## INTRODUCTION

Amorphous materials play an essential role in modern semiconductor devices (*1*), such as photoelectric conversion layers for solar cells, phase change memory, thin-film transistors for displays, microelectromechanical systems, thermoelectric devices, and gate dielectrics and interlayer dielectrics for complementary metal-oxide semiconductor technology. The behavior of these devices depends strongly on the operation temperatures, which makes thermal management an important issue that needs to be tackled to ensure their performance and reliability. Therefore, understanding the thermal transport properties of amorphous solids is extremely important to optimize the thermal design of microelectronic devices. In particular, as the size of the device components scales down to the nanometer order, this aspect becomes more important given that unusual thermal transport properties have been identified in crystalline solids with nanometer feature sizes.

However, heat transport in amorphous solids is much more complicated than that of the crystalline solids despite the fact that heat is similarly carried by atomic vibrations. Heat conduction in crystalline solids with complete periodicity is well understood in terms of mode-dependent phonon transport properties. The thermal conductivity κ of crystalline solids obtained by computational work has thus shown good agreement with measurements (*2*, *3*). Theoretical interpretation is performed on the basis of the phonon Boltzmann transport equation, which requires phonons to have well-defined group velocities. However, for the amorphous solids, only a small portion of vibrational modes with low frequencies are considered to have well-defined group velocity. Seminal work by Allen and Feldman described that the vibrational modes in amorphous materials can be classified into three categories, namely, propagons, diffusons, and locons (*4*–*6*). The nonlocalized vibration modes at low frequencies that exhibit wave-like features are called propagons. On the other hand, a large amount of density of states is occupied by the nonlocalized vibrational modes at higher frequencies called diffusons, which conduct heat in a rather diffusive manner. Allen and Feldman expressed κ as a function of mode “diffusivity” to describe diffuson transport (AF theory) (*4*, *5*). Propagons and diffusons are nonlocalized modes contributing to the heat transport, whereas locons are localized vibration mode, which does not contribute to κ.

Following the AF theory, several studies have focused on the properties of vibration modes in amorphous solids (*1*, *7*), for example, the definition of threshold frequency between the three vibration modes, the contribution of each vibration modes to the total κ of solids, and their size effects. Several criteria have been proposed to classify propagons and diffusons, such as by the vibrational mode density of states (*6*, *8*), the eigenvector periodicity (*9*), and the dynamical structure factor intensity (*10*). Theoretical studies showed that propagon contributes ~40% of the total κ of amorphous silicon and that their mean free path (MFP) extends up to 1 μm (*8*, *11*). Experimental studies have revealed that both the cross-plane κ (*12*, *13*) and the in-plane κ of amorphous films depend on the feature size of the material (*14*–*16*). These experimental studies supported that propagons have relatively long MFPs up to 1 μm and contribute up to 50% of total κ. It is also noteworthy that the propagons showed a ballistic transport feature in short distances (*16*), which is similar to phonons in crystalline solids.

The previous studies thus indicate that the thermal transport properties of amorphous solids can be controlled by fine nanostructuring in analogy with those of crystalline materials, which has been demonstrated in the form of nanowires (*17*, *18*), superlattices (*19*, *20*), and holey phononic crystals (PnCs) (*21*–*23*). Similar approaches used in “phonon engineering” (*24*) may be effective in the manipulation of thermal transport properties of amorphous materials as well. However, we still lack a quantitative understanding of nanoscale thermal transport of amorphous materials. Systematic measurements of samples with a wide variety of feature sizes, together with supportive theoretical simulation mimicking the approaches in phonon engineering, will be indispensable to fully leverage the phonon engineering techniques developed in crystalline materials.

In this study, we experimentally and theoretically investigated the thermal transport properties of amorphous silicon nitride (a-Si_{3}N_{4}) with ultrafine phononic nanostructures. The samples examined here are suspended a-Si_{3}N_{4} thin films with holey PnC structures, which have a periodic two-dimensional array of through-holes aligned at various pitch sizes ranging widely from several tens of nanometers to micrometer order. The wide variety of sample feature sizes investigated here allowed us to clearly identify the effect of nanostructuring on the thermal transport properties of a-Si_{3}N_{4}. The effect of boundary scattering on propagons and diffusons was quantitatively analyzed by a modified simulation model based on a Monte Carlo ray tracing (MCRT) (*25*) method, which was developed previously for describing the phonon transport mechanism in crystalline solids. We found that the theoretical calculation reproduces the measured κ of a-Si_{3}N_{4} well throughout the samples investigated here. The current result not only deepens our understanding of nanoscale thermal transport of amorphous solids but also demonstrates that the approaches developed in phonon engineering are highly applicable to propagon and diffuson engineering as well.

