# Optimizing Topological Switching in Confined 2D-Xene Nanoribbons via Finite-
Size Effects

Muhammad Nadeem<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>ARC Centre of Excellence in Future Low-Energy Electronics
Technologies (FLEET), University of Wollongong, Wollongong, New South Wales
2525, Australia  Institute for Superconducting and Electronic Materials
(ISEM), Australian Institute for Innovative Materials (AIIM), University of
Wollongong, Wollongong, New South Wales 2525, Australia.  Chao Zhang  School
of Physics, University of Wollongong, Wollongong, NSW 2522, Australia
Dimitrie Culcer  ARC Centre of Excellence in Future Low-Energy Electronics
Technologies (FLEET), University of New South Wales, Sydney 2052, Australia.
School of Physics, University of New South Wales, Sydney 2052, Australia.
Alex R. Hamilton  ARC Centre of Excellence in Future Low-Energy Electronics
Technologies (FLEET), University of New South Wales, Sydney 2052, Australia.
School of Physics, University of New South Wales, Sydney 2052, Australia.
Michael S. Fuhrer  ARC Centre of Excellence in Future Low-Energy Electronics
Technologies (FLEET), Monash University, Clayton, Victoria 3800, Australia.
School of Physics and Astronomy, Monash University, Clayton, Victoria 3800,
Australia.  Xiaolin Wang  ARC Centre of Excellence in Future Low-Energy
Electronics Technologies (FLEET), University of Wollongong, Wollongong, New
South Wales 2525, Australia  Institute for Superconducting and Electronic
Materials (ISEM), Australian Institute for Innovative Materials (AIIM),
University of Wollongong, Wollongong, New South Wales 2525, Australia.

###### Abstract

In a blueprint for topological electronics, edge state transport in a
topological insulator material can be controlled by employing a gate-induced
topological quantum phase transition. Here, by studying the width dependence
of electronic properties, it is inferred that zigzag-Xene nanoribbons are
promising materials for topological electronics with a display of unique
physical characteristics associated with the intrinsic band topology and the
finite-size effects on gate-induced topological switching. First, due to
intertwining with intrinsic band topology-driven energy-zero modes in the
pristine case, spin-filtered chiral edge states in zigzag-Xene nanoribbons
remain gapless and protected against backward scattering even with finite
inter-edge overlapping in ultra-narrow ribbons, i.e., a 2D quantum spin Hall
material turns into a 1D topological metal. Second, mainly due to width- and
momentum-dependent tunability of the gate-induced inter-edge coupling, the
threshold-voltage required for switching between gapless and gapped edge
states reduces as the width decreases, without any fundamental lower bound.
Third, when the width of zigzag-Xene nanoribbons is smaller than a critical
limit, topological switching between edge states can be attained without bulk
bandgap closing and reopening. This is primarily due to the quantum
confinement effect on the bulk band spectrum which increases the nontrivial
bulk bandgap with decrease in width. The existence of such protected gapless
edge states and reduction in threshold-voltage accompanied by enhancement in
the bulk bandgap overturns the general wisdom of utilizing narrow-gap and wide
channel materials for reducing the threshold-voltage in a standard field
effect transistor analysis and paves the way toward low-voltage topological
devices.

††preprint: AIP/123-QED

##  I Introduction

Two-dimensional topological insulators are promising materials for topological
quantum electronic devices where edge state transport can be controlled by a
gate-induced electric field Wray (2012); Seidel (2019). In general, edge state
transport can be controlled either by a perpendicular electric field, which
drives a topological phase transition via bulk bandgap closing and reopening
Ezawa (2013a); Liu  _et al._ (2014, 2015); Pan  _et al._ (2015); Qian  _et
al._ (2014); Zhang  _et al._ (2017); Molle  _et al._ (2017); Collins  _et al._
(2018); Nadeem  _et al._ (2021) or via inter-edge tunneling between gapped
edge states, assisted by a transverse electric field Xu  _et al._ (2019). In
the latter case, though a very weak transverse electric field is sufficient to
induce inter-edge tunneling, edge state conductance quantization may remain a
challenge constraining the geometry of topological insulator ribbons Xu  _et
al._ (2019). In the former case, a blueprint for topological quantum
electronics, the strength of the critical electric field required for
topological switching depends upon the strength of quantum mechanical
perturbations, such as the spin-orbit interaction (SOI) Kane and Mele (2005a,
b) and Bernevig–Hughes–Zhang (BHZ) mass term Bernevig, Hughes, and Zhang
(2006); König  _et al._ (2008), which reflect the bulk band topology and
therefore lead to a quantized edge state conductance. In this class, numerous
theoretical proposals for 2D topological insulator materials, which exhibit
electrically-driven topological phase transition, such as staggered sublattice
potentials Ezawa (2013a); Molle  _et al._ (2017); Nadeem  _et al._ (2021),
mirror symmetry breaking Liu  _et al._ (2014), and the Stark effect Qian  _et
al._ (2014); Liu  _et al._ (2015); Pan  _et al._ (2015); Zhang  _et al._
(2017); Collins  _et al._ (2018), have been put forward. Among these various
proposals, electric field switching has been demonstrated Collins  _et al._
(2018) in ultrathin (monolayer or bilayer) Na3Bi where the experimentally
reported bandgap of 360 meV significantly exceeds the thermal energy at room
temperature (25 meV) and the critical electric field is about 1.1 V nm-1.

Though a large topological bulk bandgap is desirable to enable quantum spin
Hall (QSH) phenomena at room temperature, one of the main challenges with such
blueprint topological switching mechanism is the requirement of
unrealistically large critical electric field to close the topological bandgap
Molle  _et al._ (2017); Vandenberghe and Fischetti (2017). For instance, the
critical electric field is of the order of 0.05 V nm-1 for silicene Molle  _et
al._ (2017), 1.0 V nm-1 for stanene Molle  _et al._ (2017), and 1.42 V nm-1
for 1T′-MoS2 Qian  _et al._ (2014). The critical electric field further
increases for heavy elemental topological insulators such as Bi/SiC Reis  _et
al._ (2017) where the experimentally reported bandgap is 800 meV. In light of
this, only recently, it is shown that the critical electric field can be
significantly reduced via the tunable Rashba effect in 2D-Xenes (G, Si, Ge,
Sn, and P, As, Sb, Bi) Nadeem  _et al._ (2021). Though the Rahsba effect is
considerably large in heavy elemental 2D-Xenes such as functionalized
bismuthene, the Rashba effect remains negligibly small for relatively lighter
group-IV (Si, Ge) and group-V (P, As) elemental 2D-Xenes Nadeem  _et al._
(2021).

Here we note that, apart from the relativistic quantum mechanical phenomenon
of SOI which plays a central role in characterizing the band topology and
limiting the critical electric field, the finite-size geometry incorporates
two additional critical phenomena: quantum confinement effects on the bulk
subbands and inter-edge coupling between spin-filtered chiral edge states. For
the study of fundamental phenomena in both laboratory and device applications,
it is crucial to investigate the fundamental topological features and the edge
state transport in the finite-size geometry of a topological insulator.
Finite-size effects have been studied for various topological insulator
materials via the thickness dependence of surface electronic dispersion in
thin films of 3D topological insulators Shan, Lu, and Shen (2010); Liu  _et
al._ (2010); Lu  _et al._ (2010) and Dirac semimetals Pan  _et al._ (2015);
Collins  _et al._ (2018) and the width dependence of edge state electronic
dispersion in 2D topological insulator materials such as HgTe Zhou  _et al._
(2008), transition metal dichalcogenides (TMDC) with 1T′ phase, TMDC-1T′ Das,
Sen, and Mahapatra (2020), and 2D-Xenes Ezawa (2006); Han  _et al._ (2007);
Son, Cohen, and Louie (2006); Brey and Fertig (2006); Ezawa and Nagaosa
(2013); Cano-Cortés, Ortix, and van den Brink (2013). However, less attention
has been devoted to finite-size effects on the gate-induced topological
switching, a central part for the working of topological electronics devices.

By studying the width dependence of electronic properties in zigzag Xene
nanoribbons (ZXNRs), it is demonstrated that both the SOI-induced barrier in
the bulk and the gate-control of quantized conductance along the edges can be
optimized via finite-size effects. It is inferred that ZXNRs are promising
materials for topological electronics with a display of unique physical
characteristics associated with the intrinsic band topology and the finite-
size effects on gate-induced topological switching. Through tight binding
calculations of band dispersion, density of states (DOS), conductance
quantization, edge state wave functions and their width dependence, we
highlight several results that are crucial for understanding fundamental
aspects and developing novel device concepts. Our findings and the analysis
presented for ZXNRs remain valid for any 2D topological insulator material
with buckled honeycomb structure terminated on zigzag edges.

