Phenomenological Analysis of Percolation Phenomena in Porous Low-k Dielectrics

1. Mass Transport and the Role of Percolated Networks in Facilitating the Diffusion of Stable Molecules, Plasma Radicals, Positronium (Ps) and Environmental Contaminants

1.1. Fabrication of Porous Low-k Films

Low-k films used in semiconductor manufacturing may exhibit constitutive porosity, which refers to inherent or intrinsic pores (voids) present within the material structure. In contrast, subtractive porosity typically describes intentionally introduced voids created during processing, such as through etching or the use of sacrificial porogens. Both types of porosity significantly influence the film’s overall dielectric properties and its suitability for specific applications in integrated circuit fabrication [7,13].

For example, materials like BD1 [15], which are deposited by PECVD with organosilane precursors without the use of sacrificial porogens, achieve reduced density and dielectric constant primarily due to a high volume of terminal organic groups that are replacing bridging oxygen atoms in silica like matrix. In sol-gel-based technologies, density reduction can be achieved during the ageing process and through the hierarchical aggregation of primary sol particles. This reduction in density can be interpreted as porosity and is measurable using advanced techniques such as ellipsometric porosimetry (EP) [13] and positron annihilation lifetime spectroscopy (PALS) [12]. According to EP measurements, such films typically exhibit the total porosity levels below 10–12%, with pore sizes comparable to that of the molecular probe used (e.g., a toluene molecule). Notable examples include the aforementioned BD1 and Aurora, developed by Applied Materials and ASM Japan in the early 2000s [16,17]. The porosity in these films is well below the predicted percolation threshold (~30%), which means they maintain robust mechanical integrity (Young’s modulus E > 10–15 GPa), show minimal degradation during plasma processing, and possess favorable electrical characteristics. These properties have enabled their successful integration into ultra-large-scale integration (ULSI) devices (Table 1) [17].

The deposition of highly porous low-k films with a dielectric constant (k) below 2.7 typically requires the use of a sacrificial porogen (also referred to as a template in sol-gel films). This porogen in PECVD processes is usually a relatively volatile organic polymer, whose precursor is co-deposited with the matrix material. While the preparation of surfactant-templated sol-gel materials predates the development of low-k dielectrics [18], integrating sacrificial porogens into PECVD processes—preferred by the microelectronics industry—posed a much greater challenge. This technology was pioneered and developed at IBM by a team led by A. Grill [19,20]. Today, it has become a standard approach widely adopted by both researchers and semiconductor manufacturers.

During processing, an initial thermal treatment of deposited films at moderate temperatures (typically 150–200 °C) stabilizes and cross-links the matrix while the porogen mainly remains intact [21]. Final porogen removal is carried out at higher temperatures (around 400–450 °C), where the polymer decomposes into small volatile fragments that can diffuse out of the film and desorb. This removal process is inherently slow (the procedure takes more than an hour), but it can be significantly accelerated using UV light, electron beams, or microwave radiation [22,23,24].

1.2. Types of Porosity and Mass Transport

Figure 1.2 illustrates various types of low-k films fabricated using different deposition techniques, along with the corresponding adsorption isotherms obtained through adsorption porosimetry. Figure 1.2a demonstrates the so-called ink-bottle effect [25,26], where the material contains large internal voids—reflected in the adsorption branch—connected to the atmosphere through narrower necks, which influence the shape of the desorption branch. Such low-k films typically exhibit very high porosity, often exceeding 50–60%, and are commonly deposited via sol-gel methods using polymeric templates with large feature sizes [26,27].

Figure 1.2b represents the typical behavior of most ultralow-k (ULK) films, which exhibit porosity levels ranging from 30% to 50% and corresponding dielectric constants (k-values) between approximately 2.5 and 2.0. The isotherm still displays a hysteresis loop, commonly interpreted as evidence of internal voids whose sizes are comparable to those of the connecting necks. The initial part of the isotherm (at near-zero relative pressure) suggests the presence of micropores, which, as already mentioned, are typically indicative of organic groups. Such films can be deposited using both PECVD and sol-gel techniques.

Figure 1.2c depicts microporous low-k films deposited without the use of sacrificial porogen; hence, the observed porosity is constitutive, originating from the intrinsic structure of the material. These films typically exhibit no hysteresis in their adsorption isotherms and have porosity levels below 15%. Representative examples include early versions of BD1 and Aurora. Some pores may be isolated—i.e., not connected to the atmosphere—and thus inaccessible to adsorbates during adsorption porosimetry. However, in the case of ellipsometric porosimetry, the volume of such closed pores can be indirectly estimated from the calculated refractive index of the matrix [13,26].

The pore size distribution, particularly in materials with multiple types of isolated voids, can be effectively assessed using PALS. However, the size of open pores can only be measured in samples with ‘capped’ pores, for instance, by depositing thin SiNx films. PALS also provides information on pore connectivity by measuring the depth from which positronium (Ps) can escape to vacuum, where it exhibits its characteristic lifetime of 142 ns. While materials shown in Figure 1.2a and 1.2b appear fully interconnected according to adsorption porosimetry, some of them show limited positronium diffusivity, as detected by PALS. This discrepancy between adsorption porosimetry and PALS data introduces uncertainty in defining the percolation threshold for mass transport in low-k films.

Extensive cross-evaluation of various low-k films deposited with different porogen concentrations (ranging from 0% to 50%) was carried out using EP and PALS techniques [28,29]. EP measurements, performed on all samples [29], revealed that most pores are accessible for adsorption and, therefore, remain interconnected even at porogen loadings well below 20%. In contrast, PALS results indicate that pore interconnectivity becomes restricted when the porogen concentration reaches approximately 20–25%, despite both EP and PALS showing similar pore size measurements. A similar discrepancy between the diffusion of stable molecules and positronium was also observed in earlier studies, where degree and length of pore interconnectivity was evaluated using a specialized technique based on the diffusion of organic solvents [30] and the results compared with PALS data [31].

Therefore, the commonly accepted idea that the percolation threshold for mass transfer is close to 30% may have a more nuanced meaning. While this threshold is theoretically predicted by percolation theory, it strictly depends on the topology and dimensionality of the system. For example, site percolation on a 3D cubic lattice yields a threshold of approximately 0.31, while continuum percolation models—such as those involving overlapping spheres or pores—can produce similar or slightly different values depending on the shape and size distribution.

In the context of low-k films, the matrix (or wall) inherently contains small voids and channels due to the relatively large size of organic components (CH₃). Moreover, as already mentioned, additional micropores of molecular size can form during the removal of porogen fragments. Stable molecules can slowly diffuse through these pores if their diameter is larger than that of the diffusing molecules. However, the situation becomes more complex with unstable species like free radicals formed in the plasma. In this case, their lifetime is drastically reduced due to collisions with the pore wall (surface recombination), leading to deactivation before they can diffuse through the pores. The depth of active radicals penetration ($\delta$) depends on effective diffusivity (D_eff) and recombination rate constant (k_r) as Equation (1.1):

$$\delta =\sqrt{{\displaystyle \frac{D_{eff}}{k_r}}},$$

(1.1)

In the case of low-k films the pore size is much less than 100 nm and therefore $D_{eff}={\displaystyle \frac{p}{\tau }}{D}_K$, where p is porosity, t is pores tortuosity and D_K is Knudsen diffusion coefficient. The depth of penetration δ, as already mentioned, is smaller than the theoretical percolation length and also depends on chemical nature of active radicals. Consequently, researchers studying the depth of low-k modification may conclude that pore permeability is limited [10].

The estimation of penetration depth (Equation (1.1)) relies on the assumption of normal diffusion (Equation (1.2)):

$$\langle x^2\left(t\right)\rangle = D_{eff} t~{\displaystyle \frac{D_{eff}}{k_r}},$$

(1.2)

where <x²(t)> is the mean squared displacement (MSD) of a particle (or probe) as a function of time t.

However, in complex branched systems, which are typical of porous low-k films, transport often exhibits subdiffusive behavior, as described by Equation (1.3):

$$\langle x^2\left(t\right)\rangle ~{\left( D_{eff} t\right)}^{\mathsf{\gamma }},$$

(1.3)

where $\gamma$is a critical subdiffusion exponent, determined primarily by the type and topology of branching. These anomalous transport properties are frequently interpreted in terms of the fractal dimension of the system, reflecting its inherent self-similarity. The branched diffusion in these materials can even reduce the depth of penetration due to small value of critical exponent $0.25<\gamma <0.5$ [32,33]. Therefore, a possible way to reduce the penetration depth is to fabricate films with smaller and more tortuous pores.

The situation becomes even more complex for positronium (Ps). In addition to accelerated surface annihilation, Ps is subject to quantum mechanical constraints that hinder its diffusion from large pores into narrow necks due to differences in ground-state energy between large voids and small constrictions [34,36]. As a result, the effective permeability of pores—and consequently the apparent percolation threshold—is lower for Ps than for reactive radicals.

For practical applications, the diffusion of active radicals is more relevant than that of Ps, as it directly reflects the depth of modification in low-k films during plasma processing. This underscores the need for a clear distinction between different definitions of percolation thresholds. Classically, the percolation threshold p_c is defined as the critical fraction of open pathways (pores, channels, or bonds) at which a connected path spanning the material first emerges, enabling macroscopic transport. In percolation theory:

• p < p_с → no continuous path exists → macroscopic transport is effectively zero.
• p > p_с → a connected network exists → transport becomes possible.

