5.2.2 Thermal Conductivity

The thermal conductivity $\kappa$ parametrizes the flux term in the heat flux equation and depicts the energy flux per area and temperature difference transported within a homogeneous material. Measurement of the thermal conductivity is conventionally based on gathering the temperature difference caused by a steady heat flow, but also more sophisticated approaches like the $3\omega$ -method can be applied in special cases [215]. The total thermal conductivity in semiconductors consists of the lattice and the electronic contribution, which is connected to the electrical conductivity by a Wiedemann-Franz law. While the lattice thermal conductivity, which is the dominant mechanism over a wide range of carrier concentrations in silicon, germanium, and several III-V semiconductors, is commonly modeled as a temperature dependent power law [133], the electron or hole contribution in n- and p-type materials gains weight in heavily doped samples. In the intrinsic range, both electrons and holes contribute to the thermal conductivity, which is referred to as the bipolar contribution. In lead telluride, these additional contributions to the thermal conductivity caused by the carrier gas play already a significant role at technically relevant carrier concentrations [121]. Based on the theoretical considerations presented in [125], the thermal conductivity dependence on the temperature and the electron concentration can be modeled as

$$\kappa = \ensuremath{\kappa_{\mathrm{300}}}\left( \ensuremath{\fr... ...n}}{\ensuremath{n_\mathrm{ref}}} \right)^{\ensuremath{\beta_\kappa}}\right) \,.$$ (5.15)

The according parameters are collected in Table 5.6.

Table 5.6: Parameter values for the lead telluride thermal conductivity model incorporating the carrier contribution.

Parameter	Value
$\ensuremath{\kappa_{\mathrm{300}}}$	$1.5486\,\mathrm{W/Km}$
${\ensuremath{\alpha_\kappa}}$	$-1.3122$
$\ensuremath{n_\mathrm{ref}}$	$4.55 \times 10^{18}\,\mathrm{cm^{-3}}$
${\ensuremath{\beta_\kappa}}$	$0.651$

In Fig. 5.3, the agreement between Bhandari's data [125] and the model is illustrated. Several measurement data available in literature [216,217,218,219] show a general decrease of the thermal conductivity for increasing temperatures with an alleviated or even reverse trend at higher temperatures caused by the increased electronic contribution due to additionally available free carriers.

Figure 5.3: Dependence of the thermal conductivity of lead telluride on the lattice temperature and carrier concentration. While red glyphs depict Bhandari's data [125], the surface denotes the modeled thermal conductivity.

The heat flux between two points with temperatures $\ensuremath{T}_1$ and $\ensuremath{T}_2$ is calculated by the integral

$$\int\limits_{\ensuremath{T}_1}^{\ensuremath{T}_2} \kappa(\ensurem... ...{n}}{\ensuremath{n_\mathrm{ref}}} \right)^{{\ensuremath{\beta_\kappa}}} \right)$$ (5.16)

assuming the carrier concentration to be constant throughout the discretization box.

Figure 5.4: Material composition dependent lattice and total thermal conductivity of Pb$_{1-x}$ Sn$_x$ Te at 300K including measurement data and model parameter sets.

In Pb$_{1-x}$ Sn$_x$ Te, alloy scattering enters as an additional mechanism and thus the thermal conductivity is drastically reduced compared to pure PbTe and SnTe. Fig. 5.4 depicts the variation of the thermal conductivity with respect to the material composition at $300\,\ensuremath{\mathrm{K}}$ . Measurement data have been collected from [220,221,222], where the latter two comprise investigations of sintered samples which explains the lower thermal conductivity values. The lattice component has been theoretically investigated by molecular-dynamics studies in [223] and by subtracting the electronic contribution

$$\ensuremath{\ensuremath{\kappa_{\ensuremath{\nu}}}}= L \sigma \ensuremath{T}$$ (5.17)

derived from the electron conductivity $\sigma$ , the Lorentz number $L$ and the temperature [220]. The model for the material composition dependent thermal conductivity in alloys reads [133]

$$\ensuremath{\kappa_{\mathrm{300}}} = \left(\frac{1-x}{\ensuremath{\kappa_{\mathrm{300}}^\ensuremath{... ...athrm{300}}^\ensuremath{\mathrm{B}}}} + \frac{(1-x)x}{C_\kappa}\right)^{-1} \,,$$ (5.18)

$${\ensuremath{\alpha_\kappa^\ensuremath{\mathrm{AB}}}} = (1-x) {\ensuremath{\alpha_\kappa^\ensuremath{\mathrm{A}}}}+ x {\ensuremath{\alpha_\kappa^\ensuremath{\mathrm{B}}}}\,,$$ (5.19)

where $x$ and $1-x$ denote the SnTe and PbTe content, respectively. $\ensuremath{\kappa_{\mathrm{300}}^\ensuremath{\mathrm{A}}}$ is the thermal conductivity for PbTe at room temperature, $\ensuremath{\kappa_{\mathrm{300}}^\ensuremath{\mathrm{B}}}$ analogously for SnTe, and $C_\kappa$ the bowing factor accounting for the alloy scattering reduction of the thermal conductivity. The exponents describing the temperature dependence are interpolated linearly between the values for the pure material constituents. The additional electronic contribution can be estimated by

$$\ensuremath{\kappa_{\ensuremath{\mathrm{tot,300}}}}= \ensuremath{\kappa_{\ensuremath{\mathrm{L,300}}}}+ A_0 + A_1 x^2$$ (5.20)

for Pb$_{1-x}$ Sn$_x$ Te based on the data published in [220]. According values for the temperature exponents $\alpha^{\ensuremath{\mathrm{A}}}$ and $\alpha^{\ensuremath{\mathrm{B}}}$ have been identified for the lattice thermal conductivity based on data published in [222] and [224]. The parameters for the material composition dependent thermal conductivity model are collected in Table 5.7.

Table 5.7: Parameter values for the material composition dependent PbSnTe thermal conductivity models.

Parameter	Value
$\ensuremath{\kappa_{\mathrm{300}}^\ensuremath{\mathrm{A}}}$	$2.2\,\ensuremath{\mathrm{W/Km}}$
$\ensuremath{\kappa_{\mathrm{300}}^\ensuremath{\mathrm{B}}}$	$3.05\,\ensuremath{\mathrm{W/Km}}$
$C_\kappa$	$0.6$
${\ensuremath{\alpha_\kappa^\ensuremath{\mathrm{A}}}}$	$-1.06$
${\ensuremath{\alpha_\kappa^\ensuremath{\mathrm{B}}}}$	$-0.27$
$A_0$	$0.5\,\ensuremath{\mathrm{W/Km}}$
$A_1$	$5.3\,\ensuremath{\mathrm{W/Km}}$

A decrease of the thermal conductivity compared to single crystals due to grain boundary scattering in sintered materials has been reported in [124,126,225] to be as high as $4-6\%$ in pure lead telluride and $11-13\%$ in disordered lead telluride alloys. However, a drastic decrease of the thermal conductivity is reported in [128] for grain sizes in the range of $0.7\,-4\,\mathrm{\mu m}$ . Furthermore, the influence of the grain size within sintered $\ensuremath{\mathrm{Pb}}_{1-x}\ensuremath{\mathrm{Sn}}_{x}\ensuremath{\mathrm{Te}}$ is investigated in [127] for hot and cold pressed materials. Recent work has dealt with the dependence of the thermal conductivity on the pressure during fabrication using high temperature and high pressure processes [226,227].

M. Wagner: Simulation of Thermoelectric Devices