Single array of magnetic vortex disks uses in-plane anisotropy to create different logic gates

TL;DR

This study demonstrates how in-plane uniaxial magnetic anisotropy in a nanodisk array can be controlled to implement various logic gates, advancing magnetic logic device design.

Contribution

It introduces a method to use in-plane anisotropy to dynamically create different logic gates within a single magnetic nanodisk array.

Findings

In-plane anisotropy direction affects energy transfer time between disks.

Magnetic interactions vary with anisotropy and influence logic gate functionality.

A single array can be configured to perform multiple logic operations.

Abstract

Using micromagnetic simulation, we show that in-plane uniaxial magnetic anisotropy (IPUA) can be used to obtain FAN-OUT, AND and OR gates in an array of coupled disks with magnetic vortex configuration. First, we studied the influence of the direction of application of the IPUA on the energy transfer time ({\tau}) between two identical coupled nanodisks. We found that when the direction of the IPUA is along the x axis the magnetic interaction increases, allowing shorter values of {\tau}, while the IPUA along the y direction has the opposite effect. The magnetic interactions between the nanodisks along x and y directions (the coupling integrals) as a function of the uniaxial anisotropy constant (K {\sigma} ) were obtained using a simple dipolar model. Next, we demonstrated that choosing a suitable direction of application of the IPUA, it is possible to create several different logic gates with a single array of coupled nanodisks

Tables

Table 1. Table 1: Logical configuration obtained with the Fan-out gate (Type 1)

Input	Output 1	Output 2	Logical
Disk 4	Disk 1	Disk 7	configuration
0	0	0	000
1	1	1	111
1	0	1	101
1	1	0	110
1	0	0	100

Equations

$$W_{int}=C_1{C}_2({\eta }_x{x}_1{x}_2+{\eta }_y{y}_1{y}_2),$$

$${\eta }_x={\eta }^{\ast }{\eta }_y=-2{\eta }^{\ast },$$

$$I_{x,y}=\frac{8\pi }{{\mu }_{0}R{M}_s^2}{\eta }_{x,y},$$

$$\text{G}\times \frac{d\text{X}}{dt}-\frac{\partial W(\text{X})}{\partial \text{X}}=0$$

$${\omega }_{1,2}^p=\sqrt{{\omega }_0^2-2p(\frac{{\eta }^{\ast }}{G}{)}^2\pm \frac{{\eta }^{\ast }}{G}{\omega }_0\sqrt{5-4p}},$$

$$f_{1,2}^{p=+1}=f_0\sqrt{1-0.0103\frac{{\lambda }^4}{d^6}\pm 0.0718\frac{{\lambda }^2}{d^3}}$$

$$f_{1,2}^{p=-1}=f_0\sqrt{1-0.0103\frac{{\lambda }^4}{d^6}\pm 0.2154\frac{{\lambda }^2}{d^3}}$$

$$\tau \sim 1/\Vert {\eta }_x+p{\eta }_y\Vert .$$

$$\tau \sim 1/{\eta }^{\ast }\Vert 1-2p\Vert$$

Full text

1 Introduction

The magnetic vortex configuration is a ground state characterized by a curling magnetization in the plane, and a small region (core vortex) in the center, where of magnetization is out the plane [1, 2]. From these characteristics, two properties are defined: the circulation C and the polarity p. The circulation is C = +1 when the curling direction is counterclockwise (CCW) and C = -1 when it is clockwise (CW). The polarity is p = +1 when the core points in the +z direction and p = -1 in the -z direction. The magnetic vortex dynamics is characterized by an eigenfrequency (gyrotropic frequency) in the sub-gigahertz range [1, 2, 3, 4]. This frequency depends on the intrinsic parameters of the material and on the ratio $\beta$ of the thickness to the radius of the disk ($\beta$ = L/R) [1].

The use of the magnetic vortices in logic gates is one of the potential technological applications in spintronics [5, 6, 7, 8]. Controlling parameters such as the polarity and circulation, different logical configurations can be obtained, as already demonstrated by Jung et al. [5], and controlling the magnetic interaction through the separation distance between the disks, it is possible to perform fan-out operations [7]. Therefore, the control of these parameters is of vital importance for the construction of logic gates. Both p as C can be controlled with rotating magnetic fields [9, 5], and the magnetic interaction can be modified without the need to change the separation distance between the disks.

