Towards ultimate impedance of phonon transport by nanostructure interface

The ultimate limit of low thermal conductivity of solids has drawn interest since 80 years ago¹ in the context of phonon gas kinetics. Suppression of heat transport (particularly at room temperature) is required for engineering applications such as thermoelectrics and thermal insulators. In semiconductors, heat is mainly carried by phonons while electronic contribution becomes significant in metals. Unlike electrons, phonons with a wide range of modes with different mean-free-paths (MFPs) and wavelengths contribute to thermal transport, which makes tuning of thermal transport in materials complex. For example, thermal transport in bulk silicon at room temperature is carried by phonons with MFPs on the orders of 10 nm–10 µm and wavelengths of 1-10 nm as shown in Fig. 1.

Nanostructures^2–9 can modulate τ_ph and v_ph in different ways in a mode-dependent fashion. For example, in analogy to Rayleigh scattering, point defects lead to scattering of high frequency phonons with a rate proportional to ω⁴, where ω is the angular phonon frequency.¹⁰ To scatter low frequency phonons, which largely contribute to thermal conductivity, introduction of interfaces or surfaces of nanostructures as in nanowires,^2,3 nanoporous thin films,^4,5 and nanocrystalline composites^6–8 is efficient. Because phonons are scattered at interfaces and surfaces, when the characteristic length, grain size or layer thickness, is smaller than MFPs, nanostructures enhance phonon scattering and thus reduce τ_ph. Furthermore, when the characteristic length of nanostructures is much smaller than phonon wavelengths, we can observe interference effects of phonons due to phonon wave nature. Although tuning of heat wave¹¹ is challenging because of extremely short wavelengths of phonons, it can lead to intriguing phenomena such as phonon localization^9,12 and zone-folding^11,13 and can be used for advanced devices such as thermal cloaking^14,15 and thermal rectification.¹⁶ 

In this review article, we present an overview and perspective of recent attempts toward ultimate impedance of phonon transport. Recently synthesized silicon (Si) nanocrystalline structures with a single-nanometer grain size showed thermal conductivity smaller than both amorphous Si and SiO₂^17,18 and reached the minimum limit proposed by Cahill et al.,¹⁹ which corresponds to modal MFPs of the half of the wavelength. We then highlight the newly developed approaches for the further designability of nanostructures with materials science combined with informatics and data science, materials informatics (MI). The optimization technique has revealed that aperiodic nanostructures can realize low thermal conductivity and consequently high thermoelectric performance. Finally, in the course of discussing future perspective toward ultimate impedance of phonon transport, we introduce recent attempts toward strain-induced tuning of coherent phonons using soft interfaces. A periodic strain field induced in carbon nanomaterials can lead to zone-folding of coherent phonons.

Reduction of thermal conductivity is demanded by different applications such as thermoelectric materials and thermal insulators. While amorphous solids generally have lower thermal conductivity than crystalline solids, amorphization also degenerates other material properties such as electrical conductivity and Young’s modulus.^20,21 In addition, amorphous structures may not be the best to realize the minimum thermal conductivity.²² For thermoelectric materials, high electrical conductivity, Seebeck coefficient, and low thermal conductivity are required simultaneously. Suppression of thermal transport of crystalline solids is, therefore, preferable.

Nanostructures have an advantage over decoupling of phonon and electron transport because the length scale of MFPs is different for electrons and phonons; e.g., for bulk Si, electron MFPs are 10-20 (50) nm for n-type doping with a concentration of 10¹⁹ (10¹⁶) cm^{−3 23} and phonon MFPs are 10 nm–10 µm as shown in Fig. 1(a). The selective enhancement of phonon scattering by interfaces in polycrystalline structures, without significant enhancement of electron scattering, benefits the thermoelectric efficiency of materials. As for mechanisms of phonon transport through interfaces, despite a number of theoretical studies with different approaches such as the nonequilibrium molecular dynamics (NEMD),^24–26 wave packet,²⁷ atomic Green’s function (AGF) method,^28–31 and Green-Kubo modal analysis,^32–34 details have not yet been fully understood due to their structural and methodological complexities. Recent experiments have shown that thermal boundary conductance of Si-Si interface varies widely, two orders of magnitude from 10² to 10⁴ MW m⁻² K⁻¹, depending on the morphology of oxidized silicon SiO_x layers at the interface. In addition, numerical simulations have shown that thermal boundary conductance between grains in a polycrystalline structure significantly affects the thermal conductivity of bulk properties.^35,36 Interfaces, therefore, provide a flexible tunability of the thermal properties of polycrystalline structures and also thermoelectric properties by balancing thermal and electrical conductivities.

