Metallic ferromagnets with ultra-low damping are highly desirable for charge-based spintronic applications. Here, we systematically investigate the magnetic dynamics of Co_{25}Fe_{75} epitaxial films with a Gilbert damping constant as low as 7.1 × 10^{−4}. The in-plane angular dependence of ferromagnetic resonance (FMR) was measured on various thicknesses of Co_{25}Fe_{75} films grown on MgO and MgAl_{2}O_{4}, from which the mechanisms for FMR linewidth broadening can be distinguished and quantified. The thickness dependencies of the magnetic anisotropy and inhomogeneous broadening of the linewidth are good indicators of crystal quality and magnetic uniformity. Additionally, it is shown that anisotropic two-magnon scattering is induced by defects at the surfaces.

Materials with low magnetic damping draw attention for their applications in microwave and spintronic devices due to their low loss and ability to generate large spin currents through ferromagnetic resonance (FMR) spin pumping. The magnetic damping in metallic ferromagnets (FMs) is typically much larger than in insulating FMs, such as yttrium iron garnet (YIG),^{1–11} due to the presence of conduction electron scattering.^{12,13} Metallic FMs with low damping and high conductivity are needed in low-loss, high-efficiency spintronic devices that require charge currents.^{14,15} While the Gilbert damping constant (*α*) of YIG can be as low as 10^{−4}–10^{−5}, it was only recently reported that the metallic FM, Co_{25}Fe_{75}, exhibits very low Gilbert damping.^{12,16} Previously, we reported^{16} that 7-nm epitaxial Co_{25}Fe_{75} films have a damping constant as low as 7.1 × 10^{−4} determined by fitting the linear region of the frequency dependence of the FMR linewidth up to 18 GHz. To further our understanding of this record-low damping metallic FM, we studied the angular dependence of FMR for various thicknesses of Co_{25}Fe_{75} films to extract a number of magnetic characteristics, such as the effective magnetization, anisotropies, and two-magnon scattering.

Co_{25}Fe_{75} epitaxial films with thicknesses (*t*) of 3, 7, 15, and 32 nm were grown on (001)-oriented MgO and MgAl_{2}O_{4} (MAO) substrates with a lattice mismatch of 3.9% and 0.4%, respectively, using ultrahigh vacuum off-axis sputtering at a substrate temperature of 300 °C. The Co_{25}Fe_{75} films were capped with a 2.8 nm Cr layer grown at room temperature to prevent oxidization. The thickness and crystalline quality were examined by X-ray reflectometry (XRR) and X-ray diffraction (XRD).

Figures 1(a) and 1(b) show the 2*θ*-*ω* XRD scans of the Co_{25}Fe_{75} films grown on MgO and MAO substrates, from which the out-of-plane lattice constants were found to be 2.875 (2.861) and 2.893 (2.876) Å for the 3 (32) nm films, respectively. The pronounced Laue oscillations for the Co_{25}Fe_{75} films as thin as 7 nm grown on MAO indicate that they are of high crystalline quality, which can be attributed to their minimal lattice mismatch.^{16} By fitting the XRR scans (see supplementary material) for the Co_{25}Fe_{75} films of 3, 7, 15, and 32 nm thicknesses, we obtain a film roughness of less than 0.6 nm.

In order to understand the mechanisms responsible for the low damping in the Co_{25}Fe_{75} epitaxial films, we measured the in-plane angular dependence of FMR for the Co_{25}Fe_{75} films at room temperature using a Bruker electron paramagnetic resonance (EPR) spectrometer in a cavity at a fixed radio frequency (rf) *f *=* *9.65 GHz. Figure 1(c) shows a schematic of the coordinate system where *θ* (*θ*_{H}) and $\varphi $ ( $\varphi $_{H}) are the out-of-plane and in-plane angles representing the orientation of the equilibrium magnetization, ** M** (applied magnetic field,

**). Previous VSM measurements found that the in-plane easy axis is along Co**

*H*_{25}Fe

_{75}[100] $(\varphi =0\xb0)$, and the in-plane hard axis is along Co

_{25}Fe

_{75}[110] ( $\varphi =45\xb0)$ due to magnetocrystalline anisotropy.

^{16}

From the in-plane FMR derivative spectra for the Co_{25}Fe_{75} (7 nm) films on MgO and MAO (see supplementary material), the resonance field (*H*_{res}) and peak-to-peak linewidth ( $\Delta H$) can be obtained by fitting to a derivative of the Lorentzian function. At some angles, the FMR derivative spectra show an additional shoulder peak [for example, Fig. S2(b) at $ \varphi H =\u221229\xb0$], likely due to inhomogeneity. However, the spectrum can still be fit to the derivative of a Lorentzian function to obtain the peak-to-peak linewidth and *H*_{res} with uncertainties no larger than 9.4% for $\Delta H$ and 1.4% for *H*_{res} as compared to fitting the individual peaks and shoulders. The resonance field decreases as the sample is rotated from the hard to easy axis, while the linewidth is the smallest along the hard axis and the largest at angles between the easy and hard axis.

