# Disorder effects of vacancies on the electronic transport properties of
realistic topological insulators nanoribbons: the case of bismuthene

Armando Pezo<EMAIL_ADDRESS>Federal University of ABC (UFABC),
09210-580, Santo André, São Paulo, Brazil  Brazilian Nanotechnology National
Laboratory (LNNano), CNPEM, 13083-970, Campinas, São Paulo, Brazil  Bruno
Focassio  Federal University of ABC (UFABC), 09210-580, Santo André, São
Paulo, Brazil  Brazilian Nanotechnology National Laboratory (LNNano), CNPEM,
13083-970, Campinas, São Paulo, Brazil  Gabriel R. Schleder  Federal
University of ABC (UFABC), 09210-580, Santo André, São Paulo, Brazil
Brazilian Nanotechnology National Laboratory (LNNano), CNPEM, 13083-970,
Campinas, São Paulo, Brazil  Marcio Costa  Instituto de Física, Universidade
Federal Fluminense, 24210-346 Niterói, Rio de Janeiro, Brazil  Brazilian
Nanotechnology National Laboratory (LNNano), CNPEM, 13083-970, Campinas, São
Paulo, Brazil  Caio Lewenkopf  Instituto de Física, Universidade Federal
Fluminense, 24210-346 Niterói, Rio de Janeiro, Brazil  Adalberto Fazzio
<EMAIL_ADDRESS>Brazilian Nanotechnology National Laboratory
(LNNano), CNPEM, 13083-970, Campinas, São Paulo, Brazil  Federal University of
ABC (UFABC), 09210-580, Santo André, São Paulo, Brazil

###### Abstract

The robustness of topological materials against disorder and defects is
presumed but has not been demonstrated explicitly in realistic systems. In
this work, we use state-of-the-art density functional theory and recursive
nonequilibrium Green’s functions methods to study the effect of disorder on
the electronic transport of long nanoribbons, up to $157\;\rm nm$, as a
function of vacancy concentration. In narrow nanoribbons, a finite-size effect
gives rise to hybridization between the edge states erasing topological
protection. Hence, even small vacancy concentrations enable backscattering
events. We show that the topological protection is more robust for wide
nanoribbons, but surprisingly it breaks down at moderate structural disorder.
Our study helps to establish some bounds on defective bismuthene nanoribbons
as promising candidates for spintronic applications.

##  I Introduction

Topological materials have been intensively studied in recent years [1, 2, 3,
4, 5, 6] unveiling interesting new physics and opening new applications
possibilities in spin-based electronic devices [7]. Of particular interest are
large band gap topological insulators (TIs), that are good candidates for the
realization of the quantum spin Hall (QSH) effect at room temperature [8, 9].
In two-dimensional (2D) QSH insulators, the edges of the sample carry metallic
states that are protected by time-reversal symmetry (TRS) [10, 11, 12, 13, 14,
15, 16] and decay exponentially into the bulk [17, 18, 19]. Moreover, theory
predicts that in nanoribbons these edge states carry dissipationless helical
spin currents. The penetration depth, that quantifies the edge states
exponential decay rate, is (roughly) inversely proportional to the band gap
[18, 20] and plays a key role in the nanoribbon transport properties. As it
happens with the length scales of 3D topological insulators [21], to fully
display the features of a TI, the nanoribbon width must be much wider than the
penetration depth $\xi$ of the edge states, otherwise they can easily
hybridize.

Among several candidates for topological materials, bismuthene, also known as
buckled or bilayer bismuthene, is of special interest due to its large
electronic band gap of $0.5\text{\,}\mathrm{eV}$, along with its structural
stability and large spin-orbit coupling (SOC) [22, 23, 24]. The existence of
charge puddles and other types of defects that could be detrimental for the
formation of a topological phase do not play an important role in bismuthene,
as experimentally demonstrated [8]. Indeed, since its experimental
realization, it was proposed that even amorphous structures of bismuthene
occurring before the annealing process [25, 26], as well as strongly
disordered systems [27], support topological states. The robustness of
bismuthene non-trivial topology makes it ideal for material design, both by
tuning the lattice constant and effective spin-orbit coupling (SOC) by
epitaxial constraint or by substitutional alloying with lighter elements, such
as Sb or As, without causing a topological transition [28, 22]. The presence
of non-magnetic defects leaves the band topology unchanged in these materials
[24, 25, 26, 27].

