5.1.2 Dielectric Constant

Figure 5.1: Relative static dielectric constant $\ensuremath {\ensuremath {\epsilon }_{\ensuremath {\mathrm {s}}}}$ for PbTe with respect to the temperature.

The dielectric constant, also called permittivity $\ensuremath{\epsilon}$ describes the relation between electric displacement and field strength. For the general case it is a tensor dependent on the frequency and external influences like magnetic fields. In isotropic materials, this tensor reduces to a scalar. The permittivity is usually given as the product of the dimensionless relative permittivity $\ensuremath{\ensuremath{\epsilon}_\ensuremath{\mathrm{r}}}$ and the vacuum permittivity $\ensuremath{\ensuremath{\epsilon}_0}$ .

Special interest is devoted to the static dielectric constant $\ensuremath {\ensuremath {\epsilon }_{\ensuremath {\mathrm {s}}}}$ and the high frequency dielectric constant $\ensuremath{\ensuremath{\epsilon}_{\infty}}$ . While the static dielectric constant enters Poisson's equation as well as the models for several scattering mechanisms, both constants are employed in the description of polar optical scattering.

The static dielectric constant of lead telluride is unusually high and is dependent on temperature as illustrated in Fig. 5.1. Experimental data are mostly available for the low temperature range [198,199,200]. Values for room temperature and higher are given in [200,194] but are not sufficient to give us a clear picture of the dependence of $\ensuremath{\epsilon}(\ensuremath{T})$ in that range. The low temperature data from Nishi [198], Tennant [200], and Dashevsky [201] can be modeled by a simple power-law

$$\ensuremath{\ensuremath{\epsilon}_{\ensuremath{\mathrm{s}}}}= 412... ...{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^{-0.6} \,.$$ (5.2)

However, data from Dalven [194] suggest a linear dependence, modeled by

$$\ensuremath{\ensuremath{\epsilon}_{\ensuremath{\mathrm{s}}}}= 428... ...th{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}\,.$$ (5.3)

An expression for $\ensuremath{\ensuremath{\epsilon}_{\ensuremath{\mathrm{s}}}}(x)$ in Pb$_{1-x}$ Sn$_x$ Te in the range $x<0.35$ is given in [202]. Furthermore, experimental data for the low temperature range are provided by Nishi in [198].

The high-frequency dielectric constant is about $11$ times lower than the static dielectric constant [194,203].

$$\ensuremath{\ensuremath{\epsilon}_{\infty}}= 38 - 5 \ensuremath{\... ...th{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}\,.$$ (5.4)

The linear models (5.3) and (5.4) are applied for the Monte-Carlo simulations presented in Section 5.4 for the high-temperature range. M. Wagner: Simulation of Thermoelectric Devices