Wiedemann-Franz equation

Total thermal conductivity is a sum of the lattice and electronic parts, K = Ki + Ke- The lattice part of the thermal conductivity describes the scattering of phonons on the vibrations of atoms, whereas the electronic part describes thermal conductivity appearing due to conduction electrons and is related to the electrical conductivity absolute temperature and Lq is the ideal Lorenz number, 2.45 X 10 Wf2K [64]. The electronic part of the thermal conductivity is typically low for low-gap semiconductors. For the tin-based cationic clathrates it was calculated to contribute less than 1% to the total thermal conductivity. The lattice part of the thermal conductivity can be estimated based on the Debye equation /Cl = 1 /3(CvAvj), where C is the volumetric heat capacity, X is the mean free path of phonons and is the velocity of sound [64]. The latter is related to the Debye characteristic temperature 6 as Vs = [67t (7V/F)] . Extracting the... 

Proceeding via a solution of either the Boltzmann equation or the Kubo formula, one obtains the Wiedemann-Franz law , relating k, p, and T ... 

The fact that the ratio of the thermal conductivity to the electrical conductivity of any metal is a constant times the absolute temperature was observed by Wiedemann and Franz and this relationship is known as the Wiedemann-Franz ratio. This relationship works because the collision time t for the electron carriers is the same in both models and cancels out when taking the ratio of the two conductivities. From Chapter 17, the classical electronic thermal conductivity was found in Equation 17.33 to be K = 4nl(fT/m n)T. The classical electrical conductivity from the Drude model is given by Equation 18.15 and the Wiede-mann-Eranz ratio becomes... 

Thermal conductivity is the most difficult quantity to understand in terms of the electronic structure. Thermal energy can be stored in vibrational normal modes of the crystal, and one can transport thermal energy through the lattice of ions. These concepts seem to be macroscopic. Therefore, one can set up suitable wave packets to treat thermal conductivity as quantized matter. In particular, electron plus induced lattice polarization can be defined as polarons. For conduction electrons, the electrical conductivity and the thermal conductivity were first observed by Wiedemann and Franz as indicated in the following equation ... 

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