Nonlocal Gilbert damping tensor within the torque-torque correlation model

An essential property of magnetic devices is the relaxation rate in magnetic switching, which depends strongly on the damping in the magnetization dynamics. It was recently measured that damping depends on the magnetic texture and, consequently, is a nonlocal quantity. The damping enters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter $\alpha$, which does not, in a straightforward formulation, account for nonlocality. Efforts were spent recently to obtain Gilbert damping from first principles for magnons of wave vector $\mathit{q}$. However, to the best of our knowledge, there is no report about real-space nonlocal Gilbert damping ${\alpha }_{ij}$. Here, a torque-torque correlation model based on a tight-binding approach is applied to the bulk elemental itinerant magnets and it predicts significant off-site Gilbert damping contributions, which could be also negative. Supported by atomistic magnetization dynamics simulations, we reveal the importance of the nonlocal Gilbert damping in atomistic magnetization dynamics. This study gives a deeper understanding of the dynamics of the magnetic moments and dissipation processes in real magnetic materials. Ways of manipulating nonlocal damping are explored, either by temperature, materials doping, or strain.

Figure 1

Schematic illustration of nonlocal energy dissipation ${\alpha }_{ij}$ between site $i$ and $j$ (red balls) represented by a power cord in a system with spin wave (gray arrows) propagation $\mathit{q}$.

Figure 2

Electronic state resolved nonlocal Gilbert damping obtained from the integrand of Eq. (3) along selected paths in the Brillouin zone for bcc Fe, fcc Co, and fcc Ni. The scattering rate used is $\mathrm{\Gamma }=0.01\mathrm{eV}$. The abscissa (both top and bottom in each panels) shows the momentum path of the electron $\mathit{k}$, where the ordinate (left and right in each panel) shows the magnon propagation vector $\mathit{q}$. The two “triangles” in each panel should be viewed separately where the magnon momentum changes accordingly (along the same path) to the electron momentum.

Figure 3

Nonlocal Gilbert damping as a function of the spectral width $\mathrm{\Gamma }$ for different reciprocal wave vectors $q$ (indicated by different colors and in units $a_0^{-1}$). Note that $q$ provided here are in direct coordinates and only the direction differs between the different elemental, itinerant magnets. The nonlocal damping is shown for bcc Fe (top) along $\mathrm{\Gamma }\to H$, for fcc Co (middle) along $\mathrm{\Gamma }\to X$, and for fcc Ni (bottom) along $\mathrm{\Gamma }\to X$. It is obtained from “Lorentzian” [Eq. (2), circles] and Green's function [Eq. (3), triangles] method. The directional dependence of $\alpha$ for $\mathrm{\Gamma }=0.01\mathrm{eV}$ is shown in the inset.

Figure 4

Real-space Gilbert damping ${\alpha }_{ij}$ as a function of the distance $r_{ij}$ between two sites $i$ and $j$ for bcc Fe, fcc Co, and fcc Ni. Both the “corrected” Kamberský (red circles) and the Kamberský (blue squares) approach is considered. The distance is normalized to the lattice constant $a_0$. The on-site damping ${\alpha }_{ii}$ is shown in the figure label. The grey dotted line indicates the zero line. The spectral width is $\mathrm{\Gamma }=0.005\mathrm{eV}$.

Figure 5

First (circles) and second-nearest-neighbor (triangles) Gilbert damping (left) as well as on-site (circles) and total (triangles) Gilbert damping (right) as a function of the spectral width $\mathrm{\Gamma }$ for the itinerant magnets Fe, Co, and Ni. In particular for Co, the results obtained from tight binding are compared with first-principles density functional theory results (gray open circles). Solid lines (right panel) show the Gilbert damping obtained for the magnon wave vectors $q=0$ (blue line) and $q=0.1a_0^{-1}$ (red line). Dotted lines are added to guide the eye. Note that since cubic symmetry is broken (see text), there are two sets of nearest-neighbor parameters and two sets of next nearest-neighbor parameters (left panel) for any choice of $\mathrm{\Gamma }$.

Figure 6

Comparing nonlocal Gilbert damping obtained by Eq. (5) (red symbols) and Eq. (4) (blue symbols) in fcc Co for $\mathrm{\Gamma }=0.005\mathrm{eV}$. The dotted line indicates zero value.

Figure 7

Nonlocal Gilbert damping as a function of the normalized distance ${\mathit{r}}_{ij}/a_0$ for a tetragonal distorted bcc Fe crystal structure. Here, $c/a=1.025$ (red circles) and $c/a=1.05$ (blue circles) is considered. $\mathrm{\Gamma }$ is put to $0.01\mathrm{eV}$. The zero value is indicated by dotted lines.

Figure 8

Evolution of the average magnetic moment $M$ during remagnetization in bcc Fe (left) and fcc Co (right) for different damping strength according to the spectral width $\mathrm{\Gamma }$ (different colors) and both, full nonlocal ${\alpha }_{ij}$ (solid line) and total, purely local ${\alpha }^{\mathrm{tot}}$ (dashed line) Gilbert damping.

Figure 9

Comparison of nonlocal damping obtained from the tight-binding method (TB) (red filled symbols), tight-binding with Edwards correction (TBe) (blue filled symbols), and the linear muffin-tin orbital method (DFT) (open symbols) for fcc Co. Two different spectral broadenings are chosen.

Figure 10

Gilbert damping $\alpha$ as a function of the spin-orbit coupling for the $d$ states in fcc Co. Lower panel shows the Gilbert damping in reciprocal space for different $\mathit{q}=\left|\mathit{q}\right|$ values (the color scheme used, represents different $\mathit{q}$-values, as shown in the inset of the figure) along the $\mathrm{\Gamma }\to X$ path. The upper panel exhibits the on-site ${\alpha }_{\mathrm{os}}$ (red dotes and lines) and nearest-neighbor ${\alpha }_{\mathrm{NN}}$ (gray dots and lines) damping. The solid line is the exponential fit of the data point. The inset shows the fitted exponents $\gamma$ with respect to wave vector $q$. The color of the dots is adjusted to the particular branch in the main figure. The spectral width is $\mathrm{\Gamma }=0.005\mathrm{eV}$.

Figure 11

Comparison of reciprocal nonlocal damping with (squares) or without (circles) corrections proposed by Costa et al. [51] and Edwards [52] for Co and different spectral broadening $\mathrm{\Gamma }$. Different colors represent different magnon propagation vectors $q$.

Figure 12

Total Gilbert damping ${\alpha }^{\mathrm{tot}}$ for fcc Co as a function of summation cutoff radius for two spectral widths $\mathrm{\Gamma }$, one in the intraband ($\mathrm{\Gamma }=0.005\mathrm{eV}$, red dots and lines) and one in the interband ($\mathrm{\Gamma }=0.1\mathrm{eV}$, blue dotes and lines) region. The dotted and solid lines indicate the reciprocal value $\alpha (q=0)$ with and without SOC corrections, respectively.