# Poole-Frenkel conduction and critical charge-to-breakdown in plasma-enhanced chemical vapor deposition silicon-rich silicon nitride films

Tania Al Moussi, Paul Lambkin, Ramji Lakshmanan, Baoxing Chen, Sombel Diaham

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## ► To cite this version:

Tania Al Moussi, Paul Lambkin, Ramji Lakshmanan, Baoxing Chen, Sombel Diaham. Poole-Frenkel conduction and critical charge-to-breakdown in plasma-enhanced chemical vapor deposition silicon-rich silicon nitride films. Thin Solid Films, 2026, 836 (15), pp.140886. 10.1016/j.tsf.2026.140886. hal-05517610

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Submitted on 18 Feb 2026

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# Poole-Frenkel conduction and critical charge-to-breakdown in plasma-enhanced chemical vapor deposition silicon-rich silicon nitride films

T. Al Moussi<sup>a, b</sup>, P. Lambkin<sup>b</sup>, R. Lakshmanan<sup>b</sup>, B. Chen<sup>c, d</sup>, S. Diaham<sup>a, b, *, e</sup>

## ARTICLE INFO

**Keywords:**

Silicon nitride

Plasma-enhanced chemical vapor deposition

Silicon-rich

Trap assisted transport

Poole-Frenkel conduction

Dielectric breakdown

## ABSTRACT

This work addresses the limited understanding of how stoichiometry influences high-field conduction and breakdown-triggering mechanisms in Silicon-rich silicon nitride (SiNₓ) thin films deposited by plasma-enhanced chemical vapor deposition (PECVD), which is critical for their use as insulating barriers for galvanic isolation in microelectronic devices operating under high electric fields. Amorphous hydrogenated a-SiNₓ:H films with x=[N]/[Si] ratios ranging from 0.86 to 1.06 were deposited by PECVD and integrated into metal–insulator–metal (MIM) structures. Electrical characterization was performed using DC polarization–depolarization measurements and AC high-voltage broadband dielectric spectroscopy over a wide frequency range, allowing extraction of current density–electric field (J–E) characteristics under high-field conditions. The experimental results show that Poole–Frenkel (PF) emission governs field-assisted conduction in Silicon-rich SiNₓ films under both DC and AC stress. PF analysis enabled extraction of the zero-field trap barrier height (φ₀) and the theoretical dielectric permittivity (ε'ᵢнео), which were compared to experimental permittivity values. The barrier height (φ₀) was found to increase from approximately 0.8 to 1.5 eV with increasing nitrogen content, while (ε'ᵢнео) exhibited a decreasing trend, consistent with experimental observations. Breakdown measurements revealed a strong dependence of the breakdown field (E<sub>BD</sub>) on composition, increasing with nitrogen incorporation. In contrast, the charge-to-breakdown (Q<sub>BD</sub>) remained within a narrow range across all compositions, indicating that dielectric failure is governed by a critical injected charge, highlighting a conduction-triggered breakdown process. These results provide design-relevant insights into the role of stoichiometry and frequency on trap-assisted conduction and breakdown behavior in Silicon-rich SiNₓ insulating layers, with direct implications for high-voltage microelectronic and galvanic isolation applications.

## 1. Introduction

Ceramic materials are widely recognized as an important class of advanced materials and are increasingly implemented in a broad range of technological applications owing to their outstanding thermal, mechanical, and electrical properties [1–3]. Among these ceramics, silicon nitride (SiNₓ) has attracted particular attention due to its versatile and tunable properties, making it suitable for demanding microelectronic and high-field insulation applications [4–6]. Their versatility makes them suitable for use in optical devices, such as antireflection coatings in solar cells [7–9] and encapsulation in inorganic LEDs, as well as in

micro-electro-mechanical systems (MEMS) [10,11], non-volatile memory elements [12], and as insulating barriers in advanced semiconductor devices [13–15]. More particular in digital isolation technologies, silicon nitride thin films have been increasingly employed as dielectric insulating barriers to provide galvanic isolation between electrically separated domains [16]. Recent studies have reported that the integration of silicon nitride (SiNₓ) dielectric barriers in digital isolator architectures can lead to a significant improvement in device lifetime, highlighting the critical role of the SiNₓ layer in governing long-term reliability under high electric field stress. When subjected to high electric fields, Silicon-rich SiNₓ films can exhibit significant leakage

