Table Of ContentInterplay of Aharonov-Bohm, chirality, and aspect ratio effects in the axial
conductance of a nanotube
Eugene B. Kolomeisky1, Hussain Zaidi1 and Joseph P. Straley2
1Department of Physics, University of Virginia, P. O. Box 400714, Charlottesville, Virginia 22904-4714, USA
2Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA
Amagneticfieldappliedalongtheaxisofananotubecancounteracttheeffectofthetubechirality
and dramatically affect its conductance, leading toa way to determine thechirality of a nanotube.
2 Theeffectoftheappliedfieldisstrongestinthelongtubelimitwheretheconductanceis(i)eithera
1 sequenceofsharp4e2/hheightpeakslocated atintegervaluesoftheflux(foran armchairtube)or
0 (ii)aperiodicsequenceofpairsof2e2/hheightpeaksforachiraltube,withthespacingdetermined
2 bythechirality. Intheshort tubelimit theconductancetakeson thevaluethatgivestheuniversal
n conductivityofanundopedgraphenesheet,withasmallamplitudemodulationperiodicintheflux.
a
J PACSnumbers: 73.50.Td,73.23.-b,73.23Ad,73.63.-b
8
2
I. CONDUCTANCE OF GRAPHENE SHEET W) and calculated both the conductivity and the Fano
] AND NANOTUBES factor of the shot noise. They found the same expres-
l sion (1) for the transmission probability except that the
l
ha Katsnelson[1]hasgivenaverygeneralandelegantex- spectrum qn was now determined by the boundary con-
- planation of the experimental observation that at zero ditions at the edges of the strip. Assuming that longitu-
s temperature an ideal graphene sheet has conductivity of dinalandtransversemomentaarenotcoupled,a general
e the order e2/h. Although earlier considerations [2] ar- conductance formula that summarizes various boundary
m
rivedatessentiallythesameconclusion,Katsnelson’sar- conditionsandaccountsforalldegeneraciescanbegiven
t. gument stands out because it clearly demonstrates that in the form [4]
a
m thefiniteconductivityofgrapheneisadirectconsequence ∞
1
of the electron dynamics, which is described by the two- G = (3)
d- dimensional massless Dirac equation. Katsnelson [1] has strip n=−∞cosh2[(n+α)πL/W]
X
n considered an undoped graphene sheet in the shape of
withα=1/2foraanedgedefinedbyaslowlyvaryingpo-
o a cylinder of length L and circumference W (with both
c length scales significantly larger than the lattice spac- tential [5], and α=0 or α=1/3 for an abrupt armchair
[ edge,dependingongeometricdetailsofthestrip[4,6]. In
ing of graphene) attached at its bases to heavily doped
the short and wide strip limit, W/L , Tworzydl o et
2 graphene leads and demonstrated by means of the Lan- →∞
al.,arrivedatthesameuniversalconductivityasKatsnel-
v dauertransmissionformula[3]thattheaxialconductivity
4 of the cylinder in the ring limit, W/L , approaches son [1] but also observed that the boundary conditions
9 the universal 4e2/πh value. The cruci→al ∞element of the haveasignificanteffectinthelongandnarrowstriplimit
3 (W/L 0): the smooth boundary (α = 1/2) yields an
analysisis anexpressionfor the transmissionprobability
0 insulat→or, while a boundary giving α=0 is a conductor.
of propagating modes in the leads [1]
.
0 We will discuss below what happens for general α and
1 1 aspect ratio.
