Danylo Babich

Laurent Cario
email: laurent.cario@cnrs-imn.fr

Benoit Corraze

Maciej Lorenc

Julien Tranchant

Roman Bertoni

Marco Cammarata

Hervé Cailleau

Etienne Janod
email: etienne.janod@cnrs-imn.fr

Artificial electro-optical neuron integrating hot electrons in a Mott insulator

Keywords: 07, 05, Mh,

Mott insulators are a class of strongly correlated materials with emergent properties important for modern electronics applications, such as artificial neural networks. Under electric field, these compounds undergo a resistive switching that may be used to build up artificial neurons. However, the mechanism of this resistive switching is still under debate and may depend on the Mott material involved. Some works suggest an electronic avalanche phenomenon, while others propose an electrothermal scenario. As electric pulses produce both Joule heating and hot carriers, disentangling their respective roles requires the use of another external stimulus. Here, an ultrashort light pulse is used to tune the number of photogenerated carriers and the energy provided to the system. In these pump-pump-probe experiments, a crystal of the Mott insulator GaTa4Se8 is simultaneously excited by electric and laser pulses while an electric probe monitors its conductivity. The study shows that the resistive switching is affected by the number of generated photocarriers rather than by the accumulation of energy deposited by the femtosecond laser.

It supports therefore a mechanism driven by a hot carriers' generation. Finally, our work opens the possibility to build up an artificial electro-optical «Mott» neuron tuned by a femtosecond laser pulse.

INTRODUCTION

Modern Artificial Intelligence (AI) technology is currently based on the emulation of the biological neural networks on classical von Neumann computers. [START_REF] Jain | Artificial Neural Networks: A Tutorial[END_REF] However, the current approach is less efficient than the mammal brain in terms of energy consumption. In that regard, one of the most challenging issues for AI is the implementation of neural network components in hardware. [START_REF] Misra | Artificial Neural Networks in Hardware: A Survey of Two Decades of Progress[END_REF][3][START_REF] Jeong | Towards Artificial Neurons and Synapses: A Materials Point of View[END_REF][START_REF] Zhu | A Comprehensive Review on Emerging Artificial Neuromorphic Devices[END_REF] The classical biological neural circuit consists of neurons interconnected by synapses that modulate the inter-neuron signal transfer. The main building block of the brain, i.e., the biological neuron, accumulates incoming signals during the "integration time" and "fires" (generates an output signal) once the integration reaches a threshold. The creation of artificial neurons requires therefore materials possessing a physical property able to implement both Integrate and Fire functionalities with a clear identification of the physical quantity that is integrated [3,[START_REF] Burkitt | A Review of the Integrate-and-Fire Neuron Model: I. Homogeneous Synaptic Input[END_REF][START_REF]A Review of the Integrate-and-Fire Neuron Model: I. Homogeneous Synaptic Input[END_REF]. One of the possible solutions, currently under investigation, is the use of a class of quantum materials, the Mott insulators, [START_REF] Stoliar | A Leaky-Integrate-and-Fire Neuron Analog Realized with a Mott Insulator[END_REF][START_REF] Adda | Mott Insulators: A Large Class of Materials for Leaky Integrate and Fire (LIF) Artificial Neuron[END_REF][START_REF] Tesler | Relaxation of a Spiking Mott Artificial Neuron[END_REF] which exhibits an insulator to metal transition (IMT) under electric field, the so-called Electric Mott Transition (EMT) [START_REF] Janod | Resistive Switching in Mott Insulators and Correlated Systems[END_REF]. The main signature of this EMT is a resistive switching occurring under electric field and which has been explained by different mechanisms depending on the investigated Mott materials. For VO2, the resistive switching was proposed to originate from the self-Joule heating generated during the electric pulse [11,12,13] . Conversely, in the Mott insulators AM4Q8, (V1-xCrx)2O3 and NiS2-xSex, the resistive switching was associated with the creation of hot carriers followed by an electronic avalanche phenomenon [START_REF] Guiot | Avalanche Breakdown in GaTa4Se8-xTex Narrow-Gap Mott Insulators[END_REF][START_REF] Diener | How a Dc Electric Field Drives Mott Insulators Out of Equilibrium[END_REF][START_REF] Stoliar | Universal Electric-Field-Driven Resistive Transition in Narrow-Gap Mott Insulators[END_REF]. As, several recent studies show impact of light on the Mott transition [START_REF] Abreu | THz Driven Dynamics in Mott Insulator GaTa 4 Se 8[END_REF][START_REF] Yamakawa | Mott Transition by an Impulsive Dielectric Breakdown[END_REF][START_REF] Giorgianni | Overcoming the Thermal Regime for the Electric-Field Driven Mott Transition in Vanadium Sesquioxide[END_REF], we have reinvestigated here the resistive switching mechanism of a GaTa4Se8 crystal thanks to a pump-pump-probe experiment with simultaneous electric and laser pulse excitation and electrical conductivity probing. Our study shows that photo-doping provided by laser pulse impacts the EMT transition dynamics. By tuning the number of photogenerated carriers and the energy provided to the system, we shed light on the physical quantity that is integrated in this Mott material [START_REF] Tranchant | Control of Resistive Switching in AM4Q8 Narrow Gap Mott Insulators: A First Step towards Neuromorphic Applications[END_REF] and establish the proof-of-concept of an artificial electro-optical "Mott" neuron.

EXPERIMENTAL DETAILS

A. Sample preparation and characterization:

GaTa4Se8 crystals were synthesized by a method described elsewhere [START_REF] Vaju | Electric-Pulse-Driven Electronic Phase Separation, Insulator-Metal Transition, and Possible Superconductivity in a Mott Insulator[END_REF], and cleaved into 200 -300 µm pieces. Сrystals used for transport measurements were contacted using 10μm gold wires and carbon paste (Electrodag PR-406), and then annealed under vacuum at 150 °C for 30 min. The low-bias resistance was measured using a Keithley 236 source-measure unit by a standard two-or four-probe technique (Figure 1 a). For the application of electric pulses, we have connected two gold wires along the crystal surface on which light should be applied. The inter-electrode distance was around 200 µm.

B. Pump -pump -probe experiments

We have applied two "pumps", a ≈ 100 µs electric pulse and a 100 fs laser pulse in a singleshot mode, on the GaTa4Se8 crystal, as we recorded the evolution of the electric resistance ("probe"). We used a Ti: Sapphire femtosecond laser synchronized with an electric pulse generator (Keithley 8114A) so that the laser output was triggered precisely with the first nanoseconds of the electrical pulse. During the pulse, the voltage and current across the sample were measured with a Tektronix DPO3034 oscilloscope associated with IeS-ISSD210 differential probes. We used an optical parametric amplifier (OPA) to tune the laser wavelength from the visible to mid-IR region. In order to control crystal temperature, we have used a nitrogen Cryostream. We carried out the experiments with the same number of absorbed photons by taking care to keep the laser spot size similar at the different wavelengths. We used a short focus (5 cm) optical lens. After the experiment on the GaTa4Se8 single crystal, we have performed knife-edge measurements to recover the spatial profile of the laser beam. It allows a reliable estimate of the beam size, which were similar for both laser energies (0.5 and 2.3 eV) and covered more than 90 % of the inter-electrode area. In these experiments, the level of photodoping depends on both the laser fluence and the impacted volume. The latter is defined by the size of the beam hitting the sample surface and a characteristic length over which the photocarrier spread inside the materials. Since photodoped carriers are subjected to applied DC electric fields E close to 3 kV / cm, this characteristic length corresponds to the diffusion length of the carriers rather than the initial absorption length (i.e., the light penetration depth). The diffusion length is indeed defined as ldiff = µ.E.τ, where µ and τ are the carriers mobility and lifetime, respectively. Considering a lifetime of 1 µs typical of AM4Q8 compounds [START_REF] Babich | Unusually Long Carrier Lifetime in a Mott Insulator Revealed by Time-Resolved Photoemission Electron Microscopy[END_REF] and a mobility representative of Mott insulators of 1 cm².V -1 .s -1 [START_REF] Petersen | Transient Terahertz Photoconductivity of Insulating Cuprates[END_REF] leads to a lower limit of 30 µm for the diffusion length. This value exceeds the typical light penetration depth by more than two orders of magnitude. Consequently, experiments conducted with similar beam sizes and density of absorbed photons, but with laser pulses at different wavelengths, will yield analogous levels of photodoping.

