Shear-strain-mediated magnetoelectric effects revealed by imaging

Large changes in the magnetization of ferromagnetic films can be electrically driven by non-180° ferroelectric domain switching in underlying substrates, but the shear components of the strains that mediate these magnetoelectric effects have not been considered so far. Here we reveal the presence of these shear strains in a polycrystalline film of Ni on a 0.68Pb(Mg1/3Nb2/3)O3–0.32PbTiO3 substrate in the pseudo-cubic (011)pc orientation. Although vibrating sample magnetometry records giant magnetoelectric effects that are consistent with the hitherto expected 90° rotations of a global magnetic easy axis, high-resolution vector maps of magnetization (constructed from photoemission electron microscopy data, with contrast from X-ray magnetic circular dichroism) reveal that the local magnetization typically rotates through smaller angles of 62–84°. This shortfall with respect to 90° is a consequence of the shear strain associated with ferroelectric domain switching. The non-orthogonality represents both a challenge and an opportunity for the development and miniaturization of magnetoelectric devices. Non-orthogonal magnetization switching is related to the shear strain associated with ferroelectric domains, with implications for magnetoelectric devices.


Macroscopic and microscopic magnetoelectric effects
All data presented in the main paper were obtained from a single sample, whose fabrication and history are described in Methods. The electrically virgin sample possessed no in-plane magnetic anisotropy (Supplementary Note 1) because the unoriented grains in the Ni film precluded a net magnetocrystalline anisotropy, and because the unoriented domains in the PMN-PT substrate precluded a net stress anisotropy 27 (macroscopic magnetic measurements are described in Methods).
Poling the substrate by applying and removing an electric field of E = -1 MV m -1 along +z (Fig. 1) created in our negative-magnetostriction film a non-volatile uniaxial magnetic anisotropy along y, and the magnitude of this anisotropy implies a y-axis compressive strain of 0.08% consistent with ref. 26 (Supplementary Note 1). On subsequently cycling the electric field at magnetic remanence, we found minima of y-axis magnetization My (Fig. 2a) and maxima of x-axis magnetization Mx  25,26,29,35,47,49,51 . The minima in My correspond to a ~50% change of magnetization, and a peak ME coupling coefficient of y = 0dMy/dE  1.610 -6 s m -1 (Fig. 2b).
Magnetic hysteresis loops that were measured along orthogonal in-plane directions at fields of E1 = 0 (Fig. 3a), E2 = +0.167 MV m -1 (Fig. 3d) and E3 = +1 MV m -1 (Fig. 3g) are also consistent with the hitherto expected 90 rotation of a global easy axis 25,26,29,35,47,49,51 created by poling. This is because both the first (E1 → E2) and second (E2 → E3) field steps appear to interconvert the hard and easy directions, as seen more clearly by comparing polar plots of magnetic-hysteresis-loop squareness at each electric field (blue data, Fig. 3c,f,i). However, we will see below that local ME measurements, and calculations based on unit-cell distortions, reveal that the switched state at E2 involves two misaligned uniaxial magnetic anisotropies rather than a single easy axis.
In order to investigate local ME effects in the same sample, we used XMCD-PEEM to obtain high-resolution vector maps (see Methods) of the in-plane magnetization direction  at E1, E2 and E3 (Fig. 3b,e,h). At E1 (Fig. 3b) and E3 (Fig. 3h), the 50 m-diameter field of view contained a small number of large domains, whose magnetizations lay approximately along y. For the intervening state at E2 (Fig. 3e), the same field of view contained a large number of small domains, whose magnetizations appear to have lain approximately along x (except for a few regions that did not switch). However, the angular resolution of the colour wheel is insufficient for visual inspection to confirm that the local magnetization typically rotated by the 90 that one would expect from our macroscopic measurements (blue data, Fig. 3c,f,i), previous macroscopic measurements 25,26,29,35,47,49,51 , and previous microscopic measurements 29,35 . Plotting the pixel magnetization directions in our vector maps on polar plots (red data, Fig. 3c,f,i) also gives the superficial impression that each electric-field step typically rotated the local magnetization by roughly the hitherto expected 25,26,29,35,47,49,51 value of 90. However, at E1 and E3 there are two slim lobes, whereas the structure of the polar plot at E2 is more complex. This observation inspires the following pixel-by-pixel comparison of our vector maps.

