Phonon engineering via microstructural defects

Real materials are not perfect, and defects will inevitably scatter phonons. Phonon MFP or relaxation time, τ, depends upon phonon scattering by microstructural defects in addition to intrinsic phonon–phonon scattering. Manipulating phonon transport by designing defects is feasible. Figure 1 shows schematic diagrams and corresponding high-resolution transmission electron microscope images of common defects in solids.^{Reference Biswas, He, Blum, Wu, Hogan, Seidman, Dravid and Kanatzidis12,Reference Kim, Lee, Mun, Kim, Hwang, Roh, Yang, Shin, Li, Lee, Snyder and Kim14,Reference Hong, Hu, Probert, Li, Lv, Yang, Gu, Mao, Feng, Xie, Zhang, Wu, Zhang, Jin, Ji, Zhang, Yuan and Zhang16–Reference Mao, Wang, Liu, Ge and Ren18}

Figure 1.

Schematic diagrams and corresponding high-resolution transmission electron microscope images of common defects in solids. (a, d) Vacancies. Reprinted with permission from Reference Reference Hong, Hu, Probert, Li, Lv, Yang, Gu, Mao, Feng, Xie, Zhang, Wu, Zhang, Jin, Ji, Zhang, Yuan and Zhang16. © 2015 Nature Publishing Group. (b, e) Dislocations. Reprinted with permission from Reference Reference Kim, Lee, Mun, Kim, Hwang, Roh, Yang, Shin, Li, Lee, Snyder and Kim14. © 2015 AAAS. (c, f) Stacking faults (SFs).^{Reference Mao, Wang, Liu, Ge and Ren18} (g, j) Grain boundaries. Reprinted with permission from Reference Reference Wu, Pei, Wang, Wu, Huang, Zhao and He17. © 2014 Wiley. (h, k) Twin boundaries.^{Reference Mao, Wang, Liu, Ge and Ren18} (i, l) Nanoprecipitates. Reprinted with permission from Reference Reference Biswas, He, Blum, Wu, Hogan, Seidman, Dravid and Kanatzidis12. © 2012 Nature Publishing Group. The insets in (e), (j), and (l) show the morphology of a dislocation, dislocation (on the grain boundary), and nanoprecipitate, respectively.

Point defects are usually thermodynamically stable due to the balance between increased crystal distortion energy and system entropy;^{Reference DeHoff19} these include vacancies, interstitials, and impurity atoms. Vacancies are the primary intrinsic point defects owing to their low formation energy.^{Reference DeHoff19} Tuning the stoichiometric composition or extrinsic methods (e.g., radiation treatment) can control the vacancy concentration in semiconductors.^{Reference Liu, Zhao, Liu, Lan, Xu, Li, Zhang, Berardan, Dragoe and Lin20,Reference Tuomisto, Ranki, Saarinen and Look21} Interstitials are high-energy defects that are less common than other point defects.^{Reference DeHoff19} In some cases, small atoms tend to occupy octahedral or tetrahedral interstitial sites of a cubic lattice or van der Waals gaps of lamellar structures.^{Reference Hu, Zhu, Liu and Zhao22} In thermoelectric materials, impurity atoms are more common due to the requirement of decreasing the thermal conductivity or optimizing the carrier concentration. Antisite defects may also occur in an ordered alloy or compound (e.g., Bi locating to Te sites in Bi₂Te₃).^{Reference Zhu, Hu, Zhao and He23}

Phonon scattering by point defects mainly arises from mass and strain fluctuations.^{Reference Callaway and von Baeyer24} Vacancies and interstitials, which are considered the strongest phonon-scattering centers, can maximize these fluctuations. Rattling fillers in skutterudites, which have the general formula RM₄X₁₂ (R is normally rare-earth elements, M is Fe, Ru, or Os, and X is P, As, or Sb), could effectively reduce κ_lat due to resonant scattering and other factors, for example, the reduced group velocity of phonons.^{Reference Yang, Xi, Qiu, Wu, Shi, Chen, Yang, Zhang, Uher and Singh25} However, the concentration of defects plays a critical role in the phonon–impurity coupling strength, and therefore, isoelectronic substitution could dramatically suppress κ_lat, especially for heavy elements.^{Reference He, Zhu, Sun, Mao, Reith, Chen, Schierning, Nielsch and Ren26}

