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# Magnetization dynamics and its scattering mechanism in thin CoFeB films with interfacial anisotropy

Edited by J. M. D. Coey, Trinity College Dublin, Dublin, Ireland, and approved February 23, 2017 (received for review August 19, 2016)

## Significance

Ferromagnetic resonance (FMR) is considered a standard tool in the study of magnetization dynamics, with an established analysis procedure. We show there is a missing piece of physics to consider to fully understand and precisely interpret FMR results. This physics manifests itself in the dynamics of magnetization and hence in FMR spectra of ferromagnets, where interfacial anisotropy is a fundamental term. Advances in ferromagnetic heterostructures enabled by developing cutting-edge technology now allow us to probe and reveal physics previously hidden in the bulk properties of ferromagnets.

## Abstract

Studies of magnetization dynamics have incessantly facilitated the discovery of fundamentally novel physical phenomena, making steady headway in the development of magnetic and spintronics devices. The dynamics can be induced and detected electrically, offering new functionalities in advanced electronics at the nanoscale. However, its scattering mechanism is still disputed. Understanding the mechanism in thin films is especially important, because most spintronics devices are made from stacks of multilayers with nanometer thickness. The stacks are known to possess interfacial magnetic anisotropy, a central property for applications, whose influence on the dynamics remains unknown. Here, we investigate the impact of interfacial anisotropy by adopting CoFeB/MgO as a model system. Through systematic and complementary measurements of ferromagnetic resonance (FMR) on a series of thin films, we identify narrower FMR linewidths at higher temperatures. We explicitly rule out the temperature dependence of intrinsic damping as a possible cause, and it is also not expected from existing extrinsic scattering mechanisms for ferromagnets. We ascribe this observation to motional narrowing, an old concept so far neglected in the analyses of FMR spectra. The effect is confirmed to originate from interfacial anisotropy, impacting the practical technology of spin-based nanodevices up to room temperature.

The magnetization dynamics is determined by the combination of intrinsic and extrinsic effects. The intrinsic contribution is governed by the fundamental material parameter, damping constant *α* (1, 2). The extrinsic counterpart is due to inhomogeneity and magnon excitations (3⇓–5) and hence structure dependent, and is enhanced in magnets with small thickness and/or small lateral dimensions (4⇓⇓–7). The contribution from each effect is usually separated by the analysis of the linewidths of ferromagnetic resonance (FMR) spectra, in which the linewidth enhancement is caused by the distributions of magnitudes and directions of the effective magnetic anisotropy, and magnon excitations. In this work, we study the temperature and CoFeB thickness dependences of the FMR linewidths in CoFeB/MgO thin films, one of the most promising material systems for high-performance spintronics devices at the nanoscale. The system possesses sizable interfacial perpendicular magnetic anisotropy (8), whose effect on the FMR linewidth is the focus of this study.

## Samples and Measurement Setups

We prepared thin Co_{0.4}Fe_{0.4}B_{0.2} layers with thickness *t* ranging from 1.4 to 3.7 nm, sandwiched between 3-nm-thick MgO layers using magnetron sputtering. The thicknesses of the layers were calibrated by transmission electron microscopy. All of the samples studied in this work possess in-plane easiness for magnetization. FMR spectra were measured both by a vector-network analyzer (VNA-FMR) using a coplanar waveguide, and a conventional method using a TE_{011} microwave cavity (cavity-FMR). The former technique enables us to measure the rf frequency *f* dependence of the spectra up to 26 GHz by applying an external magnetic field *H* either parallel (magnetic field angle *θ*_{H} = 90°) or perpendicular (*θ*_{H} = 0°) to the sample plane, and the spectra are obtained as the transmission coefficient *S*_{21} (9). The latter technique measures the spectra at a fixed *f* of 9 GHz under *H* at various *θ*_{H}, and the spectra are obtained as the derivative of the microwave absorption with respect to *H* (10). The temperature dependence of spontaneous magnetization *M*_{S} was measured by a superconducting quantum interference device magnetometer.

