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# Thickness and temperature dependence of the atomic-scale structure of SrRuO3
thin films

Xuanyi Zhang  Department of Physics, North Carolina State University. Raleigh,
NC, 27695, USA  Aubrey N. Penn  Department of Materials Science and
Engineering, North Carolina State University. Raleigh, NC, 27695, USA  Lena
Wysocki  University of Cologne, Institute of Physics II, 50937 Cologne,
Germany  Zhan Zhang  Advanced Photon Source, Lemont, IL 76019, USA  Paul H. M.
van Loosdrecht  University of Cologne, Institute of Physics II, 50937 Cologne,
Germany  Lior Kornblum  Andrew & Erna Viterbi Department of Electrical
Engineering, Technion Israel Institute of Technology, 3200003 Haifa, Israel
James M. LeBeau  Department of Materials Science and Engineering,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Ionela
Lindfors-Vrejoiu<EMAIL_ADDRESS>University of Cologne, Institute of
Physics II, 50937 Cologne, Germany  Divine P. Kumah<EMAIL_ADDRESS>Department of Physics, North Carolina State University. Raleigh, NC, 27695,
USA

###### Abstract

Due to the strong lattice-property relationships which exist in complex oxide
epitaxial layers, their electronic and magnetic properties can be modulated by
structural distortions induced at the atomic scale. The modification and
control can be affected at coherent heterointerfaces by epitaxial strain
imposed by the substrate or by structural modifications to accommodate the
film-substrate symmetry mismatch. Often these act in conjunction with a strong
dependence on the layer thickness, especially for ultrathin layers. Moreover,
as a result of these effects, the temperature dependence of the structure may
deviate largely from that of the bulk. The temperature-dependent structure of
3 to 44 unit cell thick ferromagnetic SrRuO3 films grown on Nb-doped SrTiO3
substrates are investigated using a combination of high-resolution synchrotron
X-ray diffraction and high resolution electron microscopy. This aims to shed
light on the intriguing magnetic and magnetotransport properties of epitaxial
SRO layers, subjected to extensive investigations lately. The oxygen
octahedral tilts and rotations are found to be strongly dependent on the
temperature, the film thickness, and the distance away from the film-substrate
interface. As a striking manifestation of the coupling between magnetic order
and lattice structure, the Invar effect is observed below the ferromagnetic
transition temperature in epitaxial layers as thin as 8 unit cells, similar to
bulk ferromagnetic SrRuO3.

##  I Introduction

Bulk SrRuO3 (SRO) is a 4d itinerant oxide metal with a ferromagnetic Curie
temperature ($T_{c}$) of 160 K.Cao  _et al._ (1997); Mazin and Singh (1997);
Koster  _et al._ (2012); Klein  _et al._ (1996) The system has been
investigated widely as an oxide electrode in all-oxide based devices and, most
recently, as a possible host of magnetic skyrmions manifesting a topological
Hall effect (THE).Gu  _et al._ (2019); Kimbell  _et al._ (2020); Matsuno  _et
al._ (2016); Ziese  _et al._ (2020) Research efforts to characterize the
atomic-scale structure of thin SRO films are motivated by observed thickness-
dependent metal-insulator and magnetic transitions. Xia  _et al._ (2009);
Chang  _et al._ (2009); Ishigami  _et al._ (2015) A thickness-dependent metal-
insulator transition (MIT) occurs in epitaxial SRO thin films fabricated by
molecular beam epitaxy and pulsed laser deposition below a critical thickness
of 2-4 unit cells(uc). Xia  _et al._ (2009); Ishigami  _et al._ (2015) The
suppression of ferromagnetism in thin SRO layers is attributed to a decrease
in the density of states at the Fermi level due to quantum confinement and the
possible existence of an antiferromagnetic interfacial layer for ultrathin SRO
films.Chang  _et al._ (2009) An important property of the SRO films grown on
STO(100) is their exceptionally strong perpendicular magnetic anisotropy,
preserved in films as thin as few unit cells. Schultz  _et al._ (2009); Ziese
_et al._ (2010); Wakabayashi  _et al._ (2021) This magnetic anisotropy is
directly related to the intriguing observation of humplike features in Hall
effect resistance loops of thin SRO films and heterostructures containing SRO
layers. The humps of the Hall loops have been interpreted as a signature of
the THE originating from the formation of skyrmions, as a result of a
surface/interface-induced or defects-induced Dzyaloshinskii-Moriya
interaction. This is however still under debate, as other mechanisms apart of
a THE contribution can explain the occurrence of humplike features in the Hall
loops as well.Matsuno  _et al._ (2016); Ziese and Lindfors-Vrejoiu (2018);
Sohn  _et al._ (2021); Huang  _et al._ (2020); Lu  _et al._ (2021); Wysocki
_et al._ (2020) Multiple factors are known to contribute to the anomalous Hall
effect in SRO including stoichiometry, structure, temperature and layer
thickness. Schultz  _et al._ (2009); Ziese and Lindfors-Vrejoiu (2018);
Wysocki  _et al._ (2020) Hence, a detailed understanding of the atomic-scale
properties of SRO is required to understand its complex electronic and
magnetic properties.

Due to the strong coupling between the lattice structure and the electronic
and magnetic properties of SRO, the atomic scale structure of SRO films has
been investigated theoretically and experimentally as a function of the
substrate-induced strain and the coupling of oxygen octahedra across
heterointerfaces, to uncover the origins of the thickness-dependent properties
of the system.Ziese  _et al._ (2019); Xia  _et al._ (2009); Lu  _et al._
(2013); Samanta  _et al._ (2021) For example, signatures of the THE effect
have been attributed to a substrate induced local orthorhombic to tetragonal
phase transition which may stabilize a chiral spin structure Gu  _et al._
(2019) and/or magnetic inhomogeneity arising, for example, from variations in
layer thickness.Kimbell  _et al._ (2020) Thus, understanding the atomic-scale
structural and interfacial interactions is crucial for decoupling intrinsic
and extrinsic origins of the unique physical properties of SRO thin films.

Figure 1: Schematic of structural distortions in orthorhombic SrRuO3 thin
films along pseudocubic (c) directions defined by the (001)-oriented SrTiO3
substrate. $\alpha_{rot}$, $\beta_{rot}$ and $\gamma_{rot}$ represent the
rotation angles about the [100]c, [010]c and [001]c axes, respectively. The
orthorhombic structure is also characterized by anti-parallel displacements of
the Sr planes along [001]c and [010]c directions.

