# Effect of Strain on the Band Gap of Monolayer MoS2

Raj K. Sah  Department of Physics, Temple University, Philadelphia, PA 19122
Hong Tang, Chandra Shahi, Adrienn Ruzsinszky, and John P. Perdew  Department
of Physics and Engineering Physics, Tulane University, New Orleans, LA 70118

###### Abstract

Monolayer $\mathrm{MoS_{2}}$ under strain has many interesting properties and
possible applications in technology. A recent experimental study examined the
effect of strain on the bandgap of monolayer $\mathrm{MoS_{2}}$ on a mildly
curved graphite surface, reporting that under biaxial strain with a Poisson’s
ratio of 0.44, the bandgap decreases at a rate of 400 meV/% strain. In this
work, we performed density functional theory (DFT) calculations for a free-
standing $\mathrm{MoS_{2}}$ monolayer, using the generalized gradient
approximation (GGA) PBE, the hybrid functional HSE06, and many-body
perturbation theory with the GW approximation using PBE wavefunctions
(G0W0@PBE). We found that under biaxial strain with the experimental Poisson’s
ratio, the bandgap decreases at rates of 63 meV/% strain (PBE), 73 meV/%
strain (HSE06), and 43 meV/% strain (G0W0@PBE), which are significantly
smaller than the experimental rate. We also found that PBE predicts a
similarly smaller rate (90 meV/% strain) for a different Poisson’s ratio of
0.25. Spin-orbit correction (SOC) has little effect on the gap or its strain
dependence. Additionally, we observed a semiconductor-to-metal transition
under an equal tensile biaxial strain of 10% and a transition from a direct to
an indirect bandgap, consistent with previous theoretical work.

##  I Introduction

Graphene is a typical two-dimensional (2D) layered material. Monolayer
graphene has been found to exhibit several notable features, including high
electrical conductivity, high transparency, high thermal conductivity at
ambient temperature, a high Young’s modulus, and a high specific surface area
Allen  _et al._ (2010); Soldano  _et al._ (2010). Graphene possesses
outstanding electrical properties; however, being a zero-gap material
Novoselov  _et al._ (2004) limits its application in logical circuits. In an
effort to overcome the limitations of graphene and increase its variety of
uses, scientists have turned their attention back to alternative 2D materials
that resemble graphene Ganatra and Zhang (2014); Chhowalla  _et al._ (2014).
Monolayer MoS2 has gained significant interest in recent days.

Monlolayer MoS2 has a 2D nature and a large band gap. Similar to graphene Lee
_et al._ (2008), a single-layer MoS2 has strong inherent tensile strength and
flexibility out of plane, allowing the film to withstand strains of up to
$11\%$ before rupturing Bertolazzi  _et al._ (2011). According to theoretical
predictions, the band gap of semiconducting transition-metal dichalcogenides
(TMDCs) would be significantly impacted by tensile strain, and applying
approximately 10% biaxial strain might potentially close the gap entirely
Johari and Shenoy (2012); Yue  _et al._ (2012); Shi  _et al._ (2013). Strains
can be used to control the electronic properties of two-dimensional (2D)
materials, which is important for implementation of a 2D material into
flexible electronics and next-generation strain engineering devices.
Therefore, it becomes important that the material withstands the desired
strain, and that we know the effect of the strain on the material. A recent
experimental study of an $\mathrm{MoS_{2}}$ monolayer on a mildly curved
graphite substrate has measured the relationship between band gap change and
strain in monolayer MoS2 Trainer  _et al._ (2019). Trainer et al. Trainer  _et
al._ (2019) determined the Poisson’s ratio of monolayer MoS2 to be 0.44. A
positive Poisson’s ratio has been previously reported Kang  _et al._ (2015);
Woo  _et al._ (2016a, b). Because of a positive Poisson’s ratio, the crystal
will respond to a tensile strain in one direction by compressing in the
perpendicular direction. Trainer et al. Trainer  _et al._ (2019) found that
the quasiparticle band gap decreases at the rate of 400 meV/% strain until a
nominal strain of 2.5% and at a much slower rate from 2.5% to 4.9%. In this
work, we have studied band gap change with strain for a free-standing
$\mathrm{MoS_{2}}$ monolayer theoretically, using the HSE06 Krukau  _et al._
(2006) and G0W0 Hedin (1965); Hybertsen and Louie (1985); van Schilfgaarde
_et al._ (2006) methods. We also used the PBE Perdew  _et al._ (1996) method
to compare the band gap of an unstrained monolayer MoS2 with the work of Woo
et al. Woo  _et al._ (2016a). Our results show that the band gap decreases at
a much slower rate than reported in the work by Trainer et al. Trainer  _et
al._ (2019).

Figure 1: Rectangular unit cell of monolayer MoS2. Only the S (yellow) atoms
in the plane above the plane of the Mo (purple) atoms can be seen. Those in
the plane below are obscured by those above. The lattice constant a is the
length of the rectangular unit cell in the x-direction and b in the
y-direction.

##  II Computational Methods

We used the rectangular unit cell as shown in Figure 1 for our calculations.
The rectangular unit cell helped to strictly maintain a Poisson’s ratio of
0.44. The rectangular unit cell comprises two Mo atoms and four S atoms. The
HSE06 and PBE calculations were performed using the Vienna Ab initio Package
Simulation (vasp) Kresse and Hafner (1994); Kresse and Joubert (1999). For
HSE06 and PBE calculations, we used an 16 x 16 x 2 $\Gamma$-centered k-grid to
sample the Brillouin zone. We used a 400 eV cutoff energy for the plane-wave
basis for HSE06 and 520 eV for PBE. Initially, we relaxed the lattice
parameters a and b and the internal co-ordinates of ions while keeping c fixed
at 10 Å to make a three-dimensional super cell. To apply strain, we utilized
the relaxed structure and adjusted the lattice parameters a and b to achieve
the desired strain, followed by the relaxations of internal co-ordinates of
ions. During the strain application, for HSE06 and G0W0@PBE calculations, we
maintained the Poisson’s ratio of 0.44 to remain consistent with experiment
Trainer  _et al._ (2019). If $\epsilon_{x}$ represents the strain along the
x-direction, then the strain along the y-direction ($\epsilon_{y}$) would be
$-0.44\epsilon_{x}$. If a and b are the lattice vectors before applying
strain, and anew and bnew are new lattice vectors after applying strain, then:

$\mathbf{a}_{\text{new}}=\mathbf{a}\left(1+\frac{\epsilon_{x}}{100}\right)$
(1)
$\mathbf{b}_{\text{new}}=\mathbf{b}\left(1+\frac{\epsilon_{y}}{100}\right).$
(2)

The G0W0 Deslippe  _et al._ (2012); Hybertsen and Louie (1986) calculations
were conducted in BerkeleyGW Deslippe  _et al._ (2012) by pairing with Quantum
ESPRESSO Giannozzi  _et al._ (2017). The wavefunction energy cutoff is 70 Ry
($\sim$950 eV). The energy cutoff for the epsilon matrix is 18 Ry ($\sim$240
eV). The k-point mesh of 20 × 12 × 1 is used for the rectangular cell. The
band number for summation is 220. The correction of the exact static remainder
and the 2D slab Coulomb truncation were used.

|  | HSE06 |  | G0W0@PBE +SOC |  | Experiment
---|---|---|---|---|---|---
strain ($\epsilon_{x}$)% |  | Band gap |  | $\Delta$ |  | Band Gap |  | $\Delta$ |  | Band Gap |  | $\Delta$
0.00 |  | 2.30 |  | 0.00 |  | 2.64 |  | 0.00 |  | 2.34 |  | 0.00
1.00 |  | 2.25 |  | -0.05 |  | 2.60 |  | -0.04 |  | 1.94 |  | -0.40
2.50 |  | 2.15 |  | -0.15 |  | 2.53 |  | -0.11 |  | 1.34 |  | -1.00
4.00 |  | 2.01 |  | -0.29 |  | 2.46 |  | -0.19 |  | 0.74 |  | -1.60
Table 1: The computed and experimental band gaps (in eV) and changes ($\Delta$) in the band gap relative to the gap at zero strain of monolayer MoS2. The HSE06 and G0W0@PBE results correspond to the rectangular unit cell shown in Fig. 1. Experimental values from Ref. Trainer  _et al._ (2019). strain ($\epsilon_{x}$) % |  |  | Band gap |  |  | $\Delta$ |  |  | Lattice constant
---|---|---|---|---|---|---|---|---|---
0.00 |  |  | 1.65 |  |  | 0.00 |  |  | 3.18
1.00 |  |  | 1.45 |  |  | -0.20 |  |  | 3.21
2.50 |  |  | 1.15 |  |  | -0.50 |  |  | 3.26
4.00 |  |  | 0.82 |  |  | -0.83 |  |  | 3.31
6.00 |  |  | 0.45 |  |  | -1.20 |  |  | 3.37
8.00 |  |  | 0.17 |  |  | -1.48 |  |  | 3.43
10.00 |  |  | 0.02 |  |  | -1.63 |  |  | 3.50
Table 2: band gap vs. strain and lattice parameter (in Å) calculated using PBE
for the rectangular cell shown in Fig. 1. Here, we applied uniform biaxial
tensile strain.

##  III Results and Discussion

First, we calculated the band gap of the unstrained monolayer MoS2. For this,
we relaxed lattice parameters a and b and internal co-ordinates of the ions.
Then, we used the relaxed structure to obtain band gaps. The lattice parameter
a for the unstrained relaxed structure is 3.15Å for HSE06 and 3.18 Å for PBE.
Our lattice parameter agrees with references Johari and Shenoy (2012); Pela
_et al._ (2024). We obtained band gaps of 1.65 eV, 2.30 eV, and 2.64 eV using
PBE Perdew  _et al._ (1996), HSE06, and G0W0@PBE methods, respectively, for
unstrained monolayer MoS2. These results agree with Refs. Johari and Shenoy
(2012); Pela  _et al._ (2024) for the unstrained monolayer MoS2. Our HSE06
band gap agrees with the experimental band gap of 2.34 eV obtained using STM
Trainer  _et al._ (2019), however the G0W0@PBE band gap is slightly larger.
Trainer et al. Trainer  _et al._ (2019) proposed the relation between the band
gap ($E_{\mathrm{g}}$) and strain ($\epsilon_{x}$) as in equation 3. Table 1
summarizes the band gap and change in band gap vs. strain based on relation 3
in the last two columns.

