**1. Introduction**

Using the inverse piezoelectric effect and inverse magnetostrictive effect in a multiferroic heterojunction, an electric field is able to control the magnetization switching of a uniaxial nanomagnet. **Figure 1** is a multiferroic heterojunction structure, that is, a two-layer magnetoelectric composite structural system, which is formed by magnetoelastic coupling of a magnetostrictive layer and a piezoelectric layer. The electric field-controlled nanomagnet in a multiferroic heterojunction essentially uses multi-field coupling of "electric-stress-magnetic." Applying a small voltage to the piezoelectric layer, the piezoelectric layer will produce uniaxial strain, which is transformed into a stress applying on the magnetostrictive layer by magnetoelastic coupling, causing the magnetization direction of the magnetostrictive layer to rotate perpendicular to the stress. If the magnetostrictive layer is constructed as a uniaxially shaped nanomagnet, the strain will reverse the magnetization direction of the nanomagnet to a logically "NULL" state, pointing to the hard magnetization axis, which is, the short axis direction. At this time, if the voltage is released (stress revocation), the nanomagnet will flip to a certain stable logic state (original logic state or opposite logic state). This magnetic logic device based on the regulation mode of the multiferroic heterojunction magnetoelectric effect is called

**Figure 1.** *Multiferroic heterojunction structure.*

a multiferroic nanomagnet device. Multiferroic nanomagnet device is one of the most competitive spintronic devices due to its low energy consumption and high thermal stability [1]. It represents Boolean logic "0" and "1" in the magnetization directions along the long axis of a uniaxial nanomagnet [2] and can be corresponding to different states in the magnetic tunnel junction [3].

### **2. Voltage pulse-induced magnetization switching**

The key to using the stress generated by the electric field to control the magnetization switching of the multiferroic nanomagnet is that the stress anisotropy must be larger than the shape anisotropy of the nanomagnet. An effective method to reduce the required stress anisotropy is to break the symmetry of the shape of the nanomagnet by slightly tilting the long axis of the nanomagnet to the direction of stress application. However, the effect of the tilt angle on the magnetization reversal of the nanomagnet is still inconclusive.

#### **2.1 Model**

**Figure 2** presents the voltage-controlled multiferroic heterostructure. The red arrow indicates the direction of magnetization. The polar angle (out-of-plane) and the azimuth angle (in-plane) of the magnetization are *θ* and *φ*, respectively. Note that the magnet is at an angle to the direction of the electrodes.

The magnetization dynamic of a single elliptical nanomagnet meets the Landau-Lifshitz-Gilbert Eq. (5):

$$\frac{d\overrightarrow{\mathbf{M}}}{dt} = -\gamma \overrightarrow{\mathbf{M}} \times \overrightarrow{\mathbf{H}\_{\text{eff}}} - \frac{\alpha \gamma}{M\_\circ} \left[ \overrightarrow{\mathbf{M}} \times \left( \overrightarrow{\mathbf{M}} \times \overrightarrow{\mathbf{H}\_{\text{eff}}} \right) \right] \tag{1}$$

where *α* is the damping coefficient, *M* �! is the magnetic moment vector of the nanomagnet, *Ms* is the saturation magnetization, *γ* is the return ratio, and [5]:

$$
\overrightarrow{\mathbf{H}\_{\text{eff}}} = -\frac{1}{\mu\_0 V} \frac{\mathbf{d}E}{\mathbf{d}\vec{\mathbf{M}}} \tag{2}
$$

is the effective field generated by a variety of energies (shape anisotropy energy, stress anisotropy energy, Zeeman energy, and thermal fluctuations), where *μ*<sup>0</sup> ¼ <sup>4</sup><sup>π</sup> � <sup>10</sup>�<sup>7</sup> is the vacuum permeability and *<sup>V</sup>* is the volume of each element. The stress is applied at the *y* direction, and the total energy *E*total is the sum of demagnetization energy, exchange energy, shape anisotropy energy, stress anisotropy

