Skip to main content
eScholarship
Open Access Publications from the University of California

Statistical Analysis for On-Chip Power Grid Networks and Interconnects Considering Process Variation

  • Author(s): Mi, Ning
  • Advisor(s): Tan, Sheldon X.-D
  • et al.
Abstract

With the aggressive scaling down of semiconductor VLSI devices

from 65nm to 45 nm, 32nm, the process induced variability becomes

the major design concern. The fundamental change in VLSI chip

design in current and future nodes is that what has been designed

will not agree with the products manufactured due to the

uncertainties in the manufacture processes. Even worse, the

variabilities keep growing as the technology scales down

continually. The process induced variations manifest themselves

from wafer to wafer, die-to-die and device to device within a die.

Some are systematic variabilities and some are random

variabilities, which have to leave extra margin for worst case.

The Monte Carlo method can come to the rescue by simulating the

probability of the worst case in a random way. However, it is well

known this approach is very time consuming and forbidding slow. It

is highly desirable to have more efficient statistical modeling

and simulation techniques and tools to guide the design in the

presence of uncertainties in the nanometer VLSI regime.

In this dissertation, the influence of the variability, such as

threshold voltage variation, interconnect wire height, width

variation, on the performance of power grid delivery networks and

signal interconnect circuits, are studied. First we develop a new

statistical method, which is based on concept of Hermite polynomial

chaos, to analyze power grid voltage drop variations of on-chip

power grid networks. The new approach considers both wire

variations and sub-threshold leakage current variations, which are

modeled as lognormal distribution random variables. We also

consider spacial correlation of the leakage variables by applying

orthogonal decomposition to map the correlated random variables

into independent ones before the analysis. Second, we propose a

more efficient statistical analysis approach, StoEKS, in which the

extended Krylov subspace method is used to speedup the solution

procedure of the variational circuit equations. By using the model

order reduction technique, StoEKS partially mitigates circuit-size

increasing problem associated with the augmented matrices from the

Galerkin-based spectral statistical method. Finally, we propose an

efficient method to calculate variational interconnect delay,

which is a crucial step in the statistical static timing

analysis(SSTA). We apply Numerical quadrature method based on

orthogonal polynomial representation (OPR) of statistical

variations to derive the non-linear, non-Gaussian analytic

interconnect delay models in terms of the interconnect wire width,

height variations. It can take in variational parameters in OPR

form and outputs the delays computed in OPR form again, which is

compatible with existing SSTA methods.

Main Content
Current View