Linear defects or dislocations were first introduced to explain the low fracture strength of real materials.^{Reference Hull and Bacon27} Plastic deformation is the most commonly used method to increase the dislocation density in metal.^{Reference Kuhlmann-Wilsdorf28} Due to the low density of dislocations at equilibrium in semiconductors, phonon scattering by dislocations has long been overlooked in thermoelectrics. Recently, liquid-phase compaction was used to produce low-angle grain boundaries with dense dislocation arrays, as demonstrated in Bi_0.5Sb_1.5Te₃ and Yb-filled CoSb₃.^{Reference Kim, Lee, Mun, Kim, Hwang, Roh, Yang, Shin, Li, Lee, Snyder and Kim14,Reference Meng, Liu, Cui, Qin, Geng, Cai, Fu, He, Ren and Sui29} Dislocation loops can be generated by the collapse of vacancy discs in lead chalcogenides,^{Reference Chen, Ge, Li, Lin, Shen, Chang, Hanus, Snyder and Pei30,Reference Chen, Jian, Li, Chang, Ge, Hanus, Yang, Chen, Huang, Snyder and Pei31} which was initially observed in metals.^{Reference Hirsch, Silcox, Smallman and Westmacott32}

A dislocation consists of a core and an elastic stress field, hence in principle, phonon scattering via dislocations can have two distinctive mechanisms.^{Reference Carruthers33} Scattering via dislocation cores can be considered a short-range-interaction model similar to mass-difference impurity scattering, whereas phonon scattering via the elastic field shows long-range characteristics that depend on the type of dislocation.^{Reference Carruthers33} Recently, the concept of a “dislon,” a quantized quasiparticle of dislocation that can interact with elastic waves, has been used to explore the nonperturbative nature of coupling between a dislocation and a phonon.^{Reference Li, Ding, Meng, Zhou, Zhu, Liu, Dresselhaus and Chen34}

Interfacial defects mainly include grain boundaries, stacking faults, and twin boundaries. Several nonequilibrium approaches, such as high-energy ball milling or melting spinning, in combination with quick direct-current hot press or spark plasma sintering, could be used to fabricate nanostructured materials with a high density of grain boundaries.^{Reference Poudel, Hao, Ma, Lan, Minnich, Yu, Yan, Wang, Muto, Vashaee, Chen, Liu, Dresselhaus, Chen and Ren11,Reference Xie, He, Kang, Tang, Zhu, Laver, Wang, Copley, Brown and Zhang35–Reference Su, Wei, Li, Liu, Yan, Li, Su, Xie, Zhao, Zhai, Zhang, Tang and Uher37} Stacking faults and twin boundaries are usually related to solid-state phase transformations (e.g., GeTe alloys)^{Reference Cook, Kramer, Wei, Harringa and Levin38} or close-packed crystal structures with low formation energy (e.g., copper-based materials).^{Reference Mao, Wang, Kim, Liu, Saparamadu, Tian, Dahal, Sun, Chen and Liu39,Reference Gao, Yi, Sun, Meng, Cai and Zhao40} Stacking faults or twin boundaries can also be observed after heavy plastic deformation^{Reference Mao, Wang, Liu, Ge and Ren18,Reference Liu, Wang, Gao, Mao, Geng, Shuai, Cai, Sui and Ren41} and develop due their lower formation energies than normal grain boundaries.^{Reference Lu42}

