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# Screening 0D materials for 2D nanoelectronics applications

###### Abstract

As nanoelectronic devices based on two-dimensional (2D) materials are moving
towards maturity, optimization of the properties of the active 2D material
must be accompanied by equal attention to optimizing the properties of and the
interfaces to the other materials around it, such as electrodes, gate
dielectrics, and the substrate. While these are usually either 2D or 3D
materials, recently K. Liu et al.[Nat. Electron. 4, 906 (2021)] reported on
the use of zero-dimensional (0D) material, consisting of vdW-bonded Sb2O3
clusters, as a highly promising insulating substrate and gate dielectric.
Here, we report on computational screening study to find promising 0D
materials for use in nanoelectronics applications, in conjunction with 2D
materials in particular. By combining a database and literature searches, we
found 16 materials belonging to 6 structural prototypes with high melting
points and high band gaps, and a range of static dielectric constants. We
carried out additional first-principles calculations to evaluate selected
technologically relevant material properties, and confirmed that all these
materials are van der Waals-bonded, thus allowing for facile separation of 0D
clusters from the 3D host and also weakly perturbing the electronic properties
of the 2D material after deposition.

###### keywords:

0D materials, database screening, density-functional theory, nanoelectronics

Mohammad Bagheri Hannu-Pekka Komsa*

M. Bagheri, Prof. H.-P. Komsa
Microelectronics Research Unit, University of Oulu, P.O. Box 8000, 90014 Oulu,
Finland
Email Address<EMAIL_ADDRESS>
##  1 Introduction

Two-dimensional (2D) materials are widely investigated for use in
nanoelectronics devices, especially as an ultrathin channel in next-generation
field-effect transistors and novel logic architectures [1, 2, 3], memory cells
[4, 5], memristors [6, 7], rectenna [8], and others [9]. In all of these
applications, one needs to optimize not only the properties of the “active” 2D
material, but also all the other materials around it, such as the dielectrics,
electrodes, and substrate, and the material interfaces between them. In a wide
majority of cases, either other 2D materials or common 3D materials are used
for these purposes. Recently, formation of 2D molecular crystal was
demonstrated from Sb2O3 0D material, i.e., a material consisting of weakly
bound small molecular units with no dangling bonds [10]. Using these as a
substrate or gate dielectric in 2D material field-effect transistors resulted
in high mobility, small hysteresis, low density of trap states, and high
subthreshold slope [11]. The authors called for theoretical studies to find
more suitable 0D material candidates [11, 12].

While 2D and 1D materials have been searched in many previous publications by
screening experimental and/or computational material databases [13, 14, 15,
16], to the best of our knowledge, not 0D materials. While 0D materials are
easily identified using the same dimensionality identification methods, many
of the found materials are crystalline phases of substances that are gaseous
or liquid even in atmospheric conditions, and thus not useable in the above-
mentioned applications. Finding the materials that are crystalline at
technologically relevant temperatures requires estimating the melting points,
which is computationally highly demanding.

In this Letter, we take first steps in identifying a set of promising 0D
materials for use in electronics applications, as a substrate or as a gate
dielectric. We achieve this by combining screening of the Materials Project
database and supplementing it by manual pruning of the list, most importantly
by extracting experimental melting points from the literature. We identified
16 materials belonging to 6 structural prototypes with high melting points and
high band gaps, and both high and low dielectric constants. We performed van
der Waals density-functional theory calculations to evaluate selected material
properties and also studied their interaction with the prototypical 2D
material MoS2.

##  2 Results

###  2.1 Database screening

We start by performing a database search to Materials Project (MP) [17], as
illustrated in Fig. 1(a), with the following search criteria

  * Cr1

The structure is classified to have 0D dimensionality.

  * Cr2

The structure consist of only one type of 0D cluster and the size of the
cluster is larger than 3 atoms.

  * Cr3

Energy above convex hull $E_{\rm hull}$ is less than 0.1 eV/atom.

  * Cr4

The MP entry contains calculated dielectric constant.

Figure 1:  (a) Schematic representation of the materials database screening
procedure. (b-d) Band gap vs. average static dielectric constant from (b) all
56 materials with melting poit above 100 ∘C, (c) materials with band gap
larger than 2 and dielectric constant larger than 5, and (d) materials with
band gap larger than 3.5 and dielectric constant less than 5. Marker color
indicates the melting point (see colorbar, or black if not known). Large/small
marker size indicate that the material is stable/unstable, defined as the
energy above convex hull being less/more than 10 meV/atom.

Several approaches have been proposed for the classification of structure
dimensionality [18, 14, 19]. We are using the algorithm proposed by Larsen et
al. [19], which also gives a useful measure of the degree of “0D-ness”. Upon
applying Cr1, out of 126335 materials in MP (accessed on 25th Jan 2022), 8335
were identified as 0D materials.

Criterion Cr2 is chosen for simplicity and since deposition of materials
consisting of many different clusters can be difficult to control, especially
since the deposition is ideally carried out at a relatively low temperature.
The clusters are compared only by the formula (number and types of atoms in
the cluster), not their atomic structures. Application of this criterion (on
top of Cr1) yields 2395 materials. In addition, we have required that the
number of atoms in the cluster is larger than 3. Clusters with one atom can be
trivially ignored (mostly noble gases), and most systems with 2- or 3-atom
clusters correspond to solid phases of, e.g., hydrogen halides (HF, HCl, etc.)
or simple molecules such as N2, H2O, and CO2. It is unlikely that very
promising new 0D materials can be found among these. Finally, we have required
that the “0D-ness” dimensionality score used in the classification scheme is
higher than 0.8 [19]. The threshold was found by manually inspecting the
structures with lowest score and comparing the dimensionality to that reported
in MP. The materials with low dimensionality score are borderline between
different dimensionalities and usually indicates that the bonding between the
(nearly) 0D clusters is stronger than in purely vdW-bonded systems.
Application of these criteria yields 1965 materials.

Applying the energy criterion Cr3, reduces the number of material candidates
by about a factor of two to 942.

Finally, in the last step of our database screening, criterion Cr4 yields a
marked reduction in the number of entries, down to 162. These materials are
listed in Table S1. Omitting such a large number of entries (780) may appear
unwarranted, but we will revisit this group of materials later in this paper.

In the next stage, we manually pruned the list. First, if several phases of
same material are found, we retain only the lowest energy phase, down to 138
materials. Inspection of the list also indicates that many of these materials
are liquid or gaseous under atmospheric conditions (e.g., methane, ammonia,
and various acids). In order to find materials with stable crystalline
structure at moderate processing temperatures, we collected experimentally
measured melting points for all materials where we could find one (also listed
in Table S1). After removing the materials that are known to be liquid or gas
under typical operating temperatures (melting point below 100 ∘C), we are left
with 56 entries.

The band gaps with respect to rotationally averaged static dielectric
constants, or “polycrystalline dielectric constant”, $\varepsilon_{\rm
poly}^{0}$ for these 56 materials are plotted in Fig. 1(b). In electronics
applications with relatively slowly varying fields, the static dielectric
constants are more relevant than the high-frequency one
$\varepsilon_{\infty}$. In fact, the ionic contribution to the dielectric
constant is critical for achieving large screening, as evidenced from the plot
of the band gap vs. $\varepsilon_{\infty}$ shown in Figure S1.

In order to find materials suitable for use as gate dielectrics, we extract
materials with band gap larger than 2 eV and $\varepsilon_{\rm poly}^{0}$
larger than 5. The relatively low band-gap threshold was chosen since DFT with
semilocal functionals underestimates gaps, sometimes significantly so, by upto
a factor of two. In this group, shown in Fig. 1(b), there are only 13
materials, which we analyze in detail. The atomic structures of the
corresponding 0D clusters are shown in Fig. S2 (and in Fig. 2 for the
promising candidates).

The only material with very high melting point (mp.) is Sb2O3 (656 ∘C), that
was already previously identified [10, 11]. The atomic structure consists of
Sb4O6 cages, similar to C10H16 carbon cages, called adamantane, but without
terminating H. There are few materials with higher dielectric constant, such
as HgBr, HgCl, and Re2H4O9, but those have also lower band gap and lower
melting point. Moreover, although HgBr and HgCl have fairly high mp., they are
unfortunately highly toxic. Re2H4O9 structure is similar to, but not exactly
the same as, that of perrhenic acid. It is a metal oxide coordinated with
water with mp. of 115 ∘C, which is likely too low.