This threshold is purely geometric, depending only on the connectivity of the medium. In real mass transport scenarios, particularly in porous media or low-k films, molecules or radicals have a finite lifetime τ due to reaction, decay, or adsorption. If the paths are long or tortuous, species may not survive long enough to traverse the system, introducing a dynamic, lifetime-dependent percolation threshold p_c (τ) > p_c. For very short-lived species, geometrically connected paths may fail to contribute to macroscopic transport because the species decays before reaching the outlet. As τ → ∞ (e.g., stable species like toluene in EP), the effective percolation threshold converges to the geometric threshold. For finite τ, the effective threshold shifts upward: a higher fraction of open paths is required for successful transport within the species’ lifetime.

1.3. Depth Profile of Plasma-Induced Damage

Plasma damage to porous low-k materials is a multifaceted phenomenon encompassing chemical damage from neutral radicals formed during etching and resist stripping (e.g., O, H, F), ion-induced bond breakage and densification, and the impact of VUV photons. Together, these factors create a complex scenario that necessitates thorough analysis. Plasma damage can alter the chemical composition and mechanical properties of low-k materials while also inducing electrically active defects.

1.3.1. Damage by Active Radicals

A downstream plasma reactor was used for this purpose, where the influence of ion bombardment is negligibly small. In this setup, active radicals diffuse into the pores, react with the material, and convert the low-k film into a hydrophilic state. Oxygen radicals remove CH₃ groups and form Si–OH, while hydrogen can break Si–CH₃ bonds and hydrogenate the silicon, also contributing to hydrophilicity. Fluorine atoms generated in fluorocarbon plasma can attack the Si–O backbone, generating various defects such as oxygen deficient centers.

The top graphs in Figure 1.3 show plasma damage in low-k films with porosities below the theoretical percolation threshold (31%), while the bottom graphs illustrate the behavior of films with porosities above this threshold. The difference is especially pronounced with oxygen plasma, whereas fluorine plasma shows minimal variation. Oxygen plasma causes a significant reduction in carbon concentration and an increase in oxygen concentration, leading the upper layers of the films to adopt a silica-like composition. In the 19% porous film, the depth of modification was less than 100 nm, while in the 40% porous film, it extended to approximately 250–300 nm.

In contrast, fluorine plasma primarily results in surface fluorination, with negligible difference between the 19% and 40% porous films. The fact that even the 19% porous film was affected by oxygen plasma indicates that low-k films with porosities below the percolation threshold can still allow the diffusion of oxygen radicals, likely due to the presence of narrow pore necks. The lack of porosity-dependent behavior in the case of fluorine plasma is probably due to the high recombination probability of fluorine atoms, which limits their penetration depth.

1.3.2. Damage by Ion Bombardment

Physical damage caused by ion bombardment results from energetic ions accelerated in the plasma sheath. These ions can break Si–O and Si–C bonds, collapse pores, and sputter material from the surface. As a result, densification, changes in pore structure, and degradation of electrical properties can be observed. However, the depth of ion-induced damage is normally very limited, typically affecting only the top of the surface layer (it is about or less than 5 nm). This shallow damage is the basis for using ion bombardment in pore sealing, where the densification of the surface helps protect the underlying film from further damage by active radicals.

Modern plasma etch reactors provide the ability to control both the intensity of ion bombardment and the impact of active radicals. To better understand the individual contributions of these plasma components to plasma-induced damage, a series of experiments were conducted using an ICP plasma reactor [36]. Low-k films were treated under three different conditions:

• Top Power Only (TPO) – generating plasma radicals without ion bombardment;
• Bottom Power Only (BPO) – applying a bias to the bottom electrode, resulting in strong ion bombardment with minimal radical generation;
• Top and Bottom Power (T&BP) – combining both plasma radicals and ion bombardment.

The exposure time in each condition was chosen to match the duration required to remove a specific thickness of photoresist (Figure 1.4). The results show that the reduction in Si–CH₃ bonds (the 1275 cm⁻¹ FTIR peak) under BPO and T&BP conditions is nearly the same and significantly lower than under TPO. This suggests that active radicals, particularly oxygen species, are primarily responsible for damaging the low-k films due to their ability to easily penetrate open pores. In contrast, ion bombardment in the BPO and T&BP conditions appears to partially seal the surface, thereby inhibiting the penetration of active radicals. Furthermore, the similarity in behavior between BPO and T&BP indicates that the efficiency of surface sealing by ion bombardment is independent of the presence of active radicals. It should be noted that the porous low-k films investigated in this work exhibited a porosity of 33% and a characteristic pore size of ~2.5 nm, as determined by ellipsometric porosimetry. The efficiency of surface sealing induced by ion bombardment is strongly dependent on both porosity and pore size (Figure 1.3).

1.3.3. VUV Damage

Plasma also emits vacuum ultraviolet (VUV) radiation, which possesses sufficient energy to break chemical bonds. VUV-induced damage to low-k materials has been extensively studied in the past, and significant insights have been gained through a variety of advanced characterization techniques. The most important findings have been reviewed and analyzed in Ref [22]. The effects of VUV radiation can resemble those caused by chemical damage, though they are often more complex. The extent and depth of VUV-induced damage strongly depend on the wavelength of the emitted photons. For example, when photon energy is below 6 eV, the damage is minimal. Maximum damage typically occurs in the 6–9 eV range. At higher photon energies, the degree of damage may decrease due to the high absorption coefficient of silica-based low-k materials, which limits light penetration depth.

VUV photons are capable of damaging even dense materials, although their absorption coefficients are influenced by the material’s density. Evidence of VUV-induced damage is apparent in the data presented in Figure 1.3 and 1.4. In both cases, the depth of damage observed in films with porosities below the percolation threshold (Figure 1.3) and under ion bombardment (Figure 1.4) reaches several tens of nanometers. This is notably deeper than expected, given the limited penetration of active radicals in sub-threshold porosity films and the expected surface sealing from ion bombardment. These results suggest that the observed deeper damage may be attributed to the influence of VUV photons generated in the plasma. A more detailed analysis of these effects, including the synergistic interaction between active radicals and VUV photons, has been carried out using specialized gap experiments [37,38,39] and is thoroughly reviewed in the paper [7].

The data related to effect of porosity are limited. Rakhimova et al. [40] studied systematically effects of different wavelengths on low-k films with different porosity. In Figure 1.5, the effective quantum yield was calculated based on the VUV exposure results of the low-k dielectrics with porosity ranging from 20% to 55%. The effective quantum yield in the range between 0 and 1 quantifies the efficiency of CH₃ escaping from the porous film with a possible reattachment of –CH₃ back to dangling Si-bonds. For our opinion this dependence fairly correlating with percolation behaviour is related to efficiency of removal of reaction products (CH₃ radical) detached from the silicon atoms. They can be attached back during the diffusion through interconnected pores.

The electrical properties of low-k films are also strongly influenced by VUV exposure. Nichols et al. [41,42] reported that VUV irradiation decreases dielectric failure times and charge-to-breakdown in low-k dielectrics. In addition, VUV exposure can reduce leakage currents by promoting the decomposition of photoactive compounds [43]. in a manner analogous to UV curing. Conversely, other studies have shown that VUV photons generate additional defects and distorted bonds within the dielectric matrix. Similar increases in defect density and leakage current following UV irradiation have also been reported [44]. Sinha et al. [45] further suggested that photon irradiation produces trapped charges inside low-k materials, where such charge accumulation may pose long-term reliability concerns.

1.4. Conclusions

Low–k organosilica films developed for the interconnects of ULSI devices exhibit a complex structure. Materials with k ≥ 2.7–2.8 are typically deposited without sacrificial porogens, featuring constitutional porosity (<10%) and molecular-sized pores (0.5-1 nm). Films with k < 2.7 are deposited using sacrificial porogens, containing larger voids interconnected by narrow necks within the matrix. Both types of films demonstrate measurable diffusivity of neutral molecules such as toluene and isopropyl alcohol, as assessed by ellipsometric porosimetry. However, PALS reveals that first type films have non-interconnected pores, while more porous materials (>30% porosity) exhibit interconnected pores. The percolation threshold for positronium (Ps) is observed around 25–30%, consistent with theoretical predictions for randomly embedded voids. The discrepancy between PALS and ellipsometric porosimetry data arises from Ps annihilation near pore walls and quantum mechanical constraints preventing Ps diffusion from large voids to narrow necks.

These characteristics are crucial for interpreting experimental results. For instance, the diffusion behavior of plasma-generated active radicals differs due to possibilities of recombination and interaction with pore walls. Furthermore, the depth of penetration depends on the nature of the radicals, as depicted in Figure 1.3. Therefore, understanding the percolation threshold measured by PALS is essential for assessing plasma damage. Additional efforts in process optimization, such as pore sealing, low-energy plasmas, and pulsed etching at sub-zero temperatures with pore-filling agents, are critical. Optimizing porosity and the chemistry of processing steps (e.g., hard masks, plasma conditions, repair layers) is crucial to maintaining film integrity while achieving desired k values. Techniques like pore stuffing (P4 approach) and cryogenic etching with micro-capillary condensation are also effective.

It is important to recognize that the percolation threshold exerts a substantial influence on both mechanical and electrical properties, including overall material reliability, as will be discussed in the following section. When analyzing the transport of various species, it is crucial to distinguish between the geometric and effective percolation thresholds. The geometric percolation threshold is intrinsic to the network structure and is independent of species lifetime. In contrast, the effective percolation threshold for species with finite lifetimes increases as the lifetime decreases. Consequently, in practical terms, percolation relevant to mass transport exhibits a clear dependence on species lifetime.

