Gate-Tunable Exchange Coupling Between Cobalt Clusters on Graphene

3.5 MB·English

by  Hua Chen

Gate-Tunable Exchange Coupling Between Cobalt Clusters on Graphene
Hua Chen,1 Qian Niu,1 Zhenyu Zhang,2 and Allan H. MacDonald1
1Department of Physics, University of Texas at Austin, Austin, TX 78712, USA
2ICQD/HFNL, University of Science and Technology of China, Hefei, Anhui, 230026, China
(Dated: January 23, 2013)
We use spin-density-functional theory (SDFT) ab initio calculations to theoretically explore the
possibilityofachievingusefulgatecontroloverexchangecouplingbetweencobaltclustersplacedon
a graphene sheet. By applying an electric field across supercells we demonstrate that the exchange
interaction is strongly dependent on gate voltage, but find that it is also sensitive to the relative
sublattice registration of the cobalt clusters. We use our results to discuss strategies for achieving
strong and reproducible magneto-electric effects in graphene/transition-metal hybrid systems.
3
1
0 PACSnumbers: 73.22.Pr,75.47.-m,75.30.Et
2

n I. INTRODUCTION to π-electrons on different sublattices. The RKKY
a interaction decays as r−3 at large distance r, because
J
of the suppressed density-of-states at the Dirac point of
1 Graphene1,2 is an atomically thin two-dimensional
graphene22,23,26,28. At finite carrier density the RKKY
2 gapless semi-conductor in which the carrier density can
coupling has spatial oscillations with period π/k on
] +be10v1a3ricemd−o2vebryagabtrionagd,arnadngise,afrroemma∼rka−b1ly01g3ocomd−co2ntdouc∼- top of an envelope which decays as r−2. Most exiFsting
ci tor at high carrier densities. Graphene/transition metal studies of the RKKY interactions in graphene have
s assumed magnetic moments due to point-like impurities
- hybridsystemsareattractiveforspintronicsbecausecar-
that are associated with a particular honeycomb lattice
l bon spin-orbit interactions are particularly weak3,4 in
r site and have purely phenomenological interactions.
t flat honeycomb-lattice arrays, because magnetic transi-
m These models are realized approximately in systems
tion element clusters5–7 form readily on graphene sur-
at. faces, and because of potentially attractive properties8,9 wcairtbhonmvaagcnaentciciesm30o,maeltnhtosugdhuethetose hdyefdercotgsensiagtnioifinc2a9ntolyr
of interfaces between graphene and magnetic transition
m modify the carbon sp2 bonds and hence the structural
metals. For example ultra-thin transition metal lay-
- ers on graphene are predicted10,11 to have extremely and electronic properties of graphene. Moments due
d to adsorbed magnetic transition metal atoms do not
n large magnetic metal anisotropy energies. For these rea-
distort the graphene bands as strongly but have small
o sons there has recently been considerable interest12–16
migration barriers31 due to weak adsorption energies.13
c in the magnetic and electronic properties of transition
[ The transition metal clusters on graphene on which we
metal adatoms and clusters placed on a two-dimensional
1 graphene sheet. focus are relatively immobile, however, and can be large
enough to exceed the super paramagnetic limit. These
v In this article we theoretically explore the possibility larger magnetic objects therefore have more potential
4
that the exchange coupling between separate magnetic for spintronics applications. We attempt to realistically
2
0 metal clusters on graphene can be altered electrically by describe the magnitude of cobalt cluster moments, their
5 gating. Since arrays of magnetic clusters can be real- magnetic anisotropy energies (MAE), the exchange
. ized on graphene by using a graphene/substrate moir´e coupling between the clusters and graphene, and finally
1
0 pattern17 as a template, and the magnetic clusters hy- the graphene-mediated magnetic exchange energies
3 bridize relatively strongly with graphene’s valence and between separated clusters.
1 conduction band orbitals, we anticipate gate-dependent We use first-principles supercell electronic structure
: exchange coupling between clusters which should lead to calculations based on spin density functional theory
v
gate-dependent magneto-resistance18,19 effects that are
i (SDFT) to investigate not only the RKKY coupling be-
X strong at room temperature. The goal of this work is
tween magnetic cobalt clusters deposited on graphene,
r to identify strategies for achieving strong, reproducible but also its dependence on external electric fields due to
a magneto-electriceffectsingraphene/transition-metalhy-
gating. We choose cobalt because its bulk lattice con-
brid systems.
stant is very close to that of graphene, and because thin
There is already a substantial theoretical cobaltfilmsdowntotwoorthreeatomiclayershavebeen
literature20–28 on Ruderman-Kittel-Kasuya-Yosida foundtohaveperpendicularmagneticanisotropy7,which
(RKKY) interactions between local moments coupled is preferable for spintronic applications. First, by calcu-
to graphene π-bands. It has been recognized,22 for latingtheelectronicstructureofatwo-atomic-layerthick
example, that when graphene is undoped the RKKY cobalt film on graphene, we find that there is consider-
interaction is ferromagnetic (FM) for magnetic moments ablechargetransferfromcobalttographene. Hybridiza-
coupled to π-electrons on the same graphene sublattice tion between the cobalt cluster and graphene leads to
and antiferromagnetic (AFM) for moments coupled sublattice and spin dependent shifts in graphene π-band
2
energies from which we are able to extract the essen- culations. Denserk-pointmeshes(upto79×79×1)were
tial kinetic-exchange parameters. Then we directly cal- usedtocheckaccuracyandtoperformMAEcalculations.
