In this video, we will discuss Non-Radiative Transition, the general abstract discussions about Non-Radiative Transition Rates. So if you have a defect, or deep impurities in your semiconductor, they will produce energy levels deep inside the band gaps. So here is the conduction band, this is the valence band, this is the band gap, and E's of i, Intrinsic Fermi levels sits in the middle of the band gap and somewhere nearby, somewhere in the middle of the band gap, there is a trap level, So we call it E's of t. Now these guys can do various things. It can capture electron from conduction band, or if it originally had an electron in its state, it could emit electron back to the conduction band, or it could capture holes, or emit holes into the valence band. So, these are the four processes, transitions, that are possible involving these defect states. Now, let's first consider the first process, Electron Capture. Now, in order for the electron capture to happen, we can imagine that these four parameters, four factors will influence that process. That is, first, there has to be an electron in the conduction band in order for the trap level or defect level to capture an electron. So, the density of electron or concentration of electron in the conduction band should be a factor. And the localized state, the defect state must be empty, if it is already filled with electron, then it cannot capture another electron because of the Pauli's exclusion principle prohibits two electrons to occupy the same state. So, the localized state must be empty, so the density of empty defect states should be a factor. And then, there is an electron, there is an empty state, that doesn't mean that capture will occur automatically, there is a certain probability associated with that process. So that probability that an electron passes nearby and eventually captured by the defect state, so that probability must be multiplied to it. So, the density of empty defect states is the density of the trap states or defect states times one minus f. F is the familiar Fermi-Dirac probability function, so one minus f represents the probability that electron is not at that energy. So, this product Nt times one minus F represents the density of empty localized states. The probability of electron capture is represented by this product, Thermal velocity times Sigma N which we call the capture cross section. So Vth is the thermal velocity of an electron. So this is an average velocity electron travels and sigma sub n is a cross section that represents the effectiveness of the capture process. So, if the Vth times sigma n this product represents a cylindrical volume, if you imagine these cross section is a circular cross section, then this Vth sigma n product represent a cylindrical volume, and the volume is defined by this cross section particle with this cross section moving at a speed of Vth per unit time, it will propagate a certain distance, and that volume, that cylindrical volume is equal to this product here. So, if an electron happens to be within this volume, or if a localized state happens to be in this volume, then your electron is captured. And this capture cross section can be calculated quantum mechanically, but it's a very complex process and a lot of time you don't really know the details, all the details of the trap states, these localized states, what kind of impurity it is and what kind of defect it is. So usually, it is determined by experiments. There are some techniques such as deep level transients spectroscopy that is quite effective in measuring this, and the typical value for this capture cross section is anywhere between 10 to the negative tenth, to 10 to negative fifteenth square centimeters. So, combining all these, you can express the capture rate R1 as the electron concentration times the density of empty trap Nt times one minus F, times the thermal velocity of electron times the capture cross section. The emission process, electron emission process is an inverse process of electron capture. So, you can construct the equation for R2 emission rate the same way, and the Nt is the density of the defect states, localized state, times this time f instead of one minus f. So, this Nt times f represents the density of occupied trap states, and then, for an occupied trap state, there is a finite probability that it will emit electron into the conduction band. So if you multiply that emission rate here E's of n, then you get the total transition rate, the total rate of electron emission from defects into the conduction band. Now, at thermal equilibrium the R1 and R2 rates should be equal to each other so that there is no net change in electron concentration in the conduction band. So, if you equate these two equations together, R1 and R2, expressions that we derive for R1 and R2, equate them together, then you can solve for the emission rate for electrons from a defect state, then it is given by this. Now, these are N's of i, Vth, sigma n. These are all some specific constants, specific to the material or the type of impurity or defect, but here its the exponential term is illuminating because the exponent has the Et, the defect energy trap energy minus Ei, the intrinsic formulable which is located in the middle of the band gap. So if your Et trap level is far away from from the mid gap, so in this case, closer to the conduction band, then emission rate is high. If its closer to the middle of the band gap, then the emission rate is low. So you can see that why deep impurities, impurities with an energy level deep inside the electron are more effective in holding on to electrons. And if shallower impurity, if the impurities are shallow impurities, that is the energy level is closer to the band edge, then they are highly likely to emit electrons into the conduction band and stay unoccupied. And these type of shallow impurities will not severely impact the conduction properties, electrical properties of semiconductor. However, the deep impurities which tend to hold on to electrons can significantly impact the carrier concentration itself also carrier dynamics, thereby, impacting the electrical properties of semiconductor quite profoundly. And we can do a similar analysis for hole capture and hole emission process, R3 and R4 in the diagram shown in the very first slide. And you can write down the emission rate for hole as this, and again, we have the same exponential factor this time Ei minus Et. So, if the trap energy level defect energy level is closer to the valence band, then the hole emission rate is high, the trap level defect level do not hold onto holes and do not really impact the hole conduction too much. But if the trap energy level E's of T is closer to Ei, then the hole emission rate is small and they tend to hold on to holes impacting the electrical properties due to hole conduction. Now, at equilibrium, we must have R1 is equal to R2, electron capture rate should be equal to electron emission rate and also R3 equal to R4, hole capture rate is equal to hole emission rate, so that there is no change in carrier concentration. In non-equilibrium situation of course, then these rates can be different and therefore, your carrier concentration will deviate from their equilibrium value. So there we can consider two cases. If both electron capture rate and hole capture rates are high, R1 and R3 are high, while the emission rates of electrons and holes R2 and R4 are small. What does that mean? That means, these defects, these localized states tend to capture carriers effectively but they don't really emit it back, they hold onto those carriers. And these type of the localized states are called the recombination center because if you capture an electron, and simultaneously capture a hole, then the electron and hole knit at the defect energy level and disappear, and that's a recombination process. So this type of state, localized state or defect state is called the recommendation center, but there are other cases where only R1 and R3 is substantial. So, it captures electrons effectively and not holes, or it captures holes effectively but not electrons. In those cases, you don't have recombination but these defect levels will then hold onto those electrons, and those type of states are called the trap states because carriers are trapped in their energy state.
