# Electrode effects on the observability of destructive quantum interference
in single-molecule junctions

O. Sengul  Institute for Theoretical Physics, Vienna University of Technology,
1040 Vienna, Austria  Angelo Valli  Institute for Theoretical Physics, Vienna
University of Technology, 1040 Vienna, Austria  Robert Stadler  Institute for
Theoretical Physics, Vienna University of Technology, 1040 Vienna, Austria

###### Abstract

Destructive quantum interference (QI) has been a source of interest as a new
paradigm for molecular electronics as the electronic conductance is widely
dependent on the occurrence or absence of destructive QI effects. In order to
interpret experimentally observed transmission features, it is necessary to
understand the effects of all components of the junction on electron
transport. We perform non-equilibrium Green’s function calculations within the
framework of density functional theory to assess the structure-function
relationship of transport through pyrene molecular junctions with distinct QI
properties. The chemical nature of the anchor groups and the electrodes
controls the Fermi level alignment, which determines the observability of
destructive QI. A thorough analysis allows to disentangle the transmission
features arising from the molecule and the electrodes. Interestingly, graphene
electrodes introduce features in the low-bias regime, which can either mask or
be misinterpreted as QI effects, while instead originating from the
topological properties of the edges. Thus, this first principles analysis
provides clear indications to guide the interpretation of experimental
studies, which cannot be obtained from simple Hückel model calculations.

## Introduction

In recent years it has been established, both theoretically and
experimentally, that the electron transport properties in single-molecule
junctions are dominated by quantum mechanical effects. In particular, as the
propagation of electrons can be described in terms of de Broglie waves, the
electronic transmission function exhibit quantum interference (QI) which
results in constructive (CQI) or destructive (DQI) interference patterns.
Lambert (2015) While CQI results in an enhancement of the electron
transmission probability, the hallmark of destructive QI is the presence of
antiresonances in the electronic transmission function which can significantly
reduce the conductance. Such QI effects are common in some classes of
$\pi$-conjugated systems, which are characterized by delocalized electronic
states.

As QI effects can be tailored by chemical Markussen _et al._ (2011); Guédon
_et al._ (2012) and physical Frisenda _et al._ (2016); Caneva _et al._ (2018);
Stefani _et al._ (2018); Sowa _et al._ (2018); Li _et al._ (2019, 2019)
mechanisms and can survive even at room temperature, Wang _et al._ (2020);
Markussen _et al._ (2011) they are relevant for the realization of functional
nanoelectronic applications, such as memory cells, Stadler _et al._ (2003)
logic gates, Stadler _et al._ (2004) single-molecule transistors, Stafford _et
al._ (2007); Jia _et al._ (2018) molecular switches, Markussen _et al._
(2010); Daaoub _et al._ (2020); Greenwald _et al._ (2020) and spin filters.
Saraiva-Souza _et al._ (2014); Valli _et al._ (2018, 2019) Particularly
interesting is the potential role of QI in thermoelectric devices, Stadler and
Markussen (2011); Markussen (2013); Famili _et al._ (2017); Klöckner _et al._
(2017); Miao _et al._ (2018); Sadeghi (2019); Sangtarash _et al._ (2018) where
both CQI and DQI have been proposed to improve device efficiency, as the
former can directly enhances the electrical conductivity Wang _et al._ (2020)
whereas the presence of a sharp DQI antiresonance can affect the Seebeck
coefficient. Strange _et al._ (2015); Klöckner _et al._ (2017); Miao _et al._
(2018); Grace _et al._ (2020)

Moreover, QI phenomena are not limited to single molecules, but have also been
observed in other $\pi$-conjugated systems, including a variety of graphene
nanostructures, such as nanoconstrictions, Gehring _et al._ (2016)
nanoribbons, Rosales _et al._ (2009); Calogero _et al._ (2019) nanoflakes,
Valli _et al._ (2018, 2019); Niţă _et al._ (2020) and carbon nanotubes, Tsuji
_et al._ (2007) as well as metal-organic complexes, such as porphyrin Nozaki
_et al._ (2015); Richert _et al._ (2017) and ferrocene Zhao _et al._ (2017);
Zhao and Stadler (2019); Camarasa-Gómez _et al._ (2020) molecules, and in
self-assembled monolayers of aromatic molecular cores. Wang _et al._ (2020)
The possibility to exploit QI effects in molecular junctions with graphene
functional blocks is highly desirable to pave the path towards an environment-
friendly nanotechnology with biodegradable and inexpensive materials. Graphene
has already been employed as a protective layer for metallic electrodes Hüser
and Solomon (2015) and to realize asymmetric junctions. Zhang _et al._ (2016,
2018); Tao _et al._ (2019); He _et al._ (2020) Another advantage of graphene
is that present technologies allow the realization of atomically precise
junctions, Cai _et al._ (2010); Chen _et al._ (2015); Caneva _et al._ (2018);
El Abbassi _et al._ (2019); Yang _et al._ (2020) which, at the same time,
offers mechanical stability beyond room temperature Prins _et al._ (2011) and
a higher degree of experimental reproducibility, which is crucial in order to
achieve a precise control over QI phenomena. However, it is also known that
graphene electrodes introduce additional transmission channels, which can
dominate in the low-bias regime and potentially obscure molecule-intrinsic
features. Ryndyk _et al._ (2012); Gehring _et al._ (2017) In this work we
carry out a systematic analysis which allows us to disentangle the features
arising from the molecule and the electrodes.

