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Simulation of Electron Spectra for Surface Analysis (SESSA)for quantitative interpretation of (hard) X-ray photoelectron spectra(HAXPES)
Accepted Manuscript Title: Simulation of Electron Spectra for Surface Analysis (SESSA)for Quantitative Interpretation of (Hard) X-ray Photoelectron Spectra(HAXPES) Author: Wolfgang S.M. Werner Werner Smekal Thomas Hisch Julia Himmelsbach Cedric J. Powell PII: DOI: Reference:
S0368-2048(13)00104-7 http://dx.doi.org/doi:10.1016/j.elspec.2013.06.007 ELSPEC 46146
To appear in:
Journal of Electron Spectroscopy and Related Phenomena
Received date: Revised date: Accepted date:
3-9-2012 19-2-2013 16-6-2013
Please cite this article as: Wolfgang S.M. Werner, Werner Smekal, Thomas Hisch, Julia Himmelsbach, Cedric J. Powell, Simulation of Electron Spectra for Surface Analysis (SESSA)for Quantitative Interpretation of (Hard) X-ray Photoelectron Spectra(HAXPES), Journal of Electron Spectroscopy and Related Phenomena (2013), http://dx.doi.org/10.1016/j.elspec.2013.06.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript
Simulation of Electron Spectra for Surface Analysis (SESSA) for Quantitative Interpretation of (Hard) X-ray Photoelectron Spectra (HAXPES) Wolfgang S.M. Werner ∗ , Werner Smekal, Thomas Hisch, and Julia Himmelsbach
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Institute of Applied Physics, Vienna University of Technology, Wiedner Hauptstraße 8–10, A 1040 Vienna, Austria
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Cedric J. Powell Surface and Microanalysis Science Division,
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National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8370
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(Dated: February 19, 2013)
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[email protected], fax:+43-1-58801-13499, tel:+43-1-58801-13462
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Abstract For the interpretation of photoelectron spectra and in order to obtain quantitative information on the chemical structure of surfaces, one commonly makes a number of symplifying assumptions concerning the generation of the signal electrons, such as the neglect of photoelectron elastic scattering, the anisotropy of photoelectron emission, etc. While the effects of these assumptions for
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planar surfaces and for conventional X-ray sources has been investigated in detail in the past, the combined influence of the nanomorphology, the polarisation of the incoming beam and other processes playing a role in the photoelectron escape on the angular- and energy-distribution of emitted
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photoelectrons has not been clarified to date. The National Institute of Standards and Technology
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(NIST) Database for the Simulation of Electron Spectra for Surface Analysis (SESSA) is a unique tool for interpretation of experimental data for nanostructured surfaces as well as for experimental design with photoelectron energies between 50 eV and 30 keV. SESSA has recently been modi-
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fied to allow a user to simulate x-ray photoelectron spectroscopy (XPS) spectra of nanostructured surfaces, such as surfaces covered with rectangular islands, nanowires, pyramids, spheres, layered
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spheres, etc. The effect of the nanomorphology on the emitted angular and energy distribution of photoelectrons is investigated and comparison is made of simulated data with experimental results.
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Finally, the full potential of XPS for characterising nanostructures by a consistent analysis of the angular distribution of both the photoelectron peaks and their associated inelastic loss features is explored.
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PACS numbers: 68.49.Jk, 79.20.-m, 79.60.-i
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INTRODUCTION
Quantitative interpretation of photoelectron spectra (PES) is a complex task, requiring a physical model, which in general contains a significant number of physical parameters, and which relates the spectrum measured with a particular experimental setup to the structure and composition of the sample. Commonly, a straightforward and simple physical model is
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chosen to facilitate this task in which elastic scattering of the photoelectrons is neglected, the so-called straight-line approximation (SLA). To accommodate the effects of deflections
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along the electron trajectories, at least in an approximate way, the parameters entering the model are replaced by effective values mimicking the influence of elastic scattering on
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the outcome of a given particular quantification approach [1]. A well-known example is the (effective) electron attenuation length, as an alternative to the inelastic mean free path
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(IMFP), which can be used to determine overlayer thicknesses using a simple exponential attenuation law based on the rectilinear motion model [1]. The effective value of the attenuation length in such cases needs to be calculated specifically to provide the correct thickness
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in the chosen quantification algorithm [1]. This calculation needs to be done separately for any overlayer/substrate-combination considering specific peaks in the spectra and account-
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ing diligently for the experimental geometrical arrangement and the employed x-ray source (photon energy, polarisation state and direction). Since it is impossible to perform such calculations in advance for all specific cases that might at some point be of interest for an
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analyst, it would be convenient to have at one’s disposal a tool which allows an analyst to circumvent this problem by predicting peak intensities based on a model taking into
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account the phenomena addressed above. In addition, the potential of XPS for investigation of the nanochemistry comes more and more into focus, adding to the complexity of quantification[2].
The NIST database SESSA (Simulation of Electron Spectra for Surface Analysis) [3, 4] has been developed in order to facilitate this task. It contains a full set of databases for all quantities required to quantitatively predict intensities and the spectral shape for a given experiment. Furthermore, a powerful expert system retrieves all required physical parameters from the databases once a user has specified the experimental setup. Finally, SESSA can perform a state-of-the-art simulation of the energy and angular distribution of photoelectrons using a sophisticated model for signal generation in XPS which includes multiple
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inelastic and elastic scattering of the photoelectrons, the angular dependence of the photoelectric cross section for the specified geometric arrangement of the source and the detector, the energy and polarisation state of the beam, etc. In these model calculations, the sample is assumed to consist of regions of different materials such as a layered sample with planar surface or a nanostructured surface consisting of a periodic 2-dimensional array of nanostructures such as (layered) nanospheres or islands that may be rectangular or pyramidal in
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shape. The dependence of the physical parameters on the position within the specimen is taken into account as is their energy dependence. The energy range for which the databases
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provide the required parameters is between 50 eV and 30 keV, i.e., for energies utilized for what is now termed hard x-ray photoelectron spectroscopy (HAXPES).
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In the present paper, the physical model on which SESSA is based is briefly described and some examples of applications are presented. These comprise: (1) HAXPES on a layered
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sample; (2) the influence of the angular dependent differential subshell photoionization cross section on the energy and angular distribution of detected photoelectrons when non-dipolar terms of the cross section are important; (3) The effect of the nanomorphology on measured
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(angle–resolved) spectra, providing an answer to the question of what type of information
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on the nanomorphology can be extracted from XPS measurements.
PHYSICAL MODEL EMPLOYED IN SESSA
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To simulate photoelectron peak intensities and spectral shapes, SESSA employs the so–
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called partial intensity approach [5, 6], which is based on the trivial observation that the emitted yield consists of groups of electrons, classified by the number of collisions ni they have experienced inside the solid. The (normalised) energy distribution after ni collisions, Fni (E), is readily found by means of an ni −1–fold self-convolution of the intrinsic energy distribution with the distribution of energy losses in an individual collision (the latter quantity is equal to the differential inverse inelastic mean free path (DIIMFP) [5]). It then suffices to find the numbers of electrons that experience ni collisions, the so called partial intensities Cni , thereby yielding the spectrum in the form of an inelastic collision expansion[5]: Y (E, µ) =
N X
Cni (µ)Fni (E)
(1)
ni =0
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Here µ = cos θ indicates the polar emission direction. Note that the zero-order partial intensity corresponds to those photoelectrons that escape without inelastic scattering, i.e., the zero-order partial intensity is equal to the intensity of the main (or no-loss) peak, while the higher–order partial intensities describe the shape of the inelastic background. The essential task of determining the partial intensities is accomplished in SESSA by invoking a symmetry property of the (linearised) Boltzmann equation describing the electron escape:
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Case’s reciprocity theorem for one-speed transport [7]. This theorem stipulates that the location of the source and the detector may be exchanged or, in other words, that one can
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simulate a trajectory in reverse. Starting the trajectory in the analyser and tracing back its history in the solid offers the great advantage that no trajectories are created in vain, such
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as for trajectories which would not escape into the solid angle of detection or trajectories which terminate inside the solid in a forward simulation[8]. Therefore the simulation speed
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is increased by a significant factor of at least several orders of magnitude compared to conventional (forward) simulations. The main point is that the simulation speed is entirely independent of the solid angle of detection of the employed analyser. In order to be able
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to apply this symmetry property, one has to make the common assumption that the energy dependence of the interaction characteristics on the energy loss in the vicinity of a peak is
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weak and can be neglected, a usually good approximation for the energy range of about 100 eV around the photoelectron line position.
