Localized strain profile in surface electrode array for programmable composite multiferroic devices

Using voltage-generated strain to control magnetism in miniaturized devices is an energy-efficient alternative to the conventional current-driven approach due to the Joule heating suppression. Such devices are based on a class of material–magnetoelectric multiferroics. Recent advances in multiferroic magnetoelectric composites¹ have brought us closer to applying fundamental research discoveries to a broad range of applications, including data-storage devices,^2–4 probabilistic computing platforms,⁵ voltage-tunable radio frequency microwave devices,^6,7 artificial neural networks,^8,9 and microfluidic particle and cell sorting platforms.^10–12 For multiferroic heterostructures, understanding the strain that is coupled into the magnetoelastic structures is necessary, especially any variation in strain with length scales similar to the magnetic element size.

To date, researchers have extensively studied the electric-field control of magnetism in a variety of mechanically coupled composite multiferroic heterostructures, where strain from a piezoelectric material^13–15 governs magnetism in an adjacent magnetoelastic material^16–18 due to the converse magnetoelectric effect. To drive a desirable magnetoelastic response, instead of optimizing the material properties of magnetoelastic and piezoelectric constituents, one could use patterned surface electrodes to engineer the strain. With surface electrodes, the in-plane strain is generated locally between the electrodes, with the strain profile defined by the location and the orientation of the electrodes.

In this paper, we focus on examining the induced strain distribution in the single-crystal piezoelectric [Pb(Mg_1/3Nb_2/3)O₃]_0.69-[PbTiO₃]_0.31 (PMN-PT) with surface electrodes. As opposed to piezoelectrics fully covered with electrodes, which rely on piezoelectric coefficients d₃₁, d₃₂ for in-plane anisotropic strain,¹⁹ the surface electrode design can provide more freedom in strain control due to the adjustability of the electrode arrangement. This work could serve as a roadmap/reference for designing compact, programmable electrode arrays for multiferroic applications.

Many multiferroic devices based on PMN-PT adopt a design that uses a single pair of electrodes across the entire surface. In particular, the (011)-cut PMN-PT generates in-plane anisotropic strain to alter the magnetic domain in the coupled magnetic layer.^20–23 A tensile strain is induced along [01–1], and a compressive strain is generated along [100]. The (001)-cut PMN-PT is another popular substrate choice, especially when the magnetic films require epitaxial growth, such as La_0.7Sr_0.3MnO₃ (LSMO)^24,25 or Fe_(1−x)Ga_x alloys,²⁶ with a lattice match to PMN-PT.²⁷ With an applied electric field, the substrate typically undergoes isotropic, in-plane compressive strain.^28,29 Using this approach of a single pair of electrodes across the entire substrate, there is little control over the direction of strain and no ability to individually control strain for different magnetic elements.

Alternatively, surface electrodes can be used to engineer the direction and magnitude of strain with local control. Surface electrode pairs patterned on piezoelectrics [PMN-PT,³⁰ Ba_1−xSr_xTiO₃ (BST),⁷ PZT^31–33] have shown local magnetoelectric control of Ni or FeGa elements.^31,32,34 These device prototypes have provided experimental evidence on the local strain-mediated behavior of a few micromagnets patterned between a set of electrode pairs a few hundreds of micrometers apart.

So far, little has been done to experimentally characterize the impact of neighboring electrode pairs on strain. In this work, we focus on characterizing the deviatoric strain (supplementary material S1) distribution between surface electrode pairs arranged in arrays on the (001)-cut PMN-PT using synchrotron-based scanning X-ray microdiffraction.³⁵ The goal is to understand the role of electrode arrangement on the local strain distribution between the electrodes, which will provide insight into how densely electrode arrays can be packed while maintaining control of each magnetic element. We analyze the average local axial deviatoric strain along [100] and [010] directions, in the region between the surface electrode pairs. The local strain is compared with the axial strain generated by the parallel plate electrodes, which is expected to be compressive in-plane.^28,29 Furthermore, we examine the effect on strain from electrode pair arrangement and interactions among neighboring surface electrode pairs. Finite element simulations are used to better interpret the varied regional strain behavior.

In the experiment, the three (001)-cut PMN-PT samples [1 cm $×$ 1 cm $×$ 500 $μm$ (thickness)] (TRS Technologies, Inc., State College, PA, USA) under investigation are sample A with parallel plate electrodes as a reference [Fig. 1(a)], samples B and C with 6 and 12 surface electrode pairs, respectively [Fig. 1(c)]. All of the electrodes consist of a 5 nm Ti and a 50 nm Pt layer. In sample A, both the top and bottom surfaces are uniformly covered by an electrode. For samples B and C, arrays of electrode pairs are patterned on the top surface, while the bottom surface is covered uniformly by an electrode. The samples are wirebonded to leadless chip carriers [Fig. 1(c)] before being mounted onto a printed circuit board.

