# Design centering scheme for robust SRAM cell design

**ABSTRACT** In this paper, a statistical approach for the optimal design of 6-T FinFET based SRAM cells considering the statistical distributions of gate length and silicon thickness of its transistors is presented. The corresponding statistical correlations of these two parameters are also considered. In this method, proper back-gate voltages for the SRAM transistors which maximize the yield against read, write, access and hold errors are determined. To assess the efficiency of the approach, the approach is applied to a 45 nm FINFET technology. The use of Monte-Carlo simulations shows the effectiveness of the method for increasing the yield of the FinFET SRAM cells. The proposed scheme is general and may be applied to other circuits.

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**ABSTRACT:**In this paper, an optimal approach for the design of 6-T FinFET-based SRAM cells is proposed. The approach considers the statistical distributions of gate length and silicon thickness and their corresponding statistical correlations due to process variations. In this method, a back-gate voltage is used as the optimization knob. With the help of particle swarm optimization (PSO), the back-gate voltages that maximize the yield of the SRAM array against read, write, and access time failures are found. It will be shown that, with this method, a very high yield is achieved.IEEE Transactions on Very Large Scale Integration (VLSI) Systems 11/2011; · 1.22 Impact Factor

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Proceedings of the International Conference on Computer and Communication Engineering 2008 May 13-15, 2008 Kuala Lumpur, Malaysia

978-1-4244-1692-9/08/$25.00 ©2008 IEEE

Design Centering Scheme for Robust SRAM Cell Design

Masoud Rostami, Behzad Ebrahimi, Ali Afzali-Kusha

Nanoelectronics Center of Excellence, School of Electrical and Computer Engineering,

University of Tehran, Tehran, Iran

m.rostamy@ece.ut.ac.ir

Abstract

In this paper, a statistical approach for the optimal

design of 6-T FinFET based SRAM cells considering

the statistical distributions of gate length and silicon

thickness of its transistors is presented. The

corresponding statistical correlations of these two

parameters are also considered. In this method, proper

back-gate voltages for the SRAM transistors which

maximize the yield against read, write, access and hold

errors are determined. To assess the efficiency of the

approach, the approach is applied to a 45 nm FINFET

technology. The use of Monte-Carlo simulations shows

the effectiveness of the method for increasing the yield

of the FinFET SRAM cells. The proposed scheme is

general and may be applied to other circuits.

I. INTRODUCTION

SRAM arrays are a major part of the chip area in

typical microprocessors. As memory will continue to

consume a large fraction of future designs, scaling of

memory continue to track the scaling trends of logic.

Unfortunately, decreasing the device dimensions will

lead to severe increase of parametric variation which

along with the trend of decreasing system supply and

transistor threshold voltages degrades the stability of

conventional six-transistor (6-T) SRAM cells [1].

Overcoming intra- and inter-die variations is emerging

as a challenge for VLSI designers and in recent years,

several schemes have been proposed to overcome

these challenges (see, e.g., [2], [3], [4].)

The FinFET [5] transistor structure has been

developed as an alternative to the bulk structure for its

improved scalability in submicron era [6]. The gates on

either side of its fin can be tied together or they can be

electrically isolated to allow their independent

operation; this is achieved by selectively removing the

gate material in the region directly on top of the fin. In

the tied-gates operating mode, the two gates are biased

together to switch the FinFET on/off, whereas in the

independent-gates operating mode, they are biased

independently in a way that one gate is used to switch

the FinFET on/off and the other gate is used to adjust

the threshold voltage [7] and this offers dynamic or

static performance tunability which gives the designers

a great flexibility [8]. These characteristics can be

utilized to improve the SRAM cell; as for example in

[9] authors proposed using this characteristic of

FinFET based SRAMs along with two wordlines. They

reported better cell stability with minimum area

expansion.

Considering this, a statistical design centering

scheme for robust design of FinFET SRAM cells has

been proposed in this paper; the value of back-gate

voltages of the six transistors of the cell is found in a

way that the minimum read, write, access and hold

errors are observed. In optimization of these voltages,

the statistical distribution of gate length and silicon

thickness of the transistors of the cells was also

considered. The proposed scheme is general and may

be applied to other circuits.

This paper is organized as follows. In Section II, the

technology parameters and the size of transistors used

in the simulations are discussed. A review of the

stability criteria of the SRAM cells and its different

failure mechanisms are discussed in Section III. The

proposed design methodology for the SRAM cell is

described in Section IV where the experimental results

are also discussed. Finally, the conclusion is given in

Section V.