First, spin-filtered chiral edge states in ZXNRs remain gapless and protected
against backward scattering even with finite inter-edge overlapping in ultra-
narrow ribbons, i.e., a 2D quantum spin Hall (QSH) material turns into 1D
topological metal. Such robustness in ZXNRs is deeply rooted in intertwining
between SOI-induced spin-filtered chiral modes and the intrinsic band
topology-driven energy-zero modes in pristine honeycomb lattice structures.
Furthermore, the topological protection of 1D metallic modes is a consequence
of different momentum space locations for edge state crossings (at time-
reversal invariant momenta (TRIM) $k=\pi$) and anti-crossings (around valleys
k=K/K′). Here the edge state crossing point is a Dirac point formed by edge
state gapless Dirac dispersion while the edge state anti-crossing point is a
momentum space location where the edge state spectrum becomes a massive Dirac
dispersion. This is highly contrasting from other 2D topological insulator
materials with inverted band structure, in which the edge state crossing and
anti-crossing points coexist, and in which hybridization due to inter-edge
overlapping opens an energy gap and leads to a gapped edge state spectrum Zhou
_et al._ (2008). For instance, in inverted band topological insulators such as
thin films of X2Y3 [X=Bi,Sb;Y=Te,Se] Shan, Lu, and Shen (2010); Liu  _et al._
(2010); Zhang  _et al._ (2010) semiconductors, type-I HgTe/CdTe Bernevig,
Hughes, and Zhang (2006); König  _et al._ (2007) and type-II InAs/GaSb/AlSb
Liu  _et al._ (2008); Knez, Du, and Sullivan (2011); Du  _et al._ (2015)
semiconducting quantum well structures, Na3Bi thin films Pan  _et al._ (2015);
Collins  _et al._ (2018), and monolayers of TMDC with 1T′ phase Qian  _et al._
(2014); Wu  _et al._ (2018), both edge state crossing and anti-crossing points
coexist at $\Gamma$-point.

Second, the critical electric-field required for switching between gapless and
gapped edge states reduces as the width of ZXNRs decreases, without any
fundamental lower bound. We demonstrate explicitly that such size dependence
of the threshold-voltage stems from a series of non-trivial quantum mechanical
phenomena associated with the geometric structure of ZXNRs. First, the edge
state wave functions at the crossing point are independent of the edge
termination and hence remain insensitive to electric fields. On the other
hand, edge state wave functions and the gate-induced coupling between
overlapping edge states across anti-crossing points are strongly dependent on
the particular edge termination and hence can be tuned via a gate electric
field. Second, with a decrease in width, the momentum space location of the
edge state anti-crossing points moves away from the valleys toward the TRIM
($k=\pi$). Furthermore, at particular momenta around the anti-crossing points,
the magnitude of the inter-edge overlap increases with decrease in width. As a
result, gate-induced coupling between spin-filtered chiral edge states is
enhanced as the ZXNR width is reduced. It shows that finite-size effects on
the edge spectrum play a central role in optimizing the gate-controlled edge
state transport, such that size dependence of the threshold-voltage stems from
width- and momentum-dependent tunability of the gate-induced coupling between
inter-edge spin-filtered chiral states.

Third, when the width of ZXNRs is smaller than a critical limit, quantum
confinement enhances the topological bulk bandgap and, hence, the energy
spacing between the bulk subbands and the edge states, which in turn leads to
topological switching between gapless and gapped edge states without bulk
bandgap closing. Both of these finite-size phenomena, central to the control
of edge state transport, are completely missing in wide ZXNRs: there the
critical electric-field is limited by the SOI-induced barrier, and the
topological phase transition is accompanied by bulk bandgap closing and
reopening.

It is important to note that the threshold reduction could in principle be
achieved with a built-in electric field due to static charges. However, for a
fixed SOI in ZXNRs, a simple enhancement of the built-in electric field also
reduces the topological bandgap in the QSH phase, which may be detrimental to
dissipationless quantized edge state conductance due to admixing of edge modes
with bulk subbands. In this regard, the size-dependent and momentum-dependent
tunability of gate-induced inter-edge coupling is a novel mechanism that
reduces the critical gate electric field even as the topological bulk bandgap
is enhanced by quantum confinement. On the one hand, it reduces the threshold-
voltage by lowering the SOI-induced barrier in the bulk, on the other hand it
enhances the bulk bandgap even in lighter monoelemental 2D-Xenes such that the
detrimental contributions from the bulk material to the edge current are
avoided and allows safe residing of the chemical potential within this gap.

Figure 1: Optimization of electronic properties in quantum confined ZXNRs. (a)
A ZXNR with lattice parameters. (b) Width dependence of threshold-voltage and
bulk bandgap. With decrease in the width of ZXNRs, threshold-voltage decreases
while nontrivial bulk bandgap increases. (c) Density of states and topological
phase transition in an ultra-narrow ZXNR with N = 4. In the absence of gate
electric field ($\lambda_{v}=0$), finite density of states (cyan) at zero-
energy represent the presence of protected helical edge states in QSH phase.
With increasing gate electric field, ZXNR enters into normal insulating (NI)
phase (red) with gapped edge states while passing through a critical gapless
phase (grey). (d) Quantized edge state conductance and the critical gate
electric field for N = 10, 15, and 25. Here, Rashba SOI is ignored for
simplicity. When Rashba SOI is incorporated, topological quantum field effect
further reduces the threshold-voltage and bulk bandgap. Here N represents the
number of zigzag chains and simulates the width of ZXNRs as
$W_{z}=\sqrt{3}Na_{0}/2$.

These features make quantum confined spin-orbit coupled ZXNRs special for
topological quantum devices, enabling optimal gate-controlled transport
through edge state channels via finite-size effects on the electronic
properties. The existence of gapless edge states and reduction in threshold-
voltage accompanied by enhancement in the bulk bandgap overturns the general
wisdom of utilizing narrow gap and wide channel materials for reducing the
threshold-voltage in a standard field effect transistor analysis, other than
negative capacitance mechanism Salahuddin and Datta (2008). Furthermore, the
advantage of utilizing ultra-narrow ZXNRs is multi-fold: (i) the availability
of large edge state conducting modes for enhanced signal-to-noise ratio via
multiple edge state channels, (ii) optimized geometry for topological
electronic devices where an array of ZXNRs, set apart by trivial insulating
layers/wires along vertical/lateral direction, is sandwiched between top and
bottom gates separated by top and bottom dielectrics, and (iii) low-voltage
and energy-efficient switching due to compressible subthreshold swing in ZXNRs
via topological quantum field effect Nadeem  _et al._ (2021) .

##  II Finite-size effects on ZXNRs

Figure 1(a) shows a ZXNR where the primitive lattice vectors are represented
by $a_{1}=a_{0}(1,0)$ and $a_{2}=a_{0}(1/2,\sqrt{3}/2)$, $d_{z}$ represents
the buckling length, while dashed rectangle (composing dimer line of A and B
sublattice sites) represents the unit cell for ZXNR. The width of ZXNR is
represented by the length of unit cell (dimer line),
$W_{z}=\sqrt{3}N_{d}a_{0}/4$ where $N_{d}=2N$ represents the number of sites
in the dimer line and $N$ represents the number of zigzag lines. The length of
ZXNR $L_{z}=Dl_{z}$, where D represents the number of dimer lines and $l_{z}$
is the width of a dimer line, can be written as a function of the number of
sites in the zigzag line ($N_{z}$) as $L_{z}=N_{z}a_{0}/2$.

