Theory of Quantum Transport in Carbon Nanotubes

Transcription

1 Theory of Quantum Transport in Carbon Nanotubes Tsuneya Ando Institute for Solid State Physics, University of Tokyo Roppongi, Minato-ku, Tokyo , Japan A brief review is given of electronic and transport properties of carbon nanotubes mainly from a theoretical point of view. The topics include an effective-mass description of electronic states, the absence of backward scattering except for scatterers with a potential range smaller than the lattice constant, a conductance quantization in the presence of lattice vacancies, and effects of phonon scattering. 1. INTRODUCTION Graphite needles called carbon nanotubes (CN s) were discovered recently [1,2] and have been a subject of an extensive study. A CN is a few concentric tubes of two-dimensional (2D) graphite consisting of carbon-atom hexagons arranged in a helical fashion about the axis. The diameter of CN s is usually between 20 and 300 Å and their length can exceed 1 µm. Single-wall nanotubes are produced in a form of ropes [3,4]. The purpose of this paper is to give a brief review of recent theoretical study on transport properties of carbon nanotubes. Carbon nanotubes can be either a metal or semiconductor, depending on their diameters and helical arrangement. The condition whether a CN is metallic or semiconducting can be obtained based on the band structure of a 2D graphite sheet and periodic boundary conditions along the circumference direction. This result was first predicted by means of a tight-binding model. These properties can be well reproduced in a k p method or an effectivemass approximation [5]. In fact, the method has been used successfully in the study of wide varieties of electronic properties of CN. Some of such examples are magnetic properties [6] including the Aharonov-Bohm effect on the band gap, optical absorption spectra [7,8], exciton effects [9], lattice instabilities in the absence [10] and presence of a magnetic field [11], magnetic properties of ensembles of nanotubes [12], and effects of spin-orbit interaction [13]. Transport properties of CN s are interesting because of their unique topological structure. There have been some reports on experimental study of transport in CN bundles [14] and ropes [15,16]. Transport measurements became possible for a single multi-wall nanotube [17-22] and a single single-wall nanotube [23-27]. In this paper we shall mainly discuss transport properties obtained theoretically. 2. ENERGY BANDS Figure 1 shows the lattice structure and the first Brillouin zone of a 2D graphite together with the coordinate systems. The unit cell contains two carbon atoms denoted as A and B. A nanotube is specified by a chiral vector L=n a a+n b b CP544, Electronic Properties of Novel Materials Molecular Nanostructures, edited by H. Kuzmany, et al. c 2000 American Institute of Physics /00/$

2 FIGURE 1. (a) Lattice structure of two-dimensional graphite sheet. η is the chiral angle. The coordinates are chosen in such a way that x is along the circumference of a nanotube and y is along the axis. (b) The first Brillouin zone and K and K points. (c) The coordinates for a nanotube. with integer n a and n b and basis vectors a and b ( a = b = a =2.46 Å). In the coordinate system fixed onto a graphite sheet, we have a =(a, 0) and b =( a/2, 3a/2). For convenience we introduce another coordinate system where the x direction is along the circumference L and the y direction is along the axis of CN. The direction of L is denoted by the chiral angle η. A graphite sheet is a zero-gap semiconductor in the sense that the conduction and valence bands consisting of π states cross at K and K points of the Brillouin zone, whose wave vectors are given by K =(2π/a)(1/3, 1/ 3) and K =(2π/a)(2/3, 0) [28]. Electronic states near a K point of 2D graphite are described by the k p equation [29,5]: ( ) γ(σ xˆkx +σ yˆky )F K (r)=γ( σ ˆk)F K (r)=εf K (r), F K F K (r)= A (r) FB K(r), (2.1) where γ is the band parameter, ˆk =(ˆk x, ˆk y ) is a wave-vector operator, ε is the energy, and σ x, σ y, and σ z are the Pauli spin matrices. Equation (2.1) has the form of Weyl s equation for neutrinos. The electronic states near the Fermi level can be obtained by imposing the periodic boundary condition in the circumference direction Ψ(r+L)=Ψ(r) except for extremely thin CNs. The Bloch functions at a K point change their phase by exp(ik L)=exp(2πiν/3), where ν is an integer defined by n a +n b =3M +ν with integer M and can take 0 and ±1. Because Ψ(r) is written as a product of the Bloch function and the envelope function, this phase change should be canceled by that of the envelope functions and the boundary conditions for the envelope functions are given by F K (r+l)=f K (r) exp( 2πiν/3). Energy levels in CN for the K point are obtained by putting k x =κ ν (n) with κ ν (n)=(2π/l)[n (ν/3)] and k y = k in the above k p equation as ε ν (±) (n, k)= ±γ κ ν (n) 2 +k 2 [5], where L= L, n is an integer, and the upper (+) and lower ( ) signs represent the conduction and valence bands, respectively. Those for the K point are obtained by replacing ν by ν. This shows that CN becomes metallic for ν =0 and semiconducting with gap E g =4πγ/3L for ν =±1. 326

