2. Experimental techniques
2.6 X-ray standing waves
The X-ray standing wave (XSW) technique is a quantitative structural method, capable of resolving the positions of atoms and molecules within adsorbate systems with high precision (i.e. < 0.05 Å).21, 91, 96-99 XSW takes advantage of the wavefield formed at a Bragg reflection in a crystal by monitoring the X-ray absorbance of atoms when scanning through the Bragg condition, effectively using the crystal lattice as a ruler to determine the atomic positions. Using the core level photoemission signal (see section 2.4) as a measure of the X-ray absorbance, XSW can provide surface sensitive structural information that is also specific to the different elements and chemically distinct atoms within the adsorbates.21, 91 Unlike other structural techniques that rely on diffraction (such as quantitative LEED and surface X-ray diffraction), XSW is a local probe technique that can characterise adsorbate systems that do not occupy well-defined adsorption sites or adopt long range ordered structures.21, 91 Consequently, XSW is apt for studying organic adsorbates (such as those studied in this thesis), which commonly form incommensurate overlayers on surfaces. In particular, XSW is well-suited to determining the height of adsorbate atoms above single-crystal surfaces - a property which gives important insight into the nature of the surface-adsorbate interaction. One advantage of XSW is that it provides this information without the need for the computationally expensive simulations and exhaustive fitting routines needed by other structural techniques21, 98 such as photoelectron diffraction,22 quantitative LEED19 and surface X-ray diffraction.37 Moreover, as standard DFT codes cannot provide reliable surface structures of organic adsorbate systems due to them not providing a description of long range dispersion interactions,27 experimental quantitative structural measurement techniques, such as XSW, are therefore useful to
34 The theory and applications of XSW are the subject of numerous review papers21, 98, 99 and book chapters;91, 96, 97 in this section only the fundamental principles will be discussed.
When an X-ray Bragg reflection is established in a crystal, two travelling waves, an incident wave and a reflected wave, combine to form an XSW. The intensity of this standing wavefield has periodicity equal to that of the crystallographic planes and thus has a well-defined phase relationship with the atoms in the crystal. By using a result derived from dynamical diffraction (multiple scattering) theory100, 101 this effect can be exploited. Typically, when considering Bragg reflections, kinematical (single scattering) theory is used. This considers the incoming X-ray to be a plane wave and sums the scattering events over all atoms at equivalent sites in the crystal, which at the Bragg condition102 is given by:
where λ is the X-ray wavelength, d is the scatterer plane spacing and θ is the scattering angle. The result of this is that kinematical theory predicts a delta function, where at the Bragg condition, an infinite scattered intensity is obtained, which drops to zero immediately off the Bragg condition. While kinematic theory is able to successfully show the positions at which interference maxima occur, it is unable to give the intensities or explain the shape of the observed maxima.103 Describing these effects requires treatment of multiple scattering, which was first demonstrated by Darwin using simple, but valid, geometrical arguments103 and later by dynamical scattering theory developed by Ewald104 and von Laue,105 which is able to reproduce Darwin’s results103 but also provides descriptions of the distribution of X-ray energy inside the crystal and the effect of absorption in a straightforward manner.101 The result of dynamical scattering theory that is of particular significance to the X-ray standing wave technique is that total reflection is achieved over a finite range of the incident conditions, rather than the delta function predicted by kinematical theory. This effect originates from the length of the wavevector changing over the course of scattering due to the difference in refractive index between the vacuum and the crystal, of which the latter is a medium with a complex dielectric constant.101, 106 This modifies the construction of the Ewald sphere and, in effect, creates a photonic band gap comprising a region of the incidence conditions where there are no states allowing the wave to propagate and thus corresponds to a region of total reflection. 101, 106 In the case of a non-absorbing crystal, this gives the characteristic top hat shaped Darwin reflectivity curve, with the reflectivity tailing-off rapidly either side of this region.101, 103, 106 Another key feature is that scanning across this region of total reflectivity shifts the phase the standing wave by π.21, 98, 100 Here, scanning across the total reflectivity
region is realised by changing the photon energy (and therefore the wavelength) or the incidence angle of the X-ray beam. For low-index reflections of simple, homonuclear, lattices (e.g. fcc and hcp lattices with one atom per unit cell), the low energy (or low angle) edge of the Darwin curve forms an XSW where the nodes coincide with the bulk atomic planes.21 With a phase shift of π at the high energy (or high angle) edge of the Darwin curve, the nodes move to the interstitial sites with the anti-nodes falling on atomic planes.100 This effect is illustrated in Figure 2.6.1a.
