Magnetic field dependence of Pauli spin blockade: A window into the sources of spin relaxation in silicon quantum dots

We investigate spin relaxation in a silicon double quantum dot via leakage current through Pauli blockade as a function of interdot detuning and magnetic field. A dip in leakage current as a function of magnetic field on an $\sim 40$ mT field scale is attributed to spin-orbit mediated spin relaxation. On a larger ($\sim 400$ mT) field scale, a peak in leakage current is seen in some, but not all, Pauli-blocked transitions, and is attributed to spin-flip cotunneling. Both dip and peak structure show good agreement between theory and experiment.

Figure 1

(a) Schematic of the silicon double quantum dot (Si DQD). (b) Scanning electron microscope image of the Si DQD before the top gate formation. The two side gates located next to side gate C are grounded. (c) Charge stability diagram of the Si DQD as a function of $V_{\mathrm{L}}$ and $V_{\mathrm{R}}$ at zero magnetic field, where $V_{\mathrm{ds}}=-2$ mV, $V_{\mathrm{TG}}=0.90$ V, and $V_{\mathrm{C}}=-1.72$ V. The white dotted lines are boundaries of the stable charge states. The charge numbers in the left and right QDs are $N_{\mathrm{L}}$ and $N_{\mathrm{R}}$, respectively.

Figure 2

(a) Triple point B shown in Fig. 1c with negative bias, where $V_{\mathrm{ds}}=-2$ mV, $V_{\mathrm{TG}}=0.97$ V, and $V_{\mathrm{C}}=-1.76$ V. The PSB appears only for this polarity. Here $\epsilon$ is the detuning axis. (b) The same triple point as in (a) under a positive bias ($V_{\mathrm{ds}}=2$ mV). (c) Energy diagrams of a Si DQD at the circle marked in (a) (the left diagram) and at the blue cross marked in (b) (the right diagram), where the valley degeneracy is assumed to be lifted. (d) The same diagram as (c) without an assumption that lifting of the valley degeneracy is small. Intradot and interdot tunnelings between different valleys are assumed to be weak so that the PSB is not lifted.

Figure 3

(a) Leakage current in the PSB regime as a function of magnetic field applied perpendicularly to the DQD and detuning, where $V_{\mathrm{ds}}=-2$ mV, $V_{\mathrm{TG}}=0.97$ V, and $V_{\mathrm{C}}=-1.99$ V. Inset: magnified plot of triple point C in Fig. 1c, where the arrow corresponds to the detuning axis in the main figure. (b) Current along the dashed line in (a) denoted by the squares, and the fit to the data indicated by the blue line. (c) Values of $B_{\mathrm{C}}$ extracted from the fit as a function of $V_{\mathrm{C}}$. Large $V_{\mathrm{C}}$ corresponds to a large interdot tunnel coupling $t$.

Figure 4

(a) Leakage current in the PSB regime as a function of magnetic field applied perpendicularly to the DQD and the detuning, where $V_{\mathrm{ds}}=2$ mV, $V_{\mathrm{TG}}=0.968$ V, and $V_{\mathrm{C}}=-1.925$ V. (b) Magnified plot of triple point A in Fig. 1c, where the arrow corresponds to the detuning axis in (a). (c) Current along the dashed line in (a) denoted by the circles, and the fit to the data indicated by the blue line.

Figure 5

(a) and (b) Energy diagrams of a Si DQD where $(N_L,N_R)=(2,1)$ or $(3,0)$ in (a), and $(2,2)$ or $(3,1)$ in (b). The blue and yellow lines indicate different valleys. (c) Electron configuration for the $(3,1)$ regime in a Si DQD where doubly degenerate valleys are taken into account.

Figure 6

Magnetic field dependence of the current at triple point C, where $V_{\mathrm{ds}}=2$ mV, $V_{\mathrm{TG}}=0.97$ V, and $V_{\mathrm{C}}=-1.99$ V. No dependences are observed.

Figure 7

$B_{\mathrm{C}}$ versus $t$, where $l_{\mathrm{SO}}$ and ${\Gamma }_{\mathrm{out}}$ are parameters.