Charge Transport in C$_{60}$-based Single-Molecule Junctions with Graphene Electrodes

2.9 MB·

by  Susanne Leitherer

Charge Transport in C -based Single-Molecule Junctions with Graphene Electrodes
60
S. Leitherer,1 P. B. Coto,1 K. Ullmann,2 H. B. Weber,2 and M. Thoss1
1Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials,
Friedrich-Alexander-University Erlangen-Nu¨rnberg (FAU),
Staudtstr. 7/B2, D-91058 Erlangen, Germany
2Chair of Applied Physics and Interdisciplinary Center for Molecular Materials,
Friedrich-Alexander-University Erlangen-Nu¨rnberg (FAU),
Staudtstr. 7/A3, D-91058 Erlangen, Germany
(Dated: January 12, 2017)
7 We investigate charge transport in C60-based single-molecule junctions with graphene electrodes
1 employing a combination of density functional theory (DFT) electronic structure calculations and
0 Landauertransporttheory. Inparticular,thedependenceofthetransportpropertiesontheconfor-
2 mationofthemolecularbridgeandthetypeofterminationofthegrapheneelectrodesisinvestigated.
Furthermore, electron pathways through the junctions are analyzed using the theory of local cur-
n
rents. Theresultsreveal,inagreementwithpreviousexperiments,apronounceddependenceofthe
a
transportpropertiesonthebiaspolarity,whichisrationalizedintermsoftheelectronicstructureof
J
themolecule. Itisalsoshownthattheedgestatesofzigzag-terminatedgrapheneinduceadditional
1
transportchannels,whichdominatetransportatsmallvoltages. Theimportanceoftheedgestates
1
for transport depends profoundly on the interface geometry of the junctions.

]
l
al I. INTRODUCTION been the focus of several theoretical investigations,4–6
h where phenomena such as spin transport, heat transport
- ortheroleofphononswerestudied. Thetheoreticalstud-
s Thetwo-dimensionalcarbonmaterialgrapheneischar-
e acterized by a number of outstanding properties includ- ies have alsoshownthat it iscrucial for the development
m of graphene-based nanoelectronics to control the edge
inghighelectronmobility,highthermalconductivity,and
. mechanical stability, which make it a promising material structure.7,8 Specifically, the role of zigzag edge states
at for electronic devices.1 An important example is the use of graphene has been examined in detail.9–11
m of graphene as alternative material for electrodes in na- Recent experiments have demonstrated the use of
- noelectronic applications, replacing conventional metal graphene electrodes in single molecule junctions.12–16 In
d
leads.2 This application appears particularly promising Ref.13,Ullmannandco-workersreportedonexperiments
n
o for single-molecule junctions in the field of molecular usingepitaxialgrapheneelectrodesandaC60-endcapped
c electronics, where the replacement of metal electrodes molecular bridge. The results of that study showed
[ by graphene could help to overcome several limitations. a striking similarity of the transport properties using

Due to its two-dimensional geometry, its small optical graphene and gold electrodes, demonstrating that trans-
1
absorption cross section and high thermal conductiv- portinthissystemisdominatedbythemolecularbridge
v
0 ity, graphene electrodes can facilitate the access to the itself. Furthermore strongly asymmetric current-voltage
8 junction using light fields or scanning probe tips, which characteristicsandswitchingbehaviorwerefound,which
9 willprovideawealthofexperimentalcontrolparameters. were associated to the existence of different conformers.