From the in-plane angular dependence of the resonance field, we determined the effective magnetization and anisotropy fields. Figure 2(a) shows the in-plane angular dependence of *H*_{res} for the 7 nm Co_{25}Fe_{75} films on MAO (for Co_{25}Fe_{75} films on MgO, see supplementary material). The total free energy density (*F*) for a FM film with cubic symmetry is given by^{17,18}

where $4\pi M eff $, *H*_{4⊥}, *H*_{4∥}, and *H*_{2∥} are the effective magnetization, out-of-plane cubic anisotropy field, in-plane cubic anisotropy field, and in-plane uniaxial anisotropy field, respectively. $4\pi M eff =4\pi M S \u2212 H 2 \u22a5 $, where 4π*M*_{s} is the saturation magnetization^{16} and *H*_{2⊥} is the out-of-plane uniaxial anisotropy field. $ \u2009 \varphi 2 \u2225 $ represents the direction of the in-plane hard axis for the uniaxial anisotropy term. For a perfect film, $ \u2009 \varphi 2 \u2225 $ = 45°, while in reality, $ \u2009 \varphi 2 \u2225 $ may deviate slightly from 45°. By minimizing the free energy density, we can deduce $\varphi $, which can then be used in the Smit-Beljer's approach^{19,20} for the FMR condition

Here, *γ* = 2π × 29.5 GHz/T is the gyromagnetic ratio,^{21} and we fix *θ = θ _{H} =* 90° for the in-plane measurements. Precise determination of

*γ*requires a frequency dependent FMR measurement at multiple in-plane angles. However, this is not possible for our Co

_{25}Fe

_{75}films with large in-plane magnetocrystalline anisotropy because when the applied field is not along the in-plane hard-axis,

*H*

_{res}decreases below the saturation field for the majority of our frequency range. Measuring FMR below the saturation field leads to a broadening of the linewidth and hinders the determination of

*γ*; thus, we use the reported

*γ*value from the literature.

^{16,21}By fitting to the experimental data of

*H*

_{res}using Eq. (2) for the whole range of $\varphi $

_{H}, we obtained H

_{2∥}= 0 Oe (−4 Oe), H

_{4∥}= 275 Oe (325 Oe), and $4\pi M eff $ = 2.25 T (1.96 T) for the 7 nm Co

_{25}Fe

_{75}films on MgO (MAO). Using the previously measured 4π

*M*

_{s}of 2.46 T (2.34 T) for the 7 nm film on MgO (MAO) by a SQUID magnetometer,

^{16}we calculated

*H*

_{2⊥}= 0.21 T (0.38 T).

It is worth mentioning that out-of-plane angular dependence and frequency dependence measurements have been used to fit the parameters as a method of consistency check. However, our Co_{25}Fe_{75} films exhibit a large $4\pi M eff $ which results in an out-of-plane resonance field well beyond our instrument limit. Additionally, we found^{16} that fitting the resonant field as a function of frequency yielded similar results as fitting the resonant field as a function of the in-plane angle. Therefore, in the rest of the paper, we will only analyze the in-plane angular dependence of FMR.

The angular dependencies of the FMR linewidth shown in Fig. 2(b) for the 7 nm Co_{25}Fe_{75} film on MAO (see supplementary material for the film on MgO) can be described by^{22}

due to the intrinsic Gilbert damping, inhomogeneous broadening,^{23} and two-magnon scattering (TMS). The intrinsic Gilbert damping induced linewidth can be expressed as^{19}

The detailed deduction is shown in the supplementary material. Our 7 nm Co_{25}Fe_{75} films on MgO exhibit $\alpha <$ 1.0 × 10^{−3} according to a linear fit of the frequency dependence of the total linewidth along the in-plane hard axis.^{16} In this work, all of our measurements are performed at 9.65 GHz, which allows us to reliably measure the linewidth and resonance field. For the remainder of the calculations, we use $\alpha $ = 1.0 × 10^{−3} for all thicknesses of Co_{25}Fe_{75}. Combining Eqs. (1) and (4), we obtain $ \Delta H Gilb = 4 \pi \alpha f 3 \gamma \Xi $, where Ξ is the magnetic dragging function given by^{24}

which describes the consequences of the misalignment of the magnetization from the applied field. The second contribution to Eq. (3) is due to the inhomogeneity of the magnetic film^{22}

