Ultra-slow spin waves propagation based on skyrmion breathing

Spin wave has attracted significant attention in various fields because of its rich physics and potential applications in the development of spintronics devices in the post-Moore era. However, the analog of a subluminal-like propagation in the field of spin waves has not been well discussed. Here, we theoretically demonstrate the ultra-slow spin waves propagation in a nanoscale two-dimensional ferromagnetic film in the presence of magnon-skyrmion interaction. The minimum spin waves propagation velocity was estimated to be as low as 1.8 m s−1 by adjusting the system parameters properly, and the spin waves group delay and advance are dynamically tunable via the intensity or detuning of the control field, which allows the possibility of observing superluminal- and subluminal-like spin waves propagation in a single experimental setup. These results deepen our understanding of the spin wave–skyrmion interactions, open a novel and efficient pathway to realize ultra-slow spin waves propagation, and are expected to be applied to magnetic information storage and quantum operations of magnons.


Introduction
Spin waves, the collective excitations of magnetic orders, are the non-uniform precession of magnetic moments in magnetic materials that can propagate in conductive or insulating materials.Their propagation does not involve the directional motion of charge, so they have the advantage of low energy consumption [1,2].Magnons, the quanta of spin waves, are expected to be used in place of electrons for information transmission and processing in the post-Moore era because of their characteristics of low energy consumption, high frequency, short wavelength, and distinct nonlinear effects [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].Numerous studies have shown that magnons resemble photons as bosonic quasiparticles, and many photonic phenomena, ranging from the multi-stability of cavity magnon polaritons [11,12], optical cooling of magnons [13], magnon lasing [14] to magnon-induced high-order sideband generation [16] and magnon blockade effects [17][18][19] have been proposed in the field of magnonics, an emerging discipline of solid-state physics.
Magnetic skyrmions, a new type of locally stable spin texture that has developed from a theoretical prediction to an important research frontier, have given rise to many interesting phenomena within condensed matter physics and quantum optics [20][21][22][23].For example, skyrmion spin textures have been proposed to interpret the Hall effect [21], the inversion of asymmetric bulk magnets [22], magnetic multilayers [23], and so on [24][25][26].Moreover, magnetic skyrmions possess superior topologically protected stability and nanoscale size, and also have various manipulation methods.For instance, magnetic skyrmions can be driven by spin-polarized current, electric field gradient, magnetic field, and temperature gradient [27].Therefore, it is expected to be used as a next-generation information carrier in the development of spintronic devices and integrated circuits, which has the advantages of high speed, high density, and low energy dissipation [28].Recently, the spin wave-skyrmion interactions have attracted more and more attentions and progressed enormously, for instance, the nonlinear scattering processes between the skyrmion excitation and magnons [29][30][31], the magnonic frequency combs based on nonlinear magnon-skyrmion scattering [32], the forward and transient retrograde motion of spin wave driven skyrmions [33], and the deeply nonlinear excitation of short spin waves [34].The study of magnetic skyrmion, therefore, not only reveals fundamentally new physics, but also has great technical prospects [35].
Recently, experimental studies have reported a method for measuring the high-speed magnon propagation in antiferromagnetic NiO at nanoscale distances with optical-driven terahertz emission, and a superluminal-like propagation of magnons with a velocity of ∼650 km s −1 has been observed [36].However, to the best of our knowledge, the realization of ultra-slow spin waves propagation and its control remains a challenge.It is comprehensible that faster is not always better, for example, slow light can outperform fast light in many aspects, which has been discussed repeatedly in field of optics and photonics [37].Here, we solve the challenge by investigating the spin wave propagation in a nanoscale two-dimensional ferromagnetic film in the presence of magnon-skyrmion interactions.An intriguing phenomenon, ultra-slow spin wave propagation is demonstrated theoretically, and the physical mechanism is analyzed.The analytical results of the magnon-skyrmion dynamics shows an excellent agreement with this theory.Furthermore, we discuss in detail the dependence of the velocity of spin waves propagation on drive field power and system damping, as well as the regulation of spin wave group delay/advance.

