 Let us start with a brief review of last class. In last class, we continued discussing about extrinsic semiconductors and if you remember extrinsic semiconductors are those where we add a small amount for specific impurity in order to selectively increase the concentration of either electrons on holes. Towards the end of last class, we looked at the temperature dependence of the majority carrier concentration. So this would be electrons in the case of n-type semiconductors and holes in the case of p-type semiconductors. We use the example of an n-type semiconductor and whatever discussion that we do using n-type, the same is valid for a p-type. So in this case, electrons are the majority charge carriers and there are essentially two sources of electrons. One, your electrons can come from the donor levels. These levels are close to the edge of the conduction band. At high temperatures, the electrons can also come from the valence band of the semiconductor. Putting these two information together, we did a plot of the log of the electron concentration versus 1 over t. So this is log n is 1 over t. Since this is 1 over the temperature, this represents the high temperature side and this represents the low temperature. In this plot, we said that we could divide an extrinsic semiconductor into essentially three regions. At low temperature, the concentration of electrons is dominated by those electrons that come from the donor level. So if you plot n versus 1 over t, you are going to get a straight line with a slope that was given by the ionization energy. So Ed is the ionization energy of the donor. We also saw that this is typically of the order of milli-electron volts. Since all the donor atoms are ionized, we have a regime where the electron concentration is more or less constant. And then at high temperature, we find that we have electrons that come from the valence band and your extrinsic semiconductor behaves like an intrinsic one. So this is the high temperature behavior and the source is minus e g over 2 k, where e g is the band gap. And this is usually of the order of electron volts. So we have three regimes in the case of an extrinsic semiconductor. The low temperature one, we call the ionization regime because it is dominated by ionization of the donor atoms. Then you have a saturation regime where the concentration of electrons which are the majority charge carriers is nearly a constant. And then we have an intrinsic regime where your extrinsic material behaves like an intrinsic one. We also call the temperatures corresponding to these as T s which is your saturation temperature and T i which is your intrinsic temperature. We did some calculations for the value of T s and T i in the case of silicon with 10 to the 17 donor atoms. And we got a value of T s that was approximately 30 Kelvin and at T i which is approximately 5 A T Kelvin, which means there is a wide temperature range where the concentration of the majority carriers is nearly a constant. And it is equal to the concentration of your donor or your acceptor. So we have two advantages if we dope semiconductors. The first is that the conductivity increases because the carrier concentration increases. But secondly we also have a regime where the carrier concentration is almost a constant and is independent of temperature. So today we will go further and talk about conductivity in the case of an extrinsic semiconductor and how that depends upon temperature. So we start with the equation for conductivity. We have written this earlier sigma is n e mu e plus p e mu h. This is your general equation for the conductivity. If you have an n type semiconductor, n is typically much larger than p. We saw in the case of silicon that n could be more than 7 or 8 orders of magnitude higher depending upon the dopant concentration in which case sigma would just be n e mu e. On the other hand if you have a p type semiconductor p is much larger than n in which case sigma will just be p e mu h. We have seen mu e and mu h earlier, mu e and mu h are the drift mobilities. So mu e is the mobility of the electron in the conduction band, mu h is the mobility of the hole in the valence band. We also saw earlier that mu e sigma e tau e over m e star, mu h e tau h over n h star where tau e and tau h are the time between two scattering events. Now if you want to look at the temperature dependence of conductivity, we need to look at the temperature dependence of the carrier concentration and also the temperature dependence of mobility. We have done looking at the temperature dependence of the carrier concentration which is what we did at the end of last class, the beginning of today. Now we look at how mu e and mu h depend on both temperature and in the case of extrinsic semiconductors how it depends upon the presence of these dopants. Now there is another way, we can write tau which is the time between two scattering events. Tau can be written as 1 over s v thermal times n s. So what are these terms? S represents this cross section of the scatterer v th is the thermal velocity of the electrons and n s is the concentration of the scatterers. If you plug in the units, s is the cross section area which is in meter square, v thermal is the velocity so that is meters per second, n s is the concentration of scatterers it is usually given per unit volume. So this is in meter cube so that the final units is in seconds. So to understand this if you have a larger cross section of the scatterer then you are going to have a shorter scattering time. If the electrons are in travel faster then once again they can interact between two scattering events quickly so the time will be short and if you have more number of scatterers once again the time will be short. Now the question is what are these scatterers that we talk about and how do these values depend on temperature. Let us start first with intrinsic semiconductors. If you have an intrinsic semiconductor