Quantitative Estimation of Exchange Interaction Energy Using Two-Electron Vertical Double Quantum Dots

We use Pauli-spin blockade in two-electron vertical double quantum dots to quantitatively estimate the exchange energy $J$ in a wide range of interdot level detuning $\Delta$ and fully compare it with calculations. Pauli-spin blockade is lifted via a singlet- ($S$-)triplet ($T$) transition mediated by hyperfine coupling, which abruptly occurs in our devices when the $S\mathrm{\text{−}}T$ transition energy or $J$ is compensated by the Zeeman energy. We use this feature to derive $J$ depending on $\Delta$ between the $S\mathrm{\text{−}}S$ and $T\mathrm{\text{−}}T$ resonances. The obtained $J$ versus $\Delta$ including the resonance effect is perfectly reproduced by Hubbard model calculations.

Figure 1

(a) Schematic picture of the double QD and measurement setups. The static magnetic fields $B_p$ and $B_{\mathrm{in}}$ are applied perpendicular and parallel to the QD plane. The ac magnetic field $B_{\mathrm{ac}}$ for NMR measurements is applied perpendicular to the QD plane. (b) Energy diagram for a two-electron ground state and the excited states with potential detuning $\Delta$ as a parameter. (c) Color-scale plot of $d{I}_{\mathrm{sd}}/d{V}_{\mathrm{sd}}$ in the $V_{\mathrm{sd}}-V_g$ plane measured for device $A$ at 1.5 K. The white region corresponds to the blockade. The energy diagrams of PSB and $S\mathrm{\text{−}}S$ resonance are indicated. (d) $I_{\mathrm{sd}}$ as a function of the frequency of $B_{\mathrm{ac}}$ under an in-plane magnetic field of $B_{\mathrm{in}}=300\mathrm{mT}$. NMR signals for ${}^{75}\mathrm{As}$, ${}^{69}\mathrm{Ga}$, and ${}^{71}\mathrm{Ga}$ are detected. (e) NMR signal positions as a function of $B_{\mathrm{in}}$ for the nuclei.

Figure 2

(a) In-plane magnetic field ($B_{\mathrm{in}}$) dependence of the leakage current $I_{\mathrm{sd}}$ measured at $I_{\mathrm{sd}}=0.8\mathrm{mV}$ to 2 mV in the PSB region. Each curve is offset upwards by 1 pA. (b) Schematic hysteretic behavior in the $I_{\mathrm{sd}}-B$ plane. The nuclear spins in the QDs polarize above $B=B_{\mathrm{sw}}$. (c) Zeeman effect on the $S$ and $T$ states. The $S$ state and $T_{-}$ states become degenerate at $B_{\mathrm{sw}}$. (d) $I_{\mathrm{sd}}$ as a function of $B_p$ measured at $V_{\mathrm{sd}}=2.5\mathrm{mV}$ in the PSB region. The sweep rate is 1 T/hour. Inset to (d): $I_{\mathrm{sd}}$ vs $V_{\mathrm{sd}}$ measured at $T=30\mathrm{mK}$, showing the PSB for $1.2\mathrm{mV}<V_{\mathrm{sd}}<5\mathrm{mV}$. The peak at $V_{\mathrm{sd}}\sim 6.8\mathrm{mV}$ is due to $T(1s,1s)-T(0,1s2p)$ resonance. (e) $B_{\mathrm{in}}$ dependence of $I_{\mathrm{sd}}$ measured at $V_{\mathrm{sd}}=3\mathrm{mV}$ in the PSB region. Inset to (e): $I_{\mathrm{sd}}$ vs $V_{\mathrm{sd}}$ showing PSB for $2\mathrm{mV}<V_{\mathrm{sd}}<3.5\mathrm{mV}$. The peak at $V_{\mathrm{sd}}=3.8\mathrm{mV}$ is due to $T(1s,1s)-T(0,1s2p)$ resonance.

Figure 3

$B_{\mathrm{sw}}$ plotted as a function of $\Delta /{\Delta }_T$. Right-hand side scale: Exchange energy $J$ of devices $A$ (square), $B$ (circle), and $C$ (triangle) derived experimentally and theoretically (solid line) as a function of $\Delta /{\Delta }_T$. Inset: Magnification of $B_{\mathrm{sw}}$ and $J$ for device $C$ as a function of $V_{\mathrm{sd}}$.