Thermal Conductivity in Thermoelectric Materials

Abstract

Thermal conductivity is a key parameter in identifying and developing alternative materials for many technological and temperature-critical applications, ranging from higher-temperature capability thermal barrier coatings to materials for thermoelectric conversion. The Figure of Merit (ZT) of a thermoelectric material (TE) is a function of the Seebeck coefficient (S), the electrical conductivity (σ), the total thermal conductivity (κ) and the absolute temperature (T). A highly-performing TE material should have high S and σ and low κ. Thermal conductivity has two contributions, the electronic (κE) and the lattice (κL). Various models have been developed to describe the lattice component of thermal conductivity. In this chapter, the models for the evaluation of lattice thermal conductivity will be explored, both phenomenological as well analytical models, taking into account the various phonon-scattering processes, with examples of real materials.

1. Introduction

The world’s demand for energy is increasing, along with its environmental impact. In fact, the majority of the energy actually produced by man is lost as waste heat. Only an average of about 1/3 of the energy produced by thermal and nuclear power plants, and similarly, only 30% of the energy produced in the internal combustion engine is converted to useful work [1]. If only a portion of the enormous amount of unused waste heat, generated by domestic heating, automotive exhaust and industrial processes could be recovered, it would cause a tremendous benefit in energy demands. Thermoelectricity (TE) could be of great use in energy harvesting [2, 3].

The thermoelectric performance is ranked by the figure of merit ZT (ZT = S²σT/κ), where S is the Seebeck coefficient, σ the electrical conductivity, κ the total thermal conductivity and T the absolute temperature. A highly-performing TE material should have high electrical conductivity and Seebeck coefficient and low thermal conductivity.

Thermal conductivity is a key parameter in identifying and developing alternative materials for many technological and temperature-critical applications, ranging from higher-temperature capability thermal barrier coatings to materials for thermoelectric conversion. Since the charge carriers (electrons in n-type or holes in p-type semiconductors) transport both heat and charge, thermal conductivity consists of two additive parts (κ = κ_L + κ_E), the lattice (κ_L) and the electronic (κ_E) contribution. High ZT requires low total thermal conductivity but high σ simultaneously, so, one of the more popular routes towards improving ZT has been to reduce κ_L [4].

In this book chapter, we shall explore both analytical and phenomenological models for evaluating the thermal conductivity in thermoelectric materials. In this book chapter, we shall start with the basic features and ways for the determination of the lattice thermal conductivity in thermoelectric materials, and we shall proceed with the theoretical foundations of the subject. Next, we will explore various effects that affect the lattice thermal conductivity, namely, the role of defects, of alloying, of disorder and we will conclude with the effect on nano-structuring. Each session starts with the necessary theoretical background and proceeds with application examples of the various aspects of different thermoelectric materials.

2. Thermal conductivity in thermoelectric materials

As previously stated, thermal conductivity has two contributions:

$$\kappa ={\kappa }_L+{\kappa }_E=\frac{1}{3}{C}_V\mathit{u\lambda }+{\kappa }_E$$E1

where C_V is the specific heat (at constant volume), u is the speed of sound and λ is the phonon mean free path. All these parameters are in general depend on temperature, and materials characteristics, like the Debye temperature, the mean phonon velocity and parameters which are material dependent, like the mean crystalline size, the density of defects a various scales-lengths, and the 3-phonon scattering parameters. Typically, as can be seen in Figure 1, the (lattice) thermal conductivity shows a more or less profound peak at low temperatures and a 1/T decay due to the 3-phonon scattering. At sufficiently low temperatures (below the Umklapp peak), the lattice thermal conductivity follows the increase of the C_V, with a Tⁿ dependence.

Figure 1.

Lattice thermal conductivity for CoSb₃. Data are taken from [5]. The solid line is the phenomenological model. κ_LT and κ_HT are the descriptions for low temperatures and high temperatures, respectively.

Since grain-boundary scattering dominates at this temperature region, the exponent n for an ideal crystalline solid should be equal to 3 [6]. Any deviation from the n = 3 at low temperatures, or from the 1/T dependence at high temperatures may be attributed to additional mechanisms, either to the presence of defects at low temperatures or to radiation losses at high ones.