2. Percolation Phenomena Governing Electrical Failure in Low-k Films

2.1. General Overview of Percolation Phenomena in Porous OSG Low-k Films

Percolation phenomena in porous OSG low-k dielectrics represent a critical bottleneck, where nanoscale porosity, introduced to achieve ultralow permittivity (k < 2.5), triggers catastrophic electrical failures when pore interconnectivity exceeds a critical porosity threshold p_c (p_c~25–30%). At this threshold point (Figure 2.1), isolated pore transition into percolating pathways that act as electrical short circuits, enabling rapid transport of charge carriers, moisture, and metal ions. These pathways drastically degrade dielectric integrity through three mechanisms: (i) breakdown voltage reduction by 40–60%, as electric fields concentrate at pore/skeleton interfaces [46,47] (ii) exponential leakage current surges via Poole-Frenkel (PF) emission along defect chains [48]; and (iii) accelerated reliability failures as copper diffusion and moisture ingress propagate along connected pores, collapsing time-dependent dielectric breakdown (TDDB) lifetimes [49,50]. This threshold behavior is topologically deterministic, governed by pore morphology, defect density, and interfacial chemistry, making percolation a fundamental constraint in advanced interconnect scaling.

Intrinsic material defects, such as graphitic sp² carbon residues and oxygen vacancies—both originating from incomplete porogen (template) burnout — act as conductive nodes that seed percolation paths [51]. Extrinsic integration processes (like plasma etching, CMP) amplify damage by stripping hydrophobic methyl groups (–CH₃), creating hydrophilic Si–OH bonds that adsorb moisture (k ≈ 80) [22]. This moisture enables ionic conduction via Grotthuss proton hopping [52] and facilitates copper migration, forming metallic filaments that bridge electrodes. Critically, the Weibull slope (β) of TDDB distributions flattens near the percolation threshold [47], reflecting heightened statistical dispersion in failure times due to stochastic defect clustering. Below 20 nm thicknesses, defect densities >10¹⁶ cm⁻³ guarantee percolation paths, rendering dielectric robustness irreversible [46,48]. Consequently, mitigation strategies must suppress porosity [53] and engineer network topology to delay percolation onset, as validated by cross-linked a-SiC:H films that maintain robustness at 40% porosity through optimized bond coordination [48].

Table 2.1 summarizes the trade-offs between dense and porous OSG films: while porous variants with lower k-values (< 2.5), they suffer from 40–60% lower breakdown fields and compromised mechanical strength due to percolation-driven mass transport and plasma damage susceptibility. These weaknesses underscore the delicate balance between ultra-low permittivity and electrical reliability.

The details of percolation threshold and its influence on various properties of porous OSG low-k films are presented in Table 2.2.

2.1.1. Dielectric Constant and Leakage Current vs. Porosity and Chemical Composition

Figure 2.2 shows the variation in dielectric constant and leakage current (resistivity) of OSG low-k films (a) as a function of Si–CH₃ concentration (b), adsorbed water concentration (c), and porosity (d), modified by annealing between 350 °C and 900 °C.

The first measurements were performed after annealing at 350 °C. According to FTIR data, the films annealed at this temperature were fully cured: the porogen was removed, and the porosity reached its maximum value. Therefore, any further changes observed during subsequent annealing are associated with gradual degradation of the film properties.

From the perspective of property evolution, the temperature range can be divided into three regions:

1.

350 °C – 450 °C.

The concentrations of Si–CH₃ groups (b) and adsorbed water (c) remain almost unchanged, as does the porosity. This indicates that the low-k films are still stable in this range. However, a significant change in resistivity (leakage current) is observed, increasing from 2×10¹⁵ to 12×10¹⁵ Ω·cm. This change is attributed to the formation (at 350 °C) and gradual removal of template residues, likely sp² carbon. Unfortunately, this residue is barely detectable in the FTIR spectra.

2.

450 °C – 700 °C.

This range is characterized by a pronounced degradation of low-k properties: a strong reduction in Si–CH₃ group concentration, an increase in adsorbed water content, and a marked decrease in porosity. These changes result in reduced resistivity and increased dielectric constant. The loss of Si–CH₃ groups makes the films more hydrophilic, and the resulting moisture uptake further increases both leakage current and dielectric constant. Film densification in this range also contributes to the dielectric constant increase.

3.

700 °C – 900 °C

At both 700 °C and 900 °C, the Si–CH₃ groups are completely removed, rendering the films highly hydrophilic. The highest amount of adsorbed moisture is observed in the films annealed at 700 °C. This elevated water uptake results in the lowest resistivity and highest dielectric constant among the studied samples. Annealing at 900 °C leads to a drastic reduction in porosity. Furthermore, the difference between open and closed porosity increases, suggesting that micropore collapse causes some pores to become closed. Consequently, these closed pores are inaccessible to water molecules, as confirmed by FTIR, and the total amount of adsorbed moisture decreases significantly. This reduction in water content leads to a decrease in the dielectric constant and an increase in resistivity compared to the 700 °C samples.

Effect of hydrocarbon residues adsorbed from contaminated UV curing chamber has been studied by Gerelt-Od [55] (Figure 2.3). The low-k sample (spin-on deposited OSG films with 40% porosity, Figure 2.3d) was loaded into a vacuum chamber pumped by a dry turbomolecular pump. Both porosity (Figure 2.3d) and leakage current (Figure 2.3a) remained unchanged when the sample was kept in the chamber purged with nitrogen. However, the film’s porosity decreased from 40% to 29% when the same sample was placed in the vacuum chamber without nitrogen purge. Correspondingly, the FTIR spectra showed a significant increase in hydrocarbon concentration (Figure 2.3b), along with an increase in leakage current (Figure 2.3a). Part of the adsorbed hydrocarbons was weakly bonded to pore wall and could be easily removed by nitrogen purging at 250 °C. Much stronger accumulation of hydrocarbon residues and a further increase in leakage current were observed when the sample in this vacuum chamber was exposed to VUV light (180 nm). This pronounced accumulation of hydrocarbon residues is most likely due to photoinduced desorption of hydrocarbons from the chamber walls, which increases their concentration in the gas phase and leads to their subsequent re-adsorption into the low-k film.

Nitrogen purging during VUV exposure reduced the amount of adsorbed hydrocarbons and increased the open porosity. However, the leakage current has increased even further compared to vacuum exposure alone. These phenomena can be explained by photochemical reactions, given that the photon energy of 180 nm light (6.9 eV) is sufficient to break C–H (4.3 eV) and C–C (3.6 eV) bonds in hydrocarbon films, thereby converting some residues into graphitic sp² carbon clusters and unsaturated carbon sites. These species have higher electrical conductivity and can form percolation pathways. Additionally, VUV photons can cause photoemission or excite electrons into the conduction band of the material or substrate. The resulting ionized carbon sites act as deep or shallow traps. Under an applied bias, carriers can hop or tunnel between these traps — a mechanism known as trap-assisted conduction — which manifests as increased leakage current [56,57]. In experiments involving nitrogen, VUV photons can dissociate N₂ into metastable N* species that react with radicals in the hydrocarbon film, forming C–N bonds or nitrides. However, 180 nm photons cannot effectively dissociate N₂ directly, but CH radicals or other hydrocarbon-derived intermediates may activate N₂ and incorporate nitrogen into C–N bonds via indirect photochemical pathways. This interaction can modify trap density and conduction pathways. Some studies have shown that N₂/VUV exposure promotes nitrogen doping of carbon surfaces, altering their conductivity.

The data presented in figures 2.2 and 2.3 suggest that increase of leakage current most probably related to impact of surface compounds formed on pore walls and groups that can form percolative passes when pore are fully interconnected. To prove this statement, it would be interesting to study OSG films free on surface compounds. Vanstreels el al. reported such data [58] for porogen-residue-free, highly hydrophobic films prepared using a specialized fabrication process [59]. In these films, with porosity ranging from 30% to 50%, the leakage current was found to be independent of porosity, suggesting that in residue-free films the pore wall surface no longer plays a dominant role (Figure 2.4). Instead, the leakage current seems to be governed by other factors—possibly including the presence of oxygen-deficient centers (ODCs)—and is significantly lower than in conventional porogen-residue-containing or hydrophilic OSG films. In contrast, the breakdown field remains dependent on porosity (Figure 2.4b), decreasing approximately linearly with increasing porosity.

In contrast, higher porosity lowered the breakdown voltage—from 6 MV/cm at 31% porosity to 3.5 MV/cm at 44–45%—due to fewer cage structures, increased Si–O–Si backbone strain under electric fields, and local field intensification near pores.

Leakage currents in porous OSG low-k dielectrics exhibit distinct behavior governed by porosity-dependent conduction mechanisms. Below the critical porosity threshold (p_c ≈ 25–35%), leakage is dominated by localized charge transport processes such as trap-assisted tunneling and PF emission [48,60,61]. Isolated pores (Figure 2.1a) create localized electric field enhancements at pore/skeleton interfaces due to permittivity contrast (k_pore≈1 vs k_matrix ∼3–4), but the dielectric matrix constrains long-range charge movement [60]. Electrons primarily hop between intrinsic defects (oxygen vacancies, strained Si–C bonds) and extrinsic traps (e.g., graphitic carbon residues from incomplete porogen burnout), with leakage scaling moderately with field strength via PF emission [62].

Trap-assisted conduction in low-k dielectrics originates from electronic traps, localized states within the bandgap, created by synergistic intrinsic defects and extrinsic plasma damage. Chemical imperfections like strained Si–C bonds, hydrophilic Si–OH groups (from moisture access), and non-bridging oxygen radicals (Si–O•) generate deep-level traps (>3 eV below conduction band) that capture electrons [48,53]. These defects are catastrophically amplified by plasma processing: for instance, VUV photons (147 nm) generated in Xe plasma penetrate porous networks, severing hydrophobic Si–CH₃ bonds and leaving carbon dangling bonds (•C) and oxygen vacancies (≡Si•) as shallow traps near the conduction band. Figure 2.5 [63] quantifies this damage, showing 147 nm light depletes 80% of Si–CH₃ bonds in high-porosity films, with defect density peaking at the dielectric interface. The degree and depth of UV damage depends on wavelength. Critically, isolated traps remain electrically benign until exceeding a threshold density (~10¹⁶ cm⁻³), where overlapping wavefunctions enable trap-to-trap tunneling. Beyond this point, accelerated at porosities >30% where pores concentrate defects, traps aggregate into conductive filaments via percolation [46]. This transition is stochastic but topologically inevitable where pore connectivity reduces the critical defect separation to <1 nm, allowing electrons to hop along chains spanning electrodes [47].

