Computer Simulation of Magnetoelectric Antenna and Performance Comparison with Micro-Loop

Electromagnetic radiation can be produced using functional materials such as magnetoelectric (ME) composites, in which the magnetoelasticity and piezoelasticity of material are involved. The mechanical nature of the vibrations is used to miniaturize the ME antenna to micro-scale size. The antenna performance evaluation requires a multiphysics analysis of the structure. An ME antenna design and simulation is detailed using the finite element method (FEM) in COMSOL Multiphysics® in which the structural mechanics, electrostatics, and magnetic field physics are coupled together to address the simulation needs. An antenna size of 250 μm × 50 μm × 1 μm is simulated within a static magnetic bias field of 20 mT. The nonlinear isotropic model is used for magnetostrictive material definition in which the prestress is defined by the magnetic bias. The model is linearized for radio frequency (RF) simulations to account for the AC simulation. The antenna farfield radiation pattern and the gain are computed using finite difference time domain (FDTD) by incorporating the extracted nearfield of the ME antenna in COMSOL. The simulated antenna impedance, radiation pattern and antenna gain are compared to an equivalent micro-loop magnetic antenna. In addition, electromagnetic computations are used to evaluate the coupling performance between the ME antenna and a larger loop antenna over a distance up to 20 mm in free space and biomedical tissues to address the potential of using ME antenna in medical implants for wireless communication and wireless power transfer.


