Moritz Fieback

Lizhou Wu

Guilherme Cardoso Medeiros

Hassen Aziza
email: hassen.aziza@univ-amu.fr

Siddharth Rao

Erik Jan Marinissen
email: erik.jan.marinissen@imec.be

Mottaqiallah Taouil

Said Hamdioui
email: hamdioui@tudelft.nl

R Fieback

G Cardosomedeiros

Device-Aware Test: A New Test Approach Towards DPPB

This paper proposes a new test approach that goes beyond cell-aware test: device-aware test. The approach consists of three steps: defect modeling, fault modeling, and test/DfT development. The defect modeling does not assume that a defect in a device (or a cell) can be modeled electrically as a linear resistor (as the traditional approach suggests), but it rather incorporates and captures the impact of the physical defect on the technology parameters of the device and thereafter on its electrical parameters. Once the defective electrical model is defined, a systematic fault analysis (based on fault simulation) is performed to derive appropriate fault models and thereafter test solutions. The approach is demonstrated using two memory technologies: resistive random access memory (RRAM) and spintransfer torque magnetic random access memory (STT-MRAM). The results show that the proposed approach is able to sensitize faults for defects that are not detected with the traditional approach, meaning that the latter cannot lead to high-quality test solutions as required for defective part per billion (DPPB) level. The new approach clearly sets up a turning point in testing at least for the considered two emerging memory technologies.

I. INTRODUCTION

Technology scaling has driven the phenomenal success of the semiconductor industry in delivering larger, faster, and cheaper integrated circuits with a high quality of service [START_REF] Borkar | Design challenges of technology scaling[END_REF]. Silicon technology has entered the nano-era and 5 nm transistors are being prototyped [START_REF] Loubet | Stacked nanosheet gate-all-around transistor to enable scaling beyond FinFET[END_REF]. However, it is widely recognized that defects and variability in device characteristics during the fabrication process, and their impact on the overall quality and reliability of the system represent major challenges, especially when considering high-quality levels, e.g., defective part per billion (DPPB) level [START_REF] Borkar | Microarchitecture and Design Challenges for Gigascale Integration[END_REF]. Moreover, newly-emerging failure mechanisms in the nano-era are causing the fault mode of chips to be dominated by transient, intermittent, and weak faults rather than hard and permanent faults [START_REF] Bhoj | Fault models for logic circuits in the multigate era[END_REF]. This shift in failure mechanisms may impact the way fault modeling has been done. Note that accurate fault models which reflect the real defects of new technologies are a must for developing high defect coverage test solutions. High-quality testing is a very critical step in the whole design and manufacturing chain responsible for screening out all the defective chips before they are sold, as it is the last chance to deliver the required quality and reliability to the end customer. All of these indicate the necessity and the importance of high-quality test solutions.

Testing defects in logic and memory chips underwent a long evolution process. For logic, early test methods were mainly functional and did not use any fault models. However, the increasing cost of such test approaches has led to the development of fault models (and hence structural testing) starting from the late 1970s. The most well-known fault models include stuck-at fault [START_REF] Mei | Bridging and stuck-at faults[END_REF], transition fault [START_REF] Waicukauski | Transition fault simulation[END_REF][START_REF] Cox | Stuck-open and transition fault testing in CMOS complex gates[END_REF], and bridge fault [START_REF] Ferguson | Test pattern generation for realistic bridge faults in CMOS ICs[END_REF][START_REF] Rearick | Fast and accurate CMOS bridging fault simulation[END_REF]. Despite the great success of these fault models, there was a clear need from the industry for new approaches and fault models (starting from late 1990s onwards) in order to reduce the increasing number of test escapes that customers were reporting. This led to the introduction of additional highquality approaches and models such as stuck-short and stuckopen transistor models [START_REF] Dahlgren | A fault model for switch-level simulation of gateto-drain shorts[END_REF], N-detect [START_REF] Pomeranz | On n-detection test sets and variable n-detection test sets for transition faults[END_REF], embedded multidetect [START_REF] Geuzebroek | Embedded multi-detect ATPG and its effect on the detection of unmodeled defects[END_REF], and layout-aware fault modeling [START_REF] Goel | Circuit topology-based test pattern generation for small-delay defects[END_REF]. Moreover, the increasing demand of customers for higher quality has further led to the introduction of cell-aware test [START_REF] Hapke | Cell-aware test[END_REF][START_REF] Gao | Defect-location identification for cell-aware test[END_REF]; it assumes that many escapes during testing are due to defects within a standard library cell, and therefore models defects as linear resistors (opens, shorts) at or between the terminals of each device within the library cell.

Testing memories went also through a quite similar revolution. The early memory tests (typically before 1980) can be classified as ad-hoc tests due to the absence of formal fault models and proofs [START_REF] Breuer | Diagnosis & reliable design of digital systems[END_REF]; they have the property of having a low defect/fault coverage and a very long test time, typically in the order of O n 2 , which made them unpractical for increasing memory sizes. During the early 1980s, many memory fault models have been introduced, allowing the fault coverage of a certain test to be provable while the test time is usually in order O (n); i.e., linear with the size of the memory. Some important fault models introduced in that time were stuckat faults and address-decoder faults [START_REF] Van De | Testing semiconductor memories: theory and practice[END_REF]. These are abstract fault models not based on any actual memory design or real defects. In the late 1990s, experimental results based on DPPM screening of a large number of tests applied to a large number of memory chips indicated that many detected faults cannot be explained with the well-known fault models [START_REF] Schanstra | Industrial evaluation of stress combinations for march tests applied to SRAMs[END_REF][START_REF] Van De Goor | Industrial evaluation of DRAM tests[END_REF], which suggested the existence of additional faults. This stimulated the introduction of new fault models ( SPICE simulation [START_REF] Hamdioui | An experimental analysis of spot defects in SRAMs: realistic fault models and tests[END_REF][START_REF] Vatajelu | Analyzing resistive-open defects in SRAM corecell under the effect of process variability[END_REF]: read destructive fault, write disturb fault, transition coupling fault, read destructive coupling fault, etc. Note that the cell-aware test fault modeling approach is quite similar to this as it also models defects as linear resistors (opens and shorts) at the terminals of devices in each memory cell. The above clearly shows that testing of both logic and memory assumes that physical defects in devices can be be modeled as linear resistors. Although it can be convincing for modeling opens and shorts in interconnects, this assumption has never been validated for devices. In addition, it is well known that scaling below 10 nm is giving rise to many device failure mechanisms that cannot be modeled with linear resistors. Moreover, it has been shown recently that this assumption is inaccurate for emerging technologies such as resistive random access memory (RRAM) [START_REF] Fieback | Testing resistive memories: Where are we and what is missing?[END_REF] and spin-transfer torque magnetic random access memory (STT-MRAM) [START_REF] Wu | Electrical modeling of STT-MRAM defects[END_REF], and may lead to wrong fault models. Hence, it cannot lead to high-quality test solutions. This has inspired us to develop a new device-aware test (DAT) approach which is the topic of this paper. This paper introduces device-aware testing which takes cell-aware testing one step further. Instead of using a fault model derived from injecting linear resistors in transistor-level netlists, DAT first changes the electrical model of the defective device (e.g., transistor) by incorporating the impact of the defect in the device's electrical parameters (model); these are then used to perform circuit simulation to derive the fault models and thereafter test solutions. In this paper, we introduce and demonstrate DAT for two emerging memory technologies, namely RRAM and STT-MRAM. The main contributions of the paper are as follows.

• We introduce the three-step DAT approach: defect modeling, fault modeling, and test development. One of the key differentiators is the defect modeling step which takes the physical defects into consideration and captures their impact on the electrical parameters, hence enabling accurate fault modeling. The latter systematically defines the complete (theoretical) memory fault space and thereafter systematically performs the fault analysis (using defect modeling of the first step and circuit simulation) to validate the space. This step provides insight information not only on the nature of realistic faults, but also about the best way to test them, which is used in the third step of DAT (test development). As an example, a fault resulting in a wrong read value can be easily detected with a March test as it is able to sensitize the fault, while a fault resulting in a random read value needs special design-for-testability (DfT) to guarantee its detection. • We apply DAT on RRAM and STT-MRAM and demonstrate the superiority of this approach as compared to conventional memory test approaches. DAT can model and detect some of the device defects that cannot be detected by the conventional approach. Hence, it can further reduce the amount of test escapes and can even better diagnose defects for fast yield learning. The rest of the paper is organized as follows. Section II provides brief background information on the operating principles of RRAM and STT-MRAM, respectively, as they will be used for the validation of DAT approach. Section III gives a complete view of the DAT methodology; each of the three steps is described in detail. Section IV selects the "forming defect" (representing a defect in an RRAM device) and applies the three steps of DAT approach; not only in order to show how the approach works, but also to validate its superiority. Section V does the same by then by selecting "pinhole defect" for STT-MRAM. Section VI discusses the advantages and limitations of the method and concludes the paper.

II. TECHNOLOGY BACKGROUND

This section provides the technical working principles of RRAM and STT-MRAM, respectively.

A. RRAM Fundamentals

Resistive random access memory (RRAM) is an emerging non-volatile memory technology that uses oxide-based memristors to store data [START_REF] Wong | Metal-oxide RRAM[END_REF]. The production of the RRAM devices can be integrated in the back-end-of-line (BEOL) of a standard CMOS process [START_REF] Wong | Metal-oxide RRAM[END_REF].

The RRAM device is schematically shown in Fig. 1a. It consists of two electrodes (top (TE) and bottom electrode (BE)) and a metallic-oxide between them. By applying a positive voltage to the TE that is higher than the set threshold (V set ), bonds between the metal and oxygen ions are broken and the oxygen ions are attracted to the TE, leaving behind a chain of oxygen vacancies, a conductive filament (CF). The device is now in its low resistive state R LRS (i.e., 'set' representing logic '1') . If a negative voltage is applied that is lower than the reset threshold (V reset ), the ions move back to fill the vacancies, bringing the device in its high resistive state R HRS (i.e., 'reset' representing logic '0'). The size of the CF determines the resistance of the device; wider CFs result in lower resistance and longer CFs result in higher resistance. Fig. 1b and Table I show the technology parameters that determine the resistance of the RRAM device. Its resistance has an analogue nature, i.e., it can take any value within a certain range. and 'reset' state, as well as the switching thresholds between them. From the plot, it becomes clear that the RRAM device is a non-linear device due to its hysteresis. Multiple cell designs exist for RRAMs, the most common of them are the 1T-1R and 1R design. The 1T-1R cell is depicted in Fig. 1d. By applying appropriate voltages to the bit line (BL), word line (WL), and select line (SL), the state of the device can be changed. The transistor controls the current flow through the RRAM. A 1R design does not have an access transistor, which has the benefit of smaller cell designs, but also the drawback that sneak-paths exist that unwantedly couple multiple cells [START_REF] Kannan | Sneak-path testing of memristor-based memories[END_REF].