Polycrystalline structures,^6–8 in which interfaces enhance phonon scattering, have been widely studied for thermoelectric materials. While sintering techniques^6–8,37 are preferable to synthesize a bulk material in practice, one can turn to the bottom-up approach to realize a polycrystalline structure with interfaces partially covered by oxidized layers. Recently, Nakamura et al. synthesized epitaxial Si nanocrystals (SiNCs) composed of Si nanoparticles with the same orientation connected with a nanowindow and partially separated by ultrathin (1-monolayer thick) amorphous SiO₂ layers as shown in Figs. 2(a) and 2(b).^17,18 The SiNC structure is fabricated by repeating two processes: (a) epitaxial growth of SiNCs on a SiO₂ layer and (b) formation of a monolayer amorphous SiO₂ film. During the first process, when Si layers are deposited with molecular beam epitaxy on the SiO₂, nanowindows with a diameter less than 1 nm are generated through the reaction $SiO2+Si→2SiO↑$ and become nucleation sites of SiNCs. Because of the oriented crystal orientation and nanowindows connecting Si nanoparticles, electrons may transport from a Si nanoparticle to the adjacent one without significant scattering.

Using a time-domain thermoreflectance method,^38,39 Oyake et al. measured thermal conductivities of SiNCs with different grain sizes 3, 5, and 40 nm as shown in Fig. 2(c).¹⁸ Thermal conductivity decreases with decreasing grain size; for grain sizes of 5 nm and 3 nm, thermal conductivities beat those of amorphous Si^40,41 and amorphous SiO₂,¹⁹ respectively, over the range of temperature. For 3 nm-grain size, thermal conductivities reach to the minimum thermal conductivity κ_min¹⁹ that is calculated with a mode-dependent relaxation time of ω/π, where ω is the mode-dependent frequency.

To gain insights of the extremely low thermal conductivity of SiNCs, the phonon properties of SiNCs were analyzed. Relaxation times due to phonon-phonon scattering τ_ph-ph and interfaces τ_int were analyzed with, respectively, first-principles calculation and the AGF method. Here, it is assumed that the oriented crystal orientation of SiNCs allows phonons with the wavelength larger than the grain size to exist in the sample and the phonon properties of bulk Si, namely, modal v_ph and τ_ph-ph, are applicable to SiNCs. τ_int was calculated with an analytical equation⁴² considering phonon scattering by interfaces both parallel to and perpendicular to the transport direction: $τint−1=1.12−1+3tint/4/1−tint−1vph/Lgrain$⁠, where t_int is the frequency-dependent transmissivity at interfaces calculated by the AGF method and L_grain is the grain size. The validity of the model in nanocrystalline Si has been confirmed with Monte Carlo ray-tracing simulations.³⁵ Calculated thermal conductivities and their temperature dependence were in good agreement with the experimental values for any grain sizes as shown in Fig. 2(c). Surprisingly, for the SiNC of 3 nm-grain size, $τph−1$ (⁠$=τph−ph−1+τint−1$⁠) reaches and even becomes smaller than the minimum limit of relaxation time π/ω at a low frequency (ω < 20 THz) as shown in Fig. 2(d), which means that MFPs are equivalent to or smaller than wavelengths. Although the use of phonon properties of bulk Si has not been strictly validated, the analysis results show that scattering of low frequency phonons, which dominates heat transport, due to single-nanometer scale grains realize the exceptionally low thermal conductivity.

The above mentioned studies have shown that interface nanostructures can realize exceptionally low thermal transport. Here, a question arises: What kind of nanostructures optimizes material properties? Search of the best nanostructures is, however, still challenging because of its complexity; a large number of atoms contained in nanostructures make the analysis of material properties difficult. Recently, informatics techniques combined with materials science, materials informatics (MI), have been applied to the design of novel materials such as cathode materials for lithium-ion batteries,⁴³ nitride semiconductors composed of earth-abundant materials,⁴⁴ and thermoelectric materials.^45–48 While all of these studies use the database of stoichiometric compounds to realize high-throughput screening of the best materials, another but intriguing way of MI is to create nanostructures by identifying optimal geometry that maximizes a target property.

