# Intrinsic and extrinsic plasmons in the hard x-ray photoelectron spectra of
nearly free electron metals

Mohammad Balal1, Shuvam Sarkar1, Pramod Bhakuni1,
Andrei Gloskovskii2, Aparna Chakrabarti3,4, Sudipta Roy Barman1∗ 1UGC-DAE
Consortium for Scientific Research, Khandwa Road, Indore 452001, India
2Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg,
Germany  3Raja Ramanna Centre for Advanced Technology, Indore 452013, Madhya
Pradesh, India  4Homi Bhabha National Institute, Training School Complex,
Anushakti Nagar, Mumbai 400094, Maharashtra, India

###### Abstract

Collective plasmon excitations in solids that result from the process of
photoemission are an important area of fundamental research. In this study, we
identify a significant number ($n$) of multiple bulk plasmons
($n$$\omega_{p}$) in the hard x-ray photoelectron spectra of the core levels
and valence bands (VBs) of two well-known, nearly free electron metals,
aluminum (Al) and magnesium (Mg). On the basis of earlier theoretical works,
we estimate the contributions of extrinsic, intrinsic, and interference
processes to the intensities of 1$s$ to 2$s$ core level plasmons. The
intrinsic contribution diminishes from 22% for 1$\omega_{p}$ to 4.4% for
2$\omega_{p}$, and becomes negligible thereafter (0.5% for 3$\omega_{p}$). The
extrinsic and intrinsic plasmon contributions do not vary significantly across
a broad range of photoelectron kinetic energies, and also between the two
metals (Al and Mg). The interference contribution varies from negative to zero
as $n$ increases. An asymmetric line shape is observed for the bulk plasmons,
which is most pronounced for 1$\omega_{p}$. Signature of the surface plasmon
is detected in normal emission, and it exhibits a significantly increased
intensity in the grazing emission. The VB spectra of Al and Mg, which are
dominated by $s$-like states, exhibit excellent agreement with the calculated
VB based on density functional theory. The VB exhibits four multiple bulk
plasmon peaks in the loss region, which are influenced by an intrinsic process
in addition to the extrinsic process. On a completely oxidized aluminum
surface, the relative intensity of the Al metal bulk plasmon remains nearly
unaltered, while the surface plasmon is completely attenuated.

##  I Introduction

The investigation of plasmon excitations in photoelectron spectra of free
electron metals has been a central focus of fundamental research for over four
decades, with ongoing theoretical [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22] and experimental [23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37] investigations. These investigations have
provided valuable insights into the physics of plasmons in the nearly free
electron metals. Furthermore, continuous interest in this field highlights its
relevance and potential for further advancements in our understanding of
collective electronic excitations and the development of various future
applications, such as plasmonics. The bulk and surface plasmons – observed as
loss features in the core level photoelectron spectra – are dominated by an
extrinsic process that results from the Coulomb interaction of the conduction
electrons with the photoelectron traversing through the solid from the
photoemission site to the surface. An additional contribution to the plasmon
intensity comes from an intrinsic process, where the sudden change in the
potential due to the formation of a localized core hole attracts the
conduction electrons to screen it, resulting in a plasmon oscillation. From
the variation of successive multiple bulk plasmon ($n$$\omega_{p}$) intensity
in Al 2$s$ x-ray photoemission spectrscopy (XPS), the intrinsic plasmon was
estimated to be 14-25% of the total intensity [30, 33]. Furthermore, an
interference [11, 12, 22, 30, 13, 25] between the intrinsic and extrinsic
processes, where virtual plasmons created by one are absorbed by the other,
plays an important role in determining the shape of the plasmons. Hedin [8]
pointed out that the interference term should decrease as a function of the
photoelectron kinetic energy. The coupling of the Al surface plasmon to the
oxygen adsorbed on its surface was observed in the O 1$s$ core level spectra
in XPS [35]. Biswas et al. [25] determined the relative contributions of the
intrinsic, extrinsic, and interference processes in the surface plasmon from
the XPS core level spectra of Al metal. The loss region has been studied for
Al and carbon by XPS to identify both the one electron interband transitions
and plasmon excitations [23, 26]. Bulk plasmons have been observed in the XPS
valence band (VB) spectra of Al [30] and Mg up to $n$= 2-3 [31, 34], but their
intensities were affected by presence of KLV Auger electron spectroscopy
signal and x-ray satellites. The mechanism of instrinsic plasmon production,
where, unlike the core level, the photo-hole is delocalized, was related to
many-body effects and electron-electron interaction by theory [14, 13].

