*Article* **Design and Optimization of a Micron-Scale Magnetoelectric Antenna Based on Acoustic Excitation**

**Na Li \*, Xiangyang Li, Bonan Xu, Bin Zheng and Pengchao Zhao**

Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi'an 710071, China

**\*** Correspondence: lina@mail.xidian.edu.cn

**Abstract:** The development of antenna miniaturization technology is limited by the principle of electromagnetic radiation. In this paper, the structure size of the antenna is reduced by nearly two orders of magnitude by using Acoustic excitation instead of electromagnetic radiation. For this magnetoelectric (ME) antenna, the design, simulation and experiment were introduced. Firstly, the basic design theory of magnetoelectric antennas has been refined on a Maxwell's equations basis, and the structure of the ME antenna is designed by using the Mason equivalent circuit model. The influence mechanism of structure on antenna performance is studied by model simulation. In order to verify the correctness of the proposed design scheme, an antenna sample operating at 2.45 GHz was fabricated and tested. The gain measured is −15.59 dB, which is better than the latest research that has been reported so far. Therefore, the ME antenna is expected to provide an effective new scheme for antenna miniaturization technology.

**Keywords:** antenna miniaturization; ME antenna; mason equivalent circuit model; finite element simulation; high gain

## **1. Introduction**

In recent years, with the rapid development of communication equipment miniaturization, most electronic components have achieved miniaturization. However, the size of an electrically small antenna is still larger than 1/10 of the working wavelength [1–3]. What is worse, the impedance matching is difficult, and the radiation efficiency is very low. The core reason is that the traditional small antenna is based on the working principle of the conduction current. Therefore, the inherent ohmic loss of the conduction current results in a reduction in the radiation efficiency. It suffers the platform effect when it is close to the conductive plane, the radiation Q value will increase and the antenna is difficult to match with the impedance [4,5]. Different from the traditional small antenna, the radiation timevarying field of the ME antenna is generated by rotating or oscillating an electric/magnetic dipole moment [6–8]. It not only overcomes the ohmic loss but also has a very high-quality factor. It converts the magnetic component of an electromagnetic wave into an acoustic wave and outputs it as a voltage. In turn, the piezoelectric material is voltage-driven to produce strain, which is then transmitted to the magnetic material, triggering it to magnetize and oscillate, and eventually radiates electromagnetic waves [9–11]. Based on the working principle of acoustic excitation, the advantages of the ME antenna are: (1) the size of the ME antenna can be reduced to 1/10 or even 1/100 of the size of the traditional small antenna, because the speed of sound waves compared to the speed of electromagnetic waves about five orders of magnitude slower. (2) It radiates without the conduction of an electric current; thus, it solves the problem of low radiation efficiency. (3) Its impedance can be adjusted by changing the sizes of the magnetic and the piezoelectric layers. There is no need to add an external matching network, which solves the problem of impedance matching.

Research of the ME antenna originated from the thin-film composite ME materials [8,12–16]. Greve used MEMS technology to fabricate a cantilever structure by integrating alnfecosib

**Citation:** Li, N.; Li, X.; Xu, B.; Zheng, B.; Zhao, P. Design and Optimization of a Micron-Scale Magnetoelectric Antenna Based on Acoustic Excitation. *Micromachines* **2022**, *13*, 1584. https://doi.org/10.3390/ mi13101584

Academic Editors: Seung-bok Choi and Viktor Sverdlov

Received: 27 July 2022 Accepted: 15 September 2022 Published: 23 September 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

magneto-electromechanical composite thin films onto a silicon substrate and obtained a ME coupling coefficient of up to 737 V/cm·Oe at a resonant frequency that was 200 times higher than that under non-resonant conditions [17]. However, the further improvement of the magnetoelectric coupling coefficient is limited by the influence of air damping, and the higher the resonant frequency of the antenna, the more significant the damping effect is. Subsequently, the introduction of acoustic resonators solved this problem. On the one hand, the dynamic magnetization of magnetic materials can be excited and controlled by elastic waves based on the magnetoelastic coupling effect. On the other hand, the mechanical energy can be confined in the piezoelectric layer and the magnetostrictive layer through the design of acoustic impedance. Surface and bulk acoustic resonators have very high-quality factors, which creates the basic conditions for improving the magnetoelectric coupling coefficient and enhancing the practicability of magnetoelectric composites [18,19]. In recent years, Yao et al. systematically studied the (bulk acoustic wave) BAW ME antenna by using the finite difference time domain (FDTD) method and, firstly, coupled the Maxwell's equation and Newton's equation by using the constitutive relationship of ME materials and constructed a complete mathematical model of the energy, average radiated power and radiated quality factor of the ME antenna [6]. Later, Yao generalized it to the 3D case but only in a simulation [20,21]. Domann et al. further proposed the concept of a strain-powered (SP) antenna and established an electrodynamic analytical model to describe the mechanical coupling of EM radiation of the SP antenna [7]. Nan et al. of Northeastern University (NEU) proposed a nanoelectromechanical system (NEMS) ME antenna excited by acoustic waves [8]. There are two structures: a nanoplate resonator (NPR) and thin-film bulk acoustic resonator (FBAR). However, both of them have the problems of low gain and a narrow bandwidth. Lin further proposed a ME antenna with high magnetic field sensitivity and high gain based on Nan [22]. Schneider et al. further experimentally demonstrated the working principle of near-field multiferroic antennas, and they mainly studied the near field of ME antennas and were not involved the far field [23]. Zaeimbashi proposed a novel ultra-miniaturized wireless implantable device, Nano Neuro RFID (Radio Frequency Identification) [24]. The core of this device is a NPR ME antenna array. They just put forward this design concept and did not really manufacture it, let alone carry out test experiments. Dong et al. applied ME antenna to VLF (Very Low Frequency, 3–30 KHz) [25], and they conducted the near-field testing, not far-field testing. Niu et al. studied a miniaturized low-frequency ME receiving antenna with integrated DC bias, which can achieve a higher performance than existing antennas without a DC bias [26], but they only studied the reception of the antenna, not the transmission of the antenna. Kevin used the Landau–Lifshitz–Gilbert (LLG) equation to accurately analyze the magnetoelastic coupling problem in the ME antenna [27], but they did not consider the spatially dependent electrodynamics governed by Maxwell's equation. Ren et al. demonstrated the possibility of using only one BAW-actuated ME transducer antenna for communication; however, the simulation method in this work cannot be used for modeling the far field of radiation [28]. Aiming at the problem of low gain and narrow bandwidth of the ME antenna, a new design scheme of the ME antenna is proposed in this paper based on the above works. In this paper, we design and test that, similar to the traditional antenna, the receiving and transmitting processes of the ME antenna are reciprocal. We first reviewed the working principle of the ME antenna and deduce the radiation power formula. For the structural design of the ME antenna, the equivalent circuit method was used to design the thickness of the ME antenna at 2.45 GHz. Based on the coupling of electric field, magnetic field and stress field, we used the finite element method (FEM) to simulate and obtain the far-field radiation pattern of the ME antenna. We tested the gain of the ME antenna to be −15.59 dB, which was better than the gain reported recently.

