[MUSIC] In this lecture, we're going to learn the why semiconductor has a bandgap. This is came from the physics of quantum mechanics and solid state physics. All you need to know is that silicon has the bandgap, which act like a semiconductor. But these are not directly related to the semiconductor device. Therefore, we're going to look free overview, what we learn the inner quantum mechanics and solid state physics. However, even if you don't know the detail, as long as you understand that silicone is a bandgap, you don't have any problem understand semiconductor device that we going to learn from the lecture three. Let's understand that silicon has the bandgap from a little different perspective. This is the EK diagram of the free electron. K is momentum which is the hard k. And depending on the momentum, they have the energy ways. Parabolic relationship, E equal pass care case care per two M. Assuming that electron is moving with the disk k momentum vector in silicon lattice. From the X-ray Bragg reflection, electron cannot propagate at certain K vector because they reflected back to the Bragg reflection. Those Bragg reflection condition is n lambda = 2a. So means that 2a distance difference is should be corresponding to the n lambda means those wave is cancelled. And electron movement has both property of a particle and wave. Therefore, the De Broglie equation, lambda should be 2pi per k. Combining this electron with the k vector equal npi per a cannot propagate the silicon lattice because of De Broglie diffraction. So, free electron E-k diagram is becomes like this and they have a discrete energy level where the npi per a. Because the periodic nature of a crystal structure, these are symmetric and they can move to the other k vector, where this becomes the here and this becomes the here. Then they have energy bands showing here. This becomes a here and this becomes here, another energy level here and bottom is there. Therefore, at certain k vector, E-k diagram can be drawn showing me here there is energy bandgap. This is only one direction of the K vector, electron can move different K vector and direction. Therefore, we have to super impose each direction of the K. This is the conclusion. Different to K vector, then diagram can be drawn in this graph and energy allowable state, that balance bend has a row, row stays to here in silicon case conduction. A balance band shown in here, the difference of the balance band and conduction band, this become bandgap. Compare this E-k diagram and free electrons. Another concept that we want to learn is the effective mass. Electron has own mass. However, electron in a crystal lattice, they interfere with the silicon as different, Phenomenon. To compensate those electron movement in silicon lattice, we calculated effective mass, so that we calculate this effective mass as a normal equation. Kinetic energy is 1 over 2 mv square. And momentum square per 2m, momentum is hard k. Therefore second derivative of energy becomes the hard square per mass. These mass is called the effective mass. You can tell that effective mass is the second derivative of energy. So, this is the E-k diagram, second derivative of the E by the weight vector becomes the effective mass. So this is the band diagram of the semiconductor, germanium, silicon and gallium arsenide. Interestingly, gallium arsenide has the direct to semiconductor, because the lowest conduction band, and highest valence band at the same k vector. Silicon and germanium, is the indirect semi conductor because the lowest of conduction band and highest balance bands at different k vector. Direct to semi conductor can emit to the li of a proton. Because electron can easily move from the conduction band to the balance band. However, indirect semiconductor with germanium and silicon cannot emit to the li because they have a different k vector of all conduction band and valence band. Indirect semi conductor is go through the phonon and proton compared to the direct semiconductor is only proton because the phonon is very slow phenomenon and directed at the lattice vibration with the electron and silicon atom generating the heat. So an indirect semiconductor can be used in optoelectronic semiconductor, like LED. So, let's summarize. This is the E-k diagram at k direction of 100 and then there is a bandgap. This is the band diagram in k direction 110, they have a bandgap, superimpose this, then there is allow overstate here, allow overstate here are both balanced in conduction band. There's a providence energy level showing in here, this is the bandgap above semiconductor, in contrast to the semiconductor mirror cases. There is a 100 there, can be a bandgap, 110, they can have a bandgap. But superimposed this, there it is, all allowable energy state, which doesn't have bandgap, then this becomes the metal. So, insulator has very high energy bandgap, semiconductor is moderate energy bandgap and metal has conduction band and valance band is overlap, there is no bandgap. For example, silicon semiconductor bandgap is one point one corresponding to the resistivity ten to the five ohm centimeter. Insulator silicon oxide had the band gap of 9 electron volt, resistivity is the 10 to the 16 ohm centimeter. And conductor, they don't have a band gap and very low resistivity of ten to the five ohm centimeters.