## RESULTS AND DISCUSSION

Figure 1 summarizes the scanning electron microscopy (SEM) images of our samples. a-Si_{3}N_{4} thin films were deposited on Si substrates by using a low-pressure chemical vapor deposition (LPCVD) method. Periodic through-holes with pitch *P* sizes from 60 to 1600 nm were patterned on the a-Si_{3}N_{4} films by electron beam lithography (Fig. 1, C to E), whereas those with *P* of 36 nm were fabricated by directed self-assembly lithography (Fig. 1B) (*26*). The holes of the PnC structures are aligned in a triangular pattern (Fig. 1). Here, we define the minimum neck width *n* of PnC structures as *n* = *P* − *D*, where *D* is the diameter of the holes. Note that *n* ranges from 11 to 670 nm. Detailed geometries of the PnC structures are shown in Table 1. The a-Si_{3}N_{4} films are partially suspended from the base substrate so as to avoid the influence of the substrate on the measured κ (Fig. 1A). The length, width, and thickness of the suspended bridge are 30 μm, 10 μm, and 70 nm, respectively. Al pads with thickness of 130 nm are deposited on the center and the edge of the suspended bridge structure for the thermal transport measurements. Detailed information on experimental fabrication processes is discussed in the “Fabrication of PnCs” section and in our recent report (*27*).

The thermal transport property of a-Si_{3}N_{4} PnCs was measured by a time-domain thermoreflectance (TDTR) method (*23*), which is a well-established method based on a pump-probe optical measurement. In our TDTR setup, a continuous-wave laser at a wavelength of 785 nm was used as the probe beam, whereas a quasi-continuous wave laser with a pulse duration of 4 μs at a wavelength of 852 nm was used as the pump beam. Both the pump beam and probe beam were focused on the Al pad deposited on the center of the suspended bridge structure. The probe beam was used to monitor the temperature-dependent reflectance change of the Al pad, while the pump beam was used to heat the Al pad. All measurements were carried out at 300 K. The peak power of the pump beam was set at 1 mW, while that of the probe beam was set at 30 μW. We confirmed that the temperature rise due to the laser irradiation had negligible influence on the measured κ, which was checked by carrying out the measurement at multiple laser power. The samples were placed under vacuum at a pressure of 5 × 10^{−4} Pa to exclude the influence of convection on the thermal relaxation behavior of the Al pad. The standard error was smaller than ±3% for each sample in our measurement. The TDTR signal exhibits a sharp rise in intensity, which is followed by an exponential decay. The material thermal conductivity κ_{mat} of a-Si_{3}N_{4} PnCs, which does not include the classical geometric effect of the pores, can be obtained by fitting the experimental TDTR signal with that of the TDTR signal simulated by a finite element method (FEM). The influence of pores on the effective thermal conductivity κ_{T} of the a-Si_{3}N_{4} films can be calculated accurately using FEM analysis. We used ANSYS software for the FEM analysis. More detailed information is summarized in the “Measurement of thermal conductivity” section.

The measured κ_{mat} of the bare thin films and PnCs as a function of *n* are shown in Fig. 2A. κ_{mat} of the 70-nm-thick bare films was 2.5 ± 0.2 W/mK, which was consistent with the previous study measured by the 3ω method (*28*). As shown in Fig. 2A, κ_{mat} of PnCs shows a decreasing trend with decreasing *n*. In particular, κ_{mat} shows a steep decrease in its magnitude when *n* is reduced below 20 nm, where it exhibits a substantially low value of ~1 W/mK. This result clearly demonstrates that κ_{mat} of the amorphous solids can be manipulated by the PnC nanostructures.

According to the previous studies, the propagating vibrational modes with relatively long MFP should dominate the magnitude of κ_{mat}. As is the case with phonons in crystalline solids, we assume that transport of these propagating modes in a-Si_{3}N_{4} is similarly disturbed by boundary scattering. To achieve further insight into the effect of boundary scattering, we show in Fig. 2B the measured κ_{mat} of a-Si_{3}N_{4} PnCs as a function of surface-to-volume (S/V) ratio. We see that κ_{mat} indeed decreases with increasing the S/V ratio. This is consistent with the feature reported for crystalline Si (c-Si) PnCs (*29*) where the reduction in κ was attributed to enhanced boundary scattering of phonons due to an increase in the S/V ratio. The result indicates that the reduction in κ_{mat} of the present a-Si_{3}N_{4} PnCs is indeed related to enhanced boundary scattering of propagons.