## **2. The Basic Principle of ME Antenna**

The basic principle of the ME antenna is the ME coupling effect, which is the product effect of the piezoelectric effect of piezoelectric materials and the piezomagnetic effect of magnetostrictive materials [29,30].

The working principle is shown in Figure 1. When the antenna transmits EM waves, it applies the RF electric field to the upper and lower sides of the piezoelectric layer of the resonant cavity. The mechanical resonance will generate alternating strain waves, which will then be transmitted to the magnetostrictive layer above.

**Figure 1.** The working principle of the ME antenna.

In the case of the ME antenna, the bulk acoustic resonator utilizes the p-wave mode in the body, and the sound wave propagates along the *Z*-axis. Therefore, under the onedimensional conditions, the piezoelectric constitutive equation can be rewritten as

$$\begin{cases} S\_E = s\_E T\_E + \frac{d\_E}{\varepsilon\_T} D\\ E = -\frac{d\_E}{\varepsilon\_T} T\_E + \frac{1}{\varepsilon\_T} D \end{cases} \tag{1}$$

Accordingly, the piezomagnetic constitutive equation can be written as

$$\begin{cases} S\_H = s\_H T\_H + \frac{d\_H}{\mu\_T} B \\ H = -\frac{d\_H}{\mu\_T} T\_H + \frac{1}{\mu\_T} B \end{cases} \tag{2}$$

where *E* and *H* are the electric and magnetic field intensity vectors, respectively, *D* and *B* are the electric and magnetic flux density vectors, *ε<sup>T</sup>* and *μ<sup>T</sup>* are the stress-free permittivity of the piezoelectric layer and stress-free permeability of the magnetostrictive layer, respectively, *sE* and *sH* are the mechanical compliance constants of piezoelectric layer and magnetostrictive layer, respectively and *dE* and *dH* are the strain constants of the piezoelectric layer and magnetostrictive layer, respectively. *SE* and *SH* are the strain field in the piezoelectric layer and magnetostrictive layer, respectively. *TE* and *TH* are the stress field in the piezoelectric layer and magnetostrictive layer, respectively.

The potential energy in the ME antenna is equal to the sum of the potential energy in the piezoelectric and the piezomagnetic layers. The potential energy in the piezoelectric layer mainly includes mechanical energy in the form of mechanical stress and electrical energy in the form of the electric field:

$$\begin{array}{l}\mathcal{W}\_{PE} = \frac{1}{2} \iiint \mathbf{S} \cdot \mathbf{T} \, dv + \frac{1}{2} \iiint \, \mathbf{D} \cdot \mathbf{E} \, dv\\\mathcal{W}\_{PM} = \frac{1}{2} \iiint \, \mathbf{S} \cdot \mathbf{T} \, dv + \frac{1}{2} \iiint \, \mathbf{B} \cdot \mathbf{H} \, dv\end{array} \tag{3}$$

Due to the open-circuit excitation of the piezoelectric layer (*D* = 0) after the initial current pulse drive and the weak magnetic field condition in the magnetostrictive layer (*H* ≈ 0), it can be derived from the second equations in Equations (1) and (2):

$$\begin{cases} E = -\frac{d\_E}{\varepsilon\_T} T\_E\\ B = d\_H T\_H \end{cases} \tag{4}$$

The electromechanical and magnetomechanical coupling figures of merit are given by:

$$k\_E^2 = \frac{d\_E^2}{s\_E \varepsilon\_T}, \qquad k\_H^2 = \frac{d\_H^2}{s\_H \mu\_T} \tag{5}$$

The mechanical compliance in the magnetic layer and piezoelectric layer are:

$$\begin{array}{l} s\_B = (1 - k\_H^2) s\_H \\ s\_D = (1 - k\_E^2) s\_E \end{array} \tag{6}$$

The total energy in the magnetic layer and piezoelectric layer are simplified according to the following:

$$\begin{array}{l}\mathcal{W}\_{PM} = \frac{1}{2} \iiint \mathbf{S}\_{H} \cdot T\_{H} dv + \frac{1}{2} \iiint \mathbf{B} \cdot H dv = \frac{1}{2} \iiint \mathbf{s}\_{H} T\_{H} ^{2} dv\\\mathcal{W}\_{PE} = \frac{1}{2} \iiint \mathbf{S}\_{E} \cdot T\_{E} dv + \frac{1}{2} \iiint \mathbf{D} \cdot E dv = \frac{1}{2} \iiint \mathbf{s}\_{E} \cdot T\_{E}^{2} dv\end{array} \tag{7}$$

It is assumed that both materials deform at the same time, the strain is equal and the stress is different. The stress in the piezoelectric layer is assumed to be *T*<sup>1</sup> and the stress in the piezoelectric layer to be *T*2, so the potential energy stored in the antenna can be expressed as the total potential stored in the form of mechanical stress:

$$\begin{aligned} \mathcal{W}\_P &= \mathcal{W}\_{PE} + \mathcal{W}\_{PM} = \frac{1}{2} \iiint \mathbf{s}\_E \, T\_1^2 dv + \frac{1}{2} \iiint \mathbf{s}\_H \, T\_2^2 dv \\ &= \frac{A}{2} \mathbf{s}\_E \int \mathbf{T}\_1^2 dz + \frac{A}{2} \mathbf{s}\_H \int \mathbf{T}\_2^2 dz \end{aligned} \tag{8}$$

The stress field in the piezoelectric layer is *<sup>T</sup>*<sup>1</sup> <sup>=</sup> *nT*<sup>0</sup> sin <sup>2</sup>*<sup>π</sup> <sup>λ</sup>ac z* , *<sup>T</sup>*<sup>2</sup> <sup>=</sup> *<sup>T</sup>*<sup>0</sup> sin <sup>2</sup>*<sup>π</sup> <sup>λ</sup>ac z* , *T*<sup>0</sup> is the amplitude of the stress field and n is the proportional constant of the stiffness coefficient of the piezoelectric layer and magnetostrictive layer. The device thickness is *d* = *<sup>λ</sup>ac* <sup>2</sup> , and *λac* is the wavelength of the sound waves.

