Hydrogenated Graphene Nanoribbons for Spintronics

1.1 MB·English

by  D. Soriano

Hydrogenated Graphene Nanoribbons for Spintronics
D. Soriano,1 F. Mun˜oz-Rojas,1,2 J. Ferna´ndez-Rossier,1 and J. J. Palacios1,3
1Departamento de F´ısica Aplicada, Universidad de Alicante, San Vicente del Raspeig, Alicante 03690, Spain
2Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea
3Departamento de F´ısica de la Materia Condensada,
Universidad Auto´noma de Madrid, Cantoblanco, Madrid 28049, Spain
(Dated: January 8, 2010)
We show how hydrogenation of graphene nanoribbons at small concentrations can open new
0 venues towards carbon-based spintronics applications regardless of any especific edge termination
1
orpassivation of thenanoribbons. Densityfunctional theory calculations show that an adsorbed H
0 atominducesaspindensityonthesurroundingπorbitalswhosesymmetryanddegreeoflocalization
2
depends on the distance to the edges of the nanoribbon. As expected for graphene-based systems,

n these induced magnetic moments interact ferromagnetically or antiferromagnetically depending on
a the relative adsorption graphene sublattice, but the magnitude of the interactions are found to
J strongly vary with the position of the H atoms relative to the edges. We also calculate, with the

8 help of the Hubbard model, the transport properties of hydrogenated armchair semiconducting
graphenenanoribbonsinthedilutedregimeandshowhowtheexchangecouplingbetweenHatoms

] can be exploited in the design of novel magnetoresistive devices.
l
l
a
h I. INTRODUCTION transmission electron microscopy18.
-
s In the light of the present controversy on the actual
e
m There is a widespread consensus on the large pote- possibilities of ever encounter zigzag magnetic edges, we
tial of graphene for electronic applications1,2. Theoret- propose in this work an alternative to edge-related spin-
.
t ically, graphene also holds promise for a vast range of tronics in which to exploit the recently shown controled
a applications in spintronics, although clear evidence of hydrogenationofgraphene19. Thekeyfactorhereisthat
m
magnetic graphene is, however, elusive to date. In a adsorption of atomic H in the bulk of graphene is ac-
d- broad sense, two factors may account for this elusive- companied by the appearance of a magnetic moment
n ness. First, the fact that hydrocarbons of high spin are of 1µB localized on the π orbitals surrounding each H
o knowntobehighlyreactiveand,unlessfabricatedorsyn- atom20. These magnetic moments interact with one an-
c thesized under very clean and controledconditions, they otherferromagneticallyorantiferromagnetically,depend-
[
will likely bind surrounding species with the concomit- ing on whether their respective adsorption sublattices
1 tant disappearance of magnetism3. A second reason re- (usually labeled A andB) are the same or not20–22. Sta-
v latestothefactthatthegroundstateofgrapheneisnear tistically speaking, a sublattice compensated H coverage
3 an interaction-driven phase transition into an insulating is expected unless adsorption on one sublattice is privi-
6 antiferromagnetic state4. This underlying antiferromag- leged, e.g., by the substrate. To date, however, there is
2
netic correlationspreventthe magneticmoments, evenif noevidencethatsuchanuncompensatedcoveragecanbe
1
they develop, from ordering ferromagnetically and pre- achieved. Inthe morelikelycompensatedcaseanoverall
.
1 clude the possibility of observing hysteresis in standard antiferromagnetic alignment with a total spin S = 0 is
0
magneticmeasurements. Notwithstanding,afewreports thus energetically favored over a ferromagnetic one with
0
of magnetic graphite5 andgraphene6 can be foundin re- S > 0. As a proof of principle and since we are inter-
1
: cent literature. ested in the diluted regime, we consider in this work the
v Most of recent theoretical ideas for graphene based fundamental problem of two H atoms. These are cova-
i
X spintronicsapplicationsarerootedonthemagneticprop- lently bonded to the surface of a semiconducting arm-
r erties of nanoribbons7–11 or nanographenes12–15 with chair graphene nanoribbon (AGNR) on different sublat-
a zigzag edges. All these proposals assume a very par- tices [see Fig. 1(a)]. Here the σ bonds of the edges are
ticular edge hydrogenation where H atoms passivate fully passivatedwithHso thatthey areirrelevantatlow
the σ dangling bonds, leaving all the π orbitals un- energies.
