bei48482_FM

354 Chapter Ten

T

he free-electron model of metallic conduction was proposed by Paul Drude in 1900, only

three years after the discovery of the electron by J. J. Thomson, and was later elaborated

by Hendrik Lorentz. Fermi-Dirac statistics were unknown then, and Drude and Lorentz assumed

that the free electrons were in thermal equilibrium with a Maxwell-Boltzmann velocity distri-

bution. This meant that the Fin Eq. (10.16) was replaced by the rms electron velocity rms. In

addition, Drude and Lorentz assumed that the free electrons collide with the metal ions, not

with the much farther apart lattice defects. The net result was resistivity values on the order of

10 times greater than the measured ones.

The theory was nevertheless considered to be on the right track, both because it gave the

correct form of Ohm’s law and also because it accounted for the Weidemann-Franz law.This

empirical law states that the ratio K(where  1 ) between thermal and electric conduc-

tivities is the same for all metals and is a function only of temperature. If there is a temperature

difference Tbetween the sides of a slab of material xthick whose cross-sectional area is A,

the rate Qtat which heat passes through the slab is given by

KA

where Kis the thermal conductivity. According to the kinetic theory of a classical gas applied to

the electron gas in the Drude-Lorentz model,

K

From Eq. (10.16) with Freplaced by rms,



Hence the ratio between the thermal and electric resistivities of a metal is



According to Eq. (9.15), ^2 rms 3 kTm, which gives

1.11  10 ^8 W

/K^2

This ratio does not contain the electron density nor the mean free path , so KTought to

have the same constant value for all metals, which is the Weidemann-Franz law. To be sure, the

above value of KTis incorrect because it is based on a Maxwell-Boltzmann distribution of

electron velocities. When Fermi-Dirac statistics are used, the result is

2.45  10 ^8 W

/K^2

which agrees quite well with experimental findings.

10.6 BAND THEORY OF SOLIDS

The energy band structure of a solid determines whether it is a conductor,

an insulator, or a semiconductor

No property of solids varies as widely as their ability to conduct electric current. Cop-

per, a good conductor, has a resistivity of 1.7   10 ^8 

m at room temperature,

^2 k^2



3 e^2

K



T

3 k^2



2 e^2

K



T

km^2 rms



2 e^2

mrms



ne^2 

knrms



2

K



ne^2 



mrms

1





knrms



2

T



x

Q



t

Weidemann-Franz Law

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