On the identification of pristine and defected graphene nanoribbons by phonon signatures in the electron transport characteristics

2.3 MB·English

by  Rasmus B. Christensen

On the identification of pristine and defected graphene nanoribbons
by phonon signatures in the electron transport characteristics
Rasmus B. Christensen,1 Thomas Frederiksen,2,3 and Mads Brandbyge1
1Dept. of Micro- and Nanotechnology, Technical University of Denmark,
Ørsteds Plads, Bldg. 345E, DK-2800 Kongens Lyngby, Denmark
2Donostia International Physics Center (DIPC) – UPV/EHU, Donostia-San Sebastián, Spain
3IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
(Dated: January 13, 2015)
Inspired by recent experiments where electron transport was measured across graphene nanorib-
bons (GNR) suspended between a metal surface and the tip of a scanning tunneling microscope
5
[Koch et al., Nat. Nanotechnol. 7, 713 (2012)], we present detailed first-principles simulations of
1
0 inelastic electron tunneling spectroscopy (IETS) of long pristine and defected armchair and zigzag
2 nanoribbonsunderarangeofchargecarrierconditions. Forthearmchairribbonswefindtworobust
IETSsignalsaround169and196mVcorrespondingtotheD-andG-modesofRamanspectroscopy
n
as well as additional fingerprints due to various types of defects in the edge passivation. For the
a
zigzagribbonsweshowthatthespinstatestronglyinfluencesthespectrumandthusproposeIETS
J
as an indirect proof of spin polarization.
9

PACSnumbers: 81.05.ue,73.63.-b,72.10.Di
]
l
l
a
h I. INTRODUCTION the electronic structure has been mapped out by
- STM, angle-resolved (two-photon) photo-emission and
s high-resolution electron energy loss spectroscopy.8,18,19
e Graphene as the basis of a new generation of
m electronics1,2 has been the center of much attention in Signatures of phonon excitation were observed by STM
in the differential conductance spectroscopy performed
the last years, and devices based on nanostructured
.
at graphene have been put forward. The most generic form at the zigzag termini state of AGNRs adsorbed on
Au(111), and these signatures were shown to be sen-
m of nanostructured graphene is graphene nanoribbons
(GNR),3 andotherstructures, suchasgrapheneanti-dot sitive to modifications in the local atomic geometry.20
-
d lattices4,5, can be viewed as networks of them. GNRs AGNRs have also been lifted up from the weakly
n arepotentialcandidatesformolecularwireswithtailored bonding Au(111) surface with the tip of a STM enabling
o measurements of the voltage-dependent conductance in
conductance properties. For graphene-based nanostruc-
c suspended configurations.21
tures the edges and their passivation, as well as defects
[
inside the structure, can play crucial roles for the trans- From the theoretical side density-functional the-

1 portproperties.6However,characterizationofedgepassi- ory (DFT) has been used to investigate the stabil-
v vationorstructural/chemicaldefectsischallengingespe- ity of structural and chemical reconstructions of GNR
6 cially after device fabrication. Raman spectroscopy7 can edges,22–24 together with the transport and band-
6
giveinformationaboutdefectsonlargeareasofthesam- gap engineering.6,25–28 The vibrational properties and
2
2 ple, while tip-enhanced Raman spectroscopy (TERS)8 phononbandstructurehavebeencalculatedwithempir-
0 have been used in combination with STM on GNRs. ical potentials29 and DFT.30,31 In addition, there have
. However, Raman studies involve averages over larger ar- beentheoreticalpredictions32,33 oftheRamanspectrum,
1
0 eas (> 10 nm), and does not yield information about in good agreement with experiments.14,34 For a finite
5 the impact of vibrations on transport. In that aspect AGNRtheroleofzigzagterministateshavebeenstudied
1 inelastic electron tunneling spectroscopy (IETS) serves theoretically, comparing DFT to the many-body Hub-
: as a way of performing non-destructive characterization bard model.35
v
i yieldingvibrational/phononfingerprintsofarangeofde- Inspired by the recent lifting experiments by Koch
X fects. InordertointerpretIETSexperiments,theoretical et al.,21 wehereinvestigatetheoreticallythesignalsofe-
r modeling of the inelastic signals in the electronic current ph scattering in the conductance of long GNRs between
a
due to electron-phonon (e-ph) scattering is needed. metal electrodes. Our aim is two-fold. First, we want to
GNRs have been fabricated using different strate- addresstherolephononscatteringinthetransportchar-
gies including lithographic techniques,9 chemical acteristicsofpristineGNRs. Second,wewishtocompute
synthesis,10,11 epitaxial growth12, and longitudinal detailed IETS for different GNRs under varying charge
unzipping of carbon nanotubes.13 Furthermore, several carrier concentrations and explore how different types of
groups have succeeded in atomically precise bottom-up realistic defects may modify the IETS and thus possibly
fabrication of armchair GNRs (AGNR)14,15, chiral be directly probed in transport measurements. We focus
GNRs,16 and AGNR hetero-junctions17 grown on metal on the two most generic edge types, namely armchair
surfaces. Experimentally, the vibrational properties (AGNR) and zigzag (ZGNR), and pay attention to the
have been investigated by Raman spectroscopy and effects of spin polarization in the latter case. In actual
2
(a)AGNR Dynamical (e)ZGNR Dynamical
Electrode Device Electrode Device
(b) (c) (d) (f) (g) (h)
1.5 1.5 0.0 eV
0.1 eV
1 1 1.0 eV
) )
V0.5 0.0 eV V0.5
(e 0.1 eV (e
y 0 1.0 eV y 0
g g
r r
e-0.5 e-0.5
n n
E E
-1 -1
-1.5 -1.5
-π Γ π 0 1 2 3 4 0 10 20 -π Γ π 0 2 4 6 0 10 20
kx T DOS kx T DOS
FIG. 1: (Color online) (a) Computational setup for a pristine AGNR showing electrode, device and dynamical regions. (b)
Electronic band structure (k is in units of inverse unit cell length). The different bands are colored according to symmetry
x
of the electronic states. Red: symmetric, corresponding to Fig. 2(a-b). Blue: anti-symmetric, corresponding to Fig. 2(c-d).