By introducing the known stress field function into the potential energy formula, the total potential energy of the stacked structure magnetoelectric antenna can be calculated. In the process of the ME antenna radiating outward, the piezoelectric layer is the driving source of the stress, while the piezomagnetic layer is the radiation region of the antenna, which is mainly responsible for the radiation of the magnetic field coupled by the stress field. Therefore, the expression of the radiated power of a ME antenna is introduced:

$$P = \frac{1}{2\eta\_0} \iint E\_0^2 ds\tag{9}$$

In which *E*<sup>0</sup> is the aperture electric field formed on the surface of the magnetosphere, and *η*<sup>0</sup> is the free-space wave impedance.

The radiation power formula of the ME antenna is:

$$\begin{array}{l} P = \frac{1}{2\eta\_0} \iint E\_0^2 ds = \frac{\omega^2 \hbar^2}{2\eta\_0} \iint B^2 ds = \frac{\omega^2 \hbar^2 d\_H^2}{2\eta\_0} \iint T\_2^2 ds\\ = \frac{A\omega^2 \hbar^2 d\_H^2 T\_0}{2\eta\_0} \sin^2 \left(\frac{2\pi}{\lambda\_{ac}} z\right) \end{array} \tag{10}$$

## **3. Design and Impedance Analysis of ME Antenna**

The main structure of the ME antenna designed in this paper is based on a cavitybacked FBAR. The substrate is Si, the bottom electrode material is Mo, the piezoelectric material layer is AlN and the magnetostrictive layer is FeGa (also known as the upper electrode). There is an AlN seed layer between the substrate and the lower electrode, and the specific structure and thickness are shown in Figure 2:

**Figure 2.** The left is the cavity-backed FBAR; the right is the structure and materials of the ME antenna.

In order to determine the resonant frequency and analyze the impedance characteristics of the ME antenna, the Mason equivalent circuit model is used to design the antenna circuit. At first, the structure size and material properties are equivalent to the circuit parameters, and the Mason equivalent circuit models of the piezoelectric layer, magnetostrictive layer and electrode layer are established, respectively, and then, the actual antenna structure is constructed, as shown in Figure 3.

**Figure 3.** (**a**) The cascade diagram of the ME antenna, and (**b**) the ME antenna encapsulation.

The electrical impedance of the antenna can be simulated by the circuit diagram shown in Figure 4, and the results shown in Figure 4a are the amplitude–frequency characteristic curve of the antenna, and Figure 4b is the phase–frequency characteristic of the antenna impedance curve. It can be seen from the simulation results that the ME antenna has two resonance frequencies—namely, the series resonance frequency is 2.305 GHz, and the parallel resonance frequency is 2.36 GHz.

**Figure 4.** Amplitude (**a**) and phase (**b**) of the antenna impedance.

#### **4. Finite Element Simulation and Performance Analysis of ME Antenna**

In the previous section, the effect of the antenna structure size and material properties on the electrical characteristics was simulated by using the Mason equivalent circuit model. In order to further analyze the electromagnetic radiation characteristics of the ME antenna, it is necessary to establish an accurate finite element model. As shown in Figure 5, the radius of the resonance region is 100 um, AlN is used as the piezoelectric material and FeGa is used as the magnetostrictive material.

**Figure 5.** Antenna model.

Firstly, the static approximation method is used to study the spatial distribution of stress, strain, displacement and potential, as shown in Figure 6. It can be seen that the maximum stress and strain occur at the interface between the piezoelectric and the magnetostrictive layers. It can be found that the antenna is greatly deformed in the thickness direction, and the maximum displacement can reach 0.25 μm in Figure 6c. From Figure 6d, it can be found that the voltage amplitude at the boundary is significantly higher than that at the middle position.

**Figure 6.** Distribution of the structure and electrical parameters of the ME antenna under a magnetic field. (**a**) Stress, (**b**) strain, (**c**) displacement and (**d**) electric potential.

In order to determine the optimum structural parameters of the antenna, the influence of the structural parameters on the resonant characteristics is analyzed [31].

1. Influence of substrate

For the cavity-backed FBAR structure, the backing is difficult to etch completely during practical machining. Therefore, the influence of the residual substrate thickness on the antenna performance is analyzed. As shown in Figure 7, the resonant frequency of the antenna decreases with the increase of the substrate thickness, because the increase of the substrate thickness directly results in the path of sound wave propagation along the longitudinal direction.

**Figure 7.** Influence of substrate thickness h on the admittance curve.

When the antenna substrate thickness H1 is increased from 0.1 μM to 0.7 μM, the real part of the information of admittance is analyzed based on the absolute value of admittance. It can be found that, with the increase of the substrate, more parasitic resonances are generated, the performance of the antenna is deteriorated. Therefore the substrate should be etched as cleanly as possible during actual processing to eliminate the clamping effect of the substrate.

2. Influence of electrode size

As shown in Figure 8, the electrode width has no effect on the operating frequency of the ME antenna, while the larger the electrode width is, the larger the admittance value will be. The increase in area results in an increase in the capacitance of the antenna and reduces the overall impedance of the antenna. Therefore, impedance matching can be performed by adjusting the electrode area. In the meantime, increasing the area of the electrode can reduce the parasitic mode, which can reduce the impact of the energy loss of the acoustic leakage on the antenna efficiency.

**Figure 8.** Influence of electrode width on the admittance curve.

#### 3. Influence of electrode shape

In addition to longitudinal resonance, the ME antenna also has transverse parasitic resonance, which can disperse the energy of the main resonance. Therefore, it should be avoided as much as possible. The influence of the electrode shape on parasitic resonance is analyzed in this section. Figure 9 shows a square and a pentagonal electrode. Both of them have the same electrode area, and their admittance curves are shown in Figure 10. It can be seen that the parasitic resonance of the pentagonal electrode is obviously smaller than that of the square electrode. Therefore, the electrode shape of this design is optimized as a pentagonal shape.