2 Furthermore,grapheneedgesmayreducetheuncertainty In this paper, we provide a detailed theoretical analy-
0
about the molecule-electrode contact geometry, a major sis of transport in this class of C -endcapped molecule-
. 60
1 drawback of metal electrodes.3 graphene nanojunctions. The study employs a com-
0 Transport through graphene-based nanojunctions has bination of density functional theory (DFT) and Lan-
7
dauer transport theory. In addition, the theory of lo-
1
cal currents17,18 is used to identify pathways of electrons
:
v through the molecular junctions.
i
X The article is organized as follows: In Sec. IIA, we
introduce the class of junctions investigated throughout
r
a the paper. The theoretical methodology is outlined in
Secs. IIB - IID, including electronic structure calcula-
tions, the Landauer transport approach as well as the
theory of local currents. Sec. III presents the results of
the simulations obtained for various types of junctions,
which differ in the conformation of the molecule and the
termination of the graphene electrodes. We discuss the
transport properties and their relation to the electronic
states of the molecule as well as the influence of the in-
FIG. 1: C -endcapped molecule-graphene junction.
60 terface geometry and the role of graphene edge states.
2
Additionally, local pathways of electrons in the junction VASP.25 Geometry relaxation was stopped when forces
are analyzed. wheresmallerthan0.02eV/A.Followingpreviouswork9,
transport calculations were carried out using cluster
modelsbuiltfromtheequilibriumstructuresobtainedus-
II. SYSTEMS AND METHODS ingtheperiodicDFTcalculations. Thesizesofthediffer-
ent clusters were selected to simulate as correct as possi-
A. Molecular junctions blethegrapheneleadswhilekeepingthesystemtractable
fromacomputationalpointofview. Theelectronicstruc-
ture calculations for the cluster models, necessary to
The architecture of the molecular junctions investi-
obtain the Hamiltonian for the different systems inves-
gatedisdepictedinFig.1. ItcomprisesaC -endcapped
60 tigated, were performed using TURBOMOLE26 at the
molecular bridge with two styrene units that are cova-
DFT level of theory employing the B3LYP functional27
lentlyconnectedviaa[2,2’]paracyclophanemoietyinthe
center.19,20 The molecular bridge binds via the fullerene and the def-SV(P) basis set.28 Dispersion interactions
were incorporated using Grimme’s empirical dispersion
anchor groups to two graphene leads. Based on this ar-
correction.22
chitecture, we have studied different types of molecular
junctions depicted in Fig. 2.
For the transport calculations, extended systems are
C. Model and transport theory
defined, comprising the molecule and a finite part of
the graphene electrodes. The graphene edges are sat-
To study charge transport through the nanojunctions,
urated with hydrogen atoms in order to avoid dan-
we employ the calculation scheme presented in Ref. 9.
gling bonds. The extended systems are coupled to two-
This involves the partitioning of the Hamiltonian ma-
dimensional semi-infinite graphene leads, described via
trix describing the extended system into blocks corre-
absorbing boundary conditions (see Sec. IIC). We in-
sponding to the molecular bridge, H , the left and
vestigate zigzag-terminated graphene nanocontacts as m
right contact, H and H , and the respective couplings
showninFig.2(a)aswellasarmchairtermination,Fig.2 l r
V ,V ,V ,V .29 To model the effect of the semi-
(b). The C anchor groups of the molecule are bounded ml mr lm rm
60
infinite graphene leads, we employ absorbing boundary
tothegrapheneleadsviadispersioninteractionsresulting
conditionsusingcomplexabsorbingpotentials(CAPs)of
in a distance of about 1.5 nm.
theform−iW ,α=l,r,with(W ) =w δ ,whichare
For both types of graphene edge termination, the ge- α α ij αi ij
modeled as
ometry optimization (see Sec. IIB) reveals the existence
of two structural conformations (A,B) corresponding to w =a|x −x |n (1)
αi i α
localminimaofthepotentialenergysurface,whoseener-
gies differ by 0.3 eV. The two conformations exhibit dif- On the basis of test calculations, the prefactor a was
ferent orientations of the phenylenevinylene moieties in chosen such that the results do not depend on it.30 The
thecentral[2,2’]-paracyclophane-phenylenevinylenefrag- absorbingpotentialsareaddedtotheHamiltonianofthe
ment (cf. Fig. 2 c,d). graphene contacts, such that the overall Hamiltonian H˜
reads
 
H −iW V V
l l lm lr
B. Electronic structure methods H˜ = Vml Hm Vrm . (2)
V V H −iW
rl mr r r
The equilibrium geometries of the different single
molecule junctions analyzed in this work were obtained Thereby, the CAPs are applied in the graphene contacts
using periodic density functional theory (DFT) calcu- far away from the fullerene molecules to avoid artificial
lations. Specifically, slab models consisting of two effects.
