Influence of inter-sublattice coupling on the terahertz nutation spin dynamics in antiferromagnets

Spin nutation resonance has been well-explored in one-sublattice ferromagnets. Here, we investigate the spin nutation in two-sublattice antiferromagnets as well as, for comparison, ferrimagnets with inter-and intra-sublattice nutation coupling. In particular, we derive the susceptibility of the two-sublattice magnetic system in response to an applied external magnetic field. To this end, the antiferromagnetic and ferrimagnetic (sub-THz) precession and THz nutation resonance frequencies are calculated. Our results show that the precession resonance frequencies and effective damping decrease with intra-sublattice nutation coupling, while they increase with inter -sublattice nutation in an antiferromagnet. However, we find that the THz nutation resonance frequencies decrease with both the intra-and inter-sublattice nutation couplings. For ferrimagnets, conversely, we calculate two nutation modes with distinct frequencies, unlike antiferromagnets. The exchange-like precession resonance frequency of ferrimagnets decreases with intra-sublattice nutation coupling and increases with inter-sublattice nutation coupling, like antiferromagnets, but the ferromagnetic-like precession frequency of ferrimagnets is practically invariant to the intra and inter-sublattice nutation couplings.

Spin-nutation resonance has been well-explored in one-sublattice ferromagnets. Here, we investigate the spin nutation in two-sublattice antiferromagnets as well as, for comparison, ferrimagnets with inter-and intra-sublattice nutation coupling. In particular, we derive the susceptibility of the two-sublattice magnetic system in response to an applied external magnetic field. To this end, the antiferromagnetic and ferrimagnetic (sub-THz) precession and THz nutation resonance frequencies are calculated. Our results show that the precession resonance frequencies and effective damping decrease with intra-sublattice nutation coupling, while they increase with inter -sublattice nutation in an antiferromagnet. However, we find that the THz nutation resonance frequencies decrease with both the intra and inter-sublattice nutation couplings. For ferrimagnets, conversely, we calculate two nutation modes with distinct frequencies, unlike antiferromagnets. The exchange-like precession resonance frequency of ferrimagnets decreases with intra-sublattice nutation coupling and increases with inter-sublattice nutation coupling, like antiferromagnets, but the ferromagnetic-like precession frequency of ferrimagnets is practically invariant to the intra and inter-sublattice nutation couplings.

I. INTRODUCTION
Efficient spin manipulation at ultrashort timescales holds promise for applications in future magnetic memory technology [1][2][3][4][5]. Introduced by Landau and Lifshitz, the time evolution of magnetization M (r, t), can be described by the phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion, which reads [6][7][8][9] with the gyromagnetic ratio γ, constant magnetization amplitude M 0 , and Gilbert damping parameter α. The LLG equation consists of the precession of spins around a field H and transverse damping that aligns the spins towards the field direction. While the spin precessional motion can be explained by Zeeman-like field-spin coupling, there are several fundamental and microscopic mechanisms leading to Gilbert damping [10][11][12][13][14][15][16][17][18][19][20][21][22]. When one approaches the femtosecond regime, however, the spin dynamics can not only be described by the traditional LLG dynamical equation of motion [23,24], but it has to be supplemented by a fast dynamics term due to magnetic inertia [25][26][27]. Essentially, the inclusion of magnetic inertia leads to a spin nutation at ultrashort timescales and can be described by a torque due to a double time-derivative of the magnetization i.e., M ×M [26,28]. The inertial LLG (ILLG) equation of motion has the forṁ with the inertial relaxation time η. In general, the Gilbert damping α and the inertial relaxation time η are tensors [29], however, for an isotropic system, these parameters can be considered as scalars. The emergence of spin * mondal@fzu.cz nutation has been attributed to an extension of Kamberský breathing Fermi surface model [30,31], namely, an s − dlike interaction spin model between local magnetization and itinerant electrons [32,33]. Moreover, the ILLG equation has been derived from the fundamental Dirac equation [20,29]. Note that the Gilbert damping and inertial relaxation time are related to each other as the Gilbert damping is associated with the imaginary part of the susceptibility, while the inertial dynamics are associated with the real part of the susceptibility [20,34]. The characteristic timescales of the nutation have been predicted to be in a range of 1−100 fs [25,32,35,36] and 1−10 ps [36,37]. More recently, it has been demonstrated that simple classical mechanical considerations superimposed with Gilbert dynamics naturally lead to magnetic inertial dynamics [38,39].
