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

Abel Thayil; Lasse Ermoneit; Markus Kantner; Thomas Koprucki

REVIEW 2 major objections 4 minor 1 cited by

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

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Conventional local envelope models for Si/SiGe valley splitting are non-invariant under energy-reference shifts; an exact non-local theory and a spectrally filtered approximation restore the invariance.

desk verdict Standard local envelope-function valley-splitting predictions are energy-reference dependent; this paper proves the exact non-local fix and offers a filtered approximation, but the numerical demonstration misses a shear-strain gauge check. read the letter →

arxiv 2602.14787 v1 pith:M7YAAUXX submitted 2026-02-16 cond-mat.mes-hall math-phmath.MPphysics.app-phphysics.comp-phquant-ph

Lasse Ermoneit , Abel Thayil , Thomas Koprucki , Markus Kantner This is my paper · ORCID

classification cond-mat.mes-hallmath-phmath.MPphysics.app-phphysics.comp-phquant-ph PACS 71.15.-m73.21.Fg73.63.Kv

keywords valleysplittingSi/SiGequantumwellsenvelopefunctiontheoryintervalleycouplingenergy-referenceinvariancenon-localpotentialspectralleakagespinqubits

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that the conventional local envelope-function (effective-mass) model for computing valley splitting in Si/SiGe heterostructures is not invariant under a global shift of the confinement potential, so its predicted valley splitting E_VS = 2|Δ| changes with the arbitrary choice of energy reference. The cause is spectral leakage: the local model does not confine envelope wave functions to their valley-specific sectors of the Brillouin zone, letting short-wavelength components contaminate the intervalley coupling. The paper constructs an exact multi-valley envelope theory with a non-local potential operator that enforces this band limitation, proves that its intervalley coupling Δ is strictly invariant under global potential offsets, and shows that a simple spectrally filtered local approximation restores the invariance while closely tracking the exact result in numerical tests. For sharply varying structures — atomically sharp interfaces, Ge spikes, wiggle wells — the local model's ambiguity is quantitatively large, implying that local-model valley-splitting predictions in this regime are reference-dependent and need to be re-examined.

What carries the argument

The central object is the intervalley coupling matrix element Δ = ∫∫ f_+*(r) u_{+-}(r,r') f_-(r') dr dr', whose magnitude sets the valley splitting E_VS = 2|Δ|. The argument is carried by valley-sector band limitation: the exact theory expands the microscopic wave function in Bloch factors and requires the envelope of each valley to contain only plane-wave components from its own sector S(k0) of the Brillouin zone, enforced through a truncated (sector-projected) delta function Δ_k0(r-r') that acts as a projector in the non-local potential operator. This projection ensures a global potential offset U0 appears only in the diagonal intravalley blocks and drops out of Δ; the conventional local m

What would settle it

Compute valley splitting for a Si/SiGe quantum well with a sharp 2–3 monolayer Ge spike using a fully atomistic method (tight binding or DFT) and compare with the exact non-local EFT, the projected-local model, and the local model evaluated at several reference energies. If the atomistic value disagrees with the exact non-local prediction by more than ~15% while the local model happens to match for some reference energy, the exact model would not be the correct gauge-invariant continuum limit; if the atomistic value lies inside the local model's reference-dependent band and coincides with the

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Extended reading notes

Core claim

The central claim is that the standard local multi-valley envelope model generically violates a fundamental symmetry: under a constant offset of the confinement potential, U → U + U0, the intervalley coupling transforms as Δ_loc → Δ_loc + U0 R, where R ≠ 0 whenever the local envelope has Fourier components outside its valley sector (Eqs. (37)–(38)). Consequently the valley splitting E_VS = 2|Δ_loc| predicted by the local model depends on the arbitrary choice of reference energy, which is unphysical because a constant potential should only shift intravalley energies and leave intervalley coupling invariant. The paper proves that the exact non-local envelope theory — in which envelopes are ban

Load-bearing premise

The exactness and invariance proof assumes the microscopic crystal potential is that of a perfect periodic silicon lattice and that all germanium and electrostatic effects enter through a single additive mesoscopic potential U(r); if atomically sharp SiGe interfaces require material-dependent Bloch bases or a fully atomistic potential, the invariance theorem does not automatically transfer to the physical device.

