Magnetization dynamics of Fe_{1-x}Gd_{x} thin films (0 ≤ *x* ≤ 22%) has been investigated by ferromagnetic resonance (FMR). Out-of-plane magnetic field orientation dependence of resonance field and linewidth has been measured. Resonance field and FMR linewidth have been fitted by the free energy of our system and Landau-Lifshitz-Gilbert (LLG) equation. It is found that FMR linewidth contains huge extrinsic components including two-magnon scattering contribution and inhomogeneous broadening for FeGd alloy thin films. In addition, the intrinsic linewidth and real damping constants have been obtained by extracting the extrinsic linewidth. The damping constant enhanced from 0.011 to 0.038 as Gd dopants increase from 0 to 22% which originates from the enhancement of L-S coupling in FeGd thin films. Besides, gyromagnetic ratio, Landé factor *g* and magnetic anisotropy of our films have also been determined.

## I. INTRODUCTION

Magnetization dynamics in nanoscale magnetic structures has attracted many interests due to their applications in magnetic memory and fast spintronic devices.^{1–4} Magnetic damping plays a critical role in the magnetization dynamic behaviors of magnetic media for the modification of response speed. Earlier studies found that the damping of rare-earth–transition-metal (RE–TM) films effectively increased due to the enhancement of the spin-orbital interaction.^{5–8}

It is well known that the magnetic damping can be obtained indirectly from FMR measurements. However, the relaxation mechanisms of FMR linewidth are rather complicated, which includes not only intrinsic Gilbert damping caused by spin-orbit coupling but also extrinsic contributions of inhomogeneity and defect resulting from surface/interface non-uniformity, respectively.^{9} We have prepared the Fe_{1-x}Gd_{x} films with various Gd concentration deposited on silicon substrates at room temperature by magnetron sputtering successfully.^{10} FMR linewidth for FeGd films was measured and intrinsic linewidth due to Gilbert damping was obtained, and the inhomogeneous linewidth broadening resulting from the inhomogeneity of the films was separated theoretically. It is found that the inhomogeneous linewidth broadening is an unreasonable negative value, which is likely that there are other mechanisms contributing to the linewidth, such as two-magnon scattering^{11,12} caused by defects in thin films. In this paper, focusing on the real intrinsic effect of Gd dopants on FMR linewidth, we further studied the various mechanisms on FMR linewidth of FeGd films, especially the contribution from two-magnon scattering. Extracted the two-magnon scattering linewidth from FMR linewidth by the theory of Arias and Mills,^{12,13} thus the real intrinsic linewidth and Gilbert damping parameters were obtained.

## II. EXPERIMENTAL METHOD

Fe_{1-x}Gd_{x} films (30 nm) with Ta (5 nm) buffer and capping layers were deposited on Si substrates at room temperature by magnetron sputtering with base pressure of 1.0×10^{-5} Pa, argon pressure of 0.3 Pa, and incident dc power of 30 W. To induce small uniaxial in-plane anisotropy, a 50 Oe magnetic field was applied in the film growth. The Gd content was adjusted by the number of Gd chips locating symmetrically on the surface of Fe target. The films’ composition was analyzed by energy dispersive x-ray fluorescence spectrometer (EDS) and the structure was investigated by x-ray diffraction (XRD). Magnetic static properties were measured by vibrating sample magnetometer (VSM) at room temperature. Dynamic properties were investigated by *ex situ* FMR technique using a Brucker EPR equipment (ER-200D-SRC) at fixed microwave frequency of 9.78 GHz. The sample placed at the center of a TE_{102} rectangular microwave cavity in a scanning dc magnetic field (0–14000 Oe) can be rotated so that the film plane was varied from perpendicular to the field ($\theta H=0\xb0$) to parallel to the field ($\theta H=90\xb0$). According to EDS results, the Gd atomic concentration in our FeGd films are 0, 7%, 10%, 13%, 16%, and 22%.

## III. STRUCTURES AND MAGNETIZATIONS

In FIG. 1(a), the intensity of the Fe (110) peak in the XRD patterns decreases with increasing Gd content and disappears completely at the concentration of 16% Gd, implying that the film crystal structure has been destroyed and transformed from polycrystalline to amorphous state because of the RE doping, similar to our earlier work.^{7} As shown in FIG. 1(b), the saturation magnetization ($MS$) of FeGd films reduces from 1215 emu/cm^{3} without Gd doping to 111 emu/cm^{3} at 22% Gd concentration, which agrees with the antiparallel alignment between the Gd and Fe moments. From the simple Neel model ($Ms=(1\u2212x)MFe\u2212xMGd$), two calculated lines are drawn in FIG. 1(b) for comparison, in which one stands for the magnetic moment of Gd in anti-parallel coupling to that of Fe, marked as [Fe($\u2191$)Gd($\u2193$)], and another without contribution of Gd moment, marked as [Fe($\u2191$)Gd(0)]. From experimental magnetization data, we see that the rate of decreasing $MS$ is faster than either of fitting lines. For Gd content below 10%, the moment of Gd is shown minor contribution, and antiparallel coupling to the moment of Fe exists only at *x* = 13%. For higher Gd concentration, however, $MS$ of FeGd film reduces rapidly, which may indicate the dispersion of Fe moment due to the local magnetic anisotropy of Gd, exhibiting a sperimagnetism^{14} as the blue guide line shows in FIG. 1(b).

