Electrical Control of Spin Relaxation in a Quantum Dot

We demonstrate electrical control of the spin relaxation time $T_1$ between Zeeman-split spin states of a single electron in a lateral quantum dot. We find that relaxation is mediated by the spin-orbit interaction, and by manipulating the orbital states of the dot using gate voltages we vary the relaxation rate $W\equiv T_1^{-1}$ by over an order of magnitude. The dependence of $W$ on orbital confinement agrees with theoretical predictions, and from these data we extract the spin-orbit length. We also measure the dependence of $W$ on the magnetic field and demonstrate that spin-orbit mediated coupling to phonons is the dominant relaxation mechanism down to 1 T, where $T_1$ exceeds 1 s.

Figure 1

(a) Electron micrograph of the gate geometry. Negative voltages are applied to the labeled gates while the unlabeled gate and the Ohmic leads (which are numbered) are kept at ground. The black solid (white dotted) ellipse illustrates the expected dot shape for less (more) negative $V_{\mathrm{shape}}$. The magnetic field is parallel to the $y$ axis and all voltage pulses are applied to gate LP2. (b) At $B=0$ and with no SOI, the spin-$\uparrow$ and spin-$\downarrow$ states of the ground orbital state $|g⟩$ are degenerate. Applying a magnetic field splits the spin states but phonon coupling between $|g\uparrow ⟩$ and $|g\downarrow ⟩$ is prohibited. The SOI acts as a perturbation and mixes the orbital and spin states: the perturbed spin states $|g\uparrow {⟩}_{\mathrm{SO}}$ and $|g\downarrow {⟩}_{\mathrm{SO}}$ contain excited orbital states ($|e⟩$) of the opposite spin so the perturbed states can be coupled by phonons. The SOI involves a momentum operator and requires a change in parity for coupling of different orbital states. (c) Dot energy spectrum as gate voltages are varied to change the shape of the dot. The value of $V_{\mathrm{shape}}$ is the voltage on SG1 for a given set of gate voltages.

Figure 2

(a) Three step pulse sequence for measuring the energy of the excited orbital states. (b) Examples of real-time data. The direct capacitive coupling to the pulsed gate causes the QPC to respond to the pulse sequence; electron tunneling events are evident on top of this response. The 0’s denote when an electron tunnels off the dot, while 1’s denote when an electron tunnels on. The charging pulse ($t_p=50\mu \mathrm{s}$ for this example) appears as a sharp spike between the ionization and readout periods. (c) The number of events $N_{\mathrm{ion}}$ for which the dot is empty after the charging pulse [top panel in Fig. 2b] as a function of $t_p$. The solid line is a fit to an exponential to determine the rate ${\Gamma }_{\mathrm{on}}$ at which electrons tunnel onto the dot. (d) ${\Gamma }_{\mathrm{on}}$ vs $E_p$ for $V_{\mathrm{shape}}=-850\mathrm{mV}$. The two sharp rises mark the energies when an excited state crosses the Fermi energy. (e) Energies of the excited orbital states of the dot as a function of $V_{\mathrm{shape}}$. The dashed lines are linear fits to the data.

Figure 3

(a) Top panel: $W$ vs $V_{\mathrm{shape}}$ at $B=3\mathrm{T}$. The solid, dotted, and dashed curves show fits with $A_x/A_y=0.01$, 0.25, and 1, respectively. Bottom panel: $\Delta$ vs $V_{\mathrm{shape}}$ at $B=7.5\mathrm{T}$ where the Zeeman splitting is large and can easily be determined by measuring the positions of the increases in ${\Gamma }_{\mathrm{on}}$ as the ground and excited spin states are pulsed below the Fermi energy of the lead [35]. (b) The same relaxation rate data as in Fig. 3a, plotted as a function of $E_y$. The solid line is a fit to find the spin-orbit length as discussed in the text. (c) Spin relaxation rate as a function of magnetic field for two different sets of gate voltages. Solid lines are fits discussed in the text.