Recent developments in the pressure wave propagation method

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F E A T U R E A R T I C L E

Conventional space charge distribu-tion measurement techniques can be used for studying complex materials provided adequate signal analysis is used.

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Recent Developments in the Pressure Wave Propagation MethodKey Words: Pressure-wave-propagation method, PWP method, LIPP method, space charge distribu-tion, divergent field, fillers, laminates, high spatial resolution

Introduction

Performing measurements is an essential part of understand-ing physical phenomena. Hence, numerous measuring tech-

niques have been developed to study space charge build-up in dielectrics [1]. This macroscopic accumulation of charge modi-fies the electric field and can lead to aging and premature failure of electrical insulation [2]. On the other hand, charge accumula-tion is essential to the operation of devices such as electret- or piezoelectric-based sensors [3]. It is therefore essential to study charge build-up in order to design insulating materials in which excessive charge does not accumulate, or electrets in which a sufficiently large amount of charge accumulates to ensure ad-equate device operation, and the stability of that charge in time is sufficient to ensure an acceptable device life-time.

Some space charge measuring techniques are indirect, and others are direct. In the first case, for example, external current measurements or the mirror method [4], the charge distribution is obtained through a theoretical model of the material being tested and/or through assumptions on the charge distribution it-self. As a consequence, the uniqueness of the obtained space charge distribution is uncertain [4] and only well-modeled mate-rials can provide accurate results. In the second case, charges are directly influenced by the local perturbation used to carry out the measurement. The experimental signals are generated directly by the charges in the distribution to be measured. Among direct measuring techniques, there are three main families. The first one uses thermal diffusion to perturb the charges and induces an electrical signal [5], [6]. The second one uses a pressure wave to perturb the charges and also induces an electrical signal [7], [8]. The third one uses a voltage pulse to perturb the charges and induces a pressure wave signal which is then converted into an electrical signal by a transducer [9], [10]. Although it is not nec-essary to assume an electrical model for the material under test in order to obtain its charge distribution using these measuring methods, it is necessary to understand how the measurements are performed.

This paper focuses on the measurement method using a pres-sure wave to perturb the charges. That method is commonly

called the pressure-wave-propagation (PWP) method and is pre-sented in Figure 1a. A pressure wave enters the sample through one of its interfaces. As it travels through the material at the velocity of sound, the charges encountered are mechanically displaced. The variation of the charge distribution induces a slight variation of the internal electric field which can then be measured, for example, as a current i(t) flowing through a low impedance measuring circuit connecting the sample electrodes. For a unipolar pressure pulse and a uniform planar sample, the short-circuit current is a good image of the charge distribution inside the sample—time and position being related by the veloc-ity of sound. A unipolar pressure pulse mechanically displaces the charges in one direction at the position of the pulse. It pro-duces a local current proportional to the displaced charges which is measured in the external circuit. This is illustrated in Figure 1b and c, respectively, for a sample under voltage, without and with internal charges. In the absence of internal charges (Figure 1b), the signal exhibits only two peaks. They are produced by the ca-pacitive charges induced on each electrode by the applied volt-age. In the presence of internal charges (Figure 1c), however,

Stéphane Holé LPEM/LEG, UPMC-Paris6, ESPCI-ParisTech, CNRS Paris, France

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the peak amplitudes vary with the interfacial electric fields, and the signal between the peaks directly shows where negative and positive charges have developed inside the material. Of course, the signal is only a qualitative image of the charge distribution. Attenuation and dispersion of pressure waves have to be taken into account to obtain quantitative results [11]. As the spatial resolution directly depends on the spatial extent of the gener-ated pressure pulse, [12] it must also be considered. The shorter the pressure pulse, the better the spatial resolution but the larger the attenuation and dispersion. The pressure pulse duration must therefore be matched to the sample thickness in order to obtain acceptable resolution and sensitivity throughout the sample.

One advantage of the PWP method is that measurements are carried out relatively rapidly, requiring only about 1 µs for a 2-mm-thick polyethylene sample. Another advantage is that sig-nals are of relatively large amplitude, yielding sufficient signal-to-noise ratio in a single measurement for samples up to 1-cm thick. In fact, it was the first method to deliver real-time mea-surements under pulsed voltages and 50-Hz ac voltages [13]. Since the introduction of the PWP method in 1976 [14], various techniques have been proposed to generate the pressure wave. When a laser is used to generate the pressure pulse [8], [15], the method is called the laser-induced-pressure-pulse (LIPP) method. When piezoelectric materials are used [16], [17], the

method is called the piezoelectrically-induced-pressure-pulse (PIPP) method if a pressure pulse is generated, and the piezo-electrically-induced-pressure-step (PIPS) method if a pressure step is generated.

If the material under test is planar and uniform, obtaining the charge distribution with the PWP method is relatively simple. It has been extensively and efficiently used to study dielectric materials for more than 30 years [18], [19]. However, charge injection varies rapidly with the interfacial electric field, which also depends sensitively on the electrode material and shape. The electric field increases and diverges in the vicinity of small irregularities, so that charge is more likely to be injected close to these irregularities [20], [21]. Studying a material submitted to divergent electric fields with a direct measuring method is therefore difficult. The non-uniformity of the electric field pro-duces more complex signals, which can be wrongly interpreted if analyzed as for a planar uniform sample.

The same problem arises in all other situations where the electric field is not uniform in the tested material, for instance when it includes fillers or fibers. Heterogeneous materials are increasingly used in high-voltage insulation applications in or-der to strengthen materials [22] or reduce their cost per unit vol-ume. These improvements can, however, be at the expense of impaired insulating properties and electrical aging. Therefore, space charge distribution measurements with appropriate signal analysis are required to study the evolution of charges in hetero-geneous materials.

In this paper, recent developments in complex material analy-sis and in very thin material testing using the PWP method are described. The first section is devoted to the measurement of charge distribution in material under divergent electric fields. The shape of the generated signal is explained and an analy-sis procedure is described. In the second section, heterogeneous materials are studied. The different situations encountered are shown and various analysis techniques are proposed. In the last section, improvement of the spatial resolution of the PWP meth-od in order to study ultra-thin insulating materials, such as those found in integrated circuits, is discussed. The reader can refer to [23] for more information on the PWP method applied to planar materials.