RESULTS AND DISCUSSION

According to conventional band theory, Mott insulators should be metallic. Their insulating ground state results from the on-site electron-electron Coulombian repulsion, not accounted for in the classical approach, which favors the localization of electrons on atomic sites. At equilibrium, the insulating state can be broken by pressure or charge doping. Recently, an Insulator to Metal transition leading to a resistive switching was also achieved under electric field in Mott insulators [START_REF] Taguchi | Dielectric Breakdown of One-Dimensional Mott Insulators Sr 2 CuO 3 and SrCuO 2[END_REF][START_REF] Stefanovich | Electrical Switching and Mott Transition in VO2[END_REF][START_REF] Guiot | Avalanche Breakdown in GaTa4Se8-xTex Narrow-Gap Mott Insulators[END_REF][START_REF] Janod | Resistive Switching in Mott Insulators and Correlated Systems[END_REF]. This phenomenon, also called Electric Mott Transition (EMT), was observed for numerous examples of Mott insulators. In this paper, we focus on a member of the family of narrow gap chalcogenide Mott insulators of formulation AM4Q8 (A=Ga, Ge; M = V, Nb, Ta, Mo; Q= S, Se) that exhibit a clustered lacunar spinel structure [START_REF] Yaich | Nouveaux Chalcogenures Mixtes GaMo4(XX′)8 (X =S, Se, Te) à Clusters Tetraedriques Mo4[END_REF][START_REF] Pocha | Crystal Structures, Electronic Properties, and Pressure-Induced Superconductivity of the Tetrahedral Cluster Compounds GaNb 4 S 8 , GaNb 4 Se 8 , and GaTa 4 Se 8[END_REF][START_REF] Bichler | Structural and Magnetic Phase Transitions of the V4-Cluster Compound GeV4S8[END_REF][START_REF] Guiot | Control of the Electronic Properties and Resistive Switching in the New Series of Mott Insulators GaTa4Se8-YTey (0 ≤ y ≤ 6.5)[END_REF]. Figure 1 presents the thermal dependence of the resistivity and a typical resistive switching phenomenon observed in a crystal of GaTa4Se8. For this compound, the proposed microscopic mechanism for resistive switching is based on Fröhlich's theory of dielectric breakdown [START_REF] Fröhlich | Theory of Electrical Breakdown in Ionic Crystals[END_REF][START_REF] Fröhlich | On the Theory of Dielectric Breakdown in Solids[END_REF].

The details of this theory and its application to the AM4Q8 Mott insulators were published elsewhere [START_REF] Diener | How a Dc Electric Field Drives Mott Insulators Out of Equilibrium[END_REF]. The model considers the existence of small numbers of electrons either trapped in localized levels below the conduction band or delocalized in the conduction band. These last pre-existing free carriers can be accelerated under electric field and share their energy with other electrons and the lattice thanks to, respectively, electron-electron (e-e) or electron-phonon (e-ph) interactions [START_REF] Diener | How a Dc Electric Field Drives Mott Insulators Out of Equilibrium[END_REF]. New hot carriers are generated under electric field, which increases the overall electronic temperature until a new equilibrium situation is reached between the power given to (electric power) and the power taken out from (e-ph) the electronic system (i.e. it corresponds to a two temperatures model with the electronic temperature stabilizing slightly over that of the lattice) . But once the applied electric field reaches a threshold value Eth, the equilibrium between the lattice and electronic sub-systems cannot be maintained, and the system enters into a nonequilibrium state with the number of hot carriers diverging and inducing an IMT (i.e. the resistive switching). The theory predicts a temperature-independent deviation from Ohm's law below the threshold field (E ≪ Eth) with the multiplication rate of new carriers in the conduction band 𝑛𝑛 � = 𝑛𝑛 𝑛𝑛 0 � depending only on the square of the electric field if we assume the mobility to be constant :

𝑛𝑛 � = 𝑛𝑛 𝑛𝑛 0 = 𝜎𝜎(𝐸𝐸) 𝜎𝜎 0 = 𝑒𝑒𝑒𝑒𝑒𝑒 � 𝐸𝐸 2 𝜀𝜀 𝐺𝐺 𝐸𝐸 𝑡𝑡ℎ 2 𝑒𝑒𝑒𝑒𝜀𝜀 � (1)
where 𝑛𝑛 0 and 𝜎𝜎 0 are the number of carriers and conductivity at zero field. Constants 𝜀𝜀 𝐺𝐺 , 𝑒𝑒𝜀𝜀 represent the width of the gap and of the in gap state levels, respectively, and 𝑒𝑒 is Euler's constant. In the boundary case of the prediction (E = Eth), the threshold value of the carrier multiplication rate 𝑛𝑛 � 𝑡𝑡ℎ becomes :

𝑛𝑛 � 𝑡𝑡ℎ = 𝑒𝑒𝑒𝑒𝑒𝑒 � 𝜀𝜀 𝐺𝐺 𝑒𝑒𝜀𝜀 � (2)
In our previous work, the low field conductivity of the narrow gap Mott insulators AM4Q8 was compared to these predictions using the experimental values for the gap εG and for 𝑒𝑒𝜀𝜀 (e.g. εG = 158 meV and 𝑒𝑒𝜀𝜀 = 112 meV in the case of GaTa4Se8) [START_REF] Diener | How a Dc Electric Field Drives Mott Insulators Out of Equilibrium[END_REF]. As displayed in Figure 2a, the measurements, performed at different temperatures for a GaTa4Se8 crystal, lie on the predictable master curve for the carrier multiplication rate 𝑛𝑛 � (see blue line). It fully supports the proposed model suggesting that, below the threshold field, 𝑛𝑛 � may stabilize in time, while above the threshold electric field (≡ when the system exceeds the threshold carrier multiplication rate 𝑛𝑛 � 𝑡𝑡ℎ slightly greater than 4 in our experiments) 𝑛𝑛 � can not stabilize anymore and may diverge in time. Figure 2 b presents the experimental conductivity ratio dynamics 𝜎𝜎 � = 𝜎𝜎/𝜎𝜎 0 measured under electric field for a GaTa4Se8 crystal. Below Eth (ranging from 0.3 to 7 kV/cm in GaTa4Se8 depending on the temperature [START_REF] Diener | How a Dc Electric Field Drives Mott Insulators Out of Equilibrium[END_REF]), the conductivity ratio stabilizes over time and the system remains insulating, which means the electronic temperature reaches an equilibrium. But for electric field higher than the threshold value (𝐸𝐸 > 𝐸𝐸 𝑡𝑡ℎ ) a sudden increase of the conductivity ratio appears (i.e. a volatile resistive switching occurs) as the carrier multiplication rate approaches 𝑛𝑛 � 𝑡𝑡ℎ and the system enters into an avalanche carrier multiplication regime (Figure 2 a).

According to the model, new hot carriers are accumulated in the AM4Q8 compounds under electric field. This model is therefore of great interest in the framework of neurocomputing as it suggests that the AM4Q8 compounds behave as hot carrier integrators. Similar to biological neurons that accumulate charges on their membranes until the membrane potential reaches a threshold, the Mott insulators would accumulate hot carriers until the carrier multiplication rate reaches a threshold. In order to support this scenario, we have compared the carrier multiplication rate dynamics obtained under electric pulse alone or when the electric pulse is coupled to a laser pulse in order to modify the carriers number. The setup sketched in Figure 3 was used to implement this electrooptical pump -and electric probe experiment. A freshly cleaved surface of a GaTa4Se8 single crystal was connected by two gold electrodes separated by about 200 µm to allow the use of ultrashort laser pulses with a fixed spot size slightly smaller than the inter-electrodes distance. We used a femtosecond Ti:Sapphire regenerative amplifier and an Optical Parametric Amplifier (OPA) to tune the wavelength of the optical pump between visible and infrared. The electrodes are used both to apply the electric field thanks to a pulse generator and to monitor conductivity dynamics thanks to an oscilloscope connected in parallel. The setup worked in single-shot mode with a synchronous application of the electric and laser pulses. We performed all experiments under nitrogen cryostream cooling at 85(±5) K in order to have optimal stability conditions to induce the resistive switching experiments.

In a first series of experiments, we simultaneously applied electrical voltage pulses of 60V (i.e.

exceeding the threshold value) and 800 nm (hν = 1.55 eV) femtosecond pulses, while varying the excitation density. As depicted in Figure 3a, the main idea was to tune the number of photoexcited carriers at the beginning of the electric pulse. Figure 3a shows that the concomitant application of an optical excitation drastically reduces the time delay necessary to observe the resistive switching. Moreover, this time delay decreases when the excitation density increases. These experiments already evidence a clear impact of photoexcited carriers on the resistive switching. They suggest a similar role of hot carriers generated by light or electric field.

This result supports that the photoexcited carriers are added to the hot electrons accumulated under electric field to reach the threshold carrier multiplication rate and promote the avalanche phenomenon. A possible impact of the thermal energy injected into the system (lattice and electrons) by the light pulse cannot be completely discarded based on this experiment alone as at the probed time scale both sub-systems should have reached thermal distributions (Fermi-Dirac and Bose-Einstein).

