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arXiv:2512.21930

Zeeman-like coupling to valley degree of freedom in Si-based spin qubits

S Akbar Jafari, Hendrik J Bluhm, David P Divincenzo

cond-mat.mtrl-sciquant-phSubmitted Dec 26, 2025• Announced Dec 29, 2025
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Abstract

Increasing the valley splitting in Si-based heterostructures is critical for improving the performance of semiconductor qubits. This paper demonstrates that the two low-energy conduction band valleys are not independent parabolic bands. Instead, they originate from the X-point of the Brillouin zone, where they are interconnected by a degeneracy protected by the non-symmorphic symmetry of the diamond lattice. This semi-Dirac-node degeneracy gives rise to the $Δ_1$ and $Δ_{2'}$ bands, which constitute the valley degrees of freedom. By explicitly computing the two-component Bloch functions $X_1^\pm$, using the wave vector group at the X-point, we determine the transformation properties of the object $(X_1^+,X_1^-)$. We demonstrate that these properties are fundamentally different from those of a spinor. Consequently, we introduce the term "valleyor" to emphasize this fundamental distinction. The transformation properties of valleyors induce corresponding transformations of the Pauli matrices $τ_1,τ_2$ and $τ_3$ in the valley space. Determining these transformations allows us to classify possible external perturbations that couple to each valley Pauli matrix, thereby identifying candidates for valley-magnetic fields, ${\mathsf B}$. These fields are defined by a Zeeman-like coupling ${\mathsf B}\cdot\vecτ$ to the valley degree of freedom. In this way, we identify scenarios where an applied magnetic field $\vec B$ can leverage other background fields, such as strain, to generate a valley-magnetic field ${\mathsf B}$. This analysis suggests that beyond the well-known mechanism of potential scattering from Ge impurities, there exist additional channels (mediated by combinations of magnetic and strain-induced vector potentials) to control the valley degree of freedom

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Valley-Enabled Intrinsic Dresselhaus Spin-Orbit Coupling in Silicon

Johannes L. P. Steinschuld, Hendrik J. Bluhm, Lars R. Schreiber +1 more

We develop a symmetry-based theory of spin-orbit-valley coupling in silicon that reveals an intrinsic source of Dresselhaus spin-orbit coupling independent of interfaces or external electric fields. Treating the valley degree of freedom as a symmetry-carrying quantum degree of freedom, we show that the Dresselhaus interaction is necessarily valley off-diagonal and that a bulk contribution proportional to the valley Pauli matrix $τ_1$ is symmetry allowed. Tight-binding calculations yield a bulk coupling more than an order of magnitude larger than typical interface-induced spin-orbit coupling; achieving the same energy scale through the interface-induced mechanism would require electric fields roughly 50 times larger than typical fields. We further derive the symmetry-allowed spin-valley couplings generated by magnetic-field gradients and show how they account for the valley-dependent Zeeman splitting observed in micromagnet experiments. A slight tilt of the background magnetic field out of the plane produces an additional isotropic contribution linear in $B_z$, providing an experimentally accessible signature of the corresponding coupling constant. Finally, we predict a spin-independent micromagnet-induced valley splitting in the $τ_3$ channel, which is distinct from the $τ_{1,2}$ channels generated by alloy disorder and therefore remains robust against disorder-induced cancellation. These results establish valley symmetry as a fundamental ingredient in the spin-orbit physics of silicon and provide new mechanisms for controlling and probing spin and valley degrees of freedom in silicon quantum devices.

cond-mat.mtrl-sciCross-listedarXiv:2608.29785•Sep 1, 2026

Non-perturbative theory of valley splitting in Si qubits from variational wave function: periodic effects of shear strain and asymptotic freedom in the wiggle-well potential

Johannes L. P. Steinschuld, Hendrik J. Bluhm, Seyed Akbar Jafari

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.

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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 $Δ$ 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.

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

Lasse Ermoneit, Abel Thayil, Thomas Koprucki +1 more

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.

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Anisotropic spin-valley coupling in SiMOS and Si/SiGe quantum dots

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While bulk silicon has long been understood to exhibit relatively weak spin-orbit coupling (SOC), confinement of electrons to quantum dots (QDs) at a silicon heterointerface results in significantly larger SOC. This is a concern for electron spin qubit performance, as intravalley and intervalley SOC can significantly perturb the operation of electron spin qubits. While these interactions can be harnessed to drive coherent rotations in a singlet-triplet qubit, coupling to low-lying excited valley states can lead to undesirable spin relaxation when valley splitting is on resonance with the Zeeman energy. In this work, we measure the angular dependence of the interfacial spin-orbit interaction as a function of the direction and magnitude of an applied external magnetic field in SiMOS and Si/SiGe heterostructures, two common material platforms for silicon spin qubits. We construct a physical model that accurately infers intra- and inter-valley SOC physics from fits to the data, allowing for a direct comparison between these two material systems. For the devices measured we find that, while the $g$-factor differences are comparable, the SiMOS QDs exhibit an order of magnitude larger spin-valley coupling than for Si/SiGe. Moreover, we find that the angular dependence of the spin-valley coupling is similar for both devices, with similar magnetic field orientations minimizing the spin-valley coupling. Our work points towards operational schemes for optimizing spin-valley coupling to avoid or exploit this mechanism for qubit operation.

