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Magnetoresistance - solid state physics
Physics
Kendriya Vidyalaya Hebbal
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7 Magnetoresistance
and, therefore, if a graph is plotted betweer
in Figure 7, with a slope of
le3/
lQkf {
ln
"I
ucis, from which the
value of Js can
b,
Equation (7) fits well into the expe
nation of the Schottky effect.
ln
"I
ln Jo
207
7,,
d straight line is obtained, as shown
raight line has an intercept of
ln
"Is
on
re and, therefore, confirms our expla-
ln,I:ln Jo*#G
Figure 7: Variation of ln J
with t/8.
7 Magnetoresistance
The magnetoresistance of a
crystal refers to a change in its electrical resistance in the
presence
of an applied magnetic field. This effect
is due to the fact that when a current-carrying
conductor is placed in a
magnetic field, the
electrons in the conductor experience
a force. As
a result, they do
not go exactly
in the direction
of the applied field, but
follow curved
paths.
When the direction of the applied
magnetic field is normal to that of the electron
flow, the
effect is termed as tmnsaerse rnagnetoresistance
and when the magnetic field is applied parallel
to the direction of
the electron
flow, the effect is termed as longitudinal rnagnetoresistanen.
Let, at any temperature, ft be the resistance of the
conductor in zero field a A-B be the
increase in its resistance when an external magnetic field
is applied. Then, it is found that
for relatively low fields, I is
proportional to .EI
whereas in higher fields, this ratio is
proportional to I/. Sometimes, we use fto instead of R,
where Bs is the resistance in zero field
at 0o C, as the ratio A,Rf R depends less on the
purity of the metal.
The tra,nsverse magnetoresistance is, hourever, ofgreater interest. In the presence ofa very
strong magnetic field, the transverse magnetoresistance of a metal may exhibit any one of the
following
properties:
- Its resistance may saturate2r at a value several times its value in zero field. The saturation
takes
place for all orientations of the crystal a:<is relative to the measurement axis.
- Its resista may continue
to rise even
at the highest field.
- The resistance may saturate along some crystal directions, but may not saturate in
nearby crystal directions.
2lResistance
becomes independent of the applied magnetic field.
208 Eree Electron Theory of Metals
Consider an electric field t, applied across the ends of the conductor along the r-direction
and a magnetic field Hrbe applied along the z-direction. Then, the force acting on an electron
moving in electric and magnetic fields, called Lorentzforce22, is given by
(7)
l\rlrg
Att=
then
beo
Int€g
where C
conditior
At t'
Subd
Subs
Intcl
Ifrt
is giv€o
xatll
I
tine e I
a
grur tb
####### F,:*#;
####### elt,*l
(,
"
?),]
=+
=+
F,:rrl#
elt-*l{onn,
-uzr;l
F*:*#;
####### elt,*lo,r,ll, as Hy:o
F*:#;-ep":
nullH,l
F,: *#;
ep,
####### *!
(,,ro
-,or)l
F":0, ffi€r-Ha-H,
=+
Similarly,
=+
=+
and,
=+
Fy: *#l: e
Fy:*#l
e
Fy:*#l
l',
*!{',',
[-:
@,D],
a's H,:o
####### -e[-:
nullH")
-usr^J]
' Integrating, Eqs. (7) and (7) give
dx
#:e€,t+Z"u*",
*fr:-9H,s+cz
where G
and Cz are the constants of integration whose values are to be determined.
At t
:0,
s:
U:
0 and
#,
: u*Hence, Eq. (7) grves Cr
: rnt!,o
then becomes
dfr
#:e€,tln,y+*u,
(7)
(7)
(7[)
(7)
(7)
22lorent
z force is
given by
?
"[
1(?,
?lt
210
flee Electron Theory of Metals
(#)
:
*,1+
-# (*,'n
*#
u
.
4).+l
/
a"\ e
I
t,r' e2 Hlt,rnl
\d,t/-irL
2
-
(#):*L-ffi
The current density is
given by
r*:""(#r):*["
ffil
and the electrical conductivity is
given by
(#):
*1v,,+
0
(7)
(7)
(7)
(7)
k
=+
=+
where
is a ourc
The
Equatiol
=>
This n
semicondu
a very lml
intrinsic
g
all tempet
7
The follor
For low fields,
ff
is small aad hence, can be
neglected. Moreover, the average values of
u, and uy are also zero, as the electrons have equal probability of moving along positive as
well as negative direction. Equation (7) can, therefore, be written as
_ _
J, ne2 l-
"'H?r'
o-:-
2*
L"-
n*:ry]
g126)
er
is zero, that is, when H"
:
0, let o
:
og and T
: Ts.
When the magnetic field
Fquation (7) then gives
Now,
If r N To, then using Eq.
o-oo Lr
J*
OO:::
Dt
ne2rg
^A:
As
Tg
ne2 l- e2 H?r
l+
'
I
2m
L'
I2m2c
l
ne2rg
o-oo
nezrg
o-oo r-ro
_
e2H?rs
os Ts L2m2c2rs
(7) the above equation
gives
o-oo
_
L,
_
e2 HZr&
og L2m2c
=+
og
#(#)
',
usingEq' (7'127)
ee Electron Theory
ztl
o-oo Lr
*(#)'H:
og
=+
where
o-oo
A-i(#)'
is a constant.
The resistivity p is inversely proportional to the
Equation (7) can, therefore, be written as
Ar
s'
-AH:
Tg
(7)
conductivity o, that
.
rsp o(:
o
11
Po-P
1
Tg
rllrz
p
Lr
_
s'
-AHZ
Tg
As
ar
Tg
\
Po )
pr- rs
is very small, the above equation reduces to
LP
xH?
Po
A,o
^
t
:AH!
Po'
=>
(7)
This result is derived for metals where the charge carriers are only electrous. In case of
semiconductors, where_there
are two types of ctrarg;arriers, viz. electrons and holes, we get
a very importarrt
conclusion that tra,nsverse magnetoresistance does not saturate in case of
intrinsic semiconductors in which the number of-electrons is equal to the number of holes at
au temperatures.
7 Failures of the flee Electron Theory
The following are the failures of the free electron theory:
Magnetoresistance - solid state physics
Course: Physics
University: Kendriya Vidyalaya Hebbal
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