Ferrimagnetic materials (FiM) represent a promising direction for the realization of spin-based devices since they can combine the THz ultrafast dynamics typical of antiferromagnets (AFMs) with an easier way to control the magnetic state via well-established optical and electrical methods already applied to FMs, such as the magneto-optical Kerr effect (MOKE)1 or anomalous Hall effect (AHE)2. FiMs can be modeled as AFMs by considering two antiferromagnetically-coupled sublattices3-5 but, unlike AFMs, the two magnetic sublattices can have a different magnetic moment and angular momentum. Consequently, the net magnetization and angular momentum of FiMs can be varied by changing either the FiM’s chemical composition6 or temperature7 leading to two different compensation points: the magnetic compensation point (MCP), i.e. a zero net magnetization state, and an angular momentum compensation point (AMCP), i.e., a vanishing net angular momentum compensation state. In this work, we micromagnetically analyze, by means of Petaspin micromagnetic solver5, the magnetization dynamics of a current-driving transition metal/rare earth ferrimagnet in a spin Hall geometry as a function of the uncompensation parameter of the angular moments of the two sublattices8. This approach allows us to show the crucial effect of non-uniform dynamics in the system which cannot be caught by the macrospin approximation. Particularly, we have reported the switching time as a function of the uncompensation parameter (FIG.1) for a current |J| = 1.3 TA/m2. We have shown the switching time diverges at the AMCP, where the only dynamical solution is a self-oscillation state. More interestingly, a finite discontinuity of the switching time appears near the MCP. We show that this discontinuity is related to a change of the switching mechanism, from uniform to non-uniform dynamics with a corresponding increase of the switching time in the latter case. We have additionally analyzed the effect of the interfacial Dzyaloshinskii-Moriya interaction (IDMI) on the switching time and oscillations amplitude. These are relevant characteristics for the design of magnetic memories (switching time) and spin torque nano-oscillators (oscillation amplitude) since they will determine their performances. In FIG.1, we can see the effect on the switching time of the IDMI and how it is beneficial for the use of FiMs as storage devices since it promotes shorter switching times as it advantageously assists the nucleation of new domains. In this work, we also study the self-oscillation frequency and amplitude as a function of the current density. We show that the current thresholds between the changes of dynamical state are well described by the analytical macrospin formula8. We observe a linear dependence of the frequency on current independently of the IDMI and compensation point in agreement with the analytical formula8. For example, by considering the MCP case (FIG.2), the amplitude of the oscillations decreases non-monotonically as the current increases. For the self-oscillation case, we observe that the effect of IDMI is detrimental for the applicability of FiM as THz sources because it induces a significant reduction of the self-oscillations amplitude at low currents (see Fig. 2). Besides, as we can see in Fig.2, the peak-to-peak amplitude of the oscillations non-monotonically decreases as the current increases. This is due to the non-uniform oscillation mediated by domain walls. For low currents, the domain wall periodicity is larger than the device size and, hence, the whole device oscillates with roughly the same phase As we increase the current, the periodicity decreases thus averaging the out-of-plane component of the Néel vector over the sample size reduces the value of the peak-to-peak amplitude. Consequently, we obtain the minimum amplitude when the periodicity is close to the device size. By further increasing the current, the periodicity becomes smaller and the average magnetization value increases Our results give a better insight into the dynamics excited in FiMs thus paving the way for the design of more performant ferrimagnetic memory and THz nano-oscillator devices.

