Two-electron spectrum of a silicon quantum dot

Abstract

The energy spectrum and wave functions of electrons in a single silicon quantum dot provide valuable insights into the capabilities and limitations of such a system in quantum information processing. Here we investigate the low-lying singlet and triplet configurations and spectra in a two-electron silicon quantum dot. To build toward a comprehensive understanding, we first examine the competition between Coulomb interaction and electron kinetic and confinement energy in the absence of valley-orbit coupling, as well as consequences of valley blockade in the presence of an ideal smooth interface. For realistic interfaces the variations in the magnitude and phase of valley-orbit coupling lead to inter-valley leakage, particularly when orbital splittings approach the valley splitting. In our study we particularly focus on the impact on the compositions of low-lying singlets and triplets. We find that for experimentally relevant parameter regimes the ground singlet and triplet states usually contain multiple configurations with significant weights as a result of a complicated competition among valley-orbit coupling, confinement potential, and Coulomb interaction. We further analyze the effects of an out-of-plane magnetic field on these the two-electron spectra. Our findings could have important implications for spin qubits in Si quantum dot in various contexts, such as qubit encoding and spin measurement.

Figures (13)

• Figure 1: Orbital exchange splitting: (Color online) The effect of the confinement energy on the exchange splitting in the two electrons silicon quantum dot. The valleys contributions are not included here and each curve represents the different numbers of CI basis sets included in the calculation.
• Figure 2: Orbital state compositions: (Color online) The dependence of state composition on the confinement energy for ground singlets (panels (a)-(c)) and triplets (panels (d)-(f)) is shown with an increasing number of excited orbital states in the calculations, excluding the valley states. The contributions from $|c^{sd_0}|^2$ and $|c^{p_-p_+}|^2$ are equal in panel (c). For triplets, the contributions from the $S=1$ and $S=-1$ orbitals are equal and thus represent as a single line. In panels (e) and (f), the contributions of $|c^{sp_-}|^2$ ($|c^{sp_+}|^2$) are approximately equal to $|c^{rest_-}|^2$ ($|c^{rest_+}|^2$).
• Figure 3: Finite valley splitting energy spectrum: (Color online) Energy levels of the few low-lying singlet (S$=$0) and triplet states (S$=$1) in a Si dot as a function of the valley splitting $|\Delta|$. For this calculation we choose confinement energy $E_0=1.0$ and $0.5$ meV. The results of exchange energy with $|\Delta|$ with the $SPD$ and $SPDF$ are shown in the panels (c) and (e), respectively.
• Figure 4: $\bf{E_J \left(E_0,\,|\Delta_0|\right)}$ : (Color Online) Density plot of the exchange interaction as a function of the confinement energy and the magnitude of valley-orbit coupling, with different sizes of the basis set $SPD$ and $SPDF$.
• Figure 5: One electron energy levels with step location: Dependence of the energy levels of a single electron in different orbital states as a function of the step position. The height of the step is one monolayer and the confinement energy of quantum dot is 0.5 meV. The magnitude of the valley orbit coupling is 0.1 meV when the step is outside the quantum dot