In this work, we report on the layer-resolved atomic scale structure of thin
SRO films as a function of temperature (10 - 300 K) and film thickness (1.2-17
nm), based on high resolution synchrotron X-ray crystal truncation rod
measurements (CTR) and reciprocal space maps (RSMs) and scanning transmission
electron microscopy (STEM). We observe a suppression of
orthorhombic/monoclinic distortions within 2-3 interfacial SRO unit cells for
all film thicknesses for films grown on (001)-oriented SrTiO3 substrates. The
magnitude of the oxygen octahedral distortions increases on cooling between
300 K and the ferromagnetic transition temperature (for bulk, TC= 160 K) and
remains constant below TC due to the Invar effect, a prominent example
demonstrating the strong coupling between structure and magnetic order.Kiyama
_et al._ (1996)

Bulk SRO has a GdFeO3-type orthorhombic crystal structure at 300 K with
ao=5.567 Å, bo= 5.530 Å, co=7.845 Å and $\beta=90^{o}$.Jones  _et al._ (1989)
The oxygen octahedra are rotated out-of-phase along the pseudo-cubic (c)
[100]c and [010]c axes, in-phase along the [001]c axis(the o and c subscripts
refer to the orthorhombic and cubic coordinates, respectively). In addition to
the oxygen octahedral rotations, anti-parallel Sr cation displacements are
present. Figure 1 shows the expected distortions relative to the cubic STO
lattice vectors. In Glazer notation, the oxygen octahedral rotational pattern
of orthorhombic SRO is represented as $a^{-}a^{-}c^{+}$ where the ’-’
superscript represents out-of-phase rotations and the ’+’ superscript
represents in-phase rotations.Glazer (1972); Woodward and Reaney (2005) At
high temperatures (above 570 K), bulk SRO transitions into a tetragonal phase
with in-phase rotations about the [001]c axis and no tilts about the [100]c
and [010]c axis. The tetragonal phase is represented in Glazer notation as
$a^{0}a^{0}c^{+}$.

Recent reports on the structure of SRO epitaxial thin films indicate that the
crystal structure of SRO can be controlled by strain, the thickness of the SRO
layers, the oxygen stoichiometry and the octahedral rotational pattern of the
substrate.Vailionis  _et al._ (2008); Gao  _et al._ (2016); Lu  _et al._
(2013); Aso  _et al._ (2013); Chang  _et al._ (2011) SRO films compressively
strained to cubic (001) SrTiO3 (STO) (a=b=c= 3.905 Å) are found to have a
distorted orthorhombic(monoclinic) structure with the orthorhombic c-axis,
[001]o, parallel to the in-plane cubic [010]c axis of the substrate, and the
orthorhombic[110]o parallel to the out-of-plane [001]c axis.Vailionis  _et
al._ (2011); Gao  _et al._ (2016) The preferential orientation of the
orthorhombic axis is dictated by the step terraces.Maria  _et al._ (2000); Rao
_et al._ (1997); Gan  _et al._ (1998)

##  II Experimental Details

SrRuO3 samples used in our experiment were grown by pulsed laser deposition
(PLD). The samples were grown at a substrate temperature of 650-700 oC and
oxygen pressure of 100 mTorr, with a laser fluence of 1.5-2.0 J/cm2 and a
pulse repetition rate of 5 Hz. Representative atomic force microscope (AFM)
images of the surface morphology of the 8, 16 and 44 unit cell (uc) SRO films
are shown in supplemental Figure S1. Clear one unit cell high step terraces
are resolved in the AFM images, inherent to the vicinal surface morphology of
the Nb-doped (0.5 wt.$\%$ Nb) STO(100) substrates. The substrates were
annealed at 925oC for 1 hour in air, after they had been etched for 2.5 min in
buffered HF solution.

Crystal truncation rods and half-order superstructure reflections were
measured between 10 K and 300 K for 3, 8, 16 and 44 uc thick SrRuO3 films. The
diffraction measurements were performed at the 33ID beamline at the Advanced
Photon Source using a photon energy of 16 keV ($\lambda=0.7749\AA{}$). The
X-ray beam was focused to a spot size of 50 $\mu$m - 100 $\mu$m. The
diffraction intensities were measured with a 2D Pilatus 100 K
detector.Schlepütz  _et al._ (2005) Samples were mounted in a Be-dome chamber
with a base pressure of $<1\times 10^{-5}$ Torr on a cryo-displex and the
temperature was varied from 300 K to 10 K. The CTRs and half-order reflections
were fit using the GenX genetic-based X-ray fitting algorithm to determine the
structural properties of the films.Björck and Andersson (2007); Koohfar  _et
al._ (2019)

The structural properties of the films were also characterized with aberration
corrected STEM. Samples for STEM analysis were prepared by conventional wedge
polishing followed by Ar ion milling. Imaging and spectroscopy were performed
on an aberration corrected FEI Titan G2 60-300 kV STEM operated at 200 kV. By
simultaneously acquiring annular dark field (ADF) and integrated differential
phase contrast (iDPC) images, the structure of both cationic and oxygen
sublattices can be analyzed. The revolving STEM (RevSTEM) imaging
techniqueSang and LeBeau (2014) was used to improve the signal-to-noise ratio
to allow for the resolution of the shape of the atom columns revealing
structural features beyond atom column positions. The film structure was
determined by fitting each atom column to a two-dimensional Gaussian function
allowing for a determination of the column amplitude (point image intensity),
x (in-plane) - and y (out-of-plane)-positions, as well as the x and
y-components of peak widths, $\sigma_{x}$ and $\sigma_{y}$. The local
distortions and tilting of oxygen octahedra are characterized by the oxygen
column positions and ellipticity, defined as
$E=\frac{\sigma_{x}}{\sigma_{y}}$. With the ellipticity, we characterize the
out-of-phase octahedral tilting along a column.

