Negative Spin Exchange in a Multielectron Quantum Dot

We use a one-electron quantum dot as a spectroscopic probe to study the spin properties of a gate-controlled multielectron GaAs quantum dot at the transition between odd and even occupation numbers. We observe that the multielectron ground-state transitions from spin-$1/2$-like to singletlike to tripletlike as we increase the detuning towards the next higher charge state. The sign reversal in the inferred exchange energy persists at zero magnetic field, and the exchange strength is tunable by gate voltages and in-plane magnetic fields. Complementing spin leakage spectroscopy data, the inspection of coherent multielectron spin exchange oscillations provides further evidence for the sign reversal and, inferentially, for the importance of nontrivial multielectron spin exchange correlations.

Figure 1

(a) Electron micrograph of the device consisting of a two-electron double quantum dot next to a multielectron quantum dot. The accumulation gate (colored in green) is operated at a positive voltage. Remaining gates deplete the underlying two-dimensional electron gas. Gates $V_{L/M/R}$, highlighted in red, are connected to high-bandwidth lines. (b) Charge diagrams indicating the electron occupation of the triple quantum dot as a function of $V_{L/M/R}$. Arrows indicate $\zeta$ and $\epsilon$ axes in gate voltage space. (c) Concept of the experiment. Two electrons are initialized in a singlet state in the left quantum dot. Thereafter, one of the electrons is moved to the middle dot and interacts with the multielectron quantum dot through exchange interaction $J$. At the end, readout is attained by performing a spin-to-charge conversion for two-electron spin states in the double quantum dot. (d) Implementation of the pulse sequence in terms of $\zeta$ and $ϵ$.

Figure 2

(a) $P_S$ as a function of $\zeta$, $ϵ$, and $B_{\parallel }$ for a fixed, long interaction time $\tau =150\mathrm{ns}$. The vertical dashed line indicates the crossing of $\zeta$ and $ϵ$ axes in Fig. 1. (b) Corresponding energy diagram of the spin states of a Heisenberg model, as a function of $\zeta$ and $ϵ$ for a fixed $B_{\parallel }$. States highlighted in red witness the interaction between the central probe spin and the effective MED spin, which combine into a singletlike state, $|\uparrow ⟩|S⟩$, that is above a tripletlike state, $|\uparrow ⟩|T_0⟩$, for sufficiently large $ϵ$ (negative $J$). The charge character of the ground-state transitions from ($2,0,2N+1$) via ($1,1,2N+1$) to ($1,0,2N+2$) as indicated by the background shading. The sign reversal of $J$ happens at $ϵ=0$. The Zeeman shift $|g|{\mu }_B{B}_{\parallel }$ and crossings with other states leading to spin leakage features in (a) are indicated (see the main text). Leakage from $|\uparrow ⟩|S⟩$ to the fully polarized $|\downarrow \downarrow \downarrow ⟩$ state (empty circle) is not observed in (a), because weak Overhauser gradients or spin-orbit coupling do not allow such large changes in spin projection. (c) Experimental exchange profiles for different operating points (distortions of the confining potential), identified by $V_R$ during the readout step (symbols). Black circles are extracted from (a). Solid lines are guides to the eye.

Figure 3

(a) Exchange oscillations in $P_S$ as a function of $ϵ$ and exchange time $\tau$, in the vicinity of the ($1,1,2N+1$) and ($1,0,2N+2$) charge transition. The external magnetic field is zero, and dc tuning voltages are the same as in Fig. 2 ($V_R=472\mathrm{mV}$). (b) Simulated exchange oscillations, assuming $J(ϵ)$ from Fig. 2, Gaussian low-frequency noise in $ϵ$ with a standard deviation of 0.18 mV, and a rise time of the experimental instrumentation of 0.8 ns. A contour with no net accumulated phase, $\varphi =0$, divides operating regimes where $J$ is positive and negative. Leakage out of the simulated subspace [magenta diamond in (a); see the text] was ignored in the simulation.

Figure 4

(a) Exchange oscillations as a function of $B_{\perp }$ and $ϵ$ for fixed exchange time $\tau =3.33\mathrm{ns}$ and fixed $B_{\parallel }=0\mathrm{T}$. (b) Exchange oscillations as a function of $B_{\parallel }$ and $ϵ$ for fixed exchange time $\tau =3.33\mathrm{ns}$ and fixed $B_{\perp }=0\mathrm{T}$. (c) The same as (a) but in the leakage spectroscopy regime ($\tau =150\mathrm{ns}$). Features of reduced $P_S$ correspond to mixing between $|\uparrow ⟩|S⟩$ and various other states [cf. horizontal cut of Fig. 2 at $B_{\parallel }=0$]. A small deviation between the $J=0$ feature in (c) and the $\varphi =0$ contour in (a) is likely due to the very different values of $\tau$ in combination with finite-rise-time effects of our instrumentation. (d) $J(ϵ)$ extracted for different values of $B_{\perp }$.