Among others (from high to low band gap), PH3NF5 is a Lewis acid-base adduct,
with fairly little experimental data. It has been synthesized, but expected to
have low melting point [20]. BH6N has a low mp. of 104 ∘C. No mp. was found
for Te2H2O3F4, but stable at room temperature and computationally stable.
H4CSN2, thiourea, has mp. of 182 ∘C, making it possibly interesting. HIO3,
iodic acid, has band gap and dielectric constant close to Sb2O3, but the mp.
of 110 ∘C is again low. GaHgCl4 has been synthesized and is computationally
stable. No mp. was found and it is sensitive to humidity [21]. CoH8(BrO2)2 (or
CoBr2(H2O)4) consist of hydrated CoBr2 molecules. The degree of hydration is
sensitive to temperature between 0-100 ∘C[22], which may prove problematic in
applications. We could not find mp., but high $E_{\rm hull}$ of 0.053 eV/atom
suggests poor stability. Te(HO)6, telluric acid, has mp. of 136 ∘C, which is
likely too low. Finally, As4S5 is a rare mineral called uzonite [23]. It is
related to another mineral As4S4, realgar, also with fairly similar cage-like
structure, and mp. 320 of ∘C. Thus, although the band gap and dielectric
constant are fairly low, the cage-like structure and stability of arsenic
sulfides makes them promising.

Eventually, the number of viable candidates found by this screening is small.
In addition to Sb2O3, there are only As4S5 (and maybe very similar As4S4)
(uzonite/realgar). The commonness of thiourea could make it interesting in
some cases.

For use as a substrate, the band gap is often more important than the
dielectric constant. Low dielectric constant materials (low-$\kappa$
dielectrics) are also used in separating the interconnects in circuits, o
reduce crosstalk and parasitic capacitance. Moreover, in optoelectronic
applications low dielectric constant might be desired in order to minimize
sreening by the substrate and to avoid subsequent renormalization of
fundamental band gap and/or reduction in exciton binding energy [24, 25, 26,
27]. We next focus on the large band gap ($>3.5$ eV) but small dielectric
constant materials $\varepsilon_{\rm poly}^{0}$$<5$, which yields 14 materials
shown in Figure 1(d) and the structures in Fig. S3. Two interesting candidates
can be immediately pinpointed: As2O3 and P2O5. As2O3 has the same structure as
Sb2O3, i.e., belonging to adamantanes. It has mp. of 312 ∘C and is also found
as a mineral arsenolite (and in small quantities in arsenic sulfide minerals
uzonite and realgar mentioned above). P2O5 is also reminiscent of adamantanes
but with additional terminating oxo groups. It has mp. 340 ∘C, and while this
polymorph consisting of P4O10 molecules is in principle metastable, it is
nevertheless commonly encountered. We note that P2S2O3 consists of P4O6S4
clusters similar to P2O5 but the tip O replaced by S. However, it has much
lower mp. of 102 ∘C.

Highest band gaps are found from silicon clusters. SiH6(NF2)2 is an adduct of
SiF4 and 2NH3 and computatioally stable, although mp. of only 166 ∘C. Si2H2O3
and Si2Cl2O3 cluster both have pyramidal structure consisting of four formula
units, similar to P4O10. We did not find mp., but $E_{\rm hull}=0.023$ and
$E_{\rm hull}=0.026$ eV/atom, respectively, suggests fairly poor stability.
Highest dielectric constants are found from metal halogen compounds.
FeH8(BrO2)2 has a structure similar to CoH8(BrO2)2 [28], and is similarly
sensitive to humidity and exhibits fairly poor stability with $E_{\rm
hull}=0.026$ meV/atom. MgH8(ClO6)2 (or Mg(ClO4)2(H2O)4) is a hydrated version
of magnesium perchlorate, which is a strong drying agent, which may again
prove problematic. On the other hand, it has fairly high mp. of 251 ∘Cand
computationally stable.

Otherwise, B(HO)3, boric acid, has mp. 171 ∘C, and is also found in mineral
sassolite, but dissolves in water. H4CN2O, urea, has mp. of only 133 ∘C and
thiourea has overall better properties. PNCl2 is phosphazene with 6-membered
ring with alternating P and N, but mp. of 112 ∘C is too low. PNClF and PBrNF
are similar and, though they are stable at RT, the mp. is likely low as in
other phosphorus halides [29]. CNCl, cyanuric chloride, consists of 6-membered
ring with alternating C and N, and has mp. of 144 ∘C.

Thus, from this set, we found only two clearly promising materials to be used
as a substrate: As2O3, P2O5. Possibly interesting candidates are magnesium
perchlorate and boric acid.

Finally, it is worth highlighting few specific data points in Fig. 1(b). The
material with dielectric constant over 40 is H2Pt(OH)6, platinic acid. In
crystalline form, it is experimentally stable to about 150 ∘C, after which it
starts to form PtO2 [30]. The calculated $E_{\rm hull}=0.069$ eV/atom also
suggests poor stability. There are also three high melting point materials,
shown by red markers, in Fig. 1(b): PtCl2, PdCl2, and NbI5. The Pt and Pd
chlorides in $\beta$-phase consist of octahedral cage structure with six
formula units (hexamer). PtCl2 and PdCl2 are reported to have high mp. of 581
and 679 ∘C, respectively, but those values correspond to the polymeric
$\alpha$-phase. There is a phase transition from $\beta$\- to $\alpha$-phase
at around 500 ∘C[31], which is still high. The calculated band gaps are less
than 2 eV and dielectric constant 3–4. NbI5 is a molecular dimer with mp. 543
∘C, gap 1.0 eV, and $\varepsilon_{\rm poly}^{0}$$=8.8$.

Figure 2:  Atomic structures of the cluster prototypes.

From this database search, we have ended up with 8 materials in 6 cluster
prototypes that showed high melting points and reasonably high band gap and
dielectric constant: adamantane (Sb4O6, As4O6), uzonite (As4S5), realgar
(As4S4) terminated adamantane (denoted T-adamantane, P4O10), metal halide
hexamer (MH-hexamer, Pt6Cl12, Pd6Cl12), and metal halide dimer (MH-dimer,
Nb2I10). The prototype cluster structures are visualized in Fig. 2.

We now revisit the materials for which calculated dielectric constant was not
available in MP. We represented all cluster structures as connected graphs and
then searched for graphs that are isomorphic to the prototype systems among
those where dielectric constant was not available. We stress, that this
approach only considers the connectivity/bonding network between atoms, but
ignores the atom types and thus, for example, P4O10 and P4S4O6 are isomorphic
despite having only a fraction of the oxygens replaced by sulfur. This search
yielded in total 27 structures, 3 adamantanes, 1 uzonite-like, 2 realgar-like,
3 T-adamantanes, 0 MH-hexamer, 18 MH-dimer. These are listed in Table S2. We
again pruned the list by removing low mp. and metastable systems, and those
with zero or vanishingly small band gap. It is worth noting that four of the
metal halide dimers with zero band gap also exhibited nonzero magnetic moment.
We are left with 12 additional materials: uzonite-like As4Se4 with mp. of 265
∘C; T-adamantane P4S10, Ge4S6Br4, Ge4S6I4 all have mp close to 300 ∘C; MH-
dimers Nb2Br10 and Ta2Br10 have mp. 250–300 ∘C, Nb2Cl10 has mp. close to 200
∘C, and Ta2I10 has mp. of 382 ∘C.

###  2.2 Cluster properties

From the screening study presented thus far, we are left with 16 materials in
the 6 prototypes. Since Materials Project data is calculated using PBE
functional that poorly describes van der Waals interactions, we recalculated
these systems, both crystalline and isolated clusters, using vdW-DF
functional. Selected properties are collected in Table 2.2, such as binding
energies, band gaps and dielectric constants, and additional properties are
listed in Table S3. The experimental melting point is also included for
convenience in Table 2.2.