##  III Results and Discussion

Figure 2 shows the measured specular (00L) CTRs for the 8 and 16 uc samples.
The presence of finite thickness oscillations is indicative of flat surfaces
and a chemically abrupt SRO/STO interface. The out-of-plane lattice parameters
determined from fits to the (00L) CTRs for the 8, 16 and 44 uc SRO films are
3.957, 3.955 and 3.948 Å, respectively. The increased lattice spacing compared
to the bulk pseudocubic value for SRO (a${}_{c,bulk}=3.93\AA{}$) is due to the
biaxial compressive in-plane strain imposed by the STO substrate. The films
are coherently strained to the STO substrate with an in-plane pseudocubic
lattice constant of 3.905 Å as evidenced by RSMs measured by X-ray diffraction
(Figure 3(a)).

Figure 2: Specular 00L scans for 8 and 16 unit cell thick SRO films on STO.
The plots are offset vertically for clarity. Figure 3: Reciprocal space map
around the STO [204]c Bragg peak showing the in-equivalent orthorhombic SRO
Bragg peaks for a 44 uc SRO film. (1 STO reciprocal lattice unit = 1/0.3905
nm-1) (b) Measured (blue circle) line profiles along the STO L direction in
(a) and fits (solid red lines). (c) Measured half-order rods (d) Annular dark
field images of the 44 uc SRO film along the [100]c projections with (e)
corresponding iDPC images. The inset shows a magnification of the iDPC image
with the XRD structure overlayed. (f) Measured in-phase tilting angle map
along the [100] zone axis with respective layer-resolved (g) in-phase tilt
angles and h) ellipticity profiles.

The reduced symmetry of the thickest (44 uc) SRO film relative to the cubic
Pm$\bar{3}$m STO substrate is observed in RSMs measured around the STO
(204)c,(024)c,(-204)$c$ and (0-24)c Bragg peaks. The measured RSMs are
compared in Figure 3(a) where 1 STO reciprocal lattice unit (r.l.u.) =
1/0.3905 nm-1. The different L values for the SRO (6,2,0)o and (2,6,0)o Bragg
peaks with fixed values of Qin-plane is due to the monoclinic distortion of
the SRO lattice.Vailionis  _et al._ (2011) The angle $\beta_{mon}$, between
the $a_{o}^{SRO}$ and $b_{o}^{SRO}$ axes is determined from the relation
$\beta_{mon}=90-arctan(\frac{\Delta Q_{z}}{Q_{inplane}})$Gao  _et al._ (2016).
From the RSMs we determine the monoclinic SRO lattice parameters to be
$\beta_{mon}=89.5^{o}$, $a_{o}^{SRO}=5.72\AA{}$, $b_{o}^{SRO}=5.52\AA{}$ and
$c_{o}^{SRO}=7.81\AA{}$. The lattice parameters are also confirmed from fits
to the in-equivalent (04L)c CTRs and half order peaks in Figure 3(b) and 3(c).

The RSMs for the 16 uc film around the STO $\\{204\\}_{c}$ Bragg peaks are
shown in Figure 4(a). The film Bragg peaks for the 16 uc sample are broader
along the L direction than the 44 uc sample due to the effect of the finite
thickness broadening and the increased fraction of the multiple rotational
domains. Thus, care must be taken in relying solely on RSMs in verifying the
orthorhombicity of the lattice.

The octahedral rotations about the out-of-plane [001]c axis and tilts about
the in-plane [100]c and [010]c axes can be qualitatively predicted from the
prescence/absence of half-order reflections. A c+ in-phase rotation along the
[001]c axis results in reflections of type
$\frac{1}{2}$(odd,odd,even)c.Woodward and Reaney (2005); May  _et al._ (2010)
The absence of peaks at integer L along the (3/2, 1/2, L)c rod for the 44 uc
film in Figure 3(c) indicates that the axis with in-phase rotations does not
lie along the out-of-plane [001]c axis.

Bragg peaks are expected for an $a^{+}b^{-}c^{-}$ structure for reflections of
type $\frac{1}{2}$(even,odd,even)c and $\frac{1}{2}$(odd,odd,odd)c. The (0 1/2
even)c peaks observed in Figure 3(c) indicate an a+ tilt. Based on the
observed reflections for the 44 uc film, the tilt system of the SRO on STO is
determined to be $a^{+}b^{-}c^{-}$. However, the presence of reflections of
type $\frac{1}{2}$(even, odd, odd)c signifies the presence of an
$a^{-}b^{+}c^{-}$ domain with the in-phase tilt along the [010]c direction.

Figure 4: (a) Reciprocal space maps for the 16 uc SRO/STO sample.(b) Measured
(blue circle) half order rod and fits (solid red lines).(c) Intensity of half
order peaks as the function of temperature.(d) Sr displacement and octahedral
rotation angles as functions of temperature.

Due to the four-fold symmetry of the cubic STO substrate, 4 rotational
variants of the SRO unit cell are expected with the [001]o axis aligned along
either the STO [100]c,[-100]c,[010]c or[0-10]c axis.Gao  _et al._ (2016);
Vailionis  _et al._ (2011); Siwakoti  _et al._ (2021) The half order peaks
associated with the $a^{-}b^{+}c^{-}$ domain are relatively weak for the 44 uc
sample and the fraction of the film with the $a^{-}b^{+}c^{-}$ orientation is
less than 5% of the total film volume. However, for the 16 uc sample, the
ratio of the $a^{+}b^{-}c^{-}$ domain fraction to the $a^{-}b^{+}c^{-}$ domain
is 3:1 suggesting that the film transitions into a single domain structure as
the thickness increases or slight differences exist in the miscut angles and
direction of the STO substrates. Maria  _et al._ (2000); Wang  _et al._
(2020); Kar  _et al._ (2021) Indeed topographic atomic force microscopy
investigations (supplemental Fig. S1) showed that the STO(100) substrate used
for the 16 uc sample has tilted terraces with respect to its edges, while the
one used for the 44 uc has terraces running almost parallel to the substrate
edges. It was shown that more tilted terraces promote the formation of
crystallographic domains with different in-plane orientations of the long
orthorhombic c axis.Wang  _et al._ (2020)

Parameters | 8 uc | 16 uc | 44uc
---|---|---|---
c (Å) | 3.957 | 3.955 | 3.948
$\alpha_{rot}$ (in-phase), 300 K | 0.9o | 3.4o | 5.31o
$\beta_{rot}$ (out-of-phase), 300 K | 3.3o | 6.4 o | 5.72o
$\gamma_{rot}$ (out-of-phase), 300 K | 9.9o | 9.7o | 5.21o
$\alpha_{rot}$ (in-phase),130 K | 1.7o | 4.2o | -
$\beta_{rot}$ (out-of-phase),130 K | 2.6o | 6.6o | -
$\gamma_{rot}$ (out-of-phase) 130 K | 9.6o | 9.1o | -
$Ru-O-Ru$ | 159o | 162o | 164o
$Sr\delta_{z}A$ (300 K) | 0.009 | 0.040 | 0.062
$Sr\delta_{z}A$ (130 K) | 0.035 | 0.060 | -
Table 1: Comparison of structural parameters of non-interfacial and non-
surface layers of SRO films as a function of thickness and temperature.