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Simulation of individual trajectories is achieved by means of a conventional Monte Carlo method in which the trajectory is made up of rectilinear steps, in between subsequent elastic
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collisions (inelastic collisions are accounted for retrospectively by means of Eqn. (1)).The direction of motion is updated and the procedure is repeated until the electron either enters a region made up of a different material or escapes from the surface. In the trajectoryreversal approach, each step between subsequent elastic collisions contributes to the detected intensity, but with a weight-factor determined by the travelled pathlength as well as the relative orientation of the incoming x-rays and the direction of motion along the step. We assume that, at each position along the step sources are present that emit photoelectrons; the detected photoelectrons have trajectories that are traced back to their sources. Apart from this weight factor, it is the distribution of pathlengths travelled within the solid which determines the partial intensities. For photoionisation with an isotropic cross section, the weight factors are always equal and the partial intensities can be expressed in terms of the 5
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distribution of pathlengths Q(s) in a compact form: Z∞ Cni = Q(s)Wni (s)ds,
(2)
0
where Wni (s) is the Poisson stochastic process. Equation (1) can be generalised in a straightforward manner if different types of inelastic
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scattering take place during the passage of the electron through the solid, such as collisions taking place in different regions of the solid, or the occurence of surface excitations and intrinsic excitations. While the latter two phenomena are not presently implemented
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in SESSA, it is possible to account for different total and differential inelastic-scattering characteristics in the top two layers of the specimen.
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There are generally three aspects of the above scenario of photoelectron emission that are affected by the nanomorphology of a surface: (1) The photoionization process itself may
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be subject to changes as compared to an ideal flat surface, such as a core level shift induced by the nanomorphology of the surface [9]. This phenomenon concerns the intrinsic peak
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shape, i.e., the energy distribution of the photoelectrons directly after photoionization; (2) The inelastic processes near the surface may exhibit additional modes which are intimately related to the morphology and composition of a nanostructured surface [10]; and (3) The
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pathlength of specific photoelectrons created in material A travelling in another material, say B, depends on the geometric boundaries of the considered morphology. As a consequence,
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the relative peak intensities (and the shape of the inelastic background) are essentially influenced by inelastic scattering along trajectories, and will thus also be influenced by the
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details of the considered morphology.
The additional modes of the inelastic processes near nanostructured surfaces are obviously of paramount importance for the optical properties of nanostructures and as such their study has become an important subfield of nanophotonics and plasmonics [10]. Such modes can most easily be studied with electron energy-loss spectroscopy (EELS) in the transmission electron microscope (TEM) and can be observed in EELS spectra as clearly distinguishable features correpsonding to optical excitations from the near infrared to the near ultraviolet. However, on the scale of loss features important for understanding the attenuation of photoelectrons, these effects are believed to be of minor importance and are in fact expected to be nearly impossible to detect in typical XPS spectra. The reason is that the overwhelming majority of processes attenuating the outgoing photoelectrons (and 6
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thereby affecting the relative intensities, extrinsic peak shapes, etc.) take place inside the solid, since the thickness of the surface scattering zone, where additional modes due to the nanomorphology come into play, is about an order of magnitude smaller than the relevant mean free paths for inelastic scattering inside the solid [11]. The most effective experimental means of studying these effects in the TEM is to use an aloof scattering geometry, in which the beam does not actually penetrate the specimen, or to use a supporting substrate which is
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(almost) completely transparent to the beam, such as mica. These techniques carefully aim at avoiding scattering inside the solid which would certainly obscure the spectral features
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of interest for such studies. For simulation of nanostructured surfaces with SESSA, one has just the opposite situation since the signal always originates from within the surface,
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making the phenomena addressed above difficult to distinguish in experimental spectra. On the other hand, theoretically explaining such spectral features is very difficult in general;
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one either has to resort to semi-classical theories or to a full quantum-mechanical treatment of the many-body problem involving the nearly free solid-state electrons[12, 13]. Since these effects are expected to be weak in the case of photoelectron spectra, they are presently not
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modelled by SESSA.
The influence of the spatial distribution of matter near a nanostructured surface is known
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to have a major effect on the features of the energy and angular distribution of emitted photoelectrons [14–17]. This situation comes as little surprise since typical dimensions of nanos-
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tructures are of the order of the inelastic mean free path of the employed photoelectrons, implying that details of the morphology must have a strong effect of the attenuation of pho-
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toelectron lines in a spectrum and therefore their relative intensities, angular distribution, inelastic background, etc.
In fact, the influence of morphology on spectra is sometimes so pronounced that it can even be stated that XPS spectra contain significant information on the nanomorphology of a surface. It has therefore been repeatedly emphasized in the literature (see e.g. Ref. [15]) that XPS could play a vital role in nanoscale science and technology, if only one had an efficient tool to study the influence of morphology on the energy/angular distribution of photoelectrons. The algorithm described above is completely general in terms of the shape of the boundaries between regions consisting of different materials. Therefore, it can be trivially extended to account for geometric effects caused by the nanomorphology of the surface. This is exactly 7
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what has been done in the latest version of SESSA where a user may specify the nanomorphology (layered spheres, islands) of the surface in terms of a few parameters, assuming that the surface consists of a periodic array of these structures on top of the (global) surface.
APPLICATIONS
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HAXPES of a layered sample
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Figure (1) shows measured HAXPES spectra of a Si substrate covered with a 52 nm thick film of Al [18]. This experiment was performed to investigate the question of why one can
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obtain sharp photoelectron emission microscopy (PEEM) images of deep buried layers with hard x-rays. The secondary electrons that are detected by PEEM are expected to diffuse over a broad lateral range, and therefore it is expected that a sharp image could not be
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formed when the signal is coming from deep within the sample and hard x-rays are used to reach such deep layers. The experimental results in the left panel of Figure (1) provide
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the explanation: the weak Si 1s peak of the buried layer is seen to be accompanied by a series of (multiple) plasmon loss peaks whose energy separation corresponds exactly to
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the plasmon loss in Al and is resolved by this experiment to be distinctly different from the plasmon energy loss in Si. This result implies that a comparatively large number of secondary electrons created as a result of an interaction with the incoming beam in the buried layer, are
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generated later when the photoelectron loses energy during its passage through the overlayer. The plasmons excited in this way can decay and transfer their energy and momentum to
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a solid-state electron which is emitted as a secondary electron if its energy is large enough to overcome the surface barrier [19]. Since the momentum transferred to the secondary is small for high energy electrons, its point of escape will be in the vicinity of the Si atom that emitted the photoelectron.
This is clearly an example in which the details of multiple inelastic electron scattering for a given experimental geometrical arrangement and photon beam play an important role and as such, is an ideal test case for a SESSA simulation. The comparison presented in Figure (1) indeed emphasizes the ability of SESSA to predict relative intensities and spectral shapes with satisfactory accuracy. In particular, the shape and intensity of the inelastic background accompanying the Si 1s peak (left panel) reproduces the experimental data quite well, while
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similar simulations for Al film thicknesses 10 % above and below the nominal specimen thickness yield distinctly different shapes of the inelastic background. This result not only gives confidence in the reliability of the databases in SESSA, but also suggests a simple way to measure film thicknesses using the spectral shape of the inelastic background with
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acceptable accuracy.
Angular distribution of Photoelectrons
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The angular distribution of the differential photoelectric cross section of a given subshell σx for x-rays linearly polarised along the z-axis, taking into account nondipolar terms, can
dσ x
z
i σx h βx 1 + (3 cos2 θ − 1) + (δx + γx cos2 θ) sin θ cos ϕ 4π 2
(3)
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dΩ
=
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be written in the following form [20]:
where βx is the usual dipolar asymmetry parameter and δx and γx are non-dipolar parameters characterising the asymmetry between the forward and the backward parts of the differential
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cross section along the direction of the incoming photons. Here θ is the angle between the momentum of the photoelectrons and the polarisation vector of the photons and ϕ is
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the angle between the plane of polarisation and the plane of photoelectron emission. An expression for partiallly polarised or unpolarised photon beams can be found in Ref. [20]. When core-level photoelectrons with asymmetry parameter β = 2 are photoionised by
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unpolarised radiation, the emitted intensity along the direction of the incident photons van-
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ishes. In solids, however, a finite contribution in the forward direction is always seen. This is illustrated by the filled symbols in Fig. 2 that represent the experimental data acquired by Nefedov and Baschenko [21]. In this experiment, an unpolarised Al Kα-laboratory source was used to irradiate a thin Al-specimen perpendicularly, while a rotatable analyser measured the photoelectron angular intensity from the opposite side of the sample. There are two reasons why a finite intensity is always observed in the forward and backward directions: first of all, the opening angle of the analyzer is finite, and, second, photoelectrons are deflected effectively in the Coulomb field of the nuclei. In a solid, this situation gives rise to rather small elastic mean free path lengths. Consequently, the signal electrons are efficiently deflected inside the solid and even for an infinitely small analyzer opening angle, a non-vanishing intensity is expected in the forward and backward directions. 9
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The simulation results shown in Fig. 2 are consistent with the above interpretation in that even in the rectilinear-motion model (curve labelled SLA), there is a finite probability for detecting electrons along the surface normal when the analyzer opening angle is finite. For an infinitely small opening angle, the intensity along the surface normal should disappear in the rectilinear motion model, but not when elastic scattering is considered. SESSA is able to perform a simulation for an infinitely small opening angle of the analyzer by virtue of the
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trajectory reversal technique [8].