$E_{\text{g}}=\left(-0.401\epsilon_{x}+2.343\right)\ eV.$ (3)

Figure 2: Change in the band gap $E_{\mathrm{g}}$ as a function of strain $\epsilon_{x}$ along the x-direction using the rectangular unit cell of monolayer MoS2 based on the HSE06 functional. We applied strain such that Poisson’s ratio remains 0.44. |  | Poisson’s ratio: 0.25 |  |  |  | Poisson’s ratio: 0.44
---|---|---|---|---|---|---
strain ($\epsilon_{x}$) % |  | Band gap |  |  | $\Delta$ |  |  |  | Band gap |  |  | $\Delta$
0.00 |  | 1.65 |  |  | 0.00 |  |  |  | 1.65 |  |  | 0.00
1.00 |  | 1.60 |  |  | -0.05 |  |  |  | 1.60 |  |  | -0.05
2.50 |  | 1.45 |  |  | -0.20 |  |  |  | 1.50 |  |  | -0.15
4.00 |  | 1.30 |  |  | -0.35 |  |  |  | 1.40 |  |  | -0.25
Table 3: PBE band gap (in eV) vs. strain evaluated for Poisson’s ratios 0.25
and 0.44.

Figure 3: Band structure of (a) unstrained, (b) 1%, (c) 2.5 %, and (d) 4%
strained monolayer MoS2 calculated using a rectangular cell with PBE+SOC and
G0W0@PBE+SOC methods. We applied strain such that Poisson’s ratio remains
0.44. The plot shows unstrained monolayer MoS2 to be a direct band gap
material, and 4% strained monolayer MoS2 to be an indirect band gap material
at the PBE level. However, it is still almost direct at the G0W0@PBE level.
(By K we indicate the point along the line from $\Gamma$ to X that is closest
to the symmetry point K of the hexagonal zone.)

To compare with the experimental results, we calculated the band gap of
monolayer MoS2 theoretically using the HSE06 and G0W0@PBE methods using the
rectangular unit cell as shown in Fig. 1. We used a rectangular unit cell to
easily maintain the Poisson’s ratio of 0.44 determined by Trainer et al.
Trainer  _et al._ (2019). We applied $\epsilon_{x}$ $\%$ strain along the
x-direction and $-0.44\epsilon_{x}\%$ strain along the y-direction. As
depicted in Figure 1, we aligned the x-direction along lattice vector a and
the y-direction along lattice vector b of the rectangular cell. Table 1
summarizes the band gap and the change in the band gap evaluated using the
HSE06 and G0W0@PBE methods. The change in the band gap at any strain
$\epsilon_{x}$ is evaluated relative to the band gap of the unstrained sample.
Additionally, we exchanged the x and y directions (i.e., the x-direction
aligned with the lattice vector b and y-direction along lattice vector a ) and
computed the band gap and the change in the band gap versus strain using the
HSE06 method. However, we did not observe any difference in the results. To
compare our result with the result of Ref. Trainer  _et al._ (2019), we
applied up to 4% strain. Table 1 clearly shows that the change in the band gap
vs. strain predicted by HSE06 and G0W0@PBE is much smaller than in the
experiment Trainer  _et al._ (2019). The change in the band gap predicted by
G0W0@PBE for 2.5% strain is 0.11 eV. This agrees with the result of Ref.
Conley  _et al._ (2013), but not with the result of Ref. Trainer  _et al._
(2019). We also quantified the change in the band gap per % of strain. For
this, we applied a linear fit to HSE06 band gap vs. strain data as shown in
Figure 2. The fit shows that the band gap decreases at a rate of 73 meV/%
strain, which is much smaller than in the experimental rate of 400 meV/%
strain Trainer  _et al._ (2019). A similar fit for G0W0@PBE shows a decrease
of band gap at the rate of 45 meV/% strain.

The discrepancy between theory and experiment Trainer  _et al._ (2019) in the
band gap change vs. strain in monolayer MoS2 is significant and needs to be
resolved. Ref. Trainer  _et al._ (2019) mentions that the strain is not
uniformly distributed in the sample, which is also evident from Figure 2e of
Ref. Trainer  _et al._ (2019). Ref. Trainer  _et al._ (2019) also reports the
slipping of the sample with respect to the substrate. It appears that the
strain is not uniformly transferred from the substrate to the film. If the
strain transfer process induced some ripple effect in the sample, then it is
possible that the sample has regions with unusually high strain.

A more interesting possibility is that the graphite surface, strained by the
monolayer, has a strong proximity effect on the bandgap of the monolayer.
While this could probably be tested by calculations, it is beyond the scope of
this work.

We first applied the theoretical strain by considering that monolayer MoS2 has
a Poisson’s ratio of 0.44. However, the PBE calculations of Refs. Kang  _et
al._ (2015); Xiong and Cao (2015) show much smaller values of Poisson’s ratio
of 0.25 and 0.267, respectively, for free-standing monolayer MoS2. Therefore,
we also used the Poisson’s ratio values of 0.25 and 0.44 from the PBE Ref.
Kang  _et al._ (2015) and from the experimental Ref. Trainer  _et al._ (2019),
respectively, and calculated the effect of biaxial strains maintaining these
Poisson’s ratios using the PBE functional. Table 3 shows the results. As
expected, the band gap decreases faster for the Poisson’s ratio of 0.25
compared to that of 0.44. We also quantified the rate of decrease of the band
gap by applying a linear fit to the band gap vs. strain data of Table 3. The
fitted equations for Poisson’s ratio 0.25 and 0.44 are:

$E_{\mathrm{g}}=-0.0898\epsilon_{x}\ +\ 1.6684\ eV$ (4)

and

$E_{\mathrm{g}}=-0.0633\epsilon_{x}\ +\ 1.6561\ eV,$ (5)

respectively. The rate of decrease of the band gap for the Poisson’s ratio of
0.44 is 63 meV/% strain, which is not very different from the HSE06 result of
73 meV/% strain, and for the Poisson’s ratio of 0.25 is 90 meV/% strain. This
also indicates that the value used for Poisson’s ratio affects the rate of
change of the band gap, but not dramatically. Our calculated values are not
surprisingly different from the values of 59 and 94 meV/% strain for direct
and indirect fundamental band gap, respectively, calculated using the GW0-BSE
method in Ref. Conley  _et al._ (2013). Ref. Conley  _et al._ (2013) also
reports the rate of decrease of the optical band gap using photoluminescence
spectroscopy to be nearly 45 meV/% strain. Possible errors in Poisson’s ratio
do not seem to be the reason for the large discrepancy in band gap change with
strain between theoretical and experimental results.

Figure 4: PBE band gap $E_{\mathrm{g}}$ as a function of equal strain
$\epsilon$ applied along both the x and y directions using a rectangular unit
cell of monolayer MoS2 (equivalent to a Poisson’s ratio of -1).

The band gap of 1H $\mathrm{MoS_{2}}$ monolayer decreases with increasing in-
plane tensile strain, a trend opposite to that of some other transition metal
dichalcogenides (TMDs), such as the 1T $\mathrm{TiSe_{2}}$ monolayer Pasquier
and Yazyev (2018). The decreasing trend in $\mathrm{MoS_{2}}$ can be
attributed to the effect on the band centers of the change in ligand field
induced by the in-plane strain, the change of the widths of the $d$-orbital
bands, and the change of $d$-orbital SOC splitting. A more detailed analysis
is presented in the Supplementary Information (SI).

To investigate the effect of spin-orbit correction (SOC), we also calculated
the PBE+SOC and G0W0@PBE+SOC band structures of monolayer MoS2 for no strain,
1.0% strain, 2.5% strain, and 4% strain along the x-direction with a strain
along the y-direction to maintain Poisson’s ratio of 0.44. The SOC effect is
not always significant to band structures of 2D materials. For example, the
SOC effect on the bands around the Fermi level of monolayer
$\mathrm{Ta_{2}Ni_{3}Te_{5}}$ was shown to be very small Guo  _et al._ (2022).
For 1H WSe2 monolayer, the SOC effect induces a strong spin-splitting ($\sim$
0.5 eV) around the valence band maximum Reyes-Retana and Cervantes-Sodi
(2016). Here, for the 1H MoS2 monolayer, the SOC-induced band splitting near
the valence band maximum is about 0.15 eV, consistent with a previous study
Qiu  _et al._ (2013). As shown in Figure 3a and 3b, both PBE+SOC and G0W0 show
the unstrained and 1.0% strained monolayer MoS2 to be a direct band gap
semiconductor. At 2.5% strain, PBE+SOC still shows a direct gap, while G0W0
shows that the valence band maximum (VBM) and the conduction band minimum
(CBM) located in the $\Gamma$X line and close to X are slightly misaligned,
showing a slightly indirect gap. At 4% strain, PBE+SOC shows that the local
VBM at Gamma has become the global VBM, yielding an indirect gap. Therefore,
with increasing strain this material tends to change from a direct band gap at
an indirect band gap material. Such an effect has been previously observed in
Refs. Lu  _et al._ (2012); Conley  _et al._ (2013); Johari and Shenoy (2012).

Additionally, theory predicts that 10% biaxial strain might close the band gap
of MoS2 Johari and Shenoy (2012); Yue  _et al._ (2012); Shi  _et al._ (2013).
We applied 10% strain along the x-direction and -4.4% strain along the
y-direction to maintain the Poisson’s ratio of 0.44 and found that HSE06 gives
a band gap of 1.6 eV, which is much greater than 0 eV. We also applied equal
biaxial tensile strains of 10% along both the x and y directions and observed
that HSE06 significantly reduces the band gap to 0.30 eV, but does not close
it completely. However, PBE almost closes the band gap of monolayer MoS2 for
10% equal biaxial tensile strain applied along both x and y directions. This
result is consistent with the work in Ref. Johari and Shenoy (2012). Table 2
shows strain vs. band gap and lattice parameter a of monolayer MoS2 obtained
using the PBE functional for up to 10% of equal biaxial tensile strains. Table
2 agrees with Table 1 of Ref. Johari and Shenoy (2012). We also plotted the
PBE band gap vs. strain up to 10% of strain as shown in Fig. 4. The plot is
again similar to the plot in Ref. Johari and Shenoy (2012). The plot shows
that band gap is not a linear function of strain. The rate of decrease of band
gap ($dE_{\mathrm{g}}/d\epsilon$) is:

$dE_{\mathrm{g}}/d\epsilon=0.0162\epsilon-0.2497\ eV/\%,$ (6)

where $\epsilon$ is the equal biaxial tensile strain applied. The band gap
decrease becomes slower with increase in strain.