**References:**

[1] K. Fleischer, N. Thiyagarajah, Y.-C. Lau, et. al., Phys. Rev. B, Vol. 98, 134445 (2018). [2] K. Ueda, M. Mann, C.-F. Pai, A.-J. Tan, and G.S.D. Beach, Appl. Phys. Lett., Vol. 109, 232403 (2016). [3] V. Puliafito, R. Khymyn, M. Carpentieri, et. al., Phys. Rev. B, Vol. 99, 024405 (2019). [4] L. Sánchez-Tejerina, V. Puliafito, P. Khalili Amiri, et. al., Phys. Rev. B, Vol. 101, 14433 (2020). [5] L. Sánchez-Tejerina, R. Tomasello, V. Puliafito, et. al., IEEE Magn. Lett. , Vol. 11, 1 (2020). [6] S. Woo, K.M. Song, X. Zhang, et. al., Nat. Commun. , Vol. 9, 959 (2018). [7] L. Caretta, M. Mann, F. Büttner, et. al., Nat. Nanotechnol. , Vol. 13, 1154 (2018). [8] I. Lisenkov, R. Khymyn, J. Akerman, et. al., Phys. Rev. B, Vol. 100, 100409 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e1f65e0922be4090a862a18c147e0267.jpg)
Fig. 1 Switching time as a function of the effective angular momentum uncompensation factor (top axis) of the two sublattices for a current density of |J| = 1.3 TA/m2. The black line corresponds to a dynamics without DMI, while the red line corresponds to a system with non-vanishing DMI. Orange line stands for simulations without demagnetizing field neither DMI. In blue the values of switching time are highlighted for MCP, in magenta there is the line that refers to AMCP.

![](https://assets.underline.io/uploads/markdown_image/1/image/8bed697f0af4114545008dbfb8097492.jpg)
Fig. 2 Amplitude of self-oscillations at the MCP vs applied current for vanishing and non-vanishing IDMI. The snapshots represent the spatial distribution of the 1st-sublattice magnetization at different current densities as indicated by the arrows and arbitrary times. The colors refer to the z-component of the magnetization.

 Micromagnetic understanding of switching and self-oscillations in ferrimagnetic materials

Antiferromagnets (AFMs) exhibit features, such as low magnetic susceptibility, robustness against external fields and lack of stray fields, that are favorable for the building blocks of spintronic devices [1]. This opens the way for a number of potential applications including storage with picosecond switching [2], THz oscillators [3], racetrack memory based on magnetic solitons such as domain walls (DWs) [4] or skyrmions [5]. AFM materials can be characterized by an easy-plane anisotropy which promotes the formation of vortices as well as spiral phases [6]. Vortices have been discussed theoretically in infinite films [7] and observed experimentally by imprinting techniques [8], but they have received less attention than other magnetic solitons. On the other hand, two spiral phases can be obtained statically in an easy-plane AFM with Dzyaloshinksii-Moriya interaction (DMI). For weak DMI constant, λ &lt; λNF, the Néel state is the ground state. For λ = λNF, an intermediate phase is obtained in the form of a nonflat spiral. This spiral presents a continuous rotation of the Néel vector n in the x-z plane, while its y-component oscillates around a nonzero value. At λ &gt; λF, a flat spiral is obtained with n fully rotating on the x-z plane. In this work, we theoretically study vortices in easy- plane AFMs with an interfacial DMI. We consider a stripe geometry, infinite long in the x-direction and confined in the y-direction, as this is the most suitable for applications involving shifting of magnetic information, while it will also give rise to interesting effects on the magnetic structure. We firstly calculate the magnetic ground state, which we find to be uniform (y-direction) and perpendicular to the boundaries. Then we numerically study the stabilization of a single vortex. We observe that this vortex is Néel close to its core and it gradually becomes Bloch as we go away from the core thus exhibiting a hybrid character. For this reason, we refer to it as hybrid vortex. The Néel chirality is a consequence of the interfacial DMI, while the Bloch character is due to the boundary conditions which force the Néel vector to be oriented primarily along the y-axis in the far field. The increase of λ leads an increase of the vortex core width and a faster rotation of the Néel vector towards the y-axis. We further analyze the hybrid vortex dynamics by propagating it along the x-direction at a velocity v. Figure 1(a) depicts that a propagating vortex is contracted along the x-direction, similarly to DWs [4] and it is elongated along the y-direction, similartly to skyrmions [5]. Figure 2(b) shows the width Lx of the propagating vortex in the x-axis as a function of velocity for various values of λ. We define the width of the vortex core as the distance between the positions where the z-component is 0.5. The width Lx, in the direction of propagation, closely follows the law of Lorentz-type contraction Lx = 1/√(1 − v2) (red solid line) despite that the model is not Lorentz invariant. For each λ, the vortex achieves a maximum velocity. Therefore, there is a minimum achievable vortex width which decreases with decreasing λ. When the velocity exceeds the maximum value, the hybrid vortex becomes unstable to periodic configurations, thus giving rise successively to a nonflat spiral at a critical velocity vNF, a vortex chain and a flat spiral at a critical velocity vF. We wish to stress that these are dynamical phase transitions, in contrast to the previously-mentioned static transitions. While the spiral phases are anticipated by a study of the 1D model [8], the vortex chain is a feature of the stripe geometry. In addition, no vortex lattice has been found in this system in an infinite film [8]. A theoretical explanation for the dynamical behavior is obtained and it leads to the general result that the velocity of localized excitations in chiral magnets cannot reach the spin wave velocity. Our results provide an understanding of the statics and dynamics of vortices in chiral AFMs and could be useful for the design of antiferromagnetic devices based on magnetic solitons. This work was supported by the project “ThunderSKY,” funded by the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No. 871.