One of the ways of obtaining this goal is by the application of perpendicular magnetic fields (PMF) to the plane of the disks [10]. This method may not be very effective, since the PMF distorts the core profile of the vortex, which is where the information bits are usually stored; therefore, the search for other mechanisms is an open topic. In this sense, the influence of the in-plane uniaxial anisotropy (IPUA) on the dynamics of magnetic vortices is gaining interest in recent years [11, 12]. Experimentally, the IPUA can be induced by voltage-induced strain via a piezoelectric transducer (PZT) [13, 12]. The advantage of using the IPUA, is that this does not alter the core profile of the vortex, making it a practical tool to control the dynamics of the magnetic vortex [11, 12].

When disks with magnetic vortex configuration are coupled, energy is transfered periodically between them [14]. The energy transfer is characterized by an energy transfer time parameter $\tau$. Therefore, controlling this parameter is very important to affect this transfer. In a previous work, we have shown that applying the IPUA along the x axis allows obtaining shorter energy transfer times [12].

The goal of this work is to study the influence of changing the direction of application of the IPUA in a pair of identical nanodisks, and using the IPUA in order to obtain different logical configurations with an array of coupled nanodisks. We used micromagnetic simulations and a simple analytical dipolar model in order to obtain the interactions along the x and y directions (the coupling integrals). We used the open source software Mumax [15], with cell size of 2 $\times$ 2 $\times$ 7 nm3; the magnetostrictive material used was Galfenol (FeGa) with parameters [16, 17]: saturation magnetization Ms = 1.360 $\times$ 106 A/m2, exchange stiffness A = 14 $\times$ 10*-12*  J/m and a typical damping constant $\alpha$ = 0.01. The anisotropy constant (Kσ) varied from 0 to 58500 J/m3. The magnetoelastic energy was included in the micromagnetic simulation as an uniaxial anisotropy energy [11, 12].

2 Results and discussion

2.1 Two coupled disks

2.1.1 Micromagnetic simulation

We considered a system of two coupled identical disks, located along the x-axis, with thickness L = 7 nm, diameters 256 nm, and separated by a center to center distance D (Fig. 1). In order to induce gyrotropic motion we have applied an in-plane static magnetic field in the x direction for a few nanoseconds, only on disk 1 to displace the vortex core from the equilibrium position (center of the disk), and using a large damping $\alpha=1$ for faster convergence. Then this field was removed and the IPUA was applied to both disks, a typical value of $\alpha=0.01$ was used, allowing the vortex core to perform the gyrotropic motion. The movement of the vortex core in the disk 2 is induced by the transfer of energy from disk 1. This process is repeated for each value of Kσ.

Unlike our previous work, we have also considered the case when the uniaxial anisotropy is in the direction of the y axis. The values of $\tau$ depending on the Kσ are shown in Fig. 2. This dependence of $\tau$ with Kσ is the opposite of that obtained in our previous work, where the direction of the IPUA was applied along the x-axis [12].

When the direction of application of the IPUA is along the y axis (IPUA-y), the values of $\tau$ increase, while when the direction is along the x axis (IPUA-x), $\tau$ decreases. For example, considering IPUA-x, for the case p = p1.p2 = +1 and a reduced distance d = D/R = 2.11, $\tau$ is reduced by approximately 66$\%$ from $\tau$ = 17.3 ns (Kσ = 0 kJ/m3) to $\tau$ = 5.8 ns (Kσ = 58.5 kJ/m3) and for the case p = -1, $\tau$ is reduced approximately by 40$\%$ from $\tau$ = 7.5 ns (Kσ = 0 kJ/m3) to $\tau$ = 4.6 ns (Kσ = 58.5 kJ/m3). This reduction in the $\tau$ values is appreciable and show the efficiency of using IPUA as a tool for the control of the energy transfer time, as we have already demonstrated in our previous work [12]. Now, for IPUA-y and a reduced distance d = 2.11, $\tau$ increases approximately 140$\%$, from $\tau$ = 17.3 ns (Kσ = 0 kJ/m3) to $\tau$ = 42 ns (Kσ = 19.5 kJ/m3) for the case p = +1, and an increase of approximately 48$\%$, from $\tau$ = 7.5 ns (Kσ = 0 kJ/m3) to $\tau$ = 11.2 ns (Kσ = 58.5 kJ/m3), for the p = -1 case. In every case, the values of $\tau$ were obtained from the temporal dependence of the energy density of each disk, considering that $\tau$ is the time in which the energy of disk 1 reaches the minimum value for the first time [14].

