1 Introduction

Recently, domain wall dynamics of antiferromagnets and ferrimagnets which have two antiferromagnetically coupled sublattices have drawn considerable attention for their potential applications in spintronics [1,2,3,4,5]. For example, antiferromagnets are suggested to be promising candidates for high-density and high-speed spintronic devices because of their fast magnetic dynamics and zero stray field attributed to zero magnetization and ultralow susceptibility [6,7,8,9,10,11,12,13]. However, zero magnetization also prevents experimental detection and manipulation of antiferromagnetic dynamics [14]. Alternatively, ferrimagnets, which have nonequivalent magnetic sublattices, provide another potential material platform [15,16,17,18,19,20].

In ferrimagnetic (FiM) systems such as rare-earth transition-metal ferrimagnets, there is an angular momentum compensation temperature (TA) below the Curie temperature [4, 17, 21, 22]. In the vicinity of TA, fast dynamics similar to antiferromagnets is achieved, due to the fact that magnetic dynamics of magnets depend closely on their angular momentum. Importantly, magnetic dynamics of ferrimagnets even at TA can be detected and controlled experimentally by conventional methods due to their nonzero magnetization.

Fast domain wall (DW) dynamics in ferrimagnets driven by electric current [4, 23], spin wave [16], magnetic field [17, 21], and magnetic anisotropy gradient [15] have been reported theoretically [15, 24] or experimentally [4, 16, 19]. While these control mechanisms could be used in future experiments and device design, several shortcomings have to be considered. For example, DW motion driven by current induced spin-transfer torques (STT) relies on the conduction electrons [25], which is unfavorable for energy efficiency. Moreover, a spin-polarized current induces Slonczewski-type and field-like STT on local magnetization, while the field-like STT which provides the main force in driving DW motion is much smaller than the Slonczewski-type STT. Thus, a large current density is needed to achieve a useful DW velocity.

Interestingly, earlier work demonstrates that a circularly polarized magnetic field (CPMF) can drive DW motion at a considerable speed even in ferromagnets [26,27,[28]]. Specifically, a CPMF can generate a field-like STT which is much bigger than the Slonczewski-type STT [28], different from the case of spin-polarized current. Thus, more useful STT contributes to the fast dynamics of the DW under the limitation due to the Walker breakdown [28,29,30]. In ferrimagnets, interesting DW dynamics driven by CPMF has been revealed [27]. There is a critical field frequency below/above which the DW velocity linearly increases/decreases with the frequency increasing. At TA, however, the DW cannot propagate due to the zero gyrotropic coupling between the domain-wall position and angle, similar to the case of antiferromagnetic DW.

The earlier work reveals the DW motion driven by the CPMF in ferrimagnets, while the role of Dzyaloshinskii–Moriya interaction (DMI) still deserves to be further clarified from the following viewpoints [31]. On the one hand, DMI generally appears in magnetic systems with spatial inversion asymmetry, whose magnitude can be effectively tuned by various methods [32, 33]. In ferrimagnet GdFeCo, as an example, the bulk DMI has been reported experimentally, which originates from the structural inhomogeneity along the thickness direction [34]. Importantly, DMI plays an essential role in stabilizing topological magnetic structures such as skyrmion and chiral DW which are crucial prospects for future spintronic applications [35].

On the other hand, DMI may result in a higher DW speed and more controllable dynamics [15, 36,37,38,39,40,41]. For example, earlier work demonstrates that additional DMI generates a twisted domain wall and induces a symmetry breaking, which is essential in driving antiferromagnetic DW under a CPMF [26]. Moreover, a unidirectional DW motion driven by CPMF has been revealed in antiferromagnets with DMI [42]. To some extent, notable effect of DMI on the DW dynamics in ferrimagnet driven by CPMF is expected, considering the similarity between FiM and antiferromagnetic systems. Thus, this subject is worth studying for its contribution to future experiments and applications.

In this work, we study the DW dynamics in ferrimagnet with DMI driven by the CPMF. The DW velocity is derived using the collective coordinate theory and checked by the Landau–Lifshitz–Gilbert (LLG) simulations. It is demonstrated that the DMI plays a role in modulating the dynamics similar to the net spin density. Specifically, DMI can change the DW velocity and even the motion direction, which could be used as another internal parameter to control FiM DW. Moreover, the DW dynamics in ferrimagnet is much faster than the ferromagnetic counterpart, further demonstrating the great potential of ferrimagnets in future spintronic applications.

