Magnetization profile for impurities in graphene nanoribbons
ABSTRACT The magnetic properties of graphenerelated materials and in particular the
spinpolarised edge states predicted for pristine graphene nanoribbons (GNRs)
with certain edge geometries have received much attention recently due to a
range of possible technological applications. However, the magnetic properties
of pristine GNRs are not predicted to be particularly robust in the presence of
edge disorder. In this work, we examine the magnetic properties of GNRs doped
with transitionmetal atoms using a combination of meanfield Hubbard and
Density Functional Theory techniques. The effect of impurity location on the
magnetic moment of such dopants in GNRs is investigated for the two principal
GNR edge geometries  armchair and zigzag. Moment profiles are calculated
across the width of the ribbon for both substitutional and adsorbed impurities
and regular features are observed for zigzagedged GNRs in particular. Unlike
the case of edgestate induced magnetisation, the moments of magnetic
impurities embedded in GNRs are found to be particularly stable in the presence
of edge disorder. Our results suggest that the magnetic properties of
transitionmetal doped GNRs are far more robust than those with moments arising
intrinsically due to edge geometry.

Dataset: Molecular hydrogen uptake by zigzag graphene nanoribbons doped with early 3d transitionmetal atoms
[Show abstract] [Hide abstract]
ABSTRACT: a b s t r a c t We performed ab initio densityfunctional calculations to investigate the structural, electronic and magnetic properties of nanostructures comprising singleadatoms of Sc, Ti or V adsorbed on a hydrogenpassivated zigzag graphene nanoribbon (GNR). We also investigated the affinity of the resulting doped nanostructures for molecular hydrogen. In all cases, the most stable structures featured the adatom at positions near one of the edges of the GNR. However, whereas in the most stable structures of the systems Sc/GNR and V/GNR the adatom was located above a bay of the zigzag edge, Ti/GNR was found to be most stable when the adatom was at a firstrow hole site. Adsorption at sites near one of the ribbon edges reduced drastically the average magnetic moment of the carbon atoms at that edge. On the other hand, the magnetic moments of the adatoms on the GNR, as the electronic character of the doped nanostructures, depended on the adsorption site and on the adatom species, but their absolute values were in all cases, except when Sc was at an edge bay site, greater than those of the corresponding free atoms. Our results showed that, of the three systems investigated in this paper, Ti/GNR (except when Ti is adsorbed at an edge bay site) and V/GNR appear to satisfy the criterion specified by the U. S. Department of Energy for efficient H 2 storage, as far as binding energy is concerned. We discussed in detail the differences between the adsorption of H 2 on the system Ti/GNR and the adsorption of H 2 on Tiadsorbed carbon nanotubes, which have been proposed as a highcapacity hydrogen storage media.  SourceAvailable from: Mauro S. Ferreira[Show abstract] [Hide abstract]
ABSTRACT: Magneticallydoped graphene systems are potential candidates for application in future spintronic devices. A key step is to understand the pairwise interactions between magnetic impurities embedded in graphene that are mediated by the graphene conduction electrons. A large number of studies have been undertaken to investigate the indirect exchange, or RKKY, interactions in graphene. Many of these studies report a decay rate faster than expected for a 2dimensional material and the absence of the usual distance dependent oscillations. In this review we summarize the techniques used to calculate the interaction and present the key results obtained to date. The effects of more detailed parameterisations of the magnetic impurities and graphene host are considered, as are results obtained from ab initio calculations. Since the fast decay of the interaction presents an obstacle to spintronic applications, we focus in particular on the possibility of augmenting the interaction range by a number of methods including doping, spin precession and the application of strain.01/2013;
Page 1
arXiv:1108.6199v1 [condmat.meshall] 31 Aug 2011
APS/123QED
Magnetization profile for impurities in graphene nanoribbons
S. R. Power(a), V. M. de Menezes(b), S. B. Fagan(c)and M. S. Ferreira(a)∗
(a) School of Physics, Trinity College Dublin, Dublin 2, Ireland
(b) Departamento de F´ ısica, Universidade Federal de Santa Maria, UFSM, 97105900, RS, Brazil
(c)´Area de Ciˆ encias Tecnol´ ogicas, Centro Universit´ ario Franciscano,
UNIFRA, 97010032, Santa Maria  RS, Brazil
(Dated: September 1, 2011)
The magnetic properties of graphenerelated materials and in particular the spinpolarised edge
states predicted for pristine graphene nanoribbons (GNRs) with certain edge geometries have re
ceived much attention recently due to a range of possible technological applications. However, the
magnetic properties of pristine GNRs are not predicted to be particularly robust in the presence of
edge disorder. In this work, we examine the magnetic properties of GNRs doped with transition
metal atoms using a combination of meanfield Hubbard and Density Functional Theory techniques.
