# Thermal effects on polyimide electrical conductivity: experimental and numerical analysis

Félix Dusart, Thierry Paulmier, Ludivine Leclercq, Pierre Sarrailh, Céline Michelin

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Félix Dusart, Thierry Paulmier, Ludivine Leclercq, Pierre Sarrailh, Céline Michelin. Thermal effects on polyimide electrical conductivity: experimental and numerical analysis. Journal of Physics D: Applied Physics, 2026, 59 (3), pp.035304. 10.1088/1361-6463/ae36ba. hal-05470133

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# Thermal effects on polyimide electrical conductivity: experimental and numerical analysis

To cite this article: Félix Dusart et al 2026 J. Phys. D: Appl. Phys. 59 035304

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# Thermal effects on polyimide electrical conductivity: experimental and numerical analysis

Félix Dusart* ID, Thierry Paulmier, Ludivine Leclercq ID, Pierre Sarrailh and Céline Michelin

DPHY, ONERA, Université de Toulouse, 31000 Toulouse, France

* Author to whom any correspondence should be addressed.

E-mail: felix.dusart@onera.fr

Keywords: radiation-induced conductivity, trap distribution, thermally stimulated current, charge transport, bipolar model

## Abstract

Kapton® is a widely used material on spacecraft due to its excellent thermal and mechanical properties. Despite its widespread usage, the thermal dependence of its electrical properties remains a significant concern. An experimental investigation was conducted using the SIRENE facility (ONERA, Toulouse, France) in order to characterize the effect of temperature on the electrical behavior of Kapton®. Through experimental evidence using a thermally stimulated current method, the trap distribution along the forbidden gap was highlighted. In addition, the existence of dark conductivity, unrelated to radiation exposure, was clearly identified. These physical mechanisms and characteristics were integrated into a numerical and physical conductivity model called THEMIS, which has been developed to assess charge transport in dielectrics in one dimension. The modified version of the THEMIS model was validated through experimental characterization. This work extends our understanding of charge transport in dielectric materials used in space, and suggests a validated method to model their behavior.

## 1. Introduction

Satellites operating in orbit are continuously exposed to a harsh space environment, which constantly puts the satellite mission at risk. In geostationary orbit, spacecraft are exposed to strong fluxes (up to 2·10¹⁰ electrons⁻¹·s⁻¹·cm⁻²) of high-energy electrons (from a few keV to around 1.5 MeV during storm events) [1], as well as significant temperature changes due to alternating heating and cooling from eclipses [2]. These combined phenomena can induce onboard anomalies and breakdowns. Among the materials used onboard, dielectric polymers play a critical role due to their excellent electrical insulation, thermal stability, and mechanical robustness. However, their electrical behavior under irradiation remains a significant concern. Indeed, their electrical conductivity can vary significantly as a function of the dose. This phenomenon is known as radiation-induced conductivity (RIC) [3]. The RIC dynamics are highly material-dependent, with different polymers exhibiting different conductivity behaviors under identical irradiation conditions [4]. The mechanisms governing RIC are complex and involve a range of interdependent physical and chemical processes, such as charge trapping, detrapping, and recombination. RIC is also influenced by other physical parameters, such as the system temperature and the electric field [5]. This complexity makes the accurate characterization and prediction of RIC particularly challenging. The low reliability of a sole empirical approach does not allow for a precise determination of RIC behavior in dielectric polymers under conditions representative of the space environment. The development of physical and numerical models, based on experimental high-energy tests [6, 7], is necessary to predict the charging behavior of space materials. As Kapton® is one of the most used polymers on spacecraft, from multilayer insulation on the exterior to insulation for electrical cables and printed circuit boards (PCBs) on the interior, several studies have been conducted on the RIC dynamics of this material [8–10]. So far, however, our knowledge of Kapton® temperature dependencies on the conductivity is limited, especially the distribution of defects in the material and their relationship to conductivity evolution with temperature. It is thus difficult to faithfully model Kapton® conductivity in non-isothermal
configurations. In this work, the electrical behavior of Kapton® in non-isothermal configurations was investigated. The investigation was conducted using both experimental and numerical approaches. In section 2, the SIRENE experimental facility and the thermally stimulated current (TSC) protocol used in this study are presented. The experimental results are then discussed in section 3. Next, a numerical analysis of Kapton® conductivity in non-isothermal configuration is performed. In section 4, the THEMIS (Transport of Holes and Electrons Model under Irradiation in Space) numerical and physical conductivity model used to perform the numerical analysis is presented. This model has been developed at ONERA [7, 11] to study in detail the charging aspects of irradiated dielectric materials. The model has been improved to include thermal dependencies on RIC and a hole trap distribution, allowing for accurate simulations of polymer RIC in non-isothermal configurations. The improved version of the model is then validated through experimental evidence in section 5. This work provides experimental data showcasing the electrical behavior of Kapton® in non-isothermal configuration, and suggests a validated numerical method to model it.

