probably in the start of the next class. and that is about a few examples of how we apply the continuity equation in some special and unique cases like shining light from one side, applying voltage to a sample and shining light. So if we can learn those things then we will exactly understand how continuity equation helps you get the profile of carrier delta n or delta p. okay. And that delta n or delta p will give you the drift and diffusion current that we already discussed. So after we are armed with all this, after we are familiar with all this concept, we will immediately start with PN junction. And in PN junction you will see that drift, diffusion current, the Einstein relation, the carrier statistics, everything that we have learned till now will be used in PN junction. Also whatever we learned in high school, things like Poisson equation, you know charge neutrality, all these things also will be used in PN junction. And once we know PN junction things like LED, photo detector, solar cell are very easy because they are all PN junctions okay. So we will end up the class here. Next class we will start with some examples of how to apply continuity equation and depending on time we will move to PN junction from there okay. Thank you.

So we will come to the white board now, we will see where we stopped in the last class. There was a typo in my last slide here, you know there is a type typing mistake, this actually is not. you know I define tau as the lifetime, it is actually given by 1 by sigma thermal velocity times the total trap density, not n the total trap density, there is a typing mistake there last time. So I told you that, let me come to a new slide here, I told you that electron cross section is assumed to be whole cross section, is assumed to be sigma, the thermal velocity of electron and whole is assumed to be the same and another thing is that, If this is your conduction band, this is your valence band, another thing that has been assumed actually in all the mathematics that were solved which I did not show it here is that your trap energy level, actually I am not drawing it well, your trap energy level E t is assumed to be at E i which is at mid gap, okay.

If the trap energy level is at mid gap or E i then your inner recombination rate becomes maximum. So So that is why we assume it here and that helps in the mathematics otherwise some cos hyperbolic term comes there. Anyways when you do this then you get a major recombination this is your total recombination rate, recombination rate okay. This is equal to 1 by tau into Pn minus ni square by P plus n plus 2 ni, ni is the intrinsic carrier concentration and tau as I told you is given by 1 by sigma Vth. So, if your tau is large you know if your tau is large which is a large lifetime then your recombination rate is also small it will recombine small and if your tau is very small say nanosecond or picosecond very small then your recombination rate is faster it will recombine too fast okay.

This sort of recombination and you remember this. Pn and Ni square they are not equal here because it is a non equilibrium you are shining light for example the excess carriers that are coming out because of this traps emitting in cross section and so on this is not an ideal situation here. So this actually is driving the recombination this is actually giving the resistance to the recombination here okay. So this has 3 assumptions one is that cross section of electron and hole traps are same thermal velocity is same and the third assumption is that the trap energy level is supposed to be exactly at mid gap. in which case this is valid, okay.

This is the total recombination rate, the unit is per centimeter cube per second here, okay. So suppose I take a moderately you know n type dope semiconductor, moderately n type dope semiconductor I am shining light generating some carriers, if I am generating some carriers, so in equilibrium, in equilibrium the whole concentration will be say P naught in equilibrium which is ni square by n. Okay, n i square by n, okay, right.

So, now I am going to shine light and generate some excess carriers and I am going to understand how the recombination rate depends on that, okay, how the recombination rate depends on that. So, u look at u here 1 by tau p n minus n i square by, sorry. Pn minus Ni square by P plus n plus 2 Ni. Now because it is a non-equilibrium, we are shining light generating some excess carrier, there are large numbers, you know, these are large numbers that are generating compared to Ni, much much larger, okay.

So you can conveniently ignore Ni in the top, in the bottom at least, here you can ignore bottom. So what you can do is that you can ignore 2ni here okay, you can ignore 2ni here and then I told you that you have n, it is an n-type doped semiconductor predominantly, it is an n-type doped semiconductor and you are shining light for example, you are generating excess carriers, excess electrons and holes because it is an n-type the excess electron that are generating, see excess electrons that you generate is suppose delta n. Because you are shining light that will always be equal to excess holes that you are generating but because it is n type doped this quantity is much smaller than the background n type doping so you cannot this is can be neglected in term in comparison to n type but because the p type is minority and that is given by ni square by n because p type is minority the delta p is actually larger than p okay.

