Marco Mansueto
email: marco.mansueto@cea.fr

Antoine Chavent
email: antoine.chavent@cea.fr

Ricardo C Sousa
email: ricardo.sousa@cea.fr

Liliana D Buda-Prejbeanu
email: liliana.buda@cea.fr

Ioan L Prejbeanu
email: ioan.prejbeanu@cea.fr

Bernard Diény

Bernard Dieny
email: bernard.dieny@cea.fr

Isotropically coercive free layer integration in a magnetic tunnel junction for neuromorphic applications

Keywords: spintronics, memristor, MTJ, TMR

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The exponentially increasing amount of data in the information and communication technology (ICT) world in the last decade results in a corresponding increase of electrical power consumption associated to it. In 2018, it was estimated that the ICT's amount of data generated worldwide reached 20 Zettabytes (20 bilion Terabytes). The power consumption associated to ICT is estimated to represent about 10% of the worldwide total electrical consumption [START_REF] Reinsel | IDC White Paper Data Age[END_REF]. The main causes in ICT power consumption are associated with the wireless transmission of data, displays, and inefficient computer architectures. To address this issue of ICT power consumption, the scientific community is looking to the most efficient and lowest power consuming system known, the human brain. The concept of AI is born to try to emulate the complex functionalities of the brain. Between those, the development of an architecture with massive parallelization of processing units (neurons) and memory units (synapses) seems to be a promising solution. In the last few years, several devices were proposed to mimic both neuron and synapse working principles. For the specific case of synapses, the concept of memristor, a device with a resistance value dependent on the history of the electrical charges passed through it, developed in the 70's [START_REF] Chua | Memristors-the missing circuit element[END_REF], has been exploited for the realization of devices using different technologies. An example is a device that is able to change resistance value thanks to the voltage dependent ion migration (as for example oxygen ions) in materials like TixO1-x or HfxO1-x [START_REF] Strukov | The missing memristor found[END_REF], [START_REF] Li | Review of memristor devices in neuromorphic computing: Materials sciences and device challenges[END_REF]. In this particular case, problems of endurance represent the main limitation since learning processes usually involve a large number of write cycles. In contrast, spintronic memristors do not present this problem, since the information relies on the magnetization state and no atomic motion takes place during the writing process. An example of this is a magnetic tunnel junction (MTJ) with a domain wall in the free layer that can be moved through current pulses, thus varying the proportion of MTJ area in parallel and anti-parallel configuration and correlatively reaching different resistance states [START_REF] Lequeux | A magnetic synapse: Multilevel spin-torque memristor with perpendicular anisotropy[END_REF]. Another example exploits spin-orbit torque (SOT) in an antiferromagnetic material exchange biased with an out-ofplane magnetized ferromagnet that is gradually switched [6], [START_REF] Kurenkov | Artificial Neuron and Synapse Realized in an Antiferromagnet/Ferromagnet Heterostructure Using Dynamics of Spin-Orbit Torque Switching[END_REF]. A third example uses a chain of MTJs in serial configuration, in which the MTJ can be switched one by one, thus achieving a gradual change in the total resistance [START_REF] Raymenants | Chain of magnetic tunnel junctions as a spintronic memristor[END_REF]. However, a drawback of these above mentioned spintronic devices is their relatively large size (micronic dimension) required to get several intermediate resistance states. The device we propose in this work is based on a single nanosized pillar very similar to a usual MRAM cell. The idea to achieve several resistance states is to exploit the dependence of the tunnel magnetoresistance (TMR) on the angle between the magnetization of the two ferromagnetic electrodes in an in-plane MTJ (the analyzer layer AL and the free layer FL in Fig. 1a) [START_REF] Slonczewski | Conductance and exchange coupling of two ferromagnets separated by a tunneling barrier[END_REF]. The lowest resistance value is obtained in the case of parallel configuration of the two magnetization while the antiparallel case gives the highest resistance. In between the two, the TMR is varying with the cosine of the angle as shown in Fig. 1b. The writing mechanism relies on the spin-transfer torque (STT) created by an out-of-plane spin polarized current injected in the FL layer. This polarized current is obtained thanks to an additional out-of-plane magnetized polarizer (POL) placed underneath the FL. This current generates an out-of-plane precession of the FL magnetization (shown as dashed line in Fig. 1a). The application of current pulses rather than a DC current enables clockwise or anticlockwise step by step discrete rotations of the FL magnetization achieving intermediate resistance states. The use of an additional inplane field perpendicular to the AL magnetization direction allows limiting the angular excursion of the FL magnetization to ±90° around this transverse field thus insuring the monotonicity of the resistance variation with the polarity of the applied voltage as required for a memristive behavior. A more accurate description of the functionalities of the device is given in [START_REF] Mansueto | Realizing an Isotropically Coercive Magnetic Layer for Memristive Applications by Analogy to Dry Friction[END_REF].

In this work, we present how to realize a FL able to stabilize the magnetization along different in-plane directions with isotropic thermal stability by creating an isotropically coercive free layer and how to integrate it in a complex structure as in Fig. 1a. After a nano-fabrication process, magnetic characterizations were performed to confirm the magnetically isotropic nature of the free layer.

II. ISOTROPICALLY COERCIVE FREE LAYER

A. Realization Isotropic magnetic properties were already observed in amorphous in-plane magnetized ferromagnetic systems in which the disordered crystalline structure creates a distribution of anisotropy directions all over the plane [START_REF] Dieny | Hysteresis loop and torque experiments on a random anisotropy system[END_REF]. In this work, instead, we exploit a structure based on the coupling between ferromagnet FM and thin antiferromagnet AF. It is well-known that, above a certain thickness of the AF layer, its anisotropic energy is strong enough to pin the FM magnetization along an in-plane direction thanks to the exchange bias phenomenon. However, for lower thicknesses, this AF pinning becomes too weak and the AF spin lattice gets entirely dragged due to the interfacial F/AF coupling when the F magnetization rotates or switches. This results in an increase in F coercive field while the exchange bias field is still negligible. In such system, similarly to the random anisotropy materials, the dissipation is not associated to the presence of a well-defined anisotropy axis but to numerous magnetization rearrangements at the nanometer scale occurring during the magnetization evolution. As a result, this dissipative mechanism is isotropic. From conceptual point of view, it can be described by a dry friction-like term which can be introduced in the Landau Lifshitz Gilbert equation (for more details [START_REF] Mansueto | Realizing an Isotropically Coercive Magnetic Layer for Memristive Applications by Analogy to Dry Friction[END_REF]).

