Current-driven antivortex core resonance measured by the rectifying effect

An antivortex confined in a magnetic cross-shaped wire is a topological counterpart to a vortex. The antivortex has an in-plane rotational magnetization. The magnetization orientation angle φ with respect to the x axis (see Fig. 1(a)) can be described as φ = vϕ − C(π/2) with azimuthal angle ϕ.^1–3 Here, v and C are a vorticity (v = + 1: vortex, v = − 1: antivortex) and a chirality (magnetization orientation C = ± 1), respectively. The antivortex has a nano-scale core that consists of an out-of-plane magnetization component with a polarity of P = + 1 (upward) or P = − 1 (downward).^3–7 The core strongly interacts with conduction electrons in the same as vortex due to the nonadiabatic contribution to the spin torque so-called β term,^8–10 which plays an important roll for domain wall devices.¹¹ Moreover, the single antivortex has a geometrical advantage compared with the vortex. For the electrical detection of single vortex confined in a ferromagnetic disk, electrodes must be placed close to the vortex core so that the influence of Oersted field from the electrode cannot be neglected. On the contrary, in the case of the antivortex in the cross intersection, the electrode contacting on wires can be separated from the intersection. In fact, as reported by Kamionka et al.,¹² the antivortex core dynamics can be dominantly excited by spin transfer torque. Therefore, the antivortex core is favorable for understanding the current-driven magnetization dynamics.

The antivortex core dynamics can be measured by current-induced rectifying effect. An application of a radio-frequency (rf) current excites a gyration of the antivortex core^3,13–15 in a harmonic potential of the crossed wire. Then, the rectified direct-current (dc) voltage arises at the antivortex due to the temporal oscillation of the resistance.^16–20 By measuring the frequency-domain rectified voltage, we can understand details of the magnetization dynamics and its driving torque.

In the previous studies, the antivortex core dynamics has been measured by using ∞-type and φ-type devices.^12,21,22 To precisely measure the current-driven dynamics of antivortex core, the X-type crossed wire is preferable because there is no short circuit among the wires.²³ However, the current-driven antivortex core dynamics has not been studied enough in this construction.

This study demonstrates the current-driven dynamics of a single antivortex core confined in an X-type Ni₈₁Fe₁₉ crossed wire measured using the rectifying effect. We succeeded in detecting peaks owing to the resonant oscillation of the antivortex core and found that the resonant frequency and amplitude of the peaks depend on the external magnetic field. These results are in good agreement with the micromagnetic simulation.

Figure 1(a) shows an optical micrograph of a sample consisting of Ni₈₁Fe₁₉ crossed wire and electrodes for a rectified voltage measurement. The Ni₈₁Fe₁₉ crossed wire was fabricated by electron beam lithography and a lift-off technique. The Ni₈₁Fe₁₉ crossed wire with a thickness of 55 nm and a wire width of 1.2 μm was deposited onto a thermally oxidized Si substrate, after which 115 nm-thick and 1.2 μm-wide Ti/Au electrodes were fabricated. The two ends of the crossed wire were sharpened to avoid domain wall nucleation whereas another two were widened to promote nucleation. The Cartesian coordinate system used in this study is shown in Fig. 1(a). The origin of the coordinate axes corresponds to the center of the intersection of the crossed wires.

The magnetic antivortex can be successfully formed by an external magnetic field in the following sequence. First, the magnetic field of B_x = ± 100 mT saturated the magnetization along the x axis. Afterward, the magnetic field ∓B_stop was applied opposite to the initially magnetized direction, and was reduced to 0 mT. When the magnetic field B_stop exceeds the switching field of the widened wire, a domain wall forms and propagates to the cross intersection, where the antivortex forms stochastically. In our previous study, we found that antivortex creation can be determined by the resistance increase ΔR of the cross intersection.²⁴ Figure 1(b) shows a minor hysteresis loop of the resistance at the cross intersection measured at B_stop = 7.5 mT. The resistance shows a sudden increase at B = 7 mT which leads to a resistance increase over 7 mΩ at the remanent state. Using magnetic force microscopy, it was confirmed that the antivortex formed in the cross intersection as shown in Fig. 1(c) when the resistance rose above 7 mΩ at the remanent state. In this study, the magnetization direction was fixed to a chirality C = − 1 (Fig. 1(c)).

