Singlet-triplet relaxation in two-electron silicon quantum dots

We investigate the singlet-triplet relaxation process of a two-electron silicon quantum dot. In the absence of a perpendicular magnetic field, we find that spin-orbit coupling is not the main source of singlet-triplet relaxation. Relaxation in this regime occurs mainly via virtual states, and is due to nuclear hyperfine coupling. In the presence of an external magnetic field perpendicular to the plane of the dot, the spin-orbit coupling is important and virtual states are not required. We find that there can be strong anisotropy for different field directions: parallel magnetic field can increase substantially the relaxation time due to Zeeman splitting, but when the magnetic field is applied perpendicular to the plane, the enhancement of the spin-orbit effect shortens the relaxation time. We find the relaxation to be orders of magnitude longer than for GaAs quantum dots, due to weaker hyperfine and spin-orbit effects.

Figure 1

(a) Level scheme for a doubly occupied silicon QD: the ground singlet involves only one orbital, whereas an additional orbital is needed to form the triplet and higher states, increasing the level separation. (b) Dominant processes in the quantum dot. The two rates indicated are the combined hyperfine and phonon rates, ${\Gamma }_{\mathrm{hc}\text{−}\mathrm{ph}}$, and the combined spin-orbit and phonon rates, ${\Gamma }_{\mathrm{so}\text{−}\mathrm{ph}}$. The energy separation of the ground-state singlet and first triplet is denoted by ${ϵ}_{\mathit{ST}}=E_T-E_{S_g}$, and the exchange splitting by $J$.

Figure 2

$T_{\mathit{ST}}$ as a function of $J$ for the process depicted in the inset: when levels $\mid T⟩$ and $\mid s^{\prime }⟩$ are close in energy $(J\to 0)$, the process is dominated by hyperfine coupling (fast), and as this energy splits, the process is dominated by the phonon emission, $J$ (Ref. 5). For higher energies, the dipole approximation breaks down (Ref. 27).

Figure 3

(Color online) $T_{\mathit{ST}}$ as a function of $B$ for two values of $J$: The solid lines correspond to a plausible value of $J=0.02\mathrm{meV}$ and the broken lines to $J={\epsilon }_{\mathit{ST}}=0.2\mathrm{meV}$ for comparision, (a) as a function of $B_{\parallel }$ and (b) as a function of $B_{\perp }$, including Rashba (red) and without Rashba (black). $T_{\mathit{ST}}$ increases with $B_{\parallel }$ and decreases with $B_{\perp }$.