## 2. Experimental setup and protocol

### 2.1. Experimental setup

The behavior of polyimide under irradiation has long been studied. Electronic irradiation over several days has been shown to modify the trap density and energy levels [12]. RIC has also been characterized under protonic irradiation with energies ranging from 1 to 2.5 MeV [13]. The proton penetrating depth was found to play a key role in the conductivity dynamics. Under x-ray irradiation with energies ranging from 8 to 25 keV, RIC increases with the current of the incident beam [14]. However, the thermal effects on its electrical conductivity, especially when coupled with irradiation-induced processes, have received comparatively limited attention. To study the Kapton® charging behavior under electronic irradiation under non-isothermal conditions, the SIRENE facility is used. This facility allows us to perform induced conductivity measurements of material samples, and is described in [6]. Dupont's Kapton®-HN (50 µm thick, 5 cm² samples) is used in this study. A silver coating is added onto both sides of the material using physical vapor deposition to ensure electrical contact with the electrodes of the system. Samples are placed into the irradiation chamber under a vacuum (10⁻⁶ mbar) and inserted between the aluminum electrodes on a temperature-controlled sample holder. This allows for both isothermal and non-isothermal measurements. A voltage difference is applied between each electrode to induce a constant electric field across the material. Voltage is measured on the back face of the sample, equivalent to the measurement of the electrical current on the back face of the sample, through the use of a current-to-voltage converter, as illustrated in figure 1. Temporal voltage evolution is displayed and recorded on a Sefram® DAS1400 data acquisition system. The conductivity σ is then computed from the evolution of the electrical current and the electric field across the sample:

$$ \sigma(t) = \frac{J(t)}{F} \cdot \frac{1}{S}, \qquad (1) $$

where σ is the conductivity [($\Omega \cdot m)^{-1}$], J is the measured current [A·m⁻²], F is the electric field across the sample [V·m⁻¹], and S is the surface area of the central electrode (5 cm²).

### 2.2. TSC method

Experimental measurements were conducted using the TSC method, described in [4]. The evolution of the temperature during the experiment allows us to obtain precise information about the material charge trap properties. A constant potential difference is first applied to the electrodes to create a constant electric field and induce charge movements into the material. Then, the temperature is lowered to -150 °C to reduce the thermal energy of the system. Material samples are then irradiated with a 400 keV electron beam for 1 h at 8 pA cm⁻². The 400 keV electronic irradiation is produced by a high-energy beam that produces ionization in the whole sample thickness, generating electrons in the conduction band and holes in the valence band. The electron implantation, present at lower energetic beams, can be neglected. The generated charges are eventually trapped in the physical and chemical defects of the materials. After an hour of relaxation at low temperature, the measured current is negligible. A linear temperature rise is then applied. This heating rate is slow (1 °C min⁻¹) to prevent temperature gradients within the sample, as it could modify the local properties of the sample. As the thermal energy of the system gradually increases, trapped charges are able to progressively regain access to the valence band (for holes) and conduction band (for electrons), leading to a rise in conductivity. The evolution of conductivity as a function of temperature is directly driven by the charge detrapping. The charge detrapping with the temperature is commonly referred to as thermally assisted detrapping or thermal activation [15], and
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Illustration of the experimental setup.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2.** Illustration of the evolution of experimental parameters as a function of time. The expected behavior of the electrical current is also illustrated in orange.</figcaption></figure>

is often described as a phenomenon following an Arrhenius-type behavior [16]. Figure 2 represents the evolution of each physical parameter in the TSC protocol. The applied potential illustrated at the top of the figure corresponds to the potential of the electrode in contact with the front side of the sample.

### 3. Experimental results

In this section, we present the experimental results for Kapton®-HN using the TSC method previously described in section 2.2. The evolution of conductivity with temperature is displayed in figure 3.

In figure 3, we can observe three different behaviors as a function of the temperature of the system. These three behaviors correspond to specific phenomena taking place in the dielectric, which are detailed below. The sharp increase in conductivity occurring between 125 and 150 K is a result of the detrapping of charges present in the shallow traps. This high conductivity rise indicates the presence of a high concentration of charges (electrons and holes) in shallow traps. Between 150 and 305 K, the conductivity steadily increases with the temperature. This gradual rise indicates a relatively continuous release of trapped charges. This suggests that Kapton®-HN is composed of traps whose depths are relatively close to each other. These trap energy levels can be considered sufficiently close to form a continuous distribution of traps. Previous studies have also identified an energetic trap distribution in Kapton® using modified isothermal discharge current theory [17, 18] and using a method based on TSC measurements [19]. The continuous release of trapped charges thus originates from the continuous trap distribution. An illustration of the trap distribution is displayed in figure 4.
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Typical evolution of conductivity of Kapton®-HN sample with temperature using TSC method. The applied temperature ramp is 1 °C min⁻¹. Three different behaviors can be observed in the graph, and these are delimited using vertical red lines.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Illustration of the energetic distribution of hole traps. The relationship between trap depth and the corresponding charge release temperature is indicated by dotted lines. The density of shallow traps, located close to the valence band, is greater than the density of deep traps.</figcaption></figure>

