Brewster H.D. Solid State Physics
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122
Magneto-transport Phenomena
so
that
· -
[EX
(roc
't)2
EX]_
E
lx-ao
+
2-O"OX
I +
(ro
c
't)2
I + (roc't)
and again there is no transverse magnetoresistance for a material with a single
carrier type having a spherical Fermi surface. Introduction
of
either a more
complicated Fermi surface
or
more than one type
of
carrier results in a
transverse magnetoresistance.
When the velocity distribution
of
carriers at a finite temperature
is
taken
into account, a finite transverse magnetoresistance
is
also obtained. In a similar
way, multi-valley semiconductors (having several electron or hole constant
energy surfaces some
of
which are equivalent by symmetry) can also display
a transverse magnetoresistance.
In
all
of
these cases the magnetoresistance exhibits
aB2
dependence. The
effect
of
two carrier types on the transverse magneto resistance
is
discussed
in
some detail.
We note that the
aX)'
and a
yx
terms arise from the presence
of
a magnetic
field. The significance
of
these terms
is
further addressed
in
our discussion
of
the. We note that for non-spherical constant energy surfaces, Equation must
be rewritten to reflect the fact that
m * is a tensor so that v and E need not be
parallel, even in the absence
of
a magnetic field. This point
is
clarified to
some degree in the derivation
of
magneto-transport effects given in using the
Boltzmann Equation.
THE
If
an electric current
is
flowing
in
a semiconductor transverse to an applied
magnetic field, an electric field is generated perpendicular to the plane
containing
J and
jj.
This
is
known as
the.
Because the magnetic field acts
to deflect the charge carriers transverse to their current flow, the Hall field
is
required to ensure that the transverse current vanishes.
Let
x be the direction
of
current flow and z the direction
of
the magnetic
field.
Then
the
boundary
condition
for
the
is
iy
=
O.
From
the
magnetoconductivity tensor equation we have
m?T:[
I )
· -
--
E
-ro
T:E
lv
- * 1 ( )2
(v
ex)
•
111
+
roc
T:
.
so
that a non-vanishing Hall field
£1'
=
roc'tEx
must
be
present to ensure the vanishing
of
iv'
It
is
convenient to define the
Hall coefficient .
Magneto-transport Phenomena
123
_
Ey
T.Ex(eB:
1m
*e)
R
Hall
=
--.
- =
_=-..:.---'''------'-
JxB: jx
B
:
Substitution
of
the Hall field into the expression for jx then yields
ne
2
T.
ne
2
T.[1+(ffi
c
T.)2]Ex
j x = 2
[Ex
+
ffi
c
T.
E y] =
---=--'--"--'--=---"-
m*[l+(fficT.) ] m*[1+(ffi
c
T.)2]
or
ne
2
T.
j =
--E
= crt!, E
x
m*
x c
X·
Substitution
of
this expression into the
Hal1
coefficient yields
eT.
R
Hall
= 2
m*e(ne
T.lm*)
=
nee
Magnetic field
B,
ttl
t t
(aJ
Section
-E,
perpendicular
to
taxis
;
drift velocity
just starting up.
(bJ
-E,
Section
+
+ +
+
+
+
+
+
perpendicular
to
i
axis;
drift velocitv
in
steady state.
),
YL
z x
(eJ
Fig. The standard geometry for
the:
a specimen
of
rectangular crosssection
is
placed
in
a magnetic field B
z
.
An electric field
Ex
applied across the end electrodes
causes
an
electric current density jx to flow down the bar. The drift velocity
ofthr
electrons immediately after the electric field
is
applied. The deflection
in
the y
direction
is
caused by the magnetic field. Electrons accumulate on one face
of
the
bar and a positive ion excess
is
established on the opposite face until, the transverse
electric field (Hall field) just cancels the force due to the magnetic field.
The Hall coefficient
is
important because:
124
Magneto-transport Phenomena
1.
R
Hall
depends only on the carrier density
n,
aside from universal
constants
2. The sign
of
R
Hall
determines whether conduction is by electrons
(R
Hall
< 0) or by holes
(R
Hall
> 0).
If
the carriers are
of
one type we can relate the Hall mobility
IlHall
to
R
Hall
:
e't
( ne
2
't
) ( 1 )
Il
= - =
--
e - =
creR
Hall
m* m*
nee
We define
IlHall
==
creR
Hall
and
IlHall
carries the same sign as
RHall"
The resistivity component
px/
nellHall
is
called the Hall resistivity. A variety
of
new effects can occur
in
R
Hall
when
there
is
more than
one
type
of
carrier,
as
is
commonly
the
case
In
semiconductors, and this
is
discussed
..
