H Rezgui

C Mukherjee
email: chhandak.mukherjee@ims-bordeaux.fr

Y Wang

M Deng

C Maneux

A Kumar

J Müller

G Larrieu

Nanoscale Thermal Transport in Vertical Gateall-around Junction-less Nanowire Transistors-Part II: Multiphysics Simulation



Abstract-Today, extensive research has focused on heat propagation in emerging nanoelectronic devices. With advances in fabrication of nanowire transistors, thermal management has become a critical issue in cooling strategies and conducting materials. In this paper, we present a novel multiphysics analysis of the nanoscale thermal transport in 18 nm vertical junctionless gate-allaround silicon nanowire transistors. Based on this multiphysics analysis, we developed a new computational model derived from the Guyer-Krumhansl equation for describing heat transport within the nanoscale device. Our simulations results agree well with available theoretical approaches as well as measurement data.

Index

Terms-Junctionless nanowire transistors, nanoscale thermal transport, Guyer-Krumhansl equation (GKE), thermal resistance, heat dissipation.

I. Introduction

UE to miniaturization of device dimensions, metal-oxidesemiconductor field-effect transistors (MOSFETs) continue to scale down to sub-20 nm technology nodes [START_REF] Gundu | 5-nm Gate-All-Around transistor technology with 3-D stacked nanosheets[END_REF]. In modern transistors, the charge density within the channel region is mostly controlled by a gate electrode rather than the source/drain junctions. As most of the CMOS transistors are still manufactured based on the fabrication process for drain and source junction alignment, high doping concentration is regarded as a major technological challenge for the semiconductor industry [START_REF] Kim | Investigation of device performance of fin angle optimization in FinFET and Gate-All-Around FETs for 3 nm-node and beyond[END_REF]. Both planar FETs, such as fully depleted (FD), and nonplanar FETs (e.g., FinFETs), suffer from self-heating effects (SHEs) due to intrinsic resistive scattering and random dopant fluctuations [START_REF] Sudarsanan | Impact of fin line edge roughness and metal gate granularity on variability of 10-nm node SOI n-FinFET[END_REF]. In contrast, junctionless nanowire transistors (JLNTs) can be fabricated without requiring steep gradients in doping concentration, and are therefore suitable for reducing the impact of SHE [START_REF] Ionescu | Nanowire transistors made easy[END_REF]. Moreover, JLNTs devices can be scaled down to 14 nm technology node satisfying Moore's law [START_REF] Larrieu | Vertical nanowire array-based field effect transistor for ultimate scaling[END_REF][START_REF] Colinge | Nanowire transistors without junctions[END_REF][START_REF] Mukherjee | Compact modeling of 3D vertical junctionless gate-all-around silicon nanowire transistors towards 3D logical design[END_REF]. Therefore, the emerging vertical gate-all-around (GAA) JLNTs technologies have the potential to offer long-term reliability and enhanced thermal stability.

At the nanoscale, thermal management has drawn significant interest owing to its increased physical presence in small dimensional systems [START_REF] Moore | Emerging challenges and materials for thermal management of electronics[END_REF][START_REF] Shakouri | Nanoscale thermal transport and microrefrigerators on a chip[END_REF][START_REF] Das | Opportunities in device scaling for 3-nm node and beyond: FinFET versus GAA-FET versus UFET[END_REF]. Moreover, next generations transistors are expected to feature very small channel regions, making the analysis of the thermal phenomena a significantly important technological factor [START_REF] Moore | Emerging challenges and materials for thermal management of electronics[END_REF]. Therefore, to fully anticipate the physical behavior of these emerging devices, understanding of nanoscale heat transport has become imperative, especially for controlling thermal energy with the development of nextgeneration transistors [START_REF] Shakouri | Nanoscale thermal transport and microrefrigerators on a chip[END_REF]. In modern semiconductor manufacturing process, characteristic lengths have already reached 20 nm and below [START_REF] Das | Opportunities in device scaling for 3-nm node and beyond: FinFET versus GAA-FET versus UFET[END_REF]. Therefore, thermal analyses at nanoscale could offer several strategies to optimize heat propagation and improve device lifetime. In classical regime, Fourier's law has long been used to describe the heat response of bulk materials at equilibrium state (diffusive regime) [START_REF] Fourier | Théorie analytique de la chaleur[END_REF]. In small dimensional semiconductor systems (i.e., sub-20 nm), when the characteristic length becomes comparable to the average mean free path (MFP) of the phonons, which are the major heat carriers [START_REF] Guo | Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures[END_REF], the classical Fourier's law fails to predict the thermal transport [START_REF] Guo | Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures[END_REF][START_REF] Lee | Ballistic phonon transport in holey silicon[END_REF][START_REF] Know | Thermal transport in Si and Ge nanostructure in the 'confinement' regime[END_REF]. As the size of the transistor reduces, boundary effects become more significant which influences the thermal conductivity. In most cases, phonon boundary scattering causes a large reduction of the thermal conductivity. Indeed, in nanoscale dimensions, the heat flux can propagate within a nanowire structure without heating the device in the event of ballistic transport [START_REF] Lee | Ballistic phonon transport in holey silicon[END_REF]. In general, nanoscale heat transport can be described by the phonon Boltzmann transport equations (BTE). Due to the complexity of the nanotransistors geometries, several analytical solutions of BTE are available with respect to extended irreversible thermodynamics (EIT) [START_REF] Jou | Extended Irreversible Thermodynamics[END_REF].

In emerging nanoelectronics, the efficiency of heat spreading is directly related to the thermal conductivity and interfacial energy transport. Particularly, low thermal conductance can lead to small thermal conductivities due to the phonon confinement near the interface between two different materials [START_REF] Anufriev | Quasi-ballistic heat conduction due to Lévy phonon flights in silicon nanowires[END_REF]. For nanoscale transistors, high thermal conductivity materials are therefore desirable in order to reduce the SHEs and thermal degradation [START_REF] Lai | Study on degradation mechanisms of thermal conductivity for confined nanochannel in Gate-All-Around silicon nanowire field-effect transistors[END_REF]. To study the phonon dynamics in D these devices, accurate models for the prediction of heat propagation are thus crucial compared to empirical models with fitting parameters. A first approach using a local equilibrium method based on the classical Fourier's law may lead to incorrect predictions about the heat propagation. Therefore, non-Fourier analyses based on mesoscopic methods are more suitable for investigating heat flow in small dimensional systems [START_REF] Guo | Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures[END_REF][START_REF] Lee | Ballistic phonon transport in holey silicon[END_REF][START_REF] Beardo | Hydrodynamic heat transport in compact and holey silicon thin films[END_REF]. For instance, the Guyer-Krumhansl equation (GKE) was derived from the BTE based on the Chapman-Enskog expansion for the description of wavelike heat flow at low temperatures when phonon normal (N) scattering dominate [START_REF] Guyer | Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals[END_REF]. Recently, the application of GKE has been demonstrated for probing non-Fourier heat transport at room temperature [START_REF] Beardo | Phonon hydrodynamics in frequency-domain thermoreflectance experiments[END_REF]. In the context of vertical nanowire transistors and in general for extremely small systems, GKE can thus be used to capture the phonon transport where confinement effects appear to be significant.

Following the experimental methods for studying electrothermal effects in JLNTs demonstrated in part -I of this work [START_REF] Mukherjee | Nanoscale Thermal Transport in Vertical Gate-allaround Junction-less Nanowire Transistors-Part I: Experimental Methods[END_REF], in this paper, we focus on nanoscale thermal transport in 18 nm JLNTs with a wide range of nanowire diameters and number of nanowires in parallel. We refer to the finite-element method for solving thermal transport equations. The present work offers a deeper understanding of nanoscale heat transport in emerging nanoscale systems. The paper is organized as follows: In the section II, we present the computational model and boundary conditions. In addition, we validate our proposed GKE based model by comparing the effective thermal conductivity of Si NWs with experimental data and theoretical models. Section Ⅲ describes the device geometry and test structures and discusses the validation of our proposed modeling framework by comparing the thermal resistance extracted using our model and measurement data from both of experimental JLNT and FinFET technologies. Finally, section IⅤ is devoted to conclusions and perspectives.

