Magnetization dynamics and its scattering mechanism in thin CoFeB films with interfacial anisotropy Atsushi Okadaa,1, Shikun Heb,c,1, Bo Gud, Shun Kanaia,e,f, Anjan Soumyanarayananb,c, Sze Ter Limc, Michael Tranc, Michiyasu Morid, Sadamichi Maekawad,g, Fumihiro Matsukuraa,e,f,h,2, Hideo Ohnoa,e,f,h,i, and Christos Panagopoulosb,2 a Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan; bDivison of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore; cData Storage Institute, Agency for Science, Technology and Research (A*STAR), 138634 Singapore; dAdvanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan; eCenter for Spintronics Integrated Systems, Tohoku University, Sendai 980-8577, Japan; fCenter for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan; gExploratory Research for Advanced Technology, Japan Science and Technology Agency, Sendai 980-8577, Japan; h World Premier International Research Center–Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan; and iCenter for Innovative Integrated Electronic Systems, Tohoku University, Sendai 980-0845, Japan
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.
|
CoFeB/MgO ferromagnetic resonance interfacial anisotropy
| damping | motional narrowing |
T
he 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 highperformance 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 Co0.4Fe0.4B0.2 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 www.pnas.org/cgi/doi/10.1073/pnas.1613864114
analyzer (VNA-FMR) using a coplanar waveguide, and a conventional method using a TE011 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 θH = 90°) or perpendicular (θH = 0°) to the sample plane, and the spectra are obtained as the transmission coefficient S21 (9). The latter technique measures the spectra at a fixed f of 9 GHz under H at various θH, and the spectra are obtained as the derivative of the microwave absorption with respect to H (10). The temperature dependence of spontaneous magnetization MS was measured by a superconducting quantum interference device magnetometer. VNA-FMR Fig. 1A shows typical VNA-FMR at selected values of f for CoFeB with t = 1.5 nm at temperature T = 300 K and θH = 90°. We determine the resonance field HR and linewidth (full width at half maximum) ΔH from the fitting of the modified Lorentz function (solid lines in Fig. 1A) to the VNA-FMR spectra (9). Fig. 1B shows the rf frequency dependence of HR at θH = 0° and 90°, which we use to determine the effective perpendicular anisotropy fields HKeff from the resonance condition; f = μ0γ(HR + HKeff)/(2π) for θH = 0° and f = μ0γ[HR(HR − HKeff)]1/2/(2π) for θH = 90°. Here, μ0 is the permeability in free space, and γ the gyromagnetic ratio. As shown in Fig. 1C, HKeff 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 ΔH as a function of t and T. As shown in Fig. 1D, when H is applied perpendicular to the film (θH = 0°), ΔH obeys linear dependence, 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 cuttingedge technology now allow us to probe and reveal physics previously hidden in the bulk properties of ferromagnets. 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. 1
A.O. and S.H. contributed equally to this work.
2
To whom correspondence may be addressed. Email: [email protected] or [email protected].
PNAS | April 11, 2017 | vol. 114 | no. 15 | 3815–3820
APPLIED PHYSICAL SCIENCES
Edited by J. M. D. Coey, Trinity College Dublin, Dublin, Ireland, and approved February 23, 2017 (received for review August 19, 2016)
B f = 6 GHz
1.0
12 GHz 14 GHz
= 0o
-1 0.0
0.2
0.3
H (T)
0.4
E t = 1.4 nm
16
H
12
0
1.6 nm 2.0 nm
4 H
0 0
= 0o T = 300 K 20
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= 0o
T=6K
10
8
280 K
4
t = 1.5 nm
160 K 0
1.0
20
20
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1.5
G t = 1.4 nm
16
= 90o H
8
0
10
20
H
2.0
20
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2.5
t (nm)
= 90o
16 T = 80 K 1.5 nm 170 K 1.6 nm 12 2.0 nm 8
3.7 nm T = 300 K
4 0
10
= 0o
T = 300 K
-0.8
f (GHz)
12
80 K
0 10
0
20
1.5 nm
8
t = 1.5 nm T = 300 K
F
20
12
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90o
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-0.4
90o
0.5
t = 1.5 nm T = 300 K = 90o H
H (mT)
H
0
0
0
D
C
HR (T)
Re(S21) (arb. units)
8 GHz 10 GHz
HKeff (T)
1
0
A
t= 1.5 nm 300 K
3.7 nm 80, 300 K
4 0 0
10
20
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Fig. 1. VNA-FMR. (A) Typical spectra [real part of transmission coefficient Re(S21) of coplanar waveguide] of CoFeB with thickness t = 1.5 nm as a function of rf frequency f obtained at magnetic–field angle θH = 90° and temperature T = 300 K. Solid lines are fits by the modified Lorentzian. (B) rf frequency f dependence of resonance field HR obtained at θH = 0° and 90°. (C) Thickness t dependence of effective perpendicular anisotropy fields HKeff. rf frequency f dependence of the FMR linewidths ΔH obtained at θH = 0° as a function of (D) t at T = 300 K and as a function of (E) T for CoFeB with t = 1.5 nm. Solid lines in D and E are linear fits. (F and G) Same as D and E but at θH = 90°. Dashed lines in F and G are nonlinear fits based on two-magnon scattering. arb, arbitrary.
