Conductance through a single impurity in the metallic zigzag carbon nanotube

0.14 MB·English

by  Po-Yao Chang

Conductance through a single impurity in the metallic zigzag carbon nanotube
Po-Yao Chang1 and Hsiu-Hau Lin1,2
1 Department of Physics, National Tsing-Hua University, Hsinchu 30013, Taiwan
2Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan
(Dated: January 26, 2009)
We investigate transport through a single impurity in metallic zigzag carbon nanotube and find
the conductance sensitively depends on the impurity strength and the bias voltage. It is rather
interestingthatinterplaybetweenthecurrent-carryingscatteringstateandevanescentmodesleads
to rich phenomena including resonant backward scattering, perfect tunneling and charge accumu-
lations. In addition, we also find a dual relation between the backscattered conductance and the
9
charge accumulation. At the end, relevance to the experiments is discussed.
0
0
2
Single-wall carbon nanotubes (CNTs)[1] are graphene rity potential and the bias voltage. In the absence of the

n sheet[2]inroll-upgeometrywithlargeaspectratio,mak- impurity, the two conducting channels in metallic zigzag
a
ingthemnaturally-madequantumwireswithinteresting CNT deliver the perfect conductance G = 2G , where
J 0
transport properties. Since the conducting channels in G = 2e2/h is the quantum conductance. In some en-
6 0
metallicCNTarelimited,defectsandimpurities[3,4]are ergy regime, we found the backward scattering is greatly
2
importantforrealisticconsiderationsandmayplayasig- enhanced so that one of the conducting channel is per-

