Dániel B Nagy

Philippe Claudin

Tamás Börzsönyi

Ellák Somfai

Rheology of dense granular flows for elongated particles

Keywords: numbers: 47, 57, Gc, 83, 80, Fg

We study the rheology of dense granular flows for frictionless spherocylinders by means of 3D numerical simulations. As in the case of spherical particles, the effective friction µ is an increasing function of the inertial number I, and we systematically investigate the dependence of µ on the particle aspect ratio Q, as well as that of the normal stress differences, the volume fraction and the coordination number. We show in particular that the quasi-static friction coefficient is nonmonotonic with Q: from the spherical case Q = 1, it first sharply increases, reaches a maximum around Q 1.05, and then gently decreases, reaching back its initial value for Q 2. We provide a microscopic interpretation for this unexpected behavior through the analysis of the distribution of dissipative contacts around the particles: as compared to spheres, slightly elongated grains enhance contacts in their central cylindrical band, whereas at larger aspect ratios particles tend to align and dissipate by preferential contacts at their hemispherical caps.

Rheology of dense granular flows is an active domain of research, motivated by fundamental questions on this 'complex fluid' as well as by practical needs in soil mechanics and geotechnical engineering. Since the 1950s a number of models have been suggested, including Bagnold's scaling [1,2], the theory of the rapid flow regime [3] as well as other regimes [4]. A major step in the description of the dense regime has been achieved 10-15 years ago with the development of the framework of the now so-called µ(I) rheology [5,6], which successfully describes these flows in the absence of strong spatial gradients or temporal changes. This approach has shown that the constitutive equations, which augment the conservation laws for a complete rheological description of these flows, can be formalized in terms of the 'inertial number' I = γd/ p/ρ, where γ is the shear rate, p is the pressure, d is the average grain diameter, and ρ is the density of the particles' material. This dimensionless number can be interpreted as the ratio of the characteristic time scale d/ p/ρ of microscopic rearrangements, and the macroscopic time scale 1/ γ of the deformation. In the case of rigid grains, for which the pressure is the only stress scale, the dimensional analysis tells us that the effective friction µ of the flow, defined by the ratio of the shear stress to the pressure, as well as the volume fraction φ of the granular packing, are functions of I. The shape of these functions has been determined both by simulations [7][8][9] and experiments [10,11].

The µ(I) formalism has been successfully applied in a number of flow geometries [5], recovering the Bagnold scaling in flows down an inclined plane [2], and describing chute flow [12][13][14][15], silo discharge [16], granular column collapse [17,18], and dynamic compressibility effects in spontaneous oscillatory motion [19,20]. This rheology has been extended in a number of ways taking into ac- count various effects, like cohesion [21], finite pressure or soft particles [22], and self-propelling particles [23]. Another example for extension is the description of granular suspensions [24][25][26]. In this case, a new time scale, that of the viscous dissipation, is introduced, which is captured by a new dimensionless group, the 'viscous number' J. This formalism describes Brownian suspensions as well [27], and has been incorporated to diphasic models for sediment transport [28][29][30].

Another trend in granular physics is considering shape anisotropy for the particles, see the recent review [31]

FIG. 2. (a) (d) (b) (e) (c) (f)
Top row: (a) effective friction, (b) first and (c) second normal stress differences, as functions of I. The normal stress differences are normalized by the absolute value of the imposed stress, py. The solid curves are fits of the form Eq. 1 with α = 0.4 in the range 10 -3.5 ≤ I ≤ 10 -2 . Bottom row: aspect ratio dependence of the quasi-static (I → 0) values of the same quantities: (d) effective friction, (e) first and (f) second normal stress differences. The first few points correspond to Q = 1, 1.01, 1.02, 1.05 and 1.1.

references therein. One of the fundamental results is the observation that elongated particles get oriented in shear flow [32][33][34][35]. The average orientation angle θ av , which is the angle between the average orientation of the particles and the streamlines, is nonzero; it decreases with the length-to-width aspect ratio Q of the particles, but only weakly depends on the shear rate [36]. There is an interplay between this orientational ordering and the packing fraction or the contact force network [37][38][39]. The quasistatic behavior of 2D systems has been investigated with rounded-cap rectangular particles in a biaxial set-up [37]. The fast regime (typically I > 0.1) has been explored in various 2D configurations, including a volume fraction controlled shear cell, with dumbbells [40,41]. In this paper, we study the rheological properties of assemblies of 3D frictionless spherocylinders in a pressure-controlled shear cell. We explore the range of shear rate for which the µ(I) formalism is expected to apply, i.e. from the quasi-static limit (I → 0) to the beginning of the kinetic regime I 0.1. We unexpectedly find a non-monotonic behavior of the quasi-static friction coefficient with the particle aspect ratio, a key result missed by previous studies [37,41]. We also report the emergence of normal stress differences, whose marginal presence was already noticed for 3D flows of spheres [42], but which clearly develop for elongated particles in a way qualitatively similar to those in suspensions of fibers [43,44]. A very recent experimental study of the rheology of non-colloidal sus-pensions of rigid fibers has investigated the effect of the particle aspect ratio in the range 3-15, indicating an aspect ratio independent friction coefficient, but a decreasing jamming packing fraction with increasing Q [45]. A non-monotonic dependence of the packing fraction on Q has been observed previously also in non-sheared systems [46][47][48].

Numerical setup.-To model homogenous shear flow, we used a 3D plane-Couette geometry, with periodic boundary conditions in the x (flow) and z (neutral) directions, and Lees-Edwards boundary conditions in the y (velocity gradient) direction (Fig. 1); this way undesirable effects of walls could be eliminated [49]. Instead of cylinders [50] we chose the spherocylindical shape, because of the availability of efficient numerical algorithms [51] and continuous transition to the reference spherical shape. The spherocylinders were parametrized by their length-to-diameter aspect ratio Q = /2R. The repulsive force F ij between particles i and j was proportional to their normal overlap, and we employed a viscous velocity-difference based term for dissipation:

F ij = (-k δ ij + b v c,ij • ĉij ) ĉij
, where δ ij and ĉij are the magnitude and unit direction vector of the normal overlap between the particles, and for the velocity difference v c,ij at contact the rotation of the particles were taken into account as well. There is no tangential component of the force, that would result from a Coulombic contact friction. The stiffness k of the contacts, the particle diameter 2R and density ρ were set to unity, implicitly defining the length, time and mass units of the simulation. Importantly, some polydispersity is introduced to reduce the effects of crystallization at large Q, and while we kept the aspect ratio constant, we have drawn the radii of the particles from a uniform distribution with a ratio of standard deviation to mean of 10%. The prefactor b in the dissipative term was set by specifying a given restitution coefficient for binary collision. The equations of motion were integrated by the velocity-Verlet scheme, representing particle rotations by quaternions [52].

We created the initial conditions of random particle orientation with overdamped dynamics, and afterwards sheared the system at constant shear rate γ. During shear we employed stress control, where one side of the box, L y was adjusted by a feedback loop such that the corresponding normal stress σ yy fluctuated around a specified value -p y . We have used p y = 10 -3 in these units, corresponding to the rigid limit where rescaled results become independent of p y . We kept L x and L z fixed in order to avoid the development of a singular box shape due to normal stress differences. All measurements were taken in the stationary state, reached after a deformation of γ = 25 when starting from the initial conditions, or of γ = 10 from the stationary state of a different shear rate. All quantities of interest were time-averaged at least over an additional deformation of γ = 5. In the steady state, the packing fraction as well as all stress components σ ij were homogenous, and the velocity profile linear (no shear banding). Finally, we checked that our results are independent of the integration time step, set to 1/100 of the duration of a binary collision, and qualitatively insensitive to the value of the restitution coefficient in the range 0.3-0.7, and here set to 0.5.

Results.-We measured the inertial number dependence of different quantities. In Fig. 2(a) we show the effective friction, µ = σ xy /p y vs I for spherocylinders of a few selected aspect ratios, and for spheres for reference. (Note that in the definition of I we used the value p y controlling the stress, instead of the pressure p.) Similarly to [26], we fitted the empirical form

µ(I) ≈ µ c + µ 1 I α , (1)
allowing us to extrapolate to the quasi-static friction µ c in the I → 0 limit. Best fitting is of course obtained when all three parameters are adjusted, but we then fixed the exponent to its average, α = 0.4, yielding less noisy data with a two parameter fit (µ c and µ 1 ). This value of α is in agreement with that deduced from 3D simulations of frictionless hard spheres [53], and similar to the exponent 0.5 observed for frictionless circles [54]. Fig. 2(d) displays the aspect ratio dependence of µ c , showing a surprising non-monotonic function: it rises steeply for aspect ratios slightly larger than one, takes the highest value around Q = 1.05, followed by a slow decrease -we shall give a microscopic interpretation of this behavior later. Similarly, we show the normalized first and second normal stress differences N 1 /p y = (σ xxσ yy )/p y and N 2 /p y = (σ yyσ zz )/p y in Fig. 2(b-c). Their quasi-static values, extrapolated with a fit like Eq. ( 1) (also with fixed α = 0.4), are shown in Fig. 2(e-f). As expected, the quasistatic first normal stress difference vanishes for spheres, and increases close to linearly for aspect ratios Q 2.5. The second normal stress difference is negative although 2-3 times smaller in amplitude than N 1 (even for spheres N 2c /p y 1%), and has a very strong aspect ratio dependence for nearly spherical particles.