B. Vibrating Sample Magnetometer Characterization

To determine the right AF thickness yielding this isotropically coercive behavior, a study of exchange bias and coercive field dependence on AF thickness was performed on a trilayer sample made of unpatterned Py(1nm)/IrMn(x)/Py(1nm). The sample was deposited by magnetron sputtering and annealed at 300°C for 1h30 under an in-plane field of 230mT. In Fig. 2a, it is shown that, for IrMn thicknesses around 2.1nm, while the exchange bias is still negligible, a steep increase of coercive field is observed as expected. This particular IrMn thickness was chosen for the following experiments.

Measurements at different in-plane angles (Fig. 2b) show that the magnetic properties such as coercive field and remanence are almost independent on this angle as in Fig. 2c, confirming the isotropic nature of the free layer. Note that the rounded shape of the loop is expected as a result of dry friction torque due to the F/AF coupling (see [START_REF] Mansueto | Realizing an Isotropically Coercive Magnetic Layer for Memristive Applications by Analogy to Dry Friction[END_REF]).

C. Rotating Field Characterization

The same sample was used for a rotating field experiment using a planar Hall (PHR) measurement setup [START_REF] Van Dau | Magnetic sensors for nanotesla detection using planar Hall effect[END_REF].

As expected, for large rotating field amplitude Hrot , the resistance varies as sin(2φH) (black curve in Fig. 3a). For smaller amplitudes of the field, the curve gradually shifts creating a dragg angle between the rotating field and the magnetization. For fields smaller than the coercive field, the magnetization no more follows the rotating field (as in the 13mT curve of Fig. 3a). The observed gradual increase of the amplitude PHR signal versus Hrot amplitude is also expected since the amount of spins following the field increases as Hrot amplitude increases. Fig. 3b shows the phase shifts between magnetization and field (i.e the dragg angle) versus field amplitude. This variation can be well fitted by the analytical solution obtained with the dry friction model. This is a further confirmation of the isotropic properties of the unpatterned free layer.

III. INTEGRATION IN THE FULL STACK AND FABRICATION

Here we briefly describe the material stack used for the device as in Fig. 1a and the fabrication process to pattern such device. All the thicknesses in parenthesis are in nm.

A. Full Stack

The F/AF/F trilayer constituting the device FL was inserted in the full structure as represented in Fig. 4a. The samples were deposited by magnetron sputtering at Spintec. After a seed layer of Pt, an out-of-plane synthetic antiferromagnet (SAF) was deposited by exploiting the perpendicular anisotropy of a multilayer structure of [Co (0.5)/Pt(0.25)]6 and [Co (0.5)/Pt(0.25)]3 separated by a thin layer of Ru(0.9) to exploit the peak of the antiferromagnetic RKKY coupling. The SAF is coupled to a polarizer layer of FeCoB(1.2) in contact with the first MgO barrier with a relatively low resistance*area product of 2Ωµm². The F/AF/F trilayer structure was then inserted between two FeCoB(1.6) layers in contact with the top and bottom MgO barrier. The second MgO barrier with a resistance*area product of 10Ωµm² was optimized to give a TMR as large as possible between the free layer and an in-plane magnetized synthetic antiferromagnet placed above the top tunnel barrier. This pinned top analyzer layer has the composition: FeCoB(1.2)/Ta(0.2)/Co(1)/Ru(0.9)/Co(2.5)/IrMn [START_REF] Kurenkov | Artificial Neuron and Synapse Realized in an Antiferromagnet/Ferromagnet Heterostructure Using Dynamics of Spin-Orbit Torque Switching[END_REF].

A capping layer of Cu and Pt protects the stack from oxidation and constitutes the top electrical contact to the stack.

B. Nano-Fabrication Process

The samples were annealed at 300°C for 1h30 under an in-plane magnetic field of 230mT. The field is intended to set the orientation of the in-plane analyzer magnetization. After deposition of an hard mask of Ta, e-beam lithography was used to create circular resist patterns with diameters between 20nm and 150nm. A Reactive Ion Etching machine was then used to etch the Ta layer and creates the hard mask Ta pillars (as in Fig. 4b). The magnetic stack was then etched by Ion Beam Etching at different incident angles. The etched stacks are then embedded in a planarizing resist which is subsequently partially etched to open the top electrical contacts. The contacts are then patterned as in Fig. 4c.

IV. CHARACTERIZATION RESULTS

Magnetic characterizations were performed on these samples to test the success of the fabrication process and to assess the isotropic properties of the FL at device level. An example of resistance -field loop is presented in Fig. 5a. The device has a diameter of 80nm and a total resistance consistent with the expected resistance*area product (2+10Ωµm² ). The TMR value is around 30% and the rounded shape of the loop is similar to the full sheet film one of Fig. 2b confirming the dry friction like reversal with field.

A. In-plane Field Loops

A series of hysteresis loops measurement was performed at different in-plane angles with respect to the AL layer magnetization direction. This was realized with two orthogonal couples of coils with the sample placed at their center. The two components of the field (along x-axis and yaxis) were then adjusted to obtain different in-plane angles of the field. As shown in Fig. 5b, the coercive field of the different loops do not vary while the high and low resistance states at zero field are gradually shifting towards the center. This demonstrates that the magnetic properties are again independent on the in-plane angle even at device level. The obtained variation of TMR was directly compared to the cosine variation expected in the ideal case in Fig. 5c showing good agreement.