Results and discussion
The physical model we consider is a nanoscale two-dimensional ferromagnetic system, which supports localized stable skyrmion textures due to the Dzyaloshinskii-Moriya interaction [27], as schematically shown in figure 1.The incident spin wave can cause significant deformation and fluctuations in the skyrmion profile, thereby exciting its internal modes such as the skyrmion breathing mode [29][30][31][32].Then, the skyrmion breathing mode can hybridize with the driving mode via the magnon-skyrmion scattering process and generate the difference-frequency and sum-frequency modes.Here, we focus on the frequency up-conversion process, i.e. sum-frequency mode, and Hamiltonian describing the magnon-skyrmion interaction, in quantum mechanical language, can be written as (the derivation is provided explicitly in appendix A): where h is the reduced Plank constant, ω k is the eigenfrequency of the incident spin wave mode.m m and m † m are, respectively, the annihilation and creation operators of the skyrmion breathing mode (with the frequency ω m ) via the Holstein-Primakoff approximation [38].m s (m † s ) is the annihilation (creation) operators of the sum-frequency mode with the frequency ω s = ω k + ω m .g is the coupling strength of the three-magnon scattering process, which is a frequency dependent parameter [32].The dynamic processes of the magnon-skyrmion interaction can be formulated by the Heisenberg-Langevin equation, and the transmission of the system can be obtained by using the input-output relation [39], and the resulting reads (the derivation is provided explicitly in appendix B) where κ s and κ m are the attenuations of the sum-frequency mode and the skyrmion breathing mode, respectively.∆ s(p) = ω s(p) − ω l is the detuning between the sum-frequency mode (probe field with frequency ω p ) and the microwave driving field (with frequency ω l ).
is the effective coupling strength.The rapid phase dispersion of the probe transmission , which provides a measure of the group delay or advance of the spin waves propagation when it passes through the nanoscale ferromagnetic film.Furthermore, the group delay of spin waves propagation can be obtained by d∆p [40,41].The symbol of τ g determines the peculiarity of spin wave propagation, that is, a positive and a negative symbols, respectively, imply the group delay and advance of the spin wave [42].Arguably, ultra-slow spin wave propagation owing to magnon-skyrmion scattering is a general phenomenon in spintronics, which deserves in-depth research.
To proceed, we discuss the spin wave group delay and advance effects in detail, and analyze the feasibility of achieving ultra-slow spin wave propagation using current experimental parameters [43].Figure 2(a) shows the spin wave (or magnon) group delay τ g as a function of the microwave driving field power P l at various damping constants κ m .Specifically, for κ m /2π = 8 MHz, as shown by the red line, the magnon group delay increases first and then decreases with further increasing P l , and eventually tends to τ g → 0. A maximum magnon group delay of τ g ≈ 0.9 ns was observed at P l ≈ 0.3 nW.Furthermore, the magnon group delay increases along with the reduction of the damping κ m .For example, when κ m is reduced to κ m /2π = 4 MHz and κ m /2π = 2 MHz, the obtained maximum magnon group delays are shown at points A  and B in the blue and green lines of figure 2(a), respectively.More interestingly, the microwave drive field power required to acquire maximum magnon group delay also decreases, and an optimal region is reflected on the inset of figure 2(a).We next evaluate the spin wave velocity in such ferromagnetic film, which is calculated by v s = d/τ g [44], where d is the propagation distance in the nanoscale ferromagnetic film (here d = 10 nm is chosen as the simulation value [32]).The result shows that the spin wave velocity first decreases apace with increasing the microwave driving field power P l , and after reaching a minimum, it increases slowly with increasing P l , which is consistent with the above discussion.
Further, we observe a high dependence of the magnon group delay τ g (and the corresponding spin wave velocity v s ) on the magnon-skyrmion scattering strength g and the damping rate of the skyrmion breathing mode κ m , and more clear results are shown in figure 3.In the absence of magnon-skyrmion interaction, the spin wave group delay is not observed (τ g = 0).Interestingly, the phenomenon of spin wave delay occurs when the magnon-skyrmion interaction is considered, and the spin wave group delay τ g increases as the magnon-skyrmion scattering strength g increases and the maximum values of τ g is about 3.7 ns corresponding to the magnon-skyrmion scattering strength g ≈ 4.6 MHz.Further increase in the magnon-skyrmion scattering strength, however, the spin wave group delay decreases instead of increasing.Correspondingly, the variation of spin wave velocity with the magnon-skyrmion scattering strength is shown in figure 3(c).From the above discussion we can see that with the increase of magnon-skyrmion coupling strength, the system goes through undercoupling, critical coupling and overcoupling processes in turn.In addition, we also identify the parameter regimes for the underdamped, critically damped, and overdamped behaviors, and the results are shown in figures 3(b) and (d).The minimum spin wave velocity v s ≈ 1.8 m s −1 at the damping rate κ m /2π ≈ 1 MHz (brown point in figure 3(d)), corresponding to the magnon group delay τ g ≈ 5.4 ns (as shown in figure 3(b)).Also, it is particularly important to obtain both positive and negative τ g in one experimental setup.Our theory shows that the symbol of τ g is dynamically tunable via the intensity or detuning of the control field, and the theoretical prognostication is shown in figure 4(a).Specifically, when the detuning between the magnon mode and the microwave driving field ∆ s /ω m = 1.1 (the red line of figure 4(a)), the magnon group delay is negative (called the magnon group advance).Likewise, the magnon group advance first increases abruptly with increasing the microwave driving field power P l , has a peak for P l = 0.3 nW, and gradually decreases with further increasing P l .For more intuitively describe the highly tunable features of magnon group delay and advance, the three-dimensional graph of τ g varies with the detuning ∆ s and P l is shown in figure 4(b).The red dashed curve represents the group delay τ g = 0, indicating that the spin wave propagation is not slowed or fastened when the detuning satisfies the resonance condition ∆ s = ω m .Obviously, when the detuning ∆ s /ω m < 1, a positive sign of τ g can be observed, implying that the subluminal-like spin wave propagation in ferromagnetic film.On the contrary, when the detuning ∆ s /ω m > 1, a negative sign of τ g is obtained, suggesting the appearance of superluminal-like spin wave propagation [36].Note that the superluminal-like spin wave propagation is still follows the principles of causality or special relativity [36,41].
In the previous sections, the spin wave group delay and advance can be observed by adjusting the detuning between the magnon mode and the microwave driving field.Here, we propose another method to realize magnon group delay and advance under the resonance condition ∆ s = ω m .Figure 5(a) shows τ g as a function of the control field power P l under ∆ s = ω m and ∆ p /ω m = 1.027.We observe that the transition between spin wave group delay and advance can be achieved simply by modifying the intensity of the microwave driving field.Points A, B and C in figure 5(a) represent the magnon group delays under different input powers, respectively.This phenomenon, physically, can be explained by the rapid phase dispersion of the probe transmission.Figure 5(b) plots the rapid phase dispersion of the probe transmission as a function of ∆ p at various P l .We find that when the probe frequency is close to control frequency, i.e. ∆ p /ω m = 1.027 (black dotted line in figure 5(b)), the rapid phase dispersion of the probe transmission is completely different at various P l .For example, when the control field power P l = 2 nW, the slope of the phase dispersion dϕ tp /d∆ p = 0 (point A on the red solid line), however, when the control field power is reduced to P l = 0.6 nW, the slope of the phase dispersion dϕ tp /d∆ p < 0 (point B on the blue dotted line, corresponding to group advance).Further modifying the control field power P l = 3.2 nW, a completely opposite result was observed, i.e. dϕ tp /d∆ p > 0 (point C on the green dotted line, corresponding to group delay).Our theoretical explanation is in good agreement with the numerical simulation results in figure 5(a).
Up till now, we have demonstrated the ultra-slow spin waves propagation results from the magnon-skyrmion interaction.In what follows, we will discuss the presence of the dipole-dipole interaction that may influence the ultra-slow spin waves propagation, which is very relevant in ferromagnetic systems [45].After considering the dipole-dipole interaction, the Hamiltonian of the system under the rotation-wave approximation is rewritten as follows where η is the dipole-dipole interaction strength.The transmission of the system was corrected accordingly, reads as Figure 6 shows the variation of spin wave group delay under the dipole interaction.We can see that the spin wave group delay can still be observed under the weak dipole interaction strength (compared to the magnon-skyrmion interaction strength g), as shown by the blue and green lines in figure 6(a).With the increase of the dipole interaction intensity, the ultra-slow propagation of spin wave will be greatly affected, as shown by the sky-blue line in figure 6(a).In short, the dipole interaction is very relevant in the ultra-slow propagation of spin wave, especially under the conditions of strong dipole interaction.Therefore, the proposed scheme of ultra-slow spin wave propagation based on magnon-skyrmion interaction is no longer applicable in the case of strong or ultra-strong dipole interaction (as shown in figure 6(b)), which requires further study.
Ultra-slow spin wave propagation may bring significant advances to the field of nonlinear spintronics [1], It is not difficult to imagine that when the incident spin wave propagates in a slow spin wave medium, it is compressed, and its energy density increases, thus promoting the nonlinear interaction of spin waves.Furthermore, in realizing quantum manipulation of spin wave, one of the key issues is to store the quantum states of spin wave for a sufficiently long time to enable quantum operations.Slowing or even stopping the spin wave may be the most efficient way to achieve this long storage time, and also is the key to enabling magnetic information storage [46].The first threshold faced by magnon applications in the information field is the preparation of the 'on' and 'off ' states of magnon, which is analogous to the switching effect of charge-based transistors.Research has shown that using the magnetic domain wall present in magnetic materials to modulate spin waves can form a controlled magnon switch [47,48].Spin wave group delay and group advance can also be used to construct the 'on' and 'off ' states of magnons in principle, which has important potential applications in the development of a new generation of spin logic devices.In brief, the studies of ultra-slow spin wave propagation based on skyrmion breathing will lead to various applications, especially in the fields of magnetic information storage and quantum operations of spin waves.
Finally, it is essential to discuss the feasibility of observing ultra-slow spin wave propagation under current experimental conditions.One can locally inject a spin wave in a chiral ferromagnetic thin film hosting skyrmions, and then the spin-wave group velocity can be measured and analyzed by utilizing time-resolved microfocused Brillouin light scattering spectroscopy [34,49].In addition, the variable parameters discussed in the numerical simulation can also be controlled in real samples.On the one hand, the magnon-skyrmion scattering strength is affected by the spin number in the system as well as frequency-dependent [32,50], which can be adjusted by adjusting the size of the skyrmion and the frequency of the driving field in experiments.On the other hand, experimental results show that the value of the damping is dependent on the thickness of the ferromagnetic film, and the damping coefficient increases as the thickness of the ferromagnetic film decreases [51][52][53].The annealing temperature during material preparation also affects the value of the damping [54], and furthermore the damping constant of the spin wave can be changed by alloying [55].Therefore, the simulation results give the minimum spin waves propagation velocity as low as 1.8 m s −1 that might be observed in future experiments.

