- Email: [email protected]
Validation of a novel dielectric constant simulation model and the determination of its physical parameters
Microelectronics Journal 33 (2002) 627–632 www.elsevier.com/locate/mejo
Validation of a novel dielectric constant simulation model and the determination of its physical parameters Michael G. Todd*, Frank G. Shi The Henry Samueli School of Engineering, University of California, Irvine, CA 92697-2575, USA Received 20 February 2002; revised 2 April 2002; accepted 4 April 2002
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Abstract
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Keywords: Dielectric constant; Composite; Effective medium theory; Rule of mixtures
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Precise determination of the dielectric constant of microelectronic packaging materials is critical for the performance optimization of high frequency devices. Accurate empirical data, however, is difficult to obtain and often not available for these complex composite materials. In order to provide a basis for predictive model verification, dielectric analysis of a series of model microelectronic packaging materials was conducted. Physical characteristics of the composite constituents, including the morphology of the dispersed phase as well as the dielectric constant of the dispersed and host phases, were used to define the indeterminate variables. The results of these analyses provide a systematic verification of a newly developed physical model for predicting the effective dielectric constant of complex composite systems. The model considers an interphase zone surrounding each dispersed particle having unique physical and electrical characteristics. A physical interpretation is presented to explain the indeterminate variables incorporated in this model. q 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
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Dielectric properties of packaging materials significantly influence the performance of high-speed microelectronic devices. Electrical characteristics of the microelectronic devices, such as signal attenuation, propagation velocity and cross talk, are influenced by the dielectric properties of the package substrate and encapsulation material [1]. Microelectronic packaging materials having a low dielectric constant minimize capacitive coupling effects and reduce signal delay. Materials having a low dielectric loss reduce signal attenuation. Optimization of both properties is critical to provide superior high frequency device performance [2]. Polymer based microelectronic encapsulation materials are generally comprised of micron-scale inorganic particulates mixed in a thermosetting polymer matrix. The most common types of these materials are formed by the dispersion of near-spherical shaped amorphous silica (SiO2) particles within a highly crosslinked epoxy or epoxy copolymer matrix (see Fig. 1). Polymer based
* Corresponding author. Address: Henkel-Loctite Corporation, 15051 East Don Julian Road, Industry, CA 91746, USA. Tel.: þ 1-626-968-6511; fax: þ 1-626-336-526. E-mail address: [email protected] (M.G. Todd).
microelectronic substrate materials are generally comprised of woven glass– fiber reinforcements mixed with a thermosetting polymer matrix. The most common types of these materials are formed by the impregnation of woven E-glass fibers into a brominated epoxy resin (FR4) or an epoxy – bismaleimide copolymer resin (BT) (see Fig. 2). In both cases, the volume percent of filler in the polymer matrix may vary from 0 to approximately 70%. Predicting the effective dielectric constant of such composite materials has proven to be difficult. Clausius and Mossotti performed some of the earliest work on the dielectric properties of composite materials. They derived independently a mean-field theory for a disordered system of polarizable spheres. Since this pioneering work, this subject has been the focus of intense research, with the extension of the effective medium theory (EMT) [3,4] as well as the development of other theoretical approaches, such as those based on boundary integral equations (BE theory) [5] and on Dissado –Hill’s cluster theory [6]. In most cases, composites are statistical mixtures (randomly dispersed systems) of two or more components. The true dielectric constant value of a statistical composite should lie between the values of the individual components. The most commonly used equation for predicting the dielectric constant of a two component composite system is
0026-2692/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 6 9 2 ( 0 2 ) 0 0 0 3 8 - 1
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Fig. 3. Interphase region surrounding filler particles.
Lichtenecker suggested a logarithmic law of mixing equation (Eq. (2)) to better emulate the non-linear characteristics observed in composite systems. 1c ¼ n1 log 11 þ n2 log 12
ð2Þ
h þ 2l 1c ¼ h2l "
ð13 2 12 Þð12 2 11 Þ a3 ð213 þ 12 Þð212 þ 11 Þ b3 ! ð13 2 1Þð13 2 12 Þ b3 22 ð13 þ 2Þð213 þ 12 Þ c3
ð4Þ !
h¼ 1þ2
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More recently, advancements in composite dielectric prediction have been realized through the use of advanced composite theories. For example, Rao et al. proposed an EMT model (Eq. (3)) for predicting the dielectric constant of composite polymer – ceramic materials [3]. In their model, Rao et al. included a correction factor, n, to compensate for the shape of the particulate fillers used in polymer –ceramic composites. A small value of n indicates filler particles having a near-spherical shape, while a high value of n indicates a largely non-spherically shaped particle. n is not a directly measurable parameter, however, and requires empirical determination. n1 ð11 2 12 Þ 1c ¼ 12 1 þ ð3Þ 12 þ nð1 2 n1 Þð11 2 12 Þ
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1c ¼ n1 11 þ n2 12
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the simple volumetric law of mixing as shown in Eq. (1) where 1c is the dielectric constant of the composite, 11 and 12 are the dielectric constant of the filler and matrix, respectively, and n1 and n2 are the volume fractions of the filler and matrix, respectively.
