Gate-Dependent Orbital Magnetic Moments in Carbon Nanotubes

We investigate how the orbital magnetic moments of electron and hole states in a carbon nanotube quantum dot depend on the number of carriers on the dot. Low temperature transport measurements are carried out in a setup where the device can be rotated in an applied magnetic field, thus enabling accurate alignment with the nanotube axis. The field dependence of the level structure is measured by excited state spectroscopy and excellent correspondence with a single-particle calculation is found. In agreement with band structure calculations we find a decrease of the orbital magnetic moment with increasing electron or hole occupation of the dot, with a scale given by the band gap of the nanotube.

Figure 1

(a) Dependence of $g_{\mathrm{orb}}$ on the energy ${\epsilon }_N$ of the $N$th-longitudinal electron mode of the nanotube quantum dot. $g_{\mathrm{orb}}$ has been normalized by $e{v}_{F}D/4{\mu }_B$ expected for the first electron on the nanotube and ${ϵ}_N$ are displayed in units of the band gap ${\Delta }_g$. (b) Contour plot of the graphene Dirac dispersion around a $K$ point of the Brillouin zone [upper right inset in (a)]. The first nanotube band is obtained by cutting at constant $k_{\perp }=k_g$ (red curve in inset). Black and green (horizontal) arrows illustrate the constant Fermi velocity ${\nabla }_{k}E$ and the circumferential velocity $\partial E/\partial k_{\perp }$ decreasing with increasing $k_{\parallel }$.

Figure 2

(a) Evolution of the level quartet upon rotation of the nanotube in a constant magnetic field. The model contains both disorder and spin-orbit coupling. (b) Field dependence of the energy levels in the perpendicular and parallel orientation. (d) Coulomb peaks of a nanotube quantum dot measured at 100 mK. The fourfold periodicity of the spectra is evident in the addition energies (c).

Figure 3

(a) Measured transconductance $dI/d{V}_g$ around the first electron of the shaded quartet in Figs. 2c, 2d. (b) $dI/d{V}_g$ vs $V_g$ for $V_{\mathrm{sd}}=4\mathrm{mV}$ along the green horizontal line in (a) measured in a constant magnetic field $B=3.25\mathrm{T}$ while rotating the sample. The black and white dashed lines are a fit to the single-particle model. The leftmost (rightmost) vertical line indicates parallel (perpendicular) alignment with the applied field. (c), (d) As (b) but measured in a parallel and perpendicular magnetic field, respectively. The fits in (b)–(d) yield $g_{\mathrm{orb}}=7.8$. In (b)–(d) an overall magnetic field dependent shift of the ground states are compensated, and the theoretical fits are only partly overlaid not to mask the underlying data.

Figure 4

$g_{\mathrm{orb}}$ and corresponding orbital magnetic moments ${\mu }_{orb}=g_{\mathrm{orb}}{\mu }_B$ as a function of gate, showing a decrease with electron or hole filling. Values are extracted from excited state spectroscopy (○) and Coulomb peak dependence on $B_{\parallel }$ (□). Close to the band gap a clear fourfold periodicity is absent impeding direct comparison to the model. Instead $g_{\mathrm{orb}}$ was measured for all charge states in the region indicated by the horizontal error bar with a spread in values indicated by the vertical error bar. The line is a fit to Eq. (3) assuming a constant dot length and with ${ϵ}_N(V_g)\propto (V_g-0.8\mathrm{V})$ yielding $D=5.3\mathrm{nm}$ and ${\Delta }_g=23\mathrm{meV}$.