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Journal of Physics B: Atomic, Molecular and Optical Physics
PAPER • OPEN ACCESS
Multiply-excited states and their contribution to opacity in CO2 laser-
driven tin-plasma conditions
To cite this article: J Sheil et al 2021 J. Phys. B: At. Mol. Opt. Phys. 54 035002
View the article online for updates and enhancements.
This content was downloaded from IP address 46.4.80.155 on 13/07/2021 at 18:37
Journal of Physics B: Atomic, Molecular and Optical Physics
J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 (11pp) https://doi.org/10.1088/1361-6455/abcedf
Multiply-excited states and their
contribution to opacity in CO2 laser-driven
tin-plasma conditions
J Sheil1,∗ , O O Versolato1,2 , A J Neukirch3 and J Colgan3
1
Advanced Research Center for Nanolithography, Science Park 106, 1098 XG Amsterdam, The
Netherlands
2
Department of Physics and Astronomy, and LaserLaB, Vrije Universiteit, De Boelelaan 1081, 1081 HV
Amsterdam, The Netherlands
3
Los Alamos National Laboratory, Los Alamos, NM 87545, United States of America
E-mail: j.sheil@arcnl.nl and jcolgan@lanl.gov
Received 22 September 2020, revised 17 November 2020
Accepted for publication 30 November 2020
Published 22 January 2021
Abstract
A recent study (2020 Nat. Commun. 11 2334) has found that transitions between
multiply-excited configurations in open 4d-subshell tin ions are the dominant contributors to
intense EUV emission from dense, Nd:YAG-driven (laser wavelength λ = 1.064 μm) tin
plasmas. In the present study, we employ the Los Alamos Atomic code to investigate the
spectral contribution from these transitions under industrially-relevant, CO2 laser-driven
(λ = 10.6 μm) tin plasma conditions. First, we employ Busquet’s ionisation temperature
method to match the average charge state Z of a non-local-thermodynamic equilibrium
(non-LTE) plasma with an LTE one. This is done by varying the temperature of the LTE
calculations until a so-called ionisation temperature T Z is established. Importantly, this
approach generates LTE-computed configuration populations in excellent agreement with the
non-LTE populations. A corollary of this observation is that the non-LTE populations are
well-described by Boltzmann-type exponential distributions having effective temperatures
T eff ≈ T Z . In the second part of this work, we perform extensive level-resolved LTE opacity
calculations at T Z . It is found that 66% of the opacity in the industrially-relevant 2%
bandwidth centred at 13.5 nm arises from transitions between multiply-excited states. These
results reinforce the need for the consideration of complex, multiply-excited states in
modelling the radiative properties of laser-driven plasma sources of EUV light.
Keywords: opacity, non-LTE, effective temperature, multiply-excited states
(Some figures may appear in colour only in the online journal)
1. Introduction [2–4], materials science [5, 6] and the manufacturing of inte-
grated circuits (ICs) in the semiconductor industry [7, 8]. The
Powerful sources of short-wavelength radiation play a cru- use of short-wavelength radiation in this latter application is
cial role in many high-profile scientific and industrial endeav- expected to enable the printing of sub-10 nm feature sizes on
ours [1]. Examples include high-resolution biological imaging ICs [9], thus sustaining the production of cheaper and more
powerful computer chips into the future.
∗
Author to whom any correspondence should be addressed. The key technology driving feature-size miniaturisation in
Original content from this work may be used under the terms
of the Creative Commons Attribution 4.0 licence. Any further
the semiconductor industry today is extreme ultraviolet lithog-
distribution of this work must maintain attribution to the author(s) and the title raphy (EUVL) [10–13]. As the name suggests, this technology
of the work, journal citation and DOI. uses EUV light to print features on ICs. In an industrial setting,
0953-4075/21/035002+11$33.00 1 © 2021 IOP Publishing Ltd Printed in the UKJ. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
this EUV light is generated in a laser-produced plasma (LPP) are therefore significantly more extensive than those consid-
formed by irradiating tin microdroplets with CO2 laser light ered previously. Importantly, we find that transitions between
(λ = 10.6 μm) [14, 15]. The ensuing laser–droplet interaction multiply-excited states make a substantial contribution to the
generates a plasma having moderate electron temperatures and opacity in the 5–20 nm region in plasma conditions pertinent
densities (T e ≈ 20–40 eV, ne ≈ 1018 –1019 cm−3 ) compris- to those generated in CO2 laser-driven tin plasmas.
ing large populations of Sn8+ –Sn14+ ions. Spectra recorded The current paper is organised as follows: in section 2,
from such plasmas are known to exhibit an intense, narrow- we describe the atomic structure and opacity calculations per-
band feature peaked near 13.5 nm [16–20]. Importantly, this formed with the Los Alamos suite of atomic physics and
emission feature provides optimum wavelength overlap with population kinetics codes. An examination of non-LTE- and
the 2% reflective bandwidth (bw) centred at 13.5 nm defined LTE-computed tin ionisation distributions and configuration
by molybdenum/silicon multilayer mirrors (MLMs) employed populations in CO2 laser-driven plasma conditions follows
in lithography tools [12]. This combination of (i) optimum in section 3. In section 4 we present level-resolved LTE-
emission characteristics and (ii) a debris-limited droplet-target computed opacities of Sn11+ –Sn14+ . Finally, a summary of
(key for minimising MLM contamination from plasma expan- this work is given in section 5.
sion [21]) established tin LPPs as the light source of choice for
EUVL. In fact, after more than 20 years of research and devel-
opment, EUVL has now entered the realm of high-volume 2. Atomic structure and opacity calculations
manufacturing of new-generation ICs [22].
