Complex Structure of Triangular Graphene: Electronic, Magnetic and Electromechanical Properties

0.74 MB·English

by  Motohiko Ezawa

Complex Structure of Triangular Graphene:
Electronic, Magnetic and Electromechanical Properties
Motohiko Ezawa
Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan
We have investigated electronic and magnetic properties of graphene nanodisks (nanosize triangular
graphene) as well as electromechanical properties of graphene nanojunctions. Nanodisks are nanomagnets
madeofgraphene,whicharerobustagainstperturbationsuchasimpuritiesandlatticedefects,wheretheferro-
1 magneticorderisassuredbyLieb’stheorem.Wecangenerateaspincurrentbyspinfilter,andmanipulateitby
1 aspinvalve,aspinswitchandotherspintronicdevicesmadeofgraphenenanodisks. Wehaveanalyzednan-
0 odiskarrays,whichhavemulti-degenerateperfectflatbandsandareferromagnet.Byconnectingtwotriangular
2 graphenecorners,weproposeananomechanicalswitchandarotator,whichcandetectatinyanglerotationby
measuringcurrentsbetweenthetwocorners. BymakinguseofthestraininducedPeierlstransitionofzigzag
n
a nanoribbons,wealsoproposeananomechanicalstretchsensor,inwhichtheconductancecanbeswitchoffby
J ananometerscalestretching.
Keyword: graphene, nanoribbons, nanodisks, graphene-nanodisk array, quasiferromagnet, spintronics,
9
1 nanomechanics,electromechanics

]
l I. INTRODUCTION
l
a
h
- Graphene,whichisaone-layerthickhoneycombstructure
s ofcarbon,isanamazingmaterial1.Electronsexhibithighmo-
e
m bility and travelmicrondistances withoutscattering at room
temperature. It is a very thin and strong material, showing
.
t veryhighthermalconductivity.Grapheneisnowamaintopic
a
m ofnanoscience.
Much attention has been focused on graphene nanorib- FIG.1. (a)Geometricconfigurationofatrigonalzigzagnanodisk.
-
d bons, which is a one-dimensional ribbon-like derivatives of ThenanodisksizeisdefinedbyN =Nben−1withNbenthenumber
n ofbenzenesononesideofthetrigon.Here,N =5.TheA(B)sites
graphene.Theyhavevariousbandstructuredependingonthe
o onthelatticeareindicatedbyreddots(whilecircles). Theelectron
edge and width. In particular, zigzag gaphene nanoribbons
c densityisfoundtobelocalizedalongtheedges. (b) Thenanodisk-
[ showedgeferromagnetismduetoalmostflatlow-energyband leadsystem. Ananodiskisconnectedtotherightandleftleadsby
at the Fermilevel. There are a profusionof paperson them,
tunnelingcoupling.Itmayactasaspinfilter.
1 among which we cite some of early works2–4. Another ba-
v
sic elementof graphenederivativesis a graphenenanodisk5.
2
1 It is a nanometer-scaledisk-likematerialwhich has a closed II. ENERGYSPECTRUM
6 edge. It may be considered as a giant molecule made of
3 aromatic compound. It is possible to manufacture them by
Wecalculatetheenergyspectrumofthenanodiskbasedon
. etchingagraphenesheetbyNinanoparticles6. Amongthem,
1 thenearest-neighbortight-bindingmodel,whichhasprovedto
0 trigonalzigzag nanodiskshave a novelelectric propertythat describeaccurately the electronic structureof graphene, car-
1 thereexisthalf-filledzero-energystatesinthenon-interacting bon nanotubes, graphene nanoribbons and other sp2 carbon
1 regime, as was revealed first by the tight-binding model5
materials. TheHamiltonianisdefinedby
: and then by first-principle calculations7–9. Various remark-
v
Xi asibvleelyprinopaesrteireisesooffnwaonrokdsi5s,k10s–1h2a.vNeabneoedniskinivseasltsigoarteefdererxetdento- H0 = εic†ici+ tijc†icj, (1)
r as nanoisland7, nanoflake9,13,14, nanofragment15 or graphene Xi hXi,ji
a
quantumdot16,17. where ε is the site energy, t is the transfer energy, and c†
i ij i
Nanoribbons and nanodisks correspond to quantum wires is the creation operator of the π electron at the site i. The
and quantum dots, respectively. They are candidates of fu- summation is taken over all nearest neighboring sites hi,ji.
