Electrically driven spin resonance in a bent disordered carbon nanotube

Resonant manipulation of carbon nanotube valley-spin qubits by an electric field is investigated theoretically. We develop a new analysis of electrically driven spin resonance exploiting fixed physical characteristics of the nanotube: a bend and inhomogeneous disorder. The spectrum is simulated for an electron valley-spin qubit coupled to a hole valley-spin qubit and an impurity electron spin, and features that coincide with a recent measurement are identified. We show that the same mechanism allows resonant control of the full four-dimensional spin-valley space.

Figure 1

(a) A bent carbon nanotube device. Left and right quantum dots are formed using gate voltages to create a pair of potential wells. As indicated, one gate acts as a microwave antenna for driving EDSR transitions, which can be detected via the current between two contact electrodes. The bend is parameterized by a local tangent vector $\widehat{\mathbf{n}}$. Inset: Coordinates used in this paper, including the polar angles $(\Theta ,\Phi )$ that characterize the direction of the magnetic field $\mathbf{B}$. (b) Spectrum of a single-electron nanotube quantum dot in a magnetic field parallel to $\widehat{\mathbf{n}}$, taking ${\Delta }_{\mathrm{SO}}=0.8\mathrm{meV},{\Delta }_{K{K}^{\prime }}=0.2\mathrm{meV},g_{\mathrm{s}}=2$, and $g_{\mathrm{orb}}=12$. These values are similar to those measured by transport spectroscopy [15]. The four energy eigenstates $\{\Uparrow ,\Downarrow ,{\Uparrow }^{*},{\Downarrow }^{*}\}$, are labeled, as are their high-field limits $\{K\uparrow ,K\downarrow ,K^{\prime }\downarrow ,K^{\prime }\uparrow \}$. The Kramers splitting $\Delta E$ and valley-mixing splitting ${\Delta }_{K{K}^{\prime }}$ are indicated.

Figure 2

Rabi frequency in a bent nanotube with homogeneous valley mixing $({\Delta }_{K{K}^{\prime }}$ and $\phi$ independent of $z)$ as a function of magnetic field angles [defined in Fig. 1]. (a) Rabi frequency $\Omega$ in the lower Kramers doublet, calculated according to Eq. (19) and normalized by the free-electron Larmor frequency $\hslash \omega =g_{\mathrm{s}}{\mu }_{\mathrm{B}}\left|\mathbf{B}\right|$ and the driving amplitude $\delta \theta$. (b) Rabi frequency as a function of $\Theta$ for $\Phi =0$, calculated by Eq. (19) (solid). Separate contributions from $\delta {\mathbf{B}}_{\mathrm{eff}}^V$ [12] and $\delta {\mathbf{B}}_{\mathrm{eff}}^{VV}$ are marked with dashed and dotted lines respectively. Dashed ellipses mark field angles where $\Omega =0$. (c), (d) The same plots for the upper doublet. Whereas for the lower doublet $\Omega$ vanishes at two field angles, for the upper doublet it vanishes only for $\mathbf{B}$ along $y$. Throughout this figure, we take ${\Delta }_{\mathrm{SO}}/{\Delta }_{K{K}^{\prime }}=4$ and $g_{\mathrm{orb}}/g_{\mathrm{s}}=6$, consistent with Fig. 1.

Figure 3

Rabi frequency in a straight nanotube device $(\delta \theta =0$, see inset) with inhomogeneous valley mixing, calculated according to Eq. (19) as a function of $\Theta$. The Rabi frequency is the same for both doublets. By symmetry, $\Omega$ is independent of $\Phi$. Three cases are considered: inhomogeneous valley-mixing amplitude only $\left(\delta \phi =0\right)$, inhomogeneous valley-mixing phase only $\left({\delta }_{K{K}^{\prime }}=0\right)$, and both effects combined $\left({\delta }_{K{K}^{\prime }}=\delta \phi \right)$. For comparison, $\Omega$ is normalized by $\sqrt{{\delta }_{K{K}^{\prime }}^2+\delta {\phi }^2}$ and $\hslash \omega =g_{\mathrm{s}}{\mu }_{\mathrm{B}}\left|\mathbf{B}\right|$. This figure assumes the same numerical parameters as Fig. 2.