The aforementioned intense, narrowband emission feature In the present work, we have employed the Los Alamos
observed in laser-produced tin plasmas has, to date, been national laboratory (LANL) suite of atomic structure and colli-
sion codes [42] as well as the population kinetics code Atomic
predominantly attributed to overlapping Δn = 0 transition
[43, 44] to calculate the opacities of a tin plasma at densities
arrays of the form 4p6 4dm − (4p5 4dm+1 + 4p6 4dm−1 4 f) in
and temperatures relevant to those attained in a CO2 laser-
charge states Sn8+ –Sn14+ (m = 6–0) [23–25]. Such claims
driven plasma. Given the central role that these codes play
have been supported by charge-state-resolved spectro-
in this study, we devote the following subsections to their
scopic experiments [26–29] as well as numerous theoretical
description.
efforts employing advanced atomic structure codes [30–39].
Although serving as the general consensus for many years,
a recent study of Torretti et al has challenged this view- 2.1. Atomic structure calculations
point [13]. In the most detailed tin-ion opacity calculations
The first step in the present opacity calculations is the gen-
performed to-date, it was found that transitions between
eration of all relevant atomic data (energy levels, oscillator
multiply-excited configurations in Sn11+ –Sn14+ are in
strengths, ionisation potentials, etc). In Torretti et al [13], con-
fact the dominant contributors to EUV emission in Nd:YAG-
siderable effort was devoted to benchmarking the calculated
driven (λ = 1.064 μm) tin LPPs. Surprisingly, the well-known
atomic structures with available experimental data and there-
4p6 4dm − (4p5 4dm+1 + 4p6 4dm−1 4 f) transitions play only a
fore we have utilised these atomic structures in the present
minor role; they comprise 11% of the opacity in the 5–20 nm
study. In brief, the Cats code [45] was employed in the cal-
region and 19% of the opacity in the 2% bw. Remarkably, it is
culation of level energies and absorption oscillator strengths.
transitions between complex, multiply-excited configurations This code is based on Cowan’s original atomic structure
(doubly-, triply- and quadruply-excited configurations) which code [46, 47] and exhibits a favourable feature where one
dominate the opacity spectrum. Strong support for their spec- can scale the magnitude of the radial integrals in order to
troscopic dominance was evidenced through comparisons of bring theoretical level structures into excellent agreement with
experimental tin-droplet EUV spectra with one-dimensional experimental observations. This semi-empirical approach has
radiation transport simulations. The incorporation of multiply- proven to be an indispensable tool in atomic spectroscopy,
excited transitions in these simulations was deemed crucial especially for systems exhibiting complicated level structures
for the reproduction of the experimental spectra. [29, 48–51]. In the work of Torretti et al [13], optimum
The purpose of the present paper is to elucidate the agreement between atomic structure calculations and previous
role of multiply-excited states in CO2 -driven tin plasmas experimental observations [17, 26, 27], was found by applying
which are currently used in the industry to provide light for scaling factors of 0.87/0.75 to the radial integrals associated
EUVL. The main difference between the present work and with singly-/multiply-excited energy levels. These two scal-
previous atomic modelling work of Sasaki and co-authors ing factors were adopted for all charge states considered in
[37, 40, 41] is the extensivity of the atomic structures this work. Extensive configuration sets were employed in the
employed. For example, in reference [37, 40, 41], a less exten- calculations through use of the so-called ‘2-mode’ approach.
sive number of doubly-excited states were included in their In these calculations, configuration interaction (CI) was
atomic models. Their inclusion was mainly for the purpose retained for a large number of configurations associated with
of understanding satellite transition array emission of the Δn = 0 and Δn = 1 single, double and triple excitations
form 4p6 4dm−1 nl − (4p5 4dm nl + 4p6 4dm−2 4 fnl). The config- from the n = 4 manifold of the ground-state configuration.
uration sets adopted in the present work include large numbers Additional configurations were included using intermediate
of doubly-, triply- and quadruply-excited configurations and coupling (only interactions between levels of the same con-
2J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
figuration are considered) [39]. This approach retains a signif- approach and the occupation probability formalism (this lat-
icant level of detail provided by full CI calculations however ter procedure allows plasma density effects to be included in
at a much reduced computational cost. a consistent manner). Importantly, non-ideal plasma effects,
such as plasma microfields and strong coupling are included
2.2. Opacity calculations in the free energy term. This leads to modified Boltzmann
and Saha relations from which level populations and CSDs are
Knowledge of opacities is crucial for understanding radi-
derived.
ation transport in a host of plasma environments, such as
Atomic also has the capability of modelling plasmas under
those encountered in stellar interiors, fusion devices and short-
non-LTE conditions [58, 59]. Unlike LTE plasmas, the cal-
wavelength plasma light sources [30, 52–55]. In this context,
culation of level populations in non-LTE conditions requires
an opacity quantifies how photons, travelling in a plasma, are
a microscopic viewpoint in which the rates of all relevant
absorbed and scattered through interactions with the plasma
atomic populating and depopulating processes are calculated.
constituents. For a given photon frequency ν, the opacity κTν is
The most general definition of a non-LTE plasma is one which
defined as the contribution of four mechanisms:
exhibits any characteristic departure from LTE conditions,
κTν = κBB BF FF SCAT such as the existence of a non-Maxwellian electron energy dis-
ν + κν + κν + κν . (1)
tribution function (EEDF) or a non-Planckian radiation field.