In complement to the stress behaviors, we show the volume fraction in Fig. 3(a). As for spheres, φ decreases with I as larger shear rates generate a more dilute system. Also, consistently with what has been observed for spherocylinders [46] and ellipsoids [47] in non-sheared systems, the volume fraction of elongated particles first quickly increases with Q, followed by a slow decreasethe behavior beyond Q 2.5 will be discussed later. Finally, we computed the coordination number Z. A static packing of frictionless hard spheres takes the isostatic value 6. Once the particles become elongated, due to the two extra rotation degrees of freedom per particle, this value jumps to 10. In Fig. 3(b) we plot the measured values. For spheres Z c is slightly larger than 6 due to the finite pressure, or equivalently the softness of the particles. For increasing aspect ratio Z c increases initially sharply but continuously, reaching a flat maximum around Q ≈ 1.8. The inset shows the inertial number dependence, demonstrating an expected decrease of contacts for more violent flows.

Next we look at the orientational order induced by the shear deformation. Figure 4 shows, for a few selected aspect ratios, the distribution of the angle θ between the streamlines (x axis) and the projection of the particle axis onto the x-y shear plane. For more elongated particles the distribution is narrower and its mode and mean are closer to zero. The average angle θ av is slightly off the mode due to asymmetry of the distribution. The aspect ratio dependence of θ av and the nematic order parameter S, which is defined as the largest eigenvalue of the order tensor, are shown in the inset: increasing Q results in increasing S and reducing θ av , in almost linear fashions. Note that even the smallest aspect ratio considered, Q = 1.01, has a small but finite nematic order, with θ av 45 • .

Finally we determined which region of the particles' surface is most responsible for dissipation (Fig. 5). The highest dissipation density was observed in the cylindrical band of the Q = 1.05 spherocylinder. More spherical particles showed a more homogenous distribution, but the slight enhancement of the dissipation near the cylindrical band is visible even for Q = 1.01. For more elongated particles, the dissipation becomes dominated at the hemispherical caps.

Discussion and perspectives.-The framework of the µ(I) rheology can be extended to elongated particles. The aspect ratio dependence of the rheological quantities display two remarkable features: (i) the dissipation, quantified by µ c , is maximal around Q = 1.05, and (ii) the normal stress differences, the volume fraction as well as the coordination number, behave non-monotonically for Q 2.5.

Issue (i) is closely related to the highest observed density of dissipation in the cylindrical band of the Q = 1.05 particles. We suspect that this effect is particularly strong for spherocylindrical particles, where the surface is not analytic (one of the curvatures is not continuous). An analysis similar to what is shown in Fig. 5 revealed that not only the dissipation density, but also the contact density is increased in that region: a larger than expected number of particle-pairs locked in a configuration where the contact is carried by the cylindrical region for at least one of the particles. We explain issue (ii) by the high ne- matic and partial spatial order observed for sufficiently elongated frictionless spherocylinders. While the introduction of polydispersity destroyed the crystalline order perpendicular to the streamlines, we still observe short range chains of particles which follow each other on a streamline. This can also explain why the tips of the particles become the dominant location for dissipation for the more elongated particles.

The robustness of these results should be assessed by complementary simulations, e.g. in 2D, with other shapes like ellipses, possibly including frictional contacts, as well as by experiments [45,[START_REF] Trulsson | Packing and yielding of frictional ellipses[END_REF]. A better understanding of the large Q behavior probably requires the use of larger systems, and here our data for Q = 3 may be affected by a too small size. This issue motivates further studies towards the flow of fibers and entangled materials [48], whose rich mechanics suggest interesting rheological

FIG. 1 .

 1 FIG. 1. (Color online) The simulation box, here containing 500 spherocylindrical particles of aspect ratio Q = 2 in the stationary state, is sheared in the x direction. A feedback loop adjusts Ly to ensure a controlled applied stress -py in the y direction. The shear rate is γ = v/Ly.

FIG. 3 .FIG. 4 .

 34 FIG. 3. The aspect ratio dependence of the quasi-static values of (a) the volume fraction, and (b) the coordination number. Insets: I dependences for a few Q values. Solid lines: fits of the form of Eq. 1, with respective fixed exponents 0.4 and 0.5.

3 FIG. 5 .

 35 FIG.5. (Color online) Average dissipative power per unit area visualized on a discrete mesh for four aspect ratios (see legends and color code). Dissipation at each contact is accumulated in triangular bins, such that the axes of the differently oriented particles are turned into a canonical orientation, in which the mesh is defined. These data are for I = 3.16×10 -4 .

properties.

We thank A. Favier de Coulomb, D. Kálmán, B. Andreotti, E. Clément, O. Pouliquen, A. Lindner, B. Szabó and M. Trulsson for useful discussions. This work was supported by the Hungarian National Research, Development and Innovation Office NKFIH under grant OTKA K 116036, and the Hungarian-French bilateral TéT/Balaton exchange programme. We acknowledge computational resources provided by NIIF in Hungary at Debrecen.


Aymen Ladhar
email: aymen.ladhar@st.com

Arnaud Virazel
email: arnaud.virazel@lirmm.fr

Machine Learning Support for Cell-aware Diagnosis

Keywords:

This chapter will provide an overview of the various machine learning approaches and techniques proposed to support cell-aware generation, test, and diagnosis. The chapter will focus on the generation of the cell-aware models and their usage for diagnosis. After some backgrounds on conventional approaches to generate and diagnose cell-aware defects, the chapter will present a learning-based solution to generate cell-aware models. Then, the chapter will present a ML-based cell-aware diagnosis technique. Effectiveness of existing techniques will be shown through industrial case studies and corresponding diagnosis results in terms of accuracy and resolution. The chapter will conclude by a discussion on the future directions in this field.

Introduction

The usage of Cell-Aware (CA) methodology becomes mandatory for semiconductor industry, especially for designs with the highest product quality. This is because fault models like stuck-at, transition, as well as layout-aware fault models are not accurate enough to achieve very low Defective Part Per Million (DPPM) rates and to resolve the underlying systematic yield detractors for a successful and fast yield ramp-up. Previous works on CA defect test and diagnosis can be classified into two categories. Techniques in the first category extend the application of logic algorithms to deal with transistor defects [START_REF] Ladhar | Efficient and Accurate Method for Intra-Gate Defect Diagnoses in Nanometer Technology[END_REF][START_REF] Sun | Effect-Cause Intra-cell Diagnosis at Transistor Level[END_REF]. The main weakness of these techniques is the quality of the logic fault models that do not properly describe the behavior of potential transistor defects. These methods are limited to cell-level diagnosis and cannot be used during test pattern generation. The second category of cell-internal defect test and diagnosis techniques relies on the realistic assumption that the excitation of a defect inside a cell is highly correlated with the logic values at the input pins of the cell [START_REF] Hapke | Cell-Aware Test[END_REF][START_REF] Maxwell | Cell-Aware Diagnosis: Defective Inmates Exposed in their Cells[END_REF]. For this category, a cell-internal-fault dictionary or CA model (also referred to as CA fault model or CA test model in the literature), describing the detection conditions of each potential defect affecting a cell, is used [START_REF] Liu | Parametric Fault Diagnosis for Analog Circuits Using a Bayesian Framework[END_REF][START_REF] Wang | Machine Learning-Based Volume Diagnosis[END_REF]. These techniques are more efficient and can be used to guide the test pattern generation and CA diagnosis phases. However, the main limitation of these techniques is the generation effort needed to characterize all the standard cells per technology in terms of run time, number of SPICE simulator licenses, CPU requirements and disk usage. The second limitation of this technique is the usage of simple fault models to describe silicon failures, which is not always possible for actual silicon failures. In addition, the assumption that the excitation of a defect inside a cell is highly correlated with the logic values at the input pins of the cell is not always correct mainly for unmodeled cell internal defect.

With the fast development and vast application of Machine Learning (ML) in recent years, ML-based techniques have been shown to be quite valuable for diagnosis purpose, especially in the volume diagnosis scenario [START_REF] Liu | Parametric Fault Diagnosis for Analog Circuits Using a Bayesian Framework[END_REF][START_REF] Wang | Machine Learning-Based Volume Diagnosis[END_REF]. In this chapter, the the usage of machine-learning algorithms is extended to generate the cell-aware models in a first step, then to diagnose cell internal defects.

The remainder of this chapter is organized as follows. Section 2 gives some background on conventional cell-aware generation, test, and diagnosis. Section 3 explains the machine learning cell-aware flow. Section 4 summarizes a machine learning technique to diagnose cell-aware defect. Section 5 presents some industrial case studies performed with latest ML-based diagnosis techniques. Section 6 concludes the chapter and draws some conclusions.