The local structure of the 44 uc film is investigated by STEM measurements
shown in Figure 3(d)-(h). The ADF image in Figure 3(d) indicates a chemically
abrupt interface between the SRO film and the STO substrate. The oxygen
sublattice is imaged using the iDPC technique. Along the [100]c projection
where the oxygen octahedra rotate in-phase, the tilt angles are determined
directly from the atomic positions of the oxygen atoms in the iDPC image in
Figure 3(e). Figure 3(f) shows a map of the unit-cell resolved oxygen
octahedral tilts about the [100]c axis. A depth profile of the magnitude of
the layer-averaged in-phase rotation angle is shown in Figure 3(g). There are
no in-phase rotations in the STO substrate as expected for bulk STO. The
rotation angle increases gradually from 0 to 5∘ within the three interfacial
SRO layers and remains uniform till the surface layer where a reduction
occurs. The suppressed in-phase tilt at the film-substrate interface is
consistent with fits to the X-ray diffraction data discussed below and
previous reports.Siwakoti  _et al._ (2021)

A projection along the [010]c direction where the octahedra rotate out-of-
phase results in an asymmetric smearing of the oxygen atomic columns. Thus,
the projected out-of-phase octahedral tilting is determined by the ellipticity
$E=\sigma_{x}/\sigma_{y}$ of the atomic columns, where $\sigma_{x}$ and
$\sigma_{y}$ represent the horizontal and vertical peak widths measured from
fitting the iDPC images. The layer-averaged ellipticity profile is shown in
Figure 3(h) along [010]c projection. In the AO planes, $E<1$ since the
rotations lead to O displacements in the vertical direction. Conversely, $E>1$
in the $BO_{2}$ planes due to O displacements in the vertical direction. $E=1$
corresponds to no rotations. The alternating values of $E$ about 1 in Figure
3(h) corresponds to successive SrO and $BO_{2}$ (B=Ti, Ru) planes. Some
ellipticity is measured in the STO substrate, likely due to low order
aberrations and/or rotations in the interfacial STO layers induced by the SRO
adlayer.

The iDPC images provide a direct quantitative measure of the magnitude of the
in-phase rotations about the [100]c axis. To quantitatively determine the
layer-resolved rotations about the [010]c and [001]c axes and the orthorhombic
Sr displacements, the CTR’s and half-order rods measured by synchrotron X-ray
diffraction are analyzed using the GenX genetic fitting algorithm.Björck and
Andersson (2007) To account for differences in the structure of the
interfacial 3 unit cells, separate fit (rotation angles, Sr displacements)
parameters are assigned to the interface layers and the non-interfacial
layers.

Table 1 summarizes the structural parameters determined for the SRO films as a
function of thickness at 300 K and 130 K. The measured octahedral rotation
angles for the non-interfacial layers of the 44 uc film about the orthogonal
cubic axes are $\alpha_{rot}=5.3^{o}$, $\beta_{rot}=5.7^{o}$,
$\gamma_{rot}=5.2^{o}$ in good agreement with bulk values of 6.19o, 5.97o and
5.97o respectively. A suppression of tilts and Sr displacement is found at the
interface due to the structural coupling to the cubic STO substrate which
possesses no octahedral rotations at 300 K. This is consistent with the iDPC
results in Figure 3(g). No evidence for off-center oxygen displacements in the
STO and SRO were found as has been recently reported in 4 uc SRO films. Sohn
_et al._ (2021)

For the 16 uc SRO film at 300 K, the in-phase rotation angle is suppressed to
3.4o while the rotation about c-axis increases (relative to bulk) to 9.7o. The
distorted orthorhombic structure at 300 K is in contrast to the tetragonal
a0b0c- structure reported for SRO films with thicknesses below 17 uc.Chang
_et al._ (2011) Since oxygen vacancies can stabilize the tetragonal structure,
the discrepancy is most likely related to the oxygen stoichiometry.

The evolution of the structure of the 16 uc film with temperature between 10 K
and 300 K is determined from fits to the temperature-dependent half order
rods. Representative measured half order rods and fits at 300 K for the 16 uc
film are shown in Figure 4(b). The intensity of the (0 -0.5 L)c and (0.5 -0.5
1.5)c peaks measured as a function of temperature from 300 K to 10 K for the
16 uc film are shown in Figure 4(c). The intensities of the half order peaks
associated with the in-phase octahedral tilts and Sr displacements increase as
the temperature decreases to the FM-PM transition at $\approx$150 K. Below 150
K, the intensity of the (0 -0.5 L)c peaks remains constant indicating a
freezing of the octahedral distortions in the ferromagnetic phase. El-Mellouhi
_et al._ (2011) A sharp increase in the intensity of the (0.5 -0.5 L)c peaks
is observed below 105 K where the antiferrodistortive STO phase transition
occurs. The STO phase transition involves rotation of the TiO6 octahedra in
the STO substrate leading to a doubling of the STO unit cell and the emergence
of the STO substrate $\frac{1}{2}$(odd, odd,odd)c peaks. The temperature-
dependent structural parameters for the 16 uc SRO film are summarized in
Figure 4(d). The Sr displacements in the SRO layers increase from 0.04 Å at
300 K to 0.06 Å at 150 K and the in-phase rotation angle increases from 3.3o
at 300 K to 4.2o at 150 K.