After the seminal work by Kane-Mele Kane and Mele (2005a, b) on graphene, it
has been shown that other 2D-Xene nanoribbons (Si, Ge, Sn, and P, As, Sb, Bi)
with honeycomb lattice structure are also QSH insulators Min  _et al._ (2006);
Liu, Feng, and Yao (2011); Liu, Jiang, and Yao (2011); Xu  _et al._ (2013);
Hsu  _et al._ (2015); Reis  _et al._ (2017); Li  _et al._ (2018). Among 2D
topological insulator materials, quantum spin Hall (QSH) insulators with
honeycomb lattice structure terminated on zigzag edges are a special class
where spin-filtered chiral modes are intertwined with the intrinsic band
topology of the pristine honeycomb lattice structure. In ZXNRs, the intrinsic
band topology, characterized by a non-vanishing winding number, leads to
energy-zero flat bands in the nontrivial regime of the first Brillouin zone.
The sublattice-resolved intrinsic SOI, modeled through next-nearest hopping
Kane and Mele (2005a, b), disperses these localized modes into spin-filtered
chiral edge states. To simulate the energy-zero modes in the pristine case,
spin-filtered chiral edge states in the spin-orbit coupled case, gate-induced
topological switching, and the dependence of electronic properties on the
width of ZXNRs, in general, we study the next-nearest neighbor tight-binding
model Hamiltonian Kane and Mele (2005a, b)

$H=t\sum_{\langle
ij\rangle\alpha}c_{i\alpha}^{\dagger}c_{j\alpha}+i\lambda_{so}\sum_{\langle\langle
ij\rangle\rangle\alpha\beta}v_{ij}c_{i\alpha}^{\dagger}s_{\alpha\beta}^{z}c_{j\beta}+\frac{\lambda_{v}}{2}\sum_{i\alpha}c_{i\alpha}^{\dagger}v_{i}c_{i\alpha}+i\lambda_{R}(E_{z})\sum_{\langle
ij\rangle\alpha\beta}c_{i\alpha}^{\dagger}(\textbf{s}_{\alpha\beta}\times\hat{\textbf{d}}_{ij})_{z}c_{j\beta}\;.$
(1)

where the first term is the nearest neighbor hopping generating Dirac
dispersion in the vicinity of valleys K(K′) while the second term is the
intrinsic Kane-Mele type SOI ($\lambda_{so}=\Delta_{so}/3\sqrt{3}$), which
opens nontrivial QSH bulk bandgap Kane and Mele (2005a, b) and induces
topologically protected spin-filtered chiral edge states. The third term
represents the staggered sublattice potential induced by the gate electric
field ($E_{v}=\lambda_{v}/\alpha_{v}$ where $\alpha_{v}$ is the buckling-
dependent parameter) which drives the QSH phase into a trivial insulating
phase - termed here as topological switching. The fourth term is the spin-
mixing Rashba SOI associated with the gate electric field Kane and Mele
(2005a, b); Rashba (2009), $\Delta_{R}=\alpha_{R}E_{v}$ where
$\Delta_{R}=3\lambda_{R}/2$ and $\alpha_{R}$ is a Rashba SOI parameter.

Here $c_{i\alpha}^{\dagger}(c_{i\alpha})$ is the creation (annihilation)
electron operator with spin polarization $\alpha=\uparrow,\downarrow$ on site
i, the Pauli matrix $s^{z}$ describes the electron intrinsic spin while
$s_{\alpha\beta}^{z}$ are the corresponding matrix elements describing the
spin polarization $\alpha$ and $\beta$ on sites i and j, $v_{i}=+1(-1)$ for
sublattice A (B), and $v_{ij}=\textbf{d}_{ik}\times\textbf{d}_{kj}=\pm 1$
connects sites i and j on sublattice A (B) via the unique intermediate site k
on sublattice B (A). The nearest-neighbor bond vectors $\textbf{d}_{ik}$ and
$\textbf{d}_{kj}$ connect the i (k) and k (j) sites on the A and B
sublattices.

To begin with, by numerically diagonalizing the tight binding model, we study
the finite-size effects on the pristine and spin-orbit coupled ZXNRs by
varying the width of ZXNRs. For simplicity, the width of ZXNRs
$W_{z}=\sqrt{3}Na_{0}/2$ is simulated by the number of zigzag chains (N) by
setting $a_{0}=1$. In the pristine case, as shown in figure 2 (a), the
intrinsic topology of honeycomb lattice structures leads to strongly localized
energy-zero flat edge states between valleys K and K′, $\Delta
k_{x}=K-K^{\prime}=2\pi/3a_{0}$, a nontrivial regime of the first Brillouin
zone characterized by a non-vanishing winding number. The intrinsic SOI drives
pristine ZXNRs into the QSH phase and disperses these energy-zero flat edge
states into spin-filtered chiral edge modes, as shown in figure 2 (b). Due to
the presence of both time-reversal and inversion symmetry, the edge states are
Kramers pairs, forming a fourfold degenerate Dirac point at the edge state
crossing point, TRIM, $k_{x}=\pi$. Figure 2 (c) shows that when a gate
electric field is applied, the Kramers partners split along the energy axis
while the twofold degenerate Dirac points in the spin down and spin up sectors
move toward the corners of the nontrivial regime, valleys K and K′
respectively. As a result, due to spin-valley locking, ZXNRs show a spin-
polarized semi-metallic behavior at a critical point where
$\lambda_{v}=\lambda_{v}^{c}$. When the strength of the staggered sublattice
potential exceeds a threshold limit, the Dirac points in both spin sectors are
gapped out at the anti-crossing points and the system enters the trivial
regime.

Figure 2: Finite-size effects in quantum confined ZXNRs. (a-c) Width
dependence of one-dimensional electronic band dispersion for pristine ZXNRs
hosting localized energy-zero edge states (a), spin-orbit coupled ZXNRs
hosting QSH phase (b), and the gate induced critical point in spin-orbit
coupled ZXNRs(c). In wide ZXNRs (N = 100) anti-crossing point lies at the
valley ($k_{x}=2\pi/3$) and the critical gate electric field reads
$E_{c}=2\Delta_{so}/\alpha_{v}$. In narrow quantum confined ZXNRs (N = 10, 5),
anti-crossing point moves from valley towards TRIM $k_{x}=\pi$ and the
critical gate electric field reduces from the SOI driven barrier, i.e.,
$E_{c}<2\Delta_{so}/\alpha_{v}$. Moreover, around anti-crossing points, the
energy spacing between edge states and the bulk subbands increases with
decrease in width of ZXNRs. Such an increase in the bulk bandgap shows that
topological switching is not accompanied by bulk bandgap closing and reopening
in quantum confined ZXNRs. (d) Width dependence of momentum space location of
anti-crossing point. (e,f) Density of states for ZXNRs in QSH phase with N = 5
(e) and edge state density of states for N = 1, 2, 3, 4 and 5 (f). Finite
density of state around the energy-zero level shows that edge states in the
QSH insulators with ZXNRs remain gapless even for ultra-narrow width. Here we
set $a_{0}=t=1$ and $\lambda_{R}=0$.

The bulk and edge state electronic band dispersion, obtained via numerical
diagonalization of the tight binding model, show a number of counter-intuitive
features depending upon the width of ZXNRs, which may prove to be interesting
for both fundamental and novel device applications in topological electronics.

###  II.1 From 2D QSH insulator to 1D topological metal

The trademark of spin-orbit coupled ZXNRs, i.e., spin-filtered chiral edge
states in 2D QSH sheets, remains protected even when the system becomes
effectively 1D and the QSH effect is no longer well defined. That is, as shown
in figure 2(b), (i) the spin-filtered chiral edge states remain gapless even
for ultra-narrow ribbons and (ii) the bulk bandgap increases with decrease in
width. It implies that as one makes the ZXNR narrower, it retains its
topological character, i.e., has well-defined 1D metallic modes associated
with the edges, each with spin-momentum locking, and the bulk bandgap grows.
So a narrow ZXNR remains a robust 1D topological metal, with a large energy
separation between the edge states and the bulk subbands, characterized by a
non-vanishing winding number associated with the intrinsic band topology. This
non-intuitive result in ZXNRs seems interesting and differentiates ZXNRs from
other 2D topological insulators with inverted band structure, where the effect
of quantum confinement is to push the system toward a large-gap conventional
insulator.

This observation can be understood from fundamental quantum mechanical
considerations in narrow ZXNRs: sublattice-resolved intrinsic SOI, quantum
confinement, and the longitudinal momentum-dependent inter-edge coupling.
First of all, both the nontrivial bulk bandgap and the dispersing spin-
filtered chiral edge states are indebted to the sublattice-resolved intrinsic
SOI, i.e., next-nearest hopping localizes the bulk electrons while the
electrons traversing along the edges remain effectively free. This SOI-induced
mechanism remains true, irrespective of the ZXNR width. In addition, even in
the 1D limit, as discussed below, the protection of spin-filtered chiral edge
modes is guaranteed by the vanishing inter-edge coupling at TRIM $k_{x}=\pi$.
On the other hand, the enhancement of a topological bulk bandgap is because of
the quantum confinement effect on the bulk band spectrum. As shown in figure
1(b), in the absence of gate potential, while bulk band varies as
$E_{G/B}=|2\Delta_{so}|$ in the wide ZXNRs, the bulk bandgap varies as
$E_{G/B}=|2\Delta_{so}+E_{qc}|$ in narrow ZXNRs. Here the energy gap $E_{qc}$
in the subband structure, induced by quantum confinement, is inversely
proportional to the ZXNR width Ezawa (2006); Han  _et al._ (2007); Son, Cohen,
and Louie (2006); Brey and Fertig (2006); Ezawa and Nagaosa (2013); Cano-
Cortés, Ortix, and van den Brink (2013).

