Spin effects in single-electron transport through carbon nanotube quantum dots

We investigate the total spin in an individual single-wall carbon nanotube quantum dot with various numbers of electrons in a shell by using the ratio of the saturation currents of the first steps of Coulomb staircases for positive and negative biases. The current ratio reflects the total-spin transition that is increased or decreased when the dot is connected to strongly asymmetric tunnel barriers. Our results indicate that total-spin states with and without magnetic fields can be traced by this method.

Figure 1

(Color online) (a) Gray scale plot of the differential conductance $dI∕d{V}_{\mathit{sd}}$ as a function of $V_{\mathit{sd}}$ and $V_g$ at $B=0\mathrm{T}$. The number $n$ indicates the number of extra electrons estimated from the Coulomb diamond around $V_g\sim -0.27\mathrm{V}$. $N$ is the total number of electrons in the dot and is suggested to be an even number (see text). (b) Coulomb oscillations in the gate-voltage range that corresponds to (a). Inset: Addition energy $\left(E_{\mathrm{add}}\right)$ as a function of $n$. The alternate change in the addition energy is observed, indicating the even-odd effect (Refs. 13, 25).

Figure 2

(Color online) (a) Typical Coulomb diamonds $(n=8–10)$. (b) $I\text{−}{V}_{\mathit{sd}}$ characteristics of the intersection of adjacent Coulomb diamonds in (a): $N\leftrightarrow (N+1)$ (open circles) and $(N+1)\leftrightarrow (N+2)$ (closed circles). (c) Typical Coulomb oscillations $(n=0–1)$ as a function of $V_g$ for $V_{\mathit{sd}}$ from $-2\text{to}2\mathrm{mV}$. A magnetic field of $7.0\mathrm{T}$ was applied perpendicular to the tube axis. Each peak is shifted for clarity. (d) $I\text{−}{V}_{\mathit{sd}}$ characteristics along the dashed lines (i) (open circles) and (ii) (closed circles) in (c).

Figure 3

(Color online) [(a)–(c)] Magnetic field dependence of the saturation current ratio $(I_{+}∕I_{-})$ from $n=4$ to 10 in Fig. 1a for the transition from even- to odd-$N$ state (open circles) and for the transition from odd- to even-$N$ state (closed circles). (d) Magnetic field evolution of Coulomb peaks up to $10\mathrm{T}$ from $n=4$ to 10 in Fig. 1a. $V_{\mathit{sd}}=0.1\mathrm{mV}$. The black-colored regions show the Coulomb blockaded regime, and the white-colored regions show the Coulomb peaks.

Figure 4

(Color online) [(a)–(d)] Magnetic field dependence of $I_{+}∕I_{-}$ from $n=0$ to 4 in Fig. 1a, for the transition from even- to odd-$N$ state (open circles) and for the transition from odd- to even-$N$ state (closed circles). (d) Magnetic field evolution of Coulomb peaks up to $10\mathrm{T}$ from $n=0$ to 4 in Fig. 1a. $V_{\mathit{sd}}=0.1\mathrm{mV}$. This plot is obtained by the same method that is used to obtain the plot in Fig. 3d. The dashed line connects the kink positions of two Coulomb peaks ($n=1\leftrightarrow 2$ and $2\leftrightarrow 3$). This result is similar to the four-electron shell-filling scheme; an internal spin flip may occur in this line (Refs. 7, 26).

Figure 5

Mechanisms of the $n=1\leftrightarrow 2$ transition for (a) positive $V_{\mathit{sd}}$ and (b) negative $V_{\mathit{sd}}$, and the $n=2\leftrightarrow 3$ transition for (c) positive $V_{\mathit{sd}}$ and (d) negative $V_{\mathit{sd}}$ in magnetic fields in the single-particle model. $A$ and $B$ sites represent the ground state and the first excited state, respectively. Each single-particle levels have undergone Zeeman splitting. The right barrier is thinner in our experiment; therefore, tunneling current is limited only by the tunneling rate of the thicker left barrier (see text).