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SRAM Dynamic Stability: Theory, Variability and Analysis

Wei DongDepartment of ECE

Texas A&M University

College Station, TX 77843weidong@neo.tamu.edu

Peng LiDepartment of ECE

Texas A&M University

College Station, TX 77843pli@neo.tamu.edu

Garng M. HuangDepartment of ECE

Texas A&M University

College Station, TX 77843ghuang@neo.tamu.edu

ABSTRACT

Technology scaling in sub-100nm regime has significantlyshrunk the SRAM stability margins in data retention, readand write operations. Conventional static noise margins(SNMs) are unable to capture nonlinear cell dynamics andbecome inappropriate for state-of-the-art SRAMs with shrink-ing access time and/or advanced dynamic read-write-assistcircuits. Using the insights gained from rigorous nonlin-

ear system theory, we define the much needed SRAM dy-namic noise margins (DNMs). The newly defined DNMsnot only capture key SRAM nonlinear dynamical character-istics but also provide valuable design insights. Furthermore,we show how system theory can be exploited to developCAD algorithms that can analyze SRAM dynamic stabil-ity characteristics three orders of magnitude faster than abrute-force approach while maintaining SPICE-level accu-racy. We also demonstrate a parametric dynamic stabilityanalysis approach suitable for low-probability cell failures,leading to three orders of magnitude runtime speedup foryield analysis under high-sigma parameter variations.

1. INTRODUCTIONStatic random access memories (SRAMs) provide indis-

pensable on-chip data storage and continue to dominate thesilicon area in many applications. It is projected that morethan 90% of silicon real estate will be occupied by SRAMin the future [1]. However, technology scaling in sub-100nmregime has significantly shrunk SRAM stability margins indata retention (standby mode), read and write [13]. At thesame time, the susceptibility to single event upsets (SEU)induced soft errors continues to cause concerns [4].

The SRAM performance and its variability have been ex-tensively studied in the past [59]. In [2, 10], either simu-lation based first-order models or closed-form performancemodels using simple transistor models are employed to de-rive SRAM statistical performance distributions. Mixtureimportant sampling is applied to speedup Monte-Carlo simu-

lation to more efficiently capture rare failure events [11]. Thesame objective is approached by combining extreme valuestatistics theory and data filtering in [12]. Euler-Newtoncurve tracing is proposed to find the boundary between thesuccess and failure regions for read access time yield analysisin [13]. A semi-analytical SRAM dynamical stability modelis proposed in [14], where approximated circuit equationsbased on simple device models are solved in time domain.The use of piecewise linearization of circuit equations, how-ever, can lead to inaccuracy and the challenging issue ofprocess variations is not addressed.

It is important to note that stability occupies a centralrole in SRAM operations. While the stability in standbyor hold implies proper data retention under SEUs and noiseinjection, stabilities in read and write correspond to non-destructive reads and successful writes, respectively. How-ever, the widely used static noise margins (SNMs) [5] cannotcapture the fundamental nonlinear dynamics upon whichSRAM cells operate. Although simple to obtain, SNMs as-sume the DC operation condition and are used with the

assumption that timing events have infinite time duration.Due to the intrinsic complexity, dynamic noise margins areresearched to a much less extent. In many ways, they maynot have been defined rigorously. However, dynamic stabil-ity metrics are strongly desirable because of the followingreasons. SEU and noise induced soft error analysis requiresan understanding of the duration, amplitude, and chargeof the injected noise and their interactions with the non-linear SRAM cell dynamics. Practically, reads and writesbehave in an increasingly dynamic fashion in state-of-the-art SRAM designs with shrinking access cycle time and/orread-write-assist circuitry [3, 15]. In the latter case, welltimed wordline pulses and write schemes are employed toenhance the read and write margins. Successful reads and

writes depend on precise timing control where the nonlineardynamics of SRAM cells plays a critical role. In design, bal-ance must be made between conflicting static and dynamicstability margins in hold, read and write while consideringtheir variability.

In this work, we start by developing an understanding onthe basic nonlinear dynamics of SRAM cells using rigorousnonlinear system theory. In particular, we employ the no-tion of stability boundary, or separatrix [16,17], and show itscentral role in determining SRAM dynamic stability. Usingseparatrix, new dynamic noise margins (DNMs), in a wayrelevant to basic SRAM operations, are defined. The newDNMs not only characterize the fundamental system charac-teristics behind SRAM operations, but also provide valuabledesign insights by connecting dynamic stability with key de-

sign parameters. Interestingly, the conventional SNMs arespecial cases of our more general DNMs.

