 Let us start with the brief recap of last class. Last class we looked at intrinsic semiconductors. These semiconductors are pure semiconductors and we calculated the concentration of electrons in the conduction band and holes in the valence band. We found that at any given temperatures, electrons and holes are created in pairs and in an intrinsic semiconductor n, which is the concentration of electrons is equal to the concentration of the holes and it is equal to a number n i and n i we called as the intrinsic carrier concentration. The expression for n i square root of n c n v exponential minus e g over 2 k t, n c and n v where the effective density of states at the conduction band edge and the valence band edge and e g is the band gap of the material. We also wrote down an expression for the conductivity sigma as n e mu e plus p e mu h. In the case of an intrinsic semiconductor where n is equal to p, this simplifies into mu e plus mu h. Mu e and mu h are the mobilities of the electrons and the holes. They represent the ease with which the electrons can move through the conduction band or the holes can move through the valence band. Higher the value of n, higher is the conductivity. Similarly, higher the value of mu e and mu h, higher is the conductivity. Today we will start by calculating some of these values for the carrier concentration, mobilities and conductivities and we will start with silicon. So if you have intrinsic silicon at room temperature, so temperature is 300 Kelvin. The first thing we want to calculate is the effective density of states n c and n v. We looked at the expression for n c and n v last class. n c depends upon the effective mass of the electron. So 2 pi m e star k t over h square whole power 3 over 2 can write a similar expression for the density of states in the valence band n v whole power 3 over 2. In the case of silicon m e star is 1.08 times m e, where m e is the rest mass of the electron m h star is 0.60 m e. If you use these numbers substitute them in here and also put in the values of the constants n c turns out to be 2.81 times 10 to the 25 meter cube and then n v is 1.16 times 10 to the 25 per meter cube. So these two represent the effective density of states at the conduction band edge and the valence band edge. Last class we saw that most of the electrons and holes are concentrated near the valence band edges and the conduction band edge. So we can go ahead and calculate the intrinsic carrier concentration times exponential minus e g over k t. At room temperature silicon has a band gap of around 1.1 e v. So e g is if we plug in that we get a value of n i to be 1 times 10 to the 16 per meter cube. So this represents the intrinsic carrier concentration of silicon. We write this down here n i for silicon at room temperature. We can convert this to centimeter cube. So this we convert to centimeter cube is around 1 times 10 to the 10 per centimeter cube. So we have 10 billion carriers that is electrons and holes in silicon at room temperature. Now this looks like a really large number. So let us try and put that in context. Now let us calculate the number of atoms per unit volume. So per centimeter cube of silicon. This is nothing but the density of the material rho divided by the atomic weight times our gad rose number. We substitute in the numbers. Density of silicon is 2.3 grams per centimeter cube. Atomic weight is 28 grams per mole. Avogadro's number is 6.023 times 10 to the 23. We substitute these values and evaluate. You get the number of silicon atoms per unit volume to be approximately 5 times 10 to the 22. So if you looked at the carrier concentration and said that was 10 billion, we can compare it to the number of atoms which is 10 to the 22. So 10 orders of magnitude more than 10 billion. So if you look at it, the number of carriers per atom, if you do the division, you will find that you have one electron or one hole for every 10 to the 11 atoms. And this turns out to be really small. You can go ahead and calculate the conductivity using this equation. Mu e for silicon, we saw yesterday was 1350 centimeter square per volt per second. Mu h is 450. So you can substitute for the value of ni which we have here. E is nothing but the electric charge and that is a constant. Mu e and mu h are given here. If you substitute and evaluate, you get a conductivity of 2.9 times 10 to the minus 6 ohm inverse centimeter inverse. So let me write this down again. If you have intrinsic silicon at room temperature, the conductivity sigma which we calculate is 2.9 times 10 to the minus 6 ohm inverse centimeter inverse. Resistivity rho is nothing but 1 over the conductivity. The resistivity is 3.5 10 to the 5 ohm centimeter. So it is just 1 over the conductivity. Just to compare, if you have copper which is a metal, the resistivity rho is approximately 15.7. We saw this in the first class when we wrote down resistivities for different elements. So rho is 15.7 times 10 to the minus 7 ohm centimeter. So comparing copper and silicon, silicon has a resistivity, 12 orders higher than copper. So this