II. THEORETICAL BACKGROUND

A. COMPUTATIONAL MODEL

In order to predict the hydrodynamic phonon transport at low temperature, Guyer and Krumhansl proposed a solution to the linearized BTE for nonequilibrium heat transfer [START_REF] Guyer | Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals[END_REF]. Many of the past works have focused on phonon hydrodynamics when both normal (N) and resistive (R) scattering processes have strong contributions to the physical origin of the heat response [START_REF] Beardo | Observation of second sound in a rapidly varying temperature field in Ge[END_REF]. In recent studies, the GKE was developed based on macroscopic hydrodynamic equation for nanoscale heat transport at room temperature [START_REF] Guo | Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures[END_REF][START_REF] Rezgui | Investigation of nanoscale heat transport in sub-10 nm carbon nanotube field-effect transistors based on finite element method[END_REF] where most nanoelectronic devices commonly operate. The governing equations are the GKE and the energy-conservation equation, respectively [START_REF] Guo | Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures[END_REF][START_REF] Lee | Ballistic phonon transport in holey silicon[END_REF][START_REF] Beardo | Hydrodynamic heat transport in compact and holey silicon thin films[END_REF] written as, 𝑞 𝜏 𝜅𝛻𝑇 ℓ 2𝛻𝛻. 𝑞 𝛥𝑞

𝑐 𝛻. 𝑞 0 (2) where 𝑞 is the heat flux, 𝜏 is the phonon relaxation time, 𝜅 is the bulk thermal conductivity, ℓ is the phonon MFP, 𝑐 is the heat capacity and T is the temperature. Equation (1) gives a simple analytical solution of the phonon BTE as well as a better understanding of non-Fourier heat-flow modeling. As device size decreases, non-local effects included in the Laplacian term 𝛥𝑞, become dominant. In general, the nonlocality of the heat flux gradient originates from the phonon-boundary scattering and confinement effects. According to the average phonon MFP, ℓ, heat transport regimes are classified into three regimes:

(1) ballistic regime (external boundary scattering); (2) quasiballistic regime and (3) bulk-like (diffusive) regime. In sub-30 nm silicon devices, ballistic transport governs heat flow, and it is expected that phonon-boundary scattering contributes to the reduction of the thermal conductivity [START_REF] Know | Thermal transport in Si and Ge nanostructure in the 'confinement' regime[END_REF]. For steady-state phonon transport, the GKE is expressed as 𝑞 𝜅𝛻𝑇 ℓ 𝛥𝑞 (3) The present GKE provides an enhanced formulation for investigating nanoscale heat transport beyond the classical Fourier's law.

B. Boundary conditions

At room temperature, slip boundary conditions is mandatory for the description of phonon transport in small dimensional materials [START_REF] Guo | Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures[END_REF]. Particularly, two macroscopic mechanisms occur during non-equilibrium phonon distribution: (1) the tangential heat flux close to the boundary and (2) the noncontinuous temperature variation (i.e. temperature jump) due to phonon-boundary collision [START_REF] Guo | Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures[END_REF]. Based on the Maxwell model, the slip boundary condition for the tangential heat flux at the boundary is given by [START_REF] Beardo | Phonon hydrodynamics in frequency-domain thermoreflectance experiments[END_REF] 𝑞 𝐶 ℓ∇𝑞 . 𝑛 (4) 𝑞 . 𝑛 0

(5) where, 𝑛 is the interface normal vector and 𝐶 is a constant which is expressed in terms of the specularity of the surface. In most cases, solving the GKE and slip boundary conditions requires an accurate value of the thermal conductivity, average mean free path and heat capacity that are obtained from microscopic approaches such as ab initio calculations. The computational model presented in this paper is described by the numerical scheme detailed in Fig. 1(a). Here, the GKE is solved based on the finite-element method (FEM) using COMSOL MULTIPHYSICS simulator [START_REF]COMSOL Multiphysics[END_REF]. For the steady state phonon transport, we consider that the nanowires are initially at a constant temperature, 𝑇 , and then one end of the nanowire comes in contact with the heat-source 𝑇 𝑇 ∆𝑇, where ∆𝑇 is the temperature difference as shown in Fig. 1(b).

C. Effective thermal Conductivity: Model Validation

In nanodevices, where the characteristic lengths are comparable to the MFP, ballistic phonon transport leads to a critical dependence of size effects on the effective thermal conductivity [START_REF] Know | Thermal transport in Si and Ge nanostructure in the 'confinement' regime[END_REF]. For the heat flow through a nanowire with radius 𝑟, Eq. ( 3) can be rewritten as

𝑞 𝑟 𝜅 ℓ ℓ (6)
The effective thermal conductivity (ETC), 𝜅 , is defined as [START_REF] Beardo | Phonon hydrodynamics in frequency-domain thermoreflectance experiments[END_REF] 𝜅 ∆ / [START_REF] Mukherjee | Compact modeling of 3D vertical junctionless gate-all-around silicon nanowire transistors towards 3D logical design[END_REF] Here Γ is the cross section, S is the surface area and 𝐿 is the distance between the two ends of the system. The classical Fourier's law of heat conduction can only predict thermal conductivity in bulk materials when the characteristic dimensions are larger than the phonon MFP. Here, we consider a Knudsen number defined as 𝐾𝑛 ℓ , where D is the diameter of the nanowire, to classify the nanoscale thermal transport within the nanowire [START_REF] Guo | Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures[END_REF]. For 𝐾𝑛 1, ballistic transport dominates and the thermal conductivity is significantly influenced by the confinement effects. For 𝐾𝑛 1, the thermal conductivity increases due to enhanced boundary scattering (i.e. weak confinement effect). In diffusive regime (𝐾𝑛 ≪ 1), Eq. ( 6) reduces to the Fourier's law, 𝑞 𝑟 𝜅 . Fig. 2 shows the diameter dependence of normalized thermal conductivity in Si NWs with a phonon MFP, ℓ 300 𝑛𝑚, at 𝑇 300 𝐾. According to Kwon et al., [START_REF] Know | Thermal transport in Si and Ge nanostructure in the 'confinement' regime[END_REF], we assume fully diffusive boundary scattering, i.e.,𝐶 1, to achieve an accurate comparison with the theoretical and experimental results [START_REF] Know | Thermal transport in Si and Ge nanostructure in the 'confinement' regime[END_REF] (Fig. 2). Our proposed model displays a good prediction of experimental data for Si NWs down to 30 nm. From Fig. 2, we can distinguish between three distinct thermal transport regimes: (1) a ballistic regime where the phonon boundary/interface scattering leads to a significant reduction of the thermal conductivity; (2) a quasi-ballistic regime where we expect a Lévy flight due to trapped phonons in the Si NW; (3) a diffusive regime where the effective thermal conductivity reaches the bulk value (the Knudsen number is small enough with 𝐾 0.05). The GKE gives a consistent result with the theoretical predictions and experimental data. For further validation, the GKE is compared to experimental results [START_REF] Li | Thermal conductivity of individual silicon nanowires[END_REF] and the Callaway-Holland model derived from the first law of thermodynamics based on BTE for phonon distribution [START_REF] Marconnet | Phonon conduction in periodically porous silicon nanobridges[END_REF]. This is particularly useful for investigating the impact of the boundary scattering and size effects (i.e., diameter of the nanowire) on the thermal conductivity. GKE being an analytical solution of the BTE, we compared it with the Callaway-Holland model to justify the accuracy of our proposed framework in Fig. 3, which illustrates the thermal conductivity of Si NWs as a function of the NW diameter. It can be clearly observed that the present model agrees well with the theoretical Callaway-Holland model [START_REF] Marconnet | Phonon conduction in periodically porous silicon nanobridges[END_REF] and experimental data reported by Maire et al. [START_REF] Maire | Ballistic thermal transport in silicon nanowires[END_REF]. The thermal conductivity decreases in small diameters due to phonon confinement. As system dimensions reach the quantum regime, the effect of phonon/interface scattering leads to reduced ETC. In the next section, this computational model is applied for analyzing JLNTs devices.

III. RESULTS AND DISCUSSION

A. Non-Fourier heat transport

The 3D schematic view of the state-of-the-art JLNT technology under test is illustrated in Fig. 4 (a). The key feature of the fabricated JLNT is its gate-all-around structure that ensures enhanced electrostatic control of the current between

D (nm)

Callaway-Holland model [START_REF] Marconnet | Phonon conduction in periodically porous silicon nanobridges[END_REF] Li et al. [START_REF] Li | Thermal conductivity of individual silicon nanowires[END_REF] Maire et al. [START_REF] Maire | Ballistic thermal transport in silicon nanowires[END_REF] Present GKE source and drain contacts. For the device under test, the gate is fabricated using chromium (Cr) that ensures better control of interfacial thermal transport within the nanowire and offers high thermal conductivity even at room temperature [START_REF] Beardo | Phonon hydrodynamics in frequency-domain thermoreflectance experiments[END_REF]. Moreover, the aluminum (Al) top, bottom and gate contacts are highly beneficial for ensuring weak thermal degradation at higher temperatures. In this work, the thermal properties of Al are obtained based on density functional theory (DFT) calculations with respect to BTE [START_REF] Jain | Thermal transport by phonons and electrons in aluminum, silver, and gold from first principles[END_REF]. Furthermore, a 5nm-SiO 2 gate dielectric is formed between the chromium gate and the Si nanowire channel as shown in Fig. 4 (b). Fig. 4 (c) shows the geometry of a highly p-doped single nanowire with a gate length of 18 nm leading to junctionless transition between S/D regions and the channel. More details of the technology can be found in [START_REF] Larrieu | Sub-15 nm gate-all-around field effect transistors on vertical silicon nanowires[END_REF][START_REF] Guerfi | Vertical silicon nanowire field effect transistors with nanoscale gate-all-around[END_REF]. To solve the thermal transport equations for this device configuration, the heat transport is coupled with the electrical characteristics. In this context, Eq. ( 2) is rewritten as

𝐶 𝛻𝑞 𝑃 (8)
where 𝑃 is the dissipated power defined as 𝑃 𝑉 𝐼 [START_REF] Jiang | Experimental investigation of self heating effect (SHE) in multiple-fin SOI FinFETs[END_REF]. To extract the thermal resistance of the device, the heat flux can be obtained as 𝑞 𝜅 (9) The heat flux distribution is then given by 𝑞 [START_REF] Das | Opportunities in device scaling for 3-nm node and beyond: FinFET versus GAA-FET versus UFET[END_REF] where 𝑅 is the thermal resistance in unit of (K/W). To obtain an accurate description of nanoscale thermal transport within the JLNT, the thermal properties of the materials chosen for our simulation are listed in Table Ⅰ. The heat capacity values of silicide and Al are obtained from the material library in COMSOL.

