Current address:] Dipartimento di Fisica, Sapienza Universitá di Roma,
Piazzale Aldo Moro 2, I-00185 Rome, Italy Current address: ]Department of
Physics, Southern Connecticut State University, 501 Crescent St, New Haven, CT
06515, USA

# Unexpectedly large electron correlation measured in Auger spectra of
ferromagnetic iron thin films: orbital-selected Coulomb and exchange
contributions

R. Gotter<EMAIL_ADDRESS>CNR-IOM, Istituto Officina dei Materiali, c/o
Area Science Park, SS14 km 163.5 I-34149 Basovizza-Trieste, Italy  A. Verna
Dipartimento di Scienze, Universit degli Studi Roma Tre, Via della Vasca
Navale 84, I-00146 Rome, Italy  M. Sbroscia  [  Dipartimento di Scienze,
Universit degli Studi Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy
R. Moroni  CNR-SPIN, Corso Perrone 24, I-16152, Genova, Italy  F. Bisio  CNR-
SPIN, Corso Perrone 24, I-16152, Genova, Italy  S. Iacobucci  Dipartimento di
Scienze, Universit degli Studi Roma Tre, Via della Vasca Navale 84, I-00146
Rome, Italy  CNR-ISM, Via Fosso del Cavaliere 100, 00133 Roma, Italy  F. Offi
Dipartimento di Scienze, Universit degli Studi Roma Tre, Via della Vasca
Navale 84, I-00146 Rome, Italy  S. R. Vaidya  [  CNR-IOM, Istituto Officina
dei Materiali, c/o Area Science Park, SS14 km 163.5 I-34149 Basovizza-Trieste,
Italy  A. Ruocco  Dipartimento di Scienze, Universit degli Studi Roma Tre, Via
della Vasca Navale 84, I-00146 Rome, Italy  G. Stefani  Dipartimento di
Scienze, Universit degli Studi Roma Tre, Via della Vasca Navale 84, I-00146
Rome, Italy  CNR-ISM, Via Fosso del Cavaliere 100, 00133 Roma, Italy

###### Abstract

A set of electron-correlation energies as large as 10 eV have been measured
for a magnetic 2ML Fe film deposited on Ag(001). By exploiting the spin
selectivity in angle-resolved Auger-photoelectron coincidence spectroscopy and
the Cini-Sawatzky theory, the core-valence-valence Auger spectrum of a spin-
polarized system have been resolved: correlation energies have been determined
for each individual combination of the two holes created in the four sub-bands
involved in the decay: majority and minority spin, as well as $e_{g}$ and
$t_{2g}$. The energy difference between final states with parallel and
antiparallel spin of the two emitted electrons is ascribed to the spin-flip
energy for the final ion state, thus disentangling the contributions of
Coulomb and exchange interactions.

