feature sizes get smaller, as FinFET aim to do, random variations of
device properties become increasingly important. It is generally
recognized that the most important source of variability is due to
random doping fluctuations (RDF). RDF is a result of the statistical
nature of the position and the discreteness of charge of the dopant
atoms. Whereas in past technologies the effect of the dopant atoms could
be treated as a continuum of charge, FinFETs are so small that the
charge distribution of the dopant atoms becomes ‘lumpy’ and variable
from one transistor to the next, hence the name RDF.
introduction of metal gates in advanced CMOS processes, a second highly
important source of variability has emerged arising from the formation
of the finite-sized metal grains with different lattice orientations:
random workfunction fluctuations (RWF). In this effect, each metal grain
in the gate, whose crystalline orientation is random, interacts with
the underlying gate dielectric and silicon in a different way, with the
consequence that the channel electrons no longer see a uniform gate
RDF and RWF manifest themselves as variation in the
output characteristics of FinFETs and circuits, and the systematic
analysis of these effects has become a priority for technology
development and IP design teams alike.
One of the key challenges
in simulating device variability is in describing and solving the
variability in a computationally efficient way. A brute force approach,
whereby each device sample is explicitly defined and its performance
characteristic computed leads to heavy computational burden as hundreds
of device samples must be simulated to actually sample the statistical
distribution. This brute force method, also known as atomistic, has been
highlighted in the academic literature as a vehicle to estimate the
variability effects – however, its computational burden is impractical
for industrial use.
Fortunately, a new method known as the
impedance field method (IFM) reduces dramatically the simulation time
without sacrificing the accuracy of the statistical variation output
that device and design engineers are interested in. The key concept in
IFM is to treat the randomness of the doping, metal grains and other
sources as perturbations of a reference device.
the IFM, we present its application to the static random access memory
(SRAM) cell, a component so ubiquitous and critical to the operation of
most integrated circuits that it warrants special consideration. The
reduction of SRAM cell size is constrained by, among other factors, its
electrical variability. Moreover, the transistors comprising the SRAM
cell are typically packed so closely together that they influence each
other’s performance – these are called proximity effects.
to the importance of capturing the proximity effects, recent progress in
3-D TCAD simulation now enables the simulation of the variability of
the static noise margin (SNM) of an entire SRAM cell . In this
approach, the 3-D TCAD representation of the SRAM cell includes the
details of the process so as to capture the proximity effects among the
transistors (see figure 6).
Figure 6. 3-D FinFET SRAM structure simulated with Sentaurus TCAD.
initial step in the IFM method is the computation of the full 3-D TCAD
solution for the nominal case representing the average of the
statistical distribution. Then, the perturbations describing all
important variation sources are applied and the linear response is
computed. The mathematics involved in this step is based on a so-called
impedance field, from which the name for the method is derived. One key
advantage of the impedance field approach is that it relies on the
well-established and calibrated TCAD transport models and their
calibrated parameters. Figure 7 shows the voltage transfer
characteristics (VTCs) of SRAM cell inverter. From these curves the
static noise margin for the cell can be extracted.
Figure 7. Statistical SRAM butterfly curve simulated with the IFM method.
in the aforementioned atomistic method the total simulation time is
proportional to the number of device samples, in IFM the only
computational effort is taken up by the single 3-D TCAD solution of the
reference device and the computation of the impedance fields. The
additional computation effort of the random variation sources is
negligible. This makes IFM a practical method for variability analysis
in the industry.
 Karim El Sayed et al,
“Investigation of the Statistical Variability of Static Noize Margins
of SRAM Cells Using the Statistical Impedance Field Method, IEEE Trans.
Electron Devices, Vol. 59, No. 6, June 2012, pp. 1738-1744.
About the author
Moroz is a Synopsys Scientist, engaged in a variety of projects on
modeling 3D ICs, transistor scaling, FinFETs, stress engineering, defect
engineering, solar cell design, innovative patterning, random and
systematic variability, junction leakage, non-Si transistors, and
atomistic effects in layer growth and doping. Several facets of this
activity are reflected in three book chapters and over 100 technical
papers, invited presentations, and patents. Has been involved in
technical committees at ITRS, IEDM, DFM&Y, ECS, IRPS, and ESSDERC.
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