Recently, structural optimization has been conducted for the design of nanostructures such as superlattices⁴⁹ and porous structures⁵⁰ using a machine-learning approach. Because the available database does not exist for individual nanostructures in general, the analysis of the transport properties of predicted structures and prediction of possible candidates are alternately performed as shown in Fig. 3 until a target property converges or exceeds a desirable value. This concept of the optimization can be easily applied to any targets, transport properties, quasiparticles, and nanostructures, and combination with experiments is also attractive.^51,52

In studies on structural optimization reviewed here,^49,50 Bayesian optimization⁵³ with the Gaussian process⁵⁴ implemented in the open-source library common Bayesian optimization (COMBO)⁵⁵ was adopted. Bayesian optimization has an advantage over empirical optimization in its ability to determine required data and a regression curve by machine learning without any prior knowledge of the model. It has been applied, for instance, to thermal transport in Si/Ge superlattices⁴⁹ and the thermoelectric properties of porous graphene nanoribbons (GNRs)⁵⁰ as shown in Fig. 3. Bayesian optimization requires descriptors, representations identifying each candidate structure. While how to determine the descriptor is one of the main topics of this field,⁵⁶ an intuitive binary flag was used as a descriptor; e.g., “0” and “1” denote the Si and Ge atom or layer for Si/Ge superlattices and the pristine and porous section for GNRs. Acceleration of the Bayesian optimization with better descriptors that capture the relationship between nanostructures and transport properties have been discussed in the supplementary material of the paper on GNRs.⁵⁰ The atomic Green’s function (AGF) method⁵⁷ with harmonic force constants for phonon transport and the Green’s function method based on the tight-binding approach^58,59 were employed for calculations of thermal and electron transport properties, respectively. While inelastic scattering such as phonon-phonon and electron-phonon scattering was not taken into account in these Green’s function approaches, these simulations are valid for nanoscale superlattices in these analyses whose lengths are less than MFPs of carriers. It was revealed that aperiodic perfectly smooth interfaces lead to the minimum thermal conductivity and phonon transport properties in such structures have been analyzed in detail. Furthermore, multifunctional structural design of thermoelectric GNR has been performed.

To gain insights of Si/Ge superlattice structures with small thermal conductance, first, the structure of a relatively small region was optimized. For this optimization, optimization was performed for every atom; n_atom-digit binary number was used as a descriptor for each structure, where n_atom is the number of atoms in the nanostructured region as shown in the top of Fig. 4(a). To obtain low thermal conductance with superlattice structures, Tian et al. have found that roughness at interfaces decreases thermal conductance in periodic superlattices,⁶⁰ and Qiu et al. have found that aperiodicity can further decrease thermal conductance.⁶¹ By using the Bayesian optimization, it was revealed that an aperiodic smooth superlattice was an optimal structure that minimizes thermal conductance as shown in Fig. 4(a). Although it was found that aperiodic smooth interfaces should be possible candidates of the minimum thermal conductance, optimization processes are still indispensable to reveal what kind of aperiodicity leads to the minimum thermal transport. Next, the Bayesian optimization was applied to a larger system in which a nanostructured region with different lengths (8-16 conventional unit cells of Si) were connected with a semi-infinite bulk Si at the both ends. Based on the knowledge obtained from the first simulation, a conventional unit cell was taken as a primitive unit of the descriptor. It was found that aperiodic supercells, whose aperiodicity is not intuitive, have again the minimum thermal conductance for each case as shown in the bottom of Fig. 4(a).