Hitherto, a clear relation between the structure and the QI properties of a
junction remains elusive. For this reason, the ability to predict the
occurrence of an antiresonance and to understand its dependence on the
chemical composition of the molecular junction plays a pivotal role for
developing novel technologies based on QI. A few techniques, based on
simplified model assumptions, allow to predict QI effects on the basis of
graphical, Markussen _et al._ (2010); Stuyver _et al._ (2015); O’Driscoll and
Bryce (2021) diagrammatic, Pedersen _et al._ (2015) topological, Geng _et al._
(2015); Tsuji _et al._ (2016, 2018) or symmetry Zhao _et al._ (2017); Valli
_et al._ (2018, 2019) considerations, without the need of numerically
expensive first-principles electron transport calculations. While the value of
the predictive power of such back of the envelope techniques is undeniable,
they are nevertheless unable to capture the detailed effects of the chemical
nature of all junction components, including the leads and the linker groups,
on the QI features.

In an attempt to bridge the worlds of molecular junctions and graphene with
this perspective in mind, we present a systematic analysis of the impact of
different anchor groups and electrodes on the QI features. To this end, we
consider molecule junctions with a pyrene core, which produces specific and
predictable QI patterns, and due to its nature as a polycyclic aromatic
hydrocarbon, can also be considered as a minimal graphene-like functional
block for nanoelectronics. Sangtarash _et al._ (2015) Our starting point is a
prototypical setup in which the pyrene subunit is attached to Au electrodes
with thiol anchoring groups. We explore the effect of subsequently changing
the anchoring groups and the electrodes to build a hydrocarbon junction, which
is relevant for understanding the electron transport through graphene
nanoconstrictions. Specifically, we substitute thiol- with acetylene-
terminated linkers and analyze the changes in the transport properties due to
the direct Au-C bonds and the charge redistribution within the junction. Next,
we substitute the Au electrodes with graphene nanoribbons (GNRs). While the
featureless density of states of Au electrodes close to the Fermi level have
little effect on the shape of the transmisson function, the electronic
properties of graphene nanostructures are strongly affected by the topology of
their edges. We shall see that this has profound consequences for the electron
transport through the junction, in particular concerning the observability of
DQI. By considering graphene electrodes with different size, we are able to
disentangle unambiguously the transport features that are intrinsic to the
molecule from those stemming from the chemical and topological nature of the
electrodes.

Figure 1: Molecular structures of the meta- and para-substituted pyrene with
propynylbenzene linkers. R denotes the anchoring groups, being either thiol or
acetylene.

## QI in molecular junctions with pyrene core

In the following, we consider molecular junctions where a pyrene core is
connected to the electrodes in two different configurations via
propynylbenzene linkers with thiol- and acetylene-terminated anchor groups, as
shown schematically in Fig. 1.

In a realistic scenario, one should expect that the transport properties of a
molecular junction are determined by the nature of the linkers and the anchor
groups, as well as the electrodes, besides the position of the contact atoms
on the $\pi$-conjugated molecule itself. In general, it is difficult to
establish a clear relation between the structure of its components and the QI
properties of a molecular junction. Graphical methods Stadler _et al._ (2004);
Markussen _et al._ (2010); Pedersen _et al._ (2015) allow, to some extent and
on qualitative grounds, to predict the occurrence of QI antiresonances, but
become hardly applicable with in complex systems. For this reason our analysis
of QI features of pyrene molecular junctions is twofold. First, we focus on
the pyrene core and we employ a graphical scheme and calculate the
transmission function within a simple Hückel model (SHM), to predict and
demonstrate that the aromatic molecular subunit is the source of DQI. For a
quantitative analysis we rely on density functional theory (DFT) calculations
in combination with non-equilibrium Green’s function (NEGF) formalism, which
allows us not only to verify the qualitative predictions, but also to
investigate the quantitative effect of different anchor groups and electrodes
on the QI features.

Figure 2: Analysis of QI properties of the pyrene core in the meta- (a) and
para- (b) configurations, and of the linker (c) with the graphical scheme. The
linker does not introduce sources of DQI, independently of the anchor group R.
(d) Transmission function of the pyrene junction within the Hückel model with
a wide-band limit approximation for the leads. (e) As an alternant
hydrocarbon, pyrene can be divided into starred and unstarred sublattices.

