• Nie Znaleziono Wyników

Universality of electron correlations in conducting carbon nanotubes

N/A
N/A
Protected

Academic year: 2021

Share "Universality of electron correlations in conducting carbon nanotubes"

Copied!
4
0
0

Pełen tekst

(1)

Universality of electron correlations in conducting carbon nanotubes

Arkadi A. Odintsov

NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540

and Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Hideo Yoshioka

Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands and Department of Physics, Nagoya University, Nagoya 464-8602, Japan

~Received 10 November 1998!

An effective low-energy Hamiltonian of interacting electrons in conducting single-wall carbon nanotubes with arbitrary chirality is derived from the microscopic lattice model. The parameters of the Hamiltonian show very weak dependence on the chiral angle, which makes the low-energy properties of conducting chiral nanotubes universal. The strongest Mott-like electron instability at half filling is investigated within the self-consistent harmonic approximation. The energy gaps occur in all modes of elementary excitations and estimate at 0.01–0.1 eV.@S0163-1829~99!50616-4#

Single-wall carbon nanotubes ~SWNT’s! are linear mac-romolecules whose individual properties can be studied by methods of nanophysics.1A recent demonstration of electron transport through single2 and multiple3 SWNT’s has been followed by remarkable observations of atomic structure,4,5 one-dimensional van Hove singularities,4 standing electron waves6and, possibly, electron correlations7in these systems. Moreover, the first prototype of a functional device—the nanotube field effect transistor working at room temperature—has been fabricated recently.8

Structurally uniform SWNT’s can be characterized by the wrapping vector wW5N1aW11N2aW2given by the linear com-bination of primitive lattice vectors aW65(61,

A

3)a/2, with a'0.246 nm ~Fig. 1!. It is natural to separate nonchiral arm-chair (N15N2) and zigzag (N152N2) nanotubes from

their chiral counterparts. A recent scanning tunneling mi-croscopy study4 has revealed that individual SWNT’s are generally chiral. According to the single-particle model, the nanotubes with N12N250 mod 3 have gapless energy spectrum9 and are therefore conducting; otherwise, the en-ergy spectrum is gapped and SWNT’s are insulating. There-fore, on the level of noninteracting electrons, physical prop-erties of SWNT’s are determined by their geometry.

The Coulomb interaction in one-dimensional SWNT’s should result in a variety of correlation effects due to the non-Fermi-liquid ground state of the system. In particular, metallic armchair SWNT’s are predicted to be Mott insulat-ing at half-fillinsulat-ing.10–13 Upon doping the nanotubes become conducting but still display density wave instabilities in three modes of elementary excitations with neutral total charge.13,14

Experimental observation of electron correlations still re-mains challenging since their signatures are usually masked by charging effects. Two recent transport spectroscopy ex-periments on an individual conducting SWNT~Ref. 7! and a rope of SWNT’s ~Ref. 15! produced contradicting results. The data by Tans et al.7 assumes spin-polarized tunneling into a nanotube, which in turn suggests the interpretation in

terms of electron correlations. On the other hand, the data by Cobden et al.15 fits the constant interaction model remark-ably well and shows no signatures of exotic correlation ef-fects. Since the atomic structure of particular SWNT’s stud-ied in these experiments is not known, it might be appealing to interpret the discrepancy in the results in terms of geometry-dependent many-particle properties of conducting SWNT’s.

Unfortunately, a consistent theory of interacting electrons in chiral SWNT’s is lacking, although such a theory has been recently developed for armchair nanotubes.12–14In this work we establish an effective low-energy model for conducting chiral SWNT’s and evaluate its parameters~scattering ampli-tudes! from the microscopic theory. We found very weak dependence of the dominant scattering amplitudes on the chiral angle. This allows us to introduce a universal low-energy Hamiltonian of conducting SWNT’s. According to the results of the renormalization group analysis13 the stron-gest Mott-like electron instability occurs at half-filling. We investigate this instability using the self-consistent harmonic approximation. Substantial energy gaps are found in all modes of elementary excitations. The conditions for experi-mental observation of the gaps are briefly discussed.

