Biquadratic magnetic coupling effect in CoPt/Cr/Fe₉₀Co₁₀ orthogonal structures

Since the discovery that magnetization dynamics can be controlled by the spin-transfer torque (STT), ^1,2) extensive research has been conducted on its applications in data storage and memory. In particular, STT has been employed in various cutting-edge technologies including microwave-assisted magnetic recording in hard-disk drives and magnetoresistive random-access memories. ^3–8) Recently, the emergence of spin-torque oscillations (STOs) has sparked considerable interest for their potential applications in neural networks. ^9–11) While conventional ferromagnetic transition metals exhibit STO frequencies in the order of several gigahertz, ^12–15) antiferromagnetic materials can theoretically manifest STO frequencies reaching the terahertz range. ^16–29) This assumption is grounded in various theoretical investigations, including those concerning experimental magnetic-resonance frequencies. However, the practical realization of usable STOs in antiferromagnets remains elusive, mainly because of the large exchange-coupling interactions between neighboring ions. In response to this challenge, artificial magnetic structures have been proposed as promising approaches for increasing the STO frequency. These structures encompass synthetic antiferromagnetic coupling layers ^30–32) and quasi-antiferromagnets featuring biquadratic magnetic coupling, ^33,34) offering a potential alternative for amplifying the STO frequency. In contrast, orthogonal magnetization multilayers, comprising magnetic layers with both in-plane and perpendicular magnetic anisotropies, have demonstrated high spin-transfer efficiency, rendering them suitable candidates for generating STOs with elevated frequencies and reduced power consumption. ^35–43) However, the increased efficiency, which is attributed to the diminished angle between the magnetizations of the two layers, poses a conundrum. Specifically, this limits the current–density range within which a stable STO can be sustained as frequent magnetization switching events occur.

To address this challenge, we introduce biquadratic magnetic coupling ^{33,34,44–56)} between the two magnetic layers within the orthogonal configuration. This strategic addition suppresses complete magnetization switching, thereby enhancing the stability of the STO. Biquadratic magnetic coupling is an interlayer exchange interaction between two ferromagnetic (FM) layers separated by a thin spacer. In general, the magnetic coupling energy E is expanded to a higher-order equation by considering the quadratic term as E = −A₁₂ M₁M₂ −B₁₂ (M₁M₂)², where M₁ and M₂ are the unit magnetizations in the first and second FM layers, respectively, and A₁₂ and B₁₂ are the bilinear and biquadratic coupling coefficients, respectively. A₁₂ contributes 0° or 180° magnetic coupling, whereas B₁₂ contributes +/−90° magnetic coupling. Typically, +/−90° coupling is realized under the conditions of |A₁₂| < 2|B₁₂| and B₁₂ < 0, using a suitable layer as a spacer. The values of A₁₂ and B₁₂ depend on the spacer layer material and magnetic material, respectively, and values of 0 to –0.24 erg cm⁻² for A₁₂ and −0.005 to –2.0 erg cm⁻² for B₁₂ have been reported. ^{33,34,44–56)}

In a previous study, we had elucidated the efficacy of biquadratic coupling to realize quasi-antiferromagnetic Co₉₀Fe₁₀, resulting in an augmentation of the STO frequency. ^33,34) However, this enhancement was capped at 15 GHz when subjected to a realistic current density like 10.0 × 10⁷ A cm⁻², using a multilayer configuration featuring two in-plane magnetization layers. Subsequently, through comprehensive simulations, we successfully observed that the biquadratic magnetic field H _bq exerted an influence: it broadened the attainable current range of the STOs by effectively suppressing magnetization reversal. ^42,43) This effect is particularly useful for increasing the STO frequency in orthogonal structures. In light of these simulation results, the present study embarks on an experimental endeavor aimed at validating the presence of H_bq within an orthogonal structural arrangement and assessing its potential to suppress magnetization reversal. This experimental investigation is a pivotal step toward further elucidating the fundamental underpinnings of biquadratic coupling in orthogonal structures. In addition to the abstract of the International Conference on Solid State Device and Materials, ⁵⁷⁾ we present the results of the in-plane structure below.