**References:**

[1] V. Baltz et al. Rev. Mod. Phys. 90, 015005 (2018). [2] V. Lopez-Dominguez et al. Phys. Rev. Appl. 11, 024019 (2019). [3] R. Khymyn et al. Sci. Rep. 7, 43705 (2017); V. Puliafito et al., Phys. Rev. B 99, 024405 (2019). [4] T. Shiino et al., Phys. Rev. Lett. 117, 087203 (2016); L. Sánchez-Tejerina et al. Phys. Rev. B 101, 014433 (2020). [5] A. Salimath et al. Phys. Rev. B 101, 024429 (2020); S. Komineas et al., SciPost Phys. 8, 086 (2020). [6] J. Chovan et al. Phys. Rev. B 65, 064433 (2002). [7] S. Komineas and N. Papanicolaou, Nonlinearity 11, 265 (1998). [8] J. Wu et al., Nat. Phys. 7, 303 (2011); F. P. Chmiel et al., Nat. Mater. 17, 581 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5de67f9945e19420ef3d918a0e67991e.jpg)
Figure 1: (a) Vector plot for λ = 0.4 for the propagating hybrid vortex with velocity v = 0.60. (b) Vortex core width Lx in the direction of propagation for a propagating hybrid vortex as a function of velocity v, for various values of λ, normalized to the width of a static vortex. The red solid line shows the expected result for Lorentz-type contraction. The dashed lines mark the maximum obtained velocities for the respective λ values.

 Theoretical study of hybrid vortex and dynamical phase transitions in an antiferromagnetic nanostripe

Systems in which magnetic skyrmions occur at room temperature could enable new computing architectures as well as compact and energetically-efficient storage such as racetrack memories. There are two fundamental problems that prevent the successful implementation of skyrmions in a racetrack device. A first problem is not being able to restrict the skyrmion motion along straight paths and one solution is the use of antiferromagnetically-coupled skyrmions [1]. The second problem is not having stable inter-skyrmion distances, which translates into fluctuating distances among bits. A potential solution was revealed with the first experimental observation of coexisting skyrmions and chiral bobbers [2], opening the possibility that a chain of binary data bits could be encoded by two different solitons. However, this has only been observed in B20-type single crystalline materials and at low temperatures. With our work, we present a ferro-/ferri-/ferromagnetic multilayer system that can host two distinct skyrmion phases at room temperature. The two phases represent a tubular (strong constrast) and a partial (weak contrast) skyrmion (see Figure 1), as revealed from quantitative magnetic force microscopy data complemented by micromagnetic simulations. Furthermore, the tubular skyrmion can be converted into a partial skyrmion. Given its high tunability, this hybrid system serves as a promising platform for future skyrmionic devices.