Theoretically, the spin nutation has recently been extensively discussed for one-sublattice ferromagnets [25,36,[40][41][42][43][44]. The nutation resonance has also been observed in experiments, however for two-sublattice ferromagnets [37]. A recent theoretical investigation predicts that the precession and nutation resonance frequencies may overlap in twosublattice ferromagnets [45]. The spin nutation resonance has been observed at a higher frequency than ferromagnetic resonance, e.g., while the ferromagnetic resonance occurs in the GHz regime, the nutation resonance occurs in the THz regime [37,46]. Moreover, the spin nutation shifts the ferromagnetic resonance frequency to a lower value. Although this shift is very small, the line-width of the resonance decreases, however, and thus the effective damping decreases, too.
Spin nutation effects have not yet comprehensively been discussed in two-antiparallel aligned sublattice magnetic systems (e.g., antiferromagnets, ferrimagnets). In a recent investigation, it has been predicted that the spin nutation in antiferromagnets may have much significance [46]. Due to sublattice exchange interaction, the antiferromagnetic resonance frequency lies in the THz regime, while the nutation resonance frequency has similar order of magnitude. This helps to detect the antiferromagnetic precession and arXiv:2107.07939v2 [cond-mat.mtrl-sci] 19 Jul 2021 nutation resonances experimentally as they fall in the same frequency range. Moreover, the calculated shift of the antiferromagnetic resonance frequency is stronger than that of a ferromagnet. Additionally, the nutation resonance peak is exchange enhanced [46], which is beneficial for detection in experiments. However, the previous investigation only considers the intra-sublattice inertial dynamics, while the effect of inter-sublattice inertial dynamics is unknown.
In a previous work, the LLG equation of motion with inter-sublattice Gilbert damping has been explored by Kamra et al. [47]. It was found that the introduction of inter-sublattice Gilbert damping enhances the damping [47][48][49]. In this study, we formulate the spin dynamical equations in a two-sublattice magnetic system with both intra and inter-sublattice inertial dynamics as well as inter and intra-sublattice Gilbert damping, extending thus previous work [46]. First, we derive the magnetic susceptibility with the inter-sublattice effects and compute the precession and nutation resonance frequencies. We find that the precession resonance frequency and the effective Gilbert damping decrease with the intra-sublattice nutation coupling in antiferromagnets, however, they increase with the inter-sublattice nutation. Unlike antiferromagnets, we find for ferrimagnets that the change of precession resonance frequencies is more pronounced with both intra and intersublattice nutation coupling constants in the exchange-like mode, but nearly negligible for the ferromagnetic mode.
The article is organized as follows. First, in Sec. II, we discuss the linear-response theory of spin dynamics to calculate the magnetic susceptibility with the intra and intersublattice nutation effects. In Sec. III, the precession resonance frequencies have been calculated with analytical and numerical tools for antiferromagnets (Sec. III A) and ferrimagnets (Sec. III B). We summarize the obtained results in Sec. IV.

