Single-Shot Measurement of Triplet-Singlet Relaxation in a $\mathrm{Si}/\mathrm{SiGe}$ Double Quantum Dot

We investigate the lifetime of two-electron spin states in a few-electron $\mathrm{Si}/\mathrm{SiGe}$ double dot. At the transition between the (1,1) and (0,2) charge occupations, Pauli spin blockade provides a readout mechanism for the spin state. We use the statistics of repeated single-shot measurements to extract the lifetimes of multiple states simultaneously. When the magnetic field is zero, we find that all three triplet states have equal lifetimes, as expected, and this time is $\sim 10\mathrm{ms}$. When the field is nonzero, the $T_0$ lifetime is unchanged, whereas the $T_{-}$ lifetime increases monotonically with the field, reaching 3 sec at 1 T.

Figure 1

(a) Scanning electron microscopy (SEM) image of a device identical to the one used. Quantum dots are formed at the approximate locations of the two circles. Charge sensing is performed by monitoring the current $I_{\mathrm{QPC}}$ through a nearby point contact. (b) Charge stability diagram of the double dot showing the detuning voltage $V_{ϵ}$. (c) Energies of two-electron states as a function of detuning energy $ϵ$. $T_{+}$, $T_0$, and $T_{-}$ are the (1,1) triplets; the (0,2) triplets are higher in energy. The (1,1) and (0,2) singlets $S_{11}$ and $S_{02}$ are coupled by spin-preserving, interdot tunneling. A magnetic field separates the triplet energies by $E_z=g{\mu }_{B}B$. (d) Time-averaged occupation of the (0,2) charge state $P_{02}$ at $B_{\parallel }=0$ with 5 kHz square pulses of peak-to-peak amplitude $\Delta V_{ϵ}$ applied along $V_{ϵ}$. The pulses drive (1,1)-(0,2) transitions within the dotted triangle. The suppression of $P_{02}$ above the dashed line shows where (1,1) to (0,2) tunneling is suppressed by spin blockade.

Figure 2

Single-shot initialization and readout of singlet and triplet states. (a),(b) Real-time measurements of $I_{\mathrm{QPC}}$ as the system is initialized to $S_{11}$ then read out 1.7 ms later. We identify the final state in (a) as one of the (1,1) triplets ($T_{11}$) because the (1,1) charge state survives for over 1 ms during the readout. In (b) a singlet is identified because the system tunnels quickly back to (0,2) during the readout. (c) Schematic stability diagram. The points marked are the four detuning values used in the measurements. At $B_{\parallel }>0$, $E_{\mathrm{ST}}$ is decreased by $g{\mu }_B{B}_{\parallel }$. The pulse is offset to keep the circle inside the blockaded region without changing the separation of the circle and triangle points. Dashed triangles bound the region where (1,1)-(0,2) transitions occur primarily by interdot tunneling. (d)–(g) Pulses repeatedly switch the ground state between (1,1) and (0,2) at 300 Hz. In (d)–(f) the system is often blockaded in a (1,1) triplet. With increasing magnetic field from (d) to (f), the durations of blockade increase significantly. In (g), the pulse reaches into (0,2) far enough to exceed $E_{\mathrm{ST}}$, and tunneling from (1,1) to (0,2) occurs freely for all spin states.

Figure 3

(a) Histogram of the number of times that the system is blockaded for a time $t_b$ in many measurements such as Fig. 2d. The binning resolution is the pulse period. The solid line is an exponential fit yielding a characteristic time ${\tau }_b=9.6\mathrm{ms}$ for the blockaded configuration. (b) Histogram of unblockaded times $t_u$ for the same data as (a). An exponential fit yields a characteristic time ${\tau }_u=23\mathrm{ms}$ for the unblockaded configuration. (c),(d) Histograms of $t_u$ and $t_b$ at $B_{\parallel }=250\mathrm{mT}$. There are two decays describing the blockade: at small $t_b$ the decay is similar to that at zero field (${\tau }_b^{\prime }=10\mathrm{ms}$). At long $t_b$ a slower decay dominates (${\tau }_b=28\mathrm{ms}$). We interpret the shorter time as arising from $T_0$ occupation, and the longer time as arising from $T_{-}$ occupation. (e) Fitted characteristic times as a function of magnetic field. The characteristic time of blockade due to $T_{-}$ states ${\tau }_b$ increases with field, while the contributions from $T_0$ and $S_{11}$ states ${\tau }_b^{\prime }$ and ${\tau }_u$ are field independent. (f) $T_{-}$ lifetime ${\tau }_T$, and $S_{11}\mathrm{\text{−}}{T}_0$ mixing rate at positive (negative) detuning ${\tau }_{+}$ (${\tau }_{-}$).