(c)Electronictransmissionforvaryingelectrodebroadeningdescribingthecouplingtothemetalcontacts,η=0,0.1,1eV,see
text. (d) Electronic DOS projected onto the dynamical region. Panels (e)-(h) show the similar entities for the pristine ZGNR
case.
AGNR ZGNR experiments the substrate or an applied gate potential
(a) 1eV Sym (e) 0.4eV control the Fermi level E in the ribbons. To address
− − F
this variability we scan E using a numerically effective
F
scheme enabling fast calculations of the IETS.36 We find
that the AGNR generally display two robust IETS sig-
(b)1eV Sym (f) 0.4eV nals around 169 and 196 mV corresponding to the D-
−
and G-modes of Raman spectroscopy and that a dehy-
drogenateddimerattheedgeshouldfurtherleaveaclear
defect signal at around 245 mV. For the ZGNR we find
(c) 1eV Anti-Sym (g)0.36eV that the spin polarization breaks the mirror symmetry
−
around the middle of the ribbon resulting in IETS sig-
nals from a range of modes around the D- and G-mode
energies. ForbothAGNRandZNGRdefectswhichbreak
the planar symmetry of ribbons allows for contributions
(d)1eV Anti-Sym (h)0.36eV
to the IETS from out-of-plane phonon modes.
FIG. 2: (Color online) (a)-(d) Electron transmission eigen-
channels for the clean AGNR for the valence bands at E −
Thepaperisorganizedasfollows. Firstwediscussour
E = −1 eV and for the conduction bands at E−E = 1
F F
atomisticmodelsetupforthedensityfunctionalandelec-
eV. (e)-(h) Electron transmission eigenchannels for the clean
ZGNRinthevalencebandsatE−E =−0.4eVandinthe trontransportcalculations, andoutlinetheapproachfor
F
conductionbandsatE−E =0.4eVforonespincomponent. the IETS simulations. In Sec. III we present our results
F
The eigenchannels for the other spin component are simply for pristine AGNR and ZGNR and relate their transport
mirror images around the middle of the ZGNR (not shown). properties and IETS to the band structures. In Sec. IV
Thered/blue(pink/gray)isosurfacesrepresentthereal(imag- we turn to the defected systems by considering realistic
inary)partandsignofthescatteringstatewavefunction. For possibilities of defects in the edge passivation, backbone
alleigenchannelcalculationstheelectrodebroadeningwasset
bondingmotifs,andpresenceofadatoms. Finally,asum-
to zero (η=0 eV).
mary and our conclusions are presented in Sec. V.
3
II. METHODS imentally reported differential conductance curves for
AGNR.21 The electronic states involved in the trans-
port are shown in Fig. 2 in terms of the transmission
We calculate the electronic and vibrational
eigenchannels44 in the valence and conduction bands of
structure from DFT using the academic codes
Siesta/TranSiesta.37,38 We employ the generalized the AGNR and ZGNR. Their spatial symmetry play a
significant role for the selection rules involved in the in-
gradient approximation (GGA) for the exchange-
correlation functional,39 a single-zeta polarized (SZP) elastic scattering as discussed later.
basis set for the carbon and hydrogen atoms, and use In principle, the electronic structure should be evalu-
a cut-off energy of 400-500 Ry for the real-space grid. ated at finite bias. However, without a detailed model
These choices, balancing accuracy and computational of the connection to the metal electrodes (where an im-
cost,provideagooddescriptiontoinvestigatetrendsand portant part of the voltage drop will take place) and for
general behavior of the substantial number of systems sufficientlylongsystems(inwhichtheelectricfieldwillbe
considered in this work. small), it is reasonable to use the zero-voltage electronic
structureandtosimplyassumeasymmetricvoltagedrop
The vibrational degrees of freedom, calculated by di-
over the two identical, idealized device-electrode inter-
agonalization of the dynamical matrix, and the e-ph
faces. Morespecifically,inthefollowingweconsiderthat
couplings are extracted from finite differences as imple-
mented in the Inelastica code.40–42 The armchair and the chemical potentials of the electrodes move according
zigzag GNRs considered here are shown in Fig. 1. We totheappliedbiasvoltageV andpossiblyanappliedgate
adopt the usual two-probe setup with the device region voltageVG (mimickingactualdopingorelectrostaticgat-
ingthatmodifythechargecarrierconcentrationinanex-
(D) coupled to left (L) and right (R) electrodes with
all electronic matrix elements expressed in a local basis perimentalsetup)accordingtoµL/R =EF±(cid:126)ωλ/2+VG.
set. The primitive unit cell of the AGNR (ZGNR) con-
sists of 18 (10) atoms and in our calculations this unit
cell is repeated 10 (18) times in the transport direction
A. Computational scheme for IETS
to form the scattering regions illustrated in Fig. 1(a,e).