T = (1)
1 n cosh2(q L) In addition to this sensitivity to boundary conditions
1 n
in the strip geometry, there is an analogous sensitivity
:
v where the index n labels the modes, and the azimuthal to the chirality of the graphene cylinder, i. e. to the
i wave numbers qn =2πn/W,n=0, 1, 2,... are singled way the graphene sheet is wrapped into a cylinder. In-
X ± ±
out by the condition of periodicity around the circum- deed, as written, Eq.(2) predicts that an infinitely long
ar ference. Then, accounting for the spin degeneracy, the tube (W/L = 0) possesses unit conductance per valley,
conductance of the cylinder (hereafter measuredin units i. e. it is a semimetal. However it is known that only
of the conductance quantum 2e2/h) per Dirac valley K about one-third of all possible tubes are semimetallic. It
′
(or K ) [2] is given by turns out that Eq.(2) can be generalized to account for
all kinds of tubes without affecting the main conclusion
∞ ∞
1 of Ref.[1]. The generalization is due to Ajiki and Ando
G = T = (2)
K n 2
cosh (2πnL/W) [7] and Kane and Mele [8] who observed that the long-
n=−∞ n=−∞
X X wavelength low-energy dynamics of the graphene elec-
Accounting for the presence of the two degenerate Dirac tronsoftheK-valleywhenconfinedtocylindricalsurface
valleys, the total conductance is G = 2G and the con- isdescribedbythetwo-dimensionalmasslessDiracequa-
K
ductivity σ follows from the relationship σ =GL/W. tion in the presence of a fictitious vector potential that
Tworzydl oet al. [4] extended Katsnelson’s analysis to represents the effects of the tube size and its chirality.
the case of an open strip geometry (length L and width The effect is to replace the azimuthal quantum number
2
n in Eq.(2) with n+α thus making the outcome the AB flux φ will be given by the sum of contributions
due to two Dirac valleys
∞
1
G (α)= (4)
K n=−∞cosh2[(n+α)2πL/W] G(φ,α)=GK(φ+α)+GK(φ−α) (6)
X
We note that in contrast to the fictitious flux α, the AB
similar to Eq.(3), but now G vanishes for large L/W.
K fluxφbreaksthe time-reversalsymmetrythus lifting the
An expression very similar to Eq.(4) is also encountered ′
degeneracy between the K- and K -valleys [7]. The con-
in the analysis of the conductance of a planar graphene
ductance (6) is an even function of both φ and α. The
flake having Corbino disc topology [9].
interesting implication is that an applied magnetic field
The intrinsic flux α combines the effects of winding
can cancel the effect of chirality, restoring conductivity
andcurvature,andprovidesaclassificationofnanotubes
to a nonmetallic nanotube.
[7,8]. In terms of the usual(M,N) classificationof nan-
The conductance of an infinitely long tube (W/L=0)
otubes (which explains how the graphene structure is
has been discussed by Tian and Datta [11]. In this
wrapped around the cylinder), α has a geometrical in-
case the effect of the flux and chirality on the conduc-
terpretation. The new periodicity introduced into the
tance of a tube can be anticipated by looking at the
graphene structure in forming the nanotube limits the
square of the energy eigenvalue of the Dirac equation
accessiblereciprocallatticevectorstolineswhichingen- E2(q )=~2v2[q2+(2π/W)2(n+φ α)2] where v is the
eraldonotvisittheDiracpointswherethegapvanishes; n z z ±
Fermi velocity, q is the wave vector in the axial direc-
z
α measures the distance between Dirac point and line in
tion, and the upper and lower signs refer to the K- and
units of the periodicity wavevector k⊥ = 2π/W. When K′-electrons. Since the integer part of α or φ can be
1
N M is not a multiple of 3, α is close to (insulating
− 3 absorbed into the definition of the azimuthal quantum
nanotube);otherwise[10]thenanotubesaresemiconduc-
number n, any measurable property will be a periodic
tors with
function of α or φ with unit period. Then the energy
√3π(N M)(2N2+5NM +2M2) gaps
α= − (5)
48 (N2+NM +M2)2 4π~v
∆K,K′ = φ α (7)
W | ± |
ThelargestvaluesoccurforM =0(”zigzag”nanotubes);
for M = N, α = 0 (”armchair” nanotubes, which are canbe periodically closedby the AB flux φ thus trigger-
semimetals). ingsemiconductor-semimetaltransitions[7,11]. Theway
The fictitious flux has opposite signs in the two Dirac these transitions manifest themselves in the flux depen-
valleys [7] because the flux is not due to a physicalmag- dence of the conductance is sensitive to the value of α.