##  IV Conclusion

The ability to control the electronic properties of a material by applying
strain is of critical importance, as it widens the application of 2D materials
into flexible electronics and next-generation strain engineering devices. 2D
MoS2 occupies a significant role in these applications due to its important
properties such as semiconducting nature, high tensile strength and
flexibility, and high carrier mobility. To be used in the flexible electronics
and next-generation strain engineering devices, it is crucial to understand
how strain affects the electronic properties of this material.

In this work, we theoretically studied the band gap and its change in free-
standing monolayer MoS2 using the HSE06 and G0W0 methods to compare with the
experimental finding for monolayer MoS2 on a mildly curved graphite substrate
Trainer  _et al._ (2019). We quantified the rate of decrease of band gap with
strain (that maintains Poisson’s ratio of 0.44) for monolayer MoS2. We found
the rate of decrease of band gap to be 63 meV/% strain for PBE, 73 meV/%
strain for HSE06 and 45 meV/% strain for G0W0@PBE, which are much smaller than
the experimental result of 400 meV/% strain Trainer  _et al._ (2019). The
discrepancy is important to resolve. Furthermore, we demonstrated that PBE and
G0W0@PBE predict monolayer MoS2 to be a direct band gap material, with the
potential to transition into an indirect band gap material under strain. We
also quantified the rate of decrease of band gap with strain (that maintains
Poisson’s ratio of 0.25) to be 90 mev/% strain at PBE level. Additionally, we
observed that 10% equal biaxial tensile strain can transform MoS2 from a
semiconductor to a metal at the PBE level, but not at the HSE06 level.

Finally, we briefly comment on the reason why the HSE06 energy gap in the
bandstructure of a solid is typically larger and more realistic than the PBE
gap. The PBE density functional for the exchange-correlation energy is
implemented in the original Kohn-Sham Kohn and Sham (1965) scheme in which the
exchange-correlation potential is a function of position that multiplies an
orbital, while the HSE06 hybrid of PBE with a fraction of exact exchange
depends explicitly on the occupied orbitals and is implemented in a
generalized Seidl  _et al._ (1996) Kohn-Sham scheme in which the exchange-
correlation potential is a non-local operator. PBE reasonably predicts the
bandstructure gap that would be predicted by a position-dependent
multiplicative exchange-correlation potential that yields the exact electron
density in a solid Perdew and Levy (1983); Sham and Schlüter (1983), while
HSE06, to the extent that it correctly predicts ionization energy minus
electron affinity for the solid from total energy differences, also predicts
the correct fundamental gap from its bandstructure Perdew  _et al._ (2017).
The difference between the HSE06 and PBE gaps is an estimate of the
discontinuity Perdew and Levy (1983); Sham and Schlüter (1983); Perdew  _et
al._ (2017, 1982) of the exact Kohn-Sham potential in an insulating solid as
the electron number crosses the number needed to fill a band. Another way to
understand this is to note that the electron self-energy that yields the exact
quasi-particle spectrum must be a nonlocal operator Sham and Schlüter (1983),
like the HSE06 and unlike the PBE and the exact Kohn-Sham exchange-correlation
potentials.

##  V Acknowledgements

The work of RKS was supported by the Department of Energy, Office of Science,
Basic Energy Sciences, Materials Chemistry Materials Science and Engineering,
under grant DE-DC0023356. The work of RKS and HT was partially supported by
the Temple Center for Research on Energy Materials through the Catalytic
Collaborative Funding Initiative sponsored by the Office of the Vice President
for Research at Temple. The work of JPP was supported by DOE grant DE-
DC0023356, and by the National Science Foundation, Division of Materials
Research, Condensed Matter and Materials Theory, under grant DMR-2344734. We
thank Professor Maria Iavarone for helpful discussions and suggestions.

##  VI Supplementary Information

Possible explanation for the decrease in band gap with strain.

## References

  * Allen  _et al._ (2010) M. J. Allen, V. C. Tung, and R. B. Kaner, Chem. Rev. 110, 132 (2010).
  * Soldano  _et al._ (2010) C. Soldano, A. Mahmood, and E. Dujardin, Carbon 48, 2127 (2010).
  * Novoselov  _et al._ (2004) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
  * Ganatra and Zhang (2014) R. Ganatra and Q. Zhang, ACS Nano 8, 4074 (2014).
  * Chhowalla  _et al._ (2014) M. Chhowalla, H. S. Shin, G. Eda, L. J. Li, K. P. Loh, and H. Zhang, Nat. Chem. 5, 263 (2014).
  * Lee  _et al._ (2008) C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321, 385 (2008).
  * Bertolazzi  _et al._ (2011) S. Bertolazzi, J. Brivio, and A. Kis, ACS Nano 5, 9703 (2011).
  * Johari and Shenoy (2012) P. Johari and V. B. Shenoy, ACS Nano 6, 5449 (2012).
  * Yue  _et al._ (2012) Q. Yue, J. Kang, Z. Shao, X. Zhang, S. Chang, G. Wang, S. Qin, and J. Li, Phys. Lett. A 376, 1166 (2012).
  * Shi  _et al._ (2013) H. Shi, H. Pan, Y.-W. Zhang, and B. I. Yakobson, Phys. Rev. B: Condens. Matter Mater. Phys. 87, 155304 (2013).
  * Trainer  _et al._ (2019) D. J. Trainer, Y. Zhang, F. Bobba, X. Xi, S.-W. Hla, and M. Iavarone, ACS Nano 13, 8284 (2019).
  * Kang  _et al._ (2015) J. Kang, H. Sahin, and F. M. Peeters, Phys. Chem. Chem. Phys. 17, 27742 (2015).
  * Woo  _et al._ (2016a) S. Woo, H. C. Park, and Y.-W. Son, Phys. Rev. B: Condens. Matter Mater. Phys. 93, 075420 (2016a).
  * Woo  _et al._ (2016b) S. Woo, H. C. Park, and Y.-W. Son, Phys. Rev. B: Condens. Matter Mater. Phys. 93, 075420 (2016b).
  * Krukau  _et al._ (2006) A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J Chem Phys 125, 224106 (2006).
  * Hedin (1965) L. Hedin, Phys. Rev. 139, A796–A823 (1965).
  * Hybertsen and Louie (1985) M. S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 55, 1418–1421 (1985).
  * van Schilfgaarde  _et al._ (2006) M. van Schilfgaarde, T. Kotani, and S. Faleev, Phys. Rev. Lett. 96, 226402 (2006).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865–3868 (1996).
  * Kresse and Hafner (1994) G. Kresse and J. Hafner, Phys Rev B Condens Matter 49, 14251 (1994).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys Rev B 59, 1758–1775 (1999).
  * Deslippe  _et al._ (2012) J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, Comput. Phys. Commun. 183, 1269 (2012).
  * Hybertsen and Louie (1986) M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986).
  * Giannozzi  _et al._ (2017) P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, Jr, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, J. Phys.: Condens. Matter. 29, 465901 (2017).
  * Pela  _et al._ (2024) R. R. Pela, C. Vona, S. Lubeck, B. Alex, I. G. Oliva, and C. Draxl, npj Comput. Mater. 10, 77 (2024).
  * Conley  _et al._ (2013) H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund, Jr., S. T. Pantelides, and K. I. Bolotin, Nano Lett. 13, 3626 (2013).
  * Xiong and Cao (2015) S. Xiong and G. Cao, Nanotechnology 26, 185705 (2015).
  * Pasquier and Yazyev (2018) D. Pasquier and O. V. Yazyev, Phys. Rev. B 98, 235106 (2018).
  * Guo  _et al._ (2022) Z. Guo, J. Deng, Y. Xie, and Z. Wang, npj Quantum Materials 7, 87 (2022).
  * Reyes-Retana and Cervantes-Sodi (2016) J. A. Reyes-Retana and F. Cervantes-Sodi, Scientific Reports 6, 24093 (2016).
  * Qiu  _et al._ (2013) D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Phys. Rev. Lett. 111, 216805 (2013).
  * Lu  _et al._ (2012) P. Lu, X. Wu, W. Guo, and X. C. Zeng, Phys. Chem. Chem. Phys. 14, 13035–13040 (2012).
  * Kohn and Sham (1965) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  * Seidl  _et al._ (1996) A. Seidl, A. Görling, P. Vogl, and M. Levy, Phys. Rev. B 53, 3764 (1996).
  * Perdew and Levy (1983) J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 (1983).
  * Sham and Schlüter (1983) L. J. Sham and M. Schlüter, Phys. Rev. Lett. 51, 1888 (1983).
  * Perdew  _et al._ (2017) J. P. Perdew, W. Yang, K. Burke, Z. Yang, E. K. U. Gross, M. Scheffler, G. E. Scuseria, T. M. Henderson, I. Zhang, A. Ruzsinszky, H. Peng, J. Sun, E. Trushin, and A. Görling, Proc. Nat. Acad. Sci. (USA) 114, 2801 (2017).
  * Perdew  _et al._ (1982) J. P. Perdew, R. G. Parr, M.Levy, and J. L. Balduz, Phys. Rev. Lett. 49, 1691 (1982).
  * Chhowalla  _et al._ (2013) M. Chhowalla, H. S. Shin, G. Eda, L.-J. Li, K. P. Loh, and H. Zhang, Nature Chemistry 5, 263–275 (2013).
  * Tang  _et al._ (2023) H. Tang, S. Neupane, L. Yin, J. M. Breslin, and A. Ruzsinszky, J. Mater. Chem. C 11, 4711 (2023).
  * Wilson and Yoffe (1969) J. A. Wilson and A. D. Yoffe, Adv. Phys. 18, 193–335 (1969).

Supplementary Information

Possible explanation for the decrease in band gap with strain

In the 1H MoS2 monolayer, one Mo atom and its six adjacent S atoms form a
trigonal prism (see Figure S1), and in the ligand field theory, the d-orbitals
of the transition metal atoms split into three groups of non-bonding levels
Chhowalla  _et al._ (2013). The lower split group is mainly of $d_{z^{2}}$
character and is filled, and the fundamental gap is formed by this filled
group and the unoccupied middle group.

