# Stability of destructive quantum interference antiresonances in electron
transport through graphene nanostructures

Angelo Valli  Institute for Theoretical Physics, Vienna University of
Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria  Department of
Theoretical Physics, Institute of Physics, Budapest University of Technology
and Economics, Müegyetem rkp. 3., H-1111 Budapest, Hungary  Thomas Fabian
Institute for Theoretical Physics, Vienna University of Technology, Wiedner
Hauptstrasse 8-10, A-1040 Vienna, Austria  Florian Libisch  Institute for
Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse
8-10, A-1040 Vienna, Austria  Robert Stadler  Institute for Theoretical
Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040
Vienna, Austria

###### Abstract

We investigate the stability of destructive quantum interference (DQI) in
electron transport through graphene nanostructures connected to source and
drain electrodes. The fingerprint of DQI is the presence of an antiresonance
in the transmission function, and its origin is deeply connected to the
topology of the atomic structure, which we discuss in terms of symmetry
arguments supported by numerical simulations. A systematic analysis of the
influence of system size on the transmission function reveals that the DQI
antiresonance persists for large systems in the ballistic regime and
establishes the quantum confinement gap as the intrinsic resolution limit to
detect QI effects. Furthermore, we consider the influence of disorder,
electron-electron and electron-phonon interactions, and provide quantitative
criteria for the robustness of DQI in their presence. We find that the
conductance is quite sensitive to perturbations, and its value alone may not
be sufficient to characterize the QI properties of a junction. Instead, the
characteristic behavior of the transmission function is more resilient, and we
suggest it retains information on the presence of an antiresonance even if DQI
is partially concealed or suppressed. At the same time, DQI results in a non-
linear transport regime in the current-bias characteristics that can be
possibly detected in transport experiments.

Quantum interference (QI) in electron transport is a purely quantum mechanical
phenomenon of keen interest in the field of molecular electronics. In single-
molecule junctions, experimental evidence of both destructive [1, 2, 3, 4, 5,
6] (DQI) and constructive [7, 8, 9] (CQI) quantum interference, as well as
their control [10, 11] has been extensively reported. Sharp antiresonances due
to QI with asymmetric Fano shapes and symmetric Mach-Zehnder shapes can be
found and theoretically explained [12, 13, 14, 15, 16, 17, 18]. Close to the
Fermi level, sharp antiresonances drastically affect the transport properties
[19, 20, 21, 22], and indeed, QI has been proposed as a paradigm for a wide
spectrum of technological applications, ranging from logic gates [23, 15]
single-molecule transistors [24, 25], molecular switches [5, 26], spin-filters
[27, 28, 29, 30], as well as for enhancing the performance of thermoelectric
[4, 31, 32, 8] and chemical sensing [21, 33, 34] devices.

Fundamentally, QI effects occur when the transmission of electrons across a
resistor is phase-coherent, which is realized when the length of the
transmission channel $\cal L$ is shorter than the characteristic scale (mean
free path) associated with elastic ($\lambda_{e}$) and phase-breaking
($\lambda_{\phi}$) electron scattering events, i.e., ${\cal
L}<\lambda_{e},\lambda_{\phi}$ (ballistic regime). For large ballistic
cavities of size $L$, an intuitive interpretation is provided by semiclassical
approaches like Gutzwiller’s trace formula [35]. By contrast, QI in the
context of molecular electronics rather originates from contributions between
different molecular orbitals (MO), which couple to the electrodes with similar
strength but different phases [36]. A qualitative understanding of QI can be
achieved within a single-particle picture making QI deceptively simple.
However, a careful treatment is warranted, since the loss of electronic phase
coherence is typically (but not exclusively) associated to inelastic processes
which are often neglected in theoretical calculations, such as, e.g.,
electron-electron or electron-phonon scattering.

Notwithstanding different sources of scattering, which can possibly spoil
phase coherence, QI effects can be detected under experimental conditions.
While early evidence of QI has been rather indirect [37], the dramatic
technological progress of the last few decades eventually lead to a clear
direct detection of DQI [1].

It is worth stressing that current fabrication techniques do not offer high
enough spatial resolution to reliably contact single molecules with well-
defined geometries. Instead, transport measurements of single-molecule devices
are typically performed with break-junction techniques, either in a
mechanically-controlled or an scanning tunneling microscope setup [38]. In the
simplest case, a thin gold wire is mechanically strained until the wire
breaks, creating two fresh electrode surfaces that can now be connected by
individual molecules in a self-assembled way. Consequently, the junction is
subject to configuration fluctuations, as the atomic arrangement is unknown.
The measurement is therefore repeated over a series of break-junction events
to obtain reliable statistics. Despite the statistical nature of break-
junction experiments, there are protocols to identify the suppression of the
electronic transmission due to DQI, e.g., in differential conductance
($dI/dV_{b}$) maps [1, 39, 40], or through the analysis of conductance
histograms [6, 11, 41, 2, 42] of single-molecule junctions. Remarkably, QI
effects seem also to be surprisingly stable, having been detected even at room
temperature [1, 10, 42, 9], and on length scales well beyond that of single-
molecule junctions, in systems such as, e.g., macromolecules [43] as well as
nanostructured graphene [28, 29, 44, 45, 19].

In particular, graphene represents a natural platform for high performance
nanoelectronics [46], thanks to its unique physical properties, and its
ability to form atomically precise nanostructures [47, 48, 49, 50, 45].
Antiresonances akin to those occurring in single-molecule junctions have been
predicted theoretically [51, 52, 53, 54, 55, 28, 29, 56, 57] and confirmed
experimentally [19, 58, 59, 60, 45]. QI effects can also occur in the
diffusive regime, i.e., $\lambda_{e}\ll{\cal L}<\lambda_{\phi}$, where weak
localization can arise due to the coherent superposition of random scattering
paths [61]. Since the effective mean free path of electrons in graphene
strongly depends on the local doping [62], both regimes may be relevant for a
single device at different energies.

In the present work we analyze, within a unified framework, the stability of
QI antiresonances in graphene nanostructures. We show that the the ballistic
transmission function displays a characteristic behavior within the quantum
confinement gap, that can be entirely ascribed to the existence of an
antiresonance. We consider a wide range of different mechanisms which are
detrimental to QI, including disorder, electron-electron, and electron-phonon
interactions and quantify their effect on a QI antiresonance. Our results thus
provide stringent boundaries on the possibility of detecting QI effects in an
experimentally-relevant parameter range.

##  I Topological conditions for DQI

Predicting the occurrence of QI in the electron transmission function is a
challenging task [23, 63, 64, 65, 66, 12]. A few _back-of-the-envelope_
methods have been developed [23, 63, 65, 67, 68, 14], including a graphical
scheme, which relies on a visual inspection of the molecular structure and the
topology of the Hückel (or tight-binding) Hamiltonian, and is able to predict
DQI without the need for explicit numerical simulations [23]. This graphical
scheme has been validated against density functional theory [63], it was
extended to hetero-atoms [64], non-alternant hydrocarbons such as azulene [65,
66], and further generalized in a diagrammatic fashion to calculate the
position of the antiresonances [65]. The Coulson-Rushbrooke pairing theorem
from quantum chemistry [69] has recently reconciled the graphical approach
with a MOs perspective more common for a theoretical analysis of molecular
properties [36].

The graphical scheme and the pairing theorem naturally link the sublattice
structure to DQI. The term alternant hydrocarbons refers to conjugated
hydrocarbon systems where carbon atoms can be divided into two subsets (or
sublattices) with nearest-neighbor interactions only between two atoms of the
two different subsets – i.e., any hydrocarbon system containing no odd-
membered rings. For such systems a QI antiresonance appears between the
highest occupied (HOMO) and the lowest unoccupied (LUMO) MOs [36, 70, 71, 28,
29, 57] if the contact sites belong to the same sublattice. If the contact
sites belong to different sublattices, DQI is still possible if certain
conditions are met [57], but it is in general harder to predict. These
properties have also been confirmed within the more general framework of
Green’s functions formalism, [28, 29, 56, 57, 72, 73] which also revealed a
rich QI phenomenology [73]. Since graphene nanostructures can be considered,
and even chemically synthesized [74, 47], as extended polyaromatic molecules
with an alternant structure and hydrogen-passivated edges, the sublattice
scenario naturally holds. DQI has been demonstrated numerically for
rectangular [56, 57] and hexagonal [28, 29] graphene nanostrutures. Indeed,
such an argument predicts DQI in graphene nanostructure with any size and
shape, with some _caveats_ [57, 72]. Such a topological argument greatly
simplifies the analysis, as it provides a very simple criterion to identify
which contact configurations display a QI antiresonance. It is equally clear,
however, that for larger graphene flakes, more phases accumulate, and as a
consequence, the characteristic dip due to DQI should become narrower. Indeed,
only the existence of the dip is topologically protected - there is no general
statement on its width. Additional interactions such as electron-electron or
electron-phonon coupling will further affect the phase coherence required for
quantum interference effects. Consequently, one might ask under which
realistic conditions DQI effects can still be measured. In the following, we
intend to provide a comprehensive answer to this question, investigating the
role of system size, disorder, electron-phonon, and electron-electron coupling
on the DQI dip.

Figure 1: Effects of system size on DQI. (a) Representative $N\times N$
graphene nanostructure. (b) Transmission function in the meta an para
configurations for a $4\times 4$ nanostructure. (c) Connections in the meta
and para configurations, the black and white circles indicate the sets of AO
$\\{\ell\\}$ and $\\{r\\}$ connected to the L and R leads, respectively. (d)
Evolution of the transmission function in the meta configuration with system
size. Within the gap $T(\omega)\propto\omega^{2}$ and saturates as
$\omega\rightarrow\omega^{\mathrm{DQI}}$. (e) Evolution of the HOMO-LUMO gap
$\Delta_{0}$ with system size. (f) Data collapse of the ratio ${\xi}$ between
the transmission function in the meta and para configurations, for system size
$N\gtrsim 4$.

##  II Results

We consider a graphene nanostructures with rhomboidal shape and size
$N\\!\times\\!N$, where $N$ denotes the number of rings along each edge, and
$n_{C}$ the number of C atoms in the nanostructure [see Fig. 1(a)]. As we
discuss below, our arguments are rather general and are expected to remain
valid for graphene nanostructure with other shapes (e.g., rectangular or
hexagonal).

