In this video, we will discuss radiative transition rate, and introduce minoritarial lifetime. The radiative recombination rate at thermal equilibrium is given by this equation called the van Roosbroeck-Shockley relationship. And it simply says that the equilibrium recombination rate, R naught, the subscript, o, represents thermal equilibrium, is given by the product of probability of absorbing a photon and the density of photon. Now the density of photon modes is given by the Planck's law. And the probability, P, here is directly related to the absorption quotient. And the absorption quotient, if you recall, is defined as the probability of absorbing a photon per unit distance of light propagation. So if you multiply to the absorption quotient this speed of light, c over n is the speed of light in a medium with a refractive index, n. So this product then represents the probability of absorbing a photon per unit time. Now the absorption quotient is a material parameter. And for widely used semiconductors, these are well characterized, it's a well known quantity. If you look at the energy dependence, they show distinct energy frequency dependence depending on whether they are direct bandgap semiconductors or indirect bandgap semiconductors. The direct bandgap semiconductors have a square root dependence on the photon energy. And this square root dependence should look familiar to some of you. And they are the same energy dependence as the density of states for electrons and holes in the conduction band and valence band respectively. And, in fact, this energy dependence is exactly from that, the energy dependence of the density of state. For the indirect bandgap semiconductors, we have a quadratic dependence on the light energy. And you have this extra term plus or minus phonon energy, E sub P is the phonon energy. And if you recall, in a indirect bandgap semiconductor, in order to have a radiative transition, you must involve phonons in order to compensate the momentum difference, the k difference. So depending on whether you're absorbing or emitting phonons, the plus and minus sign represents the absorption or emission of phonon process. And that has to be included in your energy equation. Now, the Planck's black body law is well known, it's shown here. And it's basically a product of photon density of state in free space times the Bose-Einstein probability function. So, with all these functions known, now you can calculate the total radiative recombination rate. Not spectral recombination rate as a function of frequency, just the total recombination rate covering all frequency. You can calculate that by simply integrating the van Roosbroeck-Shockley relationship over frequency. So the R naught is unknown. It is something that can be calculated is a known quantity. Now let's consider non-equilibrium situation. So to consider non-equilibrium situation, you must first recall that recombination process involves one electron in one hole, by definition. So the probability of recombination should be proportional to the concentration of electron in the concentration of holes. So in the most general equation for recombination rate, R, whether it's equilibrium or non-equilibrium, is something that's proportional to the np product, product of the carrier concentration. Proportionality constant here, B, is something that depends on your energy band structure. And it is called the probability of the radiative recombination. Now B can be found by just considering the special case of thermal equilibrium. So rewriting, rewrite this first equation here for the R = Bnp. For a thermal equilibrium situation then, R naught = Bn naught times p naught. But the n naught times p naught is antiproduct to thermal equilibrium is equal to the intrinsic error concentration squared through the law of mass action. And, therefore, you can solve for B and write it as our R naught divided by ni squared. So plug this thing into B here in the very first equation. Then you can rewrite R, the recombination rate, in a general non-equilibrium case as R naught equilibrium recombination rate times np product, divided by ni squared. So this is equation one, we will refer back to this later in this video. Now before we do that, just want to show you some example of these number, the B, for various different semiconductor. And they are different for different materials, obviously. Because they depend on the details of the energy band structure. However, you can see that in general you have a large B for direct bandgap semiconductor and small B many orders of magnitude smaller B, for indirect bandgap semiconductor. And this is the manifestation of this energy band structure where indirect bandgap requires the involvement of phonons in order to match momentum. Now let's look at the recombination dynamics. According to the equation given here in equation 1, here. So it is proportional to np product, and this n and p here is the carrier concentration at a general non-equilibrium case. So, let's first consider the case where somehow the carrier concentrations increased beyond the thermal equilibrium value. So then np product increases, and, therefore, r increases. When recombination rate is increased, then there are more recombinations, meaning that the carrier concentration tends to decrease. So this recombination rate works against the perturbation that drives the semiconductor away from the equilibrium. And wants to bring back the semiconductor to equilibrium situation again. And if you imagine the case where the carrier concentration is reduced below the thermal equilibrium value, then recombination rate decreases. And, effectively, increasing the carrier concentration, once again working against the perturbation. And wants to restore a equilibrium situation. Now, for a more quantitative discussion, let's assume a p-type material in a non-equilibrium situation. So the carrier concentration in this material is now different from the thermal equilibrium value. And we define the difference between the actual carrier concentration and the thermal equilibrium value as excess carrier concentration, excess electron and excess holes. So, mathematically, you can simply write it this way. So n sub p, so the subscript p here indicates p-type material. So we know that n is here is a minority carrier concentration is equal to the equilibrium value, n sub p naught plus delta n, excess carrier concentration. Same thing for holes, the hole concentration is equal to the equilibrium value plus excess hole concentration. And we can also write R, the recombination rate in general. As the equilibrium recombination rate plus delta R, the amount of recombination rate that deviates from equilibrium value. Now, plug all these into the equation 1 that we pointed out, discussed a few minutes ago. Then, you get this equation, and here we assume extrinsic semiconductor, meaning that the doping density is much greater than ni, intrinsic carrier concentration. In this case majority carrier concentration is equal to your doping density. And minority carrier concentration becomes much, much smaller than the majority carrier concentration, the equilibrium values are, that is. So np naught here is very small compared to pp naught. And we can ignore this guy, compared to this, that is. And another assumption is a low level injection condition. So this is a very important condition, very widely used. It applies to a lot of a wide range of operating conditions for semiconductor devices. What that means is that, whatever you're doing to your semiconductor, shining light, driving a current, whatever, that whatever you are doing to drive your semiconductor away from equilibrium, it causes changes in carrier concentration. So it creates this excess hole and excess electron concentration. But that excess remains small compared to your majority carrier concentration. That's the low level injection condition. If these two conditions, condition for extrinsic semiconductor, and condition for low level injection are satisfied, then, in this equation, we can ignore the delta n times delta p product and also np nought times delta p product. Because these are product of two small quantities. So if you collect the remaining terms, then you will see that delta R over R naught is equal to delta n over np naught. So this is the final equation you get. And now we're ready to introduce minority carrier lifetime, and you define your minority carrier lifetime tau sub n as this. So delta R is equal to delta n divided by tau sub n. This is the definition equation for your tau n minority carrier lifetime. Now, plug into the equation before and solve for tau sub n, then you get this equation. So this is the lifetime of minority carrier, in this case, the electron. Now, what does this mean? To see the meaning of the minority carrier lifetime, we write down a simple rate equation. So the rate off change of your electron concentration here should be equal to your delta R. And thermal equilibrium by definition carrier concentrations don't change. So R naught, thermal equilibrium recombination rate doesn't change your carrier concentration, shouldn't. But when the recombination rate deviates from it, then those excess amount, those deviated amount delta R should impact your carrier concentration. And that's what I'm saying here. So the rate of change of your carrier concentration is equal to the delta R, the variations called of recombination rate from the thermal equilibrium value. And there is a negative sign here, because recombination reduces carrier concentration. So this here is a simple first order differential equation, and you can solve it. Solution is an exponential function, and it shows an exponentially decreasing carrier concentration. And the decay constant in this case, it is the minority carrier lifetime. So the physical meaning of the binary carrier lifetime is the average time the minority carrier survive before being annihilated by radiative recombination process.