 Let us start with a brief review of last class. In last class we looked at two important concepts. The first one is called the density of states. We denoted this as G of E. Density of states refers to the number of available states for electrons to occupy. We looked at a simple model where we had a solid as a uniform three dimensional box with no potential. In such a case, we found that the density of states is directly proportional to the square root of the energy. So, thus as the energy goes up, the total number of available states for the electrons to occupy also go up. We also looked at another concept which we call the Fermi function. We denoted this by f of E. The Fermi function tells you the probability of occupation of an energy state by an electron. f of E found is 1 over 1 plus exponential E minus E f over k T. We saw that at temperature equal to 0 Kelvin. If E is less than E f, f of E is 1 which means all the levels below the Fermi energy are occupied. At E greater than E f, f of E is 0 which means above the Fermi energy all the levels are unoccupied and we saw that for all temperatures at energy E equal to E f, f of E is half. When the energy is much higher than k T, we can approximate the Fermi function by a Boltzmann function. So, this becomes exponential minus E minus E f over k T which is the Boltzmann function and this is true. So, this is what we looked at last class. We will use these concepts of density of state and Fermi function in order to calculate the electron hole concentration in semiconductors and first we will start with intrinsic semiconductors. We will start with intrinsic semiconductors, another name for this is a pure semiconductor. We will use the concepts of the density of states and Fermi function to calculate the carrier concentration that is the electron hole concentration. We will also use, we will also define concepts of electron mobility and conductivity. Once we are done this for the intrinsic semiconductors, we will move on to extrinsic semiconductors. So, what are intrinsic semiconductors? So, these are materials I will just say semiconductors within bracket that are single crystals and have no impurities or defects. We will see later why it is important that there should be no impurities or defects. Most of what we do we will reference using silicon as the material because silicon is the dominant material in today's microelectronics industry, but all the concepts that we develop can be equally applied to other semiconductors and later when we look at examples we will also look at other materials to compare and contrast with silicon, but for the most part we will deal with silicon. As opposed to intrinsic I mentioned earlier that you also have extrinsic semiconductors, these are doped semiconductors. If you look in terms of applications extrinsic semiconductors are almost always used, intrinsic semiconductors do find some applications, but not a whole lot. Mostly intrinsic semiconductors are used as optical sensors in the case of photo luminescence experiments or as sensors for x-rays in electron microscopes. So, these are sensors which are called energy dispersive x-ray analysis. Usually in those cases either intrinsic silicon or germanium is used cool to around liquid nitrogen temperatures, but for most applications extrinsic semiconductors are preferred. We will start with intrinsic first and once we develop these concepts we will go on to use them to understand extrinsic semiconductors. So, let us start with intrinsic silicon. As we saw earlier the electronic configuration of the outer shell of silicon is 3S2, 3P2. There are total of 4 electrons. The S and the P orbitals hybridize to give you 4 SP3 hybrid orbitals and we later saw that these orbitals when they form a solid give you a valence band that is completely full and a conduction band that is completely empty. And show this band diagram here you have a valence band that is completely full you have a conduction band that is empty. The y axis here refers to energy usually the energy is referenced with respect to the bottom of the valence band. So, the bottom of the valence band is given 0, E v denotes the top of the valence band and E c denotes the bottom of the conduction band. The difference between the top of the valence band and the bottom of the conduction band is the band gap we call E g. In the case of silicon at 0 Kelvin E g has a value of around 1.17 electron volts. At room temperature E g is slightly lower around 1.10 electron volts. The top of the conduction band is usually denoted as E c plus chi where chi is the electron affinity. For silicon chi has a value of around 4.05 electron volts. So, at 0 Kelvin you have a valence band that is completely full and a conduction band that is completely empty. At any temperature above 0 Kelvin we saw earlier you will have thermal excitation of electrons. So, that electrons from the valence band can move to the conduction band and leaving behind holes. So, at any temperature greater than 0 Kelvin you will have electrons in the conduction band and holes which is the absence of electrons in the valence band. We also saw that apart from thermal excitation we can also use light in order to excite carriers across the band gap. If E g is the band gap of the material the wavelength of light that is required in order to excite carriers is nothing but H c over lambda. We can do this calculation for silicon where we find that lambda is approximately 1000 nanometers and this lies in the IR region. As long as we shine light with the wavelength that is less than 1000 nanometers which means the energy will be higher than the band gap can always excite carriers from the valence band to the conduction band. So, this explains why silicon is opaque because visible light has a wavelength less than 1000 nanometers. The