Complexity-reduced simulation of circuit reliability

Abstract

A system and method for simulating an electronic circuit is disclosed. The method includes creating a finite set of circuit or device parameter points selected from within an n-dimensional parameter space. The method includes determining, for each circuit or device parameter point of the set, a corresponding response value of the performance metric and a corresponding probability of occurrence. The method includes determining, for a predetermined value of the performance metric, the total probability of occurrence.

Claims

1. A method comprising: providing a frequency error margin value and a duty cycle error margin value associated with processing a digital signal having a transient waveform that comprises a sequence of pulses, each pulse having a pulse frequency, a pulse duty cycle, and a pulse duration; identifying a buffer frequency, a buffer duty cycle, and a buffer duration of one of a plurality of sub-sequences of pulses, wherein each sub-sequence of pulses comprises one or more pulses, and wherein for the one of the sub-sequences of pulses: the buffer frequency is a first function of the pulse frequencies of the sub-sequence of pulses, such that the buffer frequency and each of the pulse frequencies of the sub-sequence of pulses differ by less than the frequency error margin, the buffer duty cycle is a second function of the pulse duty cycles of the sub-sequence of pulses, such that the buffer duty cycle and each of the pulse duty cycles of the sub-sequence of pulses differ by less than the duty cycle error margin, and the buffer duration is the sum of the pulse durations over the sub-sequence of pulses; representing the sub-sequence of pulses as a point comprising the buffer frequency, the buffer duty cycle, and the buffer duration; determining a response value of a performance metric and a probability of occurrence corresponding to the response value, based on the point; and designing an electronic circuit based on the probability of occurrence, or driving an electronic circuit with a workload selected based on the probability of occurrence.

2. The method of claim 1, further comprising: identifying a next buffer frequency, a next buffer duty cycle, and a next buffer duration of a next sub-sequence of pulses, wherein for the next sub-sequence of pulses: the next buffer frequency is the first function of the pulse frequencies of the next sub-sequence of pulses, such that the next buffer frequency and each of the pulse frequencies of the next sub-sequence of pulses differ by less than the frequency error margin, the next buffer duty cycle is the second function of the pulse duty cycles of the next sub-sequence of pulses, such that the next buffer duty cycle and each of the pulse duty cycles of the next sub-sequence of pulses differ by less than the duty cycle error margin, and the next buffer duration is the sum of the pulse durations over the next sub-sequence of pulses; and representing the next sub-sequence of pulses by the next buffer frequency, the next buffer duty cycle, and the next buffer duration, wherein the next sub-sequence of pulses starts from a pulse such that: the buffer frequency and the frequency of the pulse differ by an amount equal to or more than the frequency error margin, and/or the buffer duty cycle and the duty cycle of the pulse differ by an amount equal to or more than the duty cycle error margin.

3. The method of claim 2, wherein the buffer frequency, buffer duty cycle, and buffer duration representing each of the sub-sequences of pulses forms a compact digital waveform (CDW) point for the corresponding sub-sequence of pulses, and the method further comprises: storing a representation of the digital signal as a series of CDW points, each CDW point being representative of one of the plurality of sub-sequences of pulses.

4. The method of claim 3, wherein the buffer frequencies, buffer duty cycles, and buffer durations of the CDW points differ between two CDW points.

5. The method of claim 1, wherein the first function is an average.

6. The method of claim 5, wherein the second function is an average.

7. The method of claim 1, wherein the signal is represented in Simulation Program with Integrated Circuit Emphasis (SPICE) format.

8. The method of claim 1, wherein the signal is represented in Value Change Dump (VCD) or equivalent format.

9. The method of claim 1, wherein the buffer frequency, buffer duty cycle, and buffer duration representing each of the sub-sequences of pulses forms a compact digital waveform (CDW) point for the corresponding sub-sequence of pulses, and the method further comprises: storing a representation of the digital signal as a series of CDW points, each CDW point being representative of one of the plurality of sub-sequences of pulses.

10. The method of claim 9, wherein the buffer frequencies, buffer duty cycles, and buffer durations of the CDW points vary over time.

11. The method of claim 1, wherein the digital signal is a voltage applied to a gate terminal of a transistor as a function of time.

12. The method of claim 11, wherein the transistor is a p-channel metal—oxide—semiconductor field-effect transistor, n-channel metal—oxide— semiconductor field-effect transistor, junction field effect transistor, high electron mobility transistor, or insulated-gate bipolar transistor.

13. The method of claim 1, wherein the digital signal is a voltage applied to an input port of a logic cell of a circuit or device as a function of time, the circuit or device comprising a plurality of logic cells.

14. The method of claim 13, wherein the logic cell is a memory cell of static random-access memory (SRAM).

15. The method of claim 1, wherein the pulse frequencies and the pulse duty cycles of the sequence of pulses vary over time.

16. The method of claim 1, wherein the response value of the performance metric and the probability of occurrence corresponding to the response value are determined based on the point and on one or more additional points representing additional sub-sequences of the plurality of sub-sequences of pulses.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will now be described further, by way of example, with reference to the accompanying drawings, in which:

(2) FIG. 1 illustrates a 6T-SRAM cell used for simulation.

(3) FIG. 2 illustrates a butterfly curve for the cell illustrated in FIG. 1. From analysis of this curve during hold mode, where the word lines are set at 0V and the bitlines at VDD, a performance metric, as an example SNM margins, may be obtained.

(4) FIG. 3 illustrates a 2-dimensional slice of the performance metric response surface, e.g., SNM response surface, where V.sub.TH,1 and V.sub.TH,2 are parameters of the total parameter space ξ.fwdarw.(V.sub.TH,1, V.sub.TH,2).

(5) FIG. 4 illustrates the probability density function g(ξ) Since the response surface cannot be constructed for all possible circuit or device parameter values, an error is introduced into the final CDF calculation. This error can be reduced to acceptable levels by choosing the lower (1) and upper (u) bounds of the circuit or device parameters to be evaluated.

(6) FIGS. 5A and 5B illustrate a schematic representation of the numerical calculation of the PDF value for each contour line C.sub.i. The PDF f(C.sub.i) can be constructed by sorting all values according to C. The final CDF F(C.sub.m) can be numerically calculated by a cumulative sum of g.sub.i(ξ) sorted according to C.

(7) FIGS. 6A and 6B illustrate the method according to embodiments of the disclosed technology (FIG. 6A) with Monte Carlo methodology (FIG. 6B). The method according to embodiments of the disclosed technology creates a finite set of circuit or device parameter points with known probability density g(ξ) and computes the output parameter, e.g., performance metric, density by using a numerical integration. Monte Carlo randomly generates samples according to the circuit or device parameter distribution density and provides an estimate for the output parameter density, e.g., performance metric density; given by the number of samples inside a given output interval.

(8) FIGS. 7A and 7B illustrate that calculation of Qbar (FIG. 7A) and Q (FIG. 7B) by sweeping voltage V uses the same function f(V, V.sub.TH,PH, V.sub.TH,PD, V.sub.TH,AC), which is dependent on the threshold voltage of the PU, PD and AC transistors of the left respectively right side of the cell.

(9) FIG. 8 illustrates hold SNM distribution for time-zero VTH random variability at decreasing supply voltage VDD. The lines illustrate that the numerical propagation technique can easily reach deep into the tails. The markers in the middle are added results of Monte Carlo simulation for comparison, which shows excellent agreement, except for the tails.