We begin our analysis by studying the dependence of DQI features on system
size within the tight-binding approximation. The corresponding Hamiltonian,
describing a single $p_{z}$ AO per C atom, reads

${\cal H}_{0}=t\sum_{\langle
ij\rangle}\sum_{\sigma}c^{\dagger}_{i\sigma}c^{\phantom{\dagger}}_{j\sigma},$
(1)

where $c^{(\dagger)}_{i\sigma}$ is the annihilation (creation) operator for an
electron on site $i$ with spin $\sigma$, and $t$ is the hopping between
nearest-neighbor sites $i$ and $j$, for which we take a typical value
$t=$2.7\text{\,}\mathrm{eV}$$. Including longer-range hoppings, or deriving
them ab-initio [75], e.g., breaks the particle-hole symmetry and shifts the
position of the antiresonance within the gap [28] but does not invalidate the
conclusions of our analysis.

Since the tight-binding Hamiltonian ${\cal H}_{0}$ does not include any
scattering mechanism (neither elastic nor inelastic), in this approximation
the electronic excitations do not decay, i.e., are characterized by a lifetime
$\tau=\infty$ and mean-free paths $\lambda_{e}=\lambda_{\phi}=\infty$. As we
introduce disorder, electron-electron, or electron-phonon scattering below, we
estimate the characteristic length scales on which QI effects are suppressed.

###  II.1 Universal transport behavior due to DQI

We describe electronic transport within the Landauer formalism [76], where the
transmission function is given by

$T(\omega)=\mathrm{Tr}\Big{[}\Gamma^{L}(\omega)G^{\dagger}(\omega)\Gamma^{R}(\omega)G(\omega)\Big{]},$
(2)

in terms of the molecular Green’s function $G(\omega)$ and the coupling matrix
$\Gamma^{L/R}$ to the left ($L$) and right ($R$) leads. Let us further denote
by $\\{\ell\\}$ and $\\{r\\}$ the sets of edge AOs contacted to the $L$ and
$R$ leads, respectively. Within a wideband limit (WBL) approximation for the
leads, the diagonal coupling elements $\Gamma^{L}_{\ell\ell}$ and
$\Gamma^{R}_{rr}$ are given by energy-independent constants $\Gamma$ for edge
C atoms $\\{\ell\\}$ and $\\{r\\}$, and zero otherwise (see also SI). In the
following, we set all non-zero couplings to the leads to
$\Gamma=0.0004t\approx$1\text{\,}\mathrm{meV}$$ (or
$\approx$10\text{\,}\mathrm{K}$$, for reference) unless otherwise specified.
We have verified numerically that neither the WBL approximation [28, 77] nor
neglecting non-diagonal couplings $\Gamma_{i\neq j}$ [78, 79, 17, 80]
fundamentally change the QI properties of the system under study.

We decompose the transmission function as a sum over independent channels

$T(\omega)=\sum_{\ell,r}T_{\ell\rightarrow r}(\omega)=\Gamma^{2}\sum_{\ell
r}|G_{\ell r}(\omega)|^{2}.$ (3)

In the non-resonant transport regime, the condition for an antiresonance in
the transmission function due to DQI becomes $\Re G_{\ell
r}(\omega_{\mathrm{DQI}})=0$, for a given frequency $\omega_{\mathrm{DQI}}$
[28, 29, 73]. Indeed, we find a pronounced antiresonance at $\omega=0$ in a
$4\times 4$ graphene nanostructure (solid line in Fig. 1b) if $\ell$ and $r$
belong to the same sublattice (labeled as _meta_ configuration, Fig. 1c). If
$\ell$ and $r$ instead belong to different sublattices (labeled as _para_
configuration, Fig. 1d) there is, in general, no antiresonance, and the
transmission function within the HOMO-LUMO gap saturates at a much higher
value [dashed line in Fig. 1(b)]. These results are consistent with the
topological conditions for DQI discussed above.

In the meta configuration, we identify a _universal_ behavior of the
transmission function at $\omega=0$. We calculate the transmission function
$T(\omega>0)$ through graphene nanostructures of increasing size (Fig. 1(d))
and find a dip only in the meta configuration. (the transmission function for
$\omega<0$ remains symmetric due to the particle-hole symmetry in our model).
The evolution of the LUMO resonance at $\omega_{\mathrm{LUMO}}$ shows that the
HOMO-LUMO gap, defined as
$\Delta_{0}=\omega_{\mathrm{LUMO}}-\omega_{\mathrm{HOMO}}$, decreases with
systems size (Fig. 1(e)), in agreement with the literature [81, 82, 83]. The
transmission function saturates close to the antiresonance
$\omega\rightarrow\omega_{\mathrm{DQI}}$ (which is _pinned_ at
$\omega_{\mathrm{DQI}}=0$ due to particle-hole symmetry) while at higher
energies $|\omega|\lesssim|\omega_{\mathrm{LUMO}}-\Gamma|$, it follows the
universal behavior

$T_{\mathrm{metha}}(\omega)=\alpha_{N}\cdot 10^{N}\cdot(\omega/t)^{2},$ (4)

with $\alpha_{N}$ a size-dependent constant. Such a behavior follows naturally
from Eq. (3) as $\Re G_{\ell
r}(\omega\rightarrow\omega_{\mathrm{DQI}})\propto\omega$ [84], yet it extends
over the whole energy gap.

From the numerical data we observe that the saturation value
$T(\omega_{\mathrm{DQI}})$ is weakly size-dependent. However, at a finite
energy scale, e.g., due to experimental resolution, the transmission increases
approximately by an order or magnitude when increasing $N$ by one unit. This
is because $\Delta_{0}$ decreases with size but the coupling between molecule
and leads $\Gamma$, which determines the width of the transmission features,
remains constant. An analogous dependence of the transmission function is
observed at constant $\Delta_{0}$ by increasing $\Gamma$ (see SI). To remove
this contribution, we consider the ratio
${\xi}(\omega)=T_{\mathrm{meta}}(\omega)/T_{\mathrm{para}}(\omega)$ between
the transmission in the meta and para configurations [Fig. 1(f)]. Away from
the Dirac point, i.e., for $\left|\omega/t\right|\gg 0$ the transmission in
meta and para configurations approach each other, ${\xi}(1)\rightarrow 1$
because the position and the width of the LUMO and HOMO resonances are
(approximately) the same in the meta and para configurations, cfr. Fig. 1(b).
For energies within the HOMO-LUMO gap, all the curves $\xi(\omega)$ _collapse_
onto each other for $N\geq 4$ [see Fig. 1(f)]. The underlying reason is
related to the distribution of poles of the transmission function on the
complex plane: for frequencies sufficiently close to zero, i.e., sufficiently
far away from the other poles at finite $\omega$, the behavior around the gap
becomes universal. Hence, by disentangling the behavior induced by the
antiresonance from $\Delta_{0}$ and $\Gamma$, we can draw the following
conclusions: (i) for $N\geq 4$, the transmission $T_{\mathrm{meta}}$ within
the HOMO-LUMO gap approaches the universal form of Eq. (4), with the prefactor
$\alpha_{N}$ determined by
$T_{\mathrm{meta}}(\omega_{\mathrm{LUMO}})=T_{\mathrm{para}}(\omega_{\mathrm{LUMO}})$
(i.e., $\xi(\omega_{\mathrm{LUMO}})\rightarrow 1$), and (ii) our argument
remains valid for differently shaped graphene nanostructures, especially as
the surface-to-bulk ratio decreases. Hence, in the following we shall focus on
$4\times 4$ graphene nanostructures, which strike a good balance between
numerical cost while displaying weak-to-none finite-size effects on the
universal behavior of the transmission of Eq. (4).

Figure 2: (a) Schematic representation of the local disorder. (b,c)
Transmission function for a specific disorder realization (b) and averaged
over hundreds of realizations (c), for different values of disorder strength.
(d,e) Rescaling the energy allows to align the position of the antiresonance
$\omega_{\mathrm{DQI}}$ and of the LUMO resonance $\omega_{\mathrm{LUMO}}$ for
each disorder realization (or their average). Disorder strongly enhances the
conductance, but the characteristic $\omega^{2}$ behavior is clearly observed
over an extended energy range up to moderate disorder strength. Dashed grey
lines mark the transmission functions of the corresponding pristine para
configurations as a reference. (f) In contrast to the other transmission
features whose variance increases with the disorder strength, a statistical
analysis reveals that the size of the HOMO-LUMO gap is remarkably stable
around the pristine values (red dashed lines).

Remarkably, the universal behavior of the transmission function reflects on
the current-bias ($I-V_{b}$) characteristics, which can be directly measured
in the experiment, whereas the transmission is loosely related to the
differential conductance $dI/dV_{b}$, at least at low bias voltages
$eV_{b}=\mu_{L}-\mu_{R}$, where $\mu_{L}$ ($\mu_{R}$) is the chemical
potential in the left (right) contact. We model the bias dependence in the
Landauer-Büttiker framework, i.e.,

$I\approx\frac{e}{h}\int_{\mu_{R}}^{\mu_{L}}T(\omega)\mathrm{d}\omega,$ (5)

where $e$ denotes the electric charge and $h$ the Planck constant. Within the
HOMO-LUMO gap, we can insert the universal behavior of the transmission
function. For the meta configuration, the $\omega^{2}$ behavior of Eq. (4)
yields a non-linear characteristics $I_{\mathrm{meta}}\propto
V_{b}+t^{-2}V_{b}^{3}$, whereas in the para configuration the slowly-varying
transmission $T_{\mathrm{para}}(\omega)\approx$ const results in a linear
characteristics $I_{\mathrm{para}}\propto V_{b}$ (see SI). Hence, we identify
this specific non-linear transport regime as a fingerprint of DQI, which is
useful for the experimental characterization of a junction, besides the value
of the zero-bias conductance $G=(e^{2}/h)T(0)$. In single-molecule junctions,
with a gap in the range of a few $\text{\,}\mathrm{e}\mathrm{V}$, the
experimental resolution necessary to resolve the in-gap $I-V_{b}$
characteristics is hardly an issue. By contrast, the HOMO-LUMO gap quickly
becomes unmeasurably small for large graphene nanostructures, i.e., below 1
meV for $N=11$. Another possible limitation is the contribution to the
transmission function through the $\sigma$ channel, which is neglected in our
numerical simulation, and can in principle mask interference effects in the
$\pi$ channel. However, the $\sigma$ contribution is naturally suppressed for
longer molecules due to a faster decay with length (see, e.g., [85] and
references therein) and can therefore be expected to be less relevant upon
increasing systems size.

In summary, for larger flakes, the main restriction for observing DQI effects
is the energy scale of the HOMO-LUMO gap, which approaches the limit of
experimental resolution at $N\gtrsim 12$.

###  II.2 Disorder

For exploiting DQI in practice, robustness with respect to moderate disorder
is critical. In state-of-the-art hBN-graphene sandwich devices, bulk disorder
is of the order of a few $\text{\,}\mathrm{meV}$ [50], with a major
contribution from long-range strain modulations [86], and evidence of
ballistic transport exceeding $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ has
been reported [87].