It must be pointed out that much of the discussion present-ed below can be directly applied to the pulsed-electro-acoustic (PEA) method in which the sample is perturbed by a voltage pulse. This method is very similar to the PWP method [24].

Diverging Electric FieldsIn high-voltage insulation, metallic conductors are often em-

bedded in a smooth semi-conducting material in order to avoid interfacial irregularities with the insulator as much as possible. The electric field is increased at interfacial irregularities, and if it exceeds the dielectric breakdown strength, a local discharge may occur and damage the material. Unfortunately the semi-conducting material in which the conductors are embedded is not smooth everywhere, since cables are usually made of differ-ent segments welded together. At the junction between two seg-ments, irregularities may be present. It has been observed that breakdowns preferentially occur at such junctions [20], [21].

Figure 1. Principle of the pressure-wave-propagation (PWP) method when using a laser for generating a pressure pulse (LIPP technique). (a) General setup and typical signals under voltage (b) without internal charges and (c) with internal charges. The front and back peaks correspond to the charges on the electrodes and can be used to determine the material boundaries in the signal.

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Charge injection under divergent electric fields, such as close to a tip, was initially studied using indirect measurement tech-niques [25], [26]. However, it proved difficult to determine how much charge had been injected and how deeply it was trapped. Direct measurements using the PWP method have been carried out since the late 1990s [27].

Measuring charge injected under a divergent electric field requires specially designed samples. Most of the charge is in-jected close to an irregularity, and therefore in a very small volume, and so the PWP signal-to-noise ratio is limited. In order to increase the sensitivity, a wire is embedded between two planar electrodes. This wire is used as the irregularity in-stead of the conventional tip. Injection then occurs all along the wire, resulting in more injected charge and therefore a better signal-to-noise ratio. Further improvement can be obtained by using several wires regularly positioned parallel to the planar electrodes, as illustrated in Figure 2a. Under voltage, each wire generates a strong electric field which decreases nearly as the inverse of the distance from the wire center. In a structure con-sisting of 25-µm-diameter wires evenly spaced at 1.27 mm in-tervals (normal to the sample thickness) and positioned 365 µm from the front electrode and 835 µm from the back electrode, the electric field at the wire interfaces is 10 times larger than at the front electrode. The electric field, calculated for 2 kV ap-plied to the wires, is shown in Figures 2b and c.

Although the geometry (Figure 2a) is almost planar, the elec-tric field varies rapidly close to the wires, even in the absence of charge. Such a situation is not typical of planar uniform materi-

als, in which any spatial variation of the electric field can be attributed to the presence of charge. The simulated signal [28] presented in Figure 3 shows a large increase close to the wires when the sample does not contain any charge. It is essential to understand how this increase is generated in the absence of real charge.

The explanation of this spurious contribution to the signal can be found in the physics of the PWP method [29]. During a measurement, the pressure wave not only moves the charge inside the material, but also slightly changes the relative permit-tivity of the material. When the material is compressed, there are more polarizable molecules per unit volume, resulting in a higher number of induced dipoles per unit volume, and thus a larger permittivity. This phenomenon, known as the electrostric-tive effect, is directly responsible for a displacement current which is proportional to the time derivative of the electric field, and therefore to its spatial derivative because time and space are simply related by the velocity of sound. As a consequence, any spatial variation of the electric field produces a signal because of the electrostrictive effect. In the case of a planar uniform ma-terial, the spatial derivative of the electric field corresponds to the charge distribution and therefore the electrostrictive-effect-induced signal is not different from the signal produced by the charges themselves. In the case of divergent electric field how-ever, the spatial derivative of the electric field along the sample thickness is no longer proportional to the charges, and therefore the signal due to the electrostrictive effect and the signal due to the charges have very different profiles. Careful analysis is re-

Figure 2. Plane-wire-plane-sample geometry generating a divergent electric field. (a) sample structure. (b) electric field amplitude perpendicular to the wires, which were of 25 µm diameter, evenly spaced at 1.27 mm intervals 365 µm from the front electrode and 835 µm from the back electrode. A voltage of 2 kV was applied to the wires. (c) Electric field amplitude in the neighborhood of an embedded wire at high voltage.

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quired to obtain the correct charge distribution from the signal. Another difference between planar uniform samples and

those with a divergent electric field is that in the latter the elec-tric field is a function of at least two spatial coordinates, instead of one. Thus, one signal is no longer sufficient to determine the charge distribution at all positions. Although various measur-ing approaches have been proposed to overcome this situation [30], they are relatively complex or require considerable time to implement. It is often preferable to assume some limited de-gree of freedom for the charge distribution in order to reduce the number of solutions for a given signal. In the case of the plane-wire-plane sample described in Figure 2a, charges are injected mainly at the wire interface. Since the electric field has almost cylindrical symmetry around the wires, the injected charges should show the same geometry, as illustrated in Figure 4. The signal should then be analyzed assuming a cylindrical charge distribution around the wires. This assumption, which is not very restrictive in the case of plane-wire-plane sample geom-etry, allows the charge distribution to be determined from only one measurement.

Quantitative information on the charge distribution can be obtained by an iterative approach [28]. Initially, a cylindrical distribution of charge around the wires is assumed, as illustrated in Figure 4. Then the signal is simulated by the finite element method and compared with the measured signal. The assumed charge distribution is then modified until the simulated and mea-

sured signals match. Such an approach requires a model of the sample response and an estimate of some of the physical pa-rameters of the material being tested. These physical parameters can be obtained from a calibration measurement, i.e., a mea-surement under a moderate voltage when the sample does not contain charges. If the sample already contains charge, subtrac-tion of the measurement under moderate voltage from the mea-surement under short-circuit is also valid. Figure 5a shows the calibration measurement under 2 kV for an epoxy resin sample containing gold-coated-tungsten wires, with the geometry de-scribed in Figure 2. Oscillations at the front electrode and at the wires result from electrical mismatches in the measuring circuit [28]. The amplitude of the calibration signal, for a known quan-tity of charge on the electrodes, is used to compare simulated and measured results. The divergence of the signal close to the wires helps to determine the material electrostrictive constant. Thus various simulations with different electrostrictive con-stants ranging from 0 to −0.7 times the permittivity ε are shown in Figure 5b. The simulation for −0.5ε fits the measured signal very well, so that the electrostrictive constant of the epoxy resin under test can be estimated as −0.48ε [28]. Notice that because the wires are made of tungsten, they are hardly deformed by the pressure wave. The signal at the wires is then mainly due to the electrostrictive effect. The simulation in Figure 5b, where the electrostrictive constant is taken as 0, shows no significant varia-tions at the position of the wires. That is not the case at the front electrode, where the peak amplitude depends both on capacitive charges and on the electrostrictive effect.