with two fitting parameters: $\Delta \varphi H $ is the variation of the angle between the applied field and the inhomogeneous local crystallographic axes and $\Delta (4\pi M eff )$ is the inhomogeneity of the demagnetizing field. The last term in Eq. (3), Δ*H*_{TMS}, is from the two-magnon scattering by crystallographic defects that couple the $ k \u2225 \u2192 \u22600$ spin wave modes with the uniform mode $ k \u2225 \u2192 =0$ of the same frequency.^{25,26} This scattering from the uniform mode into spin wave modes damps the uniform mode and increases $\Delta H$. The $\Delta H TMS $ term can be expressed as^{22,26,27}

where $ \omega 0 =\gamma 4\pi M eff $, $ \varphi x i $ is the angle that describes the crystallographic axes $ x i $ ([100] or [110]) and $ \Gamma x i $ is the magnitude of the two-magnon scattering along $ x i $. According to Ref. 28, the eddy current damping (in SI unit) $ \alpha eddy = \mu 0 2 \gamma M s t 2 / 12 \rho $, where $ \mu 0 $ is the vacuum permeability and *t* and *ρ* are the thickness and resistivity, respectively, of the thin film. For Co_{25}Fe_{75} thin films, taken the maximum thickness of 32 nm as the upper limit of $ \alpha eddy $ and *ρ* = 30.1 *μ*Ω·cm, $ \alpha eddy =1.3\xd7 10 \u2212 7 $, which is three orders of magnitude smaller than the total Gilbert damping constant. Thus, the eddy current contributed damping in our measurement is negligible.

From fitting to the experimental data of $\Delta H$ in Fig. 2(b) using Eqs. (3), (4), (6), and (7), all three linewidth contributions can be obtained. Both $\Delta H Gilb $ and $\Delta H inh $ show a non-sinusoidal four-fold symmetry (note the range of 180°) and peaks between the easy ⟨100⟩ and hard ⟨110⟩ axes due to the magnetic dragging effect.^{22} We note that the magnitude of $\Delta H Gilb $ is by far the smallest among the three terms with a weak angular dependence because of the exceptionally low $\alpha $ in our Co_{25}Fe_{75} films. The two-magnon scattering, which has a much stronger magnitude and angular dependence than $\Delta H Gilb $, also exhibits four-fold symmetry, but the peaks occur along the easy axis. $\Delta H inh $ plays a dominant role in the linewidth, which is typically seen in thin films.^{22} The peak positions of $\Delta H$ are 10°–15° from the hard axes and 30°–35° from the easy axis due to that the in-plane magnetocrystalline anisotropy field, *H*_{4∥}, misaligns ** M** from the applied field when $ \varphi H $ is near the hard axes. This misalignment of

**from**

*M***significantly increases the linewidth.**

*H*We next investigate the thickness dependence of damping in Co_{25}Fe_{75} by performing the same measurements and analysis for the 3, 15, and 32 nm films on MgO and MAO. Figure 3(a) shows the angular dependence of *H*_{res} for all thicknesses of Co_{25}Fe_{75} films grown on MAO (see supplementary material for the films on MgO) with their corresponding fits, from which $4\pi M eff $, *H*_{4∥}, and *H*_{2∥} are obtained as shown in Figs. 4(a) and 4(b). The effective magnetization increases with the film thickness and saturates at 15 nm. The in-plane cubic anisotropy, *H*_{4∥}, has a similar thickness dependence and reaches a value of *H*_{4∥} = 381 and 384 Oe at a thickness of 32 nm for the films on MgO and MAO, respectively. Meanwhile, the magnitude of the in-plane uniaxial anisotropy, *H*_{2∥}, is less than 40 Oe for all samples and does not show a clear thickness dependence.

Figure 3(b) shows the angular dependence of $\Delta H$ for different thicknesses of Co_{25}Fe_{75} films on MAO (see supplementary material for the Co_{25}Fe_{75} films on MgO). As the film thickness increases, the angular dependence of $\Delta H$ transforms from four-fold symmetry to a more complicated angular dependence. This change in symmetry indicates that the two-magnon scattering, which exhibits four-fold symmetry, becomes dominant in thinner films. For thicker Co_{25}Fe_{75} films, the inhomogeneous broadening contribution dominates the linewidth. The magnitude of two-magnon scattering, which is described by