I. INTRODUCTION
M INIATURIZATION of the antenna in sensors or wireless devices without performance degradation is of great importance. However, with conventional antennas such as dipoles and loops, the antenna efficiency will dramatically decrease when they are used in smaller electrical sizes i.e. ka < 0.1 where k is the wavenumber of free space and a is the radius of the sphere encircle the antenna [1]. On the other hand, small electrical size is more interesting in compact applications such as biomedical implants in which the space is limited and low-frequency operation is of great interest. Another problem with the miniaturization of conventional lowprofile planer antennas is the current image that is created on the antenna ground plane that reduces antenna radiation efficiency, which is also known as the platform effect in the literature. To overcome this problem, instead of using a high dielectric material, one can use magneto-dielectric substrates [2], [3]. However, in the magneto-dielectric material, the problem is to provide low loss magnetic material with weak dispersion behavior at GHz frequencies. New miniaturization of the antennas would use the piezoelasticity and magnetoelasticity that has recently drawn great attention, in which the magnetoelectric (ME) effect is involved. The ME effect can be created by single-phase multiferroics or laminated composite materials, where the composite materials achieve a larger ME effect at room temperature. The ME laminated composite was first introduced in 2001, since then, strong ME effects have been discovered in many laminated composites [4]- [6].
The ME coupling strength in these devices depends on the magnetostrictive and piezoelectric phases, as well as the interface bonding layers. These structures excite acoustic waves inside the device by creating magnetoelectric coupling between magnetostrictive and piezoelectric parts. Therefore, the structure works with acoustic wave resonance instead of electromagnetic resonance. Since the acoustic wavelength is much shorter than the electromagnetic wavelength at the same operating frequency, the ME structure is expected to show a resonance at much lower frequencies than the metallic antennas in the form of a loop or a dipole. Another positive point about the ME antenna is that, unlike the conventional antennas, it does not show the platform effect since the image current acts as in-phase radiation because of the magnetic nature of the created current on the structure [7].
Many papers have studied the performance of ME antennas numerically, analytically, and experimentally [8], [9]. In [8], ME antennas based on nano-plate resonator (NPR) and thin-film bulk acoustic wave resonator (FBAR) structures have been introduced. The measurement results show a promising future for this type of antenna in miniaturizing. The potential application of ME antenna is proposed in [10], in which an array of ME antennas has been introduced for biomedical use. The simulation of the ME antenna has been implemented in COMSOL Multiphysics ® in [11] for the neafield communications in which the CST Studio ® is used to calculate the nearfield coupling. Recently, theoretical analyses on fundamental performances of ME antenna with a focus on wireless power transfer have been presented [12], [13]. The equivalent circuit models and two-port network theory have been utilized to predict the performance of the ME structures [13], where an explicit closed-form equation has been derived and validated by different sets of rigorous experiments. In [12], the compact closed form of the optimal load impedance and its corresponding maximum output power is developed for wireless power transfer purposes. In [14], the radiation Q factor of the antenna has been calculated, and the comparison has been made by a 1-D multiscale FDTD code. However, the effect of nearfield around the antenna structure, as well as the effect of the feeding network, has not been included in the calculation of the Q factor. In [15], an unconditionally stable FDTD algorithm is proposed; the model accounts for ferromagnetic resonance since it solves the Landau-Lifshitz-Gilbert equation for the spin precessional motion. However, the model does not explain the effect of prestress created by the magnetic DC bias field. In [16] a dynamic finite element model has been introduced to solve the coupling between elastic mechanics and linear electromagnetism to calculate the radiation from an ME antenna. However, the nonlinear properties of magnetostrictive material have not been taken into account. To simulate magnetostrictive materials, different nonlinear models have been introduced in the literature, such as the standard square model [17], a model based on the density of domain switching, and the hyperbolic tangent model [18].
In this paper, the nonlinear model proposed in [9] is used for magnetostrictive material since the experimental data shows that the proposed model can accurately describe the nonlinear behavior and saturation trend of magnetostrictive material. In a real scenario, the nonlinear model might not be isotropic since the material sputtering changes its properties in different crystal directions. However, for the sake of simplicity, we have considered the nonlinear isotropic model for the magnetostrictive material FeGaB which is available in the literature [8]. The simulation has been validated by experimental data provided in [19]. We examine the performance of the structure through a multiphysics finite element method which predicts the nonlinear behavior of the magnetostrictive material. We have used different physics, i.e., structural mechanics, electrostatics, and magnetic field as well as two different couplings among solid mechanics and electrostatics and magnetic field. The coupling between solid mechanics and electrostatics simulates piezo-elasticity in the piezoelectric material, and the coupling between solid mechanics and magnetic field simulates the magnetoelasticity in magnetostrictive material. Since the ME structure behaves like a magnetic dipole, we compare its nearfield and farfield performances with a micro-loop of the same size. To this end, we use COMSOL Multiphysics ® and CST Studio ® for nearfield and farfield simulations, respectively. As a use case, we have simulated the ME antenna in the biomedical medium since it offers miniaturization without affecting the antenna impedance characteristics, easy integration with the metal ground plane, and low loss in the nearfield due to the magnetic near field nature of the antenna. These features make the antenna an appropriate choice in WBAN applications such as wireless communication and wireless powering of leadless cardiac pacemaker capsules [20], deep brain stimulators [21], and under skin RFID sensor implants [22].
The paper is organized as follows. Section II introduces the finite element formulation implemented in COMSOL Multiphysics ® . Section III shows the ME antenna structure which contains AlN and FeGaB as the piezoelectric and the magnetostrictive materials, respectively. Section IV explains how the DC and AC simulations in COMSOL have been implemented together with the different physics. In section V the simulation has been validated with the measurement results from [19]. Finally, section VI discusses the different results from simulations along with the comparison between ME and micro-loop antennas.

II. FINITE ELEMENT FORMULATION
To simulate the magnetic field variation around the ME antenna, we have used two-step simulations. First, we use the nonlinear model to calculate the prestress created by DC magnetic field, which is called the bias point. Then, the dynamic variation of the magnetic field, which is a small oscillation about the bias point, has been calculated by smallsignal magnetization. To formulate the structure, different 2 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and physics has been considered as follows:

A. SOLID MECHANICS
The equilibrium equation in solid mechanics is written using a spatial formulation in terms of the Cauchy stress tensor S where F V is the body force per unit volume, ρ is the mass density, ω is the applied frequency, and u is the displacement. In the static case, the right side of (1) will be zero. In addition, the infinitesimal strain tensor (ϵ) is defined as Two different responsive materials, namely magnetostrictive and piezoelectric, are used in the structure. Their behavior is formulated in subsections (II-B) and (II-D), respectively.