culatetheexchangeinteractionbetweentwoparalleltwo- To study the indirect exchange coupling between re-
atomic-layer-thick cobalt ribbons placed on graphene. mote cobalt clusters on graphene, we constructed a su-
For the geometries we have been able to consider, we percellwithtwoparallelcobaltribbonstwo-atomic-layers
find that the exchange interactions have a typical size thickandthreeatomswide,orientedalongthezigzagdi-
∼ 10−4 eV per cobalt atom, comparable to the MAE of rection of graphene (Fig. 2). The supercell used in this
bulk cobalt (4×10−5 eV32) and thin films of cobalt on caseis25×1withthesame20˚Avacuumlayerinzˆdirec-
graphene7, but smaller than anisotropy energies which tion. These ribbon calculations used a 1×49×1 k-point
can be achieved in asymmetrical clusters.33,34 We also mesh. The lattice parameters of the cobalt ribbons were
find that exchange interaction tend to change sign when taken from the infinite 2D slab calculations mentioned
agrapheneclusterchangesitssublatticeregistration,and above without further relaxation. (We checked the in-
that the exchange interactions can be modified by gate fluence of relaxation for several cases and did not find
voltages. qualitative modification relative to the results reported
In Section II we briefly describe the methods that we on below.) The exchange coupling between the cobalt
use for these computations. For the sake of definiteness ribbonswasestimatedbycalculatingthetotalenergydif-
wehavefocusedourattentiononcobaltclustersthatare ference between spin-parallel (FM) and spin-antiparallel
two atomic layers thick and arranged in a ribbon geom- (AFM) configurations:
etry. In Section III we describe our results for the elec-
∆E =E −E . (1)
tronic structure of a bulk two-layer thick film of cobalt FM AFM
on graphene. We find that there is considerable charge
With this convention a positive ∆E corresponds to anti-
transfer from cobalt to graphene, and that hybridiza-
ferromagnetic exchange between the ribbons.
tion between the magnetic cluster and graphene leads
An external electric field across the supercells was re-
to sublattice and spin dependent shifts in graphene π-
alized by adding a saw-tooth like external potential to
band energies. In Section IV we summarize our results
the total energy functional40. We have applied electric
for the dependence of total energy on the relative spin
fields of different size in the same supercell as in Fig. 2.
orientations of separated clusters. We are able to under-
In this case the external field can produce only charge
standourmainfindingsusinganapproximatetreatment
transfer between the two cobalt ribbons and graphene.
which treats the cobalt-graphene interaction perturba-
A more realistic representation of gating action on the
tively. FinallyinSectionVwepresentourresultsforthe
graphene/transitionmetalhybridsystemcanbeachieved
gate-voltage dependence of these exchange interactions.
byaddingabilayerCuslabtothesupercellasin(Fig.9).
We find that gate fields can produce sizable changes in
The copper acts as a a charge reservoir and also screens
exchangeinteractions,insomecaseschangingtheirsigns
the part of graphene directly below the cobalt ribbons
and substantially reducing their sublattice registration
from external fields. A more detailed discussion of some
dependence. In Section VI we summarize our findings
issues involved in using VASP to simulate gates is pro-
and discuss some possible directions for future research.
vided in Appendix B.
II. METHODS III. KINETIC EXCHANGE COUPLING
BETWEEN COBALT OVERLAYERS AND
GRAPHENE π-BANDS
The DFT calculations reported on in this work were
performedusingtheprojector-augmented-wave(PAW)35
method as implemented in the Vienna ab initio simula- A. Ab Initio Spin-density-functional Theory
tion package (VASP)36–38. The Perdew-Burke-Emzerhof
generalized gradient approximation (PBE-GGA)39 was As illustrated in Fig. 1 (a), we have calculated the to-
used for the exchange-correlation energy functional. To tal energies of bilayer cobalt films adsorbed on graphene
calculate the electronic band structure of an infinite with different registries and have found that the most
graphene sheet fully covered by a two-atomic-layer-thick stable geometry is that with the C atoms in one sub-
cobalt film [Fig. 1 (a)], we used a a 20 ˚A thick vacuum lattice of graphene located directly below bottom-layer
region between neighboring supercells in the zˆ (perpen- cobalt atoms, i.e. at atop sites, and the C atoms in the
dicular to the graphene plane) direction. We fixed the othersublatticebelowthetop-layercobaltatoms, i.e. at
lattice constant at the experimental value for graphene hcp sites. The optimal separation between the cobalt
(2.46 ˚A)since the (0001) surface of bulk hcp cobalt has a overlayer and graphene is about 2.21 ˚A. After adsorb-
smalllatticemismatch(<2%). Allatomsinthesupercell tion on graphene, the magnetic moments on the cobalt
were allowed to relax until the Hellmann-Feynman force atoms in the first layer (adjacent to graphene) decrease
oneachatomwassmallerthan0.001eV/˚A.Aplane-wave from1.710µ percobaltatom,whichisclosetothebulk
B
energy cutoff of 400 eV and a 33×33×1 k-point mesh value, to 1.560 µ per cobalt atom. Meanwhile, the C
B
were used for structure relaxation and total energy cal- atoms in sublattice A (adjacent to cobalt atoms) obtain
3
a per-atom magnetic moment of 0.043 µ , antiparallel B. Kinetic Exchange Model
B
tothemagnetizationofthecobaltoverlayer,whereasthe
C atoms in sublattice B acquire a moment of 0.041 µB Our electronic structure calculations can be qualita-
per atom and parallel to the cobalt moments. There- tively described using a simple model for graphene cou-
fore the overall magnetization direction of graphene is pled to a cobalt overlayer in which hybridization and
opposite to that of the cobalt film. We have also calcu- chargetransfereffectsshifttheenergiesofbothmajority
lated the magnetocrystalline part of the MAE by eval- and minority spins on both graphene sublattices:
uating the total energy difference, including spin-orbit
coupling,betweenconfigurationswithallmomentsalong H =(cid:126)v k·τ +µ−h τ −h S −h S τ . (2)
F 0,z z z,0 z z,z z z
the zˆ direction (out-of-plane) and along the xˆ direction
(in-plane). The system is found to have perpendicular InEq.2thefirsttermontherighthandsideistheusual
magnetic anisotropy7, with a MAE of ∼0.09 meV per Dirac Hamiltonian for hopping on a honeycomb lattice
cobaltatom, whichislargerthanthatofbulkhcpcobalt with velocity vF ∼ 106 m/s and wave vectors measured
(∼0.04 meV), but still the same order of magnitude. relative to the Brillouin-zone corners, Sz = ±1/2 labels
The spin-resolved Kohn-Sham band structure of the spin, and τz = ±1 distinguishes A (under the atop site)
Co-graphene hybrid system is shown in Fig. 1 (b). The and B (under the hcp site) sublattices. The parameters
graphene bands are spin-split and the Dirac points at of this model can be identified by fitting to the energies
the K point are gapped because of the relatively strong of the bands that have the largest π-band character at
interaction with the cobalt overlayer, in agreement with the Brillouin-zone corner (K) points, which are summa-
previous results41–43. It is nevertheless clear from the rized in Table I. H is diagonal when k =0 and its four
position of the Fermi level that graphene is n-doped, eigenvalues
i.e. electrons are transferred from cobalt to graphene42.
