Spectral Correlation in Incommensurate Multi-Walled Carbon Nanotubes

5 Pages·0.22 MB·English

by  Kang-Hun Ahn

Spectral Correlation in Incommensurate Multi-Walled Carbon Nanotubes
K.-H. Ahn,1 Yong-Hyun Kim,2 J. Wiersig,3 and K. J. Chang2,4
1 School of Physics, Seoul National University, Seoul 151-747, Korea
2 Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea
3 Max Planck Institute for Physics of Complex Systems,
3
N¨othnitzer Strasse 38, Dresden 01187, Germany
0
4 School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Korea
0
(Dated: February 1, 2008)
2

n Weinvestigatetheenergyspectraofcleanincommensuratedouble-walledcarbonnanotubes,and
a findthattheoverallspectralpropertiesaredescribedbythecriticalstatisticssimilartothatknown
J intheAndersonmetal-insulatortransition. Intheenergyspectra,thereexistthreedifferentregimes

5 characterized by Wigner-Dyson, Poisson, and semi-Poisson distributions. This feature implies that
1 the electron transport in incommensurate multi-walled nanotubes can be either diffusive, ballistic,
or intermediate between them, dependingon theposition of theFermi energy.

]
l
l PACSnumbers: 03.65.-w,72.15.-v,73.23.-b
a
h
-
Carbon nanotubes have attracted great attention due At the Anderson transition point, a critical statistics
s
e to their remarkable electrical and mechanical proper- was found [14]. The short-range level correlation has
m
ties [1]. Many theoretical and experimental studies have been later described by semi-Poisson distribution [12],
. demonstratedthatsingle-wallednanotubes(SWNTs)ex- whichhasnowbeenanimportantissueinquantumchaos
t
a hibit ballistic electron conduction [2, 3], and single- [15, 16, 17].
m
molecule devices utilizing semiconducting SWNTs have In this Letter we make a spectral analysis of the en-
- been realized [4]. On the other hand, despite useful ap- ergy levels of clean incommensurate MWNTs, and find
d
plications of multi-walled nanotubes (MWNTs) [5], the that the level statistics is similar to that of the Ander-
n
o transportproperties ofMWNTs arenot wellunderstood son metal-insulator transition. The energy spectra ex-
c andevencontroversial;conductancemeasurementsusing hibit both the Wigner-Dyson and Poisson distributions
[
scanning probe microscope showed ballistic behavior [6], depending on the position of the Fermi level. In ad-