2 Model and methods

We consider a one-dimensional FiM nanowire along the z direction as depicted in Fig. 1, where two inequivalent sublattices have antiferromagnetically coupled magnetic moments μ1S1 and μ2S2, respectively, with the moment magnitude μ1,2 and normalized moment S1,2 [43, 44]. Then, we introduce the staggered vector n = (S1 − S2)/2 and magnetization vector m = (S1 + S2)/2, the gyromagnetic ratio (γ1,2), and the Gilbert damping constant (α1,2). Thus, the spin density of sublattice i is given by si = Mi/γi with γi = giμB/ℏ, where Mi is the sublattice magnetization, gi is the Landé g-factor, ℏ is the Planck constant, and μB is the Bohr magneton.

Fig. 1
figure 1

Schematic depiction of a one-dimensional ferrimagnetic nanowire along z direction with a domain wall, where applied CPMF is also shown

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The dynamics can be described by the following Lagrangian density [16, 21, 43, 45]:

$$L = s{\dot{\mathbf{n}}} \cdot ({\mathbf{n}} \times {\mathbf{m}}) + \delta_{\text{s}} {\mathbf{a}}({\mathbf{n}}) \cdot {\dot{\mathbf{n}}} - U$$
(1)

where s = (s1 + s2)/2 is the staggered spin density, δs = s1 − s2 is the net spin density, a(n) is the vector potential of a magnetic monopole satisfying ∇n × a = n, and ṅ=dn/dt. Without DMI, the potential energy density U is written as

$$U = \frac{{A_{\text{ex}} }}{2}(\nabla {\mathbf{n}})^{2} + \frac{{{\mathbf{m}}^{2} }}{2\chi } - \frac{{K_{z} }}{2}\left( {{\mathbf{n}} \cdot {\hat{\mathbf{z}}}} \right)^{2} - {\mathbf{h}}(t) \cdot {\mathbf{n}}$$
(2)

where the first and second terms are the inhomogeneous and homogeneous exchange energies, respectively, Aex is the exchange stiffness, and χ is the magnetic susceptibility. The third term is the easy-axis anisotropy along the z axis with the anisotropy constant (Kz). The last term is the Zeeman coupling with the CPMF h(t) = Mnetμ0h0(cosωt, sinωt, 0) with the net magnetization (Mnet=M1 − M2), the vacuum permeability (μ0), the magnitude (h0) and frequency (ω). The dynamic variable m can be expressed by m = sχ ṅ × n. Substituting m in Eq. (1), we obtain:

$$L = \frac{\rho }{2}{\dot{\mathbf{n}}}^{2} + \delta_{{\text{s}}} {\mathbf{a}}({\mathbf{n}}) \cdot {\dot{\mathbf{n}}} - \frac{{A_{{\text{ex}}} }}{2}(\nabla {\mathbf{n}})^{2} + \frac{{K_{z} }}{2}\left( {{\mathbf{n}} \cdot {\hat{\mathbf{z}}}} \right)^{2} + {\mathbf{h}}(t) \cdot {\mathbf{n}}$$
(3)

where ρ = s2χ denotes the inertia of dynamics. In the Lagrangian formalism, the Rayleigh function R = sαṅ2/2 with sα = α1s1 + α2s2 is introduced to describe the dissipative dynamics. For simplicity, the Gilbert damping constants of two sublattices are assumed to be α1 = α2 = α. The low-energy dynamics of the DW can be captured by its collective coordinates [44, 46, 47]: the position (q(t)) and the azimuthal angle (ϕ(t)) with time (t). The Walker ansatz for the DW is given by n(z, t) = (sech((z − q)/λ)cosϕ, sech((z − q)/λ)sinϕ, tanh((z − q)/λ)), where λ is the DW width. After applying the Euler–Lagrange equation in conjunction with the Rayleigh dissipation function, the associated equations of motion for q and ϕ are given by:

$$M\ddot{q} + G\dot{\phi } + M\dot{q}/\tau = 0$$
(4a)
$$I\ddot{\phi } - G\dot{q} + I\dot{\phi }/\tau = F$$
(4b)

where M = 2ρΑ/λ is the mass with the cross-sectional area (Α), G = 2δsΑ is the gyrotropic coefficient, I = 2ρΑλ is the moment of inertia, τ = ρ/sα is the relaxation time, and F = ΑπMneth0λsinωt is the force induced by the CPMF. It is noted that the motion equations are in consistent with those derived from the LLG equation [28]. From Eq. (4), a critical frequency ωc = π|Mnet|h0/2(sα + δs2/sα) is obtained, which differentiates ω into two regimes: the perfect synchronization regime for ω < ωc and oscillating motion regime for ω > ωc.

