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). 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). Theoretically, the proportionality constant L, known as the Lorenz number, is equal to 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. The proportionality of κ/σ with temperature was discovered by Ludvig Lorenz in 1872. Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the free electrons in 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. The current density j is observed to be proportional to the applied electric field and follows Ohm's law where the prefactor is the specific electrical conductivity. Since the electric field and the current density are vectors Ohm's law is expressed here in bold face. The conductivity can in general be expressed as a tensor of the second rank (3×3 matrix). Here we restrict the discussion to isotropic, i.e. scalar conductivity. The specific resistivity is 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 an ideal gas. The force applied to the electron by the electric field leads to an acceleration according to This would lead, however, to a constant acceleration and, ultimately, to an infinite velocity. The further assumption therefore is that the electrons bump into obstacles (like defects or phonons) once in a while which limits their free flight. This establishes an average or drift velocity Vd. The drift velocity is related to the average scattering time as becomes evident from the following relations. The value L0 = 2.44×10−8 W Ω K−2 results from the fact that at low temperatures ( T → 0 {displaystyle T ightarrow 0} 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 L = κ / ( σ T ) {displaystyle L=kappa /(sigma T)} from the theoretical Lorenz value L0: (i) other thermal carriers such as phonon or magnons, (ii) Inelastic scattering.As the temperature tends to 0K, inelastic scattering becomes weak and promotes large q scattering values (trajectory a in the figure). 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 T → 0 {displaystyle T ightarrow 0} K and the thermal conductivity would vanish κ → 0 ; L → 0 {displaystyle kappa ightarrow 0;L ightarrow 0} .At finite temperature small q scattering values are possible (trajectory b in the figure) and electron can be transported without the transport of a thermal excitation L(T) < L0.At higher temperatures, the contribution of phonon to thermal transport in a system becomes important. This can lead to L(T) > L0. Above the Debye temperature the phonon contribution to thermal transport is constant and the ratio L(T) is again found constant.