Derivation
of
the Magneto-transport Equations
from the Boltzmann Equation
The corresponding results relating J and it will now be found using the
Boltzmann equation in the absence
of
temperature gradients. We first use the
linearized Boltzmann equation given by equation to obtain the distribution
function
it. Then we will use
II
to obtain the current density J in the presence
of
an electric field it and a magnetic field B =
Bi
in the
z-direction.
In
the presence
of
a magnetic field the equation
of
motion becomes
lik
=
e(
it+~VXB).
We use, as
in
equation,j=
10
+
II
with
~
=
8/0
8E(k)
=liv8/o
8k 8E 8f
8E·
Substituting into the linearized form
of
the Boltzmann equation gives an
equation for
it
:
e(
it +
~
v x
B)-
(
nv
~
+
~
) = -
~
.
In
analogy with the case
of
zero field, we assume a solution for
fl
of
the
form
- -
8/0
II =
-e'tV·
V 8E
Magneto-transport Phenomena
125
with
where
Vis
a vector to
be
determined in analogy
with
the solution for
B =
O.
The
form
of
is motivated
by
the form suggested
by
the magnetocon-
ductivity tensor.
For
a simple parabolic band, v =
hk
I m
*,
and substitution
of
equation
gives
e _ -
8fi
e
2
t - -
8/r.o
-(vxB)·---=
=
--(vxB)'V-,
he
8k
m*e
8E
The
following equation for V is
then
obtained from equation
-
et
- - -
v·E---(vxB)·V
=v·
V
m*e
where
we
have neglected a term
E.
V
which
is small
(of
order
IEI2
if
lEI
is small). Equation 7.27
is
equivalent to
vxEx +
roc
tVxV
y
=
vxVx
vyE
y
-roctvyVx =VyVy
which can
be
rewritten more compactly as:
V.l
=
(E.l
-(etl
m *e)[ih
E.ll)(1
+
(etB
I m
*e)2r
l
Vz=E
z
where
the
notation "1-" in equation denotes
the
component
in the x - y plane,
perpendicular to
B.
This solves the problem
of
finding!l'
Now
we
can carry
out
the calculation
of
J in
equation,
using the
new
expression for fl given by Eqs., and 7.29.
With
the
more
detailed calculation
using the Boltzmann equation,
it
is clear
that
the cyclotron mass governs the
cyclotron frequency while the dynamic effective mass controls the coefficients
(ne
2
'Clm*)
in Eqs. 7.6 and 7.7.
We
discuss in §7.5
how
to calculate the cyclotron
effective mass.
TWO
CARRIER
MODEL
We
calculate both the and the transverse magnetoresistance for a two-
carrier modeL
The
geometry
under which
transport measurements are made
cJ
II
x)
imposes the
conditioni
y
=
O.
From
the magnetoconductivity tensor
of
crol~Ex
+
crOlEy
_
cr02~2Ex
+
cro2Ey
= 0
equation
iy
= 2 2 2 2
1+~1
l+~l
1+~2
1+132
where
126
Magneto-transport Phenomena
and the subscripts on
0"0,
and
~i
refer to the carrier index, i =
1,2,
so that
O"Oi
= n,e
2
T)m* and
~,
= ffiOC't
j
•
Solving equation yields a relation between
Ey
and
Ex
which defines the Hall field .
for a two carrier system. This basic equation is applicable to two kinds
of
electrons,
1\'{0
kinds
of
holes, or a combination
of
electrons and holes. The
generalization
of
equation to more than two types
of
carriers
is
immediate.
The magnetoconductivity tensor is found by substitution
of
equation into:
. _
0'01
I
Ex
+
0'01
Il3lEy
+
0'02
I
Ex
+
0'02
I132Ey
i
x
-
1+13?
1+l3f
1+13~
1+13~
In general equation
is
a complicated relation, but simplifications can be
made in the low field limit
~
«
1,
where we can neglect terms in
~2
relative to
terms in
~.
