CN105868427A - Method for rapidly calculating SRAM failure probability by adopting self-adaption grid division and slide window technology - Google Patents

Abstract

The invention belongs to the field of semiconductor manufacturability design, and specifically relates to a simulation method for rapidly calculating the SRAM failure probability by considering the deep sub-micrometer technology disturbance. In the method, the spheroid transformation is carried out in a parameter space, and the self-adaption grid division and slide window technology is adopted, the technology can greatly reduce the sampling amount, obtain the SRAM failure probability conforming to the accuracy requirement and the failure boundary information of the parameter space. The precise SPICE simulation is adopted, empirical and semi-empirical models are not depended on. In the simulation process, the precision is high, and the simulation frequency is low, so the rapid calculation is achieved. The SRAM failure probability can be obtained, and the parameter space failure boundary distribution is also obtained, the optimization design of the circuit is benefited.

Description

A kind of method using adaptive meshing algorithm and sliding window technique quickly to calculate SRAM failure probability

Technical field

The present invention relates to quasiconductor design considers the SRAM CALCULATION OF FAILURE PROBABILITY field of nanometer technology disturbance, be specifically related to a kind of method using adaptive meshing algorithm and sliding window technique quickly to calculate SRAM failure probability.

Background technology

Research display, along with Nanometer integrated circuit manufacturing process characteristic size constantly declines, process fluctuations and the performance of random deviation appreciable impact circuit.Generally, in SRAM circuit designs, SRAM memory cell typically uses minimum dimension to design so that design capacity is less；Comprising up to a million memory element in a sram chip, even if the failure probability of a memory element is the highest, the failure probability of sram chip is likely to not reach design requirement simultaneously.

Owing to the inefficacy of SRAM memory cell is a minimum probability event, failure probability according to traditional Monte Carlo method calculating storaging unit, millions of samplings are needed to be only possible to obtain an inefficacy sampled point, therefore in higher-dimension parameter space, quickly the inefficacy of calculating storaging unit belongs to extreme event emulation, is a challenging difficult problem.

In order to improve simulation velocity, it is to avoid call time-consuming SPICE emulation, propose the parsing to the modeling of SRAM memory cell behavioral scaling and semi-analytic method in the world；Wherein Aghababa etc. propose a kind of I-V characteristic curve by transistor and obtain relatively accurate SRAM memory cell read operation nargin model, then by Jacobian matrix calculus joint density distribution function；But owing to minimum probability event being estimated, the second-order effects that there is error and transistor is difficult to express with analytic expression, therefore there is limitation in this analytic method in precision.

On the other hand, owing to circuit SPICE simulation result is approved by industry, therefore method based on SPICE emulation is gradually paid attention to.The method being currently based on SPICE emulation has: importance sampling method (Important Sampling, IS), extreme value theorem method, boundary method, Spherical approach etc..Importance sampling method is a kind of method of minimum estimation of effective calculating.Kanj etc. propose the important method of sampling of mixing；Dolecek etc. estimate the side-play amount of importance sampling by minimum norm principle (norm minimization principle)；Qazi etc. propose importance sampling method based on spherical coordinates；Katayama etc. use extension ball (expanding sphere) to find most probable failed areas, then use Temporal Sampling method；Dong etc. travel through the failed areas of each dimension by the selectable Gibbs method of sampling；Sun etc. use the method for sampling of variable Sigma (scaled-sigma).Owing to SRAM memory cell has the strongest nonlinear characteristic, in all method of samplings, most crucial problem is how to find most suitable probability density function, and suitably starting sample point.The process that the above-mentioned method of sampling finds suitable probability density function is the most more complicated, and there is the difficulty that operand is increased dramatically under higher-dimension.

In terms of extreme value theorem method, by extreme value theorem, Singhee etc. estimates that the tail of SRAM memory cell performance is distributed (tail distributions) model.The method is primarily based on model and produces substantial amounts of sampled point, then filters out, by tail distributed model, the sampled point that successful sampled point, only emulation may lose efficacy, finally gives the failure probability of SRAM memory cell.Although the method computational efficiency is higher, but parameter space Failure Boundaries cannot be obtained.

In terms of boundary method, Srivastava etc. does not propose Euler-Newton curved surface tracking method and Nonlinear Curved sampling method with Gu decile.Both approaches is all based on solving the Failure Boundaries of parameter space, then calculates crash rate by parameter space failure probability density integral.Boundary method is highly effective in lower dimensional space, but amount of calculation is exponentially increased in higher dimensional space.

