D Khusyainov

S Ovcharenko

A Buryakov

A Klimov

P Pernod

V Nozdrin

E Mishina

A Sigov

V Preobrazhensky

N Tiercelin

Composite Multiferroic Terahertz Emitter: Polarization Control via an Electric Field

Electrical control of conjugate degrees of freedom in multiferroics provides the advantage of reducing energy consumption to femto-and even attojoules per switch in spintronics and memory devices. This is achieved through the development of technologies that make it possible to fabricate artificial materials with constantly improving properties. Here, we present the design, physics, and characteristics of a composite multiferroic spintronic emitter, which provides electrical control of the emitted terahertz (THz) wave polarization. The effect is due to electrical control of the magnetization in a high-quality magnetostrictive superlattice, TbCo2/FeCo, deposited on an anisotropic piezoelectric substrate. In our approach, several mechanisms are realized in the system simultaneously: the strain-mediated coupling of the magnetic and piezoelectric subsystems, which operate in the range of the spin-reorientation transition of the magnetic superlattice, and THz-wave generation in the superlattice by an optical femtosecond pulse. This provides flexibility and control of the set of parameters. We determine the magnetoelectric parameter, which is responsible for THz polarization control. Our results offer a significant fundamental insight into the physics of composite multiferroic systems that can be used for applications of multiferroicity, primarily for THz spintronic emitters. We believe that our findings represent a decisive step towards technologies for other types of spintronic and memory devices.