The first three terms represent bound–bound (photoab- As such, there are a variety of ways that a plasma can be
sorption), bound–free (photoionisation) and free–free (inverse defined as under non-LTE conditions. Consider an atomic state
bremsstrahlung) contributions to the opacity and each contain (energy level, configuration or superconfiguration) i associ-
a factor correcting for stimulated emission. The final term rep- ated with some charge state ζ in the plasma. In non-LTE
resents photon scattering. In the present work, we are mainly plasmas, the population of this state evolves according to a
concerned with the bound–bound contribution to the opac- collisional-radiative (CR) equation of the form [60, 61]
ity (since all other contributions are minor in the wavelength
dniζ
range of interest for these plasma conditions) and therefore = n jζ R jζ →iζ − niζ Riζ→jζ . (4)
we drop the superscript ‘BB’ notation in the following. The dt
jζ jζ
bound–bound opacity κν is written
Here, R jζ →iζ = RCjζ →iζ + RRjζ →iζ denotes the rate of colli-
κν = ρ−1 σνi j (ni − gi g−1
j n j), (2) sional (RCjζ →iζ ) and radiative (RRjζ →iζ ) processes connecting j
i< j
(associated with some charge state ζ ) to i. In a similar fash-
where ρ is the mass density and the summation term is known ion, rates describing i → j are denoted Riζ→jζ . The first term in
as the absorptivity. Here, σνi j is the cross section for photoab- equation (4) represents population influx into i and the second
sorption from level i to level j and ni( j) are the population den- term describes population outflux from i. A CR equation of the
sities (simply called ‘populations’ in the following) of energy form equation (4) exists for all atomic states i in each charge
levels i( j) having statistical weights gi( j) . The sum runs over state ζ present in the plasma. Coupling these states yields the
all transitions i→j associated with photon absorption at fre- so-called rate equation
quency ν. The second term in the summation is the aforemen-
tioned correction for stimulated emission. The cross section dN
= AN. (5)
for photoabsorption σνi j is given by dt
πe2 Here, N is a vector of atomic state populations (niζ ∈ N)
σνi j = f i jφ(ν), (3) and A is the so-called rate matrix containing all relevant rates
me c
RC,R
jζ →iζ,iζ→jζ .
where e, me and c are the electron charge, electron mass and The choice of atomic processes to be included in the rate
speed of light, respectively. The term f i j is the absorption oscil- matrix A is highly case-dependent; in coronal plasmas, for
lator strength for a transition
∞ i → j and φ(ν) is the line profile example, one can neglect the role of three-body recombina-
function satisfying 0 φ(ν)dν = 1. tion. Such simplifications are not possible in the current case
As one can see from equations (2) and (3), evaluation of and therefore we must consider a wide range of atomic pro-
the bound–bound opacity requires knowledge of both atomic cesses (see section 3 for more details). Fortunately, Atomic
structure (ν and f i j) and the level populations ni( j) . Evaluation has the capability of modelling a variety of (de-)populating
of this latter quantity requires knowledge of the plasma envi- processes such as electron-impact excitation/ionisation, pho-
ronment in which the atomic species is embedded. Atomic, toexcitation, photoionisation, autoionisation (AI), etc. Cross
using as input the mass density ρ and electron temperature sections for electron-impact excitation were calculated in the
T e , can calculate these level populations in LTE or non- plane-wave Born (PWB) approximation using the CATS code.
LTE plasma conditions (the radiation temperature T r can Electron-impact ionisation cross-sections were obtained from
also be specified in non-LTE calculations). The LTE-mode of the GIPPER code (photoionisation cross sections can also be
Atomic uses the ChemEOS equation-of-state (EOS) model to obtained if an external radiation field is included in the non-
derive level populations and charge state distributions (CSDs) LTE model) [62]. Atomic uses these cross sections to calculate
[56, 57]. This model is based on the free-energy-minimisation the rates which form the entries of A. Invoking a conservation
3J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
12+ 14+
Table 1. Configuration sets adopted for Sn and Sn in the non-LTE/LTE CA calculations (see main text).
Configurations are grouped according to excitation degree. Here, angular momenta l = 0 – 4(s − g), l = 1 – 4(p − g)
and l̃ = 2 – 4 (d − g), respectively.
