 In the last few classes, we have looked at electronic devices. So, we first started looking at a metal semiconductor contact, forms your short key junction or an ohmi junction. We then looked at PN junctions and then finally transistors. We saw that there were different kinds of transistors, but we focused on the metal oxide semiconductor field effect transistors because these are the ones that are commonly used in your IC circuits. In the next few classes, we are going to look at devices where there is interaction of light with the electrical properties of the device. So, we are going to look at optoelectronic devices. In these kinds of devices, we have essentially two categories. One, where you have an incident light which then create carriers in your material. So, a classic example of this is your solar cell, where the incident solar radiation is absorbed in order to give you current. You also have the other type where you inject electrons or holes into your device which then recombine in order to give you light. So, this the example would be an LED, a light emitting diode or a laser. So, before we look into these devices, let us first start by looking at the interaction of light with matter. So, the first thing we are going to look at is the phenomenon of optical absorption. So, we want to know what happens when light of a certain energy is incident on a semiconductor. In the case of a semiconductor, we define its energy by the band gap. So, EG which is the band gap of the semiconductor is nothing but the energy difference between the valence band and the conduction band. We are going to say that your semiconductor is ideal so that there are no energy states between the valence band and the conduction band. So, in this particular case, if you have light of certain energy incident on the semiconductor, if E which is the energy of the light is less than EG, then the semiconductor is said to be transparent. Of course, whenever you have interfaces, you always have scattering. So, you could have scattering of light from the surface, your semiconductor is polycrystalline. You could have scattering from the various interfaces, but overall there is no optical absorption per se. On the other hand, if E is greater than EG, then we say that your semiconductor is opaque. It will essentially absorb the light. When this thing happens, when you have incident light of energy greater than EG and light gets absorbed, it produces electron hole pairs. You can also abbreviate this as EHP, just means electron hole pairs. So, you have electrons that are produced in the conduction band and holes that are produced in the valence band. So, we have looked earlier at the concept of density of states and the Fermi function. We found that at room temperature, most of the electrons are located close to the conduction band edge and most of the holes are located close to the valence band edge, so that when you have your electron hole pairs being created, they will essentially thermalize to the edges of the valence band and the conduction band. So, let me just explain this by drawing a small schematic. So, I have my semiconductor in which I mark a valence band and a conduction band. EV is your top of the valence band and EC is the bottom of the conduction band, so that this gap is EG. So, now we have light that is incident on the semiconductor. The energy of the light is less than EG, then it is not going to be absorbed. So, if I have light of energy h mu where mu is the frequency equal to EG or slightly above EG, it is very close to the band gap. In this case, an electron goes from the valence band to the conduction band, so let me show that here. An electron goes from the valence band to the conduction band and electron goes creates a hole. So, you have an electron at the bottom of the conduction band and a hole at the top of the valence band. Now, if h mu is greater than EG, if h mu is greater than EG, which case light has an energy greater than the band gap, there is some excess energy that is left. So, this excess energy can either be manifest in the electron in the conduction band or the hole in the valence band or both. But in this case, the electron or the hole loses this excess energy by a process called thermalization where the energy is lost to the lattice and then it goes to the edge of the valence and the conduction band. So, this excess energy is lost by thermalization that is it is just transferred to the lattice. So, if I want to show that I will use the same diagram, but now I have h mu that is greater than EG. So, in this particular case, I can have a hole at the top of the valence band, but my electron can be somewhere within the conduction band. But this electron will lose energy and then ultimately come to the bottom of the conduction band. So, you start with the hole at the top of the valence band, because h mu is greater than EG, your electron goes not to the edge of the conduction band, but to some energy higher and the excess energy is lost by thermalization. You can also have a situation where you have the electron at the edge of the conduction band and you have a hole that is deep within the material and the hole loses energy and then goes to the edge of the valence band, so that is also possible. So, the point is we have optical absorption whenever the energy of the incident radiation is greater than or equal to EG, usually when h mu is just equal to EG, you have the electron at the edge of both bands, but the density is of states at the edge is 0, but we saw earlier, then even if you are k T above EG and k T at room temperature is just 25 milli electron volts, you have a large density of states. So, even for energies slightly above EG and very close to it, you will still have optical absorption. If I is the intensity of the incident radiation, I can be given in units of watts per unit area or watts per unit volume, so this is let me say watt per unit area, which is nothing but the power that is delivered to the sample and if h mu is the energy of the incident radiation, mu is nothing but the frequency, you can also write energy as h mu or h c over lambda, where lambda is the wavelength, so whether you know the frequency or the wavelength can find the energy, so h mu is the energy of the incident radiation, this is usually in joules or electron volts, then the number