So for n type semiconductor when you generate excess carrier the excess holes actually matter more than excess electrons okay. because p is minority here, okay. So I can neglect the p also here, okay.

I can neglect the p here. So what I can do is that 1 by tau p n by n minus n i square by n. So this is given by 1 by tau p.

You remember n i square by n actually is the equilibrium hole concentration. This is your equilibrium hole concentration without any shining light. equilibrium hole concentration, okay. And this is your total hole concentration that you are coming, total hole concentration because of shining light.

The difference of total hole concentration minus the equilibrium hole concentration is called the excess carrier that you are generating, excess hole concentration that you are generating which is equal to delta P. See you have a baseline P naught which is the equilibrium concentration. On top of that you are shining light and generating some excess carrier delta P.

The total is P the total whole concentration. So P minus P naught gives you delta P. So in other words if I write this equation again better here it means that U the total recombination rate is equal to 1 by tau times delta P right. So this is delta P by tau. This is a very important fundamental very important quantity okay.

It is given by centimeter cube per second. It tells you that the recombination happens as delta excess carriers by tau. How many excess carriers are there and how they are decaying, okay. So if you see if your lifetime is very short, your recombination is very fast and if you have more holes then of course your recombination also will be very fast.

This is a very important, a very very important relation. If you are told that you know your recombination rate is say 10 to the power 16 per centimeter per cube and your lifetime is say 1 nanosecond, what is the excess carrier generating? The excess carrier you are generating is delta p is equal to u times tau which is 10 to the power 16 times 10 to the power minus 9 second which is equal to 10 to the power 7 per centimeter cube of excess electron holes you are generating. So this is a very powerful relation also for electrons it will hold true in a moderately p-type dope semiconductor excess electron by that lifetime that you are doing.

This is your total recombination rate the rate at which electrons or holes are recombining. The rate at which electrons or holes are recombining is the rate at which electrons or holes are recombining. And in case I did not tell you before this entire premise this entire expression here that we have learnt here you know the entire expression that we have learnt here this recombination okay this is this is called SRH recombination statistics what does SRH stand for SRH stands for Shockley Reed these are three people actually Reed and Hall. Shockley-Reed Hall statistics gives you this recombination, this is a trap related recombination, okay. This is a trap related recombination and I am showing you a shining light example to show how the trap related recombination can actually you know take part here.

So now we know the recombination, we know also generation when you shine light you might generate carriers for that the light has to be absorbed by the semiconductor and you typically shine light that can be absorbed by the semiconductor. So the light energy has to be equal to or more than the energy of the semiconductor if you remember that. So suppose I have a semiconductor and the band gap is EG. So the light that I have to shine the energy of the light for it to get absorbed the energy of the light is h nu which is equal to hc by lambda that has to be at least greater than or equal to EG.

If it is less than the band gap then it will pass through it will not get absorbed it has to be at least the band gap of the energy or more than the band gap of the energy of the band gap. So from here I can say that lambda has to be less than equal to hc by EG. Okay, and this is simplified as you know 1242 by EG in electron volt will give you in nanometer.

What I mean is that lambda, suppose the band gap of a semiconductor is 1.2 electron volt, then the lambda minimum lambda you know is lambda less than equal to 1242 by 1.2 EV will be almost equal to how much? Almost equal to 1000, so 1000 nanometer. So, lambda less than or equal to 1000 nanometer will be absorbed in the material. So, 900 nanometer will be absorbed, 950 nanometer will be absorbed but 1050 will not be absorbed okay, because 1050 nanometer is lower in energy than this okay. So, that we should keep that in mind just in case okay.

Now we know generation when you shine light you can have generation or even you can have more thermal generation in other case we have recombination. which is because of say traps and then it is called Shockley-Hall-Reed recombination. There can be also direct band to band recombination that might emit light. That is a different thing.