I. INTRODUCTION

Multiferroics are materials in which several types of coupled order parameters are simultaneously presented: ferroelectricity, ferromagnetism, and ferroelasticity (or ferrotoroidicity) [START_REF] Eerenstein | Multiferroic and magnetoelectric materials[END_REF]. Among them, magnetoelectric coupling of magnetic and electrical order parameters is the most widespread and studied phenomenon [START_REF] Spaldin | Advances in magnetoelectric multiferroics[END_REF][START_REF] Zhou | The memory effect of magnetoelectric coupling in FeGaB/NiTi/PMN-PT multiferroic heterostructure[END_REF][START_REF] Ba | Electric-field control of skyrmions in multiferroic heterostructure via magnetoelectric coupling[END_REF][START_REF] Hu | Non-volatile ferroelectric switching of ferromagnetic resonance in NiFe/PLZT multiferroic thin film heterostructures[END_REF][START_REF] Iguchi | Magnetoelectrical control of nonreciprocal microwave response in a multiferroic helimagnet[END_REF][START_REF] Fiebig | Revival of the magnetoelectric effect[END_REF]. Magnetoelectrics, the history of which dates back to the 19th century, starting with their prediction [START_REF] Curie | Sur La symétrie dans Les phénomènes physiques, symétrie d'un champ Électrique et d'un champ magnétique[END_REF] and implementation [START_REF] Smolenskiĭ | Ferroelectromagnets[END_REF][START_REF] Rado | Observation of the Magnetically Induced Magnetoelectric Effect and Evidence for Antiferromagnetic Domains[END_REF][START_REF] Folen | Anisotropy of the Magnetoelectric Effect in, Cr2O3[END_REF], are now very close to commercial use in various devices [START_REF] Shevlin | Multiferroics and the path to the market[END_REF][START_REF] Barman | The 2021 magnonics roadmap[END_REF]. One of the most important advantages of magnetoelectric multiferroics is the ability to reduce, by several orders of magnitude, energy consumption for the switching of magnetic states in spintronic and memory devices; this decreases to femto-and even attojoules [START_REF] Roy | Hybrid spintronics and straintronics: A magnetic technology for ultra Low energy computing and signal processing[END_REF][START_REF] Peng | On the speed of piezostrain-mediated voltage-driven perpendicular magnetization reversal: A computational elastodynamics-micromagnetic phase-field study[END_REF][START_REF] Klimov | Magnetoelectric write and read operations in a stress-mediated multiferroic memory cell[END_REF][START_REF] Dusch | Stress-mediated magnetoelectric memory effect with Uni-axial TbCo2/FeCo multilayer on 011-cut PMN-PT ferroelectric relaxor[END_REF][START_REF] Tiercelin | Room temperature magnetoelectric memory cell using stress-mediated magnetoelastic switching in nanostructured multilayers[END_REF]. Despite extensive and intensive development in the last decade of both the physics and technology of magnetoelectric multiferroics [START_REF] Hu | Opportunities and challenges for magnetoelectric devices[END_REF][START_REF] Nicolenco | Strain-gradient effects in nanoscale-engineered magnetoelectric materials[END_REF][START_REF] Dmitriyeva | Magnetoelectric coupling at the Ni/Hf0.5Zr0.5O2 interface[END_REF][START_REF] Liang | Magnetoelectric materials and devices[END_REF][START_REF] González-Casal | Direct reversible magnetoelectric coupling in a ferroelectric/ferromagnetic structure controlled by series resistance engineering[END_REF][START_REF] Fan | Site-Specific spectroscopic measurement of spin and charge in (LuFeO3)m/(LuFe2O4)1 multiferroic superlattices[END_REF][START_REF] Pesquera | Large magnetoelectric coupling in multiferroic oxide heterostructures assembled via epitaxial lift-Off[END_REF][START_REF] Liou | Multiferroic materials: Extremely fast optical and nonvolatile control of mixed-phase multiferroic BiFeO3 via instantaneous strain perturbation[END_REF][START_REF] Pahuja | Latest advancement in magnetoelectric multiferroic composites[END_REF][START_REF] Safin | Electrically tunable detector of THz-frequency signals based on an antiferromagnet[END_REF], a window of opportunity remains open for discovering effects that can further expand the field of application and improve the characteristics of devices based on these materials. Recently, an area of spintronics, terahertz (THz) spintronics, has been intensively developed. This has happened because THz radiation, which, for many years, fell into an unexplored and unused gap in the full electromagnetic spectrum, finally started to be investigated, primarily due to the creation of convenient femtosecond lasers, and showed a huge range of applications [START_REF] Papaioannou | THz spintronic emitters: A review on achievements and future challenges[END_REF][START_REF] Wu | Principles of spintronic THz emitters[END_REF][START_REF] Hirai | Terahertz Emission From Bismuth Thin Films Induced by Excitation with Circularly Polarized Light[END_REF][START_REF] Nenno | Modification of spintronic terahertz emitter performance through defect engineering[END_REF][START_REF] Seifert | Spintronics XIV[END_REF][START_REF] Daneshmand | Analysis and evaluation of the effectiveness of methods for ensuring the quality of service for softwaredefined networks of the standard 5G/IMT-2020[END_REF]. Analogously to common (near-visible) optics, THz optics requires a set of THz optical elements to generate, manipulate, and detect THz radiation. Spintronic THz emitters prove to be indispensable in their characteristics due, first of all, to the width of the generated spectrum (0.5-10 THz) and compactness (they have already emerged the market [35]). A very important task in operating THz radiation is to control its polarization. At present, for this task, THz quartz wave plates are used (and are available on the market [36]). However, the absorption of quartz is quite strong in the THz range, particularly for frequencies higher than 3 THz. In addition to unwanted losses, this means that the THz radiation transmitted through the half-wave plate intrinsically has an irreducible elliptical polarization component. An efficient way to control polarization is provided by spintronic emitters. They allow for control of the polarization directly during THz generation: switch the phase, rotate the polarization, or even adjust the polarization in a more sophisticated way [START_REF] Feng | Spintronic terahertz emitter[END_REF]. Recent results show the possibility of coherent control of asymmetric spintronic terahertz emission [START_REF] Cong | Coherent control of asymmetric spintronic terahertz emission from Twodimensional hybrid metal halides[END_REF]. Since the polarization of the terahertz wave is perpendicular to the magnetization of the spintronic emitter, the magnetization should be rotated to rotate the polarization. Initially, the idea of mechanical rotation of a magnet around the emitter was proposed [START_REF] Kong | Broadband spintronic terahertz emitter with magnetic-field manipulated polarizations[END_REF]. Then an emitter was developed that rotated the polarization by simply increasingdecreasing the current in the electromagnet [START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF]; the operation is based on a spin-reorientation phase transition in magnetic nanostructures. In Ref. [START_REF] Gueckstock | Modulating the polarization of broadband terahertz pulses from a spintronic emitter at rates up to 10 KHz[END_REF], an alternating magnetic field was applied to scan the THz polarization direction. The current trend, driven by the requirements for energy efficiency and compactness of devices, is to replace current-controlled systems with voltage-controlled ones [START_REF] Cheng | Giant Electrical Modulation of Terahertz Emission in Pb(Mg1/3Nb2/3)0.7Ti0.3O3/Co-Fe-B/Pt Structure[END_REF]. The use of a composite multiferroic is a way to combine the advantages of spintronic THz emission and voltage control. Here, we present the design and characteristics of a composite multiferroic spintronic emitter, which provides electric control of polarization of the emitted THz wave in the range up to 66°. To integrate the spintronic principle of emission with the multiferroic concept of controlling the magnetization by an electric field, we use a high-quality magnetostrictive superlattice deposited on an anisotropic piezoelectric substrate, while the THz wave is generated in the superlattice by an optical femtosecond pulse. The principle of polarization control in such a structure is based on the strain-mediated coupling of magnetic and piezoelectric subsystems in the range of the spin-reorientation transition [START_REF] Dusch | Stress-mediated magnetoelectric memory effect with Uni-axial TbCo2/FeCo multilayer on 011-cut PMN-PT ferroelectric relaxor[END_REF][START_REF] Tiercelin | Room temperature magnetoelectric memory cell using stress-mediated magnetoelastic switching in nanostructured multilayers[END_REF][START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF]. By comparison of the experimental results with the theoretical model, we determine the parameter of magnetoelectrically induced anisotropy, which is responsible for the rotation of the magnetization and THz polarization. II. METHODS Pb(Mg1/3Nb2/3)0.7Ti0.3O3 (PMNPT) single-crystal substrates are commercial (011)-cut and one-side polished with a size of 5 × 5 × 0.3 mm 3 . ( TbCo2 4.4 nm/ Fe-Co 5.2 nm) × 25 multilayers with the total thickness of 240 nm are deposited on top of the substrate by rf sputtering. Sputtering is performed in a constant magnetic field, which induces the magnetic anisotropy of the "easy axis" type in the film plane. The easy axis is directed at an angle of 45° relative to the [100] crystallographic axis of the PMNPT substrate. Sputtering details are given in note 3 within the Supplemental Material [43]. Magnetic properties of the samples are studied using a vibration magnetometer (ADE EV9, MicroSense) with a resolution of 1 ÷ 5 µemu. The results are presented in Fig. S3 within the Supplemental Material [43]. For THz time-domain spectroscopy (TDS), an amplified femtosecond laser system is used with a laser pulse duration of 35 fs, wavelength of 800 nm, a pulse repetition rate of 3 kHz, and a pulse fluence of up to 1.8 mJ/cm 2 . Details are given in note 1 within the Supplemental Material [43].

