Original Article Commentary Kendrick Kay PhD, Center for Magnetic Resonance
Research, Department of Radiology, University of Minnesota, Minneapolis, MN,
USA<EMAIL_ADDRESS>G.B. was supported by the National Science and
Engineering Research Council (NSERC, Canada).

# Appreciating the variety of goals in computational neuroscience

Konrad P. Kording PhD  Departments of Bioengineering and Neuroscience,
University of Pennsylvania, Philadelphia, PA, USA  Gunnar Blohm PhD  ViKinG
Lab, Centre for Neuroscience Studies, Queen’s University, Kingston, ON, Canada
Paul Schrater PhD  Departments of Psychology and Computer Science, University
of Minnesota, Minneapolis, MN, USA  Kendrick Kay PhD  Center for Magnetic
Resonance Research, Department of Radiology, University of Minnesota,
Minneapolis, MN, USA

###### Abstract

Within computational neuroscience, informal interactions with modelers often
reveal wildly divergent goals. In this opinion piece, we explicitly address
the diversity of goals that motivate and ultimately influence modeling
efforts. We argue that a wide range of goals can be meaningfully taken to be
of highest importance. A simple informal survey conducted on the Internet
confirmed the diversity of goals in the community. However, different
priorities or preferences of individual researchers can lead to divergent
model evaluation criteria. We propose that many disagreements in evaluating
the merit of computational research stem from differences in goals and not
from the mechanics of constructing, describing, and validating models. We
suggest that authors state explicitly their goals when proposing models so
that others can judge the quality of the research with respect to its stated
goals.

###### keywords:

computational neuroscience, metascience, publication criteria

##  1 Introduction

###  1.1 Diversity of modeling goals

Models are essential for progress in neuroscience and exist in a variety of
forms and flavors. Here we will follow the definition of ‘computational model’
in Merriam-Webster Dictionary [50]: “a system of postulates, data, and
inferences presented as a mathematical description of an entity or state of
affairs”. Within neuroscience, such models come in many flavors. Models can
summarize existing data. They can jointly describe brain and behavior data.
They can express relations that can be tested in experiments and predict
successful clinical treatments. Formulating a concrete model can help uncover
hidden assumptions and help assess the suitability of hypothesized
relationships. Formulating models can provide mathematical insights and
simulating them can lead to systems that solve real-world problems.
Accordingly, there is a large community of neuroscientists who construct and
use models.

As computational neuroscientists, we became interested in the goals of
modeling when we noticed stark differences across models in different papers
and fields of neuroscience. For example, when Kendrick studies nonlinearities
in the human brain, he cares most about macroscopic measurements and model
interpretability [29]. When Paul studies representations decoded from the
brain, he cares most about interpretability and representations [11]. When
Gunnar writes a paper about linear-systems explanations of eye movements, he
cares most about behavior, mathematical simplicity, and the real-world
relevance of the task [40]. One might suspect that these differences in goals
stem merely from differences in modeling methodology. However, when Konrad
writes a paper using the same methodology as Gunnar (i.e. linear systems), he
cares most about the model being the optimal solution to a computational
problem [32]. Thus, there appears to be a diversity of modeling goals with
real impact on the way we organize our research.

Despite this diversity, outsiders often perceive computational neuroscience as
being homogeneous. What unites computational neuroscience is a commitment to
an approach that combines mathematical reasoning with computer simulations.
However, this approach is applied across a broad array of topics and within
each topic, researchers strive to achieve distinct goals. At Society for
Neuroscience, computational approaches are often corralled out of ease into a
single section despite differing goals. When experimentalists add computation
to papers or grants, they often do so without choosing a goal first. Young
scientists declare they want to do computation without first committing to a
goal. And lastly, when neuroscientists (admittedly insiders) write books, they
tend to merge computational approaches, despite vastly differing goals. This
creates the false illusion of homogeneity of a field, whereas we believe that
computational neuroscience is, rather, the accumulation of the computational
branches of many different fields.

The diversity of goals within computational neuroscience is not without
consequence: we propose that many disagreements in evaluating the merit of
computational research stem from differences in goals and not from the
mechanics of constructing, describing, and validating models. Goals affect the
way science is reviewed. They form key criteria [7, 44] that inform both
reviewers’ and editors’ decisions. Goals are implicitly invoked when consuming
and evaluating research, and therefore impact an article’s likelihood of
success. Across several disciplines, both meta-analyses and editorial comment
[8, 10, 42, 48] provide evidence that editors’ and reviewers’ preferred goals
are criteria to which authors must conform for success111Examples of these
preferences are not hard to find. For example, “Research doesn’t add value to
the journal. Sometimes the findings of a research aren’t appealing to the
journals, especially if those findings do not really contribute to any
advancement in their field. If this is the case, it’s likely that the paper
would be rejected.” [38], and “It’s boring. … The question behind the work is
not of interest in the field. The work is not of interest to the readers of
the specific journals.” [48]. Editors reject theory papers if they do not
directly explain empirical data. For example, at _PLoS Computational Biology_
, the criterion “Significant biological insight and general interest to life
scientists” often excludes theory papers that do not prioritize biological
realism.. In our considerable experience as editors, we find that disagreement
regarding what constitutes a worthwhile goal for modeling is one of the main
drivers of paper rejections. We believe the problem is that editors’ and
reviewers’ preferred goals are implicit. By making modeling goals explicit,
authors, reviewers, and editors can start to find common ground for the merits
of a paper.

###  1.2 A short list of modeling goals

To the extent that the goals we choose for modeling matter, an important open
question is: what exactly are these goals? Examining a broad range of papers
in computational neuroscience, we gleaned a variety of different modeling
goals, typically revealed in the Introduction, Methods, and Discussion
sections [7]. While it is impossible to produce an exhaustive list, we compile
here a list of the most salient and common ones [44].

  * •

Useful (can be applied to other domains). Some models of the nervous system
are also good at solving real-world problems. Models can be evaluated in terms
of how good they are at solving such problems. For example, a model of the
visual system might be able to solve challenging problems in computer vision
[21, 46]. This assumes that the modeled system in the brain is solving a
problem that also appears in technical systems.

  * •

Normative (best possible given certain assumptions). Some models provide the
optimal solutions to problems that exist in the real world [14, 13, 30, 49].
Models can be evaluated in terms of how well they represent an optimal
solution to a meaningful problem. Normative models are thus often used in
domains where behavior or neural properties are expected to be optimal or near
optimal [1, 17, 31]. For example, a model may ask how well people minimize
energy when walking [45]. Thus, we might ask whether a model supplies the
optimal solution to a computational problem faced by the brain and how similar
behavior is to these predictions. A normative model can also ask whether the
assumed principles underlying the optimality criterion are biologically
accurate. This assumes that we can understand the goals of a system and that
we gain insight if a system appears to optimize what it is expected to
optimize [4, 37].

  * •

Clinically relevant (helps healthcare). Some models produce insights that are
relevant for developing or evaluating clinical interventions. Models may be
evaluated in terms of how well they generalize to medical problems. For
example, simulating individual differences with respect to electrical
stimulation enables us to place electrodes to maximize stimulation outcome
[3]. Given the potential to reduce human suffering, there is no doubt that
clinical relevance is a meaningful goal. In order for modeling insights to
transfer to medicine, a model must be sufficiently similar to the real system.

  * •

Inspire experiments (untested assumptions, new hypotheses). Some models change
the way we think about a problem and thereby raise interesting new hypotheses
via abductive inference [28, 35]. Models can be evaluated in terms of the
richness of potential experiments they inspire. For example, a model may
suggest that spike timing may affect plasticity and therefore lead to a broad
set of tests [16, 23]. A formal model might also uncover hidden assumptions
that a field makes when considering a proposed mechanism. To inspire
experiments, a set of potential models must be small enough such that
experimental tests are meaningful.

  * •

Microscopic realism (looks like the brain). Some models describe the
microscopic properties of the brain, such as synaptic, pharmacological, and
cellular-level properties. Models can then be evaluated in terms of how well
they quantitatively describe those properties. For example, models may predict
changes in synapses over time [52]. Commitment to microscopic realism assumes
that microscopic properties can be sufficiently decoupled from macroscopic
properties such that a reductionist understanding of neural properties is
possible [24].

  * •

Macroscopic realism (looks like the brain at the population level). Some
models describe properties of brain areas and networks. Models can then be
evaluated in terms of how well they quantitatively describe those properties.
For example, models may predict the population activity of brain areas as
measured by EEG [2]. Commitment to macroscopic realism assumes that
macroscopic properties can be sufficiently decoupled from finer-scale,
distributed properties [5].

  * •

Behavioral realism (looks like real behavior). Some models can faithfully
describe and explain behavioral phenomena. Models can then be evaluated in
terms of how well they quantitatively account for behavior. For example,
models can predict the way we move our arm as a function of distance we need
to travel [26]. An approach based on behavioral realism supposes that behavior
can be understood without a deeper understanding of the brain and that compact
models of behavior are possible [25, 33, 47].

  * •

Representational (codes like the brain). Some models aim to use
representations of information that are similar to representations in the
brain. Models can then be evaluated in terms of how well they quantitatively
describe representations. For example, models predict that neurons in motor
cortex have cosine tuning [39]. Such modeling assumes that representations can
be compactly understood and are the basis of the phenomena we want to
understand [15].

  * •

Compact (few short equations). Some models can be succinctly expressed in
mathematical language and/or computer code [9, 34]. Models can then be
evaluated in terms of how well they trade off complexity against the quality
of description of the phenomena. For example, Fitt’s law can compactly
describe the balance between speed and precision during hand movements [19].
This approach assumes that the phenomenon of interest has a low-complexity
description [9].

  * •

Analytically tractable (exact solutions exist). Some models are understandable
through mathematical equations as opposed to numerical simulations. Models can
then be evaluated in terms of how well they can be analytically solved. For
example, models may allow the combination of cues with neurally realistic
properties while being analytically solvable [36]. For scientists with
mathematical training, an analytic approach provides a more generalizable
understanding compared to numerical models. An implicit assumption is that the
system of interest is sufficiently similar to the analytically tractable model
such that analyzing one provides insights into the other [41].

