# ChiBench: a Benchmark Suite for Testing Electronic Design Automation Tools

Rafael Sumitani  0009-0003-5226-1966 UFMGBrazil<EMAIL_ADDRESS>,
João Victor Amorim  0009-0001-6775-2579 UFMGBrazil<EMAIL_ADDRESS>,
Augusto Mafra  0000-0000-0000-0000 CadenceBrazil<EMAIL_ADDRESS>,
Mirlaine Crepalde  0000-0000-0000-0000 CadenceBrazil<EMAIL_ADDRESS>and
Fernando M Quintão Pereira  0000-0002-0375-1657 UFMGBrazil
<EMAIL_ADDRESS>
###### Abstract.

Electronic Design Automation (EDA) tools are software applications used by
engineers in the design, development, simulation, and verification of
electronic systems and integrated circuits. These tools typically process
specifications written in a Hardware Description Language (HDL), such as
Verilog, SystemVerilog or VHDL. Thus, effective testing of these tools
requires benchmark suites written in these languages. However, while there
exist some open benchmark suites for these languages, they tend to consist of
only a handful of specifications. This paper, in contrast, presents ChiBench,
a comprehensive suite comprising 50 thousand Verilog programs. These programs
were sourced from GitHub repositories and curated using Verible’s syntactic
analyzer and JasperTM’s HDL semantic analyzer. Since its inception, ChiBench
has already revealed bugs in public tools like Verible’s obfuscator and
parser. In addition to explaining some of these case studies, this paper
demonstrates how ChiBench can be used to evaluate the asymptotic complexity
and code coverage of typical electronic design automation tools.

Benchmark, Verilog, Testing

††ccs: Software and its engineering Compilers

##  1\. Introduction

EDA (Electronic Design Automation) tools are software applications used by
engineers in the design, development, simulation, and verification of
electronic systems and integrated circuits (ICs). These tools cover various
stages of the electronic design process, from conceptualization and design
entry to implementation, verification, and testing. Examples of EDA tools
include Cadence Jasper for formal verification111Available at
https://www.cadence.com/en_US/home/tools/system-design-and-
verification/formal-and-static-verification.html and Verible’s
tool222Available at https://github.com/chipsalliance/verible. These tools
operate on similar types of input data: programs in some Hardware Description
Language (HDL), such as Verilog, SystemVerilog, VHDL or SystemC. Thus, the
effective development and testing of such tools require benchmarks in these
languages.

#### Verilog Benchmarks (or their lack thereof)

A Benchmark Suite is a collection of programs used to test computing systems
that process such programs. There exist open source benchmark collections
tailored for EDA tools, such as the ISCAS Benchmark Circuits (Brglez et al.,
1989) (31 circuits), the MCNC Benchmark Circuits (Koźmiński, 1991) (19
circuits in the YAL format), the EPFL Combinational Benchmark Suite (Amaru et
al., 2019) (23 circuits), the RAW Benchmark Suite (Babb et al., 1997) (twelve
programs), the KOIOS collection (19 circuits implementing different neural
networks) and the Titan23 suite of 23 circuits (Murray et al., 2015). These
collections contain a small number of programs: typically less than 50. This
fact is unfortunate because, in the works of Wang and O’Boyle (2018):
“Although there are numerous benchmark sites publicly available, the number of
programs available is relatively sparse compared to the number that a typical
compiler will encounter in its lifetime.” This paper mitigates this problem,
releasing a much larger collection of Verilog circuits.

#### The Contribution of this Work

This paper describes ChiBench, an open collection of 50K Verilog programs,
mined from open-source GitHub repositories. These programs were curated in a
three-step process, which Section 2 explains: first, Verilog codes were
automatically scraped from public code repositories. In a second phase, each
program was parsed using Verible’s syntactic analyzer, to ensure compliance
with the IEEE Standard 1364 (Thomas and Moorby, 1996). Finally, these programs
were sieved via Jasper’s HDL semantic analyzer, to ensure that each circuit
contains all the required dependencies.

This paper illustrates three different usage scenarios where ChiBench has been
employed, using two tools from the Verible project as test subjects. First,
Section 3.1 demonstrates how ChiBench programs can be used to investigate the
asymptotic complexity of EDA tools. Second, Section 3.2 explains how these
programs can test EDA tools by gauging the coverage of ChiBench as a test
dataset. Finally, Section 3.3 briefly describes two bugs discovered in open-
source tools through ChiBench-based tests.

##  2\. The Construction of a Benchmark Suite

In order to build our Benchmark Suite, we have mined programs from open-source
GitHub repositories, using GitHub’s REST API 333Available at
https://docs.github.com/en/rest. We use GitHub’s API to build a list of
candidate Verilog repositories. Said list is sorted by popularity (measured as
the number of stargazers). We remove from the candidate list repositories that
are not available for public usage, due to the lack of a license. Thus, for
each repository $R$ in the sorted list, we have implemented a Python script
that proceeds as follows:

  1. (1)

Clone $R$ and locally copy all its .v files;

  2. (2)

Assigns a unique name to each .v file, based on its repository and its local
path;

  3. (3)

Remove any special characters from the file’s name to avoid encoding issues.

We repeat the above sequence of steps for all the repositories in the base
list, until reaching a predefined number of files. This threshold is set upon
calling the mining script. Currently, ChiBench provides only Verilog programs;
however, the mining script can be easily adapted to fetch files in any other
language that is tagged in GitHub, such as VHDL.

###  2.1. Curating the Data

After we have copied the necessary number of Verilog files from GitHub, we
proceed to select valid programs. To this effect, we only keep files that are
syntactically and semantically valid. Thus, this process involves passing the
files through two sieves. The first sieve, the syntax analysis, happens via
the Verible syntactic analyzer. At this stage, if Verible’s parser cannot
build an abstract syntax tree for a file, we discard it. Example 2.1
illustrates one such situation.

