France



Mechanisms for voltage control of magnetism are often based on strain or charge effects. Recently, a new approach based on electrochemical mechanisms was discovered. Due to the ionic reactions involved, it is promising especially for large and non-volatile voltage induced effects on the magnetic properties.&lt;sup&gt;[1,2]&lt;/sup&gt;&lt;br/&gt;In the charge-based or electrochemical approach, the fast majority of publications reports on Co-based thin films with perpendicular anisotropy. In our study, we focus on a distinctly different system, namely FeO&lt;sub&gt;x&lt;/sub&gt;/Fe thin films with uniaxial in-plane anisotropy. Although films with uniaxial in-plane anisotropy hold great potential for switchable artificial magnetic stray field landscapes for lab-on-a-chip or for giant magnetoresistance sensors, there are no reports of electrochemical voltage control of such films yet.&lt;br/&gt;We demonstrate that significant effects on their magnetic properties can be achieved via electrolytic gating. The detailed analysis of the interplay between electrochemical processes and hysteresis, anisotropy and domain changes allow us to reveal an oxygen-based tuning of the Néel wall interactions as the underlying mechanism.&lt;br/&gt;The thin films are composed of 3nm FeO&lt;sub&gt;x&lt;/sub&gt; and 5nm Fe. In-plane angular magnetometry measurements show the presence of a uniaxial anisotropy. Along the easy axis, the shape of the magnetization curve is close to rectangular. Nevertheless, the hysteresis along the hard axis is more rounded and the coercivity (&lt;i&gt;H&lt;sub&gt;C&lt;/sub&gt;&lt;/i&gt;) is comparable with the &lt;i&gt;H&lt;sub&gt;C&lt;/sub&gt;&lt;/i&gt; along the easy axis (red curve in Figure 1 a). This behavior is known to be caused by charged Néel domain wall interactions. These interactions block the rotation of the magnetization and lead to a large hysteresis along the hard axis.&lt;br/&gt;Upon voltage application, &lt;i&gt;H&lt;sub&gt;C&lt;/sub&gt;&lt;/i&gt; strongly decreases along every angle. In the vicinity of the hard axis, &lt;i&gt;H&lt;sub&gt;C&lt;/sub&gt;&lt;/i&gt; almost completely vanishes (blue curve in Fig. 1 a). As &lt;i&gt;H&lt;sub&gt;C&lt;/sub&gt;&lt;/i&gt; and anisotropy are strongly correlated, we investigate the anisotropy of the thin film via anhysteretic measurements. We find an increase of the uniaxial anisotropy energy density upon the voltage-triggered oxide-metallic transformation. This is an unusual finding as it is often assumed in voltage control of magnetism that &lt;i&gt;H&lt;sub&gt;C&lt;/sub&gt;&lt;/i&gt; and anisotropy are directly proportional to another. Obviously, this simple model is not applicable here. Instead, changes in the microstructure or the magnetic domain structure dominate over direct anisotropy effects.&lt;br/&gt;To evaluate how the magnetic domain structure is affected, we investigated the AC-demagnetized state, which resembles the equilibrium domain state. Upon a low voltage application (1V), and thus transforming the FeO&lt;sub&gt;x&lt;/sub&gt; to metallic Fe, the domain size significantly increases, see Fig. 1 b and c. In order to understand the domain coarsening, we approximated the change in the surface energy density for the Néel walls. This calculation shows an increase in domain-wall energy of 40% upon the oxide-metal transformation, which could explain the decrease in the number of domain walls and the associated increase in the domain size.&lt;br/&gt;In order to propose a consistent mechanism which explains the observed changes, we focused on the specific properties of the magnetically charged Néel walls. The Néel domain wall interactions becomes a significant factor, as soon as their tails overlap which is when the domain width is smaller than two times the tail width. With the extracted domain size from Fig. 1, we calculated strong interactions for the oxidized state. For the oxide to metal transformation, the interactions are significantly decreased. A comparable trend is expected and also observed in the remanent state. As a result, the tails would not overlap and we thus expect significantly fewer interactions between the walls in the reduced state. This is consistent with the observed voltage controlled deblocking.&lt;br/&gt;To demonstrate the technological relevance, we demonstrated that our electrochemical based switching mechanism allows for 180° magnetization reversal along the easy axis. Purely magnetic field reversal shows the nucleation and growth of magnetic domains (Fig. 2 (a) red curve and (b)). In comparison, during voltage assisted switching, the magnetization reversal occurs at a lower field within seconds (Fig. 2 (a) blue curve and (c)). Assuming a linear relation between the electrode diameter and the switching energy, we estimate the energy efficiency of this process to be 5fJ for an electrode diameter of 3nm. These value approaches the same range as for the so far lowest reported switching energies. Interestingly, the domain evolution is similar to the reversal process induced by a magnetic field. This indicates that the nucleation sites and the local distribution of pinning sites that affect the reverse domain growth do not change substantially during the reduction process. However, the E-induced switching proceeds at much lower magnetic field, which points to a substantially reduced pinning strength during the FeO&lt;sub&gt;x&lt;/sub&gt; to metal Fe transformation.&lt;br/&gt;The reversible and low voltage (1V) control of defect sites might be transferrable to other defect-controlled materials, such as type II superconductors or materials with specific mechanical properties.&lt;sup&gt;[3]&lt;/sup&gt;