B. Rotating Field Experiment

Rotating field experiments were performed on a similar setup as the previous one. The two couples of coils along x and y directions are supplied by 90° out-of-phase current creating a rotating field on the device of constant amplitude. The rotating frequency could be varied up to 5Hz. The results shown in Fig. 6a correspond to an applied field of 40mT and frequency of 2Hz. The resistance (Fig. 6b) is oscillating between its maximum and its minimum value with the same frequency as the rotating field one. Note that the resistance smoothly varies in quasi-sine form, passing through all the intermediate values. In contrast to this, if the same experiment is performed on a standard binary MTJ, the result of Fig. 6c is observed. The resistance values are then distributed non uniformly with two main regions in the high resistance region and in the low resistance region. The magnetization does not follow smoothly the rotating field. Due to its uniaxial anisotropy, during its rotation, the FL magnetization undergoes irreversible jumps while it crosses its hard axis direction. Also in this case, the isotropic properties of the device can be confirmed.

V. CONCLUSION AND PERSPECTIVES

In conclusion, a memristive device based on the angular variation of TMR requires the development of an isotropically coercive free layer able to stabilize the magnetization along different in-plane angles with equal thermal stability. In this study, we demonstrated the realization of such layer by exploiting the coupling at the FM/AF interface of a trilayer FM/AF/FM. The full sheet film measurements of coercivity and exchange bias at different inplane angles and the rotating field experiments confirmed the isotropic properties of the free layer. After the development of the total stack, and fabrication process, devices with diameter between 20nm and 150nm were measured with the same techniques. The hysteresis loops at different angles showed that the angular variation of TMR is, as expected, cosine-like while the coercive field is kept constant. The rotating field experiments demonstrated that the resistance can take all the intermediate values between its maximum and its minimum without any preference. This does not occur in a standard MTJ where the magnetization is stable along two opposite directions giving rise to a bimodal distribution of resistance values. These promising results bring us to the next step where the use of Spin Transfer Torque through the application of pulses will allow the discrete rotation of the magnetization. The realization of this device would represent a good option for a scalable, easily integrable and low power consumption device for neuromorphic computing.

Fig. 1 a

 1 Fig. 1 a) Schematics of the device. b) TMR variation with the in-plane angle between analyzer and free layer as in the inset and schematics of the relative resistance loops.

Fig. 2 a

 2 Fig. 2 a) Coercive field HC and exchange bias HEB dependence on IrMn thickness. b) Hysteresis loops at different in-plane angles with respect to the annealing field direction (red arrow in the inset). c) Coercive field HC and exchange bias HEB dependence on the in-plane angle φH. [10]

Fig. 4 a

 4 Fig. 4 a) Schematics of the device material stack with thicknesses in nm. b) SEM image of a hard mask after RIE etching. c) Top view of the two contacts.

Fig. 5 a

 5 Fig. 5 a) Example of resistance loop of a device with 80nm diameter. b) Example of evolution of resistance loop with the in-plane field angle. c) TMR variation with the in-plane angle in an ideal case (red line) and experimentally obtained.

ACKNOWLEDGMENT

The authors want to acknowledge the Plateforme de Technologie Amont PTA for technical help in the fabrication process. Funding from ERC Advanced grant MAGICAL 669204 is acknowledged.


Kotb Jabeur

Gregory Di Pendina

Guillaume Prenat

Liliana Daniela Buda-Prejbeanu

Bernard Dieny

Fellow

Compact modeling of a magnetic tunnel junction based on

Keywords: Magnetic tunnel junction (MTJ), modeling, MRAM, spin orbit torque (SOT), spintronics, Verilog-A

I. INTRODUCTION

C URRENT researches in spintronics are investigating a variety of magnetic memory devices, which are promising candidates to improve the capability of existing RAMs by adding one key feature, the non-volatility (NV), and replace them as a long-term purpose, once they reach the maturity stage. For the first generation of the NV MRAMs, the switching mechanism was controlled by applying external magnetic fields. The approach is known as field-induced magnetic switching (FIMS) [START_REF] Wolf | Spintronics: A spin-based electronics vision for the future: Magnetism and materials[END_REF]. It suffers from selectivity and scalability issues. The selectivity issues were partially solved by the so-called toggle approach proposed in [START_REF] Engel | A 4-Mb toggle MRAM based on a novel bit and switching method[END_REF], but field-based writing schemes remain hardly scalable below 90 nm. Then came the thermally assisted switching (TAS), which allowed solving the selectivity issues while reducing the writing energy and improve the scalability [START_REF] Prejbeanu | Thermally assisted MRAM[END_REF].

Both consist of a magnetic tunneling junction (MTJ) as a storage element and require high writing current and consequently increased power consumption and die area due to large transistors and huge write lines. In addition, magneticfield-based MTJs suffer from the lack of robustness against downscaling. Furthermore, more advanced writing scheme has been proposed to solve the latter issues; spin polarized current can be used through the junction to apply spin transfer torque (STT) [START_REF] Deng | Low power magnetic full-adder based on spin transfer torque MRAM[END_REF], [START_REF] Zhao | A scaling roadmap and performance evaluation of inplane and perpendicular MTJ based STT-MRAMs for high-density cache memory[END_REF] and switch the magnetization of the storage layer. Although the writing current has been decreased by orders of magnitude and the scalability issue is solved, two main shortcomings are still limiting the reliability and endurance of STT-MRAMs: 1) in large-scale memories, the high current density required for writing can occasionally damage the MTJ barrier and 2) since writing and reading operations share the same path (through the junction), it remains a challenge to fulfill a reliable reading without ever causing switching.

To solve the latter issues and ensure a continuous advance of MRAM, an interesting phenomena known as spin-obit coupling can be exploited to create a torque on the storage layer magnetization and switch it [START_REF] Gambardella | Current-induced spin-orbit torques[END_REF]. The two fundamental physical phenomena responsible for the spin-orbit-torque (SOT) switching mechanism are the Rashba effect [START_REF] Gambardella | Current-induced spin-orbit torques[END_REF], [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF] and the spin Hall effect (SHE) [START_REF] Liu | Current-induced switching of perpendicularly magnetized magnetic layers using spin torque from the spin Hall effect[END_REF], [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF].