In the perfect synchronization regime with ∂ϕ/∂t = ω, − η = ωt − ϕ increases from 0 to π/2 as ω increases [28]. Thus, the damping torque exerted on the DW ~ S × (S × H) which depends on the η value also increases, resulting in the linear increase of the DW velocity with ω. In the oscillating motion regime for ω > ωc, the DW spins cannot catch up with the CPMF, and ∂η/∂t = (ω2 − ωc2 )1/2 is obtained. Thus, the DW velocity can be described as:

$$\nu_{1} = \dot{q} = - \frac{{\delta_{{\text{s}}} \lambda \omega }}{{s_{\alpha } }}, \;{\text{for}}\;\omega < \omega_{\text{c}}$$
(5a)
$$\it \nu_{2} = - \frac{{\delta_{{\text{s}}} \lambda }}{{s_{{\upalpha }} }}\left( {\omega - \sqrt {\omega^{2} - \omega_{\text{c}}^{2} } } \right),\;{\text{for}}\,\omega > \omega_{{\text{c}}}$$
(5b)

Then, we consider the role of DMI in modulating the DW velocity. In the presence of DMI, the DW ansatz becomes [26]:

$$n_{x} = {\text{sech}} \left[ {(z - q)/\lambda^{\prime } } \right]\cos \left[ {(z - q)\xi + \phi } \right]$$
(6a)
$$n_{y} = {\text{sech}} \left[ {(z - q)/\lambda^{\prime } } \right]\sin \left[ {(z - q)\xi + \phi } \right]$$
(6b)
$$n_{z} = \tanh \left[ {(z - q)/\lambda^{\prime } } \right]$$
(6c)

where ξ =  − Dz/Aex with the bulk DMI magnitude (Dz), and the DW width is changed to λ′ = (Aex /(2Kz − ξ2Aex))1/2. In this case, the DMI breaks the rotation symmetry of the DW and leads to a twisted wall. Meanwhile, δs naturally breaks the symmetry of the two sublattices and generates the gyrotropic coupling between the DW position and angle. To describe the effect of the DMI on the FiM dynamics, we introduce an additional term similar to that in antiferromagnets [26] and rewrite the DW velocity as:

$$\nu_{1}^{\prime } = \left( { - \frac{{\delta_{{\text{s}}} \lambda }}{{s_{{\upalpha }} }} + \frac{{\xi \lambda^{\prime 2} }}{{1 + (\xi \lambda^{\prime } )^{2} }}} \right)\omega, \;{\text{for}}\;\omega < \omega_{\text{c}}$$
(7a)
$$\nu_{2}^{\prime } = \left( { - \frac{{\delta_{{\text{s}}} \lambda }}{{s_{{\upalpha }} }} + \frac{{\xi \lambda^{\prime 2} }}{{1 + (\xi \lambda^{\prime } )^{2} }}} \right)\left( {\omega - \sqrt {\omega^{2} - \omega_{\text{c}}^{2} } } \right),\;{\text{for}}\;\omega > \omega_{c}$$
(7b)

This equation shows that the domain-wall motion is attributed to the symmetry breaking induced by the net spin density and DMI. Moreover, the period of the DW rotation is the same as that of the CPMF for ω < ωc, which can hardly be affected by the net spin density and DMI.

To verify our theoretical derivation, we perform LLG simulations for typical rare-earth (RE) transition-metal (TM) ferrimagnets. The DW is placed initially at the center of the nanowire as depicted in Fig. 1. The corresponding model Hamiltonian is given by [48]:

$$\begin{aligned} H = & J\sum\limits_{i} {{\mathbf{S}}_{i} \cdot {\mathbf{S}}_{i + 1} } - \sum\limits_{i} {K(S_{i}^{z} )^{2} } \\ & - \sum\limits_{i} {{\mathbf{D}}_{i} \cdot ({\mathbf{S}}_{i} \times {\mathbf{S}}_{i + 1} )} - g_{i} \mu_{\text{B}} \mu_{0} \sum\limits_{i} {{\mathbf{H}}(t) \cdot {\mathbf{S}}_{i} } \\ \end{aligned}$$
(8)