<footnote><sup>a</sup> LAPLACE, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France</footnote>

<footnote><sup>b</sup> Analog Devices International, Limerick, Ireland</footnote>

<footnote><sup>c</sup> Analog Devices Incorporation, Wilmington, MA, USA</footnote>

<footnote>* Corresponding author.
E-mail address: sombel.diaham@laplace.univ-tlse.fr (S. Diaham).</footnote>
currents, which may compromise the performance and reliability of devices relying on their insulating function [17–19].

Understanding the electrical conduction behavior in these materials is therefore crucial, especially as a function of their composition and under high electric field combined to a switching frequency, to improve the next generation of devices for the design and optimization of reliable isolation barriers [20–22]. It has been shown that plasma-enhanced chemical vapor deposition (PECVD) Silicon-rich SiNₓ films contain structural irregularities, silicon (Si) dangling bonds, and hydrogen (H) impurities [23,24]. In particular, the presence of excess of silicon leads to the formation of defect states, such as Si dangling bonds. In addition, our earlier findings have shown the presence of H atoms where its concentration varies with composition, generally increasing with nitrogen content [25,26]. Si–H bonds are more prevalent in Silicon-rich films, whereas N–H bonds tend to dominate at higher [N]/[Si] ratios. These structural variations modify the local bonding environment and play a key role in determining the dominant charge transport mechanisms [27].

Previous studies have identified various DC conduction mechanisms in silicon nitride SiNₓ films, including Schottky emission, hopping conduction, and Poole–Frenkel (PF) conduction [28,29]. Among these, PF emission—a trap-assisted transport process describing field-assisted thermal excitation of trapped carriers—has been shown to dominate in Silicon-rich films under high-field DC conditions [30–33]. However, it is still not clear how the charge transport is affected under AC field conditions and how the frequency modifies the conduction mechanism through the trapping effectiveness.

This work focuses on analyzing the Poole–Frenkel conduction behavior in hydrogenated amorphous Silicon-rich a-SiNₓ:H films under DC and AC field conditions. To this end, six different silicon nitride SiNₓ compositions were synthesized and integrated into metal–insulator–metal (MIM) test structures. Their electrical properties were investigated through a combination of DC and AC high-voltage measurements. DC polarization/depolarization cycles were used to probe steady-state polarization currents, from which current–electric field (J–E) characteristics were extracted. Unlike many previous studies that focus on DC characterization only, this work extends the analysis of Poole–Frenkel conduction to include AC measurements at multiple frequencies by using high-voltage broadband dielectric spectroscopy (HVBDS), allowing for the analysis of frequency-dependent conduction behavior and offering new insights into the role of frequency in trap-assisted conduction mechanisms in Silicon-rich SiNₓ films under AC high electric fields. Morever, the charge-to-breakdown (Q<sub>BD</sub>) is assessed in to the dielectric breakdown field. By correlating the electrical conduction behavior at high field with both the silicon nitride SiNₓ composition and excitation frequency, this work contributes to the identification of the physical origins and the critical conditions where the breakdown occurs in silicon nitride SiNₓ films. It paves the way for optimal dielectric working performances for high-field applications and enhances the fundamental understanding of charge transport to get better high-field reliability in digital isolation applications relying on SiNₓ-based insulating barriers.

## 2. Experimental part

Blanket silicon nitride a-SiNₓ:H films of 1 µm-thick were deposited by PECVD in an industrial production cleanroom facility on ø200 mm Si wafers with a thickness of 300 µm. The deposition parameters were systematically varied to vary the chemical composition of the films. Silicon-rich SiNₓ films that range in composition from SiN<sub>0.86</sub> to SiN<sub>1.06</sub> were produced [26]. The film composition was determined by quantifying the atomic percentages of Si and N by field emission gun (Cameca SXFiveFE) and an electron microprobe X-Ray microanalysis (FEG-EPMA) analysis and by calculating the value of x=[N]/[Si].