2.1.2. Leakage Current in Porous Low-k Films and Reliability

The reliability consequences manifest in real-time leakage evolution (Figure 2.6) and statistical failure shifts. Leakage currents surge exponentially as defect chains lower activation barriers for PF emission; electrons hop between traps with barrier reduction Δφ ∝ √E. Residue-rich films (sp² carbon clusters) exhibit 1000× higher leakage at 3 MV/cm than clean matrices [64]. Simultaneously, breakdown strength collapses by 40–60% and electric fields intensify to >4× applied levels at trap-rich pore interfaces [65], exceeding Si–O bond strength (8–10 eV) and triggering cascading bond breakage.

Figure 2.6 captures this progression: phase I shows current decrease as traps fill; phase II reveals gradual current rise from new defect generation (Cu⁺ drift along hydrated pores); phase III marks abrupt hard breakdown when percolation paths bridge electrodes. This aligns with flattened Weibull slopes (β < 2 vs. β > 5 in dense films), reflecting unpredictable early failures [47]. Plasma damage creates a malicious cycle: each etch step generates fresh traps that interconnect existing chains [48].

At the percolation threshold (p ≥ p_c), isolated pores interconnect into continuous pathways, triggering an abrupt, exponential surge in leakage (Figure 2.1c) [48,60]. This transition enables percolative conduction, where charges bypass the matrix via low-resistance tunnels formed by connected pores or defect chains [61]. Simulations by COMSOL confirm that electric fields intensify by >4× at pore necks (4.5 MV/cm locally vs. 3.3 MV/cm applied) [62], accelerating charge injection. Moisture exacerbates this effect: hydrophilic Si–OH bonds (from plasma damage) adsorb water, dissociating into H₃O⁺/OH⁻ ions that enable ionic conduction via Grotthuss proton hopping [66]. This transforms pores into proton highways, elevating leakage by 10–100× in humid environments [67].

Moisture and ionic contaminants (Na⁺, K⁺, Cl⁻) exploit porous networks in low-k dielectrics, migrating through interconnected pores and transforming inert voids into conductive pathways. Plasma-induced hydrophilicity, caused by methyl group (–CH₃) depletion (Figure 2.5), creates hydrophilic Si–OH surfaces that chemisorb water monolayers, while capillary action draws additional moisture deeper into the film [48]. Under elevated temperature and humidity, this adsorbed water dissociates into H₃O⁺ and OH⁻ ions, enabling ionic conduction through Grotthuss proton-hopping along hydrogen-bonded networks [66]. Simultaneously, the dissolved ions (Cl⁻ from etch residues) migrate under bias, creating electrolytic currents that scale with pore connectivity. Critically, moisture amplifies pre-existing trap-assisted paths where water molecules passivate deep traps (Si–O•) but generate shallow proton traps (Si–OH₂⁺), lowering activation barriers for PF emission by up to 0.5 eV [67]. This synergy is self-propagating, humid environments accelerate Cu²⁺ diffusion along hydrated interfaces, depositing metallic clusters that further enhance local fields [68].

The reliability impacts are severe and multifaceted. Leakage currents surge by 10–100× in humid conditions, exhibiting non-linear J–E curves with characteristic humps at 1–2 MV/cm [69], a signature of ionic hopping transitions. More critically, moisture reduces breakdown fields by 30–50% through dual mechanisms: (i) the high permittivity of water (k ≈ 80) concentrates electric fields at pore interfaces, exceeding the intrinsic breakdown strength (8–10 eV) and triggering Si–O bond rupture [65]; (ii) the hydrated Cu ions form dendritic filaments along percolation paths, bridging electrodes at lower voltage [49]. TDDB lifetimes collapse exponentially with humidity, chemisorbed water (requiring >400 °C annealing for removal) increases the thermal activation energy of breakdown by 0.7 eV, while hydrophilicity flattens Weibull slopes (β) to <2 [62].

Conductive residues further amplify leakage. Graphitic sp² carbon clusters, trapped during non-optimized porogen burnout, act as ohmic paths with resistivity ∼10⁻³ Ω∙cm, independent of electric field [64]. These residues lower PF activation barriers by up to 0.5 eV and enable direct tunneling at high fields (>3 MV/cm) [70]. Plasma processing (RIE and ashing) deposits conductive carbon residues on the pores wall, primarily graphitic sp² clusters, which are formed by the incomplete combustion of organic porogens or photoresists. These residues, accumulating at pore necks and interfaces, create ohmic conduction paths independent of electric field. Wu et al. [48] quantified leakage increases of 10–100× in films with >20% carbon residue coverage, as electrons bypass the dielectric matrix via these low-resistance filaments (resistivity ~10⁻³ Ω∙cm). Critically, carbon residues synergize with trap-assisted conduction where sp² sites act as deep traps that capture charges, enhancing PF emission at low fields (<1 MV/cm), while enabling direct tunneling at high fields. This dual mechanism is worsened by pore connectivity, residues at interconnected pore junctions form conductive bridges that override the P_c at lower defect densities [46]. Similarly, the barrier layer damage during integration further injects metallic particles (Cu, Ta) into the dielectric. Therefore, the metallic contaminants for like Cu, Ta diffuse along hydrated pores during integration, precipitating as nanoclusters (<5 nm) that intensify local fields (Figure 2.7) [49] . Plasma etching residues or barrier-layer damage inject these metals, creating preferential percolation paths that reduce the critical defect density for conduction by 3× compared to pristine pores [46].

The reliability impact is severe. Above p_c, leakage currents surge exponentially, governed by interconnected defect networks rather than bulk properties [47]. This erodes dielectric integrity, accelerating TDDB and flattening Weibull slopes (β < 2) [71]. Plasma processes worsen this by expanding pores >50% and generating fresh traps (carbon dangling bonds), which interconnect existing chains [40]. Plasma processing (etching, ashing) actively reshapes pore networks through radical-driven restructuring. Reactive species (O•, H•, CF_x•) penetrate interconnected pores, scouring pore walls and severing Si–CH₃ bonds. This not only generates carbon dangling bonds (trap sites) but etches the matrix backbone, expanding pore diameters by >50% and merging isolated voids into continuous channels [40]. The resulting pore connectivity surge transforms the dielectric into a percolated network, effectively predetermining conduction paths before voltage application. Moisture exacerbates this where plasma-created Si–OH groups adsorb water, which reacts with radicals to form corrosive acids (HF from F•), accelerating matrix dissolution. Critically, 147 nm VUV photons [40] inflict deepest damage due to low absorption in OSG, enabling uniform restructuring through thick films. These plasma-expanded channels act as low-resistance highways, facilitating rapid diffusion of contaminants and lowering activation barriers for ionic conduction.

Plasma-expanded pores catastrophically degrade dielectric integrity. Leakage currents increase due to: (i) ohmic transport through connected pores; (ii) enhanced trap-assisted tunneling along damaged walls; and (iii) electrolytic currents from moisture ingress. Simultaneously, breakdown fields plummet by 40–60% as enlarged pores intensify local electric fields and reduce the effective dielectric thickness [47]. Figure 2.7 quantifies this vulnerability: high-porosity films (50%) exhibit near-unity quantum yield for bond breakage, meaning almost every photon creates a defect. This damage flattens TDDB Weibull slopes to β < 1.5 (vs. β > 5 in pristine films), indicating stochastic early failures [47]. Crucially, the plasma damage creates irreversible percolation seeds when ALD barrier precursors penetrate the expanded pores and depositing conductive Ta clusters that further elevate leakage.

2.1.3. Breakdown Voltage

Some studies have analyzed local electric field enhancements caused by porosity and their effect on the probability of breakdown, and have also discussed possible mechanisms, including the presence of moisture and active centers in the band gap [46,60,70]. Ogawa et al. [72] considered the pore as a pre-existing defect and using a percolation model to predict relative breakdown fields. They demonstrated that porosity in SiO₂ films has virtually no effect on the physics of breakdown but significantly influences the time to breakdown (TDDB). In this study, pores were considered as pre-existing defects that make it easier for conductive paths to form in dielectric material. The key conclusion is that porosity doesn’t really affect the fundamental physical processes that determine the mechanism of electrical breakdown. Nevertheless, the presence of pores significantly reduces the time to breakdown, which manifests itself in a decrease in the shape parameter (β) of the Weibull distribution. The decrease in β, in turn, is due to an increase in the statistical variability of breakdown paths, which is a direct consequence of the percolation nature of the formation of conductive channels. Thus, porosity contributes to the formation of multiple alternative paths for breakdown, thereby increasing the dispersion in the distribution of failure times. Ogawa et al. [60] also investigated the effect of porosity on leakage currents, breakdown, and time to dielectric breakdown (TDDB) in various low-k silicon-based dielectrics, including fluorinated silica (FSG), porous carbon-doped silica (OSG), and highly porous methylsilsequioxane (P-MSQ). Experimental data demonstrate that an increase in porosity (resulting in a decrease in the dielectric constant k, to 2.1–2.3) leads to a significant reduction in breakdown voltage E_BD (from ~10 for CVD-SiO₂ to ~4 mV/cm for P-MSQ). Additionally, it reduces the time to breakdown, while the breakdown kinetics (field acceleration parameter γ ≈ 4.5 cm/mV and activation energy E_a ≈ 0.5–0.66 eV) remain unchanged.

