Spin-Wave Diode and Circulator Based on Unidirectional Coupling

In magnonics, a fast-growing branch of wave physics characterized by low energy consumption, it is highly desirable to create circuit elements useful for wave computing. However, it is crucial to reach the nanoscale so as to be competitive with the electronics, which vastly dominates in computing devices. Here, based on numerical simulations, we demonstrate the functionality of the spin-wave diode and the circulator to steer and manipulate spin waves over a wide range of frequency in the GHz regime. They take advantage of the unidirectional magnetostatic coupling induced by the interfacial Dzyaloshinskii-Moriya interaction, allowing the transfer of the spin wave between thin ferromagnetic layers in only one direction of propagation. Using the multilayered structure consisting of $\mathrm{Py}$ and $\mathrm{Co}$ in direct contact with heavy metal, we obtain submicrometer-size nonreciprocal devices of high efficiency. Thus, our work contributes to the emerging branch of energy-efficient magnonic logic devices, giving rise to the possibility of application as a signal-processing unit in the digital and analog nanoscaled spin-wave circuits.

Figure 1

(a) Schematic representation of the multilayer stack and the geometry considered. The layer sequence consists of two ferromagnetic films, FM1 and FM2, separated by the nonmagnetic layer (NM). In FM2, the IDMI is induced by the proximity with the heavy metal (HM). Generally, this structure underlies the unidirectional coupling in a wide range of frequency. (b) Definition of the transmission lengths $x_{\mathrm{tr}}$ and $x_{\mathrm{tr}2}$ on the basis of the SW in the coupled FM bilayer system.

Figure 2

(a),(b) Dispersion relation of SWs as a function of wavevector $k$ in the $\mathrm{Py}$(3)/NM(5)/$\mathrm{Co}$(2)/$\mathrm{Pt}$ multilayer for the IDMI constant in the $\mathrm{Co}$ layer (a) $D=0$ and (b) $D=-0.7{\mathrm{mJ}/\mathrm{m}}^2$. For reference, we show the dispersion relation of SWs in the uncoupled $\mathrm{Pt}$/$\mathrm{Co}$ and $\mathrm{Py}$ layers with dashed and dotted lines, respectively. In the insets in (b), we show the $m_x$ amplitude of the SWs propagating in both directions at 15.2 GHz. For $D=0$, the dispersion relation is almost symmetrical with respect to $k=0$. For $D=-0.7{\mathrm{mJ}/\mathrm{m}}^2$, the IDMI breaks the symmetry leading to strongly coupled modes in the $+k$ range (insets 3 and 4) and single-layer excitation in the $-k$ range (insets 1 and 2). The highest coupling occurs in the region of overlapping of the dispersion relation of the uncoupled layers. (c) Propagation of the SW at 15.2-GHz frequency in the $\mathrm{Py}$(3)/NM(5)/$\mathrm{Co}$(2)/$\mathrm{Pt}$ multilayer. The antenna is located in the $\mathrm{Py}$ layer, below the indent in the $\mathrm{Co}$/$\mathrm{Pt}$ layer. SW propagating in the $+x$ direction transfers back and forth between $\mathrm{Py}$ and $\mathrm{Co}$ layer. The transmission to the $\mathrm{Co}$ layer in the $-x$ direction is weak, and most of the SW intensity remains in the $\mathrm{Py}$ layer.

Figure 3

Power-transfer factor $F_P$ (the vertical axis) and energy-distribution factor $F_E$ (color scale) as a function of the wavevector in the $\mathrm{Py}$(3)/NM($t$)/$\mathrm{Co}$(2)/$\mathrm{Pt}$ multilayer with the IDMI constant in the $\mathrm{Co}$ layer $D=-0.7{\mathrm{mJ}/\mathrm{m}}^2$ for three different thicknesses of the NM layer. We reach coupling parameters close to a maximum value of 1 in the range between $1.6\times {10}^7$ and $6.3\times {10}^7{\mathrm{m}}^{-1}$ (marked with dashed lines). In the negative $k$ range, the coupling is weak and reduces with the increase of the NM-layer thickness.

