Feng Gao
email: fgao@hbku.edu.qa

Abdelkrim Khelif
email: abdelkrim.khelif@femto-st.fr

Sarah Benchabane
email: sarah.benchabane@femto-st.fr

Abstract-In-liquid wave attenuation is the major factor that limits the resolution of acoustic wave biosensors. Existing acoustic wave devices rely on shear mode operation to reduce the mechanical motion transferred to the liquid. However, the energy leakage still exists through acoustic radiation. In this article, we combined phononic metasurface based on local mechanical resonators with shear horizontal surface acoustic wave (SH-SAW) to address the above issue. The shear-horizontal phononic metasurface (SH-PM) resonator is made of periodic high-aspectratio electrode (HARE). It produces a slow phononic wave that cannot propagate in water due to its low velocity, which suppresses the acoustic radiation. Furthermore, as the particle motion of the HARE is polarized in shear mode, the SH-PM also reduces the mechanical motion transfer to water. The minimization of the total acoustic energy leakage in the SH-PM results in high quality factor (Q) resonance in water. Besides, the increased surface-to-volume ratio of the HAREs also enhances the mass sensitivity of the device. Experimental results show that the SH-PM resonator achieved an in-liquid Q factor of 230 and a normalized mass sensitivity of -192. 4 𝒄𝒎 𝟐 /𝒈. The high in-liquid Q factor together with the high mass sensitivity makes the SH-PM a promising candidate for highresolution gravimetric detection of biomolecules.

Index Terms-Phononic metasurface, Biosensor, Surface Acoustic Wave, Gravimetric Sensing.

I. Introduction

The detection of biomolecules using acoustic biosensors is widely studied because of their high sensitivity, small size, lowcost and label-free characteristics [START_REF] Länge | Bulk and Surface Acoustic Wave Sensor Arrays for Multi-Analyte Detection: A Review[END_REF][START_REF] Zhang | Film bulk acoustic resonators (FBARs) as biosensors: A review[END_REF][START_REF] Durmuş | Acoustic-based biosensors[END_REF] . However, acoustic biosensors suffer from strong in-liquid acoustic wave attenuation that reduces their quality factor (Q). The current approach for addressing this issue relies on shear mode operation of the acoustic wave devices to minimize mechanical motion transferred to the liquid environment [START_REF] Song | Shear mode bulk acoustic resonator based on inclined c-Axis AlN film for monitoring of human hemostatic parameters[END_REF][START_REF] Ji | An aptamer-based shear horizontal surface acoustic wave biosensor with a CVD-grown single-layered graphene film for high-sensitivity detection of a label-free endotoxin[END_REF][START_REF] Ji | A high sensitive SH-SAW biosensor based 36° YX black LiTaO3 for label-free detection of Pseudomonas Aeruginosa[END_REF] . Nevertheless, the shear polarization cannot prevent the energy leakage through acoustic radiation. To address this problem, phononic metasurface (PM) is introduced [START_REF] Gao | Acoustic radiation-free surface phononic crystal resonator for in-liquid low-noise gravimetric detection[END_REF][START_REF] Gao | Towards Acoustic Radiation Free Lamb Wave Resonators for Highresolution Gravimetric Biosensing[END_REF] .

PM resonator is formed by substituting the thin electrode in conventional surface acoustic wave (SAW) resonator with periodic high aspect ratio electrode (HARE) acting as local mechanical resonators. The acoustic wave produced in the PM has a velocity lower than the sound velocity in water, which _______ cannot propagate in water and thus eliminates the acoustic radiation. The previously reported vertically polarized PM (VP-PM) resonator derived from the classical Rayleigh waves demonstrated the radiation suppression phenomenon [START_REF] Gao | Acoustic radiation-free surface phononic crystal resonator for in-liquid low-noise gravimetric detection[END_REF] . However, the vertical polarization together with the processinduced defects in the HARE compromised the suppression effect, leading to a limited Q factor improvement. In this article, we designed a shear horizontal PM (SH-PM) resonator that combines the advantages of shear polarization and phononic metasurface, which can simultaneously suppress the energy leakage through acoustic radiation and mechanical motion transfer. In addition, the large surface-to-volume ratio of the HARE also enhances the mass sensitivity of SH-PM biosensor.

II. Device Design and Fabrication

Fig. 1a shows a top-view SEM image of the fabricated SH-PM resonator. Thirty pairs of interdigitated transducers (IDTs) are located in the center of the device for generating the phononic wave, while 20 reflector strips are placed on the two sides of the IDTs for enhancing the resonance. The number of reflector strips used in the SH-PM resonator is significantly less than the one in conventional SAW resonators [START_REF] Morgan | Surface acoustic wave filters: With applications to electronic communications and signal processing[END_REF] . This is because the locally-resonant HARE produces phononic wave with very low group and phase velocities that suppress the energy propagation, and thus only a small amount of reflectors is needed for confining the acoustic energy. The pitch of the metasurface is 5 μm, which leads to a wavelength of 10 μm. The height of the HARE is 5.3 μm. The profile of the electrode is revealed by a tilted SEM image of the device in Fig. 1b.

The fabrication of the device is implemented by an electroplating-based process. 100-nm copper seed layer was firstly deposited on a four-inch Y-X lithium niobate wafer by sputtering. Optical photolithography is then used to create a 7μm thick photoresist mold for electroplating. Subsequently, 5.3-μm nickel layer is electroplated on the wafer surface to form the HARE. After stripping the photoresist mold, the copper seed layer between nickel structures is removed by reactive ion etching. At last, a 200-nm silicon dioxide layer is deposited and patterned to form a passivation layer for the IDTs and reflectors.

The SH-PM resonator achieves the prohibition of in-liquid wave propagation by producing an acoustic wave that is slower than the sound velocity in water through increasing the electrode height. Fig. 2 shows the phase velocity of the SH-PM resonator at different electrode heights simulated by finite element method (FEM). When the electrodes are higher than 3.7 μm, the phase velocity of the acoustic wave is slower than the sound velocity in water. To reveal the phenomena of acoustic radiation suppression, the mode shapes and acoustic pressure distributions of two SH-PM resonators and a conventional SH-SAW resonator immersed in water are simulated. Simplified unit cells are used in the simulation to reduce the computation time. Perfectly matched layers (PMLs) are exploited in the bottom of the substrate and top of the water to effectively extend their depths to infinity. Fig. 3a shows the simulation result for SH-PM with 5.3-μm electrode. The positive and negative acoustic pressures in the water displayed by the red and blue regions distribute only near the electrodes, representing the local confinement of acoustic energy. In comparison, the PM resonator with 2-μm electrode (Fig. 3b) and the SH-SAW (Fig. 3c) produces propagating acoustic waves in the water, which are seen by the alternating positive and negative acoustic pressures.

III. Performance of the SH-PM resonator

The mass sensing resolution of the SH-PM resonator is determined by its frequency noise divided by its mass sensitivity. Q factor is used to evaluate the frequency stability of the resonator. Fig. 4a shows the measured impedance of the SH-PM resonator in air and water, respectively. The minimum and maximum impedance points correspond to the resonance (𝑓 ! ) and anti-resonance frequencies (𝑓 " ), respectively. The Q factor of the device is derived from its impedance by the following equation:

𝑄 = 𝑓 2 • | 𝑑ϕ 𝑑𝑓 | (1)
where 𝑓 and ϕ are the operating frequency and impedance phase of the device, respectively. The results in Fig. 4b show that the in-liquid Q factor of the SH-PM at 𝑓 " deteriorated by only 1.34 times from 309 in air to 230 in water. For comparison, a conventional SH-SAW based on 36°-YX lithium tantalate operating at 417 MHz was fabricated using the same layout. It was found that the Q factor of the SH-SAW at 𝑓 " deteriorated We used FEM to compare the in-liquid performances of SH-PM together with SH-SAW, shear mode film bulk acoustic wave resonator (S-FBAR) and the previously reported HARE Lamb wave resonator (HARE-LWR). We chose FEM to evaluate the Q factors and mass sensitivities because the experimental values are usually not reported. The simulation models of HARW-LWR and SH-SAW were previously reported in [8]. The simulated S-FBAR comprises of 2-um aluminum nitride (AlN) sandwiched by two 200-nm aluminum electrodes. The c-axis of the AlN is titled by 30° to support the excitation of shear mode. 100-nm poly methyl methacrylate (PMMA) layer is used as the mass loading layer whose density is swept in the simulation to produce the frequency shift signal.

The simulation results in Table 1 show that SH-PM resonator exhibits the highest figure of merit (𝐹𝑂𝑀 = 𝑆 $ × 𝑄) among the four devices. The S-FBAR has the highest mass sensitivity due to its high operating frequency but a relatively low Q factor due to its in-liquid acoustic radiation. The simulated mass sensitivity of the S-FBAR is comparable to the reported mass sensitivity (1690.1 cm 2 /g) of a 1.47-GHz S-FBAR [START_REF] Mirea | Assessment of the Absolute Mass Attachment to an AlN-Based Solidly Mounted Resonator Using a Single Shear Mode[END_REF] . The mass sensitivities of SH-PM and HARE-LWR are significantly higher than the SH-SAW due to the high surface-to-volume ratio of HARE. They are also higher than the reported 272 cm 2 /g mass sensitivity of a conventional Lamb wave biosensor operating at 473 MHz 11 , despite the later has a higher operating frequency. The simulated mass sensitivity of SH-PM is higher than the experimental values because the experiments only involves the mass loading on the top part of the HARE. As the adsorption of molecules in real applications are homogenous all-over device surface, a 200-nm PMMA layer uniformly covering the metasurface was used to obtain the homogenous loading mass sensitivity in simulation. Table 1 also shows that SH-PM has the highest Q factor among the four devices due to the simultaneous reduction of acoustic radiation and mechanical motion transfer. It should be noted that the simulated Q factors are much higher than the actual devices because the process and material defects are not considered in the simulation. Simulation without material damping enables the univariate modeling of acoustic radiation effect. Although material damping can be added to the simulation model, such artificial damping does not reflect the reality as the material defects are unknown. In addition to the high performance, SH-PM also has low fabrication cost compared to S-FBAR and HARE-LWR because the later involve either a suspended film structure that requires back-etching or a distributed Bragg reflector that requires deposition of multiple layers of reflecting materials. Nevertheless, the defects in the electroplating process of the SH-PM is still a limiting factor for its Q value.