**References:**

[1] C. Navarro-Senent, A. Quintana, E. Menéndez, E. Pellicer, J. Sort, &lt;i&gt;APL Materials&lt;/i&gt; 2019, &lt;i&gt;7&lt;/i&gt;, 030701.&lt;br/&gt;[2] C. Song, B. Cui, F. Li, X. Zhou, F. Pan, &lt;i&gt;Progress in Materials Science&lt;/i&gt; 2017, &lt;i&gt;87&lt;/i&gt;, 33.&lt;br/&gt;[3] J. Zehner, I. Soldatov, S. Schneider, R. Heller, N. B. Khojasteh, S. Schiemenz, S. Fähler, K. Nielsch, R. Schäfer, K. Leistner, &lt;i&gt;Adv. Electron. Mater.&lt;/i&gt; 2020, &lt;i&gt;6&lt;/i&gt;, 2000406.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d35959871426edaba9a12be7d3a1dbc5.jpg)
Fig.1: Magnetization curves and magnetic domains in the FeOx/Fe state (red curve in (a) and domains in (b)) and in the Fe state, blue curve in (b) and domains in (c).

![](https://assets.underline.io/uploads/markdown_image/1/image/5d4968e18040b7ba8b1145609a0e0b42.jpg)
Fig.2: E-induced 180° switching of magnetization (M). (a) pathway for H- and E-induced M-reversal. (b) H-induced and (c) E-induced M-reversal.

INTERMAG 2021

 Voltage control of Néel domain wall interactions and pinning sites

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The great event is jointly organized by the Association for Computational Linguistics (ACL) and Asian Federation of Natural Language Processing (AFNLP).

Mechanisms for voltage control of magnetism are often based on strain or charge effects. Recently, a new approach based on electrochemical mechanisms was discovered. Due to the ionic reactions involved, it is promising especially for large and non-volatile voltage induced effects on the magnetic properties.[1,2] In the charge-based or electrochemical approach, the fast majority of publications reports on Co-based thin films with perpendicular anisotropy. In our study, we focus on a distinctly different system, namely FeOx/Fe thin films with uniaxial in-plane anisotropy. Although films with uniaxial in-plane anisotropy hold great potential for switchable artificial magnetic stray field landscapes for lab-on-a-chip or for giant magnetoresistance sensors, there are no reports of electrochemical voltage control of such films yet. We demonstrate that significant effects on their magnetic properties can be achieved via electrolytic gating. The detailed analysis of the interplay between electrochemical processes and hysteresis, anisotropy and domain changes allow us to reveal an oxygen-based tuning of the Néel wall interactions as the underlying mechanism. The thin films are composed of 3nm FeOx and 5nm Fe. In-plane angular magnetometry measurements show the presence of a uniaxial anisotropy. Along the easy axis, the shape of the magnetization curve is close to rectangular. Nevertheless, the hysteresis along the hard axis is more rounded and the coercivity (HC) is comparable with the HC along the easy axis (red curve in Figure 1 a). This behavior is known to be caused by charged Néel domain wall interactions. These interactions block the rotation of the magnetization and lead to a large hysteresis along the hard axis. Upon voltage application, HC strongly decreases along every angle. In the vicinity of the hard axis, HC almost completely vanishes (blue curve in Fig. 1 a). As HC and anisotropy are strongly correlated, we investigate the anisotropy of the thin film via anhysteretic measurements. We find an increase of the uniaxial anisotropy energy density upon the voltage-triggered oxide-metallic transformation. This is an unusual finding as it is often assumed in voltage control of magnetism that HC and anisotropy are directly proportional to another. Obviously, this simple model is not applicable here. Instead, changes in the microstructure or the magnetic domain structure dominate over direct anisotropy effects. To evaluate how the magnetic domain structure is affected, we investigated the AC-demagnetized state, which resembles the equilibrium domain state. Upon a low voltage application (1V), and thus transforming the FeOx to metallic Fe, the domain size significantly increases, see Fig. 1 b and c. In order to understand the domain coarsening, we approximated the change in the surface energy density for the Néel walls. This calculation shows an increase in domain-wall energy of 40% upon the oxide-metal transformation, which could explain the decrease in the number of domain walls and the associated increase in the domain size. In order to propose a consistent mechanism which explains the observed changes, we focused on the specific properties of the magnetically charged Néel walls. The Néel domain wall interactions becomes a significant factor, as soon as their tails overlap which is when the domain width is smaller than two times the tail width. With the extracted domain size from Fig. 1, we calculated strong interactions for the oxidized state. For the oxide to metal transformation, the interactions are significantly decreased. A comparable trend is expected and also observed in the remanent state. As a result, the tails would not overlap and we thus expect significantly fewer interactions between the walls in the reduced state. This is consistent with the observed voltage controlled deblocking. To demonstrate the technological relevance, we demonstrated that our electrochemical based switching mechanism allows for 180° magnetization reversal along the easy axis. Purely magnetic field reversal shows the nucleation and growth of magnetic domains (Fig. 2 (a) red curve and (b)). In comparison, during voltage assisted switching, the magnetization reversal occurs at a lower field within seconds (Fig. 2 (a) blue curve and (c)). Assuming a linear relation between the electrode diameter and the switching energy, we estimate the energy efficiency of this process to be 5fJ for an electrode diameter of 3nm. These value approaches the same range as for the so far lowest reported switching energies. Interestingly, the domain evolution is similar to the reversal process induced by a magnetic field. This indicates that the nucleation sites and the local distribution of pinning sites that affect the reverse domain growth do not change substantially during the reduction process. However, the E-induced switching proceeds at much lower magnetic field, which points to a substantially reduced pinning strength during the FeOx to metal Fe transformation. The reversible and low voltage (1V) control of defect sites might be transferrable to other defect-controlled materials, such as type II superconductors or materials with specific mechanical properties.[3]

**References:**

[1] C. Navarro-Senent, A. Quintana, E. Menéndez, E. Pellicer, J. Sort, APL Materials 2019, 7, 030701. [2] C. Song, B. Cui, F. Li, X. Zhou, F. Pan, Progress in Materials Science 2017, 87, 33. [3] J. Zehner, I. Soldatov, S. Schneider, R. Heller, N. B. Khojasteh, S. Schiemenz, S. Fähler, K. Nielsch, R. Schäfer, K. Leistner, Adv. Electron. Mater. 2020, 6, 2000406.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d35959871426edaba9a12be7d3a1dbc5.jpg)
Fig.1: Magnetization curves and magnetic domains in the FeOx/Fe state (red curve in (a) and domains in (b)) and in the Fe state, blue curve in (b) and domains in (c).

