††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Robust Subthermionic Topological Transistor Action via Antiferromagnetic
Exchange

Sagnik Banerjee  Department of Electronics and Telecommunication Engineering,
Jadavpur University, Jadavpur-700032, India  Koustav Jana  Anirban Basak
Department of Electrical Engineering, Indian Institute of Technology Bombay,
Powai, Mumbai-400076, India  Michael S. Fuhrer  School of Physics and
Astronomy, Monash University, Clayton, Victoria 3800, Australia  ARC Centre of
Excellence in Future Low-Energy Electronics Technologies (FLEET), Monash
University, Clayton, Victoria 3800, Australia  Dimitrie Culcer  School of
Physics, University of New South Wales, Sydney 2052, Australia  ARC Centre of
Excellence in Future Low-Energy Electronics Technologies (FLEET), University
of New South Wales, Sydney 2052, Australia  Bhaskaran Muralidharan
<EMAIL_ADDRESS>Department of Electrical Engineering, Indian Institute of
Technology Bombay, Powai, Mumbai-400076, India

###### Abstract

The topological quantum field-effect transition in buckled 2D-Xenes can
potentially be engineered to enable sub-thermionic transistor operation
coupled with dissipationless ON-state conduction. Substantive device design
strategies to harness this will necessitate delving into the physics of the
quantum field effect transition between the dissipationless topological phase
and the band insulator phase. Investigating workable device structures, we
uncover fundamental sub-threshold limits posed by the gating mechanism that
effectuates such a transition, thereby emphasizing the need for innovations on
materials and device structures. Detailing the complex band translation
physics related to the quantum spin Hall effect phase transition, it is shown
that a gating strategy to beat the thermionic limit can be engineered at the
cost of sacrificing the dissipationless ON-state conduction. It is then
demonstrated that an out-of-plane antiferromagnetic exchange introduced in the
material via proximity coupling can incite transitions between the quantum
spin-valley Hall and the spin quantum anomalous Hall phase, which can
ultimately ensure the topological robustness of the ON state while surpassing
the thermionic limit. Our work thus underlines the operational criteria for
building topological transistors using quantum materials that can overcome the
Boltzmann’s tyranny while preserving the topological robustness.

##  I Introduction

A fundamental challenge today in the evolution of field-effect transistors
(FETs) is the compulsory power penalty resulting from a fundamental thermionic
limit, also known as the Boltzmann’s tyranny. This relates to the steepness of
the transfer characteristics: the subthreshold swing ($SS$) [1, 2, 3], which
is conventionally restricted to 60mV/dec at room temperature. In the context
of low-power devices, it is hence paramount to innovate strategies to suppress
the $SS$, thereby ensuring a sub-thermionic operation. Several attempts have
been made to overcome this limit, popular ones include tunnel FETs [4, 5, 6],
impact ionization MOSFETs [7] and negative capacitance FETs (NC-FETs) [8, 9,
10, 11, 12, 13, 14, 15], to name a few.
The topological quantum field effect [16] (TQFE) induced by the Rashba
interaction in buckled 2D materials like 2D-Xenes has been recently shown to
potentially propel a steeper $SS$ ($SS$<kBTln(10)/q) via a faster-than-linear
translation of the topological gap with electric-field modulation. In this
context, the topological quantum field-effect transistor (TQFETs) [17, 18, 19,
20, 21, 22] should additionally feature the robust dissipationless edge modes
hosted in the topological insulator (TI) phase as means toward high ON current
and energy-efficient low-power electronics. A great deal of experimental
effort has thus already been geared towards the realization of topological
transistors [23, 24, 25]. Quantum spin Hall (QSH) materials such as group-IV
and V-Xenes with buckled 2D honeycomb lattices [26, 27, 28, 29, 30], monolayer
transition metal dichalcogenides in the 1T’ configuration [31], HgTe
nanoribbons [32] and thin films of 3D topological insulators Bi2Se3 [33, 34]
as well as Dirac semi-metals like Na3Bi [35] are among the prominent material
candidates.
The demonstration of a workable device design uniting the merits of a
dissipationless channel along with the possibility of overcoming Boltzmann’s
tyranny using the TQFE can thus underline the operational criteria for
designing TQFETs as a building block for low-power electronics. In this work,
we present such a framework for a holistic analysis of TQFETs and highlight
the engineering intricacies involved in harnessing the Rashba spin orbit
interaction (SOI) for a steep SS, while simultaneously preserving the
topological robustness of the ON state. Building on this, we propose a device
structure that utilizes the spin quantum anomalous Hall (SQAH) state induced
via an out-of-plane antiferromagnetic (AF) exchange to achieve the desired
performance.

Figure 1: Device structure and phase transitions. a) A 2D buckled honeycomb
monolayer Xene is used as the channel (C), left (L), and right (R) lead
material. The leads are colored green to distinguish them from the channel
region, where the two sub-lattices A(B) are represented in red and blue
respectively. b) Schematic representation of a dual-gated device structure.
The top and bottom gates have applied potentials $V_{A}$ and $V_{B}$
respectively so that the net potential difference across the channel material
is ($V_{A}-V_{B}$). For the symmetric bias arrangement, $V_{A}=\lambda_{v}/2$
and $V_{B}=-\lambda_{v}/2$. c, d) Band structures of the channel and leads for
the TQFET, in the ON and the OFF-state respectively. The Fermi level ($E_{f}$)
is represented by a thick solid black line. e) Phase diagram of a monolayer
Xene nanoribbon with homogeneous perpendicular electric field and
antiferromagnetic exchange field [36, 37, 38].

Employing the Keldysh non-equilibrium Green’s function (NEGF) formalism [39,
40, 41], we uncover fundamental sub-threshold limits posed by the gating
mechanism that effectuates such a transition. By presenting an in-depth
analysis of the band translations necessitated by the field effect, we
demonstrate that the thermionic limit of the $SS$ in the TQFETs, designed
according to conventional principles, is half as steep as that of the
conventional FETs i.e., kBTln(10)/q [16]. In an attempt to alleviate this
issue, we propose to engineer the gate biasing to modulate one of the bands
while restricting the other, ultimately attaining an $SS$ transcending the
thermionic limit of 60 mV/dec at room temperature. However, this also
introduces dissipative conduction modes from the bulk in the ON state,
defeating one of the desired attributes in a TQFET. As a tactical solution, we
demonstrate that the introduction of out-of-plane antiferromagnetic exchange
interaction, which can be induced via proximity coupling [42] restores the
dissipationless ON state and can effectively reap the merits expected from the
Rashba-assisted TQFET.

##  II Results and Discussion

The building block of the TQFET is the buckled hexagonal lattice structure
that forms the channel as depicted in Fig. 1(a). In the transistor setup
depicted in Fig. 1(b), an electric field ($E_{Z}$) applied perpendicular to a
buckled channel manifests as a staggered potential between the sub-lattice $A$
and $B$ of the honeycomb unit cell. The dual-gate structure helps in realizing
an electric field between the two plates, hence imparting a capacitive action.
Such a dual-gate manifestation of a topological transistor also enables a two-
fold biasing scheme: a) symmetric biasing, where equal and opposite bias
voltages are applied to the two gates, i.e., $V_{A}=-V_{B}$, and b) rigid
biasing, where the entire voltage is applied to one of the gate plates with
the other plate grounded. For the symmetric bias setup the Fermi level is
positioned as in Fig 1(c) and Fig 1(d) to ensure topological ON state
conduction.
Previous works have been centered around the electric-field driven transition,
i.e., the transition involving the QSH phase, which effectively navigates the
horizontal axis of the phase diagram in Fig. 1(e). By modulating the strength
of the AF exchange interaction $M_{AF}$, the phase transition could also track
the vertical axis. In the former, the phase transition is between the
conducting QSH phase and the insulating quantum valley Hall (QVH) phase. In
the latter, phase transitions are between an insulating quantum spin valley
Hall (QSVH) phase and a conducting spin quantum anomalous Hall (SQAH) phase
[36, 37, 38]. Unlike the former case, here the AF interaction breaks the time
reversal symmetry (TRS), resulting in spin-polarized conducting modes without
a chiral counter-propagating partner.

To analytically investigate the subthreshold physics of topological
transistors we adopt the low energy effective four-band Bloch Hamiltonian
$H_{\eta}$ in the vicinity of Dirac points $K(K^{\prime})$ as given by (1).
This Dirac Hamiltonian has been derived from the tight-binding Hamiltonian
model for a 2D buckled honeycomb lattice as elaborated in the Supplementary
Information (refer to (S1)).

$H_{\eta}=\hbar v_{f}(\eta
k_{x}\tau_{x}+k_{y}\tau_{y})\sigma_{0}+\eta\lambda_{SO}\tau_{z}\sigma_{z}\\\
+\lambda_{v}\left(E_{Z}\right)\tau_{z}\sigma_{0}+\frac{\lambda_{R}\left(E_{Z}\right)}{2}(\eta\tau_{x}\sigma_{y}-\tau_{y}\sigma_{x}),$
(1)

where $\eta=+(-)$ is the valley index denoting $K(K^{\prime})$ and, $\sigma$
and $\tau$ are the spin and pseudo-spin Pauli matrices respectively. Here,
$\hbar$ is the reduced Planck’s constant and $v_{f}$ denotes the Fermi
velocity, given by the expression $v_{f}=3ta_{o}/2$ where $t$ is the hopping
parameter and $a_{o}$ is the lattice constant. The quantities $\lambda_{SO}$,
$\lambda_{v}$ and $\lambda_{R}$ denote the strengths of the intrinsic spin-
orbit coupling, staggered sublattice potential and Rashba spin-mixing
interaction respectively. Additional details about the same can be found in
the Supplementary Information.

Figure 2: Band structures of the channel for different $\lambda_{v}$. a)
represents the QSH phase characterized by topologically protected edge states.
Here, we have assumed $\lambda_{v}=0$ and considered two cases of
$\lambda_{R}=0$ and $\lambda_{R}\neq 0$. b) depicts the transition from QSH to
QVH phase for the $\lambda_{R}=0$ case. For the $\lambda_{R}\neq 0$ case, the
system has already entered the band insulator phase. Here, we have considered
$\lambda_{v}=\lambda_{SO}$, where $\lambda_{SO}$ is the intrinsic spin-orbit
coupling strength. c) demonstrates a faster retreat of the CB in the
$\lambda_{R}\neq 0$ case compared to the $\lambda_{R}=0$ case. The VB in both
cases remains almost pinned. Here, $\lambda_{v}=1.5\lambda_{SO}$.

