In Situ Electric-Field Study of Surface E ﬀ ects in Domain Engineered Pb(In 1 / 2 Nb 1 / 2 )O 3 -Pb(Mg 1 / 3 Nb 2 / 3 )O 3 -PbTiO 3 Relaxor Crystals by Grazing Incidence Di ﬀ raction

: In this work, we present a grazing incidence X-ray di ﬀ raction study of the surface of a 0.24Pb(In 1 / 2 Nb 1 / 2 )O 3 -Pb(Mg 1 / 3 Nb 2 / 3 )O 3 -PbTiO 3 (PIN-PMN-PT) [011] poled rhombohedral single crystal. The near surface microstructure (the top several tens to hundreds of unit cells) was measured in situ under an applied electric ﬁeld. The strains calculated from the change in lattice parameters have been compared to the macroscopic strain measured with a strain gauge a ﬃ xed to the sample surface. The depth dependence of the electrostrain at the crystal surface was investigated as a function of temperature. The analysis revealed hidden sweet spots featuring unusually high strains that were observed as a function of depth, temperature and orientation of the lattice planes.


Introduction
Relaxor ferroelectric single crystals are widely used as transduction materials, due to extremely high piezoelectric coefficients d ij several times larger than those of commercial lead Pb[Zr x Ti 1−x ]O 3 (PZT) ceramics. Recently, significant attention and resources have been put into the development of novel compositions to find better candidates based on solid solution Pb(Mg 1/3 Nb 2/3 )O 3 -PbTiO 3 (PMN-xPT) and PbZn 1/3 Nb 2/3 O 3 -PbTiO 3 (PZN-PT), which are currently being used in medical ultrasound transducers and numerous SONAR transducer and sensors applications [1][2][3][4]. Two specific properties make these crystals ideal for applications: an extraordinarily high piezoelectric coefficient (d 33 > 2000 pV/m) combined with an extraordinary coupling coefficient (>0.9) achieved by "domain engineering;" i.e., cutting and poling the crystals along certain planes [3,[5][6][7][8][9][10]. In order to extend the linear voltage regime with higher temperature stability to greater than that of binary relaxors, the ternary single crystal system Pb(In 1/2 Nb 1/2 )O 3 -Pb(Mg 1/3 Nb 2/3 )O 3 -PbTiO 3 (PIN-PMN-PT) has recently been developed. This next-generation material overcomes the need for dc electrical bias to achieving high power drive, a requirement for naval projector applications.
For several decades, extensive work developing theories and experimental studies of single crystal PMN-PT and, more recently, PIN-PMN-PT has been undertaken, in order to shed light on explaining effect of residual stresses present at the surface which would depend on the nature of the atomic bonding and, hence, crystallographic orientation and surface preparation such as polishing, grinding or annealing; (b) a change in response at a free surface impacted by domain configurations; (c) a gradient in the stoichiometry (such as oxygen or cation); (d) the effect of symmetry breaking at the surface; or (e) some combination of these factors. This information is clearly very important for users of relaxor materials for MEMs or magnetoelectric composite applications, where the relevant length scale is commensurate to that of the skin effect, and would thus play a primary role in actual device performance [22]. Figure 1 illustrates the sample geometry crystallographic orientation and poling direction of an as-prepared [011] poled PIN-PMN-PT. A Poisson ratio of 0.37 was used to match to macroscopic strain along the [001] direction to the X-ray data. Figure 2 gives the schematic of how measurements were made. Figure 3 shows the typical 'butterfly loop' bulk strain characteristics of ferroelectric materials. The coercive field is~5.0 kV/cm, with a maximum strain of ∼ 4000 microstrain (µ ), yielding a piezoelectric coefficient, d 32 ∼ 1327 pm/V, which closely matches that of both the manufacturer's data sheet and an independent direct piezoelectric measurement of d 32 . Figure 4 shows typical raw data with and without field while Figure 5 shows the typical response to the electric field E for several values of α. For the grazing incidence reflections chosen (based on 4-circle geometry and physical access to certain reciprocal lattice vectors-see Figure S2 in Supplementary Materials), the variation in d spacing is shown Figure 6 for the (010), (110), (111) and the (322) planes as a function of depth and temperature at zero field. The data show the variation in d spacing with depth of penetration of the X-rays, as expected from the literature for this material system, with a more subtle variation with temperature for all reflections except the (322). The interplanar lattice spacing changed by 0.3% for the (010) planes assuming a strain-free pseudo-cubic bulk lattice parameter of 4.04 Å. The changes for the other planes are ∼0.16% for (110), ∼0.1% for the (111) and ∼0.02% for the (322) planes-though the actual structure for the bulk is expected to be rhombohedral for this composition at all the temperatures measured. Figure 7 shows contour plots of the field-induced change in strain as functions of temperature and depth for the various lattice planes.