In terms of Spherical approach, Fonseca etc. proposes the ball method of sampling, is sampled radius of a ball direction by simple Monte-Carlo, then calculates failpoint in radius directions of rays, but the precision of the method is on the low side.

The deficiency existed for said method, present inventor intends providing a kind of method using adaptive meshing algorithm and sliding window technique quickly to calculate SRAM failure probability.

List of references related to the present invention has:

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Summary of the invention

It is an object of the invention to overcome the defect of prior art, it is provided that a kind of quick SRAM failure probability emulation mode considered under nanometer technology disturbance.It is specifically related to a kind of method using adaptive meshing algorithm and sliding window technique quickly to calculate SRAM failure probability.Carrying out spheroid conversion in the inventive method in parameter space, and use adaptive meshing algorithm and sliding window, the method can significantly reduce sampling quantity, it is thus achieved that meets the SRAM failure probability of required precision.

In the present invention, first the parameter space of SRAM memory cell circuit technology disturbance is carried out spheroid compression variation；Then use adaptive stress and strain model, the parameter space of ellipsoid deflection divides different sample area, and calculate the scope of Failure Boundaries radius in sample area；Finally use sliding window method sample area is encrypted sampling and calculates, thus obtain the failure probability of SRAM memory cell, and available Failure Boundaries.

Present method be advantageous in that: (1) uses accurate SPICE to emulate, be independent of experience and semiempirical model；(2) although this method still presents between number of samples and the dimension of parameter space exponential relationship, but the exponential increase relation between number of samples and the dimension of parameter space can be slowed down；(3) while obtaining SRAM memory cell failure probability, the Failure Boundaries of parameter space can also have been obtained.(4) under conditions of reaching equal accuracy, amount of calculation (hits) is less than current main stream approach.

Concrete, the method for the quickly calculating SRAM failure probability that the present invention proposes, it is characterised in that it includes (as shown in Fig. 1 flow chart):

Input parameter:

The parameter space R of artificial circuitⁿ, parameter space dimension n, circuit parameter probability distributing density function p (x), x ∈ [x^-, x⁺], x^-、x⁺It is the dividing value up and down of parameter vector x, SRAM memory cell circuit meshwork list, stress and strain model precision thresholding α respectively₀, stress and strain model radius μ₁, divide uniformity parameters τ, sliding window radius μ₂, grid subdivision number h, sliding window size d, adaptive mesh hits N₁, hits N in sliding window₂, initial mesh divides number k.

Output result:

Parameter space Failure Boundaries radius r, call simulation times N of SPICE, SRAM failure probability.

Step 1: utilize two way classification to determine ellipsoid tranformation coefficient by SPICE emulation, and by cartesian parameter space RⁿIt is converted into ellipsoidal coordinates；

Step 2: the parameter space of ellipsoid deflection is carried out adaptive meshing algorithm, and by the inefficacy radius of the SPICE sampling each grid of simulation calculation；

Step 3: be unit (cell) by each grid uniform subdivision, unit use slip window sampling accurately solve the Failure Boundaries radius r of this unit, the Failure Boundaries radius r of all unit constitutes the Failure Boundaries of SRAM memory cell, then carries out probability integral on each unit and can get the failure probability of whole SRAM.

More specifically, in step 1 of the present invention, emulated by SPICE, use two way classification to obtain the Failure Boundaries point c of each parameter coordinate axes in parameter space_i.By iteration simulation parameter distributed area intermediate value, constantly reduce the interval of Failure Boundaries point according to the simulation result of intermediate value, finally give Failure Boundaries point；

The flow process of described two way classification comprises the steps:

Input parameter: coordinate axes x_iSpan be x_i∈[x_i ^-, x_i ⁺]；

Output parameter: Failure Boundaries point c_i；

Step 1.1: calculate parameter x_iInterval intermediate value

Step 1.2: if x_i ⁺-x_i ^-＜ ε, ε=0.01, then export c_i；

Step 1.3: simulation parameter x_iInterval intermediate value c_i；

Step 1.4: if c_iSimulation result be successfully, then x_i ^-=c_i, jump procedure 1.1；

Step 1.5: if c_iSimulation result for lost efficacy, then x_i ⁺=c_i, jump procedure 1.1.

