Abstract
Many lowthermalconductivity (κ_{L}) crystals show intriguing temperature (T) dependence of κ_{L}: κ_{L} ∝ T^{−1} (crystallike) at intermediate temperatures whereas weak Tdependence (glasslike) at high temperatures. It has been in debate whether thermal transport can still be described by phonons at the IoffeRegel limit. In this work, we propose that most phonons are still well defined for thermal transport, whereas they carry heat via dual channels: normal phonons described by the Boltzmann transport equation theory, and diffusonlike phonons described by the diffusion theory. Three physicsbased criteria are incorporated into firstprinciples calculations to judge modebymode between the two phonon channels. Case studies on La_{2}Zr_{2}O_{7} and Tl_{3}VSe_{4} show that normal phonons dominate low temperatures while diffusonlike phonons dominate high temperatures. Our present dualphonon theory enlightens the physics of hierarchical phonon transport as approaching the IoffeRegel limit and provides a numerical method that should be practically applicable to many materials with vibrational hierarchy.
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Introduction
Lowthermal conductivity (κ_{L}) crystals are of great interest in a variety of applications including thermal barrier coatings (TBC), thermoelectrics and nuclear reactors. They often show intriguing thermaltransport properties: κ_{L} decreases inversely with temperature (T) at intermediate temperatures as expected for crystals; but shows weak or even no distinct dependence on T at high temperatures, which is an anomalous, glasslike behavior^{1,2,3,4,5,6,7}. While the former can be explained within the scheme of standard phonon Boltzmann transport equation (BTE) by primarily considering threephonon scattering^{8}, the latter is still an open question. Advance in developing unified theories and numerical methods is just looming over the horizon to improve understanding the disparate κ_{L} ~ T relationship in these crystals.
Recent studies have identified that in lowκ_{L} crystals, some phonon modes have mean free path (l) shorter than the Ioffe–Regel limit^{9}, casting questions whether these modes can still be defined as phonons and how they contribute to thermal transport. Theoretical models of various sophistication have been developed to resolve this challenge^{10,11,12,13,14,15,16}. Agne et al.^{10} proposed that heat transfer in lowκ_{L} complex crystals could be reasonably described by assuming a media of diffusons according to the randomwalkbased diffusion theory, as opposed to phonons, and a model of diffusonmediated κ_{L} was proposed to better explain the experimental results particularly at high temperatures. The diffuson model by itself, however, does not fit to pure crystals where phonons are still welldefined as ensured by the periodicity of the lattice. On the other hand, other models have considered the hierarchy of vibrational modes based on the Ioffe–Regel limit. Chen et al.^{11} proposed that for weakly disordered crystals with complex unit cell (e.g., higher manganese silicides), κ_{L} could be explained by a hybrid phonon and diffuson model. The model employed a few fitting parameters and used the inelastic neutron scattering spectra to obtain an Ioffe–Regel crossoverfrequency of 20 meV, below which the vibrational modes were treated as phonons and above which were diffusons. Mukhopadhyay et al.^{12} used the scale of interatomic spacing as Ioffe–Regel criterion and proposed that for Tl_{3}VSe_{4} crystals, the phonon modes with mean free path l smaller than the Ioffe–Regel limit no longer behave as phonons, but should be replaced by hopping modes whose frequencies or eigenvectors cannot be meaningfully defined. They hence proposed a twochannel model that combines the phonon channel treated with BTE and the hopping channel calculated using Einstein’s model or Cahill’s model, yielding κ_{L} and its temperature dependence in better agreement with experimental data. However, some open questions still remain. Particularly, how do the welldefined phonons interact with the hopping modes? How to subtract the welldefined modes from the hopping channel in κ_{L} calculation? On the other hand, we note that infrared and Raman spectroscopies of some semiconductors have shown that, the zonecenter optical phonons, many of which would have extremely short or nearly zero l, still have welldefined frequencies and linewidths (scattering rates) that can be accurately predicted by firstprinciples calculations^{17,18}. This may suggest that the phonon concept for these modes is still valid, while the failure is with BTE which does not recognize the physical lowerlimit of l. We can also note that this is not the first time BTE fails for phonons; in fact, BTE was known to fail for coherent phonons in superlattices or phononic crystals; therein, it does not capture the wave effects^{19,20}.
Most recently, attempt has also been made to unify the thermaltransport theory in crystals and glasses. Simoncelli et al.^{13} have transmuted the BTE formulism into a κ_{L} equation written in terms of the phonon velocity operator with diagonal and offdiagonal elements describing the particlelike propagation of phonons and the wavelike tunneling of coherence, respectively. Applying this model in perovskite CsPbBr_{3} arrives at reasonable simulation of its glasslike κ_{L}. Meanwhile, Isaeva et al.^{14} developed a quasiharmonic Green–Kubo method, as if to generalize the Allen–Feldman model^{21,22,23,24} for amorphous systems into relaxationtimeapproximationbased BTE for crystals by expressing energy transport in a quantum mechanical fashion, and gives reasonable κ_{L} predictions for amorphous and crystalline Si. Nevertheless, further physical insights are expected to improve understanding the nature of hierarchical vibrations in the context of physically based theories.