B. Measurements

Due to the high doping in the NW channel, on-wafer DC measurements of the JLNTs under test exhibited an increase of drain current with temperature, in contrast with conventional CMOS technologies. This can be explained by the threshold voltage reduction with temperature combined with a change in the S/D access resistances due to thermionic emission through the Schottky contacts, along with negligible carrier mobility degradation that result in a monotonic increase in the drain current [START_REF] Lee | High-Temperature Performance of Silicon Junctionless MOSFETs[END_REF]. As conventional methods were not directly applicable in our case, to extract thermal resistance of the JLNTs using experimental methods, we developed several novel experimental methods using DC measurements under different temperatures along with low-frequency S-parameter measurements [START_REF] Mukherjee | Nanoscale Thermal Transport in Vertical Gate-allaround Junction-less Nanowire Transistors-Part I: Experimental Methods[END_REF]. With these DC methods, the thermal resistance is extracted using 𝑅 ∆𝑇/𝑃 for the available transistor geometries. The extracted values thermal resistances are then compared with the simulations in subsection D for the validation of the proposed modeling framework.

C. Comparison of thermal resistances in JLNTs and FinFETs

In this section, we compare the thermal resistance of JLNTs and SOI FinFETs as a first assessment of our simulation results. In order to ensure a reasonable comparison between both technologies, we consider a gate length of 90 nm and a 𝑉 = 1.5 𝑉. It is important to note that the model proposed in our method is solved under non-isothermal conditions. The interfacial thermal transport is governed by non-slip boundary condition given by Eq. ( 3). Here, the thermal resistance is extracted from the drain region where the temperature hot-spot is observed. Fig. 5 (a) compares the extracted thermal resistance as a function of the number of fins (for FinFETs) and nanowires (for JLNTs). The thermal resistance reduces with increasing number of fins and nanowires with the maximum thermal resistance 𝑅 , , observed in the drain region of the device, corresponding to fewer parallel nanowires and fins. When the number of nanowires increases, the phonon-boundary scattering becomes weak leading to enhanced reduced thermal resistance. In addition, the thermal degradation increases as the diameter reduces due to ballistic phonon transport dominating in sub 20 nm dimensions as observed from Fig. 5 (b). Interestingly, the thermal resistance observed in both p-type SOI FinFETs and JLNTs is smaller than the 𝑅 extracted for n-type SOI FinFETs. This is because of the fact that p-type devices have a higher thermal conductivity compared to n-type devices [START_REF] Jiang | Experimental investigation of self heating effect (SHE) in multiple-fin SOI FinFETs[END_REF]. In solids, electron-phonon interaction is important to understand energy flow in electronic devices. The thermal transport is affected by two types of carrier scatterings: phonon-Fig. 5. The extracted thermal resistance in SOI FinFETs [START_REF] Jiang | Experimental investigation of self heating effect (SHE) in multiple-fin SOI FinFETs[END_REF] and JLNTs as function of (a) 𝑁 /𝑁𝑊 and (b) 𝑊 / 𝐷. electron and phonon-hole scattering [START_REF] Zhou | Direct observation of large electron-phonon interaction effect on phonon heat transport[END_REF]. The reduction of thermal conductivity mostly originates from phonon-electron scattering. In N-type devices, the phonon-electron interaction strongly modifies the lattice vibration which in turn influences the thermal transport and thermodynamic properties. Therefore, the thermal resistance of N-type devices is higher than that of P-type devices due to a higher phonon-electron coupling. JLNTs with respect to fins and nanowire numbers. The temperature increases with increasing 𝑁 and 𝑁𝑊𝑠 as the dissipated power 𝑃 also increases. For 𝑁 /𝑁𝑊𝑠 2 and 6, one can observe a temperature difference, ∆𝑇, of around 8 𝐾. However, ∆𝑇 reaches 17 𝐾 for large number of fins and nanowires (𝑁 /𝑁𝑊𝑠 20 and 30). It is to be noted that SOI FinFETs have a higher temperature increase than those extracted for JLNTs, due to a more confined flow of the phonons, indicating better thermal stability for the latter case. The extracted thermal conductivity in SOI FinFETs is found to be 8 W/m/K, whereas it reaches 10 W/m/K for the JLNTs [START_REF] Larrieu | Thermal consideration in nanoscale gate-all-around vertical transistors[END_REF]. In the other words, the thermal transport mechanism is more resistive in the FinFETs.

D. Nanoscale Thermal Transport in GAA JLNTs

The GAA structure offers the advantage of better thermal stability of nanoscale devices [START_REF] Lai | Study on degradation mechanisms of thermal conductivity for confined nanochannel in Gate-All-Around silicon nanowire field-effect transistors[END_REF], [START_REF] Jain | Modeling of effective thermal resistance in sub-14 nm stacked nanowire and FinFETs[END_REF]. In order to further understand the heat dissipation in JLNTs, we carried out multiscale GKE model simulation for JLNT geometries diameter 17, 22 and 34 nm with different number of nanowires in parallel, for the operating condition 𝑉 = 𝑉 2 𝑉. Table II lists the JLNTs geometries used for simulation in this work. Fig. 7 shows the extracted thermal resistance in the JLNTs for different diameters and nanowires, comparing device simulation and measurement results from section B. The thermal simulations were performed using an extremely fine mesh with 65000 spatial grids around the transistor active area to facilitate solving the GKE model. The maximum lattice temperature is obtained from the dissipated power calculated from 𝐼 -𝑉 characteristics. As expected, the thermal resistance decreases with increasing nanowires in parallel, whereas, it increases when diameters are scaled down. In particular, for small diameters, the thermal resistance increases due to phonon confinement effects occurring as a result of the surface roughness which, in turn, results in a loss of momentum and eventually a strong resistive transport at room temperature. For 𝐷 17 𝑛𝑚, the thermal resistance decreases from 2.66 × 10 5 K/W to 7.4×10 4 K/W with increasing number of parallel NWs (𝑁𝑊 25 to 81). Similarly, for 𝐷 34 𝑛𝑚, the 𝑅 decreases from 1.12×10 5 K/W to 3.3×10 4 for 𝑁𝑊 25 and 81, respectively. As depicted in Fig. 7, our simulation results agree quite well with measurements for 𝐷 22 𝑛𝑚 and 𝐷 34 𝑛𝑚. As observed from the trend in Fig. 7, thermal resistance is significantly influenced by the NW diameter, however, for a higher number of NWs in parallel, the spreading of the thermal resistance values with diameter reduces. Fig. 8 shows the thermal conductivity of the JLNTs along the channel direction with 25 nanowires in parallel extracted from the device simulation. As expected, the lower ETC of the nanowire is directly associated to higher thermal resistance in small dimeters, which, in turn, may lead to pronounced SHEs. It is to be noted that, the ETC reaches 3.26 W/m/K and 11.6 W/m/K for 𝐷 17 𝑛𝑚 and 𝐷 34 𝑛𝑚, respectively. The reduction of the thermal conductivity is mostly caused by phonon confinement effects in the drain area (the hot-spot region). Close to the drain area, phonon transport takes place in the ballistic regime and boundary scattering takes place which influences the thermal conductivity. The phonon-boundary scattering rate is significantly high in low dimensional materials which can be held responsible for the weak value of the ETC for 𝐷 17 𝑛𝑚. In addition, for nanowires of diameter smaller than 50 nm, the Young's modulus is likely another factor governing the thermal transport. Incoherent phonon scattering can also impact the heat propagation, which further reduces the thermal conductivity [START_REF] Ravichandran | Crossover from incoherent to coherent phonon scattering in epitaxial oxide superlattices[END_REF]. In general, in sub-10 nm nanowires, incoherent phonon scattering leads to a backscattering mechanism (i.e. resistive effects) which degrades the phonon momentum and thermal conductivity. The temperature hotspots appear due to the dominance of resistive heat transport. When the thermal conductivity reduces, heat carriers (phonon) are confined in a small space which leads to an accumulation of the heat. The physical phenomena behind the confinement mechanism can be explained by the Lévy walk [START_REF] Anufriev | Quasi-ballistic heat conduction due to Lévy phonon flights in silicon nanowires[END_REF], in which the phonons are trapped in the nanowire through short flights between the boundary and the nanowire. Phonon Lévy flight has been experimentally observed in silicon nanowires which result in the reduction of thermal conductivity in the quasiballistic regime. Therefore, the increase of device temperature and the reduction of the thermal conductivity can mainly be explained by the confinement effect. Our simulations results demonstrate that the confinement effect is very weak at the source side. It is expected that the effective thermal conductivities of the devices with D=17 nm and D=22 nm are quite close. Near the source region, the heat transport is less resistive and phonons experience less boundary scattering, therefore leading to a higher thermal conductivity.