We investigate the influence of uncorrelated local random disorder focusing on
the configurations exhibiting DQI. The scope of our analysis is two-fold: (i)
we show how the transmission function changes with respect to the pristine
sample, for configurations with _static_ disorder, and (ii) we look at the
transmission function averaged over hundreds of disorder realizations (also
referred to as _dynamic_ disorder) by adding the individual transmission
traces incoherently. The disorder average is representative of the statistical
nature of experimental measurements in a break-junction setup. The combination
of the two effects allows us to understand the stability of DQI against
disorder. We find that QI is surprisingly robust up to disorder amplitudes of
$\approx$40\text{\,}\mathrm{meV}$$, which is at least one order of magnitude
above the experimental estimates for state-of-the-art devices.

The local disorder is described by adding to the tight-binding Hamiltonian (1)
the following term

${\cal H}_{\mathrm{disorder}}=\sum_{i\sigma}\epsilon_{i}n_{i\sigma},$ (6)

where $n_{i\sigma}=c^{\dagger}_{i\sigma}c^{\phantom{\dagger}}_{i\sigma}$ is
the number operator and $\epsilon_{i}$ is the on-site energy of site $i$. We
create random disorder with $\langle\epsilon_{i}\rangle=0$ and
$\langle\epsilon_{i}^{2}\rangle=A^{2}$, with $A$ the disorder amplitude.

We calculate the transmission function for different disorder realizations
(Fig. 2). Since disorder breaks the particle-hole symmetry, the energies of
the MOs, and their projection onto the C-$p_{z}$ AOs no longer fulfill the
Coulson-Rushbrooke pairing theorem. In particular, the position of the
frontier MOs ($\omega_{\mathrm{HOMO}}$ and $\omega_{\mathrm{LUMO}}$) now
depends on the specific disorder realization, shifting the characteristic DQI
dip in the transmission function randomly (Fig. LABEL:fig:effect_disorderb).
The antiresonance, which emerges from the cancellation of contributions
involving all MOs [36], is no longer pinned at the Fermi energy, i.e.,
$\omega_{\mathrm{DQI}}\neq 0$. Moreover, the cancellation of the transmission
is partial, as $\omega_{\mathrm{DQI}}$ becomes channel-dependent (see SI).
Averaging over disorder smears out the signatures of DQI Fig. 2(c). Close to
the DQI resonance, the conductance is thus effectively _enhanced_ by disorder,
both in individual realizations and on average. In this sense, we can talk
about _disorder-assisted_ transport, as the role of the disorder is to
suppress the QI effects responsible for the transmission minimum. Finally, we
note that for individual realizations the resonant transport through the MOs
remains unitary, while the average over disorder introduces an effective
decoherence and drives the system away from the ballistic regime.

In order to understand how disorder affects the universal behavior of the
transmission function, we rescale the conductance traces as a function of the
dimensionless scale
$(\omega-\omega_{\mathrm{DQI}})/(\omega_{\mathrm{LUMO}}-\omega_{\mathrm{DQI}})$,
using the corresponding value of $\omega_{\mathrm{DQI}}$ and
$\omega_{\mathrm{LUMO}}$ extracted for each individual disorder realization
[Fig. 2(d)] or their average [Fig. 2(e)]. This aligns the position of both the
antiresonance and the gap edge for both static and dynamic disorder. Focusing
on the transmission function close to the antiresonance, we find quadratic
enhancement with disorder strength, i.e., $T(\omega_{\mathrm{DQI}})\propto
A^{2}$ for both individual configurations and on average (see SI). Hence, for
any given value of the disorder strength, the transmission function
interpolates between a constant regime at $\omega\approx\omega_{\mathrm{DQI}}$
and the characteristic $\omega^{2}$ behavior at higher energies, and the
crossover scale increases with disorder strength. Above a critical threshold,
the transmission function becomes qualitatively indistinguishable from that of
the para configuration. Despite a lower conductance the effects of DQI are
lost. This scenario emerges for both individual disorder configurations and on
average. This suggests that an analysis of the conductance alone may not be
conclusive for detecting DQI while the effects on the transmission function at
finite energy and the corresponding non-linearity of the $I-V_{b}$
characteristics are more resilient to the effects of disorder (see SI).

Disorder may also be characterized by a disorder scattering length. For any
given realization of static disorder the electron transport is ballistic and
the transmission in the resonant regime (e.g., at the LUMO resonance) is
unitary, hence $\lambda_{\mathrm{disorder}}=\infty$. After averaging over the
disorder realizations the resonant transmission is reduced as [88]

$T(\omega_{\mathrm{LUMO}})=\frac{\lambda_{\mathrm{disorder}}}{{\cal
L}+\lambda_{\mathrm{disorder}}}.$ (7)

For $\lambda_{\mathrm{disorder}}\gg{\cal L}$ unitary transport is restored.
Taking as system size ${\cal L}\approx$1\text{\,}\mathrm{nm}$$, estimated as
the longest distance among all $\ell\rightarrow r$ channels, we find, e.g.,
$\lambda_{\mathrm{disorder}}\approx$1.2\text{\,}\mathrm{nm}$$ at
$A=$30\text{\,}\mathrm{meV}$$, and a relation
$\lambda_{\mathrm{disorder}}(A)\propto A^{-1}$ (see SI). Alternatively, if we
express the condition above as

$\frac{\lambda_{\mathrm{disorder}}}{{\cal
L}}=\frac{T(\omega_{\mathrm{LUMO}})}{1-T(\omega_{\mathrm{LUMO}})},$ (8)

it is possible to estimate a disorder threshold (independent on ${\cal L}$)
defined by $\lambda_{\mathrm{disorder}}\approx{\cal L}$, corresponding to a
resonant transmission reduced to half its ballistic value. We find
$A_{\mathrm{critical}}\lesssim$40\text{\,}\mathrm{meV}$$, and verified that
when approaching the critical disorder strength the $\omega^{2}$ transmission
disappears, meaning that the effects of QI are contextually lost (see SI).
Experimentally, the critical value is comparable with disorder estimates for
graphene on a SiO2 substrate but well above estimates for devices encapsulated
in hBN [50]. For individual disorder realizations, QI are lost only at much
higher values of disorder strength (i.e., above those investigated here). We
conclude that the intrinsic (i.e., static) disorder of the sample is not as
detrimental to QI as averaging over several disorder configurations, which can
be considered representative of a statistical analysis of the transport
properties over a series of break-junction configurations.

Finally, we compare the disorder average and variance to a few energy scales
relevant for electron transport, i.e., the position of the resonant
($\omega_{\mathrm{HOMO}}$ and $\omega_{\mathrm{LUMO}}$) and interference
($\omega_{\mathrm{DQI}}$) features, and the gap [Fig. 2(f)]. Since the
disorder distribution is symmetric, i.e., $\langle\epsilon_{k}\rangle=0$, the
mean value of all quantities is close to that of the pristine sample, while
their variance generally increases with disorder strength. As already
discussed, the fluctuation of $\omega_{\mathrm{DQI}}$ in the individual
transmission channels for each disorder realization is responsible for the
enhancement of the conductance, see Figs. 2(b,c,d,e). However, we also find
that the gap $\Delta$ is remarkably stable against disorder, i.e., its
variance is significantly lower than the variance of the frontier MOs position
for any value of the disorder strength. This suggests that the dominant effect
at play is rather a fluctuation of the Fermi level alignment. The data are
also compatible with a weak _increase_ of the gap, e.g., less than $2\%$ with
respect to the pristine value $\Delta_{0}$ at $A\approx\Delta_{0}$.

Overall, limiting disorder to below 10 meV — which is readily possible for
state-of-the-art substrates like hexagonal boron nitride [87] — sufficiently
limits disorder effects to still be able to observe DQI.

###  II.3 Electron-electron interaction

Electronic correlation arising from the Coulomb repulsion are believed to play
a relevant role in electron transport, when electrons are constrained in
narrow conduction channels. Unfortunately, taking into account many-body
effects in the theoretical description of electron transport is a challenging
task [38], and despite attempts to include dynamical correlations (in
different fashions) in the recent literature [89, 90, 84, 91, 28, 29, 92, 93,
94], systematic studies for correlated nanoscale quantum junctions are still
few and far between.

In the following, we add to the tight-binding Hamiltonian a screened local
Coulomb repulsion (i.e., a Hubbard interaction) described by the Hamiltonian

${\cal
H}_{\mathrm{e-e}}=U\sum_{i}n_{i\uparrow}n_{i\downarrow}-\mu\sum_{i}(n_{i\uparrow}+n_{i\downarrow}),$
(9)

where the chemical potential to be $\mu=U/2$ ensures the the Fermi energy is
located at $\omega=0$ for any value of the Coulomb repulsion. We take into
account the many-body effects due to electron-electron interaction within
real-space dynamical mean-field theory (DMFT). In a nutshell, each C atom is
mapped onto an auxiliary impurity problem (describing a single C-$pz$ AO)
self-consistently embedded in the nanostructure, which is solved self-
consistently with Lánczos exact diagonalization, [95, 96] similarly as in
previous works [97, 28, 29, 98]. All the many-body effects are enclosed in the
electronic self-energy $\Sigma_{ij}(\omega)=\Sigma_{ii}(\omega)\delta_{ij}$,
which is a diagonal matrix in the space of the carbon $p_{z}$ AOs. The
dynamical nature of the self-energy allows to simultaneously describe both
coherent and incoherent electronic excitations living on different energy
scales, giving rise to a non-trivial renormalization of the spectral features.
Real-space DMFT has been employed in the literature to describe many-body
effects in a wide range of systems, lacking translational invariance in one or
more spatial directions [99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109,
92, 110, 93, 94], including graphene nanostructures [97, 28, 29, 98, 111]. To
address the transport properties, the transmission function is evaluated from
the Landauer formula, where the Green’s function is dressed with the many-body
self-energy. This approach is a reasonable approximation when the system is
not far away from equilibrium [112, 113, 91, 114, 94] (see also SI).

The local Coulomb repulsion between $\pi$-electrons was estimated by Parr et
al. [115] to be $U=$16.93\text{\,}\mathrm{eV}$$, corresponding to $U/t\approx
6$, while recent studies suggest that a screened value, ranging between
$U/t\approx 1.6$ for graphene and $U/t\approx 1.2$ for a benzene molecule, can
supply for the missing long-range repulsion in the Hubbard model [116]. In the
following, we take $U$ as a parameter, which we vary within a reasonable range
of values in order to explore the system behaviour from the weak- to the
strong-coupling regimes. We perform zero-temperature calculations, which can
be expected to be accurate as long as thermal excitation of charge carriers
across the spectral gap are negligible, i.e., for $k_{B}T\ll\Delta_{0}$, where
$k_{B}$ denotes Boltzmann’s constant.