Inspired by the idea of vibrational hierarchy from previous models, while attempting to resolve the open questions, in this work we sparkle a different concept that the vibrational modes with very short l could still be treated as phonons with welldefined frequencies, eigenvectors, and scattering rates, but their heat conduction should be described by the diffusion theory instead of BTE. We hence propose a dualphonon theory, by treating the shortl phonons with the diffusion theory, and other normal phonons with BTE. Our theory does not introduce a different type of heatcarrying modes other than phonons, and eliminates the theoretical challenge of how they would interact with phonons and alter their scattering events. Also, for a sophisticated model, we introduce three different criteria, based on phonon mean free path, wavelength, and thermal diffusivity, to judge modebymode between the normal phonon channel and diffusonlike phonon channel, and hence avoid the doublecounting issue. The three criteria all yield consistent results that agree quantitatively with experiments, demonstrating the robustness and predictive capability of our theory. Our theory is demonstrated on La_{2}Zr_{2}O_{7}, a thermalbarriercoating (TBC) candidate material for gas turbine technologies^{25}, and Tl_{3}VSe_{4}, a potential thermoelectric material^{12}. More background of La_{2}Zr_{2}O_{7} could be found in Supplementary Note 1. Our approach is able to explain the thermal conductivity and the intriguing κ_{L} ~ T dependence over the entire temperature range, and is expected to help understand the thermal transport of such lowκ_{L} crystals. Also, our model may provide physical insights toward unifying the theories of thermal conductivity in crystals and amorphous materials.
Results
Normal phonons and diffusonlike phonons
We first calculate the phonon properties of La_{2}Zr_{2}O_{7} using the standard anharmonic lattice dynamics based on density functional theory. Immediately we find that a large percentage of vibrational modes have very small l even at room temperature, and more so at high temperatures, as shown in Fig. 1a. Similar features have been identified for Tl_{3}VSe_{4} (see Supplementary Fig. 5). Apparently, these modes cannot be treated as normal phonons, which by definition, are expected to propagate far enough to sample the periodicity of the transport media, i.e., comparable to the scale of phonon wavelength or several lattice spacing^{21,22,23}. In developing our theory, these modes are treated as diffusonlike phonons.
In order to sophisticatedly judge whether a phonon mode should be treated as normal phonon or diffusonlike phonon, we propose three criteria, as conceptually shown in Fig. 2. The first two criteria are derived from the Ioffe–Regel limit of phonons^{9}, arguing that the value of l for normal phonons should not be smaller than their wavelength (λ) (I. l–λ criterion), or the minimum interatomic spacing (a_{min}) of the lattice (II. l–a_{min} criterion), respectively. The third criterion argues that a vibrational mode should be characterized as diffusonlike if its phonon thermal diffusivity (D_{Phon}) becomes smaller than its diffuson thermal diffusivity (D_{Diff}) (III. D_{Phon}–D_{Diff} criterion). Here we note that, another reasonable criterion to define a normal phonon mode is based on its longenough relaxation time (τ) as τ >> 1/ν (ν is the vibrational frequency). For acoustic phonons approaching the longwavelengthlimit, this criterion is the same with the l–λ criterion, whereas for some phonons whose phase velocity (v_{p}) is considerably higher than group velocity (v_{g}), it is less restrictive^{23}. Therefore, the τcriterion is not considered in developing our theory.
For modeling κ_{L} in materials with such strong vibrational hierarchy, we assume a thermaltransport media with both normal phonons and diffusonlike phonons. All these phonon modes are able to transfer heat (i.e., nonlocalized), but following BTE theory for the former and diffusion theory for the latter, respectively. Our dualphonon theory combines a normal phonon channel and a diffusonlike phonon channel:
Heat conduction from both channels are derived by following the physical picture that heat is carried by atomic vibrations of a solid. There are three components to be considered: (1) the number of vibrational modes that are available to carry heat, summed up to be the total degrees of vibrational freedom; (2) the amount of heat that can be carried by each vibrational mode; (3) the propagation behavior of vibrations through the dualphonon media. We write the contribution from each channel as:
Here, i and j denote the indices of vibrational modes, sampled over the Brillouin zone (BZ). N_{Phon} and N_{Diff} are the numbers of normal phonons and diffusonlike phonons, respectively. One of the merits of our theory is that, we impose the conservation of vibrational degrees of freedom by requiring N_{Phon} + N_{Diff} = 3N, where N is the total number of atoms. As such, once a mode is characterized as a normal phonon, it cannot be treated as a diffusonlike phonon at the same time, and vice versa. C_{s} is the permode specific heat following the Bose–Einstein statistics of phonons. D_{Phon} and D_{Diff} are defined as the permode thermal diffusivities for normal phonons and diffusonlike phonons, respectively. In our present model, D_{Phon} is calculated using the phonon BTE theory; whereas D_{Diff} could be calculated based on the randomwalk theory^{10} (denoted as \(D_{{\mathrm{Diff}}}^{{\mathrm{RW}}}\)) or the Allen–Feldman theory^{23} (denoted as \(D_{{\mathrm{Diff}}}^{{\mathrm{AF}}}\)). Details are presented in the Methods section. In the present work, we only performed limited test for our dualphonon theory coupled with \(D_{{\mathrm{Diff}}}^{{\mathrm{AF}}}\) (using the III. D_{Phon}–D_{Diff} criterion), mainly due to its high computational expense.
