Surface structure determination by x-ray standing waves at a free-electron laser

The XSW experiment was performed at the free-electron laser FLASH at DESY in Hamburg [2, 38] at the PG2 beamline (see figure 2a), the only high repetition rate XFEL in operation at the time of the experiment. FLASH was operated in the multibunch mode, delivering pulse trains with a repetition rate of 10 Hz. Each bunch train consisted of 400 pulses with photon energy of 93 eV or 91.7 eV(see table A1 in appendix A) and an intrabunchtrain repetition rate of 1 MHz for FLASH1 operation only. When FLASH1 and FLASH2 were operated in parallel [39] each bunch train consisted of 320 pulses.

When measuring photoelectron spectra, space-charge effects need to be taken into account. In fact, if too many photoelectrons are emitted in a small area and in a short femtosecond time, the resulting photoemission spectrum will be energy shifted and broadened due to Coulomb repulsion [36, 37]. To avoid this, the intensity of the FEL pulses was reduced by a gas attenuator filled with 2.7 × 10⁻² mbar Xe gas as well as by solid filters (see appendix A). The monochromator was set to the first order diffraction from the 200 lines/mm plane grating with a fix-focus constant (c_ff) of 1.25 and the exit slit was set to 100 μm to have 40 meV energy bandwidth. As a result, the number of photons per pulse at the sample was on average 5 × 10⁷ with a beam size (≈30 cm behind the XFEL focus) of 150–200 μm FWHM, therefore the corresponding fluence was <1 μJ cm⁻². This fluence was about 5 orders of magnitude smaller than the single shot damage threshold of Mo/Si ML at normal incidence (83 mJ cm⁻²) [40]. Therefore radiation induced sample damage could be excluded. Moreover, initial XFEL pulses ≈60 fs FWHM long were elongated to ≈200 fs FWHM due to the pulse front tilt caused by the monochromator grating.

The measurements were carried out in the experimental chamber WESPE (Wide-angle Electron SPEctrometer) equipped with a vertical manipulator to tune the angle of incidence θ, and the electron time-of-flight spectrometer THEMIS 1000 (SPECS), provided with a four-quadrant delay line detector (Surface Concepts), to measure photoelectron spectra. The spectrometer was set to measure electrons of kinetic energy 63 eV, pass energy 40 eV and with an acceptance angle of ±6°. Given these settings, and the FEL attenuation needed to avoid space charge effects, on average 0.5–1.0 electron per FEL pulse was detected.

Since FLASH operates in the soft x-ray range (λ = 4.2–52 nm), the periodic structure generating the standing wave had to have a period of comparable size. Our samples were Mo/Si ML, consisting of 50 Mo/Si bilayers deposited on a super-polished Si substrate by means of sequential magnetron sputtering of Mo and Si in Ar atmosphere [41]. The Si substrates were placed on a rotating substrate holder above the magnetron, such that all installed substrates could be coated at the same time and all coated layers were identical. The thickness of each layer was controlled by pre-calibrated sputtering time leading to a nominal ML period of d_ML = 7.3 nm (figure 1(c)). To match the first order Bragg condition $2{d}_{{\rm{ML}}}\sin \left(\theta \right)=\lambda$, FLASH was tuned to the wavelength λ = 13.3 nm or λ = 13.5 nm, and the angle of incidence of the maximum reflectivity was θ_max = 72.5° or θ_max = 71.8° (figure 5). In order to demonstrate the structural sensitivity of the XSW technique using an XFEL, we employed 4 ML samples terminated with the top Si layer of different nominal thickness ${d}_{\mathrm{Si}}^{\mathrm{top}}$. After the deposition of the last Mo layer a system of masks was used to enable coating of the top Si layer with different thicknesses ${d}_{\mathrm{Si}}^{\mathrm{top}}$. As a result we obtained four identical periodic Mo/Si MLs terminated with nominal top Si layers of thickness 2.0 nm, 2.8 nm, 3.6 nm, and 4.3 nm, referred to as sample 1, 2, 3, and 4, respectively.