Figure 3: (a) Schematic representation of the local Coulomb repulsion. (b)
Many-body effects on the transmission function include a renormalization of
the HOMO-LUMO gap, and a redistribution of spectral weight, giving rise to
many-body resonances. (c) The universal behavior of the transmission function
in the presence of the QI antiresonance is preserved in the presence of
electron-electron interactions. (d,e,f) Evolution of the spectral gap
$\Delta$, of the average quasi-particle weight $Z$, and the average scattering
rate $\gamma$ as a function of the Coulomb repulsion $U$. The renormalization
of the gap $\Delta/\Delta_{0}$ correlates with the reduction of $Z$ and it is
mostly controlled by its value at the edges $Z^{\mathrm{min}}$. The low
scattering rate, i.e., $\gamma\ll\Gamma$, due to the lack of electronic states
within the gap, cannot drive the electron transport away from the ballistic
regime.

In Fig. 3(b,c) we show how many-body effects reflect on the transmission
function of the meta configuration. The primary effect is a renormalization of
the spectral gap $\Delta$ with respect to the tight-binding value
$\Delta_{0}$, see also Fig. 3(d). While the low-energy structure is
qualitatively identical to the one of the original tight-binding model, at
higher energy scales, the system displays a significantly richer electronic
structure, characterized by a redistribution of spectral weight between
emergent many-body resonances. Remarkably, the universal $\omega^{2}$
behavior, as well as the QI antiresonance, survive in the presence of
electron-electron interactions ranging from the weak- to the strong-coupling
regime [28, 29]. The physics close to the Fermi energy, which is relevant for
electron transport, can be rationalized —to some extent— in terms of key
parameters derived from the many-body self-energy, i.e., the quasi-particle
weight $Z_{i}$ and scattering rate $\gamma_{i}$

$\displaystyle Z_{i}$ $\displaystyle=$
$\displaystyle\Big{(}1-\frac{\partial}{\partial\omega}\Re\Sigma_{ii}(\omega)\Big{|}_{\omega=0}\Big{)}^{-1}$
(10) $\displaystyle\gamma_{i}$ $\displaystyle=$ $\displaystyle-
Z_{i}\Im\Sigma_{ii}(0)$ (11)

which account for the renormalization of spectral features, and the (inverse)
lifetime of electronic excitations, respectively. Since both depend on
position, and hence display some degree of spatial distribution (see SI), it
is also convenient to look at the spatial average, defined as ${\cal
O}=n_{C}^{-1}\sum_{i}{\cal O}_{i}$ where ${\cal O}_{i}$ is a generic
observable. In order to understand the behavior of those parameters as a
function of the Coulomb repulsion, it is useful to recall that in the non-
interactive limit ($U\rightarrow 0$) the many-body self-energy vanishes,
therefore $Z\rightarrow 1$ and $\gamma\rightarrow 0$.

Within DMFT, the renormalization of the gap of a correlated insulator (or
semiconductor) is controlled by the quasi-particle weight [117, 97, 28, 29].
In a spatially-translational system, one would expect $\Delta=Z\Delta_{0}$. At
the size considered here, the gap decreases _faster_ than the average $Z$ as a
function of the Coulomb repulsion, and its behavior correlates well with the
minimal value of $Z$ over the structure [$Z^{\mathrm{min}}$, compareFigs.
3(d,e)]. The lowest values of $Z$ correspond to edge C atoms, where the
balance between potential and kinetic energy tilts more towards the former
compared to bulk C atoms [97, 98]. The edges thus seem to control the
reduction of the gap despite the average $Z$ being closer to the bulk values,
see Fig. 3(e). Non-local correlations beyond real-space DMFT have been shown
to _enhance_ the spectral gap [104, 109]. A possible interpretation is that
the corrections arising from $\Sigma_{i\neq j}(\omega)$ renormalize the
hopping parameters responsible for the splitting between the bonding (HOMO)
and anti-bonding (LUMO) $\pi$ states. Taking into account such corrections is
extremely challenging, and lies beyond the scope of the present analysis.
However, note that our estimate of the spectral gap should be considered a
_conservative_ approximation.

Figure 4: (a) Schematic representation of the electron-phonon interaction.
(b,c) Effects of the electron-phonon coupling on the transmission function. In
the single-mode case, phonon satellite peaks are visible at integer multiples
of $\omega_{ph}=$26\text{\,}\mathrm{meV}$$ around the frontier MOs. In the
multi-mode case, the weighted average over $\omega_{ph}$ at
$300\text{\,}\mathrm{K}$ broadens the resonances, and the transmission at the
frontier MOs is no longer unitary. (d,e) Evolution of the universal behavior
of the transmission function across the weak- to the strong-electron-phonon
coupling regimes. The transmission functions of the corresponding pristine
para configurations is also given as a reference (dashed grey lines). (f,g)
Single-mode transmission function around $\omega_{\mathrm{LUMO}}$ for
$\omega_{ph}=$13\text{\,}\mathrm{meV}$$ (red line) and
$\omega_{ph}=$26\text{\,}\mathrm{meV}$$ (orange line) and
$g=$100\text{\,}\mathrm{meV}$$ compared to the result for $g=0$ (grey line).
The arrows and the dashed lines highlight the phonon satellite peaks.

The many-body scattering rate $\gamma$ is a measure of the dampening of the
electronic excitations at the Fermi energy, while at finite energy, the self-
energy induces broadening of the spectral and transport features. In a
semiconductor (or insulator) the scattering rate is dramatically suppressed
due to the lack of electronic states available for the scattering. As show in
Fig. 3(f), $\gamma$ increases with the Coulomb repulsion, yet only becomes
comparable with the interface scattering to the leads (i.e.,
$\gamma\sim\Gamma$) towards the strong-coupling regime. In this scenario, the
scattering time
$\tau_{\mathrm{e-e}}=\hbar/\gamma\approx$100\text{\,}\mathrm{fs}$$ suggests
that long-lived electronic excitations propagate through the graphene
nanostructure. Assuming a typical value
$v_{F}\approx$1\text{\times}{10}^{6}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$
for the Fermi velocity in free-standing graphene [118], the electron-electron
scattering length can be estimated as
$\lambda_{\mathrm{e-e}}=v_{F}\tau_{\mathrm{e-e}}\approx$100\text{\,}\mathrm{nm}$$,
which is significantly longer than the size of the nanostructures considered
here. This scenario is in stark contrast with Mott insulators, where the
electronic excitation within the (correlated) energy gap features an extremely
short lifetime [119, 120] and molecules in a open-shell configuration [121].

Hence, the primary effect of electronic-electron scattering is a
renormalization of the spectral gap, while the electronic transport mechanism
is dominated by coherent quasi-particles, and a QI antiresonance is clearly
observed also in the presence of a strong Coulomb repulsion.

###  II.4 Electron-phonon interaction

Molecular vibrations are another possible source of scattering for electrons.
In the literature, a strong inelastic vibrational signal has been reported
close to electron transmission resonances through graphene nanoconstrictions
[122], and the phonon-induced dephasing has been suggested to be most relevant
for molecular junctions in the meta configuration [70]. Indeed, evidence of a
partial quenching of DQI in molecular junctions due to inelastic electron-
phonon contributions has been reported both theoretically [84] and
experimentally [123, 40]. However, QI effects are not completely destroyed,
not even at room temperature, as the conductance difference between junction
in the meta and para configurations still remains substantial [84].

In the following, we analyze the effects of electron-phonon scattering on the
electronic transmission through small graphene nanoflakes. Given the
comparatively small sample size (compared to large-scale graphene devices) we
are interested in coupling through bosonic vibrational modes rather than long-
wavelength acoustic phonons [124, 125]. For the sake of simplicity, we
consider the coupling to a dispersionless such mode, so that the coupled
electron-phonon system is described by the Fröhlich Hamiltonian

${\cal H}_{\mathrm{Fr\ddot{o}hlich}}={\cal H}_{0}+{\cal H}_{\mathrm{ph}}+{\cal
H}_{\mathrm{e-ph}}$ (12)

with

${\cal H}_{\mathrm{ph}}=\omega_{ph}\Big{(}a^{\dagger}a+\frac{1}{2}\Big{)}$
(13)

and

${\cal H}_{\mathrm{e-ph}}=g(a^{\dagger}+a)\sum_{i\sigma}n_{i\sigma},$ (14)

where $a$ ($a^{\dagger})$ is the bosonic annihilation (creation) operator of a
phonon with energy $\omega_{ph}$, coupled to the local electron density
$n_{i}$ through a complex electron-phonon coupling matrix which is diagonal in
the phonon subspace, $g=|g|\exp(\imath\phi)$, where $\phi$ is a random phase.
We treat both $\omega_{ph}$ and $g$ as parameters, in order to understand the
effect of electron-phonon scattering on the QI features in different regimes.
For coupling to vibrational modes, prefactors $g=\lambda\omega_{ph}$ with
$\lambda\approx 0.7$ have been estimated [125], corresponding to
$g\approx$20\text{\,}\mathrm{meV}$$, i.e., at the lower end of the values we
investigate here. Typical values of the electron-phonon coupling for graphene
on a substrate have been estimated to be $38\text{\,}\mathrm{meV}$ on SiO2 and
$62\text{\,}\mathrm{meV}$ on hBN [126].

The expanded Hilbert space we simulate is spanned by a direct product of the
electronic and phononic degrees of freedom, i.e., $n_{C}\cdot n_{ph}$, with
$n_{ph}$ the number of phononic excitations. We compute the transmission
$T_{0\rightarrow n}(\omega;\omega_{ph})$ with a fixed $\omega_{ph}$, from an
incoming mode with $n=0$ phonons to each outgoing mode with $-n_{ph}\leq n\leq
n_{ph}$ phonons. These contributions are added fully coherently by taking the
trace in the Landauer formula over the phononic degrees of freedom. We also
sample the phase $\phi$ of the electron-phonon coupling, in order to get the
total transmission $T(\omega;\omega_{ph})$. We further extend our analysis
from a single- to a multi-mode scenario by incoherently superimposing
transmission functions $T(\omega;\omega_{ph})$ over a range of phonon
frequencies, weighted with a Boltzmann factor evaluated at
$T=$300\text{\,}\mathrm{K}$$ (i.e., $k_{B}T\approx$26\text{\,}\mathrm{meV}$$)
that takes into account the thermal occupation of the corresponding phonon
mode (for a step-by-step description of all procedures see SI).

