The Hall Effect in Conductors
The Hall Effect
LL8 Section 22
A conductor in an external magnetic field
H
Onsager’s principle doesn’t hold anymore
Instead
v.5 section 120, and v.2: Time-
reversal symmetry only if
H
-
H
1. A magnetic field breaks the symmetry of the conductivity tensor
2. Separate conductivity tensor into symmetric and antisymmetric parts.
This is always possible for a rank 2 tensor.
But
Components of s
ik
are even functions of
H
Components of
a
ik
are odd functions of
H
3. The a
ik
has only 3 components, like a vector.
Any antisymmetric a
ik
is dual to an axial vector
(no sign change under inversion)
Joule heat
Zero, since
E
(
E
x
a
)
4. Joule heating is determined by the symmetrical part of the conductivity tensor
alone.
5. External H-fields are usually weak. Expand
(
H
) in powers of
H
.
a
(
H
) is odd, so expansion contains only odd powers of
H
.
a
is an axial vector
Must be ordinary polar tensor.
Transforms like products of components of
vectors that changes sign under inversion.
H
is an axial vector
a
i
=
ik
H
k
+ …
s
ik
(
H
) is even.
Expansion of s
ik
has only even powers
Zero-field
conductivity
tensor
Symmetrical in (
i,k
)
and in (
l,m
)
First order effect of
H
-field is linear in
H
.
This term might also have a
component perpendicular to
E
.
Hall effect.
Axial vector
a
is linear in
H
.
a
i
=
ik
H
k
Hall current is perpendicular to
E
and proportional to H & E
j
i
= s
ik
E
k
+ (
E
x
a
)
i
6. The first order term in the expansion of
ik
(
H
), gives the
Hall effect.
Symmetric part
Resistivity tensor
Antisymmetric part
Inverse formula
perpendicular to
j
and proportional to H and j,
but not necessarily perpendicular to
H
The axial vector
b
is dual to b
ik
,
b
is linear in
H
for small
H
Ordinary Ohm’s law
Hall effect
Math arguments repeat
7. Inverse of j
i
=
ik
E
k
, and the resistivity tensor.
8. For isotropic conductor (e.g. cubic semiconductors) axial vectors
a
and
b
must be
parallel to
H.
(Then Hall current and Hall field are perpendicular to
H
.)
For non-isotropic conductors,
All tensors that characterize an isotropic medium
must be invariant under all rotations about
H
Likewise, symmetric parts of conductivity
and resistivity tensors must be invariant
under rotations about
H
Let
j
lie in the xz plane of an isotropic conductor
In an isotropic
conductor, the Hall
field is the only
E
-
field that is
perpendicular to
both
j
and
H
.
Hall field
In an isotropic body
Hall Constant,
can be positive or negative
9. The Hall constant
R
•
Next terms in expansion of
ik
j
k
must be
–
quadratic in
H
,
–
linear in j,
–
And be a vector
•
Only possible combinations of
H
&
j
are
and
in an isotropic body
General form of
E
=
E
(
j
)
Zero-field
term with
isotropic
resistivity.
Hall term
E
H
H
E
H
j
Quadratic terms
E
Q1
||
j
E
Q2
||
H
The Hall Effect in conductors is described in detail, showcasing how a magnetic field affects the conductivity tensor, breaking symmetry and leading to unique properties. The discussion covers the separation of the conductivity tensor, the impact of magnetic fields on conductivity components, and the manifestation of the Hall Effect. Various tensors and vectors are explored to understand the behavior of materials in external magnetic fields.
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Presentation Transcript
The Hall Effect LL8 Section 22
1. A magnetic field breaks the symmetry of the conductivity tensor A conductor in an external magnetic field H Onsager s principle doesn t hold anymore Instead v.5 section 120, and v.2: Time- reversal symmetry only if H -H
2. Separate conductivity tensor into symmetric and antisymmetric parts. This is always possible for a rank 2 tensor. But
Components of sik are even functions of H Components of aik are odd functions of H
3. The aik has only 3 components, like a vector. Any antisymmetric aik is dual to an axial vector (no sign change under inversion)
4. Joule heating is determined by the symmetrical part of the conductivity tensor alone. Joule heat Zero, since E (E x a)
5. External H-fields are usually weak. Expand (H) in powers of H. a(H) is odd, so expansion contains only odd powers of H. ai = ik Hk+ a is an axial vector H is an axial vector Must be ordinary polar tensor. Transforms like products of components of vectors that changes sign under inversion. sik(H) is even. Expansion of sik has only even powers Zero-field conductivity tensor Symmetrical in (i,k) and in (l,m)
6. The first order term in the expansion of ik(H), gives the Hall effect. First order effect of H-field is linear in H. ji = sik Ek + (E x a)i Hall effect. Axial vector a is linear in H. ai = ik Hk This term might also have a component perpendicular to E. Hall current is perpendicular to E and proportional to H & E
7. Inverse of ji = ikEk , and the resistivity tensor. Inverse formula Resistivity tensor Antisymmetric part Symmetric part Math arguments repeat The axial vector b is dual to bik, b is linear in H for small H Hall effect Ordinary Ohm s law perpendicular to j and proportional to H and j, but not necessarily perpendicular to H
8. For isotropic conductor (e.g. cubic semiconductors) axial vectors a and b must be parallel to H. (Then Hall current and Hall field are perpendicular to H.) For non-isotropic conductors, All tensors that characterize an isotropic medium must be invariant under all rotations about H
Likewise, symmetric parts of conductivity and resistivity tensors must be invariant under rotations about H
In an isotropic conductor, the Hall field is the only E- field that is perpendicular to both j and H. Hall field
9. The Hall constant R In an isotropic body Hall Constant, can be positive or negative
Next terms in expansion of ikjk must be quadratic in H, linear in j, And be a vector Only possible combinations of H & j are and in an isotropic body
General form of E = E(j) Hall term EH H EH j Quadratic terms EQ1 || j EQ2 || H Zero-field term with isotropic resistivity.