In [73] Hong et al. used modeling to show that pores can cause local field enhancement in porous low-k films, and that high-field regions around pores can create an effective conduction path for drifting charged particles, accelerating dielectric breakdown. In addition, the work [74] also demonstrated the calculation in the COMSOL environment in porous SiO₂ (porosity 30%) showing that the local field around the pores increases significantly. It was found that the combined effect of an increase in the electric field at the pore edges and a decrease in the activation energy of diffusion leads to a reduction in the time to breakdown by 26% for a relatively dense material.

It was also reported in [49,75] that the drift of Cu ions at the cap/low-k boundary is the dominant factor causing dielectric breakdown. The dielectric breakdown is associated with a critical copper concentration in the dielectric, and this concentration depends on the current conduction mechanism, i.e., Schottky or Poole-Frenkel. Since the field dependence in both conduction mechanisms is the square root of the electric field, the failure time has a √E dependence. Lloyd et al. [66] reported that the presence of Cu was not a necessary condition for failure, but its presence significantly reduced the failure time. In all the above models, the role of pores and copper in low- k dielectric breakdown is considered separately, so the physics of breakdown is still unclear. An extension of the Ogawa model was made in [46], where porosity and copper ions were taken into account. The colleagues showed (simulation in the Ansys environment) that a local maximum field (E_max) will arise inside the pore and will propagate perpendicular to the applied electric field (Figure 2.7). The maximum field was 4.5 MV/cm, which is about 1.4 times higher than the applied average field (3.3 MV/cm). This is because the electric flux is concentrated around the pores due to the low dielectric constant of the pores (k = 1).

By comparing the simulation results of the local field enhancement of pores and Cu clusters [74], it was shown that Cu clusters have a stronger field enhancement effect than pores, which significantly worsens the reliability of the dielectric under stress conditions. But under the use condition, although the ratio of E_max /E_ave is the same as under stress conditions, the difference in the absolute field value between pores and Cu clusters becomes small, and therefore its effect on the failure time of the dielectric becomes less important. Therefore, it is necessary to consider the field enhancement due to pores in the reliability characterization of porous low –k dielectrics. In fact, the field enhancement due to pores is equivalent to shortening the percolation path (or dielectric thickness) for breakdown, which did not change the physics of dielectric breakdown, but would significantly affect the failure time of the dielectric.

Figure 2.8 [71] plots the maximum allowed electric field (E_max) versus porosity for 20–60 nm films, showing how E_max decays exponentially as porosity rises. This summarizes the percolation-driven erosion of dielectric robustness and underscores the thickness-porosity interdependence critical for reliability modeling. Including the H₂ treatment’s impact on reducing intrinsic defects provides context for why some high-porosity films might outperform expectations.

Based on the King et al. [53] study, the influence of network bond percolation on electrical failure and reliability in low-k a-SiC:H films reveal a critical relationship with the average coordination number, ⟨r⟩. Electrical resistivity exhibits a distinct saturation behavior as ⟨r⟩ decreases. Specifically, resistivity increases significantly with reduced network connectivity but plateaus near ⟨r⟩≈2.4–2.5, slightly above the rigidity percolation threshold (⟨r⟩_c=2.4) observed for mechanical properties. This saturation is attributed to conduction transitioning to a site-percolation mechanism governed by electron transport between point defects or traps in the dielectric bandgap. The dominant mechanism is Frenkel-Poole (FP) conduction, where the applied electric field lowers the energy barrier for trapped electrons to hop between defect sites. The authors note that the observed critical point (⟨r⟩≈2.5) aligns more closely with site-percolation theory (predicted ⟨r⟩_c=2.3) than bond-percolation models (⟨r⟩_c=2.4), likely due to the additional energy constraints inherent to FP conduction.

From a reliability perspective, reducing ⟨r⟩ to achieve lower-k involves inherent issues. While lower ⟨r⟩ increases resistivity (beneficial for leakage reduction), it simultaneously compromises mechanical and thermal properties, creating vulnerabilities for device failure. Below ⟨r⟩_c≈2.4, interconnected nano-porosity percolates (Figure 2.9), causing abrupt deteriorations in thermal conductivity and potentially accelerating dielectric breakdown under thermal stress.

Furthermore, the films near ⟨r⟩_c exhibit homopolar bonding shifts (Si–Si, C–C) and increased terminal groups, which may introduce electronic traps that exacerbate leakage under bias. They concluded that while reduced connectivity below ⟨r⟩_c theoretically stabilizes elastic properties, pore percolation and defect-dominated conduction pose significant reliability risks, highlighting the challenge in balancing ultra-low k targets with robust electrical performance. Strategic design, such as incorporating deformable C–C–C chains or stiff, low-polarizability bonds, is suggested to mitigate these issues.

Paquette et al. [76] highlights how percolation theory governs key electrical transitions in amorphous semiconductors and dielectrics. In crystalline and amorphous semiconductors, increasing dopant concentration beyond a critical threshold (N_c) triggers a metal-insulator transition. Below N_c, dopant states are localized and conductivity is suppressed, but at N_c, these states delocalize to form percolating paths, enabling metallic conduction. This transition follows a universal scaling law: σ(0)=σ₀[(N−N_c)/N_c]^ν, where ν is a critical exponent. Interestingly, this mirrors the mechanical rigidity percolation transition described by Phillips-Thorpe theory, with dopant concentration (N) analogous to mean coordination number (<r>). Such transitions underpin emerging resistance-switching devices, where voltage-controlled percolation paths enable novel memory and logic functionalities.

In amorphous dielectrics, electrical percolation thresholds manifest through mean network coordination (<r>). For example, in hydrogenated amorphous silicon carbide (a-SiC:H), resistivity increases as hydrogen (acting as a network modifier) reduces <r>, eventually plateauing near <r>=2.4. This aligns perfectly with the critical coordination predicted for conductivity percolation in tetrahedral resistor networks, where resistor bonds below ∼39% density halt current flow. Similar plateaus in resistivity occur in hydrogenated boron carbide (a-B_x C:H) at 30–35% H content, further underscoring the role of topological constraints. Crucially, reducing <r> also lowers dielectric constant (k) in organosilicate low-k dielectrics, but simultaneously weakens mechanical properties, creating a fundamental issue governed by percolation-driven rigidity loss.

The percolation model conceptualizes the low-k dielectric as a composite where pores (k=1) act as intrinsic defects within the silica matrix. The critical percolation threshold is reached when pore density and connectivity enable a continuous conductive pathway. Figure 2.10 starkly demonstrates this threshold effect experimentally: as k-values decrease below ~2.7 (signifying increased porosity), the characteristic breakdown field (E_BD) plummets and the Weibull shape factor (β) worsens.

This confirms that porosity does not just linearly degrade properties but triggers a nonlinear collapse in intrinsic robustness near a material-specific threshold. Hyper-porous ULKs (k < 2.3) and ELK materials (k ≤ 2.0) operate perilously close to this threshold, where even minor process variations (ash damage, moisture uptake) or inherent pore connectivity can catastrophically reduce the number of defects needed to form a lethal percolation path. The model explains why these materials show such poor Weibull β values – broader failure distributions reflect the heightened statistical variability in forming critical defect clusters within the porous network.

Modeling and experimental validation: Percolation modeling resolves key conflicts in lifetime extrapolation. While traditional models (E-model, √E-model) debate field dependence, percolation provides the physical basis for observed trends (i) Area scaling: The consistent Weibull statistics (validated by Figure 11.10 and area-scaling relationships like Equation 11.4 in Ogawa et al. [46]) confirm percolation as a weakest-link mechanism. Larger areas contain more potential critical defect clusters, accelerating failure, (ii) k-value dependency: Figure 2.10 compiles experimental proof that breakdown strength collapses as k decreases. The percolation model attributes this to higher porosity increasing defect density (pores) and reducing the inter-defect spacing, easing conductive path formation and (iii) Pore structure role: The model emphasizes that not just porosity %, but pore size distribution and connectivity (open vs. closed pores) dictate percolation susceptibility. This explains why identical k-values from different processes show divergent reliability, pore topology matters critically.

Percolation theory thus transforms low-k reliability from a mere integration challenge into a fundamental materials science problem governed by defect statistics and network connectivity, with profound implications for future technology scaling.

In summary, the percolation phenomena govern catastrophic electrical failures in OSG-based low-k dielectrics, where nanoscale porosity, introduced to achieve k < 2.5, transforms from isolated voids into interconnected conduction paths once exceeding a critical threshold (25–30% porosity). This percolation transition enables five key failure mechanisms: (i) trap-assisted conduction along defect chains (Si–OH, sp² carbon); (ii) moisture-induced ionic transport through hydrated pores; (iii) ohmic leakage via carbon/metallic residues (Cu, Ta); (iv) plasma-expanded pore networks acting as low-resistance highways; and (v) electrical aging dynamically generating percolation paths under bias. Beyond the threshold, leakage currents surge exponentially, breakdown voltages drop by 40–60%, and TDDB lifetimes collapse as stochastic percolation paths bridge electrodes, flattening Weibull slopes (β < 2) and eroding reliability.

Mitigation strategies disrupt percolation physics by elevating the critical threshold:

• Material design: Optimizing network topology via cross-linking (Si–O–Si rings in a-SiC:H) delays pore interconnectivity beyond 40% porosity [53], while H₂/He plasmas remove sp² carbon residues [59,64].
• Surface engineering: Hydrophobic SAMs seal pore entrances, blocking moisture/contaminant ingress [77], while size-excluded ALD precursors prevent metal diffusion [78].
• Integration control: Pore stuffing (P4 approach) with pore protection by deposited polymers [79,80] and cryogenic etching (− 100 °C), which condenses byproducts to minimize plasma damage [79], silylation passivates Si–OH traps, and stress-graded porosity design weakens field enhancement by 30–50% [76]. These approaches collectively shift the percolation threshold, suppress defect connectivity, and restore dielectric robustness even at ULK regimes.