In [START_REF] Guo | MRAM device incorporating single-layer switching via Rashba-induced spin torque[END_REF], the Rashba spin orbit coupling effect has been utilized to model and design a non-volatile memory with higher spin torque efficiency because of the spin injection free device. In addition, the switching of perpendicularly magnetized ferromagnetic layer based on spin-orbit interactions was reported in [START_REF] Gambardella | Current-induced spin-orbit torques[END_REF] and [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF]. Then, Liu et al. [START_REF] Liu | Current-induced switching of perpendicularly magnetized magnetic layers using spin torque from the spin Hall effect[END_REF] have confirmed the attracting potential of spin-orbit interaction to build magnetic devices favorable for logic storage. Because of its threeterminal architecture, Miron et al. and Liu et al. [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF], [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF] claim that SOT switching devices can be built with independent reading and writing paths and so avoid challenges encountered with the state of the art of conventional STT MRAM, whereas keeping competitive switching currents.

As the semiconductor industry is progressively going toward hybrid CMOS integrated circuits, compact model development has become a cornerstone in the circuit/system verification tool flow. By view of this fact and in order to make in evidence the prospective of the SOT magnetic device, it becomes a fundamental step to define a compact model for the SOT device to be used for design purposes based on magnetic storage logic. To the best of our knowledge, this paper represents the first macro-model written in Verilog-A of the SOT-MTJ. The model describes the behavior of the three-terminal SOT device and includes the dynamic behavior described by the Landau-Lifshitz-Gilbert model (LLG) [START_REF] Landau | Theory of the dispertion of magnetic permeability in ferromagnetic bodies[END_REF].

To follow the variation of the SOT-MTJ resistance, Julière's model [START_REF] Julliere | Tunneling between ferromagnetic films[END_REF] as well as Simmons's model [START_REF] Simmons | Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film[END_REF] were used in the expression describing the conductance through the junction. Moreover, for an improved accuracy, we integrated the dynamic conductance given by Brinkman model [START_REF] Brinkman | Tunneling conductance of asymmetrical barriers[END_REF] and we took in consideration the dependence of magnetoresistance on bias voltage. Finally, a special interest has been given to Rashba effect and SHE torques inside the LLG equation to highlight the impact of these two factors on the dynamic of magnetization switching intensively argued in [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF]- [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF].

This paper is organized as follows. In Section II, we present the SOT-MTJ device as well as its switching mechanism. Then, in Section III, a description of the compact model is given. Section IV represents the simulation results and discussions. Finally, we conclude this paper in Section V.

II. MAGNETIC TUNNEL JUNCTION

A. Basics

The MTJs are used as the basic memory elements in MRAM cells and magnetic logic devices. The MTJ is a nanostructure composed of two ferromagnetic layers (FM), such as cobalt, iron, or nickel, separated by a thin layer of insulator typically alumina oxide or magnesium oxide, which represents the tunnel barrier. The first FM layer (hard layer)-with a pinned magnetization-acts as a reference, whereas the second FM layer (soft layer)-with a free magnetization-acts as a storage layer.

The magnetization of the storage layer can be switched between two stable states, either parallel (P) or antiparallel (AP) with respect to the reference layer. Electrons can tunnel through the thin barrier (∼1 nm) when a bias voltage is applied between the two electrodes of the device. The MTJ resistance is low (or high) for a P (or AP) magnetization configuration. These two configurations can be used to code two different logic states, logical 0 and 1, for instance (Fig. 1). This resistance variation behavior was first observed and explained in [START_REF] Julliere | Tunneling between ferromagnetic films[END_REF]. Tunnelling magnetoresistance ratio (TMR) describes the ratio between the two resistance values as

TMR = R AP -R P R P . (1)
The highest effects observed to date with aluminum oxide insulators are ∼70% at room temperature. The highest TMR recorded in literature is up to 600% at room temperature and >1100% at 4.2 K observed in junctions of CoFeB/MgO/CoFeB [START_REF] Ikeda | Tunnel magnetoresistance of 604% at 300 K by suppression of Ta diffusion in CoFeB/MgO/CoFeB pseudo-spin-valves annealed at high temperature[END_REF]. A MTJ with a large TMR value offers an accurate read of the output signal. Indeed, for memory applications, we can use a single MTJ memory cell or two MTJs, which have opposite magnetic states, to store one data bit [START_REF] Zhao | A scaling roadmap and performance evaluation of inplane and perpendicular MTJ based STT-MRAMs for high-density cache memory[END_REF]. In both cases, the fact to have a large TMR eases the reading process, which is, most of the time, based a differential approach [START_REF] Zhao | Spin-MTJ based non-volatile flip-flop[END_REF], [START_REF] Chabi | Robust neural logic block (NLB) based on memristor crossbar array[END_REF].

B. SOT-Based MTJ

As mentioned in the introduction, the first generations of the MTJ relying on the Oersted field (generated by an injected current in the writing line) suffer from high writing current, die area, and lack of scalability. Although STT-based MTJ has solved the scalability problem, the common READ/WRITE path through the junction can cause barrier damaging during the writing operation and the switching of the magnetic state during the reading operation. A representative geometry of a three-terminal memory cell with the perpendicular switching of the magnetized FM layer using the SOT writing mechanism is shown in Fig. 2. The structure consists of a three layers nanopillar on the top of a conductor (metal electrode). For clarity reasons, we present only the main layers of the MTJ stack. Additional layers and materials combination could be considered to increase the robustness of the whole structure and mitigate undesirable effects (e.g., dipole fields around the free layer, and so forth). The writing electrode materials existing in literature to fabricate the sample device are β-tantalum (β-Ta) or Pt [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF]- [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF]. The magnetic cell is written by applying a charge current via the writing line. The orientation of the storage layer magnetization is controlled by the direction of the applied charge current. Positive currents (along x) produce a spin injection current with transverse direction (along z) and spins pointing to y-direction. The injected spin current in-turn produces spin torque to align the magnet in the y or -y direction.