A local minimum of thermal conductivity (κ_lat) with respect to a layer thickness of periodic superlattice is often discussed as a unique feature of superlattices. A local minimum of κ_lat appears due to competition between zone-folding and boundary scattering at the crossover from coherent to incoherent transport;^62,63 while group velocity decreases with increasing periodicity of superlattices due to zone-folding, the phonon scattering rate increases with increasing boundary concentration, decreasing layer thickness, due to boundary scattering as schematically shown in the left panel of Fig. 4(b). The local minimum of κ_lat may be observed at ≈5 nm.^64,65 Interestingly, we can also observe a local minimum of thermal conductance K_lat in the fully coherent regime as shown in the right panel of Fig. 4(b).⁴⁹  Fig. 4(b) shows K_lat of one-dimensional (1D) superlattices as a function of the number of interfaces, n_int. The superlattice is composed of 15 light or heavy atoms terminated by a quasi-infinite light-atom chain. Blue and red marks denote all data and the minimum K_lat for each n_int, respectively, and black marks show K_lat of superlattices in which light and heavy atoms are arranged alternately. K_lat of such a periodic superlattice (K_period) decreases monotonically and converges with increasing n_int. Moreover, K_period takes the maximum value for the corresponding n_int. As for the minimum K_lat (K_min), K_min as a function of n_int takes a local minimum and finally merges K_period because the system can no longer be aperiodic for the maximum n_int. To experimentally detect the effect of aperiodicity, optimization of n_int is a key issue because the optimal n_int gives not only K_min but also a flexible tunability of K_lat, a wide range of variation of K_lat from K_min to K_period. While periodic structures have been mainly studied in previous experimental studies,^13,66,67 nanostructured aperiodic superlattices can further impede thermal transport.

To gain insights into the tunability of κ_lat/K_lat of superlattices, we analyzed the phonon properties of two types of simple atomic chains that can vibrate only along the transport direction:⁴⁹ the first and second models target Fabry-Pérot-like resonance and boundary scattering, respectively. See the caption of Fig. 4 for details of simulation models. Fabry-Pérot resonance peaks appear at higher frequency and their number increases with increasing degrees of freedom in the nanostructured region, namely, the number of atoms n_atom, as shown in the left panel of Fig. 4(c).⁶⁸ The right panel of Fig. 4(c) shows that, with increasing n_atom, thermal conductance K_lat decreases and converges quickly and κ_lat increases almost linearly. Next, we analyzed atomic chains with different n_int with a fixed length of sections. As shown in the right panel of Fig. 4(d), K_lat monotonically decreases and converges when n_int increases; κ_lat again increases almost linearly.⁶⁶ It is worth mentioning the analogy and difference of this interfacial effect on fully coherent phonons with boundary scattering effects on fully ballistic phonons. Assuming a constant phonon transmissivity at interfaces, that is, in the fully ballistic transport, we also obtain a linear correlation between κ_lat and n_int as the same as in the fully coherent transport. It is, therefore, difficult to distinguish the fully coherent transport from the fully ballistic transport with this linear correlation in the experiment. The difference between ballistic and coherent transport is that while the scattering rate of ballistic phonons by interfaces is identical for any interfaces, the effect of interfaces on coherent phonons may vary depending on the relative position of interfaces. In the coherent regime, boundaries between the nanostructured region and terminated leads should dominate phonon scattering of the whole system because difference of local density of states (LDOS) between adjacent sections is significant at the ends of the nanostructured region as shown in the left panel of Fig. 4(d).

In a fully coherent regime, the decrease and convergence of K_lat in the above two systems can be understood with the same manner as following. When the number of periodicity in the nanostructured region, corresponding to n_atom and n_int in the first model [Fig. 4(c)] and second model [Fig. 4(d)], respectively, is large enough, vibrational modes in the nanostructured region will become Bloch states and, hence, phonons are no longer scattered at the nanostructured region except for its ends.⁵⁰ Therefore, while we can derive the same relation between K_lat and n_int shown in Fig. 4(d) in a phonon particle picture, the decrease in K_lat with increasing n_int, it should be again stressed that the interfacial effect in the fully coherent regime cannot be described with a phonon particle picture because every interface is identical in the phonon particle picture. Conversely, aperiodicity of the scattering region leads to scattering of different phonon modes at each interface. Then, the local minimum of K_min of superlattices [Fig. 4(b)], giving a tunability of K_lat, can be attributed to the increase in structural degrees of freedom of the scattering region rather than the correlation between the interference effect and particle boundary scattering.

