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Spectral properties of incommensurate double-walled carbon nanotubes
Available online at www.sciencedirect.com
Physica E 22 (2004) 666 – 669 www.elsevier.com/locate/physe
Spectral properties of incommensurate double-walled carbon nanotubes Kang-Hun Ahna , Yong-Hyun Kimb , J. Wiersigc , K.J. Changb;∗ a Department
of Physics, Chungnam National University, Daejon 305-764, South Korea of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea c Max-Planck-Institut f+ ur Physik komplexer Systeme, N+othnitzer Str. 38, 01187 Dresden, Germany
b Department
Abstract Incommensurate double-walled carbon nanotubes (iDWCNTs) are one of the simplest nanostructures where we can study the role of incommensurability in quantum transport. We investigate the dynamics of electrons via spectral statistics for di1erent interwall interactions in iDWCNTs. The spectral properties of iDWCNTs are described by Poisson, Wigner–Dyson, and semi-Poisson distributions, depending on the energy regime, implying that the transport nature might be ballistic, di1usive, and intermediate between them. The spectral distribution is found to change from Poisson to Wigner–Dyson as the interwall interaction strength increases. We also analyze the characteristics of wave functions for the three di1erent energy regimes, in conjunction with the transport behavior. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.23.−b; 72.15.−v Keywords: Electron transport in mesoscopic systems; Electron conduction in metals and alloys
1. Introduction Multi-walled carbon nanotubes have very complex electronic structure, so that a direct use of their transport properties is severely hindered. If concentric carbon shells are especially incommensurate, since the Bloch theorem is no longer valid, it is di=cult to directly calculate the conductance within the Landauer approach [1]. Therefore, the spectral analysis of energy levels has proven to be a useful indirect tool to probe the nature of eigenstates in the multi-walled carbon nanotubes [2]. ∗ Corresponding author. Tel.: +82-42-8692531; fax: +82-428692510. E-mail address: [email protected] (K.J. Chang).
Recently, transport measurements using transmission electron microscope on individual double-walled carbon nanotubes (DWCNTs) were reported by Iijima’s group [3]. Owing to their precise determination of the chiral indices of DWCNTs, the physical properties of incommensurate DWCNTs now become the topic of interest in both theory and experiment. While it is not clear where the Inger-print of incommensurability will appear in current–voltage characteristics [3], there is no doubt that incommensurability will appear in conductance as well as magnetoresistance measurements. In this work, we present the results of our calculations on the spectral properties of energy levels and the nature of wave functions in incommensurate double-walled nanotubes (iDWCNTs), and discuss the
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.095
K.-H. Ahn et al. / Physica E 22 (2004) 666 – 669 1 0.8 0.6
P(s)
transport behavior of electrons based on the spectral analysis. In Section 2, we introduce the recent results of spectral statistics in iDWCNTs, and present how the spectral statistics will be a1ected by the interwall coupling. In Section 3, we analyze the characteristics of wave functions for three di1erent energy regimes.