then the electrons will scatter because of the thermal vibration of the silicon atoms. So we have thermal vibration of the atoms. So if s is the scattering cross section, s can be written as pi a square where a is the amplitude of these vibrations. So this is in the case of a simple 2D model. We can show that if your scatterers are your thermal vibration of the atoms then s is proportional to the temperature which means as temperature increases the amplitude of the vibrations increase. So a increases and correspondingly s will also increase. The next term in the equation for tau is v thermal. If you have electrons moving in the conduction band we can say that they are moving through a uniform potential region and in this case we can say that the thermal energy of the electrons is approximately the same as the kinetic energy. Typically thermal energy is given as 3 by 2 kT and the kinetic energy is nothing but one half mv square. So this is the kinetic energy. So if we use this expression v thermal is proportional to square root of temperature which means once again as the temperature increases your thermal energy increases. So the kinetic energy will also increase. So v thermal will increase. So let us put these two terms together. So we have the expression for tau which is nothing but 1 over s v thermal ns. In the case of an intrinsic silicon atom we said that the scattering is only due to the lattice vibrations. So we wrote this as 1 over pi a square v thermal and ns. We found that pi a square is directly proportional to 1 over the temperature, v thermal is proportional to square root of the temperature which means tau is proportional to 1 over T times square root of T, ns is the number of atoms and that is usually a constant. It is not going to change with temperature. So if you take these two into account this is proportional to T to the three halves with a negative sign. So tau due to the scattering of the lattice I am just going to denote it as tau L is proportional to T to the minus 3 over 2. I am going to call this now mu is equal to e tau over m and since tau is proportional to temperature minus 3 over 2 can say that mu is proportional to temperature 3 over 2. So if you have an intrinsic material where the conductivity is where the scattering is due to the vibration of the lattice atoms as the temperature increases the vibration increases. So there is more scattering and hence the mobility will go down. So I will write this as mu L to denote that it is due to lattice scattering. So as temperature increases mu L will decrease. So this makes sense because as temperature increases you have greater lattice vibrations and also you have electrons that are traveling faster. Now in the case of extrinsic semiconductors you also have this lattice scattering because you also have the silicon lattice your material is silicon. So you also have the lattice scattering and this is given by your term mu L but you also have the scattering due to your dopants whether they be donors or acceptors. I am going to call that mu i. So i is to denote the fact that these are impurities. So in the case of extrinsic semiconductors you have scattering due to the lattice. You also have scattering due to the dopants. So both of them play a role in determining the mobility of the electrons and holes. The lattice scattering term is very similar to what we did for intrinsic semiconductors. So once again mu L will be proportional to t to the minus 3 over 2. The one thing we have to see is how the dopant scattering changes with temperature. So we want to look at the dopant scattering in an extrinsic semiconductor. We call this mu i. You want to find out the temperature dependence of this term. So if you look at an extrinsic semiconductor, so whether you have an n-type or a p-type you are going to have donors and acceptors and these are ionized. Because you find that in the case of n-type the donor atom has an extra electron that goes to the conduction band in the case of a p-type. You have an acceptor atom that can accept an extra electron and create a hole. So these are ionized impurities and these can then interact with the electron. So if you have an electron with velocity being v-thermal, then the kinetic energy is one half m Me v-thermal square or one half Me star v-thermal square. We said that in the case of an intrinsic semiconductor where an electron is traveling in the conduction band, this kinetic energy is nothing but the thermal energy. So this is 3 half kT but now you have the electron having an electrostatic force of attraction with the ionized donor. So if you have an n-type you have donor atoms or donor ions with a positive charge and these can interact with the electrons. So this electrostatic force I will call it a potential energy is nothing but E square over 4 pi epsilon 0, epsilon 0 R times R. So R here is the distance between the electron and the ionized impurity. Usually in the case of extrinsic semiconductors we define a critical radius Rc where the kinetic energy and potential energy are equal. If R is less than Rc, if R is less than Rc here then the potential energy is greater than the kinetic energy and your electron will scatter. If R is greater than Rc it is the reverse. The kinetic energy will be greater than the potential energy and the electron will not scatter or the electron will escape. In these two terms we define a critical radius when the kinetic and potential energies are equal. If we do that we can say that 3 over 2 kT is equal to E square over 4 pi epsilon 0 epsilon R and Rc square sorry and Rc which means Rc is proportional to T to the minus 1. Now if you want to look at the cross section for scattering S is nothing but pi Rc square and knowing that Rc is proportional to 1 over T, S is proportional to T to the minus 2. So in the case of an extrinsic semiconductor where we are looking at the interaction of the electrons with the ionized impurities we find that the