###  II.2 Low-Voltage topological switching without bulk bandgap closing

While finite-size effects have been widely studied in 2D-Xenes Ezawa (2006);
Han  _et al._ (2007); Son, Cohen, and Louie (2006); Brey and Fertig (2006);
Ezawa and Nagaosa (2013); Cano-Cortés, Ortix, and van den Brink (2013),
effects of quantum confinement and momentum-dependent inter-edge overlapping
on the gate-induced topological switching in spin-orbit coupled ZXNRs have
received comparatively less attention. Similarly to the
$\mathbb{PT}$-symmetric case, $\mathbb{PT}$-symmetry breaking via gate
electric field also displays interesting features in narrow ZXNRs. As depicted
in figure 1(b) and 2(c), when the width of ZXNRs is smaller than a critical
limit ($W_{z}^{c}$), (i) the critical gate electric field required for
switching between gapless and gapped edge states decreases with decrease in
width, and (ii) topological switching between gapless and gapped edge state
spectrum is not accompanied by bulk bandgap closing and reopening.

First, with decreasing ZXNR width, the gate induced anti-crossing points in
the edge state spectrum move away from the valleys toward TRIM. Since the
momentum space location of anti-crossing points is directly associated with
the threshold-voltage, the threshold-voltage decreases as the anti-crossing
points move closer to the TRIM. For instance, in wide ZXNRs (N = 100), the
spin-filtered Dirac points are gapped out exactly at the valleys K/K′ and the
SOI-induced barrier for critical electric field reads
$E_{c}=2\Delta_{so}/\alpha_{v}$. On the other hand, in narrow quantum confined
ZXNRs (N = 10, 5), the edge states are gapped out before reaching to the
valleys K/K′. As a result, the critical electric field reduces significantly
from the SOI-driven barrier $E_{c}<2\Delta_{so}/\alpha_{v}$ with decrease in
the width. This trend suggests that the critical electric field has no lower
bound and any nonzero electric field can open an energy gap in the edge state
spectrum of ultra-narrow ZXNRs.

Second, the evolution of bulk subbands during topological switching from
gapless to gapped edge state spectrum looks quite different for wide and
narrow ribbons. In wide ZXNRs, during edge state evolution under the gate
electric field, the bulk bandgap closes at the valleys when
$E_{c}=2\Delta_{so}/\alpha_{v}$. At this point, the highest occupied molecular
orbital and lowest unoccupied molecular orbital (HOMO-LUMO), carrying the same
spin as that for edge states at particular valleys, of the bulk band spectrum
become valley degenerate with the edge states. The bulk bandgap reopens when
$E_{c}>2\Delta_{so}/\alpha_{v}$. It implies that the topological switching via
electric field is accompanied by a quantum phase transition between
$\mathbb{Z}_{2}$-nontrivial and $\mathbb{Z}_{2}$-trivial insulating phases
where the bulk bandgap closes and reopens at the valleys. On the one hand, in
narrow ZXNRs where bulk subbands and edge states are widely separated in
energy due to quantum confinement, transitioning between gapless and gapped
edge state spectrum occurs without bulk bandgap closing and reopening. The
closing and reopening of the bulk bandgap is not necessary in narrow ZXNRs, as
the 1D system is no longer a 2D topological insulator with a well-defined
$\mathbb{Z}_{2}$ index. Hence, no bandgap closing and reopening is needed to
switch the topology.

Such a finite-size-driven topological switching of edge state conductance,
without bulk bandgap closing and reopening, is an entirely different concept
from the previously studied quantum phase transition of the bulk band topology
induced by symmetry breaking Ezawa, Tanaka, and Nagaosa (2013); Yang  _et al._
(2013); Rachel (2016); Matsumoto  _et al._ (2020); Schindler (2020). Since the
symmetry class of ZXNRs remains unchanged, irrespective of width, the quantum
critical point for transitioning between $\mathbb{Z}_{2}$-nontrivial and
$\mathbb{Z}_{2}$-trivial should remain the same, constrained by the SOI-
induced barrier. However, apart from SOI terms, quantum confinement induces an
extra contribution to the bulk bandgap of narrow ZXNRs. Since the gate
electric field cannot manipulate the bandgap due to quantum confinement but
the only one induced by SOI, it leads to topological switching of the edge
state conductance via spin-filtered chiral modes without bulk bandgap closing.
The critical electric field reads
$E_{c}^{TS}<E_{c}^{QPT}=2\Delta{so}/\alpha_{v}$, where the superscript "TS"
and "QPT" represent topological switching and quantum phase transition
respectively. It shows that the switching without bulk gap closing/reopening
is a sheer consequence of the quantum confinement effect on the bulk band
spectrum of narrow ZXNRs and can be verified from the calculated band
dispersion (2).

Accompanying finite-size effects on the edge state spectrum and quantum
confinement effects on the bulk band spectrum, another critical phenomenon
occurs, associated with the bulk band spectrum: the Rashba effect. For a
specific width of ZXNRs, the Rashba SOI further lowers the critical gate
electric field via topological quantum field effect on the bulk band spectrum
Nadeem  _et al._ (2021). For quantum confined spin-orbit coupled ZXNRs, low-
energy single-particle electronic dispersion in the vicinity of Dirac points
reads as follows:

$E(k_{x})=\pm\sqrt{v_{F}^{2}k_{x}^{2}+v_{F}^{2}k_{n}^{2}+\frac{1}{2}\Bigg{|}6\sqrt{3}\lambda_{so}-\alpha_{v}E_{v}\Bigg{(}\frac{1}{2}+\sqrt{\frac{1}{4}+\Theta_{v(c)}\Bigg{(}\frac{2\alpha_{R}}{\alpha_{v}}\Bigg{)}^{2}}\Bigg{)}\Bigg{|}^{2}}\;.$
(2)

where $v_{F}=\sqrt{3}a_{0}t/2$ is the Fermi velocity, $\Theta_{v(c)}=1(0)$ for
valence (conduction) bands in the QSH phase while $\Theta_{v(c)}=0(1)$ for
valence (conduction) bands in the trivial phase and $k_{n}$ is the quantized
transverse momentum along the confinement direction. Such finite-size-
dependent quantization of transverse momentum divides the electronic band
dispersion into infinite set of discrete subabands indexed by quantum number n
= 1, 2, 3…. Specific to our interest in this study, the band dispersion shows
that quantum confinement induces an additional factor, $v_{F}^{2}k_{n}^{2}$,
which enhances the bulk bandgap. The discretized transverse momentum $k_{n}$
is related to the longitudinal momentum $k_{x}$ as follows Brey and Fertig
(2006):

$k_{x}=\frac{k_{n}}{\tan(k_{n}W_{z})}$ (3)

###  II.3 Role of intrinsic topology in pristine ZXNRs

The flat bands in the edge state spectrum of pristine ZXNRs are not generated
from the intrinsic electronic spectrum of 2D-sheets but are rather indebted to
the intrinsic band topology associated with the edge state wave functions. The
electronic dispersion of pristine ZXNRs shows that a critical longitudinal
momentum $k_{x}=k_{x}^{c}$ divides the momentum space regime for the extended
(trivial) and the localized (nontrivial) edge states associated with gapped
dispersing and gapless flat bands respectively. As shown in figure 2(a), the
nontrivial regime of the first Brillouin zone hosting flat bands dwindles with
decreases in the width of ZXNRs. That is, with decreasing width of the ZXNR,
the location of the critical longitudinal momentum $k_{x}=k_{x}^{c}$ moves
toward the TRIM $k_{x}=\pi$. As a result the critical longitudinal momentum
reads $k_{x}=k_{x}^{c}>K$ for narrow ZXNRs, in contrast to $k_{x}=K$ in wide
ZXNRs.