![](https://assets.underline.io/uploads/markdown_image/1/image/5d4968e18040b7ba8b1145609a0e0b42.jpg)
Fig.2: E-induced 180° switching of magnetization (M). (a) pathway for H- and E-induced M-reversal. (b) H-induced and (c) E-induced M-reversal.

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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

In the field of magnetism and spintronics chiral magnetic structures, such as spin spirals, chiral domain walls and skyrmions [1-2], are intensively investigated due to their fascinating properties such as potentially enhanced stability and efficient spin-orbit torque driven dynamics [3-4]. It has been shown that these structures are stabilized by the Dzyaloshinskii-Moriya interaction (DMI) that favours a chiral winding of the magnetisation. While the investigation of static structures and slow (ns) dynamics of chiral magnetic structures has been intensified recently, experimental studies addressing the ultimate fs-ps dynamics of the chirality have been elusive so far as ultrafast pump-probe experiments have so far concentrated on the collinear order in magnetic systems [5-7]. In this work we employ circularly polarized light pulses from an XUV free-electron laser and investigate time-resolved the evolution of the chiral order of domain walls in magnetic thin film samples by an IR pump - X-ray magnetic scattering (XRMS) probe experiment (Fig. 1a) [8]. Using samples with interfacial DMI and perpendicular magnetic anisotropy exhibiting labyrinth-like domain patterns (Fig. 1b), we measure in the same experiment both the (CL+CR) sum signal (Fig. 1d) corresponding to the ferromagnetic order in the domains (average collinear domain magnetisation mz2+mx2) and the difference signal (CL-CR) (Fig. 1e) corresponding to the average chiral order in the domain walls (chiral order correlator mz mx). In contrast to the (CL+CR) signal, phase information is preserved in the dichroic signal (CL-CR) allowing to determine the domain wall chirality (left/right-handed) and character (Bloch (helical), Néel (cycloidal)) [9-10]. Fig. 1f shows the orthoradial profile of the dichroic signal, which indeed confirms the prevalence of fully right-handed Néel-type domain walls [9-10]. This provides a tool to individually probe the time resolved dynamics of the chiral magnetic order in the domain walls [8]. In a next step we investigated the time evolution of the scattering signals upon IR laser excitation. The key step is the comparison of the collinear ferromagnetic order dynamics and the chiral order dynamics. To ascertain both, we calculated the average intensity for the sum and the difference signal as a function of the wavevector Qr for different time delays. With this the numerical integration of the radial profiles for each time delay leads to the data shown in Fig. 2a, which demonstrates the evolution of the total intensity of the sum and the difference signal as a function of delay time normalized to the unpumped total intensity. The data shown in Fig. 2b is the result of a second experimental run. We find an ultrafast intensity decrease of both signals in the sub-ps regime with similar time constants. However, a significantly faster recovery of the chiral signal in the sub-ns timescale is observed [8]. In order to explain the key finding of the experimentally observed faster recovery of the chiral signal after laser excitation we can envisage two different mechanisms: (1) a change in the size ratio between domain walls and domains caused by an increase of the domain wall width during the whole investigated time frame or (2) a faster recovery dynamics of the chiral order within the domain walls compared to the ferromagnetic order in the domains leading to a faster build-up of the chiral magnetisation [8]. We subsequently investigate the origin of the faster recovery of the chiral signal by performing numerical simulations of the scattering signal, which reproduce the experimental findings. We conclude from these that the main driver behind our experimental findings is likely mechanism (2) leading to a faster recovery of the chiral order in the domain walls in comparison to the ferromagnetic order in the domains. In the future further fundamental aspects will be studied in detail e.g. the dependence of the timescales of the chiral order build-up on the absolute strength of the DMI by varying the heavy metal layers. The better control of the DMI and the chirality of spin structures on the ultrafast timescale can finally allow for the controlled ultrafast manipulation of chiral magnetism, e.g. ultrafast writing of chiral topological objects such as skyrmions and pave the path to applications in the field of ultrafast chiral spintronics.