The binding energies per cluster are usually few eVs, which may appear quite
large, but the sizes of these clusters and the areas between them are also
significant. We estimate binding energy per area by mapping the volume per
molecule in a crystal to a sphere and taking its area. These values fall
within 6–40 meV/Å, which is comparable to the 2D material binding energies,
where majority of compounds fall within 13–21 meV/Å [32, 33]. Thus, it appears
safe to label these systems as vdW-bonded 0D materials.

Table 1:  Selected properties of promising 0D materials calculated using vdW-
DF functional: cluster formula, binding energy per area (meV/Å), band gap from
crystal (eV), HOMO-LUMO gap from cluster (eV), and static and high-frequency
dielectric constants. The experimental melting points (mp., in ∘C) are also
included for convenience.

formula | binding energy | mp. | Crystal gap | Cluster gap | $\varepsilon_{\rm poly}^{\infty}$ | $\varepsilon_{\rm poly}^{0}$
---|---|---|---|---|---|---
Sb4O6 | -30.37 | 656.0 | 2.92 | 4.09 | 3.95 | 10.92
As4S5 | -18.66 |  | 1.42 | 2.64 | 6.31 | 8.10
As4S4 | -17.29 | 320.0 | 1.38 | 2.80 | 6.80 | 7.91
As4O6 | -18.30 | 312.2 | 4.04 | 4.49 | 2.59 | 4.92
P4O10 | -13.40 | 340.0 | 5.47 | 6.01 | 1.37 | 2.41
Pt6Cl12 | -6.15 | 581.0 | 1.48 | 2.01 | 3.99 | 4.19
Pd6Cl12 | -6.50 | 679.0 | 1.09 | 1.64 | 6.25 | 6.80
Nb2I10 | -8.04 | 543.0 | 0.61 | 1.08 | 9.98 | 15.01
As4Se4 | -39.87 | 265.0 | 0.85 | 2.17 | 10.69 | 12.61
P4S10 | -12.33 | 288.0 | 2.17 | 2.97 | 3.98 | 4.60
Ge4S6Br4 | -11.46 | 305.0 | 2.14 | 2.79 | 3.40 | 4.22
Ge4S6I4 | -11.02 | 310.0 | 2.06 | 2.59 | 3.85 | 4.75
Nb2Br10 | -26.47 | 254.0 | 1.37 | 1.70 | 4.72 | 7.13
Ta2Br10 | -18.94 | 265.0 | 1.86 | 2.19 | 3.85 | 5.79
Nb2Cl10 | -17.27 | 204.7 | 2.07 | 2.33 | 3.12 | 4.99
Ta2I10 | -30.31 | 382.0 | 0.94 | 1.41 | 7.25 | 10.54

In Table 2.2, we also listed the gaps calculated using vdW-DF for crystals and
for isolated clusters. Comparing to the MP gaps listed in Tables S1 and S2,
shows that, as van der Waals interaction pulls the clusters closer together,
the band gap decreases: 0.4–0.5 eV decrement is seen for many materials,
except for As2O3 and P2O5 which remain largely unaffected. The lowering of the
band gap is in turn reflected in slight increase of the dielectric constants,
see also the plot of band gap vs. dielectric constant shown in Fig. 3(a).
Conversely, the HOMO-LUMO gaps of the isolated clusters are higher than the
crystal gaps by about 0.5–1 eV.

Figure 3:  Properties from vdW-DF calculations. (a) Band gap vs. static
dielectric constant. Marker color indicates the melting point or black if
unknown. (b) HOMO and LUMO levels of 0D clusters aligned via vacuum level. The
materials are grouped and colored according to the cluster prototype. (c) Top
and side vies of atomic structure of As2O3 on WSe2. Dashed rhombus denotes the
As2O3 surface unit cell. (d) Effective band structure of the heterostructure
model (underlying gray features) compared to the band structure of pristine
WSe2 (overlaid lines). The local DOS of As2O3 and WSe2 layers are also shown.

For nanoelectronics applications, it is important to know the alignment of the
valence band maxima and conduction band minima of the 0D materials with
respect to the band extrema of the 2D materials. Here, we evaluated this from
the HOMO and LUMO levels of isolated clusters and the results are shown in
Fig. 3(b). Adamantanes, uzonite, realgars, and some of T-adamantanes have all
suitable band edge positions compared to common 2D materials. Metal halide
hexamer and dimer band edges are deep and as a result the LUMO level of 0D
cluster are below the conduction band minima of most 2D materials.

0D materials have several natural cleaving planes, that lead to 2D layers of
clusters with different orientations and different lattices, depending on the
symmetry of the clusters and of the 3D lattice. It is not a priori clear what
will be the preferred orientation and ordering of the clusters when sputtered
on top of a particular 2D material and how to control the growth mode [12].
Investigating this by first-principles calculations is highly demanding.

To shed light on the interaction of the clusters with 2D materials, we first
studied the interaction of isolated 0D clusters placed on a monolayer MoS2.
The density of states for all 16 cases are collected in Fig. S4 and shows that
the MoS2 states are largely unaffected by the 0D cluster, evidencing that the
interaction is weak. Moreover, in Fig. S4 we also indicate the binding
energies. For the interaction area we have again transformed the volume per
cluster to a sphere (as mentioned above) and taken its great circle. The
values fall within 10–20 meV/Å, again similar to those reported above and as
reported for many 2D materials, providing evidence to the vdW-type binding
with the 2D material.

Next, as an example, we constructed one heterostructure with the surface of
WSe2 fully covered by As2O3. Due to the similarity of As2O3 and Sb2O3, the
structure of 0D clusters on the surface is also likely similar, and thus we
can adopt for As2O3 the experimentally determined structure of Sb2O3. For the
2D material we chose WSe2 due to suitable lattice constant, which allowed us
to construct heterostructure model with minimal strain. As2O3 has higher band
gap than Sb2O3 but smaller dielectric constant and thus useful as a
low-$\kappa$ dielectric or, e.g., as a substrate for studying single-photon
emitters in WSe2 [34, 35] without interference from the substrate defects. The
optimized atomic structure is shown in Fig. 3(c). The atomic structures of
both the WSe2 and As2O3 layers are very weakly perturbed by the neighboring
layer, indicating again van der Waals type interaction. Similarity of the
effective band structure of the heterostructure and the band structure of
pristine WSe2 in Fig. 3(d) demonstrates that the band structures of As2O3 and
WSe2 are effectively uncoupled. The As2O3 bands with very small dispersion are
seen at about 2 eV below valence band maximum and about 1 eV above conduction
band minimum of WSe2. Thus, charge transfer to/from the As2O3 should not occur
and we expect the electronic and optical properties of WSe2 layers to remain
largely intact.

##  3 Conclusion

In conclusion, we have extracted a list of promising 0D materials to be used
with 2D materials in nanoelectronics applications, exhibiting fairly high band
gaps, dielectric constants, and melting points. This was achieved by screening
Materials Project database, combining it with experimental melting points, and
using graph-theoretical methods to find other similar structures. We used
first-principles calculations to predict additional material properties and
studied the interaction of the 0D clusters with a prototypical 2D materials
MoS2. All these materials were confirmed to have binding energies typical for
vdW-bonded systems and to weakly perturb the electronic structure of the 2D
material.

Owing to the vdW gaps between clusters, their conductivity is expected to be
low and thus they should be fairly good insulators even though their band gaps
in the crystalline form are not very high. Consequently, “metallic” 0D
materials do not appear particularly promising as electrode materials. In
addition to their use as a gate dielectric or substrate, these 0D materials
could be used as spacer layers in 2D material superlattices [36].
Intercalation between 2D layers can also stabilize crystalline order for
clusters that have too low melting point in 3D lattice. For instance,
superlattices of 2D materials and CTAB (and other related) molecules was
demonstrated in [37]. Also, AlCl3 and CuCl2 clusters and various dimerized and
trimerized forms of them have successfully been intercalated between graphene
sheets [38]. Although materials consisting of more than one type of clusters
were ignored in this Letter for reasons of simplicity, a vast variety of such
0D molecular compounds could exist. As an example, 2D crystal made of SbI3 and
S8 molecular units was reported in Ref. [39].