  * •

Interpretable (relates directly to something the brain does). Some models are
easily interpreted with respect to how they work (e.g. what outcomes they
predict) and/or how the brain might implement the computations. Models can
then be evaluated in terms of how well humans can interpret their meaning. For
example, units in a simulated system may have receptive fields similar to
those of real neurons [6, 39]. For many scientists, prioritizing the
interpretability of a model makes the model more relatable to their way of
thinking about the brain.

  * •

Beauty (elegant). Some models may be symmetrical, balanced, or resonate well
with the way we think. Models can then be evaluated in terms of how well they
resonate intuitively with their target audience. For example, the same model
can be presented in the languages of physics, math, and biology, and can be
distinctly useful for these different communities [12, 43].

##  2 Methods

To assess modeling goals in the computational neuroscience community, we
constructed an online survey using Google Forms. Each of the authors then
contacted colleagues via personal e-mails, mailing lists, and Twitter. We
collected survey responses for approximately a month, with a survey deadline
of August 31, 2018. We told participants that we would be releasing the
responses from this survey as a public resource (with the exception of e-mail
addresses, which would be kept private). People contacted were free to decline
participation in the survey. Only adult scientists were allowed to
participate. The research was approved by the UPenn IRB (Protocol number
830156).

Figure 1: Survey demographics. Shown are counts of survey participants binned
by career stage (left) and counts of papers rated by survey participants
binned by journal (right).

The survey asked each participant to choose up to 3 papers they authored or
co-authored and to rate each paper on the 12 modeling goals described above.
Participants were instructed to submit papers representative of distinct types
of their research. The full set of survey questions and survey results are
available at https://osf.io/pqe7f/. We note that the survey results may be
useful for answering a variety of additional questions not addressed in this
paper. For example, one might be interested to compare one’s prediction of the
modeling goals held by a given researcher to the actual goals held by that
researcher. Or as another example, one might be interested to see where one’s
goals fall relative to the group norms.

For the analyses performed in this paper, ratings were aggregated across
papers (251 papers from 113 distinct authors; 22 female, 91 male). For Figure
2B, a small amount of Gaussian noise (mean 0, standard deviation 0.5) was
added to the data prior to computing summary statistics in order to avoid
discretization effects.

##  3 Results

###  3.1 A simple survey provides evidence computational researchers have
diverse goals

To empirically assess modeling goals, we conducted an informal online survey
in which we asked authors to rate their own modeling work with respect to the
goals listed above. Participants rated up to 3 of their authored papers on
each of the 12 goals, indicating the importance of each goal. We obtained
results from 113 distinct authors who rated a total of 251 papers (Figure 1).
On average, interpretability was rated as the most important modeling goal,
whereas clinical relevance was rated as least important (Figure 2B, black
bars). In addition, we found large variance of ratings across papers (Figure
2B, gray error bars), suggesting that there is, indeed, wide diversity of
modeling goals in the neuroscience community. Not surprisingly, some goals are
highly correlated (Figure 2C), such as compactness and tractability.

To better understand the underlying structure of the ratings, we subtracted
the average rating of each modeling goal and identified a lower-dimensional
space using probabilistic principal components analysis (Figure 3A). We
reconstructed the data in this lower-dimensional space and recomputed the
pairwise correlation structure (Figure 3B). Finally, we re-ordered the
modeling goals, revealing three groups or clusters (Figure 3C). One simple
interpretation of these clusters is that people are sampling independently
mixed contributions from three clusters that somewhat overlap with the
intuitive grouping of _Scientific impact_ , _Biological realism_ , and _Style_
(as shown in Figure 2A). These differences might represent different subfields
of neuroscience having different modeling goals and/or different types of
models naturally fulfilling certain goals more readily than others. However,
this lower-dimensional reconstruction accounted for only 51% of the total
variance in goal ratings. The remaining 49% of the variance reflects diversity
of individual preferences in goals. Thus, just like the contrast between
Gunnar’s and Konrad’s linear-systems models, the variability in modeling goals
between researchers appears to be high.

Figure 2: Modeling goals in the computational neuroscience community. We
conducted an informal survey on the Internet to assess the modeling goals that
different researchers held for specific papers that they authored. (A)
Histogram of results. For each of 12 modeling goals (dimensions), we plot a
histogram of the reported ratings. Values range from 1 (completely irrelevant)
to 5 (absolutely essential). Dimensions are ordered according to three
conceptual groups that seem intuitively reasonable. (B) Summary statistics.
For each dimension, we plot the median (black bars), interquartile range (gray
error bars), and bootstrapped 68% confidence interval on the median (red error
bars). (C) Pairwise correlation (Pearson’s _r_) of dimensions across all
papers.

To further illustrate diversity, we visualize the location of each paper in
the space spanned by the first three principal components (Figure 4). This
shows again that despite dimensionality reduction accounting for roughly half
of the variance, there are no discernible clusters or groups in this subspace.
In other words, there is a continuum of preferences with respect to modeling
goals that span the space. Highlighting the authors’ own papers in Figure 4
allows several additional observations: (1) Most of the authors’ papers’ goals
are fairly polarized, i.e. they reside at the edges of the space spanned. (2)
Konrad’s and Gunnar’s papers lie in diametrically opposite sides of this space
despite some overlap in modeling techniques used. The same is true for Paul’s
and Kendrick’s papers. (3) Somewhat surprisingly, Paul’s and Gunnar’s goals
(and Kendrick’s and Konrad’s goals) align fairly well despite apparently
different technical approaches used. Note that this diversity in goals and
approaches does not mean that we do not appreciate each other’s research
efforts; quite the opposite! We view this diversity as a strength (see
Discussion).

###  3.2 Limitations of the survey

Of course, this survey is not intended as a formal scientific instrument to
identify modeling goals in the field, but the results do suggest that it would
be worthwhile to invest in more systematic meta-scientific analyses. It would
be valuable to identify modeling goals for a much larger sample of papers and
to calibrate the survey and analysis methods. Our sample is small and not
randomly drawn from the population. The survey questions themselves may not
have been understood in exactly the same way by different participants, which
may have increased the apparent diversity. But the survey is sufficient to
make our point: diversity in modeling goals is real and high!

Figure 3: Groups of modeling goals. We performed probabilistic principal
component analysis to characterize the latent covariance structure of the
survey ratings. (A) Component loadings. The top three components (composing an
orthonormal basis) are shown; the order of dimensions is the same as in Figure
2. (B) Data reconstruction. We reconstruct the original data using the three
identified components and recompute the pairwise correlation as in Figure 2C.
The correlation structure is similar to that of the original data, but
stronger due to the dimensionality reduction. (C) Grouping of dimensions. We
re-plot the results of panel B, re-ordering dimensions to highlight the block
structure. Thick gray squares indicate three groups of dimensions that appear
to be present in the data.

##  4 Discussion

###  4.1 Goals matter to how modeling is done

The choice of goals matters in just about every imaginable way for modeling in
neuroscience. Modeling goals affect the overall utility and interpretation of
a model by influencing the evaluation metrics, the choice of model type, and
the way we replace models with newer, better models. For example, if
microscopic realism is required, this severely constrains the types of
modeling techniques that can be used. Some research fields have implicit
agreements on a set of desirable modeling criteria. For instance,
historically, the eye-movements field has used linear-systems theory to model
saccades: the field has been most concerned with behavioral realism,
usefulness, inspiration of experiments, and interpretability, but has not
placed much value on microscopic realism. If the field had been concerned with
microscopic realism, the linear-systems approach would have likely been
inappropriate and a different toolset—such as spiking neural-network
models—would have been used instead. Thus, getting clarity on _why_ we model
may be just as important as understanding the mechanics of _how_ to model.

Figure 4: Diversity of modeling goals within and across authors. Here we plot
all recorded paper ratings in the space defined by the first three principal
components (see Figure 3A). (A) Space defined by components 1 and 2. Each gray
dot indicates a single paper, and colored markers indicate papers from the
authors. Insets show raw ratings for example papers; in these plots, thin gray
lines indicate the mean of each dimension across all papers. Some authors are
consistent across papers they write (Schrater, Kay), whereas other authors
show more diversity in their papers (Kording, Blohm). Furthermore, there is
high diversity in modeling goals across the four authors of this paper. (B)
Space defined by components 1 and 3. Same format as panel A.

###  4.2 But what are the goals of computational neuroscience?

In shaping our list of goals of computational neuroscience, we drew from our
own experience: this is thus an opinion piece in which we try to provide a
meaningful perspective to the field. We combine 7+ years of teaching students
the broad range of modeling techniques useful in movement science and
extensively discussing modeling objectives at the Summer School in
Computational Sensory-Motor Neuroscience (CoSMo,
http://www.compneurosci.com/CoSMo/) and beyond. That being said, while goals
in neuroscience are diverse as we have shown, we cannot claim that our list is
exhaustive or that other neuroscientists would not structure it differently.
Even if our views are wrong, we hope that our humble paper will jump-start a
drive towards clarity in modeling goals in neuroscience.

###  4.3 Authors should state goals; readers should evaluate based on those
commitments

Why does it matter that different researchers have different modeling goals?
We wish to raise awareness because diversity often leads to significant
tension and misunderstandings between researchers. For example, a reviewer
might have a certain set of goals associated with a particular modeling
approach and might evaluate a given paper outside of the authors’ intentions.
This out-of-scope evaluation is one of the most frequent and frustrating
reasons for miscommunication in computational neuroscience. Its consequence is
often paper rejection. We argue that this behavior is detrimental for science.

What are some practical steps we can take? As action items, we suggest that
(1) authors should explicitly spell out their goals, ideally in multiple
instances across the Introduction, Methods, and Discussion sections [7, 44],
and (2) readers should deliberately evaluate a given paper within those
constraints. For example, an author might include a section that begins, “In
this paper, we sought to satisfy the criteria of X, Y, and Z. It is not our
goal to develop a model that exhibits properties A, B, and C for the following
reasons…” We believe that explicit characterization of goals leads to more
constructive interactions and therefore promotes scientific progress,
discovery, and societal impact.