To embody our system concepts and DNM metrics into apractical CAD tool with SPICE-level accuracy, we show howefficient system-theoretically motivated CAD algorithms canbe developed. By exploiting nonlinear system theory, a fastseparatrix tracing algorithm for SRAM cells under devicemismatch is developed. The entire separatrix can be ef-ficiently computed by running two special transistor-leveltransient simulations, achieving three orders of magnitude

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speedup over the brute-force approach. With this as a ba-sis, we further present a parametric DNM analysis approach,which for a given DNM performance target identifies theacceptance regions in the parameter space. The approachemploys Newton method based acceleration and allows forefficient dynamic stability yield analysis over wide rangeof parametric variations, speeding up expensive high-sigmaMonte-Carlo simulation by three orders of magnitude.

2. BACKGROUNDConventional SNMs in hold and read are shown in Fig. 1.

In hold, the conventional static noise margin (SNM) is deter-mined as the side of the largest square that can be inscribedbetween the mirrored DC voltage transfer curves (VTCs) ofthe cross-coupled inverters [5]. This measure corresponds tothe largest differential voltage noise that can be tolerated atthe two storage nodes. SNM in read can be defined similarlyby including the two access transistors as part of the inverterpair VTCs. SNM in read represents the largest DC voltageperturbation that can be tolerated without a state flip, ora destructive read. During write, one bit line is dischargedand the other is pre-charged. The two inverter VTCs donot form any enclosed region. SNM in write is found by

inscribing the largest square in between the two VTCs. To

WL WL

Bit Bit_B

+-

+-

Vn

Vnholdread

0 Vdd

Vdd

Figure 1: Static noise margins in hold and read.

underscore the importance of dynamic stability, we exam-ine the correlations between the SNM and our new DNMin hold as outlined in Section 4.3. The results are shownin Fig. 2. Totally 600 random samples of a 6T SRAM cellare taken, where independent Gaussian transistor thresholdvoltage variations are assumed. No strong correlation be-tween the SNM and DNM is observed. Hence, in practice,it is difficult to determine whether a better SNM always cor-responds to an improved DNM. The SNM and DNM mustbe both analyzed to provide design guidance.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Normalized Dynamic Noise Margin

NormalizedStaticNoiseMargin

Figure 2: Correlation between SNM and DNM.

3. STABILITY BOUNDARY: SEPARATRIXAn SRAM cell can be described using the MNA equations

f(x(t)) +d

dtq(x(t)) + u(t) = 0, (1)

where x(t) RN is the vector of nodal voltages and branchcurrents, u(t) RN is the input, f() and q() are nonlin-ear functions describing static and dynamic nonlinearities.

Without loss of generality, the simple 2D case (N = 2) isfocused to simplify the visual presentation. As a nonlineardynamical system, a cell has three equilibria, which satisfy

f(xe) = 0. (2)

Two of them are stable and denoted as xse, corresponding tothe zero and one states of the cell. The third one, denotedas xue , is unstable and also referred to as the saddle. Thestable manifold of an equilibrium xe is defined as [16]

Ws(xe) = {x RN| lim

t(t, x) = xe}, (3)

where (t, x) is the state trajectory that starts from x andconverges to xe.

V1

V2

(0,0)

(Vdd, Vdd)

V1

V2

(0,0)

(Vdd, Vdd)Stable equilibrium

Stable equilibrium

instableequilibrium

Stable equilibrium

Stable equilibrium

instableequilibrium

Symmetric SRAM Cell SRAM Cell w/ Mismatch

Separatrix

Figure 3: The separatrix of an SRAM cell: symmet-ric cell vs. cell with mismatch.

To understand SRAM dynamic stability, we consider howthe SRAM state transits from one stable equilibrium to theother (i.e. a state flip) under an SEU, read or write event.The stability region, or region of attraction, of a stable equi-librium is defined to be its stable manifold. There existsa stability region for each stable equilibrium in the statespace of a cell. Starting from any initial state in the sta-bility region, the transient SRAM state trajectory will beattracted towards to the corresponding stable equilibrium.The b oundary between two stability regions, or the sepa-ratrix, splits the entire state space. The importance of the

separatrix lies in that a state flip will be generated if and onlyif the amplitude and duration of the disturbance to the cellare high and long enough so that the state is pushed awayfrom the initial stable equilibrium to cross the separatrix. InFig. 3, the separatrix of a symmetric cell is contrasted withthat of an asymmetric cell with device mismatch across thetwo cross coupled inverters. The vector field (time deriva-tives of the state variables) is shown by arrowed lines. Whilethe separatrix of the former is along the well expected 45

line, the one of the latter is distorted by mismatch.