As can be seen in Fig. 2, we have considered only three values of Kσ (0, 9.75 and 19.5 kJ/m3) for IPUA-y, d = D/R = 2.11, and p = +1, For larger values of Kσ, a simple free induction decay (FID), with a decay time of approximately 100 ns and the same gyrotropic frequency (thus there is no frequency splitting), is observed in both disks. This means that the energy of disk 1 is fully transferred to disk 2 for a value of $\tau$ comparable to the decay time of the FID. During this stage, the vortex core of disk 2 performs a gyrotropic motion of negligible amplitude.

The initial amplitude of the FID in disk 1 is due to the initial displacement of the vortex core by the application of the in-plane magnetic field. During this stage, although no magnetic field is applied in disk 2, the vortex core is also displaced due to the magnetic interaction between the disks [18]. This interaction is due to the presence of surface and/or volume charges in disk 1 resulting from the displacement of the vortex core by the application of the in-plane magnetic field [19, 18]. The displacement in disk 2 is negligible in comparison to the displacement in disk 1, and is of the order of 15 nm. For IPUA-y and p = -1, we have used 58.5 kJ/m3 as the maximum value of Kσ, as already mentioned in section 1. For larger values of Kσ, the magnetic vortex configuration is not stable.

The reason why it is still observed an energy transfer, even using our maximum allowed value of Kσ, is because the magnetic interaction between the disks is stronger in comparison with the case p = +1, therefore, high values of Kσ are required to decrease the magnetic interaction between the disks.

2.1.2 Analytical dipolar model

In order to clarify the physical meaning of the influence of the IPUA on the magnetic interactions, we used a simple dipolar model, also used in our previous work [12]. The magnetic interaction energy ($W_{int}$) between two disks with magnetic vortex configuration is given by [20]:

[TABLE]

where $\eta_{x}$ and $\eta_{y}$ are the interactions between the disks along x and y directions (the coupling integrals), C1 and C2 are the circulations in disk 1 and disk 2, and $\bf{X}_{i}=(x_{i},y_{i})$, with i = 1,2 is the vector position of the vortex core of the disks.

The coupling integrals can be expressed as a sum of contributions of high order magnetic interactions, such as dipole-octupole, octupole-octupole, dipole-triacontadipole, etc. [21]. However, in our model we only consider purely dipole interaction since this is sufficient to explain the influence of IPUA on the magnetic interactions, as we demonstrated in our previous work [12]. Thus, within our dipolar model, the coupling integrals are defined as follows:

[TABLE]

or in their dimensionless forms:

[TABLE]

where $\eta^{*}=\mu_{0}\lambda^{2}M_{s}^{2}L^{2}R^{2}/4\pi D^{3}$.

The $\lambda$ parameter, which takes into account the influence of Kσ, can be obtained from a simultaneous fit of the coupling frequencies f1,2 = $\omega_{1,2}/2\pi$ (or eigenfrequencies) as has already been demonstrated by Asmat et al. [22]. In order to obtain this parameter, we used the Thiele’s equation [23]. Considering a negligible damping, this equation can be written as:

[TABLE]

where G is the gyrovector G = -Gp$\hat{z}$, G = 2$\pi\mu_{0}$LMs/$\gamma$ is the gyrotropic constant, and $\gamma$ = 2.21$\times$105  m/As is the gyromagnetic ratio; W(X) = W(0) + $\frac{1}{2}\kappa\textbf{X}^{2}$ is the potential energy and $\kappa=40\pi$M${}_{s}^{2}$L2/9R is the stiffness coefficient calculated within the side-charge-free model at L/R $<<$ 1 [1].

In our previously work [12], from the first variation of the Lagrangian expression for a pair of coupled disks based on the constant of the Thiele’s equation, we obtained analytical expression for the coupling frequencies (Eq. (11) in reference [12]), written as:

[TABLE]

with $\omega_{0}=2\pi f_{0}$, where f0 is the gyrotropic frequency for an isolated disk.

Replacing the expressions for G, $\kappa$, f0 and $\eta^{*}$ in Eq. 5, we have:

[TABLE]

[TABLE]

Eq. (6) and Eq. (7) are the analytical expressions for the coupling frequencies (f1 and f2 are the high (+) and low (-) frequencies) within the dipolar approximation for p = p1.p2 = +1 and p = p1.p2 = -1, respectively. The dependence of the eigenfrequencies on reduced distance d = D/R is shown in Fig. 3 for Kσ = 9.75 kJ/m3 and IPUA-x.

The values of lambda were obtained from the simultaneous fits to the results shown in Fig. 3.