saturated and carrying the magnetic moments. How- As shown in Sec. III, after a brief introduction to
ever, this is just one out of many possible edge realiza- the theoreticalbasics presented in Sec. II, the magnetic-
tions which range from H-free self-passivation16 to full fielddrivenferromagnetic(F)state,wheretheH-induced
H passivation17. According to the work of Wassmann et magnetic moments are aligned by the field [as shown in
al.17, relatively low H concentations at room tempera- Fig. 2(b)], can present a different resistance from that
ture suffice to completely passivate the edges, including ofthe naturalantiferromagnetic(AF) zero-fieldstate [as
the edge π orbitals responsible for the magnetic order. thatinFig. 2(a)]. Twodifferentcasesarediscussed: Infi-
The self-passivated or reconstructed zigzag edges16 are, nitesemiconductingAGNR’sandfiniteonesconnectedto
in fact, among the least energetically favorable of all, conductive graphene [see schematic piture in Fig. 1(d)].
although, interestingly, have been recently observed by ThedifferencesinconductancebetweentheFandtheAF
2
FIG.1: (a)ArmchairgraphenenanoribbonofwidthN (where
N is thenumberof dimerlines) with twoHatoms (shown in
green) adsorbed in the middle of the ribbon at a distance d
fromeachother. Theinsetshowsadetailoftheadsorptionge-
ometry. (b)Pictorical representationoftheone-orbitaltight- FIG.2: Pictoricalviewoftheantiferromagneticstate(a)and
bindingmodelwherethepresenceofHatomsissimulatedby the magnetic-field driven ferromagnetic state (b) where the
removing sites in a head-to-head configuration. (c) Same as magnetic moments localized around two hydrogen atoms are
in (b), but for the opposite ordering (tail-to-tail). (d) Same depicted by red arrows (the orientation of the arrows with
asin(b),butforafinitesemiconductingAGNRconnectedto respect to the graphene plane is arbitrary since spin-orbit
metallic nanoribbons at theedges. coupling is neglected here).
statescanbesubstantialandtranslateintoamagnetore- nˆ = cˆ+cˆ are the number operators associated to
iη iη iη
sistive response as large as 100% for distances between each π orbital with spin η. These two different lev-
the H atoms of the order of few nanometers. Practical els of approximation to the electronic structure have
implicationsoftheseresultsarediscussedinSec. IVand beenshowntoyieldsimilarresultsfornanoribbons24and
a brief summary is presented in Sec. V nanographenes13 where a full passivation by H of the σ
bonds is assumed.
The bulk adsorption geometry of a H atom has been
II. COMPUTATIONAL DETAILS obtainedbyrelaxingtheCatombondedtotheHandthe
nearestCatomsuntilthecharacteristicsp3hybridization
Forthecalculationoftheelectronicstructureofhydro- isobtained[seeinsetinFig. 1(a)]. Thebondingbetween
genatedAGNR’sweusebothab initio techniqueswithin a H atom and a C atom results in the effective removal
the local spin density approximation (LSDA), aided by of the π orbital from the low energy sector, so that the
the CRYSTAL03 package23, and a one-orbital (π) first- H adsorption is simulated by simply removing a site in
neighbortight-binding modelwherethe electronicrepul- the one-orbitalmean-field Hubbard model [see Fig. 1(b-
sionis treated by means ofa Hubbard-like interactionU d)]. Ourresults,asshownbelow,providefurthersupport
in the mean field approximation: to the use of the Hubbard model in graphene systems,
as an alternative to the computationally more demand-
Hˆ = tcˆ†cˆ +U (nˆ hnˆ i+nˆ hnˆ i)−U hnˆ ihnˆ i. ing LSDAandextendthe rangeofapplicabilityofLieb’s
X i j X i↑ i↓ i↓ i↑ X i↓ i↑
theorem25 to a wider set of situations.