This paper is itself a (rudimentary) modeling effort: we sought to
characterize the intentions or goals held by computational neuroscientists in
conducting their research and the relationship of these goals to one another.
Thus, in a sense, this is behavioral research on how brains (scientists)
function. With respect to the 12 modeling criteria, we sought to characterize
the phenomenon at the behavioral level, with some interest in identifying the
underlying representations (latent structure). Interpretability of our results
was paramount, and we sought to gather observations that may inspire further
meta-scientific efforts. On the other hand, our investigation is obviously not
intended to understand the macroscopic or microscopic neural mechanisms
underlying how scientists conduct their research. Likewise, several modeling
criteria are clearly not applicable to our study (e.g. normative, clinical).

###  4.4 Diversity of modeling goals is a strength

Modeling aims to generate insight into a phenomenon of interest. Since models
in computational neuroscience all refer to brains, one could argue that they
are guaranteed to produce synergistic answers and that the distinctions
highlighted in this paper are not that important. We think this stance is
debatable. Models positioned at different levels of biological realism
(microscopic, macroscopic, behavioral, representational) are not guaranteed to
inform each other, as distinct phenomena may emerge at different levels.
Models following different styles (compact, tractable, interpretation, beauty)
are optimizing different criteria, so a model that fulfills one criterion may
be suboptimal under other criteria. Models aimed towards a specific type of
scientific impact (useful, normative, clinical, inspire) often fail to deliver
other types of impact. Thus, it is our contention that modeling goals are
truly diverse and that models in computational neuroscience are not aimed
towards a single coherent class of answers.

Should the community attempt to converge on a single set of standards? While
this might seem appealing to some, actual diversity of modeling goals makes it
difficult to find a shared community preference function. From Arrow’s
Impossibility Theorem [18] to Harsanyi’s Aggregation Theorem [20, 27, 51],
arriving at a consistent group preference function is known to be hard, to
require special conditions, and even when possible, to require trading off
individual preferences to allow a non-unique group preference as a point on a
Pareto front. Editor and reviewer preferences for goals are unlikely to
represent such a hypothetical aggregate preference, thus leading to
idiosyncratic critiques [22]. Rather than encourage conformity to particular
preferences, appreciating goal diversity allows exploring possibilities
without getting stuck in local minima. For example, 15 years ago, some
researchers were claiming that working on machine learning was career suicide;
researchers that nonetheless persevered are now superstars in the field.

We advocate embracing diversity in modeling goals as a strength for the field.
As in other aspects of life, humanity works best by respecting and not
excluding diversity. In terms of scientific progress, diversity balances
biases, provides alternative views, encourages discussion, invigorates
problem-solving, and facilitates specialization of individual researchers,
each of whom can make distinct meaningful contributions to the field. Perhaps
one day the neuroscience community will come to consensus on a single
framework for describing and understanding the brain. But until that day
comes, embracing diversity and explicitly recognizing each other’s modeling
goals will be critical for achieving progress.

## acknowledgements

We would like to thank the reviewers and Editor J. Pillow for constructive
feedback.

## conflict of interest

The authors declare no conflicts of interest.

## References

  * Acuña and Schrater [2010] Acuña, D. E. and Schrater, P. (2010) Structure learning in human sequential decision-making.  PLoS computational biology, 6, e1001003.
  * Al-Nashash et al. [2004] Al-Nashash, H., Al-Assaf, Y., Paul, J. and Thakor, N. (2004) EEG signal modeling using adaptive Markov process amplitude.  IEEE transactions on bio-medical engineering, 51, 744–751.
  * Bai et al. [2019] Bai, S., Martin, D., Guo, T., Dokos, S. and Loo, C. (2019) Computational comparison of conventional and novel electroconvulsive therapy electrode placements for the treatment of depression.  European Psychiatry: The Journal of the Association of European Psychiatrists, 60, 71–78.
  * Barlow [1961] Barlow, H. B. (1961) Possible principles underlying the transformation of sensory messages.  Sensory communication, 1, 217–234.
  * Bennett and Hacker [2003] Bennett, M. R. and Hacker, P. M. S. (2003) Philosophical Foundations of Neuroscience, vol. 79.  Blackwell Oxford.
  * Blohm et al. [2009] Blohm, G., Keith, G. P. and Crawford, J. D. (2009) Decoding the cortical transformations for visually guided reaching in 3D space.  Cerebral Cortex, 19, 1372–1393.
  * Blohm et al. [2018] Blohm, G., Kording, K. P. and Schrater, P. R. (2018) A how-to-model guide for Neuroscience.  OSF Preprints.
  * Bornmann et al. [2010] Bornmann, L., Weymuth, C. and Daniel, H.-D. (2010) A content analysis of referees’ comments: How do comments on manuscripts rejected by a high-impact journal and later published in either a low-or high-impact journal differ?  Scientometrics, 83, 493–506.
  * Burgess [1998] Burgess, J. (1998) Occam’s razor and scientific method.  In The Philosophy of Mathematics Today, 195–214. Clarendon Press Oxford.
  * Byrne [2000] Byrne, D. W. (2000) Common reasons for rejecting manuscripts at medical journals: A survey of editors and peer reviewers.  Science Editor.
  * Carlson et al. [2003] Carlson, T. A., Schrater, P. and He, S. (2003) Patterns of activity in the categorical representations of objects.  Journal of Cognitive Neuroscience, 15, 704–717.
  * Chandrasekhar [2013] Chandrasekhar, S. (2013) Truth and Beauty: Aesthetics and Motivations in Science.  University of Chicago Press.
  * Chater and Oaksford [1999] Chater, N. and Oaksford, M. (1999) Ten years of the rational analysis of cognition.  Trends in Cognitive Sciences, 3, 57–65.
  * Chater and Oaksford [2000] — (2000) The rational analysis of mind and behavior.  Synthese, 122, 93–131.
  * Churchland and Sejnowski [1990] Churchland, P. S. and Sejnowski, T. J. (1990) Neural representation and neural computation.  Philosophical Perspectives, 4, 343–382.
  * Dan and Poo [2004] Dan, Y. and Poo, M.-M. (2004) Spike timing-dependent plasticity of neural circuits.  Neuron, 44, 23–30.
  * Dayan and Abbott [2001] Dayan, P. and Abbott, L. F. (2001) Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems.  MIT Press.
  * Fishburn [1970] Fishburn, P. C. (1970) Arrow’s impossibility theorem: Concise proof and infinite voters.  Journal of Economic Theory, 2, 103–106.
  * Fitts and Radford [1966] Fitts, P. M. and Radford, B. K. (1966) Information capacity of discrete motor responses under different cognitive sets.  Journal of Experimental Psychology, 71, 475–482.
  * Fleurbaey [2009] Fleurbaey, M. (2009) Two variants of Harsanyi’s aggregation theorem.  Economics Letters, 105, 300–302.
  * Fukushima [1980] Fukushima, K. (1980) Neocognitron: A self organizing neural network model for a mechanism of pattern recognition unaffected by shift in position.  Biological Cybernetics, 36, 193–202.
  * Garfunkel et al. [1990] Garfunkel, J. M., Ulshen, M. H., Hamrick, H. J. and Lawson, E. E. (1990) Problems identified by secondary review of accepted manuscripts.  JAMA, 263, 1369–1371.
  * Gerstner et al. [1996] Gerstner, W., Kempter, R., van Hemmen, J. L. and Wagner, H. (1996) A neuronal learning rule for sub-millisecond temporal coding.  Nature, 383, 76–81.
  * Gillett [2016] Gillett, C. (2016) Reduction and Emergence in Science and Philosophy.  Cambridge University Press.
  * Green et al. [2010] Green, C. S., Benson, C., Kersten, D. and Schrater, P. (2010) Alterations in choice behavior by manipulations of world model.  Proceedings of the National Academy of Sciences of the United States of America, 107, 16401–16406.
  * Harris and Wolpert [1998] Harris, C. M. and Wolpert, D. M. (1998) Signal-dependent noise determines motor planning.  Nature, 394, 780–784.
  * Harsanyi [1979] Harsanyi, J. C. (1979) Bayesian decision theory, rule utilitarianism, and Arrow’s impossibility theorem.  Theory and Decision, 11, 289–317.
  * Josephson and Josephson [1996] Josephson, J. R. and Josephson, S. G. (1996) Abductive Inference: Computation, Philosophy, Technology.  Cambridge University Press.
  * Kay and Yeatman [2017] Kay, K. N. and Yeatman, J. D. (2017) Bottom-up and top-down computations in word- and face-selective cortex.  eLife, 6.
  * Knill and Richards [1996] Knill, D. C. and Richards, W. (1996) Perception as Bayesian Inference.  Cambridge University Press.
  * Körding [2007] Körding, K. (2007) Decision theory: What "should" the nervous system do?  Science, 318, 606–610.
  * Kording et al. [2007] Kording, K. P., Tenenbaum, J. B. and Shadmehr, R. (2007) The dynamics of memory as a consequence of optimal adaptation to a changing body.  Nature Neuroscience, 10, 779–786.
  * Krakauer et al. [2017] Krakauer, J. W., Ghazanfar, A. A., Gomez-Marin, A., MacIver, M. A. and Poeppel, D. (2017) Neuroscience Needs Behavior: Correcting a Reductionist Bias.  Neuron, 93, 480–490.
  * Li and Vitányi [2019] Li, M. and Vitányi, P. (2019) An Introduction to Kolmogorov Complexity and Its Applications, vol. 3.  Springer.
  * Lombrozo [2012] Lombrozo, T. (2012) Explanation and abductive inference.  In Oxford Handbook of Thinking and Reasoning, 260–276. Oxford University Press.
  * Ma et al. [2006] Ma, W. J., Beck, J. M., Latham, P. E. and Pouget, A. (2006) Bayesian inference with probabilistic population codes.  Nature Neuroscience, 9, 1432–1438.
  * Mayr [2004] Mayr, E. (2004) What Makes Biology Unique?: Considerations on the Autonomy of a Scientific Discipline.  Cambridge University Press.
  * Mukherjee [2018] Mukherjee, D. (2018) 11 Reasons Why Research Papers Are Rejected.  https://blog.typeset.io/11-reasons-why-research-papers-are-rejected-3e272b633186.
  * Olshausen and Field [2004] Olshausen, B. A. and Field, D. J. (2004) Sparse coding of sensory inputs.  Current opinion in neurobiology, 14, 481–487.
  * Orban de Xivry et al. [2013] Orban de Xivry, J.-J., Coppe, S., Blohm, G. and Lefèvre, P. (2013) Kalman filtering naturally accounts for visually guided and predictive smooth pursuit dynamics.  The Journal of Neuroscience, 33, 17301–17313.
  * Parker [2012] Parker, W. S. (2012) Computer simulation and philosophy of science.  Metascience, 21, 111–114.
  * Pierson [2004] Pierson, D. J. (2004) The top 10 reasons why manuscripts are not accepted for publication.  Respiratory Care, 49, 1246–1252.
  * Russell [2019] Russell, B. (2019) Mysticism and Logic and Other Essays.  Good Press.
  * Schrater et al. [2019] Schrater, P., Kording, K. and Blohm, G. (2019) Modeling in Neuroscience as a Decision Process.  OSF Preprints.
  * Selinger et al. [2015] Selinger, J. C., O’Connor, S. M., Wong, J. D. and Donelan, J. M. (2015) Humans Can Continuously Optimize Energetic Cost during Walking.  Current biology, 25, 2452–2456.
  * Serre and Riesenhuber [2004] Serre, T. and Riesenhuber, M. (2004) Realistic Modeling of Simple and Complex Cell Tuning in the HMAXModel, and Implications for Invariant Object Recognition in Cortex.  Tech. rep., Massachusetts Institute of Technology.
  * Tao et al. [2018] Tao, G., Khan, A. Z. and Blohm, G. (2018) Corrective response times in a coordinated eye-head-arm countermanding task.  Journal of Neurophysiology, 119, 2036–2051.
  * Thrower [2012] Thrower, P. (2012) Eight reasons I rejected your article.  https://www.elsevier.com/connect/8-reasons-i-rejected-your-article.
  * Todorov and Jordan [2002] Todorov, E. and Jordan, M. I. (2002) Optimal feedback control as a theory of motor coordination.  Nature Neuroscience, 5, 1226–1235.
  * Webster [2016] Webster (2016) The Merriam-Webster Dictionary: International Edition.  Merriam-Webster.
  * Weymark [1993] Weymark, J. A. (1993) Harsanyi’s social aggregation theorem and the weak Pareto principle.  Social choice and welfare, 10, 209–221.
  * Zador et al. [1990] Zador, A., Koch, C. and Brown, T. H. (1990) Biophysical model of a Hebbian synapse.  Proceedings of the National Academy of Sciences, 87, 6718–6722.