Assuming that both low- and high-temperature regions, κ_LT and κ_HT, respectively, can be approximated by a power-law dependence, i.e. κ_LT ∼ Tⁿ and κ_HT ∼ T^m, and the lattice thermal conductivity κ_L in Figure 1, can be expressed using the phenomenological relation (κ_L)⁻¹ = (κ_LT)⁻¹ + (κ_HT)⁻¹ [7], which describes quite well the thermal conductivity κ_L for CoSb₃ [5] in the entire temperature range.

2.1 Electronic contribution to thermal conductivity

Figure 2 shows the total thermal conductivity (κ) as a function of temperature for Bi-doped Mg₂Si_0.55Sn_0.4Ge_0.05 [8]. As can be seen, as Bi-content increases, doped materials have higher thermal conductivity due to the increased electronic contribution (κ_E). The electronic contribution to thermal conductivity is given using the Wiedemann-Franz law: κ_E = L^.σ^.T, where L is the Lorenz number and σ the electrical conductivity.

Figure 2.

Total (κ) and lattice thermal (κ_L) conductivity vs. Bi content for Bi-doped Mg₂Si_0.55Sn_0.4Ge_0.05. Lines are guide to the eye. Note the underestimation in κ_L if the L = 2.44 × 10⁻⁸ WΩK⁻² (strongly degenerate case) is taken (adapted from [8]).

Lorentz number L spans from 1.44 to 2.44 × 10⁻⁸WΩK⁻² for the case of non-degenerate electron gas to the fully degenerated, respectively. Several authors use the highly degenerate value (L = 2.44 × 10⁻⁸WΩK⁻²), resulting in underestimation of the lattice thermal conductivity, which may reach up to 40% [9]. Therefore, careful evaluation of L is critical in characterising enhancements in ZT due to κ_L reduction. Though modern thermoelectric materials are narrow gap semiconductors with non-parabolic multiple bands, many authors assume single parabolic band (SPB) approximation and scattering of the charge carriers by acoustic phonon (APS), which is the most commonly considered mechanism for thermoelectric materials at high temperatures. Withing the SPB-APS consideration, both L and S are functions only of reduced Fermi level (η), and can be used for the evaluation of L (S→η→L):

$$L={\left(\frac{k_B}{e}\right)}^2\frac{2F_0\left(\eta \right)F_2\left(\eta \right)-4F_1^2\left(\eta \right)}{F_0^2\left(\eta \right)},S=\pm \left(\frac{k_B}{e}\right)\left(\frac{2F_1\left(\eta \right)}{F_0\left(\eta \right)}-\eta \right)$$E2

where F_j(η) is the Fermi integral. Kim et al. [10] developed an approximated expression for L as a function of S:

$$L=1.5+\mathit{exp}\left(-\frac{\mid S\mid }{116}\right)$$E3

where L is in units of 10⁻⁸WΩK⁻² and S in μV/K., the approximation of Eq. (3) is far better than considering L = 2.44^x10⁻⁸WΩK⁻². For a comprehensive analysis and the various source or errors for different materials of the evaluation of L, readers may refer to the original paper by Kim et al.

2.2 Bipolar contribution to thermal conductivity

Thermoelectric materials are heavily doped, narrow-gap semiconductors. The high carrier density (typically of the order of 10²⁰ cm⁻³) makes the carrier concentration practically unvaried with temperature. However, since the density of states increases to T^3/2, the reduced Fermi level gradually drops within the band gap, resulting in the contribution of the minority carriers with increasing temperature. The bipolar (electron-hole pair) thermal conductivity (κ_BP) presents a dominant contribution above 500 K in small-bandgap thermoelectrics. According to Glassbrenner and Slack [11], this contribution can be expressed as:

$${\kappa }_{\mathit{BP}}=\frac{\beta }{{\left(1+\beta \right)}^2}{\left[\frac{\mathit{Eg}}{k_{B}T}+4\right]}^2{\left[\frac{k_B}{e}\right]}^2\mathit{\sigma T}$$E4

where Eg is the band gap and β is the ratio of electron to hole mobility. Yelgel and Srivastava [12] simplified this expression to:

$${\kappa }_{\mathit{BP}}=F_{\mathit{BP}}{T}^p\mathit{exp}\left(\frac{-\mathit{Eg}}{2k_{B}T}\right)$$E5

where F_BP and p are adjustable parameters, changing with doping type. Figure 3 shows the subtraction/difference of total and electronic thermal conductivity (κ-κ_E) as a function of the inverse temperature for low Bi-doped Mg₂Si_0.55Sn_0.4Ge_0.05 [8]. These values include the bipolar (κ_BP) and the lattice contribution (κ_L) of the thermal conductivity. Lattice thermal conductivity exhibits a linear dependence vs. 1/T, and the bipolar contribution is manifested by an upturn at high temperatures (low 1/T) from the linear dependence, as proposed by Kitagawa et al. [13].