Hard x-ray photoelectron spectroscopy (HAXPES) has become a vital method to
study the bulk electronic structure of materials because of its large
inelastic mean free path (8-10 nm) [38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51]. Previous research has investigated this phenomenon for a
limited number of elements, including carbon [52], silicon [53], germanium
[54], phosphorous [55], and Al films grown on Si [56]. However, the
investigation of plasmon excitation in nearly free electron metals remains a
captivating subject due to the substantial theoretical research that has been
undertaken in this area previously [1, 5, 6, 3, 4, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 57, 22]. Notwithstanding this, theoretical
investigations of plasmons in free electron metals in the HAXPES regime are
scarce. Shinotsuka et al. [3] performed a full quantum mechanical calculation,
taking the interference term into account, using the model suggested by Hedin
et al. [10]. Fujikawa et al. have attempted to address this issue through the
application of the quantum Landau formula, which accounts for elastic
scatterings both prior to and subsequent to the loss [4]. These works show
that in the HAXPES regime, the extrinsic contribution is dominant, although
both the positive intrinsic and the negative interference terms are present.
The interference and intrinsic contributions decrease very slowly as a
function of the photoelectron energy and are therefore significant even at a
large photon energy of 10 keV. Even the surface plasmon exhibits a peak of
significant intensity at this photon energy [4]. The other interesting
theoretical result [3] is that the loss features hardly exhibit any angular
dependence i.e., the grazing versus the normal emission spectra are nearly
similar. This behavior is different compared to the low photon energy XPS
regime [25, 7, 11, 12, 3]. The above-discussed intriguing characteristics of
the plasmons, as postulated by theory [4, 3], have yet to be empirically
investigated.

HAXPES study of the core-level and VB plasmons in free electron metals is
scarce in previous literature,  despite the fact that this technique offers
unique advantages over XPS due to its deeper probing depths.  Another benefit
of HAXPES over XPS is that the former does not experience interference from
the Auger electron spectroscopy signal. A recent study on Al thin films on Si
and that of fractured Al metal reported Al 1$s$ related plasmons [56].
However, the Si(111) surface was naturally oxidized and the ex-situ introduced
Al films were inhomogeneous and showed a large Al-oxide component in the Al
1$s$ spectrum. Thus, although bulk plasmons for clean bulk Al were observed,
the thin film related Al plasmons were affected by presence of oxide and Si
related loss features.

Here, we present a detailed HAXPES study of the plasmon excitations in the VB
and core level spectra for clean Al and Mg bulk single crystals. A large
number of multiple bulk plasmons ($n$$\omega_{p}$) up to $n$= 13 is observed,
in addition to a relatively weaker surface plasmon ($\omega_{s}$). We conduct
exhaustive multi-parameter curve fitting considering the plasmons to ascertain
the variation of their intensity and shape. From the intensity variation, the
intrinsic, extrinsic, and interference contributions to $n$$\omega_{p}$ are
determined based on prior theoretical works [19, 22, 21, 57, 14]. Normal and
grazing emission properties of the bulk and surface plasmons have been
investigated and compared with an oxidized Al surface. In addition, our work
shows the existence of the intrinsic process in the VB related bulk plasmons.

##  II Methods

The experiments were carried out at the P22 beamline in PETRA III, Deutsches
Elektronen-Synchrotron, Hamburg, Germany. A post-monochromator was used to
improve the resolution and stability of the photon beam. All the spectra were
measured using 6 keV photon energy. The electron energy analyzer has an
angular acceptance angle of $\pm$15∘. The overall energy resolution, including
the source and the analyzer contribution, was set at 0.3-0.4 eV, as measured
from the Au Fermi edge that was in electrical contact with the specimens. The
details of the beamline and the endstation can be found in Ref. 58. The
measurements were carried out at $\theta$= 85∘ (referred henceforth as normal
emission) and 10∘ (grazing emission), where $\theta$ is the angle between the
analyzer axis and sample surface.

Polished and oriented Al single crystals were cleaned in-situ by sputtering
with Ar+ ions of energy 3-5 keV for the core-level studies. In the case of the
VB and some core-level studies, the Al and Mg crystals were mechanically
scraped in-situ with a diamond file. The binding energy ($E$) scale of both
the core level and valence band spectra has been defined from the Fermi level
position of Al/Mg metal (rather than from Au) because of the recoil shift [46,
40]; this is discussed later in subsection IIIC. The HAXPES core level main
peaks have been fitted using the least squares error method, where the
Doniach-Šunjić (DS) line shape represents the main peak [59]. Asymmetric
Lorentzian line shapes have been used to represent the plasmon loss peaks [25,
30], as also suggested by the theoretical 1$\omega_{p}$ and 1$\omega_{s}$ line
shapes from Ref. 3 shown in Fig. S1 of Supplementary Material (SM) [60]. The
lifetime broadening of the core level main peak, the DS asymmetry parameter,
intensities, peak positions, and the inelastic background have been varied.
The half width at half maximum of the life-time broadening and the DS
asymmetry parameter for Al (Mg) 2$s$, determined from our fitting, are 0.4
(0.32) eV and 0.1 (0.12), respectively. Each of the plasmon peaks has been
simulated by four parameters: intensity, position, and an asymmetric
Lorentzian [61] with independently varying left ($\Gamma_{L}$) and right
($\Gamma_{R}$) half width at half maximum. Besides the $n$$\omega_{p}$ peaks,
2$\omega_{s}$, 1$\omega_{p}$+1$\omega_{s}$, and 2$\omega_{p}$+1$\omega_{s}$
surface plasmons were also considered for Al 1$s$. The inelastic background
was simulated by the Tougaard method [62, 63], whereas use of the Shirley
method resulted in unphysically large value of the intrinsic plasmon
probability. As in our earlier works [25, 64], all the parameters were not
varied simultaneously to avoid unphysical solutions: for example, the
parameters for plasmons with $n$$\geq$7 were varied individually. Thus, the
number of independently variable parameters for 1$s$ (2$s$) was 68 (39).

