**3. Electric control of nanomagnetic logic gate**

The previous section introduced the electric field regulation of a single nanomagnet, and this section will continue to discuss the electric field control method for nanomagnet arrays. Information transmission and calculations in nanomagnetic logic rely on the control of nanomagnet array. The problem of efficient information transmission is well solved [6]. However, electric-controlled magnetic logic gate is still a major challenge. Imre et al*.* used five single-axis nanomagnets to build a majority logic gate [2], which made nanomagnetic logic possible. However, this logic gate requires multiple clock controls to ensure correct logic calculations. Gypens et al*.* used 19 dipole-coupled uniaxial nanomagnets to form a stable system and built a NAND (NOR) logic gate that can be accurately calculated [23]. However, this solution requires more nanomagnets, which increases the NML area. Roy uses a multiiron material to propose an ultra-low-energy NAND (OR) logic gate based on a magnetic tunnel junction [24]. However, this logic gate design requires casting multiple layers of materials, which increases the difficulty of manufacturing. Niemier et al*.* put forward a long axis tilted nanomagnet structure by using an edge-slanted nanomagnet and designed dual-input AND/OR logic gates based on it. Most studies now use this type of edge-slanted nanomagnet to achieve long axis tilted nanomagnet structures. However, there are three defects in edge-slanted nanomagnets: (1) This type of nanomagnet requires a larger size, thus increasing the NML space and introducing clock errors of the C-shape and eddy current that easily occur in large-sized nanomagnets. (2) Complex calculations caused by the irregular shape are inevitable. (3) More importantly, the irregular shape of nanomagnet increases the requirements of fabrication process.

From the above perspective, a more effective and more reliable design of basic magnetic logic gates is required to be proposed. The design should address two key issues: (1) how to eliminate C-shaped and eddy current clock errors and (2) how to reduce the complexities of calculations and fabrication process.

#### **3.1 Design and analysis**

In the previous section, the long axis tilted nanomagnet is introduced. As shown in **Figure 8(a)**, the long axis and short axis of the nanomagnet rotate from the *x* axis and *y* axis to the *x'* axis and *y'* axis, respectively. If the tilt angle that long axis makes with the direction of the electrodes is *β*, the included angles between long axis and the clock will be a larger one (90° + *β*) and a smaller one (90° *β*). When driven by no other energy, the nanomagnet will flip toward the smaller angle after the stress is released. This is because the nanomagnet has higher anisotropy along the clock than that along the long axis and will spontaneously flip to the shape anisotropy potential

(*β* = 5°) of the nanomagnet are set to eliminate the C-shaped and eddy current clock errors. As shown in the inset, the demagnetization energy curve of the tilted nanomagnet is shifted 5° to the left, where logic "1" and "0" correspond to 85° and 265°, respectively, while "NULL" (high energy state) states correspond to 175° and 355°. If the initial clock of the nanomagnet is pointing right (*φ* = 0 or 360°), after the stress is released, the tilted nanomagnet will flip counterclockwise to the side that is at a smaller angle to the long axis, which is the +*y'* direction (85°). This is because the nanomagnet needs to cross the right shape anisotropy barrier of the hard axis (see the purple box shown in **Figure 8(b)**) when turning clockwise to the *y'* direction (265°), whereas when turning counterclockwise, it is not necessary to cross the barrier. Thus the nanomagnet will flip counterclockwise to the +*y'* direction, yielding logic "1." **Figure 8(c)** gives the OOMMF simulations of the preferred magnetization of the nanomagnet with initial clock pointing left or right. As shown in the inset, if the initial state is pointing left, the tilted nanomagnet will rotate counterclockwise to logic "0," whereas if the initial state is pointing right, it will

