Calculation of dielectric constant and loss of two-phase composites

Transcription

1 JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 6 15 MARCH 2003 Calculation of dielectric constant and loss of two-phase composites Chen Ang a) Materials Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania and Department of Physics, The University of Akron, Akron, Ohio Zhi Yu, Ruyan Guo, and A. S. Bhalla Materials Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania Received 26 August 2002; accepted 18 December 2002 The field distribution, dielectric constant, and loss in a two-phase composite, in which phase A is distributed inside a square matrix of phase B, have been calculated using the finite-element method FEM. The calculation was carried out by taking into account different shapes for phase A, such as circles, triangles, and rings with different sizes. The modeling by FEM in the dielectric composite indicates that the shape for phase A has an influence on the electric-field distribution. In the case of the triangular shape of phase A, the electric flux condensed on the sharp angles of phase A. In a particular case of a ring for phase A, the electric-field distribution in the composite was quite different from that of circles/triangles due to the shielding effect of flux, and hence the dielectric constant and loss are greatly changed. The calculated values are analyzed and compared with that from the empirical Licktenecher relation. The results indicate that the FEM method can reflect the change in the shape and size of the particles of a composite and provide more reasonable results than that from the empirical Licktenecher relation American Institute of Physics. DOI: / I. INTRODUCTION Field/frequency agile materials for microwave electronics have been extensively studied in recent years in order to meet the increasing demand of radar and telecommunication systems. It requires that a dielectric material possesses both a high dc electric-field tunability K (E 0 ) (E) / (E 0 ) and a low-dielectric loss at microwave frequencies. 1 6 One of the most promising bulk materials for this objective is the composites consisting of (Sr,Ba)TiO 3 solid solutions and some oxides with low-dielectric constant and low-dielectric loss. 7,8 For a practical application, the dielectric loss is expected to be as low as possible, in addition to the requirement for the low-dielectric constant ( several hundred) and a moderate tunability. It is possible to tailor the dielectric constant and loss of the materials by adding another phase with very low-dielectric constant and loss into (Sr,Ba)TiO 3 solid solutions to form a composite. 9 For example, in the composites consisting of (Sr,Ba)TiO 3 and MgO or MgTiO 3, the dielectric constant can be adjusted by changing the ratio of MgO or MgTiO 3, and the loss can also meet some practical needs. However, in many cases, the dielectric loss is unstable, which seems to be closely related to the processing conditions of the composites. Therefore, theoretical analyses and predications on the dielectric loss in a composite are expected. In the last three decades, there were a number of reports for modeling the dielectric constant in composite materials, however the modeling of dielectric loss (tan ) was less reported. One of the excellent work might be the modeling of the dielectric constant via the finite element method FEM a Electronic mail: angchen@physics.uakron.edu combined with the Monte Carlo simulation reported by Wakino et al. 10 Based on the modeling results in a binaryphase composite material, they suggested an empirical equation v1 v0 v 1 1 v1 v0 v 2 2 v1 v0. The authors pointed out that Eq. 1 provides better agreement with the experimental data than all other equations reported in the literature. By taking into consideration the random distribution generated by the Monte Carlo simulation, the most possible average value was obtained, and it was close to the real situation in a material, and hence a most reasonable result was obtained. However, in their work, only the dielectric constant was calculated, and no dielectric loss tan was calculated. In recent years, Sareni et al. 11 have published a series of articles on the modeling of both dielectric constant and loss tan by FEM and the boundary integral equation method. They studied a period binary-phase composite material i.e., the phase A has a period distribution in matrix phase B, and reported modeling results of both and tan. However, for a random distribution of phase A in matrix phase B, only the dielectric constant was calculated. Details of the modeling of the dielectric constant and loss in composites can be found in an excellent review article. 12 A combination of FEM and the Monte Carlo method, which is expected to be close to the real case and to give better results. In this article, we report our preliminary results for the modeling of the dielectric constant and loss of composites that consists of phase A with different shapes circles and triangles distributed in a square matrix phase B. A simplified case of one circle or triangle inside a square matrix was /2003/93(6)/3475/6/$ American Institute of Physics