As depicted in figure 2(a) and 2(c), such finite-size effects on the pristine
ZXNRs are intertwined with finite-size effects on the gate-induced topological
switching in spin-orbit coupled ZXNRs. It is interesting to note that the
momentum space location of gate-induced anti-crossing points in spin-orbit
coupled ZXNRs is exactly the same as the critical longitudinal momentum
$k_{x}=k_{x}^{c}$ in pristine ZXNRs. At this point the fourfold degenerate
energy-zero flat bands in pristine ZXNRs are intrinsically gapped out by
finite-size effects while the gate-induced twofold degenerate spin-filtered
Dirac points in spin-orbit coupled ZXNRs are gapped out by the dominating gate
electric field. It implies that the reduction of critical gate electric field
in the spin-orbit coupled ZXNRs is intrinsically associated with the finite-
size effects on the nontrivial character of pristine ZXNRs rather than mere
manipulation of the intrinsic SOI. More specifically, the impact of intrinsic
band topology of pristine ZXNRs on the electronic properties of spin-orbit
coupled ZXNRs can be summarized as follows: in the nontrivial regime, while
the critical momentum space location $k_{x}^{c}$ depends on the width of ZXNR,
the strength of the critical gate electric field $E_{c}$ depends upon both the
strength of SOI and the width of ZXNRs. This effect is further demonstrated by
studying the real space wave functions for edge states, as shown below, that
the reduction in the critical gate electric field is associated with gate-
induced longitudinal momentum-dependence of inter-edge coupling in the
vicinity of $k_{x}^{c}$.

##  III Topological edge state transport

The existence and protection of spin-filtered chiral edge states in ultra-
narrow ZXNRs and low-voltage topological switching from gapless to gapped edge
states can be verified by studying the DOS, width and momentum dependence of
inter-edge overlapping, and gate-induced inter-edge coupling for ZXNRs of
various widths.

###  III.1 Density of states

The existence of spin-filtered chiral edge states in ultra-narrow ZXNRs is
revisited by analyzing the DOS and the conductance quantization in ZXNRs. In
the absence of a gate electric field, finite DOS at energy-zero represent the
gapless edge states in the QSH phase, as shown in figure 1(c), 2(e) and 2(f).
Each edge state channel contributes $e^{2}/h$ to the conductance, leading to a
quantized conductance of $2e^{2}/h$, figure 1(d), which is an important
signature of the QSH phase. The DOS remains finite at energy-zero level even
for ultra-narrow ZXNRs, figure 2(e), a direct evidence of the existence of
topological edge states in atomic-wide ZXNRs. When the gate electric field is
switched on,the energy-zero DOS disappear in the $\mathbb{Z}_{2}$-trivial
phase, figure 1(c). Furthermore, a sharp disappearance of the DOS in the
$\mathbb{Z}_{2}$-trivial phase shows that DOS measurement must be an efficient
way for determining energy gap in edge spectra.

###  III.2 Width/Momentum dependence of edge states

In order to understand how the width and longitudinal momentum-dependence of
inter-edge overlapping/coupling guarantees the protection of conducting edge
states and assists electric field-driven topological switching, we investigate
the real-space wave functions for edge states near the Fermi energy of a spin-
orbit coupled ZXNR terminated at A and B sublattice sites respectively. As
shown in figure 2, the spin-filtered chiral edge states are characterized by a
range of longitudinal momentum $k_{x}\in(2\pi/3a_{0},4\pi/3a_{0})$, defining a
nontrivial regime of the first Brillouin zone. In the vicinity of valleys
$k_{x}\approx 2\pi/3a_{0}$ and $k_{x}\approx 4\pi/3a_{0}$, as depicted in
figure 3(a)-3(c), the real-space squared wave functions decay exponentially
along the confined direction and have finite overlapping. With decrease in the
width, though the amplitude near the edges increases, overlapping between edge
states at the two sides of ZXNRs also increases. As one moves away from
valleys toward TRIM $k_{x}=\pi$, due to large probability distribution near
the edges, the amplitude of squared wave functions increases while the decay
length decreases. For example, as shown in figure 3(d)-3(f), nearly orthogonal
squared wave functions indicate that the penetration depth of exponentially
decaying edge states becomes much smaller than those around valleys K/K′.
Associated with longitudinal momentum around the TRIM $k_{x}=\pi$, as shown in
figure 3(g)-3(i), spin-filtered chiral edge states distributed near the edges
appear to be completely orthogonal and, hence, the inter-edge overlap integral
remains zero even for ultra narrow ZXNRs.

The accuracy of numerical tight binding results, describing the longitudinal
momentum dependence of edge states in quantum confined ZXNRs, can be probed by
obtaining explicit expressions for the wave functions for the spin-filtered
chiral edge states by analytically solving tight binding model Zarea, Büsser,
and Sandler (2008). Based on the nature of wave functions, the edge state
spectrum in ZXNRs can be divided into three regimes of momentum space: (i) In
region I, in the vicinity of TRIM $k_{x}\approx\pi$ where the edge spectrum
forms a fourfold degenerate Dirac point in the absence of gate electric field,
the wave functions are damped oscillatory. (ii) In region II,
$k_{x}\in[2\pi/3a_{0},\pi)\cup(\pi,4\pi/3a_{0}]$ lying between the Dirac
points, the wave functions at the edges decay exponentially along the confined
direction. (iii) In region III, away from the Dirac point
$2\pi/3a_{0}>k_{x}>4\pi/3a_{0}$, the wave functions are oscillatory in nature
and represent the localized 1D edge states due to admixing with bulk subbands.
It implies that the spin-filtered chiral edge states are formed by a
combination of wave functions in region I and II and the nature of these wave
functions changes from exponentially decaying to damped oscillatory as $k_{x}$
moves from region II to region I. It shows that numerical tight binding
calculations are consistent with analytical tight binding results in region I
and II as shown in the insets of figure 3.

Figure 3: Longitudinal momentum and width dependence of real-space squared
wave functions for the spin-filtered chiral edge states. Real-space squared
wave function for the spin-filtered chiral edge states near the Fermi energy
of a ZXNRs with longitudinal momentum lying at Dirac/valley points
$k_{x}=2\pi/3a_{0}$ (a-c), away from Dirac points in the nontrivial regime
(d-f), and in the vicinity of TRIM $k_{x}\approx\pi/a_{0}$ (g-i). The insets,
dashed curves which are consistent with analytical tight binding model
calculations, show that edge states are damped oscillatory in region I (g-i)
while the edge states decay exponentially for momentum away from TRIM, in
region II (a-f). For a fixed N, decay length of edge state decreases as
$k_{x}$ moves from valleys towards the TRIM (from top to bottom). While there
is no inter-edge overlapping in region I, finite inter-edge overlapping in
region II increases with decrease in width (a-c). Here SOI parameters are
taken as $\lambda_{so}/t=0.05$ and $\lambda_{R}=0$ in the absence of gate
potential $\lambda_{v}=0$. The horizontal axis is the confinement direction,
along y-axis of the zigzag 2D-Xenes nanoribbon here.

While tight binding dispersion and DOS confirms the existence of gapless edge
states in atom-wide ZXNRs, the analysis of width and momentum dependence of
edge state wave functions leads to the following three important outcomes: (i)
protection of 1D topological metal, (ii) gate-induced tunability of inter-edge
coupling via correspondence between various momentum space regimes and real
space edge termination, and (iii) size-dependent optimization of topological
switching.

###  III.3 Protection of 1D topological metal

The classification of edge state longitudinal momentum $k_{x}$ shows that the
edge states in ZXNRs are similar to those in a conventional quantum Hall strip
where translational symmetry is preserved along the strip. The spin-filtered
chiral edge states along the two sides of ZXNRs are associated with different
$k_{x}$, and they do not hybridize with each other even when there is a finite
overlap along the confined direction in region II Halperin (1982); MacDonald
and Středa (1984). On the other hand, in region I where the energy and
momentum of edge states around the crossing point $k_{x}=\pi$ are nearly equal
and they can possibly couple to open an energy gap, their wave functions do
not overlap in a finite space. It implies that, in the absence of gate
electric field, spin-filtered chiral edge states do not hybridize/couple even
in ultra-narrow ZXNRs.