Figure 4

Transmission-length dependence on (a) the damping constant in the $\mathrm{Co}$ layer in the $\mathrm{Py}$(3)/NM(5)/$\mathrm{Co}$(2)/$\mathrm{Pt}$ multilayer with the damping in the $\mathrm{Py}$ ${\alpha }_{\mathrm{Py}}=0.005$ and (b) the thickness of NM layer in the $\mathrm{Py}$(3)/NM($t$)/$\mathrm{Co}$(2)/$\mathrm{Pt}$ multilayer. The source of the SW of 15.2 GHz is located in the $\mathrm{Py}$ layer. In (a), the simulations (Sim) results are compared with Eqs. (13) and (14) from the coupled-mode theory (CMT). In (b), we present only the simulation results for $x_{\mathrm{tr}}$.

Figure 5

(a),(b) Propagation of the SW ($m_x$ component) in a diode at 15.2-GHz frequency in $\mathrm{Py}$(3)/NM(5)/$\mathrm{Co}$(2)/$\mathrm{Pt}$ in (a) forward and (b) reverse direction for the width of the $\mathrm{Co}$/$\mathrm{Pt}$ stripe 320 nm. In (a), the SW propagation direction is opposite to the coupling direction, so that the SW transfers weakly to the $\mathrm{Co}$ stripe, and we get a signal of high intensity in the output. In (b), the SW propagation direction is the same as the coupling direction, so that the SW transfers to the $\mathrm{Co}$ stripe, where is strongly damped, leading to the low intensity of the signal at the output. (c) The power loss in the forward and reverse direction in dependence on the frequency. In a broad frequency range of 7 GHz, the SW diode preserves its strong isolating properties. (d) The power loss in the forward and reverse direction in dependence on $\mathrm{Co}$ stripe width.

Figure 6

(a),(b) Model of the four-port SW circulator based on the multilayered structure with unidirectional coupling. Propagation of the SW ($m_x$ component) of 15.2-GHz frequency in the lossless Pt/$\mathrm{Co}$(2)/NM(15)/$\mathrm{Py}$(3)/NM(15)/$\mathrm{Co}$(2)/$\mathrm{Pt}$ circulator in the (a) noncoupling direction and (b) coupling direction. In (a), the antenna A is located in the upper-right corner. The SW transfers weakly to the $\mathrm{Py}$ layer, so it goes mainly to port P2. In (b), the antenna A is located in the upper-left corner. The SW transfers from the upper $\mathrm{Co}$ layer to $\mathrm{Py}$, and after the reflection from the right side of $\mathrm{Py}$, it transfers to the lower $\mathrm{Co}$ layer, reaching port P3 in the end. A small-amplitude signal visible at isolated ports is a result of weak direct magnetostatic coupling between $\mathrm{Co}$ layers. (c),(d) The power loss measured in the output ports in regard to the input port in dependence on the NM-layer thickness for the input located in (c) port P1 and (d) port P2. In (c), the power loss in the target port P2 is increasing and in the rest of the ports is decreasing with the increase of NM-layer thickness. In (d), the power loss in the target port P3 and port P1 is fluctuating due to the resonance in the $\mathrm{Py}$ stripe. The minima of the power loss are corresponding to the $\mathrm{Py}$ stripe width fulfilling the resonance condition. (e),(f) The power loss measured in dependence on the damping constant in the $\mathrm{Co}$ layer for the input located in (e) port P1 and (f) port P2. In (e), the power loss in port P4 is almost constant, ultimately reaching the value in the target port P2. In (f), the power loss in the target port P2 is significantly larger than in the other ports. The SW circulator is not working properly for ${\alpha }_{\mathrm{Co}}>0.025$.

Figure 7

Three-port SW circulator in (a)–(d) easy-input and (e)–(h) easy-output port P3 configurations. We present the models (a),(e) and the way of acting when the input of SWs is localized in (b),(f) port P1, (c),(g) port P2, and (d),(h) port P3. The coupling direction determines the direction in which the SW will be transmitted from one layer to another. The orange arrows directing from one layer to another denote the SW transmission.