i,j i i
(1) To calculate the transport properties we use the
The first term represents the kinetic energy with first standard Green’s function partitioning method as im-
neighbors hopping t between π orbitals. The remain- plemented, e.g., in the quantum transport package
ing terms account for the electronic interactions where ALACANT26. To this purpose, the infinite system is
3
divided into three parts, namely a central region (C),
containing the H atoms, which is connected to the right
(R) and left (L) semi-infinite clean leads. The Hamil-
tonian matrix describing the whole system is then given
by
H =H +H +H +V +V (2)
C R L LC RC
where H , H and H are the Hamiltonian matrices of
C L R
the central region, the left and the right lead, respec-
tively. V and V represent the coupling between
LC RC
the central region and the leads. In general, the non-
orthogonalityofthe basissetmustbetakenintoaccount
when writing the Green’s function of the central region:
G (E)=[ES −H −Σ (E)−Σ (E)]−1. (3)
C C C L R
The self-energies of the left (Σ ) and right (Σ ) leads
L R
accountfortheinfluence oftheseonthe electronicstruc-
ture of the central part and S is the overlap matrix in
C
this region. For the calculation of the conductance we
use the Landauer formula, G(E) = e2T(E). The trans-
h
mission function T can, in turn, be obtained from the
expression:
T(E)= Tr G† (E)Γ (E)G (E)Γ (E) , (4)
X h C R C L i
η
η
with Γ = i Σ −Σ† . Notice that since we
R(L) (cid:16) R(L) R(L)(cid:17)
areinterestedincollinearmagneticsolutions,allthema-
trices carry the spin index η, which we have not made
explicit in previous equations. All the terms in Eq. 2
must be obtained self-consistently either from a periodic
boundary condition calculation, e.g., using CRYSTAL03
in the case of the LSDA calculations, or following the
methodology in Ref.9 in the case of the Hubbard model.
TheFermienergyissettozeroand,inbothcases,global
chargeneutralityinallregionsisimposedbyshiftingthe
onsite energies as necessary.
III. RESULTS
A. Energetics
We first examine the energetics of the F (S = 1) and
AF (S = 0) states as a function of the mutual distance
d between H atoms. We first choose a semiconducting FIG.3: (a)TotalenergyreferredtoE0 [lowestenergyincase
AGNR of width N =9, where N is the number of dimer (b)]asafunctionofthedistancebetweenHatomsforthefer-
lines acrossthe ribbon, and restrictourselvesto the case romagnetic (dashed) and the antiferromagnetic (solid) state
when placed in the middle of the ribbon. Upper inset: Pic-
ofHatomsplacedondifferentsublattices[seeFig. 1(a)].
tureofananoribbonwithtwoHatoms. Lowerinset: Energy
The reasonfor this choice is three-fold. The AB (or BA)
differencebetweenbothstatesandextrapolationtolargedis-
configurations are always energetically preferred to the
tances (solid line). The vertical line in the inset denotes the
AAorBBconfigurationsforsimilardistancesbetweenH
distanceabovewhichtheenergydifferencebecomeslessthan
atoms22. Second, and most importantly for the purpose 1 meV. (b) Same as in (a), but for both H atoms near the
ofthiswork,themagneticstateoftheAB(orBA)config- sameedge. (c)Sameasin(a),butforHatomsplacedonop-
uration can be tuned by a magnetic field. Furthermore, positeedges. Herethewidthoftheribbonislarger(N =13)
asbrieflymentionedintheintroduction,evenifenergetic and E0 is theminimum energy among all the AFsolutions.
4
considerations are left aside, the AB (or BA) configura-
tion represents the simplest case of a random ensemble
of H atoms which, in average, will equally populate the
two graphene sublattices.
In Fig. 3(a) we plot the LSDA total energy for H
atoms in the middle of the AGNR as a function of d.
The S =0state is alwaysthe groundstate. This implies
antiferromagnetic coupling, except at short distances
for which the local magnetization, quantified through
Σ = hm i2, vanishes altoghether [see Fig. 4(c)].