netic field and the system still has time-reversalsymme- ThegapsinthetwoDiracvalleys(7)canbeclosedsimul-
′
try. The contribution of the electrons of the K -valley taneouslyonly if α 0 (the caseα=1/2does not occur
≡
into the conductance, GK′(α) = GK( α), is still given in nanotubes). If both valleys become ungapped at once
−
by Eq.(4) because it is an even function of chirality α. (∆K,K′ = 0), there are two open ideal conductive chan-
Thus the net conductance is given by G=2GK(α). nels, and the conductance (according to the Landauer
formulation [3, 11]) is 4e2/h, i.e. twice the conductance
quantum [15]. These conductance peaks are located at
II. EFFECTS OF AN AXIAL MAGNETIC FIELD
integer [11] (α=0) values of the AB flux φ. In all other
cases,theAB fluxφcanonlycloseoneofthe gaps(7)at
Since the chirality enters the problem as a fictitious a time. As a result there is only one conductive channel
flux, the physical properties of graphene tubes can be presentandtheconductanceisgivenbytheconductance
dramatically affected by the presence of an axial mag- quantum 2e2/h [16]. These peaks appear in pairs and
netic fieldaswasanticipatedtheoretically[7,11,12]and the pair with its center at integer flux has a separation
observed experimentally [13]. Indeed, imagine there is of∆φ=2α. Forthe 2e2/hconductancepeak,the axial
| |
an axial Aharonov-Bohm (AB) flux Φ [14] entirely con- currentis entirely due to the electronsof the single open
fined within the interior of the cylinder. Adopting the valley.
gaugewhere the vectorpotentialpoints inthe azimuthal Thisleadstothepossibilityof”αspectroscopy,”where
direction,weobservethattheelectronsexperienceavec- the magnetic field response can help index a nanotube.
tor potential A = Φ/W whose effect can be accounted Unfortunately,this is somewhatconstrainedbyavailable
for via the Peierls substitution q q + eA/~c = magnetic fields. The flux quantum is 4.138 105 Tesla
n n
(2π/W)(n + φ) where φ = Φ/Φ0 is→the dimensionless ˚A2, with the result that the size of the m×agnetic field·
flux measured in units of the flux quantum Φ0 = hc/e. B neededto see one complete AB cycle for an extremely
In present geometry the Peierls substitution is exact be- largetubeofradiusR=25˚Ais200Tesla[7]. Thisisstill
∗
cause the vector potential has constant magnitude and a very small field compared to characteristic field B of
direction in the natural coordinate system of the tube. onsetofdisruptionofnormalatomicstructure. Thefield
∗
Then, in the light of the above discussion, the net con- B canbeestimatedfromtheconditionofequalityofthe
ductance of the cylinder of chirality α in the presence of magneticlength (~c/eB)totheBohrradius~2/me2or
p
3
equivalentlyrequiringthatthereisonefluxquantumper 4
cross-sectionalarea of undisturbed atom. This gives the Π Α=1(cid:144)8 Α=0
field B∗ m2e3c/~3 which is over 105 Tesla. 3
When≃N M is not a multiple of 3, nanotubes arein- Π Α=1(cid:144)6
sulating and−have α close to 1/3. Achieving φ = 1/3 22e h 2
(i.e. having B × Area = 13Φ0) is also not possible Σ Π
with the available magnetic fields and nanotubes of ex- 1 Α=1(cid:144)4
perimental interest. However, according to (5), α can Π Α=1(cid:144)2
be significantly smaller when N M = 0 (mod 3).
−
The M = N ”armchair” nanotubes have α = 0, and 1 2 3 4 5
for other cases the curvature of the tube surface gives W
α 0.23/N 0.09˚A/R for ”zigzag” tubes (M = 0), L
and≃α 1.9/N≃2 0.16˚A2/R2 for the ”nearly armchair”
case M≃=N 3.≃For this latter case, the magnetic field Figure 1: (Color online) Conductivity versus aspect ratio
− W/L for different tube chiralities parameterized by the fic-
needed to achieve conductance at low temperatures is
B = 104T ˚A4/R4, implying that a 11 Tesla field would titious flux αas given by Eq.(10).
be adequat·e for the (8,11)nanotube (R=6.5˚A; the gap
wouldbe 0.006eV). Although the ”armchair”nanotubes
shiftstowardsW/L=0andbecomeslarger. Forα=1/2
are conducting at zero magnetic field, application of a
both terms of (8) contribute equally and we find again
longitudinal magnetic field would render them noncon-
both vanishing conductance and vanishing conductivity.
ducting.