Based on the above analysis results, the optimum structural parameters of the magnetoelectric antenna are determined, and the structural model of the magnetoelectric antenna is established. The solid mechanics and low-frequency Maxwell equation are used to calculate the inverse ME effect, and the near-field radiation is obtained. The far-field distribution is shown in Figure 11. In the COMSOL simulation, the input power is set as Pin = 1 W, and it can be calculated that the real gain Greal = 1.93 × <sup>10</sup>−13. Since impedance matching is not considered in the 3D model, the S11 of the resonance point is −0.0183 dB. Excluding the mismatch factor, the radiation efficiency can be calculated to be about 2.55 × <sup>10</sup><sup>−</sup>11.

**Figure 9.** Three-dimensional models of different electrode shapes: (**a**) square electrode and (**b**) pentagonal electrode.

**Figure 10.** Absolute values of admittance for different electrode shapes: (**a**) square electrode and (**b**) pentagonal electrode.

**Figure 11.** Far-field 3D pattern of the antenna.

## **5. Fabrication and Testing of Antenna Samples**

In order to verify the superiority of the proposed design scheme, a 2.45-GHz antenna sample is fabricated, and its radiation performance is tested. The processing flow is shown in Figure 12.

**Figure 12.** ME antenna preparation process.

The optical image of the ME antenna is shown in Figure 13.

**Figure 13.** Optical image of the ME antenna.

The test platform is shown in Figure 14.

**Figure 14.** Testing platform.

The reflection coefficient (S11) of the antenna is measured as shown in Figure 15. It can be obtained that the resonant frequency of the antenna is about 2.49 GHz, and the peak return loss is −17.3 dB.

**Figure 15.** S11 curve of the ME antenna.

Then, the S11 and S21 curves are tested in the anechoic chamber environment, where S21 indicates that the ME antenna transmits the signal, and the horn antenna receives the signal. The S11 curve is converted into the Z11 curve and put together with the S21 curve, as shown in Figure 16. The test results show that the antenna has an obvious radiation enhancement at the parallel resonance point. It indicates that the ME antenna produces obvious radiation at the mechanical resonance frequency, which verifies that the radiation of the ME antenna comes from the ME coupling of the mechanical resonance.

The S12 and S21 parameter curves in Figure 17 are basically the same, indicating that the ME antenna has reciprocity under the action of the external magnetic field of small signals. The ME antenna meets the reciprocity principle of traditional antenna.

**Figure 16.** Z11 and S21 parameters of the antenna.

**Figure 17.** S12 and S21 parameters of the ME antenna.

1. Calculation of Antenna Gain

Measured against the antenna in an anechoic chamber at the resonant frequency of 2.49 GHz, the *S*21,*<sup>a</sup>* is −50.42 dB, and the *S*21,*<sup>r</sup>* is −18.03 dB, wherein the gain *Gr* of the reference standard horn antenna is 16.8 dB, and *S*21,*<sup>a</sup>* and *S*21,*<sup>r</sup>* represent the S21 peak value of the ME antenna to be tested and the reference horn antenna test, respectively. According to the comparison method, the antenna gain is calculated to be −15.59 dB.

2. Antenna pattern

In order to fully analyze the radiation characteristics of the ME antenna, its pattern is tested. Verified by the literature, the radiation of the ME antenna can be equivalent to

a model of a magnetic dipole. Using the MATLAB simulation, the radiation pattern of the equivalent magnetic dipole of the ME antenna can be obtained, as shown in Figure 18. Further radiation power P is calculated as:2.61 × <sup>10</sup>−8*W*.

**Figure 18.** Pattern of the ME antenna: (**a**) 3D radiation pattern and (**b**) principal gain (phi = 90) pattern.

### **6. Conclusions**

The proposed ME antenna provided a new method for antenna miniaturization. In this paper, a ME antenna structure was designed, the finite element simulation was carried out on it and the sample was prepared and tested. The results showed that the radiation of the ME antenna originates from the mechanical resonance. It also shows that the ME antenna has the potential to solve the problems of difficult miniaturization and impedance matching of traditional antennas. It can be equivalent to a dipole antenna, its radiation signal comes from the ME coupling and its gain is measured to be −15.59 dB.

**Author Contributions:** N.L. and X.L. are responsible for the design of the ME antenna, the simulation in this paper, the test of the ME antenna and the writing of the paper; N.L. is responsible for supervising and suggesting the revisions; P.Z. and B.Z. is responsible for data processing and B.X. is responsible for part of the simulation and thesis revision. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the National Natural Science Foundation of China (Grant No. 51775402 and U1931139).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Magnetic Interconnects Based on Composite Multiferroics**

**Alexander Khitun**

Electrical Engineering Department, University of California Riverside, Riverside, CA 92521, USA; akhitun@engr.ucr.edu

**Abstract:** The development of magnetic logic devices dictates a need for a novel type of interconnect for magnetic signal transmission. Fast signal damping is one of the problems which drastically differs from conventional electric technology. Here, we describe a magnetic interconnect based on a composite multiferroic comprising piezoelectric and magnetostrictive materials. Internal signal amplification is the main reason for using multiferroic material, where a portion of energy can be transferred from electric to magnetic domains via stress-mediated coupling. The utilization of composite multiferroics consisting of piezoelectric and magnetostrictive materials offers flexibility for the separate adjustment of electric and magnetic characteristics. The structure of the proposed interconnect resembles a parallel plate capacitor filled with a piezoelectric, where one of the plates comprises a magnetoelastic material. An electric field applied across the plates of the capacitor produces stress, which, in turn, affects the magnetic properties of the magnetostrictive material. The charging of the capacitor from one edge results in the charge diffusion accompanied by the magnetization change in the magnetostrictive layer. This enables the amplitude of the magnetic signal to remain constant during the propagation. The operation of the proposed interconnects is illustrated by numerical modeling. The model is based on the Landau–Lifshitz–Gilbert equation with the electric field-dependent anisotropy term included. A variety of magnetic logic devices and architectures can benefit from the proposed interconnects, as they provide reliable and low-energy-consuming data transmission. According to the estimates, the group velocity of magnetic signals may be up to 10<sup>5</sup> m/s with energy dissipation less than 10−<sup>18</sup> J per bit per 100 nm. The physical limits and practical challenges of the proposed approach are also discussed.

