# Spectroscopic evidence of high temperature superconductivity in VSe2

Turgut Yilmaz  National Synchrotron Light Source II, Brookhaven National Lab,
Upton, New York, 11973, USA  Elio Vescovo  National Synchrotron Light Source
II, Brookhaven National Lab, Upton, New York, 11973, USA  Jerzy T. Sadowski
Center for Functional Nanomaterials, Brookhaven National Lab, Upton, New York,
11973, USA  Boris Sinkovic  Department of Physics, University of Connecticut,
Storrs, Connecticut, 06269, USA

(August 27, 2024)

###### Abstract

High-resolution angle-resolved photoemission experiments reveal subtle
modifications of the surface electronic structure of VSe2. Most remarkably, we
show that superconductivity can be induced in VSe2 by the right selection of
substrate and growth parameters. Evidence for the superconducting state comes
from the simultaneous detection of spectral kink, quasiparticle peak, Fermi
gap, and their evolution with the temperature. The observation of Bogoliubov-
like back-bending bands at low temperatures, signaling electron-hole pairing,
further supports the presence of superconductivity in this system. The
photoemission experiment also provides evidence for a formation of a pseudogap
state at high temperatures, characterized by the progressive quenching of the
quasiparticle peak feature coexisting with a persistent gap at the Fermi
level, a behavior reminiscent of high-Tc superconductors. We attributed the
origin of superconductivity in these VSe2 films to a modified Fermi surface
combined with the formation of van Hove singularity points with a binding
energy corresponding to the chemical potential of the system. Although Tc
cannot be accurately determined from photoemission data, observations based on
the survival temperature of the quasiparticle peak suggest that Tc could be as
high as 100 $\pm$ 5 K and substantially higher than previous reports for any
transition metal dichalcogenides.

††preprint: APS/123-QED

##  I Introduction

More than 100 years after its discovery, superconductivity remains one of the
most intrigin phenomena in material science; an apparently inexhaustible
source of scientific excitement and technological interest. The observation of
unconventional superconductivity in (La,Ba)2CuO4 a few decades ago Bednorz and
Müller (1986), with their unexpectedly high-Tc, redoubled interest in this
phenomenon. Once broken the taboo of ultra-low temperatures, the challenge of
understanding the physics of the new superconductors became inseparably
connected with the hope of achieving Tc above ambient temperature. Nowadays,
the adoption of superconducting materials as basic components in quantum
computing (Qbits) has become another driving force for research in this area
Devoret  _et al._ (2004). There is therefore a constant search for new
superconductors, each addition to the list contributing to a more complete
picture of the phenomenon while at the same time expanding the range of
potential applications. Interestingly, the two latest entries in the list of
high-Tc FeSe monolayer He  _et al._ (2013) and twisted graphene bilayers Cao
_et al._ (2018) have put a new spin in the search for room temperature and /or
topological superconductivity, suggesting that equally important to the
synthesis of new materials, is the accurate control of growth parameters and
lattice orientations in the preparation of old ones.

Among materials’ families, layered transition metal dichalcogenides (TMDCs)
are under intense scrutiny in the search for superconductivity. Although
relatively simple materials, they host an impressive variety of physics,
including Mott insulating state, charge density waves, and ferromagnetism
Manzeli  _et al._ (2017); Choi  _et al._ (2017). This richness offers a
possibility to study the interplay between superconductivity and other states
of matter. In this connection, particularly remarkable is the case of TMDC
displaying topological order, another exotic state of matter which can induce
Majorana fermions if it coexists with superconductivity Shi  _et al._ (2015);
Li  _et al._ (2021); Fu and Kane (2008). This can provide a realistic material
candidate for fault-tolerant quantum computing Lian  _et al._ (2018). However,
so far, the highest observed superconducting Tc in TMDCs is 8.8 K reported for
2M-WS2 Fang  _et al._ (2019). Such a low-Tc hinders the technological adoption
of these materials.

Here, we report on a comprehensive high-resolution angle-resolved
photoemission (ARPES) study of one TMDC , VSe2. Optimally grown films display
all the hallmarks of high-Tc superconductivity: the formation of the spectral
kink, the quasiparticle peak (QP), and the gap at the Fermi level. As a
function of temperature, the QP is observed up to 110 K while the Fermi gap
persists with decreasing size up to 150 K, indicating the presence of a
pseudogap in analogy to high-Tc superconductors. The superconducting state is
further verified by the observation of Bogoliubov back-bending bands that are
the standard signature of electron-hole pairing in the superconducting state
Matsui  _et al._ (2003). These spectral features are very sensitive to the
films’ growth condition; in particular, the substrate type / quality and the
growth temperature are crucial for inducing superconductivity in VSe2. Even
though the photoemission data do not provide a direct way of determining Tc,
the temperature evolution of the spectral line shape suggests that Tc could be
as high as 100 $\pm$ 5 K. This would be much higher than previously reported
for any TMDCs, providing a potential material candidate with easily accessible
experimental parameters to investigate the superconductivity and its
connection to the other quantum states.

Figure 1: (a1) Constant energy cut at 20 meV with respect to the Fermi level
obtained from a vacuum cleaved VSe2 single crystal. (a2) Binding energy versus
in-plane momentum map of the same sample taken in the $\overline{\Gamma}$ \-
$\overline{M}$ direction of the Brillouin zone. (a3) EDC obtained along the
dashed black line in (a2). (a4) Corresponding Se 3d peak recorded with 110 eV
photons at 10 K. Inset in (a4) is the $\mu$-low energy electron diffraction
($\mu$LEED) pattern of the same sample. (b1) - (b4) and (c1) - (c4) are the
same as (a1) - (a4) but for 5 ML VSe2 thin films grown on Gr/SiC and HOPG
substrates, respectively. The same growth parameters are adopted for the thin
films and the substrate temperatures were kept at 570 K during the growth.
Unlike other two samples, the VSe2 sample grown on HOPG displays a 3 x 3
superstructure $\mu$LEED pattern at 300 K and 85 K as indicated in the inset
of (c4). The blue arrow in (c1) shows the van Hove singularity points while
the green arrow points to a Fermi arc. Purple arrow in (c3) marks the spectral
kink. The black arrow in (c4) marks a new spectral feature as a shoulder in
the higher binding energy side of Se 3d for the film grown on HOPG. This
feature is absent in other two samples. All the ARPES maps are collected with
50 eV photons at 10 K. Figure 2: Top panels: constant energy cuts, 20 meV
below the Fermi level. Middle panels: ARPES band dispersions along
the$\overline{\Gamma}$ \- $\overline{M}$ high symmetry direction of the
Brillouin zone. Lower panels: EDCs along the dashed line in dispersion
spectra. All films were grown in identical conditions except for the substrate
temperatures set to 550 K, 570 K, 600 K, and 650 K from left to right. Spectra
were measured at 10 K with 50 eV photons from the samples grown on HOPG
substrates.

##  II Impact of the growth method and substrate on the electronic structure

To investigate the impact of the growth method on the electronic and
structural properties, we compare various form of VSe2: bulk crystal vs.
epitaxial films deposited on different substrates. These comparisons are
summarized in Fig. 1, presenting the ARPES maps, $\mu$LEED patterns, and Se 3d
core-level spectra. The left column is for the bulk single crystals
synthesized by the flux zone method. The central and right columns are for two
epitaxial films prepared by molecular beam epitaxy (MBE) under nominally
identical conditions on Gr/SiC and HOPG, respectively. At first glance, it is
apparent that qualitative differences distinguish the electronic structure of
the bulk sample from the thin films and, even more surprisingly, also of the
two films grown on the two similar substrates. The Fermi surface of the bulk
VSe2 exhibits two distinct features: an intense emission at the
$\overline{\Gamma}$-point and ellipsoidal electron pockets centered at the
$\overline{M}$-points (Fig. 1(a1)). The ellipsoidal features are attributed to
the V 3d band, while the emission at the zone center derives from Se 4p atomic
orbitals Duvjir  _et al._ (2018). The energy dispersion of the valence bands
along the $\overline{\Gamma}$ \- $\overline{M}$ direction is displayed in Fig.
1(a2). The V 3d states dominate the region close to the Fermi level while the
top of the Se 4p reaches the binding energy of about 0.2 eV at the
$\overline{\Gamma}$-point.