makes intrinsic silicon a very poor conductor and the reason for that is because we have a very low density of electrons and holes. So you have a very low carrier density and this is related to the band gap of silicon. In the case of copper which is a metal, you have empty states that are there. So both the valence band and the conduction band overlap. So you have empty states that are available for conduction. So if you have 10 to the 22 atoms, then each atom can donate one electron. So you have a high density of electrons which leads to a high conductivity. Last class, we also looked at the comparison of silicon with other semiconductors based on mobility. So we have three semiconductors. We looked at germanium, silicon and gallium arsenide. Let me write down their mobility values mu e and mu h. The units will be in centimeter square volts per second. Germanium is 3900, 1900 for the holes. Silicon is 1350 for the electrons and 450 for the holes. And gallium arsenide has a very high mobility for electrons. It is around 8500. Holes is around 400. So the question was, if you want to increase the conductivity, then what material will you choose? Based on mobility, the answer is you will choose gallium arsenide because gallium arsenide has the highest mobility. And we know that sigma is directly related to the mobility. But it turns out that the overriding factor here is Ni, which is the concentration of electrons and holes. For that, we also need the values for the band gap. So let me write the values for band gap here EG. This is an electron volts. So germanium has a band gap of around 0.66EV. This is at room temperature. Silicon is 1.10. Gallium arsenide is 1.42. So even though gallium arsenide has a higher mobility, because it has a higher band gap, we will find that the concentration of carriers is lower. Because you have a lower concentration of carriers, you will also have a lower conductivity. So let us just do the math for gallium arsenide. Once again, we need the masses, the effective masses of the electrons and holes. So Me star for gallium arsenide is 0.067 Me. Me star is 0.50 Me. So now you can calculate Nc using the same equation that we used for silicon. Nc is around 4.3 times 10 to the 23 cube Nv is 8.85. From Nc, Nv and the band gap, which is 1.42, we can calculate the intrinsic carrier concentration. This is 2.4 times 10 to the 6 per centimeter cube. Correspondingly, the conductivity is lower, which is 3.4 times 10 to the minus 9. So we find that in the case of gallium arsenide, your intrinsic carrier concentration is lower. Correspondingly, the conductivity is lower. Just to complete this table, we will also do a similar calculation for germanium. Then we can come back and fill in the values for Ni in terms of centimeter cube and also conductivity. We already did this calculation for silicon. So for silicon, we had Ni to be 10 to the 10 and then we had the conductivity for silicon to be 3 times 10 to the minus 6. We just did the calculation for gallium arsenide. So if you write the values for gallium arsenide, Ni is 2.4 times 10 to the 6. Correspondingly, conductivity is 3.4. So let us do the calculation for germanium. So then we can fill in these two values as well. So in the case of germanium, once again I need the effective masses, Me star is 0.12 Me, M H star is 0.23. So once you know Me star and M H star, we can go ahead and calculate N C and N V. So the formula is just the same. You are just substituting different numbers. N C is 1 times 10 to the 19 centimeter cube. N V is 6 times 10 to the 18. So the intrinsic carrier concentration Ni is nothing but N C N V exponential minus E G over 2 k T, which if you substitute the values, give you an Ni of 2.4 times 10 to the, sorry, 2.4 times 10 to the 13 centimeter cube. So Ni is higher in the case of germanium. Similarly, you can calculate the conductivity. The conductivity works out to be 0.0213. We can now fill in this information back in the table which we started. So let me go to the table and fill in the values for germanium. We already did for silicon and gallium arsenide. So in case of germanium, your carrier concentration is 2.3 times 10 to the 13, 2.4 times 10 to the 13 and the conductivity 0.0213. So we have three semiconductors, germanium, silicon and gallium arsenide. We found that gallium arsenide has the highest mobility. Germanium is also higher than silicon, but the main term that really dominates the conductivity is Ni, which is the carrier concentration and Ni depends upon the band gap. So higher the band gap, lower the carrier concentration correspondingly lower the conductivity. Based on this graph or based on this table, if you want to choose a material with the highest conductivity, then the preference goes to germanium. In fact, the first transistor or the very first solid state transistor that was built in Bell Labs is actually made of germanium. One of the reasons though silicon has now come to dominate the microelectronics industry is because we can actually control the conductivity and the electronic properties of silicon by doping. So