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Fig. 1. Polymer–ceramic composite.
dielectric constant of composite systems based on ‘interphase’ characteristics [7]. This model is the first that relies on the physical properties of an interphase zone that develops between the matrix and the filler phases having its own unique physical and electrical characteristics. In this model, the composite is comprised of a filler volume phase, n1, interphase zone, n2, and matrix volume phase, n3. Depending upon the dielectric properties of the interphase region, the composite material may have an effective dielectric constant different from that of the characteristics of the filler or matrix alone. Furthermore, this model, unlike other EMT models, accounts for non-linear composite characteristics due to particle – particle overlap caused by proximity at high filler loading (see Fig. 3). The Vo model is given by Eq. (4).
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Finally, Vo et al. proposed a new theory for predicting the
Fig. 2. Polymer-woven glass composite.
!# ð13 2 1Þð13 þ 212 Þð12 2 11 Þ a3 22 ð13 þ 2Þð213 þ 12 Þð212 þ 11 Þ c3 " !# ð13 2 1Þ ð213 þ 1Þm b3 j2 l¼ ð13 þ 2Þ ð13 þ 2Þð213 þ 12 Þ c3 " !# ð13 2 12 Þð12 2 11 Þ a3 j¼ 1þ2 ð213 þ 12 Þð212 þ 11 Þ b3 " !# ð13 þ 212 Þð12 2 11 Þ a3 m ¼ ð13 2 12 Þ þ ð212 þ 11 Þ b3 a3 ð1 þ kn1 Þ a3 ; ¼ ¼ n1 ; ð1 þ kÞ b3 c3 b3 ð1 2 n1 Þ ¼ n 1 þ k 1 ð1 þ kn1 Þ c3
There are two indeterminate variables incorporated in these equations: k represents a correction for the relative volume of the interphase region. This variable should, therefore, be related to the surface area of the dispersed phase as well as the interactive bonding between the matrix (host) phase and
M.G. Todd, F.G. Shi / Microelectronics Journal 33 (2002) 627–632
Sample D
52.25 47.34 0 0.42
41.80 37.87 20 0.33
26.12 23.67 50 0.21
15.67 14.2 70 0.13
the filler phase. 12 represents the dielectric constant of the interphase region. This variable should be related to the dielectric constant of the matrix (host) phase under structured conditions. The means for determining these two variables would provide the packaging engineer with a method to design and determine the dielectric constant of new composite packaging materials. In order to evaluate the efficacy of these models and to determine the actual values of the various correction factors involved in these models, a series of model polymer – ceramic composite materials were formulated. By varying the filler concentration, filler particle size (shape factor) and polymer composition, a matrix of tests was designed to determine the ability of each of the aforementioned models to predict actual test data.
