High Spin-Wave Propagation Length Consistent with Low Damping in a Metallic Ferromagnet

TL;DR

This study demonstrates ultra-low magnetic damping and long spin-wave propagation lengths in Co25Fe75 heterostructures at room temperature, highlighting their potential for magnonic applications.

Contribution

It provides the first comprehensive measurement of intrinsic damping and spin-wave propagation lengths in Co25Fe75 at room temperature, confirming their suitability for low-loss magnonic devices.

Findings

Intrinsic damping as low as 3.18×10⁻⁴ in 26 nm Co25Fe75

Spin-wave propagation length of approximately 21 μm at room temperature

Excellent agreement between damping measurements and spin-wave propagation lengths

Abstract

We report ultra-low intrinsic magnetic damping in Co$_{\text{25}}$Fe$_{\text{75}}$ heterostructures, reaching the low $10^{-4}$ regime at room temperature. By using a broadband ferromagnetic resonance technique, we extracted the dynamic magnetic properties of several Co$_{\text{25}}$Fe$_{\text{75}}$-based heterostructures with varying ferromagnetic layer thickness. By estimating the eddy current contribution to damping, measuring radiative damping and spin pumping effects, we found the intrinsic damping of a 26\,nm thick sample to be $$\alpha_{\mathrm{0}} \lesssim 3.18\times10^{-4}$. Furthermore, using Brillouin light scattering microscopy we measured spin-wave propagation lengths of up to $(21\pm1)\,\mathrm{\mu m}$ in a 26 nm thick Co$_{\text{25}}$Fe$_{\text{75}}$ heterostructure at room temperature, which is in excellent agreement with the measured damping.

Figures

Equations

$$\chi (H_0)$$

$$S_{21}(H_0)=S_{21}^0+iA\frac{\chi (H_0)}{M_{\mathrm{s}}}=S_{21}^0(1+\Delta S_{21}),$$

$${\mu }_0{H}_{\text{res}}={\mu }_0{M}_{\text{eff}}+{\mu }_0{H}_{\text{eff}},$$

$${\mu }_0\Delta H={\mu }_0{H}_{\mathrm{inh}}+2\cdot \frac{2\pi f{\alpha }_{\mathrm{G}}}{\gamma }.$$

$$\frac{L}{\chi }\equiv \tilde{L}=-\frac{2Z_{0}A}{M_{\mathrm{s}}{S}_{21}^0\omega }.$$

$${\alpha }_{\mathrm{rad}}=\frac{1}{4}\frac{\gamma {\mu }_0{M}_{\mathrm{s}}}{Z_0}{\tilde{L}}_0.$$

$${\alpha }_0+{\alpha }_{\mathrm{sp}}={\alpha }_0+2\frac{\gamma \hslash g_{\mathrm{eff}}^{\uparrow \downarrow }}{4\pi M_{\text{s}}}\frac{1}{t},$$

$$\begin{gathered} f_{\mathrm{res}}=\frac{{\mu }_0\gamma }{2\pi }\sqrt{H_0+H_{\mathrm{d}}+H_{\mathrm{k}}+M_{\mathrm{s}}\frac{1-\exp (-kt)}{kt}} \\ \times \sqrt{H_0+H_{\mathrm{d}}+M_{\mathrm{s}}\left(1-\frac{1-\exp (-kt)}{kt}\right)}, \end{gathered}$$

Full text

Itinerant ferromagnets (FM) are advantageous for spintronic and magnonic devices. They benefit from, e.g., large magnetoresistive effects and current-induced spin-orbit torques Gambardella and Miron (2011). In many magneto-resistive technologies (e.g., anisotropic magnetoresistance, giant magnetoresistance, tunnel magnetoresistance) electronic conductivity is indispensable. Moreover, due to high saturation magnetization in metallic FMs, spin-wave (SW) group velocities are in general significantly higher than in insulating ferrimagnets Wessels et al. (2016); Talalaevskij et al. (2017); Körner et al. (2017); Collet et al. (2017). High saturation magnetizations in general ease detection. Nevertheless, itinerant FMs typically have considerable magnetic damping Maksymov and Kostylev (2015); Twisselmann and McMichael (2003). This is unfavorable for many applications. For example, low damping is crucial for oscillators based on spin transfer torques and spin orbit torques as well as for achieving large spin-wave propagation lengths (SWPL) Demidov et al. (2014); Kruglyak, Demokritov, and Grundler (2010); Chumak et al. (2015). The need for thin film materials with low magnetic damping has triggered the interest in the insulating ferrimagnet yttrium-iron garnet (Y${}_{\text{3}}$Fe${}_{\text{5}}$O${}_{\text{12}}$, YIG)Hauser et al. (2016); Evelt et al. (2016); Jungfleisch et al. (2015). Although for YIG, very small total (Gilbert) damping parameters in the order of $\alpha_{\mathrm{G}}\approx 10^{-5}$, and large SWPLs of a few tens of micrometers (up to $\sim$ 25 $\mu$m) in thin films ($\sim$ 20 nm) have been reported Houchen Chang et al. (2014); Collet et al. (2017); Jungfleisch et al. (2015), its insulating properties and requirement for crystalline growth are challenges for large scale magnonic applications.