The effect of impurity location on the magnetic moment of such dopants in GNRs is investigated
for the two principal GNR edge geometries  armchair and zigzag. Moment profiles are calculated
across the width of the ribbon for both substitutional and adsorbed impurities and regular features
are observed for zigzagedged GNRs in particular. Unlike the case of edgestate induced magneti
sation, the moments of magnetic impurities embedded in GNRs are found to be particularly stable
in the presence of edge disorder. Our results suggest that the magnetic properties of transition
metal doped GNRs are far more robust than those with moments arising intrinsically due to edge
geometry.
PACS numbers:
I. INTRODUCTION
The experimental discovery of graphene has precipi
tated wideranging research to fully determine the phys
ical properties of this novel material and to pave the way
for its application in future technological devices.1–6Of
particular interest is the potential for graphenebased
spintronic devices to be realised, and thus much at
tention has been focused on determining the magnetic
properties of graphene.7The existence of spinpolarised
edge states, predicted by many theoretical works,8–11
when a graphene sheet is cut to have a socalled zigzag
edge geometry has underpinned a large number of these
proposed devices. Narrow stripes of graphene, dubbed
Graphene Nanoribbons (GNRs), with parallel zigzag
edges, predicted to have opposite spin orientation, in par
ticular are proposed.10,12,13The other principal nanorib
bon geometry, the armchair case, does not display such
interesting spin polarised edges. GNRs with zigzag and
armchair edge geometries are shown schematically in the
top panels of Fig. 1. Despite theoretical advances in the
study of GNRs, experimental validation of their prop
erties has so far been inconclusive, due to the difficulty
in patterning the edge geometries required for these ef
fects to be observed. Furthermore, the spinpolarised
edge state in zigzagedged GNRs is predicted to be highly
dependent on the edge geometry and not particularly ro
bust under the introduction of edge disorder in the form
of vacancy defects or impurity atoms.14These factors
present major obstacles in the path of utilizing the in
trinsic magnetic edge states of graphene in experimen
tally realisable devices. Another possibility that has been
proposed is the exploitation of defectdriven magnetic
moments that arise in graphene.15,16Magnetic moments
have been predicted to form around vacancies and other
defects in the graphene lattice and the possibility of engi
neering a ferromagnetic state in graphene from such mo
ments has been suggested. However, such a claim would
seem to be restricted by the implications of the Lieb
theorem,17which states that any such magnetic moments
arise from a disparity between the two sublattices of
graphene. Largescale, randomised disorder would tend
to minimise such a disparity and prevent the formation
of a ferromagnetic state. A third possibility for incorpo
rating graphene in spintronic devices lies in the doping of
graphene systems with magnetic impurity atoms. This
approach takes advantage of the indirect exchange cou
pling, often referred to as the RKKY coupling,18–20be
tween magnetic impurities embedded in a graphene sys
tem which is mediated by the conduction electrons of
the graphene host. Although this coupling is predicted
to decay rapidly in graphene sheets,21–24the quasione
dimensional nature of nanoribbons suggests that a much
longer range coupling may persist in these materials for
certain impurity configurations, in a similar manner to
that found in Carbon Nanotubes.25–27
However, unlike carbon nanotubes, whose periodic
boundary conditions ensure the equivalence of lattice
sites, the physical properties of graphene nanoribbons
are expected to display a strong dependence on the loca
tion of introduced impurities.28,29Indeed, in a previous
paper we have shown that the binding energies of im
purity atoms depend strongly on their location across
a ribbon.29This leads to impurity distributions heavily
weighted towards the edge sites  an observation con
firmed elsewhere in the literature.30–36In addition to the
binding energy, the magnitude of the magnetic moment
on an impurity atom should also depend on impurity
Page 2
2
FIG. 1: Top Panel: Schematic representation of a 4ZGNR
(Zigzagedged ribbon, left) and 7AGNR (Armchairedged
ribbon, right). Nanoribbons are labelled by the number, N, of
zigzag chains (ZGNRs) or dimer lines (AGNRs) across their
width. In each case the number of atoms in the repeated
unit cell is 2N. The possible sites for substitutional (centre
adsorbed) impurities are labelled with Arabic (Roman) nu
merals. The arrows refer to the periodicity direction.