Sn12+ Sn14+
Ground configuration Ground configuration
4s2 4p6 4d2 4s2 4p6
Singly-excited configurations Singly-excited configurations
4s2 4p6 4d + {4 f, 5l} 4s2 4p5 + {4d, 4 f, 5l}
4s2 4p5 + {4d 3 , 4d 2 4 f, 4d 2 5l} 4s1 4p6 + {4d, 4 f, 5l}
4s4p6 + {4d 3 , 4d 2 4 f, 4d2 5l}
Doubly-excited configurations Doubly-excited configurations
4s2 4p6 + {4 f 2 , 4 f 5l, 5s5l, 5p5l , 5d5l̃ , 5 f 5g} 4s2 4p4 + {4d 2 , 4d4 f, 4 f 2 , 4d5l, 4 f5l, 5s5l, 5p5l , 5d5 f, 5d5g, 5 f5g}
4s2 4p5 + {4d4 f 2 , 4d5s2 , 4d4 f5l, 4d5s5p, 4d5s5d} 4s1 4p5 + {4d 2 , 4d4 f, 4 f 2 , 4d5l, 4 f5l, 5s5l, 5p5l , 5d5 f, 5d5g, 5 f5g}
4s2 4p4 + {4d 4 , 4d 3 4 f, 4d 3 5l, 4d 2 4 f 2 } 4p6 + {4d 2 , 4d4 f, 4d5l, 4 f5l}
4s4p5 4d 4
Triply-excited configuration Triply-excited configurations
4s2 4p3 4d5 4s2 4p3 4d 3 , 4s2 4p3 4d 2 4 f, 4s4p4 4d3 , 4s4p4 4d2 4 f
equation of the form this choice of extensivity over simplicity makes the solution of
equation (5) for these level structures intractable; there are sim-
Z Ni −1
ply too many atomic levels to consider. As discussed in refer-
niζ = nion , (6) ence [13], the calculation of detailed, level-resolved opacities
ζ =0 i =0
for such complex structures is currently only possible using
where N i is the number of atomic states and nion is the ion the LTE approach in Atomic.
density allows one to solve equation (5) for the atomic state
populations niζ . These populations can then used to produce 3.1. Busquet’s ionisation temperature method
opacities [equation (2)] in a similar fashion as in the LTE mode
of Atomic. In order to circumvent computational intractabilities associ-
Finally, it should be emphasized that a consistent atomic ated with level-resolved, non-LTE opacity calculations involv-
structure has been employed in all stages of the calculation, ing multiply-excited configurations, we have employed the
ranging from the evaluation of collisional cross-sections to the ionisation temperature method of Busquet [64, 65] to effec-
derivation of atomic state populations and level-resolved opac- tively ‘mimick’ non-LTE opacities using LTE computations.
ities. This procedure ensures self-consistency in all aspects of The key observation underlying this method is that non-LTE
the calculation. CSDs (performed at some T e ) often resemble LTE CSDs
undertaken at a temperature T Z different to T e [66]. The first
step in Busquet’s approach is to identify the LTE temperature
3. Tin ionisation distributions and configuration T Z which yields optimum agreement in average charge state
populations in CO2 laser-driven plasma conditions Z between the non-LTE and LTE calculations, i.e.
The study of CO2 laser irradiation of tin targets comprises a
large body of work (see [20] and references therein). A very Z(Te )non−LTE = Z(TZ )LTE . (7)
important aspect of this work is the study of tin-ion opaci-
ties in the relevant plasma conditions [32, 37–39, 63]. Plasmas In the current context, T Z is known as the ‘ionisation
generated by CO2 laser irradiation lie in a rather unique posi- temperature’. Once T Z is determined, one then performs
tion in (ne , T e ) space; electron densities are not high enough detailed, level-resolved LTE opacity calculations at T Z .
to allow for LTE conditions to prevail, however not so low to Higher-order moments of the CSD, such as the width, are
justify the use of coronal equilibrium relations. In this respect, not considered in this method. Although this approach pro-
the atomic modelling of CO2 -driven laser–plasmas should, in vides a numerically cost-efficient means of incorporating non-
theory, employ the CR approach described in section 2. LTE opacities (and EOS quantities) into hydrodynamic codes,
As outlined in the introduction, the goal of the present Klapisch and Bar-Shalom have stressed that it should only be
work is to evaluate the role of multiply-excited states in CO2 applied to plasmas not far from pure LTE conditions [66].
laser-driven plasma conditions. As such, we must inherit the We wish to test the validity this method in the context of
complicated level structures considered in reference [13]. As the current problem. Guided by previous modelling efforts of
discussed in reference [13, 39], the use of extensive atomic Sasaki and co-workers [37, 40, 41], we have undertaken a
models for Sn11+ –Sn14+ is necessary to ensure (i) correct non-LTE configuration-averaged (CA) Atomic calculation at
wavelength positions of Δn = 0 transitions (which are sensi- ρ = 2 × 10−4 g cm−3 and T e = 36 eV. The electron density of
tive to the choice of underlying atomic structure as a result of this plasma is ne = 1.27 × 1019 cm−3 , close to the critical elec-
CI) and (ii) partition function convergence [56]. Unfortunately, tron density for CO2 laser light (ncrit,CO2 = 1.1 × 1019 cm−3 ).
