The Aharonov–Bohm effect in the carbon nanotube ring

Abstract

The Aharonov–Bohm oscillations of the energy gap and the persistent current in carbon nanotube rings (CNTRs) are investigated theoretically. By considering the rotational symmetry about the axis of CNTRs, we construct a tight-binding model for CNTRs under a magnetic flux (MF) Φ. The Φ-dependent energy gap and persistent current are calculated when the bond lengths are different between the outer atoms and inner atoms. We find that all CNTRs are semiconductors and that zigzag CNTRs exhibit paramagnetism at small magnetic flux Φ in contrast with the diamagnetism of armchair CNTRs. In addition, the effect of the vortex magnetic flux (VMF) on CNTRs is studied. The persistent current is periodic in a VMF Φ_θ with the same period fluxon Φ₀ as in a MF Φ. Furthermore, the VMF is more effective at tuning the energy gaps of the CNTRs than the MF.

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1. Introduction

Among the allotropes of carbon, carbon nanotubes (CNTs) are an important quasi-one dimensional material and are considered to be the ideal candidate for fabricating nanoelectronic and nanomechanical devices due to their unique electronic and mechanical qualities.¹ A (n,m) CNT can be obtained by rolling up a single graphene sheet along the translation vector where n and m are two integer numbers called wrapping indices, and and are the unit vectors of graphene. The radius of a (n,m) CNT is where a is the bond length between the carbon atoms in graphene.

A variant of CNTs, carbon nanotube rings (CNTRs) have attracted many scientists’ attention because they show many unusual qualities, for example a negative magnetoresistance at low temperature,² and interesting transport properties^3,4 since it was observed experimentally.⁵ Y. Ren et al. synthesized CNTRs experimentally by a floating chemical-vapor deposition method and observed additional curvature-induced Raman splitting.⁶ The transport properties of CNTRs were studied theoretically by using the Green function method⁷ and a generalized tight-binding molecular dynamics scheme.⁸ In addition, the energy spectra and the stable structures of CNTRs have been investigated extensively.^9–16 CNTRs can only be constituted of hexagons when the diameter of the ring is large, but can also have some pentagon-heptagon defects when the diameter is small.^17–19 A CNTR with no defect can be made by bending a CNT and connecting its two ends seamlessly. Obviously, the bond between the outer atoms is stretched and the bond between the inner atoms is compressed in the CNTR, which leads to different coupling of the outer atoms with the inner atoms. In previous research,^20,21 these differences were always ignored and the rotational symmetry of the small cell, only containing two atoms, was kept. Actually, these differences can break the rotational symmetry about the CNT’s axis of the small cell. But how this different coupling affects the electronic structure of CNTRs is unknown, and it is reasonable to believe that the effect is strong for CNTRs with a small ring radius.

The effect of an external magnetic field on CNTRs has been an interesting problem.^3,4,20,21 It was mentioned that as a quasi-one dimensional ring, CNTRs are significant to investigate the Aharonov–Bohm current.^20,21 The magnetic flux Φ is pierced through the CNTR and induces a persistent current. M. F. Lin and D. S. Chuu found that the persistent current is linearly dependent on the flux Φ and that semiconducting CNTRs exhibit diamagnetism in contrast with the paramagnetism of metallic CNTRs.²¹ Magnetic fluxes are always applied to the cylinder and the ring structures to study the Aharonov–Bohm effect. The vortex magnetic flux (VMF) corresponding to the vortex field along the axis of a CNT can also be applied to a CNTR. However, a report of this VMF effect has not been seen in the literature.

In this paper, we will investigate the stretched/compressed bond effect and the VMF effect on the electronic structure and the persistent current of the CNTR. The paper is organized as follows. In Sec. II, by considering the rotational symmetry about the axis of a CNTR, we construct a tight-binding model for CNTRs with stretched outer atom bonds and compressed inner atom bonds under a magnetic flux (MF). In Sec. III, the energy gap and the persistent current are investigated in the absence of a VMF for two typical CNTRs: zigzag CNTR (ZCNTR) and armchair CNTR (ACNTR). We find that all the CNTRs are semiconductors and the ZCNTRs exhibit paramagnetism in contrast with the diamagnetic ACNTRs, which is different from previous research.^20,21 In Sec. IV, we study the effect of the VMF. The persistent current and energy gap are periodic in a VMF Φ_θ with the same period as in a magnetic flux Φ. The conclusion is made in Sec. V.

