In this video, we will discuss step junction. Step junction is a pn junction in which the p-type material and n-type material has an abrupt planar boundary and both the n-side and p-side has a uniform doping profile so that if you plot the doping profile, then it looks like a step function, hence the name step junction. We adopt the depletion approximation which says that inside the depletion region, there are no mobile carriers. So, the carrier concentration is zero within the depletion region that builds up across the pn junction, across the junction. We denote the pn junction edge boundary on the n-type side as X sub n and the depletion region edge on the p side as negative X sub p. The depletion approximation is that this boundary is well-defined. So, suddenly, the carrier concentration changes from zero to the equilibrium majority carrier concentration, which equals the doping density at X equals X sub n. Same thing for the p-side, at X equals negative X sub p. The carrier concentration jumps from zero abruptly to the majority carrier concentration in the neutral p-type region, which is equal to the doping density N sub of A. Now, if you plot the net charge density then, then outside the depletion region in the quasi-neutral region, your charged density is zero. Inside the depletion region, you have a net charge density non-zero. On the n-side, you have a positive charge due to ionized donors. On the p-type side, you have a negative charge due to ionized acceptors. So, if you write out the Poisson's equation, which allows us to calculate the electrostatic potential and also electric field from any given charge distribution, then within the depletion region, because your carrier concentration is zero, p and n are both zero, the charged density is determined solely by the doping density, the net doping density, that is. If you write out these equations for n-type side of the depletion region and the p-type side of the depletion region separately, and we're going to look at the n-type side first, between zero and X sub n, then you only have donor here. So, your second derivative of potential, which is the left-hand side of your Poisson's equation, and that is equal to the negative first derivative of the electric field according to the definition of your electrostatic potential, and that is equal to the negative two times N sub D, this is the net charge density on the n-type side, divided by the permittivity of the semiconductor. You can solve this first-order differential equation easily. Because everything on the right-hand side is just constant, you get a linear function. We use the boundary condition that your electric field goes to zero at the edge of the depletion region. Why? Because outside the depletion region, you have zero charge and therefore, naturally, you expect no electric field. So, at the boundary transition point to this neutral region, your electric field should reach zero, and that's our boundary condition, and that gives us this equation for the electric field on the n-type side of the depletion region. You can do the same exercise for the p-type side of the depletion region, negative X sub p to zero, and you get this equation here. Once again, the boundary condition is that at X equals negative X sub p, your electric field goes to zero. So, what do we have now? We have an electric field profile that looks like this. So, it's a linear function, a straight line on both p-side and n-side, with an opposite sign for the slope, negative slope for the p-side and positive slope for the n-side. At the edge of the depletion region, at X equals negative X sub p, at X equals X sub n, they go to zero. At the boundary, at the junction, they should be continuous. If you plug in X equals zero on this equation and also for the equation in the previous slide, and set them equal to each other, then you get this equation here which is simply a charge neutrality condition. That is, the total negative charge building up on the p-side of the depletion region should equal to the total positive charge building up on the n-side. Now, we have an expression for electric field, and we can continue on. By integrating the electric field one more time, you can get the potential. So, the potential on the n-side is obtained by integrating the electric field equation one more time. The boundary condition here is that at X equals X sub n, the potential should be equal to the potential of the neutral n-type region, V sub n, which we derived in the previous video. Then, similarly for p-type region, you get the potential by integrating the electric field equation for the p-type depletion region. Once again, the boundary condition is that at X equals negative X sub p, it reaches V sub p, which is the potential for the neutral p-type region. Of course, the difference between the two, V sub n and V sub p is the built-in potential, and this was the equation that we derived in our previous class as well. Now, use the charge neutrality condition. Also, the potential drop on each side of the equation should be equal to this. Where do these conditions come from? It comes from the fact that the the potential drop on the n-side should simply be equal to the integration of the electric field from zero to X sub n. Likewise, the potential drop on the p-side of the