Nutation Resonance in Ferromagnets

The inertial dynamics of magnetization in a ferromagnet is investigated theoretically. The analytically derived dynamic response upon microwave excitation shows two peaks: ferromagnetic and nutation resonances. The exact analytical expressions of frequency and linewidth of the magnetic nutation resonance are deduced from the frequency dependent susceptibility determined by the inertial Landau-Lifshitz-Gilbert equation. The study shows that the dependence of nutation linewidth on the Gilbert precession damping has a minimum, which becomes more expressive with increase of the applied magnetic field.


I. INTRODUCTION
Recently, the effects of inertia in the spin dynamics of ferromagnets were reported to cause nutation resonance [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] at frequencies higher than the conventional ferromagnetic resonance. It was shown that inertia is responsible for the nutation, and that this type of motion should be considered together with magnetization precession in the applied magnetic field. Nutation in ferromagnets was confirmed experimentally only recently [2], since nutation and precession operate at substantially different time scales, and conventional microwave ferromagnetic resonance (FMR) spectroscopy techniques do not easily reach the highfrequency (sub-Terahertz) regime required to observe the inertia effect which in addition yields a much weaker signal.
Similar to any other oscillatory system, the magnetization in a ferromagnet has resonant frequencies usually studied by ferromagnetic resonance [17,18]. The resonant eigenfrequency is determined by the magnetic parameters of the material and applied magnetic field. Including inertia of the magnetization in the model description shows that nutation and precession are complementary to each other and several resonances can be generated. In this Letter, we concentrate on the investigation of the resonance characteristics of nutation.
The experimental as well as theoretically investigation of the inertial spin dynamics is at the very beginning, although it might be of significance for future high speed spintronics applications including ultrafast magnetic switching. For instance, the spin dynamics in inertial regime was recently studied in thin ferromagnetic films [2], nanomagnets [3] and single spins embedded in tunnel junctions [4][5][6].
The microscopic derivation of the magnetization inertia was performed in ref. [7][8][9][10][11]. A relation between the Gilbert damping constant and the inertial regime characteristic time was elaborated in ref. [7]. The exchange interaction, damping, and moment of inertia can be calculated from first principles as shown in [11]. The study of inertia spin dynamics with a quantum approach in metallic ferromagnets was performed in [12]. In addition, nutation was theoretically analyzed as a part of magnetization dynamics in ferromagnetic nanostructure [13,14] and nanoparticles [15]. Despite these advances, nutation resonance has not been studied well enough.
In [30], the inertial regime was introduced in the framework of the mesoscopic nonequilibrium thermodynamics theory, and it was shown to be responsible for the nutation superimposed on the precession of magnetization. Wegrowe and Ciornei [1] discussed the equivalence between the inertia in the dynamics of uniform precession and a spinning top within the framework of the Landau-Lifshitz-Gilbert equation generalized to the inertial regime. This equation was studied analytically and numerically [16,31]. Although these reports provide numerical tools for obtaining resonance characteristics, the complexity of the numerical solution of differential equations did not allow to estimate the nutation frequency and linewidth accurately. Also in a recent remarkable paper [32] a novel collective excitationthe nutation wavewas reported, and the dispersion characteristics were derived without discussion of the nutation resonance lineshapes and intensities.
Thus, at present, there is a necessity to study the resonance properties of nutation in ferromagnet, and this paper is devoted to this study. We performed the investigation based on the Landau-Lifshitz-Gilbert equation with the additional inertia term and provide an analytical solution.
It is well known that the Landau-Lifshitz-Gilbert equation allows finding the susceptibility as the ratio between the timevarying magnetization and the time-varying driving magnetic field (see for example [33,34] and references therein). This susceptibility describes well the magnetic response of a ferromagnet in the linear regime, that is a small cone angle of the precession. In this description, the ferromagnet usually is placed in a magnetic field big enough to align all atomic magnetic moments along the field, i.e., the ferromagnet is in the saturated state and the magnetization precesses. The applied driving magnetic field allows one to observe FMR as soon as the driving field frequency coincides with eigenfrequency of precession. Using the expression for susceptibility, one can elaborate such resonance characteristics as eigenfrequency and linewidth. Let us present similar expression for the dynamic susceptibility, which takes into account the nutation.

II. SUSCEPTIBILITY
The ferromagnet is subjected to a uniform bias magnetic field 0 H acting along the z-axis and being strong enough to initiate the magnetic saturation state. The small time-varying magnetic field h is superimposed on the bias field. The coupling between impact and response, taking into account precession, damping, and nutation, is given by the Inertial Landau-Lifshitz-Gilbert (ILLG) equation By performing the Fourier transform and changing the order of integration, the equation (4) becomes (5) where the integral representation of the Dirac delta function can be found. With using the delta function, the equation (5) is simplified as

III. APPROXIMATION FOR NUTATION FREQUENCY
Let us turn to the description of an approximation of the nutation resonance frequency. If we equate the denominator D  to zero, solve the resulting equation, we obtain the approximation from the real part of the roots. This is reasonable, since the numerator of    is the linear function of  , and we are interested in 1. One should choose the same sign from the  symbol in each formula, simultaneously. The real part of expression (13) gives the approximate frequency for FMR, but in negative numbers, so the sign should be inversed.

IV. PRECISE EXPRESSIONS FOR FREQUENCY AND LINEWIDTH OF NUTATION
The analytical approach proposed in this Letter yields precise values of the frequency of nutation resonance and the full width at half maximum (FWHM) of the peak. The frequency is found by extremum, when the derivative of the dissipative part of susceptibilities (9) is zero 0.
It is enough to determine zeros of the numerator of the derivative, that are given by Let us use Ferrari's solution for this quartic equation and introduce the notation: The performed analysis shows that approximate value of nutation resonance frequency is close to precise value.
A root of the nested depressed cubic equation lw y must be found in the same way as provided in (20) with the corresponding replacement of variables, i.e. subscript r is replaced by lw. The difference between two adjacent roots gives the nutation linewidth

 
The effect of the inertial relaxation time on the nutation linewidth is shown in Fig. 2. One can see that increasing of the inertial relaxation time leads to narrowing of the linewidth. This behavior is expected and is consistent with the traditional view that decreasing of losses results in narrowing of linewidth. Since the investigated oscillatory system implements simultaneous two types of motions, it is of interest to study influence Gilbert precession damping parameter  on the nutation resonance linewidth. The obtained result is presented in Fig. 3 and is valid for ferromagnets with vanishing anisotropy. One sees that the dependence of N  on  shows a minimum that becomes more expressive with increasing of the bias magnetic field. In other words, the linewidth is parametrized by magnitude of field. This nontrivial behavior of linewidth relates with the nature of this oscillatory system, which performs two coupled motions.
To elucidate the non-trivial behavior, one can consider the susceptibility (9) in the same way as it is usually performed for the forced harmonic oscillator with damping [35]. For this oscillator, the linewidth can be directly calculated from the denominator of the response expression once the driving frequency is equal to eigenfrequency. In the investigated case of magnetization with inertia, the response expression is (9) with denominators (10) and (11)

V. CONCLUSION
In summary, we derived a general analytical expression for the linewidth and frequency of nutation resonance in ferromagnets, depending on magnetization, the Gilbert damping, the inertial relaxation time and applied magnetic field. We show the nutation linewidth can be tuned by applied magnetic field and this tunability breaks the direct relation between losses and the linewidth. This for example leads to the appearance of a minimum in the nutation resonance linewidth for the damping parameter around 0.15.

 
The obtained results are valid for ferromagnets with vanishing anisotropy, investigation of general case will be published in an upcoming paper.