III. RESULTS

THz emission is generated by a linearly polarized femtosecond laser pulse incident at 45° onto a multiferroic composite structure. The structure consists of a magnetic (TbCo2/FeCo) × 25 superlattice, which is deposited onto a PMNPT 〈011〉 piezoelectric substrate. A schematic view of the structure and the configuration of magnetization and external fields regarding the magnetocrystalline axes are presented in Fig. 1(a). Magnetic hysteresis loops measured in two perpendicular directions unambiguously demonstrate an in-plane anisotropy with an anisotropy field of HA = 1.4 kOe (see Fig. S3 within the Supplemental Material [43]). To achieve the maximum and controlled rotation of the THzwave polarization when a voltage is applied to the substrate, the sample is mounted in such a way that the direction of the hard axis of magnetic anisotropy coincides with the direction of the magnetic field. A detailed sample description and the scheme of the optical experiment are presented in Sec. II. The temporal waveform and appropriate spectrum of the generated THz pulse are shown in Fig. 1(b). The measurements are performed on the substrate at a voltage of 60 V and a magnetic field of 0.7 kOe, which provides a maximal signal for each fixed pump fluence. The maximal spectral amplitude is observed at a frequency of 0.8 THz, and the spectral bandwidth is limited to 3.5 THz due to the spectral range of the detector. The THz pulse peak-to-peak signal linearly depends on the fluence of the optical pump [Fig. 1(c)]. In the THz TbCo2/Fe-Co emitter, a change in the absolute value of the magnetic field below the anisotropy field, HA, rotates polarization [START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF]. When all optical components are fixed, the Zn-Te-based detection system acts as an optical analyzer measuring the Ey-THz-field component [START_REF] Kovalev | Terahertz electro-optical detection: Optical phase or energy measurements[END_REF][START_REF] Buryakov | Generation of elliptically polarized terahertz radiation from black phosphorus crystallites[END_REF][START_REF] Brunken | Electro-optic sampling at the TESLA test accelerator: Experimental setup and first results[END_REF]. This gives a THz-amplitude hysteresis loop when the magnetic field is swept [Fig. 2 From Fig. 2, the electric field and magnetic field look interchangeable: sweeping of both gives hysteresis loops for the amplitude of the THz field. But this is not the exact case: for H = 0, no THz signal is observed at all for any voltage [Fig. 2(b), light brown line]. At the same time, small splashes of THz field are observed for U = 0, H = 1 kOe, when the magnetic field is swept [Fig. 2(a), green line]. The width of the magnetic hysteresis loop for the THz amplitude can be determined by Hloop= ±1 kOe. The width of the electric hysteresis loop for the THz amplitude can be determined by Uloop= ±60 V. The value of 60 V is found to be the optimal voltage that provides the maximal THz amplitude. This is found through the direct measurements of the THz signal on the electric field. As shown in Fig. S4 within the Supplemental Material [43], the polarization-angle-control range decreases sharply above 60 V. This is due to a sharp decrease, above this limit, of piezoelectric deformation in the PMNPT 〈011〉 ferroelectric crystal [START_REF] Wu | Principles of spintronic THz emitters[END_REF]. This limit in our sample corresponds to an electric field of 0.2 MV/m. The value of the polarization rotation angle in the scheme used can be obtained only by complete polarization measurements when the THz field polarizer is rotated (see note 1 within the Supplemental Material [43]). From these types of measurements, we first confirmed the pure rotation of polarization without any change in the amplitude of THz radiation for the range of the magnetic field, |H|<3kOe and U = 0. Then, the voltage of 60 V is set and the polarization rotation angle, α, is measured as a function of magnetic field [Fig. 3(a)]. The maximal value of α is observed at H=0.7kOe and decreases rapidly up to H=1.5kOe, followed by a very slow decrease for a stronger magnetic field (which corresponds to a saturation of magnetization at U = 0). Rotation of polarization reverses direction when the sign of the electric field is reversed [see inset in Fig. 3(a)], showing an odd voltage dependence, α(-U)=-α(U). The latter allows the total rotation to be doubled, which reaches 66° in the experiment. Figure 3(b) demonstrates the operation principle of polarization switching and the correlation between the electric hysteresis loop of the THz-emission amplitude and polarization rotation angle for the optimal conditions found. Polarization switching is accompanied by a change in the sign of the THz pulse, but without a change in the amplitude value. Notably, the THz amplitude is measured directly at the maximum of the THz signal by scanning the value of Udc of the dc voltage on PMNPT. The polarization rotation angle is the result of the fitting of polarization dependences for specific Udc values (see note 2 within the Supplemental Material [43]). Figure 4 shows the dependence of the maximum THz-pulse amplitude on magnetic field (THz magnetic hysteresis loop). The sample can be rotated around the normal to the surface (azimuthally). When a magnetic field is applied exactly along the hard axis, the dependence of the THz amplitude on the magnetic field degenerates [Fig. 4(a)], except for small peaks at a field of H=±1kOe. Azimuthal rotation by 4° from the direction of the hard axis leads to a significant increase of the THz-emission amplitude and the appearance of its ordinary hysteresis dependence on the magnetic field [Fig. 4(b), black line]. Application of a voltage to the PMNPT substrate also removes the degeneracy. For the voltage of -60 V and exact orientation of the hard axis along the magnetic field [zero azimuthal angle, red line in Fig. 4(b)], the two hysteresis loops coincide with each other. Thus, we observe the interchangeability of voltage application to the PMNPT substrate and the azimuthal rotation of the sample relative to the direction of the magnetic field.

