Empirically characterized equation- and table-based cell models have been applied in static timing analysis for decades. These models have been extended to handle a variety of environmental and circuit phenomena over the years. This has given rise to a profusion of cell models that are used to verify circuit functionality and performance. The recent invention of a second-generation of current source models shows the promise of a unified electrical cell model that comprehensively addresses most of the effects that are perceived as accuracy limiters. In this paper, we describe these accuracy limiters and present comprehensive results for a particular current source model [11].

We explore the power benefits of changing a microprocessor path histogram through circuit sizing based on statistical timing analysis and optimization (STAO) versus a deterministic timing approach that uses statistical design to establish a global guardband followed by conventional optimization (SDGG). Using an analytical modeling approach, we quantify the differences in total power between the two approaches while maintaining an equivalent performance distribution. For a relative 1σ random WID stage delay variation of 5% and representative microprocessor critical paths, the analysis indicates that the STAO approach enables ~2% power reduction over the SDGG approach. To achieve a 4% and 6% power reduction through the STAO approach, the process variation needs to increase by a factor of 2x and 4x, respectively.

Current source models have emerged as a promising technique for reducing digital cell netlists to a simpler electrical model for use in timing and other applications. The Multiport Current Source Model (MCSM) is one of the most general models in this class, which has been shown to handle multiple electrical effects including multiple-input switching (MIS) events in timing. However, this new model is hampered by two major problems: port characterization runtime and accuracy across a range of complicated cells which are deployed in advanced microprocessor design such as complex combinational cells, muxes, and sequentials. In this paper we demonstrate a significant leap in modeling accuracy and characterization runtime over the MCSM model which effectively eliminates these remaining issues. The quality of the new approach is conclusively demonstrated on a comprehensive 45nm cell library currently in use. The new approach accurately models both complex combinational as well as, for the first time, sequential cells, and puts MCSMs on the path for next generation gate level electrical analysis.

Recently, statistical static timing analysis (SSTA) has been proposed as a technique for mitigating the effects of parametric process variations on typical design metrics like performance and power (through circuit optimization). In spite of significant advances in the algorithmic state-of-the-art, the applicability of SSTA to general purpose IC design continues to be a matter of debate at various forums across the semiconductor and EDA industry. This talk will present analyses, studies, and trends which will highlight the applicability of statistical techniques at different stages of microprocessor design: skew calculation, min-delay computation, leakage estimation, bin-split prediction, and, finally, timing analysis. The increasing role of test in handling the effects of process variation will also be discussed. A case will be made that while SSTA may have applicability in the design of some products, there is a need for significant variation-related innovation to enable continued scaling.

A model for process-induced parameter variations is proposed, combining die-to-die, within-die systematic, and within-die random variations. This model is put to use toward finding suitable timing margins and device file settings, to verify whether a circuit meets a desired timing yield. While this parameter model is cognizant of within-die correlations, it does not require specific variation models, layout information, or prior knowledge of intrachip covariance trends. The approach works with a "generic" critical path, leading to what is referred to as a "process-specific" statistical-timing-analysis technique that depends only on the process technology, transistor parameters, and circuit style. A key feature is that the variation model can be easily built from process data. The derived results are "full-chip," applicable with ease to circuits with millions of components. As such, this provides a way to do a statistical timing analysis without the need for detailed statistical analysis of every path in the design

The problem of multiple-input switching (MIS) has been mostly ignored by the timing CAD community. Not modeling MIS for timing can result in as much as 100% error in stage delay and slew calculation. The impact is especially severe on stages immediately after a bank of flops, where the inputs have a high probability of arriving simultaneously. Other problems such as modeling of interconnect loads, complex (nonlinear/non-monotonic) input waveforms, power-droop impact on cell delay, nonlinear input capacitances, delay variations due to cross-capacitance, etc. are also known sources of error. In this paper, we introduce the multi-port current source model (MCSM). MCSM can efficiently handle an arbitrary number of simultaneously switching inputs, including single-input switching (SIS). Moreover, MCSM is comprehensive in that other modeling problems associated with delay and noise computation are elegantly covered. We demonstrate the applicability of MCSM on a large 65 nm industrial test-case. For cells experiencing MIS, the model yields delay and slew-rate errors within ±5% for 88.3% and 93.0% of the cases, respectively. We also present data that show that MCSM is an effective receiver model which captures active loading effects without incurring significant additional error. MCSM makes combined cell-level timing, noise, and power analysis a possibility.

