transients, where the spectral energy distribution of the transient, the
underlying efficiency of the LSST bandpasses and the cadence in each band all
play a role. For example, kilonovae are likely intrinsically redder transients
and yet our metric does not improve with a redder filter distribution. This
could be because the LSST $g$-band single visit depth is much deeper than the
redder bands, meaning the optimum for kNe detection is somewhere between the
redder and the bluer distribution and ends up being close to baseline. The
baseline filter allocation was heavily influenced by photometric redshifts so
any radical departures from it could prove detrimental. Our photometric
redshift metric actually improves slightly for some of the filter distribution
simulations, but not significantly. From our results, it is difficult to find
a compelling reason to change the distribution from baseline. While the
overall filter distribution may already be acceptable for cosmology, _when_
those filters are used can still have significant impact on transient metrics,
as discussed in Section 6.6.

Figure 11: Selected metrics, relative to their values at baseline, as a
function of varying choices of filter distributions. Note that these
simulations use a different footprint so all metrics are measured against the
baseline filter_dist simulation, and not the standard baseline. We do not show
the 3x2pt FoM or LSS metrics because they do not vary significantly here. We
find no compelling reason to vary the baseline filter distribution: different
distributions improve different probes but at the cost of others.

###  6.5 Visit pairs and exposure time

Cadence remains a critical choice when designing observing strategy for any
survey that aims to detect transients. One of the most important changes to
observing strategy simulations has been to ensure visit pairs are taken in
different filters. Figure 12 shows the dramatic decrease in the number of
supernovae detected if this is not enforced. Although taking pairs in the same
filter is more efficient, it severely degrades the overall cadence of the
survey. Taking visit pairs in different filters has been the default in
simulations since the 2018 white paper call.

Another important decision to be made is whether the visits should be made in
a single exposure ($1\times 30$ s) or separated into two exposures ($2\times
15$ s), which could help reject cosmic rays. While this decision will only be
made during commissioning, Figure 13 shows how beneficial the $1\times 30$ s
exposure would be to all major cosmology metrics.

Figure 12: Selected metrics, relative to their values at baseline, for the
baseline_samefilt_v1.5_10yrs simulation. This clearly shows that while taking
visit pairs in the same filter does not impact static science much, it
dramatically degrades the transient metrics. Figure 13: Selected metrics,
relative to their values at baseline, for the baseline_nexp1_v1.7_10yrs.db
simulation, as compared to the current v1.7 baseline called
baseline_nexp2_v1.7_10yrs.db. The only difference between these two
simulations is the choice of whether an observation is taken in two 15 s
exposures (the current baseline) as opposed to a single 30 s exposure. It is
clear that the efficiency gained through a single exposure improves all
metrics, some dramatically.

###  6.6 Cadence

Figure 14 shows how different metrics behave as a function of the median
inter-night gap. While the median inter-night gap is a good measure of overall
cadence, it gives an incomplete picture as the tails of the distribution of
inter-night gap can be quite broad. It is still useful however to quantify the
average cadence of a simulation. The behavior of the metrics correlates fairly
well with the usual area and depth trade-off: larger areas tend to produce
worse cadence overall. An interesting point is that while there are only a few
simulations that produce better cadence than baseline, there are many that can
dramatically reduce the cadence which has a serious impact on the transient
metrics and metrics that depend on depth.

Figure 15 takes the cadence analysis a step further by investigating the
impact of inter-night gap per filter on the number of supernovae detected. For
fair comparison, only simulations with a similar number of visits in WFD are
included. The average cadence varies significantly between the bands, mostly
due to the non-uniform filter distribution but also due to decisions of what
filters to use in different moon conditions. For supernova observations, the
redder bands seem to be more important than the bluer bands. We find similar
results for strongly lensed supernovae and kilonovae, although kilonovae have
less dependence on $z$-band.

Figure 14: Selected metrics, relative to their values at baseline, as a
function of the median inter-night gap of different observing strategies. To
improve readability, points that are nearby in the $x$-axis are binned with
only the mean and error bar plotted for that bin. The supernova metric prefers
shorter gaps between observations but the situation is complicated by many
other factors. Generally it is important the cadence is at least no worse than
baseline. We have highlighted specific simulations with numbered annotations.
Simulation 1 includes visits at high airmass, which is generally avoided,
indicating that high airmass observations can improve cadence without too much
degradation of metrics. Simulation 2 is when $2\times 15$ s exposures are used
instead of $1\times 30$ s. Simulation 3 is a large area footprint but removes
visits from the extragalactic area reducing cadence. List of annotations:
1-dcr_nham2_ugr_v1.5_10yrs, 2-baseline_2snaps_v1.5_10yrs,
3-footprint_newAv1.5_10yrs. Figure 15: The relative improvement over baseline
for the number of supernovae as a function median inter-night gap in each
filter. The baseline cadence is represented by star markers. In general, the
number of supernovae improve as the cadence improves, but note the relatively
small dependence on $g$ and $u$ band, indicating the higher importance of good
cadence in the redder bands. A Gaussian process has been fitted to the set of
points in each filter to help guide the eye; the mean is plotted with a solid
line and the standard deviation indicated with a coloured envelope. Also note
that we have restricted this plot to simulations with similar number of visits
in the WFD.

Figure 16 shows selected metrics as a function of cumulative season length
(how long in years a field is observed for on average). We find that as long
as the season length remains comparable to baseline, most metrics are fairly
insensitive to changes in season length.

Figure 16: Selected metrics, relative to their values at baseline, as a
function of the cumulative season length (the total amount of time a field is
observed) of different observing strategies. To improve readability, points
that are nearby in the $x$-axis are binned with only the mean and error bar
plotted for that bin. Shorter season lengths degrade strong lensing and
supernovae performance but as long as the season length does not fall much
below it’s value at baseline, these metrics do not seem to be strongly
affected. We have highlighted specific simulations with numbered annotations.
Simulation 1 is a larger area footprint which removes visits from the
extragalactic area. Simulation 2 artificially improves cadence (and hence SN
performance) by removing the mini surveys. Simulation 3 replaces some visits
with short exposures that cannot be used for SN observations thus reducing
signal-to-noise. And simulation 4 takes some observations at high airmass,
which is usually avoided. This indicates that taking some number of high
airmass observations is an effective way to increase season length without
degrading metric performance significantly. List of annotations:
1-footprint_newBv1.5_10yrs, 2-wfd_depth_scale0.99_v1.5_10yrs,
3-short_exp_5ns_5expt_v1.5_10yrs, 4-dcr_nham1_ugri_v1.5_10yrs

###  6.7 Weather and seeing

Figure 17 shows observing strategies that simulated different weather
scenarios, from realistic to impossibly optimistic. Naturally, metrics
sensitive to cadence suffer immensely when weather conditions restrict
observing time. Although these optimistic simulations are not realistic, they
do indicate that much could be gained in transient science by ensuring fields
missed by bad weather are observed as soon as possible to avoid large light
curve gaps.

Figure 17: Selected metrics, relative to their values at baseline, as a
function of different weather simulations, for an older FBS version, v1.3. The
simulations make use of real weather data and differ on what cloud coverage
would be required before observations are paused: 30% (realistic), 70%
(v1.3-like in other words similar to previous simulations), 120% (optimistic,
where the dome would never close due to bad weather) and no down time
(optimistic-no downtime, where the telescope would always observe no matter
what). While bad weather only has a small impact on static science metrics by
slightly reducing the median depth, it has enormous impact on transient
science metrics by as much as 150%. This highlights the importance of
including realistic weather in simulations to make accurate predictions of
scientific returns.