1 1
The graphene layer majority-spin Dirac point is easily µ−h − h − h , (3)
0,z 2 z,0 2 z,z
identified in the two-dimensional bands, but its minor-
1 1
ityspin-counterpartissostronglyhybridizedwithcobalt µ−h + h + h ,
d-orbitals that it is less easily identified. The K point 0,z 2 z,0 2 z,z
1 1
is at a higher energy for graphene majority spin bands
µ+h − h + h ,
than for minority spin bands, indicating an overall anti- 0,z 2 z,0 2 z,z
ferromagnetic coupling between the cobalt overlayer and 1 1
µ+h + h − h ,
graphene. This conclusion is also in agreement with the 0,z 2 z,0 2 z,z
antiparallelorientationsofthegrapheneandcobaltmag-
correspond to the four eigenvectors
netizations mentioned above.
|A↑(cid:105),|A↓(cid:105),|B ↑(cid:105),|B ↓(cid:105). (4)
The four Kohn-Sham bands with the strongest carbon
p character at the K point of Brillouin zone are bands
z
2, 3, 4, and 5 in Table I. By fitting their energies to
the SDFT band energies we can obtain the values of the
parameters:
µ = −0.622 eV (5)
h = 0.195 eV
0,z
h = −0.214 eV
z,0
h = −0.766 eV.
z,z
The model band structure calculated with these these
parameters is plotted in Fig. 1 (c).
FIG. 1: (color online). (a) Top and side views of the su-
Several comments are in order:
percell (with a 3×3 repetition in the xy plane for illustra-
(i)Thechemicalpotentialµspecifiestheenergyshiftav-
tion purpose). The larger balls represent cobalt atoms and
eragedoverspinandsublattice,whichisnegativebecause
the smaller balls C atoms. (b) Two-dimensional Kohn-Sham
electrons are transferred to graphene, in agreement with
quasiparticle band structure of the Co-graphene hybrid sys-
our previous discussion.
tem neglecting spin-orbit interactions. The blue lines illus-
trate the majority spin bands and the red lines the minority (ii) The value of h0,z is positive because the A sublattice
spin bands. The blue and red dots indicate the strength of is more strongly influenced by the cobalt overlayer than
carbonp orbitalcharacterinthemajorityandminorityspin theBsublattice,whichisexpectedsincetheAsublattice
z
states. (c) Model graphene projected band structure calcu- is directly below the cobalt atoms at the interface.
lated using Eq. 2. The model parameter values (Eq. 6) are (iii) The value of h measures the kinetic exchange
z,0
obtained by fitting to the DFT results listed in Table I. coupling between cobalt and graphene spins averaged
4
over sublattices. Its negative sign means the sublattice-
TABLE I: Orbital character of the bands in Fig. 1 (b) at
averaged magnetic coupling is AFM, also in agreement
the K point of 2D Brillouin zone. Only those having strong
with our observations in the previous subsection.
carbon p characters are listed. A and B correspond to the
z
(iv)Thespin-andsublattice-dependenttermh reflects two sublattices of graphene, as shown in Fig. 1 (a).
