IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 30, NO. 9, SEPTEMBER 2011 1349
Analysis and Design of Energy and Slew Aware
Subthreshold Clock Systems
Jeremy R. Tolbert, Student Member, IEEE, Xin Zhao, Student Member, IEEE, Sung Kyu Lim, Senior Member, IEEE,
and Saibal Mukhopadhyay, Member, IEEE
Abstract—In this paper, we analyze the effect of clock slew
in subthreshold circuits. Specifically, we address the issue that
variations in clock slew at the register control can cause serious
timing violations. We show that clock slew variations can cause
frequency targets to deviate by as much as 28% from the design
goals. Based on these observations, we recognize the importance
of clock slew control in subthreshold circuits. We propose a
systematic approach to design the clock tree for subthreshold
circuits to reduce the clock slew variations while minimizing the
energy dissipation in the tree. The combined approach, including
the wire sizing and dynamic nodal capacitance control, can
achieve better slew control (and better timing control) at lower
energy in subthreshold circuits.
Index Terms—Design automation, reliability, system analysis
mobile applications such as micro-sensors and biomedical
devices. When the primary goal is to save energy, subthreshold
logic can allow for significant power reduction by operating
at a supply voltage lower than the threshold voltage of the
devices. Even though low power is the main focus, it is still
innate for the designer to optimize secondary parameters such
as robustness and performance. As a result of these efforts,
works have been presented to optimize energy and delay,
while performing computations with minimal error –.
Additionally, efforts have been made to optimize devices such
that circuits can be operated at medium frequencies in the
order of tens to hundreds of megahertz .
In addressing the design of an optimal energy-delay sub-
threshold system, the clock network plays a significant role.
Delivering robust clock signals to hundreds or even thousands
of flip-flops requires the clock tree to be optimally designed
to handle issues of delay, skew, and jitter. In subthreshold,
RANSISTORS OPERATING in the subthreshold region
constitute an attractive technology for ultralow power
Manuscript received February 10, 2011; accepted March 14, 2011. Date
of current version August 19, 2011. This work was supported by the
National Science Foundation, under Grant CCF-0917000, the National Science
Foundation Graduate Research Fellowship, under Grant DGE-0644493, the
SRC Interconnect Focus Center, and Intel Corporation. This paper was
recommended by Associate Editor I. Bahar.
The authors are with the School of Electrical and Computer Engineer-
ing, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail:
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCAD.2011.2144595
the signal slew (10% to 90% transition time) also has ca-
pacity to affect system performance , . Additionally,
due to its high switching activity, the clock network can
contribute up to 40% of the total dynamic power . The
same trend is expected when an above threshold system
is scaled to subthreshold voltages. Thus, designing a low-
power yet robust clock tree is a critical challenge to im-
plement large scale subthreshold systems. This challenge is
increasingly difficult because subthreshold designs are always
constrained by the requirement of robustness. As the supply
voltage of digital circuits is scaled below the device threshold,
the characteristics of the transistor change. An immediate
observation is that the current in the subthreshold regime has
an exponential dependence on gate voltage, threshold voltage,
and additional parameters that are functions of the process.
This is in contrary to the above threshold design, whose
dependence has been noted to be linear or square . As
a result of these effects, small variations in the subthreshold
regime have been known to follow this exponential depen-
dence, and care has been taken to control these effects –
The purpose of this paper is to analyze the impact of clock
slew in subthreshold design and propose a technique for a
low-power slew controlled clock tree design. We examine
the inherent slew variations in a clock tree and show that
the slew variations can cause a direct increase in cycle time
computations. We propose a systematic approach to design
the clock tree for subthreshold circuits to reduce the clock
slew variations while minimizing the energy dissipation in the
tree. We will show that a tighter nodal capacitance control
is necessary to control the slew in a subthreshold clock tree,
which can increase the energy dissipation. Recognizing that
the wire resistances have a negligible effect in subthreshold
circuits, we will show proper wire sizing is necessary to
reduce the clock energy. Finally, we propose a dynamic nodal
capacitance control technique that allows larger slew at the
earlier nets of the tree while controlling it more aggressively
near the sink nodes.
The rest of this paper is organized as follows. Section II ad-
dresses the motivation. Section III provides current techniques
that can be used for low energy slew control in subthreshold.
Section IV proposes and analyzes dynamic CMAXselection as
a new approach to clock tree design. Section V presents a dis-
cussion regarding process, voltage and temperature variations.
Section VI summarizes with a conclusion.
0278-0070/$26.00 c ? 2011 IEEE
1350 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 30, NO. 9, SEPTEMBER 2011
clock tree designed using above threshold concepts. (b) Normalized output
slew contours and their dependence on input slew and total load capacitance
for an inverter.
(a) Deterministic slew variations at the sink nodes of a subthreshold
The purpose of this section is to demonstrate to the reader
the effects of clock slew, and how it can directly impact the
system cycle. The input slew of a logic gate can cause the
output delay to change in the range of 50–100% , .