If the effect of the standing wave on atomic X-ray absorption is considered when traversing the Darwin curve, the atomic positions relative to the bulk scattering planes can be extracted. When scanning across the reflectivity curve, the atomic positions go in and out of phase with the standing wavefield at different points, depending on the distance of the atoms (z) from the bulk scattering planes. Consequently, if one records the X-ray absorption as a function of energy (or angle) across the region of total reflection, the resulting profile is characteristic of z and thus gives information on the atomic positions (see Figure 2.6.1b). Since the XSW extends out beyond the surface, this effect can be exploited to probe atoms adsorbed on the surface as well as those within the crystal.21, 91, 98
In an XSW experiment, the X-ray absorption is measured indirectly by recording the photoemission, Auger electron emission or X-ray fluorescence signal.21, 91 Since the energies of these emission processes are dependent on core level binding energies, this allows absorption profiles for elements to be measured separately. Detecting X-ray fluorescence has the advantage that measurements do not need to be made under UHV conditions, whereas the two electron detection methods offer an intrinsically greater degree of surface specificity due to the short IMFP of low-energy electrons in condensed matter.78 However, only core-level photoemission is able to resolve different chemical shifts of the same element, and thus measuring the X-ray absorption in this way can provide chemical-state specific structural information.21, 91, 98 One drawback of this is that in order to obtain the X-ray absorbance from core-level photoemission, the angular resolved photoemission yield must be corrected for non-dipolar effects.107
In order to scan through the Bragg scattering condition, one must either change the X-ray wavelength (requiring a tuneable X-ray source) or the angle of incidence with respect to the scattering planes. Indeed, in early experiments which relied on fixed energy laboratory X-ray sources, the Bragg condition was traversed by ‘rocking’ the crystal substrate to change the angle of incidence.108, 109 However, this approach has some significant disadvantages, the most prevalent being that at some general angle of incidence, the range of angles over which total reflectivity is achieved (the so-called ‘rocking curve’) is very narrow, typically spanning only a minute of arc or less.21, 91 This places very stringent demands on the instrumentation, requiring a very precise sample goniometer and an X-ray beam with an angular
Figure 2.6.1 – An illustration of the fundamental principles of the XSW technique. a) A depiction of the phase relationship between the X-ray standing wave and the scattering crystal at the two extremes of the reflectivity curve. The red lines represent the antinodes of the standing wave and the dotted black lines denote the crystal scattering planes shown both within and projected beyond the crystal surface. Above each panel, a representation of the Darwin reflectivity curve is shown with the red circle indicating the point on the curve that is represented in the diagram below it. The left-hand panel depicts the onset of the reflectivity curve with the XSW the nodes coinciding with the atomic planes. The right-hand panel depicts the other side of the reflectivity curve in which the antinodes fall on the atomic planes. b) An illustration of how the X-ray absorption across the reflectivity curve is characteristic of the position relative to the scatterer planes. The X-ray absorption as a function of photon energy across the reflectivity curve is shown for four adsorbate atoms at different positions relative to the scattering planes.