*E*total ¼ *E*demagnetization þ *E*exchange þ *E*shape�anisotropy þ *E*stress�anisotropy þ *E*dissipation

For Terfenol-D as the magnetic material, the crystal anisotropy energy of is small, and thus is ignored in the calculation of the total energy. The exchange energy can also be neglected in the single domain particles of 100 nm � 50 nm � 20 nm [6]. The shape anisotropy energy of the nanomagnet can be written as [7]:

ð

is the magnetic moment vector of the nanomagnet and *H<sup>M</sup>*

^*<sup>i</sup>* � *NdyMy*

�ð Þ *μ*0*=*2 *M*

*\**

� *H<sup>M</sup> \**

^*<sup>j</sup>* � *NdzMz*

^

*E*shape�anisotropy ¼

¼ �*NdxMx*

anisotropy energy field, which can be expressed as [7]:

*H<sup>M</sup> \**

(3)

is the shape

d*V* (4)

*\**

*k* (5)

energy, and energy dissipation:

*Stress-regulated multiferroic tilted nanomagnet device [4].*

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet*

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

**Figure 2.**

where *M \**

**19**

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

**Figure 2.** *Stress-regulated multiferroic tilted nanomagnet device [4].*

is the effective field generated by a variety of energies (shape anisotropy energy, stress anisotropy energy, Zeeman energy, and thermal fluctuations), where *μ*<sup>0</sup> ¼ <sup>4</sup><sup>π</sup> � <sup>10</sup>�<sup>7</sup> is the vacuum permeability and *<sup>V</sup>* is the volume of each element. The stress is applied at the *y* direction, and the total energy *E*total is the sum of demagnetization energy, exchange energy, shape anisotropy energy, stress anisotropy energy, and energy dissipation:

$$E\_{\text{total}} = E\_{\text{demagnezization}} + E\_{\text{exchange}} + E\_{\text{shape}-\text{anisotropy}} + E\_{\text{stress}-\text{anisotropy}} + E\_{\text{dissipation}} \tag{3}$$

For Terfenol-D as the magnetic material, the crystal anisotropy energy of is small, and thus is ignored in the calculation of the total energy. The exchange energy can also be neglected in the single domain particles of 100 nm � 50 nm � 20 nm [6]. The shape anisotropy energy of the nanomagnet can be written as [7]:

$$E\_{\text{shape}-\text{anisotropy}} = \int - (\mu\_0/2) \, \overrightarrow{\mathbf{M}} \cdot \overrightarrow{\mathbf{H}}\_M \, \mathrm{d}V \tag{4}$$

where *M \** is the magnetic moment vector of the nanomagnet and *H<sup>M</sup> \** is the shape anisotropy energy field, which can be expressed as [7]:

$$\overrightarrow{\mathbf{H}}\_{M} = -\mathbf{N}\_{dx}\mathbf{M}\_{x}\hat{\mathbf{i}} - \mathbf{N}\_{dy}\mathbf{M}\_{y}\hat{\mathbf{j}} - \mathbf{N}\_{dx}\mathbf{M}\_{x}\hat{\mathbf{k}}\tag{5}$$

$$E\_{\text{shape}-\text{anisotropy}} = - (\mu\_0/2) \left( M\_s^2 V \right) \left[ N\_{dx} \cos^2 \theta \sin^2 \varphi + N\_{dy} \sin^2 \theta \sin^2 \varphi + N\_{dz} \cos^2 \varphi \right] \tag{6}$$

where *Nd* is the demagnetization factor. For elliptical shaped nanomagnets, the demagnetization factors *Ndx*, *Ndy*, and *Ndz* can be calculated through [7]:

$$N\_{dx} = \frac{\pi}{4} \left( \frac{th}{a} \right) \left[ 1 + \frac{5}{4} \left( \frac{a-b}{a} \right) + \frac{21}{16} \left( \frac{a-b}{a} \right)^2 \right] \tag{7}$$