of photons that are incident on your sample at the surface, so pH is the number of photons, so this incident at your sample is nothing but i over the energy, so over h mu, so this has the units of a number per unit area per unit time, so s is in seconds area could be centimeter square or meter square, where is the number of photons that are incident on your sample. Again if these photons have energy greater than the band gap, they are going to create your electron hole pairs, so this process can be 100 percent efficient, which means every incident photon will generate one electron hole pair or the efficiency can be less than 100, so we talked about the absorption of light by a material, specially a semiconductor and we say that light gets absorbed when the energy is greater than band gap, so this we can quantify by introducing a term called the absorption coefficient, so we define a term alpha, where alpha is called the absorption coefficient, it has units of centimeter inverse or length inverse, so it could be centimeter inverse or meter inverse and so on, so if alpha is the absorption coefficient and then i naught is the incident intensity at the surface and x is the depth within your material, then the intensity i at some depth x is i naught exponential minus alpha x, so the intensity drops exponentially as you go deeper within your sample, this is called the beer Lambert law, so this is true whether you have incident infrared radiation or visible light or UV or x rays, in all of those cases you are going to find that the intensity goes down exponentially as you go within your sample. One over alpha which has a units of length is called the penetration depth, so that at this distance i is nothing but i naught over e, which is 0.37 i naught, which means your intensity has dropped to 37 percent of the original intensity i naught, we can relate this to the band gap of a material, so that alpha is actually a function of the wavelength of light, because wavelength relates to energy and the energy can be compared to your band gap, so if you look at a semiconductor alpha is small if the energy is below e g and then alpha becomes large once your energy exceeds the band gap, we can show this by plotting the absorption coefficient as a function of energy, so in this case I have alpha on a y axis, which is in per micrometer and I am going to use a log scale, so I have 0.1, 1, 10, 100, let me just expand the scale a bit 1000, on the x axis I am going to plot the energy of the radiation in electron volts, so 0, 1, 2, 3, 4 and 5, so we will start with silicon, so silicon is an indirect band gap material and its band gap is around 1.1 electron volts at room temperature, so below 1.1 the absorption coefficient is very small and then above 1.1 it starts to increase, so this here represents the band gap, so that you have very low absorption below the band gap and very high absorption above it, you can do a similar plot for gallium arsenide, the difference between silicon and gallium arsenide is that silicon is an indirect band gap semiconductor, in this particular case we have seen earlier, if you want an electron transition from the valence band to the conduction band, the transition needs some lattice vibration or phonons to help it, which is why you have a small increase initially when the energy is more than the band gap, to make a similar plot for gallium arsenide. So, gallium arsenide is a direct band gap semiconductor with the band gap of 1.4, so initially when the energy is below 1.4 the absorption coefficient is really small, but then it starts to increase rapidly once your energy is above the band gap, so this is gallium arsenide. So, this is a direct band gap semiconductor, so that compared to silicon the rise in the absorption coefficient is really abrupt for gallium arsenide. So, so far we have talked about absorption, but once your electron and hole pairs are generated, these are dynamic, so they tend to recombine, in the case of a direct band gap semiconductor when the electron and hole recombine, the majority of the dominant mechanism by which energy is released is in the form of photons or light. So, that once the electron and hole recombines you get light, whose wavelength depends upon the band gap of the material, so in the case of gallium arsenide the energy of the light that comes out will be 1.43, in the case of silicon which is an indirect band gap material, the recombination takes place with the dominant mechanism being heat, so that the energy is lost in the lattice as lattice vibrations. So, let us now look at some numbers where we have an electron hole pair being generated in a semiconductor due to incident light. So, consider the case of an n type semiconductor for simplicity, I am just going to take the material to be silicon. I have n type silicon with my donor concentration equal to 5 times 10 to the 16 per centimeter cube. We will take the silicon to be at room temperature, so that the intrinsic carrier concentration is 10 to the 10. So, if you have the semiconductor in equilibrium, we know that n is equal to n d, this is 5 times 10 to the 16, p is just n i square over n and if you do the numbers, p is just 2 times 10 to the 3 which is much smaller than n. So, in this case we also know that n p is n i square, your law of mass action is valid which is how we calculated the concentration of p in the first place. So, in this n type semiconductor, I am going to illuminate it uniformly with light, so that I generate electron hole pairs. So, I am going to illuminate uniformly, so when I say uniformly it means that I have electron hole pairs generated within the entire material. I am not worried about gradients within it, later we will see what happens if we have gradients and the energy h mu is greater than e g, so that we have electron hole pairs. So let delta n and delta p be the excess electrons and holes and these are equal because really you have an electron and a hole pair being created at the same time. So, delta n and delta p are the excess, we define a condition which we call weak illumination in which case the excess majority carriers which is electrons is less than the carrier concentration in the bulk of the material. So, that delta n is less than n or delta n is less than n d. So in this particular example let us take delta n equal to delta