Two recent works have proposed that vacancies can spoil the topological
(elastic) transport properties of 2D TIs. Studying a generic tight-binding
model with a Hubbard mean-field term, Ref. [29] has shown that vacancy induced
localized states can give rise to local magnetic moments and destroy the
topological protection. In turn, Ref. [30] proposes a very different
mechanism. Based on another tight-binding toy model, Ref. [30] has numerically
shown that small concentrations of vacancy defects do not eliminate the
topological edge states, but can cause inter-edge state hybridization in
certain energies intervals within the topological gap [30]. For the BHZ model
[12], several works have addressed the role of strong disorder, demonstrating
a modification of the local currents and inter-edge tunneling effects [31, 32,
33, 34]. The dominant mechanism depends critically on the choice of the
schematic-model Hamiltonian, which—from the point of view of materials—is very
unsatisfactory. To better understand the interplay between edge states and
defects a material-specific systematic study using ab initio methods is in
order.

Vacancies on bismuthene show small formation energies where $sp^{2}$ bismuth
dangling bonds can be distributed around the vacancy leading to resonances in
the band gap, modifying the electronic properties of the host material.
Theoretical calculations on bismuthene show no evidence of magnetic moments
induced from vacancies [23], preserving TRS and retaining its non-trivial
topological band structure. Recent ab initio calculations demonstrate that
transport along the edges is insensitive to vacancies created in ultra-narrow
TI zigzag nanoribbons [35]. Even though these vacancies allow the development
of magnetic moments, the perfect conductance for energies within the bulk gap
is recovered provided the vacancies are passivated by hydrogen. Our study, in
turn, addresses more realistic system sizes, namely, both much wider and much
longer, unveiling a different kind of disordered-induced transport mechanism.

In this work, we investigate the electronic transport properties of buckled
bismuthene nanoribbons of realistic sizes using the full Hamiltonian in the
orbital representation obtained from density functional theory (DFT) [36, 37,
38] calculations combined with recursive nonequilibrium Green’s functions
(NEGFs) [39, 40, 41, 42]. First, we explore the backscattering mechanism due
to inter-edge hopping mediated by vacancy localized states in different
nanoribbon widths and discuss the detrimental effects in transport properties
caused by single vacancies in narrow nanoribbons. We proceed to investigate
the robustness of the topological properties by studying the conductance of
ribbons as a function of their widths, lengths (up to $157\;\rm nm$), and
vacancy concentration. We show that narrow ribbons, whose widths $w$ are
comparable with the edge states penetration depth $\xi$, present the onset of
Anderson localization effects already at low vacancy concentrations. In
distinction, for wide nanoribbons, where $w\gg\xi$, the edge states maintain a
quantized conductance in the topological gap at low disorder concentrations.
Surprisingly, we find that topological protection is destroyed already at
modest vacancy concentrations.

##  II Computational Methods

Our calculations combine the flexibility of a plane-wave basis set to obtain
the optimized structures with a localized basis for electronic transport. In
both cases, we use the Perdew-Burke-Ernzerhof (PBE) [43, 44] exchange-
correlation functional. We perform the geometry optimizations with a plane-
wave basis as implemented in the VASP package [45, 46]. In these calculations,
we employ $400\text{\,}\mathrm{eV}$ for the plane-wave expansion cutoff and
ionic potentials are described using the projector augmented-wave (PAW) method
[47]. Vacancies are modeled by removing an atom from the bulk and performing
the geometry optimization with force criterion of
$5\text{\times}{10}^{-3}\text{\,}\mathrm{eV}\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$.

The transport calculations use full ab initio DFT Hamiltonian matrices
obtained directly from the SIESTA code [48], employing atom-centered
single-$\zeta$ plus polarization (SZP) basis sets. We use the energy cutoff
for real-space mesh of $350\text{\,}\mathrm{R}\mathrm{y}$, sampling the
reciprocal space with 10 $\mathbf{k}$-points along the periodic direction of
the presented nanoribbons, which edges were hydrogen-passivated. We add
$20\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ of vacuum in both non-periodic
directions to avoid spurious interactions between periodic images. The self-
consistent SOC is introduced via an on-site approximation [49] using fully-
relativistic norm-conserving pseudopotentials [50]. The system Hamiltonian and
overlap matrices are obtained after performing a full self-consistent cycle.