We calculate the transmission function for different values of the electron-
phonon coupling $g$. In the single-mode approximation [Figs. 4(b,d)], we
observe phonon satellite peaks around the frontier MO resonances, at integer
multiples of the phonon frequency [Figs. 4(f,g)] for different values of
$\omega_{ph}$. Focusing on the transmission function close to the
antiresonance, we find that it is enhanced quadratically with the electron-
phonon coupling, i.e., $T(\omega_{\mathrm{DQI}})\propto g^{2}$, in close
analogy to the effect of disorder. Continuing the analogy, we also observe
that for a given value of the electron-phonon coupling, the transmission
function interpolates between a constant regime at
$\omega\approx\omega_{\mathrm{DQI}}$ and the characteristic $\omega^{2}$
behavior at higher energies. The crossover scale again increases with the
electron-phonon coupling. Above a critical threshold, the transmission
function becomes qualitatively indistinguishable from that of the para
configuration. In the multi-mode case, instead of generating individual phonon
satellites, the electron-phonon coupling broadens the resonances so that
electron transport through the MOs is no longer unitary as shown in Figs.
4(c,e). The effects on the antiresonance are qualitatively similar to those
observed in the single-mode scenario. The dichotomy between the effects close
to the antiresonance and at higher energy scales appears to be a generic
feature of the suppression of QI, which applies to different scattering
mechanisms.

Since in our numerical framework the electronic subsystem is no longer energy-
conserving, nor we define an electronic self-energy, we cannot estimate a
scattering length as we did in the case of disorder or electron-electron
interactions. Therefore, we employ an alternative strategy that allows us to
estimate the inelastic scattering length as a function of the electron-phonon
coupling. In a nutshell, we consider the probability distribution of
$T_{0\rightarrow n}(\omega=0;\omega_{ph})$ as a random walk in $n$. As the
number of steps increases, we can fit a Gaussian distribution with width
proportional to $\lambda_{\mathrm{e-ph}}/{\cal L}$ (the detailed procedure is
described in the SI). For instance, we find
$\lambda_{\mathrm{e-ph}}\approx$10\text{\,}\mathrm{nm}$$ at
$g=$20\text{\,}\mathrm{meV}$$ but we only reach the typical system size
$\lambda_{\mathrm{e-ph}}\approx$2\text{\,}\mathrm{nm}$$ at
$g=$100\text{\,}\mathrm{meV}$$, and a relation
$\lambda_{\mathrm{e-ph}}(g)\propto g^{-1}$, analogous to the case of disorder.
Upon comparison with the numerical data for the transmission function, this
corresponds to the coupling range in which the characteristic $\omega^{2}$
behavior of the meta configuration is no longer observable and hence the
effects of QI are contextually lost.

In summary, phonons (or, more generally, inelastic scattering) limit DQI
effects as soon as $k_{B}T$ exceeds the size of the HOMO-LUMO gap, in line
with a much more simple estimate of smearing out the conductance on this
energy scale. We therefore conclude that explicit consideration of electron-
phonon scattering does not pose an additional limit to the observability of
DQI.

##  III Discussion

The focus of the present work is the resilience of QI effects in the electron
transport through graphene nanostructures. In the ballistic transport regime,
the stability of DQI is rooted in its topological and symmetry origin, as also
established in the recent literature. Furthermore, we investigated several
effects, which can be possibly detrimental to QI, and for each we estimate a
characteristic scale above which QI effects are likely to be lost.

Our findings can be summarized as follows. (i) A size effects analysis reveals
that in the ballistic regime, the in-gap transmission function displays a
characteristic $\omega^{2}$ behavior that can be entirely ascribed to the
presence of a QI antiresonance. In this regime, the transmission function is a
universal function of the ratio between the molecule-lead coupling and the
width of the gap, i.e., $\Gamma/\Delta_{0}$. In turn, it also determines an
intrinsic resolution threshold necessary to resolve QI effects. (ii) The
dominant effect of local many-body correlations due to the Coulomb repulsion
is to renormalize the gap $\Delta<\Delta_{0}$ and, as long as thermal
excitations of electrons across the gap is negligible, the electron-electron
scattering rate is low and the effect on the QI properties is marginal. (iii)
For disorder and electron-phonon scattering, we identify a similar behavior of
the transmission function. Close to the antiresonance the transmission
function is strongly enhanced with analogous scaling laws versus disorder
strength $A$ and electron-phonon coupling $g$, whereas at higher energies the
characteristic $\omega^{2}$ behavior is more resilient. For typical values
reported in the literature, QI can be suppressed for graphene devices
deposited on substrates like SiO2 but not for cleaner devices deposited on
hBN. The electron-phonon coupling in graphene (as well as in organic
molecules) is typically low-enough that even at room temperature QI effects
can survive.

While we investigated each case independently, multiple scattering sources are
simultaneously in play. Hence, the effective electron lifetime is dominated by
the process with the highest scattering rate (i.e., with the lowest lifetime
or scattering length). Specifically, the overall scattering time is determined
like a resistance in parallel

$\frac{1}{\tau}=\frac{1}{\tau_{\Gamma}}+\frac{1}{\tau_{\mathrm{disorder}}}+\frac{1}{\tau_{\mathrm{e-e}}}+\frac{1}{\tau_{\mathrm{e-ph}}}+\ldots,$
(15)

where the characteristic times correspond to processes involving electron
scattering at the interface with the leads $\tau_{\Gamma}$, off disorder
($\tau_{\mathrm{disorder}}$), off other electronic ($\tau_{e-e}$) and phononic
($\tau_{e-ph}$) excitations, respectively, and similarly for any other
possible process not explicitly included here.

Within the present framework some effects can be combined with some additional
effort, e.g., disorder with either electron-electron or electron-phonon
interactions. For instance, we speculate that in the presence of the
electronic correlations, the disorder-driven Fermi level alignment
fluctuations could be reduced, especially for weak-to-moderate disorder
$|\epsilon_{i}|\ll U$. The argument is that the Coulomb repulsion penalizes a
spatially inhomogeneous charge distribution, [97, 98] and is expected to
compete with disorder by renormalizing the disorder potential, i.e.,
$\epsilon_{i}\rightarrow\epsilon_{i}+\Re\Sigma_{ii}(0)$ (see also SI). Other
combinations, such as the simultaneous treatment of electron-electron and
electron-phonon interactions are very challenging, and are beyond the scope of
the present analysis.

In conclusion, our analysis provides a unified theoretical ground to explore
the resilience of QI effects, and the necessary conditions for observing them
under experimental conditions. At the same time, it emerges that the
conductance alone may be insufficient for a characterization of the QI
properties of a junction, while identifying non-linear $I-V_{b}$
characteristics can reveal the presence of an antiresonance even when
partially concealed or suppressed due to the Fermi level alignment or
electronic scattering.

###### Acknowledgements.

We are thankful to A. Amaricci and M. Capone for useful discussion and for
providing the Lánczos impurity solver. [95, 96] AV and RS acknowledges
financial support from the Austrian Science Fund (FWF) project number No.
P31631. TF and FL acknowledge support from FWF (DACH proposal I3827-N36) and
WWTF project MA14-002. Preliminary work for this project was also supported
through the FWF Erwin Schrödinger fellowship J3890-N36.

## Methods

Electron transport. Within the Landauer formalism [76] the electron
transmission function is given by

$T(\omega)=\mathrm{Tr}\Big{[}\Gamma^{L}(\omega)G^{\dagger}(\omega)\Gamma^{R}(\omega)G(\omega)\Big{]},$
(16)

where is $G$ is the retarded Green’s function of the graphene nanostructure in
the AO basis

$G(\omega)=\Big{[}\omega+\imath\eta-H-\Sigma^{L}-\Sigma^{R}\Big{]},$ (17)

and $\eta$ an infinitesimal. The coupling to the leads is described in terms
of the leads’ self-energy, as

$\Gamma^{\alpha}(\omega)=\frac{\imath}{2}\Big{[}\Sigma^{\alpha}(\omega)-\Sigma^{\alpha\dagger}(\omega)\Big{]},$
(18)

which is also a matrix in the AO basis. Within the WBL approximation the
matrix elements $\Gamma^{L}_{\ell\ell}$ and $\Gamma^{R}_{rr}$ are given by an
energy-independent constant $\Gamma$ for the edge C atoms $\\{\ell\\}$ and
$\\{r\\}$, and zero otherwise. The applicability of the WBL approximation, in
this context, is ensured by the observation that the spectral properties of
the leads can modulate the transmission function and affect the Fermi level
alignment but do not affect the existence of a QI antiresonance [77, 28, 127].

Including disorder. In the presence of static disorder $\\{\epsilon_{k}\\}$,
the transmission function $T(\omega;\\{\epsilon_{k}\\})$ is evaluated
analogously as the pristine case, as the local energies enter the Green’s
function through the Hamiltonian. The disorder-averaged transmission is
obtained as an _incoherent_ average over configurations of the random
potential, i.e.,

$T(\omega)=\sum_{\\{\epsilon_{k}\\}}T(\omega;\\{\epsilon_{k}\\}).$ (19)

The transport is ballistic for each static configuration but not after the
disorder averaging.

Including electronic correlations. The effects of the electron-electron
interactions are included in the Green’s function (17) though a many-body
self-energy as

$G(\omega)=\Big{[}\omega+\imath\eta-H-\Sigma^{L}-\Sigma^{R}-\Sigma(\omega)\Big{]}.$
(20)

For each frequency, the self-energy is a matrix in the AO basis obtained
within a real-space DMFT approximation. In a nutshell, each locally
inequivalent C atom of the nanostructure is mapped onto an auxiliary Anderson
impurity model, which is solved by Lánczos exact diagonalization, [95, 96]
similarly as in previous works [97, 28, 29, 98]. The solution of the impurity
problem yields a local self-energy $\Sigma_{ii}(\omega)$, so that the elements
of the many-body self-energy are given by
$\Sigma_{ij}(\omega)=\Sigma_{ii}(\omega)\delta_{ij}$, while elements
$\Sigma_{i\neq j}$ are neglected [99, 103, 104]. The procedure is iterated
self-consistently, with an initial guess (typically zero) until convergence.

Including electron-phonon coupling. Due to the interaction with phonons, it is
necessary to consider an expanded Hilbert space spanned by a direct product of
the electronic and phononic degrees of freedom, i.e., $n_{C}\cdot
n_{\mathrm{ph}}$, with $n_{ph}$ the number of phononic excitations.

