4.1 Characterization of Materials

The overall performance of a thermoelectric generator is rated by characteristic numbers as its efficiency, total power output, and power density. However, these numbers, especially the efficiency, is limited by several parameters. Besides the geometrical impact, material parameters such as the Seebeck coefficient as well as thermal and electrical conductivities have a strong influence on both transport of carriers and phonons and thus the overall device behavior. In the sequel, the thermoelectric figure of merit, which embraces the material parameters affecting the device behavior, as well as its influence on the device efficiency are discussed.

According to Ioffe [7], the maximum conversion efficiency $\ensuremath{\eta}_{\ensuremath{\mathrm{max}}}$ of a thermoelectric generator at matched load condition $R_{\ensuremath{\mathrm{i}}} = R_{\ensuremath{\mathrm{L}}}$ is given by the product of the ideal reversible thermodynamic process' efficiency and a factor describing the energy losses within the device due to Joule heating and non-ideal thermal conductivity [11]

$$\ensuremath{\eta}_{\ensuremath{\mathrm{max}}} = \frac{\ensuremath... ...\, \frac{M - 1}{M + \ensuremath{T_{\mathrm{C}}}/\ensuremath{T_{\mathrm{H}}}} \,$$ (4.1)

where $\ensuremath{T_{\mathrm{H}}}$ and $\ensuremath{T_{\mathrm{C}}}$ denote the temperatures of the heated and the cooled end of the device, respectively and

$$M = \sqrt{1 + \frac{1}{2} Z \left( \ensuremath{T_{\mathrm{C}}}+ \ensuremath{T_{\mathrm{H}}}\right)} \,.$$ (4.2)

The averaged thermoelectric figure of merit for both legs of the device, indicated by numerical subscripts, with matched geometry

$$Z = \frac{\left(\ensuremath{\alpha}_1 - \ensuremath{\alpha}_2\right)^2}{\left( \sqrt{\kappa_1/\sigma_1} + \sqrt{\kappa_2/\sigma_2} \right)^2} \,.$$ (4.3)

This figure of merit incorporates all relevant material parameters, which are the Seebeck coefficient $\ensuremath{\alpha}$ , thermal conductivity $\kappa$ , and electric conductivity $\sigma$ . Due to the strong dependence on both temperature and the concentration of free carriers of these single parameters, the figure of merit exhibits according dependencies as well, which means that each material has its own optimum range of operation. However, in practical devices, both legs have similar material properties, and the bulk figure of merit for a certain material is conveniently defined as

$$Z = \frac{\ensuremath{\alpha}^2 \sigma}{\kappa} \,.$$ (4.4)

From a microscopic point of view, the figure of merit is influenced by both charge and heat transport as well as the coupling of these two within a semiconductor. Thus, the figure of merit follows from the band structure, lattice dynamics, and scattering mechanisms of charge carriers.

An ideal thermoelectric material is not only assured by a high figure of merit, but also by the temperature range, where these high values are achieved. In practical situations, each material has its ideal operation temperature range, thus the choice of the material is strongly affected by the intended use.

Figure 4.1: Seebeck coefficient, conductivity, thermal conductivity, and figure of merit with respect to free carrier concentration, after [121].

The free carrier concentration which is influenced by the doping in semiconductors, has a strong influence on the figure of merit. Fig. 4.1 illustrates the dependence of several material parameters on the concentration of free carriers. While increasing carrier concentrations have generally a detrimental effect on the Seebeck coefficient, the electric conductivity $\sigma$ increases due to the increased number of available carriers. On the other hand, the electric part of the thermal conductivity $\ensuremath{\kappa_{\ensuremath{\nu}}}$ becomes non-negligible at high values of the carrier concentration and the dominant thermal conductivity mechanism on the transition to metals. Both insulators and metals show superior conditions for single parameters, but accordingly poor conditions for others. Metals are characterized by generally low values of the Seebeck coefficient and comparably high thermal conductivities, which cannot be compensated by their low electric resistances. On the other hand, insulators have comparably high Seebeck coefficients, which cannot outperform the very low electric conductivities. Semiconductors are positioned in the competition region of the single parameters, and thus the resulting thermoelectric figure of merit has its maximum. This maximum is supported by still moderate Seebeck coefficients and already good electrical conductivities and limited by elevated electrical thermal conductivity in the region of high carrier concentrations. Within semiconductors, the optimum carrier concentration can be accurately controlled by proper doping concentrations.

Figure 4.2: Thermoelectric figure of merit vs. temperature for several materials used for thermoelectric devices, after [121].

Temperature dependent figure of merit data for materials commonly used in thermoelectric devices are collected in Fig. 4.2. While bismuth telluride and several ternary alloys area good choice for low temperature thermoelectrics, silicon and silicon-germanium alloys are suitable for higher temperatures. Lead telluride covers the intermediate range between bismuth telluride and silicon-germanium. For even higher temperatures, wide band gap materials such as silicon carbide and boron carbide have to be considered. The dashed line depicts the product of the figure of merit and temperature to be one. Maximum figure of merit values for several materials do not outperform this line by far which results in an accordingly limited conversion efficiency.

M. Wagner: Simulation of Thermoelectric Devices