##  7 Conclusions

Because of its effect on a wide variety of science cases, optimization of LSST
observing strategy is critical and has become an active area of research that
has already resulted in significant improvements to the scheduler and baseline
strategy. In this paper, we have introduced metrics used within the LSST DESC
to investigate the impact of observing strategy choices on cosmological
measurements with LSST. While we continue to work towards our goal of a full
DETF Figure of Merit combining all probes, which includes systematics, the
metrics introduced here are still valuable in gaining insights about observing
strategy.

###  7.1 LSST DESC recommendations for observing strategy

Here we summarize our conclusions drawn from the metrics outlined in this
paper, combined with those from the observing strategy white paper (Lochner et
al., 2018):

  * •

Footprint and area: The nominal area for the LSST WFD survey is 18,000 deg2,
but its area and definition can be changed. Lochner et al. (2018) and Olsen et
al. (2018) both proposed to shift the WFD footprint away from regions of high
Galactic dust extinction to better benefit extragalactic science and rather
have a dedicated Galactic plane survey that does not have the same
requirements as extragalactic. Figure 8 shows that all metrics decrease with
increased effective area, aside from the 3x2pt FoM, but that one simulation,
footprint_big_sky_dust, improves all metrics. This simulation proposes a large
area footprint but uses a dust extinction cut to define the boundary. However,
it should be noted that this footprint would degrade Galactic science and so,
while it should be considered the ideal for cosmology, some compromise must be
made to ensure the integrity of other science goals of LSST. Area is not the
only consideration of course; there is a natural trade-off between depth and
area, with depth (and correspondingly good cadence) being critical for
mitigating systematic effects and for transient probes. Figure 10 shows a
simple attempt to illustrate this trade-off. In Lochner et al. (2018), we
recommended a $18,000\ \rm{deg}^{2}$ footprint, that uses an E(B-V)<0.2
extinction cut, with declination limits of $-70<\rm{dec.}<12.5$, which is
similar to footprint_big_sky_dust. This footprint obtains a large area
suitable for extragalactic science and, crucially, extends the footprint North
to increase overlap with other surveys like DESI. Our work in this paper shows
that such a footprint would ensure cosmology goals with LSST are met. Of
course, dedicating $18,000\ \rm{deg}^{2}$ to the extragalactic area means
visits must be moved from the WFD to the Galactic plane to support Galactic
science goals, potentially reducing the cadence and impacting the transient
probes and some systematics. This could be mitigated with a change in strategy
such as implementing rolling cadence (see Section 7.2).

  * •

Survey uniformity: Uniformity of depth across the WFD is a strict requirement
for cosmology to avoid systematic effects being introduced by non-uniform
measurements. We have a metric to measure uniformity (described in Section
3.1.1) which finds that most strategies do not deviate significantly from
baseline in terms of uniformity, indicating the SRD requirements are being
met. However, it is worth pointing out that there is some flexibility at what
points in the survey the observations need to be uniform. Lochner et al.
(2018) proposed to ensure observations be uniform by the data releases at the
end of years 1, 3, 6, 10 to enable periodic cosmological analyses; however
these are not yet finalized and can be changed. The important point is that
any strategy should allow regular “checkpoints” where uniformity is achieved
and analysis-ready data can be released. This should naturally be
straightforward with most strategies but we note that rolling cadence (further
discussed in Section 7.2) could affect when these uniform “checkpoints” are
possible.

  * •

Dithering: The process of adding random rotational and translational offsets
when repeating an observation in a field, known as dithering, is extremely
important for cosmology systematics. Uniform coverage of 180 deg of camera
rotation angle in every field is important for reduction of camera-based PSF
systematics. All simulations used here implement translational and rotational
dithers, which help maintain survey uniformity. While some investigation
remains, we consider the current baseline dithering strategy to be largely
effective. We note, however, that it is important that any translational
dithers used in the DDFs should be as small as possible so that the depth of
the fields is not compromised.

  * •

Overlap with other surveys: As discussed in Section 6.3, overlap with other
surveys such as DESI, 4MOST:TiDES and Euclid is critical for a number of
cosmological probes. Spectroscopic data and infrared photometry can provide
better calibration and training of photometric redshifts, improved WL
constraints and more accurate de-blending of galaxies. Transient probes, in
particular supernovae, will rely heavily on spectroscopy for host galaxy
redshift identification and live spectroscopic follow-up to provide training
sets for photometric classifiers. Improving overlap with other surveys further
supports our recommendation for a larger footprint, particularly one which
extends further North to improve overlap with DESI and Euclid.

  * •

Exposure time: The LSST SRD describes a visit to a field as a pair of 15 s
exposures ($2\times 15$ s). This is designed to robustly reject cosmic rays,
but Lochner et al. (2018) reported a potential gain of 7% efficiency, as well
as improved image quality, by switching to a single exposure ($1\times 30$ s)
and using standard cosmic-ray rejection techniques (e.g., Aihara et al. 2018).
Our results show that a single exposure can improve our metrics by as much as
20% (see Figure 13). The only advantage of short exposures is that they may
mitigate saturation effects for very low redshift ($z<0.05$) SNe. However,
this advantage does not outweigh the severe degradation to the larger SNe
sample. While we strongly advocate for single 30 s exposures, we acknowledge
that the final decision about exposure times will only be taken after
commissioning tests prove that cosmic rays and satellite trails can be
rejected accurately.

  * •

Repeated visits in a night: A requirement in the SRD is that any given LSST
field must be visited at least twice within a short period of time (15–40
minutes) to ensure accurate asteroid detection and assist with orbit
characterization. There is a clear trade-off between the number of _intra-
night_ visits and _inter-night_ visits, which can heavily impact cosmology
with transient objects. Hence, Lochner et al. (2018) proposed that repeat
visit pairs be in different filters to improve transient characterisation.
This has been adopted for all the simulations used in this paper and remains
one of our strongest recommendations, as it significantly improves the
transient metrics (Figure 12).

  * •

Cadence: The return time with which each field in the WFD is observed is of
critical importance to many science cases and is one of the most difficult
factors to optimize. While the LSST SRD states that every field must be
observed on average 825 times over the survey, it does not specify how those
visits should be distributed and while each field is visited every 3 days _on
average_ , there is a large spread in this distribution. Figure 14 shows the
impact of the median internight gap on various metrics. Significant gains can
be made by changing how the filters are allocated and ensuring the $griz$
filters maintain regular cadence whatever the moon conditions. We find that
the redder filters, $riz$, are particularly important for supernova
characterisation and need to have high cadence (see Figure 15). $u$-band,
while important for photometric redshifts, does not heavily impact our
transient probes and thus does not require high cadence. Cadence and filter
allocation are expected to also be important for transient classification
which, while not included as a metric, is discussed in Section 7.2.

###  7.2 Future work

While significant gains have been made in understanding the impact of
observing strategy on cosmological constraints with LSST, there remain some
unanswered questions and active areas of research:

  * •

Rolling cadence: This is a proposed strategy that can increase the cadence in
a certain area of sky by prioritizing it for a period of time at the expense
of other areas. This would then be repeated for the initially de-prioritized
areas, allowing a pattern of rolling cadence. While rolling cadence is very
promising, it has proved difficult to simulate. The LSST SRD proper motion
requirements mean that the survey must have uniform coverage at its start and
end, meaning a rolling cadence strategy must start and stop smoothly. It also
has to roll for full observing seasons, constraining when a roll can start.
Finally, rolling cadence strategies have many parameters to optimize including
how large an area to roll at a time, for how long and whether to roll
completely or allow the remaining part of the survey to still be observed but
at a lower priority. All these complexities and constraints have made it
difficult to find a rolling strategy that works well for transient science
without impacting other science cases. Although it is difficult, we still
consider it a promising option to obtain large sky area without reducing
cadence, and are continuing to research this approach.

