Direct recovery of interfacial topography from coherent X-ray reflectivity: model calculations for a 1D interface

The inversion of X-ray reflectivity to reveal the topography of a 1D interface is evaluated through model calculations.

Abstract

The use of coherent X-ray reflectivity to recover interfacial topography is described using model calculations for a 1D interface. The results reveal that the illuminated topography can be recovered directly from the measured reflected intensities. This is achieved through an analysis of the Patterson function, the Fourier transform of the scattering intensity (as a function of lateral momentum transfer, Q _//, at fixed vertical momentum transfer, Q _z). Specifically, a second-order Patterson function is defined that reveals the discrete set of separations and contrast factors (i.e. the product of changes in the effective scattering factor) associated with discontinuities in the effective interfacial topography. It is shown that the topography is significantly overdetermined by the measurements, and an algorithm is described that recovers the actual topography through a deterministic sorting of this information.

1. Introduction  

The use of X-ray scattering to probe the structure and topography of any interface (e.g. solid–vapor, solid–liquid and solid–solid) is well established (Robinson, 1986 ▸, 1991 ▸; Fenter, 2002 ▸; Feidenhans’l, 1989 ▸). Reflection of an incoherent X-ray beam from a flat surface or interface is found only in the specular direction (i.e. corresponding to mirror-like reflection) due to its lateral translational invariance. The presence of lateral heterogeneity in the form of non-uniform topography leads to diffuse scattering that reflects the details of the topography (Sinha et al., 1988 ▸). Generally, the illumination of the incident beam spans macroscopic regions and the coherent interference between regions that are separated by distances larger than the beam coherence length is lost. As such, a complete calculation of the reflected line shape includes an ensemble average over the locally interfering regions that add coherently, and the remotely scattering regions that add incoherently (Warren, 1990 ▸; Sinha et al., 1998 ▸). Thus, the characteristics of reflected X-rays (i.e. the width and line shape) reveal the average properties of the interface (i.e. the average interfacial domain size and correlations between steps) (Warren, 1990 ▸).

The increasing availability of coherent X-ray sources provides new opportunities to probe interfacial processes. The central difference is that signals from all areas illuminated under the beam footprint (typically approximately µm’s in extent) interfere coherently. The observed scattering pattern is characterized by ‘speckles’, the high-frequency oscillations in the intensity commonly seen with coherent laser beams (Sutton et al., 1991 ▸) (Fig. 1 ▸). The use of coherent X-rays provides sensitivity to the locations, sizes and correlations between all structures under illumination. However, this information is obscured, but not eliminated, by the loss of phase information in the measured intensities (i.e. the phase problem) (Robinson et al., 1998 ▸). This phase information can be recovered using phasing retrieval algorithms to invert the scattering intensities directly to the illuminated structure (e.g. using error correction algorithms) (Robinson et al., 1999 ▸; Robinson & Miao, 2004 ▸; Gerchberg & Saxton, 1972 ▸; Fienup, 1978 ▸), but this is subject to the stochastic noise in any experimental data and the associated challenges of algorithm convergence. The measured diffraction patterns for overlapping beam spots can also be used to understand topographic structures in a ptychographic imaging mode (Claus et al., 2011 ▸; Hruszkewycz et al., 2011 ▸). This approach has an advantage with respect to direct inversion from a single diffraction pattern in that the structures of interest are sampled with a higher degree of redundancy so that they can be recovered more robustly. Coherent X-ray beams also provide a route to probe dynamics of interfaces directly through X-ray photon correlation spectroscopy (XPCS) (Sutton et al., 1991 ▸; Mokhtarzadeh & Ludwig, 2017 ▸) by comparing the statistical differences in scattering data as a function of time separation. XPCS can be used to probe stationary dynamics (e.g. diffusion) and dynamically evolving systems through the use of two-time correlation functions (Sutton et al., 2003 ▸). The ability to reconstruct a real-space structure routinely from a measured coherent diffraction would enable the possibility of simultaneously understanding the explicit evolution of structures (i.e. in real space) as well as the statistical evolution of those structures (the underlying dynamics). This, however, relies on a robust understanding of the information content in the intensities.

Figure 1.

Schematic perspective view (left) of an interface topography (with terrace height indicated by color) for a rough interface that is illuminated by a coherent beam having a footprint indicated by the dashed circle, and (right) its associated 2D scattering intensity (as a function of −π/a < Q _x, Q _y < π/a, and at a fixed Q _z = π/c; intensities are indicated by a logarithmic color scale). A schematic of the specular reflection geometry is shown (left) including the horizontal dashed line indicating the scattering plane and the specular reflection condition with the incident and reflected angles having the same angle, α.