3 3. ABSENCE OF BACKWARD SCATTERING In the presence of impurities, electronic states in the vicinity of K and K points can be mixed with each other. Therefore, we should use a 4 4 Schrödinger equation ( ) ( ) F K F K HF = εf, F =, F K A =, H = H F K FB K 0 + V. (3.1) The unperturbed Hamiltonian is given by γ(σ x k x +σ y k y ) for the K point and γ(σ x k x σ y k y ) for the K point. The effective potential of an impurity is written as [30] u A (r) 0 e iη u A (r) 0 0 u V = B (r) 0 ω 1 e iη u B (r) e iη u, (3.2) A (r) 0 u A (r) 0 0 ωe iη u B (r) 0 u B (r) where ω = exp(2πi/3). If we use a tight-binding model, we obtain the explicit expressions for the potentials as u A (r)= R A g(r R A )u A (R A ), u A (r)= R A g(r R A )e i(k K) R A u A (R A ), (3.3) where u A (R A ) is the local site energy at site R A due to the impurity potential and g(r) is a smoothing function having a range of the order of the lattice constant a and satisfying the normalization condition R A g(r R A )= R B g(r R B )=1. The similar expressions can be obtained for u B (r) and u B (r). In the vicinity of ε = 0, we have two right-going channels K+ and K +, and two left-going channels K and K. Figure 2 shows calculated scattering amplitude as a function of d/a for a Gaussian potential located at a B site and having the integrated intensity u in the absence of a magnetic field. The backward scattering probability decreases rapidly with d and becomes exponentially small for d/a 1. The same is true of the intervalley scattering. This absence of the backward scattering for long-range scatterers disappears in the presence of magnetic fields although not shown explicitly. It has been proved that the Born series for back-scattering vanish identically [30]. This has been ascribed to a spinor-type property of the wave function under a rotation in the wave vector space [31]. The absence of backward scattering has confirmed by numerical calculations in a tight binding model [32]. Because of the presence of large contact resistance between a nanotube and metallic electrode, the conductance usually exhibits a prominent effect of a single electron tunneling due to charging effects. An important information can be obtained on the effective mean free path and the amount of backward scattering in nanotubes [23-27]. In fact, the Coulomb oscillation in semiconducting nanotubes is quite irregular and can be explained only if nanotubes are divided into many separate spatial regions in contrast to that in metallic nanotubes [33]. This behavior is consistent with the presence of considerable amount of backward scattering leading to a strong localization of the wave function in semiconducting tubes. In metallic nanotubes, the wave function is extended throughout the whole region of a nanotube because of the absence of backward scattering. 327

4 Average Amplitude (units of u) (L/2πl) 2 = 0.00 K+ => K- K+ => K+ K+ => K Number of Vacancies L 3a=50, N = 5~ 13 N AB Tube Axis A B N AB Potential Range (units of a) Conductance (units of e 2/πh) - FIGURE 2. (left) Calculated effective scattering matrix elements versus the potential range at ε=0 in the absence of a magnetic field. After Ref. [30]. FIGURE 3. (right) Calculated histogram of the conductance of nanotubes with a vacancy. After Ref. [35]. 4. LATTICE VACANCIES Effects of scattering by a vacancy in armchair nanotubes have been studied within a tight-binding model [34,35]. It has been shown that the conductance at ε = 0 in the absence of a magnetic field is quantized into zero, one, or two times of the conductance quantum e 2 /π h for a vacancy consisting of three B carbon atoms around an A atom, of a single A atom, and of a pair of A and B atoms, respectively [35]. Numerical calculations were performed for about vacancies and demonstrated that such quantization is quite general [36]. Figure 3 shows a histogram of the conductance for different values of N AB, where N A and N B are the number of removed atoms at A and B sublattice points, respectively, and N AB =N A N B. This rule was analytically derived in a k p scheme later [37]. 5. PHONON SCATTERING A conductance quantization was observed in multi-wall nanotubes [38]. This quantization is likely to be related to the absence of backward scattering shown here, but much more works are necessary including effects of magnetic fields and problems related to contacts with metallic electrode before complete understanding of the experimental result. At room temperature, where the experiment was performed, phonon scattering is likely to play an important role [15,39,40]. Acoustic phonons important in the electron scattering are described well by a continuum model [40]. The potential-energy functional for displacement 328