Editorial extensions

If this is right

Local envelope-function theory is confirmed as a reliable approximation only when the confinement potential is smooth on the lattice scale; for atomically sharp interfaces, Ge spikes, or oscillating Ge profiles its valley-splitting predictions are non-unique, with the ambiguity quantified by 2|R|.
The exact non-local model defines a unique, reference-energy-invariant valley splitting that can serve as a gauge-consistent benchmark for continuum and atomistic comparisons in the sharp-interface regime.
The spectrally filtered (projected-local) model restores invariance exactly and reproduces the exact non-local results across interface widths, well widths, electric fields, and wiggle-well resonances, though it tends to overestimate the Ge-spike case.
The invariance proof extends to the full multi-band envelope system, so the energy-reference problem is intrinsic to local multi-valley EFT, not an artifact of the two-valley reduction.
For smooth heterostructures the ambiguity measure |R| is small, so existing local-model predictions for such devices remain meaningful; the issue is specific to engineered sharp profiles.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The reference-energy ambiguity likely affects other continuum multi-valley methods that do not enforce valley-sector band limitation; applying the same projection test to effective-mass models for other materials (e.g., germanium, transition-metal dichalcogenides) would reveal whether this is a general gauge-consistency condition.
A practical corollary for the qubit community: reported local-model 'agreement' with a measured valley splitting in sharp-interface devices could reflect a hidden reference-energy choice; re-evaluating those numbers with the filtered model would test whether the agreement survives.
The filtered model's overestimation for Ge spikes suggests that simple projection loses the non-local backfolding of high wave-number potential components; a frequency-dependent filter that retains the band-structure coefficients might close this gap and could be tested against the exact model in a wider parameter sweep.
The exact theory might be extended to treat random-alloy disorder in the same gauge-consistent way, since disorder potentials carry large short-wavelength components — the same leakage mechanism could contaminate disorder-averaged valley splittings in current models.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a serious look: this paper shows that the conventional local envelope-function model gives a valley splitting that depends on the arbitrary choice of reference energy, and that the exact Burt–Foreman multi-valley EFT resolves the ambiguity. The analytical core is clean, and the filtered local approximation is a practical fix that should interest anyone doing quantitative Si/SiGe qubit modeling.

What is actually new: a proof that the intervalley coupling in the sector-projected non-local model is strictly invariant under global potential shifts, and that the local model generically violates this invariance through spectral leakage between valley sectors. The numerical studies across interface width, well width, electric field, wiggle wells, and Ge spikes make the qualitative claim concrete. The projected-local (spectrally filtered) model restores invariance and tracks the exact non-local results well in most cases. Code is provided. The central theorem is not circular; it follows directly from the definitions of the sector-projected kernel.

Soft spots, in proportion. The largest is not the theory but the numerical support for it. The exact numerical model is solved in an effective 1D reduction (Appendix C). The stress-test note makes a specific point that holds up: under a global shift U⊥ → U⊥ + U0, the reduced intervalley coupling (Eq. 34) acquires a term proportional to C_{-1} ∫ e^{-i(2k0-G0)z}|f|^2 dz. Since C_{-1} = -31.8 ε_xy, that term does not vanish unless the Fourier component of |f|^2 at that wavevector is exactly zero, which the paper does not demonstrate. The authors never test U0 invariance of their numerical “exact” results in the shear-strained wiggle-well case (Fig. 5a), where ε_xy is doing real work. This does not invalidate the analytical theorem, but it leaves the quantitative demonstration incomplete. A straightforward fix would be to compute the residual ambiguity measure for the reduced exact model under shear strain.

Second, the numerical inputs are largely self-sourced: the empirical pseudopotential coefficients come from the authors’ own prior work [40], and the odd-integer coefficients’ shear-strain slopes are least-squares fits. Not disqualifying, but the quantitative numbers rest on choices not independently checked here. One atomistic benchmark (tight-binding or DFT) for a sharp-interface case would materially strengthen the claims.

Third, the modeling assumption that Ge effects can be fully absorbed into a mesoscopic U(r) with Si Bloch factors is an approximation, especially at atomically sharp interfaces. The authors flag this limitation in the text, but future versions should state it more defensively if they want to claim quantitative device-level accuracy.