We start from the standard kinetic term Hk

5(s,kW$j(kW)a2,s

(kW)a1,s(kW)1H.c.% of the tight-binding Hamiltonian for pz electrons of a graphite sheet.

16

Here

FIG. 1. The graphite lattice consists of two atomic sublattices p51,2 denoted by filled and open circles. SWNT at the anglex to the x axis can be formed by wrapping the graphite sheet along wW 5OO8vector.

RAPID COMMUNICATIONS

PHYSICAL REVIEW B VOLUME 59, NUMBER 16 15 APRIL 1999-II

PRB 59

(2)

ap,s(kW) are the Fermi operators for electrons with the spin s56 at the sublattice p56 ~Fig. 1!. The matrix elements are given byj(kW)52t(e2ikya/A312eikya/2A3cos k

xa/2), t

be-ing the hoppbe-ing amplitude between neighborbe-ing atoms. The eigenvalues of the Hamiltonian vanish at two Fermi points of the Brillouin zone, aKW with a56 and KW5(K,0), K

54p/3a.

We consider conducting chiral (N1,N2) SWNT of the

radius R5(a/2p)

A

N121N1N21N22 whose axis x

8

forms the angle x5arctan@(N22N1)/

A

3(N11N2)# with the direction of chains of carbon atoms ~x axis in Fig. 1!. Expanding Hk

near the Fermi points to the lowest order in qW5kW2aKW

5q(cosx,sinx) and introducing slowly varying Fermi fields

cpas(x

8

)5L21/2(q52pn/Leiqx8apas(qW1aKW), we obtain Hk52iv

(

pas a

e2ipax

E

dx

8

cpas]x8c2pas, ~1!

with the Fermi velocity v5

A

3ta/2'8.13105 m/s. The ki-netic term can be diagonalized by the unitary transformation

cpas5 1

A

2e 2ipax/2

(

r56 ~ra! ~12p!/2w ras ~2!

to the basiswras of left (r52) and right (r51) movers.

The Coulomb interaction has the form Hint5 1 2 p p8,

(

$a i%,ss8 Vp p8~2a¯ K! 3

E

dx

8

cpa 1sc p8a2s8 † cp8a3s8cpa4s, ~3!

with the matrix elements Vp p8~2a¯ K!51

r

(

rWp

U~rWp2rWp8!exp@22ia¯ KW~rWp2rWp8!#, ~4! corresponding to the amplitudes of intra- ( p5p

8

) and inter-( p52p

8

) sublattice forward (a¯50) and backward (a¯

561) scattering17 ~the sum is taken over the

nodes of the sublattice p of SWNT!. Here U(rW)5e2/

$k

A

a021(x

8

)214R2sin2(y

8

/2R)% is the Coulomb interaction with a short-distance cutoff a0;a,r54pR/

A

3a2is a linear density of sublattice nodes along SWNT, and a¯5(a1

2a4)/25(a32a2)/2. We will choose the parameter a0

from the requirement that the on-site interaction in the origi-nal tight-binding model corresponds to the difference be-tween the ionization potential and electron affinity of s p2 hybridized carbon.18This procedure gives a050.526a.

The forward-scattering part HF @terms with a¯50 in Eq.

~3!# of the Hamiltonian Hint can be separated into the

Lut-tinger model-like term Hr and the term Hf related to the

differenceDV(0)5Vp p(0)2Vp2p(0) between intra- and

in-tersublattice amplitudes, Hr5Vp p~0! 2

E

dx

8

r 2~x

8

!, ~5! Hf52 DV~0! 2 paa

(

8ss8

E

dx

8

cpasc 2pa8s8 † c2pa8s8cpas, ~6! where r5(pascpasc

pas is the total electron density. The

backscattering Hamiltonian HB @terms with a¯561 in Eq.