$$N\_{d\eta} = \frac{\pi}{4} \left( \frac{th}{a} \right) \left[ 1 - \frac{1}{4} \left( \frac{a-b}{a} \right) - \frac{3}{16} \left( \frac{a-b}{a} \right)^2 \right] \tag{8}$$

$$\mathbf{N}\_{\rm dz} = \mathbf{1} - \mathbf{N}\_{\rm dx} - \mathbf{N}\_{\rm dy} \tag{9}$$

where *a* is the length of the long axis, *b* is the length of the short axis, and *th* is the thickness of the nanomagnet. For a nanomagnet whose tilt angle is *β*, as shown in **Figure 2**, the short axis and long axis of the nanomagnet rotate clockwise from the *x* axis and *y* axis to the *x'* axis and *y'* axis, respectively, and *z'* axis (not shown) is still at vertical direction. The shape anisotropy field components in the new coordinate system are:

$$h'\_{\text{shape}-\text{anisotropy},\text{xx}} = -M\_\text{s}N\_{d\text{x}}\cos\left(\rho + \beta\right)\sin\theta\tag{10}$$

$$h'\_{\text{shape}-\text{anisotrop}\_{-\text{YY}}} = -M\_i N\_{dj} \sin \left(\varphi + \beta \right) \sin \theta \tag{11}$$

$$h'\_{\text{shape}-\text{anisotropy}\_{-}\text{zz}} = -M\_s N\_{dx} \cos \theta \tag{12}$$

By the coordinate rotation conversion of (10)–(12), the field components of shape anisotropy of the tilted nanomagnet on the original coordinate axes become:

$$h\_{\text{shape}\text{---anistor\\_xx}} = -M\_{\varepsilon} \left( N\_{d\text{x}} \cos \left( \varphi + \beta \right) \cos \beta + N\_{d\text{y}} \sin \left( \varphi + \beta \right) \sin \beta \right) \sin \theta \quad \text{(13)}$$

$$h\_{\text{shape}-\text{anisotrop},\text{yy}} = -\mathbf{M}\_{\text{s}} \left( -\mathbf{N}\_{\text{dx}} \cos \left( \boldsymbol{\rho} + \boldsymbol{\beta} \right) \sin \boldsymbol{\beta} + \mathbf{N}\_{\text{dy}} \sin \left( \boldsymbol{\rho} + \boldsymbol{\beta} \right) \cos \boldsymbol{\beta} \right) \sin \boldsymbol{\theta}$$

(14)

where *<sup>k</sup>* = 1.38 � <sup>10</sup>�<sup>23</sup> J/K is the Boltzmann constant,*<sup>T</sup>* = 300 K is the room temperature, *f* = 1 GHz is the frequency of thermal noise oscillations, and *G*ð Þ 0,1 represents a Gaussian function with a mean of 0 and a variance of 1. By combining

be involved in the *x* direction if it is required. The components of each coordinate

*hx* ¼ �*Ms Ndx* cosð Þ *<sup>φ</sup>* <sup>þ</sup> *<sup>β</sup>* cos *<sup>β</sup>* <sup>þ</sup> *Ndy* cosð Þ *<sup>φ</sup>* <sup>þ</sup> *<sup>β</sup>* sin *<sup>β</sup>* � � sin *<sup>θ</sup>*

*hy* ¼ �*Ms Ndx* cosð Þ *<sup>φ</sup>* <sup>þ</sup> *<sup>β</sup>* sin *<sup>β</sup>* <sup>þ</sup> *Ndy* sin ð Þ *<sup>φ</sup>* <sup>þ</sup> *<sup>β</sup>* cos *<sup>β</sup>* � � sin *<sup>θ</sup>*