The output slew of a logic gate has a strong dependence
on the device dimensions, gate and drain voltages, as well
as the load capacitance. More recently, this effect has been
designated a concern for flip-flop design in the subthreshold
region . When designed with above threshold methods,
subthreshold clock trees exhibit significant slew variations at
the sink nodes (i.e., the nodes directly connected to the latches)
that will increase the probability of timing violations. As an
example, the focus of this experimental section will be based
on a clock network designed using an above threshold, zero
skew clock tree design algorithm, with a slew control method
that limits the maximum capacitance driven by each internal
and external clock node (i.e., hereafter, referred to as the nodal
capacitance). The power supply of this design was then scaled
to below the device threshold. In this clock tree, inverting
buffers were used to reduced the number of devices and thus
save on power. The clock tree was designed using a 65nm
predictive technology model (PTM) with VTP= −378mV and
VTN= 429mV . The power supply was 300mV and the
clock tree has 267 sink nodes, each driving multiple flip-flops.
The design used to define the clock sink destinations is the
IBM r1 benchmark .
A. Deterministic (Design Induced) Slew Variations
Fig. 1(a) depicts the deterministic slew distribution at the
sink nodes for the subthreshold clock tree described above.
Deterministic or design induced variations occur as a
function of load capacitance, routing distances, and buffer
placement. Under this definition, it is unlikely that all sink
nodes will have the same slew, which results in a deterministic
variation across the chip. In an ideal case, each individual chip
would have the same spread of deterministic slew variations.
The slew at the clock sink nodes is important because they
are the control for the latches and flip-flops in a chip. The
coefficient of variation, CV, is a normalized measure of
dispersion of a probability distribution and is defined as the
ratio of standard deviation (σ) to mean (μ)
In the subthreshold clock tree the CV is 23%, showing there
is a wide distribution of slew. Well controlled CVs are in the
range of 10–15%.
This distribution of slew shown in Fig. 1(a) corresponds
to the slew variation across different sink nodes of the tree.
This slew variation is caused by variation in the output slew
of the inverter stages of the clock tree. Fig. 1(b) shows the
effect of input slew and load capacitance on the output slew
of an inverter. The line contours represent that the output
slew of an inverter has a strong dependence on the input slew
and load capacitance. This is important for clock tree paths,
because it is composed of numerous inverters, and the slew
and capacitance often vary from node to node. For a smaller
output slew, a smaller input slew and load capacitance are
required. The recovery of the slew (i.e., output slew smaller
than the input slew) through an inverter stage is an important
consideration for the clock tree path. If not controlled, the slew
can become progressively higher as signals progress down a
clock path. For example, with a load capacitance of 80fF and
input slew of 1ns, the output slew is 3X the input slew. The
slew recovery is a strong function of the load capacitance.
Fig. 1 shows that if the load capacitance is reasonable, the
inverter can recover slew even for a large input slew. Hence,
controlling the capacitance driven by each node in the clock
tree is very important to control slew propagation through the
tree and reduce the slew variation at the sink nodes.
B. Clock Slew Impact on Cycle Time
A direct motivation of this paper is the impact of clock skew
and slew on the cycle time. The schematic in Fig. 2(a) shows a
generic logic path composed of fan-out-4 NAND gates between
two registers. The minimum cycle time and the maximum
clock frequency for the above system is given by
Tmin≥ tc−q+ tlogic+ tsu− δ
where tc−q denotes the maximum propagation delay of the
register (clock-to-q), tlogicis the maximum delay of the com-
binational logic, tsuis the setup time for the registers, and δ the
clock skew. The graph in Fig. 2(c) plots three Fmaxcases for
this subthreshold path as we alter the total number of stages
(N). In each case, skew and slew requirements are as follows:
1) optimal: 0ns clock skew and 1.0ns clock slew;
2) skew only: a skew that is 5% of the optimal period and
1.0ns clock slew;
3) slew + skew: takes into account both skew and slew,
where skew is 5% of the optimal period and slew is
The 10ns slew is chosen in the experiment to reflect the
maximum slew obtained from the clock tree that was first
designed in above threshold and then operated at subthreshold
condition. As expected, the skew reduces the operating fre-
quency. When both slew and skew are considered, Fmaxcan
be reduced from 14% to 28% depending on the length of the
logic path. In essence, as subthreshold systems target higher
TOLBERT et al.: ANALYSIS AND DESIGN OF ENERGY AND SLEW AWARE SUBTHRESHOLD CLOCK SYSTEMS 1351
mission gate flip-flop used for subthreshold experiments. (c) Impact of clock
skew and clock slew on frequency. (d) Timing variations for a clock tree
designed in above threshold lowered to subthreshold.
(a) Sample logic path simulated with F04 nand delays. (b) Trans-
frequencies (i.e., as the logic path reduces), the effect of slew
on cycle time cannot be ignored.
To understand the effect of clock slew directly, we have
analyzed how these variations impact the setup time and clock-
to-q. Fig. 2(b) depicts a commonly used transmission gate flip-
flop configuration that can be used in subthreshold operation
. The plot in Fig. 2(d) shows how the slew distribution
directly affects the timing metrics described above. They have
been normalized to the delay of a fan-out-4 gate, tFO4, since
the focus of this paper is on the general behavior. When
slew variations are applied to clock, the setup time is directly
proportional to it. In severe cases the setup time can vary
by 52% worse than the best case achieved. This variation in
setup time reduces the time available to compute logic, and in
some cases will cause errors in the logic by violating the setup
requirements. For the given slew distributions, the clock-to-q
delay can be 58% worse than its best case value.