To understand the trend in the change of gap with applied tensile strain, we
analyze the band widths of the relevant d-orbitals and the spin-split of the
valence band around the K point, and the results are listed in Table S1. As
can be seen, the spin-orbit coupling (SOC) induced spin-split is almost
unchanged with an increase in strains, implying that the SOC effect may not be
obviously influenced by the strains. This is unlike the case in the bent WSe2
nanoribbons, where the non-uniform bending strain induced flexoelectric effect
can dramatically increase the SOC effect Tang  _et al._ (2023).

The band widths of the d-orbital derived valence bands just below the Fermi
level show a roughly unchanged trend with strains, and those of conduction
bands just above the Fermi level show a slightly increasing trend with
strains, as seen in Table S1. The relevant d-orbital band broadening is
believed to be the main reason for the gap’s decreasing trend from 2H-MoS2, to
2H-MoSe2, and to 2H-MoTe2, with increasing atomic number of the chalcogen
Wilson and Yoffe (1969). However, the total band width broadening of bands
around the Fermi level cannot completely explain the band gap decrease of
monolayer MoS2 with in-plane strains here. Comparing the case of strain
$\epsilon_{x}=4\%$ with the non-strained one, G0W0 gives a gap decrease of 190
meV, see Table 1 in the main text, while the change of the total band
broadening is only about 98 meV, see Table S1. So, the band gap decrease of
monolayer MoS2 with strains should also involve the relevant band-center
shifts.

The gap is also related to the separation between the band-centers of the
above-mentioned lower filled split group and the middle unoccupied group. The
separation is determined by the ligand field symmetry and strength. Figure S1
shows the trigonal prism of 1H MoS2, with the relevant angles and bond lengths
denoted. The AB direction is in the x axis, where the strain $\epsilon_{x}$ is
applied, and the y axis is perpendicular to AB. The three angles $\alpha$,
$\beta$, and $\gamma$, and bond lengths a and b in the prism under different
strains are listed in Table S2. For the unstrained case, the angles $\alpha$
and $\beta$ are equal, and the bond lengths a and b are also equal. The prism
has a three-fold rotational symmetry. This symmetrical configuration may
result in the largest ligand field splitting between the lower filled and the
middle unoccupied groups. With increasing strains, the angle $\alpha$
decreases, and the angle $\gamma$ increases, while angle $\beta$ remains
almost unchanged, and the bond length a decreases, but b increases, due to the
stretching in the x axis and compression in the y axis. The prism loses the
three-fold rotational symmetry, and the ligand field also changes its symmetry
and strength, resulting in a decrease in the splitting of the lower and the
middle groups, hence a decrease in the gap.

Figure S1: The structure of the trigonal prism of the 1H MoS2 monolayer. The direction AB is that of the x axis, and the y axis is perpendicular to AB, as in Figure 1. The three angles $\alpha$, $\beta$, and $\gamma$, and bond lengths a and b are denoted in different colors for clarity.  |  | G0W0@PBE+SOC |  |  | PBE+SOC
---|---|---|---|---|---
|  |  Band width_v |  Band width_c |  TBWB |  Spin split |  |  |  Band width_v |  Band width_c |  TBWB |  Spin split
no strain |  |  1.205 |  0.731 |  0.000 |  0.152 |  |  |  1.061 |  0.648 |  0.000 |  0.146
$\epsilon_{x}$=1.0% |  |  1.205 |  0.751 |  0.020 |  0.154 |  |  |  1.066 |  0.667 |  0.025 |  0.147
$\epsilon_{x}$=2.5% |  |  1.203 |  0.794 |  0.060 |  0.155 |  |  |  1.057 |  0.707 |  0.056 |  0.148
$\epsilon_{x}$=4.0% |  |  1.189 |  0.846 |  0.098 |  0.156 |  |  |  1.067 |  0.754 |  0.112 |  0.149
Table S1: Contributions to the decrease of bandgap in monolayer $\mathrm{MoS_{2}}$ with increasing in-plane tensile strain. TBWB stands for total band width broadening. The band widths and broadening of the d-orbital derived bands around the Fermi level and the spin-splits for 1H $\mathrm{MoS_{2}}$ monolayer under different strain conditions. In strained cases, the strain along the y axis is $\epsilon_{y}=-0.44\epsilon_{x}$. Band widthv is the band width of the four valence bands just below the Fermi level. Band widthc is the band width of the two conduction bands just above the Fermi level. Total band width broadening is referenced to the non-strained case. The spin-split is the one at the top of the valence band around the K point. Unit of entries: eV. |  | $\alpha$ |  | $\beta$ |  | $\gamma$ |  | a |  | b
---|---|---|---|---|---|---|---|---|---|---
no strain |  | 80.850 |  | 80.850 |  | 82.488 |  | 2.386 |  | 2.386
$\epsilon_{x}$=1.0% |  | 80.483 |  | 80.875 |  | 83.251 |  | 2.382 |  | 2.392
$\epsilon_{x}$=2.5% |  | 79.820 |  | 80.876 |  | 84.427 |  | 2.374 |  | 2.400
$\epsilon_{x}$=4.0% |  | 79.193 |  | 80.867 |  | 85.593 |  | 2.367 |  | 2.408
Table S2: The three angles $\alpha$, $\beta$, and $\gamma$, and bond lengths a
and b in the trigonal prism, depicted in Figure S1, of 1H MoS2 monolayer under
different strain conditions. In strained cases, the strain along the y axis is
$\epsilon_{y}=-0.44\epsilon_{x}$. Units: angle: degree; bong length: angstrom.


# ATLAS ITk Pixel Detector Overview

Lingxin Meng, on behalf of the ATLAS Collaboration
CERN
<EMAIL_ADDRESS>Corresponding author

(Talk presented at the International Workshop on Future Linear Colliders
(LCWS2021), 15-18 March 2021. C21-03-15.1.)

00footnotetext:
Copyright 2021 CERN for the benefit of the ATLAS Collaboration. Reproduction
of this article or parts of it is allowed as specified in the CC-BY-4.0
license.

## I. Introduction

In the High Luminosity (HL) era of the Large Hadron Collider (LHC), the
average number of collisions per bunch crossing (BC) will increase from the
current 30 to about 200, resulting in an increase of the peak luminosity from
1$\cdot 10^{34}$ cm-2s-1 to 5–7$\cdot 10^{34}$ cm-2s-1. The integrated
luminosity of 350 fb-1 at the end of Run 3 will accumulate more rapidly up to
4000 fb-1 until the end of the HL programme. The requirement for the pixel
detector of the ATLAS experiment [1] and its readout also becomes more
demanding: the occupancy is kept below 1% by increasing the detector
granularity with the frontend chip operating at 1 MHz trigger rate. The
innermost pixel system has to withstand a harsher radiation envirionment up to
2$\cdot$1016 n${}_{\textrm{eq}}$/cm2. This HL upgrade is planned to take place
during the Long Shutdown (LS) 3 as indicated in Figure 1.

Figure 1: Timeline of the Large Hadron Collider in the coming years. The High
Luminosity upgrade is planned for Long Shutdown (LS) 3 [2].

## II. ATLAS Inner Tracker Upgrade

For the HL upgrade of the ATLAS detector, the current Inner Detector (ID),
consisting of silicon pixel, silicon strip detectors and Transition Radiation
Tracker (TRT) is going to be replaced by an all-silicon Inner Tracker (ITk)
[3, 4], which has 13 m2 of pixel detectors with 5 billion readout channels and
160 m2 of strip detectors with 50 million readout channels. The layout of
active elements in the ITk Pixel detector in the r-z projection is shown in
Figure 2. The five-layer design provides a higher tracking coverage of up to
$|\eta|=4$, while at the same time multiple scattering is minimised by greatly
reducing the material budget compared to the current ID, as shown in Figure 3.

Figure 2: Layout schematic of active elements in the r-z projection of the ITk
Pixel detector [5].

Figure 3: Comparison of the material budget in the detector between the
current Inner Detector during Run 2 [3] and ITk [6].

Sections i, ii and iii will introduce the sensors, readout chips and pixel
modules as building blocks of the pixel detector, with which larger structures
are built and described in sections iv to vii.

### i. Sensors

The pixel sensors are required to have an efficiency of $>$98.5% before, and
$>$97% after an irradiation with a corresponding end-of-life fluence [3]. Two
technologies are used for the ITk Pixel detector: 3D sensors in the innermost
layer (L0) in triplet modules, consisting of three single frontend-sensor
assemblies on a triplet flex, and planar sensors in quad modules in all other
layers (L1–L4), consisting of four frontend chips on a quad sensor tile on a
quad flex. Both technologies use n-implant in p-substrate with a generally
thinner active thickness compared to the current pixel detector, thus charge
and hit efficiencies are saturated at lower bias voltages which lead to a
reduction of dissipated power. At the moment, a market survey is finished for
both technologies and pre-production is ongoing.

####  i.1 3D Sensors

3D sensors with a pixel size of 50$\times$50 $\upmu$m2 and 25$\times$100
$\upmu$m2 with a single collection electrode in the centre and a good
efficiency, as shown in Figure 4, are used in the endcap and barrel regions of
L0, respectively. They consist of 150 $\upmu$m active thickness plus 100
$\upmu$m support wafer. 3D sensors are intrinsically more radiation tolerant
and therefore used in the innermost layer which is only 34 mm from the beam.
It will be replaced after 2000 fb-1 with an expected fluence of 1.3$\cdot$1016
n${}_{\textrm{eq}}$/cm2 as part of the Inner pixel system replacement [3].

Figure 4: Top: pixel sizes used in 3D sensors with a single collection
electrode (red) [7]. Bottom: Efficiency maps for different pixel sizes [7].

####  i.2 Planar Sensors

Planar sensors have a pixel size of 50$\times$50 $\upmu$m2 in all regions. The
thickness is 100 $\upmu$m in L1 in the Inner System and 150 $\upmu$m in the
outer layers. There are designs with punch-through bias structure as shown in
Figure 5 left and without (not shown). The efficiency of a $4\times 4$ pixel
region (grey area) before irradiation is shown in Figure 5 right, where a
slightly lower efficiency is observed in regions with punch-through bias dots
[8].

Figure 5: Pixel matrix design with punch-through bias structure (left) and
efficiency map (right) of $4\times 4$ pixels (grey area) of a planar sensor
before irradiation [8].