Table 1 shows the x=[N]/[Si] ratio values obtained along with the atomic percentages of Si and N atoms contained in the different silicon

<caption>**Table 1**</caption>

Atomic percentages of N and Si and the [N]/[Si] ratio of the different Silicon-rich SiNₓ films measured by FEG-EPMA.

<table><thead><tr><th>Sample#</th><th>[N] atomic %</th><th>[Si] atomic %</th><th>x=[N]/[Si]</th></tr></thead><tbody><tr><td>1</td><td>51</td><td>48</td><td>1.06</td></tr><tr><td>2</td><td>50</td><td>50</td><td>0.99</td></tr><tr><td>3</td><td>49</td><td>50</td><td>0.96</td></tr><tr><td>4</td><td>48</td><td>51</td><td>0.95</td></tr><tr><td>5</td><td>48</td><td>52</td><td>0.91</td></tr><tr><td>6</td><td>46</td><td>53</td><td>0.86</td></tr></tbody></table>

nitride SiNₓ films, as measured by FEG-EPMA where the Si concentration increases and the N content decreases. For clarity and consistency with the experimental accuracy of the technique, the reported atomic percentages of Si and N are rounded to integer values. The stoichiometric ratio x=[N]/[Si] was calculated from the original (unrounded) compositional data, and its associated uncertainty was obtained by standard error propagation from the uncertainties on the N and Si concentrations. For the investigated samples, the resulting uncertainty on x is on the order of ±0.04–0.07. All x values lie well below the stoichiometric ratio of 1.33 corresponding to Si₃N₄, confirming an excess of silicon atoms in the matrix structure and a silicon-rich composition for all six SiNₓ films. MIM structures were fabricated for each SiNₓ sample and used as test devices for the electrical characterization of conduction behavior. Six MIM capacitor structures with an Au/TiW–SiNₓ–TiW/Au/TiW stack (Fig. 1(a)) were manufactured on 300 µm-thick silicon substrates under identical PECVD deposition conditions. The SiNₓ dielectric layer thickness was 1 µm, and the circular top electrodes had a diameter of 1.5 mm. All samples were processed on an industrial production line in a high-class cleanroom environment, ensuring controlled and reproducible fabrication conditions.

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** (a) SEM image of cross-sectional view of MIM silicon nitride SiNₓ test structure prepared by focused ion beam (FIB) etch. (b) Schematic illustration of the cross-section of a MIM capacitor wafer-level test structure. On the left, a digital microscope image shows the top view of the MIM structure capped with a polyimide final encapsulation layer [22]. This figure is reproduced from [26].</figcaption></figure>
Titanium–tungsten (TiW) layers were used both below and above the SiNₓ film as adhesion layers between SiNₓ and gold. The presence of TiW on both sides of the dielectric results in a symmetrized MIM structure, ensuring work-function symmetry during high-voltage electrical characterization. To enable reliable electrical contacting and to prevent mechanical damage to the SiNₓ layer during probing, a thick gold layer was electroplated on the top electrode. Electrical connections were made using probe needles contacting this reinforced gold surface. Finally, the MIM devices were encapsulated with a polyimide capping layer, which also served as an etching mask during wet etching of the SiNₓ layer to expose the bottom electrode contact (see Fig. 1(b)).

All electrical characterizations were performed using a Novocontrol Alpha-A dielectric analyzer, operated in both frequency-domain and time-domain modes. For high-electric-field measurements, the analyzer was coupled with a HVB 4000 high-voltage booster. The samples, configured as MIM devices, were mounted on a manual probe station. Electrical contact was ensured using two conductive probe needles, allowing access to the top and bottom electrodes during measurement. All measurements were carried out at room temperature (≈ 23 °C) under ambient atmosphere.

