Heat Transfer in Porous Low–k Materials: Modeling Based on Fractional Calculus and Material Structure Fractality

Heat Transfer in Porous Low–k Materials: Modeling Based on Fractional Calculus and Material Structure Fractality | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Heat Transfer in Porous Low–k Materials: Modeling Based on Fractional Calculus and Material Structure Fractality Valerii Arkhincheev, Mungunsuvd Gerelt-Od, Mikhail Baklanov This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8804757/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The influence of fractal pore structures on heat transfer in porous low –k films, focusing on organosilicate (OSG) materials with porosity near the percolation threshold, is investigated. Thermal transport deviates from classical behavior at length scales comparable to characteristic pore sizes, with effective thermal conductivity governed not only by porosity but also by the topology of the pore network. Fractal organization of the pores modifies phonon scattering, producing measurable non-classical thermal behavior. Fractional-order heat transport models capture these effects, revealing that the observed deviations arise from a broad, hierarchical distribution of transport length scales. Our results highlight that engineered multiscale pore architectures can be used to tailor thermal transport in low –k dielectric materials. Analytical expressions for the effective thermal conductivity tensor, incorporating finite pore lengths and the fractal nature of the network, are derived, providing a predictive framework for designing porous dielectrics with controlled thermal properties. Physical sciences/Engineering Physical sciences/Materials science Physical sciences/Mathematics and computing Physical sciences/Physics low–k films percolation threshold fractals heat transfer Figures Figure 1 Figure 2 Figure 3 1. Introduction The interconnects of modern ULSI (Ultra Large Scale Integration) devices use a metal conductor (Cu, Co, Ru), an ultrathin diffusion barriers (Ta/TaN as an inter-metal barrier and SiCN as an interlayer), and a porous OSG low dielectric constant (low –k ) dielectric (often referred to as silicon carbon oxyhydride (SiCOH) 1 , 2 , 3 (Fig. 1 ). While porous low –k materials effectively reduce parasitic capacitance, their extremely low thermal conductivity ( λ < 0.5 W·m⁻ 1 ·K⁻ 1 ) creates a severe heat-dissipation bottleneck. Thermal transport is further degraded by size-induced conductivity suppression in ultrathin Cu and barrier layers, and by Kapitza resistance 4 at Cu/TaN and TaN/low –k interfaces, originating from phonon spectrum mismatch. As a result, inefficient heat removal promotes localized hot spots, accelerating electromigration, dielectric breakdown, and long-term reliability degradation. Accurate characterization of thermal transport in low –k films, barriers, and interfaces is therefore essential for advanced interconnect design. 5 In low –k dielectrics, which are particularly critical for thermal management in advanced interconnects, early quantitative studies established the dominant role of nanoscale structure in limiting heat transport. The introduction of nanoscale porosity strongly reduces the effective phonon mean free path through enhanced boundary scattering, while pore size distribution, shape, and connectivity further modulate thermal conductivity by controlling the continuity and percolation of the solid backbone. As porosity increases, the suppression of thermal conductivity is therefore not governed solely by the void volume fraction, but by the progressive fragmentation of the heat-conducting network. When the characteristic structural length scales approach or fall below the dominant phonon mean free path, thermal transport transitions from a bulk-like regime to one governed by boundary scattering and solid-phase percolation, resulting in a pronounced and often non-linear reduction of thermal conductivity. The majority of available experimental techniques, reported data, and their comprehensive analysis have been systematically reviewed by Plawsky et al., 6 and Cahill et al. 5 . Table 1 summarizes representative values compiled from these reviews. Hybrid organosilicate dielectrics (silsesquioxanes) occupy an intermediate regime, exhibiting thermal conductivities comparable to xerogels and polymeric low –k films, but experience additional phonon scattering due to abrupt density and elastic mismatches at organic–inorganic interfaces, as well as defects intrinsic to sol–gel–derived networks. As the organic content increases, thermal conductivity progressively approaches polymer-like values, reflecting a trade-off between reduced defect density and diminished phonon velocity. Importantly, these trends cannot be fully captured by effective-medium or purely Euclidean transport models, as increasing porosity transforms heat conduction into a geometrically complex, scale-dependent process. 7 This observation motivates a description of thermal transport in porous low –k dielectrics that explicitly accounts for the fractal nature of the pore network and its role in anomalous, non-diffusive heat conduction. It is notable that the highest thermal conductivity values are observed for the sintered porous xerogel film 6, 8 . The sintering process reduces both matrix- and pore-induced phonon scattering by healing the microstructure: microcracks are eliminated, microporosity is reduced 9 , residual organic content is removed, and both the pore size and pore shape distributions become narrower 10 . Collectively, these changes reduce the degree of microstructural disorder and effectively shift the pore network away from a highly fractal, multi-scale architecture toward a more homogeneous morphology with a reduced fractal dimensionality. This demonstrates that thermal transport in porous xerogel films is governed not only by porosity, but critically by the fractal nature of the pore–matrix microstructure and its associated length-scale hierarchy. Table 1 Thermal conductivity versus dielectric constant for different materials used for ULSI interconnects. Dielectric material Dielectric Constant k Thermal Conductivity, λ (W/(m·K)) Note SiO 2 3.9–4.1 1.0–1.4 Dense Nearly dense SiCOH 2.7–3.0 0.4–0.7 Moderate porosity ( 25%) Polymeric (SiLK, PI) 2.6–2.9 0.1–0.3 Very low porosity Air (in pores) 1.0 0.026 Explains the strong drop in λ Liu et al. 11 also demonstrated that 50% porosity in MSQ based low –k (methylsilsesquioxane) reduces thermal conductivity λ to ~ 0.2 W·m⁻ 1 ·K⁻ 1 , revealing a fundamental trade-off between dielectric constant and thermal performance. Subsequent work showed that film thickness introduces an independent size effect: even non-porous OSG low –k materials exhibit order-of-magnitude reductions in λ when thinned below ~ 100 nm, highlighting the dominance of boundary-induced phonon scattering 6 . Later reviews and comparative studies consolidated these findings, identifying porosity, matrix disorder, and pore size distribution as the primary determinants of λ, and establishing amorphous nanofilms as fundamentally limited thermal conductors. 5, 12 Materials with ordered porosity (periodic mesoporous organosilicates, PMOs) demonstrate improved heat transfer along the channels; however, the effect of porosity is dominant. 13 Despite this progress, existing descriptions of thermal transport largely rely on effective-medium or phenomenological approaches that implicitly assume a Euclidean pore geometry characterized by a representative length scale. In a recent study 14 , the percolation-related phenomena in porous low –k dielectrics were analysed and demonstrated that electrical leakage, dielectric breakdown, and mechanical degradation are governed by distinct classes of connectivity thresholds. While this framework successfully rationalizes a wide range of experimental trends, it treats the percolating pore network primarily as a topological object described by connectivity and correlation length, leaving its geometrical nature largely unexplored. Crucially, percolation theory alone does not specify whether the connected pore network is Euclidean or scale-invariant. This distinction is essential for transport phenomena, since heat and mass transfer are fundamentally sensitive to the geometry of the conducting pathways. The present work addresses this gap by establishing the fractal nature of the pore network using adsorption data and developing a corresponding theoretical framework for anomalous thermal transport. We explore the application of fractional calculus and the comb-structure model to describe heat transport in porous low –k dielectrics, with particular emphasis on how fractal pore geometry governs effective thermal conductivity and enables new strategies for thermal property engineering. 