High-resolution multidimensional space charge measurement using elastic wave methods Ste ´ phane Hole ´ * Laboratoire des Instruments et Syste `mes, Universite ´ Pierre et Marie Curie, 10, rue Vauquelin, 75005 Paris, France Jacques Lewiner Laboratoire d’E ´ lectricite ´ Ge ´ne ´rale, E ´ cole Supe ´rieure de Physique et de Chimie Industrielles de la Ville de Paris, 10, rue Vauquelin, 75005 Paris, France Received 20 November 2000; revised manuscript received 2 April 2001; published 23 August 2001 In various fields of physics, materials can develop or exhibit a nonuniform space-charge distribution. During the last 20 years various approaches have been proposed for the direct measurement of these distributions involving optical, acoustical, or thermal processes. Most of them, however, lead only to one-dimensional distributions. In this paper we describe two approaches to obtain, in isotropic condensed matter, multidimen- sional space-charge distributions. They are based on the pressure-wave-propagation method and the electroa- coustic method. In the case of the pressure-wave-propagation method, various independent measurements are performed with plane waves at different angles of incidence. A tomographiclike algorithm is then applied to retrieve the multidimensional space-charge distribution. In the case of the electroacoustic method, independent measurements are provided from a set of piezoelectric transducers at various positions. An echographiclike algorithm is then applied to retrieve the multidimensional space-charge distribution. A theoretical model is proposed, and preliminary experimental and simulated results showing the validity of these techniques are presented. DOI: 10.1103/PhysRevB.64.104106 PACS numbers: 77.22.-d, 72.50.+b, 81.70.Cv I. INTRODUCTION In various fields of physics, materials can develop or ex- hibit a nonuniform space-charge distribution. Such a situa- tion can be found, for instance, in dielectrics stressed by an applied electric field. In this case, this may lead to dramatic effects and eventually to breakdown. In order to understand the phenomena involved and consequently to take the appro- priate countermeasures, it has been shown 1 that a nonde- structive direct measurement of the space-charge distribution in the material is of great help. Such measurements are car- ried out with optical, acoustical, and thermal probes. They are generally made with simple structures such as coaxial or plane geometries. However, it has been noticed that many poorly understood phenomena take place in heterogeneous regions. In these regions the electric field, the properties of the insulator, or both have a strong dependence on the posi- tion. This is the case, for instance, when the electrodes have irregularities or when the material contains impurities, water trees, or some gradient of chemical structures or physical properties. 2–6 Different approaches have been proposed in order to mea- sure the space charge in such complex field distributions. In a first approach, a unidimensional technique determines the charge distribution through the sample thickness at a given position on the sample surface after which the position is changed and a new measurement is performed. When the sample surface has been completely scanned, a three- dimensional space-charge distribution can be recons- tructed. 6–10 Unfortunately, this technique is slow and due to diffraction the lateral resolution is poor as compared to the resolution in thickness. In another approach, the Kerr effect is used to measure the field distribution in two 11 or three 12 dimensions. This technique gives good results in the case of fluid insulators, but can hardly be applied to diffusive or nontransparent materials. In this paper we demonstrate that it is possible to obtain the space-charge distribution, in isotropic media, in more than one dimension with an equivalent resolution in any di- rection by using either of the two different techniques. One technique relies on the pressure-wave-propagation method. An elastic wave propagating through the sample dis- places the trapped charges, which induces a current in a low- impedance-measuring circuit. This current is essentially pro- portional to the product of the charge amplitude and the displacement integrated over space. 13 In the case of a pulse shaped plane wave, the signal is proportional to the space- charge amplitude integrated over the wave front. Since the same space-charge distribution can be observed from various angles of incidence of the plane wave, a tomographiclike measurement of the charges can be performed. Utilizing a proper algorithm with the tomographiclike measurement makes it possible to retrieve the multidimensional space- charge distribution. The second technique relies on the electroacoustic method. A variation of the electric field in the insulator pro- duces a modification of the Coulombic force acting on each charge, which in turn acts as the source of elastic waves. Propagating waves proportional to the amplitude of each charge are thus generated at their position. 13 The time for each elastic wave to reach an observation point depends on the location of the initiating charge. The resulting wave am- plitude measured at this point at each time is the superposi- tion of the waves radiated from all the sources. In the case of an electric field varying as a short pulse, the wave sources in the sample detected at a given time are those for which the traveling time requested to reach the observation point is the same. By detecting the waves at a range of observation PHYSICAL REVIEW B, VOLUME 64, 104106 0163-1829/2001/6410/1041068/$20.00 ©2001 The American Physical Society 64 104106-1