Figure 3.

The effect of bipolar conduction on thermal conductivity. Note the deviation from the linear 1/T tend (adapted from [8]).

At low Bi contents, and at high temperatures, the term κ-κ_EL gradually deviates from the linear T⁻¹ relationship. This implies that the bipolar diffusion starts to contribute to the thermal conductivity. Suppression of bipolar contribution in thermal conductivity may be achieved either by higher doping or by introducing inclusions of a second phase in the lattice, which act as energy barriers that filters-out the low energy minority carriers. A comprehensive review on how energy barriers act, may be found in [14, 15].

2.3 Theoretical foundations

Advances in thermoelectric technology and materials have led to a considerable amount of computational work aiming at the more accurate simulation of electronic and phononic transport properties for both complex band-structure materials and nano-structured materials. Several techniques have been adopted and have been able to describe electronic transport taking into account the full energy dependencies of the band structure and the scattering mechanisms. They need to merge length scales and physical phenomena, i.e. from atomistic to continuum and from quantum mechanical to fully diffusive, regimes that can co-exist in a typical nanostructured thermoelectric material. They also need to describe transport in arbitrary disordered geometries. A comprehensive book has been recently published in the field of computational modelling and simulation methods for both electronic and phononic transport in thermoelectric materials, and readers should refer to it [16].

Apart from computational simulations, the theory for thermal conductivity is known for several decades now. Though more elaborate models have been introduced, our discussion will be limited to the Debye approximation for specific heat. In order to understand the mechanisms of phonon scattering, semi-classical theoretical calculation based on the modified Callaway’s model [17, 18] is considered:

$$\begin{array}{l}{\kappa }_L\left(T\right)=\frac{k_B}{2{\pi }^{2}u}{\left(\frac{T}{{\Theta }_D}\right)}^3\underset{0}{\overset{{\Theta }_D/T}{\int }}{\tau }_C\left(x\right)\frac{x^4{e}^x}{{\left(e^x-1\right)}^2}\mathit{dx}\\ {\kappa }_L\left(T\right)=\frac{k_B}{2{\pi }^{2}u}{\left(\frac{\mathrm{\hslash }}{k_{B}T}\right)}^2\underset{0}{\overset{{\omega }_D}{\int }}{\tau }_C\left(\omega \right)\frac{{\omega }^{4}exp\left(\frac{\mathrm{\hslash }\omega }{k_{B}T}\right)}{{\left(exp\left(\frac{\mathrm{\hslash }\omega }{k_{B}T}\right)-1\right)}^2}\mathit{d\omega }\end{array}$$E6

where k_B is the Boltzmann’s constant, η is the Plank constant, T is the absolute temperature, u is the average phonon-group velocity, ω_D and Θ_D the Debye frequency and temperature, x = hω/k_BT and τ_C is the phonon relaxation time. In the simplest model, the phonon relaxation time combines the scattering from Umklapp processes (τ_U), alloying (τ_Α) and grain boundaries (τ_B) according to the Matthiessen’s rule [19, 20]:

$$\frac{1}{{\tau }_C\left(T,\omega \right)}=\frac{1}{{\tau }_B}+\frac{1}{{\tau }_{{\rm A}}\left(\omega \right)}+\frac{1}{{\tau }_U\left(T,\omega \right)}$$E7

some among the aforementioned mechanisms depend only on geometrical features (τ_B), others on phonon frequency (ω) only, while Umklapp on both temperature and phonon frequency. A more elaborated model has been developed by Holland [21] that considers the contribution of thermal conductivity both from the longitudinal and the transverse phonons. It also applies two averaged phonon group velocities to describe the phonon dispersions. Following Callaway and Holland, there have been several modifications to the thermal conductivity model. These modifications are aimed to better capture the temperature dependence over a broader range and focus on achieving a better description of the phonon dispersions rather than developing new thermal conductivity models.