A value of $\lambda=2.77$ is obtained for Kσ = 9.75 kJ/m3. The same procedure is done for each value of Kσ. We obtained an increases of the $\lambda$ parameter, from $\lambda$ = 2.59 (Kσ = 0  kJ/m3) to $\lambda$ = 3.79 (Kσ = 58.5 kJ/m3) for IPUA-x. For IPUA-y 111In this case, the simultaneous fit is not possible, since for p = +1, there is no frequency splitting for some values of Kσ (subsection 2.1). Thus, we have used only the eigenfrequencies obtained for the case p = -1. we obtained a decrease of the $\lambda$ parameter, from $\lambda$ = 2.59 (Kσ = 0  kJ/m3) to $\lambda$ = 2.32 (Kσ = 58.5 kJ/m3). There are discrepancies between the values obtained from micromagnetic simulation and the fit using Eq. (6). This is expected, since our model considers only dipolar interaction, without consider high-order magnetic interactions as dipole-octupole, octupole-octupole, dipole-triacontadipole, which are important for small separation distances [21, 12]. However, the fit is well-behaved when the separation distance between the disks is larger [12].

The dependence of the coupling integral $\eta_{x}$ on Kσ and the reduced separation distance d is shown in Fig. 4. For IPUA-x, the coupling integrals increase with the increase of Kσ, while for IPUA-y, the coupling integrals decrease with the increase of Kσ. The dependence of the coupling integrals on Kσ and on the reduced separation distance d is shown in Fig. 4. In order to explain the dependence of $\tau$ on Kσ, we used the analytical expression of $\tau$ [14]:

[TABLE]

Substituting Eq. (2) in Eq. (8), we have:

[TABLE]

From Eq. (9), we have that the value of $\tau$ is inversely proportional to $\eta^{*}$. Therefore, we have that for IPUA-x, the coupling integrals increase with Kσ (Eq. (2)), thus $\tau$ decreases, and for IPUA-y, the coupling integrals decrease with Kσ, thus $\tau$ increases. This analytical approach applies to both p = +1 and p = -1, and is consistent with the results shown in Fig. (2).

Next, we used the influence of the IPUA in order to obtain several logic gates using an array of coupled nanodisks with magnetic vortex configuration.

2.2 Logic gate applications

2.2.1 Fan-out gate

We used an array of seven identical disks, as shown in Fig. 5 (Type 1), with diameter 240 nm, thickness L = 7 nm and separation distance D = 260 nm between the disks 222The disks can be arranged in different ways to achieve the desired gate. For example, in ref [7] it is shown another type of arrangement to achieve a fan-out gate.. All disks have positive polarity p = +1 and positive circulation C = +1. The reason for choosing parallel polarities is that the interaction between the disks can be decreased using IPUA, whereas this would be more difficult in the case of having antiparallel polarities (subsection 2.1). We have chosen an L-Shaped (LS) array geometry, because by coupling several LS it is possible to form a two dimensional network array consisting of nodes, where each node can serve as a nonvolatile memory bit, as has already been proposed by Zhang et al. [24].

We considered the disk 4 (blue disk) as an input and disk 1 and disk 7 as outputs (red disks). All the nanodisks have positive polarity and counterclockwise circulation.

The type of array where there is one input and two outputs is known as a fan-out gate [8, 24, 7]. Since our objective is to observe the output signals on disk 1 and disk 7, the IPUA is applied only to disks 2, 3, 5 and 6. We used in all cases a value Kσ = 29.25 kJ/m3. The input signal on disk 4 is implemented by applying a counterclockwise rotating magnetic field B(t) = B0 $\cos(2\pi f_{0}t)$ $\hat{x}$ + B0 $\sin(2\pi f_{0}t)$ $\hat{y}$, where B0 = 2 mT is a magnetic field amplitude and $f_{0}$ = 0.45 GHz is the gyrotropic frequency of one isolated disk. The same effect could be obtained with a time-dependent polarized current. The value of B0 was chosen in such a way that the vortex core is not switched. We use 1 to indicate that there is a signal and 0 to indicate that there is no signal (or negligible signal strength) [5].

First, we considered the case when the IPUA is off in all green disks, thus the signal input is propagated through the neighboring disks, reaching the 2 outputs (disk 1 and disk 7). It is well known that the signal can propagate in disk arrays due to the magnetic interaction between them [25, 14]. The intensity level of the output signals obtained by micromagnetic simulation, compared to the input signal is shown in (Fig. 6 (a-b)).