**References:**

[1] W. Legrand et al., Nat. Mater. 19, 34–42 (2020) [2] F. Zheng et al., Nat. Nanotechnol. 13, 1–7 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8658ad08d7d440f2f349350bb470ce35.jpg)
Fig. 1 Upper: Schematics of the trilayer sample consisting of two ferromagnetic skyrmion (SK) layers separated by a 3.8-nm thick ferrimagnetic (Fi) layer; SK = [Ir(1nm)/Fe(0.3nm)/Co(0.6nm)/Pt(1nm)]×5 and Fi = [TbGd(0.2m)/Co(0.4nm)]×6/TbGd(0.2nm). The weak skyrmions are depicted by the vertical pale blue rectangles penetrating through the SK layers only. The strong skyrmions are indicated by wider dark blue rectangles penetrating through all layers. Lower: 2 × 2 µm2 magnetic force microscopy image taken in a magnetic field of about 133 mT, highlighting a weak and a strong skyrmion in the lower two panels.

 Coexistence of distinct skyrmion phases observed in hybrid ferromagnetic/ferrimagnetic multilayers

The magnetic skyrmion is a particle-like topologically stable spin texture that has been observed in chiral bulk magnets and asymmetric magnetic multilayers. Skyrmions can be generated and driven through the spin-polarized electric currents in the presence of an external magnetic field. Our early results show that the skyrmions can be thermally activated and behave like Brownian particles [1], which motivates us to explore the thermal control of skyrmion dynamics. Through using on-chip heaters, we are able to thermally generate, manipulate and thermoelectrically detect skyrmions [2]. Figure 1 shows that the skyrmion nucleation from the edge of the microstructured [Ta/CoFeB/MgO]15 multilayer, upon applying an electric current pulse to the upper heater on the side that generates local joule heating. The local creation of joule heating can facilitate a domain morphological transition and nucleate skyrmions by overcoming the low energy barrier. As the temperature of the heater is increased, more skyrmions are generated, which is accompanied by a unidirectional diffusion motion from the hot to the cold regions. This is due to the interplay among the repulsive forces between skyrmions, thermal spin-orbit torques, magnonic spin torques, and entropic forces. The ability to generate and manipulate skyrmions using heat opens another avenue of exploiting skyrmions in insulating and metallic materials. Moreover, by utilizing the anomalous Nernst effect in the same device, we can obtain the Nernst voltage which develops in the presence of a thermal gradient and magnetization perpendicular to each other. Since the magnetization can be varied by the change in the total number of skyrmions. We observed several discretized steps in the time evolution of Nernst voltages, as shown in Fig. 2. Each step is 90±10 nV which is attributed to the annihilation of a single skyrmion. The thermoelectric detection of skyrmion is important because the electrical detection of skyrmions is still challenging in many materials systems, but it is key to enabling spintronic applications.

**References:**

[1] L. Zhao, Z. Wang and W. Jiang, PRL, 125, 027206 (2020). [2] Z. Wang, M. Guo and W. Jiang, Nat. Electron., 3, 672 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d0769f89c8b455dd1173267ed8956388.jpg)
Fig. 1 Optical image of the microstructured magnetic multilayer with two Ta/Pt heaters on top of a 100-nm-thick Si3N4 membrane and the consecutive X-ray images acquired from the multilayer sample after applying pulse voltages to the upper heater.

![](https://assets.underline.io/uploads/markdown_image/1/image/382300011f11c8401b110ae6a6c01013.jpg)
Fig. 2 ‘Discretized’ steps of the Nernst voltage difference ΔVANE = 90±10 nV are detected after switching off the 6 mA current and applying a small current of 3 mA.

