Transport properties of disordered carbon nanotubes with long-range Coulomb interaction
Hideo YoshiokaDepartment of Physics, Nagoya University, Nagoya 464-8602, Japan
and Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
共Received 26 March 1999; revised manuscript received 16 December 1999兲
Transport properties of disordered carbon nanotubes are investigated including long-range Coulomb inter-action. The resistivity and optical conductivity are calculated by using the memory functional method. In addition, the effect of localization is taken into account by use of the renormalization-group analysis, and it is shown that the backward scattering of the intravalley and that of the intervalley cannot coexist in the localized regime. Differences between the transport properties for the metallic state with two valleys and those with one valley are discussed.
Single wall carbon nanotubes共SWNT’s兲 are the new ma-terials that are an experimental realization of one-dimensional 共1D兲 electron systems.1 Since the SWNT is made by rolling up a graphite sheet, it is expected that fas-cinating properties different from the conventional quantum wires made from semiconductor heterostructures will be ob-served. From this point of view, transport properties of dis-ordered SWNT’s have been discussed in Refs. 2 and 3. It has been predicted that the backward scattering due to the impu-rities vanishes when the range of the impurity potential is much larger than the lattice constant, but that the backward scattering reappears when applying a magnetic field perpen-dicular to the tube axis. It should be noted that the absence of the backward scattering holds for the graphite sheet as well as any SWNT’s.
In the above studies, electron correlations have been ne-glected. The 1D nature together with the electron-electron interaction has been known to result in a variety of correla-tion effects in SWNT’s in case of short range interaccorrela-tions4–6 and of the long-range Coulomb interaction.7–10 Effects of electronic correlation in SWNT’s have been measured in the Coulomb blockade regime11as well as for Ohmic contacts.12 In the latter experiment, power-law dependences of the con-ductance as a function of temperature and of the differential conductance as a function of bias voltage have been ob-served and interpreted in terms of tunneling into clean SWNT with the interaction.
Even when the Coulomb interaction is taken into account, the conclusion of the absence of the backward scattering in Refs. 2 and 3 is not changed. However, the effects of the Coulomb interaction on the transport in disordered SWNT’s should be observable in case of shorter range impurity po-tentials. In the present paper, I will discuss the transport properties of SWNT’s with short range impurity potentials and the long-range Coulomb interaction. It is shown that the interaction gives rise to an enhanced resistivity compared to that without the interaction. In addition, the interaction leads to a power-law dependence of the resistivity as a function of temperature and modifies the power of the frequency for the optical conductivity. In the localized regime, it is found that intravalley and intervalley backward scattering cannot coexist.
The SWNT has metallic bands when the wrapping vector, wជ⫽N⫹aជ⫹⫹N⫺aជ⫺, satisfies the condition, N⫹⫺N⫺⫽0
mod 3, where aជ⫾⫽(⫾1,))a/2 with a being the lattice con-stant. The Hamiltonian of the metallic SWNT with the long-range Coulomb interaction is written by the slowly varying Fermi field,p␣s, of the sublattice p⫽⫾, the spin s⫽⫾ and the valley␣⫽⫾, as follows:10
H0⫽⫺iv0
兺
p␣s ␣ e⫺ip␣冕
dxp†␣sx⫺p␣s ⫹V¯ 共0兲 2冕
dx共x兲 2, 共1兲 where (x)⫽兺p␣sp␣s † p␣s, v0⯝8⫻105m/s, and V¯ (0) ⫽2e2/ln(Rs/R) with⯝1.4, Rsand R being the cutoff of long range of the Coulomb interaction and the radius of the tube, respectively. In Eq. 共1兲, ⫽tan⫺1关(N⫹⫺N⫺)/)(N⫹ ⫹N⫺)兴 is the chiral angle (⫽0 corresponds to an armchair nanotube兲. In the above expression, I disregard the matrix elements of the interaction of the order of a/R, which lead to energy gaps.7–10Therefore, the present theory is valid when the temperature T, or the frequency , are larger than the gaps induced by the Coulomb interaction.
