Controlling edge states of zigzag carbon nanotubes by the Aharonov-Bohm flux

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by  K. Sasaki

Controlling edge states of zigzag carbon nanotubes by the Aharonov-Bohm flux
K. Sasaki,1, S. Murakami,2 R. Saito,3 and Y. Kawazoe1
∗
1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
2Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
3Department of Physics, Tohoku University and CREST, JST, Sendai 980-8578, Japan
5
(Dated: February 2, 2008)
0
0 It has been known theoretically that localized states exist around zigzag edges of a graphite
2 ribbon and of a carbon nanotube, whose energy eigenvalues are located between conduction and
n valencebands. Wefoundthatinmetallicsingle-walledzigzagcarbonnanotubestwoofthelocalized
a statesbecomecritical,andthattheirlocalization lengthissensitivetothemeancurvatureofatube
J and can be controlled by the Aharonov-Bohm flux. The curvature induced mini-gap closes by the
9 relatively weak magnetic field. Conductance measurement in the presence of the Aharonov-Bohm
1 fluxcan give information about thecurvatureeffect and thecritical states.

]
l Prior theoretical studies on zigzag carbon nanotubes (n,0), we obtain k = 2πµ/C (C = √3a n) where
l h h cc
a clarifiedthattheycanexhibiteithermetallicorsemicon- µ (= 1, ,n) is an integ|er |an|d a| 1.42˚A is the
h ductingenergybanddependingontheirchiralvector1. It carbon-ca·r·b·on bond length. We acncal≈yze the system
-
s has been shown theoretically2 and experimentally3 that using the nearest-neighbor tight-binding Hamiltonian,
e
m evenin“metallic”zigzagsingle-walledcarbonnanotubes H = a=1,2,3 i A(Vπ +δVa)a†i+aai + h.c.. “A” (in
(SWCNTs) a finite curvature opens small energy gaps. ∈
. Therefore, any zigzag SWCNTs have finite energy gaps. thesumPmationPindex)denotesanA-sublattice,ai anda†i
t are canonicalannihilation-creationoperatorsof the elec-
a
m On the other hand, Fujita et al.4 theoretically showed tronatsitei,andsitei+aindicatesthenearest-neighbor
that localized states (edge states) emerge at graphite sites (a=1,2,3)ofsite i. We include curvature effect as
-
d zigzag edges. The edge states are plane wave modes thebonddirection-dependenthoppingintegral,Vπ+δVa.
n along the edge and their energy eigenvalues are in be- We ignore the electron spin for simplicity.
o tween the valence band and the conduction band (zero- The energy eigen equation, Ψ =E Ψ , becomes
k k
c energystates). Sincethegraphitesheetwithzigzagedges H| i | i
[
can be rolled to form a zigzag SWCNT, the edge states ǫφJ+1 =φJ +gφJ+1 (J =0, ,N 1), (1)
1 are supposed to be localized at both edges of the zigzag A B B ··· −
ǫφJ =φJ+1+gφJ (J =0, ,N 1), (2)
v SWCNT. In this case, a zigzag nanotube has not only B A A ··· −
0 bulk (extended) states with a finite energy gap but also ǫφ0 =gφ0, ǫφN =gφN, (3)
4 A B B A
zero-energylocalizededgestates. Althoughseveralprop-
4
erties of the edge state have been investigated5, physical where we define
1
0 relationship between electrical properties of bulk states
E (V +δV ) π(µ n )
5 and edge states remains to be clarified. ǫ , g 2 π 2 cos − Φ . (4)
0 In this paper, to investigate this relationship, we ≡ (Vπ +δV1) ≡ (Vπ +δV1) n
/
t study an effect of the Aharonov-Bohm (AB) flux along
a Here, we write the wave function at site (I,J) of A(B)
the metallic zigzag nanotube axis. Among the Fujita’s
m sublattice as ψJ (k) = exp(i2πIµ/n)φJ (µ) (see
edge states of metallic zigzag SWCNTs, we will show (A,B)I (A,B)
- Fig. 1). We assume δV = δV in the Hamiltonian be-
d that there exist “critical states”. Their wave functions, 2 3
cause of the mirror symmetry along the axis of a zigzag
n in particular their localization length, are sensitive to
nanotube. The effect of the AB flux is included in g
o the following two perturbations: the curvature and the
c Aharonov-Bohm(AB)flux. Theseperturbationsarenew of Eq. (4) by the replacement, µ → µ − nΦ, where
: n Φ/Φ is the number of flux quantum, Φ .
