Nanotechnology PAPER • OPEN ACCESS Thermal conductivity in disordered porous nanomembranes To cite this article: Marianna Sledzinska et al 2019 Nanotechnology 30 265401 View the article online for updates and enhancements. This content was downloaded from IP address 158.109.11.194 on 17/07/2019 at 12:35 Made open access 6 June 2019 Nanotechnology Nanotechnology 30 (2019) 265401 (8pp) https://doi.org/10.1088/1361-6528/ab0ecd Thermal conductivity in disordered porous nanomembranes Marianna Sledzinska1 , Bartlomiej Graczykowski2 , Francesc Alzina1 , Umberto Melia3 , Konstantinos Termentzidis4 , David Lacroix5 and Clivia M Sotomayor Torres1,6 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology Campus UAB, Bellaterra, E-08193 Barcelona, Spain 2 Max Planck Institute for Polymer Research, Ackermannweg 10, D-55218 Mainz, Germany 3 Department of ESAII, Centre for Biomedical Engineering Research, Universitat Politècnica de Catalunya, CIBER-BBN, Barcelona, Spain 4 Univ Lyon, CNRS, INSA-Lyon, Université Claude Bernard Lyon 1, CETHIL UMR5008, F-69621, Villeurbanne, France 5 Université de Lorraine, CNRS, LEMTA, Nancy, F-54000, France 6 ICREA-Institucio Catalana de Recerca i Estudis Avancats, E-08010 Barcelona, Spain E-mail: marianna.sledzinska@icn2.cat Received 30 October 2018, revised 27 February 2019 Accepted for publication 12 March 2019 Published 12 April 2019 Abstract In this work we study the effects of disorder on the thermal conductivity of porous 100 nm thick silicon membranes, in which the size, shape and position of the pores were varied randomly. Measurements using two-laser Raman thermometry on both non-patterned and porous membranes revealed more than a 10-fold reduction of the thermal conductivity compared to that of bulk silicon and a six-fold reduction compared to non-patterned membranes for the sample with random pore shapes. Using Monte Carlo methods we solved the Boltzmann transport equation for phonons and compared different possibilities of pore organization and its influence on the thermal conductivity of the samples. The simulations confirmed that the strongest reduction of thermal conductivity is achieved for a distribution of pores with arbitrary shapes that partially overlap. Up to a 15% reduction of the thermal conductivity with respect to the purely circular pores was predicted for a porous membrane with 37% filling fraction. The effect of the pore shape and distribution was further studied. Maps of temperature and heat flux distributions clearly showed that for particular pore placement heat transport can be efficiently blocked and hot spots can be found in narrow channels between pores. These findings have an impact on the fabrication of membrane-based thermoelectric devices, where low thermal conductivity is required. This work shows that for porous membranes with a given filling fraction the thermal conductivity can be further modified by introducing disorder in the shape and placement of the pores. Supplementary material for this article is available online Keywords: silicon nanomembrane, thermal conductivity, Monte Carlo methods, disorder 1. Introduction have shown that it can improve or add functionality to the system. For instance, disordered nanostructures on the surface Although physical models prefer to describe perfectly ordered of flower petals produce visual signals which attract bees [1]. systems, they are hard to find in nature; real systems always In photonics, photon transport and collimation can be strongly contain defects and disorder. In many systems the presence of enhanced in disordered structures [2]. Therefore, instead of disorder is considered unfavorable, however recent works trying to eliminate disorder in nano-devices in this work we 0957-4484/19/265401+08$33.00 1 © 2019 IOP Publishing Ltd Printed in the UK Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Nanotechnology 30 (2019) 265401 M Sledzinska et al intentionally introduce disorder during the fabrication process nanoholes (similar to the ones reported previously [10]), in order to study its effects in nanostructures and obtain new became a randomized lattice of deformed holes. functionalities. The two structures, hereinafter referred as Sample 1 and Two-dimensional phononic crystals (PnCs) are periodi- Sample 2, fabricated in a membrane of thickness t=100 nm, are cally patterned membranes and in such structures disorder is shown in figures 1(a) and (b) respectively. The samples were