IV. MODEL AND DISCUSSION

According to the mechanism of inverse spin-Hall effect, the polarization of THz radiation is orthogonal to the spin current polarization, which is parallel to the magnetization. In the strain-mediate composite multiferroic, the application of an electric field to the ferroelectric layer generates elastic deformation in the magnetic subsystem. Specific symmetry of the PMNPT 〈011〉 crystal provides anisotropic inplane deformation under electric field normal to the structure. Due to the effect of magnetostriction, this deformation causes uniaxial magnetoelastic anisotropy in the plane of the intermetallic multilayer. For a reason of the magnetic multilayer deposition in a magnetic field of a certain direction, the angle between the axes of magnetocrystalline and magnetoelastic anisotropy is 45°, as shown in Fig. 1. The thermodynamic potential of the magnetic system as a function of the angle φ between the magnetization M and the hard axis of the magnetocrystalline anisotropy contains Zeeman energy of the magnetization interaction with the magnetizing field H, the energy of magnetocrystalline anisotropy with effective field HA, and magnetoelastic anisotropy energy caused by piezoelectric deformation uxy: [START_REF] Eerenstein | Multiferroic and magnetoelectric materials[END_REF] where B is magnetoelastic constant. In the absence of deformations, the application of a sufficiently strong magnetic field H>HA provides the only minimum of the thermodynamic potential at φ = 0, which corresponds to the saturation magnetization along the hard axis of anisotropy. Decreasing the magnetic field H<HA creates two equivalent minima for cos φ=H/HA, symmetric with respect to φ=0. These two energetically equivalent states bring to the formation of domain structure with opposite projection of the magnetization onto the easy axis. Such structure was visualized in a similar object in the Ref. [START_REF] Klimov | INTERMAG 2006 -IEEE International Magnetics Conference (IEEE)[END_REF] Emergence of domains is reflected in degeneration of the THz signal shown in Fig. 4 (0 V, 0 deg). The deviation of the magnetic field from the hard axis by a sufficiently large angle leaves only one minimum of the thermodynamic potential, thus the spin reorientation in decreasing magnetic field occurs. This fact is reflected in Fig. 4 (0 V, 4 deg). The effect of magnetoelastic anisotropy is equivalent to the deviation of the easy axis from the normal to the magnetic field as it is seen in Fig. 4(b) (-60 V, 0°). The effect of strain on the magnetization orientation is carried out through the parameter Vs=2Buxy/(MHA). The physical meaning of the control parameter VS is the ratio of the energies of strain induced and magnetocrystalline anisotropies. The sign and value of the deformations are controlled by the electric voltage applied to the ferroelectric layer.