~3!# can be subdivided into the intrasublattice Hb

(1)

( p

5p

8

) and intersublattice Hb(2) ( p52p

8

) parts.

The dominant contribution to the forward-scattering am-plitudes Vp p8(0) comes from the long range component of

the Coulomb interaction, Vp p(0)5(2e2/k)ln(Rs/R), where Rs.min(L,D) characterizes the screening of the interaction

due to a finite length L of the SWNT and/or the presence of metallic electrodes at a distance D.12The forward-scattering differential partDV(0) and the intrasublattice backscattering amplitude Vp p(2K) can be estimated from Eq. ~4! as fol-lows, DV(0),Vp p(2K);ae2/kR ~for a0;a). Despite the

amplitudesDV(0),Vp p(2K) are much smaller than Vp p(0), they produce essentially non-Luttinger terms in the low-energy Hamiltonian which will be important in further analy-sis.

We evaluated the matrix elements~4! numerically for chi-ral SWNT’s with radii R in the range 2R/a5427 @2R/a

55.5 for ~10,10! SWNT’s#. We found that dimensionless

amplitudes 2pkR@DV(0),Vp p(2K)#/ae2 show very weak

dependence on the radius of SWNT and its chiral angle ~see Table I!. The results are sensitive to the value of the cutoff parameter a0.

The intersublattice backscattering amplitude Vp2p(2K) is

almost three orders of magnitude smaller than DV(0), Vp p(2K). This is due to the C3 symmetry of a graphite

lat-tice, which leads to an exact cancellation of the terms ~4! contributing to Vp2p(2K) in the case of a plane graphite

sheet. The matrix elements Vp2p(2K) are generally complex

due to asymmetry of effective one-dimensional~1D! intersu-blattice interaction potential~the matrix elements are real for symmetric zigzag and armchair SWNT’s!. Let us note that after the unitary transformation ~2! of the Hamiltonian H

5Hk1Hint, the chiral angle x enters only to the

intersub-lattice backscattering matrix elements.19 Due to the small-ness of these matrix elements, the low-energy properties of chiral SWNT’s are expected to be virtually independent of the chiral angle.

Neglecting the intersublattice backscattering we arrive to the universal low-energy model of conducting SWNT’s given by the Hamiltonian H5Hk1HF1Hb(1). The latter can be bosonized along the lines of Refs. 14 and 13. We introduce bosonic representation of the Fermi fields,

wras5

hras

A

2p˜a

exp

@

irqFx

8

1i~uas1rfas!

#

, ~7!

and decompose the phase variables uas,fas into symmetric

d51 and antisymmetricd52 modes of the charger and-spin s excitations, uas5(ur11sus11aur21asus2)/2 and fas5(fr11sfs11afr21asfs2)/2. The bosonic TABLE I. Scattering amplitudesDV(0), Vp p(2K), Vp2p(2K)

in units ae2/2pkR for all chiral SWNT’s with radii R in the range

2R/a54 –7. a0/a DV(0) Vp p(2K) uVp2p(2K)u 0.4 0.44265–0.44274 0.97060–0.97095 0.622.231023 0.526 0.17378–0.17395 0.53549–0.53561 0.521.631023 0.7 0.04880–0.04895 0.24778–0.24797 0.321.531023 RAPID COMMUNICATIONS

(3)

fields satisfy the commutation relation, @ujd(x1),fj8d8(x2)# 5i(p/2)sign(x12x2)dj j8ddd8. The Majorana fermionshras

are introduced14to ensure correct anticommutation rules for different species r,a,s of electrons, and satisfy

@hras,hr8a8s8#152drr8daa8dss8. The quantity qF5pn/4 is

related to the deviation n of the average electron density from half-filling, and a˜;a is the parameter of the standard exponential ultraviolet cutoff.