In the simple Callaway model, the various phonon scattering mechanisms may be expressed as follows. Note that the Holland model uses slightly different expressions for the phonon relaxation mechanisms.

Phonon scattering due to interfaces and grain boundaries (τ_B) is given by [18]:

$$\frac{1}{{\tau }_B}=\frac{4}{3}\frac{u}{L}\frac{1-t}{t}\mathit{\text{or}}\frac{1}{{\tau }_B}=\frac{u}{L_{\mathit{eff}}},L_{\mathit{eff}}=\frac{3}{4}\frac{t}{1-t}L$$E8

where t is the phonon transmissivity and L is the average grain size, or scaled to L_eff.

When alloying, the solute atom scattering is accounted by the effective medium approach using a Rayleigh-like expression as described by Klemens [22]:

$$\frac{1}{{\tau }_A\left(\omega \right)}={\rm A}{\omega }^4,{\rm A}=\frac{{\delta }^3}{4\pi u^3}\left[x\left(1-x\right)\left\{{\left(\frac{\mathit{\Delta {\rm M}}}{{\rm M}}\right)}^2+{\left(\frac{\mathit{\Delta \delta }}{\delta }\right)}^2\right\}\right]$$E9

where δ is the radius of the solute atom, M is the molar host weight, ΔM/M is the mass fluctuation, Δδ/δ is the strain field fluctuation and ε is an adjustable parameter.

The Umklapp scattering, derived from 2nd order perturbation is given by [23]:

$$\frac{1}{{\tau }_U\left(\omega ,T\right)}=B_U{\omega }^2\mathit{\text{Texp}}\left(\frac{-{\Theta }_D}{T}\right),B_U=\frac{\mathrm{\hslash }{\gamma }^2}{{\rm M}{u}^2{\Theta }_D}$$E10

where γ is the Gruneisen parameter and M is the average mass.

Figure 4 depicts the calculated thermal conductivity according to this simple model plotted against the dimensionless T/Θ_D. Two plots are given, increasing the concentration of defects (A and 5A). Thermal conductivity decreases, as expected by increasing the concentration of defects, the peak becomes broader and shifts to slightly higher temperatures, and the temperature dependence of the thermal conductivity in the low-temperature region becomes weaker. The (T/Θ_D)³ curve is also presented in figure to demonstrate the deviation of the thermal conductivity from the expected ideal T³ law at low temperatures.

Figure 4.

Theoretical model for lattice thermal conductivity. The two plots correspond to increasing the concentration of defects, A and 5A (after [7]).

However, the analysis of thermal conductivity is difficult to perform. There are in fact two main complications for accurate analysis; the first is related to the inevitable radiative losses that strongly affect accuracy of measurements at higher temperatures, and the second is related to the numerical problem itself. In order to account for the various mechanisms that limit the lattice contribution to total thermal conductivity, one has to integrate the phonon mean free path over the whole frequency region up to Debye frequency. The numerical problem refers to how to express the phonon mean free path and mean phonon velocity in terms of more readily obtainable or tabulated material parameters. Even in the case of known material parameters, such as the Debye temperature, mean phonon velocity and mean crystalline size, still, the fitting of thermal conductivity data to the integral-based model is not an easy numerical problem. Though some authors have performed the fitting to the integral relation 6, or even more complex ones [5, 24], still many authors prefer the adoption of simplified or phenomenological models, which shall be examined in the following.

2.4 The effect of point defects

Defects strongly impact both the electronic and thermal properties of solids, and defect engineering is ubiquitous in the field of thermoelectrics. Therefore, introducing point defects by doping and alloying is the historically most important and robust approach for tuning the charge carrier concentration and reducing κ_L (by scattering high-frequency phonons) [25].

Figure 5 is depicted the inverse of thermal conductivity (1/κ_L) as a function of the dimensionless T/Θ_D. Data were taken from Figure 4, for two cases of increasing defect concentration (A and 5A). As can be seen, the calculated thermal resistivity exhibits a good linear relation with temperature, for temperatures T > Θ_D/3, and the constant term in the linear trend is strongly related to the concentration of defects. The change in the slope is negligible, and this can be attributed to the unvaried value of the Umklapp term B.

Figure 5.

Calculated lattice thermal resistivity for two different concentrations of point defects A and 5A, respectively (after [7]).

