# Thermal Breakdown Modeling in Polyimide Films by Solving the Heat Balance Equation

Sombel Diaham, Dominique Mary, David Malec, Marie-Laure Locatelli

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Sombel Diaham, Dominique Mary, David Malec, Marie-Laure Locatelli. Thermal Breakdown Modeling in Polyimide Films by Solving the Heat Balance Equation. IEEE Conference on Electrical Insulation and Dielectric Phenomena (CEIDP 2013), Oct 2013, Shenzhen, China. pp.975-978, 10.1109/CEIDP.2013.6747076 . hal-03943459

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# Thermal Breakdown Modeling in Polyimide Films by Solving the Heat Balance Equation

S. Diaham¹,², D. Mary¹,², D. Malec¹,² and M.-L. Locatelli¹,²

¹Université de Toulouse; UPS, INPT; LAPLACE (Laboratoire Plasma et Conversion d'Energie);

118 route de Narbonne, F-31062 Toulouse cedex 9, France

²CNRS; LAPLACE; F-31062 Toulouse, France

**Abstract**- The theoretical thermal breakdown in polyimide (PI) films was modeled at very high temperature between 320 °C and 400 °C using the heat balance equation. Its value was extracted from the internal temperature rise versus electric field of the sample up to its thermal degradation. The originality of this study appears in the consideration, for the first time, of the real temperature dependence of the heat capacity $C_v(T)$, compared to others studies where $C_v$ is fixed at a constant value. It has been possible by measuring it by DSC in a large temperature range up to 390 °C and by interpolating higher values using polynomial fitting. Moreover, the thermal conductivity has also been measured at room temperature only but its value was assumed to be constant and remaining low like in most of polymers. As an input, a simple thermal-activated current density has been used to solve the heat balance equation in a first approach. The results show that the use as an input of a constant value of the capacity $C_v(T_0)$, where $T_0$ is the sample temperature, leads to large difference between both calculated and measured breakdown fields. In comparison, the implementation of the temperature dependence $C_v(T)$ during the solving of the model leads to closer quantitative results. Some future ways of enhancement of the model are proposed.

## I. INTRODUCTION

Polyimides (PI) are high electronic grade organic materials widely used as intermetallic insulating layer or semiconductor passivation [1]. In accordance with their excellent thermal, electrical and mechanical properties, they appear as very interesting for high temperature and high field applications. Although electrical properties of PI films have been investigated since many years, their operating limit under high electric field at high temperature remains unknown up to now. As PI are known to break down via a thermal process [2,3], it is fundamental to use them below the temperature run away when high electric fields are applied to their bulk. Moreover, even if PI films own a high dielectric strength [4], the electric field range for high temperature operation should be limited to keep the thermal balance in order to avoid the breakdown. In a previous paper, the extreme high temperature (320 °C to 400 °C) electrical conduction of PI films has been investigated highlighting a space-charge limited currents mechanism using pre-breakdown current measurements [5]. In the past, Hikita et al. have measured the pre-breakdown currents in PI films and calculated the temperature rise, but only for an initial sample temperature of 90 °C [2]. In the present paper, and based on our previous results of pre-breakdown currents [5], we have

modeled the temperature rise and runaway inside PI films for extreme temperatures by solving the heat balance equation in order to derive the theoretical thermal breakdown field. Moreover, and for the first time, a particular attention has been given in the experimental determination of all the thermal parameters (heat capacity and thermal conductivity), and their temperature dependence, for the solving of the heat balance equation. Thus, the theoretical thermal breakdown field was calculated considering the two following cases: a constant heat capacity and a temperature-dependent heat capacity for showing the influence of this parameter. Results are analyzed with the experimental breakdown fields.

## II. EXPERIMENTS AND MODEL

### A. Sample preparation and electrical measurements

PI thin films have been synthesized from biphenyltetracarboxilic dianhydride (BPDA) and p-phenylene diamine (PDA) and spin-coated on stainless steel substrates before a curing at 400 °C for 1 hour in N₂ [6]. A film thickness of 1.4 µm was obtained. Capacitor structures were achieved by evaporating in vacuum circular gold electrodes of 2.5 mm in diameter. Pre-breakdown conduction currents have been measured until the dielectric breakdown using a source meter Keithley 2410 (with an integrated high-speed 1 nA-accuracy ammeter) connected to a needle-probe station. Measurements were performed between 320 °C and 400 °C using a thermally-regulated chuck in air [5].