pPi i
The quenching ofthe magnetizationis easily understood
in terms of the formation of a spin singlet27. At the
minimum distance for which Σ 6= 0, the energy differ-
ence presents a maximum and decays exponentially for
larger d [see inset in Fig. 3(a)]. As d → ∞ the spin
clouds do not interact anymore and both F and AF so- FIG. 4: Magnetic moments (whose magnitude is represented
lutions tend to have the same energy. In summary, for by the size of the circles) on individual C atoms when the
H atoms are placed head-to-head in the middle of an N = 9
any distance between H atoms the groundstate presents
S = 0, following Lieb’s theorem25, but the overall spin armchair graphene nanoribbon at a distance d = 32.0 ˚A[(a)
ferromagnetic state and (b) antierromagnetic state] and at a
texture strongly depends on their mutual distance. We
distanced=15.4˚A closetotheedge[(d)ferromagneticstate
note that it also depends on the ordering (AB or BA)
and (e) antiferromagnetic state]. Panels (c) and (f) showthe
of the H atoms. Whereas in bulk the spin cloud associ-
magnetization(seedefinitionofΣintext)asafunctionofthe
ated to the H atom would be invariant under rotations distance between H atoms.
of 120 degrees, in a nanoribbon there is a preferential
direction along the ribbon axis which is different for H
atoms located on A and B sublattices [see Fig. 1(b-c)]. semiconducting AGNR. Placing the atoms on different
Werefertothepreferentialdirectionastheheadandthe edges allows us now to explore the energetics of differ-
tail to the opposite one. Thus, a head-to-head coupling ent magnetic coupling orientations (tail-to-tail for d>0
(AB) is expected to be much stronger than a tail-to-tail and head-to-head for d < 0) at a reasonable computa-
coupling (BA). The magnetization clouds are shown for tional cost (d is now the longitudinal distance between
an AB (or head-to-head) case in Fig. 4(a-b). It is easy H atoms). As expected, due to the strong anisotropy
to appreciate the strong directionality just alluded to. of the spin texture, the magnetic coupling strongly de-
As a consequence, when reversing the ordering of the pendsonthe orderingoftheatomsandnotonlyontheir
H atoms to a BA (or tail-to-tail) configuration [see Fig. mutual distance. While for d < 0 the energy difference
1(c)], these do not couple magnetically at any distance, between the AF and F states is large, for d>0 this dif-
exceptwheninverycloseproximityforwhichΣ=0. All ference decreases substantially and does not depend too
these results are similar to the ones reported using the much on d for the values considered. This asymmetry
one-orbital mean-field Hubbard model27. is easily understood since for d> 0(d< 0) the spin tex-
In Fig. 3(b) we present the LSDA total energy of the turesapproximatelycoupleinatail-to-tail(head-to-head)
F and AF states when the H atoms are placed near one manner. As shown below, this may have important ex-
of the edges. Again the ground state is the AF state for perimental consequences.
any distance. The proximity to the edge decreases the
localization length of the spin texture, increasing Σ [see
Fig. 4(d-e)],andthereby,decreasingthecriticaldistance B. Magnetoresistance
belowwhichthemagnetizationdisappears[seeFig. 4(f)].
The perturbation of the edge modifies the spin texture We now turn our discussion to the implications these
around each atom and, contrary to the previous case, results may have on the electrical transport. Under the
when reversing the ordering to a tail-to-tail configura- influence of a magnetic field, the hydrogenated AGNR
tion, the H atoms couple magnetically in a finite range behaves like a diluted paramagnetic semiconductor29 for
ofdistances(seebelow). Allenergies,includingthosede- smallconcentrationsofH.Atlargeconcentrations,when
picted in Fig. 3(a), are referred to E0, the lowest energy the spin density is zero everywhere, the influence of the
solution for H atoms close to the edge, corresponding field can only give rise to a minor diamagnetic response.
tothesmallestdistance. Notethatforthesamedistance Atintermediateconcentrations,wherethemagnetization
betweenHatomsthetotalenergyisalwayssmallerwhen clouds induced by the H atoms interact with each other,
these areclosertothe edge,whichreflectsthatthe bind- one can switch from the AF to the F state by apply-
ing energy is higher there by approximately 1 eV28. ing a sufficiently strong magnetic field. In analogy with
Finally we present in Fig. 3(c) the case where the the H molecule,where switching fromthe singletto the
2
H atoms are placed on opposite edges for an N = 13 triplet state modifies the orbital part of the wavefunc-
5
FIG.5: Spinresolvedtransmissionasafunctionofenergyfor
the ferromagnetic state of an armchair graphene nanoribbon
of width N =9 with two H atoms adsorbed in themiddle at
a mutual distance d = 32 ˚A calculated in the (a) local spin
density approximation and (d) with a one-orbital mean-field
Hubbard model. (b) and (e) panels show the same, but for
theantiferromagnetic state. Theresultingmagnetoresistance
in both approximations is shown in (c) and (f).
