Spin-valley blockade in carbon nanotube double quantum dots

We present a theoretical study of the Pauli or spin-valley blockade for double quantum dots in semiconducting carbon nanotubes. In our model, we take into account the following characteristic features of carbon nanotubes: (i) fourfold (spin and valley) degeneracy of the quantum-dot levels, (ii) the intrinsic spin-orbit interaction which is enhanced by the tube curvature, and (iii) valley mixing due to short-range disorder, i.e., substitutional atoms, adatoms, etc. We find that the spin-valley blockade can be lifted in the presence of short-range disorder, which induces two independent random (in magnitude and direction) valley-Zeeman fields in the two dots, and hence acts similarly to hyperfine interaction in conventional semiconductor quantum dots. In the case of strong spin-orbit interaction, we identify a parameter regime where the current as the function of an applied axial magnetic field shows a zero-field dip with a width controlled by the interdot tunneling amplitude, in agreement with recent experiments.

Figure 1

(Color online) Schematic of the spin-valley blockade setup with a carbon nanotube double quantum dot and an external magnetic field $\mathit{B}$ aligned with the tube axis. In this regime electrons are transported from source (S) to drain (D) while the DQD occupancy changes between single and double. Spots represent electrons; the figure shows the (0,1) charge configuration of the double dot. Lead-dot tunneling rates ${\Gamma }_L$, ${\Gamma }_R$ and interdot tunneling amplitude $t$ are indicated.

Figure 2

(Color online) (a) Schematic of a quantum dot in a carbon nanotube, with an external magnetic field $\mathit{B}$ aligned with the tube axis. (b) Magnetic-field dependence of the spin-orbit-split single-electron ground state sublevels of a nanotube quantum dot, obtained from diagonalizing $H_0+H_{\text{eff},\text{dis}}$ [see Eqs. (5, 9)]. Spin and valley quantum numbers of the energy levels are indicated on the right.

Figure 3

(Color online) Numerical results for the current as a function of magnetic-field-induced valley splitting for different values of interdot tunneling (shown). Further parameters: ${\Delta }_{\text{SO}}=250\mu \text{eV}$, $b_{Lx}=20\mu \text{eV}$, $b_{Ly}=10\mu \text{eV}$, $b_{Rx}=80\mu \text{eV}$, $b_{Ry}=0\mu \text{eV}$, and therefore $b_{-}^2/{\Delta }_{\text{SO}}=23.6$. Points: numerical data. Curves: the analytical formula (23) fitted to the numerical data with $n^{\ast }$ as the single fitting parameter. Lower two data sets are scaled as shown.

Figure 4

(Color online) Unperturbed two-electron states (lines) and their energies in a CNT DQD at strong spin-orbit coupling and zero magnetic field (cf. Table ). Horizontal arrangement of the states reflects charge configuration and vertical arrangement reflects spin-orbit energies. Red/gray lines: supertriplet states. Dashed red/gray lines: blocked supertriplet states. Black lines: supersinglet states. Green/light gray (blue/dark gray) arrows correspond to off-diagonal elements of the Hamiltonian, induced by disorder (interdot tunneling). (a) Up-spin and down-spin states. The indicated $\times 2$ degeneracy corresponds to the two possible spin configurations. Different spin species are uncoupled. (b) Mixed-spin states.

Figure 5

(Color online) Numerical (dots) and analytical (lines) results for the current as a function of magnetic-field-induced valley splitting for two different disorder realizations. Parameters: $t=10\mu \text{eV}$, $b_{Lx}=\text{cos}\theta \times 100\mu \text{eV}$, $b_{Ly}=\text{sin}\theta \times 100\mu \text{eV}$, $b_{Rx}=200\mu \text{eV}$, and $b_{Ry}=0\mu \text{eV}$. Circles: $\theta =\pi /4$. Boxes: $\theta =15\pi /16$.

Figure 6

(Color online) Two-electron states (lines) and their energies in a CNT DQD at strong disorder. Arrows denote tunnel couplings between (1,1) and (0,2) states. Decay rates of (1,1) states are indicated, cf. Eqs. (25, 26) in text. Here $b_R^{\left(\text{tot}\right)}>b_L^{\left(\text{tot}\right)}$. (a) States and tunnel couplings in the five-dimensional spin-triplet subspaces. The same plot refers to all three spin-triplet subspaces. (b) States and tunnel couplings in the seven-dimensional spin-singlet subspace. (cf. Table ).