###### Example 2.1.

The program in Figure 1, which specifies an 8-bit counter, will be filtered
out by the syntactic filter. It contains a missing semicolon at Line 7. Such
syntactically invalid files are uncommon in the mining process. Nevertheless,
they occur, as the repositories contain, for instance, files that are still
under development.

Figure 1. Verilog specification filtered out by our syntactic verification.

Verilog specification filtered out by our syntactic verification.

Once we remove any syntactically invalid programs, we use Jasper’s HDL
semantic analyzer to filter out any semantically invalid programs444Jasper’s
HDL semantic analyzer is triggered with the analyze built-in command.. Notice
that Jasper’s HDL analyzer also rejects invalid syntax. However, Jasper’s HDL
analyzer is more computationally expensive than Verible’s because it also
considers semantic analysis, being more restricted to Verilog language
standards. Consequently, in order to reduce the number of programs sent for
semantic analysis, we chose to filter out syntactically invalid programs
before using Jasper. Example 2.2 better explains what is the role of the
semantic analysis.

###### Example 2.2.

Figure 2 shows an example of a program that fails the semantic sieve due to a
type inconsistency. In this case, the IEEE standard forbids the declaration of
data ports with the wire type. Our data generation process considers each file
independently, thus, this error might occur. Nevertheless, our experience is
that most Verilog programs can be successfully validated as a single
compilation unit.

Figure 2. Verilog specification that fails the semantic test.

Verilog specification filtered out by our semantic verification.

###  2.2. Licensing

ChiBench only contains files that provide permissive licenses. In other words,
we remove from the public distribution of this collection any Verilog
specification that comes from either a repository without a license or that
comes from a repository whose license prevents distribution. Figure 3 shows
the number of licenses in each repository used to build ChiBench. Each
ChiBench file contains, as a comment, the URL of the repository from where it
comes, plus its license.

Figure 3. Licenses of repositories used to build ChiBench.

Licenses of repositories used to build ChiBench.

##  3\. Evaluation

The goal of this section is to demonstrate that ChiBench is a useful
collection of benchmarks. To this end, we shall investigate the following
three research questions:

RQ1::

Can ChiBench Programs be used to infer, empirically, the asymptotic complexity
of EDA tools?

RQ2::

How much coverage should we expect to obtain by using ChiBench as a dataset to
test EDA tools?

RQ3::

Are ChiBench-based tests able to uncover zero-day bugs in well-known EDA
tools?

RQ4::

How are ChiBench’s programs characterized in terms of size?

#### Experimental Setup

The evaluation described in this Section uses two tools available in the
Verible Project (Release v0.0-3622-g07b310a3, Mar 13th, 2024) as test
subjects: the parser and the obfuscator. Experiments run on an AMD Ryzen 9
5900X 12-Core 3.7GHz processor featuring Ubuntu Linux 22.04.4 LTS (kernel
6.5.0-27-generic). Coverage is measured via Clang’s source-based code coverage
feature (available in Clang 14.0.0).

###  3.1. RQ1 – Asymptotic Analysis

The asymptotic complexity of a tool is an expression that relates the running
time of that tool as a function of its input size. Determining an analytical
formula for the asymptotic complexity of a tool is a challenging problem, as
this formula is influenced by many details of that tool’s implementation.
Thus, as an alternative, to understand the asymptotic behavior of
implementations of algorithms, researchers resort to the so-called empirical
complexity analysis. In the words of Sumitani et al. (2023): “Empirical
Complexity Analysis is a branch of computer science that tries to infer the
asymptotic complexity of algorithms through the observation of multiple
executions of that algorithm with different inputs”. In this paper, we show
how ChiBench can be effectively used as a means to carry out empirical
complexity analysis.

Figure 4. The impact of the number of tokens on Verible’s parsing time. This
figure contains about 50K points—one point per program in the ChiBench
collection.

The impact of the number of tokens on Verible’s parsing time.

#### Discussion

Figure 4 relates the number of tokens in ChiBench programs with the time that
Verible’s syntactic analyzer takes to parse these programs. To perform this
experiment, we used the programs in the ChiBench collection as input to
Verible’s parser. As Figure 4 shows, the running time behavior of Verible is
linear. In this regard, Pearson’s coefficient relating running time and number
of tokens is 0.99, with a p-value of less than $2^{-16}$. Therefore, Verible’s
parser is expected to run in linear time on the number of tokens that form the
program with very strong probability.

###  3.2. RQ2 – Coverage Analysis

Coverage is a metric that quantifies the effectiveness of a test suite. In
this paper, we report coverage in two ways. Either as the number of lines in
the source code of a program that the test suite exercises. Or else as the
number of branches exercised by the test case in the binary representation of
that same program. Thus, we define the coverage ratio as either the number of
lines covered divided by the total lines of code or as the number of branches
covered divided by the total number of branches. The higher the coverage
ratio, the better the test suite. This section analyzes the coverage ratio of
ChiBench.

Figure 5. Code coverage of different tools of the Verible Framework, using the
128 largest programs from ChiBench.

Code coverage of different tools of the Verible Framework, using the 128
largest programs from ChiBench.

#### Discussion

Figure 5 presents the code coverage for Verible’s obfuscator and parser
assessed with the 128 largest programs from ChiBench. The coverage pattern for
both tools is similar; however, the parser demonstrates a notably higher
coverage. For lines covered it ranges from 10% to 19% against 4.5% to 7% for
the obfuscator. In terms of branches covered, the parser starts at 31% and
reaches 36% whereas the obfuscator begins at 11% and ends at nearly 12%. The
red lines represent the code coverage when using all programs from ChiBench.
For the parser, we achieved 42% coverage for lines and 48% for branches. In
comparison, the obfuscator reached 14% for lines and nearly 16% for branches.