2 while magnetoresistances measured for MWNTs on top dition, we demonstrate that incommensurate MWNTs
v of metallic gate indicated diffusive conduction [7, 8]. have an intermediate regime, which satisfies the semi-
8 Poissonstatistics.
In general, MWNTs have very complex electronic
7
To investigate the spectral correlation, we consider
2 structure, so that a direct use of their transport prop-
double-walled carbon nanotubes (DWNTs), because the
8 erties is severely hindered. If concentric carbon shells
0 are especially incommensurate, the Bloch theorem is no electrical conduction in MWNTs is believed to be gov-
2 erned by the outermost shells [8, 18]. We use a tight-
longer valid, and the Landauer formula for conductance
0 bindingmodelwithoneπ-orbitalpercarbonatom,which
is notapplicable because it is difficult to countthe num-
/
at ber of conducting channels [9]. In fact, several experi- successfullydescribestheelectronicstructureofDWNTs
[19, 20]. The tight-binding Hamiltonian is given by
m ments on MWNTs indicated that carbon shells have of-
ten different periodicities [10, 11]. However, electronic
d- structurecalculationshavebeenmostly focusedforcom- H =γ0 c†jci−W cos(θi′j′)e(a−di′j′)/δc†j′ci′, (1)
n mensurate multi-walled nanotubes so far. Here, instead Xi,j Xi′,j′
o
of directly calculating conductances, we investigate the
c whereγ (=-2.75eV)is the hoppingparameterbetween
0
v: spectral properties of energy levels in incommensurate intra-layer nearest neighbor sites, i and j, and W (=
MWNTs.
Xi γ0/8) is the strength of inter-wall interactions between
The spectralanalysisofenergylevelshasprovento be inter-layer sites, i′ and j′, with the distance of di′j′ and
ar a useful tool to probe the nature of eigenstates in disor- thecut-offfordi′j′ >3.9˚A.Hereθij istheanglebetween
dered conductors [12]. In the diffusive metallic regime, twoπ orbitals,ci′ is the annihilationoperatorofanelec-
the statistics of spectral fluctuations is described by the tron on site i′, a (= 3.34 ˚A) is the distance between two
Gaussian orthogonalensemble (GOE) of random-matrix carbon walls, and δ = 0.45 ˚A.
theory [13]. In this regime, the distribution P(s) of en- To extract the fluctuations from the level sequence, it
ergy spacings between nearest levels is well fitted by the is customary to map the real energy spectra {ǫ } onto
i
Wigner-Dyson surmise P (s)= πsexp(−πs2), where the unfolded spectra {E } through E = N¯(ǫ ), where
GOE 2 4 i i i
s is in units of mean level spacing ∆. In the insulating N(ǫ ) is the number of levels up to ǫ and the over-
i i
regime, where the energy levels are uncorrelated, P(s) line denotes its broadened value [21]. Here we use the
is given by the Poisson distribution, P (s) = exp(−s). Gaussian broadening scheme for averaging the energies
P
2
1 (SP) distribution [15, 16, 17];
5
P (s) = 4sexp(−2s) (2)
SP
0.8
L 1
L) ΣSP(L) = + (1−e−4L). (3)
( 2 8
Σ
0.6 The SP distribution is defined by removing every other
s) 0 level from an ordered Poisson sequence and turned out
(
P 0 5 10 tobeareferencepointforthecriticalstatisticsofseveral
0.4
L disordered systems like the Anderson model at mobility
edge [14, 23], and also of other systems without disorder
such as pseudo-integrable quantum billiards [16, 17, 24].
0.2
While P(s) carries information on short-range corre-
lation in the energy spectra, Σ(L) contains rather long-
0 range correlation. Usually Σ(L) indicates a deviation
0 1 2 3 4 from the universal value for large L, thus, the particle
s dynamics becomes nonuniversal at short time scale. For
large L, the spectral correlation is not universally semi-
Poisson but depends on the helicity of nanotubes. The
FIG. 1: The nearest energy spacing distribution P(s) for
deviation from the SP distribution gives useful informa-
energy levels between -7 and 7eV and different helicities
tion; the linear behavior for large L, i.e., Σ(L)/L → χ,
(16,5)/(23,8)(triangle),(17,2)/(16,15)(circle),(16,5)/(17,15)
(square), and (17,2)/(21,9) (diamond). The semi-Poisson represents the level compressibility, and for disordered
(dashed),Poisson (long-dashed),andGOE(dotted)distribu- metals χ was shown to be [25]
tions were also plotted for comparison. The inset shows the
numbervariance Σ(L) for (16,5)/(23,8) case. The numberof 1 D2
χ= 1− , (4)