# From Structural Stability to Electronic Flexibility: Unveiling Strain-
induced Effects in a MoS2/Perylene Orange Hybrid System

Mohammed El Amine Miloudi  0000-0002-2654-8758 Oliver Kühn*
0000-0002-5132-2961<EMAIL_ADDRESS>University of Rostock,
Institute of Physics, Albert-Einstein-Str. 23-24, D-18059 Rostock, Germany

###### Abstract

This study delves into the interaction between a monolayer of molybdenum
disulfide (MoS2) and a single Perylene Orange (PO) molecule, representative of
inorganic and organic semiconductor materials, respectively. Investigation of
the amalgamation of these materials under mechanical strains reveals
significant alterations in the electronic properties of the MoS2/PO interface.
Tensile strain induces a reduction in the bandgap, while compressive strain
initially engenders an increase, followed by a subsequent decrease. Notably,
MoS2 undergoes a transition from a direct to an indirect bandgap under both
stretching and compression conditions. These alterations stem from shifts in
the density of states and band structure adjustments resulting from lattice
deformations induced by applied strain. Remarkably, under specific compression
conditions, the MoS2/PO system manifests a transition between type II and type
I band alignments. The detailed analysis of a range of strain magnitudes
yields profound insights into the behavior of MoS2 and MoS2/PO systems under
mechanical strain, with potential implications for nano- and optoelectronics
applications.

##  I INTRODUCTION

The field of advanced optomechanical materials has witnessed substantial
growth, particularly with the advent of two-dimensional transition metal
dichalcogenides (2D TMDs) [1, 2, 3]. Among these materials, molybdenum
disulfide (MoS2) stands out due to its exceptional optical and mechanical
properties [4]. Recognized for these properties, MoS2 exemplifies distinctive
features of 2D TMDs, placing them at the forefront of innovation in state-of-
the-art optomechanical devices [3, 4, 5, 6, 7].

An important characteristic of TMDs is their layered structure, having high
mechanical stability such as to allow monolayers to sustain up to 10% strain
[8]. Applying mechanical strain significantly alters the band structure and
optical characteristics of these materials [3, 9]. This renders TMDs
susceptible for applications involving mechanical deformation [4]. In fact,
mechanical deformation has been shown to govern the energies of excitons [3,
5, 10] and phonons [11, 12] in ultrathin semiconductor layers. It leads to
alterations in phonon-mediated exciton relaxation rates, changes in the phonon
energy spectrum, and variations in the strength of exciton-phonon coupling [8,
13].

Experimental efforts have largely focused on the highly symmetrical MX2
family, where ’M’ denotes metals like Mo or W, and ’X’ includes elements such
as Se or S [3, 8, 9]. Previous studies have revealed significant effects upon
application of uniaxial deformation to MoS2 [4, 12, 14], encompassing changes
in electrical mobility, shifts in Raman lines, alterations in
photoluminescence (PL) spectra, electron transport characteristics, variations
in Schottky barrier heights, bandgap tunings, and the emergence of
piezoresistive phenomena [12, 14].

In addition to mechanical strain effects, recent research has delved into the
interaction between transition metal dichalcogenides (TMDs) and organic
molecules, leading to the emergence of hybrid TMD/organic interfaces with
novel optoelectronic properties [15, 16]. These interfaces represent a
synergistic amalgamation of the strengths of both components, offering a
diverse array of benefits across various technological applications [16, 17,
18, 19]. While organic molecules are recognized for their outstanding light-
absorption capabilities, they may confront challenges related to charge
mobility and stability. In contrast, inorganic materials excel in charge
transport but may exhibit limited light-absorption capabilities [16, 17, 18,
20, 21]. The introduction of organic molecules onto TMD surfaces serves to
enhance crucial properties such as charge carrier mobility and PL quantum
efficiency by proficiently passivating defects within the TMD structure and
mitigating disruptive electron screening effects [18, 21]. Furthermore, the
integration of organic dye molecules with MoS2 holds promise in augmenting the
optoelectronic characteristics of photodetectors. Extensive investigations
have explored the synergy between MoS2 and diverse organic molecules,
including perylene derivatives, among others, highlighting the pivotal role of
the hybrid interface for bolstering charge transfer, and modifying
photoemission [16, 21, 22, 23, 24, 25, 26]. This enhancement stems from the
intricate interaction between MoS2 and organic molecules, leading to
improvements not only in structural integrity but also in the overall
optoelectronic performance of the resultant hybrid system. To further
elucidate these advancements, the types of interfaces formed between TMDs and
organic molecules can be distinguished as being of type I or type II band
alignment. Such categorization already provides valuable guidance into the
underlying mechanisms governing charge transfer processes and exciton dynamics
or PL within these hybrid systems.

Perylene Orange (PO), i.e. N,N’-bis(2,6-diisopropyl phenyl)-3,4,9,10-perylene
tetracarboxylic diimide (C48H42N2O4), is a widely studied organic dye molecule
[27, 28, 29, 30]. Its molecular structure (cf. Fig. 3 below) comprises a
perylene tetracarboxylic diimide (PTCDI) core, featuring fused benzene rings
with four carbonyl and two imide groups. N, N’-bis(2,6-diisopropyl phenyl)
substituents are connected to the imide nitrogen atoms, each incorporating an
identical 2,6-diisopropyl phenyl group [28, 31, 32]. The planar and polycyclic
aromatic architecture results in a vibrant coloration, establishing PO as a
highly sought-after pigment and dye with versatile applications [33]. It
exhibits exceptional optical characteristics, including robust absorption and
emission properties within the visible spectrum. Consequently, it emerges as
an optimal choice for applications that require enduring and vivid coloration
[34, 35, 36].

PO utility extends beyond pigmentation to optoelectronics [34, 35]. Its
exceptional properties, including tunable absorption and emission, have
garnered interest for developing organic electronic devices [32, 33, 35, 36].
The unique combination of high PL quantum yield and efficient charge transport
capabilities in PO holds promise for applications such as organic light-
emitting diodes (OLEDs). Perylene diimide (PDI) derivatives, similar to PO,
exhibit high PL quantum yields and have proven effective in OLEDs. N-annulated
PDI with bulky side chains has demonstrated PL quantum yields over 20% in the
solid state and around 80% in solution. These materials have been successfully
used in OLEDs, achieving significant brightness and low turn-on voltages,
making them promising for red-colored OLEDs [37, 38]. Moreover, the rigid
structures of these derivatives prevent aggregation, enabling high PL quantum
yields at large dye loadings, suitable for room temperature polariton lasing
and other advanced optoelectronic applications [38].

In this study, we focused on the MoS2/PO system, elucidating the adsorption
geometry, mechanical stability, and alignment between MoS2 band structure and
PO molecular orbitals. In particular, we investigated the dependence of band
alignment on applied mechanical strain. Our findings confirmed a type II band
alignment in unstrained MoS2/PO. Application of strain is shown to allow for a
precise control of the bandgap and alignment, transitioning to type I under
specific conditions. This control provides direct influence over charge
transfer dynamics and photoemission characteristics. Our work shows the
potential for fine-tuning these attributes in the MoS2/PO system, opening up
potential flexibility in designing hybrid systems for the possibility of
breakthroughs in nano- and optoelectronics.