B. MAGNETOSTRICTIVE MATERIAL
To simulate the nonlinear and saturation behavior of the magnetostrictive material, we have used the model introduced in [9], which is available in COMSOL Multiphysics ® . The magnetostrictive constitutive relations can be expressed in the following general form [9] where H is the magnetic field strength and ε 0 ij (σ kl ), is the elastic strain produced by prestress. After applying symmetry for isotropic magnetostrictive materials, the nonlinear strain is modeled as follows where, λ s , M s and δ ij are saturation magnetostriction, saturation magnetization, and the Kronecker delta, respectively. Using (4), we can derive a linear response around a given bias point characterized by the vector M 0 = [0,M 0 ,0] along with a small-signal magnetization characterized by vector m = [m 1 ,m 2 ,m 3 ]. The bias magnetic field is defined in the y direction, As a result, (4) can be simplified as The stress in the magnetostrictive material is described as where c H is the stiffness matrix. Nonlinear magnetization in the magnetostrictive material is found from the following nonlinear implicit relation, where, for the nonlinear scalar function, the Langevin function [coth (x) − 1/x] has been used and χ m is the magnetic susceptibility in the initial linear region. For cubic crystals, the effective magnetic field intensity in the material is given by where H is the applied magnetic field, the second term in (8) is the mechanical stress contribution to the effective magnetic field intensity, and dev is the deviatoric operator.
Finally, we use Maxwell's equations to account for the change in the magnetic field as a result of the magnetization change in the magnetostrictive material.

C. MAGNETIC VECTOR POTENTIAL
Because of the divergence-free nature of the magnetic flux density, one can define magnetic vector potential A to calculate the field components. This implies that it is possible to rewrite Maxwell's equations in the following forms where B is the magnetic flux density, H is the magnetic field strength, J e is the external current density, σ is the electrical conductivity and D is the electric displacement field. Here, with the gauge transformation A = A − j ω ∇φ, φ = 0, (9) can be written as follows.

D. PIEZOELECTRIC MATERIAL
The constitutive relation in the piezoelectric material can determine the coupling between the stress and the electric field [23] as follows, where the material parameters s E , d, and ε T , correspond to the material compliance, the coupling properties, and the permittivity. To simulate the physics involving the piezoelectric material Gauss law equation should be solved as well where D is the electric displacement field and ρ V is the free electric charge density. The electric field is computed from the electric potential V as In the transmitting mode, the applied AC voltage to the piezoelectric terminal will create the electric field variation inside the piezoelectric material, which in turn creates the strain in the piezoelectric material. VOLUME 4, 2016 E. MULTIPHYSICS COUPLING Figure. 1, shows the transmitting mode of the ME antenna heterostructure, when the voltage is applied to the piezoelectric terminals, a strain wave in the piezoelectric material is created that is directly transferred to the piezomagnetic (magnetostrictive) material, this strain wave will induce a dynamic change of magnetization inside the piezomagnetic which in turn will create radiation. Conversely, in the receiving mode, the dynamic magnetic field (RF wave) creates a strain wave in piezomagnetic material, which is transferred to the piezoelectric material, this strain wave inside the piezoelectric material will then create voltage in the piezoelectric terminals. FIGURE 1: Coupling mechanism in the ME antenna. In the transmitting mode the dynamic voltage creates a dynamic magnetic field through piezoelectric and piezomagnetic effects and in the receiving mode the process is reversed.
To calculate the multi-physic effect, first, the DC magnetic bias field is applied to the magnetostrictive material to create a prestress inside the structure. To this end, a uniform background magnetic flux density has been defined. The prestress accounts for the nonlinear part of the elastic strain, which is responsible for the change of the maximum magnetostrictive strain created in the structure [9]. To calculate the prestress, we solve (1), (4), (6)-(8), (10) in which ω equals zero in (1), (10). To calculate the dynamic magnetic field intensity created by the AC voltage in the piezoelectric part of the ME antenna, two different multiphysics couplings should be solved. First, piezoelectric physics which combines solid mechanics and electrostatics in which, (1), (2), (11)-(13) will be solved. Here we assume that the electrostatic formula is valid in the time-harmonic study since the electromagnetic wavelength and skin depth are very large compared to the size of the structure. Second, magnetostrictive physics that couples solid mechanics and magnetic field, in which (1), (2), (5)-(8), (10) are involved.

