Causal Modeling and Extraction of Dielectric Constant and Loss Tangent for Thin Dielectrics

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1 Causal Modeling and Extraction of Dielectric Constant and Loss Tangent for Thin Dielectrics A. Ege Engin 1, Abdemanaf Tambawala 1, Madhavan Swaminathan 1, Swapan Bhattacharya 1, Pranabes Pramanik 2, Kazuhiro Yamazaki 2 1 Packaging Research Center, School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta GA 2 Oak-Mitsui Technologies, LLC, Hoosick Falls, New York Abstract New dielectric materials are being used for reducing electromagnetic interference (EMI) and improving signal integrity (SI). Examples include using high dielectric constant materials for decoupling and thin dielectrics for managing return currents. As the frequency of the signals being propagated through such materials increases, the frequency dependent material properties become very important. We present a method to extract the frequency-dependent dielectric constant and loss tangent of such materials using rectangular power/ground planes. We have also developed a rapid plane solver for fast extraction of material properties and a causal modeling methodology based on the vector fitting algorithm. I. INTRODUCTION Power/ground planes in electronic packaging have been typically used as a low impedance path from the integrated circuit (IC) to the SMD capacitors on the board. As the power consumption of the ICs and the frequency content of the signals increase, the impedance of the power/ground planes becomes increasingly important. Using a thinner dielectric improves the performance of the planes by decreasing the impedance. Also, a higher dielectric constant increases the charge capacity of the planes, which can help to provide charge to switching circuits or manage return currents at high frequencies. In a printed circuit board (PCB) the impedance of the planes, hence the electrical properties of the dielectric material, may not be very important at high frequencies for core decoupling. Looking from the IC, their performance is limited by the package inductance. However, for I/O decoupling, these properties become very important, since a lowimpedance plane improves the return path discontinuities (for example due to vias) of high-speed signals that exist also at the PCB level. In addition, some of these thin materials can be placed in the package, extending the frequency range of their effectiveness also for core decoupling. A high-k dielectric can also be used to design electromagnetic structures such as filters or electromagnetic bandgap (EBG) structures using a smaller area. Due to these reasons, high-frequency material properties of such thin dielectrics are required. To accurately estimate the performance of thin high-k dielectrics, the frequency-dependent dielectric constant and the loss tangent have to be extracted. One method for dielectric characterization is to use microstrip or ring gap resonators [1], [2]. This method is not suitable for thin high-k materials as the coupling through the gap becomes negligible. A time-domain reflectometry (TDR) based technique is described in [3] using a stripline, which requires additional efforts for fabrication as two sheets of the same dielectric are required for a stripline configuration. This procedure generates causal models for dielectrics using an iterative fitting of the attenuation and phase constant of the stripline. Parallel-plate resonators have been used previously for materials characterization [4], based on analytical equations for the resonance frequency and the quality factor. A similar method, the full-sheet resonance (FSR) technique, has been applied recently [5]. These techniques are based on the measurement of just a rectangular plane pair, hence they provide an easy way for dielectric characterization. Another method based on the same principles, which is called the corner-to-corner plane probing method, has been presented in [6]. This method requires only a 2-layered structure as compared to the 4-layered structure in [5], hence it is simpler in terms of fabrication of the structure. In addition, measurements can be done without introducing any via holes inside the plane pair. In this paper we extract the material properties using the corner-to-corner plane probing method. The extraction is done using a rapid solver, as opposed to analytical equations in [4] and [5], improving the accuracy of the extraction for lossy substrates. In order to demonstrate the sensitivity of the extraction method based on measurements on a power/ground plane pair, a square-shaped plane pair having 33.3mm on a side is considered. The plates are formed by using 35um thick copper planes. The nominal values of the dielectric assumed are: Dielectric constant (DK)=4, loss tangent (DF)=.25, and thickness is 6um. Fig. 1 shows the simulated variation of the magnitude of the transfer impedance between two ports that are placed on diagonally opposite corners, as the dielectric constant changes. Any variation in DK mainly changes the location of the resonance frequencies, whereas DF and thickness mainly influence the amplitude of the response. It can be inspected from this result that the response of a plane pair is very sensitive to changes in the materials properties. Based on the measurement of a plane pair, the dielectric constant and loss tangent can be extracted using analytically derived equations for the resonant frequencies. However, these equations become inaccurate when lossy conductors /7/$ IEEE