**Keywords:** synthetic multiferroic; interconnects; magnetic logic devices

## **1. Introduction**

The development of novel computational devices is well stimulated by the technological challenges and physical limits of the current complimentary metal–oxide–semiconductor (CMOS) technology [1]. Magnetic logic circuits are among the most promising approaches offering a significant reduction in consumed power by utilizing the inherent non-volatility of magnetic elements. In magnetic logic circuitry, logics 0 and 1 are encoded into the magnetization state of a nano-magnet, which may be kept for a long time without any power consumption, while the external energy is required only to perform computation (i.e., switching between the magnetization states). Though magnetic memory became a widely used commercial product a long time ago, magnetic logic is largely in its infancy. The development of energetically efficient and reliable magnetic interconnects is one of the main challenges to be overcome. Similar to electronic transistor-based circuits, where one transistor drives the next stage transistors by electric signals, magnetic logic circuits require one magnet to drive the next stage magnets by sending magnetic signals. There is a variety of possible mechanisms for magnetic signal transmission between the input and the output magnets (i.e., by making an array of nano-magnets sequentially switched in a domino fashion [2], by sending a spin-polarized current [3], by sending a spin wave [4], or by moving a domain wall [5]). There is always a tradeoff between the speed, the energy per bit, and the reliability of magnetic signal transmission. It takes either a large amount

**Citation:** Khitun, A. Magnetic Interconnects Based on Composite Multiferroics. *Micromachines* **2022**, *13*, 1991. https://doi.org/10.3390/ mi13111991

Academic Editors: Viktor Sverdlov, Seung-bok Choi and Jayne C. Garno

Received: 6 October 2022 Accepted: 12 November 2022 Published: 17 November 2022

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of energy for error-prone signal transmission, or the error probability increases with the distance due to the thermal noise, defects, and signal dispersion. The lack of amplification is one of the key issues inherent to the above-mentioned approaches. In this work, we consider composite multiferroics for magnetic interconnects, which may provide magnetic signal amplification by transferring energy between the electric and magnetic domains.

Composite multiferroics (or two-phase multiferroics) comprise piezoelectric and magnetoelastic materials, where an electric field applied across the piezoelectric produces stress, which, in turn, affects the magnetization of the magnetoelastic material. Although the study of composite multiferroics started in the 1970s [6], they have been in the shadow of the single-phase multiferroics (i.e., BiFeO3 and its derivatives [7]) for a long time. Currently, there is a resurgence of interest in composite multiferroics due to the technological flexibility in the independent variation of piezoelectric or magnetostrictive layers. The most important advantage of composite multiferroics over the single-phase ones (e.g., BiFeO3) is the larger strength of the electro-magnetic coupling, which can significantly exceed the limits of their single-phase counterparts [8]. Magnetization rotation in two-phase multiferroics was observed as a function of the applied voltage in several experimental works [9,10]. For instance, a reversible and permanent magnetic anisotropy reorientation was reported in a magnetoelectric polycrystalline Ni thin film and (011)-oriented [Pb(Mg1/3Nb2/3)O3](1 − x)–[PbTiO3] x (PMN-PT) heterostructure [9]. The application of a 0.2 MV/m electric field induces 1200 ppm strain, which, in turn, affects the magnetization of Ni film. According to our preceding work on a similar sample [11], a 0.8 MV/m electric field produces a linear response with in-plain anisotropic strains of εx = 350 μm/m and εy = −1200 μm/m. It is also important to note that the changes in magnetization states are stable without the application of an electric field and can be reversibly switched by an electric field near a critical value (i.e., 0.6 MV/m for Ni/PMN-PT). An ultra-low energy consumption required for magnetization rotation is possible because of this relatively small electric field [12]. The idea of using a stress-mediated mechanism for nano-magnet switching is currently under extensive study [13,14]. The development of multiferroics provides a new approach to spin-wave control. For instance, strain reconfigurable spin-wave transport in the lateral system of magnonic stripes was achieved [15]. It was also observed that the properties of spin-wave propagation in magnonic crystal in contact with a piezoelectric layer can be controlled by an external electric field [16]. Recently, spin-wave propagation and interaction were demonstrated in the double-branched Mach–Zehnder interferometer scheme. The use of a piezoelectric plate connected to each branch of the interferometer leads to the tunable interference of the spin-wave signal at the output section [17]. Here, we propose to utilize multiferroics in magnetic interconnects and exploit the strain-mediated electro-magnetic coupling for magnetic signal amplification. The rest of the paper is organized as follows. In Section 2, we describe the material structure and the principle of operation of the composite multiferroic interconnects. The results of numerical modeling illustrating signal propagation are presented in Section 3. The Discussion and Conclusions are given in Sections 4 and 5, respectively.

## **2. Material Structure and Principle of Operation**

The schematics of the proposed interconnect on top of a silicon wafer are shown in Figure 1A. It consists of the bottom to the top of a conducting layer (e.g., Pt), a layer of piezoelectric material (e.g., PMN-PT), and a layer of magnetoelastic material (e.g., Ni). The whole structure represents a parallel plate capacitor filled with a piezoelectric, where one plane (the bottom) comprises a non-magnetic metal and the top plate comprises a magnetoelastic metal. The top layer is the medium for magnetic signal propagation between the nano-magnets to be placed on the top of the layer. For simplicity, we have shown just two nano-magnets, which are marked as A and B in Figure 1A. The nano-magnet market A is the input element to send a magnetic signal to the receiver nano-magnet B. The spins of the nano-magnets are coupled to the spins of the ferromagnetic magnetostrictive layer via the exchange interaction. The nano-magnets are assumed to be of a special

shape to ensure the two thermally stable states of magnetization. Hereafter, we assume the magnetoelastic layer to be polarized along the *x*-axis, and the nano-magnets to have two states of magnetization along or opposite the *y*-axis. Each of the nano-magnets has an electric contact where a control voltage is applied. The bottom layer, comprising a nonmagnetic metal, serves as a common ground plate.

**Figure 1.** (**A**) Schematics of the synthetic multiferroic interconnect comprising a piezoelectric layer (PMN-PT) and a magnetostrictive layer (Ni). The structure resembles a parallel plate capacitor. An application of voltage at point A results in charge diffusion through the plates. In turn, an electric field applied across the piezoelectric produces stress, which rotates the easy axis of the magnetoelastic material. (**B**) The equivalent electric circuit—RC line, which is used in numerical simulations. (**C**) Results of numerical simulations showing the distribution of the electric field and the magnetization along the interconnect. The change of magnetization in the magnetoelastic layer follows charge diffusion.