we will see doping next. More importantly, silicon is much more abundant than germanium. So silicon is the second most abundant element on earth. It is around 27 percent. Germanium on the other hand is the 50 earth most abundant element. Its concentration is around 10 to the minus 6 percent. So this is the reason why silicon has come to dominate the microelectronics industry. The next thing I like to do is in last class, we never calculated the position of the fermi energy. So we will go ahead and calculate where the fermi energy is located in an intrinsic semiconductor. Let us go ahead and draw the band picture. We have energy on the y axis. We have a valence band. The bottom of the valence band is referenced as 0 and the top of the valence band is E v. Then we have a conduction band. The bottom of the conduction band is referenced as E c. In the top of the conduction band is E c plus chi. And this distance between valence and conduction band is E g. We also wrote down expressions for n and p. n is n c exponential E c minus E f over k t, where E f is the fermi energy. p is n v exponential minus E f minus E v. So in last class, we went ahead and multiplied these two terms in order to eliminate E f. To calculate the value of E f, that is where the fermi energy is located, the semiconductor. Let us just equate n and p. So we have n c exponential minus E c minus E f over k t is equal to n v exponential minus E f minus E v. So I have just equated n and p. So we can take natural log on both sides and then rearrange this expression to give you the value of E f. Do that E f plus E g over 2 minus one half k t on of n c over n v. So this you can get by just rearranging this expression. Since we are looking at an intrinsic semiconductor, this is usually denoted as E f i. So that this is the fermi energy of an intrinsic semiconductor. So let me rewrite this expression. So let me rewrite the expression for E f i. So E f i is E v plus E g over 2 minus one half k t non of n c over n v. Now n c and n v are also related to the effective masses of the electrons and holes. You have seen that expression before. So instead of n c and n v, we can substitute the effective mass and this expression will change as E f i is E v plus E g over 2 minus 3 by 4 k t non of m e star over m h star. So now you have essentially two equations for calculating E f i. E f i is the position of the fermi level in the intrinsic semiconductor. So let us do the calculation for silicon at room temperature. We can either use n c and n v that we calculated earlier or you can use the effective masses. So let me use the effective mass. m e star was 1.08 m e m h star is 0.60 m e. Since I am using the effective mass, I will use the second equation from which you get E f i is E v. The band gap of silicon is 1.1, 0.10 minus 3 fourths. If you simplify this expression, this gives you, we write it below, E v plus 0.54 or writing this another way E f i minus E v is 0.54. So in the case of intrinsic silicon, the fermi level is located approximately 0.54 electron volts above the valence band. So if we try to redraw our picture for the energy band gap, I will do that here. I have a valence band that is full. This is E v. We have a conduction band that is E c. E g in the case of silicon is 1.1 and E f i is 0.54 above E v. E f i which is the fermi energy is located very close to the center of the band. The actual center of the band from this value of E g is 0.55 and the fermi energy is located at 0.54. So E f i is very close to the center of the band. If m e star and m h star are the same, so if these two are the same, which means n c and n v will also be the same, then E f i will be exactly at E g over 2 or if temperature were 0, so if t were 0, then also E f i will be exactly at E g over 2. But because the effective masses are different and because the effective density of states are different, it is slightly shifted from the center of the band gap. So what does this value for E f i mean? In the case of a metal, we define fermi energy as the line that separates the highest filled states and the lowest empty state. So if you have a metal, we drew a band picture for a metal, this is your band. The case of a metal, your band was half full, so that you had filled states and you had empty states and the fermi energy separated the filled and the empty states. Now in the case of a semiconductor, we have a valence band that is completely full, we have a conduction band that is completely empty. So the question is where do we put the fermi energy? Now to look at it, there is another way of defining fermi energy. E f is related to the work function, we will call the work function psi. Now psi represents the energy that is required in order to remove an electron from a solid. So if you go back to this band picture and you take the top of the band to be vacuum level, then psi represents the energy to remove an electron from fermi level up to the vacuum level. In the case of semiconductors, we do not have electrons independently, we always have electron hole pairs. So in such a case, psi represents