The electrode couplings Γ are included on the two
L/R
first/last unit cells before folding onto D. In our treat- For a device strongly coupled to the electrodes, a cou-
ment a subset of atoms in D is allowed to vibrate. We pling between the electron current I(V) and a phonon
fixthisdynamicalregion,restrictedbytheconditionthat mode λ ideally shows up at zero temperature as a step
the e-ph couplings are fully included inside D, to the 4 discontinuityinthedifferentialconductancewhenthein-
and6centralunit-cellsfortheAGNRandZGNR,respec- elasticphononemissionprocessbecomesenergeticallyal-
tively. The corresponding e-ph couplings used to calcu- lowed,thatis,whenthechemicalpotentialdifferenceex-
latetheinelasticelectrontransportarethusexpressedin ceeds the quantum of vibrational energy, |µ − µ | =
L R
the center 6 unit-cells for the AGNR and 8 unit-cells for (cid:126)ωλ. Thus, around the emission threshold the electronic
the ZGNR. The convergence of our results with the size states involved in the scattering process are those at µ
L
of the dynamical region is addressed below. and µ . The IETS signal, conventionally expressed as
R
Wegenerallyconsidernanoribbonsthataresuspended the ratio between the second and first derivatives of the
between two metallic leads. In the case of the lifting current with respect to the voltage,
experiments,21 thesewouldcorrespondtothemetalsam-
ple surface and the STM tip. We wish here to focus on ∂2I(V)
IETS= V , (1)
the action inside the GNRs and put aside the possible ∂ I(V)
V
complications due to the detailed electronic structure of
the metals, and the metal-GNR interface in particular.
iscalculatedbyconsideringthee-phcouplingastheper-
To this end we introduce a simple model of the metal
turbationonthecurrent,evaluatedusingthenonequilib-
electrodeswithoutsubstantialelectronicfeatures: weuse
rium Green’s functions (NEGF). In the so-called lowest
semi-infinite GNRs with highly broadened states (effec-
order expansion (LOE) the inelastic part of the differen-
tivelysmearingoutenergygaps). Inpracticethisisdone tial conductance can be written as,36
by adding a finite numerical imaginary part η to the en-
ergy argument in the electrode recursion calculation.43
This scheme ensures that the phonon effects originate ∂VI(V)=γλ∂VIsym(V,(cid:126)ωλ,T) (2)
from the GNRs themselves and not from details of the +κλ∂VIasym(V,(cid:126)ωλ,T),
metal-GNR interface, which is generally unknown in the
STMexperiments. Theelectronicbandstructuresforthe where summation over the vibration index λ is assumed.
infinite ribbons, along with the transmission and den- Isym andIasym arethe“universal” (system-independent)
sity of states (DOS) are shown for η = 0,0.1,1 eV in functions that depend on the applied bias V, phonon
Fig. 1(b,c,d) and Fig. 1(f,g,h) for AGNR and ZGNR, energy (cid:126)ωλ and the temperature T. Assuming the elec-
respectively. We note that the broadened transmission tronic and phononic distribution functions are given by
spectrum [Fig. 1(d)] is quite consistent with the exper- the Fermi-Dirac and Bose-Einstein distributions, respec-
4
tively, their analytical expressions can be written as:
(a)AGNR (b)ZGNR
Isym≡G0 (cid:88)s((cid:126)ωλ+seV) (3) 3.5 6Unitcells 8Unitcells
2e
s= 3
(cid:18) ± (cid:19)
× coth (cid:126)ωλ −coth(cid:126)ωλ+seV , )2.5 4Unitcells 6Unitcells
2k T 2k T 1
B B − 2
Iasym ≡G0 (cid:90) +∞dεH{f(ε )−f(ε )}(ε) (4) (V1.5 2Unitcells 4Unitcells
2e (cid:48)− (cid:48)+ TS
−∞ E 1
×[f(ε−eV)−f(ε)], I
0.5
1Unitcells 2Unitcells
0
where G = 2e2/h is the conductance quantum, f(ε) is
0
the Fermi-Dirac function, ε(cid:48)s ≡ε(cid:48)+s(cid:126)ωλ, and H denotes 0 0.1 0.2 0.3 0 0.1 0.2 0.3
the Hilbert transform.
Bias (V)
Thesignalamplitudesγ andκ ofthesymmetricand
λ λ
antisymmetric signals in the differential conductance are FIG.3: (Coloronline)ConvergenceoftheintrinsicIETSfor
evenandoddinbias,respectively. Forasymmetricstruc- pristine (a) AGNR and (b) ZGNR as a function of the size
turetheasymmetricsignalvanishesinthewide-bandap- of the dynamical region (stated in the legends). The results
proximation (LOE-WBA)40. However, this is not guar- are normalized with respect to the number of vibrating unit
anteed in the more general treatment employed here,36 cells, i.e., we show the IETS amplitude per H4C14 segment
where the energy dependence of the electronic structure forAGNRandperH2C8 segmentforZGNR.Nogatevoltage
is applied (V =0.0 V).
is explicitly taken into account. The amplitudes γ and G
λ
κ are expressed in terms of electronic structure quanti-
λ
ties and e-ph couplings,36
III. PRISTINE GRAPHENE NANORIBBONS
γ =Tr[M A˜ (µ )M A (µ )]+ImB , (5)
λ λ L L λ R R λ
Now we first turn to the IETS results of the two pris-
κ =2ReB , (6)
λ λ tine (clean) ribbons, and in the following section to the
impact of selected defects in the IETS. As our main sys-
where B is defined as
λ tem we focus on the AGNR systems directly relevant for
the lifting experiments.21 The results for the ZGNR are
Bλ ≡Tr[MλAR(µL)ΓL(µL)Gr(µL)MλAR(µR) provided mainly as comparison and to look into the role
−M Ga(µ )Γ (µ )A (µ )M A (µ )]. (7) of chirality and in particular effects rooted in spin polar-
λ R L R R R λ L L
ization, and thus we now discuss these separately.
