Anomalous Coulomb diamonds and power-law behavior sensitive to back-gate voltages in carbon nanoscale peapod quantum dots

We report anomalous charging effect of single electrons (Coulomb diamonds) observed in carbon nanoscale peapod quantum dots that encapsulate a series of ${\mathrm{C}}_{60}$ molecules. We find that behaviors of diamonds are anomalously sensitive to back-gate voltages $\left(V_{\mathrm{bg}}\right)$, exhibiting two evidently different $V_{\mathrm{bg}}$ regions and a large polarity on $V_{\mathrm{bg}}$. In particular, we find only a sequence of one large diamond followed by three smaller ones existing around ground state. Magnetic-field dependence indicates the presence of shell filling by spin singlet to doubly degenerate electronic levels for these. The encapsulated-${\mathrm{C}}_{60}$ molecules indirectly affect this shell filling at low $V_{\mathrm{bg}}$ possibly via nearly free electrons. In contrast, they act as individual quantum dots coupled in series in high $V_{\mathrm{bg}}$ region. It directly contributes to highly overlapped very large diamonds. Moreover, we report power-law behaviors on conductance versus energy relationships observed in the same carbon nanoscale peapods. We find that the values of powers are also highly sensitive to applied $V_{\mathrm{bg}}$ with three different regions and anomalously high at high source-drain voltages. Because the power laws are found at voltages, which are the nearest outside of the above-mentioned fourfold Coulomb diamonds, correlation of the anomalous powers with orbital-related Tomonaga-Luttinger liquid is discussed.

Figure 1

(Color online) (a) Coulomb diamonds observed in a carbon nanoscale peapod quantum dot at $T=1.5\mathrm{K}$. The $z$ axis is the differential conductance with the magnitudes of which are indicated on the right side. Dotted lines at $V_{\mathrm{bg}}=\pm 1.7\mathrm{V}$ indicate boundaries for two different $V_{\mathrm{bg}}$ regions for small and large diamonds. (b) Expansion of Coulomb diamonds (red regions surrounded by the dotted lines) for $0\mathrm{V}<V_{\mathrm{bg}}<+1.7\mathrm{V}$ in (a). $n$ indicates the number of electrons confined in peapod quantum dots for each diamond. The dotted diamonds are guides to the eye. Arrows mean the $V_{\mathrm{bg}}$ points for Fig. 2.

Figure 2

(a) $V_{\mathrm{bg}}$ shift of conductance peak positions [at $V_{\mathrm{bg}}=0.11$, 0.26, 0.53, and 0.77 around $V_{\mathrm{sd}}=0\mathrm{V}$ shown by arrows in Fig. 1b] in Fig. 1b as a function of magnetic field $B$. (b) Additional energy obtained from each peak pair in Fig. 2a versus $B$.

Figure 3

The double-logarithmic plot of differential conductance $(G=d{I}_{\mathrm{sd}}∕d{V}_{\mathrm{sd}})$ divided by $T^{\alpha }$ as a function of $eV∕k_{B}T$ measured at $V_{\mathrm{bg}}=+0.4\mathrm{V}$ for three different temperatures.

Figure 4

(Color online) Relationships of $d{I}_{\mathrm{sd}}∕d{V}_{\mathrm{sd}}$ to source-drain voltage ($e{V}_{\mathrm{sd}}∕k⪢T=1.5\mathrm{K}$ measured) on doubly logarithmic scales for four different $V_{\mathrm{bg}}$ regions; (a) $V_{\mathrm{bg}}<0\mathrm{V}$, (b) $0\mathrm{V}<V_{\mathrm{bg}}<0.8\mathrm{V}$, (c) $0.8\mathrm{V}<V_{\mathrm{bg}}<1.7\mathrm{V}$, and (d) $1.7\mathrm{V}<V_{\mathrm{bg}}$. These power laws primarily appear just at the nearest-outside regions of fourfold Coulomb diamonds in Fig. 1b. Only the power law in the low $V_{\mathrm{sd}}$ region at $V_{\mathrm{bg}}=1\mathrm{V}$ appears inside of the large Coulomb diamond in fourfold diamonds. The liner lines were obtained from accurate data fitting including measurement points as many as possible and values of power $\alpha$ were exactly estimated.

Figure 5

(Color online) Dependence of power $\alpha$ on different $V_{\mathrm{bg}}$ values, estimated from Figs. 4b, 4c, 4d, in both the present peapod and the empty SWNT quantum dots. Three different $V_{\mathrm{bg}}$ regions [(1) $V_{\mathrm{bg}}<0.8\mathrm{V}$, (2) $0.8\mathrm{V}<V_{\mathrm{bg}}<1.7\mathrm{V}$, and (3) $1.7\mathrm{V}<V_{\mathrm{bg}}$ corresponding to Figs. 4b, 4c, 4d, respectively] separated by dotted lines are evident. Several values of $\alpha$ were added in addition to Fig. 4d only in region 3.