Background on Conventional Cell-Aware Generation, Test and Diagnosis

To compete in the fast-growing market of automotive ICs as well as 3D transistor era, semiconductor industries need to address new challenges across the entire design flow. In addition, the requirement to be compliant with the highest standards like the ISO 26262 goal of zero Defective Parts Per Million (DPPM), has an impact on the test and diagnosis phase of nowadays silicon failures [START_REF] Mukherjee | Defect-Oriented Cell-Aware ATPG and Fault Simulation for Industrial Cell Libraries and Designs[END_REF]. Cell internal defects are one of most critical defects that should be targeted since they are increasingly occurring in latest technology nodes. In addition, this defect type is hard to be targeted with conventional ATPG solution and fault models. Test and diagnosis of cell internal defects is possible thanks to the realistic assumption that the excitation of a defect inside a cell is highly correlated with the logic values at the input pins of the cell [START_REF] Hapke | Cell-Aware Test[END_REF]. To perform CA test and diagnosis, a CA model is needed. It is obtained through the characterization of each standard cell library with regards to all possible cell-internal defects. These defects can be either transistor defects or inter-transistor defects like open and shorts [22][START_REF] Goncalves | Integrated Approach for Circuit and Fault Extraction of VLSI Circuits[END_REF][START_REF] Goncalves | Realistic Fault Extraction for High-Quality Design and Test of VLSI Systems[END_REF][START_REF] Stanojevic | Fed-x -A Fast Bridging Fault Extractor[END_REF]. Electrical simulations are performed to generate CA models for each cell in the library. These models include information about the behavior of each defect within the cell with regards to the stimuli applied at the cell inputs.

One bottleneck of the CA flow deployment in industry is the generation effort in terms of run time and flow complexity. Indeed, it requires extensive computational efforts to characterize all standard cells of a library [START_REF] Lorenzelli | Speeding up Cell-Aware Library Characterization by Preceding Simulation with Structural Analysis[END_REF]. Fig 5 .1 represents a typical CA model generation flow. It starts with a SPICE netlist representation of a standard cell which is usually derived from a layout description, e.g., a GDSII file. The DSPF (Detailed Spice Parasitic Format) contains information about potential shorts and open defect, as well as their coordinates within the cell. This DSPF cell netlist is then used by an electrical simulator to simulate each potential cell internal defect against an exhaustive set of stimuli containing static and dynamic patterns. Once the simulation process is completed, all cell-internal defects are classified into defect equivalence classes with their detection information (required input values for each defect within each cell) and are synthetized into a CA model. Two tables or matrix exist in each CA model, the first one targets static defects whereas the second one targets dynamic defects. As standard cells may have several inputs, and thousands of cells with different complexities are used for a given technology, the generation time of CA models for complete standard cell libraries of a given technology may reach up to several months, thus drastically increasing the library characterization process cost.

Learning-Based Cell-Aware Model Generation

The reason behind the ML usage for defective cell characterization is the result of several observations made while performing electrical simulation of several cell internal defects on different standard cell libraries and technology nodes:

• Several cell internal defects are independent of the technology and transistor size, and their defective behavior depends mainly on the location of the defect and test stimuli applied on the cell under characterization [START_REF] Venkataraman | A Technique for Logic Fault Diagnosis of Interconnect Open Defect[END_REF][START_REF] Li | Diagnosis of Resistive-Open and Struck-Open Defects in Digital CMOS ICs[END_REF]. • For the same function, cell schematics are usually quite similar whatever is the technology node. • Detection tables for static and dynamic defects, in the form of binary matrices describing the detection patterns for each cell-internal defect, are ML friendly. • A large database of CA models is usually available and can be used to train a ML algorithm.

All these above-mentioned reasons intuitively revealed that CA model generation through ML could be feasible. However, the first tricky task is to be able to describe cell transistor netlist as well as corresponding cell-internal defects in a uniform (standardized) manner, so that a ML algorithm can learn and infer from data irrespective of their incoming library and technology. Indeed, similar cells (e.g., cells with same logic function, same number of inputs and same number of transistors) may be described differently in transistor-level (SPICE) netlists of various libraries (e.g., a transistor label does not always correspond to the same transistor in two similar cells coming from two different libraries), and it is therefore mandatory to standardize the description of cells and corresponding defects for the ML-based defect characterization methodology. Heuristic solutions developed to this purpose are described in Sections 3.1 and 3.2. The second challenging task is to find a solution to represent all these information / input data so that they can be ML friendly. A matrix description of cells and corresponding defects can be chosen to this purpose. Section 3.3 describes how to adapt the ML method to deal with sequential cells.

Generation of Training and New Data

The learning-based defect characterization methodology proposed in [START_REF] Hondt | A Learning-Based Methodology for Accelerating Cell-Aware Model Generation[END_REF] is used to predict the behavior of a cell affected by cell internal defects, hence avoiding costly and heavy electrical defect simulations. The proposed flow is presented in Fig. 5 Training data are made of various and numerous CA models formerly generated by relying to brute-force electrical defect simulations. For each cell in a library, the CA model is transformed into a so-called CA-matrix and filled in with meaningful information.

An overview of the CA-matrix flow is described in Fig. 5.3. It begins by rewriting the CA model in a ML friendly description. Then, it categorizes the activation conditions of each transistor with respect to input stimuli. Once the activation conditions for each transistor have been identified, transistors are renamed in a uniform way. This is a critical step in the proposed flow as it allows the usage of the training data across different libraries and technologies. Finally, the CA-matrix is created with the above information. • Cell patterns and responses. This information represents an extended representation of a NAND2 truth table (A, B: inputs, Z: output). As it can be seen, the test pattern sequence provides all the possible input stimuli that can be applied to the cell. It takes into consideration all the possible transitions that can be applied on the inputs of this cell. These stimuli must also be efficient to detect sequence-depending defects like stuck-open defects. For this reason, a fourvalued logic algebra made of 0, 1, R and F is used to represent input stimuli in the CA-matrix. R (resp. F) represents a Rising (resp. Falling) transition from 0 to 1 (resp. from 1 to 0). • Active / Passive Transistor Identification. This indicates the activation conditions of each transistor in the cell schematic. Each transistor can be in the following state: active, passive, switching to active state, switching to passive state. • Defect description. This gives information about defect locations in the cell transistor schematic. At least one logic value is used to indicates the name and the transistor port impacted by the defect. • Defect detection. This is the class of the data sample (the output of ML classifier). A value '1' ('0') means that the defect is detected (undetected) by the input pattern. The first three types of information constitute the inputs of the ML algorithm.

0 1 0 0 1 … 0 0 0 … free free 0 0 1 1 0 1 1 … 0 0 0 … free free 0 0 F 1 0 F 1 … 0 0 0 … free free 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 1 1 0 1 1 … 1 0 1 … D15 short 1 1 1 0 1 1 0 … 1 0 1 … D15 short 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
New data represent the cells to be characterized and are obtained for each standard cell from the cell description, corresponding list of defects and cell patterns. The format of a new data instance is like that of the training data, except that the class (label) of the new data instance is missing. The ML classifier is used to predict that class. As for training data, new data are grouped together according to their number of cell inputs and transistors, so that inference can be done at the same time for cells with the same number of inputs and transistors.

As highlighted in Fig. 5.2, a Random Forest Classifier is used for predicting the class of each new data instance. This choice comes from the results obtained after testing several learning algorithms (k-NN, Support Vector Machine, Random Forest, Linear, Ridge, etc.) and observing their inference accuracies. So, the first main step of our CA diagnosis flow consists in generating a Random Forest model and to train it by using the training dataset. A Random Forest Classifier is composed of several Decision Tree Classifiers, which are models predicting class of samples by applying simple decision rules. During training, a Decision Tree tries to classify data samples and its decision rules are modified until it reaches a given quality criterion. Then, the Forest averages the responses of all Trees and outputs the class of the data sample. The second main step consists in using the Random Forest Classifier to make prediction (or inference) when a new data instance must be evaluated. Prediction for a new data instance amounts to answer to the question: "Does this stimulus detects this defect affecting this cell?". Answering to this question allows obtaining a new CA model for a given standard cell

Cell and Defect Representation in the Cell-Aware Matrix

This section describes the used method to represent a transistor-level (SPICE) netlist in a CA-matrix, which is a standardized and ML friendly description. This representation must be accurate enough to clearly identify each transistor and each net of the cell transistor schematic. Furthermore, this description must also be able to correlate each transistor to its sensitization patterns and to report the output response for each pattern. Therefore, the cell description process needs several successive operations that are detailed below. Note that this process is applied to all cells in a library to be characterized.