We hypothesize that this disparity arises because the parser is a larger
feature and, consequently, calls a larger variety of functions from Verible’s
modules. In contrast, the obfuscator is a more self-contained tool. Although
these numbers seem low in principle, we remind the reader that Verible is a
large framework, which includes several tools, such as a linter, a code
formatter, and a language server. The code that forms these parts, although
part of the Verible binary library, remained largely unseen during this
experiment.

###  3.3. RQ3 – Actual Bug Reports

One of the primary goals of a benchmark suite is to uncover bugs in the
implementation of tools that read this suite. ChiBench was initially conceived
to stress-test any EDA tool and to demonstrate the effectiveness of this
collection as a bug-finding mechanism, we apply it onto open-source tools
available in the Verible Framework.

#### Discussion

We have used ChiBench to evaluate the correctness of Verible’s parser and code
obfuscator. In this case, we define as a buggy evaluation the analysis of a
program from ChiBench that results in a “crash”; that is, an execution of the
subject tool that leads to either an assertion or to an operating system
exception (such as a segmentation fault). Under these definitions, we have
observed one buggy evaluation on the parser and another on the obfuscator.
Both have been reported to the Verible’s community:

  * •

https://github.com/chipsalliance/verible/issues/2159:
Verible’s obfuscator crashes when reading a program that only contains the
pragma directive.

  * •

https://github.com/chipsalliance/verible/issues/2181:
Verible’s parser crashes instead of reporting syntax errors related to
instantiation type.

Interestingly, the second issue—which has been acknowledged as a true bug—was
not activated by a program in ChiBench itself, but by a program derived from
the suite: to maximize the diversity of ChiBench programs, we have used this
collection to generate new programs. Generation works as follows: we use the
50K programs in ChiBench to calculate the probability that each production
rule in Verible’s parser is exercised when parsing a program. Then, we use
this probabilistic grammar to generate new programs. In this case, every type
is set as logic.

###  3.4. RQ4 – Size Characterization

ChiBench programs are mined from open-source repositories; hence, we speculate
that they approximate the average Verilog program that represents typical
circuits. This section provides some characterization of such programs. To
this end, we analyze their size distribution.

#### Discussion

Figure 6 shows a density curve representing the size distribution of ChiBench
programs. We measure size as the number of tokens that the Verible lexer
produces for each Verilog file. Notice that this metric does not depend on
user-defined names. Figure 6 makes it clear that most Verilog specifications
are small. In total, ChiBench contains 50,611 Verilog programs. Out of this
lot, 66.87% contain less than 1,000 tokens, and 79.24% contain less than
2,000. The largest program in ChiBench contains 25,690,281 tokens, and the
smallest contains only 4. The average number of tokens is 33,350.9, and the
median number is 489.

Figure 6. Distribution of ChiBench programs per size, measured in number of
tokens.

Distribution of ChiBench programs per size, measured in number of tokens.

##  4\. Related Work

As we have already mentioned in Section 1, there exist already collections of
benchmarks formed by hardware specification languages (Brglez et al., 1989;
Koźmiński, 1991; Amaru et al., 2019; Babb et al., 1997; Murray et al., 2015).
However, these collections are small: never containing more than 50 circuits.
Nevertheless, with the rising popularity of large language models, this
scenario has seen changes in the last year.

As an example of this new trend, at the end of 2023, Thakur et al. (2024)
released VeriGen, a version of the CodeGen (Nijkamp et al., 2023) fine-tuned
for the synthesis of Verilog specifications. In the process of tuning CodeGen,
Thakur et al. have collected 50K Verilog circuits. However, in contrast to
ChiBench, the dataset used by Thakur et al. has not undergone any form of
filtering; hence, we do not know if these programs are semantically valid.
This dataset is publicly available555At
https://huggingface.co/datasets/shailja/Verilog_GitHub; however, each program
is a single line string. Parsing these programs automatically to reconstruct
the original Verilog specification is not possible, due to the presence of
line comments in the codes. This shortcoming is not a problem in the context
of Thakur et al.’s work, given that they are interested in building a large
language model that is based on k-grams of Verilog codes. Yet, the dataset
used to train the model could not be used, for instance, to stress-test EDA
tools. We believe that ChiBench might be useful to support the implementation
of models like VeriGen. To this effect, independent evaluations of VeriGen
have found that “A primary contributing factor to this shortfall [the
inability to uncover bugs in EDA tool] is the insufficiency of HDL code
resources for training” (Nijkamp et al., 2023). This perceived lack of
benchmarks seems to be a common issue among researchers working on large
language models for hardware specifications (Yao et al., 2024; Tsai et al.,
2024).

##  5\. Conclusion

This paper has described ChiBench, a collection of 50K Verilog programs mined
from open-source GitHub repositories. In addition to explaining the
methodology to build this suite, this paper showed how ChiBench can be
effectively used to test and analyze the behavior of electronic design
automation tools. The infrastructure used in the construction of ChiBench can
be adjusted to other languages, such as VHDL. Thus, future work might involve
maximizing the diversity of ChiBench programs to broaden coverage of EDA
tools.

#### Software

ChiBench is available at https://github.com/lac-dcc/chimera. Each benchmark
available in ChiBench provides, as a header comment, its original license, in
addition to the repository from where that specification was obtained.

## Acknowledgement

This project is sponsored by Cadence Design Systems. Additionally, the authors
acknowledge the support of CNPq (grant 406377/2018-9), FAPEMIG (grant
PPM-00333-18), and CAPES (grant “Edital CAPES PrInt”).

