Novel magnetic properties of carbon nanotubes.

An external magnetic field is found to have strong effects on the electronic structure of carbon nanotubes. A field-induced metal-insulator transition is predicted for all pure nanotubes. In a weak field, nanotubes exhibit both large diamagnetic and paramagnetic responses which depend on the field direction, the position of the Fermi energy, the helicity, and the size of the nanotube. Universal scalings are found for the susceptibility as functions of the Fermi energy, the temperature, and the size of nanotubes. These results are in agreement with experiments. PACS numbers: 61.46+w, 36.40+d Typeset using REVTEX 1 The exciting discovery of carbon nanotubes [1] stimulated a large number of theoretical studies on their electronic properties. Both tight-binding [2,3] and first principles calculations [4] predicted that nanotubes can be either metallic or semiconducting depending on their helicity and size. Several recent experiments demonstrated some unusual properties. Giant magnetoresistance and indications of a field-induced metal-insulator transition are found in transport measurements [5]. Large diamagnetic susceptibilities are found for a magnetic field both perpendicular and parallel to the tube axis [6,7]. The magnetic properties of nanotubes were studied by Ajiki and Ando [8] using the k · p perturbation method However, they found that the susceptibility χ⊥ (for the field H perpendicular to the tube axis ẑ) is three orders of magnitude larger than χ‖ (when H ‖ ẑ). This finding disagrees with experiments, where it is found that the two are comparable [6]. The k · p calculation is valid only if the Fermi energy is at the center of the band (half filling), and it only provides information about the band structure near the Fermi energy. The orbital magnetism depends on the total band energy which requires the calculation of the total band structure. In addition, in real materials it is likely that the system is not exactly at the half filling (corresponds to the case of a finite carrier density in graphite). Thus, a calculation which includes the field dependence of the complete π band is necessary to understand the magnetic response of the nanotube. In this letter we report the results of such a calculation using the tight-binding model and the London approximation. We obtained the following results. 1) A magnetic field induced metal-insulator transition is predicted for all pure nanotubes, the transition depends on the helicity, the radius R, and the magnetic field direction. 2) The weak-field magnetic susceptibility is large and increases linearly with the size of nanotubes, χ ∼ R; it can be either diamagnetic or paramagnetic and is sensitive to the position of the Fermi energy ǫF . 3) Associate with each nanotube is a unique energy scale ∆0; the scaled susceptibility χ/R is found to be a universal function of ǫF/∆0 and kBT/∆0 for each family of nanotubes. 4) For typical nanotubes, χ ∼ −250×10−6cgs/mole, and |χ| decreases with increasing T . These results are in agreement with recent experiments [6,7]. 2 We use the nearest-neighbor tight-binding hamiltonian to calculate the band structure formed by the π orbitals. This hamiltonian has been shown to be an excellent approximation for calculating the electronic structure of fullerene-related materials such as the large fullerene molecules [9], the solid fullerite and fullerides [10], and the nanotubes [2]. Including the effect of a magnetic field in such a model is straightforward in the London approximation, which has been used successfully for studying the ring current and the magnetic response of C60 and C70 molecules [11]. The symmetry of nanotubes have bees studied by several groups [2]. We follow the elegant approach of White et al. [3]. The structure of a nanotube is defined as the conformal mapping of a strip of a two-dimensional (2-D) graphitic lattice onto the surface of a cylinder. Each nanotube is uniquely characterized by a 2-D lattice vector L = n1a1 + n2a2 = [n1, n2], where a1, a2 are 2-D primitive lattice vectors. The set of integers [n1, n2] determines the geometric properties of the nanotube. The radius of the tube is R = L 2π = √ 3d0 2π √ n21 + n 2 2 + n1n2, where d0 is the C-C bond length. There are two symmetry operations: CN and S(α, h). CN is a N -fold rotation along the axis, where N is the largest common denominator of n1 and n2. The screw translation S(h, α) represents a rotation of α about the axis followed by a translation of h along the axis. The parameters h and α are determined from a 2-D lattice vector P = p1a1 + p2a2, where p1, p2 are integers which satisfy the condition p2n1 − p1n2 = N [3]. In the London approximation, the hopping between site i and site j is modified by a phase factor due to the presence of a magnetic field, Vij = V0 exp (i 2π φ0 ∫ j i A(r) · dr). Here V0 is the nearest-neighbor hopping amplitude, A(r) is the vector potential associated with the magnetic field, and φ0 = hc e is the flux quantum. For the special case of a uniform magnetic field H parallel to the tube axis ẑ, both CN and S(α, h) remain symmetry operations of the tube. In this case the hamiltonian can be solved analytically and we obtain the following band structure, ǫn(κ) = ±V0 [3 + 2 cos(δ1) + 2 cos(δ2) + 2 cos(δ1 + δ2)] 1/2 , n = 0, 1, · · · , N − 1 , δ1 = n1κ− 2πnp1 N + β(n1 + 2n2) , (1) 3 δ2 = n2κ− 2πnp2 N − β(n2 + 2n1) , β = 3d20H 4φ0 . Here κ is the pseudo-momentum associate with the screw translation S(α, h). For H = 0 we recover the result of White et al. [3]. For an idea nanotube each π orbital contributes one electron, the Fermi energy is at the center of the band, ǫF = 0. From Eq.(1) one finds that, in addition to the dependence on the helicity and the radius of the nanotube, the band gap ∆ = 2min{ǫn(κ)} varies strongly with the magnetic field. Our calculations show that ∆ is a simple period function of the flux, φ = πRH , with a period of φ0,