667
0.4
2. Spectral properties We use a tight-binding model with one -orbital per carbon atom, which successfully describes the electronic structure of DWCNTs [4,5]. The tight-binding Hamiltonian is given by † H = 0 c j ci − W cos(i j )e(a−di j )= cj† ci ; (1) i; j
i ; j
where 0 (= − 2:75 eV) is the hopping parameter between intra-layer nearest neighbor sites, i and j, and W (=0 =8) is the strength of inter-wall interactions between inter-layer sites, i and j , with the distance L Here ij is the of di j and the cut-o1 for di j ¿ 3:9 A. angle between two orbitals, ci is the annihilation L is the operator of an electron on site i ; a (=3:34 A) L distance between two carbon walls, and = 0:45 A. The two shells of DWCNTs are incommensurate if the ratio of the unit cell lengths along the tube axis is irrational. The size of the Hamiltonian matrix is about 20,000 for the (16; 5)=(23; 8) tube with the tube length of about 53 nm. In constructing the Hamiltonian matrix, we use two-fold coordinated C atoms at the edges of DWCNTs. The contribution of the C dangling bond states to the spectral statistics is removed by excluding the energy states between −0:08 and 0:08 eV. We extract the Muctuations from the level sequence in a customary way to map the real energy spectra {ji } onto the unfolded spectra {Ei } through Ei = NN (ji ), where N (ji ) is the number of levels up to ji and the overline denotes its broadened value. After unfolding, we obtain the distribution P(s) as a function of nearest energy spacing, si = Ei − Ei−1 (the mean level spacing of the unfolded spectra equals 1). For di1usive metals, the statistics of spectral Muctuations in the di1usive regime is known to be described by the Wigner– Dyson distribution or in other words the Gaussian orthogonal ensemble (GOE) of random-matrix theory [8]. In this regime, the distribution P(s) of energy spacings between nearest levels is well Itted by the
0.2 0
0
1
2
3
4
s Fig. 1. The nearest energy spacing distribution P(s) for energy levels between −7 and 7 eV and di1erent helicities (16; 5)=(23; 8) (triangle), (17; 2)=(16; 15) (circle), (16; 5)=(17; 15) (square), and (17; 2)=(21; 9) (diamond). The semi-Poisson (dashed), Poisson (long-dashed), and GOE (dotted) distributions were also plotted for comparison.
Wigner–Dyson surmise PGOE (s) = =2s exp(−=4s2 ), where s is in units of mean level spacing . In the insulating regime, where the energy levels are uncorrelated, P(s) is given by the Poisson distribution, PP (s) = exp(−s). In Fig. 1, we Ind that the spectral properties of iDWCNTs for the large energy window between −7 and 7 eV are well described by the semi-Poisson (SP) distribution, PSP (s) = 4s exp(−2s);
(2)
where the Fermi energy at the charge neutral point is set to zero. The SP distribution is deIned by removing every other level from an ordered Poisson sequence and turned out to be a reference point for the critical statistics of several disordered systems like the Anderson model at mobility edge [6,7]. While the overall spectral statistics follows the semi-Poisson distribution, the spectral properties of energy levels are found to be di1erent for small energy windows: Poisson, semi-Poisson, GOE according to the location of the energy window [2]. These results imply that the electron transport may be either ballistic or di1usive, depending on the position of the Fermi level. As the Fermi energy increases, the spectral statistics evolves as following [2]: Poisson → semi-Poisson → GOE:
668
K.-H. Ahn et al. / Physica E 22 (2004) 666 – 669 1.2 W = 0.0 W = γ0 /8 W = γ0 /4 W = γ0 /2 SP GOE
P(s)
0.8
0.4
0.0
0
1
2
3
4
s
Fig. 2. The nearest energy spacing distribution P(s) for the (7; 4)=(17; 2) iDWCNT in the energy range between −3:0 and −0:08 eV for di1erent interwall interaction strengths.
In Fig. 2, P(s) is plotted for various interwall interaction strengths. As the interwall interaction becomes stronger, P(s) get closer to the semi-Poisson distribution. For weak couplings between the two walls, the spectral correlation is weakened and the degeneracy of single-walled nanotubes, which are constituents of iDWCNTs, are not clearly lifted up. We point out that the actual distribution is Poisson-like with a peak near s = 0, as illustrated in Fig. 2, because the degeneracy still remains due to mirror symmetry in each nanotube. As the interwall interaction becomes stronger, P(s) evolves to the semi-Poisson distribution,
Fig. 3. The distribution of wave functions on the inner (above) and outer (below) walls of the (9; 1)=(17; 2) iDWCNT in the Poisson regime. Darker regions indicate higher densities.