2. Fractality of Porosity in Low Dielectrics. 2. Fractality of Porosity in Low –k Dielectrics. Organosilicate glass (OSG) based low –k materials, mesoporous silicates, and hybrid organic-inorganic compounds are characterized by a complex hierarchical pore structure with isolated, fully interconnected and fractal structure. A range of experimental techniques, including positron annihilation lifetime spectroscopy (PALS), ellipsometric porosimetry (EP), small-angle X-ray and neutron scattering (SAXS/SANS and GISAXS), and transmission electron microscopy, have demonstrated that pore distributions in many synthesized low– k materials deviate from a uniformly random arrangement. 15 , 16 , 17 Instead, the pore size distributions exhibit self-similar (scaling) organization over a specific length scale, where pore clusters form intricately branched aggregates characteristic of fractal objects. Reported fractal dimensions typically range from 2.3 to 2.8. 18 , 19 , 20 Recently developed low– k materials are periodic mesoporous organosilicates (PMOs), which may exhibit ordered porosity, incorporate organic bridges that replace a fraction of the oxygen atoms in the silica-like network, and possess terminal methyl groups on the pore wall surfaces 21 (Fig. 2 a). For the present work, we focused on ethylene bridged PMO films of the type described in Refs. 14,21 For the quantitative analysis of adsorption mechanisms in meso- and microporous samples, the Frenkel–Halsey–Hill (FHH) model was employed. 22 In this model, the interaction between the adsorbate and the surface is assumed to be dominated by long-range van der Waals forces, and multilayer adsorption governs the growth of the adsorbed layer. According to the FHH model, the dependence of surface coverage on relative pressure can be expressed as: $$\:\begin{array}{cccc}&\:\text{l}\text{n}V=\text{A}+({D}_{f}-3)\text{l}\text{n}[\text{l}\text{n}(P/{P}_{0}\left)\right]&\:&\:\end{array}$$ 1 , relates the adsorbed volume V to the relative pressure P/P 0 , with D f representing the effective fractal dimension of the surface. 23 In logarithmic coordinates, this relationship highlights linear regions corresponding to specific adsorption regimes. Deviations from ideal linearity or changes in slope reflect structural heterogeneity and fractality of the surface, which in turn influence percolation pathways and connectivity in the pore network. Figure 3 . FTIR spectra and IPA adsorption isotherms plotted in FHH coordinates. IPA adsorption on the fully hydrophilic, CH 3 -free low –k surface (700°C) exhibits linear behavior, indicating a fractal pore structure with a surface fractal dimension of D f = 2.63. The low –k film annealed at 700°C is CH 3 -free, and the blue shift of the Si–O–Si stretching band indicates the formation of a silica-like network. In contrast, IPA adsorption on CH 3 -containing low –k surfaces does not reliably reflect the intrinsic surface chemistry. The extracted fractal dimension D f depends on adsorbate–surface interactions, which are sensitive to surface chemistry. 23 For instance, IPA adsorption on SiO 2 surfaces varies with silanol content, with OH-rich surfaces yielding a stable fractal dimension of ~ 2.6. This reflects increased surface accessibility rather than changes in underlying pore geometry. Applying the FHH analysis to CH 3 -free samples annealed at 700°C gave D f = 2.63, with similar values observed for samples where CH 3 groups were removed by H 2 plasma treatment. At 300 and 400°C the surface still contains a significant density of CH 3 groups, as evidenced by FTIR (peak at 1275–1280 cm –1 ), which alters IPA–surface interactions. Under these conditions, the FHH-derived slope reflects changes in adsorbate chemistry and accessibility rather than the intrinsic surface roughness, making the extracted D f unreliable and not representative of the underlying pore geometry. The fractal dimension D f (where 2 < D f <3) quantitatively describes the degree to which the porous phase fills space. A value of D f ~ 2.3–2.8 signifies a "surface fractal", characterized by an extremely developed and rugged matrix-pore interface. This implies that the pores are not isolated but constitute a highly branching, volume-permeating network. The formation of such fractal structures results from kinetically controlled synthesis processes. Common fabrication methods for low– k materials—such as sol-gel processing, porogen removal, and plasma chemical deposition—are inherently non-equilibrium. Within these processes, nanoparticle aggregation occurs via mechanisms like diffusion-limited aggregation (DLA) or cluster-cluster aggregation (CCA), leading to fractal structures with D f < 3. Low– k materials can concurrently contain micropores (within the walls) and mesopores (between diffusion aggregates), resulting in a hierarchical, self-similar architecture. The established fractal nature of pore distribution in low –k materials, with D f ~ 2.3–2.8, is central to understanding their "structure-property" relationships. This dimension reflects a critical compromise between the goal of minimizing the dielectric constant and the concurrent requirements for mechanical integrity, operational reliability, and adequate thermal conductivity. The fractal pore structure can be effectively modeled using either the random percolation cluster model or deterministic fractal sets (e.g., the Sierpinski carpet or Cantor set 24 , 25 ). These structures are characterized by several key features: self-similarity across multiple scales, a non-integer (fractional) fractal dimension, and power-law behavior in their correlation functions. 3. Comb Structure Model for Heat Transfer in Porous Materials. The comb structure model, originally developed to describe diffusion, provides the simplest representation of a percolation cluster. It allows for an analytical justification of the subdiffusive nature of random processes in percolation systems 26 , 27 , 28 . In the present work, this model is adapted to describe heat transfer in porous low –k dielectrics. By analogy with the diffusion tensor, we therefore introduce a thermal conductivity tensor into Fourier's law \(\:{j}_{i}=-{\lambda\:}_{ij}\frac{\partial\:T}{\partial\:{x}_{j}}\) , expressed in the following form 29 , 30 .: $$\:{\lambda\:}_{ij}=\left(\begin{array}{ccc}{\lambda\:}_{1}\delta\:\left(y\right)\delta\:\left(z\right)&\:0&\:0\\\:0&\:{\lambda\:}_{2}\delta\:\left(z\right)&\:0\\\:0&\:0&\:{\lambda\:}_{3}\end{array}\right)$$ 2 , where λ 1 is the thermal conductivity along the solid skeleton of the material (X-axis); λ 2 is the thermal conductivity along the porous branch of the 1st generation (Y-axis); λ 3 is the thermal conductivity along the porous branch of the 2nd generation (Z-axis). 