Another remarkable feature observed in Fig. 2B is that κ_{mat} converges to a certain value (~1 W/mK) after a monotonic decrease when the S/V ratio is increased above 0.1 nm^{−1}, which indicates that propagons are fully suppressed, leaving only diffusons to dominate thermal transport. Here, we thus call this converged κ value of ~1 W/mK as the “diffusive limit,” which represents the κ contribution of diffusons in bulk a-Si_{3}N_{4}.

Similar reduction trend of κ in terms of S/V ratio was also reported in c-Si PnCs and nanowires. However, their κ reduction rate is much larger than that of a-Si_{3}N_{4} PnCs. The reason is that the MFP of phonons that govern κ at room temperature in c-Si is distributed in a range of 100 nm to 10 μm, whereas that of heat carriers in a-Si_{3}N_{4} is mainly distributed in a range of 1 to 100 nm (fig. S1). Therefore, phonons in c-Si are much more sensitive to boundary scatterings. Moreover, as the S/V ratio increases, localized phonons become more important to thermal transport and lastly make κ saturate, as indicated in the work of Chen *et al*. (*30*) for nanowires. Although the localization of propagons may also take place in a-Si_{3}N_{4}, our theoretical calculation implies that the saturation of κ observed here is mainly related to the suppression of propagon transport by boundary scattering. As we show in the following paragraphs, κ contributed by diffusons in bulk a-Si_{3}N_{4} was calculated to be 1.1 W/mK, which is consistent with the converged κ value shown in Fig. 2 (A and B). The good agreement supports that the heat is mainly carried by diffusons in the a-Si_{3}N_{4} PnCs that exhibit S/V ratio larger than 0.1 nm^{−1}.

Propagons are phonon-like propagating vibrational modes, and thus, we assume that the thermal conductivity contribution from propagons (κ_{P}) follow the phonon gas model as

(1)where *V* is the system volume, ω_{t} is the transition frequency of propagons and diffusons, which is determined as 4 THz (details are summarized in section S1), *v*_{s} is the appropriate sound speed, τ(ω) is the frequency-dependent relaxation time, and DOS (ω) is the vibrational density of states. *C*(ω) is the mode-dependent specific heat capacity described as

(2)where *k*_{B} is the Boltzmann constant, ℏ is the reduced Planck constant, and *T* is temperature. Here, we focus on room temperature, i.e., 300 K.

The dispersion relation of propagons is expected to be linear, similar to that of sound. Thus, their DOS is assumed to obey the Debye approximation. DOS is then described as

$$\text{DOS}(\mathrm{\omega})=\frac{3V{\mathrm{\omega}}^{2}}{2{\mathrm{\pi}}^{2}{v}_{\mathrm{s}}^{3}}$$(3)

On the other hand, the thermal conductivity contribution from diffusions (κ_{D}) can be described by the AF theory as

(4)where ω* _{i}* is the frequency of the

*i*-th diffuson mode, and

*D*(

*ω*) is the diffuson diffusivity, which can be calculated as

_{i}(5)where δ is the delta function broadened into Lorentzian, and *S _{ij}* is the heat current operator as a function of frequencies and eigenvectors, which can be calculated from the harmonic lattice dynamics theory.

The relaxation time of each vibration mode can be obtained by performing the normal mode decomposition (NMD) analysis on the phase space trajectories obtained by equilibrium molecular dynamics (MD) simulation. Since propagons are similar to phonons with low frequencies, the relaxation time of propagons can be modeled with the form similar to the following Klemens model, which is widely used and has been validated for low-frequency phonons in various kinds of materials. This is described as

$$\mathrm{\tau}(\mathrm{\omega})=B{\mathrm{\omega}}^{-2}$$(6)where *B* is a constant coefficient that incorporates the effect of scatterings and temperatures. The magnitude of *B* can be evaluated by fitting the relaxation time of propagons from NMD calculations.

To quantitatively evaluate boundary scattering of propagons and diffusons in amorphous PnCs, we need to define their MFP. For phonon-like propagons, the MFP is defined by the appropriate sound speed and relaxation time as

$$\mathrm{\Lambda}(\mathrm{\omega})={v}_{s}\mathrm{\tau}(\mathrm{\omega})$$(7)

For diffusons, the MFP is defined by their diffusive length, which can be obtained by comparing Eqs. 1 and 4

$$D(\mathrm{\omega})=\frac{1}{3}{v}_{D}^{2}\mathrm{\tau}(\mathrm{\omega})=\frac{1}{3}{v}_{D}\mathrm{\Lambda}(\mathrm{\omega})$$(8)

$$\mathrm{\Lambda}(\mathrm{\omega})=\sqrt{3D(\mathrm{\omega})\mathrm{\tau}(\mathrm{\omega})}$$(9)where *v*_{D} is the diffusive velocity of diffusons, and τ(ω) is the frequency-dependent relaxation time of diffusons obtained by the NMD calculations.

