Spin-orbit effects in carbon-nanotube double quantum dots

We study the energy spectrum of symmetric double quantum dots in narrow-gap carbon nanotubes with one and two electrostatically confined electrons in the presence of spin-orbit and Coulomb interactions. Compared to GaAs quantum dots, the spectrum exhibits a much richer structure because of the spin-orbit interaction that couples the electron’s isospin to its real spin through two independent coupling constants. In a single dot, both constants combine to split the spectrum into two Kramers doublets while the antisymmetric constant solely controls the difference in the tunneling rates of the Kramers doublets between the dots. For the two-electron regime, the detailed structure of the spin-orbit split energy spectrum is investigated as a function of detuning between the quantum dots in a 22-dimensional Hilbert space within the framework of a single-longitudinal-mode model. We find a competing effect of the tunneling and Coulomb interaction. The former favors a left-right symmetric two-particle ground state while in the regime where the Coulomb interaction dominates over tunneling, a left-right antisymmetric ground state is found. As a result, ground states on both sides of the (11)-(02) degeneracy point may possess opposite left-right symmetry, and the electron dynamics when tuning the system from one side of the (11)-(02) degeneracy point to the other is controlled by three selection rules (in spin, isospin, and left-right symmetry). We discuss implications for the spin-dephasing and Pauli blockade experiments.

Figure 1

(Color online) Sketch of the planar graphene sheet showing the honeycomb lattice structure. Nearest neighbors belong to two different sublattices $A$ and $B$. A nanotube with chirality (4,1) is formed when the sheet is rolled up along the direction of the chiral vector $\mathbf{C}$. The chiral angle $\theta$ gives the misalignment between the chiral vector and the primitive lattice vector ${\mathbf{a}}_1$. The direction perpendicular to the chiral vector defines the tube axis and is denoted $\mathbf{T}$. Within the tight-binding approximation, we assume the $z$ direction to be perpendicular to $\mathbf{T}$ and $\mathbf{C}$, i.e., the $p_z$ orbitals stick out of the plane of the figure. The Brillouin zone of the honeycomb lattice is shown in the upper right corner. The Dirac points are denoted by ${\mathbf{K}}^{\tau }$, and we choose in our calculations two inequivalent points $K^{\tau }=2\pi /a\left(\tau /3,1/\sqrt{3}\right)$ (marked in red). A graphene sheet folded into a nanotube is shown in the lower right corner.

Figure 2

Energy diagram of a symmetrical CNT-DQD with electrostatic gates inducing a potential $V\left(t\right)$ along the nanotube axis; the origin is chosen in the center of the double dot, and the energy reference point is chosen in the middle of the gap. Two dots $D_1$ and $D_2$ of length $\ell /2$ each are tunnel coupled via a middle barrier $M$ of width $d$. The bound state energy $E$ (shown by dashed line) obeys the criterion $E_g<E<V_0+E_g$.

Figure 3

(Color online) Magnetic field dependence of the energy of the bonding mode of a single-electron DQD for $k_g=-0.045{\text{nm}}^{-1}$ and $\ell =200\text{nm}$, $d=120\text{nm}$, and $V_0=3.95\text{meV}$. The ground-state energy $E_{\text{GS}}$ is close to the gap edge, $E_{\text{GS}}-E_g=1.4\text{meV}$. (a) SO coupling is absent, ${\Delta }_{\text{SO}}=0$. At $B=0$, the spectrum is fourfold degenerate. (b) SO split energy spectrum. The zero field splitting between Kramers doublets is ${\Delta }_{\text{SO}}=0.37\text{meV}$ (with ${\Delta }_1=-0.15\text{meV}$ and ${\Delta }_0=0.04\text{meV}$), see text for details.

Figure 4

(Color online) Single-particle wave functions ${\psi }_{+,\uparrow ,B}^{A\left(B\right)}\left(t\right)$ of a DQD for the same parameter values as in Fig. 3 and $B=1\text{T}$. Vertical lines sketch edges of both dots. (a) Real parts of ${\psi }_{+,\uparrow ,B}^{A\left(B\right)}\left(t\right)$ on both $A\left(B\right)$ sublattices are shown for bonding and antibonding modes, denoted by the superscripts $a$ and $ab$, respectively. At $t=0$ the former modes retain a considerable amplitude, whereas the latter vanish at the origin. Note that at each point the $A$ and $B$ components have nearly the same absolute values but opposite signs. (b) Electron density distribution in the bonding state; a nonvanishing density at $t=0$ is visible as well as a $t\to -t$ asymmetry and difference between $A$ and $B$ densities.

Figure 5

(Color online) Tunneling matrix elements ${\eta }_{\tau ,s}\left(B\right)$ plotted versus magnetic field. Their absolute values are larger for the lower Kramers doublet $\left(K_{\downarrow }^{+},K_{\uparrow }^{-}\right)$. For $B=0$, they depend on spin and isospin through the product $s\tau$. The zero-field difference in the tunneling rate originates from ${\Delta }_1\ne 0$ and vanishes for ${\Delta }_1\to 0$. Parameter values are the same as in Fig. 3b.

