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The Hall effect is one of the best known effects in (solid-state) physics. Conventionally, this phenomenon describes the occurrence of charge currents that are perpendicular to an externally applied electric field due to a time-reversal symmetry breaking magnetic field. Besides, in ferromagnetic systems, the net magnetization can break time-reversal symmetry even in the absence of a magnetic field which allows the so-called anomalous Hall effect. This effect originates from extrinsic and intrinsic contributions that are both related to the existence of spin-orbit coupling [1]. Moreover, another contribution to the Hall effect, which is known as the topological Hall effect, can exist even if spin-orbit coupling is negligible. It may occur in certain noncollinear noncoplanar magnetic textures with a nonzero scalar spin chirality like skyrmions [2, 3].&lt;br/&gt;&lt;br/&gt;However, recent works [4, 5] reported the occurrence of an anomalous Hall effect in several compensated kagome magnets (cf. Fig. 1). These materials are coplanar antiferromagnets with vanishing net magnetization, and still, a group theoretical analysis allows the existence of the effect. The large conductivities obtained via first-principle calculations have been confirmed in experiments for Mn&lt;sub&gt;3&lt;/sub&gt;Sn [6] and Mn&lt;sub&gt;3&lt;/sub&gt;Ge [7]. However, a straightforward microscopic picture for this phenomenon was still missing.&lt;br/&gt;&lt;br/&gt;In this talk, we present an explanation on a microscopic level based on tight-binding calculations and analytical considerations [8]. For coplanar kagome magnets, we show the equivalence of spin-orbit coupling and an out-of-plane tilting of the magnetic moments. The existence of spin-orbit interaction does not only break a combined time-reversal and mirror symmetry of the Hamiltonian but can be transformed to a magnetic texture that is virtually canted, whereas, the original texture remains coplanar [cf. Fig. 2(a)]. Consequently, the ‘new’ anomalous Hall effect can be interpreted as a combination of an effective anomalous and topological Hall effect due to the net magnetic moment and the net scalar spin chirality of this virtual magnetic texture, respectively.&lt;br/&gt;&lt;br/&gt;Furthermore, as we demonstrate, a noncoplanar kagome magnet with spin-orbit coupling is able to behave like a system that is virtually coplanar and with compensated spin-orbit coupling [cf. Fig. 2(b)]. In this case, the combination of mirror and time-reversal symmetry of the Hamiltonian that was broken before has been restored. A critical out-of-plane tilting angle of the real texture can be found, where the virtual texture is coplanar and the Hall effect is absent for all energies. As we show in detail, the electronic properties are determined by this virtual texture that is hidden in the Hamiltonian.&lt;br/&gt;&lt;br/&gt;In consequent investigations, the calculations have been repeated for other transport quantities like the spin Hall effect where charge currents are converted into spin currents. These results can again be related to the virtual spin texture which has, however, different consequences for the spin Hall effect. Besides, in order to simulate the experimental situation, the investigated model was extended from a two-dimensional kagome lattice, as considered here, to a more realistic model including &lt;i&gt;d&lt;/i&gt;-orbitals and kagome planes that are stacked along the out-of-plane direction.

