Theory of valley-orbit coupling in a Si/SiGe quantum dot

Electron states are studied for quantum dots in a strained Si quantum well, taking into account both valley and orbital physics. Realistic geometries are considered, including circular and elliptical dot shapes, parallel and perpendicular magnetic fields, and (most importantly for valley coupling) the small local tilt of the quantum-well interface away from the crystallographic axes. In absence of a tilt, valley splitting occurs only between pairs of states with the same orbital quantum numbers. However, tilting is ubiquitous in conventional silicon heterostructures, leading to valley-orbit coupling. In this context, “valley splitting” is no longer a well-defined concept, and the quantity of merit for qubit applications becomes the ground-state gap. For typical dots used as qubits, a rich energy spectrum emerges, as a function of magnetic field, tilt angle, and orbital quantum number. Numerical and analytical solutions are obtained for the ground-state gap and for the mixing fraction between the ground and excited states. This mixing can lead to valley scattering, decoherence, and leakage for Si spin qubits.

Figure 1

(Color online) Energy levels for an elliptical quantum dot in the parallel field geometry, including valley coupling. Dot parameters are appropriate for a single-electron spin qubit: $\hslash {\omega }_x=1.5\text{meV}$, $\hslash {\omega }_y=0.75\text{meV}$, and $v_v{\zeta }_0^2=0.5\text{meV}$. Uniform, lateral step orientation is set to $\phi =\pi /4$. Blue dot-dashed lines show the analytical approximation, obtained by restricting the analysis to the lowest two orbital levels in Eq. (46). The subband energy shift (a constant) has been dropped here for simplicity.

Figure 2

(Color online) Mixing fraction, Eq. (39), due to interorbital valley coupling. Results are shown for the same quantum-dot parameters used in Fig. 1. The solid black line is the projection of the perturbed ground state $\left(\tilde{n}=0\right)$ onto the unperturbed excited states $\left(n>0\right)$. The red dashed line shows the main contribution to ${\mathcal{F}}_{\tilde{n}=0}$ coming from just the first excited orbital $\left(n=1\right)$. Blue dot-dashed line shows the corresponding analytical result of Eq. (50), as obtained within a restricted subspace. Inset: the projection of the first excited state $\left(\tilde{n}=1\right)$ onto the unperturbed excited states $\left(n>0\right)$.

Figure 3

Crossover radius for the suppression of valley coupling caused by a tilted interface. For a quantum dot lying below the curve, valley splitting is weakly suppressed or unsuppressed. Above the curve, valley splitting is strongly suppressed.

Figure 4

Energy levels for a circular quantum dot in a perpendicular field geometry, including valley coupling. Dot parameters are appropriate for a single-electron spin qubit: $\hslash {\omega }_0=0.75\text{meV}$ and $v_v{\zeta }_0^2=0.5\text{meV}$. Uniform step orientation is set to $\phi =\pi /4$. The effective miscut tilt angles are given by (a) $\vartheta =0.03°$, (b) $\vartheta =0.3°$, and (c) $\vartheta =3°$. (b) is probably closest to experimental conditions. In (c), each curve is doubly degenerate.