which is expressed as ΔH = ΔHin + ΔHinhom = (2hα/gμ0μB)f + ΔHinhom, where h is the Planck constant and μB is the Bohr magneton (9, 11). Here, ΔHin is related to the intrinsic linewidth governed by α and ΔHinhom is the extrinsic contribution due to inhomogeneity, such as the distribution of magnetic anisotropy. The value of α determines the slope in Fig. 1D and ΔHinhom corresponds to the intercept on the vertical axis. Fig. 1D shows that α is nearly independent of t (α ∼ 0.004) and ΔHinhom increases with decreasing t. As seen from Fig. 1E for CoFeB with t = 1.5 nm, α is also nearly independent of T, whereas ΔHinhom 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 (θH = 90°) (Fig. 1F), 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 ΔHTMS; TMS is known to be activated when the magnetization angle θM from the sample normal is greater than 45° (4–6). The nonlinear frequency dependence of ΔH in Fig. 1F can be described in terms of TMS (dashed lines), assuming ΔH = ΔHin + ΔHinhom + ΔHTMS. As depicted in Fig. 1G, 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 ΔHTMS with T from 300 to 4 K is calculated to be ∼10% at most, using the measured T dependence of MS and HKeff (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). 3816 | www.pnas.org/cgi/doi/10.1073/pnas.1613864114
Cavity-FMR Fig. 2A 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 HR and ΔH (16). We fit the resonance condition to the magnetic-field angle dependence of HR shown in Fig. 2B to obtain the magnitude of the effective firstorder and second-order perpendicular anisotropy fields, HK1eff and HK2, respectively, by following the procedure in ref. 10. Fig. 2C shows the CoFeB thickness dependence of HK1eff and HK2 along with that of HKeff in Fig. 1C obtained from VNA-FMR. HK2 is nearly independent of t and its strength is a few tens of millitesla at most. HK1eff increases with decreasing t, in agreement with HKeff obtained from VNA-FMR, confirming again the presence of perpendicular interfacial anisotropy at the CoFeB/ MgO interface (8). In Fig. 2C, we plot also the values of HKeff obtained from magnetization measurements (8), which show good correspondence with those obtained from FMR measurements. Fig. 2D shows the magnetic-field angle dependence of ΔH as a function of t. For θM > 45°, where TMS is expected to be activated, ΔH is larger for CoFeB with smaller t. As shown in Fig. 2E, however, a nearly twice larger ΔH at lower T cannot be explained by the TMS contribution to ΔH. The results obtained from cavity-FMR are consistent with those from VNA-FMR, indicating that the unexpected T dependence of ΔH is not an artifact. Discussion Because the linewidth enhancement for θH = 90° is larger for thinner CoFeB (Figs. 1F and 2D), 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. 1C and 2C. Fig. 2F shows log(K(T)/K(4 K)) versus log(MS(T)/MS(4 K)) (Callen– Callen plot) (17), where MS(T) is the measured spontaneous Okada et al.
~0
1.0
o
0 t = 1.5 nm T = 300 K f = 9 GHz
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T = 300 K
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VNA-FMR, Heff K 0o,
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16
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12 8
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24o
C
0
44
32o
o
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= 88o H
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1
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4 0 0
10
f (GHz)
20
Fig. 2. Cavity-FMR. (A) Typical spectra of CoFeB with thickness t of 1.5 nm as a function of magnetic-field angle θH at temperature T = 300 K. Solid lines are fits by the derivative of Lorentzian. (B) Angle θH dependence of resonance field HR as a function of t. Solid lines are fitted lines by using the resonance condition. (C) Thickness t dependence of effective first-order perpendicular anisotropy field HK1eff and second-order anisotropy field HK2 along with the results in Fig. 1C and from magnetization measurements. Angle θH dependence of the FMR linewidth ΔH obtained as a function of (D) t at T = 300 K and as a function of (E) T for CoFeB with t = 1.5 nm. Solid lines in D and E are fitted lines. (F) Double-logarithm plot of normalized magnetic anisotropy energy density K(T)/K(4K) versus normalized spontaneous magnetization MS(T)/MS(4 K). Solid line is a linear fit. (G) Calculated ΔH as a function of rf frequency f at different temperatures by using the parameters obtained from the analyses of cavity-FMR spectra. arb, arbitrary.