s

Biswas et al*.* used two pairs of electrodes to control the nanomagnet in the experiment to achieve a reliable 180° switching [10, 11]. However, since the two pairs of electrodes have to be operated in sequence, the nanomagnet needs a longer switching time. Fashami used a timed pulse to switch the nanomagnet by 180°, which is error-free and dissipates arbitrarily small energy [12]. However, in this scheme, a hard magnet is essential to break the energy symmetry, and a long switching time is required. Recently, a method of 180° switching has been proposed, in which a repeatable 180° nanomagnet switching was induced by voltage pulses. By setting suitable amplitude, width, and period of the voltage pulse, it is possible to achieve repeatable 180° switchings without a magnetic field [13, 14]. However, although this solution can achieve repeatable magnetic switching, the first switching requires a large start-up time, making the first switching time much longer [15, 16]. In magnetic storage and logic application, the first switching is most often needed. More importantly, these studies did not consider the thermal fluctuations, which play an important role in the switching of the nanomagnet. In conclusion, fast switching of nanomagnets at room temperature is still a challenge for straintronics in the application of logic storage and computing. This section introduces a fast switching method of nanomagnets at room temperature. The structure is shown in **Figure 1** of the previous section. The authors use OOMMF software to

The authors chose PMN-PT (Pb(Mg1/3Nb2/3)O3-PbTiO3) as the piezoelectric layer

material to use its higher piezoelectric coefficient [17, 18]. And for the magnetic material, the authors chose Terfenol-D (Tb0.7Dy0.3Fe2), because the magnetocrystalline

Since (Object Oriented Micromagnetic Framework) software [20] cannot directly set the stress anisotropy energy, the authors use the uniaxial anisotropy energy acting in the direction of (�cos*β* sin*β* 0) for replace. Accordingly [6]:

> *<sup>K</sup>* ¼ � <sup>3</sup> 2

*λsσV* (22)

anisotropy can be smaller [19]. The parameters are shown in **Table 1**.

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*αkTf γμ*0*MsV*

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*αkTf γμ*0*MsV*

Gð Þ 0,1 ðÞþ*t hbias*

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet*

*hz* ¼ �*MsNdz* cos *θ* þ

��! can be obtained. A bias field can also

*G*ð Þ 0,1 ð Þ*t* (20)

*G*ð Þ 0,1 ð Þ*t* (21)

(19)

the above functions, the effective field *Heff*

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*αkTf γμ*0*MsV*

þ 3*λs=Msμ*<sup>0</sup> ð Þ*σ* sin *θ* sin *φ* þ

simulate and study the switching of nanomagnets.

þ

**2.2 Results and discussions**

s

axis are:

**21**

$$h\_{\text{shape}-\text{anisotropy}\_{-}\text{xz}} = -M\_{\text{s}}N\_{d\text{dx}}\cos\theta\tag{15}$$

The stress anisotropy energy of the nanomagnet is given by [7]:

$$E\_{\text{stress}-\text{anisotropy}} = -\frac{3}{2}\lambda\_i \sigma V \sin^2 \theta \sin^2 \rho \tag{16}$$

where 3*λs*/2 is the saturation magnetostriction and the stress *σ* is considered negative for compression and positive for tension. The stress is applied in the *y* direction, so there is only a field component in the *y* axis direction [8]:

$$h\_{\text{stress}-\text{anisotropy}\_{-\text{YY}}} = (\Im \lambda\_{\text{s}} / \mathcal{M}\_{\text{s}} \mu\_{\text{0}}) \sigma \sin \theta \sin \rho \tag{17}$$

Considering the thermal fluctuations, the effect of random thermal fluctuations can be described by a random thermal field [9]:

$$h(t) = \sqrt{\frac{2akT\hat{f}'}{\gamma\mu\_0\mathcal{M}\_sV}}\mathcal{G}\_{(0,1)}(t) \tag{18}$$