C. Deterministic (Design Induced) Timing Variations
In the previous subsections, we have discussed that: 1) a
clock tree design causes slew variations, and 2) slew varia-
tions have the potential to cause severe timing violations in
subthreshold. In this section, we make the connection that the
design of the clock tree contributes to these timing variations.
Fig. 3(a) and (b) shows the distribution of the timing metrics
that impact cycle time. Fig. 3 reiterates the concept that the
setup time and clock-to-q are worsened by clock slew, as the
deterministic variations are plotted.
Clock slew directly impacts timing metrics; a smaller slew
variation translates to smaller setup and clock-to-q variations.
in above threshold and then the power supply is scaled to subthreshold
voltages. The variations are normalized by the best case scenario.
(a) Setup time and (b) Clock-Q distributions for a clock tree designed
subthreshold. (a) shows that increasing CMAX increases slew and reduces
power. The numbers indicate the fixed CMAX in fF. (b) Impact of CMAX
constraint on wire length, energy, and buffer count.
Summary of experimental methods to reduce the slew variations in
By reducing the deterministic clock slew variations, an optimal
maximum frequency can be met.
III. Techniques for Low Energy Slew Control in
The findings from Section II show that it is necessary to
control and reduce the variations associated with clock slew for
robust subthreshold operation. The focus of this section is on
investigating techniques to design a clock tree in subthreshold
that provides a smaller slew variation.
A. Smaller CMAXRequirements in Subthreshold
Conventional, above threshold methods for controlling clock
slew currently exist and rely on limiting the maximum nodal
capacitance, CMAX a buffer can drive –. In general,
when a buffer reaches a node that exceeds the designated
CMAX, buffer insertion is required to reduce the load. In our
analysis of an above threshold clock tree scaled to operate
in subthreshold, the CMAX was defined by above threshold
methods. In above threshold, there is more available current to
drain the charge from the node; therefore we should impose
a smaller CMAX requirement for optimal subthreshold clock
trees. Fig. 4(a) shows the results of a clock tree designed in
subthreshold, with varying CMAX. Keep in mind that this figure
of merit CMAX represents the maximum nodal capacitance a
node can have, and in most cases the actual load capacitance
of a node is less than the CMAX. The results show that it is
possible to better control the slew in subthreshold reducing the
average rise slew from 6ns to 3ns. At the same time we are
1352 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 30, NO. 9, SEPTEMBER 2011
reducing the slew, we are increasing the energy by nearly 20%.
This behavior is exhibited because as we reduce the CMAX
each node can drive, the total interconnect capacitance remains
constant and we need to insert more buffers to compensate for
the reduced CMAX. Fig. 4(b) shows this trend of energy and
number of inverters as a function of CMAX. Additionally, the
total wire length of the design has small changes, because
whenever an inverter is removed a small wire segment takes
the place of the removed inverter. Note that going from a CMAX
of 100fF to 300fF, the number of inverters decreases by 60%,
but the energy only decreases by nearly 20%. A large portion
of this is unaffected energy comes from the large interconnect
capacitance. In summary to design a subthreshold clock tree,
we require a smaller CMAXthan in above threshold.
B. Minimum Wire Width in Subthreshold
The wire interconnect contributes to a significant portion of
the energy in a clock network. Reducing the capacitance in
interconnect without sacrificing delay in the clock path can
help to address this energy component. When modeling the
interconnect as a distributed RC line, the design rule of thumb
is that rc wire delays should only be considered when the line
being modeled has reached a critical length, Lcrit
where td is the gate delay, r the resistance per unit length,
and c the capacitance per unit length. Based on the above
equation, the Lcrit for a 1ns gate in 65nm (using the PTM
model) should be much greater than 4mm. This value also
assumes a minimum wire width to reduce the interconnect
capacitance. A recent work showed a 65nm subthreshold
chip with a 2.29mm × 1.86mm area . Using this as a
benchmark we could expect a worse case wire length of just
over 4mm. Recent products in 65nm technology from Intel
show similar results in regards to maximum wire lengths .
Since we expect the length of the wire to be much less than
the critical length, it may be possible to neglect the distributed
rc behavior of the wire. When L <<Lcrit, the wire can be
modeled primarily as a lumped capacitance. This is possible
in subthreshold as the device resistance is much higher than
the wire resistance and we are operating at lower frequencies.
Since rc wire delays are negligible, wire resistance does not
play a critical role in determining wire delay and all wires can
be designed with minimum width. In above threshold it is not
possible to neglect rc wire delays as the operating frequencies
are higher and driver delays are much smaller. Therefore the
wires are usually larger than minimum width.
