Strongly coupled magnon–phonon dynamics in a single nanomagnet

Polaritons are widely investigated quasiparticles with fundamental and technological significance due to their unique properties. They have been studied most extensively in semiconductors when photons interact with various elementary excitations. However, other strongly coupled excitations demonstrate similar dynamics. Specifically, when magnon and phonon modes are coupled, a hybridized magnon–phonon quasiparticle can form. Here, we report on the direct observation of coupled magnon–phonon dynamics within a single thin nickel nanomagnet. We develop an analytic description to model the dynamics in two dimensions, enabling us to isolate the parameters influencing the frequency splitting. Furthermore, we demonstrate tuning of the magnon–phonon interaction into the strong coupling regime via the orientation of the applied magnetic field.


Supplementary Note 2. Analytic Derivations
First, the equations describing the individual phonon and magnon dynamics are introduced.
Then the coupling term which is responsible for the hybridized dynamics is considered. The basis for the derivation as well as the symbolic representation has been adopted from

2d Elastic Dynamics
The elastic energy density is given by: are the components of the stiffness tensor, = ( + ) 2 ⁄ are the strain components and is the displacement vector. In order to quantify the phononic eigenmodes the elastic wave equation is solved is the Cauchy stress tensor and is the density of the material. Due to the small z dimension of the element we can consider the nanomagnet to be two-dimensional in the x and y directions ( Supplementary Fig. 2). 2 We assume a solution of the form , = ,

and
, is the dimension of the nanoelement along the x or y direction. 3 Furthermore, since each phononic mode is degenerate for every value of and we set , = √2 which ensures 2 = 2 + 2 . The in-plane angle of k is φ k = tan −1 ( ⁄ ). Because the system is isotropic, the expansion of the element due to a heat pulse from the laser is the same in the x and y directions such that 0 = 0 . The system of equations can be solved by setting the Determinant equal to zero. This yields the eigenfrequencies for the phononic system given by

Magnetic Dynamics
The magnetic dynamics in the absence of damping are given by the well known Landau-Lifshitz Where is the gyromagnetic ratio, = S , is the Saturation Magnetization and eff is the effective field. We neglect the magnetic permeability, 0 since 0 = 1 in the cgs system. As is customary, a new Cartesian frame of reference is introduced where the 3-axis points along direction of the magnetization vector, the 2-axis is in the film plane, and the 1-axis is orthogonal to the 1 and 2 directions. (Supplementary Fig. 2b) The transformation is given by The effective field is given by Where m = (

Coupled Dynamics
When magnon-phonon coupling is present there is an added term to the magnetic free energy as well as the elastic energy density. This term is related to the orientation of the magnetization components with respect to the corresponding dynamic strains. The coupling term is given by The effective field of the magnetic system is now given by = −∇ m where = + . The magnetization components are assumed to follow the spatial profile of the phononic vibrations, so we assume a plane-wave ansatz of the form = 0 ( • − ) . The system of equations is now Where is defined in the same way as . Transforming to the (1,2,3) coordinate system and keeping in mind that the system is two dimensional so that all strain components which have a zdependence can be neglected gives the following Magneto-Elastic contribution to the magnetization equations of motion.