Intrinsic origin of interfacial second-order magnetic anisotropy in ferromagnet/normal metal heterostructures

Interfacial perpendicular magnetic anisotropy, which is characterized by the first-order (K1) and second-order (K2) anisotropies, is the core phenomenon for nonvolatile magnetic devices. A sizable K2 satisfying a specific condition stabilizes the easy-cone state, where equilibrium magnetization forms at an angle from the film normal. The easy-cone state offers intriguing possibilities for advanced spintronic devices and unique spin textures, such as spin superfluids and easy-cone domain walls. Experimental realization of the easy-cone state requires understanding the origin of K2, thereby enhancing K2. However, previously proposed origins of K2 cannot fully account for experimental results. Here we show experimentally that K2 scales almost linearly with the work-function difference between the Co and X layers in Pt/Co/X heterostructures (X = Pd, Cu, Pt, Mo, Ru, W, and Ta), suggesting the central role of the inversion asymmetry in K2. Our result provides a guideline for enhancing K2 and realizing magnetic applications based on the easy-cone state.


Introduction
Magnetic anisotropy describes a magnetization-angledependent 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): where K eff 1 ¼ K 1 À 2πM 2 s À Á is the effective first-order anisotropy energy density that comprises the demagnetization energy density (with K 1 and M s being the first-order anisotropy energy density and saturation magnetization, respectively), K 2 is the second-order anisotropy energy density, and θ is the polar angle of the magnetization. The magnetic phase diagram as functions of K eff 1 and K 2 (Fig. 1a) shows several distinct magnetic states 1 . Among them, the out-of-plane state originating from perpendicular magnetic anisotropy (PMA) has been a main focus of spintronics research 2 because it offers scalable magnetic random-access memories (MRAMs) 3 .
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) MRAMs 4-6 and zerofield precession of STT oscillators 7 . Moreover, it hosts spin superfluids associated with spontaneous breaking of U(1) spin-rotational symmetry 8,9 and allows unique easycone domain wall dynamics 10 . The existence of the easycone state was experimentally verified in various layered structures 6,11,12 . However, the design window for forming a stable easy-cone state is very narrow 6,11,12 , 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 K 1 alone, the easy-cone state requires a large K 2 value; it is formed for K eff 1 < 0 and K 2 > À 1=2K eff 1 (Fig. 1a). To actively employ the easy-cone state in various applications, therefore, it is of crucial importance to find a way of enhancing K 2 , which necessitates a fundamental understanding of its origin. The origin of K 1 has long been a subject of extensive theoretical and experimental research. It was found to depend on the orbital anisotropy 13 , spin-orbit interaction of electronic structures near the Fermi level 14 or Rashba-type spin-orbit interaction at the interface associated with the inversion symmetry breaking [15][16][17] . Concerning the origin of K 2 , three mechanisms have been proposed: (1) spatial fluctuations of K 1 18 , (2) interfacial PMA combined with a gradual weakening of the exchange energy along the thickness direction 19 , and (3) the mixture of bulk magnetocrystalline cubic anisotropy and interfacial uniaxial anisotropy 20 . The first and second mechanisms predict only positive K 2 and fail to explain the negative K 2 observed in experiments 21,22 . The third mechanism predicts both signs of K 2 depending on the nature of the bulk cubic anisotropy. Our measurement of K 2 for a Pt/Co/Cu structure, however, shows that K 2 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 K 2 of the interface, none of the three aforementioned mechanisms can explain this experimental observation; thus, a new origin of K 2 must be identified.
In this study, we focus on the role of the inversion symmetry breaking in K 2 for the following two reasons.
First, recent theoretical and experimental studies indicated the important role of the inversion asymmetry in K 1 for ferromagnet-normal metal heterostructures [15][16][17] . As K 1 and K 2 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 K 1 and K 2 for Pt/Co/Cu and Pt/Co/MgO stacks over a wide range of Co thickness (t Co ) show that for both K 1 and K 2 , 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.

Sample preparation
To investigate the correlation between the inversion asymmetry and K 2 , 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 interface [23][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 currentinjection 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).