Once these calibration parameters have been obtained, charge density and radial position can be estimated using the iterative approach. Figure 5c shows a typical signal measured under short-circuit after a period of high voltage stress during which charges have been injected [27]. Two simulated signals with different cylindrical charge distributions are superimposed on the measured signal. The diameter d and amplitude ρ of the charge distribution are chosen so that the simulated signal am-plitude at the wires is close to the measured signal amplitude. It is clear that simulation 1 (Figure 5c) is the best choice, because simulated and measured signals agree fairly closely at the front interface and in the diverging field region. The accuracy of the estimated radius is better than 10 µm, due to the limited spatial extent of the pressure pulse (about 15 µm) and the large signal-to-noise ratio [12].

Heterogeneous StructuresInsulating materials are becoming more and more complex

in their composition. They can include fillers or fibers to modify their mechanical, thermal, or electrical properties or to reduce their overall cost. Such heterogeneities may, however, have dra-matic contradictory effects on space charge, sometimes acceler-ating aging [31] and sometimes reducing charge storage [32]. In the case of very small randomly and uniformly distributed fillers (i.e., fillers with a diameter much smaller than the spatial resolu-tion of the measurement method) the material can be treated as if it were uniform because there is a sufficient number of fillers under the wave front to define a unique equivalent average ma-terial at all probed positions. However, if larger fillers are not

Figure 3. Typical signal from plane-wire-plane sample geometry under voltage and without charges. For the simulation, the sample geometry described in Figure 2 was used, with 2 kV applied to the wires. The insulator is assumed to be epoxy resin and the wires of tungsten. To the left of the wires the signal increases with the electric field due to the electrostrictive effect.

Figure 4. A likely charge distribution around a wire with only two degrees of freedom. The wire diameter is d and the charge density is ρ.

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uniformly distributed, the equivalent average material depends on position and noticeable differences between layers have to be taken into account in the signal interpretation.

An effect due to differences between material layers appears in the propagation of the pressure wave itself. If too many reflec-tions occur during the propagation, the signal at a given time

would be induced by charges at very different positions and would then be very difficult to analyze. Fortunately, the reflec-tion of a plane wave on a small filler is relatively small and is spatially rapidly dispersed. Figure 6a shows how a planar pres-sure pulse behaves in epoxy resin in the presence of an isolated silica filler particle with a diameter similar to the pulse width, here about 50 µm. In this figure, the wave propagation is cal-culated using a finite-difference time-domain (FDTD) method. Three situations are shown with respect to the time at which the pressure pulse reaches the filler position. In the first situation (left), the wave front is obviously not altered by the filler. In the second situation (center), a spherical reflection is clearly visible and the wave front presents a notch (low pressure amplitude) inside the filler. The amplitude of that reflection is about 50% of the pressure pulse amplitude. In the third situation (right), the notch has almost disappeared from the wave front and the reflection amplitude is reduced to about 10 % of the pressure pulse amplitude. In the presence of more fillers laterally distrib-uted in a random manner (Figure 6b), reflections between adja-cent fillers cancel and only the reflection and transmission along the sample thickness axis remain. The plot at the bottom clearly shows that the pulse profile is not altered by the presence of the filler layer. The positive and negative reflections correspond re-spectively to the entrance and exit of the pressure from the silica filler region. The amplitude of these reflections depends on the acoustical mismatch between the resin and the filler region. If fillers are randomly distributed throughout the resin, the slight differences between the various layers of the material due to the slight variations of the number of fillers per unit volume produce very small reflections. Therefore only the transmission is impor-tant and reflections are negligible, or similar to attenuation and dispersion, and can be treated by appropriate algorithms [11], [33]. The same conclusion applies to fibers (instead of fillers). The reflection of the pressure wave greatly reduces with increas-ing roughness of the epoxy-resin/silica-fibers interface. This is illustrated in Figure 7, 20 ns after the pressure pulse entered the silica region. When the roughness is larger than the pres-sure pulse width (right), the reflection can be neglected. Con-sequently, for samples including randomly distributed fillers or fiber layers with sufficient roughness, the profile of the pressure wave is not significantly affected.

A second effect due to the differences between layers is the variation of permittivity and electrostrictive constants from one position to another. Under an applied electric field, such a spatial variation of permittivity generates apparent charge with density proportional to the field. This proportional spurious contribu-tion to the signal should not be confused with a trapped charge distribution. Conductivity mismatches between layers have been suggested as the cause of this spurious contribution [34]. While conductivity mismatches may indeed result in charge accumula-tion at the interfaces, this suggestion is not in accordance with the experimental data, because the buildup time constant for such charge is often much longer than the delay between voltage application and first measurement. The buildup time constant, which is the ratio between permittivity and conductivity, ranges from minutes to hours in most insulating materials. However, permittivity mismatches produce quasi-instantaneous spurious

Figure 5. Measurement analysis for plane-wire-plane-sample geometry. (a) Calibration measurement carried out under 2kV. The oscillations are due to electrical mismatches in the measuring circuit. (b) Simulations with various electrostrictive constants ranging from 0 to −0.7ε are superimposed on the calibration measurement. The electrostrictive constant of epoxy resin can be estimated as −0.5ε since simulation and measurement fit almost perfectly. (c) Comparison between a measurement under short-circuit with a sample containing charges and simulated signals with two different charge distributions. The first simulation fits the measurement very closely, indicating that the simulated charge distribution is very similar to the charge distribution in the real sample.

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contributions which can be observed as soon as the voltage is applied to the sample.

Heterogeneities may include charges or be piezoelectric. This leads to an additive spurious contribution to the signal, since it does not depend on the electric field. Knowing how heteroge-neities influence the signal is therefore of great importance for a correct determination of the charge distribution. For that pur-pose, specially designed test samples as shown in Figure 8a can be used. They do not contain heterogeneities close to the front electrode [35]. Depending on the nature of the heterogeneities, different situations can be encountered during measurement un-der short-circuit or a moderate applied voltage.