Identification of Active and Passive Transistors

The first step consists in identifying active and passive transistors in the cell netlist with respect to an input stimulus. This information is used to represent the transistor netlist in the matrix format. To this end, a single defect-free (golden) electrical stimulation of each cell to be characterized is performed to identify active and passive transistors for each input stimulus (test pattern) and to measure the cell response on the output pin. A NMOS (resp. PMOS) transistor is considered active if a logic-1 (resp. logic-0) value is measured on its gate terminal. Similarly, a NMOS (resp. PMOS) transistor is considered passive if a logic-0 (resp. logic-1) value measured on its gate terminal. With this information, each cell pattern can be associated to the list of active transistors in the cell. After this step, the CA-matrix contains the columns:

• Cell inputs & responses columns. They contain all input stimuli (test patterns) that can be applied to the cell, and the corresponding responses. • Transistor switching activity columns. They contain four possible values indicating if the transistor is active (1), passive (0), switching from an active state to a passive one (F) or switching from a passive state to an active one (R). Since PMOS and NMOS transistors are activated in opposite way, the '-' character is used before the PMOS values. In the partial representation of the CA-matrix of the cell shown in Fig. 5.4.b, columns A and B list all the possible input stimuli for this cell. These columns also define the length of the CA-matrix, which is equal to 2 n + 2 n .(2 n -1), n being the number of cell input pins (2 n is the number of static stimuli, 2 n .(2 n -1) is the number of dynamic stimuli). For each stimulus, active and passive information about each transistor of the cell is entered in the CA-matrix. For example, AB=00 leads to two active PMOS transistors (Px and Py) and two passive NMOS transistors N10 and N11). This matrix description of the cell netlist is clearly dependent on the transistor names and the order they are specified in the SPICE netlist. In fact, two similar cell schematics may have different transistor naming and the order of transistors in the SPICE netlist may differ from cell to cell and from one technology to another. Without an accurate naming convention of each cell transistor in the CA-matrix, any ML algorithm will fail to predict the behavior of the cell in presence of a defect. To prevent this problem, a second step in the cell description process is needed. This step consists in renaming all cell transistors independently of their initial names and order in the input SPICE netlist. The algorithm built to this objective is explained in the next subsection.

Renaming of Transistors

In the CA-matrix, two cells with the same transistor structure will have the same transistor names irrespective of their incoming library and technology. A transistor structure is a virtual SPICE netlist without specification of the connections between transistor gates, i.e., only source and drain connections between transistors are listed.

The transistor-renaming algorithm consists of the following two steps:

• Determination of branch equations: Since the transistor gate connections are not considered, the transistor structure is composed of one or more branches.

A branch is a group of transistors connected by their drain and source terminals. The entry of each branch is the set of transistor gates, and the exit is the connection net between the NMOS and PMOS transistors. A branch is connected to a power and/or a ground net. A branch equation is a Boolean-like equation describing how the transistors of the branch are connected, using Boolean-and (symbolized by '&') for serial transistors or serial groups of transistors, and Boolean-or (symbolized by '|') for parallel transistors or parallel groups of transistors.

In Fig. 5.5, the structure is composed of two branches. The two-transistors output-inverter is the simplest branch whose input is net Y and output is net Z. The inverter creates two paths between branch output and power nets, so its branch equation is (Ninv|Pinv). The equation of the second branch (NMOS branch

GND VDD VDD Z A B P x P y N 10 N 11 0 0 -1 -1 0 0 0 1 -1 0 0 1 1 0 0 -1 1 0 ... ... ... ... ... ... R R -F -F R R N11 N10 Px Py A B A B net0 (a) (b)
driving net Y) is ((N0&(N1|N2))|N3). To do not rely on any name present in the SPICE netlist, the branch equations are anonymized, i.e., a NMOS is described by '1n' and a PMOS by '1p'. The anonymized equation of the NMOS branch driving net Y in Fig. 5.5 is therefore ((1n&(1n|1n))|1n). • Sorting of branch equations: Once all the branch equations for the considered cell have been determined, they are sorted by using deterministic criteria: o Level of each branch. It is defined in ascending order with respect to the cell output (level-1 branches drive the cell output, level-2 branches drive the gates of transistors in level-1 branches, and so on and so forth), o Number of transistors in each branch -in ascending order, o Anonymized branch equation defined in alphabetical order.

Identification of Parallel Transistors

Identifying branch equations is not enough to rename all transistors of a given cell. The problem comes from parallel transistors. In fact, two or more parallel transistors share the same source and drain, which makes their identification quite difficult. For example, in Fig. 5.5, transistors N1 and N2 can be either presented as "N1|N2" or as "N2|N1". As the order of transistors in each branch will determine how transistors will be renamed, such confusing situation cannot by accepted. A solution consists in sorting transistors inside their branch according to their activity with respect to the input stimuli. The algorithm developed to this purpose proceeds as follows. For each transistor, an activity value is computed. For a cell with n inputs, the activity value is a 2 n -bit integer that represents the accumulative activation state of a transistor for all possible stimuli applied to the cell. The input stimuli range from (0,0…,0) to (1,1…,1). For each of these stimuli, the transistor is either active (1) or passive (0). The activity value is defined as a binary number, whose MSB is the activity of the transistor under input stimulus (0,0…,0) and LSB is the activity of the transistor under input stimulus (1,1…,1), with a bit significance decreasing for increasing binary value of input stimulus.

To compute the activity value, one need to know whether the transistor is active or passive for each input stimulus. This information is already presents in the CA-

GND PMOS part VDD VDD GND Z Y Y Y N0 N1 N2 N3 N inv P inv PMOS part
matrix as described in Section III.A. To illustrate this process, activity values for the transistors of the NAND2 cell in Fig. 5.4.a are given in Table 5.2.

0 0 MSB 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 LSB 0 0 1 1 Activity value 12 10 3 5 ↓ Renaming ↓ P1 P0 N0 N1
Finally, transistors of each branch are sorted by increasing activity value to give the final description of the cell in the CA-matrix. For the example in Fig. 5.4, this leads to: first NMOS transistor of the first branch is named N0, second NMOS transistor of the first branch is named N1, first PMOS transistor of the second branch is named P0, and second PMOS transistor of the second branch is named P1.

Defect representation

This section details the cell-internal defects representation in the CA-matrix in a standardized and ML friendly manner. Cell internal defects are classified into:

• Intra-transistor defects. These defects affect transistor terminals (source, drain, gate and bulk) and can be either an open defect or a short. To describe these defects, all transistor terminals are listed as a column in the CA-matrix (cf. Table I). For an open defect, a value '1' indicates that this transistor terminal is affected by the defect, '0' otherwise. For a short, a value '1' on two transistor terminals indicates that a short exists between these two terminals, '0' otherwise.

• Inter-transistor defects. These defects affect a connection(s) between at least two different transistors. Though these defects are not considered in this work, the matrix representation is flexible enough to represent them. For these defects, the same representation mechanism is used as for intra-transistor defects.

Table 5.3 is an example of defect description in the CA-matrix of the NAND2 cell example presented in Fig. 5.4. Row with red cells describes the intra-transistor short defect between drain and source terminals of transistor P1 (formerly Px). Row with blue cells describes the inter-transistor short defect between P0-source and "net0" (net0 connects N0-source and N1-drain).

TABLE 5.3: DEFECT REPRESENTATION FOR A NAND2 CELL P0_S P0_D P1_S P1_D N0_S … N1_D Comment 0 0 1 1 0 … 0 source-drain short on P1 1 0 0 0 1 … 1 net0 & P0-source short

Support of Sequential Cells

This section describes the different modifications that must be considered during the creation of the CA matrix in case of a sequential cell:

• Cell inputs and outputs: For sequential cells it is important to know the stored value inside the cell before applying the test stimuli. To this end, a new column is included in the CA matrix. It includes the output value of the cell before applying the new test stimuli. • Identification of Active, Passive, and pulsing transistors: One characteristic of sequential cells is the presence of a clock signal, which has an impact on the value applied on each transistor. In fact, clock-signal-controlled transistors can be pulsing (resp. anti-pulsing), which means a 0-1-0 (resp. 1-0-1) sequence appears on the transistor gate terminal during application of the test pattern. • Determination of branch equations: Sequential cells tend to integrate transmission gates to separate the latches from each other and from the inputs. A transmission gate is a transistor configuration acting as a relay that can conduct or block depending on the control signal. It is composed of one PMOS and one NMOS transistors in parallel (i.e., sharing drain and source), and the control signal applied to the gate of the NMOS transistor is the opposite (i.e., NOT-ed) of the signal applied to the gate of the PMOS transistor. A transmission gate directly connects the exit of a branch to the entry of another branch. As such, a transmission gate is considered as an autonomous branch of the transistor structure. The entry of such a branch is the set of transistors gates plus the exit of the previous branch, and the exit is the entry of the following branch. Each time a transmission gate is identified, the branch equation will be considered as a transmission gate and not as a PMOS with a NMOS transistors. The symbol t will be used into the branch equation.

Advanced Cell-Aware Diagnosis Based on Machine Learning

Although conventional diagnosis techniques such as "Cause-Effect" [START_REF] Waicukauski | Failure Diagnosis of Structured VLSI[END_REF] and "Effect-Cause" [START_REF] Abramovici | Fault Diagnosis Based on Effect-Cause Analysis: An Introduction[END_REF] can achieve a good resolution, in some cases (e.g., complex cells, complex failure mechanisms) the number of candidates may be too high to allow an efficient PFA (Physical Failure Analysis). This problem will be exacerbated in the future with the advent of very deep submicron (i.e., 5 nm and beyond) technologies.