Figure 1: Schematic representation of the two-probe setup studied in this
work. (a) The electrodes are located at the left (L) and right (R) sides of
the device. (b) Zoomed-in view of the scattering region (S), illustrating some
of its building blocks containing vacancies.

The electronic transport calculations are implemented following the standard
NEGF approach [39, 40, 41]. We consider a two-probe terminal setting, as
illustrated in Fig. 1(a). The left (L) and right (R) electrodes are modeled by
semi-infinite pristine zigzag bismuthene leads.

Our approach employs a decimation technique [51, 52, 53, 54, 55, 56] that
allows us to address large system sizes. The scattering region is partitioned
into building blocks connected by first neighbour interactions. Each building
block is computed through DFT. The procedure takes into account all degrees of
freedom of the system comprised of all the building blocks. The partition
scheme is depicted in Fig. 1(b). Accordingly, the system Hamiltonian is
expressed in the localized basis by the block matrix

$\mathbf{H}=\begin{pmatrix}\mathbf{H}_{L}&\mathbf{H}_{C}&0\\\
\mathbf{H}^{\dagger}_{C}&\mathbf{H}_{S}&\mathbf{H}_{C}\\\
0&\mathbf{H}^{\dagger}_{C}&\mathbf{H}_{R}\\\ \end{pmatrix}$ (1)

where $\mathbf{H}_{L}$ and $\mathbf{H}_{R}$ are the Hamiltonian matrices
describing the left and right electrodes, while $\mathbf{H}_{C}$ is the
coupling between the leads and the central region, and $\mathbf{H}_{S}$ is the
tridiagonal block matrix

$\mathbf{H}_{S}=\begin{pmatrix}\mathbf{H}_{1}&\mathbf{H}_{C}&0&\ldots&0\\\
\mathbf{H}^{\dagger}_{C}&\mathbf{H}_{2}&\ldots&\ldots&\vdots\\\
0&\vdots&\ddots&\vdots&0\\\
\vdots&\ldots&\ldots&\mathbf{H}_{N-1}&\mathbf{H}_{C}\\\
0&\ldots&0&\mathbf{H}^{\dagger}_{C}&\mathbf{H}_{N}\\\ \end{pmatrix}$ (2)

representing the scattering region (S). We take $\mathbf{H}_{C}$ also as the
coupling between each building block. We consider a uniform distribution of
vacancies within the building blocks represented in Fig. 1(b), except for a
small region of $2.175\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ at the
simulation box boundaries in the periodic direction. This is necessary to
properly connect successive building blocks by $\mathbf{H}_{C}$ as in Eq. (2).

The scattering region spin resolved retarded Green’s function [41, 57] reads

$\mathbf{G}^{r}_{S}(E)=\left(E^{+}\mathbf{S}_{S}-\mathbf{H}_{S}-\mathbf{\Sigma}_{L}^{r}-\mathbf{\Sigma}_{R}^{r}\right)^{-1}$
(3)

where $E^{+}=\lim_{\delta\to 0^{+}}E+i\delta$, $\mathbf{S}_{S}$ is the overlap
matrix, and
$\mathbf{\Sigma}^{r}_{R/L}(E)=(E^{+}\mathbf{S}_{C}-\mathbf{H}_{C})\mathbf{G}^{r}_{0,R/L}(E^{+}\mathbf{S}^{\dagger}_{C}-\mathbf{H}_{C}^{\dagger})$
are the embedding self-energies that account for the system decay width due to
the coupling with the leads. Here $G^{r}_{0,R/L}$ is the retarded surface
Green’s function of the $R/L$ electrode [58, 51]. The block structure of
$\mathbf{H}_{S}$ allows for a very efficient computation of $G^{r}_{S}(E)$
using the recursive Green’s function method (see, for instance, Ref. [55], for
a review).

In the linear response, at small bias, the zero-temperature conductance is
given by the Landauer formula $G(E_{F})=(e^{2}/h)\;T(E_{F})$, where the
transmission $T$ reads [39, 40, 41]

$T(E)=\text{Tr}\left[\mathbf{\Gamma}_{L}(E)\mathbf{G}_{S}^{a}(E)\mathbf{\Gamma}_{R}(E)\mathbf{G}_{S}^{r}(E)\right]$
(4)

where $\mathbf{G}_{S}^{a}=[\mathbf{G}_{S}^{r}]^{\dagger}$ and the decay width
matrices $\mathbf{\Gamma}_{L/R}$ are given by
$\mathbf{\Gamma}_{L/R}={i}(\mathbf{\Sigma}_{L/R}^{r}-\mathbf{\Sigma}_{L/R}^{a})$.