Laue (polychromatic) X-ray microdiffraction can be used for investigating elastic strain distribution at the micrometer-scale^22,36–38 (supplementary material S1). Recently, Lo Conte et al. used X-ray microdiffraction to map out the electrically induced axial strain distribution in the (011)-oriented PMN-PT with parallel plate electrodes,²² achieving micrometer-scale resolution at the locations with patterned magnetic microstructures [Fig. 1(a-i)]. During the microdiffraction scanning, the individual diffraction pattern is collected stepwise from a grid point (an x–y position) to provide information on lattice strain and crystal orientation. For this work, the electrically induced deviatoric strain is calculated for each step, as represented by a 10 $×$ 10 $μm$² pixel in the constructed 2D strain maps [Fig. 1(c)], by taking the deviatoric strain difference of extracted strain at a non-zero voltage and at zero voltage. In sample A, the X-ray performs a raster scan with an area of 500 $×$ 500 $μm2$ at the center of the sample. In samples B and C, the X-ray scans multiple 250 $×$ 250 $μm2$ areas (supplementary material S2). This work mainly focuses on the experimentally measured in-plane deviatoric strain components,³⁵ $εxx′$ and $εyy′$⁠, measured by the Laue method, as they are the driving mechanism for in-plane magnetization rotation or switching in numerous studies.^22,31,32,39 In the remaining part of this paper, we will refer to deviatoric strain as strain.

Before examining the regional strain profile generated by the surface electrode pairs, we conduct X-ray microdiffraction on two prepoled PMN-PT samples with parallel plate electrodes and apply voltages up to 400 V (supplementary material S3), as shown in Fig. 1(a). The first one [Fig. 1(a-i)], a (011)-cut PMN-PT studied in Lo Conte et al., generates in-plane anisotropic strain along the main crystallographic directions [100] and [01–1] as a function of the voltage. The second one (sample A, Fig. 1(a-ii)] is a (001)-cut PMN-PT, which would ideally exhibit isotropic compressive strain on a macroscopic level along the [100] and [010] directions.^28,29 For the first sample, the average-induced in-plane strain is compressive along [100] and tensile along [01–1]. For the second, the average axial strains are both compressive. Interestingly, the axial strains between nominally equivalent [100] and [010] directions are close in magnitude but not identical when the strain is examined at the microscale. The error bars in the strain–voltage plot in Fig. 1(a) also suggest that the axial strain exhibits spatial variation in the scanned regions for both PMN-PT samples. In Fig. 1(b), we show the difference between the average of two deviatoric strains as the total (in-plane) strain from both samples. This difference is the driver for controlling magnetization (i.e., strain-induced magnetoelastic uniaxial anisotropy) in previously reported strain-coupled thin-film magnetic nanostructures.^22,40 For the (011)-cut PMN-PT, it is obvious that a large strain difference is induced at 400 V, whereas for the (001)-cut, the strain difference is much lower. Ideally, we expect this strain difference to be zero, but the local inhomogeneity of strain²² and ferroelectric domains¹⁸ likely account for the non-zero average in the biaxial strain difference magnitude.

Next, we obtain regional microdiffraction scans in samples B and C for two voltage cycles from 0 to 400 V, where the first one poles the samples (supplementary material S3). During the microdiffraction experiment, a positive voltage is applied to the top surface electrodes and the bottom is grounded [Fig. 1(c)]. The regions of interest are each marked by four markers and labeled numerically for ease of reference. Figure 1(c) (Right) also demonstrates the zoomed-in reconstructed deviatoric strain maps from sample C. When a voltage is applied to the surface electrode pairs, to satisfy the compatibility conditions of strain,⁴¹ an in-plane compressive strain is produced at the surface of the PMN-PT along the y-direction (i.e., the direction of the electrode pair), and a tensile strain is generated along the x-direction.

The major difference between samples B and C is the density of electrode array pairs, where C has two more rows of electrode pairs than B and thus is more densely packed, as shown in Fig. 1(c) (supplementary material S4). Two electrode pairs located at regions 3 and 12 in both samples share the same electrode pair configuration, where the electrode pairs are separated by 400 $μ$m. Previous simulation studies have shown that at such a length scale, the surface electrode pair on a PZT could produce a highly localized strain field in regions smaller than 1 $×$ 1 $mm2.$³¹ Since the separation distance between surface electrodes is on the order of a few hundred micrometers, it becomes impossible to characterize the strain by a strain gauge. Hence, X-ray microdiffraction is crucial for characterizing the spatial strain profile as a result of varied surface electrode array configuration, including pair density, separation distance between electrode pairs, and the angle of the electrodes with respect to the substrate crystallographic direction.

From X-ray microdiffraction data, we extract the electric-field-induced deviatoric strains $εxx′$ and $εyy′$ for individual regions. Figure 2(a) shows an example of the 2D map of induced strain. As expected, the local induced strain is tensile along the x-direction and compressive along the y-direction. However, from the micrometer-scale mapping, the strain is not uniform at the micrometer scale. A finite element simulation using COMSOL Multiphysics (supplementary material S5) with the same electrode setup as in sample B also shows local anisotropic axial strain as in the experiment. The simulated induced strain mapping is shown in Fig. 2(a). For the simulation, we do not consider non-uniform strain and ferroelectric domains present in the experimental system, as suggested by the strain distribution in sample A [Fig. 1(a-ii)], so the strain variation is less pronounced (supplementary material S6).