In this simple phenomenological model for examining the effect of defects, the observed linear dependence can be understood as follows: The lattice thermal resistivity (w_L) is considered to arise from two contributions according to [26]:

$$w_L=\frac{1}{{\kappa }_L}=w_o+\mathit{\Delta w}=\mathit{\alpha T}+\beta$$E11

where the term w₀ is the lattice thermal resistivity of the ideal crystal and Δw represents a contribution due to the presence of defects. For the ideal defect-free crystal w₀ ∼ T in the high-temperature range and Δw = 0. This linear dependence has been observed in many thermoelectric materials [7, 26, 27, 28] and enables to separate the expected ideal behaviour of a defect-free crystal from that due to the presence of defects.

2.5 The effect of alloying

Solid-solution alloy scattering of phonons is a demonstrated efficient way to reduce lattice thermal conductivity. It may be viewed as a particular case of deliberately introducing defects in the structure, and since the introduced defects alter the local atomic arrangement from the ideal crystal structure, they will intuitively disrupt and affect the charge and thermal transport properties of solids. Similar to the effect of defects, in the case of alloying, the complexity of fitting the fitting of thermal conductivity data to the integral-based model has led to the development of simplified models.

The simplest phenomenological model was developed by Adacci [29], according to which, the composition dependence of thermal conductivity κ (z) in solid solutions series A_1-ZB_Z can be modelled using the harmonic mean of the thermal conductivities for the two end members κ_A and κ_B, plus an additional ‘bowing’ term C, which is introduced to account for the mass and strain fluctuation caused by the substitution of the element B into A:

$$\frac{1}{{\kappa }_L\left(z\right)}=\frac{1-z}{{\kappa }_A}+\frac{z}{{\kappa }_B}+\frac{z\left(1-z\right)}{C}$$E12

The Adacci model can be easily extended to ternary or quintenary alloys and has been employed by several authors in the evaluation of the bowing factor [7, 30]. Figure 6 depicts the lattice thermal conductivity for K₂ (Bi_1-XSb_Z)₈Se₁₃ solid solutions vs. composition at different temperatures. Dashed lines are from Adacci model. As can be seen, the lattice thermal conductivity in the solid solutions is significantly lower than at the end members, and the incorporation of bowing factor C describes very well the observed compositional dependence.

Figure 6.

Lattice thermal conductivity for K₂ (Bi_1-XSb_X)₈Se₁₃ solid solutions vs. composition at different temperatures. Dashed lines are from Adacci model (after [7]).

A more elaborated model has been developed by Klemens [31], which takes into account the mass difference and strain introduced by the alloyed material and the 3-phonon Umklapp process and predicts the ratio of κ_L for a solid solution to that of a hypothetical solid solution without mass or strain fluctuation (κ₀), as a function of a disorder parameter u:

$$\frac{{\kappa }_L}{{\kappa }_0}=\frac{{\mathit{tan}}^{-1}\left(u\right)}{u},u^2=\frac{{\left(6{\pi }^5{\Omega }^2\right)}^{1/3}}{2k_B{V}_S}{\kappa }_0\Gamma ,\Gamma =\frac{⟨\overline{{\mathit{\Delta {\rm M}}}^2}⟩}{{⟨\overline{{\rm M}}⟩}^2}+\epsilon \frac{⟨\overline{{\mathit{\Delta R}}^2}⟩}{{⟨\overline{R}⟩}^2}$$E13

where Ω is the average volume per atom, V_S is the average speed of sound and the scattering parameter (Γ = Γ_M + Γ_S) is related to the average variance in atomic mass (ΔM) and atomic radius (ΔR), with ε an adjustable parameter related to the Gruneisen parameter. For a detailed methodology on the evaluation of Eq. (13) readers may refer to [32]. In general, there will be several different types of atoms that substitute atoms in different sublattices. For the evaluation of the average variance in atomic mass, one should take into account that the k^th atom of the i^th sublattice has mass M^k_i, radius r^k_i and fractional occupation f^k_i [33]. For example, the half-Heusler ZrNiSn has 3 sublattices where substitution may occur (namely the Zr, the Ni and the Sn sublattice), so Sb doping occurs in the Sn sublattice (ZrNiSn_0.99Sb_0.01) while solid solution by Hf may occur in the Zr sublattice (ex. Hf_0.5Zr_0.5NiSn_0.99Sb_0.01).