v ingredients for cylindrical geometry, absent in the flat Φ ≡ 0 0
By solving Eqs. (1), (2) and (3), we obtain an analyt-
i graphene sheet. The main purpose of the paper is to
X ical form of the energy eigen function as
clarify the dependence of the critical states on the AB
r
flux and the relationship between the wave functions at
a 1sinJφ sin(J +1)φ
the bulk and at the edge. This dependence can be ex- φJ = + φ0, (5)
amined by a conductance measurement in the presence A (cid:18)g sinφ sinφ (cid:19) A
oftheAB flux. We note thatsuchABflux appliedalong sin(J +1)φ
φJ = φ0 (J =0, ,N), (6)
the SWCNT axis (see Fig. 1) has already been realized B (cid:18) sinφ (cid:19) B ···
in experiments6,7.
where φ satisfies
Because the unit cell is composed of two sublat-
tices, A and B, we write the wave functions as Ψ =
t(ψA(k),ψB(k))wherekisadiscretewavevector|arokuind 2cosφ= ǫ2−g2−1 κ. (7)
the tubule axis. By fixing the chiral vector1 as C = g ≡
h
2
0 1 2 N vector k satisfying g < N+1, the two localized states
I J | | N+2
haveenergieswithoppositesigns,ǫ= (e N ϕ). Each
− | |
±O
ofthe two states is localizednear both two edges. In the
e
e g left (right) edge, it is localized in the A (B) sublattice.
g d
Lefted ψψAA01II+′ 1 ψBNI+1Righte Hbeenlacregfoer;tthhewleoccaolnizseiddersttahteesleanrgetthheNnaolflotwheednafonro|tgu|b<e t1o.
ψA0IψBNI−′1 ψBNI (a) ǫ3 (b) g=− 1+ δV2−δV1
Vπ
(cid:16) (cid:17)
Φ
1
Left edge Right edge g −1 g
−2 −1 1 2
−1
FIG. 1: Lattice structure of a zigzag carbon nanotube. The
filled (open) circle indicates the A (B) sublattice. Both the
left and right edges are Fujita’s edges.
−3
The energy eigenvalue is determined by the boundary
condition of Eq. (3). Using Eqs. (5) and (6), we get
FIG.2: (a)Regionforlocalizedstatesκ2 >4shownasshaded
sin(N +1)φ+gsin(N +2)φ area in the(g,ǫ) plane. Whetherlocalized states are allowed
=0. (8)
sinφ dependsontheboundarycondition. Ifthefigureisrewritten
in terms of k and E, the empty region reproduces the well-
This equationcorrespondsto vanishing wavefunction at known band structure for the graphene sheet. (b) When we
fictitious A sites of J = N +1, i.e., φN+1 = 0. Most of ignore the curvature effect, there are states with g = −1 in
A
thesolutionsforφinEq.(8)arereal,asweexplainlater. the absence of the AB flux. The curvature effect displaces
Such realsolutions representextended states and satisfy the states onto the line g = −(1+(δV2 −δV1)/Vπ). The
κ2 4. AB flux changes the value of g, and eigenstates will make
≤ trajectorieslikethedashedcurvesinthefigure. Twoextended
In addition, there can be localized states, where φ has
states (filled circles) become localized as they run into the
an imaginary part and κ2 >4. Their localization length
shadowregion. Eigenstatesexceptforthesetwostatesremain
is proportional to the inverse of the imaginary part of
extended(open circles).