Figure 3: (a) QP-EDCs vs. temperature. (b) Symmetrized EDCs. (c) Comparison of
EDCs taken at 10 K and 100 K. (d) Fermi gap vs. temperature. The gap is
estimated from the symmetrized EDC given in (b). (e) and (f) Dispersion map
taken at 10 K and corresponding momentum integrated EDCs. The red line and
blue line in (f) mark the $\overline{\Gamma}$-point and the Fermi momentum,
respectively. The green line in f represents the inward band bending. (g) and
(h) are the same as (e) and (f) but taken at 180 K sample temperature. The
pink line in (h) marks the nearly flat dispersion in the vicinity of the Fermi
level and around the $\overline{\Gamma}$-point. (i) QP-EDCs as a function of
the angle-$\theta$ in (j) where constant energy cut 20 meV below the Fermi
level measured at 10 K is given. All the spectra are obtained with 50 eV
photons from the sample grown on HOPG substrates at 570 K substrate
temperature.

Comparing the Fermi surface of bulk sample to that of the 5 ML VSe2 film grown
on Gr/SiC, the main difference is the formation of a hexagonal-shaped hole
pocket centered at the $\overline{\Gamma}$-point. Away from the zone center,
the ellipse-shaped electron pockets centered at the $\overline{M}$-point and
the nearly -triangular hole-like pocket around the $\overline{K}$-point are
similar although less intense than in the bulk sample (Fig. 2(b1)). The
binding energy – in-plane momentum (k∥) map of the VSe2 film on Gr/SiC
exhibits an M-shape dispersive band in the vicinity of the Fermi level (Fig.
1(b2)). Energy distribution curves (EDCs) of both samples presented in Fig.
1(a3) and 1(b3), respectively, display a similar density of states with a
broad bump located in the vicinity of the Fermi level without any spectral
anomaly. The electronic structures presented here for the single crystal and
the thin film grown on Gr/SiC are consistent with the literature Duvjir  _et
al._ (2018); Feng  _et al._ (2018); Coelho  _et al._ (2019).

The case of a 5 ML VSe2 film grown on a HOPG substrate is the most interesting
one. New structures develop compared to both the bulk samples and the films on
graphene. The center of the Fermi surface assumes a star-like shape and
intense triangular pockets surround the $\overline{K}$points (Fig. 1(c1)). The
vertices of these two features connect along the $\overline{\Gamma}$ \-
$\overline{M}$ directions at about 1/3 of the way, to form van Hove
singularity (vHs) points. The six-fold symmetric ellipsoidal electron pockets
centered at the $\overline{M}$-points are also well resolved in this sample
but and are now delimited by extremely sharp momentum arcs (green arrow in
Fig. 1(c1)). Finally, a well-defined circular electron pocket is now present
close to the $\overline{\Gamma}$ point (Fig. 1(c1)). These states are
concentrated in a narrow energy window of about 25 meV in the vicinity of the
Fermi level, possibly resulting from fine tuning of the binding energy of vHs
points. Relevant information on the unique electronic structure of this sample
is contained in the dispersion map (Fig. 1(c2)). First, the V 3d derived bands
approaching the Fermi level exhibit a spectral kink (see red arrows in Fig.
1(c2)). Second, an intense QP peak is now present in the EDC, followed by a
dip and a hump of decreasing intensity towards higher binding energy (Fig.
1(c3)). Such spectral line shape is characteristic of the superconducting
state Dessau  _et al._ (1991). Therefore, our observations suggests that VSe2
can be turned into a superconductor when grown on a HOPG substrate under
proper conditions.

To further investigate the origin of the spectral differences in these
samples, we report $\mu$LEED patterns taken at 300 K and the Se 3d peaks for
the corresponding samples. The diffraction pattern of single-crystal VSe2
displays sharp spots, forming the expected three-fold diffraction pattern
(inset in Fig. 1(a4)). For the sample grown on the Gr substrate, the
diffraction spots are broader and elongated along the tangential direction,
indicating the presence of the multiple domains (inset in Fig. 1(b4)). This
broadening is consistent with what is observed in ARPES. Furthermore, Se 3d
peaks of both samples show a single doublet line shape, again somewhat broader
in the case of the film on Gr (Fig. 1(a4) and 1(b4)). Corresponding to the
unique features observed in the ARPES data, the $\mu$LEED pattern for the VSe2
sample grown on HOPG exhibits a sharp 3 x 3 superstructure at 300 K and 85 K,
and the Se 3d doublet has an additional component at higher binding energy,
marked with a black arrow in Fig. 1(c4). These features are possibly due to V
impurity intercalation as seen in a variety of similar compounds Bonilla  _et
al._ (2020). We should also emphasize that impurity intercalation can affect
the chemical potential of the system and consequently modify the binding
energy of the vHs points. This mechanism could explain the suppression of the
charge density waves in these films and enhance the superconducting $T_{c}$
Chikina  _et al._ (2020). Furthermore, absence of the any difference between
the low temperature and room temperature $\mu$LEED shows that the emergent QP
peak, spectral kink, and vHs points in VSe2 grown on HOPG cannot be associated
with the structural evolution.

##  III Impact of the substrate temperature on the electronic structure

Even though, the surface electronic and crystal structure of VSe2 samples in
thin films and bulk form have been extensively studied Duvjir  _et al._
(2018); Feng  _et al._ (2018); Coelho  _et al._ (2019), the formation of the
kink, the QP peak, and clear vHs points were not seen in previous studies. The
discrepancy between the present work and earlier studies suggests that the
growth parameters are critical in these films, close to a phase transition.
Interestingly, this is analogous to the emergence of the high-Tc
superconductivity in monolayer FeSe whose Fermi surface topology is greatly
affected by the specific annealing procedure He  _et al._ (2013). This process
boosts the Tc for the mono layer FeSe nearly six times compared to the bulk
FeSe samples. Furthermore, the enhanced Tc is observed only in FeSe films
grown on SrTiO3 substrate. Evidently, VSe2 displays a very similar behavior,
the superconducting phase being linked to the substrate type and the growth
conditions. Furthermore, the growth conditions are known to affect the surface
electronic structure and superconducting properties of cuprate superconductors
by controlling the amount of electron or hole doping Drozdov  _et al._ (2018).

Following these premises, we investigated the role of the substrate
temperature in the electronic structure of the 5 ML VSe2 films grown on HOPG.
The ARPES spectra for four samples grown at 550 K, 570 K, 600 K, and 650 K are
shown in Fig. 2. The Fermi surfaces (top panels), band dispersions (middle
panels) and EDCs (bottom panels) demonstrate that the superconducting phase
forms in a relatively narrow temperature range. At the optimal growth
temperature of 570 K, QP peak exhibits the highest intensity, and the spectral
kink is more pronounced (Fig. 2(b2) and 2(b3)). Both features weaken at lower
and higher growth temperatures (Fig. 2(a3) and 2(c3)). The ellipsoidal
electron pockets centered at the $\overline{M}$-point and the triangular
electron pockets centered at the $\overline{K}$-points are also not well
resolved in the samples grown at 550 K and 650 K (Fig. 2(a1) and 2(d1)). These
observations show that the growth is critical in the formation of the surface
electronic structure of VSe2 films. The relatively narrow temperature range
helps explaining why the superconductive phase has not been previously
observed.