magnetization and K(T) the perpendicular magnetic anisotropy energy density determined from HK1eff and HK2 using cavityFMR (10); K = MS(HK1eff − HK2/2)/2 + MS2/(2μ0). The linear behavior in Fig. 2F gives a slope m of 2.16 [the exponent m in the Callen–Callen law of K(T)/K(0) = (MS(T)/MS(0))m], consistent with the relationship between interfacial anisotropy energy density and MS 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 δHi 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 δHi is along the film normal (the in-plane component is expected to cancel out), it gives rise to a broader linewidth at larger θM, which is similar to the angular dependence of the TMS contribution. The T dependence of linewidths is also explained by the presence of δHi. The value of δHi 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 δHi, 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′ = −ΣiδHi·Siz ∼ (S/2)1/2sinθMΣkδH−k(uk + vk)(αk +α+−k) (z direction along film normal) (Materials and Methods). H′ describes the coupling between local spin Si and fluctuating field δHi, which contributes to spin relaxation and thus FMR linewidth. Here, k is the wavevector of magnon; αk, α+−k, uk, and vk are the annihilation and creation Okada et al.
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 δHi (21), the linewidth ΔHMN due to δHi at k = 0 (Kittel mode) is expressed as ΔHMN ∼ (S/2)(δHk=0)2sin2θM(H1/H2)1/2 with H1 = HRcos(θH − θM)+HK1effcos2θM − HK2cos4θM, H2 = HRcos(θH −R θM)+HK1effcos2θM −(HK2/2)(cos2θM + cos4θM), and (δHk=0)2 = dτδHk=0(t0)δHk=0(t0 + τ)e−2iπfτ. Here, t0 is the arbitrary time, and τ is the elapsed time from t0. Writing the correlation function δHk=0(t0)δHk=0(t0 + τ) = (δH)2e−jτj/τ0, where τ0 is the relaxation time of the random field δHk=0, we obtain ΔHMN ≈ ðS=2ÞðδHÞ2 τ0 sin2 θM ðH1 =H2 Þ1=2 = Γ sin2 θM ðH1 =H2 Þ1=2 , [1] for 2πfτ0
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.
|
CoFeB/MgO ferromagnetic resonance interfacial anisotropy
| damping | motional narrowing |
T
he 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 highperformance 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 Co0.4Fe0.4B0.2 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 www.pnas.org/cgi/doi/10.1073/pnas.1613864114
analyzer (VNA-FMR) using a coplanar waveguide, and a conventional method using a TE011 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 θH = 90°) or perpendicular (θH = 0°) to the sample plane, and the spectra are obtained as the transmission coefficient S21 (9). The latter technique measures the spectra at a fixed f of 9 GHz under H at various θH, and the spectra are obtained as the derivative of the microwave absorption with respect to H (10). The temperature dependence of spontaneous magnetization MS was measured by a superconducting quantum interference device magnetometer. VNA-FMR Fig. 1A shows typical VNA-FMR at selected values of f for CoFeB with t = 1.5 nm at temperature T = 300 K and θH = 90°. We determine the resonance field HR and linewidth (full width at half maximum) ΔH from the fitting of the modified Lorentz function (solid lines in Fig. 1A) to the VNA-FMR spectra (9). Fig. 1B shows the rf frequency dependence of HR at θH = 0° and 90°, which we use to determine the effective perpendicular anisotropy fields HKeff from the resonance condition; f = μ0γ(HR + HKeff)/(2π) for θH = 0° and f = μ0γ[HR(HR − HKeff)]1/2/(2π) for θH = 90°. Here, μ0 is the permeability in free space, and γ the gyromagnetic ratio. As shown in Fig. 1C, HKeff 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 ΔH as a function of t and T. As shown in Fig. 1D, when H is applied perpendicular to the film (θH = 0°), ΔH obeys linear dependence, 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 cuttingedge technology now allow us to probe and reveal physics previously hidden in the bulk properties of ferromagnets. 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. 1
A.O. and S.H. contributed equally to this work.
