Yiorgos Bontzios 1 , Michael G. Dimopoulos 2 , Alkis Hatzopoulos 1

1. A GA-based Method for Efficient Interconnect Capacitance Computation in Mixed-Signal Integrated Circuits Using Sets of Linear Charges Yiorgos Bontzios1, Michael G. Dimopoulos2, Alkis Hatzopoulos1 1Dep. of Electrical & Computer Eng., Aristotle Univ. of Thessaloniki, Thessaloniki, Greece 2Dept. of Electronics, Alexander Technological Educational Inst. of Thessaloniki, Thessaloniki, Greece

2. A GA-based Method for Efficient Interconnect Capacitance Computation in Mixed-Signal Integrated Circuits Using Sets of Linear Charges • Introduction • Problem Formulation • The proposed GA-Based Method • Simulation Results • Discussion and conclusions PresentationOutline

3. Motivation:Various techniques for calculating the interconnect capacitive coupling have been proposed • The Finite Element (FEM) and Boundary Element (BEM) methods. • But they suffer from increased memory requirements and simulation time. • The modeling of the capacitance coupling with a small number of lumped capacitances. • But their range of validity is limited to the specific type of problem that was solved.

4. Outline of the work • A Genetic Algorithm (GA) based method is introduced to overcome the disadvantages of the above methods in the capacitive coupling modeling. • The proposed method is based on the well known method of images in electromagnetism. • A set of linear charges (LC) is used in every case to produce the equipotential surface of the conductor. • The optimal placing of the charges is computed utilizing a GA.

5. A GA-based Method for Efficient Interconnect Capacitance Computation in Mixed-Signal Integrated Circuits Using Sets of Linear Charges • Introduction • Problem Formulation • The proposed GA-Based Method • Simulation Results • Discussion and conclusions

6. Equipotential surfaces two equal linear charges four equal linear charges

7. Preliminaries (1) • The main idea of the proposed method is based on the method of images, which states: • The EM field remains the same in a region of space as long as we keep the sources and the boundary conditions the same. • So, a set of charges may be placed in a space region in a way that one of the produced equipotential surfaces, will match the outer surfaces of the given conductors • The conductors may be removed and the initial problem is translated to a problem of charges, which can be explicitly solved.

8. Preliminaries (2) • The space inside every conductor is discretized to a finite number of cells. • For each one of the given conductors, a set of M points SP along its surface is selected. The SP are selected so to better “track” the given conductor surface. • More SPs are selected near the corners or at the nearest faces between the given structures and a lower number at the rest of the structure. • A set of N LCs is placed inside every structure and the potential produced at each SPi is computed

9. Problem metric • The main requirement for all the SPs is to have the same potential i.e. to belong to the same equipotential surface. To follow this requirement the RMS metric is introduced: • where:

10. Problem formulation • The interconnect capacitance computation problem may be formulated in our case as follows: • For a given set M of SP, place a set of N LC inside the conductor such that: Minimize RMS • The final capacitance value is computed by its definition formula: • where Qtot is the total net charge inside the conductor and φ1-φ2 is the potential difference between the two conductors

11. A GA-based Method for Efficient Interconnect Capacitance Computation in Mixed-Signal Integrated Circuits Using Sets of Linear Charges • Introduction • Problem Formulation • The proposed GA-Based Method • Simulation Results • Discussion and conclusions

12. Algorithm Steps (1) • Step1. Randomly construct the initial population of members or individuals . Each individual corresponds to a feasible solution. • Step2. The fitness value for each individual is computed. • Step3. Population is sorted in order to find the individual with the minimum fitness value. • Step4. Application of the selection method. The individuals are copied into the mating pool according to their fitness value (roulette wheel selection), and they are combined randomly.

13. Algorithm Steps (2) • Step5. Generation of offsprings and application of the genetic operators to each offspring in the new population with a specific probability. • Step6. The initial population is replaced by the new population and steps 2–6 are repeated until the termination criteria are met, that is the minimization of the fitness function or the maximum number of generations.

14. Chromosome encoding • The chromosome is defined as a N-size vector with N being the number of the LC used. • Each vector element (gene) of the chromosome is a triplet (xloc, yloc, charge) • xloc, yloc are the coordinates that define the location of the LC • charge is the normalized charge value {-1,+1}.

15. Mutation Operators • Three different mutation operators are used: • Single mutation: A random cell from the unoccupied ones is selected as the candidate place for a randomly selected LC to be placed into. • Double mutation: Two LC are randomly chosen and relocated to random unoccupied cells. • Part mutation: A vector (a subset) of LC is randomly selected and relocated to randomly selected unoccupied cells.

16. Fitness function • The individuals are ranked according to certain evaluation rules, which form the so called fitness function: • Vthres is a threshold factor which defines the maximum allowable percentage variance of the potential of each SPi from its normalized potential difference • Spen is a penalty value used for any SPi whose potential value exceeds the above boundaries.

17. A GA-based Method for Efficient Interconnect Capacitance Computation in Mixed-Signal Integrated Circuits Using Sets of Linear Charges • Introduction • Problem Formulation • The proposed GA-Based Method • Simulation Results • Discussion and conclusions

18. Simulation setup • The proposed algorithm has been implemented in C++. • All the test cases correspond to typical structures and arrangements of interconnects encountered in practice. • The results have been compared to the simulation data obtained by a commercial simulator based on the FEM method. • Since the results do not depend on absolute values, but rather to relative ones, they are presented in normalized units (per unit, pu) of length.

19. Performance of the proposed algorithm Fitness [pu] Number of cells Performance of the proposed algorithm as a function of its parameters for different number n={2,4,5,6} of charges

20. Interconnects with different cross-sections [Ω] Capacitance [10-12 F/m] Offset distance (d) Simulation data for the coupling capacitance between a square cross section and a rectangular cross section interconnect with offset distance

21. Interconnects with offset distance Capacitance [10-12 F/m] Offset distance (d) Simulation data for the coupling capacitance between two interconnects of equal cross section with offset distance

22. Comparison with PTM and Comsol Capacitance [10-12 F/m] Separation (s) Verification of the proposed method with the benchmark interconnect geometry from the Predictive Technology Model and with simulation data

23. A GA-based Method for Efficient Interconnect Capacitance Computation in Mixed-Signal Integrated Circuits Using Sets of Linear Charges • Introduction • Problem Formulation • The proposed GA-Based Method • Simulation Results • Discussion and conclusions

24. Discussion and Conclusions • A method to deal with the problem of the interconnect capacitance computation in mixed-signal integrated circuits is presented in this work. • Comparative simulation results are presented against a commercial tool. The capacitance values as computed by the proposed method are in close agreement with the results obtained by the commercial tool with the average difference lying in the range of 2%~5%. • One intrinsic property of the proposed method is that it can be easily parallelized and there are several straightforward ways for achieving this. • Work is under way to implement a parallel version of the proposed method, to extend the current method to the general 3D capacitance computation case and also to apply the current method to more complex structures.

25. A GA-based Method for Efficient Interconnect Capacitance Computation in Mixed-Signal Integrated Circuits Using Sets of Linear Charges Yiorgos Bontzios, Michael G. Dimopoulos, Alkis Hatzopoulos Thanks a lot for your kind attention