2. Experimental
3. Results and discussion
Dielectric constant frequency sweep spectrum were produced for each sample. An example of the test output is shown in Fig. 4 below. The frequency scans clearly show a trend of decreasing dielectric constant as a function of increasing test frequency. Due to the scope, further analysis and explanation of this trend is the subject of another publication. Table 3 lists the dielectric constant values for each of the samples measured at 1 MHz. This data clearly shows a trend towards increasing
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Model epoxy-based polymer –ceramic composite formulations were produced having filler concentrations ranging from 0 to 70%. Table 1 lists the formulation details for each sample. The diglycidyl ether epoxy resin used in these experiments has an average epoxy equivalent weight of 183. The epoxy curing agent used was methylhexahydrophthalic anhydride (MHHPA). The near-spherical amorphous silica filler used was synthetically produced with an average particle diameter of approximately 4.5 mm. The stoichiometry of these formulations was maintained at approximately 98%. Each formulation was mixed to thoroughly disperse the filler particles within the resin matrix using a dual planetary shaft stainless steel mixer assembly. The formulations were thoroughly degassed in a vacuum chamber at , 1 Torr pressure. Each sample was then packaged into HDPE containers and stored at 2 40 8C to preserve their workability. Next, a corresponding sample to #D described above was prepared having a modified epoxy polymer matrix to
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Sample C
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Epoxy MHHPA Silica Catalyst
Sample A
determine the effect of the dielectric constant of the matrix on the dielectric constant of the model composite compositions, independent of filler changes. This sample, sample #E, was comprised of an identical epoxy polymer matrix modified with 0.5 wt% silane and the same volume percent and size of filler as that of sample #D. The only difference between these compositions was the silane modifier incorporated in the epoxy matrix. Finally, a corresponding sample to #E described above was prepared having a different average particle diameter filler to determine the effect of filler size and surface area on the dielectric constant of the model composite compositions. The near-spherical amorphous silica filler used in this sample has an average particle diameter of approximately 3.5 mm. This sample, sample #F, was comprised of an identical epoxy polymer matrix modified with 0.5 wt% silane and the same volume percent of filler as that of sample #E. The only difference between these compositions was the average particle diameter of the ceramic (silica) filler. Table 2 lists the formulation details for each of these samples. Each sample was next cast into disk-shaped molds having a diameter of 150 mm and a thickness of 3.2 mm for dielectric testing. Each cured disk was thoroughly baked prior to dielectric testing. The samples were analyzed for dielectric constant and dissipation factor over a frequency range of 1 MHz to 1 GHz using an Agilent 4291B parallel plate dielectric analyzer.
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Table 1 Formulation details of composite samples A– D
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Table 2 Formulation details for composite samples E and F Sample E
Epoxy MHHPA Silica Catalyst
Resin modified
Sample F 15.67 14.2 70 0.13
Resin modified Finer particle size
15.67 14.2 70 0.13
Fig. 4. Dielectric constant frequency scan.
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Table 3 Dielectric constant of composite samples—measured at 1 MHz
A B C D – E F
0 20 50 70 100 70 70
2.9728 3.0416 3.2081 3.3975 3.80 3.3264 3.3446
Fused silica—literature value Resin modification Filler size/resin modification
Fig. 6. Dielectric constant as a function of filler content—EMT model.
This value of n is approximately 10 £ the value obtained by Rao et al. to fit their experimental data for a polymer – ceramic composite system of lead magnesium niobate – lead titanate filler dispersed in an epoxy matrix. While this model does accurately fit the experimental data, a means to independently determine the value of the correction factor, n, must be developed in order to use this model to effectively predict the dielectric constant of unknown systems. Fig. 7 shows the experimental data fit to the model proposed by Vo et al. As described earlier, this model requires the determination of two independent correction factors, k and 12. When k is set to zero, that is when the interphase region is eliminated, the Vo model degenerates to a simple law of mixtures equation. As shown in Fig. 7, there are many combinations of the two correction factors, k and 12, that were found to closely fit the experimental data. Fig. 8 shows the interdependence of k and 12 for this system. The relationship follows a second order exponential decay function (Eq. (5)): 2ðK 2 K0 Þ 12 ¼ 10 þ A1 exp t1 2ðK 2 K0 Þ þ A2 exp t2
ð5Þ
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dielectric constant as a function of increasing filler content of the composite. This is expected for composites in which the dielectric constant of the filler is greater than that of the polymer matrix. Furthermore, this data shows that the dielectric constant of the highly filled polymer composite is significantly reduced through the addition of a silane to the epoxy polymer matrix. Finally, this data suggests that, all other factors being constant, a reduction in the particle size of the ceramic filler in the polymer matrix (corresponding to an increase in filler surface area) effectively increases the dielectric constant of the composite. The dielectric constant values recorded at 1 MHz for samples A – D (shown in Fig. 5) illustrate the effect of filler concentration on the effective dielectric constant of the composite system. Also shown are the volumetric law of mixtures (Eq. (1)) and logarithmic law of mixtures models (Eq. (2)) fit to the experimental data. While the logarithmic model more closely predicts the non-linear trend expected, neither model sufficiently fits the experimental data. Clearly a more advanced model is required to accurately predict the dielectric constant of these polymer –ceramic composite materials. The dielectric constant values recorded at 1 MHz for samples A – D are again shown in Fig. 6. In this case, however, the data has been fit to the EMT model (Eq. (3)) proposed by Rao et al. A value for the correction factor, n, of 1.10 was found to closely fit the experimental data. Rao et al. offer no explanation as to the significance of the value of n.
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Dielectric constant
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Filler (wt%)
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Sample
Fig. 5. Dielectric constant as a function of filler content—volumetric mixture models.
Fig. 7. Dielectric constant as a function of filler content—Vo model.