In the above, M denotes the e-ph coupling matrix
λ
for mode λ, Gr/a the retarded/advanced unperturbed
Green’s functions, and A =GrΓ Ga the spectral den- A. Pristine armchair nanoribbons
α α
sity matrices for left/right moving states with the time-
reversed version A˜α = GaΓαGr. The purely electronic As representative of the AGNR class we have investi-
quantities are thus being evaluated at the chemical po- gated a pristine AGNR with a width of W = 7 dimers
tentials of the left/right electrodes corresponding to the (7-AGNR) corresponding to a C-C edge distance of 7.5
excitationthresholdforeachvibration. WecomputeMλ Å(seeFig.1). Itpresentsadirectsemi-conductingband
with the finite-difference scheme of Inelastica taking gap E due to the lateral confinement and can be classi-
g
thevacuumenergyasacommonreference(inabsenceof fied as a “large-gap ribbons” since p = 2 is an integer in
real metal leads to pin the Fermi energy).41 the relation W =3p+1.1 We obtain E ≈1.3 eV at the
g
Inthelocalizedatomicbasissetof Siestaalltheabove present level of approximation (DFT-GGA and SZP ba-
quantities are matrices defined in the electronic space sisset),asseenfromtheelectronicbandstructureshown
corresponding to region D. The second derivatives of in Fig. 1(b). This value is smaller than those estimated
theuniversalfunctionsinEqs.(3)-(4)aresharplypeaked experimentally (E ≈ 2.3-2.6 eV for a flat AGNR on
g
around the phonon threshold. For this reason the coeffi- Au(111)19,45 and E ≈ 2.7 eV for an AGNR suspended
g
cients γ and κ can be considered voltage-independent between surface and STM-tip21) due to the underesti-
λ λ
withtheirvaluescomputedexactlyatthethreshold. Due mation of electron-electron interaction46 which plays an
to the computational efficiency of the LOE scheme de- moreimportantroleinquasione-dimensionalGNRscom-
scribed above we are able to evaluate the IETS on a fine paredtopristinegraphene. Dielectricscreeningfromthe
grid of gate voltages V spanning a large range of rele- substratealsoinfluencessignificantlytheactualgapsize:
G
vantvaluesbetweenvalenceandconductionbandsofthe a band gap of 3.2 eV for a 7-AGNR was found to be
GNRs. lowered to 2.7 eV on a hexagonal boron-nitride (hBN)
5
(a) AGNR (b) ZGNR
)
) 1−
V
( V
(
e
t S
a T
G
E
I
Bias (V)
(c) (d)
24
VG=0.80V ×11//62 VG=0.80V
) 20 ×
1
− 1/15
V 16 VG=0.68V ××1/5 VG=0.34V
(
12
S 1/3
ET 8 VG=0.30V × VG=0.22V
I
4 1/3
VG=0.00V × VG=0.00V
0
0.1 0.15 0.2 0.1 0.15 0.2
Bias (V)
FIG. 4: (Color online) IETS signals as a function of gate
voltage for (a) pristine AGNR (4 vibrating unit cells) and
(b) pristine ZGNR (6 vibrating unit cells). Vertical dashed
lines are guides to the eye indicating the energy of the most
contributingvibrationalmodes. SpecificIETSsignalsforthe
(c) AGNR and (d) ZGNR at selected gate voltages marked
with horizontal dashed lines in panels (a) and (b). Broad-
ening originates from temperature T = 4.2 K and a lock-in
modulationvoltageV =5mV(exceptforthethinredlines FIG.5: (Coloronline)(a)Computedphononbandstructure
rms
in the lower panels with Vrms =0 mV ). for the pristine, infinite AGNR (kx is in units of inverse unit
celllength). Themagnitudeofthered,greenandbluebands
(correspondingtothethreeverticallinesinFig.4(c)),ispro-
portional to the signal size weighted overlap, (F (V =0V)
substrate using GW calculations,47 similar to the lower- nk G
in Eq. (8)), between the repeated band vector and modes
ingcalculatedfora7-AGNRonAu(111).48 Ingeneralwe
with frequencies (cid:126)ω > 180 meV, 180 > (cid:126)ω > 162 meV and
expect that underestimation of band gaps would mainly (cid:126)ω < 162 meV for red, green and blue, respectively. The
amounttoasimplescalingtheFermilevelpositionwithin red band is scaled by 0.2 compared to blue and green. (b-e)
the gap. Selected phonon band modes at Γ for the infinite structure
We first discuss the effect of the finite size of the dy- which, according to the projection, characterize the active
namical region in our treatment. Figure 3(a) shows how IETS modes. Units of kx?
