Magnon softening in a ferromagnetic monolayer: a first-principles spin dynamics study Anders Bergman, 1, 2, ∗ Andrea Taroni, 1 Lars Bergqvist, 1 Johan Hellsvik, 1 Bj¨ orgvin Hj¨ orvarsson, 1 and Olle Eriksson 1 $^{1}$Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden $^{2}$Department of Materials Science and Engineering, Royal Institute of Technology, Brinellv¨ agen 23, 100 44 Stockholm, Sweden We study the Fe/W(110) monolayer system through a combination of first principles calculations and atomistic spin dynamics simulations. We focus on the dispersion of the spin waves parallel to the [001] direction. Our results compare favorably with the experimental data of Prokop et al. [ Phys. Rev. Lett. 102 , 177206], and correctly capture a drastic softening of the magnon spectrum, with respect to bulk bcc Fe. The suggested shortcoming of the itinerant electron model, in particular that given by density functional theory, is refuted. We also demonstrate that finite temperature effects are significant, and that atomistic spin dynamics simulations represent a powerful tool with which to include these. Recent progress in experimental techniques have al- lowed the first observation of the magnon dispersion spec- trum of a single ferromagnetic monolayer $^{1}$. The measure- ments, which were performed on an atomic monolayer (ML) of Fe on W(110) using spin-polarized electron en- ergy loss spectroscopy (SPEELS), revealed magnon en- ergies that are much smaller compared to the bulk and surface Fe(110) excitations. This strong magnon soften- ing is in contradiction with theoretical predictions based on an itinerant electron model at T = 0 K $^{2,3}$. This dis- crepancy raises the possibility that a ML of Fe on W(110) may not be a simple itinerant ferromagnet, as generally assumed, and indicates a shortcoming in the theoretical understanding of low-dimensional systems. In fact, from their experimental results the authors of Ref. 1 suggested this possibility. Furthermore, a recent theoretical study by Grechnev et al. has suggested that effects of elec- tron correlations might be more important for surfaces $^{4}$. Given the fundamental nature of spin wave excitations, and their role in physical processes such as fast magne- tization reversal 5 and current induced magnetic switch- ing $^{6–8}$, it has become both timely and important to ad- dress these questions. Low-dimensional magnetic structures lack inversion symmetry. Consequently, significant anisotropic contri- butions to the magnetic ordering can arise, in the form of Dzyaloshinskii-Moriya interactions (DMI) 9,10 and mag- netic anisotropy energies. Both of these stem from rel- ativistic spin orbit coupling, and may be energetically strong enough to compete with isotropic exchange inter- actions, leading to complex magnetic ground states in systems such as Mn on W(001) 11 and Fe on W(110) $^{12,13}$. Furthermore, the interaction between the Fe monolayer and the W substrate is known to be significant, and in the W(001) case this can even lead to antiferromagnetic ordering $^{14–17}$. An additional challenge when modeling spin waves is posed by temperature: the experimental data in Ref. 1 were taken at 120 K, which corresponds to a value for T/T$_{c}$ = 0 . 5, where T$_{c}$ is the Curie tempera- ture. In this regime, finite temperature effects, which are normally excluded from density functional theory, can be expected to play an important role. We demonstrate that atomistic spin dynamics (ASD) In this expression, γ is the the electron gyromagnetic ra- tio, and b$_{i}$ is a stochastic magnetic field with a Gaussian distribution, the magnitude of which is related to the where K is the strength of the anisotropy field along the direction of e$_{K}$ , and D$_{ij}$ is the Dzyaloshiskii-Moriya vec- tor $^{9,10}$. From this Hamiltonian, the effective interaction field experienced by each atomic moment m$_{i}$ is calculated as B$_{i}$ = − ∂ H ∂ $_{m$_{i}$}$. The temporal evolution of the atomic spins at finite temperature is modeled by Langevin dy- namics, through coupled stochastic differential equations of the Landau-Lifshitz form, where i and j are atomic indices, m$_{i}$ is a classical atomic moment and J$_{ij}$ is the strength of the exchange interac- tion. Relativistic effects are taken into account by adding one or both terms from the equation We performed atomistic spin dynamics simulations 18 using the UppASD package $^{19}$. In the simulations, the isotropic exchange interactions are treated by introduc- ing the classical Hamiltonian simulations based on first principles theory provide a powerful tool for studying magnons at finite tempera- ture. Using this approach, we are able to account for most of the experimentally observed magnon softening in one ML Fe on W(110) $^{1}$. We mainly attribute the soft- ening to hybridization with the underlying W substrate combined with finite temperature effects. We show that the itinerant electron model of surface and thin film mag- netism, as given by density functional theory, does not have to be abandoned, provided dynamical aspects and finite temperature effects are considered. H = − 1 2 ∑ i = j J$_{ij}$ m$_{i}$ · m$_{j}$ , (1) H$_{SO}$ = K ∑ i ( m$_{i}$ · e$_{K}$ ) 2 + ∑ i,j D$_{ij}$ ( m$_{i}$ × m$_{j}$ ) , (2) ∂ m$_{i}$ ∂t = − γ m$_{i}$ × [ B$_{i}$ + b$_{i}$ ( t )] − γ α m m$_{i}$ ×{ m$_{i}$ × [ B$_{i}$ + b$_{i}$ ( t )] } . arXiv:1001.1669v2 [cond-mat.mtrl-sci] 26 Apr 2010