visible range is from 400 to 800 nanometers which means silicon will be able to absorb the visible light and produce electrons and holes. Similarly, Si O 2 which is glass has a band gap of approximately 10 electron volts. So, if you have Si O 2 has a band gap approximately 10 electron volts which means the wavelength that is required to excite electrons from the valence band to the conduction band is approximately 106 nanometers. So, 10 times less than that of silicon this lies in the UV region again explains why glass is transparent. So, coming back to silicon here is the picture we have at room temperature. We have electrons in the conduction band, we have holes in the valence band and these electrons and holes are said to be delocalized that is they can move through the solid. The process of formation of the electrons and holes is a dynamic process that is electrons and holes are constantly being formed at the same time electrons also fall back to the valence band and recombine with the holes. So, that they are also getting eliminated. So, the formation and recombination takes place. So, that we can say this is a dynamic process. So, we have an equilibrium concentration of electrons and holes and this concentration depends upon the temperature can say we have an equilibrium concentration of electrons and holes and this is temperature dependent when we apply an electric field these electrons and holes can essentially move. So, we will just show schematically a solid of silicon you have electrons you have holes you apply an electric field. E is the electric field electrons will go in the direction opposite to the field holes will go in the direction of the field and finally, you will have a current. The current is because the electrons move in the conduction band and the holes move in the valence band. What are the factors on which the current depends on? Let us look at that next. Current or if you want to think about it conductivity in the case of an intrinsic semiconductor depends upon 2 factors. The first one is the concentration of electrons and holes that are available. So, more the electrons and holes there are higher is the conductivity concentration of electrons and holes. We can denote conductivity by the symbol sigma and I will just say that as concentration increases your sigma will also increase. The next factor on which the conductivity depends on is how far these electrons and holes can travel before they get scattered by the lattice. So, remember you have electrons that are moving in the conduction band and you have holes that are moving in the valence band, but these are moving through a solid of silicon atoms. If you are looking at intrinsic silicon and all these atoms are vibrating which means these electrons and holes can interact with these atoms and get scattered. In the case of a semiconductor these electrons and holes are said to drift through the material are said to drift. To understand this we define a quantity that we call the mobility. Mobility is denoted by the symbol mu and the expression for mu. If you are looking at electrons can say mu e is nothing but tau e over m e star can write a similar expression for holes mu h. So, mu e depends upon a factor tau e and the effective mass of the electron mu h depends on a factor tau h and the effective mass of the holes. So, mu e and mu h are the mobilities. The factors tau e and tau h refers to the time between two scattering events. So, if tau e and tau h are large which means time between two scattering events is large. So, the electrons can travel a large distance before scattering. In such a case if these two are large your mobilities are also large and hence the conductivity is also higher. So, we can say if mu e and mu h increases and they would increase if the scattering is less the conductivity increases. So, we saw that the conductivity depends on two terms one is the concentration the other is the mobility. We can put these together to write an equation for the conductivity. So, if sigma is the conductivity then sigma is nothing but n e mu e plus p e mu h. Now, this is a very important equation which relates the conductivity to the electron concentration and the mobility. And this equation is true whether you have an intrinsic semiconductor or an extrinsic semiconductor. n and p refers to the concentration of electrons in holes which is the first factor that we saw earlier. e is the electric charge which is 1.6 times 10 to the minus 19 coulombs. So, that is a constant and mu e and mu h are the mobilities of electrons and holes. So, if you look at this expression higher the concentration. So, higher n and p your mobility is higher. Similarly, higher the conductivity is higher. Similarly, higher mu e or mu h the conductivity is higher. The typical units for mobility are meter square volts per second. So, units for mobility can either use meter square per volt per second. You will also find that some books give values in centimeter square volts per second. So, let us take a look at the mobility values for silicon. We have silicon mu e which is the mobility of the electron in the conduction band is around 1350 centimeter square volts per second. If you want to convert this to SI units you just divide by 10 to the 4. So, this is 0.1350 mu h which refers to the mobility of the holes in the valence band slightly smaller. So, around 450 centimeter square volts per second can divide this by 10 to the 4. The top one should be 1.350. Just rewrite that. So, these are the units or these are the values of conductivity the case of silicon for both electrons and holes. We define mobility as the ability of the electron to move through the material before being scattered. We also saw that the mobility depends upon the scattering time. We will use this to do some calculations for