Direct method for imaging elemental distribution profiles with long-period x-ray standing waves

A model-independent Fourier-inversion method for imaging elemental profiles from multilayer and total-external reflection x-ray standing wave (XSW) data is developed for the purpose of understanding the assembly of atoms, ions, and molecules at well-defined interfaces in complex environments. The direct-method formalism is derived for the case of a long-period XSW generated by low-angle specular reflection in an attenuating overlayer medium. It is validated through comparison with simulated and experimental data to directly obtain an elemental distribution contained within the overlayer. We demonstrate this formalism by extracting the one-dimensional profile of Ti normal to the surface for a ${\text{TiO}}_2/\text{Si}/\text{Mo}$ trilayer deposited on a Si substrate using the $\text{Ti}K\alpha$ fluorescence yield measured in air and under an aqueous electrolyte. The model-independent results demonstrate reduced coherent fractions for the in situ results associated with an incoherency of the x-ray beam (which are attributed to fluorescence excitation by diffusely or incoherently scattered x-rays). The uniqueness and limitations of the approach are discussed.

Figure 1

(Color online) Reflection and refraction of x-ray beam at a vacuum-solid interface. The case shown is of a film (refractive index $n_2$) on a substrate (refractive index $n_3$), present in air $\left(n_1=1\right)$. The incident and reflected waves are shown (in blue and magenta, respectively), along with the x-ray standing wave antinodes (horizontal black lines). The XSW is also generated within the film because of the presence of the reflection from the film-substrate interface (${\mathbf{k}}_2$ and ${\mathbf{k}}_2^R$ denote the transmitted and reflected wave vectors inside the film). The effect due to refraction is exaggerated for clarity.

Figure 2

(Color online) Model-independent formalism applied to simulated fluorescence yield data for Si/Mo bilayer (with thicknesses of 560 and $60\text{Å}$) on Si substrate, for three assumed Gaussian elemental distributions, each with $\sigma =5\text{Å}$. (A) The density profile of the bilayer system (black) used to calculate the reflectivity and phase of reflected wave ($R$ and $\nu$). The three elemental distributions shown are for Gaussian profiles centered at $50\text{Å}$ above (blue), $10\text{Å}$ below (green), and $50\text{Å}$ below (red) the top Si surface. (B) Simulated reflectivity vs $Q$ calculation for the Si/Mo bilayer at incident x-ray energy of 17 keV. (C) The simulated fluorescence yield data (symbols) and the model-independent fits (solid lines, based on electric field intensities in vacuum). The dashed black line shown for the case of the Gaussian located $50\text{Å}$ below the surface is the model-independent fit based on the use of electric field intensities inside the Si layer and including x-ray absorption in the medium as described by Eq. (12). The data are offset vertically for clarity. [(D) and (E)] The model-independently derived amplitudes and phases (symbols) as well as the $A$ and $P$ expected based on the assumed models (lines). (F) The reconstructed density profiles from Fourier-inversion of the derived $A$ and $P$.

Figure 3

(Color online) Illustration of the relative benefits of the model-independent analysis using fixed values of $A\left(Q^0\right)$ and $P\left(Q^0\right)$ within segment $\Delta Q$ at $Q^0$, versus the use of linear gradient terms to describe the variation in $A$ and $P$ within each segment. Data are simulated for a Si/Mo bilayer on Si substrate for a two-layer distribution of the element of interest consisting of Gaussians centered at and $50\text{Å}$ above the surface (shown in inset in 3B), each with $\sigma =5\text{Å}$. (A) The simulated fluorescence yield data (blue circles), and two model-independent calculations: without (dashed red line) and with (solid green line) the gradient terms of $A$ and $P$ included. In the first case, the yield is calculated for the range shown $\left(0<Q<0.08{\text{Å}}^{-1}\right)$ using $A=A\left(Q^0\right)$ and $P=P\left(Q^0\right)$. In the second case, additional slope parameters $dA/dQ$ and $dP/dQ$ are included based on the gradients of $A$ and $P$ obtained from the actual elemental structure factor at $Q^0$. [(B) and (C)] The variation in $A$ and $P$, respectively, for the actual elemental distribution (blue circles), along with the model-independent calculation with fixed values defined at $Q^0=0.04{\text{Å}}^{-1}$ (red dashed lines), and when the gradient terms are included (green lines).

Figure 4

(Color online) Model-independent analysis of measured ex situ and in situ reflectivity and Ti yield for the ${\text{TiO}}_2\text{-Si-Mo-Si}$ substrate system. (A) The reflectivity data (dots) in absolute units measured in air (magenta) and in contact with aqueous solution (blue). The model fits are shown as lines. Error bars are not shown for clarity. Typical statistical uncertainty in the reflectivity data was $\sim 0.5\mathrm{\%}$ while a minimum uncertainty of 2% based on systematic errors was assigned to the data before analysis. The in situ reflectivity data (blue symbols) is multiplied by 0.01. (B) A highlight of the reflectivity data in the $Q<0.12{\text{Å}}^{-1}$ range corresponding to the full Ti-fluorescence yield data (circles, panel C). The parameters derived from the reflectivity analysis were based on the fit of the full-range (shown in A). (C) The Ti-yield data (circles) measured in air (magenta) and in contact with an aqueous solution (blue). The vertical dashed lines show the segments into which the data were divided for the model-independent analysis (a constant segment width of $\Delta Q=0.01{\text{Å}}^{-1}$ was used). Model-independent fits (dashed lines) and model-based fits (solid lines) are shown. The blue arrow shows the range of in situ data included in the fit $\left(Q>0.05{\text{Å}}^{-1}\right)$. Inset in A shows the schematic of the trilayer sample in air.

Figure 5

(Color online) The density profiles for the ${\text{TiO}}_2\text{-Si-Mo-Si}$ substrate system based on analysis of reflectivity measurements (black lines) for (A) the ex situ analysis and (B) the in situ analysis. The horizontal black dotted lines show the expected bulk electron density of each material. These interfacial profiles include the interfacial roughness obtained by the reflectivity analysis (see Table ). The ${\text{TiO}}_2$ profiles based on the analysis of Ti-yield data are also shown for model-independent (dashed) and model-based (solid) cases, these profiles are scaled to absolute units based on the Ti coverage measured from XRF.

Figure 6

(Color online) (A) Amplitudes and (B) phases (symbols) extracted from the model-independent analysis for the sample measured in air (magenta) and in contact with an aqueous solution (blue). The $A$ and $P$ variation based on the model analysis are shown in lines. Note the difference in the extrapolated value of $A_{Q=0}$ for the ex situ and in situ results. Also shown are the $A$ vs $Q$ variation (black line) for the ex situ case if the distribution were fully coherent (i.e., with $A_{Q=0}=1$).