We apply the graphical scheme Stadler _et al._ (2004); Markussen _et al._
(2010) to the pyrene core connected in the meta- and para-configurations. The
scheme prescribes to draw a continuous path through neighboring atoms from the
left to the right lead. No DQI is predicted if such a path crosses all sites
of the molecule, or if the remaining sites can be grouped into either pairs or
closed loops (red ellipses). Representative paths for the meta- and para-
configurations are shown in Figs. 2(a,b), respectively. It is easy to convince
oneself that any possible path in the meta configuration leave at least an
unpaired site (green circle). Hence, we predict the occurrence of a QI
antiresonance in the meta configuration. Including the linkers (with either
thiol- or acetylene-termination) in the graphical scheme do not change this
conclusion, since linear chains and para-connected benzene rings do not result
in DQI, see Fig. 2(c) for the corresponding graphical analysis. Markussen _et
al._ (2010)

The graphical predictions can be easily verified by considering a SHM for the
pyrene junction. We set the energy of the C 2$\text{p}_{z}$ atomic orbitals
(AOs) to zero and assume a overlap between neighboring AOs of $\beta=2.7$ eV,
which is a typical value for graphene, Castro Neto _et al._ (2009) so that the
Hamiltonian of the SHM is proportional to pyrene’s adjacency matrix as
$[{H}_{\text{SHM}}]_{ij}=\beta{A}_{ij}$ with $A_{ij}=1$ for AOs $i$ and $j$
sharing a bond, and zero otherwise. Tsuji _et al._ (2018) The retarded Green’s
function of the molecule reads

${G}(E)=\big{[}(E+\imath\eta){I}-{H}_{\textrm{SHM}}+{\Sigma}^{L}+{\Sigma}^{R}\big{]}^{-1},$
(1)

and the Landauer transmission function Landauer (1957) is given by

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

where ${\Gamma}=\imath({\Sigma}^{\phantom{\dagger}}-{\Sigma}^{\dagger})/2$
denotes the coupling to either the left (L) or right (R) lead in terms of the
corresponding embedding self-energy. As pyrene is contacted to the leads
through a single Hückel AO, each matrix ${\Gamma}$ has a single non-zero
matrix element, $\Gamma^{L}_{\ell\ell}$ and $\Gamma^{R}_{rr}$, respectively,
which we assume to be energy independent (wide-band limit approximation). The
transmission function hence reduces to

$T(E)=\Gamma^{L}_{\ell\ell}\Gamma^{R}_{rr}|G_{\ell r}(E)|^{2}$ (3)

and displays resonances in correspondence to the energy Hückel molecular
orbitals (MOs) of pyrene. In the meta configuration, the transmission function
also exhibits an antiresonance in the middle of the HOMO-LUMO gap (i.e., at
$E=0$). The antiresonance orginates from DQI, in complete agreement with the
scenario predicted within the graphical method.

Analogous conclusions can be drawn from the Coulson-Rushbrooke pairing
theorem, Zhao _et al._ (2017) which states that for alternant hydrocarbons,
when the contact sites $\ell$ and $r$ belong to the same sublattice, the
contribution to the Green’s function coming from the MOs cancels pairwise.
Each pair $k$ is formed by an occupied and an unoccupied MO. Then, the
spectral representation of the Green’s function in the MO basis reads

$G_{\ell
r}(E)=\sum_{k=1}^{N/2}\Big{[}\frac{c^{\phantom{\ast}}_{\ell;-k}c^{*}_{r;-k}}{E+\imath\eta-\epsilon_{-k}}+\frac{c^{\phantom{\ast}}_{\ell:k}c^{*}_{r;k}}{E+\imath\eta-\epsilon_{k}}\Big{]},$
(4)

where $c_{i;k}$ is the coefficient of MO $k$ at site $i$ and $N$ the number of
MOs in the Hamiltonian. The pairing theorem predicts that for alternant
hydrocarbons each pair fulfills the conditions $\epsilon_{k}=-\epsilon_{-k}$
and that all coefficient in one sublattice have opposite sign between the
occupied and unoccupied MOs, then for each $k$ the contribution of the pair to
the sum in Eq. (4) cancels exactly at $E=0$. Zhao _et al._ (2017) In the meta
configuration, $\ell$ and $r$ belong to the same sublattice of the pyrene
molecule, as shown in Fig. 2(e), and the Coulson-Rushbrooke predicts a QI
antiresonance at the Fermi level, in agreement with the graphical scheme and
the SHM calculation.

## DFT+NEGF Electron transport calculations

The above analysis establishes clearly the occurrence of DQI depends on the
position of the contact atoms on the pyrene core. While this scenario is
qualitatively robust, we shall see in the following that the details of the
chemical bonding between the molecule and the electrodes, as well as the
physical properties of the electrodes, which can be addressed within DFT, can
drastically affect the observability of DQI in the transmission function.