The extraordinary macroscopic properties of technologically relevant materials
exploiting quantum effects, including magnetism itself, are mainly determined
by the relevance of electron-electron correlations. The development of novel
magnetic and spintronic devices, nowadays evolving more and more at the
nanoscale and at the interface level, requires an accurate comprehension of
the strongly correlated nature of $d$ or $f$ electron shells, in terms of
local and non-local Coulomb and exchange interactions, which depend on energy,
orbital, momentum and spin degrees of freedom Tusche  _et al._ (2018); Steiner
_et al._ (1992); Sánchez-Barriga  _et al._ (2009); Da Pieve (2016). Density
functional theory (DFT) Kohn and Sham (1965) and computational methods
referred as “Beyond-DFT” Mandal  _et al._ (2019) provide accurate descriptions
of the ground state of moderately-correlated materials while strongly-
correlated materials and their excited states still pose significant
challenges Himmetoglu  _et al._ (2014). These shortcomings are mitigated
within Dynamical Mean Field Theory adding, as a tuning parameter, an Hubbard
interaction energy $U$ Hubbard (1963); Himmetoglu  _et al._ (2014) and an ad-
hoc spin-spin interaction term $J$ whenever the magnetic properties have to be
taken into account Biermann  _et al._ (2005). While spin-resolved
photoemission can provide a direct measure of $J$, a wide spectrum of $U$
values is reported in the literature even for the paradigmatic case of iron,
whose behavior is at the border between localized and itinerant regimes
Antonides  _et al._ (1977); Steiner  _et al._ (1992); Pou  _et al._ (2002);
Sánchez-Barriga  _et al._ (2009); Katanin  _et al._ (2010). The microscopic
understanding of ferromagnetism as a consequence of electron correlations,
which changes according to dimensionality, remains a challenging task Stöhr
and Siegmann (2006). The measurement of the correlation energy is an elusive
task for most conventional spectroscopies, since their spectral functions are
determined by single-quasi-particle contributions. This is however not the
case for core-valence-valence (CVV) Auger spectra, which are sensitive to
correlations because of the creation of two interacting holes in the final
state and due to the short range of the two-body Coulomb operator acting on
many-particle wavefunctions. CVV Auger spectra are described by a two-particle
density of states in the valence band (VB), that corresponds to the self-
convolution of the independent-particle density of states (SCDOS) only in the
absence of correlations Weightman (1982). From the pioneering theory proposed
by Cini and Sawatzky (CS) Cini (1977); Sawatzky (1977), up to the most recent
spectral-density-approach (SDA) developed by Nolting and coworkers Nolting
_et al._ (1991), an effective electron correlation $U_{eff}$ is understood to
determine the Auger line-shape Sawatzky and Lenselink (1980); Potthoff  _et
al._ (1995); Sarma  _et al._ (1992). As $U_{eff}$ becomes comparable to the
valence band width $W$, the Auger line-shape changes from purely band-like to
atomic-like due to the presence of resonant two-hole final states. Over time,
theories have been successfully extended from closed to partially filled bands
Treglia  _et al._ (1981); Verdozzi  _et al._ (2001); Sarma and Mahadevan
(1998); Potthoff  _et al._ (1993). In such a perspective, $U_{eff}$ embeds
several electron correlation contributions, beyond the on-site Coulomb
interaction $U$, thus including the exchange interaction $J$, as well as off-
site Coulomb interaction, spin-orbit coupling and dynamical screening effects
not included in the CS theory initially formulated for closed bands Verdozzi
_et al._ (2001). $U_{eff}$ can assume different values for different two-hole
final states, as found for the weakly correlated CVV Auger spectrum of
graphite Houston  _et al._ (1986), as well as for the M4,5VV spectra of Ag
Arena  _et al._ (2001) and Pd Butterfield  _et al._ (2002). Also Auger spectra
of spin polarized systems are expected to exhibit different $U_{eff}$
depending on the spin and the band of the electrons involved in the decay
process Dürr  _et al._ (1997); Schumann  _et al._ (2010); Gotter  _et al._
(2012); Wegner  _et al._ (2000). Ferromagnetic compounds display Coulomb and
exchange interactions whose combined action is not negligible with respect to
$W$ Steiner  _et al._ (1992); Santoni and Himpsel (1991). Nevertheless, in
contrast with the correlated nature of their magnetic properties,
ferromagnetic materials exhibit almost band-like Auger spectra, thus hampering
the capability to extract information on the correlation of the two-hole final
state Yin  _et al._ (1977); Lund  _et al._ (1997), or even leading to the
conclusion that electron correlation is irrelevant Antonides  _et al._ (1977);
Sinković  _et al._ (1995). Angle resolved - Auger photoelectron coincidence
spectroscopy (AR-APECS) overcomes this deadlock by combining the selectivity
on individual core-hole states Jiang  _et al._ (2001) with the sensitivity to
the spin of the Auger final state Gotter  _et al._ (2007), thus allowing
correlated final states to be unveiled Gotter  _et al._ (2005); Cini  _et al._
(2011); Gotter  _et al._ (2009, 2011, 2013).

In this Letter we investigate the role of electron correlation in Auger
spectra of a ferromagnetic Fe thin film grown on Ag(001). Due to the non-
overlapping $d$ bands of Fe and Ag and to the small population of the Ag $sp$
bands, Fe/Ag(001) is a close approximation to a free standing 2D Fe film
Blügel (1992). The AR-APECS investigation enables to disentangle features of
the Auger spectrum that are originated from electron correlation and the
$U_{eff}$ acting on individual pairing of the final state holes, is singled
out by the help of the CS model.