With an increasing trend in the variation of the primary parameters affecting circuit performance, the need for statistical static timing analysis (SSTA) has been firmly established in the last few years. While it is generally accepted that a timing analysis tool should handle parameter variations, the benefits of advanced SSTA algorithms are still questioned by the designer community because of their significant impact on complexity of STA flows. In this paper, we present convincing evidence that a path-based SSTA approach implemented as a post-processing step captures the effect of parameter variations on circuit performance fairly accurately. On a microprocessor block implemented in 90nm technology, the error in estimating the standard deviation of the timing margin at the inputs of sequential elements is at most 0.066 FO4 delays, which translates in to only 0.31% of worst case path delay.

Starting from a model of the within-die systematic variations using principal components analysis, a model is proposed for estimation of the parametric yield, and is then applied to estimation of the timing yield. Key features of these models are that they are easy to compute, they include a powerful model of within-die correlation, and they are "full-chip" models in the sense that they can be applied with ease to circuits with millions of components. As such, these models provide a way to do statistical timing analysis without the need for detailed statistical analysis of every path in the design.

We study scaling in the context of typical block-level wiring distributions, and identify its impact on the design process. In particular, we study the implications of exponentially increasing repeater and clocked repeater counts on the algorithms and methodologies used for physical synthesis and full-chip assembly, showing that mere capacity scaling of current algorithms and methodologies is insufficient to handle the new challenges. Finally, we suggest a few approaches to tackle these challenges by constructing a case for abstract fabrics.

We present the results of scaling studies in the context of typical block-level wiring distributions, and study the impact of the identified trends on the post-RTL design process. In particular, we look at the implications of exponentially increasing repeater and clocked repeater counts on the algorithms and methodologies used for logic synthesis, technology mapping, layout, and full-chip assembly, and identify several new research problems relevant to future designs. Next, we introduce the basic principles of correct-by-construction (CbC) design. We look at some techniques for post-RTL design meeting CbC philosophy, and then construct a case for flexible, abstract fabrics. Finally, we suggest CbC approaches to tackle the new synthesis and layout challenges identified in this paper.

Recently, several algorithms for interconnect optimization via repeater insertion and wire sizing have appeared based on the Elmore delay model. Using the Devgan noise metric [6] a noiseaware repeater insertion technique has also been proposed recently. Recognizing the conservatism of these delay and noise models, we propose a moment-matching based technique to interconnect optimization that allows for much higher accuracy while preserving the hierarchical nature of Elmore-delay-based techniques. We also present a novel approach to noise computation that accurately captures the effect of several attackers in linear time with respect to the number of attackers and wire segments. Our practical experiments with industrial nets indicate that the corresponding reduction in error afforded by these more accurate models justifies this increase in runtime for aggressive designs which is our targeted domain. Our algorithm yields delay and noise estimates within 5% of circuit simulation results.

Buffers in clock trees introduce two additional sources of skew: the first source of skew is the effect of process variations on buffer delays. The second source of skew is the imbalance in buffer loading. We propose a buffered clock tree synthesis methodology whereby we first apply a clustering algorithm to obtain clusters of approximately equal capacitance loading. We drive each of these clusters with identical buffers. A sensitivity based approach is then used for equalizing the Elmore delay from the buffer output to all of the clock nodes. The skew due to load imbalance is minimized concurrently by matching a higher-order model of the load by wire sizing and wire lengthening. We demonstrate how this algorithm can be used recursively to generate low-skew buffered clock trees.

With an ever-increasing portion of the delay in high-speed CMOS chips attributable to the interconnect, interconnect-circuit design automation continues to grow in importance. By transforming the gate and multilayer wire sizing problem into a convex programming problem for the Elmore delay approximation, we demonstrate the efficacy of a sequential quadratic programming (SQP) solution method. For cases where accuracy greater than that provided by the Elmore delay approximation is required, we apply SQP to the gate and wire sizing problem with more accurate delay models. Since efficient calculation of sensitivities is of paramount importance during SQP, we describe an approach for efficient computation of the RC circuit delay sensitivities.