We hope that the our study motivates further experimental and computational
work on these materials. Based on the cluster prototypes identified here,
other 0D materials could be identified or purposefully designed, either by
substitution or by addition of functional groups. These can extend the range
of properties or provide even new functionalities, such as charge doping or
magnetism.

##  4 Methods

The material information were extracted from Materials Project we used the
REST API [40]. The structures were then fed into the dimensionality analysis
tools as implemented in ASE [41]. The graph-theoretical part was done using
NetWorkX package [42].

All density-functional theory calculations were carried out using VASP
software [43, 44], together with projector augmented plane wave method. We
adopted Hamada’s rev-VDW-DF2 functional [45, 46, 47, 48], which was found to
yield good structural properties and interlayer binding energies for 2D
materials [33]. Most other input parameters were taken from the Materials
Project [17, 40], such as the k-point meshes and the plane wave cutoff of 520
eV. Isolated molecules on MoS2 were modeled using 4$\times$4 supercell of MoS2
and a 3$\times$3 k-point mesh. For the heterostructure of As2O3 and WSe2, we
take the [111] plane of As2O3. This has lattice constant of 7.715 Å, in which
case a 3$\times$3 supercell ($23.144$ Å) matches closely with a 7$\times$7
supercell of WSe2 ($7\cdot 3.297=23.079$ Å). In the heterostructure model, we
keep the lattice constant of WSe2 fixed and compressively strain As2O3 layer
by 0.28%. We used 2$\times$2 k-point mesh during relaxation and 4$\times$4
k-point mesh in calculating density of states. Effective band structure was
calculated using BandUP code [49].

Supporting Information

Supporting Information is available from the Wiley Online Library or from the
author.

Acknowledgements

We thank CSC–IT Center for Science Ltd. for generous grants of computer time.

Conflict of Interest

The authors declare no conflict of interest.

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See pages - of si.pdf


Derivation of Instrument Requirements for Polarimetry using Mg, Fe, and Mn lines between 250 and 290 nm
Derivation of Instrument Requirements for Polarimetry

0000-0002-5084-4661]A.G. de Wijn
0000-0001-5174-0568]P.G. Judge
High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA

0000-0002-8504-8470]R. Ezzeddine
Department of Astronomy, University of Florida, 211 Bryant Space Sciences Center, Gainesville, Florida, 32611, USA

0000-0002-3234-3070]A. Sainz Dalda
Lockheed Martin Solar & Astrophysics Laboratory, 3251 Hanover Street, Palo Alto, CA 94304, USA
Bay Area Environmental Research Institute, NASA Research Park, Moffett Field, CA 94035, USA.

A.G. de Wijn

De Wijn et al.

Judge et al. (2021)2021ApJ...917...27J recently argued that a region of the solar spectrum in the near-UV between about 250 and 290 is optimal for studying magnetism in the solar chromosphere due to an abundance of Mg2, Fe2, and Fe1 lines that sample various heights in the solar atmosphere.
In this paper we derive requirements for spectropolarimetric instruments to observe these lines.
We derive a relationship between the desired sensitivity to magnetic field and the signal-to-noise of the measurement from the weak-field approximation of the Zeeman effect.
We find that many lines will exhibit observable polarization signals for both longitudinal and transverse magnetic field with reasonable amplitudes.

§ INTRODUCTION

The region between 250 and 290 of the solar spectrum contains the well-known Mg2 h and k lines but also a large number of Fe1 and Fe2 lines.
In particular, there are many strong Fe2 lines that sample various heights in the solar atmosphere.
[Judge et al., 2021] argue that these lines are highly promising for studying magnetism in the solar chromosphere based on a broad evaluation of possible diagnostics of magnetic field in the solar chromosphere.
In order to evaluate the practical use of this region of the spectrum, they analyzed the signal levels expected in the Mg2 k line by synthesizing polarized spectra using the HanleRT code [del Pino Alemán et al., 2020] over a grid of line-of-sight angles and magnetic field strengths for a given field inclination and azimuth angle.
They find that the combined Hanle and Zeeman effects produce measurable signals for this line in many geometries.
In particular, they note that diagnostics of vector field with strengths of 5 to 50 are achievable for 1observation angles greater than 45, as a result of the Hanle effect in Stokes $Q$ and $U$.

However, they do not evaluate observability of magnetic field diagnostics using the many Fe2 lines, which underpin the value of this region of the spectrum for diagnostics of magnetic field in the chromosphere.
In this paper we evaluate the signal-to-noise ratio (SNR) required to observe a signature of longitudinal or transverse magnetic field with a given field strength under the assumption of the weak-field approximation of the Zeeman effect.
We do not treat the Hanle effect in our study.
[Judge et al., 2021] note that work on the Hanle effect in the Fe2 lines is underway, and will be reported elsewhere.
However, the Stokes $V$ signal is unaffected by the Hanle effect, and therefore the signal strength of the longitudinal component of the magnetic field can be estimated using the methods in this paper.
In addition, many science cases will require observations on the disk or of magnetic field with strengths considerably exceeding the critical Hanle field strength.
In those situations, the Hanle effect has a negligible contribution to the polarization signal, and therefore the methods used in this paper apply.

We will first discuss briefly the weak-field approximation and its applicability, and then derive formulae to relate the error on a measurement of the magnetic field to the SNR of an observation for both the longitudinal and transverse components.
We will derive parameters in several different ways to illustrate possible ways one may approach a similar problem for other spectral lines.
Finally, for specific lines in this particular spectral region, we will investigate the instrumental effect of limited spectral resolution, and illustrate the method through an example calculation.

§ ANALYSIS

The Zeeman splitting of a spectral line is given by

\begin{equation}\label{eq:deltalambdab}
\Delta\lambda_\mathrm{B}=s\,B\,\lambda_0^2,
\end{equation}

where $B$ is the magnetic field strength and $\lambda_0$ is the rest wavelength of the line.
We here work with vacuum wavelengths in units of $\mathrm{nm}$ and the magnetic flux density $B$ in units of , and therefore have $s=\SI{4.67e-11}{\per\nm\per\gauss}$.

If the Zeeman splitting is much smaller than the Doppler width of the line, it is possible to apply a perturbative scheme to the radiative transfer equations and derive expressions for the circular and linear polarization signals in terms of the first and second derivative of the intensity profile, respectively [Landi Degl'Innocenti & Landi Degl'Innocenti, 1973].
[Landi Degl'Innocenti & Landolfi, 2004] note that for iron lines in the visible spectrum that sample the photosphere, this approximation is valid up to k field strengths.
At shorter wavelengths, the weak-field approximation is applicable for stronger field strengths because Zeeman splitting scales with the square of the wavelength, while Doppler broadening scales linearly.
In addition, lines that form in the chromosphere have larger Doppler broadening than those that form in the photosphere.
For these reasons we can safely assume the weak-field approximation is applicable for magnetic field up to at least k strength for the lines we consider here.

The circular and linear polarization signals as a function of wavelength are

\begin{align}
\label{eq:Vlambda}
V(\lambda)&=-\Delta\lambda_\mathrm{B}\, \bar g\,\frac{\partial I(\lambda)}{\partial\lambda}\,\cos\theta,\\
\label{eq:Llambda}
L(\lambda)&=-\frac14\Delta\lambda_\mathrm{B}^2\, \bar G\, \frac{\partial^2 I(\lambda)}{\partial\lambda^2}\,\sin^2\theta,
\end{align}

where $\bar g$ and $\bar G$ are the effective Landé factors for longitudinal and transverse magnetic field, respectively, $I(\lambda)$ is the intensity, and $\theta$ is the inclination of the magnetic field with respect to the line-of-sight.
For convenience, we write $B_\parallel=B\,\cos\theta$ and $B_\bot=B\,\sin\theta$.