The reported experiments were carried out at the ALOISA beamline of the
ELETTRA synchrotron radiation facility (Basovizza -Trieste, Italy). The
experimental setup is discussed in detail elsewhere Gotter  _et al._ (2001);
Sbroscia  _et al._ (2019). A 2 mono-layers (ML) thick Fe film was grown at a
pressure of 5$\times$10-8 Pa, by electron beam assisted evaporation onto the
Ag(001) substrate at RT, prepared with standard surface science procedures: 1
KeV Ar+ sputtering and annealing at 750 K. The number of layers was
established by the oscillations of the reflection high-energy electron
diffraction (RHEED) specular beam intensity Lüth (2010). The film, kept at 170
K, well below its Curie temperature Lang  _et al._ (2006), is ferromagnetic
and magnetized out-of-plane Schaller  _et al._ (1999). A new sample was
prepared every 12 h of beam exposure to prevent oxidation. A monochromatic,
linearly polarized beam of 253 eV photons impinged onto the sample surface at
a grazing angle of about 6∘, with the surface normal lying in the plane
defined by the photon polarization $\bm{\varepsilon}$ (the quantization axis)
and the momentum $\mathbf{k}$ vectors. Auger- and photo-electron pairs were
selected in energy and detected in coincidence within the solid angles (4∘
opening) defined by the electron analyzers. Three analyzers placed in the
$\bm{\varepsilon}\mathbf{k}$ plane collected Fe 3$p$ core photoelectrons at 0∘
and $\pm$36∘ polar angle with respect to $\bm{\varepsilon}$, with an energy
resolution of 3.2 eV; they were detuned by 1.5 eV at higher kinetic energy
with respect to the 3$p$ maximum photoemission intensity Gotter  _et al._
(2009) for collecting mainly the three photoemission lines closely packed at
the high kinetic energy side of the $3p$ sextet Tamura  _et al._ (1994); Henk
_et al._ (1999). Auger electrons resulting from the Fe M3M4,5M4,5 super-
Coster-Kronig transition have been collected at an angle of 38∘ off the
$\bm{\varepsilon}\mathbf{k}$ plane via a multichannel analyzer with an energy
resolution of 1 eV. The electron pairs detected at the three different
photoelectron emission angles allow to exploit the dichroic effect in angle
resolved APECS (DEAR-APECS) thereby providing a moderate selection of the
final state spin Gotter  _et al._ (2005). In analogy to previous AR-APECS
experiments carried out with the same geometries Gotter  _et al._ (2013), in
the present experiments the analyzer pair selecting photoelectrons emitted
along $\bm{\varepsilon}$ will be termed as antiparallel-spin (AS)
configuration, while the pairs with photoelectrons detected at $\pm$36∘ apart
from $\bm{\varepsilon}$, will be termed as parallel-spin (PS), as they detect
electron pairs with predominantly antiparallel and parallel spins,
respectively. The coincidence count rate for these experiments was of the
order of 1.3 10-1 counts per second, thus, to achieve a good statistics,
nearly 40 h of integration time was required. In the upper panel of Fig. 1 the
AR-APECS spectra as measured in AS and PS configurations are shown together
with the conventional Auger (AES) measured under identical experimental
conditions. The AR-APECS spectra show a rich multiplicity of narrow
transitions spread across an energy interval larger than twice the band width
($W$), i.e. well beyond the interval from 49 to 40 eV allowed by energy
conservation if electronic correlation were not effective Schröder  _et al._
(1985). Not even spin-resolved Auger investigations Sinković  _et al._ (1995);
Schröder  _et al._ (1985); Allenspach  _et al._ (1987) disclosed these
manifold structures, because detecting the spin of only one electron (the
Auger electron) does not allow a selectivity on the two-hole final state. For
instance, the majority spin ($\uparrow$) Auger intensity was represented by
terms proportional to the self-convolution of the majority-spin DOS
($\uparrow\uparrow$ terms) plus terms proportional to the convolution of the
majority- with the minority-spin DOS ($\uparrow\downarrow$ terms) Sinković
_et al._ (1995), that is by a mix of terms having different total spin $S$,
used to identify the final states in the LS coupling.