z,z
the property that the majority spin is higher in energy
Band No. Energy (eV) C p character cobalt d character
on the A sublattice whereas the minority spin is higher z
in energy on the B sublattice. In other words, the Co- 1 1.006 A↓: 0.069 3z2−r2 ↓: 0.671
graphene exchange coupling is AFM on the A sublattice
xz,yz↓: 0.098
butFMontheBsublattice. Tounderstandthisproperty 2 -0.151 B↓: 0.319
x2−y2,xy↓: 0.099
werefertoTableIinwhichbands1,6,7,8areidentified
as cobalt d bands that hybridize with the graphene π 3 -0.328 A↑: 0.297 3z2−r2 ↑: 0.368
bands. From the carbon p and cobalt d characters that
z xz,yz↑: 0.059
these bands carry, it can be seen that spin-splitting on 4 -0.703 B↑: 0.439
the A sublattice is because of hybridization mainly with x2−y2,xy↑: 0.016
the d3z2−r2 orbitals of cobalt (bands 1 and 6), whose 5 -1.307 A↓: 0.341 3z2−r2 ↓: 0.019
minority spin states are higher in energy than majority
6 -1.754 A↑: 0.191 3z2−r2 ↑: 0.303
spinstatesandabovetheFermilevel. Thehigherenergy
ofthecarbonmajorityspinstatesontheAsublatticecan xz,yz↓: 0.19
7 -1.965 B↓: 0.185
therefore be understood as the result of level repulsion x2−y2,xy↓: 0.054
from cobalt d3z2−r2 orbitals with the same spin. The
xz,yz↑: 0.206
same argument also applies for the B sublattice, whose 8 -3.048 B↑: 0.055
p orbitals mainly hybridize with the dxz, dyz, dxy, and x2−y2,xy↑: 0.166
z
dx2−y2 orbitals of cobalt because of symmetry. However,
bothofthetwocobaltdbands(band7and8)withthese
characterare below the Fermi energy and the π-bands
at the K point, with the minority spin band higher in
energy. Therefore level repulsion in this case results in
the higher energy of the minority spin states, i.e. in
ferromagnetic coupling.
(v) h is much larger than h because the kinetic ex-
z,z z,0
change interaction between the cobalt overlayer and the
graphene is strongly dependent on sublattice. We will
see later that this property will translate to a strong de- FIG.2: (coloronline). Topandsideviewsofthesupercell(re-
pendenceofthegraphene-mediatedexchangeinteraction peatedby4timesintheyˆdirectionforillustrationpurposes)
between two cobalt clusters on their relative registries used to calculate the magnetic coupling between two parallel
cobalt ribbons (larger blue balls) placed on a graphene sheet
with respect to the sublattices of a continuous graphene
(smaller yellow balls).
sheet.
A. Electronic Structure
In Fig. 3 (a) we show the electrostatic potential (ionic
IV. MAGNETIC COUPLING BETWEEN
potential plus Hartree potential from electrons) profile
COBALT CLUSTERS ON NEUTRAL GRAPHENE
within the graphene sheet for the system in Fig. 2. In
equilibrium, the chemical potential will shift relative to
In this section we will investigate the magnetic cou- the bands by the opposite amount. Therefore Fig. 3 (a),
pling between cobalt clusters on neutral graphene sheets withasignchangeanduptoaconstant,canbeviewedas
which are mediated mainly by their mutual influence on aplotofFermenergyrelativetotheDiracpoint. Onecan
the graphene π-bands. First we employ SDFT to study see that there is a large positive shift of chemical poten-
a relatively small system with parallel quasi-1D cobalt tial in the region directly below the two cobalt ribbons,
ribbons placed on graphene (Fig. 2) and separated by meaning the graphene is strongly n-doped at these posi-
∼ 1 nm. Then we will calculate the RKKY coupling tions. Theπ-bandelectronbarrierheightbetweencobalt-
in graphene perturbatively using the the model devel- covered and bare graphene regions is therefore about 0.5
oped above to compare with the SDFT calculation re- eV,closetothe0.622eVseparationbetweenthechemical
sults. This comparison informs perturbative estimates potentialandtheDiracpointfoundearlierfortheinfinite
of coupling which cannot be directly addressed using ab 2DCo/graphenehybridsystem. Thebarrierissmallerin
initio tools. thepresentcasebecauseseparationsbetweenneighboring
5
cobalt ribbons are not large enough for the pristine neu- in the case of complete two-layer cobalt coverage. This
tralgraphenevalue. Thisbarriercanpotentiallydecrease property is maintained in the uncovered portion of the
magnetic coupling between remote graphene clusters by graphene sheet. Opposite spin polarizations on the two
localizing electronic states more strongly in the vicinity sublattices suggests that the graphene-mediated inter-
of one particular cluster. action will be strongly sublattice dependent as in the
In Figs. 3 (b-d) we plot partial density-of-states RKKY case. This behavior is common in systems with
(PDOS) functions projected to the p orbitals of car- bipartite lattices.22,23
z
bon atoms at different points in the structure. At all
three sites the PDOS Dirac-point minima are shifted to
lower energy, indicating n-type doping over the entire
graphene sheet. The magnitude of the Dirac-point shift
decreases as one goes further away from the cobalt rib-
bons, as expected. One feature worth mentioning in the
PDOS plots is the appearance of resonant features that
are absent in pristine graphene. These features can be
identifiedas confinement effectsinthe zigzag-ribbon-like
uncovered graphene regions between the cobalt ribbons.
Weseelaterthatalthoughthesemodificationstothelin-
earDOSofgraphenedonotgreatlyinfluencetheformof FIG.4: (coloronline). Colorscaleplotofspinpolarizationas
the π-band mediated magnetic coupling, they do play a afunctionofpositionwithinthegrapheneplane,intheregion
between two cobalt ribbons with parallel spin orientations.
roleinthedependenceofthechargetransfertographene
Theverticalaxisinthisfigureisonpositionalongtheribbon
on gate field.
directionwhichhasatomicscaleperiodicity. Thepositiveand
negativespindensities(inarbitraryunits)areconcentratedon
carbonatomsonoppositesublattices. Theblackdotsindicate
the positions of C atoms.