4J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
CSD is possible using a T e = 26 eV non-LTE calculation,
however one must include an external radiation field in the
calculation characterised by T r = T e . In this way, all atomic
processes are in detailed balance. However, as detailed above,
our non-LTE calculations do not include an external radia-
tion field. In order to match our present calculations with a
T Z = 26 eV LTE calculation, we must therefore compensate
for the missing (de)excitation and ionisation channels pro-
vided by photoexcitation and photoionisation by increasing
T e above T Z . The difference T e − T Z is effectively a gauge
for how close the system is to LTE conditions; in the work
of Torretti et al [13], for example, a value T e − T Z = 2.5 eV
(where T e = 34.5 eV and T Z = 32 eV) highlights the domi-
Figure 1. Charge-state distributions for plasmas calculated in
nance of collisional rates over radiative rates at near-LTE con-
non-LTE (T e = 36 eV; shown in black) and LTE (T Z = 26 eV; ditions (ne = 1.27 × 1020 cm−3). The same cannot be said for
shown in red) conditions. The plasma mass density used in both the CO2 -relevant conditions, where a value T e − T Z = 10 eV
calculations was ρ = 2 × 10−4 g cm−3 . is indicative of truly non-LTE conditions, i.e. collisional rates
do not significantly outweigh radiative rates.
The CA approach generates appropriately averaged ener- 3.2. Population distributions and effective temperatures
gies, oscillator strengths and PWB electron-impact excita-
tion cross sections from which configuration-to-configuration The replication of a non-LTE CSD using LTE computations,
rates are derived [59]. This approach is employed in the such as that shown in figure 1, is key for justifying the use
present context to ensure computational tractability. Config- of LTE opacities as an approximation for non-LTE opacities.
urations sets employed in the CA CATS calculations for An even stronger argument in favour of this approximation
Sn12+ and Sn14+ are provided in table 1. The calculations is to establish equivalence between the non-LTE and LTE-
were performed under steady-state conditions, i.e. dN/dt = 0. computed configuration populations (the cumulative sum of
This approximation is valid for plasmas where the time taken which yields the CSD—see equation (6)). To this end, we
to generate a given CSD (through ionisation and recombi- present in figure 2 non-LTE- and LTE-computed configuration
nation processes) occurs on timescales shorter than the laser populations (Pnon−LTE, PLTE ) as a function of CA energy (Eav )
pulse duration. This is certainly true for industrial LPP EUV for Sn12+ (upper panel) and Sn14+ (lower panel). The non-LTE
sources, which are typically driven by multi-10 nanosecond- configuration population calculations are shown in figure 2(a)
long CO2 pulses. We calculated the rates of electron-impact (Sn12+) and figure 2(d) (Sn14+ ). The corresponding LTE cal-
excitation/ionisation and AI (as well as their inverse pro- culations are shown in figures 2(b) and (e). Configurations
cesses) assuming a Maxwell–Boltzmann EEDF at T e = 36 eV. associated with single-, double- and triple-electron excitations
The calculations were performed without an external radiation are shown as blue, orange and green circles, respectively. The
field, i.e. photoexcitation and photoionisation processes were ground-state configurations, fixed at Eav = 0 eV, are shown as
not considered (see subsection 3.2 below for further discus- grey circles. Perhaps the most striking feature of these calcula-
sion). The process of spontaneous emission was included in tions is the near-coincidence in non-LTE- and LTE-computed
the model. configuration populations. This is quantified in figures 2(c)
The CSD resulting from this calculation is shown in black in and (f ), where, for a given configuration C, we plot the non-
figure 1. Importantly, the relevant EUV-emitting Sn11+ –Sn14+ LTE-normalised Configuration Population Difference (CPD)
ions are produced under such conditions, with Sn12+ and defined as (PC,non−LTE − PC,LTE )/PC,non−LTE . For the majority
Sn13+ clearly the dominant species. The average charge of configurations, this quantity takes on values below 1%. This
state of this plasma is Znon−LTE = 12.6. Also illustrated in indicates a near-coincidence in non-LTE and LTE-computed
figure 1 is an LTE CA calculation (shown in red) performed at configuration populations. The few configurations where the
ρ = 2 × 10−4 g cm−3 and T e = 26 eV. These calculations yield non-LTE-normalised CPD values are above 1% all have LTE-
ZLTE = 12.6, identical to the non-LTE calculation. Also, we computed populations greater than the corresponding non-LTE
note that the CSDs are very nearly identical. An ionisation tem- populations.
perature of T Z = 26 eV is thus established. We note that the In addition to validating the use of Busquet’s approach in
ratio T e /T Z = 36/26 ≈ 1.4 is larger than that obtained by the the current context, the equivalence in configuration popu-
scaling law Te /TZ ≈ (1 + (5.36 × 1013 )Te n−1
7/2 lations reveals an interesting characteristic of the non-LTE-
e )
0.215
≈ 1.2
obtained by Busquet et al [65]. Much better agreement with computed populations in that they appear to follow a single
this scaling law is obtained in the Nd:YAG-relevant case Boltzmann-esque temperature law of the form
considered in [13] (ne = 1.27 × 1020 cm−3 , T Z = 32 eV,
E
T e = 34.5 eV), where T e /T Z = 34.5/32 ≈ 1.08 and the scal- PC,non−LTE ∝ gC exp − , (8)
Teff
ing law yields T e /T Z ≈ 1.02.
It is interesting to consider why optimum agreement in where gC = JC (2JC + 1) is the total statistical weight of con-
CSDs is obtained for T e > T Z . Exact reproduction of an LTE figuration C (J C is an energy level associated with C) and T eff is
5J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
Figure 2. Configuration populations for Sn12+ (top) and Sn14+ (bottom) calculated for a T e = 36 eV non-LTE plasma [panels (a) and (d)]
and a T e = 26 eV LTE [panels (b) and (e)] plasma. The plasma mass density used in both calculations was ρ = 2 × 10−4 g cm−3 .
