# Evolution of the spectral lineshape at the magnetic transition in Sr2IrO4
and Sr3Ir2O7

Paul Foulquier  Université Paris-Saclay, CNRS, Laboratoire de Physique des
Solides, 91405, Orsay, France.  Université Paris-Saclay, CEA, CNRS, SPEC,
91191, Gif-sur-Yvette, France  Marcello Civelli  Marcelo Rozenberg  Université
Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France.
Alberto Camjayi  Joel Bobadilla  Departamento de Física, FCEN, UBA, and IFIBA,
CONICET, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Dorothée Colson  Anne Forget  Université Paris-Saclay, CEA, CNRS, SPEC, 91191,
Gif-sur-Yvette, France  Pierre Thuéry  Université Paris-Saclay, CEA, CNRS,
NIMBE, 91191, Gif-sur-Yvette, France  François Bertran  Patrick Le Fèvre
Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin-BP 48, 91192 Gif sur
Yvette, France  Véronique Brouet  Université Paris-Saclay, CNRS, Laboratoire
de Physique des Solides, 91405, Orsay, France.

(September 3, 2024)

###### Abstract

Sr2IrO4 and Sr3Ir2O7 form two families of spin-orbit Mott insulators with
quite different charge gaps and an antiferromagnetic (AF) ground state. This
offers a unique opportunity to study the impact of long-range magnetic order
in Mott insulators. It appears to play a different role in the two families,
as there is almost no change of the resistivity at the magnetic transition
$T_{N}$ in Sr2IrO4 and a large one in Sr3Ir2O7. We use angle-resolved
photoemission to study the evolution of the spectral lineshape through the
magnetic transition. We use Ru and La substitutions to tune $T_{N}$ and
discriminate changes due to temperature from those due to magnetic order. We
evidence a shift and a transfer of spectral weight in the gap at $T_{N}$ in
Sr3Ir2O7, which is absent in Sr2IrO4. We assign this behavior to a
significantly larger coherent contribution to the spectral lineshape in
Sr3Ir2O7, which evolves strongly at $T_{N}$. On the contrary, the Sr2IrO4
lineshape is dominated by the incoherent part, which is insensitive to
$T_{N}$. We compare these findings to theoretical expections of the Slater vs
Mott antiferromagnetism within Dynamical Mean Field Theory.

Figure 1: (a) Sketch of the electronic structure expected for Sr2IrO4 and
Sr3Ir2O7. The J3/2 band is filled with 4 electrons, the J1/2 band is half-
filled and split into Lower Hubbard Band (LHB) and Upper Hubbard Band (UHB) by
$\Delta$. The gap between the tail of the bands is $\delta$. (b) Temperature
dependence of the resistivity in Sr2IrO4 either pure (blue line), or doped
with Ru (red-brown lines) or La (green line). The doping levels are indicated
on the graph. TN, determined by magnetic measurements, is indicated by an
arrow. (c) Same as (b) for Sr3Ir2O7.

##  I Introduction

The motion of one hole in an antiferromagnetic (AF) background is a central
problem for many correlated systems, among which high temperature cuprate
superconductors remain a hallmark. As moving one hole necessarily breaks some
AF bonds, the charge and spin sectors become intimately intricated. The
relative energy scale for charges are of the order of the charge gap $\Delta$,
set by Coulomb repulsion U in a Mott insulator, independently of the magnetic
order. AF order usually sets in at a lower temperature, depending on magnetic
couplings J. Most treatments of the metal-insulator transition (MIT) are
implicitely deep in the Mott insulating state and neglect the role of magnetic
order. This is typically relevant for cuprates where U>>J. However, when the
two energy scales become similar, closer to the MIT, the interplay between the
two degrees of freedom may become quite complex Hao  _et al._ (2019); Fratino
_et al._ (2017). The role of AF long-range order on the MIT has been treated
with DMFT theory in ref. Camjayi  _et al._ , 2006 and predicts the coexistence
of Mott-like and Slater-like excitations. To our knowledge, this has never
been directly compared to experiments.

Iridates offer a unique opportunity to carry over this comparison by tuning
($\Delta$, J) parameters. Indeed, the first two members of the Ruddlesden-
Popper perovskite serie, Sr2IrO4 and Sr3Ir2O7, have quite significantly
different charge gaps $\Delta$ at low temperatures, but similar magnetic
transition temperature TN. The two families can essentially be understood Kim
_et al._ (2008) as built from two filled J3/2 bands and a half-filled J1/2
band, which is split by the electronic correlation opening a gap $\Delta$ (see
sketch in Fig. 1a, J is the effective angular momentum). The Mott nature of
the insulating state in Sr2IrO4 was supported by cluster-DMFT
calculationsMartins  _et al._ (2018). Sr2IrO4 is built from single IrO2
layers, stacked with SrO layers, and $\Delta\simeq 0.6eV$ was estimated by
optical spectroscopy Moon  _et al._ (2008), STM Battisti  _et al._ (2017);
Zhao  _et al._ (2019); Sun  _et al._ (2021) and ARPES using a small electron
doping to visualize the whole gap Brouet  _et al._ (2015). For the bilayer
version Sr3Ir2O7, the gap is smaller $\Delta\simeq 0.2eV$ Moon  _et al._
(2008); Okada  _et al._ (2013); Zhao  _et al._ (2021); Brouet  _et al._
(2018). The different $\Delta$ in the two families is essentially understood
from the different effective dimensionality of the structure, leading to
larger bandwidth for the bilayer compared to single layer Moon  _et al._
(2008). There are also some more subtle differences in the electronic
structures. For example, interactions within the bilayer bands in Sr3Ir2O7
leads to a non-interacting semi-metallic electronic structure with an indirect
gap of 0.05eV de la Torre  _et al._ (2014), which is simply enlarged by
correlations Brouet  _et al._ (2018); sup .

The two compounds display slight differences also in the AF magnetic
structure. In Sr2IrO4 a transition takes place at TN=240K to a canted in-plane
AF state, where RIXS measured Heisenberg-like magnon dispersion over 0.2eV,
characterized by J=0.06eV between first neighbors Kim  _et al._ (2012a). In
Sr3Ir2O7, the magnetic transition at TN=280K gives rise to an AF order with
moments along the c-axis Dhital  _et al._ (2013). The magnetic interaction
have similar order of magnitude (J=0.09eV), but the magnon dispersion is
characterized by a large gap of 0.07eV due to pseudodipolar interactions in
this geometry Kim  _et al._ (2012b).

On one hand, the two compounds display insulating behavior in transport
measurements above TN, up to more than 600KKorneta  _et al._ (2010); Cao  _et
al._ (2002). This suggests that correlations, including short-range magnetic
correlations, which persist above TNChen  _et al._ (2015); Zhao  _et al._
(2019), are responsible for the insulating properties, rather than the
magnetic order. On the other hand, there is a rather strong temperature
evolution in optical spectroscopy, characterized by weight appearing in the
gap at high temperatures Moon  _et al._ (2009); Ahn  _et al._ (2016); Xu  _et
al._ (2020), indicative of bad metal properties. As the evolution is smooth
and the transition temperature relatively high, it is difficult to disentangle
the role of temperature and magnetic transition. These were tentatively
attributed to the temperature dependence of polaronic excitations Sohn  _et
al._ (2014), but never fully clarified. More recently, Song et al. used Ru
doping, which decreases TN, to correlate a transfer of spectral weight with TN
in Sr3Ir2O7 Song  _et al._ (2018). Another optical study in Rh-doped Sr2IrO4
observed a transfer of spectral weight to a mid-infrared peak, which was
interpreted as a spin-polaron feature from a one band Hubbard model Xu  _et
al._ (2020). From a different viewpoint, STM favors an inhomogeneous picture,
where in-gap states are observed near dopants Battisti  _et al._ (2017); Wang
_et al._ (2019) or defects Okada  _et al._ (2013) and may lead to a
percolative-like MIT Sun  _et al._ (2021). Angle-resolved photoemission could
in principle go further by resolving the gapped structure in k-space. Its
lineshape could help understand the nature of coherent and incoherent
excitations and the possible emergence of in-gap states. However, there are
few ARPES data available as a function of temperature and only for Sr3Ir2O7,
either pure King  _et al._ (2013) or doped with La Affeldt  _et al._ (2017) or
30% Ru Song  _et al._ (2018).