Then it is ellipsoidal parameter space according to formula (1) cartesian parameter spatial transformation:

$\begin{array}{c}\{\[x_1,...,x_n\]\}\⇔\{\[r,\mathrm{\θ}\]\}\\ \mathrm{\θ}={[{\mathrm{\θ}}_1,...,{\mathrm{\θ}}_{n-1}]}^T\\ x_i\∈[a_i,b_i],i=1,...,n\\ {\left(\frac{x_1}{c_1}\right)}^2+{\left(\frac{x_2}{c_2}\right)}^2+...+{\left(\frac{x_n}{c_n}\right)}^2=1\end{array}---\left(1\right)$

Wherein [x₁,...,x_n] it is cartesian coordinate, [r, θ] is ellipsoidal coordinates, a_iAnd b_iIt is cartesian parameter space boundary, c_iIt is the radius of ellipsoid, as shown in Figure 2.

In the inventive method step 2, by following sub-step, the parameter space of ellipsoid deflection carried out adaptive meshing algorithm, and by the inefficacy radius of SPICE sampling simulation calculation grid:

Step 2.1: ellipsoidal parameter space is evenly dividing into k part in each θ direction, at [the r-μ of the radial direction of each initial mesh₁, r+ μ₁] region carries out N₁Secondary sampling, and carry out SPICE emulation；

Step 2.2: iteration simulation calculation Failure Boundaries radius r；The inventive method, by following classification, carries out Failure Boundaries calculating.Adding up for the simulation result in initial mesh, simulation result is divided three classes: successfully, lost efficacy and overflowed.If occurring certain situation in Fang Zhen, then this situation is designated as " 1 ", is otherwise designated as " 0 ".Different strategies is used to revise r according to statistical result.

Table 1: the adjustable strategies of inefficacy radius r

After revising r, continue at [r-μ₁, r+ μ₁] uniform sampling N in region₁Individual sampled point, and again emulate, according to statistical result correction r, such iteration, until final statistical result is situation 3 or situation 4, iterative process figure sees Fig. 3.

Step 2.3: all grids carrying out Homogeneity to decide whether to segment further, the grid after segmentation need to be according to step 2.2 again iteration simulation calculation Failure Boundaries radius r.

When in step 2.2 4, i-th grid representation is, wherein ${\mathrm{\α}}_i^{-}=\[{\mathrm{\θ}}_{i1}^{-},{\mathrm{\θ}}_{i2}^{-},...,{\mathrm{\θ}}_{i(n-1)}^{-}\],{\mathrm{\α}}_i^{+}=\[{\mathrm{\θ}}_{i1}^{+},{\mathrm{\θ}}_{i2}^{+},...,{\mathrm{\θ}}_{i(n-1)}^{+}\]$ .If α_i ⁺And α_i ^-It is unsatisfactory for:

||α_i ⁺-α_i ^-||_∞≤α₀ (2)

The most successively fromEach dimension, this net is evenly dividing into left grid and right grid, checks that the uniformity of left and right grid success failure ratio should meet formula (3).

$|\frac{{\mathrm{blue}}_l}{{\mathrm{red}}_l+{\mathrm{yellow}}_l}-\frac{{\mathrm{blue}}_r}{{\mathrm{red}}_r+{\mathrm{yellow}}_r}|\≤{\mathrm{\τ}}_i,i=1,...,n-1---\left(3\right)$

Wherein blue_l、red_lAnd yellow_lBe left area successful, lost efficacy and overflowed the sampling number of parameter space, blue_r、red_rAnd yellow_rBe region on the right successful, lost efficacy and overflowed the sampling number of parameter space.If being unsatisfactory for formula (3), then this grid is halved in this dimension, and bisection process schematic sees Fig. 4.Then continue to repeat step 2.2 and step 2.3, as shown in step 2 in Fig. 1 at the new grid divided.

In the inventive method step 3, it is unit by following sub-step by each grid uniform subdivision, uses slip window sampling to solve the Failure Boundaries radius r of this unit, then calculate Failure Boundaries and the failure probability of SRAM:

Step 3.1: the grid d decile on each dimension θ self adaptation divided, obtains the unit after this grid subdivision, and number of unit is d^θ, have on the sliding window of h unit in each dimension and carry out N₂Secondary uniform sampling also emulates；

Step 3.2: according to N₂The SPICE simulation result of secondary sampling, adjusts the Failure Boundaries radius r of sliding window according to table 1_win, and N again₂Secondary sampling emulation, until meeting the situation 3 in table 1 or situation 4；

Step 3.3: calculate sliding window Failure Boundaries radius r according to formula (4)_win:

$r_{\mathrm{win}}=\sqrt{4r{\mathrm{\μ}}_2\frac{{\mathrm{blue}}_{\mathrm{num}}}{{\mathrm{total}}_{\mathrm{num}}}+{(r-{\mathrm{\μ}}_2)}^2}---\left(4\right)$

Wherein blue_numAnd total_numIt is successful sampled point number and all sampled point numbers in this sliding window respectively.