4. DYNAMIC NOISE MARGINSUsing the concept of stability boundary, new dynamic

noise margins (DNMs) are defined for read, write and hold.

4.1 Dynamic Noise Margin in ReadThe read DNM is defined for a given wordline pulse width

TR. To obtain the read DNM, the circuit setup correspond-ing to the read operation in Fig. 4 (a) is analyzed. Beforethe read starts, the bit and bit-bar lines are modeled as twofully charged line capacitances. Beside the cell being read,other unselected cells in the same column may draw leakagecurrents from either the bit or bit-bar lines depending on thestored value. These leakage contributions can be properlymodeled and included to capture the inter-cell interaction.

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First, the separatrix of the cell in hold, i.e. when the two ac-cess transistors are off, is analyzed (e.g. using the algorithmdescribed in Section 5). Then, starting from an initial zeroor one state, the transient state trajectory of the cell withthe access transistors turned on (Fig. 4 (a)) is simulated.The time it takes for the trajectory to reach the separatrixis denoted as Tacross. As shown in Fig. 4 (b), the read DNMis defined as

TDNM,R = Tacross TR. (4)This is an appropriate dynamic stability metric as follows.As the read process proceeds (the access transistors remainon), the cell state may be pushed away from its initial statetowards the separatrix of the cell with the access transistorsoff. Note that after the wordline goes off, the read operationends and the cell returns to hold. Hence, if the trajectorydoes indeed go across the separatrix, a state flip will begenerated after the access transistors are turned off in hold.

WLWL

VDD

x1x2

CBBCB

Cell1 0 leakage

TRTR

01

Initial charge:Q

BB=V

DDC

BB

Initial charge:Q

B=V

DDC

B

High High

(a)

V1

V2Separatrix: access trans. off

Trajectory: access trans. on

Tacross

TR

Read Margin:

Tacross - TR

(b)

Figure 4: Read dynamic noise margin: a) circuitsetup, and b) definition of read DNM.

The defined read DNM specifies the amount of read oper-ation time margin before read instability takes place. Thatis, when Tacross > TR, there exists a positive margin; whenTacross = TR, the cell is on the verge of read instability;when Tacross < TR, state-flip happens and the cell losesread stability. As can be seen, TDNM,R decreases as TR, thepulse width of the wordline signal, increases. It is easy tosee that when TR = , our read DNM will correctly pre-

dict the static read stability, i.e. whether or not the statewill be flipped if the duration of the read is infinite. On theother hand, if the conventional SNM is used to evaluate thecell read stability when TR < , which corresponds to thenormal read operation, a pessimistic estimate may be pro-duced. That is, even if the SNM predicts a state flip, but inreality, it may not happen.

4.2 Dynamic Noise Margin in WriteThe write DNM can be defined in a way analogous to

that of the read DNM, but by noting that a successful write

overwrites the SRAM state, hence producing a state flip.The cell in write mode is analyzed using the circuit setup inFig. 5 (a). One of the bit and bit-bar lines are dischargedbefore the write starts. Hence, the initial voltage of onebit/bit-bar line capacitor is set to zero. Similar to the pre-vious case, transient analysis is performed to compute thetime, Tacross, at which the state trajectory crosses the sep-aratrix of the hold mode. For a given wordline pulse width,TW, the write DNM is defined

TDNM,W = TW Tacross. (5)

WLWL

VDD

x1

x2

CBBCB

Cell0 1leakage

TWTW

01

Initial charge:QBB=0

Initial charge:QB=VDDCB

High Low

(a)

V1

V2Separatrix: access trans. off

Trajectory: access trans. on

TW

Tacross

Write Margin:TW - Tacross

(b)

Figure 5: Write dynamic noise margin: a) circuitsetup, and b) definition of write DNM.

When TW is set to , our DNM can correctly predictthe static write-ability. On the other hand, when TW < ,which corresponds to the normal write operation, however,the SNM may provide an optimistic estimate for dynamicwrite-ability. That is, even if the SNM predicts a success-ful write, in the reality, the write can actually fail. Forthe state-of-the-art SRAM designs with short access cyclesand advanced read/write timing control circuitry, the dis-tinctions between the SNMs and DNMs in read and writereveal the important role of cell nonlinear dynamics in de-termining dynamic SRAM stability.