In the context of an electric field induced topological phase transition
between the QVH and the QSH phases [43], it is important to analyze the Chern
number [44] ($\mathcal{C}$) and the nature of the edge states. For
$\lambda_{v}<\lambda_{SO}$, the QSH phase is characterized by a zero total
Chern number $\mathcal{C}$ and a non-zero spin Chern number
$2\mathcal{C}_{s}=+2$ resulting in helical edge states as illustrated in Fig.
2(a). A phase transition at $\lambda_{v}=\lambda_{SO}$, as in Fig. 2(b),
results in the QVH phase for $\lambda_{v}>\lambda_{SO}$. This band insulator
phase is characterized by both zero $\mathcal{C}$ and $2\mathcal{C}_{s}$,
leading to a trivial gap as shown in Fig. 2(c). The faster closing and
reopening of gap for the $\lambda_{R}\neq 0$ case when compared to
$\lambda_{R}=0$ case, as shown in Fig. 2, nicely illustrates the topological
quantum field effect (TQFE) switching.
The total current with components $I_{c}$ and $I_{v}$ due to the electrons in
the conduction band (CB) and holes in the valence band (VB) is given as

$\begin{split}&I_{c}=I_{co}\exp-\frac{q\left(E_{c}-E_{f}\right)}{k_{B}T}\\\
&I_{v}=I_{vo}\exp\frac{q\left(E_{v}-E_{f}\right)}{k_{B}T},\end{split}$ (2)

where $I_{co(vo)}$ is the CB (VB) current maximum, $E_{c(v)}$ represents the
CB (VB) minimum (maximum), and $E_{f}$ denotes the equilibrium Fermi energy
level. Here, $k_{B}T$ is the thermal energy at temperature $T$.
The $SS$ of a transistor during the ON-OFF state transition is hence defined
as

$SS=\left|\frac{d(log_{10}I)}{dV_{G}}\right|^{-1},$ (3)

where $I$ ($=I_{c}+I_{v}$) represents the current in response to an applied
gate voltage $V_{G}$. Substituting $I$ from (2) in (3), we get,

$SS=\frac{k_{B}T}{q}ln(10)\frac{I}{\left|I_{v}\left(\frac{dE_{v}}{dV_{G}}\right)-I_{c}\left(\frac{dE_{c}}{dV_{G}}\right)\right|},$
(4)

where 2kBTln(10)/q is the thermionic limit at temperature $T$, which reduces
to 60 mV/decade at room temperature. We then define the reduced $S^{*}$, which
is the reduced $SS$ [16] as,

$S^{*}=\frac{I}{\left|I_{v}\left(\frac{dE_{v}}{dV_{G}}\right)-I_{c}\left(\frac{dE_{c}}{dV_{G}}\right)\right|}.$
(5)

For conventional MOSFETs, it has hitherto been assumed that $S^{*}\geq 1$.
Moreover, it is anticipated in [16] that the introduction of Rashba
interactions can achieve $S^{*}\leq 1$ to overcome the Boltzmann’s limit.
However, we now demonstrate that this limit ($S^{*}$) for a TQFT is,
unfavorably, twice as much.

Figure 3: Band movement schematics and I-V characteristics for QSH-QVH
transition based FET. a) represents the band movement in an ordinary n-MOSFET
for an applied bias potential of $\Delta V_{G}$. Here, $I_{C}>I_{V}$ b)
represents the band movement in the OFF state of a topological transistor with
$\lambda_{R}=0$. Here, $I_{C}=I_{V}$. c) represents the band movement in the
OFF state of a topological transistor with $\lambda_{R}\neq 0$. Here,
$I_{C}<I_{V}$. d) is the I-V characteristics of a topological transistor based
upon phase transition from ON ($V_{GS}<0.8$V) to OFF ($V_{GS}>0.9$V) phase,
for different Rashba strengths of $\lambda_{R}=0$, $\lambda_{R}=0.3\Delta_{Z}$
and $\lambda_{R}=0.5\Delta_{Z}$. Here, $\Delta_{Z}=\lambda_{v}$. For all the
three cases, the subthreshold-swing ($SS$) remains confined to 120 mV/decade
of current.

Figure 3(a) depicts the n-MOS operation of a conventional transistor. For an
applied gate bias $\Delta V_{G}$, both the CB and the VB move equally by an
amount proportional to $\Delta V_{G}$. For an n-MOS device, this upward ascent
of the bands results in $I_{c}>>I_{v}$. In this case, both $(dE_{c}/dV_{G})$
and $(dE_{v}/dV_{G})$ are unity since both bands translate by an equal amount
($\propto\Delta V_{G}$).
For a TQFET with $\lambda_{R}=0$, however, the translation of bands in
response to $\Delta V_{G}$ is non-trivial. As depicted in Fig. 3(b), both the
bands move away from the Fermi level by an amount proportional to $\Delta
V_{G}/2$ in the OFF state. So, $(dE_{c}/dV_{G})=1/2$ and
$(dE_{v}/dV_{G})=-1/2$ for a TQFET with $\lambda_{R}=0$. Moreover, due to
symmetry in the band translation, the current components $I_{c}\approx
I_{v}\approx I/2$. This results in $S^{*}\approx 2$ and hence the $SS$ in a
standard TQFET will be restricted to 2kBTln(10)/q instead of kBTln(10)/q. At
room temperature, this translates to the thermionic limit being restricted to
120 mV/decade instead of 60 mV/decade in the conventional case.
With the introduction of the Rashba SOI term, the band translation during the
OFF state (QVH phase) is asymmetric. From the Dirac Hamiltonian elaborated in
(1), we calculate $E_{c}$ and $E_{v}$ for $\lambda_{v}>\lambda_{SO}>0$, i.e.,
in the OFF state as follows:

$\begin{split}&E_{v}=\lambda_{SO}-\lambda_{v}\\\
&E_{c}=-\lambda_{SO}+\sqrt{{\lambda_{v}^{2}}+\lambda_{R}^{2}},\end{split}$ (6)

where, $\lambda_{v}$ and $\lambda_{R}$ can be represented as $\alpha_{V}E_{Z}$
and $\alpha_{R}E_{Z}$ respectively since they are both linearly proportional
to the gate electric field $E_{Z}$. Taking derivatives with respect to $V_{G}$
we get, $(dE_{v}/dV_{G})=-1/2$ and
$(dE_{c}/dV_{G})=0.5(1+\alpha_{R}^{2}/\alpha_{V}^{2})^{1/2}$. Note that,
putting $\alpha_{R}=0$ we get same results as estimated previously for the
$\lambda_{R}=0$ case. Also, we notice that in the QVH phase, the CB translates
faster with increasing gate bias than the VB due to the effect of the Rashba
term $\alpha_{R}$. However, since $\left|E_{v}\right|<\left|E_{c}\right|$, we
expect $I_{v}$ to contribute more to the current than $I_{c}$. Thus, from (5)
we expect $(dE_{v}/dV_{G})$ to contribute more to the $SS$. Thus, the improved
$SS$ due to $(dE_{c}/dV_{G})$ is undone by the relatively smaller contribution
from $I_{c}$. This mechanism has been represented schematically in Fig. 3(c).
In Fig. 3(d), we show the evaluated subthreshold characteristics using the
NEGF approach [39, 40, 41, 45]. We set $T=300$K and $\lambda_{SO}=0.41$eV with
a hopping parameter $t=1.6$eV, consistent with first-principles calculations
[16]. We assume a finite-sized nanoribbon as the channel material, with a
device length $N_{L}=45$ and width $N_{W}=20$. The NEGF simulations reconcile
with the analytical conclusions drawn. Despite the introduction of Rashba
interactions, the $SS$ is limited to 2kBTln(10)/q or 120 mV/decade at
$T=300$K.

Figure 4: Rigid biasing approach wherein $V_{A}=V_{G}$ and $V_{B}=0$. The band
structures for varying $\lambda_{v}$ along with the corresponding Fermi levels
(elaborated in text) are depicted in (a-d). (e) describes the band-to-band
tunneling possibility when the channel Fermi level lies inside the VB, leading
to significant amount of OFF current. (f) is the I-V characteristics for a
rigidly biased TQFET with suitably positioned Fermi level for different values
of $\lambda_{R}$. The corresponding $SS$ values for $\lambda_{R}=0$,
$0.3\Delta_{z}$, and $0.5\Delta_{z}$ are 60mV/dec, 52mV/dec, and 49mV/dec
respectively

With the symmetric gate biasing, as seen in previous sections, the channel
Fermi level is pinned in the middle of the bandgap of the channel. As a
result, in the QVH phase (OFF) the channel conducts the smallest possible
current. Moreover, using symmetric gate biasing, the $SS$ cannot be improved
significantly below 120mV/dec even in the presence of Rashba SOI. Hence, an
alternate approach is required which relies upon a smart positioning of the
channel Fermi level to create an imbalance in the rate with which the two
bands move away from the channel Fermi level, as a function of the applied
gate voltage.
The easiest way to do this would be via an asymmetric gate biasing where
$V_{A}$ and $V_{B}$ corresponding to Fig. 1(b) are set to $V_{G}$ and zero
respectively. This particular gating scheme has an added advantage that it is
compatible with the modern-day transistor architectures because of it
requiring only a single gate voltage as opposed to the dual gate voltage
requirement of symmetric gate biasing. Now positioning the Fermi level at
$E=0$ (solid line in Fig. 4(a-d)) leads to an immanent problem as described
below. At $\lambda_{v}=0$, as shown in Fig. 4(a), the Fermi level lies right
at the middle of the bandgap. Because of rigid biasing in the QSH phase, only
the VB moves with varying gate voltage and after a certain point, the Fermi
level slips inside the VB. Beyond the critical field, the Fermi level remains
pinned inside the VB. As a result, the TQFET does not turn OFF in the QVH
phase, because of the VB bulk states participating in the conduction. Despite
the Fermi level in the leads being aligned to the CB, the current conduction
is still facilitated by band-to-band tunneling as depicted in Fig. 4(e).
Hence, one obtains a large OFF current which is undesirable.
The problem highlighted above arises because of the Fermi level lying inside
the VB in the QVH phase resulting in a deteriorated OFF state. One obvious way
to alleviate this issue is to align the Fermi level inside the bandgap for the
QVH phase. However, in this scenario, the Fermi level moves inside the CB in
the QSH phase (ON state). Thus, apart from the dissipationless edge modes,
even the dissipative bulk states participate in conduction. This strategy
restores the full advantage of the Rashba SOI-enabled TQFE in reducing the SS.
However, conduction no longer occurs solely through dissipationless edge
states. This may be disadvantageous, though the details of the ON state
conduction will depend on material and device parameters, and in principle
dissipative bulk states can also contribute to enhanced ON current. Thus, by
allowing dissipative bulk state conduction, it is possible to achieve
subthermionic performance with negligible OFF current, by aligning the Fermi
level as shown by the dashed lines in Fig. 4(a-d).
The obtained I-V characteristics for the above case are as shown in Fig. 4(f).
The ON current is significantly higher (> $10^{4}$) when compared to that of
the symmetric biasing case in Fig. 3(d), owing to the fact that the current-
carrying states in the ON state are now the bulk modes of the QSH phase.
Because of rigid gate biasing, as illustrated by our numerical simulations,
the $SS$ calculated is around 60mV/dec, the thermionic limit at 300K. The
inclusion of Rashba SOI further enhances the subthreshold performance with
$SS$ values of 52mV/dec and 50mV/dec for the cases of
$\lambda_{R}=0.3\Delta_{z}$ and $0.5\Delta_{z}$ respectively, thus
exemplifying subthermionic performance enabled by the introduction of Rashba
SOI for a rigidly biased TQFET with a suitably positioned Fermi level.

Figure 5: Band structures of the channel for different $\lambda_{v}$
highlights the underlying switching principle under the influence of an
antiferromagnetic term. a) represents the quantum spin valley Hall (QSVH)
phase characterized by an insulating gap for the two valleys K(K’). Here, we
have assumed $\lambda_{v}=0$ and considered two cases of $\lambda_{R}=0$ and
$\lambda_{R}\neq 0$. b) depicts the QSVH-SQAH transition for the
$\lambda_{R}=0$ case. For the $\lambda_{R}\neq 0$ case, topologically
protected edge states are prominent. Here, we have considered
$\lambda_{v}=0.5\lambda_{SO}$, $\lambda_{SO}$ being the intrinsic spin-orbit
coupling strength. c) demonstrates the onset of ON phase for both the
$\lambda_{R}=0$ and $\lambda_{R}\neq 0$ case. Here, we have chosen
$\lambda_{v}=0.8\lambda_{SO}$.

While the above strategy is a robust route to a subthermionic transistor, we
would also like to understand whether it is possible to retain both the
subthermionic SS and the dissipationless edge transport in the ON state.
Dissipationless edge transport should provide significant advantages for
certain device geometries and hence, is an indispensable feature to retain. We
thus explore ideas beyond the QSH-QVH transition and look into new topological
phases. As suggested by (4), one way to achieve $S^{*}\leq 1$ is by ensuring
that at least one of the quantities $(dE_{c}/dV_{G})$ and $(dE_{v}/dV_{G})$
exceeds unity. If, say, the CB satisfies this, then for subthermionic
performance the CB current should be the major contributor of the total
current. In other words, we can attain subthermionic performance by ensuring
$(dE_{c}/dV_{G})\geq 1$ and $I_{c}>>I_{v}$.