Figure 1. Schematic of a conventional 6-T SRAM cell.

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II. DEVICE PARAMETERS

Fig. 1 shows a typical 6-T FinFET based SRAM

cell. The nominal values of key design parameters are

summarized in Table I. For implementing the proposed

scheme, we used a 45nm FinFET technology [10]. We

assumed that the pull-up and pass-transistors have one

fin and for decreasing the probability of read error, it

was decided to have pull-down transistors with two

fins. A triple-fin FinFET is shown in Fig. 2 [2].

Table 1 NOMINAL DEVICE PARAMETERS USED FOR HSPICE

SIMULATION [10].

L (Channel Length)

tox (Oxide Thickness)

tsi (Silicon Thickness)

VDD (Power Supply)

NBODY (Channel Doping)

Hfin (Fin Height)

VTH0 nmos typical (Threshold

Voltage of NFET)

VTH0 pmos typical (Threshold

Voltage of PFET)

45 nm

15 A

8.4 nm

1 V

2e16 cm-3

60 nm

0.31 V

-0.25 V

Figure 2. Triple-fin FinFET structure [2].

Fig. 3 shows the SRAM array and its corresponding

parameters. A 128 column × 256 row memory array

which has a size of 32Kb has been assumed. The

wordlines and the bitlines are modeled as distributed

RC networks. The resistance and the capacitances of

wordline and bitline were derived from predictive

interconnect model for the 45nm technology node [10]

and is shown in Fig. 3. We did not include the sense

amplifiers in the simulations; but it is assumed that the

read operation is performed when a pre-specified

voltage difference (Δmin ≈ 0.1VDD) is produced

between the two bitlines.

III. FAILURE MECHANISM IN SRAM CELLS

A robust SRAM cell design requires a careful

balancing between different parameters in the presence

of many contradictory constraints. Here, we present a

brief review of the failure mechanisms in SRAM cells

and define some merits to assess them quantitatively.

Figure 3. SRAM circuit schematic and its parameters.

A. Read Failure

Figure 4. Unstable Read.

During the read operation of the cell shown in Fig.

1 (VL= 1 and VR=0), the voltage at node R (VR)

increases to a positive value VREAD, due to voltage

dividing between right access transistor (AR) and right

pull down transistor (NR). If VREAD is higher than

the trip point of the left inverter (VTRIP,RD), then the cell

flips while reading the cell and a read failure occurs

(Fig.4Error! Reference source not found.). If the

access and pull down transistors have the same

threshold voltage, the W/L (width/length) ratio of NR

to that of AXR will determine how high VR will rise.

The ratio is commonly referred to as the cell β ratio.

The read stability can be increased by upsizing the

pull-down transistor (increasing its fins) or decreasing

its threshold voltage by increasing its back-gate

voltage, which results in an area and leakage penalty

and/or decreasing the threshold voltage of the access

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transistors which increases the delay. For a robust

SRAM cell design, the random variations in the

strengths of different transistors should also be taken

in account.

Figure 5. Read butterfly plot of a SRAM cell with back-gate

voltages of all transistors being zero is depicted. The Read SNM

is 38.5 mV.

Usually read SNM is used as a merit for the

stability of the cell against read failures. In

construction of butterfly curves of the cell in the read

operation, the AR and PR can be considered as parallel

transistors which degrades the gain of the inverters and

hence the butterfly curves became narrower. The

maximum length of a square which can be fit in to the

butterfly curves is a good merit for the static noise

margin of the memory [11]. In Fig. 5, the read butterfly

curve of the cell are depicted while back-gate voltage

of all the transistors is zero. In this non-optimum point

of operation, the read SNM is 39 mV which is

relatively very low.

B. Access Time Failure

The cell access time (TACCESS) is defined as the time

required for producing a pre-specified voltage

difference (e.g. Δmin ≈ 0.1VDD) between the two

bitlines of the cell. If due to a parametric variation,

most importantly a threshold voltage variation, the

access time of the cell is longer than a maximum

tolerable limit (TMAX), an access time failure occurs.

The access failure is caused by the reduction in the

strength of the access and the pull-down transistors. In

other words access failure can be caused by an

increase in the threshold voltage of the AR and/or NR

transistors. Thus, the inter-die variation which changes

the parameters in the same direction may increase the

access failure, too [12]. This is different from the read-

failure which only occurs in the case of intra-die

variation or in other words in the case that the strength

of neighboring transistors may change in different

directions.