**References:**

[1] Jiang, W. J. et al. Phys. Rev. 704, 1–49 (2017). [2] Everschor-Sitte, K., Masell, J., Reeve, R. M. & Kläui, M. J. Appl. Phys. 124, 240901 (2018). [3] Fert, A., Cros, V. & Sampaio, J. Nat. Nanotech. 8, 152–156 (2013). [4] Woo, S. et al. Nat. Mater. 15, 501–506 (2016). [5] Koopmans, B. et al. Nat. Mater. 9, 259–265 (2010). [6] Battiato, M., Carva, K., & Oppeneer, P. M. Phys. Rev. Lett. 105, 027203 (2010). [7] Pfau, B. et al. Nat. Commun. 3, 1100 (2012). [8] Kerber, N. et al. Nat. Commun. 11, 6304 (2020). [9] Zhang, S. L. et al. Phys. Rev. B 96, 094401 (2017). [10] Chauleau, J.-Y. et al. Phys. Rev. Lett. 120, 037202 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0a6311ce5b2275e4d9718738f6370d87.jpg)
Figure 1: Experimental setup and diffraction images. (a) Measurement geometry: A magnetic thin film sample is pumped by an optical infrared laser pulse and probed by a circularly polarized X-ray FEL pulse. Afterwards an IR-protected CCD detector records the magnetic SAXS pattern. (b) MFM image of a typical labyrinth domain pattern of the [Ta(5.3 nm)/Co20Fe60B20(0.93 nm)/Ta(0.08 nm)/MgO(2.0 nm)]x20/Ta(1.6 nm) sample. (d) The resulting sum = CL+CR of the diffraction pattern confirms that the diffraction corresponds to the magnetic domains observed by MFM. (e) The dichroic scattering signal = CL-CR and its azimuthal dependence (f) confirms the presence of (c) right-handed chiral Néel (cycloidal) domain walls [8].

![](https://assets.underline.io/uploads/markdown_image/1/image/6eaf4721a65ec4df5813cb74ec270cf9.jpg)
Figure 2: Time dependence of total scattering intensities and second moments. Time evolution of the (a,b) radially integrated intensity from two different experimental runs and (c,d) the 2nd moments of the sum signal (ferromagnetic order) and the difference signal (chiral magnetic order) from the scan shown in (a). The solid lines in (c,d) provide a guide to the eye [8].

 Faster chiral versus collinear magnetic order recovery after optical excitation revealed by femtosecond XUV scattering INVITED

Ultrafast optical generation of chiral magnetic structure is one of the most promising and highly sought-after technology for next generation magnetic memory devices. Recent experimental demonstration of generation of skyrmions with ultrafast laser pulse [1] has revealed a new horizon in this field. However, a proper theoretical understanding of the underlying interactions taking place at different timescales as well as main stimulants of the process are still in darkness. Most of the existing studies focus on the magnetic interactions only and completely neglect the interaction between the electric field of the laser and the electronic wavefunction of the material and thus create a huge gap in the complete understanding of the laser assisted magnetization dynamics. We bridge this gap by employing a quantum-classical hybrid method [2] to study the evolution of quantum states as well as the magnetic moments on equal footing and demonstrate that how a stable spin spiral can be generated from a collinear anti-ferromagnetic spin chain with an ultrashort laser pulse [3] (Fig.1,2). We use a minimal tight binding model to define our antiferromagnetic chain with a scalar hopping and onsite magnetic exchange. The laser electric field is modelled as a Gaussian pulse which couples to the hopping. On being hit by the pulse several different mechanisms take place at different timescale starting with an instantaneous change in occupation of states resulting in a gradual buildup of effective torque which leads to the magnetization dynamics. The magnetization dynamics being a slower process compared to the quantum evolution lags by approximately 50fs behind the incident of laser. To initiate this process the initial state must deviate from an ideal collinear state. A stronger deviation shows a faster convergence to the steady state configuration. The mixing of quantum states gives rise to different emergent spin mixing interactions which drives the subsequent dynamics. The system finds its chiral state within a timescale of 1ps. The timeframe of the different steps of the spiralization process is of the same order of the timescales associated with the ultrafast demagnetization which indicates towards the similar underlying physics. After 1ps the dynamics becomes slower and the chain gradually attains a uniform steady chiral structure. We characterize this evolution in terms of the relative angle between adjacent magnetic moment and vector chirality which clearly distinguish different time frames. The relative angle and the induced chirality can be tuned by the strength of the amplitude field. We further show that these effects are fairly robust against thermal fluctuations. Our results thus reveal several salient features of optical manipulation of chirality which has not been revealed in existing literature and thus would be instrumental in their experimental realization.

**References:**

[1] F. Büttner et.al. Nat. Matter. 20, 30 (2021). [2] A. Ono and S. Ishihara, Phys. Rev. Lett. 119, 207202 (2017). M. D. Petrović, B. S. Popescu, U. Bajpai, P. Plecháč, and B. K. Nikolić, Phys. Rev. Applied 10, 054038 (2018). [3] S. Ghosh, F. Freimuth, O. Gomonay, S. Blügel and Y. Mokrousov, arXiv:2011.01670

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/87c5fef65fb042eaab838c75b45a688d.jpg)
Fig. 1: Formation of a spin spiral (solid arrows) from an collinear antiferromagnetic (transparent arrows) chain with 32 sites.

![](https://assets.underline.io/uploads/markdown_image/1/image/ed03b2ecc70a85f0fdc71272a0a68860.jpg)
Fig. 2: Time evolution of the components of magnetization over time for 1st (solid) and 17th (transparent) magnetic moment. The yellow line shows the laser pulse.