Results
whereby unusually high electrostrains were observed for certain depths, temperatures and orientation of the lattice planes. An important question to consider is whether the skin effect arises from (a) the effect of residual stresses present at the surface which would depend on the nature of the atomic bonding and, hence, crystallographic orientation and surface preparation such as polishing, grinding or annealing; (b) a change in response at a free surface impacted by domain configurations; (c) a gradient in the stoichiometry (such as oxygen or cation); (d) the effect of symmetry breaking at the surface; or (e) some combination of these factors. This information is clearly very important for users of relaxor materials for MEMs or magnetoelectric composite applications, where the relevant length scale is commensurate to that of the skin effect, and would thus play a primary role in actual device performance [22]. Figure 1 illustrates the sample geometry crystallographic orientation and poling direction of an as-prepared [011] poled PIN-PMN-PT. A Poisson ratio of 0.37 was used to match to macroscopic strain along the [001] direction to the X-ray data. Figure 2 gives the schematic of how measurements were made. Figure 3 shows the typical 'butterfly loop' bulk strain characteristics of ferroelectric materials. The coercive field is ~5.0 kV/cm, with a maximum strain of ∼ 4000 microstrain (μϵ), yielding a piezoelectric coefficient, d32 ∼ 1327 pm/V, which closely matches that of both the manufacturer's data sheet and an independent direct piezoelectric measurement of d32. Figure 4 shows typical raw data with and without field while Figure 5 shows the typical response to the electric field E for several values of α. For the grazing incidence reflections chosen (based on 4-circle geometry and physical access to certain reciprocal lattice vectors-see Figure S2 in Supplementary Materials), the variation in d spacing is shown Figure 6 for the (010), (110), (111) and the (322) planes as a function of depth and temperature at zero field. The data show the variation in d spacing with depth of penetration of the X-rays, as expected from the literature for this material system, with a more subtle variation with temperature for all reflections except the (32 2). The interplanar lattice spacing changed by 0.3% for the (010) planes assuming a strain-free pseudo-cubic bulk lattice parameter of 4.04 Å . The changes for the other planes are ∼0.16% for (110), ∼0.1% for the (111) and ∼0.02% for the (322) planes-though the actual structure for the bulk is expected to be rhombohedral for this composition at all the temperatures measured. Figure 7 shows contour plots of the field-induced change in strain as functions of temperature and depth for the various lattice planes.

Results
(a) Sample and crystallographic geometry.      was cycled with a triangular waveform to yield a constant rate of voltage with time (for accurate PE loop analysis), for each set of planes at each condition of grazing incidence and temperatures.   The calculated value of d 32~1 327 pm/V. Note this measurement was carried out after the grazing incidence experiment with an increased electric field amplitude of 6.5 kV/cm in order to observe the coercive field.
Crystals 2020, 10, x FOR PEER REVIEW 5 of 13 was cycled with a triangular waveform to yield a constant rate of voltage with time (for accurate PE loop analysis), for each set of planes at each condition of grazing incidence and temperatures.

Discussion
The present results on polished single crystals of 24%PIN-PMN-PT indicate that strain gradients persist down to hundreds of nm. Reduced dielectric and piezoelectric properties are often observed with decreasing sample size and generally correlate to surface damage as a result of lapping and polishing although the thickness effect is not solely due to mechanical-chemical polishing [24]. For instance, results from relaxor PMN-PT ceramics and crystals showed that the dielectric and piezoelectric properties were correlated with their domain size [24]. Domain sizes for PIN-PMN-PT single crystals and thin films have been shown to be dependent on large variety of factors and examples ranging from tens to a few hundreds of nm have been recorded [24][25][26][27][28][29], although tens of nm is more typical.  (1)] for different lattice planes as a function of temperature and depth from the surface of the sample (∼2 nm) into its bulk. The contours have been linearly interpolated from the data taken at three specific temperatures and five values of α. These specific temperatures were chosen as representing a range commonly used in real life applications for this class of material.