mainly discussed in context of phonon coherence and its imaged using SEM and images were further analyzed using implication on thermal transport. In a porous material thermal MATLAB software (supplementary information is available conductivity is influenced by several factors such as porosity, online at stacks.iop.org/NANO/30/265401/mmedia). The pore size and shape, temperature and the emissivity in the effective filling fraction was calculated for Samples 1 and 2 to be pore [3]. Low thermal conductivity of nanopatterned silicon of 25% and 37%, respectively. The shapes of the pores were not makes it a potential building block for future thermoelectric cylindrical, therefore we computed the distribution of the areas of devices. We have previously investigated the influence of the pores. Due to the computational reasons, in the MC model we short-range disorder in Si membrane-based two-dimensional assumed a circular distribution of the pores whilst maintaining PnCs on their GHz and THz phononic properties and showed the filling fraction and the area distribution of the samples, as that although low-frequency phonon modes are affected by shown in figures 1(c) and (d). The comparison between cumu- periodicity, their impact is not sufficient to affect thermal lative sum of the areas of the pores obtained experimentally and conductivity at room temperature (RT). For samples with a used in the simulations is shown in figures 1(e) and (f). The insets period of 300 nm we observed purely diffusive thermal in figure 1 display the distribution of the pores’ diameters transport and obtained the same value for the thermal con- assuming a circular shape. ductivity for ordered and disordered PnCs [4]. When the pore and pitch sizes are in the order of few 2.2. Two-laser Raman thermometry hundreds of nanometers, the position of each pore has been reported to be relevant at very low temperatures. Zen et al Thermal conductivity measurements were performed using showed coherent reduction of thermal conductance at sub- 2LRT, which has been successfully applied to 2D free-standing Kelvin temperatures [5]. Maire et al demonstrated the trans- plain [11, 12] and porous membranes [4, 10]. The technique, ition from coherent to purely diffusive heat conduction at shown schematically in figure 2(a), consists of using two lasers, 10 K [6]. In a paper by Anufriev et al the effect of disorder in one of which (405 nm) is a localized steady-state heat source, circular pore PnCs was further studied. In the later study the while the second laser (532 nm) measures temperature profile disorder induced a reduction of the thermal conductivity at utilizing the temperature-dependent frequency of optical pho- RT and was only measurable in the samples with high filling nons [11]. In the case of silicon we use the longitudinal optical fraction and periodicity below 200 nm [7]. phonon mode position (521 cm−1 at RT) as a calibrated temp- In this work we studied two samples in which we did not erature probe. The position of the mode depends on the temp- only vary the position of the pores and filling fraction, but erature with −43.43±0.05 K(cm−1)−1. also the size and shape distribution of the pores. Detailed In the experimental set-up both lasers were focused on sample characterization was performed using scanning elec- the samples using 50×microscope objectives with numer- tron microscopy (SEM). The thermal conductivity of the ical apertures of NA=0.55. Both samples had a non- samples was measured using two-laser Raman thermometry patterned 5×5 μm2 square in the center (figure 1(b)) where (2LRT) and the results were compared with Monte Carlo the heating laser was focused. Finally, the absorbed power (MC) simulations of phonon transport in similarly disordered Pabs was measured in situ for each sample as the difference membranes. between the incident light and the sum of transmitted and reflected light intensities by a calibrated system based on a non-polarizing cube beam splitter. All the measurements were performed in a vacuum of 5 × 10−3 mbar. 2. Results In order to calculate thermal conductivity of the mem- branes we need to solve the Fourier heat equation. Due to the 2.1. Samples small thickness of the membranes we can use the 2D solution Disordered porous membranes can be fabricated using variety for a thermally isotropic medium in steady-state: of techniques, such as nanocrystallisation, block-copolymer Pabs lithography or porous