ALGORITHM FOR ACCURATE CAPACITANCE MATRIX MEASUREMENTS OF THE MULTICONDUCTOR STRUCTURE FOR VLSI INTERCONNECTIONS

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1 ALGORITHM FOR ACCURATE CAPACITANCE MATRIX MEASUREMENTS OF THE MULTICONDUCTOR STRUCTURE FOR VLSI INTERCONNECTIONS Lech ZNAMIROWSKI Olgierd A. PALUSINSKI DEPARTMENT OF AUTOMATIC CONTROL, DEPARTMENT OF ELECTRICAL ELECTRONICS AND INFORMATICS AND COMPUTER ENGINEERING Silesian University of Technology University of Arizona Ul. Akademicka 16 TUCSON, AZ 85721, USA GLIWICE, POLAND Tel.: (520) Tel.: Fax: (520) Fax: ABSTRACT The demand for computer simulation of multiconductor interconnections in VLSI structures necessitates that such structures should be modelled. Calculations of the model capacitance matrix from measurements, for the multiconductor transmission lines which are coupled interconnections in a VLSI structures, using the "two-terminal" capacitances indirect measurement procedure is strongly corrupted by the measuring errors. Sensitivity analysis and error propagation relations demonstrates it. The paper presents alternative method for direct capacitance measurements in the multiconductor structure using active separation of the capacitance network to increase the measurements accuracy. 1. INTRODUCTION The interconnections between parts of a system in high-speed VLSI silicon structures are working as strongly coupled multiconductor transmission lines. The computer simulation of multiconductor interconnection structures necessitates that such structures should be modelled. To find the parameters of that model, one may start by determining the capacitance matrix of the multiconductor system. The remaining parameters, such as inductance or line losses, are determined depending on the type of propagated waves and physical phenomena accompanying high speed transmission [1, 10, 12]. One approach to find the capacitance matrix as a per-unit-length (PUL) capacitance matrix, is based on geometrical data and the electrostatic properties of multiconductors strip lines [4, 7, 9, 10]. The second approach is based on measurements. The capacitance matrix can be determined indirectly basing on the laboratory data and calculations [14] or directly basing on the algorithm (presented in this paper) applying active separation of the capacitance network [6, 11] for accurate capacitance matrix measurements.

2 - 2 - Consider a structure with n signal conductors and a ground conductor (denoted by #0) with cross-section presented in Fig. 1. Fig. 1. The cross-section of n-conductors geometry with reference In order to determine the PUL "two-terminal" capacitance matrix T=[T ], let us at first consider capacitance coefficients matrix i.e. the Maxwell matrix C for multilayer nonhomogeneous dielectric and n conductors (and the ground) imbedded. We have a basic equation for the matrix C =[C ] definition [4]: where: Q = C V, (1) Q T = [Q 1, Q 2,..., Q n ] - is a PUL charges vector on the n-conductor transmission line, V T = [V 1, V 2,..., V n ] - is the corresponding potentials vector. The diagonal elements of the matrix C are called coefficients of capacitance and the off-diagonal elements are called the coefficients of electrostatic induction. Properties of the matrix C are discussed in [4]. Analysing the potential differences between conductors and using "two-terminal" capacitances T, one finds [7]: T T ii = = -C n j= 1 C for i. j, (2) These "two-terminal" capacitances T have simple interpretation presented in Fig. 2. The goal of presented algorithm is determining the matrix T entries from direct measurements. Fig. 2. Capacitances equivalent circuits for n conductors with reference