As the Si-terminated ML samples were exposed to air, a native SiO₂ layer of ${d}_{{\mathrm{SiO}}_{2}}=1.2\,\mathrm{nm}$ is formed at the surface (see appendix C). This led to 4 different distances of the surface oxide from the underlying identical periodic structure. In our XSW experiments we measured the photoelectron yield of O2s core level originating from the O atoms located at the surface of the SiO₂ layer (see figure 1(c)) as a function of the incident angle θ. In this way we probed the position of the surface relative to the standing wave modulation and demonstrated the structural sensitivity of the XSW technique at an XFEL source. Based on this, it will be possible to measure changes in the electronic structure of atoms with picometer spatial accuracy at femtosecond time resolution.

A typical photoelectron spectrum measured on one of our ML samples is shown in figure 2(b). The most intense peak at about 6 eV below the Fermi level consists mainly of O2p photoelectrons plus the underlying Si valence band [42]. Our attention focuses on the O2s photoelectron peak at about 25 eV binding energy (BE). After subtraction of a Shirley background [43], the integral O2s peak area is defined as the photoelectron yield ${Y}_{\exp }\left(\theta \right)$ of the oxygen atoms in the SiO₂ layer at the sample surface, measured at a given angle θ. Each ${Y}_{\exp }\left(\theta \right)$ needs to undergo several normalization steps described in detail in appendix A. Importantly, the spectrum shown in figure 2(b) was measured in 20 min. To obtain a spectrum of similar statistics at any other XFEL, delivering hard x-ray single pulses at a maximum repetition rate of 120 Hz (for example, at the present LCLS), 9 hours of acquisition time would be needed. This makes time-resolved photoelectron spectroscopy measurements in the (sub)-ps time scale, without space charge effects, and with high statistics feasible only at high repetition rate XFELs, such as FLASH [2], the European XFEL [6] and LCLS-II [44].

The structural information of XSW measurements is contained in the photoelectron yield profile, i.e. the sequence of ${Y}_{\exp }\left(\theta \right)$ measured as the incidence angle θ is scanned through the Bragg condition. The normalized photoelectron yield profiles (appendix A) corresponding to ML samples with four nominally different top Si layers ${d}_{\mathrm{Si}}^{\mathrm{top}}$ (2.0, 2.8, 3.6, and 4.3 nm) are displayed in figure 3. The variations in yield follow from the XSW intensity variations at the top SiO₂ surface of each sample (see equation (1)). Notably, the photoelectron yield profiles in figure 3 are very different from each other and are strongly correlated with the thickness of the top Si layer. This indicates significantly different positions of the corresponding emitting oxygen atoms with respect to the standing wave modulation.

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Importantly, the XSW effect can be exploited, by simply rotating the sample and thereby tuning the angle of incidence, to change the x-ray intensity within and above the sample, in this case by a factor of 3 (figure 3), without changing any of the beamline parameters. This feature can be very useful for a fast and reproducible fine tuning of the XFEL intensity at specific sample positions.

In order to extract the exact position of the O atoms contributing to the O2s photoelectron spectra we fitted the yield profiles with the model introduced below. First, we need to determine the relation between the intensity of the XSW and the measured photoelectron yield. In general, the photoelectron yield $Y\left(\theta \right)$ of an atom at a given position z is not simply proportional to the XSW intensity ${I}_{{\rm{SW}}}\left(z\right)$ as expressed in equation (1). In fact, for angularly resolved photoelectron spectroscopy in π-polarization the photoelectric cross-section in presence of an XSW depends on the experimental geometry. Particularly important are the direction and polarization of the x-ray waves and the direction of the emitted photoelectrons [45]. For our case of π-polarization the incident and Bragg-diffracted polarization vectors e₀ and e_H lie within the scattering plane, defined by the incident and Bragg-diffracted propagation vectors k₀ and k_H, as shown in figure 2(c). Since soft x-rays are employed, in the calculation of the photoelectric cross-section higher order multipole terms can be neglected [46]. Therefore, in the dipole approximation, for an initial s-state and π-polarization geometry the angularly resolved photoelectron yield can be expressed as

