Keywords

11.1 Introduction

With bulk sensitivity being an important characteristic of HAXPES, another useful aspect of this technique appears to be the very pronounced dependence of the spectra on the polarization of the light. This can in principle be used to identify the character of the orbitals contributing to the valence band [1–4]. In fact, if this can be made quantitative, one can obtain a very detailed understanding of the electronic structure of the material under study, especially when guided by theoretical ab initio calculations. The polarization dependent HAXPES experiments reported so far [1–4] made use of phase retarders to rotate the polarization of the light. The efficiency of the retarders to generate vertically polarized light, however, is about 0.8 only [1–4]. This hampers a reliable determination of the limits of polarization dependent HAXPES due to the inaccuracies in the characterization of the efficiency of the phase retarders.

In our HAXPES project we followed a different route: instead of rotating the polarization of the light, we altered the position of the electron energy analyzer, i.e. we used two different experimental geometries , one in which the analyzer is placed horizontally in the direction of the linear polarization of the light and the other vertically and thus perpendicular to the light polarization (Fig. 11.1). This set-up has the advantage that all the spectra can be taken with the full light polarization provided by the undulator beam line. We have carried out a test experiment on NiO as a model system for strongly correlated oxides. We can identify clearly the Ni 3d spectral weight of the valence band and at the same time demonstrate the importance of the Ni 4s for the chemical stability of the compound. We have also quantified the limitations of the polarization analysis: the polarization dependence is less than expected on the basis of calculations for free atoms. We ascribe this incompleteness of the polarization dependence to the presence of appreciable side-scattering effects of the outgoing electrons, even at these high kinetic energies in the 6–8 keV range [5].

Fig. 11.1
figure 1

Experimental geometry for different angles θ between the photon beam (red, striped) electrical field vector E and the momentum of the analyzed photoelectrons (blue). E is always horizontal, in the plane of the synchrotron storage ring, the photoelectron analyzer is always mounted perpendicular to the photon beam. Top Analyzer mounted vertically, θ = 90°. Bottom Analyzer mounted horizontally, θ = 0°

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11.2 Experimental

The experiments have been carried out at the recently installed Max-Planck-NSRRC HAXPES set-up at the Taiwan undulator beamline BL12XU at SPring-8, Japan. The photon beam is linearly polarized with the electrical field vector in the plane of the storage ring (i.e. horizontal). The photon energy is about 6.5 keV. The beam is monochromatized in two stages: a diamond crystal diverts a photon beam with about 1 eV bandwidth to the HAXPES side branch and a channel-cut monochromator is introduced to produce the desired photon resolution. Several silicon single crystal channel-cut monochromators are available, allowing to select the photon resolution within the range of 75–220 meV. A pair of K-B mirrors focuses the beam on the sample position, achieving a spot size at the sample of \(40\,\upmu{\text{m}} \times 40\,\upmu{\text{m}}\). The ultra-high vacuum experimental chamber is mounted on a motorized table which allows to easily and precisely align the setup. Also the sample manipulator is fully motorized. The sample is mounted on a continuous-flow cryostat which allows to use liquid helium for cooling to 10 K.

An MB Scientific A-1 HE analyzer was used for the measurements. In the horizontal geometry, the analyzer was mounted parallel to the photon beam’s electrical field vector and perpendicular to the Poynting vector of the light. The maximum angular acceptance was limited to ±15.3° (30.5° total acceptance) by the circular opening of the first lens element. The photon energy and the overall energy resolution was determined using a gold reference sample. The Fermi level was at 6474.4 eV kinetic energy and the overall energy resolution was set to 390 meV. For the measurements in the vertical geometry, the spectrometer was mounted perpendicular to the photon beam and to its electrical field vector. The maximum angular acceptance was limited to ±8.3° (16.6° total acceptance) by the circular opening of the first lens element. A silver reference sample was used to determine the photon energy and energy resolution. The Fermi level was at 6482.4 eV kinetic energy and the overall energy resolution was set to 350 meV.