Figure 6. Voltage values for a cell, with back-gate voltages of all

transistors being zero, are depicted while accessing the memory.

The voltage difference of bitlines in this case is 96 mV.

The amount of voltage difference between the two

bitlines of the cell, at the moment that wordline

deactivates, is a good merit for the stability of the cell

in regard to access failure. In Fig. 6, the voltage values

of bitlines and other related signals while reading the

cell is depicted. We assumed for this technology that

the wordline will be high for duration of 90 ps. In the

case that the backgate voltages of all the transistors are

zero, the voltage difference between the bitlines is

0.096 V, which may be acceptable.

C. Write Failure

When writing a "0" to a cell storing "1," the node

VL becomes discharged through BL to a lower value

(VWR) determined by the voltage division between the

left PMOS pull down transistor (PL) and the left

access transistor (AL). If the voltage of VL cannot be

reduced below the trip point of the right inverter (PR −

NR) (VTRIP,WR) within the time window that the

wordline is high (TWL), then a write failure occurs, Fig.

7.

Figure 7. Unstable Write.

The variation in the device strengths due to random

variations in process parameters can increase the write

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time. For example, if the threshold voltage of PL is

reduced and that of AXL is increased, the write time

will increase and a write failure may be observed [12].

Figure 8. Voltage values for a cell in the write operation. The x

axis is the value of the bitline voltage which is supposed to be

ideally zero.

Usually, the maximum voltage value of the bitline

that is going to write '0' to a cell containing '1' is

considered a good merit for the stability of the cell in

regard to write error. In Fig.8, the voltage values of the

storage nodes in respect to the bitline's voltage, which

is supposed to be ideally zero, are depicted; the other

bitline is kept as VDD in this simulation. In the case that

the backgate voltages of pull-down, access, PL and PR

transistors are 0.4 V, 0.5 V, 0.5 V and 0.3 V

respectively, the write margin is around 0.64 V.

When there is a mismatch between right and left

sections of a cell, the write margin must be calculated

twice. One time for the case that VL=1, VR=0 and the

BLC is zero. In the second time, VL=0, VR=1 and BL

is zero. The write margin of the cell is the minimum of

these two values.

D. Hold Failure

In the stand-by mode, the VDD of the cell is lowered

to reduce the leakage power consumption [13].

However, if lowering VDD causes the data stored in the

cell to be flipped, then the cell is said to have failed in

the hold mode. As the supply voltage of the cell is

lowered, the voltage at the node storing "1" (suppose

node VL) also reduces. Meanwhile, for a low supply

voltage (when PL is not strongly "ON"), the leakage of

the pull-down NMOS (NL) reduces the voltage at node

VL, even below the supply voltage applied to the cell.

In this case, if the voltage at the node L is reduced

below the trip-point of the right inverter, then flipping

occurs and the data is lost in the hold mode (Fig. 9).

The supply voltage in the hold mode is chosen to

ensure the holding of the data under the nominal

conditions. However, the variations in the process

parameters can result in the device mismatch causing

hold failures. For example, if the threshold voltage of

NL reduces while that of PL increases (which

facilitates the reduction of the voltage at node L from

the applied supply voltage) and/or if the threshold

voltage of NR increases, while that of PR reduces

(increasing the trip-point of the right inverter), the

possibility of the data flipping in the hold mode

increases [12].

Figure 9. Unstable Hold.

The hold margin is usually taken as the static noise

margin of the cell while the cell is in the hold mode.

The butterfly curves of the cell operating in the hold

mode ( when the wordline is deactivated) is obtained

and the length's of the maximum square, that can be fit

in to it, is considered as the static noise margin of the

cell in the hold mode. The procedure is the same as

calculating the read noise margin, but this time, the

increase in the gain of the inverters makes the signal to

noise margin bigger. In the case that all the backgate

voltages of the FinFET transistors are kept at zero, the

hold margin is 0.1561 V (Fig. 10).

Figure 10. Hold Butterfly plot of a SRAM cell with back-gate

voltages of all transistors being zero is depicted. The Hold SNM

is 0.1561 V.

IV. METHODOLOGY AND RESULTS

Each transistor of the cell has few parameters that

can be changed during the design, for example its

back-gate voltage or number of its fins. Some of these

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parameters are fixed when the technology is

determined; examples of such parameters are the

doping and the gate oxide thickness. All of these

parameters have statistical variations which should be

considered during a robust design of the cell. In

addition, a robust design against a range of variation

for a specific parameter, such as power supply noise,

may require a worst case design for that parameter.