(10) FIG. 9 illustrates that aggressive BTI shifts up to 80 mV are introduced in 28 nm planar node to show the excellent fitting of discontinuities when the SNM becomes 0.

(11) FIG. 10 illustrates write trip point (WTP) distribution for time-zero variability of a 10 nm FinFET SRAM cell. Numerical propagation technique is extendable to higher dimensions, e.g., 6-D in this case for WTP distributions.

(12) FIG. 11 illustrates that the total V.sub.TH distribution can be approximated by i) only shifting the time-0 distribution by the degradation amount ΔV.sub.TH or by ii) shifting and adjusting the spread. The discrepancy iii) between the defect-centric distribution and the approximations is evident at higher quantiles.

(13) FIG. 12 illustrates hold SNM degradation for 28 nm planar SRAM simulated for time-zero variation and for BTI degradation using a Normal and defect-centric distribution.

(14) FIG. 13 illustrates the difference between Normal and defect-centric BTI at −6σ for planar and FinFET technologies.

(15) FIG. 14 illustrates hold SNM degradation at the −6 standard normal quantile taken at various V.sub.DD reduction points.

(16) FIG. 15 illustrates CDW in accordance with embodiments of the disclosed technology in comparison to the respective SPICE and VCD waveforms: signal regions with similar frequency f and duty cycle α are grouped in a single CDW point.

(17) FIG. 16 illustrates a pseudo-transient simulation set-up according to embodiments of the disclosed technology.

(18) FIG. 17 is a flow chart of a tool for producing CDW representations given SPICE and VCD waveforms, in accordance with embodiments of the disclosed technology.

(19) FIG. 18, FIG. 19 and FIG. 20 illustrate accuracy assessments of a tool for producing CDW representations given SPICE and VCD waveforms, in accordance with embodiments of the disclosed technology, whereby the tool produces CDW representations using a 1 μs bit stream.

(20) FIGS. 21A-21O illustrate accuracy assessment of the atomistic BTI model of Equation (17) for various degrees of CDW compression in accordance with embodiments of the disclosed technology.

(21) FIGS. 22A-22I illustrate benchmarking results of a single iteration of Equation (17) on bit streams of different average frequencies f.sub.avg of the bit stream. The error of the model according to embodiments of the disclosed technology is restricted to even smaller defect subsets as f.sub.avg is increased.

(22) FIG. 23A illustrates scaling defect time constants for various bias (Vdd) conditions, according to Equation (19). FIG. 23B illustrates scaling defect time constants for various temperature (T) conditions, according to Equation (20).

(23) FIG. 24 is a flow chart illustrating that, before resolving occupancy of defects with the CDW, in accordance with embodiments of the disclosed technology, time constants may be scaled.

(24) FIG. 25 illustrates application of the concept of pseudo-transient BTI/RTN simulation on a single device. Based on the characteristics of the workload (f.sub.avg), a split is made between “slow” and “fast” defects. The former class is handled with the pseudo transient simulation tool in accordance with embodiments of the disclosed technology. Defects that are too “fast” to be simulated that way, are monitored with a brief transient simulation.

(25) FIG. 26 illustrates the distinction between “fast” and “slow” defects.

(26) FIG. 27 illustrates short term device workload, which can be used for “fast” defect simulation.

(27) FIG. 28 illustrates short term BTI/RTN impact on the V.sub.th.

(28) FIG. 29 illustrates the functionality criterion when using pseudo-transient BTI analysis for the estimation of time-dependent functional yield of an SRAM circuit.

(29) FIG. 30 illustrates five different workloads, with regard to voltage supply when using pseudo-transient BTI analysis for the estimation of time-dependent functional yield of an SRAM circuit.

(30) FIG. 31 illustrates the functional yield of a memory at various instances of circuit lifetime, for five different workload classes (static and dynamic V.sub.dd configurations).

(31) FIG. 32 is a flowchart illustrating a method for simulating an electronic circuit according to the disclosed technology.

(32) The drawings are only schematic and are non-limiting. In the drawings, the size of some of the elements may be exaggerated and not drawn on scale for illustrative purposes. The dimensions and the relative dimensions do not necessarily correspond to actual reductions to practice of the invention.

(33) Any reference signs in the claims shall not be construed as limiting the scope.

(34) In the different drawings, the same reference signs refer to the same or analogous elements.

DETAILED DESCRIPTION OF CERTAIN ILLUSTRATIVE EMBODIMENTS

(35) The disclosed technology will be described with respect to particular embodiments and with reference to certain drawings, but the invention is not limited thereto but only by the claims.

(36) The terms first, second and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequence, either temporally, spatially, in ranking or in any other manner. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other sequences than described or illustrated herein.

(37) Moreover, directional terminology such as top, bottom, front, back, leading, trailing, under, over and the like in the description and the claims is used for descriptive purposes with reference to the orientation of the drawings being described, and not necessarily for describing relative positions. Because components of embodiments of the disclosed technology can be positioned in a number of different orientations, the directional terminology is used for purposes of illustration only, and is in no way intended to be limiting, unless otherwise indicated. It is, hence, to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other orientations than described or illustrated herein.

(38) Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the disclosed technology. Thus, appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment, but may. Furthermore, the particular features, structures or characteristics may be combined in any suitable manner, as would be apparent to one of ordinary skill in the art from this disclosure, in one or more embodiments.

(39) Similarly it should be appreciated that in the description of exemplary embodiments of the invention, various features of the invention are sometimes grouped together in a single embodiment, figure, or description thereof for the purpose of streamlining the disclosure and aiding in the understanding of one or more of the various inventive aspects. This method of disclosure, however, is not to be interpreted as reflecting an intention that the claimed invention requires more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive aspects lie in less than all features of a single foregoing disclosed embodiment. Thus, the claims following the detailed description are hereby expressly incorporated into this detailed description, with each claim standing on its own as a separate embodiment of this invention.

(40) Furthermore, while some embodiments described herein include some but not other features included in other embodiments, combinations of features of different embodiments are meant to be within the scope of the invention, and form different embodiments, as would be understood by those in the art. For example, in the following claims, any of the claimed embodiments can be used in any combination.

(41) It should be noted that the use of particular terminology when describing certain features or aspects of the invention should not be taken to imply that the terminology is being re-defined herein to be restricted to include any specific characteristics of the features or aspects of the invention with which that terminology is associated.

(42) In the description provided herein, numerous specific details are set forth. However, it is understood that embodiments of the invention may be practiced without these specific details. In other instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description.

(43) An electronic circuit in the context of the disclosed technology relates to a circuit assembled of individual electronic devices such as transistors, diodes, resistors, capacitors, inductors. These electronic devices are connected to one another by means of conductive paths. Electronic devices include semiconductor devices.