The principle of operation is the following. In order to send a signal from A to B, a control voltage V is applied to the nano-magnet A. The application of voltage starts the charge diffusion through the conducting plates. The equivalent circuit is shown in Figure 1B. The charge diffusion through the capacitor plates is well described by the RC model, where the resistance R and the capacitance C are defined by the geometric size and the material properties of the conducting plates and the piezoelectric layer. An electric field appears across the piezoelectric produces stress, which affects the anisotropy of the magnetostrictive

material by rotating its easy axis. It is assumed that the application of voltage rotates the easy axis from the *x*-axis towards the *y*-axis. The change of the anisotropy field caused by the applied voltage affects the magnetization of the magnetoelastic layer. There are two possible trajectories for the magnetization to follow: along or opposite the *y*-axis. The particular trajectory is defined by the magnetization state of the sender nano-magnet A (i.e., the magnetization of the ferromagnetic layer copies the magnetization of the sender nano-magnet).

In Figure 1C, we present the results of numerical modeling, showing the snapshot of the distribution of the electric field E(x) and the magnetization component My(x) through the interconnect. The details of numerical modeling are presented in the next section. Here, we wish to illustrate the main idea of using composite multiferroics as a magnetic interconnect: magnetic signals (i.e., the local change of magnetization) can be sent through large distances without degradation, as the angle of magnetization rotation is controlled by the applied voltage. The direction of signal propagation (e.g., from A to B, or vice versa) is also controlled by the applied voltages. The charging of the capacitor eventually leads to the uniform electric field distribution among the plates and the static distribution of magnetization through the magnetoelastic layer. There are several possible ways to switch output nano-magnet B. For example, it can be preset in a metastable state prior to computation (e.g., magnetization along the *z*-axis), so the magnetic signal sent by A triggers the relaxation towards one of the thermally stable states along or opposite to the *y*-axis. There may also be possible scenarios where the receiver nano-magnet is connected to two or more nano-magnets, so the final state is defined by the interplay of several incoming signals (e.g., MAJ operation). In this work, we focus on the mechanism of signal transmission only, though the utilization of composite multiferroic interconnects may further evolve the design of magnetic logic circuits similar to the ones presented in [3,4,18].

## **3. Numerical Modeling**

The model for signal propagation in the composite multiferroic combines electric and magnetic parts. The electric part aims to find the distribution of an electric field through the piezoelectric, and the magnetic part describes the change of magnetization in the magnetoelastic layer. The charge distribution is modeled via the following equation [19]:

$$R\_s \mathcal{C}\_s \frac{d^2 V(\mathbf{x}, t)}{d\mathbf{x}^2} = \frac{dV(\mathbf{x}, t)}{dt} \tag{1}$$

where *Rs* and *Cs* are the resistance and capacitance per unit length, and *V*(*x*,*t*) is the voltage distribution over the distance. The simulations start with *V*(0,0) = *V*in, and *V*(*x*,0) = 0 everywhere else through the plates.

The process of magnetization rotation is modeled via the Landau–Lifshitz equation [20]:

$$\frac{d\overrightarrow{m}}{dt} = -\frac{\gamma}{1+\eta^2}\overrightarrow{m}\times[\overrightarrow{H}\_{eff} + \eta\overrightarrow{m}\times\overrightarrow{H}\_{eff}]\tag{2}$$

where <sup>→</sup> *<sup>m</sup>* <sup>=</sup> <sup>→</sup> *M*/*Ms* is the unit magnetization vector, *Ms* is the saturation magnetization, *γ* is the gyro-magnetic ratio, and *η* is the phenomenological Gilbert damping coefficient. The effective magnetic field <sup>→</sup> *Heff* is the sum of the following:

$$
\stackrel{\rightarrow}{H}\_{eff} = \stackrel{\rightarrow}{H}\_d + \stackrel{\rightarrow}{H}\_{ex} + \stackrel{\rightarrow}{H}\_a + \stackrel{\rightarrow}{H}\_b \tag{3}
$$

where *Hd* is the magnetostatic field, *Hex* is the exchange field, *Ha* is the anisotropy field <sup>→</sup> *Ha* = (2*K*/*Ms*)( → *m*· → *c* ) → *c* (*K* is the uniaxial anisotropy constant, and <sup>→</sup> *c* is the unit vector along the uniaxial direction), and *Hb* is the external bias magnetic field. The two parts are connected via the voltage-dependent anisotropy term as follows:

$$c\_{\chi} = \cos(\theta), \ c\_{\mathcal{Y}} = \sin(\theta), \ c\_z = 0 \tag{4}$$

$$\theta = \frac{\pi}{2} \left( \frac{V(\chi)}{V\_{\pi}} \right)$$

where *Vπ* is the voltage resulting in a 90-degree easy axis rotation in the X-Y plane.

The introduction of the voltage-dependent anisotropy field (Equation (4)) significantly simplifies simulations, as it presumes an immediate anisotropy response on the applied electric field without considering the stress-mediated mechanism of the electro-magnetic coupling. Such a model can be taken as a first-order approximation. Nevertheless, this model is useful in capturing the general trends of signal propagation and can provide estimates of the maximum speed of signal propagation and energy losses. In our numerical simulations, we use the following material parameters: the dielectric constant ε of the piezoelectric is 2000; the electrical resistivity of the magnetoelastic material is 7.0 × <sup>10</sup>−<sup>8</sup> <sup>Ω</sup>·m, the gyro-magnetic ratio is *γ* = 2 × 107 rad/s, the saturation magnetization is *Ms* = 10 kG/4π; 2 K/Ms = 100 Oe, external magnetic field *Hb* = 100 Oe is along the *x*-axis, and the Gilbert damping coefficient is *η* = 0.1 for the magnetostrictive material. For simplicity, we also assumed the same resistance for the bottom and the top conducting plates. The strength of the electro-magnetic coupling (i.e., *Vπ*) is calculated based on the available experimental data for PMN-PT/Ni (i.e., 0.6 MV/m for 90-degree rotation [9]). More details on the simulation procedure can be found in [21].