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

where *<sup>k</sup>* = 1.38 � <sup>10</sup>�<sup>23</sup> J/K is the Boltzmann constant,*<sup>T</sup>* = 300 K is the room temperature, *f* = 1 GHz is the frequency of thermal noise oscillations, and *G*ð Þ 0,1 represents a Gaussian function with a mean of 0 and a variance of 1. By combining the above functions, the effective field *Heff* ��! can be obtained. A bias field can also be involved in the *x* direction if it is required. The components of each coordinate axis are:

$$\begin{split} h\_{\rm x} &= -M\_{\rm t} \big( N\_{\rm dx} \cos \left( \wp + \beta \right) \cos \beta + N\_{\rm dy} \cos \left( \wp + \beta \right) \sin \beta \big) \sin \theta \\ &+ \sqrt{\frac{2 \alpha k T \mathcal{f}}{\chi \mu\_{0} \mathcal{M}\_{\rm s} V}} \mathcal{G}\_{(0,1)}(t) + h\_{\rm bias} \end{split} \tag{19}$$

$$\begin{split} h\_{\gamma} &= -\mathsf{M}\_{\mathfrak{s}} \Big( \mathsf{N}\_{\mathrm{dx}} \cos \left( \wp + \beta \right) \sin \theta + \mathsf{N}\_{\mathrm{dy}} \sin \left( \wp + \beta \right) \cos \theta \Big) \sin \theta \\ &+ \left( \mathsf{3} \lambda\_{\mathfrak{s}} / \mathsf{M}\_{\mathfrak{s}} \mu\_{0} \right) \sigma \sin \theta \sin \varphi + \sqrt{\frac{2 \mathrm{ad} T f}{\chi \mu\_{0} \mathrm{M}\_{\mathfrak{s}} V}} G\_{(0, 1)}(t) \end{split} \tag{20}$$

$$h\_x = -M\_s N\_{dx} \cos \theta + \sqrt{\frac{2\alpha k T \overline{f}}{\gamma \mu\_0 M\_s V}} G\_{(0,1)}(t) \tag{21}$$

#### **2.2 Results and discussions**

Biswas et al*.* used two pairs of electrodes to control the nanomagnet in the experiment to achieve a reliable 180° switching [10, 11]. However, since the two pairs of electrodes have to be operated in sequence, the nanomagnet needs a longer switching time. Fashami used a timed pulse to switch the nanomagnet by 180°, which is error-free and dissipates arbitrarily small energy [12]. However, in this scheme, a hard magnet is essential to break the energy symmetry, and a long switching time is required. Recently, a method of 180° switching has been proposed, in which a repeatable 180° nanomagnet switching was induced by voltage pulses. By setting suitable amplitude, width, and period of the voltage pulse, it is possible to achieve repeatable 180° switchings without a magnetic field [13, 14]. However, although this solution can achieve repeatable magnetic switching, the first switching requires a large start-up time, making the first switching time much longer [15, 16]. In magnetic storage and logic application, the first switching is most often needed. More importantly, these studies did not consider the thermal fluctuations, which play an important role in the switching of the nanomagnet. In conclusion, fast switching of nanomagnets at room temperature is still a challenge for straintronics in the application of logic storage and computing. This section introduces a fast switching method of nanomagnets at room temperature. The structure is shown in **Figure 1** of the previous section. The authors use OOMMF software to simulate and study the switching of nanomagnets.

The authors chose PMN-PT (Pb(Mg1/3Nb2/3)O3-PbTiO3) as the piezoelectric layer material to use its higher piezoelectric coefficient [17, 18]. And for the magnetic material, the authors chose Terfenol-D (Tb0.7Dy0.3Fe2), because the magnetocrystalline anisotropy can be smaller [19]. The parameters are shown in **Table 1**.

Since (Object Oriented Micromagnetic Framework) software [20] cannot directly set the stress anisotropy energy, the authors use the uniaxial anisotropy energy acting in the direction of (�cos*β* sin*β* 0) for replace. Accordingly [6]:

$$K = -\frac{3}{2}\lambda\_\text{\prime}\sigma V\tag{22}$$


**Figure 3.**

**Figure 4.**

**23**

*magnetization. (b) Voltage pulse waveform.*

*pair axis [22].*

*Minimum stress required for the nanomagnet to rotate by more than 90° with different tilt angles* β *of electrodes'*

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet*

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

*180° switching with the stress electrodes' pair axis along the long axis of the nanomagnet. (a) Dynamic*

#### **Table 1.**

*Parameters of multiferroic heterostructure.*

The size of nanomagnet is 51 nm 102 nm 21 nm. The selection of large aspect ratio and thickness can reduce C-shaped and eddy vortex errors [21]. The mesh size of OOMMF is 3 nm 3 nm 3 nm. Magnetization toward up and down is defined as logic "1" and "0," respectively. The initial state of the nanomagnet is assumed as logic "1."