Schoen et al. recently observed ultra-low intrinsic magnetic damping in Co${}_{\text{25}}$Fe${}_{\text{75}}$ (CoFe) metallic thin films ($\alpha_{\mathrm{0}}=(5\pm 1.8)\times 10^{-4}$) Schoen et al. (2015), and Körner et al. reported PLs of $5\,\mathrm{\mu m}-8\,\mathrm{\mu m}$ in CoFe using time resolved scanning magneto-optical Kerr microscopy Körner et al. (2017). This motivated our study on sputter-deposited CoFe-based thin film heterostructures. We use broadband ferromagnetic resonance (BB-FMR) spectroscopy Kalarickal et al. (2006) in out-of-plane (OOP) geometry and Brillouin light scattering (BLS) microscopy Sebastian et al. (2015) and find intrinsic damping parameters in the lower $10^{-4}$ regime as well as SWPLs of more than $20\,\mathrm{\mu}$m. The damping is therefore comparable to YIG/heavy metal (HM) heterostructures Sun et al. (2013) and the SWPL is comparable to that of state-of-the-art YIG thin films Collet et al. (2017); Jungfleisch et al. (2015). Thin film CoFe is a promising candidate for all-metal magnonic devices, as it combines low magnetic damping with good electrical conductivity and large saturation magnetization, while enabling easy fabrication by room-temperature processing/deposition, no required annealing, polycrystalline structure, and scalability to the nanometer regime.

For BB-FMR, Ta(3 nm)/Al(3 nm)/Co${}_{\text{25}}$Fe${}_{\text{75}}$($t$)/ Al(3 nm)/Ta(3 nm) heterostructures with different thickness $t$ of the CoFe layer were sputter deposited on a thermally oxidized Si (100) substrate at an Ar pressure of $5\times 10^{-6}$ bar at room temperature. No subsequent annealing process was performed. The CoFe layer thickness was varied between 1.4 nm $<t<$ 26 nm as determined by X-ray reflectometry.

The OOP BB-FMR measurements were performed at room temperature with a vector network analyzer (VNA). This geometry was chosen to determine the intrinsic magnetic damping without further damping contributions due to magnon-magnon scattering Hillebrands and Ounadjela (2003). The samples were placed directly on a coplanar waveguide (CPW), with a $80\,\mathrm{\mu m}$ wide center conductor. For the measurements, the VNA frequency $f$ was kept constant and the microwave transmission parameter $S_{\mathrm{21}}$ was recorded as a function of applied magnetic field $H_{\mathrm{0}}$ for a range of frequencies at a VNA output power of 0 dBm. A representative set of data as measured of the real and imaginary part of $S_{\mathrm{21}}$ at 16 GHz for samples with $t=1.8\,\mathrm{nm}$ and $t=26\,\mathrm{nm}$ is shown in Fig. 1 (a) and (b).

The magnetic response of the thin film FM magnetized out-of-plane is given by the susceptibility $\chi$ which is obtained by solving the Landau-Lifshitz-Gilbert (LLG) equation Nembach et al. (2011); Schoen et al. (2015):

[TABLE]

Here, $M_{\mathrm{s}}$ is the saturation magnetization, $H_{\mathrm{res}}$ is the resonance field, $H_{\mathrm{eff}}=2\pi f/(\mu_{\mathrm{0}}\gamma)$ with $\gamma$ being the gyromagnetic ratio and $\Delta H=2(2\pi f\alpha)/(\gamma\mu_{\mathrm{0}})$ is the full width at half maximum (FWHM) linewidth of the resonance. The data in Fig 1 (a) and (b) is fitted to Berger et al. (2018)

[TABLE]

where $S_{\mathrm{21}}^{0}$ is the background transmission through the CPW without magnetic resonance peak. It is determined from the fits as a complex linear background to the data $S_{\mathrm{21}}^{0}(H_{\mathrm{0}})=S_{\mathrm{21}}^{a}+H_{\mathrm{0}}S_{\mathrm{21}}^{b}$. The factor $A$ is a complex-valued scaling parameter.