 Ultrafast electronic manipulation of antiferromagnetic spin spiral states

The observation of ultrafast and coherent control of spins in magnetic materials at room temperature [1-2] has prompted the experimental and theoretical efforts to understand the underlying mechanisms driving such magnetic phenomena. In order to understand the driving mechanisms, in this non-equilibrium regime, it is necessary to disentangle the relevant interactions (exchange, spin-lattice, electron-phonon, coulomb etc.). Stoichiometric B2 ordered epitaxial (001) FeRh undergoes a first order magnetic phase transition from antiferromagnetic (AFM) to ferromagnetic (FM) at ≈ 380K. In the AFM phase of FeRh, only Fe has net magnetic moment, whereas in the FM phase both Fe and Rh carry net moments. The phase transition is also accompanied by a ≈ 1% isotropic lattice expansion in the bulk BCC structure, and changes in electronic structure, see [3-5] and refs. therein. Since magnetic and structural dynamics can occur at different timescales, it makes FeRh an ideal candidate for disentangling relevant interaction mechanisms at different timescales. The phase transition has been extensively studied theoretically and experimentally [6-10], in thermal equilibrium (static heating) and at ultrafast time scales (fs- optical excitation), but a precise knowledge of the roles of the electronic, phononic and spin sub-systems remains elusive. We have studied the laser-driven AFM to FM phase transition in FeRh with X-ray absorption spectroscopy (tr-XAS) around the Fe L3 edge. The experiments were performed at the SCS Instrument of the European XFEL. We also performed the density functional theory (DFT) calculations to simulate the XAS spectra at different time delays including magnetic, lattice and temperature changes. Temperature modelling was done using two temperature model (2TM). In this contribution, we will discuss the changes in the electronic band structure of FeRh in the femto- and picosecond timescales, derived from the X-ray absorption spectroscopy measurements at the Fe L3 absorption and compare them with the density functional theory simulations.

**References:**

[1] E. Beaurepaire et al., Phys. Rev. Lett. 76, 4250 (1996) [2] J.-Y. Bigot et al., Nature Physics 5, 515 (2009) [3] J.-U. Thiele et al. Appl. Phys. Lett. 82, 2859 (2003) [4] A.X. Gray et al., Phys. Rev. Lett. 108, 257208 (2012). [5] C. Baldasseroni et al., Appl. Phys. Lett. 100, 262401 (2012). [6] J.-U. Thiele, M. Buess, C.H. Back, Appl. Phys. Lett 85, 2857 (2004). [7] G. Ju et al., Phys. Rev. Lett. 93, 197403 (2004) [8] I. Radu et al., PRB 81, 104415 (2010) [9] S. Mariager et al., Phys. Rev. Lett. 108, 87201 (2012) [10] F. Quirin et al., Phys. Rev. B 85, 020103(R) (2012)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/92b1687e04e78b6abfbee5f8574ccb81.jpg)
Figure-1: a: X-ray absorption (XAS) changes: (dashed line: experiment) near the Fe L3 edge before and after laser excitation at a fixed time delay of 70 fs, (solid lines: DFT theory) XAS changes corresponding to temperature of 2000K 
 
b: a: X-ray absorption (XAS) changes: (dashed line: experiment) near the Fe L3 edge before and after laser excitation at a fixed time delay of 11.5 ps and 70 us, (solid lines: DFT theory) XAS changes corresponding to net changes including temperature (400K), lattice (0.5 % expansion) and magnetic changes (AFM->FM)

 Ultrafast dynamics in electronic band structure of optically excited epitaxial FeRh: X-ray absorption spectroscopy and density functional theory approach

The relaxation of the magnetisation in magnetic materials is of great importance from both an applied and fundamental point of view. Conventional modelling of magnetisation dynamics employs the Landau-Lifshitz-Gilbert (LLG) equation [1], where the damping is included phenomenologically by the Gilbert damping term. This describes the coupling of the magnetic modes (given primarily by the atomic spin) to the non-magnetic modes (lattice vibrations and electron orbits) which are assumed to be in equilibrium. Recent studies of the dynamics induced by THz laser pulses has highlighted the necessity of understanding magnetisation relaxation beyond this assumption. In reality the spin and lattice dynamics mutually influence one another, hence it is necessary to employ a unified model of molecular and spin dynamics [2,3,4], Spin-Lattice dynamics (SLD). In the present work we employ a model where the transfer of energy and angular momentum between the lattice and the spin system (Fig. 1) is realised by the pseudo-dipolar coupling [3,4], which arises from the spin-orbit interaction and can be parameterised by magneto-elastic experiments. The spin system exchange parameters are taken from ab-initio parameterisation for BCC iron [2], while for the phonon system interactions potential we compare the Harmonic and Morse potentials. Our results in Fig. 1 show that equilibration of both sub-systems can be obtained on a sub-nanosecond timescale in both the microcanonical and canonical ensembles, the relaxation time being governed by the energy initially deposited in the system. We also observe that the equilibrium magnetisation is independent of the thermostat and by coupling the spin system only to the lattice vibrations the magnetisation temperature dependence can be reproduced without the need of a phenomenological spin damping [3]. Our model allows the evaluation of the effective magnon-phonon damping (Fig. 2) which agrees well with the values measured in magnetic insulators where they depend predominantly on magnon-phonon coupling rather than on electronic effects. The functional form of the damping variation is quadratic, in accordance with the form of the pseudo-dipolar coupling term. The magnon-phonon damping has been also calculated for an alternate form of the coupling, where the pseudo-dipolar coupling is replaced by an on-site form, i.e a Néel-like anisotropy term. The latter leads to much smaller damping values, as shown in Ref. [3], suggesting that the magnon-phonon damping can clearly have complex behaviour depending on the properties of the system, especially the coupling term. Based on this remark, no universal behaviour of damping as a function of temperature can be deduced for spin lattice models. The model developed in this work [3] opens the possibility to describe the distinct dynamics of spins and phonons, necessary for the understanding of ultrafast magnetisation dynamics experiments and the subsequent angular momentum transfer between the two subsystems. We have demonstrated that the model works well in the absence of a phenomenological Gilbert damping, which consists mainly of electronic contributions, hence the SLD model can be employed to study magnetic insulators, such as YIG, where the principal contribution to damping is via magnon-phonon interactions.

