Voltage-driven displacement of magnetic vortex cores

Magnetic vortex cores in polycrystalline Ni discs underwent non-volatile displacements due to voltage-driven ferroelectric domain switching in single-crystal BaTiO3. This behaviour was observed using photoemission electron microscopy to image both the ferromagnetism and ferroelectricity, while varying in-plane sample orientation. The resulting vector maps of disc magnetization match well with micromagnetic simulations, which show that the vortex core is translated by the transit of a ferroelectric domain wall, and thus the inhomogeneous strain with which it is associated. The non-volatility is attributed to pinning inside the discs. Voltage-driven displacement of magnetic vortex cores is novel, and opens the way for studying voltage-driven vortex dynamics.


Introduction
Magnetic vortices and skyrmions are topological structures that currently attract huge interest because their nanoscale dimensions [1,2] and inherent stability [3][4][5] render them suitable for storing and carrying information [6,7]. Moreover, they can be manipulated by static [8] and variable [9] magnetic fields, electric currents [10,11], voltage [12][13][14][15][16] and strain [17]. Magnetic vortices can display four stable states because the core can be magnetized up or down (vortex polarity), and the surrounding in-plane magnetization can curl clockwise or Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. anticlockwise (vortex chirality). Given that polarity and chirality can be independently switched by both magnetic fields and electric currents [8][9][10]18], researchers have been stimulated to pursue the prospect of magnetic media with two bits per element [19,20].
Magnetic data storage devices would be more energy efficient if data were written using electric fields instead of electric currents or magnetic fields. This technological goal forms a core aspect of the large volume of research into magnetoelectric effects [21], where voltage-controlled magnetic order has been demonstrated in different types of materials. Juxtaposing a magnetic film with another material (typically, but not exclusively a ferroelectric) can result in large room-temperature magnetoelectric effects that are mediated by strain [22][23][24][25][26], charge [27] or exchange bias [28][29][30]. These magnetoelectric effects may be identified in the magnetic film via changes of magnetization [22,28,31], anisotropy [32,33], coercivity [34] and magnetic domain wall motion  [35]. Moreover, the strategy of juxtaposition can achieve magnetoelectric control of magnetic vortices [13][14][15][16] and skyrmions [36].
Vortex core displacements have been driven by magnetic fields [9,37] and electrical currents [10], permitting core gyration and core polarity switching, but the voltage-driven motion of a vortex core has been hitherto studied only theoretically [38]. Here we report the first experimental demonstration of this phenomenon by studying polycrystalline Ni discs (diameter 2 µm, thickness 25 nm) that were grown on a ferroelectric substrate of BaTiO 3 (BTO). By imaging the magnetic structure of the discs as well as the surrounding ferroelectric domains, we find that vortex cores are displaced after ferroelectric domain switching has taken place (a → c → a). Corroborative micromagnetic simulations required just 60% of the 10 -2 uniaxial strain associated with this switching. These simulations show that the vortex core tracks the passing a-c domain wall, implying that the vortex core is translated by the inhomogeneous strain associated with the ferroelectric domain wall. The vortex core does not return to its central starting position due to pinning inside the magnetic disc. Our findings complement the studies of vortex core displacement using magnetic fields [9,37] and electrical currents [10], and should stimulate further work on the magnetoelectric control of magnetic vortices.

Experimental methods
We used electron-beam-assisted evaporation to deposit Ni discs of thickness 25 nm and diameter 2 µm on electroded substrates of 0.5-mm thick BTO in the pseudo-cubic (001) orientation (figure 1). The discs were capped with 3 nm of Cu, and formed 10 × 10 arrays with nearest-neighbour centre-tocentre separations of 6 µm. All fabrication details are given in [13], where the samples differ because the Ni discs are 1 µm in diameter (such that vortex states are magnetoelectrically annihilated rather than modified as they are here). Note that vortices are known to form in nominally unstrained discs of both sizes [39].
Photoemission electron microscopy (PEEM) with contrast from x-ray magnetic circular dichroism (XMCD) was to image used to image the magnetic order in our Ni discs, PEEM with contrast from x-ray linear dichroism (XLD) was used to image the surrounding ferroelectric domains in BTO, and PEEM with contrast from x-ray absorption (XAS) was used to image topographical/chemical contrast. The PEEM probe depth was~7 nm and the lateral resolution was typ-ically~50 nm. We show either averaged PEEM images, or magnetic vector maps that are derived from averaged XMCD-PEEM images obtained for two orthogonal sample orientations. All details are exactly as specified in [13].
Micromagnetic simulations performed using MuMax3 [40] are detailed in [13]. The resulting vector maps average the magnetization in the top 5 nm, which is similar to the~7 nm PEEM probe depth. We have neglected the relatively small cubic magnetocrystalline anisotropy of Ni (K c = − 5 kJ m −3 ) [22] because it averages out in our polycrystalline discs where the grains are small (a lateral size of~100 nm was recorded for similar films [22]). Simulations that discretized a disc into randomly oriented grains of this lateral size yielded similar results, and so our assumption of a continuous medium is reasonable.

