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Resistance of carbon nanotubes connected to reservoirs
ELSEVIER
Synthetic Metals 103 (1999) 2527-2528
Resistance
aDepartment
of carbon
nanotubes
connected
to reservoirs
H. Yoshiok& and A. A. OdintsovbC of Applied Physics, Delft University of Technology, Lorenttweg 1, 2628 CJ Delft, The Netherlands b Departtnent of Physics, Nagoya University, Nagoya 464-8602, Japan ‘Nuclear Physics Institute, Moscow State University, Moscow 119899 GSP, Russia
Abstract A phase Hamiltonian of conducting carbon nanotubes with an arbitrary chirality is given which takes account of the long range Coulomb interaction and the microscopic lattice structure. By using this Hamiltonian, the transport properties at half-filling are investigated for the case that the system is adiabatically connected to reservoirs. Keywords
: Many-body
and quasiparticle
theories,
Graphite
Single wall carbon nanotubes (SWNTs) with diameters of a few atomic distances and lengths of several micrometers can be considered as experimental realization of onedimensional (1-D) electron systems [l] . The SWNTs can be generally characterized by two integers n+, which specify the superlattice translation vector wrapping around the waist of the cylinder, ?j; = n+a’+ + n-a’- (a’& are the primitive Bravais lattice vectors, see Fig.1). Depending on the two integers, the SWNTs show metallic or semiconducting behavior [2] . Recent experiments on the transport for the individual [3] and multiple [4] SWNTs have indicated the importance of the electron-electron and/or electron-phonon interaction in these systems. The 1-D nature of the lowenergy electronic states in the SWNTs together with the interaction should result in a variety of correlation effects. Utilizing previous studies of 1-D interacting electrons, theories of correlation have been developed for armchair SWNTs (x = 0 in Fig.1) with a short range interaction [5,6] and with the realistic long-range Coulomb interaction [7-lo] . The ground state and its excitations, the transport properties, the tunneling density of states of finite systems and the persistent currents in toroidal tubes have been investigated. The above theoretical investigations focus on the armchair SWNTs. In this paper, the phase Hamiltonian of SWNTs with metallic band structure is given for an arbitrary chirality. Based on this phase Hamiltonian, the transport properties at half-filling are investigated for the case that the SWNT is adiabatically connected to reservoirs of non-interacting electrons. We begin with the tight-binding model on the honeycomb lattice shown in Fig.1, where the axis of the tube (waist) is denoted as Z’ (y’). Calculating the band structure of this model for the discrete set of allowed quantized transverse momenta leads to two gapless points at the loca-
and related
compounds
Fig. 1. A single layer of graphite. The black (white) circle denotes the sublattice p = +(-), Zk is the two primitive Bravais lattice vector, and the axis 2’ (y’) points at the direction of the tube (the waist) of the SWNT. tions, I;’ = (rtKo,O) for n+ - n- = 3q (q : integer), where I<0 = 4rf3a (a : the lattice constant). By linearizing the dispersion at the gapless points, we derive the effective kinetic energy in terms of the slowly varying Fermi fields. The Coulomb interaction can be also written by the Fermi fields. By applying the Abelian bosonization method to the Hamiltonian and introducing the phase fields, @,a and dj6, of the charge (j = p) and spin (j = g) excitation of the symmetric (6 = -t) and antisymmetric (S = -) mode between the two bands, we obtain,
WO) +2(rra)2 - cos(4qpz’
cos(4qm’ + 2Qp.t) cos 2e,-
+ 26,+) cos 28,+ + cos(4qFz’
- cm 28,- cos 28,- + cos 2e,+ cos 2e,-