For a given value of the phonon frequency $\omega_{\mathrm{ph}}$ the elements
of the Fröhlich Hamiltonian can be explicitly written as

${\cal H}=\begin{pmatrix}[l]\ddots&&&&\\\ &{\cal H}_{0}+{\cal
H}^{[-1]}_{\mathrm{ph}}&{\cal H}_{\mathrm{e-ph}}&&\\\ &{\cal
H}_{\mathrm{e-ph}}^{\dagger}&{\cal H}_{0}&{\cal H}_{\mathrm{e-ph}}&\\\ &&{\cal
H}_{\mathrm{e-ph}}^{\dagger}&{\cal H}_{0}+{\cal H}^{[1]}_{\mathrm{ph}}&\\\
&&&&\ddots\end{pmatrix},$ (21)

where ${\cal H}^{[n]}_{\mathrm{ph}}$ denoes the phonon Hamiltonian with $n$
phonons, ${\cal H}_{\mathrm{e-ph}}$ the electron-phonon coupling, and ${\cal
H}_{0}$ the tight-binding Hamiltonian in the electron subspace.

The electron-phonon coupling is a complex-valued matrix, which is diagonal in
the phonon subspace, with elements $g=|g|\exp(\imath\phi)$. Hence, for each
value $|g|$ we average the transmission function over the phases $\phi$

$T(\omega;\omega_{ph})=\frac{1}{\pi}\int_{0}^{\pi}\mathrm{d}\phi\
T(\omega;\omega_{ph},\phi),$ (22)

with

$T(\omega;\omega_{ph},\phi)=\mathrm{Tr}\Big{[}\Gamma_{L}G_{\mathrm{e-ph}}^{\dagger}(\omega)\Gamma_{R}G_{\mathrm{e-ph}}(\omega)\Big{]}.$
(23)

Note that Eq. (19) has the same form of the Landauer formula for the
electronic system, but we denote with
$G_{\mathrm{e-ph}}(\omega)=G(\omega;\omega_{ph},\phi)$ the Green’s function in
the extended Hilbert space, defined through Hamiltonian (18), and the
$\Gamma_{\alpha}$ matrices are analogously extended, so that the trace
includes both electronic and phononic degrees of freedom.

Taking the trace corresponds to adding fully coherently all transmission
contributions $T_{0\rightarrow n}(\omega;\omega_{ph})$ for a fixed value of
$\omega_{ph}$, from incoming modes with $n=0$ phonons to each outgoing mode
with $-n_{ph}\leq n\leq n_{ph}$ phonons

$T(\omega;\omega_{ph},\phi)=\sum_{n}T_{0\rightarrow
n}(\omega;\omega_{ph},\phi).$ (24)

In the weak-coupling regime, we verified that it is sufficient to restrict the
phonon emission and absorption up to $n_{ph}=4$ phonon quanta.

We extend our analysis of the transport properties from a single- to a multi-
mode scenario for the phonons by incoherently superimposing transmission
functions $T(\omega;\omega_{ph})$ over a range of phonon frequencies, weighted
with a Boltzmann factor that takes into account the thermal occupation of the
corresponding phonon mode

$T(\omega)=\frac{1}{{\cal
Z}}\int_{0}^{\Omega}\mathrm{d}\omega_{ph}T(\omega;\omega_{ph})e^{-\beta\omega_{ph}},$
(25)

where the inverse temperature $\beta^{-1}=k_{B}T$ is evaluated at
$T=$300\text{\,}\mathrm{K}$$, while $\Omega$ denotes an ultra-violet cutoff
for the phonon excitation energy, and the partition function

${\cal Z}=\int_{0}^{\Omega}\mathrm{d}\omega_{ph}e^{-\beta\omega_{ph}}$ (26)

ensures the proper normalization. Let us stress that the approximation behind
the incoherent superposition of transmission functions calculated for
different phonon energies is reasonable in the weak-coupling regime, where
phonon-phonon scattering is unlikely.

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Supporting Information:
Stability of destructive quantum interference antiresonance in electron
transport through graphene nanostructures
A. Valli,1,2,∗ T. Fabian,1 F. Libisch,1 and R. Stadler1

1Institute for Theoretical Physics, Vienna University of Technology, Wiedner
Hauptstrasse 8-10, A-1040 Vienna, Austria
2Department of Theoretical Physics, Institute of Physics, Budapest University
of Technology and Economics, Müegyetem rkp. 3., H-1111 Budapest, Hungary

## Remarks on quantum transport formalism

### Molecule-lead coupling and WBL approximation

The tight-binding Hamiltonian in the basis of a single $p_{z}$ atomic orbital
per C atom reads

${\cal H}_{0}=t\sum_{\langle
ij\rangle}\sum_{\sigma}c^{\dagger}_{i\sigma}c^{\phantom{\dagger}}_{j\sigma},$
(1)

where $c^{(\dagger)}_{i\sigma}$ is the annihilation (creation) operator for an
electron on site $i$ with spin $\sigma$, and $t$ is the hopping between
nearest-neighbor sites $i$ and $j$. Within the Landauer formalism [76], the
electron transmission function is given by

$T(\omega)=\textrm{Tr}\Big{[}\Gamma^{L}(\omega)G^{\dagger}(\omega)\Gamma^{R}(\omega)G(\omega)\Big{]},$
(2)

where the molecule retarded Green’s function

$G(\omega)=\Big{[}\omega+\imath\eta-{\cal
H}_{0}-\Sigma^{L}(\omega)-\Sigma^{R}(\omega)\Big{]}^{-1}$ (3)

with $\eta\rightarrow 0^{+}$. The coupling to the leads are given in terms of
the embedding self-energy

$\Gamma^{\alpha}(\omega)=\frac{\imath}{2}\Big{[}\Sigma^{\alpha}(\omega)-\Sigma^{\dagger\alpha}(\omega)\Big{]}.$
(4)

In order to highlight the different contributions to the transmission
function, the Landauer formula can be expressed also by expanding the trace
explicit as

$T(\omega)=\sum_{\ell\ell^{\prime}}\sum_{rr^{\prime}}\Gamma^{L}_{\ell\ell^{\prime}}(\omega)G_{\ell^{\prime}r^{\prime}}^{\dagger}(\omega)\Gamma^{R}_{r^{\prime}r}(\omega)G_{r\ell}(\omega),$
(5)

where the sets $\\{\ell,\ell^{\prime}\\}$ and $\\{r,r^{\prime}\\}$ denote the
AO connected to the left ($L$) and right ($R$) leads, respectively. Since the
QI properties are determined by the topology of the molecular bridge rather
than from the electronic structure of the leads, we make the following
assumptions:

  * (i)

that the molecule-lead coupling is diagonal in the AO basis,

  * (ii)

a wideband limit (WBL) approximation for the leads,

  * (iii)

symmetric coupling between the AO of the molecule and the $L$ or $R$ leads.

The condition (i) implies that
$\Sigma^{L}_{\ell\ell^{\prime}}(\omega)=-\imath\Gamma_{\ell\ell}(\omega)\delta_{\ell\ell^{\prime}}$,
and analogously
$\Sigma^{R}_{rr^{\prime}}(\omega)=-\imath\Gamma_{rr}(\omega)\delta_{rr^{\prime}}$,
while within the the WBL aprroximation (ii)
$\Sigma^{\alpha}(\omega)=-\imath\Gamma^{\alpha}$ is an energy-independent
constant, and finally (iii) results in $\Gamma_{\ell\ell}=\Gamma_{rr}=\Gamma$.
While the spectral properties of the leads modulate the transmission function,
they do not directly affect the QI properties of the junction. [77, 28, 127]
Within the assumptions (i)-(iii) above, the transmission function takes a
simplified form in terms of independent transmission channels $\ell\rightarrow
r$ given by

$T(\omega)=\sum_{\ell r}T_{\ell\rightarrow r}(\omega)=\Gamma^{2}\sum_{\ell
r}|G_{\ell r}(\omega)|^{2}.$ (6)

Note that cross terms can arise if we relax the assumption of molecule-lead
diagonal coupling. Elements $\Gamma_{\ell\neq\ell^{\prime}}$ result in
contributions to the transmission function of the form

$T_{\ell,\ell^{\prime}\rightarrow
r}(\omega)=\Gamma_{\ell\ell^{\prime}}\Gamma_{rr}|G_{\ell
r}(\omega)+G_{\ell^{\prime}r}(\omega)|^{2},$ (7)

thus inducing coherence effects between transmission channels, which can,
e.g., induce or suppress QI antiresonances within the HOMO-LUMO gap. [78, 79,
17, 80] However, we verified numerically that this is not the case for the
systems we investigate here, and the general conclusions we draw remain valid
also in the presence of coherent cross-channel contributions.

### Transport through a disordered system

The local disorder is described by adding to the tight-binding Hamiltonian the
term

${\cal
H}_{\mathrm{disorder}}=\sum_{i\sigma}\epsilon_{i}c^{\dagger}_{i\sigma}c^{\phantom{\dagger}}_{i\sigma},$
(8)

where $|\epsilon_{i}|$ are the randomly distributed on-site energies, with
$\langle\varepsilon_{i}\rangle=0$ and $A=\sqrt{\langle\varepsilon^{2}\rangle}$
is the disorder strength. The transmission function
$T(\omega;\\{\epsilon_{k}\\})$ for a specific realization of disorder is
evaluated as in pristine case, through the Landauer formula (2). The disorder-
averaged transmission is obtained as an _incoherent_ average over different
realizations of the random potential, i.e.,

$T(\omega)=\sum_{\\{\epsilon_{k}\\}}T(\omega;\\{\epsilon_{k}\\}).$ (9)

The transport is ballistic for each static configuration but not after the
disorder average.

### Estimate of the disorder scattering length $\lambda_{\mathrm{disorder}}$

Since the scattering off disorder is elastic, i.e., the electron energy does
not change, we can obtain an estimate the scattering length as follows. We
consider the transmission function for resonant scattering at, e.g., the HOMO
or LUMO resonance. For any static disorder configuration the transport is
ballistic, and therefore for a single open channel $T=1$, as we also verified
in the numerical simulations. Due to elastic scattering, the transmission is
reduced and given by [88]

$T_{\mathrm{resonant}}=\frac{\lambda_{\mathrm{disorder}}}{{\cal
L}+\lambda_{\mathrm{disorder}}}.$ (10)

Obviously, the ballistic limit is recovered for ${\cal
L}\ll\lambda_{\mathrm{disorder}}$. It follows that (indicating
$T_{\mathrm{resonant}}$ as $T$, for shortness)

$\frac{\lambda_{\mathrm{disorder}}}{{\cal L}}=\frac{T}{1-T}.$ (11)

For the $4\times 4$ nanostructure, we can estimate ${\cal
L}\approx$1\text{\,}\mathrm{nm}$$ as the longest distance between the any pair
($\ell,r)$ of contact C atoms. In Fig. S1 we show the reduction of the
transmission function at the LUMO resonance, and well as the estimate of
$\lambda_{\mathrm{disorder}}$ as a function of disorder strength $A$. We find,
e.g., $\lambda_{\mathrm{disorder}}\approx$1.2\text{\,}\mathrm{nm}$$ at
$A=$30\text{\,}\mathrm{meV}$$, with a dependence
$\lambda_{\mathrm{disorder}}(A)\propto A^{-1}$. Moreover, since one can expect
QI effects to be lost for $\lambda_{\mathrm{disorder}}\approx{\cal L}$ (for
which $T=1/2$), we can identify the critical value of the disorder strength
$A_{\mathrm{critical}}\approx$40\text{\,}\mathrm{meV}$$ fulfilling this
condition. For finite disorder, the depth of the DQI dip is reduced, and the
transmission no longer follows $T\propto(\omega-\omega_{\mathrm{DQI}})^{2}$
behavior. Approaching the critical disorder strength, the DQI dip vanishes as
qualitatively indicated by the arrow in Fig. S1(c).