In our experiments, we fabricated five different samples on a thermally oxidized Si wafer by DC sputtering, with all measurements reported in nanometers (nm), as schematically illustrated in Fig. 1. The sample configurations are as follows: Buffer/Ir₂₂Mn₇₈ (5)/Fe₉₀Co₁₀ (A) (2)/Cr t_Cr/Fe₉₀Co₁₀ (B) (2)/Cu 6/Fe₉₀Co₁₀ (C) (2.5)/Cap, where t_Cr is 1, 2, and 3 nm for Samples 1, 2, and 3, respectively, and Buffer/[Co (0.5)/Pt (0.5)]₄/Co (0.5)/Spacer/Fe₉₀Co₁₀ (2)/Cap where the Spacer is Cu 5 nm and Cr 1 nm for Samples 4 and 5, respectively. Samples 1–3 were designed as in-plane structures, to investigate the magnitude of the biquadratic magnetic coupling introduced by the Cr spacer layer and examine the relationship between this coupling and the thickness of the Cr film. After deposition, they were annealed with an in-plane field of 0.46 T at 270 K in vacuum for 2 h to pin the magnetization of Fe₉₀Co₁₀ (A) by an exchange coupling with the Ir₂₂Mn₇₈ layer. As a result, the magnetization of Fe₉₀Co₁₀ (B) was coupled with that of Fe₉₀Co₁₀ (A) at angles of +/− 90° through the Cr spacer layer. Notably, these samples featured an anomalous FM layer, Fe₉₀Co₁₀ (C), which served as a reference for magnetization.

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In contrast, Samples 4 and 5 were configured as orthogonal structures. Sample 4 with a 5 nm Cu spacer ensured that there was no magnetic coupling between the bottom [Co (0.5)/Pt (0.5)]₄/Co (0.5)/and top Fe₉₀Co₁₀ (2) layers. In contrast, the 1 nm Cr spacer of Sample 5 was employed to establish strong biquadratic magnetic coupling between the top and bottom layers. When an external magnetic field was applied perpendicular to the plane, the magnetization direction of the top layer aligned upward with the field. Therefore, both the samples exhibited opposing static magnetic fields, relative to the applied external field. However, when the spacer was composed of Cr, the biquadratic magnetic coupling introduced an additional field, denoted as H_bq . The direction of H_bq was opposite to that of the external magnetic field, with the anticipated effect of suppressing magnetization reversal.

The magnetization was measured at RT using a vibrating-sample magnetometer and superconducting quantum interference device. The roughness of Cr and Cu was measured using X-ray reflection (XRR) with SmartLab (Rigaku) and GenX was used to analyze the XRR profile. ⁵⁸⁾ In GenX, the figure of merit (FOM) serves as an indicator of the fitting curve's alignment with the experimental data. We achieved reasonable results by iteratively repeating XRR fittings until the FOM reached a value below 1.1 e⁻¹.

Figure 2 presents the MH curves for the in-plane–structured samples with Cr spacer thicknesses of 1, 2, and 3 nm, as the magnetic field is swept in a direction perpendicular to the Fe₉₀Co₁₀ (A) layer. Notably, the MH curves reveal two distinct magnetization reversals, each of which corresponds to a particular magnetization sequence depicted in states (1)–(4), as illustrated in Fig. 2. In the state (1), the magnetization of the sample becomes fully saturated in an alignment with the direction of the applied magnetic field. As the system evolves from the state (1) to the state (2), the magnetization within the Fe₉₀Co₁₀ (A) layer gradually reorients itself from the direction of the applied field toward the pinned direction, facilitating near-zero-field flipping of the Fe₉₀Co₁₀ (C) layer in response to the applied magnetic field. In the state (2), the Fe₉₀Co₁₀ (A) layer is fixed in one direction by the Ir₂₂Mn₇₈ layer, while the magnetization of the Fe₉₀Co₁₀ (B) layer is coupled at an angle of ±90° relative to the Fe₉₀Co₁₀ (A) layer. While transitioning from the state (2) to the state (3), the biquadratic magnetic coupling between the Fe₉₀Co₁₀ (A) and Fe₉₀Co₁₀ (B) layers dissolves, causing the magnetization to align gradually in the direction of the applied magnetic field. In the state (3), the magnetization attains saturation parallel to the direction of the applied field. Notably, the state (4) mirrors the state (2) in terms of the magnetic configuration. Importantly, the applied magnetic-field values at which magnetization reversal occurs in states (2) and (4) provide a means to evaluate the strength of the biquadratic magnetic coupling.