## VNA-FMR

Fig. 1*A* shows typical VNA-FMR at selected values of *f* for CoFeB with *t* = 1.5 nm at temperature *T* = 300 K and *θ*_{H} = 90°. We determine the resonance field *H*_{R} and linewidth (full width at half maximum) Δ*H* from the fitting of the modified Lorentz function (solid lines in Fig. 1*A*) to the VNA-FMR spectra (9). Fig. 1*B* shows the rf frequency dependence of *H*_{R} at *θ*_{H} = 0° and 90°, which we use to determine the effective perpendicular anisotropy fields *H*_{K}^{eff} from the resonance condition; *f* = μ_{0}γ(*H*_{R} + *H*_{K}^{eff})*/*(2π) for *θ*_{H} = 0° and *f* = μ_{0}γ[*H*_{R}(*H*_{R} *− H*_{K}^{eff})]^{1/2}*/*(2π) for *θ*_{H} = 90°. Here, μ_{0} is the permeability in free space, and γ the gyromagnetic ratio. As shown in Fig. 1*C*, *H*_{K}^{eff} increases monotonically with decreasing *t*, indicative of interfacial perpendicular anisotropy at the CoFeB/MgO interface (8).

Fig. 1 *D*–*G* shows the frequency dependence of Δ*H* as a function of *t* and *T*. As shown in Fig. 1*D*, when *H* is applied perpendicular to the film (*θ*_{H} = 0°), Δ*H* obeys linear dependence, which is expressed as Δ*H* = Δ*H*_{in} + Δ*H*_{inhom} = (2*hα/g*μ_{0}μ_{B})*f* + Δ*H*_{inhom}, where *h* is the Planck constant and μ_{B} is the Bohr magneton (9, 11). Here, Δ*H*_{in} is related to the intrinsic linewidth governed by *α* and Δ*H*_{inhom} is the extrinsic contribution due to inhomogeneity, such as the distribution of magnetic anisotropy. The value of *α* determines the slope in Fig. 1*D* and Δ*H*_{inhom} corresponds to the intercept on the vertical axis. Fig. 1*D* shows that *α* is nearly independent of *t* (*α* ∼ 0.004) and Δ*H*_{inhom} increases with decreasing *t*. As seen from Fig. 1*E* for CoFeB with *t* = 1.5 nm, *α* is also nearly independent of *T*, whereas Δ*H*_{inhom} increases with decreasing *T*. It is known that *α* depends on *T* through the change in resistivity with *T* (12⇓–14). The nearly temperature-independent *α* observed here may be due to the small temperature dependence of the resistivity of CoFeB (15). When *H* is in-plane (*θ*_{H} = 90°) (Fig. 1*F*), we observe a nonlinearity, which is enhanced with decreasing *t*. The nonlinearity, so far, is believed to be due to the contribution of two-magnon scattering (TMS) to the linewidth Δ*H*_{TMS}; TMS is known to be activated when the magnetization angle *θ*_{M} from the sample normal is greater than 45° (4⇓–6). The nonlinear frequency dependence of Δ*H* in Fig. 1*F* can be described in terms of TMS (dashed lines), assuming Δ*H* = Δ*H*_{in} + Δ*H*_{inhom} + Δ*H*_{TMS}. As depicted in Fig. 1*G*, the nonlinearity for CoFeB with *t* = 1.5 nm is enhanced strongly with decreasing *T* (nearly twice at 80 K compared with 300 K). This strong temperature dependence cannot be attributed to TMS, because the change in Δ*H*_{TMS} with *T* from 300 to 4 K is calculated to be ∼10% at most, using the measured *T* dependence of *M*_{S} and *H*_{K}^{eff} (4). Hence, it is imperative to consider an alternative mechanism for the nonlinearity, possibly related to the CoFeB/MgO-interface effect; notably, the nonlinearity is absent in thicker CoFeB with *t* = 3.7 nm (Fig. 1 *F* and *G*).