3. Mechanical Properties of Porous Low-k Materials

The mechanical properties of low-k films are critical to the successful integration of ULSI interconnects, as the films must withstand both the full manufacturing sequence and long-term operational conditions. Parameters such as elastic modulus, hardness, adhesion strength, and fracture toughness govern the ability of a low-k film to resist cracking, delamination, or degradation of dielectric performance during fabrication, packaging, and service. Insufficient mechanical robustness can offset the electrical advantages provided by low permittivity [82].

3.1. Mechanical Properties Versus Connectivity and Porosity

Table 3.1 summarizes the relationship between the mechanical properties of low-k films and ULSI integration requirements, associated metrics, and typical failure modes.

Table 3.1 demonstrates that integrated low-k materials exhibit diverse mechanical property requirements, which are primarily dictated by porosity. The present work reviews the relationship between Young’s modulus and porosity in the investigated films, with particular emphasis on issues associated with the percolation threshold. In porous low-k dielectrics, the Young’s modulus (E) decreases markedly with increasing porosity, as pores simultaneously reduce the fraction of load-bearing solid material and serve as stress concentrators.

Experimental studies of porosity effects in nanoporous low-k dielectrics are often complicated by the difficulty of disentangling porosity from intrinsic matrix properties. In organosilicate glass (OSG) thin films prepared using the porogen approach, porosity is controlled by adjusting the porogen loading during deposition. Removal of the porogen phase typically requires post-deposition treatments such as annealing or ultraviolet (UV)-assisted curing. Because the matrix bond network is highly sensitive to the specific conditions of UV irradiation, the porogen-removal process can modify the matrix structure to varying degrees, depending on the initial porogen content [21].

3.1.1. Mechanical Properties Versus the Matrix Connectivity (dense OSG)

In comparing the mechanical properties of OSG low-k dielectrics with those of SiO₂, the observed reduction in mechanical strength is primarily attributed to the substitution of bridging oxygen atoms with terminal methyl groups, which decreases the overall network connectivity. This effect is rationalized within the framework of Phillips–Thorpe constraint theory, which correlates the mechanical rigidity of glass networks with the number of topological constraints (bond-stretching and bond-bending interactions) relative to the available atomic degrees of freedom [83]. This theory was first applied to OSG low-k films by Gleason and co-workers, who provided a physical explanation of the percolation effect arising from changes in bond connectivity during film fabrication and its consequent influence on the mechanical properties of organosilicate glass dielectrics. [84,85].

In their model, the reduction in Young’s modulus arises from the breaking of bridging (Si–O–Si) bonds and their substitution with terminal groups such as Si–CH₃ or Si–H. Compared with Si–O bonds, Si–C bonds in OSGs are more covalent. By contrast, the large electronegativity difference between Si and O imparts partial ionic character to Si–O bonds, increasing their polarizability and, consequently, the dielectric constant. Furthermore, OSG materials introduce pseudo-porosity (free volume) by replacing the dense, cross-linked SiO₂ lattice with methyl groups bonded to Si atoms. As a result, even “dense” OSG thin films typically exhibit dielectric constants in the range of 2.7–3.3, lower than that of dense silica. Their basic structural units are “M,” “D,” “T,” and “Q” groups, corresponding to mono- (-OSi–(CH₃)₃), di- (=O₂Si–(CH₃)₂), tri- (≡O₃Si–CH₃), and quad- (SiO₄/₂) oxygen-substituted silicon atoms, respectively.

This relationship between chemical bonding and mechanical properties can be further understood through continuous random network theory and the concept of rigidity percolation. Rigidity percolation occurs when the average connectivity number—the number of network-forming bonds per networked atom—reaches approximately 2.4. The connectivity numbers for M, D, and T groups are 1.5, 2.0, and 2.4, respectively. Fully cross-linked SiO₂ consists entirely of Q groups, which have a connectivity number of 2.67. These values are obtained by dividing the total number of network-forming bonds in a structural unit by the number of networked atoms in that unit. For example, a T group contains one silicon atom bonded to three oxygen atoms. The silicon atom forms three network-forming bonds to oxygen, and each oxygen atom forms two such bonds, giving six in total. Since each oxygen is shared between two units, only half of its bonds are counted, leaving six effective network-forming bonds. The total number of atoms is one silicon plus half of the three oxygen atoms (1 + 1.5 = 2.5), yielding a connectivity number of 6 ÷ 2.5 = 2.4.

Two trends are evident: (1) mechanical strength increases sharply as the connectivity number approaches 2.35–2.4, marking the threshold for rigidity percolation, and (2) the introduction of terminal groups such as Si–CH₃ reduces connectivity, lowering mechanical strength while also reducing the dielectric constant through the creation of pseudo-porosity.

The critical connectivity threshold corresponds to a value of 2.35–2.4 (Figure 3.1). The incorporation of CH₃ groups induces a percolation-type transition, manifested as a sharp decrease in Young’s modulus (dotted line). It has been demonstrated that T-type materials (O₃–Si–CH₃) are located near the percolation threshold, depending on porosity, whereas materials containing D- or M-type groups remain below this threshold. Consequently, the Young’s modulus of “dense” OSG materials is significantly lower than that of dense SiO₂ and is strongly influenced by both the type and concentration of organic terminal groups.

In current microelectronics applications, typical low-k dielectrics are primarily methyl-terminated (T-group) OSG materials, with a methyl concentration of approximately 15–20%. These films generally exhibit Young’s modulus values in the range of 8–15 GPa (Table 3.1). Increasing the concentration of methyl groups or introducing D-type groups can enhance plasma resistance; however, this improvement comes at the expense of mechanical strength, as the additional organic substitution further reduces bond connectivity.

3.1.1. Mechanical Properties Versus Porosity (Porous OSG)

Advanced ULSI technology requires the implementation of ultralow-k dielectrics, which are primarily fabricated by introducing porosity. As previously mentioned, porous low-k films are typically prepared using the porogen approach, where porosity is controlled by varying the porogen loading. Subsequent porogen removal necessitates post-deposition treatments, such as ultraviolet (UV) light or electron beam-assisted annealing.

The matrix’s bond network is highly sensitive to specific irradiation conditions, making it difficult to directly compare low-k materials produced under different processing parameters—even when FTIR spectra indicate similar chemical compositions. Nevertheless, general trends have been reported, and comparisons of various studies suggest that mechanical properties evolve with porosity, following a percolation law with a percolation threshold between 35% and 45%.

The dependence of Young’s modulus on porosity is often described by empirical scaling laws or effective medium models. A widely used form is a power-law scaling according to Equation (3.1) [86]:

E(P)=E₀ (1−p)ⁿ,

(3.1)

E₀ – modulus of the dense matrix

p – volume fraction porosity (0–1)

n – exponent between ~2 and ~4 for silica-based films

Several theoretical approaches describe the modulus–porosity relationship:

(a) Gibson–Ashby open-cell foam model (Equation (3.2)) [87]:

E/E₀ ≈ (1−p)²,

(3.2)

This theoretical model predicts the mechanical properties of materials containing open pores (open foams), typically with a length-to-diameter ratio greater than 5. By approximating the complex foam structure as a network of interconnected pores, it enables the estimation of key properties, such as the elastic modulus. Consequently, the model is most applicable to highly porous films where porosity exceeds the percolation threshold. At lower porosity levels—particularly near or below the percolation threshold—the n-value (a critical exponent in the model) is expected to be higher, reflecting the transition in mechanical behavior as the pore network becomes less continuous.

(b) Hashin–Shtrikman bounds [88,89]

The Hashin-Shtrikman (HS) model is a micromechanical framework used to predict the effective elastic moduli of composite materials or heterogeneous systems, such as porous media or multiphase alloys. It establishes rigorous theoretical bounds (known as HS bounds) on these properties and, for certain idealized microstructures, provides exact analytical solutions. By accounting for phase interactions without requiring detailed microstructural information, the HS model bridges fundamental theoretical limits with practical material behavior.

(c) Percolation-type behavior [90,91]

Near a critical porosity p_c (usually ~0.6 for random pores), modulus follows Equation (3.3):

E ∝ (p_c−p)^t,

(3.3)

with t ≈ 3–4. This captures the rapid fall-off near the mechanical percolation threshold. Several studies support the percolation-type scaling of elastic modulus with porosity, expressed as Equation (3.3). For instance, Rice [91] proposed a generalized model E/E₀=(1−p/p_c)ⁿ, with the exponent n ranging between 0.5 and 4 depending on microstructural details. Kováčik [92] further demonstrated such behavior across various porous ceramics, fitting data to E=E₀ (1−p/p_c)^f and finding f ≈ 1.1– 1.7.

The experimental determination of Young’s modulus in thin films is typically conducted via nanoindentation, laser-induced surface acoustic waves, or ellipsometric porosimetry. Owing to their fundamentally different physical operating principles, these techniques may yield slightly divergent values, as each probes a different aspect of the film’s mechanical response. Although each method provides unique information, cross-evaluation can be performed using the results obtained from one technique.