In contrast to the thicker films, the intensity of of the half-order
reflections for the 8 uc film associated with the in-phase tilts and Sr
displacements are strongly suppressed at 300 K. The STEM analysis and the XRD
results are described in Figure S3 and S4 of the supplemental materials. The
rotation angles at 300 K away from the film-substrate interface are
$\alpha_{rot}=0.9^{o}$, $\beta_{rot}=3.3^{o}$ and $\gamma_{rot}=9.9^{o}$. The
suppression of the in-plane tilts and the enhancement of the rotation around
the c-axis relative to bulk leads to an in-plane Ru-O-Ru bond angle of 159o
which is slightly less than the value of 162∘ for bulk orthorhombic SRO. Fits
to the half-order reflections at 130 K in the ferromagnetic phase show a
slight increase in the in-phase tilt angle to 1.7 o but still less than the
expected value for bulk SRO.

Figure 5: (a) Measured half-order rods as a function of temperature for a 3 uc
SRO film. (b) Reciprocal space map along the H=1.5 K=0.5 L rod and (c) Model
structure of rotations in SRO film and interfacial STO layers.

Octahedral distortions are also observed for the thinnest 3 uc SRO sample.
Half-order Bragg peaks are observed for $\frac{1}{2}$(odd, odd, odd)
reflections for the 3 uc SRO sample while peaks associated with in-phase
rotations and Sr displacements are absent between 300 K and 10 K. A
$a^{0}a^{0}c^{-}$ tetragonal film structure can be ruled out since (1/2 1/2
$odd/2$) diffraction peaks are present as shown in Figure 5(a) which indicate
additional tilts about the [001]c and [010]c axes. The peak intensities
increase as the temperature decreases from 250 K to 150 K indicating an
increase in the magnitude of the octahedral tilts and rotations with
decreasing temperature. The tilts and rotations observed in the thinnest
sample are consistent with low energy electron diffraction results on single
unit cell thick SRO films reported by Siwakoti et al.Siwakoti  _et al._ (2021)
The peak widths along the L direction in the RSM image in Figure 5 is narrower
than width expected for a 3 uc thick film suggesting that the octahedral
rotations extend 2-3 layers into the STO substrate as it was observed for
CaRuO3 and (La,Sr)MnO3 films on (001)-SrTiO3 by STEM imaging.Siwakoti  _et
al._ (2021); Koohfar  _et al._ (2019) The induced rotations in the STO are
found to be $<$ 1o in magnitude and out-of-phase around the [010] and [001]
axes and thus, will be difficult to observe directly in ABF images. However,
the half-order reflections obtained by synchrotron X-ray diffraction
measurements are sensitive to these distortions.

The thickness and temperature-dependent structural results are summarized as
follows: 1) a suppression of orthorhombic distortions occurs within the 2-3
SRO layers at the SRO/STO interface consistent with theoretical predictionsHe
_et al._ (2010); 2) reduced A-site displacements are correlated with a
decrease in the magnitude of in-phase rotations and an increase in the oxygen
octahedral rotations about the c-axis as the film thickness is reduced from 44
uc to 8 uc; 3) freezing of rotations and A-site displacements occurs below the
paramagnetic-ferromagnetic transition temperature.

The suppression of octahedral rotations about the orthogonal in-plane axis for
the SRO layers close to the SRO/STO interface is expected due to structural
coupling with the cubic STO substrate and these results are consistent with
previous reports on the SRO/STO interface structure. We find that the
orthorhombic distortions are present, but weak in the 8 uc film, with the
magnitude of the distortions increasing with decreasing temperature.

The freezing out of distortions in the 8, 16 and 44 uc films below TC arises
due to the coupling of the magnetic moment ordering to the lattice structure.
This effect which is also known as the Invar effect is characterized by a
freezing out of the unit cell volume and octahedral rotations below the
ferromagnetic transition temperature, leading to anomalously low coefficient
of thermal expansion: This effect was observed in bulk SRO and our results
confirm the existence of the Invar effect in ultrathin strained SRO
films.Dabrowski  _et al._ (2006); Kiyama  _et al._ (1996); Bushmeleva  _et
al._ (2006)

We find that, while the 8 uc sample is close to an orthorhombic-tetragonal
transition at 300 K, the orthorhombic distortions become stronger as the
temperature decreases and the low temperature structure, where humplike
anomalies of the Hall resistance loops were observed and taken for a THE
fingerprint Gu  _et al._ (2019), is orthorhombic. Based on structural
investigations done at 300 K, Gu et. al. proposed that in their 8 uc SRO
films, the interfacial RuO6 octahedral tilting induced by a local
orthorhombic-to-tetragonal structural phase transition across the SRO/STO
interface resulted in breaking the inversion symmetry. This was further used
to account for an interfacial DMI and to explain why a THE contribution may
occur in the Hall resistance loops Gu  _et al._ (2019). Our findings are
therefore very important, because we have information about the structure at
the temperature where the physical properties are measured. The temperature
and thickness dependent structures of SRO layers is highly relevant for
understanding the magnetocrystalline anisotropy, the anomalous Hall effect and
the temperature dependence and sign of the anomalous Hall constant for
tetragonal and orthorhombic SRO phases. Kan  _et al._ (2013); Ziese  _et al._
(2019); Bern  _et al._ (2013); Samanta  _et al._ (2021)

In addition to the structural distortions observed in this work, local
variations in film thickness arising from the step-flow-growth mechanism can
lead non-uniform coercivity and humplike features in magnetoresistance
measurements which mimick the THE. Malsch  _et al._ (2020); Kimbell  _et al._
(2020) Magnetic force measurements indicate lateral variations in the
coercivity of nominally 4 uc films capped with SrIrO3/SrZrO3 layers which may
be correlated with local variations in the film thickness. For the thicker
films (8-44 uc), the surface occupation of the top 2-3 layers is slightly less
than unity, indicating that the local film thickness averaged over the X-ray
probe area (100 $\mu$m) fluctuates within this range. These variations become
more critical as the film thickness is reduced below 8 unit cells.

While this analysis assumes no structural or chemical changes occur upon the
surface exposure to ambient conditions, in some oxides such changes can be
quite significant.Caspi  _et al._ (2022) SRO films exposed to ambient
atmosphere and heated in vacuum may decompose.Shin  _et al._ (2005) However,
the strong signal originating both from the half order reflections and the
RSMs (Figure 5) indicate that even at 3 uc, a significant portion of the film
retains its high quality crystalline structure. Overoxidation and other
surface reactions could sometimes result in amorphous phases, which would not
be observable in diffraction experiments. Nonetheless, the strong diffraction
intensity (originating from inherently low-intensity reflections) indicates
that this possibility is not playing a significant role in the current case.

