Tuning the Two-Electron Hybridization and Spin States in Parallel-Coupled InAs Quantum Dots

We study spin transport in the one- and two-electron regimes of parallel-coupled double quantum dots (DQDs). The DQDs are formed in InAs nanowires by a combination of crystal-phase engineering and electrostatic gating, with an interdot tunnel coupling ($t$) tunable by one order of magnitude. Large single-particle energy separations (up to 10 meV) and $|g^{*}|$ factors ($\sim 10$) enable detailed studies of the $B$-field-induced transition from a singlet-to-triplet ground state as a function of $t$. In particular, we investigate how the magnitude of the spin-orbit-induced singlet-triplet anticrossing depends on $t$. For cases of strong coupling, we find values of $230\mu \mathrm{eV}$ for the anticrossing using excited-state spectroscopy. Experimental results are reproduced by calculations based on rate equations and a DQD model including a single orbital in each dot.

Figure 1

(a) A scanning electron microscope image of a typical DQD device where the three gates (left and right side gate and global back gate) and source and drain electrodes are indicated. (b) High-resolution transmission electron microscopy image viewed along a $⟨110⟩$-type direction of the disk-shaped ZB QD (4 nm) sandwiched between WZ (22 and 28 nm) segments (highlighted). The diameter of the single QD is 67 nm. The positions of the left and right QDs, in a simplified picture, are indicated. (c) Charge stability diagram where the electron populations of the two QDs are indicated ($V_{BG}=0\mathrm{V}$, $V_{SD}=1\mathrm{mV}$) [cooldown I]. The red arrow indicates the gate vector used in Fig. 2.

Figure 2

(a), (c), (e) Coulomb charge stability diagrams for cooldown I, recorded along the side-gate vector ${\overrightarrow{V}}_{L,R}$ crossing the two triple points indicated by the arrow in Fig. 1. The back-gate potential decreases along (a),(c),(e) {$V_{BG}=0.00$, [same as Fig. 1] $-0.75,-1.25\mathrm{V}$} effectively increasing the interdot tunnel coupling ($2t=0.9$, 2.5, $4.6\pm 0.1\mathrm{meV}$) and the singlet-triplet energy split ($J=0$, 0.6, $1.8\pm 0.1\mathrm{meV}$). $N$ denotes the total electron population of the DQDs and the arrows indicate the conductance lines associated with transport via the antibonding and triplet states. (b),(d),(f) $B$-field evolution of the ground states of the first four electrons recorded at $V_{SD}=20\mu \mathrm{V}$ along the side-gate vector illustrated by the dashed line in Fig. 2. Here, the shaded areas indicate where the singlet is ground state, and the arrows indicate the spin direction. (g) Modeled energy of $S(1,1)$ and $T(1,1)$ states as a function of the $B$ field for intermediate (blue) and strong (red) interdot tunnel coupling; parameters extracted from measurements in the back-gate regimes displayed in (c) and (e), respectively, are used in the modeling. The shaded areas indicate the singlet-ground-state regimes.

Figure 3

$B$-field evolution of transitions involving $S$ and $T$ states recorded during cooldown I at a fixed gate-potential configuration; (a),(b) same $V_{BG}$ as in Figs. 2 and 2, respectively. (a) The $2e$ states involved in the transitions are indicated. (d) Magnitude of $S/T$ anticrossing ${\mathrm{\Delta }}_{ST}^{*}$ as a function of interdot tunneling coupling $t$. Blue and red traces correspond to data recorded at two different cooldowns, I and II, respectively. Here, the overlap between the spatial distribution of the electrons in the two dots, parametrized by $t$, is increased by decreasing (blue) [increasing (red)] $V_{BG}$, as indicated in (c). The error bars reflect the width of the conductance lines. The black dashed trace correspond to results from modeling, where ${\mathrm{\Delta }}_{ST}^{*}$ has been fitted to data set I by tuning $\alpha$ in the SO-tunneling term $t_{\mathrm{SO}}=\alpha t$; a constant $\alpha =0.08$ is used for all $t$ values.

Figure 4

(a) Sketch of the energy levels at a finite external $B$ field and possible transitions between these levels for the $1e$ and $2e$ states (adapted from [26]). Here, we assume spin-conserving first-order tunneling processes with no SOI-assisted mixing between states. (b) Modeled energy of the $GS(1,1)$ and $ES(1,1)$ and the unperturbed $S$ and $T_{+}$ as a function of magnetic field, where the fraction of $S$ ($T_{+}$) in $GS(1,1)$ [$ES(1,1)$] is indicated by the color. (c) Overview of the $B$-field evolution of the $1e-2e$ transitions (cooldown III); the color markings indicate the external $B$ fields at which the Coulomb charge stability diagrams (d)–(g), enlarged on the transitions between $1e$ and $2e$ states, are recorded. (e) Arrows and roman numerals indicate the transitions illustrated in (a); red and black symbolize transitions involving antibonding and bonding $1e$ states, respectively. (f),(g) The double arrows next to the notation of the $1e$ states indicate conductance lines involving the hybridized states $GS(1,1)$ and $ES(1,1)$. (h),(i) Modeling using a $B$ field corresponding to (f) and (g), respectively, where the transitions seen in the experimental data are indicated. Additional lines corresponding to transitions involving $T_0$ and $1e-0e$ are indicated by green arrows.