### Computational details

We investigate the transport properties of molecular junctions with pyrene
core within the DFT+NEGF formalism, Brandbyge _et al._ (2002); Xue _et al._
(2002) as implemented within the Atomic Simulation Environment Larsen _et al._
(2017) (ASE) and the GPAW software package.Mortensen _et al._ (2005);
Enkovaara _et al._ (2010) The transmission function is calculated according to
Eq. (2) where the Green’s function is evaluated on the whole DFT basis set for
the scattering region. The coupling matrix $\Gamma$ is obtained from the
surface Green’s function of the leads and the couplings between the basis
function of the lead and scattering regions. The transport Hamiltonian of each
structure is evaluated in a local combination of atomic orbitals (LCAO) Larsen
_et al._ (2009) double-$\zeta$ polarized basis, with a Perdew-Burke-Ernzerhof
(PBE) parameterization for the exchange-correlation functional, and a sampling
of 0.2 Å grid spacing. For each molecular configuration we perform an atomic
optimization of the molecule in vacuum until the value of the Hellmann–Feynman
forces are below $0.05$ eV/Å.

Figure 3: Bandstructure $E(k)$ and density of states (DOS) of (a) fcc Au, and
(b) 12-zGNRs. The width of the GNR is defined as the number of C atoms along
the dimer line. The insets next to each panel show the corresponding unit cell
(dashed lines). Close to the Fermi level, Au displays a featureless DOS, while
zGNRs display non-dispersive bands and a corresponding peak in the DOS, due to
the localized states near the ZZ edges (highlighted in color in the inset).

In the case of Au electrodes, the scattering region consists of seven $6\times
4$ layers of Au(111). The linkers are connected via anchoring groups to Au
adatoms placed in the hollow position of the Au(111) surface. We take a
typical bonding distances of $d_{\textrm{Au-S}}=2.12$ Å for thiol-terminated
linkers Zhao _et al._ (2017); Stadler _et al._ (2005) and
$d_{\textrm{Au-C}}=1.92$ Å for acetylene-terminated linkers. Liu _et al._
(2013) The scattering region is sampled with a $4\times 4\times 1$ Monkhorst-
Pack mesh, and the leads are sampled with a $4\times 4\times 6$ Monkhorst-Pack
mesh, where $z$ denotes the transport direction.

In the case of graphene electrodes, GNRs typically have either a zigzag (ZZ)
or an armchair (AC) termination. We consider the GNRs with ZZ edges and denote
periodic structures with width N as N-zGNRs, following the standard
nomenclature in the literature. The dangling bonds at the graphene edges are
passivated with H atoms. For the portion of GNRs included in the scattering
region, we assume a rectangular termination, which can be expected for
atomically-precise GNRs obtained though bottom-up synthesis by polymerization.
Cai _et al._ (2010) Hence, in this configuration, at the junction interface
between the GNRs and the molecular bridge, zGNRs have a transverse edge with
AC topology (on the contrary, aGNRS have a transverse edge with ZZ topology).
For graphene electrodes, a natural choice is to connect the molecule to the
leads by replacing one of the H atoms passivating the edge on each side with
the acetylene-terminated linkers, with a bonding distance
$d_{\textrm{C-C}}=1.54$ Å. The scattering region is sampled with a $1\times
1\times 1$ Monkhorst-Pack mesh, and the leads consist of GNR unit cells
sampled with a $6\times 1\times 1$ Monkhorst-Pack mesh, where $x$ denotes the
transport direction. For infinitely-wide GNRs (semi-infinite graphene sheet)
we impose periodic boundary conditions along the in-plane direction ($y$)
transverse to the transport direction, which we sample with a Monkhorst-Pack
mesh with up to $256$ $k$-points.

## Results and Discussion

From the atomic and electronic structure perspective, Au and graphene are
completely different materials. While for Au electrodes the transmission close
to the Fermi level is dominated by a single transmission channel with
predominant Au 6s character, Cuevas and Scheer (2010); Jia and Guo (2013) the
electronic properties of graphene critically depend on the topology of the
edges. In particular, it is well known that GNRs with ZZ termination support
non-dispersive states localized near the edges. Hence, the zGNRs that we
consider in the following are metallic, while aGNRs are typically
semiconductors. Wakabayashi _et al._ (1999); Son _et al._ (2006) Moreover,
aGNR electrodes have transverse ZZ edge states that decay very slowly into the
bulk, Li _et al._ (2014, 2015) making the correct computational treatment with
first principle methods very challenging. To illustrate the properties of the
electrodes, in Fig. 3 we compare the bandstructure $E(k)$ and corresponding
density of states (DOS) of fcc Au, to those of periodic zGNR.

Another difference between Au and graphene electrodes is that GNRs with ZZ
edges have a tendency towards magnetic ordering. Hence, in order to complete
our analysis, we also discuss the effect of a spin-symmetry breaking on the
DQI properties of pyrene molecular junction.

### Au electrodes

Figure 4: (a,b) Transmission functions in the meta- and para-configurations
connected with different anchor groups to Au electrodes. The grey lines
represent the transmission function obtained with only the two pairs of MOs
closest to the Fermi level. The insets show the linkers connecting the pyrene
molecule to the Au leads. (c-f) Frontier MOs obtained by subdiagonalization of
the pyrene core and corresponding HOMO-LUMO gaps.