considered levels for (16,5)/(23,8) case is 20220. 2 d
(cid:0) (cid:1)
whereD is the multifractalexponentofthe inversepar-
2
ticipationratioanddisthespatialdimension. HereD /d
2
[22]. After unfolding, we obtain the distribution P(s) is related to the ’probability of return’ [26],
as a function of nearest energy spacing, s = E −E
i i i−1 p(t)∝t−D2/d. (5)
(the mean level spacing of the unfolded spectra equals
1), and the number variance Σ(L) = (n(L,E)−L)2 ,
Our incommensurate DWNT behave as a two-
which represents the variance of the(cid:10)number of leve(cid:11)ls
dimensional systems (d = 2) with off-diagonal disorder,
in the interval (E − L/2,E + L/2), i.e., n(L,E) =
whichhasamixedboundarybetweenperiodicandhard-
N(E+L/2)−N(E−L/2). Thebracket ··· denotesthe
wall conditions. From the inset in Fig. 1, we estimate
averagearoundthe energyE onanener(cid:10)gyw(cid:11)indowmuch
D /d to be about 0.32 for the DWNT considered here.
2
largerthanthemeanlevelspacingbutmuchsmallerthan
While p(t) ∝ t−d/2 in normal diffusion, our system ex-
E.
hibits anomalous diffusion, which was also found by pre-
The two shells of DWNTs are incommensurate if the vious wavepacketspreading calculations [20].
ratio of the unit cell lengths along the tube axis is irra- The SP distribution in Fig. 1 is mostly contributed
tional. We test many incommensurate DWNTs, where from the region of large |E|, where the density of states
both the shells are metallic or semiconducting, a semi- (DOS) is high, while the energy statistics is usually not
conductingshellis insideametallic tube, andviceversa. independent of the energy regime. Since real electron
The diameters of the inner and outer tubes are 1.5 and conductions occur near the Fermi level, it is instructive
2.2 nm, respectively, and the nanotube length is set to to examine the energy-level statistics on various energy
beabout53nm. Wefindthespectralpropertiesaresim- windows. Although we discuss the level statistics when
ilar for all the incommensurate DWNTs with different theFermilevelincreases,wefindsimilarstatisticsforthe
helicities. (See the P(s) in Figure 1). Since the spec- downward shift of the Fermi level, which usually occurs
tral properties are similar for all the tubes considered in hole-doped tubes.
here, from here on, we only present the spectral proper- In the level statistics, we remove the states lying be-
ties of the (16,5)/(23,8) DWNT where the semiconduct- tween 0 and 0.08 eV, which are associated with the lo-
ing(16,5)single-walledtubeisalignedinsidethemetallic calizedstatesnearthesampleboundaryduetothe finite
(23,8) tube. In the inset of Figure 1, Σ(L) is also shown sizeofthesystem. InFig. 2(a),P(s)isdrawnforlowen-
for the (16,5)/(23,8) nanotube. We find that P(s) and ergystates from0.08to 1.3 eV,and exhibits the Poisson
Σ(L) cannot be described by the Poissondistribution or distribution, indicating that the energy levels are uncor-
GOE, but, they are well described by the semi-Poisson related. Single-wall nanotubes, which are constituents
3
Fig. 2 Ahn, et al.
of the DWNT, are approximately described in terms of a) 1 b)
GOE
PP(s) and δ(s). In this case, the delta-function peak re- 0.8 Semi−Poisson
sults from the doubly degenerate states due to inversion Poisson
symmetry. If inter-wall interactions are absent in the s) 0.6
DWNT, a superposition of two independent spectra of P( 0.4
the nanotube constituents is also described by the Pois-
0.2
son and delta functions.
When inter-wall interactions break the degeneracy, 0
c) 1 d)
smearing out the delta-function peak, P(s) is still GOE GOE
Poisson-like, as shown in Fig. 2(a), implying that the 0.8 Semi−Poisson Semi−Poisson
energy mixing between the two carbon shells is insignif-
0.6
icant for low energies. In the low energy regime, it is P(s)
0.4
useful to use the Landauer formula,
0.2
2e2
G= T (E ), (6) 0
h i F 0 1 2 3 4 0 1 2 3 4
X
i s s
whereTi(EF)isthetransmissionprobabilityofthechan- FIG. 2: The nearest energy spacing distribution P(s) on
neliobtainedfromthebandstructurecalculationsatthe the energy windows of (a) (0.08,1.3eV) with 511 levels, (b)
Fermi energy E [9]. Our results infer that quantized (1.0,2.5eV) with 1872 levels, (c) (2.5,3.5eV) with 2696 levels,
F
conductancesobservedinMWNTs[6]maybedue tothe and (d) (3.5,4.0eV) with 888 levels.
