Nonmonotonic spin relaxation and decoherence in graphene quantum dots with spin-orbit interactions

We investigate the spin relaxation and decoherence in a single-electron graphene quantum dot with Rashba and intrinsic spin-orbit interactions. We derive an effective spin-phonon Hamiltonian via the Schrieffer-Wolff transformation in order to calculate the spin relaxation time $T_1$ and decoherence time $T_2$ within the framework of the Bloch-Redfield theory. In this model, the emergence of a nonmonotonic dependence of $T_1$ on the external magnetic field is attributed to the Rashba spin-orbit coupling-induced anticrossing of opposite spin states. A rapid decrease of $T_1$ occurs when the spin and orbital relaxation rates become comparable in the vicinity of the spin-mixing energy-level anticrossing. By contrast, the intrinsic spin-orbit interaction leads to a monotonic magnetic field dependence of the spin relaxation rate which is caused solely by the direct spin-phonon coupling mechanism. Within our model, we demonstrate that the decoherence time $T_2\simeq 2T_1$ is dominated by relaxation processes for the electron-phonon coupling mechanisms in graphene up to leading order in the spin-orbit interaction. Moreover, we show that the energy anticrossing also leads to a vanishing pure spin dephasing rate for these states for a super-Ohmic bath.

Figure 1

Schematic of a gate-tunable circular graphene quantum dot setup. A homogeneous magnetic field is applied perpendicularly to the gapped graphene sheet. A metallic gate put on top of the graphene defines the confinement potential for a single electron. The figure is not drawn to scale.

Figure 2

Magnetic field dependence of the energy difference between the perturbed three lowest-energy levels and the ground state in a circular graphene quantum dot. Our spin qubit is composed by the ground state and the first excited state with opposite spin orientations. Sequentially from bottom to top, $E_{{\gamma }_0}-E_{{\gamma }_0}$ (solid), $E_{{\gamma }_1}-E_{{\gamma }_0}$ (dashed), and $E_{{\gamma }_2}-E_{{\gamma }_0}$ (dotted-dashed). The Rashba SO interaction-induced anticrossing of the bare quantum dot states $E_{1/2,1,\uparrow }$ and $E_{-1/2,1,\downarrow }$, at $B=B^{*}$ (solid lines in the inset). The spin relaxation rate takes place between the states $|{\gamma }_0\rangle$ and $|{\gamma }_1\rangle$ (${\Gamma }_{\downarrow \uparrow }={\Gamma }_{{\gamma }_0\leftarrow {\gamma }_1}$) before the anticrossing, and between the states $|{\gamma }_0\rangle$ and $|{\gamma }_2\rangle$ (${\Gamma }_{\downarrow \uparrow }={\Gamma }_{{\gamma }_0\leftarrow {\gamma }_2}$) after the anticrossing. Notice that there can be also spin relaxation from $|{\gamma }_2\rangle$ to $|{\gamma }_1\rangle$ followed by a orbital relaxation from $|{\gamma }_1\rangle$ to $|{\gamma }_0\rangle$ after the anticrossing. Nevertheless, we neglect this contribution since we can show that ${\Gamma }_{{\gamma }_1\leftarrow {\gamma }_2}\ll {\Gamma }_{{\gamma }_0\leftarrow {\gamma }_2}$ (higher-order process). Inset: Blowup of the energy levels in the vicinity of the crossing region.

Figure 3

Magnetic field dependence of the spin relaxation time. Parameters used in the numerical evaluation are given in Table 2. Contributions from the deformation potential $g_1:\text{LA}$ (dark, dotted), bond-length change mechanism $g_2:\text{LA}$ (dark, dotted) and $g_2:\text{TA}$ (light, dashed), and the out-of-plane phonons $\mathrm{ZA}$ (light, dotted-dashed). Dark solid: The sum of all processes. The minimum in $T_1$ occurs at the energy-level anticrossing at $B^{*}$. Inset: Blowup of the low magnetic field regime. Competition between the two electron-phonon dominant mechanisms, deformation potential and bending-mode phonons. The absence of Van Vleck cancellation [26, 33, 34, 35] leads to a finite value for $T_1$ in the limit $B\to 0$.