Here, model calculations are used to evaluate the information content in the scattering data for a 1D interface in which the topography derives from discrete steps on a single-crystal substrate. It is well known that the information content in a diffraction pattern is least robust in 1D structures, and becomes increasingly overdetermined in 2D and 3D structures (Bates, 1982 ▸). Consequently, the present results explore the information content in the limit where this information is least robust. The results demonstrate that the topography of a 1D interface (i.e. defined by the location and contrast change at each interfacial discontinuity, including features due to topography and illumination) is overdetermined with respect to information that can be obtained directly from the measured intensities.

2. Interfacial model and scattering formalism  

An understanding of these relationships is obtained with a 1D model of the interfacial topography (Fig. 2 ▸). The topography of a crystalline interface can be described by a local interfacial structure within each lateral unit cell that is identical at every site except for the interface height that varies with position. The scattering from such an interface is described as a crystal truncation rod from the solid substrate for an ideally terminated lattice, with the rods oriented perpendicular to the physical interfacial plane. In the presence of topographic irregularities (i.e. steps), the reduction of the interface domain size leads to lateral broadening and diffuse scattering. A general description for any interfacial system (e.g. including solid–vacuum, solid–liquid and solid–solid interfaces) expresses the interface as a semi-infinite column of material along the surface normal direction, ρ_sub(z), that is laterally reproduced periodically for every lateral unit cell, x = na (−∞ < n < ∞, where n is an integer and a is the lateral lattice spacing). This column is identical at each site except for the height of the interface which can have values that are integral multiples of the substrate vertical lattice spacing, h(n)c (where c is the vertical lattice spacing and h is an integer). The scattering intensity is calculated from the atomic scattering factors, f_j(Q), of each atom, j, and illumination function, E, as a function of position, r, for all areas illuminated by the X-ray beam, and for each momentum transfer, Q, and is written as

where

This shows that the observable scattering intensity is completely separable between two terms: F _sub describes the internal structure of the interface (with a sum over all atoms in an infinite column of material within a single lateral surface unit cell) and F _top depends only on the interface topography, h(n), and the illumination function, E(n),

The form of the substrate structure factor has been previously described extensively (Robinson & Tweet, 1992 ▸). For simplicity, we consider the simple case of an ideally terminated surface (i.e. where all crystal layers are identical as a function of depth). This term takes on the form

where F _UC is the bulk structure factor describing the atomic arrangement with a single bulk unit cell, and F _CTR is the crystal truncation rod form factor, describing the rods of intensity that emanate from each substrate Bragg peak, oriented along the surface normal direction, due to the termination of the lattice. The more general case that includes the layer-dependent variation of the structure (e.g. surface relaxations) does not change any conclusions as all of those details remain within F _sub.

Figure 2.

1D model of (a) the topography and illumination function, (b) the associated effective scattering-factor profile at Q _z = π/c, and (c) the calculated scattering intensities [as a function of Q _x = H(2π/a) at Q _z = π/c]. Specific details of the model are provided in the supporting information.

The focus of this article is on the topographic structure factor,

Here, f _eff(n) E(n)exp[iQ_zc h(n)] is the effective topographic structure factor which corresponds to the product of the topographic phase factor, exp[iQ_zc h(n)], and the illumination factor, E(n), at each site, n. Its value is controlled by a combination of the scattering condition, Q _z, through the specific value of the interface height, h(n) c, and the illumination function, E(n), 0 ≤ E(n) ≤ 1, that describes the visibility of any site on the interfaces and consequently the integrated intensity will have the value I ₀ = ∑_n E(n)². For an illumination that is uniform over N sites, E(n) = 1, and the magnitude of the topographic structure factor is |F _top(Q _x = 0, Q _z = 0)| = N.

The topographic structure-factor magnitude can be calculated as

where f _eff*(m) = conj[f _eff(m)] is the complex conjugate of f _eff(m).

The phase factor of the topographic structure factor for the nth site is written as exp[iQ_zc h(n)] = exp[i2πLh(n)] = ϕ₀ ^{h (n)}, where ϕ₀ = exp(iQ_zc) = exp(i2πL) [where L is the vertical Bragg index and Q_z = (2π/c)L]. Consequently, the contrast between interface sites that differ in height depends only on the vertical momentum transfer, Q _z. For the specific case of scattering at the ‘anti-Bragg’ condition (i.e. halfway between Bragg peaks, at Q _z = π/c or L = 1/2), the phase factor has the value, ϕ₀ ^{h (n)} = exp[iπh(n)] = [−1]^{h (n)}. At this scattering condition, neighboring terraces separated by a single-unit-cell-high step have scattering factors that are either 1 or −1 (i.e. they are exactly out of phase), and any terraces that differ in height by an even number of layers are exactly in phase. More generally, the phase factor is a complex function of the step height and vertical momentum transfer. So for instance, at L = ¼, the scattering factor for neighboring terraces differs by factors of exp(iπ/2) = i, so that terraces at heights of h = 1, 2, 3 and 4 have structure factors with pre-factors of: 1, i, −1, −i, respectively.

