What is the Play Level of Dialogical Logic and What is it Good for.

A Pragmatist Perspective on Logical Omniscience.

(Draft 9-12-015) Shahid Rahman Univ. Lille, CNRS, UMR 8163 -STL -Savoirs Textes Langage, F-59000 MESH, Lille, France1 (shahid.rahman@univ-lille3.fr)

Some initial Reflections

When introducing equations in the way we are used to in mathematics there are two main different notions at stake. On the one hand we use equality when introducing both nominal definitions (that establish a relation between linguistic expressions -such a relation yields abbreviations) and real definitions (that establish a relation between objects within a type -this relation yields equivalences in the type). But definitions are neither true nor false, though real definitions can make propositions true. For example, the following equalities are not propositions but certainly constitute an assertion: a + 0 and a are equal objects in the set of numbers Which we can write -using the notation of chapter 2 -as: a + 0 = a : number Since it is an assertion we can formulate the following inference rule: a : number ----------------------a + 0 = a : number Once more, a real definitional equality is a relation between objects, it does not express a proposition. In other words, it is not the dyadic-predicate as found in the usual presentation of first-order logic. However in mathematics, we do have, and even need, an equality predicate. For example when we assert that a + b = b + a. In fact, we can prove it: we prove it by induction. It is proving the proposition that expresses the commutativity of equality. Thus equality expresses here a dyadic predicate.2

• Since we do not have much to add to the subject of nominal definitions, in the following, when we speak of definitional equality we mean those equalities that express a real definition.

It is the Constructive Type Theory of Per Martin-Löf that enabled us to express these different forms of equality in the object language. In recent work by Rahman and Clerbout (2015, it has been claimed that these distinctions can be seen as the result of the different forms that a specific kind of dynamic process can take when (what we call) immanent reasoning is deployed.

Immanent reasoning is the reasoning where the speaker endorses his responsibility of grounding the conclusion by rooting it in the assertion of the relevant premises made by the antagonist. In fact the point of such a kind of reasoning is that the speaker accepts the assertions of the premises brought forward by the antagonist and he has now the duty to develop his reasoning towards the conclusion based on this acceptance. We call this kind of reasoning immanent since there is no other authority that links premises and conclusion beyond the intertwining of acceptance and responsibility during the interaction. Göran Sundholm (1997) called such premises epistemic assumptions, since with them we assume that the proposition involved is known, though no demonstration backing the assumption has been (yet) produced. 3 In a recent talk in Paris, Martin-Löf (2015) provided a dialogical interpretation for them:

[…] the speaker is under an obligation, he is undertaking a certain duty when he makes the assertion, whereas the hearer has the right to trust that he can fulfil his obligation. So the speaker has a duty, whereas the hearer has a right, and right I take to be the same as permission. […] When you are giving an account of the notion of immediate inference, the notion of demonstration is not yet at your disposal. So, to say, Assume that J 1 , …, J n have already been demonstrated makes you accusable of trying to explain things in a circle. The solution to this circularity problem, it seems to me now, comes naturally out of this dialogical analysis -once you have seen it, you can go to the normal logical situation and explain things properly there also, but at least I have seen it via the dialogical analysis. The solution is that the premisses here should not be assumed to be known in the qualified sense, that is, to be demonstrated, but we should simply assume that they have been asserted, which is to say that others have taken responsibility for them, and then the question for me is whether I can take responsibility for the conclusion. So, the assumption is merely that they have been asserted, not that they have been demonstrated. That seems to me to be the appropriate definition of epistemic assumption in Sundholm's sense.

These paragraphs, deploy in a deontic language4 one of the main features of the dialogical framework: the proponent is entitled to use the opponent's moves in order to develop the defence of his own thesis. 5 According to this perspective the proponent takes the assertions of the opponent as epistemic assumptions (to put it into Sundholm's happy terminology), and this means that the proponent trusts them only because of its force, just because she claims that she has some grounds for them. 6 More generally, the conceptual links between equality and the formal rule, nowadays called by Marion/Rückert (2015) more aptly the Socratic rule, is one of the many lessons Plato and Aristotle left us concerning the meaning of expressions taking place during an argumentative process 7 .

The main aim of the recent study of Rahman/Clerbout is to show that in logical contexts the Πand Σ-rules of definitional equality can be seen as highlighting the dialogical interaction between entitlements and duties mentioned above. Under this perspective the standard monological presentation of these rules for definitional equality encodes implicitly an underlying process -by the means of which the proponent "copies" some of the opponent's choices -that provides its dialogical and normative roots. Moreover, this can be extended to the dialogical interpretation of the equality-predicate. We are tracing back, in other words, the systematic origins of the dialogical interpretation recently stressed by Göran Sundholm and Per Martin-Löf. In a nutshell

• The Socratic rule underlying immanent reason deploys the equality in action

This journey to the origins also engages us to study the whole process at the level of plays, that is, the stuff which winning-strategies (the dialogical notion of demonstration) are made of. In fact, as discussed further on, the dialogical framework distinguishes the strategy level from the play level.

• While a winning strategy for the proponent can be seen as linked to a CTT-proof with epistemic assumptions, the play-level constitute a level is reducible neither to formal truth (that amounts to the existence of a winning strategy) nor to material truth (that comes close to model-theoretic truth).

If I say "S is P" when I don't even believe it, I am lying: if I say it when I believe it but am not sure of it, I may be misleading but I am not exactly lying. When I say "I know" ,I give others my word: I give others my authority for saying that "S is P".

Sundholm's extension of Austin's remark on acts of assertion to inferences yields the following forceful formulation:

When I say therefore, I give others my authority for asserting the conclusion, given theirs for asserting the premises. 6 The recent terminology in dialogical logic, following a suggestion of Sundholm, uses the denomination "posit" rather than epistemic assumption, but this presumably is only a terminological variant. 7 In fact, Martin-Löf's discussion is a further development of Sundholm's (2013, p. 17) remark on the links between some pragmatist tenets and inferentialism, that emerge from the following insight of J. L Austin (1946, p. 171):

If I say "S is P" when I don't even believe it, I am lying: if I say it when I believe it but am not sure of it, I may be misleading but I am not exactly lying. When I say "I know" ,I give others my word: I give others my authority for saying that "S is P".

Sundholm's extension of Austin's remark on acts of assertion to inferences yields the following forceful formulation: When I say therefore, I give others my authority for asserting the conclusion, given theirs for asserting the premises.

• What characterizes the play-level (of non-material dialogues) are speech-acts of acceptance that lead to games where the proponent, when he wins a play, he might do so because he accepts some specific moves brought forward by the opponent during that play (without asking for the evidence that supports the propositions brought forward) -this leads to some kind of pragmatic-truth, if we wish to speak of truth.