We first discuss the case in which the molecule bridges Au electrodes. The
transmission as a function of energy of the incoming electrons for the pyrene
connected in the meta- and para-configurations with either thiol or acetylene
anchoring groups is shown in Fig. 4(a,b), respectively. The transmission
function is similar in all cases, with the main difference being the
antiresonance within the HOMO and LUMO gap present in the meta configuration.
This is in complete agreement with the previous analysis with the graphical
scheme, the Coulson-Rushbrooke pairing theorem, and the SHM. However, the
conductance of the junction also strongly depends on the relative positions of
the HOMO, the LUMO, and the antiresonance with respect to the Fermi level,
which we are going to analyze in detail in the following.

In all cases, the antiresonance is closer to the HOMO than to the LUMO. For
thiol-terminated linkers, $\epsilon^{\textrm{DQI}}\approx 0.12$ eV is also
close to the Fermi level $E_{F}=0$, whereas for acetylene-terminated linkers
there is a shift to higher energies, resulting in
$\epsilon^{\textrm{DQI}}\approx 0.6$ eV. The most striking difference between
the two cases is the relative position of the Fermi level within the HOMO-LUMO
gap. In order to shed light on this, we perform a Bader charge analysis Tang
_et al._ (2009) for the free molecule and the junction with both linkers,
because the Fermi level alignment is closely related to the zero-bias charge
transfer. Stadler and Jacobsen (2006); Stadler (2007, 2010); Kastlunger and
Stadler (2013) The HOMO-dominated profile in the case of acetylene linkers is
coming from the negatively charged molecule after the placement in between the
leads, which shifts the position of HOMO and LUMO to higher energies, with
respect to thiol linkers. Stadler and Jacobsen (2006); Stadler (2007) When we
analyze the case with thiol linkers, there is a clear difference in the Fermi
level alignment of the meta- and para-configurations. This can be explained by
considering the larger electron transfer from the molecule towards the
electrodes in the meta with respect to the para configuration as reported in
Table 1. In the case of acetylene linkers instead, the charge transfer is
negative, corresponding to electron transfer from the electrodes to the
molecule. However, there is no direct relation between
$\epsilon_{\textrm{HOMO}}$ and $\Delta Q$ and the molecular level position
cannot be fully explained by net charge transfer alone, but the molecular
dipole moments, can possibly account for it. The dipole moment is larger in
the meta configuration and it is orientated within the plane of the molecule
and orthogonal to the transport direction (see Table 1).

Table 1:  Position of the HOMO resonance, obtained from the subdiagonalization of the molecule including the linkers, net charge on the molecule $\Delta{Q}$, dipole moment of the free molecule $\mu$ in all configuration.  |  |  $\epsilon_{\text{HOMO}}$ (eV) |  $\Delta{Q}$ ($|e|$) |  $\mu$ (D)
---|---|---|---|---
Thiol | meta | -0.97 |  -0.39 | 0.33
| para | -0.72 |  -0.16 | 0.19
Acetylene | meta | -0.20 | -0.38 | 0.14
| para | -0.20 | -0.50 | 0.01

It is possible to qualitatively describe the QI effects taking into account
only the symmetry properties of the pyrene core. This analysis is also
consistent with our DFT calculations. We diagonalize the sub-block of the
transport Hamiltonian of the pyrene core to obtain the projected MOs. We can
calculate an approximate transmission function selecting the two pairs of MOs
closest to the Fermi level, i.e., $\textrm{HOMO}-1$ , HOMO, LUMO, and
$\textrm{LUMO}+1$ and cutting out the rest. Stadler (2009); Guédon _et al._
(2012); Zhao _et al._ (2017); Gandus _et al._ (2020) As shown in Fig. 4(a,b)
with grey lines, the contribution of these two pairs is sufficient to
qualitatively reproduce the antiresonance. The approximate transmission
function shows a good qualitative agreement with the one from the original
transport Hamiltonian but the position of the antiresonance and the Fermi
level alignment are more sensitive to such simplifications.

The symmetry of the MOs carries relevant information on QI. According to the
Coulson-Rushbrook pairing theorem, the antiresonance is a consequence of
pairwise cancellation of the contributions of the MOs to the transmission. The
frontier MOs of the pyrene core are shown in Figs. 4(c-f) for thiol- and
acetylene-anchoring group in the meta- and para-configurations. In the case of
thiol anchoring groups, the signs of the amplitudes of the eigenvector at the
positions where pyrene is connected to the linkers are opposite for both the
HOMO and the LUMO in the meta configuration. Hence, we expect a cancellation
of the contribution of the frontier MOs in the transmission function,
resulting in an antiresonance at some energy $\epsilon^{\textrm{DQI}}$, not
necessarily at the Fermi level, or mid-gap as in the SHM calculation, since
the spectrum is not particle-hole symmetric. Instead, in the para
configuration, the amplitude of the LUMO has the same sign at the contact
sites, and therefore there is no cancellation in the transmission. When we
substitute thiol- with acetylene-group as anchoring units, the same symmetry
considerations hold. This suggests that while the origin of the QI phenomena
does not depend on the nature of the linkers and anchoring groups, the energy
at which the contributions from all MOs cancels is related in a non-trivial
way to the chemical nature and the electronic structure properties of all the
components of the junction.

