Towards Time-Resolved Atomic Structure Determination by X-Ray Standing Waves at a Free-Electron Laser

We demonstrate the structural sensitivity and accuracy of the standing wave technique at a high repetition rate free-electron laser, FLASH at DESY in Hamburg, by measuring the photoelectron yield from the surface SiO2 of Mo/Si multilayers. These experiments open up the possibility to obtain unprecedented structural information of adsorbate and surface atoms with picometer spatial accuracy and femtosecond temporal resolution. This technique will substantially contribute to a fundamental understanding of chemical reactions at catalytic surfaces and the structural dynamics of superconductors.


I. INTRODUCTION
The use of renewable energies for heterogeneous catalysis imposes the understanding of catalytic processes under dynamic reaction conditions. To achieve this goal there is a need of time-resolved spectroscopy measurements, predictive theory and the development of new catalysts 1 . With the advent of x-ray free-electron lasers (XFEL) 2-6 , delivering femtosecond, extremely brilliant, and coherent pulses in the soft and hard x-ray range, it became possible to explore the ultrafast dynamics of heterogeneous catalysis using a pump-probe approach 7,8 .
Optical laser pump pulses are absorbed at the catalyst surface and trigger the reaction by electronic or phononic excitations 9 . XFEL probe pulses are used to measure time-resolved x-ray absorption and emission spectra. In this way several elementary processes, essential for understanding more complex chemical reactions, were unveiled: breaking of the bond between CO molecules and a Ru surface 10  The interpretation of these spectroscopic data relies on density functional theory (DFT) calculations. Only the comparison of measured and simulated data allows to sketch the time evolution of a chemical reaction 7 , as depicted in Fig.1(a). At the same time, a direct structural information on the position of atoms and molecules during the reaction is still missing.
Time-resolved structures of sample surfaces can be obtained in principle by means of low energy electron diffraction (LEED) 12 , reflection high energy electron diffraction (RHEED) 13,14 or surface x-ray diffraction 15 . However, all these methods require lateral long range order of the structure to be resolved. In the case of atoms and molecules involved in chemical reactions at surfaces this requirement may not be fulfilled 16 . Therefore, to measure the time-resolved structure of reactants and catalysts as the reaction proceeds at the surface, we propose to combine photoelectron spectroscopy with the structural accuracy of the x-ray standing wave (XSW) technique [17][18][19][20] and the time resolution provided by an XFEL. In this way we can obtain at the same time sensitivity to the chemical environment of the reactants, by photoelectron spectroscopy (e.g. Refs. [21][22][23] ), and to their position along the Bragg diffraction vector H, by the standing wave technique ( Fig. 1(a)). In fact, XSW proved already to be an ideal tool to determine position and geometry of adsorbates at metal surfaces 20,[24][25][26] . Importantly, the predictive quality of DFT calculations will enormously profit from the experimental structural benchmark provided by time-resolved XSW data, leading ultimately to a better understanding of the fundamental processes in heterogeneous catalysis.
In this pioneering experiment, we demonstrate the structural sensitivity and accuracy of the XSW technique combined with the ultrashort pulses of an XFEL. The XSW forms in the region of spatial overlap between two coherently coupled incoming and Bragg-diffracted x-ray plane waves 17 . This results in a periodic modulation of the x-ray intensity (with period d SW , Fig. 1(b)) along the z direction (perpendicular to the reflecting planes) described by the following equation 18 : where R (θ) is the sample reflectivity and φ (θ) the phase of the ratio E H /E 0 (θ) = R (θ) exp (iφ (θ)), with E 0 and E H the complex electric field amplitude of the incoming and Bragg-diffracted electromagnetic waves. Note that both R (θ) and φ (θ) are functions of the normal angle of incidence θ (Figs. 2(c) and 5).
The main interest of this technique lies in the inelastic scattering of the x-ray standing wave from atoms that work as a probe, leading to photoelectron or x-ray fluorescence yield. The strength of this scattering signal is proportional to the intensity of the XSW at the position of the emitting atoms. Thus, by moving the standing wave in space it is possible to obtain information about the location of the emitters along the perpendicular direction to the Bragg planes, with a spatial accuracy of about 0.01 d SW . In fact, in an XSW experiment as the normal angle of incidence θ of the incoming x-ray wave varies through the Bragg condition, the phase φ changes by π, thus the standing wave shifts along the z direction by d SW /2. Typically the position of light atoms, predominantly present in the reactants, is monitored by the photoelectron signal due to the larger cross-section as compared to fluorescence 18 . Therefore, performing XSW experiments combined with photoelectron spectroscopy at an XFEL on chemical reactions at single crystal catalysts with d SW of about fewÅ (using photon energy of few keV) may deliver structural information of the reactants with an unprecedented high spatial accuracy much below 1Å 24 and femtosecond temporal resolution.