Sensitivity-based methods for wire sizing have been shown to be effective in reducing clock skew in routed nets. However, lack of efficient sensitivity computation techniques and excessive space and time requirements often limit their utility for large clock nets. Furthermore, most skew reduction approaches work in terms of the Elmore delay model and, therefore, fail to balance the signal slopes at the clocked elements. In this paper, we extend the sensitivity-based techniques to balance the delays and signal-slopes by matching several moments instead of just the Elmore delay. As sensitivity computation is crucial to our approach, we present a new path-tracing algorithm to compute moment sensitivities for RC trees. Finally, to improve the runtime statistics of sensitivity-based methods, we also present heuristics to allow for efficient handling of large nets by reducing the size of the sensitivity matrix.

For efficiency, the performance of digital CMOS gates is often expressed in terms of empirical models. Both delay and short-circuit power dissipation are sometimes characterized as a function of load capacitance and input signal transition time. However, gate loads can no longer be modeled by purely capacitive loads for high performance CMOS due to the RC metal interconnect effects. This paper presents a methodology for interfacing empirical gate models to reduced order RC interconnect models in terms of a nonlinear iteration procedure. The delay and power are calculated with errors on the same order as those for the original empirical equations. Moreover, a linear equivalent gate model is generated which accurately captures the delays at the interconnect fan-out nodes.

Achieving near-zero skew for a large clock tree is generally at the expense of adding large amounts of metal interconnect. This added metal can significantly increase the total clock-net capacitance, thereby increasing the power dissipation in proportion. In addition, it is shown that it becomes increasingly difficult to control clock-signal skew due to metal-wiring process variations as the total clock-net capacitance increases. In this paper we demonstrate that buffer insertion can be used to reduce the total capacitance, hence the power, while generating a design which is as robust as one with no intermediate buffering. Moreover, delays are reduced substantially as well. Given an initial feasible route with constraints. On wire widths, wire sizing and buffer insertion are performed concurrently at each iteration of optimal buffer location search. Statistical process variations of the buffers, their loads, and the metal interconnect parameters are considered as part of the robust design process.

Abstract—With delays due to the physical interconnect domi-nating the overall logic path delays, circuit-level delay optimiza-tion must take interconnect effects into account. Instead of sizing only the gates along the critical paths for delay reduction, the trade-off possible by simultaneously sizing gate and interconnect must also be considered. We show that for optimal gate and interconnect sizing, it is imperative that the interaction between the driver and the RC interconnect load be taken into account. We present an iterative sensitivity-based approach to simulta-neous gate and interconnect sizing in terms of a gate delay model which captures this interaction. During each iteration, the path delay sensitivities are efficiently calculated and used to size the components along a path. I.

With an ever-increasing portion of the delay in highspeed CMOS chips attributable to the interconnect, interconnect-circuit design automation continues to grow in importance. By transforming the gate and multilayer wire sizing problem into a convex programming problem for the Elmore delay approximation, we demonstrate the efficacy of a sequential quadratic programming (SQP) solution method. For cases where accuracy greater than that provided by the Elmore delay approximation is required we apply SQP to the gate and wire sizing problem with more accurate delay models. Since efficient calculation of sensitivities is of paramount importance during SQP, we describe an approach for efficient computation of the accurate delay sensitivities.

As signal speeds increase and gate delays decrease for high-perform ance digital integrated circuits, the gate delay m odeling problem becom es increasingly m ore difficult.With scaling, increasing interconnect resistances and decreasing gateoutput im pedances m ake it m ore difficult to em pirically characteriz e gate-delay m odels.Moreover, the single-input-switching assum ption for the em pirical m odels is incom patible with the inevitable sim ultaneous switching for todays high-speed logic paths.In this paper a new em pirical gate delay m odel is proposed.Instead of building the em pirical equations in term s of capacitance loading and input-signal transition tim e, the m odels are generated in term s of param eters which com bine the benefits of em pirically derived k -factor m odels and switch-resistor m odels to efficiently: 1) handle capacitance shielding due to m etal interconnect resistance, 2) m odel the RC interconnect delay, and 3) provide tighter bounds for sim ultaneous switching.

Recognizingthat routing constraints and process variations make non-zero skew inevitable, this paper describes a novel methodology for constructing reliable low-skew clock trees.The algorithm efficiently calculates clock-tree delay sensitivities to achieve a target delay and a target skew.Moreover, the sensitivities also show that wires should be widened as opposed to lengthened to reduce skew since the former improves reliability while the latter reduces it.This paper introduces the concept of designing reliable clock nets with process-insensitive skew.