The topographic structure factor has two general characteristics. First, it is obtained by a discrete Fourier transformation calculated by the sum over each lateral unit cell, with integer index n, and an interface height that can only have values that are discrete multiples of the substrate vertical lattice spacing, h(n)c (where h = integer). Consequently, the topographic structure factor is periodic in reciprocal space, both laterally with a period of 2π/a, and vertically with a period 2π/c. That is, the contribution of the topography to the scattering intensities at any (Q _x, Q _z) = [H(2π/a), L(2π/c)] (indicated by the Miller indices, [H, L]), is identical to that for any integer multiple of the Miller indices (e.g. [H + 1, L], [H, L + 1], [H + 1, L + 1] etc.). The relevant integration region for the conserved intensity is the lateral reciprocal-space unit cell, −π/a < Q _x < π/a.

A second feature of the topographic structure factor is that it represents the scattering from a ‘pure phase object’. That is, the magnitude of the effective scattering factor, f _eff(n), is unity for any interface height within a uniformly illuminated region where E(n) = 1 {i.e. |f _eff(n)| = |exp[iQ_zc h(n)]| = |ϕ₀ ^{h (n)}| = 1}. The information about the interface height is encoded in the complex phase of the scattering factor, ϕ₀ ^{h (n)}. Pure phase objects have the property that the integrated scattering intensity (in this case, integrated laterally over Q _x) is precisely conserved for any h(n) [i.e. for a fixed illumination function, E(n)]. That is, the introduction of a variable interface topography shifts the angular distribution of the scattering intensity but does not change the total integrated scattering signal (e.g. it moves signal from the specular reflection condition for a flat interface to a non-specular ‘diffuse’ scattering for an interface roughness). In a well known optical analogy [i.e. the imaging of cells in water (Nugent et al., 2001 ▸)], such objects are invisible when viewed by a perfect imaging system and similar considerations have been applied to the direct imaging of topographic features of a surface with interfacial X-ray microscopy (Fenter et al., 2008 ▸).

These ideas are illustrated by comparing a flat interface with an interface with a single island (Fig. 2 ▸), including the interface topography and illumination functions [Fig. 2 ▸(a)], the spatial variation of the effective scattering factor [Fig. 2 ▸(b)] and the associated scattering intensities. These calculations are done as a function of Q _x at the vertical anti-Bragg condition, Q _z = π/c (L = 1/2) [Fig. 2 ▸(c)]. The illumination varies from 0 to 1 over a finite lateral range of positions, n, the effective scattering factor is zero outside the illuminated region, but has the value of 1 or −1 for the regions having heights of h = 0 and h = 1, respectively (h = 1 is found in the region having a mono-atomic step). The scattering intensities for these cases are shown in Fig. 2 ▸(c). The intensity from the flat interface exhibits regular oscillations corresponding to the width of the illumination region having ‘sharp’ edges. These are the simplest manifestation of the ‘speckles’ in coherent scattering. The same calculation of the scattering intensity of the interface with a single island is qualitatively similar except that the intensity variation is more complex and somewhat irregular, but also has a larger non-specular signal (for Q _// ≠ 0) compared with that found for the flat interface. Note also that the specular signal (i.e. at Q _// = 0) has decreased with the introduction of a terrace, providing a visual illustration that the integrated intensity is conserved.

3. Interface topography as seen by the Patterson function  

The information content in the line shape of the reflected X-rays constrains the unknown topography, and is directly visualized by the Patterson function, P(Δx). The Patterson function is defined as the Fourier transform of the observed intensities, and also represents the density–density correlation function of the density as a function of separation, Δx [equation (6)],

Using the expression for |F _top|² [equation (5)], the Patterson function for a coherently illuminated interface is

The last term represents correlations for Δx > 0, and δ(Δx) is the Kronecker delta function. When considering only positive separations, Δx > 0, and substituting f _eff(n) = exp[i2πL h(n)], the Patterson function becomes

The range of momentum transfer that is used to define the Patterson function can be as large as one Brillouin zone (−π/a < Q < π/a), but typically will be smaller due to limitations in the measurable intensities (i.e. where signals are above background). The individual sites on the interface therefore are not spatially resolved in the Patterson function (since the spatial resolution varies as δx ∼ π/Q _max > a). In this case, the Patterson function is effectively a continuous function of Δx.

