Main

In recent decades, several theoretical approaches have attempted to quantitatively predict magnetic damping in metallic systems. One of the early promising theories was that of Kambersky, who introduced the so-called breathing Fermi-surface model9,10,11. More recently, Gilmore and Stiles2 as well as Thonig et al. 12 demonstrated a generalized torque correlation model that includes both intraband (conductivity-like) and interband (resistivity-like) transitions. The use of scattering theory to describe damping was later applied by Brataas et al. 13 and Liu et al. 14 to describe damping in transition metals. A numerical realization of a linear response damping model was implemented by Mankovsky3 for Ni–Co, Ni–Fe, Fe–V and Co–Fe alloys. For the Co–Fe alloy, these calculations predict a minimum intrinsic damping of αint ≈ 0.0005 at a Co-concentration of 10% to 20%, but was not experimentally observed15.

Underlying this theoretical work is the goal of achieving new systems with ultra-low damping that are required in many magnonic and spin-orbitronics applications7,8. Ferrimagnetic insulators such as yttrium–iron–garnet (YIG) have long been the workhorse for these investigations, because YIG films as thin as 25 nm have experimental damping parameters as low as 0.9 × 10−4 (ref. 16). Such ultra-low damping can be achieved in insulating ferrimagnets in part due to the absence of conduction electrons—and, therefore, the suppression of magnon–electron scattering. However, insulators cannot be used in most spintronic and spin-orbitronic applications, where a charge current through the magnetic material is required, nor is the requirement of growth on gadolinium gallium garnet templates compatible with spintronics and complementary metal-oxide semiconductor (CMOS) fabrication processes. One proposed alternative class of materials are Heusler alloys, some of which are theoretically predicted to have damping parameters as low as 10−4 (ref. 17). Although such values have yet to be realized, damping parameters as low as 0.001 have been reported for Co2FeAl18 and NiMnSb19. However, Heusler alloys have non-trivial fabrication constraints, such as high-temperature annealing, which are also incompatible with spintronic and CMOS device fabrication constraints.

In contrast, metallic ferromagnets such as 3d transition metals are ideal candidate materials for these applications, because high-quality materials can be produced at room temperature (RT) without the requirement of annealing. However, ultra-low damping is thought to be unachievable in metallic systems, because damping in conductors is dominated by magnon–electron scattering in the conduction band, resulting in a damping parameter well over an order of magnitude higher than those found in high-quality YIG.

Inspired by Mankovsky’s theoretical prediction of ultra-low damping in the CoxFe1−x alloy system3, we systematically studied the compositional dependence of the damping parameter in CoxFe1−x alloys, including careful evaluation of spin-pumping and radiative damping contributions. Polycrystalline CoxFe1−x alloy films, 10 nm thick, were sputter-deposited at RT with Cu(3 nm)/Ta(3 nm) seed and capping layers. X-ray diffraction (XRD) reveals that the Co–Fe alloys exhibit a body-centred-cubic (bcc) phase over a Co-concentration of 0% to 60%, a face-centred-cubic (fcc) phase above 80% Co, and a mixed phase between 60% and 80% Co, in good agreement with the bulk phase diagram of this system. The damping parameter is determined from broadband ferromagnetic resonance (FMR) spectroscopy, which measures the susceptibility over frequencies spanning from 5 GHz to 40 GHz. An example of S21(H) vector-network-analyser transmission data is shown in Fig. 1a, b, together with fits to the complex susceptibility for the real and imaginary parts, respectively. The total damping parameter αtot is determined from the frequency dependence of the linewidth obtained from these susceptibility fits, according to equation (1),

where μ0 is the vacuum permeability, μB is the Bohr magneton, h is Planck’s constant, g is the Landé g-factor, and ΔH0 is the inhomogeneous linewidth.