“Poisson like” → semi-Poisson: It is interesting to note that P(s) does not reach the GOE regime by increasing the interwall interaction. We Ind that the coupling of W =0 =2 is strong enough to exhibit the semi-Poisson distribution, indicating that the natural coupling of W =0 is already in the strongly coupled regime. 3. Wave functions Figs. 3–5 show the distributions of wave functions in three di1erent energy regions of the (9; 1)=(17; 2) iDWCNT. When the Fermi energy is close to the charge neutral point, P(s) follows the Poisson distribution. In this weak coupling regime, the wave functions mostly reside on one of the carbon shells, and the phase characteristics is well retained over the tube, as shown in Fig. 3. In this case, the energy levels are
Fig. 4. The distribution of wave functions on the inner (above) and outer (below) walls of the (9; 1)=(17; 2) iDWCNT in the GOE regime. Darker regions indicate higher densities.
uncorrelated, and thus the ballistic transport is highly expected. This is analogous to the appearance of the Poisson distribution for Anderson localized states in disordered metals [8]. As the Fermi energy moves above the Poisson regime, we Ind that P(s) evolves into the semi-Poisson and then into GOE distribution. In the GOE regime, we Ind that the wave functions are severely mixed between the two tubes, and the
K.-H. Ahn et al. / Physica E 22 (2004) 666 – 669
669
iDWCNTs show the spectral distributions of Poisson, semi-Poisson, and GOE, depending on the position of the Fermi level, while overall the P(s) follows the semi-Poisson distribution. These results indicate that the nature of electron transport in multi-walled carbon nanotubes can be either ballistic or di1usive, depending on the Fermi energy. We also Ind that the spectral statistics is a1ected by inter-wall interaction strength. The wave functions of iDWCNTs mostly reside on one of the walls in the Poisson regime, resulting in the ballistic transport. In the semi-Poisson and GOE regimes, a mixing of wave functions occurs between the two walls, and the transport nature is governed by the level of mixing. Fig. 5. The distribution of wave functions on the inner (above) and outer (below) walls of the (9; 1)=(17; 2) iDWCNT in the semi-Poisson regime. Darker regions indicate higher densities.
on-site charges are quite randomly distributed with almost equal weights on both the inner and outer walls, as shown in Fig. 4. The randomness in wave function is coincident with the di1usive transport in the GOE regime. On the other hand, the transport behavior is unclear in the semi-Poisson regime. However, the distribution of wave functions in this regime (see Fig. 5) may provide an insight for understanding the transport characteristics. We Ind that there exist locally random but globally coherent patterns in the distribution, lying in between the random and coherent states. 4. Conclusions Based on the spectral analysis, we Ind that incommensurate double-walled carbon nanotubes exhibit rich behavior in the electron transport. As the Fermi energy increases above the charge neutral point,
Acknowledgements This work was supported by the MOST of Korea through the National Science and Technology Program (Grant No. M1-0213-04-0001) and the QSRC. References [1] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [2] K.-H. Ahn, Y.-H. Kim, J. Wiersig, K.J. Chang, Phys. Rev. Lett. 90 (2003) 026601. [3] M. Kociak, K. Suenaga, K. Hirahara, Y. Saito, T. Nakahira, S. Iijima, Phys. Rev. Lett. 89 (2002) 155501. [4] R. Saito, G. Dresselhaus, M.S. Dresselhaus, J. Appl. Phys. 73 (1993) 494; J.C. Charlier, J.P. Michenaud, Phys. Rev. Lett. 70 (1993) 1858. [5] S. Roche, F. Triozen, A. Rubio, D. Mayou, Phys. Rev. B 64 (2001) 121401. [6] B.I. Shklovskii, B.I. Shapiro, B.R. Sears, P. Lambridianides, H.B. Shore, Phys. Rev. B 47 (1993) 11487. [7] D. Braun, G. Montambaux, M. Pascaud, Phys. Rev. Lett. 81 (1998) 1062. [8] M.L. Mehta, Random Matrices and the Statistical Theory of Energy Levels, Academic Press, New York, 1967.
