Dispersion relation of nutation surface spin waves in ferromagnets

Inertia effects in magnetization dynamics are theoretically shown to result in a new type of spin waves, i.e. nutation surface spin waves, which propagate at terahertz frequencies in in-plane magnetized ferromagnetic thin films. Considering the magnetostatic limit, i.e. neglecting exchange coupling, we calculate dispersion relation and group velocity, which we find to be slower than the velocity of conventional (precession) spin waves. In addition, we find that the nutation surface spin waves are backward spin waves. Furthermore, we show that inertia causes a decrease of the frequency of the precession spin waves, namely magnetostatic surface spin waves and backward volume magnetostatic spin waves. The magnitude of the decrease depends on the magnetic properties of the film and its geometry.

Recently, it has been theoretically and experimentally demonstrated that the effects of inertia of magnetization should be considered in the full description of spin dynamics at pico-and femtosecond timescales [32][33][34][35][36][37][38]. The nutation motion of magnetization is a manifestation of inertia of the magnetic moments. A rigorous derivation including inertia in the Landau-Lifshitz-Gilbert equation was carried out by R. Mondal et al. in the Dirac-Kohn-Sham framework [33,34]. A relation between the Gilbert damping and the inertial characteristic time was investigated in Ref. [32]. In another approach M.-C. Ciornei et al. confirmed that inertia is responsible for nutation, and that this motion is superimposed on the precession of magnetization [38]. The influence of nutation on the dynamic susceptibility was analytically [39] and numerically [40] studied.
Despite these theoretical advances, the experimental study of inertial spin dynamics has only begun. Following the first indirect observation of inertial magnetization dynamics in 7921 NiFe and Co films [41], the first direct experimental confirmation of nutation resonance was reported by Neeraj et al. [37].
In this paper, we predict an additional effect, that is the emergence of propagating nutation surface spin waves (NSSW) in the dipole-dipole coupling limit, and the transformation of conventional precession waves to precession-nutation spin waves. We derive dispersion relation of NSSW and calculate the spectral shift of precessionnutation spin waves with respect to precession spin waves. The emergence of nutation waves due to exchange coupling rather then dipolar interaction has been proposed by Makhfudz et al. [42] recently.
In general, the following interactions must be taken into account to describe the dynamics of spin waves: Zeeman, spin-orbit, exchange, and dipole-dipole interactions. The phase shift between precessing magnetic moments propagates as a spin wave through the ferromagnet because of dipoledipole or exchange coupling ( Fig. 1(a)). Magnetic inertia effects, which are expected to contribute to dynamics of spin waves, originate from spin-orbit coupling (coupling of the spins to the lattice via the orbital moment). In magnetization dynamics, this relativistic effect is considered with different orders of approximation. In the lowest order, one obtains the Gilbert damping of magnetization precession and the gyromagnetic ratio, i.e., the relation between angular momentum and spin. In higher order approximations, magnetic inertia appears [33,34,43], and the gyromagnetic ratio must be generalized, which leads to nutation motion of magnetic moments superimposed on their precession. Taking inertia into account one finds that the deviation of localized moments will propagate through the spin system in the form of both precession and nutation motions, i.e. in ferromagnetic materials one needs to add to all "conventional" spin wave modes a high frequency wave-like motion with small amplitude caused by inertia. Additionally, waves having predominantly inertial nature appear in ferromagnetic thin films, which we call here nutation surface spin waves. Since these waves have terahertz frequencies (compared to typically GHz frequencies of other spin wave modes), they can be plotted as a small deviation on top of a "frozen" precession motion ( Fig. 1(b)). In our calculation, we work in the dipole-dipole coupling limit, which allows us to use a magnetostatic approach in which Maxwell's equations are transformed into the Walker equation [4]. To obtain the dispersion relation including inertia, we use the dynamic susceptibility derived from the Inertial Landau-Lifshitz-Gilbert (ILLG) equation [39] and substitute the result into the Walker equation.
In this paper, we consider waves propagating in thin ferromagnetic films magnetized in-plane by an external magnetic field. We focus on two particular configurations: (A) waves propagating perpendicular the external magnetic field 0 H (see Fig. 1(c), y-axis), and (B) waves propagating along 0 H ( Fig. 1(c), z-axis). The latter case (B) corresponds to backward volume magnetostatic spin waves (BVMSW), when only precession is taken into account and as n-BVMSW when precession-nutation case is considered. Similarly, for perpendicular configuration we distinguish in our nomenclature between magnetostatic surface spin waves (MSSW), i.e. in other words Damon-Eshbach modes, when inertia is neglected, and n-MSSW when inertia is included. Finally, for perpendicular configuration our calculation predicts a new type of waves -nutation surface spin waves.