The Bayesian optimization method was applied to the design of thermoelectric GNR shown in Fig. 5(a).⁵⁰ The thermoelectric figure of merit ZT is maximized by applying the Bayesian optimization: ZT = S²G_el/K_th, where S is the Seebeck coefficient, G_el is the electrical conductance, and K_th = K_lat + K_el is the thermal conductance with K_el being the electronic thermal conductance. While thermoelectric performance is a multifunction including S, G_el, and K_th, the efficiency of the Bayesian optimization did not significantly change from that for a single-function, namely, K_lat. For GNRs with the width of ≈1.7 nm, the optimization increases ZT of a zigzag-type GNR (ZGNR), Z_ZT, and an armchair-type GNR (AGNR), Z_AT, 11 and 2.7 times, respectively. Interestingly, Z_zT is approximately 20% higher than Z_AT for optimal structures although Z_ZT is 70% smaller than Z_AT for pristine structures. We can understand the effects of porous structure on the phonon transport properties of GNR basically with the same manner as the Si/Ge superlattice. Therefore, we here mainly focus on electron transport that may be more sensitive to nanostructures than phonon transport.

A detailed analysis of electronic transport properties of the optimal ZGNR gives an intriguing insight to increase ZT of GNRs by using edge states. Although ZGNRs originally have small ZT because of their small band gap, optimal arrangement of porous structures strongly suppresses electron transport at energies around the edge state in the nanostructured region as shown in Fig. 5(b). The right panel of Fig. 5(b) shows that while electron transmission function T_el(E) of resonance states near the edge state is not diminished for the periodically nanostructured system, T_el(E) is almost diminished around the edge state for the optimal structure despite the presence of states, e.g., finite DOS. The efficient suppression of T_el(E) of resonance states can be attributed to the non-uniform distribution of LDOS in the nanostructured region as shown in Fig. 5(c); while LDOS is distributed in the whole area of the periodic structure, LDOS partially disappeared in the optimal structure. The optimal structure, therefore, can significantly decrease T_el(E) around the edge state. Because step-like T_el(E) generally leads to a large ZT,⁵⁹ the optimization process allows increasing ZT of ZGNRs significantly by taking advantage of the edge states.

In this section, we show that MI gives a further designability of interface nanostructures and how aperiodicity improves target material properties. While the observation of phonon coherent effect may be challenging, it is expected that state-of-the-art fabrication techniques^66,67,69 enable a direct observation of the unique properties of aperiodic superlattices. Since MI for structural optimization is still at the initial stage, extensive progresses can be expected in different directions: (a) development of efficient descriptors and methods to determine them, (b) development of the optimization process for larger or smaller data such as Monte-Carlo tree research and transfer learning, (c) acceleration of time-consuming calculations of material properties by combining with informatics techniques, and (d) development of the platform of MI. For Bayesian optimization, while descriptors strongly correlate with target material properties and affect the optimization speed, descriptors are currently determined with intuitive or trial-and-error methods. Therefore, rigorous ways to determine a optimal descriptor are required. As for the optimization method, various optimization methods are being developed such as Monte-Carlo tree search⁷⁰ and transfer learning⁷¹ to compensate for disadvantages of the Bayesian optimization; Monte-Carlo tree search can treat finite candidates and transfer learning can help obtaining a prediction model from a small database by partially or initially adopting model parameters trained for related properties with a larger database. The use of these optimization methods should widely expand the range of application of MI. In addition, reduction of simulation time for material properties is also required. For example, while harmonic calculations with Green’ functions are powerful tools when a representative length of nanostructures is much smaller than MFPs of low-frequency phonons, anharmonic calculations, which are generally time-consuming, are also intriguing to study material properties in micro- or bulk-scale. For example, although NEMD simulations, including anharmonic effects of phonons, are widely used to calculate the thermal properties of both pure and nanostructured systems, it is difficult to implement NEMD simulations into an optimization process because of their computational cost. Conversely, informatics techniques may be able to find hidden features in initial MD results and predict target physical properties, which significantly reduces the computational time. The reduction of simulation time of anharmonic calculations, including the electron-phonon interaction as well as phonon-phonon interaction, enables us to combine an optimization process and different intriguing interfacial problems such as van der Waals interaction⁷² and in-plane thermal transport.⁷³ Another and also an important aspect of MI is the development of platform: database,^74–76 data repository,⁷⁷ and automation of series of analysis processes.⁷⁸ While, currently, there does not exist a phonon database with phonon anharmonicity, once we make a database, we can easily include anharmonic effects in optimization processes. In addition, because various unpredicted errors occur during an optimization process such as imaginary frequency of phonon dispersion, efforts against the automation of a process, including calculations, data check, and data gathering, are also important. As above, while there are still challenges to overcome, since MI is basically applicable to any physical properties, structures, and simulation methods, it can change conventional ways of materials development not only scientifically but also industrially.