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet*

Based on the preferred magnetization of tilted nanomagnet, a design of dualinput AND/OR magnetic logic gates is proposed, as shown in **Figure 9**. This design is composed of two input nanomagnets A and B, as well as one output tilted nanomagnet Out (clinched 5° clockwise), interacting via ferromagnetic coupling. The magnetization direction of the magnet Out is influenced by the ferromagnetic coupling of the input magnets A and B as well as its own preferred magnetization. As shown in **Figure 9(a)**, if the initial state is pointing left, the nanomagnet Out tends to flip to logic "0." As a consequence, when the inputs A and B are "01," "00," or "10," the output magnet rotates counterclockwise to logic "0," whereas when the inputs A and B are both "1," the output magnet rotates clockwise to logic "1," thereby yielding AND logic. If the initial state is pointing right, as shown in **Figure 9(b)**, the nanomagnet Out tends to flip to logic "1," so when the input

magnets A and B are "01," "11," or "10," the output magnet rotates

*(a) and (b) show the design of (A) AND logic gate and (B) OR logic gate based on tilted nanomagnet.*

*The initial magnetization of magnet Out is pointing left in (a) and right in (b).*

rotate counterclockwise to logic "1."

*DOI: http://dx.doi.org/10.5772/intechopen.91231*

**Figure 9.**

**29**

#### **Figure 8.**

*(a) The nanomagnet is rotated clockwise by a small angle* β*. (b) the demagnetization energy is calculated as a function of* φ*. (c) the preferred magnetization is simulated by OOMMF.*

well of the long axis. However, in the course of flipping toward the larger angle (90° + *β*), it is necessary to cross the shape anisotropy barrier of the hard axis. As a consequence, the nanomagnet tends to flip toward the smaller angle (90° *β*) without the need of crossing the shape anisotropy barrier of the hard axis.

As shown in **Figure 8(b)**, for a nanomagnet with a tilt angle *β* = 5°, the demagnetization energy is calculated by OOMMF, as a function of *φ*. For the parameters, the authors have assumed a space size of 80 nm 100 nm 20 nm, mesh size of 2 nm 2 nm 2 nm, magnet dimensions of 50 nm 100 nm 20 nm, saturation magnetization of 800 kA/m, Gilbert damping constant of 0.5, and zero magnetocrystalline anisotropy. The high aspect ratio (2:1) and the small tilt angle

#### *Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet DOI: http://dx.doi.org/10.5772/intechopen.91231*

(*β* = 5°) of the nanomagnet are set to eliminate the C-shaped and eddy current clock errors. As shown in the inset, the demagnetization energy curve of the tilted nanomagnet is shifted 5° to the left, where logic "1" and "0" correspond to 85° and 265°, respectively, while "NULL" (high energy state) states correspond to 175° and 355°. If the initial clock of the nanomagnet is pointing right (*φ* = 0 or 360°), after the stress is released, the tilted nanomagnet will flip counterclockwise to the side that is at a smaller angle to the long axis, which is the +*y'* direction (85°). This is because the nanomagnet needs to cross the right shape anisotropy barrier of the hard axis (see the purple box shown in **Figure 8(b)**) when turning clockwise to the *y'* direction (265°), whereas when turning counterclockwise, it is not necessary to cross the barrier. Thus the nanomagnet will flip counterclockwise to the +*y'* direction, yielding logic "1." **Figure 8(c)** gives the OOMMF simulations of the preferred magnetization of the nanomagnet with initial clock pointing left or right. As shown in the inset, if the initial state is pointing left, the tilted nanomagnet will rotate counterclockwise to logic "0," whereas if the initial state is pointing right, it will rotate counterclockwise to logic "1."

Based on the preferred magnetization of tilted nanomagnet, a design of dualinput AND/OR magnetic logic gates is proposed, as shown in **Figure 9**. This design is composed of two input nanomagnets A and B, as well as one output tilted nanomagnet Out (clinched 5° clockwise), interacting via ferromagnetic coupling. The magnetization direction of the magnet Out is influenced by the ferromagnetic coupling of the input magnets A and B as well as its own preferred magnetization. As shown in **Figure 9(a)**, if the initial state is pointing left, the nanomagnet Out tends to flip to logic "0." As a consequence, when the inputs A and B are "01," "00," or "10," the output magnet rotates counterclockwise to logic "0," whereas when the inputs A and B are both "1," the output magnet rotates clockwise to logic "1," thereby yielding AND logic. If the initial state is pointing right, as shown in **Figure 9(b)**, the nanomagnet Out tends to flip to logic "1," so when the input magnets A and B are "01," "11," or "10," the output magnet rotates

#### **Figure 9.**

*(a) and (b) show the design of (A) AND logic gate and (B) OR logic gate based on tilted nanomagnet. The initial magnetization of magnet Out is pointing left in (a) and right in (b).*

counterclockwise to logic "1," whereas when inputs A and B are both "0," the output magnet rotates clockwise to logic "0," yielding OR logic.