The parameter Vs directly determines the deviation angle, θ, of the easy axis: tg2θ=2Vs. Comparing data in Fig. 4(b), we determine the deviation angle equal to θ = 4° for an electric field of 0.2 MV/m, which gives VS = 0.07. Within the linear piezoelectric effect, the control parameter is expressed as Vs=B(d32-d31)e3/(MHA), where d3q are piezoelectric constants and e3 is the normal projection of the applied electric field. Using the reported parameters of the structure's constituents, B = -7 MPa [START_REF] Tiercelin | Thin film magnetoelectric composites near spin reorientation transition[END_REF], 4πM = 8 kGs (see Fig. S3 within the Supplemental Material [43]), HA = 1 kOe, d32- d31= -2.5 × 10 -9 C/N [START_REF] Wang | Complete Set of elastic, dielectric, and piezoelectric constants of orthorhombic 0.71Pb(Mg1/3Nb2/3)O3-0.29PbTiO3 single crystal[END_REF], and e3 = 0.2 MV/m, we estimate VS = 0.055, which is close to the measured value. Figure 5 demonstrates the effect of piezoelectric strain on the thermodynamic potential for two characteristic values of magnetizing field applied normally to the magnetocrystalline easy axis. The magnetic field dependence of the angle φm, corresponding to the absolute minimum of the thermodynamic potential, is shown in Fig. 3(a) by the lines. Calculations are performed for a voltage of 60 V using the anisotropy field of the structure, HA = 1 kOe. This value is obtained from data in Fig. 3 as a field of spin-reorientation instability peak for zero deviation angle and zero tension. We should note that this value is about 35% less than that measured by magnetometry (see Fig. S3 within the Supplemental Material [43]). This difference is explained by the effect of anisotropy disruption during optical pumping, as studied previously on a similar intermetallic structure [START_REF] Ovcharenko | Photoinduced spin dynamics in a uniaxial intermetallic heterostructure[END_REF]. The thermodynamic potential in the area of spin reorientation, H < 0.7 kOe, has two minima, in spite of piezoelectric deformation. The domain structure arises in this area, which is the reason why the measured value of the angle of rotation of polarization is less than that predicted in the framework of the single-domain model. Nevertheless, it is sufficient for THz-polarization control by an electric field over a wide range of 66°. We note that the thickness of our magnetostrictive multilayer is not optimized in terms of THz-emission efficiency. For the reflection mode of emission used in this work, the choice of thickness is not crucial. We have already reported on THz-emission control by a magnetic field in transition mode in a 26-nmthick TbCo2/Fe-Co multilayer [START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF], which is an order of magnitude thinner that that used in this work. However, for polarization control via an electric field, the optimization problem requires special consideration. In contrast to conventional three-layer spintronic emitters, the optimization of a composite multiferroic device should include requirements for the intensity of the THz emission as well as for controllability of the THz polarization. The latter is determined by magnetostriction, depending on the thickness of the structure.

V. CONCLUSION

We report on electric field control over the magnetization direction and, therefore, on the control over the THz-wave polarization without changing the THz-pulse amplitude. We propose a technique for the experimental determination of the strain-induced parameter, VS, which is responsible for magnetoelectric control of THz polarization. We establish a fundamental relationship between the applied electric voltage and the angle of reorientation of the magnetic anisotropy axis caused by the magnetoelectric effect. It allows us to directly measure (without any fitting or use of unknown material characteristics) the magnetoelectric parameter of THz polarization and to control the magnetization with an electric field. For the studied TbCo2/Fe-Co/PMNPT structure, the measured value of the magnetoelectric parameter is equal to VS = 0.07 for an electric field of 0.2 MV/m. The polarization-controlling THz spintronic emitter reported here has two advantages over the previously suggested use of a composite structure based on an amorphous magnetic film [START_REF] Cheng | Giant Electrical Modulation of Terahertz Emission in Pb(Mg1/3Nb2/3)0.7Ti0.3O3/Co-Fe-B/Pt Structure[END_REF]. First of all, it is based on the TbCo2/Fe-Co magnetic multilayer with a spin-reorientation transition. If a magnetic structure is used as the emitter itself, then polarization control can be provided by changing only the value (not direction) of the magnetic field (as we demonstrated in Ref. [START_REF] Khusyainov | Polarization control of THz emission using spin-reorientation transition in spintronic heterostructure[END_REF]). It is very important that the amplitude of the THz wave and its polarization rotation angle are decoupled within the range of spin reorientation transition: pure rotation is achieved with constant amplitude. If the magnetic multilayer is a part of a multiferroic structure, the same is achieved by electric field control. As in the previous case, amplitude and polarization rotation remain decoupled. Second, the observed value of the maximum angle of THz polarization rotation by the electric field is 66°. This is twice that previously reported. The limitation of the polarization-control range, in comparison with the calculated value, is explained by the appearance of a magnetic domain structure in a weak magnetizing field. The current operation voltage is quite high, but it is more than 3 times smaller than that previously used [START_REF] Cheng | Giant Electrical Modulation of Terahertz Emission in Pb(Mg1/3Nb2/3)0.7Ti0.3O3/Co-Fe-B/Pt Structure[END_REF]. Moreover, it is expected to be reduced to the commercial battery tension by reducing the technologically induced magnetic anisotropy and by reducing the PMNPT layer thickness with the use of modern nanoelectromechanical systems technologies [START_REF] Roxworthy | Nanomechanical motion transduction with a scalable localized Gap plasmon architecture[END_REF][START_REF] Zhang | Monolithic processing of a layered flexible robotic actuator film for kinetic electronics[END_REF].

