Date: Oct 19, 1998 (Monday)

Time allowed: 50 minutes.

Answer all questions.

1. Heat capacity of chain lattice. (a) Consider a dielectric crystal made up of chains of atoms, with rigid coupling between chains so that the motion of the atoms is restricted to the line of the chain. Show that the phonon heat capacity in the Debye approximation in the low temperature limit is proportional to T. (b) Suppose instead, as in many chain structures, that adjacent chains are very weakly bound to each other. What form would you expect the phonon heat capacity to approach at extremely low temperature?

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2. Semiconductor crystal. For simplicity, let us assume there is only one conduction band, one valance band, and one donor band (i.e. degeneracy = 1 for each case) in this problem. The effective densities of state of the valance (N_v) and conduction (N_c) band are given as respectively, where m_e and m_h are masses of electron and hole respectively, and T is temperature. Now consider an intrinsic semiconductor of energy gap E_g = 1 eV, m_h = 0.5 m_e, and N_c = 3 10¹⁹ cm^-3 at T=300K. (a) Calculate n and p in cm^-3, the carrier density of electron and hole respectively. (b) Calculate the Fermi energy E_F. The semiconductor is now doped with a donor impurity at a concentration of 10¹³ cm^-3. The donor ionization energy is so small that they are essentially 100% ionized (i.e. N_D⁺ = N_D). (c) What is the new hole density (p) in cm^-3? (d) What is the new Fermi energy?

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3. de Hass - van Alphen Effect. (a) Write down an expression in relating the flux quanta ₀ to fundamental physical constants , e, and c. (b) Consider an electron moving in close orbits in both k- and real space under a constant external magnet field B. Show that , where S and A are area enclosed by the orbits in k- and real space respectively. (c) The following figure shows data for De Haas-van Alphen oscillations in silver The magnetic field is along a <111> direction. The vertical axis is magnetic moment and the horizontal axis is . Estimate the ratio of the areas of the two extremal orbits responsible for the oscillations as shown in the lower figure.

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