X-ray-induced piezoresponse during X-ray photon correlation spectroscopy of PbMg1/3Nb2/3O3

X-ray illumination induces surface charging and gives rise to substantially enhanced piezoresponse in a relaxor ferroelectric captured by X-ray photon correlation spectroscopy.


ESTIMATING TWO-TIME CORRELATIONS FROM SAMPLE TILTING
We wish to estimate the two-time correlation of the speckle pattern arising from sample tilts ∆χ(t) and ∆η(t) determined from observed speckle motions on the detector of ∆i(t) and ∆ j(t).This is conveniently done by considering the reciprocal space coordinates Q h = (2π/a)H, Q k = (2π/a)K, and Q ℓ = (2π/a)L, where a = 4.04 Å is the lattice constant of cubic PMN.Sample tilts ∆η and ∆χ will produce motions at a given point on the detector of the sample-fixed reciprocal space of space will be (Hruszkewycz et al., 2012) If we assume the same ratio between the actual and ideal ξ Qh and ξ Qk that we use for the speckle size on the detector ξ , then we can write the actual FWHM speckle sizes in Q and in tilt angles as The time-dependent changes ∆Q i due to the sample rotations give an estimated two-time correlation function of the speckle of where we have introduced the coherence factor 0 ≤ β ≤ 1 to account for incomplete coherence of the incident beam and finite detector resolution.In the second and third lines, we show that the same result is obtained if we ratio the tilts ∆χ and ∆η to the speckle sizes normalized to the tilt angles ξ χ = 82 µrad and ξ η = 34 µrad, respectively, or if we ratio the speckle motions ∆i and ∆ j to the actual speckle size on the detector of ξ = 1.2 pixels, neglecting the ∆Q ℓ term under the assumption that the illuminated volume is very thin in the z direction due to a short absorption length so that ξ Qℓ is very large.

MODEL FOR DYNAMIC SURFACE TILTING FROM SMALL APPLIED FIELDS
Our hypothesis is that the sample has a steady-state electric field distribution caused by surface charging in the area of X-ray illumination owing to ejection of photoelectrons.This static field, combined with the small dynamic applied field, results in local deformation from the electrostrictive properties of the sample, giving the observed Bragg peak motion and two-time correlation functions of the speckle.