For subsequent analyses on heat propagation, we focused on a single nanowire. Fig. 9 plots the simulated distribution of internal device temperature in JLNTs for different diameters. The maximum temperature is detected near the drain region which then gradually diminishes towards the channel layer. Interestingly, the effective temperature rise with increasing diameter is the combined effect of increasing dissipated power 𝑃 and reducing thermal resistance in larger diameters. For smaller diameters, in Fig. 9 (a) and Fig. 9 (b), the heat flow is localized in the drain side which then propagates along the nanowire. However, in Fig. 9 (c) for largest diameter (34 nm), the heat accumulation spreads over the drain side only. Furthermore, temperature distributions for smaller diameters are indicative of the thermal confinement near the drain region that leads to elevated temperature higher thermal energy. Further reduction in internal device temperature can be achieved by increasing the diameter and nanowires in parallel.

IV. CONCLUSION

In this paper, we investigated nanoscale thermal transport beyond Fourier's law in GAA-JLNTs for the first time using Guyer-Krumhansl equation developed for thermal parameter extraction. In particular, we demonstrated that non-Fourier heat modeling is an efficient method for understanding heat flow in more-Moore nanoelectronic devices. Our developed simulation framework was used to investigate thermal transport in both FinFETs and JLNTs indicating better thermal stability of the latter technology. Comparison of thermal resistance of the JLNTs from measurements [START_REF] Mukherjee | Nanoscale Thermal Transport in Vertical Gate-allaround Junction-less Nanowire Transistors-Part I: Experimental Methods[END_REF] and our proposed model demonstrated excellent agreement validating the accuracy of the proposed modeling framework. Further analysis suggested that a high thermal conductivity is recommended for better control of heat dissipation and reduction of temperature hotspots in JLNTs. Moreover, JLNTs with larger diameters and higher number of nanowires in parallel clearly demonstrate a higher efficiency for heat dissipation through the 3D vertical nanowire architecture. Further investigation of electro-thermal behavior of the JLNT technology under study will focus on electron-phonon coupling and its impact on thermal transport.

Fig. 1 .

 1 Fig. 1. (a) Numerical simulation scheme for nanoscale thermal transport. (b) Schematic of phonon transport within a nanowire.

Fig. 2 .Fig. 4 .

 24 Fig. 2. Comparison of normalized thermal conductivity for silicon nanowire at room temperature. 10 100 1000 0.01

Fig. 3 .

 3 Fig. 3. Thermal conductivity as a function of NW diameter.

5 VFig. 6 .

 56 Fig.[START_REF] Colinge | Nanowire transistors without junctions[END_REF]. Extracted device temperature in SOI FinFETs[START_REF] Jiang | Experimental investigation of self heating effect (SHE) in multiple-fin SOI FinFETs[END_REF] and JLNTs as function of 𝑁 /𝑁𝑊 for 𝑉 = 1.5 𝑉.

Fig. 6

 6 Fig. 6 plots the average temperature within FinFETs and

Fig. 7 .

 7 Fig. 7. The extracted thermal resistance in JLNTs as a function of number of nanowires comparing simulation and measurements.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Effective thermal conductivity along the nanowire for different diameters.

TABLE Ⅰ

 Ⅰ

	: THERMAL PARAMETERS OF DIFFERENT MATERIALS
	Symbol Si [21]	K (Wm -1 K -1 ) 150	C (J m -3 K -1 ) 1.692×10 6	ℓ (nm) 185
	SiO 2 [23]	1.4	1.75×10 6	0.4
	PtSi [25, 33]	18.5	2.1×10 6	-
	Cr [21]	111	2.47×10 6	-
	Al [25, 27]	252	2.43×10 6	10

TABLE II -

 II FABRICATION PROCESS PARAMETERS FOR JLNTS

This work was supported by the project FVLLMONTI funded by European Union's Horizon 2020 research and innovation program under grant agreement N•101016776 and by the LAAS-CNRS micro and nanotechnologies platform, a member of the Renatech French national network.


Erwan Le Merrer
email: erwan.le-merrer@inria.fr..

Ronan Pons

Gilles Tredan

Algorithmic audits of algorithms, and the law

Keywords: Algorithmic decision-making, machine learning, audits, law 1

Algorithmic decision-making is now widespread, ranging from health care allocation to more common actions such as recommendation or information ranking. The aim to audit these algorithms has grown alongside. In this article, we focus on external audits that are conducted by interacting with the user side of the target algorithm, hence considered a black box. Yet, the legal framework in which these audits take place is mostly ambiguous to researchers developing them: on the one hand, the legal value of the audit outcome is uncertain; on the other hand, the auditors' rights and obligations are unclear. The contribution of this article is to articulate two canonical audit forms to law, to shed light on these aspects: 1) the first audit form (we coin the Bobby audit form) checks a predicate against the algorithm, while the second (Sherlock) is looser and opens up to multiple investigations. We find that: Bobby audits are more amenable to prosecution, yet are delicate as operating on real user data. This can lead to rejection by a court (notion of admissibility). Sherlock audits craft data for their operation, most notably to build surrogates of the audited algorithm. It is mostly used for acts for whistleblowing, as even if accepted as proof, the evidential value will be low in practice.

2) these two forms require the prior respect of a proper right to audit, granted by law or by the platform being audited; otherwise, the auditor will be also prone to prosecutions regardless of the audit outcome. This article thus highlights the relation of current audits with law, to structure the growing field of algorithm auditing.

Introduction

Law is supposed to hold service providers and their algorithms accountable. In particular, decision-making algorithms, now often based on Artificial Intelligence technologies, are now widespread [START_REF] Diakopoulos | Accountability in algorithmic decision making[END_REF]. They directly face users, and govern large portions of our lives (from apparently subtle decisions such as recommendations, to more life-changing ones such as criminal justice or health care allocation [START_REF] Ledford | Millions of black people affected by racial bias in health-care algorithms[END_REF]). The ethical impacts of such pervasive mechanisms are salient. Yet, to quote Robles Carrillo [START_REF] Carrillo | Artificial intelligence: From ethics to law[END_REF]: "Ethics is necessary, even indispensable, but not sufficient to meet the challenge of AI. Ethics is especially needed when regulation is lacking. Law, however, is essential. Law implies a binding legal commitment, including for instance those ethical contents that are common and/or shared and therefore reach the statute of obligatory norms." The legal perspective on algorithms, especially of online platforms, is indeed evolving into strong legal frameworks. For example, at a European level, in order to protect the fundamental rights of European residents 1 , or to frame artificial intelligence systems 2 ; this explicitly shows the willingness to better regulate algorithms.

The IT perspective and the nascent field of audits Computer scientists and engineers are designing and developing algorithms that process information and that can have an important impact on society [START_REF] Klonick | Content moderation modulation[END_REF][START_REF] Metcalf | Algorithmic impact assessments and accountability: The co-construction of impacts[END_REF]. For improving and refining these, developers operate a feedback loop on data fed as inputs to the algorithm, and on the corresponding algorithm results (output accuracy for instance). Company-internal and ethics-based audits may also be conducted by third parties [START_REF] Mökander | Operationalising ai governance through ethicsbased auditing: an industry case study[END_REF].

Considering an exterior viewpoint (the viewpoint of users or regulators) that observes or audits the behavior of remote algorithms is less frequent. A so-called black box approach to algorithms can be dated back to Moore's tests of black box automata in 1956 [START_REF] Moore | Gedanken-experiments on sequential machines[END_REF]. Relatively recent and sporadic works instead placed this viewpoint at the service of algorithmic auditing, to allow users to gain some understanding on the algorithmic decisions they are facing [START_REF] Diakopoulos | Accountability in algorithmic decision making[END_REF][START_REF] Lécuyer | Xray: Enhancing the web's transparency with differential correlation[END_REF][START_REF] Aïvodji | Fairwashing: the risk of rationalization[END_REF][START_REF] Buolamwini | Gender shades: Intersectional accuracy disparities in commercial gender classification[END_REF][START_REF] Ribeiro | why should i trust you?" explaining the predictions of any classifier[END_REF][START_REF] Chen | Peeking beneath the hood of uber[END_REF][START_REF] Ribeiro | Auditing radicalization pathways on youtube[END_REF][START_REF] Bandy | Auditing news curation systems: A case study examining algorithmic and editorial logic in apple news[END_REF][START_REF] Galdon Clavell | Auditing algorithms: On lessons learned and the risks of data minimization[END_REF][START_REF] Panigutti | Fairlens: Auditing black-box clinical decision support systems[END_REF][START_REF] Huszár | Algorithmic amplification of politics on twitter[END_REF][START_REF] Kaiser | The implications of venturing down the rabbit hole[END_REF]. In particular, these nascent forms of algorithmic audits can also constitute a prerequisite to enabling platform regulation [START_REF] Petropoulos | A european union approach to regulating big tech[END_REF] or accountability [START_REF] Falco | Governing ai safety through independent audits[END_REF]: if a state wants to enforce some behavior, means for verification are mandatory.

And the law?

There is a blind spot regarding the current development of algorithmic audit techniques: what are their legal groundings? If ethical considerations are well represented in the field, with multiple standards proposals [START_REF] Carrillo | Artificial intelligence: From ethics to law[END_REF][START_REF] Unesco | Recommendation on the ethics of artificial intelligence[END_REF][START_REF] Jobin | The global landscape of ai ethics guidelines[END_REF], little (if any) mention is made in research works of the legal consequences of the conducted audit. In a nutshell, two fundamental questions are unaddressed: i) what are the legal risks taken by an auditor, and ii) can the outcome of an audit be used against the platform in court? This lack is a structural problem, as the question to whom the discovered issues must be addressed is central.