0379-6779/99/$ - see front matter 0 1999 Elsevier Science S.A. All rights reserved. PII: SO379-6779(98)01091-l
+ 28,+) cos 20,cos 28,+
cos 20,-
)
2528
H. Yoshioka, A.A. Odintsov
J{
i Synthetic
Metals
103 (1999)
2527-2528
Eq.(3) is given by ~/(2]w~l), which leads to r-l = 2e2[n = 4e2/h. The absence of the conductance renormalization due to the interaction is in agreement with the result of Ref. + cos 20,- cos 20,- - cos(4qFz’ + 28p.f ) cos 2& [12] . The non-linear terms including cos 28,+, which cor+ cos20,+ cos24,- + cos2e,- cosad,- + cog20,- cos ad,, respond to the interaction processes that two right moving > electrons are scattered into two left movers and vice versa, (1) increase the resistance. We calculate AGp+(zl, x2;&) E G,+(sl,~~~;iw~)-GGOpS(~~1,a2;i~~) up to secondorder in the where ti36 = u,,dm (~0 : the Fermi velocity) and non-linear terms. From this formula, the following depenI(36 = dm with A,+ = 1 + {4P(O) - au(O)/4 dence of the extra resistance, Ar z r - (2e2/a)-I, on the %4~hb)l~~l(~~o), Aot = 1-{6V(O)/4+V~/,,(21))/2}/(~~0), temperature, T, the length and the radius is derived, A,= A,= 1 - {&f(O)/4 - Vpp(21~,3)/2}/(m~), and Bj6 = 1+ SI/(O)/(4nve). The quantity QF = TYZ/~ (n : deviation of the average density from half-filling) can be tuned AT cc + TL, v,+/L’>)T> vo/L . (4) by the gate voltage. Note that the expression, Eq.(l), does T2L2, vo/L >> T not depend on the chirality angle, x, explicitly. Since I<,+ N 0.2, the extra resistance increases with deFrom the Coulomb interaction on the tube with the racreasing temperature for T >> v,+/L. On the other hand, dius R, U(fl = e2/(lcv/ai +s’~ +4R2sin2(y’/2R)} (6 N for T << vp+./L, a decrease of the extra resistance is seen, a), the matrix elements, v(O), W(0) and 1.4 and 00 N which is due to the effect of the reservoirs. It should be Vpp(2K0), are given as c(O) = (L/NL) C,{U(i?) + U(4 + noted that the T2L2 dependence for w/L >> T is also pre4}/2, W(O) = (L/N~.)C,~{U(r'r)-U(ri+~},andV~~(2ICg) dicted in modulated quantum wires [13] . In deriving the = (L/NL) & U(~)ei2”o(“i cOsx--l~ JinX). Here L are NL are above formula, it is assumed that the charge gap, A,+., is the length of the tube and the number of the unit cell, remuch smaller than T. At low temperature, T << Apt, the spectively, his defined in Figl., and r’r is a Bravais lattice extra resistance is expected to increase as Ar c( exp(A,+/T). vector. The quantity, V(f;) (U(c + d’)) expresses the interThe actual value of the charge gap has been estimated in action between the same (different) sublattices. The matrix Ref. [9] and strongly depends on the choice of as. For element, p(O), which hardly depends on the chirality, has a example, in case of the armchair SWNT with n+ = 10, strongest amplitude and is estimated as 2e2/Kln(R,/R) for A P-bN lOOI< for ae = a/2 and A,+ - 10K for QO = a. R, >> R [s] where R, = min(L,D) with D being the disThe authors would like to thank G. E. W. Bauer, R. tance between the metallic electrodes. On the other hand, Egger, Yu. V. Nazarov and U. Weiss for stimulating disW(O) and Vpp(21Co) are both of the order of a/R as long cussions. The financial support of the Dutch Foundation as R >> a. Dependence of W(O) and Vpp(2K~) on the for Fundamental Research on Matter (FOM) is gratefully chirdity is also very weak. In fact, the numerical calcuacknowledged. This work is also a part of INTAS-RFBR lation shows &V(O) = 0.21 and Vppp(21Co) = 0.60 in unit of 95-1305. ae2/(2n@ for a0 = a/2 and any chirality. Note that we neglect the 2l(o-component of the interaction between the References different sublattice, Vp421cO), because it is much smaller than the others due to C’s symmetry of the graphite lattice ill A. Thess et.al., Science, 273 (1996) 483. [9] . From the above estimation of the matrix elements, we PI N. Hamada et&, Phys. Rev. Lett. 68 (1992) 1579~;-R. obtain Ir,+ N we/up+ 1: 0.2 for R, = 1OOnm and nk = 10. Saito etal., Appl. Phys. Lett. 60 (1992) 2204 ; C.T. The low energy properties of Eq.