“The theory of the thermopower of the noble metals is as uncertain as many of the experimental results which it sets out to explain,” wrote J. M. Ziman in his classic paper [Adv. Phys., 1961]. In truth, only the low temperature data in the ‘phonon drag' regime below about 0.5 of the Debye temperature are in such wide disagreement, while measurements of the ‘diffusional thermopower' above that temperature range are quite uncontroversial.
With regard to the theory, however, the calculation of the absolute thermoelectric power of even such simple one-electron metals as Li, Cu, Ag, or Au is still so unreliable that even the sign of the effect is often incorrect. To calculate the absolute thermopower SA for a material A, SA = (1/e) μ/ T = (1/eT) L12/L11, where the two transport coefficients L12 and L11 of the Onsager equations of irreversible thermodynamics have to be determined from the energy dependence of the density of states (DOS) by the theory of scattering processes. Here μ is the gradient of the electrochemical potential, or Fermi energy, of the electrons. The application of this theory to a practical example requires several simplifications and assumptions of uncertain consequences. In particular, the free electron model, while making the math more manageable, is responsible for most of the discrepancies between measured and calculated values. The problems seemed so insurmountable that further attempts have fallen out of favor during the last 40 years, despite the availability of powerful computers. Actually, only the ratio of these transport coefficients is required, not their individual values.
Horst Brodowsky and coworkers at the University of Kiel, Germany have shown that the ratio of Onsager's transport coefficients relevant to the absolute thermopower can be obtained directly. The temperature dependence of the electrochemical potential of the electrons can be readily calculated from the DOS of the material without recourse to the unrealistic free electron model. The situation is equivalent to the Wiedemann-Franz law, or the law of mass action. In both cases, the ratio of two transport or kinetic properties, difficult to calculate individually, is equal to an equilibrium quantity, which is a simple function of temperature. Brodowsky has underpinned his argument with a thermodynamic derivation of this approach, which depends on the constant electron/atom ratio along the length of the wire. This condition is fulfilled in metals and alloys, but not, unfortunately, in semiconductors.
The new approach is bound to be met with skepticism. Can the thermopower, this paradigm of irreversible thermodynamics, be calculated as an equilibrium property? Perhaps some skeptics will be swayed by the extraordinary success of the new method in calculating even very complicated curves of thermopower versus temperature (in pure metals) or thermopower isotherms versus mole fraction (in alloys). So far, the fingerprint-like agreement between measured and calculated isotherms for the Ir-Pt-Au and Rh-Pd-Ag alloys is truly remarkable, and the results for the alkali metals or for Nb-Mo alloys are also very gratifying [Z. Metallkd. (1999) 90, 111; (2000) 91, 375; (2002) 93, 1164; (2004) 95, 698]. The possibility of forging a firm connection between the thermoelectricity of metals and the databases of DOS should revive the long dormant discussion of the theory. If upheld, the approach will also throw fresh light on the properties of the electron gas in general.
From a practical point of view, the voltage EAB generated by a thermocouple with fixed low-temperature junction at T1 is related to SA and SB by SAB = SA − SB = dEAB/dT2. Temperature measurements with noble metal thermocouples in a reducing atmosphere have been plagued by a gradual loss of accuracy because components of the insulating ceramics (e.g. Si or Al) dissolve in the noble metals. With a reliable theory to predict the effect of the dissolved impurities, the performance or design of such thermocouples might be improved. The theory can also identify alloys with maximum thermopower.
The great prize, of course, would be the extension of the theory to semiconductors. These materials are the most useful for such applications as power generation (for satellites) or Peltier cooling (for picnic chests). Most research and development are now concentrated on semiconductors. Brodowsky and coworkers have suggested some theoretical lines that might permit such an extension.