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# Emergence of 6-particle “hexciton” states in WS2 and MoSe2 monolayers

J. Choi1,2, J. Li1,3, D. Van Tuan4, H. Dery4,5, S. A. Crooker1 1National High
Magnetic Field Laboratory, Los Alamos, NM 87545, USA  2Advanced
Instrumentation Institute, Korea Research Institute of Standards and Science,
Daejeon 34113, Korea  3Wuhan National High Magnetic Field Center and School of
Physics, Huazhong University of Science and Technology, Hubei 430074, China
4Department of Electrical and Computer Engineering, University of Rochester,
Rochester, New York 14627, USA  5Department of Physics and Astronomy,
University of Rochester, Rochester, New York 14627, USA

###### Abstract

When doped with a high density of mobile charge carriers, monolayer
transition-metal dichalcogenide (TMD) semiconductors can host new types of
composite many-particle exciton states that do not exist in conventional
semiconductors. Such multi-particle bound states arise when a photoexcited
electron-hole pair couples to not just a single Fermi sea that is quantum-
mechanically distinguishable (as for the case of conventional charged excitons
or trions), but rather couples simultaneously to multiple Fermi seas, each
having distinct spin and valley quantum numbers. Composite six-particle
“hexciton” states were recently identified in electron-doped WSe2 monolayers,
but under suitable conditions they should also form in all other members of
the monolayer TMD family. Here we present spectroscopic evidence demonstrating
the emergence of many-body hexcitons in charge-tunable WS2 monolayers (at the
A-exciton) and MoSe2 monolayers (at the B-exciton). The roles of
distinguishability and carrier screening on the stability of hexcitons are
discussed.

The monolayer transition-metal dichalcogenide (TMD) semiconductors such as
WSe2, MoSe2, WS2, and MoS2 host a multitude of excitonic complexes, due in
part to the spin-orbit-split nature of their conduction and valence bands at
the $K$ and $K^{\prime}$ valleys of the Brillouin zone Urbaszek:2018 ;
Muller:2018 . When they are optically allowed and possess a non-zero
oscillator strength, these bound complexes manifest as discrete resonances in
optical absorption spectra. Early studies of nominally undoped TMD monolayers
revealed pronounced absorption peaks from the fundamental electron-hole
optical transition (i.e., the $X^{0}$ neutral exciton) Splendiani:2010 ;
Mak:2010 ; Li:2014 . Subsequent studies of charge-tunable TMD monolayers
demonstrated the emergence, at lower energy, of additional strong absorption
lines when the monolayer was populated with a background Fermi sea of holes or
electrons (i.e., the $X^{\pm}$ charged excitons) Ross:2013 ; Mak:2013 ;
Wang:2017NL . Whether, and under what conditions, a $X^{\pm}$ complex is most
accurately described as a simple three-particle ‘trion’ (wherein the
photoexcited exciton binds a carrier from the Fermi sea Ross:2013 ; Mak:2013 ;
Wang:2017NL ; Kheng:1993 ; Astakhov:2002 ; BarJoseph:2005 ), or a four-
particle ‘tetron’ (a trion additionally correlated with the resulting hole
that is left behind in the Fermi sea Bronold:2000 ; Suris:2001 ; Suris:2003 ;
Combescot:2018 ; Rana:2020 ), or an ‘exciton-polaron’ (an exciton dressed by
collective excitations of the Fermi sea Efimkin:2017 ; Sidler:2016 ), remains
an active area of study and discussion Reichman:2019 ; Glazov:2020 ; Rana:2021
; Efimkin:2021 ; Fey:2020 ; Liu:2021 .

Regardless of interpretation, all of these descriptions share a common
understanding: $X^{\pm}$ are bound states arising from the interaction of a
photoexcited electron-hole (e-h) pair with the subset of carriers in the Fermi
sea that have distinguishable quantum numbers. In most conventional III-V and
II-IV semiconductors such as GaAs or ZnSe, where band extrema occur at the
single central $\Gamma$-point valley of the Brillouin zone Kheng:1993 ;
Astakhov:2002 ; BarJoseph:2005 , this means that $X^{\pm}$ forms with mobile
carriers having opposite spin to that of the photoexcited electron or hole
because Pauli exclusion prevents strong short-range interactions with same-
spin carriers. However, the multi-valley nature of monolayer TMDs expands the
basis set of available quantum numbers, and band-edge electrons and holes can
be distinguished not only by their spin (up or down) but also by their valley
degree of freedom ($K$ or $K^{\prime}$). TMD monolayers therefore permit,
under suitable conditions, photoexcited e-h pairs to interact with Fermi seas
containing more than one type of quantum-mechanically distinguishable carrier.
As demonstrated recently Li:2022 ; vanTuanPRL:2022 ; vanTuan:2022 , this leads
to qualitatively new types of multi-particle composite exciton ground states
that can be described as bound six-particle “hexcitons” (when photoexcited e-h
pairs interact with two distinguishable Fermi seas) or even eight-particle
“oxcitons” (when interacting with three distinguishable Fermi seas). These
bound many-body excitonic states have large oscillator strengths and manifest
as discrete absorption resonances in linear optical spectroscopy, and emerge
at energies even further below $X^{0}$ and $X^{\pm}$. We emphasize that these
composite hexcitons are optically-allowed ground states of the interacting
exciton-Fermi sea system, and are therefore distinct from the many types of
optically-forbidden dark excitons and trions that appear only in
photoluminescence studies Urbaszek:2018 ; Muller:2018 ; Robert:2017 ;
Malic:2018 ; He:2020 ; Yang:2022 , and moreover should not be confused with
multi-exciton complexes (such as biexcitons) that appear only at higher
photoexcitation intensity Urbaszek:2018 ; Muller:2018 ; Barbone:2018 ;
ZLi:2018 ; Chen:2018 .