Figure 2: (a) Temperature dependence of the resistivity obtained within DMFT
for strong interaction $U/D=4$ for various doping x. The magnetic transition
temperature $T_{N}$ is indicated by an arrow. (b) same as (a) for $U/D=1.7$.

In this paper, we study systematically the evolution of the ARPES lineshape
through TN in Sr2IrO4 and Sr3Ir2O7 and use La Ge  _et al._ (2011); Hogan  _et
al._ (2015) and Ru Dhital  _et al._ (2014); Yuan  _et al._ (2015); Calder  _et
al._ (2015) substitutions to tune TN. La substitutes out-of-plane for Sr (we
define x as Sr1-xLax) and induces electron doping. This leads to the reduction
of the magnetic order, which vanishes around x=0.04 Ge  _et al._ (2011); Hogan
_et al._ (2015). On the contrary, Ru substitutes in-plane for Ir (Ir1-xRux)
and, although Ru has one less electron than Ir, it seems there is an
electronic phase separation at early dopings, so that Ru dilutes the magnetic
state rather than dopes it Dhital  _et al._ (2014); Yuan  _et al._ (2015);
Calder  _et al._ (2015); Brouet  _et al._ (2021). A high concentration around
x=0.35 is required to suppress the magnetic order and induce a metallic state.

We shall qualitatively compare our results with reference theoretical results
obtained by Dynamical Mean Field Theory on the doped Hubbard Model. A detailed
quantitative comparison with iridates would require employing more realistic
material approaches, which can include short ranged spacial correlations (e.g.
the cluster extension of DMFT), spin-orbit coupling, multi-bands effects and
possibly weakly correlated bands within the ab-initio framework. However such
studies, which demand future developments Martins  _et al._ (2018); Moutenet
_et al._ (2018); Zhang  _et al._ (2013); Jeong  _et al._ (2020), are beyond
the scope of this paper which focus on the general properties of the
antiferromagnetic Slater to Mott crossover. We shall consider two different
interaction regimes: the rather weakly correlated one ($U/D$= 1.7), which is
dominated by the Slater AF mechanism, and the strongly correlated one ($U/D$=
4), dominated by the Mott localization mechanism. We shall show that as a
matter of facts many physical properties of Sr3Ir2O7 can be described with the
former Slater regime, while the Sr2IrO4 well fits the latter Mott regime. This
is especially true for transport, but also for the ARPES spectral lines,
provided we make some assumption on features related to the chemical
substitution and disorder, which can broaden the spectra and affect the Fermi
level position. Our experimental-theoretical comparison shows that Ir-based
oxides can provide a unique platform to study non-trivial correlation
phenomena, like the evolution from weak to strong correlation, the interplay
of Slater magnetism and Mott localization and the effects of doping,
temperature and disorder.

##  II Methods

Single crystals of Sr2IrO4 and Sr3Ir2O7 were grown by a standard flux growth
technique. High-purity powders of SrCO3 (99.995%), IrO2 (99%), RuO2 (99.9%)
were dried, weighed, and mixed in a glove box under argon with SrCl2 (99.5%)
flux. The mixture was loaded into a platinum crucible covered with a platinum
tip, under ambient atmosphere, and heated in a muffle furnace. For Sr2IrO4, we
used ratios 2:1:10, heated up to 1300 ∘ C and then slowly cooled down at a
rate of 10∘ C/h to 800∘ C. For Sr3Ir2O7, we used ratios 3:2:5, heated at a
rate of 190∘ C/h up to 1100∘ C for 6 hours and then slowly cooled down at a
rate of 10∘ C/h to 600∘ C, at which temperature the furnace was turned off.
Deionized water was used to dissolve the SrCl2 flux and extract the single
crystals. The crystals were platelets with larger dimensions between 0.3 and 2
mm, and with the smallest dimension along the [001] direction. The exact
composition of each studied sample has been determined via energy-dispersive
X-ray spectroscopy (EDS) measurements in several spots of the surface of
several crystals from the same batch. The structure was further refined by
x-ray diffraction. The results for nine samples of Sr3(Ir1-xRux)2O7 with x in
the 0-0.78 range are given as supplementary material.

ARPES experiments were carried out at the CASSIOPEE beamline of SOLEIL
synchrotron, with a SCIENTA R-4000 analyser, 100 eV photon energy and an
overall resolution better than 15meV.

DMFT calculations where performed on the Hubbard Model, the reference
playground to study correlated phenomena Georges  _et al._ (1996). The model
has the typical semi-circular density of states of bandwith $D$, which fixes
the energy unit throughout the paper. The DMFT is implemented by means of the
continuous time diagrammatic Quantum Monte Carlo Gull  _et al._ (2011).
Spectra are obtained via analytic continuation of the one particle propagator
performed by the Maximum entropy method Levy  _et al._ (2017). DMFT allows to
unbiasedly access the paramagnetic insulating and metallic states, as well as
the ordered antiferromagnetic insulatorCamjayi  _et al._ (2006); Fratino  _et
al._ (2017). Here we shall study how these states evolve and compete upon
doping the system with holes.

Figure 3: (a-c) Stacks of EDC spectra at X in the three indicated compounds as
a function of temperature (from lowest in blue, 20K, to highest in red, 300K).
The top spectrum (light blue) is taken at 20K after a temperature cycle. The
blue lines are in the magnetic state, the red ones in the paramagnetic state
(TN=240K for the pure, 150K doped with 20%Ru and 200K for 2% La). Dotted lines
indicate the baseline used in the fit. (d-f) Top : zoom on the spectra
superimposed for low and high temperatures, with the baselines as dotted
lines. Bottom : corresponding spectra with baseline subtracted. The arrow
indicates the size of the gap $\Delta$ expected at X (0.6eV in each case). The
two spectra are normalized to their maximum for easier comparaison of the
lineshape (high temperature spectra are magnified by the indicated amounts).
The fit used to extract position and width in Fig. 5 are shown as dotted lines
(often indistinguishable from the raw data). The model chosen for the fit (an
asymmetric Gaussian with width $L_{1}$ and $L_{2}$) is also presented.

##  III Resistivity behavior

A clear indication of the different role of the magnetic transition in the two
families is already evident in the evolution of resistivities plotted in Fig.
1. The arrows indicate TN, as determined from magnetic measurements (by SQUID
in Sr2IrO4 and neutrons in Sr3Ir2O7 Dhital  _et al._ (2014); Hogan  _et al._
(2016); sup ). There is almost no change in Sr2IrO4 at TN, while there is a
clear anomaly in Sr3Ir2O7 leading to a more conducting state in the
paramagnetic regime.

The resistivity does not follow a simple activated behavior on the full
temperature range. Fitting to an activated law between 100 and 200K gives a
gap of the same order of magnitude in the two systems, $\delta\simeq$60meV.
This is much smaller than the gap $\Delta$ previously evaluated. It can be
understood as the smallest energy distance between the tail of the peaks (see
Fig. 1a). This emphasizes the importance of in-gapped low energy states in
these systems, but also that limited information on the evolution of $\Delta$
can be extracted from resistivities alone.

This qualitative behavior is in good agreement with what is expected within a
Slater-to-Mott antiferromagnetic crossover. To show this point, we plot in
Fig. 2 the resistivity vs temperature obtained by DMFT for two values of the
interactions strength, $U/D=4.0$ (panel a), which is deep in the strongly
correlated Mott regime, and $U/D=1.7$ (panel b), which is in the weakly
correlated regime. The system is lightly doped, up to $x=0-10\%$.

These theoretical curves display a qualitative behavior in line with what is
observed in the experimental curves described above in Fig. 1. Namely, in the
correlated regime, alike to Sr2IrO4, the resistivity vs $T$ curve is rather
flat and does not display any anomaly in correspondence to the
antiferromagnetic transition temperature $T_{N}$. In sharp contrast, the
weakly correlated regime looks like Sr3Ir2O7, clearly displaying an anomaly in
the curve in line with $T_{N}$. These features disappear at higher doping
level in the absence of magnetic transition.

