densities - EE471 Course Notes Lecture 1 These notes...

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EE471 Course Notes: Lecture 1 These notes summarize the information presented in Lecture 1. Contents 1. CARRIER DENSITIES IN SEMICONDUCTORS .................................................................................................. 1 1.1 I NTRINSIC S ILICON ...................................................................................................................................................... 1 1.2 D OPED (E XTRINSIC ) S EMICONDUCTOR - N-T YPE AND P-T YPE S EMICONDUCTORS ................................................ 2 1.3 S UMMARY OF E QUATIONS .......................................................................................................................................... 5 APPENDIX A: PHYSICAL CONSTANTS .................................................................................................................... 6 APPENDIX B: PROPERTIES OF SILICON AND GAAS AT 300K AND SOME OTHERS .............................. 7 1. Carrier Densities in Semiconductors 1.1 Intrinsic Silicon "Intrinsic silicon" is a pure silicon crystal (no dopants). The discussion of intrinsic silicon developed the following (values of physical and semiconductor constants are given in the appendix. The general equation for the density of electrons (of holes) in terms of the effective density of states in the conduction band (valence band), the Fermi energy, and the thermal energy factor kT. The equations for the carrier densities in general (i.e., also for doped silicon) and values of parameters (for silicon at 300K) are n = N C " exp # E C # E f kT $ % & ( ) p = N V " exp # E f # E V kT $ % & ( ) kT = 0.0257 eV at 300 ° K N C = 2.88 " 10 19 cm -3 N C = 2.66 " 10 19 cm -3 The intrinsic carrier density n i (electrons and holes have same density in intrinsic silicon since they are created in pairs) in terms of the densities of state and the energy band gap E g = E C " E V , where E C and E V are the conduction and valence band energy levels, respectively. The equation for and value (for silicon and at 300K) of the intrinsic carrier density are n i = N C N V " exp # E g 2 kT $ % & ( ) n i 300 ° K ( ) = 9.65 " 10 9 cm -3 The "intrinsic Fermi energy" E i " E V + E g /2 (middle of band gap) as the Fermi level for intrinsic semiconductor. The "approximately equal" sign is used because we have ignored a small term (of order kT ). We will simply draw the intrinsic Fermi energy in the middle of the band gap.
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The general equation for the density of electrons and of holes in terms of the Fermi level for the given sample, the thermal energy factor, and the intrinsic silicon parameters ( n i and E i ). The equations are generated from the equations above for n and p and are fully equivalent to those equations. n = n i exp E f " E i kT # $ % & ( p = n i exp E i " E f kT # $ % & ( From either set of equations for n and p , we obtain the important result relating the density of one type of carrier to the density of the other type of carrier. n " p = n i ( ) 2 n = n i ( ) 2 p p = n i ( ) 2 n 1.2 Doped (Extrinsic) Semiconductor - N-Type and P-Type Semiconductors By doping silicon with donor atoms (from the group V elements) or with acceptor atoms (from the group III elements), the carrier densities can be changed from the intrinsic carrier densities.
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