##  III Results and Discussion

In this section we analyze the effect of vacancies on the transport properties
of bismuthene zigzag nanoribbons. We begin by discussing the topological
properties of pristine nanoribbons of different representative widths. Next,
we investigate the effect of a single-vacancy on the electronic and transport
properties of these systems. Finally, we study the conductance of these
systems for different vacancy concentrations and nanoribbon widths.

Figure 2: Electronic band structure along the $\overline{\Gamma}-\overline{X}$
direction (left panels) and the corresponding conductance $G$ (right panels)
for pristine (infinite length) bismuthene nanoribbons of widths (a) $w_{20}=$
$20\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, (b) $w_{65}=$
$65\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, and (c) $w_{110}=$
$110\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The gray energy windows
correspond to the bulk topological gap.

Let us first investigate the effect of the ribbon width on pristine systems.
Due to translation invariance these systems can be viewed as having infinite
length. In Fig. 2 we show the electronic band structure and the conductance
for bismuthene nanoribbons with 3 different representative widths. To help the
discussion, the bulk-topological gap $\Delta_{\rm TG}$, corresponding to
$$-0.1\text{\,}\mathrm{eV}$<E<$0.4\text{\,}\mathrm{eV}$$, is indicated in
grey. Figure 2(a) shows the results for a narrow nanoribbon, $w_{20}$ of width
$20\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, comparable to those of ref. [35].
The electronic states bridging the bulk-topological $\Delta_{\rm TG}$ are
split and the system displays a small gap, in distinction to the standard
picture of topological protected metallic edge states in very large systems.
Figure 2(b) shows an intermediate width nanoribbon, namely $w_{65}$, of
$65\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ width. Here, there is still a
small gap at the $\Gamma$-point, though the edge states are degenerate as
expected for a topological insulator. In addition, the top of the valence and
bottom of the conduction trivial state bands approach the corresponding bulk
energies, indicating that finite width quantization effects are much smaller
than in the previous case. Finally, Fig. 2(c) shows a wide nanoribbon,
$w_{110}$ of width $110\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, with no gap
and degenerate edge states with helical texture, as expected for a TI. Here,
the bulk-topological gap corresponds very closely to the energy interval where
one finds only edge states. In all cases, the conductance $G(E)$ is quantized
and the conductance steps are observed as expected for pristine systems [41].

The lack of topological protection in narrow ribbons is a result of strong
overlap between the states at the opposite system edges. This can be
understood in terms of the spatial localization of edge states or, more
precisely, their so-called penetration length $\xi$. The latter can be roughly
estimated from the mass term in the $k\cdot p$ Dirac Hamiltonian describing
the inverted bulk band gap as [20, 59, 60, 61]

$\xi\approx\hbar v_{F}/\Delta_{\rm TG},$ (5)

where $v_{F}$ is the Fermi velocity corresponding to the topological bands,
namely, $\hbar v_{F}=d\varepsilon_{k}/dk$. By estimating the Fermi velocities
from Figs. 2(b) and (c) we find: For the $w_{65}$ nanoribbon $v_{F}=$
$4.58\text{\times}{10}^{5}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$, that
renders $\xi=0.6\;\rm nm$, while for $w_{110}$ the Fermi velocity is
$5.79\text{\times}{10}^{5}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ and
$\xi=0.8\;\rm nm$. These estimates of $\xi$ for buckled bismuthene are
slighlty larger than the values reported in experiments [62, 8] that obtain
$\xi\approx 0.4\;\rm nm$ in SiC(0001) supported flat bismuthene. We stress
that these are different material systems. The discrepancy can be explained by
recalling that $\Delta_{\rm TG}$ of the buckled bismuthene is smaller than the
band gap of the flat one.