Possible recipients can then be the general public (e.g., through the act of whistleblowing), or justice. In both cases, legal aspects are at stake: the auditor is likely to have violated the terms of service of the audited algorithm, displayed on the platform. What are then the consequences she faces? Or the consequences of the findings in a trial? If the auditor is in her own right to perform certain actions, what are the audit steps that will prevent acceptance of scientific proof in the eyes of justice?

Contributions

In an attempt to shed light on the relation between audits and the law, we first propose to bind two novel prototypes of audit algorithms (that encapsulate state-of-the-art audit algorithms) to specific law perspectives. This constitutes an important step to structure the auditing field, as most of the recent works calling for it are still of a prescriptive nature [START_REF] Falco | Governing ai safety through independent audits[END_REF][START_REF] Raji | Outsider oversight: Designing a third party audit ecosystem for ai governance[END_REF][START_REF] Mökander | Auditing large language models: a three-layered approach[END_REF] and far from giving technical leads that will hold in future legal frameworks. Since the law is by definition a sovereign prerogative to each state in the world, we will take the French law system as an instance and example in our presentation. We will as well include European perspectives from the global law frameworks currently in progress. This yields two salient points: 1) the simplest form of audit (we coin the Bobby audit) is easily usable in court, yet it is more delicate as it leverages real data (which can be a cause of rejection if all care has not been taken regarding laws such as the GDPR: the General Data Protection Regulation 3 2) The most complex audit form (we coin the Sherlock audit) is less problematic as it crafts data to input in the algorithm of the platform; yet, it is far more difficult to bring to justice due to possibly lower probative value, leaving it in priority for whistleblowing. Finally, we review the recent proposal for a "vetted researchers" status by the European Union, and its consequences for future audits.

Two canonical algorithms capturing algorithmic audit schemes

There is a growing diversity of audits in the recent literature [START_REF] Diakopoulos | Accountability in algorithmic decision making[END_REF][START_REF] Lécuyer | Xray: Enhancing the web's transparency with differential correlation[END_REF][START_REF] Aïvodji | Fairwashing: the risk of rationalization[END_REF][START_REF] Buolamwini | Gender shades: Intersectional accuracy disparities in commercial gender classification[END_REF][START_REF] Ribeiro | why should i trust you?" explaining the predictions of any classifier[END_REF][START_REF] Chen | Peeking beneath the hood of uber[END_REF][START_REF] Ribeiro | Auditing radicalization pathways on youtube[END_REF][START_REF] Bandy | Auditing news curation systems: A case study examining algorithmic and editorial logic in apple news[END_REF][START_REF] Galdon Clavell | Auditing algorithms: On lessons learned and the risks of data minimization[END_REF][START_REF] Panigutti | Fairlens: Auditing black-box clinical decision support systems[END_REF][START_REF] Huszár | Algorithmic amplification of politics on twitter[END_REF][START_REF] Kaiser | The implications of venturing down the rabbit hole[END_REF]. Each one spans a specific behavior of a specific platform with its own methodology.

In an attempt to structure this novel field, we introduce a set of fundamental distinctions that allows to separate these audits into two broad categories that distinguish audits both on their technical approach, and on their relevance for a potential trial.

Let us take the parallel with police work, tasked to check the application of the law. On one hand, Bobby-family audits are tasked to tour the audited algorithm evaluating a well-defined characteristic of the platform, similar to a policeman tasked to tour a district to fine car parking infringements. Key to this approach is the existence of a logical predicate that very precisely defines the desirable (resp. undesirable) behavior of the audited algorithm, similar to the set of driving regulations that precisely define what is a correctly parked car. On the other hand, Sherlock-family audits target a deeper and loosely defined characterization of some aspect of the audited algorithm, similar to an inspector trying to elucidate some crime. Such an approach typically requires some interpolation to provide a general analysis based on some observed examples of the algorithm behavior.

Context and Terminology

Let's consider the following use case: an individual (or a group of) hereafter named the auditor seeks to study the behavior of some algorithm executed remotely by some platform hereafter named a target platform or the target algorithm. Auditors are completely external to the target platform, and hence can only interact with the public side of the algorithm as regular users would.

This context is tailored to represent the typical context of a black box audit, where auditors are simple users interested in understanding or evaluating the behavior of the platform they use. This context can also capture situations in which the competent authorities have no specific access to the algorithm and want to verify the compliance of this behavior with some regulations.

Concretely, we wish to capture a spectrum of use cases ranging from informal citizen-driven audits (see e.g., COMPAS) to academic research work on auditing platforms. All those situations cover the same high-level steps: an auditor writes some code to i) request the target platform (either through some API, or through its web interface directly), ii) parse and collect the target algorithm answers, and iii) publicize some analyses based on the collected data.

Formally, let A be the target algorithm. Let X (resp. Y ) be the input (resp. output) space of A. Like regular users, auditors can only submit some request x ∈ X to A, and then record the corresponding result A(x) ∈ Y .

The Bobby audit form

This is the simplest category of audits: in this form, an infraction is constituted by an input set (that is data existing in a dataset), to which corresponds a (problematic) collected output.

Propositional formula L

In the pseudo-code presented on Algorithm 1, the central component is the definition of the propositional formula L to be checked against the audited algorithm. In its definition, L encodes the desirable property one wants to observe. More precisely, L is a propositional formula defined over a set of input/output couples of the target algorithm that constitute the variables of the proposition. In L, those variables are linked by logical operators such that L is well-formed and has a truth value: L is either true or false. As an illustrating example, imagine A is the algorithm that is in use in an online flight search platform that allows users to seek and book flights. For each request (departure and destination locations belonging to the IATA list, and dates), it provides the user with an ordered list of flights f 1 , f 2 , . . .. Assume the platform operating A declares that it ranks the resulting flights according to their cost. Such an assertion can be easily converted to a propositional formula that can be evaluated over any couple of flights f i , f j : L cost (f i , f j ) := i ≤ j ⇒ cost(f i ) ≤ cost(f j ). Such declaration can be audited with a Bobby audit that regularly requests A to verify if L holds. In our example, an input for which L is violated is a couple of two returned flights f a , f b such that f a is more expensive and yet presented before f b (formally: a < b∧cost(f a ) > cost(f b )). If such violating input is found, the algorithm stops and reports this observation.

Algorithm 1 The Bobby audit

Bobby approaches in the literature

We now illustrate some examples of Bobby from the literature.

Cookies and transparency consent form auditing

The GDPR and ePrivacy Directive recently set that European users must explicitly consent to non-necessary data collection, in general stored as a consent cookie on the users' computers. The checking of this rule is easily auditable: Matte et al. [START_REF] Matte | Do cookie banners respect my choice?: Measuring legal compliance of banners from iab europe's transparency and consent framework[END_REF] implemented a crawler that i) visits a target website without any interaction and ii) detects the writing of a cookie registering consent by the target website. In that typical Bobby audit, the input space X can be all the target's webpages, and the predicate is L ="not a positive consent cookie".

The detection of "fairwashed" explanations

Online services are now increasingly proposing to explain the main factors driving some of their automated decisions. The rationale is for them to gain the trust of the general public. Nevertheless, there is a possibility that the provided explanations are faked (or coined fairwashed [START_REF] Aïvodji | Fairwashing: the risk of rationalization[END_REF]) to justify a discriminative decision. In work by Le Merrer et Tredan [START_REF] Merrer | Remote explainability faces the bouncer problem[END_REF], socalled incoherent pairs are looked for; these are two conflicting explanations that yet give the same decision, and are the sign of a fairwashed explanation by the audited algorithm. This can be written as L = {∄In = ((a, white), (a, black)) ∈ X 2 s.t. f (a, white) ̸ = f (a, black)}. This mimics associations that are performing discrimination tests at the entrance of clubs for instance.

Copyright auditing

Some forms of audits are dedicated to identifying copyright infringements by remote algorithms (i.e., algorithms being executed without permission). The audit result is a Boolean answer on whether or not the remote algorithm is indeed the one that is suspected. This relates to the field of watermarking, where an algorithm is queried, and returns specific outputs if it is indeed the one suspected of infringement [START_REF] Adi | Turning your weakness into a strength: Watermarking deep neural networks by backdooring[END_REF]. Here, the inputs used as queries are specifically designed to operate as identification keys. The predicate resemble L = {∀In ∈ K g , f (In) == g(In)}, with K g being the watermark.

Skin color or gender bias audits

Multiple studies fit in this class: to take a precise example, Buolamwini et al. [START_REF] Buolamwini | Gender shades: Intersectional accuracy disparities in commercial gender classification[END_REF] benchmark three commercial gender classifier systems with an intersectional dataset (skin color/gender). Gender classification accuracies are compared: the paper notes that e.g., classification is 8.1% to 20.6% worse on female than male subjects and 11.8% to 19.2% worse on darker than lighter subjects.