The major advantage of reducing the wire width is the
corresponding decrease in the wire capacitance. Referring back
to Fig. 4(a), we observe that reducing the wire width in a
subthreshold clock tree reduces the energy without sacrificing
slew. The above threshold clock tree used 4X minimum wires
to ensure the rc delays were handled correctly. When this
tree is lowered to subthreshold voltages, the 4X wire width
only adds capacitance and thus energy to the system. This
result can allow us to design the clock tree in subthreshold
variations at the sink nodes, while ensuring the energy does not increase.
Dynamic CMAXselection is the proposed technique to reduce slew
with minimum widths to reduce wire cap and neglect the wire
resistance compared to the driver resistance.
IV. Optimizing Subthreshold Clock Tree for
Reduced Slew and Energy With Dynamic CMAX
The previous section has provided insight into reducing
the slew (and variation) while at the same time considering
energy. While they each have their pros, alone they cannot
provide an optimal subthreshold clock tree for slew control.
As a result, we propose a new technique for optimal clock
tree design in subthreshold, based on regressive CMAX near
clock sinks. Since the proposed technique allows the CMAXto
vary dynamically from one level to another level, we refer
to this method as Dynamic CMAX instead of the conven-
tional approach (referred to as Fixed CMAX) where CMAX is
constant across all levels. We compare the slew, skew, and
energy behaviors of the clock trees designed using Dynamic
CMAX against the ones designed using Fixed CMAX. The
Fixed CMAX trees are first designed in the above threshold
voltage and the supply voltage is simply scaled to study their
performance/energy at subthreshold levels. We first explain the
basic concept of Dynamic CMAX. Next we study the behavior
of broad selection of Dynamic CMAX based clock trees to
understand what traits the desired trees have in common, and
use that information as a basis for future designs. Finally, we
summarize the results to show how tighter CMAX, lower wire-
width, and Dynamic CMAX helps design better subthreshold
clock tree compared to simply scaling the supply voltage of
a clock tree from above threshold to subthreshold level. In
essence, we reinforce the principle that new design concepts
are required for designing a clock tree in subthreshold and
simply scaling the voltage of an above threshold tree is not
A. Principles of Dynamic CMAX
From the results of Fig. 4(a), we know that we can reduce
the power in the clock network by increasing the CMAXthat
each node drives. At the other end of the spectrum, a smaller
CMAXwill control slew better. For the purposes of controlling
slew, we are most interested in reducing variations at the sink
nodes, which are directly attached to flip-flops. We still need to
control slew at other nodes, but we can relax constraint in order
to save energy (fewer or smaller buffers). To reduce energy
while achieving a proper slew at the sinks, CMAXshould vary
at each level of the clock tree, reducing the closer we get to the
sink. Additionally, if we ensure the wire widths as minimum,
we should receive more power savings. Fig. 5 summarizes
TOLBERT et al.: ANALYSIS AND DESIGN OF ENERGY AND SLEW AWARE SUBTHRESHOLD CLOCK SYSTEMS1353
namic CMAXselection. (b) Results zoomed in for a rise slew of 3–5ns. The
numbers indicate the fixed CMAX in fF. By traversing the dynamic CMAX
path, we can achieve better slew control at targeted power.
(a) Preliminary results of slew control and energy savings of Dy-
the proposed technique and the previous method with Fixed
CMAX. In the proposed technique of Dynamic CMAX, the CMAX
values selected are reduced from the source (buffer level 1)
to the sink (buffer level N). We cannot guarantee that the
CMAX values selected are the exact values that a level will
drive. We can, however, guarantee that the CMAXvalues will
limit the maximum capacitance a level can drive. Fig. 6 shows
the potential advantage that Dynamic CMAX selection has
compared to Fixed CMAXmethods. It is apparent that the axes
in the figure represent a energy versus robustness plane. In an
optimal case, there is zero slew and zero energy, leading us
toward the origin. Based on this, curves that are closer to the
origin represent a better energy-robustness tradeoff. Fig. 6(a)
shows the trend for a selection of points, while Fig. 6(b) zooms
into a region where slew is small. In Fig. 6(b), we see that it
is possible to reduce the power in the clock network by nearly
6% while maintaining the same slew, by employing Dynamic
B. CMAXSelection Trends of Clock Trees
To understand the trends of Dynamic CMAX based clock
trees, we simulated 79 trees designed using the rules of
Dynamic CMAXselection. The trees were all implemented to
deliver clock to the same r1 IBM benchmark as before, which
has 267 sinks. Additionally, there are definitions for 10 buffer
levels and thus 10 dynamic CMAX selections required. The
Dynamic CMAXvalues ranged from 100fF to 400fF in 50fF
intervals. All 79 trees were uniquely selected to provide a
broad range of dynamic clock trees, so that the results could
provide insight into general trends. The authors believe that in
lieu of all 710possible combinations, this is the best scenario
1) CMAXValue Selection and Level Placement: Fig. 7(a)
shows how dynamic CMAX values for each clock tree level
can affect slew delivered to the latches. The endpoints (for
a given level) denote the minimum and maximum ranges for
slew when the given Dynamic CMAX value has been placed
at that level. For example, of all the 79 Dynamic CMAXtrees
selected, when level 5 has a CMAXof 200fF, the final clock
slew is in the range of 3–5ns.
