Hall Effect Measurements on New Thermoelectric Materials
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Hall Effect Measurements on New Thermoelectric Materials Jarrod Short†, Sim Loo†, Sangeeta Lal†, Kuei Fang Hsu‡, Eric Quarez‡, Mercouri G. Kanatzidis‡, Timothy P. Hogan† † Electrical and Computer Engineering Department, Michigan State University 2120 Engineering Building, East Lansing, MI 48824-1226 ‡ Chemistry Department, Michigan State University East Lansing, MI 48824-1322
ABSTRACT In the field of thermoelectrics, the figure of merit of new materials is based on the electrical conductivity, thermoelectric power, and thermal conductivity of the sample, however additional insight is gained through knowledge of the carrier concentrations and mobility in the materials. The figure of merit is commonly related to the material properties through the B factor which is directly dependent on the mobility of the carriers as well as the effective mass. To gain additional insight on the new materials of interest for thermoelectric applications, a Hall Effect system has been developed for measuring the temperature dependent carrier concentrations and mobilities. In this paper, the measurement system will be described, and recent results for several new materials will be presented.
INTRODUCTION The efficiency of thermoelectric generators is well known to be dependent on the electrical conductivity, thermoelectric power, and thermal conductivity of the materials forming the n- and p- legs of the module. Analysis of thermoelectric materials based on a one band model [1] shows the importance and interrelationship of the electrical conductivity, thermoelectric power, and the electronic contribution to the thermal conductivity. It also shows the relationship between these material parameters and the position of the Fermi energy level relative to the band edge. Based on these formulae, more information can be extracted from the data by adding the measurement of Hall effect, and determining the carrier concentration. The electrical conductivity for the oneband model for the conduction band (n-type materials) gives qµ x 2k B T σ = qµ x n = 2π 2 h 2
3
2
(m ) ∗ n
3
2F
1
2
(1)
where µx is the mobility in the x direction, mn∗ is the density of states effective mass, kB is Boltzmann’s constant (8.616×10-5 eV/K), T is temperature, h , is Plank’s reduced
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constant, q is the electronic charge (1.602×10-19 C), and F 1 is the Fermi-Dirac function 2
of order ½. The Fermi-Dirac function is defined as: ∞
( )
Fi = Fi ζ ∗ =
∫( 0
and ζ ∗ =
e
xi
x −ζ ∗
(2)
) + 1 dx
ζ is the reduced chemical potential relative to the conduction band edge. k BT
From (1) it is also seen that the carrier concentration can be written as 1 2k T n = 2 B2 2π h
3
2
(mn∗ )3 2 F
1
2
.
(3)
The thermoelectric power (absolute Seebeck coefficient) for this n-type material is given by S =−
(s + 2) Fs +1 − ζ ∗ , (s + 1) Fs
kB q
(4)
and the thermal conductivity consists of an electronic contribution, κe, and a lattice contribution, κL as κ = κe + κ L k µ h 2 2k T = B x 2 B2 q 4π h
5
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