Positive magnetic susceptibility in polygonal nanotube tori

A type of large paramagnetic persistent current is revealed in polygonal carbon nanotube (CNT) tori through calculation of the persistent current-induced magnetic moment using a tight binding model. Polygonal CNT tori, constructed by apices formed at heptagonal and pentagonal defects on the inner and outer fringes, exhibit paramagnetism related to the semimetallic band structures of the periodic CNT junctions, despite the inherent diamagnetism of graphite. This type of persistent current differs from that originating from metallic band structures, and is caused by both magnetic flux in the inner hole (Aharonov-Bohm flux) and the magnetic flux penetrating the graphite plane directly (direct flux). This magnetic moment is close to that calculated based on the Aharonov-Bohm alone for a cross section that includes the region of direct flux.

Figure 1

(a) Schematic of three-dimensional structure of a polygonal CNT torus. Uniform magnetic field is applied perpendicular to the page. Unit cell labels are denoted by $n_1$. (b) Unit cell of the CNT torus. The torus is composed of six unit cells.

Figure 2

Projection map for the CNT torus. The rectangle $A$ to $B^{\prime }$ is the unit cell. Magnetic field is applied perpendicular to the page $\left(B_{\perp }\right)$. Atoms $(n_2,n_3)$ in the unit cell are labeled.

Figure 3

Dispersion relation with (dashed) and without (solid) direct flux. In the absence of a magnetic field $(B=0)$, the HOL of the torus is located at the crossing of the solid lines (gray circle). The HOL is degenerate and partially occupied. Under finite $B$, the degeneracy is lifted and only the lower level (closed circle) is occupied, while the higher level becomes vacant (open circle). The change of level induced by the $AB$ effect is indicated by an arrow.

Figure 4

Band structure of type-2 periodic CNT junction with parameters $(n_S,n_L,n_{75},n_{55})=(5,7,3,2)$. Circles, lines, and arrows have the same meanings as in Fig. 3. Dashed lines denote the condition $B=0.01{\varphi }_0∕a^2$. (Inset) Detail of the band structure near the HOL $\left(E_N(-2\pi ∕3,0)\right)$.

Figure 5

Band structure of type-3 periodic CNT junction with parameters $(n_S,n_L,n_{75},n_{55})=(3,7,1,3)$. Circles, lines, and arrows have the same meanings as in Fig. 3. Dashed lines denote the condition $B=0.01{\varphi }_0∕a^2$.

Figure 6

Magnetic moments of the polygonal tori corresponding to Fig. 4 (i.e., $(n_S,n_L,n_{75},n_{55})=(5,7,3,2)$) as a function of magnetic field $B$. $M$ and $M_{AB}$ denote the magnetic moments induced by full flux and $AB$ flux only. The full-flux magnetic moment is calculated from the total energy $U\left(B\right)$ by $M\left((j-0.5)\Delta B\right)=-(U\left(j\Delta B\right)-U\left((j-1)\Delta B\right))∕\Delta B$ with $\Delta B=0.5\times {10}^{-4}({\varphi }_0∕a^2)$ and $j=2,3,\dots ,100$. For $M_{AB}$, the total energy is calculated from the Hamiltonian with ${\gamma }^D$ in Eq. (2) is replaced with zero.

Figure 7

Ratios $M_{AB}∕M$ and $c=[(M-M_{AB})∕M_{AB}][{\mid {\vec{S}}_2\mid }^2∕({\mid \vec{S}\mid }^2-{\mid {\vec{S}}_2\mid }^2)]$ as functions of $M$ for 15 CNT tori with $M>4t{a}^2∕{\varphi }_0$, where $M$ and $M_{AB}$ denote the magnetic moments due to full flux and $AB$ flux only. The applied magnetic field $B$ is $0.75\times {10}^{-5}{\varphi }_0∕a^2$. The factor ${\mid \vec{S}\mid }^2∕{\mid {\vec{S}}_2\mid }^2$ denotes the ratio of full flux to $AB$ flux.