The results of the numerical simulations shown in Figure 1C are obtained for the interconnect comprising 40 nm of piezoelectric and 4 nm of magnetoelastic materials. The two curves in Figure 1C depict the distribution of the electric field *E*(*x*) and the projection of magnetization My(x) along the interconnect after the voltage has been applied through the nano-magnet *A*. The curves are plotted in the normalized units *E*/*E*<sup>0</sup> and *My*/*Ms*, where *E*<sup>0</sup> = *Vπ*/*d*, where d is the thickness of the multiferroic layer (40 nm). The distribution of the electric field was found by solving Equation (1). Then, the anisotropy field was found via Equation (4), and, finally, magnetization change was simulated via Equations (2) and (3). The results in Figure 1C show a snapshot taken at 0.4 ns after the voltage has been applied. In these simulations, we assumed the nano-magnet A to be polarized along the *y*-axis, and the magnetization of the interconnect beyond the nanomagnet My(0) = 0.1*Ms* due to the exchange coupling with the spins of the nano-magnet. The spins of the magnetoelastic material tend to rotate in the same direction as the spins of the sender nano-magnet A. Eventually, the Y-component of the magnetization of the interconnect saturates along the constant value, which is defined by the interplay of the anisotropy and the bias magnetic fields.

In Figure 2, we show the results of numerical modeling illustrating the dynamics of magnetization rotation in the interconnect. The curves in Figure 2 depict the evolution of local magnetization in the interconnect located 1.5 μm, 1.7 μm, and 2.0 μm away from the excitation point. The insets in Figure 2 show the initial state of magnetization of the sender nano-magnet A. In all cases, the magnetization trajectory in the interconnect repeats the initial magnetization of the nano-magnet A (e.g., the magnetization component My is positive if nano-magnet A is polarized along the *y*-axis, and the My is negative if nano-magnet A is polarized opposite to the *y*-axis). The absolute value of the final steady state is the same (about 0.5 Ms) for all six curves. These results illustrate the main idea of implementing electric field-driven multiferroic interconnects, allowing us to keep the amplitude of the magnetic signal constant regardless of the propagation distance.

**Figure 2.** Results of numerical modeling showing the normalized magnetization *MY*/*MS* as a function of time. The two sets of curves show magnetization trajectories following the initial state of the sender nano-magnet A (e.g., along or opposite to axis *y*). The black, red, and blue curves show magnetization at 1.0 μm, 2.0 μm, and 3.0 μm distance away from the starting point A.

#### **4. Discussion**

The ability to pump energy into the magnetic signal during its propagation is the most appealing property of the described interconnects. The pumping occurs via the magneto-electric coupling in the multiferroic, where some portion of the electric energy provided to the capacitor is transferred to the energy of the magnetic signal. The amplitude of the magnetic signal (i.e., the angle of magnetization rotation) is controlled by the applied voltage and saturates to a certain value as the electric field across the piezoelectric reaches its steady-state distribution. This property is critically important for logic circuit construction, allowing us to minimize the effect of structure imperfections and make logic circuits immune to thermal noise. It should be also noted that the absolute value of magnetization change in the interconnect may exceed the initial magnetization state of the sender nanomagnet. For instance, the Y component of the magnetization of the nano-magnet A may be 0.1 Ms, while the Y magnetization of the magnetic signal in the interconnect may saturate around 0.5 Ms, as illustrated by numerical modeling in the previous section. In other words, the proposed interconnects may serve as an amplifier for magnetic signals, similar to the multiferroic spin-wave amplifier described in [21]. Another important property of the proposed interconnect is the ability to control the direction of signal propagation by the applied voltage. Similar to the "All Spin Logic" approach [3], where the direction of magnetic signal is defined by the direction of spin-polarized current flow, the change of magnetization in the multiferroic interconnect follows the charge diffusion. This property resolves the problem of input–output isolation and provides an additional degree of freedom for logic circuit construction.

Energy dissipation in a two-phase magnetoelastic/piezoelectric multiferroic has been studied in [14,22,23]. According to the estimates, a single two-phase magnetoelastic/piezoelectric multiferroic single-domain shape-anisotropic nano-magnet can be switched, consuming as low as 45 kT for a delay of 100 ns at room temperature, where the main contribution to the dissipated energy comes from the losses during the charging/discharging (≈CV2) [23]. The capacitance of one-micrometer-long multiferroic interconnects comprising 40 nm of PZT and 4 nm of Ni with the width of 40 nm is about 15 fF, and the control voltage required for 90-degree anisotropy easy-axis change is 0.6 MV/m × 40 nm = 24 mV. Thus, assuming all the electric energy dissipated during signal propagation, one has 9 aJ per signal per 1 μm transmitted. It is important to note that, according to the theoretical estimates [23], the energy dissipation increases sub-linearly with the switching speed. For example, in order to increase the switching speed by a factor of 10, the dissipation needs to increase by a factor of 1.6.

The propagation of the magnetization signal involves several physical processes: charge diffusion, the mechanical response of the piezoelectric to the applied electric field, change of the anisotropy field caused by the stress, and magnetization relaxation. Thus, the total delay time *τ<sup>t</sup>* is the sum of the following:

$$
\pi\_l = \pi\_\varepsilon + \pi\_{mech} + \pi\_{mag} \tag{5}
$$

where *τ<sup>e</sup>* is the time delay due to the charge diffusion *τ<sup>e</sup>* = RC, τmech is the delay time of the mechanical response *τmech* ≈ *d*/*va*, where *d* is the thickness of the piezoelectric layer, *va* is the speed of sound in the piezoelectric, and *τmag* is the time required for the spins of magnetostrictive material to follow the changing anisotropy field. In the theoretical model presented in the previous section, we introduced a direct coupling among the electric field and the anisotropy field (i.e., Equation (4)), presuming an immediate anisotropy field response on the applied electric field. The latter may be valid for the thin piezoelectric layers (e.g., taking d = 40 nm, *va* = 1 × 103 m/s, *<sup>τ</sup>mech* is about 40 ps). We also introduced a high damping coefficient *η*, which minimizes the magnetic relaxation time *τmeag* < 50 ps. In this approximation, the speed of signal propagation is mainly defined by the charge diffusion rate. The smaller RC, the faster the charge diffusion and the lower the energy losses for interconnect charging/discharging.

In Figure 3, we show the results of numerical modeling on the speed of signal propagation for different thicknesses of the piezoelectric layer. The four curves correspond to the signal propagation in the interconnects with different PMN-PT thicknesses (e.g., 20 nm, 40 nm, 80 nm, and 200 nm), respectively. The thickness of the nickel layer is 4 nm for all cases. We also plotted a reference line corresponding to the magnetostatic spin wave with a typical group velocity of 3.1 × 104 m/s. According to these estimates, one may observe that the magnetic signal in the multiferroic interconnect may propagate faster than the spin wave at short distances (<500 nm) and slower than the spin wave at longer distances. The latter leads to an interesting question of whether or not it is possible to transmit magnetic signals faster than the spin wave in the magnetoelastic material. Although magnetic coupling does not define the speed of signal propagation, it should determine the trajectory of spin relaxation. Exceeding the speed of spin wave in ferromagnetic material may lead to a chaotic magnetic reorientation along the ferromagnetic layer. At the same time, it will limit the propagation length. Would it be possible to cascade multiferroic interconnects? This is one of many questions to be answered with further study.