In tracing the comparison between these experimental and theoretical
resistivity curves, some caveats must be taken into consideration. In the
strongly correlated regime $U/D=4$, the system immediately becomes metallic
upon doping, though the metallic character is very weak (for example, the
absolute value of the resistivity on a doped state curve for $U/D=4$ is an
order of magnitude higher than the resistivity of the weakly correlated
$U/D=1.7$ case). In a real material, such a state would likely display
insulating-like properties. Disorder, not taken into account in our theory,
can play an important role and localize a small number of carriers. This was
observed for instance in Rh-doped Sr2IrO4 Louat  _et al._ (2018). Overall, our
theory-experiment comparison enforces the interpretation that Sr2IrO4 is deep
in the Mott state, while in Sr3Ir2O7, both correlation and Slater
antiferromagnetism play a key role.

##  IV Overview of ARPES lineshapes

Fig. 3 and 4 present ARPES Energy Distribution Curves (EDC) as a function of
temperature taken at the top of the J1/2 band, located at the X point of the
reciprocal space (see supplementary material sup  for a sketch of the
electronic structure). The position of the magnetic transition is indicated by
the change of line color (blue to red). In each family, we examine three cases
: pristine compound (TN=240K for Sr2IrO4 and 280K for Sr3Ir2O7), doped with
20% Ru (TN=150K and 180K respectively) and with a few percent La [TN=200K (2%
Sr2IrO4) and 130K (2.4% Sr3Ir2O7)]. More dopings are presented in
supplementary information, these ones illustrate the universality of the
behavior. Two bands can be observed in each EDC, J1/2 around -0.2/-0.3 eV
binding energies and J3/2 around -0.8/-1eV Brouet  _et al._ (2015, 2018). At
the bottom of Fig. 3 and 4, we show superimposed spectra at low and high
temperatures. To isolate the shape of the J1/2 peak, we subtract a parabolic
baseline (bottom, see supplementary material sup  for more details). The
spectra are scaled to their total area. The high temperature spectra is
magnified by the indicated number to better compare the lineshapes.

Figure 4: Same as Figure 3 for Sr3Ir2O7 family. TN=280K for the pure, 180K
doped with 20%Ru and 130K for 2.4% La. The arrow indicating the size of the
gap $\Delta$ measures 0.3eV. In (d) a Fermi-Dirac step at 300K is shown as
thin black line. In (e), the low temperature spectrum shifted to the high
temperature one position is shown as thin blue line.

For the pure compounds, a gap $\Delta$ opens up in J1/2 at X. The Fermi level
is roughly in the middle of the gap $\Delta$ estimated by STM and optics for
the undoped systems. The peaks are quite broad (0.2eV at half maximum for
Sr2IrO4 (Fig. 3d) and 0.15eV for Sr3Ir2O7 (Fig. 4d) and the distance between
the tail of the peaks will be significantly smaller than the peak to peak
positions, in agreement with the smaller gap $\delta$ dominating the
resistivity, as indicate in Fig. 1a. For a quasiparticle (QP) excitation, the
ARPES lineshape should be lorentzian-like with a width given by the QP inverse
lifetime. However, the peak here is rather gaussian-like, asymmetric and much
broader than what would be expected for a QP. This situation is common to many
insulating oxides, as cuprates Kim  _et al._ (2002) or manganites Dessau  _et
al._ (1998). This suggests a composite nature of the line, where the spectrum
is the envelope of a distribution of excitations. A possible origin is the
formation of polarons Perfetti  _et al._ (2001); Dessau  _et al._ (1998); Shen
_et al._ (2004), as was actually proposed to explain the ARPES linewidth in
Sr3Ir2O7 King  _et al._ (2013). In this case, the shape of the peak is fixed
by the strength of electron-phonon coupling, asymmetric for low couplings and
gaussian for higher ones. As there is no obvious reason why the electron-
phonon coupling should be different in Sr2IrO4 and Sr3Ir2O7, this picture does
not explain easily the much more asymmetric lineshape of Sr3Ir2O7. Also, we
will see that the lineshape changes at TN, whereas no strong evolution of the
electron-phonon coupling would a priori be expected there. Indeed, Raman
experiments, which are sensitive to the phonon renormalization due to
pseudospin-lattice coupling, do not evidence large changes at TNGretarsson
_et al._ (2017).

Figure 5: (a) Resistivity in (doped) Sr3Ir2O7. The vertical lines indicate the
width of the magnetic transition, as seen by neutron experiments (see text).
(b) Shift of the ARPES peak position in (doped) Sr3Ir2O7 with respect to its
low temperature value, for J1/2 at X (closed symbol) and J3/2 at $\Gamma$
(crosses). Raw data are shown in the supplementary material. (c) The two
widths $L_{1}$ (towards high binding energy) and $L_{2}$ (towards low binding
energy) in (doped) Sr3Ir2O7 extracted from a fit of the J1/2 peak at X to an
asymmetric gaussian. $L_{2}$ is magnified to match $L_{1}$ value at low
temperatures, as indicated. (d) Same as (c) for (doped) Sr2IrO4. $T_{N}$
(arrow) is defined here by the onset of the ferromagnetic signal in SQUID
measurements sup .

When the samples are lightly doped, the gap essentially does not change (as
will be justified later), but the Fermi level moves inside the gap, as a
result of filling of the first available states. In Sr2IrO4 (Fig. 3), there is
no big change of the lineshape between high and low temperatures, except for a
slight broadening on both sides. In particular, there is no sudden shift at
TN, implying that the gap does not suddenly close. The J1/2 peak intensity
however is strongly supressed at high temperature. This effect is only
partially reversible and it is difficult to disentangle the role of the
temperature increase and of TN in this intensity loss.

On the contrary, in pure Sr3Ir2O7, there is a clear deformation of the
spectrum, which extends towards $E_{F}$ (see Fig. 4). The leading edge shifts
up by 50meV, but remains away from $E_{F}$. A comparison with a Fermi-Dirac
distribution at 300K (black line) suggests a remaining "pseudogap" of 50meV in
Fig.4d. Nevertheless, some residual density appears at $E_{F}$, which is
consistent with a bad metallic character. The peak maximum itself has not
moved significantly, ruling out a sudden collapse of the gap at TN. This
comparison implies that the difference between the two systems is more complex
than a gap simply closing in one case and not the other. It seems rather
related to a transfer of weight in the gap for Sr3Ir2O7 that is not present
for Sr2IrO4. In doped Sr3Ir2O7, there is in addition a clear shift of the
spectra towards $E_{F}$, both for J1/2 and J3/2, bringing weight at the edge
of $E_{F}$. As J3/2 should not be affected by a gap closure in J1/2, this
behavior suggests a shift of $E_{F}$ inside the gap. To compare the lineshape,
we shift the low temperature spectra to the high temperature one for 20% Ru
(thin blue line, shifted up by 70meV). This reveals a similar change of the
lineshape as for the pure, with a characteristic extension of the spectrum
towards $E_{F}$. The comparaison is more difficult for the La case, where the
low temperature spectrum is broader than the other ones, especially towards
$E_{F}$. This is probably due to some distribution in La content, which hides
the intrinsic lineshape at low temperatures.

##  V Change at $T_{N}$

Figure 6: Peak position as a function of temperature of the asymmetric
gaussian used to fit the J1/2 EDC at X in Sr2IrO4 (a) and Sr3Ir2O7 (b) for the
indicated dopings. (c) Sketch of the evolution at $T_{N}$, involving a
transfer of spectral weight from the Hubbard bands to in-gap states. The
initial position of $E_{F}$ for the different dopings is sketched by color
lines on the top graph, which all move to the middle of the $\Delta$ gap above
$T_{N}$. The detail of the bands above $T_{N}$ cannot be known from ARPES.

We now study how these temperature evolutions correlate with the magnetic
transition. The magnetic transition is quite broad in doped iridates, probably
proceeding through some phase separated region Dhital  _et al._ (2014). For
Sr2IrO4, we define $T_{N}$ by the onset of the ferromagnetic signal in SQUID
measurements sup . This is not possible for Sr3Ir2O7, where this signal is
very weak, and we rely on neutrons scattering performed in Ru Dhital  _et al._
(2014) and La Hogan  _et al._ (2015) doped Sr3Ir2O7. We define two
temperatures limiting the magnetic transition, $T_{N,1}$ where magnetic Bragg
peaks appear, and $T_{N,2}$ where their intensity saturates. Experimentally,
it is this latter value that corresponds to the anomaly of resistivity,
reported as TN on Figure 1. To characterize the temperature dependence of the
lineshape, we fit spectra at each temperature to an asymmetric gaussian, as
shown in Fig. 3d and 4d. This emphasizes the key evolution we have described
and limits the number of parameters (more examples of fits are shown in the
supplementary material sup ).