3 - 3 - Solutions of many problems such as some identification methods, the minimization of functionals, the summation of Fourier series with approximately specified coefficients, some techniques for semiconductor data extraction calculations or complex laboratory methods for indirect parameter estimation, strongly depend on algorithm solving the problem. In general case, it is possible to classify algorithms as ill-posed or correctly-posed depending on properties of the operator which is the fundamental of the algorithm basis according to Hadamard ill-posed operator equation definition [3]. Important method of approximate solution of ill-posed problems are regularizing algorithms [8]. In [14] was analyzed an error propagation in computing the Maxwell matrix C (the results of analysis are directly valid for "two-terminal" capacitance model i.e. the matrix T computing) of multiconductor transmmision line on surface of the wafer for VLSI interconnections using laboratory data. The measurements was realised indirectly for per-unit-length interconnections with the 2-D cross-section. As a conclusion it was pick out a great sensitivity of the results on measuring data especially for off-diagonal elements of the matrix model. Calculations of the capacitance model for multiconductor transmission line interconnections using laboratory data was simple solved, basing on measurements/calculations but the great sensitivity on the measuring data was observed (ill-posed problem). Using measurements to finding Maxwell or "two-terminal" matrices' elements requires extremely high accuracy of instrument and very carefully constructed measurement station for obtaining the acceptable results. In this paper is presented alternative algorithm for extract capacitances T, applying direct measurement method using instrumental unity-gain amplifier for n-conductor interconnections structure and a ground conductor, which fulfills active separation of the capacitance network to increase the measurements accuracy. Results of this measurements can be used in different areas of VLSI interconnections characterization [7, 13] and simulation [1, 2, 5, 12]. The Maxwell matrix C can be determined using Eq. (9) (see 3.2), basing on accurate measurements of the T matrix. 2. ALGORITHM 2.1. Notation For convenience of the algorithm description (the conductors numbered from #1 to #n, and ground conductor denoted by #0) we'll use for element T the following notation: a) for i=j: T jj T 0 j and j=1, 2,... n. b) for i j : the indexes of the T fulfill the inequality i<j (see 3.2) and i=0, 1,... n-1, j=1, 2,... n Unity-gain Voltage Amplifier The Unity-Gain voltage Amplifier (UGA) with input pin IN, output pin OUT and a common point G, is presented in Fig. 3. The voltage gain of the UGA is equal unity, so the input voltage V IN =V OUT with very high input

4 - 4 - impedance of the UGA and very low output impedance. i and j are numbers of the conductors of the multiconductor structure connected respectively to the pins G and IN of the UGA. Simultaneously, the G and IN pins are connected to the input of the capacitance meter (pins B and A in Fig. 4). The UGA always fulfills active separation because the pins IN and OUT have the same potentials, and a part of a circuit connected to the pins G and OUT is charged from the output of the UGA. Fig. 3. Unity-gain voltage amplifier 2.3. Measurement Algorithm The algorithm for the measurements of the "two-terminal" T capacitances values with active separation which fulfills condition separately measured value in a multi-conductors system with ground conductor, has a following form: Algorithm MATRIX-T-MEASUREMENT input: n; output: matrix T; begin for i=0 to n-1 do for j=1 to n do if i<j then connect i-th conductor to the G pin of the UGA; connect j-th conductor to the IN pin of the UGA; finally, shunt together all remaining conductors and connect them to the OUT pin of the UGA; measure T ; using notation ( 2.1), find suitable entry of the matrix T; end. Let us use the graph representation of the muticonductor structure (Fig. 4, 5 and 6). The nodes of the graph represent the conductors with a ground conductor and branches (bolded lines), represent the sufficient capacitances. In a 2-conductors and ground conductor system (Fig. 4) in a case of measuring the T 12 capacitance, the conductor #2 connected to the IN pin of the UGA and the ground conductor #0 connected to the OUT pin of the UGA guarantees, that capacitor T 22 has electrodes at the same potentials and is not charged. The T 11 capacitor is charged from amplifier and in consequency, the measured capacitance T 12 is separated from the structure. The inputs of the capacitance meter (B and A) "sees" only isolated capacitance T 12. Similar properties have the measuring structures presented in Fig. 5 and Fig. 6, for exemplary 3-conductors and ground structure. In general case, for a n-conductors and ground system, realising the steps of the algorithm MATRIX-T-MEA-

5 - 5 - SUREMENT we get for T capacitance measurements, the equivalent circuit presented in Fig. 7. Fig. 4. Graph representation of T 12 capacitance measurement in a 2-conductors and ground system with active separation Fig. 5. Graph representation of T 12 capacitance measurement in a 3-conductors and ground system Fig. 6. Graph representation of T 11 (T 01 ) capacitance measurement in a 3-conductors and ground system The equivalent capacitance T 2eq is not charged, and equivalent capacitance T 1eq is charged from the amplifier. The T capacitance is separated from the circuit. 3. ACCURACY OF THE MEASUREMENTS 3.1. Accuracy of the T Matrix Measurements Let us assume that the relative error of a capacitance meter (compensated, full scale) is defined as δ. The measurements of the values T entries of the matrix T when the range if instrument is properly chosen, will be nearly the value δ, because the measurements are performed directly.