That is during your LED that we will talk about. We will not talk about that right now. What is important next logically is continuity equation. Continuity.

This continuity equation will allow you to derive the solution for any kind of semiconductor situation. Okay what does continuity equation actually mean? Suppose I have a semiconductor and here I am going to take I have a semiconductor like this and I am gonna go and take a slice of this I am gonna take a slice of this okay I am going to go. I am going to take a slice of this which is a very small thickness, this is the thickness okay. I am going to take a slice of this, the slice thickness is dx and the area uniform area is A.

There is current that is flowing J of n, this point is x, this point is x plus dx. So the current at this point is J of f of x, the current that is coming out is J x plus dx. Now you might say it will not it will violate Kirchhoff's law because the current has to be continuous everywhere true but here there will be some recombination taking place within this volume that is why the current the particle rate has to be conserved not the current okay because there will be some recombination generation of carriers that will come out here in this volume if you see this volume well.

So some current is flowing here Jn of x some current is coming out here Jn x plus dx. So Within this volume the red shaded volume that I am giving the area is a ok within this volume you have a generation of gl per centimeter cube per second maybe optically it is your shining light within this volume there is also recombination the recombination is given by you know if I am talking about electron then delta n by tau. This is also centimeter cube per centimeter cube by second, so this is what at rate at which the electrons are recombining, this is the rate at which electrons are generating, electrons are flowing in, there is a current, electrons are flowing out, there is a current. So the total rate of particle flow within this block, the total rate of particle flow, electron flow I mean, the total rate of particle flow, okay, the total rate of particle flow is given by d n.

by DT the rate of flow of particle okay actually it is a position of both X and T okay the rate at which particles are flowing here and what will that rate be so actually there is a current here flow in there is a current out here flow out there is generation here here there is recombination lost also currents are particles are lost here. So, dn by dt the rate at which it is going is equal to what is equal to actually the you know if your current is flowing jn. and the current that is flowing out is Jnx plus dx this is Jnx then the rate of particle flow is actually given by 1 by Q Jnx plus dx minus Jnx which can be written as which can be written as what 1 by Q d Jnx by dx into delta x of course this is the current density So, if you want to get normalize current the total current you have to multiply the area A okay you have to multiply the area A. Similarly this is the total particle flow in flow out minus flow in okay flow out minus flow in this is the total current flow in and flow out okay and G L is the optical generation rate.

So, this also has to be multiplied by delta x times A. So, this gives you the total generation rate and delta n by tau okay. is the recombination the carrier cell losing because of recombination times delta X into A gives you the total recombination current here the rate recombination rate.

So if you look at the previous slide the total rate of particle flow is given by particle flow out minus particle flow in plus generation minus recombination. Okay so the total rate of particle flow is given by flow out minus flow in which is given by this quantity plus generation because you are generating carrier which is given by this minus recombination. which is given by this.

You see what is that flow out minus flow in that is the generation rate plus optical generation rate or any other generation rate minus recombination rate gives you the total rate of particle flow. So now I can rephrase this equation and write it very well okay. I can write dn by dt of course there will be A into delta x is equal to dn 1 by Q dNX by dx into delta X into A plus GL minus delta N by tau delta X by A.

So of course this quantity can go away, this will be cancelled out everywhere, okay. If you remember I mean this derivative of the current how does it come if you recall actually this quantity comes because of this. It is like you know if you recall in the last class about the diffusion kindle side set the same thing f of x plus delta x minus f of x is given by df x by dx into delta x that is what I am doing here.