During the time of the applied current pulse, the magnetization of the storage layer is oriented along the y-axis. At the end of the current pulse, the m z magnetization has an identical probability to be oriented up or down along the z-axis. If there is no external applied field H a , an arbitrary switching occurs that may be caused by nucleation events [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF]. The solution used by research teams working on the realization of the SOT device [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF]- [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF] is the generation of a bias magnetic field H a by adding a permanent magnet on chip or a biasing layer on top of MTJs. Thus, the magnetic field H a is permanently present on the top of the MTJ (not only when a current pulse is applied) and it contributes to the control of switching when a pulse of writing current is applied. Fig. 3 illustrates the behavior of the SOT-MTJ in the presence of a permanent magnetic field: starting from the initial magnetization, the AP or P states are obtained by applying a negative current pulse or a positive current pulse, respectively. In Fig. 3, the permanent magnetic field is negative (H a < 0). It is also possible to work with a positive H a (H a > 0). The effect of the current pulses will be simply inverted, as explained in the pseudocode in Fig. 3. More details about the switching mechanism are available in [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF]- [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF].

III. COMPACT MODELING OF THE SOT DEVICE

A. Modeling Environment and Approximations

Several MTJs compact models have already been published in the literature. While some authors used HSPICE language [START_REF] Das | A generalized HSPICE TM macro-model for pinned spin-dependent tunneling devices[END_REF]- [START_REF] Lee | Advanced HSPICE macromodel for magnetic tunnel junction[END_REF], others preferred the litheness of C language [START_REF] Prenat | CMOS/magnetic hybrid architectures[END_REF]. There are also several behavioral models Verilog-AMS or VHDL-AMS (both high-level description languages for analog or mixed signal for use with ICCAD tools) that are state-driven, such as the model written in VHDL-AMS [START_REF] Sugimura | Low-power and high-sensitivity magnetoresistive random access memory sensing scheme with body-biased preamplifier[END_REF], [START_REF] Madec | Compact modeling of a magnetic tunnel junction-Part II: Tunneling current model[END_REF] or the compact model for (MTJ) switched by thermally assisted STT (TAS + STT) developed in Verilog-A [START_REF] Zhao | A compact model for magnetic tunnel junction (MTJ) switched by thermally assisted Spin transfer torque (TAS + STT)[END_REF].

In this paper, we provide the first compact model written in Verilog-A of a SOT-MTJ. Our choice of the coding language is motivated by the capability of Verilog-A to afford a quick method of enhancing compact models to illustrate new physics of advanced processes. In addition, it is on the path to becoming the preferred compact modeling language for both academic and industrial research groups because of its flexibility to run in numerous simulators (Spectre, HSpice, ADS, Eldo, and so forth) and internal simulators of semiconductor companies. We aim to obtain a straightforward, highspeed, and precise electrical representation of the physical behavior of the SOT-MTJ device. First, we analyze the model equations along with some approximations. Then, because of close interactions with Spintec technologists, a number of associated parameters are fed into these equations. The simulator represents the established equations as equivalent circuit elements. In order to develop this model, we proceed under the macrospin approximation. We consider that the magnetization of each FM layer is uniform (single domain). Hence, it can be described by a single macroscopic magnetic moment. This hypothesis significantly abridges the mathematical analysis. The smaller the sample used, the more pertinent the macrospin assumption is.

B. SOT-MTJ Modeling Strategy

The SOT-MTJ model describes the dynamic magnetic behavior of the single domain storage layer as a function of the current density injected through the conductor underneath it.

The physical model of the SOT-MTJ device in Fig. 4(a) illustrates the layers and the signal interface for its input and output signals. There are three external nodes (S 0 , S 1 ,a n dS 2 ). 1) S 1 and S 0 represent the terminals of the writing path.

2) S 2 and S 0 represent the terminals of the reading path. The three internal connections are m x , m y ,andm z , which correspond to the components of the storage layer magnetization.

1) LLG Module: The LLG module describes the dynamic evolution of the storage layer magnetization as a function of time and applied current density (J app ). This module has one input (J app ), which represents the current applied through the WRITE line (S 1 S 0 ) of the device to control the magnetization reversal of the storage layer. The three outputs m x , m y ,andm z represent the magnetization components in a Cartesian system. This module implements the LLG equations [START_REF] Landau | Theory of the dispertion of magnetic permeability in ferromagnetic bodies[END_REF] ∂ m ∂t =-

γ 0 1 + α 2 ( m × H eff ) -α γ 0 1 + α 2 m × ( m × H eff ) (2)
where m is the unit vector along the magnetization of the soft layer, α is the Gilbert damping constant, γ 0 is the gyromagnetic ratio, and H eff is the effective field.

The first term in (2) describes the precession of the magnetization around the effective magnetic field. The second term in (2) is the Gilbert damping torque, which forces the magnetization to relax toward the effective field.

The effective field is the sum of the different magnetic fields, which acts on the storage layer magnetization

H eff = H k + H d + H R + H SHE + H a ( 3
)
where H k is the magnetocrystalline anisotropy field, H d is the demagnetizing field, H R is the Rashba field, H SHE is the SHE field, and H a is the external applied field.

1) The interaction between the magnetic moment and the crystalline lattice is responsible for a privileged direction, in which the magnetization spontaneously aligns itself in the absence of an external field. The equivalent magnetocrystalline anisotropy field is expressed as

H k = (2K u /µ 0 M s )m z z (4)
where K u is the uniaxial anisotropy constant, and M s is the spontaneous magnetization value. 2) Another source of anisotropy in the shape of magnetic nanostructures via the magnetostatic energy. The magnetization always tends to align with the longest dimension of the ellipsoid in magnetic ellipsoidal nanostructures. This effect is described in the model using the following equation:

H d =-M s (n xx m x + n yy m y + n zz m z ) (5)
where n x , n y ,a n dn z are demagnetizing tensor coefficients.