Another way to address this issue is to tune the excitation density and the photon energy of the pulse using the OPA. As sketched in Figure 3b, the main objective of these experiments is to tune the photon energy while keeping a constant number of photons. This series of experiments were conducted using another GaTa4Se8 crystal in a configuration similar to that depicted in All these findings are of great interest to build up an electro-optical Mott neuron for artificial neuromorphic devices [3,[START_REF] Zhu | A Comprehensive Review on Emerging Artificial Neuromorphic Devices[END_REF] and neuromorphic photonics [START_REF] Goi | Perspective on Photonic Memristive Neuromorphic Computing[END_REF][START_REF] Prucnal | Neuromorphic Photonics[END_REF]. As depicted in Figure 4 a, when a biological neuron receives input spikes from other neurons, they are integrated into the membrane potential. In between the incoming spikes, the membrane potential relaxes, which is known as a leaky behavior. Finally, when the membrane potential reaches a threshold value, the neuron triggers an output action potential (Firing event). The biological neuron implements, therefore, three important functionalities that are at the basis of the most used model of an artificial neuron, namely the Leaky, Integrate and Fire (LIF) artificial neuron. In a previous work, we have demonstrated that the AM4Q8 compounds, as well as other Mott insulators, like (V1-xCrx)2O3 or [Au(iPr-thiazdt)2] behave as LIF artificial neurons [START_REF] Adda | Mott Insulators: A Large Class of Materials for Leaky Integrate and Fire (LIF) Artificial Neuron[END_REF]. This behavior was observed when a crystal of Mott insulator was subjected to a train of electrical pulses. Figure 5(a-c) shows the results of other experiments intended to control the Mott neuron also by light, and for which an ultrashort laser pulse was synchronized with the first electrical pulse of the train. These experiments were conducted by applying the train of electrical pulses and laser pulses on a GaTa4Se8 crystal using a similar setup as shown in Figure 4b. The green curves in Figure 5b, represent conductivity ratio or carrier multiplication rate measured without laser.

When electric pulses of 63V are applied, the system fires (i.e. a resistive switching is observed) during the fifth pulse while no firing event is observed when pulses of 52V are used. Applying a laser pulse (10μJ/pulse, hν = 1.55 eV) at the beginning of the train of electric pulses has a strong impact on the behavior of the Mott neuron. Indeed, the firing event occurs much sooner compared to the case without optical excitation. For electrical pulses of 63V, it occurs after only two electrical pulses while it appears in the fifth pulse when using 52 V (see violet curves).

There is a striking difference as no firing event is recorded with electrical stimulus only. These experiments demonstrate a clear impact of an ultrashort light pulse on the behavior of the Mott artificial neuron. It reveals that ultrashort light pulses could be used as a new external parameter to tune the Firing time of the device. At this step the device is made with a single crystal and for this reason the LIF behavior is observed at low temperature using quite high voltage pulses.

But our previous works on Mott artificial Mott neuron have already demonstrated that the working conditions can be much improved by using thin film devices [START_REF] Adda | Mott Insulators: A Large Class of Materials for Leaky Integrate and Fire (LIF) Artificial Neuron[END_REF][START_REF] Adda | First Demonstration of "Leaky Integrate and Fire[END_REF]. Indeed the downscaling of thin film device allows to lower the operational voltage to a few volts and to increase the working temperatute towards room temperature. The experiments reported in this paper and performed on single crystals open, therefore, the door to the realization of artificial electro-optical Mott neurons made of thin films that would be promising components for neuromorphic photonics These experiments open, therefore, the door to the realization of artificial electro-optical Mott neurons which are promising components for neuromorphic photonics [START_REF] Goi | Perspective on Photonic Memristive Neuromorphic Computing[END_REF][START_REF] Prucnal | Neuromorphic Photonics[END_REF].

CONCLUSIONS

To conclude, the electro-optical pump -electrical probe measurements presented in this work When the carrier multiplication rate reaches a threshold value, the firing event (i.e., the resistive switching) occurs. Hot carriers and carrier multiplication rates are, therefore, the physical quantities of interest to control the Mott Neuron. Beyond this fundamental output, we have also shown that both light and electric field stimuli can control the dynamics of the Mott artificial neuron.

Our work demonstrates an interesting concept of electro-optical artificial Mott neuron and opens an alternative path in the emerging field of neuromorphic photonics [START_REF] Goi | Perspective on Photonic Memristive Neuromorphic Computing[END_REF][START_REF] Prucnal | Neuromorphic Photonics[END_REF].

Figure 3a . 5 ( 8 . 8

 3a588 Figure 3a. The boundary values of the photon energies used (hν) were 0.5 eV and 2.3 eV. The

  demonstrate the possibility to control the dynamics of the Electric Mott transition by ultrashort laser pulses in the Mott insulator GaTa4Se8. It further confirms that light and electric field can act synergistically to induce an insulator to metal transition in a Mott insulator. These results are compatible with an Electric Mott transition mechanism based on a carrier multiplication phenomenon whose dynamics can be modified thanks to the generation of extra photo-carriers by optical excitation. The obtained results demonstrate that Mott insulators behave under electric field as leaky integrators of hot carriers, more precisely of carrier multiplication rate.

Figure 1 :

 1 Figure 1 : (a) Resistivity versus temperature measured on a single crystal of the Mott insulator GaTa4Se8. The inset shows the crystallographic structure of GaTa4Se8 and a picture of the GaTa4Se8 single crystal connected with four gold wires. (b-d) A typical example of an Electric Mott Transition induced at 85 K by the application of a 120 µs / 90 V electric pulse on a circuit made of the GaTa4Se8 crystal in series with a 1 kΩ load resistance. The temporal evolutions of the current (b), of the sample voltage (c), and of the sample resistance (d) clearly indicate a resistive switching occurring after a time delay of ≈ 80 µs.

Figure 2 :

 2 Figure 2 : (a) Transport measurements performed at different temperatures and low electric field for a GaTa4Se8 crystal. All measurements fall on a master curve following the predicted evolution of the carrier multiplication rate 𝑛𝑛 � (Blue dotted line). (b) Conductivity ratio dynamics 𝜎𝜎 � = 𝜎𝜎/𝜎𝜎 0 measured under electric field for a GaTa4Se8 crystal for 𝐸𝐸 < 𝐸𝐸 𝑡𝑡ℎ and 𝐸𝐸 > 𝐸𝐸 𝑡𝑡ℎ .

Figure 3 :

 3 Figure 3 : Temporal evolution of the normalized conductivity 𝜎𝜎 � = σ / σ0 during a voltage pulse applied to different circuits containing a GaTa4Se8 crystal and which is synchronized or not with a single-shot 100 fs laser pulse applied at the beginning of the pulse. (a) Schematic representation of the setup and impact of laser fluence or wavelength on the number of electrons excited from the Lower (LHB) to the Upper Hubbard Band (UHB). (b) Evolution of 𝜎𝜎 �(𝑡𝑡) measured under a 60V voltage pulse and when fs laser pulses of constant energy (800 nm, hν = 1.55 eV) and of increasing fluences are applied (from 1µJ/Pulse to 10 µJ/P). (c) Evolution of 𝜎𝜎 �(𝑡𝑡) measured on a different GaTa4Se8 crystal under a 85 V voltage pulse and when fs laser pulses leading to the same number of induced photocarriers are applied with two different energies (hν = 0.5 and 2.3 eV).

Figure 4 :

 4 Figure 4 : (a) Schematic picture of a biological neuron receiving input spikes from other neurons and triggering an output action potential when the membrane potential reaches a threshold value. This behavior of the neuron membrane was first modeled by a RC circuit and led to the definition of the simplest model of artificial neuron called "Leaky integrate and Fire (LIF)" (b) Sketch of an electro-optic Mott neuron setup and graphical representation of the electronic dynamics with or without laser pulse. The artificial Mott neuron follows a similar dynamics as the LIF model but the integrated quantity is not the membrane potential (or the charge in the capacitor) but the carriers multiplication rate (n �) in the Mott device.

Figure 5 :

 5 Figure 5 : Experimental resistive switching obtained by applying trains of 25µs electric pulses of 63 V or 52 V synchronized or not with a 100 fs laser pulse. (a) Profile of applied voltage pulses, b) and c) sample conductivity time profiles measured without and with laser pulse, respectively. The measurements were performed in similar conditions to the ones of Figure 3 a.

ACKNOWLEDGEMENTS

All authors gratefully acknowledge Agence Nationale de la Recherche for financial support under grant ANR-16-CE30-0018 ("Elastica"). DB thanks University of Nantes for financial support in the framework of the IM-LED LIA (CNRS).