II. LINEAR-RESPONSE SUSCEPTIBILITY IN TWO-SUBLATTICE MAGNETS
First, in the ground state, we consider that the A sublattice magnetization is , such that we can describe the antiferromagnets (M A0 = M B0 ) and ferrimagnets (M A0 > M B0 ). We then expand the magnetization around the ground state in small deviations, For convenience, we work in the circularly polarized basis, i.e., m A/B± = m A/Bx ± im A/By , h A/B± = h A/Bx ± ih A/By , and define With the time-dependent harmonic fields and magnetizations h A/B± , m A/B± ∝ e ±iωt , we obtain the magnetic susceptibility tensor [46] with the definition of the determinant As one expects, the inter-sublattice Gilbert damping and inertial dynamical contributions arise in the off-diagonal components of the susceptibility tensor, while the intrasublattice contributions are in the diagonal component of the susceptibility [46]. Note that without inertial dynamics terms, the expression for the susceptibility is in accordance with the one derived by Kamra et al. [47]. To find the resonance frequencies, the determinant ∆ ± must go to zero, thus one has to solve the following fourthorder equation in frequency where the coefficients have the following forms The solutions of the above equation (9) result in four different frequencies in the presence of a finite external field. Two of those frequencies can be associated with the magnetization precession resonance, ω p± (positive and negative modes) that exists even without nutation. The other two frequencies dictate the nutation resonance frequencies, ω n± (positive and negative modes).

III. RESULTS AND DISCUSSION
The intrinsic intra-sublattice inertial dynamics have been discussed extensively in Ref. [46]. Essentially, the resonance frequencies and effective damping decrease with increasing intra-sublattice inertial relaxation time for antiferromagnets and ferrimagnets. Therefore, we consider a constant intra-sublattice inertial relaxation time in this work. In this section, we specifically discuss the effects of inter-sublattice nutation in both antiferromagnets and ferrimagnets.

A. Antiferromagnets
To start with, we calculate the frequency-dependent dissipated power of an antiferromagnet.
Using the expressions for the susceptibility in Eq. (8), we calculate the dissipated power in the inertial dynamics with the following definition To focus on the inter-sublattice nutation η AB = η BA = η , we set the inter-sublattice Gilbert damping to zero, i.e., α AB = α BA = 0, and choose M A0 = M B0 = 2 µ B . The exchange and anisotropy energies, magnetic moments used in the here-presented computations are comparable to a typical antiferromagnetic NiO [23,50,51] or CoO [52,53] system. However, we mention that NiO or CoO bulk crystals have biaxial anisotropy. Also, the Gilbert damping of NiO is very small α ∼ 10 −4 , i.e., less than the here-used value. In contrast, a large spin-orbit coupling in antiferromagnetic CrPt (that has ∼2 µ B Cr moments) leads to a higher Gilbert damping α ∼ 10 −2 [54,55]. Importantly, the inertial relaxation times η and η are not known in these antiferromagnetic systems. Our simulations pertain therefore to typical, selected model systems. We show the evaluated dissipated power with and without inertial dynamics for such antiferromagnet in Fig. 1. Note that the dissipated power has already been calculated in Ref. [46], however, without the inter-sublattice inertial dynamics. We can observe that while the intra-sublattice inertial dynamics decreases the precessional resonance frequencies (see the cyan lines in Fig. 1), the inter-sublattice inertial dynamics works oppositely. Note that the nutation resonance frequencies decrease with the introduction of inter-sublattice inertial dynamics.
To understand the effect of the inter-sublattice nutation terms, first, we solve the Eq. (9), considering again α AA = α BB = α, η AA = η BB = η, η AB = η BA = η , and α AB = α BA = 0. As the nutation in antiferromagnets is exchange enhanced [46], we calculate the effect of intersublattice terms on the precession and nutation frequencies, setting γ A = γ B = γ and M A0 = M B0 = M 0 for antiferromagnets. Therefore, the fourth-order equation in Eq. (9) reduces to an equation with A AFM The solution of the above equation results in precession and nutation resonance frequencies for the two modes (positive and negative). Inserting the real and imaginary parts of the solutions ω ± = Re (ω ± )+iIm(ω ± ), we numerically calculate the precession resonance frequencies and effective damping (the ratio of imaginary and real frequencies) for an antiferromagnet as a function of inter-sublattice nutation. The results are shown in Fig. 2, where the data points correspond to the numerical solutions.
with ω (0) being the solution of the above equation for α = 0 and H 0 = 0. The solutions of the above equation are rather simple and provide the two precession frequency modes (positive and negative) for antiferromagnets. Expanding the solutions of Eq. (15) up to the first order in α and H 0 , and also in first order in K/J 1, the precession resonance frequencies are obtained (neglecting the higher-order in ω (0) -terms) as Now substituting the |ω (0) | from the leading term in the frequency expression into the perturbative terms in Eq. (16), the approximate precession frequencies are obtained as .