## IV. THEORETICAL MODEL FOR FITTING FMR LINEWIDTH

The Gilbert damping is described by the phenomenological Gilbert parameter in the LLG equation:

The first term is the magnetization precession torque in the effective field, and the second term is the Gilbert damping torque with the damping parameter. $\gamma $ is the gyromagnetic ratio, ** M** is the instantaneous magnetization vector, and $MS$ is the saturation magnetization.

To fit the experimental data of FMR linewidth, we first need to fit the experimental angular dependence of resonance field *H*_{res}. The free energy density of a polycrystalline film system, ignoring the in-plane anisotropy, is given:

where, $\theta $ and $\theta H$ are the angle of the magnetization vector and applied field with respect to the film normal, respectively. The first term is the Zeeman energy, the second term is the demagnetizing energy, and the terms in brackets represent the second- and fourth-order surface perpendicular anisotropy energy. By minimizing the free energy density and then solving the LLG equation, the dispersion relation for resonance field as a function of field orientation is obtained^{7}

where, $4\pi Meff=4\pi MS\u22122K\u22a51MS$, $H\u22a5=2K\u22a52MS$.

Using Eq. (3), we obtained the fitting curves and yielded the values of the effective magnetization $4\pi Meff$, the second- and the fourth-order perpendicular anisotropy constant $K\u22a51$, $K\u22a52$, and the gyromagnetic ratio $\gamma $ in Table I.

X% Fe_{1-x}Gd_{x}
. | M_{s} (Gs)
. | $\gamma $ (10^{7})
. | g
. | $K\u22a51$ (10^{4}erg/cm^{3})
. | $K\u22a52$ (10^{4}erg/cm^{3})
. | $4\pi Meff$ (10^{4}Gs)
. | $\alpha $ (10^{-3})
. |
---|---|---|---|---|---|---|---|

0 | 1215 | 1.78 | 2.03 | − 51 | 0 | 1.6 | 1.8 |

7 | 1101 | 1.81 | 2.07 | − 21 | 0 | 1.4 | 1.5 |

10 | 1045 | 1.78 | 2.03 | − 22 | 0 | 1.4 | 1.1 |

13 | 820 | 1.93 | 2.20 | − 11 | 33 | 1.1 | 1.4 |

16 | 288 | 1.89 | 2.15 | 9.3 | 1.2 | 0.3 | 2.7 |

22 | 111 | 1.97 | 2.24 | − 1.6 | − 0.7 | 0.2 | 3.8 |

X% Fe_{1-x}Gd_{x}
. | M_{s} (Gs)
. | $\gamma $ (10^{7})
. | g
. | $K\u22a51$ (10^{4}erg/cm^{3})
. | $K\u22a52$ (10^{4}erg/cm^{3})
. | $4\pi Meff$ (10^{4}Gs)
. | $\alpha $ (10^{-3})
. |
---|---|---|---|---|---|---|---|

0 | 1215 | 1.78 | 2.03 | − 51 | 0 | 1.6 | 1.8 |

7 | 1101 | 1.81 | 2.07 | − 21 | 0 | 1.4 | 1.5 |

10 | 1045 | 1.78 | 2.03 | − 22 | 0 | 1.4 | 1.1 |

13 | 820 | 1.93 | 2.20 | − 11 | 33 | 1.1 | 1.4 |

16 | 288 | 1.89 | 2.15 | 9.3 | 1.2 | 0.3 | 2.7 |

22 | 111 | 1.97 | 2.24 | − 1.6 | − 0.7 | 0.2 | 3.8 |

For an ideal film which is homogeneous and without defects, the field-swept linewidth is simply proportional to the energy consumption due to the interaction of orbital and lattice in the material, which is influenced by L-S coupling, defined as intrinsic FMR linewidth.^{15} However, in the practice, FMR linewidth is broadened by various surface/interface non-uniformity and defects. It is necessary to account for extrinsic inhomogeneous linewidth and two-magnon scattering broadening. Thus the total FMR linewidth is the sum of the intrinsic linewidth and the extrinsic linewidth, which can be expressed as follows:

By assuming that the damping coefficient in Eq. (1) is homogeneous and isotropic, the in-plane linewidth due to Gilbert damping is derived as, $\Delta HGilb=2\alpha \omega 3\gamma $^{16} which is simply proportional to the frequency.