depletion region should be equal to the integration of the electric field from negative X sub p to zero. These two equations combined and plug that into the previous equations. They allow us to derive an expression for the X sub n, depletion region width on the n-side and the depletion region width on the p-side, X sub p. X sub n and X sub p is equal to X sub D, which is the sum of X sub n and X sub p. So, X sub D is the total depletion region width, and you can see that the X sub n, the depletion region width on the n-side, is proportional to the doping density on the opposite side. X sub p, likewise. E is proportional to the doping density on the n-side. Combine these two you get an expression for X sub v the total depletion region width, and the total depletion region width is proportional to the square root of the inverse of the doping density here, and also square root of the built-in potential. Now, if all of these equations are the results for equilibrium pn junction, if you apply a voltage to the pn junction, apply either Forward Bias or Reverse Bias, all you do is to change the potential difference between the n-side and the p-side. So, all you have to do is to replace V sub i here, they'll be built-in potential, with V sub i minus Va, va is the applied voltage. So if Va is positive, which is forward bias, then you are reducing the potential barrier between the two sides. If you are applying a reverse bias, Va is negative, then you're increasing the potential difference between the two sides, and how the carriers and depletion region respond to that dictates how the pn junction behaves under the bias which we will discuss in much more detail later. Now, before we finish, I want to have a few words on the Depletion Approximation. Now, the Depletion Approximation obviously is an approximation because it assumes a very unnatural, abrupt transition between the depletion region and the quasi neutral region. In reality, you should expect that there is a gradual transition from depletion region into the quasi neutral region. So, let's consider a potential near the n-type depletion region edge, X sub n. We define V sub v prime as the deviations of the actual potential from the neutral n-type region potential. If you're very close to the X sub n, the depletion region edge, this number should be very small. Now, the Poisson equation in this case, in the depletion approximation, we ignored n, the carrier concentration altogether, but if we relax that approximation, then we actually cannot ignore this and we have to include the carrier constant non-zero carrier concentration, and the carrier concentration is related to the potential through an exponential function. So, from this, you can simplify this into this equation here. Now, this is something that needs to be solved numerically, but if E prime, this definition here is very small, and it should be very small if you're really close to the edge of the depletion region, then we can expand this exponential function into a Taylor series and retain only the first order term linear term, and that allows us to rewrite the Poisson's equation into this form. Now, this is a second order differential equation, and the solution to this is an exponential function. So, what you see is that right at the edge of the depletion region, the potential actually varies exponentially, and this is a very fast varying function, and what that means is that the potential approaches the neutral region potential V sub n very rapidly exponentially. The characteristic length that defines the transition region, how fast is your potential approaching the neutral region potential is characterized by this quantity L sub D here, and this L sub D is defined here, and this is called the Debye length. The Debye length is a characteristic length that usually you used to define a mobile carrier distribution. So, to get how accurate or how inaccurate our depletion approximation really is, just consider a simple example of symmetric step junction, symmetric meaning that the doping density on the n-side and the doping density on the p-side are equal at 10th to the 16th, rather the mild doping density, and within the depletion approximation, the depletion region width turns out to be 210 nanometer on each side. So, the total depletion region width will be four 420 nanometer, and your Debye length is about 40 nanometer, so, about 10 percent. So, the depletion approximation you can say in this case is accurate within 10 percent which is pretty reasonable approximation, but depending on what you want to accomplish, if you need a better accuracy than that, then you have to consider the more rigorous approach rather than a simple Poisson's equation within the Depletion Approximation. So, if you relax the Depletion Approximation, the real carrier concentration and the charge density has a gradual transition as opposed to a short abrupt transition, and this transition region width is typically characterized by the Debye length once again. So, once again the electric field doesn't go to all the way to zero, is a straight line but rather has a gradual change at the edge of the depletion region as shown here. Same thing for the potential but it is not as pronounced because the potential approaches the neutral value quadratically to begin with.