• The strategy level is a level where the proponent wins accepting whatever the player might posit in every play that constitutes that strategy. Zoe McConaughey suggests that one other way to put the difference between the play and the strategic level; is that at the play-level we might have real concrete players, and that the strategic level only considers an arbitrary idealized one. 8 -this also has been suggested in a different formulation (less explicitly) by Keiff/ Rahman (2010). 9 Perhaps one should formulate it in the following way: at the strategy level, the concrete player might be replaced by an arbitrary one.

Let us focus on what we called immanent reasoning, where, as explained, the claims of the opponent are taken to be grounded, without requiring a defence for them.

However, we could also develop a similar kind of analysis for the case of the so-called

• material dialogues: here the claims of the opponent engage her to put under dialogical scrutiny that what supports her claim. As pointed out by McConaughey, once the claim has been grounded, the proponent adheres rather than accepts the claim under consideration. 10 Grounded claims of material dialogues provide the most basic form of real definitional equality, and not only of the definitional equality as displayed in logical contexts. In fact, the notion of winning in a material dialogue can be seen as the dialogical analogue of model-theoretical truth Assume also that the proponent chooses rank 1 P: A∧B →B O: A∧B P: L? P lost the play.

P lost this play, since he made two mistakes, the crucial one is that the has chosen left instead of right, and moreover since he has chosen the rank 2 he cannot repair his first bad choice, by launching a new challenge One can imagine contexts where this might happen (time restrictions for example) Now; how can we explain that? There is no truth value that explains this, nor is it explained by logical truth. Does this mean that P does not master the meaning of the conjunction? From the proof-theoretical point of view of meaning explanations, he has no such a knowledge. But the lost of the play is not incompatible with the fact that he has local knowledge on the logical constants involved; but because of some contextual constraints he cannot fully develop this knowledge in his favour or does not come to realize how to do so. It is not the case that the there is a winning strategy that is not accessible by an epistemic subject, but rather that a concrete player in some concrete situation does not find it.

In fact, the standard way to deal with such situation is to think an impossible world where the negation of logical truths can be true.

But this is the case since the pragmatist play-level of meaning is not present in most of the standard frameworks. Notice that P's losing cannot be understood as establishing that there is a situation where its negation is the case! Deductive closure is reasonable; when we are thinking strategically: when we think in the way of Kant on the formal: what was possible and what was, normatively speaking the best of the possible choices. Or whoever knows A, ought to know whatever can be inferred from A. This is what a winning strategy is about. However, but this does not mean that some concrete player in a given situation will find the winning strategy.

Logical omniscience is about pragmatic

CONCLUSION

The failure of logical omniscience is perfectly compatible with deductive closure if we link the latter with the strategy level and the failure of logical omniscience with the play level

What is the play-level?:

  The play level of immanent reasoning, is a level where the notion of winning is based on acceptances, not adherences. Those acceptances are brought forward, in a given context, by some concrete player, with the role of the proponent against another concrete player, with the role of the opponent. The winning of a play does not lead to logical truth (validity), nor to local-or model-theoretical truth. It is a level only possible in a framework with an underlying pragmatist theory of meaning

	Now here the main points

What is it good for? Play level and the failure of Logical Omniscience without contradictions:

  Assume the thesis A∧B →B8McConaughey (2015) interpretation stems from her dialogical reading of Aristotle; where, according to this reading, while e dialectics deploy the play level the syllogistic, deploy the strategy level. 9 Keiff/Rahman (2010, vol. 2, n° 66, pp. 149 à 178). 10 Zoe McConaughey suggested in a discussion at my office the 13/11/2015 that while acceptance seems to correspond to the speech act linked to epistemic assumptions, adherence corresponds to the speech act linked to material dialogues.

The paper is a draft for first contribution the ANR research program SÊMAINÔ, and is being developed in the context of the axe transversal Argumentation of STL and the research axe ADA of the MESHS.

For a thorough discussion on this issue seeGranström (2011, pp.

30-36, and pp. 63-69).

It is important to recall that in the context of CTT a distinction must be drawn between open assumptions, that involve hypothetical judgements, judgements that are true, that involve categorical judgements, and epistemic assumptions. What distinguishes open assumptions from true judgements is that open judgments contain variables,: we do not know the proof-object that corresponds to the hypothesis. Open assumptions are different from epistemic assumptions, since with the former we express that we do not know the hypothesis to be true, while the latter we express that we take it to be true. Moreover, epistemic assumptions are not part of a judgement. It is a whole judgement that is taken to be true. A hypothetical judgement; can be object of an epistemic assumption. This naturally leads to think of epistemic assumptions as related to the force of a given judgement.

Let us point out that one of the main philosophical assumptions of the constructivist school of Erlangen was precisely the tight interconnection between logic and ethics, see among others: Lorenzen/Schwemmer(1975) andLorenzen (1984).

In fact, Martin-Löf's discussion is a further development ofSundholm's (2013, p. 17) remark on the links between some pragmatist tenets and inferentialism, that emerge from the following insight of J. L Austin(1946, p. 171):


Sylvain Feruglio
email: sylvain.feruglio@upmc.fr

Thierry Courcier

Olivier Tsiakaka

Armine Karami

Annick Alexandre-Gauthier

Olivier Romain

Vincent Aimez

Paul G Charette

Patrick Pittet

Guo-Neng N Lu

P G Charrette

A CMOS Buried Quad p-n Junction Photodetector Model

Keywords: CMOS, modeling, multi-spectral sensor, optoelectronic, photodetector

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

ince the mid-twentieth century and the advent of microelectronics, the photonic devices market has been growing constantly, mainly driven by the developments of embedded systems [START_REF] Lebby | Optoelectronics: Successfully penetrating many new markets[END_REF]. They are now ubiquitous, simplifying our everyday life. However, there are always requirements for the detecting devices and systems to be more compact, to consume less power, to have greater sensitivity, while remaining manufacturable at low cost [START_REF] Zimmermann | Silicon Optoelectronic Integrated Circuits[END_REF].

In the visible and near infrared (NIR) ranges, simple p-n junction silicon photodiodes are extensively employed. The CMOS process has made it possible to increase the number of vertically-stacked junctions, making the detector structure have a spectral discriminating ability. This has opened the way to many applications, such as color image sensors, biochemical analyses, etc. .