3) The Rashba effect field and the SHE field are the two original components of the total effective field H eff , which differ the SOT-MTJ from other MTJ generations (FIMS, TAS, and STT). Their key role in the magnetic switching of FM dots has given birth to a new concept of three-terminal MRAM that we are describing in this paper. Based on the experiments and the interpretations afforded by SPINTEC technologists [START_REF] Gambardella | Current-induced spin-orbit torques[END_REF], [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF], the Rashba effect and the SHE fields are expressed according to the following equations, respectively:

H R = C R J app y ( 6
)
H SHE =-C SHE J app m × y (7)
where C R is the Rashba coefficient, C SHE is the SHE coefficient, and J app is the density of current applied in the conductor. The values of these two factors depend on the nature and the dimensions of the SOT-MTJ layers [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF], [START_REF] Haney | Current induced torques and interfacial spin-orbit coupling: Semiclassical modeling[END_REF]. 4) H a is the external applied field required to coexist with Rashba effect and SHE in order to achieve the magnetic reversal of the storage layer. It is a static magnetic field that can be generated by small permanent magnets fabricating on-chip, close or on top of each active dot. We can also envisage adding a bias layer on the top of the MTJ to maintain a static magnetic field with a component according the x-axis. Neither its value nor its exact orientation is critical for the reversal. Very small applied fields suffice to provoke reversal [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF].

In our model, we admit that the external applied field is expressed as

H a = H a x. ( 8
)
2) TMR Module: The second module of the Verilog-A model is the TMR module, which describes the magnetoresistance variation based on the orientation of the storage and reference layers (P/AP). It takes the output of the LLG module as its inputs and it depends basically on the m x and m z component of the magnetization as shown below in equation.

Following the first TMR model in [START_REF] Julliere | Tunneling between ferromagnetic films[END_REF] and [START_REF] Slonczewski | Currents, torques, and polarization factors in magnetic tunnel junctions[END_REF], the tunneling conductance is a function of the angle between the magnetizations of the two FM layers. Furthermore, the bias-dependent conductance of the device at 0 K can be derived in the framework of Simmons's model [START_REF] Simmons | Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film[END_REF] and Brinkman's model [START_REF] Brinkman | Tunneling conductance of asymmetrical barriers[END_REF]. By combining formulas given by these complementary models, we can assume that the total tunneling conductance is given by

g(V , m x , m z ) = G P 0 (1-2βV +3δV 2 ) 1+ 1-(m x cos θ mhl +m z sin θ mhl ) 2 ⎛ ⎝ tmr 0 1+ V 2 V 2 h ⎞ ⎠ (9)
where β and δ are tunnel oxide material-dependent constants, G P0 is the conductance in P magnetic configuration at 0 V and 0 K usually calculated from Simmons's model.

V represents the bias voltage applied across the device and tmr 0 is the TMR under a low bias voltage. V h is defined as the voltage at which the TMR amplitude has decreased to half of its low bias value (tmr (V h ) = tmr 0 /2). θ mhl corresponds to the angle with the z-axis of the reference layer magnetic state.

The electrical modeling of the LLG module consists of three RC circuits correlated between each other's. When it comes to the TMR module, it is considered as a simple resistor with the conductance equation depending on the output of the LLG module as explained in Fig. 4 and illustrated in [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF]. This technique of the MTJ electrical modeling has been used and detailed in [START_REF] Prenat | CMOS/magnetic hybrid architectures[END_REF]. It showed accurate results with an attractive simulation time.

IV. SIMULATION RESULTS

In this section, we present simulation results for the MTJ cell run with Spectre simulator under Cadence Virtuoso platform. We also investigate the impact of the current density, pulsewidth, and external applied field on the behavior of the MTJ dynamics. We note that the following conventions are used in the discussion of simulation results: 1) the initial magnetization state of the SOT-MTJ is in the P state, i.e., the magnetization of the reference layer and the storage layer are along the z-axis with upward orientation (at initial state, we have the perpendicular magnetization along z-axis thus m z = 1); 2) the initial external applied field H a has a negative value; and 3) negative current pulse switches the MTJ configuration into the AP state, whereas positive pulses switch the MTJ configuration into the P state in the presence of a negative H a .

A set of experimental-based parameters from [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF] and [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF] is fed into the model for simulations, among which the geometrical parameters such as the shape, the size, the initial magnetic state of the MTJ, and the other physical and technological parameters. The experiments results delivered by research teams working on SOT devices [START_REF] Gambardella | Current-induced spin-orbit torques[END_REF]- [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF] claimed that the current density threshold required for switching the magnetization from P (AP) to AP (P) has the same value (symmetrical switching). It is of the order of J app = 10 12 A•m -2 .W e admit that the SHE factor (C SHE ) is twice the value of Rashba coefficient (C R ). All this parameters can be easily tuned by the user as current experimental work proceeds.

A. Impact of Current Density and Pulsewidth on the Perpendicular Magnetization m z

To reverse the perpendicular magnetization (m z ) of the storage layer, a current is applied through the conductor stripe underneath the MTJ structure.

Depending on the width and thickness of the conductor, this current is equivalent to a current density J app , which is the determinant variable in the model to switch the magnetization m z down (negative J app ) or up (positive J app ). To find out the required dimensions for the stripe and consequently the writing current, the proper J app must be known in advance. We study in Fig. 5 the dynamic evolution of the m z component for different values of applied current density (J app ) b a s e do nt h e experimental results published for this type of device, which claim that J app must be in the order of 10 12 A•m -2 .

We choose to begin our study with a negative pulse of 3 ns and vary the current density value to observe the dynamics of switching. Fig. 5(a) shows that at J app =-1 × 10 12 A.m -2 , the spin torque is not enough to act on the magnetization of the storage layer, the initial state (m z = 1) is retained during and after the pulse. Starting from a value of J app =-2 × 10 12 A.m -2 , we can notice the manifestation of a spin torque action on the magnetization. However, because of the shortness of the applied pulse, it was not possible to attain the reversal of the magnetization. The pulse of current ends during precessions (before the stabilization of the m z ) and so we obtain and erroneous writing. If we apply a longer pulse, the magnetization occurs. This is confirmed in Fig. 5(b), where we vary the pulsewidth with a fixed current density of J app =-2 × 10 12 A.m -2 . So, when we apply large pulses, a lower current is required for switching. From Fig. 5(a), we notice that the higher is the value of J app , the smaller is the width of the pulse required for switching. For example, at J app =-4 × 10 12 A.m -2 , the width of the pulse could be only 0.3 ns instead of the 3 ns applied and could decrease further (∼0.1 ns for J app = 8 × 10 12 A.m -2 ). To sum up, based on the simulations results, at least a current density of -2 × 10 12 A.m -2 is required to make the magnetization reversal occur at large pulses (∼6 ns). For higher current density, the required pulsewidth decreases further to attain hundreds of picoseconds starting from a current density of -4 × 10 12 A.m -2 . These simulation results show a good agreement with experimental measurement [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF], where the magnetization switching is achieved at J app ∼2 × 10 12 A.m -2 with a pulsewidth of 9 ns.

