Inphysics, theWiedemannFranz lawis the ratio of the electronic
contribution of thethermal conductivity() to theelectrical
conductivity() of ametal, and is proportional to
thetemperature(T).[1]
Theoretically, the proportionality constantL, known as the
Lorenz number, is equal to
Thisempiricallaw is named afterGustav WiedemannandRudolph Franz,
who in 1853 reported that/has approximately the same value for
different metals at the same temperature.[2]The proportionality
of/with temperature was discovered byLudvig Lorenzin
1872.Qualitatively, this relationship is based upon the fact that
the heat and electrical transport both involve the freeelectronsin
the metal.
The mathematical expression of the law can be derived as
following. Electrical conduction of metals is a well-known
phenomenon and is attributed to the free conduction electrons,
which can be measured as sketched in the figure. Thecurrent
densityjis observed to be proportional to the appliedelectric
fieldand followsOhm's lawwhere the prefactor is the
specificelectrical conductivity. Since the electric field and the
current density arevectorsOhm's law is expressed here in bold face.
The conductivity can in general be expressed as atensorof the
second rank (33matrix). Here we restrict the discussion
toisotropic, i.e.scalarconductivity. The specificresistivityis the
inverse of the conductivity. Both parameters will be used in the
following.Drude(c. 1900) realized that the phenomenological
description of conductivity can be formulated quite generally
(electron-, ion-, heat- etc. conductivity). Although the
phenomenological description is incorrect for conduction electrons,
it can serve as a preliminary treatment.The assumption is that the
electrons move freely in the solid like in anideal gas. The force
applied to the electron by the electric field leads to
anaccelerationaccording to
This would lead, however, to an infinite velocity. The further
assumption therefore is that the electrons bump into obstacles
(like defects orphonons) once in a while which limits their free
flight. This establishes an average ordrift velocityVd. The drift
velocity is related to theaverage scattering timeas becomes evident
from the following relations.
Contents[hide] 1Temperature dependence 2Limitations of the
theory 3See also 4External links 5ReferencesTemperature
dependence[edit]The valueL0= 2.44108WK2results from the fact that
at low temperatures (K) the heat and charge currents are carried by
the same quasi-particles: electrons or holes. At finite
temperatures two mechanisms produce a deviation of the ratiofrom
the theoretical Lorenz valueL0: (i) other thermal carriers such as
phonon or magnons, (ii) inelastic scattering. In the 0 temperature
limit inelastic scattering is weak and promotes largeqscattering
values is favored (trajectory a in the figure). For each electron
transported a thermal excitation is also carried and the Lorenz
number is reachedL=L0. Note that in a perfect metal, inelastic
scattering would be completely absent in the limitK and the thermal
conductivity would vanish. At finite temperature smallqscattering
values are possible (trajectory b in the figure) and electron can
be transported without the transport of an thermal
excitationL(T)L0. Above theDebye temperaturethe phonon contribution
to thermal transport is constant and the ratioL(T)is again found
constant.
Sketch of the various scattering process important for the
Wiedemann-Fanz law.For references see:[3][4]Limitations of the
theory[edit]Experiments have shown that the value ofL, while
roughly constant, is not exactly the same for all
materials.Kittel[5]gives some values ofLranging fromL=
2.23108WK2for copper at 0C toL= 3.2108WK2for tungsten at 100C.
Rosenberg[6]notes that the WiedemannFranz law is generally valid
for high temperatures and for low (i.e., a few Kelvins)
temperatures, but may not hold at intermediate temperatures.In many
high purity metals both the electrical and thermal conductivities
rise as temperature is decreased. In certain materials (such as
silver or aluminum) however, the value ofLalso may decrease with
temperature. In the purest samples of silver and at very low
temperatures,Lcan drop by as much as a factor of 10.[7]In
degenerate semiconductors, the Lorenz number L has a strong
dependency on certain system parameters: dimensionality, strength
of interatomic interactions and Fermi-level. This law is not valid
or the value of the Lorenz number can be reduced at least in
following cases: manipulating electronic density of states, varying
doping density and layer thickness in superlattices and materials
with correlated carriers.[8][9]See also[edit] Drude modelExternal
links[edit] Strong violation of the Wiedemann-Franz law