We first focus on Sr3Ir2O7 where there are clear changes. In Fig. 5a, we
observe that the change in resistivity starts at $T_{N,2}$ (solid line) and
evoluates until roughly $T_{N,2}$ (dotted line). Similarly, the shift of the
peak position of J1/2 at X, shown in Fig. 5b, is precisely tied to the
magnetic transition, it starts at $T_{N,2}$ and saturates above $T_{N,1}$. We
compare it to the shift of J3/2 observed at $\Gamma$ (details are given in
supplementary sup ) and find that it exhibits a very similar temperature
evolution, although smaller by a factor 2. This suggests that the shift is at
least partly due to a motion of $E_{F}$ in the gap and not simply to a change
in the energy scales $\Delta$ and $\delta$.

Interestingly, the lineshape also changes at the magnetic transition. In Fig.
5(c), we present the two widths of the asymmetric gaussian, $L_{1}$ (towards
high binding energies) and $L_{2}$ (towards $E_{F}$), scaled to the low
temperature value. We find that they are roughly constant below $T_{N,2}$, but
$L_{2}$ increases above $T_{N,2}$, strongly diverging from $L_{1}$, which
remains constant or even decreases. This is in agreement with the evolution
described in Fig. 4, where only the low energy side of the peak changes at
high temperature. This however further demonstrates that this evolution is
triggered by magnetic ordering. A first possibility would be that this
reflects a distribution of positions arising above $T_{N}$. As we have seen
that the transition is quite inhomogeneous, it has to be considered. However,
a narrower lineshape would then be expected again, when the sample has fully
transited, contrary to our observation (the linewidth saturates but remains
broad above $T_{N,1}$). Therefore, although disorder certainly plays a role,
it cannot explain the broadening above $T_{N}$. We then assume that spectral
weight is transferred in the gap, deforming the lineshape towards $E_{F}$.

This behavior (both the shift and the spectral evolution) is intrinsic to
Sr3Ir2O7. In Fig. 5(d), we give a similar view of the evolution of the widths
in Sr2IrO4 (we could not fit reliably the widths at high temperature for high
Ru dopings, the spectral weight becoming too small to be separated from the
background). They do not point to a systematic change of lineshape at $T_{N}$.
The width broadens moderately with increasing temperature, but it is sometimes
$L_{1}$ that is largest (La doped) or $L_{2}$ (pure) or they remain similar
(Ru doped), with no clear anomaly through $T_{N}$.

In Fig. 6(a-b), we summarize the evolution of the J1/2 position in the two
families. For Sr2IrO4, there is no particular shift at TN. In the case of La,
we observe a small shift with temperature, but it is not happening at $T_{N}$
(200K in this case) and it could be due to the formation of defect states in
the gap moving $E_{F}$ to a new position. Indeed, it is not completely
reversible (see Fig. 3c). For completeness, we add the case of a small hole
doping of Sr2IrO4, obtained by 5% Rh Louat  _et al._ (2018) (TN=170K), which
confirms the absence of shift at $T_{N}$ in Sr2IrO4 family.

For Sr3Ir2O7, the shift is significant for all doped compounds and Fig. 6(b)
further reveals that all peaks seem to converge to the position of the pure at
high temperature, independently of the doping. For the pure compound at low
temperature, this position is well understood as corresponding to a Fermi
level in the middle of the Mott gap, at $\Delta$/2. This convergence then
means that the Fermi level is fixed at $\Delta$/2 at high temperatures, for
all dopings. This also implies that there remains a Mott-like energy scale
$\Delta$ above TN for both families.

Fig. 6(c) summarizes our understanding of the evolution. At low temperatures,
the position of the Fermi level is fixed by the doping. It is either near the
middle of the gap (pure compound), at the tail of UHB for electron doping (La
case) or LHB for hole doping (Rh case). For more disordered situations (Ru
case), it is found at intermediate position. Above $T_{N}$, the Mott-like gap
$\Delta$ is not closing suddenly, neither for Sr2IrO4, nor for Sr3Ir2O7. There
is a remaining LHB at $\Delta$/2 in both cases, which still dominates the
spectral weight. However, spectral weight is transferred in the $\Delta$ gap,
in a much more pronounced way for Sr3Ir2O7 than Sr2IrO4, and this transfer
starts and stops over the width of the magnetic transition. This transfer
fixes the position of the Fermi level at the center of the $\Delta$ energy
scale, overruling the previous disorder/doping dependent position and
producing a shift of the spectra in doped cases. By comparison, the absence of
shift in Sr2IrO4 appears as a sensitive sign that there is no significant
change of the in-gap structure.

To distinguish the "filling" of the Mott gap, from a "closure", we add the
case of a higher La doping of 6% in Sr3Ir2O7, where the metallic state is
realized de la Torre  _et al._ (2014); Brouet  _et al._ (2018) (dark yellow
triangles in Fig. 6(b)). Clearly, the peak is much closer from $E_{F}$ at low
temperatures in this metallic compound. We will further show in Fig. 8 that
the lineshape is also completely different, with a narrow peak.

##  VI Discussion

Figure 7: Temperature evolution of antiferromagnetic DMFT spectra (from top to
bottom) for the strongly correlated Mott case ($U/D=4$, left hand side) and
the weakly correlated Slater case ($U/D=1.7$, right hand side). In the
strongly correlated case $U/D=4$, a Mott energy scale $\Delta$ (separating the
Hubbard bands) remains well defined and line-shapes do not essentially move.
Antiferromagnetism is restored by the recovering of equal spin up (red) and
spin down (blue) spectral weights. For weak correlation $U/D=1.7$, though
Hubbard bands remain visible across TN, an antiferromagnetic gap $\delta_{S}$
closes at the Fermi level across TN, inducing a shift of spectral weight to
the Fermi energy, as expected within a Slater mechanism. This forces the Fermi
level to be located at the middle of $\Delta$.  Figure 8: (a) ARPES lineshape
at 15K for J1/2 (EDC at X with subtracted background) for Sr2IrO4 (black),
Sr3Ir2O7 (blue) and metallic Sr3Ir2O7 (doped with 6% La, orange). (b) Same at
300K. (c) Simulation of the low temperature spectra with a peak and hump
structure (see text). (d) Same for high temperature spectra. Insets show the
evolution in temperature of the peak contributions.

We now compare these experimental spectra with the theoretical expectations in
a Mott vs Slater scenario. For this purpose we show in Fig. 7 the DMFT spectra
of the Hubbard Model at small doping. On the left column we display the
strongly correlated Mott regime ($U=4D$), on the right one the weakly
interacting Slater regime ($U=1.7D$), for increasing temperature (from top to
bottom). The system is at small hole doping, $x=2.5\%$ and $x=3.5\%$, for
strong and weak correlations respectively. Spectra put into evidence the sharp
difference between the Mott and the Slater mechanism closing of the gap above
$T_{N}$.

In the Mott regime a Mott energy scale is well defined at low temperature,
marked as $\Delta$ on the top left panel, separating the LHB from the UHB.
Upon doping, a small QP peak appears at the Fermi level, at the upper edge of
the LHB. The AF is evident from the strong spectral weight differentiation
between the spin up (red dashed line) and spin down (blue dashed line)
species. This weak QP peak feature in the strong correlation regime is likely
to be washed away by disorder and impurity effects in a real material. Indeed,
we have seen with Figure 1 that lightly doped Sr2IrO4 is not immediately
metallic, contrary to this prediction, and that $E_{F}$ is not fixed to the
upper of LHB, but remains within the gap, at a position depending on doping
and/or impurities. The key point is that upon increasing temperature, the
spectra are little affected, and AF is suppressed only by recovering the
balance between the spin up and spin down spectral weights.