Using Eqs. 8 and 9, the thermal conductivity of bulk amorphous solids κ_{T} (= κ_{P} + κ_{D}) can be expressed in the form similar to that of the phonon gas model as

(10)where, *v*_{s},_{D} represents *v*_{s} for propagons and *v*_{D} for diffusons. Note that, here, we defined the effective MFP of propagons and diffusons in a unified way; however, we still can identify their contributions to κ_{T} by their unique features such as the frequency-dependent DOS and the effective dispersion relations of propagons, as shown in section S2.

The calculation of the bulk properties of a-Si_{3}N_{4} was similar to the works of Larkin and McGaughey (*8*) (detailed information is summarized in the “Calculation of bulk thermal properties of silicon nitrides” section and section S2). On the basis of this model, we have calculated the room temperature κ_{T} of bulk a-Si_{3}N_{4}. The obtained value was 2.9 W/mK, which is consistent with the measured κ reported by Sultan *et al.* (*31*), Zink and Hellman (*32*), and Ftouni *et al.* (*28*). The calculated κ_{P} and κ_{D} were identified as 1.8 and 1.1 W/mK, respectively. This indicates that propagons with long MFP can contribute to a greater part (62%) of the κ_{T} of bulk a-Si_{3}N_{4}, which is consistent with the work reported by Sultan *et al.* (*14*) (50%).

Now that we confirmed the validity of our model in bulk a-Si_{3}N_{4}, we move on to discussing the thermal transport property of a-Si_{3}N_{4} thin films and PnCs. To incorporate the effect of boundary scattering in the bulk model, we used the MCRT method, which has been a powerful tool to quantify the boundary scattering of phonons in crystalline solids (*25*). The MCRT method provides us the effective MFP of propagons and diffusons of thin films and PnCs, which, in turn, provides us the magnitude of κ_{T} of the a-Si_{3}N_{4} thin films and PnCs.

The calculated κ of the 70-nm-thick films was 2.3 W/mK, which was consistent with our measured value of 2.5 ± 0.2 W/mK. Because of boundary scattering of propagons, 21% reduction in κ was identified with respect to that of bulk a-Si_{3}N_{4}. In contrast, diffusons were not affected by the boundary of the 70-nm-thick films because of their very short MFP, which range from several angstroms to several nanometers. As mentioned above, κ_{D} of the bulk a-Si_{3}N_{4} was calculated to be 1.1 W/mK. This means that κ_{D} of the 70-nm-thick a-Si_{3}N_{4} films is 1.1 W/mK as well. One of the remarkable aspects here is that the calculated κ_{D} of thin films agrees well with the measured diffusive limit (1.01 ± 0.09 W/mK). This result supports our discussion that propagons can be drastically suppressed by the PnC structures, leaving only the diffusons to contribute to the thermal transport in a-Si_{3}N_{4} PnCs when *n* is reduced below 20 nm.

In Fig. 3, we compare our experimental results and the simulated results for various kinds of a-Si_{3}N_{4} PnCs. Here, the measured κ_{mat} was converted into κ_{T} using the Maxwell-Garnett model expressed by

(11)where φ is the porosity of the material. The validity of the Maxwell-Garnett model for our samples was confirmed using the steady-state thermal analysis module of ANSYS software. We find that κ_{T} of these PnCs is substantially reduced with respect to that of the thin films. The lowest κ_{T} obtained here was 0.26 ± 0.03 W/mK, which is comparable with typical plastic materials such as polyethylene (κ = 0.367 W/mK) (*33*). We can also see that the calculation agrees well with the measurement. This indicates that the model developed here originally based on phonon simulation can be useful to reproduce the thermal transport properties of amorphous PnCs. For most of the a-Si_{3}N_{4} PnCs, the difference between the calculated values and the experimental values is within 10%. The only exception was observed in samples with *n* = 19 nm (*P* = 100 nm and *D* = 81 nm), which showed an acceptable but relatively large 35% difference. A possible reason for the discrepancy can be attributed to the difficulty in accurately determining *n* at this small size scale.