Who benefits: anyone using local EFT for valley-splitting predictions in engineered Si/SiGe heterostructures. The gauge ambiguity is a real effect the community should know about. I would send this to peer review; the core result is solid and the numerical gap is addressable rather than fatal. Recommended: ask the authors to add the shear-strain gauge check and ideally one independent atomistic comparison before publication.

Referee Report

2 major / 4 minor

Summary. The paper develops an exact multi-valley envelope-function theory (EFT) for Si/SiGe nanostructures by combining Burt–Foreman EFT with a Brillouin-zone sector decomposition. The central analytical result is a proof that the intervalley coupling matrix element Δ is invariant under global shifts of the confinement potential, whereas the conventional local EFT generically violates this invariance due to spectral leakage between valley sectors. The paper quantifies this ambiguity numerically for several heterostructure designs (interface width, well width, electric-field asymmetry, wiggle wells, Ge spikes) and proposes a spectrally filtered projected-local approximation that is claimed to restore the invariance and to approximate the exact non-local results. The analytical proof (App. B) is clean, and the numerical method is described in detail with convergence checks and reproducible code.

Significance. If the central claim holds, the paper identifies a previously unrecognized energy-reference ambiguity in conventional local multi-valley EFT that is directly relevant to quantitative valley-splitting predictions for Si spin qubits in sharply varying heterostructures. The analytical theorem is not circular and is a genuine structural insight. The numerical benchmark of the filtered local model is useful and the GitHub code is a strength. However, the numerical implementation relies on an effective 1D reduction whose gauge-invariance properties are not established; this is load-bearing for the paper's quantitative conclusions.

major comments (2)

[IV B / App. C, Eqs. (33)–(34)] The gauge-invariance proof in App. B applies to the full multi-band model. The numerical 'exact' model, however, uses the effective 1D reduction (33)–(34). Under U⊥→U⊥+U0, Eq. (34) changes by U0 Σ_n C_n ∫ e^{-i(2k0+nG0)z}|f|² dz. Since f has support (−k0,k1), only the n=−1 term can contribute; C_{−1}=−31.8 ε_xy (Tab. II) is nonzero for the shear-strained 2k1 wiggle-well case (Fig. 5a). The paper neither proves this integral vanishes nor numerically tests U0-invariance of the reduced model's Δ. Please add such a test and, if invariance fails, either modify the reduced model or qualify the claim that the numerical results demonstrate exact invariance.
[IV D, Eq. (40)] The projected-local model is claimed to be invariant 'by construction,' but under U⊥→U⊥+U0, Eq. (40) also acquires the term U0 Σ_n C_n ∫ e^{-i(2k0+nG0)z}|\tilde f|² dz. The sector projection (39) does not by itself make this term vanish; the Fourier support argument leaves q=2k0−G0 inside the autocorrelation support of the projected envelope. The paper should either prove a cancellation or verify numerically that Δ_filtered is actually constant under U0 shifts. Without this, the benchmark in Figs. 3–5 is not evidence that the filtered model restores the invariance that the local model lacks.

minor comments (4)

[Sec. IV E 4 / Fig. 5] The shear strain ε_xy used to unlock the 2k1 resonance is not listed in Tab. I or stated in the text. Please report the value for reproducibility.
[Fig. 2(b) caption] Typo: 'is valley-sector' should read 'its valley-sector.'
[Eq. (39)] The notation e^{−𝑖𝑘_0(𝑧−𝑧′)} is typographically awkward; use standard upright or italic math consistently.
[Tab. II] For the odd-integer coefficients, it would be helpful to state the strain range over which the linear least-squares fit was performed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the gauge-invariance theorem is derived from the sector-projected envelope equations, and numerical inputs are external parameters, not fitted targets.

full rationale

The central result — invariance of the intervalley coupling under global potential shifts in the non-local model and its violation in the local model — is an analytic consequence of the definitions of the valley-sector projectors Δ_{k0} and the non-local kernel. It does not use any computed valley-splitting value as input, and no fitted parameter is renamed as a prediction. Appendix B derives the shift identity (26) directly from the projector structure and the disjointness of valley sectors; Equations (37)-(38) then quantify the local-model ambiguity from the same formalism. The numerical coefficients C_n^(2) and B_n are taken from the authors' prior empirical-pseudopotential work [40], but these are external band-structure parameters, not quantities fitted to the target predictions (E_VS or the ambiguity measure R). The shear-strain linear coefficients in Table II are fitted to numerical band-structure data, not to valley splitting. The projected-local model is constructed by sector projection, so its invariance property is a design property rather than an independently predicted outcome, but the paper explicitly benchmarks it against the exact non-local model rather than treating the construction itself as evidence. The acknowledged limitations — Ge effects are lumped into a phenomenological U and 'exact' means formally exact envelope representation — restrict the atomistic scope but do not make the derivation circular. The potential numerical-support gap raised by the effective-1D reduction under shear strain is a correctness/verification concern, not an input-output equivalence, and therefore does not constitute circularity.