φ. We examine if the boundary condition allows such a
localized state. The complex solutions of Eq. (8) can be
The critical condition, κ2 =4, separates the extended
writtenasφ=iϕorφ=π+iϕwhereϕisarealnumber.
and the localized states. We plot the lines of the critical
The former case, φ = iϕ, corresponds to κ > 2. In this
condition in the (g,ǫ) plane in Fig. 2(a). The shadow
case, Eq. (8) becomes
regions satisfy κ2 > 4, representing localized states. By
sinh(N +1)ϕ+gsinh(N +2)ϕ=0. (9) applyingthe AB flux, eachstate movesandmakesatra-
jectory in the (g,ǫ) plane. Suppose one extended state,
Dueto sinh(N+2)ϕ > sinh(N+1)ϕ,weobtain 1< located outside of the shaded region, comes across the
| | | | −
g < 0. The latter case, φ = π + iϕ, corresponds to boundaries κ2 = 4 between the empty and shaded re-
κ< 2and0<g <1. When g 1,ontheotherhand, gions. It means that the extended state turns into anlo-
− | |≥
Eq. (8) does not allow complex solutions and therefore calizedstate. Onthevergeofthetransitionthestatebe-
there is no localized state in this region. Because the comes “critical”, when g = N+1 1 and ǫ= 1 .
boundaryconditionimpliesg = e+ϕ+ (e2Nϕ)forϕ< If we assume δV = 0 and±tNh+e2re≈is±no externa±l mN+ag2-
0 ( 1<g <0) and g =e−ϕ+ −(e−2NϕO) for ϕ>0 (0< neticfield,thiscoanditionforg issatisfiedonlyinmetallic
− O
g < 1), the energy eigenvalues for localized states are zigzag nanotubes, namely, when n is a multiple of 3 and
ǫ = ±O(e−N|ϕ|), exponentially small as a function of µ1/n=1/3or2/3(seeEq.(4)). Whenitissatisfied,the
nanotube length. states with g = 1, ǫ 0 are located very close to the
By a more elaborated analysis, for a fixed k, we can critical line κ2 =±4, and≈thus can be easily controlled by
analyticallyshowthat,(i)for g N+1,allthe2(N+1) external perturbations as we see later.
| |≥ N+2
states are extended, (ii) for 0 < g < N+1, there are 2N The cylindrical geometry of nanotubes yields a fi-
N+2
extended states and two localized states with Reφ = π, nite mean curvature and induces a change of the hop-
and(iii)for N+1 <g <0,thereare2N extendedstates ping integral δV 2,8. The scaling of the curvature gives
and two loc−aliNze+d2 states with Reφ = 0. For each wave δV /V (a2 /a C 2). The values of g are then driven
a π ≈O cc | h|
3
(a) n=30 (b) n=51
away from g = 1 to
± 250 200
g ≈±(cid:18)1+ δV2V−πδV1(cid:19)+√3πnnΦ. (10) C|h 125000 |Ch|=7.3nm C|h 110500 |Ch|=12.5nm
/| 100 /|
From an experimental data of the mini-gap in metallic L 50 L 50
zigzag nanotubes3, we estimate9 δV δV =π2V /8n2.
2 1 π 0 0
− 10 20 30 10 20 30
Thus, in the absence of the AB flux, these states have
B[T] B[T]
g > 1, and are not critical but extended. However,
| |
the AB flux can make one of g within g < 1, which
| |
implies that two extended states become localized (see FIG.3: Localizationlengthofthecriticalstatefor(a)n=30
Fig. 2(b)). Other extended states remain extended even and (b) n = 51 zigzag nanotubes. The horizontal axis is a
with this external magnetic field. magneticfield,B[T],(a)7.5≤B≤30and(b)1.6≤B≤30.
OneoftheimportantquestionsiswhethertheABflux
requiredforsuchatransitionisexperimentallyaccessible
or not. We calculate the magnetic field required to close the Fermi level, only one channel (g 1) contributes
≈ −
the curvature-induced mini-gap. From Eq. (10), we find to the transport and gives a unit of quantum conduc-
n¯Φ = π/(8√3n) is necessary to close the mini-gap for tance G0. In (c), when nΦ = n¯Φ, the zero conductance
g = 1 (for g = 1, n¯ is necessary). This AB flux state disappears and the critical states start to localize.