Basic CMOS and BiCMOS processes have allowed implementation of buried double junction (BDJ) and buried triple p-n junction (BTJ) device structures [START_REF] Lu | Colour detection using a buried double p-n junction structure implemented in the CMOS process[END_REF][START_REF] Ben Chouikha | Colour detection using buried triple pn junction structure implemented in BiCMOS process[END_REF][START_REF] Lu | A dual-wavelength method using the BDJ detector and its application to iron concentration measurement[END_REF][START_REF] Alexandre | Modeling and Design of Multi Buried Junctions Detectors for Color Systems Development[END_REF][START_REF] Lyon | Eyeing the camera: into the next century[END_REF][START_REF] Pittet | CMOS LlF detection system for capillary analysis[END_REF][START_REF] Gilblom | Real-time color imaging with a CMOS sensor having stacked photodiodes[END_REF][START_REF] Pittet | CMOS absorbance detection system for capillary electrophoresis[END_REF][START_REF] Ross | A buried triple-junction self-reset pixel in a 0.35µm high voltage CMOS process[END_REF][START_REF] Feruglio | Dark Current and Signal-to-Noise Ratio in BDJ Image Sensors[END_REF][START_REF] Feruglio | A Review of CMOS Buried Double Junction (BDJ) Photodetector and its Applications[END_REF][START_REF] Yang | Integrated Colour Detectors in 0.18-µm CMOS Technology[END_REF][START_REF] Liang | Color measurement for RGB white LEDs in solid-state lighting using a BDJ photodetector[END_REF][START_REF] Schidl | LED wavelength determination sensor system with BiCMOS triple junction photodetector[END_REF][17][START_REF] Polzer | Investigation of triple-junction photodetector in 90 nm CMOS technology[END_REF][START_REF] Schidl | Investigation of transient photocurrent response of triple pn junction structure[END_REF][START_REF] Schidl | AC response of 0.35 µm CMOS triple junction photodetector[END_REF][START_REF] Schidl | CMOS chip with multi junction photo detector for sensing biomedical signals[END_REF][START_REF] Zhan | Filterless optical oxygen sensor based on a CMOS buried double junction photodiode[END_REF]. Using an advanced CMOS technology, we can design and integrate a buried quad junction (BQJ) photodetector for specific applications [START_REF] Richard | Integration of hybrid optical filter with buried quad pnjunction photo detector for multi-labeling fluorescence detection applications[END_REF][START_REF] Richard | CMOS buried Quad p-n junction photodetector for multi-wavelength analysis[END_REF][START_REF] Courcier | CMOS BQJ detector chip with integrated charge-amplifiers for fluorescence measurements[END_REF][START_REF] Courcier | BQJ Photodetector Signal Processing[END_REF][START_REF] Feruglio | Opto-electrical Modeling of CMOS Buried Quad Junction Photodetector[END_REF][START_REF] Lu | CMOS buried multi-junction (BMJ) detector for bio-chemical analysis[END_REF].

For system design, we propose in this paper an optoelectrical model of the BQJ photodetector, with the aim of allowing design optimization at device, circuit and system levels, including associated signal processing electronics. The proposed model was built with a physically-based approach, and has been integrated in EDA software for fast and accurate simulation study, based on analytical expressions.

The paper contains 6 sections. Following the introduction, the second section describes the structure of the BQJ photodetector, its operation and discusses the potential applications. The physical modeling of the photocurrent components inside the detector is presented in the third section. Section 4 proposes an equivalent circuit of the device for large-signal analysis. The fifth section is for model validation by comparing characteristics of the BQJ detector, determined from the model with results from TCAD simulations and experimental measurements. Section 6 concludes this paper.

II. BURIED QUAD JUNCTION PHOTODETECTOR

After a detailed presentation of the structure under study, its operation and applications are presented, using a proprietary high voltage CMOS process (HV-CMOS) at Teledyne DALSA Semiconductor Inc. (Bromont, Canada). A picture of the device is shown in Fig. 1.

A. Structure

The device consists of 4 vertically-stacked buried p-n junction photodiodes. Its implementation is illustrated in Fig. 2, with formation of its 4 junctions as follows:

S

Fig. 2 Simplified cross-section of the BQJ photodetector.

J 1 : shallow P + -diffusion / N-base well junction. J 2 : N-base well / P-well junction. J 3 : P-well / deep N-well junction. J 4 : deep N-well / P-epi substrate junction. The detector surface is covered with both Si 3 N 4 and SiO 2 passivation layers. The active photodetection area, A PH , corresponds to the surface of the P + -diffusion layer, located between the contact electrodes (see Fig. 2). The P-substrate is grounded and the 4 outputs correspond to the P + -diffusion, Nbase, P-well and deep N-well contacts, respectively.

B. Operation

All buried p-n junctions (J 1 to J 4 ) are reverse-biased to operate in photoconductive mode. The bias conditions are: [START_REF] Ben Chouikha | Colour detection using buried triple pn junction structure implemented in BiCMOS process[END_REF] and V 0 are the potentials respectively applied to the P + -diffusion, the Nbase well, the P-well, the deep N-well and the P-epi substrate. Thus, there are 4 output currents (I 1 to I 4 in Fig. 2):

V J1 _= V 2 -V 1 > 0 V, V J2 = V 2 -V 3 > 0 V, V J3 = V 4 -V 3 > 0 V and V J4 = V 4 -V 0 > 0 V, where V 1 , V 2 , V 3 , V
I 1 =I J1 (1-a) I 2 =I J1 +I J2 (1-b) I 3 =I J2 +I J3 (1-c) I 4 =I J3 +I J4
(1-d) where I Jθ , with θ = 1, …, 4, are the currents flowing through each junction. Under illumination, each junction current is composed of a photo-generated component (I PHθ ) and a dark component (I DCθ ):

I Jθ =I PHθ +I DCθ (2)
The device operation is based on the optical absorption properties of silicon, where the absorption depth depends on the incident light wavelength and each stacked junction has its own distinct spectral sensitivity. Indeed, shorter wavelengths are absorbed near the silicon surface, while longer ones have deeper penetrations. Consequently, the shallow junction (J 1 ) is sensitive to short wavelengths, as it collects only photogenerated carriers near the silicon surface. In contrast, the deepest junction (J 4 ) is more sensitive to red/NIR light, thanks to its collection of photo-generated carriers in the deeper region. The two other junctions (J 2 and J 3 ) are more sensitive to the middle of the visible spectrum. The 4 spectral responses resemble to bandpass filters, centered on 4 different wavelengths. Fig. 3 exhibits the normalized photocurrents contribution of the BQJ photodetector versus wavelength, λ, defined by:

N θ λ = I PHθ λ ∑ I PHθ λ 4 θ=1
(3)