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Figure 2 clearly illustrates that, among the samples tested, the one featuring a 1 nm Cr spacer exhibits the most robust biquadratic magnetic coupling. As the thickness of the Cr film increases, the strength of this coupling gradually diminishes, with the lowest value observed for 3 nm Cr. Specifically, the magnetic field at the second magnetization reversal, corresponding to the strength of the magnetic coupling, is 0.01 T, 0.006 T, and 0.005 T for the spacer of the 1 nm, 2 nm, and 3 nm Cr layer, respectively. These findings underscore the superiority of the 1 nm Cr spacer as the optimal spacer, among Samples 1–3, to achieve strong and effective biquadratic magnetic coupling. Therefore, we use the 1 nm Cr spacer for the orthogonal structure in the following. Here, the magnetic coupling depends on the roughness of the spacer. Previously, the cross-sectional TEM image and XRR analysis showed that the spacer roughness is about 0.3 nm. ^33,56) Such a small roughness is necessary to restrain the orange peel coupling, leading the smaller bilinear coupling than the biquadratic coupling.

Figure 3 presents the MH curves for the orthogonal structure samples, featuring both Cr and Cu spacers, with the magnetic field swept perpendicular to the sample plane. Notably, our observations confirm that the Co/Pt bottom layer exhibits perpendicular magnetic anisotropy with a coercivity (H_c) of approximately 0.03 T. In contrast, the saturation field for the Fe₉₀Co₁₀ top layer differs, measuring 2 and 1.6 T for the samples with Cr and Cu spacers, respectively. Also, we show the XRR data for Sample 4–5 in Fig. 4. The fitting results showed that the spacer Cr and Cu have roughness of 0.10 nm and 0.41 nm, respectively. What is important is that the roughness of the Cr spacer is as small as 0.10 nm. When the spacer is thin and its roughness is large, the magnetic coupling like the orange peel coupling caused by the roughness is not negligible. Compared to this, when the spacer is as thick as 5 nm, the magnetic coupling originated from the roughness is small enough to be ignored. Hence, it is not needed to consider the effect of the roughness on the magnetic coupling through the Cr spacer with sufficient small roughness and the Cu spacer with 5 nm thickness. Note that the deviation observed in the curve during the full reflection stage may be attributed to the effect of sample fringe because the sample size is small.

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In the high-field regime exceeding +/− 2 T [i.e. for states (1) and (3)], the collective magnetization of both the bottom and top layers becomes saturated along the z-direction. As the size of the applied field is reduced below +/− 1.6 T, state (2), the Fe₉₀Co₁₀ top layer begins to rotate, aligning itself along the easy axis within the sample plane. This clearly demonstrates the anticipated orthogonal structure of the magnetized films in the top and bottom layers.

The observed disparity in the saturation field (H_sat) for Fe₉₀Co₁₀ is attributed to the biquadratic magnetic coupling between these two magnetic layers. Specifically, we estimate the biquadratic coupling parameter (B₁₂) to be −0.6 for the Cr spacer and 0.0 for the Cu spacer, within this structural context.

Consequently, our findings confirm the successful realization of an effective field arising from biquadratic coupling in an orthogonal magnetization structure. This implies that a higher electrical current is required to induce magnetization inversion when Cr serves as the spacer, thereby broadening the range of electrical current levels capable of sustaining a stable STO through the introduction of biquadratic magnetic coupling.