2 12 1 DK=4 Nominal Plus 1% Minus 1% Ground Vias y Port Port 1 (a) x Ground via (b) x 1 9 Fig. 1. Variation of the frequency response of a plane pair with respect to dielectric constant, showing the sensitivity of the method. and substrates are considered. An accurate extraction of the material parameters can be achieved by fitting electromagnetic simulation data to measurement data. The general procedure for the extraction of the material parameters using an electromagnetic solver and measurements on a resonant structure such as a plane pair can be summarized as follows: 1) Measurement of the test structure using a vector network analyzer (VNA). 2) Initial guess for the frequency-dependent dielectric constant and loss tangent. 3) Simulation of the test structure using an electromagnetic (EM) solver. 4) Comparison of the measurement results with simulation data especially at resonant frequencies. 5) Going back to step 3 with a refined guess of the material properties until there is a sufficient match between measurement and simulation results. As the fitting process requires a lot of iterations, the efficiency of the simulator becomes very important. We apply a rapid solver [6], which provides a very fast method for simulation of such rectangular plane pairs. After extracting the dielectric constant and loss tangent, a causal model representing the permittivity needs to be developed for modeling purposes. In this paper we show how a Debye model can be generated using the vector fitting method [7]. We also apply the methodology on a well-known material, FR4, for verification of the results. II. TEST VEHICLE DESIGN AND EXTRACTION METHODOLOGY Fig. 2 shows the test vehicle design and the placement of the measurement probes for the corner-to-corner plane probing method. In this method, the measurements are taken by placing the ports on opposite corners of the board, as shown in the figure. Air coplanar probes of GSG type can be used to make Fig. 2. (a) Top view, and (b) cross section of a power/ground plane pair on the test vehicle. (c) Placement of a GSG probe on one corner the measurements. The signal tip of the probe touches the plane island in the middle, whereas the ground tips of the probe touch the surrounding plane which is shorted to the bottom plane with vias all around. This design ensures that there are no via holes in the power/ground plane pair that is to be measured. As a result of this the material properties can be extracted very accurately. In the extraction process, several points should be kept in mind for fast convergence to the correct material properties: 1) The dielectric constant mainly influences the location of the resonance frequencies. 2) The loss tangent mainly changes the amplitude of the frequency response. The main influence of the loss tangent can once again be observed at the resonant frequencies. 3) Care should be taken for accurate estimation of the dielectric thickness, as it influences the amplitude of the frequency response similar to the loss tangent. A good way of estimating the thickness (without the need for a cross-section) is at low frequencies, where the loss tangent can be extracted directly assuming the planes act purely as a capacitor. 4) The transfer impedance is the preferred frequency response to use, as it is less prone to measurement noise. The input impedance is affected by the probe inductance, which is not the case for the transfer impedance. In this way, frequency-dependent dielectric constant and loss tangent can be extracted at discrete frequencies, which can later on be fitted to a smooth function as it will be shown later in this paper. For broadband extraction of dielectric properties, low frequency data can be obtained by using an LCR-meter. At intermediate frequencies, the planes can be considered as a parallel-plate capacitor and measured using a VNA [8]. (c)

3 Assuming a parallel-plate capacitor model, dielectric constant and loss tangent extraction is straightforward. At high frequencies, the resonances need to be taken into account using an electromagnetic solver as summarized in the above list. To ensure continuous interpolation between the various frequency points, plane pair cavities of different sizes can be used to cover the entire frequency range as well. The most time consuming part of the extraction process is the EM simulation. Next section presents a fast solver that can be used for such simulations. III. RAPID SOLVER FOR A RECTANGULAR PLANE PAIR The underlying elliptic partial differential equation for the modeling of planes is a Helmholtz equation ( 2 T + k 2) u = jωµdj z, (1) where 2 T is the transverse Laplace operator parallel to the planar structures, u is the voltage, d is the distance between the planes, k is the wave number, ω is the angular frequency, and J z is the current density injected normally to the planes. Problem definition is completed by assigning homogenous Neumann boundary conditions, which correspond to assuming a magnetic wall, or an open circuit, on the periphery of the planes. One method to solve the Helmholtz equation is by applying the finite-difference scheme. The 2-dimensional Laplace operator can be approximated as 2 T u i,j =(u i,j 1 + u i,j+1 + u i 1,j + u i+1,j 4u i,j ) /h 2 (2) within an error in the order of O(h 2 ), where h is the mesh length and u i,j is the voltage at node (i, j) for the cell-centered discretization. Applying this 5-point approximation on the Helmholtz equation yields u i,j 1 + u i,j+1 + u i 1,j + u i+1,j 