16.e&h

16.e&h - T. Y. Tan Tan 16. ELECTRONS AND HOLES IN...

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T. Y. Tan Tan 1 16. ELECTRONS AND HOLES IN SEMICONDUCTING CRYSTALS Semiconducting crystals such as Si and GaAs are the base materials of manufacturing modern electronic and optoelectronic devices. A small amount (as compared to the crystal atomic density) of mobile or free electrons and holes in a semiconducting crystal constitutes the electrical charge carriers. As has been briefly noted in Chapter 14, the electrons and holes are defects resulting from some of the crystal bond electrons becoming 'free' at a non-zero temperature. This is a thermal equilibrium property of the crystal, just as the crystal also contains (atomic) point defects. From Chapter 15, we see that calculation of the thermal equilibrium point defect concentrations of metals is carried out by consider the Gibbs free energy of the system consisting of the crystal and the point defects using Eq. (15.1): G = Cg f - TS m . That is, the problem is treated as a classical one. From the point of view of quantum mechanics, this can be done because point defects are Bosons for which many can occupy the same energy level, and for a point defects species, i.e., vacancies, there exists but one energy level: g v f . The calculation of the thermal equilibrium concentrations of electrons and holes is, however, not as straightforward as that of point defects. This is because electrons are Fermions for which some aspects must be treated by quantum mechanics. Fermions such as electrons obey the Pauli exclusion principle in that each energy level can only be occupied by one particle. While compared to atomic density the number of electrons and holes exists in a semiconducting crystal is small, this number itself is nevertheless sizable. At room temperature, it reaches 10 13 cm -3 in Ge, 10 10 cm -3 in Si, and 10 7 cm -3 in GaAs. When 'doped', the concentration of the appropriate type of carrier can become larger by many orders of magnitude. This means that at least the same number of energy levels are involved for a 1 cm 3 crystal and a calculation of the charge carrier thermal equilibrium concentrations using expressions similar to Eq. (15.1) becomes quite impossible, because there will be more than 10 7 g f values with two adjacent ones differing by a negligibly small amount. Thus, the calculation of the thermal equilibrium electron and hole concentrations in a semiconductor is carried out using a different method, which involves essentially three steps. In the first, the Fermi-Dirac distribution function of electrons is derived (Chapter 4). This function states the probability of an energy level with energy E is occupied by an electron. In the second, the density of states is calculated. A state is the synonym of an energy level, and the density of the states is needed because the energy level density for the same δ E
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T. Y. Tan Tan 2 value is different at different E values. In the third, the results obtained from the first two steps are combined together to obtain the thermal equilibrium electron and hole concentrations in a
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This note was uploaded on 07/13/2011 for the course ME 218 taught by Professor Dr.tan during the Fall '11 term at Duke.

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16.e&h - T. Y. Tan Tan 16. ELECTRONS AND HOLES IN...

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