The impurity potential introduced as disorder of the atomic potential is given by the following Hamiltonian,2
Him p⫽兺p␣␣⬘s兰dxV␣␣⬘ p (x)p␣s † p␣⬘s, where V␣␣⬘ p (x) is the impurity potential at the sublattice, p, by which the elec-tron on the valley, ␣
⬘
, is scattered into the valley, ␣.13 Here I diagonalize the kinetic term in Eq. 共1兲 and move to the basis of the right-moving (r⫽⫹) and the left-moving (r⫽⫺) electrons by the unitary transformation, p␣s ⫽(e⫺ip␣/2/&)兺r(␣r)(1⫺p)/2r␣s. Then,Him pis written as follows: Him p⫽
兺
r␣s 1 2冕
dx兵关V␣␣ ⫹共x兲⫹V ␣␣ ⫺ 共x兲兴 r␣s † r␣s ⫹关ei␣V ␣⫺␣ ⫹ 共x兲⫺e⫺i␣V ␣⫺␣ ⫺ 共x兲兴 r␣s † r⫺␣s ⫹关V␣␣⫹共x兲⫺V␣␣⫺共x兲兴r␣s † ⫺r␣s⫹关ei␣V␣⫺␣⫹ 共x兲 ⫹e⫺i␣V ␣⫺␣ ⫺ 共x兲兴 r␣s † ⫺r⫺␣s其. 共2兲Here, the first共second兲 term expresses the intravalley 共inter-valley兲 forward scattering, and the third 共fourth兲 one is the
PHYSICAL REVIEW B VOLUME 61, NUMBER 11 15 MARCH 2000-I
PRB 61
intravalley共intervalley兲 backward scattering. When the range of the impurity potential is much larger than the lattice con-stant, V␣␣⫹ ⫽V␣␣⫺ and V␣⫺␣⫹ ⫽V␣⫺␣⫺ ⫽0, which leads to van-ishing of the third and fourth terms in Eq. 共2兲, i.e., the ab-sence of backward scattering, in agreement with Ref. 2. Here, I consider the case of an impurity potential with range shorter than the lattice constant by retaining finite matrix elements of the backward scattering in Eq. 共2兲. I disregard forward scattering because it does not contribute to transport. As was pointed out by Abrikosov and Ryzhkin,14 in 1D systems and in the limit of weak impurity potentials, the interaction between the electrons and the impurities can be parameterized by uncorrelated Gaussian random fields. Here, I extend this method to the present model and introduce two kinds of the random fields,(x) and(x), expressing the intravalley and the intervalley backward scattering, respectively. The fields have the probabil-ity distributions, P⫽exp兵⫺(2D1)⫺1兰dx2(x)其 and P ⫽exp兵⫺(D2)⫺1兰dx(x)*(x)其 where D1 and D2 are given by v0/1 andv0/2 with1 (2) being the scattering time due to the intravalley共intervalley兲 backward scattering. The Hamiltonian of the impurity potential is given by
Him p⫽
冕
dx共x兲兺
r␣s r␣s † ⫺r␣s ⫹冕
dx再
共x兲兺
rs r⫹s † ⫺r⫺s⫹H.c.冎
. 共3兲 Note that the intravalley 共intervalley兲 backward scattering, where the momentum transfer in the scattering process is small 共large兲, is parameterized by a real 共complex兲 field.Here, I utilize the bosonization method and introduce the phase variables expressing the symmetric (␦⫽⫹) and anti-symmetric (␦⫽⫺) modes of the charge ( j⫽) and spin ( j ⫽) excitations, j␦ andj␦. The phase fields satisfy the commutation relation, 关j␦(x),j⬘␦⬘(x