B. Impact of the External Field H a on the Magnetization Reversal

The effective field resulting from the in-plane current injection is orthogonal to the easy magnetization axis (z)a n ds o it cannot control the magnetization reversal by itself. In fact, during the time when the pulse is applied, the magnetization vector m z oscillates at first till stabilizing at the zero value. Once the pulse is completed, m z has an identical probability to be oriented up or down along the z-axis. If there is no external applied field H a , an arbitrary switching occurs that may be caused by nucleation events. Fig. 6(b) illustrates the random switching in the absence of an external magnetic field. In our case, the reversal in one direction or another when H a = 0, is probably due to numerical noise. The solution used by research teams working on the realization of the SOT device [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF]- [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF], is the generation of a permanent magnetic field H a by adding a permanent magnet on chip or a biasing layer on top of the MTJs. The field H a contributes to the control of switching when a pulse of writing current is applied as it is explained in Section II-B in Fig. 3.

Fig. 6(a) shows a proper reversal of the magnetization m z according to the current pulse in the presence of a negative field. If a negative pulse is applied, the magnetization is reversed downward (m z =-1), whereas a positive pulse the magnetization upward (m z = 1 ) .F i g .6 ( c ) shows that the response to the current direction can also be inverted by reversing the direction of H a (reversing the polarization of the magnets responsible for the generation of the static magnetic field H a ). Finally, we verified the fact that we do not require high values of applied field to reverse the magnetization as shown in Fig. 6(d), where we used three different values of H a and observed that the value of applied field is not crucial in the reversal mechanism, a small value is sufficient.

C. Magnetoresistance Variation as a Function of the Applied Current

In Fig. 7, we investigate the variation of the SOT-MTJ as a function of the current pulses applied through the writing line underneath. We fixed the bias voltage at 1 V and we observed the evolution of the current for different pulses.

For the P state (m z = 1), we notice a higher current value of 8.75 µA (i.e., lower resistance) and for the state AP (m z =-1), we notice a lower current value of 7.3 µA (i.e., higher resistance). Thus, the model describes correctly the dependence of the MTJ resistance on the state of the perpendicular magnetization m z , which in turn depends on pulses of the writing current.

D. Limitations, Advantages and Future of SOT Devices

Based on the simulation results obtained from the compact model, which is in accord with the first experimental characterizations of SOT-MTJs available in literature, we highlight the main drawbacks, the benefits, and the possible improvements concerning the SOT MRAMs. It is worth to point out that the first fabricated devices show a high current density required for the magnetization switching (J app ∼2-3 × 10 12 A.m -2 ) and consequently a high writing current (>2m A ) .

However, researchers working on the SOT concept are optimistic about the future of this device and claim that the current can be decreased with further interface engineering. In [START_REF] Miron | Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection[END_REF], the author noted that the in-plane current can be applied through a thinner lateral surface, thus the absolute current required for switching can be made very small (∼1m A ) . An attractive alternative has been already achieved by Cornell University in [START_REF] Liu | Spin-torque switching with the giant spin hall effect of tantalum[END_REF], where they fabricated a SOT device with the tantalum (Ta) as a microstrip conductor for the writing line instead of platinum. They reported a giant SHE in β-tantalum that generates spin currents intense enough to induce efficient spin-torque switching of ferromagnet at room temperature. Using tantalum, experimental results show that a current of ∼1 mA is required for switching (half the current required with platinum). We predict that it is possible to further decrease the switching current by a factor of 3, if they shrink the width of the Ta microstrip to be equal to the dimension of the long axis of the nanopillar. In addition, they suppose that a further reduction in current could be achieved by reducing the demagnetization field of the FM free layer [START_REF] Liu | Reduction of the spin-torque critical current by partially canceling the free layer demagnetization field[END_REF], [START_REF] Moriyama | Tunnel magnetoresistance and spin torque switching in MgO-based magnetic tunnel junctions with a Co/Ni multilayer electrode[END_REF].

Based on all these improvements, the required current for SOT devices could be reduced to <100 µA. At such a current value, three-terminal SOT devices would be very competitive with the efficiency of conventional STT-MTJs [START_REF] Ikeda | A perpendicular-anisotropy CoFeB-MgO magnetic tunnel junction[END_REF], [START_REF] Kishi | Lower-current and fast switching of a perpendicular TMR for high speed and high density spin-transfer-torque MRAM[END_REF]. Three-terminal SOT approach can even overstep the two-terminal STT devices since it separates the reading and the writing paths. This enhances the reliability of devices and overcomes the main challenges encountering MRAMs. Although it is still in its infancy, the three-terminal architecture of SOT devices promises efficiency and easiness for the fabrication process. Finally, a giant step toward the perfection of this innovative device could be the discovery of materials with larger the spin Hall angle.

V. C ONCLUSION

In this paper, we present the first compact model for a threeterminal MTJ nanopillar switching using the SOT approach, which is expected to enhance reliability and overcome challenges facing the conventional two-terminal STT devices. Transient simulations of the model validate its functionalities and show a good agreement with the fabricated device in literature. This model is written in Verilog-A language, compatible with the Spectre simulator of the standard Cadence design suite and with the majority of known simulators. Furthermore, improvements of the model require additional feedback from technologists working on the SOT-MTJs physics. However, already in its present form, it constitutes a very important tool for the design of complex hybrid CMOS/magnetic architectures and it enables one designer to explore the prospective of SOT devices. Using this model, a number of hybrid MTJ/CMOS logic circuits are under examination in our laboratory with inventive architectures because of the three-terminal structure of the SOT-MTJ.

