Introduction

Magnetic anisotropy describes a magnetization-angle-dependent change in magnetic energy and stabilizes the magnetization in specific directions. Its angular dependence is determined by the symmetry of the crystal or structure. In thin-film heterostructures such as ferromagnet/normal metal bilayers where the structural inversion symmetry is broken at the interface, the magnetic anisotropy is dominated by interfacial contributions, as follows (up to the second order):

$$E(\theta ) = K_1^{\mathrm{eff}}{\mathrm{sin}}^2\theta + K_2{\mathrm{sin}}^4\theta$$
(1)

where \(K_1^{\mathrm{eff}}\left( { = K_1 - 2\pi M_{\mathrm{s}}^2} \right)\) is the effective first-order anisotropy energy density that comprises the demagnetization energy density (with K1 and Ms being the first-order anisotropy energy density and saturation magnetization, respectively), K2 is the second-order anisotropy energy density, and θ is the polar angle of the magnetization. The magnetic phase diagram as functions of \(K_1^{{\mathrm{eff}}}\) and K2 (Fig. 1a) shows several distinct magnetic states1. Among them, the out-of-plane state originating from perpendicular magnetic anisotropy (PMA) has been a main focus of spintronics research2 because it offers scalable magnetic random-access memories (MRAMs)3.

Fig. 1: Phase diagram showing various magnetic states and inverse thickness dependences of K1 and K2.
figure 1

a Magnetic phase diagram as functions of \(K_1^{{\mathrm{eff}}}\) and K2, showing four different magnetic states and their energy surfaces. b, c Inverse Co thickness dependence of K1 (b) and K2 (c) for the Pt/Co (tCo)/Cu structure. The error bars in K1 and K2 were obtained from three repeated measurements. The negative slope in (c) indicates negative interfacial K2.

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Recently, interest in another state—the easy-cone state, where the equilibrium magnetization direction is tilted from the film normal and forms a cone—has increased for the following reasons. It provides improved functionalities of various spintronics devices, such as low-power operation of spin-transfer torque (STT) MRAMs4,5,6 and zero-field precession of STT oscillators7. Moreover, it hosts spin superfluids associated with spontaneous breaking of U(1) spin-rotational symmetry8,9 and allows unique easy-cone domain wall dynamics10. The existence of the easy-cone state was experimentally verified in various layered structures6,11,12. However, the design window for forming a stable easy-cone state is very narrow6,11,12, which presents a critical challenge for realizing magnetic devices utilizing the easy-cone state.

In contrast to the out-of-plane state that can form with K1 alone, the easy-cone state requires a large K2 value; it is formed for \(K_1^{{\mathrm{eff}}}\, <\, 0\) and \(K_2 \,>\, - 1/2K_1^{{\mathrm{eff}}}\) (Fig. 1a). To actively employ the easy-cone state in various applications, therefore, it is of crucial importance to find a way of enhancing K2, which necessitates a fundamental understanding of its origin. The origin of K1 has long been a subject of extensive theoretical and experimental research. It was found to depend on the orbital anisotropy13, spin–orbit interaction of electronic structures near the Fermi level14 or Rashba-type spin–orbit interaction at the interface associated with the inversion symmetry breaking15,16,17. Concerning the origin of K2, three mechanisms have been proposed: (1) spatial fluctuations of K118, (2) interfacial PMA combined with a gradual weakening of the exchange energy along the thickness direction19, and (3) the mixture of bulk magnetocrystalline cubic anisotropy and interfacial uniaxial anisotropy20. The first and second mechanisms predict only positive K2 and fail to explain the negative K2 observed in experiments21,22. The third mechanism predicts both signs of K2 depending on the nature of the bulk cubic anisotropy. Our measurement of K2 for a Pt/Co/Cu structure, however, shows that K2 is inversely proportional to the Co thickness (thus, the interface origin) and is negative for thin Co layers (see Fig. 1c and Supplementary Note 1 for details). As the third mechanism cannot account for the origin of the negative K2 of the interface, none of the three aforementioned mechanisms can explain this experimental observation; thus, a new origin of K2 must be identified.

In this study, we focus on the role of the inversion symmetry breaking in K2 for the following two reasons. First, recent theoretical and experimental studies indicated the important role of the inversion asymmetry in K1 for ferromagnet–normal metal heterostructures15,16,17. As K1 and K2 are the order-expanded coefficients of the net magnetic anisotropy [Eq. (1)], it is reasonable to expect that they share the same origin. Second, our measurements of K1 and K2 for Pt/Co/Cu and Pt/Co/MgO stacks over a wide range of Co thickness (tCo) show that for both K1 and K2, the interfacial contribution is dominant compared with the bulk contribution (Supplementary Note 1), indicating the important role of the inversion asymmetry at the interface in the anisotropy.