Step 3.4: a sliding window mobile unit in each dimension successively, retains the sampled point with a upper sliding window lap, and adds sampled point in the range of newly increasing, make the sampling number in sliding window reach N₂, repeat step 3.2 and step 3.3, until traveling through whole parameter space.The step schematic diagram that sliding window slides sees Fig. 5, and sliding window increases sampled point schematic diagram and sees Fig. 6.

Step 3.5: according to sliding window Failure Boundaries radius calculation element failure bound radius r_cell；

Each unit is by its center p₀It is expressed as Cell (p₀), wherein p₀It it is one (n-1) vector of tieing up.Each sliding window uses Win (q) to represent the most similarly, and q is the center of sliding window.The Failure Boundaries radius of sliding window is shown with the Failure Boundaries radius relationship such as formula (5) of unit and formula (6), and the inventive method carrys out the Failure Boundaries radius of computing unit parameter space by formula (7)

$r_{\mathrm{Cell}\left(p\right)}=\frac{1}{d^{n-1}}\underset{{\left|\right|q-p\left|\right|}_{\∞}\≤\frac{d-1}{2}}{\mathrm{\Σ}}{r}_{\mathrm{Win}\left(q\right)}---\left(5\right)$

$r_{\mathrm{Win}\left(q\right)}=\frac{1}{d^{n-1}}\underset{{\left|\right|q-s\left|\right|}_{\∞}\≤\frac{d-1}{2}}{\mathrm{\Σ}}{r}_{\mathrm{Cell}\left(s\right)}---\left(6\right)$

$r_{\mathrm{Cell}\left(p_0\right)}=\frac{1}{{(2d-1)}^{n-1}}\underset{{\left|\right|q-p_0\left|\right|}_{\∞}\≤d-1}{\mathrm{\Σ}}(v_1{\·v}_2\·...v_{n-1})r_{\mathrm{Cell}\left(q\right)}=g(q,p_0)\·r_{\mathrm{Cell}\left(q\right)},{\left|\right|q-p_0\left|\right|}_{\∞}\≤d-1---\left(7\right)$

Wherein、r_cell(q)、r_win(q)It is Cell (p respectively₀), the inefficacy radius of Cell (q) and Win (q), v_iIt is the i-th vector of V, V=d^*-(q-p₀), d^*=d × [1...1]^T.By sliding window method computing unit Failure Boundaries radius r_cellSchematic diagram see Fig. 7.

Step 3.6: according to Failure Boundaries radius calculation SRAM memory cell failure probability and the failure probability of whole SRAM of unit.

Failure probability by formula (8) calculating SRAM memory cell:

$P=\underset{\mathrm{\Ω}}{\∫}p\left(x\right)\·\mathrm{dx}=\underset{i=1}{\overset{M}{\mathrm{\Σ}}}\left(2\underset{j=1}{\overset{n-1}{\mathrm{\Π}}}\left(\frac{{a_i}^{+}\left[j\right]-{a_i}^{-}\left[j\right]}{2\mathrm{\π}}\right)(1-e^{-\left(\frac{r_i^2}{2}\right)}\·(r_i^4+4r_i^2+8)/8)\right)---\left(8\right)$

Wherein M is unit number, and i-th cell corner range of variables can be expressed as [a_i ^-, a_i ⁺], r_iIt it is the Failure Boundaries radius of i-th unit.

Failure probability by formula (9) calculating SRAM:

$P_{\mathrm{SRAM}}=1-{(1-P)}^{n_{\mathrm{size}}}---\left(9\right)$

Here P_SRAMBeing the failure probability of whole SRAM, P is SRAM memory cell failure probability, n_sizeIt it is the amount of storage of whole SRAM.

It is an advantage of the current invention that:

1. the present invention uses accurate SPICE to emulate, and is independent of experience and semiempirical model；

2. the simulation process precision of the inventive method is high, simulation times is few, can reach the purpose quickly calculated；

3. the inventive method obtains the optimization design of the distribution of parameter space Failure Boundaries, beneficially circuit.