p equal to 5 times 10 to the 15 per centimeter cube. So, these are the excess and this number is less than n d. So, if we have this your new electron concentration is let me call it n prime which is the nu is n plus delta n. So, n is 5 times 10 to the 16 the excess is 5 times 10 to the 15. So, the new concentration is 5.5 times 10 to the 16 which is approximately at 10 percent increase. On the other hand your new hole concentration p prime is p plus delta n, but your p originally is very small is 2 times 10 to the 3. While the number of carriers you have generated is 10 to the 15. So, it is p plus delta n which is the same as delta p. So, it is equal to 5 times 10 to the 15. So, you have an increase by 12 orders of magnitude. So, the increase in the case of illumination by light is always in the minority carriers. So, this had been a p type semiconductor the argument would have been reversed we find that there is a increase in the amount of electrons a large increase in the amount of electrons compared to holes. So, that is the increase is always in the minority carriers. When we turn off the light we have all of these excess carriers and when we have excess carriers we will find that n prime p prime is not equal to ni square because your system is not in equilibrium. So, now when I turn off my light all these excess carriers have to start to recombine. So, the real increase in concentration is in that of the holes. So, this recombination is driven by the lifetime of the holes. So, it is driven by a quantity which we call tau h which is the minority carrier lifetime. So, tau h determines how long does it take for the electrons and holes to recombine. So, that we go back to the equilibrium situation. So, it is possible to write an equation where we link the excess carriers that are generated to the amount of electron hole pairs that are created in the amount that is lost to recombination. So, let delta p be the excess holes. So, we are still looking at an n type semiconductor but if you are looking at a p type then it will just be delta n. So, let delta p be the excess holes that are created and let g p h be the rate of generation of electron hole pairs. We also defined a minority lifetime tau h. This is your whole lifetime. So, we can write an equation that links d delta p over dt which is the rate of change of the minority carriers or the excess minority carriers which is equal to how many that are generated minus how many holes that are lost due to recombination. So, this term is the rate of change of minority carriers. This one is the generation of new carriers. This one represents the recombination term. So, let us consider an example in order to understand this better. So, consider a case where I have a n type semiconductor in equilibrium. So, this is time on this axis. I am shifting the 0 slightly to the right. So, that we can look at things more clearly. So, you have a system that is at equilibrium. At time equal to 0, we turn on the light. So, that we have some electron hole pairs being created. We let the light be on for some time. So, that the system achieves some steady state and at time equal to t of, we turn off the light. So, you have illumination within this region. So, the question we want to know is what happens to the minority carrier concentration in this process. So, when there is no light, initially p and not, which is just the concentration of holes in your n type semiconductor is ni square over n d, which is much smaller than n d. So, now I have the light being turned on. So, that my time is greater than 0, but it is before it is switched off. So, it is before t of. So, in this case, I can write this equation d delta p n by d t. So, I will just take this equation. This is g p h minus delta p n over tau n. So, this is just a first order differential equation. We can integrate it and the boundary condition is at time t equal to 0 delta p n is 0. There are no excess carriers in your material. So, if I solve for this delta p n at time t and this t is between 0 and t of is nothing but tau h g p h 1 minus exponential minus t over tau h. So, when the system is reached steady state, your d p n over d t is 0. So, at steady state, the concentration of holes is nothing but tau h g p h. So, now I am going to turn off my light. We will see what happens. So, now I am going to turn off the light. So, that g p h is 0. There is no more generation of electrons or holes and the equation just reduces. This again we can solve. It is a first order differential equation. So, delta p n, so from t of, so t of is when the light is turned off is nothing but tau h g p h exponential minus t minus t of over tau h. So, this information we can put together and plot. So, this is time. So, we will first replot the light. So, we said that at time t equal to 0, we turn the light on and at time equal to t of. So, this is your illumination. So, this is g p h. So, what happens to the concentration of the holes? So, initially your whole concentration is just the equilibrium concentration. When you turn on the light, the concentration starts to increase and then it becomes a constant when you have steady state. The steady state value is g p h times tau h and then when I turn the light off, it decays exponentially until it reaches back the steady state. So far, we have looked at the case of light interaction where you have a uniform illumination. So, it is possible to generalize this argument where we have a variation in concentration both along the depth of the material and also as a function of time. So, we call this the continuity equation. So, let us take a look at it next. So, here we want to write a generalized equation for the interaction of the light with matter and how it generates electrons in holes. So, consider the case of a slab. Again for simplicity, you are going to deal with an n type material under weak illumination. So, that you can ignore the change in the majority carriers. You only change significant change will be in the minority carriers. So, I have a small portion of this material which is at a certain depth x and then I consider a small increment x plus delta x. So, this is just a portion of your slab. So, this direction is my x direction, this direction is the y direction. So, I have some flux of holes that is entering into the small incremental slab at