C. Applications

With buried multiple p-n junction (BMJ) photodetectors, it is possible to quantify at the same instant, different spectral components with a single detector. Moreover, increasing the number of vertically-stacked junctions improves the detector's spectral discriminating ability. It is particularly attractive for multispectral applications, such as color detection, light wavelength characterization and simultaneous spectral analysis of several light sources [START_REF] Lu | Colour detection using a buried double p-n junction structure implemented in the CMOS process[END_REF][START_REF] Ben Chouikha | Colour detection using buried triple pn junction structure implemented in BiCMOS process[END_REF][START_REF] Alexandre | Modeling and Design of Multi Buried Junctions Detectors for Color Systems Development[END_REF][START_REF] Lyon | Eyeing the camera: into the next century[END_REF][START_REF] Gilblom | Real-time color imaging with a CMOS sensor having stacked photodiodes[END_REF][START_REF] Ross | A buried triple-junction self-reset pixel in a 0.35µm high voltage CMOS process[END_REF][START_REF] Feruglio | Dark Current and Signal-to-Noise Ratio in BDJ Image Sensors[END_REF][START_REF] Yang | Integrated Colour Detectors in 0.18-µm CMOS Technology[END_REF][START_REF] Liang | Color measurement for RGB white LEDs in solid-state lighting using a BDJ photodetector[END_REF][START_REF] Schidl | LED wavelength determination sensor system with BiCMOS triple junction photodetector[END_REF][17]. It has been also shown that this kind of photodetector is well adapted to optical data communication of more than 100 Mbits/s [START_REF] Polzer | Investigation of triple-junction photodetector in 90 nm CMOS technology[END_REF][START_REF] Schidl | Investigation of transient photocurrent response of triple pn junction structure[END_REF][START_REF] Schidl | AC response of 0.35 µm CMOS triple junction photodetector[END_REF], where the BMJ photodetectors permit the natural demultiplexing of separate incident light sources.

In biomedical and chemical applications [START_REF] Lu | A dual-wavelength method using the BDJ detector and its application to iron concentration measurement[END_REF][START_REF] Pittet | CMOS LlF detection system for capillary analysis[END_REF][START_REF] Pittet | CMOS absorbance detection system for capillary electrophoresis[END_REF][START_REF] Feruglio | A Review of CMOS Buried Double Junction (BDJ) Photodetector and its Applications[END_REF][START_REF] Schidl | CMOS chip with multi junction photo detector for sensing biomedical signals[END_REF][START_REF] Zhan | Filterless optical oxygen sensor based on a CMOS buried double junction photodiode[END_REF][START_REF] Richard | Integration of hybrid optical filter with buried quad pnjunction photo detector for multi-labeling fluorescence detection applications[END_REF][START_REF] Richard | CMOS buried Quad p-n junction photodetector for multi-wavelength analysis[END_REF][START_REF] Courcier | CMOS BQJ detector chip with integrated charge-amplifiers for fluorescence measurements[END_REF][START_REF] Courcier | BQJ Photodetector Signal Processing[END_REF][START_REF] Feruglio | Opto-electrical Modeling of CMOS Buried Quad Junction Photodetector[END_REF][START_REF] Lu | CMOS buried multi-junction (BMJ) detector for bio-chemical analysis[END_REF][START_REF] Matos Pires | Recent Developments in Optical Detection Technologies in Lab-on-a-Chip Devices for Biosensing Applications[END_REF][START_REF] Pereira | Heterogeneous immunoassays in microfluidic format using fluorescence detection with integrated amorphous silicon photodiodes[END_REF], quantitative spectral analysis is often needed. As illustrated from Fig. 4, knowing the individual spectral response of the junctions, S θ , and the incident light source profile, G(λ), we can monitor the optical properties of a medium under test, H(λ), with: In the case of the BQJ photodetector, 4 different components can be analyzed in the same spatiotemporal conditions. Fig. 5 presents the contribution (in percent) of each output to the total photocurrent. These curves are particularly interesting for rapid diagnostics. For example, in NIR spectroscopy [START_REF] Kamrani | Low-Noise, High-Gain TIA Integrated with CMOS APD for Low-Intensity Light Detection in Near-Infrared Spectroscopy[END_REF] or in 3D image reconstruction [START_REF] Kolar | A Multishutter Time Sensor for Multispectral Imaging in a 3-D Reconstruction Integrated Sensor[END_REF], the instrumentation requires a minimum of 2 different light sources, emitted alternatively. Thanks to this kind of sensor, light emission (and their detection) can be simultaneous. Thus, the measurement of parameters of interest is correlated (done at the same time and in the same spatial, physiological and electrical conditions).

I PHθ = H λ G λ S θ λ dλ (4)
In addition, noise and parasitic signal rejection coming from light source(s) or transmission channel (H(λ)) is improved in comparison to simple photodiodes. Indeed, all spectral responses are taken in the same site and they can be based on the current ratios [START_REF] Feruglio | Dark Current and Signal-to-Noise Ratio in BDJ Image Sensors[END_REF]. For instance, in Fig. 3 and Fig. 5, oscillations due to reflection are suppressed.

Moreover, fewer additional components, such as optical filters or prism(s) to discriminate the incident light, are required for very specific application, such as fluorescence detection [START_REF] Richard | Integration of hybrid optical filter with buried quad pnjunction photo detector for multi-labeling fluorescence detection applications[END_REF]. Thus, additional cost of non-standard production's step is minimized or null.

Finally, for consumer applications such as CMOS APS image sensors, photodetectors require a minimum surface area (less than ten microns, typically) with high fill factors (ideally 100 %) [START_REF] Tournier | A 2.2-µm-Pitch Single-Transistor Charge-Modulation Pixel in a 0.13-µm CMOS Process[END_REF]. However, with this kind of photodetector, by superposition of layers (P and N) with pads, a significant area of the sensor doesn't participate to the detection. Indeed, the counterpart of this photodetector use is it low fill factor (FF) [START_REF] Webster | A silicon photomultiplier with >30% detection efficiency from 450-750nm and 11.6µm pitch NMOS-only pixel with 21.6% fill factor in 130nm CMOS[END_REF]. In Fig. 6, the FF is given for a square BQJ photodetector according to the dimension of its edges. This type of sensor is generally suitable for applications necessitating a rather large collection surface (a few tens or hundreds of microns aside). However, this must be in contrast to the fact that most conventional image sensors use an array of laterally arranged simple photodiodes, covered with different filters (at least 3), for color selectivity. As each photodetector records only one color, the incident light is rebuilt by combining the output signals of neighboring pixels. Thus, as well as the cost of filters, major drawbacks of these sensors are the increased silicon area, reduction of light gathering and color artifact [17].