The DC electrical behavior of the MIM structures was characterized using a step-wise polarization–depolarization protocol to extract the steady-state polarization current under increasing electric fields (see Fig. 2(a)). Each measurement sequence began with a 0 V hold for 500 s followed by the application of a voltage $V_i$ for 500 s. The applied voltage was increased in increments of 10 V, starting from 10 V up to 800 V. After each voltage step, the system was returned to 0 V for 500 s to monitor depolarization dynamics. This cycling sequence (0 V → $V_i$ → 0 V → $V_{i+1}$) was repeated across the entire voltage range. The polarization current was extracted at the end of each voltage hold step, where a quasi-steady-state current response for silicon nitride SiNₓ was observed, as shown in Fig. 2(b). In addition to conduction current analysis, breakdown events observed in DC measurements enabled the determination of the breakdown field $E_{BD}$ at the maximum withstand field and the associated charge-to-breakdown $Q_{BD}$ at which the breakdown occurs, assessed as follows:

$$Q_{BD} = \int_{0}^{t_{BD}} J(t)dt \qquad (1)$$

where $J(t)$ is the instantaneous current density and $t_{BD}$ the time-to-breakdown [34].

To complement the DC conduction data, high-voltage broadband dielectric spectroscopy (HVBDS) was employed to investigate the AC response of the MIM structures under varying electric field amplitudes and frequencies at room temperature. A sinusoidal AC voltage was applied using with amplitudes ranging from 1 $V_{rms}$ to 800 $V_{rms}$ at three fundamental frequencies: 100 mHz (low frequency), 50 Hz, and 100 Hz

(higher frequencies) [35,36]. The resulting AC current was measured as a function of a full-wave voltage amplitude at each frequency, from which the AC current density $J$ versus the electric field $E$ were derived. Additionally, the real part of the permittivity $\epsilon'$ of each silicon nitride SiNₓ film was also experimentally extracted from the measured impedance.

From both DC and AC J–E characteristics, the Poole–Frenkel (PF) emission model was analyzed [37]. The PF model expresses the current density $J$ as a function of the electric field $E$ according to the relation:

$$J = AE\exp\left(\frac{-q(\phi_0 - \beta_{PF}\sqrt{E})}{K_B T}\right) \qquad (2)$$

where $\phi_0$ is the zero-field potential barrier height, $k_B$ the Boltzmann's constant, $T$ the absolute temperature and $\beta_{PF}$ the Poole–Frenkel coefficient given by:

$$\beta_{PF} = \sqrt{\frac{q^3}{\pi\epsilon'\epsilon_0}} \qquad (3)$$

where $\epsilon'$ is theoretical film permittivity, $\epsilon_0$ the vacuum permittivity and $q$ the electric charge.

From J–E data obtained, $\ln(J/E)$ vs. $E^{1/2}$ were plotted for each silicon nitride SiNₓ composition and measurement frequency to evaluate the validity of the PF model and to extract the key parameters [38,39]. The intercept of the fitted curve was used to estimate the zero-field potential barrier height $\phi_0$, while the slope provides an estimation of the theoretical dielectric permittivity $\epsilon'_{theo}$. These values were then compared to the experimental permittivity $\epsilon'_{exp}$ measured by HVBDS, enabling a comprehensive evaluation of the Poole–Frenkel conduction under both static and dynamic electrical conditions.

# 3. Results and discussion

## 3.1. J-E characteristics

Fig. 3 shows the J-E characteristics of the Silicon-rich SiNₓ films for the different $x=[N]/[Si]$ ratios at $f = 100$ mHz measured by HVBDS. Two distinct regimes are clearly observed: (i) a low-field region (below ~100 V/µm) where the current density increases gradually with the electric field, and (ii) a high-field region characterized by a sharp rise in the current density [40,41]. This behavior indicates a transition to field-enhanced conduction, typical of mechanisms such as Poole–Frenkel emission. The same behavior was observed in DC measurements and at higher AC frequencies (50 Hz and 100 Hz).