turecarbon-basednanoelectronicsandspintronicsalternative Owing to their homogeneous geometrical configuration, we
to silicon devices. A nanoribbon-nanodisk complex can in maytakeconstantvaluesfortheseenergies,ε =ε andt =
i F ij
principle be fabricated, embodying various functions, only t≈2.70eV.Thereexistsoneelectronperonecarbon,andthe
byetchinga graphenesheet. Furthermore,grapheneis com- band-fillingfactoris1/2. Then,thediagonaltermyieldsjust
mon material and ecological. In this paper, exploring elec- a constant, ε N , and can be neglected in the Hamiltonian,
F C
tronic,magneticandelectromechanicalpropertiesoftrigonal whereN isthenumberofcarbonatoms.
C
zigzaggraphenenanodisks[seeFig.1],weproposesomeap- WedefinethesizeN ofananodiskbyN =N −1,where
ben
plicationofnanodisk-nanoribboncomplextonanoelectronics, N isthenumberofbenzenesononesideofthetrigonasin
ben
spintronicsandelectromechanicsdevices Fig.1(a).Itcanbeshown5thatthedeterminantassociatedwith
2
theHamiltonian(1)hassuchafactoras
det[εI −H(N )]∝εN, (2)
C
implying N-fold degeneracy of the zero-energy states. The
gapenergyisaslargeasafeweVfornanodiskswithsmallN,
where it is a goodapproximationto investigatethe electron-
electroninteractionphysicsonlyinthezero-energysector,by
projecting the system to the subspace made of those zero-
energystates.AsweshallseeinSectionV,theapproximation FIG. 2. Probability density flow for the zero-energy states in the
remainstobegoodevenforthosewithlargeN. nanodiskwithN = 7. Therepresentationofthetrigonalsymmetry
group C3v is indicated in the parenthesis. A vortex appears at the
centerofmassforthestatebelongstotheE(doublet)representation.
Thewindingnumberatthecenteris2inthestate|k+i.
III. TRIGONALSYMMETRY n
ThesymmetrygroupofatrigonalnanodiskisC3v,whichis groupedaccordingto the representationof the trigonalsym-
generatedbythe2π/3rotationc3andthemirrorreflectionσv. metrygroupC3v asfollows,
Ithastherepresentation{A1,A2,E}. TheA1 representation
TiusnhidenevAra2σrivrae.nptTruehnseedneEtrattrhieoepnrreoisstaeintnivtoaantriioca3nntaauncndqdutehirreecsm3±iarn2rdoπra/rn3etflipsehycamtsiomensehtσrivifct. AA21 ((ssiinngglleett)):: ||kknn00ii−+||−−kknn00ii,, ) for kn0 = 63nN+a3π,
6n±1
under the 2π/3 rotation. The A1 and A2 are 1-dimensional E (doublet): |k±i, |−k±i, for k± = π.
representations(singlets)andtheE isa2-dimensitionalrep- n n n 3Na
(6)
resentation(doublet).Thesepropertiesaresummarizedinthe
Thezero-energystateisindexedbythequantizedwavenum-
followingcharactertable:
beras|kαiwith(6).
n
Toseethemeaningofthewavenumberkαmoreindetail12,
C3v e 2c3 3σv wehavecalculatedtheprobabilitydensityflnow,
A1 1 1 1 (3)
A2 1 1 −1 Ai(x,y)=−iψ∗(x,y)∂iψ(x,y) (7)
E 2 −1 0
forstates|kαi,whichweshowforthecaseofN =7inFig.2.
n
Weobserveclearlyatextureofvortices.Thesevorticesmani-
The zero-energy sector consists of N orthonormal states.
festthemselvesasmagneticvorticesperpendiculartothenan-
We are able to index them by the wave number k along
odiskplanewhentheelectromagneticfieldsarecoupled.