The universal low-energy Hamiltonian of conducting SWNT’s has the following bosonized form,20

H5

(

j5r,s d

(

56 vjd 2p

E

dx

8

$K21jd ~]x8ujd!21Kjd~]x8fjd!2%1 1 2~p˜a!2

E

dx

8

$@DV~0!2Vp p~2K!#

3@cos~4qFx

8

12ur1!cos 2us12cos 2ur2cos 2us2#2DV~0!cos~4qFx

8

12ur1!cos 2ur2

1DV~0!cos~4qFx

8

12ur1!cos 2us22DV~0!cos 2us1cos 2ur21DV~0!cos 2us1cos 2us2

2Vp p~2K!cos~4qFx

8

12ur1!cos 2fs21Vp p~2K!cos 2us1cos 2fs2

1Vp p~2K!cos 2ur2cos 2fs21Vp p~2K!cos 2us2cos 2fs2%, ~8!

vjd5v

A

AjdBjd and Kjd5

A

Bjd/Ajd being the velocities of

excitations and exponents for the modes j ,d. The parameters Ajd, Bjd are given by Ar15114V¯ ~0! pv 2 DV~0! 4pv 2 Vp p~2K! 2pv , ~9! And512DV~0! 4pv 2d Vp p~2K! 2pv , ~10! Bnd511DV~0! 4pv , ~11! with V¯ (0)5@Vp p(0)1Vp2p(0)#/2. Since V¯ (0)/v;ln(Rs/R)@1 and DV(0)/v, Vp p(2K)/v

;a/R!1, the renormalization of the parameters Kjd,vjdby

the Coulomb interaction is the strongest in r1 mode. As-suming k51.4 ~Ref. 14! and Rs5100 nm we obtain Kr1

.0.2, vr1.v/Kr1 for generic SWNT’s with R;0.7 nm.12 The interaction in the other modes is weak: vjd5v, Kjd

51, up to a factor 11O(a/R).

The renormalization group analysis of armchair SWNT’s with long-range Coulomb interaction has been performed in Refs. 12–14. The modification of the parameters of the Hamiltonian ~8! by the neglected small term Vp2p(2K)

should not change the results qualitatively.21 The most rel-evant perturbation is the umklapp scattering at half-filling. In this case the nonlinear terms of the Hamiltonian~8!, which do not contain cos 2us2 scale to strong coupling and the phases ur1,us1,ur2,fs2 get locked at (ur1(m),us1(m),ur2(m),fs2(m))5(0,0,0,0) or (p/2,p/2,p/2,p/2). Therefore, the ground state of half-filled SWNT is the Mott insulator with all kinds of the excitations gapped.

To estimate the gaps quantitatively, we will employ the self-consistent harmonic approximation which follows from Feynman’s variational principle.22 We consider trial har-monic Hamiltonian of the form:

H05

(

jd vjd 2p

E

dx

8

$K21jd @~]x8ujd!21~12djsdd2!qjd 2 u jd 2 # 1Kjd@~]x8fjd! 21d jsdd2qjd 2 f jd 2 #%, ~12!

qjd being variational parameters. By minimizing the upper

estimate for the free energy F*5F01

^

H2H0

&

0, 22

one ob-tains the following self-consistent equations:

qr12 5 2Kr1 p˜a2vr1cr1$@Vp p~2K!2DV~0!#cs11DV~0!cr2 1Vp p~2K!ds2%, ~13! qr22 5 2Kr2 p˜a2vr2cr2$DV~0!cr11DV~0!cs1 2Vp p~2K!ds2%, ~14! qs12 5 2Ks1 p˜a2vs1cs1$@Vp p~2K!2DV~0!#cr11DV~0!cr2 2Vp p~2K!ds2%, ~15! qs22 5 2 p˜a2Ks2vs2ds2Vp p~2K!$cr12cs12cr2%, ~16!

where cjd5

^

cos 2ujd

&

05cos 2ujd

(m)

(g˜qa jd)Kjd, ds2

5

^

cos 2fs2

&

05cos 2fs2(m)(g˜qa s2)1/Ks2,

^

•••

&

0 denotes

aver-aging with respect to the trial Hamiltonian ~12!, and g

.0.890. Note that

^

cos 2us2

&

050, so that only the terms of the Hamiltonian~8! that scale to the strong coupling contrib-ute to Eqs. ~13!–~16!.