Conclusion
In conclusion, an intriguing phenomenon, ultra-slow spin wave propagation based on skyrmion breathing in a nanoscale ferromagnetic film, has been theoretically investigated.We have demonstrated that the magnon group delay and advance can be dynamically tuned by adjusting the control power and effective control detuning, indicating that it is possible to switch directly between superluminal-and subluminal-like spin wave propagation by modulating the intensity or detuning of the control field.Also, the physical mechanism of the magnon group delay/advance originating from the strong phase dispersion of the probe transmission is analyzed.The minimum spin wave velocity can be as low as 1.8 m s −1 in our simulation results, which is a good ultra-slow propagation signal.Our findings not only offer the simplicity of operation but also provide numerous practical advantages, such as enhancing the attractiveness of spintronic devices in magnetic information storage fields.Furthermore, our examination of ultra-slow spin wave propagation has, in our view, significantly expanded the understanding of one of the fundamental facets of spin wave science.
where ∆ k(s) = ω k(s) − ω l is the detuning between the magnon mode and the microwave driving field, and ∆ p = ω p − ω l is the beat frequency between the microwave driving field and the probe field.Under the non-depletion approximation, the external microwave driving field ⟨m k ⟩ = iε l /(−i∆ k − κ k ), which implies that the three-magnon process can be reinforced by the amplitude of ⟨m k ⟩.Note that the Hamiltonian in equation (A1).includes an interface-induced DM interaction term, which stabilizes Néel-type skyrmions (figure 1 in the main text).For other types of magnetic solitons, the form of the DM interaction in equation (A1).needs to be changed accordingly, and the above derivation process is also applicable.