  * •

Transient classification: All our transient metrics have made the explicit
assumption that transients detected with LSST can be perfectly classified,
which is obviously not true. Not only does photometric transient
classification impact the ability to constrain cosmology with SNe, kNe and
strongly lensed SNe, it is also in turn affected by the quality of the
photometric redshift estimation. Additionally, transient classification will
likely be highly sensitive to observing strategy, making a transient
classification metric critical. This metric is a challenge to develop as it
would be affected by many aspects of observing strategy, would need to
encompass a broad array of transient classes and could be somewhat dependent
on the choice of classifier used. Because photometric transient classification
for large surveys is still a young research field, not enough is known to
build a reliable emulator to use as a fast metric. Thus, to evaluate a single
observing strategy, sophisticated and expensive simulations must be run. An
important step in this direction was taken by the PLAsTiCC challenge (The
PLAsTiCC team et al., 2018), which was a community challenge that released a
set of LSST-like simulations incorporating a wide variety of transients and
variables. Work is ongoing to understand the effect of observing strategy on
existing transient classification pipelines (e.g., Lochner et al., 2016;
Möller & de Boissière, 2019) using the PLAsTiCC simulations themselves, and
also by running new simulations based on some of the observing strategies
discussed in this paper.

  * •

A third visit: There is a proposal to add an additional third visit some time
after the visit pair, repeating one of the filters used in the original pair.
The motivation is to improve early classification of transients, which will
affect all transient science and be of particular importance to spectroscopic
follow-up. However, a third observation in the same band will reduce overall
cadence and hence the number of well-measured supernovae, kilonovae and
strongly lensed transients. As stated in the previous point, we do not yet
have a reliable classification metric to evaluate this approach.
Classification performance must be carefully weighed against overall
cosmological constraints for a balance to be found. Considerations must also
be made about the spectroscopic follow-up strategy to build training sets for
classifiers. Thus, the question of whether or not to add a third visit is
actually only a single component of an important and complex analysis that is
ongoing.

  * •

DDFs: Cosmology with LSST relies heavily on the DDFs for both photometric
redshift training and particularly to provide a deep sample of high redshift
supernovae. While this paper focused on the WFD, many of the metrics developed
for the SN probe in Section 5.1.1 can be applied to evaluate different DDF
strategies. The optimal outcome for the DDFs is that they produce a well-
measured SNIa sample at significantly higher redshifts than the WFD. The
cadence for a DDF survey will be on the scale of every $\sim 2$ days rather
than every $10-15$ days, which results in higher quality light curves than
WFD. However, to improve the redshift range of supernovae detected in the
DDFs, we advocate here for two strategies: 1\. Rolling deep fields such that
in a given year, 1–2 fields can go significantly deeper than the other fields,
and 2. A significantly skewed filter-allocation towards the redder bands.

The metrics developed in this paper have been invaluable in gaining insights
about observing strategy and lay a useful foundation for ongoing work on this
challenging optimization problem.

## Acknowledgements

Author contributions are listed below. HAl: Developed WL systematics metric,
contributed MAF metric for 3x2pt Figure of Merit. Analysis and interpretation
for weak lensing metric results. Writing and discussion for sections 4 and 6.
TA: Helped develop time delay lensed quasar metric, run metrics, analysis and
discussion of Section 5.2. HAw: Provided technical assistance with
simulations. Developed MAF metrics for the extragalactic footprint and the
number of galaxies. Provided numbers for basic static metrics (e.g depth
statistics) and those for LSS. Helped with discussions, and writing sections 3
and 4. RB: Contributed to software and metric development TB: Contributed to
long time-scale microlensing efficiency understanding EG: Conceptualization,
discussion, and writing in sections 3 and 4. Edited sections 1-4 and 7. SGG:
Section 5.1.3, metric for SNe PV PG: Section 5.1: discussion, oversight and
writing. Developed a set of supernovae metrics implemented in MAF. Processed
all observing strategies. Developed real-time display
(https://me.lsst.eu/gris/OS_Videos/) to visualize the progress of observing
strategies on a daily basis to study cadence/gap effects. Implementation of
the lensed SNe Ia by galaxies metric in MAF and processing of observing
strategies (section 5.2.1). IH: Section 5.1:discussion. Writing (review and
editing) of Sections 1,2 and 5. SH: Section 5.2.1:metric, Section
5.2:validation, analysis, writing, editing, SWJ: Conceptualization and
development of observing strategies related to SN Ia (section 5.1) and rolling
cadences. Editing of sections 1 and 2. RLJ: Created simulations, technical
assistance with simulations and metric interpretation. Worked with various
groups to develop metrics and analysis. AGK: Section 5.1.3, writing/metric-
creation ML: Conceptualization, software, validation, formal analysis,
investigation, data curation, writing - original draft, writing - review and
editing, visualization, supervision RM: Discussion and writing in sections 3
and 4 PM: Section 5.2 (designed and implemented metrics, wrote text) MM:
Microlensing discussion and writing in introduction EHN: Improved seeing input
to opsim simulations (section 2.5, 6.7), general descussions and
recommendations HP: Section 5.3: conceptualization, validation, writing -
review, supervision, funding acquisition TP: Metric development and writing of
subsection 5.2.2 NR: Section 5.1: discussion and writing. Developed a set of
metrics to evaluate the expected quality on the SN light curves, on the DDF
and WFD surveys. Built a pipeline to evaluate the pre-white paper, white-paper
and fbs1.3 - fbs1.7 cadences. DR: Developed deterministic scheduler (alt
sched) as a conceptual alternative to OpSim for comparison purposes DS:
Conceptualization, writing - original draft, writing - review & editing,
visualization, supervision CNS: Software, metrics, validation, analysis and
writing of Section 5.3. CS: Developed deterministic scheduler (alt sched) as a
conceptual alternative to OpSim for comparison purposes SHS: Coordinated
strong lensing group; writing, validation, analysis and editing of Section
5.2. PY: Created survey simulations. Section 2 writing. Helped with
development of MAF metrics for Section 3, 5.

TA acknowledges support from Proyecto Fondecyt N 1190335 and the ANID-
Millennium Science Initiative Program – ICN12_009 awarded to the Millennium
Institute of Astrophysics MAS. HAw also acknowledges support by the Rutgers
Discovery Informatics Institute (RDI2) Fellowship of Excellence in
Computational and Data Science, Rutgers University & Bevier Dissertation
Completion Fellowship, Leinweber Postdoctoral Research Fellowship, and DOE
grant DE-SC009193. HAw also thanks the LSSTC Data Science Fellowship Program,
which is funded by LSSTC, NSF Cybertraining Grant #1829740, the Brinson
Foundation, and the Moore Foundation, as participation in the program has
benefited this work. This material is based upon work supported by the U.S.
Department of Energy, Office of Science, Office of High Energy Physics Cosmic
Frontier Research pro- gram under Award Numbers DE-SC0011636 and DE-
SC0010008, which supported EG and HAw. SGG acknowledges support from the
University of Rochester and the DOE Office of High Energy Physics under grant
number DE-SC0008475.