Valentin Leroy
email: valentin.leroy@u-paris.fr

Juliette Pierre

Benjamin Dollet

A toy model for the effective density of acoustic metamaterials

We propose a simple mechanical system that can be used as a toy model for calculating the effective density of acoustic metamaterials. Through analytical calculations, it gives a better understanding of how the effective density can become negative, when the system responds elastically instead of inertially. We show that this toy model can reproduce, qualitatively, the acoustical behaviour of some real acoustic metamaterials.

Introduction

Metamaterials are structured materials for which exotic behaviours in terms of wave propagation emerge from the microstructure. The most striking example is probably that of metamaterials for electromagnetic waves having a negative refractive index [START_REF] Pendry | Metamaterials in the sunshine[END_REF], which is obtained when both the permittivity and permeability of the medium are negative. In acoustics, finding structures that lead to negative parameters has also been a fruitful field of research. The literature contains many examples of acoustic metamaterials with negative bulk modulus and/or negative density [START_REF] Ma | Acoustic metamaterials: From local resonances to broad horizons[END_REF]. Among the countless resonant systems exhibiting such properties, one finds several studies on membrane or thin plate structures: simple membranes arranged in parallel or series [START_REF] Lee | Acoustic metamaterial with negative density[END_REF][START_REF] Park | Amplification of acoustic evanescent waves using metamaterial slabs[END_REF][START_REF] Varansi | The low frequency performance of metamaterial barriers based on cellular structures[END_REF][START_REF] Duan | Theoretical requirements for broadband perfect absorption of acoustic waves by ultra-thin elastic meta-films[END_REF][START_REF] Lee | Composite Acoustic Medium with Simultaneously Negative Density and Modulus[END_REF][START_REF] Fleury | Extraordinary Sound Transmission through Density-Near-Zero Ultranarrow Channels[END_REF][START_REF] Mallejac | Zero-phase propagation in realistic plate-type acoustic metamaterials[END_REF][START_REF] Mallejac | Experimental evidence of a hiding zone in a density-near-zero acoustic metamaterial[END_REF][11], decorated membranes [START_REF] Yang | Membrane-Type Acoustic Metamaterial with Negative Dynamic Mass[END_REF][START_REF]Coupled Membranes with Doubly Negative Mass Density and Bulk Modulus[END_REF][START_REF] Ma | Acoustic metasurface with hybrid resonances[END_REF][START_REF] Bongard | Acoustic transmission line metamaterial with negative/zero/positive refractive index[END_REF], thin plates loaded with small pillars [START_REF] Roca | Broadband and intense sound transmission loss by a coupledresonance acoustic metamaterial[END_REF], and also natural systems composed with liquid membranes [START_REF] Pierre | Resonant Acoustic Propagation and Negative Density in Liquid Foams[END_REF].

However, the concept of negative density is still often unsettling when first encountered, much more than the idea of a negative compressibility or a negative permittivity. This is of course because the term of density can refer to the gravitational density, which cannot be negative, or to the inertial density, which can. The main motivation of the present article is to highlight the physical mechanisms underlying the negative effective density, in order to demythify this notion.

We propose a simple system that can be used as a toy model to understand the concept of effective density in acoustics. All the calculations are straightforward, based on Newton's second law, and we give some details about their derivations (sections 2 to 4), hopefully allowing scientists that are not familiar with metamaterials to gain a physical understanding of the mechanisms leading to negative density. With this toy model one can determine under which conditions a unit cell will give rise, or not, to a negative density (section 5), including in the case of dissipation (section 6). We also show that despite its crudeness, our toy model can reproduce, at least qualitatively, the acoustic behaviour of several existing metamaterials (section 7).

Effective mass

The system that we consider is sketched in Fig. 1. In an infinite tube of section S = πR 2 , we consider an object made of an annulus (part 1) of mass M 1 and section S 1 around a disk (part 2) of mass M 2 and section S 2 , both of thickness e. Parts 1 and 2 of this object are linked by a spring of stiffness K, and the ring is connected to the tube by a spring of stiffness K . We note z = 0 the position of the left face of the object, and z = e that of its right face. The pressure difference between the left and right faces is noted ∆P and we are interested in the situation in which the system is harmonically excited: ∆P exp(-iωt).

For the simple case of a uniform mass M that slides without friction on the tube (M 1 = 0, M 2 = M , K = K = 0) the displacement U is given by: -M ω 2 U = ∆P S.

(2.1)

Note that the sign of U might seem counter-intuitive here: for ∆P > 0, one finds that U < 0, which means that a pressure force that pushes to the right results in a displacement to the left. This is what is expected for a purely inertial response of the system: the system displacement is in antiphase with the excitation force.

In the general case with two masses, we define the effective mass, M eff , as the ratio between the force ∆P S and the acceleration -ω 2 U . The average displacement is defined as U = (1x)U 1 + xU 2 , where U 1 and U 2 are the displacements of masses 1 and 2, and x = S 2 /(S 1 + S 2 ) is the surface fraction occupied by the central mass (mass 2). U 1 and U 2 are determined by solving the coupled equations It is then easy to calculate the effective mass:

-M 1 ω 2 U 1 = ∆P S 1 -K(U 1 -U 2 ) -K U 1 (2.2) -M 2 ω 2 U 2 = ∆P S 2 -K(U 2 -U 1 ). (2.3) (b) (a)
M eff = M 2 + M 1 (1 + K /K) -K /ω 2 -ω 2 M 1 M 2 /K 1 + x 2 K /K -ω 2 [x 2 M 1 + (1 -x) 2 M 2 ]/K . (2.4)
M eff is thus frequency dependent, and it also depends on the parameters of the system (M 1 , M 2 , K, K and x). To illustrate the idea of negative mass, let us consider the case of an infinitely rigid spring between the two masses (K → ∞). The system is then a M 1 + M 2 mass linked to the tube by a spring K , and Eq. (2.4) reduces to

M eff = (M 1 + M 2 ) -K /ω 2 , which is negative for ω 2 < K /(M 1 + M 2 ).
It is interesting to write Newton's second law in this case:

-(M 1 + M 2 )ω 2 U + K U = ∆P S. (2.5)
In response to our forcing pressure, there are two components now: the inertial response, that involves the mass of the system, and the elastic response, which depends on the spring. At low frequency, for ω K /(M 1 + M 2 ), the elastic force dominates, and we see that ∆P > 0 leads to U > 0 this time. The system thus responds in a way that is more natural to our everyday life experience: it moves in the direction of the applied force. However, when analysed from an inertial point of view, it gives an abnormal phase, and a negative mass. A negative mass system is thus merely a system that reacts elastically rather than inertially. Note that this situation is formally analogous to the plasma resonance in electromagnetics, in which the permittivity is negative for frequencies below the plasma resonance [19].

Designing an elementary cell in view of obtaining a negative density consists in including somehow an elastic response that will be interpreted as an abnormal inertial response. The spring K between the outer ring and the tube is a solution, but it is not the only one. As shown in Fig. 2, Eq. (2.4) can also predict the existence of a regime of negative mass even for K = 0. Here the parameters are M 1 = 9 g, M 2 = 1 g, x = 0.9, K = 1000 N/m, and K = 0. We see that the effective mass is negative from about 200 to 600 Hz. Let us look into detail the response of the system as a function of frequency. At low frequency, the two masses move together and we find M eff = M 1 + M 2 . For a given frequency, here close to 150 Hz, there is a resonance and the effective mass diverges and becomes negative. We will discuss in section 5 how this resonance frequency depends on the parameters. For now, let us focus on the physical picture. The resonance comes from the different inertial responses of the two masses (because their mass and surface are different), which leads to different displacements U 1 and U 2 and brings the elastic response K(U 1 -U 2 ) into play, such that the effective mass becomes negative. However as in the case K = 0 discussed before, the inertial response dominates the elastic one at higher frequencies, meaning that the effective mass becomes positive again. At high frequency, the asymptotic value of M eff corresponds to two masses in parallel, without connection. . Example of (a) the effective mass and (b) the amplitude of transmission for objects defined by M 1 = 9 g, M 2 = 1 g, x = 0.9, K = 10 kN/m, K = 0 and immersed in a tube filled with air (k 0 = 2πf /c 0 and Z0 = ρ 0 c 0 with c 0 = 340 m/s and ρ 0 = 1.2 kg.m -3 ). Inset: sketch of the transmission and the reflection of an incident wave through the effective system.

