 Let us start with the brief recap of what we learnt in the last class. So last class we said that based on resistivity we can classify materials into three types conductors or metals, semiconductors and insulators. We are trying to explain the vast difference in resistivity or conductivity between metals and semiconductors. So metals are so much more conductive. In order to do that we tried to develop a simple band diagram where we have atomic orbitals that come together to form molecular orbitals and at really large values of N where N is about Avogadro's number we found that these form bands. In the case of metals these bands are formed from S shells that are either incompletely filled or from S and P shells overlapping so that we always have a band that is not completely full. So we do have some empty states. These empty states basically help in conduction of electrons which is why metals are such good conductors. We also saw an example of an energy versus a bond length diagram using lithium as the example. So let me just redraw that again and from there we will move on to semiconductors. So this is from last class we looked at lithium which has an electronic configuration of 1S2 and 2S1. 1S2 forms the inner shell and it is not involved in bonding, 2S1 is the outer shell. So we can draw the energy versus bond length diagram, the 1S shell is not involved in bonding. The 2S shell involves in bonding and basically forms a band. The S shell is half filled so the band is also half filled. These are the filled states, these are the empty states and the gap and the highest filled state is called the Fermi energy. So today we will move on to semiconductor materials and see how we can draw a similar picture for semiconductors. When we talk about semiconductors the dominant material there is is silicon. Silicon has an atomic weight of 14, sorry atomic number of 14, it has an atomic weight of 28 grams per mole. Given the atomic number is 14, the electronic configuration for silicon is 1S2, 2S2, 2P6, 3S2, 3P6. So the 1S and the 2S shells are the inner shells and they are not involved in bonding. The shells that are involved in bonding are the 3S and the 3P. These are the outer shells, should be 3P0. Rearrange in order to form SP3 hybrid orbitals. So you have 2 electrons in the S, 2 electrons in the P, they hybridize to form 4 SP3 hybrid orbitals. Thus each silicon atom can form 4 bonds and these bonds are directed in the tetrahedral directions and essentially what you have is a diamond lattice. So let us look at where silicon is in the periodic table. We will concentrate on the group where silicon is. If you look at it, the group has carbon, silicon, germanium, tin and lead. Out of these 5 elements, tin and lead are metals. Carbon in the form of diamond is an insulator and more importantly the S and the P electrons in carbon can form a variety of hybridizations. So you can have SP3, SP2, SP hybridizations. Silicon and germanium are semiconductors and they have SP3 hybridization. So instead of looking at atomic orbitals in silicon, we are looking at bonding using these SP3 orbitals. So let us look at a picture for bonding for silicon. So we start with the 3S and the 3P, 2 electrons in them. We said that in silicon, they hybridize to form SP3 orbitals. So you have 4 SP3 hybrid orbitals and each of them are involved in a bond. We will consider one of these SP3 hybrid orbitals. This can bond with another orbital to give you a bonding and an anti-bonding orbital. This is the same picture that we had in the case of hydrogen. This forms sigma, sigma star, sigma is your bonding orbital, sigma star is the anti-bonding and on the other side you have another SP3 hybrid orbital. So you have 2 electrons, both electrons go to sigma and sigma star is empty. So this is the picture we have when we have 1 silicon bond being formed. Now in a solid, you are going to have a large number of these silicon bonds. So you are going to have a large number of sigma and sigma star. So these will then interact with each other and then they will give you a band. So sigma forms a band and pictorially represent this as the valence band. So the valence band is formed by the interaction of all the bonding orbitals, the valence band is full. Same way, you have a conduction band that is formed by interaction of the sigma star and that is empty, conduction band and between the valence band and the conduction band there is a gap. This gap is called the energy gap. It is denoted by the symbol EG, it is called the energy gap or sometimes the band gap. So this is the picture of silicon we have from interaction of these molecular orbitals. So we have a sigma and a sigma star. This just represents one bond. When we have a whole bunch of these bonds in a solid, they all interact with each other leading to broadening and finally we have an energy band which is the valence band is completely full. We have an energy band from sigma star that is completely empty and then you have a band gap between them. You can once again draw an energy versus bond length plot for silicon but now instead of having the atomic orbitals, we will have the sigma and sigma star. Let us do that. So I have energy on the y-axis, bond length, there is some equilibrium spacing. We have sigma. So here this sigma represents the effect of one bond. In the solid it forms a band. There are two electrons here which means you have this band is completely full. Now you also have a sigma star which is the anti-bonding. This will also form a band which is completely empty and the space between these two bands is your band gap. So this is the model for silicon. You can use the same model for germanium which lies in the same group. So again it has s and p orbitals which hybridize to form sp3 orbitals and so on. We can extend this model in the case of insulators as well but in insulators for example if you think of an insulator like sodium chloride, you have bonds being formed not by sodium and chlorine but you have Na plus and Cl minus ions. So it is the ions that are forming the bonds and the bands. So it is the same picture but depending upon the material the details will change. What are some typical values for band gap? Let us write down some typical values for band gap e.g. The units for this are in electron volts and as we saw yesterday electron volts is related to joules. In electron volt it is nothing but 1.6 10 to the minus 19 joules. So we have silicon e.g. terms of e.v. All these values are at room temperature. Carbon has a value of 1.1, 1.11, germanium is 0.67, gallium arsenide is 1.43, cadmium sulphide