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]

$$\ensuremath{v_{\rm {sl,[100]}}} = \sqrt{\frac{c_{11}}{\rho}} \,,$$ (5.5)

$$\ensuremath{v_{\rm {st,[100]}}} = \sqrt{\frac{c_{44}}{\rho}} \,,$$ (5.6)

$$\ensuremath{v_{\rm {sl,[111]}}} = \sqrt{\frac{c_{11}+2c_{12}+4c_{44}}{3\rho}} \,.$$ (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

$$c_{11} = 126.9 - 14.5 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_... ...\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^2 \,,$$ (5.8)

$$c_{12} = 2.86 + 4.74 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}\,,$$ (5.9)

$$c_{44} = 15.0 - 1.55 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}\,.$$ (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

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

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

M. Wagner: Simulation of Thermoelectric Devices