graphene leads and the molecular linker comprising 784 Within the Landauer approach, transportthrough the
(834) atoms for the zigzag (armchair) terminated junc- contact is described by the transmission function t(E),
tions were used. To prevent artificial interactions be- which for an electron with energy E is given by31
tween successive slabs, a vacuum layer with a minimum t(E)=tr(cid:8)Γ G Γ G† (cid:9). (3)
width of 12 ˚A was enforced. In all the optimizations, r m l m
the molecular bridge and the hydrogens saturating the Hereby, G denotes the Green’s function projected onto
m
grapheneedgeswererelaxed. TheDFTcalculationswere the molecular bridge,
carried out using the Perdew-Burke-Emzerhof (PBE)
exchange-correlation functional,21 including dispersion Gm(E)=(cid:2)E+−Hm−Σl(E)−Σr(E)(cid:3)−1, (4)
corrections employing Grimme’s D3 method22 and using
which involves the self energies Σ (E) that describe the
α
the projected augmented wave method23. A k-mesh of
coupling of the molecular bridge to the left and right
2×3×1 points was generated using the Monkhorst-Pack
electrodes (α=l,r), and are given by the expression
technique24 and an energy cut-off of 415.0 eV was used
after testing. These calculations were carried out using Σ (E)=V [E−H +iW ]−1V (5)
α mα α α αm
3
FIG. 2: Single-molecule junctions investigated in this work. (a) Junction with zigzag-terminated graphene electrodes
and (b) armchair-terminated graphene electrodes. (c) Side view of molecular conformer A and (d) conformer B.
Theselfenergiesarerelatedtothewidth-functionΓ (E), withtheBoltzmannconstantk , theelectrodetempera-
α B
used in Eq. (3), via ture T and their chemical potentials µ (α=l,r). For a
α
symmetric drop of the bias voltage V around the Fermi
i
Σα(E)=∆α(E)− 2Γα(E), (6) energy EF, the chemical potentials are given by
eV
where ∆α denotes the level-shift function. µα =EF ± 2 . (10)
Tocharacterizethemolecularfeaturesintransport,we
haveinadditionusedasimplerapproach,wherethecon-
ductancepropertiesarecalculatedfortheisolatedmolec- D. Analysis of transport pathways using local
ular bridge without explicit inclusion of the graphene currents
electrodes. Inthisapproach,theelectrodesaredescribed
within the wide-band approximation (WBA), where ∆α Eqs. (3) and (8) describe charge transport in terms of
is zero and Γα is replaced by a constant. Specifically, theoverallelectricalcurrentandtransmissionfunctionof
the matrix elements of the self energies Σα(E) in a local the complete junction. In order to have a more detailed
atomic orbital basis are given by description and, in particular, to analyze pathways of
charge transport in the junction, we use the method of
i
(Σ (E)) =− (Γ ) (7) localcurrents17,32. Withinthistechniquetheoverallcur-
α νν 2 α νν
rent I through a surface perpendicular to the transport
for orbitals ν corresponding to carbon atoms at the bot- direction is represented in terms of local currents via
tomoftheC , whichcoupletothegraphenesurface. In
60 (cid:88) (cid:88)
thecalculationsreportedbelow,wehaveused(Γα)νν =1 I = In(cid:48)n, (11)
eV . n(cid:48)∈MLn∈MR
Basedonthetransmissionfunctiont(E),theelectrical
where M denotes atomic sites left and right of the
L/R
current through the junction is given by the expression
chosen surface, respectively. The local contributions to
2e(cid:90) the current from site n to n(cid:48) are given by17,33
I(V)= dEt(E)(f (E)−f (E)), (8)
h l r 2e(cid:90)
I = dEK (E), (12)
n(cid:48)n h n(cid:48)n
where f (E) denotes the Fermi function for the elec-
l/r
trons in the left/right lead. It is given by with
(cid:88) (cid:88)
1 K (E)= (V G< (E)−V G< (E)). (13)
f (E)= (9) n(cid:48)n νµ µν µν νµ
α 1+e(E−µα)/kBT ν∈n(cid:48)µ∈n
4
Here,Vνµ denotesthecouplingconstantbetweenorbitals (a) 100 Conformer A zigzag
ν and µ, and G< is the lesser Green’s function, given by
on 10-2
G< =(iflGmΓlG†m+ifrGmΓrG†m). (14) missi 10-4
s
For temperature T = 0, the expression an 10-6
(cid:80) (cid:80) Tr
n(cid:48)∈ML n∈MRKn(cid:48)n can be identified with the 10-8
transmission function, Eq. (3), and thus K defines -5 -4.5 -4 -3.5 -3
n(cid:48)n
Energy [eV]
local contributions to the transmission between pairs of
atoms.