IV. Conclusion

An SH-PM resonator is designed to overcome the in-liquid acoustic attenuation issue that limits the mass sensing resolution of acoustic wave biosensors. The PM was formed by electroplated nickel HAREs acting as mechanical resonators. It produces a surface-electrode hybridized locally-resonant wave with a velocity lower than the sound velocity in water, which does not propagate in water and thus fully stops the acoustic radiation. In addition, the shear horizontal polarization of the SH-PM reduces mechanical motion transferred to water, which further improves its Q factor. Experimental results showed the Q factor and mass sensitivity of the SH-PM resonator are 4.6 and 1.64 times higher than the previously reported VP-PM resonator, which would lead to an overall equivalent sensing resolution improvement of 7.5 times. A theoretical study through FEM also shows that the SH-PM is superior compared to other acoustic biosensors, including SH-SAW, S-FBAR, and HARE-LWR. In summary, the SH-PM resonator demonstrated a novel approach for high-resolution gravimetric biosensing through the combination of suppressed acoustic radiation enabled by the phononic metasurface and reduced mechanical motion transfer enabled by the shear horizontal polarization.

Figure 1 .

 1 Figure 1. (a). Top-view SEM image of the SH-PM resonator. (b) Tilted SEM image of the HARE.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Acoustic wave velocity of phononic metasurface with different electrode height in water

by 4 .

 4 58 times from 151 in air to 33 in water. The small relative Q factor degradation of SH-PM compared to SH-SAW proves the effectiveness of acoustic radiation suppression. The electromechanical coupling coefficient (𝐾 # = 𝜋 # (𝑓 " -𝑓 ! )/4𝑓 ! ) of the SH-PM in air and in water are 14.8% and 3.26%, respectively. 𝐾 # in water is lower than that in air due to the mismatch of dielectric constant between water and LiNbO3. The normalized mass sensitivity (𝑆 $ = ∆𝑚/(∆𝑓 • 𝑓 ! )) of the SH-PM is characterized utilizing its electrode height variation across the wafer, which originates from the nonuniformity of the electroplating process. The height change is actually equivalent to the variation of additive mass on top of the electrode. The loaded mass causes variations in the elastic resonance of the HARE, the coupled SAW velocity and the electrode reflectivity, leading to a shift in the SH-PM resonance frequency. Fig.5shows the 𝑓 " distribution with the variation of electrode height. The normalized SH-PM mass sensitivities

Fig. 4 Fig. 5

 45 Fig. 4 Impedance (a) and Q factor (b) of the SH-PM resonator.

  Shear Horizontal Phononic Metasurface for In-liquid Gravimetric BiosensingFeng Gao, Member, IEEE, Abdelkrim Khelif, Sarah Benchabane, Amine Bermak, Fellow, IEEE

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Table 1 .

 1 Comparison of SH-PM with other acoustic resonators Compared to the previously reported VP-PM resonator 7 that has an in-liquid Q factor of 50.4 and mass sensitivity of -117.3 𝑐𝑚 # /𝑔, the SH-PM resonator shows 4.6-time Q factor and 1.64-time mass sensitivity improvements, which represents an equivalent 7.5-time improvement of the overall sensing resolution.

	Device	|𝑆 " | a	In-liquid Q	FOM	𝑓(𝑀𝐻𝑧)
	SH-SAW	99	3881	3.84 × 10 #	134.3
	S-FBAR	3067.4	2574	7.90 × 10 $	1273.9
	HARE-LWR	485.2	1.1× 10 #	5.34 × 10 %	131.7
	SH-PM	536.1	1.63× 10 #	8.74 × 10 %	116.4

a Unit of the normalized mass sensitivity, 𝑆 $ : cm 2 /g are extracted from the slope of the linear fitted lines, which are -184.3 𝑐𝑚 # /𝑔 (𝑅 # =0.84) and -192.4 𝑐𝑚 # /𝑔 (𝑅 # =0.81) in air and in water, respectively.

This work is funded by NPRP grant No. NPRP10-0201-170315 from the Qatar National Research Fund of Qatar Foundation. This work is also supported by the EIPHI Graduate School (contract "ANR-17-EURE-0002") and the FEMTO-ST technological facility of the French RENATECH network. The work also received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 865724)


Member, IEEE Feng Gao
email: fgao@hbku.edu.qa

Aisha M Al-Qahtani
email: aialqahtani@hbku.edu.qa

Abdelkrim Khelif
email: abdelkrim.khelif@femto-st.fr

Farid Boussaid
email: farid.boussaid@uwa.edu.au

Sarah Benchabane
email: sarah.benchabane@femto-st.fr

Yu Cheng
email: ychengad@connect.ust.hk

Omar El

Fellow, IEEE Amine Bermak
email: aber-mak@hbku.edu.qa

Towards Acoustic Radiation Free Lamb Wave Resonators for High-resolution Gravimetric Biosensing

Keywords: Acoustic resonators, Biosensors, Lamb wave, Ultrasonic transducers

A novel Lamb wave resonator with fully suppressed acoustic radiation in water is proposed for high-resolution masssensitive detection of biomolecules. The elimination of acoustic radiation is achieved by slowing down the Lamb wave to a velocity lower than the sound speed in water. This enables high-qualityfactor resonance in water and reduces the sensing frequency noise. High aspect ratio electrodes (HAREs) are used to slow down the Lamb wave. The elastic resonance and large surface area of the HAREs can also enhance the mass sensitivity of the device. The improved mass sensitivity together with the low frequency noise substantially improves the overall sensing resolution. Although reducing the plate thickness can also slow down the Lamb wave, it makes the device very fragile and not practical to use. In contrast, slowing down the Lamb wave by increasing electrode height allows the use of thick plates which is robust. In this paper, the behavior and performance of the proposed high aspect ratio electrode Lamb wave resonator (HARE-LWR) are theoretically analyzed using finite element method simulations. Optimum design parameters were found through the simulations. Reported results show that a significant figure of merit improvement was achieved by the proposed HARE-LWR design.

I. INTRODUCTION

T HE in-situ detection of biomolecules is important in applications such as disease diagnosis and food security analysis [START_REF] Vigneshvar | Recent advances in biosensor technology for potential applications-an overview[END_REF]- [START_REF] Gaudin | Advances in biosensor development for the screening of antibiotic residues in food products of animal origin-a comprehensive review[END_REF]. A key requirement in achieving insitu biosensing is to miniaturize the sensor while keeping a good sensitivity. Among various transduction techniques, mass-sensitive acoustic biosensors are popular because of their small size, fast response and high sensitivity [START_REF] Voiculescu | Acoustic wave based mems devices for biosensing applications[END_REF], [START_REF] Mujahid | An overview of high frequency acoustic sensors-qcms, saws and fbars-chemical and biochemical applications[END_REF]. Quartz crystal microbalance (QCM) [START_REF] Mannelli | Quartz crystal microbalance (qcm) affinity biosensor for genetically modified organisms (gmos) detection[END_REF]- [START_REF] Atashbar | Qcm biosensor with ultra thin polymer film[END_REF], film bulk acoustic wave resonators (FBAR) [START_REF] Zhang | Film bulk acoustic resonators (fbars) as biosensors: A review[END_REF]- [START_REF] Guo | A biosensor based on a film bulk acoustic resonator and biotin-avidin system for the detection of the epithelial tumor marker mucin 1[END_REF] and surface acoustic wave (SAW) [START_REF] Ten | Highly sensitive escherichia coli shear horizontal surface acoustic wave biosensor with silicon dioxide nanostructures[END_REF]- [START_REF] Ji | A high sensitive sh-saw biosensor based 36°yx black litao3 for labelfree detection of pseudomonas aeruginosa[END_REF] sensors are the three main types of acoustic biosensors that have been extensively studied.

Despite its relatively low sensitivity, QCM is the most popular mass-sensitive biosensor due to its simplicity and low-cost. It has become a reference tool for label-free detection of biomolecules [START_REF] Hussain | Bioapplications of acoustic crystals, a review[END_REF]. Shear mode FBAR biosensor shares the same principle as QCM but works at a much higher frequency due to its small film thickness. The high operating frequency significantly improves its mass sensitivity. However, it is more costly than QCM due to its more complex fabrication process. SAW is reported to be the most sensitive acoustic sensor in gas sensing [START_REF] Fanget | Gas sensors based on gravimetric detection -a review[END_REF], [START_REF] Gao | Dual transduction surface acoustic wave gas sensor for voc discrimination[END_REF]. This is because SAW is very sensitive to the surface changes as the energy is mostly confined around the surface. Nevertheless, applying SAW in biosensing is difficult due to the strong acoustic radiation in water. Shear horizontal SAW (SH-SAW) and love SAW sensors were developed to reduce the acoustic radiation and thus enable operation in water [START_REF] Länge | Surface acoustic wave biosensors: a review[END_REF]. Both sensors rely on the principle that particle movement of shear-waves are parallel to the water interface and thus less energy is transferred to water compared to longitudinal waves. In fact, QCM and FBAR also suffer from acoustic radiation in water. They mitigate the problem also by making the device work in shear wave mode. Although the shear wave can reduce acoustic radiation, the corresponding resonance quality factors are still significantly lower in water compared to the ones in air.

One potential way to fully suppress the acoustic radiation in water is to reduce the acoustic wave velocity to a value below the radiation limit, which is the sound speed of water. This idea was exploited in Lamb wave biosensors with very thin plates [START_REF] White | Fluid loading of a lamb-wave sensor[END_REF]- [START_REF] Vellekoop | Integrated-circuitcompatible design and technology of acoustic-wave-based microsensors[END_REF]. It was found that reducing the plate thickness can reduce the velocity of antisymmetric Lamb wave and thus suppress the acoustic radiation. However, to fully suppress the radiation, the plate thickness needs to be a few percents of the wavelength, which makes the device very fragile especially when working in water. Due to this reason, the thin plate Lamb wave biosensor has rarely been further studied. Besides the plate thickness reduction method in Lamb wave based sensors, increasing the electrode height was also reported to be effective in reducing the velocity of Rayleigh SAW in air [START_REF] Laude | Surface acoustic wave trapping in a periodic array of mechanical resonators[END_REF]. By applying the latter method to Lamb wave, we found that Lamb wave can also be slowed down by increasing the electrode height in both air and liquid environment. If the electrode is thick enough, the velocity of the Lamb wave can be reduced to below the sound velocity of water for which the acoustic radiation is fully suppressed. Because this method can be implemented with thick plates, it makes the device much more robust than the thin plate Lamb wave resonator, enabling broad practical applications of the device.