This equation has been plotted in Fig. 2 as black lines. Note that, for ω p− we show for convenience −Re(ω p− ) in Fig. 2(a) and in the following. Due to the presence of η − η in the denominator of the frequency expressions, the precession resonance frequency increases when inter-sublattice nutation is taken into account (η < η), which explains the increase in frequency in Fig. 2(a). At the limit η → η , the nutation (intra and inter-sublattice) does not play a significant role as the precession resonance frequency is decreased by a factor 1 + 4γηK M0 which is very small due to K J. Note that the two resonance frequencies are approximately 0.332 THz and 0.276 THz with α = 0 and η = η = 0, while these two frequencies are 0.322 THz and 0.266 THz with α = 0.05 and η = η = 0. The latter has been shown in Fig.  2(a) as dashed lines. Therefore, the Gilbert damping has already the effect that it reduces the resonance frequencies.
The effective Gilbert damping can be calculated using the ratio between the imaginary and real parts of the frequencies, i.e., the line width. From Eq. (17) one arrives at Note that the two resonance modes have the same effective damping. For ferromagnets, the exchange energies do not contribute and thus the effective damping remains the same as α, in the absence of magnetic inertial terms (see [45]). However, in antiferromagnets the effective damping is enhanced due to the exchange interaction by a factor , even without any inertial terms. As investigated earlier [46], the effective damping decreases with the intra-sublattice relaxation time. However, similar to the increase in frequency, the effective damping also increases with the inter-sublattice inertial relaxation time, as seen in Fig. 2(b). The analytical solution in Eq. (18) agrees excellently with the numerical solutions. Close to the limit η → η, the effective Gilbert damping in Eq. (18) one expects the effective damping to be increased by a factor , as can be seen in Fig. 2(b). Next, we discuss the field dependence of the resonance frequencies. The precession resonance frequencies and effective damping have been plotted as a function of the applied field H 0 for several inter-sublattice relaxation times in Fig. 3. As can be observed, at zero applied field, the two modes (positive and negative) coincide in antiferromagnets, a fact that can be seen from Eq. (17). However, the applied field induces the splitting of these two modes. The frequency splitting scales with γH 0 , meaning that the splitting is linear in the applied field, H 0 . On the other hand, at a constant field, the splitting also depends on the interand intra-sublattice nutation. From Eq. (17), it is clear that the splitting is reduced with intra-sublattice nutation, while it is enhanced with inter-sublattice nutation. Such a conclusion can also be drawn from the numerical solutions in Fig. 3(a). The effective damping of the antiferromagnet  remains field independent which can be observed in Fig.  3(b).
Proceeding as previously, we obtain the following nutation frequencies Note that at the limit η → 0, the nutation frequencies without the inter-sublattice coupling are recovered [46]. The dominant term in the calculated frequency is the first term in Eq. (19). With the introduction of inter-sublattice coupling η , both the numerator and denominator of the dominant frequency term decrease and therefore, the nutation frequencies approximately stay constant (with a slow  decrease) with inter-sublattice nutation when η > η as plotted in Fig. 4. However, in the limit η → η, the denominator vanishes, and thus the nutation frequencies diverge as can be seen in Fig. 4. It is interesting to note that the inter-sublattice inertial dynamics increase the precession resonance frequencies, however, decrease the nutation frequency. Such observation is also consistent with the dissipated power in Fig. 1. The damping of the inertial dynamics also shows a similar behavior: it stays nearly constant with a divergence at the limit η → η.