In this section, we introduce recent attempts toward tuning of phonon wave nature with realistic nanostructures. Because tuning of phonon wave nature at room temperature typically requires a single-nanometer scale, at least less than ≈10 nm, it is important to propose easily fabricated structures for experiments. Figure 6(a) summarizes the previously proposed nanostructures to tune phonon wave nature: superlattices^13,62 and periodic porous thin films leading to zone-folding effect, which is a representative coherent effect, nanopillar and nanojunction structures^79–82 leading to phonon localization and hybridization around the nanostructured region, red-colored regions, nanoinclusions⁸³ leading to interference resonance, and nanoscale roughness or disorders leading to Anderson localization.^12,84,85 While observation of phonon localization may be challenging,⁸⁶ nanoscale materials such as GNRs^85,87 or optimization of aperiodicity with MI^49,50 could enable the observation. Despite many difficulties, experimental observation of coherent phonons using these nanostructures is indispensable for realizing advanced devices such as phonon transistors and thermal insulators.

Among phonon coherent effects induced by nanostructures, a zone-folding effect may be the most intriguing one because it modifies a phonon dispersion in a wide range of the frequencies and preserve a phonon dispersion. However, because phonons easily lose their phase information due to disorders such as surface or interface roughness and impurities,^69,88,89 observation of coherent effects of heat phonons at room temperature is still challenging. In nanostructures, phonons start to lose their coherent features in a higher frequency or shorter wavelength region. Zone-folding effects, thus, can be observed only at a low frequency as shown in a schematic phonon dispersion in Fig. 6(a). As a result, features of coherent phonons have been captured only at a low temperature (<10 K)⁹⁰ or at a low frequency (<1 THz)⁹¹ in the experiment.

Recently, fullerene-encapsulating carbon nanotubes have been identified to be a material that can overcome the problem. This material does not have an interface between unit cells like superlattices and the periodicity is approximately 1 nm, which is sufficiently less than phonon wavelengths. Kodama et al.⁹² have developed a fabrication technique of devices that enables the measurement of thermal conductivity of suspended nanomaterials. To make measurement of many samples possible for a statistical analysis, a Si substrate covered by a thin Si_xN_y film with narrow (500-800 nm width) trenches filled with silicon oxide is prepared. This substrate enables fabrication of suspended thermal conductivity measurement devices on target carbon nanotubes (CNTs) dispersed on the substrate. With the developed device, the thermal and electrical properties of single-walled CNTs (SWNTs) and carbon nanopeapods, simply called peapods hereafter, fullerenes encapsulated a single-walled CNT (SWNT), were measured. It was found that the fullerene encapsulation improves thermoelectric performance of CNT: two-fold reduction of κ_lat and two-fold enhancement of the Seebeck coefficient without a significant change in electrical conductivity σ_el, i.e., eight times increase in the figure of merit, ZT (=S²σ_el/κ_lat). Moreover, the peak temperature of temperature-dependent κ_lat shifts to lower temperature (decreases approximately 50 K), which indicates that encapsulated fullerenes do not act as impurities; because impurity scattering mainly reduces thermal conductivity in the low temperature regime and thus does not significantly shift the peak to lower temperature. Effects of the fullerene encapsulation may be attributed to the radial expansion of the outer SWNT due to the interaction between fullerenes and SWNT.^93,94 As for electronic properties, the increase in S is simply due to the increase in the band gap of CNTs due to the radial expansion.^92,95

To gain insight into the effects of fullerene encapsulation on the phonon properties of CNTs, spectral energy density (SED) of CNTs, (10, 10) SWNTs with a diameter of 1.36 nm, and C₆₀ at (10, 10) SWNT peapods, C₆₀ fullerenes encapsulated in a (10, 10) SWNT was analyzed. The optimized Tersoff potential⁹⁶ and 6-12 Lennard-Jones potential were used to represent intralayer and interlayer interactions, respectively. For Lennard-Jones potential, the energy scale of 2.4 meV and length scale of 5.5 Å were adapted to reproduce experimentally observed deformation of outer SWNT and inner C₆₀ fullerenes and decrease in κ_lat.⁹² 