Above 305 K, the conductivity rises sharply with the temperature. Analytical models underestimate the measured RIC values at high temperature. To investigate the origin of this discrepancy, an additional measurement was conducted. The TSC protocol was applied without irradiating the sample. The conductivity evolution of this pristine sample is compared to the conductivity evolution of the irradiated sample and displayed in figure 5. The difference between the two datasets originates from radiation-induced processes. Despite the absence of irradiation on the pristine sample, an increase in conductivity with the temperature is still measured. This conductivity, which does not come from irradiation-induced processes, is referred to as dark conductivity [20, 21]. Below 305 K, this dark conductivity appears to be negligible. However, the dark conductivity rises sharply with the temperature. Its contribution to the total conductivity overcomes the radiation processes above 340 K. Two phenomena could be responsible for this sharp rise in conductivity. The first is the migration of electrons from the electrode, overpassing the potential barrier at the interface between the electrode and the sample and injecting the dielectric. Indeed, the potential barrier at the interface is lowered at high electric fields, and its overpassing is more probable at high temperatures. This phenomenon is known as the Richardson-Schottky effect [22]. However, the conductivity dynamics do not seem to fit any model of such phenomenon. Moreover, the strong temperature dependence of the measured conductivity does not originate from interface phenomena, as investigated in [23]. The second investigated origin of this change of behavior is the generation of charge carriers in the bulk of the material due to a thermal activation process modified by the electric
<figure><img src="image_8.png" /><figcaption>Figure 5. Top panel: Experimental evolution of conductivity of both pristine and irradiated samples as a function of temperature using an applied temperature ramp of 2 °C min⁻¹. Bottom panel: comparison between experimental data (no prior irradiation) and conductivity computed using the formula of Adamec and Calderwood. Dark conductivity is present in Kapton®-HN sample, and is prevalent at T ≥ 340 K (represented by the green dotted line).</figcaption></figure>

field. When the electric field and/or temperature are sufficiently high, the studied sample can spontaneously ionize, leading to a sharp rise in conductivity [24]. This theory seems to be applicable to both semiconducting and insulating solids. Based on the work of Adamec and Calderwood [25], this conductivity can be modeled as

$$ \left\{ \begin{aligned} \sigma(F, T) &= \sigma(T) \left( \frac{2+\cosh\left(\frac{\sqrt{\frac{e^3F}{\pi\epsilon}}}{2k_B T}\right)}{3} \right) \times \left( \frac{2k_B T}{eF\delta} \sinh\left(\frac{eF\delta}{2k_B T}\right) \right) \\ \sigma(T) &= \sigma_{\infty} \times \exp\left(-\frac{E_A}{k_B T}\right) \end{aligned} \right. \quad (2) $$

where δ is the jumping distance (assumed to be a constant value of 1 nm), σ∞ is the maximum conductivity as the temperature approaches infinity [Ω·m⁻¹], Eₐ is the activation energy [J], kₐ is the Boltzmann constant [J·K⁻¹], ε is the material dielectric constant [F·m⁻¹], F is the electric field [V·m⁻¹], and T is the temperature [K]. The first part of the first equation corresponds to the dependence of charge concentration on the electric field and temperature. The second term of the first equation corresponds to the dependence of charge mobility on the electric field and temperature. As shown in figure 5, this expression fits the experimental measurements conducted without prior irradiation, with coherent parameters compared to other dielectrics [26] (Eₐ = 0.79 eV, σ∞ = 6.0·10⁻⁴ (Ω·m⁻¹)). In the next chapters, we will assume the origin of this change of behavior to be due to a thermal activation process in the bulk of the material.

## 4. THEMIS conductivity model

As described in [7, 11], the THEMIS conductivity model, developed at ONERA, is a numerical tool to study in detail the charging aspects of irradiated dielectric materials; in particular, the charge transport and ionization processes. The model includes the spatial distribution, along the sample thickness, of both the implanted charge density and the ionizing dose profile, making it a 1D numerical and physical model. This dependence is essential to study dielectrics presenting heterogeneous dose profiles. The electric field dependencies on the charge dynamics have previously been integrated and validated through experimental data [11]. The presented work shows the integration of thermal aspects and an exponential trap distribution into the model, and their validation using experimental evidence.

### 4.1. Mono-trap model

The THEMIS model is based on band theory to study the charging behavior of materials under irradiation. When a material is irradiated by energetic electrons, these particles penetrate the sample and gradually lose their energy through inelastic collisions with the atoms of the material. This energy deposition induces ionization processes along their path, generating electron-hole pairs throughout the depth
<figure><img src="image_9.png" /><figcaption>Figure 6. Description of the THEMIS conductivity model. Electron (○) and hole (□) dynamics are represented. V.B. stands for valence band and C.B. for conduction band. The phenomena implemented into the model are represented by the colored arrows: generation (blue), transport (black), trapping (red), detrapping (green), recombination (orange).</figcaption></figure>

of the material. As the incident electrons slow down, they eventually populate the bottom of the conduction band. The model takes into account the implanted incident electrons, as well as the generated electrons and holes within the thickness of the material.

Free electrons in the conduction band and holes in the valence band are able to move within the material. During their transport, these charge carriers may become trapped in localized states within the bandgap. The model also considers the evolution of trapped charges, which can be thermally or electrically released (detrapping), or might recombine with carriers of opposite charge. These phenomena are illustrated in figure 6.