cond-mat.mes-hallCross-listedarXiv:2604.16713•Apr 21, 2026

Strain-Tunable Spin Filtering and Valley Splitting Coexisting with Anomalous Hall Effect in 2D Half-Metallic VSe2/VN Heterostructure: Toward a Unified Spintronic-Valleytronic Platform

Vivek Chowdhury, Ahmed Zubair

Rapid progress in valleytronics and spintronics is limited by the scarcity of two-dimensional materials that simultaneously provide robust valley splitting and strong spin selectivity. Here we showed that a van der Waals heterostructure (VSe2/VN) built from hexagonal VSe2 and hexagonal VN addressed this gap. Using first-principles density functional theory, phonon, ab initio molecular dynamics stability tests, Bader charge analysis, and Wannier-based Berry-curvature calculations, we demonstrated an energetically and dynamically stable heterostructure that exhibited interlayer charge transfer and a work function intermediate between the constituent monolayers. The electronic structure showed small indirect PBE gap (108.9 meV), with HSE06 indicating a half-metallic tendency; a sizable conduction-band valley splitting (ΔCKK' = 22.9 meV for spin-up and ΔCKK' = 61.3 meV for spin-down); and pronounced spin asymmetry, where the spin-down channel showed a wide semiconducting gap (0.64 eV) while the spin-up channel was nearly gapless. These features yielded a high zero-strain spin-filter efficiency P = 75.4%, tunable to 82.5% under +4% biaxial tensile strain. The heterostructure also supported non-zero, valley-contrasting Berry curvature, and a large anomalous Hall conductivity (peak sigmaxy = 568.33 S/cm). Importantly, mean-field estimation placed the ferromagnetic Curie temperature near room temperature at zero strain (Tc = 284.04 K), while Tc decreased to 183.9 K at +4% strain, the magnetic order remained robust to cryogenic temperatures, providing a beneficial tuning knob to balance spin-filter performance with thermal stability in device-relevant regimes. These results identified VSe2/VN as a practical, strain-tunable platform for integrated valleytronic, spintronic devices, and for exploring anomalous Hall and valley-dependent transport phenomena.