Bottom panel: Schematic of a substitutional (left) and centre
adsorbed (right) impurity in graphene.
position.
pal features of this dependence derive from the under
lying electronic structure of the GNR host. In section
II we show how the dependence can be calculated us
ing a simple tightbinding representation of the graphene
electronic structure combined with a selfconsistent mean
field description of the impurity species. In Section III,
the features of the dependence will be investigated and
the effect of the ribbon geometry and the nature of the
impurity considered. The reliability of these results will
be demonstrated by comparison to a full abinitio treat
ment of Mn inpurities embedded into graphene nanorib
bons. The nature of these features will be described com
prehensively by considering the two principal GNR ge
ometries, armchair and zigzag edges, for the case of two
impurity configurations  substitutional atoms replacing
a single carbon atom in the graphene lattice, and cen
tre adsorbed atoms sitting in the centre of a hexagon
of carbon atoms. These two configurations are shown
schematically in the bottom panels of Fig. 1. In Section
IV the effects of edge disorder on the magnitude of the
magnetic moments formed at the various impurity sites
across the ribbon are examined. We find that an edge va
cancy only has a significant effect on moments formed on
neighbouring sites, and that the moment profile quickly
returns to that of the pure ribbon when we move a few
carbon chains away from the vacancy. This illustrates the
relative robustness of magnetic moments introduced into
graphene systems through transition metal doping com
We shall demonstrate here that the princi
pared with those arising intrinsically, which are unstable
in the presence of edge disorder.14
II. MODEL
The electronic structure of the system is described us
ing a Hubbardlike HamiltonianˆH = H0+ Hint, where
H0 =?
netic energy plus a spinindependent local potential, and
Hint is the electronelectron interaction term. The op
erator ˆ c†
iµσcreates an electron with spin σ in atomic or
bital µ on site i. The graphene electronic structure is
described using a singleorbital nearestneighbour tight
binding model. The lowenergy properties of graphene
are known to be well reproduced within this frame
work. The magnetic impurities are described by a five
fold degenerate d band representing a typical transition
metal magnetic atom. Within H0, we set the onsite en
ergy of the carbon atoms to zero. The carboncarbon
nearestneighbour hopping is γCC= −2.7eV, and we set
t = γCC as our unit of energy. However a correction to
the hopping parameter at the edge of armchair ribbons,
γE
CC= 1.12γCC, is needed in order to achieve the ex
pected semiconducting behaviour for all ribbon widths.37
The exact value of the hopping parameter between the
carbon atoms of the graphene lattice and the magnetic
impurity atom, γCM, will depend on the impurity atom
chosen and can be calculated in a number of ways. While
the magnitude of this parameter can amplify the value of
the magnetic moment, it is not expected to have a signif
icant qualitative effect on the moment profile across the
width of the ribbon.
We assume that Hint is an onsite interaction which
takes place between electrons occupying the d orbitals
of the transition metal impurities only, and is neglected
elsewhere. In this case, the matrix elements of the spin
dependent part of the Hamiltonian reduce to vσ
−1
2∆µδµνσ, where ∆µrepresents the local exchange split
ting associated with orbital µ, and σ = ±1 for ↑ and ↓
spin, respectively. For simplicity we shall assume that
the onsite effective exchange integrals U are the same for
all d orbitals, whence it follows that ∆µis µindependent
and equal to ∆ = Um, where m is the local magnetic
moment. The value of m, and hence vµν, along with
the spinindependent contribution to the onsite energies
of the transition metal atoms, γMM, is calculated self
consistently within the mean field approximation under
constraints imposed by the values of the onsite exchange
integral U and the dband occupation nd. The dband oc
cupation is calculated at each stage in the selfconsistent
procedure by an integral over the density of states, which
comes directly from the diagonal element of the real
space Green function associated with the GNRimpurity
system. This can be calculated in turn from the recur
sively calculated Green function for the pristine GNR
using the Dyson equation.