##  IV Temperature dependent evolution of the electronic structure and the
energy gap

We have seen that the formation of the QP and the kink in the electronic
structure of VSe2 depend on the growth properties. Even though these features
are commonly observed in superconductors, they do not necessarily prove the
emergence of the superconducting state. In addition to these features, the
Fermi gap and its temperature dependence are also critical for identifying the
superconductivity. Fig. 3 summarizes the temperature dependences from these 5
ML VSe2 films on HOPG. Fig. 3(a) displays the evolution of the QP feature in
the temperature range 10 to 180 K. The EDCs measured along the dashed line in
Fig. 2(b2) track the evolution from a sharp coherent peak at low temperature
to a broad incoherent spectral feature at high temperature. The QP weakens
with increasing temperature but it can still be traced up to 100 K (see Fig.
3(c)). Symmetrized EDCs are given in Fig. 3(c) to visually detect and extract
the superconducting gap opening at the Fermi level. The symmetrization
procedure has been used in the analysis of superconductors to remove the
effect of the Fermi function19. The gap is observed as a spectral dip, whose
width serves to estimate the gap size. From the symmetrized EDCs, it would
appear that the gap persists up to nearly 150 K (Fig. 3(d)). This behavior of
an apparent gap persisting at higher temperatures than the QP has been
observed in high-Tc superconductors, in which a pseudogap, in addition to the
superconducting gap, plays a critical role in the electronic structure Kondo
_et al._ (2011). Thus, the coexistence of the spectral kink, QP, and the Fermi
gap strongly suggest the existence of the superconductivity in optimally grown
VSe2 samples and the temperature dependence of the gap and the QP indicates
that the Tc could be as high as 100 $\pm$ 5 K.

The gap observed in the present work has been attributed to the emergence of
superconductivity mainly in analogy with known superconductors’ behavior.
However, the origin of the gap could be due to other types of electronic
modifications. In the case of VSe2, the formation of a charge density wave
(CDW) phase comes to mind even though VSe2 is known to show CDW phase only at
1 ML Chen  _et al._ (2018). In CDW, however, the gap should be observed only
in some specific parts of the Fermi surfaces, where nesting conditions between
parallel portions of the Fermi surface are satisfied Grüner (1994). Instead,
the gap observed here is highly isotropic. As shown in the EDCs of Fig. 3(i),
the QP peak is present with the same intensity and location in all directions
(see also Fig. 3(j)). Furthermore, the observed size of the gap in our films
does not seem to be consistent with a CDW interpretation. In particular, in
single layer VSe2 the CDW gap was estimated from symmetrized ARPES spectra to
be 100 meV Kondo  _et al._ (2011), much larger than the gap reported in this
work, which is measured to be - 16 meV at 10 K. Nevertheless, $\mu$LEED
patterns taken at 85 K and 300 K do not exhibit any differences indicating the
absence of the structural transformation across the temperature range in which
QP peak, spectral kink, and the Fermi gap form. Therefore, this observation
excludes the structural origin of the gap opening.

Another strong evidence for the superconducting gap is the observation of the
Bogoliubov back-bending bands due to the particle-hole pair formation in the
superconducting state Matsui  _et al._ (2003). This can be seen in the binding
energy – in-plane momentum map taken at 10 K in Fig. 3(e) and its
corresponding momentum integrated EDCs in Fig. 3(f). Band bending towards
higher binding energy away from the Fermi momentum is marked with a green arc
in the spectrum. At low temperatures, the QP is located at the Fermi wave
vector where its binding energy is the lowest but towards the
$\overline{\Gamma}$-point it shows a systematic shift to higher binding
energy. On the other hand, the spectra taken at 180 K exhibits a flat feature
at the Fermi level without any signature of band bending (Fig. 3(g) and 3(h)).
This is an expected consequence of the disappearance particle-hole pairing in
the normal state. Therefore, we can conclude that the gap at low temperature
is indeed due to superconductivity. Unfortunately, ARPES cannot directly
detect the Tc of the superconductivity due to the coexistence of the
superconducting and pseudogap as is the case in high-Tc superconductors. Other
types of measurements such as magnetic characterization or transport
measurements are required for direct detection of the Tc. Nevertheless, the
temperature dependence of the gap and the QP indicates that the Tc could be as
high as 100 $\pm$ 5 K where the coherence spectrum is significantly
suppressed.

##  V DISCUSSION and CONCLUSION

VSe2 is a well-known TMDC whose band structure has been studied extensively.
It is therefore surprising that new, qualitatively different spectra can be
obtained from this material. Apart from the narrow growth temperature window
mentioned above, another possible explanation is the HOPG substrate used to
grow the films. Most previous ARPES investigations have been conducted on VSe2
films grown on Gr/SiC Shi  _et al._ (2015); Li  _et al._ (2021); Fu and Kane
(2008); Kondo  _et al._ (2011). The relevance of the substrate choice is one
of the main results of the present work, even though it is difficult to point
to its exact role. Gr and HOPG are very similar 2D materials which are not
likely to form a strong chemical bond with the VSe2 overlayers, also in view
of the large lattice mismatch. Neither substrates is then expected to induce a
prominent change in the band structure of the VSe2 films. Possibly a
significant difference between the two cases results from different domain
orientations and domain sizes. Compared to Gr/SiC, cleaved HOPG surfaces can
be expected to expose multiple domains, oriented in different directions.
These structural differences could also indirectly affect the impurities
distribution between the VSe2 layers, modifying the possible diffusion
channels. Structural defects can induce superconductivity or enhance the Tc of
a superconductor as discussed for the high-Tc superconductivity in graphite
and semiconducting superlattices Fogel  _et al._ (2001); Kopnin  _et al._
(2013). Therefore, one can speculate on an indirect role of the substrate,
inducing relevant modification in the VSe2 films through minor structural
differences.

It is also interesting to note that the superconductivity in VSe2 is not
completely unprecedented: a recent work reports that bulk VSe2 become
superconducting under high pressure, albeit with a Tc of only 4 K Sahoo  _et
al._ (2020). Another interesting aspect of our results is that VSe2 possibly
being a CDW material is not expected to turn a superconductor, as the two
phenomena are unlikely to coexist. In this respect, the role of the vHs points
is probably critical. For example, intercalation of impurities in TMDCs has
been shown to shift the vHs points toward the Fermi level, effectively
suppressing the CDW phase Chikina  _et al._ (2020). This mechanism has been
invoked to explain the enhancement of Tc by more than an order of magnitude in
Pd-intercalated TaSe2 Chikina  _et al._ (2020); Bhoi  _et al._ (2016); Kim
_et al._ (2018). A similar increase in the Tc was also realized in Cu-doped
PdTe2 whose origin is attributed to Cu intercalation which shifts the binding
energy of the vHs points toward the Fermi level Yan  _et al._ (2015).
Furthermore, the absence of the superconductivity in a similar compound,
PtTe2, was also directly linked to the different dispersion character of the
vHs points compared to the parent superconducting materials Kim  _et al._
(2018). These studies conclude that the binding energy of the vHs points is a
necessary component of the superconductivity in TMDCs playing a critical role
in controlling Tc.

The role of the vHs points is also widely debated for the cuprate
superconductors, whereby it has a prominent influence on the Tc or even can
induce superconductivity Newns  _et al._ (1992); Markiewicz (1997). The
assumption behind the vHs scenario to explain the observed superconductivity
is that the binding energy of the vHs points and the Fermi level are in close
proximity. This can introduce strong divergence in the quasi two-dimensional
density of states modifying the transport property of the material. This
suggests the possibility of forming superconducting pairs at higher
temperatures by pushing the vHs points to the Fermi level through doping or
strain.