Since we are reporting results for a per level basis, we do
not directly know how the remaining levels in the designated
tree have been selected. That is, given that level 5 has a CMAX
(a) final clock slew and (b) total clock tree energy.
Effect of independent dynamic CMAXvalues and their impact on the
binary clock tree network with merging node definitions. With Dynamic CMAX
selection, different CMAXvalues are selected at each buffer level, denoted by
the buffer output.
(a) Correlation of inverter count and clock tree energy. (b) Example
value of 200fF, based on the Fig. 7(a) we cannot definitely say
we know the CMAXvalues of levels 1 to 4, and 6 to 10. We
can however, indirectly know that the following holds true:
CMAX(1...4)≥ CMAX(5)≥ CMAX(6...10).
The interesting features to note for Fig. 7(a) are that when
level 5 has CMAX value of 100fF, the slew at the latches
is well controlled. This is because (5) has to hold true, and
since 100fF is the smallest cap value selected, when level 5
is 100fF, levels 6–10 are 100fF as well. This fact results in a
well controlled slew delivered to the latches.
Another interesting point to make is looking at level 10.
When the CMAXvalue selected at level 10 is 100fF, we have
smaller min/max values compared to when the CMAXvalue is
200fF. This reinforces the concept that a smaller CMAXvalue
near the sink (level 10) has the best chance to reduce the slew.
Fig. 7(b) shows an inverse trend with Dynamic CMAXvalues
based on the energy in the entire clock tree. What we notice
here is that although a smaller CMAXat level 10 can provide
lower slew at the latches, it can also increase the energy in the
entire system as well. The latter point is denoted by the range
at level 10 of 6–6.6 pJ, when CMAXis 100fF. On the contrary,
a larger CMAX value at level 10 can yield lower energy, by
sacrificing the slew delivered to the latches.
The last thing to note from Fig. 7 is that the further
away we are from the sink level (10), the more disparity
we have between min/max values for slew and energy. This
occurs when the CMAX value is 200fF, because there is
more flexibility in CMAX values for the levels near the sink.
Ultimately a CMAXvalue closer to the sink (level 10) has more
restrictions in the slew and energy than a CMAXvalue selected
1354 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 30, NO. 9, SEPTEMBER 2011
with (b) skew and (c) slew.
Indirect relationship between the clock slew (delivered to the latches) with (a) clock skew and (d) wirelength. The relationship between inverter count
near the source (levels 1–3). In general, selecting a CMAXvalue
for each node can turn designing an optimal clock tree into a
lesson in pure combinatorics. With that being said, it should be
possible to design an optimal tree using combinatorics and the
previously described relationships between CMAX, slew and
Energy is Directly Related to Inverter Count: By
changing the CMAXvalues at different levels, we are directly
altering the number of inverters in the clock tree system. From
Fig. 8(a), we recognize the relationship between the inverter
count and clock tree energy. The direct trend hints that most
of the power associated in the clock trees are attributed to the
inverters. As a reiteration, to control the number of inverters
(and thus energy) in a clock tree, we can change the CMAX
values at each level. This can be seen from the previously
described Fig. 7(b).
To explain the direct correlation, we can examine the
structure of a clock tree network. Assume (for simplicity) that
the clock distribution network is the form of the binary tree
shown in Fig. 8(b), a merging node is defined as the junction
where the clock tree splits into two directions. Buffer levels
are at the output of a buffer, where CMAXvalues need to be
defined. Recall that in Fig. 7(b), at the clock tree sink, (buffer
level 10), when the CMAX value is 100fF, it has a higher
energy than when it is 200fF. There are two factors that could
contribute to this increase in energy. First, from our binary
tree model and reducing CMAX, we know that the clock nodes
near the sink are guaranteed to have more inverters than at the
higher levels (near the source). This is based on the nature of
the binary tree and by reducing the CMAXvalues near the sink;
we are forced to introduce more inverters. Essentially, we are
introducing a large fraction of inverters by reducing CMAXat
the lowest level. Therefore, energy is strongly controlled by
the CMAXvalues, and more importantly, CMAXvalues selected
nearest to the sink.
3) Dynamic CMAX Can Reduce Slew, Skew, Wirelength:
Knowing that inverter count is directly related to the clock
tree energy, there are also indirect trends related to the clock
slew that can be considered. Fig. 9 compares how the clock
slew and inverter count are correlated to important metrics
in clock tree design: clock skew, slew, and wirelength. For
Fixed CMAX, we considered a constant CMAXfor all levels and
generated the clock tree using conventional low-skew clock
tree generation methods , . We analyzed the generated
trees to estimate the slew, skew, number of inverters, and
wirelength. The different points in the Fixed CMAXlines for
all subplots of Fig. 9 represent the trees generated by different
Fixed CMAX values. Fig. 9(a) shows that when designing
with a Fixed CMAX method, there is an inverse relationship
between clock skew and the clock slew delivered to the latches.