To date, such multi-particle hexcitons have been identified and studied only
in electron-doped WSe2 monolayers Li:2022 ; vanTuanPRL:2022 ; vanTuan:2022 .
This is due to i) the excellent optical quality of exfoliated WSe2, ii) the
ability to electrostatically dope WSe2 to high electron densities, and iii)
because composite hexcitons in WSe2 are expected at the low-energy A-exciton
optical transitions where spectral linewidths are typically much sharper than
at the higher-energy B-exciton. This latter fact arises from the positive sign
of the conduction band (CB) spin-orbit splitting in WSe2 ($\Delta_{c}>0$)
Song:2013 ; Kormanyos:2015 , which mandates that A-exciton optical transitions
photoexcite electrons to the upper CBs in $K$ and $K^{\prime}$, where Pauli
exclusion does not prevent them from interacting with both of the
distinguishable Fermi seas of electrons that occupy the two lower CBs (as
depicted in the band diagram in Fig. 1, one Fermi sea resides in the same
valley but has opposite spin, the other has same spin but resides in the
opposite valley).

However, under appropriate conditions, composite hexcitons should also emerge
in all members of the monolayer TMD family. For example, monolayer WS2 also
has a positive $\Delta_{c}$ and CB structure similar to WSe2, and therefore
hexcitons should also appear at its A-exciton under conditions of high
electron doping. In contrast, the negative $\Delta_{c}$ of monolayer MoSe2
Song:2013 ; Kormanyos:2015  precludes the existence of hexcitons at its
A-exciton (instead, only conventional $X^{\pm}$ should appear), but does allow
for the formation of hexcitons at its B-exciton transition. To date, neither
of these predictions have been explicitly tested. Here, using low-temperature
optical absorption measurements of electrostatically-gated WS2 and MoSe2
monolayers, we demonstrate and investigate the emergence of composite
hexcitons in both WS2 monolayers (at the A-exciton) and MoSe2 monolayers (at
the B-exciton). Based on the data, the roles of distinguishability and carrier
screening on the stability of hexcitons are discussed.

Figure 1: a) Schematic of the charge-tunable monolayer TMD structures. b,c)
Gate ($V_{g}$) dependent optical absorption spectra at 4 K and $B$=0 from
monolayer WSe2 and WS2, respectively, at their A-exciton band edges. Similar
to WSe2, WS2 exhibits strong $X^{0}$ absorption at charge neutrality,
conventional $X_{s,t}^{-}$ absorption at small electron density ($n_{e}$), and
the emergence at even lower energy of a strong “hexciton” absorption resonance
at larger $n_{e}$ (dashed oval, labeled as $H^{-}$). d) Band diagram of WSe2
and WS2. Extending the picture of $X^{\pm}$ as 4-particle tetrons, the
hexciton corresponds to a 6-particle correlated state that forms when a
photoexcited e-h pair couples simultaneously to both of the quantum-
mechanically distinguishable Fermi seas residing in the two lower CBs. This
occurs in WSe2 and WS2 at the A-exciton because $\Delta_{c}>0$, such that
optical transitions promote electrons to the upper CBs, where they are
distinguishable from electrons in the Fermi sea. Blue/red colors denote spin-
up/down bands. For clarity, only optical transitions in the $K$ valley are
depicted. The redshift of the hexciton ceases (yellow arrows) when the Fermi
sea begins to fill the upper CBs, at $n_{e}\approx 5(4)\times 10^{12}$ cm-2
for WSe2 (WS2).

Figure 1a depicts the charge-tunable TMD monolayer samples studied in this
work. Single monolayer flakes of WS2, MoSe2, and WSe2 were mechanically
exfoliated and sandwiched between thin slabs of hexagonal boron nitride (hBN).
Few-layer graphite flakes were used to electrically contact the monolayer, and
to serve as top and bottom gates. In this work, the top and bottom gates were
tied together and gate voltage $V_{g}$ was used to electrostatically dope the
monolayers with a background Fermi sea of electrons or holes. Dual gating
allows us to attain high carrier densities approaching 1013/cm2. Each
assembled structure was then positioned and placed directly over the core of a
single-mode optical fiber, to ensure a rigid and robust alignment between the
optical path and the doped TMD monolayer. This experimental approach Li:2020 ;
Stier:2018  mitigates the drift and vibration that can otherwise complicate
optical studies of TMD monolayers at low temperatures and in the pulsed
magnetic fields used in this work.

The sample-on-fiber assembly was then mounted on a purpose-built probe and
loaded into a liquid helium cryostat. Broadband white light from a Xenon lamp
was directed down the single mode fiber. Following transmission through the
sample the light passed through a thin-film circular polarizer and was then
retro-reflected and directed back into a multimode collection fiber. The
transmitted light was dispersed in a 300 mm spectrometer and detected by a
charge-coupled device (CCD). In this way the optical absorption from the doped
TMD monolayer was directly measured as $1-T/T_{0}$, where $T$ is the spectrum
of the transmitted light and $T_{0}$ is a reference spectrum. We note that
absorption spectra typically permit a straightforward evaluation and
visualization of exciton oscillator strengths, in comparison to reflectivity
studies where lineshapes depend sensitively on interference effects from the
surrounding layer structure.

Figure 1b shows a map of the gate-dependent absorption spectra from a WSe2
monolayer at low temperature (4K) and at zero magnetic field, in the spectral
range of its A-exciton. When $V_{g}\approx-2$V, the monolayer is at its charge
neutrality point and only the neutral exciton ($X^{0}$) absorption resonance
is observed. At increasingly negative or positive $V_{g}$, the monolayer
becomes lightly doped with mobile holes or electrons, and the well-known
$X^{\pm}$ resonances appear at energies $\approx$20-35 meV below $X^{0}$. As
extensively described in previous works, $X^{\pm}$ are optically-active bound
states arising from the interaction of the photoexcited e-h pair with the
carriers in the Fermi sea that possess distinguishable quantum numbers from
those of the photoexcited e-h pair (i.e., different spin and/or valley). Thus,
only a single $X^{+}$ resonance appears on the hole-doped side, but two
conventional $X^{-}$ resonances appear on the electron-doped side (the so-
called singlet $X_{s}^{-}$ and triplet $X_{t}^{-}$ charged excitons) because
there are two distinct Fermi seas with which to interact. The different
energies of $X_{s}^{-}$ and $X_{t}^{-}$ stem from the different amplitude of
the short-range electron-hole exchange interaction Glazov:2020 ; Hichri:2020 ;
Courtade:2017 .