The DFT calculations were performed by consider
ijµνσγµν
ijˆ c†
iµσˆ cjνσ represents the electronic ki
µν =
Page 3
3
ing a substitutional or adsorbed Mn impurity located at
different positions across the width of zigzag and arm
chair nanoribbons. These calculations were carried out
using the SIESTA code38with the systems placed in
a supercell so that the calculations were performed us
ing periodic boundary conditions. Doublezeta plus po
larization functions were employed and the exchange
correlation function was adjusted using the generalized
gradient approximation according to the parametrization
proposed by Perdew, Burke and Ernzerhof.39The inter
actions between the ionic cores and the valence electrons
were described with normconserving TroullierMartins
pseudopotentials.40The structural optimizations were
performed with the conjugate gradient approximation38
until the residual forces were smaller than 0.05 eV/˚ A.
Care must be taken when performing DFT calculations
within a periodic supercell as moment suppression can
occur if neighbouring moments prefer an antiferromag
netic alignment, but are forced to adopt a ferromag
netic alignment by the periodicity of the system.41The
proper emergence of moments in all our calculations sug
gests that antiferromagnetic alignment is not an issue
in this case, an assertion supported by the energetically
favourable ferromagnetic alignment between Mn impuri
ties in carbon nanotubes,42a similar material.
Edge disorder is considered by introducing vacancies at
edge sites in the ribbon. Within the simple model, these
vacancies are introduced by placing a very large onsite
potential (50t) on the lattice site containing the vacancy.
This is a simplified approach that does not take into ac
count lattice deformation effects that may occur when
an atom is removed, but is adequate to describe the elec
tronic effects of removing the relevant orbital from that
lattice site. It is worth highlighting that this simplifica
tion only occurs in the simple model and that the DFT
calculations discussed earlier for systems without edge
disorder fully account for relaxation.
III. MAGNETIC IMPURITIES
We present results for two magnetic impurity config
urations  the case of a substitutional atom replacing a
carbon atom and also that of a centreadsorbed impurity
sitting in the centre of a hexagon of carbon atoms. For
the latter case we assume that the hopping parameters
connecting the impurity atom to the lattice are equiva
lent for each of the neighbouring six carbon atoms. The
impurity is moved across the finite width of the ribbon
and the selfconsistent value for m is calculated at each
site. We consider both the zigzag and armchair geome
tries and a comparison is made for all cases with a full
Density Functional Theory calculation.
qualitative comparison, the quantity of interest is the rel
ative fluctuation of the magnitude of the moment around
its value at the centre of the ribbon, mc. This is given
by∆m
mc
.
For a simpler
mc≡m−mc
0.2
0
0.2
0.4
edge
centre
Impurity position
0
0.1
0.2
0.3
∆m / mc
FIG. 2: The magnetic moment profile across a 6ZGNR (top
panel) and 11AGNR (bottom panel) for a substitutional im
purity, calculated using the selfconsistent Hubbard model
discussed in the text (red squares) and also a full DFT treat
ment with Mn atoms (black circles). An excellent qualitative
match is found for the zigzag case, but the armchair match is
less convincing. As discussed in the text, this is due to lattice
distortion for impurities near the edge of the AGNR.