6 - 6 - Fig. 7. Equivalent circuit for T capacitance measurement with active separation in a n- conductors and ground system 3.2. Accuracy of the C Matrix Determined from the Direct Measurement of the Matrix T From the properties of matrix C and Eq. (2) we have: and C = C ji (3) T = T ji. (4) Thanks to the symmetry we need to consider (n 2 -n)/2+n relations instead of n 2 relations. To simplify the notation we use the following vector definitions C T = [C 11,C 12,C 13,C 14,... C 1n,C 22,C 23,C 24,... C 2n,C 33,C 34,C 35,... C 3n,C 44,... C nn ] = [C 1,C 2,C 3,... C N ] (5) and T T = [T 11,T 12,T 13,T 14,... T 1n,T 22,T 23,T 24,... T 2n,T 33,T 34,T 35,... T 3n,T 44,... T nn ] = [T 1,T 2,T 3,... T N ]. (6) Vectors C and T have N=(n 2 -n)/2+n=n(n+1)/2 entries, symbol T denotes transposition. Using relation (2) and symmetry that T =T ji, one may write: n T = ( δ -1 )C + δ Aik (7) k= 1 where ä = A ik = 1 for i = j 0 for i j, C ik for i k C ki for i > k. for i = 1, 2,..., n j = 1, 2,..., n and i j,

7 - 7 - Using relations (5) and (6) we have: T = DC (8) where matrix D is given by (7). Relation between the "two-terminal" vector T and the vector C representing elements of Maxwell matrix can be found using Eq. (8): C = D -1 T = GT (9) where G=D -1 - inverse matrix of D (N N). Elements of the matrix C are represented by vector C according with the relation (5). We can find [14] the absolute error in C i calculation (C i is i-th coordinate of the vector C): N Ci = G T j (10) where j=1 ÄT j - absolute error at measurement of the j-th element of the vector T. The relative error for coordinates of vector C has the form: From (9) one finds: Ci δ C i =. (11) C i C = G T (12) i i where G i - i-th row of the matrix G. Then using (10) and (11) we have: N C i 1 δ C i = = Gi j T j. C T (13) i G i j= 1 Using expression δ T j = T T j j for relative error of the j-th element of vector T, at last we have N 1 δ C = G ä T. i i j T j G T j (14) i j= 1 Relation (14) describes relative error in C i element calculation for given measurements of T j elements of vector T and their relative errors of measurements.

8 EXAMPLE I For a 4-conductors and ground system, connections of the active separation in the measurement circuit are presented in Figs. 8-17, accordingly with the algorithm MATRIX-T-MEASUREMENT. Fig. 8. T 11 (T 01 ) measurement in a 4-conductors and ground system Fig. 9. T 22 (T 02 ) measurement Fig. 11. T 44 (T 04 ) measurement Fig. 10. T 33 (T 03 ) measurement

9 - 9 - Fig. 12. T 12 measurement Fig. 13. T 13 measurement Fig. 14. T 14 measurement

10 Fig. 15. T 23 measurement Fig. 16. T 24 measurement Fig. 17. T 34 measurement 5. EXAMPLE II For a 6-conductor and ground system, the connections of the UGA fulfilling active separation during the measurements T matrix of multiconductor structure, are presented in Table 1.

