1. Issued: May 5, 2010 Due: May 12, 2010 (at the start of class) Suggested reading: Kasap, Chapter 5, Sections 5.1-5.7 Problems: Stanford University MatSci 152: Principles of Electronic Materials and Devices Spring Quarter, 2009-2010 Homework #5 3. Kasap Problem 5.17 (Photoconductivity and speed) 5. Kasap Problem 5.22 (Direct recombination and GaAs) 1.Kasap Problem 5.3 (Fermi level in intrinsic semiconductors) 4. Kasap Problem 5.18 (Hall effect in semiconductors) 2. Kasap Problem 5.10 (Temperature dependence of conductivity)

2. Using the values of the density of states effective masses m e * and m h * in Table 5.1, find the position of the Fermi energy in intrinsic Si, Ge and GaAs with respect to the middle of the bandgap (E g /2). An n-type Si sample has been doped with 10 15 phosphorus atoms cm -3. The donor energy level for P in Si is 0.045 eV below the conduction band edge energy. a. Calculate the room temperature conductivity of the sample. b. Estimate the temperature above which the sample behaves as if intrinsic. c. Estimate to within 20 percent the lowest temperature above which all the donors are ionized. d. Sketch schematically the dependence of the electron concentration in the conduction band on the temperature as log(n) versus 1/T, and mark the various important regions and critical temperatures. For each region draw an energy band diagram that clearly shows from where the electrons are excited into the conduction band. e. Sketch schematically the dependence of the conductivity on the temperature as log(  ) versus 1/T and mark the various critical temperatures and other relevant information. 1.Kasap Problem 5.3 (Fermi level in intrinsic semiconductors) 2. Kasap Problem 5.10 (Temperature dependence of conductivity) The Hall effect in a semiconductor sample involves not only the electron and hole concentrations n and p, respectively, but also the electron and hole drift mobilities,  e and  h. The hall coefficient of a semiconductor is (see Chapter 2), where a. Given the mass action law, pn = n i 2, find n for maximum (negative and positive R H ). Assume that the drift mobilities remain relatively unaffected as n changes (due to doping). Given the electron and hole drift mobilities,  e = 1350 cm 2 V -1 s -1,  h = 450 cm 2 V -1 s -1 for silicon determine n for maximum in terms of n i. b. Taking b = 3, plot R H as a function of electron concentration n/n i from 0.1 to 10. 4. Kasap Problem 5.18 (Hall effect in semiconductors) Consider two p-type Si samples both doped with 10 15 B atoms cm -3. Both have identical dimensions of length L (1 mm), width W (1 mm), and depth (thickness) D (0.1 mm). One sample, labeled A, has an electron lifetime of 1  s whereas the other, labeled B, has an electron lifetime of 5  s. a. At time t = 0, a laser light of wavelength 750 nm is switched on to illuminate the surface (L  W) of both the samples. The incident laser light intensity on both samples is 10 mW cm -2. At time t = 50  s, the laser is switched off. Sketch the time evolution of the minority carrier concentration for both samples on the same axes. b. What is the photocurrent (current due to illumination alone) if each sample is connected to a 1 V battery? 3. Kasap Problem 5.17 (Photoconductivity and speed)

3. Consider recombination in a direct bandgap p-type semiconductor, e.g., GaAs doped with an acceptor concentration N a. The recombination involves a direct meeting of an electron–hole pair as depicted in Figure 5.22. Suppose that excess electrons and holes have been injected (e.g., by photoexcitation), and that Δn p is the excess electron concentration and Δp p is the excess hole concentration. Assume Δn p is controlled by recombination and thermal generation only; that is, recombination is the equilibrium storing mechanism. The recombination rate will be proportional to n p p p, and the thermal generation rate will be proportional to n po p po. In the dark, in equilibrium, thermal generation rate is equal to the recombination rate. The latter is proportional to n no p po. The rate of change of Δn p is where B is a proportionality constant, called the direct recombination capture coefficient. The recombination lifetime τ r is defined by a. Show that for low-level injection, n po   n p  p po, τ r is constant and given by b. Show that under high-level injection, Δn p  p po, so that the recombination lifetime τ r is now given by that is, the lifetime τ r is inversely proportional to the injected carrier concentration. c. Consider what happens in the presence of photogeneration at a rate G ph (electron–hole pairs per unit volume per unit time). Steady state will be reached when the photogeneration rate and recombination rate become equal. That is, A photoconductive film of n-type GaAs doped with 10 13 cm −3 donors is 2 mm long (L), 1 mm wide (W), and 5 µm thick (D). The sample has electrodes attached to its ends (electrode area is therefore 1 mm × 5 µm) which are connected to a 1 V supply through an ammeter. The GaAs photoconductor is uniformly illuminated over the surface area 2 mm × 1 mm with a 1 mW laser radiation of wavelength λ = 850 nm (infrared). The recombination coefficient B for GaAs is 7.21 × 10 −16 m 3 s −1. At λ = 850 nm, the absorption coefficient is about 5 × 10 3 cm −1. Calculate the photocurrent I photo and the electrical power dissipated as Joule heating in the sample. What will be the power dissipated as heat in the sample in an open circuit, where I = 0? 5. Kasap Problem 5.22 (Direct recombination and GaAs)