Most importantly, at higher $n_{e}$ the $X_{s}^{-}$ and $X_{t}^{-}$ resonances
disappear and a new strong absorption resonance appears at even lower energy
($\approx$15 meV below $X_{s}^{-}$), indicating the emergence of a new bound
excitonic ground state with large oscillator strength. First observed in gated
WSe2 monolayers in 2013 Jones:2013  and very clearly resolved in several
subsequent studies Wang:2017NL ; Wang:2017 ; Barbone:2018 ; SuFeiShi:2020 ;
Liu:2021 , this low-energy absorption resonance – occasionally called
$X^{-\prime}$ in earlier literature – completely dominates the absorption
spectrum of WSe2 at high $n_{e}$. Moreover, it does not appear in hole-doped
WSe2 at the A-exciton, and does not appear in the A-exciton absorption
spectrum of gated MoSe2 monolayers Wang:2017NL ; Smolenski:2019 . Recently,
this absorption resonance was identified as a qualitatively new type of many-
body composite (six-particle hexciton) state, arising from the simultaneous
interaction of the photoexcited e-h pair with both of the Fermi seas that
reside in the lower CBs of monolayer WSe2 Li:2022 ; vanTuanPRL:2022 ;
vanTuan:2022 . As depicted in Fig. 1d, each of these two Fermi seas has
quantum numbers that are distinguishable from those of the photoexcited
electron (one has opposite spin, the other has opposite valley), and –
extending the picture of $X^{\pm}$ being four-particle tetrons Suris:2003 ;
Rana:2020  – the hexciton bound state comprises the photoexcited e-h pair, an
electron from each of the two distinguishable Fermi seas, and the two Fermi
holes that are left behind in the Fermi seas. As described recently, hexcitons
are the stable ground state of this interacting exciton-Fermi sea system, and
the Fermi holes not only ensure overall charge neutrality but also provide the
‘glue’ that binds the complex vanTuanPRL:2022 ; vanTuan:2022 .

Figure 2: a,b) Magnetic field evolution of the hexciton absorption resonance
in monolayer WSe2 and WS2, respectively, for both $\sigma^{+}$ and
$\sigma^{-}$ circularly polarized light, showing clear valley Zeeman
splitting, and the appearance of higher-energy absorption lines corresponding
to optical transitions to higher Landau levels ($V_{g}$=2 V). c,d) The average
energy of the $\sigma^{+}$ and $\sigma^{-}$ absorption lines reveals the
diamagnetic shift of the hexciton absorption resonance, which does not follow
a purely quadratic dependence (even at low $B$). As a point of reference, the
dashed red curves depict a $\sigma B^{2}$ shift, using a diamagnetic
coefficient $\sigma=3~{}\mu$eV/T2, which is $\approx$10$\times$ larger than
the small diamagnetic shifts of the neutral exciton in WSe2 and WS2.

The ordering of the spin and valley polarized CBs in monolayer WS2 is similar
to that of WSe2 (i.e., $\Delta_{c}$ is also positive), and optical transitions
at the A-exciton couple to the upper CBs. Consequently, optical signatures of
composite hexcitons can therefore be anticipated at high $n_{e}$ in WS2
monolayers. The gate-dependent absorption map of Fig. 1c confirms this
prediction: Strong absorption from $X^{0}$ is plainly visible at $V_{g}\approx
0$, and $X_{s,t}^{-}$ charged excitons appear at low $n_{e}$. These
conventional $X_{s,t}^{-}$ resonances in WS2 have been clearly resolved in
several recent studies Kapuscinski:2020 ; Zipfel:2020 ; Zinkiewicz:2021 ;
Robert:2021NC . Most importantly, our dual-gated structure allows a smooth
tuning to a regime of high $n_{e}$, where Fig. 1c shows that $X_{s,t}^{-}$
disappear and a new strong absorption resonance emerges at even lower energy
($\approx$15 meV below $X_{s}^{-}$). The gate-dependent absorption of
monolayer WS2 is therefore qualitatively identical to that of WSe2, albeit
with broader linewidths that are likely due to the reduced material quality of
sulfur-based TMDs. Thus, we associate the emergence of the low-energy
absorption resonance with the stable formation of 6-particle hexciton states.
Note that we were unable to dope our WS2 monolayers with mobile holes; even at
large negative $V_{g}$, only the neutral $X_{0}$ exciton was visible, likely
due to strong mid-gap pinning of the Fermi level by the larger number of
defects in sulfur-based TMDs.

Additional evidence supporting a picture of hexcitons in monolayer WS2 is the
evolution of its optical resonance in applied magnetic fields $B$. Figure 2
shows circularly polarized magneto-absorption from both WSe2 and WS2, under
conditions of large $n_{e}$ where the hexciton absorption dominates. As shown
recently Li:2022 , the hexciton resonance in WSe2 monolayers splits and shifts
with increasing $B$, and additional absorption resonances appear at higher
energy that disperse linearly with $B$ and are related to the development of
Landau levels (LLs) in the conduction and valence bands. Figure 2 shows that a
qualitatively similar $B$-dependent evolution of the hexciton peak also exists
in WS2, where additional LL-like absorption features emerge for $B>30$ T. From
the separation of these peaks we can estimate a combined electron and hole
cyclotron resonance energy of $\approx$0.45 meV/T, which is slightly larger
than obtained from WSe2 Li:2022  but in line with expectation given the
slightly lighter carrier masses in WS2 Goryca:2019 .

