The distribution of the contrast of X-ray standing waves fields in different media

Abstract

X-ray standing waves (XSW) fields generated by total external reflection at different types of interfaces have been analyzed in order to determine the vertical distribution of interference contrast. The intensity contrast between the nodes and antinodes of the XSW field is of high relevance for the interpretation of fluorescence curves. Silicon wafers covered by nanoparticles, polymer layers and a liquid film containing ions were irradiated by XSW fields of 10 to 15.5 keV. From the deviation of the experimental data from the signals simulated assuming a field of constant contrast, the relation between the node/antinode contrast and the distance from the reflecting substrate was determined. For the interface solid/air, a decrease of contrast was measured; for solid/medium interfaces, a complete fading of the interference within the measurement range was observed.

Introduction

With the increasing technological and scientific interest in structures featuring dimensions in the nanometer range, also the importance of adequate characterization methods has grown. A technique, which has proven very useful for the analysis of various types of nanostructures at surfaces and interfaces, is the X-ray standing waves (XSW) method. A beam of coherent and monochromatic X-rays, hitting a flat surface under grazing incidence, is totally reflected. In the overlap of incoming and reflected beam, an interference pattern of alternating minima (nodes) and maxima (antinodes), the X-ray standing waves field, is generated. Atoms or ions within the interference volume are excited to fluorescence depending on the local intensity I(α,z) of the XSW field. The position of the fluorescence maxima depends on the incident angle α, so by recording fluorescence spectra with a stepwise variation of α between the single measurements, the distribution of elements above the reflecting surface is scanned vertically. The XSW method has successfully been applied to the investigation of particles, layered structures, electric double layers, adsorbates and biofilms [1], [2], [3], [4], [5], [6], [7]. The advantages of the method are a high vertical resolution of few nanometers, the possibility of energy dispersive measurements including a wide range of elements and the independence of specific conditions of the sample environment (e.g. vacuum, temperature).

However, a limitation of the method which has to be taken into account is the dependence of the intensity contrast between nodes and antinodes of the XSW field from the vertical position. This is of particular relevance in the case of sample structures extending over several periods of the XSW field and will be discussed in this work.

The intensity of an ideal XSW field is given by

$$I_{\text{XSW}}(\alpha ,z)=I_0[1+R(\alpha )+2\sqrt{R}cos(2\pi z/a-\varphi \left(\alpha \right)\left)\right]$$

with the intensity of the incoming beam I₀, the reflectivity R(α) and an interference term including the phase shift ϕ(α) between the two beams [8]. The period a of the standing waves is determined by the wavelength λ and the incident angle: a = λ/(2 sinα) The variable z indicates the height above the reflector, for which the intensity is calculated. In the ideal case, the intensity of the XSW field oscillates between I_XSW(α,z) = 4I₀ in the antinodes and I_XSW(α,z) = 0 in the nodes. However, this contrast is not achieved under experimental conditions, as discussed in the following. The XSW field overlaps with the element distribution C(z) above the reflector, so the fluorescence photons are collected from the entire irradiated sample volume. Consequently, for an incident angle α the fluorescence intensity is given by

$$I_z\left(\alpha \right)=\underset{0}{\overset{z_{max}}{\int }}{I}_{\text{XSW}}\left(\alpha ,z\right)\cdot C\left(z\right)d{z}^{\prime }\text{.}$$

The lower limit of this integration is the reflecting surface at z = 0, whereas the upper limit z_max is not defined by the geometrical parameters (beam height and incident angle) of the field. Here, as shown by von Bohlen et al. [9], the limiting factor is the finite coherence of the radiation. A high coherence of the incoming radiation is necessary for a high contrast of the XSW field, as only photons with a defined phase relationship can generate an interference field. The more this phase relationship gets lost, the weaker is the contrast of the interference field. For the evaluation of an XSW-excited fluorescence intensity signal, I(α) is calculated for a first probable element distribution profile C(z) and the resulting curve is fitted to the measured data. Then, the distribution model is modified until optimal agreement between measurement and simulation is achieved.

For the generation of XSW fields, coherent and monochromatic radiation is needed. These features are typical for free electron lasers,[10] whereas the light emitted by wigglers or undulators needs further conditioning [11], [12]. For this purpose, crystals and multilayer mirrors are applied to filter out radiation of a selected wavelength λ₀ from the polychromatic (“white”) synchrotron light. However, the quality of a monochromator is limited; even single crystals emit a small bandwidth λ₀ ± Δλ of wavelengths, whereas Δλ is the width of the distribution curve. Considering only two parallel waves trains, the wavelength difference Δλ defines the longitudinal coherence length ξ_l = λ₀²/(2Δλ). The fact that ξ_l is finite also limits the height of the XSW field. Besides the not perfect monochromaticity of the beam, also the scattering of photons at the sample material confines the extent of the interference field. Each scattering process reduces the fraction of the coherent photons of the beam, so after a certain beam path length in a scattering medium all coherence will be lost.

The phenomena described above do not only limit the height of the interference region, but also influence the contrast between nodes and antinodes. The less coherent photons are available to generate an interference pattern; the weaker is the contrast of the XSW field. Instead of alternating maxima and minima, only the not modulated overlap of incoming and reflected radiation causes the excitation of fluorescence. The neglect of this effect results in the distortion of the concentration profile deduced from the respective fluorescence scan. Thus, for a correct interpretation of the measured fluorescence intensity curves, a factor F(z) describing the contrast of the XSW field has to be introduced:

$$I_{\text{XSW}}(\alpha ,z)=I_0[1+R(\alpha )+2\sqrt{R}cos(2\pi z/a-\varphi \left(\alpha \right))\cdot F(z\left)\right]$$

F(z) takes values between 1, meaning an undisturbed contrast and 0, which represents the case of a complete fading of the contrast, where the field is given by the sum of I₀ and R(α).

By the experiments described in the following, the distribution of contrast F(z) should be determined for different types of interfaces. For this purpose, the simulation procedure described above is inverted: the element distribution C(z) is known (spheres, layers etc.) and the factor F(z) of the equation describing the intensity of the XSW field is varied until the simulation agrees with the measurement.

First, the XSW field is produced on a silicon surface covered by gold nanoparticles, as principally shown in Fig. 1. This way, the fading of a standing waves field in air or vacuum should be studied. In the second case, the reflection takes place at a silicon/polymer interface to investigate the properties of an XSW field in a medium. The last example features a solid/liquid interface, where the influence of a liquid film of ca. 1 μm thickness is studied. A part of the measurements presented in this article have also been mentioned in the work of von Bohlen et al. [13], where a different approach to the problem is discussed. A similar effect is discussed in the work of Kirchner et al. [14], who investigated the effects of surface roughness on the XSW field applying different models for the inhomogeneity of the reflector.