III. STRUCTURE
The resonant bodies of the magnetoelectric antenna resonators are AlN and FeGaB heterostructures fully suspended in air, where AlN and FeGaB serve as the piezoelectric and magnetostrictive element of the ME heterostructure, respectively (see Fig. 2). Appendix I provides the material properties of FeGaB and AlN. The antenna size and different loss mechanisms are given in Table 1.

B. AC SIMUALTION
The AC simulation uses the Frequency-Domain Perturbation solver, in which, as shown in Fig. 3, the Magnetic physics and Solid Mechanic physics are coupled to get the magnetostrictive and inverse magnetostrictive effects together with the coupling between Solid Mechanics and Electrostatics physics which gives the piezoelectric and inverse piezoelectric effects. In this case, the dependent variables are magnetic vector potential A, magnetization M, displacement u and electric potential V. Here, the result of the DC simulation is used as the working point to apply a small signal study, in which, 1 mV AC signal (perturbation) is applied to the piezoelectric terminals as a voltage source. The solver type and the mesh settings are the same as DC simulation and the Infinite Element Domain node is activated as well to model an infinite medium around the structure. The simulation time and memory for a single frequency are 450 s and 35 GB, respectively.

V. VALIDATION OF SIMULATION
To validate the simulation of the ME antenna, first we use an example of the array of three ME antennas with the same size and material properties as developed in [19]. The setup is shown in Fig. 5, the array consists of three ME antennas with the size of 250 µm × 50 µm which are 12 µm apart. The power transfer efficiency (PTE) of the ME antenna array has been measured at different distances from the transmitter (T x ) coil. The structure shows three different modes of operation: along width, length, and thickness. Here, we use the thickness modes of the ME resonators since it shows higher efficiency than the width and length modes [19]. The resonance frequency for thickness mode is as follows [19] f thickness mode = 1 2t where t is the thickness of the piezoelectric, and E eq and ρ eq are the equivalent Young's modulus and equivalent density of the structure, respectively.
The thickness mode in COMSOL simulation resonates in 2.471 GHz, Fig. 4 shows the measured and simulated return loss S 11 , the measured plot is taken from [19]. The simulated S 11 resonates in a lower frequency i.e. 30 MHz  is lower than the measured resonance and it shows lower bandwidth. One possible explanation for the frequency shift  [19]. An array of ME antennas with the size of 250 µm × 50 µm which are 12 µm apart from each other (for illustration purposes, the ME size is magnified). The power transfer efficiency (PTE) has been calculated between the T x coil and the array of ME antennas at different distances between them.
could be due to tolerances of the fabrication process. The presence of the lossy materials in the measurement increases the bandwidth which may explain the high bandwidth of the measurement result compared to the simulated one. To calculate the coupling, a 37 Oe DC bias field is provided in the ME antennas through the background magnetic DC bias in order to create a prestress inside the magnetostrictive material FeGaB. In COMSOL we have simulated the coupling for a few samples for different distances between the T x coil and ME antenna. As shown in Fig. 6 the simulated results are in good agreement with the measurements. After validation, we use a single element ME antenna to simulate the antenna gain and radiation pattern as well as its the antenna nearfield performance. To make a comparison with electrical antennas, we use a micro-loop as an equivalent magnetic antenna and compare the ME and the loop performances.