Physica E 22 (2004) 666 – 669 www.elsevier.com/locate/physe
Spectral properties of incommensurate double-walled carbon nanotubes Kang-Hun Ahna , Yong-Hyun Kimb , J. Wiersigc , K.J. Changb;∗ a Department
of Physics, Chungnam National University, Daejon 305-764, South Korea of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea c Max-Planck-Institut f+ ur Physik komplexer Systeme, N+othnitzer Str. 38, 01187 Dresden, Germany
b Department
Abstract Incommensurate double-walled carbon nanotubes (iDWCNTs) are one of the simplest nanostructures where we can study the role of incommensurability in quantum transport. We investigate the dynamics of electrons via spectral statistics for di1erent interwall interactions in iDWCNTs. The spectral properties of iDWCNTs are described by Poisson, Wigner–Dyson, and semi-Poisson distributions, depending on the energy regime, implying that the transport nature might be ballistic, di1usive, and intermediate between them. The spectral distribution is found to change from Poisson to Wigner–Dyson as the interwall interaction strength increases. We also analyze the characteristics of wave functions for the three di1erent energy regimes, in conjunction with the transport behavior. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.23.−b; 72.15.−v Keywords: Electron transport in mesoscopic systems; Electron conduction in metals and alloys
1. Introduction Multi-walled carbon nanotubes have very complex electronic structure, so that a direct use of their transport properties is severely hindered. If concentric carbon shells are especially incommensurate, since the Bloch theorem is no longer valid, it is di=cult to directly calculate the conductance within the Landauer approach [1]. Therefore, the spectral analysis of energy levels has proven to be a useful indirect tool to probe the nature of eigenstates in the multi-walled carbon nanotubes [2]. ∗ Corresponding author. Tel.: +82-42-8692531; fax: +82-428692510. E-mail address: [email protected] (K.J. Chang).
Recently, transport measurements using transmission electron microscope on individual double-walled carbon nanotubes (DWCNTs) were reported by Iijima’s group [3]. Owing to their precise determination of the chiral indices of DWCNTs, the physical properties of incommensurate DWCNTs now become the topic of interest in both theory and experiment. While it is not clear where the Inger-print of incommensurability will appear in current–voltage characteristics [3], there is no doubt that incommensurability will appear in conductance as well as magnetoresistance measurements. In this work, we present the results of our calculations on the spectral properties of energy levels and the nature of wave functions in incommensurate double-walled nanotubes (iDWCNTs), and discuss the
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.095
K.-H. Ahn et al. / Physica E 22 (2004) 666 – 669 1 0.8 0.6
P(s)
transport behavior of electrons based on the spectral analysis. In Section 2, we introduce the recent results of spectral statistics in iDWCNTs, and present how the spectral statistics will be a1ected by the interwall coupling. In Section 3, we analyze the characteristics of wave functions for three di1erent energy regimes.