The case where the output signal levels are appreciable, are referred to the logical configuration 111, where the first digit refers to the input signal on disk 4, the second digit refers to the output signal on disk 1, and the third digit refers to the output signal on disk 7. Now, we used the influence of the IPUA in order to cancel the signal on any of the disks used as outputs. As shown in section 2.1, when the directions of the IPUA are perpendicular to the direction to the chain, the magnetic interaction between the disks is reduced.

To get the logical configuration 101, we applied the IPUA in the y direction, only on disk 2 and disk 3. The result is a large decrease in the level of signal on disk 1, while the signal level in disk 7 is high (Fig. 6(c-d)). The logical configuration 110 is obtained by application of the IPUA in the x direction only on disk 5 and disk 6 (Fig. 6(e-f)). Now, in order to obtain the logical configuration 100, we applied the IPUA in disks 2, 3, 5 and 6 as follows: on the disk 2 and disk 3 the IPUA is applied in the y direction while, on disk 4 and disk 5 the IPUA is applied in the x direction. With this, we were able to cancel the output signals on disk 1 and disk 7 (Fig. 6(g-h)). The logical configuration 000 is trivial, since if there is no input signal, there is no output signal.

All logical configuration obtained with the Fan-out gate are summarized in Table 1.

2.2.2 AND, OR gates

We used the Type 2 array shown in Figure 5. The disk 1 and disk 7 (blue disks) are used as inputs (input 1 and input 2, respectively) and disk 4 as output (red disk). The input signals are implemented by the counterclockwise rotating magnetic field with the same parameters used in subsection 2.2. The value of Kσ and the directions of the application of the IPUA too are the same used in subsection 2.2. In an AND gate, the following logical operations can be performed: 0 + 0 = 0, 0 + 1 = 0, 1 + 0 = 0, and 1 + 1 = 1 and in an OR gate, we have: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 1 [8].

The logical operation 0 + 1 means that we have an input signal only in disk 7, 1 + 0 means there is a input signal only in disk 1 and 1 + 1 is referred when we have input signal in both disk 1 and disk 7. Here again, the operation 0 + 0 is trivial.

OR gate: to obtain the logical operation 0 + 1 = 1, it was applied an input signal only in disk 7. This signal propagates to disk 4. Due to the magnetic interaction, the signal can also propagate to disk 1 (input disk), but to avoid this propagation, we have applied the IPUA in disk 2 and disk 3. Similarly, to obtain the logical operation 1 + 0 = 1, an input signal was supplied to disk 1 and the IPUA was applied in disk 5 and disk 6. These results can be seen in Fig. 7.

AND gate: To obtain the logical operation 0 + 1 = 0, it was applied an input signal only in disk 7. In order to avoid the propagation of the signal, the IPUA was applied in disk 5 and disk 6. In a similar way, to obtain the logical operation 1 + 0 = 0, an input signal was applied only in disk 1, and IPUA in disk 2 and disk 3. These results are shown in Fig. 7.

The logical operation 1 + 1 = 1 is the same for both AND and OR gates; this means having an input signal in disk 1 and disk 7. In this case, it is not necessary to apply the IPUA, since we want to have an output signal in disk 4. This operation can be considered trivial and is not shown here.

The advantage of using the IPUA, is that using it, we can cancel the propagation along a chosen direction. This would be impossible to do with some other external agent acting on the nanodisks and preserving the vortex configuration.

3 Conclusions

In this work, the influence of the direction of application of the IPUA in a system of two coupled identical nanodisks with magnetic vortex configuration was first studied using micromagnetic simulation. We demonstrated that it is possible to obtain shorter energy transfer times when the IPUA is along the x axis, while if the IPUA is along the y axis, the magnetic interaction between the disks is canceled, therefore the energy transfer is not efficient. Also, using a simple dipolar model, we obtained the coupling integrals depending on Kσ, and we explained why $\tau$ increases or decreases, depending on the direction of application of the IPUA.

We have created Fan-out, AND and OR gates in an array of nanodisks using the influence of the IPUA to control the propagation of the signal in a desired direction. All these results show the great efficiency of using IPUA to control the magnetic interaction and the direction of signal propagation between the disks without the need of changing the separation distance between them.

These results show the large potential of using magnetic vortices and controlled in-plane anisotropy for the design of logic gates and other devices.

ACKNOWLEDGMENTS

The authors would like to thank the support of the Brazilian agencies CNPq and FAPERJ.