DOI: 10.1039/C9MH01990A
(Focus)
Mater. Horiz., 2020, Advance Article

Ramya Gurunathan,
Riley Hanus and
G. Jeffrey Snyder*

Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA. E-mail: jeff.snyder@northwestern.edu

Received
10th December 2019
, Accepted 22nd April 2020

First published on 22nd April 2020

Solid-solution alloy scattering of phonons is a demonstrated mechanism to reduce the lattice thermal conductivity. The analytical model of Klemens works well both as a predictive tool for engineering materials, particularly in the field of thermoelectrics, and as a benchmark for the rapidly advancing theory of thermal transport in complex and defective materials. This comment/review outlines the simple algorithm used to predict the thermal conductivity reduction due to alloy scattering, as to avoid common misinterpretations, which have led to a large overestimation of mass fluctuation scattering. The Klemens model for vacancy scattering predicts a nearly 10× larger scattering parameter than is typically assumed, yet this large effect has often gone undetected due to a cancellation of errors. The Klemens description is generalizable for use in ab initio calculations on complex materials with imperfections. The closeness of the analytic approximation to both experiment and theory reveals the simple phenomena that emerges from the complexity and unexplored opportunities to reduce thermal conductivity.

Typically, experimental trends of thermal conductivity versus point defect concentration are modeled using the expression originally derived by Klemens.^{4–7} These closed-form expressions that simply use the mass and size of the defect are attractive because of their simplicity and utility for determining the source of phonon scattering and thermal conductivity suppression in a solid solution. By calculating the impact of an impurity from just its mass or size, one can uncover material design strategies to optimize the thermal conductivity for a given technological application.^{8–11} The alloy scattering model has been used to identify the dominant phonon scattering mechanisms for several alloy systems important to the field of thermoelectrics including PbTe–PbSe,^{2} Bi_{2}Te_{3}–Bi_{2}Se_{3},^{7} and Mg_{3}Sb_{2}–Mg_{3}Bi_{2}.^{12} While first-principles methods are essential to understanding the details of phonon interactions,^{13–16} the Klemens alloy scattering model describes the emergent phenomena across material systems well, even given the ostensibly limiting approximations, and therefore continues to be widely used.^{15,17–19}

The Klemens analytic model predicts the ratio of the defective solid's lattice thermal conductivity to that of the pure solid without defects (κ_{L}/κ_{0}). This ratio is a function of the disorder parameter u which depends on properties of the pure material: its lattice thermal conductivity (κ_{0}), elastic properties through its average speed of sound (v_{s})†, the average volume per atom (V), as well as a scattering parameter to capture the influence of point defects (Γ = Γ_{M} + Γ_{K}),

(1) |

(2) |

The average mass and mass variance are most easily computed by considering each element (or crystallographic site) separately.^{9,21,22} Consider a generic compound with formula unit: A1_{c1}A2_{c2}A3_{c3}…An_{cn} (e.g. Mg_{2}Sn), where An refers to the nth component (e.g. Mg, or Sn) and c_{n} refers to the stoichiometry of that component (e.g. 2 or 1). Each site An_{cn}(e.g. Sn) can be occupied by a set of atomic species i, including the host atom (e.g. Sn) and any substitutional defects (e.g. Si) with species site fraction (f_{i,n}) (e.g. 1 − x and x in Sn_{1−x}Si_{x}). Then, the average mass of the compound is given by the stoichiometry weighted average of each site average mass

(3) |

Similarly, the average mass variance of the compound is given by the stoichiometry weighted average of the all site mass variances

(4) |

The Klemens model using mass difference alone (Γ = Γ_{M}) quantitatively describes the κ_{L} trends with alloy composition for several material systems.^{3,10,13,24} The solid solution between Mg_{2}Sn and Mg_{2}Si is a case in which the Klemens mass difference model works well, and is recreated here to demonstrate use of these equations in a multiatomic system. For a given composition, the value of κ_{0}, V, and v_{s} are taken to be the linear interpolation between the values for the end-member species (Fig. 2). Here, the inputs for Mg_{2}Sn and Mg_{2}Si are, respectively, V = 25.7 and 21.5 Å^{3} for the average volumes per atom, v_{s} = 3160 and 6715 m s^{−1} for the average sound velocities, and a scattering parameter of