**References:**

[1] N. Nagaosa, J. Sinova, S. Onoda, &lt;i&gt;et al.&lt;/i&gt; &quot;Anomalous Hall effect&quot;. &lt;i&gt;Rev. Mod. Phys. &lt;/i&gt;82, 1539 (2010).&lt;br/&gt;&lt;br/&gt;[2] P. Bruno, V. K. Dugaev, and M. Taillefumier. &quot;Topological Hall Effect and Berry Phase in Magnetic Nanostructures&quot;. &lt;i&gt;Phys. Rev. Lett. &lt;/i&gt;93, 096806 (2004).&lt;br/&gt;&lt;br/&gt;[3] N. Nagaosa and Y. Tokura. &quot;Topological properties and dynamics of magnetic skyrmions&quot;. &lt;i&gt;Nat. Nanotechnol&lt;/i&gt;. 8, 899 (2013).&lt;br/&gt;&lt;br/&gt;[4] H. Chen, Q. Niu, and A. H. MacDonald. &quot;Anomalous Hall Effect Arising from Noncollinear Antiferromagnetism&quot;. &lt;i&gt;Phys. Rev. Lett.&lt;/i&gt; 112, 017205 (2014).&lt;br/&gt;&lt;br/&gt;[5] Y. Zhang, Y. Sun, H. Yang,&lt;i&gt; et al&lt;/i&gt;. &quot;Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn&lt;sub&gt;3&lt;/sub&gt;X (X=Ge, Sn, Ga, Ir, Rh, and Pt)&quot;. &lt;i&gt;Phys. Rev. B &lt;/i&gt;95, 075128 (2017).&lt;br/&gt;&lt;br/&gt;[6] S. Nakatsuji, N. Kiyohara, and T. Higo. &quot;Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature&quot;. &lt;i&gt;Nature (London) &lt;/i&gt;527, 212 (2015).&lt;br/&gt;&lt;br/&gt;[7] A. K. Nayak, J. E. Fischer, Y. Sun,&lt;i&gt; et al&lt;/i&gt;. &quot;Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn&lt;sub&gt;3&lt;/sub&gt;Ge&quot;. &lt;i&gt;Sci. Adv. &lt;/i&gt;2, e1501870 (2016).&lt;br/&gt;&lt;br/&gt;[8] O. Busch, B. Göbel, and I. Mertig. &quot;Microscopic origin of the anomalous Hall effect in noncollinear kagome magnets&quot;. &lt;i&gt;Phys. Rev. Research&lt;/i&gt; 2, 033112 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5ba5815bd43591c766a0d90db120a54b.jpg)
Fig. 1. Overview of the discussed coplanar kagome magnets. The arrows indicate the magnetic moments that are located at the basis atoms of the unit cell. The configurations can be characterized by their vector spin chiralities [(a), (b) positive, and (c), (d) negative]. Besides, each magnetic texture has different symmetry planes (time reversal T, mirror M or a combination of both) as indicated by the red lines.

![](https://assets.underline.io/uploads/markdown_image/1/image/93336328e4e3193aa019ed2b5bbc3832.jpg)
&lt;br /&gt;
Fig. 2. Within the considered model, spin-orbit coupling gives rise to a virtual magnetic texture (red arrows) that is tilted out of the kagome plane with respect to the original magnetic texture (black arrows). (a) The virtual texture of a coplanar configuration can be noncoplanar resulting in the occurrence of an effective anomalous and topological Hall effect. (b) At a critical out-of-plane canting angle of the real magnetic texture, spin-orbit coupling is compensated and the virtual magnetic texture is coplanar. In this case, the anomalous Hall effect is absent.

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 Interpretation of the anomalous Hall effect as an effective topological Hall effect in noncollinear kagome magnets

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The Hall effect is one of the best known effects in (solid-state) physics. Conventionally, this phenomenon describes the occurrence of charge currents that are perpendicular to an externally applied electric field due to a time-reversal symmetry breaking magnetic field. Besides, in ferromagnetic systems, the net magnetization can break time-reversal symmetry even in the absence of a magnetic field which allows the so-called anomalous Hall effect. This effect originates from extrinsic and intrinsic contributions that are both related to the existence of spin-orbit coupling [1]. Moreover, another contribution to the Hall effect, which is known as the topological Hall effect, can exist even if spin-orbit coupling is negligible. It may occur in certain noncollinear noncoplanar magnetic textures with a nonzero scalar spin chirality like skyrmions [2, 3]. However, recent works [4, 5] reported the occurrence of an anomalous Hall effect in several compensated kagome magnets (cf. Fig. 1). These materials are coplanar antiferromagnets with vanishing net magnetization, and still, a group theoretical analysis allows the existence of the effect. The large conductivities obtained via first-principle calculations have been confirmed in experiments for Mn3Sn [6] and Mn3Ge [7]. However, a straightforward microscopic picture for this phenomenon was still missing. In this talk, we present an explanation on a microscopic level based on tight-binding calculations and analytical considerations [8]. For coplanar kagome magnets, we show the equivalence of spin-orbit coupling and an out-of-plane tilting of the magnetic moments. The existence of spin-orbit interaction does not only break a combined time-reversal and mirror symmetry of the Hamiltonian but can be transformed to a magnetic texture that is virtually canted, whereas, the original texture remains coplanar [cf. Fig. 2(a)]. Consequently, the ‘new’ anomalous Hall effect can be interpreted as a combination of an effective anomalous and topological Hall effect due to the net magnetic moment and the net scalar spin chirality of this virtual magnetic texture, respectively. Furthermore, as we demonstrate, a noncoplanar kagome magnet with spin-orbit coupling is able to behave like a system that is virtually coplanar and with compensated spin-orbit coupling [cf. Fig. 2(b)]. In this case, the combination of mirror and time-reversal symmetry of the Hamiltonian that was broken before has been restored. A critical out-of-plane tilting angle of the real texture can be found, where the virtual texture is coplanar and the Hall effect is absent for all energies. As we show in detail, the electronic properties are determined by this virtual texture that is hidden in the Hamiltonian. In consequent investigations, the calculations have been repeated for other transport quantities like the spin Hall effect where charge currents are converted into spin currents. These results can again be related to the virtual spin texture which has, however, different consequences for the spin Hall effect. Besides, in order to simulate the experimental situation, the investigated model was extended from a two-dimensional kagome lattice, as considered here, to a more realistic model including d-orbitals and kagome planes that are stacked along the out-of-plane direction.

