Spin relaxation due to deflection coupling in nanotube quantum dots

We consider relaxation of a single-electron spin in a nanotube quantum dot due to its coupling to flexural phonon modes, and identify a new spin-orbit-mediated coupling between the nanotube deflection and the electron spin. This mechanism dominates other spin-relaxation mechanisms in the limit of small energy transfers. Due to the quadratic dispersion law of long-wavelength flexons, $\omega \propto q^2$, the density of states $dq/d\omega \propto {\omega }^{-1/2}$ diverges as $\omega \to 0$. Furthermore, because here the spin couples directly to the nanotube deflection, there is an additional enhancement by a factor of $1/q$ compared to the deformation-potential coupling mechanism. We show that the deflection coupling robustly gives rise to a minimum in the magnetic field dependence of the spin lifetime $T_1$ near an avoided crossing between spin-orbit split levels in both the high- and low-temperature limits. This provides a mechanism that supports the identification of the observed $T_1$ minimum with an avoided crossing in the single-particle spectrum by Churchill et al. [Phys. Rev. Lett. 102, 166802 (2009)].

Figure 1

Spin relaxation due to deflection coupling to bending-mode phonons. (a) Single-electron quantum-dot energy-level diagram with the parameter values of Ref. 5. The four-dimensional subspace is spanned by the states $|\tau s⟩$ labeled by the valley index $\tau =\pm 1$ and the spin projection $s=\pm 1$ along the laboratory $z$ axis, see Eqs. (2, 4). For the magnetic field $B=0$, the upper (lower) Kramers doublet includes states $|1,1⟩$ and $|-1,-1⟩$ ($|1,-1⟩$ and $|-1,1⟩$). Due to the flexon density of states singularity as $\omega \to 0$, we focus on spin-relaxation rates between levels with small energy splittings in the lower Kramers doublet (denoted by $E_{\pm }$), and in the narrow upper avoided crossing (dashed circle). (b) Gedanken experiment to illustrate the coupling mechanism. In the absence of spin-orbit coupling, the electron spin remains fixed in the laboratory frame irrespective of the nanotube’s motion. When the strength of spin-orbit coupling ${\Delta }_{\text{SO}}$ is much greater than the rate of motion $\Omega$, coupling between the spin and orbital moments keeps the electron spin aligned with the direction of the tube axis. (c) For a nanotube with constrained ends, bending-mode phonons cause spatial variations in the direction of the nanotube axis $\hat{\mathit{t}}\left(z\right)$, which locally couple to the electron spin as described above.

Figure 2

(Color online) Behavior of the spectrum in limiting regimes ${\Delta }_{K{K}^{\prime }}=0$ or $B_{\perp }=0$. (a) Spectrum for ${\Delta }_{K{K}^{\prime }}=0,B_{\perp }\ne 0$. The levels $E_1$ and $E_2$ exhibit an avoided crossing with splitting controlled by the field misalignment $B_{\perp }$. The crossing between $E_2$ and $E_3$ is exact, except when the field direction deviates very far from the tube axis (see text). (b) Spectrum for ${\Delta }_{K{K}^{\prime }}\ne 0,B_{\perp }=0$. Intervalley scattering opens an avoided crossing between levels $E_1$ and $E_3$. The exact crossing between $E_1$ and $E_2$ persists as long as ${\Delta }_{\text{SO}}\gtrsim 2\left({\mu }_B/{\mu }_{orb}\right){\Delta }_{K{K}^{\prime }}$.

Figure 3

(Color online) Longitudinal form factor ${\left|M\left(q\right)\right|}^2$ for symmetric square-well confinement as described in Appendix, with barrier separation $L_d=100\text{nm}$, for $E_g=0.03\text{eV}$ (dashed blue) and $E_g=1\text{eV}$ (solid red). For the large-gap tube, $k{L}_d\approx \pi$ and ${\left|M\left(q\right)\right|}^2$ closely follows the hard-wall limit, Eq. (A16), with strong suppression for $q{L}_d\gtrsim 2\pi$. For the small-gap tube, softer confinement results in a larger effective length $L_d^{\ast }$, $k{L}_d^{\ast }=\pi$. When plotted against $q{L}_d^{\ast }$, the profile nearly collapses onto the large-gap result (dotted line).

Figure 4

(Color online) We consider a quantum dot formed in a (small or large gap) semiconducting nanotube. The gate-induced potential $V\left(z\right)$ is taken to be $V\left(z\right)=V_0$ for $z<-L_d/2$ or $z>L_d/2$, and $V\left(z\right)=0$ for $-L_d/2<z<L_d/2$. The band gap $E_g$ arises either due to quantization of transverse motion for pure semiconducting tubes, in which case $E_g\propto 1/R$, or due to curvature of the graphene sheet, in which case $E_g\propto 1/R^2$. Electron density $n\left(z\right)$ is indicated by the red (dotted) curve.