the edge. It is a continuousparameter for an infinitely long
ThetotalwindingnumberN iscalculatedby
grapheneedge. Accordingto the tight-binding-modelresult, vortex
theflatbandemergesfor
1 A (x,y)
N = dx i =N +m−1, (8)
vortex 2π i |ψ(x,y)|2
−π ≤ak <−2π/3 and 2π/3<ak ≤π. (4) I
with m = 0,1,2,··· ,⌊(N −1)/2⌋ in the size-N nanodisk,
We focusonthe wavefunctionψ(x,y)atoneofthe A sites
where ⌊a⌋ denotes the maximum integer equal to or smaller
onanedge,andinvestigatethephaseshiftwhenwestepover
than a. We find N = 3n for k0, N = 3n+1 for
totheneighboringsite[seeFig.1(a)].ThereareN linksalong vortex n vortex
k− and N = 3n+2 for k+. The wave functionsare
one edge of the size-N nanodisk, for which we obtain the n+1 vortex n
classifiedintermsofmoduloofthetotalwindingnumber:The
phaseshiftNak,whereaisthespacingbetweentheneighbor-
wavefunctionbelongsto theE-representationandhaschiral
ingAsitesandk isthewavenumberalongtheedge. Onthe
edgemodeforN ≡ 1,2(mod3),andbelongstotheA-
otherhand,thephaseshiftisπ atthecorner. Thetotalphase vortex
representationandhasnon-chiraledgemodeforN ≡ 0
shiftis3Nak+3π,whenweencirclethenanodiskonce. By vortex
(mod3).Thewindingnumberofthevortexatthecenterofthe
requiringthesingle-valuenessofthewavefunction,itisfound
nanodiskis0,1,2inthestate|k0i,|k−i,|k+i,respectively.
tobequantizedas n n n
ak =±[(2n+1)/3N +2/3]π, (5)
n
IV. QUASIFERROMAGNET
whereit followsthat0 ≤ n ≤ (N −1)/2fromthe allowed
region(4)ofthewavenumber. The total spin of the groundstate is determinedby Lieb’s
Wemaygroupthestatesaccordingtothetrigonalsymme- theorem. The total spin is given by the difference of the
try (3). With respectto the rotation there are three elements A site and the B site, S = 21|NA−NB|, where NA and
c03,c3,c23,whichcorrespondto1,e2πi/3,e4πi/3. Accordingly, NB are number of A site and B site. Here, S = 12N in
the phase shift of one edge is 0, 2π/3, 4π/3. The state is the size-N nanodisk, where N = (N +1)(N +6)/2 and
A
3
FIG.4. Thedensityofstatesof(a)graphene,(b)graphenenanorib-
FIG.3. Thermodynamical propertiesofthenanodisk-spin system. bonand(c)graphenenanodisk. Thehorizontalaxisistheenergyε
(a)ThespecificheatC inunitofkBN. (b)Thesusceptibilityχin inunitoft.
unit of Sg. The sizeis N = 1,2,22,···210. Thehorizontal axis
standsforthetemperatureT inunitofJN/k .Thearrowrepresents
B
thephasetransitionpointTcinthelimitN →∞. as illustrated in Fig.4. Hence, together with spin degrees of
freedom,itbehavesas
D(ε)=2cN |ε|+2Nδ(ε), (9)
N =(N+2)(N+3)/2.Henceweexpectananodisktoact C
B
as a ferromagnet. The ferromagnetic ground state is robust
withacertainconstantfactorc. Thelineartermisduetothe
against perturbationssuch as randomnessand lattice defects
bulk states, and the Dirac delta function term is due to the
sinceitisassuredbyLieb’stheorem.Thisfeaturebringsouta
edge states. The important point is that the edge-state peak
remarkablecontrastbetweennanodisksandnanoribbons.The
isclearlydistinguishedfromtheDOSduetothebulkpart. It
ferromagneticorderis fragiledue to the lack of Lieb’s theo-
isenoughtotakeintoaccountonlythezero-energysectorto
reminthecaseofnanoribbons.
analyzephysicsneartheFermienergy,sincethecontribution
We investigate ferromagnetic properties by introducing
fromtheedgestatesisdominant.