Figure 6:  a) The band translation schematics in the OFF state of a TQFET is
represented which includes AF exchange interactions, when $I_{C}<I_{V}$. The
Fermi level is closer to the VB than the CB. b) The I-V characteristics based
on phase transitions in the subthreshold region for different Rashba
interactions, for $\Delta_{Z}=\lambda_{v}$. We notice a progressive
improvement in the $SS$ as $\lambda_{R}$ increases.

In the QSH-QVH transition-based TQFET with symmetric biasing, evident from
(6), it is possible to achieve $(dE_{c}/dV_{G})\geq 1$ in the presence of a
large enough Rashba SOI. However as a result of this, in the QVH phase, the CB
moves away from the Fermi level at a greater rate than the VB and we get
$I_{v}>>I_{c}$, as illustrated in Fig. 3(c), and thus there is no discernible
improvement in $S^{*}$. This happens because in the OFF state the bandgap
opens as we increase the gate voltage. For the Rashba influenced band to
contribute more to the total current, the bandgap should close as the applied
field increases. In such a scenario, the Rashba influenced band would be
closer to the Fermi level throughout the subthreshold regime (OFF state). This
will eventually lead to a subthermionic performance.
The above scenario can be realized by the addition of AF exchange interaction,
which can be incorporated via proximity coupling [42] to a 2D topological
insulator implying the addition of the term $M_{AF}\tau_{z}\sigma_{z}$ to the
low-energy Hamiltonian in (1). As represented by the points (1),(2) and (3) in
Fig. 1(e), for $M_{AF}>\lambda_{SO}$, one can realize the QSVH phase and the
SQAH phases by varying the perpendicular electric field [36, 37, 38]. For
$\lambda_{v}<M_{AF}-\lambda_{SO}$, we obtain the QSVH phase with a Chern
number $\mathcal{C}=0$ and no gapless edge states. On the other hand,
$\lambda_{v}>M_{AF}-\lambda_{SO}$ opens the SQAH phase with a Chern number
$\mathcal{C}=1$, possessing spin-polarized chiral edge states. The perfect
spin polarization of the edge currents may also provide connections to
spintronics applications. Here, the QSVH phase represents the OFF state, i.e.,
the transition is from OFF to ON as the gate voltage increases, contrary to
the scenario in the QSH-QVH transition.
Now, as depicted in Fig. 6(a), the bandgap closes in the subthreshold regime
on increasing gate voltage and the Rashba influenced VB moves closer to the
Fermi level, such that $I_{v}>>I_{c}$. In this case, if $\lambda_{R}$ is large
enough for $dE_{v}/dV_{G}$ to be greater than unity, it is possible to achieve
$S^{*}<1$. Analytically, based on the low-energy Dirac Hamiltonian in (1), and
in presence of Rashba SOI, we have the following in the QSVH phase:

$\begin{split}&E_{v}=\lambda_{SO}-M_{AF}+\sqrt{{\lambda_{v}^{2}}+\lambda_{R}^{2}}\\\
&E_{c}=M_{AF}-\lambda_{SO}-\lambda_{v}.\end{split}$ (7)

Clearly, on increasing $E_{z}$, the VB approaches the Fermi level at a greater
rate for a non-zero $\lambda_{R}$, and the VB current is the major
contributor.
These analytical findings are well supported by the simulated results in Fig.
6(b). Here the following parameters are considered: $\lambda_{SO}=0.41eV$,
$M_{AF}=1.7\lambda_{SO}$. We consider three values of $\lambda_{R}$ for
comparison: $\lambda_{R}=0$, $0.5\Delta_{z}$ and $0.8\Delta_{z}$, where
$\Delta_{z}=\lambda_{v}$. The $\lambda_{R}=0$ case, similar to the previous
QSH-QVH case, gives an $SS$ of around 120mV/dec, because of symmetric band
translation under symmetric biasing. However, upon increasing $\lambda_{R}$,
we notice a significant improvement in the $SS$, with the topological
transistor achieving $SS<$ 60mV/dec for $\lambda_{R}=0.8\Delta_{z}$. Thus the
addition of an out-of-plane AF exchange interaction to a 2D-TQFET under
symmetric biasing can attain subthermionic characteristics. One thing to note
is that the current remains constant in the ON state for an appreciable range
of gate voltage, suggesting that the dissipationless edge modes are
responsible for the ON state conduction. Adding to this, a TQFET hosting
chiral QAH edge modes is expected to be more resilient to back-scattering than
the one having helical QSH edge modes. This is because QSH requires the time-
reversal symmetry (TRS) to be preserved for maintaining its robustness.
However, this is not the case with the QAH phase which is robust even to back-
scattering by magnetic disorder, because of the absence of any time-reversal
partner for the chiral edge modes.

##  III Conclusion

Our analysis into the physics of the quantum field effect transition unraveled
that the fundamental subthreshold performance of the QSH-QVH transition is at
best half as steep as that of the conventional field-effect transistor and
that the mere introduction of Rashba interaction does not render any
additional steepness. We proposed tactical upgrades to alleviate these
drawbacks to actually steer toward the desired subthermionic performance. We
first demonstrated that a modified gating scheme could drive the topological
transition and successfully overcome the thermionic limit while sacrificing
the dissipationless nature of the ON state. We then proposed to exploit the
topological transition between the QSVH and the SQAH phase via the
introduction of out-of-plane AF exchange interaction, thereby ensuring the
topological robustness of the ON state while surpassing the thermionic limit.
Our work thus underlines the operational criteria for building topological
transistors using quantum materials that can overcome the Boltzmann’s tyranny
while preserving the topological robustness.

##  IV Methods

All the transport calculations are based on the NEGF formalism [39, 40, 41,
45], within the tight binding framework of the model Hamiltonian [18, 19, 43,
44] described in the Supplementary Information. To obtain the desired I-V
characteristics, current calculations are performed based on the Landauer
transmission formula, evaluated from the device retarded Green’s function

$I=\frac{e^{2}}{h}\int_{-\infty}^{\infty}T(E)(f(E,\mu_{L},T)-f(E,\mu_{R},T))dE,$
(8)

where $f(E,\mu,T)=(1+exp(\frac{E-\mu}{k_{B}T}))^{-1}$ is the Fermi-Dirac
distribution at Fermi energy $\mu$ and temperature $T$. The calculation of the
transmission coefficient $T(E)$ using NEGF formalism has been discussed in the
Supplementary Information.

## Acknowledgements

The author BM acknowledges the Visvesvaraya Ph.D Scheme of the Ministry of
Electronics and Information Technology (MEITY), Government of India,
implemented by Digital India Corporation (formerly Media Lab Asia). The author
BM also acknowledges the support by the Science and Engineering Research Board
(SERB), Government of India, Grant No. Grant No. STR/2019/000030, and the
Ministry of Human Resource Development (MHRD), Government of India, Grant No.
STARS/APR2019/NS/226/FS under the STARS scheme. MSF and DC acknowledge support
from the ARC Centre of Excellence in Future Low-Energy Electronics
Technologies (CE170100039).

## Supplementary Information

###  IV.1 Device Hamiltonian

For numerical calculations pertinent to our proposed topological transistor,
we consider the typical tight-binding Hamiltonian model for a 2D buckled
honeycomb lattice [46, 47], as shown in (S1) in the second quantized notation:

$\hat{H}=-t\sum_{\langle
i,j\rangle\alpha}c_{i\alpha}^{\dagger}c_{j\alpha}+i\frac{\lambda_{SO}}{3\sqrt{3}}\sum_{\langle\langle
i,j\rangle\rangle\alpha\beta}\nu_{ij}c_{i\alpha}^{\dagger}s_{\alpha\beta}^{z}c_{j\beta}+\lambda_{v}\sum_{i\alpha}c_{i\alpha}^{\dagger}\mu_{i}c_{i\alpha}+i\lambda_{R}\sum_{ij\langle\alpha\beta\rangle}c_{i\alpha}^{\dagger}(s_{\alpha\beta}\times\hat{d}_{ij})_{z}c_{j\beta},$
(S1)

where $c_{i\alpha}^{(\dagger)}$ represents the electronic annihilation
(creation) operator on site $i$ with a spin $\alpha=\uparrow(\downarrow)$, and
$\langle i,j\rangle$ and $\langle\langle i,j\rangle\rangle$ characterize the
nearest neighbour and the next-nearest neighbour hopping respectively. The
spin indices are represented with corresponding values $+1/-1$ respectively.
The first term in (S1) represents the nearest-neighbor hopping term with a
hopping strength $t$. The second term represents the intrinsic spin-orbit (SO)
coupling with strength $\lambda_{SO}$, where $\nu_{ij}=+1(-1)$ for anti-
clockwise (clockwise) next-nearest neighbour hopping with respect to the
positive $z$-axis. The third term denotes the staggered sub-lattice potential
of strength $\lambda_{v}$, with $\mu_{i}=+1(-1)$, where $i$ denotes the sub-
lattice A(B). The fourth term represents the nearest neighbour Rashba spin-
mixing interaction, with $s_{\alpha\beta}$ denoting the corresponding matrix
elements indicating spin-polarization $\alpha$, $\beta$ at lattice sites $i$,
$j$ respectively, and $\hat{d}_{ij}$ is the distance vector between lattice
sites $i$ and $j$. In the low-energy limit, the essential physics governing
the device operation can be captured using the four-band Bloch Hamiltonian
around the Dirac points $K(K^{\prime})$ as in equation (1) of the main
article.

###  IV.2 Calculation of transmission coefficient

The I-V calculations have been done using equation (8) of the main article,
which involves the term $T(E)$ i.e. the coherent transmission coefficient at a
given energy $E$. To calculate $T(E)$, we employ the NEGF formalism [39, 40,
41, 45] based on the tight-binding framework described in (S1), where $T(E)$
is evaluated using the Green’s function as:

$T(E)=\mathbf{Tr}[\Gamma_{L}(E)G^{R}(E)\Gamma_{R}(E)G^{A}(E)]$ (S2)
$[\Gamma_{L,R}(E)]=i[\Sigma_{L,R}(E)-\Sigma_{L,R}(E)^{{\dagger}}]$ (S3)
$[G^{R}(E)]=[(E+i\eta)I-H-\Sigma_{L}(E)-\Sigma_{R}(E)]^{-1}$ (S4)

where $\mathbf{Tr}$ represents the trace operation, $[\Gamma_{L(R)}(E)]$ is
the broadening matrix corresponding to the lead $L(R)$, and $[G^{R}(E)]$ and
$[G^{A}(E)]$ are the matrix representations of the retarded and advanced
Green’s functions respectively. All quantities in the above equations can be
obtained from the Hamiltonian defined in (S1) and the self-energy matrices
$[\Sigma_{L,R}]$, which are calculated recursively based on the formalism
prescribed in [43, 45].

