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Unformatted text preview: EE151 Equations: Set 1 S.K. Tewksbury Sept. 12, 1997 This set of notes summarizes the equations which you are expected to know at this point in the class. The equations relate to the basics of semiconductors (specifically silicon semicon- ductor crystals). 1 Properties of Silicon Semiconductors 1.1 Thermal Energy The thermal energy E th of free electrons and holes (mobile carriers of charge) is related to the temperature by E th = 3 · kT 2 (1) where Boltzmann’s constant k = 1 . 38 × 10- 23 J / K = 8 . 62 × 10- 5 eV / K. 1 1.2 Carrier Densities (Electrons and Holes) The density n (number of electrons per unit volume) of free electrons (not bound to silicon atoms) and the density p (number of holes per unit volume) of holes are related to the Fermi energy E f by the following equations, which hold under all equilibrium conditions. Basic Equations: Fermi Level and Band Edge: n = N c · exp [- ( E c- E f ) /kT ] (2) E f = E c- kT · ln ( N c /n ) p = N v · exp [- ( E f- E v ) /kT ] (3) E f = E v + kT · ln ( N v /p ) Basic Equations: Fermi Level and Intrinsic Fermi Level: n = n i · exp [- ( E i- E f ) /kT ] (4) E f = E i + kT · ln ( n/n i ) 1 The units in “eV” are obtained from the units in Joules by dividing by the electrom charge q = 1 . 6 × 10- 19 coulambs. 1 p = n i · exp [+( E i- E f ) /kT ] (5) E f = E i- kT · ln ( p/n i ) Basic Equations: Fermi Potential and Electrostatic Potential: n = n i · exp [+( ψ- φ ) /V T ] (6) φ = ψ- kT · ln ( n/n i ) p = n i · exp [+( φ- ψ ) /V T ] (7) φ = ψ + kT · ln ( p/n i ) Calculation of φ using these equations is not normally done. Instead, normally we work with E i and E f directly. Product of Carrier Densities n · p = N c N v · exp [- E g /kT ] (8) = ( n i ) 2 (9) Equation (8) is used only to illustrate the temperature dependence of n · p = n 2 i . Equation (9) is used to calculate the minority carrier density, given the majority carrier density. The terms in these equations are summarized in the following sections. 1.2.1 Equations 2 and 3 • N c = 2 . 8 × 10 19 / cm 3 is the effective density of states in the conduction band . • N v = 1 . × 10 19 / cm 3 is the effective density of states in the valence band . • E c- E f (always positive) is the position of the Fermi level below the conduction band E c . If E f lies above the middle of the band gap, then the semiconductor is N-Type , caused by doping with do N ors which donate a free electron to the conduction band, leaving behind a postively charged donor atom. For an N-type semiconductor, the free electron density n (the majority carrier density ) is much larger than the hole density p (the minority carrier density ). The density (atoms per unit volume) of donor atoms is represented by N d . All dopant atoms are ionized at room temperature, giving a free electron density n equal to the density of donors, i.e....
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- Spring '09
- Condensed matter physics, nd, Fermi level, carrier density