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Unformatted text preview:  24Equilibrium Carrier Concentrations in SemiconductorsTo predict quantitatively electrical behavior of a semiconductor, we need to know (a) Density of states function D(E). (b) Probability of occupancy of those states in order to evaluate carrier concentrations. Water AnalogyA very good analogy exists between water in a container and electrons in a semiconductor. Water cannot occupied the same space. (i.e. Water cannot be compressed.) Versus Electrons cannot occupied the same state (Pauli’s exclusion principle) To predict quantitatively the total amount of water in the container, we need to know (a) Crossectional area of container at various height and (b) Probability of water occupancy at various height.  25Probability of OccupancyFermiDirac Distribution FunctionThe probability that an allowed state is occupied is described by the FermiDirac distribution function: f EeEEkTf()11(1) Where EfFermi Level Energy at which probability of occupancy is ½. 111f EeEEkTf()(2) Probability of a state not being occupied Probability of a state to have a hole  26For T= 0 K All states < Efare occupied All states > Efare empty. For T> 0 K Some electrons have E> EfSome holes have E< Ef. Note that: )()(1EEfEEfffi.e. FermiDirac distribution is complementary and independent of temperature for a symmetric pair of energy about Fermi level. Boltzmann ApproximationThe FDdistribution f(E) can be approximated by the MaxwellBoltzmann distribution as: f EeEEkTf()kTEEf3(3a) kTEEfeEf)(1kTEEf3(3b) This is a good approximationfor many device applications, as we will see later. (3 kT78 meV @ room temperature)  27Density of State functionWhen an electron in the conduction band gains energy, it moves up to an E> Ec. Similarly when a hole in the valence band gains energy, it moves down to an E< Ev. From quantum mechanical considerations, it may be shown that: DEhmEEnec()() ()4233(4) = Density of state in the conduction band DEhmEEphv()() ()4233(5) = Density of state in the valence band where h= Plank‘s constant = 6.63...
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This note was uploaded on 10/07/2010 for the course EE EE05 taught by Professor Dr.hu during the Spring '10 term at American University in Cairo.
 Spring '10
 Dr.Hu

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