Φ
corres−pondstothema−gneticfieldB 2 105[T]. Thus,B Even when further AB flux is applied, there remains a
≈ ×n3 finiteconductanceatFermilevelduetothecriticalstates
is accessible if n>20. For a zigzag nanotube with C =
h
as is depicted in (d). However, because further AB flux
(n,0), the flux quantum Φ correspondsto the magnetic
0
quicklyreducesthe transmissionprobabilityfromFig.3,
field B = B /n2[T] (B 8.5 105[T]), and therefore
n 1 1
≡ × the conductance at the Fermi level decreases.
experimentally accessible magnetic flux corresponds to
w|neΦll| b≪ey1o.ndFoarninasctcaensscieb,lewemhagavneetBic9fi≈eld104[T10],2[wTh],ichanids (a) nΦ =0 (b) 0<nΦ <n¯Φ
we can attain n 10 2 at most. ≈ 2G0
Φ −
| |≈
Here,weplotthelocalizationlengthofthecriticalstate
for metallic zigzag nanotubes with different diameters. G0
We define the localization length L by |φJA|2/|φ0A|2 ≈ Vg Vg
exp( 3a J/2L),where3a /2isthedistancebetweenJ
cc cc
and−J +1 lines (Fig. 1). We obtain for n >n¯ (c) nΦ =n¯Φ (d) nΦ >n¯Φ
Φ Φ
L 1 1
= = . (11)
|Ch| 4πnΦ− 2√π23n 4BπnB − 2√π23n
In Fig. 3, we plot L/C for n = 30 taken as the Vg Vg
h
| |
largest diameter for a SWCNT and n = 51 taken as a
shellinamulti-walledcarbonnanotube(MWCNT).The critical states
curvature-induced energy gaps are E =7.4[meV] and
gap
2.6[meV], respectively, where we use V =2.7[eV]. n¯ is
π Φ
estimatedas7.1[T]forn=30and1.4[T]forn=51. Ifwe FIG. 4: Theoretical result of conductance with a metallic
neglectthe interlayerinteractionina MWCNT, a zigzag zigzag nanotube. We plot conductance as a function of gate
SWCNT in a MWCNT is the most suitable to examine voltage, Vg for several values of the ABflux.
the behavior of the critical states, because one does not
needstrongmagneticfieldtoclosethecurvature-induced The transition between extended and localized states
mini-gap. by the AB flux can be also understood in terms of the
The curvature effect and the modulation of localiza- effective-mass theory of graphene. The effective-mass
tionlengthcanbeexperimentallyprovedbyconductance theoryoflow-energydynamicsisgivenbytwoWeylequa-
measurementsasafunctionofthegatevoltageforvarious tions11, each of which represents the dynamics around
values of the AB flux. We plot the conductance for dif- the K point and K’ point in the momentum space. Cur-
ferent n in Fig. 4, where we assume ideal contacts and vature in the nanotube induces an effective gauge field
Φ
zero temperature. Such measurement is attainable, as in the Weyl equations8, and this gauge field gives the
conductance of well-contacted individual SWCNTs was mini-gap of 2(δV δV ) through the AB effect. When
2 1
measured in the ballistic regime by Kong et al10. In we apply the AB fl−ux (n < n¯ ), the energy gap of one
Φ Φ
Fig 4(a), the solid curve represents conductance in the mode opens while that of the other closes; therefore, the
absence of the AB flux, n = 0. The zero conductance energy gap closes as a total system. If further magnetic
Φ
state corresponds to the curvature-induced mini-gap. In flux (n > n¯ ) is applied, some of the extended states
Φ Φ
(b),conductancechangesaccordingtotheABflux. Near become localized at the edges and their energy becomes
4
zero,while other extended states openup the gapagain. the localized states. A typical energy scale of the charg-
Such zero-energy states cannot be obtained by solving ing energy for an edge state is E e2/L , where
c edge
≈
the Weyl equations with a uniform energy gap. Instead, L is the typical length for the edge states. This
edge
it is known that by modulating locally the energy gap, energy should be compared with that of the extended
there appear localized states called Jackiw-Rebbi (JR) states evaluated as E e2/L , where L is the sys-
c sys sys
states12. The edge states in the nanotube induced by tem size. Since L ≈L , the Coulomb energy E
sys edge c
≫
the AB flux resemble the JR states, in that they have for the edge states are much bigger than that of the ex-
zero energy and are localized; meanwhile they are differ- tendedstates,andmayhinderlocalization. Tolowerthis
ent from ours because these edge states are induced by Coulombchargingenergy,thespinsofthelocalizedstates
a uniform AB flux, and not by a local modulation. We at each edge will align ferromagnetically4.