While excellent conductance quantization, important signature in many
topological states, and its robustness are well-known features of the quantum
Hall effect, even the extensively studied 2D topological insulators Bernevig,
Hughes, and Zhang (2006); König  _et al._ (2007); Liu  _et al._ (2008); Knez,
Du, and Sullivan (2011); Du  _et al._ (2015); König  _et al._ (2008) with
inverted band structure show experimentally much more fragile conductance
quantization at low temperatures König  _et al._ (2007); Knez, Du, and
Sullivan (2011); Du  _et al._ (2015). So a question arises: is there any
advantage to QSH effect in ZXRNs or will the topological protection remain
relatively fragile? Within the accuracy of electronic dispersion, DOS,
quantized conductance, and momentum-dependence of edge state wave functions
found via numerical tight binding approximations, the topological protection
of QSH states in ZXNRs is equivalent to that for quantum Hall insulators. It
suggests that the edge states in ZXNRs are far more stable than other
topological insulator materials with inverted band structure.

As mentioned above, the answer lies in the energy and momentum space location
of conducting modes on opposite edges in a finite-size geometry. The existence
of different momentum-space locations for the edge state crossings and anti-
crossings in ZXNRs is highly contrasting from other 2D topological insulator
materials with inverted band structure, in which edge state crossing and anti-
crossing points coexist, and in which hybridization due to inter-edge
overlapping opens an energy gap and leads to a gapped edge state spectrum Zhou
_et al._ (2008). Furthermore, it is also explicitly demonstrated in section-IV
that edge states in honeycomb structures remain protected against electron-
electron Coulomb interaction which may become inevitable due to inter-edge
overlapping in narrow ribbons. In short, we are not aware of any experimental
obstacles that may cause potential threat to edge state conductance
quantization in ZXNRs. However, precise control of the zigzag edge is required
in device fabrication.

###  III.4 Momentum dependence of gate-induced inter-edge coupling

The resemblance of damped oscillatory behavior around $k_{x}=\pi$ to the one
in spin-orbit coupled armchair 2D-Xenes nanoribbons (AXNR) Zarea and Sandler
(2007) suggests that the dynamical evolution of edge states remains
independent of particular edge termination in region I. On the other hand, the
exponentially decaying wave functions in region II are directly associated
with the particular edge termination on A and B sublattice sites and, hence,
their penetration depth can be tuned via gate-induced staggered sublattice
potentials.

Figure 4: Longitudinal momentum dependence of gate-induced inter-edge
coupling. For a fixed width of N=50, effect of gate electric field on real-
space squared wave function for the spin-filtered chiral edge states
associated with longitudinal momentum $k_{x}$ lying in region II (a-f) and
region I (g-i). In the vicinity of valleys $k_{x}=2\pi/3a_{0}$ (a-c), critical
gate electric field localizes both the spin-down and spin-up sectors by
turning exponentially decaying edge states into sinusoidal form. Such gate-
induced enhancement in the penetration depth of chiral edge states and the
gate-induced inter-edge coupling leads to an energy gap in the edge state
spectrum when gate electric field exceeds critical limit. As one moves away
from Dirac point, $k_{x}\approx 5\pi/6a_{0}$ (d-f), similar evolution occurs
in spin-down sector but the spin-up sector remains always exponentially
decaying and traversing. In the vicinity of TRIM $k_{x}=\pi/a_{0}$ (g-i),
penetration depth of edge states remains insensitive to gate electric field
effect and both the spin up and the spin down chiral edge states remains
traversing along the edges even for quite large gate electric field. Here SOI
parameters are taken as $\lambda_{so}/t=0.05$ and $\lambda_{R}=0$. The
horizontal axis is the confinement direction, along y-axis of the nanoribbon
here.

To further investigate the dependence of the gate electric field effect on the
longitudinal momentum $k_{x}$, we study the gate electric field modulation of
edge state wave functions associated with various longitudinal momenta. As
shown in figure 4(g)-4(i), our numerical calculations show that the damped
oscillatory edge states in region I remain insensitive to the gate electric
field, i.e., penetration depth remains the same and both the spin up and down
edge states remain damped oscillatory and traversing along the edges even for
very large gate electric field. On the other hand, as shown in figure
4(a)-4(f), exponentially decaying edge states in region II are highly
sensitive to gate electric field effects. First of all, in the vicinity of
momentum $k_{x}\approx 5\pi/6a_{0}$ as shown in figure 4(d)-4(f), gate
electric field hybridizes spin-down edge states while the amplitude of spin-up
edge states decreases with increasing electric field but they remain
uncoupled. It implies that, spin up edge states remain exponentially decaying
and traversing while spin down edge states become sinusoidal and gapped in
this regime of longitudinal momentum, consistent with tight binding electronic
dispersion. As one moves toward the valley $k_{x}\approx 2\pi/3a_{0}$ as shown
in figure 4(a)-4(c), it can be clearly seen that gate electric field induces
coupling between overlapping exponentially decaying wave functions in both
spin up and down sectors and localizes them. The period of these exponentially
turned sinusoidal wave functions decreases with increase in the gate voltage.
Similar gate-controlled edge state dynamics appears on the other valley,
$k_{x}\approx 4\pi/3a_{0}$, but the spin character is interchanged due to
electric field-induced spin-valley locking.

Based on the edge state dynamics, we came to the following conclusion: in the
absence of gate electric field, the non-vanishing overlap between edge sates
in region II do not hybridize/couple to open energy gap as they lie at
different longitudinal momenta. However, mainly due to spin-valley locking,
the gate electric field splits fourfold degenerate Dirac point in region I and
moves the spin-polarized twofold Dirac points toward region II. The finite
inter-edge overlapping in region II allows the gate electric field to induce
coupling between spin-filtered inter-edge states and open an energy gap in the
edge state spectrum.

###  III.5 Size-dependent optimization of topological switching

In region II as shown in figure 3(a)-3(f), an increase in the overlap between
inter-edge states with decreases in width indicates the enhancement of gate-
induced inter-edge coupling in narrow ZXNRs. The gate electric field utilizes
such enhancement of the penetration depth and, hence, inter-edge overlapping
in region II to lower the critical gate electric field required for
topological switching between gapless and gapped edge states. The consistency
of both numerical and analytical tight binding results indicates that the
finite-size effect assistance in topological switching is an artifact of this
gate-induced momentum-dependent coupling between wave functions along the
edges.

The width dependence of gate-induced inter-edge coupling is also consistent
with the width dependence of tight binding electronic dispersion: with a
decrease in the width, the gate-induced anti-crossing points move away from
valleys toward TRIM and, thus, the threshold-voltage decreases. Furthermore,
figure 5 shows that there is no fundamental limit on the threshold-voltage:
For a single zigzag chain, N = 1, the edge states of both pristine and spin-
orbit coupled ZXNR form a gapless Dirac dispersion where crossing and anti-
crossing points coexist at $k_{x}=\pi$. As a result, any non-zero value of
staggered sublattice potential opens an energy gap in the edge state spectrum.

Figure 5: Size-dependent threshold-voltage. Topological switching from gapless
to gapped edge states for a ZXNR with N = 5 (a), N = 3 (b), and N = 1 (c).
Here grey lines represent edge states for pristine ZXNR, purple lines
represent gapless edge states with $\lambda_{so}=0.05t$ and $\lambda_{v}=0$,
red and cyan solid lines represent critical phases with
$\lambda_{v}=\lambda_{v}^{c}$, and red and cyan dashed lines represent gapless
edge states with $\lambda_{v}=0.22t$ (a), $\lambda_{v}=0.12t$ (b), and
$\lambda_{v}=0.02t$ (c). For N = 1, any positive threshold sublattice
potential $\lambda_{v}^{c}$ opens energy gap in the edge state spectrum. Here
we set $a_{0}=t=1$, $\lambda_{so}=0.05t$, and $\lambda_{R}=0$.

##  IV Effect of electron-electron Coulomb interactions

Furthermore, though the electron-electron Coulomb interactions become
inevitable in quantum confined ZXNRs, spin-filtered chiral edge conducting
channels may remain gapless even when both intra- and inter-edge Coulomb
interactions are present. For example, contrary to pristine ZXNRs where
Coulomb interactions lead to energy gap by lifting the fourfold degeneracy of
energy-zero flat bands in the edge state spectrum Son, Cohen, and Louie
(2006); Hikihara  _et al._ (2003); Fujita  _et al._ (1996); Yang  _et al._
(2007), it has been explicitly shown that the QSH phase in spin-orbit coupled
ZXNRs remain stabilized against intra-edge Coulomb interactions Xu and Moore
(2006). Moreover, the mass term produced by backward scattering, which may
have originated from the mixing of right and left moving chiral modes carrying
the same spin polarization and located at opposite edges of finite-size ZXNRs,
can be suppressed in the large SOI limit and, hence, the spin-filtered chiral
edge states may also remain protected against unscreened inter-edge Coulomb
interactions Zarea, Büsser, and Sandler (2008). It can be justified by a
simple argument based on the interplay between the strength of intrinsic SOI
and the screening length of Coulomb interactions: Since the decay length of
spin-filtered chiral edge states in ZXNRs is inversely proportional to the
intrinsic SOI, the overlaps between oppositely moving spin-filtered chiral
modes are suppressed with increasing SOI. It shows that, in the large SOI
limit, ultra-narrow ZXNRs can be described by the tight binding model where
both intra- and inter-edge Coulomb interactions are effectively absent. Thus,
even in the presence of Coulomb interactions, a large SOI limit renders
gapless spin-filtered chiral edge states since the reduced inter-edge overlap
diminishes the backward scattering terms.

