Spin Relaxation in Quantum Dots due to Electron Exchange with Leads

We calculate spin relaxation rates in lateral quantum dot systems due to electron exchange between dots and leads. Using rate equations, we develop a theoretical description of the experimentally observed electric current in the spin blockade regime of double quantum dots. A single expression fits the entire current profile and describes the structure of both the conduction peaks and the suppressed (“valley”) region. Extrinsic rates calculated here have to be taken into account for accurate extraction of intrinsic relaxation rates due to the spin-orbit and hyperfine spin scattering mechanisms from spin blockade measurements.

Figure 1

(a) Lateral double dot system in the spin blockaded regime. In the initial configuration, the system has one electron on the left dot $(1,0)$. Electric current through the dot flows from right to left in a sequence of steps. (b) Energy level representation of transport through the dot. The sequence of transitions via singlet states $(1,0)\to (1,1{)}_S\rightsquigarrow (2,0{)}_S\to (1,0)$ quickly transports electron from the right to the left lead, whereas current through intermediate triplet state $(1,0)\to (1,1{)}_T$ is blocked since subsequent transition to $(2,0{)}_{S(T)}$ state is forbidden by spin (energy) conservation.

Figure 2

Electron exchange with the lead in lowest tunneling orders. (a) In the first-order process, a single electron jumps on or off the dot, with electron energy in the lead equal to $E_{(1,1)}$. The detuning from the Fermi level is $\Delta =E_{(1,1)}-E_F^R$. (b) The second-order process represents double electron tunneling through a virtual state. Energy of participating electrons in the lead is close to the lead Fermi energy and can be far from the dot energy level.

Figure 3

Current through a double dot system in the spin blockaded regime. The dotted (dashed) lines show asymptotes due to first- (second-) order processes that dominate the peak (valley) and are given by Eq. (8) with ${\Gamma }_s$ replaced by ${\gamma }_1$ ($2{\gamma }_2$). Panel (d) shows a fit to measured current (circles) along a line cut of the spin blockade peak reported in 8.