In this paper, we analyzed the behavior and performance of the proposed biosensor based on high aspect ratio electrode Lamb wave resonator (HARE-LWR) through finite element method (FEM) simulations. The results aim to serve as a fundamental design reference for future experimental implementations.

The rest part of this paper is organised as follow. The conventional approach for slowing down Lamb wave through reducing the plate thickness is firstly discussed in Section II. The design of the HARE-LWR is then introduced to overcome the reliability problem of the plate thickness reduction method. In section III, the performance of the HARE-LWRs under different plate thickness, electrode height and electrode material configurations is analysed and compared by their mass sensitivity, resonance quality factor, ectromechanical coupling factor and figure of merit.

II. DEVICE DESIGN AND ANALYSIS FOR SLOW LAMB WAVE

HARE-LWR achieves the reduced phase velocity by increasing the electrode height. In comparison, the phase velocity of an conventional antisymmetric mode (A 0 ) Lamb wave resonator is reduced by decreasing its plate thickness [START_REF] White | Fluid loading of a lamb-wave sensor[END_REF]. These two devices can both achieve in-liquid operation once their phase velocity is lower than the sound velocity of water. In this section, the difference between the slowing down effects of the HARE-LWR and the conventional lamb wave resonator are studied by FEM simulation.

A. FEM modeling parameters

Fig. 1 shows the FEM simulation unit cells of the conventional lamb wave resonator and the HARE-LWR. The widths of both unit cells are set to one wavelength (λ) with periodic conditions applied to the two sides of the unit cells. The periodic conditions are equivalently repeating the unit cells in the propagating direction, making the number of interdigitated transducer (IDT) pairs infinite. The actual FEM simulation is performed with 3D models. The depths of the 3D models are set to 0.2λ with periodic boundary conditions applied to the front and back sides, which equivalently extends the model depths (acoustic aperture) to infinity. The wavelengths of both resonators are set to 10 µm. This value is chosen because it is a frequently used dimension for microfabrication with the classic photolithography method. In fact, the selection of simulation dimension does not affect the final conclusion as all the numerical calculations are scalable. On top of the devices operation environment, which is water or air. The depth of the environment is set to 4λ for both devices with the top 1λ set to perfectly matched layer (PML). The PML adsorbs all the acoustic wave propagates into it, which is equivalent to extend the depth of the environment to infinity. The conventional lamb wave resonator (Fig. 1a) is made of an aluminum nitride (AlN) plate with 200-nm-thick aluminum IDTs on top. One of the IDT finger is set to 1 volt amplitude AC electrical potential while the other is set to ground to form the electrical stimulation. The width and separation of the IDT electrodes are both half wavelength. The thickness of the AlN plate (t plate ) is set as the sweeping parameter in the simulation to study the slowing down effect. The HARE-LWR (Fig. 1b) is made of 1-µm AlN nitride film with 1-µm or 3-µm silicon dioxide (SiO 2 ) supporting layer underneath it. High aspect ratio IDTs were placed on top of the AlN for generating the hybridized slow wave. The electrical stimulation, IDT width and separation are set the same as the conventional lamb wave resonator in Fig. 1a. The height of the high aspect ratio electrode (HARE) is set as sweeping parameter to study the slowing down effect. The material of the metal electrodes is changed among nickel (Ni), gold (Au), copper (Cu) and platinum (Pt) to study the relevant impact on the device performance.

Multiple physics are involved in the simulation for the two devices. Pressure acoustics are solved in the water/air environment while solid mechanics are solved in the substrate plate and metal electrodes. Electrostatic study is applied to the whole device except the metal electrodes part. Piezoelectric effect is considered in the AlN part. Acoustic-structural boundary is applied to the interface between the device and the water/air environment to couple the pressure acoustic and solid mechanics study.

Al Al

tplate Plate thickness (μm) Fig. 2. The resonance frequency of the A 0 mode lamb wave resonator changing with plate thickness in air and in water. The black line corresponds to the frequency of the radiation limit in water.

B. Slowing down lamb wave by reducing plate thickness

Fig. 2 shows the simulated resonance frequency of the A 0 mode Lamb wave changing with the plate thickness in air and in water, respectively. The blue line in Fig. 2 shows that the Lamb wave velocity reduces with the decrease of the plate thickness (t plate ) in air. At large plate thicknesses, the wave transforms into a Rayleigh wave and propagates at the velocity of Rayleigh wave, which is about 5760 m/s. The resonator remains operational in water only when the Lamb wave velocity is slower than the sound velocity in water (radiation limit), as depicted by the orange line in Fig. 2. This happens when the plate thickness is smaller than 0.09λ. The black line marks the frequency of the radiation limit (f RL =148.1 MHz), which corresponds to a wave velocity equals to the sound speed of water. The resonance frequencies of the resonators are lower in water than in air because of the mass loading of water. At plate thicknesses smaller than 0.02λ, the resonance also disappears in water, possibly due to the insufficient piezoelectric conversion at very small plate thicknesses [START_REF] Wang | Electromechanical coupling and output efficiency of piezoelectric bending actuators[END_REF]. The acoustic pressure distribution of the 0.05λ-plate resonator at its resonance frequency (73.3 MHz) in water confirms the suppression of acoustic radiation (Fig. 3a). It can be seen that the positive (red part) and negative (blue part) acoustic pressures only exist in a small region above the resonator, which means the acoustic energy is confined in this small region. For comparison, the acoustic pressure distribution of the Lamb wave resonator with 0.2λ thickness is shown in Fig. 3b. The alternating positive and negative acoustic pressures extend all the way up till they get absorbed by the PML. This means that the acoustic energy is strongly radiative. It should be noted that a pseudo-resonance frequency (280 MHz) was selected to show the radiation of the 0.2λ- plate device as it cannot resonate in water. This frequency is calculated by subtracting the estimated frequency shift caused by water loading from the resonance frequency of the device in air.

Although the Lamb wave resonator can work in water when the plate is sufficiently thin, it is very fragile and difficult to fabricate. As shown in Fig. 2, to have resonance in water, the plate thickness of the Lamb resonator needs to be smaller than 0.09λ. To have a good quality factor, the acoustic aperture and number of IDT pairs should be at least 20λ and 20 pairs, respectively [START_REF] Yantchev | Thin film lamb wave resonators in frequency control and sensing applications: a review[END_REF]. The surface area of the film is thus at least 20λ × 20 λ. Considering the internal stress of the film and other imperfections during fabrication, the yield of this large-area thin device will be very low. Although higher plate thickness can be used when the wavelength is large [START_REF] White | Fluid loading of a lamb-wave sensor[END_REF], the aspect ratio (the plate thickness relative to the wavelength and film area) remains the same and thus the structural strength remains the same. Due to their fragility, these ultra thin plate Lamb wave resonators are rarely reported in biosensing applications.

C. Slowing down lamb wave by HAREs

To overcome the limitations of the plate thickness reduction technique, we propose the HARE-LWR that uses HAREs for slowing down the lamb wave without sacrificing the structural strength of the device. The mass loading caused by the HAREs together with the hybridization between the elastic resonance of the HAREs and the lamb wave resonance can significantly reduce the phase velocity of the HARE-LWR.

Fig. 4 shows the FEM simulated electrode-height-dependent resonance frequency (slowing down curve) of two Ni-electrode Lamb wave resonators with 2-µm and 4-µm plate thicknesses. The use of two plate thicknesses is to investigate the impact of plate thickness on the slowing down effect. The two values are based on typically available SiO 2 thicknesses of an oxidized silicon wafer, which are 1 and 3 µm. Thicker SiO 2 is also possible but will take a lot longer to grow on the silicon wafer, so they are not considered. The slowing down curve of the HARE-LWR with gold, copper and platinum electrodes are similar to that of nickel and are shown in Fig. S1-S3 of Supporting Information. It can be seen that the 2µm-plate resonator remains operational in water even when its electrode height is only 1 µm. However, the electrode height needs to increase to more than 2 µm for the acoustic radiation to be sufficiently suppressed. This is revealed by the acoustic pressure distribution of the resonators when their electrode height increases. In Fig. 5a, the positive acoustic pressure (red part) and negative acoustic pressure (blue part) of the 1-µm-electrode resonator at its resonance frequency (148.5 MHz) extend up to the PML boundary with gradually decreasing intensities. Although the amount of acoustic energy reaching the PML boundary is much smaller than that of the conventional Lamb wave resonator shown in Fig. 3b, the acoustic energy confinement is still not very effective. In contrast, the acoustic energy of the 2-µm-electrode resonator at its resonance frequency (138.1 MHz) is limited in the region about 2λ high above the electrodes (Fig. 5b). The 2λ confinement length is an estimation of the region where the acoustic pressure fields marked by red and blue are clearly visible. The resonators with 4-µm plate thickness show similar behavior. The radiation is seen to be strong at its resonance frequency (148.1 MHz) when its electrode height is 3.5 µm (Fig. 5c). In contrast, when its electrode height increases to 5 µm (Fig. 5d), the radiation at its resonance frequency (129.2 MHz) is suppressed and the energy gets confined in the region about 1.5λ from the electrodes. The relatively strong radiation of the resonators in Fig. 5a and 5c is due to the fact that their wave velocity is close to the sound speed of water. It is thus necessary to further increase the electrode height to leave a margin of about 100-200 m/s between the wave velocity of the resonator and the sound speed in water so that the acoustic radiation can be fully suppressed. In Fig. 4, the resonance frequencies of the two resonators in air are also shown for comparison. Comparing the results of Fig. 4 and Fig. 2, it can be found that the frequency shifts due to the mass loading of water in HARE-LWRs are much smaller than those in the conventional Lamb wave resonators. This smaller mass loading effect of water can potentially make the HARE-LWRs more resistant to environmental disturbances in water.

III. PERFORMANCE OF THE HARE-LWR FOR BIOMOLECULE DETECTION

The mass sensing resolution (R m ) or the detection limit (LOD) of the frequency based biosensors are determined by the following equation:

R m = S m n f (1)
where S m and n f are the mass sensitivity and the frequency noise (jitter) of the sensor, respectively. To maximize the sensing resolution, the sensor should have both high mass sensitivity and low frequency noise.