⬘
)兴⫽i(/2)sign(x ⫺x⬘
)␦j j⬘␦␦␦⬘. The Fermi field,r␣s, is expressed asr␣s⫽ r␣s
冑
2˜aexp冋
irqFx⫹ ir 2兵␣s⫹r␣s其册
, 共4兲 where ␣s⫽⫹⫹s⫹⫹␣⫺⫹␣s⫺, ␣s⫽⫹ ⫹s⫹⫹␣⫺⫹␣s⫺, and a˜⫺1 is a large momentum cutoff.15 Klein factorsr␣s are introduced to ensure correct anticommutation rules for different species r,␣,s, and sat-isfy 关r␣s,r⬘␣⬘s⬘兴⫹⫽2␦rr⬘␦␣␣⬘␦ss⬘. The spin-conserving products r␣sr⬘␣⬘s in the Hamiltonian can be represented as,7 A⫹⫹(r,␣,s)⫽r␣sr␣s⫽1, A⫹⫺(r,␣,s)⫽r␣sr⫺␣s ⫽i␣x, A⫺⫹(r,␣,s)⫽r␣s⫺r␣s⫽ir␣z, and A⫺⫺ (r,␣,s)⫽r␣s⫺r⫺␣s⫽⫺iry with the standard Pauli ma-trices i (i⫽x,y,z). The quantity qF⫽n/4 is related to the deviation n of the average electron density from half filling and can be controlled by the gate voltage. The Hamil-tonian, H0 and Him p⫽兺i⫽1,2Him pi
, is expressed by the phase variables as follows:
H0⫽
兺
j⫽,␦兺
⫽⫾ vj␦ 2冕
dx兵Kj␦ ⫺1共 xj␦兲2⫹Kj␦共xj␦兲2其, 共5兲 Him p 1 ⫽ iz 2˜a冕
dx共x兲兺
r␣s r␣exp共⫺2irqFx兲 ⫻exp兵⫺ir共⫹⫹s⫹⫹␣⫺⫹␣s⫺兲其, 共6兲 Him p 2 ⫽⫺iy 2˜a冕
dx兺
rs r exp兵⫺ir共2qFx⫹⫹⫹s⫹兲其 ⫻关共x兲exp兵⫺i共⫺⫹s⫺兲其⫹H.c.兴, 共7兲 where K⫹⫽(v⫹/v0)⫺1⫽1/冑
1⫹4V¯(0)/(v0) and Kj␦ ⫽vj␦/v0⫽1 for the others. The Pauli matrices seen in Him pi
are due to the product of Klein factors.
With the above phase Hamiltonian, I calculate the dy-namical conductivity (), which is expressed by the memory function, M (), as follows:16
共兲⫽⫺i⫹M共共0兲兲, 共8兲
M共兲⫽共
具
F;F典
⫺具
F;F典
⫽0兲/⫺共0兲 , 共9兲
where
具
A;A典
⬅⫺i兰dx兰0⬁dte(i⫺)t具
关A(x,t),A(0,0)兴典
with→⫹0, 具¯典 denotes the thermal average with respect to H, F⫽关 j,H兴 with j being the current operator, and (0) ⫽
具
j; j典
⫽0. Since the present Hamiltonian conserves the total particle number, there are no nonlinear terms includ-ing ⫹. Then the current operator is expressed by j⫽2v⫹K⫹x⫹/, which leads to (0)⫽ ⫺4v⫹K⫹/. To lowest order in D1 and D2, M () is calculated as M共兲⫽2v⫹K⫹ i⫽1,2兺
Di 共˜a兲2sin Ki 2冉
2T F冊
Ki 1 T ⫻再
B冉
Ki 2 ⫺i 2T,1⫺Ki冊
⫺B冉
Ki 2 ,1⫺Ki冊冎
, 共10兲 where K1⫽(K⫹⫹K⫹⫹K⫺⫹K⫺)/2, K2⫽(K⫹⫹K⫹ ⫹K⫺⫺1⫹K⫺⫺1)/2, Fis the cutoff energy given byv0/a˜ and B(x,y )⫽⌫(x)⌫(y)/⌫(x⫹y) with ⌫(x) being the gamma function. For→0, M() reduces to M共0兲⫽i2v⫹K⫹兺
i⫽1,2 Di v0 2冉
2T F冊
Ki⫺2⌫2共K i/2兲 ⌫共Ki兲 , 共11兲 which leads to the resistivity,⫽共0兲⫺1⫽ 2 2a˜i
兺
⫽1,2Di冉
2T F冊
Ki⫺2⌫2共K i/2兲 ⌫共Ki兲 , 共12兲 where Di⫽Di˜ /(a v0 2)⫽a˜/(li) with li⫽v0i being the mean-free path. In the present case of Ki⫽(K⫹⫹3)/2, the above expression results in ⫽B0(2T/
F)(K⫹⫺1)/2⌫2((K⫹⫹3)/4)/⌫((K⫹⫹3)/2) where B0 ⫽/2(l1⫺1⫹l2⫺1) is the resistivity for the noninteracting sys-tem in the Born approximation. It is remarkable that/B0 is