In the OOP geometry, the resonance condition for thin films is given by Kittel (1948)

[TABLE]

where $M_{\mathrm{eff}}=M_{\mathrm{s}}-H_{\mathrm{k}}$ is the effective magnetization, with the uniaxial out-of-plane anisotropy field $H_{\mathrm{k}}$. In Fig. 1 (c), we plot the determined $H_{\mathrm{res}}$ vs. the frequency $f$. From the fit to Eq. (3) (red solid lines in Fig. 1 (c)), we obtain $M_{\mathrm{eff}}$ and $\gamma$ of the specific sample.

The FWHM linewidth vs. frequency data shown in Fig. 1 (d) is fitted to

[TABLE]

Here, $H_{\mathrm{inh}}$ is the inhomogeneous linewidth broadening and $\alpha_{\mathrm{G}}$ is the phenomenological Gilbert damping parameter Woltersdorf et al. (2013); McMichael, Twisselmann, and Kunz (2003). $H_{\mathrm{inh}}$ indicates the presence of long-range magnetic inhomogeneities, which become more relevant for thinner films, but do not contribute to our $\alpha_{\mathrm{G}}$.

Several contributions to the measured total damping ($\alpha_{\mathrm{G}}$) were extracted from our data. In addition to the intrinsic damping of the magnetic material itself ($\alpha_{\mathrm{0}}$), spin pumping ($\alpha_{\mathrm{sp}}$) contributes significantly Tserkovnyak, Brataas, and Bauer (2002); Haertinger et al. (2015); Brataas et al. (2017) to the total damping in our thinner heterostructures due to the adjacent HM (Ta) layers. Furthermore, we consider additional damping contributions from eddy currents ($\alpha_{\mathrm{eddy}}$) and radiative damping ($\alpha_{\mathrm{rad}}$)Schoen et al. (2015); Berger et al. (2018). Due to these contributions, the total damping ($\alpha_{\mathrm{G}}=\alpha_{\mathrm{0}}+\alpha_{\mathrm{sp}}+\alpha_{\mathrm{eddy}}+\alpha_{\mathrm{rad}}$) depends on the FM thickness. We calculated damping due to eddy currents and measured radiative damping contributions to the total damping. The eddy current contribution is given bySchoen et al. (2015) $\alpha_{\mathrm{eddy}}=\gamma\mu_{\mathrm{0}}^{2}M_{\mathrm{s}}t^{2}/16\rho$. Here, $\mu_{\mathrm{0}}M_{\mathrm{s}}=2.35$ T (see SI) and $\rho=340\,\mathrm{n\Omega\,m}$ is the estimated weighted resistivity value of the CoFe film derived from the resistivities of iron and cobalt thin films with thicknesses of around 20 nmRaeburn and Aldridge (1978); De Vries (1988). With these values, we find an almost negligible eddy current contribution to the total damping. A quantitative determination analogous to Ref. 21 of the radiative damping is done by analyzing the magnitude of the measured inductance $L$ of all samples. The quantification of this contribution is important for BB-FMR, because it represents a damping by inductive power dissipation into the CPW and, hence, is not a property of the sample itself but depends on the setup. In possible applications like, e.g., magnonic waveguides or spin-Hall nano-oscillators, this contribution vanishes and the damping lowers by $\alpha_{\mathrm{rad}}$. With Eq. (2) above and Eq. (9) from Ref. 21, one obtains:

[TABLE]

Here, $Z_{\mathrm{0}}=50\,\mathrm{\Omega}$ is the CPW impedance. It has been shown, that $\tilde{L}=\tilde{L}_{\mathrm{0}}+\tilde{L}_{\mathrm{1}}(\omega),$ where $\tilde{L}_{\mathrm{0}}\in\mathbb{R}$ and $\tilde{L}_{\mathrm{1}}\in\mathbb{C}$, due to the effect of inverse spin-orbit torques Berger et al. (2018). We extract $L$ from the FMR measurements, and the dipolar inductance $\tilde{L}_{\mathrm{0}}$ from a fit of $\tilde{L}$ vs. $f$ for each sample. The radiative damping contribution is then given asSchoen et al. (2015)

[TABLE]

This analysis allows us to determine $\alpha_{\mathrm{rad}}$ independently of geometrical parameters of the samples or CPWs and is used to quantitatively extract the dipolar inductance without any calibration of the microwave circuit. For the thickest sample we obtain $\alpha_{\mathrm{rad}}=(4.69\pm 0.05)\times 10^{-4}$, which is comparable to previously obtained values Schoen et al. (2015, 2016). The damping including the spin pumping contribution $\alpha_{\mathrm{sp}}$ is given by