the IETS signals for the AGNR (at fixed gate voltage
V = 0.0 V) vary as a function of the size of the dy-
G
namical region, ranging from 1 to 6 unit cells. For easy voltages are shown in Fig. 4(c) for both the intrinsic
comparison, the data are normalized by the number of part (temperature broadening at T = 4.2 K) as well
vibratingunitcells. Asthesignalamplitudesinthisrep- as that one would observe employing the experimental
resentation are roughly constant we conclude that the lock-in technique (additional broadening due to a mod-
absolute IETS simply scale linearly with the active e-ph ulation voltage of V = 5 mV). We find that for the
rms
coupling region. Consequently, the magnitudes in IETS AGNR there are generally two well-defined IETS signals
may thus provide insight into the active scattering re- appearing around 169 and 196 meV, corresponding to
gioninactualexperiments. Further,aswefindthatboth the D- (ring breathing) and G- (E phonon) modes, re-
2g
IETS amplitude and shape is well converged with 4 vi- spectively,alsoobservedinRamanspectroscopy.7,9. The
bratingunitcells,wefixthedynamicalregiontothissize D-signal also has a shoulder with a local maximum at
in the following analysis. 159 meV with contributions from several modes. These
The computed IETS signals for the AGNR as a func- three distinct features are indicated with vertical lines
tion of varying gate voltage are shown in Fig. 4(a) as in Fig. 4(a,c). Shifting E inside the gap region with
F
a density plot. Specific IETS spectra at selected gate a relatively small gate voltage |V | (cid:46) 0.5 V does not
G
6
affect the IETS appreciably. However, when E comes (a)
F (b)
close to the conduction band of the AGNR the signal in-
creases by a factor of five and a small peak-dip feature 6 VG=0.5V
1.5
appear similar to the one reported for gated benzene- 5 0.0 eV
dithiol molecular contacts.36,49 Upon further gating into 1) 4 01..10 eeVV 1 V)
the conduction band the IETS signals undergo a sign re- − 0.5 e
vbeerysoanld(fraopmprpoxeaimksatteolydi0p.s5) afosrthtehetriannvsomlviessdiocnhainncnreelass.5e0s TS(V 23 SSppiinn PDoelg.en. 0 ergy(
SbiamndilaorfetffheecAtsGaNreRa.lso found by gating into the valence IE 1 VG=0.0V --10.5En
We can easily identify the most important vibrational 0 -1.5
mode vectors vλ for the IETS from the two amplitudes -1
|γλ| and |κλ| given in Eqs. (5)-(6). These modes can 0.1 0.15 0.2 0 1 2 3
further be analyzed in terms of the phonons in the in- T
Bias (V)
finite AGNR. To do so we introduce the measure F
nk
representing the overlap between modes in the finite dy-
FIG. 6: (Color online) (a) IETS signals for the pristine
namicalcellandthephononbandmodesweightedbythe ZGNR(6vibratingunitcells). Theblacklinescorrespondto
size of the IETS signal, spin-degenerate calculations while the red lines are the spin-
up components of spin-polarized calculations. Broadening
F (V )=(cid:88)|γ (V )|(cid:12)(cid:12)u (cid:16)1,eik,...,ei(N 1)k(cid:17)·v (cid:12)(cid:12)2, originatesfrom temperature T =4.2Kand modulationvolt-
nk G λ G (cid:12) nk − λ(cid:12) age V = 5 mV (full lines) or V = 0 mV (dashed lines).
rms rms
λ (b)Electronictransmissionfromspin-degeneratecalculations
(8)
with varying electrode broadening describing the coupling to
whereu isthephononbandmodeindexedbyn,andv
nk λ the metal contacts, η=0,0.1,1 eV (see also Fig. 1(g) for the
is the modes in a finite N primitive cell long dynamical corresponding spin-polarized case).
region index by λ.
The projections F (V =0V) are depicted as widths
nk G
of the phonon bands in Fig. 5(a), where the red, green
and blue colors refer to modes with frequencies in the
ranges (cid:126)ω > 180 meV, 180 > (cid:126)ω > 162 meV, and (cid:126)ω <
162 meV, respectively. In total four bands contribute to
theIETSsignalcorrespondingtothefoursignalsseenin
theintrinsicpartoftheIETSspectruminFig.4(c). The
correspondingΓ-pointphononmodesinsidetheprimitive
cell for the infinite ribbon are shown in Fig. 5(b-e).
B. Pristine zigzag nanoribbon
We next turn to our results for the pristine ZGNR
shown in Fig. 1(e). It has a width of W = 4 zigzag
“chains” (4-ZGNR)correspondingtoaC-Cedgedistance
of 7.26 Å. The breaking of sublattice symmetry for the FIG. 7: (Color online) The phonon band structure of the
ZGNR and lack of pseudo-phase result in different selec- ZGNR, (k is in units of inverse unit cell length), together
x
tion rules for the matrix elements and difference in for with the Γ-point modes. The widths of the red bands are
exampleRamansignals.33 TheZGNRgenerallypresents proportionaltotheweightfunctionF(0V)(Eq.(8)),whilethe
spin-polarizededgestatesexhibitingasmallbandgapat widthsofthebluebandsareproportionaltoF(0V)+F(0.5V).
theDFTlevel,1inourcaseE ≈0.6eV(wenotethatthis
g
gap disappears in simpler tight-binding descriptions1 or
spin-degenerate DFT calculations). The spin-polarized greaternumberofmodescontributingtotheIETSforthe
edge states play the major role for the conduction, see ZGNRresultsinbroadersignalswithsimilarmagnitudes
the spin-down eigenchannels visualized in Fig. 2(e-h). as compared to the IETS for AGNR. As for the AGNR
Since the edge states break the mirror symmetry case the IETS signal is well converged with a dynamical
with respect to the middle of the ribbon, there are region consisting of 6 vibrating unit cells [Fig. 3(b)].