Figure 1: (a) Top and side views of 2H-MoS2, where large and small spheres
represent Mo and S atoms, respectively. The figure illustrates a supercell
with the unit cell denoted by a black box. The symbols ”a” and ”b” denote the
lattice constants. (b) The isosurface of the electron localization in the
$\langle 001\rangle$ and $\langle 110\rangle$ planes, with a contour interval
of 0.1. (c) Phonon dispersion and density of states (PhDOS), and (d) band
structures obtained using the PBE and HSE06 methods for the MoS2 unit cell
(energies given with respect to the Fermi energy $E_{\rm F}$).

##  II COMPUTATIONAL DETAILS

First-principles calculations were conducted using the Vienna ab initio
simulation package (VASP), employing projector augmented wave (PAW)
pseudopotentials and the Perdew–Burke–Ernzerhof (PBE) exchange-correlation
functional [39, 40, 41]. To prevent interactions between periodic layers, a 20
Å vacuum spacing was included. Additionally, to address the influence of
dipole moments, a $z$-axis dipole correction was used. The kinetic energy
cutoff was set at 450 eV after confirming convergence, and $\Gamma$-centered
grids with dimensions of $17\times 17\times 1$ and $1\times 1\times 1$ were
used for atomic relaxation of the MoS2 unit cell and the MoS2/PO supercell,
respectively. A denser grid was adopted for the density of states (DOS) and
band structure calculations. The electronic convergence criterion was set at
$10^{-6}$ eV, while for geometry optimization the maximum force components
were converged to levels below $10^{-2}$ eV/Å. To account for long-range van
der Waals (vdW) interactions, the DFT+D3 method was incorporated into atomic
relaxation and total energy computations, effectively including pairwise
electron density interactions with empirically determined parameters [42]. The
valence electron configurations considered in these calculations are
$4d^{5}5s^{1}$ for Mo, $3s^{2}3p^{4}$ for S, $2s^{2}2p^{2}$ for C,
$2s^{2}2p^{3}$ for N, $2s^{2}2p^{4}$ for O, and $1s^{1}$ for H.

Electronic band structures of the MoS2 unit cell were performed using both the
PBE [43] and the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional methods,
with a standard mixing parameter of 0.25 employed exclusively for the HSE06
calculations [44]. This comparative calculation was performed to verify the
consistency of the bandgap nature and its position. For the MoS2/PO supercell
only PBE calculations were performed. The MoS2 supercell consists of 275
atoms, while the MoS2/PO system comprises 339 atoms. Note that we consider the
case of low PO coverage only, i.e. a single PO molecule per supercell. The
phonon dispersion was determined utilizing a $2\times 2\times 1$ supercell and
density functional perturbation theory (DFPT) [45].

Molecular orbital and frequency calculations for the isolated PO were
performed using the PBE functional with a 6-31G(d) basis set as as implemented
in the QChem code [46].

##  III RESULTS AND DISCUSSION

###  III.1 Isolated MoS2 and PO

Before presenting an in-depth exploration of the hybrid MoS2/PO system, some
results on the pristine 2H-phase MoS2 monolayer are summarized in Fig. 1.
Panel (a) shows the unit as well as the supercell. Panel (b) provides the
electron localization function (ELF), which is employed to qualitatively
analyze the nature of chemical bonding within the material. The Mo cation
sites exhibit low ELF values, while the S sites display prominent blue annular
regions. This significant contrast in ELF values indicates an ionic nature of
the chemical bonding. However, the bonding also exhibits covalent
characteristics, as evidenced by the non-spherically symmetric high ELF
regions around the S anions, which form lobes directed towards the Mo ions.
Additionally, there are lobes observed between adjacent S anions, both within
the same layer and between two layers. Therefore, the bonding in MoS2 is a
hybrid, consisting of both ionic and covalent components, with a slight
covalent interaction present between adjacent S anions.

In Fig. 1(c) the phonon dispersion along the high symmetry path ($\Gamma-
X-M-\Gamma$) within the Brillouin zone (BZ) is given for the unit cell. We
also show the corresponding phonon density of states (PhDOS). Given these
results, we note that throughout the BZ, no instances of imaginary phonon
frequencies were observed, thereby confirming the dynamic stability of the
model used for the primitive MoS2 [47]. Calculations of bond lengths reveal
Mo-S, Mo-Mo, and $d_{z}$ (S-S) distances measuring 2.41, 3.16, and 3.13 Å,
respectively. These values are in good agreement with previous experimental
and theoretical findings [48, 49, 50, 51, 52].

The electronic band structure of the unit cell is plotted in Fig. 1(d),
confirming the well known direct bandgap at the $K$ point [53, 54, 55]. As far
as the functional dependence is concerned, bandgap values of 1.65 eV and 2.1
eV are obtained for PBE and HSE06, respectively. These findings align
excellently with prior theoretical investigations and provide compelling
evidence of a strong covalent bond formation incorporating the S-3p and Mo-4d
states, thus underscoring the presence of robust hybridization within the
monolayer structure [47, 48, 49, 50, 56].

Figure 2: Comparison of PO HOMO and LUMO for gas phase (energies with respect
to vaccum level) (a) and adsorption (b) geometries. Panel (c) shows the
distribution of vibrational frequency (histogram) of PO at gas phase geometry.

Molecular orbitals of PO in gas phase are shown in Fig. 2. Specifically, panel
(a) and b) show the HOMO (highest occupied molecular orbitals) and LUMO
(lowest unoccupied molecular orbital) for the optimized gas phase geometry and
the geometry adopted upon adsorption (see also Fig. 3). It is noteworthy that
for the adsorption geometry the HOMO-LUMO gap changes marginally from 1.52 eV
to 1.43 eV only, in particular the HOMO becomes distorted and more localized.
In Fig. 2(c) the distribution of vibrational mode frequencies of PO is shown
for the gas phase geometry. Upon adsorption it is to be expected that in
particular the low-frequency part (i.e. below 20 THz) is modified, although
not qualitatively. Hence there will be substantial overlap with the bare MoS2
phonon spectrum in Fig. 1(c).

Figure 3: (a) Molecular structure of perylene orange (PO), (b) parallel and
(c) perpendicular optimized configurations of MoS2/PO hybrid structures, (d)
binding energy analysis using PBE and PBE+D3 methods, and (e) comparison of
potential energy variation with uniaxial strain in MoS2 and MoS2/PO.

###  III.2 Structural stability and mechanical properties

A comprehensive exploration was undertaken to characterize the interaction
between a monolayer of MoS2 and a single PO molecule. The molecular structure
of the molecule is illustrated in Fig. 3(a). First geometry and binding energy
of both parallel ($\parallel$) and perpendicular ($\perp$) orientations were
scrutinized. Visual representations of the relaxed, i.e. geometry-optimized,
atomic structure of the MoS2/PO system in these orientations, parallel
($\parallel$) and perpendicular ($\perp$), can be found in Fig. 3(b) and (c),
respectively. In passing we note that due to the presence of the bulky side
groups, the structure of PO distorts substantially at the surface as compared
with the isolated molecule case where the perylene core is essentially planar.

The binding energy ($E_{\rm b}$) of the MoS2/PO hybrid interface was
calculated for different distances between MoS2 and PO according to the
equation

$E_{\rm
b}=E_{\text{MoS{\hbox{_2}}/PO}}-E_{\text{MoS{\hbox{_2}}}}-E_{\text{PO}}\,.$
(1)

Here, $E_{\text{MoS${}_{2}$/PO}}$ is energy of the MoS2/PO composite, while
$E_{\text{MoS${}_{2}$}}$ and $E_{\text{PO}}$ denote the total (geometry-
optimized) energies of the isolated MoS2 and the PO, respectively. Determining
the most stable configuration and the equilibrium interlayer distance ($d_{\rm
int}$) is based on finding the minimum of $E_{\rm b}$.

In Fig. 3(d) the binding energy is plotted as a function of interlayer
distance for the MoS2/PO hybrid interface. It is clear from this figure that
the parallel configuration using PBE+D3 is more stable than the perpendicular
one with $d_{\rm int}$ of 2.37 Å. In the following, therefore, only the
parallel configuration is considered when exploring the electronic structure
and the role of strain. We also note from Fig. 3(d) that in both cases binding
is essentially facilitated by dispersion interaction (vdW) and thus of
physisorption type.

Figure 4: Comparative analysis of band structures and T(P)DOS in (a) MoS2 and
(b) MoS2/PO systems, with (c) schematic of interface band alignment (energies
given with respect to Fermi energy $E_{\rm F}$).

Beyond structural stability, we studied the mechanical properties of both the
MoS2 and the MoS2/PO systems. Our focus centered on the analysis of the
elastic coefficients, specifically $C_{11}$, $C_{12}$, and $C_{22}$. These
coefficients were extracted by fitting the unit cell energy, $U$, for a
certain set of strain values ($\epsilon_{11}$, $\epsilon_{22}$).

The elastic stiffness coefficients were computed via the following expressions
(Note that for the underlying hexagonal lattice it holds that
$C_{11}=C_{22}$.)