A. Surface Charging by X rays
Surface charging of insulating materials by X-ray illumination is a well-known effect in X-ray photoelectron spectroscopy (Moulder et al., 1992), and has been observed to shift photoelectron energies by > 100 eV (Yasuno, 2019).In particular, small X-ray beams have been shown to produce local charging with non-uniform electric fields (Tielsch and Fulghum, 1996), potentially inducing tilting in the material.The first step in modeling this behavior is to calculate the steady-state electric potential and field distributions from Xray-induced charging of the sample surface.For an insulating sample, X-ray illumination can result in surface charging because photoelectrons are ejected from the surface region.We assume that the photoelectron escape depth ℓ esc is small, of the order of 10 nm (Bras et al., 2021), and that the local free charge density is concentrated at the surface in a layer having a negligible thickness δ z.One can estimate the slow dynamics and steady-state behavior of the surface charge density σ (x, y) by writing a continuity equation for its rate of change, where the electric current densities J out and J in are the net photoelectric current density leaving the surface and the leakage current density back to the surface, respectively.These are all considered to be functions of the spatial coordinates x and y in the plane of the surface.We assume that the photoelectric current density is proportional to the absorption rate of X rays in the surface region.For simplicity we assume that each photon produces a single photoelectron with kinetic energy sufficient to escape the charged surface.We can write the net photoelectric current density as where I 0 (x, y) is the distribution of incident X-ray intensity (counts per second per unit surface area), −e is the electron charge, ℓ abs is the X-ray absorption length, and θ B is the incidence angle of the X rays.The quantity in parentheses gives the fraction of X rays absorbed within one photoelectron escape depth of the surface.(Here we neglect both the possibility of secondary electrons escaping, as well as the cutoff in net photoelectric current that will occur if the surface potential approaches the photoelectron energy.)We assume the leakage current density is proportional to the potential φ (x, y, 0) at the surface z = 0, where R leak is the inverse of the effective conductance per unit area of the leakage medium, which could be the sample or the vapor environment.
The final relationship between the surface charge and potential is based on the Poisson equation relating surface charge and surface potential, where ε 0 = 8.85 × 10 −12 C V −1 m −1 is the permittivity of free space, and ε s is the effective surface dielectric constant.
We can find a solution to these equations by expressing the surface potential φ (x, y, 0) and the incident X-ray illumination I 0 (x, y) as two-dimensional Fourier series, where φmn (0) and Î0 mn are the Fourier coefficients for wavenumbers q xm ≡ mπ/a x and q yn ≡ nπ/a y , and 2a x and 2a y are the periods in the x and y directions, respectively.(To insure that the periodic boundary conditions in x and y do not affect the results, we use values of a x and a y much larger than the illuminated area.)Since φ and I 0 have real values, the Fourier coefficients for negative m and n are the complex conjugates of those for positive m and n.The in-plane Laplacian of the surface potential is given by Thus the steady-state surface potential distribution is simply proportional to the incident X-ray intensity distribution, We can solve for the three-dimensional static potential and field distributions inside the sample, assuming that the sample is an insulator with no free charge density.We assume that the relation between the polarization P and field E is isotropic.Then the Maxwell equation 0 This implies that the potential φ (x, y, z) obeys Laplace's equation, ∇ 2 φ = 0, with the boundary condition of a specified surface potential φ (x, y, 0).This can be solved using separation of variables (Griffiths, 1999), assuming φ is the sum of terms obeying Laplace's equation that are each products of functions in the x, y, and z directions.The three-dimensional potential that obeys Laplace's equation with the correct surface potential is then (S.23) Here we assume that z is the depth into the sample, so that the sample extends in the positive z direction from the surface at z = 0.

B. Steady-State Surface Charge
For the rest of this section we will assume that the surface potential has reached it steady-state value, φ (x, y, 0) = φ ss (x, y).The x, y, and z components of the electric field vector E = −∇φ can be written using expressions analogous to Eq. (S.If we assume a form for the X-ray illumination I 0 (x, y), we can calculate the potential and field distributions.Here we  The integrated intensity in photons per second is (S.28) The Fourier coefficients of the intensity I 0 (x, y) and steadystate surface potential φ ss (x, y) are Î0 mn = πI pk σ x σ y 2a x a y exp(−q 2 xm σ 2 x /2) exp(−q 2 yn σ 2 y /2), (S.29) where φ pk = B I pk is the maximum potential at the surface.Figure S.6 shows the potential and field calculated using the parameter values in Table S.1.For a symmetric reflection geometry, we expect the width of the beam footprint to be longer in the scattering plane (x) than perpendicular to the scattering plane (y) by a factor of 1/ sin θ B .Using the experimental beam size of 5 µm FWHM and θ B = 24.4• gives σ y = 2.12 µm and σ x = 5.13 µm.For the calculations we use Fourier half-periods a x = 12σ x and a y = 12σ y and Fourier components in the range -24 to 24 for both m and n.
While the values of I tot , σ y , σ x , and θ B were determined in the experiments, and the absorption length ℓ abs is straightforward to calculate (Gullickson, 2010) for PMN at the photon energy of 7.35 keV, the photoelectron escape length ℓ esc is an estimate, and nothing is known about the value of R leak .It was simply chosen to give a peak surface potential φ pk = 10 V.
For the conditions modeled, the maximum value of the steady-state in-plane electric field is about E y = 2.9 × 10 6 V m −1 .This is much higher than the maximum applied field, and so will greatly increase the electrostrictive strain.Furthermore, the peak potential φ pk could be much higher than the 10 V assumed, since the value of R leak in Eq. (S.12) is unknown.The potential could approach the escape voltage of the photoelectrons.For a photon energy of 7.4 keV, the primary absorption edge is Pb M5 at 2.5 keV, giving a primary photoelectron energy of 4.9 keV.Thus surface charging to φ pk > 1000 V is possible.Since E y is simply proportional to φ pk , this would give E y > 2.9 × 10 8 V m −1 .