For an experiment where the nondipolar terms in the photoelectric cross section should
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be taken into account [22], the forward-backward symmetry is broken, as illustrated in the polar plot in Figure (3a) for Cu2p3/2 photoelectrons excited by 5 keV x-rays. The full solid
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(black) curve is the SESSA result, using the most realistic model in which elastic scattering and the non-dipolar terms in the cross section are considered, while the dotted (blue) curve
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represents results from model calculations in which elastic scattering was ”switched off” and the dash-dotted (red) curve is without the non-dipolar terms, again leading to a symmetric angular distribution with respect to the plane perpendicular to the incoming beam. Com-
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paring the realistic result with the case where elastic scattering is neglected, it is seen that, in the direction where the cross section exhibits its maximum, the intensity is reduced when
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elastic scattering comes into play, while in the near nodal direction it is slightly increased. This result can be understood in terms of the distribution of pathlengths travelled by pho-
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toelectrons for different emission directions which shifts towards slightly larger pathlengths when elastic scattering is switched on. According to Eqn. (2), this situation will decrease the
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zero-order partial intensities at the expense of the higher-order partial intensties; in other words, the photoelectron peak is reduced compared to the inelastic background accompanying it. For the near nodal directions, however, there is an additional contribution caused by electrons with an original direction that does not lie within the solid angle of detection of the analyser, and which are scattered into the detection angle in the course of one or more elastic collisions.
The influence of elastic scattering on the shape of the photoelectron peak can be conveniently visualised by considering the partial intensities as a function of the collision number. Such a representation is given in Figure (3b) for the three geometries indicated in Figure (3a), where we show the reduced partial intensities γni = Cni /Cni =0 corresponding to the spectrum normalised with the area of the no-loss peak (i.e., the zero order partial in10
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tensity). It can be seen that, for emission directions near the maximum of the photoelectric cross section, the partial intensities decrease monotonically with the collision order, while for a near-nodal emission direction, the first-order partial intensity is about 10% larger than for the off-nodal directions and the near-nodal partial intensities decrease slower with the scattering order. The above behaviour of the partial intensities is reflected in the spectral shapes of the
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Cu 2p3/2 photoelectron line, shown in Figure (3c) where the spectra have been normalised with the no-loss intensity. For the near-nodal emission direction (θ = 45◦ ), the inelastic
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tail is significantly enhanced with respect to the other two (off-nodal) emission directions (θ = 0, −45◦ ). It is therefore concluded that for HAXPES measurements on solid surfaces,
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the presence of non-dipolar terms in the photoelectric cross section not only gives rise to a forward-backward asymmetry in the angular distribution of the photoelectron (no-loss)
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intensity, but that there exists an asymmetry in the spectral shape in addition. We conclude this section by addressing the impact of the above findings on analyses of the intrinsic shapes of photoelectron transitions[23]. The first step of intrinsic line-shape
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analysis is the removal of extrinsic scattering effects. The partial-intensity approach provides
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a convenient algorithm to accomplish this goal by iterative application of [6, 23]: Z∞
Yk+1 (E) = Yk (E) − qk
Yk (E + T )Lk (T )dT,
(4)
0
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where Lk (T ) is the distribution of energy losses after k collisions and the coefficients qk are
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functions of the partial intensities: q1 = γ1
q2 = γ2 − q1 q1 q3 = γ3 − q1 q2 − q1 q1 q1 q4 = γ4 − q1 q3 − q2 q2 − q1 q1 q2 − q1 q1 q1 q1 ...
(5)
The subscripts of the coefficients qk in Equation (5) are the partitions of the natural numbers [24]. Note that Eqn. (4) and Eqn. (1) are not only mathematically mutual inverse operations, but also numerically, since the integration in Eqn. (4) and the self–convolution in Eqn. (1) are noise-conserving. This algorithm can be used for successive eliminations of the 11
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contributions of multiple scattering in the volume of the material, plural surface scattering and intrinsic excitations, in this way exposing the bare photoelectron transition which then yields valuable information on the structure of nearby satellites accompanying it [25, 26]. In order to yield a decisive result for the intrinsic lineshape using this procedure, one needs to account accurately for multiple inelastic bulk scattering. Since the combined influence of elastic scattering and the (asymmetric) angular distribution of the photoelectric cross
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section significantly modifies the pathlength distribution, it is mandatory to use the proper sequence of partial intensities for peak-shape analysis, which can conveniently be obtained
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using SESSA.
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Angular and Energy Distribution of Photoelectrons emitted from Nanostructured
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Surfaces
As a final application, the energy and angular distribution of photoelectrons emitted from nanostructured surfaces will be considered. SESSA simulations have been performed
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for an Au substrate covered with Pd nanoparticles of different shapes and sizes, but always with the same amount of material per unit area. The simulations were performed with
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unpolarised Al Kα radiation for an XPS configuration with a fixed angle of 54◦ between the direction of x-rays and the analyser axis, corresponding to a typical laboratory system with (off-normal) emission angles between 0 and 80◦ . Simulations were performed for Pd
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nanospheres with a radius of 3 nm, for nanocubes with essentially the same volume (with
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a height h = 4.8 nm), and for islands with a square base and smaller height, with their width increasing to maintain their volume constant, and, finally for a 1.1 nm thick planar Pd-overlayer on an Au substrate. The resulting angular distributions of the ratio of peak intensities IP d3d5/2 /IAu4f 7/2 are shown in Figure (4). Panel (a) presents the results calculated in the straight line approximation and panel (b) is the corresponding result accounting for elastic scattering. The black dashed curve labelled ”SLA” in Figure (4) corresponds to the angular distribution of a homogeneous film calculated analytically. As expected, the angular distributions for a nanosphere and a cube of the same volume (h=4.8 nm) are very similar, with significant deviations only seen at emission angles larger than about 55◦ . For smaller island thickness (and a corresponding increase of their width), the angular distributions are seen to approach the result for a planar surface consisting of a 1.1 nm Pd overlayer on an 12
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Au substrate. The corresponding angular distribution in the straight–line approximation is shown for comparison as the black dashed curve. When elastic scattering is ”switched on” in the simulations, the angular distributions exhibit a similar qualitative behaviour, but the absolute values of the peak intensity ratios are clearly different, and for emission angles larger than 60◦ the shapes of the angular distributions deviate from the SLA result. The sequence of partial intensities for the considered surface morphologies (for an emis-
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sion angle of 0◦ ) along with the corresponding spectral shapes of the Pd3d5/2 and Au4f 7/2 transitions are shown in Figure (5). For Pd, the partial intensities exhibit the expected sharp
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monotonic decrease with the collision number, while the details of the sequence of partial intensities are different depending on the nanomorphology considered. For the substrate
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signal, the low-order partial intensities are attenuated significantly by the overlayer, giving rise to a qualitatively different sequence of partial intensities (compared to the overlayer)
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and a corresponding difference in spectral shapes. These results clearly demonstrate that both the angular distribution and the spectra of photoelectrons emitted from nanostructured surfaces contains valuable information on the specimen structure. The data shown here for
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the partial intensities can be used for deconvolution of experimental data in combination
CONCLUSIONS
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with Eqn. (4) to extract such information from experimental data on the spectral shape.
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We have described three applications of the NIST SESSA database for quantitative XPS.
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This database provides data for needed physical parameters and can simulate XPS spectra for user-specified experimental conditions and sample materials. The original release of SESSA [3, 4] was designed for multilayer thin-film samples with planar surfaces and interfaces. A new version of SESSA has been designed to allow specification of nanostructured surfaces such as rectangular islands, nanowires, pyramids, spheres and layered spheres on top of a substrate consisting of planar layers of different materials. Our first example is HAXPES for a nominal 52 nm Al film on an Si surface. SESSA simulations agreed well with the experimental measurements of Kinoshita et al. [18] In our second example, SESSA simulations demonstrated the effects of elastic scattering on the angular distribution of photoelectrons emitted from Al in a configuration for which there should be zero emitted Al 2s intensity if elastic-scattering effects were negligible. The simulated distribution with elastic 13
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scattering switched on was in reasonable agreement with experiment [21]. We also showed the effects of non-dipole terms in the photoionization cross section that are relevant in HAXPES. Finally, we examined changes in the angular and energy distributions of photoelectrons from different Pd nanomorphologies on an Au substrate. We considered Pd nanospheres, Pd nanocubes, Pd nano-islands with varying shapes, and a uniform Pd film. Substantial differences were seen in the simulated Pd 3d5/2 and Au 4f7/2 photoelectron spectra for each
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morphology and in the angular distributions, particularly for emission angles greater than about 55◦ . We believe that the new version of SESSA will be a useful tool in characterizing
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different types of nanomorphologies.