In Fig. 5 we plot SDFT results for magnetic coupling
betweencobaltribbonsfordifferentedge-to-edgesepara-
tions between the ribbons and different registries with
respect to the sublattices of the continuous graphene
sheets. We first note that although both cobalt ribbons
have the same atop-hcp registry with graphene, the first
layer cobalt atom is sometimes atop an A site carbon
atomandsometimesatopaBsitecarbonatom. Thecon-
figurations of atop(A)-hcp(B) and atop(B)-hcp(A) are
degenerateforanindividualcobaltribbon,butmagnetic
coupling energies can change if one ribbon changes reg-
istry and the other does not. The strong oscillation be-
tween FM and AFM coupling in Fig. 5 is due to pre-
cisely this effect. From now on we refer to the geometry
in which the two cobalt ribbons have the same registry
FIG. 3: (color online). (a) Electrostatic potential variation
or different registries respectively as geometry AA, and
inadsorbedcobaltribbons. (b-d)Densityofstatesprojected
to the p orbitals of three carbon atoms whose positions are geometry AB.
z
indicatedbytheblackarrows. Blacklines–graphenewithad- From Fig. 5 we see that the strength of the magnetic
sorbedcobaltribbons,redlines–baregraphene. Thenegative coupling is about 1.3 meV per supercell for the AA con-
PDOSaxisplotsminoritybandvalueswhilethepositiveaxis figuration for separations between 8 ˚A and 17 ˚A. This
plots majority band values. exchange coupling is about 0.13 meV when normalized
percobaltatom,whichismuchlargerthanthe0.04meV
MAE of bulk hcp cobalt and somewhat larger than the
MAE of a 2 layer cobalt film on graphene (0.09 meV).
B. Exchange Coupling (We have also calculated the MAE of a single cobalt rib-
bon on graphene as in the present setup and the value
Wenextstudytheexchangecouplingbetweenthetwo is 0.08 meV per cobalt atom, with the easy axis along
cobalt ribbons in Fig. 2. In Fig. 4 we plot the spin the ribbon direction.) The similar strength of the MAE
density vs. position within the graphene sheet for the andtheexchangecouplingmeansthatinter-ribboninter-
case of two ferromagnetically aligned cobalt ribbons. In actions can have a substantial influence on the magnetic
the region below the cobalt ribbons, the spin polariza- configurationofclusterarrays. RKKY-likeoscillationsin
tions are opposite for the two sublattices of graphene, as the coupling are expected to have period ∼ π/k , with
F
6
k the Fermi wave vector. In the present system the than the 0.13 meV value obtained at the original cluster
F
Fermi energy E is about 0.4 eV on average in the part size. Therefore one can expect the total exchange cou-
F
ofgraphenebetweenthetwocobaltribbons, correspond- plingtoincreasesublinearlywithclustersize. Thereare
ing to a period of ∼5 nm. Therefore it is not surprising several reasons why this finding is expected. First, as we
thatwedonotseeRKKY-likeoscillationsinthesecalcu- mentioned previously, there is a large chemical potential
lations. The small coupling at distances below 5 ˚A may barrier at the cluster edge, which will weaken the influ-
be due to competition between direct exchange coupling ence of cobalt atoms deeper inside the clusters. Second,
and graphene-mediated coupling between the two cobalt when the cluster size is comparable to or larger than the
ribbons. It is not clear why there is strong variation oscillationperiodoftheRKKYinteraction,contributions
in the exchange coupling strength for the AB configura- from different parts of the cluster interfere destructively,
tion. One guess is that it is due to structural details at asweseeinthenextsubsection. Finally,sincethelargest
the boundaries of the zigzag-ribbon-like graphene region contribution to the kinetic exchange interaction between
between the two cobalt ribbons. cobalt clusters and graphene is from the cobalt atoms
closest to graphene, adding more layers of cobalt to the
clusters is expected to be less effective in increasing the
magnetic coupling.
C. Qualitative Theory of Exchange Coupling
In this subsection we will use conventional perturba-
tion theory and the model defined by Eq. 2 to calcu-
late the RKKY coupling between magnetic clusters on
graphene, and compare the result with our SDFT re-
sults. Similar calculations for the RKKY interaction in
FIG. 5: (color online). Magnetic coupling (per supercell, graphenehasbeenperformedpreviously,20–28 butmainly
which has 10 Co atoms) between two cobalt ribbons as a for the case of point-like magnetic impurities. Here we
function of ribbon separation. The interaction strength is willexplicitlyincludethesizeandshapeofmagneticclus-
the total energy difference between parallel and antiparallel
ters. Whencombinedwiththeessentialkineticexchange
spin-alignment configurations. Black squares (red dots) cor-
parameters obtained from first principles, the formalism
respond to configurations in which the cobalt atoms in the
developed in this subsection can be a useful tool for ex-
bottom layers of the two cobalt ribbons are directly above
trapolationstosystemsizesbeyondtherangewhichcov-
the same (different) sublattice(s) of graphene.
ered by SDFT calculations.
For a graphene sheet that is partially covered by two
Itisimportantforpotentialapplicationstounderstand
distinct magnetic clusters 1 and 2, Eq. 2 becomes
how these exchange couplings will change with the size
of the cobalt clusters. Due to computational power lim-
H = H +H +H (6)
itations we consider only two cases. First we increase 0 1 2
the width of the two cobalt ribbons from 3 to 4 atoms, = (cid:126)v kˆ·τ +D (r)V +D (r)V ,
F 1 1 2 2
so that there are 14 cobalt atoms in a supercell. In the
second case we add one more layer of cobalt atoms to where D (r) = 1 at positions covered by cluster 1
1(2)
the4-atom-wideribbonsincase1,sothatthenumberof (2) and zero otherwise, and V = µ −h τ −
1(2) 1(2) 0,z z,1(2)
total cobalt atoms increases to 18. The per-cobalt mag- h S −h S τ . ThereforetheRKKYinter-
z,0 z,1(2) z,z z,1(2) z,1(2)
netic coupling is 0.10 and 0.099 meV for the two cases. actionisevaluatedbycalculatingthecontributiontothe
In both cases the per-atom coupling strength is smaller totalenergyatsecondorderintheperturbationH +H :
1 2
∆E(2) =g(cid:88)(cid:90) d2k (cid:90) d2k(cid:48) f (1−f )|(cid:104)sk|(H1+H2)|s(cid:48)k(cid:48)(cid:105)|2 (7)
(2π)2 (2π)2 sk s(cid:48)k(cid:48) E −E
sk s(cid:48)k(cid:48)
ss(cid:48)
where g =2 is the valley degeneracy, s=±1 is the band neglect inter-valley transitions which add an anisotropic
index, and f is the Fermi distribution function [1 + and rapid modulation to the spatial dependence of the
sk
exp((E −µ)/k T)]−1. In keeping with the continuum RKKY interaction27,28.