The similarities between our findings and other superconductors are
indisputable. Based on these analogies, we propose that the emergent
superconductivity in VSe2 is closely related to the formation of vHs points
and their fine-tuning with respect to the Fermi level induced by impurity
intercalation. Clearly, the complete understanding of our observations and
their implications require further theoretical and experimental studies,
possibly adopting different approaches. For example, the contribution of other
bands located near or at the Fermi level may need further consideration to
describe the superconducting properties of VSe2 accurately. Finally, our work
does not suggest a new material but more importantly, shows that a well-known
non-superconducting material can be turned into a high-Tc superconductor by
manipulating its growth conditions. This is a significant step towards
understanding high-Tc superconductivity and its interplay with other states of
matter.

## ACKNOWLEDGMENTS

This research used ESM (21-ID-1, 21-ID-2) beamline of the National Synchrotron
Light Source II, a US Department of Energy (DOE) Office of Science User
Facility operated for the DOE Office of Science by Brookhaven National
Laboratory under Contract number DE-SC0012704. This work also used the
resources of the Center for Functional Nanomaterials, Brookhaven National
Laboratory, which is supported by the U.S. Department of Energy, Office of
Basic Energy Sciences, under Contract number DE-SC0012704. The initial work on
VSe2 film growth was supported by the University of Connecticut under the
UCONN-REP (Grant No. 4626510).

## APPENDIX: METHODS

###  V.1 Sample growth

Molecular beam epitaxial growth (MBE) technique was employed to grow VSe2 thin
films on HOPG and ML Gr/SiC substrates. HOPG substrates were first cleaved in
air and rapidly loaded into a high vacuum chamber where they were annealed at
850 K for 30 minutes to clean the surface prior to film deposition. ML
graphene were prepared by annealing SiC substrates to 1800 K for 15 minutes at
the XPEEM/LEEM endstation of the 21-ID-2 beamline of NSLS-II. Then, Gr/SiC
substrates were loaded to MBE chamber and annealed at 950 K for 30 minutes to
clean the surface. 5N purity Se and V with 99.8 $\%$ purity were used to grow
the samples. An e-beam evaporator was used for V deposition while resistive
heating of a ceramic crucible type evaporator was employed for Se. Deposition
rates of V and Se were kept at 0.2 $\AA$/min and 10 $\AA$/min, respectively.
Before starting the growth, sample and V are brought to growth temperatures
and kept there for 2 hours to minimize the outgassing. We have realized that
the slower growth rate can significantly enhance the quality of the ARPES
data, likely due to the well-ordered samples. The sample were annealed under
Se flux for 5 minutes at the growth temperatures of each sample. All the thin
film samples studied here are 5 ML thick as estimated by using a quartz
thickness monitor with a 15 $\%$ error bar. VSe2 bulk sample were obtained
from 2dsemiconductors company with the part number of BLK-VSe2. Samples for
ARPES and $\mu$LEED experiments were capped with 20 nm amorphous Se film
before being removed from the MBE chamber.

###  V.2 Photoemission experiments

ARPES experiments and core-levels were recorded at 21-ID-1 ESM beamline of
National Synchrotron Light Source II (NSLS-II) by using a DA30 Scienta
electron spectrometer. Se 3d peak were recorded at 10 K sample temperature
with 110 eV photons. The energy resolution during the photoemission
experiments was better than 15 meV and a spot size of 20 $\mu$m was used
during the photoemission experiments. The Fermi level position was calibrated
with respect to photoemission edge measured from Ag films deposited on the
sample surface after the ARPES experiments. Prior to photoemission
experiments, the VSe2 samples were annealed at 500 K for 30 minutes under the
ultra-high vacuum condition to remove the Se capping layer. This annealing
temperature was tested on various samples to insure it does not induce
structural or chemical changes on the sample but still high enough to remove
the capping layer.

###  V.3 $\mu$LEED

$\mu$LEED experiment was performed at X-ray photoemission electron
microscopy/low-energy electron microscopy (XPEEM/LEEM) endstation of the ESM
beamline (21-ID-2). The $\mu$LEED patterns were obtained after removing the Se
capping layers by annealing the thin films samples at 500 K.

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††institutetext:  aTheoretical Physics Department, Fermi National Accelerator
Laboratory, P.O. Box 500, Batavia, IL 60510, USA
bDepartment of Physics, University of California Berkeley, Berkeley, CA 94720,
USA
cPhysics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA
dBerkeley Institute for Data Science, University of California, Berkeley, CA
94720, USA

# Beyond 4D Tracking:
Using Cluster Shapes for Track Seeding

Patrick J. Foxa Shangqing Huangb,c Joshua Isaacsona Xiangyang Juc,
and  Benjamin Nachmanc,d<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Tracking is one of the most time consuming aspects of event reconstruction at
the Large Hadron Collider (LHC) and its high-luminosity upgrade (HL-LHC).
Innovative detector technologies extend tracking to four-dimensions by
including timing in the pattern recognition and parameter estimation. However,
present and future hardware already have additional information that is
largely unused by existing track seeding algorithms. The shape of pixel-
clusters provides an additional dimension for track seeding that can
significantly reduce the combinatorial challenge of track finding. We use
neural networks to show that cluster shapes can reduce significantly the rate
of fake combinatorical backgrounds while preserving a high efficiency. We
demonstrate this using the information in cluster singlets, doublets and
triplets. Numerical results are presented with simulations from the TrackML
challenge.

††preprint: FERMILAB-PUB-20-650-T

##  1 Introduction

Analyzing data from the Large Hadron Collider (LHC) present a hyper challenge.
A given collision event may result in hundreds of outgoing particles, each
with many features (momentum, electric charge, etc.). This hypervariate phase
space is then observed by complex multi-channel detectors that are essentially
hyperspectral cameras. The LHC detectors have millions of readout channels and
dimensionality reduction is essential for data analysis. One natural and
nearly lossless reduction is the reconstruction of charged particle
trajectories (‘tracks’). The innermost layers of the detectors at the LHC are
constructed to register the passage of charged particles without significantly
altering the particle energy or direction. In the ATLAS and CMS detectors,
this is achieved using silicon sensors that are finely segmented in one or two
directions and are called strips and pixels, respectively. We will focus on
pixels, although our methodology applies more generally.

Typically, the first step in a tracking algorithm is the construction of
seeds, which are sets of three or more hit pixel clusters that can be used to
fit charged-particle trajectories (see e.g. Ref. Aaboud:2017all ;
Chatrchyan:2014fea ). If there are $\mathcal{O}(10)$ detector layers, a given
event with $\mathcal{O}(10^{3})$ particles would have $\mathcal{O}(10^{4})$
pixel clusters and therefore about $\mathcal{O}(10^{10})$ possible seeds.
Since the number of seeds per real particle is $\mathcal{O}(1)$, the initial
purity of seeds can be as low as $\mathcal{O}(10^{-10})$. This is a
significant challenge for pattern recognition at the LHC and its high-
luminosity upgrade, the HL-LHC, where the number of particles per event will
grow even higher.

The geometric arrangement of seeds in space can be used to eliminate those
that are unlikely the result of a real particle. However, given the
significant challenge presented by the large number of seeds, it is important
to examine other non-geometric solutions. One solution is to augment tracking
detectors with precise timing information to reject combinations of clusters
from different collisions Sadrozinski:2017qpv ; CERN-LHCC-2017-027 ; CERN-
LHCC-2018-023 . This is especially effective at high luminosity where a given
proton-proton bunch crossing may result in $\mathcal{O}(100)$ individual
proton-proton collisions with a non-negligible spread in time. While timing is
a promising avenue, the existing detectors are not fast enough for this
information to be useful. Other hardware modifications have been studied, such
as linking two nearby spatial hits CERN-LHCC-2017-009 , but this is also not
available in existing detectors and is currently not possible for the
innermost detector layers due to bandwidth limitations.

