Spin waves (SWs) have been considered a promising candidate for encoding information with lower power consumption. Here, we propose dual function SW logic gates, one unit cell with two synchronized logic operation functions, based on the electric field controlling of the SW propagation in the Fe film of a Fe/BaTiO3 heterostructure by the motion of a magnetic anisotropy boundary. We show micromagnetic simulations to validate the and–or and nand–nor logic gates. Our research may find a path for simplifying integrated logic circuits using such dual function SW logic gates.

With the rapid development in the miniaturization of electronic devices, power consumption has become one important issue because of the unbearable Joule heating.1,2 To overcome this issue, seeking new charge-neutral information carriers has spurred great attention. One representative charge-neutral information carrier is spin angular momentum, the electron's another degree of freedom. Spin waves (SWs), the collective spin angular momentum excitations in a magnetically ordered material and their associated quanta, magnons, are considered the promising information carrier for the next-generation lower power consumption, higher speed, and higher density devices. The phase,3–6 amplitude,7–10 and polarization11,12 of SWs can be employed to encode information. Generally, the nanostructured SW logic gates are designed on the uniform magnetized ferromagnetic materials by using all kinds of methods to control the propagation of SWs, including the magnetic field,10 electric current,7–9 and voltage.13,14

The logic gates based on an electric field or voltage can be easily programmable and compatible with nanoscale microwave devices.15 Rana and Otani14 have proposed SW xnor and universal nand logic gates based on the voltage-controlled magnetic anisotropy (VCMA), which mainly arises from the change of an electronic occupation state near the interface between the magnetic films and nonmagnetic heavy metals controlled by the voltage.15–17 Magnetoelectric effects18–20 provide another path for electric-field control of SWs in multiferroic materials.21,22 The most representative multiferroic system is artificial multiferroic materials. The single-crystalline ferroelectric barium titanate (⁠BaTiO3⁠) membrane demonstrated super-elasticity and ultra-flexibility properties.23 This indicates that the lifetime of transformation of the ferroelectric domain may be over the generally expected for the thin membrane. Both the interfacial and bulk magnetoelectric coupling in Fe/BaTiO3 have been studied by first-principles calculations and experiments.24–26 Recently, electric field controlled spin wave propagation had been experimentally demonstrated in an artificial multiferroic Fe/BaTiO3 heterostructure.27 Magnetoelectric effects could help us to design many more functional SW logic gates based on a BaTiO3 based multiferroic structure. Here, the dual function SW logic gates, one unit cell with two synchronized logic operation functions, are proposed by controlling the motion of the magnetic anisotropy boundary (MAB) in an epitaxial Fe film on the ferroelectric BaTiO3 substrate with altering in-plane and out-of-plane polarization domains. The and–or gate and nand–nor gate are validated by micromagnetic simulation.

We perform micromagnetic simulations by MuMax328 to study the propagation of Damon–Eshbach (DE) SWs29 in the epitaxial Fe film on the ferroelectric BaTiO3 substrate. Figure 1(a) shows one unit structure in our simulation, including one 2 nm thick Fe layer, 100 nm thick BaTiO3⁠, and the bottom electrode layer. The width and length of the structure are 800 and 2000 nm, respectively. The magnetic Fe layer includes uniaxial and cubic magnetic anisotropy regions via inverse magnetostriction owing to the different strains between the Fe film and two kinds of domains in the BaTiO3 layer.30 The easy axis of uniaxial magnetic anisotropy on top of the in-plane ferroelectric a-domain is along the y axis, while the easy axes of cubic anisotropy on top of the out-of-plane ferroelectric c-domain are along the direction with 45° to the x axis, as shown by the white double head arrows. The MAB is at the middle of the Fe film (x = 0, along the y axis) and is pinned on the ferroelectric domain wall of the BaTiO3 substrate. The width of MAB in the Fe film is the same to the ferroelectric domain wall in BaTiO3 [2–5 nm (Refs. 31–33)], which is far smaller than the wavelength of SWs (several hundred nanometers). So, we neglect the width of the MAB in simulation. The parameters of the Fe film used in simulation are the following:34 saturation magnetization Ms = 1.7× 106 A/m, exchange stiffness constant Aex = 2.1 × 10−11 J/m, and Gilbert damping constant α = 0.01. The uniaxial and cubic magnetic anisotropy constants are the experiment values27 of Ku = 1.5× 104 J/m3 and Kc = 4.4 × 104 J/m3⁠. The cell size is 2 × 2 × 2 nm3⁠, which is smaller than the exchange length (⁠lex≈3.4 nm). In order to avoid SWs' reflection at the boundary, the damping is set to 1 at both ends (width = 100 nm) of the Fe film. An external magnetic field μ0Hext = 100 mT is applied to avoid the formation of the magnetic domain wall and to magnetize the Fe film along the y axis. To obtain the dispersion relation of the DE SWs in uniaxial and cubic anisotropy regions, a sinc based exciting field, h(t)=h0sinc(2πfc(t−t0))êz⁠, with μ0h0 = 10 mT, fc = 50 GHz, and t0 = 5 ns, is applied locally to a 2 × 800 × 2 nm3 central section of the Fe film, which is the SW source port (S) as indicated by the red region in Fig. 1(a). The dispersion relation35,36 of SWs can be obtained by performing a two-dimensional Fourier transform on mx (⁠mx=Mx/Ms⁠), as shown in Fig. 1(b). The curve of the left branch is the dispersion relation of SWs propagating along the negative direction of the x axis in the uniaxial anisotropy region, and the right branch is the dispersion relation of SWs propagating along the positive direction of the x axis in the cubic anisotropy region. The theoretical curves of SW dispersion relations are calculated by adopting Eqs. (S4) and (S5) in the supplementary material. The difference in ferromagnetic resonance frequencies in the two anisotropy regions determines the working frequency range for the proposed device, γμ02π(Hext−2Kcμ0Ms)(Hext+Kcμ0Ms+Ms) < f < γμ02π(Hext+2Kuμ0Ms)(Hext+2Kuμ0Ms+Ms)⁠, which is dominated by the anisotropy energy of the two regions. It is clear to show that the SWs are only excited in the cubic anisotropy region in the frequency range of 9.5 GHz < f < 14.5 GHz. To further clarify the property, we excite SWs at the source port with a sinusoidal field h(t)=h0 sin (2πft)êz with μ0h0 = 10 mT and a fixed f = 12.6 GHz. A snapshot of the Mx component of the magnetization taken at t = 4 ns is presented in Fig. 1(c). SWs only propagate in the region with cubic anisotropy.