667
0.4
2. Spectral properties We use a tight-binding model with one -orbital per carbon atom, which successfully describes the electronic structure of DWCNTs [4,5]. The tight-binding Hamiltonian is given by † H = 0 c j ci − W cos(i j )e(a−di j )= cj† ci ; (1) i; j
i ; j
where 0 (= − 2:75 eV) is the hopping parameter between intra-layer nearest neighbor sites, i and j, and W (=0 =8) is the strength of inter-wall interactions between inter-layer sites, i and j , with the distance L Here ij is the of di j and the cut-o1 for di j ¿ 3:9 A. angle between two orbitals, ci is the annihilation L is the operator of an electron on site i ; a (=3:34 A) L distance between two carbon walls, and = 0:45 A. The two shells of DWCNTs are incommensurate if the ratio of the unit cell lengths along the tube axis is irrational. The size of the Hamiltonian matrix is about 20,000 for the (16; 5)=(23; 8) tube with the tube length of about 53 nm. In constructing the Hamiltonian matrix, we use two-fold coordinated C atoms at the edges of DWCNTs. The contribution of the C dangling bond states to the spectral statistics is removed by excluding the energy states between −0:08 and 0:08 eV. We extract the Muctuations from the level sequence in a customary way to map the real energy spectra {ji } onto the unfolded spectra {Ei } through Ei = NN (ji ), where N (ji ) is the number of levels up to ji and the overline denotes its broadened value. After unfolding, we obtain the distribution P(s) as a function of nearest energy spacing, si = Ei − Ei−1 (the mean level spacing of the unfolded spectra equals 1). For di1usive metals, the statistics of spectral Muctuations in the di1usive regime is known to be described by the Wigner– Dyson distribution or in other words the Gaussian orthogonal ensemble (GOE) of random-matrix theory [8]. In this regime, the distribution P(s) of energy spacings between nearest levels is well Itted by the
0.2 0
0
1
2
3
4
s Fig. 1. The nearest energy spacing distribution P(s) for energy levels between −7 and 7 eV and di1erent helicities (16; 5)=(23; 8) (triangle), (17; 2)=(16; 15) (circle), (16; 5)=(17; 15) (square), and (17; 2)=(21; 9) (diamond). The semi-Poisson (dashed), Poisson (long-dashed), and GOE (dotted) distributions were also plotted for comparison.
Wigner–Dyson surmise PGOE (s) = =2s exp(−=4s2 ), where s is in units of mean level spacing . In the insulating regime, where the energy levels are uncorrelated, P(s) is given by the Poisson distribution, PP (s) = exp(−s). In Fig. 1, we Ind that the spectral properties of iDWCNTs for the large energy window between −7 and 7 eV are well described by the semi-Poisson (SP) distribution, PSP (s) = 4s exp(−2s);
(2)
where the Fermi energy at the charge neutral point is set to zero. The SP distribution is deIned by removing every other level from an ordered Poisson sequence and turned out to be a reference point for the critical statistics of several disordered systems like the Anderson model at mobility edge [6,7]. While the overall spectral statistics follows the semi-Poisson distribution, the spectral properties of energy levels are found to be di1erent for small energy windows: Poisson, semi-Poisson, GOE according to the location of the energy window [2]. These results imply that the electron transport may be either ballistic or di1usive, depending on the position of the Fermi level. As the Fermi energy increases, the spectral statistics evolves as following [2]: Poisson → semi-Poisson → GOE:
668
K.-H. Ahn et al. / Physica E 22 (2004) 666 – 669 1.2 W = 0.0 W = γ0 /8 W = γ0 /4 W = γ0 /2 SP GOE
P(s)
0.8
0.4
0.0
0
1
2
3
4
s
Fig. 2. The nearest energy spacing distribution P(s) for the (7; 4)=(17; 2) iDWCNT in the energy range between −3:0 and −0:08 eV for di1erent interwall interaction strengths.
In Fig. 2, P(s) is plotted for various interwall interaction strengths. As the interwall interaction becomes stronger, P(s) get closer to the semi-Poisson distribution. For weak couplings between the two walls, the spectral correlation is weakened and the degeneracy of single-walled nanotubes, which are constituents of iDWCNTs, are not clearly lifted up. We point out that the actual distribution is Poisson-like with a peak near s = 0, as illustrated in Fig. 2, because the degeneracy still remains due to mirror symmetry in each nanotube. As the interwall interaction becomes stronger, P(s) evolves to the semi-Poisson distribution,
Fig. 3. The distribution of wave functions on the inner (above) and outer (below) walls of the (9; 1)=(17; 2) iDWCNT in the Poisson regime. Darker regions indicate higher densities.