2 3476 J. Appl. Phys., Vol. 93, No. 6, 15 March 2003 Chen et al. FIG. 1. Contour of calculated electric-field E distribution from dark to light: E increases and calculated electric-field distribution along the central line of the contour from the bottom to the top in the composite consisting of one circle of phase A inside a square matrix phase B, with the area ratio of 28:72. a Phase A, 1 7; phase B, b Phase A, 1 80; phase B, 2 7. calculated for the total dielectric constant and field distribution of the composites. Further calculation on the total dielectric constant and field distribution was carried out when phase A is divided into smaller particles, while keeping the same ratio of phase A to phase B. A special case for a ring inside a square was also calculated. The FEM calculated results are analyzed and compared with that of the Licktenecher relation. II. CALCULATION METHOD: PHYSICAL MODEL AND FINITE-ELEMENT METHOD 10,11 The calculation model is based on two preconditions for the composite: 1 The composite consists of two homogeneous phases A and B, with phase A as an inclusion, which is distributed in the host matrix phase B. Phase B is in a square, and phase A has different shapes, such as circles, triangles, and rings. The composite with two-side electrodes is recognized as a parallel plate capacitor. 2 We suppose that the scale of heterogeneities in the composite is small as compared to the wavelength and skin depth within the material. In this case, the material medium is recognized as a homogeneous material, which is a quasistatic approximation. The potential distribution in a substance is described by the Laplace s equation u 0, where u is a potential distribution in a spatial domain. By dividing the domain into triangular finite elements and for each element, the calculation is carried out by interpolation of the potential u and its normal derivative u/ n with the corresponding nodal values u j j u j, u/ n j j u/ n j, where j denotes interpolation functions, which can be generally approximated as the following form: 1 2 3

3 J. Appl. Phys., Vol. 93, No. 6, 15 March 2003 Chen et al FIG. 2. Contour of calculated electric-field E distribution from dark to light: E increases and calculated electric-field distribution along the central line of the contour from the bottom to the top in the composite consisting of one triangle of phase A inside a square matrix phase B, with the area ratio of 28:72. a Phase A, 1 7; phase B, b Phase A, 1 80; phase B, 2 7. c 1 c 2 x c 3 y c 4 x 2 c 5 xy c 6 y 2. 4 For dielectric constant : The electrostatic energy for each element W e k 1/2 Sk k u/ x 2 u/ y 2 dx dy, where k is the dielectric constant of the kth element, and S k is the surface of the kth triangular element. For dielectric loss tan : The loss energy for each element e k 1/2 Sk k tan k u/ x 2 u/ y 2 dx dy, 6 where is the angular frequency of the electric field, and tan k is the dielectric loss of the kth element. A field calculation package flux2d Magsoft Corporation with the FEM calculation technology has been used to solve the Laplace s equation. The electric-field distribution, dielectric constant, and loss were calculated by assuming 1 volt of ac voltage signal being applied on the composite. 5 III. RESULTS AND DISCUSSION A. One circleõtriangle inside a square matrix with a fixed ratio A simplified two-phase composite was calculated, in which phase A in the shape of a circle or triangle is distributed in a square matrix, phase B, with a dimension of 5 5 mm 2. The of the phase A is about ten times higher or lower than that of the matrix phase B, and the ratio of phase A to phase B area or volume is 28:72. The electric-field distribution and total dielectric constant ( total ) of the composite were calculated by FEM and are shown in Figs. 1 and 2. The calculated total values of the composite are listed in Table I. The field distribution in Figs. 1 and 2 shows that the electrical field was mainly applied on the component with smaller. High-electric field is concentrated in the vicinity of the triangle corners, as shown in Fig. 2. It can be seen that the calculated is different from that calculated from the Licktenecher relation, ln v 1 ln 1 v 2 ln 2. According to the relation, for example, in case 1, the total is 40.4, different from the value of 49.2 obtained by