Under short circuit (Figure 8b), a signal is observed if the heterogeneities are piezoelectric or contain charges (Cases 2

and 3 respectively). Otherwise (Case 1) no significant signal can be detected. In the case of piezoelectric heterogeneities (Case 2), a signal is produced only when the pressure reaches the heterogeneous region. Therefore there is no significant signal at the front electrode because there are no heterogene-ities at that interface. With heterogeneities containing charges (Case 3), a signal is generated when the pressure reaches the heterogeneous region, but as charges generate an electric field and thus induce capacitive charges on the electrodes, a signal is also generated at the front electrode. Thus measurement under short circuit makes it possible to discriminate between piezo-electric, charged, or other kinds of heterogeneities. In cases 2 and 3, additive spurious contributions to the signal must be taken into account.

Figure 6. Finite-difference time-domain simulations of the propagation of a pressure pulse inside heterogeneous media at different times: 40 ns before the filler region (left), at the filler region (center), and 40 ns after the filler region (right). Amplitudes are scaled in percent with respect to the incident pressure pulse amplitude. (a) Reflection and transmission from a single silica filler in epoxy resin. At 0 ns, a large reflection is visible and the wave front presents a notch (low pressure) at the filler position. At 40 ns, the reflection amplitude has reduced by a factor 5 and the notch has almost disappeared. (b) Reflection and transmission from a random planar distribution (40% filling factor) in epoxy resin. The reflections adopt a planar geometry. At the bottom the pressure pulse profile is represented along the sample thickness axis. At 40 ns, the positive and negative reflections occurred respectively when the pressure pulse entered and exited the filler region.

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Under moderate applied voltage (Figure 8c), interfacial peaks appear due to induced capacitive charges on the electrodes. If the signal between the peaks is different from the signal under short-circuit, then the heterogeneities and the host material must have different permittivity and electrostrictive constants. Taking Case 2 in Figure 8b (piezoelectric heterogeneities), the signal under moderate voltage is not modified between the peaks (Case 2a) if heterogeneities and host material have similar permittivity and electrostrictive constants, but is modified (Case 2b) if they differ significantly. In the latter case, proportional spurious con-tributions to the signal must be taken into account.

Additive spurious contributions to the signal can be elimi-nated simply by subtracting the short-circuit signal from the signal obtained under applied voltage [36]. It is then essential to carry out measurements at the same position on the sample surface, because otherwise additive spurious contributions would not be correctly treated. Figure 9a shows the results of measurements made at different positions on the surface of a test sample containing 50-µm-mean-diameter-silica fillers. The impact area of the laser beam is shifted by 2 mm between the two measurements. Even for shifts as small as a quarter of the impact diameter, large differences can be observed. If the same impact area is used (Figure 9b), the subtraction operates cor-rectly and only the proportional contributions remain. Figure 9c shows that subtracting the measurement under short circuit from the measurement under applied voltage, and dividing the result by the applied voltage, leads to almost perfectly superimposed traces. Thus what remains after subtraction is proportional to

the electric field and corresponds to the proportional spurious contribution.

As an illustration, the results of a study of 1-mm-thick-epoxy resin containing 50-µm-mean-diameter-silica fillers are present-ed in Figure 10. These fillers are made of natural crushed sand. Some have an isotropic structure and others have the crystalline structure of quartz, and therefore they partially exhibit piezo-electricity [35]. Their large size and random mixing produce un-der short circuit a completely random signal, due to their piezo-electric properties (Figure 10a). This is the additive spurious contribution. When the sample is under voltage (Figure 10b), a difference can be observed after 17 hours of stress, indicat-ing the development of charge. However, it is unclear whether charge has built up in the resin or at the interface with the fill-ers, because the difference is of the same order of magnitude as the additive spurious contributions caused by piezoelectricity. Subtracting the measurement under short circuit from the mea-surement under applied voltage in order to eliminate the additive spurious contributions (Figure 10c) results in an almost flat trace at the beginning of the stress period, but oscillations are present after 17 hours. Because the frequency of the oscillations multi-plied by the sound velocity corresponds to the mean diameter of the fillers, it can be deduced that charges have been trapped mainly at the filler interfaces.

As far as proportional spurious contributions to the signal are concerned, various approaches can be used. Large material heterogeneities, as in the case of laminates, have to be taken into account in order to eliminate the proportional spurious

Figure 7. Finite-difference time-domain simulations of reflection and transmission for silica fibers with different surface roughness. Amplitudes are scaled in percent with respect to the pressure pulse amplitude. The reflection amplitude 20 ns after the pressure pulse entered the silica is very sensitive to the interface roughness. The greater the roughness, the smaller the amplitude, as shown by the pressure pulse profile along the sample thickness axis (bottom).

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contribution. In a first attempt, the sample was modeled by assigning various permittivity values to each layer. This is il-lustrated in Figure 11 for a 400-µm-thick-printed-circuit-board sample containing two layers of silica fibers. The permittivities are adjusted until the simulated signal under voltage (here 4 kV) approaches the experimental signal (Figure 11b). In order to un-derstand the general behavior of the material, it is not necessary to model the sample structure precisely. Although the simulated and measured signals in Figure 11b do not fit perfectly at all positions, they exhibit the same overall behavior for a limited number of sample layers. Once the sample has been modeled, various assumed charge distributions within the structure (Fig-ure 12c) are adjusted to make the simulated signals (Figure 12b) similar to the measured signal (Figure 12a). The example pre-sented in Figures 12 shows a clear polarization of the material and a charge build up at some of the internal epoxy/silica fiber interfaces [37]. Such a detailed analysis would not have been possible using the raw signal alone. However, this approach has some limitations; because the estimated charges depend on the material model, quantitative errors may occur.