“Poisson like” → semi-Poisson: It is interesting to note that P(s) does not reach the GOE regime by increasing the interwall interaction. We Ind that the coupling of W =0 =2 is strong enough to exhibit the semi-Poisson distribution, indicating that the natural coupling of W =0 is already in the strongly coupled regime. 3. Wave functions Figs. 3–5 show the distributions of wave functions in three di1erent energy regions of the (9; 1)=(17; 2) iDWCNT. When the Fermi energy is close to the charge neutral point, P(s) follows the Poisson distribution. In this weak coupling regime, the wave functions mostly reside on one of the carbon shells, and the phase characteristics is well retained over the tube, as shown in Fig. 3. In this case, the energy levels are
Fig. 4. The distribution of wave functions on the inner (above) and outer (below) walls of the (9; 1)=(17; 2) iDWCNT in the GOE regime. Darker regions indicate higher densities.
uncorrelated, and thus the ballistic transport is highly expected. This is analogous to the appearance of the Poisson distribution for Anderson localized states in disordered metals [8]. As the Fermi energy moves above the Poisson regime, we Ind that P(s) evolves into the semi-Poisson and then into GOE distribution. In the GOE regime, we Ind that the wave functions are severely mixed between the two tubes, and the
K.-H. Ahn et al. / Physica E 22 (2004) 666 – 669
669
iDWCNTs show the spectral distributions of Poisson, semi-Poisson, and GOE, depending on the position of the Fermi level, while overall the P(s) follows the semi-Poisson distribution. These results indicate that the nature of electron transport in multi-walled carbon nanotubes can be either ballistic or di1usive, depending on the Fermi energy. We also Ind that the spectral statistics is a1ected by inter-wall interaction strength. The wave functions of iDWCNTs mostly reside on one of the walls in the Poisson regime, resulting in the ballistic transport. In the semi-Poisson and GOE regimes, a mixing of wave functions occurs between the two walls, and the transport nature is governed by the level of mixing. Fig. 5. The distribution of wave functions on the inner (above) and outer (below) walls of the (9; 1)=(17; 2) iDWCNT in the semi-Poisson regime. Darker regions indicate higher densities.
on-site charges are quite randomly distributed with almost equal weights on both the inner and outer walls, as shown in Fig. 4. The randomness in wave function is coincident with the di1usive transport in the GOE regime. On the other hand, the transport behavior is unclear in the semi-Poisson regime. However, the distribution of wave functions in this regime (see Fig. 5) may provide an insight for understanding the transport characteristics. We Ind that there exist locally random but globally coherent patterns in the distribution, lying in between the random and coherent states. 4. Conclusions Based on the spectral analysis, we Ind that incommensurate double-walled carbon nanotubes exhibit rich behavior in the electron transport. As the Fermi energy increases above the charge neutral point,
Acknowledgements This work was supported by the MOST of Korea through the National Science and Technology Program (Grant No. M1-0213-04-0001) and the QSRC. References [1] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [2] K.-H. Ahn, Y.-H. Kim, J. Wiersig, K.J. Chang, Phys. Rev. Lett. 90 (2003) 026601. [3] M. Kociak, K. Suenaga, K. Hirahara, Y. Saito, T. Nakahira, S. Iijima, Phys. Rev. Lett. 89 (2002) 155501. [4] R. Saito, G. Dresselhaus, M.S. Dresselhaus, J. Appl. Phys. 73 (1993) 494; J.C. Charlier, J.P. Michenaud, Phys. Rev. Lett. 70 (1993) 1858. [5] S. Roche, F. Triozen, A. Rubio, D. Mayou, Phys. Rev. B 64 (2001) 121401. [6] B.I. Shklovskii, B.I. Shapiro, B.R. Sears, P. Lambridianides, H.B. Shore, Phys. Rev. B 47 (1993) 11487. [7] D. Braun, G. Montambaux, M. Pascaud, Phys. Rev. Lett. 81 (1998) 1062. [8] M.L. Mehta, Random Matrices and the Statistical Theory of Energy Levels, Academic Press, New York, 1967.