Current hot spot in the spin-valley blockade in carbon nanotubes

We present a theoretical study of the spin-valley blockade transport effect in a double quantum dot defined in a straight carbon nanotube. We find that intervalley scattering due to short-range impurities completely lifts the spin-valley blockade and induces a large leakage current in a certain confined range of the external magnetic field vector. This current hot spot emerges due to different effective magnetic fields acting on the spin-valley qubit states of the two quantum dots. Our predictions are compared to a recent measurement [F. Pei et al., Nat. Nanotech. 7, 630 (2012)]. We discuss the implications for blockade-based schemes for qubit initialization/readout and motion sensing of nanotube-based mechanical resonators.

Figure 1

(a) Schematic of the carbon nanotube double quantum dot transport setup showing the spin-valley blockade. The red arrow represents the external magnetic field $\mathit{B}=(B_x,0,B_z)$. Lead-dot tunneling rates ${\Gamma }_L$, ${\Gamma }_R$ and the coherent interdot tunneling amplitude $t$ are indicated. (b) Schematic of the energy levels involved in the transport cycle $(1,1)\to (0,2)\to (0,1)$. If two electrons form a triplet in the (1,1) charge configuration, then current becomes blocked due to Pauli's exclusion principle. (c) Current hot spots at finite transverse magnetic field, formed due to the complete lifting of the spin-valley blockade by short-range disorder.

Figure 2

Current hot spot for various values of the valley-mixing matrix elements ${\Delta }_{K{K}^{\prime }}^L$ and ${\Delta }_{K{K}^{\prime }}^R$ in the two dots, in the case of coherent interdot tunneling. Each row of graphs is obtained using the valley-mixing matrix element magnitudes given at the right end of the row. The difference $\Delta \phi$ of the complex phases of the valley-mixing matrix elements is shown in each graph at the top right corner. Parameters: $g_s=2$, $g_v=54$, ${\Delta }_{\mathrm{so}}=370$$\mu$eV, $t=5$$\mu$eV, ${\Gamma }_L={\Gamma }_R$. The plots are obtained by evaluating Eq. (19). The leakage current values at the marked points ($▵$, $▴$, $\blacksquare$, $\square$, $\diamond$) are discussed in the text.

Figure 3

Schematic energy diagram of the two-electron sector of the DQD Hamiltonian $H_{\mathrm{DQD}}$ at longitudinal magnetic field $B_z=0$ and zero detuning $\epsilon =0$. The solid blue arrow represents the interdot tunneling $H_t$, whereas the dashed orange lines represent the coupling matrix elements of $H_{\mathcal{B}}$ originating from the effective magnetic fields in the two dots. The latter matrix elements are typically of the same order of magnitude, $\sim \frac{g_s{\mu }_B{B}_x{\Delta }_{K{K}^{\prime }}}{{\Delta }_{\mathrm{so}}}$.

Figure 4

Leakage current as a function of transverse external magnetic field and (1,1)-(0,2) energy detuning. Parameters: $B_z=0$; further parameter values are the same as those in Fig. 2c.

Figure 5

The case of inelastic interdot tunneling. Leakage current as a function of transverse ($B_x$) and longitudinal ($B_z$) external magnetic field for various values of the valley-mixing matrix elements ${\Delta }_{K{K}^{\prime }}^L$ and ${\Delta }_{K{K}^{\prime }}^R$ in the two dots. Each row of graphs is obtained using the valley-mixing matrix element magnitudes given at the right end of the row. The difference $\Delta \phi$ of the complex phases of the valley-mixing matrix elements is shown in each graph at the bottom left corner. Parameters: $g_s=2$, $g_v=54$, ${\Delta }_{\mathrm{so}}=$370$\mu$eV. The plots are obtained by evaluating Eq. (24).