Accompanying drawing explanation

Fig. 1 is the inventive method flow chart.

Fig. 2 is the ellipsoid tranformation schematic diagram of the present invention.

Fig. 3 is that grid iteration of the present invention emulates schematic diagram.

Fig. 4 is that the present invention carries out stress and strain model schematic diagram.

Fig. 5 is sliding window slip schematic diagram of the present invention.

Fig. 6 is that sliding window of the present invention increases sampled point schematic diagram.

Fig. 7 is sliding window statistical method schematic diagram of the present invention.

Fig. 8 is sram cell circuit diagram of the present invention.

Fig. 9 is that Monte Carlo method of the present invention obtains SRAM read operation result analogous diagram.

Figure 10 is that Monte Carlo method of the present invention obtains SRAM reading Failure Boundaries analogous diagram.

Figure 11 is that the inventive method acquisition SRAM reading Failure Boundaries compares with Mote Carlo method.

Figure 12 is the grid number variation diagram that the inventive method obtains that SRAM write lost efficacy.

Figure 13 is that four kinds of methods (MNIS, MIS, SCC and SSS) of simulation times and other that the inventive method acquisition SRAM write lost efficacy are compared.

Figure 14 is that four kinds of methods (MNIS, MIS, SCC and SSS) of error rate and other that the inventive method acquisition SRAM write lost efficacy are compared.

Figure 15 is the simulation times convergence that the inventive method obtains that SRAM write lost efficacy.

Figure 16 is the error rate convergence that the inventive method obtains that SRAM write lost efficacy.

Figure 17 is different stress and strain model number d and the analogous diagram of sliding window size h that the inventive method obtains that SRAM write lost efficacy.

Figure 18 is that the inventive method obtains SRAM write Failure Boundaries point (Vthp4 Yu Vthn6 coordinate axes).

Figure 19 is that the inventive method obtains SRAM write Failure Boundaries point (Vthp4, Vthn5 and Vthn6 coordinate axes).Figure 20 is that the inventive method obtains SRAM write Failure Boundaries point (Vthp4, Vthn1 and Vthn5 coordinate axes).Figure 21 is that the inventive method acquisition SRAM keeps the four kinds of methods (MNIS, MIS, SCC and SSS) of simulation times and other lost efficacy to compare.

Figure 22 is that the inventive method acquisition SRAM keeps the four kinds of methods (MNIS, MIS, SCC and SSS) of error rate and other lost efficacy to compare.

Figure 23 is SRAM sense amplifier circuit diagram of the present invention.

Figure 24 is that four kinds of methods (MNIS, MIS, SCC and SSS) of simulation times and other that the inventive method acquisition SRAM sense amplifier lost efficacy are compared.

Figure 25 is that four kinds of methods (MNIS, MIS, SCC and SSS) of error rate and other that the inventive method acquisition SRAM sense amplifier lost efficacy are compared.

Detailed description of the invention

Now by the implementation process of instantiation, the inventive method is described.

Implement example 1

As shown in Figure 8, calculated sram cell reads to lose efficacy and Monte Carlo method and YENSS, tangential method, comparison between MIS, MNIS, SCC for the circuit that the inventive method uses.In this example, threshold voltage vt h1, Vth5 scope of M1, M5 transistor is all [0V, 0.8V], and this is a two-dimensional parameter space Solve problems.First pass through spherical coordinates conversion, as shown in formula (10).

V_th1=r × sin θ, V_th5=r × cos θ, 0≤θ ＜ 2 π (10)

Wherein θ₁And θ₂Spherical coordinates angular range.This example r initial value is 1, and θ is divided into 10 cell, the boundary curve g obtained_boundaryAs shown in formula (11).

g_boundary=g (r, θ), θ₁≤θ≤θ₂ (11)

The sram cell obtained by this method is read to lose efficacy and is compared with existing method and Monte Carlo method.As shown in Figure 9, Figure 10, the result that this method obtains compares as shown in figure 11 Monte Carlo method simulation result with Monte Carlo method.Result shows and is reaching same less than under 1% precision conditions, and method is faster 53 times than Monte Carlo, faster than other method 2.5～24 times.Experimental data shows that this method, on the premise of accuracy is guaranteed, calculates speed and promoted by a relatively large margin.

Implement example 2

The calculated sram cell of the inventive method writes the comparison between inefficacy and Monte Carlo method and MIS, MNIS, SCC.It is a sextuple parameter space Solve problems that sram cell writes inefficacy.