**References:**

 [1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [2] P.-W. Ma et al., Phys. Rev. B, vol. 78, no. 2, p. 024434, Jul. 2008. [3] M. Strungaru et al. arXiv preprint arXiv:2010.00642 (2020) (accepted in Phys. Rev. B) [4] M. Aßmann and U. Nowak, J. Magn. Magn. Mater 6 469 (2019): 217-223.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0415f635b3cb2e3e3855832e043401d0.jpg)
 
Fig.1: NVE (microcanonical – top panel) and NVT ( canonical – bottom panel) simulations for a 10 × 10 × 10 unit cell BCC Fe system. The spin system is initialised at a spin temperature TS = 0K corresponding to a saturated magnetic state in the z direction, while the lattice velocities are initialised by a Boltzmann distribution at T L = 1200 K (for NVE) and T L = 600 K (for NVT). In both cases we obtain equilibration of the two subsystems under 1ns.

![](https://assets.underline.io/uploads/markdown_image/1/image/6755da1be71fc906a6b75c124d5d162a.jpg)
 
Fig.2 Damping parameter extracted from fitting the z component of the magnetisation for two different choices of potential: HP- Harmonic Potential (green open squares) and MP-Morse Potential (black open circles) as a function of temperature - Panel a) and as a function of the coupling strength – Panel b); The effective damping in Panel a) is calculated for a constant coupling strength of C = 0.5, while in Panel b) the damping is calculated for a constant temperature of T = 300K. The black and green lines represent the average damping parameter obtained from the simulations using the Morse and the Harmonic Potentials, respectively.

 Magnon-phonon damping calculations in the spin-lattice dynamics model

Recently, the effects of inertia in the spin dynamics of ferromagnets were reported to cause nutation resonance [1-8] at the frequencies higher than the conventional ferromagnetic resonance. It was shown that spin-orbit induced inertia is responsible for nutation, which should be considered together with magnetization precession in a magnetic field. Recently nutation in ferromagnets was confirmed experimentally [1], since nutation and precession operate at substantially different time scales, and conventional microwave ferromagnetic resonance (FMR) spectroscopy techniques do not easily reach the high-frequency (sub-Terahertz) regime, required to observe the inertia effect which in addition yields much weaker signal. The interest in the ultrafast spin dynamics has inspired various theoretical studies. For instance, a relation between the Gilbert damping constant and the inertial regime characteristic time was elaborated in ref. [2]. The exchange interaction, damping, and moment of inertia can be calculated from the first principles as shown in [3]. In [4], the inertial regime was introduced in the framework of the mesoscopic nonequilibrium thermodynamics theory, and it was shown to be responsible for the nutation superimposed on the precession of magnetization. Wegrowe and Ciornei [5] discussed the equivalence between the inertia in the dynamics of uniform precession and a spinning top within the framework of the Landau–Lifshitz–Gilbert equation generalized to the inertial regime. This equation was studied analytically and numerically in ref. [6, 7]. In this digest we present new results on the frequency dependence of high-frequency magnetic susceptibility in ferromagnets taking inertia effects into account [8]. We performed calculations based on the Landau-Lifshitz-Gilbert equation with the additional inertia term and obtained an analytical solution. These results provide a basis for further studies of magnetocrystalline anisotropy and damping in magnetic materials in the sub-Terahertz range and furthermore a guideline for experimental research. Using circular variables of magnetization and magnetic field we obtained the dependence of the susceptibility on frequency for the right-hand precession, i.e. positive polarization (Fig. 1(a)). Due to the contribution of nutation, the frequency and linewidth of this resonance are slightly different from the ones of usual FMR. In Fig. 1(b) the nutation resonance possessing negative polarization is presented. Note that the polarizations of ferromagnetic and nutation resonances are reversed. In addition, we derived a general analytical expression for the linewidth of nutation resonance in ferromagnets, and will show its dependence on magnetization, the Gilbert damping, the inertial relaxation time and applied magnetic field. We find that the nutation linewidth can be tuned by the applied magnetic field, and this tunability breaks the direct relation between losses and the linewidth.

**References:**

[1] K. Neeraj, N. Awari, S. Kovalev et al., Nat. Phys. (2020), doi: 10.1038/s41567-020-01040-y [2] M. Fähnle, D. Steiauf, and C. Illg, Phys. Rev. B, Vol. 84, 172403 (2011). [3] S. Bhattacharjee, L. Nordström, and J. Fransson, Phys. Rev. Lett., Vol. 108, 057204 (2012). [4] M.-C. Ciornei, J. M. Rubí, and J.-E. Wegrowe, Phys. Rev. B, Vol. 83, 020410 (2011). [5] J.-E. Wegrowe and M.-C. Ciornei, Am. J. Phys., Vol. 80, p. 607 (2012). [6] E. Olive, Y. Lansac, and J.-E. Wegrowe, Appl. Phys. Lett., Vol. 100, 192407 (2012). [7] E. Olive, Y. Lansac, M. Meyer et al., J. Appl. Phys. 117, 213904 (2015). [8] M. Cherkasskii, M. Farle, and A. Semisalova, Phys. Rev. B 102, 184432 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/06be23fe2d97187eb18d84ec0746a005.jpg)
Fig. 1. (a) The FMR peak with nutation. (b) The nutation resonance. The calculation was performed for γ/(2π)=28 GHz T-1, μ0M0=1 T, μ0H0=100 mT, α=0.0065 and τ = 10-11 s.