$\displaystyle
C_{11}=\frac{1}{A_{0}}\frac{\partial^{2}U}{\partial\epsilon_{11}^{2}},$ (2)
$\displaystyle
C_{12}=\frac{1}{A_{0}}\frac{\partial^{2}U}{\partial\epsilon_{11}\partial\epsilon_{22}},$

with $A_{0}$ being the cell area at zero strain. Subsequently, the the Young
modulus ($Y$), shear modulus ($G$), Poisson’s ratio ($\nu$), and the bulk
modulus ($K$) specific to the 2D nature of these materials were derived
through the following relationships:

$\displaystyle Y$ $\displaystyle=$
$\displaystyle\frac{C_{11}^{2}-C_{12}^{2}}{C_{11}},\quad
G=\frac{{C_{11}-C_{12}}}{2},$ $\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{C_{12}}{C_{11}},\quad K=\frac{{C_{11}+C_{12}}}{2}.$ (3)

The analysis involved the systematic application of uniaxial strains
$\epsilon_{11}$ and $\epsilon_{22}$ along two in-plane directions, spanning
from -0.08 to +0.08, with a step size of 0.02 (for geometrical parameters see
Tab. S1 in the Supplementary Material (SM)). To assess the adherence of all
considered strains to the elastic limit, the per-atom strain energy, denoted
as $E_{\rm S}$, is computed as the difference between the energy of the
strained and unstrained state, normalized by the total number of atoms within
the cell. (Notice that the MoS2 results have been obtained for the unit cell.
A comparative discussion of the differences between unit and supercell is
given in the SM, Figs. S1 and S2.) The results presented in Fig. 3(e)
consistently demonstrate a smooth variation of $E_{\rm S}$ with strain for
both MoS2 and MoS2/PO. From this figure, one also notices that for larger
strain values the harmonic limit is no longer valid. Upon compression
repulsive interactions give rise to a steeper increase of $E_{\rm S}$, whereas
bonds are weakened upon stretching, thus leading to a slower increase of
$E_{\rm S}$ as compared with the harmonic approximation. The values for the
elastic stiffness coefficients reported below were obtained by fitting in the
range $\pm 4$%. Fits for different ranges are given in the SM, Table S2.

The mechanical properties are summarized in Table 1. Significantly, both the
MoS2 and the MoS2/PO interface satisfactorily meet the Born criteria for a
hexagonal structure, wherein $C_{11}$ exhibits a positive value, and $C_{11}$
exceeds $|C_{12}|$. These observations indicate the inherent mechanical
stability of both materials. In particular, the MoS2 demonstrates elastic
stiffness coefficients of $C_{11}=138.49$ N/m and $C_{12}=31.37$ N/m, which
are consistent with previous results [57]. For the MoS2/PO hybrid interface,
the values are $C_{11}=140.90$ N/m and $C_{12}=18.17$ N/m.

Table 1: Mesh parameter ($a$), interlayer distance ($d_{\text{int}}$), elastic constants ($C_{11}$, and $C_{12}$), Young modulus ($Y$), shear modulus ($G$), Poisson’s ratio ($\nu$), and bulk modulus ($K$) of MoS2 and MoS2/PO systems. Material |  $a$(Å) |  ${d_{\rm int}}$ (Å) |  $C_{11}$ (N/m) |  $C_{12}$ (N/m) |  $Y$ (N/m) |  $G$ (N/m) | $\nu$ |  $K$ (N/m)
---|---|---|---|---|---|---|---|---
MoS2 | 3.18 | – | 138.49 | 31.37 | 131.38 | 53.56 | 0.23 | 85.24
MoS2/PO | 3.17 | 2.37 | 140.90 | 18.17 | 138.55 | 61.36 | 0.13 | 79.53

Table 1 also reflects that both MoS2 and MoS2/PO systems manifest a high
Young’s modulus and a low Poisson’s ratio, symbolic of their remarkable
stiffness, incompressibility, and resistance to shear deformation, supported
by their shear modulus. The relatively incompressible nature of the hybrid
interface can be attributed to the presence of MoS2, having a high bulk
modulus. Furthermore, the anisotropic behavior of the hybrid interface is
influenced by the organic molecule. An important observation is the
discernible 5% variance, in Young’s modulus when comparing MoS2 with its
MoS2/PO hybrid counterpart. This deviation underscores the distinct mechanical
behavior of the hybrid interface which calls for further investigation, i.e.
using Raman or Brillouin spectroscopy to elucidate the underlying molecular
and structural dynamics associated with this difference in mechanical
rigidity. This difference in Young’s modulus indicates an altered atomic
configuration of the MoS2 layer within the hybrid material. Indeed the value
of root mean square deviation between isolated MoS2 and MoS2/PO was calculated
as 0.28 Å. This small value is in accord with the 5% change of $Y$ and the
notion of physisorption.

###  III.3 Electronic Properties

Results of the investigation of electronic properties of the MoS2 supercell
and its hybrid interface with PO are shown in Figs. 4(a) and (b),
respectively. The left and right panels of these figures give the band
structure and the total/partial (T/P) DOS. Notice that when performing MoS2
supercell simulations Brillouin zone folding occurs, causing a shift in the
direct bandgap from the $K$ to the $\Gamma$ symmetry point (see Fig. 4(a)). In
case of pristine MoS2 this is, of course, an artefact of the used supercell.
Although this artefact could be removed by band unfolding techniques [58], for
the sake of comparison with the MoS2/PO hybrid system we will use the folded
band structure for pristine MoS2.

In Fig. 4(a), the electronic properties of MoS2 around the bandgap are
governed by substantial contributions from the 4d orbitals of Mo atoms and the
3p orbitals of S atoms. This attribution is substantiated by the respective
peaks observed in both the DOS and band structure, confirming in particular
the hybridization between these orbitals at the Valence Band Minimum (VBM).

Figure 4(b) shows band structure and DOS for the MoS2/PO hybrid system. Due to
the physisorption nature of the adsorption, band gaps of the constituents can
still be identified, in addition to the overall bandgap. In particular we can
assign the bandgap $E_{\rm g1}$ which characterizes the gap between VBM and
Conduction Band Minimum (CBM) of MoS2 and the bandgap $E_{\rm g2}$ delineating
the energy difference between the HOMO and LUMO of the PO molecule. Whereas
the CBM is of Mo-4d origin, in the range of the VBM there are contribution of
both Mo-4d and C-2p orbitals. States close to the Fermi energy are of
molecular origin (mostly C-2p). In fact even the HOMO-1 is well-separated
within the MoS2 bandgap.

Figure 5: Bandgap ($E_{\rm g}$) evolution in strained MoS2 and MoS2/PO. Figure
6: Analysis of band structure and T(P)DOS in (a) MoS2 and (b) MoS2/PO under
varying uniaxial strain (energies given with respect to Fermi energy $E_{\rm
F}$).

The relative positions of the states forming the two mentioned bandgaps is
highly relevant for the optical properties of the hybrid system. The present
case can be categorized as type II band alignment, cf. Fig. 4(c). Here upon
photoexcitation, electrons in the HOMO and HOMO-1 of PO can be excited to its
LUMO. These electrons can then transfer to the CBM of MoS2, leaving behind
holes in the PO molecule. This process effectively separates the charge
carriers, reducing the likelihood of radiative electron-hole recombination and
thus enhancing charge separation efficiency. In principle, the holes in the PO
molecule HOMO and HOMO-1 can interact with the VBM of MoS2, potentially
leading to additional charge transfer dynamics. Of course, this qualitative
discussion should be taken with caution since it completely neglects many body
effects.

Such efficient charge transfers would have consequences for the
photoluminescence Mo$S_{2}$ or the fluorescence of PO, which would be quenched
due to the effective separation of electron-hole pairs and the suppression of
their recombination. This charge separation is confirmed by recent experiments
on this system, which showed completely quenched PO fluorescence [24].

Table 2: Strain-dependent electronic characteristics of MoS2: bandgap ($E_{\rm g}$), its type, and position at different strains. Strain (%) |  $E_{\rm g}$ (eV) |  $E_{\rm g}$ type |  $E_{\rm g}$ position
---|---|---|---
-8 | 1.66 | Indirect | [$\Gamma$]-[$Y$]
-6 | 1.75 | Indirect | [$\Gamma$]-[$Y$]
-4 | 1.84 | Indirect | [$\Gamma$]-[$Y$]
-2 | 1.76 | Direct | [$\Gamma$]-[$\Gamma$]
Pristine | 1.65 | Direct | [$\Gamma$]-[$\Gamma$]
+2 | 1.37 | Direct | [$\Gamma$]-[$\Gamma$]
+4 | 1.15 | Direct | [$\Gamma$]-[$\Gamma$]
+6 | 0.95 | Indirect | [$\Gamma$]-[$X$-$\Gamma$]
+8 | 0.76 | Indirect | [$\Gamma$]-[$X$-$\Gamma$]
Table 3: Strain-dependent electronic characteristics of MoS2/PO systems: $E_{\rm g1}$, $E_{\rm g2}$, bandgap ($E_{\rm g}$), and band alignment type. For an analysis of the effect of geometry alone on the HOMO-LUMO gap, see Table S3 in the SM. Strain (%) |  $E_{\rm g1}$ (eV) |  $E_{\rm g2}$ (eV) |  $E_{\rm g}$ (eV) | type
---|---|---|---|---
-8 | 1.78 | 1.31 | 1.31 | I
-6 | 1.78 | 1.35 | 1.35 | I
-4 | 1.76 | 1.37 | 1.36 | II
-2 | 1.72 | 1.37 | 1.28 | II
Pristine | 1.71 | 1.38 | 1.25 | II
+2 | 1.70 | 1.40 | 1.23 | II
+4 | 1.66 | 1.35 | 1.12 | II
+6 | 1.62 | 1.34 | 1.07 | II
+8 | 1.57 | 1.35 | 1.07 | II

###  III.4 Effect of strain on electronic properties

Tailoring electronic properties through external manipulation, such as
mechanical strain, offers great promise for advancements in the fields of
nano- and optoelectronics. In what follows, we explore the electronic
characteristics of the MoS2 supercell and the MoS2/PO hybrid interface under
the influence of uniaxial strain.

Figure 5 presents a summary of the predicted bandgap evolution ($E_{\rm g}$)
with strain, utilizing the PBE approximation for both MoS2 and MoS2/PO
systems. This figure reveals a discernible trend: when subjected to tensile
strain, both the MoS2 and MoS2/PO systems exhibit a systematic reduction of
their bandgap. Conversely, as compressive strain increases, reaching a
critical threshold of approximately -4%, the bandgap decreases with further
increasing of the compressive strain. Our findings regarding MoS2 demonstrate
a good agreement with prior experimental results [6].