Figure S1: (a) Reduction of the transmission function at the LUMO resonance
for a $4\times 4$ nanostructure, and (b) corresponding estimate of
$\lambda_{\mathrm{disorder}}$ as function of disorder strength $A$. The dashed
line indicate the critical value $A_{\mathrm{critical}}$, identified by the
conditions $\lambda_{\mathrm{disorder}}\approx{\cal L}$ and $T=1/2$. (c)
Transmission function for the pristine and the disordered system at $A\approx
A_{\mathrm{critical}}$.

### Transport in the presence of electron-electron interaction

To account for electron-electron interaction, we add to the tight-binding
Hamiltonian a screened local Coulomb repulsion (i.e., a Hubbard interaction)
described by

${\cal
H}_{\mathrm{e-e}}=U\sum_{i}n_{i\uparrow}n_{i\downarrow}-\mu\sum_{i}(n_{i\uparrow}+n_{i\downarrow}),$
(12)

where the chemical potential $\mu=U/2$ ensures the Fermi energy to be at
$\omega=0$ for any value of the Coulomb repulsion. The effects of electron-
electron interactions are included in the Green’s function though a many-body
self-energy as

$G(\omega)=\Big{[}\omega+\imath\eta-{\cal
H}_{0}-\Sigma^{L}(\omega)-\Sigma^{R}(\omega)-\Sigma(\omega)\Big{]}^{-1}.$ (13)

For each frequency, the self-energy is a matrix in the AO basis obtained
within a real-space DMFT approximation. In a nutshell, each locally
inequivalent C atom of the nanostructure is mapped onto an auxiliary Anderson
impurity model, defined by the dynamical Weiss field

${\cal
G}^{0}_{i}(\omega)=\Big{[}G_{ii}^{-1}(\omega)+\Sigma_{ii}(\omega)\Big{]}^{-1}$
(14)

with an initial guess for the many-body self-energy $\Sigma_{ii}(\omega)=0$.
Each impurity model is solved by Lánczos exact diagonalization, [95, 96]
similarly to previous works [97, 28, 29, 98]. The solution of the impurity
problem yields a local self-energy, so that the elements of the many-body
self-energy are given by $\Sigma_{ij}(\omega)=\Sigma_{ii}(\omega)\delta_{ij}$,
while elements $\Sigma_{i\neq j}(\omega)$ are neglected [99, 103, 104]. As the
many-body self-energy renormalizes the Green’s function through (13) and hence
defines a corresponding impurity model (14), the procedure is iterated self-
consistently until convergence.

Within this scheme, the electronic transmission function is evaluated through
the Landauer formula (2), where the Green’s function is renormalized by the
many-body self-energy. In principle, the interactions renormalize not only the
molecular Green’s function but also the molecular-lead coupling, resulting in
incoherent contributions (which cannot be expressed in terms of an effective
transmission function) and the electric current should be instead evaluated
with the Meir-Wingreen formula. [112] However, the incoherent contributions
are typically neglected in numerical simulations, which within a linear-
response Kubo formalism for the current-current response function, corresponds
to neglecting vertex corrections. [113, 91, 114, 94]

### Transport in the presence of electron-phonon interaction

Within the Landauer formalism, the electronic transmission function at a given
energy $T(\omega)$ is evaluated from the Green’s function $G(\omega)$.
However, electron-phonon interaction changes the electron energy. If we
restrict ourselves to a single bosonic mode with fixed energy
$\omega_{\mathrm{ph}}$, corresponding to lattice vibrations typically found in
the small graphene nanostructures considered here [125], the possible
electronic excitations become discrete and a combined treatment becomes
numerically feasible. We consider an expanded Hilbert space spanned by a
direct product of the electronic and phononic degrees of freedom, i.e.,
$n_{C}\cdot n_{\mathrm{ph}}$, with $n_{ph}$ number of phononic excitations.
The electron-phonon coupled system is described by a Fröhlich Hamiltonian

${\cal H}_{\mathrm{Fr\ddot{o}hlich}}={\cal H}_{0}+{\cal H}_{\mathrm{ph}}+{\cal
H}_{\mathrm{e-ph}}$ (15)

with

${\cal H}_{\mathrm{ph}}=\omega_{ph}\Big{(}a^{\dagger}a+\frac{1}{2}\Big{)}$
(16)

and

${\cal H}_{\mathrm{e-ph}}=g(a^{\dagger}+a)\sum_{i\sigma}n_{i\sigma},$ (17)

where $a$ ($a^{\dagger})$ is the bosonic annihilation (creation) operator of a
phonon with energy $\omega_{ph}$, which couples to the local electron density
$n_{i}$ through a complex electron-phonon coupling matrix which is diagonal in
the phonon subspace, $g=|g|\exp(\imath\phi)$, where $\phi$ is a random phase.
The Fröhlich Hamiltonian above can be explicitly as a matrix

${\cal H}=\begin{pmatrix}[l]\ddots&&&&\\\ &{\cal H}_{0}+{\cal
H}_{\mathrm{ph}}&{\cal H}_{\mathrm{e-ph}}&&\\\ &{\cal
H}_{\mathrm{e-ph}}^{\dagger}&{\cal H}_{0}&{\cal H}_{\mathrm{e-ph}}&\\\ &&{\cal
H}_{\mathrm{e-ph}}^{\dagger}&{\cal H}_{0}+{\cal H}_{\mathrm{ph}}&\\\
&&&&\ddots\end{pmatrix}$ (18)

The transmission function then is evaluated as

$T(\omega;\omega_{ph},\phi)=\mathrm{Tr}\Big{[}\Gamma^{L}(\omega)G_{\mathrm{e-ph}}^{\dagger}(\omega)\Gamma^{R}(\omega)G_{\mathrm{e-ph}}(\omega)\Big{]},$
(19)

which has the same form of the Landauer formula for the electronic system, but
with $G_{\mathrm{e-ph}}(\omega)=G(\omega;\omega_{ph},\phi)$ the Green’s
function in the extended Hilbert space, defined through Hamiltonian (18), and
the matrices $\Gamma^{L/R}$ are block diagonal, so that the trace includes
both electronic and phononic degrees of freedom. Taking the trace corresponds
to adding fully coherently all transmission contributions $T_{0\rightarrow
n}(\omega;\omega_{ph})$ with $-n_{ph}\leq n\leq n_{ph}$ phonons, at a given
$\omega_{ph}$

$T(\omega;\omega_{ph},\phi)=\sum_{n}T_{0\rightarrow
n}(\omega;\omega_{ph},\phi).$ (20)

For a vanishing electron-phonon coupling $g\rightarrow 0$, only the mode with
$n_{\mathrm{ph}}=0$ contributes to the transmission. At finite coupling, there
are also contributions from channels $n_{\mathrm{ph}}>0$. In the weak-coupling
regime, which is expected for reasonable values of the electron-phonon
coupling, we verified that it is sufficient to retain absorption and emission
processes with up to $n_{ph}=\pm 4$ phonon quanta. Finally, we average the
transmission function over the phase $\phi$, as

$T(\omega;\omega_{ph})=\frac{1}{\pi}\int_{0}^{\pi}\mathrm{d}\phi\
T(\omega;\omega_{ph},\phi).$ (21)

We extend our analysis of the transport properties from a single- to a multi-
mode phonon scenario by incoherently superimposing transmission functions
$T(\omega;\omega_{ph})$ over a range of phonon frequencies, weighted with a
Boltzmann factor that takes into account the thermal occupation of the
corresponding phonon mode

$T(\omega)=\frac{1}{{\cal
Z}}\int_{0}^{\Omega}\mathrm{d}\omega_{ph}T(\omega;\omega_{ph})e^{-\beta\omega_{ph}},$
(22)

where the inverse temperature $\beta^{-1}=k_{B}T$ is evaluated at
$T=$300\text{\,}\mathrm{K}$$, while $\Omega$ denotes an ultra-violet cutoff
for the phonon excitation energy, and the partition function

${\cal Z}=\int_{0}^{\Omega}\mathrm{d}\omega_{ph}e^{-\beta\omega_{ph}}$ (23)

ensures proper normalization. The approximation behind the incoherent
superposition of transmission functions calculated for different phonon
energies is reasonable in the weak-coupling regime, where phonon-phonon
scattering is unlikely.

### Estimate of the electron-phonon scattering length
$\lambda_{\mathrm{e-ph}}$

Since the electron-phonon scattering is not elastic, we cannot follow the same
strategy we used to estimate the scattering length in the presence of
disorder, and we instead rely on an alternative procedure. The classical
picture to our electron-phonon model is a random walk. An electron enters the
lead with an initial energy $\varepsilon_{0}$ associated with an initial
phononic occupation $n_{0}$,
$\mathinner{|{\varepsilon_{0}}\rangle}\otimes\mathinner{|{n_{0}}\rangle}$.
After a characteristic length scale $\lambda_{\mathrm{e-ph}}$, there will be,
on average, one inelastic scattering event
$\mathinner{|{\varepsilon_{0}}\rangle}\otimes\mathinner{|{n_{0}}\rangle}\rightarrow\mathinner{|{\varepsilon_{0}\mp\omega}\rangle}\otimes\mathinner{|{n_{0}\pm
1}\rangle}$. For $\mathcal{L}\gg\lambda_{\mathrm{e-ph}}$, the resulting
distribution over levels with a final energy transfer of $n\omega$ into states
$\mathinner{|{\varepsilon_{0}\mp n\omega}\rangle}\otimes\mathinner{|{n_{0}\pm
n}\rangle}$ approaches a normal distribution centered around $n=0$ (where $n$
counts the change in phononic occupation). By exploiting this analogy we
obtain an estimate on the electron-phonon scattering length
$\lambda_{\mathrm{e-ph}}(g)$. For a given coupling $g$, we compute the
distribution of the transmission to the different phonon modes as a function
of length ${\cal L}$ of an reference armchair graphene nanoribbon, Fig. S2(a),
with the same width as the actual nanostructure. At ${\cal
L}\ll\lambda_{\mathrm{e-ph}}$, the initial mode
$\mathinner{|{\varepsilon_{0}}\rangle}\otimes\mathinner{|{n_{0}}\rangle}$
represents the dominant contribution to the transmission, while for increasing
length also modes with $n\neq 0$ are populated. We fit the transmission
(averaging over initial electron energy in a window of $\pm 0.01\ \omega/t$
around $\omega=0$) for fixed $\omega_{\mathrm{p}h}=$3\text{\,}\mathrm{meV}$$
with a normal distribution ${\cal P}\\{T_{n\rightarrow
0}(\omega=0;\omega_{\mathrm{ph}})\\}$ (See Fig. S2(b)).If the variance of the
distribution $\sigma=1$, then we identify $\lambda_{\mathrm{e-ph}}$ with
${\cal L}$, while the distribution becomes broader (narrower), i.e.,
$\sigma>1$ ($\sigma<1$), for ${\cal L}>\lambda_{\mathrm{e-ph}}$ (${\cal
L}<\lambda_{\mathrm{e-ph}}$).