Pixel-by-pixel comparison
The changes of magnetization direction  are mapped for both E1 → E2 (Fig. 4a) and E2 → E3 (Fig. 4e), after excluding the small (white) areas between magnetic domain walls in the vector maps at E1 (Fig. 3b) and E3 (Fig. 3h), such that we consider only regions where the magnetization returned to its original direction. The number N of unexcluded green pixels in our vector map at E1 that underwent magnetization direction change  during E1 → E2 is plotted in Fig. 4b, while the number N of unexcluded purple pixels in our vector map at E1 that underwent magnetization direction change  during E1 → E2 is plotted in Fig. 4c. Similarly, the number N of unexcluded green pixels in our vector map at E3 that resulted from magnetization direction change  during E2 → E3 is plotted in Fig. 4f, while the number N of unexcluded purple pixels in our vector map at E3 that resulted from magnetization direction change  during E2 → E3 is plotted in Fig. 4g. Having thus considered separately each type of magnetic domain (unexcluded green and purple pixels) in our vector maps at E1 and E3, we see that the magnetization of many pixels switched by large angles  that typically fall well short of the hitherto expected 25,26,29,35,47,49,51 value of 90 (Fig. 4b,c,f,g).
Given that the two FWHM peaks in Fig. 4b (in Fig. 4c) are essentially interchanged in Fig. 4f (in Fig. 4g), we infer that the net effect of the two field steps was to switch and switch back the magnetization of many pixels by large angles of typically less than 90. To identify which pixels switched and switched back in this way, we filtered our maps of  (Fig. 4a,e) using the colour code in N() (Fig. 4b,c,f,g) to produce simplified maps of  (Fig. 4d,h). Comparison of the two simplified maps confirms that the magnetization of many pixels switched and switched back by large angles of typically less than 90 (yellow, Fig. 4i). These regions are either purple in Fig. 4d and green in Fig. 4h, or else they are green in Fig. 4d and blue/purple in Fig. 4h.
Having used our formal pixel-by-pixel comparison ( Fig. 4) to reveal that the magnetization of many pixels switched and switched back by large angles of typically less than 90, we will now investigate whether this sub-90 switching could have been directly identified from our vector maps ( Fig. 3b,e,h) and polar plots (red data, Fig. 3c,f,i). In our vector maps, pixels coloured purple at E1 and E3 were typically red/blue at E2, while pixels coloured green at E1 and E3 were typically orange/cyan at E2, thus confirming the sub-90 nature of the switching. By splitting our polar plot at E2 (Fig. 3f) in order to distinguish pixels that were green at both E1 and E3 (brown data appearing in both Fig. 5a,b) from pixels that were purple at both E1 and E3 (black data appearing in both exclude the pixels between the magnetic domain walls in our vector maps at E1 and E3. Fig. 5a compares the split polar plot at E2 with the polar plot at E1 (pink), and Fig. 5b compares the split polar plot at E2 with the polar plot at E3 (pink). Fig. 5a,b also show the modal changes of magnetization for unexcluded pixels (Fig. 4b,c,f,g), thus summarizing the key results from Figs 3,4.