Assumptions & free parameters 2 free parameters · 9 assumptions · 0 invented entities

No new physical entities are postulated. The mathematical sector-projected delta Δ_k0 is a construction, not a physical object. The main external ingredients are empirical pseudopotential coefficients from the authors' earlier paper [40] and fitted shear-strain slopes (Tab. II), which make the numerical demonstrations dependent on self-cited, fitted input. The central gauge-invariance theorem is nevertheless independent of those coefficients.

free parameters (2)

Odd-integer coupling coefficients' shear-strain slopes (C_n^(2), B_n for odd n) = e.g., C_-1^(2) = -31.8·ε_xy, C_1^(2) = -0.402·ε_xy, B_1 = 2.92·ε_xy, B_3 = -8.55e-2·ε_xy
Tab. II states the linear dependence of odd-integer coefficients on shear strain was obtained from a least-squares fit to numerical pseudopotential data; these enter the numerical Δ and valley-splitting values.
Empirical pseudopotential form factors / band-structure model from Ref. [40] = not listed; fitted to bulk Si/Ge band structure in prior work
All C_n^(2), B_n, effective masses, band offset, and k0 values are taken from the authors' own empirical pseudopotential parametrization; the present paper does not independently re-derive them.

assumptions (9)

standard math Bloch functions at fixed k0 are complete and orthonormal on lattice-periodic functions (Eqs. 8a-8b), and the first Brillouin zone is partitioned into disjoint valley sectors (Eq. (5), Fig. 1(b)).
Used to justify the ansatz (2) and the uniqueness/orthonormality of band-limited envelopes in Appendix A.
domain assumption The crystal potential V is that of a perfect periodic Si crystal; all Ge and gate effects enter only through the mesoscopic potential U(r).
Sec. II A after Eq. (1); without this, the 'exact' envelope representation is not exact for the physical heterostructure.
domain assumption Only the two low-energy valleys at ±k0 are relevant; the other four conduction-band valleys are omitted.
Sec. II B/III A; standard for biaxially strained Si/SiGe QWs, but a load-bearing truncation for the two-valley model.
domain assumption The two valleys have identical band edges E_c and identical effective mass tensors m_c.
Sec. III A; enables the two-component model and the degenerate perturbation theory.
domain assumption Remote bands can be decoupled via Schrieffer-Wolff/Löwdin renormalization, giving a single-band effective-mass equation that retains the nonlocal intervalley structure.
Eqs. (19)-(20); assumes remote-band coupling is weak, which may be questionable at atomically sharp Ge features.
domain assumption In-plane and vertical motions separate; the QD is large compared with the lattice, allowing harmonic-oscillator ground state and Gaussian selection rules (Eqs. C2-C5).
Appendix C; underpins the effective 1D Fourier-space model (33) and all numerical results.
domain assumption Empirical pseudopotential coefficients C_n^(2), B_n and material parameters from Ref. [40] are accurate for strained Si/SiGe.
Tabs. I-II are the central numerical input; they come from the authors' own prior work and are not independently verified here.
ad hoc to paper Odd-integer coupling coefficients depend linearly on shear strain, with slopes from a least-squares fit to numerical pseudopotential data.
Tab. II statements; a fitted interpolation used in the numerical Δ and in the wiggle-well/shear-strain discussion.
standard math The reciprocal-lattice combination G-G' lies inside the first Brillouin zone only for G=G' (used to kill sector-overlap terms in App. B).
Needed in the gauge-invariance proof; implicitly ignores measure-zero sector-boundary coincidences, consistent with the continuum L2 setting.