Experimental results
The XAS-PEEM image in figure 1 ). The XMCD-PEEM data for the discs reveals magnetic inhomogeneity in all but three discs, which are uniformly dark in each image. The XLD-PEEM data for the intervening BTO reveals two types of ferroelectric domain (figure 2(a)), which are not resolved after rotating by 45 • (figure 2(b)), and which reappear with inverted contrast after rotating by 90 • (figure 2(c)). The ferroelectric domains are therefore identified as a 1 -a 2 domains, with c-axis polarizations that alternate between x and y, as shown via the blue schematics that depict tetragonal unit cells.
The composite image in figure 2(c), which we repeat in figure 3(a), is modified by the application of 150 V across the BTO substrate, such that the a 1 -a 2 domains are annihilated in favour of a single c-domain ( figure 3(b)). This 90 • ferroelectric domain switching imparts uniaxial compressive stress to all Ni discs along either x or y, according to the orientation of the annihilated a domain. The switching aligns magnetic domain walls with the local compression, as expected given that domains in negative-magnetostriction Ni should be magnetized along this direction. Magnetic domain walls that were present before applying the voltage (figure 3(a)) are thus rotated by 90 • by the ferroelectric domain switching ( figure 3(b)).
Vector maps of magnetization were obtained for discs 1-3 before, during and after applying the 150 V (figure 4). Initially at 0 V, discs 1 and 2 present vortex states that are anisotropic, such that they contain the domain walls discussed earlier [13]. These domain walls are Néel walls given that the magnitude of the XMCD asymmetry remains fairly constant across them,  i.e. the magnetization remains in-plane, as expected for a magnetic material whose thickness is just 25 nm. The vortex anisotropy arises as a consequence of the tensile growth strain that is associated with the underlying a domain in the BTO substrate [41]. The anisotropic vortex state is dominated by the presence of two large regions that possess a net magnetization along the direction of compressive strain, and these two regions are inhomogeneously magnetized to yield curling on each side of the intervening Néel walls that widen on approaching the disc edge. Note that this configuration should not be mistaken for a Bloch line, which lies between homogeneous Néel walls separating homogenous magnetic domains [42]. The application of +150 V rotates the average direction of magnetization by 90 • , enhances the anisotropy of the magnetization distribution, and creates in each domain wall multiple chiral discontinuities (vortex/antivortex cores). The return to 0 V restores a magnetization distribution that is reminiscent of the initial state. However, the multiple vortex cores that were created at +150 V are displaced off-centre. This complexity (and the complexity that we will now describe for disc 3) is possible because our discs are large enough to support metastable states. Disc 3 behaves differently (figures 4(g)-(i)). It formed a single domain at 0 V, despite exceeding the critical diameter, which is 400 nm for a Ni disc of this thickness [39]. This is because the single-domain state was stablized by tensile growth strain from the BTO substrate, as seen for 1 µmdiameter discs of Ni, whose magnetization lay perpendicular to the underlying ferroelectric polarization as expected [41]. Here, the magnetization lay at~45 • to the underlying ferroelectric polarization, perhaps because the relatively small underlying a domain experienced compressive strain from ferroelectric domains elsewhere. The application of +150 V caused the disc to develop two anisotropic vortices, with the preferred magnetization along x. On returning to 0 V, vortex cores are created (annihilated) along the upper (lower) Néel wall, whose position is barely modified.