Coulomb interactionsinto the zero-energysector5. We have
Wecalculatethemagnetizationofananodiskwhenitssize
calculated specific heats and susceptibilites at temperature
islarge.WestartwiththeHubbardHamiltonian,
T in Fig.3. There appear singularities in thermodynamical
quantities as N → ∞, which represent a phase transition H = ε(k)c†kσckσ +U c†k+q↑c†k′−q↓ck′↓ck↑. (10)
atTc betweentheferromagnetandparamagnetstates, where kσ kk′q
X X
T = JN/2k . ForfiniteN, therearesteepchangesaround
c B
Tc, though they are not singularities. It is not a phase tran- Let hn↑i,hn↓i be the average numbers of the up and down
sition. However, it would be reasonable to call it a quasi- spins. Themagnetizationisgivenbyhmi = hn↑i−hn↓i. It
phasetransitionbetweenthequasiferromagnetandparamag- isdeterminedself-consistentlybytherelation
net states. Such a quasi-phase transition is manifest even in
finitesystemswithN =100∼1000. hmi= dεD(ε)[f(ε−∆)−f(ε+∆)], (11)
Thespecificheattakesnonzero-valueforT >Tc,asshown Z
inFig.3(a),whichiszerointhelimitN → ∞. Theresultin- intermsoftheFermidistributionfunction
dicates the existence of some correlationsin the paramagnet
state. On the other hand, the susceptibility χ always shows f(x)=(exp[(x−µ)/kBT]+1)−1 (12)
theCurie-Weisslowχ ∝ 1/T nearT = 0,andexhibitsalso
and
a behavior showing a quasi-phase transition at T = T , as
c
showninFig.3(b). Inthefinitesystem,theexpectationvalue U 1
∆= hmi+ h. (13)
ofSz,tot isalwayszerobecausethereisnospontaneoussym- 2NC 2
metrybreakdowninthefinitesystem,andthebehavioristhat
Substitutingtheformula(9)intotheStornerequation(11),we
ofparamagnet.
obtain
β∆ 1 π2
V. MAGNETISMOFLARGENANODISKS hmi=Ntanh 2 +cNc ∆2+ β2 3 +4Li2 −e−β∆ ,
(cid:20) (cid:26) (cid:27)(cid:21)
(cid:0) (14(cid:1))
For large N nanodisks, the band gap decreases inversely with the dilogarithmfunction Li2(x). It is difficult to solve
proportionaltothesize.Onemaywonderifouranalysisbased thisequationforhmiself-consistentlyatgeneraltemperature
only on the zero-energy sector is valid. Indeed, the size of T.Weexaminetwolimits,T →0andT →∞.
experimentallyavailablenanodisksisaslargeasN = 100∼ Forthe zerotemperature(T → 0) we obtainthe magneti-
1000. We wish to arguethatour analysisbasedon the zero- zationas
energysector isessentially correct, evenif the size N of the U2 cUh
nanodiskislargeandthebandgapbecomesverynarrow. hmi=N +c hmi2+ hmi+O(h2). (15)
4N 2
C
Near the Fermi energy, the density of states (DOS) D(ε)
consists of that of the bulk graphene and an additional peak Because |hmi| ≤ N, itfollowsthathmi = N +O(1). The
at the zero-energystates due to the edge states for N ≫ 1, contributionfromthebulkgivesanegligiblecorrectiontothe
4
FIG.5. (a) Anelectrontunnels fromtheleft leadtothenanodisk
andthentotherightlead. Thesystemisareminiscenceofametal- FIG. 6. The spin valve is made of two nanodisks with the same
ferromagnet-metal junction. (b)Only electronswiththesamespin size, whichareconnected toleads. Applying anexternalmagnetic
directionasthenanodiskspincanpassthroughthenanodiskfreely. field, we control the spin direction of the first nanodisk to be |θi
As a result, when we apply a spin-unpolarized current to the nan- andthatofthesecondnanodisktobe|↑i. Theincomingcurrentis
odisk, theoutgoing current isspinpolarizedtothedirectionof the unpolarized,buttheoutgoingcurrentispolarizedanditsmagnitude
nanodiskspin.Consequently,thissystemactsasaspinfilter. canbecontrolledcontinuously. Thisactsasaspinvalve.
total magnetization. Hence the magnetization is hmi = N, quasiferromagnethas an intermediate nature: It can be con-
and the ground state is fully poralized whenever U 6= 0. trolled by a relatively tiny currentand yet holds the spin di-
Ferromagnetism occurs irrespective of the strength of the rection for quite a long time5. Taking advantage of these
Coulomb interaction. The magnetization is propotional not propertieswehavealreadyproposedelsewhere11someappli-
toN butN. Inthissencethegroundstateofnanodiskisnot cations of graphene nanodisk-lead systems to spintronic de-
C
bulkferromagnetbutsurfaceferromagnet,whichisconsistent vices. Theyare spinfilter, spinmemory,spinamplifier, spin
withthepreviousresult. valve,spin-field-effecttransistor,spindiodeandspinswitch,
We nextinvestigatethe high temperatuerlimit (T → ∞). among which here we make a consice review of spin filter,
UsingtheTaylorexpansionofthedilogarithmfunction, spin valve and spin switch. We newly propose nanodisk ar-
raysandnanomechanicalswitch.