Figure 3: a) Gate-dependent absorption of a charge-tunable MoSe2 monolayer at
4 K. Only conventional $X^{-}$ and $X^{+}$ are observed at the A-exciton
($\sim$1.63 eV), even at large electron or hole densities, because
$\Delta_{c}<0$ and therefore only a single distinguishable Fermi sea is
available – see diagrams in panels b) and d), respectively. But at the
B-exciton ($\sim$1.83 eV), photoexcited e-h pairs couple to the upper CBs, and
interaction with both Fermi seas in the lower CBs is possible, leading to
hexciton formation at higher $n_{e}$ – see diagram in panel c). Moreover,
hexciton formation at high hole densities is always possible at the B-exciton
(for all TMD monolayers, because $\Delta_{v}\gg 0$) – see diagram in panel e).
Blue/red colors in the diagrams denote spin-up/down bands. For clarity, only
optical transitions in the $K$ valley are depicted.

Moreover, from these spectra we can extract the diamagnetic shifts of the
hexciton resonance, given by the average of the $\sigma^{+}$ and $\sigma^{-}$
absorption energies. This analysis, shown in Figs. 2c and 2d, reveals that the
diamagnetic shifts of hexcitons – even at low $B$ – deviate from the purely
quadratic behavior that is known to exist for neutral excitons in WSe2 and WS2
Stier:2018 ; Goryca:2019  (especially for WS2, where atypical diamagnetic
shifts have also been observed for charged excitons Plechinger:2016 ). This
behavior may arise from non-zero angular momentum contributions from the
particles in the hexciton complex, and will be further investigated in future
studies. For reference, however, the dotted red curves depict purely quadratic
shifts ($\sigma B^{2}$) using a large diamagnetic coefficient
$\sigma=3~{}\mu$eV/T2, which is $10\times$ larger than the known diamagnetic
shifts of the very small and tightly-bound neutral excitons Stier:2018 ;
Goryca:2019 . To the extent that these curves approximately capture the
overall shifts of the hexciton resonances, these data are qualitatively
consistent with recent calculations indicating that composite hexcitons are
several times larger in size than neutral excitons vanTuanPRL:2022 ;
vanTuan:2022

Taken together, these data provide evidence for the emergence of composite
hexciton states in electron-doped monolayer WS2, which appear at the A-exciton
owing to the positive sign of $\Delta_{c}$ and consequent ordering of the
spin-orbit-split CBs in the $K$ and $K^{\prime}$ valleys.

In marked contrast, neither electron-doped nor hole-doped MoSe2 monolayers
show any indication of hexciton formation at the A-exciton, as shown in Fig.
3a. Rather – and as also observed in earlier studies Wang:2017NL ;
Smolenski:2019 ; Liu:2021 – charge-tunable MoSe2 monolayers exhibit only a
single $X^{-}$ and $X^{+}$ absorption that appears $\approx$25 meV below the
neutral exciton $X_{A}^{0}$. This observation is consistent with the negative
sign of $\Delta_{c}$ in MoSe2 Song:2013 ; Kormanyos:2015 , which dictates that
optical transitions at the A-exciton photoexcite electrons to the lower (not
upper) CBs. As such, when mobile electrons populate the lower CBs, only a
single type of distinguishable electron exists in the Fermi sea, and many-body
hexcitons cannot form (see diagrams in Fig. 3b,d).

The ordering of the CBs in MoSe2 does, however, permit hexciton formation at
the higher energy B-exciton, where optical transitions couple to the upper CBs
(see Fig. 3c). As Fig. 3a shows, when $n_{e}$ is small, the neutral B-exciton
($X_{B}^{0}$) at 1.84 eV disappears and only a faint and diffuse absorption
remains at energies where conventional charged excitons are expected (that is,
at about 25 meV below $X_{B}^{0}$, or $\approx$1.815 eV). However, at larger
$n_{e}$ ($\approx 5\times 10^{12}$ cm-2), a stronger absorption with increased
oscillator strength clearly emerges at $\approx$1.795 eV. Its separation from
$X_{B}^{0}$ is $\approx$45 meV, which is larger than the 25 meV expected for
conventional charged excitons, but is commensurate with the larger energy
separation between hexcitons and neutral excitons that we observed in WSe2 and
WS2 (cf. Fig. 1). Notably, Fig. 3 shows that this new absorption does not
emerge until a large $n_{e}$ comparable to where the conventional $X^{-}$
trion loses oscillator strength, suggesting that it is not a conventional
trion. Therefore, although past studies of charge-tunable MoSe2 monolayers
have associated similar spectral signatures with conventional
trions/tetrons/exciton-polarons of the B-exciton Wang:2017NL ; Liu:2021 , we
argue that the low energy of the emerging absorption resonance and
(especially) its dependence on $n_{e}$ are, in fact, more consistent with the
formation and emergence of many-body hexcitons in electron-doped MoSe2
monolayers.

Furthermore, analogous signatures of hexcitons appear when the MoSe2 monolayer
is doped with high concentrations of mobile holes, $n_{h}$ (see Fig. 3a). As
depicted in the diagram of Fig. 3e, e-h pairs excited at the B-exciton will
always have two distinguishable Fermi seas of mobile holes with which to
interact, and composite hexcitons can be anticipated. Figure 3a shows that as
$n_{h}$ starts to increase, both $X_{A}^{0}$ and $X_{B}^{0}$ disappear, and a
conventional $X^{+}$ resonance appears 25 meV below $X_{A}^{0}$ when
$V_{g}=-3$ V. However, the concomitant response at the B-exciton is much
weaker, until $V_{g}$ increases up to $\approx-4$ V, at which point a strong
absorption emerges at a much lower energy of $\approx$40 meV below
$X_{B}^{0}$. This new resonance actually gains oscillator strength with
increasing $n_{h}$, while in this same doping range the conventional $X^{+}$
fades away. Based on this $n_{h}$ dependence, we rule out the possibility that
the new resonance could be, e.g., a conventional trion associated with
$X_{B}^{0}$ or an excited Rydberg state of $X_{A}^{0}$. Moreover, this new
resonance redshifts with increasing $n_{h}$, similar to the redshift observed
for hexcitons in electron-doped WSe2 and WS2 (cf. Fig. 1). These data
therefore support a picture of robust hexciton formation at the B-exciton in
hole-doped MoSe2. The resonance features are rather broad, however, likely due
to the shorter lifetime of B-excitons.

