Valleytronic full configuration-interaction approach: Application to the excitation spectra of Si double-dot qubits

A study of the influence of strong electron-electron interactions and Wigner-molecule formation on the spectra of $2e$ singlet-triplet double-dot Si qubits is presented based on a full configuration-interaction (FCI) approach that incorporates the valley degree of freedom (VDOF) in the context of the continuous (effective mass) description of semiconductor materials. Our FCI solutions correspond to treating the VDOF as an isospin in addition to the regular spin. A major advantage of our treatment is its capability to assign to each energy curve in the qubit's spectrum a complete set of good quantum numbers for both the spin and the valley isospin. This allows for the interpretation of the Si double-dot spectra according to an underlying $\mathrm{SU}(4)\ensuremath{\supset}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ group-chain organization. Considering parameters in the range of actual experimental situations, we uncover and demonstrate for a double-quantum dot that, in the (2,0) charge configuration and compared to the expected large, and dot-size determined, single-particle (orbital) energy gap, the strong $e\text{\ensuremath{-}}e$ interactions drastically quench the spin-singlet--spin-triplet energy gap ${E}_{\mathrm{ST}}^{\ensuremath{\bigoplus}}$ within the same valley, making it competitive to the small energy gap ${E}_V$ between the two valleys. We present results for both the ${E}_{\mathrm{ST}}^{\ensuremath{\bigoplus}}<{E}_V$ and ${E}_{\mathrm{ST}}^{\ensuremath{\bigoplus}}>{E}_V$ cases, which have been reported to occur in different experimental qubit devices. In particular, we investigate the spectra as a function of detuning and demonstrate the strengthening of the all-important avoided crossings due to a lowering of the interdot barrier and/or the influence of valley-orbit coupling. We further demonstrate, as a function of an applied magnetic field, the emergence of avoided crossings in the (1,1) charge configuration due to the more general spin-valley coupling, in agreement with experiments. The valleytronic FCI method formulated and implemented in this paper, and demonstrated for the case of two electrons confined in a tunable double quantum dot, offers also a most effective tool for analyzing the spectra of Si qubits with more than two wells and/or more than two electrons, in field-free conditions, as well as under the influence of an applied magnetic field. Furthermore, it can also be straightforwardly extended to the case of bilayer graphene quantum dots.