A more quantitative estimate of $\xi$ is taken from the local density of
states (LDOS) averaged over the $x,z$ directions as a function nanoribbon
transversal axis $y$, see Fig. 3. Fitting an exponential function, we estimate
the penetration length as $\xi=0.9\rm\;nm$ for the $w_{110}$ ribbon, in good
agreement with $\xi$ obtained using Eq. (5). Using $\xi$, we can also estimate
a threshold length for the LDOS decay for which the interedge states
hybridization becomes negligible. For instance, at distances $\Delta y\simeq
2.8\;\rm nm$ from the edges, the LDOS decays by $\sim 95\%$. Hence, we expect
that narrow nanoribbons with $w\lesssim 2\Delta y$ lack topological protection
due to the strong overlap of states localized at opposite system edges. The
$w_{65}$ nanoribbons are at the crossover between non-protected and
topologically protected phases.

Figure 3: Local density of states (LDOS) at the Fermi level ($E_{F}=0$) for
pristine (infinite length) $w_{110}$ nanoribbon. The LDOS is averaged over the
$x$ and $z$ directions and the y-axis corresponds to the transverse direction
along the ribbon width. The gray dashed line shows the exponential fit. The
geometry below the LDOS shows the bismuthene $w_{110}$ nanoribbon view
perpendicular to the $x$–$y$ plane.

Next, we investigate ribbons containing a single-vacancy. For a slab geometry,
we calculate the formation energy $E_{V}$ of these single vacancies systems
using the following expression [63]

$E_{V}=E_{\rm tot}-(E_{\rm pristine}+\mu_{V}N_{V})$ (6)

where $E_{\rm tot}$ is the total energy of the single-vacancy system given by
a fully relaxed DFT calculation, $E_{\rm pristine}$ is the energy for the
pristine ribbon, $N_{V}$ is the number of vacancies (in our case $N_{V}=1$),
and $\mu_{V}$ is the chemical potential to remove a bismuth atom. Here, we
take $\mu_{V}$ as the energy per atom of the pristine bismuthene monolayer.
For purposes of comparison, we simulate a $5\times 5$ supercell containing a
vacancy. By doing so, we avoid that theses vacancies interact with their
periodic images obtaining a formation energy of $1.04\text{\,}\mathrm{eV}$,
which is in agreement with previous reports [23].

Figure 4: Single-vacancy close to the edge of a bismuthene nanoribbon: (a)
Lattice structure and conductance for (b) $w_{20}$ nanoribbon, and (c)
$w_{110}$ nanoribbon. The gray region indicates the bulk topological gap. The
black dashed line corresponds to the conductance in the pristine case. All
structures consist of a single building block $N=1$ containing the vacancy
coupled to semi-infinite pristine ribbons.

Figure 4 shows the conductance of the $w_{20}$ and $w_{110}$ ribbons in the
presence of a single vacancy placed close to one of the system edges. For this
calculation, the scattering region corresponds to a single building block
$N=1$ containing the vacancy. For the $w_{20}$ ribbon the quantized
conductance is destroyed, showing that the edge states are not robust against
disorder. In turn, the $w_{110}$ ribbon shows no deviation from perfect
conductance $G_{0}=2e^{2}/h$ within the topological gap.

Several studies on a variety of 2D materials indicate that single-vacancies
give rise to localized states [64, 29]. It has been further shown that such
disorder-induced localized states cause the formation of local magnetic
moments [65, 66, 64, 67, 68] that, if close to the system edges, can be
detrimental for the topological protection, differently from dual topological
insulators [69]. Let us study the relevance of these findings to bismuthene.

Figure 5 shows the local density of states (LDOS) for the $w_{20}$, $w_{65}$
and $w_{110}$ nanoribbons calculated for the energy window
$$0.2\text{\,}\mathrm{eV}$<E<$0.3\text{\,}\mathrm{eV}$$. The strong
enhancement of the LDOS centered around the vacancy corresponds to
exponentially decaying orbitals paired after the atom relaxation. Figure 5(a)
shows that the vacancy states increase the overlap between inter-edge states,
whereas for the $w_{65}$ nanoribbon the overlap is small even in the presence
of a vacancy. For the $w_{110}$ ribbon shown in Fig. 5(c) the overlap between
the edge states and the vacancy localized states is negligible.

Figure 5: Top ($x$–$y$ axis) and lateral ($y$–$z$ axis) projections of the
local density of states (LDOS) for single vacancy in (a) $w_{20}$, (b)
$w_{65}$ and (c) $w_{110}$ nanoribbons. The isosurface value for the LDOS is
of
$0.025\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-3}\text{\,}{\mathrm{eV}}^{-1}$,
corresponding to an energy range
$$0.2\text{\,}\mathrm{eV}$<E<$0.3\text{\,}\mathrm{eV}$$. All structures
consist of a single building block $N=1$ containing the vacancy coupled to
semi-infinite pristine ribbons.

