Unformatted text preview: S x, p, t Ωp
Assuming equipartition of kinetic energy υ2 2 E
2
υx =
=
3 3 m*
so ( X xx ! ) 21
!
# E 2 fS x, p, t *
!
3m " p ( ) (xx) (xxi) (xxii) (xxiii) 4π k 2 dk . (xxiv) To evaluate this sum, write X xx ≈ ECE
656 21
21
E2 ∑ E 2 fS x, p, t = 3m* 4π 3 ∫ ( E − Fn ) 3m* Ω p
1+ e ( ) 7 k BTe Fall 2013 Mark Lundstrom For parabolic energy bands 2k 2
E=
2 m*
so (xxv) dE . 3
Using (xxvi) to change variables in (xxiv), we find (xxvi) (xxvii) ( 2m )
k dk = 3/ 2 * 2 () 3/ 2 *
2 2m
=
3m* " 2 ! 3 ( 2m ) (xxix) (xxx) (xxxi) (xxxii) ( ) ( k T ) " d"
$ 1+ e
( 2m ) ( k T ) % (7 2)F (" )
!!
* dE 5/ 2 7/2 " # "F Be 3/ 2 7/2 Be 23 5/ 2 Now use 15
Γ72=
π 8
to simplify (xxx) as () ( ) (k T )
3/ 2 E 1/ 2 !3 k B Te 3/ 2 E 5 / 2 dE
$ 1 + e( E # Fn ) kBTe *
2 2m
X xx =
3m* ! 2 ! 3 3/ 2 * Now define η = E k BTe (xxviii) η F = Fn k BTe so that (xxvii) becomes 2
=
3m* E 1/ 2 21
E2
X xx ! * 2 $
3m " 1 + e( Fn # E ) *
2 2m
X xx =
3m* ! 2 ! 3 7/2 Be F 15
! e"F , 8 where we have also assumed Boltzmann statistics. Recall that ! n = N C e F (xxxiii) . where *
1 " 2 m k BT %
NC = $
4 # ! !2 '
& 3/ 2 (xxxiv) Using (xxxiv), eqn. (xxxii) can be written as ECE
656 8 Fall 2013 Mark Lundstrom *
2 1 " 2 m k BTe %
X xx =
$
'
3m* 4 # ! ! 2 & 3/ 2 ( k T ) 15 e
2
2 (F Be = 2 " 3k BTe %
15 " 2 %
n
' k BTe 2 $ 3 '
*$
#&
3m # 2 & = 10 k BTe
W
3 m* ( ) (xxxiii) Now returning to eqn. (xviii), our current equation becomes 5
10
d "k T %
FWx = ! µ EWE x ! µ E $ B e W ' &
3
3
dx # q (xxxiv) which is (B). To re
cap, we have shown that the energy flux balance equation ∂FWx
dX
F
5q
= − xx −
WE x − Wx ∂t
dx
3 m*
τF
W can be simplified to 5
10
d ⎛k T ⎞
FWx = − µ EWE x − µ E ⎜ B e W ⎟ ⎠
3
3
dx ⎝ q
To do so, we needed to make a number of assumptions, such as parabolic energy bands, isotropic distribution function that can be described by an electron temperature, and slow variations in time. We also had to terminate the hierarchy. Rather than writing yet another balance equation for the unknown, X xx , we assumed a reasonable form of the distribution, and then evaluated X xx from (xxiii). 4) In reading the literature, you will sometimes see the near
equilibrium drift
diffusion equation written as !
!
(A) J n = nqµ nE + qDn!n or as !
!
J n = nqµ nE + q! Dn n (B) ( )...
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 Fall '14

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