Parameters of multiferroic heterostructure.
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. Compared with traditional spintronic devices based on magnetic field, multiferroic nanomagnet devices have the advantages of ultra-low consumption and high radiation resistance, showing great application potential in modern high-integrated circuits and military electronic systems. However, the difficulties of electric field control of complete magnetization reversal of the nanomagnet and nanomagnet arrays in a nanomagnetic logic gate still restrict the developments of multiferroic nanomagnet device. In this chapter, the uniaxial nanomagnets in multiferroic heterojunctions are mainly discussed. The two core problems of the electric field control of nanomagnets and nanomagnetic logic gate are well solved.
- magnetization switching
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 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 . It represents Boolean logic “0” and “1” in the magnetization directions along the long axis of a uniaxial nanomagnet  and can be corresponding to different states in the magnetic tunnel junction .
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.
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
The magnetization dynamic of a single elliptical nanomagnet meets the Landau-Lifshitz-Gilbert Eq. (5):
is the effective field generated by a variety of energies (shape anisotropy energy, stress anisotropy energy, Zeeman energy, and thermal fluctuations), where is the vacuum permeability and
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 . The shape anisotropy energy of the nanomagnet can be written as :
where is the magnetic moment vector of the nanomagnet and is the shape anisotropy energy field, which can be expressed as :
where is the demagnetization factor. For elliptical shaped nanomagnets, the demagnetization factors , , and can be calculated through :
The stress anisotropy energy of the nanomagnet is given by :
Considering the thermal fluctuations, the effect of random thermal fluctuations can be described by a random thermal field :
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 . 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 . The parameters are shown in Table 1.
Since (Object Oriented Micromagnetic Framework) software  cannot directly set the stress anisotropy energy, the authors use the uniaxial anisotropy energy acting in the direction of (−cos
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 . 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
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 magnetization of the repeatable 180° switchings in the nanomagnet, when the electrodes’ pair axis is aligned with the long axis of the nanomagnet (
If the electrodes’ pair axis is not aligned with the long axis of the nanomagnet, but is tilted by a small angle
Figure 6 shows the dynamic magnetization of the switchings and optimal voltage pulse waveform when
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
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
Secondly, the authors try to control the switching by voltage pulse of
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,
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.
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 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
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 . 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 , 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 . However, this solution requires more nanomagnets, which increases the NML area. Roy uses a multi-iron material to propose an ultra-low-energy NAND (OR) logic gate based on a magnetic tunnel junction . 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
As shown in Figure 8(b), for a nanomagnet with a tilt angle
Based on the preferred magnetization of tilted nanomagnet, a design of dual-input 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 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 :
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” (
The authors assume that the initial state of the nanomagnet Out is logic “1” (
The input nanomagnets A and B only produce small fluctuations (∼2°) in the plane and eventually return to the original logic state (
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. 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.
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.
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.
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).