4 3478 J. Appl. Phys., Vol. 93, No. 6, 15 March 2003 Chen et al. FIG. 3. Contour of calcualted electric-field E distribution from dark to light; E increases and calculated electric-field distribution along the central line of the contour from the bottom to the top in the composite consisting of three triangles of phase A inside a square matrix phase B, by keeping the same ratio 28:72 of phase A to phase B. a Phase A, 1 7; phase B, b Phase A, 1 80; phase B, 2 7. FEM. In addition, there is no difference between different shapes of cells phase A for the same ratio of the two phases according to the Licktenecher relation. From the FEM calculation, total varies for different shapes of phase A, as shown in Table I. If we compare it with that obtained from the Licktenecher relation, total shows a deviation of 18% and 11% for the cases of circle and triangle, respectively. The deviation is 7.5% if we compare the triangle shape of phase A case 3 with that of the circle shape of phase A case 1 of the composite that phase A has a lower-dielectric constant than that of phase B. For Cases 2 and 4, the deviation of total is 21% and 11% as compared with that of the Licktenecher relation. The deviation is 8.1% between the shapes of the circle and triangle. This indicates that the current calculation by the FEM gives more precise data than that of the Licktenecher relation. B. Three circlesõtriangles inside a square matrix The influence of the size of phase A on the total dielectric constant and the field distribution in the composite was studied by dividing phase A into smaller particles, while keeping the same ratio 28:72 of two phases. The dielectric constant and field distribution were calculated for three circles/triangles for phase A inside the square matrix B. The calculated total of the composite is listed in Table I. The results show that the calculated for the composite consisting of three circles for phase A is almost the same as that from the one circle of phase A. Further calculation on decreasing the particle size, like 5, 7, and 9 circles of phase A, shows a similar result, i.e., the difference between different particle sizes is within the calculation error for the same ratio phase A to B (28:72). However, for three triangles inside the square, the calculated total is different from that obtained in Sec IIIA for the case of one triangle. The deviation is 4.4% and 7.5% for cases 3 and 4, respectively. The field distribution in the composites is shown in Fig. 3. Obviously, for phase A with a triangle shape, the electric field being concentrated in the vicinity of the triangle corners is the main reason why the

5 J. Appl. Phys., Vol. 93, No. 6, 15 March 2003 Chen et al TABLE I. The calculated total of the composite consisting of phase A one/three circle/triangle/ring distributed inside a square matrix phase B. The ratio of phase A to phase B is 28:72. The results obtained from the Licktenecher relation are also shown for comparison. Case Phase A Phase B total one particle for phase A total three particles for phase A total Licktenecher relation Figure Shape 1 2 square 1 circle Fig. 1 a 2 circle Fig. 1 b 3 triangle Fig. 2 a 4 triangle Fig. 2 b 5 ring Fig. 3 a 6 ring Fig. 3 b triangle shape has difference, but the circles have no noticeable change in the total of the composite when the size of phase A changed. This implies a more precise calculation by the FEM method than that of the Licktenecher relation, which cannot reflect the change in the size and shape of phase A if the ratio remained the same. D. One ring inside a square matrix with a fixed ratio In some cases, phase A may be a ring inside phase B. We also computed the total dielectric constant and the field distribution in the composite with this structure, by keeping the same ratio 28:72 of phase A to phase B. C. Dielectric constant and loss in a composite with one triangle inside a square matrix with varied ratios In this section, we calculated and tan of the composite that consists of one triangle inside a square matrix, with varied ratios x of phase A ( 1 7, tan ) to phase B ( 2 80, tan ). When x changes from 5% 35%, the calculated and tan versus x is plotted in Fig. 4. It can be seen that and tan decrease almost linearly with increasing x below x 25%. But as x increases further and near 35% 40%, and tan quickly deviate from the linear behavior, and a percolationlike behavior occurs. A question is raised, Is this percolation-like behavior true? With a careful inspection of the field distribution in the composite, it seems that the quick drop in both and tan might be originated from the calculation errors in the FEM in this configuration, because at these ratios in this configuration, two angles of the triangle are very near the edge of the square matrix, and this may cause the calculation errors. Further calculations must be modified to solve this problem. FIG. 4. Calculated and tan of the composite consisting of one triangle of Phase A ( 1 7, tan ) inside a square matrix B ( 2 80, tan ) as a function of the ratio x of phase A to phase B. FIG. 5. Calculated electric-field distribution contour from dark to light: E increases in the composite consisting of one ring of phase A inside a square matrix phase B, with the area ratio of 28:72. a Phase A, 1 7; phase B, b Phase A, 1 80; phase B, 2 7.