III. MODELING OF THE BQJ PHOTOCURRENTS

When incident monochromatic light impinges on the active photodetection area of the BQJ photodetector, a part, Φ t , of the initial photon flow, Φ, is transmitted in the device:

Φ t λ = 1 R λ Φ (5)
where R(λ) is the reflection coefficient, precisely defined in [START_REF] Feruglio | A Review of CMOS Buried Double Junction (BDJ) Photodetector and its Applications[END_REF][START_REF] Sze | Physic of Semiconductor Devices[END_REF][START_REF] Harbecke | Coherent and incoherent reflection and transmission of multilayer structures[END_REF][START_REF] Heavens | Optical Properties of Thin Films[END_REF], which is dependent on the wavelength, λ. Light with energy, E(λ), greater than the silicon bandgap, propagates through the silicon. Though electron-hole pairs were photo-generated along the entire path depth-wise through the material, most of the photo-carriers are generated in a region δx 0 , centered at depth x 0 , according to the following generation wavelength-dependent rate equation:

G PH x, λ =Φ t λ α λ e -α λ x (6) where α(λ) is the absorption coefficient, that depends on the wavelength, but also doping concentrations and temperature [START_REF] Schmid | Optical absorption in heavily doped silicon[END_REF][START_REF] Geist | Analytic representation of the silicon absorption coefficient in the indirect transition region[END_REF].

The 4 junctions serve to collect photo-generated electronhole pairs, each resulting in a junction photocurrent. Each photocurrent has 3 different components: a drift current (I PHdrθ ) from the collection of photo-generated carriers in the depletion region and 2 diffusion currents (I PHpθ and I PHnθ ) from both adjacent quasi-neutral zones (QNZ) [START_REF] Alexandre | Modeling and Design of Multi Buried Junctions Detectors for Color Systems Development[END_REF][START_REF] Feruglio | A Review of CMOS Buried Double Junction (BDJ) Photodetector and its Applications[END_REF][START_REF] Sze | Physic of Semiconductor Devices[END_REF] (see Fig. 7):

I PHθ =I PHdrθ +I PHnθ +I PHpθ (7)

A. Drift Components

In the depletion region, the internal electric field imposed by the reverse bias separates the photo-generated carriers by preventing their recombination. As the recombination process is negligible, this leads to a drift current which, for monochromatic illumination, can be simply calculated by integrating the generation rate over the depletion layer: where W θ (where θ from 1 to 4) represent the depletion-layer widths (see Fig. 2). They are dependent on the applied voltages and the doping levels. S PHdrθ denote the drift spectral sensitivities.

I PHdrθ λ =qA PH Φ t λ G PH W θ x,λ dx =A PH S PHdrθ λ Φ λ (8)

B. Diffusion Components

In a QNZ, the newly photo-generated carriers modify the existing electron and hole densities. For the majority carriers in the zone, this variation is negligible compared to the high concentration of dopants. However, for the minority carriers, the contribution of photo-carriers generates a significant concentration gradient. This results in the displacement of charges from their point of generation to the nearest depletion region. Reaching the depletion region, they are driven by the electric field in the region, giving rise to a diffusion component. Therefore, inside the detector structure, the QNZ of the P + -diffusion, P-well and P-epi substrate layers produce 4 photocurrent components due to diffusion of denoted by I PHn1 , I PHn2 , I PHn3 and I PHn4 (see Fig. 7). Similarly, the QNZ of N-base and the deep N-well layers produce 4 photocurrents by diffusion of holes, noted I PHp1 , I PHp2 , I PHp3 and I PHp4 (see Fig. 7). It is noted that the N-base well, P-Well and deep N-well layers are common to 2 junctions, the photogenerated minority carriers are directed to the nearest depletion region. This leads to the production of 2 diffusion currents in opposite directions for each QNZ.

The calculation of these diffusion currents requires the solving of the continuity and Fick's equations. Therefore, in steady state, the following expression is valid for each QNZ:

∆j θ (λ)=B θ1 e -x/L jθ +B θ2 e x/L jθ + Φ t (λ)α(λ)τ jθ 1-α(λ)L jθ 2 e -α(λ)x (9)
where ∆j θ are the variations of minority carrier concentration in each QNZ, with j = n for electrons and p for holes; L jθ are the diffusion lengths of the minority carriers; τ jθ are the corresponding time constants; the real numbers, B 1θ and B 2θ , are defined from the following boundary conditions: (10-e) ∆p 3 x 3n =∆p 4 x 3n =∆p 3 x 4n =∆p 4 x 4n =0 (10-f) Here, S rP+ and S rPsub are the surface recombination velocities on the device's top and bottom, respectively; D nP+ and D nPsub correspond to the diffusion constant of the P + -diffusion and the P-substrate, respectively. Finally, the diffusion photocurrents, at points x', are given by:

S rP+ ∆n 1 0 =D nP+
I PHjθ λ =±qA D jθ ∂∆j θ λ ∂x x ' =A PH S PHjθ λ Φ λ (11)
where the electrons flow is considered positive, while the holes flow is negative; x' corresponds to the boundary depths presented in Fig. 7 (i.e., 0, x jθ and x sub ); S PHjθ denote the diffusion spectral sensitivities.

From equations ( 8) and ( 11), ( 7) can be rewritten as:

I PHθ λ =A PH S PHdrθ λ +S PHnθ λ +S PHpθ λ Φ λ =A PH S PHθ λ Φ λ (12)
and the total spectral responses become:

S θ λ = I PHθ λ P OPT λ = I PHθ λ A PH E λ Φ λ = E λ S PHθ λ (13)
with E(λ) = hc/λ, where h is the Planck's constant, c denotes the speed of light in vacuum, q represents the electron charge and P OPT (λ) is the optical power.

IV. EQUIVALENT CIRCUIT For fast system on chip (SoC) design, simulation tools with reliable models of all components are essential. Based on the previous considerations, associated to earlier works [START_REF] Alexandre | Modeling and Design of Multi Buried Junctions Detectors for Color Systems Development[END_REF][START_REF] Feruglio | Dark Current and Signal-to-Noise Ratio in BDJ Image Sensors[END_REF][START_REF] Feruglio | A Review of CMOS Buried Double Junction (BDJ) Photodetector and its Applications[END_REF][START_REF] Senapati | A SPICE Model for Silicon Photodiode[END_REF][START_REF] Peng | Behavior model of a CMOS process compatible photo-diode[END_REF][START_REF] Perry | Using PSPICE to simulate the photoresponse of ideal CMOS integrated circuit photodiodes[END_REF][START_REF] Ravezzi | A versatile photodiode SPICE model for optical microsystem simulation[END_REF][START_REF] Huiming | Modeling of pinned photodiode for CMOS image sensor[END_REF][START_REF] Bian | A Novel Spice Model of Photodetector for OEIC Design[END_REF], we propose an equivalent circuit of the BQJ photodetector here for large-signal analysis.