Configurations associated with single-, double- and triple-electron excitations are shown as blue, orange and green circles, respectively. The
ground-state configurations are shown in grey and are fixed at Eav = 0 eV. Non-LTE-normalised configuration population differences
(CPDs) (see text for description) are shown in panels (c) and (f). The vertical red lines indicate the ionisation potentials of Sn12+ and Sn14+ .
a so-called effective temperature [67] (the Boltzmann constant states typically decay towards high/low-lying lower states)
kB typically appearing in equations of the form equation (8) is tend to generate Boltzmann-type population distributions
here ‘absorbed’ into the definition of T eff ). [71]. This is in fact a general phenomenon of non-LTE hot
In order to evaluate T eff , we plot in figure 3 the natural plasmas [68]. Finally, it should be noted that the approach
logarithm of the ratio PC,non−LTE/gC as a function of Eav for of Bauche et al [70], does not ascribe a single effective
(a) Sn12+ and (b) Sn14+ . First-order polynomial fits to the temperature T eff to a population distribution. Rather, con-
data are shown in red. Excellent fits to the data are charac- figurations are grouped according to superconfigurations
terised by R2 values of 0.99 attained in both cases. The effec- and equation (8) is used to derive superconfiguration-
tive temperatures T eff are obtained from the inverse-negative specific temperatures. These temperatures may vary
of the slopes of the lines of best fit. As indicated in the significantly from superconfiguration-to-superconfiguration
plots, the fits yield values of T eff = 26.6 (Sn12+ ) and 26.9 eV
(see figure 1 of [70]). Although this approach differs to that
(Sn14+ ), clearly very similar to the LTE ionisation temperature
presented in figure 3 (where we have performed a ‘global’ fit
T Z = 26 eV. This close agreement between T eff and T Z is per-
to all configurations), the non-LTE-computed population dis-
haps not too surprizing given the near-coincidence in non-LTE
tributions are clearly well-described by a single-temperature
and LTE-computed configuration populations.
This observation, that non-LTE-computed configuration Boltzmann-type law.
populations follow Boltzmann-like temperature laws has been Finally, we wish to comment on the importance of a non-
studied extensively in the works of Bauche, Bauche-Arnoult zero radiation field in our non-LTE computations. As out-
and others [67–72]. The emergence of temperature laws lined in section 3, all non-LTE calculations presented thus far
is attributed to the complex interplay of atomic processes have not included an external radiation field in the underly-
defining the steady-state CR model. More specifically, the ing atomic kinetics. Indeed, identifying an external radiation
combination of (i) the microreversibility nature of colli- field appropriate for the current study is rather difficult as it
sional processes (which drives the population distributions is a purely non-local quantity, i.e. it depends on the radia-
towards LTE conditions) and (ii) correlation laws asso- tive characteristics of the plasma surrounding the studied (ρ,
ciated with spontaneous emission (high/low-lying upper T e ) plasma. Proper evaluation of an external radiation field
6J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
Figure 3. Logarithm of the statistical-weight-normalised Figure 4. Logarithm of the statistical-weight-normalised
non-LTE-computed configuration populations for (a) Sn12+ and non-LTE-computed configuration populations for (a) Sn12+ and
(b) Sn14+ . The plasma conditions correspond to ρ = 2 × 10−4 g (b) Sn14+ . The plasma conditions correspond to ρ = 2 × 10−4 g
cm−3 , T e = 36 eV and T r = 0 eV. A first-order polynomial fit to the cm−3 , T e = 33 eV and T r = 21 eV. A first-order polynomial fit to
data is shown in red. Effective temperatures T eff are obtained from the data is shown in red. Effective temperatures T eff are obtained
the slopes of the lines of best fit. from the slopes of the lines of best fit.
requires complex simulations involving the coupling of non- 4. Tin-ion opacities
LTE atomic kinetics with a model for radiation transport in
a multi-(ρ, T e , T r ) plasma. Such an endeavour is outside the We now present the results of detailed, level-resolved LTE
scope of the present work. Instead, we have performed an addi- opacity calculations at CO2 laser-driven plasma conditions.
tional non-LTE calculation where we have included a Planck- The LTE plasma conditions considered here are the same
ian radiation field (characterised by a radiation field T r ). We as those of section 3 (ρ = 2 × 10−4 g cm−3 , T Z = 26 eV).
fixed T r and tuned T e until equation (7) is established with Unlike section 3, we adopt a fully level-resolved approach
a T Z = 26 eV LTE plasma. We find that a T e = 33 eV and where Atomic is used to calculate level populations and
T r = 21 eV plasma, for example, reproduces both equation (7) bound–bound opacities. The atomic structures employed in
and the LTE CSD. Moreover, the configuration populations these calculations are the same as those considered in reference
associated with Sn12+ and Sn14+ are also found to be in excel- [13] (see section 2).
lent agreement with the LTE-computed configuration popu- The results of these calculations are presented in figure 5.