4. Fractional-Order Equations for Anomalous Heat Transfer in Porous Structures Combining Fourier's law with the continuity equation yields the heat conduction equation: $$\:\left[\frac{\partial\:}{\partial\:t}-{\lambda\:}_{1}\delta\:\left(y\right)\delta\:\left(z\right)\frac{{\partial\:}^{2}}{\partial\:{x}^{2}}-{\lambda\:}_{2}\delta\:\left(z\right)\frac{{\partial\:}^{2}}{\partial\:{y}^{2}}-{\lambda\:}_{3}\frac{{\partial\:}^{2}}{\partial\:{z}^{2}}\right]T\left(x,y,z,t\right)={Q}_{0}\delta\:\left(t\right)\delta\:\left(x\right)\:\delta\:\left(y\right)\delta\:\left(z\right)$$ 3 , where T i s the temperature and Q ₀ represents a point heat source. We seek a solution of the form 29 : $$\:T\left(k,y,z,s\right)=g\left(k,s\right)\text{e}\text{x}\text{p}(-{\lambda\:}_{y}\left|y\right|-{\lambda\:}_{z}\left|z\right|)$$ 4 , Substituting Eq. ( 4 ) into Eq. ( 3 ) allows us to determine the parameters λ y and λ z , as well as the function g ( k,s ): $$\:{\lambda\:}_{z}=\sqrt{\frac{s}{{D}_{3}}},\:{\lambda\:}_{y}=\sqrt{\frac{2{D}_{3}{\lambda\:}_{z}}{{D}_{2}}},\dots\:,\:g\left(k,s\right)=\frac{{Q}_{0}}{2{D}_{2}{\lambda\:}_{y}+{D}_{1}{k}^{2}}$$ 5 , From this solution, the mean square displacement along the X -axis is found to be: $$\:⟨{x}^{2}(t)⟩>\:\propto\:({t)}^{1/4}$$ 6 , This result indicates anomalous diffusion in the X direction. The behavior in the Y direction is also anomalous: $$\:⟨{y}^{2}(t)⟩>\:\propto\:({t)}^{1/2}$$ 7 , In contrast, diffusion along the Z axis is normal: $$\:⟨{z}^{2}(t)⟩>\:\propto\:t$$ 8 , Here, the mean square displacements (dispersions) characterize the thermal front width and its propagation speed over time. Using the method of integral equations, the heat transfer problem in porous materials can be reduced to an effective equation in the form of a fractional-order differential Eq. 2 8–31 . The dependencies in equations ( 6 – 8 ) correspond to the following fractional heat conduction equations, respectively: $$\:\frac{{\partial\:}^{\frac{1}{4}}T(x,t)}{\partial\:{t}^{1/4}}={\lambda\:}_{xx}^{eff}\frac{{\partial\:}^{2}T(x,t)}{\partial\:{x}^{2}},\:\frac{{\partial\:}^{\frac{1}{2}}T(x,t)}{\partial\:{t}^{1/2}}={\lambda\:}_{yy}^{eff}\frac{{\partial\:}^{2}T(x,t)}{\partial\:{y}^{2}},\:\frac{\partial\:T(x,t)}{\partial\:t}={\lambda\:}_{zz}^{eff}\frac{{\partial\:}^{2}T(x,t)}{\partial\:{z}^{2}}$$ 9 , Consequently, the components of the thermal conductivity tensor take an operator form, described by fractional-order differential operators 31 , 32 . The order of the fractional time derivative α has a direct physical interpretation: A first-order derivative (α = 1) corresponds to normal heat conduction (classical Fourier's law). A derivative of order α < 1 signifies subdiffusive heat transfer. The specific value of α–1, determined by the difference in the exponents in equations ( 6 ) and ( 7 ), quantifies the slowdown of heat transfer due to the fractal nature of the porous structure. 4.1. Dependence of Effective Thermal Conductivity on Low –k Material Geometry and the Control of Nanothermal Processes In real materials, pores and heat transfer channels have finite dimensions. Accurate modeling of heat transfer in porous structures must therefore account for the finite lengths of pores, denoted L 2 and L 3 , with appropriate boundary conditions: $$\:{J}_{y}\left(y=\pm\:{L}_{2}\right)=0,\:{\:J}_{z}\left(y=\pm\:{L}_{3}\right)=0$$ 10 , These conditions represent the reflection of thermal phonons at pore boundaries. Mathematically, this requires solving the heat conduction equation with reflective boundary conditions. From a physical standpoint, introducing such conditions accounts for the limited nature of thermal pathways. Solving the fractional-order equation with the boundary conditions in Eq. ( 10 ) yields the effective thermal conductivity tensor: $$\:{\lambda\:}_{ij}^{eff}=\left(\begin{array}{ccc}\frac{{\lambda\:}_{1}{\left({\lambda\:}_{3}\right)}^{1/4}}{{\left({\lambda\:}_{2}\right)}^{1/2}{L}_{2}^{2}}&\:0&\:0\\\:0&\:\frac{{\lambda\:}_{1}}{{\left({\lambda\:}_{2}\right)}^{1/2}{L}_{3}}&\:0\\\:0&\:0&\:{\lambda\:}_{3}\end{array}\right)$$ 11 , The dependence of the tensor components on pore channel lengths L 2 , L 3 indicates a means of controlling thermal processes and enabling directional heat removal. Specifically, shorter pores of a given generation lead to larger values of the corresponding effective thermal conductivity component, thereby enhancing heat removal efficiency. Consequently, incorporating finite pore lengths reveals two key principles: Reducing pore size increases the effective thermal conductivity. To structure a system of fractal pores for effective heat removal, pores of subsequent generations should be shorter than those of the previous ones: L 3 < < L 2 < < L 1 Conversely, in the case of large pore lengths L 2 →∞, L 3 →∞, a reduction in thermal conductivity is observed, and such pores act as "thermal barriers". According to Ref. 7 a nonlinear decrease of thermal conductivity with porosity in porous low-k dielectrics is not captured by commonly used effective medium models, including differential effective medium and coherent potential approximations. This behavior suggests that heat transport is increasingly influenced by the connectivity of the remaining solid framework and by interface-dominated phonon scattering, rather than by an averaged homogeneous response. Within this view, the breakdown of effective medium descriptions reflects the loss of continuous heat-carrying pathways as porosity increases. Independent characterization techniques, such as adsorption-based FTIR and porometry measurements, indicate a hierarchical and correlated pore network, often described as fractal-like, with crevices and branched pathways that dominate thermal transport. By emphasizing the role of pore connectivity and hierarchical branching, this perspective provides a physically intuitive framework for understanding the limitations of classical effective medium models in porous low-k materials. In Fig. [4], the measured thermal conductivity for samples with different porosity values is represented by circles for each material. In Ref. [33], these experimental thermal conductivity data were approximated with lines according to the DEM and CP theories. A significant difference was observed between the experimental data and the established models. The authors of [33] proposed a hypothesis regarding the role of residual porogens, which may not increase the density significantly while still affecting thermal conductivity. 5. Conclusions Heat transport in porous low –k dielectric films deviates from classical heat transfer behavior at length scales comparable to characteristic pore sizes, indicating a breakdown of the Fourier description and the emergence of scale-dependent thermal transport. The effective thermal conductivity is governed not only by the overall porosity but also by the topology of the pore network. In particular, fractal characteristics of the pore structure influence phonon scattering processes and lead to measurable modifications of thermal transport. Fractional-order heat transport models provide a consistent description of the experimental data and suggest that the observed non-classical behavior arises from a broad, hierarchical distribution of transport length scales. These results indicate that controlled multiscale pore architectures may offer a pathway for tuning thermal transport in low –k dielectric materials. Declarations Author contributions statement M.R.B. conceived the experiments, M.R.B and M.G. conducted the experiment, M.R.B. and V.A.E. analyzed the results. All authors reviewed the manuscript. Declaration of competing interest The authors declare no competing interests. Funding The authors (M.G. and M.R.B) thank the Russian Science Foundation for financial support [grant No. 23-79-30016]. Author Contribution M.R.B. conceived the experiments, M.R.B and M.G. conducted the experiment, M.R.B. and V.A.E. analyzed the results. All authors reviewed the manuscript. Data Availability All data generated or analysed during this study are included in this published article [and its supplementary information files] References Baklanov, M. R., Ho, P. S. & Zschech, E. Advanced Interconnects for ULSI Devices . (Wiley & Sons, 2012). 