2. Model

As shown in Fig. 1, a CNTR with no defects can be made by bending a long CNT containing N primitive cells of a (n,m) single wall CNT and connecting its two ends. (N,n,m) is used to label this kind of CNTR. The N primitive cells are numbered as j = 0,1,2,…,N − 1 in sequence in a counterclockwise direction. The Cartesian coordinates (x,y,z) of the torus corresponding to the CNTR can be expressed as
where θ,φ ∈ [0,2π) are two angle parameters and the CNTR radius with the carbon bond length a in the CNT. N₀ = gcd(n,m) is the greatest common divisor of n and m.

Fig. 1 A CNTR made from bending a CNT and connecting its two ends. θ and φ directions are indicated in this figure.

The structure of the CNTR with rotational symmetry group C_N around the z axis can make the wave function |ψ〉 of the electrons in the CNTR obey the theorem as follows

where l ∈ [0,N) is an integer. Based on the π electron model, the Hamiltonian of the tight-binding approximation isin which p,q = 1,2,3…N_A with the atom number of the cell N_A = 4(m² + n² + mn)/N₀, and atoms p and q belong to the primitive cells j₁ and j₂, respectively. The hopping matrix elements are the Peierls substitutions, defined as
where a MF Φ is applied to study the persistent current. The fluxon Φ₀ = h/e with the Plank constant h and the charge of an electron −e and Δφ_pq = φ_p − φ_q, Δθ_pq = θ_p − θ_q in which φ_p,q,θ_p,q are the coordinates of p,q atoms in the (θ,φ) coordinate system, respectively. The CNTR has a hexagon lattice structure in which the two basis vectors and are in (θ,φ) coordinates. The hopping parameters t_pq′ are dependent on the distances between atoms. Here we use the Harrison expression²² where the bond length between carbon atoms p and q in the same primitive cell is . The bonds between the outer atoms are stretched while those between the inner atoms are compressed. The strain range 1 − r/R ≤ d_pq/a ≤ 1 + r/R obviously belongs to (0.80,1.15).²³ In this paper, we only consider the nearest neighbor interaction and t = 3.033 eV.²¹

The energy spectra for the CNTRs (100,4,m) are shown in Fig. 2 (a) with m = 2 and (c) with m = 4. The blue solid points are for our results E_bd and the red cross points E_no are obtained from the equation in ref. 21. In order to reveal the effect of the bond length dependent hopping parameters, we show the difference ΔE = E_bd − E_no for the first energy level above the Fermi energy E_F = 0 as a function of l/N in Fig. 2(b) for the CNTR (100,4,2) and in Fig. 2(d) for the CNTR (100,4,4). The difference ΔE can reach 0.3 meV for the CNTR (100,4,2) and 100 meV for the CNTR (100,4,4). Furthermore, we find that the bond length dependent hopping parameters have a bigger impact on the other energy levels. Hence, the bond length dependent hopping parameters cannot be ignored, especially for CNTRs with small radii.

Fig. 2 (Left) The energy spectra for the CNTRs (100,4,m): (100,4,2) in (a) and (100,4,4) in (c). The blue solid points are for our results E_bd and the red cross points E_no are obtained from the equation in ref. 21. (Right) The difference ΔE = E_bd − E_no for the first energy level above the Fermi energy E_F = 0 in the CNTRs (100,4,m) as a function of l/N. m = 2 in (b) and m = 4 in (d).

The persistent current along the φ direction can be obtained from

where the sum runs over all the occupied states, the Lande factor g = 2 due to the spin degeneracy and the Fermi–Dirac distribution in which μ is the chemical potential and with Boltzmann constant k_B and temperature T.

3. The Aharonov–Bohm effect

In this paper, we focus on two typical cases: ZCNTRs (N,n,0) and ACNTRs (N,n,n) for simplicity. In this section, we study the energy gaps and the persistent currents for these two kinds of CNTR with consideration of the bond length dependent hopping parameters.