### ii. RD53 Readout Chip

The frontend (FE) readout chip is developed by the RD53 collaboration [9],
consisting of 24 institutes from Europe and the United States, as a common
readout chip for ATLAS and CMS pixel detectors. It evolved from the RD53A
prototype [10], consisting of three different frontend flavours, into ITkPix
[11] for ATLAS with 400$\times$384 pixels using differential frontend, and
CROC for CMS wtih 432$\times$336 pixels using linear frontend. ITkPixV1, shown
in Figure 6 on a single chip card (SCC), was submitted in March 2020 and is
currently being verified. Submission of the final version, ITkPixV2, is
foreseen for the end of 2021.

Apart from an issue with the time-over-threshold (ToT) latch which is fixed in
version 1.1, Figures 7, 8 and 9 show a small selection of verifications tests.
The chips main reference current and voltage are stable within 4% over a large
temperature range of up to 120 ∘C (Figure 7. The eye-diagramme of command data
recovery (CDR) and PLL in Figure 8 shows an improved jitter from RD53A with
29.94 ps RMS to ITkPixV1 with 11.2 ps RMS [12]. X-ray irradiation campaignes
are ongoing with ring oscillators as test structures. Figure 9 compares high
dose rate of 5 Mrad/h (blue) with a low dose rate of 25 krad/h (red) on the
CLK gates of lengths 0 and 4 [12]. While it is faster to accumulate ionising
dose at end-of-life with a high rate, low rate of ionising radiation that
reflects realistic irradiation conditions in the detector generates more
degradation at the same accumulated dose, which can be seen from the steeper
slope. Studying both rates could allow an estimate of scalability of the
degradation.

Table 1: Technical specifications of ITkPix [3, 11]. Chip size |  $20\times 21\,$mm2
---|---
Pixel size |  $50\times 50\,\upmu$m2
Hit rate | 3 GHz/cm2
Trigger rate | 1 MHz
Latency | 12.8 $\upmu$s
Data rate | 5.12 Gbit/s
Low threshold | 600 e
Radiation tolerance | 500 Mrad
Low power | 4 $\upmu$A/pixel
Figure 6: An ITkPixV1 chip on a single chip card [13]. Figure 7: Main
reference current and voltage of the ITkPixV1 show only a change of less than
4% over 120 ∘C temperature range [13]. Figure 8: Eye diagrammes of command
data recovery (CDR) and PLL from RD53A (left) to ITkPixV1 (right). The jitter
improved from 29.94 ps RMS to 11.2 ps RMS, respectively [12]. Figure 9: X-ray
irradiations comparing high (blue) and low (red) dose rate on ring oscillators
of lengths 0 (left) and 4 (right) on the ITkPixV1 [12].

### iii. Pixel Module

There are linear and round triplet modules in the innermost layer L0 for the
barrel and rings, and quad modules for layers 1–4 making the largest part of
the detector. About 10000 modules are going to be assembled for the ITk Pixel
detector by hand during the next four years. Bare, hybridised sensor-frontend
assemblies are glued onto module flexes using custom built tools, where glue
is distributed by means of a laser-cut stencil and the thickness of the glue
layer is controlled via precision shims as shown in Figure 10.

Every assembled module has to pass a quality control procedure which involves
visual inspection, metrology and electrical testing at different temperatures.
Currently, the RD53A module programme is ongoing for institutes to set up
infrastructure and gain experience in assembly, quality control and system
test. Also first ITkPixV1 digital quad modules using prototype flexes are
being tested with custom readout and power adapter boards as shown in Figure
11. These first tests verified the flex design but also allow access to probe
multi-chip features like command forwarding, data aggregation etc.

Figure 10: Custom tools for pixel quad module assembly. Figure 11: One of the
first ITkPixV1 digital quad module on a prototype flex PCB being tested.

### iv. Serial Powering

Readout chips on a triplet or quad module are powered in parallel using up to
8 W of power. By powering up to 14 [14] of those modules in series in a chain,
the amount of required cables is reduced and thus material budget in the
detector. This is made possible through shunt-LDO (SLDO) powering mode of the
frontend chip by providing the chip with a constant and redundant power
budget. Experience in operating serial powering chains was gained through
demonstrators based on FEI4 modules [15]. Studies on the first serial powering
chain with RD53A quad modules is ongoing as shown in Figure 12 and chains with
ITkPix modules are in preparation.

Figure 12: Serial powering test with RD53A quad modules connected to pseudo
Type-0 services and end-of-stave (EoS) card [16]. Figure 13: Schematic of the
data path [7].

### v. Data Transmission and Services

Each frontend chip is capable of transmitting data via 4 links with 1.28
Gbit/s, resulting in up to 16 links on a quad module. The activity of a
frontend chip depends on the hit rate and thus the position of the module in
the detector. When full data rate is not required, the number of physical
outgoing links per module can be reduced by aggregating data within the module
through the frontend chips. The number of required data links can vary from 9
lines per triplet module in the innermost layer down to 1 line per quad module
in L3. These links are shown in Figure 13 as kapton-copper flexes (Type-0
services) between module and patch panel 0 (PP0). From PP0 data is transmitted
through custom Twinax cables to the optoboard [17], where electrical signal is
recovered by the Gigabit Receiver Chip [18] and is aggregated and converted to
optical signal through the low-power Gigabit Transceiver (lpGBT [19]) and VTRx
[20] before being sent off detector to the readout.

### vi. Mechanical Support

In the detector, modules are attached onto and supported by lightweight carbon
fibre composite structures, as shown in Figure 14. The Inner System uses a
quarter shell concept for the barrel and endcap regions with different designs
for the endcap rings (a). For the Outer System, the Barrel region (b) consists
of longerons for the flat region and inclined half rings towards higher
$|\eta|$, and Outer Endcap consists of modules that are attached onto half
rings embedded in half shells (c).

(a)

(b)

(c)

Figure 14: Lightweight local support carbon fibre structures with pixel
modules for the Inner System (a), Outer Barrel (b) and Outer Endcap (c) [21,
22].

### vii. System Test

System test with realistic module staves and infrastructure is ongoing for all
subsystems of the pixel detector, using all components introduced above.
Figure 15 shows an example of the system test infrastructure for Outer Barrel
at CERN, consisting of a CO2 cooling plant, a dry air unit, racks with power
supplies, detector control system (DCS), an interlock matrix, and a light-
tight and thermally insulated testbox with a source stage.

The demonstrator is a full-size, 1.6 m long prototype stave that is loaded
with FEI4-based modules [23]. The module arrangement is indicated in Figure 16
top: with 14 flat quad modules in the centre, 7 on each A-side and C-side, and
13 and 16 inclined dual modules towards high $|z|$ on the A-side and C-side,
respectively. The modules are connected in 6 serial powering chains. The
middle plot shows a noise level of all modules, measured warm and cold with
CO2 cooling at $17\,^{\circ}$C and $-25^{\circ}$C, respectively. As expected
the noise level decreases with lower temperature. A source scan of the C-side
quad modules performed at $-25^{\circ}$C is shown in the bottom plot by
running two 90Sr sources on the stage over all modules. The quad modules on
the C-side were fully functional on the module level. After integration onto
the stave some frontend chips lost communication which is mostly due to issues
related to cables, except for the chip on BM_03 which shows a perfect source
scan when warm [23]. The path leading to the cold source scan of the
demonstrator provided valuable experience and demonstrated a working
constellation of system infrastructure.

Figure 15: System test infrastructure at CERN SR1. From left to right: CO2
plant, dry air unit, racks with power supplies and interlock system, and light
tight and thermally insulated testbox [24].

Figure 16: Results from the Outer Barrel demonstrator using prototype FEI4
modules. Top: layout of the modules. Middle: noise level of the modules
measured warm and cold with CO2 cooling at $17\,^{\circ}$C and $-25^{\circ}$C,
respectively. Bottom: source scan of the C-side quad modules at $-25^{\circ}$C
using 90Sr sources [23].

## III. Summary

This paper has given a brief overview of the status of the pixel system of
ATLAS Inner Tracker upgrade. Module components, sensor, frontend chip and
module flex are being finalised. Module institutes are set up and being
qualified for assembly and testing so the RD53A module programme can progress
with full speed. With these modules, the gained experience with serial
powering tests and fully functional FEI4 prototype systems preparing for the
next generation demonstrators with the first RD53A modules to validate every
aspect of the ITk Pixel detector development.