For magnet Out, whose magnetization is interacted by inputs A and B, the dipole–dipole interaction energy writes [7]:

$$E\_{\text{dipole}} = \frac{\mu\_0 M\_s^2 V^2}{4\pi R^3} \left[ \begin{pmatrix} -2\cos\varphi\_A \sin\theta\_A \cos\varphi \sin\theta\\ +\sin\varphi\_A \sin\theta\_A \sin\varphi \sin\theta + \cos\theta\_A \cos\theta \end{pmatrix} \right.$$

$$+ \left. \begin{pmatrix} -2\cos\varphi\_B \sin\theta\_B \cos\varphi \sin\theta\\ +\sin\varphi\_B \sin\theta\_B \sin\varphi \sin\theta + \cos\theta\_B \cos\theta \end{pmatrix} \right] \tag{23}$$

where *R* is the separation between the centers of neighbor nanomagnets and the magnetization angles of the input magnets are labeled with subscripts A and B.

#### **3.2 Results and discussions**

Only OR logic gate is discussed in this section. For shape symmetry, the results will be same for AND logic gate; on account of which, it is not discussed here for clarity. In order to obtain OR logic gate, an initial clock pointing right is necessary. However, whether the clock direction is pointing left or right cannot be controlled simply by the stress. The magnetization vector only tends to be perpendicular to where the stress is applied. Fortunately, for the nanomagnet tilted clockwise by 5°, the direction of initial clock will be determined by the initial magnetization direction of the nanomagnet. As mentioned in Section II, there is no need of crossing the hard axis barrier for the magnet when flipping clockwise. As a consequence, a nanomagnet whose initial state is logic "1" (*φ* = 90°) tends to flip clockwise under the stress applied in the *y* direction. It is worth mentioning that if it is not possible to know the initial state of the tilted nanomagnet, a clock pointing right can be obtained by adding a biasing magnetic field pointing right (a stress of 45 MPa and a bias magnetic field of 500 Oe).

The authors assume that the initial state of the nanomagnet Out is logic "1" (*φ* = 90°, *θ* = 90°). A stress of 90 MPa is applied to nanomagnet Out for 3 ns. As shown in **Figure 10(a)–(d)**, the nanomagnet flips to "NULL" after the stress has been applied for 1.8 ns. Note that the magnetization of "NULL" state here does not exactly correspond to *φ* = 0. Rigorously, it makes a certain angle (*φ* = 7°) with the *x* axis. This is because the stress field component in the –*y* direction and the field component of shape anisotropy in the +*y* direction yield a stable equilibrium, so that the magnetization vector of magnet Out is stably deviated from the *x* axis. If *φ* < 10°, the field component of shape anisotropy energy in the +*y* direction is much smaller than the stress field component, thus not affecting calculation result. Inputs "00," "01," "10," and "11" are read in at 2.9 ns. After the stress has been released for 0.9 ns (*t* = 3.9 ns), magnet Out will flip to a stable logic state. When the inputs are "01," "10," and "11," magnet Out is logic "1" (*φ* = 88°), whereas when the inputs are "00," the magnet Out is logic "0" (*φ* = �92°), successfully yielding OR logic. Note that the nanomagnet Out does not flip to the long axis (*φ* = 85° or *φ* = �95°) under the interaction of the ferromagnetic coupling of the input nanomagnets.

The input nanomagnets A and B only produce small fluctuations (�2°) in the plane and eventually return to the original logic state (*φ* = 90° or *φ* = �90°) under the interaction of the ferromagnetic coupling of nanomagnet Out. The angular variations of *θ* are similar in the four situations. Situation of inputs "10" is specially shown in **Figure 10(e)** and **(f)**. The polar angles (out-of-plane) of initial and final states of the three magnets are all *θ* = 90°. Magnets A and B produce smaller fluctuations (�2°) than magnet Out (�33°), as shown in **Figure 10(e)**.