independent of the scattering strength. For typical nanotubes with K⫹⯝0.2, the resistivity is enhanced compared to that without the Coulomb interaction and shows a temperature dependence as⬀T(K⫹⫺1)/2⯝T⫺0.4.17In the presence of the long-range Coulomb interaction, the phase expressing the symmetric charge fluctuation,⫹, becomes rigid compared to the noninteracting case, which is expressed by the fact of K⫹ less than unity. Such a rigid phase is easily pinned by the impurity potential. Thus, the long-range Coulomb inter-action enhances the resistivity. From the asymptotic behavior of Eq. 共10兲 forⰇT, the optical conductivity for high fre-quencies is calculated as ()⬀(K⫹⫺5)/2⬃⫺2.4, which decays faster than that for the noninteracting case, () ⬀⫺2. I note that the above two results of the resistivity and the optical conductivity do not depend on the filling, qF. The umklapp scattering has been known to be another origin of resistivity. The umklapp scattering leads to ⬀T(2K⫹⫺1)
⯝T⫺0.6,8,9 for TⰇv
0qF and ()⬀(2K⫹⫺3)⯝⫺2.6 for
Ⰷv0qF. Though the powers due to the umklapp scattering are similar to those given by the impurity scattering, it is possible to separate them by tuning the gate voltage.
Next I take into account of the effects of localization by the renormalization-group共RG兲 method. Following Giamar-chi and Schulz,18averaging over the random fields,(x) and
(x), I obtain the action, Sim p, corresponding to the impu-rity potential as follows:
Sim p1 ⫽⫺D1 2
冉
4 a˜冊
2冕
dx冕
0  dd⬘
⫻兵cos共2qFx⫹⫹兲cos⫹sin⫺cos⫺ ⫻cos共2qFx⫹⫹
⬘
兲cos⫹⬘
sin⫺⬘
cos⫺⬘
⫹sin共2qFx⫹⫹兲sin⫹cos⫺sin⫺⫻sin共2qFx⫹⫹
⬘
兲sin⫹⬘
cos⫺⬘
sin⫺⬘
⫺2 cos共2qFx⫹⫹兲cos⫹sin⫺cos⫺ ⫻sin共2qFx⫹⫹⬘
兲sin⫹⬘
cos⫺⬘
sin⫺⬘
其, 共13兲 Sim p2 ⫽⫺D2冉
2 ˜a冊
2冕
dx冕
0  d d⬘
e⫺i(⫺⫺⬘⫺)⫻兵sin共2qFx⫹⫹兲cos⫹cos⫺ ⫻sin共2qFx⫹⫹
⬘
兲cos⫹⬘
cos⫺⬘
⫹cos共2qFx⫹⫹兲sin⫹sin⫺ ⫻cos共2qFx⫹⫹⬘
兲sin⫹⬘
sin⫺⬘
⫹i关sin共2qFx⫹⫹兲cos⫹cos⫺ ⫻cos共2qFx⫹⫹⬘
兲sin⫹⬘
sin⫺⬘
⫺cos共2qFx⫹⫹兲sin⫹sin⫺⫻sin共2qFx⫹⫹
⬘
兲cos⫹⬘
cos⫺⬘
兴其, 共14兲 where j␦⫽j␦(x,), ⬘
j␦⫽j␦(x,⬘
), j␦⫽j␦(x,), andj
⬘
␦⫽j␦(x,⬘
). It should be noted that the interaction pro-cesses generated from the equal-time component in Eqs.共13兲 and共14兲 are disregarded. The results given in the following do not qualitatively changed even when the such processes are included because such operators are less divergent than Sim pi . By calculating the various correlation functions to the lowest order of D1and D2, we have the following RG equa-tions: D1⬘
⫽兵3⫺共K⫹⫹K⫹⫹K⫺⫹K⫺兲/2其D1, 共15兲 D2⬘
⫽兵3⫺共K⫹⫹K⫹⫹K⫺⫺1⫹K⫺⫺1兲/2其D2, 共16兲 Kj⫹⬘
⫽⫺共D1/X1⫹D2/X2兲K2j⫹uj⫹, 共17兲 uj⫹⬘
⫽⫺共D1/X1⫹D2/X2兲Kj⫹uj⫹ 2 , 共18兲 Kj⫺⬘
⫽⫺共D1Kj⫺ 2 /X 1⫺D2/X2兲uj⫺, 共19兲 uj⫺⬘