5 ω B ω(k z ) n=0 n=1 n>1 Resistivity (units of ρ (T )) A B η π/6 (Armchair) π/12 (Chiral) 0 (Zigzag) 0 1 k z R Temperature (T/T B ) FIGURE 4. (left) Frequencies of phonons obtained in the continuum model. FIGURE 5. (right) The resistivity of armchair (solid line) and zigzag (dotted line) nanotubes in units of ρ A (T B ) which is the resistivity of the armchair nanotube at T =T B, and T B denotes the temperature of the breathing mode, T B = hω B /k B. u=(u x,u y,u z ) is written as U[u] = dxdy 1 2 ( B(u xx +u yy ) 2 + µ [ (u xx u yy ) 2 +4u 2 xy] ), u xx = u x x + u z R, u yy = u y y, 2u xy = u x y + u y x, (5.1) where the term u z /R is due to the finite radius R of the nanotube. The parameters B and µ denote the bulk modulus and the shear modulus for a graphite sheet. Figure 4 shows phonon dispersions calculated in the continuum model (a small out-of-plane energy proportional to the curvature has been added although not important in the electron scattering). The electron-phonon interaction is given for the K point by V el ph = ( V1 V 2 V 2 + V 1 ), V 1 =g 1 (u xx +u yy ), V 2 =g 2 e 3iη (u xx u yy +2iu xy ), (5.2) where g 1 is the deformation potential and g 2 describes the effective potential due to bond-length modification. The diagonal deformation-potential term does not contribute to the backward scattering as in the case of impurities and only the smaller off-diagonal term remains. Figure 5 shows calculated temperature dependence of the resistivity, where ρ A (T ) is the resistivity of an armchair nanotube given by g 2 2 k BT/2e 2 hv 2 F Rµ with v F = γ/ h. The resistivity of an armchair CN is same as that obtained previously [39] except for a difference in g 2. ACKNOWLEDGMENTS The author acknowledges the collaboration with T. Nakanishi, H. Matsumura, R. Saito, H. Suzuura, and M. Igami. This work was supported in part by Grants-in-Aid for Scientific Research and Priority Area, Fullerene Network, from Ministry of Education, Science and Culture in Japan. 329