Measurement of magnetic anisotropy
The magnetic anisotropies (K 1 and K 2 ) were characterized by the anomalous Hall effect (AHE) in a standard fourprobe 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 inplane current (I x = 5 mA) along the x direction and sensing the Hall voltage induced along the y direction. The external magnetic field (H ext ) was applied at a polar angle (θ H ) of 80°t o facilitate coherent magnetization behavior (Fig. 2a). The generalized Sucksmith-Thompson method was used to accurately determine the effective firstand second-order anisotropy fields (denoted as H eff K1 ¼ 2K 1 =M s À 4πM s and H K2 ¼ 4K 2 =M 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 · H ext )]: The AHE results were normalized with respect to the anomalous Hall voltages to obtain m z − H ext curves (Fig.  2b), and then αH ext was plotted with respect to 1 À m 2 z to extract H eff K1 and H K2 from the intercept and slope, respectively [Eq. (2) and Fig. 2c]. We observed a slight misalignment in θ H from its nominal value (mostly within 2°), which was adjusted to maximize the linearity of the αH ext vs. 1 À m 2 z plot. To confirm the accuracy of the anisotropy constants, the measured m z − H ext curves were compared with those from macrospin simulations using the obtained H eff K1 and H K2 values as inputs (Fig. 2b). The M s values of the continuous samples were measured using a vibrating sample magnetometer. The anisotropy constants were then obtained from the relationships K 1 ¼ M s H eff K1 =2 þ 2πM 2 s and K 2 = M s H K2 /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, K 1 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 literature [27][28][29] . We choose these material parameters because of their potential correlation with the inversion asymmetry or Rashba effect at the Co/X interface [30][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 K 1 and ΔW (≡W X − W Co ). K 1 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 transfer 33 , 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 asymmetry 27 . We also plot K 1 as a function of ΔW measured for our samples by UPS (denoted as ΔW meas ) (see Fig. 3d, Methods, and Supplementary Note 2 for details) and find a similar correlation between the two parameters (K 1 and ΔW meas ) with a correlation coefficient of 0.78. This result shows that the inversion asymmetry at the interface plays an important role in the K 1 of Pt/Co/X heterostructures. Figure 3e-h shows the results for K 2 , which are similar to those for K 1 shown in Fig. 3a-d. The correlation coefficients for K 2 are −0.59, −0.51, and −0.18 for literature values of ΔW, χ, and ξ, respectively. Similar to K 1 , K 2 exhibits meaningful correlations with ΔW and χ. The correlation coefficient of K 2 with ΔW meas is substantially improved to −0.94 (Fig. 3h), suggesting a strong correlation. Importantly, K 2 changes its sign depending on the type of material X but still shows an almost linear correlation with ΔW meas . According to theoretical work, the Rashba spin-orbit coupling is known to be proportional to the surface potential (seen by electrons) and electron density distribution 34 . Since the surface potential of a metal-metal interface is equivalent to ΔW meas , the good correlation observed in Fig. 3h suggests that the inversion asymmetry is an intrinsic origin of K 2 in Pt/Co/X heterostructures. We call it intrinsic because this mechanism is distinct from the first (spatial fluctuations of K 1 18 ) and second (interfacial PMA combined with a gradual weakening of the exchange energy along the thickness direction 19 ) mechanisms, which are extrinsic. Furthermore, our simple tight-binding model calculation with Rashba spin-orbit coupling supports this conclusion, as it shows that K 2 can have both positive and negative signs depending on the band filling even though K 1 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 structures 17 .
The linear correlation between K 2 and ΔW meas , 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 level 13,16 . The following explanation can be given to understand the observed linear correlation. The effective electric field (E 0 ) 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 E 0 , dE, with a resultant E 0 value of E 0 + dE. Since the magnetic anisotropy is quadratic in the Rashba parameter (i.e., E 0 + dE) 16 , the variation of the magnetic anisotropy with a differing normal metal can be simply expressed as 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 K 2 , 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. , ξ (c), and ΔW meas (d) for Pt/Co/X stacks with various X elements. e-h Correlation results for K 2 , similar to those shown in (a-d). The values of ΔW, χ, and ξ were taken from the literature [27][28][29] , and those of ΔW meas were obtained in this study via UPS measurements. The error bars of K 1 , K 2 , and ΔW meas were obtained from three repeated measurements, whereas those of ΔW represent the standard deviations of reported values.
First, strong Rashba splitting was observed at metal-oxide interfaces 31,32 . Our ΔW meas 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 ΔW meas among all the metallic Co/X interfaces. Second, MgO is widely adopted in various spintronic devices 3 . For a Pt/Co (1.0 nm)/MgO stack, we obtain a K 1 of 1.47 × 10 7 erg/cm 3 and K 2 of 2.61 × 10 6 erg/cm 3 . Compared with the all-metallic structures, the K 2 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 K 2 . However, previously proposed mechanisms 18-20 not considering the role of the inversion asymmetry are unable to explain the enhanced K 2 (see Supplementary Note 7 for details). Nonetheless, we note that the simple linear correlation between K 2 and ΔW meas describes the enhanced K 2 of the Pt/Co/MgO structure only qualitatively, not quantitatively. Extrapolation of the linear line in Fig. 3h gives a K 2 value of approximately 0.27 × 10 6 erg/cm 3 , which is significantly smaller than the measured value of 2.61 × 10 6 erg/cm 3 . A similar behavior is observed for K 1 ; in this case, the extrapolation gives a value of 0.55 × 10 7 erg/cm 3 , which is approximately one third of a measured value of 1.47 × 10 7 erg/cm 3 . These deviations may indicate that ΔW meas 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 K 2 allows the easy-cone state to be formed in Pt/Co/MgO structures at t Co near the spin reorientation transition 1 . The formation of the easy-cone state is validated by both vibrating sample magnetometry and AHE measurements (Supplementary Note 8). The . We find that θ c can be engineered by controlling t Co (Fig. 4b), which is beneficial for device applications of the easy-cone state.

Conclusion
We investigated the origin of K 2 in Pt/Co/X heterostructures and found that inversion asymmetry plays an important role in K 2 . Among the material parameters considered in this study, the work-function difference at the Co/X interface shows the strongest correlation with both K 1 and K 2 . Replacing the metallic X layer with MgO, whose interface with Co has a strong inversion asymmetry, we obtain greatly enhanced K 2 , allowing the easycone state. The intrinsic origin of K 2 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.