Different dimensionality trends in the Landau damping of magnons in iron, cobalt, and nickel: Time-dependent density functional study

Paweł Buczek, Arthur Ernst, and Leonid M. Sandratskii

Phys. Rev. B 84, 174418 – Published 18 November 2011

Abstract

We study the Landau damping of ferromagnetic magnons in Fe, Co, and Ni as the dimensionality of the system is reduced from three to two. We resort to the ab initio linear response time dependent density functional theory in the adiabatic local spin density approximation. The numerical scheme is based on the Korringa-Kohn-Rostoker Green's function method. The key points of the theoretical approach and the implementation are discussed. We investigate the transition metals in three different forms: bulk phases, freestanding thin films, and thin films supported on a nonmagnetic substrate. We demonstrate that the dimensionality trends in Fe and Ni are opposite: in Fe the transition from bulk bcc crystal to Fe/Cu(100) film reduces the damping whereas in Ni/Cu(100) film the attenuation increases compared to bulk fcc Ni. In Co, the strength of the damping depends relatively weakly on the sample dimensionality. We explain the difference in the trends on the basis of the underlying electronic structure. The influence of the substrate on the spin-wave damping is analyzed by employing Landau maps representing wave-vector-resolved spectral density of the Stoner excitations.

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Received 29 September 2011

DOI:https://doi.org/10.1103/PhysRevB.84.174418

Authors & Affiliations

Paweł Buczek^*, Arthur Ernst, and Leonid M. Sandratskii

Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle (Saale), Germany

^*pbuczek@mpi-halle.mpg.de

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Vol. 84, Iss. 17 — 1 November 2011