Hierarchy of lattice vibration in La_{2}Zr_{2}O_{7}
The hierarchy of lattice vibrations for La_{2}Zr_{2}O_{7} is shown in Fig. 1. We see that the vibrational modes that are judged as diffusonlike phonons account for 75.35%, 23.82%, 65.42%, and 69.73% of the total number of modes at T = 300 K, respectively, according to our proposed judging criteria I, II, and III coupled with the randomwalk theory, and III coupled with the Allen–Feldman theory. The fractions rise to 98.00%, 80.23%, 91.25%, and 93.55% at T = 1500 K. As illustrated in Fig. 3, this behavior originates either from small v_{g}, especially for highfrequency vibrations, or from small τ, indicative of intense scattering among those modes. The increased population of diffusonlike phonons at higher temperatures is presumably due to increased phonon scattering.
To better visualize the degree of the vibrational hierarchy, the vibrational density of states (DOS) for normal phonons vs. diffusonlike phonons are illustrated in Fig. 1b, d, f, h. There is a crossover from normal phonondominated states at the lowfrequency range to diffusonlike phonondominated states at highν range, and the crossover shifts to lowerν at higher temperatures. Take the case of II. l–a_{min} Criterion as an example. At T = 300 K, normal phonons dominate below ν ~ 13 THz; whereas for T = 1500 K, diffusonlike modes exhibit the first peak at ν ~ 2 THz, and quickly becomes dominant at above ν ~ 5 THz. This could be understood from increased scattering at higher temperatures, resulting in suppressed τ and l values for all vibrational modes. Furthermore, detailed comparisons among the three criteria show that, the number of vibrational modes assigned as diffusonlike phonons based on the randomwalk theory increases in the order of II. l–a_{min} Criterion, III. D_{Phon}–D_{Diff} Criterion, and I. l–λ Criterion, and the major difference comes from assignment of the vibrational modes in the frequency range 2 < ν < 13 THz. On the one hand, the fact that a larger number of vibrations are assigned as diffusonlike phonons by Criterion III than Criterion II could be understood from the interplay of low v_{g} · l and high ν for certain modes. On the other hand, the difference between Criterion I and Criterion II is expected to be more pronounced for zonecenter longwavelength vibrations, as certain modes have a_{min} < l < λ hence they are assigned as normal phonons by the l–a_{min} Criterion while as diffusonlike phonons by the l–λ Criterion. Moreover, as shown in Fig. 1f, the ratio N_{Diff}/N_{Phon} appears to increase with the vibrational frequency under Criterion III coupled with the randomwalk theory, but the trend is not so clear for the results derived from the Allen–Feldman theory (Fig. 1h). Such difference could be understood from the different frequency dependence of \(D_{{\mathrm{Diff}}}^{{\mathrm{RW}}}\) vs. \(D_{{\mathrm{Diff}}}^{{\mathrm{AF}}}\). The randomwalk theory assumes that heat is transferred through successful random jumps of independent oscillators within a period of oscillation to a distance related with the number density of atoms, and thus arrives at \(D_{{\mathrm{Diff}}}^{{\mathrm{RW}}} \propto \nu\) throughout the spectrum. The Allen–Feldman theory, on the other hand, is rooted in the thermal correlation of each vibrational mode and arrives at varied frequency dependence of \(D_{{\mathrm{Diff}}}^{{\mathrm{AF}}}\). See Supplementary Note 2 and Supplementary Fig. 1 for more details.