In the Slater regime, we can still identify a Mott energy scale $\Delta$,
separating the LHB and the UHB, though, with respect to the correlated Mott
regime, this scale is now renormalized to a smaller value than the onsite
interaction $U$. Much more pronounced QP peaks appear now at the edges of the
Hubbard bands and they define a small gap $\delta_{S}$ around the Fermi level.
This gap is this time opened by the AF mechanism (besides the spin up and spin
down spectral weight differentiation), separating spin up and spin down QP-
like peaks. This is the key difference with respect to the Mott regime, for
which the QP peak, though weak, is at the Fermi level even at small doping and
T<TN. Upon increasing the temperature, while the Mott $\Delta$ scale is only
slightly affected, the low $\delta_{S}$ energy scale collapses at the
transition temperature $T_{N}$. At the same time the spin up and spin down
spectral weight imbalance disappears. The distinctive feature is that with the
closing of the $\delta_{S}$ energy scale, a sharp increase of spectral weight
appears at the Fermi level explaining the "anomaly" observed in the
resistivity.

We now attempt a direct comparison between the experimental spectra and the
two theoretical scenarios. In Fig. 8(a-b), we compare three different ARPES
lineshapes (a baseline has been subtracted) for large gap $\Delta$ (Sr2IrO4),
smaller gap (Sr3Ir2O7) and a metallic case (Sr3Ir2O7 with 6% La). We find that
the lineshape at low temperature in Sr2IrO4 ($L_{1}$=0.15 and $L_{2}$=0.08eV)
is broader and more symmetric than in Sr3Ir2O7 ($L_{1}$=0.12 and
$L_{2}$=0.05eV). It narrows down significantly in the metallic state, where a
peak with $L$=0.025eV dominates. Although none of the spectra exhibits a well
resolved peak-hump structure, as for the theoretical spectra, it may be
present but hidden by broadening. Assuming this, the different lineshapes mean
that the hump dominates in Sr2IrO4, the peak and the hump have similar
contributions in Sr3Ir2O7 (building a more asymmetric shape) and the peak
dominates in doped Sr3Ir2O7, as sketched in Fig. 8(c) for low temperatures. As
there is no well defined separation between peak and hump, it is of course
impossible to refine the fitting further. However, it gives a simple and
natural way to explain the evolution from the more insulating state to the
more metallic situation, by simply transferring weight from the broad hump to
the peak. Such a decomposition was actually already proposed to describe the
insulator to metal evolution of the lineshape in Sr3Ir2O7 doped with La He
_et al._ (2015).

The very interesting point is now to use this underlying structure to better
understand the evolution at $T_{N}$. The experimental high temperature spectra
after background subtraction are compared in Fig. 8(b). The absence of
sensitivity of Sr2IrO4 to $T_{N}$ can be explained if the incoherent hump is
not sensitive to the magnetic order. This is in agreement with the theoretical
scenario for the strongly correlated case, where most of the weight correspond
to the Hubbard bands. In Sr3Ir2O7, most of the changes are due to the
evolution of the coherent peak. More specifically, to explain a more symmetric
lineshape, the coherent peak has to shift towards $E_{F}$, as sketched in the
inset of Fig.8(d). This is in agreement with the closure of the $\delta_{S}$
gap described before in the theory of the weakly correlated case. This
suggests to identify the $\delta$ and $\delta_{S}$ energy scales. We note that
the shift in Sr3Ir2O7 is only indirectly due to the closing of $\delta_{S}$.
The main driving force for the shift is the relocation of the Fermi level at
the middle of the $\Delta$ energy scale. Interestingly, there is a clear
difference in theory for this position in the Mott and Slater cases at small
dopings. In the Mott case, the QP peak forms at the edge of the Hubbard band
without a gap. In contrast, in the Slater case, the QP forms at the edge of
the remaining "Slater" gap $\delta_{S}$, around $\Delta$/2. In the
experimental case, it seems that as long as a gap is open (either $\Delta$ or
$\delta_{S}$), extrinsic degrees of freedom (disorder/impurities/dopant) may
fix the position of the EF within the gap. Indeed at low temperatures, the
peak poistions are similar in Sr2IrO4 and Sr3Ir2O7 for the same dopings. On
the other hand, as soon as the low energy gap $\delta_{S}$ closes, EF is fixed
to the theoretical position, inducing a shift in the weakly correlated case.

In Fig. 1, there is obviously a continuity between the metallicity found at
high temperatures in the compounds keeping a magnetic ground state and the
completely metallic ones. This decomposition bridges the two behaviors. The
evolution in metallic Sr3Ir2O7 implies a rather large broadening of the peak
with increasing temperature. Consequently, it is not well resolved from the
hump anymore at high temperatures and the metallic nature of this spectrum is
not obvious. Indeed, the lineshape for Sr3Ir2O7 pure and doped with 6% La are
very similar (Fig. 8b). This is why identifying the nature of the metallic
state at high temperatures was difficult. This decomposition offers a
qualitative understanding, but, here again, any more advanced fitting is
impossible as it is difficult to separate not only the coherent part from
incoherent part, but also the incoherent part from the background at high
temperatures.

##  VII Conclusion

Iridates offer a unique opportunity to study the crossover from Mott to Slater
behaviors as a function of correlation strength. Although there are many
examples of Mott oxides with magnetic transitions Kovaleva  _et al._ (2004);
Gorelov  _et al._ (2010), there is often an orbital degree of freedom, which
complicates the analysis of the mere impact of long-range magnetic order on
the Mott state. This is not the case in iridates, where the filled J3/2 states
do not take an active part in the transition. Here, we have studied the
evolution of ARPES lineshapes corresponding to the half-filled J1/2 across the
temperature driven magnetic transition in Sr3Ir2O7 and Sr2IrO4 compounds,
using different dopings to establish the universality of the behavior. We have
then compared our results to theoretical expectations within the Dynamical
Mean Field Theory.

We argue that iridates are intermediately correlated systems, where the ARPES
lineshape is formed by a continuity between incoherent Hubbard-like features
and coherent QP-like excitations. In the more weakly correlated compound
Sr3Ir2O7, a key change of the spectral lineshape at the magnetic transition is
a broadening towards $E_{F}$, which we attribute to a shift of the coherent
weight towards $E_{F}$. This result agrees with the closing of a coherent
"Slater" gap at the magnetic transition expected by DMFT in the weakly
correlated regime. This coherent weight coexists with incoherent features,
centered at the position of Hubbard bands $\pm\Delta$/2. In contrast, we find
that the behavior of Sr2IrO4 agrees more with the Hubbard-Mott scenario. The
coherent part of the lineshape is small and no clear evolution is observed
through TN, as expected within the DMFT description. This decomposition of the
spectra implies the existence of two energy scales, the large Mott-like gap
$\Delta$, essentially fixed by short-range magnetic correlations and the small
coherent gap $\delta_{S}$, fixed by the long range magnetic order. This is
consistent with the absence of correlation between the large charge gap and
the short-range magnetic order observed in Sr2IrO4 by spin-polarized STM Zhao
_et al._ (2019).

We further find that in the weakly correlated case, the closing of the small
magnetic gap $\delta_{S}$ redefines the position of the Fermi level in the
middle of the Hubbard bands. This drives a shift of the whole spectrum towards
this position, exactly starting and stopping over the width of the magnetic
transition. This view is quite different from the Lifshitz-like transition
that was proposed before, where the shift of the band would bring weight to
the Fermi level Song  _et al._ (2018). In future experimental and theoretical
investigations, the role of disorder, also induced by the chemical
substitution, should be investigated to fully describe the lineshapes and
their behavior as a function of doping and temperature. The width of the
magnetic transition, as well as the spectral changes observed by STM near
defects and/or dopants Okada  _et al._ (2013); Sun  _et al._ (2021); Wang  _et
al._ (2018), implies that there is some degree of heterogeneity in these
systems, which certainly plays a role in the absence of well resolved coherent
peak in ARPES.

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# Relativistic effects on stopping power of plasmas with heavy ions

A. M. P. Mendez  J. Chacón-Gijón  J. Vázquez-Moyano  M. D. Barriga-Carrasco

###### Abstract

This work investigates the electronic stopping power for protons of a tungsten
plasma at various electron densities and temperatures. The study employs a
dielectric formalism to model the stopping power due to free and bound
electrons, considering relativistic and non-relativistic effects. The bound
electron stopping power is modeled using the shellwise local plasma
approximation (SLPA) introduced by Montanari and Miraglia. The ionization
state of the plasma is also examined, revealing its impact on the bound
electron contribution. We compare our results with the T-Matrix approach and
the Li-Petrasso model for free electrons in combination with SLPA calculations
for the bound electron ones. Results demonstrate the significance of bound
electron stopping power, particularly for plasma with high-$Z$ ions. This
investigation contributes valuable insights into plasma physics and fusion
energy research, providing essential data for future experiments and
simulations.