Improving diagnosis efficiency at the transistor level (i.e., CA diagnosis) is therefore mandatory.

Previous works on Cell-Aware (CA) fault diagnosis focusing on logic cells can be classified into three approaches. The first approach converts a transistor-level netlist into an equivalent gate-level netlist by means of complex transformations rules [START_REF] Fan | A Novel Stuck-At Based Method for Transistor Stuck-Open Fault Diagnosis[END_REF]. Then, on the equivalent gate-level netlist, any classical fault diagnosis approach can be applied. The main drawback of this approach is that the set of transformation rules depends on the targeted defect and, thereby, the non-modeled defects may not be diagnosed. The second approach is based on the "Cause-Effect" paradigm [START_REF] Fan | A Novel Stuck-At Based Method for Transistor Stuck-Open Fault Diagnosis[END_REF][START_REF] Hapke | Cell-Aware Production Test Results from a 32-nm Notebook Processor[END_REF]. The transistor-level netlist of a cell is exploited, in order to inject the targeted defects. Therefore, a defect dictionary is created by transistor-level simulations and the defect signatures of all the defects affecting the cells in the library are stored in this defect dictionary. Then, during fault diagnosis the defect signature of all defects affecting a suspected cell is compared with the observed failures to obtain a list of candidates inside the cell. These approaches can be further classified depending on the "accuracy" of the injected defects and the simulation "precision". In [START_REF] Hapke | Cell-Aware Production Test Results from a 32-nm Notebook Processor[END_REF], a large number of defects are simulated at transistor-level using SPICE. For a given defect, different resistance values are simulated, in order to be as accurate as possible. This approach leads to more precise results but it requires a huge simulation time. To reduce the simulation time and the fault dictionary size while keeping a high resolution, authors in [START_REF] Fan | A Novel Stuck-At Based Method for Transistor Stuck-Open Fault Diagnosis[END_REF] propose to exploit layout information, in order to consider only realistic defects. For example, for each cell, only the realistic, potential net bridging defects and via open defects are extracted and then simulated. Then, the identified set of realistic defects is simulated at transistor-level. The third intra-cell fault diagnosis approach is based on the "Effect-Cause" paradigm [START_REF] Sun | Intra-Cell Defects Diagnosis[END_REF]. All the existing diagnosis techniques depend on the targeted fault models or defects. In [START_REF] Sun | Intra-Cell Defects Diagnosis[END_REF], the main goal is to achieve a resolution close to the transistor-level. However, instead of explicitly considering defects at transistor-level, the idea is to exploit the knowledge of the faulty behavior induced by the defects.

Unfortunately, CA fault diagnosis resolution is typically far from the ideal due to circuit complexity. A mean to achieve this goal is to use supervised learning algorithms to determine suspected defects. Supervised learning is now used in numerous classification problems where the knowledge on some data can be used to classify a new instance of such data. This section summarizes the latest developments in the field of CA diagnosis based on supervised learning.

Preliminaries and Test Scenarios

Several learning-guided solutions for CA diagnosis have been proposed recently in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF]. All solutions are based on a Bayesian classification method for accurately identifying defect candidates in combinational standard cells of a customer return. Choosing one solution over another depends on the test scenario (test sequence, test scheme, test conditions) considered during the diagnosis phase and selected according to the types of targeted defects and failure mechanisms.

The test scenarios in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF] are sketched in Table 5. [START_REF] Maxwell | Cell-Aware Diagnosis: Defective Inmates Exposed in their Cells[END_REF]. In [START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF], two distinct processes were developed to diagnose static and dynamic defects separately. In [START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF], a basic scan testing scheme used to apply static CA test sequences is considered, so that stuck-at faults plus static intra-cell defects are targeted during diagnosis. In [START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF], a fast sequential testing scheme used to apply dynamic CA test sequences is considered, so that transition faults plus dynamic intra-cell defects are targeted during diagnosis. Note that [START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF] is just a combination of [START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF] and [START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF], i.e., two testing schemes, one static and one dynamic, and two CA diagnosis flows, one static and one dynamic, are considered independently. The main limitation of the solutions in [START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF] is the required a priori knowledge of the type of targeted defects in the CUT. In other words, a test engineer needs to know what type of defects is screening before choosing between [START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF] or [START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF].

To deal concurrently with all types of defects that may occur and without any a priori knowledge of the targeted defect type, a new implementation of the CA diagnosis flow was proposed in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF]. Note that [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF] is a fully extended version of [START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF]. Authors assume a test scenario in which two test sequences (static and dynamic) are used successively, each one considering a dedicated testing scheme, i.e., basic scan and fast sequential. First, a static CA test sequence generated by a commercial cell-aware ATPG tool is applied to the CUD. This sequence targets all cell-level stuck-at faults plus cell-internal static defects, considering that these defects are not covered by a standard stuck-at fault ATPG. A standard (low speed) scan-based testing scheme is used to this purpose. Next, another option of the cellaware ATPG is used to generate a dynamic CA test sequence that targets cell-level transition faults plus intra-cell dynamic defects not covered by a standard transition fault ATPG. In this case, an at-speed Launch-On-Capture (LOC) scheme (also called fast sequential) is used during test application. To construct the comprehensive flow described in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF], a new framework was set up in which specific rules were defined to achieve a high level of effectiveness in terms of diagnosis accuracy and resolution. The proposed method was based on a Gaussian Naive Bayes trained model to predict good defect candidates. This method is summarized in the next subsection.

In [START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF], a new version of the CA diagnosis flow was proposed, assuming a test scenario in which both static and dynamic defects can be diagnosed owing to a single dynamic CA test sequence applied at-speed. This scenario may happen when such a test sequence has been generated to target transition faults plus cell-internal dynamic defects and appears to also cover the required percentage of stuck-at faults plus cellinternal static defects (or, more generally, satisfies the test coverage specifications). In this case, note that only one (dynamic) datalog is generated after test application and can further be used for diagnosis purpose. Nevertheless, both static and dynamic defects are considered in this scenario. As only dynamic instance tables are manipulated, the representation of training and new data is simplified, i.e., a single type of feature vector is used, without no loss of information and hence without decreasing the quality of the training and inference phases. After investigating several ML algorithms and observing their inference accuracies in [START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF], a Bayesian classification method has been chosen for the learning and inference phases in [START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF]. So, the first main step of the CA diagnosis flow consists in generating a Naive Bayes (NB) model and to train it by using a training dataset. In this step, training data are used to incrementally improve the model's ability to make inference. The training dataset is divided into mutually exclusive and equal subsets. For each subset, the model is trained on the union of all other subsets. Some manipulations, such as grouping data by considering equivalent defects or removing data instances of undetectable defects, are also done during this phase. Once training is complete, the performance (accuracy) of the model is evaluated by using a part of the dataset initially set aside. More details about performance evaluation as done in this framework can be found in [START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF].

Learning-Based Cell-Aware Diagnosis Flow

The second main step consists in implementing the NB classifier by using a Gaussian distribution to model the likelihood probability functions and use this classifier to make prediction when a new data instance has to be evaluated. The next subsections detail the various steps of the CA diagnosis flow, which is able to deal with any type of cell-internal defect (i.e., static and dynamic) that may occur.

Generation of Training Data

Training data are generated for each type of standard cell existing in the CUD during an off-line characterization process done only once for a given cell library. These data are extracted from CA views provided by a commercial CAD tool that contain all characterization results for a given cell type. These results are provided in the form of a fault dictionary containing, for each defect within a cell, the cell input patterns detecting (or not) this defect. An example of training dataset, as used in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF] and containing six instances for an arbitrary two-input cell, is shown in Fig. Dynamic defects can be detected not only by dynamic patterns, but also by static patterns applied using a basic scan testing scheme, provided that: i) at least one transition has been generated at the cell inputs between the next-to-last scan shift cycle and the launch cycle, and ii) the delay induced by the defect is large enough to be detected (these are the detection conditions of a dynamic defect modeled by a New data

Model training

Inference Bayesian Model

Cell library CUD Instance Tables Generation Characterization

Training data

Classifier

Good defect candidates

stuck-open or a gross delay fault). For this reason, the value '0.5' is assigned to each dynamic defect (D11, D12, D13) for all related static cell-patterns, meaning that such a defect is detectable or not depending on whether or not the above conditions are satisfied. As only dynamic test sequences are considered in [START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF], the representation of training data as used in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF] could be simplified without losing information and decreasing the quality of the training phase. This comes from the observation that a static defect is a particular case of dynamic defect (e.g., a full open is a resistive open with an infinite value of the resistance), and that all static cell-patterns for a given defect are embedded in its whole set of dynamic cell-patterns. Indeed, a dynamic defect requires a two-vector test pattern (V1V2) in which the values of V1 and V2 have to be properly defined for the defect to be detected. Conversely, only the value of V2 is significant for a static defect to be detected by such pattern, irrespective of the value taken by V1. When looking at Fig. 5.7, one can see that P1={00} is embedded in P6={0F}, P11={F0} and P12={FF}, and the same for P2, P3 and P4. Similarly, one can see that static defect D2 is detectable by P1 and P4, and hence by P6, P8, P10, P11, P12, and P15. So, by "compacting" a training dataset as shown in Fig. 5.8, in which only dynamic cell-patterns are considered, one can see that all meaningful information is still contained in this set, while redundant ('0' and '1' values in the first four columns of Fig. 5.7) or insignificant ('0.5' values in the same columns for dynamic defects) information is removed. More generally, such compact format for training data makes so that only one type of feature vector (dynamic) is used for both types of defects.