### B. Thermal breakdown model

The thermal breakdown model is one of the main theories used to explain the breakdown process in polymers submitted to a DC electric field. It has been investigated by several researchers from both theoretical and experimental points of view [7-11]. In general, a solid dielectric fails thermally when the rate of energy or heat transfer to the lattice, due to an applied field F, exceeds the rate of heat absorption and heat dissipation to the external medium. Such a condition leads to catastrophic thermal runaway. The process is then governed by the heat balance equation:

$$C_v \frac{\partial T}{\partial t} - \text{div}(\lambda_{th} \text{grad } T) = jF \quad (1)$$

with the current continuity condition,
where C_v is the heat capacity of the material per volume unit, λ_th is the thermal conductivity, j is the electrical conduction current density, F is the electrical field, t is the time, ∂T/∂t is the time derivative of the temperature, and grad T is the space gradient of the temperature. Considering the case, when the sample length L and breadth are much larger than its thickness d (which is the case: d/L=5.6×10⁻⁴) and assuming that λ_th is independent of temperature (very low variation for polymers), Eq. (1) can be reduced to a one-dimensional form, as followed:

$$C_v \frac{\partial T}{\partial t} - \lambda_{th} \frac{\partial^2 T}{\partial x^2} = jF \quad (2)$$

If the heat dissipation from the dielectric surface, whose temperature is T, to the surroundings, maintained at ambient temperature T₀ (here between 320 °C and 400 °C), obeys the Newton's law of cooling and if the system is symmetrical with respect to the midplane, the boundary conditions are:

T = T_m at x=0 (midplane),

T = h(T - T₀) at x=±d/2 (edges of the sample),

$$2 \int_{0}^{d/2} F dx = V_{app}$$

where V_app is the applied potential difference, h is the heat transfer coefficient of air to metal (h=10 W/m²K), and T_m the maximum temperature (chosen at 600 °C, i.e. the degradation temperature of BPDA/PDA PI films [6]).

In a first approach, we have only considered in the present study the thermal-activated current form as the input term:

$$j = j_0 \exp\left(-\frac{\phi_t - \phi}{kT}\right) E^2 \quad (3)$$

where j₀ is a constant, (φ_t-φ) is the activation energy and k is the Boltzmann's constant. Previously, activation energy was estimated around 1.15 eV [13].

### C. Thermal parameter measurements

The thermal conductivity has been measured at room temperature on PI films using the photo-thermal method and a value λ_th=0.21 W/mK was obtained [12]. The heat capacity per volume unit C_v=ρC_p was determined, on 5 mg of peeled films, in two steps: (i) the measurement of the specific heat capacity C_p per mass unit by differential scanning calorimetry between 20 °C and 390 °C and (ii) the measurement of the volume density ρ by helium pycnometry at room temperature (ρ=1.48 g/cm³). Considering the low thermal expansion of the BPDA/PDA PI structure (3 ppm/°C), we can consider as realistic that the temperature variation of ρ is negligible due to the low mechanical deformation of the films in the investigated range. Consequently, the temperature dependence of C_v has been calculated proportionally to C_p (i.e. C_v(T)=ρC_p(T)).

### A. Temperature dependence of C_v(T)

Fig. 1 shows the temperature dependence of C_v measured on PI films between 25 °C and 390 °C. It is possible to observe that the heat capacity exhibits a strong increase with increasing temperature. Indeed, C_v is increased by more than a factor 2 in the whole temperature of investigation. Thus, while C_v is around 1.3×10⁶ J/m³K at room temperature, its values is close to 2.8×10⁶ J/m³K at 390 °C. In previous studies, all the authors always modeled the heat balance equation considering the heat capacity as a constant, probably obtained at room temperature [2,3,8,9,11]. Moreover, Zebouchi et al. has pointed out in the case of PET the possible influence that this parameter could have on the obtained difference between experimental and modeling results [11]. For PI studies reported in the literature, all the authors used a constant value of C_v either of 1.6×10⁶ or 2×10⁶ J/m³K [2,3,9] that corresponds to values at temperatures below our range of investigation.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Temperature dependence of the heat capacity C_v. The solid line corresponds to a 3rd order polynomial fit.</figcaption></figure>

In the following, we decided to study the influence of the temperature dependence of the heat capacity on the theoretical breakdown field and the modeled temperature in the PI films.

### B. Thermal breakdown modeling at C_v constant

Fig. 2 presents the electric field dependence of the current density and the modeled temperature using the constant experimental values C_v(T₀) for each temperature. It is observed that the modeled currents, using a simple thermal-activated conduction only, are in the range of the ones obtained experimentally. Moreover, the theoretical breakdown field extracted from the temperature rise when the temperature limit is reached is between 8.77 and 4.85 MV/cm, for sample initial temperature from 320 to 400 °C. Compared to experimental values of the breakdown field, such values appears as 2-3 times higher (cf. Table 1). Indeed, for PI films the
experimental breakdown field is between 2.9 and 2.2 MV/cm, for sample initial temperature from 320 to 400 °C [4].