Other available measurements are the energy deposition and the structure of
pixel clusters. In this paper we analyze the impact on seed purity of using
cluster shape information. The pixelation in tracking detectors is
sufficiently fine that multiple pixels are often activated from a single
charged particle. A hit is a pixel that has registered a signal above
threshold and a collection of nearest neighbor hits are grouped together into
a cluster. The shape of a pixel cluster contains useful information about the
direction of the charged particle and offers a complementary source of
information to the spatial location of the cluster centroid that is currently
used for seed selection111Cluster shapes have been proposed for applications
beyond centroid estimation in the forward region Peilian ; VIEL2016254 , but
they can be useful more generally.. This is illustrated schematically in Fig.
1. Thin layers of semiconductor provide the sensor material and with a strong
applied electric field, ionized electrons and holes can be collected and
registered with local readout electronics. In some cases, these pixels may
record a measurement of the deposited energy. This information has a poor
intrinsic resolution from straggling effects Landau:1944if ; Bichsel:1988if ;
Vavilov:1957zz  and is typically digitized with a small number of bits. While
this digitized information has many useful purposes (see e.g. Ref. 1710.02582
), the binary hit map carries the most important information and is the
dominant source of additional information considered in this paper.

Figure 1: A schematic diagram of a proton-proton collision producing a charged
particle that traverses layers of a pixelated tracking detector. In reality, a
given collision would result in many charged particles, but only one is
indicated for illustration.

We propose to use neural networks to optimally combine local information about
pixel cluster shapes with global geometric information to improve seed
selection. Neural networks have been recently studied for cluster seeding
Choma:2020cry ; Dietrich:2019qif ; DiFlorio:2018res , but there has not yet
been a systematic study of how much information local cluster structure adds
to seed selection performance. We compare classifiers trained to distinguish
true seeds from noise seeds using local and/or global information.
Furthermore, we study neural networks applied to 1-, 2-, and 3-cluster seeds.
An $n$-cluster seed is a set of $n$ clusters that are used for constructing
track candidates. Even with a single cluster, one can use the information
inside a pixel cluster to determine information about the underlying charged
particle trajectory.

This paper is organized as follows. Section 2 introduces the TrackML trackml ;
Amrouche:2019wmx  dataset, which we employ as a surrogate for actual LHC data,
and defines the variables used in the analyses. The results for one-, two-,
and three-layer pixel clusters are presented in Sec. 3. The paper ends with
conclusions and outlook in Sec. 4.

##  2 Simulation

The TrackML dataset uses top quark pair production from proton-proton
collisions as a representative process for track reconstruction at the LHC. In
order to emulate a realistic occupancy for the high-luminosity LHC, a Poission
random number (with mean 200) of minimum bias events are overlaid on top of
the $t\bar{t}$ collisions. This leads to an average of about 10,000
particles/event. The hard-scatter and minimum bias events are both simulated
using Pythia Sjostrand:2006za ; Sjostrand:2007gs .

The TrackML detector is a set of concentric cylindrical layers of pixelated
sensors (i.e. pixel layers) complemented by a set of circular disks (i.e.
strip layers) to ensure nearly $4\pi$ coverage in solid angle. For the study
presented in this paper, only the barrel pixel layers are used. The pitch size
of these pixel layers is 50 $\mu\text{m}$ in the direction perpendicular to
the beam and 56.25 $\mu\text{m}$ in the beam direction. The coordinates of
clusters are determined as the charged-weighted average over the constituent
hit locations.

Collisions occur near the geometric center of the simulated detector with
Gaussian profiles that have mean zero and standard deviation 5.5 mm in the
longitudinal direction (global $z$) and $15$ $\mu$m in the transverse
directions (global $x$ and $y$). The A Common Tracking Software (Acts) toolkit
andreas_salzburger_2020_3741401  provides the simulation engine to propagate
particles through a detector similar to ATLAS Aad:2008zzm  or CMS
Chatrchyan:2008aa  at the LHC. Particle trajectories are deterministic, except
for multiple Coulomb scattering, ionizing energy loss, and radiation energy
loss as emulated by the Fast Track Simulation Package in Acts. Noise is
simulated by randomly adding additional hits (15% of total hits).

Figure 2: A sample of clusters in the inner most four barrel detector layers.
The $x$-$y$ view is on the left and the $y$-$z$ view is on the right.

A subset of clusters from one event are shown in Fig. 2 which illustrates the
cylindrical nature of the inner barrel layers, and the clusters from a single
particle are shown in Fig. 3. Trajectories are approximately helical and
manifest as arcs of circles in $x$-$y$ and straight lines in $r$-$z$. Due to
the small beamspot in $x$-$y$, the path in $x$-$y$ goes through the origin. In
contrast, there is a non-trivial spread in the beamspot in $z$; in the right
plot of Fig. 3, the trajectory does not visually go through the origin.

Figure 3: A representative example of pixel clusters from a single particle,
in the $x$-$y$ plane (left) and the $z$-$r$ plane (right).

In addition to the global position of the hit, there are the local coordinates
that describe the location of the hit within a given module. These local
coordinates are defined such that the $u$, and $v$ direction are along the
active surface of the given module, with $u$ perpendicular to the beam
direction, $v$ along the beam direction, and $w$ is the coordinate normal to
the surface (see Fig. 1). The module defines the location of the hit using a
pixel matrix, where the pixels in the $u$ direction are said to be in channel
zero ($ch0$), and those in the $v$ direction are said to be in channel one
($ch1$). The number of pixel hits in $ch0$ and $ch1$ are denoted by $n_{ch0}$
and $n_{ch1}$ respectively. The cluster lengths in the $u$ direction ($\Delta
u$) and in the $v$ direction ($\Delta v$) are given by multiplying the pitch
sizes of the cells by $n_{ch0}$ and $n_{ch1}$ respectively. Therefore, the
cluster shape is defined by [$\Delta u$, $\Delta v$, $\Delta w$], where
$\Delta w$ is the width of the module.

In many instances, more than one pixel will have charge deposited within it.
In order to reconstruct the hit location, a weighted average of the charge
deposited information is used, i.e.:

$\vec{h}_{\mathrm{local}}=\frac{1}{\sum_{i}q_{i}}\sum_{i}q_{i}\,\vec{p}_{i},$
(1)

where the sum goes over all pixels with deposited charge above a threshold,
$q_{i}$ is the charge deposited, and $\vec{p}_{i}=(u_{i},\,v_{i},\,w_{i})$ is
the local position of the pixel in millimeters. After reconstructing the local
hit position information, the hit position is translated to global coordinates
within the detector. This is achieved through the use of a rotation matrix and
a translation vector:

$\vec{h}_{\mathrm{global}}=U_{\text{rot}}\,\vec{h}_{\mathrm{local}}+\vec{x}_{\text{trans}},$
(2)

where both the translation vector, $\vec{x}_{\text{trans}}$, and rotation
matrix, $U_{\text{rot}}$, are module dependent as illustrated in Fig. 4. In
what follows we make use of $\vec{h}_{\mathrm{local}}$,
$\vec{h}_{\mathrm{global}}$ and $\vec{p}_{i}$.

Figure 4: Example of transformation from local coordinates to global
coordinates. Reproduced from the TrackML challenge webpage trackml .