4u i,j + h 2 k 2 u i,j = h 2 jωµdj z, (3) which gives a system of linear equations when applied to all nodes on the plane. Obtaining the frequency response of a plane pair requires the solution of a large sparse linear system that is obtained by applying (3) on all nodes on the plane and considering the boundary conditions. For the general case of a plane pair with an arbitrary shape, this involves the solution of a linear equation system [9]. For solid rectangular planes, however, this solution can be obtained using a rapid method based on the Fourier transform [1]. Applying the discrete cosine transform to (3) yields h 2 jωµdĵz û m,n = (2 cos πm I + 2 cos πn J 4+h2 k 2 ), (4) where û m,n is the discrete cosine transform of u i,j. Hence the procedure for solving the Helmholtz equation for rectangular planes using the discrete cosine transform can be summarized as: Compute the discrete cosine transform of the right hand side ( h 2 jωµdĵz). Compute û m,n from (4). Inverse transform û m,n to obtain the solution u i,j. The discrete cosine transform ensures that homogeneous Neumann boundary conditions (i.e., open circuit boundary conditions) hold at the boundaries. This rapid solution method does not require the solution of a linear equation system. Hence it provides a very efficient way of analyzing solid rectangular planes. As an example, more than a 5X speedup in simulation time could be obtained using the rapid solver compared to the conventional solution using a sparse matrix solver for a 5x5 discretization. This method can also be extended to planes with arbitrary contours using the segmentation method or the capacitance matrix method. IV. EXTRACTION OF MATERIALS PROPERTIES A high-k material with a thickness of 14um was characterized using the corner-to-corner plane probing method together with the rapid solver. The frequency-dependent dielectric constant and loss tangent of this high-k material was extracted in three steps: Using an LCR meter, parallel-plate capacitor approximation, and the rapid solver. In the frequency range of 1KHz-1MHz, DK and DF was extracted using an LCR meter. A. Parallel-Plate Capacitor Approximation In the frequency range of 1MHz-1MHz, DK and DF was extracted using a VNA and a parallel-plate capacitor model. For this approach a small capacitor is desired so as to increase the resonant frequency. In this example, a small parallel-plate having a square shape of 4.4mm on a side was used. Based on a parallel-plate model, the input impedance (Z11) can be used to obtain the capacitance (C) and conductance (G) as: Z 11 = R + jωl + 1 G + jωc, (5) where R, L, G, and C represent the the probe resistance, probe inductance, dielectric conductance, and capacitance, respectively. Fig. 3 shows the correlation of this model with measurements. As expected, this model can only capture the first resonance of the frequency response. The probe parasitics (R and L) were extracted from measured Z11 in the frequency range between the first resonance frequency and the first anti-resonance frequency (from about 2GHz to 4GHz), as the behavior in this frequency range is mainly inductive. In this frequency range, it was assumed that the third term of the sum in (5), which represents the capacitance of the dielectric, is negligible compared to the first two terms. R was extracted as 2mOhm and L was extracted as 37pH. C and tan δ (where G = ωc tan δ) were then fitted to the measurement results around the resonance frequency as 335pF and.3 using an iterative process. These parameters can be most easily extracted at low frequencies below the first resonance frequency.

4 Measurement RLGC model.25.2 Good fit around 1GHz DF is different at 2GHz Measurement Rapid solver mag(z11) [Ohm] x 1 9 Fig. 3. Parallel-plate capacitor approximation: Measured and modeled input impedance of a square-shaped parallel-plate capacitor with 4.4mm on a side Fig. 4. Comparison between the rapid solver and measurement results (DK=27.6 at 1GHz, DF=.3) B. Application of the Rapid Solver In the frequency range of 1MHz-1GHz and beyond, DK and DF can be extracted using a VNA and the rapid solver. In this approach, a larger parallel-plate is used, which had dimensions of 33.3mm x 33.3mm for this example. The dielectric constant and loss tangent were fitted at the resonance frequencies by running multiple simulations with the rapid solver. Around those resonance frequencies, it was assumed that the loss tangent was approximately constant in a limited bandwidth. A minimum-phase (hence causal) function that has these properties is given by ε(ω) =as 2δ/π, (6) where a is an arbitrary positive constant, s is the Laplace variable, and tan δ is the loss tangent [11]. Transfer impedance (Z12) was used to conduct the parameter extraction, since it is not affected by the probe inductance and resistance. Fig. 4 shows the comparison between the rapid solver and the magnitude of measured Z12, where the dielectric properties were specified as DK=27.6 at 1GHz and DF=.3 to the rapid solver. It can be seen that at around 1GHz, the rapid solver matches excellently with the measurements. At higher frequencies, there is more attenuation in the measurements. Hence, the loss tangent seems to be increasing with frequency, and another fit is required around that frequency. In this way, the DK and DF were extracted for all resonance frequencies of the structure up to 1GHz. V. CAUSAL MODEL DEVELOPMENT An important property of the complex permittivity function is that it should be consistent with the Kramers-Kronig