For both geometries, an analyzer entrance slit/aperture set was used which limited the angular acceptance in the energy dispersive direction to ±1.65°. Applying the Helmholtz-Lagrange relation [6], this corresponds to an acceptance of about ±3.5° at the lens entrance. The acceptance angle in the direction parallel to the analyzer entrance slit (i.e. perpendicular to the energy dispersive direction) is determined by the circular opening of the first element as specified above. A NiO single crystal from SurfaceNet, Germany, was used (vertical geometry: as introduced, horizontal: cleaved in situ). It was aligned in a grazing incidence and near normal emission geometry. The temperature was 300 K, the pressure in the measurement chamber was \(2\mbox{-}3 \times 10^{ - 8} \,{\text{mbar}}\).

We have also performed X-ray photoemission (XPS) measurements for the NiO valence band using a Scienta SES-100 electron energy analyzer and a Vacuum Generators twin crystal monochromatized Al-\(K_{\alpha }\) \(h\nu = 1486.6\,{\text{eV}}\) source. The overall energy resolution was set to 350 meV, as determined using the Fermi cutoff of a silver reference. The pressure in this spectrometer was \(2 \times 10^{ - 10} \,{\text{mbar}}\) during the measurements, and the NiO single crystal was cleaved in situ to obtain a clean surface.

11.3 Core Levels

Figure 11.2 displays the HAXPES Ni 2p and O 1s core level spectra of NiO taken with the photoelectron momentum parallel (red curves) and perpendicular (blue curves) to the polarization vector of the light. The parallel versus perpendicular spectra are normalized with respect to the Ni 2p main peak intensity. No corrections have been made to the spectra apart from a constant background subtraction. The line shape of the Ni 2p spectra is essentially identical to the ones published using also lower photon energies [7–12]. The O 1s spectrum shows a narrow single line demonstrating that the NiO sample is clean and of good quality. The relevant information that is contained in Fig. 11.2 is that the O 1s intensity is much lower for the perpendicular (blue curve) than for the parallel (red curve) polarization. This is qualitatively in agreement with the observations for s-orbitals in experiments in which the polarization has been varied using a phase retarder [1–4].

Fig. 11.2
figure 2

HAXPES (photon energy 6.5 keV) spectra of NiO Ni 2p and O 1s core levels with θ = 90° (vertical) and θ = 0° (horizontal)

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Making use of the fact that the degree of the photon polarization is identical for the two experimental geometries, we now can be quantitative concerning the physics underlying the change of the O 1s intensity with the photoelectron momentum. The expression for the angular dependence for the differential photoionization cross section is given by Trzhaskovskaya et al. [13, 14]:

$$\frac{{{\text{d}}\sigma }}{{{\text{d}}\varOmega }} = \frac{\sigma }{4\pi }\left[ {1 + \beta P_{2} (\cos \theta ) + (\gamma \mathop {\cos }\nolimits^{2} \theta + \delta )\sin \theta \cos \varphi } \right]$$
(11.1)

where σ is the subshell photoionization cross section and \(P_{2}\) the second Legendre polynomial. θ is the angle between the photoelectron momentum and the polarization vector of the light, φ the angle between the photon momentum vector and the plane spanned by the photoelectron momentum vector and the electrical field vector. \(\beta\), \(\gamma\), and \(\delta\) are the angular distribution parameters . In our setup, we have for the horizontal analyzer θ = 0° (φ undefined) and for the vertical analyzer θ = 90° and φ = 90°. The expression  (11.1) now simplifies to

$$\frac{{{\text{d}}\sigma }}{{{\text{d}}\varOmega }} = \frac{\sigma }{4\pi }\left\{ {1 + \beta P_{2} (\cos \theta )} \right\}$$
(11.2)

or

$$\frac{{{\text{d}}\sigma }}{{{\text{d}}\varOmega }} = \frac{\sigma }{4\pi }\left\{ {1 + \frac{\beta }{2}(3\mathop {\cos }\nolimits^{2} \theta - 1)} \right\} = \frac{\sigma }{4\pi }\left\{ {1 + \beta \left( {\frac{1}{4} + \frac{3}{4}\cos 2\theta } \right)} \right\}.$$
(11.3)

A polar plot of this angular dependence is shown in Fig. 11.3 for several values of β.