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[7] K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, MA, 1967). [8] W. H. Gries and W. S. M. Werner, Surf. Interface Anal. 16, 149 (1990).
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[15] D. R. Baer and M. H. Engelhard, J. Electron Spectrosc. Rel. Phen. 178, 415 (1950). [16] H. P. C. E. Kuipers, H. C. E. van Leuven, and W. M. Visser, Surf. Interface Anal. 8, 235
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(1986).
[17] M. Mohai and I. Bertoti, Surf. Interface Anal. 44, 1130 (2012). [18] T. Kinoshita, E. Ikenaga, J. Kim, S. Ueda, M. Kobata, J. R. Harries, K. Shimada, A. Ino, K. Tamasaku, Y. Nishino, et al., Surf. Sci. 601, 4754 (2007). [19] W. S. M. Werner, W. Smekal, H. Winter, A. Ruocco, F. Offi, S. Iacobucci, and G. Stefani, Phys. Rev. B78, 233403 (2008). [20] P. S. Shaw, U. Arp, and S. H. Southworth, Phys. Rev. A54, 1463 (1996). [21] O. A. Baschenko, G. V. Machavariani, and V. I. Nefedov, J. Electron Spectrosc. Rel. Phen. 34, 305 (1984). [22] M. Novak, N. pauly, and A. Dubus, J. Electron Spectrosc. Rel. Phen. 185, 4 (2012).
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[23] W. S. M. Werner, Phys. Rev. B52, 2964 (1995). [24] S. Ahlgren and K. Ono, Notices of the Am. Math. Soc. 48, 978 (2001). [25] M. Novak (2012).
Ac ce
pt
ed
M
an
us
cr
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[26] W. S. M. Werner, T. Cabela, J. Zemek, and P. Jiricek, Surf. Sci. 470, 325 (2001).
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cr
ip t Si 1s
Exp. SESSA
an
us
Plasmons (Al)
d=57 nm
1950 1900 1850 Binding Energy (eV)
1800
1700
1650 1600 1550 Binding Energy (eV)
1500
ed
2000
Intensity (arb. units)
d=52 nm
M
Intensity (arb. units)
d=47 nm
FIG. 1: Si 1s (left panel) and Al 1s (right panel) core photoelectron spectra of an Si substrate cov-
pt
ered with 52 nm of Al measured with linearly polarised photons with an energy hν = 7936 eV[18]. The solid symbols indicate the experimental results [18] and the curves represent the results of the
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SESSA simulations for the indicated film thicknesses.
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2
Experiment SESSA SLA TA
1.8 1.6
ip t cr
1.2 1 0.8
us
2s
IAl /IAl
2p
1.4
an
0.6 0.4
-60
-40
-20 0 20 emission angle (degrees)
40
60
80
pt
ed
0 -80
M
0.2
FIG. 2: Ratio of the Al 2s and Al 2p photoelectron intensities as a function of the emission angle.
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Circles: experimental results by Nefedov and Baschenko [21]. Solid (black) curve: SESSA simulation based on Mott elastic-scattering cross sections. Dash-dotted (blue) curve: SESSA simulation based on an isotropic transport cross section (TA). Dotted (red) curve: SESSA simulation based on the rectilinear motion model (straight-line approximation). In the experiment, the excitation was with Al Kα x-rays that illuminate the back side of a thin Al specimen. In the present simulations, a semi–infinite Al specimen was assumed to be irradiated with Al Kα x-rays at normal incidence. The half-polar opening angle of the analyzer was taken to be 6◦ . The fact that the polar analyzer opening angle is finite is responsible for a finite value of the plotted ratio at normal emission in the straight-line approximation (SLA).
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Page 18 of 22
(a)
Cu2p3/2 Nondip. Terms Neglected SLA
Cu, hν=5 keV
Geo. I, θ=0o
Geo. III, θ=-45o Geo. II, θ=45o
ε
ip t
hν (b)
cr
0.9 0.8
0.6 0.5
an
Geo. I, θ=0o 0.4 Geo. II, θ=45o Geo. III, θ=-45o 0.3 0 5 10 15 Number of inelastic collisions ni
pt
ed
Geo. I, θ=0o Geo. II, θ=45o Geo. III, θ=-45o
20
M
(c) Intensity (arb. units)
us
0.7
i
γn (arb. units)
Reduced partial intensity
1
Ac ce
3650 3700 3750 3800 3850 3900 3950 4000 4050 Kinetic energy (eV)
FIG. 3: (a) Polar plot of the angular distribution of Cu 2p3/2 emitted photoelectron intensity for a Cu surface irradiated with polarised x-rays incident along the surface. The polarisation vector ε is directed along the surface normal. Dotted (blue) curve: straight line approximation (SLA); Dash-dotted (red) curve: neglect of the non-dipolar terms; Solid (black) curve: account taken of nondipolar terms as well as elastic scattering. (b) Reduced Cu 2p3/2 partial intensities for the three geometries indicated in (a); (c) Cu 2p3/2 spectra for the three geometries and conditions indicated in (a) normalised to the same intensity for the no-loss peak.
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Page 19 of 22
(a) Straight Line Approximation (SLA)
(b) With elastic scattering
4
4 Planar SLA Spheres Islands
Planar
2
h=2.6 nm
1.5
h=4.8 nm
7/2
2.5
h=1.5 nm h=1.9 nm
3
5/2
1
2.5 2
h=2.6 nm h=4.8 nm
1.5
10 20 30 40 50 60 Emission angle (degrees)
Cubic
0.5 0 70
0
10 20 30 40 50 60 Emission angle (degrees)
70
cr
0
h=1.5 nm h=1.9 nm
1 Cubic
0.5 0
Planar
ip t
5/2
IPd3d /IAu4f
7/2
3
Planar SLA Spheres Islands
3.5
IPd3d /IAu4f
3.5
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FIG. 4: Emission angular distribution of peak intensity ratios IP d3d5/2 /IAu4f 7/2 for an Au substrate covered with Pd nanoparticles of different shapes and sizes, maintaining global coverage constant.
an
We plot the ratio of the Pd 3d5/2 intensity to the Au 4f7/2 intensity as a function of the photoelectron emission angle for (a) use of the straight line approximation (SLA) and (b) with elastic scattering
Ac ce
pt
ed
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included in the SESSA simulations.
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(b)
1.3
1.1
1.2
1.5 2.6
1.1 1
h=4.8 nm
0.9 0.8 0.7
Au4f7/2 0
(d)
2 4 6 8 10 Number of inelastic collisions ni
Spheres Islands Planar
an
1.1 1.5 2.6 h=4.8 nm h=4.8 nm 2.6 1.5 1.1 1050 1100 1150 Kinetic Energy (eV)
M
Au4f7/2
1200
1250
1300 1350 1400 Kinetic Energy (eV)
1450
ed
1000
ip t
1.4
Pd3d5/2
Spheres Islands Planar
Spheres Islands Planar
1.5
cr
i
1.6
us
(c)
1 Spheres 0.9 Islands 0.8 h=4.8 nm Planar 0.7 2.6 0.6 1.5 0.5 1.1 0.4 0.3 0.2 0.1 Pd3d5/2 0 0 2 4 6 8 10 Number of inelastic collisions ni
Reduced partial intensity γn
Reduced partial intensity γn
i
(a)
FIG. 5: (a) and (c) Reduced partial intensities and (b) and (d) spectral shapes (normalised to the
Ac ce
an emission angle of 0◦ .
pt
no-loss intensity) for the Pd 3d5/2 and the Au 4f7/2 transition for different nanomorphologies at
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Highlights
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HAXPES spectra of an overlayer system have been calculated and compared with experiment. The angular distribution of photoelectrons including non-dipolar terms has been calculated for homogeneous specimens.
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pt
ed
M
an
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The angular and energy distribution of photoelectrons emitted from nanostructured surfaces with different nanomorphology has been calculated and the ability to distinguish different morphologies using XPS is discussed.