VI. ME ANTENNA VERSUS MICRO-LOOP
The performance of the ME antenna is studied in the following subsections in which we present the input impedance, equivalent magnetic current, farfield, nearfield, as well as a use case in biomedical implans. The nearfield simulations are performed in COMSOL Multiphysics ® and for the farfield simulation such as gain and radiation pattern we use CST Studio ® . We show that the ME antenna performs are much better than a micro-loop of an equivalent size (see Fig. 7), either in farfield radiation in free space, antenna nearfield for wireless communication and wireless powering as well as in the biomedical implant use case.

A. INPUT IMPEDANCE
To calculate the input impedance, we have applied an AC voltage to the piezoelectric material in COMSOL. Electro- static simulations are used that is valid in the AC simulation as well since the piezoelectric thickness compared to the wavelength is very small. In the ME antenna, different loss mechanisms such as mechanical damping in piezoelectric and magnetostrictive material as well as the electrical loss affects the input impedance, these losses are given in Table  1. The real and imaginary parts of the input impedance of the ME antenna are plotted in Fig. 8. The micro-loop, with the size of λ/500, shows a very small resistivity which makes it difficult to match to 50 Ω impedance. In Fig 9. the return loss of the ME antenna with magnetic DC bias 20 mT as well as the micro-loop are shown, the reference impedance for calculating the return loss is 50 Ω. As shown, the ME antenna has much lower return loss than the micro-loop, the antenna resonates at 2.4705 GHz and 2.477 GHz, and the input impedances in these frequencies are 28.6+j47 Ω and 51.6+j46 Ω, respectively. 6 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and   Fig. 7.

B. EQUIVALENT MAGNETIC CURRENT
With the stress and strain created in the magnetostrictive material, the ME antenna produces an equivalent magnetic current. This current creates the variation of the magnetic field intensity around the ME antenna which is shown in Fig.  10. a. To simulate the AC magnetic field intensity around the structure, first, we apply the magnetic DC bias to the ME antenna to calculate the bias point in which the nonlinear (4) is solved together with (1), (6)- (8), (10) where ω is equal to zero in (1), (10). After calculating the bias point, we use the linearized equation (5) along with (1)-(2), (6)-(8), (10)- (13) to calculate the AC change. Then, we apply AC voltage to the piezoelectric terminal, the AC voltage creates AC vibration in the piezoelectric material. Subsequently, this vibration creates AC magnetostrictive strain which in turn results AC magnetic field intensity around the antenna. In Fig. 10. b, the equivalent magnetic current M=−∇ × E has been plotted, as shown, the current superposition is in the same direction as the magnetic DC bias since the dynamic variation of the magnetic current acts as the perturbation to the DC magnetic bias. As we know from the image theory, Fig. 10. c, the equivalent magnetic current, and its image are in the same direction when the magnetic current is parallel to the perfect electric conductor (PEC) ground plane. Therefore, in this case, the equivalent magnetic current will be doubled. Therefore, as discussed previously, the ME antenna does not have the platform effect. To study the radiation pattern of the ME antenna we need to extract the nearfield of the structure. We calculate the nearfield around the ME antenna using FEM method in COMSOL Multiphysics ® then we extract and use the calculated nearfield as a source of Maxwell's Equations in CST Studio ® to calculate the gain of the ME antenna. According to the surface equivalence theorem, if the materials and the surrounding environment are linear, a structure generating electromagnetic fields can be replaced by a set of electric and magnetic surface current densities existing on a surface surrounding the structure [24]. Even though the materials being used in the antenna's structure are not linear, the structure can be assumed linear due to the linearization of the nonlinear model of the magnetostrictive material. After linearization, small-signal perturbation as a voltage has been applied to the piezoelectric. If we consider the ME and microloop antennas as Fig. 7, in which the bias direction for the ME antenna is assumed to have the same direction as the equivalent magnetic dipole moment of the micro-loop, they both will have the same radiation pattern because both antennas behave as the magnetic dipole. The farfield simulation in the CST shows that both antennas are omnidirectional in the azimuth direction, the gain of the antennas with respect to the elevation angle is shown in Fig. 11. The gain for ME antenna biased with 20 mT and micro-loop antenna are -61.25 and -70.3 dB, respectively. However, in practice, we will have the realized gain which is -63.5 and -84 dB for the ME and micro-loop antennas, respectively. The realized gain accounts for the antenna impedance matching as well. Therefore, the ME antenna shows almost 20 dB better performance than the micro-loop with the same size because the micro-loop, with the size of λ/500, shows a great mismatch to 50 Ω load, whereas the ME antenna shows a better matching to 50 Ω load (see Fig. 9).