spread that is much smaller than the width of the rocking curve in order for the standing wave effect to be observed. Furthermore, the crystalline substrate must have a high degree of perfection with a
2 3
1
4
Single crystal substrate Adsorbed atoms Photon energy R ela tiv e X -r ay ab sor b an ce hv Anti-nodes fall on interstitial sites Anti-nodes fall on bulk lattice planes
hv
Tuning hv
Phase shift of π
a)
mosaicity smaller than the rocking curve width. This limits the variety of materials accessible to the XSW technique, with only a few covalently bonded semiconductor materials (such as Si) able to be produced with a sufficient degree of perfection.21, 91 It transpires, however, that this limitation can be overcome by conducting XSW experiments at normal incidence and by scanning through the Bragg scattering condition by tuning the X-ray wavelength, which is the basis of the NIXSW technique. These effects are best illustrated if one considers the underlying principles of XSW, which are presented below, closely following the discussion included in the cited XSW review papers and book chapters.21, 91, 97, 98
Taking the incident X-ray flux amplitude to be unity, the intensity of the standing wavefield at a perpendicular distance (z) from lattice planes spaced by dH is given by the modulus squared of the sum of the incident and reflected waves such that:
where EH and E0 are the amplitudes of the incident and reflected X-rays respectively. The X-ray reflectivity amplitude can be expressed in terms of the geometrical structure factors 𝐹𝐻 and 𝐹𝐻̅ for the respective reflections defined by H and –H as:
Here, η is a term that describes how much the scattering conditions differ from the midpoint of the Darwin reflectivity curve. If the Bragg scattering condition is traversed by varying the angle of incidence, the value η relates to the angular deviation (Δθ) from the Bragg angle (θB) such that:
where P is the polarisation factor which is equal to unity for σ-polarisation and cos(2θB) for π-polarisation, F0is the structure factor for forward (000) scattering and Γ is given by:
𝐼 = |1 + (𝐸𝐸𝐻 0) exp ( −2𝜋𝑖𝑧 𝑑𝐻 )| 2 [2.6.2] 𝐸𝐻 𝐸0 = − ( 𝐹𝐻 𝐹𝐻̅) 1 2 [𝜂 ± (𝜂2− 1)12] [2.6.3] 𝜂 =−∆𝜃 sin 2𝜃𝐵+ 𝛤𝐹0 |𝑃|𝛤(𝐹𝐻𝐹𝐻̅) 1 2 [2.6.4]
where e and m are the electronic charge and mass respectively, V is the volume of the unit cell, c is the speed of light in vacuum and ε0 is the permittivity of free space. Equation 2.6.3 shows that for a non-absorbing crystal, in which the structure factors are all real, unity reflectivity is achieved over a finite region of Δθ, which corresponds to values of η in the range of -1 to +1. This defines the width of the Darwin curve and can be written as:
Equation 2.6.3 also highlights that the centre of the reflectivity curve does not correspond to the Bragg condition described by kinematical scattering but instead is offset by an amount defined by the forward scattering structure factor F0:
Equation 2.6.6 also has important implications that can be exploited to circumvent the demanding requirements of crystal perfection and instrumentation needed for XSW experiments. As previously discussed, for an arbitrary angle of incidence, the width of the Darwin curve is very narrow. However, if the experiment is conducted with the incident beam at, or near, normal incidence to the scattering planes, one reaches a turning point in the Bragg equation, where the rate of change to the scattering condition as a function of θ becomes zero and thus the width of the reflectivity curve increases significantly.109, 110 In fact, equation 2.6.6 suggests for a Bragg angle 𝜃
𝐵= 9 ° (and thus sin 2𝜃𝐵= )
that the width of the rocking curve tends to infinity, though, in reality, the approximations that underpin this equation are not valid under this condition. Nonetheless, at normal incidence the width of the rocking curve becomes comparatively large (of the order of one degree) and thus the NIXSW technique is accessible to less perfect crystal substrates such as metals and metal oxides.
𝛤 = 𝑒 2𝜆2 4𝜋2𝜀 0𝑚𝑐2𝑉 [2.6.5] 𝛥𝜃𝑟𝑎𝑛𝑔𝑒 = 2 (|𝑃|𝛤(𝐹𝐻𝐹𝐻̅) 1 2) sin 2𝜃𝐵 [2.6.6] 𝛥𝜃𝑜𝑓𝑓𝑠𝑒𝑡 = 𝛤𝐹0 sin 2𝜃𝐵 [2.6.7]
In order to operate at normal incidence, the X-ray wavelength must be tuned and therefore NIXSW experiments are reliant on synchrotron radiation. One advantage of using a tuneable photon source is that one can relinquish the need for a precise sample goniometer by keeping the sample fixed and scanning through the Bragg condition by changing the photon energy (and thus the X-ray wavelength) instead of the angle of incidence. The variation of the X-ray standing wavefield when using this approach can be described by modifying equation 2.6.4 to give η as a function of energy deviation (ΔE):
for which the energy range of the Darwin reflectivity curve is:
In order to extract structural information from the XSW technique, one must consider how the X-ray absorbance of atoms changes, depending on their position, when scanning through the Bragg condition. In an XSW experiment, the relative X-ray absorbance of an atom is directly related to the local intensity of the standing wavefield, which at a perpendicular distance z relative to the Bragg planes is given by:
where R is the reflectivity, ϕ is the phase angle of the standing wave, which relates to the displacement from the nominal Bragg condition and dh is the Bragg plane spacing. In theory, one could find the spacing of the adsorbate atoms relative to the lattice planes by finding the value of z that gives the best fit to the experimental data. However, in practice the adsorbate atoms are unlikely to adopt a single discrete spacing throughout the structure, even in the most ideal cases, due to thermal vibrations. Furthermore, in more complex systems the adsorbate atoms may occupy two or more sites with distinctly different spacings from the scattering planes. To account for this, a distribution of z values must be considered. This can be implemented into equation 2.6.10 to give the expected X-ray absorbance by integrating over all absorbers within the distribution, which gives:
𝜂 =−2(∆𝐸/𝐸) sin2𝜃𝐵+ 𝛤𝐹0 |𝑃|𝛤(𝐹𝐻𝐹𝐻̅) 1 2 [2.6.8] 𝛥𝐸𝑟𝑎𝑛𝑔𝑒 = 𝐸|𝑃|𝛤(𝐹𝐻𝐹𝐻̅) 1 2 sin2𝜃 𝐵 [2.6.9] 𝐼 = 1 + 𝑅 + 2√𝑅 cos (𝜙 − (2𝜋𝑧𝑑 𝐻)) [2.6.10]
where f(z)dz gives the fraction of absorbers at a spacing of z within a range of dz, defined such that
∫ 𝑓(𝑧)𝑑𝑧 = 1.𝑑𝐻
0 Alternatively, this can be written as:
This now gives two structural fitting parameters, which are the coherent fraction (fco) and the coherent position (D). In effect, the coherent position is the average distance of the atoms above the nearest scattering plane and the coherent fraction is commonly regarded as a measure of the disorder, although, more exactly, it is a measure of the distribution of atomic positions. In the ideal case of a single adsorption site with no static or dynamic disorder, the coherent fraction is unity and the coherent position is the atomic position relative to the Bragg planes. In practice, thermal vibrations inevitably reduce the coherent fraction to a value less than unity, such that values of 0.8 (or more) still correspond to the occupation of a single site. Lower values commonly arise due to multiple site occupation with interpretation of the coherent position and fraction then requiring model calculations of different possible distributions of the occupied sites. An extreme example of this is that if two equally-occupied sites have heights that differ by exactly half of a lattice spacing the coherent fraction is zero, highlighting that a low coherent fraction does not necessarily imply a high degree of disorder. When interpreting XSW results, it is useful to consider the following:
where the coherent fraction and position describe the magnitude and phase of the distribution of absorber position perpendicular to the scattering planes respectively. One convenient way to visualise this is by constructing an Argand diagram. Figure 2.6.2 depicts an Argand diagram for a system in which the absorbing atoms occupy two distinctly different sites. The two red vectors are the components that correspond to each atomic site which each have a phase angle (2πz/dH) and an amplitude f(z). Here
𝐼 = 1 + 𝑅 + 2√𝑅 ∫ 𝑓(𝑧)𝑑𝐻 0 cos (𝜙 − ( 2𝜋𝑧 𝑑𝐻)) 𝑑𝑧 [2.6.11] 𝐼 = 1 + 𝑅 + 2𝑓𝑐𝑜√𝑅 cos (𝜙 − ( 2𝜋𝐷 𝑑𝐻 )) [2.6.12] 𝑓𝑐𝑜exp (2𝜋𝑖𝐷 𝑑𝐻 ) = ∫ 𝑓(𝑧) 𝑑𝐻 0 𝑒𝑥𝑝 (2𝜋𝑖𝑧 𝑑𝐻 ) 𝑑𝑧 [2.6.13]
the amplitude will relate not only to the static and dynamic disorder of each site but will also represent the relative occupancy of each site. Summing these two component vectors yields a vector (blue arrow) of magnitude fco with a phase angle (2πz/dH) corresponding to the overall coherent position.
Figure 2.6.2 – Argand diagram illustrating the contributions of two distinctly different sites to the overall coherent fraction and coherent position. The red vectors of magnitude fn correspond to the two absorber sites. Summing the two red vectors gives the blue vector, for which the magnitude corresponds to the overall coherent fraction (fco) and the phase angle relates to the overall position (D).
As discussed previously, the X-ray absorbance cannot be obtained directly for the different atomic species within the structure and instead it is typically measured indirectly in an element and chemical- state specific fashion by recording the photoemission yield. One complication of this is that the photoemission yield only reflects the intensity of the XSW if the photoemission cross section in the direction of the detector is equal for both the incident and reflected X-ray beams.21, 111 In general, this is only true in situations where the dipole-approximation can be used to quantitatively describe the photoemission process. The dipole-approximation is only valid if the spatial variation of the