Our ab initio fully relativistic calculations show that, regardless of the
superlattice size, type of basis-set used in the calculation, and position of
the vacancy (bulk or nanoribbon edge), there is no indication of a defect-
induced magnetic moment, more precisely, we find $\mu<10^{-4}\rm\mu_{B}$.
Interestingly, when the SOC is artificially turned off, our spin polarized
calculations give a magnetic moment of $0.8\;\rm\mu_{B}$ around the vacancy.
This suggests that the local orbital hybridization, mainly $s$ and $p$, caused
by the spin-orbit interaction is probably responsible for the spin-
polarization quench in bismuthene. Besides, the low energy electronic
properties of bismuthene are dominated by $p$-orbitals, and in this material,
there is no evidence of strong interaction effects due to localized electrons;
therefore the non-magnetic ground state is adequately described by DFT [70,
71, 23, 17]. Based on these results, we rule out vacancy induced magnetic
moments as a mechanism [29] to hinder the topological protection in
bismuthene.

We now study the conductance $G$ as a function of vacancy concentration. This
is done by modelling nanoribbons as a sequence of $N$ building blocks with a
given concentration $n_{\rm V}$ of randomly placed vacancies, as described in
Sec. II. The calculation of the electronic properties of the individual
building blocks is the main computational bottleneck of our study. Smaller
building blocks optimize the computation time, but their lengths $\ell$ have
to be large enough to correctly describe the vacancy-induced localized states.
We find that $\ell=$17.4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ gives
accurate results with a feasible computational time even for the wider ribbon.
We choose $N=90$ to address nanoribbons of realistic sizes. Here, the
considered scattering region has a total length of
$156.6\text{\,}\mathrm{nm}$.

Figure 6 shows that vacancies, even in small concentrations, have a strong
effect on the transport properties of bismuthene nanoribbons. In all cases,
the pristine conductance quantization is destroyed outside the topological gap
$\Delta_{\rm TG}$. For instance, for $n_{\rm V}\approx 0.2\%$, one observes
the onset of localization ($G/G_{0}\ll 1$) for $w_{20}$ and a very strong
suppression of $G$ for $w_{110}$, when compared to the pristine case. For
energies outside $\Delta_{\rm TG}$, bismuthene is an ordinary material and the
results can be interpreted in terms of standard disorder-induced
backscattering mechanisms. In what follows we discuss the behavior of $G$ for
energies within the topological phase.

Figure 6: Conductance $G$ (in units of $e^{2}/h$) as a function of the energy
$E$ (in eV) for several vacancy concentrations for the $w_{20}$, $w_{65}$, and
$w_{110}$ nanoribbons. The ribbon widths and vacancy concentrations are
indicated in each panel. The blue line corresponds to $G$ in the presence of
disorder, while the black dashed one stands for $G$ in the pristine case. The
gray region indicates the topological gap. In all cases the length is
$157\text{\,}\mathrm{n}\mathrm{m}$.

Figures 6(a) to 6(d) correspond to the narrow width nanoribbon $w_{20}$ with
vacancy concentration $n_{V}$ ranging from $0.22\;\%$ to $0.72\;\%$. As
discussed above, these systems are not topological insulators. Interestingly,
for increasing $n_{V}$ the $w_{20}$ ribbons behave as trivial insulators for
energies outside the topological gap, but the conductance is less suppressed
inside $\Delta_{\rm TG}$. A similar behavior has been studied in graphene [72,
73] and MoS2 nanoribbons [74, 75]. The large momentum transfer necessary to
enable backscattering processes, as inferred from the trivial pristine
nanoribbon electronic band structure in Fig. 2(a), preserves the conductance
in narrow energy intervals. See, for instance [72, 73] for detailed
discussions.

