In this video, we will discuss base-emitter current. Now, in the previous video, what we calculated or we derived is an expression for electron current flowing through the device. So, electrons are injected from the emitter into the base and out to collector. So, that is the collector current as shown in this diagram. So, this diagram here shows all current components that can flow in your BJT. So, the current that goes from collector through base out into emitter, which represents the electrons flowing from emitter through base and out to collector, that component is B component, B component is the collector current i sub c. There are additional current that can flow in the device but not reach the collector, and that is a collector flowing only between emitter and base as shown here, and there can be three different components here. But before we discuss these individual components that make up the base-emitter current, notice that the collector current here depends on the voltage, forward-biased voltage between emitter and base. So, your emitter and base represents the control terminal, and your collector terminal represents the output terminal. So, any current that flows between base and emitter represents an input current, and in order to have a small input power, you would like to minimize the current between base and emitter. That obviously leads large current gain and therefore, a desirable transistor performance. The base-emitter current has three components as I mentioned here. So, they are the recombination of injected electrons with holes in the quasi neutral region, so that's this component here. Electrons are injected and recombine with holes in the base. Second component is the recombination within the depletion region or space charge region between base and emitter, and the last component is the hole injection the base is a p-type. So, holes from p-type will be injected into n type. So, there is a hole injection current. These are the three components. Most transistors that are based on silicon have a very very small recombination in the space charge region. Silicon is really pure, very very high quality. So, the recombination due to any defects or anything like that can be ignored. So, we can ignore the second component the recombination within the space charge region. The recombination within the base region can be expressed by this equation here. So, this is basically the integration over recombination rate, so this numerator here, n, is the actual carrier concentration, minority carrier concentration and the ni squared divided by NA is the equilibrium minority carrier concentration. So, the numerator here simply represents the excess carrier concentration, and divide it by the carrier lifetime Tau sub n is a recombination rate. So, if you integrate the recombination rate across the base region, that is the total current recombination current. In order to minimize this recombination current, obviously, you want to minimize the base region. So, make the base region very narrow, very short. In the case of uniform doping, you can write down the expression for these excess minority carrier concentration. That will be simply given by this, and so you can do the integration and you end up with this simple equation for the recombination current. The loss of carriers in the base region due to recombination is characterized by a parameter called base transport factor, and we write it as Alpha sub t. It's defined as the total electron injection minus the recombination, divided by the total electron injection. So, this basically the numerator represents the part of the current that survived that did not recombine divided by the total injection current. So, you want this number to be as close to one as possible. In the uniform doping case, we derived an expression for isobar b in the previous slide, use the equation that we derived in the previous video, and you can write down a simple expression for the base transport factor as shown here. You can obviously improve this by reducing x sub b the base region width, and it also depends on the diffusion length of the minority carriers. This factor is once again, for the uniformly doped case, and it will be better if you do a non-uniform doping. Now, the last third and last component of the base-emitter current is the hole injection from base, and this is usually the biggest factor in typical transistors. The hole injection current, we already derived this expression for the ideal p-n diode so we just take that from the p-n junction discussion. If the emitter region is long, long compared to the diffusion length of the holes, then we use diffusion length here in the denominator. If the emitter region is very short, then of course, we need to replace this with the actual emitter region width x sub e. In most integrated circuit, the emitter length is small but they are usually doped non-uniformly. So, when you have a non-uniform doping, you will have a built-in electric field. Because of the built-in electric field, you will have both drift current and the diffusion current within the emitter region. So, first, you need to know the built-in electric field due to non uniform doping, and if you recall our discussion on the non-uniform doping, the built-in electric field is proportional to the gradient of your doping density. So, when the base emitter junction is forward-biased, then the total field within the emitter region is this built-in field plus any additional field due to you're biasing. So, the total hole current can be expressed by this drift current where the carrier concentration will be increased from the equilibrium value by this extra excess current due to hole injection and your electric field of course is the built-in the field times any additional field due to biasing, and then the second term here is the diffusion current. Once again, we separated out the carrier concentration into the equilibrium, plus the excess. So, you can simplify this into this equation here and you can rewrite this equation for the hole current as a derivative of the excess hole concentration times the doping density in the middle region. So, when the emitter length is very short compared to, once again, the diffusion length of holes, then the hole current will be independent of the position just like the short base diode, and so we can integrate from x of n which is the depletion region edge of the emitter side, and then x sub e which is the end of the emitter region, then you can obtain this equation here and we use the result of the step junction from which we remember this excess carrier concentration is simply depends exponentially on the applied voltage, forward-biased voltage. So, from this we finally arrive at this expression for the current within a hole current in the emitter due to hole injection from base into emitter. Now, the effectiveness of emitter junction in injecting electrons. So, you want your emitter base junction to primarily inject electrons and not really inject any holes on the other direction. So, the effectiveness of that is characterized by emitter injection efficiency. Gamma is defined as the electron injection current divided by the total current, which is the sum of electron injection current plus the hole injection current. From the equation that we derived here, your Gamma can be written here in the case of the uniform doping and it depends on the ratio of the diffusion length, but also, it depends on the ratio of the gamma number or the doping density in the base and the emitter region. Finally, we know two possibilities for deviation from the equation that we have derived here for the emitter injection efficiency. First is when you are doping is really heavy, then there is an effect called the bandgap narrowing. So, due to the larger density of dopants, the effective bandgap of your semiconductor becomes smaller. When your bandgap is smaller, then obviously, the intrinsic carrier concentration is larger. Therefore, that leads to a higher hole injection from base into emitter. Also, when your carrier concentration is very high, then there is an additional recombination mechanism called OJ recombination that becomes very very effective. So, this leads to a very large recombination rate and that leads to the breakdown of the short diode approximation that we have used in deriving the hole current in the emitter region. If you remember, the short diode approximation is to ignore any recombination in the region, and that's a good approximation when your primary recombination mechanism is the Shockley-Hall-Read recombination. But if you have something like all the recombination, which becomes very efficient at high doping density, then obviously that assumption can break down and the emitter injection efficiency will deviate from the nice expression that we derived in the previous slide.