Coulomb interaction and valley-orbit coupling in Si quantum dots

The valley-orbit coupling in a few-electron Si quantum dot is expected to be a function of its occupation number $N$. We study the spectrum of multivalley Si quantum dots for $2\le N\le 4$, showing that, counterintuitively, electron-electron interaction effects on the valley-orbit coupling are negligible. For $N=2$ they are suppressed by valley interference, for $N=3$ they vanish due to spinor overlaps, and for $N=4$ they cancel between different pairs of electrons. To corroborate our theoretical findings, we examine the experimental energy spectrum of a few-electron metal-oxide-semiconductor quantum dot. The measured spin-valley state filling sequence in a magnetic field reveals that the valley-orbit coupling is definitively unaffected by the occupation number.

Figure 1

(a) Energy diagram of the four lower spin-valley single-particle levels in the quantum dot. The Zeeman splitting is assumed to be smaller than the valley-orbit coupling. A tunnel barrier separates the dot from a two-dimensional electron gas (2DEG), acting as a reservoir. (b) Device architecture used for the experiments. A single-lead quantum dot (top) is capacitively coupled to a single-electron transistor charge sensor (bottom). Regions filled in orange represent electron layers; nonfilled regions are tunnel barriers. The readout current signal $I_{\text{SET}}$ is sensitive to the dot's charge state via capacitive effects.

Figure 2

Eigenvalues of $H_{\mathrm{valley}}$ as functions of ${\Delta }_{ee}$ for a quantum dot with $N=2$ electrons, with ${\Delta }_0=0.05$ meV. At large ${\Delta }_{ee}$ the ground state is a spin triplet (green line).

Figure 3

(a) Energy diagrams of the dot when voltage pulses are superimposed to the dc bias. When a single-particle level is near the reservoir's Fermi level, an electron can be loaded into the dot during the upper phase of the pulse (solid lines) and unloaded during the lower phase (dashed lines). (b) Magnetic field dependence of the spin-valley states filling for $N\le 3$, where $D_{\pm }$ label single-electron valley eigenstates and$S,T_{\downarrow \downarrow }$ label spin states, consistent with Eq. (2). Data points show where the differential sensor's signal $\frac{d{I}_{\mathrm{SET}}}{dE}$ is maximized. Solid lines are guides for the eye with slopes $+\frac{g{\mu }_B}{2}$ in red and $-\frac{g{\mu }_B}{2}$ in green. Boxes of different colors indicate different valley occupancies. Blue (black) arrows represent the spin state of the current (previous) electron addition(s). Sets of data for different charge transitions are arbitrarily shifted in energy for clarity. (c) Energy diagram of the one-electron spin-valley states evolution in magnetic field. At the field value $B=B_k$ the two valley states cross, and Zeeman and valley-orbit splittings coincide. Dashed lines indicate states that are not probed in the experiments.