M.G. Todd, F.G. Shi / Microelectronics Journal 33 (2002) 627–632
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Fig. 8. Valid combinations of k and 12 for Vo model. Fig. 9. Results of Eq. (5) used in Vo model to fit experimental data.
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found for each sample. The value of k corresponding to sample F is approximately 24% greater than that of sample E, which corresponds closely with the theoretical difference of 29%. Through this process, a single real value of both k and 12 were determined for samples E and F. Eq. (5) was used to similarly determine singular real values of k and 12 for samples A –D. A summary of the results of this work is shown in Table 4. These values of k and 12 were substituted into Eq. (4) to confirm the validity of the results. Fig. 9 shows the results of this work; the values of k and 12 determined from Eq. (5) provided a very good fit to the experimental data, with x 2 values ranging from 7.8 £ 1026 to 7.2 £ 1025.
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This decay function may be used to determine any valid combination of k and 12 to fit the experimental data. As the interphase region grows in volume (larger k ), the dielectric constant value of the interphase region diminishes, eventually reaching a value of the polymer phase itself, 2.97, which correlates perfectly with the experimental data from sample A. Realize, of course, that the value of 12 should not fall outside the dielectric constant values of the filler or the polymer matrix. Therefore, the real value 12 are limited to the range of 2.97– 3.80 as shown in Fig. 8. This data, however, is insufficient to positively determine the single real value of k and 12. Therefore, the experimental dielectric constant values of samples E and F were each fit to the Vo model. Using Eq. (5), valid combinations of k and 12 were determined for each system. Finally, since the only difference in composition between sample E and sample F is the ceramic filler particle size, the following conclusions are made:
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1. The composition of the polymer matrix and the composition of the ceramic filler are identical in samples E and F; therefore the value of the dielectric constant of the interphase, 12, is the same for each sample. 2. The ceramic filler particle size of sample E and F is different; therefore the filler surface area and corresponding interphase volume, k, is different for each sample. The filler surface area of sample F is calculated to be 29% greater than that of sample E based on the reported particle diameters. Therefore, the interphase volume, k, of sample F should be approximately 29% greater than that of sample E.
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Consequently, by setting the value of 12 in Eq. (5) for samples E and F equal to each other, a single value of k was
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Table 4 Calculated values of k and 12 for experimental data
k 12
4. Conclusions
The effective dielectric constant of a series of microelectronic packaging materials was fit to several numerical models developed to predict the dielectric constant of composite materials. Advanced numerical models based on EMT were shown to satisfactorily predict the experimental data when correction factors were empirically determined. A method was introduced to determine singular real solutions to the values of k and 12 in the Vo EMT model. The values of k for the experimental composite formulations were shown to correlate to the relative surface area of the dispersed phase of the composite. The values of 12 for the experimental formulations were shown to be proportional to the dielectric constant of the polymer matrix. Future work will reveal the precise relationship between the values of k and 12 and the physical/electrical characteristics of the individual phases of the composite.
Acknowledgments
Samples A –D
Sample E
Sample F
20.6335 3.138
20.6335 3.190
20.5128 3.190
This work at UCI is supported by Henkel-Loctite Electronic Materials. The authors would like to thank Hung Vo for his EMT model contributions, Michael Bell,
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Kathryn Costello and Armando Rodriguez for their formulation and characterization efforts, and Lee Fehr and Maury Edwards for their continued support of advancements in microelectronic packaging materials development.
[4]
[5]
References [6]
[7]
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[1] P. Garrou, I. Turlik, Multichip Module Technology Handbook, McGraw-Hill, New York, 1998. [2] P. Gonon, A. Sylvestre, J. Teysseyre, C. Prior, Dielectric properties of epoxy/silica composites used for microelectronic packaging, and their dependence on post-curing, Journal of Materials Science: Materials in Electronics 12 (2001) 81–86. [3] Y. Rao, J. Qu, T. Marinis, C.P. Wong, A precise numerical prediction
of effective dielectric constant for polymer–ceramic composite based on effective-medium theory, IEEE Transactions on Components and Packaging Technologies 23 (4) (2000). Q. Xue, Effective-medium theory for two-phase random composites with an interfacial shell, Journal of Materials Science and Technology 16 (4) (2000). B. Sareni, L. Krahenbuhl, A. Beroual, C. Brosseau, Ab initio simulation approach for calculating the effective dielectric constant of composite materials, Journal of Electrostatics 40 (1997). L. Dissado, R. Hill, A cluster approach to the structure of imperfect materials and their relaxation spectroscopy, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 390 (1798) (1983) 131– 180. H. Vo, F. Shi, Towards model-based engineering of optoelectronic packaging materials: dielectric constant modeling, Microelectronics Journal (2002) 33 (5) (2002) 409–415.