II. DISPERSION EQUATIONS AND WAVE CHARACTERISTICS
The ferromagnetic film, magnetic field and coordinate system are shown in Fig. 1(c). The film with thickness L is placed in an external magnetic field 0 H strong enough to saturate the magnetization of the film. We assume that the exciting magnetic field is small 0 , << hH and the static magnetization vector 0 M and external magnetic field 0 H are aligned.
The Maxwell's equations in magnetostatics are written as 0, where m is the response of the magnetization to the small driving magnetic field. Equ. (1) allows to introduce the magnetic potential using , y =Ñ h substitute this potential into equation (2) and obtain the Walker's equation  (6) where c ¢ is the real dispersive part of , c and a c¢ is the real dispersive part of the anti-diagonal component of the dynamic susceptibility tensor. The effect of inertia of the magnetization is introduced by the dynamic susceptibility deduced from the ILLG equation. The detailed derivation for a Cartesian coordinate system can be found in the Supplemental Material [44]. In the following sections, we focus on the dispersion relations for spin waves propagating in perpendicular (A) and parallel (B) direction to the magnetic field.

A. Perpendicular configuration
For spin waves propagating in perpendicular direction to the external magnetic field the equation (6) since z k should be equal to zero in this configuration. The substitution of the susceptibility expressions (equations (S7)-(S11) from Ref. [44]) into (7) We employ Ferrari's method for finding the solutions of equation (8), and introduce the notation: The bi-quartic equation (8) has four roots describing the relationship of frequency and wavenumber, i.e. different dispersion branches. The first branch corresponds to ferromagnetic resonance (FMR). The second one is the n-MSSW branch (Fig. 2(b) and (e)). The third and fourth branches are complex conjugates and describe NSSW, the real and imaginary parts are plotted in Fig. 2(b) and (c).
The n-MSSW branch is given by the expression  and for MSSW are the same parameters except 0 a = and 0. t = The difference is 115 MHz or 0.7 percent, and it can be clearly seen in the Fig 2(e). The decrease of frequencies of n-MBVW is caused by nutation, which is a rotation of magnetization in the opposite direction compared to the precession [44].
Due to the fact that the lower spectrum limit of the MSSW corresponds to upper spectrum limit of the BVMSW, we discuss the spectral red-shift of the lower limit of n-MSSW below.
The dispersion branch of NSSW is complex, and the real part of the branch describing propagation of the waves is determined by 12 The imaginary part corresponding to the inherent losses of NSSW is written as The NSSW have inherent losses, since these waves exist only if 0. at ¹ This fact is the direct consequence of their inertial nature -these waves do not exist, if one neglects inertia and damping.
The magnetostatic potential of spin waves is concentrated close to the surface of the ferromagnetic film if the magnetic field is applied perpendicular to the wavevector, hence in this configuration conventional spin waves and nutation spin waves are surface waves. However, NSSW have lower group velocity than the precession waves that can be seen from the Fig. 2(a) (23) We substitute the susceptibility expressions (S7)-(S11) from Ref. [44] and employ numerical damped Newton's method for finding the roots of the algebraic equation (23) to calculate the relations between frequency and wavenumber in both precession and precession-nutation cases. These relations demonstrate a set of dispersion branches, and the first three branches are plotted in Fig. 3. It is clearly seen from the Fig. 3, that the dispersion of n-BVMSW is shifted relatively to the BVMSW. To investigate this, we compare the spectral limits of the precession and precession-nutation waves. Note that the volume waves exist only in the frequency range where 10. c ¢ +£ This condition is a consequence of equation (5) We repeat the procedure for finding the solutions using Ferrari's method. The spectrum limits of n-BVMSW must be found in the same way as provided in (15)- (17) with the corresponding replacement of variables, i.e. subscript s is replaced by v, which denotes volume waves. Thus, the upper limit of the spectrum is determined by the expression

III. CONCLUSION
We theoretically predict the emergence of nutation surface spin waves due to magnetization inertia in the dipolar coupling limit for in-plane magnetized ferromagnetic thin films, propagating perpendicular to the direction of the external magnetic field. These waves are backward waves and propagate at terahertz frequencies with a group velocity lower than the velocity of conventional spin waves. Inertia leads to a red-shift of precession-nutation spin waves compared to precession spin waves. The upper spectral limit of the dispersion branches of the precession-nutation waves undergo a greater shift then the lower spectral limit.