In a second approach, the sample is directly modeled using a calibration measurement. It can be shown [38], [39] that, if

there are no significant reflections of the pressure pulse inside the sample, the electric field can be estimated simply by inte-grating the signal over time. Performing such integration with the calibration measurement yields the permittivity distribution because, without space charges, the electric displacement is uni-form in a planar sample and thus the electric field varies exactly as the inverse of the permittivity distribution. If the permittivity distribution is uniform, then a given charge density will yield the same signal amplitude whatever its position. If the permittivity distribution is not uniform, however, then a given charge density will be overestimated or underestimated depending on its posi-tion. Integrating over time two measurements on the same sam-ple, the first for an unknown charge distribution and the second for no internal charge (calibration), and dividing one signal by the other, will reduce the estimation errors, i.e., an overestimate (underestimate) will almost compensate an overestimate (under-estimate). A subsequent differentiation with respect to time will yield a corrected signal that can be interpreted as if the sample were uniform [38]. This approach was tested on a 400-µm-thick-printed-circuit-board sample to which 6 kV had been applied for 30 minutes. Measurements and integration of these measure-ments over time are respectively shown in Figures 13a and b.

Figure 8. Influence of heterogeneities in a test sample. (a) Test samples. (b) Signal obtained under short-circuit. (c) Signal obtained under moderate applied voltage, assuming Case 2 in (b).

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The measurement at 0 min is taken as the calibration measure-ment. Expected variations between the interfacial peaks, due to permittivity variations, are seen. The integration over time of the calibration measurement also shows these variations, from which it is easy to see where charge will be overestimated or underestimated. Dividing the integration over time by the in-tegrated calibration measurement corrects for the proportional spurious contribution. Differentiating with respect to time then yields a trace which can be interpreted as if the sample were uniform (Figure 13c). In addition to the polarization indicated by the large variation of the peaks, charges have developed close to the electrodes, probably at the epoxy/silica fiber interfaces as formerly deduced from iterative simulations.

The integration of the measured signals over time is not al-ways straightforward, because the electronic response of the

measuring setup may have slightly deformed the signals. Inte-grations with different time constants τ, for instance by using a convolution with exp(−t/τ) instead of the Heaviside function, lead to quite different integrated signals. However, dividing the integrated measurements by the integrated calibration measure-ment eliminates most large errors, and the resulting signals are expected to be accurate. Figure 14a shows the integration over time of the 0 min calibration measurement in Figure 13a, using a convolution product with the function exp(−t/τ) for various time constants τ. These time constants were chosen proportionally to the optimal time constant τ

opt, here 185 ns, for which the inte-

grated calibration presents no drift. Figure 14b shows the results of applying the correction procedure to the signal at 30 min (Fig-ure 13a) for each time constant. As expected, the charge distri-bution inside the material does not vary greatly, even with errors as large as a factor of 2 in the choice of time constant [39].

Very Thin Insulating MaterialThe development of space charge in a material is often de-

pendent on the amplitude of the electric field to which it is submitted, typically larger than a few kilovolts per millimeter.

Figure 9. The random nature of the signal for 50-µm-mean-diameter-silica fillers in a test sample. (a) For different laser beam impact areas, spurious contributions may be very different for the same electrical condition, in this case under short-circuit. (b) If the laser beam impact area does not change between measurements, then additive spurious contributions are the same under short-circuit and under applied voltage. (c) A simple subtraction of the measurement under short-circuit from the measurement under applied voltage eliminates all additive spurious contributions, and a subsequent normalization with respect to applied voltage shows that only proportional spurious contributions remain.

Figure 10. Space charge measurements in 1-mm-thick-epoxy resin containing 50-µm-mean-diameter-silica fillers randomly distributed throughout the sample volume. Measurement at 50° C (a) under short circuit and (b) at the beginning and after 17 h under 10 kV. (c) Measurement under short circuit subtracted from the measurement under applied voltage in order to eliminate the additive spurious contributions.

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Thus problems related to space charge buildup often occur in the insulation of integrated circuits, since their small thickness leads to large electric field even under low voltage. Although some integrated sensors and actuators take advantage of charge buildup [40], it usually has a negative impact. Besides damaging materials, charge modifies the voltage threshold of field effect transistors and can lead to malfunctioning of the whole device.

Measuring charge distributions in very thin material is diffi-cult [41]. The only available method (etch-off method) consists in measuring the surface potential of the material while chemi-cally removing small layers [42], [43]. A good spatial resolution can then be obtained, but the etch-off method has the great dis-advantage of being destructive. In addition, the time evolution of space charge distributions cannot be studied. A nondestruc-tive measurement method with sufficient spatial resolution is required.

In the case of the PWP method, high spatial resolution im-plies a broad bandwidth. As an example, because time and space are linked by the velocity of sound, 50 nm resolution in silicon dioxide implies a 10 ps pulse duration.

Laser pulses with duration in the range of femtoseconds can be used in the thermoelastic regime to generate picosecond-du-ration pressure pulses [44]. When the laser pulse hits the sample front electrode the electrode quickly warms up, and because heat diffusion is negligible during the short pulse, most of the mechanical constraints are perpendicular to the sample surface, as illustrated in Figure 15a. This generates an internal pressure pulse which propagates in both directions of the sample axis (Figure 15b). As the part of the pulse propagating toward the free electrode interface is rapidly totally reflected, the overall pressure profile is bipolar (Figure 15c). This bipolar pressure pulse is equivalent to a unipolar material displacement pulse whose extent determines the spatial resolution [12] which can be achieved using the PWP method. Figure 16a shows a measure-ment of the interfacial mechanical displacement as a function of

time when a 100-fs laser pulse hits a 250-nm-thick aluminum electrode deposited on a 200-nm-thick silicon nitride sample on a silicon substrate. The measurement was made using a Sag-nac interferometer [45]. A large step corresponding to thermal expansion appears first, followed by mechanical displacement pulses. These pulses come from the reflections of the bipolar pressure pulse at the internal interfaces of the sample (Fig-ure 16b). The first peak is due to the reflection of the pulse at the interface between the aluminum electrode and silicon nitride, the second to the reflection at the back electrode of the sample (silicon substrate), and the third peak to the second reflection at the interface between the aluminum electrode and silicon ni-tride. The duration at mid-amplitude of these pulses is of the order of 11 ps, corresponding to a spatial resolution of 55 nm in silicon dioxide and 100 nm in silicon nitride. The resolution can be improved by at least a factor of 2 by using 10-fs laser pulses [46]. However the resolution improvement is not proportional to the reduction of the laser pulse duration, because the laser

Figure 11. Laminate model of a 400 µm-thick-printed-circuit-board sample. (a) A priori model with various relative permittivities for each main region. (b) Simulated and measured signals under 10 kV/mm without charges.

