Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.
•• The metal-insulator transition in silicon. If the concentration of donor impurities in silicon is large enough for the donor state wave functions to overlap, the donor states will hybridize to form a partially filled band of delocalized wave functions, and the heavily doped silicon will become a metal (that is, it will conduct at T = 0 K).The Bohr radius of a donor impurity is about 3 nm in silicon and about 8 nm in germanium. Estimate the donor concentrations at which silicon and germanium undergo the metal–insulator transition.
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