(44) Various kinds of electronic devices and device technologies may be investigated by means of embodiments of the disclosed technology. Whilst of particular application to p- and n-channel FETs (MOSFETs), any other kind of electronic device for which the characteristics are influenced by capture and emission events covering both fabrication time or during operation, may be considered. Every effect in such a device that can be modelled by capture and emission events can be simulated by the disclosed technology. This includes bipolar junction transistors (BJTs), junction field effect transistors (JFETs), high electron mobility transistors (HEMTs), insulated-gate bipolar transistors (IGBTs) and so on. The disclosed technology is, however, not restricted to transistors as devices; for example various kinds of diodes known in the art may also be considered. Embodiments of the disclosed technology may be applied to any device were there are effects that degrade the devices in a partly statistical partly deterministic manner, depending on the workload

(45) Embodiments of the disclosed technology relate to workload-dependent reliability simulations of complex circuits and systems. In the context of the disclosed technology, “simulation” means a transient simulation over a given time period. During such simulation, the parameters of the technology are fixed for the active and passive elements that are used in the electronic circuit to be simulated. Hence a given parameter file is used with a predetermined set for all components (NMOS slow, PMOStyp, Rmin, Ctyp, etc.). The environmental conditions, e.g., temperature and supply voltage, determine the workload, and may be fixed for the entire simulation. Therefore, one process implementation may be simulated in a set of particular environmental conditions, such as at one predetermined temperature condition, and at a particular power supply. Nevertheless, such simulations can be repeated for different process parameters and/or for different environmental conditions, e.g., temperatures or power supply values, to get a more complete overview of the simulation results. For instance, one could wish to see the effect of a spike/dip on the supply or ground line, and can introduce this in the simulation by taking another V.sub.dd value. Furthermore, environmental conditions, e.g., temperature and/or supply voltage, may vary across the overall simulation time. Only within a single CDW segment, in accordance with particular embodiments of the disclosed technology, the environmental conditions are kept constant.

(46) A first aspect is a method for simulating an electronic circuit. The method includes creating a finite set of circuit or device parameter points selected from within an n-dimensional parameter space. The set of circuit or device parameter points is a subset, i.e. less than all points of the total amount of possible circuit or device parameter points within that parameter space. The method further includes, for each circuit or device parameter point of the set, determining a corresponding response value of the performance metric and a corresponding probability of occurrence, and determining, for a predetermined value of the performance metric, the total probability of occurrence.

(47) Due to their minimum device dimensions thus subject to variability, SRAM cells are excellent for a high sigma evaluation case study. Therefore, embodiments will be explained and elucidated by referring to SRAM cells and circuits as particular examples of electronic circuits. This, however, is not intended to be limiting for the disclosed technology, which can be applied to any type of electronic circuit. The disclosed methods and devices are particularly useful for simulation of large electronic circuits, for instance electronic circuits including at least thousands of gates, for instance tens of thousands, hundreds of thousands, or millions of gates.

(48) Furthermore, features and advantages are explained with reference to BTI as one particular example of a time-varying phenomenon that is critical to reliability. This, however, is not intended to be limiting for the disclosed technology. The methods and devices according to embodiments of the disclosed technology apply also to other workload-dependent reliability simulations of complex circuits and systems.

(49) FIG. 1 depicts a 6T-SRAM cell 100 used, as an example only, for simulation. Such cell is currently the most commonly used 6-T SRAM structure for embedded memory. For the sake of description, both commercial 28 nm planar and research-grade FinFET 14 nm and 10 nm nodes will be evaluated. The exemplary illustrated SRAM cell 100 includes four core transistors: two PMOS Pull UP transistors PU1, PU2 and two NMOS Pull Down transistors PD1, PD2. Access NMOS transistors AC1, AC2 are used to read and write the value Q, Qbar stored by the core transistors PU1, PU2, PD1, PD2 using the Word Lines WL and the Bit Lines BL, BLbar. Access and core transistors are assumed to share their bulk connections which are tied to the supply rails VDD and VSS. Exemplary transistor and SRAM sizing parameters for the SRAM cell 100 under study are given in TABLE 1, which lists transistor and SRAM parameters used for planar and bulk FinFET technology.

(50) TABLE-US-00001 TABLE 1 BULK FinFET Tech node [nm] 28 14 10 A.sub.VT [mV μm] 2 1.3 1 σ.sub.0,PU [mV] 35 22 19 σ.sub.0,PD [mV] 30 16 14 σ.sub.0,AC[mV] 35 22 19 sizing ratio 2-2-3 1-1-2 1-1-2 V.sub.DD [V] 1 0.8 0.7 V.sub.TH0 [V] 0.6 0.25 0.25 T.sub.INV [nm] 1.5 1.3 1.2 V.sub.OV/T.sub.INV [MV/cm] 2.7 4.2 3.7 A.sub.n [V] 3.5e−2 3.1e−2 3.1e−2 A.sub.p [V] 1.9e−2 2.0e−2 2.0e−2 γ.sub.n 4.8 5.2 5.2 γ.sub.p 2.5 3 3 n.sub.n 0.203 0.158 0.158 n.sub.p 0.186 0.173 0.173

(51) BTI ΔV.sub.TH is simulated using distribution (1) with the powerlaw components (Y, η) and prefactors A. The mean shift <ΔV.sub.TH> follows a power-law dependency
μ.sub.ΔV.sub.TH(t)=<ΔV.sub.TH(T)>=AV.sub.OV.sup.γt.sup.n  (4)

(52) where t is the total DC stress time and V.sub.OV is the overdrive voltage calculated as
V.sub.OV=V.sub.G−V.sub.TH=V.sub.DD−V.sub.TH  (5)

(53) Correlation between the time-zero and time-dependent variances have been observed experimentally and can be used to estimate the average impact per defect η by first calculating the time dependent variance as

(54) $\begin{array}{cc}{\sigma }_{\Delta V_{\mathrm{TH}}}^2\left(t\right)=\frac{.Math.\Delta V_{\mathrm{th}}\left(t\right).Math.}{100\mathrm{mV}}{\sigma }_{V_{\mathrm{TH}0}}^2& \left(6\right)\end{array}$

(55) To quantify the impact of BTI stress, static margins may be used to determine a performance metric. In the example illustrated, not intended to be limiting for the disclosed technology, the performance metric is the static noise margin (SNM), which may be calculated from a butterfly curve analysis as illustrated in FIG. 2, and as described by Seevinck E. et al. in “Static-noise margin analysis of MOS SRAM cells”, Solid-State Circuits, IEEE Journal of Solid State Circuits, vol. 22, no. 5, pp. 748-754, October 1987, incorporated herein by reference. SNM may be estimated graphically as the length of a side of the largest square that can be embedded inside the lobes of the butterfly curve. For determining the SNM, in the example illustrated, the internal nodes Q and Qbar of the SRAM cell 100 as illustrated in FIG. 1 are independently swept from 0 to VDD, and their complementary nodes Qbar and Q, respectively, are monitored. Here the SNM is defined as the minimum of the two largest squares, each fitting inside a lobe of the butterfly curve. The shape of the butterfly curve is highly dependent on the V.sub.TH of all SRAM transistors. SNM may be estimated during read and hold cases.

(56) Taking this into account, the SNM becomes a 6-dimensional non-analytical function. Numerical representation of the response surface (FIG. 3) may be determined by a simulator, such as a SPICE (simulation program with integrated circuit emphasis) level simulator, and may be used to calculate the SNM response surface.

(57) In embodiments of the disclosed technology, other performance metrics could for instance be, the disclosed technology not being limited thereto, write margin, circuit delay, dynamic and/or leakage power.

