Competition of lattice and spin excitations in the temperature dependence of spin-wave properties

The interplay of magnons and phonons can induce strong temperature variations in the magnetic exchange interactions, leading to changes in the magnetothermal response. This is a central mechanism in many magnetic phenomena, and in the new field of Spin Caloritronics, which focuses on the combination of heat and spin currents. Boson model systems have previously been developed to describe the magnon-phonon coupling but, until recently, studies rely on empirical parameters. In this paper, we propose a first-principles approach to describe the dependence of the magnetic exchange integrals on phonon renormalization, leading to changes in the magnon dispersion as a function of temperature. The temperature enters into the spin dynamics (by introducing fluctuations) as well as in the magnetic exchange itself. Depending on the strength of the coupling, these two temperatures may or may not be equilibrated, yielding different regimes. We test our approach in typical and well-known ferromagnetic materials: Ni, Fe, and Permalloy. We compare our results to recent experiments on the spin-wave stiffness, and discuss departures from Bloch's law and parabolic dispersion.

Figure 1

A schematic showing the methodology used in this paper. The green squares represent the use of a code or method, the circles represent processing of data from a code/method, and the arrows show the transfer of data with the text next to the arrow representing what data is being transferred. All the symbols are explained in the Methods section.

Figure 2

Py phonons energies calculated within DFPT in the harmonic approximation from VCA compared to experimental results from Ref. [40].

Figure 3

Temperature-dependent Heisenberg exchange integrals (times reduced radius cubed) vs reduced radius and for different values of the phonon temperature. (a) Fe (top) and Ni (bottom) (b) Py. Three interactions are present: Fe-Fe (top), Ni-Ni (middle), and Ni-Fe (bottom).

Figure 4

Temperature dependence of the exchange interactions for the first-four neighbor shells shown explicitly for the Fe-Fe, Ni-Ni, and Ni-Fe interactions in Permalloy.

Figure 5

Left: SW dispersion relation in Permalloy for a range of effective phonon temperatures calculated (a) within LSWT (no magnetic disorder) and (c) within R-ASD with both magnon and phonon contributions. Right: temperature dependence of the SW stiffness within LSWT (b) and using ASD (d) with renormalized exchange integrals (green circles) and with ${\mathcal{J}}_{ij}^{\rho \eta }$ values for $T_p=0\mathrm{K}$ (blue triangles). Dashed lines with open symbols in (d) are stiffness values extracted using a fit to Bloch's law (which underestimate, as found in Ref. [28]), the red crosses with error bars in (d) represent the fit of $D\left(T\right)$ in Ref. [28]. Phonon thermal effects are invisible in the experimental SW stiffness, which probes only the very long wavelength limit.

Figure 6

Temperature-dependent magnetization curves of the two sublattices in Py: Fe (green circle) and Ni (blue triangles), calculated using ASD (open symbols) and temperature-dependent (closed symbols) exchange constants (R-ASD). Full lines are Taylor expansion fits [47] to extract the Curie temperatures (vertical lines). The grey area shows the range of Curie temperatures found experimentally [43, 48], while the red vertical lines represent the Curie temperature from ASD (dashed) and R-ASD (full).

Figure 7

Temperature-dependent exchange stiffness for three different nonequilibrium cases: (1) fixed magnetic (spin thermostat) temperature of $T_{\mathrm{m}}=100$ K and a varying phonon temperature, $T_{\mathrm{p}}$ (blue square points), (2) fixed phonon temperature $T_{\mathrm{p}}=100$ K and a varying magnetic (red diamond points), and (3) equal magnetic and phononic temperature (green circle points). The error bars are estimated from the error in the nonlinear least-squares fitting procedure.