As the goal with training data is to provide a distinct feature vector for each data (defect), it is important to be able to distinguish between static and dynamic defects with such a new format of the training dataset. Let us consider two defects D1 and D11 where D1 is static and detectable by {00} and D11 is dynamic and detectable by {F0} (note that {00} is the second vector of {F0}). As can be seen in Fig. 5.8, these two defects can easily be distinguished since their training data instances (or feature vectors) are different. The consequence of using such a new format for training data (and hence for new data as will be shown later on) is not an improved accuracy or resolution, but rather a simplified manipulation of feature vectors.

Generation of Instance Tables

An instance table is a failure mapping file generated for each suspected cell by using information contained in the tester datalog. It describes the behavior (pass / fail) of the cell for each cell-pattern occurring on its inputs during test of the CUD. The generation process of instance tables is sketched in Fig. 5.9. First, CA test patterns are applied to the CUD. These test patterns are obtained from a commercial CA test pattern generation tool that targets intra-cell defects. Next, a datalog containing information on the failing test patterns and corresponding failing primary outputs is obtained. From this datalog and the circuit netlist, a logic diagnosis is carried out (still using a commercial tool) and gives the list of suspected cells. From this list and the datalog information, one can finally generate an instance table for each suspected cell. Note that in case several test sequences, e.g., one static and one dynamic, are used for diagnosis of the CUD, the generation process is repeated so as to produce static and dynamic instance tables for all suspected cells. This is the case in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF]. The format of a static instance table is illustrated in Fig. 5.10 for a given twoinput NOR cell and two static cell-patterns. In this example, the first part of the file gives information on how the cell is linked to other cells in the circuit, while the second part represents, respectively, the pattern number, the pattern status (failing, passing), and the cell output Z with the associated fault model for which exercising conditions are reported. These conditions shown right below each cell-pattern in Fig. 5.10 represent the stimulus arriving at the cell inputs during the shift phase (before '-') and applied during the launch cycle (after '-'). For example, cell-pattern 2 consists in applying a 1 on input A and B, and failing in detecting a stuck-at 1 on Z.

Generation of New Data

New data are generated after post-processing of instance tables. They are composed of various instances, each of them being associated to one suspected cell in the CUD and represent a feature vector that characterizes the real behavior of the cell during test application. From each new data instance, one can extract one or more defect candidates that must be classified as good or bad candidate with a corresponding probability to be the root cause of failure. This classification is done by comparing the new data instance with the training data of the corresponding suspected cell and identify those training data instances that match (or not) with the new data instance.

The formats of a new data instance as used in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF] and [START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF] are illustrated in Fig. 5.11 and Fig. 5.12 respectively. This format is quite close to the format of a training data instance but has a different meaning. In each instance, the value '1' ----------------------------------------------------------------NOR Cell -NR2NHVTX1 - -------------------------------------------------------------- -------------------------------------------------------------- ---------------------------------------------------------------(resp. '0') is associated to a failing (resp. passing) cell-pattern Pi for a given defect candidate, meaning that the candidate is actually detectable (resp. undetectable) by the cell-pattern Pi at the output of the cell during test of the CUD and hence can (cannot) be the real defect. In such instance, the value '0.5' is associated to a cellpattern for a given defect candidate when this pattern cannot appear at the inputs of a suspected cell during real test application with an ATE. The median value '0.5' was chosen to avoid missing information in new data instances while not biasing the features of these data.

- Z Output L412/C1381A A Input U59/Z B Input U28/Z -
- Pattern 1 PASSING Z: stuck-at-0 Z 000011111111111 -1 A 111100000000000 -0 B 000000000000000 -0 Pattern 2 FAILING Z: stuck-at-1 Z 011100000000000 -0 A 000011111111111 -1 B 100011111111111 -1 -

Diagnosis of Defects in Sequential Cells

All the work carried out in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF] was about diagnosis of defects occurring in combinational standard cells of a customer returns. However, defects in SoCs may also occur in sequential standard cells of logic blocks. This section shows how the previous diagnosis flow can handle sequential cells and related defects by adding new information to the training dataset [START_REF] Hondt | A Comprehensive Framework for Cell-Aware Diagnosis of Customer Returns[END_REF].

The two main differences between a combinational cell and a sequential cell are that i) the latter has a clock input pin and ii) the fact that the previous logic value of a sequential cell output can affect the current output value of the cell. To take this difference into account, each cell-pattern for a sequential cell is considered as a tuple in which the first value represents the input clock signal (pulsing or not), the second value is associated to the main input of the cell (e.g., D), and the third value is associated to a virtual input pin representing the previous value of the output pin of the cell (e.g., Q). Note that in case of sequential cells with multiple real inputs (e.g., D flip-flop with a D, Scan-In, Scan-Enable and Clock input signals), the cellpattern representation is expanded accordingly. In each tuple, the first value is either U (i.e., pUlse) or 0, depending on whether or not there is an active clock signal. The second value can be 0, 1, R or F. The third value can only be static (i.e., 0 or 1). An example of training dataset for all defect types (static and dynamic) that may occur in a sequential cell is shown in Fig. 5.13. Note that the CA views used during the generation of training data do not contain information about cell-patterns with non-pulsing clock signals (i.e., none of the cell internal defects can be detected at the cell output without clock pulse). Consequently, the training data do not include such cell-patterns as can be observed in the example of Fig. 5.13. Note also that instance tables of sequential cells may contain cell-patterns with no transition on the main inputs of the cell. To allow the ML algorithm understanding this information, the solution consists in including static cell-patterns (e.g., P1 to P4 in Fig. 5.13) in the training data of sequential cells.

Applications on Industrial Cases

This section shows the experimental results of ML usage for CA model generation as well as CA diagnosis.

CA Model Generation Results

The CA model generation is performed with a python program. The ML algorithms were taken from the publicly available python module called scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF]. The dataset was composed of 1712 standard cells coming from standard cell libraries developed using three technologies (C40 (446 cells), 28SOI (825 cells) and C28 (441 cells)). All these cells already had a CA model generated by a commercial tool. The CA-matrix is generated for each cell. The method was experimented in two different ways. First, the ML model was trained and evaluated using cells belonging to one technology. Second, the model was trained on one technology and evaluated it on another one.

Predicting defect behavior on the same technology

The ML model is first trained on cells of 28SOI standard cell libraries. As mentioned earlier, cells were grouped according to their number of transistors and inputs. For m cells available in each group, the ML model is trained over m-1 cells and evaluate its prediction accuracy on the m-th cell. A loop ensured that each cell is used as the mth cell. On average, a group contains 8.6 cells. In the following, all possible open and short defects (static and dynamic) in each cell are considered. Results presented below report the prediction for open defects. Results achieved for short defects are similar. Table 5.5 presents the prediction accuracy achieved for open defects. Non-empty boxes report the average prediction accuracy obtained for a group of cells. Empty boxes mean that there is zero or one cell available and that the group cannot be evaluated. A grey background indicates that the maximum prediction accuracy in this group is 100%, i.e., the ML model can perfectly predict the defective behavior of at least one cell. In contrast, white background indicates that no cell was perfectly predicted in that group (all prediction accuracies are less than 100%). For example, let us consider the bold value in Table 5.5. One has 24 cells having 4 inputs and 24 transistors: i) 15 cells are perfectly predicted (100% accuracy), which leads to a green background, ii) the prediction accuracy for the 9 remaining cells ranges from 99.82% to 99.99% and iii) the average prediction accuracy over all 24 cells is 99.97%.

These results show that the ML model can accurately predict the behavior of a cell affected by a given defect and that our method could be used to generate CA models. The goal of the next subsection is to leverage on existing CA models to generate CA models for a new technology.

Predicting defect behavior on another technology

Experiments are also conducted on cells belonging to two different technologies.

Evaluation was slightly different compared to the previous one. Here, the ML model was trained over all available cells of a given technology and the evaluation was done on one cell of another technology. A loop was used to allow all cells of the second technology to be evaluated. Cells were grouped according to their number of inputs and transistors. 1. After investigating on this point, one can noticed that the behavior of most of the cells (68% of cells) is accurately predicted (accuracy > 97%), while accuracy for few cells is quite low. This phenomenon is discussed later in this section.

To verify the efficiency of the method when different transistor sizes are considered, the ML model was trained over the 28SOI standard cells and used to predict the behavior of C40 cells. Table 5.7 shows the prediction accuracy achieved on open defects of the C40 cells after training on the 28SOI cells. Results are averaged over all cells in each group (same number of inputs and transistors). This time, 80% of cells are accurately predicted (accuracy > 97%), proving that our MLbased characterization methodology could be used to generate CA models for a (large) part of cells of a new technology.