## References

  * Lundstrom [2006] M. Lundstrom, in  _2006 IEEE international SOI Conferencee Proceedings_ (IEEE, 2006) pp. 1–3.
  * Sakurai [2004] T. Sakurai, IEICE transactions on electronics 87, 429 (2004).
  * Sarkar  _et al._ [2015] D. Sarkar, X. Xie, W. Liu, W. Cao, J. Kang, Y. Gong, S. Kraemer, P. M. Ajayan, and K. Banerjee, Nature 526, 91 (2015).
  * Kobayashi [2018] M. Kobayashi, Applied Physics Express 11, 110101 (2018).
  * Newns  _et al._ [1998] D. Newns, J. Misewich, C. Tsuei, A. Gupta, B. Scott, and A. Schrott, Applied Physics Letters 73, 780 (1998).
  * Banerjee  _et al._ [2008] S. K. Banerjee, L. F. Register, E. Tutuc, D. Reddy, and A. H. MacDonald, IEEE Electron Device Letters 30, 158 (2008).
  * Gopalakrishnan  _et al._ [2004] K. Gopalakrishnan, P. B. Griffin, and J. D. Plummer, IEEE Transactions on electron devices 52, 69 (2004).
  * Íñiguez  _et al._ [2019] J. Íñiguez, P. Zubko, I. Luk’yanchuk, and A. Cano, Nature Reviews Materials 4, 243 (2019).
  * Rusu  _et al._ [2010] A. Rusu, G. A. Salvatore, D. Jimenez, and A. M. Ionescu, in  _2010 international electron devices meeting_ (IEEE, 2010) pp. 16–3.
  * Krivokapic  _et al._ [2017] Z. Krivokapic, U. Rana, R. Galatage, A. Razavieh, A. Aziz, J. Liu, J. Shi, H. Kim, R. Sporer, C. Serrao, _et al._ , in  _2017 IEEE International Electron Devices Meeting (IEDM)_ (IEEE, 2017) pp. 15–1.
  * Saeidi  _et al._ [2017] A. Saeidi, F. Jazaeri, F. Bellando, I. Stolichnov, G. V. Luong, Q.-T. Zhao, S. Mantl, C. C. Enz, and A. M. Ionescu, IEEE electron device letters 38, 1485 (2017).
  * Si  _et al._ [2018] M. Si, C.-J. Su, C. Jiang, N. J. Conrad, H. Zhou, K. D. Maize, G. Qiu, C.-T. Wu, A. Shakouri, M. A. Alam, _et al._ , Nature nanotechnology 13, 24 (2018).
  * Wang  _et al._ [2019] X. Wang, P. Yu, Z. Lei, C. Zhu, X. Cao, F. Liu, L. You, Q. Zeng, Y. Deng, J. Zhou, _et al._ , Nature communications 10, 1 (2019).
  * McGuire  _et al._ [2017] F. A. McGuire, Y.-C. Lin, K. Price, G. B. Rayner, S. Khandelwal, S. Salahuddin, and A. D. Franklin, Nano letters 17, 4801 (2017).
  * Fuhrer  _et al._ [2022] M. S. Fuhrer, M. T. Edmonds, D. Culcer, M. Nadeem, X. Wang, N. Medhekar, Y. Yin, and J. H. Cole, ArXiv: 2201.05288 (2022).
  * Nadeem  _et al._ [2021] M. Nadeem, I. Di Bernardo, X. Wang, M. S. Fuhrer, and D. Culcer, Nano Letters 21, 3155 (2021).
  * Gilbert [2021] M. J. Gilbert, Communications Physics 4, 1 (2021).
  * Ezawa [2014] M. Ezawa, New Journal of Physics 16, 065015 (2014).
  * Ezawa [2013a] M. Ezawa, Applied Physics Letters 102, 172103 (2013a).
  * Shi  _et al._ [2021] B. Shi, H. Tang, Z. Song, J. Li, L. Xu, S. Liu, J. Yang, X. Sun, R. Quhe, J. Yang, _et al._ , Nanoscale 13, 15048 (2021).
  * Vandenberghe and Fischetti [2017] W. G. Vandenberghe and M. V. Fischetti, Nature communications 8, 1 (2017).
  * Xue [2011] Q.-K. Xue, Nature nanotechnology 6, 197 (2011).
  * Tian  _et al._ [2014] J. Tian, C. Chang, H. Cao, K. He, X. Ma, Q. Xue, and Y. P. Chen, Scientific reports 4, 1 (2014).
  * Zhu  _et al._ [2013] H. Zhu, C. A. Richter, E. Zhao, J. E. Bonevich, W. A. Kimes, H.-J. Jang, H. Yuan, H. Li, A. Arab, O. Kirillov, _et al._ , Scientific reports 3, 1 (2013).
  * Liu and Ye [2011] H. Liu and P. D. Ye, Applied Physics Letters 99, 052108 (2011).
  * Min  _et al._ [2006] H. Min, J. Hill, N. A. Sinitsyn, B. Sahu, L. Kleinman, and A. H. MacDonald, Physical Review B 74, 165310 (2006).
  * Liu  _et al._ [2011] C.-C. Liu, W. Feng, and Y. Yao, Physical review letters 107, 076802 (2011).
  * Xu  _et al._ [2013] Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S.-C. Zhang, Physical review letters 111, 136804 (2013).
  * Hsu  _et al._ [2015] C.-H. Hsu, Z.-Q. Huang, F.-C. Chuang, C.-C. Kuo, Y.-T. Liu, H. Lin, and A. Bansil, New Journal of Physics 17, 025005 (2015).
  * Reis  _et al._ [2017] F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass, W. Hanke, R. Thomale, J. Schäfer, and R. Claessen, Science 357, 287 (2017).
  * Qian  _et al._ [2014] X. Qian, J. Liu, L. Fu, and J. Li, Science 346, 1344 (2014).
  * Fu  _et al._ [2014] H.-H. Fu, J.-H. Gao, and K.-L. Yao, Nanotechnology 25, 225201 (2014).
  * Shan  _et al._ [2010] W.-Y. Shan, H.-Z. Lu, and S.-Q. Shen, New Journal of Physics 12, 043048 (2010).
  * Liu  _et al._ [2010] C.-X. Liu, H. Zhang, B. Yan, X.-L. Qi, T. Frauenheim, X. Dai, Z. Fang, and S.-C. Zhang, Physical review B 81, 041307 (2010).
  * Collins  _et al._ [2018] J. L. Collins, A. Tadich, W. Wu, L. C. Gomes, J. N. Rodrigues, C. Liu, J. Hellerstedt, H. Ryu, S. Tang, S.-K. Mo, _et al._ , Nature 564, 390 (2018).
  * Xu and Jin [2017] Y. Xu and G. Jin, Phys. Rev. B 95, 155425 (2017).
  * Zhang  _et al._ [2020] C.-X. Zhang, X.-L. Lü, and H. Xie, Journal of Physics D: Applied Physics 53, 195302 (2020).
  * Ezawa [2013b] M. Ezawa, Phys. Rev. B 87, 155415 (2013b).
  * Datta [1997] S. Datta,  _Electronic transport in mesoscopic systems_ (Cambridge university press, 1997).
  * Meir and Wingreen [1992] Y. Meir and N. S. Wingreen, Physical review letters 68, 2512 (1992).
  * Haug  _et al._ [2008] H. Haug, A.-P. Jauho, _et al._ ,  _Quantum kinetics in transport and optics of semiconductors_ , Vol. 2 (Springer, 2008).
  * Bhattacharyya  _et al._ [2021] S. Bhattacharyya, G. Akhgar, M. Gebert, J. Karel, M. T. Edmonds, and M. S. Fuhrer, Advanced Materials 33, 2007795 (2021).
  * Basak  _et al._ [2021] A. Basak, P. Brahma, and B. Muralidharan, Journal of Physics D: Applied Physics 55, 075302 (2021).
  * Jana and Muralidharan [2021] K. Jana and B. Muralidharan, ArXiv: 2107.13318 (2021).
  * Datta [2018] S. Datta,  _Lessons from Nanoelectronics_ , 2nd ed. (World Scientific, 2018).
  * Kane and Mele [2005a] C. L. Kane and E. J. Mele, Physical review letters 95, 146802 (2005a).
  * Kane and Mele [2005b] C. L. Kane and E. J. Mele, Physical review letters 95, 226801 (2005b).


[1,2,11]S.Ilieva 1]Sezione INFN di Napoli,Napoli,80126,Italy 2]Università di
Napoli “Federico II”,Napoli,80126,Italy 3]Affiliated with an institute covered
by a cooperation agreement with CERN 4]Sezione INFN di
Bologna,Bologna,40127,Italy 5]Università di Bologna,Bologna,40127,Italy
6]Institute of Physics,Lausanne,1015,Switzerland 7]Physik-
Institut,Zürich,8057,Switzerland 8]Hamburg University,Hamburg,22761,Germany
9]European Organization for Nuclear Research (CERN),Geneva,1211,Switzerland
10]Laboratory of Instrumentation and Experimental Particle Physics
(LIP),Lisbon,1649-003,Portugal 11]Faculty of Physics,Sofia
University,Sofia,1164,Bulgaria 12]Università degli Studi di
Cagliari,Cagliari,09124,Italy 13]University of Leiden,Leiden,2300RA,The
Netherlands 14]Taras Shevchenko National University of Kyiv,Kyiv,01033,Ukraine
15]University College London,London,WC1E6BT,United Kingdom 16]Università di
Napoli Parthenope,Napoli,80143,Italy 17]Sungkyunkwan University,Suwon-
si,16419,Korea 18]Institut für Physik and PRISMA Cluster of
Excellence,Mainz,55099,Germany 19]Humboldt-Universität zu
Berlin,Berlin,12489,Germany 20]Università del Sannio,Benevento,82100,Italy
21]Sezione INFN di Bari,Bari,70126,Italy 22]Università di
Bari,Bari,70126,Italy 23]Middle East Technical University
(METU),Ankara,06800,Turkey 24]Università della Basilicata,Potenza,85100,Italy
25]Departamento de Física,Santiago,4860,Chili 26]Imperial College
London,London,SW72AZ,United Kingdom 27]Millennium Institute for Subatomic
physics at high energy frontier-SAPHIR,Santiago,7591538,Chile 28]Departamento
de Física,Serena,La,Chile 29]Ankara University,Ankara,06100,Turkey
30]Department of Physics Education and RINS, Gyeongsang National
University,Jinju,52828,Korea 31]Gwangju National University of
Education,Gwangju,61204,Korea 32]Nagoya University,Nagoya,464-8602,Japan
33]Center for Theoretical and Experimental Particle Physics,Santiago,Chile
34]Korea University,Seoul,02841,Korea 35]Toho University,Chiba,274-8510,Japan
36]Niels Bohr Institute,Copenhagen,2100,Denmark 37]Present address: Faculty of
Engineering,Kanagawa,221-0802,Japan 38]Constructor
University,Bremen,28759,Germany