## References

  * [1] The ATLAS Collaboration.  The ATLAS Experiment at the CERN Large Hadron Collider.  Journal of Instrumentation, 3(08):S08003–S08003, Aug. 2008.
  * [2] Project schedule | HL-LHC industry, Mar. 2021.  https://project-hl-lhc-industry.web.cern.ch/content/project-schedule.
  * [3] ATLAS Collaboration.  Technical Design Report for the ATLAS Inner Tracker Pixel Detector.  Technical Report CERN-LHCC-2017-021. ATLAS-TDR-030, CERN, Geneva, Sep. 2017.
  * [4] Technical Design Report for the ATLAS Inner Tracker Strip Detector.  Technical report, CERN, Geneva, Apr 2017.
  * [5] ITk Pixel Layout Updates.  Technical Report ITK-2020-002, CERN, Geneva, Apr. 2020.  https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PLOTS/ITK-2020-002/.
  * [6] Updated ITk Material Distributions: Overall material budget and breakdown for recent ITk simulations, Jan. 2020.  https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PLOTS/ITK-2019-001.
  * [7] Claudia Gemme.  The ATLAS ITk detector for HL-LHC.  Vertex 2020, Oct. 2020.  https://indico.cern.ch/event/895924/contributions/3968850.
  * [8] ITk Planar Pixel Sensor MS, Feb. 2020.  https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PLOTS/ITK-2020-001.
  * [9] RD53 Collaboration, May 2021.  {http://rd53.web.cern.ch/rd53},.
  * [10] Maurice Garcia-Sciveres.  The RD53A Integrated Circuit.  Technical Report CERN-RD53-PUB-17-001, CERN, Geneva, Oct. 2017.
  * [11] Maurice Garcia-Sciveres.  RD53B Manual.  Technical Report CERN-RD53-PUB-19-002, CERN, Geneva, Mar. 2021.
  * [12] Flavio Loddo.  RD53 pixel chip developments for the ATLAS and CMS High Luminosity LHC.  TREDI 2021, Feb. 2021.  https://indico.cern.ch/event/983068/contributions/4223182.
  * [13] Timon Heim.  First test results from the ITkPixV1 pixel readout chip.  Vertex 2020, Oct. 2020.  https://indico.cern.ch/event/895924/contributions/3968862.
  * [14] Tobias Flick.  Module and System test Development for the Phase-2 ATLAS ITk Pixel Upgrade.  VCI 2019, Feb. 2019.  https://indico.cern.ch/event/716539/contributions/3246646/.
  * [15] Maurice Garcia-Sciveres et al.  The FE-I4 Pixel Readout Integrated Circuit.  Technical Report ATL-UPGRADE-PROC-2010-001, CERN, Geneva, Jan. 2010.
  * [16] Florian Hinterkeuser.  Development and Evaluation of Prototypes for the ATLAS ITk Pixel Detector.  ICHEP 2020, Jul. 2020.  https://indico.cern.ch/event/868940/contributions/3814693.
  * [17] Meghranjana Chatterjee.  Opto-System Prototypes And Preliminary Tests For ATLAS ITk Pixel Detector.  ACES 2020, May 2020.  https://indico.cern.ch/event/863071/contributions/3856114.
  * [18] C. Chen et al.  Characterization of a gigabit transceiver for the ATLAS inner tracker pixel detector readout upgrade.  Journal of Instrumentation, 15(03):T03005–T03005, Mar. 2020.
  * [19] lpGBT Design Team.  lpGBT Documentation.  Technical report, May 2020.  https://twiki.nevis.columbia.edu/twiki/pub/ATLAS/SliceTestboard/lpGBT.pdf.
  * [20] Jan Troska et al.  The VTRx+, an Optical Link Module for Data Transmission at HL-LHC.  In PoS(TWEPP-17), volume 313, page 048, 2018.
  * [21] Susanne Kühn.  Results of larger structures prototyping for the Phase-II upgrade of the pixel detector of the ATLAS experiment.  Pixel 2018, Dec. 2018.  https://indico.cern.ch/event/669866/contributions/3235377.
  * [22] Peter Sutcliffe.  ATLAS Pixel Endcap Structure.  Forum on Tracking Detector Mechanics 2019, Jun. 2019.  https://indico.cern.ch/event/775863/contributions/3413590.
  * [23] Stefano Zambito.  ATLAS Inner Tracker Pixel Outer Barrel Demonstrator System Tests with Serial Powering.  TREDI 2021, Feb. 2021.  https://indico.cern.ch/event/983068/contributions/4229352.
  * [24] Lingxin Meng.  Module and System Test Development for the Phase-2 ATLAS ITk Pixel Upgrade.  TWEPP 2019, Sep. 2019.  https://indico.cern.ch/event/799025/contributions/3486430.


. ††thanks: corresponding author<EMAIL_ADDRESS>
# Observation of Fermi arcs and Weyl nodes in a non-centrosymmetric magnetic
Weyl semimetal

Anup Pradhan Sakhya  Department of Physics, University of Central Florida,
Orlando, Florida 32816, USA  Cheng-Yi Huang  Department of Physics,
Northeastern University, Boston, Massachusetts 02115, USA  Gyanendra Dhakal
Department of Physics, University of Central Florida, Orlando, Florida 32816,
USA  Xue-Jian Gao  Department of Physics, Hong Kong University of Science and
Technology, Hong Kong, China  Sabin Regmi  Department of Physics, University
of Central Florida, Orlando, Florida 32816, USA  Xiaohan Yao  Department of
Physics, Boston College, Chestnut Hill, MA 02467, USA  Robert Smith
Department of Physics, University of Central Florida, Orlando, Florida 32816,
USA  Milo Sprague  Department of Physics, University of Central Florida,
Orlando, Florida 32816, USA  Bahadur Singh  Department of Condensed Matter
Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai
400005, India  Hsin Lin  Institute of Physics, Academia Sinica, Taipei, Taiwan
Su-Yang Xu  Department of Chemistry and Chemical Biology, Harvard University,
Cambridge, MA, USA  Fazel Tafti  Department of Physics, Boston College,
Chestnut Hill, MA 02467, USA  Arun Bansil  Department of Physics, Northeastern
University, Boston, Massachusetts 02115, USA  Madhab Neupane  Department of
Physics, University of Central Florida, Orlando, Florida 32816, USA

###### Abstract

Weyl semimetal (WSM), a novel state of quantum matter, hosts Weyl fermions as
emergent quasiparticles resulting from the breaking of either inversion or
time-reversal symmetry. Magnetic WSMs that arise from broken time-reversal
symmetry provide an exceptional platform to understand the interplay between
magnetic order and Weyl physics, but few WSMs have been realized. Here, we
identify CeAlSi as a new non-centrosymmetric magnetic WSM via angle-resolved
photoemission spectroscopy (ARPES) and first-principles, density-functional
theory based calculations. Our surface-sensitive vacuum ultraviolet ARPES data
confirms the presence of surface Fermi arcs as, the smoking gun evidence for
the existence of the Weyl semimetallic state in CeAlSi. We also observe bulk
Weyl cones in CeAlSi using bulk-sensitive soft-X-ray ARPES measurements. In
addition, Ce 4f flat bands are found near the Fermi level, indicating that
CeAlSi is a unique platform for investigating exotic quantum phenomena
resulting from the interaction of topology, magnetism and electronic
correlations.

The recent discovery of Weyl fermions as exotic quasiparticles in Weyl
semimetals (WSMs) has generated tremendous research interest in condensed
matter physics and materials science, due to their intimate link between the
concepts of different fields of physics and its technological applications Wan
; Burkov ; Franchini ; Bahadur ; Weng ; Chen ; Xu ; Ding ; Lv ; Vanderbilt .
In a WSM, the conduction and valence bands disperse linearly in three-
dimensional (3D) momentum space and the point where they intersect is known as
the Weyl node Lv ; Alidoust ; Felser . The essential prerequisite for a WSM is
that either the crystal inversion (CI) symmetry Weng  or the time-reversal
(TR) symmetry Wan  of the crystal must be broken. The TaAs-family of
transition-metal pnictides Weng ; Xu , MoTe2 and WTe2 family transition-metal
dichalcogenides Soluyanov ; Sun ; Deng ; Kaminski  belong to the class of WSMs
with broken CI symmetry. The TR symmetry broken WSMs have been discovered
recently in ferromagnets such as Co3Sn2S2, Co2MnGa, YbMnBi2 Shen ; Shekhar ;
Kim ; Hasan ; Borisenko . However, when both the symmetries are broken, a rare
WSM arises which provides an exciting opportunity to understand the
correlation between magnetism and Weyl fermions Chang ; Singh .

Figure 1: Crystal structure and electronic structure of CeAlSi. (a) Unit cell
of CeAlSi where the red, teal, and blue colored solid spheres denote the Ce,
Al and Si atoms respectively. (b) Bulk Brillouin zone (BZ) and the projected
(001) surface BZ. (c) Calculated bulk bands along the high symmetry directions
without SOC and (d) with the inclusion of SOC. (e) Configuration of the 40
Weyl nodes in the bulk BZ created upon the inclusion of SOC. (f) Projection of
the Weyl nodes on the (001) surface BZ in one quadrant highlighting all the
Weyl nodes. The red (blue) dots denote the Weyl node with chirality +1 (-1)
respectively. (g) Schematics comparing the three types of Weyl nodes appearing
upon the inclusion of SOC. Figure 2: Fermi surface as a function of photon
energy. Experimental Fermi surface measured along the (001) direction at 15 K
using a photon energy of (a) 30 eV (b) 40 eV (c) 50 eV and (d) 60 eV. The
spectral intensity is integrated within 20 meV of the chemical potential. (e)
Calculated Fermi surface on (001) surface where blue (red) spectral weight
denotes bulk (surface) states. (f) Schematic illustration showing Fermi arcs
(black lines) emerging from W3′ or W3′′ pair along the
$\overline{\Gamma}-\overline{M}$ direction. The experimental data were taken
at the SSRL beamline 5-2 at a temperature of 15 K.

Recently, the RAlGe and RAlSi (R = Rare earth elements) materials, which are
noncentrosymmetric, have been proposed as a new family of magnetic WSMs, where
the Weyl nodes are tunable due to intrinsic magnetic order LaAlGe ; CeAlGe ;
LaAlSi ; CeAlGer ; NdAlSi ; CeAlSi ; PrAlGe . These families of compounds can
exhibit different types of magnetic ground states and host Type-I or Type-II
Weyl fermions. CeAlSi, a material from the RAlSi family, was recently shown to
be a noncentrosymmetric ferromagnetic WSM that exhibits an anisotropic
anomalous Hall effect between the easy and hard magnetic axes Singh . The
large ferromagnetic domains observed below Tc = 8.3 K, makes this material
exciting as the Weyl nodes can be generated and manipulated by breaking both
the CI and TR symmetries, making it an ideal material to explore the interplay
between Weyl electrons and magnetism.

Figure 3: (a) Fermi surface and (b-d) constant energy contour plots at 15 K
measured using a photon energy of 40 eV where the Fermi arcs can be seen along
the $\overline{\Gamma}–\overline{M}$ direction. (e-g) Photon energy dependent
ARPES along the horizontal line (black dotted colour in Fig. 3a). Similar cuts
were taken for Fig. 3 (f,g) (not shown). Negligible kz dispersion is observed
for the Fermi arc. Data were taken at the SSRL beamline 5-2 at a temperature
of 15 K. Figure 4: Observation of Chiral charges in CeAlSi. Zoomed FS (a) and
the curvature plot (b) obtained from ARPES measurement with a photon energy of
40 eV. Cuts of interest are illustrated with horizontal line labeled Cut 1 and
blue loops labeled 1 and 2 with the starting/end points marked by the vertical
blue lines. (c) Measured band dispersion along horizontal Cut 1. (d) Second
derivative plot of (c). (e) ARPES measured band dispersion along the loop 1.
Loop 1 encloses the termination point of the measured Fermi arc and shows a
single left-moving chiral mode, corresponding to an enclosed Chern number n
$=$ -1 and (f) Band dispersion along the loop 2 showing a single right-moving
chiral mode, corresponding to a Chern number n = +1. (g) Calculated energy
dispersion around $k_{y}$ = 0.34 Å-1 showing the chiral mode. (h) Band
dispersion measured along the $\overline{\Gamma}-\overline{M}$ direction and
its corresponding integrated energy distribution curves showing the presence
of Ce 4f states near the Fermi level at various photon energies as noted on
the figure.