Modelling
The lowest energy state for unstrained Ni discs of our size would be an isotropic vortex [39], as we have confirmed for ourselves, but this was not observed here because of the growth strain. Assuming this strain to be ε y = 2 × 10 −3 resulted in an anisotropic vortex ( figure 5(a)) that matches well the initial state of disc 1 ( figure 4(a)) and captures the curling and wall widening discussed above. We previously found that BTO produces this growth strain in Ni discs of the same thickness and half the diameter [13], resulting in an anisotropy constant K = − 3 2 Eε y λ = − 10 kJ m −3 that has the correct order of magnitude for Ni films on BTO [22].
The a → c switching at +150 V figure 3(b) is represented by switching the tensile strain of ε y = 2 × 10 −3 to a compressive strain of ε y = − 4 × 10 −3 (K = 20 kJ m −3 ). This produces an anisotropic vortex ( figure 5(b)) that corresponds to the state of disc 1 at +150 V ( figure 4(b)). In agreement with experiment, the vortex is not annihilated, and its preferred magnetization direction is rotated by 90 • . Moreover, the multiple vortex/antivortex cores that we observed arise in our model ( figure 5(c)) by accessing a state that is just 0.2% higher in energy than the ground state with a single vortex core (figure 5(b)). The observation of multiple vortex/antivortex cores is therefore energetically reasonable. Note that the Néel wall containing these cores (figures 5(c) and (f)) should not be confused with a cross-tie wall, in which the energy cost of a 180 • domain wall is avoided by the presence of 90 • domains.
On returning to 0 V, imposing the y-axis strain associated with c → a switching recovers the initial vortex state in our simulations, but the observed displacement of the core cannot  ) and (e), and show a multivortex state whose energy is 0.2% higher. The magnitude of the local magnetization is denoted M. be achieved by varying the magnitude of this homogeneous strain (which only modifies the degree of anisotropy [13]). We will therefore proceed to understand the observed core displacement by considering the inhomogeneous strain associated with the passage of an a-c wall during the switching process (figure 6), neglecting spin precession effects because ferroelectric domain motion is slower than spin dynamics. Regions of the disc that overlie the a domain experience the tensile growth strain along y, whereas regions of the disc that overlie the c domain experience the compressive strain along y. Motion of the negligibly thick wall displaces the vortex core, which is repelled by the strain (and resulting magnetic anisotropy) gradient, thus reducing the exchange energy. The vortex core displacement is reversible, except for the final step (figures 6(j) and (k) that recovers the state we simulated with ε y = 2 × 10 −3 ( figure 5(a)). Similar simulations of the multivortex state reveal that the cores are initially displaced and compressed (figures 7(a)-(g)) until they coalesce into a single core (figures 7(g) and (h), which then (figures 7(i)-(k)) mimics the single core that is described in figure 6.
The non-volatile vortex core displacement that we observed for discs 1-2 (figure 4) can be understood by considering both the inhomogeneous strain associated with the ferroelectric domain wall, and pinning associated with the ferromagnetic domain walls in the anisotropic vortex state. In the absence of pinning, the passage of the ferroelectric domain  wall would recover the ground state with a vortex core at the disc centre. However, we may infer that the wall is pinned by defects, such that the vortex core does not return to the centre figure 6(j) and (k)).
The magnitudes of the strains that reproduce our experimental data are similar to the strains that created and annihilated vortices in 1 µm-diameter discs of Ni on BTO [13], and yet here the vortices are not annihilated. Replacing the tensile growth strain of 2 × 10 −3 with the compressive strain of −4 × 10 −3 implies that the a → c switching produced a compressive strain of −6 × 10 −3 , which is 60% of the 10 -2 uniaxial strain calculated from the BTO lattice parameters for 90 • ferroelectric domain switching.
In summary, we have used XMCD-PEEM to study anisotropic magnetic vortices in 25 nm-thick Ni discs of diameter 2 µm, and we have used XLD-PEEM to study the surrounding BTO substrate. Voltage-driven ferroelectric domain switching served to modify-not annihilate-the vortices, whose cores underwent non-volatile displacement. Using micromagnetic simulations, we attribute this displacement to the inhomogeneous strain associated with the transit of a ferroelectric domain wall, and we attribute the non-volatility to the pinning of the ferromagnetic walls in the anisotropic vortices. Our demonstration of voltage-driven core motion is novel, and could pave the way for time-resolved studies of voltage-driven core dynamics. Such studies would require epitaxial discs of excellent crystal quality, and fast, repeatable pulses of inhomogeneous strain.