3.1 The energy gap

It is well known that whether the CNTR is a semiconductor or a metal depends on the energy gap between the highest occupied state (HOS) and the lowest unoccupied state (LUS). In ref. 21, the authors pointed out that the ZCNTRs (N,3L,0) and the ACNTRs (3L,n,n) are metals for a positive integer L, while the others are semiconductors without an external field, but the difference of hopping parameters due to the different bond lengths was ignored. However, the bond length dependent hopping parameter does exist in CNTRs and induces an obvious change of the electronic structure.

The energy gaps of the ZCNTRs (N,n,0) depend on the MF Φ as shown in Fig. 3. The blue solid line is our data taking into consideration the bond length dependent hopping parameters’ difference, and red dashed line is the result from ref. 21. Fig. 3(a–d) are the energy gaps of the ZCNTRs (99,4,0), (100,4,0), (99,6,0) and (100,6,0), respectively. One sees that the bond length dependent hopping parameters can change the gaps of the ZCNTRs (N,n,0). Compared with the red line, the gap of the ZCNTR (99,4,0) decreases by 9.1 meV (3.000 × 10⁻³t) while the gap of the ZCNTR (100,4,0) decreases by 9 meV (2.967 × 10⁻³t). It can be seen that the bond length dependent hopping parameters make the ZCNTR energy gap smaller when n ≠ 3L and larger when n = 3L.

Fig. 3 The energy gap as a function of MF Φ for the ZCNTRs (N,n,0). The blue solid line is for our results and the red dashed line is obtained from the equation in ref. 21. (a) N = 99,n = 4. (b) N = 100,n = 4. (c) N = 99,n = 6. (d) N = 100,n = 6.

Notably, when Φ = 0 and n = 3L, our results show that a gap still exists in the ZCNTRs. This is in contrast to ref. 21. For the ZCNTR (99,6,0), this gap is about 9.7 meV and the gap is 9.5 meV for the ZCNTR (100,6,0). Hence, all the ZCNTRs are semiconducting. It must be pointed out that our conclusion is different from the previous research where some ZCNTRs were metals.²¹

Moreover, the magnitude of gaps’ scale can be tuned by the MF. It is about 1 meV for the two ZCNTRs (99,4,0), (100,4,0) and is 100 meV for the other two ZCNTRs (99,6,0), (100,6,0). We find that the energy gap is insensitive to the MF for the ZCNTRs (N,n ≠ 3L,0), while for the ZCNTRs (N,n = 3L,0), the gap can reach a maximum when the flux Φ = 0.5Φ₀ and still exists without the flux.

The ACNTR (N,n,n) energy gaps vs. the MF Φ are plotted in Fig. 4. The blue solid line and the red dashed line represent the same as in Fig. 3. Fig. 4(a–d) are the energies of the ACNTRs (99,4,4), (100,4,4), (99,6,6) and (100,6,6), respectively. Obviously, the bond length dependent hopping parameters’ difference can also impact on the gaps of the ACNTRs and opens a gap when Φ = 0 and N = 3L, which is different from the conclusion, no gap, in ref. 21. This gap is about 5.45 meV for the ACNTR (99,4,4) and is 11.4 meV for the ACNTR (99,6,6). With fixed N = 3L, the gap increases with n in the case of zero flux.

Fig. 4 The energy gap as a function of MF Φ for the ACNTRs (N,n,n). The blue solid line is for our results and the red dashed line is obtained from the equation in ref. 21. (a) N = 99,n = 4. (b) N = 100,n = 4. (c) N = 99,n = 6. (d) N = 100,n = 6.

The energy gap vanishes for the ACNTR (99,4,4) when Φ = ±0.0166Φ₀ and for the ACNTR (99,6,6) when Φ = ±0.0373Φ₀. As shown in Fig. 4(b and d), the ACNTRs (100,4,4) and (100,6,6) are semiconductors without MF. The highest occupied state and the lowest unoccupied state meet each other at the Fermi energy for (100,4,4) when Φ = ±0.35Φ₀ and for (100,6,6) when Φ = ±0.37Φ₀ and then the CNTRs become metals. Moreover, the magnitude of gap tuned by the MF are 100 meV for these four CNTRs. We see that, contrary to previous research, all ACNTRs (N,n,n) are semiconductors at zero MF and there exist some MFs which make the gap vanish for all ACNTRs.