where an odd index i represents a site for TM and an even i represents a site for RE. The exchange coupling J = Aexa/4, and the anisotropy constant K = Kza3/2. Di = D0\(\hat{\text{z}}\) is the uniform bulk DMI vector, and H(t) = h0(cosωt, sinωt, 0) is the CPMF. The atomistic LLG equation is given by [17, 19]:

$$\frac{{\partial {\mathbf{S}}_{i} }}{\partial t} = - \frac{{\gamma_{i} }}{{1 + \alpha^{2} }}{\mathbf{S}}_{i} \times ({\mathbf{H}}_{{{\text{eff}},i}} + \alpha {\mathbf{S}}_{i} \times {\mathbf{H}}_{{{\text{eff}},i}} )$$
(9)

where Heff,i =  − (1/μi)∂H/∂Si is the effective field at site i with the magnetic moment (μi). We use the following simulation parameters: J = 7.5 meV, K = 0.01J, μBμ0h0 = 0.005J, the lattice size a = 0.4 nm, the damping constant αRE = αTM = 0.01, and the gyromagnetic ratio γRE = 1.76 × 1011 rad·s−1·T−1 and γTM = 1.936 × 1011 rad·s−1·T−1 (the Landé g-factors gRE = 2 and gTM = 2.2 [21, 25]). Moreover, the magnetic moments MRE and MTM are listed in Table 1.

Table 1 Used magnetic transition-metal moments (MTM), rare-earth moment (MRE), and net spin density (δs) in simulations. Parameter 5 coincides with angular momentum compensation point TA, where net spin density vanishes (δs = 0)
Full size table

3 Results and discussion

First, we study the dynamics of the ferrimagnetic DW in the absence of DMI to give a direct comparison. Figure 2a shows the theoretically calculated (solid lines) and numerically simulated (symbols) DW velocities as functions of ω for various δs. At TA for δs = 0, the DW can hardly be driven by the CPMF. For a finite δs, v linearly increases with ω for ω < ωc, while v decreases as ω further increases from ωc. It is worth noting that a similar behavior of the transition from phase locking (ω < ωc) to phase unlocking (ω > ωc) is observed in the ferromagnetic DW motion driven by the AC force [49]. Thus, there is a maximum velocity vc at ωc, which is estimated to be:

$$\nu_{\text{c}} = - \frac{{\delta_{\text{s}} \lambda \omega_{\text{c}} }}{{s_{{\upalpha }} }} = - \frac{\uppi }{2}\frac{{\left| {M_{\text{net}} } \right|h_{0} \lambda \delta_{\text{s}} }}{{\delta_{\text{s}}^{2} + s_{{\upalpha }}^{2} }}$$
(10)

where ωc shifts toward the low ω side with the increase of |δs|. The simulated data coincide well with the calculations for ω < ωc and slightly deviate from the calculations for ω > ωc. As discussed earlier, an oscillating motion is driven for ω > ωc and a finite acceleration is induced. However, the acceleration is simply neglected in deriving the velocity, resulting in the deviation between simulations and theory.

Fig. 2
figure 2

Simulated (symbols) and calculated (solid lines) velocities as functions of a ω for various δs and b vc at critical frequency (ωc) as a function of δs for various α

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To some extent, the ferrimagnetic DW can be regarded as a combination of a head-to-head DW and a tail-to-tail DW on which the driving torques compete with each other. For δs = 0, the driven torques are well cancelled, and the DW hardly be driven. For a finite δs, one of the driven torque conquers the other one, resulting in the DW motion. The sign of the net torque depends on the sign of δs, and the DW motion is reversed when the δs sign is altered. As a result, a positive/negative δs leads to a negative/positive velocity, as shown in Fig. 2a.

In addition, the dependence of vc on δs is noticed in Eq. (10), which provides useful information in choosing potential materials for application. Figure 2b gives vc as a function of δs for various damping constants (α). It is clearly shown that the velocity quickly increases to a peak with the |δs| increasing, and then decreases gradually to the ferromagnetic limitation. The peak position is estimated to be |δs| ≈ sα which shifts toward low |δs| side as α decreases. More importantly, the peak value ~ Mneth0λπ/4sα is one order magnitude larger than that in ferromagnetic counterpart ~ h0λ/(1 + α2), demonstrating again the ultrafast dynamics of ferrimagnets. Taking GdFeCo as an example, the parameters are MTM = 440 kA·m−1, MRE = 400 kA·m−1, Aex = 50 pJ·m−1, Kz = 0.5 MJ·m−3, and αTM = αRE = 0.01, which gives the velocity ~ 350 m·s−1 at the critical frequency of 6 GHz for h0 = 50 mT.

