Negative exchange interactions in coupled few-electron quantum dots

It has been experimentally shown that negative exchange interactions can arise in a linear three-dot system when a two-electron double quantum dot is exchange coupled to a larger quantum dot containing on the order of one hundred electrons. The origin of this negative exchange can be traced to the larger quantum dot exhibiting a spin tripletlike rather than singletlike ground state. Here we show using a microscopic model based on the configuration interaction (CI) method that both tripletlike and singletlike ground states are realized depending on the number of electrons. In the case of only four electrons, a full CI calculation reveals that tripletlike ground states occur for sufficiently large dots. These results hold for symmetric and asymmetric quantum dots in both Si and GaAs, showing that negative exchange interactions are robust in few-electron double quantum dots and do not require large numbers of electrons.

Figure 1

Illustration of the three-dot system, where the $L$ and $M$ dots form a two-electron double quantum dot and $R$ is the multielectron quantum dot. In the main text we analyze the effective exchange interaction $J$ between the middle ($M$) and right ($R$) quantum dots.

Figure 2

(a) Schematic of the energy levels and charge configuration for the three-dot system. Here $\mathrm{\Delta }{E}_i$ is the difference between two orbitals on the multielectron quantum dot. The schematic only shows the tunnel coupling between the middle and right dot, where $t_{MR,N+1}$ and $t_{MR,N+2}$ are the tunneling amplitudes between the middle dot's single orbital and the lowest two orbitals above the frozen core in the multielectron quantum dot. (b) Eigenenergy spectrum, calculated as a function of the detuning $\epsilon$, for the three-dot system at the transition between $(1,1,2N+1)$ and $(1,0,2N+2)$ charge configurations. The eigenstates $|D_{+1/2}\rangle$ ($|D_{-1/2}\rangle$) and $|Q_{+1/2}\rangle$ ($|Q_{-1/2}\rangle$) are almost degenerate and the exchange energy $J$ is the difference between the tripletlike state $|D_{\pm 1/2}\rangle$ and the singletlike state $|D_{\pm 1/2}^{\prime }\rangle$.

Figure 3

Exchange energy vs frequency difference of the elliptical potential.