A. Mass sensitivity of the HARE-LWR

The mass sensitivity of the HARE-LWR was analyzed by FEM simulation. A 200-nm polymer layer (Polymethyl methacrylate, PMMA) was used as the mass sensing layer. It uniformly covers the surface of the HARE-LWR as shown in the insert of Fig. 6a. To evaluate the mass sensitivity of the device, we changed the density of the sensing layer and recorded the corresponding frequency shift. The variation of the density is an emulation of the mass loading effect when the sensor absorbs biomolecules. The mass sensitivity is calculated by the following equation:

S m = | ∆f t∆ρ | ( 2
)
where ∆f is the frequency shift before and after mass loading, t is the thickness of the sensing layer, ∆ρ is the density variation of the sensing layer. Fig. 6a shows the simulated electrode-height-dependent mass sensitivities of the HARE-LWRs with different electrode materials and plate thicknesses.

The mass sensitivity of all the devices exhibit similar behavior.

After the electrode height exceed the limit which the devices can operate in water, the mass sensitivities of all the devices increase rapidly with the increasing of the electrode height and reach maximum points. A gradual decrease is then observed with the further increase of the electrode height. Table I summarises the maximum mass sensitivities S m-max of all HARE-LWR configurations and their corresponding electrode height (h ele-max ) and operating frequency (f max ). Among the four analysed electrode material, Cu-electrode HARE-LWR has the best mass sensitivity which is closely For the nickel electrodes, the mass sensitivities of two HARE-LWRs with different plate thicknesses were simulated. They showed almost the same maximum mass sensitivity, but the 4µm-plate HARE-LWR reaches the maximum sensitivity at a larger electrode height of 4.6 µm compared to the 2.2 µm electrode of the 4-µm-plate HARE-LWR. This is because the original phase velocity of the 4-µm-plate HARE-LWR is faster than that of the 2-µm-plate HARE-LWR, it thus requires higher electrodes to sufficiently slow down its phase velocity. Despite the electrode material and height differences of the HARE-LWRs, it is interesting to find that all the devices reaches the maximum mass sensitivity at similar operating frequencies around 130 MHz. This phenomenon suggest that we can change the electrode configuration to improve the device reliability or adapt it to sensing applications that have requirement on electrode material while keeping the same sensitivity at the same operating frequency.

To investigate the origin of the high mass sensitivity of the HARE-LWR, we simulated the mass sensitivity of the Cuelectrode HARE-LWR when the sensing film coverage on the device surfaces changes. The plate thickness and electrode height of the HARE-LWR are both 4µm. The frequency shifts of the impedance curves together with the sensing layer coverage diagram are shown in Fig. S4 of supporting information. The mass sensitivities of the device with sensing film covering all the surface, the electrode surface and the bottom surface are 63.87, 62.61 and 2.78 (Hz/(ng • cm 2 )), respectively. This means the high mass sensitivity of the HARE-LWR is mainly contributed by the elastic resonance of the HAREs. The contribution from the mass loading of the bottom surface is very small.

B. Quality factor and FOM of the HARE-LWR

To determine the resolution, we also need to evaluate the frequency noise of the sensor. However, the frequency noise is a parameter associated to the complete sensing system. It is affected by the noise from the detection circuits. We thus use the quality factor of the resonator to evaluate the frequency noise caused by the sensor part. Resonators with higher quality factors will exhibit a lower frequency noise of the final sensing system. The quality factor calculated from the simulated electrical impedance is given by the following equation:

Q = f 2 d∠Z df (3)
where f and Z are the working frequency and electrical impedance of the device respectively. Fig. 6b shows the calculated quality factor of the HARE-LWRs at different electrode heights. For all the HARE-LWRs, the quality factor is relatively low at small electrode heights due to the incomplete suppression of the acoustic radiation. When the electrode height further increases, the radiation is fully eliminated and thus the quality factor improves dramatically. After the rapid quality factor improvement, the quality factors follow different trends depending on the plate thickness and electrode material. For the 2-µm-plate Ni-electrode HARE-LWR, the quality factor decreases slightly and then increases again with the increasing of the electrode height. For the 4-µm-plate HARE-LWR based on nickel, copper and platinum, the quality factor saturates and fluctuates with the increasing of electrode heights. The quality factor of the 4-µm-plate gold electrode however fluctuates a while and then continues to increase with the further increase of electrode height. The Q factor behaviour difference of these HARE-LWR configurations is possibly linked to their different electromechanical coupling factor (K 2 ) changing trends as shown in Fig. 6c. The coupling factor is calculated by the following equation [START_REF] Konno | Scaln lamb wave resonator in ghz range released by xef 2 etching[END_REF]:

K 2 = π 2 4 ( f s f p ) f p -f s f p (4)
where f s and f p are the resonance and anti-resonance frequencies, respectively. The increase, decrease or fluctuation of the quality factor are inversely linked to the variation of K 2 . This is because the bandwidth of the resonator is inversely proportional to K 2 . As a result, the quality factor improves because it is inversely proportional to the bandwidth.

To evaluate the sensing resolution of the device, we define the figure of merit (FOM) of the device as the mass sensitivity multiplied by the quality factor (F OM = S m × Q). The calculated FOM is shown in Fig. 6d. It can be found that the FOMs of all the HARE-LWRs increase rapidly and reach maximums at the beginning due to the simultaneous increase of the quality factor and the mass sensitivity. Table II summarises the maximum FOM of all the HARE-LWRs and their corresponding electrode height and operating frequency. After the rapid increase, the FOMs of the 4-µm-plate HARE-LWRs based on nickel, copper and platinum decrease because the decrease of mass sensitivity. In contrast, the quality factor of the 2-µm-plate Ni-electrode and the 4-µm-plate Cu-electrode HARE-LWRs keep a saturated level as the decrease of mass sensitivity is compensated by the evident increase of quality factor. In reality, this FOM saturation may not happen as the quality factor increase at large electrode height can be less than the simulated results because higher electrodes can accumulate more physical defects in the fabrication process which will compromise the quality factor. It is thus better to choose an electrode height that is just high enough to fully suppress the acoustic radiation to achieve the best FOMs for all the HARE-LWR configurations.

C. Comparison between HARE-LWR and SH-SAW

To demonstrate the advantage of our design, we compared the device with conventional shear mode SAW biosensors based on SH-SAW. As the operation frequency affects the mass sensitivity, the devices in comparison are scaled to proper wavelengths so that they all work around 133 MHz. The electrode thickness of the SH-SAW resonators was set to 0.02λ. The same polymer, PMMA, with the thickness of 0.02λ, is used to simulate their mass sensitivity. Table III summarizes the simulated performance data of all the devices in comparison. For HARE-LWR, the 4-µm-plate Cuelectrode configuration is chosen as it shows the best FOM. Fig. 6e shows the resonance frequency shifts of the SH-SAW resonator based on 38°Y-cut lithium tantalate when the density of the PMMA film is increased by one and two percent. Both 38°Y-cut lithium tantalate (LT38) and AT-cut quartz (QAT) are used as the SH-SAW substrates. Lithium tatalate is reported to be one of the best piezoelectric substrate for SH-SAW because of its small dielectric mismatch with water, which reduces the acoustic attenuation [START_REF] Länge | Surface acoustic wave biosensors: a review[END_REF], [START_REF] Shiokawa | Surface acoustic wave sensors[END_REF].

Quartz is used as SH-SAW substrate as it is reported to have higher mass sensitivity [START_REF] Mitsakakis | Parametric study of sh-saw device response to various types of surface perturbations[END_REF]. The FEM simulation results in Table III confirm that the mass sensitivity of SH-SAW based on AT-cut quartz (25.9Hz/(ng/cm 2 )) is indeed higher than that of 38°Y-cut lithium tantalate (13.3Hz/(ng/cm 2 )). Nevertheless, their mass sensitivity is both lower than the maximum sensitivity of the 4-µm-plate Cu-electrode HARE-LWR (63.9Hz/(ng/cm 2 )). Although quartz substrate shows higher mass sensitivity, it is a less favorable substrate for SH-SAW because it has a very low relative dielectric constant ( r = 3.8). This results in a large dielectric mismatch with water ( r = 80) that leads to strong acoustic attenuation and thus low quality factor resonance in water [START_REF] Länge | Surface acoustic wave biosensors: a review[END_REF]. The simulated quality factor of SH-SAW on AT-cut quartz (57) is a lot lower than that on lithium tantalate substrate (3881). Despite the higher mass sensitivity, the FOM of SH-SAW on quartz is significantly lower than that of lithium tantalate due to the lower quality factor on quartz. When compared with the 4µm-plate Cu-electrode HARE-LWR, the quality factor of SH-SAW based on lithium tantalate is still 28.3 times lower. This is because of the strong acoustic radiation in liquid as shown in Fig. 6f. Although the shear polarized plate motion of the SH-SAW resonator can minimize the mechanical energy transferred to the water molecules, the acoustic radiation is not eliminated as the wave velocity is above the radiation limit. The FOM of the SH-SAW based on lithium tantalate is 135 times lower than the maximum FOM of the 4-µm-plate Cu-electrode HARE-LWR. The above comparison shows the superior performance of the proposed HARE-LWR compared to the SH-SAW resonators.

D. Discussion

Except the above theoretical investigations, other factors can also affect the performance of the HARE-LWR biosensor. One of it is the fluid motion produced by the movement of the HAREs and the piezoelectric plate. In a lamb wave delay line configuration, it was reported that the particles can be moved towards a fixed direction if only one side of the IDT is stimulated [START_REF] Moroney | Fluid motion produced by ultrasonic lamb waves[END_REF]. This constant particle movement can change the regional concentration of the target analytes and affect the sensor operation. However, the particles are trapped instead of being moved when a standing wave is formed by turning on both sides of the IDTs. The proposed HARE-LWR works in a single port configuration that also generates a standing wave and thus has the same trapping effect on the particles. As the particle trapping does not change the regional concentration of the analytes, the fluid motion generated in the HARE-LWR thus does not affect its sensing performance.