Lattice thermal conductivity
The κ_{L} values calculated from an iterative solution of BTE coupled with anharmonic lattice dynamics are shown in Fig. 4a. Both phonon–phonon scattering and isotope scattering are considered in our calculations, but the latter has little impact on κ_{L} values and their temperature dependence. At room temperature, the calculated κ_{L} shows reasonable agreement with experimental data^{26,27,28}. At higher temperatures, it yields κ_{L} ~ T^{−1} dependence as expected from the dominance of phonon–phonon Umklapp scattering, whereas experimental data goes nearly temperatureindependent. The calculated κ_{L} = 0.55 W·m^{−1}·K^{−1} at T = 1500 K is only onethird of the measured value κ_{L} = 1.5 W·m^{−1}·K^{−1}, and is even lower than the reported hightemperature limit (κ_{min} = 1.2 W·m^{−1}·K^{−1}) for La_{2}Zr_{2}O_{7} crystals^{29,30}. Moreover, we notice that including the stateoftheart fourphonon scattering^{31,32,33} is expected to give further reduction of κ_{L} at high temperature and results in strongerthanT^{−1} temperature dependence, and thus could not bridge the gap between experimental and BTEderived results in our case. Guided by the Ioffe–Regel limit of phonons^{9}, we recalculate κ_{L} within the framework of phonon BTE theory, by assuming a lowerbound of l for all modes, i.e., the l values are manually set to a_{min} if l < a_{min}. This trial yields κ_{L} = 0.61 W·m^{−1}·K^{−1} at T = 1500 K, showing inadequate correction to the previous results. Such significant gap between the calculated temperaturedependent κ_{L} and experimental values indicate that the conventional BTE theory becomes invalid for heat conduction in La_{2}Zr_{2}O_{7}, and a different physical picture is needed to describe the transport behavior of the interesting smalll or smallD_{Phon} phonon modes.
Next, we calculate κ_{L} of La_{2}Zr_{2}O_{7} using the twochannel model proposed by Mukhopadhyay et al.^{12}, by combining a phonon conduction channel (κ_{phonon}) and a hopping channel (κ_{hop}) from modes of l < a_{min}. In this model, κ_{phonon} is calculated from phonon BTE theory, by excluding the contribution from smalll vibrations; whereas κ_{hop} is calculated either via Cahill’s formula^{34,35} or Einstein’s formula from which the Cahill’s model is derived. Cahill’s formula requires v_{g} of acoustic phonons as input parameters, which could be extracted from phonon dispersions of La_{2}Zr_{2}O_{7} (shown in Supplementary Fig. 2): 3837 and 3969 m·s^{−1} for transverse acoustic phonons (TA1 and TA2), and 6872 m·s^{−1} for longitudinal acoustic phonons (LA). Einstein’s formula requires defining the Einstein temperature (θ_{E}) for the oscillators, which could be estimated from the calculated lowtemperature specific heat of La_{2}Zr_{2}O_{7} (shown in Supplementary Fig. 3): θ_{E} = 180 K, in moderate agreement with reported values^{36}. Using these parameters as input, the total κ_{L} (κ_{phonon} + κ_{hop}) are calculated to be 1.90 and 0.92 W·m^{−1}·K^{−1} at T = 1500 K by employing Cahill’s formula and Einstein’s formula, respectively. The corresponding temperature dependencies are κ_{L} ~ T^{−0.31} and κ_{L} ~ T^{−0.65} above T = 1000 K. Now the experimental values of κ_{L} fall within the range provided by the twochannel model, and the temperature dependence shows improved agreement. As shown in Fig. 4a, the range in between might be originated from uncertainties in defining the θ_{E} values, which has also been pointed out in ref. ^{12}. Besides, we note that in order to preserve the total number of vibrational modes, the normal phonon modes need to be subtracted from the hopping channel, which is difficult to do when incorporating the Cahill’s formula in the twochannel model.
The results of our dualphonon theory for La_{2}Zr_{2}O_{7} are shown in Fig. 4a. Our model coupled with the randomwalk theory gives weakened κ_{L} ~ T dependence as T increases: κ_{L} ~ T relationship calculated using I. l − λ criterion (II. l–a_{min} criterion; III. D_{Phon}–D_{Diff} criterion) weakens from κ_{L} ~ T^{−0.69} (κ_{L} ~ T^{−0.72}; κ_{L} ~ T^{−0.67}) for 300 K < T < 500 K, to κ_{L} ~ T^{−0.47} (κ_{L} ~ T^{−0.40}; κ_{L} ~ T^{−0.47}) for 500 K < T < 1000 K, and to κ_{L} ~ T^{0.29} (κ_{L} ~ T^{−0.24}; κ_{L} ~ T^{−0.30}) for T > 1000 K, reasonably reproducing the flatteningout behaviors of the experimental κ_{L} ~ T data. Partial contributions from normal phonons (\(\kappa _{\mathrm{L}}^{{\mathrm{Phon}}}\)) and diffusonlike phonons (\(\kappa _{\mathrm{L}}^{{\mathrm{Diff}}}\)) to the total κ_{L} are plotted in Fig. 4b. It is shown that, \(\kappa _{\mathrm{L}}^{{\mathrm{Diff}}}\) starts to dominate over \(\kappa _{\mathrm{L}}^{{\mathrm{Phon}}}\) at around T = 700, 1000, and 800 K based on Criterion I, II, and III, respectively. Clearly, the role of diffusonlike phonons for the heat conduction of La_{2}Zr_{2}O_{7} is more significant at hightemperature ranges. Besides, the calculated \(\kappa _{\mathrm{L}}^{{\mathrm{Phon}}}\) decreases in the order of criteria II, III, and then I, while the trend is reversed for \(\kappa _{\mathrm{L}}^{{\mathrm{Diff}}}\). This result is consistent with the percentage of diffusonlike phonons assigned out of all the vibrational modes (inset Fig. 4b). Evidently, despite using three different criteria to judge normal phonons vs. diffusonlike phonons, the calculated κ_{L} values as well as the κ_{L} ~ T dependence show quite similar features, demonstrating the robustness of our approach. Moreover, our dualphonon theory is also validated on Tl_{3}VSe_{4}, a potential thermoelectric material with ultralow κ_{L} that was used as the model materials in ref. ^{12}. See Supplementary Note 3, Supplementary Figs. 4–7, and Supplementary Table 1 for detailed results and discussions.