alumina templates. In order to fabricate k=- , (1 ) dT the disordered structures here we have modified the fabrica- 2pt d (ln (r )) tion process based on electron beam lithography (EBL) and reactive ion etching (RIE) on free-standing membranes where r is the distance from the heating spot, t is the mem- (Norcada Inc.) [8]. EBL systems are designed to be robust to brane thickness, k is the thermal conductivity and T is the external noise in order to reproduce the design with the temperature. highest accuracy with reduced shot-noise influence [9]. To The non-patterned silicon membrane with t=100 nm intentionally introduce disorder in our structures, we utilized a was measured as a reference sample. The temperature profile noise-assisted EBL, where strong external vibrations distort at the absorbed power of Pabs =2.18 mW is shown in the beam path. As such, a perfectly periodic targeted design of figure 2(b). The heating laser is placed at r=0, where the 2 Nanotechnology 30 (2019) 265401 M Sledzinska et al Figure 1. SEM images of (a) Sample 1 with 25% filling fraction and (b) Sample 2 with 37% filling fraction and the detail of the heating island located in the center of the sample. Scale bar 5 μm. Theoretical samples used for MC simulations corresponding to (c) Sample 1 and (d) Sample 2. Red and blue bars correspond to the hot and cold side of the sample, respectively. Scale bar 1 μm. Normalized cumulative sum of the areas of the pores in (e) Samples 1, (f) Sample 2; circles correspond the experimentally measured distributions from SEM, red curves are the modeled distributions derived from figures (c) and (d). Insets correspond to the cumulative sum of the diameters, assuming circular pores for a given area distribution. temperature increase is the highest and the temperature decay 3 Wm−1K−1. This represents more than a two-fold reduction is probed between −500 and 500 μm from the center. The compared to bulk silicon at RT [13]. shaded parts are taken into consideration as we are interested Secondly, the two porous samples were measured. The in thermal conductivity near RT. Weak temperature depend- temperature profiles for Samples 1 and 2 are shown in ence of k enables a straightforward procedure for its exper- figures 2(d) and (f), respectively, for a given Pabs. The imental evaluation. Following equation (1), k can be extracted values of the intrinsic thermal conductivity taking determined from the slope of T(ln(r)). The solid line in into account the porosity for these samples were 19±3 and figure 2(c) is the T(ln(r)) linear fit of the experimental 11±3 Wm−1K−1 for the Sample 1 and 2, respectively data (black circles) from which we obtained k=60± (figures 2(e) and (g)). This means that thermal conductivity in 3 Nanotechnology 30 (2019) 265401 M Sledzinska et al Figure 2. (a) Schematic of the two-laser Raman thermometry set-up and an example of a temperature profile of the sample. Temperature profiles and temperatures as a function of ln(r) for: (b), (c) non-patterned, 100 nm thick membrane. (d), (e) Disordered porous membranes with filling fraction of 25%. (f), (g) Disordered porous membranes with filling fraction of 37%. Figure 3. Pore distribution used in Monte Carlo simulations. Sample 1: (a) no overlap (100%), (b) 85% possible overlap (c) 55% possible overlap. Sample 2: (d) no overlap, (e) 85% possible overlap (f) 55% possible overlap. Red and blue rectangular represent the hot and the cold thermostats respectively. the disordered porous membrane configuration is lowered by already proven to be very efficient and reliable. The MC one order of magnitude as compared to bulk silicon at RT. solution of BTE used in this study is derived from a pre- viously developed simulation tool that allows the modeling of plain membranes [14] and porous membranes [15]. In the latter work, mono-dispersed spherical pores or aligned/ 3. Monte Carlo simulations staggered cylindrical pores were considered. The main evol- The Boltzman transport equation (BTE) can be handled with ution of the MC tool in the present work was to make possible different techniques, particularly those developed around time the modeling of poly-dispersed cylindrical pores which are and space discretization (finite volumes, finite elements, etc) allowed to overlap, mimicking the disordered porous mem- and which are efficient for simple geometries. Creating a branes achieved experimentally (figure 3). All the basics of mesh for