(58) In embodiments of the disclosed technology, the response surface may be used for calculation of output statistics. FIG. 3 illustrates the response surface for the particular example of performance metric SNM(ξ) as function of finite set (hence a limited number of) circuit or device parameter points of the parameter space ξ. The parameter space ξ is n-dimensional, the number of dimensions depending on the number of circuit or device parameters taken into account for the simulation. FIG. 3 represents only a 2-dimensional slice of the SNM response surface, where V.sub.TH,1 and V.sub.TH,2 are parameters of the total parameter space ξ. The corresponding SNM Probability density function (PDF) can be evaluated by propagating the PDF g(ξ) of the circuit or device parameters via the output response surface SNM(ξ)(FIG. 4). The SNM probability density f(x) can be evaluated by integrating g(ξ) over all n parameters of the n-dimensional parameter space (hence most likely over more than only V.sub.TH,1 and V.sub.TH,2 illustrated in FIG. 3 and FIG. 4) for SNM(ξ)=x:

(59) $\begin{array}{cc}f\left(x\right)={\int }_{x}g\left(\xi \right)d\xi & \left(7\right)\end{array}$

(60) By integration of the PDF, one can obtain the cumulative distribution function (CDF) of the performance metric SNM:

(61) $\begin{array}{cc}F\left(x\right)={\int }_{-\infty }^X{\int }_{X}g\left(\xi \right)d\xi dx& \left(8\right)\end{array}$

(62) Solving this problem has no analytical approach, however, since the response surface has no closed form solution and is evaluated using a simulator like for instance SPICE simulation. Simulation of the response surface in a finite amount of time requires truncating the response surface. This introduces an error on the final SNM CDF and has to be considered. The error depends on the range taken for the circuit or device parameter domain, hence on the lower bound and upper bound of parameter values taken into account. This error, as illustrated in FIG. 4 is, for independent circuit or device parameters is given by

(63) $\begin{array}{cc}\mathrm{err}=1-\underset{i=1}{\overset{n}{.Math.}}{\int }_{l_i}^{u_i}{f}_i\left(x\right)\mathrm{dx}& \left(9\right)\end{array}$

(64) This error can be reduced to very low levels by suitable selection of lower bounds and upper bounds of the plurality of parameter values, and by suitable distribution of the selected parameter values between the lower and upper bounds, making it possible to reach deep into tails.

(65) In alternative embodiments of the disclosed technology, rather than using the response surface for calculating the output statistics as explained above, a mathematically equivalent calculation method may be used. One embodiment thereof is by making use of line integrals. However, calculating line integrals for a large enough set of SNM values can be computationally quite intensive. In accordance with embodiments of the disclosed technology, a numerical method may be used to calculate the CDF using a similar methodology but circumventing the cumbersome line integrals.

(66) Shown in FIGS. 5A and 5B are SNM PDF points calculated using equation (7) and placed on the SNM axes, i.e. they are sorted according to the SNM value. The PDF points f(C.sub.i) can be unwrapped to their original contributions being the PDF points g(ξ) for all values of circuit or device parameter points ξ where SNM(ξ)=C.sub.i. For an equidistant discrete parameter space (e.g., response surface lookup table), equation (8) can be rewritten as

(67) $\begin{array}{cc}F\left(C_m\right)=\underset{i=0}{\overset{m}{.Math.}}g\left({\xi }_i\right)& \left(10\right)\end{array}$

(68) which is a mathematically equivalent calculation performed by sorting all circuit or device parameters PDF points g(ξ.sub.i) by their corresponding output value C.sub.i (FIGS. 5A and 5B). For a non-equidistant discrete input space, proper bookkeeping is needed where weights w.sub.i need to be attributed to g(ξ.sub.i) to calculate the SNM CDF by the sum

(69) $\begin{array}{cc}F\left(C_m\right)=\underset{i=0}{\overset{m}{.Math.}}{w}_{i}g\left({\xi }_i\right)& \left(11\right)\end{array}$

(70) The weights w.sub.i are proportional to the n-dimensional volume of the circuit or device parameter points ξ.sub.i. The methodology can thus make advantage of optimal meshing schemes for the parameter space, e.g., adaptive meshing.

(71) The above embodiment is explained for the case where all circuit or device parameters PDF points g(ξ.sub.i) are sorted by their corresponding output value C.sub.i. Hereto, the response values C.sub.i may be grouped based on value so that response values that do not deviate more than a per-determined error margin from one another are grouped in a same bin. The bins may then be ranked, for instance from high to low or vice versa. Determining the total probability of occurrence of a predetermined value of the performance metric may then include selecting one of the bins, and for the selected bin, summing all probabilities of occurrence.

(72) In alternative embodiments, rather than sorting the circuit or device parameters PDF points g(ξ.sub.i) by their corresponding output value C.sub.i, the output values C.sub.i may each be simply taken and compared to a reference value. In this case, equation (8) can be rewritten as
F(SNM)=Σ.sub.i=0.sup.mg(ξ.sub.i) for Ci<SNM  (12)

(73) Determining the total probability of occurrence of a predetermined value of the performance metric in this case may include comparing each response value with the predetermined response value, and for all response values lower than the predetermined response value, summing the probabilities of occurrence.

(74) FIGS. 6A and 6B compare the methodology according to embodiments of the disclosed technology with the standard MC methodology where various similarities can be observed. Both methodologies are numerical integration techniques for calculating the output CDF. Monte Carlo randomly generates samples according to the circuit or device parameter distribution density and provides an estimate for the output parameter density, e.g., performance metric density, given by the number of samples inside a given output interval.

(75) A method according to embodiments of the disclosed technology (illustrated in FIG. 6A) creates a finite set of circuit or device parameter points with known probability density g(ξ) and computes the output parameter density, e.g., performance metric density, by using a numerical (e.g., Riemann-Stieltjes) integration. Compared to Monte Carlo (illustrated in FIG. 6B) and extensive MC approaches (e.g., importance sampling, quasi MC), the method according to embodiments of the disclosed technology allows for extensive control over the accuracy of the output distribution by controlling the circuit or device parameter points. The methodology according to embodiments of the disclosed technology can provide for a uniform accuracy of the output distribution across a wide sigma range, compared to MC where the accuracy decreases exponentially when moving towards higher sigmas.

(76) Experiments

(77) Using the methodology in accordance with embodiments of the disclosed technology, as described hereinabove, the SRAM structure 100 has been simulated with the parameters as given in TABLE 1. When simulating Hold SNM it is sufficient to create the response surface as a function of the threshold voltages of the four core transistors PU1, PU2, PD1, PD2, since in this mode of operation the access transistors AC1, AC2 are off and will have no impact on the butterfly curves. Contrary, for Read SNM calculation, where the Word Lines WL are high, the access transistors AC1, AC2 are on and do impact the butterfly curves. Here the response surface has to be calculated as a function of the threshold voltages of all six transistors PU1, PU2, PD1, PD2, AC1, AC2. Furthermore, margins to assess the writeable of the cell like for instance Write Trip Point (WTP) analysis also requires calculating the response surface as function of all six transistor's threshold voltage.

(78) Reducing the dimensionality of the problem using sensitivity analysis is desirable, since the number of simulation points (i.e. the finite set of circuit or device parameter points) to create the response surface increases rapidly for higher dimensions. Specifically for SRAM analysis a significant reduction can be achieved by exploiting the symmetry of the cell as illustrated in FIGS. 7A and 7B. For a symmetric SRAM cell the SNM response surface can be written as

(79) $\begin{array}{cc}\begin{array}{c}\mathrm{SNM}=h\left(V_{\mathrm{TH},\mathrm{PU}1},V_{\mathrm{TH},\mathrm{PD}1},V_{\mathrm{TH},AC1},V_{\mathrm{TH},\mathrm{PU}2},V_{\mathrm{TH},\mathrm{PD}2},V_{\mathrm{TH},AC2}\right)\\ =h^{*}(f\left(V_{\mathrm{TH},\mathrm{PU}1},V_{\mathrm{TH},\mathrm{PD}1},V_{\mathrm{TH},AC1}\right),\\ f\left(V_{\mathrm{TH},\mathrm{PU}2},V_{\mathrm{TH},\mathrm{PD}2},V_{\mathrm{TH},AC2}\right))\end{array}& \left(13\right)\end{array}$

(80) This allows simulation of one side of the SRAM cell instead of two for function ƒ(V, V.sub.TH,PU, V.sub.TH,PDV.sub.TH,AC) and reconstruction of the response surface using equation (13). This reduces the 6-dimensional problem of the experiment to a 3-dimensional problem.