$$\begin{eqnarray}&&Y\left(\theta \right)=1+{g}^{2}R\left(\theta \right)+2g\sqrt{R\left(\theta \right)}{F}_{{\rm{c}}}\cos \left(\phi \left(\theta \right)-2\pi {P}_{{\rm{c}}}\right),\end{eqnarray} \tag{ 2 }$$

where $g=\cos {\theta }_{{\rm{H}}}/\cos {\theta }_{0}$ is the polarization factor [47], with θ₀ and θ_H the angles between the polarization directions e₀ and e_H and the direction of the emitted electrons n_p (see insets in figure 2(c)). In equation (2) the coherent position P_c is defined as ${P}_{{\rm{c}}}=\langle z\rangle /{d}_{{\rm{SW}}}$, with $\langle z\rangle$ being the average position of the emitting atoms contributing to the photoelectron yield, and the coherent fraction F_c indicates the distribution width of the emitters around their average position $\langle z\rangle$. Equation (2) is accurate if the distribution of atoms contributing to $Y\left(\theta \right)$ does not extend for more than one standing wave period d_SW and is located at the sample surface, because in that case the damping of photoelectrons due to the inelastic mean free path can be neglected.

In our case, the measured photoelectron yield results from oxygen atoms in the top SiO₂ layer extending for ${d}_{{\mathrm{SiO}}_{2}}=1.2\,\mathrm{nm}$ (appendix C) below the surface z_surf, with z = 0 defined at the top of the Mo layer (see figures 1(c) and 6). Because of the inelastic mean free path, photoelectrons emitted from atoms below the surface at z < z_surf and at a given angle α with the surface will contribute less to $Y\left(\theta \right)$ by a factor ${{\rm{e}}}^{-\left({z}_{\mathrm{surf}}-z\right)/\left({\lambda }_{{\rm{I}}}\cdot \sin \alpha \right)}$ [48], where λ_I is the electron inelastic mean free path (appendix A) and α is the angle between the electron detection direction ${{\bf{n}}}_{{\rm{p}}}$ and the sample surface (figure 2(c)). As a result, the fit model for ${Y}_{\exp }\left(\theta \right)$ can be expressed as:

$$\begin{eqnarray}&&{Y}_{\mathrm{model}}\left(\theta \right)=\ 1+{g}_{1}R\left(\theta \right)+2{g}_{2}\sqrt{R\left(\theta \right)}\,\displaystyle \frac{{\int }_{{z}_{{\rm{surf}}}-{d}_{{{\rm{SiO}}}_{2}}}^{{z}_{{\rm{surf}}}}{{\rm{e}}}^{-\left({z}_{{\rm{surf}}}-z\right)/\left({\lambda }_{{\rm{I}}}\cdot {\rm{\sin }}\alpha \right)}\cos \left(\phi \left(\theta \right)-2\pi z/{d}_{{\rm{SW}}}\right){\rm{d}}{z}}{{\int }_{{z}_{{\rm{surf}}}-{d}_{{{\rm{SiO}}}_{2}}}^{{z}_{{\rm{surf}}}}{{\rm{e}}}^{-\left({z}_{{\rm{surf}}}-z\right)/\left({\lambda }_{{\rm{I}}}\cdot {\rm{\sin }}\alpha \right)}{\rm{d}}{z}}.\end{eqnarray} \tag{ 3 }$$