# Measurement of the muon flux at the SND@LHC experiment

R.Albanese  0000-0003-4586-8068 A.Alexandrov  0000-0002-1813-1485 F.Alicante
0009-0003-3240-830X A.Anokhina  0000-0002-4654-4535 T.Asada
0000-0002-2482-8289 C.Battilana  0000-0002-3753-3068 A.Bay
0000-0002-4862-9399 C.Betancourt  0000-0001-9886-7427 D.Bick
0000-0001-5657-8248 R.Biswas  0009-0005-7034-6706 A.Blanco Castro
0000-0001-9827-8294 V.Boccia  0000-0003-3532-6222 M.Bogomilov
0000-0001-7738-2041 D.Bonacorsi  0000-0002-0835-9574 W.M.Bonivento
0000-0001-6764-6787 P.Bordalo  0000-0002-3651-6370 A.Boyarsky
0000-0003-0629-7119 S.Buontempo  0000-0001-9526-556X M.Campanelli
0000-0001-6746-3374 T.Camporesi  0000-0001-5066-1876 V.Canale
0000-0003-2303-9306 A.Castro  0000-0003-2527-0456 D.Centanni
0000-0001-6566-9838 F.Cerutti  0000-0002-9236-6223 M.Chernyavskiy
0000-0002-6871-5753 K.-Y.Choi  0000-0001-7604-6644 S.Cholak
0000-0001-8091-4766 F.Cindolo  0000-0002-4255-7347 M.Climescu
0009-0004-9831-4370 A.P.Conaboy  0000-0001-6099-2521 G.M.Dallavalle
0000-0002-8614-0420 D.Davino  0000-0002-7492-8173 P.T.de Bryas
0000-0002-9925-5753 G.De Lellis  0000-0001-5862-1174 M.De Magistris
0000-0003-0814-3041 A.De Roeck  0000-0002-9228-5271 A.De Rújula
0000-0002-1545-668X M.De Serio  0000-0003-4915-7933 D.De Simone
0000-0001-8180-4366 A.Di Crescenzo  0000-0003-4276-8512 R.Donà
0000-0002-2460-7515 O.Durhan  0000-0002-6097-788X F.Fabbri
0000-0002-8446-9660 F.Fedotovs  0000-0002-1714-8656 M.Ferrillo
0000-0003-1052-2198 M.Ferro-Luzzi  0009-0008-1868-2165 R.A.Fini
0000-0002-3821-3998 A.Fiorillo  0009-0007-9382-3899 R.Fresa
0000-0001-5140-0299 W.Funk  0000-0003-0422-6739 F.M.Garay Walls
0000-0002-6670-1104 A.Golovatiuk  0000-0002-7464-5675 A.Golutvin
0000-0003-2500-8247 E.Graverini  0000-0003-4647-6429 A.M.Guler
0000-0001-5692-2694 V.Guliaeva  0000-0003-3676-5040 G.J.Haefeli
0000-0002-9257-839X C.Hagner  0000-0001-6345-7022 J.C.Helo Herrera
0000-0002-5310-8598 E.van Herwijnen  0000-0001-8807-8811 P.Iengo
0000-0002-5035-1242 0000-0001-9204-2563<EMAIL_ADDRESS>A.Infantino  0000-0002-7854-3502 A.Iuliano  0000-0001-6087-9633 R.Jacobsson
0000-0003-4971-7160 C.Kamiscioglu  0000-0003-2610-6447 A.M.Kauniskangas
0000-0002-4285-8027 E.Khalikov  0000-0001-6957-6452 S.H.Kim
0000-0002-3788-9267 Y.G.Kim  0000-0003-4312-2959 G.Klioutchnikov
0009-0002-5159-4649 M.Komatsu  0000-0002-6423-707X N.Konovalova
0000-0001-7916-9105 S.Kovalenko  0000-0002-8518-2282 S.Kuleshov
0000-0002-3065-326X H.M.Lacker  0000-0002-7183-8607 O.Lantwin
0000-0003-2384-5973 F.Lasagni Manghi  0000-0001-6068-4473 A.Lauria
0000-0002-9020-9718 K.Y.Lee  0000-0001-8613-7451 K.S.Lee  0000-0002-3680-7039
S.Lo Meo  0000-0003-3249-9208 V.P.Loschiavo  0000-0001-5757-8274 S.Marcellini
0000-0002-1233-8100 A.Margiotta  0000-0001-6929-5386 A.Mascellani
0000-0001-6362-5356 A.Miano  0000-0001-6638-1983 A.Mikulenko
0000-0001-9601-5781 M.C.Montesi  0000-0001-6173-0945 F.L.Navarria
0000-0001-7961-4889 S.Ogawa  0000-0002-7310-5079 N.Okateva
0000-0001-8557-6612 M.Ovchynnikov  0000-0001-7002-5201 G.Paggi
0009-0005-7331-1488 B.D.Park  0000-0002-3372-6292 A.Pastore
0000-0002-5024-3495 A.Perrotta  0000-0002-7996-7139 D.Podgrudkov
0000-0002-0773-8185 N.Polukhina  0000-0001-5942-1772 A.Prota
0000-0003-3820-663X A.Quercia  0000-0001-7546-0456 S.Ramos
0000-0001-8946-2268 A.Reghunath  0009-0003-7438-7674 T.Roganova
0000-0002-6645-7543 F.Ronchetti  0000-0003-3438-9774 T.Rovelli
0000-0002-9746-4842 O.Ruchayskiy  0000-0001-8073-3068 T.Ruf
0000-0002-8657-3576 M.Sabate Gilarte  0000-0003-1026-3210 Z.Sadykov
0000-0001-7527-8945 M.Samoilov  0009-0008-0228-4293 V.Scalera
0000-0003-4215-211X W.Schmidt-Parzefall  0000-0002-0996-1508 O.Schneider
0000-0002-6014-7552 G.Sekhniaidze  0000-0002-4116-5309 N.Serra
0000-0002-5033-0580 M.Shaposhnikov  0000-0001-7930-4565 V.Shevchenko
0000-0003-3171-9125 T.Shchedrina  0000-0003-1986-4143 L.Shchutska
0000-0003-0700-5448 H.Shibuya  0000-0002-0197-6270 S.Simone
0000-0003-3631-8398 G.P.Siroli  0000-0002-3528-4125 G.Sirri
0000-0003-2626-2853 G.Soares  0009-0008-1827-7776 J.Y.Sohn
0009-0000-7101-2816 O.J.Soto Sandoval  0000-0002-8613-0310 M.Spurio
0000-0002-8698-3655 N.Starkov  0000-0001-5735-2451 I.Timiryasov
0000-0001-9547-1347 V.Tioukov  0000-0001-5981-5296 F.Tramontano
0000-0002-3629-7964 C.Trippl  0000-0003-3664-1240 E.Ursov  0000-0002-6519-4526
A.Ustyuzhanin  0000-0001-7865-2357 G.Vankova-Kirilova  0000-0002-1205-7835
V.Verguilov  0000-0001-7911-1093 N.Viegas Guerreiro Leonardo
0000-0002-9746-4594 C.Vilela  0000-0002-2088-0346 C.Visone
0000-0001-8761-4192 R.Wanke  0000-0002-3636-360X E.Yaman  0009-0009-3732-4416
C.Yazici  0009-0004-4564-8713 C.S.Yoon  0000-0001-6066-8094 E.Zaffaroni
0000-0003-1714-9218 J.Zamora Saa  0000-0002-5030-7516 [  [  [  [  [  [  [  [
[  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [
[  [  [  [

###### Abstract

The Scattering and Neutrino Detector at the LHC (SND@LHC) is a compact
standalone experiment, which started taking data at the beginning of Run 3 of
the LHC. The experiment is designed to perform measurements with neutrinos
produced in proton-proton collisions at the LHC in an energy range between 100
GeV and 1 TeV, and a previously unexplored pseudo-rapidity range of
$7.2<\eta<8.4$. The detector, located 480 m downstream of the ATLAS
interaction point in the TI18 tunnel, comprises a veto system followed by an
830 kg target mass consisting of tungsten plates, interleaved with nuclear
emulsion plates and Scintillating Fiber (SciFi) electronic trackers, and then
a calorimeter and a downstream muon system (DS). The measurement of the muon
flux in the TI18 tunnel using the nuclear emulsion tracker and the electronic
detectors, SciFi tracker and downstream muon detector, is reported here. The
muon flux per integrated luminosity through an 18$\times$18 cm2 area in the
emulsions is $1.5\pm 0.1(\textrm{stat})\times 10^{4}\,\textrm{fb/cm}^{2}$. The
measured muon flux per integrated luminosity through a 31$\times$31 cm2
central SciFi area is

$2.06\pm 0.01(\textrm{stat})\pm 0.12(\textrm{sys})\times
10^{4}\,\textrm{fb/cm}^{2}$,

while for the downstream muon system the flux is

$2.35\pm 0.01(\textrm{stat})\pm 0.10(\textrm{sys})\times
10^{4}\,\textrm{fb/cm}^{2}$

for a 52$\times$52 cm2 central detector region. The total relative uncertainty
of the electronic detectors measurements is 6 $\%$ for the SciFi and 4 $\%$
for the DS measurement. The Monte Carlo simulation prediction of these fluxes
is 20-25 $\%$ lower than the measured values.

###### keywords:

muon flux, SND@LHC, LHC

##  1 Introduction

The SND@LHC detector [1] is designed to perform measurements with high energy
neutrinos (100 GeV–1 TeV) produced in proton-proton collisions at the LHC in
the forward pseudo-rapidity region $7.2<\eta<8.4$. It is a compact, stand-
alone experiment located 480 m away from the ATLAS interaction point, (IP1) in
the TI18 tunnel, where it is shielded from collision debris by around 100 m of
rock and concrete. The signal events for the experiment are neutrino
interactions and searches for dark matter scatterings. The majority of
recorded events however consist of muons arriving from the particles produced
in proton-proton collisions at IP1. Most of these muons are vetoed when
entering the detector. Nevertheless, in rare cases muons can either enter the
detector without being vetoed, generating showers via bremsstrahlung or deep
inelastic scattering, or interact in the surrounding material, producing
neutral hadrons entering the apparatus and possibly mimicking neutrino
interactions. Since muons reaching the detector location are the main source
of background for the neutrino search, a dedicated muon flux measurement at
the SND@LHC detector location was performed.

##  2 Detector

The SND@LHC detector consists of a hybrid system with a $\sim 830$ kg target
made of tungsten plates interleaved with nuclear emulsions and electronic
trackers, followed by a hadronic calorimeter and a muon system (see Figure 1).
The electronic detectors provide the time stamp of the neutrino interaction,
preselect the interaction region, tag muons and measure the electromagnetic
and hadronic energy, while the emulsion detectors provide excellent vertex
reconstruction. A veto system is located upstream of the target region and is
used to tag muons and other charged particles entering the detector from the
IP1 direction.

The veto system comprises two parallel planes of scintillating bars read out
on both ends by silicon photomultipliers (SiPMs). Each plane consists of seven
1 $\times$ 6 $\times$ 42 cm3 vertically stacked bars of plastic scintillator.

The target section contains five walls. Each wall consists of four units of
emulsion cloud chambers (ECC) [2] and is followed by a scintillating fiber
(SciFi) [3] station for tracking and electromagnetic calorimetry. More
practical details regarding the layout of the emulsion detector are given in
Section 6.1.

Each ECC unit is a sequence of 60 nuclear emulsion films, with an area of 19.2
$\times$ 19.2 cm2 and approximately 300 µm thick, interleaved with 59 tungsten
plates, 1 mm thick. Each unit weighs approximately 41.5 kg, totaling about 830
kg for the whole target. The submicrometric spatial resolution of the nuclear
emulsions allows for very efficient tracking of all the charged particles
produced in high energy neutrino interactions, despite their small angular
separation due to the large boost. The efficiency for detecting tau neutrino
interactions based on a separation of more than 10 µm between the primary and
secondary vertices of the tau lepton is expected to be around 50 $\%$ [1].

Each SciFi station consists of two 39 $\times$ 39 cm2 planes, alternating in
$x$ and $y$. Each plane comprises six densely packed staggered layers of 250
µm diameter polystyrene-based scintillating fibers read out by SiPM arrays.
The single particle spatial resolution in one plane is of order of $\sim$150
µm and the time resolution for a particle crossing both $x$ and $y$ planes is
about 250 ps.

The muon system and hadronic calorimeter consists of two parts: upstream (US),
the first five stations, and downstream (DS), the last three stations (see
Figure 1). Each US station consists of 10 stacked horizontal scintillator bars
of 82.5 $\times$ 6 $\times$ 1 cm3, similar to the veto detector, resulting in
a coarse $y$ view. A DS station consists of two layers of thinner bars
measuring 82.5 $\times$ 1 $\times$ 1 cm3, arranged in alternating $x$ and $y$
planes, allowing for a spatial resolution in each axis of less than 1 cm. The
eight scintillator stations are interleaved with 20-cm-thick iron blocks. In
combination with SciFi, the muon system and hadronic calorimeter act as a
coarse sampling calorimeter ($\sim$9.5 hadronic interaction lengths
$\lambda_{\mathrm{int}}$ in the US detector), providing the energy measurement
of hadronic jets with an expected resolution around 20$\%$ for typical jets
associated to neutrino interactions [1]. The spatial resolution of the DS
detector together with the SciFi tracker allows for the identification of muon
tracks traversing the complete detector.

A right-handed coordinate system is used, with $z$ along the nominal proton-
proton collision axis and pointing away from IP1, $x$ pointing away from the
centre of the LHC, and $y$ vertically aligned and pointing upwards.

All signals exceeding preset thresholds are read out by the front-end
electronics and clustered in time to form events. A software noise filter is
applied to the events online, resulting in negligible detector deadtime and
negligible loss in signal efficiency. Events satisfying certain topological
criteria, such as the presence of hits in several detector planes, are read
out at a rate of around 5.4 kHz at the highest instantaneous luminosity
achieved in 2022 of 2.5 $\times$ 1034 cm-2 s-1.

