Solid State Electronics Homework #2
2.1 The work function of a material refers to the minimum energy required to remove an electron from the material. Assume that the work function of gold is 4.90 eV and that of cesium is 1.90 eV. Calculate the maximum wavelength of light for the photoelectric emission of electrons for gold and cesium.
2.2 Calculate the de Broglie wavelength, l = h/p, for: (a) an electron with kinetic energy of (i) 1.0 eV, and (ii) 100 eV. (b) A proton with kinetic energy of 1.0 eV.
2.3 According to classical physics, the average energy of an electron in an electron gas at thermal equilibrium is 3kT/2. Determine, for T = 300 K, the average electron energy (in eV), average electron momentum, and the de Broglie wavelength.
2.5 It is desired to produce x-ray radiation with a wavelength of 1 Å. (a) Through what potential voltage difference must the electron be accelerated in vacuum so that it can, upon colliding with a target, generate such a photon ? (Assume that all of the electron’s energy is transferred to the photon.) (b) What is the de Broglie wavelength of the electron in part (a) just before it hits the target?
2.6 Calculate the energy of an electron in the hydrogen atom (in units of eV) for the first four allowed energy levels.
2.7 Show that the most probable value of the radius r for the 1s electron in a hydrogen atom is equal to the Bohr radius a0.
2.9 The solution to Schrödinger’s wave equation for a particular situation is given by. Determine the probability of finding the particle between the following limits: (a) 0 £ x £ a0/4, (b) a0/4 £ x £ a0/2, and (c) 0 £ x £ a0.
2.12 Consider a three-dimensional infinite potential well. The potential function is given by V(x) = 0 for 0 < a, 0 < y < a, 0 < z < a, and V(x) = ¥ elsewhere. Start with Schrödinger’s wave equation, use the separation of variables technique, and show that the energy is quantized and is given by
where nx = 1, 2, 3,…, ny = 1, 2, 3,…, nz = 1, 2, 3,…..
2.13 Evaluate the transmission coefficient for an electron of energy 2.2 eV impinging on a potential barrier of height 6.0 eV and thickness 10-10 m. Repeat the calculation for a barrier thickness of 10-9 m. Assume that Equation (E.27) in Appendix E is valid.
2.16 Consider Figure 2.8b, which shows the energy-band splitting of silicon. If the equilibrium lattice spacing were to change by a small amount, discuss how you would expect the electrical properties of silicon to change. Determine at what point the material would behave like an insulator or like a metal.
2.17 The bandgap energy in a semiconductor is usually a slight function of temperature. In some cases, the bandgap energy versus temperature can be modeled by
where Eg (0) is the value of the bandgap energy at T = 0 K. For silicon, the parameter values are Eg (0) = 1.170 eV, a = 4.73 ´ 10-4 eV/K and b = 636K. Plot Eg versus T over the range 0 £ T £ 600 K. In particular, note the value at T = 300K.
2.19 (a) Determine the total number of energy states in silicon between Ec and Ec + kT at T = 300 K. (b) Repeat part (a) for GaAs.
2.20 (a) Determine the total number of energy states in silicon between Ev and Ev – kT at T =300 K. (b) Repeat part (a) for GaAs.
2.22 Find the ration of the effective density of states in the conduction band at Ec + kT to the effective density of states in the valence band at Ev – kT.
2.23 Plot the Fermi-Dirac probability function, given by Equation (2.29), over the range -0.2 £ (E - EF) £ 0.2 eV for (a) T = 200 K, (b) T = 300 K, and (c) T = 400 K.
2.24 (a) If EF = Ec, find the probability of a state being occupied at E = Ec + kT. (b) If EF = Ev, find the probability of a state being empty at E = Ev – kT.
2.25 Determine the probability that an allowed energy state is occupied by an electron if the state is above the Fermi level by (a) kT, (b) 3 kT, and (c) 6 kT.
2.32 Consider the energy levels shown in Figure P2.32. Let T = 300 K. (a) If E1 – EF = 0.30 eV, determine the probability that an energy state at E = E1 is occupied by an electron and the probability that an energy state at E = E2 is empty. (b) Repeat part (a) if EF – E2 = 0.40 eV.
2.35 Assume the Fermi energy level is exactly in the center of the bandgap energy of a semiconductor at T = 300 K. (a) Calculate the probability that an energy state at E = Ec + kT/2 is occupied by an electron for Si, Ge, and GaAs. (b) Calculate the probability that an energy state at E = Ev – kT/2 is empty for Si, Ge, and GaAs.
2.37 Calculate the energy range (in eV) between fF (E) = 0.95 and fF (E) = 0.05 for EF = 7.0 eV and for (a) T = 300 K and (b) T = 500K.
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