Page 22 of 22
S0368-2048(13)00104-7 http://dx.doi.org/doi:10.1016/j.elspec.2013.06.007 ELSPEC 46146
To appear in:
Journal of Electron Spectroscopy and Related Phenomena
Received date: Revised date: Accepted date:
3-9-2012 19-2-2013 16-6-2013
Please cite this article as: Wolfgang S.M. Werner, Werner Smekal, Thomas Hisch, Julia Himmelsbach, Cedric J. Powell, Simulation of Electron Spectra for Surface Analysis (SESSA)for Quantitative Interpretation of (Hard) X-ray Photoelectron Spectra(HAXPES), Journal of Electron Spectroscopy and Related Phenomena (2013), http://dx.doi.org/10.1016/j.elspec.2013.06.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript
Simulation of Electron Spectra for Surface Analysis (SESSA) for Quantitative Interpretation of (Hard) X-ray Photoelectron Spectra (HAXPES) Wolfgang S.M. Werner ∗ , Werner Smekal, Thomas Hisch, and Julia Himmelsbach
ip t
Institute of Applied Physics, Vienna University of Technology, Wiedner Hauptstraße 8–10, A 1040 Vienna, Austria
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Cedric J. Powell Surface and Microanalysis Science Division,
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National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8370
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(Dated: February 19, 2013)
∗
[email protected], fax:+43-1-58801-13499, tel:+43-1-58801-13462
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Abstract For the interpretation of photoelectron spectra and in order to obtain quantitative information on the chemical structure of surfaces, one commonly makes a number of symplifying assumptions concerning the generation of the signal electrons, such as the neglect of photoelectron elastic scattering, the anisotropy of photoelectron emission, etc. While the effects of these assumptions for
ip t
planar surfaces and for conventional X-ray sources has been investigated in detail in the past, the combined influence of the nanomorphology, the polarisation of the incoming beam and other processes playing a role in the photoelectron escape on the angular- and energy-distribution of emitted
cr
photoelectrons has not been clarified to date. The National Institute of Standards and Technology
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(NIST) Database for the Simulation of Electron Spectra for Surface Analysis (SESSA) is a unique tool for interpretation of experimental data for nanostructured surfaces as well as for experimental design with photoelectron energies between 50 eV and 30 keV. SESSA has recently been modi-
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fied to allow a user to simulate x-ray photoelectron spectroscopy (XPS) spectra of nanostructured surfaces, such as surfaces covered with rectangular islands, nanowires, pyramids, spheres, layered
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spheres, etc. The effect of the nanomorphology on the emitted angular and energy distribution of photoelectrons is investigated and comparison is made of simulated data with experimental results.
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Finally, the full potential of XPS for characterising nanostructures by a consistent analysis of the angular distribution of both the photoelectron peaks and their associated inelastic loss features is explored.
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PACS numbers: 68.49.Jk, 79.20.-m, 79.60.-i
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INTRODUCTION
Quantitative interpretation of photoelectron spectra (PES) is a complex task, requiring a physical model, which in general contains a significant number of physical parameters, and which relates the spectrum measured with a particular experimental setup to the structure and composition of the sample. Commonly, a straightforward and simple physical model is
ip t
chosen to facilitate this task in which elastic scattering of the photoelectrons is neglected, the so-called straight-line approximation (SLA). To accommodate the effects of deflections
cr
along the electron trajectories, at least in an approximate way, the parameters entering the model are replaced by effective values mimicking the influence of elastic scattering on
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the outcome of a given particular quantification approach [1]. A well-known example is the (effective) electron attenuation length, as an alternative to the inelastic mean free path
an
(IMFP), which can be used to determine overlayer thicknesses using a simple exponential attenuation law based on the rectilinear motion model [1]. The effective value of the attenuation length in such cases needs to be calculated specifically to provide the correct thickness
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in the chosen quantification algorithm [1]. This calculation needs to be done separately for any overlayer/substrate-combination considering specific peaks in the spectra and account-
ed
ing diligently for the experimental geometrical arrangement and the employed x-ray source (photon energy, polarisation state and direction). Since it is impossible to perform such calculations in advance for all specific cases that might at some point be of interest for an
pt
analyst, it would be convenient to have at one’s disposal a tool which allows an analyst to circumvent this problem by predicting peak intensities based on a model taking into
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account the phenomena addressed above. In addition, the potential of XPS for investigation of the nanochemistry comes more and more into focus, adding to the complexity of quantification[2].
The NIST database SESSA (Simulation of Electron Spectra for Surface Analysis) [3, 4] has been developed in order to facilitate this task. It contains a full set of databases for all quantities required to quantitatively predict intensities and the spectral shape for a given experiment. Furthermore, a powerful expert system retrieves all required physical parameters from the databases once a user has specified the experimental setup. Finally, SESSA can perform a state-of-the-art simulation of the energy and angular distribution of photoelectrons using a sophisticated model for signal generation in XPS which includes multiple
3
Page 3 of 22
inelastic and elastic scattering of the photoelectrons, the angular dependence of the photoelectric cross section for the specified geometric arrangement of the source and the detector, the energy and polarisation state of the beam, etc. In these model calculations, the sample is assumed to consist of regions of different materials such as a layered sample with planar surface or a nanostructured surface consisting of a periodic 2-dimensional array of nanostructures such as (layered) nanospheres or islands that may be rectangular or pyramidal in
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shape. The dependence of the physical parameters on the position within the specimen is taken into account as is their energy dependence. The energy range for which the databases
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provide the required parameters is between 50 eV and 30 keV, i.e., for energies utilized for what is now termed hard x-ray photoelectron spectroscopy (HAXPES).
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In the present paper, the physical model on which SESSA is based is briefly described and some examples of applications are presented. These comprise: (1) HAXPES on a layered
an
sample; (2) the influence of the angular dependent differential subshell photoionization cross section on the energy and angular distribution of detected photoelectrons when non-dipolar terms of the cross section are important; (3) The effect of the nanomorphology on measured
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(angle–resolved) spectra, providing an answer to the question of what type of information
ed
on the nanomorphology can be extracted from XPS measurements.
PHYSICAL MODEL EMPLOYED IN SESSA
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To simulate photoelectron peak intensities and spectral shapes, SESSA employs the so–
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called partial intensity approach [5, 6], which is based on the trivial observation that the emitted yield consists of groups of electrons, classified by the number of collisions ni they have experienced inside the solid. The (normalised) energy distribution after ni collisions, Fni (E), is readily found by means of an ni −1–fold self-convolution of the intrinsic energy distribution with the distribution of energy losses in an individual collision (the latter quantity is equal to the differential inverse inelastic mean free path (DIIMFP) [5]). It then suffices to find the numbers of electrons that experience ni collisions, the so called partial intensities Cni , thereby yielding the spectrum in the form of an inelastic collision expansion[5]: Y (E, µ) =
N X
Cni (µ)Fni (E)
(1)
ni =0
4
Page 4 of 22
Here µ = cos θ indicates the polar emission direction. Note that the zero-order partial intensity corresponds to those photoelectrons that escape without inelastic scattering, i.e., the zero-order partial intensity is equal to the intensity of the main (or no-loss) peak, while the higher–order partial intensities describe the shape of the inelastic background. The essential task of determining the partial intensities is accomplished in SESSA by invoking a symmetry property of the (linearised) Boltzmann equation describing the electron escape:
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Case’s reciprocity theorem for one-speed transport [7]. This theorem stipulates that the location of the source and the detector may be exchanged or, in other words, that one can
cr
simulate a trajectory in reverse. Starting the trajectory in the analyser and tracing back its history in the solid offers the great advantage that no trajectories are created in vain, such
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as for trajectories which would not escape into the solid angle of detection or trajectories which terminate inside the solid in a forward simulation[8]. Therefore the simulation speed
an
is increased by a significant factor of at least several orders of magnitude compared to conventional (forward) simulations. The main point is that the simulation speed is entirely independent of the solid angle of detection of the employed analyser. In order to be able
M
to apply this symmetry property, one has to make the common assumption that the energy dependence of the interaction characteristics on the energy loss in the vicinity of a peak is
ed
weak and can be neglected, a usually good approximation for the energy range of about 100 eV around the photoelectron line position.
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Simulation of individual trajectories is achieved by means of a conventional Monte Carlo method in which the trajectory is made up of rectilinear steps, in between subsequent elastic
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collisions (inelastic collisions are accounted for retrospectively by means of Eqn. (1)).The direction of motion is updated and the procedure is repeated until the electron either enters a region made up of a different material or escapes from the surface. In the trajectoryreversal approach, each step between subsequent elastic collisions contributes to the detected intensity, but with a weight-factor determined by the travelled pathlength as well as the relative orientation of the incoming x-rays and the direction of motion along the step. We assume that, at each position along the step sources are present that emit photoelectrons; the detected photoelectrons have trajectories that are traced back to their sources. Apart from this weight factor, it is the distribution of pathlengths travelled within the solid which determines the partial intensities. For photoionisation with an isotropic cross section, the weight factors are always equal and the partial intensities can be expressed in terms of the 5
Page 5 of 22
distribution of pathlengths Q(s) in a compact form: Z∞ Cni = Q(s)Wni (s)ds,
(2)
0
where Wni (s) is the Poisson stochastic process. Equation (1) can be generalised in a straightforward manner if different types of inelastic
ip t
scattering take place during the passage of the electron through the solid, such as collisions taking place in different regions of the solid, or the occurence of surface excitations and intrinsic excitations. While the latter two phenomena are not presently implemented
cr
in SESSA, it is possible to account for different total and differential inelastic-scattering characteristics in the top two layers of the specimen.