membranes with a disordered distribution of pore phonon transport in nanostructures (drift and scattering of size and location is challenging and make finite difference carriers) with the MC method can be found in the previously techniques very complex to use. In such a complex geome- cited papers. Concerning physical inputs, the dispersion trical configuration, a MC technique based on energy carrier properties of bulk silicon, assuming isotropic dispersion displacement and scattering is more appropriate and has relations, were used. Similarly, phonon lifetimes were derived 4 Nanotechnology 30 (2019) 265401 M Sledzinska et al Table 1. Summary of the Monte Carlo and experimental results for the cases of varying possible overlap percentage of the pores, resembling sample 1 and sample 2. Sample 1 Sample 2 −1 −1 −1 −1 k (Wm K ) Porosity (%) k (Wm K ) Porosity (%) Overlapping (%) MC (300 K) 59.3±1.3 0 59.3±1.3 0 — MC 35.6±0.1 25 30±0.1 37 No MC 33.1±0.5 26.5 27.4±0.1 38.1 85 MC 32.9±0.1 25.5 27.5±0.2 38 70 MC 32.9±0.1 25.5 25.7±0.6 38 55 MC 33.4±0.1 25.1 27±0.6 36.5 40 MC (400 K) 24.6±0.1 25.5 20.7±0.15 38 55 MC (450 K) 21.9±0.1 25.5 18.5±0.15 38.5 55 Experiment 60±3 0 60±3 0 — Experiment 19±3 25 11±3 37 — from bulk according to the Holland formalism [16]. The expected as large porosity makes the overall behavior of the membrane thickness is taken to be 100 nm (as in the experi- structure diffuse regarding the phonon transport and increas- ment) and random distributions of holes for a porosity of 25% ing this diffuse behavior by pore scattering does not affect the and 37% with overlapping and non-overlapping of holes was calculations. Lastly, the impact of pore placement in the studied. Diffuse reflections are assumed from pore boundaries membrane, meaning either a uniform or a non-uniform dis- and membrane’s surfaces. The width of the samples was tribution on the resulting thermal conductivity was studied. varied from 10 to 40 μm and length of the porous part was set This is undoubtedly the most complicated issue to deal with, to 2 μm, as shown in figure 3. The temperatures set at the hot as it can change the value of the thermal conductivity dras- and cold end were 302 K and 298 K, respectively. In order to tically, up to an order of magnitude depending on the pore reduce the uncertainty, the results were averaged over 16 spatial distribution. Two extreme cases were applied: the pore different membrane configurations for each porosity and can be uniformly located and therefore thermal transport is overlapping parameters. only ruled by the porosity and specularity at pore interface, or First of all, thermal conductivity of a plain 100 nm thick pores can be non-uniformly placed and locally increase the membrane was calculated to be 59.3±1.3 Wm−1K−1, in resistance to phonon transport due to the fact that some areas good agreement with the value obtained experimentally. Then of the membrane have increased apparent porosity. The effect we proceed to calculate the thermal conductivity of the porous of the pore placement will be further discussed in this section. membranes, assuming circular shape of the pores (non- Apart from the pore placement, it was previously repor- overlapping) and same filling fractions as the experimental ted that different shapes of pores (circular, square and trian- samples. The results obtained for samples 1 and 2, at RT, are gular) would give varying thermal conductivity, however of 35 and 30 Wm−1K−1, respectively. these simulations were made for pores of a significantly smaller size of about a few nanometers [19]. In the case of a disordered lattice of square pores the k values were 30% 4. Discussion lower than that of their aligned counterparts [20]. Therefore, from MC simulations reported previously there is a clear To the best of our knowledge, there is only one paper dealing connection between the position/shape of the pores and the with disordered porous membranes with similar geometric thermal conductivity. parameters. In the work by Wolf et al numerical simulations In our work, the main difference between the exper- were carried out on porous thin films with thickness of imental samples and the modeled samples is the shape of the 100 nm and pore diameter of 50 nm, with different pore pores. While in the model we assume a circular shape for roughness (from diffuse to specular), and random pore loca- computational simplicity, the actual shape of the pores in the tions