Qualitatively speaking, the phonon–boundary scattering rate by interfacial defects is evaluated by the boundary misfit angle, namely coherent, semicoherent, and incoherent.^{Reference Klemens43} For example, the phonon relaxation time has an inverse correlation with the square of the grain-boundary misfit angle for low-angle grain boundaries.^{Reference Klemens43} Thus, stacking faults, twin boundaries, and low-angle grain boundaries have a smaller impact on thermal transport compared to conventional high-angle grain boundaries as they do not reflect phonons effectively. Given the microstructural complexity and other unknowns, it is difficult to give a quantitative picture of scattering strength in bulk materials merely based on experiments. Nevertheless, detailed computations have provided a clear microscopic understanding of phonon scattering by interfacial defects in specific systems (e.g., SiGe alloys).^{Reference Minnich44}

Volume defects are 3D aggregates of atoms or vacancies, including precipitates and voids. Traditional metallurgical science (e.g., nucleation and growth or spinodal decomposition) enables better understanding and control of the morphology, size, and density of precipitates in the matrix.^{Reference Callister and Rethwisch45} Nucleation and growth methods require low or no precipitate solubility in the solid state, but complete solubility in the liquid state. Due to the final increased system free energy, interfacial energy and strain energy codetermine the final morphology of a precipitate.

Here, it should be mentioned that AgPb_mSbTe_2+m, known as the LAST alloy, has endotaxially embedded Ag-Sb–rich nanoprecipitates, resulting in a ultrahigh ZT of ∼2.2 at 800 K.^{Reference Hsu, Loo, Guo, Chen, Dyck, Uher, Hogan, Polychroniadis and Kanatzidis46} This strongly motivated the development of nanostructured thermoelectric materials with in situ nanoprecipitates.^{Reference Makongo, Misra, Zhou, Pant, Shabetai, Su, Uher, Stokes and Poudeu47–Reference Zhao, Zhang, Wu, Tan, Pei, Xiao, Chang, Wu, Chi, Zheng, Gong, Uher, He and Kanatzidis49} Spinodal decomposition, having a miscibility gap in the phase diagram, is a typical “uphill” diffusion process without the requirement to overcome the thermodynamic barrier, leading to the formation of a uniform-sized, periodic fine microstructure. More importantly, the phase interfaces generally remain coherent or semicoherent; meanwhile, the separate regions with the same crystal structure only hold different chemical compositions.^{Reference Callister and Rethwisch45} Specifically, PbTe/GeTe, PbTe/PbS (Ti, Zr, Hf) NiSn, and La_0.8Ti_0.1Ga_0.1Fe₃CoSb₁₂, show the typical morphology of spinodal decomposition,^{Reference He, Girard, Kanatzidis and Dravid50–Reference Zhang, Chere, Wang, Kim, He, Cao, Dahal, Broido, Chen and Ren54} but there are few concrete investigations on the role of spinodal decomposition in phonon scattering.

Recently, it was found that in p-type filled skutterudites with spinodal decomposition, the sharp reduction in κ_lat is caused by the strain-field scattering of high-energy phonons.^{Reference Meng, Cai, Liu, Li, Geng and Sui52} In certain circumstances, some nanoscale second phases (e.g., SiC,^{Reference Li, Tan, Li, Liu, Li, Li, Zou and Wang55} carbon nanotubes,^{Reference Kim, Choi, Shin, Moon, Koo, Lee and Ha56} and reduced graphene oxide^{Reference Zong, Hanus, Dylla, Tang, Liao, Zhang, Snyder and Chen57}) are also intentionally incorporated into the parent phase to suppress κ_lat or enhance mechanical properties. For porous structures, the reduction in κ_lat depends not only on the volume fraction of pores, but also on their aspect ratio and spatial distribution.^{Reference Smith, Alzina, Bourret, Nait-Ali, Pennec, Tessier-Doyen, Otsu, Matsubara, Elser and Gonzenbach58} Normally, the existence of low volatile phases or a hollow precursor forms a porous 3D structure,^{Reference Schlichting, Padture and Klemens59} but the validity of enhancing ZT is still in debate owing to the counterbalancing effect of degrading electrical conductivity.^{Reference Lee, Vashaee, Wang, Dresselhaus, Ren and Chen60} Recently, a porous architecture containing nano- to micrometer-size irregularly shaped and randomly oriented pores in rare-earth-free skutterudites, ∼Co_23.4Sb_69.1Si_1.5Te_6.0, was shown to give rise to a high ZT ∼ 1.6, showing more than a 100% increase as compared to the pristine sample.^{Reference Khan, Kobayashi, Tang, Yamauchi, Hasegawa, Mitome, Xue, Jiang, Tsuchiya, Golberg, Bando and Mori61}