Large-signal operation corresponds to the case where the device operates either in steady-state (i.e., DC) or in a transient state. Small-signal and noise models can be deduced from this equivalent circuit [START_REF] Feruglio | Dark Current and Signal-to-Noise Ratio in BDJ Image Sensors[END_REF][START_REF] Feruglio | A Review of CMOS Buried Double Junction (BDJ) Photodetector and its Applications[END_REF] for various computational analyses (AC, noise, sensitivities, etc.). Thus, it is possible to perform accurate and rapid simulation at different levels (component, circuit or system).

The proposed equivalent circuit of the photodetector for large-signal operation is shown in Fig. 8, where the block on the right is the physical equivalent circuit of the BQJ photodetector. In contrast, the block on the left hasn't physical reality. It is an equivalent modeling to take into account the intrinsic opto-electrical response speed of the device. Thus, we have:

with:

τ θ ≈2.2 τ nθ ²+τ drθ ²+τ pθ ² (15)
where the τ nθ are the diffusion times of the photogenerated electrons in the P-type QNZ; the τ pθ are the diffusion times of the photo-generated holes in the Ntype QNZ (the both previously defined by τ jθ ); the τ drθ are the photo-carrier transit times in the depletion layers. The drift component has a faster response than the diffusion one, as the charge collecting process differ. Yet, in function of the nature of the incident light, photo-carriers are generated mostly in a specific zone, which imposes the most part of this time τ θ . In the worst case, they conduct to both rise and fall times of some nanoseconds. Note that there are also the usual purely electrical time constants, associated with impedance connected to the BQJ outputs and the junction capacitances, C Jθ (block on the right in Fig. 8). I PHθ are the photocurrent sources. In Fig. 8, they are current sources controlled by the voltage across C eqθ . Thus, we have in the frequency domain:

I PHθ λ,f = K Vθ 1+j f f cθ -3dB A PH S PHθ (λ)V eqθ (λ) (16)
where K Vθ =1 A/V and S PHθ is defined in [START_REF] Feruglio | Dark Current and Signal-to-Noise Ratio in BDJ Image Sensors[END_REF]. R κ , with κ = 1, …, 5, are the parasitic series resistances of the ohmic contacts and the bulk resistances of the QNZ. They are non-linear elements that depend on the applied bias and reduce the device efficiency. Their values (typically, some ohms) depend mainly on the device layout via both the junctions' area and the number of contacts on each layer. R κ can be estimated before the device realization, either manually or from parasitic extraction with EDA software, based on the technology design kit [START_REF] Baker | CMOS: Circuit Design, Layout, and Simulation[END_REF]. C jθ are the junction capacitances. As the transit capacitances are negligible in reverse-bias polarization, only these capacitances C jθ are taken into account. They are well defined from the following equation:

C Jθ = C J0θ 1+ V Jθ V biθ M θ (17)
Here, C J0θ are the effective zero-bias junction capacitances, taking into account both surface and peripheral contributions; V Jθ are the reverse voltage biases applied to junctions; V biθ correspond to the built-in potentials; M θ are dimensionless technological parameters that depend on the doping profiles of the junction.

Note that, assimilating these physical capacitances to parallel-plate capacitors, it is possible to have a fairly reliable estimate of the associated depletion layer width (W θ ) and its maximum electrical field, from Gauss' law (both parameters being necessary for the calculation of the BQJ currents, both in dark conditions and under illumination). I DCθ are the dark current sources, the ubiquitous parasitic leakage current flowing through a p-n junction. They impose a physical limit on the minimum detectable signal [START_REF] Feruglio | Dark Current and Signal-to-Noise Ratio in BDJ Image Sensors[END_REF]. In the voltage range of interest, the involved physical mechanisms are:

• Shockley-Read-Hall (SRH) generation-recombination (GR) process: present in each space-charge region, this mechanism produces a thermal generation current under reverse-bias polarization. At room temperature, it is generally the dominant contribution for moderately doped junctions with low and moderate reverse bias voltage. In the device structure under study, these dark current components, I DC_SRHθ , are measurable in all junctions (mainly for J 2 to J 4 ). Their contribution is given by: I DC_SRHθ ≈qA Jθ W θ 2τ gθ [START_REF] Polzer | Investigation of triple-junction photodetector in 90 nm CMOS technology[END_REF] where A Jθ represents the effective junction areas; the τ gθ denote the effective GR lifetimes [START_REF] Schroder | Carrier Lifetimes in Silicon[END_REF]. This kind of current is directly proportional to V Jθ at the power M θ (i.e., V Jθ M Jθ ). • Diffusion mechanism: for a low-doped p-n junction, with a small reverse voltage bias and a junction temperature slightly higher than room temperature, the thermally generated minority carriers move by diffusion from both neutral regions of a junction to its depletion layer, where they recombine. They produce a diffusion current (I dθ ), explained by Shockley [START_REF] Sze | Physic of Semiconductor Devices[END_REF]. However, as the device is a P + NPNP structure, a mutual coupling between neighboring junctions exists and an interdependency of diffusion currents flowing through the junctions is expected in particular bias conditions (similarly to the Ebers-Moll model of bipolar transistors). The analytical expression is obtained through equations ( 9) and ( 11), with Φ = 0 photon.cm -2 .s -1 , equations (10-a) and (10-b) and the following new boundary conditions:

∆n 1 x 1p =n 0P+ exp - q KT V J1 -1 (19-a) ∆n 2 x 2p =∆n 3 x 2p =n 0Pwell exp - q KT V J2 -1 (19-b) ∆n 2 x 3p =∆n 3 x 3p =n 0Pwell exp - q KT V J3 -1 (19-c) ∆n 4 x 4p =n 0Psub exp - q KT V J4 -1 (19-d) ∆p 1 x 1n =∆p 2 x 1n =p 0Nbase exp - q KT V J1 -1 (19-e) ∆p 2 x 2n =∆p 3 x 2n =p 0Nbase exp - q KT V J2 -1 (19-f) ∆p 3 x 3n =∆p 4 x 3n =p 0Nwell exp - q KT V J3 -1 (19-g) ∆p 3 x 4n =∆p 4 x 4n =p 0Nwell exp - q KT V J4 -1 (19-h)
where n 0P+ , n 0Pwell , n 0Psub , p 0Nbase , p 0Nwell are the minority carrier concentrations in each QNZ of the structure at equilibrium; K is Boltzmann's constant; T is the absolute temperature. For a reverse bias higher than ~100 mV, this component is independent of V Jθ .

