1. Potential Step Quantum Physics 2002 Recommended Reading: Harris Chapter 5, Section 1

2. Transmission and Reflection of Light incident wave reflected wave transmitted wave Consider a ray of light incident on a glass slide. Some of the incident light will be reflected and some transmitted. If I 0 is the intensity of the incident light (number of photons pre unit area per second), I R theintensity of the reflected light and I T the intensity of the transmitted light then we define. and from conservation of energy R+T = 1 assuming no absorption in the glass.

3. Transmission and Reflection of Matter Waves According to deBroglie hypothesis matter behaves as a wave. So does a matter wave divide into a reflected wave and a transmitted wave when it encounters an abrupt change in potential? Need to distinguish between particles moving in different directions. In 1-Dimension we need to be able to distinguish between particles moving to the left and those moving to the right For a free particle of mass m, the solution to the Schrodinger equation are plane waves (see previous lecture) particle moving to the right, +x direction particle moving to the left, -x direction the time dependent part ( f(t) = exp(-i  t) )is the same for both particles so we can leave it out of the analysis

4. x U = 0 U = U 0 E = K.E. III 0 U K.E. E = K.E + U 0 Potential Step Two possibilities: (1) total energy E of particle is greater than the potential energy U 0. (2) total energy E of particle is less than the potential energy U 0. x U = 0 U = U 0 E < U III 0 U E = K.E. x > 0 is classically forbidden region

5. Experimental Realisation of a Potential Step x U = 0 U = U 0 E = K.E. III 0 U K.E. 0 Volts -U 0 Volts -e E = K.E + U 0 If E is > U 0 then electron is transmitted through the electrodes If E < U0 then electron will be reflected electron with total energy E

6. Energy Bands in Solids In a semiconductor we have allowed energy bands, I.e. Valence band and Conduction band separated by a Band gap. Band gap Conduction Band Valence band Energy Different semiconductors have different band gaps. For example: Si E G = 1.11 eV Ge E G = 0.66 eV GaAs E G = 1.43 eV InAs E G = 0.33 eV If we grow one semiconductor material on top of another we can get a discontinuity in the potential. An electron moving through the material may be reflected or transmitted when it encounters this discontinuity. This will effect the ‘transport properties’ of the device. InAsGaAs

7. Potential Step E > U 0 x U = 0 U = U 0 E = K.E. III 0 U K.E Solution in Region I: E Solution where 2

8. Region II: Solution where x U = U 0 III 0 3 4 No term because there is no particle incident from the right.

9. Continuity Conditions Only one boundary, at x = 0. Match wavefunction and derivative at x = 0. Solve for B and C in terms of A. 6 5 78

10. Reflection and Transmission Coefficients What happens a particle when it encounters a potential step? The particle does not split into two fragments, but rather it has a certain probability of being transmitted and a probability of being reflected. A better way to view this problem is to consider a beam of particles incident on the potential discontinuity. Then a certain fraction of them will be transmitted and the rest will be reflected. For example if there are 1000 particles per second in the incident beam and 400 are reflected per unit time while 600 are transmitted per unit time then: Total number of particles is conserved.

11. To find the reflection and transmission probabilities, we need to find the ratios of the numbers per unit time in the corresponding beams The number per unit time follows directly from  2. where v is the velocity of the particle Number per unit time The Transmission probability T, is then defined as Transmission Probability

12. Reflection Probability Similarly the Reflection probability R, is defined as Using these definitions and the values of B and C in terms of A we find 9 10 Recall and so

13. Reflection and Transmission Coefficients Things to note about equations 9 and 10: 1. R + T = 1 2. R is non-zero, so some particles are reflected even though they have E > U 0   non-classical behaviour, wavelike property. 3. When E = U 0, T = 0 and R = 1. 4. When E >> U 0,  R = 0 and T = 1, so at very high energy all of the particles are transmitted

14. Reflection and Transmission Coefficients E/U 0 R, T R+T=1 R T

15. Potential Step E < U 0 x U = 0 U = U 0 E = K.E. III 0 Region I: 11 Solution where 12 same as previous case

16. Region II: Solution where x U = U 0 III 0 13 14 No term because as note that  is always positive

17. Only one boundary, at x = 0. Match wavefunction and derivative at x = 0. Solve for B and C in terms of A. 16 15 17 18 Continuity Conditions

18. Reflection Coefficient For E < U 0 we get total reflection and no transmitted ‘wave’, this is the same as the classical case BUT there is a difference between this and the classical case! Reflection Coefficient 19 Transmitted current density?

19. 7. What happens in the region x > 0 Quantum mechanically there is a finite probability that the wavefunction penetrates the classically forbidden region, x > 0  finite probability of finding particle in region II (x > 0). This probability is a maximum at x = 0 and falls off exponentially to small values as x becomes large. The extent of the penetration into the classically forbidden region is governed by the constant . We define the penetration depth as 20 21 The closer E is to U 0 the greater is  and the slower is the decay of the wavefunction.

20. Question: What happens if we have a potential barrier instead of a potential step? x U = U 0 III 0 E x U = U 0 III 0 E III Again 2 cases to consider 1) E > U 0. Allowed both in both classically and quantum mechanics 2) E < U 0. Classically forbidden but allowed in quantum systems. QUANTUM TUNNELING.