Efficient Reluctance Extraction for Large-Scale Power Grid with High- Frequency Consideration

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 105 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. Power wire Via Ground wire Figure 1. A two-layer structure of P/G grid. P/G Grid Structure • Several metal layers, mesh structure • Along either X-axis or Y-axis, alternatively. • Power wires interlaced with ground wires. • 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. z z x x Basic Idea of Block Reuse Fig 3. The reluctance is different • Extended for reluctance extraction • The idea can not be directly applied • The reluctance affected by environment Wires on different layers are denoted by diamond and ellipse marks. 1 2 1 2 (a) (b) 1 1 2 2 (c) (d)

15. Basic Idea of Block Reuse • 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.

16. pitch block1 block2 block3 z x Basic Idea of Block Reuse • 3 overlapped blocks. • Geometric is identical. • The results reused for other blocks. • Proper block position and size, the error induced may be very limited. Figure 4. The division of 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 Table 1: Error distribution of loop inductance for the fist case • 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

22. Other ThreeExamples • 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 * 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!