Another factor that can affect the HARE-LWR's performance is the damping caused by the electrodes. It was reported that the electrode material can affect the insertion loss of conventional SAW filters [START_REF] Johnsen | Improved metallization for saw filters on linbo 3[END_REF]. One might question that using HAREs can increase the damping caused by the electrodes and thus decrease the quality factor of the HARE-LWR. This is true for the HARE-LWR operating in air as the electrode induced damping can be a significant part of energy loss. However, in the case of the HARE-LWR biosensor operating in liquid, the damping caused by the electrode is much less than the damping through acoustic radiation. In fact, the HARE-LWRs without sufficient electrode height cannot resonate in water because of the strong damping caused by acoustic radiation. Although the HARE increases the energy loss in the electrodes, it is a necessary trade-off to enable the operation of the device in water. We compared the quality factor impact caused by the electrode damping and the acoustic radiation through the simulation of a SH-SAW based on 36°Y lithium niobate whose aluminum electrode is with an isotropic loss factor (η s ) of 0.01. SH-SAW is chosen for the comparison as it can operate in liquid with acoustic radiation. The simulation results shows that the SH-SAW operate in water with the electrode damping has a quality factor of 3593. If we set the damping of the electrode to zero (η s = 0), the quality factor becomes 3901, which means the electrode damping introduced 8% of quality factor drop for the SH-SAW. If we let the same device with electrode damping work in air, the quality factor becomes 43680, which means the acoustic radiation results in 11.2 times quality drop. This comparison proves that acoustic radiation is a much stronger damping source than the loss in the electrodes. The simulated spectrum of the quality factor in this comparison are given in Fig. S5 of supporting information.

IV. SUMMARY

A HARE-LWR biosensor with high in-liquid quality factor and mass sensitivity is proposed. Although conventional A 0 mode Lamb wave biosensors can also achieve non-radiative operation in water by reducing their plate thickness to a very thin level, they become very fragile and thus not suitable for practical use. In contrast, the proposed HARE-LWR uses HAREs to slow down the wave while allowing for the use of thick plates, which significantly improve the structural reliability of the device. Through FEM simulations, we analyzed the slowing down behavior of the Lamb wave in HARE-LWR resonators. It was found that the resonator will enter a semileaky region after its acoustic velocity is slowed down to just below the sound velocity of water. The device is still operational in this semi-leaky region but has a relatively low quality factor. When the electrode height is further increased, the acoustic radiation becomes fully suppressed. The quality factor of the resonator is thus dramatically improved. The mass sensitivity of the resonator also reaches its maximum just after the semi-leaky region. Further increase of the electrode height reduces the mass sensitivity due to the simultaneous decrease of the resonance frequency. To evaluate the mass sensing resolution of the sensor, we defined the FOM as the mass sensitivity multiplied by the quality factor. The FOMs of the HARE-LWRs with different electrode materials and electrode heights all reach their optimal value just after the semi-leaky region. By comparing the proposed HARE-LWR biosensor with the widely used SH-SAW biosensor working at the same frequency, it was found that the HARE-LWR provides a significant 135 times FOM improvement compared to the SH-SAW. In conclusion, the HARE-LWR biosensor combines the benefits of high mass sensitivity, high quality factor and high reliability. To our knowledge, it is the first nonradiative acoustic resonator that can work reliably in water. This work has the potential to help substantially push the performance boundary of mass-sensitive acoustic biosensors.

Fig. 1 .

 1 Fig. 1. FEM simulation unit cell (cross-section) of (a) the conventional Lamb wave resonator and (b) the HARE-LWR.

Fig. 3 .

 3 Fig. 3. (a) Lamb wave resonator with 0.05λ plate thickness showing resonance with fully suppressed acoustic radiation in water. (b) Lamb wave resonator with 0.2λ plate thickness showing resonance with strong acoustic radiation in water.

Fig. 4 .

 4 Fig. 4. The slow-down curves of the Ni-electrode HARE-LWR with 2 and 4 µm thickness. The black line corresponds to the frequency of the radiation limit in water.

Fig. 5 .

 5 Fig. 5. (a-b). Displacement (enlarged) and acoustic pressure distribution of the 2-µm-plate Ni-electrode HARE-LWR at 1 and 2 µm electrode height. (c-d). Displacement (enlarged) and acoustic pressure distribution of the 4-µm-plate Ni-electrode HARE-LWR resonator at 3.5 and 5 µm electrode height.

Fig. 6 .

 6 Fig. 6. (a) Mass sensitivity (Sm), (b) Quality factor (Q), (c) Electromechanical coupling factor and (d) Figure of merit (F OM = Sm × Q) of the HARE-LWR biosensors. The insert in 5a shows how the mass sensing layer covers the surface of the HARE-LWR biosensor. (e). Resonance frequency shift of the SH-SAW biosensor based on 36°Y LiTaO 3 . (f) Acoustic radiation of the SH-SAW biosensor at resonance frequency.

TABLE I MAXIMUM

 I

	Ni	2	62.2	2.2	129.96
	Ni	4	62.4	4.6	132.32
	Au	4	39.1	2	132.20
	Cu	4	63.9	4	130.09
	Pt	4	34.5	2	131.79

MASS SENSITIVITY OF THE HARE-LWRS

Electrode t plate S m-max h ele-max fmax Material (µm) (Hz/(ng • cm 2 )) (µm) (M Hz)

followed by the Ni-electrode HARE-LWR. In comparison, HARE-LWRs based on gold and platinum are less sensitive.

TABLE III COMPARISON

 III OF THE HARE-LWR AND THE SH-SAW BIOSENSORS Unit of Sm: Hz/(ng/cm 2 )

	Device	Sm a	Q	FOM	λ(µm) f (M Hz)
	HARE-LWR	63.9 1.10 × 10 5 6.97 × 10 6	10	131.7
	SH-SAW-LT38 13.3	3881	5.16 × 10 4	30	134.3
	SH-SAW-QAT 25.9	57	1476	38	134
	a

ACKNOWLEDGMENT

This work is funded by NPRP grant No. NPRP10-0201-170315 from the Qatar National Research Fund (a member of Qatar Foundation). The findings herein reflect the work, and are solely the responsibility of the authors.

Abdelkrim Khelif received his M.S. in fundamental physics and Ph.D. in Material sciences from the University of Lille, France in 1993 and 1998, respectively. He joined the FEMTO-ST Institute in 2002 as a full-time researcher at Centre National de la Recherche Scientifique (CNRS) to develop phononic crystals and acoustic metamaterials. He has extensive experience in design, optimization, and simulation of phononic structures. Due to his achievements in the field of phononic crystal, he has been involved as the symposium session chair, technical committee member, and invited speaker in several conferences in the last 5 years. He is also a frequent reviewer in this field for multiple physics and acoustic journals (Physical Review Letter, B, E, Applied Physics Letters, Journal of Applied Physics, and so on). He has more than 80 journal papers and several invited, and contributed conference and seminar presentations. He also has 4 patents in phononic crystal and is the recipient of Bronze medal awards from CNRS for his work in phononic crystal in 2007. Omar El-Agnaf received his Ph.D. in Biochemistry in 1997 at the Queen's University Belfast, UK. From 1997 to 1999, he worked as a post-doctoral scientist at the School of Biology and Biochemistry, Queen's University Belfast. Following this, he undertook an additional two-year post-doctoral position at St. George's Medical School in London. In 2001, he was awarded a research fellowship from the Parkinson's Disease Society-UK to establish his research group at Lancaster University. After completing his research fellowship in 2004, he was invited to join the College of Medicine at UAE University and was appointed to the Biochemistry Department. In September 2014, he joined HBKU at Qatar Foundation, and in 2016 he was appointed Acting Executive Director of the Qatar Biomedical Research Institute. He is frequently invited as a speaker at international scientific and clinical meetings and is currently a member of the editorial boards of several international journals. He has published nearly 120 refereed articles, with citations exceeding 9,000 (to date), and possesses an h-index of 52.

Farid

Amine Bermak received his M.S. and Ph.D. degrees in electrical and electronic engineering from Paul Sabatier University, Toulouse, France, in 1994 and 1998, respectively. In 2002, he joined the Electronic and Computer Engineering Department, The Hong Kong University of Science and Technology (HKUST), where he was a professor and the ECE associate head for research and postgraduate studies. Over the last decade, He has acquired a significant academic and industrial experience. He is currently with Hamad Bin Khalifa University, Qatar Foundation, Qatar. He has published over 250 articles in journals, book chapters, and conference proceedings, and designed over 30 chips. He has served on the editorial board of the IEEE Transactions on Very Large Scale Integration Systems, the IEEE Transactions on Biomedical circuits and Systems, the Journal of Low Power Electronics and Applications, and Frontiers in Neuromorphic Engineering. He is an IEEE fellow and distinguished lecturer.


Ahcène Zehnati

Marwân-Al-Qays Bousmah
email: marwan-al-qays.bousmah@ird.fr

Mohammad Abu-Zaineh

Public-private differentials in health care delivery: the case of cesarean deliveries in Algeria

Keywords: Public-private differentials, Physician dual practice, Algeria, Cesarean delivery JEL Classification I11, I18, K32

Akin to other developing countries, Algeria has witnessed an increasing role of the private health sector in the past two decades. Our study sheds light on the public-private overlap and the phenomenon of physician dual practice in the provision of health care services using the particular case of cesarean deliveries in Algeria. Existing studies have reported that, compared to the public sector, delivering in a private health facility increases the risk of enduring a cesarean section. While confirming this result for the case of Algeria, our study also reveals the existence of public-private differentials in the effect of medical vari-ables on the probability of cesarean delivery. After controlling for selection in both sectors, we show that cesarean deliveries in the private sector tend to be less medically justified compared with those taking place in the public sector, thus, potentially leading to maternal and neonatal health problems. As elsewhere, the contribution of the private health sector to the unmet need for health care in Algeria hinges on an appropriate legal framework that better coordinates the activities of the two sectors and reinforces their complementarity.