## Cavity-FMR

Fig. 2*A* shows typical cavity-FMR spectra at *T* = 300 K for CoFeB with *t* = 1.5 nm. Fitting the derivative of the Lorentz function to the spectra gives *H*_{R} and Δ*H* (16). We fit the resonance condition to the magnetic-field angle dependence of *H*_{R} shown in Fig. 2*B* to obtain the magnitude of the effective first-order and second-order perpendicular anisotropy fields, *H*_{K1}^{eff} and *H*_{K2}, respectively, by following the procedure in ref. 10. Fig. 2*C* shows the CoFeB thickness dependence of *H*_{K1}^{eff} and *H*_{K2} along with that of *H*_{K}^{eff} in Fig. 1*C* obtained from VNA-FMR. *H*_{K2} is nearly independent of *t* and its strength is a few tens of millitesla at most. *H*_{K1}^{eff} increases with decreasing *t*, in agreement with *H*_{K}^{eff} obtained from VNA-FMR, confirming again the presence of perpendicular interfacial anisotropy at the CoFeB/MgO interface (8). In Fig. 2*C*, we plot also the values of *H*_{K}^{eff} obtained from magnetization measurements (8), which show good correspondence with those obtained from FMR measurements. Fig. 2*D* shows the magnetic-field angle dependence of Δ*H* as a function of *t*. For *θ*_{M} > 45°, where TMS is expected to be activated, Δ*H* is larger for CoFeB with smaller *t*. As shown in Fig. 2*E*, however, a nearly twice larger Δ*H* at lower *T* cannot be explained by the TMS contribution to Δ*H*. The results obtained from cavity-FMR are consistent with those from VNA-FMR, indicating that the unexpected *T* dependence of Δ*H* is not an artifact.

## Discussion

Because the linewidth enhancement for *θ*_{H} = 90° is larger for thinner CoFeB (Figs. 1*F* and 2*D*), it is natural to consider that its mechanism is related to an interfacial effect in CoFeB/MgO. The most pronounced interfacial effect is the presence of interfacial perpendicular anisotropy as seen in Figs. 1*C* and 2*C*. Fig. 2*F* shows log(*K*(*T*)*/K*(4 K)) versus log(*M*_{S}(*T*)*/M*_{S}(4 K)) (Callen–Callen plot) (17), where *M*_{S}(*T*) is the measured spontaneous magnetization and *K*(*T*) the perpendicular magnetic anisotropy energy density determined from *H*_{K1}^{eff} and *H*_{K2} using cavity-FMR (10); *K* = *M*_{S}(*H*_{K1}^{eff} *− H*_{K2}*/*2)*/*2 + *M*_{S}^{2}*/*(2μ_{0}). The linear behavior in Fig. 2*F* gives a slope *m* of 2.16 [the exponent *m* in the Callen–Callen law of *K*(*T*)*/K*(0) = (*M*_{S}(*T*)*/M*_{S}(0))^{m}], consistent with the relationship between interfacial anisotropy energy density and *M*_{S} reported for CoFeB/MgO systems (18⇓–20). Hence, the temperature dependence of anisotropy in our CoFeB films is also governed by the interfacial anisotropy. Therefore, the linewidth broadening with decreasing *t* and *T* is expected to be due to random thermal fluctuation δ*H*_{i} in the anisotropy field at interfacial site *i* of a magnetic atom. Indeed, phonons can induce thermal fluctuations of interfacial anisotropy through vibrations of interfacial atoms. Because δ*H*_{i} is along the film normal (the in-plane component is expected to cancel out), it gives rise to a broader linewidth at larger *θ*_{M}, which is similar to the angular dependence of the TMS contribution. The *T* dependence of linewidths is also explained by the presence of δ*H*_{i}. The value of δ*H*_{i} fluctuates with time due to the thermal fluctuation of phonons, the frequency of which increases with increasing *T*, resulting in motional narrowing by averaging the randomness of δ*H*_{i}, albeit an increase in its amplitude with increasing *T* (21).