Figure 6

(Color online) Detuning dependence of the energy spectrum of a single-electron DQD. Tunnel matrix elements ${\eta }_{\tau ,s}$ from Fig. 5 are used for the respective states, other parameters as in Fig. 3. (a) Both Kramers doublets [identified by line styles (and color), same as Fig. 3] split into bonding and antibonding modes. The numbers denote energy level multiplicity. (b) Level splitting by magnetic field $B=1\text{T}$. Because of the dominating effect of the Aharonov-Bohm flux, $K^{-}$ components with both $s=\uparrow ,\downarrow$ move upwards. The magnitudes of the avoided crossings at $\epsilon =0$, seen both in (a) and (b), are controlled by ${\eta }_{\tau ,s}$.

Figure 7

(Color online) Magnetic field dependence of the Coulomb matrix elements $U_{02}$ in a single QD with dielectric constant $\kappa =1$. Parameters are the same as in Fig. 3b. The $B$ dependence is strong for isospin-polarized states $|{\Phi }_3^{02}⟩$ and $|{\Phi }_4^{02}⟩$, with the opposite signs of the slope for $\tau =\pm$. For isospin-unpolarized states the $B$ dependences are weak. The energies $U_{02}^{\uparrow }$ and $U_{02}^{\downarrow }$ nearly coincide. For $B=0$, the largest difference in $U_{02}$ energies is achieved for the states with both electrons belonging to the lower or upper Kramers doublet. The matrix elements $U_{02}$ strongly influence the position of the (11)-(02) degeneracy point in Figs. 9, 10 below.

Figure 8

(Color online) Magnetic field dependence of the energy spectrum of a two-electron single QD. Coulomb interaction is screened by $\kappa =10$; other parameters are the same as in Fig. 3. (a) Without SO coupling the spectrum is sixfold degenerate at $B=0$, and its $B$ dependence originates mostly from the coupling of the orbital and spin magnetic moments to the field. Wave functions can be represented as spin singlets—isospin triplets and spin triplets—isospin singlets. (b) With SO coupling the spectrum is split at $B=0$. The level crossing at finite $B$ results in a ground state change from two electrons populating the lower Kramers doublet to two isospin polarized electrons. Numbers with arrows denote the energy that corresponds to a particular state among $|{\Phi }_{1,\dots ,6}^{02}⟩$.

Figure 9

(Color online) Two particle spectrum at $B=0$ as a function of detuning $\epsilon$ demonstrating a gradual transition between the (11) and (02) configurations. Parameter values are the same as in Fig. 3b, dielectric constant $\kappa =10$. Hybridized bonding and antibonding states are designated as ${\Phi }_i^B={\alpha }_i{\Phi }_{i+}^{11}+{\beta }_i{\Phi }_i^{02}$ and ${\Phi }_i^{AB}={\alpha }_i{\Phi }_{i+}^{11}-{\beta }_i{\Phi }_i^{02}$, respectively. The coefficients ${\alpha }_i,{\beta }_i>0$ depend on $\epsilon$ and were found from numerical diagonalization. Antisymmetric $|{\Phi }^{11}⟩$ states that practically do not hybridize with $|{\Phi }^{02}⟩$ states are designated as $|{\Phi }_{i-}^{11}⟩$. Slashes indicate the states that are either exactly (Kramers) or nearly degenerate; all of them are spin (isospin) polarized. The states $|{\Phi }_5⟩$ and $|{\Phi }_6⟩$ are split by SO coupling. Remarkably, the ground states of (02) and (11) are not tunnel coupled, and the circle highlights the crossing point between the states $|{\Phi }_6^B⟩$ and $|{\Phi }_{6-}^{11}⟩$

Figure 10

(Color online) Same as in Fig. 9 for a magnetic field $B=1\text{T}$. Again, bonding and antibonding states are denoted as ${\Phi }_i^B$ and ${\Phi }_i^{AB}$, respectively. All degeneracies are removed. Four $|{\Phi }_{13–16}^{11}⟩$ states and six $|{\Phi }_{i-}^{11}⟩$ that practically do not hybridize with $|{\Phi }^{02}⟩$ states are shown as ascending lines. The circle highlights the intersection of the $|{\Phi }_3^B⟩$ hybridized state with $|{\Phi }_{3-}^{11}⟩$. See text for details.

Figure 11

(Color online) Magnetic field dependence of the energy spectrum of a two-electron double dot for $\epsilon =2\text{meV}$ where the admixture of both (02) and (20) configurations is negligibly small. The Coulomb interaction is screened by $\kappa =10$ and other parameters are the same as in Fig. 3. Numbers with arrows denote particular $|{\Phi }_{1\pm ,\dots ,6\pm ,14,\dots ,16}^{11}⟩$ states. Relative magnitudes of the different level splittings originating from the tunneling and Coulomb interaction are distinctly seen. At $B=0$, the dominating splitting comes from ${\Delta }_{\text{SO}}$ depending on whether one or both electrons belong to the upper or lower Kramers doublet; other contributions are smaller by one order of magnitude. The strong $B$ dependence is controlled by the isospin through ${\mu }_{orb}$ and a weaker one by the spin through ${\mu }_B$. The splittings of bonding and antibonding levels are weak, and their sign is controlled by the prevalence of the Coulomb contribution over tunneling.