##  3 Results

The following section explores the information available from a single (Sec.
3.1), from a pair (Sec. 3.2), and from a triplet (Sec. 3.3) of pixel
cluster(s). To illustrate the potential gains from local cluster shape
information, we will use deep neural networks trained using three feature
sets. First, we will train neural networks to only consider the spatial
information contained in the global $x$,$y$,$z$ coordinate per pixel cluster
(based on the cluster centroid). Next, we consider networks trained using only
local information about the pixel cluster shape. Finally, both sets of
features are combined to demonstrate the optimal combination of both local and
global information. As will be shown below, the shape of the cluster can help
to narrow down the search region by providing pointing information on where
the next hit in the subsequent layer should be. The similarity of cluster
shapes between layers when considering pairs or triplets can also reject
incorrect pairings. By using this information, the number of seeds that need
to be considered can be reduced, improving subsequent steps in the seeding
algorithm.

All neural networks are multilayer perceptrons with fully connected layers
using the rectified linear unit (ReLU) activiation function between layers.
These networks are optimized by the gradient-based stochastic algorithm Adam
adam .

###  3.1 Single clusters

For a single pixel cluster, the networks are trained to predict the direction
of the particle in the frame defined by the local detector unit. We found that
choosing to use the momentum defined in the module reference frame instead of
attempting to predict the direction in a globally defined coordinate system
improves the accuracy of the network’s predictions. The neural networks
consist of five layers using $[256,128,64,32]$ units in the hidden layers. The
final output is a 3-vector prediction for the particle momentum,
$\vec{p}_{\text{pred}}$, at the location of the hit in the frame defined
locally by the module. Here, it is safe to neglect the effect of the magnetic
field since we are looking at the inner most layers which are close enough
together that the impact of the magnetic field from one layer to the next does
not modify the momentum in a significant manner. The networks are trained with
a cosine-proximity loss function, which minimizes the angle, $\Delta\theta$,
between the true momentum $\vec{p}_{\text{true}}$ and the predicted momentum
$\vec{p}_{\text{pred}}$,

$\cos\Delta\theta=\frac{\vec{p}_{\text{pred}}\bm{\cdot}\vec{p}_{\text{true}}}{|\vec{p}_{\text{pred}}||\vec{p}_{\text{true}}|}~{}.$
(3)

Thus, the normalisation of the network’s output is unimportant. It would be
interesting to see if the network could be trained to predict the direction
and magnitude of the momentum, but this is a more difficult task. Since the
change in momentum is small between consecutive hits in the inner layers this
additional information could be used to further reduce the combinatorics when
constructing doublets.

The dimension of the input vector to the neural networks depends upon which
set of features are being considered. There are three inputs for the spatial
information ($x$-$y$-$z$) and eight inputs associated with the cluster shape.
In reality, a cluster may have charge deposits in more than 50 pixels, or as
few as one. However, to encode the shape of the cluster in fewer variables we
first find the convex hull of the hits in the cluster using the Graham scan
algorithm GRAHAM1972132 . We then define a bounding box around the convex hull
by identifying the two extremum pixels. The eight input variables for the
cluster are then given as the position of these two extremum pixels (four
variables), the charge deposited in these pixels (two variables), and the
cluster shape ($\Delta u$, $\Delta v$). Although this information contains
redundancy we found it was beneficial with respect to training.

After the training is complete, the networks are compared using three
different metrics. The first metric is to compare the cosine of the angle
between the true direction of the particle and the predicted direction from
the network. Since we are only considering the inner most layers for this
network, the subsequent hit should be further from the collision point than
the initial. Therefore, if the network predicts the momentum to be towards the
center, we can flip the direction without introducing additional errors. The
different networks are compared in Fig. 5 for this metric. While the shape-
only network has a sharper peak at $\cos(\Delta\theta)=1$ than the spatial-
only network, the shape-only network has a bump at
$\cos(\Delta\theta)\in[-0.25,0.75]$ degrading the overall performance of the
shape-only network in comparison to the spatial-only network. Furthermore, by
combining the spatial and the shape information, there is a significant
improvement in the ability to predict the true direction of the particle. The
most interesting aspect is related to clusters in which the track does not
fall within a single value of channel zero (row) on the detector (see Fig. 1).
In the case of only using shape information, when the cluster falls along a
single row, the network is unable to predict in which direction the hit went
through the layer. This introduces the bump in the prediction for the shape
information in Fig. 5 due to the degeneracy over possible angles consistent
with the width and breadth of the cluster shape. Giving the network access to
the information about the global position of the hit allows it to determine
the most likely direction of the particle. This results in the improvement
seen in the combined prediction. However, if we remove the single row cluster
shapes (which make up approximately 30% of all hits), then the network’s
performance is improved even further as shown by the gray (combined) and
orange (shape) curves.

Figure 5: The number of counts versus the cosine between the true direction
and the predicted direction for hits in the tracker, using spatial (green),
shape (red), and combined (blue) information. Furthermore, if the cluster
shape is required to have a width in channel 0 greater than 1, the results for
the shape and combined are updated to the orange and gray lines respectively.
Figure 6: The fraction of events with a predicted direction less than
$\Delta\theta$ away from the true direction. The results are shown using the
spatial information only (green), the shape information only (red), and the
combined (blue). Furthermore, if the cluster shape is required to have a width
in channel 0 greater than 1, the results for the shape and combined are
updated to the orange and gray lines respectively.

The second metric is to consider the fraction of events for which the
predicted direction falls within a cone of a given angle from the true
direction. The smaller the cone required the better, since this would help to
reduce the number of combinations to consider during the seeding process. The
results are shown in Fig. 6. Again, the shape-only network has more
predictions close to the true direction, but has a very long tail as compared
to the spatial information. By combining the two however, significant
improvement is made by both having approximately 50% of all predicted
directions within $2\degree$ of the true direction, but by also removing the
long tail. In comparison, the spatial only does not reach a 50% containment
until almost $10\degree$ and the shape only does not reach 50% until about
$5\degree$. In the case of removing single row cluster shapes, we can see that
the network is able to predict all of the hit directions to within $2\degree$
of the true direction.

Figure 7: The fraction of hits in the next layer that fall within a cone are
shown against the probability that the true hit is contained within that cone.
This was obtained by averaging over all track hits in layers 2 and 4 from 100
events. The curves are parameterized by the opening angle of the cone. The
results for spatial information only (green), shape information only (red),
and combined (blue/gray) are shown considering all hits (left) and taking into
account that the hit must be moving away from the collision point (right). The
results shown here are for the innermost layer to the next layer. Requiring
that the width in channel 0 is greater than 1 moves the combined curve to the
gray curve shown, and moves the shape only to the orange curve.

The final metric is the comparison of the probability that the true hit is
within a cone of a given size compared to the fraction of all the hits in the
event that lie inside the cone. The angular opening of the cone parameterizes
the curve defining this metric. The results are shown in Fig. 7, which was
made by considering 100 events each with about 1000 hits in the innermost
layer and looking only at the hits in the innermost layer to the next layer
out. The left plot shows the results if the outward direction of the hit from
the interaction point is not taken into account, while the right plot takes
the direction into account. If we take a working point with 95% probability of
finding the true hit inside the cone and considering that the hit has to be
moving outwards, then we have approximately a 38.5% of all hits within the
cone for the combined network, which improves to 4.60% when only cluster
shapes with a channel 0 width greater than 1 are considered. The accuracy
degrades slightly when neglecting the outward direction to 41.1% and 9.4%
respectively. The effect of using wider clusters is only important in the
region of high probability of finding the hit within the cone. If instead of a
95% chance to find the hit we use a 50% chance, then the fraction of hits
within the cone is 0.13% and 0.1% for the combined and combined with channel 0
greater than 1 respectively. If this network is used to produce a triplet
seed, we expect the reduction in possible seeds to scale as the fraction of
hits in the cone squared. Table 1 shows the fraction of hits within the cone
for all results at a 50% and 95% probability of finding the hit in the cone.

| Fraction of hits in cone
---|---
Network | 50% Prob. Working Point | 95% Prob. Working Point
Spatial | 0.07 | 0.75
Cluster | 0.006 | 0.70
Combined | 0.001 | 0.39
Cluster ($n_{ch0}>1$) | 0.003 | 0.09
Combined ($n_{ch0}>1$) | 0.001 | 0.05
Table 1: Efficiency and purity numbers at two possible working points, based
on Figure 7.