The extrinsic contributions depend on the sample inhomogeneity that is associated with the local variation of magnetization, or internal magnetic fields. In the regime of large and strong inhomogeneity, the local inhomogeneous fields are larger than the intrinsic exchange and dipole fields. Inhomogeneous linewidth broadening which can be expressed as follows:^{7}

where $\Delta \theta H$ represents the spread in the orientations of the crystallographic axes among various grains, and $\Delta 4\pi Meff$ represents the magnitude of the inhomogeneity of local demagnetizing field.

Another extrinsic linewidth mechanism, referring to two-magnon scattering, is usually essential in ultrathin ferromagnets. Previous research has shown that two-magnon scattering model is valid for weak and small inhomogeneity,^{17} where the ** k** = 0 magnon excited by FMR scatters into degenerate states of magnons with wave vectors

*k$\u2260$*0. This process requires that the spin-wave dispersion is allowed for degenerate states and the scattering centers in the sample such as defects with random spatial character. The defects scatter the FMR spin waves with zero wave vector into the manifold of degenerate modes. The fundamentals of the two-magnon mechanism were recently reviewed,

^{11–13,18–20}and the existence of two-magnon scattering has been demonstrated in many systems.

^{21,22}Two-magnon scattering linewidth $\Delta HTMS$ can be expressed as based on theoretical description

^{12,13}and the free energy density model (Eq. (2)):

where $\Gamma (H,\theta H)$ is the fitting value changed with value and orientation of external field.

The experimental values of FMR linewidth reflect the effects of both extrinsic linewidth and intrinsic Gilbert linewidth. The equality values between the out-of-plane $\Delta H(0\xb0)$ and in-plane $\Delta H(90\xb0)$ linewidths present a clear evidence that the FMR linewidth is determined exclusively by the intrinsic contribution due to the Gilbert damping. However, in reality, it probably happens that $\Delta H(0\xb0)\u2260\Delta H(90\xb0)$. For $\Delta H(0\xb0)>\Delta H(90\xb0)$, such a major deviation is related to the sample inhomogeneity resulting from a specific granular microstructure characteristic.^{23} For the case of $\Delta H(0\xb0)<\Delta H(90\xb0)$, usually encountered in ultrathin films with a good structural order,^{24–26} the intrinsic Gilbert and extrinsic two-magnon contributions are usually more than sufficient to describe experimental data.^{12} Note that the extrinsic inhomogeneous linewidth and two-magnon scattering broadening is not referring to the real damping but seen only in the measurement of FMR linewidth. The Gilbert damping constant $\alpha $ is the intrinsic damping related to energy loss which is important in determining magnetization dynamics and switching time, hence extracting real damping $\alpha $ is of great importance.

## V. RESULTS AND DISCUSSION

Fitting the applied field angular dependence of resonance field experimental data by Eqs. (2) and (3), the fitting curves are plotted in FIG. 2(a) and match the experimental data well, from which values of the effective magnetization $4\pi Meff$, the second- and the fourth-order surface perpendicular anisotropy constant $K\u22a51$, $K\u22a52$, and gyromagnetic ratio $\gamma $ are extracted (Table I). Obviously, the resonant field perpendicular to the film plane decreases as Gd content increases, by contrast, it increases in parallel geometry in FIG. 2(b), which is caused by the decrease of the effective magnetization as shown by the material parameters in Table I. Meanwhile we found there is a leap in resonance field when Gd content *x* increases from 13% to 16%, which might be related to the film structure since the film crystal structure has been transformed from polycrystalline to amorphous state as confirmed by XRD in FIG. 1(a). From Table I, we see that the effective magnetization $4\pi Meff$ decreased from $1.6\xd7104$ Gs to $0.2\xd7104$ Gs with 0-22% Gd doping, which is consistent with saturation magnetization $MS$, thus the effective field and can be adjusted by the change of Gd doping. It is interesting to note that the evolution of the surface perpendicular uniaxial anisotropy (PA) constants $K\u22a51$ and $K\u22a52$ with increasing Gd concentration displays a spin reorientation behavior (Table I). The undoped film shows an easy-plane anisotropy (as both $K\u22a51$ and $K\u22a52$ are negative). For the 13% Gd-doped film, the PA witnesses an easy cone behavior (0<$\u2212K\u22a51$ <2 $K\u22a52$), that is $\theta 0=sin\u22121\u2212K\u22a512K\u22a52=9.6\xb0$ with the lowest PA energy on the shell of the cone.