(81) Shown in FIG. 8 is the methodology according to embodiments of the disclosed technology applied to the calculation of the Hold SNM (i.e. Word Lines WL=0V) distribution with time-zero variability at decreasing supply voltage VDD. Excellent matching of MC samples is obtained at low quantiles but the accuracy goes up to the 7-8 sigma quantile by choosing a wide enough input range for creating the response surface according to equation (9). Due to the flexible set-up, various supply voltages can be simulated to acquire the minimum retention voltage under local variability.

(82) Since the SNM is defined as the minimum of the two Seevinck squares that can be fit inside the butterfly curve, it will at some point collapse to zero. This will happen abruptly when the two voltage transfer curves stop overlapping. Consequently, the SNM response surface has a discontinuity at SNM=0 which will propagate to the SNM distribution. However, discontinuities of the response surface are inherently handled which is crucial for numerous circuit output performance parameters. Shown in FIG. 9 is the simulation of the Hold SNM for a 28 nm planar SRAM, where aggressive BTI shifts up to 80 mV are introduced to show the excellent fitting of discontinuities when the SNM becomes 0. It is thus clear that, with a method in accordance with embodiments of the disclosed technology, this discontinuity can be accurately handled. The methodology according to embodiments of the disclosed technology can also be applied to higher input dimensions, e.g., the case of Write Trip Point (WTP) analysis as shown in FIG. 10, where the access transistors need to be taken into account.

(83) FIG. 11 illustrates that the total V.sub.TH distribution can be approximated by (i) only shifting the time-zero distribution by the degradation amount ΔV.sub.TH or by (ii) shifting and adjusting the spread. The discrepancy between the defect-centric distribution and the approximation is evident at higher quantiles. As thus shown in FIG. 11 the defect-centric BTI distribution deviates from a normal distribution for high sigmas. To verify the importance of correctly simulating the tail of the BTI distribution, as can be done in accordance with embodiments of the disclosed technology, Hold SNM is simulated for a 28 nm planar SRAM cell using an approximating Normal distribution as illustrated in FIG. 12. Using a Normal distribution for the BTI degradation can cause significant error compared to the defect-centric case and will moreover increase further for higher quantiles. Upon reducing the supply voltage, higher sensitivity for this discrepancy is observed at lower VDD as shown in FIG. 14 which can severely underestimate the minimum supply voltage for data retention. Planar technologies show a higher discrepancy compared to FinFET due to the higher intrinsic variability. FinFET technologies on the other hand show higher sensitivity of the discrepancy as function of supply voltage.

(84) Using the method according to embodiments of the disclosed technology, fast sensitivity analysis for worst case −6 quantile can be performed for supply voltage lowering and as a function of stress time using equation (4), as shown in FIG. 13. Here FinFET technology proves to have a higher robustness for SRAM cell hold stability at lower voltages. BTI degradation results in an increased data retention voltage needed to maintain a pre-determined SNM margin.

(85) In accordance with further embodiments of the disclosed technology, the probabilities of occurrence of the circuit or device parameter points are time dependent. Simulating a probability of occurrence of a value of a performance metric of an electronic circuit, for instance such simulations as described in more detail above, can be applied for a sequence of time instances in a potentially very long time period, such as e.g., over a time period as long as a few minutes to a few hours. The time period over which the simulations can be carried out can be at least 10.sup.8 clock cycles, for instance 10.sup.12 clock cycles or more. The method then includes repeating the steps of determining the probability of occurrence of the circuit or device parameter points, and determining, for a predetermined value of the performance metric, the total probability of occurrence.

(86) Each simulation, i.e. each evaluation of the response value of the performance metric is dependent on the active workload (value of input signals) applied to the electronic circuit at that time instance. Each workload signal can be represented by a waveform over time, as for instance illustrated in FIG. 15. The computational complexity of a simulation model is tightly coupled to the granularity of the signal activity representation for each simulated device of the electronic circuit. Moreover, given the variability of supply voltage and temperature across modern integrated circuits (ICs), also full temperature and bias dependency of the BTI effect are desirable in the simulation.

(87) In accordance with embodiments of the disclosed technology, a waveform representation called compact digital waveform (CDW) can be used. Being of tunable accuracy, this format exploits optimally the accuracy versus complexity tradeoff. It can be used for fast BTI simulations.

(88) For ease of explanation, a transient waveform is assumed, which in the most detailed case is represented as a SPICE signal. Up to now, SPICE has been considered as the industry standard for IC simulation, in which in a transient simulation mode, signals are adaptively discretized across time. It can be assumed that a SPICE signal is the most accurate signal representation.

(89) In embodiments of the disclosed technology, it is proposed to group consecutive signal regions that feature a similar frequency f and duty cycle α figures and that occupy a duration Δt, at particular environmental conditions such as supply voltage V.sub.DD and temperature T. Each region of the signal that adheres to this property is represented by a single CDW point with coordinates (f,α,Δt). To derive these points, the initial signal is scanned, which for the sake of simplicity is represented by a Value Change Dump (VCD) equivalent. Such VCD representation creates an abstraction of voltage signals to logic levels.

(90) It will be assumed that the VCD format includes only the high (Vdd) and low (Vss) levels; however, the VCD standard does support multiple Vdd representations, and thus, the input can be perceived as a succession of low and high voltage levels, e.g., a sequence of AC pulses that have, in the general case, varying f and a values. As the digital signal is scanned, a buffer accumulates pulses that satisfy Equation (14):
|ƒ.sub.pulse−ƒ.sub.buffer|<εƒ{circumflex over ( )}|α.sub.pulse−α.sub.buffer|<ε.sub.α  (14)

(91) When a digital pulse is come across that cannot be appended (i.e., does not satisfy Equation (14)), the buffer is flushed to a single CDW point and re-initialized to receive the next pulses. E.sub.f (Hz) and ε.sub.α (p.u.) are user-defined error margins. These margins enable the tunable accuracy of the CDW approximation. The crudest approximation is achieved with ε.sub.f.fwdarw.+∞ and ε.sub.α.fwdarw.1, since all digital pulses are then appended to the buffer. Conversely, for the finest approximation is set ε.sub.f.fwdarw.0 and ε.sub.α.fwdarw.0. With this configuration, each individual AC pulse constitutes a single CDW point, in the worst case. Using the CDW format, any signal may be represented by a series of periodic activities, each having a period, a duty cycle and a duration over which the activity is periodically repeated. A method according to embodiments of the disclosed technology may include such grouping of a series of distinct periodic activities with similar period and duty cycle by temporally distinguishing such regions.