††journal: Fusion Engineering and Design

[UCLM] organization=E.T.S.I. Industrial, Universidad de Castilla-La Mancha,
city=Ciudad Real, postcode=13071, country=Spain

[CONICET] organization=IAFE, CONICET and Universidad de Buenos Aires,
city=Buenos Aires, postcode=C1428EGA, country=Argentina

##  1 Introduction

There are two main approaches to fusion energy: magnetic confinement fusion
(MCF) and inertial confinement fusion (ICF). Due to its properties, tungsten
is a plasma-facing material candidate, which will be directly exposed to the
target explosion. Several works have investigated the possibility of using
tungsten lines to allow plasma diagnosis using spectroscopy [1, 2, 3]. Other
studies have examined the capability of high-$Z$ dopants as stabilizers of the
fuel [4, 5]. Because the presence of tungsten in fusion plasmas is
indisputable, the characterization of the energy loss processes for ions in
plasma with tungsten is vital.

During laser-driven ICF implosion experiments, laser pulses ablate the surface
of a layered target and implode the target [6]. The laser-driven shocks and
the spherical convergence compress the fuel, creating extreme temperatures and
pressures. The high temperatures and densities cause the fuel to fuse,
producing energetic $\alpha$ particles and neutrons. The implosion generates a
relatively low-temperature and high-density shell and a high-temperature and
medium-density hotspot center. The $\alpha$ particles created from the
reaction in the target center carry the necessary energy to heat the plasma
further. As the particles travel through the plasma, they deposit their
energies. The energy loss per unit length, also called stopping power, is
directly involved in determining the effectiveness of this process.

In this work, we aim to study the electronic stopping power for protons of
tungsten plasmas at various electron densities and temperatures. Although
heavy-ion plasma measurements have gone as heavy as Xe [7, 8], we
theoretically examine a very heavy-ion plasma such as tungsten, which features
a nuclear charge $Z=74$. Studying the stopping power of such plasma may be
vital for understanding the ion energy deposition and further ignition of
W-doped ICF plasmas.

The electronic stopping power model and calculations are obtained considering
the mean energy loss per unit length contribution of free and bound electrons
separately. For the free electrons, we use a dielectric model for electron
quantum plasmas in thermal equilibrium that introduces a temperature
dependence. Because elements ionize with increasing temperatures, we computed
the stopping power for protons due to the bound electrons of all the tungsten
ions that should form in the plasma within a particular temperature range. For
this, we also implemented the dielectric formalism with a dielectric function
that relies on electronic structure values. We computed the electronic
structure of the tungsten ions using two frameworks, a relativistic and a non-
relativistic approach; then, we examined the effects of employing these models
in the final stopping power results. We also compare our calculations with
values from various theoretical models for each type of electron contribution.

##  2 Stopping power of free electrons

We employ the dielectric formalism to model the stopping power for protons due
to the free electrons of a tungsten plasma. For a bare ion of charge $Z_{P}$
moving at velocity $v$, the stopping power due to the free electrons is given
by

$S=\frac{2}{\pi}\frac{Z_{P}^{2}e^{2}}{v^{2}}\int_{0}^{\infty}\frac{dk}{k}\int_{0}^{kv}d\omega\,\omega\operatorname{Im}\left[\frac{-1}{\varepsilon(k,\omega)}\right].$
(1)

where $\epsilon(k,\omega)=\epsilon_{1}(k,\omega)+i\epsilon_{2}(k,\omega)$ is
the dielectric function proposed by Arista and Brandt [9]. The real and
imaginary parts of this function are, respectively,

$\epsilon_{1}(k,\omega)=1+\frac{\chi_{0}^{2}}{4z^{3}}[g(u+z)-g(u-z)],$ (2)

with

$g(x)=-g(-x)=\int_{0}^{\infty}\frac{ydy}{e^{Dy^{2}-\eta}+1}\ln\left|\frac{x+y}{x-y}\right|\,,$
(3)

and

$\epsilon_{2}(k,\omega)=\frac{\pi\chi_{0}^{2}}{8z^{3}}\theta\ln\left(\frac{1+\exp\left[\eta-D(u-z)^{2}\right]}{1+\exp\left[\eta-D(u+z)^{2}\right]}\right),$
(4)

where $\chi_{0}^{2}=1/(\pi k_{F}a_{0})$, $\theta=k_{B}T/E_{F}$, $D=1/\theta$,
and $\eta=\beta\mu=\mu/(k_{B}T)$. The usual reduced variables are
$u=\omega/(kv_{F})$ and $z=k/(2k_{F})$, with $E_{F}=\hbar^{2}k_{F}^{2}/(2m)$
being the Fermi energy, and $k_{F}$ the corresponding wave vector, $a_{0}$ is
the Bohr radius, and $k_{B}$ is Boltzmann’s constant.

This research group has proposed other dielectric functions for free electrons
in the past [10, 11, 12], but they are much more complex to deal with and have
similar results to the Arista dielectric function in the present calculations.

##  3 Stopping power of bound electrons

The contribution of the bound electrons of the tungsten ions to the stopping
power is also modeled with a dielectric function approach: the shellwise local
plasma approximation (SLPA) introduced by Montanari and Miraglia [13]. The
SLPA formulation for the stopping power cross-section by a bare ion of charge
$Z_{P}$ moving at velocity $v$ in the atomic cloud of $nl$ subshell electrons
is given by

$S_{nl}=\frac{2}{\pi}\frac{Z_{P}^{2}e^{2}}{v^{2}}\int_{0}^{\infty}\frac{dk}{k}\int_{0}^{kv}\omega\operatorname{Im}\left[\frac{-1}{\varepsilon_{nl}(k,\omega)}\right]d\omega\,.$
(5)

Within the SLPA, the imaginary part of the inverse dielectric function is
expressed as

$\operatorname{Im}\left[\frac{-1}{\varepsilon_{nl}(k,\omega)}\right]=\int\operatorname{Im}\left[\frac{-1}{\varepsilon^{\mathrm{SLPA}}\left(k,\omega,\rho_{nl}(r),E_{nl}\right)}\right]\mathbf{dr},$
(6)

where $\rho_{nl}$ is the orbital radial density and $E_{nl}$ is the binding
energy of the $nl$ subshell. In this work, we used the Levine-Lindhard
dielectric function [14], i.e., $\varepsilon^{\mathrm{SLPA}}=$
$\varepsilon^{\mathrm{LL}}\left(k,\omega,\rho_{nl},E_{nl}\right)$, which was
initially proposed to describe ionization threshold or excitation gaps. Levine
and Louie [14] included explicitly the energy gap $E_{nl}$ within the Lindhard
dielectric function as follows:

$\operatorname{Im}\left[\frac{-1}{\varepsilon^{\mathrm{LL}}\left(k,\omega,E_{nl}\right)}\right]=\operatorname{Im}\left[\frac{-1}{\varepsilon^{\mathrm{L}}\left(k,\omega_{g}\right)}\right]\Theta\left(\omega-\left|E_{nl}\right|\right)\,,$
(7)

where $\varepsilon^{\mathrm{L}}(q,\omega)$ is the Lindhard dielectric function
[15], $\Theta(x)$ is the Heaviside step function, and
$\omega_{g}=\sqrt{\omega^{2}+E_{nl}^{2}}$. Following Lindhard and Scharff
[16], in the formulation proposed by Montanari and Miraglia [13], the orbital
radial density is introduced in the dielectric function as a local electron
density with a plasmon frequency given by
$\omega_{p}^{nl}(r)=\sqrt{4\pi\rho_{nl}(r)}$.

##  4 Results

This section presents our results for the stopping power of a tungsten plasma
for protons. We analyze plasmas with two electron densities $n_{fe}=1\times
10^{22}$ cm-3 and $1\times 10^{23}$ cm-3 at three temperatures: $T=$ 10 eV, 30
eV, and 60 eV. We obtained the present results by assuming independent free
and bound electron contributions to the stopping power. The free electron
component is modeled with the temperature-dependent dielectric formalism
detailed in Section 2. The corresponding bound electron contribution to the
stopping power is also modeled within the linear response formalism, which
includes electronic structure data of the ions. Because we consider a plasma
composed of heavy atoms, we should consider including relativistic corrections
for describing the most inner-bound electrons. We compare our results with
non-relativistic ones to account for these corrections within the same
formalism.