Transmission

In the previous section, we imagined a situation in which we were able to excite the system with a given force ∆P S. In practice, the excitation is rather the result of an incoming wave. Let us consider this more realistic situation. It will also turn out to be a good intermediate calculation for the next section, in which we look at an effective continuous medium. The inset in Fig. 2b shows the situation that we want to calculate: an incident pressure wave P exp[i(k 0 z -ωt)] impinges the object and leads to a reflected wave rP exp[i(-k 0 z -ωt)] and a transmitted one tP exp[i(k 0 zωt)], where k 0 is the wavevector for the medium filling the tube (air, for example). Note that we consider only plane waves propagating in the tube. It imposes that the wavelength remains larger than the diameter of the tube, otherwise other modes could be excited, especially when the two parts of the objects exhibit different displacements. For a tube of radius R = 5 cm filled with air, the frequency must be lower than 2 kHz.

Let us calculate the coefficients r and t, which will allow us to determine ∆P . The total pressure field on each side of the object can be written (dropping the time dependence) p(z) = P e ik0z + rP e -ik0z for z < 0,

p(z) = tP e ik0z for z > e.

On the other hand, the displacement field is given by

u(z) = i ωZ 0 P (e ik0z -re -ik0z ) for z < 0, (3.3)
u(z) = i ωZ 0 tP e ik0z for z > e, (3.4)
where Z 0 is the impedance of the surrounding medium. At this point it is important to introduce a condition on the thickness of the object. We will consider that the wavelengths, in both the host medium and in the constitutive material of the object, are much larger than the thickness e of the object. Hence, we may discard wave propagation in the object, and we can consider that its left and right faces are moving with the same displacement: u(z = 0) = u(z = e) = U . It gives a first relationship between r and t:

1 -r = te ik0e . (3.5)
Note that exp(ik 0 e) is close to one, but we keep this term because it will be useful in section 4.

The second relation is obtained by writing Newton's second law for the object (directly with its effective mass given by Eq. (2.4)):

-M eff ω 2 U = P S 1 + r -te ik0e . (3.6)
Putting Eqs. (3.4)-(3.6) together, we finally obtain:

t = e -ik0e 1 + ωM eff 2iZ0S , (3.7
)

r = 1 -te ik0e . (3.8)
Eq. (3.7) is known as the Mass law, valid for an obstacle that is thinner than the wavelength. It shows that low frequency sound is hard to block, because |t| → 1 for ω → 0. A high value of surface mass (M eff /S) is necessary for efficiently blocking low frequency noise. This is actually a challenge for metamaterials: having a large effective mass with a light and thin system. Fig. 2b shows the amplitude of the transmission in our example. There is a clear minimum at resonance (when the effective mass diverges) and a maximum when the mass goes through zero.

Continuous medium

Now that we know the acoustical behaviour of our elementary cell, we can build a metamaterial. The idea is simply to stack several objects one after the others. For simplicity let us consider that the stacking is periodic, with a distance d from one layer to the other, as illustrated in Fig. 3a.

Between the layers, we have the same medium as in the rest of the tube (e.g., air), with k 0 and Z 0 . As shown in Appendix A, at the low frequency limit (k 0 d 1), the N layers behave as an effective homogeneous liquid of thickness L = N d, with an effective compressibility χ eff and an effective density ρ eff given by

χ eff χ 0 = 1 + i k 0 d 1 -t -r t , (4.1)
ρ eff ρ 0 = 1 + i k 0 d 1 -t + r t . (4.2)
Inserting the values (3.7) for t and (3.8) for r, we obtain:

χ eff = 1 - e d χ 0 , (4.3)
ρ eff = 1 - e d ρ 0 + M eff Sd . (4.4)
We can check that these expressions correspond to a mixing law, in which an effective property A eff is given by an average of the properties of its components (A 0 , A 1 ), with a weight corresponding to the volume fraction:

A eff = (1 -Φ)A 0 + ΦA 1 ,
where Φ is the volume fraction of component 1 in the unit cell. In our case, the volume fraction is given by e/d. The effective compressibility (4.3) is a mixture law with χ 1 = 0, because the inclusion is incompressible (we assumed that both faces of the object had the same displacement). As for Eq. ( 4.4), it also takes the form of a mixture law if we define a density ρ 1 = M eff /(Se) for the inclusion. The multilayer has thus the same qualitative behaviour as its constitutive elements, as shown in Fig. 3b, with a resonance followed by a regime of negative density.

Frequency range of negative mass given by the toy model

We now study in more details the predictions of the toy model. We take advantage of its simplicity to derive analytical predictions for the frequency ranges where the effective mass is negative (henceforth called "negative bands" for simplicity), and which sets of parameters maximise these ranges. Since we saw that ρ eff was an affine function of M eff (eq. (4.4)), all our conclusions on the effective mass will also apply on the negative density, provided the density of the host medium is low (as for air). In order to explore the conditions for a negative effective mass, we introduce four dimensionless parameters describing our system:

x = S 2 S 1 + S 2 , (5.1)
µ = M 2 M 1 + M 2 , (5.2)
κ = K K , (5.3
)

Ω 2 = M 1 + M 2 K ω 2 , (5.4)
with x the surface fraction occupied by the central cylinder (already introduced in Sec. 2), µ the mass fraction carried by the central mass, κ the ratio between the stiffness of the two springs and Ω a dimensionless frequency. Using these dimensionless parameters on can rewrite eq. (2.4) as:

m eff ≡ M eff M 1 + M 2 = µ + (1 -µ)(1 + κ) -κ/Ω 2 -µ(1 -µ)Ω 2 1 + x 2 κ -[(1 -µ)x 2 + µ(1 -x) 2 ]Ω 2 , (5.5)
where m eff is the dimensionless effective mass.

(a) Case K = 0

First, let us look at the simple case where κ = 0. The dimensionless effective mass becomes

m eff = 1 -µ(1 -µ)Ω 2 1 -[(1 -µ)x 2 + µ(1 -x) 2 ]Ω 2 .
(5.6)

In Fig. 4a we plot the dimensionless effective mass of Fig. 2a Moreover, Eq. (5.6) shows immediately that the effective mass cancels out for one characteristic frequency, Ω 0 , and presents one frequency resonance Ω r0 :

Ω 2 0 = 1 µ(1 -µ) , (5.8
)

Ω 2 r0 = 1 (1 -µ)x 2 + µ(1 -x) 2 .
(5.9)

Finally, at high frequency, the effective mass does not depend on the stiffness of the spring between the two masses. It takes the form of two masses in parallel:

lim Ω→∞ m eff ≡ m∞ = x 2 µ + (1 -x) 2 1 -µ -1
.

(5.10)

As illustrated in Fig. 4a, the effective mass is negative between Ω r0 and Ω 0 . To appreciate the width of the negative band, we study Ω r0 /Ω 0 , which will be small for a large negative band. Interestingly, this ratio is equal to the high frequency limit of the effective mass, and it is almost symetrical in x and µ:

Ω 2 r0 Ω 2 0 = m∞ = 1 + (x -µ) 2 µ(1 -µ) -1
.

(5.11) Fig. 4b proposes a visualisation of how (Ω r0 /Ω 0 ) 2 depends on x and µ. The symmetry is visible, and one can conclude that a large negative band can be obtained either with a small µ and a large x, or with the opposite, a small x and a large µ. We will see in Section 7 that both cases have been explored in experimental systems.

(b) Case K = 0

Let us now consider the effect of the spring between the tube and the annulus. Since the analysis is more involved than in the previous subsection, most calculations are deferred to Appendix B. The behaviour of the effective mass is now much richer, as exemplified in Fig. 5.

From eq. (5.5), at low frequency, on can get a first expression:

lim Ω→0 m eff = - κ (1 + x 2 κ)Ω 2 ,
(5.12)

In this limit the effective mass is no longer the sum of the two masses. The negative sign of the limit (5.12) shows that the elastic response conferred by the spring of stiffness K dominates the behaviour of the system at low frequency. This is essentially different from the limit (5.7), taken from a case where the stiffness K is absent, and where elastic effects dominate only at intermediate frequency, as the two masses start moving at different amplitudes and phases.