in the nanostructure [17]. Among the different outputs fabricated sample is undefined. In order to partly reproduce of the work by Wolf et al three main conclusions were drawn. this randomness in form, holes were allowed to overlap, Firstly, increasing porosity leads to thermal conductivity whilst maintaining the filling fraction constant (figures 1(c) reduction: a three-fold reduction was calculated from unpat- and (d)). Overlapping is introduced by weakening the lim- terned to 35% porosity. This has been observed in several itation introduced in the previous simulations that the minimal studies dealing with nanoporous materials, and the results are distance between pores is defined when their edges touch, i.e. in the same range as that of those in our work [6, 7, 18]. the center-to-center distance is equal to Ri+Rj where Ri and Secondly, increasing pore roughness, assuming that pores Rj are the pore radii. This distance was instead varied between diffusely scatter phonons, also reduces the thermal con- 40% and 85% of the previously set limit of Ri+Rj (figure 3). ductivity; this is also observed in our simulation with a lim- The results for different percentage of allowed overlapping ited impact upon increasing porosity. Once again, this is are summarized in table 1. For sample 1 there is a slight 5 Nanotechnology 30 (2019) 265401 M Sledzinska et al Figure 4. Simulated map of (a)/(d) temperature distribution, (b)/(e) heat flux and (c)/(f) volume distribution (white area present no porosity, while dark area are representative of high porosity) in Sample 1/Sample 2, assuming 55% of hole overlap. The white circles correspond to the overlapping pores present in the samples. decrease of about 2 Wm−1K−1 in thermal conductivity for a to its higher porosity. Another example of a phonon bottle- 55% of overlapping. More significant results are obtained neck is marked with red arrow in figure 4 (f). In that region, for sample 2. In this case we obtained a decrease of constituted of various holes either overlapping or placed close 4.5 Wm−1K−1 for 55% overlapping. In both simulated to each other, it can be seen that the heat flux is almost zero. samples, k is seen to reach a minimal value at 55% of over- This effect is robust in temperature. We have performed the lapping, and increases with a further decrease of the center- simulations at 200, 300 and 400 K and for all three tem- to-center distance. This seems to be related to the optimal peratures we can see the same phonon blocking behavior (see value for overlapping giving maximum blocking of the direct supplementary information). paths of heat flow. The effect of phonon bottlenecks on thermal transport We have further studied the influence of the pore pla- has also been recently discussed by Romano et al [20]. Even cement on the heat transport in porous membranes. For this though the simulations were performed for much smaller we calculated maps of temperature and heat flux for Samples structures (periods of 10 nm) the results were similar to the 1 and 2, assuming the 55% hole overlap configuration. The ones shown here. The formation of high-flux heat regions is simulation domain is of 2.4 μm by 20 μm which is important irregular as it depends on the pore configuration and the for nanoscale heat transport calculations, even for MC density of pores along the heat flux direction has a significant simulations. In this domain we have considered a discretiza- influence on thermal conductivity. The disordered circular tion of 12 by 20 cells as a reasonable ratio between spatial crystals were found to have average thermal conductivity resolution and the number of sampled phonons (between values of up to 15% lower than that of their aligned coun- 30 000 and 60 000 distinct energy bundles) which have to terparts. This is in line with what was observed in our sam- be followed during all the simulation time steps (typically ples, where some parts of the membrane exhibit higher pore 250 000 time steps of 1 ps). density than others. This naturally affects the resulting k both In figures 4(a) and (d) the temperature distributions are in experimental and MC simulations. shown. For both samples the temperature is not homogenous We have calculated the temperature dependence of the within the sample. The same applies to the heat fluxes, shown in figures 4(b) and (e). Instead, both temperature and heat flux thermal conductivity of porous samples with 55% overlap in