Optimization of the X-ray incidence angle in photoelectron spectrometers

The interplay between X-ray reflectivity, X-ray absorption and photoelectron attenuation in the photoelectron emission process is analyzed. The optimal X-ray incidence angle to maximize the photoelectron signal is evaluated in a wide VUV to hard X-ray energy range.


Introduction
X-ray photoelectron spectroscopy (XPS) experiments are in general characterized by a disparity of some two orders of magnitude between the relatively large X-ray attenuation depth and relatively small photoelectron escape depth. Most of the photoelectrons are excited therefore at a depth much larger they can elastically escape from, and on their way to vacuum dissipate in a series of inelastic scattering events to form the secondary-electron cascade background that carries little spectroscopic information. Obviously the way to gain the elastic signal will be to deposit more of the X-ray energy closer to the surface, which can be achieved by using grazing-incidence angles.
The process of the X-ray-excited production of photoelectrons was cast into an exact numerical framework in 1972 in the seminal work of Henke (1972). Henke showed that a significant increase of the elastic (no-loss) photoelectron yield can be achieved with grazing X-ray incidence angles approaching the total external reflection (critical) angle c . These results have received further theoretical developments (Fadley, 1974) including generalization to multilayer structures (Chester & Jach, 1993;Fadley et al., 2003) as well as extensive experimental verification (Hayashi et al., 1996;Kawai et al., 1995).
Here we analyze the interplay between the X-ray reflectivity, X-ray absorption and photoelectron escape processes with particular attention given to the effects of the photon spot size with respect to the analyzer field of view (FOV). The optimal X-ray incidence angle to achieve the maximal XPS intensity gain is determined in a wide energy range from VUV to hard X-rays. These results bear immediate implications for optimization of the experimental geometry of (angle-resolving) XPS spectrometers.

Formalism
We will first recap the basic formalism describing the X-ray excited photoelectron current I PE ð; Þ as a function of the X-ray grazing-incidence angle and photoelectron emission angle relative to the surface normal (see Fig. 1a). According to the Beer-Lambert law, the electromagnetic field intensity S(x) in the media exponentially decreases with depth x as S(x) = AðÞ exp½Àx=dðÞ, where d() is the electromagnetic field penetration depth (perpendicular to the surface) and A ð Þ is a normalization coefficient. The latter is defined by the condition that R 1 0 SðxÞ dx, expressing the total absorption in the media, obeys the complementarity principle and is thus proportional to 1 À R, where R is the X-ray reflection coefficient. Performing this integral and equating it to 1 À R, we immediately obtain AðÞ / ½1 À RðÞ=dðÞ and Then, neglecting the photoexcitation matrix elements and the photoelectron refraction important only at low energies, the photoelectron intensity dI PE ð; Þ originating from a layer with a thickness dx placed at depth x is proportional to the power absorbed in the layer SðxÞ dx multiplied by the photoelectron transmission through the overlayer expðÀx= cos Þ, where is the photoelectron attenuation length (Powell et al., 1999). Note that in contrast to d() taken perpendicular to the surface and having the meaning of depth, is taken along the photoelectron path and has the meaning of length. Integration of dIð; Þ over the depth yields which evaluates to

View factor
The above formalism implied that the analyzer collected all photoelectrons emerging at the sample. Now, the above expression (3) should be multiplied by a geometrical view factor Vð; Þ which is defined by the relation of the incidentbeam cross section b and the analyzer FOV f (see Fig. 1a) in their projections to the sample, b= sin and f = cos , respectively. Obviously, with a less grazing and more grazing , when the inequality b= sin < f = cos arises we have full photoelectron acceptance and Vð; Þ is identically equal to 1.
In the opposite case the acceptance is FOV-limited and Vð; Þ is equal to the ratio of the FOV and beam projections, The full-acceptance (FA) regime implies that all photoelectrons emerging throughout the X-ray footprint on the sample are intercepted by the analyzer FOV; this is typical of the current synchrotron sources delivering a beam focused to some 10 mm and below. The FOV-limited (FOVL) regime (often referred to as the overfilled analyser slit) implies the loss of the photoelectrons outside the analyzer FOV; this is typical of the laboratory X-ray or older synchrotron sources with their spot being of the order of 1 mm. Our formalism for the FOVL regime is equivalent to that of Henke (1972)