## 3.2. Poole–Frenkel conduction

Poole–Frenkel plots $\ln(J/E)$ vs. $E^{1/2}$ of field-dependent leakage

<figure><img src="image_5.png" /><figcaption>Fig. 2. DC polarization–depolarization protocol applied to MIM structures of PECVD Silicon-rich SiNₓ films. (a) Applied voltage waveform illustrating successive field-stress cycles with incremental voltage steps and defined polarization/depolarization intervals. (b) Transient polarization $I_{Pol}$ and depolarization $I_{Depol}$ current responses highlighting the extraction of the steady-state $I_{Cond}$ currents at the end of each bias plateau for SiN₀.₉₉ film, used for subsequent J–E analysis.</figcaption></figure>

<figure><img src="image_6.png" /></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 3. Current density–electric field *J*–*E* characteristics of silicon nitride SiNₓ films with varying *x*=[N]/[Si] ratios under AC field measured at 100 mHz.</figcaption></figure>

currents are shown in Fig. 4, corresponding to DC, 100 mHz, 50 Hz, and 100 Hz measurements, respectively. In all cases, the high-field region exhibits a clear linear trend, confirming the relevance of the PF mechanism. The linear portion appears more pronounced and extends over a broader field range in Silicon-rich compositions, reflecting enhanced trap-assisted conduction. In contrast, films with higher *x* values require stronger fields to enter the linear regime, indicating larger potential barriers and fewer available trap states. At higher electric fields, a noticeable change in the slope of the ln(*J*/*E*) vs. *E*¹/² plots is observed for all SiNₓ compositions. This nonlinear behavior suggests that additional conduction mechanisms may become active under strong electric stress

[42]. In particular, the slope change could be related to trap saturation, enhanced carrier injection, or the formation of space charges within the dielectric layer [43].

The zero-field potential barrier $\phi_0$ and theoretical relative permittivity $\epsilon'_{theo}$, derived from fitting the high-field of each PF plot of Fig. 4, are shown in Fig. 5 as a function of composition and for all the measurement conditions (DC and AC at 100 mHz, 50 Hz, and 100 Hz). A strong compositional dependence is observed in both properties. The zero-field potential barrier $\phi_0$, calculated from the linearized Eq. (2), increases with nitrogen content across all measurement regimes, indicating that charge carriers require a higher field to escape from trap states in the lower Silicon-rich compositions [38]. This trend directly reflects the presence of deeper traps in films with higher $x$=[N]/[Si] ratios, which is consistent with a denser and more cross-linked Si–N network [44,45]. Conversely, more Silicon-rich films ($x$ = 0.86–0.91) exhibit lower barrier heights $\phi_0$, indicative of shallower trap depths, which facilitate easier carrier emission into the conduction band under high electric field. This is in agreement with the reported increase in Si–H bonds and silicon dangling bonds in Silicon-rich PECVD SiNₓ, which are known to create shallow localized states that promote field-enhanced conduction [46]. The highest $\phi_0$ values are observed under DC conditions between 1.07 eV and 1.47 eV, suggesting that dynamic AC excitation may assist further in trap depopulation, effectively reducing the field intensity required for carrier emission into the conduction band and leading to lower apparent barrier heights between 0.6 eV and 0.8 eV for $f$ = 100 Hz [47]. The uncertainty associated with the extracted Poole–Frenkel barrier height $\phi_0$ was obtained from the linear regression of the Poole–Frenkel plots. Depending on composition and excitation frequency, the resulting error on $\phi_0$ ranges from approximately $\pm$0.003 eV to $\pm$0.05 eV.