B. STT-MRAM Fundamentals

The magnetic tunnel junction (MTJ) is the core of STT-MRAM, as it is the data-storing element. As shown in Fig. 2a, an MTJ device is composed of two ferromagnetic layers sandwiching an ultra-thin insulating MgO layer called tunnel barrier (TB). The top ferromagnetic layer is called free layer (FL), its magnetization can be switched by a spin-polarized current flowing through it. There are several key technology parameters that significantly impact the STTinduced switching behavior for the magnetization in the FL. They are the saturation magnetization M s and the magnetic anisotropy field H k of the FL, and the potential barrier height φ of the TB [START_REF] Wu | Electrical modeling of STT-MRAM defects[END_REF]. In contrast, the magnetization in the bottom ferromagnetic layer is pinned to a certain direction. Therefore, the bottom layer is usually referred to as pinned layer (PL). Due to the tunneling magneto-resistance (TMR) effect [START_REF] Khvalkovskiy | Erratum: Basic principles of STT-MRAM cell operation in memory arrays[END_REF], the MTJ's resistance is low when the magnetization in the FL is parallel to that in the PL. The resistance is high when in anti-parallel configuration. The TMR ratio is defined by: TMR=(R AP -R P )/R P , where R AP and R P are the resistances in the anti-parallel and parallel states, respectively. To evaluate the resistivity of MTJ devices, the resistance-area (RA) product is commonly used in the MRAM community, as it is independent of the device size. In summary, RA, TMR, φ, M s , and H k are critical technology parameters of the MTJ device, as listed in Table II.

Fig. 2b shows the most widely-adopted STT-MRAM cell design, namely the bottom-pinned 1T-1MTJ cell, and its corresponding control voltages during write and read operations. The cell includes an MTJ device and an NMOS selector; it has three terminals similar to RRAM, as illustrated in the figure .  For STT-MRAMs, R P , R AP , I c , and t w are four key electrical parameters determining the electrical behavior of MTJ devices [START_REF] Wu | Electrical modeling of STT-MRAM defects[END_REF]. I c is the critical switching current, and t w is the average switching time.

Device Model Defective Device

Fig. 3: Generic device defect modeling flow.

III. DEVICE-AWARE TEST

Traditional memory testing assumes that a device defect can be modeled as a linear resistor in series or in parallel with the device. However, it has been shown that this approach is not accurate at least for emerging memory technologies such as RRAM [START_REF] Fieback | Testing resistive memories: Where are we and what is missing?[END_REF] and STT-MRAM [START_REF] Wu | Electrical modeling of STT-MRAM defects[END_REF], resulting in incomplete or inaccurate fault modeling; hence escapes. Device-Aware Test (DAT) aims at solving this problem, and setting up a step toward meeting DPPB-level requirement. First, the device defects are physically modeled and their electrical behaviors are incorporated into device models. Second, the model is integrated in a memory simulation platform in order to analyze the impact of the defect on memory behavior; this is done in a systematic manner by validating a pre-defined fault framework/space using SPICE Simulation. The results of this step provide insights on the nature of realistic faults, which is used in order to develop optimal and appropriate test solutions (e.g., March test, DfT). Next, these three steps are described in detail. Note that these steps will be applied in Section IV and Section V to RRAM and STT-MRAM, respectively.

A. Device Defect Modeling

Inaccurate defect modeling may result in poor fault models, thereby limiting the effectiveness of proposed test solutions and DfT designs, not only in terms of defect coverage but also in terms of test time. For example, a test targeting a fault model that does not represent any real defect will not increase the defect coverage while still consuming test time. To accurately model physical defects, the device model should incorporate the way the defect impacts the technology parameters (e.g., length, width, density) and thereafter the electrical parameters (e.g., the critical switching current) of the device [START_REF] Wu | Electrical modeling of STT-MRAM defects[END_REF]; this is exactly what device defect modeling does. Fig. 3 shows the flow of such modeling approach; its inputs are 1) the electrical model of a device, and 2) the defect under investigation. The output is an optimized (parameterized) model of the defective device. Note that in general a device can be a FinFET Paper X.Y INTERNATIONAL TEST CONFERENCE transistor, an STT-MRAM device, an RRAM device, a PCM device, etc. The approach consists of three steps: 1) Physical defect analysis and modeling. Given a set of physical defects D = {d 1 , d 2 , ..., d n } that may take place during the manufacturing process of the device, each defect d i has to be analyzed to fully understand the defect mechanism and identify its impact on each (key) technology parameter of the device. Due to such a defect, one or more (defect free) technology parameter T p df will be modified, resulting in what we refer to as an effective technology parameter T p eff . This can be described by the following abstract function:

T p eff (S i ) = f i (T p df , S i ) (1)
where T p df is the defect-free technology parameter, f i is a mapping function corresponding to defect d i (i∈ [1, n]), and

S i = {x 1 , x 2 , • • • , x t } is a set of parameters representing the size or strength of defect d i .
2) Electrical modeling of the defective device. In this step, the impact of the altered technology parameters from Step 1 on each of the key electrical parameters of the device is identified. The resulting electrical parameters are therefore qualified to describe the electrical behavior of the defective device with defect d i . One way to perform this, is by modifying the defect-free device electrical model and converting it into a defect-parameterized model by integrating Equation (1) of each involved technology parameter. This step gives a raw defective device model with the effective electrical output parameters.

3) Fitting and model optimization. To guarantee the effectiveness and the accuracy of the defective device model, the model needs to be calibrated. Therefore, real-world defective devices need to be measured. If any physical or electrical parameters of the defective model do not accurately match the characterization data, it is necessary to keep optimizing the device model until an acceptable accuracy is obtained. By performing silicon data fitting and model optimization, we can derive an optimized defective device model, which enables accurate circuit simulation for fault modeling.

B. Fault Modeling

The second DAT step is fault modeling. In this step, the defect models from the previous step are used to analyze the behavior of a memory in the presence of defects. The results from this analysis are used to develop a high-quality test. First, we define the fault space that describes all possible faults, and classify them. Second, we present the fault analysis methodology that determines which faults from the space are realistic for the defect under consideration; i.e., which faults can be sensitized in the presence of such a defect.

1) Fault Space and Classification: In this work, we limit the analysis to static and dynamic single-cell faults [START_REF] Van De Goor | Functional memory faults: a formal notation and a taxonomy[END_REF]. A static fault is defined as a fault that can be sensitized by performing at most one operation, while a dynamic fault is sensitized by more than one operation. If more than one cell is involved in the fault, the fault is called a coupling fault. A strong fault can be systematically described using the fault primitive (FP) notation [START_REF] Van De Goor | Functional memory faults: a formal notation and a taxonomy[END_REF]. An FP describes the difference between the observed and expected memory behavior, denoted as a three-tuple S/F/R where: Table III lists all single-cell static FPs and their names. The naming of the FPs follows the following scheme:

FP = {read impact} {behavior} {initial} {F } (2)
Here, read impact is applicable only if a read sensitizing operation results in a faulty read: incorrect (I), or random (R) read values. behavior describes the behavior of the faulty cell: it specifies the nature of the operation (read (R) or write (W)) and the resulting fault effect (destructive (D), transition (T), or nothing). For example, 'WDF' means write destructive fault. initial describes the initial state of the cell and F is the value that is stored in the cell after S is performed. For example, RRDF0 1 = 0r0/1/? is the Random Read Destructive Fault that puts the cell into '1' during a read '0', and returns a random read value at the output. State faults are an exception to this scheme because no sensitizing operation is performed. Their names follow this scheme: FP = SF {initial} {F } . For dynamic faults, the name of an FP gets the prefix ndwhere n denotes the number of operations in S and the FP is based on the last operation in S, e.g., 1r1w0/L/-is 2d-WTF1 L . Memory faults can be classified, as shown in Fig. 5, into two types: strong and weak faults. Strong faults are functional faults that can always be sensitized (and may be detected) by applying a sequence of operations and can cause functional errors; e.g., all FPs of Table III are strong faults. In contrast, weak faults do not result in FPs, but they cause parametric faults, e.g., a reduction in bit line current during a read operation. Note that these faults cannot be detected with any sequence of operations as they do not cause any functional errors. Obviously, these faults need to be also detected as they cause reliability problems (e.g., shorter lifetime, higher infield failure rate). Depending on the effort needed to detect them, faults can be further divided into easy-to-detect and hard-to-detect faults. The detection of easy-to-detect faults can be simply guaranteed by applying write and read operations, e.g., by using a March test. The detection of hard-to-detect faults, however, cannot be guaranteed by just March tests and their detection requires additional effort; e.g., use of a special circuity such as DfT. Note that strong faults consist of easy-todetect and hard-to-detect faults, while weak faults are all hardto-detect. Examples of strong hard-to-detect faults are random read faults such as RRF1 1 and RRF0 0 . For example, in an STT-MRAM with a small defect, the bit-line current during a read may be very close to the reference current of the sense amplifier causing random behaviour between devices.

2) Fault Analysis Methodology: Once the defect is modeled and the framework of faults is defined, the validation of the faults can be performed using a systematic circuit simulation approach. In this paper we will restrict ourselves to singlecell fault analysis as the case studies we will show for RRAM and STT-MRAM involve single-cell defects. Our fault analysis consists of seven steps [START_REF] Van De Goor | Functional memory faults: a formal notation and a taxonomy[END_REF]: 1) circuit generation, 2) defect injection, 3) stimuli generation, 4) circuit simulation, 5) fault analysis, 6) fault primitives identification, and 7) defect size sweeping and repetition of steps 2 to 6 till all sizes are covered. Note that in our case, defect injection means changing the electrical model of the device (e.g., RRAM or STT-MRAM) with the defective device model obtained in step 1 of DAT, while defect size sweeping means changing the size of the defect which also modifies the electrical parameters of the defective device model. Fig. 6 shows the methodology of fault analysis that enables us to get more insight on the nature of realistic faults and the way to test them. Given a list of defects and ranges of their sizes, the seven steps of the fault analysis should be first performed for the validation of static single-cell FPs of Table III (i.e., n≤1). The result will be a set of FPs classified into easy-to-detect faults and hard-to-detect faults, associated with the size/range of the defect/parameters. In case that no FP is sensitized in the presence of a defect, the fault is considered to be weak and it is added to the list of hard-to-detect faults. Next, all defects that resulted in hard-todetect faults will be further analyzed, but then using dynamic fault analysis, starting at n=2. Some defects leading to hardto-detect faults can trigger now easy-to-detect faults; e.g., S=0w0 causes a weak fault, while S=0w0w0 causes an easyto-detect strong fault. Once 2 operation single-cell dynamic fault analysis is done, we can redo similar analysis for n=3 for defects that resulted in hard-to-detect faults. The process can be repeated by extending S each time with one operation till the considered n max is reached. Each step in the process aims at reducing the hard-to-detect fault set and increasing the easyto-detect fault set; this is an important step towards not only optimizing test cost but also towards improving the overall product quality. The final results will be a set of faults that can be easily detected by the generation of March tests, and another set that needs special attention in order to guarantee their detection (e.g., DfT, special tests, etc.).