Appendix B. Evolution equation and spin wave group delay
According to the Heisenberg-Langevin equation, the dynamic processes of the system can be formulated by the following coupling equations: where □ • Φ = (dm m /dt, dm s /dt) T , Φ = (m m , m s ) T , σ = (0, −i) T , Γ = [ √ κ m m in (t), √ κ s a in (t) ] T , and Here κ m is the damping rate of the skyrmion breathing mode and G = g⟨m k ⟩ is the effective coupling strength.m in (t) and a in (t) are the quantum noise terms of the system, which can be safely dropped under the semiclassical approximation [39].Following the perturbation method of depicting the magnon-skyrmion scattering process, the coupling equation (B1) can be solved analytically, that is, the solutions of m m and m s can be written as a sum of the steady-state value and the fluctuation via a factorization assumption, i.e. m m = M 0 + δm m , and m s = A 0 + δm s .Therefore, we can get the fluctuation equation for δm m and δm s , as where □ • Ψ = (dδm m /dt, dδm s /dt) T , Ψ = (δm m , δm s ) T .Such fluctuation equation can be solved analytically by using the ansatz: δm m = M + e −i∆pt + M − e i∆pt and δm s = A + e −i∆pt + A − e i∆pt , where the higher-order frequency components are safely ignored due to the fact that these components contribute little in the perturbative regime.Substituting the ansatz into the fluctuation equation, we can obtain the solution of interest: Using the input-output relation, the transmission of the system is given by t p = 1 − κ s A + /ε p , i.e. (equation (2) in the main text) The rapid phase dispersion of the probe transmission Φ tp (∆ p ) = arg[t p (∆ p )] and the group delay of spin waves propagation can be obtained by using the following two approaches