2
To whom correspondence may be addressed. Email: [email protected] or [email protected].
PNAS | April 11, 2017 | vol. 114 | no. 15 | 3815–3820
APPLIED PHYSICAL SCIENCES
Edited by J. M. D. Coey, Trinity College Dublin, Dublin, Ireland, and approved February 23, 2017 (received for review August 19, 2016)
B f = 6 GHz
1.0
12 GHz 14 GHz
= 0o
-1 0.0
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0.3
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E t = 1.4 nm
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12
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= 0o
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10
8
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HKeff (T)
1
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Fig. 1. VNA-FMR. (A) Typical spectra [real part of transmission coefficient Re(S21) of coplanar waveguide] of CoFeB with thickness t = 1.5 nm as a function of rf frequency f obtained at magnetic–field angle θH = 90° and temperature T = 300 K. Solid lines are fits by the modified Lorentzian. (B) rf frequency f dependence of resonance field HR obtained at θH = 0° and 90°. (C) Thickness t dependence of effective perpendicular anisotropy fields HKeff. rf frequency f dependence of the FMR linewidths ΔH obtained at θH = 0° as a function of (D) t at T = 300 K and as a function of (E) T for CoFeB with t = 1.5 nm. Solid lines in D and E are linear fits. (F and G) Same as D and E but at θH = 90°. Dashed lines in F and G are nonlinear fits based on two-magnon scattering. arb, arbitrary.
which is expressed as ΔH = ΔHin + ΔHinhom = (2hα/gμ0μB)f + ΔHinhom, where h is the Planck constant and μB is the Bohr magneton (9, 11). Here, ΔHin is related to the intrinsic linewidth governed by α and ΔHinhom is the extrinsic contribution due to inhomogeneity, such as the distribution of magnetic anisotropy. The value of α determines the slope in Fig. 1D and ΔHinhom corresponds to the intercept on the vertical axis. Fig. 1D shows that α is nearly independent of t (α ∼ 0.004) and ΔHinhom increases with decreasing t. As seen from Fig. 1E for CoFeB with t = 1.5 nm, α is also nearly independent of T, whereas ΔHinhom 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 (θH = 90°) (Fig. 1F), 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 ΔHTMS; TMS is known to be activated when the magnetization angle θM from the sample normal is greater than 45° (4–6). The nonlinear frequency dependence of ΔH in Fig. 1F can be described in terms of TMS (dashed lines), assuming ΔH = ΔHin + ΔHinhom + ΔHTMS. As depicted in Fig. 1G, 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 ΔHTMS with T from 300 to 4 K is calculated to be ∼10% at most, using the measured T dependence of MS and HKeff (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). 3816 | www.pnas.org/cgi/doi/10.1073/pnas.1613864114
Cavity-FMR Fig. 2A 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 HR and ΔH (16). We fit the resonance condition to the magnetic-field angle dependence of HR shown in Fig. 2B to obtain the magnitude of the effective firstorder and second-order perpendicular anisotropy fields, HK1eff and HK2, respectively, by following the procedure in ref. 10. Fig. 2C shows the CoFeB thickness dependence of HK1eff and HK2 along with that of HKeff in Fig. 1C obtained from VNA-FMR. HK2 is nearly independent of t and its strength is a few tens of millitesla at most. HK1eff increases with decreasing t, in agreement with HKeff obtained from VNA-FMR, confirming again the presence of perpendicular interfacial anisotropy at the CoFeB/ MgO interface (8). In Fig. 2C, we plot also the values of HKeff obtained from magnetization measurements (8), which show good correspondence with those obtained from FMR measurements. Fig. 2D shows the magnetic-field angle dependence of ΔH as a function of t. For θM > 45°, where TMS is expected to be activated, ΔH is larger for CoFeB with smaller t. As shown in Fig. 2E, however, a nearly twice larger ΔH at lower T cannot be explained by the TMS contribution to ΔH. The results obtained from cavity-FMR are consistent with those from VNA-FMR, indicating that the unexpected T dependence of ΔH is not an artifact. Discussion Because the linewidth enhancement for θH = 90° is larger for thinner CoFeB (Figs. 1F and 2D), 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. 1C and 2C. Fig. 2F shows log(K(T)/K(4 K)) versus log(MS(T)/MS(4 K)) (Callen– Callen plot) (17), where MS(T) is the measured spontaneous Okada et al.