Equation (3) represents the sum of photoelectron yield contributions between ${z}_{\mathrm{surf}}-{d}_{{\mathrm{SiO}}_{2}}$ and z_surf weighted by the inelastic mean free path factor. Since the polarization factor g in equation (2) depends on the direction of the emitted photoelectrons, the acceptance angle ±6° of the time-of-flight spectrometer needs to be taken into account. Therefore, the polarization factors g₁ and g₂ in equation (3) are defined as follows:

$$\begin{eqnarray}&&{g}_{1}\left(\theta \right)=\displaystyle \frac{1}{12{\cos }^{2}{\theta }_{0}}{\int }_{-6}^{+6}{\cos }^{2}\left({\theta }_{{\rm{H}}}+{\theta }_{{{\bf{n}}}_{{\rm{p}}}}\right){\rm{d}}{\theta }_{{{\bf{n}}}_{{\rm{p}}}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{g}_{2}\left(\theta \right)=\displaystyle \frac{1}{12\cos {\theta }_{0}}{\int }_{-6}^{+6}\cos \left({\theta }_{{\rm{H}}}+{\theta }_{{{\bf{n}}}_{{\rm{p}}}}\right){\rm{d}}{\theta }_{{{\bf{n}}}_{{\rm{p}}}},\end{eqnarray} \tag{ 5 }$$

where ${\theta }_{{{\bf{n}}}_{{\rm{p}}}}$ indicates the emission angle relative to θ_H. The polarization factors ${g}_{1}\left(\theta \right)$ and ${g}_{2}\left(\theta \right)$ depend on the grazing angle of incidence θ via the angle ${\theta }_{{\rm{H}}}=215^\circ -2\theta$ (figure 2(c)).

To apply equation (3) it is necessary to know the reflectivity $R\left(\theta \right)$, the phase $\phi \left(\theta \right)$ of the complex electric field amplitude ratio ${E}_{{\rm{H}}}/{E}_{0}$, and the period of the standing wave d_SW. These parameters could be easily calculated if the exact structure of our ML samples was known. To determine these parameters, grazing incidence x-ray reflectivity (GIXR, λ = 0.154 nm) and extreme ultraviolet reflectivity (EUVR, λ = 13.5 nm) measurements were performed using respectively a laboratory Cu K_α source (PANalytical Empyrean) at the University of Twente and the Metrology Light Source synchrotron radiation at the Physikalisch-Technische Bundesanstalt (PTB) in Berlin on the same samples probed with XSW at FLASH. GIXR and EUVR data are reported in figures 4 and 5 together with the corresponding fitting curves (obtained as described below) and phase calculations resulting from the structural model in figure 6. Note that EUVR data were planned to be recorded in situ at FLASH along with XSW measurements. However, a technical problem with our in-vacuum photodiode prevented us from doing so. Nevertheless, the consistency of our XSW, XPS, EUVR and GIXR data measured within three months from the samples production is guaranteed by the long time stability of ${\mathrm{SiO}}_{2}$ capped Mo/Si ML [49]. In general, a simultaneous measurement of reflectivity and XSW data should be obtained as this will extend this technique also to less stable ML and in situ grown superlattices, e.g. perovskites.

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The data analysis consisted of several steps and to facilitate their comprehension a visual representation is given by the flowchart in figure 7. First, using the assumption independent approach [50] GIXR measurements from sample 2 with ${d}_{\mathrm{Si}}^{\mathrm{top}}=2.8$ nm were analyzed. The Mo/Si bilayer in the repetitive part of the ML and the top Mo/Si bilayer were divided in 30 sub-layers of Mo${}_{1-x}$Si_x, where x was a fitting parameter. The best fit model was parameterized introducing Mo and Si layers, Mo${}_{1-x}$Si_x interlayers, and sinusoidal transition layers between them for the simultaneous fit of GIXR and EUVR data as described in [51]. Second, the data of samples 1, 3, and 4 were fitted using the same 49 Mo/Si bilayers derived from sample 2. The periodic ML structure was identical for all samples as a result of the coating procedure. The only fitting parameters were: the thickness of the top Si layer ${d}_{\mathrm{Si}}^{\mathrm{top}}$, the thickness of the top ${\mathrm{SiO}}_{2}$ layer ${d}_{{\mathrm{SiO}}_{2}}$, the thicknesses of the transition layers from Si to SiO₂ and from SiO₂ to vacuum, the densities of the top Si and SiO₂ layers, and the total period thickness d_ML. The simultaneous best fit of GIXR and EUVR data [51] provided a structural model for each of the four samples with different ${d}_{\mathrm{Si}}^{\mathrm{top}}$. The real part of the refractive index δ for the top Mo/Si bilayers and the SiO₂ above is displayed in figure 6. The resulting structural parameters of the identical ML periodic structure are d_Mo = 3.3 nm and d_Si = 4.0 nm, leading to a standing wave periodicity of d_SW = 7.3 nm.