The conductivity $\sigma$ is computed from the density of free electrons and holes as

$$ \sigma(x, t) = e \times (n(x, t) \cdot \mu_n + p(x, t) \cdot \mu_p). \quad (3) $$

Here, $e$ is the elementary charge, and $n, \mu_n$ and $p, \mu_p$ are respectively the density and mobility of free electrons and free holes. The parameters of the equations defining the THEMIS model are detailed in table 1. The charge densities vary as follows:

$$ \left\{ \begin{aligned} \frac{\partial n(x,t)}{\partial t} &= g(x, t, T, F) - \frac{\nabla \cdot j_0}{e} - \nabla \cdot J_n - \alpha_1 n p_t - \alpha_2 n p - \frac{n}{\tau_n} + \frac{n_t}{\tau_{nt}(T,F)} \\ \frac{\partial n_t(x,t)}{\partial t} &= -\alpha_3 n p + \frac{n}{\tau_n} - \frac{n_t}{\tau_{nt}(T,F)} \\ \frac{\partial p(x,t)}{\partial t} &= g(x, t, T, F) + \nabla \cdot J_p - \alpha_3 n p - \alpha_2 n p - \frac{p}{\tau_p} + \frac{p_t}{\tau_{pt}(T,F)} \\ \frac{\partial p_t(x,t)}{\partial t} &= -\alpha_1 n p_t + \frac{p}{\tau_p} - \frac{p_t}{\tau_{pt}(T,F)} \end{aligned} \right. \quad (4) $$

where $n_t$ and $p_t$ correspond to the density of trapped electrons and holes, respectively, $g$ is the electron-hole pair generation rate, and $j_0$ is the implantation current. $J_n$ and $J_p$ are the electron and hole currents, $\alpha_i$ is a recombination coefficient, and $\tau_n, \tau_{nt}, \tau_p, \tau_{pt}$ are the trapping and detrapping characteristic times for electrons and holes.

Currents $J_n$ and $J_p$ are described as

$$ \left\{ \begin{aligned} J_n &= J_n^{\text{Convection}} + J_n^{\text{Diffusion}} = n\mu_n F + D_n \nabla n \\ J_p &= J_p^{\text{Convection}} + J_p^{\text{Diffusion}} = p\mu_p F + D_p \nabla p. \end{aligned} \right. \quad (5) $$
The electric field and thermal effects on detrapping rates are taken into account through the Arrhenius and Poole–Frenkel theories:

$$ \tau_t = \tau_{t0} \times \exp \left( \frac{E_A - \sqrt{\frac{e^3 F}{\pi \epsilon}}}{k_B T} \right) \quad (6) $$

with $E_A$ being the activation energy [J], $\epsilon$ the dielectric permittivity [F · m⁻¹], and $k_B$ the Boltzmann constant [J · K⁻¹]. The generation rate $g$ includes the electric field and temperature dependencies from Onsager’s dissociation theory [27]. Indeed, the electric field and temperature affect the geminate recombination [28], impacting the free charge generation rate. The implementation of these dependencies into THEMIS is detailed in [11].

Despite integrating numerous physical phenomena into the THEMIS model, a noticeable difference remains between the simulation results and experimental data under non-isothermal conditions. In the low-temperature range ($T < 270$ K), the computed conductivity remains far below the expected value. On the other hand, the conductivity rises sharply in the high-temperature range, overestimating the experimental data. This limit originates from the assumptions inherent in the model’s construction; specifically, the assumption that charge carriers are trapped in a single energetic depth leads to a binary configuration, where conductivity is either suppressed (at low temperatures and electric fields, with all charges remaining trapped) or significantly enhanced (at high temperatures and electric fields, with all charges escaping their traps). However, this binary framework fails to capture the experimentally observed gradual increase in conductivity with the rising temperature, which suggests a more complex behavior. To be able to accurately model the experimental data, the model was improved by implementing a distributed energetic profile of charge traps.

## 4.2. Exponential trap distribution model

Charge transport models featuring an exponential distribution of traps have already been the subject of previous developments, as detailed in [29, 30]. Some of these models with an exponential trap distribution have also been specifically developed to study polyimides, as in [8, 31, 32]. The exponential trap distribution law is commonly used in the study of RIC aspects in dielectric polymers (Polyimide or Polyetheretherketone, for example), as it can be considered to be the most appropriate trap representation [33, 34]. However, as yet, no 1D bipolar charge transport model featuring an exponential trap distribution and taking into account charge generation and implementation, trapping, detrapping, recombination, and diffusion has been developed. To accurately represent the electrical behavior of dielectric materials under conditions representative of the space environment, it is essential to precisely depict the charge transport and evolution in the material by implementing such phenomena into the models, as their contribution to RIC appears to be significant [6]. The THEMIS model also features a dependence on the electric field and thermal aspects to represent the space environment as faithfully as possible. This implementation is important because the conductivity of dielectric polymers strongly depends on the temperature and electric field evolution [11].

A previous study conducted at ONERA [35] showed that in Kapton®-HN, conductivity is mostly governed by the hole dynamics. In the cited study, the electric potential of a low-energetic 20 keV irradiated polyimide sample was measured using a Kelvin probe. The low-energetic irradiation allows electrons to be implanted in the sample close to the surface, and avoid any bulk ionization of the sample. The results show that at the end of irradiation, the potential remains at a high level with no potential relaxation over time. This confinement of the potential reflects the absence of movement of electrons in the material, their quick trapping and permanent trapping (no detrapping), and thus their low contribution to RIC. If the electrons were mobile, they would diffuse in the material, lowering the potential level. In the present case, there was no need to apply such distribution to the electrons, as their contribution to RIC appears negligible under the applied experimental conditions. The modification of the model is thus applied exclusively to hole traps. In this model, the trap distribution is considered to be constant in the material thickness. The energetic trap density thus depends solely on the energetic depth, as follows [34]:

