Electronic excitation spectrum of metallic carbon nanotubes

We have studied the discrete electronic spectrum of closed metallic nanotube quantum dots. At low temperatures, the stability diagrams show a very regular fourfold pattern that allows for the determination of the electron addition and excitation energies. The measured nanotube spectra are in excellent agreement with the theoretical predictions based on the nanotube band structure. Our results permit the complete identification of the electron quantum states in nanotube quantum dots.

Figure 1

(a) Low-energy band structure of a metallic SWNT. In a finite-length SWNT, the wave vector $k$ is quantized along the tube axis which leads to a set of quantized energy levels separated by $\Delta$ in each band. $\delta$ is the mismatch between the two bands. (b) Schematic diagram of the device geometry. (c) Meaning of $J$ (left) and $dU$ (right). The exchange energy favors spin alignment and $dU$ is the extra charging energy associated with placing two electrons in the same energy level. (d), (e), and (f) Conductance as a function of gate voltage in the linear-response regime at 4 K for three different CVD-grown samples. The NT lengths are 500, 680, and 760 nm, respectively.

Figure 2

(Color online) Sample A—Current as a function of $V$ and $V_G$ at $T=300\mathrm{mK}$. Current goes from −40 nA to $+40\mathrm{nA}$. (b) Values of the parameters for three different groups of four (see text). (c) The differential conductance $(dI∕dV)$ for the first group from (a). Black is zero and bright is $>12\mu \mathrm{S}$. Lines running parallel to the diamond edges correspond to discrete energy excitations. The excitation energies corresponding to the dotted (white) arrows have been used to deduce the model parameters. The predicted excitations are indicated by (yellow) normal arrows. (d) Calculated spectrum for sample A. The stars correspond to the normal (yellow) arrows in (c) and $x$ corresponds to the dotted (white) arrow. The white diagrams indicate the spin filling of the ground state.

Figure 3

(Color online) Differential conductance of samples $B$ (a) and $C$ (c) as a function of $V$ and $V_G$ measured at 300 mK. Black represents $dI∕dV\sim 0$, while the lighter tones correspond to a higher conductance. The dashed lines in (c) indicate the excited states together with inelastic cotunneling. (b) Obtained parameters for sample $B$. (d) Electron quantum states of the NT QD. The numbers on the left denote the ground-state (GS) number of electrons in the last occupied shell. The left column indicates the GS electron configuration. (Note that the two-electron GS is degenerate.) The columns on the right denote the excited-state (ES) configuration. Up to four ES’s are visible in the large Coulomb diamonds.[22] The dotted (red) arrow in the second ES for one electron corresponds to an electron excited from the lower shell.