Based on a similar argument, our findings provide another ground: In the QSH
phase, spin-filtered chiral states associated with a longitudinal momentum
$k_{x}=\pi$ in region I remain protected against backward scattering due to
vanishing inter-edge overlap. On the other hand, in the presence of a gate
electric field, unscreened inter-edge Coulomb interactions may also assist
topological switching by inducing an energy gap due to finite inter-edge
coupling between edge states associated with longitudinal momentum $k_{x}$
lying in region II of the Brillouin zone, similar to the finite-size effect.
In passing, unlike the finite-size effect, which is characterized by critical
longitudinal momentum $k_{x}^{c}$ and remains the same in both pristine and
spin-orbit coupled ZXNRs, the effect of Coulomb interactions in pristine ZXNRs
is completely different from spin-orbit coupled ZXNRs.

##  V Low-voltage topological quantum devices

The analogy between the rich momentum-dependent behavior of edge states and
the gate-controlled inter-edge coupling in the spin-orbit coupled ZXNRs leads
to the following two phenomena which are critical for topological quantum
devices: (i) In the absence of gate electric field and hence Rashba SOI, spin-
filtered chiral edge edge states with fourfold degenerate Dirac point at TRIM
$k_{x}=\pi$ remain gapless even for ultra-narrow ZXNRs. Vanishing inter-edge
coupling across the crossing point guarantees that the spin-filtered chiral
edge states (enabling dissipationless and quantized conductance) remain
topologically protected against backscattering and hence the deviation from
conductance quantization - a figure of merit in QSH materials. (ii) Since the
gate electric field splits fourfold Dirac point at TRIM and moves spin-
filtered twofold Dirac points toward valleys, gate-induced coupling due to
finite overlap between spin-filtered inter-edge states across anti-crossing
points assists in opening the energy gap in the spin-filtered chiral edge
states and lowers the critical gate electric field. This artifact of the
finite-size effect, dwindled nontrivial regime of the Brillouin zone without
affecting the bulk band topology and reduced critical gate electric field
without affecting the quantized edge state conductance in the QSH phase,
provides an ideal platform for devising energy-efficient low-voltage
topological quantum devices.

To exemplify the advantages of utilizing spin-orbit coupled ZXNRs for
computing technologies, we explicitly demonstrate the working principle of a
topological quantum field effect transistor (TQFET) and compare its critical
functionalities with a MOSFET. Unlike a MOSFET, where a conventional
semiconductor is utilized as a channel material and conduction is enabled via
bulk electronic states, TQFET configures a topological insulator material as a
channel in which the dissipationless current is carried by topologically
protected edge modes. In a blueprint for a TQFET, the gate electric field
tunes a topological insulator material from a topological insulating phase
(on-state) to a conventional insulating phase (off-state), a phenomenon known
as topological switching. In other words, the topological switching mechanism
relies on transitioning between gapless (on-state) and gapped (off-state) edge
modes. Such a gate electric field-driven topological switching is a
fundamentally different mechanism compared to a traditional carrier inversion
in conventional semiconducting switching devices. Figure 6 shows a schematic
representation for a TQFET configuring quantum confined ZXNR as a channel
between the source and drain. First, the existence of 1D gapless edge states
in the ultra-narrow ZXNRs promises the availability of large edge state
conducting modes for enhanced signal-to-noise ratio via multiple edge state
channels, figure 6(a). It allows optimized geometry for a TQFET where an array
of ZXNRs, set apart by trivial insulating layers/wires along vertical/lateral
direction, is sandwiched between top and bottom gates separated by top and
bottom dielectrics.

Figure 6: Topological quantum field effect transistor. (a) Schematic
representation for a TQFET configuring multiple quantum confined ZXNRs
allowing conduction between source and drain. (b) Topological switching driven
by gate electric field which tunes a ZXNR form on-state with gapless edge
modes ($\lambda_{v}<\lambda_{v}^{c}$) to off-state with gapped edge modes
($\lambda_{v}>\lambda_{v}^{c}$). Here VG is the gate-voltage, VDD is the
supply-voltage, and ID is the source-to-drain current.

Second, the reduction in threshold-voltage with decrease in the channel width,
even though the topological bulk bandgap increases, overturns a general wisdom
of utilizing narrow gap and wide channel materials for reducing threshold-
voltage in a standard field effect transistor analysis. For example, in a
blueprint topological transistor where topological switching is implemented
via bulk bandgap closing and reopening, materials with large bulk bandgap
require an unrealistically large threshold-voltage Molle  _et al._ (2017);
Vandenberghe and Fischetti (2017). Moreover, TQFET with quantum confined ZXNRs
is in high contrast to MOSFET in which width-dependence of the threshold-
voltage Vth depends upon the isolation technique used for transistor
fabrication Tsividis and McAndrew (2011): the effective threshold-voltage in a
narrow channel device increases with decreases in width when the transistor is
made using the LOCOS (local oxidation of silicon) process while decreases with
decrease in width when the transistor is made in a shallow-trench-isolation
(STI) process. That is, unlike the size dependence of threshold-voltage on
isolation technique in MOSFET, the reduction of threshold-voltage in a TQFET
is an intrinsic property of ZXNRs associated with topological and quantum
mechanical functionalities. It suggests that, along with vastly different
conduction and switching mechanisms, the technological aspects required for
fabricating a TQFET with ZXNRs also radically differ from those of MOSFETs:
There is no fundamental requirement of specialized technological/isolation
techniques for a low-voltage TQFET with an energy-efficient switching
mechanism.

Third, the reduction in threshold-voltage becomes important for reducing the
supply voltage (VDD) in a low-voltage switching device if the subthreshold
swing is compressible. That is, power dissipation Ionescu and Riel (2011) P
$\approx$ IOFFVDD3 can be reduced while maintaining the device performance
(ION) by simultaneous scaling down $V_{th}$ and $V_{DD}$ and, thus, keeping
the overdrive constant ($\propto$(VDD \- Vth)2). In a MOSFET, incompressible
subthreshold swing leads to an exponential increase in IOFF in the transfer
characteristics Ionescu and Riel (2011). That is, for every 60 mV at room
temperature, there is more than tenfold increase in IOFF. In contrast to
MOSFETs, this is not a problem in a TQFET where subthreshold swing can be
tuned via topological quantum field effect Nadeem  _et al._ (2021), a combined
effect of electric field and tunable Rashba SOI that allows overcoming the
“Boltzmann’s tyranny”. Power dissipation can be lowered by reducing the
threshold-voltage via geometric optimization of quantum confined ribbons of
QSH materials while keeping subthreshold swing subthermal via strain
engineering, buckling parameterization, tuning inter-orbital hopping, and
normalization of intrinsic atomic SOI. In summary, quantum confined ZXNRs with
optimized geometry may prompt the progress of topological computing
technologies with vastly lower energy consumed per operation than CMOS
technologies and greet the Moore’s trajectory of transistor miniaturization,
doubling transistors per chip and doubling the processing power every two
years.