Nanoindentation measurements of Young’s modulus for a comprehensive set of low-k films are plotted in Figure 3.2 as a function of porosity, the latter quantified by ellipsometric porosimetry. While a universal curve derived from the Gibson-Ashby equation (3.2) provides a satisfactory formal fit to the entire dataset, a superior description is obtained by modeling the data with two separate curves. The left curve (1) characterizes the low-porosity regime, where mesopores remain discrete and unconnected. In this region, the reduction in Young’s modulus is attributed not to pore interconnection but to a decrease in the connectivity number of the solid matrix. In region 2, the connectivity remains relatively constant despite the introduction of some porosity. This is because the matrix can locally relax and redistribute stresses, allowing the network to accommodate small pores without significant loss of connectivity. Comprehensive theoretical details on the porosity dependence of stiffness are presented in the Supporting Information (SI). Once porosity reaches a critical threshold in region 3, percolation pathways begin to form, and the connectivity rapidly decreases as the continuous pore network dominates the structural behavior. The percolation transition for mechanical properties (p_c) is identified near 30% porosity, a threshold consistent with that reported for the dielectric breakdown field.

The percolation threshold (p_c) is the critical fraction of occupied sites or bonds at which a connected path spans the entire system, forming an infinite connected cluster (the percolation cluster) [93]. Below p_c all clusters are finite with probability 1; at and above p_c an infinite cluster exists with probability 1 according to Equation (3.4):

$$P\left(p\right)=\{\begin{array}{c}0, for p<p_c\\ >0, for p>p_c\end{array},$$

(3.4)

The value of p_c depends on lattice geometry and the type of percolation problem. For example, in a triangular lattice, percolation occurs at ~30% site occupancy, whereas in a square lattice it requires >50% occupancy. In porous structures, the threshold can be even higher. Different formulations of the problem—site percolation, bond percolation, or continuum percolation—also influence the threshold value.

As already mentioned, generally Young’s modulus depends on many different factors related not only porosity but also their shape, type of ordering etc. Several papers are related to these issues [31,94,95,96].

For instance, Yeap et al. [94] examined a series of OSG low-k films fabricated by both sol-gel and PECVD methods. A key finding was the determining role of pore topology in enabling the synthesis of dielectrics with an ultralow dielectric constant alongside a relatively high Young’s modulus. To elucidate the relationship between porosity, pore topology, and elastic modulus, a finite-element modeling approach was employed. This theoretical framework, when integrated with experimental nanoindentation (NI) data on elastic modulus, provided a powerful tool for predicting the pore morphology of diverse OSG films.

This pore structure prediction was validated using PALS. The porosity threshold for the transition from a closed-pore (non-overlapping) to an open-pore (overlapping) structure, as measured by PALS, was found to be consistent with the value derived from the finite-element model and nanoindentation analysis

It was demonstrated that the dependence of the elastic moduli of PECVD-deposited OSG films on porosity is more accurately described by the randomly overlapping spherical solid model, whereas Sol–Gel OSG films fabricated via self-assembling (PMO) technology are better represented by a model corresponding to ordered, non-overlapping pores.

Li Han et al. [96] employed nanoindentation and the double cantilever beam technique to systematically examine the influence of porogen loading on the stiffness and cohesive fracture energy of a series of porous organosilicate glass (OSG) thin films. The films were synthesized via PECVD with controlled porosities between 7% and 45%.

The research resolved how porogen loading simultaneously alters the film’s network structure and porosity, and it delineated the distinct contributions of each factor to the mechanical properties. Experimental stiffness results were validated against micromechanical models and finite element simulations, confirming a high sensitivity to porosity. This correlation indicates that substantial improvements in mechanical integrity are achievable through strategic optimization of the pore architecture. A linear reduction in cohesive fracture energy with increasing porosity was also observed. This relationship supports a failure mechanism governed by fracture propagation along a planar path through the pore network.

Vanstreels et al. [31] evaluated effect of pore structure of porous low-k films on the measured elastic modulus mechanical properties of various low-k films with different porosity and pore size using various techniques such as nanoindentation, laser-induced surface acoustic wave spectroscopy (LAwave), and EP. The link between the elastic response of these nanoporous materials and their internal pore structure was investigated using positronium annihilation lifetime spectroscopy (PALS), EP, and diffusion experiments. For films with porosity above 25%, the broad band UV cured and narrow band UV cured films follow a similar linear trend and a clear transition is found near 25% of porosity. This transition correlates very well with the pores connectivity threshold (percolation threshold) that was observed by PALS technique and agrees well with the reported values of other nanoporous dielectric films in the literature. PALS measurements indicated that pores are interconnected (percolated) starting from 31% porosity. In addition to PALS, diffusion experiments were performed on nanoporous thin films using isopropanol as a probe molecule. From these measurements, diffusion coefficients were determined. The results showed that molecular diffusion of isopropanol occurred at porosity values well below the percolation threshold determined by PALS. This difference highlights that the percolation threshold depends on the diffusing species: stable molecules like isopropanol can traverse narrow necks connecting internal voids, whereas the PALS threshold reflects the interconnection of larger mesopores accessible to positronium.

3.2. Stiffness Enhancement Strategies in Low-k and Hybrid Glasses

Achieving high mechanical stiffness while maintaining low density and low dielectric permittivity is a central challenge in the design of low-k and hybrid glassy materials for microelectronics and nanotechnology applications. Various strategies have been explored, including molecular-level network design, hyperconnected precursors, organic cross-linking, and bond percolation control. Computational modeling, sol–gel synthesis, and plasma-enhanced deposition techniques have all contributed to understanding how network connectivity, precursor geometry, and non-affine deformations influence stiffness and other material properties. In the following, we review key studies that illustrate how these approaches enable the rational design of ultrastiff low-density glasses.

Figure 3.3 presents the fundamental building blocks of key dielectric materials used in interconnects. The classic SiO₂ structure is shown in (a), with its tetrahedral coordination (Si bonded to four O atoms) and oxygen atoms bridging two silicon atoms. By replacing these bridging oxygen atoms with terminal methyl groups, a methyl-terminated low-k film is formed (b). This process reduces the material’s density and Young’s modulus, as the terminal groups decrease the network’s connectivity.

Han Li et al. [96] performed molecular dynamics simulations showing that organosilicates can be made substantially stiffer than amorphous silica while maintaining a lower mass density. This enhanced stiffness is achieved by incorporating organic cross-links that replace bridging oxygen atoms in the silica network (Figure 3.3(c)). The study explores how both organic cross-linking and terminal groups influence the stiffness and density of OSG materials. To this end, the authors used systematic modeling and analysis, examining structural properties (density, bond angles, and bond lengths) as well as mechanical properties (bulk modulus, shear modulus, and Young’s modulus) as functions of composition and network connectivity. The incorporation of organic bridging groups is expected to improve mechanical properties due to the higher bending rigidity of Si–CHₓ–Si bonds compared with Si–O–Si bonds. Another anticipated advantage is that OSG materials containing bridging organic groups can form films with ordered porosity, known as periodic mesoporous organosilicates (PMOs).

An even more aggressive strategy was proposed by Burg et al. [97], who demonstrated networks with silicon coordination numbers greater than four, achieving theoretical Young’s moduli of up to 48 GPa (Figure 3.3 (d)). Using 1,3,5-silylbenzene precursors, they designed a hyperconnected network in which each silicon atom bonds to as many as five neighboring silicon atoms, exceeding the typical tetravalent coordination. This hyperconnectivity arises from a combination of Si–O–Si linkages and rigid benzene bridges, which raise the mean silicon network connectivity beyond the conventional limit of four (Figure 3.3a). To validate the computational model, a 1,3,5-benzene-based glass was synthesized via sol–gel chemistry from 1,3,5-(triethoxysilyl)benzene, yielding a film with a density of 1.52 g cm⁻³—significantly lower than fully dense silica (~2.2 g cm⁻³). Most silicon atoms adopt T-group configurations (three oxygen and one carbon neighbor), while 10.8% form Q-groups (four oxygen neighbors). Precursor dissociation produced a glass comprising ~2:1:1 molar ratios of 1,3,5-benzene, 1,3-benzene, and SiO₂ structural units. Whereas optimized organosilica glasses deposited via PECVD typically achieve E≈16 GPa with ≤10 at.% carbon, the studied sol–gel glass contained 36 at.% carbon and exhibited E ≈ 23±0.5 GPa at the same density. The high stiffness was attributed to rigid benzene bridges, which preserved network connectivity, whereas flexible precursors generated network terminations that lowered both connectivity and mechanical performance. Steric hindrance from bulky terminal groups (e.g., benzene) similarly reduced stiffness by disrupting network continuity.

King et al. [98] investigated the effect of network bond percolation on the properties of amorphous hydrogenated silicon carbide (a-SiC:H) thin films with both high and low dielectric constants. They examined how variations in the mean bond coordination number <r> influence mechanical, thermal, electrical, and optical properties, using constraint theory and bond percolation theory. The a-SiC:H films were deposited via PECVD, with deposition parameters adjusted to control hydrogen content and C/Si stoichiometry. This work provides experimental validation of constraint theory and bond percolation concepts for a-SiC:H, advancing the understanding of structure–property relationships in amorphous materials. These findings also inform the optimization of low-k materials for integrated circuits, where balancing low dielectric permittivity with high mechanical strength is essential. Based on their results, the authors proposed strategies such as incorporating stronger network bonds (e.g., Si–C instead of Si–O) or controlling weak van der Waals interactions to enhance structural integrity.

3.3. Non-Affine Deformations Control

The third research direction involved introducing hyperstiff precursor molecules with distinct geometrical characteristics to suppress nonaffine deformations. Kilic et al. [99,100,101] demonstrated that precursor design—specifically short bridging groups, molecular planarity, and symmetry—reduces nonaffine deformations and enhances stiffness. As a result, hyperconnected organosilicate networks formed from these precursors exhibit elastic properties superior to fully dense silica.

Molecular dynamics and constraint analyses identified the features that render precursors hyperstiff, offering the first quantitative evaluation of nonaffine deformations in hybrid organosilicate glasses. This represents a new framework to assess how both precursor geometry and network connectivity govern mechanical response.