Fig. 1 .

 1 Fig. 1. Resistance variation of the MTJ according to the storage layer magnetization state.

Fig. 2 .

 2 Fig. 2.Schematic view of the three-terminal SOT device and the two independent paths for READ and WRITE operations.

Fig. 3 .

 3 Fig. 3. Switching dependence on the external applied field H a and the current direction.

Fig. 4 .

 4 Fig. 4. Modeling strategy of the three-terminal SOT-MTJ. (a) Physical view of the SOT-MTJ (cross section). (b) Block diagram of the compact model.

  Fig. 4(b) gives a block diagram showing all the SOT-MTJ module connections. The whole block diagram is composed of two principal modules: 1) the LLG module and 2) the TMR module.

Fig. 5 .

 5 Fig. 5. Study of the dynamic behavior of the perpendicular magnetization m z according to the variation of the current density J app and the pulsewidth. (a) Evolution of m z according to the current density J app at 3-ns pulse. (b) Evolution of m z according to the pulsewidth at J app =-2 × 10 12 A•m -2 .

Fig. 6 .

 6 Fig. 6. Study of the dynamic behavior of the perpendicular magnetization m z according to the variation of the sign and value of the external applied field H a . (a) Negative value of H a . (b) Absence of H a . (c) Opposite behaviour of m z depending on the sign of H a . (d) Effect of increasing H a magnitude.

Fig. 7 .

 7 Fig. 7. Validation of the magnetoresistance variation according to writing current pulses.

ACKNOWLEDGMENT

The authors would like to thank G. Gaudin for fruitful discussions. This work was supported by the framework of the Spot Project under Grant n°318144 through the European Commission under the Seventh Framework Programme.


T he goal of machine learning, broadly speaking, is to design a computer code -the eponymous machine -capable of discovering meaningful structure in data. The last decade saw a game-changing revolution unfold in this field: with the development of deep neural networks 1 , tasks that were considered inaccessible to automated learning became possible. This prompted fierce competition in the artificial intelligence market, but it also brought promise to many areas of data-intensive fundamental science -with physics being no exception. Current machine-learning systems are not yet able to divine the laws of general relativity from planetary data, but they are able to reliably recognize human faces, detect objects in photographs and even beat world champions of Go 2 . And now, writing in Nature Physics, two groups have used artificial neural networks to recognize different phases of matter and localize associated phase transitions 3,4 .

Juan Carrasquilla and Roger Melko 3 used a type of neural network commonly used to classify images. To illustrate how it works, consider a set of images of cats and dogs, each labelled 1 or 0, respectively. The network then uses this set and the labels to construct a function that takes pixels of each image as an input and returns an output of 1 for cats and 0 for dogs. If this function is later given a previously unseen image, it should return an output corresponding to the presence of a dog or cat in the new image. In Carrasquilla and Melko's study the labels dog and cat were replaced by the low-temperature phase and high-temperature phase of a condensed-matter model. The training set was not given by pixels of an image, but by an equilibrium confi uration of the model obtained from Monte Carlo simulations. In another paper, Evert van Nieuwenburg and co-workers 4 showed that if the set of confi urations can be ordered on a linerepresenting the temperature, for examplethen these phases can be learned without even knowing the labels. Both works offer an exciting perspective on studying unexplored phases of matter and phase transitions in systems with unknown order parameters.

It should be stressed that saying one applied machine learning to a given problem is about as generic as saying that one used numerical simulations. It is clear to every researcher in physics that there are many kinds of numerical simulations. Depending on the system and the question of interest, it requires insight and experience to fi d the right numerical simulation and carry it out with sufficient care in order to be able to truly advance our understanding of a given problem. The same is true for applications of tools of machine learning.

A wide range of tools stemming from machine learning have recently been used in physics-related studies. Carrasquilla and Melko represented phases of matter with a two-layer feed-forward neural network. They also showed that in some cases, including Ising lattice gauge theory, this architecture fails, and instead a convolutional neural network succeeds. Van Nieuwenburg et al. used a similar two-layer feed-forward neural network, but showed that a much more basic method of principal component analysis succeeds in some of the examples. A different application of machine learning in physics aims at reducing the dimension of a Hilbert space by representing the wavefunction using a restricted Boltzmann machine, with weights obtained through reinforcement learning 5 . Another group recently showed that a support vector machine can be used to classify which particles in a glassy system are susceptible to rearrangement 6 . Decision forests have been used to classify metals from insulators based on the hybridization function, combined with kernel ridge regression to predict correlation functions in manybody physics 7 . For the quantum systems considered by van Nieuwenburg et al., the neural network was trained on the entanglement spectrum instead of directly on the wavefunction. This is a type of data preprocessing that was exploited in another recent study predicting the quantum energies of molecules, using sparse linear regression on approximated electron densities after preprocessing with the so-called scattering transform 8 .

Clearly, this new trend brings with it a host of machine-learning terms that an average physicist encounters only very rarely. We can, however, anticipate that soon all these terms will be taught in undergraduate physics along with Monte Carlo methods and the Fourier transform. When it comes to a new set of tools, it is not only important for the physics community to learn about them, but also for it to build an intuition and understanding of which of the tools are applicable to which question and system. Machine learning is not a magic boxin order to be valuable in physics its results need to be interpreted and validated. So far the fi st studies have opened up a set

MACHINE LEARNING

New tool in the box

A recent burst of activity in applying machine learning to tackle fundamental questions in physics suggests that associated techniques may soon become as common in physics as numerical simulations or calculus.

Lenka Zdeborová

M a c h i n e l e a r n i n g of interesting directions with hand-picked examples of successful applications, but systematic studies are needed to better understand the range of validity and applicability of each of the approaches 9 . One might also hope that such a systematic understanding-driven effort will feed back into the machine-learning community where new theoretical insights about various architectures will be welcome. ❐

Lenka Zdeborová is at the Institut de Physique Théorique, CNRS, CEA, Université Paris-Saclay, F-91191, Gif-sur-Yvette, France. e-mail: lenka.zdeborova@cea.fr