The resonance of the antivortex core was measured from the frequency domain dc voltage through the rectifying effect.^16–20 Figure 1(d) shows a schematic diagram of the measurement circuit. The rf current with a power of 2.5 mW was injected into the crossed wire. The rectified voltage was measured by the nano-voltmeter connected to the bias tee which can separate a dc and rf voltage. The frequency domain rectified voltage was measured in the range from 50 to 150 MHz. An external magnetic field was applied at an angle ϕ in the range from B = 0 to the annihilation field of antivortex. Figure 2 shows the rectified voltage spectra of the current-driven antivortex core resonance measured at ϕ = 45 ^∘ (Fig. 2(a)), 135^∘ (Fig. 2(b)) 225^∘ (Fig. 2(c)) and 315^∘ (Fig. 2(d)). Note that, peaks representing a gyration of the antivortex core were observed at approximately 100 MHz (see rectangular areas of the red broken line in Fig. 2).

To confirm that these peaks arise from the antivortex core resonance, we calculated the temporal evolution of the antivortex core using the Object Oriented MicroMagnetics Framework (OOMMF).²⁵ Figure 3(a) shows the magnetization configuration of the antivortex confined in the crossed wire with a width of w = 1.2 μm and a thickness of t = 55 nm. The size of the numerical grid is 5 × 5 × 5 nm. The material parameters are typical for Ni₈₁Fe₁₉ : the damping constant is α = 0.01, the saturation magnetization is M_s = 1.08 T, and the exchange stiffness constant is A = 1.3 × 10⁻¹¹ J/m. The initial state was set to the antivortex configuration with C = − 1. We calculated the temporal evolution of the magnetization after applying a step change in the magnetic field from 0 to 1 mT along the x axis. Red and blue curves in Fig. 3(b) show the x and y components of the volume-averaged magnetizations M_x and M_y, respectively. It is noted that M_y has a phase difference of +π/2 with respect to M_x. This is clear evidence that the antivortex core gyrates in the film plane. Figure 3(c) shows the Fourier spectrum of the temporal evolution of M_x. The spectrum has a local maximum at 120 MHz. This result is in good agreement with the experimental result of resonance frequency f_r = 100 MHz, therefore, the peaks of the rectified spectra arise from the antivortex core resonance.

From Figs. 2(a)-2(d), we can determine three aspects of the nature of the resonance peaks: first, that the peak intensity is proportional to the external field B; second, that increasing the external field leads to a red shift of the resonance frequency f_r, which gradually decreases from 93 to 72 MHz owing to an increase in the external field from B = 3 to 7 mT as shown in Fig. 2(b); third, that the intensity of the rectified spectra drastically changes when the antivortex is annihilated. For example, the rectified spectra in Figs. 2(c) and 2(d) change to complicated shapes at 6 and 9 mT, respectively. Moreover, the resonance peaks in Figs. 2(b) and 2(d) vanish at 9 and 6 mT, respectively.

To discuss these tendencies in detail, we fitted the resonance spectra in Fig. 2(b) using the following function:

Here, f, f_r, and Δ are the frequency of the rf current, resonance frequency, and full-width at half maximum, respectively. The coefficients A_SL and A_AL are the amplitudes of the symmetric- and asymmetric-Lorentzian components, respectively. Figure 4(a) shows the field dependence of the symmetric-Lorentzian amplitude A_SL at the field orientation ϕ = 135 ^∘. The example of the fitted curve is shown in the inset of Fig. 4(a); the amplitude, A_SL, increases with increasing the external field in the range 3 mT <B < 7 mT. The increase in A_SL with B is attributed to the change in the magnetic structure of the antivortex. The rectified voltage is generated by the multiplication of the rf current and a temporally varying resistance whose amplitude depends on the position of the antivortex core owing to the anisotropic magnetoresistance effect. The resistance increases with the movement of the antivortex core along the rf current direction although the displacement of the antivortex core normal to the rf current direction leads to a decrease of the resistance. When the antivortex core gyrates around the center of the intersection, the rectified voltage is not observed because the resistance oscillates with frequency 2f. When the antivortex core moves from the center of the intersection, the resistance consists of the superposition of the temporally varying components with frequencies f and 2f. The frequency f component generates the rectified voltage. In particular, symmetry breaking of antivortex core gyration is essential for the electrical detection of the gyration. Consequently, the amplitude of the rectified spectra increases with the external magnetic field owing to the displacement of the antivortex core from the center of the intersection.

