Suppressed phonon conduction by geometrically induced evolution of transport characteristics from Brownian motion into Lévy flight

Despite extensive research on quasi-ballistic phonon transport, anomalous phonon transport is still observed in numerous nanostructures. Herein, we investigate the transport characteristics of two sets of samples: straight beams and nanoladders comprising two straight beams orthogonally connected with bridges. A combination of experiments and analysis with a Boltzmann transport model suggests that the boundary scattering within the bridges considerably dictates the distribution of phonon mean free paths, despite its negligible contribution to the net heat flux. Statistical analysis of those boundary scatterings shows that phonons with large axial angles are filtered into bridges, creating dead spaces in the line-of-sight channels. Such redistribution induces Lévy walk conduction along the line-of-sight channels, causing the remaining phonons within the bridges to exhibit Brownian motion. Phonon conduction in the nanoladders is suppressed below that of the straight beams with equivalent cross-sectional areas due to trapped phonons within the bridges. Our work reveals the origin of unusual thermal conductivity suppression at the nanoscale, suggesting a method to modulate phonon conduction via systematic nanostructuring. Despite extensive previous research, the suppression in phonon conduction at the nanoscale still calls into questions on the interaction of phonons with various sources of boundary scatterings. In this work, a combination of Boltzmann transport model and the experiments finds that the bridges contribute to phonon mean free paths proportional to its volume fraction despite its negligible contribution to net heat flux. A statistical analysis of boundary scattering reveals that transport characteristics of phonon evolves from Brownian motion to Lévy walk due to phonons trapped within the bridges.


Introduction
Understanding the nanoscale phonon conduction mechanism is essential in a wide range of semiconductor applications, such as nanoelectronics 1,2 , optoelectronics 3,4 , and energy conversion devices 5,6 . In the context of phonon conduction, nanostructures can be broadly categorized based on the uniformity of the cross-sectional area along the predominant direction of heat flow. Typical examples of structures with uniform cross-sectional areas include thin films 7-10 , nanobeams 11 , and smooth nanowires [12][13][14][15] , which have line-of-sight (LOS) channels for heat flow. These nanostructures show a significant reduction in thermal conductivity compared to their bulk counterparts due to increased boundary scattering. Such suppressed thermal transport in nanostructures with uniform cross-sections has substantially contributed to establishing a microscopic phonon conduction model 16,17 . However, numerous nanostructures show anomalous heat conduction with increasing complexity, calling into question the suppression mechanisms of phonon transport 18,19 .
It is generally accepted that the LOS channel is a primary heat conduction path, and its cross-section dictates the thermal conductivity of nanostructures. With the increased complexity of nanostructures that have a non-LOS channel volume, phonon conduction is observed to deviate from that of the LOS channel [20][21][22][23][24] . A key feature associated with such suppression in conduction is that phonon flux is disturbed due to the interactions of phonons with geometrical perturbations. The length scale of this perturbation ranges from angstroms to hundreds of nanometers, which is on the order of the phonon mean free path (MFP). Various reduction mechanisms have been suggested to explain the suppressed thermal conductivity of these complex nanostructures, with a focus on the interplay between phonons and nanostructures. For example, previous reports on nanowires with crosssectional areas disordered in the range of a few nanometers 25,26 as well as on phononic films 20,27,28 with periodic nanoscale holes suggest that this suppression can be associated with the wave-like effects of phonons, atomic defects, and backscattering 21,29 . On the other hand, in fishbone structures, i.e., a nanobeam with orthogonally protruded pillars, or in corrugated nanowires with a series of periodically folded structures along the sidewall, the reduction in thermal conductivity is attributed to the Sharvin resistance in the ballistic regime 24 and Lévy walk transport characteristics 22,30 . In addition, it has been further shown that in bare nanoladder structures 31 , a nanobeam with nanobridges and a single row of pores and bridges constitute a thermally dead volume 23 . The abovementioned reduction mechanisms are still under debate, calling for further studies on the origin of the phonon reduction mechanism in complex nanostructures beyond the conventional phonon transport mechanism that follows Brownian motion.
Herein, we investigate the evolution of thermal transport characteristics along LOS channels in nanoladders with a periodically varying cross-sectional areas at room temperature. Specifically, we prepare two sets of samples with~78 nm thick single crystal silicon: (1) a set of nanobeams with varying cross-sectional areas and (2) a set of two identical nanobeams connected with a series of orthogonally placed bridges resembling ladders. With respect to the predominant heat direction, the thermal transport in nanoladders consists of two regimes, line-ofsight (LOS) channels and bridges, which are connected orthogonally, while the thermal transport in nanobeams consists of a single LOS channel regime. A nanoladder is considered as a series of repeating unit cells, as illustrated in Fig. 1a. Phonons contributing to the net transfer across the unit cell can be categorized into two groups: those traveling directly across the unit cell and those fully thermalized within the bridges. To quantify the relative contribution of each component to the thermal conduction, we deliberately designed our samples by modulating the cross-section ratios of LOS channels to bridges, ranging from~1 to 5.29. Using a combination of the Boltzmann transport equation and the Monte Carlo approach, we model the phonon transport in our samples and find that the mean free path is determined by the relative volume ratio of LOS channels to bridges, corresponding to geometrical inhomogeneity. We further analyze a statistical distribution of phonon free paths, i.e., the traveling distance between scattering events, to investigate their transport characteristics as well as their impact on resultant mean free paths.