The simplest topography is one in which a flat, uniform sample is illuminated with a uniform beam that illuminates N_I sites, and where exp[i2πLh(n)] = 1 [i.e. h(n) = 0]. In this case, the self-correlation for all sites under illumination has the value P(0) = N_I. The number of correlations decreases linearly with increasing separation, and there is only one correlation at separations, Δx = (N_I − 1)a, and consequently P[(N_I − 1)a] = 1. It is also apparent that P(Δx ≥ Na) = 0 as no two sites are illuminated at distances larger than N_I [Fig. 3 ▸(a)].

Figure 3.

(a) The Patterson function as a function of the lateral separation calculated from the intensities for a flat interface and an interface with an island (red and blue lines, respectively). The extended Patterson maps for the two interfaces, (b) flat and (c) with one island, are shown, as indicated in the inset to (b). This image shows the product of the effective scattering factors for any two sites, n ₁ and n ₂ (indicated by color), as a function of their separation Δn = n ₂ − n ₁. The variation of the effective scattering factor, f _eff(n), is shown below the abscissa in (b) and (c). The arrows indicate the relationship between effective scattering factor at two sites and their product (yellow and white arrows indicate sites that have the same versus opposite effective scattering factors, respectively). The topography and illumination of two structures are indicated above (b) and (c) as red dashed and black solid lines, respectively.

A similar calculation for the interface with a single island is only slightly more involved. The correlations include both positive and negative scattering factors, exp[i2πLh(n)] = ±1 (calculated for L = ½). In this case the Patterson function is unchanged for the self-correlation term, P(0) = N, but it decreases faster with increasing separations than the Patterson function for the flat interface, with regions that have fixed slope separated by discontinuities. However, the Patterson function at sufficiently large separations becomes the same as that for the flat interface, and in between these two extremes it can have either negative or positive values, as shown [Fig. 3 ▸(a)].

4. Extended Patterson maps  

The systematics of this behavior for any arbitrary structure can be visualized more explicitly by plotting an ‘extended Patterson map’ that shows the contributions to the Patterson function from each pair of sites within the illumination region [Figs. 3 ▸(b) and 3 ▸(c)]. This shows the variation of the effective scattering factor for each site below the abscissa [from equation (4)], and product of the effective scattering factors (represented by color) for each pair of sites site, n_i and n_j [as indicated by the yellow arrows, Fig. 3 ▸(b)]. The effective scattering factors are arranged in a triangular structure with each cell corresponding to a specific pair of sites. Since this is calculated for the anti-Bragg condition, the effective scattering factors can only have values of ±1 within the illuminated region, and the product of scattering factors for each pair of sites is either 1 or −1. In this representation, each horizontal row shows the individual correlations for all pairs of sites at a given lateral separation, Δx = (n_i − n_j)a. The sum of these values along a given row (i.e. at a fixed separation Δx) corresponds to the value of the Patterson function at that separation, Δx [Fig. 3 ▸(a)]. Consequently, the apex of this triangle corresponds to the product of effective scattering factors between the two sites at the edges of illumination.

This representation provides a simple understanding of how topography influences the shape of the Patterson function. The Patterson function for a flat surface has the form P(Δx) = Na − Δx [red line, Fig. 3 ▸(a)]. From the extended Patterson map, it is seen that this is because the product of the effective scattering factors is equal to 1 for any two sites within the illuminated region for a flat surface. Also the number of pairs of sites at a given separation decreases with the separation distance, as indicated by the lateral width of the triangular region at a given separation, Δx [red regions, Fig. 3 ▸(b)]. For instance, the illumination of N surface sites includes N − 1 pairs of sites separated by a single unit cell, a, but only 1 pair of sites separated by Na (i.e. the edges of the illumination region). For separations larger than the illumination region, the extended Patterson map and the Patterson function have the value P(Δn > N_I) = 0 [green region, Fig. 3 ▸(b)] since no sites separated by distances larger than the illumination region contribute to the scattering intensity.

A similar consideration provides insight into the Patterson function for a surface with topography, in this case illustrated by the inclusion of a single island, with h(n) = 1, within the illumination region. This changes the effective scattering factors for sites within the island to −1, while that for the rest of the surface retains the value of 1. The values in the extended Patterson map for any pair of sites that are at the same height have the value of 1 (either within the island, where −1 × −1 = 1, or on the terrace where 1 × 1 = 1), shown as red regions in Fig. 3 ▸(c) (indicated by yellow arrows). In contrast, any two sites that are separated by a single step results in an effective scattering-factor value of −1 (indicated by white arrows). This leads to bands of contributions to the extended Patterson map with negative correlations [blue regions, Fig. 3 ▸(c)]. The presence of these negative correlations leads to the larger negative slope in the Patterson function at small separations [Fig. 3 ▸(a)] with respect to the flat surface. Meanwhile, the Patterson function is unaffected by the presence of the island at the largest inter-site separations since all of the sites at those separations have the same height.