C. Test Development

The results of the fault analysis facilitate the development of high-quality and efficient test solutions. All easy-to-detect faults can be detected by applying appropriate test algorithms.

Paper X.Y INTERNATIONAL TEST CONFERENCE

List of defects 1. Pinhole: 0-100 % 2. Forming: 1-100 µA 3. . . .

Fault analysis n operations

Easy-to-detect faults 1. > 0.79%: Developing optimized algorithms starts by identifying the minimal detection conditions for each of the faults and thereafter compiling them in test algorithms. This will guarantee 100% fault coverage of easy-to-detect faults at minimal test cost. One can also incorporate a DfT scheme in order to further optimize the test time; e.g., a DfT that enables the test of many faults simultaneously, parallel testing, etc. Hard-to-detect faults, however, require special attention. Special DfT schemes and tests may be required. Examples are: DfT schemes that may directly measure the bit line swing, modify the operation conditions such as weak write operations [START_REF] Haron | DfT schemes for resistive open defects in RRAMs[END_REF], stress tests, etc. The aim is to maximize the fault coverage for these faults while keeping the test cost economically affordable.

IV. DEVICE-AWARE TEST FOR RRAM

In this section we apply the DAT approach on RRAM by following the three steps discussed in the previous section. However, first we describe RRAM manufacturing defects and select a representative defect.

A. RRAM manufacturing defects

The fabrication process flow of an RRAM is depicted in Fig. 7a [START_REF] Fieback | Testing resistive memories: Where are we and what is missing?[END_REF] and their associated defects are listed in Table IV; a more detailed overview can be found in [START_REF] Fieback | Testing resistive memories: Where are we and what is missing?[END_REF]. The process starts with manufacturing transistors on the wafer in the front-end-of-line (FEOL) production phase. Then, the lower metal interconnection layers are deposited in the BEOL phase. RRAM devices are typically constructed between two metal layers (e.g., M4 and M5) as depicted in Fig. 7b [START_REF] Grossi | Fundamental variability limits of filament-based RRAM[END_REF]. After the devices are fabricated, the remaining metal layers are deposited. The devices do not have a conductive filament (CF) yet, therefore an initial CF forming step needs to be performed in order to achieve a functional device. In this paper we focus on defects that result from this step.

During the forming step, an initial CF is generated in the RRAM device's oxide. The conditions of this step have a strong impact on the performance of the device and therefore this step may result in defects. A few observations of the forming conditions can be made: higher forming currents (I form ) result in lower device resistance with less variation [START_REF] Grossi | Fundamental variability limits of filament-based RRAM[END_REF][START_REF] Raghavan | Performance and reliability trade-offs for high-κ RRAM[END_REF], and variations in the forming current lead to more resistive variations [START_REF] Kalantarian | Controlling uniformity of RRAM characteristics through the forming process[END_REF]. Variations in the device geometry and oxide defect density also affect the forming step [START_REF] Govoreanu | 10x10nm2 Hf/HfOx crossbar resistive RAM with excellent performance, reliability and low-energy operation[END_REF]. Two defects can result from the forming step: overforming, when the CF is too large, and non-forming, when no or only a tiny CF is formed.

F E O L M e t a l l i z a t i o n B E p a t t e r n i n g B E p l a n a r i z a t i o n O x i d e d e p o s i t i o n C a p d e p o s i t i o n T E d e p o s i t i o n P a s s i v a t i o n M e t a l l i z a t i o n C F F o r m

i n g BEOL (a) Processing flow [START_REF] Fieback | Testing resistive memories: Where are we and what is missing?[END_REF][START_REF] Grossi | Fundamental variability limits of filament-based RRAM[END_REF][START_REF] Govoreanu | 10x10nm2 Hf/HfOx crossbar resistive RAM with excellent performance, reliability and low-energy operation[END_REF]. (b) Cross-section TEM [START_REF] Grossi | Fundamental variability limits of filament-based RRAM[END_REF].

Fig. 7: General manufacturing process of RRAM.

B. Forming Defect Modeling

In this section, we model the forming defect using DAT approach and show how the conventional approach use linear resistor to model the defect.

1) DAT Approach: For the DAT approach, we relate the input parameters of the RRAM device model (such as in [START_REF] Li | A SPICE model of resistive random access memory for large-scale memory array simulation[END_REF]) to the forming current, thus incorporating the physics of the forming step, that could result in overforming or non-forming, into the electrical model. The model can be included in a netlist to observe its electrical effects.

Physical defect analysis and modeling. The forming current is directly related to the shape of the CF, i.e., it affects the key parameters shown in Fig. 1b. It is shown that l CF and φ T have the strongest impact on the resistance of the device [START_REF] Raghavan | Performance and reliability trade-offs for high-κ RRAM[END_REF]. Therefore, these parameters are used to model the forming effects of the device. To include the stochastic variation of the l CF , an additional parameter ∆l CF (that sets the strength of this variation) is included. These parameters are used to model the forming effects of the device. The physical defect modeling step can be denoted mathematically as:

l CF,eff (I form ) = a 1 exp (b 1 • R µ (I form )) + c 1 exp (d 1 • R µ (I form )) , (3)
φ T,eff (I form ) = a 2 exp (b 2 • R µ (I form )) + c 2 exp (d 2 • R µ (I form )) , (4)
∆l CF,eff (I form ) = a 3 exp (b 3 • R σ (R µ )) + c 3 exp (d 3 • R σ (R µ )) . (5)
Here,

a k , b k , c k and d k (k ∈ {1, 2, 3}) are fitting parameters. R µ (I form ) = f (I form )
, where f (I form ) is a cubic Hermite interpolation of I form to the median resistance in [START_REF] Grossi | Fundamental variability limits of filament-based RRAM[END_REF], and R σ (R µ ) is given by Equation ( 1) in [START_REF] Grossi | Fundamental variability limits of filament-based RRAM[END_REF].

Electrical modeling of the defective device. The RRAM device model in [START_REF] Li | A SPICE model of resistive random access memory for large-scale memory array simulation[END_REF] takes l CF , φ T , and ∆l CF as input parameters. These three parameters dictate the switching behavior and the resistance of the RRAM device, and thus are well suited to model the effects of forming on the device's electrical behavior. When the resulting model is simulated in a netlist, the effects on the electrical parameters, e.g., resistance, switching speed and thresholds, can be analyzed.

Paper X.Y INTERNATIONAL TEST CONFERENCE 6 R sd R pd

Deviceaware

Defect Free Resistive Fig. 8: Device-aware and resistive defective device models. TABLE V: Faults sensitization using device-aware vs resistive defect model.

Fitting and model optimization. In this step, the three alterable parameters are calibrated so that the defective behavior of the RRAM device corresponds with measurements of real devices such as in [START_REF] Grossi | Fundamental variability limits of filament-based RRAM[END_REF]. To realize this, we first analyze the influence of l CF and φ T on the mean resistance. These parameters are then fitted against the measurements in [START_REF] Grossi | Fundamental variability limits of filament-based RRAM[END_REF] and thus are linked to I form . The effect of ∆l CF is analyzed and fitted in a similar fashion. We vary I form between 5 µA and 34.1 µA.

2) Conventional Approach: The conventional resistive defect modeling approach models the forming defect as a resistor that is either parallel (R pd ) or in series (R sd ) with a defect-free RRAM device. The difference with the device-aware defect models is shown in Fig. 8. The strength of a resistive defect is represented by the resistance value; we sweep the resistance of both R pd and R sd from 1 Ω and 100 MΩ in our simulations.

C. Fault Modeling

This step consists of fault analysis based on the use of the electrical models generated in the first step. We start with the static fault analysis for both defect models. Next, we analyze the dynamic faults for the DAT and traditional approach.

As a forming defect impacts a single RRAM device (cell, see Fig. 1d), we only analyze single-cell faults. The possible single-cell static faults are those listed in Table III; the dynamic fault space can be constructed by following the definitions in Section III-B.

We start the fault analysis by analyzing static faults. Table V lists the static faults that were sensitized both with the DAT approach as well as the conventional (conv.) approach for all I form , R pd , and R sd . Fig. 9 summarizes the unique faults that are sensitized by both approaches and their overlapping faults. The figure clearly shows the difference between the approaches. The unique DAT faults (6 out of 8 of the realistic faults which corresponds to 75%) may lead to test escapes in case tests are used based on the conventional defect model. On top of that, the conventional defect model approach triggers 9 faults which are not realistic, hence leading to a waste of test time. Note that only 2 common faults are observed between both approaches. We continue the fault analysis with two case studies in which we increase the length of S, i.e., we sensitize dynamic faults. Fig. 10 shows the fault class and FP for the strong faults that were observed for varying I form on the same row of their sensitizing operation. The sequences were chosen to illustrate that more strong faults are sensitized with increasing length of S. The longer the sensitizing sequence, the stronger the fault becomes. Note that the faults are still hard-todetect faults (name bold faced in the figure). This can be explained by the fact that lower I form results in increased RRAM device resistance (both R LRS and R HRS ), or even non-forming defects. Due to this increase, the cells are unable to switch in the valid '1' region and instead switch into the 'U' region, while cells that have to switch into the '0' region end up in the 'H' region, as illustrated by the FPs. Note that despite the faults being strong hard-to-detect, they can still be caught easier than weak faults. The figure shows further that the ranges of fault types are interrupted. This is caused by the stochastic behavior of the filament growth and rupture, sometimes bringing the cell in an unpredicted state.