Device fabrication
We fabricate silicon nanostructures using silicon-oninsulator (SOI) wafers (Soitec Inc.) comprising añ 340 nm thick silicon layer and a 1 µm thick buried oxide layer (BOX). We employ thermal oxidation and consecutive oxide removal through a wet etching process to decrease the thickness of the silicon layer to~78 nm. As an electrical passivation layer, an~25 nm thick Al 2 O 3 layer is deposited using atomic layer deposition (ALD). To pattern both nanobeam and nanoladder structures, a window of 10 μm × 20 μm is patterned between membranes by photolithography and etched using wet processes. Using electron beam lithography, the nanostructures are patterned, and a dry etching process follows. A serpentine Pt heater is patterned onto the insulation layer of each membrane by using a combination of electron beam lithography. A lift-off process is applied after the deposition of Cr and Pt layers of~5 and 40 nm thicknesses using electron beam metal evaporation. To suspend the structure, the BOX layer under the surrounding area of the membranes and legs is etched using RIE and consecutive gaseous hydrogen fluoride (HF) etching. For the conversion of thermal conductance into thermal conductivity and the uncertainty analysis, dimension measurement with scanning electron microscopy (SEM) is conducted after the fabrication process.

Sample design and electrothermal characterization
As shown in Fig. 1b-i, we carefully prepare the two following sets of samples: (1) straight beams and (2) nanoladders (consisting of two straight beams placed in parallel and connected orthogonally using bridges). For the straight beam samples, we vary the aspect ratios of the cross-sections from~0.9 to~13.9 by modulating the width w LOS from~70 nm to~970 nm while keeping the thickness t at~78 nm. Similarly, for the nanoladder samples, we vary the widths of the two straight beams from~70 nm to~370 nm, and~70 nm by~78 nm-sized bridges are placed between these beams with a periodicity of 200 nm. Here, the bridge width l bridge is~70 nm. Accordingly, the volume ratio of LOS channels to bridges ranges from~0.48 to~9.2 in our nanoladders, and both sets of samples have identical LOS channels. The length of both sets of samples is 10 μm, which is long enough to ensure diffusive phonon transport along the LOS channels.
The thermal conductivity of our samples is characterized using an electrothermal characterization method with two suspended membranes, and this method has been applied for measuring the thermal conductivities of numerous nanomaterials. The detailed methodology is well documented elsewhere 14,32,33 . We note that both the samples and membranes are monolithically fabricated using both electron-beam lithography and photolithography (see Supporting Information for fabrication details). Heat is generated via Joule heating using serpentine metal structures, which imposes a finite temperature difference across the sample, and the associated temperatures on both sides are measured using resistive thermometry. Given the heat generation and associated temperature differences, the thermal conductivity of the samples can then be calculated by numerically solving a heat equation. We note that the heat equation captures the geometrical contribution in the diffusive transport regime.
We assume uniform thermal conductivities across the nanostructured samples (see Supporting Information for uncertainty analysis). The experiment is performed in vacuum to minimize convective heat loss.