x. So, I am going to call it j h which is the flux of holes that is entering the small portion and I have a flux that is leaving. I am going to call it j h plus a delta j h. So, delta j h can be positive or negative if it is positive. We have some generation of holes within the slab. If it is negative, we have some loss of holes. We also have some incident light g p h which creates electron hole pairs and we also have a situation where we have recombination of electrons in holes. So, in this process generates new holes and the recombination basically leads to loss of new holes. So, in this particular case the rate of change of hole concentration within this portion delta x. So, rate of change of hole concentration within delta x. So, delta x is a small region of the slab is nothing but 1 over a delta x. So, a is the area. So, that a delta x is the volume minus a delta g h over e. So, delta g h is the change in flux as you go through the region. So, a will essentially cancel. So, delta j h according to this equation is negative and this whole thing is positive. There is an increase in the hole concentration. On the other hand if it is positive, there is a decrease in the hole concentration. So, this just denotes the rate of change of hole concentration. So, we can relate this to how the concentration of holes actually change. So, d P n or dou P n over dou t is nothing but minus 1 over e dou j h over dou x. So, it is just this term that is written here minus the holes that are lost due to recombination. So, P n minus P n naught which is your excess holes. So, divided by dou h plus the holes that are generated which is g P h. So, this equation is called your continuity equation. So, it is similar to a diffusion equation that you will write in the case of diffusion of elements within say a metal or an alloy. In this particular case P is a function of both x and time. So, that the concentration of holes changes with time because you have generation and recombination and it also changes with distance because you have some flux within the material. So, the continuity equation is a generalized equation. When we think of say uniform illumination which we just saw before. So, if you have uniform illumination in that particular case there is no flux within the material. So, that dou j h over dou x is 0. So, P will only change as a function of time. So, that you have d P over d t is g P h minus delta P over tau h. So, this is the equation that you wrote. So, we get back the original equation that we wrote. So, uniform illumination is then just a specific case of the generalized continuity equation. So, let us look at another case where instead of uniform illumination you just have illumination at the surface. So, consider the case of a steady state continuity equation where I have an n type semiconductor. I have incident light only on the surface. There is always going to be some absorption. So, I depict a small region of with x naught in which the light is absorbed. So, this is an n type material. We know that we are going to get some excess concentration of holes. I am only showing the holes. There are also electrons and these holes will essentially diffuse within the material. So, there are also electrons in here, but I am only showing the minority carriers. So, you have light incident on the surface absorbed within some distance x naught which are creating your electron hole pairs which then diffuse through the material. So, we are considering the system at steady state. So, that dou delta p n over dou t is 0. We also have a system of illumination only at the surface. So, that g p h is 0. So, we can go back and rewrite the continuity equation. In this particular case the equation simplifies h over dou x is minus delta p n over tau n minus delta p n over tau h. So, you have a hole current that is set up mainly due to diffusion. There is a generalized equation which relates the whole flux to both diffusion current and drift current. The generalized equation j h is E p mu h epsilon x minus E d h d p over dx. So, the first term is due to drift of the holes within an electric field. The second term is due to diffusion. If we ignore the drift current so that this part we ignore, we can equate this flux to this equation so that when we differentiate what we get dou j h over dou x is minus d h. So, in this particular case for weak illumination we can write an equation for the change in the concentration of holes as we go along this length. So, let me just mark this to be x. So, that delta p n is delta p n naught which is how many electron hole carriers are generated at the surface times exponential minus x over l h. So, l h is called the diffusion length which is just diffusion coefficient times the recombination time. So, l h is the diffusion length. So, delta p n naught which is the number of carriers that are generated at the surface is equal to x naught g p h square root of dou h times exponential minus d h. So, it is related to how many carriers are generated within the region x naught where your light is absorbed. So, we can write a similar equation for the electrons and then we can plot how this carrier concentration goes as a function of distance. So, for electrons you can write a similar equation says del n of x which is your x's electrons that are generated naught exponential minus x over l e where l e is the diffusion length of the electrons. So, delta p n naught which is the number of holes generated at the surface by delta n n naught is nothing but square root d e over d h. So, delta n n naught is x naught g p h square root of dou h over d e. So, if you plot these x's concentrations as a function of distance again in the case of majority carriers the change in concentration is not much. So, this is delta n n naught going as a distance within the material on the other hand for a minority carriers the slope is much more. So, this is delta p n naught. So, this is what happens when you have illumination at the surface you have electron and hole pair generated which then diffuse within the material in order to give you a diffusion current. So, today we have looked at some aspects of the interaction of light with the semiconductor material. So, starting next class we will start to look at some devices and examples of where we can use this in the case of electronic materials.