Lec4-5

Lec4-5 - Subject_04: Outline Carrier Statistics in...

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Carrier Statistics in Semiconductors I * Semiconductor Density of States * Electron and hole concentrations * Extrinsic carrier concentrations Subject_04: Outline
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Semiconductor Density of States Electrical current in semiconductors is carried by ELECTRONS and HOLES in the conduction and valence bands at concentrations that VARY with TEMPERATURE * To quantitatively determine the NUMBER of electrons and holes at any given temperature we need expressions for the DENSITY OF STATES in the different bands DERIVATION OF DENSITY OF STATES FOR FREE ELECTRONS:
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Semiconductor Density of States In SEMICONDUCTORS we note that the density of states must VANISH completely inside the forbidden gap * With these considerations we write the density of states for the conduction and valence bands as * As is conventional we have defines the TOP of the VALENCE BAND ( E v ) as the ZERO reference of energy so that the bottom of the conduction band ( E c ) lies at E c = E g ) 1 . 3 ( )) ( 2 ( 1 ) ( 2 / 1 3 * 3 2 g e c E E m E g ) 2 . 3 ( ) 2 ( 1 ) ( 2 / 1 3 * 3 2 E m E g h v FORBIDDEN GAP, E g g ( E ) E E c = E g E v = 0 g c ( E ) g v ( E ) VALENCE BAND CONDUCTION BAND
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Semiconductor Density of States While the density of states gives the NUMBER of states in a given energy range to determine the number of these that are OCCUPIED we introduce the FERMI FUNCTION f ( E ) * This function gives the PROBABILITY that a state at given energy E is OCCUPIED by an electron at a particular temperature T * At T = 0 this function takes the form of a STEP with its edge at the FERMI ENERGY E F but at higher temperatures the step ROUNDS since some states below E F that were filled at T = 0 EMPTY as the electrons that filled them move to HIGHER energy states ) 3 . 3 ( 1 ) / ) exp(( 1 ) , ( T k E E T E f B F
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With the density of states we can compute the NUMBER of electrons and holes present in the conduction and valence bands at any temperature * Beginning with the CONDUCTION band then at a given energy E the number of states that are occupied by electrons in UNIT VOLUME of material is just * To obtain the TOTAL number of electrons that occupy states in the conduction band we simply need to INTEGRATE Eq. 3.4 over ALL energies in this band Note that because of the definition of the density of states the computed number here is for UNIT volume of material Semiconductor Density of States ) 4 . 3 ( d 1 ) / ) exp(( 1 ) ( d ) ( ) ( E T k E E E g E E f E g B F c c g c g c E E B F c E E c E T k E E E g E E f E g n ) 5 . 3 ( d 1 ) / ) exp(( ) ( d ) ( ) (
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Semiconductor Density of States Similar arguments can be made to determine the HOLE concentration in the valence band at any temperature * We begin by noting that the probability of a HOLE state being occupied at any temperature is the corresponding probability that the state is EMPTY * In the VALENCE
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Lec4-5 - Subject_04: Outline Carrier Statistics in...

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