### Graphene electrodes

We examine now the case in which the pyrene molecule is connected to graphene
electrodes via acetylene-terminated linkers. The schematic representations of
the junctions with GNR electrodes are shown in Fig. 5(a) where the edges
transverse and parallel to the transport direction are highlighted in color.
As anticipated, zGNRs have AC edges transverse to the transport direction. In
order to assess the effect of edge size and topology on DQI, we consider zGNRs
with different widths.

Figure 5: Transmission functions for pyrene junctions in meta- and para-
configurations with zGNR electrodes of different width. The width of the zGNR
is defined as the number of C atoms along the dimer line. Each inset
highlights the low-energy features of the transmission function close to the
Fermi level.

Since with Au electrodes the QI properties arise from the position of the
contact atoms in the pyrene, it is reasonable to expect similar features also
for graphene electrodes, and an antiresonance is found indeed only for the
meta-connected structures. In contrast to the Au electrodes, in which the
transmission is HOMO-dominated, here the Fermi level is located close to the
middle of the HOMO-LUMO gap and the antiresonance is below $E_{F}$. This is
not surprising considering that with GNR electrodes, the junction is hydro-
carbon throughout and the charge distribution between the molecule and the
leads is expected to be uniform. Alqahtani _et al._ (2018)

In all transmission functions showed in Fig. 5, there are features at the
Fermi level, which were absent in the case of Au electrodes. The most natural
explanation is that these features stem from graphene edge states. Indeed, the
shape of the transmission reflects the presence of both ZZ and AC edges, in a
way that depends on the width and the topology of the electrodes. In Figs.
5(b), we observe a resonance at the Fermi level, which indicates a
transmission mechanism involving the edge states of the 12-zGNRs. The
transmission feature is significantly broad due to the high DOS of the zGNR
leads. Some feature, albeit not a resonance is also observe in Fig. 5(c), for
the 24-zGNRs. We speculate the overlap between the edge state and the states
of the molecular bridge decreases by increasing the width of the zGNR.
Eventually, for $\infty$-zGNR, the transmission function in Fig. 3(c) recovers
a _V-shape_ , reminiscent of the DOS of bulk graphene, but it exhibits a gap
at the Fermi level (see inset) due to the presence of the transverse edge of
the electrode, which has AC character.

## Effect of DQI on the I-V characteristics

In an experimental setup, the current-voltage (I-V) characteristics is
measured rather than the transmission function. In order to draw conclusions
on the observability of DQI effects, we evaluate the electric current (per
spin) as

$I=\frac{e}{h}\int_{\infty}^{\infty}dE\ T(E)\
\big{[}f_{S}(E)-f_{D}(E)\big{]},$ (5)

where $e$ is the electric charge and $h$ the Planck constant. The Fermi
distribution function of the source (S) and drain (D) electrodes are given by

$f_{S/D}(E)=\frac{1}{1+\exp[(E-V_{S/D})/k_{B}T]},$ (6)

where $k_{B}$ is the Boltzmann constant and $V_{b}=V_{S}-V_{D}$ is the
symmetric bias drop between source and drain. In the calculations, we assume
the thermal broadening of the Fermi distribution to be $k_{B}T=5$ meV and the
current is evaluated neglecting the bias dependence of the transmission
function, i.e., $T(E,V_{b})\approx T(E)$. In the following, we also look at
the differential conductance $G=dI/dV_{b}$ as a function of both bias
($V_{b}$) and gate ($V_{g}$) voltages.

Figure 6: I-V characteristics (a,d,g) and differential conductance
$G=dI/dV_{b}$ as a function of gate ($V_{g}$) and bias ($V_{b}$) voltages
(b,c,e,f,h,i) for all junctions. For Au electrodes, DQI suppresses electron
transport in the meta configuration in the case of thiol anchoring groups,
while for acetylene groups it is dominated by resonant tunneling though the
HOMO and QI has no visible effects. For graphene electrodes, the effect of DQI
is partially masked due to the presence of the zGNR edge states but it can be
enhanced by applying a gate voltage $V_{g}=240$ mV to align the Fermi level to
the antiresonance.

The calculated I-V characteristics and the differential conductance maps for
the junctions with Au and zGNR electrodes reveal the qualitative relationship
between the transmission and the transport features. The results are presented
in Fig. 6. In the case of Au electrodes, the outcome is quite interesting. For
all combinations of contact configurations, the HOMO-LUMO gap is very similar,
while the junctions with acetylene linkers exhibit an overall much higher
current compared to the thiol ones, reaching the order of a few $\mu$A, see
Figs. 6(a,d). This is related to the close proximity of the HOMO to the Fermi
level and its wider peak in the transmission function for this linker type.
Schwarz _et al._ (2014) As a consequence, the meta- and para-configurations
display very similar I-V characteristics since the effect of DQI is negligible
with respect to the resonant contribution of the HOMO. The current displays
clear steps in correspondence to the MOs entering the bias window, as
indicated in Fig. 6(d). For thiol-terminated linkers, the combination of two
effects, i.e., the Fermi level being closer to the middle of the HOMO-LUMO gap
and closer to the antiresonance, results in a large difference in the I-V
characteristics of the meta- and para-configurations. In Figs. 6(b,c), we show
the map of the differential conductance, where the effects of DQI is clearly
visible, by comparing the meta- and para-configurations for this combination
of electrodes and anchoring groups. The bright lines in the I-V
characteristics correspond to the resonances of the MOs.