This example Failure Boundaries radius r initial value is 1, and the initial value of initial mesh number k is 4, mesh radius thickness μ₁=0.1, cell radius thickness μ₂=0.05, it is evenly dividing and judges factor τ=0.2, minimum grid α₀=π/4, grid cutting unit number h=3, sliding window size d=3.The border hypersurface g obtained_boundaryAs shown in formula (12).

g_boundary=g (r, θ), θ=[θ₁,θ₂,θ₃,θ₄,θ₅] (12)

0≤θ₁,θ₂,θ₃,θ₄＜ 2 π, 0≤θ₅＜ π

The sram cell obtained by this method is write inefficacy and is compared with existing method and Monte Carlo method, and result shows and reaching same less than under 1% precision conditions, faster than other method 1.7～36 times.As shown in figure 12, with the comparative result of other method as shown in Figure 13, Figure 14, this Algorithm Convergence is as shown in Figure 15, Figure 16, h and d is on the impact of algorithm as shown in figure 17 for this method grid iterations.Experimental data shows that this method, on the premise of accuracy is guaranteed, calculates speed and promoted by a relatively large margin.Method is on the premise of obtaining crash rate simultaneously, has obtained the information of Failure Boundaries as shown in Figure 18, Figure 19, Figure 20, has provided the foundation for circuit optimization, and this is not available for other method.

Implement example 3

The calculated sram cell of the inventive method keeps the comparison between inefficacy and Monte Carlo method and MIS, MNIS, SCC.Sram cell keeps inefficacy to be a sextuple parameter space Solve problems.

In this example, Failure Boundaries radius r initial value is 1, and the initial value of initial mesh number k is 4, mesh radius thickness μ₁=0.1, cell radius thickness μ₂=0.05, it is evenly dividing and judges factor τ=0.2, minimum grid α₀=π/4, grid cutting unit number d=3, sliding window size h=3, supply voltage is 750mV.

The sram cell obtained by this method is kept losing efficacy and compares with existing method and Monte Carlo method, result shows and is reaching same less than under 1% precision conditions, faster than other method 1.7～36 times, the comparison of this method and other method is as shown in Figure 21, Figure 22.Experimental data shows that this method, on the premise of accuracy is guaranteed, calculates speed and promoted by a relatively large margin.Method is on the premise of obtaining crash rate simultaneously, has obtained the information of Failure Boundaries, has provided the foundation for circuit optimization.

Implement example 4

Comparison between the inefficacy of the inventive method calculated SRAM sense amplifier and MIS, MNIS, SCC and SSS, SRAM sense amplifier circuit diagram is as shown in figure 23.It is an octuple parameter space Solve problems that SSRAM sense amplifier lost efficacy.

In this example, Failure Boundaries radius r initial value is 1, and the initial value of initial mesh number k is 4, mesh radius thickness μ₁=0.1, cell radius thickness μ₂=0.05, it is evenly dividing and judges factor τ=0.2, minimum grid α₀=π/4, grid cutting unit number d=3, sliding window size h=3.

The SRAM sense amplifier obtained by this method was lost efficacy and compared with existing method, and result shows and reaching same less than under 1% precision conditions, and faster than other method 1.7～5 times, the comparison of this method and other method is as shown in Figure 24, Figure 25.Experimental data shows that this method, on the premise of the octuple space and accuracy are guaranteed, calculate speed and promoted by a relatively large margin.

Claims (4)

1. use the method that adaptive meshing algorithm and sliding window technique quickly calculate SRAM failure probability, It is characterized in that, carry out probability statistics based on SPICE accurate simulation result, comprising:

Input parameter:

The parameter space R of artificial circuitⁿ, parameter space dimension n, circuit parameter probability distributing density function P (x), x ∈ [x^-, x⁺], x^-、x⁺It is the dividing value up and down of parameter vector x, SRAM memory cell electricity respectively Road network table, stress and strain model precision thresholding α₀, stress and strain model radius μ₁, divide uniformity parameters τ, cunning Dynamic windows radius scope μ₂, grid subdivision number h, sliding window size d, adaptive mesh hits N₁, sliding Hits N in dynamic window₂, initial mesh divides number k；

Output result:

Parameter space Failure Boundaries radius r, call simulation times N of SPICE, SRAM failure probability；

Pass through following step:

Step 1: utilize two way classification to determine ellipsoid tranformation coefficient by SPICE emulation, and cartesian parameter is empty Between RⁿIt is converted into ellipsoidal coordinates；

Step 2: the parameter space of ellipsoid deflection is carried out adaptive meshing algorithm, and is sampled by SPICE The inefficacy radius of each grid of simulation calculation；

Step 3: be unit (cell) by each grid uniform subdivision, uses slip window sampling essence on unit Really solving the Failure Boundaries radius r of this unit, it is single that the Failure Boundaries radius r of all unit constitutes SRAM storage The Failure Boundaries of unit, then carries out probability integral on each unit and can get the failure probability of whole SRAM.