The extended Patterson map also illustrates how the observed scattering intensities are sensitive to a combination of the structure of interest (i.e. the topography within the illumination region) and the details of the illumination (Fig. 4 ▸). It is apparent that small changes in illumination position [solid and dashed green curves, Fig. 4 ▸(a)] lead to significant changes to the calculated intensities [Fig. 4 ▸(b)] in spite of the simplicity of the structure of interest (i.e. a single island). These changes can be illustrated through the Patterson function analysis [Fig. 4 ▸(c)]. These calculations show that changes to the illumination position for a fixed illumination shape (solid blue versus dashed green lines) or the illumination shape for a fixed illumination position (solid blue versus gray dashed lines) lead to systematic changes to most aspects of the Patterson function. These appear as changes to the slope at inflection points in the Patterson function.

Figure 4.

(a) Spatial variation of the effective scattering-factor profile (blue, green and gray) for a topographic profile with three choices of illumination (blue and green profiles have the same sharp illumination function that differs only in its position, the gray profile differs from the blue profile only in the sharpness of the illumination function edges). Specific spacings in the structure are indicated (arrows). (b) Scattering intensities for profiles in (a) (green and gray intensities are offset by factors of 0.1 and 0.01, respectively), (c) Patterson functions, P(Δx), and (d) second-order Patterson functions, P ₂(Δx), for the structures in (a). Discrete Fourier components in (d) are labeled with blue, black and red arrows, corresponding to specific features in (a), corresponding to the island size (blue), the size of the illumination function (red), and separations between the island and illumination edges (black). Only those features indicated by black arrows move with a change in the illumination function position (green line). Specific details of the model are provided in the supporting information.

5. Direct characterization of topography and illumination  

Further insights into the information content in the Patterson function can be evaluated by expressing the topographic structure factor as an effectively continuous profile:

(where the integral is over the illuminated interface). This expression can be transformed through integration by parts to

where the sum is over all sites, m, where there is a discontinuity, Δf _eff, in the effective scattering factor (e.g. at steps and edges of the illumination region). A second-order Patterson function, P ₂(Δx), can be defined based on the Fourier transform of Q _x ² I(Q _x), which is equivalent to the second derivative of the Patterson function [calculated from either the topography or the Fourier transform of the intensities, equations (6)–(8)]:

Here, the function G[Δx − (X _m − X _n)] ∫ exp[iQ_x(X _m − X _n)]exp(iQ_xΔx) dQ _x describes the contribution to P ₂(Δx) due to a pair of features located at X_m and X_n, with peaks at their separations, Δx = (X _m − X _n) (including the self-correlation term at Δx = 0). For data having a finite Q range (e.g. for 0 < Q_x < Q _max) this function exhibits peaks having the form sin{Q _max[Δx − (X _m − X _n)]}/{Q _max[Δx − (X _m − X _n)]} whose width is controlled by the limits of integration (i.e. proportional to the resolution of the data, ∼π/Q _max). This function becomes the Kronecker delta function in the limit of Q _max → ∞.

Consequently, the second-order Patterson function, P ₂(Δx) [Fig. 4 ▸(d)], consists of a discrete set of peaks corresponding to the separations between discontinuities in the effective scattering factor (e.g. steps and the edges of the illumination regime), rather than the separation and effective scattering factor for every site (as Fig. 3 ▸). For the case of a single island illuminated with a sharp beam [Fig. 4 ▸(a), blue lines] the measured intensities contain six distinct Fourier components, one corresponding to the island size (indicated by the blue bold arrow), another associated with the illumination size [Fig. 4 ▸(a), red dashed arrow] and an additional four spacings associated with the distances separating the edges of the island from those of the illumination [Fig. 4 ▸(a), black arrows]. These Fourier components are clearly distinguished in P ₂(Δx) [Fig. 4 ▸(d)]. This assessment is further confirmed by observing P ₂(Δx) for the two different illumination conditions. Changes in the beam position [green lines, Fig. 4 ▸(a)] alter the Fourier components associated with the separations between the edges of the island and illumination regions [black arrows, Fig. 4 ▸(d)], while those corresponding to the island and illumination sizes are unaffected [blue and red arrows, respectively, Fig. 4 ▸(d)]. Similarly, the inclusion of ‘soft’ edges in the illumination [Figs. 4 ▸(a), 4 ▸(d), gray lines] suppresses all contributions associated with the illumination shape, leaving only those Fourier components that are associated with the intrinsic topography, as expected. This leads to a much simpler scattering pattern that is dominated by a single Fourier component associated with the island size [gray line, Fig. 4 ▸(b)].