Ω - Ω R 0 1 0 1 x µ A A-B-A (c) (a) A-C-A A-C 0 4 1 2 3 Ω Ω - Ω R Ω + (c) (b) C: Ω 0 4 2 -1 0 1 Ω - Ω + Ω R
In contrast, at high frequency, the effective mass remains independent of the two stiffnesses. With or without springs the effective mass remains the same quantity, given by (5.10):

lim Ω→∞ m eff ≡ m∞ (5.13)
To study the frequency dependence of the effective mass, we can recast (5.5) as:

m eff = - κ (1 + x 2 κ)Ω 2 (1 -Ω 2 /Ω 2 -)(1 -Ω 2 /Ω 2 + ) (1 -Ω 2 /Ω 2 r ) , (5.14)
which shows that the effective mass now cancels for two characteristic frequencies Ω ± , and that one frequency resonance Ωr still appears:

Ω 2 ± = 1 2µ(1 -µ) µ + (1 -µ)(1 + κ) ± [µ -(1 -µ)κ] 2 + (1 -µ)[1 + µ + 2(1 -µ)κ] ,
(5.15) and:

Ω 2 r = (1 + x 2 κ)Ω 2 r0 .
(5.16)

The factorised expression (5.14) shows that one can expect three changes of sign for the effective mass, hence two negative bands. If the three frequencies Ω -, Ω + and Ωr are ordered as: Ω 1 < Ω 2 < Ω 3 , the two negative bands appear at frequencies below Ω 1 , and between frequencies Ω 2 and Ω 3 .

Since Ω -< Ω + from their definition (5.15), we must compare Ωr to the two frequencies Ω ± to determine the bounds of the negative bands in terms of these three frequencies. Three cases are possible: (a) Ω -< Ωr < Ω + , where one negative band is below Ω -, and one between Ωr and Ω + (Fig. 5a); (b) Ωr < Ω -< Ω + , where one negative band is below Ωr, and one between Ω -and Ω + (Fig. 5b); (c) Ω -< Ω + < Ωr, where one negative band is below Ω -, and one between Ω + and Ωr (Fig. 5c). Moreover, degenerate cases are possible when Ωr = Ω -or Ωr = Ω + . Then (5.14) shows that both the resonance and the second negative band vanish, and that a single low-frequency band remains present; this degenerate case is illustrated in the inset of Fig. 5c. Now that the different cases (a), (b) and (c) have been presented, we must predict in which range of parameters (x, µ, κ) they are found. First, it is worth noting from (5.15) and (5.16) that lim κ→0 Ω -= 0, while lim κ→0 Ωr = Ω r0 > 0 and lim κ→0 Ω + = Ω 0 > Ω r0 where Ω 0 and Ω r0 are given by (5.8) and (5.9), hence the condition (a) Ω -< Ωr < Ω + is fulfilled for sufficiently low values of κ, whatever x and µ. Hence, for any fixed couple of values (x, µ), we study whether the ordering changes as κ varies between 0 and ∞. The result of such a study is summarised in Fig. 5d (more details are given in Appendix B), where in the blue region the condition (a) is met whatever the value of κ, while in the other regions, varying κ for fixed x and µ results in changes of ordering of Ωr relative to Ω ± .

Dissipation

We now consider the effect of dissipation. For simplicity, we restrict ourselves to adding an imaginary part to the stiffness K = Kr + i K i , as if there were a dashpot between the two moving masses.

In the simplest case K = 0, the effective mass is given by m eff = mr + i m i with:

mr = (1 -Ω 2 /Ω 2 0 )(1 -Ω 2 /Ω 2 r0 ) + α 2 (1 -Ω 2 /Ω 2 r0 ) 2 + α 2 , ( 6.1)
m i = α 1 Ω 2 r0 - 1 Ω 2 0 Ω 2 (1 -Ω 2 /Ω 2 r0 ) 2 + α 2 , (6.2)
where the dimensionless parameter α is the ratio between the imaginary part and the real part of the complex stiffness :

α ≡ K i Kr . (6.3)
Figure 6a shows how the system considered in figure 4 is modified by dissipation into account. The negative band is reduced by the presence of dissipation. As expected for a resonance, the imaginary part presents a maximum around the frequency resonance decreasing with α. Interestingly, the frequency of this maximum slightly increases with α (Ω 2 c = Ω 2 r0 √ 1 + α 2 ). The effective mass cancels for two frequencies Ω α± up to a critical value of α. From the numerator of . Eq. ( 6.1), one gets that the negative band vanishes for:

αc = 1 2 Ω 0 Ω r0 - Ω r0 Ω 0 , (6.4)
The width of the negative frequency band is quantified by Γ = (Ω α+ -Ω α-)/Ω α+ . This quantity Γ is represented in Fig. 6b for a fixed value of x.

In the general case where both stiffnesses K and K are present, it is easy to show that the main effect of dissipation is to suppress the second negative band (see Sec. 5(b)) for a large enough value of α. The proof is presented in Appendix C; however, there is no analytical prediction of αc, contrary to Eq. (6.4) in the case κ = 0.

Comparison with real systems

Despite its crudeness, the toy model that we propose can reproduce, at least qualitatively, the acoustic behaviours reported in some experimental systems. We will discuss three types of systems.

A first example is the negative density obtained by placing a series of membranes [START_REF] Lee | Acoustic metamaterial with negative density[END_REF][START_REF] Mallejac | Zero-phase propagation in realistic plate-type acoustic metamaterials[END_REF] in a tube. Each elastic membrane, attached to the tube, confers a stiffness K to the system, and has a mass M 1 , leading to a plasma-like resonance, as discussed in [19], below which the effective density is negative.

Two other examples associate a membrane and a heavier structure: the decorated membrane resonators (DMR), and the liquid foams. Interestingly, each case illustrates one of the two favourable configurations which we identified for the emergence of a negative band: a heavy central part with a limited surface (large µ and small x) or a light one with a large surface (small µ and large x).

(a) Membranes

Due to its ability to bring an elastic response to the system, a membrane is a good candidate for obtaining a negative effective density. One of the first experimental evidences of a negative density was brought by Lee et al. [START_REF] Lee | Acoustic metamaterial with negative density[END_REF] who placed a series of membranes in a tube and demonstrated the existence of a negative density below a critical frequency fc. Figure 7a shows the phase velocity v that they measured in their system, as a function of frequency. As v = (ρχ) -1/2 , the passage through ρ = 0 is associated to a peak in velocity. Injecting eq. ( 2.4) into eq. (4.4) for K → ∞, we obtain the following law for the effective density of the system:

ρ eff = 1 - e d ρg + M -K /ω 2 Sd (7.1) = ρ 1 - ω 2 c ω 2 , (7.2)
with ρ = (1 -e/d)ρg + ρme/d and ω 2 c = K /(Sdρ ) (ρg is the density of the gas, and ρm that of the membrane). The experimental system was made with thin membranes (e = 10 µm, ρm = 940 kg/m 3 ) in a R = 1.5 cm tube, with a separation of d = 7 cm. We can thus evaluate ρ = 1.34 kg/m 3 . For K , we adjust its value to match the cutoff frequency fc = 720 Hz, which leads to K = 6210 N/m. As shown in Fig. 7a, with these values of the parameters, the measured velocity is well captured by the toy model.

Metamaterials with density close to zero have attracted attention recently because they could open the way to narrow and long channels capable of efficiently coupling wave guides [START_REF] Fleury | Extraordinary Sound Transmission through Density-Near-Zero Ultranarrow Channels[END_REF]. Within this perspective, a recent study by Malléjac et al. [START_REF] Mallejac | Zero-phase propagation in realistic plate-type acoustic metamaterials[END_REF] has conducted careful experiments on a system of thin plates in a tube. The measured effective density is reproduced in Fig. 7b, the open symbols showing the real part of the density, and the black ones the imaginary. The data were obtained with plates (e = 102 µm, ρm = 1400 kg/m 3 ) with a spacing of d = 1 cm in a 1.5 cm tube. The material of the plates was characterized by a Young modulus of E = E 0 (1 + iβ), with E 0 = 4.6 GPa and β = 0.13, and a Poisson ratio of 0.4.

Adapting the toy model to the mass of the plates and the geometry of the system, we take a total mass M = 101 mg, which leads to ρ = 15.5 kg/m 3 . For the spring, we select the value that leads to the correct cutoff frequency of 413 Hz: K = 748 N/m. And we add an imaginary part of α = 13% to account for the dissipation. The prediction of the toy model with these parameters is shown in Fig. 7b. As in the experiments, it suggests that the imaginary part of the density is not negligible. It fails, however, at reproducing the correct trend for the increase of density with frequency. This suggests that the toy model, which has only one degree of freedom, cannot capture the whole behaviour of the plates. The interest of the toy model in this case is limited to its ability to reproduce qualitatively the observed phenomenon.

(b) Decorated membrane resonators

Loading a membrane with a heavier element leads to what is often called a decorated membrane resonator (DMR). A typical example is shown in the inset of figure 8: a latex membrane (in yellow) is stretched on a rigid frame, and loaded with a plastic disk (in grey). Figure 8a is extracted from reference [START_REF] Ma | Acoustic metasurface with hybrid resonances[END_REF] and shows the experimental transmission as a function of frequency, obtained with such a system (black line). Two peaks are visible, with a very low minimum of transmission in between. These features can be interpreted in terms of an effective mass (red dashed curve), which goes through zero for the peaks of transmission, and increases abruptly at the resonance, corresponding to the minimum of transmission.