In the limiting case of interest, uDV(0)u,uVp p(2K)u!v

and Kr1!1, the solution of Eqs. ~13!–~16! can be found in a closed form, giving rise to the following estimates for the gaps Djd5vjdqjd in the energy spectra:

Dr15 vr1 g˜a

S

2g2Vr1 pvr1

D

1/~12Kr1! , ~17! Dr25uDV~0!uV r1 Dr1, ~18! Ds15uVp p~2K0!2DV~0!u Vr1 Dr1, ~19! RAPID COMMUNICATIONS

(4)

Ds25

uVp p~2K0!u

Vr1 Dr1, ~20!

with Vr15$@DV(0)2Vp p(2K)#21@DV(0)#2

1@Vp p(2K)#2%1/2. In the above expressions we used the

ap-proximation, v0/vr15Kr1 and v0/vjd5Kjd51 for s6

and r2 modes. The formulas ~17!–~20! indicate that the largest gap occurs inr1 mode, albeit all four gaps are of the same order for realistic values of the matrix elements ~see Table I!. The gaps decrease as Djd}(1/R)1/(12Kr1)

.(1/R)5/4with the tube radius. This should be contrasted to

the 1/R dependence of wide semiconductor gaps and 1/R2 dependence of narrow deformation induced gaps23expected from the single-particle picture.

In Fig. 2 we present numerical results for the gapsDjdfor

the cutoff parameter a050.526a. The data for somewhat larger and somewhat smaller values of a0 indicate possible

variation of the gap Dr1 due to uncertainty in the short-distance cutoff of the Coulomb interaction. The gaps can be loosely estimated at Djd;0.01–0.1 eV for typical SWNT’s

with R.0.7 nm. Due to the gaps in the spectrum of bosonic elementary excitations, the electronic density of states should disappear in the subgap region and display features at the gap frequencies and their harmonics. Both signatures should be observable by means of the tunneling spectroscopy.

Why have the gaps not been observed in the experiments?4,7,15This might be due to the effect of metallic electrodes. The difference in the work functions of the elec-trodes~Au, Pt! and the nanotube results in a downward shift of the Fermi level of the nanotube by a few tenths of an eV.4 This causes substantial deviation Dn54qF/p;1 nm21 of

the electron density in SWNT from half-filling, at least in the vicinity of the electrodes. Therefore, we expect the gap fea-tures to be observable in the layouts with well separated source and drain contacts. The piece of nanotube between them should be well isolated from any conductor.

In conclusion, we have developed an effective low-energy theory of conducting chiral SWNT’s with long-range Cou-lomb interaction. The many-particle properties of SWNT’s are found to be virtually independent of the chiral angle. The universal Hamiltonian ~8! of conducting SWNT’s is intro-duced. The Mott-like energy gaps in the range of 0.01–0.1 eV should be observable at half-filling.

The authors would like to thank B.L. Altshuler, G.E.W. Bauer, R. Egger, Yu.V. Nazarov, and N. Wingreen for stimulating discussions. The financial support of the Royal Dutch Academy of Sciences~KNAW! is gratefully acknowl-edged. One of us~A.A.O.! acknowledges the kind hospitality of the NEC Research Institute.

1A. Thess et al., Science 273, 483~1996!.

2S. J. Tans et al., Nature~London! 386, 474 ~1997!. 3M. Bockrath et al., Science 275, 1922~1997!.

4J. W. G. Wildo¨er et al., Nature~London! 391, 59 ~1998!. 5

W. Clauss, D. J. Bergeron, and A. T. Johnson, Phys. Rev. B 58, R4266~1998!.

6L. C. Venema, Science 283, 52~1999!.