Discussion
The present results on polished single crystals of 24%PIN-PMN-PT indicate that strain gradients persist down to hundreds of nm. Reduced dielectric and piezoelectric properties are often observed with decreasing sample size and generally correlate to surface damage as a result of lapping and polishing although the thickness effect is not solely due to mechanical-chemical polishing [24]. For instance, results from relaxor PMN-PT ceramics and crystals showed that the dielectric and piezoelectric properties were correlated with their domain size [24]. Domain sizes for PIN-PMN-PT single crystals and thin films have been shown to be dependent on large variety of factors and examples ranging from tens to a few hundreds of nm have been recorded [24][25][26][27][28][29], although tens of nm is more typical.
The current studies probe down to the top few hundred nm, depths larger than typical domain sizes. Thus, the measured strain response then should mostly exclude the extrinsic domain reorientation contribution for the reflections measured at larger α values. The macroscopic field-strain response of the relaxor class of material arises from the intrinsic electrostrictive coefficient and its piezoelectric response. The two responses are often reported as simply the piezo response, but the observation can be differentiated via the quadratic response to field observed for electrostriction compared to linear dependence of polarization with field for the piezo effect. In relaxor ferroelectrics both physical processes are generally responsible for their actual response [26,30]. The field dependence of the strain response from several of the measured lattice planes ( Figure 5) displays a mix of both behaviors and is more complex than the bulk behavior ( Figure 3). A double loop hysteretic response is observed in all plots in positive and negative going voltage excursions indicating ferroelectric behavior superimposed on a linear response indicative of an electrostrictive response. Note that at the larger depths (lower panel, Figure 4), it is clear that local structure is distinct from that of the surface.
An additional factor that affects the strain response from different planes is based on the degree to which those planes are affected by domain reorientation under an applied electric field (Figure 4, upper panel). This particular sample was poled along the [011] direction with the local polarization aligned with the cubic [111] directions, which may rotate under electric and stress fields, to reduce the domain density (from multi domain to single domain for example-see [12]). In the ideal engineered crystal of this type, there are two equivalent [111] polarization directions within the crystal. In the present case, there is zero applied stress with only thermal and electrical excitation (stress-induced phase changes are the subject of a future paper, though elements are reported here [31]). Our data identifies lattice reflections near the free surface (<200 nm) that exhibit a higher degree of strain compared to others. While there is a weak depth dependence of (332) planes, the other showed significant changes with depth with (010) > (110) > (111) at 40 • C.
The strain dependence on temperature also presents some interesting features: for the (010) and (110) reflections, we note higher temperatures generally feature larger strains, which supports the thermodynamic view linking increased domain mobility with increasing temperature [6]. An interesting anomaly rests with the (111) reflection data, however, with a peak in strain at high temperature/shallow depth and low temperature/high depth (Figure 7). The absolute values of strain are perhaps less important than the relative changes for any given reflection since the behavior is determined by the full tensor response of the crystal. For example, planes aligned parallel to the specular surface representing the d 31 coefficient and those planes parallel to the long axis of the sample [100] direction representing the d 32 coefficient have an inherently different response. In these experiments, none of the grazing incidence planes are aligned to those primary directions, so a direct piezoelectric coefficient calculation was not possible. However, the specular reflection (011) corresponds with the out of plane d 31 coefficient, which, coupled to the strain gauge data, means a value of the piezoelectric d 32 coefficient can be calculated from the appropriate Poisson ratio. The strain associated with the (110) planes closely matches that measured with the strain gauge data. The (110) planes are inclined at an angle of 60 • to the electric field direction ([011] X3) and 30 • to the [100] X2 axis, and so the resolved strain within that set of planes with respect to the sample surface is expected to most closely match d 32 of the crystal, which is ∼ (1600 µ )/(cos60 • sin60 • ) ∼ 3700 µ , close to the value of ∼ 4000 µ , measured by the strain gauge along X1 direction. We note that the analysis here is highly simplified and resolving strains into different directions from known lattice parameter changes is non-trivial, but this serves as a good first approximation.