As shown in Figure 1, there is a characteristic chirality
α = 1/4 beyond which there is no maximum at all and
c
III. EFFECTS OF FINITE LENGTH the conductivity is a monotonically increasing function
of the aspect ratio W/L.
According to Figure 1, the tube limit corresponds to
The sharpness of the semiconductor-semimetaltransi-
W/L < 2α. This will give a nonconducting tube at low
tions discussed above is determined by the aspect ratio
temperatures. This also describes the sensitivity to ap-
W/L. Consider the case where there is no applied mag-
plied magnetic field: when φ α > 2W/L the con-
netic field. In the ring limit (W/L ), the sum in | − |
→ ∞ ductance will be suppressed, while for φ α < W/πL
Eq.(4) can be approximated by an integral with the re- | − |
the conductance will be close to the ideal value G 1.
sultGK(α)≃(1/π)(W/L)whichleadstoafiniteuniver- For a 1 micron-long nanotube with circumference≃30˚A
sal conductivity σ0 = 2GK(α)L/W = 2/π [1, 4]. This this means that the flux has to be set to within 0.3% to
is chirality-independent because the quantization of the
observe restoration of conductance (if α = 0); but this
azimuthal motion is irrelevant in the ring limit. 6
means it will take a 16T field to render an ”armchair”
In the tube limit (W/L 0), the discreteness of the
→ nanotube of these dimensions nonconducting. We note
transverse wavevectors is relevant. The sum in Eq.(4)
that currently available nanotubes have aspect ratios in
is well approximated by the n = 0 and n = 1 terms
− the W/L<0.79 range [17].
(assuming 0<α61/2):
ThesameFiguregivestheconductivityofthegraphene
2 2 strip with the aspect ratio of 2L/W (compare Eqs. (3)-
G(0,α) 2 + 2 (8) (6)). The α=0 andα=1/2plots werepreviouslygiven
≃ cosh (2παL/W) cosh [(1 α)2πL/W]
− [4].
In the case α = 0, this simplifies to G(0,0) 2 + WhenW/LislargeitisusefultousethePoissonsum-
≃
8exp( 4πL/W)wherethe firstterm representsthe con- mationformula[18]to thesumoverninEq.(4)withthe
−
ductance of an infinite tube and the second term is the results for the net conductance
correction due to finite aspect ratio. The conductivity W2 ∞ ncos(2πnα)
σ = GL/W L/W is infinite. Since in the ring limit G(0,α)=2G (α)= (9)
(W/L )≃the conductivity approaches 2/π, the con- K L2 sinh(πnW/2L)
ductivi→ty ∞is a monotonically decreasing function of the n=X−∞
aspect ratio for α=0, as shown by Tworzydl o et al. [4] and conductivity
in their analysis of Eq.(3) for α=0.
∞
Turningonfinitechiralityormagneticfield,nomatter 2 2W ncos(2πnα)
σ(α)= + (10)
how small, has a dramatic effect on the conductance of π L sinh(πnW/2L)
n=1
a long tube. For 0 < α < 1/2 the first term of Eq.(8) X
dominates and we find exponentially small conductance respectively. We now observe that for arbitrary aspect
G(0,α) 4exp( 4παL/W); the tube is an insulator ratio W/L the effect of chirality manifests itself in peri-
≃ −
withagapoforderαR. Weobservethattheconductivity odic oscillations of the conductivity about σ0 =2/π; the
σ(0,α) 4(L/W)exp( 4παL/W) has a maximum for aspectratiodeterminestheamplitudeoftheconductivity
≃ −
W/L 4πα. As α 0, the conductivity maximum oscillations.