Retaining the lowest power in terms in
~
then yields
E =
[0'01131
+ 0'02P2]
Y
0'01
+
0'02
jx =
(0"01
+
0"02)E
x
'
We thus obtain the following important relation for the Hall coefficient
which is independent
of
magnetic field
in
this limit
IlIO'ol +
1l20'02
c(
0'01
+ 0'02)2
where we have made use
of
the relation between
~
and the mobility
/..I.
~
=
/..I.B/C
=
e'tBI(m~
c) = ffic't.
This allows us to write R
HaII
in terms
of
the Hall coefficients
Ri
for each
of
the two types
of
carriers
since
13
B =
RHallO',
where for each carrier type we have
Magneto-transport Phenomena
and
1
R=-
, n,eic
127
where i =
1,2.
We note
in
equation that
e,
=
±Iel
where
lei
is
the magnitude
of
the charge on the electron. Tht:s electrons and holes contribute with opposite
sign to
R
Hall
in
equation. When more than one carrier type
is
present, it is not
always the case that the sign
of
the Hall coefficient
is
the same as the sign
of
the majority carrier type. A minority carrier type may have a higher mobility,
and the carriers with high mobility make a larger contribution per carrier to
R
Hall
than do the low mobility carriers.
The magnetoconductivity for two carrier types
is
obtained from equation
upon substitution
of
equation and retaining terms in
~
2. For the transverse
magnetoconductance we obtain
CJB(B)-CJB(O)
CJB(O)
which is an average
of
~
1 and
P2
weighted by conductivity components
a
OI
and a
02
. But since
~p/p
=
-~ala
we obtain the following result for the
transverse magnetoresistance
~p
p
=
We note that the magnetoconductivity tensor yields no longitudinal
magnetoresistance for a spherical two-carrier model.
CYCLOTRON
EFFECTIVE MASS
To calculate the magneto resistance and explicitly for non-spherical Fermi
surfaces, we need to derive a formula for the cyclotron frequency
roc
= eBI
(m~
c) which is generally
applicabl~for
non-spherical Fermi surfaces. The
cyclotron effective mass can be
den!rmined
in
either
of
two ways. The first
method
is
the tube integral method for a schematic
of
the constant energy
surfaces at energy
E and E +
~
which defines the cyclotron effective_ mass
as
1
4
11dK
1124
dK
m*-
-
----
c
2rc
Ivl
2rc
laE
I
akl
128
Magneto-transport Phenomena
where dK is an infinitesimal element
of
length along the contour and we can
obtain
m~
by direct integration. For the second method, we convert the line
integral over an enclosed area, making use
of
!~.E
= I1k(I1Elak) so that
tz2
1
112
M
111*
=
---rf(l1k)dK
=---
c
2n
M'j
2rc
I1E
where M is the area
ofthe
strip indicated by the separation
M.
Therefore we
obtain the relation
h
2
BA
m*=--
c
2n
BE
which gives the second method for finding the cyclotron effective mass.
For
a
spherical
constant
energy
surface,
we
have
A =
rrk2
and
E
(f)
=
(h2~)/(2m*)
so that m* = m*. For an electron orbit described by an
ellipse in reciprocal space (which
is
appropriate for the general orbit in the
presence
of
a magnetic field on
an
ellipsoidal constant energy surface at wave
vector
kB
along the magnetic field) we write
h
2
k
2
h
2
k
2
h
2
k
2
E(f)=
__
1 +
__
2 =
__
0
1.
2ml
2m2
2mo
which defines the area A enclosed by the constant energy surface as
A = rrklk2 =
rrk~
~mlm2/mo
where (k,2/m;) = (kJ
Imo).
Then substitution in equation gives
m~=
~mlm2.
This expression for m
~
gives a clear physical picture
of
the relation
between
m~
and the electron orbit on a constant ellipsoidal energy surface in
the presence
of
a magnetic field. Since finding the electron orbit requires
geometrical calculation for a general magnetic
tield orientation, it is more
convenient for computer computation to use the relation
_
[
detiiz*
)112
m~-
b'
-*
b'
·m
.
for calculating
m~for
ellipsoidal constant ent:rgy surfaces
where;;.
iiz*
.;;
is
the effective mass component along the magnetic field, det
ni*
denotes the
"-
determinant
of
the effective mass tensor
m*,
and b is
avector
along
the
magnetic field. One can show that for this case the Hall mobility
is
given by
et
IlHall =
-*
me
Magneto-transport Phenomena
and
B
IlHall -
==
roc't =
~
C
so that IlHaIl involves the cyclotron effective mass.