**References:**

[1] N. Nagaosa, J. Sinova, S. Onoda, et al. "Anomalous Hall effect". Rev. Mod. Phys. 82, 1539 (2010). [2] P. Bruno, V. K. Dugaev, and M. Taillefumier. "Topological Hall Effect and Berry Phase in Magnetic Nanostructures". Phys. Rev. Lett. 93, 096806 (2004). [3] N. Nagaosa and Y. Tokura. "Topological properties and dynamics of magnetic skyrmions". Nat. Nanotechnol. 8, 899 (2013). [4] H. Chen, Q. Niu, and A. H. MacDonald. "Anomalous Hall Effect Arising from Noncollinear Antiferromagnetism". Phys. Rev. Lett. 112, 017205 (2014). [5] Y. Zhang, Y. Sun, H. Yang, et al. "Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn3X (X=Ge, Sn, Ga, Ir, Rh, and Pt)". Phys. Rev. B 95, 075128 (2017). [6] S. Nakatsuji, N. Kiyohara, and T. Higo. "Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature". Nature (London) 527, 212 (2015). [7] A. K. Nayak, J. E. Fischer, Y. Sun, et al. "Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn3Ge". Sci. Adv. 2, e1501870 (2016). [8] O. Busch, B. Göbel, and I. Mertig. "Microscopic origin of the anomalous Hall effect in noncollinear kagome magnets". Phys. Rev. Research 2, 033112 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5ba5815bd43591c766a0d90db120a54b.jpg)
Fig. 1. Overview of the discussed coplanar kagome magnets. The arrows indicate the magnetic moments that are located at the basis atoms of the unit cell. The configurations can be characterized by their vector spin chiralities [(a), (b) positive, and (c), (d) negative]. Besides, each magnetic texture has different symmetry planes (time reversal T, mirror M or a combination of both) as indicated by the red lines.

![](https://assets.underline.io/uploads/markdown_image/1/image/93336328e4e3193aa019ed2b5bbc3832.jpg)
 
Fig. 2. Within the considered model, spin-orbit coupling gives rise to a virtual magnetic texture (red arrows) that is tilted out of the kagome plane with respect to the original magnetic texture (black arrows). (a) The virtual texture of a coplanar configuration can be noncoplanar resulting in the occurrence of an effective anomalous and topological Hall effect. (b) At a critical out-of-plane canting angle of the real magnetic texture, spin-orbit coupling is compensated and the virtual magnetic texture is coplanar. In this case, the anomalous Hall effect is absent.

technical paper

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

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.