With the Fixed CMAXmethod, smaller CMAXvalues are used
everywhere to achieve smaller slew. For a given buffer size
a tighter CMAX at all levels requires more inverters in the
tree. We also observe that a tighter CMAXconstraints results
in larger deterministic skew. Note that if random process
variation is considered, larger number of inverters also implies
more sources of variations, which could further increase the
random variation in skew. In summary, when designed using
Fixed CMAX, there exists an inverse relationship between slew,
and skew: when the CMAX is less, slew is larger and skew
is smaller. Fig. 9(b) and (c) supports the observed inverse
trend of skew and slew for Fixed CMAXtrees. Hence, from a
design perspective, a tradeoff is required when using the Fixed
CMAX method, as it is only possible to achieve low slew or
low skew. The observed trends for a Dynamic CMAXtree are
different. As with the Fixed CMAXmethod, the slew reduces
with increasing the number of inverters (assuming same buffer
sizes for a single design). However, by employing various
Dynamic CMAX topologies, we can increase the number of
inverters in a design, and still achieve a reduced skew design.
Essentially, Dynamic CMAX method allows more degrees of
freedom in placement of the inverters as nodes at different
levels have different CMAXconstraints. Consequently, for same
number of inverters, there exist several different instances of
clock tree (for same design) with varying skew [Fig. 9(b)]. For
example, in Fig. 9(b) when the number of inverters is nearly
600, a Dynamic CMAXtree can achieve a reduced slew as low
as 2ns, or as high as 20ns. That is, with an optimal selection
of Dynamic CMAXvalues for different levels, it is possible to
achieve low slew and low skew by reducing the CMAX near
the sink, and increasing the CMAXnear the source [Fig. 9(a)].
Fig. 9(d) shows how the wirelength is correlated with the
slew rate delivered to the latches. The increase in the number
of inverters with an increase in the slew can be explained
based on the absence of inverters. As stated before, when the
CMAX values are large, there are fewer inverters since each
inverter has a larger limit of capacitance it can drive. With
fewer inverters the slew will tend to increase, because there
is less control. Additionally, with fewer inverters there will
be more wirelength to compensate for the removed inverters.
TOLBERT et al.: ANALYSIS AND DESIGN OF ENERGY AND SLEW AWARE SUBTHRESHOLD CLOCK SYSTEMS1355
compared with the original clock tree. The new slew distribution has a coefficient of variation of 0.1579, better than the original 0.2309 of Fig. 1. (c) Clock
tree (4) routed using dynamic CMAXand combined methods.
(a) Summary of combined approaches to reduce deterministic slew variations in subthreshold. (b) Dynamic CMAXdeterministic slew distribution
Summary of Fixed and Dynamic CMAXValues With Combined Slew Reduction Techniques
If we compare the Fixed CMAXand Dynamic CMAXselection
trends there are a few points to consider. First, they both follow
the explained trend above that a larger slew is correlated with
a longer wirelength. Second, for smaller slew targets, it is
possible to achieve a smaller wirelength using Dynamic CMAX
selection compared to Fixed CMAX. This can be seen in the
figure where the points are located around 3ns of slew.
4) Explanation of Outliers: During our preceding discus-
sion of the trends of dynamic clock trees, we have neglected
the outliers that do not follow the trends. These points can be
seen from the dotted circles in Figs. 8 and 9. Since Dynamic
CMAX can only limit the maximum capacitance a level can
drive, it has no bearing on the minimum capacitance. If for
we cannot assure that
C6≥ C8≥ C7≥ C9≥ C10
where Cidenotes the actual capacitance at the level after the
clock tree has been routed. Recall from Fig. 1(b) that when
the actual capacitance does not follow the trend in (7), the
slew is not well controlled and can become unpredictable. An
algorithm to control the upper and lower capacitance values
would assure optimal design.
5) Limitations of Buffer Reduction: In Section IV-B2, we
learned the direct correlation between energy and inverter
count. While it is true that shallower trees may be better for
skew as described in , there may be secondary effects of
slew depending on the size of the circuit. In an attempt to
reduce the overall power by continuing the trend of reducing
the number of inverters, we designed a 1-buffer H-tree to
investigate the limitations of buffer reduction. A comparison of
this design and the previously described designs can be seen
in Table II. We recognize that the skew is improved using
a 1-buffer H-tree compared to the previous options, but the
slew has increased drastically. In this design, the large size of
the IBM r1 benchmark circuit has contributed to a long wire
length and thus extra capacitance. Additionally, a large buffer
has been introduced in the 1-buffer H-tree to accommodate this
and the energy is still greater than that of the dynamic CMAX
tree. In essence, the 1-buffer H-tree is best for minimum skew
and is appropriately used for small-scale designs. However, for
large-scale designs, using a clock-tree with dynamic CMAX
design, it is possible to achieve a near optimal tree with good
slew, skew, and minimum energy.