Indeed, owing to the large 100s-of-meV spin-orbit splitting of the valence
bands ($\Delta_{v}$) that exists in all monolayer TMD semiconductors, robust
hexciton formation at the B-exciton should emerge when any TMD monolayer
semiconductor is doped with a high density of mobile holes (see diagram in
Fig. 3e). Furthermore, because $\Delta_{v}$ is large, the Fermi sea of holes
never occupies the valence bands from which the photoexcited e-h pair
originates, and the photohole always remains distinguishable from every hole
in the Fermi sea. Hexcitons under such conditions should therefore remain
robust and should redshift due to the primary effects of bandgap
renormalization Scharf:2019  up to very large values of $n_{h}$. While in this
work we have studied B-excitons only in MoSe2, we note that the seminal work
of Wang et al. Wang:2017NL  clearly revealed a strong absorption below
$X_{B}^{0}$ in heavily hole-doped WSe2 monolayers, that redshifted with
increasing $n_{h}$ and did not broaden, consistent with hexciton formation.

An important insight gained from these various results is that the decay and
broadening of excitonic complexes is likely governed more by the quantum-
mechanical distinguishability of the photoexcited e-h pair (in relation to the
carriers in the Fermi sea), than by screening from the Fermi sea. For example,
in the case of electron-doped WSe2 and WS2 shown in Fig. 1, the redshifts of
the hexciton resonances in WSe2 and WS2 cease, and their decay/broadening
begins, only when electrons begin to populate the upper CBs. Beyond this
point, the photoexcited electron is no longer distinguishable from every
electron in the Fermi sea, and the Pauli exclusion principle dictates that
when it is introduced into the (now occupied) upper CB, it must scatter away
those mobile electrons having similar spin and valley quantum numbers
vanTuan:2022 . This exchange scattering process leads to the decay and
broadening of the hexciton resonance. Hexciton resonances therefore retain
their amplitude and narrow linewidth when the charge density increases, as
long as the photoexcited e-h pair remains distinguishable from all carriers in
the Fermi sea. In hole-doped TMD monolayers, the large $\Delta_{v}$ ensures
distinguishability of e-h pairs excited at the B-exciton, and therefore a
robust stability of hexcitons even to very large $n_{h}$.

Moreover, screening by mobile carriers does not seem to be the primary driving
force for broadening of the hexciton resonance, as inferred by the observation
that the conventional $X^{\pm}$ resonances begin to lose oscillator strength
at much smaller carrier density. For example, Fig. 3 shows that the
conventional trions of MoSe2 around 1.63 eV decay when $|V_{g}|\gtrsim 5$ V.
Concomitantly, however, the type-B hexciton in hole-doped conditions around
1.8 eV neither broadens nor decays at the same (and even higher) $n_{h}$. A
similar behavior was also measured for a wider $n_{h}$ range by Liu et al.
Liu:2021 , where the type-B resonance around 1.8 eV maintains a large
oscillator strength and continues to redshift long after the $X^{+}$ resonance
at the A-exciton decays. This suggests that screening is not the primary cause
of decay and broadening, because the Coulomb potential cannot selectively
weaken the attraction between particles of one complex species but not of
another.

In summary, composite hexcitons are expected in all members of the monolayer
TMD family. When doped with a high density of electrons, hexcitons can emerge
at the A-exciton resonance (as for the case of WSe2 and WS2) or the B-exciton
resonance (as for the case of MoSe2) depending on the sign of $\Delta_{c}$ and
the consequent ordering of the CBs. For hole-doped TMD monolayers, hexcitons
should always emerge at the B-exciton, as investigated here for MoSe2 and as
suggested by earlier spectroscopic data from both WSe2 and MoSe2 Wang:2017NL ;
Liu:2021 . To the best of our knowledge, this has not yet been studied in
hole-doped WS2 or MoS2. As a final point of discussion, we note that electron-
doped MoS2 monolayers represent an interesting case: While early theory
suggested that $\Delta_{c}$ was small and negative (implying a CB ordering
similar to MoSe2 Kormanyos:2015 ), more recent experimental work indicates an
opposite CB ordering Robert:2020 ; Park:2022 , particularly when exciton
effects are taken into account, making it more akin to that of WS2 and WSe2.
In this case, a hexciton resonance can be expected to emerge in electron-doped
MoS2 at the A-exciton, at an energy below that of the conventional
$X_{s,t}^{-}$ charged excitons. Indeed, various recent studies have revealed
additional optical resonances emerging in electron-doped MoS2 monolayers
Klein:2021 ; Klein2:2021 ; Roch:2019 , although in some cases it was
associated with exotic ferromagnetic order. Looking forward, we anticipate
that composite 6-particle hexcitons (and, in high magnetic fields, 8-particle
oxcitons) can provide a rich platform to study novel many-body effects and
inter-valley correlations within a strongly interacting exciton-Fermi sea.

We thank X. Marie for helpful discussions, and we acknowledge support from the
Los Alamos LDRD program and the US Department of Energy (DOE) “Science of 100
T” program. The National High Magnetic Field Lab is supported by National
Science Foundation DMR-1644779, the State of Florida, and the US DOE. Work at
the University of Rochester was supported by the DOE Basic Energy Sciences,
Division of Materials Sciences and Engineering under Award No. DE-SC0014349.

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