The theoretical permittivity $\epsilon'_{theo}$ values (see Fig. 5(b)), calculated from Eq. (3) from the PF slope for each condition, although small deviations are observed at higher frequencies, the theoretical permittivity

<figure><img src="image_8.png" /><figcaption>Fig. 4. Poole–Frenkel analysis ln(*J*/*E*) versus *E*¹/² of SiNₓ films with varying *x*=[N]/[Si] ratios under (a) DC bias and AC field at (b) 100 mHz, (c) 50 Hz and (d) 100 Hz.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Correlation between the zero-field potential barrier $\phi_0$ (a), the theoretical permittivity $\epsilon'_{theo}$ (b) in the SiN$_x$ films extracted from Poole-Frenkel analysis and the $x=[N]/[Si]$ ratio under DC and AC (100 mHz, 50 Hz, and 100 Hz) conditions. Error bars represent the uncertainty associated with the dielectric spectroscopy measurements and data processing procedure for each condition.</figcaption></figure>

extracted from the PF slope show an overall decrease with increasing x for all frequencies investigated. The uncertainty on $\epsilon'_{theo}$ was estimated and the resulting error bars are shown in the figures and ranges from approximately $\pm0.02$ to $\pm1.8$. The highest values are found in the most Silicon-rich compositions for $x = 0.86$ and $x = 0.91$, which is consistent with the presence of a denser trap network and more polarizable bonding configurations, such as Si-H. These features contribute to an enhanced dielectric response, particularly at low frequencies, where slow polarization mechanisms are active. In contrast, the less Silicon-rich films, which are characterized by strong, rigid Si-N and N-H bonds, exhibit lower permittivity due to limited bond reorientation and reduced overall polarizability [48,49].

Fig. 6 presents the relative deviation between the dielectric permittivity values extracted from Poole-Frenkel fitting ($\epsilon'_{theo}$) and those measured experimentally by HVBDS ($\epsilon'_{exp}$) at three AC frequencies (100 mHz, 50 Hz and 100 Hz). Overall, the comparison between the theoretical and experimental values is consistent across the full range of compositions, supporting the relevant applicability of the PF model for describing the field-enhanced conduction behavior in SiN$_x$ films. At 100 mHz, the relative error remains acceptable and relatively stable across compositions, indicating good agreement between the PF-extracted and experimentally measured permittivity in the low-frequency regime, where both dipolar and interfacial polarization processes are active. At higher frequencies (50 Hz and 100 Hz), the relative error is further

<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Relative deviation between the theoretical permittivity extracted from the Poole-Frenkel model and the experimentally measured permittivity as a function of $x=[N]/[Si]$ of SiN$_x$ films for AC measurements at 100 mHz, 50 Hz and 100 Hz.</figcaption></figure>

reduced, suggesting improved consistency between theoretical and experimental values. These results confirm that the PF framework remains a valid and reliable approach for analyzing the dielectric behavior of Silicon-rich SiN$_x$ films under high electric fields across a broad range of frequencies and compositions.

### 3.3. Dielectric breakdown and charge-to-breakdown

Fig. 7 illustrates the variation of the dielectric breakdown field $E_{BD}$ and the corresponding charge-to-breakdown $Q_{BD}$ as a function of the [N]/[Si] ratio under DC stress for the SiN$_x$ films. A strong compositional dependence is observed for $E_{BD}$, which increases with increasing nitrogen content. The most Silicon-rich films SiN$_{0.86}$ and SiN$_{0.91}$ exhibit the lowest breakdown fields, in the range of 320–340 V/µm, whereas the less Silicon-rich film SiN$_{1.06}$ shows significantly enhanced dielectric strength, with breakdown fields around 500 V/µm. This behavior reflects the influence of microstructural and chemical characteristics on the dielectric performance of the SiN$_x$ films and is consistent with the results obtained from Poole-Frenkel analysis, where lower $x=[N]/[Si]$ ratios were shown to have lower potential barrier heights, indicating shallower trap depths. The earlier breakdown observed in these films can thus be attributed to the easier activation and emission of carriers from shallow traps under increasing electric field. In contrast, for the higher $x=[N]/[Si]$ ratios, which exhibit higher values and thus deeper trap states, require stronger fields to initiate carrier release and reach breakdown [50,51]. The lowest $x=[N]/[Si]$ ratios are also known to contain higher defect densities due to excess silicon-related bonding configurations. These defects, including Si dangling bonds, contribute to increased leakage currents and reduced dielectric strength through localized field enhancement and trap-assisted carrier injection [52].