Figure 1: In the upper panel AR-APECS spectra of 2 ML Fe/Ag(001) as measured
in the PS (red open circles) and in the AS (green open triangles)
configurations are shown together with their estimated integral background
(dashed-dotted lines); the continuous lines are guides for the eye. The black
dotted line is the conventional Auger electron spectrum (AES), simultaneously
collected during the coincidence measurement. In the lower panel the dichroic
effect in AR-APECS (DEAR-APECS) (violet open squares) is defined as the
difference between PS and AS spectra divided by the semi-sum averaged over the
spectrum energy interval (from 30 to 50 eV); the continuous line is a guide
for the eyes.

The DEAR-APECS Gotter  _et al._ (2009) reported in the lower panel of Fig. 1,
that is the difference between the PS and the AS spectra, is more pronounced
in the low kinetic energy region (from 30 to 37 eV). The CS model foresees an
Auger lineshape consisting of a manifold of closely-spaced sub-bands
contributions on the high energy side of the spectrum, accompanied by sharper
features at lower kinetic energy due to resonant two-hole states Cini (1977);
Sawatzky (1977); Verdozzi  _et al._ (2001). A previous AR-APECS investigation
on the Fe MVV spectrum of a 3ML film of Fe/Cu(001), although performed with a
moderate 2 eV energy resolution, has associated the low-energy region of the
spectrum to resonant two-hole final states with a single average correlation
energy of 2.7 eV, estimated by applying the Cini formula (Cini, 1977, formula
(14)) to a DFT computed DOS of a Fe impurity in a Cu jellium Gotter  _et al._
(2012). After the removal of an integral background due to energy losses
suffered by the Auger electrons Seah (2001), the AS and PS spectra reported in
Fig. 2 unravel a manifold of features that are interpreted as individual two-
hole correlation resonances. The relative maxima at 32.4, 34.9 and 37.6 eV,
which are prominent in the PS configuration and strongly reduced in the AS
one, can be ascribed to the parallel spin of the two emitted electrons; vice
versa, the antiparallel spin character can be attributed to the structure at
36.6 and 40.3 eV, which are dominant in the AS configuration. The data
analysis reported in the following, builds on the Fe DOS calculated in DFT-
LSDA by Rhee Rhee (2005) for a 3 ML Fe ultrathin film on Ag(100) and for bulk
Fe, the latter being in very good agreement with a recent (DFT-LSDA)
calculation Da Pieve (2016). The 2 ML-thick film here investigated has been
modelled using Rhee’s DOS of the surface (top) and interface (bottom) layers;
this latter weighted by the effective mean free path of the emitted electron
pair Werner  _et al._ (2005). The dashed-dotted line in Fig. 2 is the
self(mutual)-convolution of the density of states (SCDOS) summed over all the
spin-resolved occupied $e_{g}$ and $t_{2g}$ bands and broadened by the
experimental resolution; it describes the band-like Auger line-shape in
absence of electron correlation, and it accounts for the measured intensity
only from the onset up to the maximum of the AR-APECS intensity, without
fitting any of the sharp features. If the atomic multiplet of Fe is taken into
account (see figures and ref. sup, ), the contribution of 16 multiplet terms
over an energy interval of 13 eV does not cover the full width of the measured
AR-APECS spectra; not even if to such an atomic multiplet, the CS model is
applied, as successfully done in the case of Pd and Ag MVV Auger spectra
Butterfield  _et al._ (2002); Arena  _et al._ (2001); Weightman (1982).
Therefore the CS model was applied to each possible pairing of the individual
spin and orbital components of the theoretical DOS as derived by Rhee Rhee
(2005) and a set of $U_{eff}$ values were used as free parameters. In
particular the Cini formula Cini (1977) provide a formulation of the two-
particle spectral density as a simple functional of one-particle DOS, suitable
to be implemented in a fitting procedure. A least square fitting procedure has
been simultaneously applied to both PS and AS AR-APECS spectra with a
tentative assignment of the manifold features made on the basis of the
following considerations: I. according to the Hund’s rule, final states of the
ion left behind corresponding to the emission of two ($\uparrow\uparrow$)
electrons have a lower total spin, so a higher binding energy has to be
expected with respect to final states corresponding to the antiparallel spin
($\uparrow\downarrow$) of the two emitted electrons; II. the above assessment
also comply with the fact that majority spin bands are almost filled bands
while minority ones are open, hence higher $U_{eff}$ are expected for final
states where both holes are created in the majority spin bands
($\uparrow\uparrow$); III. due to the small hybridization between $e_{g}$ and
$t_{2g}$ bands, $e_{g}$ behaves like a Luttinger (localized) electron liquid
while $t_{2g}$ behaves like a Fermi (itinerant) liquid Katanin  _et al._
(2010); hence Auger transitions involving $e_{g}$ states should be credited
for larger $U_{eff}$; IV. the electron correlation between two holes created
in the same sub-band is larger with respect to the case when they are created
in different sub-bands; V. the ($\downarrow\downarrow$) contribution to the
Auger intensity is neglected in view of the smallness of the resulting SCDOSs.
Such a rationale, together with the possibility to identify the three
($\uparrow\uparrow$) contributions (red filled peaks in Fig. 2), that are
predominant in the PS configuration, and the four ($\uparrow\downarrow$)
contributions (green filled peaks in Fig. 2), that are relatively more intense
in the AS configuration, allowed to set the initial guess of $U_{eff}$ values
in the fitting procedure. A chi-square likelihood test of the best fit
procedure provided the set of $U_{eff}$ parameters shown in Table 1 and the
resulting Auger line-shapes are the continuous thick lines in Fig. 2.