Materials and methods

Sample preparation

To investigate the correlation between the inversion asymmetry and K2, we examine various sputtered Pt/Co/X stacks, with X = Pd, Cu, Pt, Mo, Ru, W, and Ta. The stacks investigated in this study had the structure of Si substrate (wet-oxidized)/Ta (5 nm)/Pt (5 nm)/Co (1 nm)/X (3 nm)/Ta (3 nm) and were fabricated using an ultrahigh-vacuum magnetron sputtering system with a base pressure of 8 × 10−8 Torr. All metallic layers were deposited under an Ar pressure of 2 × 10−3 Torr. The Ta under- and upper-layers were introduced to improve the surface roughness and prevent the oxidation of the stacks, respectively. For X = Ta, Pt (3 nm) was used as the upper layer. Pt/Co/MgO (2 nm) stacks were also prepared, followed by postannealing at 400 °C for 30 min to maximize the interfacial PMA at the Co/MgO interface23,24,25. Details regarding the fabrication and annealing are provided in Supplementary Note 5. The continuous samples were patterned into a Hall bar structure via photolithography and inductively coupled plasma etching. The current-injection line and the voltage branch had dimensions of 5 μm (width) × 35 μm (length). A 50-nm-thick Pt layer was deposited on top of the patterned structure as a contact pad for magnetotransport characterization (Fig. 2a).

Fig. 2: Measurement of magnetic anisotropy.
figure 2

a Schematic showing the Hall bar device used for the magnetic anisotropy measurements, together with an optical microscopy image (upper right). The Hall voltage was measured while injecting an in-plane current (Ix) along the x direction. The Hext was applied along θH = 80° to facilitate coherent magnetization behavior. b mz–Hext plots for Pt/Co/X heterostructures. The symbols and dashed lines indicate the results of AHE measurements and macrospin simulations, respectively. c αHext vs. \(1 - m_z^2\,\) plots converted from the results in (b). The solid lines represent the linear fittings to the data.

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Measurement of magnetic anisotropy

The magnetic anisotropies (K1 and K2) were characterized by the anomalous Hall effect (AHE) in a standard four-probe Hall geometry. The Hall bar device was mounted on a rotatable sample stage placed in the gap of an electromagnet. The AHE measurements involved injecting an in-plane current (Ix = 5 mA) along the x direction and sensing the Hall voltage induced along the y direction. The external magnetic field (Hext) was applied at a polar angle (θH) of 80° to facilitate coherent magnetization behavior (Fig. 2a). The generalized Sucksmith–Thompson method was used to accurately determine the effective first- and second-order anisotropy fields (denoted as \(H_{{\mathrm{K1}}}^{{\mathrm{eff}}} = 2K_1/M_{\mathrm{s}} - 4\pi M_{\mathrm{s}}\) and \(H_{{\mathrm{K}}2} = 4K_2/M_{\mathrm{s}}\), respectively)26. The key to this method is the use of the following equations, which can be derived from the total magnetic energy equation [Eq. (1), considering the Zeeman energy (−M · Hext)]:

$$\alpha H_{{\mathrm{ext}}} = H_{{\mathrm{K1}}}^{{\mathrm{eff}}} + H_{{\mathrm{K2}}}\left( {1 - m_z^2} \right),$$
(2)
$$\alpha \equiv \frac{{m_z\sin \theta _{\mathrm{H}} - \sqrt {1 - m_z^2} \cos \theta _{\mathrm{H}}}}{{m_z\sqrt {1 - m_z^2} }}.$$
(3)

The AHE results were normalized with respect to the anomalous Hall voltages to obtain mz − Hext curves (Fig. 2b), and then αHext was plotted with respect to \(1 - m_z^2\) to extract \(H_{{\mathrm{K1}}}^{{\mathrm{eff}}}\) and HK2 from the intercept and slope, respectively [Eq. (2) and Fig. 2c]. We observed a slight misalignment in θH from its nominal value (mostly within 2°), which was adjusted to maximize the linearity of the αHext vs. \(1 - m_z^2\) plot. To confirm the accuracy of the anisotropy constants, the measured mz − Hext curves were compared with those from macrospin simulations using the obtained \(H_{{\mathrm{K1}}}^{{\mathrm{eff}}}\) and HK2 values as inputs (Fig. 2b). The Ms values of the continuous samples were measured using a vibrating sample magnetometer. The anisotropy constants were then obtained from the relationships \(K_1 = M_{\mathrm{s}}H_{{\mathrm{K1}}}^{{\mathrm{eff}}}/2 + 2\pi M_{\mathrm{s}}^2\) and K2 = MsHK2/4. All measurements were performed at room temperature.