So I can do that dn by dt is equal to 1 by q d Jn x dn by dt by dx plus gl minus delta n by tau this is called continuity equation I will make it simplified also again little bit more this is called continuity equation that tells you how currents will actually flow. Now you look at this equation this term carefully current look at the term current jn it has two component electron current one is diffusion current one is drift current. Diffusion current is given by Q dn dn by dx drift current is given by Q mu n field. Suppose there is no field then this component goes okay then what remains is Jn of x will be equal to Q dn dn by dx. You substitute that in the continuity equation if you remember the continuity equation dn by dt is equal to 1 by Q of x of this which is q dn dn by dx plus gl minus delta n by tau what it will be dn by dt is equal to dn sorry here I will write dn you have two derivative now this is one there is already one so this square N x actually it will be N of x comma t, but I will just write it as N for example by d x squared.

plus g L minus delta N by tau. If it is in steady state, if we talk about steady state, in steady state there is no time variation, there is no time variation which means this quantity is 0 in steady state this quantity will be 0 then what will happen there then you will have dn d square n by dx square plus gl minus delta n by t is equal to 0. Now this n is actually the background concentration n0 I mean the equilibrium concentration n0 which is constant this is constant plus the excess carrier concentration that you are generating whatever means it is and dn by dx. will be equal to d of this but this is constant right so the derivative will also be only delta nx by dx. So I can write this expression again as dn d square delta nx by dx square plus gl minus delta nx by tau is equal to 0. Okay this is a simplified form of continuity equation assuming there is no field because if there is a field there is a drift component that I have ignored here okay. If there is no optical generation then this quantity will go to 0 if there is no optical generation then your equation will look like dn d square delta n by delta x square is equal to delta n by tau.

You see this is a second order differential equation. You can write it as del square n by d square equal to delta n by dn into tau n and this quantity can be written as ln square in other words I can write ln is equal to dn by half it is called the diffusion mean diffusion length mean free path of diffusion you can say it is or you can just say it is a diffusion length actually that is the average length over which electrons and holes or electrons will diffuse okay. So I can write it as d square delta n by d square x equal to delta n by ln square.

This is it is like you know d square y by dx square equal to y by some constant. How do you solve it? Actually it has a generic solution which is delta n x will be equal to some a1 e to the power x by ln plus a2 e to the power minus x by ln.

You have to take some boundary condition and do things like that, but we can do two approximation one is there can be two situations not approximation two situations where we can approximate one is that number one is that your the semiconductor you are talking about the length of the semiconductor okay the length of the semiconductor is much larger than the diffusion length ln and in one case the length of the semiconductor is much shorter than the diffusion length ln okay. Because the length of the semiconductor will come because you are talking about a finite block of semiconductor in which you are going to recombine the things. So, in one case the semiconductor can be much larger than the diffusion length, in one case the semiconductor length can be much smaller than the diffusion length, in which case this can be a actually this solution this solution will give you a hyperbolic equation, a hyperbolic sine cosine equation ok.

We do not have to memorize or you know derive that directly here, but you can simplify that actually when you have. Two unique situation when either the length of the semiconductor is much larger than the diffusion length and or the length of the semiconductor is much smaller than the diffusion length. So if your length of the semiconductor is much smaller than the is much larger than the diffusion length then you know this is a solution delta n x will have an exponential decay it will basically it will decay exponentially as e to the power minus x by ln which means this is suppose delta n.

Excess carrier concentration at x equal to 0 it will be here over distance x it will decay exponentially like this. This is excess carrier remember this is excess carrier not the total carrier. The total carrier will be the excess carrier plus the baseline equilibrium concentration okay.

This excess carrier will decay exponentially like this as e to the power minus x by ln okay e to the power minus x by ln. And if the semiconductor length is smaller than the diffusion length then If the semiconductor length is smaller than the diffusion length then this excess carrier delta n of x at x it will decay linearly it will decay linearly. So it will be like n of if this is point is sorry if this point is a delta n of 0 then it will be something like delta n of 0 1 minus it is something like the x by the length of the semiconductor l for example. So it will decay linearly.

And if it is a short diode, it is a short sort of a material or very short, we call it a short diode or something. It is a long diode or long semiconductor. It will decay exponentially. It will decay exponentially. And this has a lot of important consequences for device.

If the delta x decays like that, then the current, the diffusion current will also go as J dn into d delta n by dx. And if this decays exponentially then this also will decay exponentially which means the diffusion current also will decay exponentially, the diffusion current also will decay exponentially in this case it is a straight line the derivative of this is constant so the diffusion current also will be constant. And in this case the diffusion current will decay exponentially.