Also shown in Fig. 4 are the results from our dualphonon theory coupled with the Allen–Feldman model. The percentage of diffusonlike phonons assigned out of all the vibrational modes using the III. D_{Phon}–D_{Diff} criterion coupled with \(D_{{\mathrm{Diff}}}^{{\mathrm{AF}}}\) is generally consistent with the results derived with \(D_{{\mathrm{Diff}}}^{{\mathrm{RW}}}\), and the calculated values of \(\kappa _{\mathrm{L}}^{{\mathrm{Phon}}}\) from the two routes show minor differences. Moreover, the κ_{L} ~ T dependence calculated with \(D_{{\mathrm{Diff}}}^{{\mathrm{AF}}}\) gives κ_{L} ~ T^{−0.61} for 300 K < T < 500 K, κ_{L} ~ T^{−0.42} for 500 K < T < 1000 K, and κ_{L} ~ T^{−0.27} for T > 1000 K, consistent with that derived using \(D_{{\mathrm{Diff}}}^{{\mathrm{RW}}}\). These results demonstrate the robustness of the dualphonon theory. Evidently, imposing the dualphonon theory coupled with Allen–Feldman model tends to yield higher \(\kappa _{\mathrm{L}}^{{\mathrm{Diff}}}\), and thus slightly higher values of total κ_{L}, as compared with those derived with the randomwalk picture. This is actually inherited from the different frequencydependent formulas of \(D_{{\mathrm{Diff}}}^{{\mathrm{RW}}}\) vs. \(D_{{\mathrm{Diff}}}^{{\mathrm{AF}}}\) as we have discussed above. Interestingly, at least for La_{2}Zr_{2}O_{7}, the randomwalkbased thermal diffusivities in our dualphonon theory perform better than the Allen–Feldman theory. Meanwhile, this route requires far less computational expense, which is potentially favorable for highthroughput κ_{L} predictions. Nevertheless, more works are required in the future to test this issue using a larger material pool.
Discussion
We can see that our dualphonon theory emphasizes the physics of hierarchical phonon transport in lowκ_{L} materials, by treating all vibrational modes within the phonon picture yet with different thermaltransport behaviors, i.e., normal phonons with the BTE theory and diffusonlike phonons with the diffusion theory. Basically, our dualphonon theory proposes a conceptual change that the vibrational modes are still phonons upon approaching the Ioffe–Regel limit, and the phonon frequency, eigenvector, and relaxation time could be rigorously described by first principles. Specifically, the identified diffusonlike phonons with small l or small D_{Phon} still fall in the physical picture of phonons with welldefined scattering rates that can be reliably predicted from first principles; just that their heat conduction cannot be treated using the scheme of mean free path or BTE. In this way, the scattering and thermal transport of diffusonlike phonons are decoupled. The three proposed physicsbased judging criteria, without relying on any fitting parameters, enable a permodebased judging between normal phonon and diffusonlike phonon channels, and give consistent κ_{L} predictions in agreement with experimental data, demonstrating the robustness of our approach. Moreover, while this paper was under review, we learned that progressive efforts have been made to unify thermal transport in crystals and amorphous materials, which involve sophisticated analytical and mathematical derivations, yet the corresponding physical insights are still under development^{13,14}. In parallel with these efforts, our dualphonon theory provides physical insights in understanding the vibrational hierarchy of crystals having low and glasslike κ_{L}, and leads up to a κ_{L}prediction method with good robustness and favorable computational expense.
