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Unformatted text preview: approximation assuming a constant relaxation time, and a small electric field, but no concentration gradient. Use the result to derive an equation for the drift current. Solution: BTE: !f
!f
!f
$f
+ "x
+F
=#
!t
!x
! px
%m
ECE
656 3 (i) Fall 2013 Mark Lundstrom 10/30/13 steady
state, spatially uniform, constant scattering time: !f
#f
="
! px
$0 (ii) (iii) (iv) %
$f (
! f = " ' # 0 Zq
E $ px * x
&
) (v) (vi) F ! f = "# 0 F $f
$ px F = + ZqE x %
$f (
f = f0 + ! f = f0 " ' # 0 Zq
E $ px * x
&
) This is a displaced Maxwellian with the displacement in momentum being: pdx = (! 0 Zq )E x (vii) (viii) The average drift velocity is pdx # q" 0 Z &
=
E = µE x , m* % m* ( x
$
'
so the mobility of these charged particles is: !x = " q! %
µ = Z $ *0 ' , #m & (ix) (x) and the drift current is J nx = nZq ! x = n Zq µE x J nx = n ( Zq ) µE x
" q! %
µ = Z $ *0 '
#m & 4b) Solve the BTE in the relaxation time approximation assuming a constant relaxation time, and a small concentration gradient, but no electric field. Solution: In this case, the steady
state BTE becomes: ECE
656 4 Fall 2013 Mark Lundstrom !x 10/30/13 "f
$f
=#
"x
%0 ! f = "# 0$ x % f0
%x (xi) 4c) Use the result from 4b) to derive an equation for the diffusion current. Solution: J nx = (
'f +
1
1
" ( Zq )# x$ f = ! " ( Zq )# x * %& 0# x ' x0  !
!
!k
)
,
k J nx = !...
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This document was uploaded on 01/15/2014.
 Fall '14

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