A. Substitutional Atoms
Fig 2 shows the magnetic moment fluctuation as func
tion of impurity location for substitutional impurities
across the width of a 6ZGNR and a 11ANGR . For
the case of zigzag ribbons, we first note an excellent
qualitative match between the simple model and the full
DFT calculation, from which we infer that the underly
ing mechanism for the variation in the magnetic moments
across the ribbon width is the electronic structure of the
nanoribbon. The position dependence arises from quan
tum interference effects caused by the boundary condi
tions imposed on the electronic stucture of graphene in
the form of the edges of the nanoribbons. Furthermore,
we note that the parameters γCM, ndand U which char
acterise the magnetic species in our simple model can
be altered to achieve a better numerical fit, but do not
affect the qualitative form of this plot. The pattern ob
served is a jagged, sawtooth style curve, characteristic of
properties measured across the width of zigzag ribbons
and a similar feature can be seen in the binding energies
of impurities.29This feature is a sublattice effect which
arises from the degeneracy breaking that occurs between
the two sublattices of graphene when a zigzag edge is
formed. The sublattices are represented schematically
in Fig. 1 by black or white circles. Each edge of the
ribbon is occupied by sites entirely from one of the sub
lattices, and that sublattice is “dominant” on that half of
the ribbon. For the case of impurity magnetic moments
on zigzag ribbons, this effect manifests itself in creating
larger moments on impurities located on the dominant
sublattice on either side of the ribbon. In other words,
impurity atoms on a black site on the side of the ribbon
Page 4
4
with black edge sites will have larger moments than their
neighbouring white sites. Focusing on a single sublattice,
we find that the trend across the ribbon width is for the
largest moment to arise on the dominant edge site for
that sublattice, to decrease as the impurity is moved to
wards the centre of the ribbon and to reach its minimum
value at the sites neighbouring the opposite edge. We
note the all the features discussed here arise in both the
simple model and the DFT results, confirming that this
is not simply an artifact of our simple model.
The corresponding plots for the armchair case do not
agree with each other as convincingly. The tightbinding
model is found to underestimate the value of the edge
moment found by the DFT calculation. This is because
the tightbinding calculation does not take into account
the distortions in the honeycomb lattice that arise when a
substitutional impurity is introduced near the edge of an
AGNR. The relaxed structures (not shown here) for the
two impurity sites nearest the ribbon edge are found to
be considerably perturbed compared to the pristine rib
bon and also to the relaxed structures corresponding to
the other impurity sites. The shape of the tightbinding
plot for AGNRs is also found to be more dependent on
the parameterisation of the impurity than in the zigzag
case. This issue will be explored further in the case of
centreadsorbed impurities. This suggests that the mo
ment profile across AGNRs is not as robust as that ob
served in the ZGNR case, and will vary somewhat ac
cording to the magnetic species chosen. However, both
tightbinding and DFT models find that the edge im
purity sites lead to larger magnetic moments than the
central ones. Fig. 2 also reveals that the sublattice ef
fect noted in zigzagedged ribbons is absent in the case
of armchair edges. This is explained from a cursory in
spection of Fig. 1 where it is obvious that the degeneracy
between black and white lattice sites is unbroken by the
imposition of armchair edges. The value of the magnetic
moment approaches mc much quicker for AGNRs, and
only minor deviations from it are observed away from
the edges of the ribbon, whereas in ZGNRS significant
deviations are still present deeper into the ribbon.
The dramatic increase observed in the magnetic mo
ment in impurities near the edge of zigzag ribbons is con
sistent with the presence of a localised edge state at the
Fermi energy. This state results in a large peak in the
density of states at the Fermi energy. Such a peak pro
vides favourable conditions for moment formation under
the Stoner Criterion41, and indeed if an intrinsic electron
electron interaction is considered in an undoped ZGNR,
will lead to the formation of the spin polarised edges as
discussed in the Introduction.
B. Centre adsorbed atoms
Fig. 3 shows the magnetic moment fluctuation for a
centreadsorbed impurity at various sites across a GNR,
calculated again using both the simple model and a full
0.1
0.05
0
edge
centre
Impurity position
0.02
0
∆m / mc
FIG. 3: The magnetic moment profile across a 8ZGNR (top
panel) and 11AGNR (bottom panel) for a centreadsorbed
impurity impurity. The black circles show the results from a
full DFT treatment with Mn atoms, whereas the red squares
and blue triangles show the results from the selfconsistent
Hubbard model with γCM = γCC and 0.7γCC respectively.
We note this parameter does not affect the qualitative features
of the moment profile across the ZGNR, but alters that across
the AGNR significantly.