The calculation of the electronic properties of the bulk Al and Mg metals has
been performed using the all-electron full potential linear augmented
planewave (FPLAPW) method, including scalar relativistic corrections, by
employing the density functional theory (DFT) based WIEN2k[65] program
package. For the exchange-correlation functional, we consider the generalized
gradient approximation over the local density approximation, given by Perdew,
Burke, and Ernzerhof (PBE) [66]. Energy cut-off values of about 15 Ry have
been used in the calculations. The charge density cut-off is 14. Further, we
have used 10 as the maximum value of angular momentum for the ($l,m$)
expansion of wave function and density. The convergence criteria for energy
and charge have been taken to be 10-5 Ry and 0.001 e-1, respectively. The
total density of states (DOS) is calculated from the Kohn-Sham orbitals, and
its integration up to the Fermi level gives the total number of valence
electrons. The partial DOS (PDOS) is obtained by projecting the total DOS for
each orbital ($s$, $p$, $d$). For Al and Mg with delocalized states, touching
muffin-tin sphere was taken to minimize the discrepancy between the total DOS
and the sum of all the PDOS.

To calculate the theoretical VB, the angular momentum projected PDOS was
multiplied by the corresponding photoemission cross sections per electron
calculated for 6 keV photon energy by Trzhaskovskaya et al. [67]. In the case
where an energy level is not occupied and so the cross section is not
calculated, the closest value is assumed, as in our previous work [50, 68].
For example, for Mg $p$ PDOS, the cross-section is taken to be the same as
that of Al $p$. Furthermore, the partial contributions are added and
multiplied by the Fermi function and subsequently convoluted with a Gaussian
function representing the instrumental resolution. Additionally, an energy
dependent Lorentzian broadening given by $\mu\times$$E$ has been used, where
$E$ is the binding energy and $\mu$ is taken to be 0.15 eV [69, 70].

Figure 1: (a) Al 1$s$ core level spectrum (red open circles) of Al taken at
normal emission ($\theta$= 85∘) showing multiple bulk plasmon peaks
$n$$\omega_{p}$ ($n$= 1-13, black arrows). The loss energy scale (the top
horizontal axis) is defined with respect to the main peak. The latter is
truncated to show the plasmon peaks on an expanded scale. The spectrum for
$n$= 7 to 13 is shown separately in a magnified scale. The fitted curve
(black) and each of the $n$$\omega_{p}$ peaks are shown. (b) The normalized
intensity of the bulk plasmons ($b(n)$) fitted with Equation 1 that includes
all the terms i.e., extrinsic, intrinsic, and interference (blue curve) and
with only the extrinsic term (black dashed curve). The corresponding residuals
of the fitting are shown in the top panel in same line type. (c) $\Gamma_{R}$,
$\Gamma_{L}$, and (d) $\kappa$ (= $\Gamma_{L}$/$\Gamma_{R}$) as a function of
$n$.

##  III Results and Discussion

###  III.1 Multiple bulk plasmons in Aluminum core level spectra

Al 1$s$: The Al 1$s$ core level spectrum in Fig. 1(a) shows multiple bulk
plasmon peaks ($n$$\omega_{p}$) with $n$ as large as 13, as is evident from
the magnified spectrum. These plasmons appear on the higher binding energy
($E$) side i.e. the loss region of the main peak at 1559.3 eV. The first bulk
plasmon peak (1$\omega_{p}$) appears at $\omega_{p}$ loss energy, and the
subsequent $n$$\omega_{p}$ multiple plasmons appear at equal $\omega_{p}$
intervals. The positions of the plasmon peaks thus provide an average estimate
of $\omega_{p}$ to be 15.3 eV; a similar value is obtained from Al 2$s$
plasmons in Fig. 2(a). The reasons that large number of multiple bulk plasmons
could be observed are large photoemission cross-section of the $s$ states at
high photon energy (6 keV), the large flux of the photon beam, the $n$= 7-13
plasmon region recorded with larger dwell time, and the absence of any other
photoelectron or Auger peaks that could obfuscate the plasmons. The
1$\omega_{s}$ surface plasmon is also distinctly visible, as shown by the blue
arrow. The behavior of this surface plasmon will be discussed further in
subsection III B. Bulk plasmons up to $n$= 7 and the 1$\omega_{s}$ surface
plasmon the 1$s$ spectrum of Al metal were reported recently [56], but
quantitative analysis their shapes and intensities were not performed.

Here, based on rigorous multi parameter curve fitting, the normalized
intensities [$b(n)$, i.e., area under $n$$\omega_{p}$ divided by that of the
main peak] and shapes [left ($\Gamma_{L}$) and right ($\Gamma_{R}$) half width
at half maximum of an asymmetric Lorentzian [61]] have been determined.
Besides taking a realistic shape of the bulk plasmons justified by theory in
subsection III B, inclusion of weak surface plasmon contributions such as
2$\omega_{s}$(red arrow), 1$\omega_{p}$+1$\omega_{s}$ (black tick), and
2$\omega_{p}$+1$\omega_{s}$ (red tick), improves the quality of the fitting.
We find that $b(n)$ determined from the area of the plasmons obtained from
fitting (shaded with different colors) decreases with $n$, as shown by the red
open circles in Fig. 1(b). This variation can be explained by Equation 1, as
discussed later. Figure 1(c) portrays the asymmetric shape of $n$$\omega_{p}$,
where $\Gamma_{R}$ is smaller compared to $\Gamma_{L}$. Both show a nearly
similar increasing trend with $n$. The larger width of $(n+1)$$\omega_{p}$
compared to $n$$\omega_{p}$ is because the former is excited by the latter and
can be regarded as the self-convolution of the latter [37]. The asymmetry can
be quantified by the ratio $\Gamma_{L}$/$\Gamma_{R}$ (= $\kappa$), its
variation with $n$ is shown in Fig. 1(d).

