PHT.301 Physics of Semiconductor Devices  

Extrinsic semiconductorsIn the section on intrinsic semiconductors we found that the conductivity of an intrinsic semiconductor depends exponentially on temperature and that at room temperature intrinsic semiconductors are rather poor conductors. It is possible to dope semiconductors with impurity atoms that improve the conductivity dramatically and makes the conductivity nearly constant as a function of temperature near room temperature. Doped semiconductors are called extrinsic semiconductors. Reading: Singh 2.6  2.9 or Thuselt 2.3
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The donor energies are the differences of the donor levels E_{d} to the bottom of the conduction band E_{c}. The acceptor energies are the differences of the acceptor levels E_{a} to the top of the valence band E_{v}. Doped semiconductorsFor a doped semiconductor, the density of electrons in the conduction band is, the density of holes in the valence band is, the density of ionized donors is, and the density of ionized acceptors is, The factor of 4 is valid in the formula for the acceptors if the semiconductor has a light hole and a heavy hole band as Si and Ge do. The four quantities n, p, N_{d}, and N_{a} can only be determined if the Fermi energy, E_{f}, is known. Typically, E_{f} must first be determined from the charge neutrality condition, n + N_{a}^{} = p + N_{d}^{+}. The Fermi energy can be found by solving the charge neutrality condition numerically. One way to do this is to program the formulas for n, p, N_{d}^{+}, and N_{a}^{} in a spreadsheet. Then choose a temperature and calculate n, p, N_{d}^{+}, N_{a}^{} for every value of the Fermi energy between E_{v} and E_{c}. For one of these E_{f} values, the charge neutrality condition will be satisfied. When n + N_{a}^{} and , p + N_{d}^{+} are ploted as a function of E_{F}, the Fermi energy is where the two lines cross. A plot like the one below can be generated by pressing the button below the plot. The Fermi energy can be calculated as a function of temperature by determining where n + N_{a}^{} = p + N_{d}^{+} for every temperature. A plot like the one below can be generated by pressing the button below the plot. Normally it is not necessary to determine E_{f} numerically and the following approximation is sufficient. ntype N_{d} > N_{a}: n = N_{d}  N_{a} ptype N_{a} > N_{d}: p = N_{a}  N_{d} 