Results and discussion
Experimental results Figure 2a shows that as the cross-sectional area of an LOS channel is decreased, the thermal conductivity decreases from~50 to~30 W m −1 K −1 and from~45 tõ 33 W m −1 K −1 for straight beams and nanoladders, respectively. Note that for straight beams, we find that as the aspect ratio is increased, the thermal conductivity approaches the thin-film limit, verifying the validity of our measurements 34 . The monotonic decrease in the thermal conductivity of both sets of samples indicates that the cross-sectional area of the LOS channel is the predominant factor that dictates the overall thermal conductivity in our samples, as discussed in previous studies [35][36][37] . However, we find a crossover of the thermal conductivity between the two sets despite the identical cross-sections of the LOS channels. The thermal conductivity of the nanoladder samples is smaller than that of the straight beam samples when w LOS > l bridge , whereas the relative magnitude is reversed when w LOS < l bridge . While the crossover is found nearly within experimental uncertainty, it is worth investigating the crossover using a Boltzmann transport model.

Boltzmann transport model
To better understand the differences in phonon conduction between straight beams and nanoladders, we model the thermal conductivity based on the Boltzmann transport equation as 38,39 where i is the phonon mode, q is the phonon wavevector, C V is the volumetric heat capacity, v is the group velocity, and τ is the relaxation time. Born-vo n Karman sine type dispersion relation is used. The phonon mean free path Λ is defined as Λ = v × τ and can be derived using Matthiessen's rule as τ ¼ ðτ À1 U þ τ À1 I þ τ À1 B Þ À1 , where τ U is the Umklapp scattering rate defined as τ À1 U ¼ Aw 2 TexpðÀ B T Þ and τ I is the impurity scattering rate defined as τ À1 I ¼ Dw 4 . The boundary scattering rate τ B is estimated by simulating phonon particles using Monte Carlo schemes. We consider the effects of internal phonon-phonon scattering and impurity scattering, and to determine the parameters A, B and D, we fit Eq. (1) to the experimental data of straight beams. For the best fit, A, B and D are given as 1.21 × 10 −19 sK −1 , 151 K and 2.54 × 10 −45 s 3 , respectively (see Supporting Information for detailed Boltzmann transport model). We apply the fitted value for internal scattering to predict the thermal conductivity of the nanoladders, and the fit shows agreement with our experimental data withiñ 2%. This agreement ensures that the model captures the characteristics of phonon transport in our samples, as seen in Fig. 2a. More importantly, the model prediction clearly shows the crossover of the thermal conductivity between nanoladders and straight beams as a function of LOS width.
We further investigate the dependence of the phonon mean free path on boundary scattering, which dictates the thermal conductivity at the given length scale of our samples. We plot the boundary scattering mean free path Λ Boundary in Fig. 2b, which shows a clear crossover behavior between the nanoladders and straight beams, similar to the observed thermal conductivity. On the one hand, when w LOS > l Bridge , the Λ Boundary in nanoladders is smaller than that in straight beams as shorter MFPs are introduced from the bridges. On the other hand, when w LOS < l Bridge , the Λ Boundary in the nanoladders asymptotically saturates at a constant value as w LOS is decreased, while it decreases monotonically in their straight beam counterparts. The asymptotic limit of the nanoladders is mainly dictated by the geometrical dimension of the bridge, which introduces a constant scattering crosssection. This indicates that the Λ Boundary in nanoladders is nearly determined by the volumetric contribution of the LOS channels and bridges in the heterogeneous structures. Particularly at w LOS ≪ l Bridge , Λ Boundary is predominantly dictated by the critical dimension of the bridge, not that of the LOS channel, which is a primary heat flow channel. As such, the critical dimension of the bridge serves as a source of free paths, while the contribution to the net heat flux from bridges is negligible.