Analysis and Discussion

The cells for which the defect characterization methodology gives excellent prediction accuracy as well as those for which the prediction accuracy was quite low are firstly analyzed. Then, the limitation of the detailed method for CA model generation is investigated. After running several experiments on different configurations using one fault model at a time, one noticed the following behaviors:

• Accuracy for most of the cells is excellent, i.e., more than 97% prediction accuracy for 70% of cells. In this case, the CA model generated by ML fit the real behavior achieved with electrical simulation. • Accuracy for few cells (30%) is quite low and the ML prediction is not accurate.

For the first cell category with good prediction score, cells have been analyzed manually to identify why they led to good results. The analysis showed that all these cells had at least one cell in the training dataset with the same transistor structure or a similar one.

For the second cell category -cells leading to poor prediction accuracy -the manual analysis showed that they have: i) new logic functions that do not appear in the cells of the training dataset, or ii) a transistor configuration which is completely new when compared to cells in the training dataset.

Hybrid flow for CA model generation

Considering the above analysis, it appears that the ML-based CA model generation flow cannot be used for all cells in a standard cell library to be characterized. A mixed solution, which consists in combining ML-based CA model generation and conventional (simulation-based) CA model generation, should be preferably used. This is illustrated in the following. The flow sketched in Fig. 5.14 is proposed for accelerating the CA model generation. Typically, when the CA model for a new cell is needed, one first check if the ML-based generation will lead to high-quality CA models. This is done by analyzing the structure of the new cell and check whether the training dataset contains a cell with identical or similar structure (as presented in V.B). If the ML algorithm is expected to give good results, the new cell is prepared (representation in a CA-matrix) and submitted to the trained ML algorithm.

The output information is then parsed to the desired file format. Conversely, if the ML algorithm is expected to give poor prediction results, the standard generation flow presented in Fig. 5.14 is used to obtain the CA model. A feedback loop uses this new simulated CA model to supplement the training datasets and improve the ML algorithm for further prediction. Surprisingly, the structural analysis revealed that only 50% (205 cells) could be evaluated using the ML-based generation part of the flow. This shows that there is still room for further improvement of the structural analysis in this flow, and hence get better performance of the ML-based CA model generation process. To conclude, experiments have been carried out on a reasonable size (1712) of standard cell population. Considering that more than 10000 cells have usually to be characterized for a given technology, the hybrid flow in Fig. 5.14 is expected to provide even better results, especially owing to the reinforcement training that uses simulation generated models for supplementing the training datasets, and hence reduce the number of electrical simulations.

CA diagnosis results

The CA diagnosis flow described in Section 4.2 and targeting defects in both combinational and sequential cells of CUD has been implemented in a Python program. For validation purpose, authors in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF][START_REF] Hondt | A Comprehensive Framework for Cell-Aware Diagnosis of Customer Returns[END_REF] have experimented the proposed flow in three different ways:

• First, they conducted experiments on ITC'99 benchmark circuits with defect injection campaigns targeting combinational cells in each circuit. Various results are reported in [START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF][START_REF] Hondt | A Comprehensive Framework for Cell-Aware Diagnosis of Customer Returns[END_REF] to show the superiority of the framework when compared to commercial diagnosis solutions. • Next, they considered a test chip developed by STMicroelectronics and designed using a 28 nm FDSOI technology, and they conducted two defect injection campaigns targeting sequential cells [START_REF] Hondt | A Comprehensive Framework for Cell-Aware Diagnosis of Customer Returns[END_REF]. Results are reported in subsection 5.2.1 and demonstrate the effectiveness of the diagnosis framework. • Finally, they considered a customer return from STMicroelectronics and performed a silicon case study with a real defect subsequently analyzed and identified during PFA. Results are reported in subsection 5.2.2.

Simulated Test Case Studies

Authors in [START_REF] Hondt | A Comprehensive Framework for Cell-Aware Diagnosis of Customer Returns[END_REF] conducted experiments on a silicon test chip developed by STMicroelectronics and designed with a 28 nm FDSOI technology. The test chip is only composed of digital and memory blocks, and one PLL. The digital blocks are made of 3.8 million cells. Other features (number of primary inputs, primary outputs, and scan flip-flops) are given in Table 5.8. A first simulated case study was done with a static defect injection campaign. All possible static defects were successively injected into three scan flip-flops (SFF) of a single full-scan digital block. This block was tested with a static CA test sequence achieving a stuck-at + static CA fault coverage of 100%. The average numbers of passing and failing test patterns are given in Table 5.9. Results obtained after executing the CA diagnosis flow and averaged over all defect injections have shown an accuracy of 100% (the injected defect was always reported in the list of suspects) and a resolution of 1.25. The resolution ranges between 1 and 3, and Fig. 5.15 shows the distribution of this resolution with respect to the total number of simulated cases. As can be seen, in most of the cases, the number of suspects is equal to 1 (perfect resolution). A second simulated case study with another defect injection campaign was performed on the same test chip. All possible dynamic defects were successively injected into three scan flip-flops of a single full-scan digital block. This time, a dynamic CA test sequence was applied and achieved a transition + dynamic CA fault coverage of 89.8%. The average number of failing test patterns was 7.9. Again, the results obtained after executing the CA diagnosis flow and averaged over all defect injections have shown an accuracy of 100%. The average resolution obtained for dynamic defect injection experiments was 1.37. Again, the resolution ranged between 1 and 3, and in most of the case, the number of suspects was equal to 1.

Silicon Test Case Studies

Next, a silicon case study was performed on a customer return designed with a 28 nm FDSOI technology from STMicroelectronics [START_REF] Hondt | A Comprehensive Framework for Cell-Aware Diagnosis of Customer Returns[END_REF]. The test conditions used to run the experiments were as follows: a nominal supply voltage of 0.83 V, a scan test frequency of 10 MHz, a launch-to-capture clock speed (for the dynamic CA test sequence application) adjusted with respect to the nominal clock frequency of the circuit, and a temperature of 25°C. The process was considered as typical. The CA diagnosis flow was experimented and the following results were obtained. Initially, the circuit failed on the tester after application of the static CA test sequence when applied at the nominal voltage. This information was stored in a "static" datalog. The above diagnosis results were provided to the Failure Analysis team of STMicroelectronics, who made a PFA in the past on this customer return based on the results found by their in-house intra-cell diagnosis tool. The result obtained with the CA diagnosis flow was validated as defect D62 was found to be the real defect. This was done after performing a polysilicon level inspection on the layout of the cell (c.f. Fig. 5.16) and observing the failure analysis cross-sectional view.

Conclusion and Discussion

This chapter has provided an overview of the various machine learning approaches and techniques proposed to support cell-aware generation, test, and diagnosis from leading-edge research in this domain. In a comprehensive form, it proposes a compendium of solutions existing in this field. More in detail, the chapter has presented, after some backgrounds on conventional approaches to generate and diagnose cellaware defects, learning-based solutions to generate CA models and CA diagnosis. Experiments on silicon test cases have been done to validate those solutions and demonstrate their efficacy in terms of accuracy and resolution.

Results of this chapter prove the appropriateness of learning-based methods to solve the problem of CA models generation and CA diagnosis. The learning-based solutions to generate CA models was evaluated using two scenarios: i) the model was trained and evaluated using cells belonging to one technology, and ii) the model was trained on one technology and evaluated it on another one. Those evaluations have shown that the ML-based CA model generation flow cannot be used for all cells in a standard cell library to be characterized. A mixed solution, Hybrid flow for CA model generation, which consists in combining ML-based CA model generation and conventional (simulation-based) CA model generation, should be preferably used. The learning-based solution to generate CA diagnosis has been experimented in three different ways: i) on ITC'99 benchmark circuits with defect injection campaigns targeting combinational cells, ii) using a test chip developed by STMicroelectronics and designed using a 28 nm FDSOI technology with defect injection campaigns targeting sequential cells, and iii) using a customer return from STMicroelectronics with a real defect subsequently analyzed and identified during PFA. In all cases, the learning-based solution to generate CA diagnosis succeeds in identifying the good defect candidate compared to commercial diagnosis solutions.

Fig. 5 . 1

 51 Fig. 5.1 Conventional cell-aware model generation flow

. 2 .

 2 It is based on supervised learning that takes a known set of input data and known responses (labeled data) used as training data, trains a model to classify those data, and then uses this model to predict (infer) the class of new data.

Fig. 5 . 2

 52 Fig. 5.2 Generic view of the ML-based defect characterization flow

Fig. 5 . 3

 53 Fig. 5.3 Generic view of the CA-matrix creation flow

Fig. 5 .

 5 Fig. 5.4 represents an example of a NAND2 cell with its CA-matrix representation.In the partial representation of the CA-matrix of the cell shown in Fig.5.4.b, columns A and B list all the possible input stimuli for this cell. These columns also define the length of the CA-matrix, which is equal to 2 n + 2 n .(2 n -1), n being the number of cell input pins (2 n is the number of static stimuli, 2 n .(2 n -1) is the number of dynamic stimuli). For each stimulus, active and passive information about each transistor of the cell is entered in the CA-matrix. For example, AB=00 leads to two active PMOS transistors (Px and Py) and two passive NMOS transistors N10 and N11).