$$ M(E) = \frac{M_0}{E_1} \times \exp \left( -\frac{E}{E_1} \right), \quad (7) $$

where $M(E)$ is the energetic density of the trap at depth $E$, $M_0$ is the total density of traps, and $E_1$ is the energy related to the distribution. In the model, the trap depths are discretized into a finite set of N elements {$E^0$; ...; $E^{N-2}$; $E^{N-1}$}. The total density of traps between two consecutive discretizations {$E^k$; $E^{k+1}$}
<figure><img src="image_10.png" /><figcaption>Figure 7. Description of the modified THEMIS model using the same legend as in figure 6. Hole traps now feature an exponential distribution of density as a function of their depth, illustrated in blue. The black crosses illustrate the discretization of the trap distribution.</figcaption></figure>

is then computed using equation (8):

$$M^k = \int_{E^k}^{E^{k+1}} M(E) dE. \qquad (8)$$

This approach is similar to considering a set of N discrete energy levels, as displayed in figure 7. To approximate the behavior of a continuous trap distribution, it is necessary to use a sufficiently smooth discretization. Based on our results, we recommend using at least 20 discrete levels to ensure accurate modeling. Once the discretization of trap distribution is computed, the charge dynamics in each trap can be determined.

Charge trapping at a certain depth (defined as the kth discretization of the distribution) depends on the amount of available traps ($M^k - p_t^k$) at this depth compared to the total amount of available traps across all energetic levels ($M_0 - p_t^{tot}$). Charges are computed as

$$ \left\{ \begin{aligned} \frac{\partial n(x,t,T,F)}{\partial t} &= g(x,t,T,F) - \frac{\nabla \cdot j_0}{e} - \nabla \cdot J_n - \alpha_2 np - \sum_k (\alpha_1 np_t^k) - \frac{n}{\tau_n} + \frac{n_t}{\tau_{nt}(T,F)} \\ \frac{\partial n_t(x,t,T,F)}{\partial t} &= -\alpha_3 n_t p + \frac{n}{\tau_n} - \frac{n_t}{\tau_{nt}(T,F)} \\ \frac{\partial p(x,t,T,F)}{\partial t} &= g(x,t,T,F) + \nabla \cdot J_p - \alpha_2 np - \alpha_3 n_t p + \sum_k \left( -\frac{p}{\tau_p} \cdot \left( \frac{M^k - p_t^k}{M_0 - p_t^{tot}} \right) + \frac{p_t^k}{\tau_{pt}^k(T,F)} \right) \\ \frac{\partial p_t^k(x,t,T,F)}{\partial t} &= -\alpha_1 np_t^k + \frac{p}{\tau_p} \cdot \left( \frac{M^k - p_t^k}{M_0 - p_t^{tot}} \right) - \frac{p_t^k}{\tau_{pt}^k(T,F)} \end{aligned} \right. \qquad (9) $$

where $\alpha_1 np_t^k$ represents the recombination rate between free electrons and holes trapped in the kth discretization of the distribution. The variables used in equations (3)–(9) can be found in table 1.

The improved THEMIS model features temperature and electric field dependence on charge dynamics, as well as a trap distribution along the forbidden gap. This distribution follows an exponential law in which deeper traps are progressively less populated. In addition, a high density of shallow traps, mainly attributed to physical defects in the material, is considered. These shallow traps, located close to the band edges, strongly influence the apparent mobility of charge carriers. Indeed, charge carriers move through the materials via successive trapping and detrapping events [36] (illustrated in figure 8).
<table><caption>**Table 1.** Description of the parameters used in THEMIS. The values are given for a temperature of 293 K and an electric field of 10⁷ V m⁻¹. The value of the hole detrapping time is the average value across the whole distribution.</caption><thead><tr><th>Symbol</th><th>Designation</th><th>Unit (SI)</th><th>Value</th></tr></thead><tbody><tr><td colspan="4">Parameters common to both models</td></tr><tr><td>n</td><td>Free electron density</td><td>m<sup>-3</sup></td><td>—</td></tr><tr><td>n<sub>t</sub></td><td>Trapped electron density</td><td>m<sup>-3</sup></td><td>—</td></tr><tr><td>p</td><td>Free hole density</td><td>m<sup>-3</sup></td><td>—</td></tr><tr><td>p<sub>t</sub></td><td>Trapped hole density</td><td>m<sup>-3</sup></td><td>—</td></tr><tr><td>τ<sub>n</sub></td><td>Trapping characteristic time of free electrons</td><td>s</td><td>0.2</td></tr><tr><td>τ<sub>nt</sub></td><td>Detrapping characteristic time of trapped electrons</td><td>s</td><td>4.6 · 10<sup>35</sup></td></tr><tr><td>τ<sub>p</sub></td><td>Trapping characteristic time of free holes</td><td>s</td><td>8</td></tr><tr><td>τ<sub>pt</sub></td><td>Detrapping characteristic time of trapped holes</td><td>s</td><td>57</td></tr><tr><td>μ<sub>n</sub></td><td>Mobility coefficient of free electrons</td><td>m<sup>2</sup> V<sup>-1</sup> s<sup>-1</sup></td><td>8.00 · 10<sup>-14</sup></td></tr><tr><td>μ<sub>p</sub></td><td>Mobility coefficient of free holes</td><td>m<sup>2</sup> V<sup>-1</sup> s<sup>-1</sup></td><td>3.45 · 10<sup>-15</sup></td></tr><tr><td>D<sub>n</sub></td><td>Diffusion coefficient of free electrons</td><td>m<sup>2</sup> s<sup>-1</sup></td><td>10<sup>-27</sup></td></tr><tr><td>D<sub>p</sub></td><td>Diffusion coefficient of free holes</td><td>m<sup>2</sup> s<sup>-1</sup></td><td>10<sup>-28</sup></td></tr><tr><td>α<sub>1</sub></td><td>Recombination rate between free electrons and trapped holes</td><td>m<sup>3</sup> s<sup>-1</sup></td><td>5 · 10<sup>-26</sup></td></tr><tr><td>α<sub>2</sub></td><td>Recombination rate between free electrons and free holes</td><td>m<sup>3</sup> s<sup>-1</sup></td><td>5 · 10<sup>-26</sup></td></tr><tr><td>α<sub>3</sub></td><td>Recombination rate between trapped electrons and free holes</td><td>m<sup>3</sup> s<sup>-1</sup></td><td>5 · 10<sup>-26</sup></td></tr><tr><td>E<sub>Aμ</sub></td><td>Mean activation energy of the extended states</td><td>eV</td><td>3.5 · 10<sup>-2</sup></td></tr><tr><td>g</td><td>Electron–hole pair generation rate</td><td>m<sup>-3</sup> s<sup>-1</sup></td><td>—</td></tr><tr><td>j<sub>0</sub></td><td>Implantation current</td><td>C s<sup>-1</sup> m<sup>-2</sup></td><td>—</td></tr><tr><td colspan="4">Parameters specific to the trap distribution model</td></tr><tr><td>E<sub>1</sub></td><td>Characteristic energy of the trap distribution</td><td>eV</td><td>0.18</td></tr><tr><td>M<sub>0</sub></td><td>Total density of traps across all energetic levels</td><td>m<sup>-3</sup></td><td>10<sup>26</sup></td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Figure 8.** Representation of the mobility of charges through successive trappings and detrappings. The bottom rectangular cuboid represents the conduction band, featuring numerous traps (represented by cylinders). The spherical electron moves by successive trappings and detrappings, represented by red arrows.</figcaption></figure>