(l) have been investiWhite et& Phys. Rev. B 47 (1993) 5485. gated in Ref. [9] . Especially, for half-filling (qF = 0), the S.J. Tans et& Nature 386 (1997) 474 ; S.J. Tans [3] ground state is a Mott-insulator and all the excitation from etal., preprint. it are gapped. [4] M. Bockrath et&, Science 275 (1997) 1922 ; C. L. The Mott-insulating behavior can be seen in the transKane et-al, Europhys. Lett 41 (1998) 683 ; D.H. Cobport properties. We discuss the temperature dependence of den et& 1998, cond-mat preprint 9804154. the resistance of the SWNT adiabatically connected to the E51L. Balents and M. P. A. Fisher, Phys. Rev. B 55 (1997) reservoirs of non-interacting electrons concentrating on preR11973. cursors of a Mott-transition. According to linear response Yu. A. Krotov et&, Phys. Rev. Lett. 78 (1997) 4245. 14 theory, the resistance, r, is given by [ll] , [71 R. Egger and A. 0. Gogolin, Phys.-Rev. Lett. 79 (1997) 5082. r-l = (2e/nj2 W,Gpt (El, x2; w~)(~,++~ , (2) C. b31 L. Kane etal., Phys. Rev. Let?. 79 (1997) 5086. whereG,+(xr,r2;iWn) = J0’dreiwn7 (T,Bp+(ol, 7)B,,+(r2,0)). PI H. Yoshioka and A.A. Odintsov, 1998, cond-mat In the absence of non-linear terms in Eq.( I), G,+ (x, r’; iwn) preprint 9805106. is given by the solution of the differential equation, m A.A. Odintsov et.al., 1998, cond-mat preprint 9805164. 1111R. Shankar, Int. J. Mod. Phys. B 4 (1990) 2371. [-a= ~vP+(x)l~(Pt (x)1 3, + ~~2/~%tw~Pt WI] P21 D.L. Malsov and M. Stone, Phys. Rev. B 52 (1995) xGEt (x, x’; iwn) = nb(~ - x’) , R.5539; V.V. Ponomarenko, ibid, R8666; I. Saf? and (3) H.J. Schulz, ibid, R17040. where (~C~+(X),V~+.(I)) is (lip+,vp+) in the SWNT and 1131A.A. Odintsov et&, Phys. Rev. 3 56 (1997) R12729. (1, ve) in the reservoirs. In the limit tin + 0, the solution of +
YPPWO) 2(7ra)2
dE’
- cos(4q&r’
+ 28,+) cos 2#,+
Synthetic Metals 103 (1999) 2527-2528
Resistance
aDepartment
of carbon
nanotubes
connected
to reservoirs
H. Yoshiok& and A. A. OdintsovbC of Applied Physics, Delft University of Technology, Lorenttweg 1, 2628 CJ Delft, The Netherlands b Departtnent of Physics, Nagoya University, Nagoya 464-8602, Japan ‘Nuclear Physics Institute, Moscow State University, Moscow 119899 GSP, Russia
Abstract A phase Hamiltonian of conducting carbon nanotubes with an arbitrary chirality is given which takes account of the long range Coulomb interaction and the microscopic lattice structure. By using this Hamiltonian, the transport properties at half-filling are investigated for the case that the system is adiabatically connected to reservoirs. Keywords
: Many-body
and quasiparticle
theories,
Graphite
Single wall carbon nanotubes (SWNTs) with diameters of a few atomic distances and lengths of several micrometers can be considered as experimental realization of onedimensional (1-D) electron systems [l] . The SWNTs can be generally characterized by two integers n+, which specify the superlattice translation vector wrapping around the waist of the cylinder, ?j; = n+a’+ + n-a’- (a’& are the primitive Bravais lattice vectors, see Fig.1). Depending on the two integers, the SWNTs show metallic or semiconducting behavior [2] . Recent experiments on the transport for the individual [3] and multiple [4] SWNTs have indicated the importance of the electron-electron and/or electron-phonon interaction in these systems. The 1-D nature of the lowenergy electronic states in the SWNTs together with the interaction should result in a variety of correlation effects. Utilizing previous studies of 1-D interacting electrons, theories of correlation have been developed for armchair SWNTs (x = 0 in Fig.1) with a short range interaction [5,6] and with the realistic