Figure 1: Ferromagnetic resonance spectra, measured by means of FMR and the resulting linewidth as a function of frequency.
figure 1

a,b, Real and imaginary parts, respectively, of the S21(H) transmission parameter (open black circles) with the complex susceptibility fit (red lines). c, Linewidths (symbols) are plotted versus the frequency for Co, Fe, Co20Fe80 and Co25Fe75. The uncertainties in the linewidths were obtained by means of the standard method for the determination of confidence limits on estimated parameters for nonlinear models under the assumption of Gaussian white noise. The lines are error-weighted fits to equation (1), which are used to determine both the total damping αtot and the inhomogeneous linewidth broadening for each alloy.

Full size image

The measured total damping αtotversus alloy composition for 10 nm films is plotted in Fig. 2. αtot shows a clear minimum of (2.1 ± 0.1) × 10−3 at a Co-concentration of 25%. However, as a result of the measurement geometry used and the structure of the sample, there are several extrinsic contributions to αtot that are independent of αint.

Figure 2: Total measured damping with radiative and interfacial contributions.
figure 2

Total damping αtot (red circles with lines), spin-pumping αsp (grey line) and radiative αrad (green line) contributions to the damping, the calculated intrinsic damping αint (black squares with lines), as well as the extrapolated bulk value of the damping (blue open circles) are plotted as a function of Co-concentration. The errors of αtot are determined from 95% confidence intervals of the error-weighted linewidth fits. The crystal structure of the alloys, obtained from XRD, is signified by the colour regions in the plot.

Full size image

The first contribution—the result of the inductive coupling of the precessing magnetization and the co-planar waveguide (CPW)—is radiative damping αrad (ref. 20). The FMR system is designed and optimized to couple microwaves into the ferromagnet, and therefore, by virtue of reciprocity, the system is efficient at coupling microwaves out of the ferromagnet. For very thin films or films with low saturation magnetization, αrad is typically not a significant contribution to the total damping, and can be ignored. However, in the present case, αrad must be accounted for in the analysis owing to the combination of a very high saturation magnetization and the exceptionally small value of αint. As described in the Supplementary Section, we calculate and experimentally validate the contribution of αrad to the total damping, which is plotted in Fig. 2.

The second non-negligible contribution to the total damping is the damping enhancement due to spin pumping into the adjacent Cu/Ta layers. The spin-pumping contribution αsp can be determined from the thickness dependence of (αtot − αrad) because it behaves as an interfacial damping term21. Indeed, we measured the thickness dependence of (αtot − αrad) for many alloy samples to quantify and account for αsp (see Supplementary Information), which is shown in Fig. 2.

Contributions from eddy-current damping20 are estimated to be smaller than 5% and are neglected. Finally, two-magnon scattering is minimized in the perpendicular geometry used in this investigation, and its contribution is disregarded22.

The total measured damping becomes αtot = αint + αrad + αsp, allowing the intrinsic damping αint to be determined, which is presented in Fig. 2. For many values of αint, the contributions of αsp and αrad are of similar magnitude, showing the importance of accounting for these contributions. For the samples for which we have full thickness-dependent data, the intrinsic damping can also be determined from the y-intercepts found in the data of Supplementary Fig. 4. We also include these data in Fig. 2 (blue circles), which show that both approaches to remove the effects of spin pumping are consistent. For 25% Co, αint now exhibits a sharp minimum in damping of (5 ± 1.8) × 10−4, which is astonishing for a conductor. Indeed, values of αint < 0.001 have been measured only in ferrimagnetic insulators23.

These results raise the question why αint can be so low in the presence of conduction electrons. To gain a deeper understanding, we performed electronic structure calculations for CoxFe1−x within a full-relativistic, multiple-scattering approach (Korringa–Kohn–Rostoker method24, KKR) using the coherent potential approximation (CPA)25,26 over the entire range of compositions (see Methods). Several representative examples are given in Fig. 3a.