SH and SHS thank the European Research Council (ERC) for support under the
European Union’s Horizon 2020 research and innovation programme (LENSNOVA:
grant agreement No. 771776)

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics Cosmic Frontier Research
program under Award Numbers DE-SC0011636, which supported SWJ. AGKs work is
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics, under contract No. DE-AC02-05CH11231. ML acknowledges support
from the South African Radio Astronomy Observatory and the National Research
Foundation (NRF) towards this research. Opinions expressed and conclusions
arrived at, are those of the authors and are not necessarily to be attributed
to the NRF. RM and H. Almoubayyed are supported by the Department of Energy
grant DE-SC0010118. This document was prepared by DESC using the resources of
the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359 CS,
RB and HVP’s work was partially supported the research environment grant
“Gravitational Radiation and Elec- tromagnetic Astrophysical Transients
(GREAT)” funded by the Swedish Research Council (VR) under Dnr 2016- 06012 and
the research project grant ‘Understanding the Dynamic Universe’ funded by the
Knut and Alice Wallenberg Foundation under Dnr KAW 2018.0067. TP acknowledges
the financial support from the Slovenian Research Agency (grants I0-0033,
P1-0031, J1-8136 and Z1-1853). SHS thanks the Max Planck Society for support
through the Max Planck Research Group.

The DESC acknowledges ongoing support from the Institut National de Physique
Nucléaire et de Physique des Particules in France; the Science & Technology
Facilities Council in the United Kingdom; and the Department of Energy, the
National Science Foundation, and the LSST Corporation in the United States.
DESC uses resources of the IN2P3 Computing Center (CC-IN2P3–Lyon/Villeurbanne
- France) funded by the Centre National de la Recherche Scientifique; the
National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under Contract No. DE-AC02-05CH11231; STFC DiRAC HPC Facilities, funded
by UK BIS National E-infrastructure capital grants; and the UK particle
physics grid, supported by the GridPP Collaboration. This work was performed
in part under DOE Contract DE-AC02-76SF00515. This paper has undergone
internal review in the LSST Dark Energy Science Collaboration. The authors
thank the internal reviewers, who were Ariel Goobar, Joshua Meyers and Samuel
Schmidt. The authors also thank Jean-Charles Cuillande and Christopher
Frohmaier for providing the footprint maps for Euclid and 4MOST respectively.

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##  Appendix A Metric descriptions and transformations

Table 5: Metrics used in this paper. The first column describes the metric, the second indicates how the metric ($x$) is transformed such that it can be directly compared with other metrics and interpreted as “larger is better”. The third column indicates the original metric value at baseline and the last column directs the reader to the section where the metric is described. Metric description | Transform | Value at baseline | Section in paper
---|---|---|---
Y1 median $i$-band coadded depth | $1/\big{(}10^{-\frac{x}{2.5}}\big{)}^{2}$ | 25 | 3.1
Y3 median $i$-band coadded depth | $1/\big{(}10^{-\frac{x}{2.5}}\big{)}^{2}$ | 26 | 3.1
Y6 median $i$-band coadded depth | $1/\big{(}10^{-\frac{x}{2.5}}\big{)}^{2}$ | 26 | 3.1
Y10 median $i$-band coadded depth | $1/\big{(}10^{-\frac{x}{2.5}}\big{)}^{2}$ | 26 | 3.1
Y1 $i$-band coadded depth stddev | $1/\big{(}10^{-\frac{x}{2.5}}\big{)}^{2}$ | 0.16 | 3.1
Y3 $i$-band coadded depth stddev | $1/\big{(}10^{-\frac{x}{2.5}}\big{)}^{2}$ | 0.13 | 3.1
Y6 $i$-band coadded depth stddev | $1/\big{(}10^{-\frac{x}{2.5}}\big{)}^{2}$ | 0.12 | 3.1
Y10 $i$-band coadded depth stddev | $1/\big{(}10^{-\frac{x}{2.5}}\big{)}^{2}$ | 0.12 | 3.1
Y1 area for static science (deg2) | $x$ | 1.5e+04 | 3.1
Y3 area for static science (deg2) | $x$ | 1.5e+04 | 3.1
Y6 area for static science (deg2) | $x$ | 1.5e+04 | 3.1
Y10 area for static science (deg2) | $x$ | 1.5e+04 | 3.1
PZ outlier fraction (low z) | $1-2x$ | 0.0019 | 3.2
PZ stddev (low z) | $1/x^{2}$ | 0.0096 | 3.2
PZ outlier fraction (high z) | $1-2x$ | 0.044 | 3.2
PZ stddev (high z) | $1/x^{2}$ | 0.022 | 3.2
WL + LSS DETF Figure of Merit | $x$ | 36 | 4.1.1
WL Average visits | $x$ | 6.1e+02 | 4.1
LSS systematics FOM for Y1 | $x$ | 0.94 | 4.2.2
LSS systematics FOM for Y10 | $x$ | 0.98 | 4.2.2
Y1 $N_{gal}$ at 0.66<z<1.0 | $x$ | 4.5e+08 | 4.2.1
Y10 $N_{gal}$ at 0.66<z<1.0 | $x$ | 9.5e+08 | 4.2.1
Number of SNe Ie with z$\textless$zlim(faint) | $x$ | 4.5e+04 | 5.1.1
Faint SNe Ia redshift limit | $x$ | 0.17 | 5.1.1
Number of SNe Ia with z$\textless$zlim(med) | $x$ | 1.6e+05 | 5.1.1
SNe Ia r-band Signal-to-Noise Ratio | $x$ | 0.59 | 5.1.2
SNe Ia r-band redshift limit | $x$ | 0.31 | 5.1.2
Medium SNe Ia redshift limit | $x$ | 0.27 | 5.1.1
SN Ia Peculiar Velocities | $x$ | 1.4 | 5.1.3
Number of SNe Ia lensed by galaxies | $x$ | 8.7 | 5.2.1
Number of lensed quasars | $x$ | 3.3e+02 | 5.2.3
Number of SNe Ia lensed by clusters | $x$ | 0.18 | 5.2.2
GW170817-like kNe Counts 10yrs | $x$ | 97 | 5.3.1
kNe MAF Mean Counts | $x$ | 34 | 5.3.1
kN Population Counts 10yrs | $x$ | 40 | 5.3.1

##  Appendix B List of observing strategy simulations

All simulations used in this paper, as well as those that have been explicitly
excluded. We have focused on the v1.5 FBS simulations but indicate where some
have been excluded, usually due to bugs or a variation in the simulation that
artificially changes a metric and is thus not directly comparable to the
baseline. We highlight which figures each simulation is used in, noting
particularly that Figure 15 is restricted to simulations with similar total
number of visits. We do not indicate the list of simulations used in the
figures in Sections 4 and 5 since it is explicitly indicated there.