4.1. Simulation model

In our simulations, the STO was investigated by solving the Landau–Lifshitz–Gilbert (LLG) equation, which incorporates an STO term. The effective field (H_eff ) guiding the dynamics of the system is composed of the following constituents: H_eff = H_K + H_exf + H_ex + H_ST + H_bl + H_bq , where H_K represents the magnetic anisotropy field; H_exf denotes the influence of the external magnetic field; H_ex characterizes the exchange coupling field determined by the exchange stiffness constant A; H_ST accounts for the stray field dependent on the saturation magnetization of the magnetic material; H_bl signifies the bilinear exchange field determined by the bilinear coefficient A₁₂; and H_bl designates the biquadratic exchange field determined by the biquadratic coefficient B₁₂. When considering two FM layers, denoted as i and j, with normalized magnetizations m_i and m_j , respectively, the biquadratic exchange field (H_bq ) can be expressed as follows: ⁵⁹⁾

$$\begin{eqnarray}&&\begin{array}{l}{{\bf{H}}}_{{\rm{bq}},i}=\displaystyle \frac{2{B}_{12}}{{M}_{s,i}{d}_{i}}\left[\begin{array}{l}{m}_{i,x}{m}_{j,x}^{2}+{m}_{j,x}\left({m}_{i,y}{m}_{j,y}+{m}_{i,z}{m}_{j,z}\right)\\ {m}_{i,y}{m}_{j,y}^{2}+{m}_{j,y}\left({m}_{i,x}{m}_{j,x}+{m}_{i,z}{m}_{j,z}\right)\\ {m}_{i,z}{m}_{j,z}^{2}+{m}_{j,z}\left({m}_{i,x}{m}_{j,x}+{m}_{i,y}{m}_{j,y}\right)\end{array}\right]\\ \,\times \,{\rm{with}}\,i\ne j,\end{array}\end{eqnarray} \tag{ 1 }$$

where M_s,i denotes the saturation magnetization of the magnetic material in the top layer. The xy plane and z-axis are in the in-plane and perpendicular directions, respectively, as shown in Fig. 5.

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In the orthogonal configuration, we designate the top and bottom FM layers as i and j, respectively, each exhibiting distinct magnetic anisotropy characteristics: i with in-plane anisotropy and j with perpendicular anisotropy. In this context, the biquadratic exchange field for layer i, denoted by H_bq,i , is formulated as follows:

$$\begin{eqnarray}&&\begin{array}{c}{{\bf{H}}}_{{\rm{bq}},i}=\displaystyle \frac{2{B}_{12}}{{M}_{s,i}{d}_{i}}\left[\begin{array}{c}0\\ 0\\ {m}_{i,z}\end{array}\right].\end{array}\end{eqnarray} \tag{ 2 }$$

This relationship indicates that the direction of H_bq aligns with m_j when B₁₂ > 0 and opposes m_j when B₁₂ < 0. Given that the biquadratic magnetic coupling becomes prominent under condition B₁₂ < 0, the direction of H_bq counters m_j when the biquadratic exchange magnetic coupling manifests. The STT term within the LLG equation aligns parallel to m_j . Consequently, the biquadratic exchange coupling H_bq assumes a direction opposite to that of the STT. Considering that H_bq plays a dominant role within the effective field H_eff , along with H_ST , the biquadratic exchange coupling H_bq emerges as a pivotal factor in averting complete magnetization reversal from an in-plane to a perpendicular orientation, thereby promoting the stability of STO. Consequently, the magnetic moment of the top layer is expected to exhibit gyroscopic motion around the z axis. Here, we did not consider the Joule heating effect on magnetization dynamics.

In our proposed model, the orthogonal configuration comprises a structure with a 2 nm thick FePt bottom layer, 2 nm spacer, and Co₉₀Fe₁₀ top layer. The purpose that we selected FePt is to obtain a large H_k, which is essential for maintaining the perpendicular magnetization of the bottom layer. However, in our experimental setup, due to the challenges associated with growing FePt at high temperatures and its more complex preparation process, we opted for the alternating sputtering of Co/Pt to create the bottom layer. This Co/Pt layer also exhibits a large H_k, enabling the maintenance of perpendicular magnetization within an orthogonal structure. The device has a disk-shaped geometry with a diameter of 320 nm. The FePt bottom layer exhibits perpendicular anisotropy, characterized by a coercivity (H_k ) of 15000 Oe and saturation magnetization (M_s) of 800 emu cc⁻¹. ^60,61) In contrast, the top layer exhibits in-plane magnetic anisotropy. Specifically, Co₉₀Fe₁₀ exhibits a H_k of 35 Oe, M_s of 1450 emu cc⁻¹, and damping constant (α) of 0.035. ³³⁾

For simplicity, we set the bilinear coefficient A₁₂ to 0 and biquadratic coefficient B₁₂ to −0.6, where B₁₂ corresponds to an experimentally reported value for the spacer Cr. Both A₁₂ and B₁₂ represent values based on previous reports.