The focus of this study centers on Fig. 6, which in panel (a) offers a
detailed analysis of the band structure, as well as the partial and total DOS
of MoS2 under varying magnitudes of uniaxial strain. Pristine MoS2 shows a
direct $\Gamma\rightarrow\Gamma$ bandgap which is also present for small
strain values. At strain values $\leq-4$% and $\geq 6$% the band structure
features an indirect bandgap. Under compressive deformation conditions, the
VBM localizes at the $\Gamma$ symmetry point, while the CBM is positioned at
the $Y$ point. Conversely, under +6% and +8% strains the VBM and CBM positions
converge to $\Gamma$ and ($\Gamma$-$X$), respectively. This distinct bandgap
behavior, observed under both compressive and tensile strains, significantly
deviates from the direct bandgap characteristics exhibited under all other
uniaxial strain percentages. Details regarding the bandgap energies value,
type, and specific positions under different strain conditions are compiled in
Table 2.

The origin for the observed behavior can be traced to transformative effects
of uniaxial strain on the 2D Bravais lattice of a MoS2 monolayer. In its
pristine state, MoS2 has a hexagonal lattice. The application of uniaxial
strain, whether tensile or compressive, prompts a deformation of this
hexagonal lattice, leading to shifts in the interatomic distances and the
angles between lattice vectors (refer to Table S4). As a result, the once-
hexagonal lattice assumes an oblique lattice structure

Finally, we point out that, across all strain conditions, the dominant
contributions in both the valence and conduction bands emanate from the Mo-4d
and S-3p orbitals. This unvarying pattern underscores the significance of
these particular orbitals in shaping the electronic landscape of MoS2 across a
spectrum of strain magnitudes.

Fig. 6(b) presents the band structure and DOS for the MoS2/PO system
associated with the behavior in Fig. 5; specific numbers can be found in Tab.
3. Comparing Tabs. 2 and 3 we notice that the total band gap variation is
about 59% and 21% of its maximum value for MoS2 and MoS2/PO, respectively. The
maximum value is attained at -4% strain. Inspecting the separate bandgaps,
$E_{\rm g1}$ and $E_{\rm g2}$, the variation is 12% and 6%, respectively. For
the CBM-VBM gap the maximum shifts to large strain values (-6%), whereas the
largest HOMO-LUMO gap is observed at +2%. Notably, the Mo-4d and S-3p orbitals
of MoS2 play a substantial role in shaping the CBM and VBM of MoS2, while the
C-2p orbital is pivotal in the formation of the HOMO and LUMO across all
examined systems, both strained and unstrained. Further the type of bandgap
does not change, i.e. it always features a direct transition.

Comparing the evolution of the bandgaps with strain one should take into
account that due to the physisorptive binding the PO molecule is only weakly
affected by strain. Note that the situation would change upon increasing the
coverage, i.e. when PO molecules would interact. Thus it is essentially the
MoS2 band structure that is shifting relative to the PO MOs, with the HOMO
positioned close to the Fermi energy. Still, the variation of $E_{\rm g1}$ in
MoS2/PO is only 20% of that in MoS2, i.e. the hybrid interface makes the MoS2
band structure less susceptible to strain.

Pristine MoS2/PO has a type II band alignment, which doesn’t change when
applying positive strain. However, the increase of the band gap $E_{\rm g1}$
with increasing negative strain leads to a situation where the LUMO shifts
into the MoS2 band gap, i.e. for strain values exceeding -6% our results
predict a change to type I band alignment. Upon photoexcitation this would
enable efficient electron transfer from the CBM to the LUMO, impacting PL in
this system.

##  IV CONCLUSION

Using Density Functional Theory we have investigated PO physisorbed onto
monolayer MoS2, putting emphasis on the mechanical and electronic properties
of this hybrid system. We found that binding of PO is preferentially in
parallel (on-top) orientation and facilitated by van der Waals interaction.
Upon binding the structure of PO is substantially distorted, without
considerably affecting the HOMO-LUMO gap.

MoS2/PO is mechanically stable with key parameters changing only modestly as
compared with bare MoS2. For instance Young’s module increases by about 5%.
The mechanical stability suggested to explore the influence of strain on the
electronic properties of the hybrid system. As a consequence of the non-
covalent binding as well as the low-coverage assumption, molecular orbitals of
PO are only marginally affected by strain. But even the band gap evolution of
the MoS2 part of the hybrid system with strain is less pronounced in bare
MoS2. Still the combination of strain effects on the constituents of the
hybrid interface resulted in a qualitative change of the electronic
properties, i.e. a switch between type II and type I band alignment upon
compressive strain, with all its consequence for the photophysical properties.

The present investigation advances our comprehension of the electronic
properties and structural behavior of MoS2 vs. MoS2/PO under mechanical
strain, furnishing insights for future endeavors into novel device
applications.

###### Acknowledgements.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - SFB 1477 ”Light-Matter Interactions at Interfaces”,
project number 441234705”. We thank T. Korn and J. Schröer (University of
Rostock) for their insightful comments on the manuscript.