II. EXPERIMENTAL DETAILS
Typically XSW experiments are carried out at synchrotron radiation facilities in order to profit from the photon energy tunability in the soft and hard x-ray range, allowing to match the period of multilayers [27][28][29][30][31] and single crystals 20 , and from the high flux in a small bandwidth that enables a fine scan of the Bragg condition. At the same time, the narrow bandwidth ∆λ ensures that the longitudinal coherence l c ∝ λ 2 /∆λ 32 is much larger than the optical path length difference between the two interfering waves. All these advantages are preserved at free-electron laser facilities. In addition, femtosecond FEL pulses enable studies of ultrafast dynamics up to few tens of femtosecond which could not be reached by ∼ 100 ps synchrotron pulses. At the same time, when measuring photoelectron spectra at an XFEL, due to the high intensity and ultrashort x-ray pulses, vacuum space-charge effects need to be considered 33,34 . To avoid them, the XFEL intensity needs to be reduced, while preserving the short pulse duration, leading to a detection of about one electron per XFEL pulse (limited by our spectrometer, see section II B). As a consequence, in order to measure time-resolved photoelectron spectra to probe sub-ps to ps dynamics with good statistics and in a reasonable amount of time, a high repetition rate XFEL is necessary.
A. Free-electron laser parameters 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 33,34 . 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 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 multilayer period of d M L = 7.3 nm ( Fig. 1(c)). To match the first order Bragg condition 2d M L sin (π/2 − θ) = λ, FLASH was tuned to the wavelength λ = 13.5 nm and the normal angle of incidence of the maximum reflectivity was θ max = 17.5 • (Fig. 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 top Si . 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 top Si . 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 SiO 2 layer of d SiO 2 = 1.2 nm 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 SiO 2 layer (see Fig. 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. Photoelectron spectra
A typical photoelectron spectrum measured on one of our ML samples is shown in Fig.   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 40 . Our attention focuses on the O2s photoelectron peak at about 25 eV binding energy. After subtraction of a Shirley background 41 , the integral O2s peak area is defined as the photoelectron yield Y exp (θ) of the oxygen atoms in the SiO 2 layer at the sample surface, measured at a given angle θ. Each Y exp (θ) needs to undergo several normalization steps described in detail in Appendix A. Importantly, the spectrum shown in Fig. 2(b) was measured in 20 minutes. 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 good statistics feasible only at high repetition rate XFELs, such as FLASH 2 , the European XFEL 6 and LCLS-II 42 .

B. Photoelectron yield profiles
The structural information of XSW measurements is contained in the photoelectron yield profile, i.e. the sequence of Y exp (θ) measured as the incidence angle θ is scanned through the 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 (Fig. 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.

C. Photoelectron yield fit model
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 (θ) of an atom at a given position z is not simply proportional to the XSW intensity I SW (z) as expressed in Eq. (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 44 . For our case of π-polarization the incident and Bragg-diffracted polarization vectors e 0 and e H lie within the scattering plane, defined by the incident and Bragg-diffracted propagation vectors k 0 and k H , as shown in Fig. 2(c). Since soft x-rays are employed, in the calculation of the photoelectric cross-section higher order multipole terms can be neglected 45 . Therefore, in the dipole approximation, for an initial s-state and in π-polarization geometry the angularly resolved photoelectron yield can be expressed as where g = cos θ H / cos θ 0 is the geometrical factor, with θ 0 and θ H the angles between the polarization directions e 0 and e H and the direction of the emitted electrons n p (see insets in Fig. 2(c)). The coherent position is P c = z /d SW , with z the average position of the emitting atoms contributing to the photoelectron yield, and the coherent fraction is F c indicating the distribution width of the emitters around their average position z . Eq.
(2) is accurate if the distribution of atoms contributing to Y does not extend for more than one layer and it 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 SiO 2 layer extending for d SiO 2 = 1.2 nm (Appendix C) below the surface z surf , with z = 0 defined at the top of the Mo layer (see Figs. 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 (θ) by a factor e −(z surf −z)/(λ I ·sin α) 46 , where λ I is the electron inelastic mean free path (Appendix A) and α is the angle between the electron detection direction n p and the sample surface (Fig. 2(c)). As a result, the fit model for Y exp (θ) can be expressed as: Eq. (3) represents the sum of photoelectron yield contributions between z surf − d SiO 2 and z surf weighted by the inelastic mean free path factor. Since the geometrical factor g in Eq.
(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 geometrical factors g 1 and g 2 in Eq. (3) are defined as follows: where θ np indicates the emission angle relative to θ H . The geometrical factors g 1 (θ) and g 2 (θ) depend on the normal angle of incidence θ via the angle θ H = 35 • + 2θ (Fig. 2(c)).