This phenomenon is described as the multiple-trapping model [15]. In this model, it is considered that charges are released from their traps via thermal activation, analogous to the Poole–Frenkel process.

$$ \mu = \mu_0 \times \exp \left( \frac{\sqrt{\frac{e^3 F}{\pi \epsilon} - E_{A\mu}}}{k_B T} \right), \qquad (10) $$

where $\mu_0$ is the initial mobility [m²·V⁻¹·s⁻¹], and $E_{A\mu}$ is the mean activation energy of the extended states [J].
<figure><img src="image_12.png" /><figcaption>Figure 9. Comparison between numerical simulation and experimental data on the evolution of conductivity with temperature. The mono-energetic trap model (top panel) and exponential trap distribution (bottom panel) implemented in THEMIS were used.</figcaption></figure>

## 5. Numerical results and analysis

In this section, we present the results of numerical simulations and compare them with the experimental data (displayed in figure 9). The experimentally measured conductivity is assumed to be composed of two contributions: dark conductivity and RIC. Numerically, the conductivity is computed as the sum of these two components (equation (11)), according to Fowler theory [37]. This assumption has already been made in previous studies on dielectrics in space environments, including polyimides such as Kapton® [1, 20], and showed matching results with the experimental data,

$$ \sigma_{\text{tot}}(t) = \sigma_{\text{dark}}(t) + \sigma_{\text{RIC}}(t). \qquad (11) $$

The total conductivity is computed using the THEMIS model for the determination of RIC and the Adamec and Calderwood parameters for the dark conductivity. The values of the parameters used in the Adamec and Calderwood expression [24] (the activation energy and conductivity value as the temperature tends to infinity) were extracted from the experimental data, and are presented in figure 5.

Figure 9(a) highlights the limits of the mono-energetic trap model. The conductivity increase with temperature observed numerically does not match the experimental data. Numerically, in the low-temperature range, the conductivity remains far below the expected values. This observed behavior originates from the inability of charges to detrap. In the high-temperature range, every trapped hole is prone to detrapping, causing a sharp increase in conductivity and overestimating the expected values. The modifications brought to the THEMIS model reveal a significant improvement in the capacity to model conductivity in non-isothermal configuration. The steady increase in temperature can now be modeled accurately. The extracted values of the Adamec and Calderwood model parameters allow for an accurate prediction of conductivity at high temperatures. In the next part, the results from the exponential trap distribution model will be discussed.

