5.1.3 Elastic Moduli and Sound Velocities

Beside the mechanical relevance of the elastic moduli, they are related to the sound velocities dependent on their direction relative to the crystallographic axes. Average values of the sound velocities enter several scattering models applied in Monte Carlo simulations [204]. In literature, data on both elastic moduli as well as sound velocities in specified directions are commonly given. In order to allow a comprehensive comparison, available data are collected in Tables 5.3 and 5.4, where the missing values are calculated using the according relations [205]

$\displaystyle \ensuremath{v_{\rm {sl,[100]}}}$   $\displaystyle = \sqrt{\frac{c_{11}}{\rho}} \,,$ (5.5)
$\displaystyle \ensuremath{v_{\rm {st,[100]}}}$   $\displaystyle = \sqrt{\frac{c_{44}}{\rho}} \,,$ (5.6)
$\displaystyle \ensuremath{v_{\rm {sl,[111]}}}$   $\displaystyle = \sqrt{\frac{c_{11}+2c_{12}+4c_{44}}{3\rho}} \,.$ (5.7)

A comprehensive disquisition on the coherence between sound velocities and second order elastic moduli in different directions is given in [205]. Throughout the calculations, a mass density of $ \rho = 8241
\,\ensuremath{\mathrm{kg}}\,\ensuremath{\mathrm{m}}^{-3}$ [193] has been used.

Measurement data and corresponding quantities of lead telluride at room temperature are collected in Table 5.3, where measurement data [196,193,188] are summarized and extended by recently presented results of first-principle approaches [206].

Table 5.3: Summary of elastic constants and corresponding sound velocities for PbTe.
$ c_{11}$ $ c_{12}$ $ c_{44}$ PbTe $ \ensuremath{v_{\rm {sl,[100]}}}$ $ \ensuremath{v_{\rm {st,[100]}}}$ $ \ensuremath{v_{\rm {sl,[111]}}}$
[GPa] [GPa] [GPa] Data Refs. $ [\rm {m/s}]$ $ [\rm {m/s}]$ $ [\rm {m/s}]$
105.3 7.0 13.22 exp. [196] 3575 1267 2639
108.0 7.7 13.43 exp. [193] 3620 1277 2677
107.2 7.68 13.00 exp. [193] 3607 1256 2657
104.0 4.37 13.00 exp. [188] 3552 1256 2581
107.4 7.8 12.90 calc. [207] 3610 1251 2657
115.7 4.2 14.3 calc. GGA [206] 3747 1317 2708


The temperature dependence of the elastic constants and sound velocities has been investigated by Houston [193] for the low temperature range between $ 4.2\,\ensuremath{\mathrm{K}}$ and $ 300\,\ensuremath{\mathrm{K}}$ .

Based on data presented in Tables 5.3 and 5.4, analytical expressions for the second-order elastic moduli of lead telluride have been derived using polynomial ansatzes of second and first order, respectively. The according expressions read

$\displaystyle c_{11}$   $\displaystyle = 126.9 - 14.5 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_...
...\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^2 \,,$ (5.8)
$\displaystyle c_{12}$   $\displaystyle = 2.86 + 4.74 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}\,,$ (5.9)
$\displaystyle c_{44}$   $\displaystyle = 15.0 - 1.55 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}\,.$ (5.10)


Table 5.4: Temperature dependence of elastic constants and corresponding sound velocities.
$ \ensuremath{T_{\mathrm{L}}}$ $ c_{11}$ $ c_{12}$ $ c_{44}$ $ \ensuremath{v_{\rm {sl,[100]}}}$ $ \ensuremath{v_{\rm {st,[100]}}}$ $ \ensuremath{v_{\rm {sl,[111]}}}$
$ [\rm {K}]$ $ [\rm {GPa}]$ $ [\rm {GPa}]$ $ [\rm {GPa}]$ $ [\rm {m/s}]$ $ [\rm {m/s}]$ $ [\rm {m/s}]$
0 126.05 4.28 14.91 3911 1345 2803
10 125.8 4.38 14.89 3907 1344 2802
20 125.41 4.41 14.86 3901 1343 2799
30 124.96 4.47 14.84 3894 1342 2796
50 124.06 4.48 14.75 3880 1338 2787
100 121.39 4.76 14.49 3838 1326 2764
150 118.5 5.23 14.23 3792 1314 2742
200 115.09 6.04 13.97 3737 1302 2721
250 111.78 6.73 13.71 3683 1290 2699
303.2 107.99 7.65 13.44 3620 1277 2676


Corresponding expressions for the temperature dependence of the average longitudinal and transversal sound velocities have been determined by applying Ridley's formalism [208] as

$\displaystyle \ensuremath{v_{\rm {sl}}}$   $\displaystyle = 3297 - 170 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\...
...\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^2 \,,$ (5.11)
$\displaystyle \ensuremath{v_{\rm {st}}}$   $\displaystyle = 2016 - 121 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\...
...\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^2 \,.$ (5.12)

M. Wagner: Simulation of Thermoelectric Devices