The Klemens model uses either only the mass difference (Γ_Μ) or both the mass and the strain fluctuations (Γ_M and Γ_S). In practice, for a given composition the values of κ₀, Ω and V_S are calculated as linear interpolation between the end-members’ values. Figure 7 depicts the variation of κ_L for Zr_XTi_1-XNiSn solid solutions. Results are shown as solid lines, and the smooth variation of Γ_M and Γ_S are shown in the insert. As can be seen, mass fluctuation is a major, but not the only mechanism to describe the composition dependence of the lattice thermal conductivity in Zr_XTi_1-XNiSn, and stain is required to model the lattice thermal conductivity at room temperature [34].

Figure 7.

Evolution of lattice thermal conductivity in Zr_XTi_1-XNiSn (modified from [34]).

2.6 The effect of disorder

The majority of phonon transport has been derived from studies of homogenous crystalline solids, where the atomic composition and structure are periodic. In crystalline solids, the solutions to the equations of motions for the atoms (in the harmonic limit) result in-plane wave modulated velocity fields for the normal modes of vibration. However, it has been known for several decades that whenever a system lacks periodicity, either compositional or structural, the normal modes of vibration can still be determined (in the harmonic limit), but the solutions take on different characteristics and many modes may not be plane wave modulated.

In the case of a highly disordered material, or for non-crystalline solid, Cahill and Pohl [35, 36] developed a model for the temperature dependence of thermal conductivity by adopting, the Einstein model for heat conduction [37] (individual oscillators vibrated independently one from the other) and the Debye model for lattice vibrations, by dividing the material into regions of size λ/2, which oscillate with frequency ω = 2πυ/λ. This model has been successfully applied to amorphous SiO₂ [36] and many other disordered materials. Similar to Eq. (6), in this case, thermal conductivity κ_D, can be written as [35]:

$${\kappa }_D\left(T\right)=\frac{k_B}{2{\pi }^{2}u}{\left(\frac{\mathrm{\hslash }}{k_{B}T}\right)}^2\underset{0}{\overset{{\omega }_L}{\int }}{\tau }_D\frac{{\omega }^{4}exp\left(\frac{\mathrm{\hslash }\omega }{k_{B}T}\right)}{{\left(exp\left(\frac{\mathrm{\hslash }\omega }{k_{B}T}\right)-1\right)}^2}\mathit{d\omega }$$E14

where τ_D is the phonon relaxation rate for the highly disordered material, defined as τ = π/ω, and ω_L is the limiting (transition) frequency, much similar defined as ω_D. comparing Eqs. (6) and (14), it can be easily seen that the thermal conductivity in a highly disordered material, κ_D, even in the case where ω_L ≈ ω_D, is substantially lower than κ_L, and results in a monotonous increase, reaching a constant value, κ_ΜIN = κ_Dmax, at high temperatures. This is the minimum thermal conductivity.

Eq. (6) does not obviously apply in the case where the phonon mean free path λ becomes of the order of one-half of the phonon wavelength (λ/2 = πυ/ω). Since the phonon means free path, λ = λ (ω, T), depends on the values of parameters of the various phonon-scattering mechanisms, this situation is plausible even in a crystalline material with high concentration of defects and low phonon-phonon interaction strength. In the general case, the phonon relaxation rate should be [38]:

$$\tau \left(\omega ,T\right)=\left\{\begin{array}{c}{\tau }_C\left(\omega ,T\right),0\le \omega \le {\omega }_0\\ \frac{\pi }{\omega }=\frac{\lambda }{2},{\omega }_0\le \omega \le {\omega }_D\end{array}\right.$$E15

where ω₀ is the transition frequency from crystalline to ‘localised’ contribution. Since completely localised states do not contribute to thermal conductivity, we assume a ‘weak localization’ and excitations can hop from one site to the next diffusively, much similar to what Cahill and Pohl had considered in the case of heat conduction in amorphous materials [38].