The charge-to-breakdown $Q_{BD}$ values are presented in Fig. 7 for the six Silicon-rich SiN$_x$ films under DC electrical stress. Despite the clear compositional dependence observed in the breakdown field $E_{BD}$ the variation in across the different $x=[N]/[Si]$ ratios are relatively limited. All samples exhibit breakdown when the accumulated charge density $Q$ reaches a critical threshold $Q_{BD}$ between approximately $10^1$ and $10^2$ C/cm$^2$, indicating that $Q_{BD}$ lies within the same order of magnitude regardless of SiN$_x$ film composition, and suggesting the existence of a composition-independent critical charge limit that governs the dielectric failure in these films. Such behavior is consistent with the $Q_{BD}$ hypothesis, which suggests that breakdown occurs when a fixed quantity of charge has been injected into the dielectric, independent of the instantaneous electric field [53,54]. This mechanism is particularly relevant for materials like amorphous SiN$_x$, which contain a distribution of intrinsic defects and trap states as discussed before. In the more Silicon-rich films, this threshold is reached more rapidly due to enhanced charge injection and trap filling under lower fields, whereas in the less Silicon-rich films, deeper traps and a more robust bonding
<figure><img src="image_11.png" /><figcaption>Fig. 7. Breakdown field $E_{BD}$ (left axis) and charge-to-breakdown $Q_{BD}$ (right axis) as a function of the $x=[N]/[Si]$ ratio of SiN$_x$ films under DC electrical stress.</figcaption></figure>

network delay charge accumulation. These results emphasize the fundamental distinction between the breakdown field $E_{BD}$, which reflects the dielectric film’s structural and chemical resilience to instantaneous high-field stress, and the charge-to-breakdown $Q_{BD}$, which represents the material’s intrinsic tolerance to cumulative charge injection and long-term electrical degradation [55].

## 4. Conclusion

In this work, the charge transport mechanism in PECVD Silicon-rich SiN$_x$ films was investigated across a range of $x=[N]/[Si]$ ratios through electrical conduction analysis under both DC and AC high field conditions with various frequencies. Analysis of the J–E characteristics confirmed that the Poole–Frenkel emission is the dominant conduction mechanism under high-field in both regimes. The extracted zero-field potential barrier height $\phi_0$ was found to decrease with increasing the silicon content, while the theoretical dielectric permittivity increased, consistently with the structural changes in the bonding network. In addition, it appears that a higher frequency of AC field reduces further the barrier height $\phi_0$ that charge carriers have to overcome for being detrapped and reemitted into the conduction band. Breakdown analysis under DC step-stress revealed that while the breakdown field $E_{BD}$ decreases significantly with increasing silicon content due to a weaker network stability and an increased defect density, the charge-to-breakdown $Q_{BD}$ remains relatively constant across the SiN$_x$ compositions. This suggests the presence of a critical accumulated charge in the SiN$_x$ film for triggering the dielectric failure, in agreement with the $Q_{BD}$ hypothesis. This study demonstrates how compositional tuning affects both the conduction and breakdown characteristics, providing a foundation for optimizing SiN$_x$ dielectrics for high-field and frequency-sensitive applications.

### CRediT authorship contribution statement

T. Al Moussi: Writing – original draft, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. P. Lambkin: Writing – review & editing, Visualization, Validation, Resources, Project administration, Methodology, Funding acquisition, Formal analysis, Conceptualization. R. Lakshmanan: Writing – review & editing, Visualization, Validation, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. B. Chen: Writing – review & editing, Visualization, Validation, Resources, Project administration, Funding acquisition,

Formal analysis, Conceptualization. S. Diaham: Writing – review & editing, Visualization, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Sombel Diaham reports financial support was provided by Analog Devices BV. Sombel Diaham reports financial support was provided by University of Toulouse. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgment

The authors would like to thank the members of the isolation team at Analog Devices for their contributions and the European Union through the Marie Skłodowska-Curie Actions (MSCA-IF, H2020 program) for financial funding in the frame of the PRISME project (grant N°846455, 2019–2021).

### Data availability

Data will be made available on request.

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