We have not attempted to balance the trade-off between efficiency ($\epsilon$)
and purity to find an optimal working point for the single cluster NN approach
but instead conclude this section with a few observations. One possibility
could be to work at a lower efficiency value, which results in a smaller
number of background hits in the cone, but to apply the direction prediction
on consecutive layers and compare them. Possibly allowing working with lower
efficiency but taking into account both forward and backward predictions.
Alternatively, the single hit network could be used to predict the particle
directions for all hits within consecutive layers and compare these directions
to find mutually consistent sets.

For hit patterns involving more than a single line of hit pixels the combined
network is remarkably adept at correctly predicting the particle’s true
direction, which potentially motivates training specialized networks for
different classes of pixel clusters. One such specialized network would be the
Bayesian Neural Network doi:10.1162/neco.1992.4.3.448 , in which the network
would be trained to predict both the track direction and an uncertainty on
that direction. Even for the single row hit patterns there is additional
information, which we have not utilised, in the fraction of charge deposited
in each pixel. Finally, the output from this single hit network can be fed to,
or used in tandem with, another network to determine which of the possible
hits in the cone are true track partners of the original hit, this will be
addressed in the subsequent sections.

###  3.2 Cluster Doublets

Doublets are composed of two clusters coming from consecutive layers.
Constructing doublets is the first step towards reconstructing a track, which
is a crucial step not only in traditional track reconstruction algorithms
Aaboud:2017all  but also in deep learning based algorithms Farrell:2018cjr ;
Ju:2020xty ; choma2020track . We now discuss our doublet neural network, which
was trained to separate true doublets from fake doublets. The list of possible
doublets may come from all possible pairs of clusters in consecutive layers,
or may be from the output of the direction prediction discussed in the
previous section. Doublets formed by clusters coming from the same charged
particle are true doublets; those formed otherwise are fake doublets. We
define the purity as the ratio of true doublets to total doublets in the given
selection. Similarly, the efficiency is defined as the ratio of the number of
true doublets that pass selection requirements to the total number of true
doublets input to the selection process. We demonstrate that the neural
network trained with the combined information of cluster spatial and shape
information results in a higher doublet purity for a given doublet efficiency.
The reason for this is that the cluster shape of two clusters coming from the
same track tend to have similar structures. As an illustration, Figure 8 and 9
show the cluster shapes of a pair of clusters coming from the same track (left
two panels) and a pair of clusters coming from different tracks (right two
panels) in two different kinematic regions.

Figure 8: Cluster shape of two clusters coming from the same track (left two)
and two clusters coming from different tracks (right two) for two clusters
forming a doublet with $p_{\text{T}}<0.5$ GeV and $|\eta|<1$ . Note that the
origin (bottom left) of $ch0$ and $ch1$ has been shifted to the position where
the cluster is located, for clarity. The pixel intensity demonstrates the
charge density deposited in the pixel, but this information is not given to
the NN. Figure 9: Cluster shape of two clusters coming from the same track
(left two) and two clusters coming from different tracks (right two) for two
clusters forming a doublet with $p_{\text{T}}>1$ GeV and $2.1<|\eta|<3$ . Note
that the origin (bottom left) of $ch0$ and $ch1$ has been shifted to the
position where the cluster is located, for clarity. The pixel intensity
demonstrates the charge density deposited in the pixel, but this information
is not given to the NN.

The doublet neural network is composed of four layers of neurons with the
sizes of [128, 64, 32, 1]. The outputs of the last neuron are transformed by a
sigmoid function and are used as the doublet scores: the network’s prediction
for the probability that the doublet is a true doublet. To balance the number
of true and fake doublets in the training, true doublets are constructed from
100 events and fake doublets are from a single event using all possible
combinations of the hits in the two innermost consecutive layers. The doublet
scores are quantitatively compared with the labeling by using the binary
cross-entropy loss function. The smaller the loss value is, the better the
predictions agree with the ground truths.

Table 2: Input variables for training the doublet neural network. The definition of the variables can be found in the text.  Group | Variables
---|---
spatial only |  $\Delta r$, $\Delta\phi$, $\Delta\phi/\Delta r$, $\Delta z$, $z_{0}$, $\Delta\eta$
shape only |  $\Delta\eta^{\prime}$, $\Delta\phi^{\prime}$

The doublet neural network is trained separately with the same training events
but different sets of input variables as summarized in Table 2. ‘Spatial only’
variables are derived from the cluster’s global position in a cylindrical
coordinate system. $\Delta r$, $\Delta\phi$, $\Delta z$ are the differences of
the two hits in $r$, $\phi$ and $z$, $\Delta\eta$ is the difference in
pseudorapidity222Pseudorapidity $\eta=-\ln(\tan{(\theta/2)})$, where $\theta$
is the polar angle relative to the beamline. and $z_{0}$ is the intercept of
the line connecting the two hits at $r=0$. ‘Shape only’ variables are derived
from the cluster shape described in Section 2. The length of the cluster is
given by $L=(\Delta u^{2}+\Delta v^{2}+\Delta w^{2})^{1/2}$. We then use the
rotation matrix in Eq. (2) to transform the cluster shape to a coordinate
system aligned with the global coordinates, in which the shape is [$\Delta
u^{\prime}$, $\Delta v^{\prime}$, $\Delta w^{\prime}$]. In the new coordinate
system, we introduce two new variables, $\eta^{\prime}$ and $\phi^{\prime}$,
that are related to the fraction of the length of the cluster along the beam
axis and its orientation in the plane perpendicular to the beam, respectively:

$\eta^{\prime}=\frac{1}{2}\log\frac{L-\Delta w^{\prime}}{L+\Delta
w^{\prime}}~{},\quad\quad\phi^{\prime}=\tan^{-1}\frac{\Delta
v^{\prime}}{\Delta u^{\prime}}~{}.$ (4)

The quantities $\eta^{\prime}$ and $\phi^{\prime}$ are calculated for each
cluster and the differences between the two clusters, $\Delta\eta^{\prime}$,
$\Delta\phi^{\prime}$, are used as inputs to the network. Redundant spatial
information is fed to the network to gain accuracy and convergence time.

The neural network based selections are compared with the cut-based
selections333$|\Delta\phi$/$\Delta r|<0.0006$ rad/mm and $|z_{0}|<100$ mm.
from Ref. Ju:2020xty , which have an efficiency and purity of $43\%$ and
$3\%$, respectively. This comparison is shown in Figure 10. For a purity of
3%, using the neural network trained with cluster spatial information yields
an efficiency that doubles the one by using the cut-based selection.
Furthermore, adding the clustering information into the neural network
increases the efficiency relatively by about 20% to 0.98. The three
differently trained neural networks are compared exclusively for doublets in
different $\eta$ and $p_{\text{T}}$ regions. The $\eta$ and $p_{\text{T}}$ of
each doublet are obtained from a helix fit of the doublet assuming the doublet
starts from the origin. The helix fit is implemented with the conformal
mapping method Hansroul:1988wa . Figure 11 shows the comparison of the doublet
purity for a given doublet efficiency of 97% for the three doublet neural
networks. For low-$p_{\text{T}}$ doublets, the clustering information is as
important as the spatial information in the central $\eta$ region and becomes
more important in the high $\eta$ region. For all kinematic regions, combining
the clustering information with the spatial information significantly boosts
the doublet purity at this high doublet efficiency working point. Finally, the
output from doublet network can be used to construct triplet clusters as
described in the next section.