The application of the methodology to traditional resistive defects is shown in Fig. 11 for R sd . Again, strong hardto-detect faults are marked bold faced while easy-to-detect faults in regular font. Due to the space limitations, we omit showing the results for R pd . The figure also shows that the fault coverage increases when the length of S is increased. For example, for a defect size of R = 5.01 kΩ both strong hard-to-detect faults (for the sequence S=1r1w0) as well as strong easy-to-detect faults (for the sequence S=1r1w1w0) can be observed. The first sequence leads to a 2d-WTF1 U strong hard-to-detect fault, while the second sequence enhances the faulty behavior and causes a strong easy-to-detect 3d-WTF1 1 fault. For comparison, Fig. 11 also shows the same sequences that were shown in Fig. 10. A difference can be seen here which is that the resistive defect model is unable to switch to the '0' state with increasing resistance, while the device-aware defect model shows that the device is still switching between the states. This difference is caused by the fact that the series resistor reduces the voltage over the RRAM device, and hence, the switching threshold is never reached.

From the above, it follows that the DAT approach and the conventional approach result in the sensitization of different faults. The device-aware model is always able to show the switching of the cell, while the resistive defect model only shows switching behavior for a limited defect range, thus incorrectly modeling this defect. Therefore, using an inappropriate defect model would lead to low-quality tests that detect non-existing faults and miss existing ones. Further, it follows that the analysis methodology is able to increase the fault coverage by extending the length of S. This extension may have defects turn the fault behaviour from weak hard-todetect faults into strong hard-to-detect ones, and strong hardto-detect faults into strong easy-to-detect ones, thus increasing their detection probability.

D. Test Development

The results from the previous step are used to develop an RRAM test. In the fault modeling step we have observed that faults caused by these defects are related to the RRAM device being in a wrong state (i.e., 'U', 'L' or 'H'), causing hard-todetect faults. Therefore, a DfT scheme is required that focuses on detecting if a cell is in one of these states. Hamdioui et al. in [START_REF] Hamdioui | Testing open defects in memristor-based memories[END_REF] have presented a Short Write Time and Low Write Voltage DfT scheme that can be used to detect if the resistance of a cell is in the 'U' state. Note that modifications to this scheme would allow the detection of cells in the 'L' and 'H' state as well. In contrast, the R sd defect model sensitizes many strong easy-to-detect faults, e.g. IRF1 1 and WTF1 1 , that are not realistic. Although they may be easily detected by the (w1, r1) element in a March test, testing for them would still increase test cost unnecessary. Note that the faults sensitized by the R sd may still be applicable to model resistive open defects.

V. DEVICE-AWARE TEST FOR STT-MRAMS

In this section, we first describe STT-MRAM manufacturing defects with a particular emphasis on pinhole defects. Thereafter, we apply the DAT methodology to pinhole defects.

A. Manufacturing Defects

The STT-MRAM manufacturing process mainly consists of the standard CMOS fabrication steps and the integration of MTJ devices into metal layers. Fig. 12a shows the bottom-up manufacturing flow and Fig. 12b the vertical multi-layer structure of STT-MRAM cells [START_REF] Song | Highly functional and reliable 8Mb STT-MRAM embedded in 28nm logic[END_REF]. Based on the manufacturing phase, STT-MRAM defects can be classified into front-end-ofline (FEOL) and back-end-of-line (BEOL) defects. As MTJs are integrated into metal layers during BEOL processing, BEOL defects can be further categorized into MTJ fabrication defects and metalization defects. Table VI list all potential defects.

Among these defects, pinhole defects in the MgO tunnel barrier are seen as one of the most important defects that may occur in STT-MRAMs [START_REF] Zhao | Failure analysis in magnetic tunnel junction nanopillar with interfacial perpendicular magnetic anisotropy[END_REF][START_REF] Wu | Pinhole defect characterization and fault modeling for STT-MRAM testing[END_REF]. A pinhole defect forms due to unoptimized deposition processes [START_REF] Zhao | Failure analysis in magnetic tunnel junction nanopillar with interfacial perpendicular magnetic anisotropy[END_REF]. This causes the formation of metallic shorts in the MgO tunnel barrier, probably due to diffusion of Boron into the MgO barrier or other metallic impurities [START_REF] Mukherjee | Role of boron diffusion in CoFeB/MgO magnetic tunnel junctions[END_REF]. As a result, it leads to a degradation of both RA and TMR parameters. Moreover, measurement data in [START_REF] Wu | Pinhole defect characterization and fault modeling for STT-MRAM testing[END_REF] also suggests that a small pinhole grows in area over time because of Joule heating and an electric field across the pinhole circumference. Therefore, if even small pinhole defects are not detected during manufacturing tests, they might cause an early breakdown in the field.

B. Pinhole Defect Modeling

For the conventional resistor-based defect modeling approach, a pinhole defect is modeled as a series resistor R sd or a parallel resistor R pd , as it is the case for modeling the forming defect in RRAM. Next, we present how pinhole defects are modeled by the DAT approach in the following three steps.

Physical defect analysis and modeling. With comprehensive theoretical studies and experimental characterizations for pinhole defects in various MTJ devices as well as our above analysis, it is clear that RA and TMR are the two key technology parameters that are significantly impacted by a pinhole defect [START_REF] Zhao | Failure analysis in magnetic tunnel junction nanopillar with interfacial perpendicular magnetic anisotropy[END_REF][START_REF] Wu | Pinhole defect characterization and fault modeling for STT-MRAM testing[END_REF]. Thus, we model the effect of pinholes on these two technology parameters as follows [START_REF] Wu | Electrical modeling of STT-MRAM defects[END_REF].

RA eff ph (A ph ) = A A(1-A ph ) RA df + A•A ph RA bd (6)
TMR eff ph (A ph ) = TMR df • RA eff ph (A ph ) -RA bd RA df -RA bd (7)
where A ph ∈[0, 1] is the normalized pinhole area with respect to the cross-sectional area A of the MTJ device. RA df and TMR df are the defect-free MTJ's RA and TMR parameters (i.e., when A ph =0), respectively. RA bd is the resultant RA after breakdown. Electrical modeling of the defective device. We then integrate Equations (6-7) into our defect-free MTJ compact model which has been calibrated by the measurement data of good devices (presented in [START_REF] Wu | Pinhole defect characterization and fault modeling for STT-MRAM testing[END_REF]). In this way, we convert Paper X.Y INTERNATIONAL TEST CONFERENCE the defect-free MTJ model into a defective MTJ model which is able to predict the electrical impact of pinhole defects on the MTJ device. Furthermore, the pinhole size is tunable by changing the input argument A ph .

Fitting and model optimization. In this step, we perform electrical characterizations for both good MTJ devices and devices with suspected pinhole defects. By fitting to the measured silicon data, we can further optimize our pinholeparameterized MTJ compact model. By stressing a device with a suspected pinhole defect and curve fitting method, we obtained RA bd =0.41 Ω • µm 2 for our devices. The fitting and model optimization results are presented in [START_REF] Wu | Pinhole defect characterization and fault modeling for STT-MRAM testing[END_REF]. It is clear that the simulation results with our proposed defective MTJ model match the measured silicon data in terms of resistance and switching voltage.

C. Fault Modeling

In this section, we apply the proposed fault modeling methodology to pinhole defects. Similar to what we did for RRAM previously, we first performed fault analysis with the DAT approach. Thereafter, we used the conventional approach to do fault analysis and compared both approaches.

Table VII shows that sufficiently large pinholes (A ph >0.62%) make the MTJ device fall into the resistance range of '0' state or even 'L' state; the corresponding fault primitives are listed in the table. As the pinhole gets smaller (A ph ∈[0.08%,0.61%]), it transforms R P into 'L' state and R AP into 'U' state. Depending on the exact MTJ resistance in the AP state, the readout value can be in three cases: 1) '0', 2) random ('?'), and 3) '1'. In Case 1) where R AP is significantly larger than the resistance of the reference cell (i.e., A ph ∈[0.36%,0.61%]), the readout value of the device in AP state is '0'. In Case 2) where R AP is close to the resistance of the reference cell (i.e., A ph ∈[0.033%,0.35%]), the readout value can be random. In other words, the read operation is unstable, and therefore both '0' and '1' are possible readout values. In Case 3) where R AP is much smaller than the resistance of the reference cell (i.e., A ph ∈[0.08%,0.32%]), the readout is '1'. As the pinhole area becomes smaller between 0.05% to 0.07%, the expected '1' state transforms to a 'U' state, while the expected '0' state remains correct. If the pinhole size is smaller than 0.04%, the device behaves normally, leading to no deterministic faults.

In contrast, we also performed fault modeling based on the injection of the R sd and R pd into a defect-free netlist; the simulation results are also shown in Table VII. Comparing the fault modeling results based on the two defect modeling approaches reveals the following.

• The faulty behavior of the memory due to a pinhole defect cannot be caught with the conventional resistor-based defect modeling approach. It is clear in the table that the FPs sensitized by our proposed pinhole defect model are not observed in the simulation results with the conventional approach. This is because the MTJ device is considered as a black box for the conventional approach. Thus, only '0' and '1' states are seen in the simulations. However,

(r0) 3.1 k-∞ Ω IRF0 0 , WTF00, WTF11 (r0)
Parallel resistor

R pd 0-1.1 kΩ IRF1 1 , WTF00, WTF11 (r1) 1.1 k-3.1 kΩ IRF1 1 , WTF11 (
r1) 3.1 k-∞ Ω Fault-free DfT needed
our simulations and measurement data clearly show that pinhole defects can lead the device to states 'U' and 'L'. This means that relying on the traditional approach for fault modeling and test development may result in low quality test solutions, meaning higher number of escapes. • The conventional approach results in some fault primitives which are not applicable to STT-MRAMs (i.e., not found with our approach based on a calibrated model for the pinhole defect). For example, using a series resistor R sd results in IRF0 0 , while using a parallel resistor R pd results in WTF0 0 . This may lead to tests targeting nonexisting faults, meaning a waste of test time.

D. Test Development

Based on our simulation results with the calibrated pinhole defect model, it is clear that the larger the pinhole, the larger its fault effect, and hence the easier it is to detect it. As shown in Table VII, a pinhole defect with a specific range of defect sizes can cause multiple faults. However, any test that is able to detect one of these faults can guarantee the detection of this specific pinhole defect. For example, when the pinhole area A ph is larger than 0.79%, there are eight sensitized fault primitives. Among these FPs, SF1 L (marked with bold font in the table) can simply be detected by a read '1' operation, because they are strong easy-to-detect faults. Thus, (r1) is the detection condition in a March algorithm for a pinhole with A ph >0.79%. The detection conditions for different pinhole sizes are listed in the last column of Table VII.