Before studying voltage pulse-induced 180° switching, the first step is to ensure that the magnetization direction of the nanomagnet is able to rotate by more than 90° (below *x* axis). **Figure 3** shows that minimum stress is required for the nanomagnet to rotate by more than 90° when the stress is applied in different directions (0 < *β* < 10°) [22]. A small *β* can reduce the required stress, which makes it easier for the nanomagnet to rotate by more than 90°. However, as *β* increases, the required stress also increases. This is because the stress tends to make the magnetization direction perpendicular to the axis of the electrodes pair, i.e., to flip to the *x'* axis. When *β* is larger, the *x'* axis will also make a larger deflection angle with the *x* axis. This makes nanomagnet difficult to rotate by more than 90°. Even so, when 0 < *β* < 7°, the required stress is less than the scheme with the electrodes' pair axis along the long axis of the nanomagnet (*β* = 0).

In the second step, optimal voltage pulse should be set to make the switching time as short as possible. The authors apply a stress of 100 MPa to the nanomagnet (voltage pulse peak of 225 mVs), which is sufficient for the nanomagnet to rotate by more than 90°. **Figure 4** shows the optimal waveform setting and dynamic

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

**Figure 3.**

*Minimum stress required for the nanomagnet to rotate by more than 90° with different tilt angles* β *of electrodes' pair axis [22].*

#### **Figure 4.**

*180° switching with the stress electrodes' pair axis along the long axis of the nanomagnet. (a) Dynamic magnetization. (b) Voltage pulse waveform.*

**Figure 5.**

*Repeated 180° switching with* β *= 5° under room temperature. (a) Dynamic magnetization. (b) Voltage pulse waveform.*

and the minimum interval time,*T* = *t*width + *t*interval = 0.447 ns, and the maximum switching frequency *f* = 1/*T* = 2.110 GHz. The time that the nanomagnet completes the initial switching is the same as the minimum switching cycle: *t*initial = *T* = 0.474 ns.

*Fast 180° switching with* β *= 5°. (a) Dynamic magnetization. (b) Voltage pulse waveform.*

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet*

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

The authors continue to calculate the minimum time and maximum switching frequency for the nanomagnet to complete the 180° switching. The voltage pulse

As shown in **Figure 7**, since there is no start-up time, the minimum initial switching time of the nanomagnet with *β* > 0 is significantly smaller than that of the nanomagnet with *β* = 0. The minimum pulse width decreases as *β* increases. For *β* > 6°, although the minimum pulse width continues to decrease as *β* increases, the minimum interval increases in the meanwhile. When 4° < *β* < 9°, the minimum total initial switching time is small and the maximum switching frequency is also larger than that of **Figure 5** (*β* = 0). Based on the above factors, *β* should be chosen to be around 5°. So nanomagnets will have less required stress, larger switching

Although voltage pulse-induced magnetization switching is very energy efficient, the possibility of operating at room temperature remains to be discussed, which plays an important role in the switching. In this section, the switching of the nanomagnet at room temperature (300 K) is calculated. Since OOMMF software could be computationally expensive and time-consuming to simulate the switching at room temperature, the authors use the mathematical stress model to calculate the

Firstly, the authors apply a stress of 100 MPa to the electrodes and observe the dynamic magnetization of the nanomagnet. The magnetization rotates by more than 90° at 0.1844–0.3470 ns and is most close to logic "0" at 0.2574 ns, meaning that *t*width = 0.2574 ns. If the stress is removed at 0.2574 ns, the nanomagnet will flip to logic "0" at 0.7427 ns, meaning that *t*interval = 0.8856 ns. Obviously, at room

temperature, both *t*width and *t*interval are much larger than that at 0 K.