It is interesting to observe the effect of edge topology of the graphene
electrodes on the I-V characteristics. The transmission features close to the
Fermi level, originating from the zGNR edge states, appear in both meta- and
para- configurations. As a consequence, in both cases we observe a metallic-
like I-V characteristics and QI effects do not play an important role.
However, for devices based on graphene, and in general 2D-materials, gating is
a viable strategy to improve device performances. Specifically, in order to
enhance the effects of DQI, one can introduce an external electric field to
align the Fermi level to the antiresonance. Applying a gate voltage
$V_{g}=240$ mV to the junctions with zGNR electrodes, results in a strongly
suppressed current, see grey dashed line in Fig. 6(g). This allows one to
clearly distinguish the I-V characteristics of the configuration with DQI from
the other. In Figs. 6(h,i) we show the map of the differential conductance.
The resonances corresponding to the edge states are clearly visible for both
meta- and para-configurations while suppression due to DQI is clearly
identified as a single dark spot at $V_{g}=240$ mV.

## Effects of edge magnetism in graphene

While the experimental evidence of magnetism in graphene nanostructures Magda
_et al._ (2014); Slota _et al._ (2018) is still controversial, from the
theoretical point of view it is well established that zGNRs have a tendency
towards magnetic order. When the spin symmetry is lifted, the GNR orders
according to a Néel antiferromagnetic (AF) pattern, which results in
ferromagnetically (FM) aligned ZZ edges, due to the local sublattice
imbalance. Opposite edges display opposite magnetization, yielding a global
singlet (S=0) and thus fulfilling Lieb’s theorem. The magnetic moments are
rapidly suppressed towards the bulk, as well as at AC defects and corners,
where the atoms from different sublattices are paired. The magnetic structure
of zGNRs has been widely discussed in the literature, and it has been shown to
remain qualitatively unchanged also when a local electron-electron
interactions is included explicitly within DFT+U, Kabir and Saha-Dasgupta
(2014); Ganguly _et al._ (2017) a mean-field Hubbard model, Fernández-Rossier
and Palacios (2007); Hu _et al._ (2017) and within more sophisticated many-
body techniques. Feldner _et al._ (2011, 2010); Valli _et al._ (2016, 2018);
Baumann _et al._ (2020); Phùng _et al._ (2020) Hence, the following discussion
can be regarded as of general relevance for the interplay between DQI and edge
magnetism.

Figure 7: Bandstructure $E(k)$ and density of states (DOS) of AF 12-zGNR. The
bandstructure of the two spin polarizations is degenerate in all bands. The
arrows mark the relevant energy splittings along the $\Gamma-Z$ path of the
one-dimensional Brillouin zone. The inset in the side panel show the
corresponding unit cell (dashed lines). The color code represents the size of
the local magnetic moments with up (red) and down (blue) polarization.  Figure
8: (a,c) Spin-ordered structures with the pyrene molecule in the meta
configuration and magnetic zGNR electrodes. The color code represents the size
of the local magnetic moments with up (red) and down (blue) polarization.
(b,d) Spin-resolved transmission functions (red and blue dashed lines)
compared to those of spin-paired calculations (grey solid lines). The presence
of the AF gap $\Delta_{AF}$ results in a strong suppression of the
transmission function close to the Fermi level, as also highlighted in each
inset. See text for discussion.

For the sake of completeness, in Fig. 7 we show the bandstructure of periodic
12-zGNR and the spatial distribution of the magnetic moments in the Néel AF
state. Note that the bandstructures for the two spin states are degenerate in
all bands. Son _et al._ (2006) The characteristic flat bands at the Fermi
level, observed in Fig. 3(b), disappear in favor of an AF gap $\Delta_{AF}$.
The band splitting at the $Z$ point of the one-dimensional Brillouin zone
(i.e., $k=\pi/a$, with $a$ the lattice constant) is approximately $0.5$ eV and
weakly width-dependent, while the narrower splitting along the $\Gamma-Z$ path
is inversely proportional to the zGNR width, in agreement with the literature.
Son _et al._ (2006)