Fig 1 .

 1 Fig 1. Sample structure and THz pulse characterization. (a) Scheme of the composite multiferroic emitter: orientation of the axes of magnetocrystalline (easy axis) and magnetoelastic (ME axis) anisotropy, external magnetic field H, magnetization M, THz-wave electric field E; ¯P, ferroelectric crystal polarization axis normal to the structure plane; φ, the angle between magnetization M and the hard axis (φ = α). Back electrode is made of copper. (b) Temporal waveform and spectrum (inset) of the generated THz pulse. (c) Dependence of THz peakto-peak amplitude on the pump fluence.

  (a)]. The slopes of the hysteresis loops are opposite because the magnetization deviates in opposite directions from the y axis under a voltage of the opposite sign [see Fig. 1(a)]. If the electric field rotates polarization, analogous THz-amplitude hysteresis loops should be observed when the electric field is swept. This is exactly what is obtained in our measurements [Fig. 2(b)].

Fig 2 .

 2 Fig 2. Dependences of the THz amplitude Ey (a) on the magnetic field strength for different voltages and (b) on the voltage for different values of magnetic field strength (THz-amplitude magnetic and electric hysteresis loops, respectively).

Fig 3 .

 3 Fig 3. THz polarization rotation (a) α versus magnetic field applied along the hard axis at a substrate voltage of 60, -60 V. Line corresponds to theoretical calculation. Inset, THz polarization angle at two different angular regions relative to the easy axis direction. (b) α versus voltage applied to the substrate at a fixed magnetic field of 0.7 kOe (points, right scale) and corresponding maximum amplitude of the THz pulse (line, left scale).

Fig 4 .

 4 Fig 4. Interchangeability of voltage and azimuthal sample position. Dependences of the maximum THz pulse amplitude on magnetic field for (a) zero voltage and exact hard-axis orientation; (b) two azimuthal sample positions (0, 4°) and two values of applied voltage (0 and -60 V). Delay time is fixed at t = 0 ps.

Fig 5 .

 5 Fig 5. Magnetization switching between two minima. Thermodynamic potential of the magnetic subsystem at a magnetic field strength H = HA = 1 kOe and H = 0.7 kOe when applying a voltage of -60 V (solid lines) and +60 V (dashed-dotted lines).

ACKNOWLEDGMENTS

The work is supported by Russian Science Foundation (Grant No. 20-12-00276). Fabrication of the samples was performed in the frame of French RENATECH network.


M Tellache

E Rixrath

P Chabrand

C Hochard

M Pithioux

S Wendling-Mansuy

Numerical simulation of an osteoporotic femur: Before and after total hip arthroplasty

Keywords: Osteoporosis, total hip arthroplasty, 3D-reconstruction, finite element model

The bone strength depends on its mineralization and its geometry, which depend themselves on the supported solicitations. The bone optimizes its mass and geometry in order to improve its strength. The optimization process is called bone remodeling. This phenomenon can be deteriorated by metabolic diseases like osteoporosis. This disease weakens the bone structure and causes bone fractures. Among those fractures, femoral neck fractures (hip articulation) are the most recurrent and involve the replacement of the entire hip articulation by a mechanic articulation (Total Hip Arthroplasty). In this paper, finite element models were developed to evaluate, firstly, the stress distribution within osteoporotic human femur bone tissue and secondly, the influence of the perturbation of the stress distribution by Total Hip Arthroplasty on its first stability. The geometry of the femur and the prosthesis was obtained by helicoid scanner acquisition. The bone was considered as two separate types of tissue: cortical bone and cancellous bone. The cortical bone was separated from the trabecular bone by apparent density threshold. In the case of osteoporotic femur the results obtained from the simulations suggest that the mechanism of load transmission is pertubated by the bone remodelling. The degradation of trabecular architecture causes high stresses in the antero-inferior zone of the cortical bone. For the femur with hip prosthesis, the results showed that the implant has significant effects on the stress distribution within bone tissue. High stresses, due to the implant, weak the bone tissue in the lateral zone of the proximal dyaphisis and in the medial zone of the distal part at the end of the stem.

Introduction

The osteoarticular structure holds an important place in the locomotor apparatus. It is essentially composed of bones. The structure of bone is designed to support the solicitations produced by daily activities and locomotion. In order to increase its strength and resist to daily solicitations, it adapts its mineralization and geometry by bone remodeling process. The bone remodeling can be deteriorated by metabolic diseases like osteoporosis which deteriorates bone strength. In Europe, the most consequences of osteoporosis are about 1700 bone fractures per day (W.H.O). Among these fractures, femoral neck fractures are the most recurrent and are the cause of quality of life decrease and rate of mortality increase. In general, femoral neck fracture is at the origin of the replacement of the entire hip articulation by Total Hip Arthroplasty (THA). The number of those implants tends to increase in the next decades due to the expectation of life, but also to the augmentation of the THA in youger patients.