(92) The values f.sub.buffer and α.sub.buffer are defined in Equation (15) for a buffer containing n pulses and are computed every time a new digital AC pulse is appended to the buffer

(93) $\begin{array}{cc}{f}_{\mathrm{buffer}}=\frac{1}{n}\underset{i\in \mathrm{buffer}}{.Math.}{f}_i.Math.{\alpha }_{\mathrm{buffer}}=\frac{1}{n}\underset{i\in \mathrm{buffer}}{.Math.}{\alpha }_i.& \left(15\right)\end{array}$

(94) The concept of the waveform approximation according to embodiments of the disclosed technology, as explained above, has been implemented in a tool, which produces CDW representations given SPICE or VCD waveforms. The functionality of this tool can be seen in the flowchart of method 1700 in FIG. 17.

(95) In decision block 1710, method 1700 identifies SPICE, VCD, and CDW formats. Method 1700 routes SPICE waveforms from block 1710 to block 1720, which converts SPICE waveforms to VCD waveforms, given V.sub.dd and V.sub.ss margins from block 1740. Method 1700 routes VCD waveforms from blocks 1710 and 1720 to block 1730, which converts VCD waveforms to CDW representations, given ε.sub.f and ε.sub.a from data block 1750. Data block 1760 illustrates CDW representations from decision block 1710 or block 1730.

(96) A qualitative example of the produced CDWs can be seen in FIG. 15. In order to assess the accuracy of the tool according to embodiments of the disclosed technology, a random 1 μs bit stream of 1 GHz frequency is used. The CDW representation is compared against the respective SPICE waveform. The error of signal approximation according to Equation (16) is defined
e.sub.s(T)=V.sub.SPICE(t)−V.sub.CDW(t)  (16)
and the mean (FIG. 18) and standard deviation (FIG. 19) values are derived. Given that the buffer of the tool according to embodiments of the disclosed technology averages-out the f and a values of the grouped pulses, reduced impact of the tool on the mean error of the approximation is expected (FIG. 18). However, the standard deviation (FIG. 19) is affected much more, given that a cruder approximation misses both voltage transitions and regions of constant voltage level.

(97) Apart from the accuracy of the CDW representation according to embodiments of the disclosed technology from a signal integrity point of view, it is relevant to consider the achieved compression in the number of points it uses for representation. FIG. 20 illustrates the number of points produced for various choices of the ε.sub.f and ε.sub.α error margins. The number of points used by any signal representation is directly coupled to the CPU time required for electronic circuit simulation, as in the case of atomistic or defect-centric BTI/RTN simulation. The reason is that each point corresponds to a single iteration of the respective simulation model. The aggressive reduction in the number of such points in accordance with embodiments of the disclosed technology is one further step towards a computationally feasible simulation. The reason for that is that (many) fewer individual time point simulations will need to be done in order to obtain the time-dependent probability of occurrence of a value of a performance metric of the electronic circuit. This is achieved over an extended lifetime of the target circuit, across which the CDW representation spans.

(98) In the following, the impact is explored that aggressive CDW compression has on the accuracy of simulation modeling.

(99) Given the compression capabilities of a CDW according to embodiments of the disclosed technology, it is important to explore its usability for the target goal, namely to execute the electronic circuit simulation, for instance atomistic BTI/RTN simulation, over strides of V.sub.gs workload. As an example, in accordance with embodiments of the disclosed technology, a version of the atomistic BTI model will be altered and reformulated as follows: Starting from the premise of first order kinetics, the capture probability of a gate stack defect follows the ODE of Equation (17). The general solution returns the capture probability (P.sub.c), after n V.sub.gs pulses of specific frequency f and duty cycle a have been applied (equivalent duration Δt=n/f). The proposed model fully retains workload memory through the initial condition P.sub.c0. Parameters a and b are functions of f and α:

(100) 0$\begin{array}{cc}\frac{{\mathrm{dP}}_c}{\mathrm{dt}}={\mathrm{aP}}_c+b.Math.P_c=\left(P_{c0}+\frac{b}{a}\right)e^{\mathrm{an}}-\frac{b}{a}.& \left(17\right)\end{array}$

(101) The initial condition P.sub.c0 is absent from the previous version of the AC BTI model. Hereinafter, P.sub.c0≠0 will be assumed, thus introducing memory to the AC BTI model (i.e., workload dependency over time). Each iteration of the general solution of Equation (17) is used as initial condition for the next model evaluation. The tool implementing the compact digital waveform approximation, presented hereinabove, identifies regions of similar frequency f and duty cycle α in the V.sub.gs (i.e., a CDW representation is produced). Then, at each (f,α,Δt) point, a single iteration of Equation (17) is evaluated, representing the impact of the entire duration of that CDW segment. The latter can potentially be very long, encompassing a huge amount of individual time points. In this way, the CDW representation according to embodiments of the disclosed technology reduces aggressively the number of model iterations required for the calculation of P.sub.c. That way, time-dependent circuit simulation, for instance BTI simulation, becomes computationally feasible over long time spans (see FIG. 15). In FIG. 16 can be seen how the CDW format is combined with the proposed BTI/RTN atomistic model of Equation (17). Given the compression capabilities of the CDW representation (see FIG. 20), the number of model iterations is aggressively reduced, thus decreasing the CPU time required for simulation.

(102) A method according to embodiments of the disclosed technology then only has to be applied to the CDW segments or primitives and not to the individual time instances. Or stated in other words: the number of simulations to be carried out can be reduced, thus improving (reducing) the time required to simulate the electronic circuit.

(103) In what follows, as an example, BTI evaluation is performed using CDW in accordance with embodiments of the disclosed technology.

(104) The accuracy, from a signal integrity point of view, of the concept of the waveform approximation according to embodiments of the disclosed technology, has been assessed hereinabove. Given the model of Equation (17), the accuracy of a CDW approximation in terms of evaluation of capture probability Pc will now be evaluated.

(105) For experiments, 100 random bit streams with an average frequency (f.sub.avg) of 1 GHz and a duration of 1 μs have been used. Given each signal, an error metric for BTI evaluation is defined, according to Equation (18):
e.sub.p=P.sub.c,SPICE−P.sub.c,CDW  (18)
where P.sub.c,SPICE is the capture probability for a defect at the end of the signal based on SPICE evaluation (e.g., using Equation (19)). P.sub.c,CDW is a similar capture probability, using a CDW file (for a specific pair of εf and εα) and the model of Equation (17).

(106) $\begin{array}{cc}{P}_c\left(t+t_s\right)=\frac{{\tau }_{\mathrm{ei}}}{{\tau }_{\mathrm{ei}}+{\tau }_{\mathrm{ci}}}+\left[P_c\left(t\right)-\frac{{\tau }_{\mathrm{ei}}}{{\tau }_{\mathrm{ei}}+{\tau }_{\mathrm{ci}}}\right]\exp \frac{-t_s\left({\tau }_{\mathrm{ci}}+{\tau }_{\mathrm{ei}}\right)}{{\tau }_{\mathrm{ci}}{\tau }_{\mathrm{ei}}}& \left(19\right)\end{array}$

(107) Equation (19) is the core of a transient workload dependent implementation of the atomistic BTI model as described for instance by B. Kaczer et al. in “Atomistic approach to variability of bias-temperature instability in circuit simulations”, in Proc. IEEE IRPS, 2011, pp. XT.3.1-XT.3.5.

(108) Five different cases are used, ranging from “fast” (τ.sub.cH=10.sup.−11 s) to “slow” (τ.sub.cH=10.sup.−3 s) defects. That way, the mean, standard deviation and maximum of e.sub.p across the 100 signals can be calculated for various values of (ε.sub.f, ε.sub.α) and for the five different defects (FIGS. 21A-21O).