Measurement of work function

To measure the work functions of metals and MgO, ultraviolet photoelectron spectroscopy (UPS) measurements were performed for separately prepared stacks of Si substrate (wet-oxidized)/X (5 nm) (including Co). The UPS measurements were performed using He I radiation (hv = 21.2 eV) from a gas-discharge lamp. The base pressure of the chamber was 2 × 10−8 Torr. Prior to the measurement, Ar ion sputtering was performed to remove any native oxides formed during the exposure to air. The metallic films were sputtered repeatedly until the Fermi edge was observed. More details on the measurement of the work function and the photoemission spectra are provided in Supplementary Note 2.

Results and discussion

In Fig. 3a–c, K1 is plotted as a function of the work function (W), electronegativity (χ), and spin–orbit coupling constant (ξ), all of which are taken from the literature27,28,29. We choose these material parameters because of their potential correlation with the inversion asymmetry or Rashba effect at the Co/X interface30,31,32. To estimate the strength of the correlation, we calculated Pearson’s r for all the plots. Pearson’s r is close to ±1 (0) for a strong (weak) correlation. We obtain correlation coefficients of 0.82, 0.63, and 0.07 for the plots in Fig. 3a–c, respectively, indicating the strongest correlation between K1 and ΔW (≡WX − WCo). K1 also appears to be correlated with χ (Fig. 3b). This is expected because the difference in χ between two elements is proportional to the charge transfer33, which could be driven by the potential gradient at the Co/X interface in our samples. We note that this correlation feature is in accordance with a recent experimental observation for the interfacial Dzyaloshinskii–Moriya interaction originating from inversion asymmetry27. We also plot K1 as a function of ΔW measured for our samples by UPS (denoted as ΔWmeas) (see Fig. 3d, Methods, and Supplementary Note 2 for details) and find a similar correlation between the two parameters (K1 and ΔWmeas) with a correlation coefficient of 0.78. This result shows that the inversion asymmetry at the interface plays an important role in the K1 of Pt/Co/X heterostructures.

Fig. 3: Correlation of magnetic anisotropies with material parameters.
figure 3

a–d K1 as a function of ΔW (a), χ (b), ξ (c), and ΔWmeas (d) for Pt/Co/X stacks with various X elements. e–h Correlation results for K2, similar to those shown in (a–d). The values of ΔW, χ, and ξ were taken from the literature27,28,29, and those of ΔWmeas were obtained in this study via UPS measurements. The error bars of K1, K2, and ΔWmeas were obtained from three repeated measurements, whereas those of ΔW represent the standard deviations of reported values.

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Figure 3e–h shows the results for K2, which are similar to those for K1 shown in Fig. 3a–d. The correlation coefficients for K2 are −0.59, −0.51, and −0.18 for literature values of ΔW, χ, and ξ, respectively. Similar to K1, K2 exhibits meaningful correlations with ΔW and χ. The correlation coefficient of K2 with ΔWmeas is substantially improved to −0.94 (Fig. 3h), suggesting a strong correlation. Importantly, K2 changes its sign depending on the type of material X but still shows an almost linear correlation with ΔWmeas. According to theoretical work, the Rashba spin–orbit coupling is known to be proportional to the surface potential (seen by electrons) and electron density distribution34. Since the surface potential of a metal-metal interface is equivalent to ΔWmeas, the good correlation observed in Fig. 3h suggests that the inversion asymmetry is an intrinsic origin of K2 in Pt/Co/X heterostructures. We call it intrinsic because this mechanism is distinct from the first (spatial fluctuations of K118) and second (interfacial PMA combined with a gradual weakening of the exchange energy along the thickness direction19) mechanisms, which are extrinsic. Furthermore, our simple tight-binding model calculation with Rashba spin–orbit coupling supports this conclusion, as it shows that K2 can have both positive and negative signs depending on the band filling even though K1 is positive (i.e., PMA) (Supplementary Note 3). It is worth noting that the linear correlation, observed in Fig. 3d, h, persists even with the structural disordering taken into account (Supplementary Note 4). This indicates that the effect of the inversion asymmetry on the magnetic anisotropy can still be valid in intermixed thin-film structures17.