Table 6: List of all simulations used in this paper, which baseline simulation it is compared against and which figures the simulation was used in (aside from early figures in the paper where the simulation is explicitly indicated). Simulation name | Comparison baseline | Included in which figures
---|---|---
weather_0.3_v1.4_10yrs | weather_0.3_v1.4_10yrs | 17
weather_0.7_v1.4_10yrs | weather_0.3_v1.4_10yrs | 17
weather_1.2_ndt_v1.4_10yrs | weather_0.3_v1.4_10yrs | 17
weather_1.2_v1.4_10yrs | weather_0.3_v1.4_10yrs | 17
agnddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
alt_dust_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
alt_roll_mod2_dust_sdf_0.20_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
baseline_2snaps_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
baseline_samefilt_v1.5_10yrs | baseline_v1.5_10yrs | 12
baseline_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 13, 12, 16, 17, 15, 10
bulges_bs_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
bulges_bulge_wfd_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
bulges_cadence_bs_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
bulges_cadence_bulge_wfd_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
bulges_cadence_i_heavy_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
bulges_i_heavy_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
dcr_nham1_ug_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
dcr_nham1_ugr_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
dcr_nham1_ugri_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
dcr_nham2_ug_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
dcr_nham2_ugr_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
dcr_nham2_ugri_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
descddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
filterdist_indx1_v1.5_10yrs | baseline_v1.5_10yrs | 11
filterdist_indx2_v1.5_10yrs | baseline_v1.5_10yrs | 11
filterdist_indx3_v1.5_10yrs | baseline_v1.5_10yrs | 11
filterdist_indx4_v1.5_10yrs | baseline_v1.5_10yrs | 11
filterdist_indx5_v1.5_10yrs | baseline_v1.5_10yrs | 11
filterdist_indx6_v1.5_10yrs | baseline_v1.5_10yrs | 11
filterdist_indx7_v1.5_10yrs | baseline_v1.5_10yrs | 11
filterdist_indx8_v1.5_10yrs | baseline_v1.5_10yrs | 11
footprint_add_mag_cloudsv1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
footprint_big_sky_dustv1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
footprint_big_sky_nouiyv1.5_10yrs | baseline_v1.5_10yrs | Excluded
footprint_big_skyv1.5_10yrs | baseline_v1.5_10yrs | Excluded
footprint_big_wfdv1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
footprint_bluer_footprintv1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
footprint_gp_smoothv1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
footprint_newAv1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
footprint_newBv1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
footprint_no_gp_northv1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
footprint_standard_goalsv1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
footprint_stuck_rollingv1.5_10yrs | baseline_v1.5_10yrs | Excluded
goodseeing_gi_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
goodseeing_gri_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
goodseeing_griz_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
goodseeing_gz_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
goodseeing_i_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
greedy_footprint_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
roll_mod2_dust_sdf_0.20_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
rolling_mod2_sdf_0.10_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
rolling_mod2_sdf_0.20_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
rolling_mod3_sdf_0.10_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
rolling_mod3_sdf_0.20_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
rolling_mod6_sdf_0.10_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
rolling_mod6_sdf_0.20_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
short_exp_2ns_1expt_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
short_exp_2ns_5expt_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
short_exp_5ns_1expt_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
short_exp_5ns_5expt_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
spiders_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
third_obs_pt120v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
third_obs_pt15v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
third_obs_pt30v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
third_obs_pt45v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
third_obs_pt60v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
third_obs_pt90v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
twilight_neo_mod1_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
twilight_neo_mod2_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
twilight_neo_mod3_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
twilight_neo_mod4_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
u60_v1.5_10yrs | baseline_v1.5_10yrs | Excluded
var_expt_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 10
wfd_depth_scale0.65_noddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.65_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.70_noddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.70_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.75_noddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.75_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.80_noddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 15, 10
wfd_depth_scale0.80_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.85_noddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 15, 10
wfd_depth_scale0.85_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 15, 10
wfd_depth_scale0.90_noddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.90_v1.5_10yrs | baseline_v1.5_10yrs |  8, 14, 9, 16, 15, 10
wfd_depth_scale0.95_noddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.95_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.99_noddf_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
wfd_depth_scale0.99_v1.5_10yrs | baseline_v1.5_10yrs |  8, 9, 16, 10
barebones_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
baseline_nexp1_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
combo_dust_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
ddf_heavy_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
dm_heavy_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
even_filters_alt_g_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
even_filters_altv1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
even_filters_g_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
even_filtersv1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
mw_heavy_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
rolling_exgal_mod2_dust_sdf_0.80_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
rolling_fpo_2nslice0.8_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
rolling_fpo_2nslice0.9_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
rolling_fpo_2nslice1.0_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
rolling_fpo_3nslice0.8_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
rolling_fpo_3nslice0.9_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
rolling_fpo_3nslice1.0_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
rolling_fpo_6nslice0.8_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
rolling_fpo_6nslice0.9_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
rolling_fpo_6nslice1.0_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | Excluded
ss_heavy_v1.6_10yrs | baseline_nexp1_v1.6_10yrs | 10
baseline_nexp1_v1.7_10yrs | baseline_nexp2_v1.7_10yrs | 13
baseline_nexp2_v1.7_10yrs | baseline_nexp2_v1.7_10yrs | 13


# Strain Induced Enhanced Photocatalytic Activities in Layered Two Dimensional
C2N/MoS2 Heterostructure: A Meta-GGA Study

Soumendra Kumar Das1, Lokanath Patra2, Prasanjit Samal1, Pratap K Sahoo1
1School of Physical Sciences, National Institute of Science Education and
Research (NISER) Bhubaneswar, HBNI, Jatni, Khurda-752050, Odisha, India.
2Department of Mechanical Engineering, University of California Santa Barbara,
CA, 93106, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The improved photocatalytic water splitting using 2D materials has
technological importance for economically viable renewable energy. The present
study focuses on the effect of uniaxial, biaxial, and vertical strain on the
energy gap and band edge positions of C2N/MoS2 van der Waals heterostructures
through first-principles density functional theory using PBE and SCAN
functionals. The calculations establish that SCAN functional provides
comparatively much better results as compared to the PBE for the band gap and
band alignment study. The heterostructure exhibits a type-II band alignment
which is beneficial for the efficient separation of charge carriers. For a
good photocatalyst, the band edge positions should straddle the water redox
potentials. It is observed that for both compressive and tensile vertical
strain, the water redox potential values lie within the valence band maximum
(VBM) and conduction band minimum (CBM) of the heterostructure. On the other
hand, for uniaxial and biaxial strain, the system can be used as a useful
photocatalyst only for larger compressive strain, whereas for tensile strain,
the energy gap between VBM and CBM keeps on decreasing and lie within the
water oxidation/reduction potential. Our study also establishes that the meta-
GGA SCAN functional shows similar results as compared to the computationally
expensive hybrid HSE functionals. The present work can be extremely useful for
experimentalists to design artificial heterostructure devices for better
performance in photocatalytic water splitting.

††:

Keywords: Photocatalytic water splitting, DFT, Van der Waals heterostructures,
strain, Type-II Band alignment.

##  1 Introduction

The emergence of photocatalytic water splitting has been a successful
technology to meet the demand for the energy crisis and environmental
pollution created by our fast-growing economy. The development of high-
performance photo-catalytic materials to create hydrogen by using solar energy
has been a serious focus of research for many years [1, 2]. The key factor for
achieving highly efficient photocatalysts (PCs) is that the band gap should be
larger than the water redox potentials. More specifically, the conduction band
minimum (CBM) of the PCs should be above the H+/H2 potential and the valence
band maximum (VBM) should be below H2O/O2 potential simultaneously, thus
requiring a minimum band gap of 1.23 eV [3]. In addition, literature reports
have established the importance of co-catalysts for boosting the electron-hole
separation and improving the reaction kinetics [4]. Under such circumstances,
two-dimensional materials like graphene, hexagonal boron nitride (h-BN) mono
layers, transition metal dichalcogenides (TMDCs), C3N4, C2N, etc, have created
a lot of interest, in meeting the demand, because of their novel electronic,
thermal and optoelectronic properties. In particular, MoS2 has a direct band
gap (2 eV), and high carrier mobility in the form of a single monolayer, which
makes it an important candidate for photocatalytic and photovoltaic
applications [5, 6]. Similarly, the porous C2N monolayer is found to be a
direct band gap semiconductor with a gap of 1.96 eV [7]. Zhao et al. have
adopted a 2D/2D polymeric Z-scheme heterostructure by using a pair of
ultrathin g-C3N4 nanosheets in order to provide H2\- and O2\- evolving
photocatalysts through the strategy of electrostatic self-assembly. Using Pt
and Co(OH)2 as co-catalysts, the heterostructure achieved a solar-to-hydrogen
efficiency of 1.16 % which originates due to the formation of direct Z-scheme
charge transfer pathway through the interface between H2\- and O2\- evolving
components [8]. It has been suggested that the use of C2N and MoS2 can be
highly efficient for photocatalytic study and also can be complementary to the
use of graphene and h-BN [9].