FIG. 3 (a)–(f) shows the angular dependences of FMR linewidth on out-of-plane field orientation $\theta H$ with increasing Gd content. The solid circles are experimental data and the lines are fitting curves. The FMR linewidth decreases with increasing Gd content by more than an order compared with Fe film. There is a peak of linewidth maximum at $\theta H\u224810\xb0$ for Gd content less than 13% as shown in FIG. 3 (a)–(d). However, only a monotonic upward trend is observed for Gd content between 16% and 22% (FIG. 3 (e)–(f)). For all of the films, We note that, for all samples, the linewidth at 0° is always narrower than that at 90°, indicating that the extrinsic two-magnon contributions is very important in the FMR linewidth fitting. For Gd content is less than 13%, the larger value of linewidth might be originated from a deviation of magnetization, thus the inhomogeneous linewidth broadening needs to be taken into account. By substituting above parameters obtained from resonance field fitting into Eq. (4), the calculated dependence of the FMR peak-to-peak linewidth on $\theta H$ are obtained, as shown by the solid lines in FIG. 3.(a)–(f), which agree with the experimental data well.

By extracting the two-magnon scattering linewidth from FMR linewidth using Eq. (6)^{12,13} and inhomogeneous linewidth broadening using Eq. (5), the intrinsic linewidth and real intrinsic damping are obtained in FIG. 3.(a)–(f) and Table I, respectively. From the fitting, it is found that the inhomogeneous linewidth $\Delta Hinhom$ plays a rather important part in peak-to-peak linewidth as lower Gd doping till 13%, in which the direction deviation of magnetization dominates linewidth as shown in Fig. 4(a). In pure Fe film, inhomogeneous linewidth $\Delta Hinhom$ shows the largest contribution, even exceeds the value of intrinsic linewidth. The contribution of two-magnon scattering broadening is dominant in the total linewidth in our FeGd films, much broader than intrinsic Gilbert FMR linewidth. As expected, relatively high surface/interface roughness for samples prepared by magnetron sputtering and more defects resulting from more Gd dopants are the origin of the inhomogeneity broadening. Similar situation has been reported before in other systems.^{22}

Furthermore, two-magnon scattering broadening is zero when applied magnetic field near the film normal, so FMR linewidth includes only intrinsic Gilbert broadening $\Delta HGilb$ and inhomogeneous linewidth $\Delta Hinhom$. When $\theta H$ is larger than critical angle $\theta TMScrit$, i.e., $\theta >\pi /4$, two-magnon scattering is active as plotted in FIG. 3(a)–(f), which is consistent with the theory,^{13} and experimental reports before.^{21} $\theta TMS$ is about 6° for 0-10% Gd doping and increases from 8° to 32° with further doping rates. Two-magnon scattering linewidth shows an upward trend with the increase of $\theta H$ as Gd content is from 13% to 22%.

FIG. 4(b) shows that the real intrinsic damping constant $\alpha $ decreases slightly from 0.018 to 0.011 at x < 13%, in agreement with other reports about low Gd doping (x < 10%) which has little effect on damping.^{27,28} In contrast, $\alpha $ rises monotonically to 0.038 when the doping continues increasing to 22%. The strong damping in heavy Gd doping films can be explained as following. In alloys, Gd is in excited states other than its ground *S* state with *L*=0, thus significantly strengthens the L-S coupling and results in higher damping constant. This is also supported by the observation of a strong increase in the orbital-to-spin moment ratio of Fe with increasing Gd concentration from X-ray magnetic circular dichroism (XMCD) measurements.^{29}

## VI. CONCLUSION

In summary, magnetization dynamics of FeGd thin films have been studied by FMR. Out-of-plane angle dependence of resonance field and linewidth have been fitted by the free energy of the structure and the LLG equation from which we found that FMR linewidth contains additional two-magnon scattering component for FeGd thin films, which contributes dominantly to broadening linewidth. Besides, an inhomogeneous linewidth broadening plays rather an important part in the total linewidth, especially in pure Fe film, which mainly comes from the direction deviations of magnetization. By extracting the extrinsic broadening terms due to two-magnon scattering and inhomogeneity from the FMR linewidth, the intrinsic damping of FeGd thin films was successfully derived from the remaining linewidth. The damping constant rises from 0.011 to 0.038 with the Gd concentration increases from 0 to 22% because of the enhancement of L-S coupling in FeGd thin films. Determination of the intrinsic damping would be very significant in the research and development of spintronic devices.

## ACKNOWLEDGMENTS

This work is supported by NSFC (Nos. 61427812, 51571062, 11504047, 61306121 and 11364015), NSF of Jiangsu Province of China (No. BK20141328).