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arXiv:2604.13435

SiGe/Si(111)/SiGe heterostructure for Si spin qubits with electrons confined in L valley of conduction band

Takafumi Tokunaga, Hiromichi Nakazato

quant-phSubmitted Apr 15, 2026• Announced Apr 16, 2026
arXivHTMLPDF

Abstract

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.

Comments
11 pages, 10 figures

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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 ΔΔΔ 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 ΔΔΔ 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 ΔΔΔ 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.
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.
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 μeV200\,μ\text{eV}200μeV to above 1 meV1\,\text{meV}1meV - offering new opportunities for engineering robust and switchable silicon qubits.
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^{73}73Ge. Our work shows that Si/SiGe heterostructures with 3~--~4~nm wide quantum wells and 73^{73}73Ge and 29^{29}29Si concentrations of 50 ppm should support average valley splittings EvE_{v}Ev​~>>>~500~μμμeV and spin dephasing times T2∗T_2^*T2∗​ exceeding 15~μμμs assuming an effective quantum dot area of 700 nm2^22. In addition, sharper Si/SiGe interfaces in general result in larger valley splittings and longer spin dephasing times.
Strained germanium (ε\varepsilonε-Ge) quantum wells on metamorphic SiGe buffers have enabled advanced hole-based spin qubit devices. Alternatively, unstrained Ge with lattice-matched strained silicon-germanium (ε\varepsilonε-SiGe) barriers eliminates the need for metamorphic buffers altogether. The ground state character of both these platforms is predominantly heavy-hole (HH) with a largely anisotropic spin response. We propose and study an alternative heterostructure, lattice-matched to Ge, in which both the SiGe quantum well and barriers are tensile strained, with their composition contrast providing the band offset for confinement and the tensile strain stabilizing a light-hole (LH) ground state. We show large spin-orbit coupling (SOC), both linear and cubic, along with a significantly more isotropic spin response compared to strained HH qubits. We also study the decoherence properties of the proposed device, showing an appreciable gain in the quality factor compared to their HH counterparts. Finally, we propose a bilayer heterostructure that allows for electrical switching between HH and LH ground state character.
Direct phonon spin-lattice relaxation of an electron qubit bound by a donor impurity or quantum dot in SiGe heterostructures is investigated. The aim is to evaluate the importance of decoherence from this mechanism in several important solid-state quantum computer designs operating at low temperatures. We calculate the relaxation rate 1/T1{1/T}_{1}1/T1​ as a function of [100] uniaxial strain, temperature, magnetic field, and silicon/germanium content for Si:P bound electrons and quantum dots. The quantum dot potential is much smoother, leading to smaller splittings of the valley degeneracies. We have estimated these splittings in order to obtain upper bounds for the relaxation rate. In general, we find that the relaxation rate is strongly decreased by uniaxial compressive strain in a SiGe-Si-SiGe quantum well, making this strain an important positive design feature. Ge in high concentrations (particularly over 85%) increases the rate, making Si-rich materials preferable. We conclude that SiGe bound electron qubits must meet certain conditions to minimize decoherence but that spin-phonon relaxation does not rule out the solid-state implementation of error-tolerant quantum computing.
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.