(109) A direct correlation exists between how “fast” a defect is and the average frequency of the V.sub.gs bit stream (f.sub.avg). More specifically, the defects that are significantly “slower” in comparison to the average frequency of the bit stream are accurately modeled, regardless of the compression imposed by the concept of the waveform approximation according to embodiments of the disclosed technology (FIGS. 21A-21I). The last two defects (FIGS. 21J-21O) have time constants comparable to the bit stream's period (1/f.sub.avg). The proposed model is unable to follow the capture probability P.sub.c of these defects, irrespective of ε.sub.f and ε.sub.α values.

(110) The relation between the average frequency f.sub.avg of the bit stream and the defect time constants has been identified earlier; however, no comprehensive study has been performed for the accuracy limitations of the AC BTI/RTN model. Given that a version of this model, enhanced with workload memory, is the kernel of the simulation methodology according to particular embodiments of the disclosed technology, its accuracy limitations for each model iteration can be exhaustively explored. More specifically, it is important to identify for which defects and under which conditions the accuracy of a model according to embodiments of the disclosed technology is acceptable.

(111) In order to conclude on the accuracy limitations of an approach according to embodiments of the disclosed technology, first a set of 400 different defects is created, with time constants (τ.sub.cH, τ.sub.cL, τ.sub.eH, τ.sub.eL) uniformly distributed in the logarithmic scale. The defects are arranged in a two dimensional plane according to min-km, {τ.sub.cH, τ.sub.cL} and min {τ.sub.cH, τ.sub.eL}.

(112) Three sets of bit streams with average frequency f.sub.avg equal to 1 GHz, 100 MHz and 10 MHz are used for each set. Each set contains 100 signals, each one with a duration of 1 μs. Each signal is CDW-approximated with maximum compression (ε.sub.f.fwdarw.+∞ and ε.sub.α.fwdarw.1) and e.sub.p is calculated according to Equation (18). That way, μ{e.sub.p}, σ{e.sub.p} and max{e.sub.p} can be calculated for each defect and for each case with the selected average frequencies f.sub.avg. The results are shown in FIGS. 22A-22I. It is relevant to note that these results represent the accuracy of the proposed model for a single iteration of Equation (17). However, they also provide useful insight on the accuracy limitations of the proposed model, when using it for an arbitrary number of iterations.

(113) For all average frequency values f.sub.avg, a subset of defects can always be found that cannot be accurately modelled. As the average frequency f.sub.avg is increasing, this subset is reduced and an acceptable accuracy of the model according to embodiments of the disclosed technology spreads to more defect time scales. For the case of f.sub.avg=1 GHz, the model according to embodiments of the disclosed technology can accurately simulate all defects that satisfy min {τ.sub.cH, τ.sub.cL}≥10.sup.−7 s and min {τ.sub.cH, τ.sub.eL}≥10.sup.−7 s.

(114) With dynamic voltage and frequency scaling (DVFS) being increasingly common among modern ICs, it is relevant to also include the impact of voltage bias on the BTI simulation. Furthermore, aggressive integration may also cause significant thermal gradients across an IC. As a result, the BTI simulator can also be aware of the local temperature conditions. It has been previously demonstrated that the time constants of BTI/RTN defects are affected by the operating bias (V.sub.dd) and temperature conditions. Based on state-of-the-art measurements, two orthogonal scaling rules of a time constant are proposed, in view of temperature and bias variations (see Equations (20) and (21)):

(115) $\begin{array}{cc}\left({\tau }_0\right)=\log \left\{\tau \right\}=2\times \left(V_{\mathrm{dd},0}-V_{\mathrm{dd}}\right)+\log \left\{{\tau }_0\right\}& \left(20\right)\\ \left({\tau }_0\right)=\log \left\{\tau \right\}=\frac{T_0-T}{50}+\log \left\{{\tau }_0\right\}& \left(21\right)\end{array}$

(116) These rules constitute a first order approximation of time constant scaling, as observed in measurements from real devices. It can be seen that in each case, the new time constant (τ) depends on its original value (τ.sub.0), as well as the original and new value of bias (V.sub.dd,0 and V.sub.dd) or temperature (T.sub.0 and T), respectively. The original temperature (T.sub.0) and bias (V.sub.dd,0) values refer to the conditions under which the time constant measurement was performed on a sample of devices from a specific technology node. The scaling factors (−2 for bias and − 1/50 for temperature) are also technology dependent. An example of bias and temperature scaling (applied individually) appears in FIG. 23A and FIG. 23B, respectively, for an initial temperature T.sub.0=125° C. and an initial bias condition V.sub.dd,0=1 V.

(117) It is easy to prove that the scaling operators and maintain a commutative composition, thus they can be applied in an arbitrary sequence. Equations (20) and (21) require an extension of the CDW representation in order to include bias and temperature information. Thus, before the occupancy of the defects is resolved, the time constants can be scaled appropriately (see, for example, the flowchart of method 2400 in FIG. 24). The flowchart includes data block 2410 which provides a CDW representation, and data block 2420, which provides initial instantiated defects. In block 2430, method 2400 initializes loop counter i, for a loop including blocks 2440, 2450, and 2460. In block 2440, method 2400 scales time constants with Ti and V.sub.dd,i from data block 2410. In block 2450, method 2400 estimates BTO with ƒ.sub.i, g.sub.i, and Δt.sub.i from data block 2410. In block 2460, method 2400 increments loop counter i.

(118) It should be noted that temperature scaling affects all four time constants. Bias affects only the time constants that refer to a non-zero voltage, namely τ.sub.cH and τ.sub.eH for the nMOS and τ.sub.cL and τ.sub.eL for the pMOS.

(119) FIG. 25 illustrates a flowchart of method 2500, of the application of the concept of pseudo-transient BTI/RTN simulation on a single device. Based on the characteristics of the workload (ƒ.sub.ang) 2530, a split is made between “slow” and “fast” defects. The former class is handled with the pseudo transient simulation tool 2550 in accordance with embodiments of the disclosed technology. Defects that are too “fast” to be simulated that way are monitored with a brief transient simulation 2570.

(120) From the results of the accuracy assessment of BTI evaluation using CDW, it can be seen that, given the time constants of a defect 2520 and the workload of the respective device 2510, it can be identified whether this defect is suitable for pseudotransient simulation 2550 (see FIG. 16).

(121) If that is the case, V.sub.gs intervals can be stridden over using a CDW approximation 2540 in accordance with embodiments of the disclosed technology. This course of action (pseudo-transient simulation 2550) involves a few iterations of the proposed BTI/RTN model and comes at a very small CPU overhead. If the defect is too “fast” for the pseudotransient model, a more detailed simulation 2570 (with a SPICE or VCD waveform 2560) is deemed necessary. However, its duration need not cover the entire device lifetime: the impact of fast defects can be seen within a small time interval. For the subset of the “fast” defects, a margin of V.sub.th short-term variation can be derived that can be superimposed to the long term BTI/RTN impact derived by the pseudo-transient model. These impacts can be combined in block 2580, resulting in graph 2590 showing long-term impact and short-term variations.