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Fits of EUVR data of each sample are reported in figure 5 together with the calculation of the corresponding phase $\phi \left(\theta \right)$, based on Abeles matrix formalism [52]. The large and broad reflectivity peak with maximum of 61.4% at θ_max = 72.5° results from the Mo/Si ML, while the smaller side peaks, so-called Kiessig fringes, result from the interference of x-ray waves reflected at the vacuum-surface interface and ML-substrate interface. As the angle θ crosses the Bragg condition the phase $\phi \left(\theta \right)$ experiences a total variation of π, corresponding to a total shift of the XSW by d_SW/2, hence leading to the photoelectron yield modulations reported in figure 3. The phase term $\phi \left(\theta \right)$ was calculated at the top of the SiO₂ layer, therefore at different positions with respect to the periodic ML structure (figure 6). This results into rigid phase shifts going from sample 1 to 4 as it is evident from the corresponding scales in figure 5.

The good quality of EUVR and GIXR curve fits enables us to employ the corresponding $R\left(\theta \right)$ and $\phi \left(\theta \right)$ functions to fit the experimental yield data ${Y}_{\exp }\left(\theta \right)$ by means of the model in equation (3). The results summarized in figure 3 show that ${Y}_{\mathrm{model}}\left(\theta \right)$ describes very well our measured data. Two fit parameters were employed: the position of SiO₂ surface z_surf and the angular offset to account for the slightly different angular scales of reflectivity and photoelectron yield measurements. The surface of the SiO₂ layer z_surf in samples 1 to 4 was found to be respectively at 2.59 ± 0.12 nm, 3.58 ± 0.06 nm, 4.43 ± 0.04 nm, and 5.76 ± 0.10 nm above the top Mo layer, while the angular offset was of about 1.5°.

The increase of z_surf going from sample 1 to 4 follows directly from the larger ${d}_{\mathrm{Si}}^{\mathrm{top}}$ leading to an increasing distance of the sample surface from the periodic ML structure as illustrated in figure 6. In this way we demonstrate the structural sensitivity of the XSW technique using XFEL pulses. In particular, the small error bars (appendix B) of z_surf (< 0.15 nm) indicate the high spatial accuracy of the measured SiO₂ surface positions.

The intensity of a Self-Amplified Spontaneous Emission (SASE) XFEL varies from pulse to pulse with variations up to approximately 20%, therefore it is necessary to normalize each electron yield data point ${Y}_{\exp }\left(\theta \right)$ by the corresponding XFEL intensity. As a reference for the XFEL intensity we consider the ion signal of a gas monitor detector [38] located directly after the undulators of FLASH (figure 2(a)). The normalization factor was calculated as the sum of the ion signal over the entire acquisition run. In this way, not only we normalize by the incoming XFEL intensity but also by the acquisition time.

