Semiclassical model for thermoelectric transport in nanocomposites

Nanocomposites (NCs) has recently been proposed and experimentally demonstrated to be potentially high-efficiency thermoelectric materials by reducing the thermal conductivity through phonon-interface scattering and possibly by increasing the Seebeck coefficient through energy-selective carrier scattering (low-energy filtering) or quantum-size effect of electrons. In this paper, we develop a Boltzmann transport equation based semiclassical electron transport model to describe the thermoelectric transport processes in semiconductor NCs. This model considers multiband transport of electrons and holes with both the intrinsic carrier scatterings and the carrier-interface scattering. A relaxation-time model is developed for carrier-interface scattering. After fitting the model with bulk thermoelectric alloys that gives reasonable material input parameters for bulk alloys, which are close to handbook values, the model is further validated by comparing the modeled thermoelectric properties with recently reported measurement values of thermoelectric properties in high efficiency NCs. The model is then applied to predict thermoelectric properties of both the particle-host-type and the particle-particle-type semiconductor NCs such as the $p$-type $\left({\text{Bi}}_{\text{y}}{\text{Sb}}_{2\text{-y}}{\text{Te}}_3\right)\text{−}\left({\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3\right)$ NCs and the $n$-type $\left({\text{Mg}}_2{\text{Si}}_{\text{y}}{\text{Ge}}_{1\text{-y}}\right)\text{−}\left({\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}\right)$ NCs. The dependence of thermoelectric transport coefficients on the size of nanoconstituent, doping concentration and temperature are studied. Our study could shed some light to optimally design high-efficiency thermoelectric NCs which could contribute to solar-thermal utilization or waste heat recovery.

Figure 1

(Color online) (a) Nanoconstituents (NPs or NWs) embedded in a host (matrix) material, noted as particle-host-type NCs. In this type of NCs, any two points in the matrix can be connected without encountering any nanoconstituents. (b) Mixture of different kinds of nanoconstituents, noted as particle-particle-type NCs. In this type of NCs, two arbitrary points in two separated nanostructures of the same material cannot be connected without encountering the other kind of nanoconstituent. (c) Schematic of the random trajectory of a carrier transmitting through or scattered by NPs one by one with transmission probability $P$ and reflection probability $R=1-P$. (d) Schematic of the open path that the carrier does not meet any interfaces. This open path can exist only in particle-host-type NCs. In a particle-particle-type NCs, there is no such open path because of the absence of topological connectedness.

Figure 2

(Color online) Schematic of the cross section of the band structure of NP and the filtering effect. Here $E_c\left(E_v\right)$ is the conduction (valence) band edge and $\Delta E_c\left(\Delta E_v\right)$ is the conduction (valence) band edge difference of two materials. $E_g$ is the band gap of the matrix material, $d$ is the size of NP, and $r$ is the radial axe of a spherical coordinate. High-energy carriers can transport through an energy barrier and the low-energy carriers are scattered back.

Figure 3

(Color online) Calculated temperature-dependent TE transport properties for the state-of-the-art $p$-type ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ alloy and its corresponding NCs, in comparison with the recently reported experimental data in Ref. 31: (a) electrical conductivity. (b) Seebeck coefficient. See Table for fitting parameters.

Figure 4

(Color online) Calculated temperature-dependent TE transport properties for the $n$-type ${\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}$ alloy, in comparison with experimental data in Ref. 55: (a) electrical conductivity. (b) Seebeck coefficient. See Table for fitting parameters.

Figure 5

(Color online) Size dependence of the lattice thermal conductivity of ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ and ${\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}$ alloys and their NCs with different volumetric fractions.

Figure 6

(Color online) The dependence of power factor and ZT on the effective energy barrier height for particle-host-type NCs with ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ as matrix material for different volumetric fractions. The calculations use $T=373\text{K}$, $n_d=6n_0$, and $n_0={10}^{19}/{\text{cm}}^3$ as input parameters.