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Kings and M.A. Haque, Thermal Conductivity Measurement of Low-k Dielectric Films: Effect of Porosity and Density Journal of electronic materials, Volume 43, pages 746–754, (2014) (2014), DOI: 10.1007/s11664-013-2949-5 Additional Declarations No competing interests reported. Supplementary Files Supplementarymaterialstopaper.docx SupplementaryData.xlsx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8804757","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":595021129,"identity":"9579d283-86ec-4d05-b716-59a8f67822cb","order_by":0,"name":"Valerii Arkhincheev","email":"data:image/png;base64,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","orcid":"","institution":"Ton Duc Thang University","correspondingAuthor":true,"prefix":"","firstName":"Valerii","middleName":"","lastName":"Arkhincheev","suffix":""},{"id":595021130,"identity":"c1091b94-74fe-466d-ba0c-657f6ef8854c","order_by":1,"name":"Mungunsuvd Gerelt-Od","email":"","orcid":"","institution":"MIREA—Russian Technological University (RTU MIREA)","correspondingAuthor":false,"prefix":"","firstName":"Mungunsuvd","middleName":"","lastName":"Gerelt-Od","suffix":""},{"id":595021132,"identity":"dcc73a2e-9185-4de4-b7fb-0a4a00116be5","order_by":2,"name":"Mikhail Baklanov","email":"","orcid":"","institution":"MIREA—Russian Technological University (RTU MIREA)","correspondingAuthor":false,"prefix":"","firstName":"Mikhail","middleName":"","lastName":"Baklanov","suffix":""}],"badges":[],"createdAt":"2026-02-06 08:54:49","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8804757/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8804757/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103222226,"identity":"a6acdee2-7c13-4ca6-a802-98a5949ef6fd","added_by":"auto","created_at":"2026-02-23 10:29:53","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":284593,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSchematic illustration of a modern ULSI interconnect structure\u003c/strong\u003e comprising Cu metal lines, TaN and SiCN barrier layers, and a porous low-\u003cem\u003ek\u003c/em\u003e dielectric, representative of multilevel (10–15) metallization stacks. Heat generation in these structures is highly localized; however, heat is transported predominantly along the metal lines, while vertical heat dissipation is strongly suppressed by the small diameter of the vertical pillars, the low thermal conductivity of the surrounding dielectrics, and significant interfacial thermal resistances. Despite its small thickness, the SiCN layer constitutes a critical thermal bottleneck in the vertical heat-flow path (Kapitza resistance).\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-8804757/v1/e36ed5218ea4ca52086bafbe.png"},{"id":103222228,"identity":"db312f80-72e0-4b1b-be98-35ee0082fddc","added_by":"auto","created_at":"2026-02-23 10:29:53","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":154851,"visible":true,"origin":"","legend":"\u003cp\u003eStructural and theoretical representations of porous low\u003cem\u003e–k\u003c/em\u003e dielectrics. \u003cstrong\u003e(a)\u003c/strong\u003e An idealized three-dimensional representation of a fragment of a porous PMO low–\u003cem\u003ek\u003c/em\u003e dielectric, illustrating ordered porosity with methyl groups localized on the pore wall surfaces. Exposure to 700\u0026nbsp;°C or to hydrogen plasma removes the methyl groups, leading to hydroxylated pore walls (Figure 3). In practice, however, these materials also exhibit pore connectivity across multiple length scales.\u003csup\u003e16,21\u003c/sup\u003e \u003cstrong\u003e(b)\u003c/strong\u003e Idealized comb-structure model used to describe transport on a fractal pore network. The long branch along the X axis coincides with the direction of the PMO channel shown in Fig. 2a, with the backbone representing long-range connected pathways and the transverse branches mimicking multiscale dead-end pores and side channels revealed by adsorption measurements. Transport along this structure is intrinsically non-Euclidean and gives rise to anomalous diffusion and heat conduction governed by the effective spectral dimension. The figure highlights the transition from real porous morphology to an abstract geometrical model capturing the essential features responsible for anomalous thermal transport in low\u003cem\u003e–k\u003c/em\u003e dielectrics.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-8804757/v1/b87c5f621a49298564fc7096.png"},{"id":103222227,"identity":"d7fd6ccd-eb6a-4cd3-9ac6-4e437845cb24","added_by":"auto","created_at":"2026-02-23 10:29:53","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":219174,"visible":true,"origin":"","legend":"\u003cp\u003eFTIR spectra and IPA adsorption isotherms plotted in FHH coordinates. IPA adsorption on the fully hydrophilic, CH\u003csub\u003e3\u003c/sub\u003e-free low\u003cem\u003e–k\u003c/em\u003e surface (700\u0026nbsp;°C) exhibits linear behavior, indicating a fractal pore structure with a surface fractal dimension of \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e = 2.63. The low\u003cem\u003e–k\u003c/em\u003e film annealed at 700\u0026nbsp;°C is CH\u003csub\u003e3\u003c/sub\u003e-free, and the blue shift of the Si–O–Si stretching band indicates the formation of a silica-like network. In contrast, IPA adsorption on CH\u003csub\u003e3\u003c/sub\u003e-containing low\u003cem\u003e–k\u003c/em\u003e surfaces does not reliably reflect the intrinsic surface chemistry.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-8804757/v1/3341c8a3ace854ec68ec18e4.png"},{"id":104403111,"identity":"5096e624-cfa3-40e3-b52d-8a8c5da80ced","added_by":"auto","created_at":"2026-03-11 12:17:30","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1341242,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8804757/v1/97517e62-e700-4fd7-9f39-47b436efb237.pdf"},{"id":103222229,"identity":"078cff4f-f5ba-40e3-a335-ddddc206d5f9","added_by":"auto","created_at":"2026-02-23 10:29:54","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":156874,"visible":true,"origin":"","legend":"","description":"","filename":"Supplementarymaterialstopaper.docx","url":"https://assets-eu.researchsquare.com/files/rs-8804757/v1/f492d7de9569597e9c516acf.docx"},{"id":103222230,"identity":"1bcd77ad-9d1e-4739-83b9-fbd39ec951a0","added_by":"auto","created_at":"2026-02-23 10:29:54","extension":"xlsx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":219070,"visible":true,"origin":"","legend":"","description":"","filename":"SupplementaryData.xlsx","url":"https://assets-eu.researchsquare.com/files/rs-8804757/v1/3f0eedeae870c5849000fb90.xlsx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Heat Transfer in Porous Low–k Materials: Modeling Based on Fractional Calculus and Material Structure Fractality","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eThe interconnects of modern ULSI (Ultra Large Scale Integration) devices use a metal conductor (Cu, Co, Ru), an ultrathin diffusion barriers (Ta/TaN as an inter-metal barrier and SiCN as an interlayer), and a porous OSG low dielectric constant (low\u003cem\u003e\u0026ndash;k\u003c/em\u003e) dielectric (often referred to as silicon carbon oxyhydride (SiCOH) \u003csup\u003e1\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e2\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e3\u003c/sup\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). While porous low\u003cem\u003e\u0026ndash;k\u003c/em\u003e materials effectively reduce parasitic capacitance, their extremely low thermal conductivity (\u003cem\u003eλ\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.5 W\u0026middot;m⁻\u003csup\u003e1\u003c/sup\u003e\u0026middot;K⁻\u003csup\u003e1\u003c/sup\u003e) creates a severe heat-dissipation bottleneck. Thermal transport is further degraded by size-induced conductivity suppression in ultrathin Cu and barrier layers, and by Kapitza resistance \u003csup\u003e4\u003c/sup\u003e at Cu/TaN and TaN/low\u003cem\u003e\u0026ndash;k\u003c/em\u003e interfaces, originating from phonon spectrum mismatch. As a result, inefficient heat removal promotes localized hot spots, accelerating electromigration, dielectric breakdown, and long-term reliability degradation. Accurate characterization of thermal transport in low\u003cem\u003e\u0026ndash;k\u003c/em\u003e films, barriers, and interfaces is therefore essential for advanced interconnect design.