 Thermal generation, manipulation and thermoelectric detection of skyrmions

Magnetic skyrmions are topologically-protected magnetization textures characterized by a non-trivial topology [1]. They have been observed in bulk materials [2] and asymmetric multilayers [3,4] in presence of a finite Dzyaloshinskii-Moriya interaction (DMI). The manipulation (nucleation, shifting, and detection) of skyrmions usually occurs via electrical currents [3], however, thermal effects have been also used to reshuffle skyrmions for low-power unconventional applications [5]. Recently, thermal gradients have been applied to magnetic multilayers where thermally-generated skyrmions unidirectionally diffuse from hot regions to cold regions [6]. This observation has been explained through the combination of repulsive forces between skyrmions, thermal spin-orbit torques, magnonic spin torques as well as entropic forces. Here, inspired by Ref. [6], we study, by means of micromagnetic simulations, the effect of thermal gradients on skyrmion motion from a fundamental point of view. We have previously demonstrated [7] that temperature induces a variation of the magnetic parameters (exchange, interfacial DMI, perpendicular anisotropy constants) which reduce with temperature following atomistically-computed scaling relations. The set of scaled values of those parameters can be used to deterministically simulate the effect of thermal fluctuations instead of a stochastic thermal field [7]. With this in mind, we consider a linear thermal gradient applied along the x-axis (Fig. 1) with the hot and cold regions placed on the left and right side, respectively. We firstly study a 1-repetition ferromagnetic sample with a number of cells Ncx=300, Ncy=300, and Ncz=1 where a Néel skyrmion is initially placed at the center of the sample. We analyze the effect of a linear gradient of only one parameter at time. The skyrmion exhibits two velocity components that follow the generalized Thiele equation [8], where the component parallel to the gradient direction depends on the parameter varied. In particular, the perpendicular anisotropy and exchange gradients move the skyrmion from the cold to the hot region, therefore to the region where those parameters are smaller. On the contrary, the saturation magnetization (magnetostatic field) and the DMI shift the skyrmion from the hot to the cold region, thus towards the region where those parameters are larger. Afterward, we study the effect of the gradients all at once, observing an overall skyrmion motion from the cold to the hot region (Fig. 1). Furthermore, we study the effect of the skyrmion profile on the thermal gradient-driven skyrmion motion. In magnetic multilayers, the skyrmion is characterized by a thickness-dependent profile which gives rise to the so-called hybrid skyrmion [4]. Therefore, we study the effect of the thermal gradients on a single hybrid skyrmion as a function of the number of ferromagnetic repetitions. Fig.1 shows a comparison of the skyrmion trajectories, where a dependence on the number of repetitions is observed. Our results can be useful for a deeper understanding of the effect of thermal gradients and therefore for the design of low-power thermally-driven skyrmion applications. This work was supported by the project “ThunderSKY,” funded by the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No. 871.

**References:**

[1] G. Finocchio et al., J. Phys. D. Appl. Phys. 49, 423001 (2016). [2] X. Z. Yu et al., Nature 465, 901 (2010). [3] S. Woo et al., Nat. Mater. 15, 501 (2016). [4] W. Li et al., Adv. Mater. 1807683, 1 (2019). [5] J. Zázvorka et al., Nat. Nanotech. 14, 658 (2019). [6] Z. Wang et al., Nat. Electron. 3, 672 (2020). [7] R. Tomasello et al. Phys. Rev. B 97, 060402 (2018). [8] R. Tomasello et al. Phys. Rev. B 98, 024421 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e7af58671e85206fb87cfe24d058a58f.jpg)
Figure 1: skyrmion trajectory under a thermal gradient (gradients of the parameters all at once) in a magnetic multilayer for different ferromagnetic repetitions, as obtained by micromagnetic simulations. The hot and cold regions are located on the left and right side, respectively.

 Micromagnetic study of thermal gradient-driven skyrmion motion in magnetic multilayers

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Micromagnetic understanding of switching and self-oscillations in ferrimagnetic materials

Theoretical study of hybrid vortex and dynamical phase transitions in an antiferromagnetic nanostripe

Coexistence of distinct skyrmion phases observed in hybrid ferromagnetic/ferrimagnetic multilayers

Thermal generation, manipulation and thermoelectric detection of skyrmions

Micromagnetic study of thermal gradient-driven skyrmion motion in magnetic multilayers

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