# Investor-patent networks as mutualistic networks

Théophile Carniel
Université Paris Cité, CNRS, LIED UMR 8236, F-75006 Paris, France
Agoranov, Paris, France

&Léo Cazenille
Université Paris Cité, CNRS, LIED UMR 8236, F-75006 Paris, France

&Jean-Michel Dalle
Agoranov, Paris, France
Sorbonne Université, Paris, France

&José Halloy
Université Paris Cité, CNRS, LIED UMR 8236, F-75006 Paris, France
<EMAIL_ADDRESS>

###### Abstract

Venture capital investments in startups have come to represent an important
driver of technological innovation, in parallel to corporate- and government-
directed efforts. Part of the future of artificial intelligence, medicine and
quantum computing now depends upon a large number of venture investment
decisions whose robustness against increasingly frequent crises has therefore
become crucial. To shed light on this issue, and by combining large-scale
financial, startup and patent datasets, we analyze the interactions between
venture capitalists and technologies as an explicit bipartite patent-investor
network. Our results reveal that this network is topologically mutualistic
because of the prevalence of links between generalist investors, whose
portfolios are technologically diversified, and general-purpose technologies,
characterized by a broad spectrum of use. As a consequence, the robustness of
venture-funded technological innovation against different types of crises is
affected by the high nestedness and low modularity, with high connectance,
associated with mutualistic networks.

_K_ eywords Mutualistic networks $\cdot$ Venture Capital $\cdot$ Technological
Innovation $\cdot$ Robustness

##  1 Introduction

Due to the increased role of startups in various technological fields, from
biotechnology to artificial intelligence or quantum computing [1], venture
capital has de facto become an important driver of technological innovation
[2, 3, 4, 5]. Indeed, because of their inability to self-finance during the
early years of their operations, startups need to rely on the investments that
they themselves receive from specialized investors called venture capitalists
(VCs). In this context, the relationships between VC funding and innovation
have gradually become a topic of interest, mostly approached through patent
data [6, 7, 8, 9]. This evolution has occurred in a world where crises have
become more and more frequent [10], increasing the need to analyze the
resilience of socio-economic systems [11]. These studies have notably
preliminarily shown that VC funding could be negatively impacted by local and
global crises, be they financial [12], health [13] or geopolitical [14].
However, it is quite surprising that, although the VC network has been an
active topic of study for the past 15 years [15, 16], and although [4] had
pioneered a complex network approach focused on robustness, the direct
interactions between VCs and the innovations they fund, based explicitly on
the patents filed [17] by the startups funded, have not been explicitly
studied, both in their own right and in relation to the robustness of the
network they constitute. This VC-patent interaction network is bipartite, with
nodes of a first class (VCs) interacting with nodes of a second class
(patents) through investments in startups that file the patents.

In this study, considering the line of analysis of financial markets suggested
by [18, 19] and echoing also the approaches that have started to directly
address economic complexity [20], we combine large-scale financial datasets on
the rounds of VC funding received by startups with patent data to explicitly
analyze this bipartite investor-patent network and its emergent structure. We
identify clusters of investors and clusters of patents and observe that their
bipartite network is topologically mutualistic, i.e. that the structure of the
network shares metric characteristics with mutualistic networks in ecology.
This is due to the prevalence of investors whose financial incentives make
them diversify their portfolios with respect to technological innovations in
order to reduce the risks taken [21] and to the existence of a large number of
general-purpose technologies, i.e. technologies whose use spreads widely
across economic sectors [22].

With respect to the robustness of this network, we analyze its nestedness [23]
and modularity [24] metrics, as they have been developed by the ecological and
physical literature. Nestedness measures the existence of a matryochka-like
structure of interaction, where specialist nodes interact with nodes that the
generalist nodes also interact with. Modularity estimates the propensity of
nodes in a module (a set of nodes allocated to the same group) to interact
with nodes in the same module. Both metrics have been linked to the system
response to perturbations. We find the investor-patent network to be strongly
nested and weakly modular, which is consistent with its topologically
mutualistic nature. As a consequence, this network is characterized by
distinct responses to different system perturbations [25]. Crises that affect
investors randomly or specialized investors tend to have relatively little
impact, due to the redundant nested structure of the network, whereas events
that target generalist investors tend to present a higher risk for related
technological innovations [26].

## Objectives

We study the interactions between investors and the technologies developed by
startup companies in their portfolios. To do so, we combine financial and
patent data in a network analysis framework. We first present the methodology
used to build this bipartite network. We then analyze its topology and, using
metrics developed by the literature in ecology and physics, notably nestedness
and modularity, we discuss the implications of this topology for its
robustness against crises.

##  2 Materials and methods

Figure 1: Workflow presenting the approach followed in this article. (A)
Investor and company information is extracted from Crunchbase. (B) Patent data
from USPTO, CIPO and WIPO is extracted and matched with the Crunchbase company
information. (C) $16$ investors communities are retrieved using a similarity
metric between pairs of investors. (D) The bipartite network between investors
and the patents of the companies they invested in is built. (E) NLP-based
topic modeling of patents is performed on their abstracts and $98$ patent
clusters are retrieved. (F) The investor community-patent cluster graph is
built based on the investor-patents bipartite network by combining the results
from steps (C), (D) and (E), i.e. by aggregating investors into their investor
communities and patents into their patent clusters on the investor-patents
bipartite network. (G) The biadjacency matrix of the investor community-patent
cluster graph is extracted to quantitatively visualize the interaction
patterns and compute network structure metrics. (H) Network structure metrics
(connectance, nestedness, modularity) are computed using the biadjacency
matrix to study the topology of the network and the properties deriving from
it.

Figure 1 presents the approach followed in this article. Using financial and
patent databases, we retrieve information concerning startups, the patents
they own and the investors that financed them. Investor communities are then
identified using a pairwise investor similarity methodology from the
literature and patent clusters are identified using Topic Modeling methods.
These aggregate-level descriptions are then used to build a bipartite graph
linking investor communities to patent clusters by using the startups as the
bridge between the two (startups can be linked both to the patents they own
and to the investors that financed them). We then use the biadjacency matrix
of the network to quantitatively investigate its structure through network-
level structural metrics frequently used in the ecological literature.

###  2.1 Datasets

The startup dataset used for this study was extracted through the
Crunchbase111https://www.crunchbase.com API on February 14, 2023. It contains
information on $2\ 597\ 998$ startups (name, headquarters location, creation
date, sectoral tags), $396\ 506$ funding events (funded startup, date of the
funding round, investors involved, funding amount, investment stage), $241\
489$ investors (name, creation date, investor type, headquarter location) and
$1\ 631\ 627$ individuals (name, professional experiences, academic education,
company board memberships and advisory roles). We removed the Software and
Other sectors from the 47 original sectoral tags as they were found to be
redundant, highly non-specific and over-represented (representing a combined
total of roughly $13\%$ of all tags in the dataset, first and fourth tags in
terms of number of occurrences). We filtered out all companies founded before
January 1st, 1998 to remove all companies that were not startups and all
companies for which geographical information was not available. Funding rounds
that were not VC funding (such as debt financing or grants) were also filtered
out as they are carried by other actors than VCs. Since we focus on the
interactions between investors and technological innovation, using companies
as linking agents between both, we filtered out all companies that did not
raise funds. After applying these filters, $234\ 358$ companies remained in
our dataset.

The patent dataset also supplied by Crunchbase and
IPqwery222https://ipqwery.com) contains a total of $15\ 713\ 946$ patents from
WIPO333World Intellectual Property Organization, USPTO444United States Patent
and Trademark Office and CIPO555Canadian Intellectual Property Office with
their title, abstract, filing date, owner identification, and International
Patent Classification (IPC) codes. It provides a matching with the startup
dataset that links patent owner IDs to Crunchbase company IDs, allowing us to
determine the patent portfolios of startups. We filtered out all patents filed
before January 1st, 2000 and all patents that were not owned by companies from
our filtered startup dataset, resulting in a final dataset of $835\ 763$
patents.

###  2.2 Networks

To study the structure of interactions between investors and technological
innovations, we consider a network where investors interact with technologies
based on their investments in startups and on the patents owned by startups :
simply, an investor and a patent are linked if the investor has financed the
startup owning the patent. At this level of granularity, the interactions
between patents and investors are however too sparse to study fundamental
behaviors. Since individual patents are known to belong to classes or clusters
associated with different fields of technological innovation [17], and since
investors belong to different types [27], we cluster investors and patents in
order to aggregate them into coarser-grained communities.