 Voltage control of Néel domain wall interactions and pinning sites

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 ferrimagnetic order and voltage-assisted spin texture writing by solid-state hydrogen gating

Exchange bias is a well-known and very useful effect that occurs due to an interfacial coupling between adjacent thin films of an antiferromagnet and a ferro(ferri)magnet [1]. Controlling exchange bias in an antiferromagnet/ferro(ferri)magnet thin film system by application of a gate voltage is much sought-after for spintronic applications. However, since the exchange bias effect is related to the ordering of spins in both the antiferromagnet and the ferro(ferri)magnet, and electric fields have little influence over the spin order, electrical control of exchange bias is very hard to achieve practically. Recently, ionic gating of magnetic heterostructures, where a gate voltage is used to drive ions (O2-, H+ etc.) in or out of the active layers, has yielded very interesting and promising results. Due to the size of the H+ ion, protonic gating is potentially faster and more reversible compared to gating with other ions and is also able to cause large changes in magnetic properties in the underlying active layers, such as modulation of perpendicular magnetic anisotropy in ultra-thin Co films [2]. The utility of protonic gating was further demonstrated when it was shown that ferrimagnetic order in GdCo metal thin films can be dynamically controlled by this method [3]. GdCo is comprised of two sub-lattices, one made up of Gd atoms and one of Co atoms, and the magnetic moments in each sub-lattice point in opposite directions, hence resulting in ferrimagnetic order. Experimental and theoretical analyses show that the protons being inserted into the GdCo reduce the effective magnetic moment of the Gd atoms, and by starting from a Gd-dominated magnetic state, the overall magnetization can be switched 180 degrees through proton gating. Harnessing this functionality, we have shown, in this work, that exchange bias toggling is possible when we incorporate the GdCo ferrimagnet in an exchange biased heterostructure with NiO serving as the antiferromagnet. Using sputter deposition we have engineered a stack with the structure NiO(33)/Pd(1)/GdCo(10)/Pd(10)/GdOx(30)/Au(3) on thermalized Si, which has fully shifted loops, i.e. the exchange bias field is larger than the coercive field (HEB&gt;HC), as depicted in Fig. 1(a). When a sufficiently high (&gt;1.5V) positive voltage is applied to the Au top electrode (Pd serves as the bottom electrode), atmospheric water is split and the proton is transported through the GdOx electrolyte, into the GdCo(10)/Pd(10) layers. Starting from a Gd-dominated state of the GdCo, upon enough insertion of hydron ions after applying +1.8V, the dominant sublattice of GdCo switches from Gd to Co, as expected from previous work, but is now also accompanied by a toggling of the exchange bias direction, as shown in Fig. 1(b). The effect is fully reversible, as shown in Fig. 1(c). We can understand this phenomena when we realize that although the gating does nothing to the relative spin orientation in either the ferrimagnet or the antiferromagnet, the ferrimagnet’s magnetization direction depends not only on the spin orientation of the sublattices but also on the relative magnitude. The magnetization direction flips 180 degrees during gating but the relative spin orientation across the interface remains the same. Essentially, if initially the pinned magnetization is +Mz (meaning HEB is -ve) then after gating the pinned magnetization will be -Mz necessarily making HEB +ve. To the best of our knowledge, this is the simplest demonstration of exchange bias toggling to date, since our system does not rely on any complex materials or growth processes and works at room temperature, as opposed to some previous works [4,5]. Due to the small size of the proton, our method is highly reversible, and initially we have shown full cycling toggling of the exchange bias up to 10 times, where the hysteresis loop remains fully shifted throughout the process (Fig. 2) as evidenced by HEB/HC being over 1. We believe our simple but robust method to toggle exchange bias can serve as a platform for demonstrating more functionalities and/or devices and can have broad implications for spintronics in general. This work is supported in part by SMART, one of seven centers of nCORE, a Semiconductor Research Corporation program, sponsored by National Institute of Standards and Technology (NIST).

**References:**

[1]. Y. Fan et al., Nature Nanotechnology, Vol. 8, p. 438 (2013) [2]. A.J. Tan et al., Nature Materials, Vol. 18, p. 35 (2019) [3]. M. Huang et al., manuscript under review [4]. S. Wu et al., Physical Review Letters, Vol. 110, p. 067202 (2013) [5]. X. He et al., Nature Materials, Vol 9, p. 579 (2010)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/36bc1892deb385267892bf4aee4bb5dd.jpg)
Fig. 1. MOKE hysteresis loops of the structures described in the text in the (a) initial state, (b) after applying +1.8V for 180s and (c) after applying 0V for 80s. The MOKE signal follows the Co magnetic moments, which is why in the Gd dominated state (a) and (c) the loops are inverted

