Small and device-dependent valley splittings remain a key challenge for electron spin qubits in silicon (Si), directly limiting qubit fidelity, device uniformity, and the scalability of Si-based quantum processors. In silicon-germanium (Si-Ge) heterostructures, this problem can be addressed through engineering of the epitaxial layer stack. Several heuristic strategies have been proposed to enhance the energy gap between the two nearly degenerate valley states in strained $\mathrm{Si}/\mathrm{Si}$-Ge quantum wells (QWs), e.g., sharp $\mathrm{Si}/\mathrm{Si}$-Ge interfaces, Ge spikes, or oscillating Ge concentrations within the QW. Here, we develop a systematic variational optimization approach to compute optimal Ge concentration profiles that enhance selected properties of the intervalley coupling matrix element. Our free-shape optimization framework is augmented by realistic technological constraints to ensure feasibility of the resulting epitaxial profiles and is based on an effective-mass envelope-function theory accounting for strain and compositional alloy disorder. Previously proposed heterostructure designs are recovered as special cases of the constrained optimization problem. Our main result is a novel heterostructure design, which we refer to as the modulated wiggle well, providing both a large deterministic enhancement of the valley splitting and a strong suppression of disorder-induced variability. In addition, this design enables wide electrical tunability of the valley splitting—from approximately $200\mathrm{\mu }\mathrm{eV}$ to above 1 meV—offering opportunities for engineering robust and switchable silicon qubits.

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• Inverse problems
• Optimization problems
• Quantum information with solid state qubits
• Valley degrees of freedom
• Disordered alloys
• Quantum dots
• Semiconductor compounds
• Spin qubits