Figures 6(e) to 6(h) correspond to $w_{65}$ nanoribbons of intermediate width
and $n_{V}$ ranging from $0.06\,\%$ to $0.36\,\%$. For energies outside
$\Delta_{\rm TG}$ the conductance is strongly suppressed with respect to the
pristine case, as expected for a trivial disordered system. In contrast to the
$w_{20}$ case, for $n_{V}\lesssim 0.36\,\%$ the system does not display any
Anderson localization features. More interestingly, the conductance is close
to $G_{0}$ within the topological bandgap. The edge states are not fully
protected by topology: there are several energy intervals where $G$ is
strongly suppressed. The size of such intervals and the $G$ suppression grow
with increasing $n_{V}$. Interedge backscattering processes are induced by
disorder due to the hybridization of the edge states with the randomly
distributed vacancy-induced localized states in the ribbon. For weak disorder,
$G/G_{0}<1$ only in narrow energy windows that depend on the disorder
configuration. This mechanism has already been shown to destroy topological
protection for such narrow nanoribbons [35] and for wide ones in a tight-
binding toy model with much larger vacancy concentrations, $n_{V}\simeq 2\%$
[30].

Let us now address the case of wide nanoribbons, namely, $w\gg\xi$. Figures
6(i) to 6(l) correspond to $w_{110}$ nanoribbons with $n_{V}$ ranging from
$0.13\,\%$ to $0.48\,\%$. As in the previous cases, $G$ decreases with
increasing $n_{V}$ for energies outside $\Delta_{\rm TG}$. In distinction, the
edge states are topologically protected by time-reversal symmetry and
$G=G_{0}$ over most of the bulk band gap energies. Although topological
protection is more robust than in the previous cases, when $n_{V}\gtrsim
0.30\,\%$, $G/G_{0}$ is strongly suppressed for certain energy intervals
within $\Delta_{\rm TG}$, see Figs. 6(k) and (l), indicating the presence of
vacancy-induced inter-edge backscattering processes.

These results suggest that the number of vacancy-induced states necessary for
an effective inter-edge hybridization of the system edge states increases with
the nanoribbon width $w$. For different building block assemblies we also
observe a similar qualitative behavior, namely, a perfect conductance $G_{0}$
over most of the topological gap with a strong suppression on narrow energy
intervals as a function of vacancy concentration.

##  IV Conclusions and Outlook

In this paper we have studied the robustness of the conductance quantization
against vacancy disorder in large scale nanoscopic buckled bismuthene
nanoribbons at the QSH phase.

We have found that vacancies in bismuthene give rise to non-magnetic mid-gap
localized states, ruling out local magnetic moments as a mechanism to destroy
topological protection as suggested by Ref. [29]. We have shown that these
vacancy-induced mid-gap states can give rise to inter-edge scattering
processes. Elastic backscattering is enabled when such states are close in
energy and the defect concentration is sufficiently large so that the vacancy-
induced states overlap, creating an inter-edge backscattering channel. The
backscattering processes depend on the edge state penetration depth, vacancy
concentration, and nanoribbon width. The interplay of these quantities has
been qualitatively discussed in QSH tight-binding models [30] and within DFT
for narrow systems [35]. Here, we have established the presence of vacancy-
induced inter-edge backscattering processes in bismuthene nanoribbons of
realistic sizes using ab initio techniques.

Our calculations show different transport behavior for bulk and edge states,
the first demonstrating localization effects and the latter showing robust
topological response for low vacancy concentrations. At moderate $n_{V}$
values, topological protection is destroyed even for wide ribbons $w\gg\xi$.

Our findings are also applicable to other materials in the QSH regime. Since
the penetration depth is material dependent, we conclude that it is possible
to engineer different samples with QSH electronic transport behavior by a
suitable tuning of the vacancy concentration. Our findings suggest an
interesting application for a spintronic device, the level of doping or the
width of the device modulates the edge states degeneracy, therefore providing
the ON/OFF states for a transistor switch. This might be obtained by changing
the chemical potentials on different leads or by applying a gate voltage. In
particular, for the topological $w_{65}$ and $w_{110}$ nanoribbons,
characterized by a pronounced drop in the conductance becoming broader for
energies around the valence band as the level of vacancy concentration
increases, allowing the possibility of reaching high ON/OFF ratios. The
mechanism to modify the conduction in these nanoribbons does not rely on
changing the topology of its band structure as it would happen by applying an
electric or magnetic field.

###### Acknowledgements.

This work is partially supported by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001; FAPESP (Grants
19/04527-0, 16/14011-2, 17/18139-6, and 17/02317-2); CNPq (Grant
308801/2015-6), INCT-Nanocarbono, and FAPERJ (Grant E-26/202.882/2018). The
authors acknowledge the Brazilian National Scientific Computing Laboratory
(LNCC) for computational resources of the SDumont computer.

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