Nuclear spin dynamics in double quantum dots: Fixed points, transients, and intermittency

Transport through spin-blockaded quantum dots provides a means for electrical control and detection of nuclear spin dynamics in the host material. Although such experiments have become increasingly popular in recent years, interpretation of their results in terms of the underlying nuclear spin dynamics remains challenging. Here we examine nuclear polarization dynamics within a two-polarization model that supports a wide range of nonlinear phenomena. We point out a fundamental process in which nuclear spin dynamics can be driven by electron shot noise; fast electric current fluctuations generate much slower nuclear polarization dynamics, which in turn affect electron dynamics via the Overhauser field. The resulting intermittent, extremely slow current fluctuations account for a variety of observed phenomena that were not previously understood.

Figure 1

Nuclear dynamics and intermittent current fluctuations driven by shot noise of the spin-blockaded current through a double dot. (a) Hyperfine spin exchange with nuclei mediates transitions between two-electron triplet and singlet states, relieving blockade and producing dynamical nuclear polarizations (DNPs) $s_L$ and $s_R$ in the left and right dots. (b) A phase portrait of DNP trajectories, with a fixed point (DNP steady state) positioned near the main diagonal, where current is low due to spin blockade (see Fig. 3). (c) Typical simulated current trace showing the effects of steady state DNP fluctuations [pink traces in (b) and (c)]; see the Appendix. Switching between quiet and noisy regions results from excursions into the dark stripe $s_L\approx s_R$, marked by the shaded regions in (c).

Figure 2

Time-dependent current, from experiment (Ref.7) (a), and from simulation (b), showing a long buildup lasting several tens of seconds followed by steady-state intermittent fluctuations (blue traces). Fluctuations can be suppressed by a small change of detuning (gate voltage) which moves a fixed point away from the sensitive region $s_L\approx s_R$ (red traces). (c) Fourier spectra of the experimental and simulated noisy traces, evaluated in the marked steady-state regions. The spectra display a roughly $1/f^{\alpha }$ dependence with ${\alpha }_{\mathrm{expt}}\approx 2.5$ and ${\alpha }_{\mathrm{simul}}\approx 2$.

Figure 3

(a) Energy level diagram showing hybridization of $S_{\mathrm{tot}}^z=0$ levels [see Eq. (1)] and relevant transitions for spin-blockaded transport. (b) Current as a function of nuclear polarizations $s_L$ and $s_R$ on the left and right dots. Current is suppressed for $s_L\approx s_R$ due to the vanishing coupling between $|T_0\rangle$ and ${|(1,1)}_S\rangle$. Small roman numerals indicate bands of high current due to resonance between blocked and unblocked levels, shown in (a). Primed labels indicate analogous configurations with $|T_{\pm }\rangle$ reversed. (c) Simulated trajectory of polarization arcing through the smooth region of the current diagram with steady state near the sensitive region $s_L\approx s_R$, similar to the schematic in Fig. 1. Arrows indicate the direction of the flow (6). Inset: Distorted zoom of a narrow strip containing the diagonal, magnified in the transverse direction to display the arc-shaped trajectory and steady-state fluctuations.

Figure 4

Current during magnetic field sweep $dB/dt<0$. (a) Experimental field sweeps for several values of $\Delta$ [see Eq. (1)] in the regime described Fig. 3(d) in Ref.7 ($\Delta =0$ could be only approximately located experimentally). Curves are offset by 0.1 pA for clarity. (b) Simulated field sweeps in the field regime indicated by the dotted box in (a), showing bistable current similar to experimental results. In both cases, the threshold for the change from noisy to quiet current moves to higher field with increasing detuning. To treat the case of $B\approx 0$, a $5\times 5$ formulation akin to that of Ref.17 must be used.