Figure 3: Calculated electron density of states (DOS) and its comparison to the intrinsic damping.
figure 3

a, Electronic structure of bulk CoxFe1−x. The DOS is shown for several Co-concentrations, as indicated. Note that all alloy compositions are aligned to a common Fermi energy, EF, at zero energy to facilitate comparison. b, The intrinsic damping (black squares, left axis) is compared to the theory in Mankovsky et al. 3, for a temperature of 0 K (blue line) and for a temperature of 300 K for pure Fe (blue star). The errors of αint were obtained by summing the errors of αtot and αsp. The calculated DOS at the Fermi energy n(EF) is plotted on the right axis (red line). The y-offset of n(EF) is chosen deliberately to demonstrate that the concentration-dependent features of the damping directly correlate to features of n(EF). We cannot exclude concentration-independent contributions to the damping, which are accounted for by the 0.4 eV−1 y-offset.

Full size image

The d-states (peak in the DOS below EF) for pure Fe are not fully occupied. Consistent with the rigid band model27, the d-states shift to lower energies when the concentration of Co increases, and become fully occupied at 25% Co, coinciding with the minimum in n(EF) shown in Fig. 3a, which originates from the hybridization between majority Fe eg and minority Co t2g states.

Ebert et al. 1 and Lounis et al. 28 suggested that αint is proportional to n(EF) in the breathing Fermi-surface model (that is, intraband transitions) in the cases of a minimally varying spin–orbit coupling (SOC) (as is the case for the CoxFe1−x system) and small electron–phonon coupling2,29. Alternatively, interband transitions become significant only if bands have a finite overlap due to band broadening, caused for example by coupling to the phonons. As a result, interband transitions are suppressed at low temperature and energy dissipation becomes dominated by intraband transitions. Our RT measurements of CoxFe1−x satisfy this ‘low-temperature’ condition because the electron–phonon coupling is <20 meV for pure bcc Fe, and <30 meV for pure hcp Co. Band broadening due to disorder is about 15 meV for the eg states that dominate at EF (50 meV for the t2g states) in Co25Fe75 and varies up to 55 meV for the eg states (150 meV for the t2g states) over the whole range of composition. These calculations show that the band-broadening effect at RT is too small to provide significant interband damping, consistent with the almost perfect proportionality between n(EF) for all alloy compositions in the bcc phase (0% to 60% Co). Such a proportionality requires an offset of 0.4 eV−1, which originates from the fact the n(EF) is a superposition of all states, some of which do not contribute significantly to the damping.

The calculations of αint by Mankovsky3 (included in Fig. 3b) show a minimum value of αint ≈ 0.0005 between 10% and 20% Co, which differs in location from the sharp minimum we find at 25% Co in both the experimental data and the calculated values of n(EF). Remarkably, with the exception of pure Fe, all calculated values of αint (at 0 K) agree with our results within a factor of approximately two. Furthermore, the agreement is greatly improved for pure Fe when a finite temperature of 300 K is included (see Fig. 3b). Although not perfect, the agreement between those calculations and our results is compelling, and provides the critical feedback needed for further refinement of theory. In this regard, it is also worth noting the recent work by Turek et al. 30, since it gives a minimum for the calculated damping parameter at a concentration of 25% Co in close agreement with the measured damping parameter in Fig. 3 as well as the minimum in the DOS curve.

We therefore demonstrate and conclude that αint is largely determined by n(EF) in the limit of intraband scattering. Second, our work shows that a theoretical understanding of damping requires an accurate account of all contributions to the damping parameter. Furthermore, the FeCo system studied here is unique from an electronic structure point of view, in the sense that the two elements have similar spin–orbit coupling and our calculations show that any enhanced broadening of electron states at the Fermi level due to alloying is small. The only remaining significant contributing factor to the damping is n(EF). Because all the other factors are relatively insensitive to the alloy composition and structure, the deep minimum in n(EF) for Co25Fe75 leads to the low intrinsic damping that we measure. If the theoretical explanation put forth here to explain low damping holds in general, it is natural to use data-mining algorithms to screen larger groups of materials to identify further low-damping systems. Examples of such studies to identify new materials for use, for example, in scintillators have been published31, and the generalization to applications in magnetization dynamics is straightforward.