is extracted from Figs. 3(b) and S4(b) (see supplementary material) and shown in Fig. 4(c).^{20} Figure 4(c) reveals a 1/*t*^{2} dependence of the two-magnon scattering, confirming that the two-magnon scattering is due to an interfacial interaction as theoretically predicted.^{25,29} The strength of the two-magnon scattering is proportional to the quadratic of the scattering potential which has a 1/*t* dependence if it is from an interfacial interaction. Figure 4(d) shows the ratio of the three damping contributions for different thicknesses of Co_{25}Fe_{75} films on MAO (see supplementary material for the Co_{25}Fe_{75} film on MgO). The percentage of each damping term is determined by the ratio of their maximum linewidths from the in-plane angular dependent FMR measurements. The Gilbert damping, inhomogeneous broadening, and two-magnon scattering have different symmetries with respect to the in-plane angle, providing a direct comparison of the contributions from each damping mechanism. Table I shows the distribution of the localized crystallographic axes, $\Delta \varphi H $, for different thicknesses of Co_{25}Fe_{75} films on MgO and MAO as obtained from the fits. The Co_{25}Fe_{75} films grown on MAO have smaller Δ $\varphi $_{H} and smaller two-magnon scattering intensity than the films on MgO because the films on MAO have much better crystalline quality and less defects.

Thickness Cr/Co_{25}Fe_{75}(t nm)
. | MgO . | MAO . |
---|---|---|

$\Delta \varphi H $ (deg) . | $\Delta \varphi H $ (deg) . | |

3 | 0.01 ± 0.01 | 0.03 ± 0.01 |

5 | 0.08 ± 0.01 | 0.02 ± 0.01 |

7 | 0.08 ± 0.01 | 0.03 ± 0.01 |

10 | 0.05 ± 0.01 | 0.00 ± 0.01 |

15 | 0.02 ± 0.01 | 0.00 ± 0.01 |

32 | 0.03 ± 0.01 | 0.00 ± 0.01 |

Thickness Cr/Co_{25}Fe_{75}(t nm)
. | MgO . | MAO . |
---|---|---|

$\Delta \varphi H $ (deg) . | $\Delta \varphi H $ (deg) . | |

3 | 0.01 ± 0.01 | 0.03 ± 0.01 |

5 | 0.08 ± 0.01 | 0.02 ± 0.01 |

7 | 0.08 ± 0.01 | 0.03 ± 0.01 |

10 | 0.05 ± 0.01 | 0.00 ± 0.01 |

15 | 0.02 ± 0.01 | 0.00 ± 0.01 |

32 | 0.03 ± 0.01 | 0.00 ± 0.01 |

As shown in Table I, for Co_{25}Fe_{75} films on MgO, the distribution of crystallographic axes first increases from 0.01° at 3 nm to 0.08 at 7 nm and then decreases to 0.02–0.03 above 15 nm. We attribute this variation to the strain relaxation of the films. For the first few nm of Co_{25}Fe_{75} on MgO with a lattice mismatch of 3.9%, the Co_{25}Fe_{75} layer is strained to the MgO lattice. As the film thickness increases, Co_{25}Fe_{75} begins to relax by incorporating dislocations; as a result, $\Delta \varphi H $ increases. The film reaches the critical thickness at ∼7 nm, where it relaxes to the bulk lattice constant of Co_{25}Fe_{75}. Above the critical thickness, the quality of the Co_{25}Fe_{75} films improves until it reaches an equilibrium level at ∼15 nm. Thus, the Co_{25}Fe_{75} films on MgO have the largest $\Delta \varphi H $ at ∼7 nm due to strain relaxation. For Co_{25}Fe_{75} film on MAO with a mismatch of 0.4%, $\Delta \varphi H $ starts at 0.02°–0.03° at thin films and essentially vanishes above 7 nm.

In summary, the high quality single crystal Co_{25}Fe_{75} films with record low Gilbert damping for metallic FMs provide an ideal platform for exploring spin dynamics and spin transfer torque phenomena. The in-plane angular FMR results show a dramatic change in both the resonance field and linewidth near the hard axes. The modeling of the linewidth data reveals the mechanisms that contribute to the broadening of the linewidth for various Co_{25}Fe_{75} thicknesses, especially in the case of two-magnon scattering.

See supplementary material for deduction of damping terms; X-ray reflectometry scans and in-plane FMR derivative spectra of the Co_{25}Fe_{75} films; angular dependencies of the resonance field and linewidth for Co_{25}Fe_{75} films on MgO; relative contributions from Gilbert damping, inhomogeneous broadening, and two magnon scattering as a function of the thickness for the Co_{25}Fe_{75} films on MgO; FMR spectra of the substrate and other control samples; and temperature dependence of the FMR linewidth.

This work was supported by National Science Foundation under Grant No. DMR-1507274 (sample growth and characterization and data analysis) and by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-FG02-03ER46054 (FMR measurements). Partial support for this work was provided by the Center for Emergent Materials, an NSF-funded MRSEC, under Grant No. DMR-1420451 (magnetization measurements).