Figure 4(b) shows the field dependence of the resonance frequency f_r with ϕ = 135 ^∘. The resonance frequency decreases with increasing the external field in the range 3 mT <B < 7 mT. This is attributed to a higher-order contribution of the potential well in the crossed wire. By considering the magnetostatic energy, the potential energy of the antivortex core can be phenomenologically written in the following form; U = (κ + κ′r²) r²/2.^3,14,26 Here, r is the position of the antivortex core and κ is the phenomenological spring constant of the antivortex core. A coefficient κ′ is the higher-order contribution of the potential well. The potential energy U can be described as a harmonic potential when the displacement of the antivortex core from the center of intersection is small compared with the wire width. When the displacement of the antivortex core is large, we cannot neglect the higher-order contribution κ′r² to the resonant frequency.²⁷ In this case, the resonance frequency f_r can be determined by the effective spring constant κ_eff = κ + κ′r² ∝ f_r. Consequently, the resonance frequency changes owing to the displacement of antivortex core r driven by the external field. From the experiment, we found that the higher-order contribution κ′ is negative because the resonance frequency f_r decreases with the increase of the the antivortex core displacement. This suggests that the potential gradient is suppressed when the antivortex core gets closer to the wire. In the case of vortex in a circular disk, the sign of κ′ is positive^28,29 which is opposite to our result. The positive κ′ of vortex is attributed to the side of the circular disk which exerts a strong restoring force on the vortex. The increase in restoring force enhances the gradient of potential well, which results in positive κ′. On the other hand, the negative κ′ of antivortex is attributed to the shape of the crossed wire. In the Ni₈₁Fe₁₉ wire, the restoring force along the longitudinal direction of the wire is weaker than that along the transverse direction. Therefore, the restoring force is suppressed when the antivortex core moves from the center of crossed wire and gets closer to the four wires, which results in negative κ′.

At the annihilation field of the antivortex, the rectified spectra drastically changed as indicated by dotted spectra in Fig. 2. In this figure, the annihilation field, B_an, depends on the field orientation angle ϕ. This can be explained by the difference in the switching fields between the sharpened and widened wires. Figure 5 shows the schematic magnetic domains of the crossed wire with C = − 1 after the application of the annihilation field B_an. Figures 5(a)-5(d) are the magnetic domains corresponding to the field orientations of ϕ = 45 ^∘, 135^∘, 225^∘, and 315^∘, respectively. The application of the annihilation field B_an switches the magnetization of the yellow wires, as shown in Fig. 5. Then, the antivortex is subsequently annihilated. It is noted that the switching field of the widened wire is lower than that of the sharpened wire. Namely, B_an with ϕ = 45 ^∘ and 315^∘ (Figs. 2(a) and 2(d), respectively) are lower than the others. In fact, the switching fields of the widened and sharpened wires measured from the resistance change were estimated as B_sw,wide = 5 ± 1 mT and B_sw,nar = 11 ± 3 mT, respectively. This result agrees well with the annihilation field B_an in Fig. 2. Consequently, the antivortex core resonance must be detected below the annihilation field B_an. In particular, in the case where the external field is applied so as to switch the sharpened wire, the resonance peak can be measured in a wide range of the external fields.

In summary, we have demonstrated the resonance of a single antivortex core in a crossed Ni₈₁Fe₁₉ wire using the current-induced rectifying effect. The resonance frequency agreed with the eigen frequency of the antivortex core calculated by OOMMF. From the field-dependence of the resonant spectra, we evaluated the annihilation field of the antivortex and higher-order contribution of the potential well in the crossed wire. Furthermore, using the crossed wire with an asymmetrically shaped wire end, we succeeded in stabilizing the antivortex over a wide range of external magnetic field. These results will provide a significant basis for the study of current-driven antivortex core dynamics.

This study was supported by a Grant-in-Aid for JSPS Fellows.