≃ →
4
In the ring limit, W/L , the sum in Eq.(10) is
→ ∞ Α=0
well-approximatedbytheconstantandn=1termswith
the result 2
2 4W πW
σ(α) + exp cos(2πα) (11)
≃ π L − 2L 1
(cid:18) (cid:19)
We observe that as W/L the limiting conductivity
→∞
of 2/π is approached from above if 0 6 α < 1/4 and
-1 0 1
from below if 1/4 6 α 6 1/2. This demonstrates that
α =1/4 as suggested by Figure 1. Α=1(cid:144)9
c
For the casewhen there is an appliedfield the Poisson 2
transformationcanagainbeusedtogivearepresentation
more suitable to finite W/L:
GH2e2L
h 1
∞
W 2 2W ncos(2πnφ)cos(2πnα)
G(φ,α)= +
L π L sinh(πnW/2L)
n=1 !
X (12) -1 0 1
We now see that the combined effects of the AB flux
Α=1(cid:144)3
φ andchiralityα manifestthemselvesinoscillations(pe-
2
riodic in both φ and α) of the conductance about the
value of 2W/Lπ corresponding to the universal conduc-
tivity of graphene σ0 =2/π; the shape and amplitude of
1
theoscillationisdeterminedbytheaspectratioW/L. In
the ring limit W/L the conductance is dominated
→ ∞
by the constant term of Eq.(12) and the leading finite
aspect ratio correction (if α = 1/4) is supplied by the -1 0 1
6
n=1 term:
Φ
2W 4W2 πW
G(φ,α) + exp cos(2πφ)cos(2πα)
≃ πL L2 − 2L Figure 2: (Color online) Conductance versus dimensionless
(cid:18) (cid:19) (13) AB flux φ for different tube chiralities parameterized by fic-
titious fluxαand aspect ratio W/L as given byEq.(12). As-
Our results summarizing the effects of chirality and AB
pect ratios are color-coded according to W/L = 0 (grey),
flux on the conductance of semimetallic (α = 0), in-
W/L=1/5 (orange),W/L=1(red),andW/L=3(brown).
sulating (α = 1/3) and semiconducting (mimicked by
α=1/9) tubes are shown in Figure 2.
Since the conductance represents directly measurable
quantity,wehopethatourconclusionswillbeexperimen- Theconductancecouldalsobemonitoredwithoutcon-
tally tested. The chirality of a tube can be determined tacts. An electric field E parallel to the axis of a con-
by measuring the dependence of the conductance on the ducting nanotube of length L will give a dipole moment
3
AB flux of a not very short cylinder, and a conducting of order L E (in Gaussian units). Then light polarized
”armchair” tube can be rendered nonconducting. parallel to the axis of a bundle of oriented nanotubes
Theissueofseparationoftheeffectswepredictedfrom will be scattered, provided that its frequency ω is low
those due to the leads is an important one. First we enoughforthechargeseparationtotakeplace. Foranan-
note that there is a series of conductance measurements otube with the Landauerconductance 2e2/h, this means
[13, 15–17] and an expertise on how to deal with the ω < 106Hz m2/L2 or ω < 1018Hz for micron-length
·
end effects is growing. For example, Javey et al. [17] tubes. But if the conductance is suppressed by a mag-
found that the resistance of a nanotube was rather close netic field, so is the rate of charge transport, and then
to the expected 6.5 kΩ; they estimated that the leads there is muchless scatteringat intermediate frequencies.
andcontactscontributednomorethan0.6kΩ(andmuch
less at low temperatures). We predict that the tube will
go from conducting to nonconducting (or the other way
around) in a magnetic field of specified size (the field
IV. ACKNOWLEDGMENTS
doesnothavetobe alignedbutthe partthatisnotaxial
will not contribute). The end effects are in series with
this; it will complicate telling what the conductance of We thanks D. R. Strachan for sharing with us his ex-
the undisturbed part of the tube is, but the change in perimental expertise. This work was supported by the
conductance should be straightforwardto observe. US AFOSR grant FA9550-11-1-0297.
5
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