Figure 2: (a) Al 2$s$ core level spectrum taken at normal emission showing
multiple bulk plasmon peaks $n$$\omega_{p}$ ($n$= 1-7, black arrows) and a
surface plasmon peak (1$\omega_{s}$, blue arrow). The black curve through the
data points (red open circles) and each of the $n$$\omega_{p}$ peaks (shaded
by different colors) are obtained from the least squares curve fitting. (b)
$b(n)$ fitted with Equation 1 (blue curve) and with only the extrinsic term
(black dashed curve), the respective residuals of the fitting are shown in the
top panel. (c) $\Gamma_{L}$, $\Gamma_{R}$, and (d) $\kappa$
(=$\Gamma_{L}$/$\Gamma_{R}$) as function of $n$.

Al 2$s$: Figure 2(a) shows the Al 2$s$ spectrum at normal emission, showing
multiple plasmon peaks $n$$\omega_{p}$ up to $n$= 7 (compared to $n$= 13 for
Al 1$s$) because of the relatively lesser photoemission cross section of 2$s$
compared to 1$s$. The intensity of the former is $\sim$9 times less, as shown
by the Al survey spectrum in Fig. S2 of SM [60]. For the same reason, although
1$\omega_{s}$ surface plasmon is also distinctly visible at 10.4 eV loss
energy (blue arrow), the multiple surface plasmons are not apparent. Another
reason for smaller $n$ is that Ar 2$p$ and 2$s$ peaks at $E$= 242.2 and 320.3
eV, respectively (Fig. S2 of SM [60]) interfere because a small amount of Ar
is implanted during the sputtering process [71, 72]. In particular, Ar 2$p$
peak coincides with the 8th plasmon, and the former has a larger intensity.
Additionally, the Al related plasmons excited by the Ar 2$p$ photoelectrons
[73] overlap with $n>$7 plasmons. $b(n)$ determined from multi parameter curve
fitting is shown in Figure 2(b). Figure 2(c) shows that the asymmetry of the
Al 2$s$ plasmons is qualitatively similar to that of the Al 1$s$ :
$\Gamma_{R}$ is smaller than $\Gamma_{L}$, and both increase with $n$. The
asymmetry in the plasmon line shape given by $\kappa$ decreases from about 2.7
($n$= 1) to about 1.8 for $n$$\geq$2-3.

Bulk plasmon intensity variation: The decrease of $b(n)$ with $n$ has been
studied theoretically by different groups [21, 57, 22, 19] to ascertain the
contributions of the different processes connected to the bulk plasmon. Based
on perturbation theory arguments, considering only the extrinsic process, it
was suggested that $b(n)$ should vary as $\alpha^{n}$, where $\alpha$ is a
measure of the extrinsic plasmon probability [21, 57, 22, 19]. But the black
dashed curve that is obtained using this expression shows a large systematic
deviation, as shown by the residual in the top panel of Fig. 1(b) in same line
type. This shows that, in addition to the extrinsic process, the intrinsic and
interference processes could also be significant. So, a modified Langreth
equation [25, 22, 21, 57] that considers all three terms has been used to
represent $b(n)$ as follows:

$b(n)=\alpha^{n}\sum_{m=0}^{n}\frac{[(\beta-c\chi)/\alpha]^{m}}{m!}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1)$

where $\beta$ is the measure of the probability of the intrinsic process for
the $n^{\rm th}$ plasmon [74]. $\chi$ is the probability of the interference
process [21, 57] given by the product of intrinsic and extrinsic plasmon
probabilities [75], where $c$ is the proportionality constant [25]. If the
above expression is used to fit the experimental $b(n)$ by varying $\alpha$,
$\beta$, and $c$, the quality of the fitting (blue curve) improves
substantially, which is evident from a reduction in chi-square error by a
factor of about 2.5 and comparison of the residuals (blue solid and black
dashed lines) in the top panel. From the fitting, we obtain $\alpha$=
0.54$\pm$0.05 and $\beta$= 0.19$\pm$0.1. The experimental value of $\alpha$ is
in good agreement with the theoretically suggested value of 0.5 by Chang and
Langreth [21, 57]. The interference term (black dot-dashed curve) is found to
be finite but negative, with a rapidly decreasing contribution that becomes
nearly zero for $n$$>$3.

