In bulk silicon the conduction band has six equivalent minima ("valleys"). A vertical confining field at a Si/SiO₂ or Si/SiGe interface lifts this to two low-lying valleys along the growth axis, separated by the valley splitting energy ΔEv — the qubit's second-most-important energy scale after the Zeeman splitting, because a spin qubit that leaks into the wrong valley loses its readout fidelity.
Δz = n · a/4 (atomic step, a = 0.543 nm Si lattice constant)
k0 = 0.82 · 2π/a ≈ 9.49 nm⁻¹ (valley wavevector offset from the X point)
ΔEv(F, n) = ΔE0 · √(F/F0) · |sin(k0 · Δz)|
w(F) = w0 · (F0/F)^(1/3) (Fang-Howard confinement width)
P(v⁺) = 1 / (1 + exp(ΔEv / kB·T))
- Vertical field F — a stronger field presses the electron harder against the interface (Fang-Howard confinement, the width panel shrinks), which increases the intervalley coupling amplitude.
- Interface step — real interfaces are never atomically flat under the dot; a step of even one monolayer shifts the interference phase between the two valley plane waves, so ΔEv oscillates — sometimes almost vanishing — as the step count changes. This single-monolayer sensitivity is the main reason valley splitting varies wildly between nominally identical qubits.
- Temperature — the two valley states form a real two-level system; at finite T the excited valley gets thermally populated with probability P(v⁺) from the Boltzmann factor above. When ΔEv is small this "valley leakage" competes with — and can ruin — spin-qubit initialization and readout.
- New device — silicon qubits fabricated side by side can land on very different points of the same oscillation because the interface disorder pattern under each dot is random at the atomic scale; this redraws that disorder and its fixed interference phase.
This is a simplified, pedagogical version of the interference model used to explain valley-splitting variability in Si/SiGe and Si-MOS spin-qubit devices (e.g. Friesen-type interference of the two valley Bloch states off a sharp interface) — real devices also depend on alloy disorder, dot size and strain, which this model omits for clarity.