D. Reflectivity data
To apply Eq. (3) it is necessary to know the reflectivity R (θ), the phase φ (θ) of the complex electric field amplitude ratio E H /E 0 , and the period of the standing wave d SW .
These parameters could be easily calculated if the exact structure of our multilayer samples was known. To determine these parameters, grazing incidence x-ray reflectivity (GIXR, Fits of EUVR data of each sample are reported in Fig. 5 together with the calculation of the corresponding phase φ (θ). The large and broad reflectivity peak with maximum of 61.4% at θ max = 17.5 • 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 φ (θ) 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 Fig. 3. The phase term φ (θ) was calculated at the top of the SiO 2 layer, therefore at different positions with respect to the periodic ML structure (Fig. 6). This results into rigid phase shifts going from sample 1 to 4 as it is evident from the corresponding scales in Fig. 5.

E. Discussion
The good quality of EUVR and GIXR curve fits enable us to employ the corresponding R (θ) and φ (θ) functions to fit the experimental yield data Y exp (θ) by means of the model in Eq. (3). The results summarized in Fig. 3 show that Y model (θ) describes very well our measured data. Two fit parameters were employed: the position of SiO 2 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 SiO 2 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 top Si leading to an increasing distance of the sample surface from the periodic ML structure as illustrated in Fig. 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 SiO 2 surface positions. to approximately 20%, therefore it is necessary to normalize each electron yield data point Y exp (θ) by the corresponding XFEL intensity. As a reference for the XFEL intensity we consider the ion signal of a gas monitor detector 35 located directly after the undulators of FLASH (Fig. 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.
b. Normalization by the filter transmission. 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 I. Therefore, in order to have consistent data within the same yield profile each electron yield Y exp (θ) needs to be normalized by the corresponding filter transmission. We kept a Si 3 N 4 filter 500 nm thick throughout all the measurements, while we alternated two ZrB 2 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 I, where also the corresponding photon energy is reported.  c. Normalization by the inelastic mean free path factor. Since the electron yied Y exp (θ) 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 I (z, θ) = e −(z surf −z)/(λ I ·sin α) , where α (θ) is the angle between the surface and the direction n p of photoemitted electrons towards the detector (Fig. 2(c)). Since the angle between k 0 and n p is 55 • , it follows that α = 35 • + 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 (Figs. 1(c) and 6), and z surf is defined as the position of the SiO 2 surface. The normalization factor accounting for λ I is given by The component P 2 relates to Si atoms with an oxidation state larger than 0 (Si in bulk) and smaller than +4 (Si in SiO 2 ). These Si atoms form a transition layer SiO x between Si and SiO 2 layers. Since the BE shift of P 2 with respect to P 1 is smaller than 1.2 eV we conclude that SiO x mainly consists of Si atoms with +1 or smaller oxidation state 59 , therefore close to bulk-like Si.
To determine the thickness of the SiO 2 layer, the photoelectron yield of bulk-like Si is defined as the sum of the yield of component P 1 and P 2 , Fig. 7(b)). The electron yield of Si, within the layer y (with y = Si, SiO 2 ), can be expressed as where: σ Si is photoionization cross section, N y = ρ y N A /M y is the number density (in atoms cm −3 ) of the element/compound y, ρ y is the density (in g cm        Normalized photoelectron yield of bulk-like Si (Y Si , black) and of SiO 2 (Y SiO 2 , red).