A magnetic domain wall (DW) is the transition region that separates two uniformly magnetized domains [1]. These magnetic configurations are interesting due to fundamental physics, but also due to potential technological applications [2, 3]. In fact, during the last decades DWs have been at the core of theoretical and experimental studies which have provided with a deep understanding of different spin-orbit coupling phenomena [4–8]. For instance, straight stacks where an ultra-thin ferromagnetic (FM) layer is sandwiched between a heavy metal (HM) and an oxide (Ox), present perpendicular magnetic anisotropy (PMA) and therefore, the domains are magnetized along the out-of-plane direction of the stacks: up ( + u ⃗ z ) or down ( − u ⃗ z ) . DWs in these HM/FM/Ox stacks adopt an homochiral configuration due to the Dzyaloshinskii-Moriya interaction (DMI) [5, 7, 9]. Adjacent DWs have internal magnetic moments along the longitudinal direction ( m ⃗ D W = ± u ⃗ x ) , and the sense is imposed by the sign of the DMI, which in turns depends on the HM [5]. For left-handed stacks such as Pt/Co/AlO, up-down (UD) and down-up (DU) DWs have internal moments with m ⃗ D W = − u ⃗ x and m ⃗ D W = + u ⃗ x , respectively [5]. These DWs are driven with high efficiency by injecting electrical currents along the longitudinal direction of the HM/FM/Ox stack [4]. Due to the spin-Hall effect [5], the electrical current in the HM generates a spin polarized current which exerts spin-orbit torques (SOTs) on the magnetization of the FM layer, and drives series of homochiral DWs which are displaced along the longitudinal direction (x-axis). DW velocities of V _DW ∼ 500 m/s have been reported upon injection of current densities of J _HM ∼ 1 TA/m² in Pt/Co/AlO [4]. Consequently, these stacks have been proposed to develop highly-packed magnetic recording devices, where the information coded in the domains between DWs can be efficiently driven by pure electrical means. Both UD and DU DWs move with the same velocity along straight stacks, but some implementations of these memory or logic devices would require to design 2D circuits, where straight parts of HM/FM/Ox stack are connected each other with curved or semi-rings sections. However, recent experimental observations [10] and theoretical studies [11] have pointed out that adjacent UD and DU DWs move with different velocity along curved HM/FM/Ox stacks, which is detrimental for applications because the size of the domain between adjacent DWs changes during the motion, with the perturbation of the information coded therein. Therefore, other systems must be proposed in order to design reliable 2D circuits for DW-based memory and logic devices.

Other stacks with materials and/or layers with antiferromagnetic coupling, such as synthetic antiferromagnets (SAF) and ferrimagnetic (FiM), have proven to outperform FM in terms of current-driven DW dynamics [11–15]. Ultrafast magnetization dynamics in the THz regime, marginal stray field effects and insensitivity to external magnetic fields are other significant advantages of materials with antiferromagnetic coupling with respect to their FM counterparts. As conventional antiferromagnets (AFs), FiM alloys are also constituted by two specimens, typically a rare earth (RE) and transition metal (TM), that form two ferromagnetic sublattices antiferromagnetically coupled to each other. GdFeCo, GdFe, or TbCo are archetypal FiM alloys, with the RE being Gd or Tb and the TM being FeCo or Co. In contrast to AFs with zero net magnetization, the magnetic properties of FiMs, such as magnetization and coercivity, are largely influenced by the relative RE and TM composition (or equivalently, temperature). This fact offers additional degrees of freedom to control the current-driven DW velocity. The spontaneous magnetization of each sublattice M _S , _i can be tuned by changing the composition of the FiM and/or the temperature of the ambient (T) [13, 14]. For a given composition of the FiM (RE _x TM_1−x ), there are two relevant temperatures below the Curie threshold. One is the magnetization compensation temperature (T _M ) at which the saturation magnetization of the two sublattices are equal ( M s 1 T M = M s 2 ( T M ) ) , so the FiM behaves as a perfect antiferromagnetic material, with zero net magnetization and diverging coercive field. The other is the temperature at which the angular momentum compensates, T _A , at which M s 1 T A / γ 1 = M s 2 ( T A ) / γ 2 , where γ _i is the gyromagnetic ratio of each sublattice (i:1,2 for 1:TM and 2:RE). As the gyromagnetic ratio depends on the Landé factors (g _i ) which are different for each sublattice, the angular compensation temperature T _A is in general different from the magnetization compensation temperature (T _M ). Consequently, the FiM have a net magnetization at T _A , so conventional techniques used for FMs can be also adopted to detect the magnetic state of FiM samples [16]. Moreover, recent experimental observations have evidenced that the current-driven DW velocity along straight HM/FiM stacks can be significantly optimized at the angular momentum compensation temperature (T = T _A ), with velocities reaching V _DW ∼ 2000 m/s for typical injected density current of J _HM ∼ 1 TA/m² along the HM underneath [14]. The DW velocity drops either below (T < T _A ) and above (T > T _A ) angular momentum compensation. Note that alternatively to tuning the temperature for a fixed composition x of the FiM alloy RE _x TM_1−x , even working at room temperature (T = 300 K) the DW velocity peaks at a given composition where angular momentum compensates [13]. Therefore, both studies, either fixing the composition (x) and changing temperature of the ambient (T), or fixing the ambient temperature and modifying the FiM composition are equivalent for our purposes of DW dynamic. Although the current-driven DW motion (CDDWM) along HM/FiM stacks suggests their potential for memory and logic applications, previous studies have been mainly focused on straight FiM strips [13–15]. The further develop of novel DW-based devices also requires to analyze the dynamics of DWs along HM/FiM with curved parts which would connect straight paths to design any 2D circuit. Such investigation of the dynamics along curved is still missing, and it is the aim of the present study.