is 2.42, zinc oxide is 3.37, SiO2 is 9. So we have some different values of band gap. So if you look at both semiconductors and insulators the difference between them is the difference in the band gap. So typically if e.g. is less than 3 electron volts we call a material a semiconductor and if e.g. is greater than 3 electron volts we call the material an insulator. Now this is not a very rigorous separation the 3 electron volts comes out because 3 electron volts corresponds to the end of the visible region. For example if you want to convert energy into wavelength the case of electromagnetic radiation e is nothing but hc over lambda where h is the Planck's constant, c is the velocity of light and lambda is the wavelength so I will write it here. If you use this if your energy is 3 electron volts then lambda is approximately 400 nanometers. So any energy greater than 3 electron volts puts you in the ultra violent region energy less than 3 electron volts puts you in the visible region. So this is used as a sort of marker for differentiating semiconductors and insulators but as I mentioned earlier it is not very strict. Based upon this all these materials silicon, germanium, gallium arsenide, carmium sulphide are semiconductors zinc oxide is close to this value of 3 but it is on the higher side and silicon dioxide is much higher so these are considered to be insulators. So we looked at some values for band gap of semiconductors. So let us now look at some classifications of semiconductors. So there are different ways of classifying semiconductors. So one particular method of classification is called elemental and compound semiconductors. So this distinction is fairly straightforward. So those semiconductors that are elements are essentially your elemental semiconductors but you can also have compounds that have semiconductors that is whose band gap lies less than 3 electron volts these are compound semiconductors. There are a lot of different types of compound semiconductors you have 3 phi's you have 2 6's to understand them let us take a look at the periodic table. We would not look at the entire periodic table but focus ourselves in the area where we have the groups 4, 3 and 5 if you do that so group 2 you have 3a, 4a so let me just write down the elements. So I am writing the elements from group 2, 3, 4, 5 and 6 group 5 and then group 6. So this is just a portion of the periodic table. Now if you look at it group 4 has silicon and germanium which are classic elemental semiconductors. So most prominent example silicon and germanium. Carbon we said in the form of diamond is an insulator. So diamond has a band gap of around 5.5 ev so that carbon is really an insulator. You can also form semiconductors by using elements from 3 and 5. Those are your 3 phi's you can also form 2 and 6 those are your 2 6's. So some examples of 3 phi and 2 6's can have gallium arsenide which is the most dominant compound semiconductor gallium nitride, gallium phosphide, indium stebite so on. So all of these are formed by putting elements from here and from here. So gallium arsenide, gallium phosphide, gallium nitride and so on. Can also have elements from 2 and 6. Again some examples are zinc sulfide, zinc selenide, cadmium sulfide, cadmium selenide and so on. If you look at the bonding character in these compound semiconductors 3 phi's are mainly covalent mainly have covalent bonding. They do have some ionic character in them. The main bonding is covalent in the case of 2 6 the main bonding is ionic though there is some covalent character in them. So this is one way of classification where we have elemental and compound. There is also another way of classifying semiconductors. So here we can classify semiconductors as direct and indirect band gap semiconductors. We have a direct band gap and an indirect band gap. So how do we understand the difference between these two? So let us go back to our model for semiconductors. We have a valence band that is completely full. We have a conduction band that is empty and we have a band gap in between them. So you have electrons that are there in the valence band. These can be excited to the conduction band. This electron excitation can either occur because of thermal energy or because of optical energy. So what happens when we have an electron in the conduction band? Now this electron wants to come back to the valence band so that it loses whatever energy it has gained and this energy loss can occur in two ways. So when the electron comes back to valence band it can release the energy in two ways. One can release it as photons or it can release it as heat. In the case of direct band gap semiconductors this energy release is dominantly in the form of photons. In the case of indirect band gap semiconductors the energy release is dominantly as heat. So let us look at some examples of direct and indirect band gap semiconductors. In the case of direct we have materials like gallium arsenide, cadmium selenide, cadmium sulphide. The case of indirect band gap semiconductors silicon and germanium which are our two elemental semiconductors are both indirect band gap semiconductors. Another example gallium phosphide. Direct band gap semiconductors have applications in the case of optoelectronic devices they can be used in LEDs, solar cells, photovoltaics and so on. So that is one of their advantages they not only have applications on the electronic side they also have applications on the optical side. To understand the difference between direct and indirect band gap semiconductors some more we can also look at the band structure. So what I will do is give you a very simple picture of the band structure in the case of semiconductors. In any element your electron in the solid can be treated as a wave and when you have a wave a wave always has something called as a wave vector and the wave vector is denoted by k. So thus for materials you can construct what are known as E versus k diagrams. This is typically done for both the valence band and the conduction band. So the difference between a direct band and an indirect band gap semiconductors is a difference between how their E versus k diagrams look. So let us look at an example of silicon which is an indirect band gap semiconductor and then gallium arsenide which is a direct band gap semiconductor. So let us draw this E versus k for silicon. I am only drawing a very simplified picture to explain the difference between direct and indirect. So this is the conduction band, this is the valence band, there is a band gap between the valence band and conduction band, this