The linear correlation between K2 and ΔWmeas, however, appears to be somewhat unreasonable, considering that the magnetic anisotropy is known to originate from the second-order perturbative treatment of spin–orbit coupling near the Fermi level13,16. The following explanation can be given to understand the observed linear correlation. The effective electric field (E0) formed at the metal-metal interface will be large when the work-function difference between the two metals (e.g., Co and normal metal) is large. The use of a different normal metal will result in the modulation of E0, dE, with a resultant E0 value of E0 + dE. Since the magnetic anisotropy is quadratic in the Rashba parameter (i.e., E0 + dE)16, the variation of the magnetic anisotropy with a differing normal metal can be simply expressed as \(\left( {E_0 + dE} \right)^2 \,\approx E_0^2 + 2E_0\,dE\). It is then possible to explain our experimental results (Fig. 3d, h) that the modulation of the work function can give rise to the linear variation of the magnetic anisotropy.

The correlation result suggests that a large negative ΔW results in a large positive K2, which is needed to form the easy-cone state. For experimental realization, we replace the metallic X layer with a MgO layer (see Supplementary Note 5). We choose MgO for the following two reasons. First, strong Rashba splitting was observed at metal–oxide interfaces31,32. Our ΔWmeas value at the Co/MgO interface is consistent with this expectation: it is −0.36 eV (Supplementary Note 5), which is more negative than the value (−0.25 eV) for the Co/Ta interface, which exhibits the most negative ΔWmeas among all the metallic Co/X interfaces. Second, MgO is widely adopted in various spintronic devices3. For a Pt/Co (1.0 nm)/MgO stack, we obtain a K1 of 1.47 × 107 erg/cm3 and K2 of 2.61 × 106 erg/cm3. Compared with the all-metallic structures, the K2 of the Pt/Co/MgO structure is larger by an order of magnitude, which is in accordance with our conclusion in this work; the inversion asymmetry is an intrinsic origin of K2. However, previously proposed mechanisms18,19,20 not considering the role of the inversion asymmetry are unable to explain the enhanced K2 (see Supplementary Note 7 for details). Nonetheless, we note that the simple linear correlation between K2 and ΔWmeas describes the enhanced K2 of the Pt/Co/MgO structure only qualitatively, not quantitatively. Extrapolation of the linear line in Fig. 3h gives a K2 value of approximately 0.27 × 106 erg/cm3, which is significantly smaller than the measured value of 2.61 × 106 erg/cm3. A similar behavior is observed for K1; in this case, the extrapolation gives a value of 0.55 × 107 erg/cm3, which is approximately one third of a measured value of 1.47 × 107 erg/cm3. These deviations may indicate that ΔWmeas is not the sole factor determining the inversion asymmetry for a metal–oxide interface. A recent experimental work combined with a first-principles study found that the asymmetric charge-density distribution (or the charge transfer) at a metal–oxide interface has a larger effect on the Rashba splitting than the work-function difference (or the potential gradient)32.

This large and positive K2 allows the easy-cone state to be formed in Pt/Co/MgO structures at tCo near the spin reorientation transition1. The formation of the easy-cone state is validated by both vibrating sample magnetometry and AHE measurements (Supplementary Note 8). The \(K_1^{{\mathrm{eff}}}\) and K2 values for the Pt/Co (1.8–2.0 nm)/MgO structures are overlaid on a magnetic phase diagram (Fig. 4a). The cone angle (θc) is estimated according to the relationship \(\theta _{\mathrm{c}} = \sin ^{ - 1}\left( {\sqrt { - K_1^{{\mathrm{eff}}}/2K_2} } \right)\). We find that θc can be engineered by controlling tCo (Fig. 4b), which is beneficial for device applications of the easy-cone state.

Fig. 4: Pt/Co/MgO structure with easy-cone state.
figure 4

a Magnetic phase diagram in the \(K_1^{{\mathrm{eff}}}\)−K2 plane, with the \(K_1^{{\mathrm{eff}}}\)and K2 values indicated at several tCo values in nanometers. b Plot of θc vs. tCo. The error bars of \(K_1^{{\mathrm{eff}}}\), K2, and θc were obtained from three repeated measurements.

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Conclusion

We investigated the origin of K2 in Pt/Co/X heterostructures and found that inversion asymmetry plays an important role in K2. Among the material parameters considered in this study, the work-function difference at the Co/X interface shows the strongest correlation with both K1 and K2. Replacing the metallic X layer with MgO, whose interface with Co has a strong inversion asymmetry, we obtain greatly enhanced K2, allowing the easy-cone state. The intrinsic origin of K2 revealed in this study will contribute to the control of its values and therefore allow various easy-cone states suitable for a wide variety of spintronic applications.