Here we theoretically explore the CDDWM along curved HM/FiM stacks by means of micromagnetic (μm) simulations. Our modeling allows us to account for the magnetization dynamics in the two sublattices independently. We explore the CDDWM below, at and above the angular momentum compensation (AMC) for different curved samples, with different widths and curvatures, and considering the realistic spatial distribution of the injected current along the HM. In particular, we will infer and isolate the relevance of different aspects governing such dynamics, as the role of the non-uniform current and other purely geometrical aspects of the curved shape. This work completes previous studies on straight samples [11, 13–15], and will be practical for designing more compact and efficient DW-based devices. The rest of the paper is organized as follows. In Section 2 we describe the numerical details of the micromagnetic model along with the material parameters and the geometrical details of the evaluated samples. Section 3 presents the micromagnetic results of the CDDWM in different scenarios. Firstly, exploring the role of the FiM sample width (w) for a fixed the curvature (ρ, given by the inverse of the average radius, ρ = 1/r _e ), and secondly fixing the width and varying the curvature. After that, we present results which allow us to infer the role of non-uniform current and geometrical aspect (w, ρ) comparing curved and straight samples. The main conclusions are summarized in Section 4.

CDDWM is numerically studied here along curved HM/FiM stacks as schematically shown in Figure 1, where r _i and r _o are the inner and outer radius r _o respectively, and r _e = (r _o + r _i )/2 is the mean effective radius. w and t _FiM are the width and the thickness of the FiM respectively. The relaxed magnetization configuration of the sublattice i = 1, shown in Figure 1 (opposite configuration in sublattice i = 2), serves as the initial state to study the CDDWM upon of current injection along the HM underneath. The temporal evolution of the magnetization of each sublattice is given by the Landau-Lifshitz-Gilbert equation (LLG) [17], d m ⃗ i ( t ) d t = − γ 0 , i m ⃗ i ( t ) × H ⃗ e ff , i + α i m ⃗ i × d m ⃗ i ( t ) d t + τ ⃗ S O T , i , (1) where here the sub-index i stands for i: 1 and 2 sublattices respectively. γ _i = g _i μ _B /ℏ and α _i are the gyromagnetic ratios and the Gilbert damping constants, respectively. g _i is the Landé factor of each layer, and m ⃗ i r ⃗ , t = M ⃗ i / M s , i is the normalized local magnetization to its saturation value (M _s , _i ), defined differently for each sublattice: M _s , _i (i: 1, 2). In our micromagnetic model the FiM strip is formed by computational elementary cells, and within each cell we have two magnetic moments, one for each component of the FiM. The respective effective field ( H ⃗ e ff , i ) acts on the local magnetization of each sublattice ( m ⃗ i r ⃗ , t ) , and it is the sum of the magnetostatic, the anisotropy (PMA), the DMI and the exchange fields [11, 15]. The magnetostatic field on each local moment in the sublattice is numerically computed from the average magnetization of each elementary cell using similar numerical techniques as for the single FM case (see [11, 15]). We checked that the demagnetising field has a marginal influence in the simulation results compared to other contributions to the effective field. For the PMA field, the easy axis is along the out-of-plane direction (z-axis), and the anisotropy constants for each sublattice are K _u , _i (PMA constant). D _i is the DMI parameter for each sublattice i: 1, 2 [11, 15]. The exchange field of each sublattice includes the interaction with itself (intra-lattice exchange interaction, H ⃗ e x c h , i ) and with the other sublattice (inter-lattice exchange interaction, H ⃗ e x c h , 1 2 ). The inter-lattice exchange effective field is computed as for a single FM sample, H e x c h , i = 2 A i μ 0 M s , i ∇ 2 m ⃗ i , where A _i is the intralattice exchange parameter. The inter-lattice exchange contribution H ⃗ e x c h , 1 2 to the effective field H ⃗ e ff , i , acting on each sublattice is computed from the corresponding energy density, ω e x c h , i = − B i j m ⃗ i ⋅ m ⃗ j , where B _ij [in (J m⁻³)] is a parameter describing the inter-lattice exchange coupling between sublattices (here, we used the notation i: 1 and j: 2).