Figure 7

(Color online) Size dependence of (a) electrical conductivity and Seebeck coefficient, (b) power factor, (c) total thermal conductivity, and (d) ZT for particle-host-type NCs with ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ as matrix for different barrier heights $E_b$ and volumetric fractions $x$. The calculations use $T=373\text{K}$, $n_d=6n_0$, and $n_0={10}^{19}/{\text{cm}}^3$ as input parameters. For comparison, the electrical conductivity in the base ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ alloy material is $0.761\times {10}^5/\Omega \text{m}$, the Seebeck coefficient is $222.5\mu \text{V}/\text{K}$, the power factor is $37.1\mu \text{W}/\text{cm}{\text{K}}^2$, total thermal conductivity is 1.25 W/mK, and the ZT is 1.13.

Figure 8

(Color online) Carrier density dependence of (a) electrical conductivity and Seebeck coefficient, (b) power factor, (c) total thermal conductivity, and (d) ZT for particle-host-type NCs with ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ as matrix material for different barrier heights $E_b$ and volumetric fractions $x$ in comparison with ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ bulk alloy. The calculations use $T=373\text{K}$ and $d=10\text{nm}$ as input parameters, $n_0={10}^{19}/{\text{cm}}^3$.

Figure 9

(Color online) Temperature dependence of ZT for particle-host-type NCs with ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ as matrix material for barrier heights $E_b$ and volumetric fractions $x$ in comparison with ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$ bulk alloy. We choose $n_d=6n_0$, $n_0={10}^{19}/{\text{cm}}^3$, and $d=10\text{nm}$ as input parameters since it is the optimal case we found in Fig. 7d. The dashed curve is the reported high ZT experimental data from Ref. 31.

Figure 10

(Color online) Barrier height dependence of power factor and ZT for particle-host-type NCs with ${\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}$ as matrix materials for different volumetric fractions $x$. The calculations use $T=800\text{K}$, $n_d=3n_0$, and $n_0={10}^{19}/{\text{cm}}^3$ as input parameters.

Figure 11

(Color online) Size dependence of (a) electrical conductivity and Seebeck coefficient, (b) power factor, (c) total thermal conductivity, and (d) ZT for particle-host-type NCs with ${\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}$ as matrix materials for different barrier heights $E_b$ and volumetric fractions $x$. The calculations use $T=800\text{K}$, $n_d=3n_0$, and $n_0={10}^{19}/{\text{cm}}^3$ as input parameters. For comparison, the electrical conductivity in the base ${\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}$ alloy material is $1.14\times {10}^5/\Omega \text{m}$, the Seebeck coefficient is $153\mu \text{V}/\text{K}$, the power factor is $26.6\mu \text{W}/\text{cm}{\text{K}}^2$, total thermal conductivity is 3.32 W/mK, and the ZT is 0.64.

Figure 12

(Color online) Carrier density dependence of (a) electrical conductivity and Seebeck coefficient, (b) power factor, (c) total thermal conductivity, and (d) ZT for particle-host-type NCs with ${\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}$ as matrix material for different barrier heights $E_b$, and volumetric fractions $x$. The calculations use $T=800\text{K}$ and $d=10\text{nm}$ as input parameters, $n_0={10}^{19}/{\text{cm}}^3$.

Figure 13

(Color online) Temperature dependence of ZT for particle-host-type NCs with ${\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}$ as matrix materials for different barrier heights $E_b$ and volumetric fractions $x$.

Figure 14

(Color online) Comparison of carrier density dependence of (a) electrical conductivity and Seebeck coefficients, and (b) power factor, for particle-host-type NCs and particle-particle-type NCs. $n_0={10}^{19}/{\text{cm}}^3$.

Figure 15

(Color online) Temperature dependence of MFP in (a) ${\text{Bi}}_{0.5}{\text{Sb}}_{1.5}{\text{Te}}_3$-based NCs and (b) ${\text{Mg}}_2{\text{Si}}_{0.6}{\text{Ge}}_{0.4}$-based NCs in both the host materials and nanoconstituents in NCs with $d=10$ and 20 nm, and the corresponding bulk materials.