\u003csup\u003e5\u003c/sup\u003e\u003c/p\u003e \u003cp\u003eIn low\u003cem\u003e\u0026ndash;k\u003c/em\u003e dielectrics, which are particularly critical for thermal management in advanced interconnects, early quantitative studies established the dominant role of nanoscale structure in limiting heat transport. The introduction of nanoscale porosity strongly reduces the effective phonon mean free path through enhanced boundary scattering, while pore size distribution, shape, and connectivity further modulate thermal conductivity by controlling the continuity and percolation of the solid backbone. As porosity increases, the suppression of thermal conductivity is therefore not governed solely by the void volume fraction, but by the progressive fragmentation of the heat-conducting network. When the characteristic structural length scales approach or fall below the dominant phonon mean free path, thermal transport transitions from a bulk-like regime to one governed by boundary scattering and solid-phase percolation, resulting in a pronounced and often non-linear reduction of thermal conductivity. The majority of available experimental techniques, reported data, and their comprehensive analysis have been systematically reviewed by Plawsky et al.,\u003csup\u003e6\u003c/sup\u003e and Cahill et al.\u003csup\u003e5\u003c/sup\u003e. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e summarizes representative values compiled from these reviews. Hybrid organosilicate dielectrics (silsesquioxanes) occupy an intermediate regime, exhibiting thermal conductivities comparable to xerogels and polymeric low\u003cem\u003e\u0026ndash;k\u003c/em\u003e films, but experience additional phonon scattering due to abrupt density and elastic mismatches at organic\u0026ndash;inorganic interfaces, as well as defects intrinsic to sol\u0026ndash;gel\u0026ndash;derived networks. As the organic content increases, thermal conductivity progressively approaches polymer-like values, reflecting a trade-off between reduced defect density and diminished phonon velocity. Importantly, these trends cannot be fully captured by effective-medium or purely Euclidean transport models, as increasing porosity transforms heat conduction into a geometrically complex, scale-dependent process.\u003csup\u003e7\u003c/sup\u003e This observation motivates a description of thermal transport in porous low\u003cem\u003e\u0026ndash;k\u003c/em\u003e dielectrics that explicitly accounts for the fractal nature of the pore network and its role in anomalous, non-diffusive heat conduction.\u003c/p\u003e \u003cp\u003eIt is notable that the highest thermal conductivity values are observed for the sintered porous xerogel film \u003csup\u003e6,\u003c/sup\u003e\u003csup\u003e8\u003c/sup\u003e. The sintering process reduces both matrix- and pore-induced phonon scattering by healing the microstructure: microcracks are eliminated, microporosity is reduced \u003csup\u003e9\u003c/sup\u003e, residual organic content is removed, and both the pore size and pore shape distributions become narrower \u003csup\u003e10\u003c/sup\u003e. Collectively, these changes reduce the degree of microstructural disorder and effectively shift the pore network away from a highly fractal, multi-scale architecture toward a more homogeneous morphology with a reduced fractal dimensionality. This demonstrates that thermal transport in porous xerogel films is governed not only by porosity, but critically by the fractal nature of the pore\u0026ndash;matrix microstructure and its associated length-scale hierarchy.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThermal conductivity versus dielectric constant for different materials used for ULSI interconnects.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDielectric material\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDielectric Constant \u003cem\u003ek\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eThermal Conductivity, λ (W/(m\u0026middot;K))\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNote\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSiO\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.9\u0026ndash;4.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.0\u0026ndash;1.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eDense\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNearly dense SiCOH\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.7\u0026ndash;3.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4\u0026ndash;0.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eModerate porosity (\u0026lt;\u0026thinsp;25%)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePorous ultralow\u003cem\u003e\u0026ndash;k\u003c/em\u003e SiCOH\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.0\u0026ndash;2.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2\u0026ndash;0.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eHigh porosity (\u0026gt;\u0026thinsp;25%)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePolymeric (SiLK, PI)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.6\u0026ndash;2.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.1\u0026ndash;0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eVery low porosity\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAir (in pores)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.026\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eExplains the strong drop in λ\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eLiu \u003cem\u003eet al.\u003c/em\u003e\u003csup\u003e11\u003c/sup\u003e also demonstrated that 50% porosity in MSQ based low\u003cem\u003e\u0026ndash;k\u003c/em\u003e (methylsilsesquioxane) reduces thermal conductivity λ to ~\u0026thinsp;0.2 W\u0026middot;m⁻\u003csup\u003e1\u003c/sup\u003e\u0026middot;K⁻\u003csup\u003e1\u003c/sup\u003e, revealing a fundamental trade-off between dielectric constant and thermal performance. Subsequent work showed that film thickness introduces an independent size effect: even non-porous OSG low\u003cem\u003e\u0026ndash;k\u003c/em\u003e materials exhibit order-of-magnitude reductions in λ when thinned below ~\u0026thinsp;100 nm, highlighting the dominance of boundary-induced phonon scattering\u003csup\u003e6\u003c/sup\u003e. Later reviews and comparative studies consolidated these findings, identifying porosity, matrix disorder, and pore size distribution as the primary determinants of λ, and establishing amorphous nanofilms as fundamentally limited thermal conductors.\u003csup\u003e5,\u003c/sup\u003e\u003csup\u003e12\u003c/sup\u003e Materials with ordered porosity (periodic mesoporous organosilicates, PMOs) demonstrate improved heat transfer along the channels; however, the effect of porosity is dominant.\u003csup\u003e13\u003c/sup\u003e Despite this progress, existing descriptions of thermal transport largely rely on effective-medium or phenomenological approaches that implicitly assume a Euclidean pore geometry characterized by a representative length scale. In a recent study \u003csup\u003e14\u003c/sup\u003e, the percolation-related phenomena in porous low\u003cem\u003e\u0026ndash;k\u003c/em\u003e dielectrics were analysed and demonstrated that electrical leakage, dielectric breakdown, and mechanical degradation are governed by distinct classes of connectivity thresholds. While this framework successfully rationalizes a wide range of experimental trends, it treats the percolating pore network primarily as a topological object described by connectivity and correlation length, leaving its geometrical nature largely unexplored.\u003c/p\u003e \u003cp\u003eCrucially, percolation theory alone does not specify whether the connected pore network is Euclidean or scale-invariant. This distinction is essential for transport phenomena, since heat and mass transfer are fundamentally sensitive to the geometry of the conducting pathways. The present work addresses this gap by establishing the fractal nature of the pore network using adsorption data and developing a corresponding theoretical framework for anomalous thermal transport. We explore the application of fractional calculus and the comb-structure model to describe heat transport in porous low\u003cem\u003e\u0026ndash;k\u003c/em\u003e dielectrics, with particular emphasis on how fractal pore geometry governs effective thermal conductivity and enables new strategies for thermal property engineering.