In this article, by performing vacuum ultraviolet (VUV) and soft X-ray (SX)
angle-resolved photoemission spectroscopy (ARPES) measurements, we report a
systematic study of the electronic structure of CeAlSi. The surface sensitive
VUV ARPES results confirm the presence of the Fermi arcs, signatures of the
WSMs on the (001) surface of CeAlSi. Bulk sensitive SX measurements reveal the
presence of Weyl nodes in the paramagnetic phase. The experimental results are
well corroborated by our density functional theory (DFT) results. We thus
establish CeAlSi as a non-centrosymmeric topological WSM materials, platform
for investigating the interplay between Weyl phases and magnetism.

CeAlSi is a semimetallic material that crystallizes in the tetragonal
structure with space group I41md (No. 109) with lattice constants a = 4.25 Å,
and c = 14.58 Å Singh , as illustrated in Fig. 1(a). The crystal structure has
two vertical mirror planes as well as two vertical glide mirror planes, but
lacks a horizontal mirror plane thus breaking the inversion symmetry. Fig.
1(b) shows the three-dimensional bulk Brillouin zone (BZ) of CeAlSi, and the
projected two-dimensional BZ onto the (001) surface. The experimental lattice
constants are used for the present DFT calculations and the calculated bulk
band structures along the high symmetry directions without spin-orbit coupling
(SOC) and with SOC are presented in Fig. 1(c) and Fig. 1(d), respectively. We
can clearly see the presence of small hole pockets at the $\Sigma$,
$\Sigma_{1}$ high symmetry points and along the $\Sigma_{1}$-Z direction
suggesting the material to be semimetallic. The Weyl nodes in the kz = 0 plane
are pinned by the C2T symmetry. When SOC is included, there are four pairs of
W1, W3′ and W3′′, as well as eight pairs of W2 Weyl nodes, as required by the
C4v point group of the paramagnetic CeAlSi. The nodal lines on the kx = 0, ky
= 0 mirror-invariant plane are stabilized by the mirror symmetry mx, my, as
the touching conduction and valence bands have opposite eigenvalues under the
mirror operation. The 8 Weyl nodes on the kz = 0 plane are referred to as W1,
whereas the remaining 16 Weyl nodes away from this plane are referred to as
W2. Furthermore, each W3 (spinless) Weyl node breaks into two W3′ and W3′′
(spinful) Weyl nodes with the same chirality. As a result, there are a total
of 40 Weyl nodes in the paramagnetic phase and the distribution of Weyl nodes
in the bulk BZ and the (001) surface are presented in Fig. 1(e) and 1(f),
respectively Chang . The presence of Weyl nodes at various energies with
respect to EF is shown in Fig. 1(g).

Figure 5: Observation of bulk Weyl cones in CeAlSi. (a) Experimental FS on the
kz = 0 plane obtained from SX-ARPES measurement with a photon energy of 498
eV. Band dispersion along (b) Cut 1, (c) Cut 2 and (d) Cut 3 showing the
linear dispersion of the Weyl cones. (e) DFT calculated bulk Fermi surface.
(f) DFT band dispersion to show the dispersion of bulk W3′ and (g) W3′′ Weyl
cones.

In order to unveil the Weyl semimetallic phase in CeAlSi, we have presented
the results from ARPES measurements on the (001) surface in Figs. 2-5. Fermi
surface (FS) maps obtained from VUV based ARPES measurements with different
photon energies are presented in Figs. 2(a-d). To distinguish the surface and
bulk contributions from various bands, we have performed photon energy
dependent ARPES experiments using a broad photon energy range from 30-60 eV,
covering more than two BZs along the kz direction. The measurements have been
performed at a temperature of 15 K i.e., in the paramagnetic phase. Along the
(001) surface, the fourfold symmetry is broken as the fourfold rotations are
screw-rotations, thus the surface bands are predicted to be anisotropic along
the $\overline{\Gamma}$-$\overline{X}$ and $\overline{\Gamma}$-$\overline{Y}$
directions. This anisotropic feature can be seen clearly from our ARPES data
[Fig. 2(a-d)]. The FS shows a circular pocket at the $\overline{\Gamma}$ point
and mustache-like pockets at the $\overline{X}$ point. These pockets disperse
quite strongly with photon energy suggesting their bulk 3D nature. In addition
to this, pockets are also observed at the $\overline{Y}$ point and along the
$\overline{\Gamma}-\overline{M}$ point. These pockets do not disperse with the
change in photon energy suggesting their 2D nature. Interestingly, arc-like
features can be observed along the $\overline{\Gamma}-\overline{M}$ direction,
which do not alter their dispersion with change in photon energy confirming
their surface origination. The features observed from the ARPES experiments
are in very good agreement with the DFT calculated FS, where the blue color
intensity denotes the contribution from the bulk bands and the red color
indicates the contribution from the surface bands. A schematic of the Fermi
arc connecting the Weyl points as observed from DFT is presented in Fig. 2(f).
In Fig. 3(a-d), we present the FS map and the constant energy contour plots
taken at various binding energies noted on top of each plot. The circular
pocket at the $\overline{\Gamma}$ point contracts with increasing binding
energy suggesting the electron-like nature of the associated bands. The
pockets at the $\overline{X}$ and $\overline{Y}$ points grow at higher binding
energy indicating their hole-like nature. To better understand the nature of
the arc-like feature along the $\overline{\Gamma}-\overline{M}$ direction, we
have performed photon energy dependent studies. The band dispersion energy-
momentum cut through the arc-like state (as shown by the black dotted line in
Fig. 3a for FS obtained at different photon energies) is presented in Fig.
3(e-g) which suggests negligible dispersion with the change in photon energy
from 40 eV to 60 eV, indicating its surface origin. The DFT results and the
experimental ARPES analysis strongly suggests the arc-like state is a
topological Fermi arc.
To conclusively prove that the arcs observed along the
$\overline{\Gamma}-\overline{M}$ direction are indeed the Weyl Fermi arcs, we
have examined the signature of projected chiral charge on the surface
corresponding to a Chern number on the bulk taking into account the bulk-
boundary correspondence between the bulk Weyl fermions and surface Fermi arcs
Wan ; Belopolski . As can be seen from Fig. 4(c) along the Cut 1, which lies
on the $k_{y}$ = 0.34 Å-1 plane, we can observe two counter propagating edge
modes. These edge modes are connected through a mirror plane $k_{x}$ = 0 and
can be visualized in the second derivative plot presented in Fig. 4(d) as
well. We have also considered closed loops in the surface BZ along one of the
Fermi arcs as shown in Fig. 4(a) by blue circles designated as loop 1 and loop
2. By analysing the energy-momentum dispersion for loop 1, we detect one left-
moving chiral mode which disperses towards EF, thus unambiguously showing
chiral charge -1 on the associated bulk manifold, Fig. 4(e). Similarly for
loop 2, we detect a right-moving chiral mode suggesting chiral charge of +1,
Fig. 4(f). In this way, using only the surface sensitive ARPES measurements,
we unambiguously confirm that the arc-like features are indeed the Weyl Fermi
arcs and measure the chiral charges in CeAlSi through the bulk-boundary
correspondence Wan ; Belopolski . The calculated bulk band dispersion for a
particular value of $k_{y}$ = 0.34 Å-1 presented in Fig. 4(g) also shows the
chiral mode clearly and is in agreement with the observed ARPES results. In
Fig. 4(h), we have presented the ARPES measurements along the
$\overline{\Gamma}-\overline{M}$ direction and its integrated energy
distribution curves where we can clearly observe a sharp dispersionless peak
at $\sim$ -0.3 eV below Fermi energy which corresponds to the so-called Kondo
resonance peak and is labeled as 4f${}^{1}_{7/2}$ kondo  suggesting that
electronic correlation is important in the system. The 4f${}^{1}_{5/2}$ peak
which lies near the Fermi level is not observed in the experiments. Further
temperature dependent and resonant photoemission experiments should be
performed to understand the importance of Kondo physics in this system.
We have also performed bulk-sensitive SX-ARPES studies on CeAlSi to
demonstrate the existence of the bulk Weyl cones and the Weyl nodes as shown
in Fig. 5. The SX-ARPES increases the relative spectral weight of bulk vs
surface features because the photoelectrons excited by soft X-rays have a
longer escape depth than the photoelectrons excited by VUV light. This SX-
ARPES data can be readily compared with the corresponding bulk band
calculations. In Figs. 5(a) and 5(e), we present the FS obtained from SX-ARPES
measurements at 15 K on the kz = 0 plane with a photon energy of 498 eV and
using DFT calculations, respectively, which are in good agreement with each
other. To observe the linear dispersion of the bulk Weyl cones, energy
dispersion maps along the various horizontal and vertical paths as shown in
Fig. 5(a) that intersect the experimentally observed Fermi arcs and calculated
position of the Weyl fermions are performed. In Fig. 5(b-d), we clearly
observe two linearly dispersing Weyl cones along Cut 1 and Cut 2 (white line
in Fig. 5(a)) that correspond to the two nearby Weyl nodes. The bulk band
dispersions shown in Fig. 5(b-d) along cut 1 to cut 3 are consistent with the
DFT calculations presented in Fig. 5(f, g), which further confirms the
presence of Weyl cones along the cut directions with Weyl nodes lying very
close to EF.
In conclusion, our systematic study of the electronic band structure of CeAlSi
using both VUV and SX ARPES at 15 K reveals it as a CI symmetry-breaking Weyl
semimetal. By utilizing VUV ARPES, on the (001) surface, we identified and
confirmed the presence of the Fermi arcs, the smoking gun evidence of the Weyl
semimetallic state. From SX-ARPES measurements, we were able to resolve
linearly dispersing conical features that correspond to the lower part of the
Weyl cone. The experimental observations are well reproduced by our first-
principles, DFT-based computations, further confirming the presence of Fermi
arcs and Weyl cones in our experimental results. A recent report suggested
that the breaking of TRS in this material will shift the position of the Weyl
nodes in the BZ Singh . The ARPES data which has been presented in this work
is performed at 15 K which is in the paramagnetic phase i.e, the Weyl nodes
are generated due to the broken inversion symmetry. CeAlSi therefore is a WSM
with broken CI inversion symmetry. Having a magnetic transition at around 8.2
K, this system provides an excellent platform to study and understand the
tuning of the Weyl points and the associated states with magnetism Singh .The
experimental confirmation for the shifting of the Weyl nodes due to the
breaking of TRS is not yet there and remains a further point of our research
interest. However, the present study unambiguously confirms the Weyl Fermi
arcs and the Weyl nodes in the paramagnetic phase arising from the breaking of
inversion symmetry and will be useful for future studies as it can be used for
comparison with those of the magnetic regime. Therefore, our study will open
up a new avenue to study and unveil the theoretically suggested tuning of the
Weyl nodes with the onset of magnetism which is currently one of the most
exciting topics in topological physics.
.