Despite the extensive use of C2N and MoS2, there are some challenges as well
for the application of these materials for photocatalytic study. The charge
distribution of the valence band maximum (VBM) and conduction band minimum
(CBM) states for these systems are not well separated in space resulting in
reduced light absorbing efficiency because of the recombination of the
photoinduced electrons and holes [10, 7]. Therefore, attempts have been made
to use van der Waals heterostructures to fix the issues. The electronic
properties of C2N/InSe heterostructure are found to be greatly affected by
vertical strain and electric field. Without any electric field, the
heterostructure possesses a type-II band alignment with an indirect band gap
of 1.34 eV at an equilibrium interlayer distance of 3.325 Å. Application of an
electric field or a change in interlayer distance results in a transition from
type-II to type-I band alignment and indirect to direct band gap in this
heterostructure [11].

Band gap and band offset engineering at C2N/MSe2 (M = Mo, W) interface have
established that the heterostructure possesses a narrow indirect band gap with
type-II band alignment which is favourable for the photogenerated electron-
hole pairs. The application of vertical strain and electric field strongly
modulate the magnitude of band gap values and band offsets, but the type-II
band alignment nature remains preserved [12]. Strain engineering plays a
significant role in tuning the electronic properties and photocatalytic
performance of 2D heterostructures. Wang et al. have studied the effect of
strain on the electronic structure of C2N/MTe (M= Ga, In) through DFT
calculations which exhibit excellent optical properties with good structural
stability. It was observed that for C2N/GaTe heterostructure, the exciton-Bohr
radius remains unaffected by the application of strain. On the other hand,
compressive strain reduces the exciton-Bohr radius in C2N/InTe system. The
power conversion efficiency shows an increase up to 22.1% for C2N/GaTe with 4%
strain and 19.8% for C2N/GaTe heterostructure with 6% strain [13]. Han et al.
reported that the catalytic efficiency of C2N/SiH heterojunction can be
effectively adjusted by the application of -2$\%$ and +4$\%$ biaxial strain
for the hydrogen evolution reaction (HER) and oxygen evolution reaction (OER),
respectively [14]. The Cs3Bi2I9/C2N heterostructure exhibits a charge
distribution across the whole structure due to the difference in the work
function between the two monolayers and a charge transfer at the interface due
to the formation of an internal electric field [15]. The photocatalytic study
in MoS2/ZnO heterostructure shows an indirect band gap with type-II band
alignment, a significant built-in potential of 7.42 eV, and a valence band
offset of 1.23 eV across the interface. The photogenerated carriers are
localized in different layers and can effectively generate hydrogen energy. In
contrast, the MoSe2/ZnO heterostructure possesses a type-I band alignment with
a direct band gap of 1.80 eV, built-in potential around 3.64 eV, and a valence
band offset of 0.34 eV [16].

Despite the extensive studies on C2N based van der Waals heterostructures,
significant progress has not been achieved both theoretically and
experimentally. As a typical case, reports on the C2N/MoS2 heterostructures
are very scarce and a systematic investigation of the physical properties of
this system is still lacking. In this article, we report the effect of
vertical, uniaxial, and biaxial strain on the photocatalytic water splitting
performance of C2N/MoS2 van der Waals heterostructures through first-
principles electronic structure calculations.

##  2 Results and Discussion

The schematic of the crystal structure of C2N monolayer, MoS2 layer, and
C2N/MoS2 heterostructure is illustrated in Fig. 1. In C2N monolayer, twelve
C-N bonds are alternately connected to the six C-C bonds in such a way that
the periodic holes are present without any atom in the middle of the lattice.
The in-plane bond length is estimated to be around 1.43 and 1.47 Å for the C-C
bond and 1.34 Å for the C-N bond, respectively which is also close to the
previously reported values [17]. The optimized lattice constant using PBE
functional is estimated to be 8.33 Å and 3.19 Å for the C2N and MoS2
monolayer, respectively which is in reasonable agreement with the reported
experimental and theoretical results [18, 19]. The heterostructure is
constructed by making a supercell of (3$\times$3$\times$1) for C2N and
(8$\times$8$\times$1) for MoS2 layer, respectively. The corresponding lattice
mismatch between the two monolayers is around 1.9$\%$ which is within the
acceptable range. The heterostructure after ionic relaxation results in a
corrugated structure (Fig. 1d), very similar to silicene. The initial van der
Waals gap (3.5 Å) between the layers reduces to 3.373 Å with an optimized
lattice constant of 25.47 Å (Fig. 1c, d).

Figure 1: Schematic representation of the top view of isolated (a) C2N
monolayer, (b) MoS2 monolayer, (c) side view and (d) top view of C2N/MoS2
heterostructure. The schematic symbols of the C, N, Mo, and S atoms are
indicated as blue, red, green, and brown colors, respectively.

The calculated electronic structure of free-standing C2N monolayer, MoS2 using
the PBE functional are given in Fig. 2a,b. The band dispersion for both the
material indicates the presence of valence band maximum (VBM) and conduction
band minimum (CBM) at the same k value in the Brillouin zone. In addition, for
both structures, the VBM remains close to the Fermi level as compared to the
CBM. This confirms that both C2N and MoS2 monolayers are p-type direct band
gap semiconductors and the calculated energy gap values are estimated to be
1.735 eV and 1.645 eV, respectively. It is interesting to note that the energy
band dispersion in C2N monolayer is relatively flat where as MoS2 shows highly
dispersive bands both in VB and CB regions. Hence, it is expected that the
MoS2 system will have a relatively low electron and hole effective mass along
the high symmetry path ($\Gamma$-M-K-$\Gamma$) as compared to that of C2N
monolayer. Therefore, the electron and hole mobilities of MoS2 would be larger
than that of the C2N monolayer, since the effective mass is inversely
proportional to the carrier mobility. Figure 2e illustrates the electronic
structure of C2N/MoS2 heterostructure which preserves the direct band gap
behaviour of the isolated monolayers. The VBM is situated close to the Fermi
level suggesting that the charge carriers are of p-type. However, the value of
the energy gap is comparatively reduced and becomes 1.353 eV. We note that the
ideal band gap for a semiconductor to use more visible light is around 1.5 eV
[20].

Figure 2: Electronic band structure of (a) C2N, (b) MoS2 monolayer, (c)
C2N/MoS2 heterostructure, (d-f) The total and projected density of states for
the corresponding system calculated using PBE functional. The dashed line in
each figure indicates the Fermi level.

It is well known that the PBE functional severely underestimates the energy
gap of the system. Again calculations involving the hybrid density functional
like HSE-06, B3LYP, etc are highly computationally expensive which is beyond
the computational resources available to us. Therefore we have further
investigated the systems with SCAN meta-GGA functional. The obtained results
are consistent with the experiment as well as previously reported simulated
results. The contribution of different orbitals at the band edges is evident
from the total and orbital projected density of states analysis as given in
Fig. 2b-f. From Fig. 2b, it is clear that the valence band of C2N is dominated
by the C ‘2p’ states with a significant contribution from N ‘2p’ states as
well. The CBM minimum is also populated by the hybridization of the ‘p’ states
of C and N atoms. The partial DOS analysis of MoS2 establishes that the VBM is
highly dominated by Mo ‘4d’ states with a relatively less population than S
‘3p’ states. On the other hand, the CBM is mainly populated by Mo ‘4d’ states.
The projected DOS for the heterostructure indicates that the VBM is populated
by the states from MoS2 and the CBM is dominated by the C2N.