7S. J. Tans, M. H. Devoret, R. J. A. Groeneveld, and C. Dekker,

Nature~London! 394, 761 ~1998!.

8S. J. Tans, A. R. M. Verschueren, and C. Dekker, Nature

~Lon-don! 393, 49 ~1998!.

9We do not discuss here the minigaps due to the curvature of the

nanotube surface, see Ref. 23.

10L. Balents and M. P. A. Fisher, Phys. Rev. B 55, R11 973~1997!. 11Yu. A. Krotov, D.-H. Lee, and Steven G. Louie, Phys. Rev. Lett.

78, 4245~1997!.

12C. Kane, L. Balents, and M. P. A. Fisher, Phys. Rev. Lett. 79,

5086~1997!.

13

H. Yoshioka and A. A. Odintsov, Phys. Rev. Lett. 82, 374~1999!.

14R. Egger and A. O. Gogolin, Phys. Rev. Lett. 79, 5082~1997!;

Eur. Phys. J. B 3, 281~1998!.

15D. H. Cobden et al., Phys. Rev. Lett. 81, 681~1998!. 16P. R. Wallace, Phys. Rev. 71, 622~1947!.

17This definition is different from the conventional one since it

involves the Fermi points rather than the directions of motion of scattered electrons.

18

See, e.g., E. Moore, B. Gherman, and D. Yaron, J. Chem. Phys.

106, 4216~1997!.

19The transformed matrix elements V

p2p(2a¯K)e2ia¯pxare invariant

underpn/3 rotations in chiral angle xn5x01pn/3, which

cor-respond to equivalent representations of a nanotube.

20Equation~8! has the same form as the phase Hamiltonian for a

two-leg Hubbard-type ladder @H. Lin et al., Phys. Rev. B 58, 1794~1998!#, provided that the difference in definitions of the fieldsuj2andfj2 ( j5r,s) in terms of densities of right and left movers in two energy bands is taken into account.

21We have checked this explicitly for ~10,10! SWNT’s at

half-filling and away from it.

22R. P. Feynman, Statistical Mechanics~Benjamin, Reading, MA,

1972!.

23C. L. Kane and E. J. Mele, Phys. Rev. Lett. 78, 1932~1997!.

FIG. 2. The energy gaps Djd for the modes r1, s2, s1,

r2 at a050.526a ~lines marked by crosses, from top to bottom!

and for the mode r1 at a050.4a ~triangles! and at a050.7a

~squares!. The energy is in units \v/a˜.2.16 eV for a˜5a. RAPID COMMUNICATIONS

Cytaty

Powiązane dokumenty

Furthermore, we carried out Raman spectroscopy of as-made and treated CNT films and compared the results with the DFT calculations of halogen doped CNTs to get

W przypadku gdy osoby brane pod uwagę zamieszkiwały przez długi czas na terytorium danego kraju i założy- ły tam rodzinę, należy zaakceptować fakt, że członkowie rodziny

Odwołując się do koncepcji francuskiej analizy dyskursu, omawiała między innymi: cechy wypowiedzi charakterystyczne dla wywiadu jako gatunku dyskursu, konstruowanie obrazu

A crucial step for this library-based control scheme to guarantee stabil- ity is the assessment of the stabilizing region of the state space for each LQR gain.. This information

Wykorzystywanie  Internetu  obecne  jest  w  strategiach  politycznych  na  różnorodnym  obszarze  terytorialnym.  Od  obszarów 

Strength test results for obtained polyethylene specimens are presented graphically as a plots of the mass content of the nanofiller HNT with the addition (5%) of a

They then move a distance of 250m on level ground directly away from the hill and measure the angle of elevation to be 19 ◦.. Find the height of the hill, correct to the

Furthermore, thanks are due to Paweł Potoroczyn, one time Director of the Polish Cultural Institute of London and subsequently Director of the Adam Mickiewicz