[TABLE]

where $g_{\mathrm{eff}}^{\uparrow\downarrow}$ is the effective spin mixing conductanceSchoen et al. (2016). We substract $\alpha_{\mathrm{rad}}$ and $\alpha_{\mathrm{eddy}}$ from the measured total damping $\alpha_{\mathrm{G}}$ (see Fig. 2 (a) and (b)) and plot the remaining damping $\alpha_{\mathrm{0}}+\alpha_{\mathrm{sp}}$ as a function of 1/$t$ in Fig. 2 (b) together with the total damping $\alpha_{\mathrm{G}}$. From a linear fit (Eq. (7)) to $\alpha_{\mathrm{0}}+\alpha_{\mathrm{sp}}$, we obtain $g_{\mathrm{eff}}$ and $\alpha_{\mathrm{0}}$. Herefore, we use $M_{\mathrm{s}}$ as above and $\gamma/2\pi=28.65\,\mathrm{GHz/T}$. The fitted $g_{\mathrm{eff}}=(5.5\pm 0.3)\times 10^{18}\,\mathrm{m^{-2}}$ is in agreement with literature valuesSchoen et al. (2015). The $y$-intercept indicating the extrapolated intrinsic damping yields $\alpha_{\mathrm{0}}=(0.91\pm 1.69)\times 10^{-4}$ hence, the intrinsic damping is below the sensitivity of our approach. For the thickest sample $t=26\,\mathrm{nm}$ shown in Fig. 2 (a), we obtain $\alpha_{\mathrm{0}}=(3.18\pm 0.48)\times 10^{-4}$ (see SI for details). Within the errors, this value lies close to the extrapolated value and is the lowest intrinsic damping for a thin film ferromagnetic metal reported so far. We attribute the slightly reduced intrinsic $\alpha_{\mathrm{0}}$ compared to Ref. 30 to the use of a different seed layer, which has a substantial impact on the damping of CoFeEdwards, Nembach, and Shaw (2019).

The low damping properties of the CoFe heterostructures, in combination with the high saturation magnetization are expected to result in long PLs of dipolar SWs. We use microfocused BLSSebastian et al. (2015) to study the SW propagation in patterned CoFe samples, which are schematically depicted in Fig. 3 (a) and (b).

For our experiments, we fabricated patterned stripes of a Pt(3 nm)/Cu(3 nm)/Co${}_{\text{25}}$Fe${}_{\text{75}}$($t$)/Cu(3 nm)/Ta(3 nm) heterostructure using laser (sample A) and electron beam (sample B) lithography, sputter deposition and a subsequent lift-off process. This stack sequence was used as lower in-plane damping was observed compared to the samples containing Al. Below, we present data on only two samples with a thickness of $t=5\,\mathrm{nm}$ and a width $w=1.5\,\mathrm{\mu m}$ for sample A and $t=26\,\mathrm{nm}$ and $w=5\,\mathrm{\mu m}$ for sample B, respectively. An aluminum antenna was placed on top of the CoFe strip to drive spin dynamics via a microwave drive applied to the antenna. For sample A we used a simple aluminum strip optimized for excitation of the uniform (FMR) mode, whereas for the sample B we used a CPW antenna optimized for an efficient excitation of SWs with wave number $k\leq 2\,\mathrm{\mu m^{-1}}$.

In order to compare the uniform FMR-mode linewidths of extended and patterned films, we used sample A in backward volume geometry and placed the laser spot close to the antenna, where the FMR mode is dominantly excited. We recorded BLS spectra for several magnetic fields for each frequency. The BLS intensity is integrated and the signal sum is then plotted vs. the external magnetic field in Fig. 3 (c). The FWHM-linewidth $\Delta H$ is determined by fitting a Lorentzian (red line). We then compared the fitted linewidth to the measured in-plane BB-FMR linewidth of a blanket film, deposited simultaneously with the structured BLS sample. In the in-plane configuration the total damping increases due to magnon-magnon scattering Hillebrands and Ounadjela (2003); Arias and Mills (1999) and possible anisotropic damping Chen et al. (2018); Seib, Steiauf, and Fähnle (2009); Steiauf and Fähnle (2005); Safonov (2002). As shown in Fig. 3 (d), the linewidths $\mu_{\mathrm{0}}\Delta H$ determined from BB-FMR (black sybmbols) and BLS (blue symbols) are very similar, indicating that the damping properties are not affected by the patterning, as expected in a lift-off process with micrometer feature sizes.