scattering cross section area S is proportional to T to the minus 2. This we got by looking at the balance between the kinetic energy of the electron and the potential energy of attraction between the electron and the ionized impurity. As temperature increases the electron can travel faster which means the kinetic energy becomes higher. So it becomes easier for it to escape the influence of the ionized impurity which means the scattering cross section S goes down. Here we saw that v thermal is proportional to T to the half. This is the same argument that we use in the case of an intrinsic semiconductor. Putting these two together tau due to scattering from the impurity. So it is tau i is equal to 1 over S v thermal ni. Once again ni is the concentration of impurities which is a constant. It is not a function of temperature. So this is proportional to 1 over T to the minus 2 in the case of S and then square root of T is proportional to T to the 3 half. So tau i which is the time due to scattering because of impurities is proportional to T to the 3 half. Mu i is nothing but E tau i for Me star. So mu i is proportional to half which means the contribution of the impurities to mobility is such that as the temperature increases mu i increases. And this is because as the temperature increases your electrons can move faster and so they can easily escape the electrostatic force. So in the case of an extrinsic semiconductor we have two contributions to the mobility. One is from the lattice and we saw earlier that this is proportional to T to the minus 3 half. The other is from the impurity or your dopant and these are ionized. So I will write these as impurity ions where mu i is proportional T to the 3 over 2. So we have two contributions both of which have an opposite dependence on temperature. If you look at the total mobility it will be dominated by whichever scattering process has the lower value. So we can write 1 over mu e as 1 over mu l plus 1 over mu i. So this is in the case of an extrinsic semiconductor where we have contribution due to both the impurity atoms mu i and the lattice mu l. So let us put this together and then do a plot of the conductivity versus 1 over temperature. Then let me rewrite the equation for the conductivity, sigma is n e mu e plus p e mu h. So in this equation the first thing we look at is the carrier concentration. So on a log scale this is 1 over T. Since this is 1 over T this n is the high temperature end. This is your low temperature end. If you plot log of the carrier concentration then this is similar to what we had before. So at high temperature you have an intrinsic contribution. You have a saturation region where the concentration is nearly a constant. Then you have a low temperature region where it is due to ionization of your donors or acceptors. But this is log n or log p. If you then plot log of the mobility we find that at low temperature mobilities are dominated by scattering due to the impurities because at low temperature your lattice vibrations are very small. So the mobility is dominated by impurity scattering and as temperature increases your mobility increases. Another mu i is proportional T to the 3 over 2. If you do that then this is the contribution due to the scattering of the ionized impurities. So this is log mu at low temperatures so it is dominated by impurities. At high temperatures scattering is dominated by the silicon lattice in which case mu l is proportional to T to the minus 3 over 2 so that as the temperature increases mu l goes down. So this is log mu l at intermediate temperatures both the scattering from the lattice and the scattering from the impurities will be more or less equal. So the first dotted line represents n. The second dotted line or dashed line represents log of mu. If we put both these together we will get the conductivity and if you plot the conductivity which is the solid line we get a curve like this. So this is log of sigma versus 1 over T. We just erase this section and plot it slightly better. So this is log of sigma. So once again we have 3 regions at low temperatures we have an ionization region. At high temperatures we have an intrinsic region and then there is a region in the middle where sigma does not change much with temperature and that is your saturation region. Another fort for the saturation region is your extrinsic region. So doping gives us two advantages. One is that it increases the conductivity but at the same time there is also a temperature region where both the carrier concentration and hence the conductivity is almost a constant and is independent of temperature. In the case of silicon this saturation region or the extrinsic region is around room temperature or I should say includes room temperature which means if you have doped silicon then your conductivity is almost a constant in and around room temperature. This is very important when we try and form devices later with these extrinsic semiconductors. So this is the carrier concentration as a function of temperature. So how does the carrier concentration in the case of extrinsic semiconductors change with doping concentration? So let me just plot log of mu versus doping concentration. So I want to plot how mu e or mu h changes with concentration of the dopants. So this is mu e can be mu h. We will use the example of silicon but this is again true for any other semiconductor. This is doping concentration square. So we increase the doping concentration. The overall mobility decreases because now you have more scattering from these ionized impurities. So let me make the plot 100, this is your y axis. Concentration is 10 to the 15. So if you plot mu e starts at a high value in the case of silicon at room temperature mu e for intrinsic silicon is around 1350 centimeter square per volts per second. So let me just write down the units here centimeter square per volts per second. So it starts of high. Initially your mobility is