###  4.1 Temperature dependence of the ionization state of the plasma

When the temperature in the plasma increases, tungsten ionizes; the ionization
state of the target changes, and correspondingly, the fraction of free and
bound electrons varies. To account for this phenomenon, we considered the
flychk code [17], which allows us to compute the temperature dependence of the
mean ionization state $\bar{q}$ of the ion at a given electron density. The
results obtained in this section are later used to compute the bound and free-
electron stopping powers.

Figure 1: (color online) Mean ionization charge of the plasma composed of
tungsten in terms of the temperature for electron densities ranging from
$1\times 10^{22}$ cm-3 to $1\times 10^{23}$ cm-3.

The temperature dependence of the ionization state of the ions composing the
plasma is shown in Fig. 1 for electron densities ranging from $1\times
10^{22}$ cm-3 to $1\times 10^{23}$ cm-3. As seen in the figure, the mean
charge state of the ion for the lowest electron density at 1 eV is close to 0.
In contrast, this value is close to 4 for the highest electron density, i.e.,
for a plasma with high density, the plasma rapidly loses its most weakly bound
electrons. The $\bar{q}$ values at $T=$ 10 eV range from 1.7 to 4.3, while for
30 eV, the mean charge is close to 5.5. In a plasma at 60 eV, the results for
$\bar{q}$ spread broadly from 10 for $n_{fe}=1\times 10^{23}$ cm-3 to 17.8 for
$n_{fe}=1\times 10^{22}$ cm-3. In this temperature value, the dependence of
the ionization state with $T$ has been reversed, i.e., in a plasma with higher
electron density, the ions require more energy to lose bound electrons.

###  4.2 Stopping power of free electrons

Figure 2: (color online) Free electron stopping power for protons of a plasma
with $n_{fe}$ electron density at 10 eV, 30 eV, and 60 eV: present results
(solid lines), Li-Petraso model (dashed lines), and T-Matrix (dash-dotted
lines).

We implemented the random phase approximation with the Arista dielectric
function (ADF) given by Eqs. (2) and (4) to model the free electron stopping
power for protons of a plasma. Our results are illustrated in Fig. 2 with
solid lines for the cases discussed before, i.e., plasma with electron
densities 1$\times 10^{22}$ cm-3 (left panel) and 1$\times 10^{23}$ cm-3
(right panel) at temperatures of 10 eV, 30 eV, and 60 eV. We compare our
calculations with two widely used methods: the Li-Petrasso model [18] and the
T-Matrix approach [19]. The Li-Petrasso stopping power model is a perturbative
model that considers short-range plasma interactions (collisions). This
approximation is valid for the entire range of projectile-electron impact
parameters. The T-Matrix method relies on a binary collision approach in the
general kinetic equation, which describes short-range interactions between the
projectile and the plasma electrons, including strong collisions. Our results
for each of the nine cases agree quite well with both theories, with our
stopping maxima calculations lying between or close to one of them. The low
energy stopping power is generally underestimated for the 10 eV and 30 eV
plasma with $n_{fe}=1\times 10^{23}$ cm-3. These differences arise since the
dielectric formalism relies on a perturbative approach, which requires the
velocity of the incident projectile to be much larger than the velocity of the
electrons. Moreover, the stopping power of the plasma decreases as the
temperature increases. Given that the thermal velocity of these electrons
increases, the energy transfer from the projectile to electrons is more
complicated.

###  4.3 Stopping power of bound electrons

We implemented the shellwise local plasma approximation (SLPA) to compute the
bound electron contribution to the stopping power for protons of plasmas at
various temperatures and densities. As established in Section 3, the SLPA also
relies on the dielectric formalism, featuring a dielectric function that
depends on the electronic structure of the target, i.e., the orbital radial
densities and binding energies. Therefore, we must compute these values for
all tungsten ions with ionization states from 0 to +19, as suggested by the
flychk results illustrated in Fig. 1.

####  4.3.1 Atomic structure of tungsten ions

We are examining ions with $Z=74$; therefore, we include relativistic effects
for correctly describing the electronic structure of the most inner electrons.
We will compare the present relativistic calculations with non-relativistic
ones to later analyze the influence of these effects on the bound electrons
and total stopping power results.

We considered two self-consistent electronic structure theories: the Hartree-
Fock and the Dirac-Fock methods, which are non-relativistic (NR) and
relativistic (R) approaches. We employed the hf code by C. Fischer [20] and
the fac code by M. Gu [21], respectively, to implement these techniques.

Figure 3: Binding energies of neutral tungsten: R, relativistic calculations
(solid line); NR, non-relativistic values (dashed line and $\times$ symbols);
experiments in solids ($\blacktriangle$ symbols).

Figure 4: (color online) Relativistic (solid lines) and non-relativistic
binding (dotted lines) energies of tungsten ions from neutral to W7+.
Experiments in solid tungsten are given for comparison.

First, we examine the orbital binding energy results for neutral tungsten.
These results are illustrated in Fig. 4. The non-relativistic and relativistic
values are compared with measurements taken in solid tungsten. As expected,
the inner-shell electrons are bound tighter in the R theory than in the NR
one, with differences of around 10%. Although these differences are
significant, the contribution of these electrons to the stopping power will
later be seen to be negligible. Therefore, the corrections to the binding
energies of the inner shells are not crucial for the total stopping power
calculation. However, while relativistic corrections are included for
describing the most inner shells, these effects affect the electrons of the
outer shells more noticeably. The NR results agree with experimental
measurements in a solid within 45%, while these differences drop to about 8%
for the R calculations. The improvement of the most-outer electrons (sub-
valence) description is a by-product of considering the relativistic
corrections in the Hamiltonian, most likely carried out by the
orthonormalization condition of the orbitals. It is essential to acknowledge
these effects since outer shell electrons are the ones that, as will be seen
later, contribute the most to the stopping power.

To examine the effects of including relativistic corrections in the binding
energies of tungsten ions, we present the binding energies of the first eight
elements of its isonuclear sequence in Fig. 4. The binding energies of the
most inner electrons slowly vary as the isonuclear sequence grows, with the
electrons becoming more bound as the atom gets ionized. However, the behavior
of valence and subvalence electrons ($4f$, $5d$, and $6s$) changes
significantly as the ionization of tungsten increases. The differences between
the relativistic and non-relativistic binding energies range from 3% to 10%
and 9% to 140% for the $6s$ and $5d$ shell, respectively. For the $4f$
electrons, these discrepancies vary from 22% to 80% for the first six elements
of the isonuclear sequence and from 5% to 21% in the remaining cases. These
differences will become relevant in the stopping power results presented in
the following section.

####  4.3.2 Stopping power

The atomic structure calculations in the previous section present $nlj$
splitting, and one may be prone to consider the stopping power due to each
group of spin-oriented electrons within $nl$ quantum numbers separately.
However, for each subshell, the collisional time can be estimated as $\Delta
t\approx\langle r\rangle_{nlj}/v$, where $\langle r\rangle_{nlj}$ is the mean
radius of orbital $nlj$ and $v$ is the velocity of the incident proton. Then,
considering the Heisenberg principle, the $nlj$ split is unresolved in the
energy region where the electrons contribute to the stopping power and the
$nl$ electrons must be considered together (intra-shell screening). The same
effect is considered for shells with similar binding energy and parity (inter-
shell screening). This approach has been successfully implemented to describe
the stopping power of protons colliding on many heavy solid targets, such as
Hf, W, Au, Pb, and Bi [22, 23, 24, 25, 26].

Figure 5: (color online) SLPA stopping power for protons of the first eight
elements of tungsten’s isonuclear sequence. Relativistic (solid lines) and
non-relativistic (dotted lines) results include inter-shell screening of the
$4f$-$5p$ and $5d$-$6s$ electrons.