In Eq. 1, τ ⃗ S O T , i are the SOTs acting on each sublattice, which are related to the electrical current along the HM ( J ⃗ H M ) . Based on preliminary studies [18], here we assume that τ ⃗ S O T , i is dominated by the spin Hall effect (SHE), so τ ⃗ S O T , i = − γ 0 H S L m ⃗ i × m ⃗ i × σ ⃗ where H S L = ℏ θ S H , i J H M 2 e μ 0 M s t F i M [19]. ℏ is the Planck constant, and θ _SH , _i is the spin Hall angle, which determines the ratio between the electric current and the spin current (J _s = θ _SH J _HM ) for each sublattice. σ ⃗ = u ⃗ J × u ⃗ z is the unit vector along the polarization direction of the spin current generated by the SHE in the HM, being orthogonal to both the direction of the electric current u ⃗ J and the vector u ⃗ z standing for the normal to the HM/FiM interface. For a longitudinal current ( u ⃗ J = u ⃗ x ) , the spin current is polarized along the transverse direction, σ ⃗ = − u ⃗ y . For curved samples where the current density J ⃗ H M = J H M r u ⃗ J has azimuthal direction ( u ⃗ J = − u ⃗ ϕ ) , the direction of the polarization is radial, σ ⃗ = u ⃗ J × u ⃗ z = u ⃗ r , as shown in Figure 1. A potential difference is applied between the ends of the curved track to inject current in the right circulation. Therefore, a gap of 25 nm is also modelled, leading to a split ring shape for the strip (see inset in Figure 1). The spatial distribution of current as a function of the radial coordinate (r _i < r < r _o ) is taken from [11, 20], and it depends on the width (w) and the radial distance (r) as J H M r = J 0 w / ( r ⁡ log 1 + w / r i ) , where J ₀ is the nominal, uniform current density, in an equivalent straight strip of same cross-section (w × t _HM , where t _HM is the thickness of the HM strip).

In order to illustrate the current-driven DW dynamics along curved HM/FiM stacks we fix t _FiM = 6 nm, and samples with different widths (w) and radii (r _e ) were evaluated. The following common material parameters were adopted for the two sublattices i: 1, 2: A _i = 70 pJ/m, K _u , _i = 1.4 × 10⁶ J/m³, α _i = 0.02, D _i = 0.12 J/m², θ _SH , _i = 0.155. The strength of the antiferromagnetic coupling between the sublattices was fixed to B _ij ≡ B ₁₂ = − 0.9 × 10⁷ J/m³. The gyromagnetic ratios (γ _i = g _i μ _B /ℏ) are different due to the different Landé factor: g ₁ = 2.05 and g ₂ = 2.0. The saturation magnetization of each sublattice M _s , _i can be tuned with the composition of the FiM and/or with the temperature of the ambient (T). Here, we assume the following temperature dependences for each sublattice: M s , i T = M s , i ( 0 ) 1 − T T C a i , where T _C = 450 K is the Curie temperature of the FiM, M s , 1 0 = 1.4 × 10 6 A/m and M s , 2 0 = 1.71 × 10 6 A/m are the saturation magnetization at zero temperature, and a ₁ = 0.5 and a ₂ = 0.76 are the exponents describing the temperature dependence of the saturation magnetization of each sublattice. The temperature at which the net saturation magnetization vanishes [M _s ,₁(T _M ) = M _s ,₂(T _M )] is T _M = 241.5 K, and the angular momentum compensation temperature corresponding to M _s ,₁(T _A )/g ₁ = M _s ,₂(T _A )/g ₂, is T _A = 260 K. We evaluate the CDDWM below, at and above the angular momentum compensation adopting three representative temperatures: T = 220 K < T A , T = 260 K = T _A and T = 300 K > T A . Samples were discretized using a 2D finite difference scheme using computational cells with Δx = Δy = 0.2 nm and Δz = t _FiM . Several tests were carried to certify that the presented results are free of discretization errors.