Fig. 11.3
figure 3

Polar plot of the cross section angular dependence for various values of β

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Ideally, in the atomic limit, β is 2 for s orbitals, so that for the perpendicular orientation, i.e. θ = 90°, the intensity for the O 1s spectrum is expected to be zero. This is clearly not the case. Normalized to the Ni 2p intensities, the intensity ratio of the O 1s signal taken with perpendicular and parallel orientation is 0.16. Cross-section calculations for atomic orbitals provide a β value of about 0.98 for Ni 2p at \(h\nu = 6.5\,{\text{keV}}\) [13, 14] (angular distribution parameters are interpolated from tabulated values), so that the Ni 2p intensity ratio for vertical versus horizontal orientation should be close to 0.258 using (11.3). The real experimental O 1s intensity ratio is therefore 0.16 × 0.258 = 0.041. This is substantially larger than the expected value of 0.008 using (11.3) for the interpolated β value of about 1.952 for O 1s at \(h\nu = 6.5\,{\text{keV}}\) [13, 14].

To identify the origin of this discrepancy we will first investigate the effect of the acceptance angle of the electron energy analyzers. We first of all note that the third term in the sum in (11.1) still can be omitted for the two geometries used in our experiment when we consider the effect of the acceptance angles: for the horizontal geometry, the acceptance angle in both the analyzer entrance slit and energy dispersive directions enters as variations of the θ angle symmetrically around θ = 0, and will not integrate a non-zero value for the third term since sin θ is an odd function of θ. For the vertical geometry, the acceptance angle in the analyzer entrance slit direction enters as variations in the φ angle symmetrically around φ = 90°. Since cos φ is an odd function around φ = 90°, the third term will not show up. The acceptance angle in the energy dispersive direction then also becomes irrelevant. We still can therefore use the simpler (11.3).

In order to calculate the cross-section average over the acceptance angle, we integrate (11.3) from θ − α to θ + α and divide by 2α, where ±α denotes the acceptance angle. We arrive at:

$$\frac{{{\text{d}}\sigma }}{{{\text{d}}\varOmega }} = \frac{\sigma }{4\pi }\left\{ {1 + \beta \left( {\frac{1}{4} + \frac{3}{4}\cos 2\theta \frac{\sin 2\alpha }{2\alpha }} \right)} \right\}$$
(11.4)

The relevant acceptance angle for the horizontal geometry is determined essentially by the acceptance along the analyzer entrance slit direction, i.e. ±15.3°, since this is much larger than the acceptance along the energy dispersive direction, i.e. ±3.5°. For the vertical geometry on the other hand, the acceptance along the analyzer entrance slit direction does not alter θ, so the relevant number here is the acceptance along the energy dispersive direction, i.e. ±3.5°. Using these numbers and (11.4), we have a theoretical Ni 2p intensity ratio for vertical versus horizontal orientation of 0.264 and 0.01 for the O 1s. The real experimental O 1s intensity ratio is 0.16 × 0.264 = 0.042. Clearly, this is much larger than expected on the basis of the cross-section tables including the effect of the analyzer acceptance angles. In fact, in order to obtain a value of 0.042 we need for the analyzer in the vertical geometry acceptance angles of the order of ±18°, which is physically not possible with the ±8.3° acceptance dictated by the opening of the first lens. Apparently, we have to conclude that the acceptance angles of the analyzer do not form the limiting factor for the suppression of the O 1s signal in the vertical orientation.