Figure 4

Energy levels and EDSR spectra of two coupled valley-spin qubits. (a)–(c) Simulated energy levels for parallel (a) and perpendicular (b) magnetic field, and of field angle at constant magnitude (c). For all plots, the field is in the $x-z$ plane, i.e., $\Phi =0$. States are labeled $\tilde{S},\widetilde{AS}$ according to their longitudinal symmetry at $B=0$ (see text). Inset to (b) shows schematically the quantum dot arrangement. (d)–(f) Simulated EDSR transition spectra for the same conditions. Selected transitions (marked with arrows in the energy spectra) are highlighted. Corresponding $g$ factors obtained from the line slopes are marked. In (e), (f), $P_{\mathrm{S}}$ is scaled in parts of the plots to make the low-frequency resonances clearer. (g)–(i): Experimental EDSR transport spectra measured in [15]. Simulations assume ${\Delta }_{\mathrm{SO}}=0.8\mathrm{meV},g_{\mathrm{orb}}=12$ for both quantum dots, ${\Delta }_{K{K}^{\prime }}=0.2\mathrm{meV}$ and $\phi =0$ for the electron quantum dot, and valley-mixing strength ${\Delta }_{K{K}^{\prime }\mathrm{h}}=0.6\mathrm{meV}$ and phase ${\phi }_{\mathrm{h}}=0.005\pi$ for the hole quantum dot. Values for ${\Delta }_{\mathrm{SO}},g_{\mathrm{orb}}$, and ${\Delta }_{K{K}^{\prime }}$ correspond to values deduced from dc transport in Ref. [15]. Values for ${\Delta }_{K{K}^{\prime }\mathrm{h}}$ and ${\phi }_{\mathrm{h}}$ are chosen to approximately match EDSR spectra in the same device, requiring ${\Delta }_{K{K}^{\prime }\mathrm{h}}$ to be slightly higher than indicated by dc transport. Exchange interaction is set to $J_0/h=7\mathrm{GHz}$ to match the observed zero-field splitting.

Figure 5

Energy spectrum (a) and EDSR transition spectra (b)–(d) of coupled valley-spin qubits and an impurity spin [schematic in inset of (a)]. We take for the impurity $g$ factor $g_{\mathrm{I}}=2$ and coupling strength $J_{\mathrm{I}}=-1$ GHz. Other parameters are as in Fig. 3. Magnetic field is parallel with the nanotube in (a) and (b), perpendicular in (c), and has magnitude $\left|\mathbf{B}\right|=0.016\mathrm{T}$ in (d). Zero-field energy gaps ${\Delta }_{+}/h=6.8\mathrm{GHz}$ and ${\Delta }_{-}/h=0.8\mathrm{GHz}$, are indicated. Energy spectra corresponding to (c) and (d) are shown in Fig. 7.

Figure 6

(a) Energy differences between valley-spin states as a function of parallel magnetic field [parameters as in Fig. 1]. (b) Fidelity of qubit rotations on each of the transitions in (a). Equal driving via all channels is assumed, characterized by a single parameter $\delta \equiv \delta \theta ={\delta }_{K{K}^{\prime }}=\delta \phi$. For this plot, $\delta =0.1$. (c) Minimum fidelity among the six transitions, plotted as a function of $B$ and $\delta$. For all values of $B$, fidelity is highest for weak driving, where individual transitions can be precisely addressed. The fidelity drops at values of $B$ where transitions in (a) become near degenerate.

Figure 7

Energy levels for two coupled valley-spin qubits and an impurity spin, for the same parameters as Fig. 5. (a) is calculated as a function of $B$ perpendicular to the nanotube, (b) as a function of field angle for $B=0.016$ T. The transitions highlighted in Figs. 5 and 5 are indicated.