(a) Schematic of the simulation model. The Fe film is grown on BaTiO3 with the alternating in-plane polarization a-domain and out-of-plane polarization c-domain. The origin of the coordinate system is set at the center of the Fe film. (b) The image plot of the dispersion relation of the SWs. The dashed lines are theoretically calculated results. (c) A snapshot of the Mx component of the magnetization taken at t = 4 ns, y = −1 nm with a fixed f = 12.6 GHz.

(a) Schematic of the simulation model. The Fe film is grown on BaTiO3 with the alternating in-plane polarization a-domain and out-of-plane polarization c-domain. The origin of the coordinate system is set at the center of the Fe film. (b) The image plot of the dispersion relation of the SWs. The dashed lines are theoretically calculated results. (c) A snapshot of the Mx component of the magnetization taken at t = 4 ns, y = −1 nm with a fixed f = 12.6 GHz.

An out-of-plane electric field controlling of ferroelectric domain wall motion can realize the movement of the MAB.34 When the electric field is along the direction of the polarization in the c-domain, the c-domain will expand while the a-domain shrinks by the lateral domain wall motion. Suppose the electric field is inverse to the direction of the polarization in the c-domain. In that case, the c-domain will shrink, and the a-domain expands. So, the electric field can drive the ferroelectric domain wall's motion, thus driving the concurrent motion of the MAB. The velocity of the MAB, that is, the velocity of the ferroelectric domain wall driven by an electric field pulse, can be up to 1000 m/s.37,38 Because the wall velocity is still smaller than the speed of sound in BaTiO3 [vsound = 2000 m/s (Ref. 39)], the strain state can be considered quasistatic during the motion of the ferroelectric domain wall, and the strength and symmetry of the magnetic anisotropy are invariant during the MAB back and forth. The velocity v of the MAB is implemented by shifting the MAB over one discretization cell (δx = 2 nm) during each time window δt=δx/v⁠. In the simulation, the velocity of MAB is 1000 m/s under the electric field 31.4 MV/m obtained by applying 3.14 V perpendicular voltage, which is inferred from previous experimental results.37 

Figure 2(a) shows the motion of the MAB driven by the voltage pulse, and Figs. 2(b)–2(d) show the schematics of the model and the snapshots of Mx when the MAB at x = −600, −100, and 400 nm, respectively. The SWs with f = 12.6 GHz are excited from the source port. The position of the MAB is driven to x = −600 nm by using the positive electric voltage (1.4–2 ns) shown in Fig. 2(a). As expected, the SWs are blocked in an uniaxial anisotropy region shown by the snapshot of SWs taken at t = 5 ns in Fig. 2(b). After applying one negative pulse voltage (7–7.5 ns), the MAB moves to the position x = −100 nm. The SWs only propagate in the cubic anisotropy region from the snapshot of Mx taken at t = 10.5 ns, shown in Fig. 2(c). When another 0.5 ns negative voltage pulse is applied (12.5–13 ns), the MAB moves to the position x = 400 nm, and the SW source port is out of the cubic anisotropy region. There are no SWs in the Fe film. Thus, the transmission and block of SWs in the Fe film can be controlled by voltage through driving the motion of the MAB. Based on this unique property, we propose the design of dual function SW logic gates.

(a) The position variation of the MAB with the voltage pulse. (b), (c), and (d) are the schematics of model and the snapshots of Mx taken at y = −1 nm when MAB at x = −600, −100, and 400 nm [adopting the coordinate system in Fig. 1(a)], respectively. The dashed lines mark the position of the MAB.

(a) The position variation of the MAB with the voltage pulse. (b), (c), and (d) are the schematics of model and the snapshots of Mx taken at y = −1 nm when MAB at x = −600, −100, and 400 nm [adopting the coordinate system in Fig. 1(a)], respectively. The dashed lines mark the position of the MAB.

Figure 3(a) is the schematic of the dual function and–or gate, which includes two input ports (A and B), two output ports (C and D), and a source port of SWs. Initially, the MAB is set at x = 400 nm parallel to the y axis. The positions of input A and B ports are arbitrary on top of the Fe film. In order to make the two input ports independent, the B port includes a time delay unit, which can delay the voltage pulse for 0.5 ns. The output ports C and D take the x component of magnetization at regions (498 nm < x