4.3 Dynamic Noise Margin in HoldThe DNM in hold characterizes the data retention prop-

erty of the cell under SEUs and noisy operating condition.Due to the scope limitation, the hold DNM is only brieflyintroduced. As in Fig. 6, the DNM may be examined by in-jecting a current disturbance into the cell. Compared withthe use of noise voltage disturbance in the SNM [5], model-ing the disturbance as an injected current more physicallyreflects the nonlinear dynamic nature of the cell. The DNMshall be evaluated by considering both the amplitude andduration of the current disturbance. Depending on thesetwo factors, the state trajectory in hold may cross the sep-aratrix, leading to instability. As in read, the conventional

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SNM may provide a pessimistic stability estimate in holdwhen blindly applied under dynamic noise injection.

T0

I0

V2

V1

(v1(T0), v2(T0))

Separatrix:Sep(v1, v2) =0

Transient trajectory

Figure 6: Hold dynamic noise margin.

5. TRACING SEPARATRIXConsider a brute-force approach for finding the separatrix

in hold as shown in Fig. 7 (a). The entire state space ofthe SRAM cell is sampled using a dense grid. Then, eachgrid point is used as an initial condition in transistor-leveltransient simulation. Starting from each initial condition,the SRAM state trajectory may be attracted eventually toone of the stable equilibria, i.e., the sampled point (initialcondition) in the state space falls into the stability region ofthe corresponding equilibrium. If a large number of samplesare taken, the separation between points falling into the twostability regions forms the separatrix. The number of tran-sient runs required in this approach, however, makes it timeconsuming. The high cost of the brute-force approach makesthe statistical dynamic stability analysis, where parametervariations are considered, even more difficult.

X1

X2

(0,0)

(Vdd, Vdd)Stable equilibrium

Stable equilibrium

saddle

Separatrix

(a)

X1

X2

(0,0)

(Vdd, Vdd)Stable equilibrium

Stable equilibrium

saddle

Separatrix

(b)

Figure 7: Finding the separatrix: a) brute-forcedense sampling, and b) fast stable-manifold tracing.

To derive a much more efficient system-theoretical alter-native, denote the stability boundary of the two stability re-gions as A. For SRAM cells, the following stability bound-ary theorem exists [16,17]:

Theorem 1. The separatrix (stability boundary) of thetwo stable equilibria is the stable manifold of the unstableequilibrium (saddle point), i.e.,

A = Ws(xue ), (6)

where xue is the unstable equilibrium on the boundary of A.

Importantly, Theorem 1 implies that to find the stabilityboundary, one shall identify the saddle on the stability bound-ary and then find its stable manifold. From the view pointof Stable Manifold Theory [18], the stable eigenvector of thelinearized system at an equilibrium is tangent to its corre-sponding stable manifold. This implies that by starting ina neighborhood of xue , the desired stable manifold can be

found by integrating the system equations along the direc-tion of the stable eigenvector backward in time. Integrationbackward in time prevents the state trajectory to convergeto any of the equilibrium points and enforces the trajectoryto stay on the stability boundary.

The above theoretical results provide a foundation for thefollowing highly efficient separatrix tracing algorithm.

Algorithm 1 Fast Stable-Manifold Separatrix Tracing

1: Compute the equilibria (xse and xue ) for the cell;2: Find the stable eigenvector, us, for the saddle xue : the stable

eigenvector corresponds to the stable eigenvalue at the saddle;3: Choose two initial conditions:

x0{1,2}

= xue us, is small;

4: Perform two transient simulations from initial conditionsx0{1,2} for the modified dynamical system:

f(x(t)) d

dtq(x(t)) = 0; (7)

5: Output the separatrix as:

A = {x|x = M(t, x01) or M(t, x

02), t = [0, Tmax]}, (8)

where M is the state trajectory of the modified system, andTmax is the maximum duration of the two transient runs.

The tracing algorithm is illustrated in Fig. 7 (b). In con-trast to the original system in (1), integration backward intime is reflected by negating the differential term in themodified system in (7). Intuitively, negating the differen-tial term reverses the vector field, as shown in the dashedarrow lines in Fig. 7 (b) (compare with the original field inFig. 3). Note that along the separatrix, the direction of thevector field points away from xue . Elsewhere in the statespace, the vector field points towards the separatrix. Sincethe two transient simulations do not start from equilibriumxue , but from a neighborhood of it, the state trajectorieswill move. Due to the new vector field, they do not con-verge to the saddle nor the two stable equilibrium points,

but are actually forced to stay on the separatrix, making itpossible to efficiently find the entire separatrix by using onlytwo transient runs. In practice, step 2 of the algorithm canbe bypassed as long as the two initial conditions are per-turbed from the saddle along opposite directions. Here, theexact stable eigenvector us is not required since the unsta-ble components dissipate fast as we integrate backward intime [16, 18]. Another practical issue is to find the saddle.Usually the saddle can be found directly in one DC analy-sis with suitable choice of the initial guess and convergencecontrol. We have also found that the saddle could be foundrather reliably through one transient simulation of the mod-ified dynamic system (7) starting from an initial condition.