FIG.6: Tunnelingmagnetoresistanceforfourdifferentatomic
tion, here the electronic structure will be indirectly af- H configurations. Two H atoms in a head-to-head configura-
fected by the field even if its direct influence on the or- tion located in the middle of an armchair graphene nanorib-
bital wave function is neglected. This change reflects in bon of length L = 73.7 ˚A with a width of N = 9 (a) and
N = 15 (b), and two H atoms located near one edge in a
the spin-resolved conductance as shown in Fig. 5(a-b)
head-to-head (c) and in a tail-to-tail (d) configuration for a
for d = 32 ˚A (the dashed line represents the conduc-
ribbon width of N =9 and length of L=57.0 ˚A.
tance of the clean AGNR). The different total transmis-
sion for the F and AF solutions, resulting from adding
the twospinchannels,resultsinapositive magnetoresis-
tance (MR), MR = G −G /G +G , at energies [see Fig. 1(d)]. (Note that in the one-orbital mean-field
F AF F AF
near the bottom and top of the conduction and valence Hubbard approximation AGNR’s of width N = 3m−1,
bands, respectively [see Fig. 5(c)]. Similar results are where m is an integer, are metallic, being semiconduct-
obtained(notshown)fordifferentintermediatedistances ing otherwise). In what follows and in the light of the
between H atoms. The right panels in Fig. 5 show the previous results, we restrict ourselves to the one-orbital
resultsobtainedfromtheone-orbitalmean-fieldHubbard mean-field Hubbard model.
model for U =t=3 eV. Apart from the recovery of the In our proposed AGNR heterostructure the difference
particle-hole symmetry, the results are remarkably simi- in the zero-bias tunnel conductance between the F and
lar, validating the use of the latter model for transport AF states is now responsible for the appearance of tun-
calculations in hydrogenated graphene. neling MR (TMR), as shown in Fig. 6. Notice that
One should note, however,that since the chemical po- unlike conventional TMR, where the magnetic elements
tentialliesinthe middle ofthe gap,the energyrangesat areintheelectrodes,inourproposalmagnetismisinthe
which MR couldmanifest itself are notrelevantin linear barrier. Panels (a) and (b) in Fig. 6 correspond to H
responsetransportforinfiniteAGNR’s. Afinitebiascal- atoms placed in the middle of a semiconducting AGNR
culation may reveal the MR obtained at those energies, of length L = 73.7 ˚A and width N = 9 and N = 15,
but this is a non-trivial task beyond the scope of this respectively. Both cases refer to head-to-head configura-
work. The application of a gate voltage is not a prac- tions. The obtained TMR changes sign with d, but it
tical alternative either since it implies a deviation from is alwaysnegligibly small. On the contrary,when placed
chargeneutrality which would fill up or empty the local- neartheedge[Fig. 6(c)],theTMRispositiveandreaches
ized states hosting the unpaired spins and kill the mag- muchlargervalues. ThisresultisforanN =9,L=57.0
netization. Instead,weproposetoexplorethepossibility ˚A AGNR. When the H atoms are now placed in a tail-
of MR at zero bias by considering finite AGNR’s con- to-tail configuration near the edge for the same AGNR,
nected at the ends to conductive graphene. This is done the TMR becomes negative and reaches values of up to
inourcalculationsbyconsideringametallic AGNRwith 100%[see Fig. 6(d)]. As a final example we showin Fig.
a narrower section in the middle of appropriate width 7 the TMR for the case where the H atoms are placed
6
0 zipped metallic carbon nanotubes31, as suggested
-10 N=15 in Ref.32. A third possibility consists in gating
) selectively a semiconducting AGNR as previously
%
( -20 done for nanotubes33.