We also calculated the energy gaps of many other ZCNTRs and ACNTRs as shown in Fig. 5. We can see that all the ZCNTRs and ACNTRs are semiconductors in the absence of an external field. For the conventional metal ZCNTRs (N,n = 3L,0) and ACNTRs (N = 3L,n,n) without an external flux, there is still a gap opened by the broken rotational symmetry along the θ direction, which results from the difference in bond length between the outer atoms and inner atoms. This is different to the previous study.²¹ However, for ACNTRs, there exists some MFs which can make the gap vanish. In addition, the gap is independent of the direction of the MF for all CNTRs. This can be understood from the eigenfunction of the Hamiltonian where is the complex conjugate of . Obviously, the eigenenergy meets E(l,Φ) = E(N − l,−Φ). Furthermore, the gap decreases with the increase in the length of the CNTR when the flux Φ = 0. It is reasonably believable that as N → ∞, the ZCNTRs (N,n,0) and the ACNTRs (N,n,n) will become metals.

Fig. 5 The energy gap as a function of the MF Φ for the ZCNTRs (N,n,0) and the ACNTRs (N,n,n).

As mentioned above, the bond length dependent hopping parameters can break the rotational symmetry and change the gap of the CNTRs (N,n,m). This can be understood as follows. The carbon nanotube ring (CNTR) has a hexagon lattice structure in the (θ,φ) coordinate system. The bond length dependent hopping parameters mean that deformation of the hexagon lattice exists. This deformation breaks the rotational symmetry and can also be described by the gauge field (A_θ,A_φ).²⁴ The gauge fields for two valleys are the opposite of each other. At the low energy approximation, the energy spectra of carriers can be given as

where τ = ±1 labels the valley and p_θ and p_φ are discrete variables related to N, n and m due to the periodic boundary conditions along the θ and φ directions. From the above equation, one can see that the gauge field can obviously change the gaps of the CNTRs (N,n,m) and is dependent on N, n and m whether the gaps become larger or smaller.

3.2 The persistent current

Because of the breakdown of the rotational symmetry along the θ direction due to the bond length dependent hopping parameters, the degeneracy of the energy should be eliminated compared with ref. 21. The bond length dependent hopping parameters can also affect the energy and persistent current.

Using eqn (3), one can deduce that the persistent current of the ZCNTRs (N,n ≠ 3L,0) with a very big gap is very small. We investigate the persistent current of the ZCNTRs (N,n = 3L,0), and the ACNTRs (N = 3L,n,n) and (N ≠ 3L,n,n). The persistent current vs. the MF is shown in Fig. 6 at temperature T = 0 K. The electrons only occupy the electronic states with E_l ≤ μ = E_F = 0 (E_F is the Fermi energy). We set I₀ = gt/Φ₀. The blue line, the red dashed line and green dashed dotted line are the persistent currents for the CNTRs (99,6,0), (99,6,6) and (100,6,6), respectively. All the currents are periodic in the MF with period Φ₀ and are antisymmetric about Φ = Φ₀/2. The curve of the persistent current for the ZCNTR (99,6,0) is more smooth than those for the ACNTRs (99,6,6) and (100,6,6). We see there exist two jumps in the persistent current of the ACNTRs (99,6,6) and (100,6,6) within one period. These are due to the discontinuity of the slope of the curve of the energy level upon the flux. At small MF, the ZCNTR (99,6,0) exhibits paramagnetism in contrast with the diamagnetism of the ACNTRs (99,6,6) and (100,6,6). The magnetic field induced by the persistent current is parallel to the external field corresponding to the small flux in the ZCNTR (99,6,0), while it is antiparallel to the external field in the ACNTRs (99,6,6) and (100,6,6). Because the slope of the curve of the first energy level below the Fermi energy upon flux is at a maximum, the direction of the persistent current mainly depends on the current of the electrons occupying the first energy level below the Fermi energy.