11.4 Valence Band

In order to investigate to what extent these findings also affect HAXPES experiments on the valence band and especially the quantitative analysis of their polarization dependence, we also set out to collect the valence band spectrum of NiO. Figure 11.4 displays the HAXPES spectrum taken with the vertical and horizontal geometries, together with the spectrum taken with the standard XPS [15] using monochromatized \(h\nu = 1486.6\,{\text{eV}}\) photons, as reference.

Fig. 11.4
figure 4

XPS [15] (photon energy 1.487 keV) and HAXPES (6.5 keV) spectra of NiO valence band with θ = 90° (vertical) and θ = 0° (horizontal). LDA total and partial density of states from FPLO band structure calculations

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Starting with the XPS, we would like to note that this spectrum is very similar to the ones reported earlier in the literature [16–18] and that it shows the characteristic features labeled as A, B, C, and D, which are all related to the multiplet structure of the Ni 3d spectral weight [16–18]. The HAXPES spectra, interestingly, have different lineshapes with clear extra features labeled S1 and S2. In the vertical geometry, the intensities of these extra features are modest, but in the horizontal geometry, the features dominate the spectrum.

To identify the origin of the S1 and S2 structures, we have performed LDA band structure calculations using the full-potential code with the basis set of local atomic-like orbitals (FPLO) [19]. The resulting total and partial densities of states (DOS) are depicted in the bottom panel of Fig. 11.4. Of particular interest is the Ni 4s partial DOS. We can clearly observe that the peak positions of the Ni 4s and O 2p bands coincide very well with structures S1 and S2. Indeed, looking at the tables [13, 14], the ratio of the subshell photoionization cross section of the Ni 4s relative to that of the Ni 3d increases from 0.12 at \(h\nu = 1486.6\,{\text{eV}}\), to 0.99 at \(h\nu = 5\,{\text{k}}{\text{eV}}\), and to 3.61 at \(h\nu = 10\,{\text{keV}}\), meaning that the contribution of the Ni 4s to the spectrum is negligible in XPS and that it becomes significant in the HAXPES experiment. The finding that HAXPES is much more sensitive to valence s orbitals is consistent with earlier reports and has been used to identify the partial density of states of the s orbital in semiconductors and metals [20–25].

This result in fact shows that HAXPES unveils an important part of the chemical bonding between the Ni and the O in the formation of NiO, namely the hybridization between the Ni 4s conduction band and the O 2p valence band. This provides an extra energy gain on top of the bonding energy due to the Ni 3d and O 2p hybridization. This Ni 4s − O 2p hybridization is analogous to that of Mg 3s and O 2p in MgO [26, 27]. We would like to note that it is rather surprising that the LDA calculation can explain so well the experimentally observed structure S1. It is well known that due to the strongly correlated nature of the Ni 3d electrons the description of the electronic structure by theory is still a great challenge. Features A, B, C, and D require an explanation that includes at least aspects of configuration interaction and atomic multiplet effects [16–18]. The Ni 3d states and their hybridization with the O 2p will therefore result in an O 2p spectral weight that may be significantly different from the LDA O 2p partial DOS. The Ni 4s that couples to this O 2p may consequently show spectral weight at quite different energies than predicted by the LDA. It is therefore invaluable that the Ni 4s states can indeed be made visible experimentally. From the fact that the LDA does give a reasonable match to the experiment for the Ni 4s we can apparently deduce that the O 2p states that hybridize with the Ni 4s have a rather weak mixing with the Ni 3d. In hindsight, we may indeed see a justification for this in the form of a rather small Ni 3d and very high O 2p partial DOS in the LDA band that form the S1 structure.