6 3480 J. Appl. Phys., Vol. 93, No. 6, 15 March 2003 Chen et al. The field distribution in the composite is shown in Fig. 5. The calculated of the composite is shown in Table I. Compared with the results obtained for the circle/triangle, the calculated for the composite consisting of a ring inside the square is quite different. The differences are 19% and 23% between the two case, respectively. The obvious difference in and tan is originated from the change in the electric-field distribution in the composite due to the shielding effect of the electric flux. When the ring with higher inside the square matrix with lower Fig. 5 a. The flux is shielded beyond the ring, and hence the dielectric constant is increased. In the opposite case, the flux condenses on the ring area, and this leads to a lower dielectric constant Fig. 5 b. It is obviously a very interesting way to construct some special structures to tailor the materials in order to obtain desirable dielectric constant, high tunability, and low loss. This structure is also reminiscent of the so-called grain-boundary or core-shell structure capacitors. Our calculation on the ring structure here might be a very suitable scenario for it. V. CONCLUSIONS In this preliminary study, the field distribution, dielectric constant, and loss in a composite consisting of phase A with different shapes and sizes inside a square matrix of phase B have been computed using the finite-element method. The following results are obtained: 1 Compared with the empirical Licktenecher relation, the present calculation on and loss is more precise and reasonable, because this method calculated the stored electrostatic energy and energy loss based on the modeling of the electric-field distribution in the composite. 2 Compared with phase A with a circle shape, the electric field is condensed on the angle area for phase A with a triangle shape, and hence the dielectric constant and loss are different although they have the same area ratio. 3 For the ring shape of phase A, the electric-field distribution in the composite is greatly changed due to the shielding effect of flux, and hence the dielectric constant and loss are obviously changed. These results indicate that the FEM is a useful method to compute the dielectric properties of composites. Especially, based on the modeling of different shapes and sizes of phase A and B, some special structures can be designed in order to obtain desirable dielectric properties, for example, high-dc bias tunability and low loss. This method may be also used to calculate the dielectric properties of the so-called grainboundary or core-shell structure capacitors. However, further work on implementing the random distribution of phase A in the matrix phase B is needed. ACKNOWLEDGMENTS The authors would like to thank Dr. L. E. Cross for his stimulating discussion. The authors C.A. and Z.Y. also acknowledge useful comments by Dr. P. Gujrati. This work was supported by DARPA under FAME and MetaMaterials projects. 1 O. G. Vendik, Ferroelectrics 12, ; O. G. Vendik, I. G. Mironenko, and L. T. Ter-Martirosyan, Microwaves RF 33, D. Galt, J. Price, J. A. Beall, and R. H. Ono, Appl. Phys. Lett. 63, ; D. Galt, J. Price, J. A. Beall, and T. E. Harvey, IEEE Trans. Appl. Supercond. 5, A. B. Kozyrev, T. B. Samoilova, A. A. Golovkov, E. K. Hollmann, D. A. Kalinikos, V. E. Loginov, A. M. Prudan, O. I. Soldatenkov, D. Galt, C. H. Mueller, T. V. Rivkin, and G. A. Koepf, J. Appl. Phys. 84, F. W. Van Keuls, R. R. Romanofsky, D. Y. Bohman, M. D. Winters, F. A. Miranda, C. H. Mueller, R. E. Treece, T. V. Rivkin, and D. Galt, Appl. Phys. Lett. 71, Y. Liu, B. Acikel, A. S. Nagra, T. R. Taylor, P. J. Hansen, J. S. Speck, R. A. York, Integr. Ferroelectr. 39, A. Chen, A. S. Bhalla, Ruyan Guo, and L. E. Cross, Appl. Phys. Lett. 76, ; 78, ; A. Chen, A. S. Bhall, and L. E. Cross, Phys. Rev. B 64, F. W. Van Keuls, C. H. Mueller, R. R. Romanofsky, J. D. Warner, F. A. Miranda, H. Jiang, Integr. Ferroelectr , C. H. Mueller, F. W. Van Keuls, R. R. Romanofsky, F. A. Miranda, J. D. Warner, C. L. Canedy, R. Ramesh, Integr. Ferroelectr , D. Garcia, R. Y. Guo, and A. S. Bhalla, Ferroelectr. Lett. Sect , K. Wakino, T. Okada, N. Yoshida, and K. Tomono, J. Am. Ceram. Soc. 76, B. Sareni, L. Krahenbuhl, A. Beroual, and C. Brosseau, J. Appl. Phys. 80, ; 80, ; 81, ; C. Brosseau, A. Beroual, and A. Boudida, a 88, E. Tuncert, Y.V. Serdyuk, and S. M. Gubanski, arxiv: cond-mat/ , 14 Nov 2001.