Figure 8 shows the temperature dependence of thermal conductivity for crystalline K₂Sb₈Se₁₃ thermoelectric material. K₂Sb₈Se₁₃ has a complex and highly anisotropic structure and morphology, large unit cell, low crystal symmetry and structural site occupancy disorder [39]. As can be seen in Figure 8, K₂Bi₈Se₁₃ has a very low thermal conductivity with a maximum value ∼2.5 W/Km at the low-temperature peak and ∼ 0.6 W/Km at the room temperature. It is, therefore, interesting to apply the mixed model presented above in a material with such low thermal conductivity: dashed lines show the contributions of crystalline and localised models, while solid line corresponds to mixed model, which describes the lattice thermal conductivity very well, in the entire temperature range.

Figure 8.

Temperature dependence of thermal conductivity for K₂Sb₈Se₁₃. Dashed lines show the contributions for crystalline and localised models (after 37]).

In order to evaluate the contributions of the various scattering mechanisms and the transition from the crystalline to the region of ‘weak localization’, the phonon mean free path, λ (ω), is presented in Figure 9 as a function of phonon frequency, at 15 K and at 100 K, along with the different scattering contributions. As expected, at low temperatures, at low temperatures λ(ω) is limited by grain size at low frequencies and by defects (impurities) at high, while 3-phonon scattering contributes in the mid-frequency range and is manifested by the Normal process. At 100 K, the contribution of grain size is shirked into much lower portion of phonon frequency spectrum, and the Umklapp process prevails over the Normal one. Interestingly, the ‘localised’ lower dashed curve intersects the phonon mean free path at high frequencies, signalling the transition of the crystalline to the regime of ‘weak localization’. In Figure 10 summarizes the phonon scattering processes that limit lattice thermal conductivity for K₂Sb₈Se₁₃ at the entire temperature range: four domains of phonon scattering processes can be assigned within the framework of Debye theory [38].

Figure 9.

Calculated phonon mean free path (cirlces) and different scattering contributions as a function of phonon frequency for K₂Sb₈Se₁₃ at: (a) 15 K, and (b) 100 K (after [38]).

Figure 10.

Summary of phonon processes that limit the lattice thermal conductivity in K₂Sb₈Se₁₃ (after [38]).

2.7 The effect of nano-structuring

Nano-structuring has been proved to be an effective way for the reduction of lattice thermal conductivity. Relaxation time of phonon scattering due to the nano-scale inclusions is given by a Mathiessen-type combination of short-wavelength and long-wavelength scattering [40]:

$${\tau }_{\mathit{NP}}^{-1}\left(\omega \right)=u{\left({\sigma }_S^{-1}+{\sigma }_L^{-1}\right)}^{-1}{N}_{\mathit{NP}},{\sigma }_S=2\pi R^2,{\sigma }_L=\frac{4\pi R^2}{9}{\left(\frac{\mathit{\Delta \rho }}{\rho }\right)}^2{\left(\frac{\mathit{\omega R}}{u}\right)}^4$$E16

where N_NP is the density of the nanoparticles, σ_S and σ_L are the scattering cross-sections for short wavelength (geometrical) and long wavelength (Rayleigh-type), R is the average particle radius, ρ is the medium density and Δρ is the density difference between the particle and matrix materials.

Figure 11 shows the effect of nano-structuring for Mg₂Si (top data) and the combination of alloying/nano-structuring of the pseudo-ternary (Mg₂Si)_1-X-Υ (Mg₂Sn)_X (Mg₂Ge)_Y (bottom). Solid lines represent the theoretical model without (top curve) and with (bottom) alloying, respectively. Dashed lines represent the corresponding models with nano-structuring. Clearly, the experimental points are better described by the lower dashed curve (with nanophases), indicating that an additional phonon scattering mechanism is required.

Figure 11.

Lattice thermal conductivity models for Mg₂Si and for Mg₂Si_0.55Sn_0.4Ge_0.05. Note the effect of alloying and of nano-structuring in the reduction of lattice thermal conductivity.