long-range Coulomb interaction [7-lo] . The ground state and its excitations, the transport properties, the tunneling density of states of finite systems and the persistent currents in toroidal tubes have been investigated. The above theoretical investigations focus on the armchair SWNTs. In this paper, the phase Hamiltonian of SWNTs with metallic band structure is given for an arbitrary chirality. Based on this phase Hamiltonian, the transport properties at half-filling are investigated for the case that the SWNT is adiabatically connected to reservoirs of non-interacting electrons. We begin with the tight-binding model on the honeycomb lattice shown in Fig.1, where the axis of the tube (waist) is denoted as Z’ (y’). Calculating the band structure of this model for the discrete set of allowed quantized transverse momenta leads to two gapless points at the loca-
and related
compounds
Fig. 1. A single layer of graphite. The black (white) circle denotes the sublattice p = +(-), Zk is the two primitive Bravais lattice vector, and the axis 2’ (y’) points at the direction of the tube (the waist) of the SWNT. tions, I;’ = (rtKo,O) for n+ - n- = 3q (q : integer), where I<0 = 4rf3a (a : the lattice constant). By linearizing the dispersion at the gapless points, we derive the effective kinetic energy in terms of the slowly varying Fermi fields. The Coulomb interaction can be also written by the Fermi fields. By applying the Abelian bosonization method to the Hamiltonian and introducing the phase fields, @,a and dj6, of the charge (j = p) and spin (j = g) excitation of the symmetric (6 = -t) and antisymmetric (S = -) mode between the two bands, we obtain,
WO) +2(rra)2 - cos(4qpz’
cos(4qm’ + 2Qp.t) cos 2e,-
+ 26,+) cos 28,+ + cos(4qFz’
- cm 28,- cos 28,- + cos 2e,+ cos 2e,-
0379-6779/99/$ - see front matter 0 1999 Elsevier Science S.A. All rights reserved. PII: SO379-6779(98)01091-l
+ 28,+) cos 20,cos 28,+
cos 20,-
)
2528
H. Yoshioka, A.A. Odintsov
J{
i Synthetic
Metals
103 (1999)
2527-2528
Eq.(3) is given by ~/(2]w~l), which leads to r-l = 2e2[n = 4e2/h. The absence of the conductance renormalization due to the interaction is in agreement with the result of Ref. + cos 20,- cos 20,- - cos(4qFz’ + 28p.f ) cos 2& [12] . The non-linear terms including cos 28,+, which cor+ cos20,+ cos24,- + cos2e,- cosad,- + cog20,- cos ad,, respond to the interaction processes that two right moving > electrons are scattered into two left movers and vice versa, (1) increase the resistance. We calculate AGp+(zl, x2;&) E G,+(sl,~~~;iw~)-GGOpS(~~1,a2;i~~) up to secondorder in the where ti36 = u,,dm (~0 : the Fermi velocity) and non-linear terms. From this formula, the following depenI(36 = dm with A,+ = 1 + {4P(O) - au(O)/4 dence of the extra resistance, Ar z r - (2e2/a)-I, on the %4~hb)l~~l(~~o), Aot = 1-{6V(O)/4+V~/,,(21))/2}/(~~0), temperature, T, the length and the radius is derived, A,= A,= 1 - {&f(O)/4 - Vpp(21~,3)/2}/(m~), and Bj6 = 1+ SI/(O)/(4nve). The quantity QF = TYZ/~ (n : deviation of the average density from half-filling) can be tuned AT cc + TL, v,+/L’>)T> vo/L . (4) by the gate voltage. Note that the expression, Eq.(l), does T2L2, vo/L >> T not depend on the chirality angle, x, explicitly. Since I<,+ N 0.2, the extra resistance increases with deFrom the Coulomb interaction on the tube with the racreasing temperature for T >> v,+/L. On the other hand, dius R, U(fl = e2/(lcv/ai +s’~ +4R2sin2(y’/2R)} (6 N for T << vp+./L, a decrease of the extra resistance is seen, a), the matrix elements, v(O), W(0) and 1.4 and 00 N which is due to the effect of the reservoirs. It should be Vpp(2K0), are given as c(O) = (L/NL) C,{U(i?) + U(4 + noted that the T2L2 dependence for w/L >> T is also pre4}/2, W(O) = (L/N~.)C,~{U(r'r)-U(ri+~},andV~~(2ICg) dicted in modulated quantum