Electrical Pump-and-Probe Study of Spin Singlet-Triplet Relaxation in a Quantum Dot

Spin relaxation from a triplet excited state to a singlet ground state in a semiconductor quantum dot is studied by employing an electrical pump-and-probe method. Spin relaxation occurs via cotunneling when the tunneling rate is relatively large, confirmed by a characteristic square dependence of the relaxation rate on the tunneling rate. When cotunneling is suppressed by reducing the tunneling rate, the intrinsic spin relaxation is dominated by spin-orbit interaction. We discuss a selection rule of the spin-orbit interaction based on the observed double-exponential decay of the triplet state.

Figure 1

(a) SEM picture of the device together with a schematic of the measurement set-up. (b) dc current $I$ as a function of $V_{\mathrm{L}}$ and $B$ measured at $V_{\mathrm{sd}}=0.15\mathrm{mV}$ in the first cooldown. $I$ for the even $N$ data is divided by 10. Evolution of the current peak is sketched around the transition fields (triangles). (c) $dI/d{V}_{\mathrm{L}}$ at $V_{\mathrm{sd}}=1.2\mathrm{mV}$ taken at the dashed square region shown in (b). ${\Delta }_{\mathrm{ST}}$ scale shows the region where the spin relaxation measurement is conducted. (d) $dI/d{V}_{\mathrm{L}}$ at $V_{\mathrm{sd}}=1.0\mathrm{mV}$ measured in the second cooldown. The vertical dashed lines denote spin transitions of the ground state. A blue circle marks an anticrossing between two singlet states. Intensity for the high $B$ data is enhanced by multiplying some smooth numerical function.

Figure 2

(a) $n_{\mathrm{t}}$ as a function of the initialization time $t_{\mathrm{l}}$ with ${\Delta }_{\mathrm{ST}}=300\mu \mathrm{eV}$ ($B=0.6\mathrm{T}$ in the first cooldown). The inset shows the energy diagram of the initialization process. (b) $n_{\mathrm{t}}$ as a function of the wait time, $t_{\mathrm{h}}$. The inset shows the energy diagram of the relaxation process.

Figure 3

(a) Log-log plot of ${\tau }_{\mathrm{s}}$ as a function of ${\Gamma }_{\mathrm{tot}}$. ${\tau }_{\mathrm{s}}$ for inter-LL is measured at $B=0.6\mathrm{T}$ (${\Delta }_{\mathrm{ST}}=300\mu \mathrm{eV}$) in the 1st cooldown, while that for intra-LL is measured at $B=1.2\mathrm{T}$ (${\Delta }_{\mathrm{ST}}=380\mu \mathrm{eV}$) in the 2nd cooldown. The solid lines are fitted to the data. (b) The coefficient $\alpha$ for the cotunneling component as a function of ${\Delta }_{\mathrm{ST}}$. The curve is calculated with an effective tunneling rate, ${\Gamma }_{\mathrm{tot}}^{*}=\beta {\Gamma }_{\mathrm{tot}}$.

Figure 4

(a) ${\Delta }_{\mathrm{ST}}$ dependence of the excitation probability $P$ for the inter-LL relaxation (1st cooldown). (b) ${\Delta }_{\mathrm{ST}}$ dependence of the ${\tau }_{\mathrm{s}}$ for the inter- and intra-LL relaxations. Data with open symbols are obtained by fitting a single exponential function, while the solid circle represents the fast component of the double-exponential function. The dotted lines are guides for the eye. (c) A logarithmic plot of $n_{\mathrm{t}}$ vs $t_{\mathrm{h}}$ with ${\Gamma }_{\mathrm{tot}}=2.0\times {10}^8{\mathrm{s}}^{-1}$ at ${\Delta }_{\mathrm{ST}}=300\mu \mathrm{eV}$ ($B=0.6\mathrm{T}$). The solid line is an exponential function fitted to the data. (d) A logarithmic plot of $n_{\mathrm{t}}$ vs $t_{\mathrm{h}}$ with ${\Gamma }_{\mathrm{tot}}=2.7\times {10}^8{\mathrm{s}}^{-1}$ at ${\Delta }_{\mathrm{ST}}=380\mu \mathrm{eV}$ ($B=0.55\mathrm{T}$). The solid line is a double-exponential function fitted to the data. The dotted and dashed lines are the fast and slow components, respectively. The inset schematically shows allowed and forbidden transitions from the triplet sublevels to the singlet state.