Combining the last three rows in Table VII, it is clear that any march tests including the element (w1,r1) can guarantee the detection of a pinhole defect with A ph >0.36% as an easy-to-detect fault. However, for smaller pinhole defects, March tests cannot guarantee their detection, because the defect causes hard-to-detect faults. As small pinhole defects grow in area over time due to the accumulated Joule heating, they would cause an early breakdown in the field if not detected during manufacturing tests [START_REF] Wu | Pinhole defect characterization and fault modeling for STT-MRAM testing[END_REF]. This calls for DfT designs or stress tests dedicated to detecting a tiny pinhole defect. One possible solution is to subject the STT-MRAM to a hammering write '1' operation sequence with elevated voltage or prolonged pulse width to deliberately speedup Paper X.Y INTERNATIONAL TEST CONFERENCE the growth of pinhole defects, thereby causing easy-to-detect faults. However, this approach is prohibitively expensive for high-volume testing. In addition, the amplitude and duration of the hammering write pulse need to be carefully tuned to avoid any inadvertent destruction of good devices while maintaining an acceptable test effectiveness and efficiency.

VI. DISCUSSION AND CONCLUSION

In this paper we have presented the device-aware test approach which consists of three steps: defect modeling, fault modeling, and test development. In contrast to conventional based resistive testing, DAT leads to accurate fault models and thereby enables high-quality (towards DPPB-level) test. Based on the observations, we conclude the following.

Test Escape Reduction and Quality Improvement: As we demonstrated for both RRAM and STT-MRAM previously, our proposed DAT approach results in more accurate fault models which reflect the physical defects. Many faults sensitized using our approach are unique and not observed by the conventional resistor-based defect modeling approach. Hence, our approach clearly reduces the number of test escapes and increases the test quality.

Efficient Yield Learning: Modeling the defects accurately and creating a fault dictionary for them may speed up the yield learning process significantly. As each defect can be modeled separately using device-aware testing, instead of using resistive defect models for all defects, unique fault signatures can be created for each defect. This improves the yield learning curve, as the defects can be more accurately diagnosed based on their fault signatures.

Test Time Optimization: Nowadays, companies are spending a lot of time on functional test (or system test) to compensate for the fault coverage due to the limitations of traditional fault modeling and testing. The DAT approach allows for the development of appropriate and efficient structural tests, which can be applied at manufacturing stage; hence, significantly reducing the expensive test time spend on board testing.

General Applicability: DAT is not limited to emerging memories; the approach can also be applied in the test generation for other circuits, e.g. SRAMs and logic, as well as for other kinds of devices, such as FinFETs and PCM devices. Future work should focus on applying the DAT approach there as well.

Fig. 1 :

 1 Fig. 1: RRAM device technology.

Fig. 2 :

 2 Fig. 2: (a) Simplified MTJ device organization, (b) 1T-1MTJ cell.

Fig. 4 :

 4 Fig.4: Measured resistance distribution of R P and R AP for 60 nm MTJ devices, suggesting the existence of states 'L', '0', 'U', '1', and 'H'.

Fig. 5 :

 5 Fig. 5: Fault classification.

Fig. 6 :

 6 Fig. 6: Fault analysis methodology.

Fig. 9 :

 9 Fig. 9: Static fault sensitization overlap.

Fig. 10 :FPFig. 11 :

 1011 Fig. 10: RRAM forming defect faults.

  Processing flow. (b) Cross-section TEM[START_REF] Song | Highly functional and reliable 8Mb STT-MRAM embedded in 28nm logic[END_REF].

Fig. 12 :

 12 Fig. 12: General manufacturing process of STT-MRAM.

TABLE I :

 I RRAM key parameters.

	Technology Parameters	Electrical Parameters
	tox	Oxide thickness	Vreset	Reset threshold
	lCF	CF length	Vset	Set threshold
	lgap	Gap length	RHRS	Reset resistance
	φT	CF top width	RLRS	Set resistance
	φB	CF bottom width	tH→L	HRS to LRS switching delay
			tL→H	LRS to HRS switching delay

TABLE II :

 II STT-MRAM key parameters.

	Technology Parameters	Electrical Parameters
	RA	Resistance-area product	R P	Resistance in P state
	TMR	Tunneling magneto-resistance ratio	R AP	Resistance in AP state
	H k	Magnetic anisotropy field of the FL	Ic	Critical switching current
	Ms	Saturation magnetization of the FL	tw	Average switching time
	φ	potential barrier height of the TB
		Technology parameters	Physical defect
		(e.g., length, width, density)	characteristics
		1. Physical defect modeling
		Effective technology parameters
		Electrical equations
		(e.g., I ON , I OFF , R eq )
		2. Electrical modeling of defective device	Defect measurement
		Defective Device Model	data
		3. Fitting and model optimization
		Optimized Defective
		Device Model

  S denotes the operation sequence that sensitizes the fault. A sequence takes the form of S=x 0 O 1 x 1 . . . O n x

	inside
	the shaded box represent good devices in accordance
	with the above design specifications. However, there is
	also a large number of devices outside the specification
	due to some defects or extreme process variations. These
	are: (1) extreme low resistance state 'L', 2) extreme
	high resistance state 'H', and 3) undefined state 'U'.
	Note that the definitions of states '0' and '1' for STT-
	MRAM differ from RRAM, where state '0' stands for
	high resistance while '1' for low resistance. Measurement
	data suggesting the existence of the five states for RRAM
	can be found in [25, 28, 29].

• n , where x∈{0, 1} and O∈{r, w}; 0 and 1 denote logical cell values, r and w denote a reading and a writing operation. If n ≤ 1, the fault is static, else dynamic. • F describes the value that is stored in the cell after S is performed. For traditional charge-based memories, e.g., SRAM, there exist only two digital states, i.e., F ∈{0, 1}

[START_REF] Van De Goor | Functional memory faults: a formal notation and a taxonomy[END_REF]

. However, emerging memory technologies like RRAM and STT-MRAM use a resistive storage element; pre-defined resistance ranges determine the logic state of the cell. Due to defects or extreme process variations, the state of such devices can be be outside these ranges, hence the need of the definition of other (faulty) resistance states. Fig.

4

presents the measured resistance distribution of a large number of 60 nm MTJ devices; it shows that F ∈{0, 1, U, L, H}, as will be explained next. Each point in the figure represents a device whose R P is shown on the x-axis and R AP on the y-axis. From a design perspective, the nominal R P is 2 kΩ and the nominal R AP is 5 kΩ; this assures a good read reliability with TMR = 150%. A 3σ of the nominal values is used to define the resistance ranges of the two states 0 and 1. As shown in the figure, the points • R describes the output of a read operation if S is a read operation. R ∈ {0, 1, ?, -} where ? denotes a random read value (e.g., the sensing current is very close to sense amplifier reference current), and '-' denotes that R is not applicable, i.e., when S is a write operation.

Paper X.Y INTERNATIONAL TEST CONFERENCE

TABLE III :

 III Single-cell static fault primitives.

	#	S	F	R	FP notation	Name	#	S	F	R	FP notation	Name
	1	1	0	-	1/0/-	SF10	27	1r1	1	?	1r1/1/?	RRF11
	2	0	1	-	0/1/-	SF01	28	0r0	0	?	0r0/0/?	RRF00
	3	1	U	-	1/U/-	SF1U	29	1r1	0	0	1r1/0/0	IRDF10
	4	0	U	-	0/U/-	SF0U	30	0r0	1	1	0r0/1/1	IRDF01
	5	1	L	-	1/L/-	SF1L	31	1r1	U	0	1r1/U/0	IRDF1U
	6	0	L	-	0/L/-	SF0L	32	0r0	U	1	0r0/U/1	IRDF0U
	7	1	H	-	1/H/-	SF1H	33	1r1	L	0	1r1/L/0	IRDF1L
	8	0	H	-	0/H/-	SF1H	34	0r0	L	1	0r0/L/1	IRDF0L
	9	1w0	1	-	1w0/1/-	WTF11	35	1r1	H	0	1r1/H/0	IRDF1H
	10	0w1	0	-	0w1/0/-	WTF01	36	0r0	H	1	0r0/H/1	IRDF0H
	11	1w0	U	-	1w0/U/-	WTF1U	37	1r1	0	1	1r1/0/1	RDF10
	12	0w1	U	-	0w1/U/-	WTF0U	38	0r0	1	0	0r0/1/0	RDF01
	13	1w0	L	-	1w0/L/-	WTF1L	39	1r1	U	1	1r1/U/1	RDF1U
	14	0w1	L	-	0w1/L/-	WTF0L	40	0r0	U	0	0r0/U/0	RDF0U
	15	1w0	H	-	1w0/H/-	WTF1H	41	1r1	L	1	1r1/L/1	RDF1L
	16	0w1	H	-	0w1/H/-	WTF0H	42	0r0	L	0	0r0/L/0	RDF0L
	17	1w1	0	-	1w1/0/-	WDF10	43	1r1	H	1	1r1/H/1	RDF1H
	18	0w0	1	-	0w0/1/-	WDF01	44	0r0	H	0	0r0/H/0	RDF1H
	19	1w1	U	-	1w1/U/-	WDF1U	45	1r1	0	?	1r1/0/?	RRDF10
	20	0w0	U	-	0w0/U/-	WDF0U	46	0r0	1	?	0r0/1/?	RRDF01
	21	1w1	L	-	1w1/L/-	WDF1L	47	1r1	U	?	1r1/U/?	RRDF1U
	22	0w0	L	-	0w0/L/-	WDF0L	48	0r0	U	?	0r0/U/?	RRDF0U
	23	1w1	H	-	1w1/H/-	WDF1H	49	1r1	L	?	1r1/L/?	RRDF1L
	24	0w0	H	-	0w0/H/-	WDF0H	50	0r0	L	?	0r0/L/?	RRDF0L
	25	1r1	1	0	1r1/1/0	IRF11	51	1r1	H	?	1r1/H/?	RRDF1H
	26	0r0	0	1	0r0/0/1	IRF00	52	0r0	H	?	0r0/H/?	RRDF0H
							Faults
					Strong				Weak
					Easy-to-detect		Hard-to-detect

TABLE IV :

 IV RRAM defect classification[START_REF] Fieback | Testing resistive memories: Where are we and what is missing?[END_REF].

	FEOL		BEOL
	Transistor	Interconnection	Memristor
	patterning proximity	opens	electrode roughness
	line roughness	shorts	polish variations
	polish variations	line roughness	varying defect density
	dielectric variations		dimensional variations
	random dopants		material redeposition
	anneal		overforming
	strain		non-forming
	gate granularity

TABLE VI :

 VI STT-MRAM defect classification.