Validation of a novel dielectric constant simulation model and the determination of its physical parameters Michael G. Todd*, Frank G. Shi The Henry Samueli School of Engineering, University of California, Irvine, CA 92697-2575, USA Received 20 February 2002; revised 2 April 2002; accepted 4 April 2002
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Abstract
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Keywords: Dielectric constant; Composite; Effective medium theory; Rule of mixtures
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Precise determination of the dielectric constant of microelectronic packaging materials is critical for the performance optimization of high frequency devices. Accurate empirical data, however, is difficult to obtain and often not available for these complex composite materials. In order to provide a basis for predictive model verification, dielectric analysis of a series of model microelectronic packaging materials was conducted. Physical characteristics of the composite constituents, including the morphology of the dispersed phase as well as the dielectric constant of the dispersed and host phases, were used to define the indeterminate variables. The results of these analyses provide a systematic verification of a newly developed physical model for predicting the effective dielectric constant of complex composite systems. The model considers an interphase zone surrounding each dispersed particle having unique physical and electrical characteristics. A physical interpretation is presented to explain the indeterminate variables incorporated in this model. q 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
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Dielectric properties of packaging materials significantly influence the performance of high-speed microelectronic devices. Electrical characteristics of the microelectronic devices, such as signal attenuation, propagation velocity and cross talk, are influenced by the dielectric properties of the package substrate and encapsulation material [1]. Microelectronic packaging materials having a low dielectric constant minimize capacitive coupling effects and reduce signal delay. Materials having a low dielectric loss reduce signal attenuation. Optimization of both properties is critical to provide superior high frequency device performance [2]. Polymer based microelectronic encapsulation materials are generally comprised of micron-scale inorganic particulates mixed in a thermosetting polymer matrix. The most common types of these materials are formed by the dispersion of near-spherical shaped amorphous silica (SiO2) particles within a highly crosslinked epoxy or epoxy copolymer matrix (see Fig. 1). Polymer based
* Corresponding author. Address: Henkel-Loctite Corporation, 15051 East Don Julian Road, Industry, CA 91746, USA. Tel.: þ 1-626-968-6511; fax: þ 1-626-336-526. E-mail address: [email protected] (M.G. Todd).
microelectronic substrate materials are generally comprised of woven glass– fiber reinforcements mixed with a thermosetting polymer matrix. The most common types of these materials are formed by the impregnation of woven E-glass fibers into a brominated epoxy resin (FR4) or an epoxy – bismaleimide copolymer resin (BT) (see Fig. 2). In both cases, the volume percent of filler in the polymer matrix may vary from 0 to approximately 70%. Predicting the effective dielectric constant of such composite materials has proven to be difficult. Clausius and Mossotti performed some of the earliest work on the dielectric properties of composite materials. They derived independently a mean-field theory for a disordered system of polarizable spheres. Since this pioneering work, this subject has been the focus of intense research, with the extension of the effective medium theory (EMT) [3,4] as well as the development of other theoretical approaches, such as those based on boundary integral equations (BE theory) [5] and on Dissado –Hill’s cluster theory [6]. In most cases, composites are statistical mixtures (randomly dispersed systems) of two or more components. The true dielectric constant value of a statistical composite should lie between the values of the individual components. The most commonly used equation for predicting the dielectric constant of a two component composite system is
0026-2692/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 6 9 2 ( 0 2 ) 0 0 0 3 8 - 1
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Fig. 3. Interphase region surrounding filler particles.
Lichtenecker suggested a logarithmic law of mixing equation (Eq. (2)) to better emulate the non-linear characteristics observed in composite systems. 1c ¼ n1 log 11 þ n2 log 12
ð2Þ
h þ 2l 1c ¼ h2l "
ð13 2 12 Þð12 2 11 Þ a3 ð213 þ 12 Þð212 þ 11 Þ b3 ! ð13 2 1Þð13 2 12 Þ b3 22 ð13 þ 2Þð213 þ 12 Þ c3
ð4Þ !
h¼ 1þ2
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More recently, advancements in composite dielectric prediction have been realized through the use of advanced composite theories. For example, Rao et al. proposed an EMT model (Eq. (3)) for predicting the dielectric constant of composite polymer – ceramic materials [3]. In their model, Rao et al. included a correction factor, n, to compensate for the shape of the particulate fillers used in polymer –ceramic composites. A small value of n indicates filler particles having a near-spherical shape, while a high value of n indicates a largely non-spherically shaped particle. n is not a directly measurable parameter, however, and requires empirical determination. n1 ð11 2 12 Þ 1c ¼ 12 1 þ ð3Þ 12 þ nð1 2 n1 Þð11 2 12 Þ
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1c ¼ n1 11 þ n2 12
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the simple volumetric law of mixing as shown in Eq. (1) where 1c is the dielectric constant of the composite, 11 and 12 are the dielectric constant of the filler and matrix, respectively, and n1 and n2 are the volume fractions of the filler and matrix, respectively.