cond-mat.mtrl-sciCross-listedarXiv:2511.06069•Nov 11, 2025
Increasing the valley splitting in Si-based heterostructures is critical for improving the performance of semiconductor qubits. This paper demonstrates that the two low-energy conduction band valleys are not independent parabolic bands. Instead, they originate from the X-point of the Brillouin zone, where they are interconnected by a degeneracy protected by the non-symmorphic symmetry of the diamond lattice. This semi-Dirac-node degeneracy gives rise to the Δ1Δ_1Δ1​ and Δ2′Δ_{2'}Δ2′​ bands, which constitute the valley degrees of freedom. By explicitly computing the two-component Bloch functions X1±X_1^\pmX1±​, using the wave vector group at the X-point, we determine the transformation properties of the object (X1+,X1−)(X_1^+,X_1^-)(X1+​,X1−​). We demonstrate that these properties are fundamentally different from those of a spinor. Consequently, we introduce the term "valleyor" to emphasize this fundamental distinction. The transformation properties of valleyors induce corresponding transformations of the Pauli matrices τ1,τ2τ_1,τ_2τ1​,τ2​ and τ3τ_3τ3​ in the valley space. Determining these transformations allows us to classify possible external perturbations that couple to each valley Pauli matrix, thereby identifying candidates for valley-magnetic fields, B{\mathsf B}B. These fields are defined by a Zeeman-like coupling B⋅τ⃗{\mathsf B}\cdot\vecτB⋅τ to the valley degree of freedom. In this way, we identify scenarios where an applied magnetic field B⃗\vec BB can leverage other background fields, such as strain, to generate a valley-magnetic field B{\mathsf B}B. This analysis suggests that beyond the well-known mechanism of potential scattering from Ge impurities, there exist additional channels (mediated by combinations of magnetic and strain-induced vector potentials) to control the valley degree of freedom
We develop a symmetry-based theory of spin-orbit-valley coupling in silicon that reveals an intrinsic source of Dresselhaus spin-orbit coupling independent of interfaces or external electric fields. Treating the valley degree of freedom as a symmetry-carrying quantum degree of freedom, we show that the Dresselhaus interaction is necessarily valley off-diagonal and that a bulk contribution proportional to the valley Pauli matrix τ1τ_1τ1​ is symmetry allowed. Tight-binding calculations yield a bulk coupling more than an order of magnitude larger than typical interface-induced spin-orbit coupling; achieving the same energy scale through the interface-induced mechanism would require electric fields roughly 50 times larger than typical fields. We further derive the symmetry-allowed spin-valley couplings generated by magnetic-field gradients and show how they account for the valley-dependent Zeeman splitting observed in micromagnet experiments. A slight tilt of the background magnetic field out of the plane produces an additional isotropic contribution linear in BzB_zBz​, providing an experimentally accessible signature of the corresponding coupling constant. Finally, we predict a spin-independent micromagnet-induced valley splitting in the τ3τ_3τ3​ channel, which is distinct from the τ1,2τ_{1,2}τ1,2​ channels generated by alloy disorder and therefore remains robust against disorder-induced cancellation. These results establish valley symmetry as a fundamental ingredient in the spin-orbit physics of silicon and provide new mechanisms for controlling and probing spin and valley degrees of freedom in silicon quantum devices.
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 zzz-independent coupling Vsτ2V_sτ_2Vs​τ2​, 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⁡(kminL)∣|\sin(k_{\rm min}L)|∣sin(kmin​L)∣ dependence. The variational result agrees with exact diagonalization to within ∼1%\sim1\%∼1% 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 Vwcos⁡(kwz)τ1V_w\cos(k_wz)τ_1Vw​cos(kw​z)τ1​, the valleyor representation reveals a crossover near the kw=2k1k_w=2k_1kw​=2k1​ 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.
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.
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.
While bulk silicon has long been understood to exhibit relatively weak spin-orbit coupling (SOC), confinement of electrons to quantum dots (QDs) at a silicon heterointerface results in significantly larger SOC. This is a concern for electron spin qubit performance, as intravalley and intervalley SOC can significantly perturb the operation of electron spin qubits. While these interactions can be harnessed to drive coherent rotations in a singlet-triplet qubit, coupling to low-lying excited valley states can lead to undesirable spin relaxation when valley splitting is on resonance with the Zeeman energy. In this work, we measure the angular dependence of the interfacial spin-orbit interaction as a function of the direction and magnitude of an applied external magnetic field in SiMOS and Si/SiGe heterostructures, two common material platforms for silicon spin qubits. We construct a physical model that accurately infers intra- and inter-valley SOC physics from fits to the data, allowing for a direct comparison between these two material systems. For the devices measured we find that, while the ggg-factor differences are comparable, the SiMOS QDs exhibit an order of magnitude larger spin-valley coupling than for Si/SiGe. Moreover, we find that the angular dependence of the spin-valley coupling is similar for both devices, with similar magnetic field orientations minimizing the spin-valley coupling. Our work points towards operational schemes for optimizing spin-valley coupling to avoid or exploit this mechanism for qubit operation.
Rapid progress in valleytronics and spintronics is limited by the scarcity of two-dimensional materials that simultaneously provide robust valley splitting and strong spin selectivity. Here we showed that a van der Waals heterostructure (VSe2/VN) built from hexagonal VSe2 and hexagonal VN addressed this gap. Using first-principles density functional theory, phonon, ab initio molecular dynamics stability tests, Bader charge analysis, and Wannier-based Berry-curvature calculations, we demonstrated an energetically and dynamically stable heterostructure that exhibited interlayer charge transfer and a work function intermediate between the constituent monolayers. The electronic structure showed small indirect PBE gap (108.9 meV), with HSE06 indicating a half-metallic tendency; a sizable conduction-band valley splitting (ΔCKK' = 22.9 meV for spin-up and ΔCKK' = 61.3 meV for spin-down); and pronounced spin asymmetry, where the spin-down channel showed a wide semiconducting gap (0.64 eV) while the spin-up channel was nearly gapless. These features yielded a high zero-strain spin-filter efficiency P = 75.4%, tunable to 82.5% under +4% biaxial tensile strain. The heterostructure also supported non-zero, valley-contrasting Berry curvature, and a large anomalous Hall conductivity (peak sigmaxy = 568.33 S/cm). Importantly, mean-field estimation placed the ferromagnetic Curie temperature near room temperature at zero strain (Tc = 284.04 K), while Tc decreased to 183.9 K at +4% strain, the magnetic order remained robust to cryogenic temperatures, providing a beneficial tuning knob to balance spin-filter performance with thermal stability in device-relevant regimes. These results identified VSe2/VN as a practical, strain-tunable platform for integrated valleytronic, spintronic devices, and for exploring anomalous Hall and valley-dependent transport phenomena.