Figure 12. Space charge profiled for a 400 µm-thick-printed-circuit-board sample). (a) Evolution of the measured signal obtained using the PEA method [37] under 10 kV/mm. (b) Evolution of the simulated signal assuming various charge distributions. (c) The assumed charge distributions.

May/June 2009 — Vol. 25, No. 3 17

beam slightly penetrates the electrode, generating a mechanical constraint over the penetration depth. The penetration depth de-pends on the electrode material and on the laser wavelength, and is typically of the order of 10 nm in aluminum at a wavelength of 800-nm [44]. This places a lower limit on the width of the pres-sure pulse which can be generated using ultra-short laser pulses, and thus on the ultimate spatial resolution.

When the bipolar pressure pulse propagates inside the sam-ple, it perturbs the charge through which it passes, and an elec-tromagnetic wave whose amplitude depends on the density of the perturbed charge at the position of the pressure wave is gen-erated. This electromagnetic wave corresponds to the signal and has picosecond time variations. At such a time scale, conven-tional measuring equipment (e.g., oscilloscopes) cannot be used. Electro-optic sampling is a possible solution. An electro-optic crystal, for instance ZnTe because of its good figure of merit, is placed as close as possible to the sample. The refractive index

of an electro-optic crystal depends on the ambient electric field, in this case the electromagnetic wave generated by the charges. Figure 17a shows a cross-section of the index ellipsoid of an iso-tropic crystal without ambient electric field. It is a circle, which means that a laser beam propagating perpendicularly to the fig-ure plane travels in the crystal at the same velocity whatever its polarization (Figure 17b). In the presence of an ambient electric field the isotropic crystal is slightly deformed and its index el-lipsoid is an ellipse. As a consequence, a laser beam polarized along the x-axis travels at a different velocity than one polarized along the y-axis (Figure 17c). This difference induces a phase shift (Pockels effect) which can be measured to determine the ambient electric field [47]. If the laser beam is pulsed, the ambi-ent electric field is probed each time a pulse travels through the crystal. Hence, the ambient electric field can be probed at pico-second intervals by using femtosecond laser pulses and a delay line. The laser pulse duration gives the sampling aperture, and

Figure 13. Direct correction of proportional spurious contributions in a 400-µm-thick printed circuit board under 6kV for 30 min. (a) measurements, (b) integration of the measurements over time. (c) time derivative of the quotient integrated measurement/integrated calibration. (The calibration measurement was the measurement at t = 0 min).

Figure 14. Effect of time constant errors on the integration procedure. (a) Calibration measurement integrated using different time constants scaled with respect to the optimal time constant τ

opt. (b) Correction for each time constant applied to

the measurement after 30 minutes of polarization.

Figure 15. Principle of the generation of ultra short pressure pulses. (a) Localized thermal heating. (b) Internal stress and (c) resulting bipolar pressure wave.

18 IEEE Electrical Insulation Magazine

the delay line increases the time at which the laser pulses sample the electric field generated by the charges.

The phase shift induced by the signal can be measured by an interferometer. In the case of a Sagnac interferometer (Fig-ure 18), the measuring laser beam is split into two probe beams propagating in opposite directions and hitting the electro-optic crystal at different times. The first probe beam (solid line) hits the crystal before the generation of the pressure wave by the laser pump beam and is used as a reference. The second probe beam (dashed line) hits the crystal during or after the generation

of the pressure pulse, depending on the setting of the delay line. The interference of these two probe beams at the photo-diodes then gives the difference between during and before the sample perturbation by the pressure wave. Because the delay between the two probe beams is relatively short, typically less than 1 ns, the Sagnac interferometer is very insensitive to vibrations. In ad-dition, the interference can be optimized by using half and quar-ter wavelength plates. The two probe beams are then in quadra-ture and any small phase shifts can be measured at the optimal point of operation. The signal produced by the interferences is however very small. A very large number of measurements must be made, using a lock-in technique, to obtain a sufficient signal-to-noise ratio [46].

This measuring technique has not yet yielded a complete space charge distribution signal. Generation and measurement of ultra-short pressure pulses have been experimentally demon-strated, but the measurement of the electric field generated by the charge displacements is still under study. Calculations of the signal sensitivity show that the PEA method should yield a bet-ter signal-to-noise ratio than the PWP method.

ConclusionIn addition to measurement of space charge profiles in uni-

form planar material, the pressure-wave-propagation method has been recently used to study more complex materials. Research is continuing to improve its spatial resolution sufficiently to allow its use in the study of integrated dielectrics.

In the case of complex materials exhibiting either a very di-vergent electric field or a heterogeneous structure, the signal is not due only to the presence of charges. Other spurious contri-butions have to be taken into account. Thus the variation of the electric field caused by its divergence induces a signal contri-bution originating in the electrostrictive effect, directly propor-tional to the applied electric field. Such a proportional spurious contribution arises also in the case of heterogeneous materials for which the permittivity changes from one point to another. An additive spurious contribution has also to be taken into ac-count in heterogeneous materials, because heterogeneities may include charges or be piezoelectric. Correction procedures have been described to compensate for these spurious contributions, by subtracting measurements, or modeling the sample and per-forming iterative simulations, or calibrating the spurious con-tributions to directly eliminate their effects. At the end of the correction procedure, the signal can be treated as if it had been obtained from a uniform planar material.

As far as the spatial resolution improvement is concerned, a significant change in the measuring setup is required. The neces-sary signal bandwidth is much broader, and conventional equip-ment can no longer be utilized. A scheme based upon a Sagnac interferometer is under study. A femtosecond laser pulse is used to produce picosecond-duration pressure pulses which perturb the charges encountered during propagation. The electromag-netic wave subsequently generated is probed by an electro-optic crystal, using a second fs-duration laser pulse. The whole evo-lution of the signal is measured using a delay line. It has been shown that a resolution of 50 nm in silicon dioxide can be ob-tained with 100-fs laser pulses.

Figure 16. Interfacial mechanical displacement produced by ultra-short bipolar pressure pulses. (a) Measurement using a Sagnac interferometer and (b) origin of the reflections.

Figure 17. Pockels effect in electro-optic crystals. (a) Cross-section of index ellipsoid depending on the ambient electric field. (b) In absence of ambient electric field, x- and y-polarized laser beams travel at the same speed. (c) Under an ambient electric field, the x-polarization travels faster than the y-polarization and a phase shift ∆φ appears.