D. NEARFIELD COMMUNICATION
To calculate the return loss and the coupling between the ME antenna and receiver (R x ) loop we use a similar setup like the one shown in Fig. 5. However, here, we use a single element ME antenna and different biases are applied to the ME antenna as a background field. The distance between the ME antenna and R x loop is fixed at 5 mm. In addition, the mismatch effect of the R x antenna has been de-embedded from the coupling efficiency by using the following formula, where S 22 is the reflection from the R x loop and S 21 is the transmission coefficients between the ME antenna and R x loop, respectively. The same setup is also used to calculate the coupling between the micro-loop and the same R x loop. The return loss is shown in Fig. 12 with reference impedance 50 Ω, as shown, the ME antenna shows two resonances that are only about 5 MHz apart from each other. However, we are interested in the first resonance which occurs in the lower frequency since the coupling shows better performance for the first one. The magnitude and location of the resonances depend on the magnetic DC bias applied to the ME antenna because the bias changes the prestress created inside the magnetostrictive material which in turn changes the resonance frequency. Unlike the ME antenna, the return loss for the micro-loop in Fig. 12 shows that the high percentage of the power reflects from the input port and it is difficult to make a practical matching-network for the micro-loop. In Fig. 13 the coupling between the ME antenna and R x loop for different DC magnetic biases has been plotted together with the coupling between micro-loop and the same R x loop. As shown, with the DC magnetic bias of 20 mT, the coupling with the ME antenna is 13 dB better than the micro-loop with the same size. The coupling first increases with higher magnetic bias but after saturation of the magnetostrictive material the coupling starts to decrease. The best coupling which is achieved here is with 20 mT, and the performance of the ME antenna with bias as small as 5 mT is better than the micro-loop.
As discussed in section VI.B the bias direction defines the direction of the equivalent magnetic dipole of the ME antenna, the DC bias must be in line with the AC magnetic field of the R x loop to provide maximum efficiency. If they are perpendicular the efficiency drops significantly. To show this, we have simulated four different cases shown in Fig. 14 (a) together with their coupling efficiency curve as shown in Fig. 14 (b). The aligned bias with the AC magnetic field of the R x loop i.e. case I and case II, show maximum efficiency with slightly different values, while the perpendicular biases have minimum coupling i.e. case III and case IV. In this paper, we have studied case I since it has been used in [19] and we validated the simulation results with the measurements in this reference.
FIGURE 13: Coupling between the ME antenna and R x loop for different DC magnetic biases along with the coupling between micro-loop and the same R x loop.