fact that the Fermi level is close to the charge neutrality
point, E ≈ 0. However, it is usually very difficult to
F
regime,becausethesequenceofstatisticswithincreasing
find experimentally the location of the Fermi level.
energy is Poisson→SP→GOE→SP .
As the Fermi level is shifted further, since the mean
In MWNTs on top of metallic gates, the details of
level spacing becomes smaller and the inter-wallinterac-
contacts,localexcessivecharges,anddepletionofcharge
tion matrix elements become more significant, the inter-
carriers may change the position of the Fermi level [28].
wall interactions induce a more significant mixing of the
Basedon ourresults, we guessthat in conductance mea-
energy levels between the carbon shells. For higher en-
surements[2, 4, 8], the Fermilevelis shifted to the GOE
ergystates,wefindthatP(s)followstheSPdistribution,
regimeortothecriticalregime(closetotheGOE)where
as shown in Figs. 2(b) and (d). The quantum conduc-
conductivity is properly described by the weak localiza-
tance in this regime still remains a challenging problem.
tion formula.
Inparticular,itisanontrivialbutimportanttasktofind
The calculated variances over every 1000 consecutive
the tube length dependence of quantum conductance in
levels are compared with the densities of states for two
this regime. Recent calculations [27] showed negligible
different DWNTs in Fig. 3. The size of fluctuations of
inter-walltunnelingcurrentsforthetubelengthof1µm,
the level spacing is dictated by the spectral statistics;
but it is questionablewhether one canuse the Bardeen’s
formalismforDWNTswherethecoherentback-tunneling ∞
betweencarbonshellsisnon-negligible. Accordingtothe (s− s )2 = ds(s−1)2P(s) (7)
Z
weaklocalizationformula,δσ ∝− dtp(t),withEq. (5), (cid:10) (cid:10) (cid:11) (cid:11) 0
1 4
conductances may depend on the tRube length. However, = 1, , −1 (Poisson, SP, GOE).
2 π
the weak localization formula considers only the inter-
ference of the pair of time-reversal paths, which is not Itis expectedthat the effect ofinter-wallinteractionson
guaranteed in the critical regime. the level spacing is most significant when the mean level
In Fig. 2(c), one can see that the critical statistics spacing ∆ is minimum near E =±γ =±2.75eV, where
0
evolves to the GOE as going to the higher energy re- themaximumDOSoccursonagraphenesheet. Onecan
gion,where P(s)is quite closeto the Wigner-Dysonsur- see (s− s )2 = 4−1≈0.27nearE =±γ =±2.75eV,
π 0
mise. We also find the GOE-like statistics for both P(s) whi(cid:10)ch is(cid:10)a(cid:11)sign(cid:11)ature of GOE distribution. The variances
andΣ(L)inalltheincommensurateDWNTstestedhere. largerthan1 near E =0 are due to the fact that P(s) is
The appearance of the GOE-like behavior indicates that notcompletelyPoisson-likebecausethedegeneratelevels
allthesymmetriesexceptfortime-reversalsymmetryare of each nanotube layer are not completely lifted up.
effectivelybroken,andthecorrespondingelectronmotion We find that a transition from the Poisson-liketo SP-
is ergodic. In this regime, the conductivity of MWNTs like statistics occurs at lower energies as the diameter of
may be described by the weak localization theory. One nanotubes increases. One of the reasons for this trend
should also note that the weak localization theory is ap- seems that the DOS is enhanced for nanotubes with
plicableforenergieshigherthanthoseinthe firstcritical larger diameters. The transition energy is close to 1 eV
4
Fig. 3 Ahn, et al.
energywindow,whiletheoverallstatesarewelldescribed
bytheSPdistribution. Thisresultsindicatethatthena-
ture of electron transportin multi-walled nanotubes can
1
s) be either ballistic, diffusive,or critical,depending onthe
2−<s>)> arb. unit pusousiatliowneaokf tlhoecaFliezramtiiolnevceolr.reItctiisonquisesrteiolenveadntwthoetehxeirsttinhge
<(s 0.5 S ( experiments. The Coulomb blockade oscillation in nan-
O
otube dots is suggestedtoinvestigatethe spectralstatis-
D
tics in this work.
0 The authors acknowledge J. Ihm, J. Yu, B. L. Alt-
shuler, G. Montambaux, K. Richter, S. Evangelou, H.-S.
Sim, G. Cuniberti, and H.-W. Lee for useful discussions.
This work was supported by QSRC and Brain Korea 21
1 s) project.
2−<s>)> arb. unit
<(s 0.5 S (
O
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5
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