R
M -30
T • From the LSDA calculations we note that the en-
-40 ergy difference between the F and AF states, or
-500 exchange coupling energy, can be as large as tens
ofmeVforshortdistances,particularlyforhead-to-
N=21
-2
headconfigurations. However,asfarasmagnetore-
)
% -4 sistive properties is concerned, exchange coupling
(
R energies above 1 meV are of no practical use since
M -6
theZeemanenergygainperspininamagneticfield
T
-8 B is0.067gµ B meVT−1andfieldshigherthan10
B
Tarehardlyaccesibleinthelab. TheAF-Ftransi-
-10
tionisthereforepracticallyforbiddenbelowd≈65
0 N=27
˚A andd≈40˚A intheexamplesshowninFig. 3(a)
)
% -0.5 and (b), respectively. At larger distances the fields
(
R necessary to induce the AF-F transition can be as
M -1
small as needed, but also is the associated MR as
T
-1.5 shown in Fig. 6(a) and (c). This is not a problem,
however,when the H atoms are coupledtail-to-tail
-2
-5 0 5 10 15 20 as, e.g., in the cases where they are located on op-
d (Å) posite edges. There,as shownin Fig. 7, a TMR as
highas≈50%canbe obtainedatreasonablysmall
exchange coupling energies [see inset in Fig. 3(c)].
FIG.7: Tunnelingmagnetoresistanceasafunctionofthelon-
gitudinal distance between H atoms for three different arm-
• The third caveat relates to the stability of the
chair graphene nanoribbon widths. The H atoms are located
atomic configurations. Given the tendency for
on opposite edges of theribbon.
H atoms to form lowest-energy non-magnetic
aggregates28, the existence of magnetically active
on the opposite edges of an L=57.0 ˚A AGNR for three dilute ensembles ofadsorbedHatoms requirescer-
different widths. Large (and negative) values are also tain conditions. For instance, working at reason-
obtained for the tail-to-tail configurations in the range able low temperatures prevents H atoms from di-
of distances explored. As can be appreciated, the TMR fusingafteradsorption34. Anotherpossibilityisone
logically decreases with the ribbon width. intrinsic to AGNR’s: The binding energy is larger
forHatomsclosetotheedgethaninthebulkofthe
nanoribbon. Relatedtothisisthefactthatthemo-
IV. PRACTICAL CONSIDERATIONS bility ofHdecreasessignificantlynearthe edge,re-
ducing the possiblity of formation of lowest-energy
non-magnetic aggregates near the edge28. As we
A critical assesment of the practical consequences of
have shown, it is precisely the hydrogenation near
the results presented is due at this point:
the edge that favors the appearance of sizable MR
• First, the TMR results presented above have been and thus an actual experimental verification.
obtained for a particular type of AGNR het-
erostructure and rely on the existence of metal-
lic AGNR’s. Calculations where the relaxation of V. SUMMARY
the atomic structure on the edges is considered re-
veal that a gap always opens even for the nom- In summary, we have shown, as a proof of princi-
inally metallic armchair nanoribbons30, compro- ple, that hydrogenated AGNR’s at small concentrations
mising the applicability of these nanoribbons as can exhibit magnetoresistive properties without invok-
electrodes. Graphene-based metallic leads, how- ing purely edge-related physics. Our results, which are
ever, can be found in recent literature. For in- deeplyrootedinLieb’stheorem,providefurtherevidence
stance, zigzag graphene nanoribbons with passi- for the wide applicability of this theorem beyond the
vated edges are metallic17. The physics described bipartite lattice Hubbard model for which was demon-
in this work does not rely on the edge termination strated. We have also shown that hydrogenation near
ofthenanoribbonsandcouldbereproducedinfully theedgepresentsadvantageswithrespecttobulkhydro-
passivated zigzag nanoribbons as well. Another genationbothfromenergeticandelectronicstandpoints.
alternative can be based on using partially un- Thisaspect,especifictonanoribbons,mightfavortheuse
7
of these ones for spintronics applications, as opposed to CSD2007-00010). D.S. acknowledges financial support
using large flakes of graphene. Although our conclusions from Consejo Superior de Investigaciones Cient´ıficas
are based on the simplest case of two H atoms, noth- (CSIC).
ingseemstopreventmagnetoresistancefromoccuringfor
emsembles of many H atoms. This, however, still needs
to be supported by a more extensive statistical analysis
which is beyond the scope of this work.
Acknowledgments
This work has been financially supported by MICINN
of Spain (Grants MAT07-67845 and CONSOLIDER
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