Interestingly, this paper first constructs a dataset of faces that have balanced gender and skin types. Assuming this dataset is standardly recognized as a good benchmark for face classification, one could imagine a Bobby approach that targets any face classification algorithm using as input D : the standardized dataset. To implement the predicate function from Algorithm 1, classification results obtained on D could be compared for instance against a 60% disparity ratio [START_REF] Feldman | Certifying and removing disparate impact[END_REF]. For any partition of D into a gender/skin type subset D s and its complementary D s, one has to compute the target algorithm's accuracy: a s = 1/|D s |. i,label(i)∈Ds A(i) == label(i). For each partition s covered by the dataset, one then evaluates the predicate L s = as as > 0.6.

Diversity in search engine results

Urman et al. [START_REF] Urman | Auditing source diversity bias in video search results using virtual agents[END_REF] track several search engines, to audit source diversity and search concentration. This is achieved by submitting a static list of 62 keywords. As for bias, the final predicate can take the form of a simple rule such as one where the diversity at search engine B must be at least 0.6 the one at B for instance.

Limits of Bobby

The Bobby forms of audits are bound to verify a predicate L over an input space X. Given an input budget N (i.e., the number of different input queries sent to A), three outcomes are possible. Among them, two are to be considered as potential limitations.

Either some input c ∈ X violating L is found (i.e., L(c) is false). In this case, an infraction has been found, and simply exhibiting c and its corresponding answers A(c) is sufficient to establish the infraction to L committed by A. Either no input violating L was found, and two sub-cases are to be distinguished:

1) N > |X|: the whole input space of A is exhausted, and no violation has been found. It is then legitimate to conclude that L is respected by A. Unfortunately, current algorithms have input spaces that are either unbound (e.g., with inputs being floats) or have a size orders of magnitude larger than typical auditing budgets N (e.g., few hundred queries for an input size of 3 × 224 × 224 corresponding to images [START_REF] Maho | Surfree: a fast surrogate-free blackbox attack[END_REF]).

2) N < |X|: no violation of L has been found within the budget N . In this quite common case, the auditor is left with a statistical guarantee but no definite answer. While the precise nature of the statistical guarantee depends on the specifics of the study (e.g., how the input space is sampled using the available input dataset), such an assertion typically translates that the empirical probability PL(c) of finding an input c that violates L is less than 1/N .

A second limitation of Bobby resides in the production of inputs to query the target algorithm. We here stress that inputs from X (l. 3) can belong to a dataset (e.g., an image dataset in [START_REF] Buolamwini | Gender shades: Intersectional accuracy disparities in commercial gender classification[END_REF]), or be formed by public data. Thus, this line hides a wide variety of situations that are both heterogeneous with respect to the technical difficulty of generating X, and heterogeneous with respect to legal consequences.

Regarding the technical difficulty first: target algorithms working on simple inputs, like a text request on a search engine, might be queried with datasets that are easily collected. On the other hand, targeting algorithms taking more complex inputs like a video, a resume or a medical record, is certainly a challenge for the auditor to constitute these datasets.

A similar legal heterogeneity also resides in constituting datasets. Consider a flight search engine whose input is awaited to be by airport names along with some future travel date: both are public data that the auditor can rightfully use. On the other hand, testing a job recommendation engine that would match candidate resumes with job offers might require the auditor to submit resumes found on the web for which privacy rights (among others) exist and requires specific conditions to be processed (i.e., consent). In this second case, a judge might consider that the auditor had no right to use these data, and hence refuse to consider the conclusions drawn.

A third limitation of Bobby is the reliance on a propositional formula L: while some desirable behaviors of target algorithms can easily be converted into propositional formulas (e.g., if declared age is below 9, do not show ads), some others are intrinsically impossible to convert to such a logical statement (e.g., if declared age is below 9 do not propose shocking videos). This limits the applicability of Bobby audits to some specific, well-defined and desired properties.

The Sherlock audit form

This second form of audit algorithms is more elaborate, flexible and does not focus on the verification of a single propositional formula. These audits target a different set of infractions that, instead of relying on the collected outputs alone, rather relate to the general behavior of the audited algorithm. To pursue the parallel with policemen: building the case for, say, a murder requires our policeman to come up with a complete narrative (including motives, absence of alibi, etc.).

A Sherlock audit also needs to collect interaction sessions that characterize the target behavior (i.e., sequences of input and output pairs), and based on these examples, to interpolate on the behavior of the target algorithm. A typical Sherlock audit thus contains two phases: a first phase that builds a local model of the target algorithm (hereafter named a surrogate), and a second phase that analyses the surrogate to extract its desired properties.

The algorithm is presented in Algorithm 2. The input crafting operation (l. 3) is here central: it pertains to a general plan to extract specific information from A, in order to create an accurate surrogate for A on the auditor's machine. This local surrogate S is then analyzed locally. While Bobby audits simply evaluate a predicate L, the analysis of S is much more open-ended (ranging from identifying shocking corner cases to characterizing the internal logic and comparing against other equivalents of A). To capture this diversity in a compact way, we define the set of Acceptable situations the auditor would refer to when conducting such an analysis. As an illustrating example, consider the same flight search platform, driven by algorithm A. Let us in this case assume the target platform does not declare anything on the techniques A relies on to rank flights F (c) = f 0 , . . . , f j (maybe merely using the term "relevance"). A typical Sherlock audit task would be to study and understand how A ranks its results. Hence, an approach here could consist in collecting many example rankings F (c 1 ), F (c 2 ), . . ., and study without any prior different factors (cost, but also duration, number of layovers, departure time, company) that could explain (correlate) with the observed rank of flights.

Algorithm 2 The Sherlock audit

A typical use case for such a task would be to show that A deliberately favors the flights of some company they are in business with. This example relates to the historical case of SABRE, American Airline's flight reservation system, that used "screen science" to favor its own flights over its competitors by systematically presenting competitors on the second page of the search results [START_REF] Sandvig | Auditing algorithms: Research methods for detecting discrimination on internet platforms. Data and discrimination: converting critical concerns into productive[END_REF].

With Sherlock, and as opposed to Bobby, the input data can be fully crafted. This means that there is no prerequisite for a dataset; data can be forged with the objective of triggering some specific behavior for the remote algorithm. This is precisely what line 3 in Algorithm 2 builds on: new input/output pairs are used to retrain a surrogate, which outputs will become more and more similar to the ones of the audited algorithm.

Reduction to Sherlock

We now list a set of notorious research works that fall into the Sherlock audit form.

Surge price forecasting for Uber

We refer to the paper entitled "Peeking Beneath the Hood of Uber" [START_REF] Chen | Peeking beneath the hood of uber[END_REF], where authors rely on some data capture (measured supply, demand, estimated waiting times and surge prices) to fit three linear regression models. Their aim is to predict the surge multiplier in the next 5-minute interval. The inputs are crafted from using several smartphones, for in particular bringing a variety of locations in the inputs to the algorithm.

This fits directly the core of Algorithm 2, where a surrogate is trained from the queried data, so that after the query budget is over, the surrogate is used to perform a final test.

Tracking action consequences in outputs with XRay

The audit by Lécuyer et al. [START_REF] Lécuyer | Xray: Enhancing the web's transparency with differential correlation[END_REF] creates fake accounts to make them interact with the audited platform (Gmail for instance), and detect which data input (e.g., an email) has likely triggered a particular output (e.g., a received ad in Gmail). Distinct ads on each account are tracked. A correlation engine is run, in order to associate inputs and outputs. To that end, the placement of inputs on given accounts is crucial to be able to properly infer associations.

A Bayesian model is proposed as a surrogate to simulate the audited service, by computing the probability to observe certain outputs given targeting associations.

Explaining ML decisions with LIME

The goal of LIME [START_REF] Ribeiro | why should i trust you?" explaining the predictions of any classifier[END_REF] is to explain the decisions of a machine learning model in the vicinity of a given input x, by training sparse linear surrogate models as explanations. These models are thus local surrogates. Input samples are drawn uniformly at random around x, to obtain a perturbed dataset (along with its labels returned by the remote model).

COMPAS

A celebrated example of Sherlock audits is the analysis of COMPAS 4 , an algorithm used by judges, probation and parole officers to assess a criminal defendant's likelihood of becoming a recidivist. This study was used to whistleblow on the bias present in the audited models.

Limits of Sherlock

While Sherlock audits can in principle target any algorithm, it comes at a price: first, a greater cost, both the number of human intervention required to exploit obtained results and in the amount of requests such results usually require. Second, it leads to a weakened power as the conclusions are ultimately drawn from a model whose interpolation capabilities can always be questioned.

Sherlock audit forms first usually require more human intervention in their design and exploitation. This can first be explained from a purely computational perspective: Bobby audits are bound to extract a binary information from the target (i.e., L is true or false); hence literally extracting the minimal amount of information, while Sherlock audits are supposed to extract much more information. Consider for instance the airline ranking audit: while the Bobby information only verifies the statement issued by the platform, Sherlock is supposed to come up with a narrative identifying the behavior of the target by generalizing a handful of observations. Typically involved steps are: identify potential functions corresponding to the observed behavior, test and validate potential functions, confirm or infirm each one, confirm conclusions drawn on the surrogate with the real target, and so on. Such steps are time-consuming, and involve a wide variety of skills such as exploratory analysis and statistics.