##  IV Conclusion

In summary, the atomic-scale structure of SRO films was investigated as a
function of temperature and film thickness. The ultrathin films (3 and 8 uc
thick) show the most pronounced structural changes with respect to the bulk
structure. An increase of the magnitude of the oxygen octahedral rotations and
Sr displacements is observed as the temperature is reduced from 300 K to the
ferromagnetic transition, below 130 K. The freezing out of the octahedral
distortions observed below 130 K in the films as thin as 8 unit cells is
associated with the Invar effect, which is known to occur in bulk
ferromagnetic SRO, as a consequence of the coupling between the crystal
structure and ferromagnetic ordering. The thickness-dependent structure of the
SRO films may be related to kinetic and thermodynamic effects during
nucleation of the orthorhombic structure of SRO thin film on the cubic STO
substrate. Further structural investigations shall enable an in depth
understanding of the unique properties of SRO thin films, with the particular
motivation of shedding light on the intriguing magnetotransport properties
reported for ultra-thin films.

## Acknowledgments

The authors acknowledge financial support by the US National Science
Foundation under Grant No. NSF DMR-1751455. This work was performed in part at
the Analytical Instrumentation Facility (AIF) at North Carolina State
University, which is supported by the State of North Carolina and the National
Science Foundation (award number ECCS-2025064). The AIF is a member of the
North Carolina Research Triangle Nanotechnology Network (RTNN), a site in the
National Nanotechnology Coordinated Infrastructure (NNCI). This material is
based upon work supported by the National Science Foundation under Grant No.
DGE-1633587. We thank René Borowski and Silvia de Waal for etching the STO
substrates and Regina Dittmann and Felix Gunkel for access to the PLD system
at FZ Jülich. I.L.-V. acknowledges the financial support from the German
Research Foundation (DFG) for project no. 403504808 within SPP2137 and for
project no. 277146847 within SFB1238 (project A01). L. K. thanks German
Israeli Foundation for financial support (GIF Grant no. I-1510-303.10/2019).
Use of the Advanced Photon Source was supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, under Contract No.
DE-AC02-06CH11357.

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Towards an integrated determination of
proton, deuteron and nuclear PDFs

Tanjona R. Rabemananjara

Department of Physics and Astronomy, Vrije Universiteit, NL-1081 HV Amsterdam
Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands

> We present progress towards a unified framework enabling the simultaneous
> determination of the parton distribution functions (PDFs) of the proton,
> deuteron, and nuclei up to lead $(^{208}\rm{Pb})$. Our approach is based on
> the integration of the fitting framework underlying the nNNPDF3.0
> determination of nuclear PDFs into that adopted for the NNPDF4.0 global
> analysis of proton PDFs. Our work paves the way toward a full integrated
> global analysis of non-perturbative QCD – a key ingredient for the
> exploitation of the scientific potential of present and future nuclear and
> particle physics facilities such as the Electron-Ion Collider (EIC).

> PRESENTED AT
>
>
>
>
> DIS2023: XXX International Workshop on Deep-Inelastic Scattering and Related
> Subjects,
> Michigan State University, USA, 27-31 March 2023
>

## Introduction.

Future measurements at the recently approved Electron-Ion Collider (EIC) [1,
2] and at the High-Luminosity Large Hadron Collider (HL-LHC) [3] will perform
key measurements that can pin down the dynamics of Quantum Chromodynamics
(QCD). By colliding (polarized) beams with proton or deuteron or heavier
nuclei for a range of center-of-mass energies, these experiments will be
fundamental in understanding how partons are distributed in momentum and
position spaces within a proton and how the parton distribution of nucleon
bound within nuclei are modified w.r.t. their free-nucleon counterparts.

In order to fully interpret the precision-level measurements offered by such
experiments, the theoretical determination of both the unpolarized proton and
nuclear parton distribution functions, henceforth referred to as (n)PDFs,
needs to be improved. While the determination of the free-proton PDFs has seen
quite an advancement in recent years [4] – owing mostly to the availability of
a broad experimental dataset and theoretical improvements in higher-order
calculations – the current status in the determination of nuclear PDFs is
somewhat less advanced. In addition, albeit both proton and nuclear PDFs are
simultaneously constrained by datasets where one of the targets or projectiles
is not a free-state proton, their extractions are usually performed in a
separate manner.

On one hand, it has been well understood that measurements involving deuteron
and heavier nuclear targets play a significant role in disentangling the
proton’s quark and antiquark distributions, and in separating the up and down
PDFs for large momentum fractions in which searches for physics beyond the
Standard Model (BSM) are relevant [5]. In the past, different approaches have
been adopted to account for nuclear corrections in proton PDF fits, each with
its own motivations and limitations. The approach adopted in the NNPDF
methodology [6, 7] consists in adding the uncertainties due to nuclear effects
as an extra contribution to the theory covariance matrix [8, 9, 10].

On the other hand, all hadronic datasets included in the determination of
nuclear PDFs involve a free-proton in the initial state. Most nuclear PDF sets
are therefore determined assuming a fixed proton baseline which can be
considered as a theoretical bias whose effect is difficult to estimate. In the
nNNPDF methodology [11, 12, 13], although the $A=1$ dependence (with $A$
representing the atomic mass number) is also fitted in the same footing as
$A\neq 1$ by means of a neural network, a fixed proton baseline is still
required to enforce the $A\to 1$ limit, therefore introducing a potential bias
in that the back-reaction of the fitted data on the proton PDf is ignored.
Such a constraint is imposed at the level of the $\chi^{2}$ as a penalty by
means of a Lagrange multiplier.

Motivated by the need for a consistent and concurrent extractions of the
unpolarized proton, deuteron, and heavier nuclear PDFs, we present here work
in progress towards an “integrated fit” (see also [14] for a first attempt) in
which the atomic mass number dependence is smoothly parametrized from $A=1$ to
$A=208$ by fitting simultaneously to proton, deuteron, and heavier nuclear
datasets, removing the need for imposing a boundary condition for the $A=1$
limit. A similar idea was applied to PDFs and fragmentation functions by the
JAM collaboration in [15, 4]. Our approach is based on the integration of the
fitting framework underlying the nNNPDF3.0 determination of nuclear PDFs into
that adopted for the NNPDF4.0 global analysis of proton PDFs [7]. As a first
attempt to apply our methodology, we perform fits in which only deep inelastic
scattering (DIS) processes are included using NNLO QCD calculations with
perturbative charm.