In order to evaluate the transport properties and the interplay between DQI
and magnetism, we perform spin-polarized calculations for both the leads and
the scattering region with the pyrene molecule contacted in the meta
configuration. The structures considered and the corresponding spin-polarized
transmission functions are shown in Fig. 8. As expected, the transmission
functions in Figs. 8(b,d) display a sizable AF gap at the Fermi level, which
depends on the width of the zGNR. Son _et al._ (2006) Weak spin-polarization
effects are observed close to the Fermi level, where the transport mechanisms
involves the states localized at the ZZ egdes. Instead, the features
attributed to MOs with a strong pyrene character barely differ from those
observed in the spin-paired calculations, as the molecule remains non-
magnetic. The transmission functions display weakly spin-dependent DQI (sDQI)
antiresonances, appearing either inside (for 12-zGNRs) or outside (for
24-zGNRs) the AF gap, as clearly shown in the corresponding insets. It is
interesting to note that DQI results in a suppression of the transmission even
inside the magnetic gap, since DQI is associated to an actual zero of the
Green’s function, rather than just a suppression of spectral weight. Pedersen
_et al._ (2014); Valli _et al._ (2018, 2019) In the limit of $\infty$-zGNR,
magnetism is expected to become irrelevant, since the ZZ edges and hence the
associated sublattice imbalance disappear.

From our analysis, we conclude that the existence of DQI induced
antiresonances is not directly affected by the edge magnetism on the graphene
electrodes. However, the possibility of identifying DQI unambiguously remains
conditional to the details of the magnetic order, which is controlled by the
width and the topology of the GNR electrodes. Let us also note that in the
cases discussed above, the position of the sDQI antiresonances is almost
independent on the spin polarization, since the pyrene core is not magnetic.
Instead, if the source of QI itself is magnetic, one can expect also a spin-
dependent splitting of the DQI energy, $\epsilon^{\textrm{DQI}}_{\sigma}$, as
reported in recent studies. Valli _et al._ (2018, 2019); Phùng _et al._ (2020)

## Conclusions and outlook

We presented a thorough analysis of the effects of different anchoring groups
and electrodes on the DQI properties in pyrene based single-molecule
junctions. Simple theoretical approaches allow to reliably predict the
occurrence or absence of DQI in specific contact configurations because it
stems from molecular symmetry. However, the role of the chemical nature of all
components of the junction for the observability of DQI features can only be
unraveled with DFT+NEGF calculations.

In our study, we find that the linkers play an important role with Au
electrodes, as the charge redistribution within the junction determines the
relative energy position of the antiresonance with respect to the Fermi level
and the frontier MOs. This eventually determines to which extent DQI affects
the current-voltage characteristics of the junction. For pyrene junctions, the
choice of Au electrodes allows to clearly observe DQI in the electronic
transmission function, as the electrode do not introduce additional transport
features.

In the case of graphene, the electronic properties at the Fermi level are
strongly dependent on the width of the electrodes and the topology of the
edges. As a consequence, the electrodes can modulate the transport properties
of the junction and introduce transmission oscillations close to the Fermi
level, which may obscure or even resemble QI antiresonances. Gehring _et al._
(2017) We were able to disentangle the features arising from the molecule and
those arising from the electrodes, and we traced back the origin of each
transmission feature to the topology of the edges. In this respect, we provide
clear indications to guide the interpretation of experimental results, which
is generally difficult, especially when the energy scales of the molecule and
that of the electrodes become comparable. We also stress that one of the
advantages of graphene over metal electrodes is a better control on the
position of the Fermi level. This can be achieved by gating, which allows to
maximize the effect of DQI (or of the edge states for that matter) on the
current-voltage characteristics and is therefore of pivotal importance for
technological applications exploiting QI.

Finally, we considered the effects of magnetic order in graphene electrodes,
which arises due to the local sublattice imbalance at the ZZ edges. Since
magnetism originates at the edge states, it mainly affects the electronic and
transport properties in an energy window around the Fermi level. For zGNRs, a
transmission gap of magnetic origin in the electrodes can potentially prevent
a clear observation of DQI. For the junctions we considered, the AF spin-
ordered pattern resulting in a global singlet state, and the transmission
displays a negligible spin polarization. In the presence of a net magnetic
moment one would expect a significant spin splitting of the transmission of
the majority and minority spin channels.

While the present analysis revolves around prototypical single-molecule
junction setups, it also poses the basis for future studies. When the focus
shifts to complex graphene-based devices, such as nanoconstrictions Gehring
_et al._ (2016) or junctions with graphene quantum dots, Valli _et al._ (2018,
2019); Niţă _et al._ (2020) the source QI may differ in the presence of the
electrodes with multiple contacts. The additional contacts increase the number
of potential transmission channels and QI patterns can emerge within
individual or between different channels, thereby increasing the complexity of
the analysis. Junctions with multiple anchors can display either a low- or a
high- conductance, depending on the connectivity of the molecule, Markin _et
al._ (2020) which can be electrochemically controlled. Darwish _et al._
(2012); Grunder _et al._ (2007) A systematic investigation of multiple-contact
junctions can open new avenues to exploit QI effects in the framework of
graphene nanoelectronics.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgments

We thank G. Gandus, G. Kastlunger, and J. Völkle for helpful discussions. We
acknowledges financial support from the Austrian Science Fund (FWF) through
project P 31631. Calculations have been performed on the Vienna Scientific
Cluster (VSC), project No. 71279.

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