sk B
modelweareusingtodescribethegrapheneπ-bands,we
7
The eigenfunctions of H are: in which D is the Fourier transform of D (r).
0 q,1(2) 1(2)
Therefore Eq. 7 becomes
(cid:18) (cid:19)
(cid:104)r|sk(cid:105)= √1 e−iθk eik·r ≡F eik·r (8)
2 s sk
where θ = arctan(k /k ). H can be written as a
k y x 1(2)
Fourier integral:
(cid:90) d2q
H (r)= eiq·rD V (9)
1(2) (2π)2 q,1(2) 1(2)
∆E(2) = 1g(cid:88)(cid:90) d2k (cid:90) d2q (f −f )|Fs†(cid:48)k+q(Dq,1V1+Dq,2V2)Fsk|2. (10)
2 (2π)2 (2π)2 sk s(cid:48)k+q E −E
sk s(cid:48)k+q
ss(cid:48)
By substituting Eq. 8 and the spin-dependent terms in factor multiplying the sublattice-dependent term is ∼10
V into |F† (D V +D V )F |2, and keeping times larger than that the factor which multiplies the
1(2) s(cid:48)k+q q,1 1 q,2 2 sk
only the cross terms between D V and D V , we sublattice-independent term. Therefore the RKKY in-
q,1 1 q,2 2
obtain teraction between cobalt clusters should be strongly de-
pendent on their registration with respect to the sublat-
|F† (D V +D V )F |2 = (11) ticesofgraphene,agreeingwithourobservationfromthe
s(cid:48)k+q q,1 1 q,2 2 sk
1 SDFT results.
(D∗ D +c.c.)·{ h2 [1+ss(cid:48)cos(θ −θ )]
q,1 q,2 2 z,0 k k+q The integration over k and the summation over bands
1 in Eq. 10 can be performed explicitly at T = 0 K.
+ h2 [1−ss(cid:48)cos(θ −θ )]τ τ }S S ,
2 z,z k k+q z,1 z,2 z,1 z,2 (We summarize calculation details in Appendix A.) The
RKKY energy, written as an integral over q, is
in which the first term in the curly brackets is
sublattice-independentandthesecondtermissublattice-
dependent. Here τz,1(2) are ±1 depending on which ∆E(2) = gh2z,0 (cid:90) d2q (D∗ D +c.c.)Π (q)(12)
graphene sublattices the clusters are directly above. For RKKY 16(cid:126)v (2π)2 q,1 q,2 z,0
F
conciseness we set Sz,1Sz,2 → 1/4 from now on. Note gh2 (cid:90) d2q
that the cross terms between h and h vanish be- + z,z (D∗ D +c.c.)Π (q)τ τ ,
z,0 z,z 16(cid:126)v (2π)2 q,1 q,2 z,z z,1 z,2
cause unperturbed graphene has spatial inversion sym- F
metry, and τ changes sign under spatial inversion. Us-
z
ing the values of h and h obtained previously, the where
z,0 z,z
(cid:115) 
q k k (cid:18)2k (cid:19)2 q 2k q
Πz,0(q) = −8 − πF + 2Fπ  1− qF + 2k arcsin qFΘ(q−2kF)+ 8Θ(2kF −q), (13)
F
q k q 2k q
Π (q) = −Λ+ F − arcsin FΘ(q−2k )− Θ(2k −q), (14)
z,z 4 π 2π q F 4 F
Θ(x) is the Heaviside step function, and Λ is the Dirac willhaveafasterdecaywithdistancethanthatfromthe
model’sultravioletcutoff. NotethatbothΠ (q)andits sublattice-dependent part.23
z,0
first derivative are continuous at q = 2k . In contrast,
F
Π (q) has a discontinuous first derivative at q = 2k ,
z,z F
similar to the behavior of 2-dimensional electron gas.