	FEOL	BEOL
	Transistor fabrication	MTJ fabrication	Metalization
	Material impurity	Pinholes in TB	Open vias/contacts
	Crystal imperfection	Extreme thickness variation of TB	Irregular shapes
	Pinholes in gate oxides	MgO/CoFeB interface roughness	Big bubbles
	Shifting of dopants	Atom inter-diffusion	Small particles
		Redepositions on MTJ sidewalls
		Magnetic layer corrosion
		Magnetic coupling

TABLE VII :

 VII Single-cell static fault modeling results.

	Defect Model	Value	Sensitized Fault Primitive	Detection Condition
		0-0.04%	Fault-free
		0.05-0.07%	SF1U, WDF1U, WTF0U, RDF1U
		0.08-0.32%	SF0L, SF1U, WDF0L, WDF1U, WTF0U, WTF1L, RDF0L, RDF1U	DfT needed
	Pinhole area	0.33-0.35%	SF0L, SF1U, WDF0L, WDF1U, WTF0U, WTF1L, RDF0L, RRDF1U
	A ph	0.36-0.61%	SF0L, SF1U, WDF0L, WDF1U, WTF0U, WTF1L, RDF0L, IRDF1U	(r1)
		0.62-0.78%	SF0L, SF1 0 , WDF0L, WDF10, WTF00, WTF1L, RDF0L, IRDF10	(r1)
		>0.79%	SF0L, SF1L, WDF0L, WDF1L, WTF0L, WTF1L, RDF0L, IRDF1L	(r1)
	Series resistor R sd	0-310 Ω 310-3.1 kΩ	Fault-free IRF0 0	DfT needed


Loris Di Cairano
email: l.di.cairano@fz-juelich.de

Riccardo Capelli
email: riccardo.capelli@polito.it

Ghofrane Bel-Hadj-Aissa
email: ghofrane.belhadjaissa@gmail.com

Marco Pettini
email: marco.pettini@cpt.univ-mrs.fr

Topological origin of protein folding transition

Keywords: numbers: 87, 15, -v; 02, 40, -k

In this paper, a geometrical and thermodynamical analysis of the global properties of the potential energy landscape of a minimalistic model of a polypeptide is presented. The global geometry of the potential energy landscape is supposed to contain relevant information about the properties of a given sequence of amino acids, that is, to discriminate between a random heteropolymer and a protein. By considering the SH3 and PYP protein-sequences and their randomized versions it turns out that in addition to the standard signatures of the folding transition -discriminating between protein sequences of amino acids and random heteropolymer sequences -also peculiar geometric signatures of the equipotential hypersurfaces in configuration space can discriminate between proteins and random heteropolymers. Interestingly, these geometric signatures are the "shadows" of deeper topological changes that take place in correspondence with the protein folding transition.

The protein folding transition takes place in systems with a small number of degrees of freedom (very far from the Avogadro number) and in the absence of a symmetry-breaking phenomenon.

Nevertheless, seen from the deepest level of topology changes of equipotential submanifolds of phase space, the protein folding transition fully qualifies as a phase transition.

I. INTRODUCTION

The study of the Hamiltonian dynamical counterpart of phase transitions combined with the geometrization of Hamiltonian dynamics (where the natural motions are identified with geodesics of suitable Riemannian manifolds) has led to find that at the roots of the phase transitions phenomena there are some peculiar changes of the topology of certain submanifolds of phase space. More precisely, the relevant mathematical objects [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF] are the potential level sets Σ V N v := {V N (q 1 , . . . , q N ) = v ∈ R} in configuration space, and, equivalently, the balls

{M V N v = V -1 N ((-∞, v])} v∈R bounded by the Σ V N v .
Both geometry and topology of these objects can affect microscopic dynamics and macroscopic thermodynamics of the modelled physical system. In fact, when the ball M V N v=E = {(q 1 , . . . , q N ) ∈ R|V N (q 1 , . . . , q N ) < E} is endowed with the metric tensor g J = 2[E -V (q)]dq i ⊗ dq k then its geodesics are the natural motions given by qi = -∇ i V (q), and the geometry of the manifold (M V N E , g J ) determines the properties of order and chaos of the microscopic dynamics [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Casetti | Riemannian theory of Hamiltonian chaos and Lyapunov exponents[END_REF].

On the other hand, a relationship also exists between macroscopic thermodynamics and the topology of the same objects, M V N v , which is expressed by [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF] S N (v) = (k B /N ) log

M V N v d N q = k B N log   vol[M V N v \ N (v) i=1 Γ(x (i) c )] + N i=0 w i µ i (M V N v ) + r(N, v)   , (1)
where S is the configurational entropy, v is the potential energy, and the µ i (M V N v ) are the Morse indexes (in one-to-one correspondence with topology) of the manifolds M V N v=E ; in square brackets: the first term is the result of the excision of certain neighborhoods of the critical points of the interaction potential from M V N v ; the second term is a weighed sum of the Morse indexes, and the third term is a smooth function of N and v. On the basis of Eq.( 1) one can infer that major topology changes with v of the submanifolds M V N v , associated with sharp changes of the potential energy pattern of at least some of the µ i (M V N v ), can affect the v-dependence of the entropy S N (v) and thus of its derivatives. Therefore, at least for a broad class of physical systems, it has been hypothesized that phase transitions stem from a suitable change of the topology of the potential level sets Σ V N v and, equivalently, of the manifolds M V N v , when v, playing the role of the control parameter, takes a critical value v c . This hypothesis is at the ground of a theoretical framework composed of exactly solvable models [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF][START_REF] Casetti | Geometric approach to Hamiltonian dynamics and statistical mechanics[END_REF] and two theorems [START_REF] Franzosi | Theorem on the origin of Phase Transitions[END_REF][START_REF] Franzosi | Topology and Phase Transitions I. Preliminary results[END_REF][START_REF] Franzosi | Topology and Phase Transitions II. Theorem on a necessary relation[END_REF] stating that an equilibrium phase transitions are necessarily induced by suitable topological transitions in configuration space. Apart from its purely theoretical meaning, this topological approach has been proved interesting to investigate the origin of phase transitions in the absence of a symmetry breaking mechanism, thus in the absence of an order parameter, as is the case of a gauge model [START_REF] Pettini | On the origin of phase transitions in the absence of symmetry-breaking[END_REF] and of the Kosterlitz-Thouless transition in the 2D-XY model [START_REF] Bel-Hadj-Aissa | Geometrical and topological study of the Kosterlitz-Thouless phase transition in the XY model in two dimensions[END_REF].

In the present work we aim at applying the topological approach to the phase transition occurring in systems with a constitutively small number of degrees of freedom, that is, much smaller than the Avogadro number. In fact, the physical phenomenon of phase transitions is observed also in nanoscopic and mesoscopic systems, that is, also at very small numbers of degrees of freedom; this circumstance is at odds with the thermodynamic limit dogma stemming from the Yang-Lee theory [START_REF] Yang | Statistical theory of equations of state and phase transitions I. Theory of condensation[END_REF]. The transition phenomenon tackled in what follows is the protein folding transition.

Protein folding is a very important and challenging open question in molecular biology.

In fact, it is well-known that the sequence of amino acids uniquely determines the native state (i.e., the compact configuration the protein assumes in physiological conditions) and understanding how the information contained in the sequence is translated into the threedimensional native structure is the core of the protein folding problem. All the naturally selected proteins generally fold to a uniquely determined native state, but a generic polypeptide does not, and is considered a random heteropolymer.

Following the line of [START_REF] Lorenzo | Curvature of the energy landscape and folding of model proteins[END_REF][START_REF] Lorenzo | Geometry of the energy landscape and folding transition in a simple model of a protein[END_REF], instead of linking the folding properties to the energy landscape by locating the minima of the landscape and the saddles joining them, or by undertaking the folding funnel approach [START_REF] Onuchic | Toward an outline of the topography of a realistic protein-folding funnel[END_REF], we focus on global property of the energy landscape which can be easily numerically computed through time averages along dynamical trajectories.

II. DEFINITION OF THE MODEL AND MD CALCULATIONS

For both the proteins studied in this system (SH3 and PYP) we generated a C α -based Gōmodel [START_REF] Clementi | Topological and energetic factors: what determines the structural details of the transition state ensemble and "en-route" intermediates for protein folding? An investigation for small globular proteins[END_REF] via the SMOG2 [START_REF] Jeffrey K Noel | SMOG 2: a versatile software package for generating structure based models[END_REF] implementation, starting from the experimental structures obtained from the Protein Data Bank (1FMK [START_REF] Xu | Three-dimensional structure of the tyrosine kinase c-Src[END_REF] for SH3 and 3PHY [START_REF] Düx | Solution structure and backbone dynamics of the photoactive yellow protein[END_REF] for PYP). In this model, only the C α atom of every amino acid is considered and the potential is defined as:

U (Γ, Γ 0 ) = bonds K r (r -r 0 ) 2 + angles K θ (θ -θ 0 ) 2 + dihedrals K (n) ϕ (1 + cos(n(ϕ -ϕ 0 )))+ + i<j-3 ε native ij 5 σ ij r ij 12 -6 σ ij r ij 10 + i<j ε n-nat σ nn r ij 12 (2)
where Γ 0 is the initial experimental structure, and Γ is the current system conformation;

similarly, r 0 , θ 0 , and ϕ 0 are the reference values for all the bonds, angle and dihedrals in the model, while r, θ, and ϕ are their value in the conformation Γ. In our implementation the dihedral potential is a sum of 2 terms for every 4 adjacent C α atoms, with periods n = 1 and n = 3. The force constants for bonded interactions in our implementation are K r = 200ε/ Å2 , K θ = 40ε/rad 2 , K ϕ = ε, and ε = 1 kJ/mol. In non-bonded interaction, native contacts are defined as all the C α pairs that have a mutual distance smaller than a threshold (here defined as 10 Å) in the reference configuration Γ 0 , and a distance along the chain of 3 amino acids. All the pairs that do not satisfy these conditions are considered as non-native contacts and their interaction is given only by a repulsive term (last term in Eq.2). σ ij is chosen so that the minimum of the potential is at the distance r ij measured in the reference conformation Γ 0 , while σ nn = 4 Å. Energy terms for non-bonded interaction are ε native ij = ε and ε n-nat = ε.

To compare this protein-like model with a polymer model that does not have a welldefined folding minimum, we generated 2 random heteropolymer models starting from the initial Gō models. We removed from the original potential almost all the bonded interaction (keeping only the bonds between the residues), and we scrambled the non-bonded interaction matrices, namely

U RMD (Γ, Γ 0 ) = bonds K r (r -r 0 ) 2 + i<j-3 εnative ij 5 σ ij r ij 12 -6 σ ij r ij 10 + i<j ε n-nat σ nn r ij 12 (3)
where εnative ij is the scrambled interaction matrix.