Figure 6(b) shows an outlook of a peapod and SEDs of (10, 10) SWNT and peapods. SEDs in Fig. 6(b) show that the fullerene encapsulation leads to softening of longitudinal modes and zone-folding. At the Brillouin zone edge of k_za_lat/π = 1, the frequency of longitudinal modes decreases from 42 to 40 THz. The zone-folding effect can be clearly observed at SED of peapod at a low frequency [see the bottom right panel of Fig. 6(b)]; phonon dispersions with the fourth-periodicity of the one of SWNTs are generated due to the fullerene encapsulation. The zone-folding can be observed up to approximately 40 THz, whereas clear phonon dispersion can no longer be observed above 40 THz due to strong phonon scattering. Group velocity, therefore, decreases in the wide range of frequencies. At the frequency of 5-10 THz, significant flattening of long-wavelength phonons can be observed. Consequently, both softening of longitudinal modes and zone-folding decrease v_ph and, thus, κ_lat of CNT. As for the peak shift of κ_lat due to the fullerene encapsulation, the phonon softening partially contributes to the shift because the phonon softening leads to the shift of the temperature-dependent heat capacity profile to lower temperature. However, the change in heat capacity can qualitatively describe the half of the decrease in the peak temperature. The remaining cause may be attributed to the increase in Umklapp phonon scattering. The zone-folding effect, particularly flattening of phonon bands, should increase the phase space of Umklapp scattering. As a result, the shift of the peak temperature of κ_lat to lower temperature originates from the shift of the heat capacity profile to lower temperature and the enhancement of Umklapp scattering.

Because carbon nanopeapods do not have interfaces between primitive unit cells and their periodicity is sufficiently less than phonon wavelengths, carbon nanopeapods may be an ideal platform to study coherent phonons, zone-folding effect. In addition, because the novel strategy to induce a new periodicity, introduction of a periodic strain field, may be repeatable, the introduction of strain can be used for phononic devices such as a thermal switch. Tuning of phonon wave nature opens many possibilities of applications not only for reducing thermal conductivity.

In this review article, we introduced recent progresses towards ultimate impedance of thermal transport. While SiNC structures demonstrated that single-nanometer grains realize extremely low relaxation time due to boundary scattering,¹⁸ the following question has arisen: What is the optimal structure for low thermal conductivity? We, therefore, introduced MI techniques for structural optimization that provides further designability and unveils the best nanostructures.^49,50 While we have so far gained knowledge on material properties in a bottom-up manner by using relatively simple systems, MI allows one to directly analyze the properties of the best nanostructure. MI for structural optimization is at the initial stage, and we expect extensive and rapid progresses in near future. Finally, we introduced that utilization of strain and a soft interface can induce a single-nanometer periodicity leading to the zone-folding effect. The phonon strain-engineering provides more degrees of freedom to suppress thermal transport; it may realize far lower thermal conductivity than the minimum limit predicted based on phonon gas kinetics. In addition to theoretical predictions, experimental observation of phonon coherent nature is expected. Recent improvement of both first-principles techniques for phonon properties^97–101 and thermal measurement of nanomaterials^92,102 have deepened our knowledge on nanoscale thermal transport. In the next decade, accurate predictions and measurements of effects of nanostructures and interactions between phonons with other quasiparticles such as electrons and magnons^37,103 should become possible, which would further expand the possibilities of phonon engineering.

The studies reviewed in this paper were carried out in collaboration with Takafumi Oyake, Lei Feng, Takuma Shiga, Masayuki Isogawa, Yoshiaki Nakamura, Shenghong Ju, Zhufeng Hou, Koji Tsuda, Masaki Yamawaki, Takashi Kodama, and Kenneth E. Goodson. This work was partially supported by CREST “Scientific Innovation for Energy Harvesting Technology” (Grant No. JPMJCR16Q5) and Grant-in-Aid for Scientific Research (B) (Grant No. JP16H04274) from JSPS KAKENHI, Japan.