2. the method as described in claim 1, is characterized in that, is emulated by SPICE in described step 1, Parameter space use two way classification obtain the Failure Boundaries point c of each parameter coordinate axes_i；Divided by iteration simulation parameter Cloth interval intermediate value, constantly reduces the interval of Failure Boundaries point, finally gives mistake according to the simulation result of intermediate value Effect boundary point；

Then it is ellipsoidal parameter space according to formula (1) cartesian parameter spatial transformation:

$\left\{\right[x_1,...,x_n\left]\right\}\⇔\left\{\right[r,\mathrm{\θ}\left]\right\}$

θ=[θ₁,...,θ_n-1]^T

x_i∈[a_i,b_i], i=1 ..., n (1)

${\left(\frac{x_1}{c_1}\right)}^2+{\left(\frac{x_2}{c_2}\right)}^2+...+{\left(\frac{x_n}{c_n}\right)}^2=1$

Wherein [x₁,...,x_n] it is cartesian coordinate, [r, θ] is ellipsoidal coordinates, a_iAnd b_iIt is cartesian parameter space boundary, c_iIt it is the radius of ellipsoid.

3. the method as described in claim 1, is characterized in that, in described step 2, uses following sub-step pair The parameter space of ellipsoid deflection carries out adaptive meshing algorithm, and by SPICE sampling simulation calculation grid Inefficacy radius:

Step 2.1: ellipsoidal parameter space is evenly dividing into k part in each θ direction, at each initial mesh [the r-μ of radial direction₁, r+ μ₁] region carries out N₁Secondary sampling, and carry out SPICE emulation；

Step 2.2: iteration simulation calculation Failure Boundaries radius r；By following classification, carry out Failure Boundaries meter Calculate；Adding up for the simulation result in initial mesh, simulation result is divided three classes: successfully, lost efficacy and overflow Go out；If occurring certain situation in Fang Zhen, then this situation is designated as " 1 ", is otherwise designated as " 0 "；By following table 1, use different strategies to revise r according to statistical result；

Table 1: the adjustable strategies of inefficacy radius r

After revising r, continue at [r-μ₁, r+ μ₁] uniform sampling N in region₁Individual sampled point, and again carry out Emulation, according to statistical result correction r, such iteration, until final statistical result is situation 3 or situation 4 is Stop 3；

Step 2.3: all grids are carried out Homogeneity to decide whether to segment further, after segmentation Grid need to be according to step 2.2 again iteration simulation calculation Failure Boundaries radius r；

To the situation 4 in step 2.2, i-th grid representation isWherein If α_i ⁺And α_i ^-It is unsatisfactory for:

||α_i ⁺-α_i ^-||_∞≤α₀ (2)

The most successively fromEach dimension, this net is evenly dividing into left grid and right grid, checks left and right net The uniformity of lattice success failure ratio should meet formula (3)；

$|\frac{{\mathrm{blue}}_l}{{\mathrm{red}}_l+{\mathrm{yellow}}_l}-\frac{{\mathrm{blue}}_r}{{\mathrm{red}}_r+{\mathrm{yellow}}_r}|\≤{\mathrm{\τ}}_i,i=1,...,n-1---\left(3\right)$

Wherein blue_l、red_lAnd yellow_lBe left area successful, lost efficacy and overflowed the sampled point of parameter space Number, blue_r、red_rAnd yellow_rBe region on the right successful, lost efficacy and overflowed the sampled point of parameter space Number；If being unsatisfactory for formula (3), then this grid is halved in this dimension；Then continue at the new grid divided Continuous repetition step 2.2 and step 2.3.