relations. One of the models that satisfies this condition is the Debye model. In this section, the extracted parameters for DK and DF at different frequencies are used to develop a Debye model using the vector fitting method. The Debye model is given as: K c i ε(ω) =ε +. (7) s a i=1 i In order for this function to be realizable in terms of an RC circuit, the coefficients c i should be positive whereas a i should be negative numbers. Such a realizable function would be minimum-phase, and hence satisfy the Kramers-Kronig relations. The high frequency asymptote of this function can be found as ε. The resulting Debye model can be used to obtain a causal function that best represents the measurement data. Fig. 5 shows the results of the Debye model obtained using the vector fitting algorithm. The algorithm was started with 5 real poles. A higher number of poles could give a better fit. However, as an example, starting the algorithm with 6 poles resultedinsomenegativec i or positive a i coefficients, making the resulting model not realizable using an RC network with positive elements. VI. CHARACTERIZATION OF FR-4 For verification of the method, the dielectric properties of a well-known PCB material, FR-4, were also extracted. A square-shaped parallel-plate capacitor with a length of 22.2mm on a side was used for this example. Fig. 6 shows the comparison between the rapid solver and the measurements. The extracted dielectric thickness was 98um. Dielectric constant and loss tangent were extracted as 4.4 and.25 at 1GHz. For the frequency-dependent behavior of these parameters, (6) was used. Since the loss tangent of FR-4 is relatively flat in the shown frequency range, using (6) provided a good agreement with measurement results. The extracted parameters for FR-4 match well with the expected properties of this material. VII. CONCLUSION As the integrated circuit (IC) technology is evolving to make smaller and faster switching circuits possible, advanced

5 DF DK Debye Vectfit (a) Debye Vectfit (b) Fig. 5. Comparison of the Debye model obtained through vector fitting with measurement results Rapid solver materials are also being developed that can be used at the package and printed circuit board (PCB) levels. Examples are thin dielectrics that can have low or high dielectric constants. Materials properties of such dielectrics at high frequencies become very critical for RF or high-speed digital signals. This paper presented a methodology to extract the dielectric constant and loss tangent of materials from S-parameter measurements using a rapid plane solver. In addition to the material properties, the dielectric thickness can also be extracted using this methodology. The extracted dielectric constant and loss tangent at discrete frequency points can then be fitted to a causal model that is consistent with Kramers-Kronig relations. This was achieved using the vector fitting method to obtain a higher order Debye model for the example of a high-k material. REFERENCES [1] R. K. Hoffmann, Handbook of Microwave Integrated Circuits. Artech House Microwave Library, [2] X. Fang, D. Linton, C. Walker, and B. Collins, Dielectric constant characterization using a numerical method for the microstrip ring resonator, Microwave and Optical Technology Letters, vol. 41, no. 1, pp , Apr. 24. [3] A. Deutsch, T.-M. Winkel, G. Kopcsay, C. Surovic, B. Rubin, G. Katopis, B. Chamberlin, and R. Krabbenhoft, Extraction of ε r(f) and tan δ(f) for printed circuit board insulators up to 3 GHz using the short-pulse propagation technique, IEEE Trans. Adv. Packag., vol. 28, no. 1, pp. 4 12, Feb. 25. [4] L. S. Napoli and J. J. Hughes, A simple technique for the accurate determination of the microwave dielectric constant for microwave integrated circuit substrates (Correspondence), IEEE Trans. Microwave Theory Tech., vol. 19, no. 7, pp , July [5] N. Biunno and I. Novak, Frequency domain analysis and electrical properties test method for PCB dielectric core materials, in DesignCon 23 East, Boston, MA, June 23. [6] A. E. Engin, A. Tambawala, M. Swaminathan, S. Bhattacharya, P. Pramanik, and K. Yamazaki, Dielectric constant and loss tangent characterization of thin high-k dielectrics using corner-to-corner plane probing, in Proc. Electrical Performance of Electronic Packaging, Scottsdale, AZ, Oct. 26, pp [7] B. Gustavsen and A. Semlyen, Rational approximation of frequency domain responses by vector fitting, IEEE Trans. Power Delivery, vol. 14, no. 3, pp , July [8] P. Muthana, M. Swaminathan, R. Tummala, P. Raj, E. Engin, L. Wan, D. Balaraman, and S. Bhattacharya, Design, modeling and characterization of embedded capacitor networks for mid-frequency decoupling in semiconductor systems, in IEEE International Symposium on Electromagnetic Compatibility, Chicago, Aug. 25. [9] A. E. Engin, K. Bharath, M. Swaminathan, M. Cases, B. Mutnury, N. Pham, D. N. de Araujo, and E. Matoglu, Finite-difference modeling of noise coupling between power/ground planes in multilayered packages and boards, in Proc. Electronic Components and Technology Conference, May 26. [1] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C : the art of scientific computing - 2nd ed. Cambridge University Press, 22. [11] A. E. Engin, W. Mathis, W. John, G. Sommer, and H. Reichl, Closedform network representations of frequency-dependent RLGC parameters, International Journal of Circuit Theory and Applications, vol. 33, pp , Nov x 1 9 Fig. 6. Comparison between the rapid solver and measurement results for FR4 (DK=4.4 at 1GHz, DF=.25)