####  2.2.1 Investor communities

We detect investor communities following the methodology described in [27]. We
select investors with 60 or more investments bringing the number of investors
down to $2017$ and, for each of them, build the $5$ characteristic
distributions as described in [27] : temporal, amount, geographic, series and
sectoral investment distributions. We compute a similarity metric between all
pairs of investors to build a weighted similarity network where all investors
are linked and the edge weights correspond to the pairwise similarities. As
all nodes are linked to each other in a pairwise similarity network, the
network is then pruned to reduce link density (removing weak links to
transform our highly mixed community structure into a simpler, lowly mixed
community structure [28]) to run a community detection algorithm [29]. This
yields $c=16$ investor communities as shown in Fig. 2A. The community results
are in line with those presented in [27], with novel communities emerging as
the dataset used was extracted more recently. [27] have shown that the
investor communities provided by this methodology are stable with regard to
perturbations to the characteristic distributions of individual investors,
suggesting that the underlying investor clustering is robust.

####  2.2.2 Patent clusters

We apply topic modeling to our patent dataset following the methodology
described in [30, 31]. In order to thematically cluster similar patents
together based on their textual abstracts, we create vector representations
(embeddings) of individual patents using the PatentSBERT [32] model
specialized in patent modeling. Each patent is thus represented by a
768-dimensional embedding. We then create a low-dimensional representation of
all embeddings using parametric UMAP [33]. Using these UMAP vectors in
conjunction with HDBSCAN [34], a density-based spatial clustering algorithm,
we perform the clustering of individual patents. The algorithm works in two
phases : first, a clustering is performed by identifying regions of high
density and grouping the points in these regions together and a hierarchical
approach is then taken to return a flat clustering able to take into account
the variable cluster densities. As spatial density-based algorithms perform
better when the dimensionality of the data is not too high, the dimensionality
reduction step is performed to improve the performance of the HDBSCAN
clustering. This process results in $p=98$ patent clusters. To characterize
the patent clusters, we perform a keyword extraction procedure for each
cluster using c-TF-IDF, extracting the n-grams that are representative of each
patent cluster.

####  2.2.3 Investor-patent network

We allocate each startup to the patent cluster most represented in its patent
portfolio. Investors that did not invest in any startup that holds a patent
and startups that do not own patents are removed, resulting in $1937$
investors and $12\ 007$ startups after filtering. We then build the weighted
investor-patent network based on the funding events between investors and
startups : the weight of the edge connecting an investor community and a
patent cluster corresponds to the number of times members of the investor
community have invested in startups allocated to the patent cluster. The
resulting weighted investor-patent network is bipartite since it is composed
of two different classes of nodes, with nodes of one class (in our case,
investor communities) being only linked to nodes of the other class (here,
patent clusters). Other examples of bipartite networks in socio-economic
research include the country-product network [35] or the country-food
production network [36]. Specific metrics have been developed to characterize
bipartite networks [37] that are presented in the following section.

###  2.3 Network metrics

####  2.3.1 Nestedness

Nestedness [23] is a structural property of bipartite networks that
characterizes to what extent specialist species interact with subsets of the
species generalist species interact with, meaning that in nested networks,
specialist-specialist interactions are infrequent. Mutualistic networks, (i.e.
networks where both species involved in an interaction have a net benefit such
as plant-pollinator or seed dispersal networks have been shown to be
significantly nested [23]. Possible explanations such as system tolerance to
species extinctions have been suggested as a reason for the prevalence of
nestedness in mutualistic systems [26], but the origin of nestedness in these
networks remains an open question [38]. Furthermore, the nested architecture
of networks has been shown to be positively correlated with structural
robustness (studied for instance through the lens of species extinction in
ecology) when it is assumed that species with lower degree are more at risk of
extinction, meaning that nested networks are maximally robust when the least
linked species (specialists) become extinct but more fragile when the most
linked species (generalists) face systematic extinction [26]. Following up on
these findings, [39] have shown that maximally nested networks maximize the
structural stability of mutualistic systems and that most observed networks
were close to this optimum architecture.

Nestedness measures have been widely studied as one of the key metrics
characterizing interactions in bipartite networks. Measuring it, however, has
been a topic of ongoing investigations [40, 41, 42, 43], and a number of
different methods have been developed [44]. Among the latter, several
nestedness metrics such as the Atmar & Patterson temperature or the overlap
and decreasing fill (NODF) [40] do not take into account the quantitative
nature of the interaction matrix, reducing it to a binary interaction matrix.
Since the nature of our data allows us to access detailed information about
the frequency of interaction between nodes, link weights span several orders
of magnitudes and we therefore opt for the spectral radius $\rho$ [45] metric,
a nestedness measure that can precisely handle weighted networks. This is of
particular relevance as it has been shown that networks could be thought to be
nested in binary form, but were eventually found not to be nested when
accounting for weighted interactions [45].

####  2.3.2 Bipartite modularity

The modularity $Q$ of a network is a structural measure of how frequently
nodes in defined subgroups of the network (modules) interact with each other
compared to their frequency of interactions with nodes of other subgroups. The
adjacency matrix of a modular network thus presents blocks of dense
interactions between nodes of a given subgroup, and few links with nodes of
other subgroups. Here, we use a modularity measure developed specifically to
take into account the bipartite nature of the network of study [24].

####  2.3.3 Connectance

The connectance of a network is defined as the number of realized links
divided by the number of possible links in the network. This structural metric
has been shown to be linked to network complexity, degree distribution and
network stability [46, 47]. Furthermore, in bipartite ecological networks, the
level of connectance of the network has been shown to impact the relationship
between modularity and nestedness [48], with low connectance networks
displaying a positive correlation between modularity and nestedness and
networks with a high connectance value displaying a negative correlation
between modularity and nestedness.

##  3 Results

###  3.1 Investor communities

Starting from $2017$ investors, we apply the clustering methodology described
in the methods section. We obtain 16 investor communities (Fig. 2A) described
in Table 1 in the appendix. The communities are relatively few in number and
are heterogeneous in size (the smallest one, C.13, is comprised of 19
investors and the largest, C.04, is comprised of 239 investors). Community
C.00 is composed of investors specialized in the Health Care sector, C.01 of
historic investors that have been active relatively homogeneously throughout
the whole period of study , C.02 of generalist investors capable of investing
different amounts at different stages without displaying a strong sectoral
specialization, C.03 of early-stage cryptocurrency investors that started
being active around 2020, C.04 of United Kingdom (UK) and Germany (DE)-focused
investors, C.05 of late-stage and private equity (PE) investors, C.06 of
early-stage and business angels (BAs), C.07 of a specific type of very early-
stage investors called accelerators, C.08 of Canada-focused early-stage
investors and incubators, C.09 of France-focused investors with a slight
preference for the Health Care sector, C.10 of China-focused investors, C.11
of early-stage actors that started being active around 2013, C.12 of investors
that started being active in 2014 capable of investing throughout all stages
of the VC cycle, C.13 of Latin America (Brazil, Mexico, Colombia)-focused
investors, C.14 of India and Southeast Asia (SEA)-focused investors (India,
Indonesia, Singapore), C.15 of Japan-focused investors.

These investor communities present relatively straightforward identities, with
some of them being mainly defined by their geography of investments (e.g. C.10
and C.14), others being defined by their sectors of investment (e.g. C.00 and
C.03), others by their stage of investment (e.g. C.05 and C.07), others by
their temporal patterns of investment (e.g. C.01 and C.11) and others by a
combination of several dimensions (e.g. C.08 with a mix of the stage and
geographical dimensions and C.09 with a mix of sectoral and geographical
dimensions). Our previous work [27] has shown these clustering results to be
robust to the decimation of the characteristic dimensions used to compute the
similarities.

###  3.2 Patent clusters

Individual patents are grouped into clusters (Fig. 2B), and we extract
keywords and apply labels to describe the resulting patent clusters in Table 2
in the appendix. The size of the patent clusters is strongly heterogeneous,
with the smallest patent cluster containing 226 patents and the largest
cluster containing $82\ 513$ patents. Our patent clusters cover a wide range
of specific technologies, the top 5 clusters by number of patents being :
cluster $53$ (Pharmaceutical compositions/therapy) with $82\ 513$ patents,
cluster $41$ (Wireless Communication Technology) with $51\ 139$ patents,
cluster $55$ (Image Processing & Autonomous Vehicles) with $41\ 507$ patents,
cluster $4$ (Pharmaceutical Compound Therapy) with $30\ 981$ patents and
cluster $64$ (Semiconductor Device Fabrication) with $20\ 700$ patents.
Examples of other patent clusters include cluster $12$ (Seismic Survey
Techniques, $322$ patents), cluster $38$ (Nucleic Acid Analysis, $14\ 600$
patents) and cluster $15$ (Social Media Content, $424$ patents).

