4.2 Optimization of Device Performance

In the previous section, the suitability of several materials to certain temperature ranges as well as the basic material properties responsible the maximum device performance have been carried out. In the following, correlations between several parameters are highlighted and furthermore strategies to improve device efficiency and performance are evolved on the basis of these material parameters.

From the definition of the figure of merit (4.4), it is obvious that the thermal conductivity has to be lowered while maintaining high values of electric conductivity and Seebeck coefficient in order to achieve high $Z$ values. Since several parameters strongly depend on the carrier concentration as illustrated in Fig. 4.1, it is an interesting task to determine the ideal carrier concentration for a certain material and a certain temperature [122]. For non-degenerated materials, the Seebeck coefficient is given by

$$\ensuremath{\ensuremath{\alpha}_{\ensuremath{\nu}}}= \mp \frac{k_... ...emath{r_\nu}}- \ln \frac{\ensuremath{\nu}}{\ensuremath{N_\mathrm{c,v}}} \right)$$ (4.5)

where the carrier concentration is denoted by $\ensuremath{\nu}$ , the scattering parameter ${\ensuremath{r_\nu}}$ , and the effective density of states for the conduction and valence bands $\ensuremath{N_\mathrm{c,v}}$ , respectively. The sign is negative for electrons and positive for holes. This expression ignores the phonon-drag effect, whose impact is normally pronounced in relatively low temperature ranges. Furthermore, the carrier concentration enters the expression for the electric conductivity

$$\sigma_{\nu} = \mathrm{q}\ensuremath{\nu}\ensuremath{\ensuremath{\mu}_\nu}$$ (4.6)

as well as the electric component of the thermal conductivity. The latter is expressed introducing the Lorentz number $L_0$ by

$$\ensuremath{\kappa_{\ensuremath{\nu}}}= L_0 \left( \frac{k_\ensuremath{\mathrm{B}}}{\mathrm{q}} \right)^2 \sigma \ensuremath{T}\,.$$ (4.7)

Inserting (4.5), (4.6), and (4.7) into equation (4.4) and after some rearrangement, one obtains

$$Z = \left( \frac{5}{2} + {\ensuremath{r_\nu}}- \ln \frac{\ensurem... ...ath{\nu}\ensuremath{\ensuremath{\mu}_\nu}} + L_0 \ensuremath{T} \right)^{-1}\,.$$ (4.8)

Setting the derivative with respect to $\ensuremath{\nu}$ of the figure of merit to zero, the equation for the optimum carrier concentration is obtained as [123]

$$\ln \frac{\ensuremath{\nu}}{\ensuremath{N_\mathrm{c,v}}} + 2 \lef... ...mathrm{L}}}} \ensuremath{T}\ensuremath{\nu}= \frac{1}{2} + {\ensuremath{r_\nu}}$$ (4.9)

assuming that the Lorentz number does not depend on the carrier concentration. Ignoring the carrier contribution in the expression for the thermal conductivity, which holds true for low to moderately doped semiconductors, the optimum carrier concentration is given dependent on the scattering parameter [122] as

$$\ensuremath{\nu}_{\ensuremath{\mathrm{opt}}} = \ensuremath{N_\mathrm{c,v}}\exp \left( \frac{1}{2} + {\ensuremath{r_\nu}}\right) \,.$$ (4.10)

This result has to be handled carefully, since the scattering parameter itself is doping dependent. While for relatively low impurity concentration, scattering by phonons is dominant and thus ${\ensuremath{r_\nu}}=-1/2$ , high doping concentrations increase scattering by ionized impurities and shift ${\ensuremath{r_\nu}}$ to a value of $3/2$ . However, relatively high doping concentrations are necessary to obtain good thermoelectric performance [122].

The maximum efficiency for a thermoelectric generator made of Si$_{0.7}$ Ge$_{0.3}$ has been derived to be $12.1\,\%$ in [19]. Furthermore, a reduction of the lattice thermal conductivity without side effects on the electric properties has been calculated to result in a total efficiency of $23.3\,\%$ , which is far from today's practically realized generators.

Optimization of the figure of merit by influencing the carrier concentration is quite limited due to the interdependency of Seebeck coefficient and electric conductivity as well as the electric part of the thermal conductivity due to their common dependence on the carrier concentration. In contrast to this, the lattice thermal conductivity incorporates potential to increase the figure of merit independently of the carrier concentration. The key to obtain elevated figures of merit are low lattice thermal conductivities in order to keep the heat flux throughout the device low. In terms of microscopic processes, it is favorable to achieve higher phonon scattering rates while not affecting the carrier ones.

In semiconductor alloys, additional scattering is introduced by the internal lattice disorder. This mechanism is especially pronounced in alloys with a rather large difference between the single masses of the components. For example, the thermal conductivity of SiGe with a germanium content of only $10\,\%$ is lower by a factor of $6$ compared to pure silicon [95]. At the same time, the electrical properties show only minor changes.

Furthermore, the introduction of sintered materials causes increased scattering of phonons at the grain boundaries resulting in accordingly limited phonon mean free paths. The preparation as well as the influence of different grain sizes on the thermal conductivity of lead tin telluride are reported in [124,125,126,127,128]. Similar results are collected for SiGe in [129]. Grain sizes have to be in a range where the electrical conductivity is not significantly affected by additional scattering. Thus, promising combinations seem to be carefully designed sintered semiconductor alloys.

A continuation of this concept is incorporated in the idea of low-dimensional structures as well as directed search for novel materials with accordingly low thermal conductivities. Recent research focuses on both nanostructures [112,113,114,115,116,117,118,119,120] as well as emerging materials such as clathrates [109,110], where their large molecular structure ensures low thermal conductivities. In nanostructures, thermal conductivity is held low by affecting phonon transport with low-dimensional structures. In a certain geometrical range, the electrical properties are almost not affected compared to bulk materials, while thermal conductivity is decreased by various size effects.

M. Wagner: Simulation of Thermoelectric Devices