This is only about 1/5 of **Figure 4**.

**Figure 6.**

**25**

peak is controlled to be a constant 225 mVs.

frequency, and shorter initial switching time.

switching of the nanomagnet at room temperature.

magnetization of the repeatable 180° switchings in the nanomagnet, when the electrodes' pair axis is aligned with the long axis of the nanomagnet (*β* = 0). **Figure 5(a)** is the dynamic magnetization of repeatable 180° switchings in the nanomagnet. It can be seen from the inset that the nanomagnet requires the voltage to be applied for a long period of time (1.6 ns) before it can enter the switching cycle. This is because the nanomagnet has equal probability of reaching either orientation when the stress is applying along the long axis. Therefore, the nanomagnet will enter a magnetization direction selection period before it can be flipped. This start-up time greatly increases the first switching time of the nanomagnet. **Figure 5(b)** shows the optimal voltage pulse waveform for the nanomagnet switching. The minimum start-up time of the nanomagnet *t*start-up = 1.600 ns, the minimum voltage pulse width *t*width = 0.180 ns, and the minimum pulse interval time *t*interval = 0.320 ns. The minimum switching period of the nanomagnet is the sum of the minimum voltage pulse width and the minimum interval time,*T* = *t*width + *t*interval = 0.500 ns, and the maximum switching frequency *f* = 1/*T* = 2.000 GHz. The time that the nanomagnet completes the initial switching is the sum of the minimum start-up time and the minimum switching period: *t*initial = *t*starting + *T* = 2.070 ns.

If the electrodes' pair axis is not aligned with the long axis of the nanomagnet, but is tilted by a small angle *β*, the nanomagnet will have a tendency to select where to flip. For *β* > 0, nanomagnets tend to flip clockwise. This allows the nanomagnet to require no start-up time during the first switching, greatly increasing the efficiency of the initial switching.

**Figure 6** shows the dynamic magnetization of the switchings and optimal voltage pulse waveform when *β* = 5°. The nanomagnet has no start-up time and directly enters the switching cycle. The minimum voltage pulse width *t*width = 0.162 ns, and the minimum pulse interval time *t*interval = 0.312 ns. Therefore, the minimum switching period of the nanomagnet is the sum of the minimum voltage pulse width *Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet DOI: http://dx.doi.org/10.5772/intechopen.91231*

**Figure 6.** *Fast 180° switching with* β *= 5°. (a) Dynamic magnetization. (b) Voltage pulse waveform.*

and the minimum interval time,*T* = *t*width + *t*interval = 0.447 ns, and the maximum switching frequency *f* = 1/*T* = 2.110 GHz. The time that the nanomagnet completes the initial switching is the same as the minimum switching cycle: *t*initial = *T* = 0.474 ns. This is only about 1/5 of **Figure 4**.

The authors continue to calculate the minimum time and maximum switching frequency for the nanomagnet to complete the 180° switching. The voltage pulse peak is controlled to be a constant 225 mVs.

As shown in **Figure 7**, since there is no start-up time, the minimum initial switching time of the nanomagnet with *β* > 0 is significantly smaller than that of the nanomagnet with *β* = 0. The minimum pulse width decreases as *β* increases. For *β* > 6°, although the minimum pulse width continues to decrease as *β* increases, the minimum interval increases in the meanwhile. When 4° < *β* < 9°, the minimum total initial switching time is small and the maximum switching frequency is also larger than that of **Figure 5** (*β* = 0). Based on the above factors, *β* should be chosen to be around 5°. So nanomagnets will have less required stress, larger switching frequency, and shorter initial switching time.

Although voltage pulse-induced magnetization switching is very energy efficient, the possibility of operating at room temperature remains to be discussed, which plays an important role in the switching. In this section, the switching of the nanomagnet at room temperature (300 K) is calculated. Since OOMMF software could be computationally expensive and time-consuming to simulate the switching at room temperature, the authors use the mathematical stress model to calculate the switching of the nanomagnet at room temperature.