the scattering time. So, consider the case of silicon and we will only talk about the electrons can use a similar calculation for the holes. Consider the case of silicon and mu e is 1350. We also said earlier that mu is related to this scattering time. The expression was e tau e over m e star. Just rearranging the terms gives you tau e. So, all I did was take this term here and then bring e down. The case of silicon m e star is around 0.26 m e where m e is the mass of an electron and m e has a value. So, we can substitute the numbers in here to get the value for tau e. So, 1350 and multiply by 10 to the minus 4 convert to SI units. If you do the math tau e works out to be 2 times 10 to the minus 13 seconds or 0.2 picosecond where 1 picosecond is 10 to the minus 12 seconds. So, this time 0.2 picoseconds represents the time between 2 scattering events for an electron in silicon that is moving through the conduction band. So, tau e refers to the time between 2 scattering events and since we are talking about the electron, it is for the electron in the conduction band. We also like to calculate the distance the electron travels between these 2 scattering events or the distance the electron travels in this time of 0.2 picoseconds. If we look at an electron with a mass given by m e and velocity that is given by V th, then the kinetic energy is nothing but 1 half m v square. This is the kinetic energy of an electron. In case of most solids, you can say that the kinetic energy is approximately equal to the thermal energy. So, this is approximately equal to 3 by 2 k t where k is the Boltzmann's constant. k has the value 1.38 times 10 to the minus 23 joules per Kelvin. Equating this expression, we can find a value for V thermal or the velocity of the electron, which is nothing but square root of 3 k t over m e. You can again plug in all the numbers. We are doing this calculation at room temperature. So, temperature is 300 Kelvin. If you do this, V thermal works out to be 1.16 times 10 to the 5 meters per second. So, we calculated V thermal to be 1 times 10 to the 5 meters per second. Earlier, we saw that the time between two scattering events was 2 times 10 to the minus 13 seconds or 0.2 picoseconds. So, the distance traveled by the electron between two scattering events is nothing but V thermal times tau, which if you do the math is around 2.33 times 10 to the minus 8 meters or approximately 23 nanometers. To put this into perspective, you can say that the lattice constant for silicon is 0.54 nanometers. So, the distance traveled in terms of the lattice constant is 23 by 0.54, which if you see is approximately 43 unit cells. So, the time between two scattering events is really small. So, the time is of the order of picoseconds, but because your electrons have such a high velocity, their velocity is around 10 to the 5 meters per second. We find that the electron actually travels a substantial distance. So, it travels nearly 40 unit cells before it undergoes another collision and then scattering. Mobility is usually a function of temperature. It is also can be thought of as a material property as long as you do not add any impurities. Later when we look at extrinsic semiconductors, you will find that increasing the concentration of the dopants decreases the mobility. You can compare the mobility of silicon with some other semiconductors to see the values. I will leave the expression for sigma up here. We saw that the mobility of an electron in silicon was around 1350 centimeter square per volt per second. This is for silicon. If you had germanium, germanium has a slightly higher mobility. Mu e is 3900. That is the value for germanium. Gallium arsenide is even higher. If you want to improve the conductivity of a material and you are asked to choose materials in order to have higher conductivity, just based upon the mobility values, the choice would be gallium arsenide because gallium arsenide has the highest value 8500, which is nearly 6 times or 7 times higher than that of silicon. But if you look at the expression for sigma, it not only depends upon the mobility, it also depends upon the concentration of electrons and holes. So, the next thing we will do is to calculate the electron and hole concentration as a function of temperature. So, in last class, we looked at the concept of the density of states and the Fermi energy. Putting those two concepts together, we can say that the concentration of electrons in the conduction band is nothing but the integral over the entire width of the band of the density of states in the conduction band. I call the G C B of E times F of E D E. To put this into words, if you want to find the number of electrons in the conduction band at any given temperature, it is going to depend on the number of states that are available in the conduction band, which is G C of E, the density of states and the probability that those states are occupied by the electron, which is F of E. If you look at a conduction band, the width of the conduction band, we saw earlier goes from E C to E C plus chi, where chi is the electron affinity. So, goes from E C to E C plus chi G C B, which is the density of states times the Fermi energy D E. Now, we want to calculate the density of states and silicon is a real crystal, but we can make use of the assumption of a u solid with a uniform potential, which we did earlier. For a solid with a uniform potential, the density of states E is given by 8 pi square root of 2 by h cube M E star, which is the effective mass of the electron in the conduction band times E to the half. So, this expression we saw earlier for a solid with a uniform potential, a 3 D solid and we will use that expression to find the density of states for electron in the conduction band of silicon. Since, we are looking at only the conduction