lations. This is demonstrated in figure 4, where, in a similar Panels (a)–(d) illustrate the contributions to the opacity from
vein to figure 3, we plot the natural logarithm of the ratio the four charge states Sn11+ –Sn14+ . In a similar fashion to
PC,non−LTE/gC as a function of Eav for (a) Sn12+ and (b) Sn14+ . Torretti et al [13], the different contributions to the opac-
Effective temperatures obtained from the slopes of the lines of ity are plotted in a cumulative fashion; photoabsorption from
best fit (shown in red) are T eff = 26.6 eV for both Sn12+ and levels in the ground manifold to singly-excited levels are
Sn14+ , clearly in excellent agreement with the data presented shown in blue (we call this the ‘ground opacity’); the summa-
in figure 3. The introduction of a non-zero radiation field tion of photoabsorption from singly-excited levels to doubly-
in these computations therefore does not break the observed excited levels (‘first opacity’) and the ground opacity is shown
Boltzmann-esque temperature law once equivalence between by the line bounding the orange shaded area; the summa-
the non-LTE and LTE-computed CSDs is established. tion of photoabsorption from doubly- and triply-excited levels
7J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
Figure 6. Panels (a)–(d): contributions from the ground, first and
higher opacities in the 2% bw in Sn11+ –Sn14+ . The total opacity in
the 2% bw, obtained by summing the contributions in panels
Figure 5. Panels (a)–(d): opacities of the ions Sn11+ –Sn14+ . The (a)–(d), is shown in panel (e).
opacities are plotted in a cumulative fashion. Photoabsorption from
levels of the ground manifold to singly-excited levels is shown in
blue (‘ground opacity’). The summation of photoabsorption from
singly-excited levels to doubly-excited levels (‘first opacity’) and the any other transition in Sn11+ –Sn14+ . There are two reasons
ground opacity is shown by the line bounding the orange shaded for this spectral dominance. First, as illustrated in figure 2,
region. The line bounding the green shaded region is the summation
of photoabsorption from doubly- and triply-excited levels (‘higher
a significant fraction of Sn12+ and Sn14+ ions are in their
opacity’), the ground opacity and first opacity. The total opacity, ground-state configurations. This also true for Sn13+ . Sec-
obtained by summing the contributions of Sn11+ –Sn14+ , is shown in ondly, this LS-allowed transition exhibits a large oscillator
panel (e). The inset in panel (e) shows the opacity in a 2% bw strength. This combination, which appears multiplicatively in
centred at 13.5 nm. the first term of equation (2), is the origin of this large opacity
value. As in the case of Sn11+ and Sn12+ , higher-order opaci-
(‘higher opacity’), the ground opacity and first opacity is ties are seen to contribute at wavelengths above 13.5 nm, how-
shown by the line bounding the green shaded area. The grey ever only on the order κ ≈ 10 × 105 cm2 g−1 . Moving to the
region in each panel represents a 2% bw centred at 13.5 nm. closed 4p subshell Sn14+ ion we observe an interesting shift
First, examining the ordinate of figures 5(b) and (c), one in opacity dominance away from the ground opacity to the
readily concludes that Sn12+ and Sn13+ are the dominant con- first opacity. The ground opacity in this wavelength region is
tributors to the opacity in this plasma. This can be under- dominated by the transition 4p6 1 S0 –4p5 4d 1 P1 , which has a
stood from the CSD presented in figure 1, where these two calculated wavelength of 13.332 nm. This is in excellent agree-
charge states accumulate nearly 80% of the total ion popula- ment with experimental identifications of this transition at
tion. Although the majority of the opacity in the Sn12+ ion λ = 13.344 nm [29] and λ = 13.343 [18, 74]. This transition
can be attributed to the ground opacity, the first and higher is clearly much weaker than the 4p6 4d 2 D5/2 − 4p5 4d2 2 F7/2
opacities contribute to the spectrum at wavelengths larger than transition in Sn13+ . This is because (i) the 4p6 1 S0 level in
13.5 nm. The same is true for Sn11+ . One of the most strik- Sn14+ has a smaller population than the 4p6 4d 2 D5/2 level in
ing features of these opacity spectra is the spectral dominance Sn13+ and (ii) the transition in Sn14+ involves energy levels
of a transition in Sn13+ [see figure 5 (c)]. This transition is having low angular momenta (J lower,upper = 0, 1) and therefore
the subvalence excitation 4p6 4d 2 D5/2 −4p5 4d2 2 F7/2 . It has a has a lower oscillator strength than the aforementioned tran-
calculated wavelength of 13.286 nm, in good agreement with sition in Sn13+ . As is clear from figure 5(d), photoabsorp-
experimental identifications of this transition at 13.3014 [17], tion from singly-excited levels is the main source of opacity
13.3020 [73] and 13.301 nm [29]. The opacity of this transi- in Sn14+ . Importantly, all of the opacity in the industrially-
tion is κ ≈ 83 × 105 cm2 g−1 , nearly a factor 3 higher than relevant 2% bw (the grey shaded region) in Sn14+ origi-
8J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
this figure that the population of multiply-excited (doubly-
, triply- and quadruply-excited) states is much higher in the
T e = 32 eV plasma than in the T e = 26 eV plasma. Examples
of this ratio for the 4p5 4d3 , 4p4 4d4 and 4p3 4d5 configurations
in Sn12+ are shown in blue, orange and green, respectively.
Interestingly, the increased population of the multiply-excited
states in the Nd:YAG-relevant plasma is a natural by-product
of the higher temperatures needed to generate a CSD equiv-
alent to that generated under relevant CO2 -laser-driven tin
plasma conditions.