Fig. 6 The variation of the persistent current with the MF Φ for the CNTRs (N,n,m) at temperature T = 0 K. I₀ = gt/Φ₀. The blue line, the red dashed line and green dashed dotted line are for the CNTRs (99,6,0), (99,6,6) and (100,6,6), respectively.

The variation of persistent current with MF is plotted at different temperatures in Fig. 7. The blue solid line, the red dashed line and the green dashed dotted line are the persistent currents at temperatures T = 0 K, T = 5 K and T = 10 K, respectively. Fig. 7(a) is for the ZCNTR (98,6,0) while Fig. 7(b) is for the ACNTR (98,6,6). For temperatures T > 0 K, the electrons will occupy the states above the Fermi energy. As shown in this figure, as the temperature increases, the amplitude of the current oscillation decreases. Due to thermal broadening, the persistent current jumps become smooth as shown for the ACNTR (98,6,6) in Fig. 7(b). However, the temperature does not destroy the periodicity of the persistent current with MF, which results from the electronic structures keeping their periodicity. The persistent current is also antisymmetric about Φ = Φ₀/2. In addition, it is shown in Fig. 3 and 4 that the persistent current of the electrons occupying the first energy level below the Fermi energy in the ZCNTR (98,6,0) is opposite to that of the ACNTR (98,6,6) at small flux. Hence the ZCNTR (98,6,0) exhibits paramagnetism in contrast to the ACNTR (98,6,6) at small flux.

Fig. 7 The same plot as Fig. 6 but at different temperatures. I₀ = gt/Φ₀. The blue solid line, the red dashed line and the green dashed dotted line are at temperatures T = 0 K, T = 5 K and T = 10 K, respectively. (a) N = 98,n = 6,m = 0. (b) N = 98,n = 6,m = 6.

Therefore, from Fig. 6 and 7, we find that the ZCNTRs (N,n,0) with small gaps exhibit paramagnetism in contrast to the ACNTRs (N,n,n). However, in ref. 21 where the difference of the hopping parameters are ignored, the metal ZCNTRs and ACNTRs exhibit paramagnetism, in contrast to the diamagnetism of the semiconducting ZCNTRs and ACNTRs. The conclusion of ref. 21 is only valid for N → ∞. In addition, the persistent currents show an Aharonov–Bohm oscillation for all the CNTRs at different temperatures and are antisymmetric about Φ = Φ₀/2. Furthermore, the amplitude of the persistent current for the ACNTRs (N,n,n) is larger than those of the ZCNTRs (N,n,0). The reason is that the persistent current of the electrons occupying the first energy level below the Fermi energy for the ACNTRs (N,n,n) is larger than that of the ZCNTRs (N,n,0).

4. The effect of VMF

As mentioned above, the MF-dependent energy gap and the persistent current of CNTRs have been studied. The magnetic flux is always applied to the ring and cylinder structures. So a VMF corresponding to the vortex field along the axis of a CNT can also be applied to CNTRs. When both the MF and VMF are applied to CNTRs, a phase factor is introduced into t_pq of our model using the Peierls substitution

In this section, we mainly study the effect of the VMF on the energy gap and the persistent current along the φ direction.

The energy gaps of the ZCNTRs (99,4,0), (99,6,0) and the ACNTRs (99,6,6), (100,6,6) as the function of the VMF Φ_θ are displayed in Fig. 8(a–d), respectively. The blue solid line, the red dashed line, the green dashed dotted line and the black dotted line stand for the energy gaps for Φ = 0, Φ = 1/6Φ₀, Φ = 1/3Φ₀ and Φ = 1/2Φ₀, respectively. We see that the energy gaps are periodic in the VMF with period Φ₀ for all the CNTRs.

Fig. 8 The energy gap as a function of the VMF Φ_θ for the (N,n,m) CNTRs. The blue solid line, the red dashed line, the green dashed dotted line and the black dotted line are for the different MFs Φ = 0, Φ = 1/6Φ₀, Φ = 1/3Φ₀ and Φ = 1/2Φ₀, respectively. (a) N = 99,n = 4,m = 0. (b) N = 99,n = 6,m = 0. (c) N = 99,n = 6,m = 6. (d) N = 100,n = 6,m = 6.