wires [13] . In deriving the = (L/NL) & U(~)ei2”o(“i cOsx--l~ JinX). Here L are NL are above formula, it is assumed that the charge gap, A,+., is the length of the tube and the number of the unit cell, remuch smaller than T. At low temperature, T << Apt, the spectively, his defined in Figl., and r’r is a Bravais lattice extra resistance is expected to increase as Ar c( exp(A,+/T). vector. The quantity, V(f;) (U(c + d’)) expresses the interThe actual value of the charge gap has been estimated in action between the same (different) sublattices. The matrix Ref. [9] and strongly depends on the choice of as. For element, p(O), which hardly depends on the chirality, has a example, in case of the armchair SWNT with n+ = 10, strongest amplitude and is estimated as 2e2/Kln(R,/R) for A P-bN lOOI< for ae = a/2 and A,+ - 10K for QO = a. R, >> R [s] where R, = min(L,D) with D being the disThe authors would like to thank G. E. W. Bauer, R. tance between the metallic electrodes. On the other hand, Egger, Yu. V. Nazarov and U. Weiss for stimulating disW(O) and Vpp(21Co) are both of the order of a/R as long cussions. The financial support of the Dutch Foundation as R >> a. Dependence of W(O) and Vpp(2K~) on the for Fundamental Research on Matter (FOM) is gratefully chirdity is also very weak. In fact, the numerical calcuacknowledged. This work is also a part of INTAS-RFBR lation shows &V(O) = 0.21 and Vppp(21Co) = 0.60 in unit of 95-1305. ae2/(2n@ for a0 = a/2 and any chirality. Note that we neglect the 2l(o-component of the interaction between the References different sublattice, Vp421cO), because it is much smaller than the others due to C’s symmetry of the graphite lattice ill A. Thess et.al., Science, 273 (1996) 483. [9] . From the above estimation of the matrix elements, we PI N. Hamada et&, Phys. Rev. Lett. 68 (1992) 1579~;-R. obtain Ir,+ N we/up+ 1: 0.2 for R, = 1OOnm and nk = 10. Saito etal., Appl. Phys. Lett. 60 (1992) 2204 ; C.T. The low energy properties of Eq.(l) have been investiWhite et& Phys. Rev. B 47 (1993) 5485. gated in Ref. [9] . Especially, for half-filling (qF = 0), the S.J. Tans et& Nature 386 (1997) 474 ; S.J. Tans [3] ground state is a Mott-insulator and all the excitation from etal., preprint. it are gapped. [4] M. Bockrath et&, Science 275 (1997) 1922 ; C. L. The Mott-insulating behavior can be seen in the transKane et-al, Europhys. Lett 41 (1998) 683 ; D.H. Cobport properties. We discuss the temperature dependence of den et& 1998, cond-mat preprint 9804154. the resistance of the SWNT adiabatically connected to the E51L. Balents and M. P. A. Fisher, Phys. Rev. B 55 (1997) reservoirs of non-interacting electrons concentrating on preR11973. cursors of a Mott-transition. According to linear response Yu. A. Krotov et&, Phys. Rev. Lett. 78 (1997) 4245. 14 theory, the resistance, r, is given by [ll] , [71 R. Egger and A. 0. Gogolin, Phys.-Rev. Lett. 79 (1997) 5082. r-l = (2e/nj2 W,Gpt (El, x2; w~)(~,++~ , (2) C. b31 L. Kane etal., Phys. Rev. Let?. 79 (1997) 5086. whereG,+(xr,r2;iWn) = J0’dreiwn7 (T,Bp+(ol, 7)B,,+(r2,0)). PI H. Yoshioka and A.A. Odintsov, 1998, cond-mat In the absence of non-linear terms in Eq.( I), G,+ (x, r’; iwn) preprint 9805106. is given by the solution of the differential equation, m A.A. Odintsov et.al., 1998, cond-mat preprint 9805164. 1111R. Shankar, Int. J. Mod. Phys. B 4 (1990) 2371. [-a= ~vP+(x)l~(Pt (x)1 3, + ~~2/~%tw~Pt WI] P21 D.L. Malsov and M. Stone, Phys. Rev. B 52 (1995) xGEt (x, x’; iwn) = nb(~ - x’) , R.5539; V.V. Ponomarenko, ibid, R8666; I. Saf? and (3) H.J. Schulz, ibid, R17040. where (~C~+(X),V~+.(I)) is (lip+,vp+) in the SWNT and 1131A.A. Odintsov et&, Phys. Rev. 3 56 (1997) R12729. (1, ve) in the reservoirs. In the limit tin + 0, the solution of +
YPPWO) 2(7ra)2
dE’
- cos(4q&r’
+ 28,+) cos 2#,+