Experimental Section
A rectangular bar of 0.24 PIM-PMN-PT with dimensions 4 × 4 × 12 mm 3 was supplied by CTS Corporation, USA. For sample details see Table S1 in Supplementary Information Figure S1).The macroscopic strain was recorded along [100] with a strain gauge attached to the sample surface (Supplementary Information Figure S3). Sample electrodes were made by depositing evaporated gold onto the [110] faces (perpendicular to the sample surface normal). See Figure 1, Figures S1 and S2 for more details of sample orientation and GIXRD geometry. After the surfaces are ground and polished to ∼ 1 4 µm, the sample was mounted with low melting temperature wax onto an insulating substrate. High-voltage wiring was affixed to the sample jig and connected to a TREK 610E HV amplifier. Ferroelectric current as a function of applied voltage was measured with a virtual earth current amplifier, and the charge was then calculated through numerical integration [32]. Details are provided in reference [33], and shown in Figure S3.
The diffracted intensity was recorded with a PILATUS3 X 300K imaging camera from Dectris (Baden-Daettwil, Switzerland). The applied voltage, ferroelectric current and X-ray intensity were all measured simultaneously and recorded with a data acquisition system based on the ESRF-designed MUSST card. The card can simultaneously record X-ray intensities and analogue voltages up to 24 bit resolution with a sampling rate of 40 kHz [33]. The full in situ capabilities at the XMaS beamline, ESRF, for characterizing ferroelectrics, developed by Electrosciences and the University of Liverpool, include electric field, temperature, frequency, magnetic field and stress. The experimental methodology is graphically represented in Figure 2.
The penetration depth of the X-rays normal to the surface is proportional to the sin α for angles greater than the critical angle, the angle of total external reflection, which is related to the wavelength-dependent refractive index of the sample (see Figures S2 and S6 in Supplementary  Information). The X-ray energy was 12.4 keV (wavelength = 1 Å) and measurements were done with 0.05 • < α < 0.5 • , corresponding to a penetration depth ranging between 2 and 200 nm, which is an order of magnitude lower than that of previously published data on similar materials systems. More details can be found in Supplementary Information.
The scattering geometry only permitted a selection of accessible reflections as defined in Figure 1. A crystal representation of the surface of the sample is shown in Figure 1b, with the [110] specular normal shown as the top surface atom cut, and the indices of the planes used in this study also labelled. The camera pixel coordinate frame was translated to HKL pseudocubic Miller indices by a program written in Python which extensively uses various PyMca (v5.4.2) modules [34]. Additional data analysis and visualization were performed on the HKL indexed images, with code written in IDL V 8.7.2 (Harris Geospatial Solutions, Inc., Broomfield, CO, USA) although data were interpolated only once within the HKL conversion process. Detailed analysis of all reflections and at all values for α and temperature was carried out following the approach outlined in the Supplementary Information. The 3D HKL volume data sets were then analysed to calculate the d spacing d HKL E at E = 4 kV/cm for each value of temperature, HKL and α. These data were then plotted as the electrostrain δ relative to the d spacing at E = 0 (d HKL 0 ) for each α value. That is as plotted in Figures 5 and 7.

Conclusions
The work has probed the strain associated with a variety of reflections taken in grazing incidence geometry for a single crystal of [011] poled rhombohedral PIN-PMN-PT relaxor ferroelectric as a function of electric field and temperature. The strain calculated from changes in lattice parameters have been compared to the macroscopic strain measured with a strain gauge affixed to the sample surface. A measurable depth dependence of lattice parameter and calculation of in operando strain identifies those planes which have a higher sensitivity to field, indicating an opportunity for optimizing strain interfaces in, for example, energy harvesting magnetoelectric composites. Application of mechanical stress affects the crystallographic strain dramatically [12,31], which would have significant impact on the strain transfer optimization across an interface in magnetoelectric composites, for example, details of which will be reported in a future publication. The origin of this observed skin effect due to (a) residual stress, (b) domain configuration, (c) stoichiometry or (d) symmetry breaking is yet to be confirmed, but is likely to be a combination of all such mechanisms. Finally, for future studies, the methodologies for in operando study of electric-field-driven structural phase changes in these materials by synchrotron X-ray diffraction have been established for grazing incidence geometry, and work will focus on development of machine learning tools to accurately follow the diffraction peaks as a function of field temperature, and depth, to augment and accelerate experimental data analysis and phase identification, stability and link to the material's functional performance.
Supplementary Materials: The following are available online at http://www.mdpi.com/2073-4352/10/9/728/s1, Table S1: Composition and some properties of samples measured; Figure S1: Two coordinate systems used in the study. specular face in right hand diagram; Figure S2: Grazing Incidence X-Ray diffraction for PIN-PMN-PT, Figure S3: Attachment of the strain gauge (RS Pro 2.5mm gauges 120 Ω 632-146) to the single crystal using cyanoacrylate adhesive, leaving sufficient space for the X-Ray beam off this specular surface; Figure S4