used since it is not possible to consider each charge carrier as a separate entity. The Wiedemann–Franz law can also be exactly derived using the non-interacting electron picture, and so in materials where the Wiedemann–Franz law fails (such as superconductors), the Mott relations also generally tend to fail. The formulae above can be simplified in a couple of important limiting cases: Mott formula in metals In semimetals and metals, where transport only occurs near the Fermi level and changes slowly in the range , one can perform a Sommerfeld expansion , which leads to This expression is sometimes called "the Mott formula", however it is much less general than Mott's original formula expressed above. In the free electron model with scattering, the value of is of order , where is the Fermi temperature, and so a typical value of the Seebeck coefficient in the Fermi gas is (the prefactor varies somewhat depending on details such as dimensionality and scattering). In highly conductive metals the Fermi temperatures are typically around 104 – 105 K, and so it is understandable why their absolute Seebeck coefficients are only of order 1 – 10 μV/K at room temperature. Note that whereas the free electron model predicts a negative Seebeck coefficient, real metals actually have complicated band structures and may exhibit positive Seebeck coefficients (examples: Cu, Ag, Au). The fraction in semimetals is sometimes calculated from the measured derivative of with respect to some energy shift induced by field effect. This is not necessarily correct and the estimate of can be incorrect (by a factor of two or more), since the disorder potential depends on screening which also changes with field effect. Mott formula in semiconductors In semiconductors at low levels of doping, transport only occurs far away from the Fermi level. At low doping in the conduction band (where , where is the minimum energy of the conduction band edge), one has . Approximating the conduction band levels' conductivity function as for some constants and , whereas in the valence band when and , The values of and depend on material details; in bulk semiconductor these constants range between 1 and 3, the extremes corresponding to acoustic-mode lattice scattering and ionized-impurity scattering. In extrinsic (doped) semiconductors either the conduction or valence band will dominate transport, and so one of the numbers above will give the measured values. In general however the semiconductor may also