The osteoporosis weakens the bone strength essentially by the reduction of bone mass which is due to bone structure deterioration [START_REF] Hajjar | Osteoporosis for the Home Care Physician, Part 1: Etiology and Current Diagnostic Strategies[END_REF].. The bone structure is reduced by a degradation of trabecular architecture and also by cortical envelop width decrease, and its porosity is increased (Bell et al., 1999). In the same manner, the first stability and durability of hip implants depends on bone strength which depends on its power to adapt its structure and mineralization to new stress distribution. For this, the analysis of stress distribution in bone before and after THA was a major stake and required studies on bone tissue, biomaterials as well as problems of interface between biomaterials and bone tissue.

Our aims are to study the stress distribution within human femur bone tissue and also, the influence of the perturbation of the stress distribution by hip arthroplasty on its first stability. For this, two models of the same human femur were developed. The first represents a 3D finite element model of the femur without prosthesis. The second represents the same finite element model of the femur with a cemented stem to simulate a total hip arthroplasty in the bone environment. This work falls under multidisciplinary project in collaboration with medical and biological teams, in order to study the rupture mechanisms of bone related to strength loss, and also the orthopaedic device stability in their deteriorated environment.

Method

The finite element generation was performed from the geometry of a fresh human donor femur. This femur was obtained from the Laboratory of Anatomy of the Faculty of Medicine of Mediterranean University, Marseilles (responsible: P. Champsaur), and was conserved by Winkler intravenous injection and frozen at -20°C. The geometry of the femur was obtained by helicoid scanner acquisition (General Electric LightSpeed Pro 16) from the Medical imagery service of LaTimone hospital, Marseilles (0,625 mm native cuts thickness, 140 Kv). The scanner was used to generate voxel finite element model by CT2FEM (CT2FEM, 1996). This method allows to assign to each generated voxel, a density described by grey level from CT scans. Thus, each element has an effective bone density. The apparent density of the bone is calculated according to the method described by [START_REF] Taylor | Determination of orthotropic bone elastic constants using FEA and modal analysis[END_REF] The bone was considered as two separated tissues, cortical bone and cancellous bone. the cortical bone was separated from the trabecular bone by apparent density threshold. For an apparent density greater than 0.2g/cm3, the bone was considered compact, under it was considered spongy (Bessho et al 2006). The cortical bone was considered transversely isotropic [START_REF] Huiskes | A detail comparison of experimental and theoretical stress-analyses of a human femur[END_REF][START_REF] Katz | The elastic anisotropy of bone[END_REF][START_REF] Pithioux | Lois de comportement et modèles de rupture des os long[END_REF] in spite of the experimental results which show a certain orthotropy.The Cancellous bone was assumed to present a large-scale isotropy [START_REF] Brown | Mechanical property distributions in the cancellous bone of the human proximal femur[END_REF], with strong variability according to studied region. This variation is strongly dependant on the orientation. of trabeculae. In this model, The compact bone elastic properties were calculated from density by a power law [START_REF] Taylor | Determination of orthotropic bone elastic constants using FEA and modal analysis[END_REF]. The grey level reported in scanners files was related to apparent density by a linear interpolation. The apparent density was then used to calculate elastic properties of cortical bone. Thus, the elastic properties of the k th element were given functions of its apparent density (ρ app;k ), the maximum apparent density in femur (ρ app;max ), the maximum Young modulus in the i th direction of human bone (E i;max ) and the maximum shear modulus in the ij th plane of human bone (G ij;max ).

2 max 2 , max , app k app i k i E E ρ ρ = ; 2 max 2 , max , app k app ij k ij G G ρ ρ = [1]
The maximum values of the elastic modulus are shown in Tab. 1. The transverse isotropy principal axis coincided with the diaphysaire axis at shaft level; with the cervical axis at femoral neck level, and with an axis in-between the two precedent axis at trochanter level. The spongy part was considered isotropic.

Table 1. Maximal values of elastic properties in human bone (Pithioux M., 2002).

The femur model was used to generate the cemented total hip replacement by introducing an implant into femoral diaphyse. The titanium prosthesis was simulated by a homogeneous isotropic elastic solid (E = 110000 MPa, ν = 0.3). The prosthesis was fixed in bone by an acrylic cement, represented by an isotropic elastic solid (E = 2200 MPa, ν = 0.3). The interface bone-cement was supposed to be bonded while a frictional contact was assumed between cement and stem. The Coulomb's law was used with a friction coefficient of µ = 0.1 [START_REF] Nuño | Static coefficient of friction between stainless steel and PMMA used in cemented hip and knee implants[END_REF]..The stem surface was considered as master and the ciment surface as slave surface.