] nificant role in transport at nanoscale. One of the recent fectly blocked, leading to resonant backward scattering.
l
l breakthroughsisthetechniqueofcovalentattachments[5] Ontheotherhand,insomeotherenergyregime,theper-
a
h toCNTs,makingthefabricationofsingle-moleculejunc- fect conductance is recovered, signaling the conspiracy
- tions possible. It is rather remarkable that one can use from the evanescent modes to make the impurity poten-
s
the conductance to monitor and manipulate the cova- tial transparent. In the following, we would elaborate on
e
m lent attachments to the targeted CNT. On the theo- thedetailderivationsthatprovidesafirmgroundforthe
retical side, vacancy defect or doped boron/nitrogen in rich physics of the conductance in CNTs.
.
t
a CNTs[6, 7] show conductance suppression due to en- In the following calculations, we concentrate on the
m hanced backward scattering. metallic zigzag CNT that is well described by the tight-
- Inspired by these experimental and theoretical stud- binding model,
d
n ies,weinvestigatethedifferentialconductancethrougha (cid:88)(cid:16) (cid:17)
H=−t c†c +c†c +V c†c , (1)
o single impurity in the metallic zigzag CNT. Note that i j j i 0 0 0
c there are many types of defects including pentagon- (cid:104)i,j(cid:105)
[
heptagonpairs,vacanciesandchemisorptionadatoms[8].
where t is the hopping amplitude between nearest neigh-
1 For quantitative comparison with experiments, the first-
bors and V is the strength of the impurity potential
v principlesapproachesaremoreappropriate. Ontheother 0
3 located at the origin. Since no spin-flip scattering is
hand,torevealthecloseconnectionsbetweentheconduc-
4 present, the spin just gives rise to a factor of two and
tance, the transmission matrix and the current-induced
1 is suppressed in the above expression.
4 charge accumulation, a phenomenological approach may
The above Hamiltonian can be solved by the general-
. prove to be useful. As long as the impurity potential
1 ized Bloch theorem[9] where the momentum is allowed
is short-ranged, the detail profile is not crucially im-
0 to be complex due to the presence of the impurity po-
9 portant and can be captured by the strength of the
tential. In the following, we sketch how this is done
0 pseudo-potential, which is taken to be a delta-function
in metallic zigzag CNT. In the absence of the impurity,
v: for simplicity. These simplifications allow us to com- the transverse momentum is a good quantum number,
i pute the scattering states in the metallic CNT and con-
X struct the transmission matrix exactly. Making use of km = 2πm/N with m = 1,2,...,N and N is the num-
ber of unit cells around the CNT. Performing a partial
r
the Landauer-Bu¨ttiker formula, we can compute the dif-
a Fouriertransformationforthewavefunctioninthetrans-
ferential conductance at different energies. verse direction, Φ(x,y)=(cid:80) Φ (x)eiky, the CNT is de-
k k
Atfirstglance,thismayseemtobeasimplecalculation composedintoN decoupledeffective1Dchainswithdefi-
leading to trivial results – one expects the conductance nitetransversemomentumk . Ingeneral,thewavefunc-
m
is close to perfect for weak impurity strength, while it tion after partial Fourier transformation take the form,
should get strongly suppressed for a strong impurity po- Φ (x) ∼ φ zx. For |z| = 1, these solutions correspond
k k
tential. Thisnaiveexpectationisnotcorrectsinceimpu- to the familiar plane waves. However, in the presence of
rities in honeycomb lattice cause strong resonant states the impurity potential, |z|=(cid:54) 1 solutions are also allowed
and non-trival evanescent modes. As a result, the con- and need to be taken into consideration. For given en-
ductancesensitivelydependsonthestrengthoftheimpu- ergy (cid:15) and momentum k , z satisfies the characteristic
m
2
G/G
0
2
1.5
0
G (a)
1
E/t
0.4 0.2 0.0 0.2 0.4
G/G E t
0
2.0
e
sit
1.5 ct
e
ef
d
G01.0 (b) he FIG.2: (Coloronline)Chargeaccumulationatperfectblock-
t ing G = G in the (9,0)-zigzag CNT with V /t = 1. The
n 0 0
o radius of the shaded area represents the charge density with
0.5 sity the defect site labeled in green (lighter gray).
n
e
D
0.0
-00..44 -00..22 00..00 00..22 00..44
Et onecanalwaysformanappropriatelinearcombinationof
E/t
thetwoconductingchannelssothatoneofthemremains
intact from scattering. This ensures that the conduc-
FIG. 1: (Color online) (a) Conductance versus energy for tance is always large than G . For weak impurity po-
0
(9,0)-zigzag CNT with different impurity strength V0/t = tential V /t = 1, the conductance is close to the perfect
0
1,5,50showninblue(dark),red(gray)andgreen(lightgray)
value G ∼ 2G at most energies. However, there exist
respectively. (b) The energy-dependent conductance (upper 0
two sharp dips down to exactly G = G where one of
curve) and the charge density on the impurity site (lower 0