May/June 2009 — Vol. 25, No. 3 19

References[1] T. Mizutani, “Space charge techniques and space charge in polyethylene,”

IEEE Trans. Dielectr. Electr. Insul., vol. 1, pp. 923–933, 1994. [2] Y. Zhang, J. Lewiner, C. Alquié, and N. Hampton, “Evidence of strong

correlation between space-charge buildup and breakdown in cable insula-tion,” IEEE Trans. Dielectr. Electr. Insul., vol. 3, pp. 778–783, 1996.

[3] F. Bauer, “Ferroelectric PVDF polymer for high pressure and shock com-pression sensors,” in Int. Symp. Electrets, Melbourne, Australia, 2002, pp. 219–222.

[4] H. J. Wintle, “Analysis of the scanning electron microscope mirror method for studying space charge in insulators,” J. Appl. Phys., vol. 86, pp. 5961–5967, 1999.

[5] R. E. Collins, “Analysis of spatial distribution of charges and dipoles in electrets by a transient heating technique,” J. Appl. Phys., vol. 47, pp. 4404–4408, 1976.

[6] S. B. Lang and D. K. Das-Gupta, “Laser-intensity-modulation method: A technique for determination of spatial distributions of polarization and space charge in polymer electrets,” J. Appl. Phys., vol. 59, pp. 2151–2160, 1986.

[7] P. Laurenceau, G. Dreyfus, and J. Lewiner, “New principle for the deter-mination of potential distributions in dielectrics,” Phys. Rev. Lett., vol. 38, pp. 46–49, 1977.

[8] C. Alquié, G. Dreyfus, and J. Lewiner, “Stress wave probing of electric field distributions in dielectrics,” Phys. Rev. Lett., vol. 47, pp. 1483–1487, 1981.

[9] T. Maeno, H. Kushiba, T. Takada, and C. M. Cooke, “Pulsed electro-acoustic method for the measurement of volume charge in e-beam irradi-ated PMMA,” in Conf. Electrical Insulation and Dielectric Phenomena, Amherst, NY, 1985, pp. 389–397.

[10] J. B. Bernstein, “Analysis of the electrically stimulated acoustic-wave method for observing space charge in semi-insulating films,” Phys. Rev. B, vol. 44, pp. 804–814, 1991.

[11] T. Ditchi, C. Alquié, and J. Lewiner, “Broadband determination of ultra-sonic attenuation and phase velocity in insulating materials,” J. Acoust. Soc. Am., vol. 94, pp. 3061–3066, 1993.

[12] S. Holé, “Resolution of direct space charge distribution measurement methods,” IEEE Trans. Dielectr. Electr. Insul., vol. 15, pp. 861–871, 2008.

[13] S. Holé, C. Alquié, and J. Lewiner, “Measurement of space charge distri-butions in insulators under rapidly varying voltage,” IEEE Trans. Dielectr. Electr. Insul., vol. 4, pp. 719–724, 1997.

[14] P. Laurenceau, J. Ball, G. Dreyfus, and J. Lewiner, “Une méthode nou-velle de détermination de la distribution spatiale des potentiels dans les diélectriques,” C. R. Acad. Sci. Paris, vol. 283, pp. 135–138, 1976.

[15] G. M. Sessler, J. E. West, and R. Gerhard-Multhaupt, “Measurement of charge distribution in polymer electrets by a new pressure-pulse method,” Polym. Bull., vol. 6, pp. 109–111, 1981.

[16] W. Eisenmenger and M. Haardt, “Observation of charge compensated zones in polyvinylidenefluoride (PVDF) films by piezoelectric acoustic step-wave response,” Solid State Commun., vol. 41, pp. 917–920, 1982.

[17] R. Gerhard-Multhaupt, G. M. Sessler, J. E. West, K. Holdik, M. Haardt, and W. Eisenmenger, “Investigation of piezoelectricity distributions in poly(vinylidene fluoride) by means of quartz or laser generated pressure pulses,” J. Appl. Phys., vol. 55, pp. 2769–2775, 1984.

[18] R. Gerhard Multhaupt, “Analysis of pressure-wave methods for the non-destructive determination of spatial charge or field distributions in dielectrics,” Phys. Rev. B, vol. 27, pp. 2494–2503, 1983.

[19] P. Rain and J. L. Auge, “Influence of the electrode metal on the injec-

Figure 18. Principle of high resolution PWP method. A femtosecond laser pulse (pump beam) creates a picosecond-duration pressure wave inside the sample. The electromagnetic wave generated by the charges encountered by the pressure wave during its propagation modifies the properties of the electro-optic crystal. A femtosecond measuring laser beam is split into two probe beams. The first and second probe beams (solid and dashed lines respectively) travel through the electro-optic crystal before and after the generation of the pressure wave. The interference of the two probe beams at the photodiodes yields a signal proportional to the electromagnetic field generated by the charges at the instant the laser beam hits the electro-optic crystal. A delay line modifies this time and thus allows the whole evolution of the signal to be measured on a picosecond time scale.

20 IEEE Electrical Insulation Magazine

tion of space charges in PET and epoxy resin,” in IEEE Int. Conf. Solid Dielectrics, Winchester, UK, 2007, pp. 466–469.

[20] T. Baumann, B. Fruth, and H. R. Zeller, “Field-enhancing defects in polymeric insulators causing dielectric aging,” IEEE Trans. Electr. Insul., vol. 24, pp. 1071–1076, 1989.

[21] Z. Zheng and S. Boggs, “Defect tolerance of solid dielectric transmission class cable,” IEEE Electr. Insul. Mag., vol. 21, pp. 35–41, 2005.

[22] M. Harada, M. Morimoto, and M. Ochi, “Influence of network chain orientation on the mechanical property of epoxy resin filled with silica particles,” J. Appl. Polym. Sci., vol. 87, pp. 787–794, 2003.

[23] J. Lewiner, S. Holé, and T. Ditchi, “Pressure wave propagation methods: A rich history and a bright future,” IEEE Trans. Dielectr. Electr. Insul., vol. 12, pp. 114–126, 2005.

[24] S. Holé, T. Ditchi, and J. Lewiner, “Non-destructive methods for space charge distribution measurements: What are the differences?” IEEE Trans. Dielectr. Electr. Insul., vol. 10, pp. 670–677, 2003.