Figure 1 .
Figure 1.Schematic illustration of a nanoscale 2D-ferromagnetic film.The skyrmion breathing mode can be excited by introducing an incident spin wave mode (sketch maps of spin wave and skyrmion breathing mode on the left).Energy level diagram of nonlinear magnon-skyrmion scattering process.

Figure 3 .
Figure 3. (a) and (b) The magnon group delay τg as a function of the magnon-skyrmion scattering strength g and the damping rate of the skyrmion breathing mode κm, respectively.(c) and (d) are the corresponding spin wave velocities.The damping rate of the skyrmion breathing mode κm/2π = 2 MHz in (a) and (c), and the magnon-skyrmion scattering strength g = 4.5 MHz in (b) and (d).The power of the microwave driving field P l = 0.07 nW, and the other parameters are the same as those in figure 2.

Figure 4 .
Figure 4. (a) The magnon group delay τg as a function of the microwave driving field power P l at various detuning constants ∆s/ωm = 0.9 and 1, respectively.(b) The three-dimensional graph of the group delay τg varies with ∆s and P l .The red dashed curve indicates the magnon group delay τg = 0.The other parameters are the same as those in figure 2.

Figure 5 .
Figure 5. (a) The magnon group delay τg as a function of the microwave driving field power P l under the condition of ∆s = ωm and ∆p/ωm = 1.027.(b) The rapid phase dispersion Φ of the probe transmission as a function of the beat frequency ∆p at various microwave driving field power P l .The other parameters are the same as those in figure 2.

Figure 6 .
Figure 6.(a) The magnon group delay τ g as a function of the microwave driving field power P l under the different dipole-dipole interaction strength η.(b) The magnon group delay τ g as a function of the dipole-dipole interaction strength η.The illustrations in (a) and (b) are the corresponding spin wave velocities.The magnon-skyrmion scattering strength g = 4.5 MHz and the power of the microwave driving field P l = 0.3 nW in (b).The other parameters are the same as those in figure 2.