Introduction

Ensuring universal access to reproductive health while reducing under-five child mortality rate by two-thirds and maternal mortality rate by three-quarters have been amongst the United Nations 2015 Millennium Development Goals (MDG 4 and MDG 5). The recently adopted Sustainable Development Goals (SDGs 2015(SDGs -2030) ) have urged all countries to intensifying their efforts to further reduce the global maternal mortality ratio to less than 70 per 100,000 live births, to end preventable deaths of newborns and children under 5 years of age and to reduce neonatal mortality to at least as low as 12 per 1,000 live births and under-5-mortality to at least as low as 25 per 1000 live births by 2030 (United Nations, 2015). Akin to other developing countries, Algeria-an upper-middle-income country undergoing a rapid demographic and epidemiological transition-has made a remarkable progress in this domain, with significant reductions in both infant mortality rate (from 39.7‰ in 1990 to 21.9‰ in 2015) and maternal mortality rate (from 216 per 100,000 live births in 1990 to 140 in 2015) (World Health Organization, UNICEF, UNFPA, World Bank Group, and United Nations Population Division, 2015).

Over the past 2 decades, Algeria has witnessed an increasing role of the private health sector and physician dual practice in health care delivery. Although encouraged by the government in order to reduce spatial inequalities in health, this increasing complementarity between public and private health sectors may lead to adverse medical and economic effects. This article contributes to the literature on the public-private differential and physician dual practice in health care delivery using the case of cesarean deliveries in Algeria. We investigate the factors influencing the cesarean section deliveries using data from the latest Multiple Indicator Cluster Survey (MICS) conducted in 2012-2013. This study examines whether the determinants of cesarean delivery differ according to the place of delivery (whether public or private).

The case of cesarean delivery is interesting in its own right. The cesarean section rate has grown considerably over the last decade, especially in developing countries [START_REF] Flamm | Cesarean section: A worldwide epidemic?[END_REF][START_REF] Porreco | The cesarean birth epidemic: Trends, causes, and solutions[END_REF]. [START_REF] Stanton | Levels and trends in cesarean birth in the developing world[END_REF] estimate cesarean rates in the developing world at 12% with variable regional rates from 3 to 26%. Arab countries are no exception to this "epidemic of cesarean sections" [START_REF] Betrán | The increasing trend in caesarean section rates: Global, regional and national estimates: 1990-2014[END_REF][START_REF] Khawaja | Hospital-based caesarean section in the Arab region: An overview[END_REF][START_REF] Mikki | Caesarean delivery rates, determinants and indications in Makassed Hospital, Jerusalem 1993 and 2002[END_REF]. This issue is particularly important form both medical and economic perspectives. The strong increase in cesarean sections raises a question about the determinants of this practice. On the medico-clinical level, professional recommendations to assist health care professionals in decision-making on cesarean section have been developed [START_REF] Di Renzo | Cesarean delivery: A comprehensive illustrated practical guide[END_REF]. However, such recommendations are rather indicative, leaving greater discretion to health care professionals to decide whether or not to practice a cesarean section. Furthermore, there is no consensus on an "optimal rate" of cesarean sections, even if the World Health Organization (1985Organization ( , 2015) ) and others (Betrán et al., 2015;[START_REF] Ye | Searching for the optimal rate of medically necessary cesarean delivery[END_REF] consider a rate of 10-15% above which cesarean sections may be deemed medically unjustified and may not necessarily be associated with a reduction in maternal and neonatal mortality. A variety of non-medical factors pertaining to the demand for medical care (e.g., socio-economic and socio-demographic characteristics of women) and the supply of medical care (e.g., characteristics of the health care system) are shown to play an important role in the decision to rescue to cesarean sections [START_REF] Baubeau | La pratique des césariennes: Evolution et variabilité entre 1998 et 2001[END_REF]. 1Cesarean indications are often medically unjustified. Although cesarean section is a common procedure, it is not uncommon to observe a complication that darkens the fetal and maternal prognosis. However, the argument generally put forward by the doctor is the fetal and/or maternal rescue. In reality, is this the case? Health professionals do not communicate enough with mothers about the risks incurred after the cesarean section. It is naive to think that all information disseminated by doctors is unbiased and expresses an opinion based on their experience [START_REF] Penna | Cesarean section for non-medical reasons[END_REF]. The costs of the negatives effects of cesarean section may be significant in comparison with normal delivery [START_REF] Filippi | After surgery: The effects of lifesaving caesarean sections in Burkina Faso[END_REF]. Research between 2000 and 2005 shows evidence of very small numbers of women requesting a cesarean section [START_REF] Mccourt | Elective cesarean section and decision making: A critical review of the literature[END_REF]. In economic terms, the significant remuneration differences between cesarean delivery and vaginal delivery may explain the high incidence of cesarean section.

The remainder of the paper is organized as follows. The following Section features the Algerian health care system. Section 3 presents the methodology and the dataset used in the econometric analysis. The results are presented in Sect. 4 and discussed in Sect. 5 along with some recommendations.

The Algerian context

The Algerian health care system is a dual public-private sector. The Ministry of Health, Population and Hospital Reform (MHPHR) runs the public sector and regulates the rapidly evolving private-for-profit sector (MSPRH, 2001). Health care expenditures are mainly funded through general tax revenues, social security funds and out-of-pocket payments. Available data indicate that total health expenditure (THE) represented 7.2% of GDP in 2014. Although the share of THE in GDP has significantly increased over the last 2 decades (from 3.7% in 1995 to 7.2% in 2014), the share of public health expenditure in THE has fallen from 77.4% in 2001to 72.7% in 2014(World Health Organization, 2017). This reflects a gradual shift of health financing towards private sources. As a result the share of health care expenditure borne by households was as high as 26.5% in 2014 [START_REF] Lamri | Comptes Nationaux de la Santé. Programme PASS[END_REF][START_REF] Tlilane | Healthcare funding problems in Algeria[END_REF]. The relatively high share of household direct expenditures on health care can also be explained by the small share of medical costs covered by the social security, especially for private sector health services. Although in the case of Algeria no empirical evidence has been made available to date on the effect of health care expenditure on household welfare, reported evidence from neighboring countries with similar socio-economic characteristics alert on the relatively high financial burden and the risk of catastrophic and impoverishing health expenditures associated with ill-health [START_REF] Abu-Zaineh | Appraising financial protection in health: The case of Tunisia[END_REF][START_REF] Makhloufi | Have health insurance reforms in Tunisia attained their intended objectives[END_REF]. From a macroeconomic perspective, [START_REF] Bousmah | Medicine and democracy: The importance of institutional quality in the relationship between health expenditure and health outcomes in the MENA region[END_REF] show that the increasing share of private health spending in total health spending in the Middle-East and North-Africa region is negatively associated with health outcomes when institutional quality is weak.

In Algeria, de facto complementarity between the public and the private providers with a phenomenon of dual practice have informally developed from an attempt to raise practitioners' remunerations. De jure complementarity between the two sectors have been endorsed by the "décret exécutif n° 99-236 du 19 octobre 1999". These dual practices have further been reinforced by a new legislation ("circulaire n° 001 MSPRH/MIN du 31 mars 2010"). Accordingly, public-sector's practitioners are allowed to combine public-sector work with a fee-for-service in the private sector-the so-called "profitable activities". These reforms are meant to tackle the spatial inequalities in the distribution of health care services across the different regions. Consequently, the MHPHR has started to contract private sector practitioners to cover areas that have deficits of doctors in certain specialties. The private health care sector encompasses private physicians and private clinics with or without hospital beds. The role and number of private physicians has grown rapidly during the last 2 decades particularly in some specialties such as gynecology, ophthalmology or otorhinolaryngology. In 2012, private-public ratios for these three specialties were estimated at 75:25%, 70:30% and 65:35%, respectively [START_REF] Zehnati | Analyse économique de l'émergence et du développement d'une offre privée de soins en Algérie[END_REF]. The deficit of specialist physicians in the public sector is more pronounced for gynecology.

Free access to different types of health care services is guaranteed to all citizens by the Algerian Constitution (Article 66). In practice, however, patients' health care pathways vary depending on financing and reimbursement modalities and on the physician's referral strategies [START_REF] Zehnati | Analyse économique de l'émergence et du développement d'une offre privée de soins en Algérie[END_REF]. The decree of 1999 allowing practitioners to work in both sectors has offered p hysicians m ore l iberty t o r efer p atients a nd d ecide o n t heir t herapeutic itineraries. [START_REF] Zehnati | Analyse économique de l'émergence et du développement d'une offre privée de soins en Algérie[END_REF] reports that referral practices tend to be driven by practitioners' financial incentives, who seek to maximize their gains through modifying patients' therapeutic pathways. For instance, in the particular case of birth delivery, the study shows that about 72% of cesarean section surgeries in two districts of Algeria, Algiers and Béjaia, took place at the private sector clinics, representing about 50% of their revenue. Given the large public-private differentials in terms of remuneration, public sector hospitals may be regarded as a source of generating a fee-for-service private clientele [START_REF] Zehnati | Les raisons de la double activité des médecins: Le cas de l'Algérie[END_REF]. Anecdotal evidence on heterogeneous referral practices (preferential and quicker referrals) are often reported. Patients generally face two entry barriers: a relational barrier to get appropriate access to the needed services in the public sector and a financial barrier to access the private sector [START_REF] Zehnati | Analyse économique de l'émergence et du développement d'une offre privée de soins en Algérie[END_REF]. The rapid growth of private health providers and the public-private dual practice may have increased the financial burden of health care.

Material and methods

Data

This paper uses the latest available data from the MICS conducted in Algeria in 2012-2013(UNICEF, 2015)). The MICS is a national representative survey that provides detailed data on maternal and infant health, in addition to a set of socio-demographic and socio-economic characteristics of households. A sample of 28,000 Algerian households, distributed according to seven regions, has been selected using a two-stage stratified cluster-random sampling procedure.

This study uses individual-level data pertaining to adult women of reproductive age (15-49). The target population comprises a sub-sample of 5278 adult women who delivered in a public or a private health facility in the 2 years prior to the survey. The variable of interest is a binary variable taking on a value of 1 if a woman had a cesarean delivery and zero otherwise. A set of explanatory variables, which are shown to be potentially important associates with cesarean delivery, are identified and included in the analysis [START_REF] Stivanello | Determinants of cesarean delivery: A classification tree analysis[END_REF]. We would expect cesarean section to be positively associated with privatesector delivery, complications during pregnancy, maternal age and socio-economic level. The number of prenatal visits is predicted to be positively associated with the probability of cesarean delivery. Of note, a high number of prenatal visits is likely to act as a proxy for complications during pregnancy uncaptured by the medical factors included in the model. Finally, we would expect very small and very large infants to be more likely to be delivered by cesarean (U-shaped relationship between infant size and cesarean delivery). Table 1 provides a detailed description of the main variables used in our study. On all deliveries, about 15.3% were cesarean, and 7.4% took place in a private health facility, 34.98; 30.28; 23.57; 11.18 as shown in Table 1. As far as public-private differential is concerned, data shows that while only 7% of women who delivered in a public health facility had a cesarean section, about 53% of deliveries taking place in a private health facility were cesarean.