EFFECTIVE MASSES FOR ELLIPSOIDAL FERMI SURFACES
129
The effective mass
of
carriers
in
a magnetic field
is
complicated by the
fact that several effective mass quantities are
of
importance. These include
the cyclotron effective mass
m
~
for electron motion transverse to the magnetic
field and the longitudinal effective mass
m
1]
for electron motion along the
magnetic field
m1]=
b·m*·b
obtained
by
projecting the effective mass tensor along the magnetic field. These
motions are considered
in
findingA, the change
in
the electron distribution
function due to forces and fields.
Returning to Equation the initial exposition for the current density
calculated by the Boltzmann equation, we obtained the Drude formula
_ 2 ( 1 )
cr
=
ne
't
m * .
Thereby defining the drift mass tensor
in
an electric field. Referring to
the magnetoresis tance and magneto-conductance tensors, we see the drift term
(ne
2
't
1m
*)
which utilizes the drift mass tensor and terms
in
(roc
't)
which utilizes
the effective cyclotron mass
m~.
Where the Fermi surface for a semiconductor
consists
of
ellipsoidal carrier pockets, then the drift effective mass components
are found
in
accordance with the procedure outlined
in
for ellipsoidal carrier
pockets. We conveniently use equation to determine the cyclotron effective
mass for ellipsoidal carrier pockets.
DYNAMICS
OF
ElECTRONS
IN
A
MAGNETIC
FIELD
We relate the electron motion on a constant energy surface
in
a magnetic
field to real space orbits. Consider first the case
of
B =
o.
At a given k value,
E(k)
and
v(k)
are specified and each
of
the quantities
is
a constant
of
the
motion, where
v(k)
=
(lIh)[oE
(k
)/0
k].
If
there are no forces on the system,
E
(k
) and
v(
k ) are unchanged with time. Thus, at any instant
of
time there
is
an equal probability that an electron will
be
found anywhere on a constant
energy surface. The role
of
an external electric field
jj;
is
to change the k
vector on this constant energy surface according to the equation
of
motion
130
Magneto-transport Phenomena
hi:
=
eE
so that under a force
eE
the energy
of
the system
is
changed.
Fig. Schematic diagram
of
the motion
of
an electron along a constant energy (and
constant
k=)
trajectory in the presence
of
a magnetic field in the z-direction.
In
a constant magnetic field (no electric field), the electron will move on
a constant energy surface
in
i:
space in an orbit perpendicular to the magnetic
field and following the equation
of
motion
.
e(_
B-)
hi:
= -
vx
c
where we note that
IV.LI
remains unchanged along the electron orbit. The
electrons will execute the indicated orbit
at
a cyclotron frequency
roc
given by
roc
=
eBlm~
c where
m~
is the cyclotron effective mass.
For high magnetic field
s,
when
roc't»
1,
an electron circulates many
times around its semiclassical orbit before undergoing a collision.
In this limit,
there is interest
in
describing the orbits
of
carriers
in
r space. The solution to
the semiclassical equations
is
f:
= v =
(1
I h
)8E
I 8i:
hk
=
e[
E + (1/
c)v
x
B]
- B
(B-
-)
r = r
--
·r
.L
B2
in
which r =
~I
+
r.L
. We also note that
Bxhk
= e[BxE+(l/c)Bx(vxB)]
Magneto-transport Phenomena
=
e[BxE+(lIe)B
2
v-(B.v)B]
Making use
of
Equation, we may write
.
en
-
-'.
C - -
r1-
=
-2
Bxk
+2(E
x B)
eB B
which upon integration yields
where
c - -
OJ
=
2(E
x
B).
B
131
From equation we see that the orbit
in
real space
is
It/2 out
of
phase
with the orbit
in
reciprocal space.
c
Fig. The electron orbits
in
metallic copper indicates only a
few
of
the many types
of
orbits an electron can pursue in k-space when a uniform magnetic field
is
applied to
a noble metal. (Recall that the orbits are given
by
slicing the Fermi surface with
planes perpendicular to the field.) The figure displays (a) a closed electron orbit; (b)
a closed hole orbit; (c)
an
open hole orbit, which continues
in
the same general
direction inde finitely
in
the repeated-zone scheme.
For the case
of
closed orbits, after a long time
tIt""
'to
Then the second
term
OJ
t
of
equation will dominate, giving a transverse or Hall current