fewer symmetry-forbidden inelastic transitions between ForZGNRstheringbreathingisforbiddenbysymme-
the scattering states for the ZGNR. Thus, we expect a try, thus the IETS is generally characterized by trans-
widerrangeofmodestocontributetotheIETSsignalas verse and longitudinal modes. To explore the impact
comparedtotheAGNRcase. Indeedthisisinagreement of spin-polarization on the ZGNR-IETS we compare in
withthefindingsshowninFig.3(b)andFig.4(b,d). The Fig. 6 the results from both spin-degenerate and spin-
7
(a)Clean (b)1H-edge (c)2H-edge
(d)1F-edge (e)8H-free (f)1C-broken
(g)2C-broken (h)4C-broken (i)Cu-adatom
FIG.8: (Coloronline)TopandsideviewsofthedynamicalregiondescribingthevariousAGNRdefectstructures. (a)Pristine
AGNR. (b) One extra H atom on one of the edges. (c) Two extra H atoms on one of the edges. (d) One H atom replaced by
a F atom. (e) Dehydrogenated edge where 4 H atoms have been removed from each side. (f) One broken C-C bond. (g) Two
broken C-C bonds. (h) Four broken C-C bonds. (i) Cu adatom in a hollow site on the edge.
polarized calculations. Without gate voltage (V = 0 inevitable occur. For example, if the AGNRs are syn-
G
V) the IETS display opposite signs due to the spin- thesized from a precursor molecule, involving heating
induced gap. Only a single peak contributes to the and dehydrogenation, as reported by Cai et al.14 and
spin-degenerate IETS while several peaks contribute to Blankenburg et al.,15 there is a chance that the reaction
the spin-polarized IETS. Even if the ZGNR is tuned is incomplete and some of the C-C bonds between the
by V = 0.5V to become metallic and the two treat- precursor molecules do not form. Also there is a chance
G
mentsthenshowthesameoverallsigninIETS,thespin- that a part of the final AGNR will have dehydrogenated
polarized IETS persists to show a much richer structure. edges or are passivated by two hydrogen atoms. Finally,
This difference suggests that IETS could be a way to defects may be introduced on purpose by locally dosing
indirectly observe spin-polarized edge states. a high current from the tip of a STM.20
Projecting the modes contributing to the IETS onto
the phonon band modes further underlines how several
bands with different symmetries contribute to the spin- A. Defects in AGNRs
polarizedIETS,whileonlyacoupleofbandscontributes
to the spin-degenerate IETS, see Fig. 7. Again we use
In Fig. 8 we show the structures of pristine AGNR
Eq. (8) for this characterization, where the overlap for
alongwith8differentdefectconfigurationswhichwehave
V =0.0 V corresponds to the red color and the overlap
G considered. These include four defects in the edge passi-
forV =0.5Vcorrespondstothedifferencebetweenthe
G vationasfollows: Asingleedgesidewithanextrahydro-
blue and red color in Fig. 7, respectively. It is clear that
gen atom [1H-edge, Fig. 8(b)], two edge sides with each
spin-polarization permits more modes to contribute to
an extra hydrogen atom [2H-edge, Fig. 8(c)], one hydro-
the IETS. In contrast to the spin-degenerate case, where
genreplacedbyafluorineatom[1F-edge,Fig.8(d)],and
thesymmetricelectronicstates(withrespecttothemid-
a dehydrogenated edge with 4 hydrogen atoms removed
dle of the ribbon) only can couple to the symmetric vi-
fromeachside[8H-free,Fig.8(e)]. Wehavealsoconsid-
bration modes, the symmetry lowering of the electronic
ered defects in the atomic structure in the form of one,
states by spin-polarization opens up also for scattering
two, or four broken C-C bonds [1C-broken, 2C-broken,
also via odd modes.
4C-broken, Fig. 8(f)-(h)] as well as a Cu adatom on the
AGNR [Cu-adatom, Fig. 8(i)]. For all these systems the
entiredynamicalregionwasrelaxed,i.e.,thepartsofthe
IV. DEFECTED GRAPHENE NANORIBBONS AGNRs shown in Fig. 8.
Defects may influence the IETS signal in two ways.
In this section we address the modification and new First, a defect can have a direct impact by changing the
signals in IETS that arise due to various defects in the vibrationaldegreesoffreedom. Inorderforthechangein
GNR. Regardless of the fabrication method, defects will thevibrationalspectrumtogiveasignalintheIETS,the
8
this effective number of conductance eigenchannels.51,52
(a) Clean (b) 1H-edge (c) 2H-edge
We can now discuss how the different defects modify the
4
electronic properties. From Fig. 9(b)-(i) we notice that
not all defects change the elastic transmission, and fur-
thermore, a change in elastic transmission needs not be
2
unique for a specific defect.
Instead, IETS may provide a additional fingerprint in
0 the current that can be used to identify the type of de-
(d) 1F-edge (e) 8H-free (f) 1C-broken fect. Figure 10 shows the computed IETS as a function
4 of gate voltage for the 8 different defects. As for the
S
O cleanstructure,thetwopeaksat169and196meVcorre-
D sponding to the D- and G- Raman modes are dominant
,
TT1 2 for a range of gate values for all the structures. Another
T,
feature,whichispresentinallthesystems,istheappear-
ance of several signals close to the band onsets. In the
0
followingsubsectionswediscussinmoredetailthetrans-
(g) 2C-broken (h) 4C-broken (i) Cu-adatom
port characteristics with the different types of defects in
4
AGNRs.
