Effective time-reversal symmetry breaking in the spin relaxation in a graphene quantum dot

We study the relaxation of a single electron spin in a circular gate-tunable quantum dot in gapped graphene. Direct coupling of the electron spin to out-of-plane phonons via the intrinsic spin-orbit coupling leads to a relaxation time $T_1$ which is independent of the $B$ field at low fields. We also find that Rashba spin-orbit induced admixture of opposite spin states in combination with the emission of in-plane phonons provides various further relaxation channels via deformation potential and bond-length change. In the absence of valley mixing, spin relaxation takes place within each valley separately and thus time-reversal symmetry is effectively broken, therefore inhibiting the Van Vleck cancellation at $B=0$ known from GaAs quantum dots. Both the absence of the Van Vleck cancellation as well as the out-of-plane phonons lead to a behavior of the spin-relaxation rate at low magnetic fields which is markedly different from the known results for GaAs. For low-$B$ fields, we find that the rate is constant in $B$ and then crosses over to $\propto B^2$ or $\propto B^4$ at higher fields.

Figure 1

(Color online) (a) The two states of a spin qubit (blue solid arrows) reside in the same valley, as opposed to a Kramers qubit (empty red arrows), formed by a Kramers pair related by time-reversal symmetry $\left(T\right)$. While in single-valley semiconductors such as GaAs these two cases are identical, in graphene the Kramers qubit involves states in different valleys ($K$ and $K^{\prime }$). (b) The $B$-field orientation is given with spherical coordinates $\theta$ and ${\varphi }_B$ relative to the normal to the graphene plane. The propagation direction of the emitted phonon (red wavy arrow) is described by the angle ${\varphi }_q$.

Figure 2

(Color online) Log-log plot of the spin-relaxation time $T_1$ as a function of an external $B$ field perpendicular to the plane $\left(\theta =0\right)$ defined by the graphene sheet. The radius of the dot is $R=25\text{nm}$ and $\Delta =U_0=260\text{meV}$. The individual relaxation channels are the coupling to LA in-plane phonons via deformation potential $\left(g_1\right)$ and the coupling to LA and TA phonons via bond-length change $\left(g_2\right)$. Also shown are the direct coupling to the out-of-plane phonons with quadratic (ZA) and linear $\left({\text{ZA}}^{\prime }\right)$ dispersion. The red dotted, blue dashed, and solid black lines represent the sum of all four processes. For the red (blue) curve, a quadratic (linear) dispersion relation is assumed, while for the black curve a crossover from linear to quadratic is assumed (see text). Inset: dependence of the relaxation rate on the inclination angle $\theta$ of the $B$ field.