It would be interesting to explore future opportunities to expand and advance our proposed dualphonon theory. First, the current study does not include the effects of fourphonon scattering or phonon renormalization, which are important effects addressed elsewhere but are beyond the scope of the present study, partly due to the very high computational cost for La_{2}Zr_{2}O_{7} (with complex and large unit cell). Fourphonon scattering is important at high temperature for all materials and even at room temperature for strongly anharmonic materials and materials with exceptionally weak threephonon scattering^{31,32,33}. Phonon renormalization is important for strongly anharmonic materials, especially those exhibiting dynamic instability^{37,38,39,40}. Indeed, consideration of fourphonon effect and the temperatureeffect might arrive at more accurate simulation of the phonon group velocities and scattering, leading to better placement of normal phonons vs. diffusonlike phonons, and more accurate κ_{L} and its temperature dependence.
Second, the primary scope of our dualphonon theory is to differentiate the thermaltransport behaviors of normal phonons against diffusonlike phonons in crystals based on whether or not they are illdefined in the space scale (in terms of small l or D_{Phon}), given that they fall within the phonon picture in terms of scattering. Interestingly, we find that some vibrational modes in La_{2}Zr_{2}O_{7} and Tl_{3}VSe_{4} exhibit 1/τ > ν; in other words, their lifetime is too short and thus should be considered illdefined in the time scale. Similar behavior has been universally observed in crystals having certain types of perturbation or disorder^{12,13,41,42,43,44}. As shown in Fig. 5a, c, the number of modes that are illdefined in the time scale (1/τ > ν) account for 0.21% for La_{2}Zr_{2}O_{7} and 11.58% for Tl_{3}VSe_{4} at T = 300 K, which are much smaller than the percentages of the illdefinedinspace phonons. They rise to 45.36% at T = 1500 K for La_{2}Zr_{2}O_{7} and 59.71% at T = 500 K for Tl_{3}VSe_{4}, respectively, though. We then further evaluate the effects of these modes on the scattering rates of phonons that are welldefined in the time scale (1/τ < ν), by calculating the percentages of scattering rates of welldefinedintime modes contributed by threephonon scattering processes involving at least one illdefinedintime mode. Figure 5b, d shows that the percentages are mostly below 20% for La_{2}Zr_{2}O_{7} and below 60% for Tl_{3}VSe_{4} at T = 300 K, which rise to mostly above 70% at T = 1500 K for La_{2}Zr_{2}O_{7}, and above 90% at T = 500 K for Tl_{3}VSe_{4}. Therefore, although our approach makes a step closer to a sound theory and appears to agree with experiments, whether we can treat scattering by these illdefinedintime phonons with standard anharmonic lattice dynamics still remains a challenging open question, especially at high temperatures. Future studies are warranted on how to better understand the interactions between the welldefinedintime phonons and illdefinedintime phonons in our model and other lately developed models^{13,14}.
The present results have important implications for a wide variety of lowκ_{L} materials, including thermal barrier coatings (TBC), thermoelectrics, and nuclear materials. For these lowκ_{L} materials, there might be large number of diffusonlike phonons characterized by small l or small D_{Phon}, and the significant contribution from \(\kappa _{\mathrm{L}}^{{\mathrm{Diff}}}\) to total κ_{L} manifest with increased temperature, and eventually dominate over the contribution from \(\kappa _{\mathrm{L}}^{{\mathrm{Phon}}}\) when approaching the hightemperature limit. In the context of our analysis on the TBC material La_{2}Zr_{2}O_{7}, the diffusonlike phonons mainly stem from low v_{g} and/or high scattering rates, which could be further linked with the complexity of crystal structure and the heterogeneity of interatomic bonds. Such structural characteristics result in foldingin of phonon dispersion, suppression of phonon frequencies and thus group velocities, and serious tangling and scatterings among lowfrequency acoustic and optical phonons (see Supplementary Note 4 and Supplementary Fig. 8 for details). This mechanism is expected to be a signature for many TBC candidates, i.e., oxide ceramics with complex crystal structure and vibrational hierarchy, probably including other RE_{2}Zr_{2}O_{7} pyrochlores^{25}, silicates^{45,46}, and acuminatesilicates^{47} etc. In this sense, our proposed dualphonon model is expected to inspire deeper understanding and practical calculation methodology of κ_{L} for TBC materials (see Supplementary Note 5 for details).
In summary, a dualphonon theory is proposed for the κ_{L} of crystals with vibrational hierarchy, by considering normal phonons described by the BTE theory and diffusonlike phonons described by the diffusion theory. Three physicsbased criteria are used to judge modebymode between normal phonons and diffusonlike phonons. Applying this theory on La_{2}Zr_{2}O_{7} and Tl_{3}VSe_{4} shows that κ_{L} is mainly contributed by normal phonons at low temperatures, whereas by diffusonlike phonons at high temperatures. This theory successfully predicts the flatteningout of κ_{L} ~ T trend upon temperature increment, in much better agreement with experiments than the conventional BTE theory. Meanwhile, it resolves the limitations of other existing models and leads to a computational procedure showing promising applicability. The improvement of heat conduction theory in lowκ_{L} crystals will provide important insights in the development of TBC materials, thermoelectric materials, and nuclear materials. Future studies can include a comparison of our approach with those in refs. ^{13,14}, a consideration of how to combine the mathematical rigor of those approaches and the physical insights from our approach and other heuristic models, and a better understanding of the interactions between the welldefinedintime phonons and illdefinedintime phonons.
