1. (207) In a particular semiconductor there are 10" donors/cm' with an ionization energy E, of 1 meV and an effective mass 0.01 m. (a) Estimate the concentration of conduction electrons at 4 K.(107) (b) What is the value of the Hall coefficient?(107) Assume no acceptor atoms are present and that E >T. 2. (207) (a) Both electrons and holes are present in a semiconductor. Derive the Hall coefficient in the semiconductor in terms of electron concentration n, hole concentration p, electron mobility 4., hole mobility 4, and electron charge e. In the derivation neglect terms of order B.(107) (6) A germanium sample shows no Hall effect (no Hall voltage or field). The mobility of electrons in germanium is 3600 cm? Ns and that of holes is 1800 cm/s. What fraction of the current of the sample is carried by holes? (107) 3. (207) In a semiconductor, the valence band is given by E = (1eV)(cosk a +cosk,a +coska), and the conduction band is given by E = 8eV -(len)(coska +cosk, a)-(2 e)cosk_a. (a) What is the band gap for the semiconductor? (57) (b) What is the effective mass tensor for electrons? (57) (c) Explain how the effective mass for the conduction band might be measured. (57) (d) An electron that had occupied the state k =(0,0,1) in the valence band is excited to the conduction band. What are the energy and velocity of the hole that remains? (57) 4. (207) A sample of silicon was doped with 16x10cm atoms of As. Assuming that all impurities are ionized, estimate the concentrations n and p of charge carriers(107) and find the position of the Fermi level from the conduction band edge E, at room temperature (302 K)(107). Use m = 1.026m and m; = 0.591m, where m is the free electron mass. 5. (207) Consider optical absorption in the diagram below by vertical transitions from the band with dispersion E. =-#k/2m to the conduction band valence E = E +h***/2m (a) Find the photon energy ho for optical absorption to occur from states at k, using the reduced mass m of the electron-hole pair created, defined by (102) mm (b) Derive the joint density of states of electron-hole pairs (that is, the density of pairs of states in the valence and conduction bands connected by vertical transitions of frequency o) using the result obtained in (a)(107) Conduction band Electron removed Valence band