Numerical examples and analysis
We will now use the above formalism in practical calculations. We restrict ourselves to the normal emission = 0. In this case the formulas (3)-(4) reduce to with VðÞ = 1 for the FA regime and VðÞ = ð f =bÞ sin for the FOVL regime. The calculations were performed for the paradigm metal Cu at h = 400 eV. The numerical values of R and d were taken from the X-ray database readily availably on the Web (Henke et al., 1993). The photoelectron energy was taken to be equal to h as relevant for the valence-band XPS. The corresponding was taken as the inelastic mean-free path 1 and calculated according to the TPP-2M formula (Powell et al., 1999) using the NIST Standard Reference Database implemented in the program IMFPWIN (NIST, 2011). The FOVL calculations assumed f = b and therefore VðÞ = sin . Fig. 1(b) shows the R() and d() curves of the X-ray database values, as well as the I PE () curves calculated for the FA and FOVL regimes. We will now discuss the general trends seen in this figure.
(i) The region of less grazing away from c . Simple geometrical considerations give here d() / sin . Furthermore, d() >> and R() ' 0. This simplifies the formula (5) to I PE ðÞ / V ð Þ= sin . For the FA regime, VðÞ = 1 and the remaining 1= sin dependence reflects the gradual increase, when going to more grazing angles, of the power absorbed in   the surface region where the photoelectrons are coming from. For the FOVL regime, the V ð Þ / sin factor reflecting the analyzer FOV overfilling compensates this trend to constant I PE ðÞ, in agreement with Henke's results (Henke, 1972). Therefore, the intensity gain with more grazing can be achieved in this region only under the FA experimental conditions.
(ii) The region near c . Here d() in (5) sharply reduces when going to more grazing angles to dramatically increase I PE . The counter-trend is the increase of R to reduce the total absorption and thus I PE . These opposite trends form the I PE peak identifying the optimal incidence angle opt . In the FA regime the intensity gain is dramatic. In the FOVL regime the factor V ð Þ / sin moderates the gain to a factor of $ 1.5, in agreement with the previous theoretical and experimental results (Henke, 1972;Kawai et al., 1995), and slightly shifts opt to less grazing angles. In the following we will concentrate on the FA regime, as it is more effective and relevant for modern synchrotron instrumentation.
To assess the universality of the above picture, the calculations were extended to another two paradigm materials, the semiconductor GaAs and strongly correlated material Sr 2 RuO 4 , and to two very different h values, 50 eV and 1500 eV. The results are shown in Fig. 2. As we have seen above, the R() and d() angular dependences combined with form the pronounced I PE peak at opt near c . In the lowenergy case the peak appears at less grazing ; it is broad and less pronounced compared with I PE at 45 . With increase of h the peak becomes more grazing, dramatically scales up in amplitude, and sharpens. These general trends appear to be fairly material-independent.

Optimal X-ray incidence angle
The above calculations were extended to determine the opt angle maximizing I PE (see Fig. 1b) in a wide energy range from VUV photons of h = 50 eV to hard X-rays of 3.5 keV. Fig. 3(a) shows the central result of our evaluation: the calculated energy dependences of opt for our three paradigm materials. The dependences drop from less grazing values of the order of 20 at the 50 eV end to very grazing values of about 1 at the 3.5 keV end. The physics of this behavior becomes clear from Fig. 4, which presents the h dependences of d taken at = 45 , far away from c (note its sharp drop at the transition-metal 2p absorption edges), compared with the energy dependences of . The increase of d through the shown energy range is about an order of magnitude stronger than that of . The need to concentrate the absorbed X-ray power in a better balance with forces opt to become more grazing with h. Working in the same direction is also the evolution of R(), whose onset shifts with h towards more grazing angles. Returning to Fig. 3(b), we also note that the intensity gain achieved at opt increases dramatically with h. Fig. 4 also shows the h dependences of d at opt . Due to opt becoming progressively more grazing, d( opt ) flattens compared with d at 45 . It is interesting to note that, somewhat counterintuitively, the normal-emission I PE maximum at opt in general does not balance d and . This manifests the modulating effect of R() which reduces the absorbed X-ray power towards more grazing . The balance between d and improves, however, at the high-energy end. In this case d starts to affect the probing depth of the XPS experiment on equal footing with .

