noc19-cy16-Lecture 59 - Band Gap and Optical Properties

Now, I will go to the fourth lecture of this last week of this course and in this lecture, I will show how to use the band structure to understand some very basic optical properties of materials. In particular, I will be looking at what is the condition on the band structure in order to have materials that can be used for optical emission like light emitting diodes or for optical absorption like solar cells. So, these two are very important modern materials and we will be looking at the optical properties of these materials. So, we will go to week 12, lecture 4, it will be band gap and optical properties. Now, the most important concept that decides the optical properties is the nature of the band gap, whether it is direct or indirect. So, we look at materials that have a band gap and we will ask the question is the band gap direct or indirect. So, what is the meaning of a direct band gap? So, if you have a band structure, let me draw a band structure, I mean this is some wave vector axis and you have the energy as a function of wave vector. This is k along some direction and the band structure will have, let us say, it has various bands and let us look at a band structure like this and now we see that in this structure there is a possibility of a band gap in such a material and we see that all through this region, there are bands all the way from here to here, you have some band. So, if you go all the way down here, you have this band, then you have these two bands, you have one or more bands in this region. Similarly, you have one or more bands for these set of energies and so you have, so these are two bands in the material and this is the band gap. When we analyze this a little further, when we analyze the bands a little further, then we get some more information about these bands, about the different bands and often we see that the bands here, for example, they are sometimes referred to as the valence band and these are referred to as the conduction band and then there is a band gap in the material. This is very common for lot of the semiconducting materials and so the semiconductors or insulators that have a band gap, often this above the band gap, the states are called conduction band. So, this band is called the conduction band and this is called the valence band. Typically, this is a very common feature of the structure of lot of materials. Now, we would not go into the details of why this is called valence band and why this is called conduction band, but essentially, these materials typically have some band gap and there is a conduction band and a valence band. Now, this point is the lowest energy in the conduction band. This point is referred to as the bottom of the conduction band and this point is referred to as the top of the valence band and this band gap is basically the energy difference between this state and this state. So, this is the band gap and it is denoted by the symbol EG. So, the band gap is denoted by EG. Now, we notice that the top of the conduction band and the top of the valence band are not located directly below each other. So, we see that they are located at different k. So, in this case, when the top of valence band and bottom of conduction band have different k and different wave vectors, this is referred to as an indirect band gap. Now, you could have another material, where things are slightly different. So, again, E versus E versus k, you might have a relation. Now, the bottom of the conduction band and the top of the valence band are directly below each other. So, they are at the same k. Now, such a band gap is called a direct band gap. Direct band gap is nothing but the opposite of the indirect band gap, where the top of the valence band and the bottom of the conduction band are directly at the same wave vector. Now, what is the implication of this? What we will see is that if you want an optically active material that can be used for optical devices, you need a direct band gap. We will see why that should be the case. Now, we said earlier that k vector is referred to as the crystal momentum and the typical range of values of k in the first Brevon zone. The typical value is of order 2 pi by a. It is the lattice parameter. So, this is the typical value of the crystal momentum and a is typically a few angstroms. So, depending on the value of a, it will be about 1 to 5 angstrom inverse, depending on what the value of a is. It may be 1 to 2 angstrom inverse. So, a is typically about 3 to 6 angstroms and 2 pi is about 1 to 2 angstrom inverse. It is the typical value of the crystal momentum. What that means is that if there is a photon, the energy of the photon is given by Planck's constant times the frequency of the photon. So, photon of frequency nu has energy h nu and there is another relation called the De Broglie relation, which says that the wavelength of the photon is the Planck's constant divided by the momentum of the photon. So, this implies that the momentum of the photon is the Planck's constant divided by the wavelength of the photon. So, I should emphasize that it is not just k that is the crystal momentum. There is a h bar in this and I will just say that k is of order. This is the typical size of k. So, coming back to the momentum of the photon, this is h by lambda. So, if you look at this value, the typical photon has visible light. So, if lambda is in visible range and let me write this momentum in a slightly different way. So, I will write this as h bar into 2 pi by lambda. So, if lambda is in visible range, then basically lambda is of order of you know visible range is about 500 nanometers. That is about 5000 angstrom and what you can see is that p for the photon is h bar times 2 pi by 5000. So, it is h bar times something like 10 power minus 3 angstrom. So, this is a typical value. So, it is of order h bar times 10 power minus 3. So, 2 pi by 5000, 2 pi is about 6. So, 6 by 5 is of