E. BIOMEDICAL APPLICATIONS
In order to study the performance of the ME antenna in the biomedical medium, we consider the coupling-setup in the tissue environment as Fig. 15. The ME antenna and microloop are located in the hollow cube with a 0.5 mm edge size created in the muscle. The muscle size is 10 mm × 10 mm × 25 mm and the R x coil with a 4 mm diameter is located as an on-body receiver with a 0.25 mm distance from the muscle surface. The surrounding muscle has σ = 1.753 and ε = 52.7 + j0.24 at 2.47 GHz [25]. We consider two scenarios, first, the coupling between the ME antenna and the R x loop, and second, the coupling between micro-loop and the same R x loop. The ME and micro-loop antennas are shown in Fig. 7, the ME antenna has 20 mT magnetic DC bias and micro-loop is made of copper with σ = 5.96 ×10 7 .
In Fig. 16 the coupling in the tissue environment versus the distance between the T x and the R x is shown. The coupling for the first scenario is at least 10 dB better than the second scenario, for both the air and muscle embedded antennas. : Rotation of the ME antenna with different orientation of DC magnetic bias with respect to AC magnetic field of the R x loop and their coupling plot. The DC bias is 20 mT and for illustration purposes, the ME size is magnified. (a) four different cases of the ME antenna bias and orientation; case I: bias along width and parallel with AC field, case II: bias along length and parallel with AC field, case III: bias along width and perpendicular to AC field, case IV: bias along length and perpendicular to AC field. (b) the coupling result for case I-IV. The ME antenna is located in the hollow cube with a 0.5 mm edge size created in the muscle and the R x loop with a 4 mm diameter is located as an on-body receiver with a 0.25 mm distance from the muscle surface.
As shown in Fig. 16 the coupling with using ME or the micro-loop in the muscle, at a close distance (<7mm) to the reference loop is reduced compared to the air, due to the antenna nearfield loading by the lossy muscle tissues. However, VOLUME 4, 2016 9 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3183107 beyond 7mm distance, the coupling in the muscle scenario is higher than the air because the embedded antenna has a larger physical size compared to the antenna in free space [26]. In addition, in the simulation, the return loss for the ME antenna when surrounded by the muscle is almost the same as when surrounded by air which accounts for the fact that in the ME antenna contribution of the mechanical loss in the input impedance is the dominating part as the structure resonates in its mechanical thickness mode. Therefore, conductivity and dielectric loss from muscle do not create a significant shift in the resonance frequency. In biomedical implants in which the electrical conductivity of the medium around the antenna varies due to the surrounded tissues, the ME antenna offers better impedance matching since the mechanical mode does not change by the electrical conductivity or dielectric loss of the medium. Therefore, the ME antenna is a promising alternative for biomedical implants. However, it has some limitations such as design and simulation complexity and high manufacturing costs since the technology is still immature and mainly used in the research community.

VII. CONCLUSION
Magnetoelectric (ME) material is a new technology with significant potential in antenna miniaturization. ME antenna modeling is the first step toward designing complex antenna geometries. In this paper, we have focused on the COMSOL Multiphysics ® modeling and simulations of the antenna. The nonlinear isotropic model is considered for magnetostrictive material to simulate the effect of DC magnetic bias which has been simulated through the stationary solver in COMSOL, then the perturbation solver has been applied to account for the dynamic oscillation of the structure in the resonance frequency. It is shown that the ME antenna behaves like a magnetic dipole similar to a micro-loop with the same radiation pattern but with approximately 20 dB higher re-alized gain. The generated magnetic dipole moment is in parallel to the magnetic DC bias of the structure. Unlike the micro-loop, with the size of λ/500, which acts like a short-circuit at 2.471 GHz, the input impedance of the ME antenna depends on the acoustic resonance in the structure and it offers better matching to a 50 Ω load impedance. Moreover, the coupling performance of the ME antenna is evaluated and compared to the micro-loop which performs 10 to 13 dB better than the micro-loop in both air and muscle tissue at different distances from an R x loop. As a result, the ME antenna shows a promising future in both farfield and nearfield communication devices due to having a low profile, low return loss, low structural loss, high gain and coupling performance. Therefore, the ME antenna can be a good choice for different biomedical applications where the antenna efficiency and miniaturization are the main goals.

APPENDIX I
The material properties for magnetostrictive and piezoelectric materials are as follows,

A. MAGNETOSTRICTIVE
FeGaB has been used as a magnetostrictive material, to use the nonlinear isotropic model of FeGaB the following parameters have been considered in the simulation, [8] To model the piezoelectric material parameters such as s E , d, and ε T , which correspond to the material compliance, the coupling properties, and the permittivity are needed. These quantities are tensors of rank 4, 3, and 2, respectively. The AlN material properties are as follows, This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3183107