C. Summary and Results of Dynamic CMAX
In this section, we summarize the effects of using tighter
CMAX, reduced wire-width, and Dynamic CMAX. As stated
before, we implemented several subthreshold clock trees in
65nm CMOS, based off the PTM Model at VDD= 300mV,
and the best were selected for comparison. Fig. 10(a) repre-
sents a plane of transitions as a different technique is employed
at each number to reduce the slew variations. The clock tree
“1” represents the base-line of comparison (i.e., a standard
clock tree designed in above threshold using Fixed CMAX =
250fF), scaled to subthreshold voltages. The details of CMAX
values used for these trees are shown in Table I. The results are
depicted for rise slew information, but a similar trend exists
for fall slews. The techniques to reduce energy and slew were
applied as follows:
1) (1 to 2) the CMAXwas reduced from 250 to 100fF and
the tree was redesigned using Fixed CMAX;
2) (2 to 3) the wire width was reduced from 4X to 1X
minimum sized, and then redesigned with Fixed CMAX;
1356 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 30, NO. 9, SEPTEMBER 2011
Methodology Comparison for IBM r1 Benchmark
and the skew variations are reported as a worst case for each Monte Carlo simulation.
Monte Carlo simulation of varying threshold voltages for clock tree. (a) Energy. (b) Slew. (c) Skew. The slew variations are reported as an average
supply voltage variations.
Supply voltage sweep and its impact on clock tree. (a) Energy. (b) Slew. (c) Skew. At low voltages, the Dynamic CMAX tree is less sensitive to
3) (3 to 4) Dynamic CMAXselection was employed with a
minimum wire width.
With a starting point of “1” and ending point of “4,” this
proves that it is possible to reduce the slew without increasing
power in a subthreshold clock tree. Fig. 10(a) and (b) shows
that our final clock tree, “4,” has smaller slew variations
compared to the original above threshold design, scaled to
subthreshold. Fig. 10(b) shows that the clock tree designed
with Dynamic CMAX selection has reduced the variations
in slew from the range of 3–10ns to the range of 3–6ns.
Additionally, the coefficient of variation for the new clock
tree is 0.1579 compared to the larger dispersion of 0.2309
[Fig. 1(a)]. Lastly, Fig. 10(c) shows the final clock tree
designed with Dynamic CMAXmethods, implemented for the
r1 IBM testbench.
V. Process, Voltage, and Temperature Variations
Random slew variations are induced by process variations
and have the potential to be more severe in designs because
they add to deterministic slew variations. The prior work on
the area of subthreshold design such as  has shown that in
the subthreshold region of operation the local variability due
to effects like random dopant fluctuations can dominate the
device threshold, VTH, variability. Therefore, we have consid-
ered local variability in the clock buffers while simulating the
random slew variations.
Using the learned techniques to design a dynamic clock tree,
we have proven it is possible to reduce the slew variations
without an energy penalty in the clock tree. While this has
remained a focus of this paper, maintaining a stable design un-
der process, voltage and temperature variations are extremely
important in subthreshold designs. In this section we study the
effect of using Dynamic CMAXon the impact of PVT variation
on the robustness and energy of the clock tree.
To model the effects of process variations in subthreshold
we have applied an independent variation to the threshold
voltage of each transistor in the clock tree. A 5000 point Monte
Carlo simulation was performed using a Gaussian distribution
with 3σ value of +/−10% of the nominal VTH. Fig. 11(a)
and (b) depicts the clock tree energy and average slew as a
result of the MC simulation under process variations. There is
a clear advantage to designing a tree using dynamic CMAX
selection, because it maintains a reduced energy and slew
TOLBERT et al.: ANALYSIS AND DESIGN OF ENERGY AND SLEW AWARE SUBTHRESHOLD CLOCK SYSTEMS1357
Fig. 13. (a) Energy, (b) Slew, and (c) Skew response to a temperature sweep shows that a dynamic clock tree is the preferable design.
under variations. Fig. 11(c) demonstrates Monte Carlo the
skew variations at all process corners. The thing to note here
is that with a smaller energy and slew, the dynamic clock tree
has similar or improved skew for all corners.
A supply voltage sweep of +/−20% of the nominal VDD
will provide a broad range to examine the clock tree response.
Fig. 12(a) shows how the resulting supply voltage sweeps
impacts the clock tree energy, with the dynamic clock tree
always more efficient than the scaled tree. Fig. 12(b) is more
interesting, because it shows how the average slew has a non-
linear dependence on voltage. Additionally, we can see at
lower voltages, the slope of the dynamic clock tree is less
than that of the scaled tree. This means that in subthreshold,
the slew is less sensitive to supply voltage variations when a
clock tree is designed with a Dynamic CMAX methodology.
Fig. 12(c) shows the skew dependence on voltage. In general,
dynamic CMAXappears to be more robust across subthreshold
supply variations. Fig. 13 examines how temperature variations
affect the clock energy, slew and skew. We can observe that the
dynamic CMAXhas lower energy, slew and skew even under
different temperature conditions.
In this paper, we explained how the design of a subthreshold
clock tree can impact the slew variations, which will in turn
corrupt the flow of data in a logic path. This notion provided
the motive to design an optimal subthreshold clock tree with
slew control. We concluded that the following guidelines
should be used when designing an optimal clock tree in
subthreshold with slew control. The maximum allowable nodal
capacitance should be small in subthreshold, minimum wire
sizes should be used at all times, and the maximum nodal
capacitance can be controlled dynamically to allow more slew
propagation near the root of the tree while saving power.