We now analyze quantitatively the polarization dependence of the Ni 4s contribution to the spectrum. With the spectra normalized to the Ni 3d peak height at 2 eV binding energy, i.e. peak A, in the experiment and normalized to the Ni 3d peak height at 1 eV binding energy in the calculation, we can make an estimate for the relative contribution of the Ni 4s to the spectra. After broadening the calculated DOS with a 0.35 eV Gaussian to account for the experimental resolution and 1.9 eV Lorentzian for the life time (full width at half maximum values), we find for the vertical geometry that a multiplication by a factor of 10 of the Ni 4s DOS gives a reasonable match for the intensity of structure S1 and S2. For the horizontal geometry, we need a multiplication factor of approximately 110. From the cross section tables we expect the Ni 3d to have an intensity ratio for vertical versus horizontal orientation of about 0.58. Here we make use of (11.4), with β = 0.39 for Ni 3d at \(h\nu = 6.5\,{\text{keV}}\) [13, 14]. The effect of the analyzer acceptance angle is included. We then find experimentally that the ratio for the Ni 4s polarization dependence is given by 10/110 × 0.58 = 0.053. This is a very large contrast which demonstrates the power of polarization dependence to identify the contribution of s orbitals to the valence band. Using the polarization dependence rather than the photon energy dependence of the photoionization cross section has the advantage that one does not also vary the probing depth as is the case when comparing HAXPES with XPS.

Yet, on the basis of atomic cross-section calculation, we expect even a larger contrast, namely a Ni 4s intensity ratio of about 0.004 with β = 1.984 for Ni 4s [13, 14]. The reduction of the Ni 4s is thus much less than can be expected on the basis of the atomic calculations including the effect of the analyzer acceptance angles. This finding is consistent with the above described observations for the O 1s. Again, in order to obtain the value of 0.053 we need somehow effective acceptance angles of the order of ±18° for the measurement, and this is clearly not related to the instrumental conditions. A possible explanation for this non-perfect polarization dependence can perhaps be found in the (quasi-elastic) scattering processes which take place following the creation of the photoelectron [28–31]. Although forward scattering will be dominant at high kinetic energies, a non-negligible amount of side scattering processes will make it possible to change the propagation direction of part of the photoelectrons such that they enter the analyzer which they otherwise will not be able to do. The effective acceptance angle of the order of ±18° that we would need, as mentioned above, reflects this process. We note that experiments using 700 eV photons on VO2 reveal only a weak polarization dependence for the O 1s [32] as shown in Fig. 11.5, supporting the explanation on the basis of such scattering processes: the kinetic energy of the photoelectrons (ca. 170 eV) is so low, that side or even back scattering events dominate, leading to an almost complete loss of the relationship between the propagation direction of the photoelectron upon creation and the direction of the photoelectron when it enters the analyzer.

Fig. 11.5
figure 5

PES (photon energy 700 eV) spectra of VO2 core levels with θ = 90° (vertical) and θ = 0° (horizontal). The suppression of the O 1s is much weaker than at higher energies

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11.5 Conclusions

Although the polarization dependence in the HAXPES experiments is not as much as we have expected on the basis of atomic cross-section calculations, i.e. a reduction to 0.042 or 0.053 instead of 0.01 or 0.004, we nevertheless can state that it is extremely large: the reduction is from 1.00 to 0.042 or 0.053. In other words, we can achieve a contrast of about 20:1 for s orbitals. This is more than sufficient for determining the character of the orbitals forming the valence band of solids as we have shown for the case of NiO. Based on the results of these experiments, we now have modified our HAXPES set-up such that two photoelectron energy analyzers are mounted simultaneously: one analyzer is placed horizontally (in the direction of the linear polarization of the light) and the other vertically (and thus perpendicular to the light polarization). Figure 11.6 shows a photograph of this new set-up. Although a simultaneous measurement of both polarizations is even possible, our aim is more to facilitate high-quality polarization dependence experiments without the need to interrupt the measurement to rotate the set-up, which usually is accompanied by stopping the vacuum pumps or even breaking the vacuum.

Fig. 11.6
figure 6

HAXPES endstation at SPring-8 Taiwan Beamline 12XU sideline

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