Shape-sensitive Pauli blockade in a bent carbon nanotube

Motivated by a recent experiment [F. Pei et al., Nat. Nanotechnol. 7, 630 (2012)], we theoretically study the Pauli blockade transport effect in a double quantum dot embedded in a bent carbon nanotube. We establish a model for the Pauli blockade, taking into account the strong $g$-factor anisotropy that is linked to the local orientation of the nanotube axis in each quantum dot. We provide a set of conditions under which our model is approximately mapped to the spin-blockade model of Jouravlev and Nazarov [O. N. Jouravlev and Y. V. Nazarov, Phys. Rev. Lett. 96, 176804 (2006)]. The results we obtain for the magnetic anisotropy of the leakage current, together with their qualitative geometrical explanation, provide a possible interpretation of previously unexplained experimental results. Furthermore, we find that in a certain parameter range, the leakage current becomes highly sensitive to the shape of the tube, and this sensitivity increases with increasing $g$-factor anisotropy. This mutual dependence of the electron transport and the tube shape allows for mechanical control of the leakage current, and for characterization of the tube shape via measuring the leakage current.

Figure 1

Pauli blockade in a bent carbon nanotube. (a) Double quantum dot in a bent CNT in a homogeneous magnetic field $\mathit{B}$. The direction of the magnetic field is characterized by the polar and azimuthal angles $\theta$ and $\phi$. The geometry of the CNT is characterized by the deflection angle $\alpha$. (b) Level diagram in the inelastic interdot tunneling regime [cf. Eq. (17)]. Dashed lines represent matrix elements mixing the triplet states with the singlet $\tilde{S}$. (c) Theoretical result for the magnetic anisotropy of the leakage current $I(\phi ,\theta )$ at $B=0.5$ T, showing features similar to the measured data (Fig. S7 in the Supplementary Information of Ref. [1]). Dashed grey line shows the position of the antiresonance as described by the analytical formula Eq. (22). Parameter values: $\alpha =3^{\circ }$, $g_{\parallel }=32$, $E_{\tilde{S}}\equiv t^2/\Delta =5\mu \mathrm{eV}$, $g_{\perp }^{\left(L\right)}=1.125$, $g_{\perp }^{\left(R\right)}=0.75$, $\gamma =0$. See Sec. 4 for more details.

Figure 2

Magnetic anisotropy of the leakage current: dependence on transverse $g$ factors and the phase $\gamma$. The transverse $g$ factors $g_{\perp }^{(L,R)}$ are shown at the top of each vertical block. The phase $\gamma$ is shown at the right end of each horizontal block. Further parameters: $B=0.5$ T, $\alpha =3^{\circ }$, $g_{\parallel }=32$, and $E_{\tilde{S}}=0$.

Figure 3

Magnetic anisotropy of the leakage current: antiresonance and shape-sensitivity. The transverse $g$ factors $g_{\perp }^{(L,R)}$ are shown at the top of each vertical block. The deflection angle $\alpha$ characterizing the tube shape is shown at the right end of each horizontal block. Further parameters: $B=0.5$ T, $\gamma =0$, $g_{\parallel }=32$, and $E_{\tilde{S}}=5\mu \mathrm{eV}$.

Figure 4

Shape-dependent leakage current in a bent carbon nanotube double quantum dot. The three curves correspond to three different settings of the transverse $g$ factors. By decreasing the transverse $g$ factors, the peak width decreases, i.e., the current becomes more sensitive to the nanotube shape. Parameter values: $B=0.5$ T, $\gamma =0$, $g_{\parallel }=32$, and $E_{\tilde{S}}=5\mu \mathrm{eV}$.