陳中平臺灣大學：電子工程學研究所王芝宇Wang, Chih-YuChih-YuWang2007-11-272018-07-102007-11-272018-07-102006http://ntur.lib.ntu.edu.tw//handle/246246/57717The ever-increasing complexity of nano-scale ULSI design has made parasitic capacitance extraction more and more challenging. Efficient and accurate capacitance calculation is needed urgently for delay evaluation especially for deep sub-micron designs. With the wide-spread application of multiple-layer interconnect structure, the BEM based discretization of dielectric-dielectric interfaces not only significantly increases the execution time and memory consumption, but also gives rise to instability and inaccuracy during the surface charge density calculation. In this thesis, we propose a fast multiple-dielectric capacitance extraction algorithm, MimCap, by integrating the truncated image method (TIM) into a hierarchical sparse capacitance extraction approach. The truncated image method approximates the dielectric-dielectric interface by a finite series of image charges and hence avoids the undesirable dielectric-dielectric interface process. Extensive experimental results demonstrate that MimCap is not only fast but also accurate. By preserving only the first 1~3 orders of image charges, MimCap exhibits over 100X speedup compared with FastCap while maintaining percentage error around 2%.1 Introduction . . . . . . . . . . . . . . . . . .1 1.1 Motivation . .. . . . . . . . . . . . . . . . 1 1.2 Capacitance Extraction . . . . . . . . . . . 2 2 Capacitance Extraction . . . . . . . . . . . . .4 2.1 Background . . . . . . . . . . . . . . . . . 4 2.1.1 CapacitanceMatrix . . . . . . . . . . . . . 5 2.1.2 Potential CoefficientMatrix . . . . . . . . 6 2.1.3 Some Solving Algorithms . . . . . . . . . . 9 2.2 Multiple Dielectrics Condition . . . . .. . . 10 2.2.1 Potential in DielectricMedium . . . . . . . 10 2.2.2 Boundary Condition . . . . . . . . . . . . 11 2.2.3 Larger Linear System . . . . . . . . . . . 12 2.3 BasicMethod of Image . . . .. . . . . . . . . 13 2.3.1 Potential without Image Charge . . . . . . 13 2.3.2 Potential with Image Charge . . . . . . . . 14 3 Algorithms . . . . . . . . . . . . . . . . . . .16 3.1 Method Applied review . . . . . . .. . . . . 16 3.2 ImageMethod for Single Interface . . . . . . 17 3.2.1 Free Charge in Dielectric . . . . . . . . . 17 3.2.2 Electric Field Intensity at Interface . . . 19 3.2.3 Induced Polarization Charge at Interface .. 20 3.2.4 Image Charge . . . . . . . . . . .. . . . . 21 3.2.5 Potential Evaluation with Image Charge . .. 22 3.2.6 Perspective of Wave Propagation . . . . . . 23 3.3 ImageMethod forMultiple Interfaces . . . . .. 24 3.3.1 Infinite Series of Image Charges . . . . . 24 3.3.2 Geometry of Dielectrics . . . . . . . . . ..28 3.3.3 The First Order Image . . .. . . . . . . . .29 3.3.4 The Second Order Image . . . . . . . . . . .30 3.3.5 The High Order image . . . . . . . . . . . .32 3.3.6 Truncation of Image Charges . . . . . . . . 32 3.4 Integration with Implicit CongruenceMethod .. 33 3.4.1 Hierarchical Partitioning . . . . . . . ... 33 3.4.2 Panel Bases . . . . . .. . . .. . . . . . . 34 3.4.3 Exchange of Panel Bases . . .. . . . . . . .35 3.4.4 Modified Potential Coefficient with Image charges .. . . . . . . . . . . . . . .. . . . . . 35 4 Experimental Results . . . . . . . . . . . . .. 38 4.1 Environment . . . .. . . . . . . . . . . . .. 38 4.2 ImageMethod Testing . . . . . ... . . . . . . 39 4.2.1 Geometry of Test Case . . . . . . . . . . .39 4.2.2 Result of Test . . . . . . . . . . . . . . .39 4.3 Bus Cases Testing . . . . . . . . . . . . . .40 4.3.1 Bus Case Geometry . . . . . . . . . . . . .40 4.3.2 Result of Test . . . . . . . . . . . . . . .41 4.4 Timing andMemory Usage . . . . . . . . . . . .42 5 Conclusion . . . . .. . . . . . . . . . . . . . 461757481 bytesapplication/pdfen-US電容萃取多層介質capacitance extractionmultiple dielectricsimage methodthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/57717/1/ntu-95-R93943118-1.pdf