**Figure 3.** Results of numerical modeling illustrating the speed of signal propagation in the synthetic multiferroic interconnect. Shown are several curves corresponding to different thicknesses of the PMN-PT layer (20 nm, 40 nm, 80 nm, and 200 nm). The blue line is the reference data for the Magnetostatic Surface Spin Wave (MSSW) with a group velocity of 3.0 <sup>×</sup> 104 m/s.

Finally, we wish to compare the main characteristics of different magnetic interconnects and discuss their advantages and shortcomings. Moving a domain wall is a reliable and experimentally proven method for magnetic signal transmission [24]. A domain wall propagates through a magnetic wire, as long as an electric current or an external magnetic field is applied, and remains at a constant position if the driving force is absent. This property is extremely useful for building magnetic memory (e.g., the "racetrack" memory [25]). The speed of domain motion may exceed hundreds of meters per second if the driving electric current has a sufficiently large density (e.g., 250 m/s at 1.5 × <sup>10</sup><sup>8</sup> A/cm<sup>2</sup> from [25]). Slow propagation speed and high energy per bit are the main disadvantages of the logic circuits' exploding domain wall motion.

The interconnects made from the sequence of nano-magnets are relatively faster and less power consuming, where the nearest neighbor nano-magnets are coupled via the dipole–dipole interaction (the so-called Nano-Magnetic Logic (NML) [2]). Experimentally realized wires formed from a line of anti-ferromagnetically ordered nano-magnets show a signal propagation speed up to 103 m/s with an internal (without the losses in the magnetic field generating contours) power dissipation per bit of approximately tens of atto Joules [18]. There is a tradeoff between the speed of signal propagation and the dissipated energy. The slower the speed of propagation, the lower the energy dissipated within the interconnect. The main shortcoming of the nano-magnet interconnect is associated with reliability, as the thermal noise and fabrication-related imperfections can cause errors in signal transmission and the overall logic functionality of the NML circuits [26].

Interconnects exploiting spin waves may provide signal propagation with the speed of 10<sup>4</sup> m/s–10<sup>5</sup> m/s. At the same time, the amplitude of the spin-wave signal is limited by the several degrees of magnetization rotation, in contrast to the complete magnetization reversal provided by the domain wall motion or NML. The amplitude of the spin wave decreases during propagation (e.g., the attenuation time for magnetostatic surface spin waves in NiFe is 0.8 ns at room temperature [27]). The unique advantage of the spinwave approach is that the interconnects themselves can be used as passive logic elements exploiting spin-wave interference. The latter offers an additional degree of freedom for logic gate construction and makes it possible to minimize the number of nano-magnets per logic circuit [4].

The All-Spin Logic (ASL) proposal suggests the use of spin-polarized currents for nano-magnet coupling [3]. This approach allows for much greater defect tolerance, as the variations in the size and position of input/output nano-magnets are of minor importance. It is also scalable, since shorter distances between the input/output cells would require less spin-polarized currents for switching. According to theoretical estimates [28],

ASL can potentially reduce the switching energy-delay product. The major constraint is associated with the need for the spin-coherent channel, where the length of the interconnects exploiting spin-polarized currents is limited by the spin diffusion length.

The described magnetic interconnects based on composite multiferroics combine high transmission speed (as fast as the spin waves) with the possibility of transmitting large amplitude signals (up to 90 degrees of the magnetization rotation). As we stated above, the main appealing property of the proposed interconnect is the ability to keep constant the amplitude of the magnetization signal. All these advantages are the result of using the electro-magnetic coupling in multiferroics, allowing us to pump energy from the electric to the magnetic domain. Based on the presented estimates, the energy per transmitted bit may be as low as several atto Joules per 100 nm of transmitted distance. From a practical point of view, the implementation of composite multiferroic interconnects is feasible, as it relies on the integration of well-known materials (e.g., PMN-PT and Ni) and can be integrated on a silicon platform. However, the dynamics of the electro-mechanical-magnetic coupling in composite multiferroics remain mainly unexplored. The expected challenges are associated with the limited scalability, as the thickness of the piezoelectric should be sufficient to generate the stress required for anisotropy change. The quality of the interface between the piezoelectric and magnetostrictive layers is another important factor to be considered. The

inevitable structure imperfections should be below the magnetization reversal threshold (e.g., as defined by Equation (4)). In Table 1, we have summarized the estimates on the main characteristics of different magnetic interconnects and outlined their major advantages and shortcomings.


**Table 1.** The estimates on the main characteristics of different magnetic.

\* Signal propagation speed is determined by the charge diffusion and decreases with the distance. \*\* The estimates for 103 m/s propagation speed and include only for the energy dissipated inside the magnetic interconnect (without considering the energy losses in the magnetic field generating contours).

## **5. Conclusions**

In summary, we considered a novel type of magnetic interconnect exploiting electromagnetic coupling in two-phase composite multiferroics. According to the presented estimates, composite multiferroic interconnects combine the advantages of fast signal propagation (up to 10<sup>5</sup> m/s) and low power dissipation (less than 1 aJ per 100 nm). The most appealing property of the multiferroic interconnects is the ability to pump energy into the magnetic signal and amplify it during propagation. A voltage-driven magnetic interconnect may be utilized in nano-magnetic logic circuitry and provide an efficient tool for logic gate construction. The fundamental limits and practical constraints inherent to twophase multiferroics are associated with the efficiency of stress-mediated coupling at high frequencies. There are many questions related to the dynamic of the stress-mediated signal propagation, which will be clarified with a further theoretical and experimental study.

**Funding:** The National Science Foundation (NSF) under Award # 2006290, Program Officer Dr. S. Basu.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All data generated or analyzed during this study are included in this published article.

**Acknowledgments:** This work was supported in part by the National Science Foundation (NSF) under Award # 2006290, Program Officer S. Basu.

**Conflicts of Interest:** The authors declare no conflict of interest.

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