Orbital magnetic moments in pure and doped carbon nanotubes

The unusual band structure of carbon nanotubes (CNs) results in their remarkable magnetic properties. The application of magnetic field $\mathbf{B}$ parallel to the tube axis can change the conducting properties of the CN from metallic to semiconducting and vice versa. Apart from that, B induces (via the Bohm-Aharonov effect) orbital magnetic moments ${\mu }_{\mathrm{orb}}$ in the nanotube. These moments are studied both in pure and hole- or electron-doped CNs, isolated or in a circuit. Remarkably, ${\mu }_{\mathrm{orb}}$ in pure CNs depend uniquely on their original conducting properties, length, and temperature but do not depend on the nanotube radius or the particular chirality. In doped nanotubes the magnetic moments can be strongly altered and depend on the radius and chirality. Temperature can even change their character from diamagnetic at low $T$ to paramagnetic at high $T$. A general electron-hole asymmetry increasing with the doping is found.

Figure 1

Orbital magnetic moment per Å in a single-wall nanotube of radius 25 Å, for different chiralities. All metallic nanotubes have the same (up to 1%) ${\mu }_{\mathrm{orb}}\left(\varphi \right)$. The same is true for the semiconducting nanotubes. The change of the magnetic moments between $T=0\mathrm{K}$ and $T=300\mathrm{K}$ is very small.

Figure 2

The reciprocal lattice of graphene, with momentum states and first Brillouin zones of Zigzag (5,0), Armchair (3,3), and Chiral (4,1) nanotubes doped to $-0.6\gamma$. The Fermi contours are the thick triangular loops and the thick straight lines correspond to $k_{\perp }=\mathrm{const}$ lines. In undoped nanotubes all momentum states in the Brillouin zone enter into the sum from Eq. (9) and ${\mu }_{\mathrm{orb}}\left(\varphi \right)$ is chirality independent. In doped nanotubes only those states which lie below the Fermi level (outside the loops) contribute to the sum. Note that the number of missing states (the missing fragments of $k_{\perp }=\mathrm{const}$ lines) is different in A, Z, and Ch cases, which results in the chirality dependence of ${\mu }_{\mathrm{orb}}\left(\varphi \right)$.

Figure 3

The magnetic moment per Å in a semiconducting nanotube as a superposition of two metallic moments shifted by $\pm {\varphi }_0∕3$.

Figure 4

The dependence of the magnetic moment per unit length on the nanotube radius in zigzag nanotubes ranging from (6, 0) to (120, 0), at half-filling and at two values of doping. The magnetic moment at $E_F=-{\gamma }_{-}$ is divided by 4 so as to fit in the plot.

Figure 5

The Brillouin zone and momentum states of an arbitrary armchair (white dots) and zigzag (black dots) nanotube of similar radius, at $\varphi =0$. The background is the contour plot of $E_{\mathbf{k}}$ from Eq. (11). The Fermi contour is marked by thick solid lines. (a) Full Brillouin zone, undoped nanotube $({\mu }_{\mathrm{chem}}=0)$. The Fermi surface reduces to two points at the vertices of the hexagon. (b) The neighborhood of the Fermi point $(2\pi ∕\left(3\sqrt{3}\right),2\pi ∕3)$, ${\mu }_{\mathrm{chem}}=-0.16\gamma$. The contour is circular and both nanotubes respond with almost the same magnetic moments (see Fig. 7), momentum lines cross the Fermi contour at identical angles. (c) The neighborhood of the Fermi point $(2\pi ∕\left(3\sqrt{3}\right),2\pi ∕3)$, ${\mu }_{\mathrm{chem}}=-0.6\gamma$. The Fermi contour loses the rotational symmetry and magnetic moments in zigzag and armchair nanotubes differ (see Fig. 10). (d) Full Brillouin zone, ${\mu }_{\mathrm{chem}}=-\gamma$. The Fermi contour is a closed hexagon, with two sides parallel to the momentum lines in the zigzag nanotube. Currents in this tube are very strongly enhanced (cf. Fig. 9), in others suppressed (cf. Fig. 8).

Figure 6

The comparison between (top) ${\mu }_{\mathrm{chem}}=\mathrm{const}$ and (bottom) $N_e=\mathrm{const}$ conditions on the conical part of the dispersion relation, i.e., for $\mid {\mu }_{\mathrm{chem}}\mid ⩽0.3{\gamma }_{+}$. The left column shows the orbital magnetic moment per Å (color scale) in armchair (15,15); the right column shows chiral semiconducting (15,14) nanotubes.

Figure 7

The orbital magnetic moment per unit length (grayscale) for $\mid {\mu }_{\mathrm{chem}}\mid ⩽0.3\gamma$, for metallic (top row) armchair (15,15), chiral (19,10), and zigzag (24,0) nanotubes, and for semiconducting (bottom row) nanotubes with similar chiralities: (15,14), (19,9), and (25,0). $R=10.2\mathrm{\AA }$, $T=0$. The Fermi contour around a Fermi point in this regime is almost circular, which accounts for the similarities between the three cases.

Figure 8

The magnetic moment per unit length in a nearly armchair $(38,36)\times (-4028,4101)(R=25\mathrm{\AA })$ nanotube. $E_F=0$, $\pm 0.3{\gamma }_{\pm }$, $\pm {\gamma }_{\pm }$, for ${\mu }_{\mathrm{chem}}=\mathrm{const}$, at $T=0\mathrm{K}$.

Figure 9

The magnetic moment per unit length in a zigzag $(64,0)\times (-2347,4694)(R=25\mathrm{\AA })$ nanotube, at $E_F=0$, $\pm 0.3{\gamma }_{\pm }$, $\pm {\gamma }_{\pm }$, both for ${\mu }_{\mathrm{chem}}=\mathrm{const}$, at $T=0\mathrm{K}$. The magnetic moment at ${\mu }_{\mathrm{chem}}=\pm {\gamma }_{\pm }$ is divided by 10 so as to fit into the plot.

Figure 10

The magnetic moment per unit length (grayscale) in (7,7) armchair (left) and (12,0) metallic zigzag (right) hole-doped nanotubes $(R\approx 5.2\mathrm{\AA })$. The range of chemical potential is from $-{\gamma }_{+}$ to ${\gamma }_{+}$. The part corresponding to an electron-doped nanotube is not symmetrical to the hole doped. At a chemical potential lower than $\sim -0.3\gamma$ (lower part of the plots), the Fermi contour ceases to be circular and the ${\mu }_{\mathrm{orb}}$ patterns become different for different chiralities. The straight line marks ${\mu }_{\mathrm{chem}}=0$.

Figure 11

The magnetic moment per unit length in a nearly armchair $(38,36)\times (-4028,4101)$. The doping is fixed at $0.3{\gamma }_{+}$. The temperature changes the character of ${\mu }_{\mathrm{orb}}$ from diamagnetic (small $T$) to paramagnetic (high $T$).