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summary-d - Summary EF Metal band overlap Insulator or...

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Summary S-19 Metal band overlap Insulator or semiconductor 5.3 Electrons in External fields Effective mass: F m dt v d = 1 j i ij k k E m = 1 1 (band curvature) F : external force m * : describes forces arising from periodic crystal potential Parabolic shape (e.g., at zone boundaries) m * = const. E F
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Summary S-20 Holes Charge: + e Effective mass: - m e m p > 0 for holes in valence band, because m e < 0 ( q m dt v d = 1 v p = -v e same sign in current: j = qnv Change in Fermi distribution in applied field or temperature gradient collisions k gradient T in r t f f grad k f grad v t f = + + 0 # Boltzmann equation Relaxation approximation: ) ( ) ( ) ( 0 k k f k f t f collisions τ = exponential decay of perturbation Collisions bring distribution back to equilibrium: f - f 0 = [ f ( t=0 )- f 0 ] e -t/ τ
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Summary S-21 Linearization: grad f grad f 0 Electrical conductivity k e F & h = = ( , friction term due to collisions stationary shift of free electron Fermi sphere ) ) ( ( ) ( 0 43 42 1 h k k e k f k f δ τ ( + = = zone Brillouin k d k f k v e j 3 3 ) ( ) ( 4 π σ ( n m E e F = ) ( 2 τ σ analogous to j=env n ( T ) = const for metals Only electrons near Fermi surface can gain energy in field.
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