Methodology

The methodology employed involves three steps. First, a baseline probit model of cesarean delivery is estimated. Secondly, we estimate a bivariate probit model to investigate whether the decisions about the mode of delivery and the place of delivery are correlated. Note that the descriptive analysis indicates that these two outcomes are strongly positively correlated ( = 0.29 , p < 0.001 ), thus, a joint-specification of the two binary variables is called for. A bivariate probit model jointly estimates two binary probit models while allowing for correlation between the error terms of both equations [START_REF] Cameron | Microeconometrics using Stata[END_REF]. A test for error term correlation (Wald test of = 0 ) is then per- formed. Such joint estimation with a bivariate probit model is needed in the case where the hypothesis that = 0 is rejected. Thirdly, we examine whether the factors associated with cesarean delivery differ according to the place of delivery. To do so, we need to estimate the determinants of the mode of delivery separately for the two sectors (deliveries in a public versus private health facility). The aforementioned correlation between the mode and the place of delivery alerts on the potential presence of self-selection: women are unlikely to be randomly distributed across places of delivery (public versus private). We suspect that giving birth in a private health facility is positively correlated with cesarean delivery. Scrutinizing the factors associated with cesarean delivery according to the place of delivery requires, therefore, controlling for potential selection. The latter is addressed here using the propensity score matching (PSM) technique [START_REF] Garrido | Methods for constructing and assessing propensity scores[END_REF]. The PSM allows to adjust for pre-treatment observable differences between the treatment group (women who delivered in a private facility) and the control group (women who delivered in a public facility). Using a probit model, we estimate in a first step the propensity scores of the treatment group on the set of explanatory variables except the mode of delivery. Then, a kernel matching procedure is implemented, with cesarean delivery specified as outcome variable, using the Epanechnikov kernel function with a bandwidth of 0.06 and the conditional treatment probability (the propensity scores) estimated in the first step. As a result, women who delivered in the public or private sector are matched based on their propensity scores. Finally, two probit models are estimated separately on the two sub-samples of women with the same set of explanatory variables, and using the matching weights from the propensity scores. It is thus possible to compare the results of the two models. It is worth noting that direct comparisons of coefficients from different nonlinear regression models are not straightforward [START_REF] Mood | Logistic regression: Why we cannot do what we think we can do, and what we can do about it[END_REF]. Although our methodology allows correcting for selection bias due to observable differences between the treatment and comparison groups, potential omitted cluster-level confounders might threaten the direct comparisons of coefficients from both models [START_REF] Arpino | Propensity score matching with clustered data. An application to the estimation of the impact of caesarean section on the Apgar score[END_REF]. As we are in the presence of very small-sized clusters, existing methods to reduce such potential bias are not applicable here. We thus acknowledge the potential presence of omitted cluster-level confounders. Note that the results of this final step are discussed mainly in terms of sign and significance. Lastly, it is worth mentioning that in all regressions, standard errors are clustered at the primary sampling unit (PSU) level to account for the possible autocorrelation in the residuals.

Results

Results of the analysis of the factors associated with the risk of having a cesarean delivery are presented in Table 2. As shown, the probability of cesarean section is significantly positively associated with the place of delivery. The average marginal effect (AME) of the place of delivery indicates that giving birth in the private sector increases the probability of cesarean section by 33.9 percentage points (p < 0.001). The probability of cesarean delivery also increases with the number of prenatal visits, with an AME of 0.052 for ten or more prenatal visits (p < 0.10). By contrast, the probability of cesarean section tend to decrease with the birth order of the newborn, being the lowest for a birth order of 5 or more (AME = -0.179, p < 0.001). Expectedly, the probability of cesarean delivery appears to be lower for the average size of a newborn compared with very small, larger than average and very large. Among the other medical factors, women who had, during the pregnancy, a high blood pressure, a facial or body edema, or gestational diabetes appear to be more exposed to have a cesarean section. Nonetheless, no significant association is found for women who had, during the pregnancy, vaginal bleeding, vaginal infection or burning urination. Place of living appears to be a contributor to the probability of having cesarean delivery with women living in urban areas and in the Northwest regions being at higher risk of having a cesarean delivery than their rural and north-center region counterparts. 2The probability of cesarean section tend also to increase with maternal age and for mothers who had ever experienced the death of a child. Compared with women in the poorest wealth quintile, those in the 3rd and the 4th quintiles have a 2.9% (p < 0.10) and 4.6% (p < 0.01) lower probability of enduring a cesarean section, respectively, while no difference is observed for those in the 2nd and the highest quintiles. Finally, more formally educated women have a markedly higher probability of enduring a cesarean section, pointing to the existence of a socio-economic gradient.

Table 3 shows the results for the bivariate probit model of enduring a cesarean section and delivering in a private health facility estimated using the same set of explanatory variables as before. AMEs, computed for the joint probability of success (Pr(cesarean delivery = 1, delivery in a private health facility = 1)), are also reported. Overall, the coefficient estimates for the cesarean section equation are broadly similar to those obtained using the binary probit model. More importantly, result of the Wald test, which allows to estimate and test for the potential correlation between the error terms of the two equations, strongly rejects the hypothesis that = 0 . The estimate for the correlation coefficient is = 0.556 and the chi-squared test of 196.20, showing that this estimate is significantly different from zero. This indicates that the decisions about the mode of delivery and the place of delivery are jointly influenced by unobservable factors, which are positively related to the mode and the place of delivery. Also note that, compared with the previous model (Table 2), the socio-economic gradient is now observed for both wealth and education: the richer and the more formally educated, the higher the joint probability of both enduring a cesarean section and giving birth in the private sector.

The previous findings motivate the use of the PSM technique in order to account for potential selection bias before estimating separately the determinants of cesarean delivery in the two sectors. In a probit model, we, first, estimate the propensity scores of the treated group (women giving birth in a private health facility) on the same set of explanatory variables, except the mode of delivery. Results show that the balancing property is satisfied. 3The resulting propensity scores are then used to perform the matching. A test assessing the comparability between the two matched samples is performed. Results show that most of the pre-matching bias is considerably reduced after matching. 4 Then, the matching weights from the propensity scores are used to estimate separately a model explaining the factors associated with the risk of having a cesarean section as per the public and private sectors. Results, which are presented in Table 4 are broadly similar for both sub-samples with regard to the birth order, the number of prenatal visits, the maternal age as well as the wealth quintile. These results are also in line with those of the baseline model (Table 2). Quite interestingly, the results reveal a substantial public-private differential concerning a number of medical factors. In particular, compared to smaller or larger newborns' size, the average size of newborns is significantly negatively associated with the probability of enduring a cesarean delivery at a public sector facility, while no significant association is found for the case of the private sector. Also of note, the probability of enduring a cesarean delivery appears to be higher in the public sector for women who had, during their pregnancy, gestational diabetes or a facial or body edema. However, none of these variables emerge to be significant for cesarean section delivery taking place in the private sector. The only complication during the pregnancy which increases the probability of a cesarean section in both public and private sectors is high blood pressure.

Discussion

This study sheds light on the public-private overlap and the phenomenon of dual practice in the provision of health care services using the particular case of cesarean deliveries in Algeria. Unless appropriately regulated, the public-private overlap coupled with physician dual practice may hinder the efficient provision of health care services and impose additional financial and non-financial barriers on households. This may also have adverse consequences on population health. In this study, we show that cesarean deliveries in the private sector are not determined only by medical factors, which goes against the WHO guidelines (World Health Organization, 1985Organization, , 2015)).

Our study has several limitations. First, the absence of data on other medical-related factors (e.g., fetal presentation at delivery, previous cesarean or previous negative birth experience) is a practical limitation. Their inclusion would have allowed us to better capture the medical factors influencing the probability of cesarean section. Second, we acknowledge the presence of a potential recall bias that is usually associated with selfreported data, for instance regarding infant size at birth or complications during pregnancy. However, we can reasonably assume that the mother's recall bias would be lower in surveys relying on a shorter recall period. For instance, [START_REF] Ngandu | Does adjusting for recall in trend analysis affect coverage estimates for maternal and child health indicators? An analysis of DHS and MICS survey data[END_REF] showed that the negative association between coverage estimates for maternal and child health indicators and the proportion of missing data was higher in the Demographic and Health Surveys, which rely on a 5-year recall period, than in the MICS, based on a 2-year recall period. Finally, our data do not allow us to quantify the potential inability to perform a cesarean section due to a lack of resources in the public sector. However, previous studies of the Algerian health system did not highlight shortages of health professionals or + p < 0.10; *p < 0.05; **p < 0.01; ***p < 0.001. Clustered standard errors computed at the primary sampling unit (PSU) level in parenthesis. dy/dx for factor levels is the discrete change from the base level. Regressions are weighted by the propensity scores computed previously to account for sample selection bias equipment for maternal health in the public sector [START_REF] Zehnati | Analyse économique de l'émergence et du développement d'une offre privée de soins en Algérie[END_REF]. Therefore, we do not expect this phenomenon, if any, to influence the results. The hypertrophy of the private medical sector is an essential characteristic of the Algerian health system. The private sector is very much present in large urban areas as well as in the underserved regions. However, its role remains limited in the economically less developed regions such as the western highlands and the south, where the public sector dominates. Our results support the hypothesis that cesarean sections taking place at the private facilities are rather driven by non-medical factors. The absence of appropriate regulations, the overloaded public facilities and the dual practice of public sector physicians all contribute to such medically-unjustified practices. Clearly, existing guidelines for cesarean sections, though necessary, remains insufficient to limit such practices, particularly in the context of the rapidly growing private medical sector, the inadequate supply in the public sector and the high fertility rates. Appropriate legislations that reinforce the public-private complementarity and the surveillance of private medical practices are required in order to improve maternal and neonatal care delivery in Algeria.