Fig. 5 . 4

 54 Fig. 5.4 Example standard cell NAND2: (a) cell transistor schematic and (b) partial CA-matrix representation

Fig. 5 . 5

 55 Fig. 5.5 Schematic example

Figure 5 .

 5 Figure 5.6 is a generic view of the learning-based CA diagnosis flow utilized in[START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF]. It is based on supervised learning that takes a known set of input data and known responses (labeled data) used as training data, trains a model, and then implement a classifier based on this model to make predictions (inferences) for the response to new data.After investigating several ML algorithms and observing their inference accuracies in[START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF], a Bayesian classification method has been chosen for the learning and inference phases in[START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF]. So, the first main step of the CA diagnosis flow consists in generating a Naive Bayes (NB) model and to train it by using a training dataset. In this step, training data are used to incrementally improve the model's ability to make inference. The training dataset is divided into mutually exclusive and equal subsets. For each subset, the model is trained on the union of all other subsets. Some manipulations, such as grouping data by considering equivalent defects or removing data instances of undetectable defects, are also done during this phase. Once training is complete, the performance (accuracy) of the model is evaluated by using a part of the dataset initially set aside. More details about performance evaluation as done in this framework can be found in[START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF].The second main step consists in implementing the NB classifier by using a Gaussian distribution to model the likelihood probability functions and use this classifier to make prediction when a new data instance has to be evaluated. The next subsections detail the various steps of the CA diagnosis flow, which is able to deal with any type of cell-internal defect (i.e., static and dynamic) that may occur.

Fig. 5 . 6

 56 Fig. 5.6 Generic view of the cell-aware diagnosis flow used in[START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF]

5 . 7 .

 57 Each instance is associated to a static defect (D1, D2, D3) or a dynamic defect (D11, D12, D13). A 1 (0) indicates that defect Di is detectable (not detectable) at the output of the cell when the cell-level test pattern Pj is applied at the inputs of the cell. Cell-level test patterns (called cell-patterns in the sequel) are static (one input vector -P1 to P4 in Fig.5.7) or dynamic (two input vectors -P5 to P16 in Fig.5.7 in which R (F) indicates a rising (falling) transition at the cell input respectively). For an n-input cell, there exists 2 n static cell-patterns and 2 n .(2 n -1) dynamic cell-patterns.

Fig. 5 . 7

 57 Fig. 5.7 Example of training dataset for all defect types in a two-input cell as used in[START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF]

Fig. 5 . 8

 58 Fig. 5.8 Example of training dataset for all defect types in a two-input cell as used in[START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF]

Fig. 5 . 9

 59 Fig. 5.9 Generation flow of instance tables

Fig. 5 .

 5 Fig. 5.10 Example of static and dynamic instance tables

Fig. 5 .Fig. 5 .

 55 Fig. 5.11 Format of a new data instance for a two-input cell as used in [C-H]

Fig. 5 .

 5 Fig. 5.13 Example of training dataset for all defect types (static and dynamic) in a sequential cell. The pin order is clock-data-previous output.

Fig. 5 .

 5 Fig. 5.14 Hybrid flow for CA model generation.

Fig. 5 .

 5 Fig. 5.15 Distribution of the resolution with respect to the simulated cases

Fig. 5 .

 5 Fig. 5.16 Layout view of the suspected cell and the incriminated transistor. Yellow circles indicate defect candidates and red mark indicates actual observed defect

  • CA models may change with respect to test conditions and Process Voltage Temperature (PVT) corners. In fact, CA model generation for the same cell with different test conditions may exhibit slight differences. Few defects can be of different types (i.e., static, or dynamic) or may have different detection patterns. Since CA models are generated for specific test conditions and can be used with different ones, it may lead to inaccurate characterization. This inaccuracy is usually allowed in industry since it is marginal. This indicates that one can also tolerate few error percentages in our ML-based prediction.• Very simple CA models are used to emulate short and open defects, for which resistance values are often identical for all technologies.

Table 5 .

 5 1 shows an example of a training dataset for a NAND2 cell. It is composed of four types of information:

TABLE 5

 5

				.1: EXAMPLE OF TRAINING DATASET FOR A NAND2 CELL
		Cell inputs &		Transistor switching	Defect	About defect	Defect
		responses			activity		description	detection
	A	B	Z	N0	N1	P0 … N1_D N1_G N1_S … name type	fZ
	0

TABLE 5 .

 5

			2: ACTIVITY VALUES FOR A NAND2 CELL
				old names
	A	B	Comment	Px	Py	N10	N11

TABLE 5 .

 5 4: TEST SCENARIOS CONSIDERED IN[START_REF] Mhamdi | A Learning-Based Cell-Aware Diagnosis Flow for Industrial Customer Returns[END_REF][START_REF] Mhandi | Towards Improvement of Mission Mode Failure Diagnosis for System-on-Chip[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Automotive Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhamdi | Cell-Aware Defect Diagnosis of Customer Returns Based on Supervised Learning[END_REF][START_REF] Mhandi | Learning-Based Cell-Aware Defect Diagnosis of Customer Returns[END_REF][START_REF] Mhandi | Cell-Aware Diagnosis of Customer Returns Using Bayesian Inference[END_REF]

	Test Scenarios

TABLE 5 . 5 :

 55 PREDICTION ACCURACY FOR CELLS IN THE SAME TECHNOLOGY

	Prediction			Number of inputs
	accuracy (%)	2	3	4	5	6
		6	99.98	99.99
		8	99.91	99.96	99.91
		9		100.0
		10	99.98	99.81	99.96
		12	99.72	99.73	100.0	99.91	99.93
		14	99.70	99.56	99.83	99.92	99.96
	Number of transistors	16 18 20 22 24 26	99.99 99.99 100.0 100.0 100.0	100.0 99.94 99.98 99.84 99.84 99.70	99.94 100.0 99.98 99.97 100.0	99.73 99.62	99.98 99.85 99.89
		28	99.49	99.98	100.0	99.88	99.81
		30	99.75	100.0	100.0
		32	100.0	100.0			99.98
		42		100.0
		44		100.0
		46		99.81
		47		99.98	99.95

TABLE 5 .

 5 6: AVERAGE PREDICTION ACCURACY FOR CELLS IN DIFFERENT TECHNOLOGIES

	Prediction			Number of inputs
	accuracy (%)	2	3	4	5	6
		6	98.21	99.47
		8	94.56	96.86	99.00
		9
		10	94.69	96.01	99.27
		12	87.73	98.05	99.10		99.76
		14	85.69	97.35	98.75
	Number of transistors	16 18 20 22 24 26	91.74 88.18 90.29 78.73 87.91 87.24	96.28 94.37 96.88 98.92	99.20 98.37 99.37		99.79
		28	88.18	98.68
		30			97.52
		32	88.73	95.6
		42
		44
		46
		47

Table 5 .

 5 [START_REF] Wang | Machine Learning-Based Volume Diagnosis[END_REF] shows the prediction accuracy achieved on open defects of the C28 cells after training on the 28SOI cells. Results are averaged over all cells in each group (same number of inputs and number of transistors). The average prediction accuracies are globally lower compared to those of Table

TABLE 5 .

 5 7: AVERAGE PREDICTION ACCURACY FOR CELLS WITH DIFFERENT TRANSISTOR SIZE

	Prediction			Number of inputs
	accuracy (%)	2	3	4	5	6
		6	100.0	99.80
		8	87.39	99.14	99.03
		9		97.19
		10	92.07	95.49	99.32	98.46
		12	91.71	98.07	99.24	98.47	99.46
		14	90.1	95.84	98.63	98.79	99.52
	Number of transistors	16 18 20 22 24 26	91.17 88.5 83.87 87.26 93.96 87.52	93.59 97.15 97.73 99.34 97.55	99.23 97.14 97.15 98.98 98.58 99.04	97.74 98.94 98.44 98.84 99.02	99.59 99.63 99.92
		28	98.19		98.79	99.31	99.44
		30			99.13	99.37	99.58
		32	92.91			98.92	99.78
		42
		44	92.03	98.82
		46		99.23
		47		98.29	99.76

TABLE 5 .

 5 8: MAIN FEATURES OF THE SILICON TEST CHIP

	#cells	#PIs	#POs	#SFF
	3.8M	97	32	17.5k

TABLE 5 .

 5 9 AVERAGE PATTERN COUNT IN INSTANCE TABLES OF THE FIRST SIMULATED CASE STUDY

  Then, a logic diagnosis gave a short list of suspected cells among which a six-input SFF cell made of 56 transistors and having a Reset, an Enable, a Test-Input and Test-Enable input pins. The cell contains 758 potential short or open defects. A static instance table was then generated for this suspected cell and contained 5 failing and 75 passing cell-level test patterns. From the new data generated after post-processing of this instance table, the NB classifier identified four suspected defects among which defect D62 (a short between the gate and source of NMOS 19).