To formulate the motional narrowing, we apply the Holstein–Primakov, Fourier, and Bogolyubov transformations to the spin-deviation Hamiltonian (22), and obtain *H*′ = −Σ_{i}δ**H**_{i}*·***S**_{iz} *∼* (*S/*2)^{1/2}sin*θ*_{M}Σ_{k}δ*H*_{−k}(*u*_{k} + *v*_{k})(*α*_{k} +*α*^{+}_{−k}) (*z* direction along film normal) (*Materials and Methods*). *H*′ describes the coupling between local spin **S**_{i} and fluctuating field δ*H*_{i}, which contributes to spin relaxation and thus FMR linewidth. Here, **k** is the wavevector of magnon; *α*_{k}, *α*^{+}_{−k}, *u*_{k}, and *v*_{k} are the annihilation and creation operators for magnon and their coefficients after the Bogolyubov transformation; and *S* is the magnitude of spin. Adopting the Redfield theory to obtain the relationship between spin-relaxation time and δ*H*_{i} (21), the linewidth Δ*H*_{MN} due to δ*H*_{i} at *k* = 0 (Kittel mode) is expressed as Δ*H*_{MN} *∼* (*S/*2)(δ*H*_{k}_{=0})^{2}sin^{2}*θ*_{M}(*H*_{1}*/H*_{2})^{1/2} with *H*_{1} = *H*_{R}cos(*θ*_{H} *− θ*_{M})+*H*_{K1}^{eff}cos^{2}*θ*_{M} *− H*_{K2}cos^{4}*θ*_{M}, *H*_{2} = *H*_{R}cos(*θ*_{H} *− θ*_{M})+*H*_{K1}^{eff}cos2*θ*_{M} *−*(*H*_{K2}*/*2)(cos2*θ*_{M} + cos4*θ*_{M}), and (δ*H*_{k}_{=0})^{2} = ∫dτδ*H*_{k}_{=0}(*t*_{0})δ*H*_{k}_{=0}(*t*_{0} + τ)e^{−2iπfτ}. Here, *t*_{0} is the arbitrary time, and τ is the elapsed time from *t*_{0}. Writing the correlation function δ*H*_{k}_{=0}(*t*_{0})δ*H*_{k}_{=0}(*t*_{0} + τ) = (δ*H*)^{2}e^{−|τ|/τ0}, where τ_{0} is the relaxation time of the random field δ*H*_{k=0}, we obtain*fτ*_{0} << 1.

We fit*T* in Fig. 2*E* (3, 10). Each contribution in the right-hand side of Eq. **2** is determined by the parameters in parentheses. The values of *M*_{S}, *α*, *H*_{K1}^{eff}, and *H*_{K2} and their temperature dependence are determined experimentally; *M*_{S} from magnetization measurements, *α* obtained from VNA-FMR measurements at *θ*_{H} = 0° (Fig. 3*A*), and *H*_{K1}^{eff} and *H*_{K2} from cavity-FMR measurements. We calculate Δ*H*_{in} from Δ*H*_{in} = *α*(*H*_{1} + *H*_{2})|d*H*_{R}*/*d[(*H*_{1}*H*_{2})^{1/2}]| (3). The contribution from Δ*H*_{inhom} is expressed as Δ*H*_{inhom} = |d*H*_{R}*/*d*H*_{K1}^{eff}|Δ*H*_{K1}^{eff}+|d*H*_{R}*/*d*H*_{K2}|Δ*H*_{K2}, and Δ*H*_{K1}^{eff} and Δ*H*_{K2} are adopted as fitting parameters (10). To describe the contribution from TMS, we adopt the expression in ref. 4. The TMS contribution at 300 K is determined from the best fit of Eq. **2** to the experimental result at 300 K with two defect-related fitting parameters, *A* and *N*, which reflect the size and density as well as the aspect ratio of the defects, respectively. The *T* dependence of TMS is calculated using the *T* dependence of *M*_{S} and *H*_{K}^{eff}, assuming the exchange stiffness constant *A*_{S}(*T*) ∝ [*M*_{S}(*T*)]^{2} and *T*-independent *A* and *N* (20). As described before, the calculated TMS contribution changes by ∼10% at most with decreasing *T* from 300 to 4 K. For the Δ*H*_{MN} contribution, we adopt Γ as an adjustable parameter. The fit agrees with the experimental results as shown by the solid lines in Fig. 2*E*. Fig. 2*G* depicts the calculated linewidths as functions of *f* and *T* using parameters obtained from the analyses, reproducing the results in Fig. 1*G*. We note that the difference in the detected areas between VNA-FMR (∼0.1 mm^{2}) and cavity-FMR (∼20 mm^{2}) may result in the small difference observed in the magnitude of Δ*H* (compare Figs. 1*G* and 2*G*), due to inhomogeneity. The *T* dependence of Γ is shown in Fig. 3*B*. Although the observed functional form is unknown, it may be due to several contributions, such as the *T* dependence of the magnitude of δ*H*_{i} and the magnon and phonon lifetimes (23). δ*H*_{i} is expected to be determined by the *T* dependence of the thermal lattice expansion coefficient and the resultant lattice mismatch between CoFeB and MgO (24), whereas the magnon lifetime may be due to the exchange/stiffness constants at the interface (25, 26).