Figure 2: Fe M3VV AR-APECS spectra after inelastic background subtraction in AS (top panel, empty green circles) and PS (bottom panel, red empty circles) configurations. In both panels, the dashed-dotted curve is the SCDOS (no electron correlation) and the red (green) peaks are $\uparrow\uparrow$ ($\uparrow\downarrow$) spin components calculated by the Cini formula. The green and red solid lines are the fitting curves to the experimental data for AS and PS configuration, respectively. Hole paring | $e_{g}^{\uparrow}e_{g}^{\uparrow}$ | $t_{2g}^{\uparrow}t_{2g}^{\uparrow}$ | $e_{g}^{\uparrow}e_{g}^{\downarrow}$ | $e_{g}^{\uparrow}t_{2g}^{\uparrow}$ | $t_{2g}^{\uparrow}t_{2g}^{\downarrow}$ | $t_{2g}^{\uparrow}e_{g}^{\downarrow}$ | $e_{g}^{\uparrow}t_{2g}^{\downarrow}$ | SCDOS
---|---|---|---|---|---|---|---|---
Kinetic Energy [eV $\pm$0.1 eV] | 32.6 | 34.9 | 36.7 | 37.7 | 40.0 | 42.7 | - | 44.4
$U_{eff}$ [eV $\pm$0.1 eV] | 11.0 | 8.7 | 7.5 | 5.20 | 4.3 | 1.0 | 0.0 | 0.0
Table 1: $U_{eff}$ values for different pairings of the two emitted electrons,
as obtained from the simultaneous fitting of the Fe M3VV AR-APECS spectra in
the AS and PS configuration of Fig. 2. The kinetic energy position of each
component is also reported.