~0
1.0
o
0 t = 1.5 nm T = 300 K f = 9 GHz
2.1 nm 0.5 1.5 nm
T = 300 K
-0.8
VNA-FMR, Heff K 0o,
0.4
0.6
0
30
D
E 20
F
20
0
2.1 nm T = 300 K
2.6 nm
0
120 K 200 K T = 300 K t = 1.5 nm
30
60
(deg) H
90
0
30
60
90
(deg) H
90o
1.5
0.9 t = 1.5 nm 300 K
0.8 0.9
2.0
t (nm)
2.5
G 20
0 0
1.0
4K
K(T)/K(4 K)
t = 1.5 nm 10
90
4K 1
10
60
(deg) H
H (mT)
0.2
HK2
MH, HKeff
-0.4
0.0
H (T) 0
H (mT)
Cavity-FMR Heff , K1
T = 80 K
16
160 K 300 K
12 8
t = 1.5 nm = 90o H
0
-1 0.0
0.0
HK (T)
24o
C
0
44
32o
o
HR (T)
= 88o H
T = 300 K t = 2.6 nm
1
MS(T)/MS(4 K)
4 0 0
10
f (GHz)
20
Fig. 2. Cavity-FMR. (A) Typical spectra of CoFeB with thickness t of 1.5 nm as a function of magnetic-field angle θH at temperature T = 300 K. Solid lines are fits by the derivative of Lorentzian. (B) Angle θH dependence of resonance field HR as a function of t. Solid lines are fitted lines by using the resonance condition. (C) Thickness t dependence of effective first-order perpendicular anisotropy field HK1eff and second-order anisotropy field HK2 along with the results in Fig. 1C and from magnetization measurements. Angle θH dependence of the FMR linewidth ΔH obtained as a function of (D) t at T = 300 K and as a function of (E) T for CoFeB with t = 1.5 nm. Solid lines in D and E are fitted lines. (F) Double-logarithm plot of normalized magnetic anisotropy energy density K(T)/K(4K) versus normalized spontaneous magnetization MS(T)/MS(4 K). Solid line is a linear fit. (G) Calculated ΔH as a function of rf frequency f at different temperatures by using the parameters obtained from the analyses of cavity-FMR spectra. arb, arbitrary.
magnetization and K(T) the perpendicular magnetic anisotropy energy density determined from HK1eff and HK2 using cavityFMR (10); K = MS(HK1eff − HK2/2)/2 + MS2/(2μ0). The linear behavior in Fig. 2F gives a slope m of 2.16 [the exponent m in the Callen–Callen law of K(T)/K(0) = (MS(T)/MS(0))m], consistent with the relationship between interfacial anisotropy energy density and MS 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 δHi 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 δHi is along the film normal (the in-plane component is expected to cancel out), it gives rise to a broader linewidth at larger θM, which is similar to the angular dependence of the TMS contribution. The T dependence of linewidths is also explained by the presence of δHi. The value of δHi 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 δHi, 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′ = −ΣiδHi·Siz ∼ (S/2)1/2sinθMΣkδH−k(uk + vk)(αk +α+−k) (z direction along film normal) (Materials and Methods). H′ describes the coupling between local spin Si and fluctuating field δHi, which contributes to spin relaxation and thus FMR linewidth. Here, k is the wavevector of magnon; αk, α+−k, uk, and vk are the annihilation and creation Okada et al.
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 δHi (21), the linewidth ΔHMN due to δHi at k = 0 (Kittel mode) is expressed as ΔHMN ∼ (S/2)(δHk=0)2sin2θM(H1/H2)1/2 with H1 = HRcos(θH − θM)+HK1effcos2θM − HK2cos4θM, H2 = HRcos(θH −R θM)+HK1effcos2θM −(HK2/2)(cos2θM + cos4θM), and (δHk=0)2 = dτδHk=0(t0)δHk=0(t0 + τ)e−2iπfτ. Here, t0 is the arbitrary time, and τ is the elapsed time from t0. Writing the correlation function δHk=0(t0)δHk=0(t0 + τ) = (δH)2e−jτj/τ0, where τ0 is the relaxation time of the random field δHk=0, we obtain ΔHMN ≈ ðS=2ÞðδHÞ2 τ0 sin2 θM ðH1 =H2 Þ1=2 = Γ sin2 θM ðH1 =H2 Þ1=2 , [1] for 2πfτ0