$b(n)$ for Al 2$s$ has been fitted using Equation 1, as shown by the blue
curve in Fig. 2(b). The residual in the upper panel demonstrates the good
quality of the fitting. On the other hand, the fit is worse if only the
extrinsic term is considered. From the fitting, we find that $\alpha$=
0.53$\pm$0.05 and $\beta$= 0.18$\pm$0.1. Thus, the extrinsic and intrinsic
contributions are similar between 2$s$ and 1$s$, although for the former, the
kinetic energy ($E_{k}$) of the electrons is larger (5831 eV) compared to the
latter (4390 eV). Al 2$s$ recorded by XPS, where $E_{k}$ is relatively smaller
(1133 eV), gave $\alpha$ (= 0.46 eV) and $\beta$ (= 0.22 eV) [25] that are not
significantly different from the HAXPES values. This shows that, contrary to
the expectation that the extrinsic process might be dominant at larger $E_{k}$
due to larger inelastic mean free path of the photoelectrons, all three
processes have nearly similar contributions over a large range of $E_{k}$ from
1133 eV ($\approx$ 1 keV) to 5831 eV ($\approx$ 6 keV). This is in agreement
with the theoretical study by Shinotsuka et al. [3] that shows hardly any
change in the different processes in the 1$\omega_{p}$ bulk plasmon of Al in
the $h$${\nu}$ range of 2 to 5 keV.

The interference term (black dot-dashed curve) also shows similar trend. Its
contribution at $n$= 1 is also nearly similar: -0.14$\pm$0.05 for 1$s$ and
-0.17$\pm$0.05 for 2$s$ using HAXPES, while it is reported to be -0.2 for 2$s$
from XPS [25]. This is in agreement with the calculation by Shinotsuka et al.
[3], who found that the interference term does not decrease in the HAXPES
regime, in disagreement with the prediction by Hedin [8].

Figure 3: (a) Mg 2$s$ core level spectrum taken at normal emission showing
multiple bulk plasmon peaks $n$$\omega_{p}$ ($n$= 1-7, black arrows) and a
surface plasmon peak (1$\omega_{s}$, blue arrow, also shown in the inset). The
black curve through the data points (red open circles) and each of the
$n$$\omega_{p}$ peaks (shaded by different colors) are obtained from the least
squares curve fitting. (b) $b(n)$ fitted with all three terms [extrinsic,
intrinsic, and interference, (blue curve)]. The residual of the fitting is
shown in the top panel. (c) $\Gamma_{L}$, $\Gamma_{R}$, and (d) $\kappa$
(=$\Gamma_{L}$/$\Gamma_{R}$) as function of $n$.

###  III.2 Multiple bulk plasmons in Magnesium core level spectra

Mg 2$s$ spectrum of clean Mg metal at normal emission, as depicted in Fig.
3(a), also exhibits multiple plasmon peaks denoted by $n$$\omega_{p}$ ($n$=
1-7). $\omega_{p}$ of Mg, determined from the average separation of the
plasmon peaks from the main peak, is 10.6 eV. It is considerably smaller
because of the smaller electron number density of Mg, which has 2 electrons in
the outer shell compared to 3 electrons in Al. A weak surface plasmon is
observed at 7.5 eV loss energy (blue arrow). Using curve fitting as in case of
Al, $b(n)$ for Mg has been determined [Fig. 3(b)]. The residual (blue curve)
in the top panel of Fig. 3(b) shows good quality of the fitting using Equation
1. The values of $\alpha$ and $\beta$ turn out to be 0.55$\pm$0.05 and
0.18$\pm$0.1, respectively.  These values are nearly similar to Al.  The
interference term for Mg (black dot-dashed curve), as in the case of Al, is
negative for $n$= 1 and decreases to almost zero for $n$$\geq$ 3\. Figure 3(c)
shows that the asymmetry of the Mg 2$s$ plasmons given by $\kappa$ decreases
from about 4 ($n$= 1) to below 2 for $n$= 3 [Fig. 3(d)]. This was also
observed for Al 2$s$. Larger $\kappa$ i.e., asymmetry in the 1$\omega_{p}$
plasmon shape compared to $n$$\geq$2 is because of the sizable intrinsic and
interference plasmon contributions that are more asymmetric in shape compared
to the extrinsic plasmon [7].

The $\alpha$ and $\beta$ values obtained for Mg 2$s$ are nearly similar to
those from Al 1$s$ and 2$s$, average representative values for these free
electron metals being $\alpha$= 0.54$\pm$0.05 and $\beta$= 0.18$\pm$0.1.
Considering these values, the ratio of the intrinsic to extrinsic plasmon
probabilities ($e^{-\beta}\beta$:$\alpha$ from Equation 1) for 1$\omega_{p}$
turns out to be 0.28:1. For 2$\omega_{p}$ the probability ratio
($e^{-\beta}\beta^{2}/2!:\alpha^{2}$) is 0.05:1, whereas for 3$\omega_{p}$ it
is 0.005:1 ($e^{-\beta}\beta^{3}/3!:\alpha^{3}$) . Thus, the intrinsic plasmon
contribution decreases rapidly with $n$ from 22% ($n$= 1) to 4.4% ($n$= 2),
and becomes negligible thereafter (0.5% for $n$= 3).

Figure 4: Normal ($\theta$= 85∘) and grazing emission ($\theta$= 10∘) (a, b)
Al 2$s$ HAXPES core level spectra showing the surface plasmon peak
(1$\omega_{s}$) peak along with 1$\omega_{p}$ for polished smooth surface
compared with XPS data from Ref. 25. (c, d) Al 2$s$ HAXPES spectra for scraped
rough surface in normal and grazing emission, respectively. The fitted curves
including the plasmon shapes, the inelastic background (bg) and the residual
of fitting (top panel) are shown.