CCoonnffoorrmmeerr BB zziiggzzaagg
(b) 100
III. RESULTS AND DISCUSSION on 10-2
si
mis 10-4
Using the methodology outlined above, we have char- s
acterized the transport properties of the molecular junc- Tran 10-6
tions. In the following, we analyze the transport prop- 10-8
erties based on the transmission function, the relevant -5 -4.5 -4 -3.5 -3
Energy [eV]
molecular orbitals and the local contributions to the
transmission. A special emphasis is on the asymmetry FIG. 3: Transmission function (blue line) for
of the transport characteristics with respect to bias po- zigzag-terminated single-molecule junctions. The energy
larity, observed in the experimental investigations, and levels of the extended systems are depicted as grey
theroleofthedifferentconformationsaswellasthetype lines. The black dotted line represents the Fermi level.
of termination of the graphene electrodes. (a) Conformer A and (b) conformer B.
A. Transport properties and molecular orbital (e,f). The overall transmission value of these peaks is
analysis very low, the corresponding contribution to the current
is thus negligible.
Fig.3showsthetransmissionfunctionofthejunctions ThetransmissionfunctionofconformerBinFig.3(b)
with zigzag-terminated graphene electrodes for the two also shows a double-peak structure at −5.1 eV due to
conformations of the molecular bridge (A and B), de- states localized on the molecular bridge between the C60
picted in an energy range around the Fermi level, which moieties. The corresponding molecular orbitals are de-
in our simulation is identified with the center of the picted in Fig. 5 (a,b). The analysis of these orbitals for
HOMO-LUMO gap. In addition, the energy levels are thetwoconformersrevealsthattheylocalizeonopposite
depictedasgreylines. Thetransmissionsexhibitaseries sides of the molecule, with those of conformer B being
ofpeakstructures, whichcanbeattributedtoindividual slightly more delocalized over the bridge. The compar-
electronic levels of the extended junction as is outlined ison of the transmission functions in Fig. 3 (a) and (b)
in the following. demonstrates that the energy levels of these states are
The transmission function of conformer A (cf. Fig. significantly closer for conformer B. The closer lying en-
3 (a)) shows two narrow peaks below the Fermi level, ergy levels in combination with a more pronounced delo-
at E = −5.9 and E = −5.1 eV, which are associated calization and thus higher transmission values result in
withthemolecularlevelsHOMO-8andHOMO-9(ofthe an overall larger contribution of these double peak res-
extended molecular bridge) localized on the molecular onances to transport in conformer B compared to that
bridge between the two fullerenes. The corresponding in conformer A. Despite having equivalent edge states
molecular orbitals are shown in Fig. 4 (a). Due to the at the Fermi level, no edge-induced transport contribu-
weak molecule-lead couplings, the resonance peaks are tions are present in the junction of conformer B. As dis-
very narrow. Close to the Fermi level, graphene edge cussedindetailbelow,thisisrelatedtoadifferenceinthe
states (cf. Fig. 4 (b)) cause a broad peak structure with molecule-leadcoupling. Thetransmissionpeakstructure
a maximum transmission value of t = 0.01, which will related to the molecular states on the fullerenes appears
be discussed in more detail in Sec. IIID. Although this atsimilarenergiesasinconformerA,anditscontribution
is a rather low transmission, these contributions would to transport is also negligible.
