1. Efficient Reluctance Extraction for Large-Scale Power Grid with High- Frequency Consideration Shan Zeng, Wenjian Yu, Jin Shi, Xianlong Hong Dept. Computer Science & Technology, Tsinghua University, Beijing 100084, China

2. Importance of Inductance Extraction for P/G Grid  More than thousand million transistors  Working frequency: multiple giga-hertz (GHz)  Power consumption increases exponentially  Capture the potential problems of power integrity  Accurate modeling and dynamic simulation of the power/ground (P/G) grid critical for VLSI circuit design and verification.

3. Importance of Inductance Extraction for P/G Grid  Modeling the inductive effect of on-chip and off-chip interconnects is another research focus for current nano-scale VLSI chip.  Conventional RC model is not enough  Resistance copper, capacitance Low-k material  Denser geometries, growing complexity of interconnect structures bring challenges to on-chip inductance modeling and extraction

4. Main Difficulty One major difficulty: unknown return path. The partial element equivalent circuit (PEEC) model The resulted inductance matrix is dense. Simply truncating would make the system unstable Prevents inductive modeling of large-scale interconnect structures, such as the P/G grid.

5. Introduction of K The reluctance matrix K is the inverse of inductance matrix L, introduced in [Devgan ICCAD ’ 00] (1) K has the locality similar to capacitance. Later works show circuit simulation has great advantage in both speed and accuracy. [Du ASP-DAC ’ 05] proved:  the sparsified partial reluctance matrix is positive definite  the circuit simulation is stable

6. Previous Works Considering High Frequency Effect [Luk ASP-DAC ’ 04]: necessity of considering high-frequency effect extension of double-inversion on DATE ’ 01. [Wei ICCCAS ’ 05]: extend to admittance at ultra high frequency obtain inductance and resistance. [Zhang ASP-DAC ’ 06]: direct extraction, combined with window technique avoid double-inversion computation We improved and reinforced through calculating frequency-dependent resistance in 2007

7. Structure Based Idea P/G grid extraction problem: large scale [Shi TCAD ’ 07]: a pattern idea to accelerate the DC simulation the geometry characteristics topology similarity to sub-matrix regularity. divided the whole P/G grid into blocks reuse of resistance elements among blocks Not sufficient, dynamitic simulation with capacitance and inductance required. Inductance extraction is very time consuming. Brought the idea in extraction

8. Main Contribution Structure regularity exploited, locality property of reluctance. Block division, reuse scheme. Combined with frequency-dependent reluctance and resistance extraction Inductive modeling with high-frequency effect. up to 10 5 of wire segments several to tens of times faster than existing methods preserving high accuracy.

9. Overview of Window-Based Extraction window-based method, main steps: 1. For conductor i, select window Wi; 2. Calculate the mutual reluctances within Wi, conductor i and conductors outside is set to 0; 3. Execute the above steps for every conductor, fill reluctances, column by column, 4. Generate a symmetric reluctance matrix

10. High-frequency Effects Not considering the high-frequency effects: inverting the inductance matrix, based on (1). Considering the high-frequency effects: conductors meshed into filaments. The frequency-dependent reluctance can be extracted, collaborated with the window technique. The flow will not change, the intra-window extraction (i.e. the 2nd step) becomes complicated.

11. P/G Grid Structure Several metal layers, mesh structure Along either X-axis or Y-axis, alternatively. Power wires interlaced with ground wires. Power wire Ground wire Via Figure 1. A two-layer structure of P/G grid. Connected through vias, which cut the wires into small metal segments.

12. P/G Structure In a certain metal layer, the same width, and the same pitch The evenly distributed metal wires, evenly distribution of vias. If irregular in later design stages, regularization process can be performed to make the distribution of P/G wires similar [Shi TCAD ’ 07]. In this paper, the regularity is taken advantage, for high-frequency reluctance and resistance extraction.

