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arXiv:2602.14787

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

Lasse Ermoneit, Abel Thayil, Thomas Koprucki, Markus Kantner

cond-mat.mes-hallmath-phphysics.app-phphysics.comp-phquant-phSubmitted Feb 16, 2026• Announced Feb 17, 2026

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.

Journal reference: Phys. Rev. B 113, 245306 (2026)
DOI: 10.1103/md2x-s44y

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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.

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 (SiGe) 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 Si/SiGe quantum wells (QWs), e.g., sharp Si/SiGe 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\,μ\text{eV}

to above

1\,\text{meV}

- offering new opportunities for engineering robust and switchable silicon qubits.

Valley splitting sets the energy scale at which spin and valley degrees of freedom hybridize in silicon quantum-well qubits, but its sensitivity to interface structure makes it difficult to predict. Using the valleyor basis, we formulate a two-band effective-mass model for intervalley coupling in a finite quantum well and develop an analytical framework for treating non-perturbative effects of shear strain and wiggle-well potentials. For a

z

-independent coupling

V_sτ_2

, corresponding to uniform shear strain, we construct a variational state from the exact plane-wave valleyors and a hard-wall envelope. This yields closed-form expressions for the two lowest states and their splitting, including the characteristic oscillatory

|\sin(k_{\rm min}L)|

dependence. The variational result agrees with exact diagonalization to within

\sim1\%

for realistic couplings, and provides a simple prescription for tuning the shear strain away from the nodes to enhance the valley splitting. We further show that finite barrier heights primarily renormalize the effective width of the quantum well while preserving the oscillatory dependence. For a wiggle-well coupling

V_w\cos(k_wz)τ_1

, the valleyor representation reveals a crossover near the

k_w=2k_1

resonance from wiggle-dominated splitting to orbital quantization. Remarkably, at high wiggle-well amplitudes, the empty-box energy scale predominantly dictates the near-resonant valley splitting, while the wiggle-well potential enters only as a sub-leading correction. This behavior can be viewed as asymptotic freedom for spin qubits. Away from resonance, the splitting exhibits a broadly peaked resonance with diffraction-like side-lobes as a function of detuning.

Single electron spins can be used to encode and process information in semiconductor quantum devices. Progress has been hindered by materials challenges, such as the small energy splitting between low-lying valley states and hyperfine coupling to nuclear spins. Here we use density functional theory to optimize the valley splitting and spin dephasing time in realistic Si/SiGe heterostructures. Reductions in the Si quantum well width generally increase the valley splitting. However, in narrow quantum wells, a larger fraction of the electronic wavefunction resides in the SiGe buffer layers, which increases the hyperfine coupling with spinful

^{73}

Ge. Our work shows that Si/SiGe heterostructures with 3~--~4~nm wide quantum wells and

^{73}

Ge and

^{29}

Si concentrations of 50 ppm should support average valley splittings

E_{v}

~

>

~500~

μ

eV and spin dephasing times

T_2^*

exceeding 15~

μ

s assuming an effective quantum dot area of 700 nm

^2

. In addition, sharper Si/SiGe interfaces in general result in larger valley splittings and longer spin dephasing times.

Achieving valley splittings well in excess of the thermal energy of electrons and avoiding valley excitations is essential for the consistent initialization, operation and readout of gate-defined Si spin qubits. In this work, we present a device-level optimization strategy for pushing valley splittings to between 1 and 5 meV, well beyond values reported in nearly all previous theoretical studies. Using device-scale simulations that incorporate atomistic alloy disorder through a 1D tight-binding theory, we demonstrate that our proposed approach yields large valley splittings with a tight distribution across disorder realizations, a key requirement for reproducible qubit performance at scale. The approach rests on an unorthodox Si/SiGe heterostructure design combining a narrow quantum well, a small Ge spike, and a pure-Ge cap. We corroborate these predictions with targeted atomistic density functional theory calculations. These results offer a clear path forward for scalable Si/SiGe spin qubit devices and, if realized experimentally, effectively eliminate valley splitting as an existential problem for large scale SiGe-based quantum processors.

In Si(111) crystals, a strong biaxial tensile strain applied within the (111) plane is considered to shift the lowest energy point of the conduction band from the

Δ

valley to the L valley. Electrons confined in this L valley experience a splitting of their quadruply degenerate energy levels into an undegenerate single-level ground state (L1) and a triply degenerate excited state (L3). The energy of the single-level ground state is sufficiently low relative to the energies of the L3 valley and the

Δ

valley, making it optimal as a two-level system for a qubit. Using deformation potential theory and incorporating quantum effects from electron confinement in the SiGe/Si(111)/SiGe structure, we determine the value of the biaxial tensile strain causing the shift of the conduction band energy minimum from the

Δ

valley to the L valley, along with the corresponding Ge concentration. We also calculate the critical thickness for the plastic relaxation of the Si quantum well under this large biaxial tensile strain and examine the feasibility of realizing it as a SiGe/Si(111)/SiGe heterostructure.

As silicon spin qubit chips are increasing in qubit number and area, methods for the screening of qubit related material parameters become vital. Here we demonstrate the two-dimensional mapping of small variations of the electron g-factor of quantum dots formed in planar Si/SiGe quantum wells with precision better than

10^{-3}

and with nanometer lateral resolution. We scan the electron g-factor across a 40 nm

\times

400 nm area and observe two g-factors per QD site which obey a striking symmetry and bimodal distribution across the area. These two g-factors relate to valley states of the electron in the quantum dot in agreement with a recent theoretical model. Using conveyor-belt shuttling of entangled electron spin pairs, complementary to the mapping of the local valley-splitting, we map the g-factor. We compare g-factor and valley splitting maps measured on the same device, and extract the complex intervalley coupling parameter along the shuttle trajectories applying a theoretical model of g-factor dependence on intervalley coupling. These maps will allow unprecedented insights into the spin-valley dynamics during qubit manipulation, readout and shuttling and serve as a benchmark for the engineering of Si/SiGe heterostructures for large-scale quantum chips.