Coherent phonon transport is generally highlighted when we discuss the thermal transport properties of periodic nanostructures. Evidence of the band folding effect or impact of periodicity on thermal transport, which all indicate the presence of coherent phonon transport, has been observed in various crystalline nanomaterials. Band folding has also been observed in amorphous superlattices by Koblinger *et al.* (*34*). However, the presence of coherent transport of propagons in amorphous phononic materials still remains as an open question. Here, our theoretical calculations showed that the measured κ of a-Si_{3}N_{4} phononic materials can be reproduced by the particle-based Boltzmann transport equation. This indicates that coherent thermal transport is not substantial in a-Si_{3}N_{4} PnCs at 300 K even when *P* is reduced down to 36 nm. However, we believe that coherent transport of propagons can become important at low temperatures as observed in crystalline PnCs (*35*), where low-frequency propagons dominate heat transport.

Now that we have demonstrated the validity of our model in amorphous PnCs, we investigate how the mode-dependent transport properties of propagons and diffusons are affected by the nanostructures. In Fig. 4A, we compare the MFPs of bulk a-Si_{3}N_{4} and those of a-Si_{3}N_{4} PnCs with *P* = 200 nm and *D* = 175 nm at 300 K. We see that the MFP of both propagons and diffusons is drastically suppressed in a-Si_{3}N_{4} PnCs. For propagons in the PnCs, the MFPs are substantially reduced from that of the bulk case (10 to 100 nm) to sub–10 nm. Even for diffusons, the MFPs are reduced to about 16% of the bulk value, which is only a few angstroms. An important feature we noticed here is that the minimum MFP of propagons is around 10 nm. This indicates that propagon transport can be substantially disturbed by the PnC structures if we can reduce *n* to a few nanometers. The steep decrease in κ_{mat} below *n* of 20 nm shown in Fig. 2A can be understood well by the feature observed in Fig. 4A.

The shortened effective MFP leads to a substantial reduction in κ. Figure 4B shows a comparison of the frequency-dependent κ spectra between bulk a-Si_{3}N_{4} and a-Si_{3}N_{4} PnCs. The κ spectra of propagons and diffusons in PnCs are substantially reduced (nearly 90 and 80%, respectively) from those of bulk counterparts. As a result, κ_{P}, κ_{D}, and κ_{T} are reduced to 0.15, 0.19, and 0.34 W/mK, respectively, which is almost one order of magnitude smaller than κ_{P} (1.1 W/mK), κ_{D} (1.8 W/mK), and κ_{T} (2.9 W/mK) of the bulk case. The calculated κ_{P} and κ_{D} for all the a-Si_{3}N_{4} PnCs are summarized in Fig. 4 (C and D), respectively. Because of the boundary scattering of propagons, the reduction in κ_{P} from the bulk value can exceed 70% even for samples with a large *n* of 665 nm. For samples with *n* below 20 nm, κ_{P} is smaller than 0.2 W/mK, which indicates that 90% of propagons are scattered. For diffusons, whose MFP varies from 0.5 to 10 nm in the bulk case, boundary scattering can still lead to a 40 to 80% reduction in κ_{D}, depending on the *P* and *n*. This indicates that diffusive boundary scattering due to the PnC structure can reduce the transmittance of diffusons. These results suggest that PnC structures can not only effectively scattering propagons but also severely backscatter diffusons. The current findings not only reveal the mechanisms of thermal conductivity reduction in amorphous PnCs but also suggest that amorphous PnCs can realize the ultimate reduction of heat conduction of amorphous solids. Moreover, our analysis suggests that propagons and diffusons can be unified treated as quasiparticles with defined propagating or diffusive lengths, i.e., the MFPs, in the boundary scattering process, which indicates that in terms of boundary scatterings, there is no fundamental difference between propagons and diffusons.

**Acknowledgments: **We thank M. Fujikane, H. Tamaki, and T. Kanno for helpful discussions. **Funding:** This research was funded by Panasonic Corporation. This work was supported by Grant-in-Aid for Scientific Research (A) (Grant No. 19H00744) from JSPS KAKENHI. Y.L. thanks the Fellowship (Grant No. JP18J14024) from JSPS. The calculations in this work were partially performed using supercomputer facilities of the Institute for Solid State Physics, The University of Tokyo. **Author contributions:** N.T. measured the thermal transport properties. N.T. and K.T. constructed the thermal reflectance optical system. Y.L. and J.S. performed theoretical calculation. C.Z., E.M.A., and P.F.N. fabricated the samples. K.T., Y.N., and J.S. directed the project. N.T. and Y.L. wrote the paper with inputs from all authors. **Competing interests:** The authors declare that they have no financial or other competing interests. **Data and materials availability:** All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors. Correspondence and requests for materials should be addressed to N.T. (tambo.naoki{at}jp.panasonic.com) and J.S. (shiomi{at}photon.t.u-tokyo.ac.jp).