π2 β2∆2 β3∆3 Spinfilter: We consideralead-nanodisk-leadsystem[see
Li2 −e−β∆ =−12 +β∆log2− 4 + 24 +··· , Fig.1(b)],whereanelectronmakesatunnellingfromtheleft
(cid:0) (cid:1) (16) lead to the nanodisk and then to the right lead. This sys-
wefind tem is a reminiscenceof a metal-ferromagnet-metaljunction
[see Fig.5]. If electrons in the lead has the same spin di-
β∆ 4∆ β∆3
hmi=Ntanh +cN log2+ +··· . (17) rection as the nanodisk spin, they can pass through the nan-
c
2 β 6 odisk freely. However, those with the opposite direction
(cid:20) (cid:21)
feelalargeCoulombbarrierandareblocked(Pauliblockade
Theleadingtermisthesecondterm,andhencethemaincon- [Fig.5(c)])11. As a result, whenwe applya spin-unpolarized
tributioncomesfromthebulk. Thesolutionisonlyhmi = 0
currenttothenanodisk,theoutgoingcurrentisspinpolarized
forwhich∆=0. Thereisnomagnetizationathightempera-
tothedirectionofthenanodiskspin. Consequently,thissys-
ture.
temactsasaspinfilter.
A comment is in order. We have assumed that the mag-
Spin valve: A nanodisk can be used as a spin valve, in-
netization axis is fixed and only longitudinal fluctuations of
ducing the giant magnetoresistance effect. We set up a sys-
themagneticmomentstakeplace. Ingeneral,spin-wave-like
temcomposedoftwonanodiskssequentiallyconnectedwith
fluctuationsaredominantattheedgesofgraphene18,because
leads [see Fig.6]. We apply external magnetic field, and
theyaregaplessGoldstonemodes.Onthecontrary,thereexist
control the spin direction of the first nanodisk to be |θi =
nogaplessGoldstonemodesingraphenenanodisks,because
cosθ |↑i + sinθ |↓i, and that of the second nanodisk to be
the edge is finite and closed. Furthermore, when the length 2 2
|0i = |↑i. We inject an unpolarized-spincurrentto the first
of the edge is very small, spin-wave-like fluctuations have a
nanodisk. Thespinoftheleadbetweenthetwonanodisksis
largegap.Hence,ourapproximationisvalidfornanodisks.
polarizedintothedirectionof|θi. Subsequentlythecurrentis
filtered to the up-spinoneby the second nanodisk. The out-
goingcurrentfromthesecondnanodiskisIout =Icosθ. We
↑ 2
VI. APPLICATIONFORSPINTRONICDEVICES cancontrolthemagnitudeoftheup-polarizedcurrentfrom0
toI byrotatingtheexternalmagneticfield. Thesystemactas
The nanodisk-spinsystem is a quasiferromagnet,which is aspinvalve.
an interpolating system between a single spin and a ferro- Spin switch: We consider a chain of nanodisksand leads
magnet. It is easy to control a single spin by a tiny current connectedsequentially[seeFig.7].Withoutexternalmagnetic
but it does not hold the spin direction for a long time. On field, nanodisk spins are oriented randomly due to thermal
the other hand, a ferromagnet is very stable, but it is hard fluctuations,andacurrentcannotgothroughthechain.How-
to control the spin direction by a tiny current. A nanodisk ever,whenandonlywhenauniformmagneticfieldisapplied
5
FIG.7. Achainofnonodisksandleadsactsasaspinswitch. With-
out external magnetic field, nanodisk spins are oriented randomly
due to thermal fluctuations, and a current cannot go through the
chain. However, as soon as auniform magnetic field is applied to
all nanodisks, the direction of all nanodisk spins become identical
andacurrentcangothrough.