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Fig. 1. Polymer–ceramic composite.
dielectric constant of composite systems based on ‘interphase’ characteristics [7]. This model is the first that relies on the physical properties of an interphase zone that develops between the matrix and the filler phases having its own unique physical and electrical characteristics. In this model, the composite is comprised of a filler volume phase, n1, interphase zone, n2, and matrix volume phase, n3. Depending upon the dielectric properties of the interphase region, the composite material may have an effective dielectric constant different from that of the characteristics of the filler or matrix alone. Furthermore, this model, unlike other EMT models, accounts for non-linear composite characteristics due to particle – particle overlap caused by proximity at high filler loading (see Fig. 3). The Vo model is given by Eq. (4).
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Finally, Vo et al. proposed a new theory for predicting the
Fig. 2. Polymer-woven glass composite.
!# ð13 2 1Þð13 þ 212 Þð12 2 11 Þ a3 22 ð13 þ 2Þð213 þ 12 Þð212 þ 11 Þ c3 " !# ð13 2 1Þ ð213 þ 1Þm b3 j2 l¼ ð13 þ 2Þ ð13 þ 2Þð213 þ 12 Þ c3 " !# ð13 2 12 Þð12 2 11 Þ a3 j¼ 1þ2 ð213 þ 12 Þð212 þ 11 Þ b3 " !# ð13 þ 212 Þð12 2 11 Þ a3 m ¼ ð13 2 12 Þ þ ð212 þ 11 Þ b3 a3 ð1 þ kn1 Þ a3 ; ¼ ¼ n1 ; ð1 þ kÞ b3 c3 b3 ð1 2 n1 Þ ¼ n 1 þ k 1 ð1 þ kn1 Þ c3
There are two indeterminate variables incorporated in these equations: k represents a correction for the relative volume of the interphase region. This variable should, therefore, be related to the surface area of the dispersed phase as well as the interactive bonding between the matrix (host) phase and
M.G. Todd, F.G. Shi / Microelectronics Journal 33 (2002) 627–632
Sample D
52.25 47.34 0 0.42
41.80 37.87 20 0.33
26.12 23.67 50 0.21
15.67 14.2 70 0.13
the filler phase. 12 represents the dielectric constant of the interphase region. This variable should be related to the dielectric constant of the matrix (host) phase under structured conditions. The means for determining these two variables would provide the packaging engineer with a method to design and determine the dielectric constant of new composite packaging materials. In order to evaluate the efficacy of these models and to determine the actual values of the various correction factors involved in these models, a series of model polymer – ceramic composite materials were formulated. By varying the filler concentration, filler particle size (shape factor) and polymer composition, a matrix of tests was designed to determine the ability of each of the aforementioned models to predict actual test data.
2. Experimental
3. Results and discussion
Dielectric constant frequency sweep spectrum were produced for each sample. An example of the test output is shown in Fig. 4 below. The frequency scans clearly show a trend of decreasing dielectric constant as a function of increasing test frequency. Due to the scope, further analysis and explanation of this trend is the subject of another publication. Table 3 lists the dielectric constant values for each of the samples measured at 1 MHz. This data clearly shows a trend towards increasing
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Model epoxy-based polymer –ceramic composite formulations were produced having filler concentrations ranging from 0 to 70%. Table 1 lists the formulation details for each sample. The diglycidyl ether epoxy resin used in these experiments has an average epoxy equivalent weight of 183. The epoxy curing agent used was methylhexahydrophthalic anhydride (MHHPA). The near-spherical amorphous silica filler used was synthetically produced with an average particle diameter of approximately 4.5 mm. The stoichiometry of these formulations was maintained at approximately 98%. Each formulation was mixed to thoroughly disperse the filler particles within the resin matrix using a dual planetary shaft stainless steel mixer assembly. The formulations were thoroughly degassed in a vacuum chamber at , 1 Torr pressure. Each sample was then packaged into HDPE containers and stored at 2 40 8C to preserve their workability. Next, a corresponding sample to #D described above was prepared having a modified epoxy polymer matrix to
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Sample A
determine the effect of the dielectric constant of the matrix on the dielectric constant of the model composite compositions, independent of filler changes. This sample, sample #E, was comprised of an identical epoxy polymer matrix modified with 0.5 wt% silane and the same volume percent and size of filler as that of sample #D. The only difference between these compositions was the silane modifier incorporated in the epoxy matrix. Finally, a corresponding sample to #E described above was prepared having a different average particle diameter filler to determine the effect of filler size and surface area on the dielectric constant of the model composite compositions. The near-spherical amorphous silica filler used in this sample has an average particle diameter of approximately 3.5 mm. This sample, sample #F, was comprised of an identical epoxy polymer matrix modified with 0.5 wt% silane and the same volume percent of filler as that of sample #E. The only difference between these compositions was the average particle diameter of the ceramic (silica) filler. Table 2 lists the formulation details for each of these samples. Each sample was next cast into disk-shaped molds having a diameter of 150 mm and a thickness of 3.2 mm for dielectric testing. Each cured disk was thoroughly baked prior to dielectric testing. The samples were analyzed for dielectric constant and dissipation factor over a frequency range of 1 MHz to 1 GHz using an Agilent 4291B parallel plate dielectric analyzer.