An interpretation of the Stokes profiles effectively combines the signal over a wavelength range $\Delta\lambda$.
We are therefore interested in the integrated absolute signals $\mathcal{V}$ and $\mathcal{L}$,

\begin{align}
\mathcal{V}&=\int_{\Delta\lambda}\left|V(\lambda)\right|\nonumber\\
&=s\,\bar g\,\lambda_0^2\,\left|B_\parallel\right|\,\int_{\Delta\lambda}\left|\frac{\partial I(\lambda)}{\partial\lambda}\right|\,\mathrm{d}\lambda\label{eq:VV},\\
\mathcal{L}&=\frac14\,s^2\,\bar G\,\lambda_0^4\,B_\bot^2\,\int_{\Delta\lambda}\left|\frac{\partial^2 I(\lambda)}{\partial\lambda^2}\right|\,\mathrm{d}\lambda\label{eq:LL}.
\end{align}

Our goal is to evaluate the expected errors $\sigma_{B\parallel}$ on $B_\parallel$ and $\sigma_{B\bot}$ on $B_\bot$ in some way that can be readily estimated, such as a function of the SNR of the intensity measurement that can be determined from a flux budget calculation.
We therefore also define the integrated intensity signal,

\begin{equation}\label{eq:II}
\mathcal{I}=\int_{\Delta\lambda}I(\lambda)\,\mathrm{d}\lambda,
\end{equation}

and note that the uncertainties $\sigma_\mathcal{V}$ of $\mathcal{V}$ and $\sigma_\mathcal{L}$ of $\mathcal{L}$ are related to the uncertainty $\sigma_\mathcal{I}$ of $\mathcal{I}$ through the modulation efficiencies $\epsilon_I$, $\epsilon_Q$, $\epsilon_U$, and $\epsilon_V$ in Stokes $I$, $Q$, $U$, and $V$ [del Toro Iniesta & Collados, 2000],

\begin{align}
\sigma_\mathcal{V}&=\frac{\epsilon_I}{\epsilon_V}\,\sigma_\mathcal{I},\label{eq:sigmaVV}\\
\sigma_\mathcal{L}&=\frac{\epsilon_I}{\epsilon_L}\,\sigma_\mathcal{I},\label{eq:sigmaLL}
\end{align}

where we have written $\epsilon_L$ for the modulation efficiency of the linear polarization signal of interest.

Since Eq. <ref> relates $\mathcal{V}$ to $B_\parallel$, we can express the uncertainty $\sigma_{B\parallel}$ as the uncertainty of $\mathcal{V}$, and subsequently of $\mathcal{I}$ by substituting Eq. <ref>,

\begin{align}
\sigma_{B\parallel}&=\frac1{s\,\bar g\,\lambda_0^2}\,\left(\int_{\Delta\lambda}\left|\frac{\partial I(\lambda)}{\partial\lambda}\right|\,\mathrm{d}\lambda\right)^{-1}\,\sigma_\mathcal{V}\\
&=\frac1{s\,\bar g\,\lambda_0^2}\,\frac{\epsilon_I}{\epsilon_V}\,\left(\int_{\Delta\lambda}\left|\frac{\partial I(\lambda)}{\partial\lambda}\right|\,\mathrm{d}\lambda\right)^{-1}\,\sigma_\mathcal{I}.\label{eq:sigmaBparsigmaI}
\end{align}

1Note that we have used the property of the WFA that $I$ does not depend on $B$.
If we can express the integral of $|\partial I(\lambda)/\partial\lambda|$ in terms of $\mathcal{I}$, then $\sigma_{B\parallel}$ can be expressed in the SNR of the intensity measurement, $\mathcal{I}/\sigma_\mathcal{I}$.

Equivalently, propagating the error in $B_\bot$ through Eq. <ref> and substituting Eq. <ref>, we find

\begin{align}
\sigma_{B\bot}&=\frac2{s^2\,\bar G\,\lambda_0^4}\,\frac1{B_\bot}\,\left(\int_{\Delta\lambda}\left|\frac{\partial^2 I(\lambda)}{\partial\lambda^2}\right|\,\mathrm{d}\lambda\right)^{-1}\,\sigma_\mathcal{L}\\
&=\frac2{s^2\,\bar G\,\lambda_0^4}\,\frac1{B_\bot}\,\frac{\epsilon_I}{\epsilon_P}\,\left(\int_{\Delta\lambda}\left|\frac{\partial^2 I(\lambda)}{\partial\lambda^2}\right|\,\mathrm{d}\lambda\right)^{-1}\,\sigma_\mathcal{I}\label{eq:sigmaBbotsigmaI}.
\end{align}

In this case, we want to express the integral of $|\partial^2 I/\partial\lambda^2|$ in terms of $\mathcal{I}$.
We note, however, that $\sigma_{B\bot}$ is a function of $B_\bot$, and therefore the same SNR in $\mathcal{I}$ will yield a different measurement error depending on the strength of the field being measured.
Notably, $\sigma_{B\bot}$ is infinite for $B_\bot=0$.

In practice, $\mathcal{I}$ will be the sum of a series of discrete measurements.
Each pixel samples the signal weighted with some point spread function, which causes cancellation of some amount of signal.
This effect is discussed in more detail in Sect. <ref>.
We assume here that the instrument is a spectrograph that samples the spectrum critically with a resolution $R$.
However, the analysis for a different type of instrument, such as a wavelength-tunable imager, is analogous.
The number of measurements $N$ that spans the wavelength range $\Delta\lambda$ is given by

\begin{equation}\label{eq:NfromR}
\end{equation}

There are several ways to approach expressing the first and second derivatives of $I(\lambda)$ in terms of itself.
We will examine a simple approximation of the gradient in terms of the intensity and a characteristic wavelength interval, and a numeric calculation from simulated data or from observations of the intensity spectrum.

Blue lines: synthetic spectrum in two NUV windows that contain Fe1 and Fe2 lines of interest.
Orange lines: measured flux from the 1983 Air Force Geophysics Laboratory (AFGL) balloon measurement.
The synthetic spectrum shows very good agreement with the measured spectrum for the Fe1 and Fe2 lines.

§.§ Characteristic Wavelength Interval

This simple approximation equates the gradient as the ratio of the intensity and some characteristic wavelength interval,

\begin{equation}\label{eq:Ityp}
\left|\frac{\partial I(\lambda)}{\partial\lambda}\right|\approx\frac{I(\lambda)}{\Delta\lambda}.
\end{equation}

Substituting Eq. <ref> in Eq. <ref>, we find

\begin{equation}
\sigma_{B\parallel}=\frac{\Delta\lambda}{s\,\bar g\,\lambda_0^2}\,\frac{\epsilon_I}{\epsilon_V}\,\frac{\sigma_\mathcal{I}}{\mathcal{I}}.\label{eq:sigmaBparallel}
\end{equation}

We approximate the pixel SNR,

\begin{equation}\label{eq:isigmaiapprox}
\sigma_I\approx\frac1{\sqrt{N}}\,\sigma_\mathcal{I}.
\end{equation}

If the measurement noise is dominated by photon statistics, as is often the case, that property is preserved also for the average intensity $I$, i.e., $\sigma_I$ is approximately the square root of $I$, and therefore the SNR $I/\sigma_I$ is the average SNR of the measurement over the wavelength range $\Delta\lambda$.
Substitution of Eqs. <ref> and <ref> in Eq. <ref> yields

\begin{equation}
\sigma_{B\parallel}=\frac1{s\,\bar g\,\lambda_0}\,\sqrt{\frac{\Delta\lambda}{2R\,\lambda_0}}\,\frac{\epsilon_I}{\epsilon_V}\,\frac{\sigma_I}{I}.
\end{equation}

We apply this method to estimate the sensitivity of the Mg2 h line.
For this line, we have $\bar g=\num{1.33}$, and $\lambda_0=\SI{280}{\nm}$.
We estimate $\Delta\lambda=\SI{0.03}{\nm}$ (equivalent to a velocity of ±15, see Eq. <ref>).
We now find

\begin{equation}
\sigma_{B\parallel}=\num{7.58e5}\,\frac1{\sqrt{R}}\,\frac{\sigma_I}{I}~\si{\gauss}.
\end{equation}

For example, if we evaluate this equation for a instrument with $R=\num{30000}$ and require $\sigma_{B\parallel}\le\SI{8}{\gauss}$, we find that the SNR in $I$ must be at least 547.