###

It has been established that strain plays a significant role in efficiently
tuning the electronic, optical, and photocatalytic properties of van der Waals
heterojunctions. In this section, we studied the effect of vertical strain by
changing the interlayer distance between the C2N and MoS2 monolayers. However,
we would like to mention that experimentally, the interlayer distance can be
effectively varied by changing pressure with a scanning tunneling microscopy
tip [21], through vacuum thermal annealing [22], inserting a dielectric BN
layer inside the van der Waals gap of the heterostructure [23], using diamond
anvil cells [24]. The vertical strain ($\Delta D$) can be defined as $\Delta
D=\frac{d-d_{0}}{d_{0}}\times 100$, where $d_{0}$ and $d$ are the interlayer
separation between C2N and MoS2 under equilibrium and strained configurations,
respectively. The effect of vertical strain on the band gap of the
heterostructure is given in Fig. 3a. It is observed that with an increase in
tensile strain along the ‘z’ direction, the band gap value shows an increasing
trend and exhibits a linear variation. Instead, when the compressive strain
between the layer is increased, the band gap decreases and follows a linear
relationship with the direction of applied strain. The band evolution is
further analysed through the strain-dependent projected density of states
(PDOS) as shown in supporting information S1. It is clear that the compressive
strain increases the population of Mo ‘4d’ states for the VBM near the Fermi
level. The CBM is mainly composed of a mixture of ‘p’ states of C2N and MoS2.
With an increase in tensile strain both Mo ‘4d’ and C2N, MoS2 ‘p’ states move
away from the Fermi level resulting in a larger band gap. The band gap
calculation is further analysed using the meta-GGA SCAN functional which
indicates an enhanced band gap value as compared to the PBE result for the
entire range of compressive and tensile strains considered. The band gap
variation for the compressive strain shows a nearly linear behaviour
consistent with the PBE result. Similarly, for the tensile configuration, the
energy gap exhibits an increase in value up to 3% strain and then remains
constant with further expansion in interlayer separation.

Figure 3: Band gap tuning of C2N/MoS2 heterostructure under (a) vertical, (b)
uniaxial, and (c) biaxial strain using PBE and SCAN functional

The present heterostructure is also investigated with the application of
uniaxial and biaxial strain in order to study the influence of in-plane
orbital overlapping. The PBE result for the band gap evolution as a function
of uniaxial strain is represented in Fig. 3b. It is interesting to note that
the band gap remains nearly linear for larger compressive strain. As the
compressive strain decreases from -5% to -1% the band gap shows a mild
increase in value. The application of tensile strain along the uniaxial
direction shows a different trend as compared to the compressive one. The band
gap shows a systematic decrease in value with the increase in tensile strain
from 0 to +5%. The reduction in band gap may be due to the decrease in orbital
overlapping between the ‘p’ states of C and N and the ‘4d’ states of Mo. The
corresponding partial density of state (See supporting information Figure S2a,
b) suggests that tensile strain increases the DOS near the Fermi level for Mo
‘4d’ and C, N ‘p’ states. Similarly, the band gap study was also executed by
applying biaxial strain from -5% to +5% range. The system shows a completely
different behaviour as compared to the result for vertical strain. The linear
variation of the band gap for the compressive strain region remains preserved
as similar to the case of uniaxial strain with a linear variation with a
slowly decreasing trend. However, if we apply tensile strain then the band gap
decreases much more rapidly. The corresponding PDOS analysis indicates that
for -5% compressive strain, the VBM is mainly composed of C and N ‘2p’ states,
and CBM is populated by ‘p’ states of N atoms. When the strain increases from
-5% to +5%, Mo ‘4d’ states become the highest occupied band near the Fermi
level. The CBM comes closer to the Fermi level and resulting in the reduced
band gap. The band gap calculation is further analysed using SCAN functional
for the uniaxial and biaxial strain configuration which provides an enhanced
band gap value as compared to the PBE functional. However, the trend in
variation of the band gap remains almost the same (Fig. 3b, c).

The essential requirement for any material for efficient photocatalytic water
splitting application is to identify appropriate band edge positions relative
to the hydrogen evolution reaction (HER) and oxygen evolution reaction (OER)
potentials of the water. The CBM of the material should lie above the
reduction reaction potential and the VBM should lie below the oxidation
reaction potential of water. To estimate the band edge positions relative to
the oxidation and reduction potential, the absolute energy position of the CBM
and VBM are calculated with respect to the vacuum level. The vacuum level for
the material is obtained by calculating the local potential and then taking
the average of the constant potential region. In the present study, we have
investigated the photocatalytic activity of the C2N/MoS2 heterostructure by
applying strain in the uniaxial, biaxial, and vertical directions. We note
that the reduction potential (E${}_{H^{+}}/$H2) and oxidation potential
(E${}_{O_{2}}/$H2O) of water with respect to vacuum level are -4.44 and -5.67
eV, respectively [25]. The band alignment is then calculated by plotting the
appropriate band edge position with respect to the vacuum level as a function
of strain.

Figure 4: Band edge position of C2N/MoS2 heterostructure as a function of
vertical strain with respect to vacuum potential using (a) PBE and (b) SCAN
functional. The horizontal solid lines represent the water redox potentials

The calculated band edge position of the CBM and VBM under the influence of
vertical strain, using PBE functional is presented in Fig. 4a. It was observed
that without applying any strain, the heterostructure exhibits band edge
positions within the water redox potentials of the water. This result
indicates that the sample is not useful to carry out photocatalytic water
splitting. The sample is subjected to compressive (tensile) vertical strain by
reducing (increasing) the interlayer separation between the C2N and MoS2
monolayers, respectively. It was observed that by applying compressive strain,
the band edge positions got improved as compared to the zero strain case.
However, both the CBM and VBM lie within the water redox potential values.
Applying tensile strain, the VBM and CBM move closer to the water redox
potential values but still lie below them. Therefore the PBE result indicates
that the application of -5 to +5% vertical strain does not enhance the band
edge positions to be useful for photocatalytic water splitting. However, we
would like to mention that PBE functional severely underestimates the band gap
and band edge positions. Therefore the calculations are further executed using
meta-GGA SCAN functional (Fig. 4b). The result indicates a significant
improvement as compared to that of the PBE functional. It was observed that,
without applying any strain, the VBM comes below the oxidation reaction
potential, whereas the CBM moves up but still lies below the reduction
potential value. The application of compressive strain up to -2% positions the
VBM well below the oxidation potential of water but the CBM still lies below
the reduction potential. On further increase in the compressive strain from -3
to -5%, both the CBM and VBM lie outside the range of water redox potentials
hence enhancing the photocatalytic activities of the sample. When the
heterostructure is subjected to vertical tensile strain from +1% to +5%, both
the CBM and VBM straddle the water redox potential range hence enhancing the
photocatalytic response under visible light. There fore we conclude that the
present heterostructure exhibits efficient charge separation for
photocatalytic water splitting under tensile strain and larger compressive
strain.

The sample is further subjected to uniaxial and biaxial strain to analyse the
photocatalytic performance. The uniaxial strain is applied by increasing or
compressing the in-plane lattice constant along the X- direction keeping the Y
value fixed. Similarly, the biaxial strain is applied by increasing or
decreasing the X-Y lattice constant values simultaneously. It is to be noted
that the interlayer separation is kept at its equilibrium value i.e 3.373 Å.
The detailed graphical representation of band alignment under uniaxial strain
is given in the Supporting information S3 a. It was observed that for PBE
functionals the CBM and VBM both lie within the redox potential range for the
entire range of uniaxial strain from -5 to +5%, thus indicating that uniaxial
strain does not produce a significant improvement to be useful for
photocatalytic water splitting.