 Nutation resonance in ferromagnets

Ferrimagnetic insulators (FIs) host an array of attractive properties for spintronics research and potential applications. Some prominent features are tunable saturation magnetization, perpendicular anisotropy, low damping, and highly ordered single crystal structure that could be beneficial for obstacle-free domain wall and skyrmion motion. However, electrically manipulating and detecting the magnetization in FIs, which is a prerequisite for spintronic devices, has been impossible for a long time due to the lack of physical phenomena to do so. Recent advances in spin current generation and detection have provided the relevant tools to detect the magnetization vector of FIs by purely electrical means [1,2]. Consequently, the research into MIs has gained significant momentum in the past decade. In this talk, we will discuss some of the recent efforts in controlling and detecting the magnetization vector via spin currents in FIs with perpendicular magnetic anisotropy. First, we will briefly overview the experiments of current-induced switching and electrical detection of magnetization vector in a FI, thulium iron garnet, with perpendicular anisotropy [3,4]. We will then discuss the emergent interfacial chiral magnetism, and current-driven dynamics of domain walls in rare-earth-based FIs. We will show that chiral magnetic interactions stabilize Néel-type domain walls in ultrathin FIs that can be propelled by spin-orbit torques with velocities ~0.8 km/s [5]. Finally, we will discuss a new thermoelectric effect that allows electrical detection of the out-of-plane magnetization component in FIs in a nonlocal device geometry [6]. We will show that, by using an engineered temperature gradient, we can convert a thermally-induced spin current in a FI into an electrical signal across a Pt detector that is proportional to the perpendicular component of the magnetization.

**References:**

[1] Uchida et al. Appl. Phys. Lett. 97, 172505 (2010) [2] Nakayama et al. Phys. Rev. Lett. 110, 206601 (2013). [3] Avci et al. Nat. Mater. 16, 309 (2017) [4] Avci et al., Appl. Phys. Lett. 111, 072406 (2017) [5] Avci et al. Nat. Nanotech. 14, 561 (2019) [6] Avci et al. Phys. Rev. Lett. 124, 027701 (2020)

**Images:**

 Spin current-driven control and detection of magnetization in ferrimagnetic insulators INVITED

While classical spintronics has traditionally relied on ferromagnetic metals as spin generators and spin detectors, spin-orbitronics exploits the interplay between charge and spin currents enabled by the spin-orbit coupling (SOC) in non-magnetic systems. The realization of magnetization switching induced by in-plane current injection in heavy metal/ferromagnetic heterostructures [1] has drawn increasing attention to spin-orbitronics, leading to the advent of spin-orbit torques magnetoresistive random access memories (SOT-MRAM). Compared to heavy metals, oxide 2D electron gases have emerged as alternative spin-orbitronics material systems. They benefit from an efficient spin-charge interconversion through the direct and inverse Edelstein effects, which appear at their interfaces where the broken inversion symmetry induces a Rashba SOC [1]. Recently, we have demonstrated an enhancement of the spin-to-charge conversion efficiency by two orders of magnitude in SrTiO3-based 2D gas compared to conventional heavy metals [2], along with a non-volatile electric-control of the spin-to-charge conversion [3]. While the sign and efficiency of the SOTs are fixed by the stack of materials in conventional SOT devices, achieving an electric-control of the mirror charge-to-spin conversion would be of great interest for developing reconfigurable SOT-MRAM and logic gates, offering the possibility of active manipulation of the torque by electric field for building new architectures. Here we report electric-field control of spin orbit torques with electrical remanence in a perpendicular ferromagnet - SrTiO3 system. Results are shown from a 1 µm wide Hall bar device made of a CoFeB ferromagnetic layer with perpendicular magnetization on top of a SrTiO3 substrate with a thin Ta insertion to protect the 2D electron gas at the interface (Figure 1). The perpendicular magnetization is achieved using an MgO capping layer that ensures high interfacial magnetic anisotropy along with compatibility for integration in magnetic tunnel junctions for SOT-MRAM. To modulate the 2D gas properties, voltage is applied to the devices via a back-gate. Non-volatile electric-field control of the sheet resistance is achieved with 1150% contrast (Figure 1c), with two switchable and remanent high and low resistivity states of the 2D electron gas. Spin-orbit torques effective fields are further measured using second harmonic Hall methods. A remanent electric-field control of the SOT efficiency is demonstrated, with inversion of the sign of the SOT anti-damping-like effective field (Figure 2). Anti-damping-like and field-like torque effective fields per 2D current density of 1.6 and 0.1 mT. A. cm-1 respectively are reached in the 2D gas low resistivity state. These results are consistent with a combination of both intrinsic modulation of the SOT efficiency together with extrinsic modulation due to the non-volatile electric-control of the current injection in the 2D gas. The non-volatile control of the SOT effective field is further evidenced in Figure 2b, which displays reproducible inversion of the SOT torques after initializing with negative or positive voltage pulses of ±130 V, opening the way to reconfigurable spin-orbit torque memory and logic gate architectures.

**References:**

[1] I. M. Miron et al., Nature 476, 189–193 (2011). [2] E. Lesne et al., Nature Materials, 15, 1261–1266 (2016). [3] D. C. Vaz et al., Nature Materials, 18, 1187–1193(2019). [4] P. Noel et al., Nature 580, 483–86 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/005ef5c4d7565a59c77e5faa647c55c7.jpg)
Schematic of the sample and magnetotransport properties. (a) Structure of the sample and measurement geometry. (b) Out-of-plane magnetic field dependence of the anomalous Hall resistance at zero electric field (Vg = 0 V). (c) Gate dependence of the sheet resistance at zero magnetic field after initializing the ferromagnet (FM) in the up (blue) and down (red) states. All data have been measured at 10K.