§.§ Observed or Simulated Intensity Spectra

The estimation in Sect. <ref> is obviously crude as it relies on a good estimate of $\Delta\lambda$, which is difficult without some prior knowledge of the line profile.
A much better estimate can be derived from intensity spectra that were observed or computed with a numerical radiative transfer code such as RH [Uitenbroek, 2001] or TURBOSPECTRUM [Alvarez & Plez, 1998, Plez, 2012] using a model atmosphere.
We can numerically calculate the integral of the absolute value of the gradient as a fraction of $\mathcal{I}$,

\begin{equation}\label{eq:gamma}
\gamma=\frac1{\mathcal{I}}\,
\int_{\Delta\lambda}\left|\frac{\partial I(\lambda)}{\partial\lambda}\right|\,\mathrm{d}\lambda,
\end{equation}

Using again Eqs. <ref> and <ref>, we now have

\begin{align}
\sigma_{B\parallel}&=\frac1{s\,\bar g\,\lambda_0^2}\,\frac1\gamma\,\sqrt{\frac{\lambda_0}{2\Delta\lambda}}\,\frac{\epsilon_I}{\sqrt{R}\,\epsilon_V}\,\frac{\sigma_I}{I}.
\end{align}

We now define the sensitivity factor for the error on the longitudinal field,

\begin{equation}\label{eq:biggamma}
\Gamma=\gamma\,s\,\bar g\,\lambda_0^2\,\sqrt{\frac{2\Delta\lambda}{\lambda_0}}
\end{equation}

that captures the properties of a spectral line.
Lines with larger $\Gamma$ have higher sensitivity to longitudinal field, i.e., a requirement for a particular sensitivity of $B_\parallel$ can be met with a measurement with lower SNR.

Similarly for the transverse component of the magnetic field we can numerically calculate the integral of the second derivative,

\begin{equation}\label{eq:chi}
\chi=\frac1{\mathcal{I}}\,
\int_{\Delta\lambda}\left|\frac{\partial^2 I(\lambda)}{\partial\lambda^2}\right|\,\mathrm{d}\lambda,
\end{equation}

and find

\begin{equation}
\sigma_{B\bot}=\frac1{s^2\,\bar G\,\lambda_0^4}\,\frac1\chi\,\sqrt{\frac{2\lambda_0}{\Delta\lambda}}\,\frac1{B_\bot}\,\frac{\epsilon_I}{\sqrt{R}\,\epsilon_L}\,\frac{\sigma_I}{I}.
\end{equation}

Therefore, we define the sensitivity factor for the error on the transverse field,

\begin{equation}\label{eq:bigchi}
\mathrm{X}=\chi\,s^2\,\bar G\,\lambda_0^4\,\sqrt{\frac{\Delta\lambda}{2\lambda_0}}.
\end{equation}

We use the above procedure to calculate $\Gamma$ and $\mathrm{X}$ for the lines listed in [Judge et al., 2021] from a synthetic spectrum and observations from the IRIS mission [De Pontieu et al., 2014].
Figure <ref> shows the synthetic spectrum calculated with TURBOSPECTRUM from a custom Sun-like MARCS model atmosphere [Gustafsson et al., 2008] and using a line list adopted from the VALD database [Piskunov et al., 1995, Ryabchikova et al., 2015].
This spectrum was calculated under the assumption of local thermal equilibrium 1(LTE), but shows very good agreement with measured spectra for the Fe1 and Fe2 lines, such as the AFGL balloon measurements [Hall & Anderson, 1991] shown also in Fig. <ref>.
1The synthesis includes 102691 lines from 115 atomic and 10 molecular species between 256 and 285.

1The MARCS model atmosphere does not include a chromospheric temperature rise.
However, using a more realistic atmospheric model does not necessarily result in a more realistic synthetic spectrum.
A model atmosphere like FAL-C [Fontenla et al., 1993] would result in emission peaks in the cores of strong lines that are not observed in the AFGL spectra.
These peaks are the result of the assumption of LTE that is not valid in the line cores, and the peaks would not be present if non-LTE physics (e.g., scattering and partial redistribution) were included in the spectral synthesis.

1We can expect the intensity in the cores of strong lines to saturate in this synthesis (see, e.g., the upper-left panel of Fig. <ref>).
The resultant line core profile is nearly flat and exhibits only a small gradient and second derivative, and hence little circular or linear polarization signal is produced in the presence of magnetic field.
Therefore, the analysis presented here based on this synthesis will produce lower values of $\Gamma$ and $\mathrm{X}$ than one based on a synthesis that incorporates pertinent non-LTE effects and uses a more realistic atmospheric model.

IRIS observes the solar spectrum around the Mg2 h and k lines that also includes the Mn1 lines used by [Ishikawa et al., 2021] to infer longitudinal magnetic field from data from the CLASP2 flight [Narukage et al., 2016, Tsuzuki et al., 2020].
We choose a data set of NOAA AR12957 taken on March 4, 2022 around 10 UT.
This observation contains a region of plage, the edge of a sunspot, and some more quiet areas.
The left panel of Fig. <ref> shows the intensity of the Mg2 k core.
We could compute the longitudinal and transverse sensitivity factors for every pixel in the map.
However, we would overestimate the factors due to measurement noise that creates spurious signals in first and second derivative of the intensity profile.
We therefore use “representative profiles” (RPs) from the IRIS$^2$ database [Sainz Dalda et al., 2019].
An RP is an average of many similar line profiles, and therefore has very low noise, so that we can accurately determine the first and second derivative of the intensity.
There are 160 RPs in this map.
More than half the pixels are represented by the most popular 25 RPs, and only 3 RPs represent less than 100 pixels each.

We note that we can express $\Delta\lambda$ in terms of a velocity $v$,

\begin{equation}\label{eq:deltalambdav}
\Delta\lambda=2\,\frac{v}{c}\lambda_0.
\end{equation}

We use $v=\SI{12.5}{\km\per\s}$ for the Fe1, Fe2, and Mg2 lines, and $v=\SI{7.5}{\km\per\s}$ for the Mn1 lines.
These velocities give reasonable integration intervals and more or less correspond to typical sound speed estimates for the chromosphere and photosphere.
The values for $\Gamma$ and $\mathrm{X}$ are not strongly dependent on the choice of $v$, since the wings of the lines tend not to contribute significant polarization signal (see Fig. <ref>).
However, other nearby spectral lines may contribute spurious polarization signal in the wavelength window.
We therefore limit the window to the nearest local maximum of the intensity spectrum to reduce contamination by other lines (see Fig. <ref> panels in the middle row left and center column).

Longitudinal and transverse field sensitivity factors for prominent lines in the  solar chromospheric spectrum.
$\Delta\lambda$ denotes the integration window.
Landé g-factors are computed using LS coupling.
Wavelength in air is included for easy reference against Table 1 in [Judge et al., 2021].

$\lambda_0$     $\lambda_{0,\mathrm{air}}$     $\Delta\lambda$
Ion     1c|$\log\tau_0$
$g$     1c|$\Gamma$
$G$     1c|$\mathrm{X}$

()     ()     (±)
256.331     256.408     -11/[retain-explicit-plus] +7     Fe2     -2.54     1.21     1.17     1.46     2.45     minor

256.425     256.502      -5/[retain-explicit-plus] +8     Fe2     -2.87     1.10     1.21     1.18     2.86     major

256.768     256.845     ±11     Fe2     -3.30     0.83     0.94     0.56     1.15

257.514     257.591     ±11     Fe2     -4.33     1.30     1.77     1.65     3.03     minor

257.869     257.946     -11/[retain-explicit-plus] +6     Fe2     -3.32     1.33     1.53     -0.01     0.01     major

258.335     258.413     ±11     Fe2     -3.19     1.47     1.62     1.74     3.59

258.665     258.743     ±11     Fe2     -2.03     1.50     0.73     2.25     1.00     minor

259.132     259.210     -11/[retain-explicit-plus]+10     Fe2     -4.47     1.50     1.71     2.25     4.27     severe

259.232     259.309      -8/[retain-explicit-plus]+11     Fe2     -3.24     1.49     1.61     2.03     4.08     minor

259.451     259.528      -8/[retain-explicit-plus]+11     Fe2     -4.02     2.17     1.80     4.52     7.14     minor

259.915     259.992     ±11     Fe2     -1.93     1.50     0.73     2.25     0.75

260.018     260.095     ±11     Fe2     -1.39     1.56     0.42     2.42     0.30     minor