###  III.3 Surface plasmon behavior in the Al 2$s$ core level spectra

This subsection is dedicated to an in-depth analysis of the surface plasmon.
Figure 4(a, b) shows the 1$\omega_{s}$ along with 1$\omega_{p}$ in Al 2$s$
spectra of a polished surface cleaned by Ar ion sputtering. Comparison of the
1$\omega_{s}$ intensity between normal emission in panel a and grazing
emission in panel b shows a sizable enhancement in the latter. From the curve
fitting, we quantify this by the normalized intensity of 1$\omega_{s}$ i.e.,
$s(1)$ (the intensity of 1$\omega_{s}$ divided by the main peak). $s(1)$
increases from 0.13 in normal to 0.44 in grazing emission. However, a direct
comparison with the XPS spectra from Ref. 25 [Figs. 4(a,b)] shows the surface
plasmon is weaker in HAXPES for both normal and grazing emission. This is
attributed to diminished extrinsic contribution due to larger velocity ($v$)
of the photoelectron [25, 35]. From Fig. 4(a, b), $b(1)$ decreases from 0.6 in
normal to 0.43 in grazing emission, which was related to the variation of
extrinsic bulk plasmon contribution with emission angle in theory [32]. The
intensity ratio $R$ (= 1$\omega_{s}$/1$\omega_{p}$) is larger in the grazing
(1 i.e., the surface plasmon is as intense as the bulk plasmon) compared to
normal emission (0.2).

The considerable increase of $s(1)$ and $R$ in the grazing emission discussed
above is in disagreement with the theoretical results of Shinotsuka et al. [3]
for HAXPES, where hardly any difference in the calculated 1$\omega_{p}$ and
1$\omega_{s}$ was observed between normal and grazing emission (Fig. S1 of SM
[60]). Nevertheless, to obtain an estimate of $R$ from theory, we have fitted
the theoretical spectrum using asymmetric Lorentzian shapes for both
1$\omega_{p}$ and 1$\omega_{s}$ from Ref. 3. A reasonably good fit of the
theoretical shape [3] with the asymmetric Lorentzians in Fig. S1 of SM [60]
post-priori justifies the use of the latter to fit the experimental plasmon
shapes in Figs. 1-4. From the area under the fitted curves, the theoretical
value of $R$ turns out to be close to 0.2, unchanged for both normal and
grazing emission. While $R$= 0.2 is in excellent agreement with the
experimental normal emission value (0.2), it is conspicuously underestimated
for the grazing emission, where the experimental value is 1. Enhancement of
$s(1)$ in grazing emission has also been reported for XPS. For example, in Al
2$s$ from XPS, a ten times increase in $s(1)$ has been reported in the grazing
emission, which was explained by an increase in $v$ parallel to the surface
[25, 35]. This effect was possibly not considered in Ref. 3. Note that in Fig.
4, the shape of the surface plasmon is found to be highly asymmetric, with
$\Gamma_{R}$ considerably smaller than $\Gamma_{L}$. Their ratio $\kappa$ is
4.9 (7.9) in grazing (normal) emission (Fig. 4). The asymmetric shape of the
surface plasmon as well as the bulk plasmons is in agreement with Ref. 3 and
the earlier theoretical works [11, 12, 13].

Figures 4(c,d) show an interesting result that for rough surface obtained by
scraping the polished surface with a diamond file, the enhancement of the
surface plasmon in the grazing emission is not as pronounced: $s$(1)= 0.14 in
the normal and 0.17 in grazing emission. This is because the emission angle is
not well defined for the rough surface. Comparison of Figs. 4(a,c) shows that
both the bulk and surface plasmon intensities in normal emission are however
nearly similar between the smooth [$b(1)$= 0.6, $s(1)$= 0.14] and the rough
surface [$b(1)$= 0.58, $s(1)$= 0.14].

###  III.4 Plasmons in the core level spectra of a fully oxidized Al surface

In Fig. S3(a) of SM [60], a fully oxidized Al crystal surface, shows oxide and
metal-related components in the Al 1$s$ spectrum. In spite of being fully
oxidized, the metal peak is dominant in normal emission compared to the oxide-
related peak at 2.7 eV higher $E$ because of the large electron mean free path
in HAXPES (80 Å for Al 1$s$ at 6 keV [76]). The thickness of the oxide film
estimated from the relative intensity variation of the metal- and the oxide-
related peaks with $\theta$ [77] turn out to be 48$\pm$2 Å. In normal
emission, the metal-related bulk plasmon is observed at 15.3 eV loss energy
with $b(1)$= 0.5. At the grazing emission in Fig. S3(b) of SM, although the
oxide-related peak is highly enhanced in intensity (shown truncated),
surprisingly, the 1$\omega_{p}$ corresponding to the metal underneath is still
observed with the same $\omega_{p}$ of 15.3 eV and a $b(1)$ value of 0.6. Both
these values of $b(1)$ in oxidized Al are close to that of the clean Al metal
[$b(1)$= 0.6, Fig. 1(b)]. Thus, the normalized intensity of the bulk plasmon
is hardly affected by the oxide layer formed on the surface. This is because
both the metal’s main peak and the bulk plasmon are similarly attenuated by
the oxide layer. However, in contrast, the surface plasmon is completely
attenuated because the metal surface is fully covered by the oxide film.