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There are generally three aspects of the above scenario of photoelectron emission that are affected by the nanomorphology of a surface: (1) The photoionization process itself may
an
be subject to changes as compared to an ideal flat surface, such as a core level shift induced by the nanomorphology of the surface [9]. This phenomenon concerns the intrinsic peak
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shape, i.e., the energy distribution of the photoelectrons directly after photoionization; (2) The inelastic processes near the surface may exhibit additional modes which are intimately related to the morphology and composition of a nanostructured surface [10]; and (3) The
ed
pathlength of specific photoelectrons created in material A travelling in another material, say B, depends on the geometric boundaries of the considered morphology. As a consequence,
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the relative peak intensities (and the shape of the inelastic background) are essentially influenced by inelastic scattering along trajectories, and will thus also be influenced by the
Ac ce
details of the considered morphology.
The additional modes of the inelastic processes near nanostructured surfaces are obviously of paramount importance for the optical properties of nanostructures and as such their study has become an important subfield of nanophotonics and plasmonics [10]. Such modes can most easily be studied with electron energy-loss spectroscopy (EELS) in the transmission electron microscope (TEM) and can be observed in EELS spectra as clearly distinguishable features correpsonding to optical excitations from the near infrared to the near ultraviolet. However, on the scale of loss features important for understanding the attenuation of photoelectrons, these effects are believed to be of minor importance and are in fact expected to be nearly impossible to detect in typical XPS spectra. The reason is that the overwhelming majority of processes attenuating the outgoing photoelectrons (and 6
Page 6 of 22
thereby affecting the relative intensities, extrinsic peak shapes, etc.) take place inside the solid, since the thickness of the surface scattering zone, where additional modes due to the nanomorphology come into play, is about an order of magnitude smaller than the relevant mean free paths for inelastic scattering inside the solid [11]. The most effective experimental means of studying these effects in the TEM is to use an aloof scattering geometry, in which the beam does not actually penetrate the specimen, or to use a supporting substrate which is
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(almost) completely transparent to the beam, such as mica. These techniques carefully aim at avoiding scattering inside the solid which would certainly obscure the spectral features
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of interest for such studies. For simulation of nanostructured surfaces with SESSA, one has just the opposite situation since the signal always originates from within the surface,
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making the phenomena addressed above difficult to distinguish in experimental spectra. On the other hand, theoretically explaining such spectral features is very difficult in general;
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one either has to resort to semi-classical theories or to a full quantum-mechanical treatment of the many-body problem involving the nearly free solid-state electrons[12, 13]. Since these effects are expected to be weak in the case of photoelectron spectra, they are presently not
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modelled by SESSA.
The influence of the spatial distribution of matter near a nanostructured surface is known
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to have a major effect on the features of the energy and angular distribution of emitted photoelectrons [14–17]. This situation comes as little surprise since typical dimensions of nanos-
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tructures are of the order of the inelastic mean free path of the employed photoelectrons, implying that details of the morphology must have a strong effect of the attenuation of pho-
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toelectron lines in a spectrum and therefore their relative intensities, angular distribution, inelastic background, etc.
In fact, the influence of morphology on spectra is sometimes so pronounced that it can even be stated that XPS spectra contain significant information on the nanomorphology of a surface. It has therefore been repeatedly emphasized in the literature (see e.g. Ref. [15]) that XPS could play a vital role in nanoscale science and technology, if only one had an efficient tool to study the influence of morphology on the energy/angular distribution of photoelectrons. The algorithm described above is completely general in terms of the shape of the boundaries between regions consisting of different materials. Therefore, it can be trivially extended to account for geometric effects caused by the nanomorphology of the surface. This is exactly 7
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what has been done in the latest version of SESSA where a user may specify the nanomorphology (layered spheres, islands) of the surface in terms of a few parameters, assuming that the surface consists of a periodic array of these structures on top of the (global) surface.
APPLICATIONS
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HAXPES of a layered sample
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Figure (1) shows measured HAXPES spectra of a Si substrate covered with a 52 nm thick film of Al [18]. This experiment was performed to investigate the question of why one can
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obtain sharp photoelectron emission microscopy (PEEM) images of deep buried layers with hard x-rays. The secondary electrons that are detected by PEEM are expected to diffuse over a broad lateral range, and therefore it is expected that a sharp image could not be
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formed when the signal is coming from deep within the sample and hard x-rays are used to reach such deep layers. The experimental results in the left panel of Figure (1) provide
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the explanation: the weak Si 1s peak of the buried layer is seen to be accompanied by a series of (multiple) plasmon loss peaks whose energy separation corresponds exactly to
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the plasmon loss in Al and is resolved by this experiment to be distinctly different from the plasmon energy loss in Si. This result implies that a comparatively large number of secondary electrons created as a result of an interaction with the incoming beam in the buried layer, are
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generated later when the photoelectron loses energy during its passage through the overlayer. The plasmons excited in this way can decay and transfer their energy and momentum to
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a solid-state electron which is emitted as a secondary electron if its energy is large enough to overcome the surface barrier [19]. Since the momentum transferred to the secondary is small for high energy electrons, its point of escape will be in the vicinity of the Si atom that emitted the photoelectron.
This is clearly an example in which the details of multiple inelastic electron scattering for a given experimental geometrical arrangement and photon beam play an important role and as such, is an ideal test case for a SESSA simulation. The comparison presented in Figure (1) indeed emphasizes the ability of SESSA to predict relative intensities and spectral shapes with satisfactory accuracy. In particular, the shape and intensity of the inelastic background accompanying the Si 1s peak (left panel) reproduces the experimental data quite well, while
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Page 8 of 22
similar simulations for Al film thicknesses 10 % above and below the nominal specimen thickness yield distinctly different shapes of the inelastic background. This result not only gives confidence in the reliability of the databases in SESSA, but also suggests a simple way to measure film thicknesses using the spectral shape of the inelastic background with
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acceptable accuracy.
Angular distribution of Photoelectrons
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The angular distribution of the differential photoelectric cross section of a given subshell σx for x-rays linearly polarised along the z-axis, taking into account nondipolar terms, can
dσ x
z
i σx h βx 1 + (3 cos2 θ − 1) + (δx + γx cos2 θ) sin θ cos ϕ 4π 2
(3)
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dΩ
=
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be written in the following form [20]:
where βx is the usual dipolar asymmetry parameter and δx and γx are non-dipolar parameters characterising the asymmetry between the forward and the backward parts of the differential
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cross section along the direction of the incoming photons. Here θ is the angle between the momentum of the photoelectrons and the polarisation vector of the photons and ϕ is
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the angle between the plane of polarisation and the plane of photoelectron emission. An expression for partiallly polarised or unpolarised photon beams can be found in Ref. [20]. When core-level photoelectrons with asymmetry parameter β = 2 are photoionised by
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unpolarised radiation, the emitted intensity along the direction of the incident photons van-
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ishes. In solids, however, a finite contribution in the forward direction is always seen. This is illustrated by the filled symbols in Fig. 2 that represent the experimental data acquired by Nefedov and Baschenko [21]. In this experiment, an unpolarised Al Kα-laboratory source was used to irradiate a thin Al-specimen perpendicularly, while a rotatable analyser measured the photoelectron angular intensity from the opposite side of the sample. There are two reasons why a finite intensity is always observed in the forward and backward directions: first of all, the opening angle of the analyzer is finite, and, second, photoelectrons are deflected effectively in the Coulomb field of the nuclei. In a solid, this situation gives rise to rather small elastic mean free path lengths. Consequently, the signal electrons are efficiently deflected inside the solid and even for an infinitely small analyzer opening angle, a non-vanishing intensity is expected in the forward and backward directions. 9
Page 9 of 22
The simulation results shown in Fig. 2 are consistent with the above interpretation in that even in the rectilinear-motion model (curve labelled SLA), there is a finite probability for detecting electrons along the surface normal when the analyzer opening angle is finite. For an infinitely small opening angle, the intensity along the surface normal should disappear in the rectilinear motion model, but not when elastic scattering is considered. SESSA is able to perform a simulation for an infinitely small opening angle of the analyzer by virtue of the
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trajectory reversal technique [8].
For an experiment where the nondipolar terms in the photoelectric cross section should
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be taken into account [22], the forward-backward symmetry is broken, as illustrated in the polar plot in Figure (3a) for Cu2p3/2 photoelectrons excited by 5 keV x-rays. The full solid
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(black) curve is the SESSA result, using the most realistic model in which elastic scattering and the non-dipolar terms in the cross section are considered, while the dotted (blue) curve
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represents results from model calculations in which elastic scattering was ”switched off” and the dash-dotted (red) curve is without the non-dipolar terms, again leading to a symmetric angular distribution with respect to the plane perpendicular to the incoming beam. Com-
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paring the realistic result with the case where elastic scattering is neglected, it is seen that, in the direction where the cross section exhibits its maximum, the intensity is reduced when
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elastic scattering comes into play, while in the near nodal direction it is slightly increased. This result can be understood in terms of the distribution of pathlengths travelled by pho-
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toelectrons for different emission directions which shifts towards slightly larger pathlengths when elastic scattering is switched on. According to Eqn. (2), this situation will decrease the
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zero-order partial intensities at the expense of the higher-order partial intensties; in other words, the photoelectron peak is reduced compared to the inelastic background accompanying it. For the near nodal directions, however, there is an additional contribution caused by electrons with an original direction that does not lie within the solid angle of detection of the analyser, and which are scattered into the detection angle in the course of one or more elastic collisions.