(5) |

Fig. 2 Lattice thermal conductivity for the full composition range of the solid solution between Mg_{2}Sn and Mg_{2}Si. Red and black data points are experimental thermal conductivity measurements,^{25,26} while the blue U-curve is the prediction from the mass difference Klemens model and the dotted black line comes from first principles T matrix scattering theory.^{13} Finally, the κ_{0} curve interpolates linearly between the two end-members. The fit helps identify mass-difference scattering as the dominant effect in this system, as it explains the trend without needing to invoke other scattering or softening mechanisms. |

Using these inputs, the full κ_{L} versus composition trend shown in Fig. 2 is calculated without fitting parameters, and shows good correspondence with experimental measurements. As a result, one can conclude that the contribution of the mass difference term in point defect scattering is the dominant effect for this materials system and explains the experimental results without having to invoke other scattering or lattice softening mechanisms. This result is consistent with the fact that the cell volume of Mg_{2}Sn_{(1−x)}Si_{x} is fairly constant with changes in composition x, leading to negligible mechanical strain contributions.

For vacancy scattering, the perturbation to lattice energy emerges from both the missing mass and the missing bonds to its neighbors. Klemens suggests that the scattering parameter Γ for this case can be modeled as a mass difference scattering with , where M_{vac} is the mass of the vacant atom. This leads to a ∼10 × stronger scattering parameter than a typical point defect.^{27} Indeed, vacancy scattering has been demonstrated to induce a large reduction in thermal conductivity in several thermoelectric compounds,^{28–35} although the enhanced scattering effect of vacancies is often overlooked. Recent data analysis suggests that the same mass difference model describes interstitial defect scattering as well.^{8,36,37}

The mass difference model captured in eqn (3) and (4) follows the recommendation originally proposed by Berman et al., and is suggested here for its conceptual clarity. Several discussions, including those of Klemens,^{4,7,21,38,39} describe this model as being equivalent to a monatomic lattice approximation, which involves a summation of the atoms in the unit cell into one large, vibrating mass. This alternate description of a compound has led to ambiguity in the meaning of the volume V. A misinterpretation has resulted in some studies over-approximating the mass scattering effect by a factor equal to the number of atoms in the unit cell. Typically, however, a cancellation of errors due to an underestimation of the effect of vacancies allows the broader conclusions of the studies about the importance of point defect scattering in a materials system to remain valid.^{8,29–31,35}

As mentioned previously, the strain due to a defect that is larger or smaller than the host atom perturbs the lattice energy through its potential energy term. Therefore, the force constant variance (ΔK^{2}) is typically expressed through the average variance in atomic radius (ΔR^{2}) scaled by a fitting parameter (ε). As before, the atomic radius variance on the nth site is defined from the atomic radius of the ith species which may occupy that site, R_{i,n}, and the average atomic radius of the site, . Although there exist theoretical models^{40} or heuristical correlations^{20} for ε, it is considered here as an adjustable parameter that typically varies between 1–500 in order to fit the experimental data.

(6) |

The simple mass and volume perturbations can be generalized for materials with complex phonon dispersions or even non-crystalline materials. Tamura defines a similar mass difference perturbation parameter for each phonon eigenstate (e_{k}(s)) that can be implemented in numerical Boltzmann transport equation solvers for thermal conductivity.^{13,15,18,19,23,41–45} In many thermal conductivity solvers, the Tamura model is the standard treatment for isotope-phonon scattering in pure compounds.^{43,44} The mass difference parameter in the Tamura model (Γ^{T}_{M}) is given as a sum over all the s atom sites in a simulation, where i again labels the species that may occupy site s including the host and impurity atoms

(7) |

The description of scattering here is general enough that it could be used to describe the perturbation induced to any vibrational mode. Therefore, in addition to plane wave phonons, which are only strictly defined in periodic crystals, the vibrational modes of amorphous solids, codified in the Allen and Feldman formalism as diffusons, locons, and propagons, are describable within the same alloy scattering theory.^{46–48}

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