## References

  * Dai  _et al._ [2019] Z. Dai, L. Liu, and Z. Zhang, Adv. Mater. 31, 1805417 (2019).
  * Peng  _et al._ [2020] Z. Peng, X. Chen, Y. Fan, D. J. Srolovitz, and D. Lei, Light Sci. Appl. 9, 190 (2020).
  * Niehues  _et al._ [2022] I. Niehues, T. Deilmann, J. Kutrowska-Girzycka, A. Taghizadeh, L. Bryja, U. Wurstbauer, R. Bratschitsch, and J. Jadczak, Physical Review B 105, 205432 (2022).
  * Castellanos-Gomez [2016] A. Castellanos-Gomez, Nature Photonics 10, 202 (2016).
  * Roldán  _et al._ [2015] R. Roldán, A. Castellanos-Gomez, E. Cappelluti, and F. Guinea, Journal of Physics: Condensed Matter 27, 313201 (2015).
  * Plechinger  _et al._ [2015] G. Plechinger, A. Castellanos-Gomez, M. Buscema, H. S. J. Van Der Zant, G. A. Steele, A. Kuc, T. Heine, C. Schüller, and T. Korn, 2D Materials 2, 015006 (2015).
  * Mueller and Malic [2018] T. Mueller and E. Malic, npj 2D Materials and Applications 2, 29 (2018).
  * Niehues  _et al._ [2018] I. Niehues, A. Blob, T. Stiehm, R. Schmidt, V. Jadriško, B. Radatović, D. Čapeta, M. Kralj, S. M. De Vasconcellos, and R. Bratschitsch, 2D Materials 5, 031003 (2018).
  * Carrascoso  _et al._ [2021] F. Carrascoso, H. Li, R. Frisenda, and A. Castellanos-Gomez, Nano Research 14, 1698 (2021).
  * Schmidt  _et al._ [2016] R. Schmidt, I. Niehues, R. Schneider, M. Drüppel, T. Deilmann, M. Rohlfing, S. M. De Vasconcellos, A. Castellanos-Gomez, and R. Bratschitsch, 2D Materials 3, 021011 (2016).
  * Zhu  _et al._ [2013] C. R. Zhu, G. Wang, B. L. Liu, X. Marie, X. F. Qiao, X. Zhang, X. X. Wu, H. Fan, P. H. Tan, T. Amand, and B. Urbaszek, Physical Review B 88, 121301 (2013).
  * Conley  _et al._ [2013] H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund, S. T. Pantelides, and K. I. Bolotin, Nano Letters 13, 3626 (2013).
  * Niehues  _et al._ [2020] I. Niehues, P. Marauhn, T. Deilmann, D. Wigger, R. Schmidt, A. Arora, S. Michaelis De Vasconcellos, M. Rohlfing, and R. Bratschitsch, Nanoscale 12, 20786 (2020).
  * John  _et al._ [2020] A. P. John, A. Thenapparambil, and M. Thalakulam, Nanotechnology 31, 275703 (2020).
  * Takahashi  _et al._ [2012] M. Takahashi, C. Figus, L. Malfatti, Y. Tokuda, K. Yamamoto, T. Yoko, T. Kitanaga, Y. Tokudome, and P. Innocenzi, NPG Asia Materials 4, e22 (2012).
  * Habib  _et al._ [2020] M. R. Habib, W. Wang, A. Khan, Y. Khan, S. M. Obaidulla, X. Pi, and M. Xu, Advanced Theory and Simulations 3, 2000045 (2020).
  * Huang  _et al._ [2018a] Y. L. Huang, Y. J. Zheng, Z. Song, D. Chi, A. T. S. Wee, and S. Y. Quek, Chemical Society Reviews 47, 3241 (2018a).
  * Habib  _et al._ [2018] M. R. Habib, H. Li, Y. Kong, T. Liang, Sk. Md. Obaidulla, S. Xie, S. Wang, X. Ma, H. Su, and M. Xu, Nanoscale 10, 16107 (2018).
  * Padgaonkar  _et al._ [2020] S. Padgaonkar, J. N. Olding, L. J. Lauhon, M. C. Hersam, and E. A. Weiss, Acc. Chem. Res. 53, 763 (2020).
  * Cho  _et al._ [2019] K. Cho, J. Pak, S. Chung, and T. Lee, ACS Nano 13, 9713 (2019).
  * Bettis Homan  _et al._ [2017] S. Bettis Homan, V. K. Sangwan, I. Balla, H. Bergeron, E. A. Weiss, and M. C. Hersam, Nano Letters 17, 164 (2017).
  * Khan  _et al._ [2020] Y. Khan, S. M. Obaidulla, M. Rezwan Habib, Y. Kong, and M. Xu, Applied Surface Science 530, 147213 (2020).
  * Obaidulla  _et al._ [2020] S. M. Obaidulla, M. R. Habib, Y. Khan, Y. Kong, T. Liang, and M. Xu, Advanced Materials Interfaces 7, 1901197 (2020).
  * Völzer  _et al._ [2023] T. Völzer, A. Schubert, E. Von Der Oelsnitz, J. Schröer, I. Barke, R. Schwartz, K. Watanabe, T. Taniguchi, S. Speller, T. Korn, and S. Lochbrunner, Nanoscale Advances 5, 3348 (2023).
  * Kong  _et al._ [2022] Y. Kong, S. M. Obaidulla, M. R. Habib, Z. Wang, R. Wang, Y. Khan, H. Zhu, M. Xu, and D. Yang, Materials Horizons 9, 1253 (2022).
  * Huang  _et al._ [2018b] Y. Huang, F. Zhuge, J. Hou, L. Lv, P. Luo, N. Zhou, L. Gan, and T. Zhai, ACS Nano 12, 4062 (2018b).
  * Ehli  _et al._ [2009] C. Ehli, C. Oelsner, D. M. Guldi, A. Mateo-Alonso, M. Prato, C. Schmidt, C. Backes, F. Hauke, and A. Hirsch, Nature Chemistry 1, 243 (2009).
  * Mattson  _et al._ [2018] M. A. Mattson, T. D. Green, P. T. Lake, M. McCullagh, and A. T. Krummel, The Journal of Physical Chemistry B 122, 4891 (2018).
  * Huang  _et al._ [2011] C. Huang, S. Barlow, and S. R. Marder, The Journal of Organic Chemistry 76, 2386 (2011).
  * Gong  _et al._ [2018] H.-X. Gong, Z. Cao, M.-H. Li, S.-H. Liao, and M.-J. Lin, Organic Chemistry Frontiers 5, 2296 (2018).
  * Chen  _et al._ [2015] S. Chen, P. Slattum, C. Wang, and L. Zang, Chemical Reviews 115, 11967 (2015).
  * Cao and Yang [2022] J. Cao and S. Yang, RSC Advances 12, 6966 (2022).
  * Schaack  _et al._ [2022] C. Schaack, A. M. Evans, F. Ng, M. L. Steigerwald, and C. Nuckolls, Journal of the American Chemical Society 144, 42 (2022).
  * Li  _et al._ [2008] Y. Li, L. Tan, Z. Wang, H. Qian, Y. Shi, and W. Hu, Organic Letters 10, 529 (2008).
  * Hwang  _et al._ [2015] Y.-J. Hwang, T. Earmme, B. A. E. Courtright, F. N. Eberle, and S. A. Jenekhe, Journal of the American Chemical Society 137, 4424 (2015).
  * Li  _et al._ [2023] L. Li, J. Xu, W. Luo, K. Zhong, X. Zhao, Y. Hu, Z. Yuan, and Y. Chen, Journal of Materials Chemistry A 11, 24136 (2023).
  * Dayneko  _et al._ [2021] S. V. Dayneko, E. Cieplechowicz, S. S. Bhojgude, J. F. Van Humbeck, M. Pahlevani, and G. C. Welch, Materials Advances 2, 933 (2021).
  * Sabatini  _et al._ [2019] R. P. Sabatini, B. Zhang, A. Gupta, J. Leoni, W. W. H. Wong, and G. Lakhwani, Journal of Materials Chemistry C 7, 2954 (2019).
  * Kresse and Furthmüller [1996a] G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996a).
  * Kresse and Furthmüller [1996b] G. Kresse and J. Furthmüller, Physical Review B 54, 11169 (1996b).
  * Blöchl [1994] P. E. Blöchl, Physical Review B 50, 17953 (1994).
  * Grimme  _et al._ [2010] S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, The Journal of Chemical Physics 132, 154104 (2010).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Review Letters 77, 3865 (1996).
  * Heyd  _et al._ [2003] J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal of Chemical Physics 118, 8207 (2003).
  * Baroni  _et al._ [2001] S. Baroni, S. De Gironcoli, A. Dal Corso, and P. Giannozzi, Reviews of Modern Physics 73, 515 (2001).
  * Epifanovsky  _et al._ [2021] E. Epifanovsky, A. T. B. Gilbert, X. Feng, J. Lee, Y. Mao, N. Mardirossian, P. Pokhilko, A. F. White, M. P. Coons, A. L. Dempwolff, Z. Gan, D. Hait, P. R. Horn, L. D. Jacobson, I. Kaliman, J. Kussmann, A. W. Lange, K. U. Lao, D. S. Levine, J. Liu, S. C. McKenzie, A. F. Morrison, K. D. Nanda, F. Plasser, D. R. Rehn, M. L. Vidal, Z.-Q. You, Y. Zhu, B. Alam, B. J. Albrecht, A. Aldossary, E. Alguire, J. H. Andersen, V. Athavale, D. Barton, K. Begam, A. Behn, N. Bellonzi, Y. A. Bernard, E. J. Berquist, H. G. A. Burton, A. Carreras, K. Carter-Fenk, R. Chakraborty, A. D. Chien, K. D. Closser, V. Cofer-Shabica, S. Dasgupta, M. de Wergifosse, J. Deng, M. Diedenhofen, H. Do, S. Ehlert, P.-T. Fang, S. Fatehi, Q. Feng, T. Friedhoff, J. Gayvert, Q. Ge, G. Gidofalvi, M. Goldey, J. Gomes, C. E. González-Espinoza, S. Gulania, A. O. Gunina, M. W. D. Hanson-Heine, P. H. P. Harbach, A. Hauser, M. F. Herbst, M. Hernández Vera, M. Hodecker, Z. C. Holden, S. Houck, X. Huang, K. Hui, B. C. Huynh, M. Ivanov, Á. Jász, H. Ji, H. Jiang, B. Kaduk, S. Kähler, K. Khistyaev, J. Kim, G. Kis, P. Klunzinger, Z. Koczor-Benda, J. H. Koh, D. Kosenkov, L. Koulias, T. Kowalczyk, C. M. Krauter, K. Kue, A. Kunitsa, T. Kus, I. Ladjánszki, A. Landau, K. V. Lawler, D. Lefrancois, S. Lehtola, R. R. Li, Y.-P. Li, J. Liang, M. Liebenthal, H.-H. Lin, Y.-S. Lin, F. Liu, K.-Y. Liu, M. Loipersberger, A. Luenser, A. Manjanath, P. Manohar, E. Mansoor, S. F. Manzer, S.-P. Mao, A. V. Marenich, T. Markovich, S. Mason, S. A. Maurer, P. F. McLaughlin, M. F. S. J. Menger, J.-M. Mewes, S. A. Mewes, P. Morgante, J. W. Mullinax, K. J. Oosterbaan, G. Paran, A. C. Paul, S. K. Paul, F. Pavošević, Z. Pei, S. Prager, E. I. Proynov, Á. Rák, E. Ramos-Cordoba, B. Rana, A. E. Rask, A. Rettig, R. M. Richard, F. Rob, E. Rossomme, T. Scheele, M. Scheurer, M. Schneider, N. Sergueev, S. M. Sharada, W. Skomorowski, D. W. Small, C. J. Stein, Y.-C. Su, E. J. Sundstrom, Z. Tao, J. Thirman, G. J. Tornai, T. Tsuchimochi, N. M. Tubman, S. P. Veccham, O. Vydrov, J. Wenzel, J. Witte, A. Yamada, K. Yao, S. Yeganeh, S. R. Yost, A. Zech, I. Y. Zhang, X. Zhang, Y. Zhang, D. Zuev, A. Aspuru-Guzik, A. T. Bell, N. A. Besley, K. B. Bravaya, B. R. Brooks, D. Casanova, J.-D. Chai, S. Coriani, C. J. Cramer, G. Cserey, A. E. DePrince, III, R. A. DiStasio, Jr., A. Dreuw, B. D. Dunietz, T. R. Furlani, W. A. Goddard, III, S. Hammes-Schiffer, T. Head-Gordon, W. J. Hehre, C.-P. Hsu, T.-C. Jagau, Y. Jung, A. Klamt, J. Kong, D. S. Lambrecht, W. Liang, N. J. Mayhall, C. W. McCurdy, J. B. Neaton, C. Ochsenfeld, J. A. Parkhill, R. Peverati, V. A. Rassolov, Y. Shao, L. V. Slipchenko, T. Stauch, R. P. Steele, J. E. Subotnik, A. J. W. Thom, A. Tkatchenko, D. G. Truhlar, T. Van Voorhis, T. A. Wesolowski, K. B. Whaley, H. L. Woodcock, III, P. M. Zimmerman, S. Faraji, P. M. W. Gill, M. Head-Gordon, J. M. Herbert, and A. I. Krylov, The Journal of Chemical Physics 155, 084801 (2021).
  * Chaudhuri  _et al._ [2023] S. Chaudhuri, A. Bhattacharya, A. Das, G. Das, and B. Dev, Applied Surface Science 626, 157139 (2023).
  * Ramakrishna Matte  _et al._ [2010] H. S. S. Ramakrishna Matte, A. Gomathi, A. K. Manna, D. J. Late, R. Datta, S. K. Pati, and C. N. R. Rao, Angewandte Chemie International Edition 49, 4059 (2010).
  * Liu  _et al._ [2013] D. Liu, Y. Guo, L. Fang, and J. Robertson, Applied Physics Letters 103, 183113 (2013).
  * Rawat  _et al._ [2018] A. Rawat, N. Jena, D. Dimple, and A. De Sarkar, Journal of Materials Chemistry A 6, 8693 (2018).
  * Huang  _et al._ [2021] M. Huang, A. Dinesh, and S. Wu, Physical Chemistry Chemical Physics 23, 15364 (2021).
  * Cappelluti  _et al._ [2013] E. Cappelluti, R. Roldán, J. A. Silva-Guillén, P. Ordejón, and F. Guinea, Physical Review B 88, 075409 (2013).
  * Lebègue and Eriksson [2009] S. Lebègue and O. Eriksson, Phys. Rev. B 79, 115409 (2009).
  * Mak  _et al._ [2010] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 105, 136805 (2010).
  * Splendiani  _et al._ [2010] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, Nano Lett. 10, 1271 (2010).
  * Shi  _et al._ [2018] Z. Shi, Q. Zhang, and U. Schwingenschlögl, ACS Applied Energy Materials 1, 2208 (2018).
  * Dong  _et al._ [2017] L. Dong, J. Lou, and V. B. Shenoy, ACS Nano 11, 8242 (2017).
  * Krumland and Cocchi [2021] J. Krumland and C. Cocchi, Electron. Struct. 3, 044003 (2021).