Methods
Calculation details of the dualphonon theory
In this study, the permode specific heat is calculated by following the Bose–Einstein statistics for phonon:
where k_{B} is the Boltzmann constant; ћ is the reduced Planck constant; T is the absolute temperature; ω is the angular frequency of a vibrational mode; V is the volume of simulated unit cell.
D_{Phon} is calculated using the phonon BTE theory, by incorporating the phonon group velocity (v_{g}), mean free path (l), and relaxation time (τ) resulted from the scattering process. D_{Diff} could be calculated by following the randomwalk scheme proposed in Agne et al.’s model^{10}, which assumes that heat is transferred by discrete jumps between independent harmonic oscillators. Its linear dependence with respect to ω could be understood from the diffusive picture, as if oscillators jumping at more steps per timeinterval contribute to higher rates of energy transfer. Basically, the diffusion term D_{Diff} has the units of D_{Phon} (m^{2}·s^{−1}), while avoiding independent definition of v_{g} and l in the diffuson picture.
where n is the number density of atoms of the unit cell. In this study, moderesolved vibrational parameters (v_{g}, l, λ, ω, τ, etc.) are gathered from outputs of DFTbased harmonic and anharmonic lattice dynamics calculations. Scattering information (τ and l) is gathered from iterative solution of the BTE; however, we tested that using parameters derived under the relaxationtime approximation (RTA) would have negligible influence on the present results.
For comparison, D_{Diff} could also be calculated using the Allen–Feldman formula^{23}, which is grounded in the Green–Kubo theory and is at the reach of ab initio simulations.
where S_{jp} is the heatcurrent operator measuring the thermal coupling between vibrational mode j and p based on their frequencies and spatial overlap of eigenvectors, and could be calculated from harmonic lattice dynamics. δ is the Dirac Delta function that could be approximated using Lorentzian broadening of width greater than the average mode frequency interval (Δ_{avg}).
where the wave vector K is summed over the Brillouin zone; e is the corresponding eigenvector; α and β denote the Cartesian directions. R_{s} is the distance between each unit cell (labeled s) and the basis unit cell (labeled 0) within a periodic supercell system; and R_{kk′} is the distance between atom k and atom k′ within a unit cell. \(D_{\beta \alpha }^{k^{\prime} k}\left( {0,s} \right)\) is the dynamical matrix element, derived from the secondorder interatomic force constant (\({\mathrm{\Phi }}_{\beta \alpha }^{k^{\prime} k}\left( {0,s} \right)\)).
where m_{k} and m_{k′} are the masses of the atom k and k′.
Computation details for first principles
Density functional theory (DFT) calculations on La_{2}Zr_{2}O_{7} are performed using the projected augmented wave (PAW) method^{48} as implemented in Vienna Ab Initio Simulation Package (VASP)^{49} with electronic exchange and correlations treated in the localized density approximation (LDA)^{50}. The wave functions are expanded in a planewave basis with the kinetic energy cutoff of 600 eV, and Monkhorst–Pack^{51} Γcentered kmesh of 7 × 7 × 7 is used to sample the Brillouin zone (BZ). Cell parameters and internal atomic positions are fully relaxed until the total energy and maximum ionic Hellmann–Feynman forces converge to 1 × 10^{−10} eV and 1 × 10^{−4} eV/Å, respectively. Lattice parameters of La_{2}Zr_{2}O_{7} (in cubic symmetry, space group \(Fd\bar 3m\)) are optimized to be 10.66 Å, in reasonable agreement with experimental data of 10.78 Å^{52}. Computational details for Tl_{3}VSe_{4} are presented in Supplementary Note 6.