C. Time to Establish Steady-State
An estimate of the typical time needed to establish the steady-state field can be obtained from Eq. (S.20) using the parameters in Table S.1 by substituting in the typical wavenumbers q xm = 2π / σ x , q yn = 2π / σ y .If we use the bulk dielectric constant ε = 2.5 × 10 4 for PMN for the surface dielectric constant ε s and a surface layer thickness δ z equal to the photoelectron escape depth ℓ esc , then the typical time constant is about 4 s.Another estimate can be obtained by con-sidering the time it takes for the surface charge σ to reach the saturation polarization for PMN, P sat ≈ 0.3 C m −1 .Using a maximum surface charging rate of −J out (0, 0) = e I tot 2πσ x σ y ℓ esc ℓ abs sin θ B + ℓ esc (S.31) gives a charging time τ sat ≡ −P sat / J out (0, 0) ≈ 5 s.

D. Surface Tilting due to Electrostriction
In this section we calculate the displacement field and local tilting due to a small, uniform, dynamic applied field in the y direction, ∆E y , in the presence of the static field distribution from the X-ray illumination, calculated above.The displacement field is related to the total strain, which in turn is the sum of the stress-free strain from the electrostrictive response to the electric field, and the elastic strain that arises to make the total strain compatible with a displacement field and to match the stress boundary conditions.We first obtain relations for the change in the stress-free strain components as a function of ∆E y .We then use an ansatz to obtain the change in the displacement field that satisfies the required strain and stress conditions.The local tilts resulting from this changing displacement field can then be calculated.The relations between the strain, stress, and displacement components are linear, so the changes in these quantities due to the dynamic applied field have the same relationships as the total quantities.
We model the PMN as a linear dielectric with electrostriction.For small fields, the polarization components are proportional to electric field, where ε is the relative dielectric constant.This linearity is known to break down for fields above about 5 × 10 5 V m − 1 (Hoover et al., 1997), as the effective dielectric constant decreases with field and the polarization saturates and becomes hysteretic.For simplicity, we will use the linear relationship Eq. (S.32), and discuss the limitations of this model elsewhere.
The stress-free strain components are quadratically related to polarization by (Stephenson and Elder, 2006) where Q 11 , Q 12 , and Q 44 are the electrostrictive coefficients for a cubic material.These quadratic relations have been found to hold even for polarization values beyond the validity of Eq. (S.32) (Li et al., 2014).Here we use the six-index Voigt notation for the components of the strain and stress tensors (Love, 1944).For a small dynamic applied field in the y direction ∆E y , the change in the stress-free strain components are given by The total strain is the sum of the stress-free strain and the elastic strain e T = e SF + e E .Likewise, the changes in these quantities due to the applied field are related by ∆e T = ∆e SF + ∆e E .The components of elastic strain e E i j are linearly related to the components of stress X i j (Love, 1944).The same relations apply to the changes in these quantities, We need total strain components that are compatible with a displacement field, and also match the stress boundary conditions at the surface.There are differential relations between the displacement field components and the total strain components (Love, 1944).Since these are linear, the same relations apply between the changes in these quantities due to the applied field, Boundary conditions arise on the stress because its components in the z direction must be zero at the surface.These also apply to the changes in stress due to the applied field, e.g.∆X zz = ∆X yz = ∆X xz = 0 at z = 0.In addition, the displacements, strains, and stresses must tend to zero at large distances from the illuminated area.
We can use an ansatz for the displacement field change ∆u that can be made to satisfy these conditions.We assume the same type of in-plane Fourier series with z-dependent components as in the solution for φ , Eq. (S.22), with where A x , A y , and A z are real constants that will be chosen below to satisfy the boundary conditions.The Fourier components of the changes in total strain can be evaluated using For the boundary conditions ∆X zz = 0, ∆X yz = 0, and ∆X xz = 0 to be met at all positions (x, y) in the plane of the surface, the z = 0 value of their Fourier components must be zero for all (m, n).Substituting the Fourier components ∆ e E i j mn (z) = ∆ e T i j mn (z) − ∆e SF i j mn (z) from Eqs. (S.24)-(S.26),Eqs.(S.45)-(S.50)and Eqs.(S.72)-(S.77)into Eqs.(S.57)-(S.62), the boundary conditions ∆X zzmn (0) = 0, ∆X yz mn (0) = 0, and ∆X xzmn (0) = 0, respectively, are satisfied for all (m, n) when which gives The change in the tilt of the surface around the x axis in response to the dynamic applied field is ∆χ = d∆u z /dy.The The change in the tilt ∆χ has maximum magnitude at x = y = z = 0 which can be evaluated as This is linearly proportional to applied dynamic field ∆E y and the peak static field from the X rays, φ pk , and inversely proportional to the beam width σ x , σ y .The predicted surface tilt around the y axis in response to the dynamic applied field is ∆η = d∆u z /dx.This is zero at the center x = y = 0, owing to the assumed symmetry of the static and dynamic electric fields.
Table S.2 shows the symmetries of the various functions calculated.There are four types of symmetries, determined by the parities in the x and y directions.The parities of the reciprocal space functions f (q x , q y ) are the same as those of the real space functions f (x, y).Because the real-space functions are all real, the corresponding reciprocal space functions have either real or imaginary type as shown.Each of the components of vectors or tensors has its own symmetry, as shown.The symmetries of the dynamic function components produced by ∆E y differ from those of the corresponding static function components.All of the types of strains (total, stress-free, and elastic) have the same symmetries of their components.Only the functions that are even in both x and y, such as ∆χ, are non-zero at the center of the illuminated area, x = y = 0. Other functions, such as ∆e T zz , that are zero at  (Ahart et al., 2007;Hoover et al., 1997;Uchino et al., 1980).Q 44 was calculated from the value of Q 33 = 1.15 × 10 −2 m 4 C −2 for the (1 1 1) direction (Lee et al., 1999), using Q 44 = (3Q 33 − Q 11 − 2Q 12 )/2 (Li et al., 2014).
the center will have minimal influence on the diffracted beam, so for example we do not predict significant dynamic radial motion of the Bragg peak.There should be significant static radial motion due to e T zz .