This analysis illustrates that the information needed to describe a complex, step-wise, continuous topographic profile is substantially reduced, and includes only the locations, x _i, and changes in the effective scattering factor, Δf _eff−I, at each site, i, where there are discontinuities in the effective scattering factor, Δf _eff. This will appear in the second-order Patterson function as discrete peaks having separations, Δx _ij = x _i − x _j, and amplitudes proportional to Δf _eff−i Δf* _eff−j. This represents a dramatic reduction of the information needed to describe the interface topography, as compared with the effective scattering factor at every site on the interface as would be used in the traditional Patterson function. {For completeness, we note that an additional multiplicative factor in equation (11) can be included with Δf _eff to account for structures that are not atomically sharp (e.g. a step with unresolved roughness or the edge of the illumination region) by multiplication with a Debye–Waller-like factor, exp[−½(Qσ_i)²], where σ_i is the r.m.s. roughness of that feature. This will appear as a broadening of each peak in the differential Patterson function, with a peak width of σ_ij = (σ_i ² + σ_j ²)^1/2. This factor is not shown for simplicity.}

The magnitude of any peak in P ₂(Δx) [Fig. 4 ▸(d)] is defined by the product of the change in the effective scattering factor, Δf _eff(x _i)Δf _eff(x _j), for the two steps at separation, Δx = x _j − x _i [Fig. 4 ▸(a)]. This leads to three subsets of peaks in P ₂(Δx) defined by their magnitudes. The discontinuity in the effective scattering factors at an island edge (at the anti-Bragg condition, L = ½) has a value of ±2 (corresponding to effective scattering factors of 1 and −1 across the step), which is larger than that near the illumination edge of ±1 (e.g. from 1 to 0, or 0 to 1). The strongest Fourier components derive from correlations between steps (i.e. topography), while the Fourier component associated with edges of the illumination region is the smallest, and those associated with the interference between topography and illumination have intermediate values.

The sign of the peaks in the second-order Patterson function, P ₂ (Δx), provides information about the relative phase of the discontinuities at a separation, Δx. Unlike X-ray scattering from atoms, where the density is positive-definite, the changes in the effective scattering factor can be either positive or negative for ‘up’ and ‘down’ steps (i.e. for L = ½). Here, the peaks are positive when the effective scattering factors for the two edges have opposite polarity. Consequently, the peaks associated with the island size and illumination size appear as positive peaks [at separations of 50 and 240, respectively, Fig. 4 ▸(b)] while the cross terms between the island edge and the illumination edge can appear as either positive or negative peaks.

These ideas are illustrated for a slightly more complex structure in which there are two islands on an otherwise flat terrace using both a sharp and diffuse illumination function [Fig. 5 ▸(a)] leading to changes in the calculated intensities [Fig. 5 ▸(b)], as well as the Patterson and second-order Patterson functions [Figs. 5 ▸(c), 5 ▸(d)]. As expected from the above discussion, the second-order Patterson function shows a larger number of peaks than that for a single island, corresponding to the larger number of features that are correlated (four step edges and two illumination edges). Fourier components that are entirely from the topography [blue arrows, Figs. 5 ▸(a), 5 ▸(d)] have magnitudes of either +1 or −1, with positive peaks observed when the changes to the effective scattering factors for the two edges have opposite polarity [i.e. up and down steps; solid blue lines, Figs. 5 ▸(a), 5 ▸(d)] and negative when the steps have the same polarity [i.e. up and up steps; dashed blue lines, Figs. 5 ▸(a), 5 ▸(d)]. Smaller peaks are seen for the cross terms between topography and illumination (with a magnitude of 0.25) and one term is associated with the illumination edges. Similarly, the phase of the correlations between the topography and illumination (gray arrows) also shows either positive or negative values [Fig. 5 ▸(d)]. The correlations associated with the edges of the illumination size range have a positive phase because the changes of the effective phase factor at the edges of the illuminated region are assumed to have the same height and therefore have opposite polarities. We note that this term will have a negative sign if there is a net change in the interface height by one layer at the edges of illumination (at L = ½). Therefore the Patterson analysis through the second-order Patterson function, P ₂ (Δx), reveals substantial insights into the details of the interface topography.

Figure 5.

Results for a two-island structure. (a) Spatial variation of the effective scattering factors for a two-island structure with a sharp (blue) and diffuse (gray) illumination function. Specific spacings in the topography are indicated (blue arrows). (b) Scattering intensities (gray profile offset by a factor of 0.1), (c) Patterson function, P(Δx), and (d) second-order Patterson functions, P ₂(Δx), are shown for the two illuminations (blue and gray lines). Arrows in (a) and (d) correspond to spacings associated with the separation between topographic steps (blue), the size of the illumination function (red), and separations between steps and the edges of the illumination (gray). Specific details of the model are provided in the supporting information.