Figure 1: Schematic layout of the SND@LHC detector. The pseudo-rapidity $\eta$
values quoted on the figure of the front detector view show the $\eta$ limits
for particles hitting the lower left and the upper right corner of the ECC.
The side view includes an illustration of a passing-through muon hitting the
veto planes, traversing the target tracker, the hadronic calorimeter and also
the downstream muon system. For a view of the $x-y$ ranges covered by the
SciFi and the muon DS system see Figures 10 and 11.

##  3 Data and Monte Carlo simulations

###  3.1 Data sample

The analyzed emulsion data covers the first months of SND@LHC detector
operation with protons in the LHC: from the 7${}^{\textrm{th}}$ May 2022 to
the 26${}^{\textrm{th}}$ July 2022. Emulsion films integrate the signals from
all particles passing through the target in the integration period. Since they
are not time-sensitive, in order to optimize their usage, only one brick out
of twenty was instrumented in the period of LHC beam commissioning and low
luminosity. Analysis of this data is presented in Section 6. The total
integrated luminosity for that time is 0.5 fb-1.

Data collected later-on during the 13.6 TeV proton run and “Stable beams” mode
of the accelerator are used for the electronic-detectors analysis. For the
full duration of the 2022 proton run, the recorded integrated luminosity for
SND@LHC is 36.8 fb-1, comprising $95\%$ of the total 38.7 fb-1 delivered
luminosity at IP1, as reported by the ATLAS collaboration [4]. At the time of
this publication, the available luminosity data for the 2023 LHC proton run
does not feature the absolute luminosity calibration obtained using the van
der Meer method [5, 6]. Based on the present information, the 2023 delivered
integrated luminosity is 31.9 fb-1, while the recorded one for SND@LHC is 31.8
fb-1. Until fully calibrated luminosity is provided, the muon flux measurement
with 2023 data is subject to relatively large uncertainties. For the presented
muon flux measurement, two SND@LHC runs from 2022 are therefore selected.
Information about them is outlined in Table 1.

Table 1: List of the two selected SND@LHC 2022 data runs. The runs are chosen to have large event counts, high delivered luminosity, isolated LHC bunches of Beam 2 passing without collisions, and different LHC filling schemes. The latter are available at the LHC Programme Coordination web page [7]. The calibrated luminosity at IP1 for the 2022 LHC run is provided by the ATLAS collaboration [4]. LHC fill number |  integrated luminosity $[$fb${}^{-1}]$ |  mean number of interactions at IP1 per bunch crossing |  SND@LHC run number |  recorded events by SND@LHC $[10^{6}]$ |  date, year 2022 |  duration $[$h$]$
---|---|---|---|---|---|---
8088 | 0.337 | 35.2 | 4705 | 71 | 3 Aug | 12.5
8297 | 0.529 | 45.4 | 5086 | 101 | 20 Oct | 19.8

The LHC filling scheme specifies which bunches cross at different interaction
points and which bunches of Beam 1 and Beam 2 are circulating in the LHC
without collision.111By convention the clock-wise circulating beam is denoted
Beam 1, while the counter clock-wise circulating one is denoted Beam 2 [8].
Independent of whether a circulating particle is in a colliding bunch or not,
it can possibly interact while passing through LHC machine elements. The
filling scheme for LHC fill 8297 is a mixed one with alternating 25 ns bunch
trains and eight bunches and four empty slots (8b4e) trains, which suppresses
the formation of electron clouds in the LHC arcs and reduces beam-gas
interactions [9]. The filling scheme for LHC fill 8088 is pure Batch
Compression Merging and Splitting (BCMS) [10], which was baseline for the 2022
LHC run.

The filling scheme information is synchronized with the SND@LHC event
timestamp, which is necessary since the detector is 480 m away from IP1. The
phase adjustments for both beams are done by finding the phase shift with
maximum overlap with SND@LHC event rates. For Beam 1, it means matching the
highest event rates to IP1 collisions. The synchronized bunch structure
provides input on whether an SND@LHC event is associated with IP1 collisions.
Moreover, it allows to unambiguously identify events originating from Beam 1
bunches which were not colliding in IP1, and Beam 2 bunches which were not
colliding in IP2. For the muon flux measurement, the contributions of Beam 1
and Beam 2 interacting with the LHC machine elements or residual gas are
subtracted from the total number of recorded muons associated with IP1
collisions. The selected runs, see Table 1, correspond to LHC fills with
comparatively large numbers of isolated non-colliding Beam 2 bunches, and also
non-colliding Beam 1 bunches, to facilitate that extraction. Further details
of this procedure are available in Reference [11].

###  3.2 Simulations

Muon production in proton-proton collisions at the LHC is simulated with the
Fluka Monte Carlo code [12, 13]. DPMJet (Dual Parton Model) [14, 15] is used
for the $pp$ event generation. The propagation of collision debris in the LHC
towards the SND@LHC detector is done using the LHC Fluka model [16, 17].
Particle transport is stopped at a virtual scoring plane of 1.8$\times$1.8 m2
area, located in the rock about 60 m upstream of SND@LHC, 419 m from IP1 [1].
The Fluka simulation is prepared by the CERN’s SY-STI team. It consists of 200
$\times$ 106 $pp$ collisions with LHC Run 3 beam conditions and a downward
crossing angle of $-$160 µrad on the vertical plane, corresponding to the
collider configuration in 2022 [18].

The propagation of muons from the scoring plane to the detector and their
interactions along the way are modelled with a Geant4 simulation [19] of
SND@LHC and its surroundings.

##  4 Track finding and fitting methods for the electronic detectors

Most muons traveling from IP1 towards the SND@LHC detector leave straight
tracks in the latter owing to a lack of magnetic field in the apparatus. Two
track-finding methods were implemented in the SND@LHC software framework. One
of them makes use of a custom track-finding solution that minimizes the
residuals between measured points and a straight-line track candidate, denoted
“simple tracking (ST)”. The other tracking approach employs the Hough
transform [20, 21] pattern recognition method and is referred to as “Hough
Transform (HT)”. In both cases, the track fitting is done using the Kalman
Filter [22, 23] method in the GENFIT package [24, 25].

The muon flux measurement in the electronic detectors is performed
independently in two detector sub-systems, the SciFi and the DS part of the
muon system. The reasons are the very different granularity of their sensitive
elements and the DS acceptance being 2.4 times larger than the SciFi one,
which is important as the muon flux increases in vertical direction, see
Section 7.

Since all SciFi planes and vertical DS planes are read out on one side, their
hits have two well defined spatial coordinates provided by the detector
geometry. Then, track building in these sub-systems is done separately in the
horizontal $x-z$ and vertical $y-z$ plane. The final three-dimensional track
is built combining the two two-dimensional tracks. Events with multiple track
candidates per projection can also be reconstructed, the procedure being
described in Reference [11].

The tracking efficiency in each detector and for each of the tracking methods
is estimated using the data and is summarized in Table 2. The tag-and-probe
technique is used. Reconstructed tracks in the SciFi and reconstructed tracks
in the DS detector are extrapolated to a reference vertical plane at a set $z$
position and the $x-y$ coordinates of their intersection points with the
reference plane are compared to one another. To determine the SciFi tracking
efficiency, reconstructed tracks in the DS system are used as tags, and for
the DS tracking efficiency, SciFi tracks serve as tags. The efficiency is
determined in 1 $\times$ 1 cm2 individual $x-y$ bins and the average of these
estimates is considered the efficiency over the full range.

Table 2: SciFi and DS tracking efficiencies for simple and Hough transform tracking procedures, see Section 4 for an explanation of the difference between ST and HT for SciFi and DS. system | tracking algorithm | tracking efficiency
---|---|---
SciFi |  simple tracking Hough transform |  $0.868\pm 0.009$ $0.956\pm 0.007$
DS |  simple tracking Hough transform |  $0.937\pm 0.007$ $0.944\pm 0.009$

##  5 Angular distribution in the electronic detectors

Most muons reaching the SND@LHC detector leave traces at small angles with
respect to the SND@LHC’s $z$ axis, see Figures 2 and 3 where the central peak
corresponds to a distant source at IP1. This peak has large tails due to
multiple scattering along the 480-m path from IP1 to the TI18 tunnel. The few
structures at negative slopes originate from beam-gas interactions in the
beamline. As shown in Figure 4, after selecting SND@LHC events in sync with
LHC bunches corresponding to non-colliding Beam 2 bunches and no present
bunches of Beam 1 (B2noB1), practically all reconstructed tracks have negative
$x-z$ slopes. Moreover, track direction studies based on detector hit timing
show that particles having reconstructed tracks in such B2noB1 events enter
the detector from the back, see Figure 5. Therefore, the origin of these
particles must be downstream of the DS stations. The reconstructed slopes
point to elements in the LHC half-cells 12-17, with angles approaching zero
for more distant sources.

Figure 2: SciFi track slopes of reconstructed tracks built using the Hough
Transform method. The slopes in the horizontal $x-z$ plane
($\tan\theta_{\mathrm{xz}}$) and in the vertical $y-z$ plane
($\tan\theta_{\mathrm{yz}}$) are derived from the differences of the track
coordinates between the first and the last track point in the detector. Data
from SND@LHC Run 4705.
Figure 3: DS track slopes of reconstructed tracks built using the Hough
Transform method. The slopes in the horizontal $x-z$ plane
($\tan\theta_{\mathrm{xz}}$) and in the vertical $y-z$ plane
($\tan\theta_{\mathrm{yz}}$) are derived from the differences of the track
coordinates between the first and the last track point in the detector. Data
from SND@LHC Run 4705. Figure 4: SciFi track slopes for data events in sync
with B2noB1 LHC bunches. Tracks are built using the Hough Transform method.
Data from SND@LHC Run 4705. Figure 5: Track separation by particle propagation
direction using simple tracking SciFi tracks. The black curve shows tracks
reconstructed in all events. The blue distribution is for tracks in events of
non-colliding Beam 1 bunches and no present bunches of Beam 2 (B1noB2). The
cyan curve is for events of non-colliding Beam 2 bunches and no present
bunches of Beam 1 (B2noB1) bunches. Tracks of particles moving from the DS
side towards the SciFi, denoted backward-going tracks, must have $1/v\,-\,1/c$
values around $-2/c$ or -0.067 (cm/ns)-1. For each track, the value $1/v$ is
obtained from a straight-line fit to a two-dimensional plot of position
difference versus timing difference between each track points and the most
upstream, first, track point in the SciFi detector. Data from SND@LHC Run
4705.

A closer look at the central region of the main peak in Figure 6 shows two
slightly shifted sub-peaks. The angular distance in the $x-z$ plane between
these sub-maxima is about 5 mrad, see Section 6.2 for the emulsion data
result. The Fluka simulations reproduce the same structure [11] and show that
muons in both sub-peaks originate from interactions of particles produced at
IP1 and particle (pion and kaon) decays at various locations. Among the more
interesting cases, a contribution comes from interactions in the area where
the clock-wise circulating LHC proton beam starts to be bent in the LHC
circular orbit. That is approximately the quadrupole magnet of half-cell 9,
which is 345 m away from IP1. Another case are interactions in the area where
the nominal IP1 collision axis intersects the ground outside the curved LHC
tunnel in the direction of the TI18 tunnel.

Figure 6: Data SciFi track slopes in the horizontal plane. The region around a
few milliradian is selected to show the two central sub-peaks of the main
peak, which are fit with two Gaussian functions. The mean and sigma values of
the combined fit are reported.

##  6 Analysis of the emulsion detector data

As discussed in Section 2, the SND@LHC detector is a hybrid system, employing
both electronic detectors and emulsions. Data is initially analysed
independently for the two components, leading to unbiased samples of
reconstructed tracks. Then, comparing the obtained distributions can further
characterize the particles traversing the detector, in majority muons from the
proton collisions.

###  6.1 Detector layout and track reconstruction

A layout of the detector configuration, referred to as “Emulsion Target 0” is
shown in Figure 7. The ECC is located in the third wall, in the position
closest to the line of sight in the transverse plane.