FIG.9. (a)Illustrationofananomechanical junctionmadeoftwo
triangulargraphenecorners.Thereareonandoffstates,switchedby
arotationbetweentwocornes. (b)Illustrationofananomechanical
rotator made of two corners and a central rhombus rotating rather
freely.
The conductance is determined by the overlap integral of
π-electronsbetweentwocorners,whichisgivenby
FIG. 8. (a) Illustration of one-dimensional nanodisk arrays. (b)
Bandstructureofone-dimensionalnanodiskarrays.(c)Illustrationof
|hp cosθ+p sinθ |p i|=cos2θ. (18)
two-dimensionalnanodiskarrays. z y z
When the two planes are parallel (θ = 0), the overlaptakes
themaximumvalueandπ-electronscangothroughthe con-
to all nanodisks, the direction of all nanodisk spins become tact. This is the on state. When the two planesare orthogo-
identicalanda currentcangothrough. Thusthesystemacts nal (θ = π/2), the overlaptakes the minimumvalue and π-
as a spin switch, showing a giant magnetoresistance effect. electronscannotgothroughthecontact. Thisistheoffstate.
Theadvantageofthissystemisthatadetailedcontrolofmag- Theangleischangednanomechanically.Thesystemactsasa
neticfieldisnotnecessaryineachnanodisk. nanomechanicalswitch. Itcoulddetecttheangleverysensi-
Nanodisk arrays: We investigate nanodiskarrays, which tivelyandbeusefulfordetectnanomechanicaloscillations.
are materialswhere nanodisksare connectedin one-or two- Byconnectingtwonanomechanicaljunctions,wecancon-
dimentions.Thesestructureshavealreadybeenmanufactured structananomechanicalrotator[Fig.9(b)],wheretwocorners
by etching a graphenesheet by Ni nanoparticles6. We show aresuspendedmechanicallywhilethecentralrhombusrotates
an example of a trigonal-zigzag-nanodisk array sharing one ratherfreely.Thisstructuremaybeusefultodetectmolecular
zigzagedgeasinFig.8(a). We showthecorrespondingband dynamics. When molecules contact the rotator, they are de-
structureinFig.8(b). ItisintriguingthatthereareN-foldde- tectedbyrotatingitandchangingthe resistancebetweenthe
generateperfectflatbandsinthenanodiskwithsizeN. This twocorners.
factisconfirmedbyLeib’stheorem. Eachnanodiskhasspin Peierls instability: We connect two triangular corners
N/2 and makes ferromagnetic coupling between two nan- with a zigzag nanoribbon [see Fig.10(a)]. This structure
odisks. Inthesame waywecanmaketwo-dimensionalnan- has already been manufactured experimentally6. Polyacety-
odiskarrays. Itistobeemphasizedthattheyshowferromag- lene has the Peierls instability: Carbons with conjugate
netism, and not quasiferromagnetism, though they are made bondsare spontaneusly deformedinto alternating single and
ofnonmagneticmaterials. Theperfectflatbandwillberobust double bonds. We expect the Peierls instability to occur
evenwhenelectroninteractionsareintroduced. alsoingraphenenanoribbons[Fig.10(b)],becausegrapahene
Nanomechanical switch: We construct a nanomechan- nanoribbonisanaturalextensionofpolyacetylene3. (Wede-
ical switch contacting two graphene trigonal corners [see notethewidthofa nanoribbonbyW with W = 0forpoly-
Fig.9(a)].Weassumetheanglebetweentwocornersisθ. This acetylene and W = 1 for polyacene.) However, as we now
angleis tunedbyan externalmechanicalforce. The carbon- show,thePeierlsinstabilitywillnotoccurinnanoribbonswith
skeletonstructureismadeofσ-bonds,andisveryrigidexcept thewidthW >2. Onthecontrary,itispossibletoinducethe
forthisrotationaldegreeoffreedom. Peierlstransitionmanuallybystreachingananoribbon.Mak-
6
of the band sturacture for W ≥ 2. A tiny gap ∆ opens at
k = π. The logarithmplot of the band gap ∆ as a function
of the width W is shown in Fig.10(d). The gap decreases
exponentiallyasafunctionofthewidth,∝10−1.7W. Thegap
ofthecaseW =1(polyacene)is49meV,andthatofthecase
W =2is0.8meV.