So they are very important things actually that you know this how the decay has a lot of role to play actually whether it is a long diode or it is a short diode and this decay of excess carriers comes from solving the continuity equation. What are the assumptions we made? We made some very big assumptions. One of them is that there is no field and that is why you are able to do Derive this equation if there is a field then you know there will be drift component that we are neglecting and also we are neglecting that optical generation rate.

Only in that case we are able to put this equation up, the second order differential equation that we are solving to get this decaying profile. So we will see in real devices actually when you inject carriers you know you will have this excess carriers will decay. And you are injecting electron somehow, you are injecting electron, so this electron will decay exponentially like this with position or if it is a long one, if it is a short one it will decay linearly, right because the electrons will decay exponentially like this the derivative also will decay exponentially which means the diffusion current also will decay exponentially. So there are important things actually, important implications for real device applications that will come how you know when we do it, for example if you have a P type semiconductor you are shining light here. We will do this all in the next class of course, but if you shine light here you will generate excess carriers here, this excess carriers also will diffuse.

So that is one thing, you might shine light everywhere uniformly. if you shine light everywhere uniformly then everywhere it will be generated then nothing will diffuse because everywhere the concentration will be same. Remember in this case there is no field it is only the diffusion of excess carriers that is happening. So continuity equation is very important and you have seen that continuity equation we have conveniently neglected the drift current if you can if you want to take the drift current then this expression will have an extra term because if you recall that the.

The way we had derived it is this we had neglected, so if you take that then in this expression you have to also take that expression. So then it will become a little longer expression and that can be done of course but. not we should not do it right now if and when the necessity arises then we can definitely do that okay.

So the things that we have learned right now okay the things that we have learned right now I can tabulate it again here we have learned about the sorry we have learned about the Shockley-Ried Hall statistics and how recombination takes place I told you the total recombination rate delta n by tau there is a tau is a lifetime this is the excess carrier you have this is the recombination rate you have. you might shine light and generate some carriers which is optical generation rate. In the presence of all of this you can actually solve and get the continuity equation and this continuity equation will be our very important starting point in many of the semiconductor devices.

This continuity equation tells you that you know how you solve for the carrier movement or the carrier distribution or the current flow. It will basically depend on dn, d square delta nx for n type, for p type it will be delta p. plus GL optical minus delta N by tau is equal to 0 in steady state, in steady state. So similarly for holes you will get this for P type and this is without consideration of field and in any case you can always solve this equation to get the carrier profile delta N. Once you get the carrier profile delta N you can get QDND of this you will get the diffusion current okay and not only that if you get delta N by multiplying mu times field times Q.

You can also get drift current. So both drift diffusion current will come out. So that is why we need to solve this equation, okay.

That is why we need to solve this equation here, okay. So we will end the lecture here today. We have solved continuity equation and we now can move to the next topic after this which is you know more or less P-N junction. We will discuss that in the next class and before P-N junction we start P-N junction. One small thing we have to learn in probably in the start of the next class.

and that is about a few examples of how we apply the continuity equation in some special and unique cases like shining light from one side, applying voltage to a sample and shining light. So if we can learn those things then we will exactly understand how continuity equation helps you get the profile of carrier delta n or delta p. okay. And that delta n or delta p will give you the drift and diffusion current that we already discussed.

So after we are armed with all this, after we are familiar with all this concept, we will immediately start with PN junction. And in PN junction you will see that drift, diffusion current, the Einstein relation, the carrier statistics, everything that we have learned till now will be used in PN junction. Also whatever we learned in high school, things like Poisson equation, you know charge neutrality, all these things also will be used in PN junction. And once we know PN junction things like LED, photo detector, solar cell are very easy because they are all PN junctions okay. So we will end up the class here.

Next class we will start with some examples of how to apply continuity equation and depending on time we will move to PN junction from there okay. Thank you.

Transcript for:Understanding Recombination in Semiconductors

Transcript for:
Understanding Recombination in Semiconductors