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Pith. "Pith review of Exact Multi-Valley Envelope Function Theory of Valley Splitting in Si/SiGe Nanostructures." pith.science (2026). https://pith.science/paper/M7YAAUXX

@misc{pith2026260214787,
  author       = {Pith},
  title        = {Pith review of: Exact Multi-Valley Envelope Function Theory of Valley Splitting in Si/SiGe Nanostructures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7YAAUXX}},
  note         = {Machine review of arXiv:2602.14787}
}

read the original 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.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Silicon-Germanium Heterostructures with Enhanced Valley Splitting for Spin Qubits
cond-mat.mes-hall 2026-07 conditional novelty 6.0 of 10

An unorthodox Si/SiGe heterostructure (narrow well + dilute Ge spike + pure-Ge cap) is predicted to yield 1-5 meV valley splittings with tight alloy-disorder distributions.

Reference graph

Works this paper leans on

58 extracted references · cited by 1 Pith paper

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The model (21) has beenwidely employed in the silicon qubit literature [17, 18, 40, 45, 46]

as 𝑢k0(r) 2≈1, one arrives at the local envelope equation model 𝐸𝛼𝑓 loc 𝑐,k0,𝛼(r)= ˆ𝐻loc 0 + ˆ𝐻loc 1 𝑓 loc 𝑐,k0,𝛼(r) (21) with ˆ𝐻loc 0 =− ℏ2 2∇·𝑚 −1 𝑐,k0∇+𝐸 𝑐,k0+𝑈(r), ˆ𝐻loc 1 = ∑︁ k′ 0≠k0 e−i(k0−k′ 0)·r𝑢∗ k0(r)𝑈(r)𝑢k′ 0(r). The model (21) has beenwidely employed in the silicon qubit literature [17, 18, 40, 45, 46]. In this model, the envelopes are typica...
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Instead, the two components are related by complex conjugation ˆ𝐻+ 0 ∗ = ˆ𝐻− 0,(25) whichfollowsfrom𝑢 ∗ k− 0,k− 0 (r,r′)=𝑢 k+ 0,k+ 0 (r,r′),seeEq.(17). B. Invariance under Reference Energy Shifts The physics of the coupled two-valley system (22) must be invariant under a constant offset of the confinement potential (choice of reference energy) by an arbit...
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By a straightforward computation, we obtain 𝑢k0,k′ 0(r,r′) 𝑈→𝑈+𝑈0 −−−−−−−−→𝑢k0,k′ 0(r,r′)+𝑈 0𝛿k0,k′ 0Δk0(r−r ′), (26) see Appendix B for details

matrix elements of the mesoscopic potential𝑢 k0,k′ 0(r,r′) transform differently under the global shift. By a straightforward computation, we obtain 𝑢k0,k′ 0(r,r′) 𝑈→𝑈+𝑈0 −−−−−−−−→𝑢k0,k′ 0(r,r′)+𝑈 0𝛿k0,k′ 0Δk0(r−r ′), (26) see Appendix B for details. Consequently, the valley-sector projection ensures that the offset𝑈0 shifts only the diagonal (intravalley...
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Figure 3(a) compares the valley splitting obtained from the exact non- local model, the conventional local approximation, and the projected-local (filtered) model

Interface Width We first considera conventional QW with𝑋(𝑧)=𝑋 QW(𝑧) in a vertical electric field and vary the width𝜎of the Si/SiGe interface (here we assume𝜎=𝜎 𝑢 =𝜎 𝑙). Figure 3(a) compares the valley splitting obtained from the exact non- local model, the conventional local approximation, and the projected-local (filtered) model. For smooth interfaces, t...
[6]

Quantum Well Width Next, we compute the valley splitting of a conventional QW in an electric field as a function of the well widthℎ, see Fig. 3(c). For thin QWs, the confinement potential varies rapidly and the slowly varying approximation underlying the local model breaks down [37]. Correspondingly, the local prediction becomes strongly energy-reference ...
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For a spatially symmetric heterostruc- ture, the results must satisfy𝐸 VS(𝐹) =𝐸 VS(−𝐹)

Electric Field Dependency Figure 4 shows the valley splitting as a function of the ver- tical electric field𝐹for a mirror-symmetric Ge concentration profile (𝜎𝑢 =𝜎 𝑙). For a spatially symmetric heterostruc- ture, the results must satisfy𝐸 VS(𝐹) =𝐸 VS(−𝐹). In the simulationstheQWisplacedasymmetricallywithinthecom- putational domain (upper interface at𝑧=0),...

