Wiedemann–Franz law
In physics, the Wiedemann–Franz law states that the ratio of the electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is proportional to the temperature (T).
This empirical law is named after Gustav Wiedemann and Rudolph Franz, who in 1853 reported that κ/σ has approximately the same value for different metals at the same temperature.
[3] Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the free electrons in the metal.
Since the electric field and the current density are vectors Ohm's law is expressed here in bold face.
Paul Drude (c. 1900) realized that the phenomenological description of conductivity can be formulated quite generally (electron-, ion-, heat- etc.
Although the phenomenological description is incorrect for conduction electrons, it can serve as a preliminary treatment.
[4] The assumption is that the electrons move freely in the solid like in an ideal gas.
The further assumption therefore is that the electrons bump into obstacles (like defects or phonons) once in a while which limits their free flight.
is the mean free path of the electrons, and is the average speed of the particles in the gas.
This is in fact due to 3 mistakes that conspired to make his result more accurate than warranted: the factor of 2 mistake; the specific heat per electron is in fact about 100 times less than
[5] After taking into account the quantum effects, as in the free electron model, the heat capacity, mean free path and average speed of electrons are modified and the proportionality constant is then corrected to
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 ratio
from the theoretical Lorenz value L0: (i) other thermal carriers such as phonons or magnons, (ii) Inelastic scattering.
For each electron transported, a thermal excitation is also carried and the Lorenz number is reached L = L0.
Note that in a perfect metal, inelastic scattering would be completely absent in the limit
At higher temperatures, the contribution of phonons to thermal transport in a system becomes important.
In many high purity metals both the electrical and thermal conductivities rise as temperature is decreased.
In the purest samples of silver and at very low temperatures, L can drop by as much as a factor of 10.
[10] 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 the following cases: manipulating electronic density of states, varying doping density and layer thickness in superlattices and materials with correlated carriers.
[11][12] [13] In 2011, N. Wakeham et al. found that the ratio of the thermal and electrical Hall conductivities in the metallic phase of quasi-one-dimensional lithium molybdenum purple bronze Li0.9Mo6O17 diverges with decreasing temperature, reaching a value five orders of magnitude larger than that found in conventional metals obeying the Wiedemann–Franz law.
[14] A Berkeley-led study in 2016 by S. Lee et al. also found a large violation of the Wiedemann–Franz law near the insulator-metal transition in VO2 nanobeams.
In the metallic phase, the electronic contribution to thermal conductivity was much smaller than what would be expected from the Wiedemann–Franz law.
The results can be explained in terms of independent propagation of charge and heat in a strongly correlated system.