The second limiting aspect is the number of requests. Indeed, training a local surrogate model naturally exhibits a trade-off between the training set size (i.e., the number of requests issued to the target algorithm) and the accuracy of the resulting surrogate. Hence, to achieve good accuracy, large volumes of input data are often required. As a result, means to automate the generation of inputs to query the target are often necessary. A positive side effect is that generated data will not be protected (unlike personal data). However, such an automation might not be easy, or require considerable human intervention.

The law perspective on audits

We now discuss the interplay of audits with the law (as summarized in Figure 1).

C o m p l e x A p p r o a c h

Fig. 1 Overview of the perspectives presented in this article: a given situation (auditor, target algorithm and offense) defines both possibilities in terms of legal outcomes and in terms of auditing approaches. In favorable situations, law defines a clear predicate that can simply be tested online. When no such predicate exists, auditors have to infer more about the target. Relying on such inferences in a court case is more complex, hence those audits are often used to leverage public opinion through whistleblowing.

Legal issues regarding the identity of the auditor

The legal status of the auditor has key consequences on two aspects of the auditing process : 1) the legal risks incurred by the auditor and 2) the judicial opportunities for the audit results. First, a regulatory authority, commissioned by the law to evaluate the compliance of online algorithms to specific regulations, is granted specific powers, which is removing all legal risks a regular person would face. Likewise, a contract between the auditor and the audited algorithm's operator allows a private auditor to realize some actions which would represent a violation of legal terms for third parties. Researchers do not benefit from specific legal protection yet. Auditing an algorithm without authorization of a national authority nor the consent of the owner (or operator) of the audited algorithms exposes the auditor(s) to lawsuits.

Second, the relationship between the auditor and the audited algorithm also plays an important role in which ways audit results can be used. If it turns out that the audit results of the targeted algorithm highlight illegal content or behavior, this does not allow the auditor to take legal action by himself. Indeed, to start legal action, the law requires a "standing", which is a legal term referring to the existence of an interest in the claim for the claimant 5 . In other words, the claimant (the auditor in our context) must gain a benefit or avoid a loss through the court action. And this is often not the case for most of the research activities. The possibility that remains is then to take the audit results to a national authority or to an association that can act on behalf of the people it represents, having therefore interest to start legal action (see for instance actions by consumer associations).

Legal issues regarding the inputs used to audit

From a legal perspective, a clear split exists based on how the inputs used by the two audit forms are chosen. While Sherlock fully crafts its inputs for the purpose of its investigations, Bobby uses preexisting data (i.e., existing pictures or user profiles) as a basis for its audit.

Nowadays, existing data of all types can be protected by plenty of legal texts. The GDPR [START_REF]du Parlement Européen et du Conseil du 27 Avril 2016 Relatif à la Protection des Personnes Physiques à L'égard du Traitement des Données à Caractère Personnel et à la Libre Circulation de Ces Données[END_REF] presents the rules of protection on personal data 6 . Every action made on personal data is called processing 7 these data. Furthermore, there are special categories of personal data which are subject to additional protections to be processed (health, national security or even private data matters). Using existing data as inputs for an audit exposes the auditors to the risk of processing personal data, while fully crafted do not. Therefore, Bobby can lead to more legal obligations and legal risks 8 for the auditors whereas Sherlock avoids it. Data can also be protected, regarding its economic value, as the content of a database [33], but also as an artistic creation through a copyright law (e.g., as a picture, a drawing). Again, using existing data can lead to process protected data, exposing auditors to legal risks.

These are just two examples of the many regulations applicable to data. From health data to passengers' data, their processing without compliance with legal obligations can expose to important economic sanctions, if not criminal ones in specific situations.

In short, the following counter intuition holds: Bobby audits, which look easier to design than Sherlock audits, expose the auditor to more legal risks because of the processing of existing data.

Scientific proof vs legal evidence

Admissibility of the proof

When scientists perform an investigation or an experiment, they produce results that will be evaluated by their peers within their research community.

5 Legal requirement in article 31 of the French code of civil procedure.

6 GDPR defines personal data as "any information relating to an identified or identifiable natural person" which is an extremely wide definition.

7 Article 4 of the GDPR defines processing as "any operation or set of operations which is performed on personal data", e.g. collection, recording, consultation, alteration, use, etc. 8 Violation of the GDPR can lead to administrative fines up to 20 million of euros or 4% of the total worldwide annual turnover.

The results are evaluated depending on the criteria of the community and this will decide the impact on the field. Nevertheless, the acceptance by a scientific community has no influence on the judge's decision during a case. Law has its own criteria when it comes to the admissibility of legal evidence. It is a specific discipline in law studies called procedural law. Depending on the legal system the auditors are operating in, those can be written explicitly or not. To illustrate the notion of "admissibility", we are going to take the example of the French legal system. In France, the admissibility of evidence depends on the loyalty of the establishment of the evidence. The "loyalty of proof" principle is a recurring principle there. However, the specification of this principle will change with the field of law the auditor is working in. As an example, the loyalty of proof principle is different in criminal law and in administrative law, and different in civil law. Even within these legal domains, the principle could differ from one action to another. Law is all about contexts and exceptions.

The legality of evidence

In particular, the legality of the evidence principle is described in the French Civil Code 9 . It means that evidence obtained illegally cannot be used in court afterward. For instance, elements found in a computer or any IT system without the proper authorization 10 to do so will not be accepted by the judge, even with proofs of the defendant guilt. Indeed, a piece of evidence obtained by means of an unfair process is inadmissible. Some exceptions to this principle exist in criminal law or in labor law.

Acceptation of evidence

The acceptance of the found evidence relies on the interpretation of the judge. The infringement(s) must be necessary and proportionate 11 to the purpose, i.e., finding evidence to support a claim. Necessary evidence means the illegal or disloyal evidence brought by the claimant is their only solution to support their claim. In other words, was the infringement or disloyalty necessary to support the claim? Then the judge will perform the proportionality test. She will balance the opposed side's rights and liberties that have been violated by the audit establishing the evidence and the evidence stakes for the claimant. This analysis of the necessity and the proportionality is solely based on facts. It means no generalization can be made out of the decisions given by the judges already 12 . In the end, if the judge qualifies the proof as necessary and proportionate, the evidence will be admitted in court.

9 Article 9 : "Each party has the burden of proving in accordance with the law the facts necessary for the success of its claim." 10 About the sanctions of fraudulent access in an IT system, see art 323-1 of the French penal code.

11 These notions of necessity and proportionality of legal evidence have been admitted in the first place by the European Court of Justice. For more information, see J. Van Compernolle, "Les exigences du procès équitable et l'administration des preuves dans le procès civil", RTDH 2012. 429.

Consequences on the choice of the auditing form

This concern echoes the inputs used by the two forms of audits and the way the audit was conducted. When someone is auditing without proper authorization, not only does the author get responsible for the violation of rights realized during the process, but also she is making the admissibility of the proof in court more difficult. If the purpose of the audit is to sue a platform, one has to be extremely cautious about the violation realized during this audit. Otherwise, the potential legal consequences of an investigation will be jeopardized. Designing an auditing algorithm that makes the less severe violation of the law is important to increase the chances of admissibility by satisfying (at least theoretically) the condition of necessity. The fact that a better-suited audit algorithm could have been used to search for the targeted element, will reduce the chances to see the evidence being admitted in court.

The probative value of the audit results

We have discussed the admissibility of scientific proof in court. But being admissible in the court is not related to the level of importance given to the scientific proof. Legal rules distinguish between the admissibility of the evidence and its evidential value. Admissibility means whether or not the elements will be accepted in the list of evidence, while the value of the proof means the value given in the court by the judge of the evidence that has been accepted beforehand. The probative value of an element can be written explicitly in the law or be let to the sovereign discretion of the judges.

Intrinsic limitation

Beforehand, algorithms face a fundamental limit regarding probative value. Algorithmic audits of algorithms target technical elements, written in code, which are an interpretation of a natural language requirement, law or others, made by the auditors. Translating unclear terms of a legal text enforces an interpretation from the auditor: it is subjective and may not be adequate to the state of law and interpretation of the judge. Audits will always succeed more in identifying simple and explicit illegal situations, which are clearly defined in legal texts or by court decisions. In consequence, the output probative value can always be challenged on the basis that it does not represent what the law definition meant in the first place: this is an inherent limit to all audits to be aware of before taking legal risks, while auditing an algorithm.

The probative value of Bobby

Bobby allows for automatic research through a digital environment which could have been done manually (by a lot of persons tasked to do so): the result can be found again by a person if a confirmation of the existence of a given output is required. There is one positive and one negative legal consequence regarding this. Starting with the negative one, the Bobby audit is useful in court only when manifest evidence is needed. By "manifest" evidence, the law refers to illegal content or situation which are noticeable. There is no need for further justification for the claimant in addition to the audit results. The utilization of Bobby audits in the goal of legal proceeding is limited by the fact that they can only be found in existing apparent elements in the environment. However, because those elements are existing and are apparent, their correctness is guaranteed therefore their probative value should remain equal to the same evidence brought by hand. Using Bobby audits should have consequences on the admissibility of the evidence but should not impact its evidential value.