To understand such a discrepancy between experimental and model value, and independently of the electrical conduction model we chose, we have introduced in the heat balance equation a more realistic model of the heat capacity, i.e. a temperature dependent $C_v(T)$.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Conduction current density (a) and temperature rise (b) versus applied electric field at $C_v(T_0)=const$. Scatters correspond to experimental currents. Solid lines correspond to modeled current and temperature.</figcaption></figure>

<table><caption>Table 1. Experimental and theoretical electrical breakdown field at different temperatures. Experimental values are extracted from [4]. Theoretical values were obtained from the thermal breakdown model (Eqs. 2 and 3).</caption><thead><tr><th colspan="2">Experiment</th><th colspan="2">Thermal breakdown model</th></tr><tr><th>$T_0$ (°C)</th><th>$F_{b \text{exp}}$ (MV/cm)</th><th colspan="2">$F_{b \text{model}}$ (MV/cm)</th></tr><tr><th></th><th></th><th>$C_v(T_0)$</th><th>$C_v(T)$</th></tr></thead><tbody><tr><td>320</td><td>2.9</td><td>8.77</td><td>4.85</td></tr><tr><td>340</td><td>2.7</td><td>7.38</td><td>4.62</td></tr><tr><td>360</td><td>2.5</td><td>6.46</td><td>4.38</td></tr><tr><td>380</td><td>2.4</td><td>5.54</td><td>3.92</td></tr><tr><td>400</td><td>2.2</td><td>4.85</td><td>3.69</td></tr></tbody></table>

### C. Thermal breakdown modeling considering the temperature dependence $C_v(T)$

On this basis, the modeling has been completed by integrating the real temperature dependence $C_v(T)$ in the heat balance equation for the calculation of the breakdown field. As the measurements of $C_v(T)$ was limited to 390 °C due to DSC setup restrictions, we have extrapolated the values of $C_v(T>390°C)$ using a 3rd order polynomial fit up to the PI thermal degradation and integrate it in the numerical model. It is probable that the modeled $C_v(T)$ will not fit perfectly to the real experimental $C_v(T)$ up to such high temperatures due to progressive ignition of degradation phenomena (e.g. thermolysis of PI) but it highlights the impact of the increase of this parameter with temperature on the current density, internal sample temperature an breakdown field.

For each initial temperature $T_0$ of the material, the initial conditions of the heat capacity was taken at $C_v(T_0)$. At each iteration $i$ of the solving program, the new value of $T_{i+1}$ of the material is calculated using the values of $j_i$ and $C_{v,i}$ at a temperature $T_i$.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Conduction current density (a) and temperature rise (b) versus applied electric field considering the temperature dependence of $C_v(T)$. Scatters correspond to experimental currents. Solid lines correspond to modeled current and temperature.</figcaption></figure>
Fig. 3 presents the electric field dependence of the current density and the modeled temperature using now $C_v(T)$. As we can observe, there is a strong tightening of the modeled current density and of the calculated internal temperature before the breakdown with an earlier threshold field compared to the previous case with $C_v(T_0)$. Moreover, a decrease in the calculated breakdown field is also obtained, which becomes in the case of $C_v(T)$ quantitatively closer to the experimental values. The difference is now between 1.63 and 1.85 times higher than the experimental breakdown field.

Nevertheless, the observed differences between theoretical and experimental values of the breakdown field is still large in the case of using $C_v(T)$. In the next steps of our modeling, some enhancements will be integrated, among which:

- a complete modeling of the term $C_v(T)$ up to 600 °C and based on real measurements of the heat capacity in this very high temperature range,

- the use of the real temperature dependence of $\lambda_{th}$,

- the use of a space-charge limited current model for the current density with a field dependent mobility of the charge carriers, which may greatly influence the electrical breakdown process in PI (the field dependent mobility may be demonstrated by the transient current curves analysis),

- the implementation of the filamentary conduction current process in the heat balance equation, since that Hikita et al. reported that the breakdown occurring in PI films was due to filamentary thermal breakdown [14]. This mechanism could result in an increase of thermally activated current flowing through the filament formed in the material under electrical stresses and to decrease the theoretical breakdown field obtained in this study (in our case the current is considered as homogenous through all the electrode up to the breakdown).

## IV. CONCLUSION

In summary, the theoretical thermal breakdown in polyimide (PI) films was modeled at very high temperature between 320 °C and 400 °C using the heat balance equation. Its value was extracted from the internal temperature rise versus electric field of the sample up to its thermal degradation. The originality of this study appears in the consideration, for the first time, of the real temperature dependence of the heat capacity $C_v(T)$, compared to others studies where $C_v$ is fixed at a constant value. It has been possible by measuring it by DSC in a large temperature range up to 390 °C and by interpolating higher values using polynomial fitting. Moreover, the thermal conductivity has also been measured at room temperature only but its value is assumed to be constant and remaining low like in most of polymers. As an input, a simple thermal-activated current density has been used for the solving of the heat

balance equation in a first approach. The results show that the use as an input of a constant value of the capacity $C_v(T_0)$, where $T_0$ is the sample temperature, leads to large difference between the calculated and measured breakdown field. In comparison, the implementation of the temperature dependence $C_v(T)$ in the solving of the heat balance equation leads to closer quantitative results. Some future ways of enhancement of the model are proposed.

## REFERENCES

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