Competition of phonon and magnon effects in the temperature dependence of spinwave stiffness

Magnons are the elementary magnetic excitations in ordered solids. Understanding such collective excitations is important for a number of technologically relevant fields, such as, magnonics [1] or spin caloritronics [2]. The central interactions in spin caloritronics are the couplings of phonons with electrons and spin degrees of freedom. Furthermore, understanding the effects of temperature on the phonon and spin degrees of freedom adds a further complexity. In the present work we have developed a multiscale model of ferromagnetic materials and demonstrate the effect of temperature dependent phonon displacements on the magnons spectra. Our results show that the for Fe and Ni the effect of phonon displacements acts to reduce the Curie temperature, whereas for Permalloy the opposite is true due to an increased long-ranged exchange interaction. This increased exchange interaction results in an increasing spin-wave stiffness with increasing temperature, overcoming the usual decrease due to magnon softening. To determine the effects of increasing the phonon temperature we have developed a multiscale model whereby we begin by calculating the thermal displacement of phonons, $\sqrt{\langle u^2(T) \rangle}$, calculated using the phonopy software package [3] using electronic ground state and phonon properties determined using the ABINIT software package [4]. Then the exchange constants are determined using the SPRKKR package [5]. Finally, we use linear spin wave theory to determine the effect of the phonon temperature on the exchange alone, demonstrating an increasing frequency of the acoustic magnon branch. We take into account the thermal effects of the magnetic system through the use of the atomistic spin dynamics approach. Magnon softening due to thermal effects demonstrates a more modest increase in the exchange stiffness (over the purely phononic effect), however, an overall increase is still observed. \newline \newline [1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Physics, {\bf 11}, 453–461 (2015). \newline [2] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature Materials {\bf 11}, 391 (2012). \newline [3] Atsushi Togo and Isao Tanaka, Scr. Mater., {\bf 108}, 1-5 (2015) \newline [4] X. Gonze \textit{et al.} Computer Physics Communications {\bf 180}, 2582-2615 (2009). \newline [5] T. Huhne \textit{at al.} Physical Review B, {\bf 58}, 10236 (1998).