![](https://assets.underline.io/uploads/markdown_image/1/image/9e2388e262906d2be322591c09b882ed.jpg)
Electric-field control of spin orbit torques with electrical remanence. (a) Spin-orbit torques anti-damping-like effective field as a function of the gate voltage for j2D = 4 A.cm-1. (b) Normalized second harmonic Hall resistance at electrical remanence VR after initializing with negative (blue) or positive (red) voltage pulses of ±130 V. Dashed and dotted lines are linear fits, yielding SOT effective fields BAD = +0.29, -0.20, +0.32 and -0.18 mT. A. cm-1, respectively.

 SrTiO3 system

The presence of a Dzyaloshinskii-Moriya interaction (DMI) in perpendicular magnetized thin films stabilizes Néel type domain walls of fixed chirality, as opposed to the Bloch walls that are formed in the absence of a DMI [1]. Néel walls of fixed chirality have been shown to be driven efficiently in the same direction by electric currents through spin-orbit torques [2,3]. Later, a large tuning of the DMI with an applied voltage was demonstrated [4], which can be used for example for electric field control of magnetic domain wall motion via the modulation of the DMI [5]. Here, we demonstrate an alternative mechanism for the control of domain wall type: Using micromagnetic simulations and analytical modeling, we show that the presence of a uniaxial in-plane magnetic anisotropy can also lead to the formation of Néel walls in the absence of a DMI. It is possible to abruptly switch between Bloch and Néel walls via a small modulation of both the in-plane, but also the perpendicular magnetic anisotropy [6]. This opens up a route towards efficient electric field control of the domain wall type with tiny applied voltages, as the magnetic anisotropy can be modulated via the direct voltage controlled magnetic anisotropy (VCMA) mechanism [7], or via magnetoelastic anisotropy induced through coupling to a piezoelectric element [8].

**References:**

1. A. Thiaville, S. Rohart, E. Jué, V. Cros, and A. Fert, EPL (Europhysics Letters) 100, 57002 (2012). 2. K.-S. Ryu, L. Thomas, S.-H. Yang, and S. S. P. Parkin, Nat. Nanotech. 8, 527–533 (2013) 3. S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater., 12, 611–616 (2013) 4. T. Srivastava, et al., Nano Letters 18, 4871 (2018) 5. T. Koyama, Y. Nakatani, J. Ieda, and D. Chiba, Science Advances 4 (2018) 6. K. J. A. Franke, et al., in preparation (2021) 7. T. Maruyama, et al., Nat. Nanotech. 4, 158 (2009). 8. S.Li, et al., J, Appl. Phys. 117, 17D702 (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5815f993e947e36c366298e73960792b.jpg)
In a perpendiular magnetized thin film, domain walls (DWs) are generally of Bloch type (bottom left). The presence of a DMI (top left), stabilises Néel walls. Alternatively, and in-plane magnetic anisotropy can also form Néel walls (bottom right). An increase in the perpendicular magnetic anisotropy switches the DW type back to Bloch (top right).

 Control of Magnetic Domain Wall Type using Anisotropy Modulations

Gate voltage control of magnetic order is desirable for spintronic device applications [1], [2], but 180° magnetization switching is not straightforward since electric fields do not break time-reversal symmetry. Ferrimagnets have two sublattices with opposing magnetic moments, and by changing the relative amplitude of the magnetization of the two sublattice, magnetic reversal can be achieved. Here we show that ferrimagnetic order in rare earth-transition metal thin films can be dynamically controlled by voltage through solid-state hydrogen gating. We focus on GdCo ferrimagnetic structures with perpendicular magnetic anisotropy, and show that the dominant lattice of GdCo ferrimagnetic thin films at room temperature can be reversibly toggled by hydrogen loading/unloading into the material controlled using a solid-state hydrogen pump [3]. The loading/unloading of hydrogen is mediated by a positive gate bias applied to the top electrode generates protons through hydrolysis of ambient moisture in the air, as shown previously [3]. We find that hydrogen injection into the GdCo leads to a significant decrease in the compensation temperature. With a fixed perpendicular bias field, we show that a gate voltage can induce 180° magnetic switching of the sublattices by shifting the magnetic compensation temperature above and below room temperature. The localized nature of the hydrogen gating effect allows for gate-voltage generation of ferrimagnetic spin textures such as chiral domain walls in racetrack devices. The work provides a means to tune ferrimagnetic spin textures and dynamics, and provides a new direction in magnetoionic control of magnetism for data storage and computing.

**References:**

[1] F. Matsukura, Y. Tokura, and H. Ohno, “Control of magnetism by electric fields,” Nat. Nanotechnol., vol. 10, no. 3, pp. 209–220, 2015. [2] E. Y. Tsymbal, “Electric toggling of magnets,” Nat. Mater., vol. 11, no. 1, pp. 12–13, 2011. [3] A. J. Tan et al., “Magneto-ionic control of magnetism using a solid-state proton pump,” Nat. Mater., vol. 18, no. 1, pp. 35–41, 2019.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a025e553f973e3a2bdc508401f7c457e.jpg)
Coercivity (Hc) versus temperature showing compensation temperature change of GdCo by solid-state ionic gating.

![](https://assets.underline.io/uploads/markdown_image/1/image/e240923efec7bf678cdf53c00f3dffa0.jpg)
Reversed domain generator based on hydrogen gating: a sequence of wide field MOKE contrast images showing the movement of the generated reversed domain.

Voltage control of Néel domain wall interactions and pinning sites

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Voltage control of Néel domain wall interactions and pinning sites

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Micromagnetic understanding of switching and self-oscillations in ferrimagnetic materials

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