As mentioned before, the inertial relaxation times η and η are not known in for typical antiferromagnetic systems. Notwithstanding, we obtain the general result that the precession resonance frequencies decrease with intra-sublattice inertial dynamics, however, increase with inter-sublattice inertial dynamics. Thus, to experimentally realize the signature of inertial dynamics, an antiferromagnet with a higher ratio of intra to inter-sublattice inertial relaxation time (η/η 1) is better suited.

B. Ferrimagnets
Next, we consider a ferrimagnetic system where the magnetic moments in the two sublattices are different, i.e., M A0 = M B0 . In this case, the analytical solution of Eq. (9) becomes cumbersome. The main reason is that +2γH 0 . For antiferromagnets, the magnetic moments in the two sublattices are exactly the same, i.e., M A0 = M B0 and thus, within the approximation of J M 0 H 0 , we find Ω A = Ω B which simplifies the analytical solution of Eq. (9). Thus, we numerically solve the Eq. (9) to calculate the precession and nutation resonance frequencies for ferrimagnets. We consider the case where M A0 = 10 µ B and M B0 = 2 µ B , reminiscent of rare-earthtransition-metal ferrimagnets as GdFeCo [2,3] or TbCo [56][57][58]. However, we emphasize that the inertial relaxation times η and η are not known for these materials. The calculated precession frequencies are shown in Fig. 5(a). The effect of intra-sublattice inertial dynamics has already been studied in Ref. [46]. For ferrimagnets, the negative frequency mode appears to have a higher frequency (i.e., larger negative) than the positive one. However, both precession frequencies decrease with intra-sublattice relaxation time, η [46]. We, therefore, have set the intra-sublattice relaxation time η to 100 fs and vary the inter-sublattice relaxation time η < η. The upper precession resonance mode ω p−the exchange-like mode -increases with the inter-sublattice relaxation time η , while the ferromagnetic-like mode ω p+ shows a very small increase. Thus, for ferrimagnets, the change in precession frequencies is more significant in the exchange-like mode than in the ferromagnetic-like mode. At the limit η → η, the precession resonance frequencies almost coincide with the resonance frequencies calculated at η = η = 0, meaning that the inertial dynamics do not play any role for the precession resonance frequency. The latter can clearly be seen in Fig. 5(a). These observations are similar to the antiferromagnet as discussed earlier. The nutation resonance frequencies in Fig. 5(b) again decline with the inter-sublattice relaxation time showing a divergence at the limit η → η. However, one can notice here two distinguishable nutation resonance frequencies unlike almost a single-valued nutation frequencies of antiferromagnets.

IV. SUMMARY
In summary, we have formulated a linear-response theory of the ILLG equations for antiferromagnets with inter-and intra-sublattice inertial dynamics. The calculation of the susceptibility tensor shows that the intra-sublattice terms appear in the diagonal elements, while the inter-sublattice terms appear in the off-diagonal elements. The dissipated power contains a precession resonance peak in the sub-THz regime for antiferromagnets, however, the introduction of inertial dynamics causes another peak, a nutation resonance peak at a higher, few THz frequency. Moreover, we observe that the inter-sublattice inertial dynamics work oppositely to the intra-sublattice inertial one. By finding the poles of the susceptibility, we calculate the precession and nutation resonance frequencies. While the precession resonance frequencies decrease with intra-sublattice relaxation time, the inter-sublattice inertial dynamics have the opposite effect. In fact, we observe that the magnetic inertia does not have any effect on the antiferromagnetic precession resonance at the limit η → η. On the other hand, the THz nutation resonance frequency decreases slightly with the introduction of inter-sublattice inertial dynamics, however, showing a divergence at the limit η → η. Our derived analytical theory explains such inter-sublattice contributions. Finally, for ferrimagnets, we find a similar behavior for the inter-sublattice inertial dynamics. However, the precession resonance frequency of the exchange-like mode depends significantly on the nutation couplings in contrast to that of the ferromagnetic-like mode that is practically independent of the nutation constants.