more or less a constant and then as your doping concentration increases the mobility starts to drop. So this is for electrons. We can do a similar plot for the holes. Once again when you have intrinsic silicon the whole mobility is around 450 centimeter square per volts per second. So if you do a similar plot for holes again it stays a constant but then it starts to drop down. In all the extrinsic semiconductors we have considered so far, you always had the situation N A or N D whether these are acceptors or donors is much greater than N I which is your intrinsic carrier concentration. This means that we can say that the electron concentration is equal to N D or the hole concentration is equal to N A. We also only consider situations where N A and N D are much smaller than the effective density of states N C and N V. So this allows us to think of these impurities as localized states in the band gap. So your donors form localized states just below the conduction band and the acceptors form localized states just above the valence band. These semiconductors are called non-degenerate semiconductors. This is true for almost all carrier concentrations that we normally encounter but if you have really high doping values so that N A and N D are comparable to N C and N V we can no longer think of them as localized states but we say that the dopants form energy bands. So these semiconductors are called degenerate semiconductors. Because your dopants form energy bands it is also possible that these bands can overlap with the conduction band or the valence band. So degenerate semiconductors have high conductivity because they have a high concentration of impurities but they also behave very similar to metals. For most of the devices that we discuss we will essentially deal with non-degenerate semiconductors but for some cases we will find that we can also have high doping and we will also have degenerate semiconductors. The last topic that I want to cover when we are talking about electronic materials are amorphous semiconductors. So far when you have talked about an intrinsic semiconductor or when we talked about an extrinsic semiconductor we always consider materials that are single crystals which are perfect and have no defects. This is because defects in a material will introduce states in the band gap and will change the electronic properties. Defects will cause defect states. We saw earlier that these states can either be shallow. Shallow states are those that are close to the band edges. They can be close to the conduction band edge or they can be close to the valence band edge or they can be deep states. So these defect states can essentially modify the electronic properties. So defects can be good or bad. For example if you are trying to look for higher conductivity then defect states can essentially act as traps and trap electrons in holes and then reduce the conductivity. In that case defect states are essentially bad. On the other hand in the case of some direct band gap semiconductors you can have defect states in them that can modify the optical properties. In such cases defect states are good. So depending upon the applications we may or may not want defect states. An amorphous semiconductor is an extreme example of a semiconductor the large number of defect states. An amorphous material has essentially no long range order which means there is a large density of dangling bonds. The dangling bonds are formed when you have one silicon atom which has one electron but there is no opposite silicon atom to supply another electron to form the bond. So amorphous materials are characterized by having a large density of dangling bonds and these dangling bonds basically act as defect states in the band gap. So if you were to draw the band picture in the case of a crystalline material. So this is a crystalline material let us take crystalline silicon. So we will have a conduction band we will have a valence band that is your band gap. So if you have a perfect crystalline material there are no electronic states in the band gap then you have a density of states in the conduction band and a density of states in the valence band. Now if you have an amorphous semiconductor because you have a large density of dangling bonds there will be a large density of defect states in the band gap and these defect states are localized states. So if I were to draw the band diagram for an amorphous material. So again l mark e c and e v. We will have a large density of states in the bands but within the band gap we will also have localized states. So these represent the conduction band and the valence band. So within the conduction band and valence band we will have extended states but we will also have localized states within the band gap. In the case of an amorphous semiconductor we no longer talk about a band gap because we have all of these localized states. So we call the distance between the conduction band and the valence band as a mobility gap. Now all these localized defect states can also take part in conduction. They can essentially trap the electrons and holes. So in the case of an amorphous semiconductor conduction takes place through a hopping mechanism where we think of the electrons and holes hopping through all of these localized defect states. So this drastically brings down the mobility and hence the conductivity. We have crystalline silicon mu e is usually around 1350 or 1400. On the other hand mu e for amorphous can be as low as 1. So you can have nearly 3 orders of difference in magnitude in the mobility in the case of amorphous and crystalline materials. So with this we are done with the electronic materials part of the course. So we have looked at intrinsic and extrinsic semiconductors. Next we will look at devices and in devices we will have to form junction between these materials. So the next thing we will do is to look at junctions and that we will look starting from next class.