Technologies can be thought of as roughly being linked to three overarching
groups : hardware-based, Health Care-related and software-based, each with its
own specific challenges and constraints. Patented technologies can of course
draw elements from several of these general fields, but we manually allocated
each patent cluster to one of the three groups based on their label and
keywords. We then colored the patent cluster nodes in Fig. 3 following this
allocation : Manufacturing in green ($38$ patent clusters), Information
Technology (IT) in blue ($35$ patent clusters), and Health Care in red ($24$
patent clusters). Even though there is some heterogeneity in the number of
patent clusters in each group, all three groups are well-represented.

Figure 2: A. Pruned investor similarity network. Each node corresponds to an
investor, and its color corresponds to the investor community it is allocated
to. All investors can be grouped into 16 communities that define types of
"investor species" (C.00 to C.15, presented in Fig. 3 and Table 1). B.
Projected latent space (2 dimensions) of the patent data. Each point
represents a patent and its color corresponds to its cluster allocation. The
clustering defines 99 clusters, 98 thematic clusters and one unlabeled
cluster. Cluster -1 (in black) corresponds to unlabeled data points.

###  3.3 Investor-patent network

A bipartite network linking investor communities and patent clusters (Fig. 3)
is built using investor communities and patent clusters. The degree
distribution of the network is highly heterogeneous (truncated power law) both
for investor communities and patent clusters, meaning that a small number of
nodes have a large number of connections to other nodes while most others have
a low number of connections. Broad-scale networks (networks exhibiting a
truncated power law) are commonly found in abiotic and biotic systems, and are
the result of finite size effects of the studied underlying network. The tail
of the distribution (nodes with high degree) by definition has few
observations, and as real processes are often bounded by the constraints of
the system (in our case, a finite number of funding events), a bounded
distribution is better suited to the system.

The biadjacency matrix associated with the bipartite network is reordered
following the descending node degree on both investor community and patent
cluster nodes (upper-left packing). Community C.02 is the most active investor
community, with a fairly diversified patent portfolio. Community C.00 is the
second most active community, with a strong specialization in Health Care-
related patent clusters (clusters 53, 57, 4 and 38). On the other end of the
matrix, communities C.13 and C.03 are the least active, with C.13 showing no
specific pattern and C.03 showing IT and finance-related patent activity
(clusters $26$, $36$, $30$ and $96$). The nested interaction pattern of the
network is visible, with a strong density of interaction in the upper-left
corner of the matrix and few interactions in the lower-right corner of the
matrix. We visually observe that the nested structure, when rearranging the
biadjacency matrix by descending order of degree, is imperfect in part due to
the specificity of community C.00. Indeed, since we work with quantitative
rather than binary data, it boasts both a high degree (high number of
interactions) and a high specialization (relatively few patent clusters with
which it interacts).

Figure 3: The investor community-patent cluster bipartite network. Square
nodes represent investor communities and circle nodes patent clusters. Node
sizes are a function of the node degrees. Link weights are normalized for each
investor community by the maximum edge weight of the investor community, and
the edge width shown is the logarithm of the normalized weight. A brief
description of investor communities is provided under each investor community
label, and a more extensive description is available in Table 1 in the
appendix. Nodes were positioned following the 4 modules obtained by the
bipartite modularity algorithm, and node label colors correspond to the module
they were allocated to. Patent clusters are colored following a manual
allocation of the high-level technological field they deal with (red for
Health Care, blue for Information Technology, green for Manufacturing).

###  3.4 Connectance

The measured connectance of our network is $C=0.72$, a high value compared to
ecological bipartite networks. As there are in theory no forbidden
interactions (interactions that are structurally impossible in a network for
physiological or phenological reasons) in our system and as our study covers a
long period of time, this high value is not surprising. The magnitude of the
connectance has strong implications on other structural network metrics such
as degree distribution, nestedness and modularity.

###  3.5 Modularity

Using a modularity-based community detection algorithm, we measure the
modularity value of our network and retrieve $4$ modules. The measured
modularity of our network is $Q_{m}=0.19$, meaning that the network is weakly
modular (modularity ranges from $-1$ to $1$, with negative values
corresponding to anti-modular networks and positive values to modular
networks). The $4$ retrieved modules are shown by text color for patent
cluster nodes and investor communities in Fig. 3. Three of these modules show
a strong technological focus (module $1$ around Manufacturing, module $2$
around Information Technologies and module $3$ around Health Care), with the
fourth one (module $0$) containing a mix of technologies. We also computed the
normalized modularity $\overline{Q}$ of a number of bipartite ecological
networks and compared it to the normalized bipartite modularity of our network
$\overline{Q}_{m}$ (results not shown here). We find that our modularity is
lower than most networks it was compared to, potentially due to the different
underlying nature of this socio-economic network compared to ecological
networks.

###  3.6 Relevance tests and ecological metrics

Relevance tests for the nestedness and modularity of our network are
performed, and the investor community-patent cluster network is found to be
significantly more nested (Fig. 4A, $\rho_{m}=5662$, $\rho_{null}=3799\pm
295$, mean $\pm$ std, $z_{\rho}=6.32$) and significantly less modular (Fig.
4B, $Q_{m}=0.19$, $Q_{null}=0.61\pm 0.02$, $z_{Q}=-21$) compared to the null
model. This specific network topology has strong implications on the
properties of the network, notably in terms of robustness to external
perturbations such as species extinction and in terms of species diversity.
The statistically high nestedness and low modularity (compared to the null
models) of the interaction structure between investor communities and patent
clusters is in line with previous findings in the literature as nestedness and
modularity have been shown to be anticorrelated for networks with high
connectance [48]. We also perform this analysis on a network where
interactions are weighted by total funding amounts rather than number of
interactions, and, even though the ordering of investor communities by degree
is different, we find similar results in terms of nestedness and modularity
(results not shown here).

Figure 4: Statistical relevance tests for the nestedness and the modularity of
the network. (A) Statistical relevance test for the nestedness $\rho_{m}$ (red
vertical line) of the investor community-patent cluster network compared with
$5\ 000$ iterations of a null model (blue histogram) keeping the qualitative
interaction structure but shuffling the quantitative interaction structure. We
see that our network is significantly more nested compared to networks
generated by the null model. (B) Statistical relevance test for the modularity
$Q_{m}$ (green vertical line) of the investor community-patent cluster network
compared with $5\ 000$ iterations of the null model (blue histogram). We see
that our network is significantly less modular compared to networks generated
by the null model. (C) Binarized representation of the biadjacency matrix.
Investor communities correspond to the rows, patent clusters to the columns.
The rows and columns are reordered by descending marginals (sums of the value
of the row or column), yielding an upper-left packed matrix. The nested
structure is displayed, with more generalist investor communities (bottom rows
of the matrix) interacting only with a subset of the patent clusters the
generalist species (top rows of the matrix) interact with.

##  4 Discussion

We observe that the bipartite network of interactions between startup
investors and the patents in which they indirectly invest exhibits an emergent
topological mutualism. This mutualistic topology, commonly found in ecological
systems, results here from the presence of many investors whose generalist
nature is induced by their portfolio diversification strategies, on one side
of the network, and of general-purpose technologies with a broad spectrum of
use, on the other. On the investor side, portfolio diversification is a
fundamental and basic idea of modern portfolio theory [49]. Investors are
statistically better off if and when they diversify the risks they take among
their investments: typically here, by supporting startups that develop
different kinds of technological innovations. On the other side of the
network, general purpose technologies [50] are technologies characterized by
their pervasiveness because they are used as inputs by many downstream
sectors: here, by numerous startups with many different products that benefit
from investments from many different types of investors. Although we do not
address here the complexity of economies properly speaking and notably trade
networks [51], this result is closely related to the studies that have shed
new light on the workings of socio-economic systems within the framework of
economic complexity, and shares with them the rationale according to which the
understanding of many societal issues implies to look at the systemic
interactions that produce them [20, 52]. In addition, but with a less
pronounced relevance with respect to our approach, it should also be noted
that another line of investigation has also attempted to draw parallels
between the study of mutualistic systems and economic issues by introducing
economic market effects to explain the evolution and stability of mutualistic
interactions in ecological systems [53, 54].

Topologically mutualistic networks have been shown to be significantly nested
[23], a property that has been related to network robustness both for socio-
economic systems and in ecology. Both literatures concur that the observed
nested structure of the bipartite matrices describing topologically
mutualistic networks contribute to their robustness and stability (e.g. [55,
44]) and study the vulnerabilities of such systems, with the general
conclusion that a nested system reacts very differently to perturbations
depending on which types of nodes they affect.