order 1 and you have 1 by 1000. That is 10 power minus 3 angstrom inverse, whereas for the crystal momentum. So, h bar k is of order h bar times about 1 angstrom 1 to 2 angstrom inverse. So, basically the conclusion we have is that momentum of photon is typically much smaller than momentum of crystal. So, I should not say momentum of crystal. I should call it the crystal momentum. So, momentum of photon is much smaller than the crystal momentum and this has an implication. If there is an electron that is there in the valence band and it absorbs the photon and it goes to the conduction band, then basically the change in momentum is very small. So, change in momentum on going from the conduction band to the valence band is typically very small. So, that means change in crystal momentum on absorbing or emitting photon is very small. That means an absorption or emission of a photon is not going to change the crystal momentum. So, what does this imply? That implies that if you have a material with a direct band gap, I will just focus on the band gap. So, I will just focus on this part of the band structure. If it were to make a transition from the bottom of the conduction band to the top of the valence band, then it would emit some light. Now, what we said is that this transition does not change the momentum. So, if it emits light, it is not going to change its momentum. So, if it has a direct band gap, then it can actually emit this light. Whereas, on the other hand, if it has an indirect band gap, if you have a material with an indirect band gap, now if there is a transition from the top of the conduction band to the bottom of the valence band, this involves a large change in crystal momentum. So, this is a large change in crystal momentum. So, there is a large change in crystal momentum on going from the bottom of the conduction band to the top of the valence band and large change in crystal momentum. So, this is given out as lattice vibrations or heat. So, that means light will not be emitted during such a process. During such a process, because the crystal momentum is changing a lot, there will be large lattice vibrations and lot of heat will be generated or lot of heat will be generated during this process, but you would not get any light emission. Whereas, in this case, there is no change in the case of a direct band gap. There is no change in the crystal momentum. So, in this case, there is a possibility of light emission. So, light emission is possible in this case. So, the typical light emitting materials are materials that have a direct band gap. One example is gallium arsenide. This has a direct band gap. The direct band gap of gallium arsenide is 1.424 electron volts. So, gallium arsenide has a band gap of 1.424 electron volts. Now, if an electron makes a transition from the bottom of the conduction band to the top of the valence band, then it will emit a photon. The wavelength of the photon will be given by the energy divided by h c. This energy is in electron volts. When you put the values of the Planck's constant, the speed of light and the energy converted to joules, you will get a wavelength of 1.424 electron volts. This works out. If you work out the answer, you will get 870 nanometers. So, for gallium arsenide, gallium arsenide can work as an LED that emits light at 870 nanometers and this is near IR. So, the visible range is about 380 to 640. So, 870 is in the IR range, in the infrared range. On the other hand, if you look at gallium nitride, that has a large band gap that is 3.4 electron volts. So, this is the band gap. So, the band gap is 3.4 electron volts and now if you calculate lambda, you will get something like 380 nanometers and this is very close to the violet or just in the UV region. So, this is violet slash UV region and by manipulating these materials, you can get light emitting diodes of the entire spectrum of light and these are used in everyday applications including all the computer screens, all the smart phone screens, etcetera. So, what we see is that light emission is usually done by materials like gallium arsenide and gallium nitride and by manipulating them, you can get the whole visible range. Now, what about light absorption? So, light absorption is the exact opposite of emission and usually direct band gap materials are most efficient like efficient light absorption and this is used in solar cells. Now, it turns out that for commercial reasons, silicon has an indirect band gap and is often used in solar cells mainly because it is available everywhere. Silicon is an abundant material. So, it is cheaply available and it is still used in solar cells. Now, it is much less efficient and typically, if you want to make a photovoltaic cell using silicon, you need 100 micrometers thick silicon crystals. In fact, for photovoltaic, mostly this polycrystalline silicon is used, but you need it to be at least 100 micrometer thick. Now, gallium arsenide, on the other hand, is a direct band gap. It is very efficient, but it is expensive. Gallium is not a very abundant material. So, it is quite expensive and very efficient and for the same light absorption efficiency, you need only 1 to 10 micrometer gallium arsenide crystal. So, the light absorption efficiency of gallium arsenide is much better than that of silicon, but nevertheless, silicon is still used because it is so widely available. However, if you are making like thin film photovoltaics, for some application, you need a very thin film photovoltaic. Then, you use materials like gallium arsenide. There are other materials also like cadmium telluride and so on. So, these are materials that are used in very special cases, where you need thin film photovoltaics, but by and large, silicon is still widely used in solar cells. With this, I will conclude this lecture. This is the fourth lecture of week 12. In the next lecture, I will summarize what we learnt in this week and then do a few practice questions. Thank you.