6. DYNAMIC STABILITY YIELDThe proposed separatrix tracing algorithm enables DNManalysis through efficient computation of the separatrix, whichis central to dynamic stability. Nevertheless, statistical SRAManalysis via direct Monte-Carlo simulation may be still pro-hibitively expensive. Large SRAM memories may have tensof millions of or even more cells. To achieve a good yield forthe entire SRAM system, each SRAM cell must be designedto tolerate wide range of parametric variations such as ran-dom dopant fluctuation induced threshold voltage variation.Cell-level failures are rare events with very low probability

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and accurate yield estimate requires a very large numberof Monte-Carlo samples. To further improve the efficiencyof dynamic stability yield analysis, we proposed a Newtonmethod accelerated parametric analysis technique.

6.1 Problem FormulationWithout loss of generality, the proposed yield analysis is

described using the read DNM as an example. For a givenwordline pulse width TR and a target read DNM Ttarget, us-

ing (4), the read DNM yield is given as the following prob-ability

YDNM,R = P r{TDNM,R(p) Ttarget}

= P r{Tacross(p) Ttarget + TR}

= P r{Tacross(p) Tacross,0}, (9)

where p RM is a set ofM parameter variations with p = 0corresponding to the nominal condition, and Tacross,0 =Ttarget + TR is a constant. Instead of performing Monte-Carlo simulation, DNM yield can be computed more effi-ciently by identifying the acceptance region in the parameterspace, accept, i.e., the parameter subspace that correspondsto SRAM cells meeting the DNM specification Ttarget. YDNM,Rcan be then simply computed as the probabilistically weightedvolume or area of the acceptance region

YDNM,R =

P

g(p)f(p)dp, g(p) =

1 if p accept0 otherwise

(10)where f() is the PDF of p.

For ease of presentation, the identification of the accep-tance region in a two-dimensional parameter space (M = 2)is considered, which is shown in Fig. 8 (a). From the nomi-nal design at the center of the acceptance region, four initialdirections are selected to search for the boundary of the ac-ceptance region, accept, in each quadrant

accept = {p | Tacross(p) = Tacross,0}. (11)

Possible initial directions can be along the 45

, 135

, 225

and 315 lines. It shall be noted that, in principle, it ispossible that the search along a direction does not reach anyboundary, indicating the acceptance region is not closed.

p1

p2

FailureRegion

Accept.Region

Tacross (p) = Tacross,0

SearchDirections

(a)

il

it

in

ii nl

=+1

1+it

boundary

di

di+1

SearchLength l

StartingPoint

(b)

Figure 8: DNM yield analysis: a) find the accep-tance region boundary, b) search along a direction.

6.2 Acceleration via Newton MethodThe search for accept is detailed in Fig. 8 (b), where

the search direction is adapted. A complete search cycle isshown in the figure. Starting from an initial point in the

parameter space, a search along a unit-length vector li is

initiated to find a point di at accept. Then, the tangentdirection, ti, and normal direction, ni, of the acceptance re-gion boundary are evaluated at di. The search moves over ashort distance along ti. To move back to accept, the search

continues along the new search direct li+1, which is set to beni. The procedure repeats till a set of points on accept arefound. This general principle of acceptance region boundarytracing is similar to the idea in [13]. In our case, however,the search direction adaption as described above is used toensure a reliable and fast tracing of accept. More impor-tantly, complications arisen specifically from the dynamicstability analysis must be properly handled as below.

The key step in the above procedure is to find a point

on the boundary along search direction l starting from aninitial point p0. Such p satisfies the line equation p = p0 +

l l, where l is the search length. For p accept, thenonlinear scalar equation in l must be satisfied

(l) Tacross( p0 + l l) Tacross,0 = 0. (12)

To speedup the solution of the above equation using Newtonmethod, the key task is to compute the derivative (l)/l,which is the focus of the subsequent discussion.