Hybrid organic–inorganic glasses occupy a unique space between oxides and polymers, yet their inherently low density and brittleness limit mechanical robustness and hinder device integration. The proposed molecular design strategy simultaneously optimizes network connectivity and precursor geometry, enabling ultrastiff hybrid glasses that exceed the stiffness of dense silica while preserving low density.

Hyperconnectivity was achieved by increasing silicon connectivity beyond its coordination number of four, up to five through three oxygen bonds and two additional carbon bridges within the precursor [97,99]. This dual exploitation of hyperstiffness and hyperconnectivity constitutes a new design paradigm. Unlike traditional approaches that inevitably increase density by raising cross-link density, this method enhances stiffness via precursor geometry. The resulting networks open opportunities for the integration of robust, lightweight hybrid glasses into microelectronic, optical, and energy technologies.

Nevertheless, the presence of rigid fragments alone does not guarantee overall material rigidity. To maintain hydrophobicity, methyl terminals must constitute at least 12–15% of the material. Consequently, the rigid fragments are often embedded in a weakly rigid matrix at concentrations below the percolation threshold [102], limiting their ability to reinforce the network. Ensuring system-wide stiffness requires the formation of a percolating cluster of rigid fragments. Therefore, understanding the mechanisms of percolation cluster formation—including characteristics such as self-similarity and fractality near the percolation threshold—is essential. This knowledge enables the purposeful synthesis of elastic percolation clusters, providing a pathway to maximize rigidity in hybrid glasses.

4. Conclusions

In this work, we reviewed percolation-related phenomena in porous low-k dielectrics and demonstrate their importance for mass transport, electrical conductivity, mechanical properties, and dielectric breakdown. A brief summary of the main processes is provided below.

1.

Percolation Threshold for Leakage Current and mass transfer.

Leakage current is governed by charge carriers, typically electrons, hopping or tunneling through connected low-density regions comprising pores and damaged dielectric around pores. Current conduction requires only a minimal percolating path connecting the electrodes. This threshold is usually reached at relatively low porosity or damage levels, as a single percolating cluster is sufficient to sustain leakage. Organosilicate glass (OSG)-based low-k films prepared with a sacrificial porogen generally contain micropores, thought to arise from the substitution of bridging oxygen atoms in the silica network with sterically larger methyl groups (Figure 1.2). Additional microporosity may also be introduced during porogen removal. PALS measurements indicate that these micropores typically appear as closed [12,29], although they may still allow diffusion of stable molecules [29,30]. Consequently, they can form interconnected pathways for impurities such as adsorbed water, residual porogen species, or oxygen-deficient centers likely generated during UV curing. Exponential increases in leakage current via Poole–Frenkel (PF) emission along defect chains further accelerate reliability failures, as copper diffusion and moisture ingress propagate through connected pores, sharply reducing time-dependent dielectric breakdown (TDDB) lifetimes. These observations suggest that the percolation thresholds for leakage current and mass transport may substantially overlap.

2.

Percolation Threshold for Dielectric Breakdown

Dielectric breakdown differs fundamentally from leakage current. While leakage arises from the earliest formation of a conductive path, breakdown corresponds to the catastrophic failure of insulating capability across the device area. Breakdown involves:

• local defect accumulation and hot-spot formation,
• field enhancement at pore edges,
• charge trapping, and
• progressive filament growth.

For breakdown to occur, multiple “near-percolating” paths across a wide area must interact. Thus, the effective percolation threshold for breakdown is higher than for leakage current: the material can tolerate more porosity before the breakdown field collapses. In other words, breakdown requires a dense network of connected pores and weak links, rather than a single tenuous path.

This statistical distinction explains why leakage current increases at a lower porosity threshold, whereas degradation of the breakdown voltage becomes significant only near the bulk-scale percolation limit [46,64]. Although many studies report that both leakage current and breakdown voltage increase with porosity, an illustrative example highlighting the distinction between the two was provided by [58]. In that study, films were fabricated using a tailored process developed by Urbanowicz [59] that effectively suppressed porogen residues and eliminated adsorbed water. As a result, the materials exhibited an unusual response: the leakage current showed no measurable dependence on porosity, whereas the breakdown field decreased roughly inversely with porosity. These findings support the interpretation that leakage current is predominantly governed by localized, impurity-assisted conduction pathways along pore walls, while dielectric breakdown is controlled by extended pore connectivity approaching the percolation threshold.

3.

Mechanical Percolation

Mechanical percolation in porous low-k dielectrics differs from electrical percolation. Properties such as elastic modulus, hardness, and fracture toughness degrade once pores coalesce into a connected network. Unlike leakage current, which requires only one conductive path, mechanical properties depend on the integrity of the global load-bearing skeleton. As a result, the threshold for mechanical failure is closer to that for dielectric breakdown than for leakage current, since both involve system-spanning connectivity of the pore network.

Even more complex behavior is observed in OSG low-k dielectrics. The characteristic double slope in the dependence of Young’s modulus on porosity can be explained by the presence of two distinct percolation thresholds (Figure 3.2). The first threshold occurs at low porosity and is associated with the replacement of bridging oxygen atoms by terminal methyl groups, which decreases the network connectivity in the OSG matrix [84,85]. This mechanism also accounts for the significantly reduced initial Young’s modulus values (E₀) compared with dense SiO₂ (15-20 GPa for dense OSG instead 70–80 for SiO₂). The second slope corresponds to the percolation threshold related to porosity itself, which poses a major challenge for achieving mechanically robust ultra-low-k (ULK) materials.

To address the problem of the intrinsically low Young’s modulus at near-zero porosity (E₀), several strategies have been proposed. The most common involve replacing bridging oxygen atoms with carbon-containing groups [96] or incorporating hyperconnected units such as benzene rings [97]. However, these modifications alone cannot ensure sufficient hydrophobicity, necessitating the additional introduction of terminal methyl groups. As a result, achieving a simultaneous improvement in both hydrophobicity and mechanical integrity remains a significant challenge in the development of OSG-based ultralow-k dielectrics. Importantly, the presence of multiple percolation thresholds underscores that both transport and mechanical properties in porous low-k materials are fundamentally controlled by network connectivity effects, providing a unifying framework for understanding their performance and guiding future material design.

4.

Experimental Trends

Experimentally, increasing porosity leads to distinct percolation-driven changes:

• Leakage current rises sharply at relatively low porosity.
• Breakdown voltage and mechanical modulus collapse abruptly at higher porosity, typically in the range of 20–30% (depending on pore size and connectivity), just above the percolation threshold for open pores.

Thus, the percolation threshold for mechanical properties aligns more closely with that for dielectric breakdown, whereas leakage current reflects the earliest onset of connectivity, as it is shown in the schematic diagram in Figure 4.1.

5.

Distinct Percolation Classes

The percolation transition induces several classes of physical phenomena:

• Transport phenomena: mass diffusion, electrical conduction, and leakage current.
• Mechanical phenomena: degradation of stiffness and strength.
• Nonlinear phenomena: dielectric breakdown.

At least three distinct percolation thresholds may therefore exist, each associated with a cluster of unique topology and backbone structure. The geometry of these clusters—and hence the thresholds—depends on the underlying physics of the phenomenon.

6.

Correlation Length and Timescales

The structure of a percolation cluster is characterized by the correlation length:

$$L_c\left(p\right)\propto {\left|p-p_c\right|}^{-\mathsf{\nu }}, p\to p_c,$$

(4.1)

where ν is the critical exponent of the correlation length. dependent on dimensionality and structure. Percolation clusters include branched conductive paths and dead ends, which affect transport.

For mass transport, the relevant timescale is the diffusion time:

$$t_{L_c}={\displaystyle \frac{L_c^2}{D}},$$

(4.2)

where D is the diffusion coefficient and L_c is the correlation length of the conductivity cluster. The impulse relaxation time τ_p, which governs conductivity, is linked to diffusion via the Einstein relation:

$$𝑞𝐷=𝜇𝑘𝑇,\mu ={\displaystyle \frac{q{\tau }_p}{m}},$$

(4.3)

Thus, diffusion and conductivity can be described within a unified framework, characterized by equivalent timescales.

Leakage current is directly related to connectivity. As porosity, defects, or absorbed moisture approach the percolation threshold, leakage current can increase by several orders of magnitude, since isolated conductive regions merge into low-resistance pathways. This process is governed by the same correlation length L_c.

7.

Elastic and Breakdown Percolation

Mechanical stiffness is governed by a distinct elastic percolation cluster with correlation length $L_y^c$. This cluster exhibits a three-dimensional, foam-like architecture that differs from the quasi-1D conduction cluster.

For dielectric breakdown, an additional length scale arises:

$$\alpha ={\displaystyle \frac{qEL}{W}}, L_E={\displaystyle \frac{W}{qE}},$$

(4.4)

where q is the elementary charge, E the electric field, and W a characteristic energy (e.g., W=kT). Breakdown occurs when the characteristic system length L becomes comparable to L_E,. The relevant comparison is whether L_E couples more strongly to L_c (conductivity cluster) or $L_y^c$ (stiffness cluster).

Breakdown pathways are typically facilitated by structural defects (voids, cracks, impurities) or agglomerated conductive particles, which form branched, foam-like chains that act as preferential channels. These differ from conduction clusters, where zigzag quasi-1D paths hinder breakdown propagation. Because stiffness percolation clusters provide three-dimensional connectivity, breakdown pathways more closely coincide with stiffness pathways. This leads to the approximate equivalence:

$$L_y^c \approx L p_c^{\gamma }\approx p_b,$$

(4.5)

These theoretical considerations are consistent with experimental observations of dielectric breakdown in porous low-k materials.