\u003c/p\u003e"},{"header":"2. Fractality of Porosity in Low Dielectrics.","content":"\u003cdiv class=\"Heading\"\u003e2. Fractality of Porosity in Low\u003cem\u003e\u0026ndash;k\u003c/em\u003e Dielectrics.\u003c/div\u003e \u003cp\u003eOrganosilicate glass (OSG) based low\u003cem\u003e\u0026ndash;k\u003c/em\u003e materials, mesoporous silicates, and hybrid organic-inorganic compounds are characterized by a complex hierarchical pore structure with isolated, fully interconnected and fractal structure. A range of experimental techniques, including positron annihilation lifetime spectroscopy (PALS), ellipsometric porosimetry (EP), small-angle X-ray and neutron scattering (SAXS/SANS and GISAXS), and transmission electron microscopy, have demonstrated that pore distributions in many synthesized low\u0026ndash;\u003cem\u003ek\u003c/em\u003e materials deviate from a uniformly random arrangement.\u003csup\u003e15\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e16\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e17\u003c/sup\u003e Instead, the pore size distributions exhibit self-similar (scaling) organization over a specific length scale, where pore clusters form intricately branched aggregates characteristic of fractal objects. Reported fractal dimensions typically range from 2.3 to 2.8.\u003csup\u003e18\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e19\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e20\u003c/sup\u003e\u003c/p\u003e \u003cp\u003eRecently developed low\u0026ndash;\u003cem\u003ek\u003c/em\u003e materials are periodic mesoporous organosilicates (PMOs), which may exhibit ordered porosity, incorporate organic bridges that replace a fraction of the oxygen atoms in the silica-like network, and possess terminal methyl groups on the pore wall surfaces \u003csup\u003e21\u003c/sup\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea). For the present work, we focused on ethylene bridged PMO films of the type described in Refs. \u003csup\u003e14,21\u003c/sup\u003e For the quantitative analysis of adsorption mechanisms in meso- and microporous samples, the Frenkel\u0026ndash;Halsey\u0026ndash;Hill (FHH) model was employed.\u003csup\u003e22\u003c/sup\u003e In this model, the interaction between the adsorbate and the surface is assumed to be dominated by long-range van der Waals forces, and multilayer adsorption governs the growth of the adsorbed layer.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAccording to the FHH model, the dependence of surface coverage on relative pressure can be expressed as:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{cccc}\u0026amp;\\:\\text{l}\\text{n}V=\\text{A}+({D}_{f}-3)\\text{l}\\text{n}[\\text{l}\\text{n}(P/{P}_{0}\\left)\\right]\u0026amp;\\:\u0026amp;\\:\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003erelates the adsorbed volume \u003cem\u003eV\u003c/em\u003e to the relative pressure \u003cem\u003eP/P\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e, with \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e representing the effective fractal dimension of the surface.\u003csup\u003e23\u003c/sup\u003e In logarithmic coordinates, this relationship highlights linear regions corresponding to specific adsorption regimes. Deviations from ideal linearity or changes in slope reflect structural heterogeneity and fractality of the surface, which in turn influence percolation pathways and connectivity in the pore network.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. FTIR spectra and IPA adsorption isotherms plotted in FHH coordinates. IPA adsorption on the fully hydrophilic, CH\u003csub\u003e3\u003c/sub\u003e-free low\u003cem\u003e\u0026ndash;k\u003c/em\u003e surface (700\u0026deg;C) exhibits linear behavior, indicating a fractal pore structure with a surface fractal dimension of \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e = 2.63. The low\u003cem\u003e\u0026ndash;k\u003c/em\u003e film annealed at 700\u0026deg;C is CH\u003csub\u003e3\u003c/sub\u003e-free, and the blue shift of the Si\u0026ndash;O\u0026ndash;Si stretching band indicates the formation of a silica-like network. In contrast, IPA adsorption on CH\u003csub\u003e3\u003c/sub\u003e-containing low\u003cem\u003e\u0026ndash;k\u003c/em\u003e surfaces does not reliably reflect the intrinsic surface chemistry.\u003c/p\u003e \u003cp\u003eThe extracted fractal dimension \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e depends on adsorbate\u0026ndash;surface interactions, which are sensitive to surface chemistry.\u003csup\u003e23\u003c/sup\u003e For instance, IPA adsorption on SiO\u003csub\u003e2\u003c/sub\u003e surfaces varies with silanol content, with OH-rich surfaces yielding a stable fractal dimension of ~\u0026thinsp;2.6. This reflects increased surface accessibility rather than changes in underlying pore geometry. Applying the FHH analysis to CH\u003csub\u003e3\u003c/sub\u003e-free samples annealed at 700\u0026deg;C gave \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e=\u003c/em\u003e2.63, with similar values observed for samples where CH\u003csub\u003e3\u003c/sub\u003e groups were removed by H\u003csub\u003e2\u003c/sub\u003e plasma treatment. At 300 and 400\u0026deg;C the surface still contains a significant density of CH\u003csub\u003e3\u003c/sub\u003e groups, as evidenced by FTIR (peak at 1275\u0026ndash;1280 cm\u003csup\u003e\u0026ndash;1\u003c/sup\u003e), which alters IPA\u0026ndash;surface interactions. Under these conditions, the FHH-derived slope reflects changes in adsorbate chemistry and accessibility rather than the intrinsic surface roughness, making the extracted \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e unreliable and not representative of the underlying pore geometry.\u003c/p\u003e \u003cp\u003eThe fractal dimension \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e (where 2\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e \u0026lt;3) quantitatively describes the degree to which the porous phase fills space. A value of \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e ~ 2.3\u0026ndash;2.8 signifies a \"surface fractal\", characterized by an extremely developed and rugged matrix-pore interface. This implies that the pores are not isolated but constitute a highly branching, volume-permeating network. The formation of such fractal structures results from kinetically controlled synthesis processes. Common fabrication methods for low\u0026ndash;\u003cem\u003ek\u003c/em\u003e materials\u0026mdash;such as sol-gel processing, porogen removal, and plasma chemical deposition\u0026mdash;are inherently non-equilibrium. Within these processes, nanoparticle aggregation occurs via mechanisms like diffusion-limited aggregation (DLA) or cluster-cluster aggregation (CCA), leading to fractal structures with \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e \u0026lt; 3. Low\u0026ndash;\u003cem\u003ek\u003c/em\u003e materials can concurrently contain micropores (within the walls) and mesopores (between diffusion aggregates), resulting in a hierarchical, self-similar architecture.\u003c/p\u003e \u003cp\u003eThe established fractal nature of pore distribution in low\u003cem\u003e\u0026ndash;k\u003c/em\u003e materials, with \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e~\u003c/em\u003e2.3\u0026ndash;2.8, is central to understanding their \"structure-property\" relationships. This dimension reflects a critical compromise between the goal of minimizing the dielectric constant and the concurrent requirements for mechanical integrity, operational reliability, and adequate thermal conductivity. The fractal pore structure can be effectively modeled using either the random percolation cluster model or deterministic fractal sets (e.g., the Sierpinski carpet or Cantor set\u003csup\u003e24\u003c/sup\u003e,\u003csup\u003e25\u003c/sup\u003e ). These structures are characterized by several key features: self-similarity across multiple scales, a non-integer (fractional) fractal dimension, and power-law behavior in their correlation functions.