E. Results for PMN
We have calculated the distributions of the components of dynamic stress, strain, and displacement and the dynamic tilt, using the parameter values in Tables S.1 and S.3.The distributions are shown in Figs.S.7 -S.11.For these parameters, the net effective electrostrictive coefficient in Eq. (S.84) has a value of c 11 + 2c 12 = −2.8× 10 −3 m 4 C −2 , (S.85) giving a peak-peak tilt of χ pp = +5.2× 10 −5 for a peak-peak field of E pp = 3.5 × 10 4 V/m, which is the maximum peakpeak amplitude of the applied field under our experimental conditions.Because we use an inward-pointing z axis, positive ∆E y corresponds to a tilt of the surface normal towards +y, which is the direction of negative applied potential.
Because the stress-free strain distributions follow those of the electric field, they are confined to the region near the illuminated surface.However, some of the components of total strain, stress, and displacement extend further in x, y, or z.The region of significant tilt ∆χ is confined to the region near the illuminated surface.The effective penetration depth of the 7.4 keV X rays is ℓ abs sin θ B /2 = 1.2 × 10 −6 m, accounting for absorption of both the incident and exit beam.Since this is smaller than the beam size, the illuminated volume has an average tilt similar to the maximum value.The average of the χ pp distribution weighted by the effective X-ray intensity I 0 (x, y) exp(−2z/ℓ abs sin θ B ) is 1.2 × 10 −5 at E pp = 3.5 × 10 4 V m −1 .Note that the predicted motions of the sample surface in the in-plane directions ∆u x and ∆u y are much smaller than the illuminated area, and so would not contribute to the speckle decorrelation.