is E g. In this particular plot E is on the y axis and then k along the x axis. So in the case of silicon the top of the valence band and the bottom of the conduction band are a different values of k, top of the valence band and the bottom of the conduction band have different values of k. This means that any electron transition from the conduction band to the valence band is accompanied by phonons which are essentially heat quanta so that any transition in the case of silicon from the conduction to the valence band the energy is released as heat. This makes it an indirect band gap semiconductor. Let us look at gallium arsenide which is a direct band gap semiconductor. So once again this is the conduction band, this is the valence band, this is E, this is k, here the top of the valence band and the bottom of the conduction band have the same value of k. So any transition from the conduction band to the valence band can be accompanied by release of energy as a photon and this makes it a direct band gap semiconductor. In the case of gallium arsenide the band gap is 1.42 ev, this energy is released as a photon so the wavelength of the photon using the same formula that we saw before it is nothing but hc over eg which is approximately 870 nanometers. This lies in the IR region if you had a material like zinc oxide its band gap is around 3.4 ev and the wavelength will lie in the uv region. So if you have different materials with different band gaps and all of these are direct then their optical transitions will have different wavelengths. So let us go back to the picture of a semiconductor you have a valence band that is full and you have a conduction band that is empty. So at any temperature you can have electrons from the valence band being excited to the conduction band. When an electron goes from the valence band to the conduction band you essentially have an absence or a loss of electron. This represents the absence of an electron is called a hole. So a hole is nothing but a mathematical construct that we use to denote the absence of electrons in the valence band. So in any semiconductor you have electrons in the conduction band and you have holes. Holes are denoted by h or h plus in the valence band. These electrons and holes are essentially free to move so they are called delocalized. So when you apply an electric field to a semiconductor the electrons can move in the conduction band the holes can move in the valence band and this gives rise to current. So along with this concept of electrons and holes you also introduce a new concept called electron effective mass. You also have a whole effective mass but the argument is very similar. Let us say you have a free electron or an electron in free space and you apply an electric field E. Electric field usually goes from positive to negative. So the electron travels in the direction opposite to the field. For such an electron the force acting on it is nothing but the charge times the electric field which is equal to the mass of the electron times the acceleration. So the acceleration is equal to the electric field divided by the mass of the electron. Now this is the picture for an electron that is in free space. What about an electron in a solid? Now you have a solid can take this to be silicon there is an electron here. Once again you apply an electric field going from positive to negative. So the electron will accelerate in the direction of the field. Now if you try to write the acceleration of the electron you have the term E which is the same as before. So that is the effect of the external field but you also have a solid where you have a lot of atoms. And an electron can feel the influence of all of these atoms. So you not only have an external field you also have an internal field. This we will write as sum of all the internal fields. So F internal reflects the effect of all the atoms on the electron that is moving in a solid. So the acceleration now is complicated because you also have to take into account the effect of the atoms. In order to simplify this picture we replace this expression with this Me by another term which we call the effective mass. So this F internal and Me are replaced by this term Me star. Me star is called the effective mass of the electron. So this represents the effect of all the atoms in the solid on the movement of the electron. It is important to note that there is no change in the actual mass of the electron. We are not changing the mass. You are only replacing the effect of the solid by using another term called Me star. What is true for electrons is also true for holes because a hole is nothing but an absence of an electron. So we also have a term called MH star which we call effective mass of the hole. Now the effective mass term will basically affect the conductivity of a material. Lower the value of Me star and MH star higher is the conductivity. One way we can think about it is that if these values are lower then the influence of the lattice is less and so you have higher conductivity. Let us look at some typical values for Me star and MH star. Here we will start with metals. We look at metals like copper, silver and gold. So these metals are called nearly free electron metals. In this we can describe the behavior of electrons in them. Typically the valence electrons are a nearly free electron model. This means that these electrons do not see the influence of the nucleus. They are delocalized and they are free to move through the entire metal. In such a case we would expect Me star and MH star to be close to 1. That is what we get. Copper, silver and gold Me star over Me. Silver is again 0.99 which is close to 1. Copper is also close to 1. Gold is similar. Let us look at semiconductors. So we will first look at the electron. We have silicon. Typical value is 1.09, germanium which is around 0.55, gallium arsenide 67, zinc oxide 0.29. Can also look at the effective mass of the holes. So this represents the influence of all the atoms on the movement of a hole in the valence band. Now you have MH star H. This is around 1.15 for silicon, it is around 0.37 for germanium 0.45, gallium arsenide around 1.21 for zinc oxide. So these values of the effective mass depend upon the band structure. So the effective mass depends upon the E versus K plots that we saw earlier. So in the next class we will look at a couple of concepts, one of which is called the density of states. Then we will also look at a concept called the Fermi energy. We will use these to calculate the electron hole concentrations in semiconductors and then proceed from there.