##  VI Experimental realization and device fabrication

Though a full experimental exploitation of 2D-Xenes is yet to be explored for
device applications, mainly due to several challenges imposed by restricted
synthesis methodologies, substrate effects, and stability issues, an
experimental confirmation of the electronic properties and the buckled
structure for 2D-Xenes has been realized through angle-resolved photoemission
spectroscopy (ARPES) and scanning tunneling microscopy (STM), respectively.
Furthermore, the epitaxial growth of both group-IV Molle  _et al._ (2017) and
group-V Khan  _et al._ (2021) 2D-Xenes has been realized on different
substrates. For instance, the atomic arrangement of group-IV buckled 2D-Xenes
has been detailed through epitaxy of silicene on metallic (Ag(111), Ir(111),
and ZBr2) Lalmi  _et al._ (2010); Vogt  _et al._ (2012); Lin  _et al._ (2012);
Feng  _et al._ (2012); Chiappe  _et al._ (2012); Fleurence  _et al._ (2012);
Meng  _et al._ (2013) and semiconducting (MoS2) Chiappe  _et al._ (2014)
substrates, STM studies of germanene on metallic (Au(111), Pt(111), Al(111)
and Hex-AIN) Dávila  _et al._ (2014); Li  _et al._ (2014); Bampoulis  _et al._
(2014); Derivaz  _et al._ (2015); D’Acapito  _et al._ (2016) and
semiconducting (MoS2) Zhang  _et al._ (2016a) substrates, and epitaxial
stanene on Bi2Te3 substrates Zhu  _et al._ (2015). The epitaxial synthesis has
also been extended to group-V 2D-Xenes, for instance, a monolayer of
phosphorene on Au(111) substrate Zhang  _et al._ (2016b) showing silicene-like
semiconducting character.

While the tight binding model with Kane-Mele type SOI describes a hypothetical
freestanding ZXNRs with poor chemical stability, a substrate supporting the
epitaxial synthesis is highly desired for real-world applications. However,
the supporting substrate brings other challenges into play due to concurrent
bonding interactions. For instance, a metallic substrate short-circuits the
edge states of interest and destroys the topological protection. On the other
hand, semiconducting MoS2 substrate can stabilize 2D-Xenes with protected edge
states; however, 2D-Xenes render into a metal due to compressive strain
Chiappe  _et al._ (2014); Zhang  _et al._ (2016a). Similar problems persist
for stanene on Bi2Te3 substrate Zhu  _et al._ (2015).

Recently, it has been shown that epitaxially deposited bismuthene on the
insulating silicon carbide substrate SiC(0001) is a large bandgap QSH
insulator where structural and electronic properties have been confirmed by
STM and ARPES measurements Reis  _et al._ (2017). However, as compared to
freestanding buckled Bi(111) bilayers, considerably larger lattice constant of
5.35 Å stabilizes Bi/SiC into an energetically favorable planar honeycomb
configuration Hsu  _et al._ (2015). With all its interesting aspects, the
planar honeycomb configuration is not desirable for gate-induced topological
switching. Furthermore, it is predicted that both As and Sb are plagued by
similar problems Li  _et al._ (2018).

This experimental odyssey of 2D-Xenes promises that the fabrication of
topological devices is just a step away. The possible first experimental step
to corroborate our main prediction for device integration, scientific studies,
and technological applications is the synthesis of buckled ZXNRs with
protected edge states on a weakly interacting semiconducting substrate. In
this direction, the growth of functionalized 2D-Xene sheets on a suitable
semiconducting substrate would be a promising development that can bring an
obvious benefit for the realization of low-voltage topological devices Nadeem
_et al._ (2021). For instance, functionalized bismuth monolayers BiX Song  _et
al._ (2014) and Bi2XY Zhou  _et al._ (2018) where X/Y = H, F, Cl, and Br
stabilize with quasi-planar/low-buckled structure and the strong on-site SOI
opens the topological bandgap at the Dirac points formed by low-lying $p_{x}$
and $p_{y}$ orbitals. Furthermore, recently corroborated first-principle
calculations show that gate-controlled topological quantum phase transition
between different topological states can be realized in functionalized bismuth
monolayer Zhou  _et al._ (2021).

##  VII Conclusion

It is demonstrated that, in a finite-size geometry, ZXNRs display unique
physical characteristics associated with their intrinsic band topology and the
finite-size effects such as longitudinal momentum-dependent inter-edge
overlapping between spin-filtered chiral edge states and the quantum
confinement effect on the bulk band spectrum. While the damped oscillatory
modes around the edge state crossing momentum remain completely orthogonal and
guarantee protected spin-filtered chiral edge states even in ultra-narrow
ribbons, enhanced gate-induced inter-edge coupling between exponentially
decaying edge states around the anti-crossing points reduces the gate electric
field required for topological switching between gapless and gapped edge
states. In addition, quantum confinement enhances the SOI-induced bandgap in
the nontrivial phase and leads to topological switching without bulk bandgap
closing. On the one hand, it reduces the threshold-voltage by lowering the
SOI-induced barrier in the bulk; on the other hand, it enhances the bulk
bandgap even in lighter monoelemental 2D-Xenes such that the detrimental
contributions from the bulk material to the edge current are avoided and
allows safe residing of the chemical potential within this gap. Furthermore,
similar to wide ZXNRs, the Rashba effect enhances the bandgap in the trivial
phase. Hence, a large nontrivial bulk bandgap by quantum confinement effect to
decouple the conducting edge states from bulk subbands and a large trivial
bandgap by the Rashba effect to overcome thermal excitation makes quantum
confined narrow ZXNRs ideal for engineering energy-efficient low-voltage
topological quantum devices.

The proposed mechanism for optimizing topological switching and devising
concepts for topological electronics is applicable to all 2D-Xene sheets
ranging from silicene to bismuthene as well as other 2D topological insulators
with honeycomb lattice structure. In principle, the threshold-voltage depends
upon the momentum space location of anti-crossing points in the edge state
spectrum, which is the same for both pristine and spin-orbit coupled ZXNRs.
Quantitatively, the threshold value depends upon both the strength of
intrinsic SOI and the width of ZXNRs. In addition, the width of ZXNRs,
$W_{z}=\sqrt{3}Na_{0}/2$, depends upon both the number of zigzag lines $N$ and
the lattice constant $a_{0}$. Increasing lattice constants, ranging from 2.46
Å for graphene to 5.35 Å for bismuthene Reis  _et al._ (2017), suggest that
the critical width ($W_{z}^{c}$) would be different for different 2D
topological insulator sheets, even if the number of zigzag lines $N$ is fixed.
Furthermore, a wide range of the intrinsic SOI, from 0.00057 meV for graphene
to 435 meV for bismuthene Reis  _et al._ (2017), indicates that the threshold-
voltage would be different for different 2D topological insulators sheets,
even if the width of the ribbon is fixed.

Considering the wide applicability and generality of the proposed mechanism,
we presented a generalized mechanism that clearly indicates how tight binding
electronic dispersion, DOS, and the penetration depth of edge state wave
functions and, thus, the threshold-voltage depend on the number of zigzag
chains. In wide ZXNRs, with $W_{z}>W_{z}^{c}$, the SOI ($\Delta_{so}$) induced
barrier imposes a limit on the threshold voltage
($\lambda_{v}^{c}=2\Delta_{so}$), which could be unrealistically large for
2D-Xenes. However, when $W_{z}<W_{z}^{c}$, the threshold-voltage decreases
with decrease in width ($\lambda_{v}^{c}<2\Delta_{so}$). A qualitative width
dependence study shows that the threshold-voltage can be lowered without any
fundamental limit. For instance, in ultra-narrow ribbons, N=1 say, any non-
zero electric field can switch the edge state conductance by opening an energy
gap in the edge state spectrum. Considering a large variation in the strength
of intrinsic SOI and size of lattice constants for 2D-Xenes, we did not
mention any single threshold limit as a figure of merit for a topological
transistor. Rather, we highlighted that a topological transistor is more
flexible to tunability of critical parameters than its conventional
counterpart, MOSFET. More defined engineering heads-up toward the manufacture
of an ideal topological transistor can be extracted for a specific 2D-Xene
configured as a channel material.

Similar to the gate voltage, finite-size effects can also be employed to tune
the exchange interaction in 2D magnetic topological insulators Ezawa (2013b,
c, 2015a); Högl  _et al._ (2020); Li  _et al._ (2013); Liang, Wu, and Hu
(2013); Zhou  _et al._ (2018, 2021); Shabbir  _et al._ (2018); Nadeem  _et
al._ (2020) and optical probing of topological signatures in 2D materials Xu
_et al._ (2020). For example, the critical regime in an antiferromagnetic
topological insulator can be optimized to design a topological spin transistor
via gate-induced topological switching of edge state spin transport. This
study may also be generalized to study other topological phases such as
topological superconductors San-Jose  _et al._ (2015); Ezawa (2015b). In a
finite-size geometry, Majorana bound states localized along the edges of 2D
topological superconductors Potter and Lee (2010) can be decoupled from bulk
states for robust information processing.

###### Acknowledgements.

This research is supported by the Australian Research Council (ARC) Centre of
Excellence in Future Low-Energy Electronics Technologies (FLEET Project No.
CE170100039), Australian Research Council (ARC) Professional Future Fellowship
(FT130100778), and funded by the Australian Government.

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