In the next set of experiments, we investigate the SWPL of sample B (see Fig. 3 (b)). Here, the magnitude of the external magnetic field was fixed at $\mu_{\mathrm{0}}H_{\mathrm{0}}=43$ mT, while the field was applied perpendicular to the CoFe strip (Damon-Eshbach geometry). The BLS intensity was recorded as a function of position ($x$,$y$) over the CoFe strip. The BLS intensity decay in $x$ direction (i.e. the BLS intensity averaged over the width of the strip in order to suppress mode-beating effects Pirro et al. (2011); Demidov et al. (2009); Clausen et al. (2011)) is shown in Fig. 3 (e) for $f=9.5$ GHz. The SWPL $\lambda_{\mathrm{prop}}$ is extracted by a fit to $I=I_{\mathrm{0}}\exp(-2x/\lambda_{\mathrm{prop}})$ Demidov et al. (2016) and plotted vs. $f$ in Fig. 3 (f). From our experiments, we extract a maximum SWPL of $(21\pm 1)\,\mathrm{\mu m}$, well exceeding previously obtained results for FeNi alloys Yamanoi et al. (2013) and CoFe Körner et al. (2017) and very comparable to values found for YIG thin films Collet et al. (2017); Jungfleisch et al. (2015). The red curve is the theoretical prediction, based on the analytical Kalinikos-Slavin model detailed below and using the magnetic parameters determined by in-plane BB-FMR ($\mu_{\mathrm{0}}M_{\mathrm{s}}=2.35$ T, $\mu_{\mathrm{0}}M_{\mathrm{eff}}=2.29$ T, $\alpha_{\mathrm{G}}-\alpha_{\mathrm{rad}}=3.92\times 10^{-3}$, $g=2.051$) for a co-deposited reference sample (see SI).

Starting with a simplified version of Kalinikos and Slavin’s SW dispersion for the modes with $\mathbf{k_{\mathrm{x}}}\perp\mathbf{M}$Kalinikos and Slavin (1986); Liensberger et al. (2019)

[TABLE]

we calculated the group velocity $v_{\mathrm{g}}=2\pi\partial f_{\mathrm{res}}/\partial k$. Here, $k=\sqrt{k_{\mathrm{x}}^{2}+k_{\mathrm{y}}^{2}}$ is the in-plane wave vector of the travelling SW and $\mu_{\mathrm{0}}H_{\mathrm{k}}=\mu_{\mathrm{0}}M_{\mathrm{eff}}-\mu_{\mathrm{0}}M_{\mathrm{s}}=-60\,\mathrm{mT}$ is the effective interface anisotropy field. The calculation of the transversal wave vector component $k_{\mathrm{y}}=0.31\,\mathrm{\mu m^{-1}}$ due to geometrical confinement was shown to be non-trivial and is used as a fitting parameter, as in Ref. 45. The resonance linewidth is given byStancil and Prabhakar (2009) $\Delta\omega=\alpha\mu_{\mathrm{0}}\gamma(M_{\mathrm{eff}}/2+H_{\mathrm{0}}+H_{\mathrm{d}})$ and the lifetime of the SW is $\tau=1/\Delta\omega$. Here, $\alpha=\alpha_{\mathrm{G}}-\alpha_{\mathrm{rad}}$. The SWPL is $\lambda_{\mathrm{prop}}=v_{\mathrm{g}}\tau$. The demagnetization field in $y$-direction was set to $\mu_{\mathrm{0}}H_{\mathrm{d}}=-18$ mT, as required for matching Eq. (8) to the SW dispersion obtained by phase-resolved $\mu$BLS Sebastian et al. (2015) (see Fig. 3 (g)). This value for $H_{\mathrm{d}}$ is in good agreement with the demagnetization ($\mu_{\mathrm{0}}H_{\mathrm{d}}\approx-12$ mT) obtained for an ellipsoid with the axes corresponding to the CoFe-stripe dimensions Osborn (1945). We find excellent agreement between this model and our experimental data in Fig. 3 (f).

In summary, our sputter-deposited Co${}_{\text{25}}$Fe${}_{\text{75}}$ layers exhibit a record low intrinsic damping for metallic thin film ferromagnets of $\alpha_{\mathrm{0}}\lesssim 3.18\times 10^{-4}$ in OOP geometry. The damping properties of extended films are maintained for micropatterned films, and spin-wave propagation lengths are in very good agreement with the properties extracted from BB-FMR. The low magnetic damping, together with the high saturation magnetization, lead to spin-wave decay lengths of more than 20 $\mathrm{\mu m}$ at room temperature, which are the highest reported so far in itinerant magnetic systems. This property makes Co${}_{\text{25}}$Fe${}_{\text{75}}$ a promising material for all-metal spintronic and magnonic devices, compatible with semiconductor technology.