After the gas monitor detector and before the monochromator of PG2 beamline at FLASH there is a gas absorber and several solid filters that can be used to reduce the XFEL intensity. The pressure of Xe in the gas absorber was always kept constant to 2.7 × 10⁻² mbar, hence normalization by the corresponding attenuation factor is not necessary. In contrast, some of the solid filters were used and changed during the acquisition of electron yield data of the same yield profile as reported in table A1. Therefore, in order to have consistent data within the same yield profile each electron yield ${Y}_{\exp }\left(\theta \right)$ needs to be normalized by the corresponding filter transmission. We kept a Si₃N₄ filter 500 nm thick throughout all the measurements, while we alternated two ZrB₂ filters with thickness 431 nm and 200 nm. The last two filters were used either both in series or only one of the two as indicated in table A1, where also the corresponding photon energy is reported.

Since the electron yield ${Y}_{\exp }\left(\theta \right)$ is measured at different angles of incidence θ, the number of photoelectrons that can leave the surface and reach the detector varies depending on the effective number of atomic layers crossed by the photoelectrons. The factor denoting the damping of the emitted O2s photoelectrons due to the corresponding inelastic mean free path λ_I is ${\rm{I}}\left(z,\theta \right)={{\rm{e}}}^{-\left({z}_{\mathrm{surf}}-z\right)/\left({\lambda }_{{\rm{I}}}\cdot \sin \alpha \right)}$, where $\alpha \left(\theta \right)$ is the angle between the surface and the direction ${{\bf{n}}}_{{\rm{p}}}$ of photoemitted electrons towards the detector (figure 2(c)). Since the angle between k₀ and ${{\bf{n}}}_{{\rm{p}}}$ is 55°, it follows that α = 125° − θ. Following the notation used in the article, the coordinate z indicates positions perpendicular to the sample layers with z = 0 at the top of the last Mo layer, z > 0 above it (figures 1(c) and 6), and z_surf is defined as the position of the SiO₂ surface. The normalization factor accounting for λ_I is given by

$$\begin{eqnarray}&&{N}_{\lambda }\left(\theta \right)={\int }_{{z}_{\mathrm{surf}}-{{d}}_{{\mathrm{SiO}}_{2}}}^{{z}_{\mathrm{surf}}}{{\rm{e}}}^{-\left({z}_{\mathrm{surf}}-z\right)/\left[{\lambda }_{{\rm{I}}}\cdot \sin \left({125}^{^\circ }-\theta \right)\right]}{\rm{d}}{z},\end{eqnarray} \tag{ A.1 }$$

where ${d}_{{\mathrm{SiO}}_{2}}=1.2\,\mathrm{nm}$ is the average thickness of the SiO₂ layer, as measured by x-ray photoelectron spectroscopy (appendix C). The normalization factor ${N}_{\lambda }\left(\theta \right)$ was calculated for each of the angle of incidence θ at which experiments were performed. The inelastic mean free path λ_I = 4.4 Å results from the interpolation of tabulated values obtained from the TPP-2 formula [62] and it was calculated for O2s photoelectrons with kinetic energy 62 eV going through SiO₂.

We need to consider that for θ < 90° the footprint of the XFEL at the sample will increase in the horizontal direction by a factor $1/\sin \theta$, thus a larger number of photoelectrons will be detected. To account for this geometrical factor, ${Y}_{\exp }\left(\theta \right)$ is normalized by $1/\sin \theta$.

Finally, each yield profile is normalized by the photoelectron yield measured away from Bragg condition. From simulations of yield profiles it results that independently from the position of the emitting atoms (for any P_c) for θ < 63° the yield ${Y}_{\exp }\left(\theta \right)$ (normalized to the intensity of the incoming x-ray electric field $| {E}_{0}{| }^{2}$) differs from 1 by less than 1%. Therefore, the yield data of each sample are normalized by the corresponding average of ${Y}_{\exp }\left(\theta \right)$ for θ < 63°. In the case of sample 1 with nominal ${d}_{\mathrm{Si}}^{\mathrm{top}}=2.0\,\mathrm{nm}$, since experimental data are available only for θ > 65°, this normalization factor is an additional fit parameter (appendix B).