Consider an ntype semiconductor with donor concentration Nd

Consider an n-type semiconductor with donor concentration Nd. In this problem we will determine the dependence of the conductance electron density, n, on temperature in the low temperature regime. Suppose that T is so low that thermal excitations of electrons from the valence band can be ignored. Moreover, assume that Ec - mu >> kT so that the electrons in the conduction band may be treated as non-degenerate (classical Boltzmann). However, since the chemical potential may be very close to the donor levels the population of the donor levels must be described using the full Fermi distribution. (a) Charge neutrality requires that n = N^+_d, where is the number of ionized donors. By solving an equation for e^betamu, show that n=1/2no^e -beta(E_c-E_d)(-1+squareroot 1+4(n_d/n_o)e^beta(E_c-E_d), where E_c - E_d is the difference in energy between the bottom of the conduction band and the donor level and n0 = 2(mkT/2pih^2)3/2. Determine n in the limits in which 4(N_d/n_0)e^beta(E_c-E_d)1. (c) Sketch n vs beta. Include in addition to the two limits studied in this problem the behavior at higher temperatures, when excitation of electrons from the valence band becomes important. Your plot should have three clearly identifiable regimes of temperature. Determine the slope of the curve in each regime and the temperatures that separate them.

Solution

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