The harmonic and anharmonic interatomic force constants (IFCs) are calculated via the realspace finite displacement difference method, where 2 × 2 × 2 supercells containing 88 atoms are constructed, and Monkhorst–Pack kmesh are set as 3 × 3 × 3. The phonon frequencies and eigenvectors are obtained using the Phonopy^{53} package, by diagonalizing the dynamical matrix constructed from the harmonic IFC matrices, and sampling on a 21 × 21 × 21 qmesh. These are typical settings for the lattice dynamics calculations of rareearth pyrochlore systems^{25}. The anharmonic IFCs are obtained using the thirdorder scripts^{54}. Interatomic interactions up to the twelfth nearest neighbors (12th NN) are taken into account, corresponding to a cutoff radius (r^{cutoff}) of 7.72 Å; whereas interactions beyond this range are taken to be zero. In fact, increasing the r^{cutoff} from the 5th NN (corresponding to r^{cutoff} = 5.19 Å) to the 12th NN could sufficiently converge the roomtemperature κ_{L} within 7%. It is noteworthy that La_{2}Zr_{2}O_{7} is a complex oxide ceramic with longrange interatomic interactions^{55}, which might lead to strong dependence of anharmonic IFCs and κ_{L} on the number of neighbor shells. Qin et al.^{56} reported that for such material systems, the extent to an adequate inclusion of longrange effect could be estimated by looking into how the interaction strength changes with increased distance between an atomic pair, based on analyzing the root mean square (RMS) of the elements of the harmonic IFC tensor (Frobenius norm):
where ϕ_{mn} is the harmonic IFC between atom m and n; and \({\phi _{mn}^{\alpha \beta }}\) is the harmonic response of the force for atom m on the αdirection resulted from the displacement of atom n on the βdirection. Following this approach, the RMS(ϕ_{mn}) for all atomic pairs are analyzed as a function of the interatomic distance. It shows that for La_{2}Zr_{2}O_{7}, setting the truncation at the 12th NN is expected to include most strong interatomic interactions up to RMS(ϕ_{mn}) = 2, and beyond this range it decays below RMS(ϕ_{mn}) < 0.15. For these reasons, we chose to include up to the 12th NN interatomic interactions in the present calculations to pursue higher precision.
The lattice thermal conductivity (\(\kappa _{\mathrm{L}}^{{\mathrm{Phon}}}\)) is calculated by the iterative solution of the BTE as implemented in the ShengBTE^{54} package, with integrations using a 16 × 16 × 16 qmesh. The convergence of κ_{L} with respect to the size of qmesh is tested, and the results show that increasing the qmesh up to 25 × 25 × 25 yields less than 2% difference to the roomtemperature κ_{L} of La_{2}Zr_{2}O_{7}. Furthermore, nonanalytical corrections are applied to the dynamical matrix to take into account longrange electrostatic interactions, based on calculations of Born effective charges (Z*) and dielectric constants (ε) via density functional perturbation theory (DFPT)^{57}.
The Allen–Feldman calculations are performed with the same 2 × 2 × 2 supercell and 16 × 16 × 16 qmesh, to ensure a consistent comparison between D_{Phon} vs. \(D_{{\mathrm{Diff}}}^{{\mathrm{AF}}}\) on a permode basis. The Delta function in Eq. (8) is broadened into the Lorentzian form \(\frac{{\eta /\pi }}{{\left( {\omega \left( j \right)  \omega \left( p \right)} \right)^2 + \eta ^2}}\). After convergence tests (details in Supplementary Note 7 and Supplementary Fig. 9), the Lorentzian broadening factor η is set to be 3.3Δ_{avg} in our study, where Δ_{avg} is the average mode frequency interval (Δ_{avg} = 0.35 THz for La_{2}Zr_{2}O_{7}). Due to the high computational expense, we use permode thermal diffusivities on 1000 out of the 4096 wave vectors for our dualphonon theory under the Criterion III coupled with Allen–Feldman formula. A convergence test shows that including up to 2000 wave vectors causes only marginal difference (<1%) on the calculated fraction of diffusonlike phonons (N_{Diff}/N_{Total}) and the final κ_{L} values.
Code availability
Vienna Ab Initio Simulation Package (VASP) is available at www.vasp.at; Phonopy package is available at https://phonopy.github.io/phonopy; ShengBTE code is available at https://bitbucket.org/sousaw/shengbte; thirdorder scripts are available at https://bitbucket.org/sousaw/thirdorder. The custom codes used in this work are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge fruitful discussions with Dr. Lucas Lindsay at the Oak Ridge National Laboratory. Y.L. and X.Y. acknowledge the support from the China Scholarship Council. Y.L. acknowledges support from the research project funded by Shenyang National Laboratory for Materials Science. T.F. acknowledges support from the project entitled “Models to Evaluate and Guide the Development of Low Thermal Conductivity Materials for Building Envelopes” funded by Building Technologies Office (BTO), Office of Energy Efficiency & Renewable Energy (EERE) at the Department of Energy (DOE). X.R. acknowledges the partial support from the Defense Advanced Research Projects Agency (Award No. HR00111520037).
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X.R. and Y.L. conceived and designed the research. X.R. supervised the research. Y.L. performed the DFT calculations, analyzed the data, prepared the figures and manuscript, with assistance and discussions from X.Y., T.F., and J.W. X.R. and T.F. revised the paper. All authors discussed the results and contributed in writing the paper.
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Luo, Y., Yang, X., Feng, T. et al. Vibrational hierarchy leads to dualphonon transport in low thermal conductivity crystals. Nat Commun 11, 2554 (2020). https://doi.org/10.1038/s4146702016371w
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DOI: https://doi.org/10.1038/s4146702016371w
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