Statistical analysis of phonon free paths
We next statistically analyze the boundary scattering free path distributions of both the nanoladder and straight beam sample sets. We consider only boundary scattering, as it is the predominant scattering mechanism at the given Fig. 3 Statistical analysis of boundary scattered phonons. a Probability density function (PDF) of phonon free path in nanoladders. The red and blue colors correspond to the phonon free path in bridge and in LOS channel, respectively (w LOS = 20 nm, thickness t = 78 nm, and bridge length l Bridge = 70 nm). Note that the PDF peaks whenever free path coincides with the dimension of the characteristic features present in nanoladders. Each peak can be described in term of the characteristic length L C and shape parameter α, which determines the decay profile of free paths. b α versus the ratio of l Bridge to w LOS in nanoladders. The dark solid gray color indicates the infinitely thick limit where thickness-dependent diffuse boundary scattering becomes negligible. For α > 2, phonon transport is diffusive (red shaded region), but as α crosses 2, phonons display Lévy walk transport characteristics (blue shaded region). Fig. 3a, the free path distribution is expressed in terms of the probability density function (PDF), which shows peaks at the characteristic dimensions of the nanoladder and the straight beam: w LOS and thickness t. Note that we choose w LOS = 20 nm to emphasize the role of the LOS channel in the free path distributions, as the peaks for the LOS channel and the bridge are close to each other when w LOS = 70 nm. We observe that the PDF of the free paths for the nanoladder samples is dramatically suppressed near 20 nm compared to that for the straight beam samples. In contrast, the PDF is enhanced at~70 nm (=l Bridge ) compared to that for the straight beam samples. Such changes in the PDF are mainly due to the phonons being trapped in the bridges, which occurs mostly for phonons traveling at relatively large axial angles with respect to the predominant direction of heat flow along the LOS channels 22 . We note that a nonzero probability is found below 20 nm, corresponding to potential phonon travel paths at a corner.