the channel is completely blocked. We found the perfect
curve) with V /t=5 are plotted together for comparison.
0
blocking is directly tied up with the resonant backward
scattering with the evanescent modes. Though the con-
equation, ductance merely comes from the scattering states, the
evanescent modes are also induced during the impurity
1
tt (z+ )+(t2+t2 −(cid:15)2)=0, (2) scattering process. In Fig. 2, the charge accumulation
m z m
at perfect blocking energy is shown with relatively weak
where tm =2tcos(km/2). By forming appropriate linear impurity strength V0/t=1. It is remarkable that such a
combinations of all possible solutions of z, the scatter- large charge accumulation can be created by a weak im-
ing states can be constructed analytically and thus the purity potential on one site. By increasing the impurity
transmissionmatrixisobtained. Sincethereareonlytwo strength, the resonant backward scattering is broadened
scattering channels in a metallic zigzag CNT, the 4×4 buttheperfectblockingG=G0 remains. Finally,strong
S-matrix is impurity strength brings the system to the unitary limit
where the particle-hole symmetry is restored.
(cid:20)r((cid:15)) t(cid:48)((cid:15)) (cid:21)
S((cid:15))= , (3) Another interesting feature is the tunneling conduc-
t((cid:15)) r(cid:48)((cid:15))
tance becomes perfect at some energies. This may not
look very surprising for weak impurity strength. How-
wherer,r(cid:48),t,t(cid:48) are2×2matrices. Finally,theLandauer-
ever, even for V /t=50, the strong impurity potential is
Bu¨ttiker formula gives the conductance as tracing over 0
still transparent at some particular energies as shown in
the square of the transmission matrix,
Fig. 1. How can the scattering states ignore the strong
G((cid:15))=G Tr(cid:2)t†((cid:15))t((cid:15))(cid:3). (4) impurity scattering and maintain the conducting chan-
0
nel perfect? To understand this phenomena, we plot the
The transmission matrix can be derived analytically, in- conductanceandtheparticledensityontheimpuritysite
volving complicated expressions which can be evaluated together in Fig. 1(b) for comparison. When the den-
by numerical means. sity on the impurity site is large, the conductance has
In Fig. 1, the numerical results are shown. First of a dip, while when the density is zero, the conductance
all, since the impurity potential only exists at the origin, is perfect. This correlation can be understood from the
3
Gb / G0 obtain the key relation between the tunneling amplitude
and the T−matrix, t(ω) = 1−iπρ(ω)T(ω). Making use
1.0
oftherelationS =1−2πiρT andtheunitarityS =e2iδ,
0.8 the tunneling amplitude is t(ω) = cosδ eiδ. It is clear
that G =G (1−cos2δ) and η =δ so that the relation
b 0 t
0.6 is established.
Our findings here echo with the recent experiment[10]
0.4
that the defect in CNT can turn opaque in the Coulomb
blockade regime or transparent at other bias voltages.
0.2
Furthermore, there exist controlled methods to fabricate
thedefects[11]systematically,ormanipulatetheeffective
0.0
0.4 0.2 0.0 0.2 0.4 impurity strength[12] by a voltage pulse from the tip of
EE/tt
theatomicforcemicroscopeonCNT.Therefore, theres-
onant backward scattering, perfect tunneling and charge
FIG. 3: Dimensionless backscattered conductance gb ≡ accumulationsdiscussedherearerelevantandcanbeob-
Gb/G0 and sin2ηt versus energy plotted in solid and dashed served in experiments on transport through the single
lines. Theidenticalmatchofthesetwocurvesagreeswiththe
defect in CNT.
analytic proof shown in the text.
We acknowledge supports from the National Science
Council in Taiwan through grants NSC-96-2112-M-007-
Lippmann-Schwinger equation – with vanishing density 004 and NSC-97-2112-M-007-022-MY3. Financial sup-
on the defect site, the scattering state remains the same portsandfriendlyenvironmentprovidedbytheNational
as the unperturbed wave function and thus the impurity CenterforTheoreticalSciencesinTaiwanarealsogreatly
potential looks transparent. appreciated.
Thesingle-impurityscatteringconsiderhereisthedual
junction for the tunneling setup through a disconnected
quantum dot. In the tunneling junction, the Friedel sum
rulerelatestheconductancetothephaseofthetransmis-
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(cid:88)
G (ω)=G0 (ω)+ G0 (ω)T G0 (ω), (6) [9] T.Pereg-BarneaandH.-H.Lin,Europhys.Lett.69,791
xx(cid:48) xx(cid:48) xx1 x1x2 x2x(cid:48) (2005).
x1,x2
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nitude of the momentum and ρ(ω) is the density of
states. Note that the tunneling amplitude t(ω) =
G (ω)/G0 (ω) is the ratio of the Green’s functions in
xx(cid:48) xx(cid:48)
the asymptotic limit, x → ∞ and x(cid:48) → −∞, where the
Dysonequationtakesthesimpleformaftersomealgebra,
G (ω) = G0 (ω)[1−iπρ(ω)T(ω)]. By comparison, we
xx(cid:48) xx(cid:48)