M.N. is supported by the Air Force Office of Scientific Research under Award
No. FA9550-17-1-0415, the Air Force Office of Scientific Research MURI (Grant
No. FA9550-20-1-0322), and the National Science Foundation (NSF) CAREER award
DMR-1847962. The work at Northeastern University was supported by the Air
Force Office of Scientific Research under award number FA9550-20-1-0322, and
it benefited from the computational resources of Northeastern University’s
Advanced Scientific Computation Center (ASCC) and the Discovery Cluster. This
material is based upon work supported by the Air Force Office of Scientific
Research under award number FA2386-21-1-4059. S.Y.X. is supported by NSF
Career under award No. DMR-2143177. We thank D. Lu and M. Hashimoto for
beamline assistance. Use of the Stanford Synchrotron Radiation Lightsource,
SLAC National Accelerator Laboratory, is supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences under Contract No.
DE-AC02-76SF00515. We thank Jessica L. McChesney for beamline assistance. This
research used resources of the Advanced Photon Source, a U.S. Department of
Energy (DOE) Office of Science User Facility at Argonne National Laboratory
and is based on research supported by the U.S. DOE Office of Science-Basic
Energy Sciences, under Contract No. DE-AC02-06CH11357.

APPENDIX: Methods
1\. Single crystal growth CeAlSi single crystals were grown by a self-flux
method in both regular alumina crucibles and the Canfield crucible sets
canfield . In both methods, the starting materials were weighed in the ratio
Ce:Al:Si = 1:10:1, placed inside a crucible in an evacuated quartz tube,
heated to 1000 °C at 3 °C/min, stayed at 1000 °C for 12 h, cooled to 700 °C at
0.1 °C/min, stayed at 700 °C for 12 h, and centrifuged to decant the residual
Al flux.

2\. Angle-resolved photoemission spectroscopy Angle-resolved photoemission
spectroscopy (ARPES) was performed at Stanford Synchrotron Radiation
Lightsource (SSRL), Menlo park, CA. Measurements were carried out at a
temperature of 15 K. The pressure in the UHV was maintained at better than
1$\times$10-10 torr. The angular and energy resolution was set better than 0.2
°and 15 meV, respectively. Measurements were performed employing the photon
energies in the range of 30 eV-150 eV using horizontal and vertical
polarization. Soft X-ray ARPES measurements were performed at the end station
sector 29-ID in APS Beamline equipped with a hemispherical Scienta R4000
electron analyzer. The energy and angular resolutions were set better than 200
meV and 0.1 degree respectively. The samples were cleaved at 20 K and the
pressure was better 1$\times$10-10 torr.

3\. DFT calculations Density functional theory (DFT) calculations were
performed using the experimental lattice parameters (a = 4.252 Å; c = 14.5801
Å) and the projector-augmented-wave (PAW) method implemented in the Vienna ab
initio simulation package (VASP) VASP . The exchange-correlation effects were
included by using the generalized gradient approximation (GGA). The spin-orbit
coupling (SOC) was included self-consistently Kresse ; Perdew . An on-site
Coulomb interaction was added for Ce f -electrons within the GGA + U scheme
with Ueff = 6 eV. A Wannier tight-binding Hamiltonian was obtained from the ab
initio results using the VASP2WANNIER90 interface, which was subsequently used
in our topological properties calculations Marzari .

## References

  * (1) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).
  * (2) A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).
  * (3) Z. Wang, Y. Sun, X. Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012).
  * (4) B. Singh, A. Sharma, H. Lin, M. Z. Hasan, R. Prasad, and A. Bansil Phys. Rev. B 86, 115208 (2012).
  * (5) H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, Phys. Rev. X 5, 011029 (2015).
  * (6) X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, Phys. Rev. X 5, 031023 (2015).
  * (7) S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. L. Zhang, R. Sankar, G. Q. Chang, Z. J. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. K. Wang, A. Bansil, F. C. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, Science 349, 613 (2015).
  * (8) B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X 5, 031013 (2015).
  * (9) B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, H. Ding, Nat. Phys. 11, 724-727 (2015).
  * (10) J. Liu, and D. Vanderbilt, Phy. Rev. B 90, 155316 (2014).
  * (11) S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian, T.-R. Chang, H. Zheng, V.N. Strocov, D.S. Sanchez, and G. Chang et al., Nat. Phys. 11, 748 (2015).
  * (12) B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys. 8, 337 (2017).
  * (13) A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Nature (London) 527, 495 (2015).
  * (14) Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan, Phys. Rev. B 92, 161107(R) (2015).
  * (15) K. Deng, G. Wan, P. Deng, K. Zhang, S. Ding, E. Wang, M. Yan, H. Huang, H. Zhang, and Z. Xu et al., Nat. Phys. 12, 1105 (2016).
  * (16) Y. Wu, D. Mou, N. H. Jo, K. Sun, L. Huang, S. L. Bud’ko, P. C. Canfield, and A. Kaminski, Phys. Rev. B 94, 121113(R) (2016).
  * (17) Q. Wang, Y. Xu, R. Lou, Z. Liu, M. Li, Y. Huang, D. Shen, H. Weng, S. Wang, and H. Lei, Nat. Commun. 9, 3681 (2018).
  * (18) E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y. Yang, D. Liu, A. Liang, Q. Xu, J. Kroder, V. Süß, H. Borrmann, C. Shekhar, Z. Wang, C. Xi, W. Wang, W. Schnelle, S. Wirth, Y. Chen, S. T. B. Goennenwein, and C. Felser, Nat. Phys. 14, 1125 (2018).
  * (19) K. Kim, J. Seo, E. Lee, K.-T. Ko, B. S. Kim, B. G. Jang, J. M. Ok, J. Lee, Y. J. Jo, W. Kang, J. H. Shim, C. Kim, H. W. Yeom, B. I. Min, B.-J. Yang, and J. S. Kim, Nat. Mater. 17, 794 (2018).
  * (20) I. Belopolski, K. Manna, D. S. Sanchez, G. Chang, B. Ernst, J. Yin, S. S. Zhang, T. Cochran, N. Shumiya, H. Zheng, B. Singh, . Bian, D. Multer, M. Litskevich, X. Zhou, S.-M. Huang, B. Wang, T.-R. Chang, S.-Y. Xu, A. Bansil, C. Felser, H. Lin, and M. Z. Hasan, Science 365, 1278 (2019).
  * (21) S. Borisenko, D. Evtushinsky, Q. Gibson, A. Yaresko, K. Koepernik, T. Kim, M. Ali, J. van den Brink, M. Hoesch, A. Fedorov, E. Haubold, Y. Kushnirenko, I. Soldatov, R. Schafer, and R. J. Cava, Nat. Commun. 10, 3424 (2019).
  * (22) G. Chang, B. Singh, S.-Y. Xu, G. Bian, S.-M. Huang, C.-H. Hsu, I. Belopolski, N. Alidoust, D. S. Sanchez, and H. Zheng et al., Phys. Rev. B 97, 041104(R) (2018).
  * (23) H.-Y. Yang, B. Singh, J. Gaudet, B. Lu, C.-Y. Huang, W.-C. Chiu, S.-M. Huang, B. Wang, F. Bahrami, and B. Xu, Phys. Rev. B 103, 115143 (2021).
  * (24) S.-Y. Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Belopolski, D. S. Sanchez, X. Zhang, G. Bian, and H. Zheng, Sci. Adv. 3, e1603266 (2017).
  * (25) H. Hodovanets, C. J. Eckberg, P. Y. Zavalij, H. Kim, W.-C. Lin, M. Zic, D. J. Campbell, J. S. Higgins, and J. Paglione, Phys. Rev. B 98, 245132 (2018).
  * (26) K. Zhang, T. Wang, X. Pang, F. Han, S.-L. Shang, N. T. Hung, Z.-K. Liu, M. Li, R. Saito, and S. Huang, Phys. Rev. B 102, 235162 (2020).
  * (27) P. Puphal, V. Pomjakushin, N. Kanazawa, V. Ukleev, D. J. Gawryluk, J. Z. Ma, M. Naamneh, N. C. Plumb, L. Keller, R. Cubitt, E. Pomjakushina, J. S. White, Phys. Rev. Lett. 124, 017202 (2020).
  * (28) J. Gaudet, H.-Y. Yang, S. Baidya, B. Z. Lu, G. Y. Xu, Y. Zhao, J. A. Rodriguez, C. M. Hoffmann, D. E. Graf, D. H. Torchinsky, P. Nikoli, D. Vanderbilt, F. Tafti, C.L. Broholm, arXiv:2012.12970 (2020).
  * (29) B. Xu, J. Franklin, H.-Y. Yang, F. Tafti, I. Sochnikov, arXiv:2011.06168 (2020).
  * (30) D. S. Sanchez, G. Chang, I. Belopolski, H. Lu, J.-X. Yin, N. Alidoust, X. Xu, T. A. Cochran, X. Zhang, Y. Bian, S. S. Zhang, Y.-Y. Liu, J. Ma, G. Bian, H. Lin, S.-Y. Xu, S. Jia, M. Z. Hasan, Nature Communications 11, 3356. (2020).
  * (31) I. Belopolski, S.-Y. Xu, D. S. Sanchez, G. Chang, C. Guo, M. Neupane, H. Zheng, C.-C. Lee, S.-M. Huang, G. Bian, N. Alidoust, T.-R. Chang, B. Wang, X. Zhang, A. Bansil, H. -T. Jeng, H. Lin, S. Jia, and M. Z. Hasan, Phys. Rev. Lett. 116, 066802 (2016).
  * (32) X.-G. Zhu, Y. Liu, Y.-W. Zhao, Y.-C. Wang, Y. Zhang, C. Lu, Y. Duan, D.-H. Xie, W. Feng, D. Jian, npj Quantum Materials 5, 1-9 (2020).
  * (33) P. C. Canfield, T. Kong, U. S. Kaluarachchi, and N. H. Jo, Philos. Mag. 96, 84 (2016).
  * (34) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * (35) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * (36) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * (37) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).