6. Recovery of interface topography from intensities  

The present 1D model provides a framework in which to understand whether an actual topography can be understood uniquely from the observed intensities, in the least favorable limit of 1D structures. Here, it will be assumed that the intensities are fully sampled with sufficient Q resolution (i.e. pixel size) and Q range (i.e. spatial resolution) to resolve all of the relevant features of the data to address the question of whether this can be done in principle. The impact of some of these assumptions will be discussed, below, for the more general case of realistic data.

The above discussion can be generalized to evaluate the explicit information content for an arbitrary topographic structure. Consider a topographic profile that consists of N discrete features (e.g. N − 2 topographic steps and 2 edges of a sharp beam, or N steps and a diffuse beam shape). This structure is defined by the N − 1 positions (rather than N because the absolute position of the structure is not determined), as well as the N magnitudes of the effective density contrast, and the N signs of the effective density contrast for a total of 3N − 1 unknowns. These unknowns are compared to the information from the second-order Patterson function, P ₂(Δx). A topographic structure with N discrete features will exhibit N(N − 1)/2 discrete spacings that each constitute an independent constraint on the structure (i.e. Δx _ij = x _j − x _i). Additional observables include the same number of magnitudes and signs for each peak in the Patterson function (e.g. |P ₂(Δx _ij)| = |Δf_i||Δf_j|), for a total number of independent observables of 3N(N − 1)/2. Comparing the observables to unknowns, the information in P ₂(Δx) is expected to be sufficient to uniquely solve the actual topographic profile when 3N(N − 1)/2 > 3N − 1, a condition that is satisfied for N ≥ 3. That is the intensities from a 1D topographic structure are sufficient to constrain the actual topography for any non-trivial structure with N > 3. For example, a structure with N = 10 has 45 distinct Fourier components in P ₂(Δx) that provide 135 independent constraints, but this structure is defined by only 29 unknowns and so such a structure can be overdetermined by a factor of >4 when the scattered intensities are fully sampled.

The recovery of the actual topography from the experimental observables can be achieved through a deterministic analysis of P ₂(Δx). The information that can be derived from P ₂(Δx) can be organized into two vectors, each with length N(N − 1)/2, containing the spacing and contrasts from the peaks in P ₂(Δx),

where ΔX _i and ΔF_i are the step separations and contrasts, respectively, for a given (but unknown) pair of steps, i. The values in each vector are ordered based on the inter-step separations, ΔX _i. The key concept of this approach is the realization that the values in each vector can be arranged in an extended Patterson map for the second-order Patterson function, P ₂(Δx), to reveal the actual arrangements of the spacings, Δx _ij = x _i − x _j, and contrasts, Δf_iΔf_j, of the features in P ₂(Δx) (e.g. Fig. 6 ▸). The extended Patterson map provides a highly constrained structure that tests the consistency with all of the observations (i.e. both separations and contrasts). This process is outlined here briefly, but is shown explicitly for the case of a six-step topographic structure, similar to that in Fig. 6 ▸, in the supporting information. The solution is seeded by the observation that the largest observed separation is uniquely associated with that of the two most extreme features, i.e. ΔX _N(N−1)/2 = Δx _1,N = x _N − x ₁ and therefore ΔF _N(N−1)/2 = Δf ₁Δf_N. Similarly, the second longest spacing, ΔX _{N(N−1)/2−1}, can be assigned to ΔX _{N(N−1)/2−1} = Δx _1,N−1 = x _N−1 − x ₁ and therefore its contrast is: ΔF _{N(N−1)/2−1} = Δf ₁Δf _N−1. [Note that the assignment of this spacing to Δx _1,N−1 rather than Δx _N−1,N is arbitrary, but it eliminates the ambiguity due to Friedel’s law, i.e. the inability of scattering data to distinguish between mirror images of a structure (Als-Nielsen & McMorrow, 2001 ▸).] Since Δx _1,N = Δx _1,N−1 + Δx _N−1,N, one of the observed spacings must take on the value, ΔX _j = Δx _N−1,N = Δx _1,N − Δx _1,N−1, and that feature will have a contrast of ΔF_j = Δf _N−1Δf_N. Continuing this logic, the extended Patterson map is assembled by considering all possible arrangements of the largest unassigned spacing, X _i, consistent with the available observations. Each assignment constitutes a potential ‘seed’ structure that is extended to subsequent generations of assignments and tests until all possible solutions consistent with the data are consistent. These seed structures can be organized into a ‘decision tree’ that is extended until it is revealed whether or not they are consistent with experimental observations. Given that the information in the extended Patterson function is significantly over-constrained, the solutions can be expected to be unique. In any case, the presence of any non-trivial solutions will be apparent, if they exist. In the example shown in the supporting information, the N = 6 step structure is identified by only considering nine seed structures in four generations of tests, out of the (N−1)! = 120 different possible arrangements of the step separations, Δx _i,i+1. Inclusion of the observed contrasts in the process provides additional constraints, and can be used to distinguish between structures that are not distinguished by the spacings although the observed contrast factors provide further confirmation that the derived model satisfies all of the constraints of the Patterson function and the simulated intensities.

