Module 40: Tunneling Lecture 40: Step potentials

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1 Module 40: Tunneling Lecture 40: Step potentials V E I II III 0 x a Figure 40.1: A particle of energy E is incident on a step potential of hight V > E as shown in Figure The step potential extends from x = 0 to x = a, the potential is zero on either side of the step. The particle has wavefunction ψ I (x,t) = e iet// h [ A I e ipx/ h + B I e ipx/ h] (40.1) in region I to the left of the step. The first pat A I e ipx/ h represents the incident particle ie. travelling along +x direction the second part B I e ipx/ h the reflected particle travelling along the x axis. In classical mechanics there is no way that the particle can cross a barier of height V > E. In quantum mechanics the particles s wave function penetrates inside the step in region II we have ψ II (x,t) = e iet/ h [ A II e qx/ h + B II e qx/ h]. (40.2) In region III the wave function is ψ III (x,t) = e iet/ h [ A III e ipx/ h + B III e ipx/ h] (40.3) where the term A III e ipx/ h represents a particle travelling to the right B III e ipx/ h represents a particle incident from the right. In the situation that 219

2 220 CHAPTER 40. TUNNELING we are analyzing there are no particles incident from the right hence B III = 0. In quantum mechanics the wave function does not vanish in region II. As shown in Figure 40.1 the incident wave function decays exponentially in this region, there is a non-zero value at the other boundary of the barrier. As a consequence there is a non-zero wavefunction in region III implying that there is a non-zero probability that the particle penetrates the potential varrier gets through to the other side even though its energy is lower than the height of the barrier. This is known as quantum tunneling. It is as if the particle makes a tunnel through the potential barrier reaches the other side. The probability that the incident particle tunnels through to the other side depends on the relative amplitude of the incident wave in region I the wave in region III. The relation between these amplitude can be worked out by matching the boundary conditions at the boundaries of the potential barrier. It is now possible to fabricate microscopic potential wells using modern semiconductor technology. This can be achieved by doping a very small regions of a semiconductor so that an electron inside the doped region has a lower potential than the rest of the semiconductor. An electron trapped inside this potential well will have discrete energy levels E 1, E 2, etc. like the ones calculated here. Such a device is called a quantum well photon s are emitted when electron s jump from a higher to a lower energy level inside the quantum well. The wave function its x derivative should both be continuous at all the boundaries. This is to ensure that the Schrodinger s equation is satisfied at all points including the boundaries. Matching boundary conditions at x = 0 we have ψ I (0,t) = ψ II (0,t) (40.4) ( ) ( ) ψi ψii = (40.5) x x x=0 x=0 We also assume that the step is very high V E so that q = 2m(V E) 2mV (40.6) we also know that p = 2mE, so p E q = V 1 (40.7) Applying the boundary conditions at x = 0 we have A I + B I = A II + B II (40.8) ip (A I B I ) = q (A II B II ). (40.9)

3 40.1. SCANNING TUNNELING MICROSCOPE 221 The latter condition can be simplified to A I B I = iq p (A II B II ) (40.10) Applying the boundary conditions at x = a we have A II e qa/ h + B II e qa/ h = A III e ipa/ h (40.11) q [ A II e qa/ h B II e qa/ h] = ipa III e ipa/ h. (40.12) The latter condition can be simplified to A II e qa/ h B II e qa/ h = ip q A IIIe ipa/ h (40.13) Considering the x = a boundary first using the fact that p/q 1 we have A II e qa/ h B II e qa/ h = 0 (40.14) which implies that Using this in equation (40.11) we have B II = e 2qa/ h A II A II (40.15) A III = 2e ipa/ h e qa/ h A II (40.16) Considering the boundary at x = 0 next, we can drop B II as it is much smaller than the other terms. Adding equations (40.8) (40.10) we have ( ) as q/p 1, this gives us 1 + iq p Using this in equation (40.16) we have The transmission coefficient A II = 2A I (40.17) A II = ip q 2A I. (40.18) A III = 4i p q e ipa/ h e qa/ h A I. (40.19) T = A III 2 A I 2 = 16 p2 q 2e 2qa/ h (40.20) gives the probability that an incident particle is transmitted through the potential barrier. This can also be expressed in terms of E V as T = 16 E V e 2a 2mV / h (40.21) The transmission coefficient drops if either a or V is increased. The reflection coefficient R = 1 T gives the probability that an incident particle is reflected.

4 222 CHAPTER 40. TUNNELING + Figure 40.2: 40.1 Scanning Tunneling Microscope The scanning tunneling microscope (STM) for which a schematic diagram is shown in Figure 40.2 uses quantum tunneling for its functionaing. A very narrow tip usually made of tungsten or gold of the size of the order of 1Å or less is given a negative bias volatge. The tip scans the surface of the sample which is given a positive bias. The tip is maintained at a small distance from the surface as shown in the figure. Figure 40.3 shows the potential experienced by an electron respectively in the sample, tip the vacuum in the gap between the sample the tip. As the tip has a negative bias, electron in the tip is at a higher potential than in the sample. As a consequence the electrons will flow from the tip to the sample setting up a current in the circuit. This is provided the electrons can tunnel through the potential barrier separating the tip the sample. The current in the circuit is proportional to the tunneling transmission coefficient T calculated earlier. This is extremely sensitive to the size of the gap a. Vacuum x Figure 40.3: In the STM the tip is moved across the surface of the sample. The current in the circuit differs when the tip is placed over different points on the sample. The tip is moved vertically so that the current reamins constant as it scans

5 40.1. SCANNING TUNNELING MICROSCOPE 223 across the sample. This vertical displacement recorded at different points on the sample gives nn image of the surface at the atomic level. Figure 40.4 shows an STM image of a graphite sample. Figure 40.4:

Lecture 5. Potentials

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