Figure S. 1
Figure S.1 Extracted two-time matrices in speckle shifts in (a) ∆i and (b) ∆ j, for applied field E pp = 3 × 10 4 V/m and incident X-ray intensity I tot = 5.5 × 10 9 cps.Shifts in pixels are proportional to shifts in ν or δ , with one pixel in i or j corresponding to 84 or 55 µrad in ν or δ , respectively.

Figure
Figure S.1 shows examples of the two-time matrices of the shifts in pixels ∆i min (t 1 ,t 2 ) and ∆ j min (t 1 ,t 2 ) that minimize the difference between speckle patterns at two times in the field cycle.Figure S.2(a) and (b) show the values from each row of these matrices.The curves in Fig. S.2(a) all have similar shapes, consistent with the extracted differences of all time pairs ∆i min (t 1 ,t 2 ) arising from the average timedependent shift ∆i(t) shown in Fig. S.2(c).The curves shown in Fig. S.2(b) are noisier and have different shapes.Thus the extracted pair differences in this direction are only approximately attributable to the average time-dependent shift ∆ j(t) shown in Fig. S.2(c).
Figure S.1 shows examples of the two-time matrices of the shifts in pixels ∆i min (t 1 ,t 2 ) and ∆ j min (t 1 ,t 2 ) that minimize the difference between speckle patterns at two times in the field cycle.Figure S.2(a) and (b) show the values from each row of these matrices.The curves in Fig. S.2(a) all have similar shapes, consistent with the extracted differences of all time pairs ∆i min (t 1 ,t 2 ) arising from the average timedependent shift ∆i(t) shown in Fig. S.2(c).The curves shown in Fig. S.2(b) are noisier and have different shapes.Thus the extracted pair differences in this direction are only approximately attributable to the average time-dependent shift ∆ j(t) shown in Fig. S.2(c).
Figure S.2 (a) and (b) Estimates of speckle pattern shifts ∆i(t) and ∆ j(t) from each row of matrices in Fig. S.1.(c) Average shifts ∆i(t) and ∆ j(t) obtained by summing columns of matrices.

Figure S. 3
Figure S.3 Maximum and minimum two-time correlation values (actual and estimated from Bragg and speckle motions) vs. (a) applied field at I tot = 5.5 × 10 9 cps, (b) incident X-ray intensity at E pp = 3 × 10 4 V/m.For estimated values, maximum contrast β is set to unity to provide offset from actual.
Figure S.4 Measured change in sample angles determined from speckle and Bragg motions as a function of time during field cycle, for different incident X-ray intensities.Peak-to-peak applied field was E pp = 3 × 10 4 V/m.
Figure S.5 Measured change in sample angles determined from (a) speckle and (b) Bragg motions as a function of time during field cycle, for carbon-coated sample.Peak-to-peak applied field was E pp = 3.5 × 10 4 V/m, incident X-ray intensity was I tot = 5.5 × 10 9 cps.

Figure S. 6
Figure S.6 Steady-state potential φ and electric field components E x , E y , and E z at surface (right) and inside sample (left) using φ pk = 10 V.

Figure S. 7
Figure S.7 Components of the dynamic stress-free strain tensor ∆e SF at the surface (right) and inside sample (left) at ∆E y = 1.75 × 10 4 V m −1 and φ pk = 10 V.

Figure S. 8
Figure S.8 Components of the dynamic total strain tensor ∆e T near the surface (right) and inside sample (left) at ∆E y = 1.75 × 10 4 V m −1 and φ pk = 10 V.

Figure S. 9
Figure S.9 Components of the dynamic stress tensor ∆X near the surface (right) and inside sample (left) at ∆E y = 1.75 × 10 4 V m −1 and φ pk = 10 V.

Figure
Figure S.10 Components of the dynamic displacement vector ∆u at the surface (right) and inside sample (left) at ∆E y = 1.75 × 10 4 V m −1 and φ pk = 10 V.

Table S .
1 Parameter values used in calculating steady-state potential and field due to X-ray illumination.

Table S .
3 Parameter values used in calculating stress, strain, and displacement distributions due to dynamic field