Here we briefly describe the integrated fitting framework, emphasizing on the
its main differences w.r.t the (n)NNPDF approaches. We then study the
stability of the integrated fits based on the NNPDF4.0 default hyperparameters
from which all the subsequent results are derived. We assess the impacts of
the integrated fitting methodology on the proton and nuclear PDFs by comparing
the results with the reference (n)NNPDF determination. Conclusions are drawn
in the last section.

## Methodology.

As mentioned above, the integrated fitting methodology incorporates the
nuclear PDF parametrization from nNNPDF3.0 into the determination of free-
proton PDF using the NNPDF4.0 methodology. The reason for this is twofold.
First, albeit the nNNPDF3.0 is also based on deterministic minimization
algorithm it does not provide some of the advanced machine learning techniques
that the NNPDF4.0 possesses – such as the automatic tuning of the
hyperparameters using the $k$-folding procedure [6, 7]. In addition, the
NNPDF4.0 methodology also provides ways to carefully validate the resulting
PDF uncertainties via the closure and future test approaches. Nonetheless, the
two determinations share a large number of similarities, one among which is
the parametrization of the (n)PDFs in the evolution basis.

The main modification to the NNPDF4.0 methodology to account for the nuclear
fit consists in parametrizing the
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}$-dependence
of the (n)PDFs during the fit. That is, the relation between the output of the
neural network and the (n)PDFs is given by:

$xf^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}_{k}(x,Q_{0};\theta)=\eta_{k}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}x^{1-\alpha_{k}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}}(1-x)^{\beta_{k}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}}\mathrm{NN}_{k}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}(x,Q_{0};\theta),$
(1)

where $k$ runs over the elements of the PDF in the evolution basis,
$\mathrm{NN}_{k}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}(x;\theta)$
is the $k$-th output of the neural network, $\theta$ indicates the full set of
neural network parameters,
$\eta_{k}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$
is the normalization corresponding to the $k$-th PDF, and
${{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$
represents the atomic mass number of the proton/deuteron/nucleus. Note that in
principle the preprocessing exponents
$\alpha_{k}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$
and
$\beta_{k}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$
should also depend on
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}$. The
way in which such a dependence on the atomic mass number
${{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$ is
propagated through the fitting framework is illustrated in Fig. 1.

$\\{x_{n}^{(k)},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}\\}$Neural
Net$x^{\alpha}$
$(1-x)^{\beta}$$\tilde{f}_{j}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}(x^{(k)}_{n})$normalization⋮${f}_{j}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}(x^{(k)}_{1})$${f}_{j}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}(x^{(k)}_{n})$$\otimes$$\otimes$$\otimes$FK${}_{1k}^{j,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$FK${}_{nk}^{j,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$⋮$\mathcal{O}_{1}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$$\mathcal{O}_{n}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}}$⋮
Tr/Vl split $\chi^{2}_{\rm tr}$$\chi^{2}_{\rm val}$

Figure 1: A modified version of Fig. 3.2 from [6] with the additional
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}A}$-dependence.

In order to test the integrated framework, we focus in the present study on
fits to DIS datasets only which include measurements on proton, deuteron, and
nuclear targets as comprised in the (n)NNPDF determinations. As theory inputs,
we use NNLO QCD calculations provided by the new theory pipeline used in the
NNPDF framework [16]. And in order to be able to compare to previous nNNPDF
PDF releases, we take the approach in which charm is generated perturbatively.
The fitting scale and the kinematic cuts are taken to be the same as in the
default NNPDF4.0 methodology, refer to [6, 7] for more details.

## Training stability.

The baseline hyperparameters used in the present analysis for the integrated
fit are the same as the ones used in the default NNPDF4.0 methodology
extracted from a $k$-folding hyperoptimization procedure. These are listed in
Table 3.7 of [6] with the difference that now the maximum number of epochs is
larger to account for the longer training. Owing to the more complex two-
dimensional parameter space $(x,A)$ to be fitted [17] – as opposed to the one-
dimensional space relevant for a free-proton PDF determination in which only
the $x$-dependence is parametrized – one indeed expects the integrated fit to
converge slower.

Table 1 compares the fit quality between a free-proton fit determined using
the default NNPDF4.0 methodology and a combined proton and nuclear fit
determined using the integrated framework. Displayed are the total
experimental $\chi^{2}_{\rm exp}$ per data point for all the datasets entering
each determination, the average experimental $\langle\chi^{2}_{\rm
exp}\rangle$ over the replica, and the experimental training and validation
error functions $\langle E_{\rm tr}\rangle$ and $\langle E_{\rm val}\rangle$.

Figure 2: Distribution of the training $\chi^{2}$ values over the different
replicas for the default NNPDF4.0 (left) and integrated methodologies (right).
The red square represents the mean value over the replicas.

Figure 3: Distribution of the training lengths over the replicas that passed the postfit selection criteria for the default NNPDF4.0 (left) and integrated methodologies (right). Methodology | $\chi^{2}_{\rm exp}$ | $N_{\rm dat}$ |  $\langle\chi^{2}_{\rm exp}\rangle$ |  $\langle E_{\rm tr}\rangle$ |  $\langle E_{\rm val}\rangle$ |  Tr. Lengths
---|---|---|---|---|---|---
NNPDF4.0 | 1.291 | 1788 |  1.336$\pm$0.028 |  2.328$\pm$0.087 |  2.440$\pm$0.180 |  9200$\pm$3200
Integrated | 1.322 | 3161 |  1.357$\pm$0.030 |  2.360$\pm$0.088 |  2.460$\pm$0.170 |  15700$\pm$6400
Table 1: Comparison of the fit quality between the free-proton only and
integrated fits.