⫽⫺共D1Kj⫺/X1⫹D2Kj⫺1⫺/X2兲uj⫺ 2 , 共20兲 where⬘
denotes d/dl with dl⫽d ln(a˜⬘
/a˜) (a˜⬘
is the new cutoff parameter兲, X1⫽u⫹K⫹/2
u⫹K⫹/2u⫺K⫺/2u⫺K⫺/2, X2 ⫽u⫹K⫹/2
u⫹K⫹/2u⫺1/2K⫺u⫺1/2K⫺, and j⫽ or . The initial conditions for the above RG equations are as follows,
Di(0)⫽Di˜ /(a v0 2), K
⫹(0)⫽u⫹⫺1⫽K⫹, and K⫹(0) ⫽K⫺(0)⫽K⫺(0)⫽u⫹(0)⫽u⫺(0)⫽u⫺(0)⫽1. The above equations together with the initial conditions indicate that K⫺⫽K⫺ and u⫺⫽u⫺. In addition, the RG equa-tions are invariant under the transformation, D1↔D2 and Kj⫺↔K⫺1j⫺. Therefore, I can discuss the case of D1(0) ⬎D2(0) without loss of generality.
By solving the RG equations numerically and using Eq. 共12兲, the resistivity is obtained as a function of temperature.19 I show the results for D1(0)⫽1/300 and
D2(0)⫽1/600 and those for D1(0)⫽1/300 and D2(0) ⫽1/3000 in Fig. 1. Surely, the resistivity is enhanced by the effects of the localization for low temperature. The inset
FIG. 1. Dependence on 2T/Fof the resistivity,, normalized
by B0, which is derived by the RG analysis for 共1兲 D1(0)
⫽1/300 and D2(0)⫽1/600 and 共3兲 D1(0)⫽1/300 and D2(0)
⫽1/3000. The dotted lines express the resistivity given by the per-turbation theory for 共2兲 D1(0)⫽1/300 and D2(0)⫽1/600 and 共4兲
D1(0)⫽1/300 and D2(0)⫽1/3000. Here, K⫹⫽0.2 is used and the
white circle shows the temperature corresponding toD1⯝1, below which the perturbative RG analysis breaks down. Inset: The ratio, 2/1as a function of 2T/F. The numbers,共1兲-共4兲, express the
same cases as the main figure.
shows the ratio,2/1, as a function of temperature, where
1 (2) is the resistivity due to the intravalley共intervalley兲 scattering. In the present case where the intravalley scatter-ing is stronger than the intervalley one,2/1decreases with decreasing temperature. For the opposite case, 1/2 de-creases. The result is not obtained by the perturbation theory 共dotted lines in the inset in Fig. 1兲, and then characteristic for the localized regime. It indicates that the two kinds of the backward scattering cannot coexist for this regime. As is seen in Eqs.共6兲 and 共7兲, the intravalley 共intervalley兲 scatter-ing pins the phases, ⫹, ⫹, ⫺, and ⫺ (⫹, ⫹, ⫺, and ⫺). Since the conjugate vari-ables, ⫺ and⫺, or ⫺ and⫺ cannot be pinned at the same time, the localization due to the two kinds of the scattering cannot happen simultaneously. In the present analysis, the quantitative discussion on the localized regime and crossover towards it have not been done. Therefore, more detailed investigation are needed for understanding of disordered SWNT’s with the Coulomb interaction.