11 Table 1 Connections of the UGA pins to the conductors for a 6-conductors and ground system Connections of the UGA pins to the conductors Measured capacitances G IN OUT T 11 (T 01 ) #0 #1 #2, #3, #4, #5, #6 T 22 (T 02 ) #0 #2 #1, #3, #4, #5, #6 T 33 (T 03 ) #0 #3 #1, #2, #4, #5, #6 T 44 (T 04 ) #0 #4 #1, #2, #3, #5, #6 T 55 (T 05 ) #0 #5 #1, #2, #3, #4, #6 T 66 (T 06 ) #0 #6 #1, #2, #3, #4, #5 T 12 #1 #2 #0, #3, #4, #5, #6 T 13 #1 #3 #0, #2, #4, #5, #6 T 14 #1 #4 #0, #2, #3, #5, #6 T 15 #1 #5 #0, #2, #3, #4, #6 T 16 #1 #6 #0, #2, #3, #4, #5 T 23 #2 #3 #0, #1, #4, #5, #6 T 24 #2 #4 #0, #1, #3, #5, #6 T 25 #2 #5 #0, #1, #3, #4, #6 T 26 #2 #6 #0, #1, #3, #4, #5 T 34 #3 #4 #0, #1, #2, #5, #6 T 35 #3 #5 #0, #1, #2, #4, #6 T 36 #3 #6 #0. #1, #2, #4, #5 T 45 #4 #5 #0, #1, #2, #3, #6 T 46 #4 #6 #0, #1, #2, #3, #5 T 56 #5 #6 #0, #1, #2, #3, #4 6. CONCLUSIONS The algorithm for accurate "two-terminal" matrix measurements (per-unit-length) of the multiconductor transmission lines for VLSI interconnections model has been presented. The algorithm uses active separation approach significantly increasing the accuracy of measurements in contrary to indirect determining the capacitance matrix. Furthermore, the Maxwell matrix computing from T matrix can be determined with significantly smaller error propagation than in the case of the indirect measurements of the matrix T entries. The presented algorithm depending on principle of measurement performed with the instrument connected to the pins B and A (higher frequency signal, pulse excitation), can be applicated for extended characterization of the

12 multiconductor transmission lines in VLSI interconnections modeling. Important application in this area is accurate measurement such phenomena as dielectric absorption [12] which plays significant role in high speed structures and thermal effects. 7. REFERENCES [1] F. Y. Chang, "Transient Analysis of Lossless Coupled Transmission Lines in a Nonhomogeneous Dielectric Medium," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-18, No. 9, September 1970, pp [2] A. R. Djordjevic, T. K. Sarkar, R. F. Harrington, "Time-Domain Response of Multiconductor Transmission Lines," Proc. of the IEEE, Vol. 75, No. 6, June 1987, pp [3] J. Hadamard, Le problème de Cauchy et les équations aux dérivée partielles lineaires hyperboliques, Hermann, Paris [4] D. W. Kammler, "Calculation of Characteristic Admittances and Coupling Coefficients for Strip Transmission Lines," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-16, No. 11, November 1968, pp [5] O. A. Palusinski, J. C. Liao, J. L. Prince, A. C. Cangelaris, "Simulation of Transients in VLSI Packaging Interconnections," IEEE Trans. on Comp., Hybrid, and Man. Technology, Vol. 13, No. 1, March 1990, pp [6] K. Reiss, "Accurate Measurement of Small Coupling Capacitance under Influence of Large Ground Capacitances," (Private communication), University of Karlsruhe, Karlsruhe [7] A. E. Ruehli (Ed.), Circuit Analysis, Simulation and Design. Advances in CAD for VLSI, Vol. 3, Part 2, North- Holland, Amsterdam [8] A. N. Tikhonov, A. V. Goncharsky (Eds.), Ill-Posed Problems in the Natural Sciences, Mir Publishers, Moscow [9] W. T. Weeks, "Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface," IEEE Trans. on Microwave Theory and Techn., Vol. MTT-18, No. 1, January 1970, pp [10] C. Wei, R. F. Harrington, J. R. Mautz, T. K. Sarkar, "Multiconductor Transmission Lines in Multilayered Dielectric Media," IEEE Trans. on Microwave Theory and Techniques, Vol. 32, No. 4, April 1984, pp [11] R. Zielonko, A. Krolikowski, Diagnostic-Measurement Methods of Analog Electronic Circuits (in Polish), Wydawnictwa Naukowo-Techniczne, Warsaw [12] L. Znamirowski, "Dielectric Absorption Model Identification for Microstrip Lines in VLSI Structures," Mixed Design of Integrated Circuits and Systems - Proceedings of the 4-th Int. Workshop, MIXDES'97, June, Poznan [13] L. Znamirowski, "Scaling of Coupled Multiconductor Interconnections in VLSI/ULSI Structures," Mixed Design of Integrated Circuits and Systems - Proceedings of the 5-th Int. Conference, MIXDES'98, June, Lodz [14] L. Znamirowski, O. A. Palusinski, "Analysis of Error Propagation in Computing the Maxwell Matrix of Multiconductor Structure from Measurements," IEE Proceedings - Science, Measurement and Technology, Vol. 145, No. 3, May 1998.