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Table 1 Formulation details of composite samples A– D
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Table 2 Formulation details for composite samples E and F Sample E
Epoxy MHHPA Silica Catalyst
Resin modified
Sample F 15.67 14.2 70 0.13
Resin modified Finer particle size
15.67 14.2 70 0.13
Fig. 4. Dielectric constant frequency scan.
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Table 3 Dielectric constant of composite samples—measured at 1 MHz
A B C D – E F
0 20 50 70 100 70 70
2.9728 3.0416 3.2081 3.3975 3.80 3.3264 3.3446
Fused silica—literature value Resin modification Filler size/resin modification
Fig. 6. Dielectric constant as a function of filler content—EMT model.
This value of n is approximately 10 £ the value obtained by Rao et al. to fit their experimental data for a polymer – ceramic composite system of lead magnesium niobate – lead titanate filler dispersed in an epoxy matrix. While this model does accurately fit the experimental data, a means to independently determine the value of the correction factor, n, must be developed in order to use this model to effectively predict the dielectric constant of unknown systems. Fig. 7 shows the experimental data fit to the model proposed by Vo et al. As described earlier, this model requires the determination of two independent correction factors, k and 12. When k is set to zero, that is when the interphase region is eliminated, the Vo model degenerates to a simple law of mixtures equation. As shown in Fig. 7, there are many combinations of the two correction factors, k and 12, that were found to closely fit the experimental data. Fig. 8 shows the interdependence of k and 12 for this system. The relationship follows a second order exponential decay function (Eq. (5)): 2ðK 2 K0 Þ 12 ¼ 10 þ A1 exp t1 2ðK 2 K0 Þ þ A2 exp t2
ð5Þ
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dielectric constant as a function of increasing filler content of the composite. This is expected for composites in which the dielectric constant of the filler is greater than that of the polymer matrix. Furthermore, this data shows that the dielectric constant of the highly filled polymer composite is significantly reduced through the addition of a silane to the epoxy polymer matrix. Finally, this data suggests that, all other factors being constant, a reduction in the particle size of the ceramic filler in the polymer matrix (corresponding to an increase in filler surface area) effectively increases the dielectric constant of the composite. The dielectric constant values recorded at 1 MHz for samples A – D (shown in Fig. 5) illustrate the effect of filler concentration on the effective dielectric constant of the composite system. Also shown are the volumetric law of mixtures (Eq. (1)) and logarithmic law of mixtures models (Eq. (2)) fit to the experimental data. While the logarithmic model more closely predicts the non-linear trend expected, neither model sufficiently fits the experimental data. Clearly a more advanced model is required to accurately predict the dielectric constant of these polymer –ceramic composite materials. The dielectric constant values recorded at 1 MHz for samples A – D are again shown in Fig. 6. In this case, however, the data has been fit to the EMT model (Eq. (3)) proposed by Rao et al. A value for the correction factor, n, of 1.10 was found to closely fit the experimental data. Rao et al. offer no explanation as to the significance of the value of n.
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Fig. 5. Dielectric constant as a function of filler content—volumetric mixture models.
Fig. 7. Dielectric constant as a function of filler content—Vo model.
M.G. Todd, F.G. Shi / Microelectronics Journal 33 (2002) 627–632
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Fig. 8. Valid combinations of k and 12 for Vo model. Fig. 9. Results of Eq. (5) used in Vo model to fit experimental data.