\u003c/p\u003e"},{"header":"3. Comb Structure Model for Heat Transfer in Porous Materials.","content":"\u003cp\u003eThe comb structure model, originally developed to describe diffusion, provides the simplest representation of a percolation cluster. It allows for an analytical justification of the subdiffusive nature of random processes in percolation systems \u003csup\u003e26\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e27\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e28\u003c/sup\u003e. In the present work, this model is adapted to describe heat transfer in porous low\u003cem\u003e\u0026ndash;k\u003c/em\u003e dielectrics. By analogy with the diffusion tensor, we therefore introduce a thermal conductivity tensor into Fourier's law \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{j}_{i}=-{\\lambda\\:}_{ij}\\frac{\\partial\\:T}{\\partial\\:{x}_{j}}\\)\u003c/span\u003e\u003c/span\u003e, expressed in the following form \u003csup\u003e29\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e30\u003c/sup\u003e.:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}_{ij}=\\left(\\begin{array}{ccc}{\\lambda\\:}_{1}\\delta\\:\\left(y\\right)\\delta\\:\\left(z\\right)\u0026amp;\\:0\u0026amp;\\:0\\\\\\:0\u0026amp;\\:{\\lambda\\:}_{2}\\delta\\:\\left(z\\right)\u0026amp;\\:0\\\\\\:0\u0026amp;\\:0\u0026amp;\\:{\\lambda\\:}_{3}\\end{array}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e is the thermal conductivity along the solid skeleton of the material (X-axis); \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e is the thermal conductivity along the porous branch of the 1st generation (Y-axis); \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e is the thermal conductivity along the porous branch of the 2nd generation (Z-axis).\u003c/p\u003e"},{"header":"4. Fractional-Order Equations for Anomalous Heat Transfer in Porous Structures","content":"\u003cp\u003eCombining Fourier's law with the continuity equation yields the heat conduction equation:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\left[\\frac{\\partial\\:}{\\partial\\:t}-{\\lambda\\:}_{1}\\delta\\:\\left(y\\right)\\delta\\:\\left(z\\right)\\frac{{\\partial\\:}^{2}}{\\partial\\:{x}^{2}}-{\\lambda\\:}_{2}\\delta\\:\\left(z\\right)\\frac{{\\partial\\:}^{2}}{\\partial\\:{y}^{2}}-{\\lambda\\:}_{3}\\frac{{\\partial\\:}^{2}}{\\partial\\:{z}^{2}}\\right]T\\left(x,y,z,t\\right)={Q}_{0}\\delta\\:\\left(t\\right)\\delta\\:\\left(x\\right)\\:\\delta\\:\\left(y\\right)\\delta\\:\\left(z\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eT i\u003c/em\u003es the temperature and \u003cem\u003eQ\u003c/em\u003e₀ represents a point heat source.\u003c/p\u003e \u003cp\u003eWe seek a solution of the form \u003csup\u003e29\u003c/sup\u003e:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:T\\left(k,y,z,s\\right)=g\\left(k,s\\right)\\text{e}\\text{x}\\text{p}(-{\\lambda\\:}_{y}\\left|y\\right|-{\\lambda\\:}_{z}\\left|z\\right|)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eSubstituting Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) into Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) allows us to determine the parameters \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e\u003cem\u003ez\u003c/em\u003e\u003c/sub\u003e, as well as the function \u003cem\u003eg\u003c/em\u003e(\u003cem\u003ek,s\u003c/em\u003e):\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}_{z}=\\sqrt{\\frac{s}{{D}_{3}}},\\:{\\lambda\\:}_{y}=\\sqrt{\\frac{2{D}_{3}{\\lambda\\:}_{z}}{{D}_{2}}},\\dots\\:,\\:g\\left(k,s\\right)=\\frac{{Q}_{0}}{2{D}_{2}{\\lambda\\:}_{y}+{D}_{1}{k}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eFrom this solution, the mean square displacement along the \u003cem\u003eX\u003c/em\u003e-axis is found to be:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:⟨{x}^{2}(t)⟩\u0026gt;\\:\\propto\\:({t)}^{1/4}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eThis result indicates anomalous diffusion in the \u003cem\u003eX\u003c/em\u003e direction.\u003c/p\u003e \u003cp\u003eThe behavior in the \u003cem\u003eY\u003c/em\u003e direction is also anomalous:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:⟨{y}^{2}(t)⟩\u0026gt;\\:\\propto\\:({t)}^{1/2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eIn contrast, diffusion along the Z axis is normal:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:⟨{z}^{2}(t)⟩\u0026gt;\\:\\propto\\:t$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eHere, the mean square displacements (dispersions) characterize the thermal front width and its propagation speed over time.\u003c/p\u003e \u003cp\u003eUsing the method of integral equations, the heat transfer problem in porous materials can be reduced to an effective equation in the form of a fractional-order differential Eq.\u0026nbsp;2\u003csup\u003e8\u0026ndash;31\u003c/sup\u003e. The dependencies in equations (\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e) correspond to the following fractional heat conduction equations, respectively:\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:\\frac{{\\partial\\:}^{\\frac{1}{4}}T(x,t)}{\\partial\\:{t}^{1/4}}={\\lambda\\:}_{xx}^{eff}\\frac{{\\partial\\:}^{2}T(x,t)}{\\partial\\:{x}^{2}},\\:\\frac{{\\partial\\:}^{\\frac{1}{2}}T(x,t)}{\\partial\\:{t}^{1/2}}={\\lambda\\:}_{yy}^{eff}\\frac{{\\partial\\:}^{2}T(x,t)}{\\partial\\:{y}^{2}},\\:\\frac{\\partial\\:T(x,t)}{\\partial\\:t}={\\lambda\\:}_{zz}^{eff}\\frac{{\\partial\\:}^{2}T(x,t)}{\\partial\\:{z}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eConsequently, the components of the thermal conductivity tensor take an operator form, described by fractional-order differential operators \u003csup\u003e31\u003c/sup\u003e\u003csup\u003e,\u003c/sup\u003e\u003csup\u003e32\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe order of the fractional time derivative α has a direct physical interpretation: A first-order derivative (α\u0026thinsp;=\u0026thinsp;1) corresponds to normal heat conduction (classical Fourier's law). A derivative of order α\u0026thinsp;\u0026lt;\u0026thinsp;1 signifies subdiffusive heat transfer. The specific value of α\u0026ndash;1, determined by the difference in the exponents in equations (\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e) and (\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e), quantifies the slowdown of heat transfer due to the fractal nature of the porous structure.\u003c/p\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e4.1. Dependence of Effective Thermal Conductivity on Low\u003cem\u003e\u0026ndash;k\u003c/em\u003e Material Geometry and the Control of Nanothermal Processes\u003c/h2\u003e \u003cp\u003eIn real materials, pores and heat transfer channels have finite dimensions. Accurate modeling of heat transfer in porous structures must therefore account for the finite lengths of pores, denoted \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e, with appropriate boundary conditions:\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\:{J}_{y}\\left(y=\\pm\\:{L}_{2}\\right)=0,\\:{\\:J}_{z}\\left(y=\\pm\\:{L}_{3}\\right)=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eThese conditions represent the reflection of thermal phonons at pore boundaries. Mathematically, this requires solving the heat conduction equation with reflective boundary conditions. From a physical standpoint, introducing such conditions accounts for the limited nature of thermal pathways.\u003c/p\u003e \u003cp\u003eSolving the fractional-order equation with the boundary conditions in Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e) yields the effective thermal conductivity tensor:\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}_{ij}^{eff}=\\left(\\begin{array}{ccc}\\frac{{\\lambda\\:}_{1}{\\left({\\lambda\\:}_{3}\\right)}^{1/4}}{{\\left({\\lambda\\:}_{2}\\right)}^{1/2}{L}_{2}^{2}}\u0026amp;\\:0\u0026amp;\\:0\\\\\\:0\u0026amp;\\:\\frac{{\\lambda\\:}_{1}}{{\\left({\\lambda\\:}_{2}\\right)}^{1/2}{L}_{3}}\u0026amp;\\:0\\\\\\:0\u0026amp;\\:0\u0026amp;\\:{\\lambda\\:}_{3}\\end{array}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003eThe dependence of the tensor components on pore channel lengths \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e indicates a means of controlling thermal processes and enabling directional heat removal. Specifically, shorter pores of a given generation lead to larger values of the corresponding effective thermal conductivity component, thereby enhancing heat removal efficiency.