The results confirm that magnets A and B will remain stable during the switching of magnet Out. The magnetization track of the magnet Out presents two obvious

*Apply a stress of 90 MPa to magnet out for 3 ns. (a)–(d) show dynamic magnetization of the azimuth angle* φ *of (a) input "00," output "0"; (b) input "01," output "0"; (c) input "10," output "0"; and (d) input "11," output "1." when the input is "10," (e) and (f) show (e) dynamic magnetization of the polar angle* θ *and*

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet*

*DOI: http://dx.doi.org/10.5772/intechopen.91231*

**Figure 11** shows the simulation of our design of OR logic gate calculated by OOMMF using the data in **Table 1**. The other parameters are set as follows: space

size = 800 nm 200 nm 20 nm, and mesh size = 5 nm 5 nm 5 nm.

energy states, as can be seen from **Figure 10(f)**.

*(f) magnetization track of the nanomagnet out.*

**Figure 10.**

**31**

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet DOI: http://dx.doi.org/10.5772/intechopen.91231*

#### **Figure 10.**

*Apply a stress of 90 MPa to magnet out for 3 ns. (a)–(d) show dynamic magnetization of the azimuth angle* φ *of (a) input "00," output "0"; (b) input "01," output "0"; (c) input "10," output "0"; and (d) input "11," output "1." when the input is "10," (e) and (f) show (e) dynamic magnetization of the polar angle* θ *and (f) magnetization track of the nanomagnet out.*

The results confirm that magnets A and B will remain stable during the switching of magnet Out. The magnetization track of the magnet Out presents two obvious energy states, as can be seen from **Figure 10(f)**.

**Figure 11** shows the simulation of our design of OR logic gate calculated by OOMMF using the data in **Table 1**. The other parameters are set as follows: space size = 800 nm 200 nm 20 nm, and mesh size = 5 nm 5 nm 5 nm.

**Figure 11.** *Simulation results of OR logic gate by OOMMF.*

The initial clock is pointing right and the inputs are "10," "01," "00," and "11." Only when the inputs are "00," the output becomes "0"; otherwise the output is "1," yielding OR logic as expected.

Unlike designs based on slanted nanomagnet, basic logic gates based on tilted nanomagnet have three advantages: (1) This tilted magnet design allows high aspect ratio (2:1) nanomagnets to be used; as a consequence of which, less C-shaped and eddy current clock errors will occur; (2) regular-shaped tilted nanomagnet reduces the requirements of fabrication process; and (3) the regular shape provides great convenience in numerical calculation.

#### **3.3 Conclusion**

In this section, a design of AND/OR logic gates is proposed based on tilted placement of nanomagnet. The mathematical model of the design is established, and the correctness is verified by the OOMMF software. This scheme can provide a more efficient and reliable basic logic unit for NML design. However, in the experimental preparation, there may be fabrication errors in tilting the placement of the nanomagnet. To reduce the process fabrication error, stress electrodes may be tilted so that the stress will also make an angle with the long axis of the nanomagnet.

## **4. Conclusions**

In this chapter, the multiferroic heterojunction is introduced into the field of spintronics. By utilizing the inverse piezoelectric effect and the inverse magnetostrictive effect in the multiferroic heterojunction, the weak electric field can be used to accurately synchronize the storage and processing of the magnetic logic signal of the uniaxial nanomagnet. Multiferroic nanomagnets are considered to be a strong competitor for post-CMOS devices due to their natural nonvolatility, high radiation resistance, and ultra-low power consumption. In this chapter, the multiferroic nanomagnet device is taken as the research object, and the research on the two key problems of fast nanomagnet rapid reversal magnetization reversal and nanomagnetic logic gate is carried out. The research results have great innovation and application background.

**Author details**

**33**

Jiahao Liu\* and Liang Fang\*

Technology, Changsha, China

provided the original work is properly cited.

Institute for Quantum Information, State Key Laboratory of High

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet*

*DOI: http://dx.doi.org/10.5772/intechopen.91231*

Performance Computing, College of Computer, National University of Defense

\*Address all correspondence to: rae20121220@163.com and lfang@nudt.edu.cn

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

### **Acknowledgements**

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 61832007) and the National Key R&D Program of China (Grant No. 2018YFB1003304).

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet DOI: http://dx.doi.org/10.5772/intechopen.91231*
