# Magnetic Interconnects Based on Composite Multiferroics

## Abstract

**:**

^{5}m/s with energy dissipation less than 10

^{−18}J per bit per 100 nm. The physical limits and practical challenges of the proposed approach are also discussed.

## 1. Introduction

_{3}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., BiFeO

_{3}) 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(Mg

_{1/3}Nb

_{2/3})O

_{3}](1 − x)–[PbTiO

_{3}] 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 Section 4 and Section 5, respectively.

## 2. Material Structure and Principle of Operation

## 3. Numerical Modeling

_{s}and C

_{s}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) = Vin, and V(x,0) = 0 everywhere else through the plates.

_{s}is the saturation magnetization, γ is the gyro-magnetic ratio, and η is the phenomenological Gilbert damping coefficient. The effective magnetic field ${\overrightarrow{H}}_{eff}$ is the sum of the following:

_{d}is the magnetostatic field, H

_{ex}is the exchange field, H

_{a}is the anisotropy field ${\overrightarrow{H}}_{a}=\left(2K/{M}_{s}\right)(\overrightarrow{m}\xb7\overrightarrow{c})\overrightarrow{c}$ (K is the uniaxial anisotropy constant, and $\overrightarrow{c}$ is the unit vector along the uniaxial direction), and H

_{b}is the external bias magnetic field. The two parts are connected via the voltage-dependent anisotropy term as follows:

_{π}is the voltage resulting in a 90-degree easy axis rotation in the X-Y plane.

^{−8}Ω·m, the gyro-magnetic ratio is γ = 2 × 107 rad/s, the saturation magnetization is M

_{s}= 10 kG/4π; 2 K/Ms = 100 Oe, external magnetic field H

_{b}= 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., V

_{π}) 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].

_{0}and My/Ms, where E

_{0}= V

_{π}/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 nano-magnet My(0) = 0.1M

_{s}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.

## 4. Discussion

^{2}) [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.

_{t}is the sum of the following:

_{e}is the time delay due to the charge diffusion τ

_{e}= RC, τmech is the delay time of the mechanical response τ

_{mech}≈ d/v

_{a}, where d is the thickness of the piezoelectric layer, v

_{a}is the speed of sound in the piezoelectric, and τ

_{mag}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, v

_{a}= 1 × 10

^{3}m/s, τ

_{mech}is about 40 ps). We also introduced a high damping coefficient η, which minimizes the magnetic relaxation time τ

_{meag}< 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.

^{4}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.

^{8}A/cm

^{2}from [25]). Slow propagation speed and high energy per bit are the main disadvantages of the logic circuits’ exploding domain wall motion.

^{3}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].

^{4}m/s–10

^{5}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 spin-wave 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].

## 5. Conclusions

^{5}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 two-phase 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