For phonon scattering via spherical nanoprecipitates, the relaxation time interpolates between long (geometrical scattering) and short (Rayleigh limit) wavelength scattering regimes. For the scattering cross section, the mass or force constant between nanoprecipitates and matrix dominates in the Rayleigh limit while the path length and the associated phase lag matter in the near geometrical scattering regime.^{Reference Kim and Majumdar62} Kundu et al. pointed out the different roles of lighter and heavier nanoprecipitates in phonon scattering, where higher scattering rates can be achieved for heavier nanoprecipitates at low frequencies.^{Reference Kundu, Mingo, Broido and Stewart63} For the best distribution of spherical nanoparticles, previous calculations show that large size distributions block broad-spectrum phonon transport.^{Reference Kim and Majumdar62} Zhang et al. recently reported that a discrete size distribution at a few well-chosen radii will be more effective.^{Reference Zhang and Minnich64} For nanoparticles with other shapes, a qualitative description for phonon scattering could be achieved by means of more elaborate treatments due to the complexity of the geometry.^{Reference Zuckerman and Lukes65}

The critical role of hierarchical micro-/nanostructures

Recently, both theoretical and experimental results on semiconductors have proven that phonons with a broad MFP spectrum contribute to κ_lat.^{Reference Minnich, Johnson, Schmidt, Esfarjani, Dresselhaus, Nelson and Chen9,Reference Tian, Garg, Esfarjani, Shiga, Shiomi and Chen66} Nano-microstructural defects with different dimensions are targeted to scatter phonons of comparable sizes. The effective scattering rate $\tau _{eff}^{ - 1}$ can be evaluated by Matthiessen’s rule, which assumes scattering channels are independent of each other: $\tau _{eff}^{ - 1} = {\sum _I}\tau _I^{ - 1}$, where $\tau _I^{ - 1}$ represents the relaxation time of a specified type of defect I.^{Reference Klemens67,Reference Callaway68}

Different defects have different phonon-scattering rates. Point defects have a phonon-scattering rate $\tau _{PD}^{ - 1}\~{\omega ^4}$, where ω is phonon frequency, thus showing they are preferable to scattering the phonons with high frequencies that could be analogous to Rayleigh scattering. Dislocations act as scattering centers mainly for mid-frequency phonons, where the phonon-scattering rates arising from the dislocation core and strain field are given by $\tau _{DC}^{ - 1}\~{\omega ^3}$ and $\tau _{DS}^{ - 1}\~\omega$, respectively. The grain-boundary scattering rate is independent of phonon frequency, $\tau _{GB}^{ - 1}\~{\omega ^0}$, when the misfit angle is less than 15° and has stronger interactions with low-frequency phonons.

Apart from these scattering centers, nanoprecipitates could scatter phonons in a wider frequency range, $\tau _{NP}^{ - 1} \sim {\omega ^0} + {\omega ^4}$. Therefore, a large suppression of κ_lat could be realized by hierarchical micro-/nanostructures due to the scattering of phonons over broad frequencies.^{Reference Biswas, He, Blum, Wu, Hogan, Seidman, Dravid and Kanatzidis12} This strategy was first proposed in the PbTe-SrTe system that includes atomic-scale lattice disorder, nanoscale endotaxial precipitates, and microscale grain boundaries, which pushes ZT to greater than 2.^{Reference Biswas, He, Blum, Wu, Hogan, Seidman, Dravid and Kanatzidis12} This strategy is also applicable to other material systems (e.g., α-MgAgSb^{Reference Liu, Mao, Sui and Ren69,Reference Liu, Wang, Mao, Geng, Shuai, Wang, He, Cai, Sui and Ren70} ) and can accelerate breakthroughs in thermoelectric science.^{Reference Su, Wei, Li, Liu, Yan, Li, Su, Xie, Zhao, Zhai, Zhang, Tang and Uher37,Reference Zhao, Qiu, Song, Blichfeld, Eikeland, Ren, Ge, Iversen, Shi and Chen71}

Challenges

It is now generally recognized that the defects previously discussed are important tools for improving ZT, however, simulation tools are not yet at a level to guide the rational design of defects. The improvements in ZT so far are mostly based on trial and error. Some key challenges from a simulation and modeling perspective are discussed here.

