Nonreciprocal magnon propagation in a noncentrosymmetric ferromagnet ${\text{LiFe}}_5{\text{O}}_8$

In noncentrosymmetric materials, the relativistic effect extensively modifies the energy band of magnons as well as that of electrons. With use of microfabricated microwave antennae, we have demonstrated that the propagation of magnons with large momentum is nonreciprocal in a noncentrosymmetric ferromagnet ${\mathrm{LiFe}}_5{\mathrm{O}}_8$. The nonreciprocity is clearly explained by the effect of the asymmetric magnon band originating from the relativistic Dzyaloshinskii-Moriya interaction. This result may pave a new path to designing a magnonic device based on the relativistic band engineering.

Figure 1

(a) Noncentrosymmetric crystal structure of ${\mathrm{LiFe}}_5{\mathrm{O}}_8$. (b) Magnetization curves for the experimentally used ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ sample for various magnetic field directions. Inset illustrates the sample shape and crystal orientation. (c) Sketch of the expected magnon dispersion in the ferromagnetic state of ${\mathrm{LiFe}}_5{\mathrm{O}}_8$. The ferromagnetic exchange and Dzyaloshinskii-Moriya interactions are dominant in the “exchange” region for the magnon propagation, while the magnetic dipole-dipole interaction is dominant in the “dipolar” region.

Figure 2

(a) Illustration of experimental setup for the absorption of uniform microwave. (b) and (c) Microwave absorption spectra $\mathrm{\Delta }{S}_{11}$ for (b) ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ and (c) ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ in various magnetic fields along $\left[1\overline{1}0\right]$ at room temperature. (d) Magnetic field dependence of peak frequency for ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ (open circle) and ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ (closed circle). Solid lines are the results of fitting with the theoretical formula Eq. (2).

Figure 3

(a) Experimental setup of nonreciprocal magnon propagation. The colored region indicates the Au/Ti electrodes. Meander microwave antennae are connected to two ports of the network analyzer. The rectangular sample is put on the top of the two antennae. (b) Illustration of the spatially oscillating magnetic fields induced by the electric current in the meander antenna.

Figure 4

(a) Sketch of the experimentally used meander antenna. The gray thick line shows the electric path. The arrows along the electric path indicate the electric current. (b) $y$-direction current $j_y$ as a function of $x$ at $y=0$. (c) Square of Fourier transformed current $|j_y{\left(k\right)|}^2$ at $y=0$. (d) The electric current along $x{j}_x$ as a function of $x$ at $y=-b$. (e) Square of Fourier transformed current $|j_x{\left(k\right)|}^2$ at $y=-b$.

Figure 5

(a) The microwave transmittance $S_{12}$ of the ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ sample-placed microwave circuit shown in Fig. 3 at 0.15 and 1.0 T ($\theta =0^{\circ }$). (b) The microwave transmittance owing to magnon propagation $\mathrm{\Delta }{S}_{12}$ at 0.15 T deduced from the difference of $S_{12}$ from the 1.0 T data. The inverted triangle shows the FMR mode. (c) The microwave transmittance $S_{12}$ of the ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ sample-placed microwave circuit shown in Fig. 3 at 0.15 and 1.0 T ($\theta =0^{\circ }$). (d) The microwave transmittance owing to magnon propagation $\mathrm{\Delta }{S}_{12}$ at 0.15 T deduced from the difference of $S_{12}$ from the 1.0 T data. The inverted triangle shows the FMR mode.

Figure 6

(a)–(d) Microwave transmittances owing to magnon propagation from port 2 to port 1 ($\mathrm{\Delta }{S}_{12}$) and from port 1 to port 2 ($\mathrm{\Delta }{S}_{21}$) for (a) and (b) ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ and (c) and (d) ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ are plotted as a function of microwave frequency at room temperature. Magnitude of the magnetic field is 0.15 T. The field direction is along $\theta =0^{\circ }$ in (a) and (c) and $\theta ={180}^{\circ }$ in (b) and (d). FMR ($k\sim 0$) signals are represented by closed triangles. (e) and (f) Nonreciprocity of microwave transmission $S_{12}-S_{21}$ of (e) ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ and (f) ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ at ${\mu }_{0}H=\pm 0.15$ T and $\theta =0^{\circ }$.

Figure 7

(a) and (b) Angular dependence of $\mathrm{\Delta }{S}_{12}$ and $\mathrm{\Delta }{S}_{21}$ for (a) ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ and (b) ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ at ${\mu }_{0}H=0.15$ T.

Figure 8

Integrated intensities of nonreciprocity $\mathrm{\Delta }{I}_{12}$ for ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ (open circle) and ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ (closed circle) are plotted as a function of the angle of magnetic field. The solid lines are merely guides to eyes.

Figure 9

(a) Illustration of the geometry used in the theoretical calculation of magnetostatic wave. The plate shows the sample crystal and [111], $\left[\overline{1}\overline{1}2\right]$, and $\left[1\overline{1}0\right]$ are the crystal orientation. (b) The dispersion relations of magnetization wave in ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ in the case of metallized boundary condition. The index $n$ classifies the magnon mode branches with different spatially oscillating patterns along the normal direction of the sample plate. The arrow shows the wavelength that can be excited by the antenna, $k=2\pi /(20\mu \mathrm{m}$). (c) Comparison of the $n=1$ dispersion relations of magnetostatic wave in ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ for free and metallized boundary conditions. The arrow shows the momentum $k$ excited by the microwave antenna, $k=2\pi /\left(20\mu \mathrm{m}\right)$. Inset shows the dispersion relation in the low $k$ region. (d) The $n=1$ dispersion relations of ${\mathrm{LiFe}}_5{\mathrm{O}}_8$ (solid lines) and ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$ (dashed lines). The metallized boundary condition is employed.