Bends in nanotubes allow electric spin control and coupling

We investigate combined effects of spin-orbit coupling and magnetic field in carbon nanotubes containing one or more bends along their length. We show how bends can be used to provide electrical control of confined spins, while spins confined in straight segments remain insensitive to electric fields. Device geometries that allow general rotation of single spins are presented and analyzed. In addition, capacitive coupling along bends provides coherent spin-spin interaction, including between otherwise disconnected nanotubes, completing a universal set of one- and two-qubit gates.

Figure 1

(Color online) Spectrum of low-energy states of a nanotube quantum dot as a function of magnetic field along axial $\left(B_{\parallel }\right)$ and transverse $\left(B_{\perp }\right)$ directions, for typical device values (${\Delta }_{K{K}^{\prime }}=25\mu \text{eV}$ and ${\Delta }_{\text{SO}}=170\mu \text{eV}$) from Ref. 6. The (yellow) box marks the low-field Kramers doublet, or qubit, for single-electron (mod four) occupancy. Spin-orbit coupling leads to a large effective $g$ factor for axial fields, $B_{\parallel }$, and small effective $g$ factor for transverse fields, $B_{\perp }$. Inset shows axial and transverse projections of applied field.

Figure 2

(Color online) Effective fields along a bend. (a) Illustration of how the magnitude, $B^{\ast }$, (red) and angle, ${\phi }^{\ast }=\angle \left({\mathbf{B}}^{\ast },{\mathbf{x}}^{\prime }\right)$, (blue) relative to local $\left(x^{\prime },y^{\prime }\right)$ coordinates of the local effective magnetic field depend on angle, $\phi$, between the applied field, ${\mathbf{B}}_{\text{ext}}$ (black) and the local transverse direction. Nanotube radius, $R$, and bend radius, $r$, are indicated. (b) Values for angle (blue) and normalized magnitude (red) of effective field, ${\mathbf{B}}^{\ast }$ as a function of angle, $\phi$, for realistic parameters (Ref. 6) with $g_{\parallel }=10$ and $g_{\perp }=1$. Dashed line (red) indicates $B^{\ast }/B_{\text{ext}}=1$.

Figure 3

(Color online) Gate-driven spin resonance on bend. (a) The geometry for EDSR for quantum dot confined along a bend. Spin in the quantum dot in the adjacent straight region is not sensitive to electric fields. An oscillating gate voltage produces an oscillating transverse field, $B_1^{\ast }$, which on resonance drives efficient qubit rotation around $B^{\ast }$. (b) Normalized static field, $B^{\ast }$, (red) and oscillating transverse field, $B_1^{\ast }$ as a function of angle, $\phi$, of the applied field measured in local dot coordinates, for $g_{\parallel }=10$ and $g_{\perp }=1$. Resonance frequency, $g_s{\mu }_B{B}^{\ast }/h$, depends on dot position, allowing frequency tuning and multiplexing. The scale of the transverse field scales with $\delta y^{\prime }/r$, the ratio of the axial excursion due to the oscillating gate voltage to bend radius, $r$.

Figure 4

(Color online) Moving between straight segments provides rapid spin rotation. (a) Two quantum dots (labeled $a$ and $b$) on straight segments with a bend of angle $2\theta$ between them. This geometry provides fast qubit rotation without an oscillating field while keeping spins in straight segments protected from stray electric fields. A spin quantized in the effective field ${\mathbf{B}}_a^{\ast }$ of dot $a$, when moved nonadiabatically to dot $b$, encounters an effective field, ${\mathbf{B}}_b^{\ast }$, that makes a sizable angle, $\eta$, with ${\mathbf{B}}_a^{\ast }$. The spin then precesses about ${\mathbf{B}}_a^{\ast }$ at a frequency $g_s{\mu }_B{B}_b^{\ast }/h$ before being returned nonadiabatically to dot $a$. Shown is the symmetric case, $\phi =0$, where the applied field is transverse to the nanotube axis at its apex. The condition for nonadiabatic transfer is easily met for small $g_{\perp }$ (see text). (b) The angle, $\eta$, between effective field directions in dots $a$ and $b$ as a function of the bend angle, $\theta$, for $\phi =0$ and $g_{\parallel }/g_{\perp }=2$ (lower, blue) and $g_{\parallel }/g_{\perp }=10$ (upper, purple). Inset: for $g_{\parallel }/g_{\perp }>5.8$ pairs of values of $\theta$ for which effective fields in $a$ and $b$ are orthogonal, $\eta =\pi /2$.

Figure 5

(Color online) Schematic of electrostatic coupling via a common gate (red) between two quantum dots formed along bends in two nanotubes. Spin-spin interaction in the quantum dots creates a two-qubit gate that is part of a universal gate set. Applied field, ${\mathbf{B}}_{\text{ext}}$, and local effective fields, ${\mathbf{B}}^{\ast }$ and ${\mathbf{B}}^{\ast \ast }$, along with local axes, $x^{\prime }$ and $y^{\prime }$, are shown for one of the two nanotubes.