Electronic Transport Spectroscopy of Carbon Nanotubes in a Magnetic Field

We report magnetic field spectroscopy measurements in carbon nanotube quantum dots exhibiting fourfold shell structure in the energy level spectrum. The magnetic field induces a large splitting between the two orbital states of each shell, demonstrating their opposite magnetic moment and determining transitions in the spin and orbital configuration of the quantum dot ground state. We use inelastic cotunneling spectroscopy to accurately resolve the spin and orbital contributions to the magnetic moment. A small coupling is found between orbitals with opposite magnetic moment leading to anticrossing behavior at zero field.

Figure 1

(a) Schematic band structure of a CNT near its energy gap. Black lines represent the one-dimensional energy dispersion relation, $E(k_{\parallel })$, at $B=0$ ($k_{\parallel }$ is the wave vector along the CNT axis). The valence (conduction) band has two degenerate maxima (minima). Size quantization in a finite length CNT results in a set of discrete levels with both spin and orbital degeneracy. The degeneracy is lifted by a magnetic field parallel to the CNT. The 1D subbands (and the corresponding levels) at finite $B$ are represented by red and blue dotted (solid) lines. Only the orbital splitting of the energy levels is shown in this figure. (b) Linear conductance, $G$, vs gate voltage, $V_G$, taken at $T=8\mathrm{K}$. Inset: Device scheme.

Figure 2

(a) $G$ vs $B$ on a color scale at $T=0.34\mathrm{K}$ for the $V_G$ range shown in Fig. 1b. Red, white, and blue indicate high, intermediate, and low $G$, respectively. The conductance range is 0 (dark blue) to $3e^2/h$ (dark red). (b) Zoom-in of (a). Here $G$ ranges from 0 (dark blue) to $2e^2/h$ (dark red). The green dashed lines highlight the evolution of the Coulomb peaks with $B$. They are labeled as $A{A}^{\prime }$, $B{B}^{\prime },\dots$, $F{F}^{\prime }$. These divide the plot in different Coulomb blockade regions indicated by the number of electrons in the last two shells (white numbers 0 to VI). The high-$G$ regions (indicated by yellow dashed lines) in between Coulomb peaks are due to the Kondo effect. Orange numbers indicate the spin in each region. On the right side, the $G$ is multiplied by 20, so that the triplet-singlet transition is clearly seen along $F_1{G}_1$.

Figure 3

Diagrams representing the orbital and spin configuration in the different regions of Fig. 2b. Each row follows the $B$ evolution of a given Coulomb peak. Each section in a given row shows two diagrams (separated by a double arrow). These represent the ground state configuration in the two regions separated by the corresponding segment of the Coulomb peak evolution with $B$. In the diagrams, different colors refer to different shells: orange, blue, and black for $n=1$, 2, and 3, respectively (in our discussion here, $n=3$ is always filled). The last added electron is displayed in red, all the others in black. Note that, for each diagram, the final state is the same as the initial state for the diagram immediately below (as indicated by the connected green dashed circles). We use solid (dotted) lines to represent levels with positive (negative) ${\mu }_{\mathrm{orb}}$, shifting down (up) with $B$ (Zeeman splitting is neglected because it is an order of magnitude smaller than the orbital splitting). The slope corresponding to the $B$ evolution of the Coulomb peaks is also indicated under the double arrow. Top right inset: qualitative energy spectrum of the CNT QD as a function of $B$ (Zeeman splitting neglected). Since ${\Delta }_{1,2}\ne {\Delta }_{2,3}$, the energy spectrum is not periodic and the $G(B,V_G)$ pattern in Fig. 2 exhibits fourfold symmetry only for $B<2\mathrm{T}$.

Figure 4

(a) Color plot of the differential conductance, $dI/dV$, vs bias, $V$, and $B$, measured in the center of the Coulomb diamond [see (c)] at $T=30\mathrm{mK}$. The yellow dashed lines indicate the traces shown in the top and bottom insets. Insets: (top) $dI/dV$ trace taken at $B=80\mathrm{mT}$, showing the onset of interorbital IC and a small zero-bias peak due to ordinary spin $1/2$ Kondo effect. The vertical axis scale spans from 0.02 to $0.08e^2/h$. (bottom) Same as top inset, but at $B=0.7\mathrm{T}$, showing the six IC steps. (b) Numerical derivative of the $dI/dV$ plot in (a). The two inner lines result from Zeeman splitting of the Kondo peak at $B=0$. The outer lines represent the $B$ evolution of the spin-split orbital levels. (c) $dI/dV$ vs $V$ and $V_G$, for a single electron in a shell at $B=80\mathrm{mT}$. (d) Calculated $B$ dependence of the IC spectrum for a single electron in a spin degenerate level for two coupled orbitals. Red (blue) lines indicate upward (downward) steps in $dI/dV$ with increasing $V$.