Studies of the bipartite network of interactions between designer and
contractor firms in the New York City garment industry [56, 57], following
Uzzi’s seminal work [58] on the sources and consequences of embeddedness [59],
have typically highlighted the fact that since the nested architecture of
mutualistic networks implies that nodes contribute heterogeneously to their
vulnerability, the removal of some nodes that specially contribute to the
global nestedness of the network is consequentially associated with stronger
vulnerabilities. In a similar vein, [55] have studied the Boulogne-sur-Mer
Fish Market, focusing on the bipartite interactions between buyers and
sellers, and studied its resistance to perturbations that would affect high-
degree buyers or sellers, with the conclusion that the auction part of this
market was more robust. The theory of economic complexity has concurrently
associated nestedness and the dynamics of industrial ecosystems [51] notably
in relation to the resilience of economies to external shocks [60, 52].

In ecology, where the nested network structure tends to minimize competition
between species and support greater biodiversity [61], maximizing structural
stability [39], perturbations impacting generalist species have been shown to
lead to faster species depletion at the network level [26] by isolating
specialist species due to the nested structure.

Furthermore, and with respect to the stability of nested bipartite networks,
ecological studies have further shown that nested interaction networks emerge
by considering an optimization principle aimed at maximizing species abundance
[62] and that nested mutualistic interactions boost equilibrium population
densities and increase the resilience of communities [63] : typically, when
analyzing the short-term dynamics following a strong population perturbation,
mutualist networks are associated with an ability to replete affected
communities when species numbers fall dangerously low [63].

With respect to the investor-patent network, the attrition of generalist
communities of investors should therefore be associated with a potentially
severe impact on the entire network, an impact that could put technological
fields and the associated emerging technologies at risk. Perturbations
targeting generalist investors such as communities C.02 (investors capable of
investing different amounts at different stages without a strong sectoral
specialization) and C.06 (early-stage and business angels) stand out as the
highest vulnerabilities, which highlights the role that these investors play
for the system as a whole and for the diversity of technological fields of
innovation that receive funding. In addition to these two communities, using
funding amounts to weight the bipartite network links instead of the number of
interactions, community C.05 (late-stage and private equity investors) also
stands out as the most generalist investor community in this nested bipartite
network, an observation which could be of special relevance since this
community has been subject to a decrease in activity since 2022 [64, 65, 66].
Such a perturbation, as it affects a community crucial to the nested patent-
investor network, could be expected to have not only quantitative
consequences, as is generally foreseen, but also qualitative ones, putting
fields of technological innovation at risk.

In parallel, investor communities such as C.00 (investors specialized in the
Health Care sector) and C.03 (early-stage cryptocurrency investors) act as
specialist investors in the bipartite network, whose emergence could be
related to the need for specific skillsets in these sectors: for instance, the
Health Care sector (that corresponds here to module 3 in Fig. 3) is well-known
to be associated with very specific regulation and R&D cycles.

A relevant question here is whether the number of technological and sectoral
specificities will increase in relation to the development of so-called deep
techs, a category that includes medtechs, quantum computing or artificial
intelligence, each of which is associated with specific and emerging
regulations. Such a phenomenon could lead to an increased number of specialist
investor communities, which would in turn increase the modularity of the
patent-investor network, a phenomenon negatively correlated with network
robustness as mentioned above but which could also mitigate perturbations
affecting key generalist investors. A more modular structure of the bipartite
patent-investor network could result in fields of technological innovations
being dependent on a limited number of investor communities for their funding,
but conversely less dependent upon generalist investors.

Finally, and with respect to future research, startup databases are known to
under-represent early-stage funding rounds compared to later-stage ones due to
an easier tracking of the latter. Although we do not expect such a bias to
affect the results of our study, complementary analyses on reduced but more
exhaustive datasets could further clarify this issue. In addition, VC funding
tends to target novel, emerging and potentially disruptive technologies, while
others are funded by a more varied panel of investments which could also
warrant more comprehensive investigations, notably innovations that spin-off
from academia in the context of its specific set of institutions and
incentives [67]. Further studies could also attempt to directly assess the
robustness of the investor-startup network with respect to technological
diversity [35, 68] when facing different types of crises.

##  5 Conclusion

In this work, by leveraging large-scale financial, startup and patent
datasets, we have built a bipartite network directly linking investors and
technologies. Using network metrics, we have found this network to display an
emergent topological mutualism, associated with a heterogeneous degree
distribution, a significant nestedness and a significantly lower modularity
compared to null models. This has relevant implications for the robustness of
the ability of startups and investors to support technological innovation when
facing crises. We notably expect the system to react differently depending on
perturbations. In particular, perturbations affecting investor communities
that contribute strongly to the nestedness of the patent-investor network
could have a far-reaching impact on technological innovation.

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Appendix

Investor community | # of investors | Brief community description
---|---|---
C.00 | 110 | Health Care investors
C.01 | 132 | Historic investors
C.02 | 200 | Generalist investors active whole period
C.03 | 27 | Cryptocurrency investors
C.04 | 239 | EU-focused investors (UK and DE)
C.05 | 102 | Late-stage investors and PE
C.06 | 236 | Early-stage and BAs
C.07 | 189 | Accelerators
C.08 | 80 | Canada-focused investors
C.09 | 71 | France-focused Health Care-focused investors
C.10 | 122 | China-focused investors
C.11 | 158 | Early-stage post-2013 investors
C.12 | 201 | "New-generation" post-2013 investors
C.13 | 19 | Latin America-focused investors
C.14 | 78 | India and SEA-focused investors
C.15 | 53 | Japan-focused investors

Table 1: Description of investor communities. UK stands for United Kingdom, DE
for Germany, PE for Private Equity, BA for Business Angel, SEA for Southeast
Asia. "Historic" investors are investors that have been active for a long
period of time, since the late 1990s-early 2000s. "Generalist" investors are
investors that do not display a significant sectoral focus, investing in all
types of sectors and related technologies. "Cryptocurrency" investors are
investors strongly specialized in cryptocurrencies and related financial
sectors. "Late-stage" investors focus on the later stages of VC financing
(series B and onwards), typically investing very large amounts. "Early-stage"
investors focus on early stages of VC financing (pre-seed, seed and series A),
investing relatively small amounts. "Business Angels" are individuals who
invest their own money in startups, usually in early-stage rounds and low
amounts. "Accelerators" are a specific type of early-stage investors that
usually operate by selecting batches of companies for a short period,
providing them with small amounts of money and an intensive mentoring program
of a few months focused on developing specific aspects of the company.
"Post-2013" investors are investors that started being active (or greatly
increased their activity) around the 2013 period, where VC financing
experienced sudden and significant growth.