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Semiconductors, I.T.R. Chapter PIDS. 2011. Available online: http://www.itrs.net (accessed on 7 January 2011).
- Cowburn, R.P.; Welland, M.E. Room temperature magnetic quantum cellular automata. Science
**2000**, 287, 1466–1468. [Google Scholar] [CrossRef] [PubMed] - Behin-Aein, B.; Datta, D.; Salahuddin, S.; Datta, S. Proposal for an all-spin logic device with built-in memory. Nat. Nanotechnol.
**2010**, 5, 266–270. [Google Scholar] [CrossRef] [PubMed] - Khitun, A.; Wang, K.L. Non-Volatile Magnonic Logic Circuits Engineering. J. Appl. Phys.
**2011**, 110, 034306–034310. [Google Scholar] [CrossRef] [Green Version] - Nikonov, D.E.; Bourianoff, G.I.; Ghani, T. Proposal of a Spin Torque Majority Gate Logic. IEEE Electron Device Lett.
**2011**, 32, 1128–1130. [Google Scholar] [CrossRef] [Green Version] - Vanrun, A.; Terrell, D.R.; Scholing, J.H. Insitu Grown Eutectic Magnetoelectric Composite-Material. 2. Physical-Properties. J. Mater. Sci.
**1974**, 9, 1710–1714. [Google Scholar] [CrossRef] - Wang, J.; Neaton, J.B.; Zheng, H.; Nagarajan, V.; Ogale, S.B.; Liu, B.; Viehland, D.; Vaithyanathan, V.; Schlom, D.G.; Waghmare, U.V.; et al. Epitaxial BiFeO3 multiferroic thin film heterostructures. Science
**2003**, 299, 1719–1722. [Google Scholar] [CrossRef] - Eerenstein, W.; Mathur, N.D.; Scott, J.F. Multiferroic and magnetoelectric materials. Nature
**2006**, 442, 759–765. [Google Scholar] [CrossRef] - Wu, T.; Bur, A.; Zhao, P.; Mohanchandra, K.P.; Wong, K.; Wang, K.L.; Lynch, C.S.; Carman, G.P. Giant electric-field-induced reversible and permanent magnetization reorientation on magnetoelectric Ni/(011) [Pb(Mg
_{1/3}Nb_{2/3})O_{3}]_{(1−x)}–[PbTiO_{3}]_{x}heterostructure. Appl. Phys. Lett.**2011**, 98, 012504–012507. [Google Scholar] - Shabadi, P.; Khitun, A.; Wong, K.; Amiri, P.K.; Wang, K.L.; Andras, C.A. Spin wave functions nanofabric update. In Proceedings of the IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH-11), San Diego, CA, USA, 8–9 June 2011; pp. 107–113. [Google Scholar]
- Balinskiy, M.; Chavez, A.C.; Barra, A.; Chiang, H.; Carman, G.P.; Khitun, A. Magnetoelectric Spin Wave Modulator Based On Synthetic Multiferroic Structure. Sci. Rep.
**2018**, 8, 1–10. [Google Scholar] [CrossRef] [Green Version] - Shabadi, P.; Khitun, A.; Narayanan, P.; Mingqiang, B.; Koren, I.; Wang, K.L.; Moritz, C.A. Towards logic functions as the device. In Proceedings of the 2010 IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH 2010), Anaheim, CA, USA, 17–18 June 2010. [Google Scholar] [CrossRef] [Green Version]
- D’Souza, N.; Atulasimha, J.; Bandyopadhyay, S. An Ultrafast Image Recovery and Recognition System Implemented With Nanomagnets Possessing Biaxial Magnetocrystalline Anisotropy. IEEE Trans. Nanotechnol.
**2012**, 11, 896–901. [Google Scholar] [CrossRef] - Roy, K.; Bandyopadhyay, S.; Atulasimha, J. Energy dissipation and switching delay in stress-induced switching of multiferroic nanomagnets in the presence of thermal fluctuations. J. Appl. Phys.
**2012**, 112, 023914. [Google Scholar] [CrossRef] [Green Version] - Grachev, A.A.; Sheshukova, S.E.; Nikitov, S.A.; Sadovnikov, A.V. Strain reconfigurable spin-wave transport in the lateral system of magnonic stripes. J. Magn. Magn. Mater.
**2020**, 515, 167302. [Google Scholar] [CrossRef] - Grachev, A.A.; Matveev, O.V.; Mruczkiewicz, M.; Morozova, M.A.; Beginin, E.N.; Sheshukova, S.E.; Sadovnikov, A.V. Strain-mediated tunability of spin-wave spectra in the adjacent magnonic crystal stripes with piezoelectric layer. Appl. Phys. Lett.
**2021**, 118, 262405. [Google Scholar] [CrossRef] - Grachev, A.A.; Sadovnikov, A.V.; Nikitov, S.A. Strain-Tuned Spin-Wave Interference in Micro- and Nanoscale Magnonic Interferometers. Nanomaterials
**2022**, 12, 1520. [Google Scholar] [CrossRef] - Niemier, M.T.; Bernstein, G.H.; Csaba, G.; Dingler, A.; Hu, X.S.; Kurtz, S.; Liu, S.; Nahas, J.; Porod, W.; Siddiq, M.; et al. Nanomagnet logic: Progress toward system-level integration. J. Phys. Condens. Matter
**2011**, 23, 493202. [Google Scholar] [CrossRef] - Solin, P. Equations of Electromagnetics. 2006; 269–318. [Google Scholar]
- Landau, L.; Lifshitz, E. Theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion
**1935**, 8, 153–169. [Google Scholar] - Khitun, A.; Nikonov, D.E.; Wang, K.L. Magnetoelectric spin wave amplifier for spin wave logic circuits. J. Appl. Phys.
**2009**, 106, 123909. [Google Scholar] [CrossRef] [Green Version] - Fashami, M.S.; Roy, K.; Atulasimha, J.; Bandyopadhyay, S.; Markwitz, A. Magnetization dynamics, Bennett clocking and associated energy dissipation in multiferroic logic. Nanotechnology
**2011**, 22, 155201, Erratum in Nanotechnology**2011**, 22, 309501. [Google Scholar] [CrossRef] - Roy, K.; Bandyopadhyay, S.; Atulasimha, J. Hybrid spintronics and straintronics: A magnetic technology for ultra low energy computing and signal processing. Appl. Phys. Lett.
**2011**, 99, 063108. [Google Scholar] [CrossRef] [Green Version] - Allwood, D.A.; Xiong, G.; Faullkner, C.C.; Atkinson, D.; Petit, D.; Cowburn, R.P. Magnetic domain-wall logic. Science
**2005**, 309, 1688–1692. [Google Scholar] [CrossRef] - Parkin, S.S.P.; Hayashi, M.; Thomas, L. Magnetic domain-wall racetrack memory. Science
**2008**, 320, 190–194. [Google Scholar] [CrossRef] [PubMed] - Spedalieri, F.M.; Jacob, A.P.; Nikonov, D.E.; Roychowdhury, V.P. Performance of Magnetic Quantum Cellular Automata and Limitations Due to Thermal Noise. IEEE Trans. Nanotechnol.
**2011**, 10, 537–546. [Google Scholar] [CrossRef] - Covington, M.; Crawford, T.M.; Parker, G.J. Time-resolved measurement of propagating spin waves in ferromagnetic thin films. Phys. Rev. Lett.
**2002**, 89, 237202. [Google Scholar] [CrossRef] [PubMed] - Behin-Aein, B.; Sarkar, A.; Srinivasan, S.; Datta, S. Switching energy-delay of all spin logic devices. Appl. Phys. Lett.
**2011**, 98, 123510. [Google Scholar] [CrossRef]