260.787     260.865     ±11     Fe2     -1.99     1.50     0.82     2.24     1.28     minor

261.185     261.263     -11/[retain-explicit-plus] +6     Fe2     -4.23     1.90     2.30     3.58     9.01     minor

261.265     261.343     ±11     Fe2     -1.79     1.59     0.73     2.52     0.83

261.460     261.538     ±11     Fe2     -2.20     1.50     0.94     2.12     1.54

261.840     261.918     ±11     Fe2     -2.36     1.66     1.13     2.75     2.42

262.119     262.197     -11/[retain-explicit-plus] +5     Fe2     -3.75     1.87     2.31     3.49     9.82     major

262.245     262.323     ±11     Fe2     -2.76     3.34     3.10     11.15     17.84

262.645     262.724     ±11     Fe2     -2.08     1.50     0.98     2.25     1.75

262.907     262.986     ±11     Fe2     -2.21     1.50     1.04     2.12     1.79

263.183     263.262     -11/[retain-explicit-plus]+10     Fe2     -2.02     1.50     0.85     2.24     1.44     major

263.210     263.289     ±11     Fe2     -1.97     1.50     0.81     2.25     1.41     major

264.191     264.269      -6/[retain-explicit-plus]+11     Fe2     -4.62     1.87     2.61     3.39     13.36     severe

268.431     268.510      -4/[retain-explicit-plus] +2     Fe2     -4.99     1.84     0.64     3.34     2.76     severe

269.363     269.443      -8/[retain-explicit-plus]+11     Fe2     -4.40     1.93     3.42     3.64     10.36     major

271.018     271.099      -4/[retain-explicit-plus]+11     Fe2     -4.30     2.10     3.53     4.17     19.21     severe

271.521     271.602     ±11     Fe2     -3.09     1.50     1.53     2.25     3.82

271.751     271.831     ±5     Fe2     -3.11     1.33     2.83     1.55     8.63     severe

271.984     272.064     ±11     Fe1     -4.45     1.25     0.92     1.54     1.77     minor

272.171     272.251     ±11     Fe1     -4.76     1.17     1.07     1.32     2.31

272.439     272.519      -8/[retain-explicit-plus]+11     Fe1     -5.22     1.00     1.08     0.85     1.67     minor

272.569     272.649     -11/[retain-explicit-plus] +4     Fe2     -3.05     1.20     1.38     1.03     2.48     severe

272.835     272.916     ±11     Fe2     -3.05     1.50     1.47     2.25     4.19     minor

273.154     273.235     ±11     Fe2     -3.19     0.80     0.97     -0.29     0.64

273.326     273.407      -9/[retain-explicit-plus]+10     Fe2     -4.92     1.45     2.63     1.60     5.32     major

273.778     273.859      -6/[retain-explicit-plus]+11     Fe2     -3.22     1.50     1.61     2.16     4.88     severe

273.812     273.893     ±11     Fe1     -5.12     2.00     2.08     3.26     7.55

274.036     274.117     ±11     Fe2     -3.31     1.43     1.05     2.04     2.02     minor

274.322     274.403      -5/[retain-explicit-plus]+11     Fe1     -5.01     1.67     1.55     2.52     5.56     severe

274.401     274.482     ±11     Fe2     -2.81     0.50     0.47     0.24     0.39

274.488     274.569     ±11     Fe1     -5.51     2.50     2.78     6.27     15.09

274.729     274.810     ±11     Fe2     -2.59     0.90     0.77     0.80     1.15

274.779     274.861     ±11     Fe2     -2.66     1.37     1.17     1.88     2.54

274.999     275.081     -11/[retain-explicit-plus] +6     Fe2     -3.02     1.20     1.03     1.44     2.84     severe

275.013     275.095      -8/[retain-explicit-plus]+11     Fe2     -2.38     1.07     0.80     1.14     1.69     severe

275.030     275.112      -5/[retain-explicit-plus]+11     Fe2     -3.24     0.00     0.00     0.00     0.00     severe

275.095     275.177     ±11     Fe1     -5.04     1.58     1.65     2.38     5.64     minor

275.655     275.737     ±11     Fe2     -2.17     1.17     0.85     1.35     1.23

275.714     275.796      -4/[retain-explicit-plus]+11     Fe1     -5.52     2.00     1.71     3.99     8.45     severe

276.263     276.344     -12/[retain-explicit-plus] +7     Fe2     -3.25     1.50     1.89     2.16     5.01     major

276.975     277.057      -7/[retain-explicit-plus]+12     Fe2     -3.08     1.50     2.26     2.25     5.84     severe

277.355     277.437      -6/[retain-explicit-plus] +5     Fe2     -3.14     1.50     2.86     2.25     9.38     severe

277.616     277.698      -6/[retain-explicit-plus] +4     Fe2     -6.06     1.34     2.36     0.44     1.84     severe

279.564     279.647     ±7     Mn1     -5.65     1.98     0.64     1.80     0.42     minor

279.635     279.718     ±12     Mg2     0.00     1.17     0.27     1.33     0.13

279.909     279.992     ±7     Mn1     -5.79     1.70     1.01     3.74     1.44

280.191     280.273     ±7     Mn1     -5.96     0.84     0.48     2.85     1.09

280.353     280.435     ±12     Mg2     -0.30     1.33     0.31     1.33     0.15

Results are given in Table <ref> and shown in Figs. <ref> and <ref>.
The line profiles were visually evaluated and qualitatively categorized as suffering from blends with varying severity given in the “Blend” column.
The $\Gamma$ and $\mathrm{X}$ values of a line with a major or severe blend are likely affected and overestimate the true values, as the blends create additional gradients.
The values for the Mg2 and Mn1 lines in the table are derived from the spectrum of the most popular RP that represents 10697 pixels (3.2% of the FOV).
We processed each RP, and map the longitudinal and transverse sensitivity factors back to pixels in the FOV, shown in the center and right panels of Fig. <ref>, respectively.
Figure <ref> shows the cumulative probability density functions for the longitudinal and transverse sensitivity factors.
The most popular RP is around the 70% and 85% percentiles for $\Gamma$ and $\mathrm{X}$, respectively.

f3fig:tblinesIntensity, and first and second derivative of intensity for selected lines.

We select a series of Fe2 lines, a single Fe1 line, and the Mg2 h and k lines for closer study.
We pick Fe1 and Fe2 lines that are preferentially unaffected by blends, sample 1at optical depths down to the photosphere in roughly equal steps of $\log\tau$, have large $\Gamma$ and $\mathrm{X}$, and are nearby one another in the spectrum.
This selection procedure results nine lines in two distinct wavelength regions of interest: between Fe2 260.018 and Fe2 262.245, and between Fe1 274.488 and Mg2 h 280.353.
Figure <ref> shows the synthetic spectra for these lines, together with the first and second derivatives.

IRIS map used to calculate the longitudinal and transverse sensitivity factors for Mg2 and Mn1 lines.
Left panel: intensity in the core of the Mg2 k line.
Center panel: longitudinal field sensitivity factor $\Gamma$.
Right panel: transverse field sensitivity factor $\mathrm{X}$.

f5fig:mgIIhk_cdfCumulative distribution function of the longitudinal and transverse sensitivity factors $\Gamma$ and $\mathrm{X}$ for the Mg2 h and k lines.
The $\Gamma$ and $\mathrm{X}$ values and the cumulative probabilities for the most popular RP (also given in Table <ref>) are indicated by vertical and horizontal dotted lines, respectively.

§ INSTRUMENTAL EFFECTS

In practice, all measurements will be affected by instrumental effects.
As already mentioned in Sect. <ref>, $\mathcal{I}$ will be the sum of a series of discrete measurements that sample the signal weighted with some line spread function (LSF).
We evaluate here how this affects the required measurement sensitivity.
For the sake of simplicity, we assume each measurement is affected by the same LSF $\rho(\lambda)$.
Equations <ref>, <ref>, and <ref> then become

\begin{align}
\mathcal{I}'&=\int_{\Delta\lambda}(\rho\ast I)(\lambda)\,\mathrm{d}\lambda,\\
\gamma'&=\frac1{\mathcal{I}'}\,
\int_{\Delta\lambda}\left|\left(\rho\ast\frac{\partial I}{\partial\lambda}\right)(\lambda)\right|\,\mathrm{d}\lambda,\\
\chi'&=\frac1{\mathcal{I}'}\,
\int_{\Delta\lambda}\left|\left(\rho\ast\frac{\partial^2 I}{\partial\lambda^2}\right)(\lambda)\right|\,\mathrm{d}\lambda,
\end{align}

where $\ast$ denotes convolution.
It is straightforward to implement this in the numerical analysis presented in Sect. <ref>.