To get a better result, the strain calculation is repeated for the
heterostructure using SCAN functional and is given in Supporting information
S3b. It was observed that the application of larger uniaxial strain (-3 to
-5%) puts the CBM and VBM outside the water redox potential range. But the
tensile strain reduces the band edge separation even below the unstrained
case. The VBM lies below the oxidation potential value but the CBM finds its
position below the reduction potential. Therefore we found that the sample can
be useful for photocatalytic water splitting for larger compressive uniaxial
strain whereas the tensile strain does not produce an enhancement in the
photocatalytic response. The present C2N/MoS2 heterostructure is also
investigated under biaxial strain using both PBE and SCAN functionals (See
Supporting information S4). The results are quite similar to that of the
uniaxial strain. Like previous results, PBE functional does not produce a
better result for the photocatalytic activity. A larger compressive strain
puts the CBM above the reduction potential but the VBM lies above the
oxidation potential of water, thus not providing a useful result for our
purpose. Using SCAN functional, it was observed that from -2 to -5%
compressive strain both the CBM and VBM straddle outside the water redox
potential. On the other hand, tensile strain decreases the band edges position
and put the CBM below the reduction potential of water. Therefore, the
heterostructure can be used for photocatalytic studies for larger uniaxial and
biaxial compressive strains.

Figure 5: Charge density difference of C2N/MoS2 heterostructure as a function
of vertical (a) -5% compressive (b) 0% (c) +5% tensile strain

In order to illustrate the charge transfer process between the C2N and MoS2
monolayers during the formation of the heterostructure, the charge density
difference is executed using PBE functional. The charge density difference is
calculated by subtracting the charge density of the individual C2N and MoS2
monolayers from the C2N/MoS2 heterostructure.The charge density difference for
the unstrained heterostructure is illustrated in Fig. 5b. In all our charge
density figures, the cyan color indicates the charge depletion and the yellow
color represents the charge accumulation process. From Fig. 5b, we observe
that the charge accumulation mainly occurs in the interface region and
partially in MoS2 monolayers, whereas most of the charge depletion occurs in
the top C2N and bottom MoS2. The change in charge density under vertical
strain is given in Fig. 5.

From the Fig.5a, we observe that with an increase in compressive vertical
strain by 5%, there is an equal proportion of charge accumulation and
depletion close to the MoS2 layer whereas the charge depletion mainly occurs
in the region close to C2N layer. The charge density given in cyan color near
the C atom in the C2N monolayer indicates a charge depletion. Similarly, the
charge density given in yellow color near the S atom in the MoS2 layer
indicates a charge accumulation. Hence this observation indicates that there
is a charge transfer from the S atom of MoS2 to the C atom of C2N monolayer.
The intensification of the charge transfer process under large compressive
strain indicates an enhanced interaction between the two monolayers. The
strain-induced charge transfer in C2N based heterostructures is consistent
with other literature reports [26, 27, 28]. When the interlayer separation in
increases to achieve a strain around +5%, the charge depletion from the C2N
layer almost disappears and most of the charge accumulation occurs in the
interface region close to MoS2. In other words, there is no appreciable charge
redistribution close to the MoS2 layer. This may be the influence of the van
der Waals gap between the two monolayers which varies as the structure moves
from a compressive strain state to a tensile strain state.

The effect of uniaxial compressive and tensile strain on the charge transfer
process is illustrated in supplementary information Fig. S5. It indicates that
there is no appreciable change in the charge density difference under uniaxial
compressive and tensile strain (up to 5%). On the other hand application of
biaxial strain indicates a significant influence on the charge density under
compressive and tensile strains. From the Supplementary information Fig. S6,
we observe that as the system is subjected to -5% compressive strain the
charge depletion occurs in the C atom of the top layer. With system changes
from large compression to large tensile strain state, the depletion near the C
atom increases indicating more charge transfer from the S atom of MoS2 to the
C atom of C2N layer. Therefore we observe that both vertical and biaxial
strain affects the charge transfer process significantly as compared to the
uniaxial strain state.

##  3 Conclusion

In summary, we have studied the photocatalytic performance of C2N/MoS2
heterostructure as a function of uniaxial, biaxial, and vertical strain
configuration through first-principles DFT calculations. The unstrained
heterostructure possesses a direct band gap of 1.35 eV, where the VBM is
populated by Mo ‘d’ states and the CBM is contributed by ‘C’ and ‘N’ ‘p’
states. The calculated position of CBM and VBM of individual monolayers
indicates that the heterostructure possesses a type-II band alignment which is
beneficial for charge separation across the two monolayers and preventing
their recombination process. The SCAN functional provides better results for
the band gap and band edge positions calculation as compared to that of the
PBE result. Using the C2N/MoS2 heterostructure as a prototype, we have also
found that the meta-GGA SCAN functional shows similar results as compared to
the computationally expensive hybrid HSE functionals. The estimated CBM and
CBM position as a function of vertical strain indicates that the water
reduction and oxidation potential values lie within the band gap region with
respect to the vacuum level for larger compressive and tensile strain. In
contrast, the system exhibits good photocatalytic performance only for larger
compression for uniaxial and biaxial strain states whereas the tensile strain
reduces the separation between the VBM and CBM within the water redox
potential value. The charge density difference indicates a significant charge
transfer for vertical and biaxial configuration as compared to the uniaxial
state. The present study can help researchers to reduce the computational cost
by considering the meta-GGA functionals over HSE for electronic structure
calculations of similar systems. Our calculation will also be extremely useful
for designing artificial strained heterostructure for the experimental
community for better device application for photocatalytic water splitting.

##  4 Computational Methods

The first principles electronic structure calculations were performed using
density functional theory (DFT) with projector augmented-wave method [29] as
implemented in the Vienna ab initio simulation package (VASP) [30]. The
Perdew-Burke-Ernzerhof (PBE) [31] parametrization-based generalized gradient
approximation (GGA) was chosen for the exchange-correlation functional. To
accurately describe the interaction between the layered structures, we have
included the van der Waals correction method (DFT-D2) presented by Grimme
[32]. Initially, the gap between C2N and MoS2 monolayers is kept at 3.5 Å
which is further optimized before running the electronic structure
calculation. As a benchmark, the system is further studied using strongly
constrained and appropriately normed (SCAN) meta-GGA functionals to get more
accurate results as compared to the PBE functional and also to be consistent
with the reported experimental data. A vacuum layer of 20 Å was selected along
the ‘Z’ direction to avoid interaction between the adjacent layers. The plane
wave expansion cut-off was chosen to be 520 eV. The Brillouin zone integration
was performed using a $\Gamma$-centered (9$\times$9$\times$1) k-mesh for the
structural relaxation and electronic structure calculation of the isolated C2N
and MoS2 monolayers. For the C2N/MoS2 heterostructure containing 354 atoms,
the geometry optimization and electronic structure calculation are performed
using a $\Gamma$ centered (2$\times$2$\times$1) k-point samplings. All the
structures are allowed for relaxation to get the optimized atomic position
until the total forces acting on each atom are less than 0.02 eV/Å.

## Data availability statement

The additional data that support the findings of this article are available in
the Supplementary Information.

## Acknowledgments

The authors would like to thank the National Institute of Science Education
and Research (NISER), Department of Atomic Energy, Government of India, for
funding the research work through project number RIN-4001. The authors
acknowledge the high-performance computing facility at NISER.

## Data availability statement

The additional data that support the findings of this article are available in
the Supplementary Information.

## Conflict of interest

The authors have no conflicts to disclose.

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