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found for each sample. The value of k corresponding to sample F is approximately 24% greater than that of sample E, which corresponds closely with the theoretical difference of 29%. Through this process, a single real value of both k and 12 were determined for samples E and F. Eq. (5) was used to similarly determine singular real values of k and 12 for samples A –D. A summary of the results of this work is shown in Table 4. These values of k and 12 were substituted into Eq. (4) to confirm the validity of the results. Fig. 9 shows the results of this work; the values of k and 12 determined from Eq. (5) provided a very good fit to the experimental data, with x 2 values ranging from 7.8 £ 1026 to 7.2 £ 1025.
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This decay function may be used to determine any valid combination of k and 12 to fit the experimental data. As the interphase region grows in volume (larger k ), the dielectric constant value of the interphase region diminishes, eventually reaching a value of the polymer phase itself, 2.97, which correlates perfectly with the experimental data from sample A. Realize, of course, that the value of 12 should not fall outside the dielectric constant values of the filler or the polymer matrix. Therefore, the real value 12 are limited to the range of 2.97– 3.80 as shown in Fig. 8. This data, however, is insufficient to positively determine the single real value of k and 12. Therefore, the experimental dielectric constant values of samples E and F were each fit to the Vo model. Using Eq. (5), valid combinations of k and 12 were determined for each system. Finally, since the only difference in composition between sample E and sample F is the ceramic filler particle size, the following conclusions are made:
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1. The composition of the polymer matrix and the composition of the ceramic filler are identical in samples E and F; therefore the value of the dielectric constant of the interphase, 12, is the same for each sample. 2. The ceramic filler particle size of sample E and F is different; therefore the filler surface area and corresponding interphase volume, k, is different for each sample. The filler surface area of sample F is calculated to be 29% greater than that of sample E based on the reported particle diameters. Therefore, the interphase volume, k, of sample F should be approximately 29% greater than that of sample E.
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Consequently, by setting the value of 12 in Eq. (5) for samples E and F equal to each other, a single value of k was
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Table 4 Calculated values of k and 12 for experimental data
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4. Conclusions
The effective dielectric constant of a series of microelectronic packaging materials was fit to several numerical models developed to predict the dielectric constant of composite materials. Advanced numerical models based on EMT were shown to satisfactorily predict the experimental data when correction factors were empirically determined. A method was introduced to determine singular real solutions to the values of k and 12 in the Vo EMT model. The values of k for the experimental composite formulations were shown to correlate to the relative surface area of the dispersed phase of the composite. The values of 12 for the experimental formulations were shown to be proportional to the dielectric constant of the polymer matrix. Future work will reveal the precise relationship between the values of k and 12 and the physical/electrical characteristics of the individual phases of the composite.
Acknowledgments
Samples A –D
Sample E
Sample F
20.6335 3.138
20.6335 3.190
20.5128 3.190
This work at UCI is supported by Henkel-Loctite Electronic Materials. The authors would like to thank Hung Vo for his EMT model contributions, Michael Bell,
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Kathryn Costello and Armando Rodriguez for their formulation and characterization efforts, and Lee Fehr and Maury Edwards for their continued support of advancements in microelectronic packaging materials development.
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References [6]
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[1] P. Garrou, I. Turlik, Multichip Module Technology Handbook, McGraw-Hill, New York, 1998. [2] P. Gonon, A. Sylvestre, J. Teysseyre, C. Prior, Dielectric properties of epoxy/silica composites used for microelectronic packaging, and their dependence on post-curing, Journal of Materials Science: Materials in Electronics 12 (2001) 81–86. [3] Y. Rao, J. Qu, T. Marinis, C.P. Wong, A precise numerical prediction
of effective dielectric constant for polymer–ceramic composite based on effective-medium theory, IEEE Transactions on Components and Packaging Technologies 23 (4) (2000). Q. Xue, Effective-medium theory for two-phase random composites with an interfacial shell, Journal of Materials Science and Technology 16 (4) (2000). B. Sareni, L. Krahenbuhl, A. Beroual, C. Brosseau, Ab initio simulation approach for calculating the effective dielectric constant of composite materials, Journal of Electrostatics 40 (1997). L. Dissado, R. Hill, A cluster approach to the structure of imperfect materials and their relaxation spectroscopy, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 390 (1798) (1983) 131– 180. H. Vo, F. Shi, Towards model-based engineering of optoelectronic packaging materials: dielectric constant modeling, Microelectronics Journal (2002) 33 (5) (2002) 409–415.