rowsep=0pt  patent cluster # of connections cluster label 1-grams 2-grams 0 74 Video Displaying Technology video | format | stream | frames | media video stream | video data | video content 1 186 Location-based Wireless Technology location | wireless | positioning | mobile | satellite mobile device | mobile station | wireless device 2 62 Cancer Treatment Therapies cancer | treating | inhibitor | combination | treatment treating cancer | methods treating | combination therapy 3 77 Fluid Valve Assembly valve | piston | fluid | chamber | pressure valve assembly | pressure tube | shock absorber 4 2801 Pharmaceutical Compound Therapy compounds | formula | thereof | derivatives | diseases pharmaceutical compositions | pharmaceutically acceptable | compounds formula 5 1938 Power Electronics Circuit power | voltage | circuit | output | signal power supply | clock signal | input signal 6 15 Wireless Network Technology network | wireless | access | mobile | service wireless network | network access | wireless device 7 232 Chemical Reaction Engineering catalyst | process | gas | stream | carbon gas stream | carbon dioxide | stream comprising 8 1104 Multimedia Streaming Services media | content | video | playback | audio media content | video content | playback device 9 862 Speech Processing Technology audio | speech | sound | voice | microphone audio signal | audio data | speech recognition 10 149 Pharmaceutical Formulations & Dosage pharmaceutical | formulations | composition | release | oral pharmaceutical composition | pharmaceutically acceptable | dosage form 11 132 Microbial Acid Production acid | production | microorganisms | microbial | amino amino acid | method producing | non naturally 12 60 Seismic Survey Techniques seismic | sensor | measurement | subsurface | acoustic seismic data | seismic trace | acoustic signal 13 41 Virus, Vaccine, Antigen virus | vaccine | protein | recombinant | vectors present invention | invention relates | nucleic acid 14 966 Online Advertising Services advertisement | advertising | ad | campaign | advertiser advertising campaign | advertising content | web page 15 119 Social Media Content content | item | social | user | online content item | social networking | social media 16 1 Vehicle Braking Systems brake | disc | caliper | disk | lining disc brake | brake disc | brake caliper 17 266 Cloud Computing Services application | computing | cloud | software | service cloud computing | computing environment | virtual machine 18 106 Augmented Reality Displays light | display | pixel | image | eye display device | light emitting | image light 19 1247 Motor Vehicle Assembly rotor | motor | shaft | assembly | vehicle steering column | aerial vehicle | motor vehicle 20 363 Video Compression Technology video | block | coding | prediction | picture video data | video coding | motion vector 21 200 Agricultural Management & Yield crop | agricultural | yield | plant | field agricultural field | crop yield | management zones 22 31 Location-based Tracking Technology location | mobile | tracking | geo | fence mobile device | tracking device | location information 23 111 Multi-Tenant Database database | application | custom | tenant | object access permissions | multi tenant | mechanisms methods 24 91 Network Management & Security network | traffic | service | policy | proxy communication network | network element | network agent 25 625 Battery Electrochemistry Technology fuel | electrolyte | battery | anode | electrode fuel cell | lithium ion | active material 26 4869 Payment & Transaction systems payment | transaction | merchant | card | account point sale | systems methods | mobile device 27 102 Web Page Management web | page | content | tab | user web page | context menu | tabs tab 28 220 Neural Network Technology neural | training | network | input | output neural network | convolutional neural | training neural 29 314 Ultrasound Medical Imaging ultrasound | imaging | tissue | image | ultrasonic ultrasound imaging | ultrasound device | ultrasound data 30 1713 Identity Authentication Technology authentication | key | identity | user | server user authentication | public key | private key 31 1204 Medical Monitoring Devices physiological | patient | monitoring | heart | blood heart rate | blood pressure | vital signs 32 68 Software Test Platform application | software | platform | test | file live multi | multi tenant | sdk platform 33 111 Footwear Assembly Tools tubular | upper | footwear | portion | projectile tubular element | projectile casing | entangling projectile 34 285 Polymer Composition Formulations composition | polymer | weight | comprising | containing composition comprising | invention relates | present invention 35 32 Heat Dissipation Technology heat | cooling | air | thermal | coolant heat dissipation | heat sink | heat exchanger 36 3542 Data Storage Systems storage | memory | cache | file | data data storage | encoded data | dispersed storage 37 102 HVAC Climate Control temperature | hvac | thermostat | energy | setpoint energy consumption | setpoint temperature | ambient temperature 38 2298 Nucleic Acid Analysis nucleic | sample | acid | dna | sequencing nucleic acid | methods compositions | invention provides 39 189 Medical Neural Stimulation stimulation | nerve | tissue | electrical | electrode electrical stimulation | nerve stimulation | peripheral nerve 40 30 Patient Support Equipment support | patient | deck | frame | foot patient support | support apparatus | hospital bed 41 2063 Wireless Communication Technology wireless | ue | station | transmission | channel base station | wireless communication | user equipment 42 723 Social Networking Platform social | networking | users | online | content social networking | user social | online social 43 977 Steel Cutting/Coating steel | material | sheet | surface | coating steel sheet | method producing | cutting edge 44 25 Light Imaging Technology light | image | imaging | lidar | sensor image sensor | light field | light pulses 45 43 Electronic Connectors connector | electrical | electronic | plug | housing electrical connector | electronic device | connector includes 46 280 Magnetic Sensor Devices sensor | magnetic | field | sensing | current magnetic field | magnetic sensor | field sensor 47 890 Organic LED/Solar solar | layer | light | emitting | photovoltaic light emitting | solar cell | emitting device 48 0 Integrated Circuit Devices circuit | transistor | voltage | integrated | semiconductor integrated circuit | semiconductor integrated | circuit includes 49 228 3D Printing Technology printing | ink | dimensional | build | printer dimensional printing | imprint lithonetworky | print head

rowsep=0pt  patent cluster # of connections cluster label 1-grams 2-grams 50 62 Food Composition & Protein protein | food | composition | soy | product soy protein | protein solution | food product 51 37 Solar Energy Conversion solar | photovoltaic | dc | power | inverter photovoltaic power | dc power | solar power 52 32 Content Delivery Network cdn | content | delivery | server | origin content delivery | delivery network | network cdn 53 10004 Pharmaceutical compositions/therapy compositions | invention | acid | cells | present present invention | invention relates | invention provides 54 95 Web Content Management content | folder | web | collection | item collection folder | content management | content item 55 8393 Image Processing & Autonomous Vehicles image | vehicle | object | display | camera image data | aerial vehicle | unmanned aerial 56 1464 Cybersecurity & Threat Detection security | malware | threat | malicious | risk malware detection | security platform | anomalies threats 57 2954 Medical Devices and Implants distal | catheter | implant | tissue | end distal end | proximal end | devices methods 58 67 Medical Stimulation Devices stimulation | tissue | ultrasound | nerve | transcutaneous transcutaneous stimulation | adipose tissue | electrical stimulation 59 386 Pharmaceutical Treatment Methods treating | treatment | administering | disease | compositions methods treating | compositions methods | pharmaceutically acceptable 60 41 Fluid Management Systems flow | valve | tubular | pipe | fluid tubular element | tubular section | flow path 61 234 Electronic Messaging Platform message | notification | messaging | user | email electronic message | agent performance | contact information 62 175 Drug Delivery Devices needle | drug | dose | syringe | delivery delivery device | drug delivery | piston rod 63 47 Optical Imaging and Analysis radiation | imaging | ray | detector | optical absorption data | light source | radiation source 64 576 Semiconductor Device Fabrication semiconductor | layer | substrate | region | gate semiconductor device | semiconductor substrate | dielectric layer 65 1104 Healthcare Information Systems patient | health | medical | healthcare | care health care | medical information | patient data 66 18 Acoustic-Piezoelectric Devices piezoelectric | acoustic | resonator | idt | surface acoustic wave | piezoelectric plate | piezoelectric element 67 94 Messaging & Collaboration message | messaging | user | chat | conversation instant messaging | user device | electronic message 68 62 Stem Cell Research cells | stem | pluripotent | progenitor | differentiation stem cells | pluripotent stem | progenitor cells 69 145 Gambling Games game | wager | player | gambling | entertainment entertainment game | real world | hybrid game 70 31 Internal Combustion Engine piston | valve | damper | crankshaft | engine connecting rod | control valve | compression ratio 71 108 Media Content Recommendation content | item | media | user | ratings content item | media content | content based 72 1275 Fluid & Gas Systems gas | fluid | liquid | chamber | air exhaust gas | heat exchanger | compressed air 73 56 Touch Sensing Technology touch | sensing | capacitive | sensor | capacitance touch sensor | touch sensitive | touch panel 74 474 Lidar Optical Technology optical | light | lidar | laser | beam light source | light beam | optical signal 75 43 Surgical Robotics Cluster surgical | instrument | robot | tool | effector surgical instrument | end effector | surgical tool 76 129 Biosensor Analysis Technology sample | analyte | electrode | test | biosensor working electrode | liquid sample | flow cell 77 14 Nucleic Acid Biotechnology nucleic | rna | acid | expression | sequence nucleic acid | control elements | promoter control 78 878 Optical Networking Technology optical | light | laser | waveguide | wavelength optical signal | optical fiber | light source 79 27 Gaming & Accessory game | player | battle | gaming | server game content | server device | game program 80 30 Vehicle Tire Monitoring tire | vehicle | sensor | pressure | door tire pressure | pressure sensor | door lock 81 28 Television Program Guide television | program | guide | interactive | schedule program guide | television program | interactive television 82 164 Semiconductor Memory Devices memory | semiconductor | layer | cell | bit memory cell | memory device | semiconductor memory 83 540 Antibody Therapy Research antibodies | antibody | binding | anti | cd antigen binding | binding fragments | present invention 84 127 Measurement Sensor Technology sensor | measuring | acoustic | measurement | gas gas sensor | data sheet | tank floor 85 61 Fluid Pump Devices fluid | pump | blood | gas | breast breast pump | breathing gas | piezo air 86 4816 Search Engine Technology search | query | document | results | queries search results | search query | search engine 87 107 Fluid Analysis Technology sample | fluid | flow | chamber | cartridge flow cell | fluid sample | analytical instrument 88 23 Image Forming Devices toner | sheet | member | forming | roller image forming | forming apparatus | main body 89 172 Microfluidic Devices/Systems microfluidic | fluid | sample | droplet | channel microfluidic device | microfluidic channel | cell processing 90 99 Location Tracking Technology location | mobile | devices | geonetworkic | determination mobile device | location information | location data 91 55 Serial Bus Protocol bus | serial | clock | signal | bit serial bus | clock signal | digital data 92 24 Wireless Power Management power | wireless | transmit | consumption | communication power control | transmit power | power consumption 93 49 Fluid Dispensing Devices dispensing | container | dispenser | outlet | liquid dispensing apparatus | dispensing device | liquid medicine 94 100 User Interface Design interface | user | ui | networkical | application user interface | networkical user | computing device 95 45 RNAi Therapy Cluster expression | compositions | gene | agents | treating compositions methods | double stranded | invention provides 96 1380 Networking & Traffic Management network | packet | traffic | routing | node network traffic | network device | virtual network 97 850 Telecommunication Services & Devices telephone | voice | caller | message | calls telephone number | calling party | called party

Table 2: Names of the patent clusters according to their ngrams. Cluster
labels were inferred from the top 20 1-grams and top 20 2-grams. The number of
connections represents the number of connections between the patent cluster
and all investor communities in the bipartite network.