First-principles simulation of electron transport

Ideally, one would design nanostructures such as grain size to be in between electron and phonon MFPs such that interfaces scatter more phonons than electrons. Knowing the MFP distributions of electrons and phonons in intrinsic materials is thus important. Although first-principles-based phonon simulation tools are becoming mature,^{Reference Esfarjani, Chen and Stokes10} electron simulation tools are just becoming available since the recent development of Wannier interpolation, a complete set of orthogonal functions used in solid-state physics, for electron–phonon coupling matrices.^{Reference Giustino, Cohen and Louie72,Reference Sjakste, Vast, Calandra and Mauri73} There are limited reports on fully first-principles-based simulations of electron thermoelectric transport properties, although simulations based on the constant relaxation time approximation are routinely reported.^{Reference Sun, Shuai, Ren and Singh74} In Figure 2, we compare electron and phonon MFP distributions in several representative materials.^{Reference Tian, Garg, Esfarjani, Shiga, Shiomi and Chen66,Reference Liu, Zhou, Liao, Singh and Chen75–Reference Song, Liu, Zhou, Ding and Chen79} More detailed studies show that nanostructures can also be effective for manipulating phonon and electron transport in some materials even when they are comparable to both the electron and phonon MFPs, since some long MFP electrons are disadvantageous to the Seebeck coefficient.^{Reference Minnich, Dresselhaus, Ren and Chen15}

Figure 2.

Normalized accumulation of thermal (dashed) and electrical conductivity (solid lines) for four representative materials. The blue, red, green, and gray lines represent the mean free path (MFP) spectra of SnTe,^{Reference Liu, Zhou, Li, Ding, Song, Liao, Fu and Chen78} GaAs,^{Reference Qiu, Tian, Vallabhaneni, Liao, Mendoza, Restrepo, Ruan and Chen76,Reference Luo, Garg, Shiomi, Esfarjani and Chen77} Si,^{Reference Esfarjani, Chen and Stokes10} and PbTe,^{Reference Tian, Garg, Esfarjani, Shiga, Shiomi and Chen66,Reference Song, Liu, Zhou, Ding and Chen79} respectively. The electrical conductivities are calculated at an n-type carrier concentration of 1 × 10²⁰, 1 × 10¹³, 1 × 10¹⁴, and 1 × 10¹⁷ cm^–3 for SnTe, GaAs, Si, and PbTe, respectively. The differences between phonon and electron MFPs provide the possibility to independently manipulate their transport behavior.

Conclusions and outlook

In this article, we summarized the formation conditions and phonon-scattering mechanisms of nano-microstructural defects in bulk thermoelectric materials, ranging from point defects to volume defects. The role of all-scale hierarchical nano-microstructures on impeding phonon transport is highlighted. We also discussed the main challenges for the roles of defects or microstructure from the perspective of simulations.

The thermal stability of nonequilibrium defects may become a critical issue during long-term practical applications. In addition to phonon engineering by defects, it is essential to pay attention to the effects of carrier concentration and carrier-scattering mechanisms,^{Reference Mao, Wu, Song, Shuai, Liu, Pei and Ren83,Reference Liu, Mao, Peng, Zhou, Gao, Sui, Pei and Ren84} which may counteract the beneficial contributions of the phonon portion and even deteriorate the final ZT. Therefore, delicate manipulation of the nano-microstructure to construct highly ordered hierarchical structures would provide a greater possibility of simultaneously optimizing electrical and thermal properties, which would open the path toward the realization of developing high-performance thermoelectric materials.