Therefore one can expect that the contribution to the
Graphene’sRKKYinteractioncanbeobtainedbyset-
RKKY interaction from the sublattice-independent part
ting D (r)=δ(r) and D (r)=δ(r−R). The k R(cid:29)1
1 2 F
8
limit is same geometry as in our SDFT calculations. The distri-
bution functions for this case are
(cid:32) (cid:33)
gh2 gh2 1
JRKKY(R)= 128πz(cid:126),0vF − 64π(cid:126)z,vzFτz,1τz,2 · R3,(15) D1(r) = Θ(cid:18)x+w+ d2(cid:19)Θ(cid:18)−x− d2(cid:19) (17)
(cid:18) (cid:19) (cid:18) (cid:19)
when graphene is undoped, and d d
D (r) = Θ −x+w+ Θ x− ,
2 2 2
J (R)=−gh2z,zkF · sin(2kFR)τ τ (16)
RKKY 16π2(cid:126)v R2 z,1 z,2 where d is the distance between the inner edges of the
F
tworibbons,andwisthewidthofthetworibbons. Their
when graphene is doped. When carriers are present the Fourier transforms are
dominant contribution is the sublattice-dependent part,
wgrhaipchheinseoissciullnadtoorpyedin, tshpeacoescainlldatdoerycaytesrmas vRa−n2is.heWshbeen- Dq,1 = qi (cid:104)eiqxd2 −eiqx(d2+w)(cid:105)·2πδ(qy). (18)
x
cause of the k prefactor, and the leading order terms i (cid:104) (cid:105)
monotonicallyFdecay as R−3. Dq,2 = q e−iqx(d2+w)−e−iqxd2 ·2πδ(qy). (19)
x
Next we use Eq. 12 to calculate the RKKY-like inter-
action between two cobalt ribbons on graphene with the Therefore,
2
D∗ D +c.c.= {2cos[q (d+w)]−cos(q d)−cos[q (d+2w)]}·2πLδ(q ). (20)
q,1 q,2 q2 x x x y
x
In deriving the above equation we have used the relation rical sense the cobalt ribbon amounts to 3 unit cells of
graphene. w may be treated as a fitting parameter in
L
δ2(q )=δ(0)δ(q )= δ(q ), (21) applications of our approximate theory. The magnetic
y y 2π y
coupling for the AB geometry (τ τ =−1 in Eq. 12),
z,1 z,2
where L is the length of the system in y direction. We which we did not show in Fig. 6, can be obtained simply
can then carry out the integration in Eq. 12 numerically. by subtracting the sublattice-dependent part from the
Below we will compare the results from this model cal- sublattice-independent part. As we mentioned before,
culations to the SDFT results. Note that the interaction the anomalous oscillation of magnetic coupling for the
energy in SDFT is the difference between spin-parallel AB geometry in Fig. 5 probably has a structural origin
andspin-antiparallelconfigurationsofthetwocobaltrib- that is not captured by this simple model.
bons. Therefore the model results below are all double Knowing that our model can capture the essential
E in Eq. 12. physics of the graphene-mediated magnetic coupling be-
RKKY
Fig. 6 shows the magnetic coupling from our model tween cobalt clusters relatively well, we can now explore
for the AA geometry, which correspond to τ τ = 1 the large separation limit which cannot be easily ad-
z,1 z,2
in Eq. 12. One can see that the order of magnitude dressedbyfirst-principlesmethods. FirstinFig.7(a)we
agrees very well with the SDFT results in Fig. 5, and plot the magnetic coupling for the AA geometry vs. rib-
thetrendwithchangingdistanceisalsowellreproduced. bonseparationforseveralcarrierdensities. Onecannow
We have chosen E to be 0.4 eV, which is the average of see the spatial oscillation between AFM and FM inter-
F
thegraphenechemicalpotentialunderthecobaltribbons actions which appears only beyond the separation range
(∼0.6 eV) and that in the center between the two cobalt covered in Fig. 5. From the figure we see that not only
ribbons (∼0.2 eV). The agreement would be improved if the periodicity, but also the amplitude of the oscillation,
wesuedthefactthatthedopinglevelofthegraphenere- depends on the doping level. This behavior is consistent
gionbetweenthetwocobaltribbonsincreasesasthetwo with the asymptotic RKKY interaction Eq. 16.
ribbonsapproachtoeachother. InFig.6(b)weassumed InFig.7(b)weplotmagneticcouplingdividedbyrib-
simplelineardependenceofE withdandtheagreement bon width w, which is proportional to the magnetic cou-
F
with Fig. 5 is remarkably improved. We note here that pling per cobalt atom. It is interesting to see that when
there is some arbitrariness in determining the width of wisverylarge(24grapheneunitcellsinthezigzagdirec-
the cobalt ribbons w since there is no sharp boundary of tion, equivalent to about 50 ˚A), the magnetic coupling
the portion of the graphene region which interacts with is strongly suppressed. This behavior can be understood
the cobalt ribbon. Here we chose w to be 4 unit cells by considering in terms of destructive superposition be-
of graphene to account for the residue influence at the tween different parts of the ribbon, when the scale of
edges of the cobalt ribbons, although in a pure geomet- the clusters is close to the oscillation period. In addi-
9
FIG. 7: (color online) (a) RKKY coupling between cobalt
FIG. 6: (color online) (a) Model results for RKKY-like cou-
pling between cobalt ribbons. E =0.4 eV, (cid:126)v =5.96 eV·˚A, ribbons at large separations, for several carrier densities. (b)
F F
L=2.46˚A,w=8.51˚A.(b)Sameas(a)butwithE increasing RKKYcouplingdividedbyribbonwidthwforseveralwidths.
F
w is expressed in terms of the number of graphene unit cells
linearly as d decreases.
alongthezigzagdirectionacrossthecobaltribbon. wisfixed
at 4 in (a) and E is fixed at 0.4 eV in (b).
F
tion, since the period of the RKKY oscillation increases
with decreasing k , the coupling for the same large clus-
F
terswillbelesssuppressedasthegrapheneislessdoped, can be seen that electrons are transferred from graphene
which we have also verified. Fig. 7 (b) also confirms our to Co, and that an out-of-plane polarization is induced
discussionontheeffectivenessofincreasingthemagnetic inthegraphenesheetitself. Theamountofchargetrans-
coupling by preparing larger clusters. Therefore a gen- ferred from the graphene plane decreases as one moves
eral criterion for real applications is that the linear size awayfromthecobaltribbons,inagreementwiththeelec-
of the clusters should be around or below π , which is trostaticpotentialprofileshowninFig.3(a). Inthisway
half of the RKKY period. 2kF onedecreasesthegraphenecarrierdensitynotonlyinthe
bare regions of graphene, but also in the regions covered
by the cobalt ribbons.