# Site-specific symmetry sensitivity of angle-resolved photoemission
spectroscopy in layered palladium diselenide

M. Cattelan  Elettra Sincrotrone Trieste, SS 14, Km 163.5 AREA Sci. Pk., 34149
Trieste, Italy  C. J. Sayers  Dipartimento di Fisica, Politecnico di Milano,
20133 Milano, Italy  D. Wolverson  University of Bath, Ctr. Nanosci. and
Nanotechnol., Dept. Phys., Bath BA2 7AY, UK  E. Carpene
<EMAIL_ADDRESS>IFN-CNR, Dipartimento di Fisica, Politecnico di
Milano, 20133 Milano, Italy

###### Abstract

Two-dimensional (2D) materials with puckered layer morphology are promising
candidates for next-generation opto-electronics devices owing to their
anisotropic response to external perturbations and wide band gap tunability
with the number of layers. Among them, PdSe2 is an emerging 2D transition-
metal dichalcogenide with band gap ranging from $\sim 1.3$ eV in the monolayer
to a predicted semimetallic behavior in the bulk. Here we use angle-resolved
photoemission spectroscopy to explore the electronic band structure of PdSe2
with energy and momentum resolution. Our measurements reveal the
semiconducting nature of the bulk. Furthermore, constant binding-energy maps
of reciprocal space display a remarkable site-specific sensitivity to the
atomic arrangement and its symmetry. Supported by density functional theory
calculations, we ascribe this effect to the inherent orbital character of the
electronic band structure. These results not only provide a deeper
understanding of the electronic configuration of PdSe2, but also establish
additional capabilities of photoemission spectroscopy.

Transition metal dichalcogenides (TMDs) host highly attractive properties for
fundamental studies of novel physical phenomena and for applications ranging
from opto-electronics to sensing at the nanoscale tmd1 ; tmd2 . Among all
TMDs, those based on noble metals (Pd, Pt) have received less attention
because of their high cost, until the recent discovery of a layer-controllable
metal-to-semiconductor transition oye ; noble1 ; noble2 ; noble3 , which
motivated their investigation in the last few years. Similar to the
extensively investigated black phosphorus bp1 ; bp2 ; bp3 ; bp4 , with a band
gap varying from 0.3 eV in the bulk to 1.5 eV in the monolayer bpgap , PdSe2
is characterized by an in-plane puckered structure resulting in an anisotropic
response to external stimuli, such as light shg , electric field field  and
strain strain ; strain2 . In addition, a linear dependence of the band gap
with the number of layers has been observed oye ; gap2 : the monolayer is
predicted to have an indirect gap of 1.3 eV omega  which monotonically
decreases to 0 eV as the thickness exceeds $40-50$ layers, suggesting
semimetallic behavior. However, unlike black phosphorous and the majority of
TMDs which share hexagonal crystal structures, the low-symmetry pentagonal
atomic arrangement of PdSe2 gives rise to exotic thermoelectric, mechanical
and optical properties penta1 ; penta2 ; penta3 .

Here, we investigate the electronic structure of PdSe2 for the first time by
angle-resolved photoemission spectroscopy (ARPES). We clarify the
semiconducting nature of the bulk by direct measurements of the electronic
bands in reciprocal space. Furthermore, we reveal a previously unexplored
sensitivity of photoemission to the site-specific crystal symmetry. In
particular, constant binding-energy cuts of the surface-projected Brillouin
zone (BZ) disclose the dominant chemical/orbital character of the metal atoms
on the top-most valence band, while the effect of the chalcogen species
becomes relevant at binding energies exceeding 1 eV. This finding is
corroborated by plane wave density functional theory (DFT) calculations
employing Perdew-Burke-Ernzerhof (PBE) functional.

Figure 1: Unit cell (a), top (b) and side (c) views of PdSe2 crystal
structure. The red links in panel (a) indicate the bonds related to Se2
dimerization.

The crystallographic morphology of PdSe2 is sketched in Fig.1. Its stable
configuration is orthorhombic with Pbca space group (#61) and experimental
lattice parameters $a=0.575$ nm, $b=0.587$ nm and $c=0.77$ nm pbca , see
Fig.1a. Layers are normal to the c-axis. The top view (panel b) shows the
characteristic pentagonal atomic arrangement of the monolayer, while the side
view (panel c) reveals its puckered structure. Each PdSe2 layer is formed by
three atomic planes: Pd atoms in the middle are covalently bound to four Se
atoms located on the top and bottom sub-layers. In contrast with other TMDs
where the metal atom has $+4$ oxidation state, PdSe2 adopts the $+2$ oxidation
kem . This is achieved via (Se2)2- dimerization (see red links in Fig.1a) that
provides further bonding between the top and the bottom chalcogen sub-layers.

Fig.2a shows the measured band structure of PdSe2 along selected high symmetry
directions, following the orthorhombic unit cell notation shown in the inset
(see Methods for experimental details). The top of the valence band (VB) is
located at the center of the BZ and is adjacent to the Fermi level $E_{F}$.
The large spectral weight at the $S$ point ($E-E_{F}\simeq-3$ eV) suggests the
intersection of several bands.

Figure 2: (a) Electronic band structure of PdSe2 measured by ARPES ($h\nu=74$
eV). The inset shows a sketch of the first BZ. (b) $k$-resolved DOS obtained
with DFT calculations along the same path in reciprocal space used in panel a.
(c) DFT mapping of the CB (top) and VB (middle) on a 2D portion of the first
BZ: dashed black lines mark the Fermi level $E_{F}$. The 2D ARPES intensity
map at $E_{F}$ (bottom) reveals no signature of the CB.

Fig.2b reports the $k$-resolved density of states ($k$-DOS) obtained with DFT
calculations along the same symmetry lines adopted in panel a (see Methods for
computational details). The main features of the occupied states are
reproduced with remarkable adherence to the experimental data, in particular
the dispersion of the top VB and the large DOS at the $S$ point and
$E-E_{F}\simeq-3$ eV, that in fact hosts an intricate overlap of several
electronic bands. The coloured rectangles in panels a and b highlight
additional features common to both measured and calculated band structures.
Our DFT analysis identifies PdSe2 as a semimetal: the top graph in Fig.2c
shows the false-colour energy landscape of the lowest conduction band (CB) on
a 2D portion of the BZ. The minimum is located close to the $YS$ line
(internal coordinates $[k_{x},k_{y}]\simeq[0.33,0.43]$ at energy
$E-E_{F}\simeq-25$ meV). The middle graph shows the energy landscape of the
highest VB, where its maximum occurs at the $\Gamma$ point and energy
$E-E_{F}\simeq+55$ meV. Hence, there is an overall negative gap of
approximately 80 meV. In contrast to the calculations, no evidence of electron
pockets due to the CB crossing the Fermi level has been observed
experimentally, as shown in the lowest graph of Fig.2c that reports the
surface-projected ARPES map at the Fermi level. Thus, we conclude that PdSe2
is most likely a semiconductor in the bulk (See Methods - Fig.5 \- for
additional evidence). This result is in agreement with recent optical
measurements opt1 ; opt2 , but diverges from calculations that predict
semimetallic behaviour oye ; omega ; nscale ; hami . While it is known that
DFT generally underestimates band gaps in semiconductors dftgap ; dftgap2 ,
the fact that VB and CB are well separated in reciprocal space preserves their
individual electronic features, regardless of the computed value of the band
gap. In particular, the occupied states (experimentally observed by ARPES) are
reproduced by DFT with remarkable accuracy.

Figure 3: Constant binding-energy ARPES maps at (a) $E-E_{F}=-2.3$ eV and (c)
$E-E_{F}=-0.4$ eV. Red and blue dashed lines correspond to the surface-
projected orthorhombic and face-centered orthorhombic (fco) BZs, respectively
(see panels d-f). (b) The selected energies are shown on the measured band
structure. (d) Orthorhombic unit cell of PdSe2 with the corresponding BZ. The
orange-shaded area sketches the surface-projected BZ (SBZ). (f) fco unit cell
describing the Pd arrangement with the corresponding BZ and its surface
projection (blue-shaded area). (e) Comparison of orthorhombic and fco SBZs.