6.2.1 Derivation of Newton DerivativesThe separatrix in hold may be specified as a general non-

linear separatrix line equation in state variables x = [x1, x2]T

and the parameter variation l

Sep(l, x1, x2) = 0. (13)

In practice, after the separatrix is computed through theproposed tracing algorithm, it can be represented as a piece-wise linear function connecting the points along the tracingtrajectories. For any given guess for l, Tacross can be com-puted by performing a transient simulation of the cell inread from an stable equilibrium initial condition to checkwhether (12) is satisfied or not. This is shown in Fig. 9 (a).At Tacross, the transient trajectory satisfies (13), specifically

Sep (l, x1(Tacross(l), l), x2(Tacross(l), l)) = 0.(14)

X1

X2 Separatrix: Sep( l, X1, X2) =0

Derivatives alongseparatrix

Derivativesalong trajectory

(a)

t

X

0 t1 t2 tn

l

ltx

),(1

l

ltx

),( 2

l

ltx n

),(

(b)

Figure 9: Derivative computation: a) key deriva-tives for Tacross, b) parametric transient sensitivities.

Here, both the state variables and Tacross depend on l.If (12) is not satisfied, (l)/l = Tacross/l shall becomputed such that Newton method can be used to correctl. Differentiating both sides of (14) w.r.t l gives

Sep

l+

Sep

x1

x1Tacross

Tacrossl

+Sep

x1

x1l

+

Sep

x2

x2Tacross

Tacrossl

+Sep

x2

x2l

= 0, (15)

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or equivalently

(l)

l=

Tacrossl

=

Sep

l+ Sep

x1

x1l

+ Sepx2

x2l

Sepx1

x1Tacross

+ Sepx2

x2Tacross

. (16)

6.2.2 Computation of Newton Derivatives

In (16),

Sep

x1 and

Sep

x2 are the sensitivities of the separa-trix line equation, where the transient trajectory intersectsthe separatrix. They can be readily computed once the sepa-ratrix is traced out using Algorithm 1. x1

Tacrossand x2

Tacrossare the time derivatives of the transient trajectory, again atthe intersection b etween the transient trajectory and theseparatrix (Fig. 9 (a)), hence they are readily available fromthe simulated transient waveforms.

The key remaining components in (16), Sepl

, x1l

andx2l

, are the derivatives of the separatrix line equation andstate trajectory (at Tacross) w.r.t the parameter variationl evaluated at the intersection. Note that in this work theseparatrix line equation Sep() is formed as a piecewise linearfunction connecting the points along the traced transienttrajectories. Hence, it suffices to compute the sensitivities

of the transient trajectory w.r.t l and then use the chainrule to finally get Sep

l. Therefore, for any of Sep

l, x1

land

x2l

, what remains to be discussed is to compute transientresponse parametric sensitivities at certain time instance.It turns out that this can be achieved by accumulating theparametric sensitivities throughout the transient analysis,a technique employed in time-domain shooting method forsteady-state RF simulation [13,19], as shown in Fig. 9 (b).

Consider the parameter form of the MNA equations

f(x(t), p) +d

dtq(x(t), p) + u(t) = 0. (17)

Using a standard numerical integration method, say Back-ward Euler, in transient analysis, (17) is discretized over a

set of time points [t0

, t1

, t2

, , tK ]

f(x(ti), p) +q(x(ti), p) q(x(ti1), p)

ti ti1+ u(ti) = 0. (18)

Differentiating (18) w.r.t a parameter pj , pj p, leads toq

x x

pj+ q

pj

ti

q

x x

pj+ q

pj

ti1

ti ti1+

f

x

x

pj+

f

pj

ti

= 0. (19)

Define: Gi =fx|ti , Ci =

qx|ti , di =

fpj

|ti , ei =q

pj|ti ,

si =x

pj|ti , and t = ti ti1. (19) can be written as

si =

Citi

+ Gi

1

Ci1si1

ti+ e

i1 eiti

di

. (20)

Note that di and ei can be obtained by computing the sensi-tivities of device model evaluations. The matrix inversion in(20) can be facilitated by reusing the matrix factorization inthe last Newton iteration of the transient simulation for eachti. Hence, by accumulating the transient response sensitivi-ties starting from time t0 according to (20), sensitivities atany other time points can be rather efficiently computed aspart of the transient analysis. The sensitivities w.r.t. other

parameters in p can be computed individually in the samefashion. Denote the sensitivity vector of state x1 (x2) w.r.tall the parameters at time ti as s1,i (s2,i), the scaler sensi-

tivity along the search direction l is simply s1,i,l =

lT s1,i

(s2,i,l

= lT s2,i).The aforementioned parametric dynamic stability analysis

can be extended to include more transistor parameter vari-ations, which amounts to search the acceptance boundaries

in a higher dimensional parameter space. The same Newtonmethod acceleration can be applied with a more involvedsearch direction control.

