Fig. 3: Valley-splitting simulations.

a Average concentration profile obtained from APT data (quantum well A). b Typical, randomized Ge concentration profile, derived from a. c Envelope function ψ_env(z), obtained for the randomized profile in b (grey curve), and the corresponding concentration fluctuations weighted by the envelope function squared: \({\delta }_{{x}_{l}}|{\psi }_{{{{{{{{\rm{env}}}}}}}}}({z}_{l}){|}^{2}\) (blue). Here, the wavefunction is concentrated near the top interface where the concentration fluctuations are also large; the weighted fluctuations are therefore the largest in this regime. d Distribution of the intervalley matrix element Δ in the complex plane, as computed using an effective-mass approach, for 10,000 randomized concentration profiles. The black marker indicates the deterministic value of the matrix element Δ₀, obtained for the experimental profile in a. e Histogram of the valley splittings from tight-binding simulations with 10,000 randomized profiles. The same profiles may be used to compute valley splittings using effective-mass methods; the orange curve shows a Rice distribution whose parameters are obtained from such effective-mass calculations (see Methods). f Schematic Si/SiGe quantum well with Ge concentrations ρ_W (in the well) and ρ_b = ρ_W + Δρ (in the barriers), with a fixed concentration difference of Δρ = 25%. g Distribution of valley splittings obtained from simulations with variable Ge concentrations, corresponding to ρ_W ranging from 0 to 10%, and interface widths 4τ = 5 ML (red circles), 10 ML (blue triangles), or 20 ML (orange squares), where ML refers to atomic monolayers. Here, the marker describes the mean valley splitting, while the darker bars represent the 25-75 percentile range and the lighter bars represent the 5–95 percentile range. Each bar reflects 2000 randomized tight-binding simulations of a quantum well of width W = 120 ML. The magenta diamond at zero Ge concentration shows the average measured valley splitting of quantum well A. In all simulations reported here, we assume an electric field of E = 0.0075 V/nm and a parabolic single-electron quantum-dot confinement potential with orbital excitation energy ℏω = 4.18 meV and corresponding dot radius \(\sqrt{\hslash /{m}^{*}\omega }\).