lead to a considerable current at low voltages, because
thepeakissignificantlybroaderthanthoseofthemolec-
ular resonances. Above the Fermi energy at around B. Role of asymmetry
E = −3.0 eV, a more complex resonance structure is
found, which can be related to molecular states local- The experimental results presented in Ref. 13 show
ized on the fullerene anchor groups, as shown in Fig. 4 a pronounced dependence of the current-voltage charac-
5
(a)
-4.9
HOMO
-5 HOMO-1
V]
e -5.1
y [
g
er -5.2
n
E
-5.3
-5.4
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
Electric field x 108 [V/m]
(b) - Negative Field (c) + Positive Field
100
- 0.5 + 0.5
- 1.5 + 1.5
n 10-2 - 3.0 + 3.0
o
si
s
mi
s10-4
n
a
Tr
10-6
-5.4 -5.3 -5.2 -5.1 -5 -4.9-5.4 -5.3 -5.2 -5.1 -5 -4.9
Energy [eV] Energy [eV]
FIG. 6: (a) Field dependence of HOMO and HOMO-1
FIG. 4: Molecular orbitals of conformer A with zigzag
of the isolated molecular bridge in conformation A. (b)
terminated graphene electrodes.
Field-dependence of the transmission function for a field
of -0.5, -1.5 and -3·10−8 V/m (c) and for positive fields.
geneous electric field along this direction (cf. SI of Ref.
13) is included in the DFT calculations. Thereby, as an
estimate, the bias voltage was converted to an electric
field using the fullerenes center of mass distance of 2.4
nm (electric field = bias voltage / 2.4 nm). Due to com-
putationallimitations,wehaverestrictedthecalculations
to the isolated molecule without explicit inclusion of the
FIG. 5: Molecular orbitals of conformer B with zigzag
graphene electrodes. The transmission functions for fi-
terminated graphene electrodes.
nite fields are obtained using the WBA for the coupling
to the leads as described in Eq. (7). In this way, the ef-
fect of the electric field in the junction for different bias
teristics on the polarity of the bias voltage, i.e. asym- voltage polarities is modeled and the polarizing effect of
metric IV curves. In Ref. 13, it could be established the applied bias can be included, which is important for
that the asymmetry is a result of contributions from the the asymmetries in the transport properties.
states localized on the molecular bridge between the two The analysis for conformer A is presented in Fig. 6.
fullerenes. These states are HOMO-8 and HOMO-9 of Thedependenceoftheenergylevelsontheexternalelec-
the extended molecular bridge (Fig. 4 (a) and Fig. 5), tric field shows a typical avoided crossing with a mini-
which correspond to HOMO and HOMO-1 of the iso- mumoftheenergylevelspacingatafinite,positivefield.
