It is known that lattice vacancies in silicon can seriously affect the electrical properties of semiconductors. Suppose that a transistor is being made with thin wafers that are produced by vapour deposition onto an inert substrate. The process is successful in producing single crystals of silicon, but the concentration of lattice vacancies is always very high (10^15 vacancies/cm^3) just after deposition. Attempts to remove the vacancies by heating the wafers to just below the melting temperature have failed because the films cool so quickly after the anneal that the vacancy concentration is still too high. It is suggested that the wafers be heated to just below the melting temperature of silicon and that they be slowly cooled to a temperature where the equilibrium vacancy concentration falls below 10^12 vacancies/cm^3. Required:
a. Calculate the temperature to which the crystals should be slowly cooled. The formation energy for vacancies in silicon is 2.3 eV/vacancy explain why it ils all right to rapidly cool the wafers once they have reached the critical temperature.
b. These crystal initially contain 10^15 vacancies/cm^3. Suppose that the crystals are cooled to a critical temperature (at which the equilibrium vacancy concentration is 10^12 vacancies/cm^3) and that the excess vacancies annihilate, causing an evolution of heat. Estimate the temperature rise that would be expected if the energy at each vacancy is 2.3 eV and the heat capacity is about 6.4 cal/mole-°C.