the average energy that is required to remove an electron. So because we have electrons and holes, electrons in the conduction band and holes in the valence band and because these are linked, we find that the fermi energy in a semiconductor is located within the band gap because it represents the average energy to remove the electron. If as I said earlier, the effective mass of the electrons and the holes were the same, the fermi energy will be located exactly at the middle of the gap. But because ME star and MH star are different, it is slightly shifted in the case of silicon. So let us go back to an intrinsic semiconductor. We want to increase the conductivity of a material. In the case of an intrinsic semiconductor, we cannot change the carrier concentration at any given temperature because the carrier concentration depends upon the band gap. So what we will do is to see the effect of temperature on N i. We will go back to this expression for the carrier concentration N c N v exponential minus E g over 2 k T. Question is what happens when we increase the temperature? The temperature comes in two terms. N c and N v also depend on temperature. So if we increase temperature, N c and N v will also increase. But the dominant term where temperature plays a role is in the exponential term. So as temperature increases, exponential minus E g over 2 k T will drop because temperature is in the denominator. So both these effects tend to increase the intrinsic carrier concentration. So if you put these together, N i will increase as temperature increases. But the dominant term is the exponential term. So consider the case of silicon where we have two temperatures T 1 and T 2 and there are two intrinsic carrier concentrations N 1 and N 2. In this particular case, you can calculate the ratio of N 1 and N 2 by substituting them in this equation and also taking the temperature dependence of N c and N v into account. We will give you T 1 over T 2 exponential minus E g 2 k. So all I did was substitute for N 1 and N 2 in this expression and then divide. Let me just rewrite that expression N 1 over N 2 by T 1. So if we have silicon and we increase the temperature from 300 Kelvin to 600 Kelvin, so we double the temperature. We want to know what the change in carrier concentration is. At 300 Kelvin which is room temperature, N 1 has a value of 10 to the 10 per centimeter cube. We want to know what N 2 is. So we can substitute in the values and evaluate this expression. So we get N 2 which is the carrier concentration temperature equal to 600 is 1.17 times 10 to the 15. So by doubling the temperature from 300 to 600 Kelvin, we have increased the carrier concentration from 10 to the 10 to 10 to the 50. So nearly 5 orders of magnitude. So if you look at an intrinsic material or an intrinsic semiconductor, the only handle we have in changing the conductivity as long as we keep the material the same is temperature. To increase the conductivity, we have to increase temperature. The problem though is most devices we know operate near room temperature or at room temperature. So really we cannot increase the temperature beyond the normal operating temperature to increase conductivity. It is for this reason that people go ahead and dope or add impurities to an intrinsic semiconductor in order to form extrinsic semiconductors. This increases the concentration of carriers and thus increases the conductivity. So with this, we are done with intrinsic semiconductors. In next class, we will start to look at extrinsic semiconductors. Before we do the transition, let me just briefly recap few points for intrinsic semiconductors that we will carry forward. So we saw that in the case of an intrinsic semiconductor N is equal to P. So electron and hole concentration is the same which is equal to N i. This generally can be written as N P equal to N i square. This is called the law of mass action. In the case of an intrinsic semiconductor, this relation N P equal to N i square is very trivial because N is equal to P. But later when we look at extrinsic semiconductors, we will find that this relation is true. So you can increase N, but at the same time you will have to decrease P. If you increase P, you will have to decrease N. You cannot increase both of them at the same time. We also saw that the intrinsic carrier concentration is a function of temperature and it depends upon the band gap where N i is square root of N c N b exponential minus E g over 2 k T. We saw that silicon, the dominant material in the semiconductor industry has an N i which is very small. It is only 10 to the 10 per centimeter cube, which means the conductivity was also very small. The value for the conductivity was the only way we could increase the conductivity. If you still want to keep intrinsic silicon was to increase the temperature, but increasing the temperature is not practical because the devices have to work at room temperature. So the next thing we will see is to how to increase the conductivity by doping and these are extrinsic semiconductors which we will start in next class.