DFT approach (black cirlces). Within the simple model
approach we consider two values of γCM, the hopping
parameter between the impurity atom and surrounding
lattice sites. The values considered are γCM= γCC(red
squares) and γCM = 0.7γCC (blue triangles). For the
case of ZGNRs (top panel), we note the impressive qual
itative match between the models. Furthermore we note
that the change in hopping parameter does not effect
the qualitative shape of the plot, but can be used to
yield a better fit. We also note that, unlike the sub
stitutional impurity considered earlier, the sublattice ef
fect is no longer present. This is because the impurity
is no longer strongly associated with a particular sub
lattice, but instead binds to three carbon atoms from
each, which has the effect of averaging out any sublat
tice dependent effects. The general trend of a monotonic
increase in the magnetic moment of the Mn impurity is
noted as it is moved towards the centre of the ribbon.
This is in stark contrast to the result for substitutional
impurities, where the largest moment is observed at the
edge and, for the dominant sublattice, the moment de
creases as the impurity is moved towards the centre of
the ribbon. The discrepancy can be explained by the
fact that the centreadsorbed Mn impurity induces fluc
tuations in the magnetic moments on nearby sites in the
graphene lattice. Edge atoms are particularly suscepti
ble to magnetic moments due to the localised state dis
cussed before, and thus have larger induced deviations
in their moments than the others. Consequently, centre
adsorbed impurities atoms at the edge of a ZGNR tend
to induce large moment deviations on the edge sites, re
sulting in a smaller moment on the impurity atom it
self. This is verified by examining the spindensity plots
Page 5
5
FIG. 4: Spin density plots showing up (blue) and down (red)
spin densities near a centre adsorbed impurity on the edge
hexagon (left) and nexttoedge hexagon (right) on a 8ZGNR.
The isosurface used was 0.001e/Bohr3.
from the DFT calculation for the case of centreadsorbed
impurities near the edge of a ZGNR. Fig 4 shows the
spindensity plots corresponding to an impurity on the
edge hexagon (left) and nexttoedge hexagon (right). It
is clear that the centreadsorbed impurity nearest the
edge introduces a much larger disturbance to the values
of the magnetic moments on surrounding sites than the
more centrally located impurity. In fact the magnetic
edge states are seen to be essentially unperturbed by the
latter. In contrast to these DFT calculations, the sim
ple model does not account for the intrinsic magnetic
edge states on ZGNRs. However a similar moment pro
file is recovered as the magnetic impurity atom induces
moments, rather than fluctuations of existing moments,
on the surrounding lattice sites and these are found to
be significantly larger for the case of the centreadsorbed
impurity nearest the ribbon edge.
However, in zigzag ribbons there is an additional type
of adsorption site which consists of an impurity atom
bound to two edge sites and the site between them.
This configuration, which we shall label “edgeadsorbed”
(EA), is illustrated schematically in Fig.
be viewed as an impurity atom connecting to half a
hexagon of the graphene lattice. As this site only oc
curs at the edge, we cannot study the position depen
dence of it. However, DFT calculations reveal that a
larger moment arises here than for the centreadsorbed
ato locamted nearest the ribbon edge and furthermore
that the edgeadsorbed configuration is also more en
ergetically favourable than any of centreadsorbed sites.
This is clear from the righthand side panels of Fig. 5.
The upper panel shows the moment fluctuation for the
EA case compared with those of the centre adsorbed lo
cations across a 8ZGNR. The bottom panel plots the
segregation energy function, β =
cases. This quantity, introduced in Ref. [29], plots the
relative deviation of the binding energy of an impurity on
a GNR, EB, around its value at the centre of the ribbon,
Ec
B. The edgeadsorbed case is found to be the most en
ergetically favourable. A spin density plot for this type of
impurity is shown in the bottom left panel of Fig. 5 and
5. It can
EB−Ec
Ec
B
B, for the same
0.02
0.01
0
∆ m / mc
EA
edge
Impurity position
centre
1
0.5
0
β
FIG. 5: The edge adsorbed impurity discussed in the text,
shown schematically in the top left panel. The spin density
plot for this configuration on a 8ZGNR is shown in the bot
tom left panel. The right hand side panels show the moment
fluctuation (top) and segregation energy function (bottom)
for this configuration calculated using the DFT approach,
compared to those for the centreadsorbed locations across
the width of a 8ZGNR.
reveals that this impurity configuration has a less dra
matic effect on the moments of nearby edge sites than
the centreadsorbed case on the edge hexagon, consistent
with larger moment found for the EA configuration.