To determine the bound electron contribution to the stopping power for protons
when interacting with tungsten ions of mean ionization charge $\bar{q}$, we
first determined the stopping power due to bound electrons of the first 20
elements of the isonuclear sequence of tungsten. In Fig. 5, we present the
results for the first eight ions. These values include the inter-shell and
intra-shell shell screening for the relativistic (solid lines) and non-
relativistic (dotted lines) calculations. These results correspond to
implementing the SLPA with the relativistic and non-relativistic electronic
structure data, i.e., $\rho_{nl}$ and $E_{nl}$, presented in the previous
section. For the first ions (W I, II, and III), the stopping maxima for the
relativistic calculations are larger than the non-relativistic ones by about
15%. Furthermore, the proton energies corresponding to these stopping maxima
are shifted by around 33% to lower values as these effects are considered.
However, as the ionization increases, the differences in the amplitude of the
stopping cross sections are reduced to about 5% without maxima shifts.
Remarkably, the amplitude of the stopping power due to bound electrons
decreases by more than half when the ion loses only three electrons; the
number of electrons in the atom goes from 74 to 71, yet the stopping maximum
decreases by about 55%. We obtain this result because tungsten loses its
weakly bound electrons; these electrons are far from the screened nuclei and
strongly interact with the incoming projectile. Then, as the ions lose their
outer electrons, only the most strongly bound remain, which do not interact
with the projectile as much. The stopping power due to remaining electrons
becomes less than 10% than in the case of neutral tungsten when the ion loses
only a couple of ten electrons.

As established in Section 4.1, the mean ionization charge of tungsten varies
according to temperature and density, and the $\bar{q}$ values were shown to
be fractional. We performed a cubic spline interpolation between the stopping
power results of the closest ions to obtain the energy loss per unit length
due to the $Z_{T}-\bar{q}$ bound electrons. Indeed, the bound electron density
depends on the plasma temperature through $\bar{q}$ and is computed as
$n_{be}=(Z_{T}-\bar{q})\,n_{fe}/\bar{q}$. For a plasma with density
$n_{fe}=1\times 10^{22}$ cm-3 at $T=10$ eV, the mean charge state of tungsten
is $\bar{q}=1.7$. To compute the corresponding bound electron stopping power,
we utilized the results shown in Fig. 5 for W+ and W2+, with a bound electron
density of $n_{be}=4.2\times 10^{23}$ cm-3.

Figure 6: (color online) Bound electron stopping power for protons of plasmas
with $n_{fe}$ electron density at 10 eV, 30 eV, and 60 eV: relativistic (R,
solid lines) and non-relativistic (NR, dotted lines) SLPA results, EWPM
(dashed lines), and Zimmerman model (dash-dotted lines).

Finally, the present results for the bound electron contribution to the
stopping power for a tungsten plasma with electron densities 1$\times 10^{22}$
cm-3 (left panel) and 1$\times 10^{23}$ cm-3 (right panel) at temperatures
$T=$ 10 eV, 30 eV, and 60 eV are shown in Fig. 6. The relativistic (solid
lines) and non-relativistic (dashed lines) values also decrease as the
temperature increases. However, in contrast to the free electron stopping
results, the bound electron stopping power varies differently as the plasma
temperature increases with constant electron density. The stopping power due
to bound electrons of a plasma with $n_{fe}=1\times 10^{22}$ cm-3 decreases
more rapidly as the plasma temperature increases than in the case of a plasma
with $n_{fe}=1\times 10^{23}$ cm-3. As previously discussed, these differences
are related to the mean ionization state $\bar{q}$ of the ion predominant at a
given temperature.

The SLPA results in Fig. 6 are compared with values from two independent
models: the extended wave-package model (EWMP) proposed by Archubi & Arista
[27] and the Zimmerman model [28]. The EWPM is a wave-packet approach with the
same binding energy threshold criteria [29] as the SLPA does. The Zimmerman
model uses a Bethe-like expression with a mean ionization potential expression
for bound electrons. The EWPM calculations have been performed with non-
relativistic binding energies, momentum parameters for neutral tungsten, and
without intra-shell screening. In most cases, our non-relativistic
calculations and EWPM values agree well, with stopping power predictions below
the relativistic ones. As expected, all the results agree with the Zimmerman
model for proton impact energy greater than 100 keV.

###  4.4 Total stopping power

We combine the free electron and the bound electron contribution to obtain the
total electronic stopping power for protons of a plasma. We illustrate our
results in Fig. 7 for all the electron densities and temperatures examined
thus far. We compare the present results with the T-Matrix approach and the
Li-Petrasso model for free electrons in combination with the SLPA for bound
electrons. We also included the free electron ADF values to discuss the
importance of considering the bound electron stopping power in these
calculations.

Figure 7: (color online) Total stopping power for protons of a plasma with
electron density $n_{fe}$ at 10 eV, 30 eV, and 60 eV. The free electron
stopping models of Fig. 2 are combined with relativistic SLPA results: SLPA +
ADF (solid lines), SLPA + Li-Petraso model (dashed lines), and SLPA + T-Matrix
(dash-dotted lines). We included the free electron stopping power results of
the Arista dielectric function formalism (thick solid lines) for comparison.

As the temperature increases, our total stopping power results in Fig. 7
decrease, following the same trend as the free and bound electron
contributions discussed in Sections 4.2 and 4.3.2. The free electrons
contribute entirely to the total stopping power for slow protons at all
temperatures and densities. For a plasma with electron density $n_{fe}=1\times
10^{22}$ cm-3, the bound electron contribution to the total stopping power
becomes relevant only for incident proton energies greater than 5 keV and 40
keV for plasmas with temperatures of 10 eV and 30 eV, respectively. For the
lowest-temperature plasma, the bound electrons constitute nearly 70% of the
total stopping power around the maxima. However, for a plasma at 60 eV, the
stopping power is almost entirely due to the free electrons. For a plasma with
a free electron density of $n_{fe}=1\times 10^{23}$ cm-3, the bound electron
contribution becomes relevant at higher proton energies, i.e., 20 keV, 30 keV,
and 80 keV for the respective increasing plasma temperatures considered. The
relative contribution to the total stopping in this case is minor compared to
the previously discussed case, constituting only around 30% of the total
stopping at the maxima. In this scenario, the ion has been ionized four times,
losing its most weakly bound electrons. The mean charge of the ion in this
plasma slowly varies in this temperature range, going from 4.3 to 10, and so
does the bound electron stopping power; however, as the plasma features high
bound electron densities, the stopping power due to these electrons is not
negligible even at temperatures of 60 eV.

##  5 Summary

This work presents relativistic results for the electronic stopping power for
protons of a tungsten plasma at temperatures $T=10$ eV, $30$ eV and $60$ eV
and free electron densities $n_{fe}=1\times 10^{22}$ cm-3 and $1\times
10^{23}$ cm-3. We obtained total values by combining stopping power
calculations due to free and bound electrons. The interaction of incident
projectiles and free and bound electrons was modeled within the dielectric
formalism, which includes collective and binary collisions. This approach
relies on the perturbative approximation, and it is valid for incident
projectile velocities much larger than the mean velocity of the electrons
interacting with the projectile and for projectiles with smaller nuclear
charges than the target.

The free electron interaction with the projectiles was modeled with a quantum
plasma dielectric function [9]. This approach allows one to model the stopping
power of a plasma at any degeneracy. The energy loss per unit length for the
projectile due to the bound electrons was modeled with the shellwise local
plasma approximation [13]. The present free and bound electron stopping power
results were compared with other models. In most cases, all the theoretical
models agree.

We found the bound electron contribution to stopping power can sometimes be
significant, if not determinant, in high-$Z$ plasmas. For certain electron
densities and temperature combinations, the stopping power of the plasma near
the maxima can dominated by the bound electrons, or it can be negliglible. In
other cases, the bound electron stopping power can be significant across the
entire temperature range. As a final mark, bound electron contributions to the
stopping power of plasmas must be addressed in cases where high-Z ions are
present, examining the specific density and temperature of the plasma under
consideration. In the future, these calculations could be extended to heavy
and multi-charged projectiles [30, 31, 32].

##  6 Acknowledgements

This work was financed by the Junta de Comunidades de Castilla-La Mancha under
contract No. SB-PLY/19/180501/000105, and the Spanish Ministerio de Ciencia e
Innovación under contracts PID2019-110802GB-I00 and PID2019-110678GB-I00, and
FEDER.

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