Furthermore, we fit Eq. **2** to the angle dependence of the linewidth for CoFeB with different *t*. Here, we treat *α*, Δ*H*_{K1}^{eff}, Δ*H*_{K2}, TMS parameters, and Γ as adjustable parameters. The CoFeB thickness dependence of *α* is shown in Fig. 3*C*. The magnitude of *α* is ∼0.004 independent of *t* (closed circles), and consistent with the values obtained from VNA-FMR spectra at *θ*_{H} = 0° (open circles). If we neglect Δ*H*_{MN} in the analyses of cavity-FMR linewidths, *α* increases with decreasing *t* (squares). It is therefore essential to include Δ*H*_{MN} in FMR analyses to obtain accurate values of *α* in thin CoFeB/MgO when *θ*_{H} ≠ 0°. Crucially, for the present CoFeB/MgO systems, we design the stacks to suppress the spin-pumping effect by sandwiching CoFeB by two MgO layers (27, 28). If we replace one or two sides of the adjacent MgO with Ta, *α* increases rapidly with decreasing *t* due to spin pumping (triangles and diamonds). The *t* dependence of Γ in Fig. 3*D* attests to its interfacial origin.

From systematic FMR studies, we have shown that interfacial anisotropy in thin-film CoFeB/MgO has a strong effect on the spectral linewidth. This effect is explained in terms of motional narrowing, which is commonly neglected in the analysis of FMR spectra. The present investigation demonstrates that great care must be taken in the study of FMR in magnetic architectures with interfacial anisotropy, a fundamental property for spin-based device applications. In addition, the result is expected to bring a new concept to spintronics devices using phonon–magnon coupling through the interfacial anisotropy (29, 30).

## Materials and Methods

### Sample Preparation.

Films were deposited on a thermally oxidized Si substrate by ultrahigh-vacuum magnetron sputtering. The stack structure, from substrate side, is Ta (5)/ CuN (30)/ TaN (20)/ Ta (5)/ [CoFeB(0.6)/ Ru (3)]_{2}/ Ta (2)/ CoFeB (0.6)/ MgO (3)/ CoFeB(*t* = 1∼2.6)/ MgO (3)/ Ta (5)/ Ru (5)/ CuN (15), where numbers in parentheses are thickness in nanometers. The three 0.6-nm-thick CoFeB layers were inserted to improve the quality of the films above them (31, 32), and are expected to exhibit superparamagnetic behavior (33). We confirmed they do not affect the FMR spectra using a reference measurement on a stack in their absence. Because they are expected to exhibit very different anisotropy, due to different interfaces and thickness (compared with thick CoFeB, which is of interest here) their resonances are not detected in the temperature, frequency, and field range investigated. The FMR active layer is CoFeB sandwiched between two MgO layers. The CoFeB thickness *t* is varied from 1 to 2.6 nm over 8-inch wafer using wedged-film deposition. The actual thicknesses of CoFeB are calibrated from cross-sectional images obtained using a transmission electron microscope. We prepared also a reference sample with thick CoFeB namely, *t* = 3.7 nm.

### VNA-FMR.

Most of the measurements were performed using a home-built instrument with applied magnetic fields up to 0.55 T (up to 1.4 T at 300 K) and frequencies up to 26 GHz. A separate home-built FMR dipper probe was used to measure the sample with *t* = 1.5 nm under a perpendicular magnetic field at temperatures down to 4.2 K. FMR spectra were acquired by sweeping the external magnetic field (9).