Figure 6.

(a) The structure of an extended map of the second-order Patterson function, consisting of spacings (left) and contrast factors (right) for each peak in P ₂(Δx). These are shown for a representative topography (c) and associated effective scattering-factor profile (b) as a function of position, x = na. The numbers in the extended Patterson maps (a) that are below the horizontal lines are the actual positions, x _j, and contrasts, Δf_i, of each step in the structure, while those above the line are the step spacings, Δx _i,j = x _j − x_i, and the product of the step contrast factors, ΔF _i,j = Δf_i Δf_j for each pair of steps (i, j), as revealed by the second-order Patterson function, P ₂(Δx _i,j). The effective scattering factor is calculated at the anti-Bragg condition, L = ½. In this representation, the full profile is described by the discontinuities of the structure, including step spacings. Here, it is implicitly assumed that the illumination function is diffuse and therefore it does not contribute significantly to the second-order Patterson function.

One source of increased ambiguity derives from the possibility of ‘degeneracies’ in the Patterson function [i.e. two or more features in P ₂(ΔX) that have values that are either the same or are unresolved due to the finite range of the data]. The presence of such degeneracies can be assessed directly since the number of peaks in a fully resolved second-order Patterson function, P ₂(Δx), is N(N − 1)/2 when there are N features in the density profile. Consequently it is immediately apparent that degeneracies must be present whenever the number of features does not correspond to N(N − 1)/2. The presence of such features decreases the number of independent constraints, but their presence is expected to be tolerated since the degree to which the profile is over-constrained is both substantial and increases with increasing N. Degeneracies will increase the number of structures that need to be considered during the sorting of the structure into an extended Patterson function since it will be necessary to consider not only the next largest unassigned spacing, but also the previous largest assigned spacing, at each generation of seed structures.

7. Discussion  

The model calculations described here provide a demonstration that the intensities from a 1D topographic structure fully constrain the actual topographic profile except for the inherent ambiguities associated with scattering measurements (e.g. the insensitivity to the absolute position of the structure, and the inability to distinguish between a given topography and its mirror image). Note that this latter ambiguity can be resolved through the ptychographic approach of comparing the measured intensities for two overlapping beam spots.

The present calculations did not include any inherent background scattering or other practical limits to the measured dynamic range that can be achieved in an experiment. Experimental data of the reflected beam shape will necessarily be limited in the range of Q over which the data are measurable. Consequently the features in P ₂(Δx) will have a finite width, σ_Δx, defined by lateral spatial resolution, δx, which is controlled, in turn, by the lateral momentum transfer range, ΔQ _//, through the relation, σ_Δx ≃ δx = π/ΔQ _//. As such, any step spacing that is smaller than the resolution will not be resolved, and will appear as a degeneracy in the observed spacings for P ₂(Δx).

An implicit assumption in this analysis is that the shape of the reflected beam is over-sampled by the detector. That is, the effective sampling interval of the intensities, δQ _//, is smaller than the reciprocal-space period associated with the beam size, δQ _// << 2π/(Na). Incomplete sampling of the scattered intensities will lead to distortions in the Patterson function. For example, this may show up as a reduction in the sensitivity to the highest-frequency Fourier components and distortion of the relative amplitudes in the Patterson maps, presumably reducing the usefulness of the observed contrasts as quantitative constraints for the determination of the interfacial topography.

8. Conclusions  

These model calculations consider the information in scattering data from a topographic profile consisting of discrete steps on a 1D single-crystal substrate. The results demonstrate that the topography is substantially overdetermined by the measured intensities. The approach that was used here is unlikely to be practical for the more realistic scenario of determining the full topography of a 2D surface, since the correlations between steps become more complex. However, it is known that scattering intensities from structures become increasingly overdetermined with higher dimensionality (Bates, 1982 ▸), and that the 1D structures, considered here, represent the limit in which this information is least robust. Consequently, it can be inferred that the topography of a real 2D surface may be even more highly determined than the examples presented here.

Funding Statement

This work was funded by U.S. Department of Energy, Office of Science grant DE-AC02-06CH11357.