###  III.5 Multiple bulk plasmons in the valence band spectra of Al and Mg

The HAXPES valence band (VB) spectrum of Al metal, along with the loss region
shows, multiple bulk plasmons $n$$\omega_{p}$ up to $n$= 4 (black arrows)
[Fig. 5(a)]. The data were recorded for a scraped surface because, for the Ar+
ions sputtered surface, the Ar 3$p$ signal arising from implanted Ar bubbles
[71, 72] appears at 9.3 eV and interferes with the VB, as shown in Fig. S4 of
SM. In Fig. 5(b), the theoretical VB – calculated using the PDOS shown in Fig.
5(c) – exhibits excellent agreement with the experiment: All the features,
such as a peak near the Fermi level at $E$= 1 eV (blue arrow), a dip at 2 eV
(blue tick), a weak feature centered at 3 eV (black arrow), and the broad hump
around 6 eV (green arrow), are visible at nearly identical energies and
possess similar relative intensities. The VB is largely dominated by the
$s$-like states because of their large photoemission cross-section for hard
x-rays [39, 40, 51, 78]. The small contribution from $p$-like states is rather
featureless. It should be noted that the interstitial ($int$) contribution to
the total DOS is significant because a substantial part of the electron
density lies outside the atomic sphere that is used to calculate the PDOS
[Fig. 5(c)]. Another point to note is that the PDOS of Al previously
calculated using the KKR method [79] differs considerably from our calculation
[e.g., compare Fig. 5(c) with the inset of Fig. S5 in SM [60]]. We used the
PDOS from Ref. 79 to calculate the VB and find that it differs significantly
from the experiment (Fig. S5 of SM [60]): a peak at $\sim$5 eV is evident.
This is absent in the experimental spectrum, while the dip at $\sim$2 eV (blue
tick) is overestimated. These observations show the shortcomings of the
previous calculation, which is possibly related to unphysical fluctuations on
the lower energy side and was performed with constant potential outside the
Muffin tin sphere [79].

In the case of Mg metal, the VB spectrum also shows multiple bulk plasmons
$n$$\omega_{p}$ up to $n$= 4 with $\omega_{p}$$\sim$10.5 eV [black arrows in
Fig. 5(e)]. The VB, shown on an expanded scale in Fig. 5(f), exhibits
excellent agreement with the theoretical VB, which is calculated using the
PDOS shown in Fig. 5(g). An earlier augmented plane wave calculation [80] of
the PDOS for Mg agrees well with our calculation. A peak close to the Fermi
level at 0.3 eV (blue arrow), a dip at 0.7 eV (blue tick), and a broad hump
around 3 eV (green arrow) are visible at similar energies and exhibit
comparable relative intensities [Fig. 5(f)]. Here also, as in the case of Al,
the VB is largely dominated by the $s$-like states.

Figure 5:  HAXPES valence band (VB) of (a) Al and (e) Mg metals along with the
bulk plasmon loss peaks ($n$$\omega_{p}$ up to $n$= 4 marked by arrows)
measured in normal emission after an inelastic background subtraction. The
simulated plasmon line shapes are shown by violet dashed curves, and their
addition is shown by a black curve. (b) Al and (f) Mg VBs are compared with
the calculated VB (black curve), the $s$ and $p$ partial contributions are
shown. The calculated VBs are shown overlaid in panels a, e and are used to
simulate the plasmons (see text). The DFT calculated (c) Al and (g) Mg total
DOS, $s$, $p$ and $d$ PDOS, and the interstitial ($int$) contribution. The
Fermi edges of (d) Al and (h) Mg have been compared to Au (black lines are fit
to the experimental data); the horizontal double-sided arrow shows the recoil
shift. Figure 6: $b_{\rm VB}(n)$ as a function of $n$ for Al and Mg HAXPES VB
compared with those extracted from Mg XPS VB [34] and the theoretical
calculation by Penn [14] for Mg with intrinsic and without intrinsic (i.e.,
only extrinsic) plasmon.

The plasmon loss spectra of VB have been calculated for Mg and Na by Penn
[14], which drew on the work by Langreth [81]. Unlike the core level intrinsic
plasmon that is created by a localized core hole in the final state, the
photo-hole in the VB is delocalized. So, the mechanism of intrinsic plasmon
production in the VB was proposed to be related to electron-electron
interaction: a second electron creates the intrinsic plasmon via its
interaction with the conduction electrons and scatters into the hole left by
the photoexcited electron [14]. Although energy dependence and interference
effect were not considered in theory [14], remarkable agreement was obtained
with the XPS spectra [34] only when both extrinsic and intrinsic plasmons were
considered. On the other hand, the plasmon intensities were significantly
underestimated if the intrinsic plasmon was not considered. This established
the presence of intrinsic plasmons in the XPS VB plasmon in nearly free-
electron metals [14].

To compare theory and experimental results, the shapes of the $n^{\rm th}$ VB
plasmons for both Al and Mg have been simulated by shifting the calculated VB
by $n$$\omega_{p}$ towards higher $E$, and allowing the intensity and an
additional Gaussian broadening to vary [Figs. 5(a,e)]. We find that unlike the
core level plasmons, asymmetric broadening is not required, and the FWHM of
the Gaussian remains nearly unchanged with $n$ (2 eV and 2.5 eV for Al and Mg,
respectively). The $b_{\rm VB}(n)$ values of Mg have also been extracted for
the theoretical calculations by Penn [14] for XPS and the the XPS spectrum
published in the literature [34] (Fig. S6 of SM [60]). This has been done by
shifting the calculated XPS VB by $n$$\omega_{p}$ and broadening it by a
Gaussian, as discussed above. Figure S6 of SM [60] shows good agreement
between the calculated and experimental XPS VB. The former is obtained from
the PDOS in Figs. 5(c,g) using the photoemission cross-sections for AlKα
(1486.6 eV) radiation [67].