Due to the several combination of parameters to consider in our study, we divided this section in three sub-sections: (A) The study on the influence of the strip width (w); (B) the study on curvature ( ρ = r e − 1 ) ; and (C) same studies for a straight strip with identical material parameters, to explore by comparison the effects of curvature on the DW dynamics. In parts (A) and (B), scenarios for three different temperatures, T ₁ = 220 K, T ₂ = 260 K, T ₃ = 300 K are considered, to study the DW motion below the AMC (T ₁), at the AMC (T ₂) and above the AMC (T ₃). We also define and refer to T ₃ = 300 K as for “room temperature” in our study. Note that a change in temperature only affects M _S in our model, therefore it has equivalent effects to changing material composition [13]. In addition to DW velocity, we also characterize the inertial motion of the DW as a function of current density. As an example, Figure 2 shows typical results of the DW position and its velocity in a ring-like strip (w = 256 nm and r _e = 384 nm) under a density current J _HM = 2 TA/m² and at T = 260 K. Qualitatively similar results are obtained at T = 220 K and T = 300 K (not shown). Insets show the (clockwise) DW displacement as a function of time for one sublattice (i = 1).

We have provided a study on DW motion in curved FiM strips, particularised to one of the two strongly coupled sub-lattices, for three different temperatures, and as a function of geometrical parameters for a HM/FiM/Ox multilayer structure. We observe an absence of tilting of the DW and domain distortion at different temperatures, 40 K above and below the angular momentum compensation temperature.

Width and curvature effects on the DW velocity are discussed. Besides contributions from a non-uniform J(r), there is an overall significant influence from the shape of the strip itself on DW velocity. This implies that, for a fixed temperature (or composition), DW velocity can be optimised by optimising the geometrical parameters of the curved strip. The relative differences between a DU and a UD walls are marginal in general. In other words, geometrical factors affect them almost equally, which is positive for a robust transmission of a bit encoded in an Up or Down domain between two adjacent DWs. With reducing current, differences in velocities between curved and straight strips are still minimised at the expense of slower DWs. Similar effect is observed as width or curvature is increased. This is beneficial for designing intricate 2D circuit tracks combining curved and straight sections, while preserving DW velocities still larger than those found in their FM counterparts. Also, DWs in a curved FiM strip show a negligible inertia in contrast to their FM counterparts (τ _FiM ≪ τ _FM ) for all the explored scenarios. The DWs start to move and stop almost immediately (τ _FiM ∼ 0.02 ns) after the application or removal of current.

Considering the obtained results altogether and assuming T ≠ T _A , which will be most of the experimental cases at room temperature (T = 300 K), our study allows us to conclude that narrow enough FiM strips are ideal candidates for designing curved tracks for 2D spintronic circuits of an arbitrary shape based on CDDWM, where bits are encoded in domains separated by walls. This is due to very fast rise and fall times (τ _r ∼ τ _f ≪ 0.1 ns), high velocities (V _DW > 1000 m/s) and negligible distortion of the two types of DWs (UD and DU) in all the scenarios explored in this work. Greater DW terminal velocities and smaller time responses in curved FiM strips than those in their FM counterparts are obtained. These results can help in the further research, development and improvement of FiM-based spintronic circuitry that may require compactness and high-speed functionality with high robustness to DW (and/or domain) distortion.