On the other hand, near the sink-nodes the maximum nodal
capacitance should be reduced to better control the slew.
We presented a systematic approach combining the above
three guidelines for subthreshold clock tree design that has
the potential to reduce the timing metric variations by 50% or
more while maintaining the power advantage of subthreshold
design. Additionally, the flip-flop design will also have an
impact on how severe the timing variations are. By co-
designing these efforts, it is will be possible to further reduce
the variations in slew, thus reducing the probability of timing
The clock routing algorithm used in this paper includes
two major steps: 1) abstract tree generation, and 2) slew-
aware buffering and embedding. Given a set of clock sinks,
we first generate an abstract tree based on the method of
means and medians algorithm . The objective of abstract
tree generation is to decide the connection among the sink
nodes, internal nodes, and the clock source while minimizing
the wirelength. Then, the routing topology and geometric
locations of all the nodes are determined by a two-phase
slew-aware buffering and embedding method. Our method
follows the classic deferred-merging and embedding flow 
in the above-threshold clock network design. But the major
difference is we insert buffers during the clock routing as well.
We first visit the abstract tree by a bottom-up manner. For a
pair of nodes, we create a set of feasible candidate solutions for
their parent node, including the merging distances and merging
styles. This bottom-up phase aims at generating zero-skew
solutions, and inserting buffers so that loading capacitance
of each buffer does not exceed the user-specified maximum
value (CMAX). The second phase is to choose the optimum
solution among the candidates by visiting the abstract tree in
a top-down order. The outcomes are the entire clock routing
topology with the exact locations of the internal nodes, buffers,
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Jeremy R. Tolbert (S’08) received the B.S. degree
in electrical engineering from the University of
Michigan, Ann Arbor, in 2007, and the M.S. degree
in electrical and computer engineering from the
Georgia Institute of Technology, Atlanta, in 2011.
He is currently working toward the Ph.D. degree in
electrical and computer engineering from the School
of Electrical and Computer Engineering, Georgia
Institute of Technology.
His current research interests include low-power
circuits and systems, techniques for robust sub-
threshold design, and energy-efficient processing for mobile computing.
Mr. Tolbert is currently sponsored by the Graduate Research Fellowship of
the National Science Foundation.
Xin Zhao (S’07) received the B.S. degree from
the Department of Electronic Engineering, Tsinghua
University, Beijing, China, in 2003, and the M.S.
degree from the Department of Computer Science
and Technology, Tsinghua University, in 2006. She
is currently pursuing the Ph.D. degree from the
School of Electrical and Computer Engineering,
Georgia Institute of Technology, Atlanta.
Her current research interests include computer-
aided design for very large scale integration circuits,
especially physical design for low power, robustness,
and 3-D ICs.
Ms. Zhao was the recipient of the Best Paper Award Nomination at the
International Conference on Computer-Aided Design in 2009.
Sung Kyu Lim (S’94–M’00–SM’05) received the
B.S., M.S., and Ph.D. degrees from the Department
of Computer Science, University of California, Los
Angeles, in 1994, 1997, and 2000, respectively.
He joined the School of Electrical and Computer
Engineering, Georgia Institute of Technology, At-
lanta, in 2001, where he is currently an Associate
Professor. He is the author of Practical Problems in
VLSI Physical Design Automation (Springer, 2008).
His current research interests include the architec-
ture, circuit design, and physical design automation
for 3-D ICs.
Dr. Lim received the National Science Foundation Faculty Early Career
Development Award in 2006. He was on the Advisory Board of the ACM
Special Interest Group on Design Automation (SIGDA) from 2003 to 2008
and received the ACM SIGDA Distinguished Service Award in 2008. He has
served the technical program committees of several conferences on electronic
design automation, including the ACM Design Automation Conference and
the IEEE International Conference on Computer-Aided Design. He has been
leading the Cross-Center Theme on 3-D Integration for the Focus Center
Research Program since 2010.
Saibal Mukhopadhyay (S’99–M’07) received the
B.E. degree in electronics and telecommunication
engineering from Jadavpur University, Kolkata, In-
dia, in 2000, and the Ph.D. degree in electrical and
computer engineering from Purdue University, West
Lafayette, IN, in 2006.
He is currently an Assistant Professor with the
School of Electrical and Computer Engineering,
Georgia Institute of Technology, Atlanta. Prior to
joining the Georgia Institute of Technology, he
was with the IBM T. J. Watson Research Center,
Yorktown Heights, NY, as a Research Staff Member and worked on high-
performance circuit design and technology-circuit co-design focusing pri-
marily on static random access memories. His current research interests
include analysis and design of low-power and robust circuits in nanometer
Dr. Mukhopadhyay was a recipient of the NSF CAREER Award in 2011,
the IBM Faculty Partnership Award for 2009 and 2010, the SRC Inventor
Recognition Award in 2009, the SRC Technical Excellence Award in 2005,
the IBM Ph.D. Fellowship Award for 2004 to 2005, the Best in Session Award
at 2005 SRC TECNCON, and the Best Paper Award at the 2003 IEEE Nano
and 2004 International Conference on Computer Design.