hall effect
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The Hall Effect
LL8 Section 22
A conductor in an external magnetic field H
Onsager’s principle doesn’t hold
Instead
v.5 section 120, and v.2: Time-reversal symmetry only if H -H
Separate conductivity tensor into symmetric and antisymmetric parts.This is always possible for a rank 2 tensor.
But
aik has only 3 components, like a vector. Any antisymmetric aik is dual to an axial vector, which has no sigh change under inversion
Joule heat
Determined for given E only by the symmetrical part of sik.
External H-fields are usually weak.Expand s(H) in powers of H.
a(H) is odd, so it contains only odd powers of H.
axial
Ordinary polar tensor, product of components that change sign under inversion, x -> -x, etc.
axial
sik (H) is even. Expansion of sik 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.
Inverse formula
Symmetric part
Resistivity tensor
Antisymmetric part
Math arguments repeat
• First term is ordinary Ohm’s law• Second term gives Hall effect– The axial vector b– dual to bik
– Linear in H for small H
For isotropic conductor, including cubic semiconductors
Axial vectors a and b must be parallel to H
More generally,
Symmetry: 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
x
In an isotropic conductor, the Hall field is the only E-field that is perpendicular to both j and H.
• Next terms in expansion of rikjk must be – quadratic in H,– linear in j,– And be a vector
• Only possible combinations of H & J are
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