13. Basic Idea of Block Reuse [Shi TCAD ’ 07] a pattern idea for DC simulation Explores the geometry characteristics Translates topology similarity to sub-matrix regularity. Divided into blocks on the X-Y plane (see Fig. 2) Reuse of resistance elements among blocks. Fig. 2 The X-Y plane partition of P/G grid with overlapped blocks

14. Basic Idea of Block Reuse Extended for reluctance extraction The idea can not be directly applied The reluctance affected by environment x z Fig 3. The reluctance is different (a) (b) (d)(c) x z 1 1 22 Wires on different layers are denoted by diamond and ellipse marks. 1212

15. Mutual impedance of perpendicular conductors negligible, the reluctance interaction among metal wires along same direction considered. Reluctance for wires along Y-axis and describe the block partition along X-axis. Fig. 3 and 4 shows the side view of two-layer Y- direction P/G wires for extraction. Assume power wire and ground wire appear in pair and their distance is the same. Only plot the P wires. Basic Idea of Block Reuse

16. Proper block position and size, the error induced may be very limited. Figure 4. The division of blocks block 1 block 2 block 3 x z pitch 3 overlapped blocks. Geometric is identical. The results reused for other blocks.

17. Basic Idea of Block Reuse Wires along X-axis handled with similar procedure Whole reluctance matrix generated.

18. Algorithm Flow For X-direction, determine the block division from the Y-Z plane view; Y-direction similarly, obtain the blocks on the X-Y plane; Extract the reluctances for the X-direction wires and Y-direction wires within the middle block, respectively; If considering high-frequency effect, both reluctance and resistance are obtained; Assemble the extraction results to obtain two global matrices, one for X-direction wires and the other for Y-direction wires; combine the two matrices to obtain the whole reluctance matrix.

19. Algorithm Analysis Reduces the number of conductors to that within one block. Speedup ratio: approximate to the ratio of the number of segments in the whole P/G grid over that in a block. The number of wires within block obtained may approximate to the number of P/G wires. degrade to window-based algorithm. suitable for number of wire within block is small.

20. Numerical Results The proposed algorithm implemented as PG_extractor, for frequency-dependent reluctance and resistance extraction considering the regular P/G grid structure. Compared with the DRRE (direct reluctance and resistance extraction) [Zhang ’ 06, Zeng ’ 07] and the impedance extractor FastHenry [Kamon TMTT ’ 94 ] developed by MIT. [Zhang ’ 06]M. Zhang, W. Yu, et al., “ An efficient algorithm for 3-D reluctance extraction considering high frequency effect, ” ASP-DAC, 2006. [Zeng ’ 07]S. Zeng, W. Yu, et al., “ Efficient extraction of the frequency-dependent K element and resistance of VLSI interconnects, ” Acta Electronica Sinica, 2007 (in Chinese).

21. Numerical Results: the First Example Four layers,1830 segments. Upper two layers: 10 P wires and 10 G wires, pitch: 6.36  m Lower two layers: 16 P wires and 16 G wires, pitch: 4.23  m. 3×3 blocks, each block: Upper two layers: 6 P wires, 6 G wires The lower two: 10 P wires. 10 G wires Table 1: Error distribution of loop inductance for the fist case Error distribution of loop inductance (%) <3%3%- 6% >6 % PG_extractor vs Fast- Henry[14] 93.95.75.70.4 DRRE vs FastHenry 98.71.30 PG_extractor vs DRRE 97.62.10.3

22. Other Three Examples Similar structure, different wire pitches and number of wires. Segment numbers: 4810, 11156 and 102674, 10GHz, segment in upper two layers partitioned into 3  3 filaments The second case: lower two layers: 25 P wires, 25 G wires, upper two layers: 17 P wires, 17 G wires, 6×6 blocks 99% of loop inductances have discrepancy within 3%.

23. Numerical Result Table 2 Time comparison CaseSegment #FastHenry * DRREPG_extractorSpeedup * 11830885655.618.53.0 24810--109.919.25.7 311156--428.741.910 4102674--5034.6109.246 * The speedup is with respect to DRRE * FastHenry is not able to extract the impedance for the three larger cases, due to the limitation of CPU time and memory usage

24. Conclusion Exploit the regularity of P/G grid, Technique of block division, blocks with similar inner structure, reuse scheme Efficient window-based method. Handle large-scale P/G grid structure with high accuracy and efficiency. In the future extending for specific P/G grid structures, investigating the regularity of reluctance matrix for accelerating dynamic simulations.

25. Thank you!