2
1. Edge passivation
0
-1 0 1 -1 0 1 -1 0 1 Considering defects in the edge passivation [Fig. 8(b-
Energy (eV) e)] the gap in the transmission is essentially unchanged
[Fig. 9(b-e)], except for the 1H-edge structure where a
zero-energy resonance appears in the DOS and trans-
FIG. 9: (Color online) Electronic properties of the AGNR
structures shown in Fig. 8. The total transmission is shown mission [Fig. 9(b)]. This new peak can be attributed
withblacklines. TheratioT/T ,whereT isthetransmission to tunneling via a mid-gap state which appears due to
1 1
originatingfromthemosttransmittingeigenchannelisshown the local breaking of sub-lattice symmetry.1 Thus, if a
withgreendashedlines(thisratiogivesalowerboundtothe H atom is added to the neighboring C atom [2H-edge,
number of contributing eigenchannels). The DOS for the C Fig. 8(c)] the peak disappears [Fig. 9(c)]. The addition
atoms in the dynamical region is shown with red lines (offset ofoneortwoHatomsonthesamesidealsoresultsinthe
by 3 units).
closing of one transmission channel between the valence
and conduction bands as shown in Fig. 9(b,c). Concern-
ing the vibrational degrees of freedom, the addition of
newvibrationsmustcoupletothecurrent,andpreferably extra hydrogen to the edge results in new vibrational
havefrequencieswhichdonotcoincidewithonesalready modesaround330meVfor1H-edgeandaround343and
giving IETS signals for the pristine ribbons. Second, a 353 meV for 2H-edge, clearly outside the bulk phonon
defect can substantially change the electronic structure band (ranging up to ∼200 meV) of pristine AGNR.53
and thereby have an impact on the e-ph couplings as- Comparing the IETS in Fig. 10(a-c) we find that only
sociated with the active modes or even the transmission 1H-edge gives a signal which differs significantly from
eigenchannels of the pristine ribbons, e.g., changing a the pristine case. Figure 10(k) shows specific IETS for
peak in the IETS to a dip (and vice versa) or enhancing selected gate voltages for 1H-edge. Here, at V = 0.2V
G
asymmetric contributions via Eq. (4). (topgreencurve)weseehownewsignalsappearatlarge
The electronic properties of the pristine AGNR is voltages: Forpositivebiaspolaritytwosignalsappearat
showninFig.9(a). ThecarbonDOSprojectedtothede- 330 and 365 meV, respectively, while for negative bias
viceregion(redcurve)revealsagapasexpectedfromthe polarity only an asymmetric signal around −365 meV is
band structure [Fig. 1(b)], which is significantly broad- present. Thesignalat330meVisduetovibrationsofthe
ened from the coupling to the metallic electrodes. The H [Fig.11(b)],whilethesignalat365meV[Fig.11(a)]is
2
two valence and two conduction bands in the considered due to the H atom on the neighboring C atom. Further,
energy range naturally explain that the total transmis- the amplitude of the signals around 169 and 196 meV is
sion (black curve) is bound below a value of 2. Further, also found to depend on bias polarity.
the ratio T/T < 2 (green dashed line), measuring the Gatingontothezero-energyresonancefor1H-edgethe
1
minimum number of contributing channels where T is IETS signal [middle red curve in Fig. 10(k)] is dom-
1
the transmission of the most transmitting eigenchannel, inated by large asymmetric signals for low energy vi-
shows that both channels play a role for the transport, brations due to the contribution from κ and Eq. (4).
λ
at least away from the edges of the direct band gap. We note that κ changes sign with bias polarity for this
λ
Measurements of shot noise may provide insights into approximately left-right symmetric structure. This can
9
IETS (V 1)
−
(j) Clean (k) 1H-edge
15
10
5
0
-5
(l) 8H-free (m) 1C-broken
)15
) 1−
V( V10
(
te S 5
a T
G
E 0
I
-5
(n) 4C-broken (o) Cu-adatom
15
10
5
0
-5
-0.2 0 0.2 -0.2 0 0.2
Bias (V)
Bias (V)
FIG. 10: (Color online) (a-i) IETS as a function of gate voltage V for the pristine and defected AGNR structures shown in
G
Fig. 8 . (j-o) IETS for six selected structures at three specific gate values (dashed horizontal lines in panels a-i). The curves
are offset with the most negative gate value at the bottom (black curves) and the most positive at the top (green curves). (j)
Clean AGNR at gate values V =−0.3, 0.0, and 0.8 V. (k) 1H-edge at V =−0.3, 0.0, and 0.2 V. (l) 8H-free at V =−0.3,
G G G
0.0, and 0.6 V. (m) 1C-broken at V = −0.3, 0.0, and 0.3 V. (n) 4C-broken at V = −0.3, 0,0, and 0.3 V. (o) Cu-adatom at
G G
V =−0.3,0.0,and0.3V.DottedverticallinesareguidestotheeyeofcharacteristicIETSsignalscorrespondingtothemodes
G
in Fig. 11
.
be seen from the red IETS curve in Fig. 10(k) which is 244meV[Fig.10(l)]matchingtheH-freemodemeasured
roughly an odd function of the bias voltage. In close by Raman.34 We find that this signal is robust as it ap-
proximity of the zero-energy resonance a characteristic pears in the whole range of gate values. When gating
“X-shape” isobservedinthegate-dependentIETS,while into to the valence band a new signal appears around
away from it the signals approach that of the pristine 43 meV [V ≈ −0.8 V in Fig. 10(e)] originating from a
G
AGNR [Fig. 10(b)]. low energy edge vibration [Fig. 11(g)].