• Tunneling effect: it is a quantum effect observed in highly doped junctions, even with moderate reverse bias voltage. In such conditions, the band-to-band tunneling (BBT) is not negligible. Its contribution is formulated from the conventional Zener tunneling equation: [START_REF] Schidl | AC response of 0.35 µm CMOS triple junction photodetector[END_REF] where C bbtθ and E 0 are constants that are highly dependent on the device material; the E MAXθ are the maximum electrical fields in each depletion zone. Typically, for V Jθ higher than a few volts, this current varies mainly as exp(V Jθ γ ), where γ is a real number. More complete information about dark currents can be found in [12-13, 35, 49-51].

I DC_tθ ≈qV Jθ C bbtθ E MAXθ 3/2 e - E 0 E MAXθ

V. MODEL VALIDATION

We have performed model validation by comparing the device's characteristics computed from our developed model with the ones by both TCAD simulations and measurements. Dark current has been also analyzed in order to identify the dominant physical mechanism.

We have measured various BQJ detectors, fabricated with the Teledyne DALSA Semiconductor High-Voltage 0.8 µm CMOS technology (shown in Fig. 1). All measurements were performed by direct probing of the detectors on the wafer. For all devices, the active area (A PH ) is 200 µm x 200 µm, with a P + -diffusion area of 206.2 x 206.2_µm², 220.4 x 220.4 µm² for the N-base layer, 233.6 x 233.6 µm² for the P-well layer and 249.8 x 249.8 µm² for the deep N-well layer (i.e., A Jθ ). According to Fig. 2, the depth of the junctions are x 1 = 0.3 µm, x 2 = 1.35 µm, x 3 = 3.5 µm and x 4 _=_11 µm.

The experimental BQJ spectral responsivity characteristics are obtained by illuminating the device with a focused spot (diameter of 100 µm) with a broad-spectrum source, and with 20 nm band-pass optical filters (FWHM). Due to material at our disposal, they are realized over the 400-1100 nm range, with a wavelength step of 25 nm (except between the two first points, at 405 and 450 nm). Thus, note that, due to these filters, each measurement point is an average on the bandwidth of the used filter. The optical power at the output of the filter in each distinct spectral band is determined with a calibrated power-meter (Model 1830-C + 818UV Si detector, Newport Corp.). Current measurements in reverse bias conditions are carried out simultaneously on all BQJ's outputs with a source meter unit (SMU) mainframe (HP4142B, Keysight Tech.), where each measurement point is the average of several acquisitions. Then, the photocurrents are deduced from (1), after subtraction of the dark currents components.

Simulations have also been performed using the Taurus Medici semiconductor modeling package (Synopsys Inc.). To reduce the computation time, the detector structure is modeled with idealized cylindrical symmetry. The spectral characteristics of the detector are computed for incident light ranging from 400 nm to 1000 nm, with a variable wavelength step between 20 and 30 nm, close to the employed step during measurement and, also, in order to obtain the best tradeoff between precision and computational load.

Moreover, we have implemented our model firstly using MATLAB (Mathwork Inc.) for fast analysis of our development. Photocurrents, I PHθ , are calculated according to their analytical expressions, especially. All the key parameters, such as α(λ), are also modeled, and the dimensions of all regions are determined. The I PHθ calculations involve determination of various parameters including physical constants, process parameters of the employed technology, temperature and bias voltages.

In a second time, these same equations, in association to the proposed equivalent circuit, have been introduced in Questa ADMS (Mentor Graphics Inc.) with the VHDL-AMS language, as in [START_REF] Quiquerez | A VHDL-AMS package for microsystems polychromatic optical modeling[END_REF]. Thus, for future perspectives, the implemented model can be easily employed in a circuit and system simulation environment, where other component's models, coming from various libraries (SPICE, VHDL-AMS, etc.) can be plugged on it, for instance. This is particularly attractive for rapid design of SoC, where the BQJ photodetector is used. Note that TCAD tools are rather used for simulating semiconductor performance, based on a drawing structure. It is now possible to couple them with SPICE-like simulator. However, the computation time is quite long and its realization can be complex, with usual convergence problems.

Fig. 9 represents the spectral responses of the BQJ detector obtained from the 3 approaches in the following conditions: at room temperature and for V Jθ =1.5 V. Fig. 9a to 9d correspond to each of the 4 junctions respectively, and Fig. 9e represents the sum of the 4 spectral responses, with a spectral response of a commercialized photodiode (FDS025, Thorlabs Inc.), added for comparison. The oscillations affecting the spectral characteristics of the BQJ detector are due to self-interferences of reflections from different dielectric layers on top of the detector surface (air/Si 3 N 4 /SiO 2 /Si). They are more pronounced in the proposed model, compared to the TCAD simulations and measurements, because narrower wavelength bands were used. Indeed, with our model's results in Fig. 9, a wavelength step of 1 nm has been chosen because, as our simulation time is short (some seconds, compared to many minutes with Taurus Medici), this low step permits to perfectly reproduce the oscillation effect due to reflection. In contrast to measurement and TCAD simulation, where we have a subsampling effect, the constraint imposed by the Nyquist-Shannon theorem is respected here and we avoided the averaging effect. Consequently, our results show larger amplitude variation and the oscillation effect is more visible than with measurement and TCAD data. However, note that this step is easily modifiable in our program. All spectral responses exhibit good agreement between the proposed model and TCAD simulations and measurements. The mean relative error on the spectral responses from our model, compared to measurements, is between 6 % and 17 %, depending on the considered junction. Factors, such as uncertainties on some basic parameters (such as doping concentrations, depth of junctions, recombination effect at the SiO 2 /Si interface, etc.) and precision of the absorption model [START_REF] Schmid | Optical absorption in heavily doped silicon[END_REF] (around 10 %), may explain the observed differences. Changes between our model and TCAD results can be attributed to the following aspects: in addition to the different wavelength step, which conducts to an oscillation effect that seems different and the amplitude's differences, the absorption coefficient model differs. With Taurus Medici, we have employed a basic model [START_REF] Synopsis | Taurus -Medici User Guide[END_REF][START_REF] Synopsis | Sentaurus Device User Guide[END_REF][START_REF] Rajkanan | Absorption Coefficient of Silicon for Solar Cell Calculations[END_REF]. In our case, a custom and tested model, described in [START_REF] Feruglio | A Review of CMOS Buried Double Junction (BDJ) Photodetector and its Applications[END_REF], is used. Moreover, the numerical method, associated the TCAD meshing, could accentuated these differences.