![](https://assets.underline.io/uploads/markdown_image/1/image/ee59777b4e38f83110d076127ab1a339.jpg)
Fig. 2. Reversible toggling of exchange bias in sample with fully shifted loops, where the voltage was toggled between +1.8V and 0V with period of 40s. Note that the erratic oscillations in EB towards the latter cycles are due to noise during measurement of individual hysteresis loops

 Exchange Bias Toggling in GdCo/NiO Thin Film System by Solid-State Hydrogen Gating

Magnetic random-access memory (MRAM) has recently emerged as a promising way for next generation memory due to its nonvolatility, low power consumption and high density. How to achieve energy-efficient magnetization switching is one of the main challenges for MRAM. Recent studies propose spin orbit torque (SOT) as a promising method for data writing [1]. However, an external magnetic field along current direction is usually needed to realize deterministic switching. It can be overcome by using antiferromagnet/ferromagnet structures in which exchange bias can replace external field [2]. Voltage controll of magnetic anisotropy (VCMA) is another candidate method for lowering down the energy height by inducing electron accumulation at the ferromagnet/oxide interface[3]. Combination of VCMA and SOT effect may lead to decrease of power consumption by a gate voltage lowering the energy barrier height. For example, Yoda et.al propose a new spintronics-based memory employing the VCMA effect as a bit selecting principle and the SOT effect as a writing principle. [4]. Back et al reported that a gate voltage can be used to manipulate both perpendicular magnetic anisotropy and spin orbit torque[5]. Hence, it is essential to make it clear whether the voltage-controlled spin-orbit effective torques (VCSOT) or voltage controlled magnetic anisotropy causes the decrease of critical current, or both. In this work, we investigate VCSOT effect and VCMA effect in the IrMn/CoFeB/MgO structures. The film structures were thermally oxidized Si substrate/Ta(2)/IrMn(5)/CoFeB(0.94)/MgO(2.5)Al2O3(5), (with film thickness in nm in the parentheses). The films were patterned into Hall bar by ion beam etching (IBE) and lithography as shown in Fig. 1(a). Fig. 1(b) shows the micrograph of the Hall bar and the measurement setup. Furthermore, to explore voltage-gated SOT switching, a gate voltage is applied at the top electrode and the switching loops are obtained with no external field applied as shown in Fig. 1(d). Figure 1(f) presents critical current densities with 5mT field under different gate voltages. The critical current density Jc shows a decreasing tendency with increasing VMgO. These results indicate that a gate voltage can be used to effectively modulate SOT-driven perpendicular magnetization switching and this is a feasible writing method for low energy consumption. Next, spin-orbit effective fields at the different voltage were measured by using a second harmonic method as shown in Fig. 2 (a). The damping-like torque efficiency decreases with positive voltage increasing, whereas the field-like torque efficiency shows weaker voltage dependence. These results demonstrate that VMgO can modulate spin torques in IrMn/CoFeB/MgO system. The damping-like torque efficiency under VMgO of -0.4 V is 1.5 times larger than that under VMgO of 0.4 V. A larger JC is required for the magnetization switching under positive voltage due to the reduction of θSHE. Through VCSOT effect, positive voltage tends to increase Jc, which is reverse to the experimental results in Fig. 1(f). In-plane hysteresis loops at different gate voltages are obtained as shown in Fig. 2(b). The dependence of anisotropy energy Keff(VMgO) on gate voltage is shown in Fig.2 (c). By linear fitting, Keff decreases by 55% when VMgO changes from -0.6V to 0.6V. According to theoretical research from Ref.[6], the Jc for SOT switching is proportional to Keff. Therefore, the decrease of Keff will lead to significant reduction of Jc. Figure 2(d) shows the interfacial anisotropy coefficient Ki. We can see that the Ki changes linearly with VMgO due to VCMA effect. From the slope of the linear fitting, the VCMA coefficient ξ of 34.3 fJ/Vm is obtained for IrMn/CoFeB/MgO structure. Therefore, VCSOT effect is not the reason for the reduction of Jc in the voltage-gated SOT switching. The decrease of Jc in the voltage-gate SOT switching is mainly caused by the reduction of PMA due to VCMA effect. In summary, we demonstrate field-free switching in the IrMn/CoFeB/MgO structures and show the critical switching current can be modulated by the control voltage. Moreover, VCSOT effect and VCMA effect are explored. We measured spin-orbit effective torques under different voltage. We concluded that through VCSOT effect, positive voltage tends to increase Jc, which is reverse to the experimental results. Next, we measured anomalous Hall resistance at different voltage to explore VCMA effect. From quantitative calculation, the critical switching current at different voltage is consistent with experiments. Therefore, we concluded VCSOT effect hindered the reduction of Jc, while the VCMA effect made a main contribution to the Jc reduction. This work provides comprehensive understanding of voltage-gated spin torque switching.