Figure 7: Layout of the SND@LHC emulsion target. The brick instrumented
emulsions is highlighted, showing a layout of its inner structure along its
depth.

The emulsion development was performed from August the 3rd to the 10th, then
the films were shipped to the University of Naples, where they were scanned
with automated optical microscopes, derived from the European Scanning System
[26, 27], upgraded to improve scanning speed and resolution [28, 29, 30, 31].

After the scanning, data reconstruction was performed with the FEDRA ROOT C++
libraries [32], in order to align the films in one single reference system for
the ECC brick, and to reconstruct the passing tracks in the emulsion target.
The reconstruction of the emulsion data is reported in detail in Reference
[33].

###  6.2 Angular distribution and muon flux in the emulsion tracker

The one-dimensional angular distribution of the tracks reconstructed in the
ECC brick is shown in Figure 8: the presence of a second peak in the $x-z$
component can be seen. Fitting the distribution of the angle in the $x-z$
plane with a two-Gaussian function results in a distance between the peaks of
$\sim$ 6mrad. The topology of the two peaks confirms the distribution
presented for the electronic detectors in Figure 6.

Figure 8: Angles of reconstructed tracks, in the $x-z$ plane, fitted with a
two-Gaussian function, whose mean and sigma values are reported.

The spatial density of the reconstructed tracks, after correcting for the
tracking efficiency [33], is shown in Figure 9. In the region represented
within the red border, the measured track density is $7.7\pm
0.6~{}(\textrm{stat})\times 10^{3}$ cm-2. Given the luminosity integrated in
the emulsion target during the exposure time, the track density corresponds to
a measured muon flux of $1.5\pm 0.1~{}(\textrm{stat})\times 10^{4}$ fb/cm2.

Figure 9: Position distribution of reconstructed tracks at the most upstream
film in each 1□ cell, corrected for reconstruction efficiency. The red border
represents the region considered for measuring the average density. The
coordinates on the axes are local coordinates on the face of the brick.

##  7 Muon flux in the electronic detectors

For the electronic detectors, the muon flux is determined as the number of
reconstructed tracks in a given SND@LHC run per corresponding IP1 integrated
luminosity and unit detector area. Similarly to the emulsion-data analysis, a
correction for the tracking efficiency is applied, being defined as the
inverse of the efficiency.

The muon flux in each electronic detector is estimated in a pre-selected inner
geometrical area rejecting regions of non-uniform tracking efficiency [11].
The considered area for the SciFi, 31$\times$31 cm2, ranges between
$-42\,\textrm{cm}\leq x\leq-11\,\textrm{cm}$ and $18\,\textrm{cm}\leq y\leq
49\,\textrm{cm}$, see Figure 10. For the muon DS detector, the selected area,
52$\times$52 cm2, is between $-54\,\textrm{cm}\leq x\leq-2\,\textrm{cm}$ and
$12\,\textrm{cm}\leq y\leq 64\,\textrm{cm}$, see Figure 11. The increase of
the muon flux with vertical position present in the data is also observed in
the Monte Carlo simulation [11] and is deemed to be due to the complex
magnetic field in the LHC.

Figure 10: Reconstructed SciFi tracks $x-y$ profile at the upstream detector
face. Tracks are built using the Hough Transform method. The distribution is
normalized to unit integral. Note the increase in the muon flux with vertical
$y$ position. The red border represents the region considered for the SciFi
muon flux measurement. Data from SND@LHC Run 4705.
Figure 11: Reconstructed DS tracks $x-y$ profile at the upstream detector
face. Tracks are built using the Hough Transform method. The distribution is
normalized to unit integral. Horizontal stripes of lower counts in the central
part of the detector originate in bar inefficiencies that are improved for the
2023 run. It is worth noting the increase in the muon flux with vertical $y$
position. The red border represents the region considered for the DS muon flux
measurement. Data from SND@LHC Run 4705.

The central value of the final muon flux result per electronic detector is the
average of the calculated fluxes for the two SND@LHC runs when using the Hough
transform tracking. The reason HT is selected is to maximize the tracking
efficiency (see Table 2), thus minimizing the correction. There are a few
percent variations in the results per detector when comparing estimates for
the two tracking methods (see Table 3). The simulation of particle transport
through the detector shows large-angle scattering of low-energy muons in the
heavy material between tracking stations affects the tracking methods
differently and contributes to this effect. Because of this dependence, the
choice of tracking method introduces a bias in the flux result. The latter
cannot be assessed with simulations since the energy spectrum of muons is not
perfectly identical between the data and the Monte Carlo simulation, including
particle transport through about 100 m of rock upstream of the detector. For
this reason, a systematic uncertainty per detector is estimated as the
difference of the muon fluxes obtained for tracks built with the simple
tracking and with the Hough Transform method (see the last column of Table 4).

Table 3: Muon flux for the two selected SND@LHC runs. Estimates with SciFi and DS data tracks built using simple tracking and the Hough transform are reported separately. The statistical uncertainty for each case is quoted. The detector regions are defined in the text. system | tracking algorithm | muon flux [104 fb/cm2]
---|---|---
run 4705, LHC fill 8088
SciFi |  simple tracking Hough transform |  $2.20\pm 0.01$ $2.05\pm 0.01$
DS |  simple tracking Hough transform |  $2.43\pm 0.01$ $2.38\pm 0.01$
run 5086, LHC fill 8297
SciFi |  simple tracking Hough transform |  $2.20\pm 0.01$ $2.06\pm 0.01$
DS |  simple tracking Hough transform |  $2.41\pm 0.01$ $2.34\pm 0.01$

Additionally, since the tracking efficiency directly enters the muon flux
estimate, a systematic uncertainty is attributed to its assessment. The
uncertainty of the efficiency evaluation, see Section 4, is taken as triple
the standard deviation of tracking efficiency values over 1$\times$1
cm${}^{2}~{}x-y$ detector coordinate bins, see Table 2. The tracking
efficiency uncertainty is propagated to the muon flux result.

The third source of systematic uncertainty to the flux measurement is the
integrated luminosity for an LHC fill, whose value is used to normalize the
muon flux. The ATLAS collaboration reports a 2.2$\%$ uncertainty in the
integrated luminosity for data recorded in 2022 [4]. This uncertainty is
propagated to the flux result.

A breakdown of the systematic uncertainties per source is reported in Table 4
for the SciFi and the downstream muon system DS. For the SciFi the dominant
source of uncertainty is the choice of tracking method, while for the DS muon
detector it is the tracking efficiency. The total systematic uncertainty is
the quadrature sum of the uncertainties for all sources.

Table 4: Relative magnitude of the considered systematic uncertainties for the SND@LHC muon flux measurement. The total uncertainty derives from the uncertainty of the luminosity measurement, the uncertainty related to slight fluctuations of the tracking efficiency in different $x-y$ detector regions, and the dependence of the tracking performance on the incident muon energy. system |  luminosity uncertainty $[\%]$ |  tracking efficiency $[\%]$ |  choice of tracking method $[\%]$
---|---|---|---
SciFi | 2.2 | 2.2 | 4.8
DS | 2.2 | 2.9 | 2.0

The final SciFi and DS muon flux per integrated luminosity and in the areas
defined in the text above are reported in Table 5, together with the
statistical and systematic uncertainties. The DS muon flux is larger than the
SciFi one, given the non-uniform flux in the vertical direction, see Figures
10 and 11, and the difference in the system’s geometrical acceptance. The
total relative uncertainty of the flux results is 6 $\%$ for the SciFi
measurement and 4 $\%$ for the DS one.

A separate muon flux estimate taking into account the same geometrical $x-y$
regions in both detectors, the one defined for SciFi, is reported in Table 6.
The relative difference between the two measurements is 2$\%$. It is smaller
than the total relative uncertainty of individual detector measurements.

The muon flux measured with the SciFi detector is higher with respect to the
muon flux measured in the analyzed ECC brick, see Section 6.2, due to the
vertical gradient of the flux. In order to perform a reliable comparison, the
data from the SciFi electronic detector in the same region in the transverse
plane of the ECC has been selected. The resulting muon flux is $1.6\pm
0.01(\textrm{stat})\pm 0.10(\textrm{sys})$ fb/cm2, consistent with the
measurement in emulsions.

Table 5: Muon flux in the SciFi and the DS detectors using the Hough Transform method. Statistical and systematic uncertainties are reported. The considered systematic uncertainty sources are the uncertainty of the luminosity measurement, the slight fluctuations of the tracking efficiency in different $x-y$ detector regions, and the dependence of the tracking performance on the incident muon energy. The detector regions are defined in the text. system |  |  | muon flux [104 fb/cm2] |
---|---|---|---|---
SciFi |  |  | $2.06\pm 0.01(\textrm{stat.})\pm 0.12(\textrm{sys.})$ |
DS |  |  | $2.35\pm 0.01(\textrm{stat.})\pm 0.10(\textrm{sys.})$ |
Table 6: Muon flux in the SciFi and the DS detectors using the Hough Transform method and considering identical detector area in range $-42\,\textrm{cm}\leq x\leq-11\,\textrm{cm}$ and $18\,\textrm{cm}\leq y\leq 49\,\textrm{cm}$. system |  |  |  muon flux [104 fb/cm2] same fiducial area |
---|---|---|---|---
SciFi |  |  | $2.06\pm 0.01(\textrm{stat.})\pm 0.12(\textrm{sys.})$ |
DS |  |  | $2.02\pm 0.01(\textrm{stat.})\pm 0.08(\textrm{sys.})$ |

###  7.1 Monte Carlo simulation expectation

In Table 7 data and simulation electronic detectors flux estimates are
compared in two cases. In the first one the efficiency corrected muon flux
estimated from the data is compared to the MC generated flux of muons which
should leave tracks in the detector. In the second case, the data muon flux
without tracking efficiency correction is compared to the MC simulation muon
flux, estimated using reconstructed MC tracks. For the latter, the Hough
transform tracking is used. In all cases, the muon flux estimates for data are
between 20–25 $\%$ larger than those for the simulations.

For the systematic uncertainty of the MC simulation flux prediction,
contributions related to the Geant4 particle propagation simulation stage are
considered. Assessment of the Fluka sample uncertainty is not included. The
assessed systematic uncertainty factors are the composition of the rock
between the scoring plane and the TI18 tunnel and the precision of the
relative alignment between the scoring plane and the SND@LHC apparatus. The
relative systematic uncertainty for the predicted flux is about 10$\%$.