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Figure 1
Examples of spin-flip spectra in bcc iron, atomic units, for different momenta along (100) direction. The largest eigenvalues of respectively enhanced and KS loss matrix are shown. Low-energy spin-wave peaks have simple Lorentzian shapes small linewidths and carry substantial spectral power. Above the critical energy of $82 m e V$ corresponding to $q_{c} = 0.35 \frac{2 π}{a}$ wave vector the spin-wave band enters abruptly a region of dense Stoner continuum. The spin-wave peaks become broad, acquire irregular non-Lorentzian shapes reflecting the energy dependence of the density of Stoner continuum, and carry much smaller spectral weight.Reuse & Permissions
Figure 2
Spin waves of bcc iron obtained using LRTDDFT implementation described in this paper. Solid circles () correspond to the maximum of spin-wave peak, while the error bars denote full width at half maximum (FWHM) of the peak. Solid line denotes spin-wave energies obtained from magnetic force theorem (MFT). Strong increase of Landau damping in all directions in the Brillouin zone (spin-wave disappearance effect) is seen for spin-wave energies above $82 m e V$ . High-energy spectrum along $Γ HN$ directions is dominated by Stoner excitations and cannot be represented by a simple Lorentzian. Representative examples of the spectral power functions are given in Fig. 1.Reuse & Permissions
Figure 3
The largest eigenvalues of loss matrix associated with unenhanced (KS) susceptibility for a selected momentum in bulk bcc Fe. There are several eigenvalues of comparable magnitude corresponding to transitions between different KS orbitals. On the contrary, the loss matrix associated with the enhanced susceptibility features only one eigenvalue, identified as a single spin-wave mode of the monatomic material.Reuse & Permissions
Figure 4
Spin waves of Fe(100) monolayer free-standing and supported on Cu(100) surface. Solid lines correspond to $ω_{0} (q)$ , while the width of the shaded region to FWHM. The spin dynamics of the freestanding and supported monolayers differ weakly. From Ref. 41.Reuse & Permissions
Figure 5
Spin waves of Fe(110) monolayer freestanding and supported on W(110) surface. Solid lines correspond to $ω_{0} (q)$ , while the width of the shaded region to FWHM. The surface state of W(110) leads to a qualitative change in the spin dynamics of the absorbed film. From Ref. 41.Reuse & Permissions
Figure 6
Spin waves of $2 ML$ Fe/W(110). Two magnon modes can be discerned in the whole two-dimensional Brillouin zone. Experimental points come from Ref. 105.Reuse & Permissions
Figure 7
Examples of spin-flip excitation spectrum in hcp Co, atomic units, for two momenta along $Γ A$ direction. Panels (a) and (b) present the two dominant eigenvalues of the loss matrix, $L χ (q, ω)$ ; the eigenvalues correspond to the acoustic and optical spin-wave branches. As expected from symmetry arguments, they become degenerate for $q_{z} = \frac{π}{c}$ . Panels (c) and (d) show the imaginary part of the Fourier-transformed susceptibility, $Im χ_{K K} (q, ω)$ , for momenta inside and outside the first Brillouin zone. The momenta differ by a reciprocal lattice vector. $Im χ_{K K} (q, ω)$ is probed in the neutron scattering experiments. By varying the momentum one can access the optical and acoustic spin wave branch.Reuse & Permissions
Figure 8
Spin waves of of hcp cobalt. (a) Position of the peak $ω_{0} (q)$ obtained from the enhanced susceptibility. The energies of spin waves are degenerated within the numerical error along $ALHA$ directions. The experimental energies come from Refs. 98 and 116. (b) Adiabatic magnon spectra obtained by means of magnetic force theorem. (c) Half-widths at half maximum; FWHM $\equiv 2 β (q)$ .Reuse & Permissions
Figure 9
Spin waves of fcc Co. Solid circles () correspond to $ω_{0} (q)$ , while the error bars denote full width at half maximum (FWHM) of the peak. Solid line denotes spin-wave energies obtained from MFT. Solid triangles () stand for the experimental estimation of the dispersion relation by Balashov (Ref. 120).Reuse & Permissions
Figure 10
Spin waves of Co(100) monolayer free and supported on Cu(100) surface. Solid circles () correspond to $ω_{0} (q)$ , while the error bars denote full width at half maximum (FWHM) of the peak. Solid line denotes spin-wave energies obtained from MFT.Reuse & Permissions
Figure 11
Spin waves of fcc nickel. Solid circles () correspond to $ω_{0} (q)$ , while the error bars denote full width at half maximum (FWHM) of the peak. Solid line denotes spin-wave energies obtained using MFT and STSM; the two methods differ significantly for this system. Most of the spectrum can be described using Lorentzian distribution function, except most energetic spin waves along $[ξ ξ 0]$ direction. An example of such a irregular spectrum is presented in Fig. 12.Reuse & Permissions
Figure 12
An example of spin-flip excitation spectrum in Ni, $- L {[χ (q, ω)]}_{λ}$ , atomic units. Only eigenvalue of the loss matrix dominates. One clearly sees the energy dependence of the Stoner continuum reflected in the behavior of enhanced susceptibility.Reuse & Permissions
Figure 13
Two examples of spin-flip excitation spectrum in Ni, atomic units, along (100) direction. The nonmonotonous dependence of the damping with the magnon momentum is clearly seen. Additionally, close to the $X$ point in the Brillouin zone a clear coexistence of the spin-wave peak and a Stoner continuum peak can be observed.Reuse & Permissions
Figure 14
(a) Double-peak feature in bulk fcc Ni. (b) Behavior of the corresponding Kohn-Sham susceptibility.Reuse & Permissions
Figure 15
Magnon energies in fcc Ni in the limit of small momenta obtained from dynamic susceptibility () and MFT (line). In the limit both methods yield identical results. This mathematical identity (Refs. 57, 85) provides a neat check for our numerics.Reuse & Permissions
Figure 16
Spin waves of Ni(100) monolayer free and supported on Cu(100) surface. Solid circles () correspond to $ω_{0} (q)$ , while the error bars denote FWHM. Solid line denotes spin-wave energies obtained from MFT. Spin waves with large momenta are not well-defined excitations in the case of the absorbed film, while for the freestanding film they exist in the whole Brillouin zone.Reuse & Permissions
Figure 17
Example of spectra in $1 ML$ Ni(100) and $1 ML$ Ni/Cu(100), atomic units. $q = (0.5, 0) 2 π / a$ corresponds to $\bar{X}$ point in the Brillouin zone. In the supported monolayer, for large wave vectors, the Stoner continuum becomes very broad (in contrast to the freestanding film) and the Landau damping completely washes out any sharp spin-wave features.Reuse & Permissions
Figure 18
Landau maps of $1 ML$ Ni(100) and $1 ML$ Ni/Cu(100): intensity of Stoner transitions with momentum $q = (0.5, 0) 2 π / a$ ( $\bar{X}$ point) and energy $ω_{0} = 400 m e V$ in the Ni layer resolved for different final $k$ vectors in the first Brillouin zone. The Stoner states cause the damping of magnons presented in Fig. 17.Reuse & Permissions
Figure 19
Evaluating the first (a) and the fourth term (b) in Eq. (36) around the valence band. Crosses ( $\times$ ) denote the positions of the fermionic poles $θ_{n}$ . Part of the integration is transformed into the sums over fermionic frequencies (small circular contours around selected $θ_{n}$ ). The integration weighted with the Fermi-Dirac distribution function[128] can be approximated close to the chemical potential $μ$ using Sommerfeld expansion. This way of integration allows one to avoid evaluation of KKRGF close to its singularities originating from the valence states; the valence band is marked with thick orange line. The sum of the integrals a, b, and the one given by Eq. (B2) gives joint contribution of the first and fourth term in Eq. (36).Reuse & Permissions
Figure 20
The structure of the integral in Eq. (B3).Reuse & Permissions
Figure 21
The analytic structure of transverse susceptibility and schematic presentations of different analytic continuation schemes. For particular $q$ the singularities of KS susceptibility form Stoner continuum (SC). The additional singularities introduced into the enhanced susceptibility by the Dyson equation, i.e., spin waves, can appear outside the continuum; such magnons cannot decay via Landau mechanism. On the contrary, when the SW pole appears in the Stoner continuum it acquires a finite lifetime manifested by an apparent shift of the pole into the lower complex semiplane. The analytic continuation in the case of the temperature susceptibility (a) involves the reconstruction of the real-time dynamics based on the values given in points located on the imaginary axis; it is in general unstable. (b) Nearly real axis calculations lead to much better stability and accuracy of the analytic continuation.Reuse & Permissions

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