Wenextcomparetheenergygainfromthisgapwiththeen-
ergycostfromtheelasticenergyofalatticedeformation.The
ground-state energy difference between distorted and undis-
torted structures is very tiny in polyacene20. On the other
hand, the elastic energycost is proportionalto the widthW,
andhencetheelasticenergycostbecomeslargerthanthegap
energygainforwidernanoribbons.ThePeierlsinstabilitywill
notoccurspontaneouslyinnanoribbonswithW >2.
We have also estimated how the band gap ∆ depends on
the externalforce. The transfer integralschange proportion-
ally to the external force. For simplicity we have set t1 =
(1−0.06λ)t and t2 = (1+0.08λ)t. It takes about10GPa
fordeformation0.01tforgraphene,whileittakesabout1GPa
fordeformation0.01tfornarrowgraphenenanoribbons21. We
FIG. 10. (a) Illustration of a configuration where a single ziazag
showthelogarithmplotoftheλdependenceofthebandgaps
nanoribbon is connected by two triangular graphene corners. (b)
inFig. 10(e).
Illustration of the Peierls instability of graphene nanoribbons with
W = 1 and W = 3, where W represents the width. Single and Weproposeanapplicationoftheabovesystem. Bystrech-
double bonds alternatingly appears. (c) The band structure around ingananoribbonalongtheribbondirection,horizontalbonds
k =πofapolyacenewiththePeierlsinstability. (d)Thelogarithm are stretched and vertical bonds are shrinked. The resultant
plotofthebandgap∆,log ∆/t,ofnanoribbonswithvariousW. structureisresembletothedeformedstructureinducedbythe
10
(e)Thelogarithmplotofthebandgap∆,log ∆/t,ofnanoribbons Peierlstransition.WemaycallitastraininducedPeierlstran-
10
asafunctionoftheappliedforceδforvariousW. sition.Whenwestretchthisstructure,thebandgapopensand
theconductanceatthezeroenergybecomeszerobythestrain
inducedPeierls-transition. Ontheotherhand,withouttheex-
inganadvantageofthisproperty,weproposeananomechan-
ternalmechanicalforce,nanoribbonswithW > 2isgapless
icalswitchsensor.
andthesystemisconductive.Wecanswitchtheconductance
The model Hamiltonian is the Su-Schrieffer-Heeger-like
fromontooffbystretchingthesystem. Thesystemactsasa
model19,
nanomechanicalswitch sensor detecting nanoscale displace-
ment.
H0 = tijc†i,σcj,σ. (19)
hXi,ji
VII. CONCLUSIONS
In this model, when the Peierls instability occurs, the trans-
fer integraltakes two values correspondingto the single and
A nanodiskcan be used as a spin filter just as in a metal-
doublebonds,andotherwiseittakesonevaluecorresponding
ferromagnet-metaljunction. Anovelfeatureisthatthedirec-
to the conjugatebond. We are able to calculatethe gapana-
tionofthespincanbecontrolledbyexternalfieldorspincur-
lytically in the case of polyacene (W = 1), where the band
rent. We havenewlyproposednanodiskarraysandnanome-
structurearoundk=πisgivenby
chanicalswitch. These nanodisk-nanoribboncomplexstruc-
turewillopenanewfieldofnanoelectronics,spintronicsand
1
ε(kx)=±2 −t2+ 4t21+5t22+8t1t2coskx . (20) nanoelectromechanicspurelybasedongraphene.
(cid:18) q (cid:19)
Thebandgapisdeterminedbysubstitutingk =πas Acknowledgements
x
I am very grateful to Y. Takada, H. Tsunetsugu, B.K.
∆=2ε(π)= −t2+ 4t21+5t22−8t1t2 ∼4δt1δt2. Nikolic and N. Nagaosa for fruitefuldiscussions. Thiswork
(cid:18) q (cid:19) (21) wassupportedinpartbyGrants-in-AidforScientificResearch
We show ε(kx) in Fig.10(c) by taking the values t1 = t − fromthe Ministryof Education,Science, SportsandCulture
δt1 = 0.94t for single bondsand t2 = t+δt2 = 1.08t for No. 22740196and21244053.
double bonds. We have carried out a numerical estimation References
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