Table 7: Data/Monte Carlo simulation muon flux comparisons for SciFi and DS selected fiducial ranges. The statistical and systematic uncertainties for each case are quoted. The difference between data and simulations and its total uncertainty are given in the last column. Tracks are reconstructed using the Hough Transform method. The selected detector regions are $-42\,\textrm{cm}\leq x\leq-11\,\textrm{cm}$ and $18\,\textrm{cm}\leq y\leq 49\,\textrm{cm}$ for SciFi and $-54\,\textrm{cm}\leq x\leq-2\,\textrm{cm}$ and $12\,\textrm{cm}\leq y\leq 64\,\textrm{cm}$ for the muon DS system. system | sample |  muon flux $[10^{4}$ fb/cm${}^{2}]$ |  1$-$$\frac{\textrm{sim}}{\textrm{data}}$ $[\%$]
---|---|---|---
on the level of eff. corrected data and generator MC flux
SciFi |  data sim |  $2.06\pm 0.01(\textrm{stat.})\pm 0.12(\textrm{sys.})$ $1.60\pm 0.05(\textrm{stat.})\pm 0.19(\textrm{sys.})$ | 22 $\pm$ 9
DS |  data sim |  $2.35\pm 0.01(\textrm{stat.})\pm 0.10(\textrm{sys.})$ $1.79\pm 0.03(\textrm{stat.})\pm 0.15(\textrm{sys.})$ | 24 $\pm$ 9
on the level of reconstructed tracks, no eff. corrections
SciFi |  data sim |  $1.97\pm 0.01(\textrm{stat.})\pm 0.07(\textrm{sys.})$ $1.52\pm 0.05(\textrm{stat.})\pm 0.17(\textrm{sys.})$ | 23 $\pm$ 8
DS |  data sim |  $2.22\pm 0.01(\textrm{stat.})\pm 0.07(\textrm{sys.})$ $1.75\pm 0.03(\textrm{stat.})\pm 0.16(\textrm{sys.})$ | 21 $\pm$ 10

##  8 Conclusion

The muon flux in the SND@LHC detector location is measured using the three
independent tracking detectors: the emulsion cloud chamber(ECC), the SciFi
tracker and the downstream (DS) muon detector. The muon flux per integrated
luminosity through an 18$\times$18 cm2 area of one ECC brick is $1.5\pm
0.1(\textrm{stat})\times 10^{4}\,\textrm{fb/cm}^{2}$. The measured muon flux
using the SciFi and a 31$\times$31 cm2 area between $-42\,\textrm{cm}\leq
x\leq-11\,\textrm{cm}$ and $18\,\textrm{cm}\leq y\leq 49\,\textrm{cm}$ is
$2.06\pm 0.01(\textrm{stat})\pm 0.12(\textrm{sys})\times
10^{4}\,\textrm{fb/cm}^{2}$. With the DS muon detector and a 52$\times$52 cm2
area, the muon flux is $2.35\pm 0.01(\textrm{stat})\pm
0.10(\textrm{sys})\times 10^{4}\,\textrm{fb/cm}^{2}$. The difference between
the estimates is due to a vertical gradient of the flux and the different
considered detector regions. The total relative uncertainty of the electronic
detectors results is 6 $\%$ for the SciFi and 4 $\%$ for the muon DS
measurement. When considering the same geometrical area for the SciFi and the
ECC brick, or for the SciFi and the DS detector, the measured muon fluxes are
in excellent agreement, given their uncertainties.

The analyzed data samples cover the 2022 LHC proton run and the measurements
are relevant for the downward beam-crossing configuration at $-$160 µrad. For
different LHC settings, the muon flux is expected to change and is subject to
a new evaluation. Meanwhile, efforts to better understand the origin of the
differences between data and simulation are ongoing.

##  9 Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staffs at CERN and at other SND@LHC institutes for their
contributions to the success of the SND@LHC effort. We acknowledge and express
gratitude to our colleagues in the CERN SY-STI team for the fruitful
discussions regarding beam losses during LHC operations. In addition, we
acknowledge the support for the construction and operation of the SND@LHC
detector provided by the following funding agencies: CERN; the Bulgarian
Ministry of Education and Science within the National Roadmap for Research
Infrastructures 2020–2027 (object CERN); ANID—Millennium
Program—$\rm{ICN}2019\\_044$ (Chile); the Deutsche Forschungsgemeinschaft
(DFG, ID 496466340); the Italian National Institute for Nuclear Physics
(INFN); JSPS, MEXT, the Global COE program of Nagoya University, the Promotion
and Mutual Aid Corporation for Private Schools of Japan for Japan; the
National Research Foundation of Korea with grant numbers 2021R1A2C2011003,
2020R1A2C1099546, 2021R1F1A1061717, and 2022R1A2C100505; Fundação para a
Ciência e a Tecnologia, FCT (Portugal), CERN/FIS-INS/0028/2021; the Swiss
National Science Foundation (SNSF); TENMAK for Turkey (Grant No.
2022TENMAK(CERN) A5.H3.F2-1). M. Climesu, H. Lacker and R. Wanke are funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project
496466340. We acknowledge the funding of individuals by Fundação para a
Ciência e a Tecnologia, FCT (Portugal) with grant numbers CEECIND/01334/2018,
CEECINST/00032/2021 and PRT/BD/153351/2021. We thank Luis Lopes, Jakob Paul
Schmidt and Maik Daniels for their help during the construction.

## References

  *   * Ahdida et al. [2021] Ahdida, C., et al.: SND@LHC - Scattering and Neutrino Detector at the LHC. https://cds.cern.ch/record/2750060, Geneva (2021)
  * Ariga et al. [2020] Ariga, A., Ariga, T., De Lellis, G., Ereditato, A., Niwa, K.: Nuclear emulsions. In: Fabjan, C.W., Schopper, H. (eds.) Particle Physics Reference Library, vol. 2: Detectors for Particles and Radiation, pp. 383–438. Elsevier, New York (2020)
  * LHCb Collaboration [2014] LHCb Collaboration: LHCb Tracker Upgrade Technical Design Report. http://cds.cern.ch/record/1647400 (2014)
  * [4] Preliminary analysis of the luminosity calibration of the ATLAS 13.6 TeV data recorded in 2022. Technical report, CERN, Geneva (2023). http://cds.cern.ch/record/2853525
  * van der Meer [1968] van der Meer, S.: Calibration of the effective beam height in the ISR. Technical report, CERN, Geneva (1968). https://cds.cern.ch/record/296752
  * Grafström and Kozanecki [2015] Grafström, P., Kozanecki, W.: Luminosity determination at proton colliders. Progr. Part. Nucl. Physics 81, 97–148 (2015) https://doi.org/10.1016/j.ppnp.2014.11.002
  * [7] LHC Programme Coordination. https://lpc.web.cern.ch/. Accessed September 2023
  * Brüning et al. [2004] Brüning, O.S., Collier, P., Lebrun, P., Myers, S., Ostojic, R., Poole, J., Proudlock, P.: LHC Design Report. CERN Yellow Reports: Monographs. CERN, Geneva (2004). https://doi.org/10.5170/CERN-2004-003-V-1 . https://cds.cern.ch/record/782076
  * Iadarola et al. [2017] Iadarola, G., Bartosik, H., Belli, E., Carver, L.R., Dijkstal, P., Li, K.S.B., Mether, L., Romano, A., Rumolo, G.: MD421: Electron cloud studies on 25 ns beam variants (BCMS, 8b+4e). https://cds.cern.ch/record/2260998 (2017)
  * Damerau et al. [2013] Damerau, H., Findlay, A., Gilardoni, S., Hancock, S.: RF Manipulations for Higher Brightness LHC-Type Beams (2013). https://cds.cern.ch/record/1595719
  * Ilieva [2023] Ilieva, S.I.: Measurement of the muon flux at SND@LHC (2023). http://cds.cern.ch/record/2859193
  * Battistoni et al. [2015] Battistoni, G., et al.: Overview of the FLUKA code. Annals Nucl. Energy 82, 10–18 (2015) https://doi.org/10.1016/j.anucene.2014.11.007
  * Ahdida et al. [2022] Ahdida, C., et al.: New Capabilities of the FLUKA Multi-Purpose Code. Front. in Phys. 9, 788253 (2022) https://doi.org/10.3389/fphy.2021.788253
  * Roesler et al. [2000] Roesler, S., Engel, R., Ranft, J.: The Monte Carlo event generator DPMJET-III. In: International Conference on Advanced Monte Carlo for Radiation Physics, Particle Transport Simulation and Applications (MC 2000), pp. 1033–1038 (2000). https://doi.org/10.1007/978-3-642-18211-2_166
  * Fedynitch [2015] Fedynitch, A.: Cascade equations and hadronic interactions at very high energies. PhD thesis, KIT, Karlsruhe, Dept. Phys. (November 2015). https://doi.org/10.5445/IR/1000055433
  * Lechner et al. [2019] Lechner, A., et al.: Validation of energy deposition simulations for proton and heavy ion losses in the CERN Large Hadron Collider. Phys. Rev. Accel. Beams 22(7), 071003 (2019) https://doi.org/10.1103/PhysRevAccelBeams.22.071003
  * Prelipcean et al. [2022] Prelipcean, D., Biłko, K., Cerutti, F., Ciccotelli, A., Di Francesca, D., García Alía, R., Humann, B., Lerner, G., Ricci, D., Sabaté-Gilarte, M.: Comparison Between Run 2 TID Measurements and FLUKA Simulations in the CERN LHC Tunnel of the Atlas Insertion Region. JACoW IPAC 2022, 732–735 (2022) https://doi.org/10.18429/JACoW-IPAC2022-MOPOMS042 . http://cds.cern.ch/record/2839993
  * Centanni [2023] Centanni, D.: Muon Background simulation at SND@LHC (2023). https://cds.cern.ch/record/2868911
  * Agostinelli et al. [2003] Agostinelli, S., et al.: GEANT4–a simulation toolkit. Nucl. Instrum. Meth. A 506, 250–303 (2003) https://doi.org/10.1016/S0168-9002(03)01368-8
  * Hough [1962] Hough, P.V.C.: Method and means for recognizing complex patterns. U.S. Patent 30696541962 (1962)
  * Leavers [1992] Leavers, V.F.: Computer Vision: Shape Detection, pp. 1–18. Springer, London (1992). https://doi.org/10.1007/978-1-4471-1940-1_1 . https://doi.org/10.1007/978-1-4471-1940-1_1
  * Kalman [1960] Kalman, R.E.: A New Approach to Linear Filtering and Prediction Problems. J. Basic Eng. 82(1), 35–45 (1960) https://doi.org/10.1115/1.3662552
  * Fruhwirth [1987] Fruhwirth, R.: Application of Kalman filtering to track and vertex fitting. Nucl. Instrum. Meth. A 262, 444–450 (1987) https://doi.org/10.1016/0168-9002(87)90887-4
  * Rauch and Schlüter [2015] Rauch, J., Schlüter, T.: GENFIT — a Generic Track-Fitting Toolkit. J. Phys. Conf. Ser. 608(1), 012042 (2015) https://doi.org/10.1088/1742-6596/608/1/012042 arXiv:1410.3698 [physics.ins-det]
  * Höppner et al. [2010] Höppner, C., Neubert, S., Ketzer, B., Paul, S.: A novel generic framework for track fitting in complex detector systems. Nucl. Instrum. Meth. A 620(2), 518–525 (2010) https://doi.org/10.1016/j.nima.2010.03.136
  * Arrabito et al. [2006] Arrabito, L., et al.: Hardware performance of a scanning system for high speed analysis of nuclear emulsions. Nucl. Instrum. Meth. A 568, 578–587 (2006) https://doi.org/10.1016/j.nima.2006.06.072 arXiv:physics/0604043
  * Armenise et al. [2005] Armenise, N., et al.: High-speed particle tracking in nuclear emulsion by last-generation automatic microscopes. Nucl. Instrum. Meth. A 551, 261–270 (2005) https://doi.org/10.1016/j.nima.2005.06.072
  * Alexandrov et al. [2015] Alexandrov, A., et al.: A new fast scanning system for the measurement of large angle tracks in nuclear emulsions. JINST 10(11), 11006 (2015) https://doi.org/10.1088/1748-0221/10/11/P11006
  * Alexandrov et al. [2016] Alexandrov, A., et al.: A new generation scanning system for the high-speed analysis of nuclear emulsions. JINST 11(06), 06002 (2016) https://doi.org/10.1088/1748-0221/11/06/P06002
  * Alexandrov et al. [2017] Alexandrov, A., et al.: The Continuous Motion Technique for a New Generation of Scanning Systems. Sci. Rep. 7(1), 7310 (2017) https://doi.org/10.1038/s41598-017-07869-3
  * Alexandrov et al. [2019] Alexandrov, A., De Lellis, G., Tioukov, V.: A Novel Optical Scanning Technique with an Inclined Focusing Plane. Sci. Rep. 9(1), 2870 (2019) https://doi.org/10.1038/s41598-019-39415-8
  * Tyukov et al. [2006] Tyukov, V., Kreslo, I., Petukhov, Y., Sirri, G.: The FEDRA Framework for emulsion data reconstruction and analysis in the OPERA experiment. Nucl. Instrum. Meth. A 559, 103–105 (2006) https://doi.org/10.1016/j.nima.2005.11.214
  * Iuliano [2023] Iuliano, A.: Measurement of the muon flux with the emulsion target at SND@LHC (2023). https://cds.cern.ch/record/2868917