More insight on the electronic structure of PdSe2 is achieved by inspecting
the ARPES iso-energetic maps. Fig.3a shows the photoemission spectral weight
measured on the reciprocal plane at constant energy $E-E_{F}=-2.3$ eV (in
Fig.3b it corresponds to the red horizontal line cutting the band structure).
Horizontal and vertical axes are expressed in units of $\pi/a$ and $\pi/b$,
respectively. The red dashed lines identify the edges of the orthorhombic BZ.
The spectral weight clearly resembles the symmetry of the BZ with peaks
(yellow colour) at $\bar{X}$ and $\bar{Y}$ points and valleys (blue colour) at
$\bar{\Gamma}$. For clarity, each $\bar{\Gamma}$ point has been labeled with
the corresponding in-plane Miller indices $(hk)$. Fig.3c shows a photoemission
map at $E-E_{F}=-0.4$ eV, i.e. closer to $E_{F}$ (in Fig.3b, it corresponds to
the blue horizontal line cutting the band structure). The observed elliptical
shapes are the surface-projected paraboloids of the top-most VB. By comparing
Fig.3a and Fig.3c, one can deduce that all $\bar{\Gamma}$ points with Miller
indices satisfying the relation $h+k=$ odd are missing in panel c. Although
surprising at first glance, this observation finds a straightforward rationale
by assuming a different crystal symmetry.

We will unfold this concept referring to Fig.3d-f. Panel d recalls the
orthorhombic unit cell of PdSe2 and the corresponding BZ. It is known that
ARPES measurements of solids probe the so-called surface-projected BZ (SBZ)
based on energy and momentum conservation arpes . In panel d the SBZ is
represented by the orange-shaded area. A closer inspection of the unit cell
reveals that Pd atoms arrange on a face-centered orthorhombic (fco) lattice,
as evident in panel f: if chalcogen atoms were absent, the corresponding first
BZ would be the one sketched on the right-hand side bz  (notice the similarity
with the BZ of the standard face-centered cubic lattice) am . Panel e compares
the SBZs of the orthorhombic (panel d) and the fco (panel f) cells with
identical lattice parameters. It can easily be verified that the following
relations hold among wavevectors: $\bar{\Gamma}\bar{X}=\pi/a$,
$\bar{\Gamma}\bar{Y}=\pi/b$, $\bar{\Gamma}\bar{C}=2\pi/a$,
$\bar{\Gamma}\bar{D}=2\pi/b$.

Returning to the photoemission map of Fig.3c, the blue dashed lines identify
the SBZ of the fco unit cell and each elliptical shape is centered at the
$\bar{\Gamma}$ point of the face-centered lattice. Since Pd atoms arrange on
an fco lattice, our ARPES analysis suggests that the top-most VB originates
predominantly from Pd orbitals with little contribution from Se.

Figure 4: (a) Difference between Pd $4d$-projected and Se $4p$-projected
$k$-DOS. Blue and red colour represent Pd and Se character, respectively. (b)
Electron density (iso-surface value at 0.04 e$/\AA^{3}$) at the four selected
points of the band structure labeled in panel a.

In order to support this hypothesis, we have computed the orbital-projected
$k$-DOS and wave functions at selected points of the band structure. Fig.4a
reports the difference between Pd $4d$-projected and Se $4p$-projected $k$-DOS
along the same path of the reciprocal space employed previously in Fig.2 (see
Methods for more details on Fig.4). Blue or red colour represents the dominant
Pd or Se character, respectively. The highest VB shows a strong Pd $4d$
character with spectral weight $\sim 3$ times larger than Se $4p$ and marginal
contributions from other atomic orbitals. On the other hand, at $E-E_{F}=-2.3$
eV (i.e. the same as Fig.3a), the dominant orbital character varies with the
specific position on the BZ. Here, it is of interest to visualize the wave
functions in real space: Fig.4b displays the electron density at four
representative points of the band structure, labeled by circles in panel a.
The iso-surface value is taken at 0.04 e$/\AA^{3}$. On the highest VB (labels
1 and 2) the electron density is centered at Pd atoms with $d_{z^{2}}$-like
symmetry. At $E-E_{F}=-2.3$ eV (labels 3 and 4) the charge distribution
depends on the position within the BZ: at the $S$ point (label 3) it is
dominated by Se $4p$ orbitals note , while at $\Gamma$ (label 4) it is
combination of Pd $4d$ states. These results support our hypothesis that the
same single orbital of Pd (i.e. $4d_{z^{2}}$) shapes the top-most VB.
Recalling that Pd atoms form an fco lattice, this symmetry is retained also in
reciprocal space, as revealed by our ARPES data (see Fig.3c). A simple tight
binding approach leads to the same conclusion and is reported in the Methods
section. At larger binding energies both Pd and Se orbitals contribute to the
band structure, exhibiting the standard orthorhombic symmetry of Fig.3a.

In conclusion, we have measured the electronic band structure of bulk PdSe2 by
angle-resolved photoemission spectroscopy. Within the experimental accuracy,
our data confirm its semiconducting nature with a minimum band gap of 50 meV
(i.e. the instrumental resolution) since no evidence of conduction band across
the Fermi level has been observed, while all electronic dispersive features
below $E_{F}$ are well-reproduced by our DFT calculations. Furthermore, we
have demonstrated a remarkable sensitivity of the ARPES technique to site-
specific symmetries of the electronic structure. This finding can be pivotal
in tuning the electronic properties of PdSe2-based heterostructures nano ,
analogous to the observed dependence of the gap on the band character of
MoS2/graphene nano2 ; nano3 . Moreover, we envisage that the chemical
selectivity of ARPES allows a fine-tuning of the electronic properties. For
example, chemical substitution of metal atoms chem  will give rise to specific
changes in the VB related both to doping and to modifications in the surface
symmetry, to which ARPES will be sensitive. We believe this implementation is
not limited to PdSe2, but it applies to a much wider class of compounds with
complex crystal structures and can help clarify the subtle interactions
related to correlated electronic phases, such as metal-insulator transitions,
charge density waves and superconductivity cdw1 ; cdw2 .

Acknowledgments
EC acknowledges funding from the Italian PRIN Grant 2017BZPKSZ. Some of the
computational work was performed on the University of Bath’s High Performance
Computing Facility.

Author Contributions
E.C. and C.J.S. proposed the experiments. M.C. performed the ARPES
measurements. M.C., C.J.S and E.C. analysed the experimental data. D.W. and
E.C. performed the DFT calculations. E.C. wrote the manuscript with input from
all authors.

Competing interests
The authors declare no competing interests.

METHODS

ARPES. Bulk PdSe2 crystals were obtained commercially from HQ Graphene hq .
Samples were cleaved in ultra-high vacuum ($10^{-10}$ mbar) to expose a clean
surface. Photoemission measurements were performed at the Spectromicroscopy
beamline of the Elettra light source using 74 eV linearly polarized radiation
focused to an approximately 0.6 $\mu$m diameter spot by a Schwarzschild
objective elettra  and incident at 45∘ with respect to the sample. The energy
and momentum resolution of the hemispherical electron analyzer are $\sim 50$
meV and 0.03 Å-1, respectively. The sample temperature was maintained at 95 K.

DFT. The electronic band structure of PdSe2 was computed with the Quantum
Espresso package qe . Exchange-correlation was considered using the Perdew-
Burke-Ernzerhof functional revised for solids (PBEsol). Van der Waals
interaction among PdSe2 layers was included using the semiempirical Grimme’s
DFT-D2 correction grimme . Atoms were allowed to relax until the residual
forces were below 0.0026 eV/Å. Cutoff energy of 60 Ry and $8\times 8\times 6$
$k$-point mesh were used. The iso-surface rendering in Fig.4b was performed
with the VESTA software vesta .

Absence of CB evidence in ARPES data.

Figure 5: ARPES constant energy maps at $E-E_{F}=-0.5$ (bottom), 0 (middle)
and $+0.5$ eV (top), revealing no sign of the CB.

Fig.5 reports three constant-energy ARPES maps of PdSe2 across the Fermi level
($E-E_{F}=+0.5$, 0, $-0.5$ eV, top to bottom). Each map employs a logarithmic
intensity scale ranging from 5% to 100% of the respective maximum. Even well
above $E_{F}$ we have detected no sign of electron pockets originating from
the CB, as DFT calculation would predict. Although we are not able to
determine the value of the band gap with static ARPES, our data uphold the
semiconducting nature of bulk PdSe2.

Orbital character of the band structure (Fig.4a).