Considering all these parameters for a robust design of

a cell with a brute force search needs a huge amount of

computation efforts which in some cases, it may be not

feasible. To overcome this challenge, we propose a

scheme which is explained next.

For reaching to the optimum yield against read,

write, access and hold errors; statistical design

centering was performed. There are lots of methods for

doing the job; depending on the application, the best

algorithm should be selected. A good review of these

methods can be found in [14]. First, for giving a good

intuition to the reader, the Director’s simplicial

approximation method [15][16] is briefly explained in

section A. Then the probabilistic variables of SRAM's

cell is discussed in section B, and according to their

characteristics a proper statistical design centering

method is selected and applied in section C.

A. Director's Method

In this technique, starting from some points in the

acceptable area of parametric space, with linear search

at least n+1 nodes in the boundary of the desired space

is found, n is number of the parameters. The result of a

specific function is considered to be acceptable when it

is larger or equal to a specific value (e.g., in the write

operation of the SRAM, write margin must be bigger

than 0.4 V). The boundary of the desired space for a

function is defined as the collection of the points at

which the function has the result equal to the minimum

value of the condition while the other functions must

completely satisfy their constraints. (For example the

write function boundary is the collection of the point

that the write margin is exactly 0.4 and read, access

and hold criteria are all satisfied completely.) After

finding some points in the boundary of the space, a

convex hull with these points is constructed. The

largest polyhedron inscribed within it gives an

approximation of the desired space. Using a linear

search, more points on the boundary is found and the

convex hull is updated. The process thus provides a

monotonic lower bound on the yield. The center and

the radius of the hypersphere can be used to determine

the centered optimal point and its tolerance,

respectively.

B. SRAM's Probabilistic Variables

As shown in [17], there exists a statistical

dependency between the gate lengths of adjacent

devices. The channel length and the silicon thickness

of submicron FinFETs have also been considered as

the major sources of the parametric variations [4].

Usually, the variance for these two parameters is

considered to be one tenth of the designed value [4].

These two kinds of variations, which stem from

imperfections in the lithography process, usually have

a Gaussian distribution and their values show a

statistical dependency between each others [3][17].

Note that the closer the transistors are, the greater their

statistical dependence is. In [3], the channel width

covariance of neighboring transistors was considered

to be 1. In other words, it is almost impossible to have

a transistor with large channel length in the

neighborhood of a transistor with small channel length.

If it is assumed that the transistors of SRAM cell have

a perfect statistical dependence, a major source of

intra-die variation will be solved. In other words, there

will be no mismatch between the six transistors and the

cell will be almost always stable, which is a trivial

solution. In this paper, the covariance of L and Tsi of

FinFET transistors in a cell has been assumed to be 0.5

and the statistical dependence between adjacent cells

was assumed negligible.

In any SRAM cell, we have three different

transistors that their relative strength should be

optimized. In this work, we used the double gate

FinFETs where the back-gate voltages of these three

transistors are used in our design centering technique.

To consider the imperfection in the circuit that

produces these three voltages, we also considered a

variance of 0.05V for each of them. A summery of

statistical variables of this design can be found in

Table 2 which implies a design space with 18

(6Transistor×3variable) dimensions.

TABLE 2. MEAN, VARIANCE, AND COVARIANCE OF THE

STATISTICAL VARIABLES USED IN THE DESIGN.

Parameter Mean

L 45nm

Tsi 8.4nm

Vbg X1,..,X6

Variance

4.5nm

0.84nm

0.05V

Covariance

0.5

0.5

0.0

C. Seifi's Method

The Director’s simplicial approximation method has

some shortcomings. First, it was indirectly assumed

that all of the variables have independent statistical

behaviors. Second, this technique also requires the

convexity of the acceptable region which may not be

always the case, thus other feasible methods should be

used. Recently, many research efforts have been

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devoted to the optimal design of integrated circuits

considering the statistical variation. (e.g. [18], [19])

These techniques do not assume independent statistical

behaviors. The technique proposed in [18] considers

the parameters as dependant Gaussian variables. Since

in this work, we make the same assumptions for our

parameters, we used its proposed technique.

