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Thermal Design, 2nd Edition
book

Thermal Design, 2nd Edition

by HoSung Lee
June 2022
Intermediate to advanced
928 pages
29h 26m
English
Wiley
Content preview from Thermal Design, 2nd Edition

Appendix GHall Factor

A magnetic field Hz in the z‐direction applied perpendicularly to an electric current ix in the x‐direction will produce an electric field Ey in the y‐direction. Then the Hall coefficient RH is defined by

(G.1)upper R Subscript upper H Baseline equals StartFraction upper E Subscript y Baseline Over i Subscript x Baseline upper H Subscript z Baseline EndFraction

In order to obtain the Hall coefficient, we have to solve the Boltzmann transport equation under the relaxation time approximation with the magnetic field. The exact solution, assuming a parabolic single band, eventually leads to the electrical conductivity, Seebeck coefficient, and electronic thermal conductivity, where the detailed derivation can be found in Putley [1] and Goldsmid [2]. Here just the results are presented. The electrical conductivity is

(G.2)sigma Subscript x Baseline equals StartFraction e squared Over upper T EndFraction StartFraction upper X 0 squared plus upper Y 0 squared Over upper X 0 EndFraction

The Seebeck coefficient is

(G.3)alpha Subscript x Baseline equals StartFraction 1 Over italic e upper T EndFraction left-parenthesis StartFraction upper X 1 upper X 0 plus upper Y 1 upper Y 0 Over upper X 0 squared plus upper Y 0 squared EndFraction minus upper E Subscript upper F Baseline right-parenthesis

The electronic thermal conductivity is

(G.4)k Subscript l comma x Baseline equals StartFraction 1 Over upper T squared EndFraction left-parenthesis upper X 2 plus StartFraction upper X 0 upper Y 1 squared minus 2 upper X 1 upper Y 1 upper Y 0 minus upper X 1 squared upper X 0 Over upper X 0 squared plus upper Y 0 squared EndFraction minus upper E Subscript upper F Baseline right-parenthesis

where

(G.5b)StartLayout 1st Row 1st Column upper Y Subscript s 2nd Column equals minus StartFraction 2 upper T Over 3 m Subscript d Superscript asterisk Baseline EndFraction integral Subscript 0 Superscript infinity Baseline StartFraction beta dot upper E Superscript s plus 1 Baseline dot tau dot g Over 1 plus beta squared EndFraction StartFraction partial-differential f 0 Over partial-differential upper E EndFraction italic d upper E EndLayout
(G.5c)

τ is the carrier relaxation time, and g is the carrier density of states. Now we ...

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