In Fig. 8(a), the energy gap of the ZCNTR (99,4,0) is sensitive to the VMF Φ_θ while this gap is not sensitive to the MF as shown in Fig. 3(a). The energy range which can be tuned by the VMF has a positive correlation with 1/r while that tuned by the MF has a positive correlation with 1/R. The VMF can also tune the magnitude of the gaps’ range up to 1 eV for the other three CNTRs while the MF only tuned the gap up to 0.1 eV. Hence the VMF is more effective at tuning the gaps of CNTRs for all ZCNTRs.

When the ZCNTR’s (99,4,0) energy gap at a minimum, the corresponding VMF Φ^min_θ ≠ 0, while Φ^min_θ = 0 for the other three CNTRs. Around the VMF Φ^min_θ, the gaps behave differently at different MFs Φ, which is especially obvious for the ACNTRs (99,6,6) and (100,6,6). The lengths of the ACNTRs (99,6,6) and (100,6,6) are larger than those of the other two ZCNTRs, so the impact of different MFs on the energy gap is larger than for those of the other two ZCNTRs. Far away from Φ^min_θ, the same gaps persist at all the different MFs Φ for these four CNTRs and are linearly dependent on the VMF applied to the ZCNTRs. Furthermore, the gaps of all the CNTRs are independent of the direction of the VMF. This can also be easily understood from the eigenfunction of the Hamiltonian with the symmetry .

Because the VMF can affect the electronic structure of all CNTRs strongly, it can also be applied to modulate the persistent current. The Φ_θ-dependent persistent currents of the ZCNTRs (99,4,0), (99,6,0) and the ACNTRs (99,6,6), (100,6,6) are shown in Fig. 9 at different MFs. The blue solid line, the red dashed line, the green dashed dotted line and the black dotted line are for the MFs Φ = 0, Φ = 1/6Φ₀, Φ = 1/3Φ₀ and Φ = 1/2Φ₀, respectively. The persistent currents are periodic in the VMF Φ_θ with a period Φ₀. When Φ = 0 and Φ = 0.5Φ₀, the persistent currents are zero for all the CNTRs. This is because for Φ = 0 and Φ = 0.5Φ₀, the currents of all the electrons occupying different energy levels are always zero. For a non-zero current with Φ = 1/6Φ₀ and Φ = 1/3Φ₀, the VMF Φ^max_θ ≠ 0 corresponds to the maximum value of the current for the ZCNTR (99,4,0) while Φ^max_θ = 0 for the other three CNTRs (99,6,0), (99,6,6) and (100,6,6). Furthermore, Φ^max_θ = Φ^min_θ, which means that the same VMF makes the gap a minimum and the persistent current a maximum for the CNTRs. Far from Φ^max_θ within a single period, the energy gap becomes very large and insensitive to the MF. Hence the persistent current almost vanishes. Thus the persistent current is also an even function of the VMF.

Fig. 9 The persistent current as a function of the VMF Φ_θ for the (N,n,m) CNTRs with different MFs Φ. (N,n,m), Φ and the different lines in (a–d) are the same as those in Fig. 8.

Hence, the VMF is more effective at tuning the gap of the CNTRs than the MF, especially for the ZCNTRs (N,n ≠ 3L,0). In addition, the VMF corresponding to the maximum persistent current makes the gap its minimum. Furthermore, the two states of the CNTRs (N,n,m), non-zero current and zero current, tuned by the VMF, can be used for magnetic storage.

5. Conclusion

In this paper, on account of the rotational symmetry around the axis of a CNTR, we constructed a tight-binding model of CNTRs under a MF, considering the difference of the hopping parameters due to the stretch in bond lengths of the outer atoms and the compression in bond lengths of the inner atoms. Based on this model, we investigated the Aharonov–Bohm oscillations of the energy gaps and the persistent current of two typical CNTRs: ZCNTRs and ACNTRs.