To evaluate the experimental strain distribution and variation for the 625 pixels, we created violin plots⁴² [Fig. 2(b)] for both the axial deviatoric strains and their difference, $εyy′$ − $εxx′$⁠. To account for the experimental noise during X-ray microdiffraction scanning, apart from fitting Laue peaks with the XMAS software, we adopt the nearest neighbor technique for outlier removal (supplementary material S7) with less than 0.6% of the pixels removed for any image. Additionally, we report the regional average strain in Fig. 2(c) from both experiment and simulation. The differential deviatoric strain achieved in localized regions on the (001)-cut PMN-PT using the surface electrodes in this work is similar to that of the global anisotropic strain profile in the (011)-cut PMN-PT from Fig. 1(a-i). However, with the locally controllable strain of our surface electrodes, one could, for example, actuate individual microscale magnetic components. Furthermore, the surface electrodes can be used to engineer the differential deviatoric strain, as opposed to the case of Fig. 1(a-i), which relies on having the appropriate material and crystalline cut, thus limiting the set of material choices.

Also observed from Figs. 2(b) and 2(c), the average strain for each direction and the strain difference vary by region. Such differences in the local strain profile suggest a collective effect from the electrode pair separation distances (400–600 μm), the electrode pair rotation of 11.25° in the bottom row, and the location of the region in the sample. Next, we resort to simulation for providing additional insight into the effect of separation gap distance and angle using parametric sweeps. The simulation results in Fig. 3(a) present the biaxial deviatoric strain difference as a function of the electrode pair separation gap distance d. For the ranges studied experimentally, the simulations show decreasing strain in both x- and y-directions as the gap distance narrows. The corresponding axial strains measured experimentally for regions 1–3 in sample B are marked in Fig. 3(a). In terms of the strain variation with electrode angles, an increase in the electrode pair tilting angle with respect to the x-direction slightly decreases the strain along the [100] and [010] directions, as shown in Fig. 3(b).

We also compare the local strain results to the anisotropic axial strain generated globally in the parallel plate (011)-cut PMN-PT substrate [Fig. 1(b)]. It is observed that the locally induced strain in the six regions (1–3, 10–12) is similar in magnitude as in the (011)-cut PMN-PT, a significant increase from the nearly isotropic compressive strain in sample A with parallel plate electrodes.

We use axial deviatoric strain ratio⁴³ (⁠$εyy′$/$εxx′$⁠) to compare the strain behavior across different regions in a sample and similar regions across samples. Different ratios reflect variations in regional piezoelectric coefficients, which allow one to access a diverse range of strain responses at a given applied voltage. First, we compare the axial strain ratio in all regions in sample B (supplementary material S6) to conclude that the strain is not strictly confined between electrode pairs; rather, it can affect the strain distribution outside of the region. Next, we compare the axial strain ratio to see how the variation of the electrode pair density affects the strain behavior in local regions. In particular, we compare the axial strain ratio in regions 3 and 12 from samples B and C since those regions have identical electrode designs (i.e., same gap distance and angle). Similar to sample B, the voltage-induced axial strain profile from the prepoled sample C (see the supplementary material S9) shows tensile and compressive strain along the x- and y- directions, respectively.

From both the experimental and simulation results (Fig. 4), the $εyy′$/$εxx′$ ratio from sample C is higher than that in sample B. This observation implies that the presence of additional rows of electrode pairs in the middle in sample C vs sample B leads to a higher axial strain ratio. It suggests that the presence of denser electrode pairs along the y-direction plays a non-negligible role in making the regional strain profile more anisotropic along the y- vs x-directions. Overall, the X-ray microdiffraction results on the micrometer-scale level reveal that the local strain can be affected collectively by the three factors investigated in this work: local electrode separation distance, angle of placement with respect to the crystallographic directions, and closeness to neighboring electrodes. With the current surface electrode design, the samples generate localized strain with tunable axial strain magnitudes.

In conclusion, X-ray microdiffraction provides a distinct opportunity to map out the local deviatoric strain in ferroelectric PMN-PT in the areas of interest between surface electrodes with micrometer-scale resolution. An in-depth understanding of the spatial distribution of regional strain is crucial, particularly for driving arrays of strain-coupled magnetic microstructures in multiferroic systems. We characterize and analyze the strain profile in PMN-PT resulting from both parallel plate electrodes and patterned electrode arrays. The results highlight the effect of electrode geometry on both the local and global scales. In particular, we examined local strain in multiple regions from (001)-cut PMN-PT samples with surface electrodes, and the average axial strain response is consistent with predictions from piezoelectric simulations. This systematic study also highlights the influence of electrode pair geometry, including the separation distance of the pair, angle of the pair, and neighboring electrode pair compactness on local and regional strain.

See the supplementary material that contains more details on the X-ray microdiffraction, experiment vs simulation results, outlier removal for strain mapping, voltage-induced axial strain ratio, and information on strain variation among samples (supplementary material S1–S9).