lated molecule. Thereby it was assumed (cf. SI of Ref. Asaconsequenceofthebehavioroftheenergylevels,the
13) that in the experimental setup used, the Fermi level double peak structure in the transmission function also
in graphene is lowered to a similar value of that of gold depends sensitively on the electric field, as illustrated in
(about -5 eV), such that the Fermi level lies in the prox- Fig. 6 (b) for negative and (c) for positive field. Specifi-
imity of the states. cally, for positive fields, the peaks between −4.9 to −5.4
Here, we extend the analysis of Ref. 13 with respect eV are closer and the transmission is considerably en-
to the asymmetry of the IV curves. To this end, we hanced,whileforanegativefield,thepeaksaremoresep-
haveinvestigatedtheresponseoftheelectroniclevelsrel- arated and the transmission is significantly lower. Tak-
evantfortransport(HOMOandHOMO-1oftheisolated ing into account additional broadening due to coupling
molecule)toexternalelectricfieldsappliedalongtheline to vibrations, this is in agreement with the experimental
connectingthecentersofmassofthefullerenes. Ahomo- resultswhichshowsforlowertemperaturesadoublepeak
6
structure with a lower current for one bias polarity but Conformer A armchair
a single peak structure with higher current for the other (a) 100
bias polarity. on 10-2
si
As discussed in Ref. 13, the field dependence of the mis 10-4
energy levels of conformer B is equivalent to that of con- s
former A for fields of equal strength but applied in op- an 10-6
Tr
positedirection. Therefore, atransitionbetweenthetwo 10-8
conformations will revert the bias polarity dependence -5 -4.5 -4 -3.5 -3
Energy [eV]
of the IV characteristics. The inversions of the asym-
metries in the IVs, observed in the experiment during
Conformer B armchair
consecutive voltage sweeps, can be assigned to confor- (b) 100
mational changes of the molecule, namely a transition
on 10-2
betweenconformerAandB.Itisinterestingtonotethat si
for the molecular states on the fullerenes (cf. Fig. 4 (c)), mis 10-4
s
no distinct bias polarity dependence could be found. an 10-6
Tr
10-8
-5 -4.5 -4 -3.5 -3
C. Junction with armchair termination Energy [eV]
Graphene nanoribbons exhibit electronic states local-
FIG. 7: Transmission function (blue line) for armchair
ized on the zigzag edge with energies located near the
terminated single-molecule junctions. The black dotted
Fermi level.34 In contrast, the armchair edge does not
line represents the Fermi level. (a) Conformer A and
support such localized states. In molecule-graphene
(b) conformer B.
nanojunction, localized edge states can contribute to
transport due to the interaction with adjacent molecular
resonances.9,10 An example is the junction of conformer
Awithzigzagterminatedgrapheneleads, whichexhibits
significantcontributionsduetoedgestates, cf.Fig.4(b).
Here, we analyze for comparison the transport prop-
erties of graphene-molecule junctions with armchair ter-
significantly different adsorption energies, depending on
mination of the graphene leads, as shown in Fig. 2 (c). the orientations, of C on graphene.35
60
The geometry optimization again yields two conforma-
tionsAandBwithanenergydifferenceof∼0.3eV.The
transmission functions for the two conformers with arm- Fig. 8 shows the interface geometry for the two con-
chairterminationoftheelectrodesarepresentedinFig.7. formers. In the junction of conformer A, the molecule
The results show that the contributions of the molecular bindsonthelefthandsideviaahexagondirectlyoriented
states related to HOMO and HOMO-1 (at about −5.1 tothecenterofagraphenehexagon,which,usingtheno-
eV) of the isolated molecule and those induced by the tation of Ref. 35, corresponds to a hexagon/hole geome-
states localized on the fullerenes (at about −3.0 eV) are try. Theinterfaceontherighthandsidecloselyresembles
almostindependentonthetypeofgraphenetermination. apentagon/holegeometry(cf.Fig.8(a)). Forconformer
However, the contributions from edge states close to the B(Fig.8(a))thelefthandsideinterfacecorrespondstoa
Fermienergy,whicharepresentinthezigzag-terminated dimer/bridgegeometry,whereasattherighthandsideit
junctionofconformerA,aremissinginthearmchairter- is a hexagon/hole geometry. While for the hexagon/hole
minatedjunctionforbothconformers. Intheexperiment and pentagon/hole orientation stronger binding energies
of Ref. 13, the existence of a well-defined zigzag termi- were found, the dimer/bridge geometry was observed to
nation is rather unlikely and thus edge-state transport be less stable. In accordance with Ref. 35, this will lead
channels are expected to be less relevant. to a lower coupling in conformer B, preventing the hy-
bridization of edge states to become conducting states.