Figure 12a shows the phonon mean free path (MFP) as a function of the phonon frequency (ω/ω_D), for the various phonon scattering mechanisms encountered. Solid lines refer to RT, dashed at 500 K while the dotted line corresponds to the model without nano inclusions. The various acting mechanisms that are reducing the lattice thermal conductivity are shown in the corresponding figure caption. At low frequencies, phonon MFP is limited by the grain boundary (GB) scattering, resulting in a flat phonon MFP, for frequencies up to 0.01ω_D. At high frequencies (>0.1ω_D) phonon MFP is governed by a combination of alloying and Umklapp processes. As only the Umklapp scattering is temperature dependent, this mechanism will squeeze more and more the phonon MFP as the temperature increases, resulting in the decrease of lattice thermal conductivity. In the absence of nano-structuring, the total phonon MFP is smoothly changing from the GB scattering to alloy/Umklapp scattering as shown by the dotted curve. However, when nano-structuring is encountered, a considerable amount of mid-range phonons is scattered by nano inclusions (black curves). The two nano-scattering regimes, the geometrical (σ_S) and the Rayleigh-type (σ_L) are evident. Figure 12b shows the effect of concentration of the nano-particles in the phonon MFP at RT. As the concentration of nano-precipitates increases, the geometrical (flat regime) contribution is lowered, and the Rayleigh-type (declining regime) is moving towards lower frequencies. As a result, the nano-inclusions will filter out a more significant portion of the mid-range phonons.

Figure 12.

(a) phonon means free path for Mg₂Si_0.55Sn_0.4Ge_0.05 at two temperatures, and (b) the effect of nano-precipitates density.

As becomes evident, the effect on nano-structuring has a strong contribution to the geometrical features. Another geometrical-type contribution comes from the grain size (reduced grain size results in reduction of the lattice thermal conductivity). The interplay of the two parameters, namely the grain size and nano-precipitate concentration in the calculated lattice thermal conductivity is depicted in Figure 13 for Mg₂Si and for Mg₂Si_0.6Sn_0.4. Note that both horizontal axes are in log-scale. Despite the difference in absolute scale, which originated mainly from alloying and the difference in Debye temperature, sound velocity and average mass to a lesser degree, both plots share common characteristics.

Figure 13.

The interplay of grain size and nano-concentration in the calculated lattice thermal conductivity for Mg₂Si (a) and Mg₂Si_0.55Sn_0.4Ge_0.05 (b).

Lattice thermal conductivity, as expected, is higher for large grain size and at low nano-precipitate concentration, and lower on the opposite side. The reduction of the calculated κ_L is more than 5 or 4-fold for the cases of the binary and pseudo-binary compounds. Furthermore, the two parameters are interconnected, and for a given grain size there is an optimal value for nano-precipitate concentration, and vice-versa. This optimal pair of parameters lies in the ‘diagonal’ curve. For values to the left of this optimal-pair curve, lattice thermal conductivity is reduced mainly due to grain size, regardless of value of the nano-precipitate concentration, as indicated by the lines parallel to N-nano axis. For values to the right of the optimal-pair curve, the opposite happens and lattice thermal conductivity is governed by the nano-precipitate concentration (lines parallel to grain-size axis).

This observation may lead to a general approach to the optimization of lattice thermal conductivity in thermoelectric materials, by controlling both the nano- as well as the micro-structural features. For materials grown from melt, nano-structuring may occur using thermodynamically driven phase segregation during cooling of the melt such as spinodal decomposition or nucleation and growth [41]. The size and the concentration of the nano-precipitates can be affected by cooling conditions, however, is difficult to control. Fine-tuning of the grain size, on the other hand, may be effectively done by ball-milling and the control of the milling conditions is much easier [42]. Therefore, tuning the grain size of the resulting compound may give the optimal pair values for the nano- and the micro-structural features.

3. Conclusion

In this book chapter, we have explored both analytical and phenomenological models for evaluating the thermal conductivity in thermoelectric materials. We have started with the basic features and ways for the determination of the lattice thermal conductivity in thermoelectric materials and proceeded with the theoretical foundations of the subject. Next, we have explored various effects that affect the lattice thermal conductivity, namely, the role of defects, alloying, of disorder and we will conclude with the effect on nano-structuring. In each session, we provided both the necessary theoretical background and proceeded with application examples of the various aspects of different thermoelectric materials.

Acknowledgments

E. H. acknowledges the financial support from the project “Design and implementation of innovative lift’s air-conditioning systems by using thermoelectric devices” (Project code: KΜP6-0074109) under the framework of the Action “Investment Plans of Innovation” of the Operational Program “Central Macedonia 2014 2020”, that is co-funded by the European Regional Development Fund and Greece. Th.K. acknowledges M-Era.Net project ‘MarTEnergy’ funded by the Cyprus Research and Innovation Foundation (P2P/KOINA/M-ERA.NET/ 0317/04).