Figure 6: $k$-DOS of (a) Pd $4d$ orbitals and (b) Se $4p$ orbitals. (c)
Comparison of Pd $4d$ (blue) and Se $4p$ (red) $k$-DOS at the $\Gamma$ point
and their difference (black-dashed).

Pd $4d$ and Se $4p$ electrons determine the valence and conduction states of
PdSe2. It is instructive to ”visualize” the specific orbital character and in
particular the Pd-Se duality along high symmetry lines of the BZ. Here, we
employ a simple colour-coded approach: the $k$-resolved density of states
projected on Pd $4d$ and Se $4p$ are shown in Fig.6a and Fig.6b, respectively.
Taking the difference between the data of these two graphs we obtain Fig.4a,
where blue and red colours encode positive (Pd) and negative (Se) values. As
an example, the Pd and Se $k$-DOS (the latter is represented on the negative
abscissa) and their difference at the $\Gamma$ point are shown in Fig.6c.
Notice, in particular, that the Pd-projected $k$-DOS at the Fermi level (i.e.
the VB top) is approximately 3 times larger than Se, as claimed in the main
manuscript.

Tight binding approach and matrix element effect. In a regular MX6 octahedral
complex ($O_{h}$ symmetry) the five outer $d$ orbitals of the transition metal
M arrange into the high-energy, double-degenerate $e_{g}$ and the low-energy,
triple-degenerate $t_{2g}$ states cf , as sketched in Fig.7a.

Figure 7: (a) Regular MX6 octahedron with the corresponding crystal field
splitting of $d$ orbitals into $e_{g}$ and $t_{2g}$ states. (b) Elongated
PdSe6 octahedron along the c-axis: the degenerate $e_{g}$ states split into
HOMO and LUMO.

A closer look at the crystal structure of PdSe2, shown in Fig.7b, reveals that
each Pd is surrounded by four Se atoms belonging to the same monolayer and two
apical atoms (in yellow colour) belonging to the nearest upper and lower
layers. The six chalcogen atoms form an octahedron elongated along the
$c$-axis ($D_{4h}$ symmetry). This distortion lifts the degeneracy of the
$e_{g}$ states, resulting in the $d_{z^{2}}$ orbital being energetically more
favorable than the $d_{x^{2}-y^{2}}$ cf ; cf2 ; ec1 . As we already pointed
out, in PdSe2 the oxidation state of Pd is $+2$ and its electronic
configuration is therefore $4d^{8}$: six of these electrons fill the $t_{2g}$
states completely, while the remaining two occupy the $d_{z^{2}}$ orbital,
leaving $d_{x^{2}-y^{2}}$ empty. $d_{z^{2}}$ is therefore the highest occupied
molecular orbital (HOMO) forging the top of the VB, while $d_{x^{2}-y^{2}}$
represents the lowest unoccupied molecular orbital (LOMO) contributing the
bottom of the CB, in agreement with our own calculations (Fig.4b) and other
recent work ec2 .

Figure 8: Pd atoms of a single PdSe2 layer using (a) the fco unit cell, (b)
the orthorhombic cell. Tight-binding VB dispersion employing (c) the fco cell,
(d) the orthorhombic cell. (e) Equivalence of (c) and (d) after considering
matrix element effect. (f) Top-VB simulation on the $xy$ reciprocal plane
(compare with the experimental data of Fig.3c). The Miller indices refer to
the orthorhombic cell: the missing VB peaks at (10) and (01) endorse the fco
symmetry.

We can now elucidate the symmetry features of the top-VB observed by
photoemission using a simple 2D tight binding approach that involves only the
HOMO. Metal atoms of a PdSe2 monolayer arrange on a rectangular lattice as
shown in Fig.8. For simplicity we will assume a square lattice of side $a$. If
the fco unit cell is used (panel a), a single site $A$ is sufficient to
describe the crystal. On the other hand, if the orthorhombic unic cell is
assumed (panel b) two sites $A$ and $B$ must be considered: each atom $A(B)$
is surrounded by four nearest neighbors $B(A)$. In the fco case, a tight
binding approach is straightforward. Let $|R_{A}\rangle$ be the Wannier wave
function at the specific lattice site $R$ (i.e. the $d_{z^{2}}$ orbital). The
ovelap integral is $\gamma=\langle R_{A}|U|0_{A}\rangle$ ($U$ is the periodic
lattice potential), where $R$ runs over the 4 nearest neighbors $(+a/2,+a/2)$,
$(+a/2,-a/2)$, $(-a/2,+a/2)$, $(-a/2,-a/2)$ and the eigenvalue of the
hamiltionian $\mathbb{H}$ is
$E(k)=E_{A}+\gamma\sum_{n.n.}e^{ikR}=E_{A}+4\gamma\cos(k_{x}a/2)\cos(k_{y}a/2)$,
with $E_{A}=\langle R_{A}|\mathbb{H}|R_{A}\rangle$. Fig.8c shows the resulting
VB dispersion along the $x$-axis ($k_{y}=0$). If the orthorhombic cell is
used, two Wannier wave functions $|R_{A}\rangle$ and $|R_{B}\rangle$ form the
basis of the tight binding hamiltonian:

$\displaystyle\mathbb{H}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}\
E_{A}&h\\\ h^{*}&E_{B}\end{pmatrix}$ (1)

It can easily be verified that the hopping term between sites $A$ and $B$ is
$h=4\gamma\cos(k_{x}a/2)\cos(k_{y}a/2)$ like in the previous fco case. Since
sites A and B are equivalent, it also follows that $E_{A}=E_{B}$. Thus, the
two eigenvalues are
$E_{\pm}(k)=E_{A}\pm|4\gamma\cos(k_{x}a/2)\cos(k_{y}a/2)|$. Fig.8d depicts
$E_{\pm}(k)$ along the $x$-axis ($k_{y}=0$). The corresponding eigenstates are
$(\pm\frac{h}{|h|},1)$ and the generic wave function at the lattice site $R$
reads matrix :
$|R_{\pm}\rangle=\pm\frac{h}{|h|}e^{ikR_{A}}|R+R_{A}\rangle+e^{ikR_{B}}|R+R_{B}\rangle$.
In the free electron final state approximation (here, the final state
$|k_{f}\rangle$ is a plane wave with wavevector $k_{f}$), the photoemission
matrix element $\mathbb{M}$ is expressed as the Fourier component of the tight
binding orbital $|0_{\pm}\rangle$ matrix , i.e.

$\displaystyle\mathbb{M}\propto\langle k_{f}|0_{\pm}\rangle$ $\displaystyle=$
$\displaystyle\langle
k_{f}|(\pm\frac{h}{|h|}e^{ikR_{A}}|R_{A}\rangle+e^{ikR_{B}}|R_{B}\rangle)$ (2)
$\displaystyle=$ $\displaystyle\pm\frac{h}{|h|}e^{ikR_{A}}\langle
k_{f}|R_{A}\rangle+e^{ikR_{B}}\langle k_{f}|R_{B}\rangle$ (3) $\displaystyle=$
$\displaystyle\pm\frac{h}{|h|}e^{i(k-k_{f})R_{A}}\langle
k_{f}|0_{A}\rangle+e^{i(k-k_{f})R_{B}}\langle k_{f}|0_{B}\rangle$ (4)

which, using the momentum conservation ($k=k_{f}$), leads to the following
photoemission intensity:

$I_{\pm}\propto\left|\mathbb{M}\right|^{2}\propto\left|\pm\frac{h}{|h|}\langle
k_{f}|0_{A}\rangle+\langle k_{f}|0_{B}\rangle\right|^{2}=\left|\langle
k_{f}|0_{A}\rangle\right|^{2}+\left|\langle
k_{f}|0_{B}\rangle\right|^{2}\pm\frac{h}{|h|}2\Re(\langle
k_{f}|0_{A}\rangle\langle 0_{B}|k_{f}\rangle)$ (5)

that simplifies to $I_{\pm}\propto 2\left|\langle
k_{f}|0_{A}\rangle\right|^{2}\left(1\pm\frac{h}{|h|}\right)$ since
$|0_{A}\rangle=|0_{B}\rangle$. At this point we notice that $\frac{h}{|h|}=$
sgn$[\cos(k_{x}a/2)\cos(k_{y}a/2)]$. Thus, $I_{\pm}\propto(1\pm$
sgn$[\cos(k_{x}a/2)\cos(k_{y}a/2)]$. Referring to Fig.8d, it is easily
verified that the previous equation completely suppresses the photoemission
intensity of one eigenvalue $E_{\pm}(k)$ depending on the values of $k_{x}$
and $k_{y}$. Fig.8e-f show the results: as expected, the fco band structure of
Fig.8c and the experimental data of Fig.3c are recovered since the use of two
equivalent sites is redundant and the appropriate unit cell is the fco.

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