The LSF of a spectrograph depends on its specific configuration [Casini & de Wijn, 2014].
The LSF of a typical spectrograph operating in Littrow condition that also satisfies the “pixel-matching” condition, i.e., the projected width of the slit is equal to the width of a camera pixel, is approximately Gaussian after accounting for sampling.
We therefore choose to model the LSF as a Gaussian function.

Signal loss factors are shown as a function of spectral resolution for a collection of Fe1 and Fe2 lines and the Mg2 lines in Fig. <ref>.
Stokes $V$ signals are less affected than linear polarization signals at reasonable spectral resolution.
Linear polarization signal loss varies considerably from line to line, but generally lines that form deeper in the atmosphere have narrower profiles that require higher spectral resolution to achieve the same loss factor.
The Fe2 260.018 line is affected by a blend that at spectral resolutions below about 40000 starts to contaminate the polarization signal, causing a spurious rise in the $L$ signal loss factor.

§ EXAMPLE SNR CALCULATION

We now show an example using the above calculations to derive measurement requirements, i.e., SNR on Stokes $I$, for a hypothetical instrument that observes the nine spectral lines previously selected.

Substituting $\gamma'$ and $\chi'$ for $\gamma$ and $\chi$ in Eqs. <ref> and <ref> to account for instrument spectral resolution yields

\begin{align}
\Gamma'&=2\,\gamma'\,s\,\bar g\,\lambda_0^2\,\sqrt{\frac{v}{c}}\label{eq:biggammaprime}\\
\mathrm{X}'&=\chi'\,s^2\,\bar G\,\lambda_0^4\,\sqrt{\frac{v}{c}}.\label{eq:bigchiprime}
\end{align}

The errors on the longitudinal and transverse magnetic field are given by

\begin{align}
\sigma_{B\parallel}&=\frac1{\Gamma'}\,\frac{\epsilon_I}{\sqrt{R}\,\epsilon_V}\,\frac{\sigma_I}{I},\label{eq:sigmabparallel}\\
\sigma_{B\bot}&=\frac1{\mathrm{X}'}\,\frac1{B_\bot}\,\frac{\epsilon_I}{\sqrt{R}\,\epsilon_L}\,\frac{\sigma_I}{I}.\label{eq:sigmabbot}
\end{align}

f6fig:SNR_lossSignal loss factor of SNR resulting from instrumental smearing as a function of the instrument spectral resolution.

We assume the instrument has a spectral resolution of 30000 and calculate $\Gamma'$ and $\mathrm{X}'$ for spectral lines of interest.
A reasonable estimate for a near-optimal, balanced modulator is $\epsilon_I/\epsilon_V\approx\epsilon_I/\epsilon_L\approx1.8$ <cit.>.
We thus set the factors $\epsilon_I/(\sqrt{R}\,\epsilon_V)$ and $\epsilon_I/(\sqrt{R}\,\epsilon_L)$ to be equal to 0.01.
Finally, we have to estimate $B_\bot$ in order to evaluate Eq. <ref>.
We choose a stepped function for $B_{\bot}$ starting at 200 at the top of the chromosphere to 50 in the photosphere based on simulations of a magnetic flux rope and a sheared arcade (M. Rempel, private communication).
As an example, we calculate the required SNR $\sigma_I/I$ required to detect the transverse field and a 20 longitudinal field with $2.5\,\sigma$ significance.
The results are summarized in Table <ref>.
We note that the SNR values are the average pixel SNR over the spectral window between the vertical dotted lines in the panels of Fig. <ref>.
Generally, we observe that lines that form higher in the atmosphere require higher SNR.
To demonstrate that these SNR requirements are achievable, we also calculate the SNR of a 12 integration by such a hypothetical instrument with a 30 aperture, 2.5% throughput, and 1 spatial resolution.
The deep Fe2 line at 260.018 drives the instrument requirements.
The Mg2 lines achieve higher SNR than that line because of the increased intensity in the 2V and 2R peaks.

Required SNR to detect magnetic field with $2.5\sigma$ significance.
Case 1: detection of transverse field with strength $B_\bot$.
Case 2: detection of $B_\parallel=20~\mathrm{G}$.
The Instrument column lists the projected performance of a hypothetical instrument.
See text for details.

$\lambda_0$     Ion
Case 1     1c|Case 2
260.018     Fe2     -1.39     150     539     419      553

261.185     Fe2     -4.23     50     185     127      771

261.265     Fe2     -1.79     150     263     228      644

261.460     Fe2     -2.20     100     351     184      581

262.119     Fe2     -3.75     75      80     112      955

262.245     Fe2     -2.76     100      26      65      700

274.488     Fe1     -5.51     50     108      76      971

279.635     Mg2      0.00     200     818     768     1593

280.353     Mg2     -0.30     200     739     724     1436

§ CONCLUSION

We have investigated the lines in the wavelength region between about 250 and 290 identified by [Judge et al., 2021] as being optimal for studying magnetism in the solar chromosphere.
We have derived equations and procedures to quantify the sensitivity of spectral lines to magnetic field through the Zeeman effect based on observed or synthetic intensity spectra.
While we have applied these methods to the spectral region suggested by [Judge et al., 2021], they are applicable to any spectral line.

An example calculation shows that observations with an instrument with a spectral resolution of 30000 need to reach achievable Stokes-$I$ SNR ratios of a few hundred, which are achievable for an instrument with a 30 aperture and 2.5% throughput at 1 spatial resolution with an integration time of 12.
We thus conclude that this region of the solar spectrum in the near-UV yields observable polarization signals with suitable diagnostic potential for studies of chromospheric magnetism through the Zeeman effect.

We have not evaluated the interpretability of observations of these lines.
That work requires a more complex approach of synthesizing spectra from known model atmospheres, degrading those spectra as if they were observed by a hypothetical instrument, and attempting to recover the model parameters through interpretation using the WFA or with inversion codes like DeSIRe [Ruiz Cobo et al., 2022], STiC [de la Cruz Rodríguez et al., 2019], or TIC [Li et al., 2022].
The Mg2 lines have been studied and used for diagnostics of chromospheric magnetism in recent years <cit.>
These efforts should be continued and expanded to include the Fe1 and Fe2 lines identified in this work.
1A first step in this direction was recently taken by [Afonso Delgado et al., 2023], who studied the magnetic sensitivity of Fe2 between 250 and 278 using a many-level model atom and realistic physics using the HanleRT code.
They note that observations of the solar spectrum are required to study the effects of UV line blanketing and validate the atomic data, in particular the rate of inelastic collisions with electrons.
We assert based on the results presented in this paper that this region of the solar spectrum holds great promise and instrumentation to observe it should be developed.

CRediT author statement
Investigation: A.G. de Wijn, P.G. Judge, R. Ezzeddine, A. Sainz Dalda;
Writing—original draft preparation: A.G. de Wijn;
Writing—review and editing: P.G. Judge.

This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement No. 1852977.
AdW acknowledges support by the National Aeronautics and Space Administration under Grant 80NSSC21K1792 issued through the Heliophysics Flight Opportunity Studies program.
R.E. acknowledges support from NSF grant AST-2206263.
IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre.
This work has made use of the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna.
The following acknowledgements were compiled using the Astronomy Acknowledgement Generator (<https://astrofrog.github.io/acknowledgment-generator/>).
This research has made use of NASA's Astrophysics Data System,
NumPy [van der Walt et al., 2011],
matplotlib, a Python library for publication quality graphics [Hunter, 2007],
SciPy [Virtanen et al., 2020],
and the IPython package [Perez & Granger, 2007].

§ COMMENTS

This version of the article has been accepted for publication after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at <https://doi.org/10.3847/1538-4357/ace041>.

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