ssp_18 - 5 Continued 5-66 Application to electrical...

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5-66 5 Continued
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5-67 Application to electrical conductivity field acceleration of conduction electrons stationary state Collision reestablishment of (i.e., 0 = t f ) equilibrium distribution at ε = 0 (friction term) 0 = + st field t f t f Moreover: grad v f = 0 (no T - gradient) e F k = = & h (see eq. 5.3.5) e [ ] ) ( / ) ( ) ( 0 k k f k f f grad k τ = (5.3.17) ) ( ) ( ) ( 0 k e k f k f τ + = ) ( k f grad k (5.3.18)
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5-68 ) ( ) ( ) ( 0 k e k f k f τ + = ) ( k f grad k (5.3.18) Solution by iteration: 1) f ( k ) = f 0 f 1 ( k ) 2) f 1 ( k ) in (5.3.17) f 2 ( k ) 3) .... n n n a k f = ) ( Generally term linear in sufficient (e.g. Ohm’s law j = σ ) ) ( ) ( ) ( 0 k e k f k f τ h + grad k f 0 ( k ) (5.3.18) Linearized Boltzmann equation holds for not too high fields (not too large deviations from equilibrium). (5.3.18) ) ( ( ) ( 0 k e k f k f τ + ) (5.3.19) Moderate fields stationary distribution shifted by Collision τ ( k ) ) ( k e τ h
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5-69 Representation in k-space = ( x , 0, 0) Fig. 5.36 The effect of a costant electric field x on the k -space distributin of quasifree electrons: a) The Fermi sphere of the equilibrium distribution [ - - - - - - , centered at (0,0,0)] is displaced in the stationary state by an amount δ k x = -e τ x / . b) The new Fermi distribution f ( E ( k )) only differs significantly from the equilibrium distribution f 0 ( - - - - - - ) in the vicinity of the Fermi energy (Fermi radius).S
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5-70 Fig. 5.37 a , b . electron scattering processes in k -space. The dashed circle represents the Fermi surface in thermodynamic equilibrium ( = 0). Under the influence of an electric field x and for a constant current, the Fermi surface is displaced as shown by the full circle.
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