From Table 1 one finds that in the integrated fit a good description of the
experimental data which includes nuclear datasets is achieved, with a total of
$\chi^{2}_{\rm exp}=1.322$ per data point. Looking at the $\chi^{2}$-breakdown
we see that the total experimental $\chi^{2}_{\rm exp}(A=1)=1.192$ on the
proton datasets is much smaller than the value obtained from the free-proton
only fit. This might potentially be a sign of an unbalanced fit in that the
integrated fit is slightly overlearning along the proton direction – which
contains much more data points – while underlearning along the nuclear
directions. Such an instability could be seen when looking at the distribution
of the training $E^{(k)}_{\rm tr}$ and validation $E^{(k)}_{\rm val}$ errors
evaluated over the different MC replicas as shown in Fig. 2. In general, one
expects that the values of $E^{(k)}_{\rm tr}$ are smaller than those of
$E^{(k)}_{\rm val}$, however, as opposed to the case of the free-proton fit,
very few points are located below the diagonal line. This observation is
further strengthened by looking at the distribution of training lengths as
shown in Fig. 3 in which one can see that not only most of the replicas
reached the maximum number of iterations but also exhibit a bimodal
distribution. This indicates that in order to further stabilize the integrated
fits, a $k$-folding hyperoptimization – on the combined proton, deuteron, and
nuclear datasets – is necessary to find the best combinations of
hyperparameters. Given that this is an exhaustive task, we henceforth, present
results based on the default hyperparameters.

Figure 4: Comparison of the free-proton PDFs determined using the default
NNPDF4.0 methodology and the integrated methodology in which the nuclear PDFs
are also determined simultaneously. Solid and dashed bands correspond to 68%
c.l. and one-sigma uncertainties, respectively.

Figure 5: Distance between the central values (left) and the uncertainties
(right) of the integrated free-proton PDF determination w.r.t the proton PDF
determined using the NNPDF4.0 methodology. The results are shown for all PDF
flavors at $Q^{2}=1.7~{}\rm{GeV}^{2}$.

## Towards integrated (n)PDFs.

Figure 6: Comparison between the nNNPDF and integrated approaches for nuclear
PDF determination. The results are shown for ${}^{2}\rm{D}$, ${}^{56}\rm{Fe}$,
and ${}^{208}\rm{Pb}$ for the up and gluon nPDFs at
$Q^{2}=1.65~{}\rm{GeV}^{2}$. The results are normalized to the central values
of the PDFs determined from the nNNPDF approach. The solid and dashed bands
represent the 68% c.l. and one-sigma uncertainties, respectively.

We now study the impacts of the integrated determination by comparing the
resulting (n)PDFs with the proton and nuclear PDFs determined using the
NNPDF4.0 and nNNPDF methodology respectively. We recall that the comparisons
shown henceforth are for DIS-only fits with NNLO QCD theory and perturbative
charm.

Let us first examine the proton-PDFs determined using the integrated approach.
In Fig. 4 we show the comparison of the NNPDF4.0 and integrated approaches for
the up and gluon PDFs as a function of $x$ at the fitting scale
$Q_{0}=1.65~{}\rm{GeV}^{2}$. The results are normalized to the central values
of the PDFs determined in the integrated approach. The solid and dashed
uncertainty bands represent the 68% confidence level interval and the one-
sigma error, respectively. In general, there is an excellent agreement between
the proton PDFs determined using the default NNPDF4.0 methodology and the
integrated approach. Marked wiggles can be observed in the integrated
determination which could be connected to the slight overfitting discussed in
the previous section. The consistency between the two approaches is further
supported by the distance plots shown in Fig. 5. Displayed are the absolute
and variance distance measured from the default NNPDF4.0 determination on the
flavor basis as a function of $x$ at $Q^{2}=1.7~{}\rm{GeV}^{2}$. As we can
see, the differences in both the central values and uncertainties are well
within one sigma.

We now turn to the comparisons of the deuteron and heavier nuclear PDFs. Fig.
6 displays the comparisons between the nNNPDF and integrated approaches for
the deuteron (${}^{2}\rm{D}$), and two heavier nuclei – namely Iron
(${}^{56}\rm{Fe}$) and lead (${}^{208}\rm{Pb}$). The results are shown as a
function of $x$ for the up and gluon nPDFs at $Q^{2}=1.65~{}\rm{GeV}^{2}$.
Similar to the previous results, the solid and dashed bands represent the 68%
c.l. and one-sigma uncertainties, respectively. All results are normalized to
the the central values of the integrated nPDFs. The two determinations are
overall in excellent agreement within the uncertainties although in general
the nPDFs determined using the integrated approach yield larger uncertainties
and more fluctuations. This can be markedly seen for instance for the gluon
PDF of the deuteron. More pronounced differences are observed for both the up
and gluon PDF of the lead in which the integrated method yield much smaller
error at medium- and large-$x$.

These comparisons demonstrate that albeit the issues regarding the stability
of the integrated fit – which is directly linked to the choice of
hyperparameters – the framework is working and is capable of reproducing the
reference fits in which the proton and nuclear PDFs are separately determined.

## Conclusions and outlook.

We presented a framework in which the proton, deuteron, and heavy nuclear
parton distributions are simultaneously determined without the need for
imposing a boundary condition to reproduce the $A=1$ limit. It is based on the
integration of the framework underpinning the nNNPDF3.0 determination of the
parton distribution of nucleons bound within nuclei into that adopted in the
NNPDF4.0 methodology for the determination of free-proton PDFs.

It was shown that the (n)PDFs extracted from the integrated approach are
consistent with the reference (n)NNPDF determinations. However, using the
default NNPDF4.0 set of hyperparameters did not yield desirable results due to
the sign of instability in the training. After all, the underlying nature of
the neural network architecture has drastically changed due to the additional
parametrization of the atomic mass number $A$. Therefore, a $k$-folding-based
scan of the parameter space must be performed in order to obtain the best
combination of hyperparameters.

From the physics’ point of view, in order to achieve a reliable separation
between quark flavors, we must extend the experimental datasets to include
hadronic processes. All the ingredients should be indeed available to perform
a global nuclear PDF fit with NNLO QCD calculations with fitted charm. Once
this is done, the next steps will be to include estimation of missing higher
order uncertainties [18, 19, 20], and to provide an approximate N3LO (n)PDF
integrated determination [21, 22].

## Acknowledgments.

The author is grateful to Juan Rojo for the careful reading of the manuscript.
The author also wishes to thank the collaborators from the Netherlands
eScience Center (NLeSC) for discussions during the development of the project.
This work is supported by an Accelerating Scientific Discoveries grant of the
NLeSC.

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