Figure 6 shows $b_{\rm VB}(n)$, which is the area of the $n^{\rm th}$ plasmon
divided by the area of the VB. $b_{\rm VB}(n)$ for Mg and Al HAXPES VB are
close, as shown by the red and blue curves. $b_{\rm VB}(1)$ for Mg HAXPES
(0.87) and XPS (0.9) are quite similar to each other and to the theoretical
value (0.9) of Mg that includes the intrinsic and extrinsic plasmons (black
filled circle-dashed line). On the other hand, if intrinsic plasmon is not
considered, the contribution of only the extrinsic process is significantly
less (0.6) (gray filled square-dashed line). Similarly, $b_{\rm VB}(2)$ and
$b_{\rm VB}(3)$ for both Mg HAXPES and XPS are closer to the theoretical
values that include both intrinsic and extrinsic plasmons. Thus, our data
provide evidence of the intrinsic plasmon production in the VB related
plasmons of Mg measured by HAXPES.

###  III.6 Recoil effect in the valence band spectra of Mg

The recoil effect is well known in the HAXPES core level spectra of light
elements. When a photoelectron of mass $m$ is emitted with a large kinetic
energy ($E_{kin}$) from an atom of mass $M$, the photoelectron delivers a
recoil energy ($E_{R}$) to that atom, which is given by
$E_{R}$$\approx$$\frac{m}{M}$$\times E_{kin}$ [46, 40]. The photoelectrons
lose this energy, $E_{R}$, resulting in a recoil shift towards higher $E$.
This effect has also been observed in the VB of light metals such as Al
(atomic mass 27 u) through a shift of the Al metal Fermi edge compared to that
of Au [46]. Au, being of high mass (197 u), hardly exhibits this effect.
Recently, such a recoil shift in the VB has been reported in a light complex
metallic alloy $\beta$-Al3Mg2 [82]. Here in Fig. 5(d), the recoil shift of Al
is, however, smaller (85 meV) compared to 120 meV reported using 7.94 keV
[46]. This can be explained by the proportionality of the recoil shift with
$E_{kin}$ i.e., $\frac{6~{}\rm keV}{7.94~{}\rm keV}\times$120 meV= 90.7 meV,
which is close to 85 meV. Due to marginally smaller mass of Mg (24.3 u)
compared to Al, the recoil shift is found to be slightly larger [90 meV, Fig.
5(h)].

##  IV Conclusion

Multiple bulk plasmons ($n$$\omega_{p}$) in the valence band (up to $n$= 4)
and core level spectra (up to $n$= 13) of free electron metals such as Al and
Mg have been studied by hard x-ray photoelectron spectroscopy (HAXPES). Based
on earlier theoretical works [19, 22, 21, 57, 14], estimates of the extrinsic,
intrinsic, and interference processes that contribute to the core level
plasmon intensities are obtained from exhaustive multi parameter curve
fitting. The probabilities of the extrinsic and intrinsic processes in the
core level plasmons are nearly similar for the two free electron metals,
$\alpha$ and $\beta$ being 0.54$\pm$0.05 and 0.18$\pm$0.1, respectively. These
values imply that, as $n$ increases, the intrinsic plasmon contribution
decreases while the interference plasmon changes from negative to zero.
Despite the larger inelastic mean free path in HAXPES, the intrinsic,
extrinsic, and interference processes remain nearly unchanged across a broad
kinetic energy range (1–6 keV). This finding is consistent with previous
theoretical research [3]. The 1$\omega_{p}$ line shape is more asymmetric
because of the presence of sizable intrinsic and interference terms. The
1$\omega_{s}$ surface plasmon has been identified in both Al and Mg. Its
intensity increases significantly in the grazing emission compared to the
normal emission. In the case of oxidized aluminum, the surface plasmon is
completely suppressed, while the bulk plasmon intensity is almost unaffected
compared to the metal’s main peak.

Multiple plasmons are observed in the loss region of the Al and Mg valence
bands; the shape of the VB is in excellent agreement with the DFT-based
calculated valence band. Comparison with the theoretical calculation [14]
after extracting the relative VB plasmon intensities establish the existence
of the intrinsic plasmon, besides the extrinsic plasmon. The recoil shift
observed in both Al and Mg VB is related to the kinetic energy of the
photoelectron and the mass of the atom.

##  V Acknowledgment

The HAXPES experiments were carried out at PETRA III of Deutsches Elektronen-
Synchrotron, a member of Helmholtz-Gemeinschaft Deutscher Forschungszentren.
Financial support by the Department of Science and Technology, Government of
India, within the framework of the India@DESY collaboration is gratefully
acknowledged. We are thankful to C. Schlueter, R. Choubisa, and D. Sébilleau
for support and encouragement. The Computer division of the Raja Ramanna
Centre for Advanced Technology is thanked for installing the DFT codes and
providing support throughout. We thank I. Schostak for skillful technical
support.

∗e-mail address<EMAIL_ADDRESS>
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