We named the 2 systems obtained from the initial SH3 and PYP models RMDa and RMDb, respectively.

All the molecular dynamics simulations were then performed using GROMACS [START_REF] James | GROMACS: High performance molecular simulations through multilevel parallelism from laptops to supercomputers[END_REF] version 2019.6 (compiled in double precision), with a Langevin integrator, with γ = 1 ps -1 , and a time step of 0.5 fs. We initially performed a short equilibration run (10 ns) to relax and thermalize the structure at the target temperature. After this initial equilibration, we performed a 100 ns-long simulation with the same parameters. To exhaustively explore the folding curve, we performed a large number of simulations at different temperatures (note that in a Gō model energy units, and consequently temperature units, are arbitrary), namely:

• For SH3 we performed 1 simulation every 0.25 K between 135 and 161 K; every 1 K between 75 K and 135 K and from 161 to 200 K; and every 2 K from 200 K to 250 K for a total of 229 simulations.

• For PYP we performed 1 simulation every 0.25 K between 145 and 160 K; every 1 K between 75 K and 145 K and from 160 to 200 K; and every 2 K from 200 K to 250 K for a total of 196 simulations.

• For the 2 random energy models, we performed 1 simulation every 5 K from 75 K to 250 K, for a total of 36 simulations.

From these production runs we computed the gyration radius using PLUMED 2.5 [START_REF] Bonomi | Promoting transparency and reproducibility in enhanced molecular simulations[END_REF][START_REF] Gareth A Tribello | PLUMED 2: New feathers for an old bird[END_REF],

and all the other observables needed using the GROMACS suite. From the potential energies at different temperatures we computed the system heat capacity (C v ) with a multiple histogram method [START_REF] Tiana | Equilibrium properties of realistic random heteropolymers and their relevance for globular and naturally unfolded proteins[END_REF].

III. OBSERVABLES AND METHODOLOGY

We sample the value of the characteristic observables along the dynamical trajectories at fixed temperatures of the minimalistic model defined in [START_REF] Das | Balancing energy and entropy: A minimalist model for the characterization of protein folding landscapes[END_REF]. Minimalistic models are those where the polymer is described at a coarse-grained level as a chain of N beads, where N is the number of amino acids; no explicit water molecules are considered and the solvent is taken into account only by means of effective interactions among the monomers.

The thermodynamic and geometrical observables are evaluated along the trajectories run at fixed temperatures. Indicating with ... the time average along the trajectories, we analyze: (i) the radius of gyration R gyr in function of the temperature T ; (ii) the specific

C V = E 2 tot -E tot 2 KB T 2 , ( 4
)
in function of the temperature, where E tot is the total energy and KB is the normalized Boltzmann constant defined as KB = N N A K B (being N A the Avogadro constant and K B the Boltzmann constant); (iii) the dimensionless variance of the Ricci-scalar curvature R(q)

as a function of the temperature

σ R = ( R 2 -R 2 )/N R /N , (5)
where

R(q) = Lap V ∇V 2 - T r[(Hess V ) 2 ] ∇V 2 + 2 Hess V ∇V 2 ∇V 4 -2 ∇V, Hess V ∇V ∇V 4 . ( 6
)
(iv) the temperature T in function of the canonical ensemble energy defines as

= KB T /2 + V /N , (7)
where we recall that V is the total potential filed; (v) the entropy of the system evaluated using

S = 2 KB T ( ) d , (8)
The units are the standard GROMACS ones, i.e., outlining similar results and we here report only two of them for the sake of simplicity. The randomization is implemented using the SH3 coarse grained potential described in [START_REF] Das | Balancing energy and entropy: A minimalist model for the characterization of protein folding landscapes[END_REF] and randomly permuting the parameters involved in the model: this way, we can get a sort of random heteropolymer starting from the good folding sequence of SH3.

The simulation are performed using the GROMACS software [22][23][24][25][26][START_REF] Páll | Proc. of EASC 2015 LNCS[END_REF] and each trajectory is run until 1µs with 0.01ps temporal steps. Averages and fluctuations are evaluated over 1000 outputs for each fixed temperature simulation. The run temperatures are taken, after some tests, in the folding range with an interval of 5K between each trajectory.

IV. GEOMETRICAL SIGNATURES OF TOPOLOGICAL CHANGES

Let us associate to the protein a configuration space, M , which coincides with the Euclidean space R 3N where N is the number of the system particles. Thus, by setting n = 3N , we have M = R 3N and, therefore, M can be equipped with the Euclidean metric tensor, g Γ .

More precisely, by introducing an orthonormal basis

{∂ q i } n i=1 ⊂ M whose dual basis {dq i } n i=1
is defined by the relation dq i (∂ q j ) = δ i j , the Euclidean metric tensor is:

g M := δ ij dq i ⊗ dq j . ( 9
)
Given a regular Hamiltonian function V : M -→ R, one defines the potential level sets to be the following collection of sets:

Σ V := q ∈ M | V (q) = V . (10)
Therefore, the metric tensor on M takes the form:

g M = d V ⊗ d V + (g Σ V ) ij dy i dy j (11)
so that M can be reordered as follows:

M = V Σ V , (12)
where the symbol is meant as the formal union on all possible level sets and {y 1 , . . . , y

3N -1 } Σ V
is the frame of coordinates on the potential level set Σ V .

In order to study the geometry of the level set Σ V , we introduce the unit normal vector to

Σ V : ν := ∇ g M V ∇ g M V g M (13)
where ∇ g M := (∂ q 1 , . . . , ∂ q n ) and • g M are, respectively, the gradient operator and the norm with respect to the metric tensor given in Eq. ( 9) and, from now on, we will omit to specify the metric unless necessary.

Thus, the curvature properties of the level sets can be analyzed through the introduction of the Weingarten operator [START_REF] Bel-Hadj-Aissa | Geometrical and topological study of the Kosterlitz-Thouless phase transition in the XY model in two dimensions[END_REF]:

W ν (X) := -∇ X ν, ∀ X ∈ T x Σ V ( 14
)
where T x Σ V is the tangent space to the level set Σ V on the point x ∈ Γ.

The geometric observables to which we are interested in since they can be directly connected to the topology properties of Σ V are the dispersion of the principal curvatures σ(k i ) 2 and the scalar curvature of the potential level sets.

The dispersion of principal curvature σ(k i ) 2 is defined by [START_REF] Bel-Hadj-Aissa | Geometrical and topological study of the Kosterlitz-Thouless phase transition in the XY model in two dimensions[END_REF]:

σ(k i ) 2 := T r[W 2 ν ] n -1 - (T r[W ν ]) 2 (n -1) 2 (15)
whereas the scalar curvature of Σ V is [START_REF] Zhou | A simple formula for scalar curvature of level sets in Euclidean spaces[END_REF]:

R Σ V := T r[W ν ] 2 -T r[W 2 ν ] (16)
In order to have an expression for such observables we need to explicitly compute the traces of the Weingarten operator and of its squared.

For what concerned the trace, we have:

T r[W ν ] = Lap V ∇V - ∇V, Hess V ∇V ∇V 3 (17)
where Lap V and Hess V are, respectively, the Laplacian and the Hessian of the potential function V with respect to g M . We note that we treat the Hessian as a linear application

Hess V : R n -→ R n .
The trace of the squared of the Weingarten operator is [START_REF] Bel-Hadj-Aissa | Geometrical and topological study of the Kosterlitz-Thouless phase transition in the XY model in two dimensions[END_REF]:

T r[W 2 ν ] = T r[(Hess V ) 2 ] ∇V 2 + ∇V, Hess V ∇V 2 ∇V 6 -2 Hess V ∇V 2 ∇V 4 (18)
Thus, by means of the Pinkall's theorem [START_REF] Pinkall | Inequalities of Willmore type for submanifolds[END_REF], we can connect the dispersion of principal curvatures to the topology [START_REF] Bel-Hadj-Aissa | Geometrical and topological study of the Kosterlitz-Thouless phase transition in the XY model in two dimensions[END_REF]:

σ 2 (k i ) µ := Σ V σ 2 (k i ) dµ Σ V dµ = Σ V (σ 2 (k i )) n/2 dη 2/n -r(Σ V ) = V ol(S n ) n i=1 i n -i n/2-i b i (Σ V ) 2/n -r(Σ V ) ( 19
)
where dη is the normalized Riemannian measure on the level set Σ V defined by

dη := dµ Σ V dµ , (20)
and V ol(S n ) is the volume of the unit n-sphere, r(Σ V ) is a small correction and b i (Σ V ) are the Betti numbers.

A further information is given by the scalar curvature of the potential level sets. In fact, the scalar curvature is the sum of all the sectional curvatures and, therefore, the fluctuations of the scalar curvature can be related to the fluctuations of scalar curvature and one can exploit Overholt theorem [START_REF] Overholt | Fluctuation of sectional curvature for closed hypersurfaces[END_REF]:

∆(sectional) > vol(S N 1 ) N k=0 b k (Σ V ) 2 vol(Σ V ) 2/N (21)
The variance of the scalar curvature R(Σ V ) is

∆ 2 (scal) = R 2 (Σ V ) -R(Σ V ) 2 N (N -1) ∆(sectional) (22)
and

R n (Σ V ) = Σv R n (Σ V ) dµ ∇V Σv dµ ∇V , (23)

V. AVERAGES OF GEOMETRIC OBSERVABLES

We now discuss how we extrapolated information about the topology associated to the potential landscape of the studied systems from the MD data.

Hence, our aim is that to understand how the phase transition, i.e., the protein folding, is guided by a change of topology of the potential level sets. Therefore, we have to connect the topological properties, namely, average of observables with respect to the geometric measure, with thermodynamic properties, namely, average of observable with respect to a statistical ensemble.

Let us start, then, noticing that the MD simulation have been performed in the canonical ensemble, this implies that the all the thermodynamics average has to be done with respect to the canonical partition function. Thus, we introduce the configurational canonical partition function:

Z(n, T ) := M e -V (q)/T dq. ( 24
)
where the Boltzmann constant K B has been equated to one.

The foliation introduced in Eq. ( 12) allows to rewrite the integral above as follows:

Z(n, T ) = n ∞ 0 e -nv/T Σv dµ ∇V dv, (25)
where dµ is the induced metric on the potential level set Σ v, V = nv, i.e., v is the value of the potential per degrees of freedom.