Subsequently, we investigate the roles of various internal parameters in modulating the DW dynamics. For example, the DW width λ ~ a(J/2K)1/2 depends on the anisotropy and determines the DW velocity. Moreover, the DW energy which is estimated to be ~ 2(2JK)1/2 also affects the DW dynamics, and a higher DW energy decreases the mobility of the DW. Thus, a larger K leads to a lower velocity, as confirmed in Fig. 3a which presents the velocity as a function of K for various δs. For a fixed δs, the velocity decreases with the increase of K. Furthermore, the DW mobility also depends on the damping constant (α), and an enhanced damping reduces the wall mobility. Figure 3b shows the velocity as a function of α for various δs, which demonstrates the linearly increase of velocity with 1/α.

Fig. 3
figure 3

Simulated (empty symbols) and calculated (solid lines) DW velocities as functions of a anisotropy constant (K) and b 1/α for various δs

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Next, we investigate the influence of DMI on the DW dynamics, noting that the DMI can be modulated through tuning the thickness of the ferrimagnetic layer [34] or atomic-scale modulation of interfaces [50] in experiments. Without DMI, the DW at TA can hardly be driven by the CPMF because of the rotation symmetry. The DMI breaks the symmetry and leads to a twisted DW which can be efficiently driven by the CPMF, as shown in Fig. 4a which gives the calculated and simulated DW velocities as functions of ω for various Dz for δs = 0. The theoretical calculations based on Eq. (7) agree with the simulations, confirming the validity of the theoretical analysis. With the increase in Dz, the velocity is significantly increased for a fixed ω, and the critical frequency ωc shifts toward the low ω side. To some extent, the DMI plays a role in modulating the DW dynamics similar to that of δs, which is attributed to the symmetry breaking. As a result, the DMI competes with or cooperates with δs in modulating the DW mobility, which depends on the signs of δs and Dz.

Fig. 4
figure 4

Simulated (symbols) and calculated (solid lines) velocities as functions of ω a for various Dz for δs = 0, and b for various δs with Dz = 0.02J

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In Fig. 4b, we present the DW velocity as a function of ω for various δs for Dz = 0.02J It is clearly shown that for a negative δs, the velocity is decreased due to the DMI, demonstrating the competition between δs and Dz. Reversely, for a positive δs, the DW motion is enhanced by the DMI. As a matter of fact, Eq. (7) demonstrates that a DMI with a same/opposite sign as/to that of δs speed up/down the DW motion, consistent with the simulations. In other words, DMI breaks the symmetry of the DW, which modulates the coupling between the DW tilt angle (ϕ) and the DW position (q). Thus, the DW dynamics in ferrimagnets notably depends on the DMI, as revealed in this work.

Recently, various methods in modulating DMI have been revealed in experiments [32, 33, 51,52,53,54]. Thus, DMI could be an efficient parameter in modulating DW dynamics in ferrimagnets driven by CPMF, especially in the synchronization regime for ω < ωc. In the synchronization regime, the DW velocity has a linear relation with δs for a fixed Dz, as shown in Fig. 5a which presents the DW velocity as a function of δs for various Dz for ω = 0.02 γJ/μs. It is clearly shown that the DMI not only modulates the DW speed, but also can change the motion direction. In addition, the velocity linearly depends on Dz, which is demonstrated in Eq. (7) and confirmed in our simulations. In Fig. 5b, a linear dependence of velocity on Dz for every studied δs is demonstrated. This behavior indicates again that Dz and δs play similar roles in modulating the DW dynamics in ferrimagnets driven by the CPMF.

Fig. 5
figure 5

Simulated (symbols) and calculated (solid lines) velocities as functions of a δs for various Dz, and b Dz for various δs for ω = 0.02 γJ/μs

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4 Conclusion

In summary, we have studied theoretically and numerically the domain wall dynamics in ferrimagnet with additional DMI driven by a circularly polarized magnetic field. Similar to the net spin density (δs), DMI is revealed to efficiently modulate the DW motion, which is related to the symmetry breaking. It is demonstrated that the two factors compete with or cooperate with each other in modulating the DW dynamics, which depends on their signs. In addition, it is demonstrated again that the domain wall dynamics in ferrimagnets is much faster than that in ferromagnets, confirming again the great potential of ferrimagnets in future applications. Thus, this work unveils another parameter of controlling the DW dynamics driven by a rotating field, benefiting future experiment designs and spintronic applications.