### Cavity-FMR.

The sample was placed in a TE_{011} microwave cavity, where microwave frequency *f* = 9 GHz was introduced. We measured the external magnetic-field *H* dependence of the FMR spectrum (derivative microwave absorption spectrum) by superimposing an ac magnetic field (1 mT and 100 kHz) for lock-in detection. The sample temperature was controlled from 4 to 300 K using a liquid He flow cryostat (10, 16).

### Motional Narrowing.

The magnetic energy in the CoFeB film is described by a Heisenberg Hamiltonian *H*_{H} with anisotropy term *D* and external magnetic field *H*_{0},

To study the linewidth in FMR due to the fluctuating field δ*H*_{i} at interfaces, we consider the perturbation Hamiltonian,

where we consider δ*H*_{i} along the normal of the interfaces (*z* direction) because its in-plane components are expected to cancel out. We study the case for tilted magnetization direction to in-plane direction along *x*, and define the direction of magnetization **M** as the ζ-axis. By rotating by *θ*_{M} about the *y* axis, we convert *x-* and *z* axes of Cartesian coordinate system to ξ- and ζ-axes. In the ξ−ζ plane, the spin operator in the *z* direction is given by

and Eq. **4** is rewritten as

Assuming a constant longitudinal spin component (along the ζ-direction) *S*_{iζ}, only the transverse spin component (along the ξ-direction) *S*_{iξ} contributes to FMR. Hence, a Hamiltonian contributing to FMR is expressed as

By the Holstein–Primakoff transformation with the creation and annihilation operators *a*^{+}_{i} and *a*_{i}, and the magnitude of spin *S*, the transverse component is written as (22, 34)

and Eq. **7** becomes

The Fourier transformation with wavevector **k** and number of interfacial sites *N* gives

Then, Eq. **9** is expressed as

Eq. **3** is diagonalized by the Bogolyubov transformation (22, 35),

and Eq. **12** transforms to

Because we are interested in the FMR mode, we consider the **k** = 0 mode in Eq. **15**,

Adopting standard process to obtain the magnon dispersion, the coefficients in the Bogolyubov transformation in Eq. **16** are obtained as

with

According to Redfield theory (21, 36), the linewidth of FMR (**k** = 0) mode is proportional to the spectral density of fluctuating fields δ*H*_{k=0},

where the time-correlation function of fluctuating field δ*H*_{k=0} at interfaces is given by (21)

Because only the *z* component of δ*H* is relevant, we assume the single relaxation lifetime τ_{0} of the fluctuating fields at interfaces. τ_{0} is of the same order of the inverse of the phonon frequency––much larger than the resonance frequency *ω*, and thus *ωτ*_{0} << 1. The integration over time *τ* is calculated as

Finally, we obtain the following linewidth for the FMR mode due to the fluctuating fields at interfaces:

which is Eq. **1** in the main text.

## Acknowledgments

The work at Tohoku University was supported in part by Grants-in-Aid for Scientific Research from Ministry of Education, Culture, Sports, Science and Technology (MEXT) (26103002) and from JSPS (16H06081 and 16J05455), R&D project for Information and Communication Technology (ICT) Key Technology of MEXT, Impulsing Paradigm Change through Disruptive Technologies Program (ImPACT) program of Council for Science, Technology and Innovation (CSTI), Japan Society for the Promotion of Science (JSPS) Core-to-Core Program, as well as the Cooperative Research Projects of Research Institute of Electrical Communication (RIEC). The work in Singapore was supported by the Ministry of Education (MoE, Academic Research Fund Tier 2 Grant MOE2014-T2-1-050), the A*STAR Pharos Fund (1527400026), and the National Research Foundation (NRF), NRF-Investigatorship (NRFNRFI2015-04). The work at Japan Atomic Energy Agency was supported in part by Grants-in-Aid for Scientific Research from JSPS (Grants 26247063, 25287094, 15K05192, and 16H01082) and from MEXT (26103006 and 26247063).

## Footnotes

↵

^{1}A.O. and S.H. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: f-matsu{at}wpi-aimr.tohoku.ac.jp or christos{at}ntu.edu.sg.

Author contributions: A.O.S.H., B.G., S.K., A.S., S.T.L., M.T., M.M., S.M., F.M., H.O., and C.P. performed research; A.O., S.H., A.S., B.G., F.M., and C.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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