**References:**

[1] M. Miron, K.Garello and G. Gaudin., Nature, vol. 476, no. 7359, pp. 189–193. (2011). [2] S.Z. Peng, D. Q. Zhu and W S Zhao., Adv. Electron. Mater. Vol. 5 pp. 1100134. (2019). [3] W.G. Wang, M. Li and C.L.Chien., Nat.Mater, vol. 11, no. 1, pp. 64–68. (2012). [4] H. Yoda, N. Shimomura and A. Kurobe., IEEE Electron Devices Meet. (IEDM). pp. 27.6.1-27.6.1. (2016). [5] L. Chen, M.Gmitra and C.H.Back., Nat. Electron., vol. 1, no. 6, pp. 350–355, (2018). [6] K.S. Lee, S.W. Lee and K.J. Lee., Appl. Phys. Lett. vol.102, pp. 112410. (2013).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1ed57da223949489824ad933c4957076.jpg)
Fig.1 (a) Schematic of IrMn/CoFeB/MgO structures. (b) Micrograph of the Hall bar and the measurement setup. (c) SOT-driven magnetization switching with various in-plane external field. Voltage-gated SOT switching under different gate voltages with (d)no external field and (e)5 mT in-plane external field. (f) Critical current density Jc dependence on voltage applied to MgO (VMgO) with 5 mT field applied.

![](https://assets.underline.io/uploads/markdown_image/1/image/177e5ff0550d0f2846103ad2879572c8.jpg)
Fig.2 (a) The dependence of spin-orbit effective torques on voltage in MgO layer (VMgO). The error bars are standard deviations. (b) Anomalous Hall resistance (RHall) under different gate voltages (VMgO) (c) Keff and (d) Ki dependence on VMgO.

 Experimental demonstration of voltage-gated spin-orbit torque switching in antiferromagnet/ferromagnet structure

The voltage control of magnetic anisotropy (VCMA) is an active field of research, as it is a promising route towards the realization of low power spintronic devices. Among a few proposed VCMA mechanisms, voltage-induced oxygen migration (i.e., magneto-ionic effect) has demonstrated large modulation of magnetic anisotropy. In this work, we study the voltage control of the interfacial perpendicular magnetic anisotropy (PMA) in a series of Pt/Co/MOx (M=Al and Tb) trilayers [1]. The stacks were patterned by electron beam lithography and ion beam etching into stripes (1 – 50 µm width) then covered with a 10 nm thick ZrO2 dielectric layer and a 6 nm thick Pt top electrode to form capacitor-like structures. We show a large modification of interfacial magnetic anisotropy (β&gt;2000fJ(Vm)) accompanied by a non-volatile effect at room temperature (RT). This modification is explained in terms of voltage-induced oxygen migration towards/away from the Co/MOx interface. The Co films can be reversibly switched from an under-oxidized state with in-plane anisotropy (IPA) to an optimal oxidized state with maximum PMA, to an over-oxidized state under applied voltage (see figure). The switching time depends on the amplitude of the applied voltage. Here, we demonstrate that the switching time can be reduced from several seconds down to few milliseconds at RT by adjusting the voltage amplitude.

**References:**

[1] A. Fassatoui, S. Pizzini, Phys. Rev. Appl. 14, 064041 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9867cc6713f62186cd8b1b84462308ad.jpg)
Variation of the coercivity of a Pt/Co/TbOx/ZrOx/Pt device during the application of different gate voltages: (a) for positive voltages showing the transition from an over-oxidised state, to max PMA, to under-oxidised state with in-plane anisotropy; (b) transition from under-oxidised to over-oxidised state with negative voltage. In both cases, increasing slightly the voltage gives rise to large acceleration of the electric-field effect.