  10; *p < 0.05; **p < 0.01; ***p < 0.001. Clustered standard errors computed at the primary sampling unit (PSU) level in parenthesis. Marginal effect for factor levels is the discrete change from the base level

Table 2

 2 Results of the probit model of the factors associated with the risk of enduring a cesarean delivery

	Probit model (dependent variable = cesarean delivery)	Coefficient estimates	AMEs
	Place of delivery (ref. = public health facility)
	Private health facility	1.159***	0.339***
		(0.079)	(0.028)
	Birth order (ref. = 1)
	2	-0.274***	-0.068***
		(0.059)	(0.015)
	3	-0.625***	-0.135***
		(0.077)	(0.016)
	4	-0.684***	-0.144***
		(0.095)	(0.018)
	5 or more	-0.959***	-0.179***
		(0.112)	(0.018)
	Number of prenatal visits (ref. = ≤ 4)
	≥ 5 and ≤ 9	0.128*	0.026*
		(0.050)	(0.010)
	≥ 10	0.247*	0.052 +
		(0.118)	(0.027)
	Infant size at birth (ref. = very small)
	Smaller than average	0.032	0.007
		(0.127)	(0.028)
	Average	-0.222*	-0.043 +
		(0.113)	(0.024)
	Larger than average	0.290*	0.071*
		(0.123)	(0.028)
	Very large	0.286 +	0.070 +
		(0.153)	(0.037)
	Complications during the pregnancy
	Vaginal bleeding	-0.107	-0.020
		(0.084)	(0.015)
	High blood pressure	0.458***	0.108***
		(0.079)	(0.021)
	Facial or body edema	0.193*	0.041*
		(0.079)	(0.018)
	Vaginal infection	-0.095	-0.018
		(0.084)	(0.015)
	Burning urination	0.008	0.002
		(0.088)	(0.017)
	Gestational diabetes	0.466**	0.112**
		(0.155)	(0.043)
	Region (ref. = north center)
	Northeast	0.157	0.032
		(0.097)	(0.020)
	Northwest	0.238*	0.050*
		(0.094)	(0.020)

Table 2

 2

	(continued)
	Probit model (dependent variable = cesarean delivery)	Coefficient estimates	AMEs
	Highlands center	-0.018	-0.003
		(0.090)	(0.017)
	Highlands east	-0.078	-0.014
		(0.092)	(0.017)
	Highlands west	0.058	0.011
		(0.088)	(0.017)
	South	0.017	0.003
		(0.092)	(0.018)
	Urban area (ref. = rural)	0.106 +	0.021 +
		(0.062)	(0.012)
	Maternal age	0.040***	0.008***
		(0.005)	(0.001)
	Mother ever had a child who died	0.238*	0.052*
		(0.098)	(0.023)
	Mother's wealth quintile (ref. = 1st (poorest) quintile)
	2nd quintile	-0.007	-0.001
		(0.072)	(0.015)
	3rd quintile	-0.143 +	-0.029 +
		(0.081)	(0.017)
	4th quintile	-0.238**	-0.046**
		(0.086)	(0.017)
	5th (richest) quintile	-0.119	-0.025
		(0.096)	(0.020)
	Mother's formal education (ref. = elementary school or less)
	Middle school	0.093	0.017
		(0.061)	(0.011)
	High school	0.226***	0.045***
		(0.064)	(0.013)
	Higher	0.265**	0.054**
		(0.085)	(0.018)
	Constant	-2.212***
		(0.205)
	Log pseudolikelihood	-1902.713
	Wald test p value	0.0000
	Pseudo R 2	0.158
	N	5278

Table 4

 4 Factors associated with the risk of enduring a cesarean section according to the place of delivery

	Probit models (dependent vari-	Public health facility		Private health facility
	able = cesarean delivery)	Coefficient estimates AMEs	Coefficient estimates AMEs
	Birth order (ref. = 1)
	2	-0.201*	-0.051*	-0.040	-0.013
		(0.079)	(0.020)	(0.188)	(0.063)
	3	-0.594***	-0.128*** -0.739**	-0.254***
		(0.102)	(0.021)	(0.230)	(0.074)
	4	-0.571***	-0.124*** -0.709*	-0.244**
		(0.122)	(0.024)	(0.287)	(0.094)
	5 or more	-0.891***	-0.167*** -0.901**	-0.306**
		(0.135)	(0.021)	(0.342)	(0.107)
	Number of prenatal visits (ref. = ≤ 4)
	≥ 5 and ≤ 9	0.094	0.020	0.256 +	0.089 +
		(0.061)	(0.013)	(0.150)	(0.052)
	≥ 10	0.283*	0.065 +	0.631 +	0.214*
		(0.143)	(0.036)	(0.342)	(0.109)
	Infant size at birth (ref. = very small)
	Smaller than average	-0.043	-0.010	0.372	0.128
		(0.156)	(0.037)	(0.382)	(0.130)
	Average	-0.273*	-0.058 +	0.261	0.090
		(0.136)	(0.033)	(0.331)	(0.113)
	Larger than average	0.249 +	0.066 +	0.420	0.144
		(0.148)	(0.037)	(0.361)	(0.123)
	Very large	0.277	0.075	0.203	0.070
		(0.186)	(0.050)	(0.431)	(0.148)
	Complications during the pregnancy
	Vaginal bleeding	-0.146	-0.030	0.014	0.005
		(0.112)	(0.021)	(0.229)	(0.078)
	High blood pressure	0.474***	0.121***	0.590*	0.196*
		(0.107)	(0.031)	(0.287)	(0.089)
	Facial or body edema	0.296**	0.070**	0.018	0.006
		(0.098)	(0.026)	(0.232)	(0.080)
	Vaginal infection	0.047	0.010	-0.137	-0.047
		(0.115)	(0.025)	(0.230)	(0.079)
	Burning urination	0.013	0.003	0.264	0.090
		(0.121)	(0.026)	(0.242)	(0.081)
	Gestational diabetes	0.396*	0.099 +	0.805	0.253 +
		(0.196)	(0.056)	(0.533)	(0.140)
	Region (ref. = north center)
	Northeast	0.124	0.028	0.736**	0.256**
		(0.121)	(0.027)	(0.262)	(0.088)
	Northwest	0.130	0.029	0.718*	0.250**
		(0.113)	(0.025)	(0.292)	(0.097)
	Highlands center	-0.033	-0.007	0.142	0.050
		(0.116)	(0.024)	(0.281)	(0.098)

Table 4

 4 (continued)

	Probit models (dependent vari-	Public health facility		Private health facility
	able = cesarean delivery)	Coefficient estimates AMEs	Coefficient estimates AMEs
	Highlands east	-0.223 +	-0.042 +	0.367 +	0.129 +
		(0.125)	(0.024)	(0.217)	(0.075)
	Highlands west	0.067	0.015	0.164	0.057
		(0.114)	(0.025)	(0.325)	(0.114)
	South	-0.065	-0.013	0.214	0.075
		(0.112)	(0.023)	(0.317)	(0.111)
	Urban area (ref. = rural)	0.084	0.018	0.078	0.027
		(0.082)	(0.017)	(0.213)	(0.073)
	Maternal age	0.043***	0.009***	0.039*	0.013*
		(0.007)	(0.001)	(0.016)	(0.005)
	Mother ever had a child who died 0.306*	0.073*	-0.207	-0.071
		(0.122)	(0.032)	(0.310)	(0.106)
	Mother's wealth quintile (ref. = 1st (poorest) quintile)
	2nd quintile	-0.016	-0.004	-0.078	-0.025
		(0.083)	(0.019)	(0.285)	(0.091)
	3rd quintile	-0.120	-0.027	-0.387	-0.129
		(0.097)	(0.022)	(0.302)	(0.098)
	4th quintile	-0.220*	-0.047*	-0.721*	-0.245*
		(0.104)	(0.023)	(0.304)	(0.099)
	5th (richest) quintile	-0.089	-0.020	-0.545 +	-0.184 +
		(0.111)	(0.025)	(0.313)	(0.102)
	Mother's formal education (ref. = elementary school or less)
	Middle school	0.097	0.018	0.050	0.017
		(0.073)	(0.013)	(0.207)	(0.072)
	High school	0.335***	0.070***	-0.140	-0.048
		(0.077)	(0.016)	(0.222)	(0.077)
	Higher	0.318**	0.066**	0.250	0.086
		(0.107)	(0.023)	(0.237)	(0.081)
	Constant	-2.324***		-1.366*
		(0.269)		(0.645)
	Log pseudo-likelihood	-149.926		-234.408
	Pseudo R 2	0.107		0.129
	Wald test p value	0.0000		0.0003
	N	4889		389

Demand-side factors include the role of social network(Leone, Padmadas, & Matthews,

2008) or the fear of vaginal delivery[START_REF] Fritel | Epidémie de césariennes et crainte du traumatisme obstétrical[END_REF], while supply-side factors are mainly related to financial motives[START_REF] Cavalieri | Financial incentives and inappropriateness in health care: Evidence from Italian cesarean sections[END_REF][START_REF] Gruber | Physician fees and procedure intensity: The case of cesarean delivery[END_REF][START_REF] Gruber | Physician financial incentives and cesarean section delivery[END_REF][START_REF] Johnson | Physicians treating physicians: Information and incentives in childbirth[END_REF][START_REF] Mcguire | Physician response to fee changes with multiple payers[END_REF][START_REF] Milcent | Tarification hospitalière et pratique médicale[END_REF][START_REF] Triunfo | The effect of physicians' remuneration system on the caesarean section rate: The Uruguayan case[END_REF].

This may be explained by the fact that the majority of clinics are located in the main urban centers. In 2015,

34.5% of medical and surgical clinics were concentrated in the cities of Algiers and Oran(MSPRH, 2017). Also, public health facilities, when located in the highlands or in southern regions, often lack obstetrician-gynecologists[START_REF] Zehnati | Analyse économique de l'émergence et du développement d'une offre privée de soins en Algérie[END_REF].

The results, which are not shown here for reasons of space, are available from the authors upon request.

Before and after the matching, and for each variable, we perform a t-test of mean equality between the two groups. The results, which are not shown here for reasons of space, are available from the authors upon request.

Acknowledgements This work has been carried out thanks to the support of the A*MIDEX project (No. ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR). We are also grateful to four anonymous reviewers and to researchers from the AHEAD research network for helpful comments.