The influence of elastic scattering on the shape of the photoelectron peak can be conveniently visualised by considering the partial intensities as a function of the collision number. Such a representation is given in Figure (3b) for the three geometries indicated in Figure (3a), where we show the reduced partial intensities γni = Cni /Cni =0 corresponding to the spectrum normalised with the area of the no-loss peak (i.e., the zero order partial in10
Page 10 of 22
tensity). It can be seen that, for emission directions near the maximum of the photoelectric cross section, the partial intensities decrease monotonically with the collision order, while for a near-nodal emission direction, the first-order partial intensity is about 10% larger than for the off-nodal directions and the near-nodal partial intensities decrease slower with the scattering order. The above behaviour of the partial intensities is reflected in the spectral shapes of the
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Cu 2p3/2 photoelectron line, shown in Figure (3c) where the spectra have been normalised with the no-loss intensity. For the near-nodal emission direction (θ = 45◦ ), the inelastic
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tail is significantly enhanced with respect to the other two (off-nodal) emission directions (θ = 0, −45◦ ). It is therefore concluded that for HAXPES measurements on solid surfaces,
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the presence of non-dipolar terms in the photoelectric cross section not only gives rise to a forward-backward asymmetry in the angular distribution of the photoelectron (no-loss)
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intensity, but that there exists an asymmetry in the spectral shape in addition. We conclude this section by addressing the impact of the above findings on analyses of the intrinsic shapes of photoelectron transitions[23]. The first step of intrinsic line-shape
M
analysis is the removal of extrinsic scattering effects. The partial-intensity approach provides
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a convenient algorithm to accomplish this goal by iterative application of [6, 23]: Z∞
Yk+1 (E) = Yk (E) − qk
Yk (E + T )Lk (T )dT,
(4)
0
pt
where Lk (T ) is the distribution of energy losses after k collisions and the coefficients qk are
Ac ce
functions of the partial intensities: q1 = γ1
q2 = γ2 − q1 q1 q3 = γ3 − q1 q2 − q1 q1 q1 q4 = γ4 − q1 q3 − q2 q2 − q1 q1 q2 − q1 q1 q1 q1 ...
(5)
The subscripts of the coefficients qk in Equation (5) are the partitions of the natural numbers [24]. Note that Eqn. (4) and Eqn. (1) are not only mathematically mutual inverse operations, but also numerically, since the integration in Eqn. (4) and the self–convolution in Eqn. (1) are noise-conserving. This algorithm can be used for successive eliminations of the 11
Page 11 of 22
contributions of multiple scattering in the volume of the material, plural surface scattering and intrinsic excitations, in this way exposing the bare photoelectron transition which then yields valuable information on the structure of nearby satellites accompanying it [25, 26]. In order to yield a decisive result for the intrinsic lineshape using this procedure, one needs to account accurately for multiple inelastic bulk scattering. Since the combined influence of elastic scattering and the (asymmetric) angular distribution of the photoelectric cross
ip t
section significantly modifies the pathlength distribution, it is mandatory to use the proper sequence of partial intensities for peak-shape analysis, which can conveniently be obtained
cr
using SESSA.
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Angular and Energy Distribution of Photoelectrons emitted from Nanostructured
an
Surfaces
As a final application, the energy and angular distribution of photoelectrons emitted from nanostructured surfaces will be considered. SESSA simulations have been performed
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for an Au substrate covered with Pd nanoparticles of different shapes and sizes, but always with the same amount of material per unit area. The simulations were performed with
ed
unpolarised Al Kα radiation for an XPS configuration with a fixed angle of 54◦ between the direction of x-rays and the analyser axis, corresponding to a typical laboratory system with (off-normal) emission angles between 0 and 80◦ . Simulations were performed for Pd
pt
nanospheres with a radius of 3 nm, for nanocubes with essentially the same volume (with
Ac ce
a height h = 4.8 nm), and for islands with a square base and smaller height, with their width increasing to maintain their volume constant, and, finally for a 1.1 nm thick planar Pd-overlayer on an Au substrate. The resulting angular distributions of the ratio of peak intensities IP d3d5/2 /IAu4f 7/2 are shown in Figure (4). Panel (a) presents the results calculated in the straight line approximation and panel (b) is the corresponding result accounting for elastic scattering. The black dashed curve labelled ”SLA” in Figure (4) corresponds to the angular distribution of a homogeneous film calculated analytically. As expected, the angular distributions for a nanosphere and a cube of the same volume (h=4.8 nm) are very similar, with significant deviations only seen at emission angles larger than about 55◦ . For smaller island thickness (and a corresponding increase of their width), the angular distributions are seen to approach the result for a planar surface consisting of a 1.1 nm Pd overlayer on an 12
Page 12 of 22
Au substrate. The corresponding angular distribution in the straight–line approximation is shown for comparison as the black dashed curve. When elastic scattering is ”switched on” in the simulations, the angular distributions exhibit a similar qualitative behaviour, but the absolute values of the peak intensity ratios are clearly different, and for emission angles larger than 60◦ the shapes of the angular distributions deviate from the SLA result. The sequence of partial intensities for the considered surface morphologies (for an emis-
ip t
sion angle of 0◦ ) along with the corresponding spectral shapes of the Pd3d5/2 and Au4f 7/2 transitions are shown in Figure (5). For Pd, the partial intensities exhibit the expected sharp
cr
monotonic decrease with the collision number, while the details of the sequence of partial intensities are different depending on the nanomorphology considered. For the substrate
us
signal, the low-order partial intensities are attenuated significantly by the overlayer, giving rise to a qualitatively different sequence of partial intensities (compared to the overlayer)
an
and a corresponding difference in spectral shapes. These results clearly demonstrate that both the angular distribution and the spectra of photoelectrons emitted from nanostructured surfaces contains valuable information on the specimen structure. The data shown here for
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the partial intensities can be used for deconvolution of experimental data in combination
CONCLUSIONS
ed
with Eqn. (4) to extract such information from experimental data on the spectral shape.
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We have described three applications of the NIST SESSA database for quantitative XPS.
Ac ce
This database provides data for needed physical parameters and can simulate XPS spectra for user-specified experimental conditions and sample materials. The original release of SESSA [3, 4] was designed for multilayer thin-film samples with planar surfaces and interfaces. A new version of SESSA has been designed to allow specification of nanostructured surfaces such as rectangular islands, nanowires, pyramids, spheres and layered spheres on top of a substrate consisting of planar layers of different materials. Our first example is HAXPES for a nominal 52 nm Al film on an Si surface. SESSA simulations agreed well with the experimental measurements of Kinoshita et al. [18] In our second example, SESSA simulations demonstrated the effects of elastic scattering on the angular distribution of photoelectrons emitted from Al in a configuration for which there should be zero emitted Al 2s intensity if elastic-scattering effects were negligible. The simulated distribution with elastic 13
Page 13 of 22
scattering switched on was in reasonable agreement with experiment [21]. We also showed the effects of non-dipole terms in the photoionization cross section that are relevant in HAXPES. Finally, we examined changes in the angular and energy distributions of photoelectrons from different Pd nanomorphologies on an Au substrate. We considered Pd nanospheres, Pd nanocubes, Pd nano-islands with varying shapes, and a uniform Pd film. Substantial differences were seen in the simulated Pd 3d5/2 and Au 4f7/2 photoelectron spectra for each
ip t
morphology and in the angular distributions, particularly for emission angles greater than about 55◦ . We believe that the new version of SESSA will be a useful tool in characterizing
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different types of nanomorphologies.
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[1] C. J. Powell and A. Jablonski, Nucl. Instr. Meth. Phys. Res. A601, 54 (2009). [2] S. Techane, D. R. Baer, and D. G. Castner, Anal. Chem. 83, 6704 (2011). [3] W. Smekal, W. S. M. Werner, and C. J. Powell, Surf. Interface Anal. 37, 1059 (2005). [4] (2005), http://www.nist.gov/srd/nist100.cfm.
ip t
[5] W. S. M. Werner, Phys. Rev. B 71, 115415 (2005). [6] W. S. M. Werner, Surf. Interface Anal. 23, 737 (1995).
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[7] K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, MA, 1967). [8] W. H. Gries and W. S. M. Werner, Surf. Interface Anal. 16, 149 (1990).