In the results for adsorbed Mn impurities on an AGNR
in the bottom panel of Fig. 3 we notice that for the DFT
result, and the simple model calculation with γCM =
γCC the edge hexagonal site has a smaller moment on it
than the other sites. However the deviation in the value
of the edge moment, and indeed of the moment at any
site, from mcis far smaller than in the ZGNR case. The
moment profile in this case is essentially flat, with only
minor deviations from mcacross the width of the ribbon.
Examining the case of γCM = 0.7γCC reveals that the
shape of the profile across the ribbon is less robust than
the zigzag case as the edge moment here is found to be
slightly larger than mc. Of the cases examined, the effect
is weakest here and does not appear to be very robust.
Thus the position dependence of magnetic impurities is
smallest for adsorbed impurities on AGNRs and cannot
be deemed a significant effect.
IV. EDGE DISORDER
In this section the robustness of the features discussed
in the previous section will be examined in the presence
of edge vacancy defects. This is an important point to
consider when comparing impuritydriven magnetic mo
ments in GNRs to those arising intrinsically due the edge
states, which have been shown to be particularly vulner
able to edge disorder.14For each of the cases discussed in
Page 6
6
1
23
d
0
0.1
0.2
∆m / m0
1
234
0
0.04
FIG. 6: Effects of a single edge vacancy on the magnetic
moments of substitutional magnetic impurities on a 6ZGNR
(left) and 11AGNR (right). The top images show schemat
ically the edge vacancy and the possible sites of magnetic
impurities across the width of the ribbons. For each ribbon
we consider sites at the edge (black, circle), next to the edge
(red, square), centre (green, triangle) and opposite edge (blue,
inverted triangle) of the ribbon. The graphs underneath show
how the fluctuation in the magnetic moment of an impurity
at each of these sites under the introduction of an edge va
cancy, relative to the moment the impurity would have in the
absense of an edge vacancy. This is plotted as a function
of distance between the edge vacancy and the unit cell con
taining the magnetic impurity, where the distance is given in
unit cells. The shaded area in each of the ribbon schematics
contains one unit cell of that ribbon.
the previous section we examine the effect of a single edge
vacancy on the magnitude of a nearby moment, calcu
lated with the meanfield Hubbard model approach. The
distance between the magnetic impurity and the edge va
cancy is varied to examine the range of this effect.
Fig. 6 shows the effect of an edge vacancy on the mag
netic moments of substitutional impurities in a 6ZGNR
(left) and an 11AGNR (right). To show the range of the
effect we plot the relative change in the moment when
an edge vacancy is introduced as a function of distance
between the edge vacancy and the unit cell containing
the magnetic impurity atom. Note that in this case we
are plotting the fluctuation of each moment relative to
its value at its current position in a system without edge
defects, not relative to its value at the centre of the rib
bon as was shown previously. This plot is shown for a
number of possible sites for the magnetic impurity across
the width of the ribbon, namely the edge atom on the
same side as the vacancy, the site next to the edge, a
site at the centre of the ribbon and a site at the opposite
edge, as shown schematically in the upper panels.
For the ZGNR, the first point to note is that the only
sites that show a considerable change in their moments
are the first two cases. The edge site has a slight reduc
tion in the value of its moment, whereas the site next
to the edge and belonging to the opposite sublattice to
24
6
d
0
0.02
∆m / m0
1
234
0.02
0
FIG. 7: The effect of an edge vacancy on the magnetic mo
ment of centreadsorbed magnetic impurity atoms on an 8
ZGNR and 11AGNR. The notation in the schematics and
graphs is the same as for Fig. 6. For the zigzag case, the result
for an edgeadsorbed impurity (star, purple) is also shown.
the edge has a significant increase in magnetic moment.
