In previous lecture, we learned about crystal of the silicon. Crystal is the diamond structure which is you learn in the sophomore of a crystallography. Now we're going to learn why silicon has a band gap and act like semiconductor. This lecture is related with quantum mechanics and solid state physics. If you want to learn more detailledly, please report to those lectures. We're going to overview briefly. In 1920, many observation revealed that atom and electron did not obey the classical laws of mechanics. New approach called quantum mechanics was proposed. First, Heisenberg uncertainty principle, which is the Delta x times Delta P which is momentum is over the upper two, or it can be expressed, Delta energy times Delta T is over to the Harper two, means that simultaneous measurement of position and momentum, or energy and time are inherently inaccurate. Instead of finding the exact number, we must look for the probability of finding electron at certain position. To solve this equation, Schrodinger made the Schrodinger wave equation in 1926. This equation showing down here is from the energy. Total energy is addition of the kinetic energy plus potential energy using quantum operator. Momentum is this, and energy is this, inserting this quantum operator to here, then final Schrodinger wave equation becomes like this. Energy of this come to here, becomes the energy of here. Kinetic energy is momentum square per 2m, P inserted here becomes like this. This is the potential. Here, Psi is wave function called Psi. This is the Laplacian. Psi squared becomes the probability that we want to find. This time-dependent to Schrodinger equation can translate to the time-independent equation. If you're removing T, then it becomes like this. Let's assume the particle in a box. This slide shows that, if electron in certain boxes that has the potential barrier is zero at zero to L, and potential is infinite at X equal zero, potential X equal L, infinite potential. Because of the infinite potential, electron cannot go over this barrier. They stay in a box. Where is the electron in this particle in a box? Can you predict them accurately, or we have to approaching improbability using Schrodinger equation? Let's solve the Schrodinger equation. What is the probability of electron finding here? First, we define the boundary condition. Boundary condition in electron from zero to L, potential is zero, and potential X equals zero, and L potential is infinite. Starting from the time-independent to Schrodinger equation and insert to this boundary condition 1 to 3, then potential should be zero. Schrodinger equation becomes this. To satisfy this equation, wave function of Psi should be a format of the A sine KX because the second derivative of a Psi should there be a exactly same form of the sine because the one first derivative becomes a cosine and second derivative becomes a sine, so the equation satisfying this to zero, then Psi, should be format of the sine KX. K should be root to 2ME per ha. Another boundary condition, which is the boundary condition two potential is infinite at X equal L, then there is no probability finding the electronic X equal. L, because potential is so high, electrons cannot be in this x equal L. So probability is size care and probability is zero, then this Psi format should be zero, and K equal L. To Psi becomes a zero at x equal L, then k should there be in pi format. So k equal n pi per L and the upper equation k equal root to 2mE, if you're combining them, then equation becomes like this, energy becomes with this equation. This equation means that particle in a box or electron in this box, has a discrete energy level depending on any core, 1, 2, 3, 4, 5. Also, another meaning that Psi square is probability. Therefore, depending on the n 1, 2, 3, then Psi becomes a sine format n equal one, this format, n equal two, this format, n equal three, this format. Psi square becomes the probability for n equal to square becomes the upper level. Then this is the probability. So particle in a box, electron in a box, has a discreet energy level and probability of finding electron is highest at n equal two in this region. This is the approach of quantum mechanics.Then let's serve silicon lattice. Silicon lattice has a silicon atom in the center, and there's 14 electron in outside. To solve Schrodinger equation in silicon atoms, you will have to define the potential with this equation means that close the silicon atom potential is very high and far from the silicon lattice, potential becomes almost zero. Psi wave function should be defined by the three factor of r radius Zeta M pi on 3D format of the wave equation. Solving Schrodinger equation with this is very complicated. I will not go over this. This is in solid state physics. The conclusion is that after solve Schrodinger equation, there will be restriction equal n equal 1, 2, 3, l, m and s, and silicone has a discrete energy level. This is the silicon atoms. What happened, silicone becomes the silicon lattice that has done two FCC inside. Then this is the one silicon lattice. There is another silicon atom, and there is another silicon atom. When they fall apart, they have discrete energy level. As we predicted in previous slide, when they become close together, then the energy level is interacting each other, and energy level makes the discrete energy level of anti-bonding energy level, bonding energy level. For example, here, S energy level becomes the anti-bonding energy level and bonding energy level. P becomes the anti-bonding energy level and bonding energy level. This area is allowable energy state, and this area is also allowable energy state. This is called the conduction band, this is called a valance band. Between the conduction band and valance band, there is ban gap that electron cannot have this energy level. This is why semiconductor has a band gap. If there is a band gap, that becomes the semiconductor. A little bit more complicated, equation in solid state physics, they serve silicon lattice without Kronig-Penny model that silicone is located, lattice showing in here. Because of the silicon lattice, there is the energy barrier potential, and they define the boundary condition of energy barrier. Let's say that this area potential is zero, this certain energy barrier. Then using the Kronig-Penny model solving the Schrodinger equation, they served that silicone as a band gap. As I said, this might sound very difficult, but even if you don't know this material, you will not have any problem of understand silicon device. But let's focus that it is okay that if you understand that there is a silicon band gap using quantum mechanics or solid the physics, that's all you need to understand.