Thus, the canonical average, • C of a generic (geometric) observable, A : M -→ R is

A C (n, T ) := 1 Z(n, T ) ∞ 0 n e -nv/T Σv A(Σ v) ∇V dµ Σv dv, (26)
The MD simulation that we performed allows to sample any observable along trajectories, therefore, the canonical average can be equivalently measured along the dynamics [START_REF] Bel-Hadj-Aissa | Geometrical Aspects in the Analysis of Microcanonical Phase-Transitions[END_REF]:

A C (n, T ) = 1 t lim t→∞ t 0 A(τ ) dτ. (27)
Let us now go back to the the Pinkall theorem ( 19) that we recall below:

σ 2 (k i ) geo := Σv σ 2 (k i ) dµ Σv dµ (28)
In this case, we should average the dispersion of the principal curvature with respect to a geometric measure but, because of the canonical average, we cannot achieve this expression at a finite number of degrees of freedom. Nevertheless, it is possible to get as closer as possible proceeding as follows.

Let us denote by Λ 2 the dispersion of the principal curvature obtained evaluating Eq. ( 15)

along the trajectory, namely, the data obtained from the right-hand side of Eq. ( 27). Theoretically, they are associated to the canonical average, thus, the best approximation of the numerator in Eq. ( 28) is given by:

∇V Λ 2 C = 1 Z(n, T ) ∞ 0 n e -nv/T Σv Λ 2 dµ dv, (29)
whereas for the denominator, we have:

∇V C = 1 Z(n, T ) ∞ 0 n e -nv/T Σv dµ dv, (30)
Now, by dividing Eq. ( 29) by ( 30), we get:

∇V Λ 2 C ∇V C = ∞ 0 n e -nv/T Σv Λ 2 dµ dv ∞ 0 n e -nv/T Σv dµ dv , (31)
At this step, it is worth noting that, for large values of n, the canonical measure narrows around the potential level set Σ v(T ) , where v(T ) is the average potential function per degree of freedom and so the largest contribution to the canonical partition function is given by Σ v(T ) which is nothing but the equivalence of ensembles.

Hence, this means that, heuristically, in the thermodynamic limit, the partition function reduces to

Z(n, T ) ≈ n e -nv(T )/T Σ v(T ) dµ Σ v(T ) ∇V , (32)
and the average in Eq. ( 31) reads:

∇V Λ 2 C ∇V C n→∞ -→ Σ v(T ) Λ 2 dµ Σ v(T ) Σ v(T ) dµ Σ v(T ) ≈ σ 2 (k i ) geo , (33)
By looking at Eq. ( 31), we conclude that, at relatively low number of degrees of freedom, there exists a collection of potential level sets which contribute to the canonical measure and the exponential function e -nv/T weights all these level sets. The resulting effect is a For what concerned the measure of the scalar curvature in Eq. ( 23), this can be directly achieved by applying the definition (26), that is:

dispersion
R C (n, T ) = ∞ 0 n e -nv/T Σv R ∇V dµ dv ∞ 0 n e -nv/T Σv dµ ∇V dv , ( 34
)
and for a large number of particles, we have:

R C (n, T ) n 1 -→ Σv R ∇V dµ Σv dµ ∇V ≈ R (35)
For such an observable the dispersion of the representative points does not appear and, as anticipated above, this is due to the fact that we do not divide the average of the scalar curvature by ∇V C .

VI. RESULTS

In Figure 2 the radius of gyration is reported for the different sequences of the SH3 and PYP proteins, respectively. It is evident that only the sequences of the good folders SH3 and PYP exhibit the bifurcation pattern typical of the folding transition. In Figure 3 the specific heat and the caloric curve are reported for the SH3 protein and display the typical patterns of a phase transition. In particular the inflection point of the caloric curve is typical of a second order phase transition [START_REF] Bachmann | Thermodynamics and Statistical Mechanics of Macromolecular Systems[END_REF][START_REF] Qi | Classification of Phase Transitions by Microcanonical Inflection-Point Analysis[END_REF]. In Figure 4 the specific heat and the caloric curve are reported for the PYP protein and also in this case display the typical patterns of a phase transition. However, the pattern of the caloric curve -in concordance with the sharp drop of the gyration radius shown by Figure 2 -could be compatible with a first order phase transition [START_REF] Bachmann | Thermodynamics and Statistical Mechanics of Macromolecular Systems[END_REF][START_REF] Qi | Classification of Phase Transitions by Microcanonical Inflection-Point Analysis[END_REF]. Remarkably, the thermodynamic signatures of a phase transition, independently of its order, are lost in the case of the randomized sequences of amino acids as shown in the same figures.

In Figure 5 the total scalar curvature and the total variance of the scalar curvature of the equipotential level sets in configuration space are reported as functions of the temperature, normalized to the folding transition temperature, for the SH3 protein. Both quantities show a kink in correspondence to the folding transition which disappears in the randomized sequence. The same phenomenology is shown in Figure 6 for the PYP protein.

Finally, in Figures 7 and8 the dispersions of principal curvatures of the equipotential level sets in configuration space are reported for the SH3 and PYP proteins, respectively, and for the randomized sequences. This quantity displays well evident peculiar patterns when plotted as a function of the potential energy value per degree of freedom. These patterns are less clear when plotted against temperature, even though the presence of cusps can be guessed by means of different polynomial fittings of the points below and above the folding transition temperature, respectively.

In order to understand what do we learn from the patterns of the geometrical quantities reported as functions of the potential energy and of temperature, let us first consider that the shape of the specific heat depends on the shape of the entropy according to the relation

C v = -(∂S/∂E) 2 (∂ 2 S/∂E 2 ) -1 stemming from C v = (∂T (E)/∂E) -1 with T (E) = (∂S/∂E) -1 .
Then, related with the formula reported in Eq.( 1), we also have [START_REF] Pettini | Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics[END_REF] S

N (E) = k B N log Σ N E dµ ∇H (36)
k B N log vol(S N -1 1 ) N i=0 b i (Σ N E ) + r 1 (E) + r 2 (E) ,
where r 1 (E), and r 2 (E) are smooth functions, b i (Σ N E ) are the Betti numbers of the energy level sets, dµ is the measure on the level set, and S N -1 1 stands for a hypersphere of unit radius. From this formula it can be understood that some "abrupt" change in the topology of the energy level sets can affect both the shape of the caloric curve T = T (E) and of the specific heat through the energy variation of S N (E). Now, the scalar curvature is the sum of sectional curvatures so that its variance contains the variance of the sectional curvatures [START_REF] Pettini | On the origin of phase transitions in the absence of symmetry-breaking[END_REF], thus the quantity

∆(sec) > vol(S N 1 ) N k=0 b k (Σ E ) 2 vol(Σ E ) 2/N (37)
in strict analogy with Eqs.( 21) and (22), detects topology changes of the energy level sets in phase space. Therefore, the jumps in the patterns of the total scalar curvature and the total variance of the scalar curvature reported in Figures 5 and6 just probe some kind of "abrupt" change in the topology of the energy level sets. Similarly, and complementary to this, the potential energy patterns of the dispersion of the principal curvatures of the equipotential level sets reported in Figures 7 and8 probe some kind of "abrupt" change in the topology of these submanifolds of configuration space, and thus also of phase space, after Pinkall's theorem relating the dispersion of the principal curvatures of a manifold with a weighted sum of its Betti numbers as given in Eq. [START_REF] Gareth A Tribello | PLUMED 2: New feathers for an old bird[END_REF].

VII. DISCUSSION

By considering a minimalistic model of the SH3 and PYP proteins, besides the standard signatures of the folding transition, the computation of suitable geometric quantities of the equipotential hypersurfaces in configuration space and of the energy hypersurfaces in phase space of these molecules, respectively, allows to probe topological changes of both families of hypersurfaces. The computation of the same geometric quantities for randomized versions of the correct sequences of the SH3 and PYP proteins yielded monotonic patterns as functions of the potential energy density, or of the total energy density, manifestly discriminating between proteins and random heteropolymers. Remarkably, the peculiar geometric signatures found in correspondence with the protein folding transition are the "shadows" of some peculiar and sharp topological change of the mentioned submanifolds of configuration space and of phase space. The protein folding transition takes place in systems with a small number of degrees of freedom (very far from the Avogadro number) and in the absence of a symmetry-breaking phenomenon, however, considered from this topological perspective, the protein folding transition fully qualifies as a phase transition.

The global geometry/topology of both total energy and potential energy landscapes is found to contain relevant information about the properties of a given sequence of amino acids, that is, to discriminate between a random heteropolymer and a protein.

  [T ] =K, [E tot ] = [V ] =kJ/mol, [R gyr ] =nm and [ KB ] =kJ/mol K. We analyze the src-Src homology 3 protein domain (SH3, PDB code ) (see left-hand panel of FIG.1), of 57 amino acids; 2 random sequences of the same 57 amino acids (RDMa,b); and the photoactive yellow protein (PYP) (see right-hand panel of FIG.1) composed by 125 amino acids. We stress how the simulations are run for several random sequences

FIG. 1 .

 1 FIG. 1. Cartoon representation of SH3 (left-hand panel) and PYP (right-hand panel).

  of the points of the representative curve of σ 2 (k i ) which is evident in the SH3 protein since its degrees of freedom are n SH3 = 171. As soon as n increases, such a dispersion shrinks as one can perceive by comparing the plots of σ 2 (k i ) associated to SH3 with that of PYP (n P Y P = 375) in Figs.[?]. A further contribution to such a dispersion can be attributed to the presence of the quantity ∇V C .

FIG. 2 .FIG. 3 .FIG. 4 .FIG. 5 .FIG. 6 .FIG. 7 .FIG. 8 .

 2345678 FIG.2. Plots of the gyration radius for the different sequences. It is evident how only the good folders SH3 and PYP exhibit a temperature dependence typical of the folding transition (upper panels) which are lost for the randomized sequences (lower panels). T f SH3 and T f P Y P identify the folding transition of the SH3 and PYP proteins, respectively.

ACKNOWLEDGEMENTS

The authors are indebted with Lapo Casetti, Mary Anne Rohrdanz, Lorenzo Mazzoni, Lorenzo Boninsegna, Nakia Carlevaro and Cecilia Clementi for inspiring discussions and suggestions. This work has been done within the framework of the project MOLINT which has received funding from the Excellence Initiative of Aix-Marseille University-A*Midex, a French 'Investissements d'Avenir' Programme. Ghofrane Bel Hadj Aissa thanks the support by the QuantERA, ERA-NET Co-fund 731473 (Project Q-CLOCKS), Italy.