Substituting a H atom with a F atom (1F-edge) is
seentohavevirtuallynoeffectintheIETSofFig.10(d).
This suggests that a significant change in the chemical 2. Structural defects
composition directly involving the π-electronic system is
required in order to obtain a signal although the vibra-
The electronic transmission in GNRs is mediated by
tions are influenced by the heavier passivation.
the carbon π system. Thus if a C-C bond fails to be
Such a significant change in the passivation occurs formed during GNR synthesis or if it is broken again
for instance by removing four H atoms on each side at a later stage, a large effect can be expected for the
(8H-free), giving rise to four very narrow peaks in the electronic conduction properties. This impact is indeed
DOS around the conduction band, [Fig. 9(e)]. These revealed in Fig. 9(f-h). Breaking one or two bonds re-
correspond to very localized dangling-bond states on sults in the formation of two in-gap states which, broad-
the dehydrogenated dimers and therefore do not show ened by the electrodes, make the gap appear smaller.
up in the transmission. However, the dehydrogenated The IETS signals for the 1C-broken and 2C-broken in
edges give rise to localized vibrations outside the range Fig. 10(f,g,m) have the same two signals at 169 and 196
of the pristine vibrational spectrum.53 The in-phase vi- meV as for the clean ribbon. However, the relative am-
bration of the dehydrogenated C dimers at the armchair plitudes are interchanged such that the "D"-peak is now
edges [Fig. 11(f)] gives rise to an extra IETS peak at slightly more intense than the "G"-peak.
10
(a)1H-edge (b)1H-edge (c)8H-free (d)8H-free
~ω=365meV ~ω=330meV ~ω=244meV ~ω=43meV
(e)4C-broken (f)4C-broken (g)4C-broken
~ω=367meV ~ω=50meV ~ω=26meV
FIG.11: (Coloronline)Visualizationofthemostcontributingdefect-inducedvibrationalmodestotheIETSsignalsindicated
byverticallinesinFig.10(j-o). (a-b)Thetwohydrogensignalsfor1H-edge. (c)Localizededgemodeatthecarbondimersfor
the 8H-free. (d) Delocalized edge mode for the 8H-free. (e) Hydrogen mode from the zigzag edge of 4C-broken. (f-g) Defect
modes for 4C-broken.
Breaking four C-C bonds [4C-broken, Fig. 8(h)], re- theπ-orbitals. However,aroundtheonsetoftheconduc-
sulting in constrictions of single C-C bonds, totally alter tion band the IETS signals in Fig. 10(i,o) is dominated
the DOS which is now dominated by three sharp peaks bylargeasymmetricsignalswithsignificantcontributions
as seen in Fig. 9(h). The corresponding IETS signals from out-of-plane phonons. These modes come into play
are shown in Fig. 10(h,n). In the proximity of the zero- due to breaking of the planar symmetry by the adatom.
energy resonance a broad range of signals at low vibra- Also note that by gating of E within the gap these sig-
F
tional energies appears (red curve in panel n) as well as naturesoftheadatomdisappear,cf.thelowerblackcurve
acharacteristic“X-shape” inthegateplot(panelh)sim- in Fig. 10(o).
ilartothatof 1H-edge. Gatingawayfromtheresonance
we observe two additional robust IETS signals at 27 and
50 meV resulting from vibrations localized at the defect
B. Defects in ZGNRs
[Fig. 11(d,e)].
Let us next consider a series of defects for the zigzag
graphene nanoribbon. In Fig. 12 we show the atomic
3. Adatoms
structures of pristine ZGNR along with 8 different de-
fect configurations. We consider the following defects in
Transition metals are typically used for growth of the edge-passivation: A single edge with an extra hy-
graphene or as a substrate for the bottom-up synthe- drogen [1H-edge, Fig. 12(b)], one hydrogen is replaced
sis of GNRs. Thus it is of interest to consider the effect by either a F atom [1F-edge, Fig. 12(c)], an OH group
of adatoms of this type on GNRs. A Cu adatom on [1OH-edge, Fig. 12(f)], or a NO group [1NO -edge,
2 2
graphene adsorbs preferentially in the on-top position.54 Fig. 12(g)]. We also consider defects in the form of
However, positioning Cu such that it breaks the axial a Cu adatom [Cu-adatom, Fig. 12(d)] or a Li adatom
symmetryofourAGNR,wefindthatitismoststablein [Li-adatom, Fig. 12(e)]. Finally, we also study the ef-
ahollowsiteattheedge[Cu-adatom,Fig.8(i)]. TheDOS fect of a structural defect in form of a 57 reconstruc-
andtransmissioninFig.9(i)revealan-typedopingeffect tion [R57, Fig. 12(h)] and a substitutional defect where
shifting E close to the conduction band while leaving a C atom next to the edge is replaced by a Si atom
F
the two transmission channels inside the gap relatively [Si-substitute, Fig. 12(i)]. For all these systems the
intact. entire dynamical region was relaxed, i.e., the parts of
For the pristine GNR the e-ph couplings of the out-of- the ZGNRs shown in Fig. 12 using spin-polarized treat-
plane vibrations are suppressed due to the symmetry of ments. The spin degrees of freedom σ =↑,↓ generalizes