[25] C. Laurent, C. Mayoux, and S. Noel, “Dielectric breakdown of polyethyl-ene in divergent field: Role of dissolved gases and electroluminescence,” J. Appl. Phys., vol. 54, pp. 1532–1539, 1983.

[26] T. Hibma and H. R. Zeller, “Direct measurement of space charge injection needle electrode into dielectrics,” J. Appl. Phys., vol. 59, pp. 1614–1620, 1986.

[27] O. Naz, T. Ditchi, J. Lewiner, and C. Alquié, “Study of charge injection in insulators submitted to divergent fields,” IEEE Trans. Dielectr. Electr. Insul., vol. 5, pp. 2–8, 1998.

[28] O. Paris, J. Lewiner, T. Ditchi, S. Holé, and C. Alquié, “A finite element method for the determination of space charge distributions in complex geometry,” IEEE Trans. Dielectr. Electr. Insul., vol. 7, pp. 556–560, 2000.

[29] S. Holé, T. Ditchi, and J. Lewiner, “Influence of divergent fields on space charge distribution measurements using elastic methods,” Phys. Rev. B, vol. 61, pp. 13528–13539, 2000. In Eq. (21) Bk should be Ek and in Eq. (47) εj should be ε j.

[30] S. Holé, T. Ditchi, and J. Lewiner, “High resolution multidimensional space charge measurement using elastic wave methods,” Phys. Rev. B, vol. 64, art. no. 104106, 2001.

[31] T. Takada, Y. Hayase, Y. Tanaka, and T. Okamoto, “Space charge trapping in electrical potential well caused by permanent and induced dipoles for LDPE/MgO nanocomposite,” IEEE Trans. Dielectr. Electr. Insul., vol. 15, pp. 152–160, 2008.

[32] O. Gallot-Lavallee, G. Teyssedre, C. Laurent, and S. Rowe, “Space charge behaviour in an epoxy resin: The influence of fillers, temperature and electrode material,” J. Phys. D: Appl. Phys., vol. 38, pp. 2017–2025, 2005.

[33] S. Twomey, “On the numerical solution of fredholm integral equations of the first kind by the inversion of the linear system produced by quadra-ture,” J. ACM, vol. 10, pp. 97–101, 1963.

[34] H. Tanaka, Y. Ohki, K. Fukunaga, T. Maeno, and K. Okamoto, “Space charge distributions in glass fibre/epoxy resin composites under dc 10 kV/mm electric field,” J. Phys. D: Appl. Phys., vol. 40, pp. 1489–1496, 2007.

[35] S. Holé, A. Sylvestre, and S. Rowe, “The influence of filler particles on space charge measurements,” J. Phys. D: Appl. Phys., vol. 37, pp. 1869–1876, 2004.

[36] S. Holé, A. Sylvestre, C. Guillermin, P. Rain, and S. Rowe, “Space charge measurement on filled epoxy,” in Int. Conf. Solid Dielectrics, pages 154–157, Toulouse, France, 2004, pp. 154–157.

[37] S. Holé, L. A. Dissado, M. N. Ajour, and J. C. Fothergill, “Space charge behaviour in epoxy laminates under high constant electric field,” J. Phys. D: Appl. Phys., vol. 38, pp. 2890–2899, 2005.

[38] S. Holé, L. A. Dissado, and J. C. Fothergill, “Direct space charge distribu-

tion estimation in laminates,” in Les Matériaux du Génie Électrique, Toulouse, France, 2008, pp. 103–106.

[39] S. Holé, L. A. Dissado, J. C. Fothergill, and J. Lewiner, “Direct analysis of heterogeneous insulators with the PWP and the PEA methods,” in Int. Symp. Electrets, Tokyo, Japan, 2008, p. 75.

[40] M. Ichiya, F. Kasano, H. Nishimura, J. Lewiner, and D. Perino, “Elec-trostatic actuator with electret,” IEICE Trans. Electron., vol. E78C, pp. 128–131, 1995.

[41] R. J. Fleming, “Space charge profile measurement techniques: Recent advances and future directions,” IEEE Trans. Dielectr. Electr. Insul., vol. 12, pp. 967– 978, 2005.

[42] E. H. Nicollian and J. R. Brews, MOS (Metal Oxide Semiconductor) Physics and Technology. John Wiley & Sons, Hoboken, NJ, 1982

[43] M. Kohno, T. Kitajima, H. Okada, S. Hirae, and T. Sakai, “Noncontact measurement of sodium ions in silicon oxide,” Jpn. J. Appl. Phys., vol. 41, pp. 2266–2275, 2002.

[44] B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci., vol. S6, pp. 444–448, 1996.

[45] G. Dagher, S. Holé, and J. Lewiner, “A preliminary study of space charge distribution measurements at nanometer spatial resolution,” IEEE Trans. Dielectr. Electr. Insul., vol. 13, pp. 1036–1041, 2006.

[46] G. Dagher. Mesure des charges d’espace à l’échelle nanométrique: Ap-plication à la micro-électronique. Ph.D. thesis, Université Pierre et Marie Curie, Paris, France, 2008.

[47] Q. Wu and X. C. Zhang, “Free-space electro-optic sampling of terahertz beams,” Appl. Phys. Lett., vol. 67, pp. 3523–3525, 1995.

Stéphane Holé, born in 1968 at Pontoise, France, studied electronics and instru-mentation at Université Pierre et Marie Curie–Paris 6, Paris, France. He joined the Laboratoire d’Électricité Générale of École Supérieure de Physique et de Chimie Industrielles, Paris, France, to study an in-strument for measuring fast development of space charges in insulators under rapid voltage variations. He received his Ph.D. in

1996. He held the position of Maître de Conférences (assistant professor) at Université Pierre et Marie Curie, in the Labora-toire des Instruments et Systèmes d’Ile de France from 1997 to 2007, and in the Laboratoire Photons Et Matière since 2007. He teaches solid state physics, electronics, and sensor physics. He conducts research on various topics such as space charge in in-sulators and semiconductors (main topic), micro-scale thermal imaging, and automotive sensors. He received the Jack Hol-lingum Award in 2002 and 2004, and obtained his Habilitation in 2007. His current address is: UPMC-LPEM-CNRS / ESPCI-LEG, 10 rue Vauquelin, 75005 Paris, France, e-mail: [email protected].

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