The first notable observation of the numerical results concerns the homogeneity of conductivity throughout the whole material's thickness (figure 10). This uniformity is attributed to the use of the 400 keV electron beam both in the experimental setup and the numerical modeling. The usage of such a high-energy electron beam allows for a uniform ionization of the sample in the whole material's thickness. The contribution of electron implantation in the sample at such energy is negligible. In the following part, the numerical results will be displayed at an arbitrary thickness of $x = 25\mu m$.
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Numerical evolution of conductivity as a function of time and material thickness. The irradiation phase and relaxation phase are each one hour long.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Numerical evolution of the density of charges located in traps as a function of temperature using 25 distribution discretisations. The color of each curve indicates the depth range of each discretisation. For clarity matters, only the ten deepest discretisations are displayed.</figcaption></figure>

## 5.1. Temperature rise phase

A major result of this study lies in the good agreement observed between the experimental and simulated conductivity during the temperature rise phase (figure 9). This agreement demonstrates the relevance of the model modifications, and confirms its ability to accurately depict Kapton®-HN behavior under irradiation. Below 340 K, the RIC+bulk results overlap the RIC-only data, showing the negligible contribution of bulk conductivity to the total conductivity. During this phase, the numerical results solely depend on the THEMIS (RIC-only) model, which is in good agreement with the experimental data. The addition of a trap distribution in the model allows for continuous detrapping as the temperature rises, as experimentally observed (illustrated in figure 11). The inclusion of dark conductivity through Adamec and Calderwood's formula, as presented in 2.2, allows for faithful simulations throughout the whole temperature range.

Figure 11 illustrates the progressive detrapping of holes with the temperature rise. Once the temperature of the system is sufficient, holes escape from their trap and regain access to the valence band. Some of these released holes get re-trapped at deeper levels. As a consequence, a rise in hole density can be observed at the deepest levels as the temperature soars up. The charges trapped in the deepest levels are then able to detrap as the temperature keeps on rising, as displayed in figure 11. However, the total density of trapped charges decreases (by $2 \cdot 10^{19} \text{ m}^{-3}$ in the whole temperature range), showcasing the continuous detrapping of trapped charges with the temperature.
<figure><img src="image_15.png" /><figcaption>**Figure 12.** Thermal evolution of free hole density across every energetic level (left panel) and hole mobility (right panel).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 13.** Thermal evolution of free electron density (left panel) and trapped electron density (right panel).</figcaption></figure>

The rise of conductivity with temperature is the direct image of the product of continuous detrapping of holes, and mobility rise with temperature. Both of these parameters are represented in figure 12. In the whole temperature range, both the free hole density and hole mobility rise, by a factor of 6.387 and 10.165, respectively. The product of the rise of both parameters (64.92) is identical to the rise factor of conductivity over the whole temperature range.

In the numerical simulations, electrons are considered to play a minor role in the conductivity dynamics. They were thus considered to be getting trapped quickly in deep states, for long periods of time, as can be seen in figure 13. The slight decrease of trapped electrons during relaxation corresponds to the recombination with holes, which then appears to play a minor role in the conductivity evolution.

Additional TSC measurements were conducted using the same experimental protocol as described in section 2.2 but applying different temperature ramps of 1 °C min⁻¹, 2 °C min⁻¹, and 3 °C min⁻¹. These experiments were conducted to test the THEMIS exponential trap distribution model capacity in different non-isothermal configurations. The results are displayed in figure 14.

The numerical results show that the sum of the THEMIS RIC model and Adamec and Calderwood's dark conductivity model allows the Kapton®-HN behavior to be reproduced in multiple non-isothermal configurations without having to change any input parameters. In the low-temperature range (T < 300 K), the numerical simulation fully matches the experimental data. However, in the high-temperature range (T ≥ 300 K), a slight difference between the numerical and experimental results can be observed. Below 300 K, the contribution of dark conductivity to the total conductivity is negligible. The very good correlation between the numerical and experimental data at low temperatures thus depends solely on the RIC model, which demonstrates its excellent ability to reproduce Kapton® behavior in different non-isothermal configurations. At high temperatures, the total conductivity dynamics depend mainly on the dark conductivity model. This model considers the generation of free charges from a thermal activation process happening in the bulk of the material. In the model, the recombination of the generated charges from the thermal activation processes is not included. As a result, there is no temporal dependency on dark conductivity. However, the temporal dependency plays a major role
<figure><img src="image_17.png" /><figcaption>**Figure 14.** Comparison between numerical simulation of the evolution of conductivity with temperature and experimental data for different temperature ramps. Identical input parameters were used for all simulation results displayed in this figure.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 15.** Numerical evolution of conductivity as a function of time. The transition between irradiation and relaxation phases is illustrated by the dotted green line.</figcaption></figure>

in the conductivity evolution, as can be observed in figure 14. Adamec and Calderwood's model nevertheless remains a good approximation of the material's high-temperature behavior, with the dynamics modeled in agreement with the experimental data.

## 5.2. Irradiation and relaxation phases

The temporal evolution of the conductivity is displayed in figure 15, during the irradiation and relaxation phases for an applied ramp of 1 °C min⁻¹. Each phase lasted for 1 h (3600 s) and was performed at 123 K. These phases occurred before applying the temperature ramp.

During the irradiation and relaxation phases, the addition of RIC and bulk models overlaps with the results from the sole RIC model, highlighting the negligible contribution of dark conductivity to the total conductivity, caused by the low temperature of the system (−150 °C). Indeed, the computed contribution of dark conductivity to the total conductivity at this temperature is $\sigma_{\text{dark}}|_{-150^{\circ}\text{C}} = 5 \cdot 10^{-34}(\Omega \cdot \text{m})^{-1}$. As the temperature increases, the dark conductivity sharply rises, as observed experimentally. The computed conductivity dynamics are similar to those observed experimentally. However, the amplitude of the computed conductivity is three times superior to the experimental data during irradiation. In the model, the recombination rate $\alpha$ is assumed to be constant. However, external effects might play a role in the evolution of the rate, such as the temperature or the electric field of the system. Moreover, the trapping rate is assumed to be constant across the whole trap distribution. However, traps of different
<figure><img src="image_19.png" /><figcaption>**Figure 16.** Left panel: Evolution of trapped hole density over time. The value 'trapping' is displayed when the density rises between two consecutive times at the same depth. The value 'detrapping' is displayed when the density at a specific depth decreases. The irradiation shutdown is shown by the yellow line. Overall, the sum of trapped charges at every depth rises during irradiation and relaxation (right panel).</figcaption></figure>

depths are not necessarily of the same nature (physical or chemical defects, for example), and therefore undoubtedly do not share the same physical properties.