 Large Reversible Voltage Manipulation of Interfacial Magnetic Anisotropy in Pt/Co/Oxide Multilayers

Controlling magnetic properties of materials with electric fields (EFs) is of remarkable importance in spintronics applications due to its potential for much lower power consumption devices. Much like the power efficiency, two other sought-after features to incorporate are non-volatility and reversibility. This has been demonstrated with magneto-ionics, where EFs induce the migration of mobile ionic-species towards/away from the magnetic interface [1–4]. However, several works conducted in a variety of magneto-ionic materials have revealed important issues like irreversibility, the need of elevated temperatures to recover the initial states and chemical and structural degradation [4–6]. Recent studies have also focused on the differences in magneto-ionic reversibility between various oxides [7], pointing out the importance of the mechanisms governing ionic conduction. In this study, we unravel a higher degree of complexity in an HfO2 based device and demonstrate the coexistence of two distinct magneto-ionic regimes. For this purpose, we chose amorphous Ta/CoFeB/HfO2 thin films grown by magnetron sputtering with an ionic-liquid gate. It is shown that a gate voltage drives the anisotropy of the system from an underoxidized state exhibiting in-plane anisotropy (IPA) to an optimum oxidation level resulting in perpendicular anisotropy (PMA) and further into an overoxidized state resulting IPA in a non-volatile manner. The IPA (underoxidized) to PMA regime is found to be highly irreversible and is significantly faster than the PMA to IPA (overoxidized) regime which is fully reversibile under the same gate voltages. In addition, the effective damping parameter also shows a marked dependence with the anisotropy evolution in different regimes; IPA to PMA regime going from 0.029 to 0.012 and only a modest increase to 0.014 in the PMA to IPA regime. The existence of two magneto-ionic regimes has been linked to a difference in the chemical environment of oxygen species added to under/over oxidized layers. Our results show that different magneto-ionic regimes can coexist and that their identification is of great importance for tuning the properties and the design of high performance spintronics devices based on magneto-ionics.

**References:**

[1] U. Bauer, S. Emori, and G. S. D. Beach, Nat. Nanotech. 8, 411 (2013) [2] U. Bauer, L. Yao, A. J. Tan, and G. S. D. Beach, Nat. Mater. 14, 174 (2015) [3] A. J. Tan, M. Huang, C. O. Avci, and G. S. D. Beach, Nat. Mater. 18, 35 (2019) [4] L. Herrera Diez, Y. Liu, D. Gilbert, M. Belmeguenai, J. Vogel, S. Pizzini, E. Martinez, A. Lamperti, J. Mohammedi, A. Laborieux, et al., Phys. Rev. Appl. 12, 034005 (2019) [5] C. Bi, Y. Liu, T. Newhouse-Illige, M. Xu, M. Rosales, J. W. Freeland, O. Mryasov, S. Zhang, S. G. E. te Velthuis, and W. G. Wang, Phys. Rev. Lett. 113, 267202 (2014) [6] D. A. Gilbert, A. J. Grutter, E. Arenholz, K. Liu, B. J. Kirby, J. A. Borchers, and B. B. Maranville, Nat. Commun. 7, 12264 (2016) [7] A. Fassatoui, Phys. Rev. Appl. 14, 064041 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/739c64ac3b9dc0fe2b2d809ba80bb186.jpg)
Fig. 1: Graphic representation of the Ta/CoFeB/HfO2 magneto-ionic stack covered with an [EMI-TFSI] ionic liquid gate. The gate voltage induces motion of oxygen species in HfO2 that drive anisotropy changes in the CoFeB layer.

![](https://assets.underline.io/uploads/markdown_image/1/image/a9cfb45146f55322b6c3cafa5270f8f3.jpg)
Fig. 2: A progressive oxidation is induced upon exposure to a gate voltage GV=-2V. Different exposure times t drive the system from IPA (a, initial state) through regime I (IPA to PMA) (b) into PMA (c) and back to IPA through regime II (PMA to IPA) (d).