The role of shear strain
The changes of local magnetization in our Ni film are typically less than 90 because ferroelectric domain switching in rhombohedral PMN-PT generates not just the well known normal strains, but also shear strains that have surprisingly been hitherto unappreciated. We will now quantify the and +0.26%) by rotating the basis through ∓27.4 in the plane to which the polarization had switched, such that polarization switching into the x-y plane at E2 created magnetic easy axes in the film at the complementary angles of 62.6 to y (non-vertical grey arrows, Fig. 6f). Specifically, regions of the film that switched with positive shear strain (due to the formation of P 3  domains, left panel in Fig. 6d) developed easy axes at +62.6 to y, while regions of the film that switched with negative shear strain (due to the formation of P 4  domains, right panel in Fig. 6d) developed easy axes at -62.6 to y. The creation and subsequent destruction of these easy axes at 62.6 to y explains the sub-90 magnetic switching that we identified as our key finding via XMCD-PEEM vector maps. A full explanation of our magnetic vector maps appears in Supplementary Note 6.
The misaligned magnetic easy axes at E2, which lie in spatially distinct regions as explained above, are directly evidenced by the paired modal switching angles in Fig. 4b,c,f,g (where ferroelectric domain populations determine peak magnitudes). The two modal angles of 62° and -64° approximately match our predicted values of 62.6°, while six modal angles adopt larger values.
Departures from the predicted values are attributed to strain-mediated interactions between the many ferroelectric domains that are present at all fields, and the prevalence of large-angle switching implies that these interactions tend to suppress the shear strains of interest (because the limiting case of normal strain yy with no shear strain xy would favour the hitherto expected 25,26,29,35,47,49,51 rotations of 90°). Note that the misaligned magnetic easy axes at E2 are not apparent from our macroscopic measurements (blue data, Fig. 3f) because the sum of the projections of these nearby easy axes is larger along the bisecting x axis than it is along either one of these easy axes themselves, such that macroscopic measurements misleadingly imply a single magnetic easy axis along x.
The macroscopic ME effects that we report would be larger if the local magnetization were to rotate by 90° in the absence of any shear. Despite this, our peak ME coupling coefficient (y  1.610 -6 s m -1 , Fig. 2b) Fig. 3a, and reports 1.6×10 −5 s m -1 for a virgin effect measured indirectly while sweeping temperature). We attribute our large ME coupling coefficient to two factors, namely the use of a substrate in which ferroelectric domain switching produces large changes of strain 44 , and the use of a magnetically soft magnetostrictive film with no in-plane anisotropy prior to poling (Supplementary Note 1). For completeness, note that our peak ME coupling coefficient is similar to a prediction 54 of 1.86×10 −6 s m -1 for epitaxial Ni on a ferroelectric substrate whose piezoelectric response was parameterized without ferroelectric domain switching.

Outlook
Our observation that electrically driven shear strain is responsible for sub-90 magnetic switching has immediate implications for the performance of ME random access memory ( our macroscopically measured ME effects exceed the best values on record 21,30 . In future, ME effects mediated by shear-strain components could be exploited to realize nanoscale devices that independently store electrically and magnetically written data. More generally, the shear strains that we have identified in our multiferroic heterostructure represents a new twist in the study of ME effects, thus echoing the 'magnetic twist for ferroelectricity' that was coined elsewhere in the context of multiferroic materials 4 .      Fig. 4 images), and after distinguishing at E2 the pixels that were purple at both E1 and E3 (black data) from the pixels that were green at both E1 and E3 (brown data). Peripheral outer (inner) arcs identify FWHM intervals for each peak in N() by data colour (position on the colour wheel). Modal values of  from Fig. 4b,c,f,g are rendered using the colour wheel, likecoloured arrow lengths are arbitrary.

Macroscopic magnetic measurements. We used a Princeton Measurements Corporation vibrating
sample magnetometer with a bespoke probe 21 whose wiring permitted the application of electric fields.
Magnetic vector maps. Raw images were obtained in zero applied magnetic field on beamline I06 at Diamond Light Source, where we used an Elmitec SPELEEM-III microscope to map secondary-electron emission arising from circularly polarized x-rays that were incident on the sample surface at a grazing angle of 16°. The probe depth was ~7 nm, and the lateral resolution in our 50 m-diameter field of view was typically ~135 nm (corresponding to pixels that represent We averaged 100 raw XMCD-PEEM images to obtain a single XMCD-PEEM image for each of two orthogonal sample orientations. These two images were combined in order to yield vector maps of in-plane magnetization, after correcting for drift and distortion via an affine transformation that was based on topographical images of x-ray absorption for each sample orientation. Each of these topographical images was obtained by averaging all raw images that had been obtained on resonance with left and right-polarized light.