depend on the position in the sample. In figures S4 and S5 in the range of 250–500 K (see figures S8 in supplementary supplementary information we show the cross sections of the information). For these simulations the temperature gradient temperature and heat flux maps in different points of the remains equal to 4 K, i.e. ±2 K around the average temper- samples. ature of the thin film. As expected thermal conductivity Both temperature and heat flux are related to the map of decreases accordingly to the evolution of the phonon popu- density, shown in figure 4 (c) and (f). In the low density lation and the scattering processes, mostly Umklapp pro- regions (i.e. regions where more holes are present) the heat cesses, as temperature increases. As the experimental results flow is efficiently blocked and the resulting temperature from are an average of the points taken in the range of temperatures linear variation between hot and cold thermostats. At such between the 500 and 350 K, we can compare them with the locations, we can even see ‘hot spots’ (green arrow in figure 4 simulated results at 400 K, which are 24.6±0.1 and (d)). The regions with low density can be understood as a 20.7±0.15 Wm−1K−1, and 450 K, which are 21.9±0.1 phonon bottlenecks. A phonon bottleneck has been defined as and 18.5±0.15 Wm−1K−1, for samples with 25% and 37% a set of pores representing the highest local resistance to porosity, respectively. These values are in reasonable agree- phonon transport [20]. This is more visible for Sample 2, due ment with the experimental results. The possible reasons for 6 Nanotechnology 30 (2019) 265401 M Sledzinska et al the small differences in the experimental and calculated Acknowledgments values are discussed in the further section. One effect, which accounts for the lower k of exper- The ICN2 is funded by the CERCA program/Generalitat de imental samples, is the roughness and the possible native Catalunya. ICN2 is supported by the Severo Ochoa program oxides at the edges of the pores. We have estimated that from Spanish MINECO (Grant No. SEV-2017-0706). We the roughness for samples fabricated using EBL and RIE to acknowledge the financial support from the Spanish MINECO be below 5 nm [4]. However, it is more pronounced in the project PHENTOM (FIS2015-70862-P). We acknowledge the samples reported in this work, being approximately 7 nm. financial support from the French ANR with the project This roughness comes from the particular fabrication process MESOPHON (ANR-15-CE30-0019). BG acknowledges the applied here. It would be logical to assume that this layer support from the Foundation for Polish Science (Homing/ inside the pores has lower thermal conductivity than the 2016-1/2 and First Team POIR.04.04.00-00-5D1B/18-00) and ERC AdG SmartPhon (Grant No. 694977). crystalline silicon of the membrane. Verdier et al have showed that amorphous layers on the pores’ surface can drastically reduce the overall thermal conductivity of the system [21]. They affect thermal transport through possible ORCID iDs localized modes that do not contribute to the energy transport in the material. Therefore we believe that the partly amor- Marianna Sledzinska https://orcid.org/0000-0001- phous layer adds to the overall low thermal conductivity. 8592-1121 Finally, from the SEM images we also note that there is a Bartlomiej Graczykowski https://orcid.org/0000-0003- non-negligible amount of small pores (approx. 50 nm dia- 4787-8622 meter) which are not fully etched-through the whole thickness Francesc Alzina https://orcid.org/0000-0002-7082-0624 of the membrane (figures A1). They were not taken into Umberto Melia https://orcid.org/0000-0003-3033-0505 account while calculating the filling fraction, but they con- Konstantinos Termentzidis https://orcid.org/0000-0002- tribute to the partial reduction of the thermal conductivity, as 8521-7107 their phonon scattering efficiency is larger than big pores. David Lacroix https://orcid.org/0000-0001-6067-8524 Clivia M Sotomayor Torres https://orcid.org/0000-0001- 9986-2716 5. Conclusions In this work we studied two disordered samples with filling References fractions of 25% and 37%, for which we measured the thermal conductivity to be 19±3 Wm−1K−1 and 11± [1] Moyroud E et al 2017 Disorder in convergent floral nanostructures enhances signalling to bees Nature 550 3 Wm−1K−1, respectively. Using MC simulations we have 469–74 shown that by introducing disorder not only in the