\u003c/p\u003e \u003cp\u003eConsequently, incorporating finite pore lengths reveals two key principles:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eReducing pore size increases the effective thermal conductivity.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eTo structure a system of fractal pores for effective heat removal, pores of subsequent generations should be shorter than those of the previous ones: \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e\u0026lt;\u0026thinsp;\u0026lt;\u0026thinsp;L\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e\u0026lt;\u0026thinsp;\u0026lt;\u0026thinsp;L\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eConversely, in the case of large pore lengths \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u0026rarr;\u0026infin;, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sub\u003e\u0026rarr;\u0026infin;, a reduction in thermal conductivity is observed, and such pores act as \"thermal barriers\". According to Ref.\u003csup\u003e7\u003c/sup\u003e a nonlinear decrease of thermal conductivity with porosity in porous low-k dielectrics is not captured by commonly used effective medium models, including differential effective medium and coherent potential approximations. This behavior suggests that heat transport is increasingly influenced by the connectivity of the remaining solid framework and by interface-dominated phonon scattering, rather than by an averaged homogeneous response. Within this view, the breakdown of effective medium descriptions reflects the loss of continuous heat-carrying pathways as porosity increases. Independent characterization techniques, such as adsorption-based FTIR and porometry measurements, indicate a hierarchical and correlated pore network, often described as fractal-like, with crevices and branched pathways that dominate thermal transport. By emphasizing the role of pore connectivity and hierarchical branching, this perspective provides a physically intuitive framework for understanding the limitations of classical effective medium models in porous low-k materials.\u003c/p\u003e \u003cp\u003eIn Fig. [4], the measured thermal conductivity for samples with different porosity values is represented by circles for each material. In Ref. [33], these experimental thermal conductivity data were approximated with lines according to the DEM and CP theories. A significant difference was observed between the experimental data and the established models. The authors of [33] proposed a hypothesis regarding the role of residual porogens, which may not increase the density significantly while still affecting thermal conductivity.\u003c/p\u003e \u003c/div\u003e"},{"header":"5. Conclusions","content":"\u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eHeat transport in porous low\u003cem\u003e\u0026ndash;k\u003c/em\u003e dielectric films deviates from classical heat transfer behavior at length scales comparable to characteristic pore sizes, indicating a breakdown of the Fourier description and the emergence of scale-dependent thermal transport.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe effective thermal conductivity is governed not only by the overall porosity but also by the topology of the pore network. In particular, fractal characteristics of the pore structure influence phonon scattering processes and lead to measurable modifications of thermal transport.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eFractional-order heat transport models provide a consistent description of the experimental data and suggest that the observed non-classical behavior arises from a broad, hierarchical distribution of transport length scales. These results indicate that controlled multiscale pore architectures may offer a pathway for tuning thermal transport in low\u003cem\u003e\u0026ndash;k\u003c/em\u003e dielectric materials.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eAuthor contributions statement\u003c/h2\u003e \u003cp\u003eM.R.B. conceived the experiments, M.R.B and M.G. conducted the experiment, M.R.B. and V.A.E. analyzed the results. All authors reviewed the manuscript.\u003c/p\u003e \u003c/p\u003e\u003cp\u003e \u003ch2\u003eDeclaration of competing interest\u003c/h2\u003e \u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eThe authors (M.G. and M.R.B) thank the Russian Science Foundation for financial support [grant No. 23-79-30016].\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eM.R.B. conceived the experiments, M.R.B and M.G. conducted the experiment, M.R.B. and V.A.E. analyzed the results. All authors reviewed the manuscript.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eAll data generated or analysed during this study are included in this published article [and its supplementary information files]\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003e Baklanov, M. R., Ho, P. 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Kings and M.A. Haque, Thermal Conductivity Measurement of Low-k Dielectric Films: Effect of Porosity and Density Journal of electronic materials, Volume\u0026nbsp;43,\u0026nbsp;pages 746\u0026ndash;754, (2014)\u003c/div\u003e\u003cdiv id=\"Par70\" class=\"Para\"\u003e(2014), DOI: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1007/s11664-013-2949-5\u003c/span\u003e\u003cspan address=\"10.1007/s11664-013-2949-5\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e \u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"low–k films, percolation threshold, fractals, heat transfer","lastPublishedDoi":"10.21203/rs.3.rs-8804757/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8804757/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe influence of fractal pore structures on heat transfer in porous low\u003cem\u003e\u0026ndash;k\u003c/em\u003e films, focusing on organosilicate (OSG) materials with porosity near the percolation threshold, is investigated. Thermal transport deviates from classical behavior at length scales comparable to characteristic pore sizes, with effective thermal conductivity governed not only by porosity but also by the topology of the pore network. Fractal organization of the pores modifies phonon scattering, producing measurable non-classical thermal behavior. Fractional-order heat transport models capture these effects, revealing that the observed deviations arise from a broad, hierarchical distribution of transport length scales. Our results highlight that engineered multiscale pore architectures can be used to tailor thermal transport in low\u003cem\u003e\u0026ndash;k\u003c/em\u003e dielectric materials. Analytical expressions for the effective thermal conductivity tensor, incorporating finite pore lengths and the fractal nature of the network, are derived, providing a predictive framework for designing porous dielectrics with controlled thermal properties.\u003c/p\u003e","manuscriptTitle":"Heat Transfer in Porous Low–k Materials: Modeling Based on Fractional Calculus and Material Structure Fractality","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-23 10:29:44","doi":"10.21203/rs.3.rs-8804757/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"6dbe8339-36ad-4627-8ba3-6201fa82d064","owner":[],"postedDate":"February 23rd, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":63320163,"name":"Physical sciences/Engineering"},{"id":63320164,"name":"Physical sciences/Materials science"},{"id":63320165,"name":"Physical sciences/Mathematics and computing"},{"id":63320166,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-03-07T00:08:45+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-23 10:29:44","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8804757","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8804757","identity":"rs-8804757","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}