The probative value of Sherlock

On the other hand, Sherlock audits provide interpolations that are most of the time hidden from the user. A working condition for a Sherlock audit is to create a representation of the audited algorithm. Because by definition the audited algorithm is not open to the public, the correctness of the representation is not guaranteed. Every statement made about the surrogate model is an assumption. As an illustration, we can take the example of someone who tries to assess the potential discrimination bias against women within a hiring decision system [START_REF] Raghavan | Mitigating bias in algorithmic employment screening: Evaluating claims and practices[END_REF]. This potential inaccuracy, inherent to all statistical tools, will have consequences regarding the probative value given by the judge to the audit results. This new question has not been addressed by courts yet. Therefore it is not possible to propose an evidential value to Sherlock audits. It could become strong evidence, like DNA in a paternity test 13 , or it could be a more contextual, secondary evidence as DNA in criminal proceedings 14 . It is crucial to realize that the use of an audit algorithm, with all the stateof-the-art elements of the scientific procedure followed, does not guarantee that the output will be considered as important evidence from a court, compared to a testimony or other kind of evidence. The consideration of the importance of potential inaccuracy and postulates in audits from a user perspective will take time for lawmakers or judges.

In that light, Sherlock audits are good candidates to trigger ethical debates, around findings that may be deemed inappropriate in some societies. When these debates exhibit consensus and a compatible nature, obligatory norms may follow [START_REF] Carrillo | Artificial intelligence: From ethics to law[END_REF]. Independently of norms, ethical misconducts may be reported through whistleblowing [START_REF] Boot | The Ethics of Whistleblowing[END_REF], which we now discuss.

The whistleblowing alternative

According to the Council of Europe, "whistleblowing is about bringing into the open information on activities that have harmed or threaten the public interest. People blow the whistle because they believe that these activities should be stopped and remedial action taken". Whistle-blower regulation initiatives emerge in some national law 15 . In 2019, 6 years after the Snowden scandal, the European union voted a directive on the protection of whistle-blowers [START_REF]the European Parliament and of the Council of 23 October 2019 on the Protection of Persons Who Report Breaches of Union Law[END_REF]. The directive grants a legal protection for people who "acquired information on breaches in a work-related context [...] 16 .

In the situation of audit algorithms, work-related context means the whistleblower had, has or is going to have a work relationship with the owner of the controversial audited algorithm. In June 2022, the Council of Europe released a report 17 assessing the impact of the directive. Whistleblower protection is also part of the public ethics framework adopted by the Council of Europe in 2020. 18 3.4 The promising role of academics in platform auditing

The "vetted researchers" status

Nowadays, auditing the algorithms of platforms without specific grants exposes the auditor to several legal risks. However, this existing situation might end soon e.g., in Europe with the regulation on a single market for digital (Digital Services Act or DSA for short) brought by the European Commission [START_REF]laying down harmonised rules on artificial intelligence (artificial intelligence act) and amending certain union legislative acts com/2021/206 final[END_REF].

The proposal provides guarantees for civil society, including researchers, for the control of platforms. Beyond the creation of a trusted flaggers status, the European proposal enforces the "very large online platforms" 19 to provide access to data to vetted researchers.

To be qualified as a vetted researcher, a person needs to satisfy four conditions: a) to be "affiliated with academic institutions", b) to be "independent from commercial interests", c) to have and prove their "expertise in the fields related to the risks investigated or related research methodologies" and d) to commit and to be able to "preserve the specific data security and confidentiality". Access to the data will not be asked directly to the very large platforms targeted by the research but through the Digital Services Coordinator, a national authority designated by each European member state 20 .

This vetted researcher's status is announced with the following purpose: the identification and understanding of systemic risks created by very large online platforms. Those systemic risks can be related to the functioning of the platforms or the use of services. Article 26 gives the list of these risks, therefore the description of the purposes that can be pursued by vetted researchers: a) "dissemination of illegal content", b) "any negative effects for the exercise of the fundamental rights to respect for private and family life, freedom of expression and information, the prohibition of discrimination and the rights of the child" 21 , c) "intentional manipulation [...] with an actual and foreseeable negative effect on the protection of public health, minors, civic discourse, or actual or foreseeable effects related to electoral processes and public security". This new legal status is an important first step forward for the algorithm audit community and a call for the involvement of civil society in digital issues.

Limitations of the vetted researchers status

The European Council gave its final approval to the Regulation of the DSA in October 2022, but months can pass before its application. The vetted researchers is a status created to legally allow researchers, mostly academics, to identify and understand the systemic risks existing in very large online platforms. First, the mentioned systemic risks are not limited to very large online platforms: illegal content, discrimination or privacy are issues that can also be found in smaller online platforms. Second, these risks are not limited to online platforms at all and can exist in offline applications as well. Researchers may still perform audits before the DSA is applied, or in a context that is out of the vetted researchers' scope of action. In these situations, the guidelines described in this article are still to consider.

Conclusion

In the context of widespread decision-making algorithms, and their significant impact on populations, we structured audit forms not in relation to ethical concepts (that require social interpretations [START_REF] Johnson | Ethical ai is not about ai[END_REF] and then are harder to grasp in general as seen in [START_REF] Brown | The algorithm audit: Scoring the algorithms that score us[END_REF]), but with regards to law. We have discussed that the outcome of audits -that we fit into two categories (Bobby and Sherlock)do not necessarily conforms to what is a proper building of a case. Multiple precautions must be carried out before and during the audit, in order for evidence to be considered so that they have an impact in practice. A central objective also is to avoid the auditor to be prosecuted.

There is an increasing number of approaches to auditing algorithms. In particular, there are some variations of Bobby that use real inputs to create synthetic ones, which will in turn be used against the algorithm [START_REF] Mahmood | Back in black: A comparative evaluation of recent state-of-the-art black-box attacks[END_REF]. This makes the boundary between Bobby and Sherlock more difficult to distinguish. As an auditor, it is necessary to continuously monitor the development of auditing techniques and evolving laws.

Finally, we stress that the absence of proof of bias in an audited algorithm does not give any certainty. This echo the so-called diesel gate, and more technically the possible temptation for fairwashing [START_REF] Aïvodji | Fairwashing: the risk of rationalization[END_REF], where the audited algorithm can "sandbox" the auditor into an acceptable vision of its operation. In such a case, more advanced regulation is needed for a proper impact on decision-making algorithms.

1

  Proposal for a REGULATION OF THE EUROPEAN PARLIAMENT AND OF THE COUNCIL on a Single Market For Digital Services (Digital Services Act) and amending Directive 2000/31/EC COM/2020/825 final. 2 Proposal for a REGULATION OF THE EUROPEAN PARLIAMENT AND OF THE COUNCIL LAYING DOWN HARMONISED RULES ON ARTIFICIAL INTELLIGENCE (ARTIFICIAL INTELLIGENCE ACT).

  Input: A an algorithm to audit. A : X → Y , L a propositional formula over an input dataset X L = {x1 . . . x l } ⊂ X and corresponding outputs Y l = {A(x1), . . . , A(x l )} ⊂ Y Output: True if the behavior is illegal, False otherwise infraction = False for X not exhausted do Pick x1, . . . , x l Collect Y1 = A(x1), . . . , Y l = A(x l ) if not L(X1, Y1), . . . or L(X l , Y l ) then infraction = True ; // A does not verify L break end return infraction ; // Boolean on violation of L

  Input: A: an algorithm to audit. A : X → Y , N : A budget (maximum number) of queries I ← find input to A /* Build a surrogate S to A */ while n < N do craft a new In ∈ I interact with A through In; collect A(In) S ← Retrain(S ∪ (In, A(In)) end /* S is now a constructed surrogate of A. Analyze S */ return Analysis(evidence) is Acceptable /* Return false if some violations are found */

16 Article 4 "

 4 personal scope of the European directive" 17 Evaluation report on Recommendation CM/Rec(2014)7 on the protection of whistleblowers https://www. coe.int/en/web/cdcj/activities/protecting-whistleblowers 18 Guidelines of the Committee of Ministers of the Council of Europe on public ethics (2020), E.h Section. 19 Term defined by article 25 of the DSA as "online platforms which provide their services to a number of average monthly active recipients of the service in the Union equal or higher than 45 million [...]." 20 Article 38 of the Digital Services Act.

  13 On the importance of DNA in a paternity test, see, among others: Cour de cassation, civile, Chambre civile 1, 25 septembre 2013, 12-24.588, Inédit, 2013. and Cour de cassation, civile, Chambre civile 1, 25 septembre 2013, 12-24.588, Inédit, 2013.

[START_REF] Bandy | Auditing news curation systems: A case study examining algorithmic and editorial logic in apple news[END_REF]

Because DNA does not provide all the elements necessary to establish guilt, its usefulness and utilization are actually limited. See Julie Leonhard, " La place de l'ADN dans le procès pénal ", Cahiers Droit, Sciences & Technologies, 9, 2019, 45-56.

[START_REF] Galdon Clavell | Auditing algorithms: On lessons learned and the risks of data minimization[END_REF]

In France, a legal protection is granted in 2016 through a law for transparency and against corruption.

Règlement (UE) 2016/679.

COMPAS stands for "Correctional Offender Management Profiling for Alternative Sanctions". About the 2016 analysis: https://www.propublica.org/article/how-we-analyzed-the-compas-recidivism-algorithm.

G. Lardeux, "Le droit à la < preuve > : tentative de systématisation", RTD civ. 2017. 1.

Statements and Declarations

The authors declare there are no competing interests with this work.