The above results of the metallic state with two valleys (2v state兲 are compared to the transport properties of the
metallic state with one valley共1v state兲, which is realized by applying the parallel magnetic flux of (i⫾1/3)hc/e 共i:inte-ger兲 to the insulating tube.20In this case, the intervalley scat-tering is neglected. In addition, the exponent of the charge mode is given by K⫽1/
冑
1⫹2V¯(0)/(v0). Note that the difference in the coefficient of V¯ 共0兲/(v0) between K⫹and Kresults from that in the number of conducting bands. On the other hand, K⫽1. The resistivityand the optical con-ductivity 共兲 for high frequencies are, respectively, given by ⫽B01v(2T/F)K⫺1⌫2关(K⫹1)/2兴/⌫(K⫹1) and()⬃K⫺3, with B0 1v⫽/l
1 being the resistivity for the 1v state by the Born approxmation. Thus the behaviors in
the 1v state are different from those in the 2v state. Since
⌫2关(K
⫹1)/2兴/⌫(K⫹1)⬎⌫2关(K⫹⫹3)/4兴/⌫关(K⫹⫹3)/ 2兴⬎1 and K⫺1⬍(K⫹⫺1)/2⬍0 as a function of V¯(0), the effects of the Coulomb interaction are stronger in the 1v
state than in the 2v state. From these results, it is expected
that the resistivity共optical conductivity兲 of multiwall carbon nanotubes, in which both the metallic and insulating carbon nanotubes exist, shows different temperature共frequency兲 de-pendence in cases without parallel magnetic flux and with that of hc/(3e).
In conclusion, I investigated the transport properties of the disordered SWNT’s with Coulomb interaction. I found that the interaction enhances the resistivity, leads to a power-law dependence of the resistivity as a function of tempera-ture, and modifies the power of the frequency of the optical conductivity. In addition, it is shown that the intravalley and the intervalley backward scattering cannot coexist in the lo-calized regime. Differences between the 2v and 1v state
were also discussed. For away from half-filling, the gaps induced by the Coulomb interaction, ⌬, are given as ⌬/F ⬃e⫺c/u,7,8 with u⬃a/(2R) and c⬃O(1), and thus very small for typical SWNT’s. Therefore, the results obtained in this paper will be well observed for away from half filling with TⰆv0qF or Ⰶv0qF. The increase of the resistance with decreasing temperature observed in SWNT’s 共Ref. 21兲 may be due to the impurity scattering because the difference in the work function of the metallic electrode and the tube results in a downward shift of Fermi level of the nano-tube by a few tenths of an eV.22
The author would like to thank T. Ando, G. E. W. Bauer, Yu. V. Nazarov, A. A. Odintsov, Y. Tokura, A. Furusaki, H. Suzuura, and A. Khaetskii for stimulating discussions. This work was supported by a Grant-In-Aid for Scientific Re-search 共11740196兲 from the Ministry of Education, Science, Sports and Culture, Japan.
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13Correspondence between the impurity potential of the present
pa-per and those of Ref. 2 is as follows: V␣␣⫹⫽uA, V␣⫺␣⫹
⫽uA⬘, V␣␣⫺⫽uB, and V␣⫺␣⫺ ⫽uB⬘.
14A. A. Abrikosov and I. A. Ryzhkin, Adv. Phys. 27, 147共1978兲. 15The cutoff parameter a˜ is considered to be of the order of the
radius of the tube because only the lowest subbands are taken into account in the present discussion关T. Ando 共private commu-nication兲兴.
16W. Go¨tze and P. Wo¨lfle, Phys. Rev. B 6, 1126共1972兲. 17The same exponent has been obtained in Refs. 7 and 8. 18T. Giamarchi and H. J. Schulz, Phys. Rev. B 37, 325共1988兲. 19T. Giamarchi, Phys. Rev. B 44, 2905共1991兲; 46, 342 共1992兲. 20H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 1255共1993兲; Physica
B 201, 349共1994兲.
21C. L. Kane et al., Europhys. Lett. 41, 683共1998兲. 22J. W. G. Wildo¨er et al., Nature共London兲 391, 59 共1998兲.