Moreover, as a consequence of the different orientations
D. Interface geometry of the fullerenes relative to the graphene surface, also
the coupling to the edges differs. On the right hand
The fact that edge-state induced transport contribu- side in conformer A, a pentagon is oriented towards the
tionsarefoundinthejunctioninconformerAwithzigzag graphenesurface,whichissurroundedbyhexagonspoint-
termination, but not in conformer B, can be related to ing towards the edge with zigzag termination. On the
a difference in the interface geometry and the resulting other hand, in conformer B a hexagon is pointing to the
coupling between the molecule and the graphene leads. surface,whereasahexagonandapentagoncoupletothe
These investigations build on previous experimental and zigzag edge. In the latter case, a weaker interaction is
theoretical studies for related systems, which revealed expected.
7
through the occupied molecular levels. In this case, the
transmission pathway leads from the left electrode into
the C anchor group, from where it goes through the
60
molecular wire into the other C and finally reaches
60
the right graphene electrode. On the left hand side, the
pathway enter the molecule via a C hexagon, and on
60
the right hand side via a pentagon, in correspondence
with the C -graphene interface shown in Fig. 8 (a).
60
Consideringthecorrespondingmolecularorbital(cf.Fig.
4(a)), the non-vanishing local transmission components
are mainly between atoms which exhibit significant elec-
tron density. The pathway at energy E follows a
HOMO-9
similar pattern (not shown).
Fig. 9 (c,d) shows the local transmissions through
the unoccupied molecular resonances, in particular
E . Since the electron density of LUMO+8 is
LUMO+8
concentrated on the fullerenes (cf. Fig. 4(c)), the local
transmissions are spread over the C moieties.
60
Aquitedifferentpatternisobtainedforthelocaltrans-
missionevaluatedattheenergyoftheedge-inducedpeak
around the Fermi level, depicted in Fig. 9 (e,f). In this
energy region, the local transmission exhibits a complex
pattern involving pronounced circular contributions in
thegraphenecontacts,transmissionswhicharebackscat-
tered at the edges as well as direct transitions from one
lead to the other. Transmissions into the molecule are
FIG. 8: Interface geometry of (a) conformer A and (b) vanishingly small, as shown in the side view of the junc-
conformer B. In (a), the interface on the left hand side tion in Fig. 9 (e) and involve contributions between the
corresponds to a hexagon/hole geometry, while on the graphene and the side chain of the molecule. Small con-
right hand side it resembles a pentagon/hole geometry. tributions along the molecule are not seen because their
In (b), the left hand side interface corresponds to a value is lower than the cutoff of 10%.
dimer/bridge geometry, and at the right hand side to a While local ring-shaped current structures are a well
hexagon/hole geometry. knownphenomenoningraphenenanoribbons36,37,direct
transitions between graphene leads have not been ana-
lyzed so far. Test calculations, which artificially neglect
E. Local transmission paths the molecular states, do not show these direct transi-
tions, thus demonstrating that they are induced by the
hybridization between the edge states and the molecular
Finally, we analyze the pathways of electrons in the
states. This finding can be further confirmed by a qual-
molecule-graphene junctions. To this end, the local
itative consideration of the potential barrier. Without
transmissions as defined in Eq. (13) are calculated at
molecule, the potential barrier for an electron to tunnel
energies, where the total transmission is maximal. The
through the junction with a vacuum gap of 1.5 nm is
results for conformer A and zigzag-terminated graphene
too high to give noticeable transmission values. With
leads are shown in Fig. 9 for a selection of representative
molecule in conformation A attached, however, the en-
energies. Thereby, local transmissions between all atoms
ergybarrierisdecreasedsignificantlythusfacilitatingdi-
are depicted as arrows. The color code ranging from
rect tunneling transitions.