We see that the effective density of this system is very similar to what was obtained in figure 2a for our toy model. It is tempting to go a step further and look if we can obtain a quantitative comparison. The membrane is 0.2 mm thick, with a density of 980 kg/m 3 and a radius of 14 mm. The disk has a weight of 125 mg and a radius of 14 mm [START_REF] Ma | Membrane-type acoustic metamaterials[END_REF]. Identifying the disk (and the part of membrane beneath) with mass M 2 in the toy model, we obtain M 2 = 173 mg and M 1 = 104 mg, which corresponds to µ = 0.62. The geometry leads to x = 0.1. Chosing K and K is less straightforward because the elasticity of a membrane cannot be reduced to a simple set of two springs. By tuning the parameters, we found that K = 1.1 kN/m and κ = 0.53 lead to the effective mass shown in figure 8b (dashed red plot). Comparison with the experimental results is qualitatively good, with the same range of frequencies of the two negative regimes. However, note that the density is in arbitrary units in figure 8a, meaning that we cannot quantitavely compare the agreement. We can do it with the transmission coefficient (black line), and it turns out to be quite different from the experimental one, with less sharp peaks and minimum. As in Fig 7b, it probably comes from the fact that the real membrane has a much richer dynamics than what can be simulated with one mass and two springs. Despite this lack of quantitative agreement, the toy model seems to capture some of the physical effects at play in the real system. It might be interesting for helping in designing new DMR, guided by an analytical expression.

(c) Liquid foams

Liquid foams are made of such a high volume fraction of gas that the bubbles are not spherical anymore and adopt polyhedral shapes, as illustrated in the inset of figure 9. The liquid is then contained in two different structures: the channels and the films. In 2014, we reported experimental evidence that the acoustic density of a liquid foam could be negative. Figure 9 shows an experimental result, adapted from [START_REF] Pierre | Resonant Acoustic Propagation and Negative Density in Liquid Foams[END_REF], with the real (white symbols) and imaginary (black symbols) parts of the density as functions of frequency. The foams had a liquid volume fraction of Φ = 11% and a bubble median radius of R 0 = 40 µm. We proved that this negative band was found at any other median radius R, provided a rescaled frequency f = f (R/R 0 ) 1.5 was used (see [START_REF] Pierre | Resonant Acoustic Propagation and Negative Density in Liquid Foams[END_REF] for details). We see in Figure 9 that the real part of the density (open symbols) takes negative values between 100 and 400 kHz. This figure is strikingly similar to figure 6a, which corresponds to the toy model in the K = 0 case. At first sight, this case of the toy model seems far from a real situation, because it consists in considering a mass that can slide without friction in a tube. But the very particular structure of the liquid foam leads to a behaviour that is similar to this idealised situation. When an acoustic wave propagates through the liquid foam, it encounters soap films and liquid channels. Most of the liquid is in the channels, the films being very thin elements. The films carry the elastic response, due to their surface tension. In total, the acoustic behaviour of the foam can thus be modelled as an heavy mass (the liquid channels) around a light one of large relative surface (the films) with a restoring force.

Let us see if we can determine the parameters of the toy model that will reproduce the measurements reported in figure 9. Equation (4.4) for the effective density can be rewritten as a function of the liquid volume fraction: Princen [21]. It depends on the liquid fraction, and for Φ = 11%, one finds that x = 0.2. The total mass M 1 + M 2 can be estimated by looking at the mass of a unit cell, whose volume is given by the median radius of the bubbles in the foam: M 1 + M 2 = Φ × 4 3 R 3 0 ρ . We find a mass of the order of 3 × 10 -11 kg. The other parameters are tuned to recover the same frequencies for the beginning and the end of the negative regime, and we obtain K = 0.18 N/m, α = 1 and µ = 10 -3 . Figure 9 shows the complex effective density given by the toy model for this set of parameters. We see that it compares well to the experimental results, even quantitatively to a certain extent: we obtain good orders of magnitude for the real and imaginary parts of the density. Note that the level of dissipation is far from being negligible (α = 1). According to our analytical study of the model, the negative band survives to this dissipation because parameter µ is particularly low (see figure 6b).

ρ eff = (1 -Φ )ρg + Φ M eff Se = (1 -Φ )ρg + Φ ρ m eff . ( 7
Liquid foams appear as quite unusual acoustic metamaterials because they combine several characteristics that are in principle not favourable to the emergence of a negative band: they are polydisperse, disordered, and highly dissipative. The negative band is robust because foams are intrinsically constituted of large but light elements (the films) elastically attached to small but massive elements (the liquid channels).

Conclusion

In this article, we have introduced a simple toy model, constituted at most of two masses, two springs and one dissipative item, to clarify the occurrence of negative effective density in acoustic metamaterials. We have shown three different sets of results. (i) We have made explicit how to derive a transmission coefficient from the single unit of the toy model, and how, when coupled to the ambient medium, its behaviour can be modelled at the macroscopic level as an effective continuous medium. (ii) With a detailed study of the toy model, we have clarified for which frequency range a negative mass can be expected. (iii) We have compared our predictions to real experimental metamaterials and found a good agreement, at least qualitatively.

Beyond its didactic interest, this toy model might be a useful tool to design acoustic metamatarials. Note that it could be generalised to systems with resonances in density and compressibility, by adding masses in series. More complex systems could then be modelled, such as the double DMR [START_REF]Coupled Membranes with Doubly Negative Mass Density and Bulk Modulus[END_REF], or the "acoustic sandwiches" [START_REF] Bolton | Sound transmission through multi-Panel structures lined with elastic porous materials[END_REF].

C. Suppression of the second negative band for large dissipation

To study dissipation, we substitute K = Kr + iK i in the expression (2.4) of the effective mass. After some algebra, this yields:

m eff = - κ (1 + x 2 κ)Ω 2 (1 -Ω 2 /Ω 2 -)(1 -Ω 2 /Ω 2 + ) + iα(1 -Ω 2 /κ) 1 + iα -Ω 2 /Ω 2 r , ( C 1)
provided κ is redefined as κ = K /Kr [compare to (5.3)]. Apart from this redefinition, the other quantities are defined as previously, in particular Ω ± is given by (5.15) and Ωr by (5.16). Hence at large dissipation, the effective mass has the following limit:

lim α→∞ m eff = - κ (1 + x 2 κ)Ω 2 1 -Ω 2 /κ Ω 2 .
This large-dissipation effective mass displays a single, low-frequency negative band, for Ω 2 < κ (and its imaginary part disappears). Intuitively, the two masses are then forced to move at the same amplitude, which suppresses the effect of the spring constant K. Hence, the main effect of dissipation is to make the second negative band studied in Sec. 5(b) vanish for a certain critical value αc(x, µ, κ) of the dissipation parameter α. However, Eq. (C 1) is too complex to allow for an analytical prediction of αc.

Figure 1 .

 1 Figure 1. Schematic views of the system which we propose as a toy model. (a) An annulus of mass M 1 and surface S 1is around a disk of mass M 2 and surface S 2 , in an infinite tube of surface S = S 1 + S 2 = πR 2 . (b) The two masses are connected via a spring K, and the annulus is linked to the tube with a spring K . The effective mass of this system can be established by calculating how it moves when excited by an overpressure ∆P exp(-iωt).

Figure 2

 2 Figure 2. Example of (a) the effective mass and (b) the amplitude of transmission for objects defined by M 1 = 9 g, M 2 = 1 g, x = 0.9, K = 10 kN/m, K = 0 and immersed in a tube filled with air (k 0 = 2πf /c 0 and Z0 = ρ 0 c 0 with c 0 = 340 m/s and ρ 0 = 1.2 kg.m -3 ). Inset: sketch of the transmission and the reflection of an incident wave through

Figure 3 .

 3 Figure 3. (a) A succession of layers, characterised by their transmission and reflection coefficients (t,r) is equivalent, at low frequency, to a homogeneous medium with an effective compressibility χeff and an effective density ρeff, given by Eqs. (4.1) and (4.2). (b) Effective density for a multilayer made of an elementary cell as in Fig. 2 (M 1 = 9 g, M 2 = 1 g, x = 0.9, K = 10 kN/m, K = 0), for e = 1 mm and d = 10 mm.

Figure 4 .

 4 Figure 4. Case κ = 0. (a) Dimensionless effective mass as a function of the dimensionless frequency Ω for µ = 0.1 and x = 0.9. (b) Ratio between the two bounds of the negative band as function of the surface fraction occupied by the central cylinder x and the mass fraction carried by the central mass µ. The smaller the ratio, the larger the negative band.

  rspa.royalsocietypublishing.org Proc R Soc A 0000000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5 .