7. EXPERIMENTAL RESULTSThe proposed techniques are implemented using C/C++

as part of a SPICE-like simulation environment on a Linuxserver with 3.0GHZ clock frequency and 2GB memory. Weconsider a 6-T SRAM cell structure with 1V supply voltage,as shown in Fig. 10(a), under various parameter settings.

7.1 Separatrix TracingFirst, consider the case where the cell is fully symmetric.

As in Fig. 10(b), the saddle is found to be at (0.4229363V,

0.4229363V). The separatrix is obtained using the proposedseparatrix tracing method via two transient runs. As ex-pected, the separatrix is along the 45 line, which verifiesthe correctness of our algorithm. Next, we consider two cellswith transistor parameter variations across the two invertersand show their separatrice in Fig. 11(a) and Fig. 11(b). Inthe former case, the threshold voltages of N-type transistorsM1 and M3 are both increased by 20% and the thresholdvoltages of P-type transistors M2 and M4 are decreased by20%. The separatrix is still along the 45 line, however,the saddle moves to (0.5123445V, 0.5123445V). For the sec-ond case in Fig. 11(b), the effective channel lengths andthe threshold voltages of M1 and M2 are simultaneouslydecreased by 30% and those of M3 and M4 are increasedby 30%. Due to the mismatch effects, the separatrix is no

longer a straight line and the saddle moves to (0.3612088V,0.4596336V).

(a) A 6-T SRAM cell

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X1

X2

separatrix

equilibrium point

(b) separatrix

Figure 10: Separatrix of a symmetric cell.

Since the main cost of our tracing algorithm comes fromthe two transient runs, the separatrix tracing method is veryefficient compared with the brute-force method. The separa-trix tracing time is less than 1 minute. But if we run 100x100brute-force transient simulations to sample the state spaceto obtain the separatrix at 1% accuracy, the total runtimeis about 38 hours. Hence, the proposed method providesmore than 2000X speedup. To verify the accuracy of the

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X1

X2

equilibrium point

separatrix

(a) Case 1 : VT variations

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X1

X2

equilibrium point

separatrix

45 degree line

(b) Case 2 : VT and Leff vari-ations

Figure 11: Separatrix under device variations.

tracing algorithm, we randomly select 20 points close to theseparatrix in the state space. We run transient simulationsby taking these points as initial conditions. Each transientstate trajectory ends up at the correct stable equilibriumpoints without crossing the separatrix.

7.2 Dynamic noise margin analysisSeveral DNM analyses for read, write and hold are con-

ducted, where the initial SRAM state is at x1 = 1.0V andx2 = 0V in Fig. 10(a).

7.2.1 Read DNM

We consider an asymmetric 6-T SRAM cell for read DNManalysis. We first trace the separatrix in hold. In the readmode, with the access transistors being turned on, start-ing from a stable equilibrium, a transient run is used tocompute the time at which the state trajectory crosses theseparatrix, or Tacross. It is found to be 8.71ns. Then, con-sider four worldline turn-on times TR1 = 8.72ns, TR2 =8.70ns, TR3 = 8.20ns, TR4 = 5.00ns. According to our readDNM definition, the read DNMs for these four cases are0.01ns, 0.01ns, 0.51ns and 3.71ns, respectively. Next, weuse the transient simulations to verify our read DNM results.The simulation trajectories under the four worldline turn-ontimes are shown in Fig. 12. As expected, in the first case astate flip is produced, indicated by a negative read DNM. Ineach of the other three cases, there is no read instability (thestate trajectory moves back to the initial stable equilibriumafter the read operation ends).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X1

X2

SeparatrixT

R1

TR4

TR3

TR2

Figure 12: Verification of read DNMs.

7.2.2 Write DNM

For the same cell, we further analyze write DNMs whenthe wordline pulse width is set to be TW1 = 0.038ns, TW2 =0.042ns, TW3 = 0.053ns, TW4 = 0.065ns, respectively. Forthe write, the time for the trajectory to reach the separatrix,Tacross, is found to be 0.040ns. Therefore, the write DNMs

are 0.002ns, 0.002ns, 0.013ns and 0.025ns, respectively.Again, we use brute-force transient simulations to verify ourresults, as shown in Fig. 13. It can be seen that in the firstcase, no state flip is produced, indicating a write failure,which is correctly predicted by our negative DNM margin.In all other three cases, the cell is successfully written, con-sistent to the computed positive write DNMs.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X1

X2

Separatrix

TW1

TW4

TW3

TW2

Figure 13: Verification of write DNMs.