4. the method as described in claim 1, is characterized in that, in described step 3, uses following sub-step to incite somebody to action Each grid uniform subdivision is unit, uses slip window sampling to solve the Failure Boundaries radius r of this unit, then The Failure Boundaries of calculating SRAM and failure probability:

Step 3.1: the grid d decile on each dimension θ self adaptation divided, after obtaining this grid subdivision Unit, number of unit is d^θ, have on the sliding window of h unit in each dimension and carry out N₂Secondary uniformly adopt Sample also emulates；

Step 3.2: according to N₂The SPICE simulation result of secondary sampling, adjusts the mistake of sliding window according to table 1 Effect bound radius r_win, and N again₂Secondary sampling emulation, until meeting the situation 3 in table 1 or situation 4；

Step 3.3: calculate sliding window Failure Boundaries radius r according to formula (4)_win:

$r_{\mathrm{win}}=\sqrt{4r{\mathrm{\μ}}_2\frac{{\mathrm{blue}}_{\mathrm{num}}}{{\mathrm{total}}_{\mathrm{num}}}{(r-{\mathrm{\μ}}_2)}^2}---\left(4\right)$

Wherein blue_numAnd total_numIt is successful sampled point number and all sampled points in this sliding window respectively Number；

Step 3.4: a sliding window mobile unit in each dimension successively, retains and a upper sliding window The sampled point of mouth lap, and in the range of newly increasing, add sampled point, make the sampled point in sliding window Number reaches N₂, repeat step 3.2 and step 3.3, until traveling through whole parameter space；

Step 3.5: according to sliding window Failure Boundaries radius calculation element failure bound radius r_cell；

Each unit is by its center p₀It is expressed as Cell (p₀), wherein p₀It is that one (n-1) ties up Vector；Each sliding window uses Win (q) to represent the most similarly, and q is the center of sliding window；Sliding The dynamic Failure Boundaries radius of window is shown with the Failure Boundaries radius relationship such as formula (5) of unit and formula (6), Failure Boundaries radius by formula (7) computing unit parameter space

$r_{\mathrm{Cell}\left(p\right)}=\frac{1}{d^{n-1}}\underset{{\left|\right|q-p\left|\right|}_{\∞}\≤\frac{d-1}{2}}{\mathrm{\Σ}}{r}_{\mathrm{Win}\left(q\right)}---\left(5\right)$

$r_{\mathrm{Win}\left(q\right)}=\frac{1}{d^{n-1}}\underset{{\left|\right|q-s\left|\right|}_{\∞}\≤\frac{d-1}{2}}{\mathrm{\Σ}}{r}_{\mathrm{Cell}\left(s\right)}---\left(6\right)$

$r_{\mathrm{Cell}\left(p_0\right)}=\frac{1}{{(2d-1)}^{n-1}}\underset{\left|\right|q-p_0{\left|\right|}_{\∞}\≤d-1}{\mathrm{\Σ}}(v_1\·v_2\·...v_{n-1})r_{\mathrm{Cell}\left(q\right)}=g(q,p_0)\·r_{\mathrm{Cell}\left(q\right)},{\left|\right|q-p_0\left|\right|}_{\∞}\≤d-1---\left(7\right)$

Whereinr_cell(q)、r_win(q)It is Cell (p respectively₀), the inefficacy radius of Cell (q) and Win (q), v_iIt is V I-th vector, V=d^*-(q-p₀), d^*=d × [1...1]^T；

Step 3.6: according to the Failure Boundaries radius calculation SRAM memory cell failure probability of unit and whole The failure probability of SRAM；

Failure probability by formula (8) calculating SRAM memory cell:

$P=\underset{\mathrm{\Ω}}{\∫}p\left(x\right)\·\mathrm{dx}=\underset{i=1}{\overset{M}{\mathrm{\Σ}}}\left(2\underset{j=1}{\overset{n-1}{\mathrm{\Σ}}}\left(\frac{{a_i}^{+}\left[j\right]-{a_i}^{-}\left[j\right]}{2\mathrm{\π}}\right)(1-e^{-\left(\frac{{r_i}^2}{2}\right)}\·({r_i}^4+4{r_i}^2+8)/8)\right)---\left(8\right)$

Wherein M is unit number, and i-th cell corner range of variables can be expressed as [a_i ^-, a_i ⁺], r_iIt it is i-th list The Failure Boundaries radius of unit；

Failure probability by formula (9) calculating SRAM:

$P_{\mathrm{SRAM}}=1-{(1-P)}^{n_{\mathrm{size}}}---\left(9\right)$

Described P_SRAMBeing the failure probability of whole SRAM, P is SRAM memory cell failure probability, n_sizeIt is The amount of storage of whole SRAM.