dimensions. As seen in
To further obtain quantitative information on the transport characteristics, we interpret the PDFs using the Pareto distribution expressed as 40 where L C = min(w LOS ,l bridge ,t) is the length scale of the smallest characteristic dimension present in the nanostructure, X is the free path, and α is the shape parameter that reflects the phonon transport mechanism: ballistic transport for α = 1, Lévy walk for 1 < α < 2, and Brownian motion for α = 2. We note that the Pareto distribution has been studied in Bose-Einstein processes, such as thermal transport in SiGe and InGaAs alloys [41][42][43] as well as in silicon nanowires at low temperatures 22 . Using the Pareto distribution framework, we fit the simulated PDFs of the free paths in the nanoladder samples and extract the shape parameter α as a function of the ratio of the predominant critical dimensions, w LOS to l Bridge . We find that for w LOS > l Bridge , α is larger than 2, and α is smaller than 2 when w LOS < l Bridge , as seen in Fig. 3b. Note that this analysis focuses on the PDF near the smallest characteristic dimension present in the nanostructure to avoid interplay with other larger characteristic dimensions. We can therefore deduce that the phonon transport in the bridge region is dominated by Brownian motion (α = 2), while the LOS channel region displays Lévy walk characteristics (α < 2). We note that heat conduction typically follows Brownian motion. For example, in the case of infinitely thick samples, the bridge region shows perfect Brownian motion, as indicated by α = 2. Furthermore, we find that straight beams also show Brownian motion (α = 2) as opposed to the Lévy walk characteristics shown by the LOS channels in nanoladders. These observations suggest that the characteristics of phonon transport are converted from Brownian motion to Lévy flight along the LOS channel in our nanoladders by bridging the parallel channels. We estimate the contribution of the Lévy walk character to the phonon mean free path E(X) using a Pareto distribution as where X is the free path, α is the shape parameter, and L c is the characteristic length for the boundary scattering. We note that α = 2 corresponds to the Brownian motion regime, while 1 < α < 2 corresponds to the Lévy walk regime. As shown in Fig. 4, the degree of boundary scattering contribution to the mean free path depends on the phonon transport characteristics, i.e., α. This suggests that phonons displaying Lévy walk characteristics are likely to have a longer mean free path than those displaying Brownian motion characteristics. We further visualize the phonon paths along the LOS channels in both nanoladders and straight beams by simulating 3 × 10 4 phonons in a 3D computational space, which are projected onto a two-dimensional plane, as seen in the inset of Fig. 4. It is noteworthy that the collective set of phonon trajectories renders dead space at the entrance of the bridges. As such, phonons propagating along the LOS channel are likely to form an artificially corrugated structure, with those propagating at narrow axial angles following Lévy walk characteristics 22,44 . Given the statistical analysis above, we suggest that phonons propagating with broad axial angles are dominantly trapped within the bridges in our nanoladder and are dictated by the boundary scattering therein. As w LOS is decreased, more phonons with large-angle scattering are likely to be Fig. 4 Mean free path normalized to the critical dimension L C as a function of the shape parameter α. The region shaded in red color indicates the Lévy walk regime (α < 2) while Brownian motion is found at α = 2. The inset shows simulated phonon trajectory in both Lévy walk (left) and Brownian motion (right) regimes. We note that the inset for Lévy walk corresponds to α = 1.73.
trapped in the bridges. The remaining phonons with narrow axial angles contribute to the decrease in the shape parameter as well as the increase in the mean free path in the LOS channel. However, the overall thermal conductivity of the nanoladder is suppressed as the majority of phonons are trapped in Brownian motiondominated bridges; this suppression becomes further pronounced as the critical dimension of the bridge is comparable to or smaller than that of the LOS channel.
Finally, we extend our discussion on the impact of geometrical heterogeneity on phonon transport in the context of previous studies on other nanostructures, such as nanomeshes 20,21 and fishbone nanowires 24,45 , as shown in the inset of Fig. 5. These structures can be decomposed into the LOS channel and the other channel, which is orthogonally aligned with respect to the LOS channel, and the corresponding critical dimensions are set to w // and l ⊥ , respectively. Figure 5 shows the thermal conductivity of the abovementioned nanostructures normalized to that of the LOS channel as a function of the volumetric ratios of the nanostructures to that of the LOS channel. For the nanomeshes, the critical dimensions are the same l ⊥ = w // , and the thermal conductivity decreases with increasing volumetric density of the non-LOS channel. For the fishbone nanowires with l ⊥ > w // , the thermal conductivity decreases with increasing volume ratio, V Total to V LOS , despite the increasing volumetric contribution of the non-LOS channel. In both cases, as the critical dimension of the non-LOS channel is larger than that of the LOS channel, the volumetric change fails to induce the suppression with increasing volume ratio of the non-LOS channel. Given these structures, phonon conduction through the LOS channel is also likely to show Lévy-walk characteristics, increasing the mean free paths along the channel. The abovementioned factors fail to explain the reduction. As such, the further suppression with increasing volume of the non-LOS channel is due to the increasingly trapped phonons with the non-LOS channel, which are forced to follow Brownian motion within the regime.

Conclusions
In summary, we investigate the phonon transport mechanisms in silicon nanoladders and nanobeams. We observe a crossover in the thermal conductivity between the nanoladders and straight beams as a function of the volumetric ratio of bridges to LOS channels. A model prediction based on Boltzmann transport suggests that the non-LOS channel is a major contributing source of phonon boundary scattering despite its negligible contribution to the net heat flux. Furthermore, a statistical analysis of the distributions of free paths suggests that the bridges convert the transport characteristics of the LOS channel. We quantitatively identify this conversion using a Pareto-distribution framework as phonons traversing along the LOS channel follow the Lévy walk process, while those trapped in bridges show diffusive behavior. As a result, phonons in the LOS channels of the nanoladders have a relatively long mean free path as bridges capture large-angle phonons with respect to the axial direction. Finally, we extend our observation to other nanostructures with orthogonal geometric obstructions, such as nanomeshes and fishbone nanowires. This work answers a long-lasting question regarding the interplay between phonons and nanostructures, contributing to a comprehensive model for complex nanostructures.