band, E here must be written with respect to the bottom of the conduction band. So, we will modify this expression G C B and write it as 8 pi square root of 2 by h cube M E star 3 over 2 and instead of E, we will write it as E minus E C the whole power half. So, that you are writing the energy with respect to the bottom of the conduction band f of E is 1 over 1 plus exponential E minus E f over k t. Usually in the case of silicon and you will actually showed explicitly later on E minus E f will be of the order of electron volts or a point greater than 0.5 electron volts k t is usually of the order of milli electron volts. So, we can approximate the Fermi function as a Boltzmann function and write this exponential minus E minus E f over k t. So, this expression approximates to this one. The case of the silicon semiconductor our limits where the width of the band that goes from E C to E C plus chi. For ease of integration, we can replace the E C plus chi term with infinity. This is also true because we will find that most of the electrons are located very close to the bottom of the conduction band. So, we can take the top of the conduction band to be infinity and we would not lose much in the model. So, we will replace E C plus chi with infinity. Let me then rewrite this expression again. If you rewrite this n which is the number of electrons in the conduction band goes from the base of the conduction band E C up to infinity. The density of states times the Fermi function d E where the density of states is given by this expression m e star whole power 3 over 2 E minus E C whole power half and f of E we will use the Boltzmann's approximation and write it as exponential E minus E f over k t. We can substitute these two in this expression and do the integration. I will not show the integration explicitly, but write down the final result. When we do this, we get n is equal to n C which is a constant at a given temperature times exponential minus E C minus E f over k t, n C is nothing but 2 times 2 pi m e star k t over h square whole power 3 over 2. This term n C is called the effective density of states at the conduction band edge. So, n depends upon the effective density of states at the conduction band edge times an exponential function. Whatever derivation we did for electrons we can equally do for holes in the valence band. I would not do the derivation for holes, but I will just write down the expression. For holes p which is the concentration of holes in the valence band is n v which is a constant times exponential minus E f minus E v over k t, n v is 2 2 pi m h star. So, instead of m e it is m h star k t over h square whole power 3 over 2. This is the effective density of states at the valence band edge. So, we have two expressions one for electrons and one for holes. These expression one for electrons and one for holes give you the concentration of electrons in the conduction band and holes in the valence band for a given semiconductor as a function of temperature. Let me just rewrite those two expressions n is n C exponential is n v exponential minus E f minus E v over k t. We can mark the position of E c and E v in the band diagram. So, if you draw this you have a valence band. The base of the valence band is set as 0. The top of the valence band is E v then you have a conduction band. The base of the conduction band is E c. The top of the conduction band is E c plus chi. And the difference between the valence band and the conduction band is the band gap. So, here is your expression as long as we know all these values n c n v E c E v. We can calculate the concentration of electrons and holes. If you look at these expressions the Fermi energy term E f is in there, but I have not marked where the Fermi energy is. So, if you do not know where E f is you would not be able to calculate n n p. So, let us go ahead and eliminate E f by multiplying n n p together. If you do that n p will be n c n v exponential minus E c minus E v over k t. All I have done is multiply these two together. E c minus E v if you see from this band diagram this whole thing is E c this is E v. So, E c minus E v is nothing but the band gap. So, this I can replace n c n v exponential minus E g over k t. In the case of an intrinsic semiconductor your electrons are created because they are excited from the valence band to the conduction band. And whenever an electron is created a hole is also created. So, in the case of intrinsic semiconductors n which is the concentration of electrons is equal to p which is the concentration of holes. And this is usually denoted as n i n i is called the intrinsic carrier concentration. So, for an intrinsic semiconductor n equal to p equal to n i. So, if you make use of the fact that n is equal to p then we can rewrite the expression n p is nothing but n i square which is the square of the intrinsic carrier concentration that is equal to n c n v exponential minus E g over k t. And then taking square root you get n i is square root of n c n v exponential minus E g over 2 k t. This is the expression for the carrier concentration. So, the concentration of electrons and holes in an intrinsic semiconductor as a function of temperature. We saw that the conductivity equation it is nothing but any mu E plus p E mu h. In the case of an intrinsic semiconductor n is equal to p equal to n i. So, we can take this term outside. So, it is n i E mu E over mu h. So, the conductivity depends on the sum of the mobilities of both the electron and the hole the case of an intrinsic semiconductor. So, this is where we will stop for today. In the next class we will start by calculating these values for the intrinsic carrier concentration and the conductivity for silicon. Then we will look at how changing the material will change these values. Then we will proceed from there.