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Wiggle Well OscillatingGeconcentrationprofileswithintheQW,known aswiggle wells, can lead to strong enhancements of the in- tervalley coupling matrix element when the modulation wave numberisresonantwiththevalleywave-vectorseparation[16– 19,23,40,47]. Withinthecontinuumapproach...
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Figure 5(b) shows the resulting valley splitting as a function of the spike position𝑧sp

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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[2] [1]

The model (21) has beenwidely employed in the silicon qubit literature [17, 18, 40, 45, 46]

as 𝑢k0(r) 2≈1, one arrives at the local envelope equation model 𝐸𝛼𝑓 loc 𝑐,k0,𝛼(r)= ˆ𝐻loc 0 + ˆ𝐻loc 1 𝑓 loc 𝑐,k0,𝛼(r) (21) with ˆ𝐻loc 0 =− ℏ2 2∇·𝑚 −1 𝑐,k0∇+𝐸 𝑐,k0+𝑈(r), ˆ𝐻loc 1 = ∑︁ k′ 0≠k0 e−i(k0−k′ 0)·r𝑢∗ k0(r)𝑈(r)𝑢k′ 0(r). The model (21) has beenwidely employed in the silicon qubit literature [17, 18, 40, 45, 46]. In this model, the envelopes are typica...

[3] [2]

Instead, the two components are related by complex conjugation ˆ𝐻+ 0 ∗ = ˆ𝐻− 0,(25) whichfollowsfrom𝑢 ∗ k− 0,k− 0 (r,r′)=𝑢 k+ 0,k+ 0 (r,r′),seeEq.(17). B. Invariance under Reference Energy Shifts The physics of the coupled two-valley system (22) must be invariant under a constant offset of the confinement potential (choice of reference energy) by an arbit...

[4] [3]

and the intervalley (k0 ≠k ′

[5] [4]

By a straightforward computation, we obtain 𝑢k0,k′ 0(r,r′) 𝑈→𝑈+𝑈0 −−−−−−−−→𝑢k0,k′ 0(r,r′)+𝑈 0𝛿k0,k′ 0Δk0(r−r ′), (26) see Appendix B for details

matrix elements of the mesoscopic potential𝑢 k0,k′ 0(r,r′) transform differently under the global shift. By a straightforward computation, we obtain 𝑢k0,k′ 0(r,r′) 𝑈→𝑈+𝑈0 −−−−−−−−→𝑢k0,k′ 0(r,r′)+𝑈 0𝛿k0,k′ 0Δk0(r−r ′), (26) see Appendix B for details. Consequently, the valley-sector projection ensures that the offset𝑈0 shifts only the diagonal (intravalley...

[6] [5]

Figure 3(a) compares the valley splitting obtained from the exact non- local model, the conventional local approximation, and the projected-local (filtered) model

Interface Width We first considera conventional QW with𝑋(𝑧)=𝑋 QW(𝑧) in a vertical electric field and vary the width𝜎of the Si/SiGe interface (here we assume𝜎=𝜎 𝑢 =𝜎 𝑙). Figure 3(a) compares the valley splitting obtained from the exact non- local model, the conventional local approximation, and the projected-local (filtered) model. For smooth interfaces, t...

[7] [6]

Quantum Well Width Next, we compute the valley splitting of a conventional QW in an electric field as a function of the well widthℎ, see Fig. 3(c). For thin QWs, the confinement potential varies rapidly and the slowly varying approximation underlying the local model breaks down [37]. Correspondingly, the local prediction becomes strongly energy-reference ...

[8] [7]

For a spatially symmetric heterostruc- ture, the results must satisfy𝐸 VS(𝐹) =𝐸 VS(−𝐹)

Electric Field Dependency Figure 4 shows the valley splitting as a function of the ver- tical electric field𝐹for a mirror-symmetric Ge concentration profile (𝜎𝑢 =𝜎 𝑙). For a spatially symmetric heterostruc- ture, the results must satisfy𝐸 VS(𝐹) =𝐸 VS(−𝐹). In the simulationstheQWisplacedasymmetricallywithinthecom- putational domain (upper interface at𝑧=0),...