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[9] H.-G. Boyen, A. Ethirajan, G. Kstle, F. Weigl, P. Ziemann, G. Schmid, M. G. Garnier, M. Bttner, and P. Oelhafen, Phys. Rev. Lett. 94, 016804 (2005).
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[10] F. J. Garcia de Abajo, Rev. Mod. Phys 82, 209 (2010).
[11] W. S. M. Werner, W. Smekal, F. Salvat-Pujol, Z. Halavani, S. Pfleger, J. Rastl, and
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C. Eisenmenger-Sittner, Appl. Phys. Lett. 98, 193111 (2011). [12] J. Vinson, J. J. Rehr, J. J. Kas, and E. L. Shirley, Phys. Rev. B 83, 115106 (2011).
165111 (2006).
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[13] A. P. Sorini, J. J. Kas, J. J. Rehr, M. P. Prange, and Z. H. Levine, Physical Review B 74,
[14] A. Frydman, D. G. Castner, M. Schmal, and C. T. Campbell, J. of Catalysis 157, 133 (1995).
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[15] D. R. Baer and M. H. Engelhard, J. Electron Spectrosc. Rel. Phen. 178, 415 (1950). [16] H. P. C. E. Kuipers, H. C. E. van Leuven, and W. M. Visser, Surf. Interface Anal. 8, 235
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(1986).
[17] M. Mohai and I. Bertoti, Surf. Interface Anal. 44, 1130 (2012). [18] T. Kinoshita, E. Ikenaga, J. Kim, S. Ueda, M. Kobata, J. R. Harries, K. Shimada, A. Ino, K. Tamasaku, Y. Nishino, et al., Surf. Sci. 601, 4754 (2007). [19] W. S. M. Werner, W. Smekal, H. Winter, A. Ruocco, F. Offi, S. Iacobucci, and G. Stefani, Phys. Rev. B78, 233403 (2008). [20] P. S. Shaw, U. Arp, and S. H. Southworth, Phys. Rev. A54, 1463 (1996). [21] O. A. Baschenko, G. V. Machavariani, and V. I. Nefedov, J. Electron Spectrosc. Rel. Phen. 34, 305 (1984). [22] M. Novak, N. pauly, and A. Dubus, J. Electron Spectrosc. Rel. Phen. 185, 4 (2012).
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[23] W. S. M. Werner, Phys. Rev. B52, 2964 (1995). [24] S. Ahlgren and K. Ono, Notices of the Am. Math. Soc. 48, 978 (2001). [25] M. Novak (2012).
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[26] W. S. M. Werner, T. Cabela, J. Zemek, and P. Jiricek, Surf. Sci. 470, 325 (2001).
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cr
ip t Si 1s
Exp. SESSA
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us
Plasmons (Al)
d=57 nm
1950 1900 1850 Binding Energy (eV)
1800
1700
1650 1600 1550 Binding Energy (eV)
1500
ed
2000
Intensity (arb. units)
d=52 nm
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Intensity (arb. units)
d=47 nm
FIG. 1: Si 1s (left panel) and Al 1s (right panel) core photoelectron spectra of an Si substrate cov-
pt
ered with 52 nm of Al measured with linearly polarised photons with an energy hν = 7936 eV[18]. The solid symbols indicate the experimental results [18] and the curves represent the results of the
Ac ce
SESSA simulations for the indicated film thicknesses.
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Page 17 of 22
2
Experiment SESSA SLA TA
1.8 1.6
ip t cr
1.2 1 0.8
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2s
IAl /IAl
2p
1.4
an
0.6 0.4
-60
-40
-20 0 20 emission angle (degrees)
40
60
80
pt
ed
0 -80
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0.2
FIG. 2: Ratio of the Al 2s and Al 2p photoelectron intensities as a function of the emission angle.
Ac ce
Circles: experimental results by Nefedov and Baschenko [21]. Solid (black) curve: SESSA simulation based on Mott elastic-scattering cross sections. Dash-dotted (blue) curve: SESSA simulation based on an isotropic transport cross section (TA). Dotted (red) curve: SESSA simulation based on the rectilinear motion model (straight-line approximation). In the experiment, the excitation was with Al Kα x-rays that illuminate the back side of a thin Al specimen. In the present simulations, a semi–infinite Al specimen was assumed to be irradiated with Al Kα x-rays at normal incidence. The half-polar opening angle of the analyzer was taken to be 6◦ . The fact that the polar analyzer opening angle is finite is responsible for a finite value of the plotted ratio at normal emission in the straight-line approximation (SLA).
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Page 18 of 22
(a)
Cu2p3/2 Nondip. Terms Neglected SLA
Cu, hν=5 keV
Geo. I, θ=0o
Geo. III, θ=-45o Geo. II, θ=45o
ε
ip t
hν (b)
cr
0.9 0.8
0.6 0.5
an
Geo. I, θ=0o 0.4 Geo. II, θ=45o Geo. III, θ=-45o 0.3 0 5 10 15 Number of inelastic collisions ni
pt
ed
Geo. I, θ=0o Geo. II, θ=45o Geo. III, θ=-45o
20
M
(c) Intensity (arb. units)
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0.7
i
γn (arb. units)
Reduced partial intensity
1
Ac ce
3650 3700 3750 3800 3850 3900 3950 4000 4050 Kinetic energy (eV)
FIG. 3: (a) Polar plot of the angular distribution of Cu 2p3/2 emitted photoelectron intensity for a Cu surface irradiated with polarised x-rays incident along the surface. The polarisation vector ε is directed along the surface normal. Dotted (blue) curve: straight line approximation (SLA); Dash-dotted (red) curve: neglect of the non-dipolar terms; Solid (black) curve: account taken of nondipolar terms as well as elastic scattering. (b) Reduced Cu 2p3/2 partial intensities for the three geometries indicated in (a); (c) Cu 2p3/2 spectra for the three geometries and conditions indicated in (a) normalised to the same intensity for the no-loss peak.
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Page 19 of 22
(a) Straight Line Approximation (SLA)
(b) With elastic scattering
4
4 Planar SLA Spheres Islands
Planar
2
h=2.6 nm
1.5
h=4.8 nm
7/2
2.5
h=1.5 nm h=1.9 nm
3
5/2
1
2.5 2
h=2.6 nm h=4.8 nm
1.5
10 20 30 40 50 60 Emission angle (degrees)
Cubic
0.5 0 70
0
10 20 30 40 50 60 Emission angle (degrees)
70
cr
0
h=1.5 nm h=1.9 nm
1 Cubic
0.5 0
Planar
ip t
5/2
IPd3d /IAu4f
7/2
3
Planar SLA Spheres Islands
3.5
IPd3d /IAu4f
3.5
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FIG. 4: Emission angular distribution of peak intensity ratios IP d3d5/2 /IAu4f 7/2 for an Au substrate covered with Pd nanoparticles of different shapes and sizes, maintaining global coverage constant.
an
We plot the ratio of the Pd 3d5/2 intensity to the Au 4f7/2 intensity as a function of the photoelectron emission angle for (a) use of the straight line approximation (SLA) and (b) with elastic scattering
Ac ce
pt
ed
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included in the SESSA simulations.
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Page 20 of 22
(b)
1.3
1.1
1.2
1.5 2.6
1.1 1
h=4.8 nm
0.9 0.8 0.7
Au4f7/2 0
(d)
2 4 6 8 10 Number of inelastic collisions ni
Spheres Islands Planar
an
1.1 1.5 2.6 h=4.8 nm h=4.8 nm 2.6 1.5 1.1 1050 1100 1150 Kinetic Energy (eV)
M
Au4f7/2
1200
1250
1300 1350 1400 Kinetic Energy (eV)
1450
ed
1000
ip t
1.4
Pd3d5/2
Spheres Islands Planar
Spheres Islands Planar
1.5
cr
i
1.6
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(c)
1 Spheres 0.9 Islands 0.8 h=4.8 nm Planar 0.7 2.6 0.6 1.5 0.5 1.1 0.4 0.3 0.2 0.1 Pd3d5/2 0 0 2 4 6 8 10 Number of inelastic collisions ni
Reduced partial intensity γn
Reduced partial intensity γn
i
(a)
FIG. 5: (a) and (c) Reduced partial intensities and (b) and (d) spectral shapes (normalised to the
Ac ce
an emission angle of 0◦ .
pt
no-loss intensity) for the Pd 3d5/2 and the Au 4f7/2 transition for different nanomorphologies at
27
Page 21 of 22
Highlights
ip t
HAXPES spectra of an overlayer system have been calculated and compared with experiment. The angular distribution of photoelectrons including non-dipolar terms has been calculated for homogeneous specimens.
Ac ce
pt
ed
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an
us
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The angular and energy distribution of photoelectrons emitted from nanostructured surfaces with different nanomorphology has been calculated and the ability to distinguish different morphologies using XPS is discussed.
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