note that relatedstates can appear when we introduce a
Inaddition to the edge states,the bulk states canalso
local geometrical deformation in the nanotube9.
haveinterestingphenomenadue tothe AB flux inSWC-
Throughoutthis paper,we haveexaminedzigzagnan-
NTs. Namely, the AB flux induces a persistent current
otubes having two Fujita’s edges. Even for zigzag nan-
around the tube axis, and gives an orbital magnetic mo-
otubeswithopenedges,onecanconsiderothercaseswith ment14. The persistent current is caused by a splitting
one or two edges being the Klein’s edge13 If one of the of the van Hove singularities by the AB flux15. Quite
edgesisKlein’sedgeandthe otheris Fujita’s edge,there
recently, the splitting was observed in semiconducting
arestilllocalizedstates,whereastheirpropertiesaredis-
nanotubes as the splitting of the first-subbandmagneto-
tinct from the previous case for two Fujita’s edges. Such absorption peak6 and also in small-bandgap (not curva-
ananotubecanbemadebyattachingA-sitesattheright
ture related) nanotubes as a temperature dependence of
edgeofthezigzagnanotubeinFig.1. Theboundarycon- conductance7. These experiments were intended to ob-
dition for the right edge is ǫφN+1 = φN. For localized
A B servethattheABfluxcanmakeasymmetrybetweentwo
states we obtain ǫ = 0 and φJ = ( g)Jφ0, φJ = 0
A − A B energybandscomposedoftheextended(bulk)wavefunc-
(J = 0, ,N). Thus, the localized states have exactly
tions near the Fermi level. By means of doping in addi-
···
ǫ=0 for every value of g, and they have amplitude only
tiontotheABflux,aninterferencebetweenmanyenergy
on the A sublattice. Such states are localized near the bands16 takes place. This interference affects magnetic
right (left) edge when g > 1 (g < 1). Furthermore,
and transport properties of the system, and is helpful to
| | | |
there are no transitions between localized and extended
get insight into the bulk electronic states.
states,evenwhenwe applyamagnetic field. Meanwhile,
Insummary,theAharonov-Bohmeffectofcarbonnan-
when g passes g = 1, the localized state with ǫ = 0
comes on the critica±l boundary of κ2 = 4 in Fig. 2(a), otubeswillbesuitedtoexaminenotonlythebulkelectri-
cal properties but also the properties of the edge states.
andthe correspondinglocalizedstatewill movefromthe
We point out that there are two critical states in metal-
right edge to the left or vice versa, and can be detected
lic zigzag nanotubes. Although the critical states be-
experimentally. Onthe otherhand,whenbothedgesare
comeextendedduetothecurvatureeffectinthe absence
the Klein’s edges, the situation is somewhat similar to
of magnetic field, the Aharonov-Bohm flux can make a
the case with two Fujita’s edges. For this case, localized
transition from the extended states into localized edge
statesarerealizedwhen g >1insteadof g <1,andthe
| | | | states. This transition can be seen as a characteristic
AB flux induces a transition from localized to extended
feature of conductance.
states.
Similar phenomena are expected also in nanotubes K. S. acknowledges support form the 21st Century
with other chiral structures except for armchair nan- COE Program of the International Center of Research
otubes. Nakada et al. showed numerically that localized and Education for Materials of Tohoku University. S.
states appear not only in the zigzag edges but also in M. is supported by Grant-in-Aid (No. 16740167) from
edges with other shapes5. We expect that such localized the Ministry of Education, Culture, Sports, Science
stateswillundergotransitionwithextendedstatesinthe and Technology (MEXT), Japan. R. S. acknowledges
presence of the AB flux. a Grant-in-Aid (Nos. 13440091 and 16076201) from
FinallyletusmentiontheCoulombchargingenergyfor MEXT.
∗ Email address: [email protected] ence 292, 27 (2001).
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