6 REFERENCES 1. Iijima, S., Nature (London) 354, 56 (1991). 2. Iijima, S., Ichihashi, T., and Ando Y., Nature (London) 356, 776 (1992). 3. Iijima, S., and Ichihashi T., Nature (London) 363, 603 (1993). 4. Bethune, D.S., Kiang, C.H., de Vries, M.S., Gorman, G., Savoy, R., Vazquez, J., and Beyers, R., Nature (London) 363, 605 (1993). 5. Ajiki, H., and Ando, T., J. Phys. Soc. Jpn. 62, 1255 (1993). 6. Ajiki, H., and Ando, T., J. Phys. Soc. Jpn. 62, 2470 (1993) [Errata, 63, 4267 (1994)]; 65, 505 (1996). 7. Ajiki,H., and Ando,T., Physica B 201, 349 (1994); Jpn.J.Appl.Phys.Suppl. 34, 107 (1995). 8. Ajiki, H., and Ando, T., Jpn. J. Appl. Phys. Suppl. 34, 107 (1995). 9. Ando, T., J. Phys. Soc. Jpn. 66, 1066 (1997). 10. Viet, N.A., Ajiki, H., and Ando, T., J. Phys. Soc. Jpn. 63, 3036 (1994). 11. Ajiki, H., and Ando, T., J. Phys. Soc. Jpn. 64, 260 (1995); 65, 2976 (1996). 12. Ajiki, H., and Ando, T., J. Phys. Soc. Jpn. 64, 4382 (1995). 13. Ando, T., J. Phys. Soc. Jpn. 69, No. 6 (2000). 14. Song, S.N., Wang, X.K., Chang, R.P.H., and Ketterson, J.B., Phys. Rev. Lett. 72, 697 (1994). 15. Fischer, J.E., Dai, H., Thess, A., Lee, R., Hanjani, N.M., Dehaas, D.D., and Smalley, R.E., Phys. Rev. B 55, R4921 (1997). 16. Bockrath, M., Cobden, D.H., McEuen, P.L., Chopra, N.G., Zettl, A., Thess, A., and Smalley, R.E., Science 275, 1922 (1997). 17. Langer, L., Bayot, V., Grive, E., Issi, J.-P., Heremans, J.P., Olk, C.H., Stockman, L., Van Haesendonck, C., and Brunseraede, Y., Phys. Rev. Lett. 76, 479 (1996). 18. Kasumov, A.Yu., Khodos, I.I., Ajayan, P. M., and Colliex, C, Europhys. Lett. 34, 429 (1996). 19. Katayama, F., Master thesis (Univ. Tokyo, 1996). 20. Ebbesen, T.W., Lezec, H.J., Hiura, H., Bennett, J.W., Ghaemi, H.F., and Thio, T., Nature (London) 382, 54 (1996). 21. Dai, H., Wong, E.W., and Lieber, C.M., Science 272, 523 (1996). 22. Kasumov, A.Yu., Bouchiat, H., Reulet, B., Stephan, O., Khodos, I.I., Gorbatov, Yu.B., and Colliex, C., Europhys. Lett. 43, 89 (1998). 23. Tans, S.J., Devoret, M.H., Dai, H.-J., Thess, A., Smalley, R.E., Geerligs, L.J., and Dekker, C., Nature (London) 386, 474 (1997). 24. Tans, S.J., Verschuren, R.M., and Dekker, C., Nature (London) 393, 49 (1998). 25. Cobden, D.H., Bockrath, M., McEuen, P.L., Rinzler, A.G., and Smalley, R.E., Phys. Rev. Lett. 81, 681 (1998). 26. Tans, S.J., Devoret, M.H., Groeneveld, R.J.A., and Dekker, C., Nature (London) 394, 761 (1998). 27. Bezryadin, A., Verschueren, A.R.M., Tans, S.J., and Dekker, C., Phys. Rev. Lett. 80, 4036 (1998). 28. Wallace, P.R., Phys. Rev. 71, 622 (1947). 29. Slonczewski, J.C., and Weiss, P.R., Phys. Rev. 109, 272 (1958). 30. Ando, T., and Nakanishi, T., J. Phys. Soc. Jpn. 67, 1704 (1998). 31. Ando, T., Nakanishi, T., and Saito, R., J. Phys. Soc. Jpn. 67, 2857 (1998). 32. Nakanishi, T., and Ando, T., J. Phys. Soc. Jpn. 68, 561 (1999). 33. McEuen, P.L., Bockrath, M., Cobden, D.H., Yoon, Y.-G., and Louie, S.G., Phys. Rev. Lett. 83, 5098 (1999). 34. Chico, L., Benedict, L.X., Louie, S.G., and Cohen, M.L., Phys. Rev. B 54, 2600 (1996). 35. Igami, M., Nakanishi, T., and Ando, T., J. Phys. Soc. Jpn. 68, 716 (1999). 36. Igami, M., Nakanishi, T., and Ando, T., J. Phys. Soc. Jpn. 68, 3146 (1999). 37. Ando, T., Nakanishi, T., and Igami, M., J. Phys. Soc. Jpn. 68, 3994 (1999). 38. Frank, S., Poncharal, P., Wang, Z.L., and de Heer, W.A., Science 280, 1744 (1998). 39. Kane, C.L., Mele, E.J., Lee, R.S., Fischer, J.E., Petit, P., Dai, H., Thess, A., Smalley, R.E., Verschueren, A.R.M., Tans, S.J., and Dekker, C., Europhys. Lett. 41, 683 (1998). 40. Suzuura, H., and Ando, T., Mol. Cryst. Liq. Cryst. (in press); Physica E (in press). 330