 5 Figure 5. Plots of the effective mass as a function of frequency, showing the three different qualitative behaviours depending on the position of the resonance frequency Ωr respective to Ω ± : (a) case Ω -< Ωr < Ω + ; (b) case Ωr < Ω -< Ω + ; (c) case Ω -< Ω + < Ωr. The values of the parameters chosen to plot these graphs are: (a) x = 1/3, µ = 1/3, κ = 1; (b) x = 0.1, µ = 0.9, κ = 101.4; (c) x = 0.65, µ = 0.55, κ = 3. The inset of (c) shows a degenerate case where Ωr = Ω + [values of the parameters: Inset:κ = κ -= 0.2 (d) x = 0.65, µ = 0.55, κ = κ -= 0.2, where κ -is defined by (B 6)]. (d) Diagram showing which of the three cases occur as κ varies at given values of x and µ. In blue, case (a) Ω -< Ωr < Ω + happens whatever κ. In orange, as κ increases, there is a first transition from (a) to case (b) Ωr < Ω -< Ω + , and a second one back to (a). In green, as κ increases, there is a first transition from (a) to case (c) Ω -< Ω + < Ωr, and a second one back to (a). In red, as κ increases, there is a single transition from (a) to (c). The three symbols , • and in panel d indicate the values of x and µ corresponding to the plots in panels a, b and c, respectively.

  rspa.royalsocietypublishing.org Proc R Soc A 0000000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 6 .

 6 Figure 6. Influence of the dissipation for κ = 0. (a) Real part (continuous lines) and imaginary part (dash lines) for the same case as in Fig. 4 (x = 0.1, µ = 0.9), but with dissipation, for two different values of α: α = 0.1 (black lines) and α = αc (red lines). (b) Width of the negative frequency band Γ = (Ω α+ -Ω α-)/Ω α+ (where Ω α± are the zero-mass frequencies) for x = 0.9. The red dashed line indicates the extinction of the negative frequency band (Eq. (6.4)).

Figure 7 .

 7 Figure 7. Comparison between experimental results obtained with membranes and the toy model. (a) Phase velocity as a function of frequency for a series of 10 µm-thick membranes separated by d = 7 cm in a 3.2 cm-radius tube (Fig. 5 of Ref. [3] -black square). With M = 30 mg and K = 6210 N/m, the toy model gives a good agreement (solid green line). Adding dissipation (dashed green line) lowers the velocity peak. (b) Effective density for 102 µm-thick plates separated by d = 1 cm in a 1.5 cm-radius tube (Fig. 4 of Ref. [9] -white circle for the real part and black circle for the imaginary). The toy model prediction is plotted for M = 101 mg, K = 738 N/m, and a dissipation of α = 0.13 (solid blue line for the real part and dashed blue line for the imaginary).

  rspa.royalsocietypublishing.org Proc R Soc A 0000000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 8 .

 8 Figure 8. Comparison between experimental results obtained with a decorated membrane [14] (a) and our toy model (b).Both panels show the transmission (black curves) and the effective mass (red dashed curves) as function of frequency.The inset is a drawing of the experimental setup. Data and drawing on the left panels are extracted from figure3in the review by Ma and Ping-Sheng[START_REF] Ma | Acoustic metamaterials: From local resonances to broad horizons[END_REF]. For the toy model, we plot Eq. (2.4) for parameters M 1 + M 2 = 277 mg, K = 1.1 kN/m, x = 0.1, µ = 0.62, and κ = 0.53.

3 )Figure 9 .

 39 Figure 9. Comparison between experimental results obtained with a liquid foam and the toy-model. The real and imaginary parts of the effective density are shown as functions of the frequency (rescaled frequency f for the experiments, see text for details). The inset is a photograph of a liquid foam, showing the distribution of the liquid in two different structures: the films and the channels. Data are extracted from figure 5 in our previous article [17]. For the toy model, we plot Eq. (2.4) for parameters M 1 + M 2 = 3 × 10 -11 kg, K = 180 mN/m, x = 0.2, µ = 0.001 and α = 1.

  rspa.royalsocietypublishing.org Proc R Soc A 0000000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  .3) rspa.royalsocietypublishing.org Proc R Soc A 0000000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . To calculate m eff , the values of M 1 + M 2 , K(1 + iα), x and µ are required. The surface fraction occupied by soap films in a liquid foam has been studied by

Acknowledgements. We acknowledge invaluable discussions with the participants of the ANR project Samousse.

A. Effective medium

In this Appendix we show that N layers characterised by their transmission and reflection coefficients (t, r) are equivalent, at low frequency, to a homogeneous medium with effective compressibility and density. The idea is to demonstrate that the N -layer system gives the same total reflection and transmission coefficients, R and T , as an effective medium. Let us consider (see Fig. 3a) a homogeneous medium of thickness L = N d, with wavenumber k and impedance Z. The total reflection and transmission coefficients are given by:

where y = (Z -Z 0 )/(Z + Z 0 ). We want to show that the transmission and reflection coefficients through the multi-layer system can be written in the same form, with Z and k being effective quantities that can be expressed as functions of r and t.

Instead of handling Eq. (A 1), we will consider the ratio:

We will show that the multi-layer system can satisfy (A 2) and (A 3) if k and y are related to r and t by

)

We follow a recursive sequence. For the initialisation (N = 1), we consider one object with a thickness d of the host medium around it. We then obtain For the recursivity, we want to show that if T N verifies (A 2) and R N /T N verifies (A 3), then T N +1 and R N +1 /T N +1 also do. Going from N to N + 1 layers is simply done by a multiple reflection calculation, adding the extra layer on the left. We obtain [START_REF] Brekhovskikh | Waves in Layered Media[END_REF]

which leads to

It can be checked that when (A 4) and (A 5) are injected into those equations, we obtain

as expected.

From equations (A 4) and (A 5), we can calculate k and Z as functions of r and t. We will only consider the large-wavelength case, for which k 0 d 1 and kd 1. The calculations are easier when using ρ and χ instead of k and Z, using the following relationships:

and we finally obtain equations (4.1) and (4.2):

Interestingly, we find that if t = 1 + r, only the compressibility is affected by the layers, while if t = 1 -r, only the density is. This is analogous to the case of three-dimensional inclusions, whose monopolar response affects the compressibility, and dipolar one the density.

B. Details about how diagram of Fig 5d is obtained

In this section we present some details to determine the position of frequency resonance Ωr comparatively to zero-mass frequencies Ω ± . The relation depends on the stiffness ratio κ for a fixed couple (x, µ) ∈ E = [0, 1] 2 . We must solve the equation Ω 2 ± = Ω 2 r . From Eqs. (5.15) and (5.16), it is equivalent to:

which can be rewritten as a second order polynomial in κ: A 2 κ 2 + A 1 κ + A 0 = 0, with: The sign of the coefficient A 1 is negative in between the two dashed lines. The coloured areas refer to position of Ωr respectively to Ω ± (see figure 5d).

and:

The discriminant of this equation is thus given by:

and its roots by:

Only positive roots are physical. A necessary condition is that the discriminant must be positive, else the roots are complex. From (B 5), we find that ∆ < 0 in two regions, labelled as such, in the parameter space (x, µ) ∈ [0, 1] 2 (Fig. 10). If the discriminant is positive, the sign of the real roots is determined by the signs of A 0 , A 1 and A 2 , since κ -κ + = A 0 /A 2 and κ -+ κ + = -A 1 /A 2 . From (B 2), A 0 is positive, while from (B 4), the sign of A 2 is the sign of the quantity x 2 -(x 2 + 2x -1)µ, which is positive except in the region denoted E + 2 in Fig. 10, which is bounded by the bottom curve of equation µ = x 2 /(x 2 + 2x -1) for 1/2 ≤ x ≤ 1. Hence in this region, there is one positive root, which is κ -. For κ < κ -, we find that the ordering Ω -< Ωr < Ω + holds, while for κ > κ -, we find Ω -< Ω + < Ωr. In the other regions, there are either no positive roots or two positive roots, respectively if A 1 is positive or negative. The sign of A 1 is given by the term in bracket in (B 3); it is found that A 1 < 0 between the two dashed curves in Fig. 10, and A 1 > 0 elsewhere. Hence, in the region denoted E 1 , the two roots are negative, while the two roots are positive in regions E - 2 and E 3 (Fig. 10), with κ -< κ + . In these two regions, we find that the ordering Ω -< Ωr < Ω + holds for κ < κ -and κ > κ + . For κ -< κ < κ + , we find that Ω -< Ω + < Ωr in region E - 2 , and Ωr < Ω -< Ω + in region E 3 .