With this method, we can estimate
$\lambda_{\mathrm{e-ph}}\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for
$g=$0.2\text{\,}\mathrm{meV}$$, and
$\lambda_{\mathrm{e-ph}}\approx$100\text{\,}\mathrm{nm}$$ for
$g=$2\text{\,}\mathrm{meV}$$, with a linear relation
$\lambda_{\mathrm{e-ph}}(g)\propto g$. We validate the procedure against the
numerical simulations, and we observe that for values of the electron-phonon
coupling strength at which the estimated scattering length approaches the size
of the nanostructure, the effects of DQI on the transmission functions are
lost.

Figure S2: (a) Auxiliary graphene nanoribbon with width ${\cal L}$. (b)
Schematic histogram of the contributions $T_{n\rightarrow
n}(\omega=0;\omega_{\mathrm{ph}})$ and a gaussian distribution fit (solid
black line) to extract the variance $\sigma$ to estimate
$\lambda_{\mathrm{e-ph}}$.

## Additional results

### Dependence of the transmission on $\Delta_{0}$ and $\Gamma$

Within the tight-binding approximation, besides the nearest-neighbor hopping
$t$, the coupling to the leads $\Gamma$ and the HOMO-LUMO gap $\Delta_{0}$,
are the other two energy scales of the problem. We have shown that in the meta
configuration, the transmission function displays a characteristic behavior
$\propto\omega^{2}$ within the gap, arising due to the existence of a QI
antiresonance. In Figs. S3(a,b), we show that the transmission functions has a
similar dependence on $\Delta_{0}$ and on $\Gamma$. In order to have a better
comparison, we align the LUMO resonances transmission function for different
system size. In both cases, each curve corresponds to a different ratio
$\Gamma/\Delta_{0}$. Instead, in Fig. S3(c) we show two curves corresponding
to nanostructures with different sizes, but with approximatively the same
ratio $\Gamma/\Delta_{0}$. The data collapse in the whole energy window where
$T(\omega)\propto\omega^{2}$ demonstrates that it is a universal curve which
depends only on the ratio $\Gamma/\Delta_{0}$. However, the saturation value
as $\omega\rightarrow\omega_{\mathrm{DQI}}$ is not universal and depends on
$\Gamma$.

Figure S3: Transmission function (a) through $N\times N$ nanostructure against
$\omega/\omega_{\mathrm{LUMO}}$ for $\Gamma=0.0004t$, (b) through a $4\times
4$ nanostructure for different values of the molecule-lead coupling $\Gamma$,
and (c) for nanostructures of different size but the same ratio
$\Gamma/\Delta_{0}$.

### Suppression of DQI: antiresonance

One of the most evident effects of disorder and of electron-phonon scattering
is the enhancement of the transmission function close to the antiresonance, as
DQI is progressively lost, as shown in Fig. 2(b,c,d,e) in the manuscript. We
analyze the dependence of $T(\omega_{\mathrm{DQI}})$ in both scenarios to
extract its dependence on the control parameter. In Fig. S4(a) we show that
$T(\omega_{\mathrm{DQI}})\propto A^{2}$ as a function of disorder strength for
both static and dynamic disorder. The main difference, is that the enhancement
is significantly stronger (about two orders of magnitude) after averaging over
the disorder realizations. In Fig. S4(b) we show that that
$T(\omega_{\mathrm{DQI}})\propto g^{2}$ as a function of the electron-phonon
coupling, and the enhancement appears to be intermediate between those of
static and dynamic disorder.

Figure S4: Transmission function at the antiresonance
$T(\omega_{\mathrm{DQI}})$ through a $4\times 4$ nanostructure, versus (a)
disorder strength and (b) electron-phonon coupling. QI effects are suppressed,
as $T(\omega_{\mathrm{DQI}})$ is enhanced quadratically as a function of both
control parameters.

### Suppression of DQI: $I-V_{b}$ characteristics

Within our numerical framework, the electron current (per spin) is evaluated
from the transmission function as

$I=\frac{e}{h}\int_{-\infty}^{\infty}\mathrm{d}\omega\
T(\omega;eV_{b})\Big{[}f\Big{(}\omega-\frac{eV_{b}}{2}\Big{)}-f\Big{(}\omega+\frac{eV_{b}}{2}\Big{)}\Big{]},$
(24)

with the Fermi distribution function for the electrodes given by

$f(\omega)=\frac{1}{1+\exp(\omega/k_{B}T)}$ (25)

where $e$ denotes the electric charge, $h$ the Planck constant, $k_{B}$ the
Boltzmann constant, and $V_{b}$ is the symmetric bias drop between the source
and the drain. The characteristic behavior of the transmission function
reflects on the $I-V$ characteristics. This is easiest to evaluate assuming
$T(\omega,V_{b})\approx T(\omega)$, and $\omega\ll k_{B}T$ so that the Fermi
distribution function can be approximated with its derivative, hence
restricting the energy integral within the bias window

$I\approx\frac{e}{h}\int_{-eV_{b}/2}^{eV_{b}/2}\mathrm{d}\omega\ T(\omega).$
(26)

Therefore, we can identify two transport regimes, which can be identified far-
enough from the resonant transport condition, i.e.,
$|\omega|\lesssim|\omega_{\mathrm{LUMO}}-\Gamma|$. In the para configuration,
$T(\omega)\sim\mathrm{const.}$ within the HOMO-LUMO gap, and the integral
yields $I\propto V_{b}$. In the meta configuration, the transmission saturates
close to the antiresonance, i.e., $T(\omega)\sim\mathrm{const.}$ for
$\omega\approx\omega_{\mathrm{DQI}}$ but at higher energies there is a
crossover to a non-linear transport regime, as $T(\omega)\propto\omega^{2}$
and the integral yields $I\propto V_{b}^{3}$.

Figure S5: Typical behavior of the I-V characteristics through $4\times 4$
nanostructure in the meta (solid lines) and para (dashed lines) configurations
in the presence of (a) electron-electron interactions, (b) dynamical disorder,
and (c) electron-phonon scattering. The characteristic linear $I\propto V_{b}$
and non-linear $I\propto V_{b}^{3}$ regimes are indicated by dashed lines, as
labelled.

The two transport regimes above can be clearly identified in the numerical
simulations, as shown in Fig. S5. Specifically, Fig. S5(a) shows the $I-V_{b}$
characteristics for different values of the local Coulomb repulsion $U$. The
net effect of electron-electron interaction is to renormalize the gap, as can
be observed from the shift of the current plateau to lower bias voltages,
which also results in a weak enhancement of the current at low $V_{b}$.
However, neither the linear regime in the para configuration, nor the
crossover behavior from the linear to the non-linear regime in the meta
configuration, are qualitatively modified upon increasing $U$. The scenario is
different in the case of disorder and electron-phonon coupling, which are
shown in Figs. S5(b,c), respectively. The suppression of QI manifest in (i) an
enhancement of the current at low $V_{b}$, and (ii) an extension of the linear
regime to higher $V_{b}$. However, the non-linear regime is not destroyed
until critical values of the control parameters, i.e., the disorder strength
or the electron-phonon coupling. The critical disorder strength is
$A_{\mathrm{critical}}\approx$40\text{\,}\mathrm{meV}$$, in agreement with the
estimates obtained from the analysis of the scattering length
$\lambda_{\mathrm{disorder}}$.

### Energy distribution of Green’s function zeros

The main effect of static disorder is to shift and broaden the QI
antiresonance, see Fig. S6(a). We shed some light on the mechanism behind this
effect. For diagonal molecule-lead coupling and the WBL approximation, the
condition for DQI in a given channel $\ell\rightarrow r$ is then connected to
the existence of a zero of a specific Green’s function component $G_{\ell
r}(\omega)$. Since $\Im G_{\ell r}(\omega)\approx 0$ far away from resonant
transport, i.e., for $|\omega-\omega_{k-\mathrm{MO}}|\gg\Gamma$, where
$\omega_{k-\mathrm{MO}}$ is the energy of the $k$-th MO, the condition then
reduces to

$\Re G_{\ell r}(\omega_{\mathrm{DQI}})=0.$ (27)

In Figs. S6(b,c) we plot the individual contributions of all transmission
channels $\\{\ell\\}\rightarrow\\{r\\}$. The curves in color (grey) solid
lines correspond to the channels marked by color (grey) connectors in Fig.
S6(d). In the pristine system, despite the individual contributions to the
transmission function are different, all channels display a node in
$\omega_{\mathrm{DQI}}=0$. In the presence of disorder there is a distribution
of nodes at different energies. This is shown explicitly by plotting $\Re
G_{\ell r}(\omega)$, as shown in Figs. S6(e,f) for the selected transmission
channels marked in color in Fig. S6(d).

Figure S6: (a) Transmission function through the pristine $4\times 4$
nanostructure and for a specific disorder configuration, as labelled. (b,c)
Individual transmission channel contributions. The color (grey) lines
correspond to the channels $\ell\rightarrow r$ shown in panel (d). (e,f) Real
part of the Green’s function $\Re G_{\ell r}(\omega)$ for pristine (e) and
disordered (f) configurations for selected channels, marked in color in panel
(d). In the pristine system, all nodes (empty black circles) are at
$\omega_{\mathrm{DQI}}=0$, resulting in a narrow antiresonance, while disorder
results in a distribution of nodes (empty red cicles) at different energies,
responsible for a partial cancellation of the transmission.