In this video, we will solve the potential barrier problem. Potential barrier problem, is given by this potential profile shown here on the right, the potential is zero, outside this barrier region, which we defined between z equals 0, and z equals L. L here defines the width of this potential barrier, and the potential height is given as V naught, and once again, the particle is instant from the left with an initial energy E, and the energy E is assumed to be greater than zero, it can be either large or greater than the barrier height. Similar to the potential step problem that we solved in the previous module, we write down the Schrodinger equation, time-independent Schrodinger equation in each region. Now we have three regions. We have region 1, where z is negative, in this either region left of the barrier, and of course, there is the barrier regions, z between zero and L, and then region 3, is the right side of the barrier. Without writing down the time-independent Schrodinger equations for each region, we take advantage of the knowledge that we already learned from the potential step problem, and write down the solutions like this. In region 1, there is the incident wave propagating to the right, with a wave vector k naught, which is defined as shown here, and then there is the left propagating wave with the same wave vector k naught, and this represents the reflected wave, we already know, and so we denote the quotient as R, and then in region 3, right side of the barrier, we only have the right propagating wave, which is the transmitted wave, so we denote the coefficients as t. Inside the barrier region, we have both left and right propagating wave, because there can be multiple reflections at the two interfaces, and we specify their amplitude as A and B respectively, the wave vector inside of area region, is given by k as shown here. We note that we already normalized all the coefficients with respect to the incident wave, the amplitude of the incident wave, what's set to equal to one, and that means all the other coefficients have been normalized with respect to the amplitude of the incident wave, as we have found in the potential step problem, and of course, I already mentioned that the left propagating wave in region 3, is omitted because the problem does not allow that to exist in this case. To determine the coefficients, we, again, impose the boundary conditions at those two interfaces, there is a left interface and the right interface, at z equals 0 and z equals L, and we require continuity of the wave function and the derivative at those two positions, so we get these four equations here, and we can work out the four equations with four unknowns. We can solve this with some significant algebra, you will find that R and T and A and B can be written in this form, where the factor mu here, is defined as the ratio of the two wave vectors, k over k naught, which is simply given by this equation here, square root of 1 minus V naught over E. Now, we have found the full solution, and then now we can look at the transmission probability, which is given by T square here. First, let's look at the case where the instant energy is smaller than the barrier height. In this case, k, the wave vector, wave number inside of area region becomes imaginary, and therefore mu is also imaginary, and therefore there is no propagating wave inside the barrier. Inside the barrier, the wave exponentially decays. Despite that, the barrier has a finite width, and therefore the barrier region ends before the wave function fully decays to zero, which means that there is always a non-zero finite amplitude for the transmitted wave. The transmission probability is always positive, always greater than zero, and this phenomenon, this non-classical phenomenon, is called tunneling, sometimes quantum mechanical tunneling. The transmission probability increases monotonically as shown here in this region with increasing energy, which makes sense intuitively understandable. As I mentioned, the wave is evanescent, exponentially decaying inside the barrier region. When the energy is greater than the barrier height, both the wave number k and Mu are real and positive number, so we now have a propagating wave even inside the barrier region. The transmission probability interestingly oscillates. It shows this oscillatory behavior as shown in this graph, it reaches unity at particular energy. These energies are given by this equation here, and n here is integer. This condition, when the transmission probability reaches one is the condition for resonant tunneling. The fact that there is this oscillation, the fact that the transmission probability is less than one in most cases, except for a particular set of energies, is also a non-classical behavior. Finally, I make a note that this solution to what we have done for potential barrier applies equally well to the negative value of V naught. For negative value of V naught, we have finite potential well, as opposed to finite potential barrier. The difference between this problem, potential barrier problem with negative barrier height and the finite potential well problem that we discussed before, the difference is that previously in the finite potential well problem, we were only looking for bound states. We were only looking for states with energy less than the depth of the potential well. We found a finite number of allowed energy levels inside the potential well in that case. In this problem, potential barrier problem with negative barrier height, this is a finite potential well problem with energy greater than the potential depth. This represents an unbound system. This is a scattering problem essentially as opposed to finding energy levels of a confined system. That's the difference. I want to close this video with a few examples of tunneling. The tunneling phenomenon resulted in many Nobel Prizes in physics. One of them is scanning tunneling microscope. Here is a diagram or a drawing of a scanning tunneling microscope. It basically involves a very sharp, atomically sharp metal tip. That metal tip is brought very close together to a surface of the specimen that you want to investigate. When these gap between the tip and the sample surface is small enough, then there is a substantial probability of electron tunneling from one to the other, producing a current. By measuring a current, you can find an information about the distance between the sample and the tip. By modulating, by providing a feedback with this measure of the current, you can actually scan this tip at a constant distance across the sample surface, obtaining an atomically precise, atomically accurate surface morphology. Let's look at the tunneling probability for a typical situation. We're talking about electron tunneling, so the mass of the electron is 9.1 times 10 to the negative 31st power kilogram. Barrier height is given by the energy difference between inside and outside the metal tip, and that difference is typically called a work function of the metal. For a typical metal, that's several electron bolts. Let's say that barrier height is five electron volts. The barrier width or the thickness of the potential barrier, this is the distance between the tip and the sample surface. Let's set that to a small number, 0.5 nanometers in this case. The electron's kinetic energy, it's defined by the applied voltage that you apply to this apparatus. Let's say that that's about one electron volt. Using the formula that we derived in the previous slide, we can calculate the tunneling probability to be about 10 to the negative fourth, 0.1 percent. That is small, but it is actually a substantial probability and it leads to a measurable current in this case. Scanning tunneling microscope is a widely used microscopy technique that allows you to obtain atomic-scale resolution. It's a very powerful imaging technique. In contrast, let's consider a case for a man running in Boulder, Colorado. Here is a beautiful Boulder campus of the University of Colorado. We're sitting at the foot of these beautiful Flatirons peaks. Let's consider a man jogging in Boulder and assuming that that person has a certain weight, mass, and a certain speed, kinetic energy is assumed to be about 3,000 joules. This mountain here on the backside, Flatirons, is approximated with a rectangular barrier gravitational potential. The barrier height, which is given by the altitude difference between the Boulder campus and the top of this mountain peak, it's about 800 meters, and that leads to about five times 10^5 joules in terms of gravitational potential energy. The width of this mountain between Boulder, and there is a big lake in the backside, is about five kilometers. You can see that this is a rather flat and very wide potential barrier and yet the barrier height is over of magnitude greater than the initial kinetic energy of the particle or a person running. If you calculate the tunneling probability in this case, well, this is a very small number, 10 to the negative 10 to the 75th power. The exponent has 75 zeros. That's what I meant. It's a very small number. There hasn't been a new report of man tunneling through the mountains in Boulder area or anywhere else as far as I can tell and I do not expect that to happen anytime soon. Now lastly, I want to introduce a semiconductor device that is used to modulate current. It's called the resonant-tunneling diode. It has this double potential barrier, and in-between, there is a potential well. In the potential well, there is a confined state or allowed energy level. By controlling the applied voltage to this device, you can change the relative alignment between the conduction band of the semiconductor region outside the barrier and the energy level of this confined state in between the two barriers. When they are aligned, there is a resonant tunneling. The tunneling probability increases by many orders of magnitude, rapidly increasing the current. When they are misaligned, then the current becomes very, very small. That's how you modulate the current. Also, this device produces negative differential resistance as well. This is one example of using the quantum mechanical tunneling phenomenon for a useful electronic device.
