Unformatted text preview: 56 11 Fall 2013 Mark Lundstrom ! Ex
#
# Ey
" $)'
L
&=+
& + ( 'T
%* 'T , ! J x $
.#
& 'L . # J y &
"
%
 10/27/13 (****) where !L = "L
1
=
2
" +"T "n !T = 2
L µB
"T
= n z 2
" +"T
"n
2
L The final result, !E
x
#
# Ey
" $
1
1)
+
&=
& ' n + ( µ n Bz
%
* µn Bz , ! J x $
.#
& 1 .# J y &
%
" shows that the longitudinal magnetoresistivity is independent of B while the longitudinal magnetoconductivity (from (via)) depends on the B
field. Comparing to prob. 5), we see that we get the same result for the magnetoresistivity without assuming a small B
field. 6e) Show that for small B
fields, eqn. (via) can be written as !
!!
!
J n = ! nE  ! n µ n E " B ( ) (vii) Note that while this analysis is simpler than solving the BTE, by beginning with !
an average electron with an average momentum, p , we have missed the averaging of the distribution of momenta which leads to a non
unity Hall factor, rH . Solution: Simply expand out (vii) and show that it gives (via) for small B. ! Jx $
'n
#
&=
# J y & 1 + ( µn Bz )2
"
% ( µn Bz , ! E x
.#
1
.# E y
" )1
+
+ µn Bz
* )1
( µn Bz
/'n +
1
+ µn Bz
*
!
!!
= ' nE  (' n µn )E 0 B ,! E x
.#
.# E y
" $
&
&
% $
&
&
%
ECE
656 12 Fall 2013 Mark Lundstrom 7) 10/27/13 Hall factors are important to consider when analyzing experiments. Answer the following questions. 7a) Derive an expression for the Hall factor in 3D and show that for ionized impurity scattering, it gives rH = 1.93 . Solution: The definition of the Hall factor is: 2
rH ! " m 2 "m (i) Assume power law scattering: ! m = ! 0 ( E k BTL ) s Recall that the average scattering...
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 Fall '14

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