Fig. 2: Quantum dot variability.

a To simulate a Si/SiO₂ interface atomistically, we use a virtual lattice approximation (see Methods). This allows us to emulate the electronic properties of SiO₂—which does not have a regular lattice structure—in a simulated material with the same lattice structure as silicon. We then define the interface between the silicon lattice and its oxide with a simulated rough surface dividing the atomic sites between the two materials. b Three-dimensional visualisation of a CMOS quantum dot at the Si/SiO₂ interface simulated atomistically. The blue arrow represents the vector spin 〈σ〉 averaged across the spin-orbitals of all the atoms in the quantum dot. c In-plane visualisation of the variability in the 7 quantum dots inside the purple rectangle in Fig. 1b. The 5-nm-diameter cyan circle is a static reference to compare the wavefunctions in different simulations. Black asterisks represent the centre of each quantum dot 〈r〉 = 〈ψ∣r∣ψ〉. d Variability distribution of dot centres. e Visualisation of the valley oscillations parallel to the [001] lattice orientation. (see Methods section). The oscillation wavelength is \(\frac{4\pi }{{k}_{0}}\), where \({k}_{0}=0.82\frac{2\pi }{{a}_{0}}\) is the wavevector of the conduction band minima in the silicon crystal. f Valley splitting distribution of the 49 quantum dots versus the electric field. Box plots indicate the median (middle line), 25th, 75th percentile (box) and 5th and 95th percentile (whiskers) as well as outliers (single points). We compare our results with experimental data measured in device F and with measurements in ref. ³⁹. Error bars indicate the standard deviation for the measured value. The electric fields are obtained from COMSOL simulations. The inset figure shows the correlation between the logarithm of the valley splitting versus the centre of the dot in the z-axis. The valley splitting is reconverted to magnetic field units in the right axis to compare it with the Zeeman splitting at different fields. g Distribution of valley phases versus the valley splitting. For convenience, we define ϕ_v = 0^∘ as the point with the highest density of valley phases. The colour code represents the value of E_z as in (f). Source data of (f, g) are provided in the Source Data file.