An unexpected spread of $U_{eff}$ values ranges from almost vanishing values
(cases $e_{g}^{\uparrow}t_{2g}^{\downarrow}$ and
$t_{2g}^{\uparrow}e_{g}^{\downarrow}$) to figures much larger than any
reported observation (cases $e_{g}^{\uparrow}e_{g}^{\uparrow}$ and
$t_{2g}^{\uparrow}t_{2g}^{\uparrow}$). It is worth noting that the difference
in $U_{eff}$ between transitions involving identical band combinations, but
with parallel and antiparallel spin of the two emitted electrons, corresponds
to the spin-flip energy of an electron in the doubly ionized final state of
the ion left behind. A spin-flip energy is used to define the exchange-
splitting in the Stoner model, which, in the case of Fe, amounts to about 2 eV
when calculated for the ground states of neutral Fe or measured by
photoemission experiments Stöhr and Siegmann (2006). Here, the quantities
$U_{eff}(t_{2g}^{\uparrow}t_{2g}^{\uparrow})-U_{eff}(t_{2g}^{\uparrow}t_{2g}^{\downarrow})=$
4.4 $\pm$ 0.2 eV, and
$U_{eff}(e_{g}^{\uparrow}t_{2g}^{\uparrow})-U_{eff}(e_{g}^{\uparrow}t_{2g}^{\downarrow})=$
5.2 $\pm$ 0.2 eV, can be identified as the energy necessary to flip a $t_{2g}$
electron in two different electronic configurations; the latter value is
larger by an amounts of 0.8 $\pm$0.2 eV because the spin-flip is paired with a
higher number of $t_{2g}$ electrons. These $t_{2g}$ spin-flip energies result
in turn higher with respect to the ones associated to the $e_{g}$ orbital,
according with a higher $d$-electron occupation number of the $t_{2g}$ sub-
bands with respect to $e_{g}$ ones Rhee (2005); in detail
$U_{eff}(e_{g}^{\uparrow}e_{g}^{\uparrow})-U_{eff}(e_{g}^{\uparrow}e_{g}^{\downarrow})=$
3.5 $\pm$ 0.2 eV, and
$U_{eff}(t_{2g}^{\uparrow}e_{g}^{\uparrow})-U_{eff}(t_{2g}^{\uparrow}e_{g}^{\downarrow})=$
4.2 $\pm$ 0.2 eV, with the latter value larger by an amount of 0.7 $\pm$0.2 eV
because the spin-flip is paired with a higher number of $e_{g}$ electrons. The
two values of 0.82 and 0.72 eV, are equal within the experimental uncertainty
and provide a final consistency of the assignments of $U_{eff}$ made so far.
The very large values here found for $U_{eff}$ are therefore due to the
combined action of the on-site Coulomb interaction and the spin-flip energies,
these latter being experimentally singled out for the first time in a CVV
Auger spectrum of a spin polarized system. The capability to reveal itemized
experimental values of the electron correlation allows to test more recent
theoretical models Wegner  _et al._ (2000), that can benefit of the
exploitation of multi-band Hamiltonians to better determine the details of
electron correlation in magnetic systems, and from which the three-particle
Green function, needed to calculate the Auger line-shape, can be developed. In
conclusion, by AR-APECS measurements on an ultrathin ferromagnetic Fe film the
CVV Auger spectrum of a spin polarized system has been fully resolved, giving
access with unprecedented accuracy to electron correlation effects, due to
Coulomb interaction and spin coupling of the valence holes created in the
final state. By exploiting the Cini-Sawatzky theory, a full set of electron
correlation energies $U_{eff}$ has been determined. Noticeable differences
among the $U_{eff}$ associated to the different pairings of the spin-polarized
sub-bands involved in the Auger decay have been found. $U_{eff}$ values, much
larger than the ones found in the literature, are responsible for the manifold
of two-holes resonances that remained undetected by conventional Auger. The
$U_{eff}$ used to describe Auger line-shapes, when acting on a spin polarized
system, has been understood as due to two effects: the first is a Coulomb
correlation energy, which depends on the orbitals involved in the Auger decay;
the second is a spin-flip energy characterizing the energy difference of final
states associated to parallel and antiparallel spin of the emitted electrons.

## Acknowledgements

Financial supports through: SIMDALEE2 Marie Sklodowska Curie
FP7-PEOPLE-2013-ITN Grant # 606988, PRIN 2015 NEWLI Protocol 2015CL3APH, and
ELETTRA users support are acknowledged. The authors are grateful for the
support of the staff of the ALOISA beamline of the ELETTRA synchrotron. One of
us (R.G.) is grateful to D.D. Sarma for the fruitful discussion. The authors
are deeply indebted to Fabiana Da Pieve for enlightening discussions.

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