Kondo effect in carbon nanotubes at half filling

In a single state of a quantum dot the Kondo effect arises due to the spin-degeneracy, which is present if the dot is occupied with one electron $(N=1)$. The eigenstates of a carbon nanotube quantum dot possess an additional orbital degeneracy leading to a fourfold shell pattern. This additional degeneracy increases the possibility for the Kondo effect to appear. We revisit the Kondo problem in metallic carbon nanotubes by linear and nonlinear transport measurement in this regime, in which the fourfold pattern is present. We have analyzed the ground state of CNTs, which were grown by chemical vapor deposition, at filling $N=1,N=2$, and $N=3$. Of particular interest is the half-filled shell, i.e., $N=2$. In this case, the ground state is either a paired electron state or a state for which the singlet and triplet states are effectively degenerate, allowing in the latter case for the appearance of the Kondo effect. We deduce numbers for the effective missmatch $\delta$ of the levels from perfect degeneracy and the exchange energy $J$. While $\delta \sim 0.1\text{–}0.2$ (in units of level spacing) is in agreement with previous work, the exchange term is found to be surprisingly small: $J\lesssim 0.02$. In addition we report on the observation of gaps, which in one case is seen at $N=3$ and in another is present over an extended sequence of levels.

Figure 1

(a) Linear response conductance $G$ as a function of back-gate voltage $V_g$ of a SWNT device with contact separation $L\sim 300\mathrm{nm}$ (edge-to-edge of reservoirs), measured at $T=300\mathrm{mK}$ and in a magnetic field of $B=0$ (solid curve) and $B=5\mathrm{T}$ (dashed curve). A clear clustering in four peaks is observed (pronounced in magnetic field), which suggests a single-electron shell pattern with fourfold degeneracy. Charge states corresponding to a filled shell (inset) are labelled as 0 or 4. (b,c) Corresponding gray-scale plots of the differential conductance $dI∕d{V}_{\mathit{sd}}$ (darker more conductive) at $B=0$ (b) and $B=5\mathrm{T}$ (c) as a function of gate $V_g$ and source-drain voltage $V_{\mathit{sd}}$. In the first shell, high conductance Kondo ridges (visible at $V_{\mathit{sd}}\sim 0$) are observed for charge states 1 and 3, whereas they appear for states $1^{\prime },2^{\prime }$, and $3^{\prime }$ in the second shell. The Kondo ridges clearly split in the applied magnetic field.

Figure 2

Illustration of the state-filling scheme for one $(N=1)$ and two $(N=2)$ excess electrons. In (a) the level-mismatch $\delta$ and the exchange energy $J$ are zero, whereas these parameters are nonzero in (b). PE denotes a paired-electron state, and S (T) denotes the singlet (triplet) two electron state. The Kondo effect may arise in three cases: obviously for the spin-$\frac{1}{2}$ with one excess electron $(N=1)$ and if $N=2$ for the spin-1 triplet state, but also for the case for which $\delta =J=0$, i.e., when the singlet and triplet states are degenerate (ST state).

Figure 3

Nonlinear differential conductance $dI∕d{V}_{\mathit{sd}}$ as a function of $V_{\mathit{sd}}$ at $V_g=\mathrm{const}$ deduced from the data shown in Fig. 1b at zero magnetic field. (a) and (b) correspond to the states $N=1$ and $N=3$, whereas (c) corresponds to the half-filled shell $N=2$. All three $dI∕d{V}_{\mathit{sd}}$ cuts have been placed in the middle of the charge state. The visible excitation peaks occur at energy $\Delta$ and are due to inelastic co-tunneling through the excited state. The relevant states are illustrated in the respective insets on the right. The gray curve in (a) has been measured in a magnetic field of 5 T. Arrows point to ${\Delta }_Z=g{\mu }_{B}B=0.58\mathrm{meV}$ using $g=2$.

Figure 4

(a) Nonlinear differential conductance $dI∕d{V}_{\mathit{sd}}$ as a function of $V_{\mathit{sd}}$ taken from the data shown in Fig. 1b at a fixed gate-voltage corresponding to $N=2$. The thick (thin) curve was measured in a magnetic field of $B=5\mathrm{T}(B=0\mathrm{T})$. (b) and (c) illustrate the magnetic-field dependence of the first two excited states at $N=2$. The two cases are drawn approximately to scale using the fact that $J$ is either $\approx 0$ (PE ground state) or $\approx 2\delta$ (T ground state) deduced from the data of Figs. 1, 3.

Figure 5

The Kondo resonance is observed to be offset with respect to the bias voltage $V_{\mathit{sd}}$ for the mixed-valence state with filling $\frac{1}{4}(N=1^{\prime })$ and $\frac{3}{4}(N=3^{\prime })$, whereas it is centered at $V_{\mathit{sd}}=0$ at half-filling. (a) shows the respective differential conductance at constant gate voltage corresponding to part (b), which reproduces the second shell of Fig. 1b. Arrows emphasize an additional asymmetry, discussed in the text.

Figure 6

(a) Linear response conductance plotted as a function of the gate voltage $V_g$ and (b) differential conductance $dI∕d{V}_{\mathit{sd}}$ (darker more conductive) plotted as a function of $V_g$ and $V_{\mathit{sd}}$ for an another SWNT device with length $L\sim 800\mathrm{nm}$ contacted by palladium. The shell pattern of four electrons each extends over five shells $(A-E)$. The Kondo effect occurring at half-filling is marked with $\alpha$, while $\beta$ corresponds to the singlet ground state. and $O$ points to an anomaly, a strong gap-feature arising for a three electron state. The nonlinear $dI∕d{V}_{\mathit{sd}}$ through the middle of this state is shown in (c).

Figure 7

(a) Differential-conductance plot of a SWNT device with contact separation $L\sim 1\mu \mathrm{m}$ at $T=300\mathrm{mK}$ ($\text{maximum}\text{conductance}=e^2∕h$, black). Coulomb blockade diamonds are clearly seen. In addition gaps appear near zero bias. (b) $dI∕d{V}_{\mathit{sd}}$ as a function of $V_{\mathit{sd}}$ and at constant $V_g$ along the corresponding lines, shown in (a). Conductance as a function of gate voltage taken at $V_{\mathit{sd}}=0$ (full) and $V_{\mathit{sd}}=1.5\mathrm{mV}$ (dashed). The shaded region corresponds to (a).