In the case of the normalized responses of Fig. 3, as R(λ) is the same for all junctions, introducing ( 12) in (3), we have:

N θ λ = S θ λ ∑ S θ λ 4 θ=1 (21)
Therefore, oscillations due to reflection are suppressed and the mean relative error between measurement and model on these normalized responses is then improved (between 3 % and 12_%, rather than 6 % and 17 %, previously).

Finally, note that changing the bias conditions from 0 to 5 V does not significantly affect the spectral responses (less than 5_%). Thus, the choice of these biases is mainly imposed by considering dark current contributions and optimization of response times, through W θ . Therefore, to tune the spectral responses for a particular application, adjustment of doping and depth of each junction are the best approach. For a defined fabrication process, adding a metal grid structure at the surface of the photodetector to shift the region of high sensitivity of the photodetector is also an interesting possibility [START_REF] Yang | Integrated Colour Detectors in 0.18-µm CMOS Technology[END_REF]. Dark current characteristics at room temperature have also been measured. They are presented in Fig. 10, where we can define 3 bias ranges on these I/V characteristics, according to the main physical mechanisms: 0 < V Jθ < 0.5 V: both SRH generation and diffusion mechanisms contribute to the dark current. V Jθ > 0.5 V: I DCθ is proportional to V Jθ M Jθ . Consequently, the SRH generation is predominant. V J1 > 3.5 V: the dark current is proportional to the exponential of the junction voltage (exp(V J1 γ )) in the highly doped junction J 1 . Thus, the BBT effect is predominant in this range. Also, note that J 4 shows the largest dark current contribution for reverse biasing voltages up to 3.5 V. This is because J 4 is the lowest doped junction and its depleted zone is the widest. Moreover, it has also the largest surface area in the sidewall region, with more significant surface thermal generation.

Junction capacitances of the device have also been measured independently (the studied junction is reversed bias, while the others are left floating), with HP4280A. Results are presented on Fig. 11. For a 3 V reverse bias, the model of ( 17) is in agreement with the measurements on J 2 and J 4 , with less than 5 % deviation. For J 3 , the junction capacitance is overestimated by about 17 %. Finally, for a reverse bias greater than 1 V, the surface junction capacitance decreases greatly, which may be explained by a breakdown of J 1 in these bias conditions. Indeed, in the study of dark currents, Fig. 10 illustrated the sudden increase of the current with a contribution of tunneling effect.

11 Evolution of junction capacitance with reverse bias.

VI. CONCLUSION

A buried quad p-n junction photodetector has been designed and integrated in CMOS technology. This photodetector achieves many improvements over conventional photodiodes, such as analysis of different spectral components in the same spatiotemporal conditions, with a higher common mode rejection. However, it could not be implemented for high resolution imagers, since the fill factor would decrease dramatically with the pixel size. Nevertheless, it could bring special features, such as high sensitivity and high spectral selectivity for scientific and biomedical applications.

For the fast optimized development of its associated analog front-end and signal processing, a physically-based BQJ photodetector model has been developed. It has been integrated in EDA software for component, circuit, system level simulations. A good agreement of the proposed model is obtained with TCAD simulations and measurements for both electrical (dark currents and junction capacitances) and optoelectrical (spectral responses) features.

The perspectives of this work are a statistical study of the main parameters of this model. The design of SoC (ASIC), based on this BQJ photodetector, is also in process. The frontend is then designed and optimized thanks to this model, associated to SPICE-like simulation.

Fig. 1

 1 Fig.1The BQJ sensor studied.

Fig. 4

 4 Fig. 4 Illustration of the BMJ photodetector interest in biomedical and chemical fields, such as ELISA test.

Fig. 3

 3 Fig. 3 Relative photocurrents contribution of the BQJ photodetector versus wavelength (measurement result).

Fig. 6

 6 Fig.6Fill factor of a square BQJ photodetector versus its size.

Fig. 7

 7 Fig. 7 Photocurrent components in the BQJ structure.

Fig. 5

 5 Fig. 5 Relative contribution of each BQJ photocurrent to the total optoelectrical response versus wavelength (measurement result).

  ∆n 4 x sub =D nPsub ∂∆n 4 ∂x x=x sub (10-b) ∆n 1 x 1p =∆n 2 x 2p =∆n 3 x 2p =0 (10-c) ∆n 2 x 3p =∆n 3 x 3p =∆n 4 x 4p =0 (10-d) ∆p 1 x 1n =∆p 2 x 1n =∆p 1 x 2n =∆p 2 x 2n =0

Fig. 8

 8 Fig. 8 Large-signal equivalent circuit of the BQJ photodetector. The V eqθ are the equivalent voltage sources controlled by Φ(λ) (i.e., image of the incident radiation), where V eqθ (λ)_= K ϕθ Φ(λ), with K ϕθ = 1 V.cm².s/photon. The R eqθ C eqθ are the opto-electrical time constants of the equivalent networks representing the photodetector. They model the response's time to generate photocurrents when an incident light suddenly reaches the surface of the photodetector. They are approximated by a first order model and defined by the cut-off frequencies [18-20, 41-47]: f cθ -3dB = 1 2πR eqθ C eqθ= 2.4 2πτ θ

Fig. 9

 9 Comparison of simulated spectral responses from the BQJ model with measured data and TCAD simulation (Fig.9ato 9d correspond to the spectral response of the 4 junctions and Fig.9eis the total spectral response).

Fig. 10

 10 Fig. 10 Dark current characteristic of the 4 junctions.

Sylvain

  Feruglio (M'06) received the B.Sc. and M.Sc. degrees in electronic, electrotechnic, and automatism and the master's degree in electronics, option instrumentation, and systems from the Université Pierre et Marie Curie (UPMC) -Paris 6, Sorbonne Université (France), in 1999, 2000, and 2001, respectively, and the Ph.D. degree in noise computation in integrated active pixel image sensors from LISIF, UPMC, in 2005. He continues his activities through a postdoctoral position with IMEP-LAHC, Institut National Polytechnique de Grenoble (France), which concerns the study and the characterization of new CMOS and SOI technologies. Since 2007, he is currently with LIP6 in the System on Chip department. His research interests include integrated sensors and electronics, noise analysis and signal integrity, mainly applied to image sensors and biomedical engineering.

ACKNOWLEDGMENT

The authors wish to thank Teledyne DALSA Semiconductor Inc. for help in the manufacture of the BQJ photodetector and for the extraction of technological parameters.

This work was performed within the Labex SMART supported by French state funds managed by the ANR within the Investissements d'Avenir program under reference ANR-11-IDEX-0004-02.

This work was performed within the Labex SMART supported by French state funds managed by the ANR within the Investissements d'Avenir program under reference ANR-11-IDEX-0004-02.