Our simulation employed a magnetic unit cell of size 5 nm × 5 nm × 2 nm, and the calculations were conducted on a grid comprising 64 × 64 × 3 cells. The exchange stiffness constant A was set to 1 × 10⁻⁶ erg cm⁻¹ and the gyromagnetic ratio γ was 1.76 × 10⁷ Oe⁻¹ s⁻¹. The simulation time step (dt) was set as 10 fs.

In the initial phase, we computed the magnetization relaxation following saturation along the y axis. Subsequently, current was supplied to the layers, originating from the top and proceeding to the bottom layer.

4.2. Simulation results and discussion

We present the simulated time-domain magnetization precessions when B₁₂ is 0 and −0.6 in Figs. 6(a)–6(c) and the corresponding Fast Fourier transform (FFT) spectra in Figs. 6(d)–6(f). The applied electrical current densities are 10.0 × 10⁷ (a), 15.0 × 10⁷ (b), and 20.0 × 10⁷ A cm⁻² (c). At a relatively low current density of 10.0 × 10⁷ A cm⁻², as shown in Figs. 6(a) and 6(d), a stable STO can be achieved regardless of whether B₁₂ is set to 0.0 or −0.6. However, upon increasing the current density to 15.0 × 10⁷ A cm⁻², as depicted in (b) and (e), the STO becomes unstable at B₁₂ = 0.0 owing to magnetization switching, whereas the STO persists at B₁₂ = −0.6. With a further increase in the current density to 20.0 × 10⁷ A cm⁻², as shown in (c) and (f), the STO becomes small and eventually the magnetization reverses in both cases of B₁₂ at 0.0 and −0.6. Additionally, the sample with B₁₂ = −0.6 exhibited a more stable STO with an almost immutable M_x /M_s , a higher quality factor Q, and higher frequency than that with B₁₂ = 0.0. These results indicate that, by introducing biquadratic magnetic coupling, we can expand the range of the electrical current density to achieve a stable STO and increase the frequency. Thus, higher current densities can be employed, leading to higher STO frequencies. Therefore, a spacer material with high biquadratic magnetic coupling is required for a stable HF STO.

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Our investigation into the magnetic properties of the biquadratic magnetic coupling within the Fe₉₀Co₁₀/Cr/Fe₉₀Co₁₀ layer revealed a noteworthy dependence on the thickness of the Cr spacer. In the descending order of coercivity, we observed the following trend: Cr 1 nm > Cr 2 nm > Cr 3 nm. This compelling finding positioned Cr as a promising candidate for achieving HF current-driven magnetization oscillations. Furthermore, we successfully fabricated multilayer structures featuring orthogonal structures, confirming the presence and efficacy of biquadratic magnetic coupling. This implied that a larger electrical current was required to induce magnetization inversion when Cr was used as the spacer. This augmented the electric-current region capable of sustaining a stable STO. Through a comprehensive numerical investigation, we conducted an in-depth exploration of the STO within magnetic multilayers featuring orthogonal magnetization configurations. By strategically introducing biquadratic magnetic coupling, we achieved a significant expansion in the range of electric current levels conducive to a stable STO. In summary, our study not only advances our understanding of STO within orthogonal magnetization structures but also underscores the pivotal role of biquadratic magnetic coupling in enhancing the operational range of stable STOs. These findings hold significant promise for the development of the next-generation spintronic devices and their applications.

This study was supported by the Japan Society for the Promotion of Science, KAKENHI (Grant No. JP19K04471, JP22H01557), Micron Foundation, MEXT Initiative to Establish Next-Generation Novel Integrated Circuit Centers (X-NICS), The Center for Spintronics Research Network (Osaka), and JST, the establishment of university fellowships towards the creation of science and technology innovation, Grant No. JPMJFS2132.