During the relaxation phase, irradiation is stopped for an hour. As a consequence, the generated free charges quickly get trapped in the potential wells, resulting in a sharp drop in conductivity. This behavior is observed both experimentally and numerically. Numerically, a global increase in trapped charges should be observed when relaxation occurs. The evolution of trapped holes is displayed in figure 16.

The left panel of figure 16 shows the evolution of trapped holes over time, during the irradiation ($t < 3600$ s) and relaxation ($3600 < t > 7200$ s) phases. During irradiation, the density of trapped holes rises over time, the trapping rate being higher than the detrapping one due to the low temperature level. During relaxation, one can clearly observe two distinct behaviors, which are dependent on the trap depth. In the shallow trap zone, the density of charges decreases during relaxation. On the contrary, in the deep trap zone, the density of trapped holes increases during relaxation. Despite the low temperature of –150°C applied, thermal energy still remains in the system. This thermal energy, assisted by the presence of an electric field, enables the liberation of charges from their trap up to a certain energy limit. Below this limit, traps are not sufficiently deep to hold charges under the given temperature and electric field conditions. The amount of trapped charge thus decreases. These released charges are then rapidly re-trapped at deeper levels, where they are more strongly confined. As a result, the amount of trapped charge increases in this region. The limit between the two behaviors is located at around 0.2 eV. The right image of figure 16 represents the evolution of the total trapped charge across every trap over time. One can observe that during relaxation, the total density of trapped holes increases, resulting in a decrease in conductivity over time. In the simulations, the density of traps is considered to be great enough to hold every charge with no saturation effect. Considering saturation effects in the simulation would lead to an overestimation of the density of free charges, directly related to an overestimation of the conductivity. By considering the saturation effect to be negligible across every energetic level, the trapped charge density $p_t$ is much lower than the density of available traps M. In figure 16, the plotted total density of trapped charges reaches at most $p_t^{tot} \approx 4 \cdot 10^{21} \text{ m}^{-3}$. The total density of traps here is considered to be $M_0 = 10^{26} \text{ m}^{-3}$, based on the parameters used by Tyutnev et al [31].

## 6. Conclusion

This study investigated the non-isothermal behavior of electron-irradiated Kapton® samples through TSC experiments. The results show that the conductivity of irradiated Kapton® is composed of two primary components: RIC and dark conductivity, which are both temperature-dependent. Dark conductivity does not originate from radiation-induced processes, but is instead considered to come from a thermal activation process in the bulk of the material. At lower temperatures, the total conductivity is mainly influenced by RIC dynamics. The RIC temperature dependence originates from thermal-assisted detrapping of charges (following an Arrhenius-type behavior [16]), playing a role in both charge evolution and transport. Dark conductivity, on the other hand, mostly affects the total conductivity dynamics at high temperature, as its contribution to the total conductivity appears to be negligible below 50°C. The experimental investigation also brings into evidence an energetic distribution of traps in Kapton®. The
THEMIS numerical and physical model was used and modified to integrate the physical phenomena highlighted experimentally. This unidimensional model allowed us to irradiate dielectric samples with an electronic beam featuring either a mono-energetic or a distributed energy spectrum. The model uses an electron implantation source term to account for the behavior of low-energy electrons in the dielectric. To simulate the temperature ramp, the model allows temperature variations during the simulation. A hole trap distribution was implemented in the model, with the density of traps considered to decrease exponentially with the trap depth [34]. The temperature effect on charge transport was introduced in the model. Charge mobility is indeed considered to originate from successive trapping-detrapping processes, taking place in the extended states of the valence and conduction bands [9, 15]. The effects of dark conductivity were also taken into account, and yielded a good agreement with the experimental data. Thermal contributions of the THEMIS model were validated using experimental data. This work successfully validates the thermal contributions of the THEMIS model, enabling accurate numerical modeling of Kapton® conductivity in non-isothermal configurations, as confirmed by the experimental evidence presented in this work. Although differences persist in the irradiation and relaxation phases, the model now allows us to faithfully reproduce the experimental measurements during the temperature rise phase, which was not possible with the previous mono-trap model. The remaining discrepancies between the experimental data and numerical simulation might originate from the trapping and recombination rates. Indeed, they are considered to be constant values. To further improve the model, the effects of the temperature and electric field on the trapping and recombination rates should be studied in detail as they play a role in the molecular dynamics of the material and are closely related to trapping and recombination phenomena.

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Acknowledgments

The authors would like to acknowledge the CNES for its financial support of this work through the funding of the doctoral research.

## ORCID iDs

Félix Dusart ID 0009-0000-8248-4034

Ludivine Leclercq ID 0000-0002-1808-9918

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