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Phys 682 Semiconductor Optics Fall 2013 Homework #1 due Wednesday, November 13, 2013 1. (10 pts) (a) Starting from the Kramers-Kronig relations for the real and imaginary parts of susceptibility, derive Kramers-Kronig relations for the real and imaginary parts of the refractive index, assuming that ||<<1. (b) Using results of part (a), consider the following. A solid has a single absorption band of width centered at frequency 0 such that the extinction coefficient is () = 0 at 0/2 < < 0+/2 and 0 otherwise. Calculate the refractive index at low frequencies. Simplify your expression by assuming that 0>>. 2. (10 pts) InP has a bandgap of 1.35 eV at room temperature. At 775 nm, the absorption coefficient is 3.5 x 106 m-1. Estimate the transmission through a 1 m InP sample at 620 nm. Assume that the sample is made with anti-reflection coated surfaces. 3. (5 pts) Consider p-type Si with a carrier density of 1.4x1016 cm-3 and hole effective mass of 0.37 m0, where m0 is the free electron mass. What is the plasma frequency of such a material? What do you expect from reflection and transmission of this semiconductor at optical frequencies? 4. (20 pts) The band parameters of the four-band model are given for GaAs in the table attached. (a) Calculate the k vector of the electron excited from the heavy-hole band to the conduction band in GaAs when a photon of energy 1.6 eV is absorbed at 300 K. What is the corresponding value for the light-hole transition? (b) Calculate the wave vector of the photon inside the crystal. How does it compare with the wave vector of the electron? The refractive index of GaAs at 1.6 eV is 3.7. (c) Calculate the ratio of the joint DOS for the heavy- and light-hole transitions. (d) What is the wavelength at which transitions from the split-off hole band become possible? 5. (20 pts) Work through the Nature 1991 paper by Cullis and Canham. (a) Describe what photoluminescence (PL) you expect to see in the bulk Si (b) Now describe what PL response the authors saw in their experiments (c) What is the physics behind their observations?