location, [2] Hsieh P, Chung C, McMillan J F, Tsai M, Lu M, but also in the shape of the pores we can effectively reduce Panoiu N C and Wong C W 2015 Photon transport enhanced thermal conductivity. This effect is more pronounced for by transverse anderson localization in disordered superlattices Nat. Phys. 11 268 samples with high filling fraction. According to the simula- [3] Sugawara A and Yoshizawa Y 1962 An experimental tions, for the sample with 37% filling fraction a decrease in investigation on the thermal conductivity of consolidated thermal conductivity of approximately 15% can be achieved porous materials J. Appl. Phys. 33 3135–8 by introducing 55% of overlap between the pores. Using MC [4] Wagner M R, Graczykowski B, Reparaz J S, El Sachat A, Sledzinska M, Alzina F and Sotomayor Torres C M 2016 simulations we have also calculated distributions of temper- Two-dimensional phononic crystals: disorder matters Nano ature and heat flux for disordered samples with 55% hole Lett. 16 5661–8 overlap. In these samples both heat flux and temperature do [5] Zen N, Puurtinen T A, Isotalo T J, Chaudhuri S and not have a uniform distribution. From the simulations we Maasilta I J 2014 Engineering thermal conductance using a two-dimensional phononic crystal Nat. Commun. 5 3435 could identify the particular regions which efficiently block [6] Maire J, Anufriev R, Yanagisawa R, Ramiere A, Volz S and heat transport. This further supports the statement that we can Nomura M 2017 Heat conduction tuning by wave nature of tune the heat transport by different pore arrangements. phonons Sci. Adv. 3 e1700027 We have shown using numerical methods, that for a [7] Anufriev R, Ramiere A, Maire J and Nomura M 2017 Heat guiding and focusing using ballistic phonon transport in given porosity of the sample, we can vary the thermal con- phononic nanostructures Nat. Commun. 8 15505 ductivity by varying the position and shape of the pores. This [8] Sledzinska M, Graczykowski B, Alzina F, Santiso Lopez J and could have an important impact on the fabrication of porous Sotomayor Torres C M 2016 Fabrication of phononic membranes for thermoelectric applications. We have also crystals on free-standing silicon membranes Microelectron. Eng. 149 41–5 discussed other possible factors which can influence the [9] Kruit P and Steenbrink S W H K 2006 Shot noise in electron- thermal conductivity of the experimental samples, such as beam lithography and line-width measurements Scanning roughness and the presence of not fully etched-through pores. 28 20–6 7 Nanotechnology 30 (2019) 265401 M Sledzinska et al [10] Graczykowski B et al 2017 Thermal conductivity and air- [16] Holland M G 1964 Phonon scattering in semiconductors mediated losses in periodic porous silicon membranes at from thermal conductivity studies Phys. Rev. 134 high temperatures Nat. Commun. 8 415 A471–80 [11] Reparaz J S, Chavez-Angel E, Wagner M R, Graczykowski B, [17] Wolf S, Neophytou N and Kosina H 2014 Thermal Gomis-Bresco J, Alzina F and Sotomayor Torres C M 2014 conductivity of silicon nanomeshes: effects of porosity and A novel contactless technique for thermal field mapping and roughness J. Appl. Phys. 115 204306 thermal conductivity determination: two-laser raman [18] Anufriev R, Maire J and Nomura M 2016 Reduction of thermometry Rev. Sci. Instrum. 85 034901 thermal conductivity by surface scattering of phonons [12] Sledzinska M et al 2016 Thermal conductivity of MoS2 in periodic silicon nanostructures Phys. Rev. B 93 polycrystalline nanomembranes 2D Mater. 3 035016–22 045411 [13] Maycock P D 1967 Thermal conductivity of silicon, [19] Romano G and Grossman J C 2014 Toward phonon-boundary germanium, III–V compounds and III–V alloys Solid⋅State engineering in nanoporous materials Appl. Phys. Lett. 105 Electron. 10 161–8 033116 [14] Lacroix D, Joulain K and Lemonnier D 2005 Monte Carlo [20] Romano G and Grossman J C 2017 Phonon bottleneck transient phonon transport in silicon and germanium at identification in disordered nanoporous materials Phys. Rev. nanoscales Phys. Rev. B 72 064305 B 96 115425 [15] Jean V, Fumeron S, Termentzidis K, Tutashkonko S and [21] Verdier M, Termentzidis K and Lacroix D 2016 Crystalline- Lacroix D 2014 Monte Carlo simulations of phonon amorphous silicon nano-composites: nano-pores and nano- transport in nanoporous silicon and germanium J. Appl. inclusions impact on the thermal conductivity J. Appl. Phys. Phys. 115 024304 119 175104 8