Firstly, the authors apply a stress of 100 MPa to the electrodes and observe the dynamic magnetization of the nanomagnet. The magnetization rotates by more than 90° at 0.1844–0.3470 ns and is most close to logic "0" at 0.2574 ns, meaning that *t*width = 0.2574 ns. If the stress is removed at 0.2574 ns, the nanomagnet will flip to logic "0" at 0.7427 ns, meaning that *t*interval = 0.8856 ns. Obviously, at room temperature, both *t*width and *t*interval are much larger than that at 0 K.

fluctuation is a great challenge. This method overcomes the start-up time of the initial switching by rotating the stress electrodes' pair axis by a small angle from the long axis of the nanomagnet. Using OOMMF software for simulation, the optimal voltage pulse waveform to control the 180° switchings of the nanomagnet is calculated, and the influence of electrodes' pair axis tilt angle *β* is studied. The results show that when the tilt angle *β* is about 5°, the nanomagnet has lower switching frequency, shorter initial switching time, and less required stress. Repeated

*Electric Field-Induced Magnetization Reversal of Multiferroic Nanomagnet*

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

switching at room temperature is calculated by mathematical model. The switching time is longer under the influence of thermal fluctuations. These findings will provide possible guidance for straintronic devices in the application of magnetic

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

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

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

reduce the complexities of calculations and fabrication process.

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

storage and logic.

of fabrication process.

**3.1 Design and analysis**

**27**

**Figure 7.** *Minimum switching times of the nanomagnet changes with the tilt angles of the electrode pair axis.*

Secondly, the authors try to control the switching by voltage pulse of *t*width = 0.2574 ns and *t*interval = 0.8856 ns at room temperature. Unfortunately, the nanomagnet succeeds to be switched to logic "0," but never return back to logic "1" again. Under the influence of thermal fluctuations, the nanomagnet needs to remain in a stable logic state for a longer period of time before it can be switched again.

Thirdly, *t*width = 0.3 ns (it can be chosen from 0.1844 to 0.3470 ns) and *t*interval = 1 ns are given to gain repeated switchings. As shown in **Figure 5**, the nanomagnet converts back and forth between two logic states at room temperature successfully.

The switching cycle is 1.3 ns. One thing that must be pointed out is that the simulations assume the ideal voltage pulse waveform. The effects of the rising and falling edges of the actual voltage pulse are not considered. Besides, through the calculation of the model, *t*interval can be long enough but *t*width can be only chosen from 0.1844 to 0.3470 ns. Actually, pulse width needed for 180° switching will not be a constant in the presence of thermal noise. The pulse width error of less than 0.2 ns will be a challenge under room temperature, which is the disadvantage of this scheme. These will need to be further studied in the subsequent experimental work.

Due to the symmetry, setting the initial logic as "0" or setting the electrodes' pair axis, a clockwise deflection will get the same result, which is not described in this paper for clarity.

#### **2.3 Conclusion**

The efficient 180° switching of the magnetization direction of the nanomagnet is the key to straintronic devices in the application of magnetic storage and logic. The voltage pulse-induced repeatable 180° switching is a fast and low energy consumption scheme, but the initial switching requires a large start-up time and thermal

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

fluctuation is a great challenge. This method overcomes the start-up time of the initial switching by rotating the stress electrodes' pair axis by a small angle from the long axis of the nanomagnet. Using OOMMF software for simulation, the optimal voltage pulse waveform to control the 180° switchings of the nanomagnet is calculated, and the influence of electrodes' pair axis tilt angle *β* is studied. The results show that when the tilt angle *β* is about 5°, the nanomagnet has lower switching frequency, shorter initial switching time, and less required stress. Repeated switching at room temperature is calculated by mathematical model. The switching time is longer under the influence of thermal fluctuations. These findings will provide possible guidance for straintronic devices in the application of magnetic storage and logic.