bluetoorangeindicatesthelocaltransmissionvalue. An
arrow is only drawn when this value exceeds 10% of the
maximallocaltransmissionvalue,whichforclarityisnor-
malized for each energy. Although the sum of all local IV. CONCLUSIONS
transmissions across a line vertical to the transport di-
rection is preserved in the calculations, this conservation We have investigated charge transport in C -based
60
is not fully apparent in the plots due to the 10% of the single-molecule junctions with graphene electrodes.
transmission contributions which are not shown. It is These type of molecule-graphene junctions were re-
also noted that the transmission probability is absorbed centlystudiedexperimentallyusingepitaxialsinglelayer
in the leads beyond a certain region due to the CAP ap- graphene.13 Ourtheoreticalstudywasbasedonacombi-
plied. nationofDFTelectronicstructurecalculationsandLan-
We first consider the local transmissions at E , dauer transport theory. Furthermore, the theory of lo-
HOMO-8
depicted in Fig. 9 (a,b), as an example for transport cal currents was used to analyze pathways of electrons
8
✭❜✮
✭❛✮
☞
✁
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✥(cid:0)✟ ✡☛✓
✥(cid:0)✞ ✡☛✒
✥(cid:0)✝ ✡☛✑
✥(cid:0)✆ ✡☛✏
✥(cid:0)☎ ✡☛✎
✥(cid:0)✄ ✡☛✍
✥(cid:0)✂ ✡☛✌
✥(cid:0)✁
✡☛☞
✥
✡
✭❞✮
✭❝✮
★
✗
✦✧✰
✕✖✤
✕✖✣ ✦✧✯
✕✖✢ ✦✧✮
✕✖✜ ✦✧✭
✕✖✛ ✦✧✬
✕✖✚ ✦✧✫
✕✖✙ ✦✧✪
✕✖✘ ✦✧✩
✕✖✗
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✕
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✱✲✻
✱✲✺ ✼✽❅
✱✲✹ ✼✽❄
✱✲✸ ✼✽❃
✱✲✷ ✼✽❂
✱✲✶ ✼✽❁
✱✲✵ ✼✽❀
✱✲✴ ✼✽✿
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FIG. 9: Local transmissions (Top and side-view) at (a,b) E , (c,d) E , (e,f) E (top view plotted
HOMO-8 LUMO+8 LUMO
without molecule).
through the junctions. The results of electronic struc- larity,whichwasalsofoundinexperiment. Furthermore,
ture calculations show that the molecular bridge can ex- the different interface geometry of the two conformers
ist in two different conformations, which are relevant for causes a significant difference in molecule-lead coupling.
chargetransport. Inaddition,wehaveconsideredmolec- As a consequence, edge-induced transport channels are
ular junctions with zigzag and armchair termination of only present in one of the conformers.
the graphene electrodes. As mentioned in the introduction, a major advantage
The analysis of the transport characteristics in terms of graphene as material for electrodes in molecular junc-
of molecular orbitals and local transmissions reveals the tions if the possibility to access the junction with light
existence of two different types of transport channels. to monitor and control the nonequilibrium state of the
The first type includes transport through molecular or- junction. The theoretical description of this scenario is
bitals, whichareeither localizedonthemolecular bridge an equally interesting and challenging extension of the
or on the C anchor groups. Thereby, the latter contri- present work.
60
bution is of less relevance in the junctions investigated.
In junctions with zigzag-termination, furthermore, edge-
state induced transport channels contribute. The results ACKNOWLEDGMENTS
also show that, for the molecule-based transport chan-
nels, the two different conformers exhibit a reversed de- This work has been supported by the the Deutsche
pendence of the transport characteristics on the bias po- Forschungsgemeinschaft (DFG) through the Cluster of
9
Excellence ’Engineering of Advanced Materials’ (EAM) (LRZ),andJu¨lich(JSC)isgratefullyacknowledged. The
and SFB 953. Generous allocation of computing time molecules were provided by Augustin Molina-Ontoria
at the computing centers in Erlangen (RRZE), Munich and Nazario Martin.
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