However, the first point in this curve corresponds to a
site neighbouring the edge vacancy. Excluding this, the
largest deviation in magnetic moment does not exceed
5%. However, for all positions, the moment reverts very
quickly back to its value without the vacancy when it
is moved further away down the ribbon. This suggests
that a single edge vacancy will have very little effect on
the moments of magnetic impurities located more than
a lattice spacing or two away. The AGNR case is quite
similar. Moving away from the vacancy the deviations in
the moments again become very small. It is clear that
significant deviations in the moments of substitutional
impurities are not seen outside the immediate vicinity
of the edge vacancy in either ribbon geometry. Similar
results are shown for the adsorbed cases in Fig. 7. The ef
fect here is even smaller than for the substitutional case,
with fluctuations of less than 2% at distances greater
than two unit cells away from the edge vacancy for all
impurity types considered, including the edgeadsorbed
case in ZGNRs.
A single edge vacancy has been shown not to have a
significant effect on the magnetic moments of transition
metal impurities in a GNRs. In fact, even the introduc
tion of an extended edge defect, consisting of a length
of ribbon to either side of the magnetic impurity with
a certain concentration of edge vacancies, does not con
siderably affect the impurity moments unless there is an
edge vacancy in their immediate vicinity. We conclude
that magnetic moments introduced into GNRs by tran
sition metal doping are particularly stable and robust in
the presence of edge disorder. In particular, the striking
moment profiles seen for magnetic impurities in ZGNRs
will not be significantly perturbed by the introduction of
a reasonably strong extended edge disorder. This point is
Page 7
7
edge
centre
Impurity position
0
0.2
∆m / mc
FIG. 8: Schematic showing region of 6ZGNR with a magnetic
impurity and edge disorder consisting of atoms removed ran
domly from the edge zigzag chain of the ribbon (top panel)
and the the resulting moment fluctuations (bottom panel).
The red squares indicate the position dependent moment fluc
tuations in a ribbon without edge disorder, whilst the black
squares correspond to the average fluctuations taken over
fifty edgedisorder configurations, with the standard devia
tion shown by the error bars.
illustrated quite clearly in Fig. 8 for the case of substitu
tional impurities on a 6ZGNR, the same case considered
in the upper panel of Fig. 2. The moment profile for the
pristine case is shown as calculated with the meanfield
Hubbard approach (red squares). Also shown is the mo
ment profile for a system with a disordered region with
a length of 100 unit cells to either side of the magnetic
impurity (black circles). Within this disordered region,
carbon atoms from the edge zigzag chain at either edge of
the ribbon are removed with a probability of 10%. This
plot shows the result of an average over 50 such configu
rations, with the error bars on each point indicating the
standard deviation. The moment profile is seen to not
vary significantly from that of pristine case, demonstrat
ing clearly the robustness of the moment profile.
V. CONCLUSIONS
We have demonstrated the features of the magnetic
moment variation of a magnetic impurity as its loca
tion is varied across the width of a graphene nanorib
bon. For zigzagedged nanoribbons we found an excellent
agreement between the simple selfconsistent Hubbard
model and a more complete ab initio treatment. Fur
thermore the qualitative features of the resulting moment
profile remained constant for different parameterisations
describing the magnetic impurity, suggesting that they
hold for a wide range of magnetic species. For substitu
tional impurities, a nonmonotonic behaviour connected
to the sublattices of the graphene atomic structure was
identified. For this type of impurity, a larger moment was
found on impurities located on the edge site of a ZGNR.
For impurities adsorbed onto the centre of a hexagon of
the graphene lattice, a monotonic increase of the moment
magnitude as the impurity was moved towards the centre
of the ribbon was found. However an additional impu
rity type, consisting of an impurity atom connecting to
three edge atoms at a zigzag edge was found to have a
larger moment than one connected to the edge hexagon.
It was also noted to be more energetically favourable. For
armchairedgednanoribbons, the moment profile features
were noted to be less robust than for the zigzag case.
However the fluctuations of the moment value around
that at the ribbon centre were also found to be smaller.
For both edge geometries and impurity configurations,
we showed that an edge vacancy did not have a signifi
cant effect on the moment of a magnetic impurity located
more than one or two lattice spacings away. Furthermore,
we demonstrated that the distinctive moment profile for
substitutional impurities on a zigzagedged ribbon was
robust in the presence of an extended edge disorder. In
light of these findings, we argue that the magnetically
doped nanoribbons may provide a route to applications
previously envisaged for nanoribbons with intrinsic mag
netic ordering, which is less stable in the presence of ex
perimentally imposed constraints such as imperfect edge
geometry.
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