The two models simulated one leg stance cycle of normal walking by quasistatic loading (Bergmann et al., 1993).. The load case included joint load and muscles actions. Muscles actions including abductors groups, Vastus-Lateralis and Tensor-Fascia-Lata, were applied to the anatomical attachment regions. Components of the forces are given in Tab. 2.

Numerical simulation of an osteoporotic femur 5

The finite element model of the femur was meshed with 85624 hexahedral 8 nodes elements corresponding to three parts (the neck, trochanters, and shaft), each of them was composed of 28 parts corresponding to material groups with same elastic properties. The two bounded part of the prosthetic femur model were meshed with 395073 tetrahedral elements. The metallic stem was meshed with 64833 tetrahedral elements. The joint load was applied on femoral and implant head at the most upper node of the femoral head prosthesis according to Yoshida et al. 2002. The ABAQUS (Hibbitt, Karlsson and Sorensen, Inc.) was used to solve the problem and analyzed the stress distribution in the femur and THA.

Results

Frontal cut view of Von Mises stresses distribution in the cortical and cancellous regions of whole femur and THA are shown in are Fig. 2. For the whole femur, the load applied to the femoral head was transmitted from trabeculae network to the inferior face of femoral neck. In this region, the Von Mises stress ranged between 22 MPa and 26 MPa. In the cancellous bone of the femoral neck, the stress is less important than in the cortical one. (~10 MPa). For the THA model, the Von Mises stress in the medial side of the diaphysis varying from 5 MPa under the neck of the stem to 28 MPa at the end of the stem. In the lateral side, the Von Mises stress varying from 24 MPa in the trochanter region to 7 MPa at the end of the stem. In the ciment, the Von Mises stress varying from 8 MPa at the top of cement to 13 MPa at the end of the stem. The contact formulation in the THA model allowed as estimating the micro-slip, frictional shear, and contact stress, between the stem and the cement (see figure below). The stress was concentred in medial and lateral face of the contact surface. Ranged between 4 MPa and 7 MPa, and was highest at proximal lateral region. The relative slip corresponding to this level of stress was ranged between 80 µm and 100 µm.

Conclusion

In this study two finite model were developed in order to study the stress distribution in human femur before and after hip replacement. The first model, represents a voxel based finite element model of human femur generated from medical imaging. This model relates bone mineralization to elastic properties with a power law [START_REF] Taylor | Determination of orthotropic bone elastic constants using FEA and modal analysis[END_REF] in order to simulate the transverse isotropy behaviour of human bone. The second model used the same femur in order to simulate, the first stability of a cemented prothesis in contact with bone, after total hip arthroplasty.

The stress distribution calculated by the first model showed an overloading of the inferior region of cortical neck. This is probably due to osteoporosis degradation of trabecular network which ensures the load transfer and repartition over all the cortical envelop of the femoral neck. Beck et al 1999., found that femoral neck was subject to micro-structural degradation of compact tissue by essentially increasing bone porosity and reducing shell thickness. The combination of cortical bone degradation and it overloading can explain the femoral neck fractures due to osteoporosis.

In the same manner, the second model showed that the hip replacement perturbs stress distribution and involves overloading of the great trochanter region and the distal end of the stem, as soon as, underloading of the medial proximal region introduce bone mass lose and reduce the stem stability. The same observations are mentioned by Rakotomanana et al., Furthermore, in contact surface between stem and cement, the micromotion predicted was comparable to those found by Raminaraka et al., with a greater friction coefficient (0.4). Rakotomanana et al., found a micromotion value of 50 µm with a friction coefficient of 0.6.

Finaly, these models studied only the first stability without take in consideration the bone adaptation. For this, in the next step, bone remodelling process will be include in our model in order to study the long term stability and osteo-integration of orthopaedic devices in their environment.

Acknowledgements

The authours want to thank professor P. Champsaur and Doctor T. LeCorroller (Anatomy laboratory of medecin faculty, Mediterranean university) for having provided the bones ans realized tomography acquisitions. We also thank S. Seguinel for her assistance in the drafting of this poster.

References :

Bell K. L., Garrahan N., Kneissel M., Loveridge N., Grau E., Stanton M., Reeve J., « Cortical and Cancellous Bone in the Human Femoral Neck: Evaluation of an Interactive Image Analysis System », Bone, vol. 19, n° 5, November 1996, p. 541-548.

Bergmann G., Graichen F., Rohlmann A., « Hip joint loading during walking and running, measured in two patients », Journal of Biomechanics, vol. 26, Issue 8, , August 1993, p. 969-990.

Figure 1 .

 1 Figure 1. (a) Maillage hexaédrique du fémur non prothésé, (b) maillage tétraédrique du fémur avec PTH cimentée.

Fx

Figure 2 .

 2 Figure 2. Frontal cut view of the Von Mises stress distribution (a) in osteoporotic femur (b) in THA.

Figure 3 .

 3 Figure 3. Frontal cut view of the relative slip between stem and ciment (a) and frictional shear stress (b) in THA model.