Download - Hall Effect
![Page 1: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/1.jpg)
The Hall Effect
LL8 Section 22
![Page 2: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/2.jpg)
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
![Page 3: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/3.jpg)
Separate conductivity tensor into symmetric and antisymmetric parts.This is always possible for a rank 2 tensor.
But
![Page 4: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/4.jpg)
![Page 5: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/5.jpg)
aik has only 3 components, like a vector. Any antisymmetric aik is dual to an axial vector, which has no sigh change under inversion
![Page 6: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/6.jpg)
Joule heat
Determined for given E only by the symmetrical part of sik.
![Page 7: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/7.jpg)
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
![Page 8: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/8.jpg)
sik (H) is even. Expansion of sik has only even powers
Zero-field conductivity tensor Symmetrical in (i,k) and in (l,m)
![Page 9: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/9.jpg)
First order effect of H-field is linear in H.
This term might also have a component perpendicular to E.
![Page 10: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/10.jpg)
Inverse formula
Symmetric part
Resistivity tensor
Antisymmetric part
![Page 11: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/11.jpg)
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
![Page 12: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/12.jpg)
![Page 13: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/13.jpg)
For isotropic conductor, including cubic semiconductors
Axial vectors a and b must be parallel to H
More generally,
![Page 14: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/14.jpg)
Symmetry: All tensors that characterize an isotropic medium must be invariant under all rotations about H
![Page 15: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/15.jpg)
Likewise, symmetric parts of conductivity and resistivity tensors must be invariant under rotations about H
![Page 16: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/16.jpg)
Let j lie in the xz plane
x
![Page 17: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/17.jpg)
In an isotropic conductor, the Hall field is the only E-field that is perpendicular to both j and H.
![Page 18: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/18.jpg)
![Page 19: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/19.jpg)
• 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
![Page 20: Hall Effect](https://reader030.vdocuments.us/reader030/viewer/2022032612/568132eb550346895d99a747/html5/thumbnails/20.jpg)