Practical considerations
The above gain in I PE of a few tens of times is huge. However, it requires extreme grazing angles, and can only be fully realised in the FA regime, i.e. while the analyser FOV intercepts a whole light spot that blows up proportionally to (sin ) À1 . Below we give some simple considerations to maintain or at least stay close to this regime at grazing angles.
(i) The experimental geometry of the synchrotron-based XPS facilities should take into account the elliptical cross section of the incident beam with its relatively small vertical size b V and large horizontal size b H . This means that the sample should be taken to grazing incidence by rotation around the horizontal axis to increase the smaller b V rather than the larger b H . In this case the measurement plane (MP) formed by the incident beam and analyzer lens axis (see Fig.  1a) is vertical. This geometry has been implemented, for example, at the highly efficient soft X-ray ARPES facility at the ADRESS beamline (Strocov et al., 2010) of the Swiss Light Source.
(ii) The analyzer FOV is determined by the operation mode of the analyzer lens (Må rtensson et al., 1994;Wannberg, 2009) and on the opening and orientation of the analyzer slit. For the magnification (essentially imaging) modes the FOV is just the slit dimension s divided by the lens magnification M, f = s/M. Obviously operation at grazing angles will benefit from low magnifications. Furthermore, the analyzer slit should be oriented in the MP, because in this case s will be determined by the relatively large slit length (usually around 20 mm) compared with its relatively small width (around 200 mm). With the modern synchrotron sources and vertical MP geometry, the FA regime is in this case practically unlimited in grazing angles. The transmission lens modes normally deliver even larger FOVs compared with the magnification modes, but its determination does not obey the simple imaging considerations because the photoelectrons collected at the same point at the slit can originate from different points on the sample. The most restrictive on the spot size are the angleresolving (ARPES) operation modes, because the best angular resolution is ensured within a FOV of the order of only 100 mm.
(iii) The XPS analyzers are normally mounted fixed at one of the flanges of the vacuum chamber. The angle between the analyzer lens and the incident light therefore stays fixed. In this case it is reasonable to optimize near the low-energy end of the required h range because at higher energies will sit on the more gradual right-hand side of the I PE peak (see Fig. 2). With the soft X-ray energy range starting at $ 300 eV, this yields opt ' 8 (see Fig. 3a). For the ARPES measurements with f ' 100 mm, the FA regime will then require b V = f/sin opt ' 14 mm, which is hardly a problem for current synchrotron instrumentation. With the hard-X-ray energy range starting at $ 2.5 keV, we arrive at opt ' 1.3 . For the ARPES measurements in this region (see, for example, Gray et al., 2011) the FA regime will be limited by b V ' 2.3 mm, which already requires aggressive focusing of the incident beam.
We also note that an additional advantage of a grazing is a reduction of the inelastic secondary electron background, because the concomitant decrease of d reduces the secondary electron background originating from photoelectrons excited in the sample depth beyond and inelastically scattered on their way to the surface (see, for example, Kawai et al., 1995).

Conclusion
We have analysed the interplay between the X-ray reflectivity, X-ray absorption depth and the photoelectron attenuation length in the photoelectron emission process. With increase of research papers 520 Vladimir N. Strocov X-ray incidence angle in photoelectron spectrometers J. Synchrotron Rad. (2013). 20, 517-521 Figure 3 (a) Energy dependences of the opt optimal angle. (b) Corresponding I PE intensity gain (normalized to I PE at 45 ) in the FA regime. opt becomes more grazing with increasing energy, accompanied by a dramatic increase of the I PE gain.

Figure 4
Energy dependences of d at = 45 (solid lines) compared with (dashed lines). The penetration of X-rays increasing with energy faster than that of photoelectrons forces the decrease of opt in Fig. 3. Also shown is d at the energy-dependent opt (dotted lines). energy from the VUV to hard X-rays, the optimal X-ray incidence angle opt delivering maximal XPS signal becomes progressively more grazing, from a few tens of degrees to about 1 . This is accompanied by an intensity gain at opt , increasing from insignificant to a factor of a few tens as long as the experiment stays in the FA regime with the whole X-ray footprint on the sample intercepted within the analyzer FOV. These trends are fairly material-independent. The practical utilization of the intensity gain at opt by the (synchrotronbased) XPS spectrometers in general requires a vertical measurement plane with in-plane analyzer slit orientation and, particularly for ARPES experiments towards the hard X-ray energies, focusing of the incident X-ray beam down to a few micrometers.