(122) In order to verify the aspects of this methodology, it has been tested on a single device. First, this device has been populated with defects and these have been arranged in the min {τ.sub.cH, τ.sub.cL} versus min{τ.sub.eH, τ.sub.eL} plane (FIG. 26). A bit stream with f.sub.avg=1 GHz is assumed as device workload. Given the results of FIGS. 22A-22I, it can be assumed that 10.sup.−7 s is the boundary between fast and slow defects. For the “fast” defects, a sample SPICE waveform may be used, representative of the device workload (FIG. 27). With a brief simulation that covers the “fast” time constants, the short-term impact of the respective defects can be obtained (FIG. 28). For the long-term impact of BTI/RTN, a CDW representation of the device lifetime (until 10.sup.8 s) is created using 100 points. As a result, each point has f.sub.avg=1 GHz and Δt=106 s. What varies between CDW points is the duty cycle α, which is visualized in FIG. 27. The long term BTI/RTN impact is calculated using the model of Equation (17) and can be combined with the worst case impact of “fast” defects (FIG. 28).

(123) The device workloads presented above are artificial. However, they can be easily derived from industry standard tools when handling realistic circuits and applications. A brief SPICE simulation of the target circuit can produce a representative workload for the “fast” defect simulation. α values for different circuit operation modes can be easily derived using timing analysis tools. The above methodology has no interdevice dependencies and can be generalized to a larger device inventory. This data parallelism is highly compatible with the multi-/many-core paradigm observed in the computing infrastructure of modern design houses.

(124) Given that the CDW signal representation according to embodiments of the disclosed technology leads to an acceleration of electronic circuit simulation, e.g., BTI simulation, across device lifetime, it is good to reflect on the usability of the proposed framework for realistic circuits. As an example, the accurate estimation of functional yield of a circuit over three years of operation is explored.

(125) A subset of an embedded static random access memory (SRAM) is used, isolating a representative path, from a memory cell to the SRAM's input/output ports. This exemplary circuit contains 63 devices and it supports reading and writing of the cell. A sample of this design is defined as functionally correct if a zero value can be written and then read successfully (FIG. 29). The functional yield of the target circuit is estimated, namely the percentage of samples that exhibit correct functionality. Derivation of such metrics is of major importance at design time and many methodologies exist solving this problem. Given the inherent time- and workload-dependency of the proposed model according to embodiments of the disclosed technology, the functional yield can be estimated at different instances of circuit lifetime.

(126) The evolution of V.sub.th is perceived for each device according to Equation (22):
V.sub.th(t)=V.sub.th,TZ+ΔV.sub.th,TD(t), where: V.sub.th,TZ˜Norm(μ.sub.0,σ.sub.0) and ΔV.sub.th,TD˜AtomisticModel.  (22)

(127) V.sub.th,TZ represents the time-zero component of V.sub.th fluctuations, whereas V.sub.th,TD represents the time-dependent counterpart. The former component is constant throughout the device lifetime but varies across device samples. In the context of the current invention, the latter component is attributed solely to BTI. V.sub.th,TD is a function of time and varies across different device samples. The atomistic model for V.sub.th,TD is configured according to TABLE 2 and is solved in a pseudo-transient way. Table 2 lists atomistic model configuration parameters for yield analysis of the target SRAM circuit, according to BTI/RTN measurement as disclosed by M. Toledano-Luque et al. in “Response of a single trap to AC negative bias temperature stress”, Proc. IEEE IRPS, 2011, pp. 4A.2.1-4A.2.8, and the 90 nm predictive technology model disclosed at http://ptm.asu.edu/.

(128) TABLE-US-00002 TABLE 2 Parameter Distribution V.sub.th,TZ Time-Zero pFET V.sub.th,TZ~Norm (−0.339 V, −0.033 V) Time-Zero nFET V.sub.th,TZ~Norm (0.397 V, 0.04 V) ΔV.sub.th,TD (t) Defects per pFET N.sub.p~Pois (λ = 10.sup.11 × Area[cm.sup.2]) Defects per nFET N.sub.n~Pois (λ = 6.7 × 10.sup.10 × Area[cm.sup.2]) Time Const. log.sub.10 {τ*.sub.pV}~Unif (a = −12, b = 12) ΔV.sub.th per ΔV.sub.th~Exp (η = 5), Defect (mV) for pFETs and nFETs alike *p: process, either capture (c) or emission (e) V: voltage, either high (H) or low (L)

(129) To emphasize on the workload memory/dependency of the proposed flow, five different cases are assumed in terms of voltage supply for the test case circuit (FIG. 30). Such settings cover the V.sub.dd parameter of the CDW. It is assumed that the test case circuit operates at a 1 GHz frequency, executing iteratively the functionality criterion test of FIG. 29 (namely the write 0.fwdarw.read 0 sequence of operations). Knowledge of this activity can provide the α and f parameters of the CDW for each device of the circuit using the waveform approximation tools according to embodiments of the disclosed technology. The inspected lifetime is about 3 years (10.sup.8 s) and is split in intervals of 2.5×10.sup.7 s (which is used as the Δt parameter of the CDW). Finally, the temperature is fixed at a reference value of 50° C. for all the simulations.

(130) The functional yield of the circuit is estimated for all five V.sub.dd cases and the 95% confidence interval is also provided based on 150 Monte Carlo iterations for each case. The execution time has been measured at roughly 3 hours of processing per 150 Monte Carlo iterations on a desktop computer with an Intel Core 2 Quad processor operating at 2.66 GHz. Each iteration starts with initialization of V.sub.th,TZ (from the normal distribution) and of the gate stack defects (based on the atomistic model). At each time step, V.sub.th,TD is evaluated per device and the circuits functionality is tested.

(131) For the sake of brevity, the impact of the “fast” defects on the functional yield analysis has been ignored. Apart from being covered by previous work, we have refrained from transient BTI/RTN simulations, since they can be incorporated to the time-zero (and ever present) variations of V.sub.th by adding a “safety” margin. Significant work exists in experimentally delivering yield and failure metrics for SRAMs, strictly due to “fast” defect activity. However, until now and to the best of the inventors' knowledge a time- and workload-dependent yield analysis of memory circuits as in accordance with embodiments of the disclosed technology has never been performed with the atomistic BTI model for the entire circuit lifetime.

(132) The yield estimations of FIG. 31 provide useful hints towards time-zero and time-dependent variability mitigation. It can be observed that at a constant V.sub.dd, the target circuit degrades aggressively within the first 2.5×10.sup.7 s lifetime interval. Afterwards, the yield shows minimal fluctuation. This is expected, considering that the defect capture time constants are uniformly distributed across the logarithmic scale. It is obvious that constant voltage supply is unable to keep the yield of the circuit at acceptable levels. On the contrary, a dynamic V.sub.dd configuration (namely Case 5 of FIG. 31) leads to a more acceptable yield profile. Finally, the BTI amplification due to elevated voltage supply appears to be overwhelmed by the reliability enhancement due to increased overdrive.

(133) The proposed pseudo-transient, atomistic simulation scheme for BTI according to embodiments of the disclosed technology is a major enable for a thorough yield analysis. By retaining the time dependency of the atomistic model, the evolution of functional yield can be monitored throughout the circuit lifetime. Based on the model's workload dependency, dynamic alterations to the circuits workload can be tested and their impact on the functional yield can be observed. Such capabilities are very welcome for the design of reliable ICs, especially at aggressively downscaled technologies.

(134) The previous description of the disclosed examples is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these examples will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other examples without departing from the spirit or scope of the invention. As will be recognized, certain embodiments of the invention described herein can be embodied within a form that does not provide all of the features and benefits set forth herein, as some features can be used or practiced separately from others. The scope of the invention disclosed herein is indicated by the appended claims rather than by the foregoing description. All changes which come within the meaning and range of equivalency of the claims are to be embraced within their scope. Thus, the present invention is not intended to be limited to the examples shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.