V. GATE CONTROL OF EXCHANGE One question which may be raised at this point is
COUPLING whether or not the exchange coupling between cobalt
and graphene will be influenced by the electric field.
Since the RKKY coupling in graphene has a strong To this end we have calculated the spin polarization in
dependence on the Fermi energy (Eq. 16 and Fig. 7), a graphene sheet fully covered by a 2-layer cobalt film
which in turn can be altered by electric gates, we expect [Fig1(a)]underelectricfieldsupto0.8V/˚Aanddidnot
that the magnetic coupling between cobalt clusters can find a significant change. Therefore the field dependence
be conveniently tuned by gating. In this section we will of the exchange coupling between graphene and cobalt
studythechangeofthemagneticcouplingbetweencobalt is not an issue in the range of electric fields considered
ribbons on graphene with external electric fields. We here.
have relegated some general remarks on how to simulate Next we study the field dependence of the magnetic
electric gates in supercell calculations to Appendix B. coupling between the two cobalt ribbons at specific sep-
arations between them. In Fig. 9 (a) we plot magnetic
coupling vs. electric field for two cobalt ribbons sepa-
A. Freestanding Co-graphene in an Electric Field ratedby∼15˚A,anddifferentregistrieswiththegraphene
sublattices. One can see that both the sign and magni-
By directly applying a electric field along the zˆ direc- tude of the magnetic coupling can be tuned by electric
tion in the supercell of Fig. 2, we can change the Fermi fields. It is also interesting to notice that for both the
energyinthegraphenebytransferringelectronsfromthe AA and AB configurations the coupling has a similar
cobalt ribbons to graphene and vice versa. In Fig. 8 we sublinear dependence on electric field. Using the sim-
show the charge transfer within the supercell after ap- ple model explained in the previous section, we found
plying a 0.2 V/˚A electric field along the −zˆdirection. It that the coupling changes almost linearly with E from
F
10
FIG.8: (coloronline). Chargedensitydifference(inanx−z
plane)betweenasystemsubjectedtoa0.2V/˚Aelectricfield FIG. 9: (color online). (a) Dependence of magnetic coupling
along the −zˆ direction, and a system with no electric field. between two cobalt ribbons on external electric field at two
Positive and negative values (in arbitrary units) correspond different separations. Blue squares (red dots) correspond to
to accumulation and depletion of charge, respectively. The theconfigurationthatthetwocobaltwiressitabovethesame
black dots (triangles) indicate the positions of C (Co) atoms (different)graphenesublattice(s),withaseparationof15.0˚A
in the plane. (14.3 ˚A). A negative value of field strength means that the
field is along the −zˆ direction. (b) Density of states (spin-
up plus spin-down) projected to the p orbital of a C atom
z
E = 0.2 eV to 0.4 eV, which is roughly the range of in the center of the supercell for several different external
F
electric field strengths and the AA configuration in (a). The
E shift produced by the electric fields in our DFT cal-
F
inset blows up the details around E .
culations [Fig. 9 (b)]. Therefore the nonlinearity should F
come from the field dependence of the Fermi energy of
graphene. Inequilibriumtheexternalpotentialdifference
betweencobaltandgraphene(eEdwheredisthespatial decreasestheeffectivecapacitance,butmostlyduetothe
separation) should be balanced by the electric potential smallverticalseparationbetweenthetwosystems,which
due to charge redistribution and the Fermi energy shift makes graphene’s quantum capacitance effect dominant.
ofgraphene(i.e.,thequantumcapacitanceofgraphene). It is clear that an external electric field does not ade-
This screening physics can be described crudely using a quately model the influence of a remote gate. In the
simple parallel plate capacitor model: next subsection we will use an alternative supercell to
bettersimulatearealisticgatinggeometry,andfindthat
edcE ·dE
ed·dE = F F +dE (22) this tactic brings additional benefits.
C F
where c is the proportionality constant for the lin-
ear dependence of graphene DOS on E , and c =
F
gvgs =0.018 eV−2˚A−2 in pure graphene, C/d is the
2π((cid:126)vF)2
geometric capacitance of the graphene/cobalt bilayer,
and dE and dE are electric field and Fermi energy dif-
F
ferentials. The solution of this differential equation is
√
2e2cd2C·E+C2+2ecd·const−C
E = , (23)
F ecd
which explains the slower-than-linear dependence of E
F
on E. Of course this argument relies on the assumption
thatthedensityofstatesofgraphenearoundE islinear
F
in energy. By looking at Fig. 9 (b) one can see that this
assumption is actually reasonable, although the effective
value of c may be different from that in pure graphene
value due to the confinement-induced resonances.
Finally in Fig. 10 (a) we plot magnetic coupling vs.
theseparationbetweenthetwocobaltribbonsforseveral
electric field strengths. The corresponding result from
the model in Sec. IVC is plotted in Fig. 10 (b). Reason-
able agreement for the E = −0.4 V/˚A case is obtained FIG. 10: (color online). (a) Magnetic coupling between two
by taking E = 0.36 eV, which means this extremely cobalt ribbons in the AA configuration vs. separation, under
F
different electric fields. (b) Results obtained using the model
large electric field is only able to shift E by 0.04 eV on
F in Sec. IVC.
average. The small number is partly due to the incom-
plete coverage of the cobalt ribbons on graphene, which