The dependencies of the gap and the persistent current on the MF Φ were investigated and compared with those of previous research. One finds that all the ZCNTRs and ACNTRs are semiconductors without an external flux and that ZCNTRs exhibit paramagnetism at small MF in contrast to the ACNTRs, which is different to the previous work. This means that the difference of the hopping parameters cannot be ignored. Moreover, the gaps and persistent currents are periodic in the MF with period Φ₀. The effect of temperature on the persistent current was also studied. The temperature can only decrease the amplitude of the Aharonov–Bohm oscillation but cannot destroy the periodicity of the persistent current.

A VMF was applied to the CNTRs and introduced into our model. The effect of the VMF on the gaps and persistent currents was investigated. The gaps and persistent currents are also periodic in the VMF with period Φ₀. The magnitude of the gaps’ range tuned by the VMF is much larger than those tuned by the MF. The VMF is more effective at tuning the gap of the CNTRs than the MF, especially for the ZCNTRs (N,n ≠ 3L,0) whose gaps are not sensitive to the MF. In addition, comparing the variation of the gap and the persistent current with the VMF, we find that the VMF Φ^max_θ corresponding to the maximum of the persistent current makes the gap its minimum. Furthermore, far away from the VMF Φ^max_θ, the persistent current almost vanishes. The two states of CNTRs, the non-zero and zero current tuned by the VMF, can be used for magnetic storage.

Acknowledgements

We thank Dr B. H. Gong and X. Chen for helpful discussions. This work was supported by NSFC with grants 11074196, 11374237 and 11304241.

References

1. J.-C. Charlier, X. Blase and S. Roche, Rev. Mod. Phys., 2007, 79, 677–732.
2. H. R. Shea, R. Martel and P. Avouris, Phys. Rev. Lett., 2000, 84, 4441–4444.
3. R. Haddon, Nature, 1997, 388, 31–32.
4. A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Y. H. Lee, S. G. Kim and A. G. Rinzler et al., Science, AAAS, Weekly Paper edn, 1996, vol. 273, pp. 483–487.
5. J. Liu, H. Dai, J. H. Hafner, D. T. Colbert and R. E. Smalley, Nature, 1997, 385, 780–781.
6. Y. Ren, L. Song, W. Ma, Y. Zhao, L. Sun, C. Gu, W. Zhou and S. Xie, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 113412.
7. M. Eroms, L. Mayrhofer and M. Grifoni, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 075403.
8. S. Lisenkov, A. N. Andriotis, I. Ponomareva and M. Menon, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 113401.
9. D.-H. Oh, J. Mee Park and K. S. Kim, Phys. Rev. B: Condens. Matter Mater. Phys., 2000, 62, 1600–1603.
10. P. Liu, Y. W. Zhang and C. Lu, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 115408.
11. S. Zhang, S. Zhao, M. Xia, E. Zhang and T. Xu, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 68, 245419.
12. J. Han, Chem. Phys. Lett., 1998, 282, 187–191.
13. M. Huhtala, A. Kuronen and K. Kaski, Comput. Phys. Commun., 2002, 147, 91–96.
14. L. Liu, C. S. Jayanthi and S. Y. Wu, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 64, 033412.
15. O. Hod, E. Rabani and R. Baer, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 67, 195408.
16. V. Meunier, P. Lambin and A. A. Lucas, Phys. Rev. B: Condens. Matter Mater. Phys., 1998, 57, 14886–14890.
17. B. I. Dunlap, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 1933–1936.
18. S. Itoh, S. Ihara and J.-i. Kitakami, Phys. Rev. B: Condens. Matter Mater. Phys., 1993, 47, 1703–1704.
19. S. Ihara, S. Itoh and J.-i. Kitakami, Phys. Rev. B: Condens. Matter Mater. Phys., 1993, 47, 12908–12911.
20. S. Latil, S. Roche and A. Rubio, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 67, 165420.
21. M. F. Lin and D. S. Chuu, Phys. Rev. B: Condens. Matter Mater. Phys., 1998, 57, 6731–6737.
22. W. A. Harrison, Elementary electronic structure, World Scientific, 1999.
23. B. Gong, X.-H. Zhang, E.-H. Zhang and S.-L. Zhang, Mod. Phys. Lett. B, 2011, 25, 823–830.
24. K.-i. Sasaki and R. Saito, Prog. Theor. Phys. Suppl., 2008, 176, 253–278.

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