7.2.3 Hold DNM

To understand the cell stability in hold, a noise currentwith an amplitude of 135A is injected into one of the stor-

age nodes. The duration is varied as: TH1 = 1.685ns,TH2 = 1.683ns, TH3 = 1.654ns, TH4 = 1.460ns. To charac-terize the hold DNM, again the separatrix in hold (withoutnoise injection) is traced. Then, with the noise injection,a transient simulation is performed to find the separatrixcrossing time Tacross to be 1.684ns. The hold DNM maybe defined as the difference between Tacross and each TH .Given this, for TH1, the hold DNM is negative, suggestinga state flip, which is confirmed by the transient simulationshown in Fig. 14. For each of the other three cases, thehold DNM is positive and hence no state flip happens, asconfirmed in the same figure.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X1

X2

SeparatrixT

H1

TH2

TH3

TH4

Figure 14: Verification of hold DNMs.

7.3 Parametric DNM analysisThe read and write DNM variations under independent

Gaussian random transistor threshold voltage variations are

considered. The nominal threshold voltages for NMOS andPMOS transistors are 0.3V and -0.28V, respectively. Theinitial SRAM state is also at x1 = 1.0V and x2 = 0V(Fig. 10(a)).

In the first example, we consider the VT variations of tran-sistors M1 and M5 in Fig. 10(a). Ttarget + TR in (9) is setto be 0.3ns and accordingly the acceptance region boundaryis:

accept = {(Vth1, Vth5) | Tacross(Vth1, Vth5) = 0.3ns},(21)

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where Vth1 and Vth5 represent the threshold voltages of tran-sistors M1 and M5. The traced boundary is shown in Fig. 15.

0.2 0.25 0.3 0.35 0.40.2

0.25

0.3

0.35

0.4

Vth1(V)

V

th5(V) Failure

Region

Accept.Region

Tacross

(Vth1,Vth5)=0.3ns

Figure 15: Read DNM acceptance region - case 1.

Next, we consider the variations of the threshold voltagesof transistors M3 and M4. The traced boundary is plotted inFig. 16. To verify the correctness of the traced boundaries,

0.46 0.462 0.464 0.4660.44

0.42

0.4

0.38

0.36

0.34

0.32

0.3

Vth3(V)

Vth4(V

)

Accept.Region

FailureRegion

Tacross

(Vth3,Vth4)=0.6ns

Figure 16: Read DNM acceptance region - case 2.

we select 50 points close to the boundary from each side inthe parameter space shown in Fig.15. For each parametercorner, we simulate the SRAM cell to obtain time Tacross.The simulated Tacross values of the points left to the bound-ary are close to 0.3ns and larger. The Tacross values of thepoints right to the boundary are also close to 0.3ns, but

smaller.Since we have obtained the boundary of acceptance region,the yield of read DNM can be efficiently evaluated. How-ever, if we use Monte Carlo simulation to evaluate the readDNM yield, to cover the wide range of variations (e.g. 6 orbeyond), the number of samples required can be huge. OneMonte-Carlo run on average takes 10 seconds. Assuming onemillion samples are needed, then a total of 107seconds willbe required. For the above two cases, the proposed methodtakes 63 and 57 minutes, respectively, hence providing a run-time speedup of three orders of magnitude.

For statistical write DNM analysis, the threshold voltagevariations of M3 and M4 are considered. The specified writeDNM is setup so that Tacross should be at most 0.2ns. Thetraced boundary, as shown in Fig. 17, is computed with a

similar efficiency, by using 71 minutes.

8. CONCLUSIONSIn this work, nonlinear system theory is adopted to pro-

vide a basis for rigorous understanding of SRAM dynamicstability. Using the concept of stability boundary, new DNMmetrics are defined for read, write and hold. Nonlinear sys-tem theory is further exploited to develop fast SPICE-levelCAD algorithm for tracing the separatrix. A fast Newton-based DNM yield analysis is also presented.

Vth3(V)

Vth4(V)

0.2 0.25 0.3 0.35 0.4

0.4

0.35

0.3

0.25

0.2

Tacross

(Vth3,Vth4)=0.2ns

Accept.Region

Failure Region

Figure 17: Write DNM acceptance region.

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