5. Conclusion
Figure 7. Ratio of the partition function factor for T e = 32 and
26 eV plasmas (shown in black). The ratio for the 4p5 4d3 , 4p4 4d 4 In conclusion, we have investigated the role played by
and 4p3 4d5 configurations in Sn12+ are shown in blue, orange and multiply-excited states in industrially-relevant CO2 laser-
green, respectively.
driven tin plasma conditions. First, we employed Busquet’s
ionisation temperature method to match the charge state dis-
tribution of a non-LTE plasma with an LTE computation. We
nates from higher-order opacities, the first opacity being the find excellent agreement between the non-LTE-computed con-
dominant contributor. figuration populations and LTE calculations for both Sn12+
The total opacity, obtained by summing the contributions and Sn14+ . Moreover, we identify the existence of Boltzmann-
in figures 5(a)–(d), is shown in figure 5(e). As expected, most like temperature laws in the non-LTE population distributions.
of the higher-order opacity contributions are located at wave- In the second part of this study we performed detailed, level-
lengths above 13.5 nm. The inset illustrates the contributions to resolved LTE opacity calculations at plasma conditions per-
the opacity in the 2% bw. Higher-order opacities clearly make taining to CO2 laser-driven plasma conditions. Approximately
a significant contribution to this industrially-relevant spectral 66% of the opacity in the 2% bw centred at 13.5 nm was found
region. In essence, they tend to ‘fill out’ regions between the to arise from transitions between multiply-excited states. This
ground opacities. The charge-state-specific opacity contribu- result highlights the important role played by multiply-excited
tions in the 2% bw are illustrated in figures 6(a)–(d), with states in industrially-relevant plasma conditions. This work
their combined opacity contribution illustrated in figure 6(e). paves the way for the generation of detailed LTE opacity tables
Importantly, only 34% of the opacity arises from the ground at numerous (ρ, T e , T r ) non-LTE plasma conditions for incor-
opacity. Of the remaining 66%, approximately 42% originates poration in radiation-hydrodynamic simulations. Such simu-
from the first opacity and 24% is associated with the higher lations will enable reliable predictions of EUV emission from
opacity. This result clearly highlights the important spectral LPPs and are therefore key for guiding experimental efforts in
role played by multiply-excited transitions in CO2 laser-driven characterising and optimising laser-driven plasma sources of
plasma conditions. EUV light.
Finally, we wish to compare these results with the Nd:YAG-
relevant case considered in [13]. The plasma conditions con-
sidered in reference [13] were ρ = 2 × 10−3 g cm−3 and Acknowledgments
T e = 32 eV. This LTE calculation yielded ne = 1.27 ×
1020 cm−3 (an order of magnitude higher than the current case) We would like to thank W van der Zande, W Ubachs and R
and an average charge state Z similar to the present case. Hoekstra for their useful comments on the paper. This project
Comparing figure 5(e) with figure 3(e) of [13], we see that, has received funding from European Research Council (ERC)
relative to the ground opacity, the higher-order opacities in Starting Grant No. 802648 and is part of the VIDI research
the Nd:YAG case make a significantly larger contribution than programme with Project No. 15697, which is financed by the
in the present case. Comparing Maxwell–Boltzmann distri- Netherlands Organization for Scientific Research (NWO). Part
butions at T e = 32/26 eV, one observes that the former has of this work has been carried out at the Advanced Research
a larger fraction of higher-energy electrons which can colli- Center for Nanolithography (ARCNL), a public-private part-
sionally populate high-lying levels associated with doubly-, nership of the University of Amsterdam (UvA), the Vrije
triply- and quadruply-excited configurations. As both calcula- Universiteit Amsterdam (VU), NWO and the semiconduc-
tions were performed in LTE, the populations of the levels are tor equipment manufacturer ASML. Part of this work was
described by the so-called ‘partition function factor’, which supported by the ASC PEM programme of the US Depart-
for some level J is written gJ exp(−EJ /T e ). To enable a com- ment of Energy through the Los Alamos National Labora-
parison between both cases, we plot in figure 7 the ratio of tory. Los Alamos National Laboratory is operated by Triad
the partition function factor for T e = 32 and 26 eV (shown National Security, LLC, for the National Nuclear Security
in black) using a continuous energy variable E. The statisti- Administration of US Department of Energy (Contract No.
cal weight gJ naturally drops out of this ratio. It is clear from 89233218NCA000001).
9J. Phys. B: At. Mol. Opt. Phys. 54 (2021) 035002 J Sheil et al
ORCID iDs [34] MacFarlane J J, Rettig C L, Wang P, Golovkin I E and
Woodruff P R 2005 Emerging Lithographic Technologies IX
J Sheil https://orcid.org/0000-0003-3393-9658 (International Society for Optics and Photonics SPIE) vol
5751 ed R S Mackay pp 588–600
O O Versolato https://orcid.org/0000-0003-3852-5227 [35] Koike F and Fritzsche S 2007 Radiat. Phys. Chem. 76 404
J Colgan https://orcid.org/0000-0003-1045-3858 [36] de Gaufridy de Dortan F 2007 J. Phys. B: At. Mol. Opt. Phys.
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[37] Sasaki A, Sunahara A, Furukawa H, Nishihara K, Fujioka S,
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