In this algorithm, given the covariance matrix of the

input parameters (can be constructed by Table 2), the

statistical distance of any nominal point in the

parametric space from the surface of the desired space

(tolerance) can be found. For doing that, at first using

the iterative Eq. 1, the point in the boundary of

equation (x*) that has the minimum distance to the

nominal point (μ) is found by the following iterative

equation.

kk

k

kTk

x

G CG

C is the covariance matrix of the input parameters,

G is the value of the first derivative of the condition

and g(.) is the condition equation, itself. Finally βk

should be found by the below equation.

()

[()]G CG

The above equation converges quickly and yields x*

and β which the later is the minimum statistical

distance of nominal point from the boundary of the

condition. For example, if the distance (β) of the

nominal point from the surface of the condition is 1, 2

and 3, the yield of the system in respect to that

condition will be 52.4%, 93.2% and 99.6%,

respectively.

As it can be seen from the above equations, the

value of the first derivative of the condition in some

points is required, which can be easily calculated by

numerical solutions from the value of the main

function.

Repeating the above procedure for all the

conditions, the minimum distance of any point from

the boundaries of all the conditions is found; which we

called them βread, βwrite, βaccess and βhold. For reaching to

the maximum yield, a cost function is defined (Eq. 3).

exp(max(

max(,3)

writewriteaccess

αβα++

Eq. 3

αread, αhold, αwrite and αaccess are the relative weight of

read, hold, write and access errors. In this paper, it was

assumed that the all of them have the same importance,

so the weight coefficients were taken equal.

For implementing the proposed method, a code in

MATLAB changes the parameters of circuit model in

1

1/2

]

()

[()()

kk

kTk

CG CG g x

G CG

β

μ

+=−−

Eq. 1

1/2

kTk

k

kTk

xG

μ

β

−

= −

Eq. 2

,3)

max(

max(

,3))

,3)

readreadholdhold

access

β

COST

αβαβ=+

several HSPICE netlists as is required by the design

centering algorithm. Then the HSPICE simulator is

called and the transient analysis is performed to check

whether read, write, or hold error have been occurred

and if they are not occurred their corresponding noise

margin is calculated. For doing this task, the result of

transient analysis are first loaded to MATLAB with the

help of HSPICE toolbox, and then read, write, access

and write margins are calculated automatically. These

margins are the results of complex functions that

should be optimized, by changing the backgate

voltages.

Thus, the problem converts to finding the maximum

of the cost function in the design space. Any linear or

genetic search can be applied for solving this kind of

max-min problems. The genetic algorithm, however, is

usually faster [20]. So, for optimization, the value of

the cost function is then taken as the fitness function of

the genetic algorithm. Every combination of the six

back-gate voltages is considered as a member of the

population. After around six generations, the algorithm

converges to the solution with the maximum yield.

Note that in the 6-T structure, we have only three

different transistors. This, however, does not mean that

the design centering can be performed in a nine-

dimensional space. The reason is that the values of the

similar transistors may change in different direction in

the manufacturing process. This symmetry however

leads to only three different final centered (optimal)

values. In other words, the desired space is

symmetrical to three different superplanes. Finally, it

should be noted that there exist other sources of

parametric variation besides the channel length and the

silicon thickness. For the state of the art technology,

they however may be neglected [4]. One of these

sources is random dopant fluctuations in the channel.

The effect of this fluctuation is minimized due to the

fact that the channel in the double gate devices is

undoped or mostly-undoped [21]. In the cases that this

effect should be considered, it should be remembered

that, in the state of the art technologies, this effect

causes the threshold voltage not to have a Gaussian

distribution [21] it is shown that the distribution of

threshold voltage mostly resembles the discrete

Poisson distribution [22]. For these cases, other

proposed methods such as [23] should be used.

In Table 3, the results of the proposed technique for

the optimal values of the back gate voltages of the

transistors can be found, with this optimal value the

yield of the cell will be around 82%. The results were

also double-checked by a rigorous Monte-Carlo

simulation with more than 5 times of computation

effort.

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TABLE 3. BACK GATE VOLTAGES OF THE OPTIMAL CELL (NOTE

THAT PD TRANSISTORS HAVE 2 FINS)

Back Gate Voltage

Pull Up

Pull Down

Pass Gate

Value

0.87V

0.26V

0.85V

V. CONCLUSIONS

A method for optimizing the design of double-gate

FinFET based SRAMs was presented. The method

used a two-stage optimization process to maximize the

yield. It was shown that by modulating the back-gate

voltage of the three kinds of transistors in the SRAM

cell, one can reach a very high yield against read,

write, access and hold errors. The final results have

also been verified by Monte-Carlo simulation.

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