Exact Multi-Valley Envelope Function Theory of Valley Splitting in Si/SiGe Nanostructures

Abstract

Valley splitting in strained Si/SiGe quantum wells is a central parameter for silicon spin qubits and is commonly described with envelope-function and effective-mass theories. These models provide a computationally efficient continuum description and have been shown to agree well with atomistic approaches when the confinement potential is slowly varying on the lattice scale. In modern Si/SiGe heterostructures with atomically sharp interfaces and engineered Ge concentration profiles, however, the slowly varying potential approximation underlying conventional (local) envelope-function theory is challenged. We formulate an exact multi-valley envelope-function model by combining Burt-Foreman-type envelope-function theory, which does not rely on the assumption of a slowly varying potential, with a valley-sector decomposition of the Brillouin zone. This construction enforces band-limited envelopes, which satisfy a set of coupled integro-differential equations with a non-local potential energy operator. Using degenerate perturbation theory, we derive the intervalley coupling matrix element within this non-local model and prove that it is strictly invariant under global shifts of the confinement potential (choice of reference energy). We then show that the conventional local envelope model generically violates this invariance due to spectral leakage between valley sectors, leading to an unphysical energy-reference dependence of the intervalley coupling. The resulting ambiguity is quantified by numerical simulations of various engineered Si/SiGe heterostructures. Finally, we propose a simple spectrally filtered local approximation that restores the energy-reference invariance exactly and provides a good approximation to the exact non-local theory.

References in corpus (26)

• Silicon Quantum Electronics
• A >99.9%-fidelity quantum-dot spin qubit with coherence limited by charge noise
• Semiconductor Spin Qubits
• Electron spin coherence exceeding seconds in high purity silicon
• Computing with spin qubits at the surface code error threshold
• Fast universal quantum control above the fault-tolerance threshold in silicon
• Two-qubit silicon quantum processor with operation fidelity exceeding 99%
• Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting
• Valley Splitting Theory of SiGe/Si/SiGe Quantum Wells
• Probing single electrons across 300 mm spin qubit wafers
• Physical mechanisms of interface-mediated intervalley coupling in Si
• SiGe quantum wells with oscillating Ge concentrations for quantum dot qubits
• Multivalley effective mass theory simulation of donors in silicon
• Practical Strategies for Enhancing the Valley Splitting in Si/SiGe Quantum Wells
• 12-spin-qubit arrays fabricated on a 300 mm semiconductor manufacturing line
• Giant anisotropy of spin relaxation and spin-valley mixing in a silicon quantum dot
• Valley splittings in Si/SiGe quantum dots with a germanium spike in the silicon well
• Low disorder and high valley splitting in silicon
• Enhanced Valley Splitting in Si Layers with Oscillatory Ge Concentration
• Strategies for enhancing spin-shuttling fidelities in Si/SiGe quantum wells with random-alloy disorder
• Multi-valley envelope function equations and effective potentials for P impurity in silicon
• Coupling conduction-band valleys in SiGe heterostructures via shear strain and Ge concentration oscillations
• Linear and planar molecules formed by coupled P donors in silicon
• Theory of Valley Splitting in Si/SiGe Spin-Qubits: Interplay of Strain, Resonances and Random Alloy Disorder
• Large Stark Effect for Li Donor Spins in Si
• Valley physics in the two bands $\mathbf{k}\cdot \mathbf{p}$ model for SiGe heterostructures and spin qubits