**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.

**Figure 2.**Results of numerical modeling showing the normalized magnetization M

_{Y}/M

_{S}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.

**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 × 10

^{4}m/s.

Domain Wall | MCA | Spin Wave | ASL | Multiferroics | |
---|---|---|---|---|---|

Mechanism of coupling | Domain wall motion | Dipole–dipole coupling | Spin waves | Spin polarized current | Magnetization signal |

Speed of propagation | 10^{2} m/s | 10^{3} m/s | 10^{4} m/s–10^{5} m/s | * 10^{5} m/s | * 10^{5} m/s |

Energy dissipated per bit transmitted | >1000 aJ | ** 1 aJ | 0.1 aJ | N/A | 1 aJ |

Main advantage | Non-volatile, can be stopped at any time and preserve its position | Internal dissipated energy approaches zero at the adiabatic switching | Computation in wires—additional functionality via wave interference | Scalable, defect tolerant | Fast signal propagation, signal amplification |

Main disadvantage | Slow and energy consuming | Effect of thermal noise increases with the propagation distance | Propagation distance is limited due to the spin-wave damping | Propagation distance is limited by the spin diffusion length | Limited scalability |

^{3}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).

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**MDPI and ACS Style**

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

**AMA Style**

Khitun A.
Magnetic Interconnects Based on Composite Multiferroics. *Micromachines*. 2022; 13(11):1991.
https://doi.org/10.3390/mi13111991

**Chicago/Turabian Style**

Khitun, Alexander.
2022. "Magnetic Interconnects Based on Composite Multiferroics" *Micromachines* 13, no. 11: 1991.
https://doi.org/10.3390/mi13111991