Measuring acoustic mode resonance alone as a sensitive technique to extract antiferromagnetic coupling strength

We have studied static and dynamic magnetic properties of a general asymmetric trilayer system using numerical simulations. For ferromagnetic, ${90}^{\circ }$, and antiferromagnetic coupling, the magnetizations of the two magnetic layers exhibit one, two, and three phases with increasing external field, respectively. The total magnetization and ferromagnetic resonance accordingly follow these phases of the magnetization vectors. The resonance condition is related to the interlayer coupling strength in such a way that a larger coupling constant yields a higher value of $f_{\mathrm{res}}/H$, where $f_{\mathrm{res}}$ is the resonance frequency at the external magnetic field $H$. Based on the simulation results, it is proposed that measurements of the acoustic mode resonance alone at unsaturated conditions provide a sensitive and accurate technique to extract the antiferromagnetic coupling strength. The technique is demonstrated experimentally with the broadband ferromagnetic resonance measurements of two trilayer films with weak and strong coupling strengths. The technique offers an efficient and sensitive method for antiferromagnetic coupling strength extraction, yielding coupling constant values with a precision of better than 0.03 erg/${\mathrm{cm}}^2$. Also, separation of the bilinear and biquadratic coupling contributions is possible with the technique.

Figure 1

The geometry and coordinate system of the film.

Figure 2

Simulated results of magnetization orientations ${\varphi }_{\text{A}}$ and ${\varphi }_{\text{B}}$ (a), magnetization (b), and FMR resonance frequency (c) versus in-plane magnetic field for different coupling strengths. The magnetization values of the two FM layers are $4\pi M_{\text{A}}=22$ kG and $4\pi M_{\text{B}}=9.5$ kG. Different coupling constants are represented by different colors of the curves, with the legend shown in (a). Solid and dashed lines in (c) correspond to acoustic and optic mode resonance curves, respectively. The red curve is covered by the magenta one in (c) as they overlap with each other. In (a) and (b), curves for all four sets of coupling constants overlap.

Figure 3

Simulated results of magnetization orientations ${\varphi }_{\text{A}}$ and ${\varphi }_{\text{B}}$ (a), magnetization (b), and FMR resonance frequency (c) versus in-plane magnetic field for different strengths of antiferromagnetic coupling. The magnetization values of the two FM layers are $4\pi M_{\text{A}}=22$ kG and $4\pi M_{\text{B}}=9.5$ kG. Different coupling constants are represented by different colors of the curves, with the legend shown in (a). In (c), the solid and dashed curves correspond to the acoustic and optic branches, respectively, of the dispersion relation.

Figure 4

(a) FMR resonance frequency versus magnetic field for a Si/${\mathrm{SiO}}_2$/FeNi/Ru/FeCo/Ta trilayer with a Ru thickness of 10 Å. Circles correspond to experimental data and curves to simulated results with solid curves representing the acoustic branch dispersion relation and dashed curves the optic branch dispersion relation. The simulation parameters for FeNi layer are $4\pi M_{\mathrm{FeNi}}=9.2$ kG and $g_{\mathrm{FeNi}}=2.10$ and for FeCo layer are $4\pi M_{\mathrm{FeCo}}=21$ kG and $g_{\mathrm{FeCo}}=2.18$, measured using single layer thin films of each material in previous studies [17, 34]. The inset shows three example FMR spectra measured with the magnetic fields 0.43 kOe (brown), 0.53 kOe (cyan), and 0.78 kOe (green). (b) A magnified view of (a) at unsaturated conditions. $\mathrm{\Delta }{J}_1$ and $\mathrm{\Delta }{J}_2$ refer to the deviation of coupling constants with respect to the best fitted $J_1$ and $J_2$ values (shown by the thick red curve).

Figure 5

(a) FMR resonance frequency versus magnetic field for a Si/FeCo/Ru/FeNi/Ru trilayer with a Ru thickness of 10 Å. Circles correspond to experimental data and curves to simulated results with solid curves representing the acoustic branch dispersion relation and dashed curves the optic branch dispersion relation. The simulation parameters for FeNi layer are $4\pi M_{\mathrm{FeNi}}=9.2$ kG and $g_{\mathrm{FeNi}}=2.10$ and for FeCo layer are $4\pi M_{\mathrm{FeCo}}=22$ kG and $g_{\mathrm{FeCo}}=2.10$, measured using single layer thin films of each material in previous studies [17, 35]. The inset shows a magnified view of the acoustic branch, with only the vertical axis zoomed. $\mathrm{\Delta }{J}_1$ and $\mathrm{\Delta }{J}_2$ refer to the deviation of coupling constants with respect to the best fitted $J_1$ and $J_2$ values (shown by the thick red curve).

Figure 6

Magnetization versus magnetic field. Circles correspond to experimental data and the solid curve is a fit of the macrospin model to the experimental results yielding the coupling constants $J_1=-0.18$ and $J_2=-0.025$ erg/${\mathrm{cm}}^2$. The right bottom inset shows the angles between the magnetic field and the magnetizations in the FeCo $\left({\varphi }_{\text{A}}\right)$ and FeNi $\left({\varphi }_{\text{B}}\right)$ layers. The left top inset is a magnified view of the main graph at unsaturated conditions. A few attempts with different coupling constants are also shown in this inset, with the violet curve corresponding to $J_1=-0.21$ erg/${\mathrm{cm}}^2$ and $J_2=-0.025$ erg/${\mathrm{cm}}^2$, the blue one to $J_1=-0.15$ erg/${\mathrm{cm}}^2$ and $J_2=-0.025$ erg/${\mathrm{cm}}^2$, the green one to $J_1=-0.18$ erg/${\mathrm{cm}}^2$ and $J_2=-0.05$ erg/${\mathrm{cm}}^2$, and the olive one corresponding to $J_1=-0.18$ erg/${\mathrm{cm}}^2$ and $J_2=0$ erg/${\mathrm{cm}}^2$.