Methods

Sample preparation.

The samples were deposited by DC magnetron sputtering at an Ar pressure of approximately 0.67 Pa (5 × 10−3 torr) in a chamber with a base pressure of less than 5 × 10−6 Pa (4 × 10−8 torr). The alloys were deposited by co-sputtering from two targets, with the deposition rates calibrated by X-ray reflectometry (XRR). The repeatability of the deposition rates was found to be better than 3% variation over the course of this study. For all deposited alloys, the combined deposition rate was kept at approximately 0.25 nm s−1, to ensure similar growth conditions. Furthermore, the Co20Fe80 and the Co25Fe75 samples were replicated by depositing from single stoichiometric targets, to prove the reproducibility of the results. Samples with a thickness of 10 nm were fabricated over the full alloy composition range, and additional thickness series (7 nm, 4 nm, 3 nm and 2 nm) were fabricated for the pure elements and select intermediate alloy concentrations (20% Co, 25% Co, 50% Co and 85% Co).

X-ray diffraction measurement.

The crystal structure of the alloys was determined by X-ray diffraction (XRD) using an in-plane geometry with parallel beam optics and a Cu Kα X-ray source. The in-plane geometry allows the signal from the Co–Fe alloys to be isolated from the high-intensity signal coming from the silicon substrate. These measurements yield both the in-plane lattice constants and the crystal structure, as shown in the Supplementary Methods, Section 1. The deposition rates were calibrated by XRR using the same system as configured for the out-of-plane geometry.

Superconducting quantum interference device (SQUID) measurement.

We measured the in-plane hysteresis curves at 300 K to determine the magnetic moment of the sample. Sample were first diced with a precision diamond saw such that an accurate value of the volume of the sample could be calculated. The saturation magnetization MS for all alloy samples is then determined by normalizing the measured moment to the volume of Co–Fe in the sample. These values are shown in the Supplementary Methods.

VNA-FMR measurement.

The FMR measurement used a room-temperature-bore superconducting magnet, capable of applying static magnetic fields as high as 3 T. An approximately 150 nm poly(methyl methacrylate) (PMMA) coat was first applied to the samples to prevent electrical shorting of the co-planar waveguide (CPW) and to protect the sample surface. Sample were placed face down on the CPW and microwave fields were applied in the plane of the sample, with frequencies that ranged from 10 GHz to 40 GHz. A vector network analyser (VNA) was connected to both sides of the CPW and the complex S21(H) transmission parameter was determined. The iterative susceptibility fit of S21(H) was done using the method described by Nembach and colleagues32. To minimize the fit residual, all fits were constrained to a field window that was three times the linewidth around the resonance field. We verified that this does not change the results, but reduces the influence of measurement noise on the error bars of the fitted values.

Calculation of the DOS.

Electronic ground state calculations have been performed by a full-relativistic multiple-scattering Green’s function method (Korringa–Kohn–Rostoker method24, KKR) that relies on the local spin-density approximation (LDA) to density functional theory. We used Perdew–Wang exchange correlation functionals33,34,35,36.

In our multiple-scattering theory, the electronic structure is described by scattering path operators τij (i, j lattice site indices)24, where, in the spin-angular-momentum representation, we consider angular momenta up to lmax = 3 and up to 60 × 60 × 60 points in reciprocal space. The substitutional alloys are treated within the coherent potential approximation (CPA)37,38. Co impurities in the Fe host lattice are created in the effective CPA medium by defect matrices. The CPA medium is described by scattering path operators. The site-dependent potentials are considered in the atomic sphere approximation (potentials are spherically symmetric within muffin-tin spheres and constant in the interstitials).

The DOS is obtained from the integrated spectral density24

with the small positive energy η. The integration in reciprocal space k runs over the first Brillouin zone ΩBZ.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.