Figure 10: Doublet efficiency versus doublet purity for different trained
doublet neural networks and the cut-based doublet selection. The definition of
doublet efficiency and purity can be found in the text.  Figure 11: Comparison
of the purity of the selected doublets for a given signal efficiency of 97% in
different $\eta$ and $p_{\text{T}}$ bins of the doublet candidates for the
same neural network architecture trained with three different sets of input
variables: only the hit locations in blue, only the cluster shape information
in orange and the combination of the two in green. The gray band indicates the
statistical uncertainty.

###  3.3 Cluster Triplets

Triplets are composed of three hits with each coming from one of the three
consecutive innermost layers444In practice, one could allow for gaps, but we
consider the consecutive case for simplicity.. As with the doublet case, the
triplet neural network was trained to separate true triplets from fake
triplets.

As shown in the previous section, the seeding performance depends strongly on
the seed kinematic properties. Cluster triplets are sorted into $p_{T}$ and
$\eta$ bins using the same helical fit, only now with three instead of two
points. For real triplets, the true momentum is known, but to directly compare
with fake triplets, the same reconstruction algorithm is used in both cases.
Due to the large number of potential fake triplets, only a fraction of fake
triplets from one event are used for training. Specifically, true triplets
from about 1000 events are used for the signal and about 0.2% of the possible
fake triplets from one event are used for the background. This results in
approximately balanced classes for the training; for the final metrics, event
weights are applied to reproduce the natural abundance of each class.

As with the doublets, three sets of features are used: ‘spatial’, ‘shape’, and
‘combined’. Each set of features are calculated from the same collection of
three hits taken from three consecutive layers, with one hit coming from each
layer. A spatial triplet is a group of twelve numbers: the first six entries
are calculated by the differences in spatial coordinates of two hits on the
first two consecutive layers whereas the second six entries are obtained in
the same process but with two hits on the second two consecutive layers.
Figure 12 shows normalized histograms of each one of the first six entries of
a spatial triplet. The features are the same as the doublet case from the
previous section.

Figure 12: A subset of the spatial triplet features with all the parameters
calculated from the first two consecutive layers. Solid line for true signal
and dotted line for fake background. Each feature is standardized to have zero
mean and unit variance and thus are dimensionless.

The second set of features use only local information about the cluster shape.
This information is summarized using the length of the clusters along the $v$
direction ($n_{ch1}$) in units of number of pixels. Other features of the
cluster shapes may be useful, but these features already contain significantly
useful information. The combination of the spatial and shape features gives 15
numbers for the ‘combined’ feature set. Each feature set is processed by a
neural network of the same architecture. These neural networks are composed of
four layers with [128, 64, 32, 1] neurons and are trained with the binary
cross entropy loss function. There are a large number of fake triplets that
can be easily eliminated and make it difficult for the triplet network to
learn. In order to improve the training efficacy, we first apply a threshold
on a doublet network. In particular, we set a threshold on the doublet network
so that we keep 99% of the true doublets for each feature set.

Neural networks trained using the three different feature sets are compared in
Fig. 13. For the evaluation of the purity and efficiency, weights are applied
to correct the number of true and fake triplets to match their expected
natural abundance. The cluster information alone is not very effective at
eliminating fake triplets, but the additional information significantly
improves the efficiency for a fixed purity when combined with the cluster hit
coordinates. For example, at a purity of 30%, the efficiency increases from
about 20% to nearly 90%. The jaggedness is a result of the limited statistics.

Figure 13: Efficiency versus purity for the three neural networks. Blue for
the spatial features alone, orange for the shape features alone, and green for
the combined features. At a fixed efficiency, the greater the purity the
better the performance.

A differential perspective of the three networks is presented in Fig. 14,
grouping the results in bins of seed $p_{T}$ and $\eta$. For the fixed
efficiency of 97%, the combined feature set has the best purity except in
three bins where the purity is statistically consistent with unity. For each
$p_{T}$ bin, all three triplet neural networks perform better at higher
$|\eta|$ than at lower $|\eta|$ because the cluster lengths are longer and so
the shape information is more useful. Overall, the purity from networks
trained only using cluster information is very low ($10^{-7}-10^{-5}$), but
when combined with spatial information, the purity is everywhere above 1% and
in some cases is statistically consistent with 100%. This improvement in the
purity of triplet seeds should allow for a drastic improvement in overall
track reconstruction time.

Figure 14: Comparison of the purity of the selected triplets for a given
signal efficiency cut at 97% in different $\eta$ and $p_{\text{T}}$ bins. Blue
for the spatial feature alone, orange for the cluster feature alone, and green
for the combined feature. The error bars represent the statistical
uncertainty. High $\eta$ bins in the range of [2.1, 3] are not included due to
the limited number of triplets passing the selection.

##  4 Conclusions and Outlook

This paper has presented a series of studies that show how local pixel cluster
shape information can be useful for performing track cluster seeding. The
additional information is useful when considering single clusters, pairs of
clusters, as well as triplets of clusters. For single pixel clusters, shape
information is used to estimate the direction of the charged particle. This
vector estimate can be used to significantly reduce the number of hits in
subsequent layers that need to be considered when building doublets and
triplets. As expected, this information is most useful when the pixel cluster
is composed of more than one row in the $z$ direction. Compared to using
spatial information alone, the combined information reduces the number of hits
inside a cone around the predicted trajectory by about a factor of ten. For
pairs of pixel clusters, one can use their relative orientation combined with
the local cluster shapes to reject fake doublets that are not from a single
charged particle. The expected gains depend on momentum of the particles. For
example, for a signal efficiency of 97%, the purity increases from about
0.5-1% to 2-5% at $p_{T}<500$ MeV and from about 3% to 6-10% for $p_{T}>1$
GeV. There are also significant improvements for triplets of pixel clusters,
with the most significant improvements projected to be at low $p_{T}$ and at
high $|\eta|$ where many order of magnitude improvement in purity is possible
at a fixed efficiency. Future studies could combine the approaches designed
for single clusters, pairs of clusters, and triplets of clusters to further
optimize the classification and computational performance.

There are a variety of extensions that would be interesting to pursue in the
future. For example, a direct comparison between the new approaches and
traditional seeding algorithms would be an important step to understand the
gains in practice. The spatial baseline shown here uses all of the same
information as traditional algorithms, but is expected to be better because it
already uses a neural network to optimally process the hit positions. With
innovations in experiment-agnostic tools like Acts, a direct comparison may be
possible in the near future and even larger gains than those shown here are
expected. Beyond comparing to traditional methods, it would also be
interesting to extend the approaches here to include even more information
(e.g. pixel cluster charge and precision timing). It would also be interesting
to apply these techniques to long-lived particles (LLPs). The single cluster
NN trained only on shape information is insensitive to the origin of the
track. On the other hand, all the other networks studied use information about
the track origin and an extension to LLPs would require more study, with an
appropriate data set. Innovations in deploying neural networks on accelerators
like FPGAs Duarte:2018ite  could also make the implementation of these seeding
algorithms fast enough to fit into trigger time budgets. These innovations and
more may significantly improve track seeding for the HL-LHC era and beyond.

## Acknowledgments

We would like to thank Maurice Garcia-Sciveres for many useful discussions and
Claudius Krause, Jack Collins, and Zhen Liu for collaboration in the early
stages of this project. We also thank Maurice, Claudius, Jack, and Zhen for
detailed feedback on the manuscript. This work was supported by the U.S.
Department of Energy, Office of Science under contract DE-AC02-05CH11231. This
manuscript has been authored by Fermi Research Alliance, LLC under Contract
No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science,
Office of High Energy Physics.

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