 Reversible and Irreversible Magneto-ionic Regimes in Ta/CoFeB/HfO2

Antiferromagnetic spintronic devices have the potential to greatly outperform their current ferromagnetic counterparts due to their robustness to external fields, ultrafast dynamics and potential for high data density storage. Mn2Au is one of the most promising materials for antiferromagnetic spintronics due to its very high Néel temperature, moderate in-plane anisotropy and layered two sublattice spin structure [1]. Here we present atomistic spin simulations that allow us to estimate the temperature dependent magnetic properties of Mn2Au single crystals. We introduce a thermodynamically consistent spin Hamiltonian guided by previous ab-initiocalculations [3], which includes easy plane and rotational anisotropy terms. Temperature scaling of the anisotropy parameters is crucial for the performance optimization of antiferromagnetic devices, from the scale of magnetic domains to stability under thermal activation [2]. However, scaling in antiferromagnet has been barely studied due to the inherent difficulties to measure antiferromagnetic order parameter, n. We find a scaling of the easy plane anisotropy with the sublattice magnetization of n2.32. This is consistent with its two-ion origin [4], whereas the rotational anisotropy presents a n10dependence, consistent with the Callen-Callen scaling in ferromagnets of a 4thorder anisotropy. The crystal structure is shown in Fig. 1 showing two distinct Mn sites with a 180 degree relative orientation. We obtain quantitative agreement with experimental measurements of the temperature dependence of the order parameter by using a thermal-bath temperature rescaling, which has been demonstrated to work in ferromagnets. Our model provides a critical temperature (Neel) of 1256 K from the ab initiovalues of the exchange interactions. Due to the presence of four easy directions, the domain wall structure of 90 degrees and 180 degrees are stable in Mn2Au. We find a narrow domain wall width of 23.13 +/- 0.085 nm at zero temperature in both cases as shown in Fig. 2. Applying thermal spin fluctuations yields a domain wall width that increases with temperature, up to 25 +/- 2 at 300K and to 53 +/- 18 nm at 800K, matching well with experimental results of Sapozhnik et al[5]. In the case of a 180 degree wall the two separate 90 degree walls are weakly bound and separate at high temperatures. Further analysis of the domain wall width could provide information about the scaling of the exchange stiffness in antiferromagnets, a pivotal parameter for the construction of hierarchical multiscale models in magnetism, which are missing for antiferromagnets. When an electrical current is applied to the system along the [110]direction a spin orbit torque (SOT) is generated by the flow of electrons through the system interacting with the Au atoms. These SOTs cause the domain wall to move. For a Neel anti-damping SOT field of 0.05 T we find a domain wall velocity of 250 m/s at 0K, which is different to the usually fast dynamics displayed by anti-ferromagnetic materials.This is because the SOT applied is precessional in nature and both sublattices have precessional motion in the same direction. Applying a field-like SOT yields the usual fast motion of the domain walls. The atomistic spin model [6] used here to compute thermal properties of Mn2Au provide new insights on the temperature dependent magnetic properties – magnetic order and anisotropies and domain wall widths in Mn2Au. The understanding of the macroscopic parameters is necessary for the developments of multiscale models of Mn2Au that could greatly help to the development of novel antiferromagnetic spintronic devices.

**References:**

[1] V. Barthem et al Nature comminucations 4 (2013) [2] L Rózsa, S Selzer, T Birk, U Atxitia, U Nowak Physical Review B 100, 064422 (2019) [3] A. B. Shick et al. Phys. Rev. B 81, 212409 (2010) [4] R.F.L. Evans et al. Phys. Rev. B 102, 020412(2020) [5] A. A. Sapozhnik, Phys. Rev. B 97, 134429 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c38d47266d2475804a86e9579cb756b5.jpg)
Magnetic unit cell of Mn2Au. The Mn atoms (atoms with arrows) form two anti-parallel sublattices aligned along the [110] and [-1-10] crystal directions. The gold coloured atoms represent Gold.

![](https://assets.underline.io/uploads/markdown_image/1/image/22f100ca89efd90f63a1ac9868813590.jpg)
90 degree and 180 degree domain wall profiles.

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